FRAUNHOFER INSTITUTE FOR SOLAR ENERGY SYSTEMS ..... also be the key to change the characteristic of a photonic crystal such that an advantage is.
PHOTONIC CONCEPTS FOR SOLAR CELLS DISSERTATION A thesis submitted for the degree of “Doktor der Naturwissenschaften” (Dr. rer. nat.) by
Ian Marius Peters from Blaubeuren June 2009
FACULTY OF MATHEMATICS AND PHYSICS ALBERT-LUDWIGS-UNIVERSITY FREIBURG IM BREISGAU prepared at
FRAUNHOFER INSTITUTE FOR SOLAR ENERGY SYSTEMS Supervised by Prof. Dr. Eicke R. Weber
PHOTONIC CONCEPTS FOR SOLAR CELLS DISSERTATION A thesis submitted for the degree of “Doktor der Naturwissenschaften” (Dr. rer. nat.) by
Ian Marius Peters from Blaubeuren June 2009
FACULTY OF MATHEMATICS AND PHYSICS ALBERT-LUDWIGS-UNIVERSITY FREIBURG IM BREISGAU prepared at
FRAUNHOFER INSTITUTE FOR SOLAR ENERGY SYSTEMS Supervised by Prof. Dr. Eicke R. Weber
Dekan: Leiter der Arbeit: Referent: Koreferent: Tag der mündlichen Prüfung:
Prof. Dr. Kay Königsmann Prof. Dr. Eicke Weber Prof. Dr. Eicke Weber Prof. Dr. Oliver Waldmann 28.07.2009
Table of contents
TABLE OF CONTENTS
1
INTRODUCTION
2
1.1
Composition of the work
2
PHOTONIC CRYSTALS
2.1
Introduction
2.2
The photonic band structure
11
2.3
Different Photonic Crystals
17
2.4
Summary of the chapter
24
3
SIMULATION OF PHOTONIC STRUCTURES
25
3.1
Simulation methods
25
3.2
Reticolo Code 2D
30
3.3
Convergence considerations
35
3.4
Comparison to other simulation methods
40
3.5
Summary of the chapter
43
7
9 9
i
Table of contents
4
SPECTRALLY SELECTIVE PHOTONIC CRYSTALS
46
4.1
Photovoltaic concepts
47
4.1.1
The advanced fluorescent concentrator concept
47
4.1.2
Spectrum Splitting
52
4.2
4.3
4.4
Considerations of principles
53
4.2.1
Considerations for the fluorescent concentrator concept
53
4.2.2
Considerations for the spectrum splitting concept
59
Optimized spectrally selective photonic structures
61
4.3.1
1D crystals
61
4.3.2
3D crystals
64
Results of the simulations of the enhanced light guiding efficiency inside the 65
fluorescent concentrator 4.5
4.6
Experimental results
67
4.5.1
Fluorescent concentrator system efficiency
67
4.5.2
Spectrum splitting system
75
Summary of the chapter & perspective
78
ii
Table of contents 5
DIFFRACTIVE STRUCTURES
81
5.1
Photovoltaic concepts
82
5.1.1
Back side gratings for solar cells with a flat front surface
82
5.1.2
Back side gratings for solar cells with a pyramidally textured front surface
82
5.2
5.3
5.4
5.5
5.6
Considerations of principles
85
5.2.1
Potential of gratings in a silicon solar cell
85
5.2.2
Maximum path length enhancement for concepts with a discrete change of the light direction inside the PV device
86
5.2.3
The binary grating
88
5.2.4
The blazed grating
90
5.2.5
Semi-analytical model to calculate the effect of a grating
92
5.2.6
Rigorous model to calculate the effects of a grating
93
Different types of gratings
95
5.3.1
The binary grating
95
5.3.2
The trapezoidal shape
96
5.3.3
The blazed grating
97
5.3.4
The pyramidal grating (2D)
98
Simulation results
99
5.4.1
Optimization of the grating period
99
5.4.2
Optimization of the grating depth
103
5.4.3
Calculated absorption profile
105
5.4.4
Summary
109
Experimental results
111
5.5.1
Grating shape
111
5.5.2
Comparison of measured and simulated results
112
Summary of the chapter & perspective
113
iii
Table of contents
6
ANGULARLY SELECTIVE PHOTONIC CRYSTALS
6.1
Angularly selective optical elements in PV
116
6.1.1
Basic idea
117
6.1.2
Scattering processes in PV converters
118
6.1.3
PV concepts with angular confinement
120
6.2
6.3
6.4
116
Considerations of principles
122
6.2.1
Path length enhancement
122
6.2.2
Efficiency limit for systems with angular confinement
126
6.2.3
Angular-dependent characteristic of Bragg-like systems
131
6.2.4
Other mechanisms to control angular selectivity
135
Angularly selective photonic structures
136
6.3.1
1D crystals
136
6.3.2
2D crystals
139
6.3.3
3D crystals
140
Experimental results
143
6.4.1
Absorption enhancement
143
6.4.2
Enhancement of the quantum efficiency
149
6.5
Summary of the chapter & perspective
7
SUMMARY & CONCLUSIONS
150
153
APPENDIX A
SIMULATION OF OPTICAL THIN FILMS
160
APPENDIX B
THE RCWA METHOD
167
APPENDIX C
IMPLEMENTATION INTO “RETICOLO CODE 2D”
175
APPENDIX D
ABBREVIATIONS, SYMBOLS, CONSTANTS
181
BIBLIOGRAPHY
190
PUBLICATIONS
197
ACKNOWLEDGMENTS
201
iv
1
Introduction
1 Introduction The story told in this thesis… The story told in this thesis is a story about two protagonists. These protagonists are the electron and the photon. One scene of this story is the field of photovoltaic energy conversion (PV), the art of extracting electrical energy from sunlight. A huge part of this field is related to silicon solar cells. As silicon solar cells have developed from solid state physics, the language spoken in PV is mainly the language of the electron. Another scene of this story is optics, and the language of the photon is spoken here. Typically, the two scenes and their languages are separated. However, at some point people have started to use the language of electrons to describe phenomena of photons. At this point another scene appeared, and this scene is the field of photonic crystals. Photonic crystals evoke extraordinary effects. Using them, photons can be refracted, deflected, diffracted and reflected. They can be retarded or redirected. In short, photonic crystals can make photons dance to their tune. The story of this thesis begins with photonic crystals and moves towards PV. The plot will be about how to use photonic crystals, or, more generally, interference structures to improve solar cells. Metaphorically speaking, this work is meant to deepen the relation between photons and electrons.
Photovoltaics and light trapping… The overarching story, to which this work shall be a tiny contribution, is the story about renewable energies and the supplying of mankind with electricity. Photovoltaics are one part of renewable energies. It is, in fact, the only part that has the potential to durably supply enough energy [Gra03] to mankind. Each hour the sun delivers to earth the amount of energy used by humanity in a whole year [Web09]. Against this overwhelming availability of easily accessible energy stands the limited availability of fossil and nuclear fuels combined with all the problems due to political power [Cre08], pollution, proliferation and the perils of catastrophic climate change. However, the contribution to primary energy made available by renewable energies is worldwide below 5% and the fraction delivered by photovoltaic and solar thermal devices is vanishingly small (source: AGEB). Still, prices for solar electricity are dropping steadily and more PV plants are continuously being installed. To harness the energy of a photon, it has to generate an electron-hole pair within a solar cell. The reason to use photonic structures is that some photons do not. The question one has to ask oneself is therefore: why do not all photons create electron-hole pairs? And an answer to this question is: for different reasons. One reason is that, due to a refractive index contrast of air and solar cell material, some photons are reflected. Another reason is that, due to a low absorption coefficient, not all photons are absorbed and leave the solar cell system again. A third reason is that when radiative processes occur within the solar cell system, the system itself emits radiation. An optical element should therefore serve one of two purposes: it should assure that photons enter the solar cell device or it should prevent photons from leaving. These purposes can be subsumed in the term ‘light trapping’. Light trapping in a solar cell allows either to increase the efficiency by enlarging the number of used photons, or, maybe even more importantly, it allows decreasing the solar cell thickness. Many light trapping concepts aim for an enlargement of the path length of light inside the solar cell. This may be illustrated when looking at Figure 1.1. The penetration depth is a measure of how thick a crystalline silicon solar cell needs to be to absorb most of the light with a certain wavelength. For λ = 1100nm, the penetration depth is t = 3mm. If the
2
Introduction
10
7
10
6
10
5
10
4
10
3
10
2
absorption coefficient [1/m]
10
4
10
3
10
2
10
1
10
0
10
-1
penetration depth [μm]
400
600
800
1000
penetration depth L [µm]
absorption coefficient α [1/m]
solar cell itself shall be thinner than t = 3mm yet still shall absorb all light, the path length of the light inside the solar cell needs to be enlarged. Typical contemporary crystalline silicon solar cells today have a thickness of t ≈ 250µm which requires a path length enhancement of a factor of 12. Over time, wafer thicknesses are shrinking (not least because of the shortage and the price of solar grade silicon; one of the great topics of 2008) and consequently light trapping is becoming more and more important. Wafer thicknesses of t ≈ 30µm are already common objects in the laboratories. These wafers require a path length enhancement of a factor of 100. This high path length enhancement makes high demands on the light trapping and new light trapping concepts are indispensable.
1200
wavelength λ [nm]
Figure 1.1: Absorption coefficient and penetration depth for light in silicon. Optical properties of silicon are found in [Gre95]. The penetration depth is a measure of how long the path of a photon in silicon must be to be absorbed. At the band edge of silicon at λ = 1100nm, the required thickness of the solar cell is t = 3mm. If the solar cell is thinner than 3mm and all light should still be absorbed nevertheless, the pathlength inside the solar cell needs to be enhanced. This is done by light trapping.
Photonic crystals and sunlight… There are plenty of ways to improve a solar cell. The way developed in this thesis makes use of a special kind of optical instrument; the photonic crystal. The field of photonic crystals was started only some 20 years ago. The basic idea was to transcribe concepts from solid state physics into optics. This idea and similar approaches resulted in a whole new class of materials with amazing physical properties, the so-called “metamaterials”. Examples for properties of metamaterials are the suppression of spontaneous emission [Yab87] or a negative refractive index [Pen00]. The properties of metamaterials are rather defined by their structure than by their composition. In the case of photonic crystals, this structure is a periodical configuration of the refractive index profile with a certain characteristic scale. The topic of photonic crystals has experienced much attention in the last few years and many different photonic crystals with different optical properties have been investigated. When considering the interaction of light and matter, one possible classification concerns the proportion of the relative typical sizes of the device under consideration and the light wavelength. Three areas may be distinguished here. If the light wavelength is much smaller than the typical variation length of the device, we find ourselves in the regime of geometrical optics. All devices resident in this regime may be described completely by ray tracing. In the second region the light wavelength is much larger than the typical variation length of the device. This region is the regime of effective media. Light may not resolve the structures present and again geometrical optics apply, though a lot happens for the material parameters that have to be used. The third (and by far most exciting region) is the one where typical
3
Introduction
variation length of the device and the light wavelength are of the same scale. This is the regime of wave optics and interference. This is also the regime that will be considered in this work.
Figure 1.2: Illustration of the three regimes. If the wavelength λ is much smaller than the typical length of the system Λ (Λ » λ), ray optics applies. If the wavelength is much larger than the typical length of the system (Λ « λ), effective medium theories apply. Interference effects apply if the typical length of the system and the wavelength of light are on the same order of magnitude (Λ ≈ λ). This regime will be considered in this work. At this point it is appropriate to give a more sophisticated definition of what constitutes the optical instruments considered in this work. The most important commonality is marked by the affiliation to the regime where the typical scales of the considered structure and the light wavelength are similar. A consequence of this relative size is the presence of distinct interference effects and interference marks the second important commonality. A third one is marked by the fact that the considered structures consist of dielectric materials. This choice of material delimits the considered structures from luminescent materials, i.e. materials that are themselves a light source. A difference between the investigated structures is branded by the effect interference induces. One effect is diffraction; the emission of light into several orders with certain efficiencies and certain directions. Another effect is constructive or destructive interference which causes an increased reflection or transmission. A further important issue is periodicity though not all structures considered were periodical. With the terms introduced here, it is possible to make the following classification: •
Photonic crystals are ideal structures that are strictly periodic and to which a certain crystal structure can be assigned to. The optical properties of photonic crystals are mainly generated by constructive and destructive interference resulting in a photonic band structure (see chapter 2). Diffractive effects may occur in addition but are not necessary.
•
Photonic structures differ from photonic crystals by the absence of a strict periodicity. However, often photonic structures show a partial or local periodicity. Strictly speaking, photonic crystals with a finite thickness such that effects from the finiteness are detectable are also photonic structures. In that sense, all real photonic crystals are photonic structures. The optical properties of photonic structures are mainly generated by constructive and destructive interference, resulting in a photonic band structure (see chapter 2). Diffractive effects may occur additionally but are not compulsory.
•
Diffractive structures are structures that are strictly periodic but may be of finite expansion. The optical properties of diffractive structures are mainly generated by diffraction. The main difference of diffractive structures to photonic structures is that the photonic band structure is of no importance for the diffractive properties. In that sense, diffractive structures exist that are no photonic structures (an example would be a grating)
4
Introduction A strict separation of the different kinds of structures is very difficult and overlaps occur, such as when a photonic crystal is also used as a diffractive structure. However, considering each application it should be possible to assign every structure discussed in this work to one group rather than to another. Altogether, these structures will be referred to as interference structures. As typical structure size and wavelength are correlated it is interesting to take a look at the wavelengths relevant for solar cells. In the optimum case, all light emitted by the sun is used. The solar spectrum initially is a black body spectrum at a temperature of 5800K. Because of absorption in the atmosphere, the spectrum is altered when it reaches the earth. The alteration depends on the length of the path the light has to travel through atmosphere, and the length depends on the zenith angle ζ 1. This factor is considered in the so-called air mass spectrum. The air mass (AM) is given by AM ≈ 1/cos ζ. The standard spectrum used for terrestrial photovoltaics today is the AM1.5 spectrum which corresponds to a zenith angle of ζ = 48.2°. For this angle the global radiation power is 1000 W/m². The AM1.5 spectrum is shown in Figure 1.3
Figure 1.3: Air mass 1.5 spectrum. This spectrum impinges on the earth under a zenith angle of 48.2°. Shown is the part of the spectrum between λ = 300nm and λ = 3000nm. The visible part of the spectrum is located between ca. λ = 380nm and λ = 750nm
Improving solar cells… The most efficient single junction solar cells today are produced from silicon. Silicon has a band edge of 1.1 eV which corresponds to a wavelength of λ ≈ 1100nm. The spectral range of interest for a silicon solar cell is therefore between λ ≈ 300nm (below this wavelength no solar radiation arrives at the earth’s surface) and λ ≈ 1100nm. The “visible range” for a silicon solar cell is therefore a good deal broader than the range visible for human eyes and covers almost two duplications in frequency. This wide range already constitutes a big challenge when thinking about the application of interference structures for solar cells. The ratio between structure size and wavelength defines the effects of the optical structure, yet the wavelength range that needs to be considered is very broad. It is in fact so broad that it is very difficult to achieve desired effects for the whole spectrum visible to the solar cell. This problem is even more hightened when thinking about multijunction solar cells that cover a wavelength range up to λ ≈ 2000nm. Corresponding to that spectral range, the typical size of the optical structures is also in the range of several 100nm. 1
The zenith angle ζ is defined as the angle of a point against the perpendicular related to the geocenter
5
Introduction
A very simple example for an interference structure that is used to improve solar cells is an antireflection coating. The antireflection coating is a thin layer or a stack of thin layers that induce a destructive interference for light reflected at the different surfaces of the coating. The effect of this destructive interference is a reduction in the overall front surface reflection. Antireflection coatings are commonly used and provide a considerable improvement of solar cells. An example for an interference structure that could increase the absorption is a grating introduced into a solar cell. Light that is diffracted by the grating has a longer path through the cell and is therefore absorbed more efficiently, consequently increasing the quantum efficiency of the solar cell. These approaches are very simple examples for the effects interference structures can have. Much more complex structures are possible that induce a number of other beneficial effects. To summarize the contents of this thesis: solar cells need to be made more efficient, if photovoltaics are to play a considerable role in the provision of electrical energy. Concepts that feature the application of special kinds of optical structures, i.e. structures that use interference effects are one way to do this, but few of these concepts (referred to as photonic concepts) have yet been realized. In this work I want to plumb the depths of what is possible with these interference structures. Photonic concepts will be investigated and evaluated, theoretically and experimentally. Some of these concepts will aim towards an enhancement of the path length in the solar cell. Examples for such concepts are the integration of diffractive gratings at the rear side of crystalline silicon solar cells or light trapping with angularly selective filters. Other concepts will aim towards a reduction of loss mechanisms, like the application of spectrally selective filters on fluorescent concentrators to eliminate escape cone losses. A further topic will be spectrum splitting. Spectrum splitting is the process of distributing solar radiation onto solar cells with different band gaps. Finally, some concepts will aim for a principal enhancement of the solar cell efficiency. One example for such a concept is the suppression of radiative recombination with angularly selective optical elements. With the concepts and considerations discussed in this thesis, I hope to be able to give an indication of a possible direction that could be taken to improve solar cells and to add some new aspects to the story of photons and electrons.
6
Introduction
1.1 Composition of the work The composition of this work is not a classical one. Typically, an introduction into the topic is given followed by a description of the theoretical background, the theoretical or experimental approach that has been developed, results and, finally, a discussion. In this work, I have chosen another composition. The reason to proceed like I have done is founded in the spread of concepts investigated. Between these concepts many differences exist that limit the number of common issues, which can be covered in a separate chapter. To accomplish this I decided to classify the different topics by certain characteristics of the optical element used. Namely these characteristics are spectral selectivity, angular selectivity and diffraction. This classification resulted in a trichotomy represented by chapters 4, 5 and 6. Each chapter starts with an introduction of the PV concepts that feature the respective photonic concept. The photonic concepts are adapted to special requirements of the respective PV concepts; therefore even in one chapter different PV concepts may be addressed. The requirements of the PV concepts define what is investigated in each of the three chapters. The second part in all three chapters concerns considerations of theoretical nature to determine the possible effects of the optical structures on the PV elements. Following that, simulated and, if possible, experimental results are given. A summary of the results and an outlook on further activities is given at the end of all chapters and also at the end of the thesis. The contents of chapter 2 and 3 are common issues needed in all three chapters like basic issues and a description of the simulation methods. The most important results and an outlook are given in chapter 7. The topic of chapter 2 is the photonic crystal. The most important characteristic of a photonic crystal is the photonic band structure. The theoretical background of the photonic band structure is introduced and the concept of a stop gap is discussed. A stop gap is a region defined by certain photon frequencies and certain relative orientation of photon propagation and photonic crystal, in which no propagation of the photon is allowed within the crystal. These stop gaps form the basis on which photonic crystals for certain applications are designed. Different photonic crystals are introduced and their characteristics are investigated to illustrate the characteristics discussed. The introduced examples include the rugate filter and the opal, two structures that have been used frequently for an application on photovoltaic concepts for different purposes. In chapter 3 the simulation methods are introduced that have been used to calculate the optical properties of different interference structures. These methods are the approach of characteristic matrices and the rigorous coupled wave analysis (RCWA). Of these methods the theoretical background is described. The approach of characteristic matrices has been implemented following this description, for the RCWA the MATLAB based “Reticolo Code 2D”, which was developed at the University of Paris, has been used. The RCWA method is per se rigorous, however the implemented structure is approximated and therefore also the results of the simulations are approximations. For this reason convergence considerations are presented for the parameters that constitute the approximation. To conclude this chapter, the optical properties of a photonic structure calculated with the RCWA method are compared to the results obtained from other methods. In chapter 4 photovoltaic concepts are introduced that include spectrally selective photonic structures. One of these concepts is the advanced fluorescent concentrator concept. Here spectrally selective photonic structures are used to reduce losses caused by the escape cone of total internal reflection. Another concept is on the topic of spectral selection for solar cells with different band gaps. Different spectrally selective optical filters are used in front of solar cells to transmit all light intended for the solar cell in question and reflect all light intended for other solar cells. For both concepts, considerations of principles concerning the demands of the filters are presented and optimization methods to design appropriate filters
7
Introduction
are discussed. For the fluorescent concentrator, an analytical model is described and used to predict the effect of the different filters on the light guiding efficiency. Promising filters are introduced and characterized that have been produced following the optimization. To conclude this chapter, measurements of the efficiencies of the solar cells used for the different applications are presented that demonstrate the beneficial effects achieved with spectrally selective optical filters. In chapter 5 photovoltaic concepts are introduced that use diffractive structures. One approach here is the introduction of rear side gratings into silicon solar cells and especially silicon solar cells with a front side texture. The aim of using diffractive structures is to relocate internal radiation so that it propagates in a direction alongside the solar cell or close to. To estimate the potential of gratings, considerations of principles concerning the maximum possible path length enhancement with a structure that induces a discrete change of the direction of the internal radiation are given, and how such a discrete change may be realized with a grating is discussed. Following that two models are introduced that have been used to calculate the optical effects of the grating on a solar cell. Different types of gratings are introduced and the presented models are used to characterize and optimize the solar cell with gratings theoretically. To conclude the chapter, first experimental results of gratings introduced in silicon wafers are given. In chapter 6 photovoltaic concepts are introduced that use angularly selective photonic structures. The concept of angular selectivity and angular confinement is introduced and discussed. Two main effects are expected from angular selectivity. One is the light trapping effect and the other is a suppression of radiative recombination, which influences the thermodynamic properties of a solar cell. Concepts that use angular selectivity always need a large angular spreading of the radiation coming from the PV element. Mechanisms are introduced that result in such an angular spreading. Considerations of principles concerning the concepts are given, such as the theoretical possible path length enhancement and the theoretical efficiency limit of PV systems with angular confinement. Also discussed are mechanisms that result in an angular-dependent characteristic of photonic crystals. One effect discussed in detail here is the Bragg effect. The angular-dependent reflection characteristic of different photonic crystals is characterized theoretically and experimentally. To conclude the chapter, experimental results are presented demonstrating an increased absorption and quantum efficiency induced by angularly selective photonic structures. The results of this work are summarized in chapter 7 and the most important results are recapitulated.
8
2 Photonic crystals In the introduction, a short discussion was given about the commonalities shared by all the structures considered in this work. Also discussed was what characterizes a photonic crystal, a photonic structure and a diffractive structure. An important point here was periodicity and the role played by interference. This was of course not an exhaustive discussion, and part of it shall be taken up here. This chapter is dedicated to photonic crystals, meaning therefore, following the definition given above, infinite, perfectly periodical structures in which interference is used to create spectral ranges of high reflection and transmission. The theory of photonic crystals forms the basis of the understanding of all structures used in this work, though none were real photonic crystals (at least, none were infinite). Photonic crystals are ideal structures. Photonic structures are non-ideal structures based on photonic crystals. A lot of the work done for the application of solar cells was to cope with effects originating from non-ideality that generates loss mechanisms. On the other hand, non-ideality may sometimes also be the key to change the characteristic of a photonic crystal such that an advantage is created. This has been done e.g. to reduce reflections in a spectral region where they were undesirable or to increase the reflection range. How the actual photonic structure looks like will be discussed directly at the point where its application is described (which is sensible because the desired characteristics depend on the application). In this chapter, first the concept of photonic crystals will be introduced (section 2.1) and the phenomenon of the photonic band structure will be discussed (section 2.2). Following that, some photonic crystals will be introduced and their characteristics as well as the characteristics of the corresponding photonic structures will be presented (section 2.3).
2.1 Introduction The basic idea of photonic crystals is an analogy of the dispersion of an electron in a solid state crystal and a photon in a photonic crystal. In a solid state crystal, the atom rumps form a periodic potential. As a consequence of the periodic potential, the dispersion of the electron in the crystal may be given in a band diagram which displays forbidden and allowed energy states for the electron depending on the direction of motion of the electron in the crystal [Iba90]. Typically, band structures are associated with solid state crystals, however quasi-crystalline and amorphous materials may also have a band structure. An analogue situation may be realized for a photon in a photonic crystal. Here the photon plays the role of the electron and the potential of the atom rumps is substituted by a spatial variation of the refractive index. In such a material, concepts of solid state physics may be applied to photons, like the band structure. A first formulation of this idea is found as early as 1972 by Bykov [Byk72]; however, the field of research started with two independent publications of Eli Yablonovitch [Yab87] and Sajeev John [Joh87] who first calculated the optical properties of photonic crystals. Photonic crystals are classified by their dimensionality. 1D, 2D and 3D photonic crystals are distinguished depending on the number of directions in space in which the structure is periodical (see Figure 2.1). 1D photonic crystals correspond to multilayer stacks. The simplest example here is the Bragg stack, an optical element, the concept of which has been long known. However, the implications of photonic crystals exceed the borders of classical optics and a photonic crystal is more than a multidimensional Bragg stack.
9
Photonic crystals
Figure 2.1: Schematic diagram of the different kinds of photonic crystals. In this example two different refractive indices n1 and n2 are ordered in one to three different directions creating a 1D, 2D or 3D photonic crystal. Important is the periodic variation of the refractive index, but one is neither limited to two different refractive indices nor in the function describing the spatial distribution [Joa95]. Whereas the period length in a solid state crystal is defined by the distance of the atoms in the crystal lattice (e.g. NaCl has a lattice constant of Λ = 560 10-12m), the period length of photonic crystals is related to the wavelength of the considered light. In the wavelength regime relevant for PV applications, typically lattice constants are in the range of several 100nm and are thus three orders of magnitudes larger than solid state crystals. More precisely, a structure will act as a photonic structure if the period length of the refractive index profile is in the same range as the considered light wavelength. Such structures are found in nature e.g. in wings of butterflies [Zha09], in the feathers of peacocks [Zi03], in opal gems [San64] and in the stinger of the sea mouse [Par01] (Figure 2.2). On the other hand it is also possible to realize artificial 3D photonic structures. One of the first artificial photonic structures was the “Yablonovite” [Yab91], a structure in which holes were drilled in a 120° symmetry. The Yablonovite and similar photonic structures, like the “woodpile” structure [Lin98], were used to realize photonic effects in the infrared part of the spectrum. It was, however, impossible to produce these structures for an application in the visible part of the spectrum as no appropriate production methods existed. Even for the comparably large periods these structures featured, the production methods were slow and extremely laborious so that no crystals could be produced on large areas. To obtain photonic structures in the visible range, several techniques exist. One involves lithography methods like holography [Cam00] or two photon lithography [Cum99]. The other uses the self-organization of monodisperse spherical particles. The best known representative of these kinds of photonic crystals is the opal [Xia00] (see section 2.3). This photonic crystal is named opal, because it has the same structure as the gem. In the gem, the refractive index profile is formed by silica areas with different water content (and therefore different refractive index), whereas in the artificial opal, the refractive index contrast is the one of sphere / surrounding area. Opals play an important role, because they may be produced on large areas with comparably little effort. The application of different photonic structures to PV applications will be the topic of chapters 4, 5 and 6. Quite a lot of other possible applications have been proposed. A lot of them can be found in the literature. The most prominent example here is the photonic crystal fiber [Bir96], which may transport light without the use of total internal reflection. Other examples are the superprism [Pra03], the photonic crystal laser [Pai98], and resonator with extremely high Q-factors [Aka03]. Such high-Q cavities could be used to detect Rabi splitting [Yos04], a phenomenon which exceeds classical optics.
10
2.2 The photonic band structure
Figure 2.2: Examples for photonic structures in nature. Butterfly wing, picture taken by Michael Apel, peacock feathers, picture by J. Hempel, opal, picture by Noodle snacks, sea mouse, picture by Michael Maggs. All pictures are under the creative commons license, attribution ShareAlike 2.5 resp. 3.0. Butterfly wing and sea mouse represent 2D photonic crystals, peacock feather and opal represent 3D photonic crystals.
2.2 The photonic band structure One concept originating from solid state physics and applied to photonic crystals is the already mentioned photonic band structure. The presence of a photonic band structure can be seen as the attribute that constitutes a photonic crystal. Regions with a high refractive index act as scattering centers and are the origin of partial waves that interfere with the incident wave. Standing waves are formed within the photonic crystal. The energy maxima of the corresponding wave functions are either located in a region with a high or a low refractive index. The energy of the allowed modes splits and a complex dispersion relation is obtained which results in the band structure. Within the band structure, so-called stop gaps exist. These stop gaps are regions where no states for a photon are allowed and are therefore the analogue to a band gap in a solid state crystal (for this reason they are sometimes also called the “photonic band gap” or PBG). For a material with a periodicity in all three spatial directions and a sufficiently large refractive index contrast, the stop gaps of all directions may superpose and form a complete stop gap, which would prevent the propagation of light with each polarization and the corresponding wavelengths in all directions within the crystal. Complete band gaps have been demonstrated theoretically for a number of photonic crystals. Examples are the inverted opal [Söz92], the diamond structure [Gar02], and the woodpile structure [Ho94]. However, to my knowledge, no photonic structure with a complete band gap has been realized yet on an area large enough to be interesting for solar applications.
11
Photonic crystals
In contrast to the complete band gap, so-called pseudogaps also exist that prevent the propagation of light only in certain directions. Pseudogaps are common and exist in every photonic crystal. For PV applications, pseudogaps are much more interesting and in all applications considered in this work pseudogaps were used. A photonic band structure with photonic band gaps is shown in Figure 2.3.
Figure 2.3: Band structure of a photonic crystal. In this example the photonic crystal consists of spheres with the refractive index of n = 3.5 in a diamond lattice in a surrounding with the refractive index n=1. Marked are a complete stop gap and a pseudogap. How the band structure is derived and what the symbols mean is discussed in this subsection. It has to be said that a complete prevention of the propagation of light only applies to perfect photonic crystals with an infinite expansion. For finite photonic structures, the propagation is not forbidden but considerably restricted. A PBG in a finite crystal will manifest as a region of increased reflection. The optical properties of photonic crystals allow the construction of optical elements with special characteristics, like spectrally (see chapter 4) or angularly selective (see chapter 6) filters. Yet photonic crystals exhibit other salutary attributes. The stop gap allows for the suppression of spontaneous emission, a characteristic which may be used to construct low threshold lasers [For05] or to influence emission processes e.g. in upconverting materials [Gar08]. In this section it will be described how some principles of solid state physics are transferred into optics. This transformation is the basis of the formulation of the theory of photonic crystals. The analogies will lead to the photonic band structure, the description of the dispersion relation of a photon in a photonic crystal. The descriptions given here are based on [Joa95] and [Mat05]. Maxwell’s equations The phenomena of classical electromagnetism are described by four equations that were formulated first by Maxwell in 1864 [Max 73]. They describe the sources and the mutual dependence of the electric and magnetic field. The quantities Maxwells’ equations describe r r r r are: the magnetic flux density B (r , t ) , the electric induction density D(r , t ) , the electric r r r r r charge density ρ (r , t ) and the current density j (r , t ) . H (r , t ) is the strength of the magnetic r r field and E (r , t ) is the strength of the electric field. The relation between these quantities is given by
12
2.2 The photonic band structure r r ∇B ( r , t ) = 0
(2.1)
r r r ∇D(r , t ) − ρ (r , t ) = 0
(2.2)
r r r r ∂ r r ∇ × H (r , t ) − j (r , t ) − D(r , t ) = 0 ∂t
(2.3)
r r ∂ r r ∇ × E (r , t ) + B(r , t ) = 0 ∂t
(2.4)
For electromagnetic (EM) fields in media, equations (2.1) - (2.4) need to be complemented by the material equations (2.5) - (2.7).
s r r r r D(r , t ) = ε 0 ⋅ ε (r ) ⋅ E (r , t )
(2.5)
r r r r r B(r , t ) = μ 0 ⋅ μ (r ) ⋅ H (r , t )
(2.6)
r r r r r j (r , t ) = σ (r ) ⋅ E (r , t )
(2.7)
r The electric conductivity is denoted σ (r ) . ε0 and μ0 are the permittivity and the r r permeability of free space, ε (r ) and μ (r ) are material-dependent relative permittivity and permeability. The last three expressions are second degree tensors. In the general case, equations (2.5) - (2.7) are much more complex and the dependencies may differ considerably from the case described here. The given equations are valid for an instantaneous and linear response of the materials to an electromagnetic field. Furthermore, r r r all media shall be isotropic. The parameters ε (r ) , μ (r ) and σ (r ) then become scalars (denoted ε, μ and σ) and are constant within the different materials. A more precise description of this topic is found in the book of Born & Wolf [Bor99]. The wave equation of the magnetic field is derived by rotating equation (2.3).
r ⎛ 1 r ⎞ 1 ∂ 2 H (r , t ) ∇ × ⎜⎜ r ∇ × H (r , t ) ⎟⎟ = − 2 c ∂t 2 ⎝ ε (r ) ⎠
(2.8)
Assuming harmonic waves, the time dependence may be separated by
r r H (r , t ) = H (r ) exp(iωt )
(2.9)
and consequently
⎛ 1 r ⎞ r ω2 ∇ × ⎜⎜ r ∇ × H (r ) ⎟⎟ = − 2 H (r ) c ⎝ ε (r ) ⎠
(2.10)
From equation (2.10), a Hermitian operator may be defined that acts on the magnetic field, r has the real eigenvalues ω2/c2 and of which the eigenfunctions H (r ) form an orthogonal system of functions. This operator is
13
Photonic crystals
⎛ ⎞ 1 Θ = ⎜⎜ ∇ × r ∇ × ⎟⎟ ε (r ) ⎝ ⎠
(2.11)
The operator shows remarkable analogies to the Hamilton operator describing the electric properties of solid state crystals.
r h2 2 H =− ∇ + V (r ) 2m
(2.12)
Both operators are e.g. invariant to translations. For this reason, well-known concepts from solid state physics may be transferred to photonic crystals. The main difference between the description of a photon in a photonic crystal and an electron in a solid state crystal is the vectoral character of the electromagnetic wave. The operator of the electric field analogue to that of equation (2.11) is not Hermitian. As being Hermitian is a basic condition for the calculation shown, it is not possible to deal with the electric field in the same way as with the magnetic field. The way to proceed here is to solve the problem for the magnetic field and afterwards calculate the electric field from the magnetic field with
r ⎛ − ic ⎞ r E (r ) = ⎜⎜ r ⎟⎟∇ × H (r ) ⎝ ωε (r ) ⎠
(2.13)
Another interesting point arises from the scalability of Maxwell’s equations. Because of the scalability, a change in the period does not result in new solutions, but gives the same solutions with a frequency scaled accordingly. In other words: a solution for a photonic crystal, once found, stays valid for all systems with the same ratio of period / frequency. For this reason, the dimensionless parameter a/λ is used, the so called “normalized frequency” with a the lattice constant of the photonic crystal and λ the vacuum wavelength of the considered light. The reciprocal lattice An ideal photonic crystal consists of an infinite periodic configuration of the refractive r r r index and is translational invariant. The crystal is defined by the lattice vectors a1 , a 2 , a 3 . A unit cell of a given crystal is constructed from the lattice vectors. The smallest volume containing integer values of the lattice constants is called Wigner-Seitz cell. For the calculation of the band structure, the reciprocal lattice is used. The reciprocal lattice
r r
r
is also constructed from lattice vectors b1 , b2 , b3 . These reciprocal lattice vectors are calculated by
r v r a 2 × a3 b1 = 2π r r v a1 ⋅ (a 2 × a3 )
(2.14)
r v r a3 × a1 b2 = 2π r r v a 2 ⋅ (a3 × a1 )
(2.15)
r v r a ×a b3 = 2π r 1 r 2v a3 ⋅ (a1 × a 2 )
(2.16)
14
2.2 The photonic band structure Analog to the Wigner-Seitz cell, the first Brillouin zone is defined by the reciprocal lattice vectors. The solution of equation (2.10) is restricted to the first Brillouin zone. The obtained dispersion relation is given as a function of the wave vector k on the first Brillouin zone. The issue is depicted in Figure 2.4. If the first Brillouin zone contains symmetries, the solution space may be further reduced to a so-called irreducible Brillouin zone. a)
b)
Figure 2.4: Example for a 2D lattice with lattice vectors a1 and a2 and the Wigner-Seitz cell (a). Also shown is the corresponding reciprocal lattice with the reciprocal lattice vectors b1 and b2 and the first Brillouin zone. Calculation of the band structure The method to calculate the dispersion relation of photons in a photonic crystal uses an expansion into plane waves. The spatial information about the structure of the photonic r crystal is contained in the periodic function of the dielectric constant ε (r ) . The photonic
r
crystal is periodic with regard to all lattice vectors ai . Consequently to the dielectric function, the following condition applies.
r
r
r
ε (r ) = ε (r + ai )
(2.17)
To obtain the band structure, the inverse of the dielectric function is expanded into a Fourier series in the reciprocal lattice.
rr 1 κ vr exp(iv r ) r =∑ r ε (r ) v
(2.18)
In order to proceed, the Floquet-Bloch theorem ([Flo83], [Blo28]) is needed. The Floquetr Bloch theorem says that the solutions ψ (r ) of a Hermitian and periodic eigenvalue problem
r r r V (r ) = V (r + ai ) have to be pseudoperiodical. A pseudoperiodical function consists of a r term u (r ) which is periodical with the same period as the original function and a plane rr r wave exp(ik r ) . The values of a pseudoperiodical function at a certain point r and a point r r relocated by one period r + ai therefore only differ by a phase factor. These functions are the so called Bloch functions.
r
r
rr
r
r
r
ψ (r ) = u (r ) ⋅ exp(ik r ) with u (r ) = u (r + ai )
(2.19)
Because of the translation invariance, the solution space can be restricted to the first Brillouin zone. Different solutions are marked with an index n. The eigenstates of the
r
r
magnetic field H (r ) are now characterized by the wave vector k and the index n.
15
Photonic crystals
rr r r r r r r H (r ) = H kr ,n (r ) = u kr ,n (r ) exp(ik r )
(2.20)
The eigenfunctions in this equation are expanded into a Fourier series
(
r v r r r r v H (r ) = ∑ H kr ,n (v ) exp i (k + v )r v v
)
(2.21)
Inserting equation (2.18) and (2.21) into (2.10), the following equation is obtained
r v r v ⎛ ω kr ,n r v v ( k + v ) × ( k + v ) × H r (v ' ) = ⎜ κ ∑vv v −v ' k ,n ⎜ c ⎝
2
⎞ ⎟ H r (vr ) ⎟ k ,n ⎠
(2.22)
To proceed from here, one needs to identify the eigenvalues ω kr ,n . The eigenvalues depend
r
on the chosen k vector. The solutions for a certain path in the Brillouin zone are assembled into a band diagram. The band diagram for a face centered cubic crystal is given in Figure 2.5.
Figure 2.5: Band structure of a face centered cubic (fcc) photonic crystal of air spheres in a surrounding medium with a dielectric constant of ε = 12.25. The band structure was calculated using the MIT program MPB [Joh01]. The inset shows the first Brillouin zone of the cubic crystal [Wik08]. Also shown are photonic band gaps that may occur in the band structure. A complete gap (here a/λ ≈ 0.8) exists between the 8th and the 9th band and a pseudogap (Γ-L, a/λ ≈ 0.5) between the 2nd and the 3rd band In the figure, a path on the Brillouin zone containing all points of high symmetry was paced. The points of high symmetry have a certain notation (X, U, L, Γ…), which is given on the x-axis. These points correspond to a certain relative orientation of crystal and light and are
r
therefore equivalent to k vectors. On the y-axis, the solutions ω kr , n constitute bands that are given in terms of the normalized frequency a/λ. The sum in equation (2.22) contains an infinite number of summands. A high number is especially needed to reproduce a sharp contrast on the boundary between two materials. Typically the summation is aborted if the solution shows satisfactory convergence. The calculation of photonic band structures in this work was performed using the excellent MIT photonic band gaps program [Joh01], which is commonly used today to calculate band structures.
16
2.3 Different Photonic Crystals
2.3 Different Photonic Crystals In this section examples of photonic crystals will be introduced. These examples cover many of the photonic structures used in the simulations and experiments in this work. Additionally the distributed Bragg reflector (DBR) has been added as an illustration for the described effects on a very simple example. The investigated crystals may be classified by their dimension. The DBR and the rugate filter are 1D photonic crystals, the checkerboard structure is a 2D photonic crystal and opal and diamond structure are 3D photonic crystals. For all photonic crystal the corresponding band structure is given. Additionally optical characteristics are given for the corresponding photonic structures. Typically, photonic crystals and corresponding photonic structures are not considered separately and also in this work the separation will not be abided strictly. Still it is useful to keep the difference at the back of one’s mind. The distributed Bragg reflector The simplest example for a photonic crystal is the distributed Bragg reflector. A DBR consists of two layers A and B with the refractive index nA and nB that are ordered periodically in the scheme ABAB…. The band structure for such a filter is given in Figure 2.6.
normalized frequency (a/λ)
1.5 1.2 0.9 0.6 0.3 0.0
X
crystallographic orientation
Γ
Figure 2.6: Band structure of a DBR. The refractive indices are n1 = 1.5 and n2 = 2. Photonic band gaps and the corresponding reflection may be located at the x-point for certain frequencies and all odd multiples of these frequencies The wavelength λ0 for which the DBR shows reflection peaks for normal incidence is determined by the thickness of the single layers d1 and d2. The so-called Bragg condition is defined by
λ0 = 4 ⋅ ni ⋅ d i , i = 1,2
(2.23)
Additionally to the band structure, the reason for the increased reflection may be explained by a simple argument: at every surface, a part of the incident light is reflected. Due to the chosen thicknesses, the reflected light at each period interferes constructively, thus creating the stop band. The same argument holds for the corresponding photonic structures. Here the stop gap marks a region of high reflectance. It is straightforward that the more layers are used, the higher the reflectance becomes. The characteristic of a DBR is calculated with the transfer matrix method (see section 3.1) and is shown in Figure 2.7
17
Photonic crystals
b)
a)
wavelength [nm] 100
1.9
reflectance
refractive index
2.0
1.8 1.7
200
300
400
600 1000
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
1.6
0.2
0.2
1.5
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.0x10
0.0 15
2.5x10
15
2.0x10
z[µm]
15
1.5x10
15
1.0x10
15
5.0x10
14
frequency [Hz]
Figure 2.7: Characteristic of the DBR, designed for a reflection at λ0 = 1000nm. Figure 2.7a shows the refractive index profile of the filter. The filter shown consists of two layers with the refractive index nA and nB that have thicknesses according to equation (2.23). In the depicted filter, ten periods have been used. Figure 2.7.b shows the reflection characteristic of the filter. Areas of high reflectance appear for the designed frequency ν0 and the odd multiples of this frequency, 3ν0, 5ν0, 7ν0 … The Bragg condition for reflection in a DBR is not only satisfied for the design frequency ν0 but also for harmonic conditions that occur for all odd multiples of this frequency 3ν0, 5ν0, 7ν0… A second effect occurring is the sidelobes that cause reflections outside the peak region. These sidelobes are caused by the periodicity of the structure and depend on the number of layers used. Rugate Filter
normalized frequency (a/λ)
Higher bands in photonic crystals are suppressed if the refractive index profile is smoothed from a discrete to a continuous function of the refractive index. One graded version of the DBR is called rugate filter [Ber89]. The rugate filter features a sinusoidal variation of the refractive index [Bov93]. The band structure of a rugate filter is given in Figure 2.8.
1.2
Rugate
1.0
Bragg
0.8 0.6 0.4 0.2 0.0
X
crystallographic orientation
Γ
Figure 2.8: Band structure of a rugate filter. The refractive indices are n1 = 1.5 and n2 = 2. Photonic band gaps and the corresponding reflections may be located at the x-point for certain frequencies and all odd multiples of these frequencies.
18
2.3 Different Photonic Crystals The period length Λ of one sinus of the rugate filter designed for a reflection at the design wavelength λ0 is given by
Λ=
λ0
(2.24)
2⋅n
In this equation, n is the average refractive index of the rugate filter. The average refractive index n is found in the refractive index profile of the rugate filter, which is given by
⎛ 2π ⎞ n( z ) = n + Δn ⋅ sin ⎜ ⋅ z⎟ ⎝ Λ ⎠
(2.25)
To calculate the characteristic of the rugate filter, the transfer matrix method has also been used (see section 3.1). The continuous refractive index is approximated by a discrete function of very thin layers. The characteristic of a rugate filter is shown in Figure 2.9. b)
a)
wavelength [nm] 100
1.9
reflectance
refractive index
2.0
1.8 1.7 1.6
200
300
400
600 1000
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
1.5 0.0
0.5
1.0
1.5
2.0
2.5
3.0
z[µm]
3.0x10
0.0 15
2.5x10
15
2.0x10
15
1.5x10
15
1.0x10
15
5.0x10
14
frequency [Hz]
Figure 2.9: Characteristic of the rugate filter, designed for a reflection at λ0=1000nm. Figure 2.9a shows the refractive index profile of the filter. The refractive index is varied between n1 = 1.5 and n2 = 2, with n = 1.75 and δn = 0.25. According to equation (2.24), Λ = 285nm. 10 periods have been used for the filter. Every period has been approximated by 100 layers. The reflection peak at the design wavelength remains, but the multiple reflections that occurred for the DBR have vanished. For the finite rugate filter, another harmonic reflection occurs at the half design wavelength. Checkerboard structure The checkerboard structure consists of two materials that are ordered in the checkerboard symmetry (Figure 2.10a). To calculate the reflection characteristic of such a structure, the RCWA method (see section 3.1) was used. The reflection characteristic for a checkerboard structure is given in Figure 2.10b. The checkerboard structure is an example for a 2D photonic structure. Some significant differences exist between 1D and 2D photonic crystals. In the 2D case, the equations for the band structure for different polarizations decouple, so that now two different band structures are needed. The band structures for the checkerboard photonic crystal are shown in Figure 2.11.
19
Photonic crystals
a)
b) 1.0
TE TM
reflectance [%]
0.8 0.6 0.4 0.2 0.0 1.4
1.6
1.8
2.0
2.2
2.4
wavelength μm
Figure 2.10: Schematic sketch of the checkerboard symmetry (a) and reflection characteristic of a checkerboard structure (b). The refractive indices are n1 = 1 and n2 = 2. The period was Λ = 1µm. b) TM-polarization normalized frequency (a/λ)
normalized frequency (a/λ)
a) TE-polarization 2 1.00 0.75 0.50 0.25 0.00
X
M
Γ
M
X
crystallographic orientation
1.00 0.75 0.50 0.25 0.00
X
M
Γ
M
X
crystallographic orientation
Figure 2.11: Band structure of the checkerboard structure. For the calculation of the band structure, the same parameters as given in Figure 2.10 were assumed.The equations for the description of the different polarizations decouple so that now two band structures are needed. As an example for the reflection of the checkerboard structure, the pseudogap at the M-point has been chosen (red circles).The insert shows the first Brillouin zone of the checkerboard structure. Opal The opal is an example for a 3D photonic crystal. It consists of spheres ordered in a closest package (Figure 2.12a). The growth direction of the opal typically is the 111 or L direction. Generally speaking, there are two crystal structures that differ in the stacking sequence. Shown is the face centered cubic (fcc) structure, in which the layers are stacked in the sequence ABC... The other structure is the hexagonal closest package (hcp), in which the sequence is ABAB... Both systems show similar optical properties [Jia99]. Real opals are a mixture of hcp and fcc. As the fcc-structure is thermodynamically slightly preferred, opals are typically modelled as fcc-structures.
2
TE stands for transversal electric, TM for transversal magnetic. Alternativley, TE is also called s-polarization and TM is called p-polarization. These two modes are typically given for polarization conditions. Every other polarization may be constructed from these two. If nothing is said about polarization, typically (TE+TM)/2 is used.
20
2.3 Different Photonic Crystals
b)
a)
Figure 2.12: Schematic sketch of the opal structure a) and first Brillouin zone b).The schematic sketch is given in top view, which corresponds to normal incidence or Γ-L direction in the Brillouin zone. Two kinds of opals are distinguished, normal opals and inverted opals. For normal opals the refractive index of the spheres is higher than the refractive index of the surrounding medium. For inverted opals, it is the other way around. Inverted opals have several principal advantages over normal opals. For one, the band structure of both crystal systems is different. The most conspicuous difference is that the inverted opal features a complete stop gap for a sufficient refractive index contrast (Δn > 2.8:1) [Söz92]. The normal opal features no complete stop gap. The second advantage of inverted opals is that for the inversion, a variety of materials is available with higher refractive indices. The refractive index contrast of crystal and surrounding is of crucial importance for the optical characteristics. The inverted opal shows more pronounced photonic effects, and consequently thinner optical elements may be created with similar optical properties compared to normal opals (see Figure 4.19). The reality of filling processes attenuates this advantage, however [Bus98][Gal05][Gal06]. Band structures of the normal and the inverted opal are shown in Figure 2.13. For all 3D photonic crystals, the band structure is given as an average over TE and TM polarization. a) normal opal
b) inverted opal normalized frequency [a/λ]
normalized frequency [a/λ]
1.2 1.0 0.8 0.6
relative gapwidth: 5.6%
0.4 0.2 0.0
X
U
L
Γ
X
W
K
1.4 1.2 1.0 0.8
0.4 0.2 0.0
crystallographic orientation
relative gapwidth: 8.3%
0.6
X
U
L
Γ
X
W
K
crystallographic orientation
Figure 2.13: Band structure of a normal opal (a) and an inverted opal (b), each with a refractive index contrast of 1.5:1. The fcc structure was calculated using the MPB program. The photonic band gap is more pronounced for the inverted opal. Theoretically, this could be used to achieve a certain reflectance with a thinner crystal, however, experimental reality is different and the inverted opal has no real advantages over the normal opal.
21
Photonic crystals
A major advantage of the opal is that it is produced by a self-organizing process which allows production of opals on large areas with comparably little effort and potentially little cost. For this reason great interest has been paid to possible applications of the opal. The reflection characteristic of an opal simulated with the RCWA method (see section 3.1) is shown in Figure 2.14.
reflection [%]
100 80 60 40 20 0
400
500
600
700
800
wavelength [nm]
. Figure 2.14: Simulated reflection characteristic of an opal. The opal consists of spheres with a diameter D = 254.7nm and a refractive index of n = 1.5. The surrounding medium is air. The opal consists of 21 sphere layers. The reflection peak corresponds to the pseudogap between the 2nd and the 3rd band at the L-point.
relative gap width [%]
16
relative gap width gap centre position
14 12
0.88 0.84 0.80
10
0.76
8 6
0.72
4
0.68
2
0.64
0 1.00
1.25
1.50
1.75
2.00
2.25
gap centre position [a/λ]
To illustrate the dependence of the band structure on the refractive index, the center position of the pseudogap between the 2nd and the 3rd band at the L point and the gap width was investigated (Figure 2.15). For the position of the gap, a linear dependence is expected. The gap width increases with increasing refractive index and converges or even decreases for very high refractive indices.
0.60 2.50
refractive index n
Figure 2.15: Gap width and gap center position of an inverted opal depending on the refractive index of the surrounding. The refractive index of the spheres was n = 1. The relative gap width is the distance of the 2nd to the 3rd band at the L-point divided by the gap center position.
22
2.3 Different Photonic Crystals Diamond structure Another 3D photonic crystal is the diamond structure. Like the opal, this structure consists of spheres in a cubic lattice. For this reason the diamond has the same Brillouin zone as the opal. The difference between the opal and the diamond is that the diamond has two spheres in the irreducible Wigner-Seitz cell, whereas the opal has only one. a) b)
Figure 2.16: Schematic sketch of the diamond structure a) and first Brillouin zone b).Top view on the cube corresponds to the Γ-L direction in the Brillouin zone. Techniques exist to produce diamond structures, such as micro robotic [Gal02] or interference lithography [Cam00]. However, none of these techniques are capable of producing diamonds on a larger scale. The diamond structure has a lot of interesting properties, though. This may already be seen in the band structure (Figure 2.17). The relative width of the pseudogap between the 2nd and the 3rd band in Γ-L direction is broader than for the opal. The same holds for the corresponding gap in the Γ-X direction.
normalized frequency (a/λ)
1.5
1.0
0.5
0.0
X
U
L
Γ
X
W
K
crystallographic orientation
Figure 2.17: Band structure of the diamond structure. The considered diamond consists of spheres with a refractive index of n = 1.5. The surrounding medium is air. The band structure is similar to that of the opal, however some photonic characteristics are stronger pronounced. This manifests itself in broader widths of the band gaps (red circles) The consequence of a broader band gap is that for a desired effect thinner crystals may be used. The diamond, therefore, stands here as an example for a photonic crystal that could offer solutions for problems arising with opals, if methods like interference lithography could be applied to photovoltaic concepts. The spectral reflection characteristic of the diamond structure is given in Figure 2.18.
23
Photonic crystals
reflection [%]
0.8
0.6
0.4
0.2
0.0 0.40
0.45
0.50
0.55
0.60
wavelength [nm]
Figure 2.18: Simulated reflection characteristic of the diamond structure. The diamond consists of spheres with a diameter D = 173nm and a refractive index of n = 1.5. The surrounding medium is air. The diamond consists of 40 sphere layers. The reflection peak corresponds to the pseudogap between the 2nd and the 3rd band at the L-point.
2.4 Summary of the chapter In this chapter I have introduced photonic crystals. Per definition in this work, photonic crystals are ideal materials that exhibit a photonic band gap, a region within certain frequencies and for certain k-vectors in which no states for a photon exist within the photonic crystal. The convention in this work is to call non-ideal photonic crystals photonic structures. Photonic structures exist in nature but may also be produced artificially with different techniques. An important property of photonic crystals is a periodicity of the refractive index with a period length in the range of the considered light. In such an arrangement, phenomena occur for photons that are similar to the ones occurring for electrons in a solid state crystal. In that sense, a photonic crystal is a crystal for photons as a solid state crystal is a crystal for electrons. A very important example of such a phenomenon is that the dispersion relation of a photon in a photonic crystal may be displayed in the band diagram. In section 2.2 the mathematical tools needed to understand the concept of the band structure were introduced and the analogy to the band structure of solid state physics was drawn. The heart and soul of this analogy is the similarity of the operator, acting on the magnetic field derived from Maxwell’s equations (for the photonic band structure), and the Hamilton operator, acting on the periodic potential of the atom rumps (for the solid state crystal). The most important commonality of these operators is that they are both Hermitian. The most important difference is the vectorial character of the EM wave. A solution has to be found for the magnetic field and the electric field has to be calculated from the magnetic field, because the operator on the electric field, derived in the same way, would not be Hermitian. In section 2.3, different photonic crystals are introduced. To these crystals the formalism from the previous section is applied and the band structure is calculated. Additionally, the optical properties are investigated. The exemplary photonic crystals are the ones that were most frequently used in the scheme of this work. Among them are the rugate filter and the opal.
24
3 Simulation of photonic structures In this chapter I will introduce the simulation methods that were used for the calculation of the optical properties of photonic structures. The two methods that have been used most frequently are the approach of characteristic matrices and the rigorous coupled wave analysis (RCWA). Both methods are introduced in section 3.1. The approach of characteristic matrices is well known and is applied commonly for PV systems, such as for the calculation of antireflex layers. Most 1D photonic structures were simulated with this method. With the RCWA method, the response of almost any kind of periodic structure to incident light is calculated. The code which has been used in the scheme of this work is based on MATLAB and is called “Reticolo Code 2D”. The simulation of photonic structures with this code is discussed in section 3.2. In the RCWA method, a given structure is approximated for several reasons that are discussed in section 3.3. The corresponding parameters have been varied to investigate the convergence. Finally, in section 3.4, a comparison is given to other simulation methods and differences in the results of the different simulation methods are discussed.
3.1 Simulation methods For the simulation of the structures discussed in this work most frequently, two simulation methods have been used. The first is the approach of characteristic matrices. This method is used to calculate the optical properties of thin films and assemblies of thin films. The technique is well established and has been known for a long time. For structures more complex than a thin film system, the rigorous coupled wave analysis (RCWA) was used. This method is discussed in some more detail, as the interpretation of the results of this simulation method demands comprehension of the method and the implementation. Simulation of optical thin films The description of the theory of optical thin films follows an approach of McLeod [Mac011]. The theory of optical thin film starts with the description of the phenomenon called “Newton Rings”, described by Robert Boyle and Robert Hooke in ca. 1660. About 150 years later, in 1801, Thomas Young enunciated the principle of the interference of light. This theory was later (ca. 1820) framed into an elegant theory of diffraction formulated by Fresnel [Ser70], who combined the interference theory from Young with the Huygens idea of light propagation. At about the same time (1817), Joseph Fraunhofer [Fra88] produced the first antireflection coatings. The formulation of the work of J.C. Maxwell’s A treatise on electricity and magnetism [Max73] and his system of equations in 1873 completed the theoretical framework needed for the description of optical thin films. Further important results were the experimental verification of Fresnel’s reflection law by Lord Rayleigh in 1886 [Ray86] and the development of the Fabry-Pérot interferometer in 1899 [Fab99]. An optical thin film is a film with a thickness d which is smaller than the coherence length of the considered light. The coherence length, therefore, marks a natural boundary for the validity of this method. Single thin films or assemblies with a thickness considerably larger than the coherence length may not be considered with this method. For a complete derivation of the method see Appendix B. Here the method is only sketched, with the situation described by the method in Figure 3.1.
25
Simulation of photonic structures
Figure 3.1: Plane wave incident on a thin film The basis of the method forms the approach that the change of the electromagnetic field throughout a thin film can be expressed in the form of a 2x2 matrix.
i sin δ / η ⎤ ⎡ Eb ⎤ ⎡ E a ⎤ ⎡cos δ ⎢ ⎥=⎢ ⎥⎢H ⎥ ⎦⎣ b ⎦ ⎣ H a ⎦ ⎣iη sin δ cos δ
(3.1)
In this equation, Ea and Ha represent the electric and magnetic field at the front interface of the thin film, Eb and Hb the fields at the rear interface. The factor δ is a phase factor and η is the so-called tilted optical admittance. The phase factor is defined by
δ =
2π ⋅ N ⋅ d ⋅ cosθ
(3.2)
λ
with N the complex refractive index of the film. The optical admittance depends on the polarization of the incident light and is given by
ηTE =
ε0 N cosθ μ0
ηTM =
ε0 1 N μ0 cosθ
(3.3)
When considering an assembly of thin films, each thin film is represented by a characteristic matrix and the effect of the electromagnetic field of the assembly is calculated by multiplying the individual matrices (each with corresponding phase factor and admittance).
i sin δ r / η r ⎤ ⎞⎟ ⎡1 ⎤ ⎡ B ⎤ ⎛⎜ r = q ⎡cos δ r = ⎥ ⎟ ⎢η ⎥ ⎢C ⎥ ⎜ ∏ ⎢iη sin δ cos δ ⎣ ⎦ ⎝ r =1 ⎣ r r r ⎦ ⎠⎣ m ⎦
(3.4)
In equation (3.4), unlike in equation (3.1), the characteristic matrix is defined independently of the electromagnetic field. The entries B and C here define the characteristic matrix of the assembly of thin films. Knowing the effect of an assembly of thin films on the electromagnetic field, the optical properties of the assembly may be calculated by 2
η B−C , R= 0 η0 B + C
T=
4η 0 Re(η m )
η0 B + C 26
2
,
A = 1− R −T
(3.5)
3.1 Simulation methods The RCWA method The method described here originates from the problem of determining the optical properties of diffractive structures. First attempts here have been made by Burcardt [Bur66] and Kogelnik [Kog69]. The gratings referred to in their works were sinusoidal modulations of the refractive index in a thick resist. Because of the, for this problem, simple form of the grating, the mathematical effort was manageable and relatively uncomplicated. Straightforward solutions could be obtained. It has to be mentioned that in the coupled wave formulation of Kogelnik, only one diffracted wave is allowed and second derivatives are neglected. From these approaches, two different procedures were distinguished. In the rigorous coupled wave analysis (RCWA), different modes that couple while they propagate in the medium are considered. In the Fourier modal method (FMM), each Fourier mode propagates independently through a medium that is described by an effective refractive index. The coupled wave formalism as formulated by Kogelnik provides a comparably simple formulation and a good physical insight into wave diffraction, but it is not exact. The modal theory, on the other hand, is inherently accurate but its formulation is rather complicated. As Magnusson and Gaylord could show [Mag78], both formulations are equivalent and rigorous if an infinite number of modes / orders of diffraction are considered. For a complete derivation of the RCWA method, see Appendix C. Here the method is only sketched. A first remark here has to be made concerning the denomination of dimensions. A difference exists between the denomination for photonic structures and for problems implemented into the RCWA method. A photonic structure is called 3D if the variation of the refractive index occurs in all three spatial dimensions. 2D photonic crystals have a constant profile in one direction and the refractive index for 1D photonic crystals is not varied in two spatial dimensions. For a problem implemented in the RCWA method, the denomination of the dimensionality refers to the number of spatial directions in which the structure may be periodic and has an infinite expansion. A photonic structure with a finite thickness, like the opal, would therefore be considered a 3D photonic structure (because the refractive index varies in all three spatial dimensions), but would be implemented into a 2D code (because the expansion in one direction is not infinite). The problem which shall be solved by the RCWA is sketched in Figure 3.2.
Figure 3.2: Sketch of an arbitrary 1D grating and nomenclature. The approach is to first look for solutions in region I and III and then to describe the solution in region II as a superposition of the solutions in region I and III and matching the boundary conditions. It is important that each polarization has to be considered separately, as the approach is slightly different (for TE polarization the electric field is used, for TM
27
Simulation of photonic structures
polarization the magnetic field). The sketch here shall be given for TE polarization and only for the 1D problem; the generalization is found in Appendix C. The mathematical formulation of the problem is to find solutions for the characteristic scalar Helmholtz equation for Ey. All other field components may be derived from Maxwell’s equations.
1 ∂² 1 ∂² Ey + E y + (iσω + εε 0ω ²) E y = 0 μ 0 ∂x ² μ 0 ∂z ²
(3.6)
These must satisfy the transition conditions at the boundaries (field components and their deviations must be continuous). An approach here is to construct this solution out of plane waves. Each plane wave than corresponds to one discrete order of diffraction. When a plane wave interacts with a grating, the only difference between the wave function at different periods is the phase. The field Ey must therefore be a pseudoperiodic function and may be written as
E y ( x + Λ, z ) =E y ( x, z ) exp(ik x Λ )
(3.7)
To obtain a periodic function this function may be transformed into
Fy ( x, z ) = E y ( x, z ) exp(−ik x x)
(3.8)
with Fy(x, z) a periodic function with the period Λ. As Fy(x, z) is periodical, the function can be expanded into a Fourier series, and consequently Ey can be written as
E y ( x, z ) =
m =+ ∞
∑F
m = −∞
m
⎛ 2πimx ⎞ ( z ) exp⎜ ⎟ exp(ik x x) ⎝ Λ ⎠
(3.9)
with Fm(z) the Fourier coefficients of the function Fy(x, z). Every term of this sum is identified with one mode. The incident wave defines kx and the periodicity defines the different discrete x-components of all further possible k-vectors of the single modes. These k-vectors will in the following be denoted as k xm = k x + 2πm / Λ . By inserting equation (3.9) into (3.5), the following equation is obtained
Fm ( z )(iσμ 0ω + εε 0 μ 0ω ² − k xm ² ) + Fm′′ ( z ) = 0
(3.10)
In region I and III the solutions of this function are obtained in a straightforward manner. With σI = σIII=0 and εε0µ0 =n²/c², the solutions of equation (3.10) are of the form
Fm ( z ) = Fm+ exp(ik zm z ) + Fm− exp(−ik zm z )
(3.11)
with 2
k zm =
n²ω ² 2 − k xm c²
With this approach, the field Ey in region I and III is given by
28
(3.12)
3.1 Simulation methods
E yI ( x, z ) = E yinc ( x, z ) +
E yIII ( x, z ) =
m =+ ∞
m =+ ∞
∑ (F
m = −∞
∑ (F
m = −∞
+ m
+ m
exp(ik zm z ) + Fm− exp(−ik zm z )) exp(ik xm x)
exp(ik zm z ) + Fm− exp(−ik zm z )) exp(ik xm x)
(3.13)
(3.14)
In equations (3.13) and (3.14), Eyinc is the corresponding component of the incident field. As a result of the Fourier expansion, the single plane waves occurring in equation (3.13) and (3.14) have discrete directions that depend on the angle of incidence and the period, but not on the exact geometry of the surface. To obtain the solutions in region II, the modulated structure is separated into N sufficiently thin layers that are parallel to the x-y plane. Sufficiently thin here means that σn(x ,z) and εn(x, z) in each layer n may be regarded as invariant in the z-direction. Initially, a separation approach is used for the directions
E yn ( x, z ) = X n ( x) Z n ( z )
(3.15)
and equation (3.5) is formulated for each direction separately
Z n′′ ( z ) + g n2 Z n ( z ) = 0
(3.16)
X n′′ ( x) + (iσ n ( x) μ 0ω + μ 0 ε 0 ε n ( x)ω ² − g n2 ) X n ( x) = 0
(3.17)
with the separation constant gn. For Zn(z) the solution is straightforward and is of the form
Z n ( z ) = a n exp(ig n z ) + bn exp(−ig n z )
(3.18)
Following the discussion given earlier, Xn(x) has to be pseudoperiodic and can therefore be expanded into a Fourier series
X n ( x) =
m =∞
∑X
m = −∞
nm
exp(ik xm x)
(3.19)
To determine Xn(x) the term iσ n ( x) μ 0ω + μ 0 ε 0 ε n ( x)ω ² needs to be expanded into a Fourier series as well. This is possible because σ n (x) and ε n (x) are per definition periodic functions with the period Λ.
iσ n ( x) μ 0ω + μ 0 ε 0 ε n ( x)ω ² =
p =∞
∑a
p = −∞
np
⎛ 2πipx ⎞ exp⎜ ⎟ ⎝ Λ ⎠
(3.20)
To obtain an equation for each Xnm equations (3.19) and (3.20) have to be inserted into (3.17) and terms with the same exponent must be combined.
g n2 X nm =
p =∞
∑ (a
p = −∞
nm − p
29
− k xp2 δ mp )X np
(3.21)
Simulation of photonic structures
The set of equations for each layer may be written in form of an infinite quadratic matrix
Mˆ n with the entries a nm − p − k xp2 δ mp
r r Mˆ n X n = g n2 X n
(3.22)
2 This is a typical Eigenvalue problem and the qth Eigenvalue g nq and the corresponding
r
Eigenvector X nq can be calculated with numerical methods integrated e.g. in MATLAB. By
r
using equations (3.18) and (3.19), every Eigenvector X nq may be used to form a solution of the homogenous differential equation (3.5)
E ynq ( x, z ) =
m =∞
∑X
m = −∞
nqm
exp(ik xm x)(a nq exp(ig nq z ) + bnq exp(−ig nq z ))
(3.23)
The complete field in each layer is than given as the sum over all of these functions
E yn ( x, z ) =
q, m = ∞
∑X
q , m = −∞
nqm
exp(ik xm x)(anq exp(ig nq z ) + bnq exp(−ig nq z ))
(3.24)
Up until now, the Fourier model method (FMM) and the RCWA were equivalent. The difference between the methods lies in the interpretation of the single terms. From the Fourier modal method, for every layer a set of plane waves is obtained that each see different effective refractive indices. In the RCWA on the other hand, modes propagate in each layer and the z-dependence of the mth mode is traced. In the z-direction, the function depends on the amplitudes of the (m+1)th and (m-1)th mode; only neighbouring modes couple. Equation (3.24) is a rigorous description of the field Ey. In the numerical implementation, infinite sums and matrices limit the accuracy of the method. The problem of infinite sizes is solved by cutting the matrices. With this procedure, information is lost and it must be admitted that a marginal amount of energy is transported in the modes that are cut away. The calculation time scales with the size of the matrix in the third power, which is a problem if many modes have to be considered. A rule of thumb is to use a number of modes that is twice the number of propagable modes. In summary, the discussed method is appropriate to the calculation of periodic structures with a period in the range of the considered light wavelength. The larger the period becomes, and the higher the refractive index in the structure is, the more modes have to be considered for an accurate result. This leads to a rapid growth in the time needed for the calculation or a decreased accuracy of the result. In this context, metallic gratings with a period larger than the considered light wavelength are especially problematic to calculate.
3.2 Reticolo Code 2D For the implementation of photonic structures, a RCWA code [Moh95] has been used that was written at the University of Paris [Lal98]. This code is called “Reticolo Code 2D”. Some aspects of the implementation of photonic structures into the code need to be introduced to delineate where and how the calculation is an approximation. Details about how the implementation is performed are given in Appendix D.
30
3.2 Reticolo Code 2D Definition of the structure The first step is to find a unit cell in a rectangular representation. It is beneficial to find a unit cell that is as small as possible, because the computational effort increases with increasing unit cells. This will be done for the example of the face centered cubic structure (fcc). Two representations of this system are shown in Figure 3.3. b) a)
Figure 3.3: Two different representations of the fcc structure. Figure 3.3a shows the straightforward representation with a cubic unit cell and additional atoms at the face centers. Figure 3.3b shows the unit cell in the hexagonal representation. Both representations have disadvantages for the implementation in the “Reticolo Code 2D”. Figure 3.3a is not the smallest possible unit cell. Especially disadvantageous is additionally that the direction in which fcc crystals crystallize is not the normal direction. Figure 3.3b. is the smallest unit cell but is not representeable in a rectangular system. Both representations shown in Figure 3.3 cannot be used for an implementation in the “Reticolo Code 2D”. Figure 3.3b is not representeable in a rectangular unit cell and Figure 3.3a. represents the wrong crystallization direction when considering real fcc structures. These examples illustrate the difficulties emerging when implementing an arbitrary crystal system into the “Reticolo Code 2D”. The unit cell needed for the implementation is shown in Figure 3.4.
Figure 3.4: Unit cell for an implementation of the fcc structure into the “Reticolo Code 2D”. Three layers of spheres have to be considered and a rectangular cut through these layers with the ratio 1 : 3 defines the unit cell.
31
Simulation of photonic structures
Simulation of the optical far field The complete information given as a result of the RCWA simulation cannot be surveyed easily. Fortunately, not all of the information is needed at once for a given problem. For this reason, it is a delicate task to choose a certain representation or a set of representations that in combination provide an accessible overview of the results needed for a certain problem. This is why some effort was invested into representations. The topic of this part is to introduce some representations that are not commonly used and to discuss the abilities and disabilities of these representations. Typically, simulations are performed to obtain a certain characteristic. One or several input parameters are varied and the effect on one or several output parameters is observed. One very important characteristic is the spectral RTA (reflection / transmission / absorption) characteristic. This characteristic typically is given in an x-y diagram with the wavelength on the x-axis and the RTA on the y-axis. The total reflected / transmitted efficiency for one polarization is calculated by summing up the efficiencies of each reflected wave. To obtain the total RTA, the average of the values obtained for both polarizations is used, i.e. (TE+TM)/2. Figure 3.5 shows a typical RTA diagram. In the context of diffractive structures, it is important to notice that this representation contains no information about the diffracted orders, i.e. the direction of the reflected or transmitted light.
reflection / transmission / absorption [%]
100 80 60
reflection transmission absorption
40 20 0 300
400
500
600
700
800
wavelength [nm]
Figure 3.5: RTA characteristic of an opal inverted with TiO2 calculated with “Reticolo code 2D”. The calculated magnitudes of R and T are the sum over the efficiency in each order and the average of both polarizations. The absorption A is given as A= 1-R-T. If the diffractive properties of the crystal are examined other representations are needed, depending on the relevant information. In solar cell systems, an interesting piece of information is the polar angle of the light. The polar angle defines, for example, the path length of the light in the solar cell. For diffracted light, the intensity diffracted into the different orders is also of interest. This information is combined and represented in the polar / intensity plot (Figure 3.6). Typically in the polar / intensity, the intensity and polar angle for a certain spectral range are given. This representation gives an overview of the polar angles and the relative intensities in the different orders.
32
3.2 Reticolo Code 2D
polar component of transmission
90 10
0
10
-1
10
-2
120
60 30
150
180 10
-2
10
-1
10
0
0
330
210 240
300 270
Figure 3.6: Polar / intensity plot. The polar component of the transmitted / reflected orders is plotted against the intensity. In the example, the transmission of an opal with spheres of a diameter D = 290nm and a refractive index of n = 1.5 on a substrate with refractive n = 1.5 is shown. The different points mark the results for different wavelengths of the incident light (λ =300nm – 700nm). For 2D and 3D photonic structures, it is sometimes not sufficient to have an overview of the polar angle but also over the azimuthal angle. Both angles together define the direction of the transmitted or reflected orders. This information may be represented by a polar / azimuthal plot which plots the direction of the transmitted / reflected radiation for diffuse incidence. The polar / azimuthal plot contains three parameters, the polar component represented by r, the azimuthal component represented by δ, and the intensity represented by a colour. In the representation, r = 0 corresponds to a polar angle of θ =0° and r = R corresponds to a polar angle of θ =90°. A linear dependence is chosen so that R/2 corresponds to θ =45°. The representation is therefore not a projection on the hemisphere. A polar / azimuthal plot is shown in Figure 3.7. Typically, the polar azimuthal plot is given for incidence from the complete hemisphere and a single wavelength of incidence.
Figure 3.7: Polar / azimuthal plot. This plot shows the transmission of an opal on a glass substrate for incident light from the complete hemisphere. The 0th order (direct transmission) is refracted and is transmitted under angles smaller than θ =41°. A first order transmission occurs with little intensity (blue regions). The sixfold symmetry of the fcc structure is perceptible for the first order of transmission.
33
Simulation of photonic structures
The polar / azimuthal plot is also used for another purpose. Diffractive processes cause light incident under certain directions to be coupled into the crystal. The effect of this coupling is that the reflection of the crystal may be reduced even for great angles of incidence. The reflective properties of a surface for a certain wavelength may also be represented by a polar / azimuthal plot. In this case, the polar and azimuthal component defines the direction of the incident light and the colour gives the complete reflection. Such a representation is shown in Figure 3.8. The representation shows the reflectance characteristic for a single wavelength of incidence.
Figure 3.8: Polar azimuthal representation of the reflectance depending on the direction of incidence. The direction of incidence is defined by the polar and the azimuthal component. The colour represents the reflection summed up over all orders. The red circle marks a region where the opal couples in light under very high angles and therefore shows a low reflection compared to Fresnel reflection under this angle. Simulation of the optical near field Additionally to the optical far field also the optical near field is calculated with the RCWA method. The optical near field corresponds to solutions in region II (Figure 3.2). The information about the near field contains the intensity of all six field components (Hx, Hy, Hz. Ex, Ey, Ez.). As an example, the simulated near field inside an opal structure is plotted in Figure 3.9. The distribution of the optical near field is interesting for solar cell applications, as the light distribution inside the solar cell defines the generation of charge carriers. The information in the optical near and far field also marks the range of what is possible with the RCWA method in the field of PV applications. Optical properties as the magnitude of reflection or absorption can be predicted. The angular characteristic, including diffractive effects, can be obtained and the distribution of the generation of charge carriers can be gained.
34
3.3 Convergence considerations
Figure 3.9: Simulation of the z-component of the electric field inside an opal calculated with the RCWA method. The plot shows four periods in the x direction and seven layers of spheres in the z-direction.
3.3 Convergence considerations In principle, the RCWA method is rigorous in the calculation of the EM field. However, the obtained solutions are approximations for several reasons: The first reason is that the number of considered modes or Fourier components (nfur) is finite. In practice, one is limited by the computational time and the available memory. Often a compromise between accuracy and time invested has to be made. The second reason lies in the assembly of a given structure in “Reticolo Code 2D”. Here it must be distinguished between the decomposition in the z-direction and the composition of inclusions in the x-y plane. The decomposition in the z-direction lies in the principle of the RCWA method. A given structure is separated into layers that are considered to have a binary refractive index profile. Consequently, the structural profile in the z-direction may only be described by a step function. The accuracy with which a structural profile in the zdirection is described depends on the number of layers (nd) used. It is important in this context that most photonic structures considered in this work consist of spherical forms. An implementation of such a crystal will therefore always be an approximation. The assembly of an inclusion in the x-y plain is rooted directly in “Reticolo Code 2D”. In the “Reticolo Code 2D” it is only possible to implement rectangular shapes. This means that other forms like circles have to be constructed out of rectangles. This is done by combining an increasingly higher number of rectangles (ndi) to approximate a given structure. To study the three factors influencing the accuracy of the obtained result, a simple exemplary system to perform convergence calculations was chosen. This simple system is a photonic structure consisting of spheres ordered in the simple cubic lattice. The spheres implemented had a refractive index of n = 1.5, which was chosen because most photonic structures consisted of materials with a refractive index in this range (different polymers, SiO2), and a diameter of D = 400nm, which is also in the range of diameters typically used. The crystal under consideration consisted of twelve layers of spheres.
35
Simulation of photonic structures
a)
c)
b)
Figure 3.10: Approximations used when calculating the EM field with the RETICOLO Code.Figure 3.10a shows the decomposition of a binary structure (dashed line) into a Fourier series. In the example, three Fourier modes are used. Figure 3.10b shows the approximation of a circular structure by equidistant layers. In the example, nine layers are used to approximate the circle. Figure 3.10c shows the approximation of a circular structure by rectangles in the x-y plane. In the example, three rectangles are used to approximate the circle. The procedure to specify parameters for which convergence is claimed is problematic because these parameters always depend on the period length of the structure L, the refractive index n, and the wavelength λ. The exemplary system was chosen such that the size and the refractive index are in the same range as for most of the simulations performed. If period length, considered wavelength, and refractive index differ considerably from the ones considered here, the convergent parameters need to be adapted. Variation of the number of Fourier components (nfur) In a first simulation the number of Fourier components / modes (nfur) was varied. In the 3D code it is possible to vary the number of Fourier components for each the x- and the ydirections independently. However, for the sake of simplicity, an independent variation was abdicated and the number of components was varied simultaneously. If in the text it is said that three Fourier components are considered, it is always meant that three Fourier components in each, the x - and the y - direction are considered. All parameters varied influence position and form of the reflection peak. For this reason, the convergence of the spectrum around the reflection peak has been considered. To evaluate the convergence, several characteristics are observed. For one, the form of the entire spectrum is investigated. The reason to investigate the convergence of a spectrum and not just a single wavelength is that the characteristic of a single wavelength is not significant. The insignificance is caused by a change of the optical properties of the investigate photonic structure induced by a variation of the parameter. An example for this mechanism is the change of the average refractive index with the variation of the resolution of the implemented structure.. The properties of the photonic crystal determine its spectral behaviour. In an extreme case that means that within a variation of one parameter different photonic crystals are observed that may have completely different spectral characteristics. This effect cannot be observed when considering only a single wavelength but requires an observation of the full spectral characteristics. The convergence of a reflection peak, for example, is much better suited to evaluation, because spectral shifts and convergence are plainly visible. It must be noted, however, that for a spectrum the accuracy of the calculation with otherwise equal parameters is least accurate for the shortest considered wavelength. In the spectrum, therefore, convergence may occur for long wavelengths earlier than for short wavelengths. The number of Fourier components that need to be considered depends on the considered wavelength with nfur ~ 1/λ. The result of the convergence consideration of the spectrum against the number of used Fourier components is shown in Figure 3.11. For the given parameters, the spectrum
36
3.3 Convergence considerations converges very fast, and even if only one Fourier component is considered, the result is similar to the convergent one. Other characteristics apart from the form of the spectrum that may be considered for the convergence and that are more easily quantifiable are the magnitude of reflection in the reflectance and the full width at half maximum (FWHM) of the reflection peak. These characteristics are not connected to a fixed wavelength and are therefore not affected by spectral shifts. The characteristics of the corresponding variations are shown in Figure 3.11. a)
b) 160
maximum reflectance FWHM
93.5
155
93.0
150
92.5
145
92.0
0
2
4
6
8
10
FWHM [nm]
reflection [%]
94.0
140
# Fourier components
Figure 3.11: Dependence of the form of the reflection spectrum (a) and the magnitude of the reflectance as well as the FWHM (b) on the number of Fourier components considered. The other parameters for this simulation were: Number of layers for the hemisphere nd = 10, number of rectangles to form a circle ndi = 10. The maximum reflectance converges very fast to a value of R = 92.5%. Even if only one Fourier component is considered, the relative error for the maximum reflection is below sR = 1%. The FWHM shows no variation at all for the given resolution. Note here that due to the discrete number of points considered for each calculation, the FWHM may only change in steps of 10nm. The magnitude of the reflection peak is therefore the more accurate measure. These results show that for the investigated simple system the number of considered Fourier components is not critical. This is because of the low refractive index contrast, the absence of absorption and the comparably simple unit cell constituting this system. The low index contrast and the absence of absorption also hold for most of the other 3D crystals that were investigated, especially the opals. However, the unit cell of the opal is more complex so more Fourier components need to be taken into account. For systems with higher refractive index contrasts and especially systems with absorption, the number of Fourier components needed increases drastically. This was especially the case for gratings implemented into silicon solar cells. In these cases, a decision about the number of Fourier components used had to be made from case to case. The necessity to keep the number of Fourier components as low as possible emerges from the time t consumed for the calculations. Dependence for the time proportional to the number of Fourier components to the power of six is expected 3. The dependence on the time consumed is shown in Figure 3.12
3
That is, for every direction, the time depends on the 3rd power of the number of Fourier components taken into account. This results in a dependence on the 6th power for both directions.
37
Simulation of photonic structures
6th order polynomial time consumed
time consumed [a.u.]
2000 1500 1000 500 0 0
2
4
6
8
10
# Fourier components
Figure 3.12: Dependence of the time consumed on the number of Fourier components considered. The theoretical expectation of time dependence proportional to the 6th power of the number of considered Fourier components is roughly confirmed. Variation of the number of segments in z-direction (ndi) In a second simulation series the number of layers (ndi) out of which the hemisphere is constructed was varied. The number of layers defines the accuracy with which a hemisphere is represented in the z-direction. In the simulation, all layers had the same thickness. Again, I have considered the influence on the entire spectrum as well as the magnitude of the reflectance and the FWHM of the reflection peak. The effect of the number of segments to create a hemisphere on the entire spectrum is shown in Figure 3.13a. The result is refined when looking at the magnitude and the FWHM of the reflection peak. These characteristics are shown in Figure 3.13b. b) a)
maximum reflectance FWHM
reflection [%]
96
180
94 160 92 140
90 88
FWHM [nm]
200
98
0
5
10
15
20
120
# segments z-direction
Figure 3.13: Dependence of the form of the reflection characteristic (a) and the magnitude of the reflection peak, as well as the FWHM, on the number of segments in the z-direction under consideration. The number of segments gives the number of layers that are used to form a hemisphere, as the complete sphere is constructed from the hemisphere by a simple symmetry. The number of layers to construct the complete sphere is therefore twice ndi. The other parameters used are nfur = 4, nd = 20. For one segment, the reflection peak is strongly delayed compared to the rest of the figure. This is because for only one segment, the obtained crystal consists of cylindrical holes through the crystal and the system is better described as a tetragonal 2D crystal. This 2D crystal shows a different band structure, which explains the delayed reflection peak. With an increasing number of segments, the form of the spectrum converges quickly. Maximum reflectance and FWHM also converge fast, and for more than ten segments, the deviation
38
3.3 Convergence considerations from the convergent value become negligible. Consequently, for the investigated sphere sizes, the hemisphere should be approximated by ca. 20 layers. For the given sphere size, this corresponds to a thickness of the layers of dz = 20nm and an optical thickness of d z ⋅ n = 30nm or ca. 1/30 of the incident wavelength. This value of the optical thickness is used as a benchmark for the decomposition of a structure in the z-direction. Again, it must be considered that convergence occurs at different speeds for different wavelengths. The needed thickness scales with 1/λ. The given optical thickness of 30nm represents a value for orientation for systems with comparable parameters as the ones considered. Additionally, the time consumption for an increasing number of layers used was examined. The expectation here is a linear dependence of the time consumed on the number of segments. The results of the respective calculations are shown in Figure 3.14.
time consumed [a.u.]
400
time consumed
350 300 250 200 150 100 50 0
0
5
10
15
20
# segments z-direction
Figure 3.14: Dependence of the time consumed on the number of segments in the zdirection. The expectation of linear time dependence is roughly confirmed. Variation of the number of segments in x-y plane (nd)
100
180
80
160
60
140
40
maximum reflectance FWHM
20 0
120
FWHM [nm]
reflection [%]
Finally the convergence for the number of segments to construct a circle in the x-y plane (nd) was investigated. The number of segments here defines the number of rectangles used to construct a circle (see Figure 3.10). Also for this consideration, the form of the entire spectrum, the maximum reflectance and the FWHM have been investigated. Figure 3.15 shows the results of the simulated form of the spectrum, the magnitude of the reflection peak, and the FWHM. a) b)
100
0
5
10
15
20
25
30
80
# segments x/y direction
Figure 3.15: Dependence of the form of the reflection characteristic and the magnitude of the reflection peak, as well as the FWHM, on the number of segments in the x-y plane considered. The other parameters used are nfur = 4, ndi = 10. The number of segments was increased in steps of one from one to 20 and in steps of five from 20 to 30.
39
Simulation of photonic structures
Form of the spectrum, maximum reflectance and FWHM converge quickly, and for more than ten segments, the deviation from the convergent value becomes negligible. It is more difficult to translate this result to other period lengths or refractive indices than it is for the parameter ndi. For this reason, both values ndi and nd were coupled in the simulations. As a rule of thumb, the same number of segments in the x-y direction was used as in the zdirection. To conclude this section, the time consumption depending on the number of segments in the x-y direction is shown in Figure 3.16
time consumed [a.u.]
5000
time consumed exponential fit
4000
y =exp(a+b*x+c*x^2) a 3.51086 ±0.07971 b 0.26434 ±0.00724 c -0.00321 ±0.00016
3000 2000 1000 0 0
5
10
15
20
25
30
# segments x/y direction
Figure 3.16: Dependence of the time consumed on the number of segments in the x-y direction. The characteristic is described by a corrected exponential function in the considered region. It is difficult to predict theoretically the time consumed for this variation.
3.4 Comparison to other simulation methods Comparison to band structure calculations From the band structure calculation the position of the reflection peak of the opal is given as the centre frequency between the 2nd and the 3rd band. As this value is given in terms of a normalized frequency a/λ, a linear dependency is expected between peak position and sphere diameter. The peak position is defined by
λ peak (d ) ≈
d 2 ⋅ 0.6343
(3.25)
The characteristic factor 0.6343 is obtained from the band structure calculations for an opal made of spheres with a refractive index nS = 1.5. The factor 2 is the ratio between the sphere diameter and the lattice constant of the fcc crystal. To compare the RCWA method to the band structure calculations, one possibility is to reconstruct the characteristic factor 0.6343 from the RCWA simulation. As with the RCWA method, the result only depends on the ratio λ/Λ and the same linear dependence is expected. However, a peak position may only be calculated with limited accuracy; therefore, the dependence of the peak position on the sphere diameter was calculated for different sphere parameters and fitted linearly. This has been done for statistical reasons; the position could as well have been obtained from a single calculation. From the analysis of the line, the characteristic value for the RCWA simulation was obtained and is given by 0.6335 ± 0.0008. RCWA simulation and band structure simulation are therefore in good accordance with one another.
40
3.4 Comparison to other simulation methods Comparison to other simulation programs
norm. reflectance [%]
To evaluate the results of the RCWA methods, comparisons of the simulation results to other methods and measurements have been made. The simulations for the comparison have been performed at the University of Jena. The applied methods were the Fourier Modal Method (FMM, see section 3.1), the Korringa - Kohn - Rostoker (KKR) method [Ste98],[Ste02], and a model based on the finite difference time domain (FDTD) [Yee66] approach. The production and measurement of the opal was performed at MLU Halle. The result of the comparison is shown in Figure 3.17. .
100
measurement RCWA simulation KKR FDTD FMM
80 60 40 20 0 500
600
700
800
900
1000
wavelength [nm] Figure 3.17: Comparison of the reflection characteristic of an opal calculated with different simulation methods and compared to a measurement. All methods are in good accordance for the form of the reflection peak. Differences occur in the neighbouring regions. A first point which is noticeable is that the width of the measured reflection peak is considerably smaller than the width of the simulated ones. The decreased width is a consequence of imperfections in the opal. This phenomenon has been observed for several opals. Apart from that, the measured and simulated characteristics are in relative good accordance with each other. The measured opal was of very high quality, as can be seen with the presence of Fabry-Pérot oscillations and the comparably low reflection outside the peak regions, effects that would be influenced by scattering. For the simulation methods, form and position of the reflection peak are similar. However, several differences appear in the region next to the reflection peak. All methods implemented at the University of Jena are in very good accordance and it is not clear why the RCWA method, being completely equivalent to the used FMM method, showed differing results. The difference became clear when the attempt was made to match all simulation conditions. The exact implementation of the opal depends on more than the three parameters described. In fact, a first task was to find out which parameters really needed to be predefined.. In the simulations performed in Jena, exactly the same system was implemented in all three simulations used there. The implementation into the RCWA that I performed, on the other hand, differed in small ways from the one performed in Jena. Comparing the differences resulted in the finding that geometry factors play a major role in the reflectance characteristics outside the peak region. One example here is the stacking sequence. It is of importance if the order is ABC or CBA (see Figure 2.12a). Another point is that a finite opal may not be composed completely of unit cells, but on the beginning and the end if the crystal, the boundary conditions have to be defined. A third point is the number of sphere layers considered. The geometry of the system varies depending on
41
Simulation of photonic structures
whether a complete ABC stack is considered or not. There are probably even more points to consider; these remarks were meant to point out the large parameter space for making small changes in the geometry that all influence the reflection characteristics. As an example, the variation of the reflection characteristic for small changes in the geometry is shown in Figure 3.18. 100
rerlectance [a.u.]
80
stacking ABC stacking CBA other boundary
60 40 20 0 500
550
600
650
700
750
800
wavelength [nm]
Figure 3.18: Reflection characteristics of an opal for different geometries. On the one hand, the stacking sequence was changed (black and green line), and on the other hand, the geometry at the boundaries was varied from a simple unit cell to a complete sphere. All of these changes have a considerable effect on the reflection characteristics, especially outside the peak region. The impact of different geometries has not been considered in full detail and future work is needed here. As in the real opal, being imperfect, different geometries are realized; it is very hard to decide which geometry should actually be considered. To conclude, the RCWA method is a good method for the simulation of the reflection peak, but for regions outside the reflection peak, especially if different geometries are possible, the results should be interpreted with great care. 4 Evaluation of the RCWA method The RCWA (or FMM) method describes the distribution of the electromagnetic field interacting with a periodic structure by a superposition of plane waves, which is a very general description. For this reason, the RCWA method is capable of simulating a lot of different structures. On the other hand, no adaptation to special cases is possible, and for certain problems other methods are to be preferred. The RCWA method is very effective for the calculation of structures with a 1D periodicity and small periods. An example for such a case would be a dielectric grating with a period of below Λ = 1µm. The accuracy of the method is limited by the number of Fourier modes considered and by the concinnity of the composition of the structure. A rigorous result is obtained for an infinite number of Fourier modes and an infinite decomposition of the structure. For a finite number of each, the result becomes an approximation that is the less accurate, the fewer modes and decomposition steps are used. Numerically, the accuracy of the results is limited by two factors; the increase of size of the calculated matrices, which consumes memory, and the time needed to calculate these matrices. For the time consumption, the number of Fourier modes considered is the most critical factor. The time consumption increases with the 3rd power with the number of considered Fourier modes for each dimension (i.e. for a 1D problem with the 3rd power, for a 2D problem with the 6th power). For the number of 4 To handle the interpretation of the obtained results with great care is not only true for the RCWA method, but for all other named methods as well.
42
3.5 Summary of the chapter layers into which a structure is decomposed, the time consumption increases linearly. As one matrix is calculated for each layer, the memory consumption also increases linearly. These dependencies are principal and occur for all RCWA programs. A peculiarity of the “Reticolo Code 2D” is the construction of a circular inclusion within one layer. A circular inclusion is constructed out of rectangles and the time consumption increases with an exponential function. Practically, the time consumption limited the application of the program in most cases, as for the calculation of spectra many calculation points were needed. However, the memory requirements are also a serious limitation. Several simulations broke down because of shortages in available memory. The needed memory grows rapidly, both with the needed Fourier components and the decomposition steps. For visible light, the calculation of dielectric structures becomes problematic for periods larger than 1µm in the 2D case and periods larger than 5µm 5 in the 1D case. Compared to other simulation methods, the biggest advantage of the RCWA method is the flexibility and the possibility to implement nearly every structure. In that regard it is comparable with the FDTD method. FDTD is even more flexible than the RCWA method in the sense that even factors such as light sources within a structure can be implemented, which is impossible in the RCWA. FDTD is also extremely time- and memory-consuming, and the more complex a structure becomes, the more difficult becomes the simulation. The advantage of RCWA over FDTD is the inherent consideration of periodicity because of the Fourier expansion. The general flexibility of the RCWA method may be considered a disadvantage because other methods are better adapted to special symmetries. For example, when considering opals, the use of spherical waves instead of plane waves is of obvious advantage and the calculation is much faster than with the RCWA method. The disadvantage is also obvious; structures that contain non-spherical parts are problematic or impossible to implement. Methods going into this direction are the KKR method or other scattering matrix algorithms.
3.5 Summary of the chapter In this chapter I have introduced the simulation methods that have been used to simulate photonic structures. The first method is the approach of characteristic matrices, which was used to calculate the optical properties of assembled thin film systems. Such thin film systems may be used to describe all 1D photonic structures. This method is well-known and established in the context of PV systems. The second method is called rigorous coupled wave analysis (RCWA). The RCWA method describes the distribution of the electromagnetic field interacting with a periodic structure by a superposition of plane waves. This method is a very effective tool to simulate dielectric structures with periods below Λ = 1µm.Both simulation methods were introduced in section 3.1. The RCWA code, which has been used to simulate the given photonic structures, is MATLAB-based and was written at the University of Paris. It is called “Reticolo Code 2D”. The topic of section 3.1 is the implementation of photonic structures into the “Reticolo Code 2D” with a detailed description of input and output parameters and an exemplary program. With the code, information may be obtained about the optical near field and the optical far field of a structure under illumination. One topic of the section was therefore how the relevant information may be extracted and illustrated. Per se, the RCWA is a rigorous method. However, several parameters influence the accuracy of the obtained result. In section 3.3 I therefore investigated the convergence characteristics of these parameters. The first parameter is the number of Fourier components (nfur) considered. For a truly rigorous result, an infinite number of Fourier 5
These values are, of course, a rule of thumb because the limitations do not depend on the exact period length but on the number of Fourier components and the decomposition. However, the way I see it, the given periods are illustrative boundaries for the periods that can be reasonably considered.
43
Simulation of photonic structures
components would be needed. Naturally, an infinite number of Fourier components can’t be considered. The limiting factor for the number of components that can be considered is the computational time. Here a dependence on the 6th power is expected and was verified. In this work dielectric structures with periods smaller than Λ = 1µm in the visible range and near infrared (λ = 400nm – λ = 1200nm) were considered. For such structures and wavelengths and using the magnitude of the reflection peak as a measure, convergence of more than 99.9% was reached if six Fourier components were considered. In most cases already the use of three components lead to good results. This estimation of needed Fourier components given here is restricted to the mentioned dielectric structures with small periods and the mentioned wavelength range. If larger periods, other wavelengths or other materials (especially metals) are used the number must be adapted and may be much larger. The second parameter analyzed is the resolution with which a given structure is decomposed in the z-direction (ndi). This resolution is defined over the thickness of a single layer. The result for this parameter, again for the defined parameters, was that the optical thickness (thickness times the refractive index) of a single layer should be in the range of 30nm. This thickness must especially be scaled with 1/λ when using considerably different wavelengths. The third parameter was the number of rectangles that are used to represent a circle (nd). For the considered structure 10 rectangles lead to a good result. However, as this number depends on the size of a circle, the number of considered rectangles was coupled to the thickness of the single layers. Also this parameter needs to be scaled with 1/λ when using considerably different wavelengths. The results of these convergence considerations were used as benchmarks for the simulations of the photonic structures in this work. Finally, in section 3.4, I compared the RCWA method to other simulation methods. The comparison to the band structure calculation showed that the peak position of the reflection peak is in very good accordance with the gap centre position in the band structure. This is even true for very small sphere diameters. A comparison with other simulation methods (FMM, KKR and FDTD) showed a good agreement concerning the form and the position of the reflection peak but showed considerable differences outside the peak regions. These differences can be explained by different geometrical conditions in the different simulation methods. These geometrical conditions, however, were not investigated exhaustively and future work is needed here. Finally I want to give an evaluation of the RCWA method as it is realized in “Reticolo Code 2D”. In this work, the RCWA method was used as well for the simulation of optical far field to obtain the spectral and angular characteristics as for the simulation of the optical near field. For both applications proper results were obtained in a reasonable time that also showed a good agreement with measured results. These results were especially obtained for 3D photonic structures made of dielectric materials (SiO2, TiO2, polymers…) and 2D diffractive structures made of semiconductors (Si) in the submicrometer regime for incident light with a wavelength in the visible and infrared. The simulated results were especially accurate for the simulation of reflection peaks, however, outside the peak region, the result is extremely sensitive to the exact geometry and oscillations there must be interpreted with great care. Careful treatment is also required, considering that the simulated results are always for a single wavelength under the assumption of perfect coherence. As for PV applications white light is needed often several simulations with variations of period length and wavelength need to be done to identify coherence effects. Problems occurred both for memory and time, if the structure size was increased into the µm regime (always considering the mentioned wavelengths). An exemplary 3D structure considered could not be simulated any more for a period of more than Λ = 1µm. An exemplary 2D structure could be simulated up to a period of ca. Λ = 5µm. Another disadvantage of the method is that it is very difficult to implement non-ideality. If this needs to be done, the unit cell must be increased, and small divergences from periodicity need to be implemented. This can only be performed in a very limited way. To some extent I have
44
3.5 Summary of the chapter tried to include scattering into the method by superimposing the simulated spectrum and a spectrum simulated under the assumption of Mie- and Rayleigh scattering. At least some effects occurring in real photonic structures may be explained in that way. Impossible is e.g. the inclusion of light sources inside the photonic structure.
45
4 Spectrally selective photonic crystals In photovoltaic systems, spectral selectivity is typically used to guide a selected part of the solar spectrum to a specific solar cell that is especially efficient in this part of the spectrum. The most prominent example for such a concept is stacking solar cells with decreasing band gaps to absorb light with increasing wavelength successively. This concept works very well but has certain constraints as are the need for current matching and compatible atom lattices. Another concept featuring spectral selectivity is the fluorescent concentrator. The spectral selection here is provided by the emission characteristic of a dye. The disadvantage of this concept is that the collection efficiency of these concentrators is hitherto not satisfying. A third example that recently has enjoyed much attention, is the spectral selection with prisms. The approach of this concept is to distribute light onto several independent solar cells with different band gap energies. Highly efficient systems, exceeding η = 40% were announced using this technique. In this chapter I will investigate spectrally selective filters for applications in solar cell systems. The characteristic of a spectrally selective filter is that it is transparent for a certain spectral range and mirroring for another spectral range. The advantage of using spectrally selective filters is that they provide a very efficient method of spectral selection and, as they are independent components, allow for freedom in the concept design. In this chapter I will investigate two concepts. The first concept concerns the already mentioned fluorescent concentrator. Here, spectrally selective filters are used to increase the ability of the concentrator to transport light to a solar cell and therefore improve its collection efficiency. The second concept concerns spectrum splitting. The approach here is similar to the one described for the prism. Light is spectrally separated and distributed onto different solar cells that are laid out for different parts of the spectrum. In that way current matching and stacking problems are avoided. One advantage of using filters over using a prism is that, with a filter, a spatial separation of light is not connected with a separation of frequencies. Another advantage is the already mentioned freedom in system design. Both concepts, fluorescent concentrator and spectrum splitting, differ considerably concerning the demands on the applied filters. For this reason both concepts are handled separately. In most sections, a separate subsection exists for each concept. In section 4.1 the named PV concepts are introduced and described in more detail including the role of the spectrally selective filter. In section 4.2 the issue of the demands on the filters is addressed. When considering spectral selectivity, the question must be answered of how distinct the characteristic of the filter needs to be specified. Typically this question is twofold: first, how much reflectance is needed for a certain amount of light to reach the solar cell, and second, how much reflectance may be tolerated in the spectral range of transparency? The topic of section 4.3 is the actual design of photonic structures showing spectral selectivity. Among the many different possible photonic systems, three examples have been chosen for which experiments have been performed. These examples are the rugate filter (see section 2.3), the band stop filter, a 1D photonic structure with a discrete and non-periodic refractive index profile, and the opal (see section 2.3). An extensive analytical consideration about the light guiding ability of a fluorescent concentrator with a spectrally selective filter is given in subsection 4.2.1. The results of these considerations are applied to the filters discussed in section 4.3 and the expected performance is given in section 4.4. In section 4.5 experimental results are presented for both concepts. For the fluorescent concentrator concept, the enhancement of the light guiding efficiency is investigated and compared to the results of the model. For the spectrum splitting concept the measured quantum efficiency was compared to the predicted maximum constituted by the performance of the filters.
46
4.1 Photovoltaic concepts
4.1 Photovoltaic concepts In this section I will introduce and discuss two examples in which spectral selectivity is used in photovoltaic concepts. In the first example, the advanced fluorescent concentrator concept is discussed. Spectrally selective optical elements are used to eliminate the main loss mechanism here. As much of my work was done in regard to fluorescent concentrators, they capture the major part of this chapter and are discussed in much more detail than the other example. The second example concerns spectrum splitting. Spectrally selective filters here are used to direct incident radiation onto solar cells with different band edges. The goal of this concept is to illuminate each solar cell so as to produce the maximum total efficiency.
4.1.1 The advanced fluorescent concentrator concept Fluorescent or luminescent concentrators are a concept well-known since the late seventies [Web76, Goe77]. These devices allow for the concentration of sunlight without tracking systems. In addition, they not only utilize direct radiation but also the diffuse part of sunlight. Fluorescent concentrators were investigated intensively in the early eighties [Wit81, Sey89]. Research in those days aimed at saving costs by using the concentrator to reduce the need for expensive solar cells. However, several problems led to reduced research interest. First, the organic dyes used had only relatively narrow absorption bands. Secondly, although the organic dyes showed high quantum efficiencies of the absorption and reemission process above 95% in the visible range of the spectrum, quantum efficiencies remained at 50% and lower in the infrared. Furthermore, the dyes that were sensitive in the infrared were unstable under long-term illumination. Reabsorption of the emitted light due to overlapping absorption and emission spectra further reduced efficiencies [Wit81]. Based on conceptual progress and new materials, several groups (see e.g. Refs. [Luq04, Slo07, Deb07]) are currently reinvestigating the potential of fluorescent concentrators. Fluorescent Concentrators A fluorescent concentrator consists of a transparent matrix material like PMMA (polymethylmethacrylate) or another synthetic material. To this material a fluorescent dye is introduced. As shown in Figure 4.1, the dye absorbs light in a certain spectral absorption range and emits it in another spectral range. The photo-luminescent light is spectrally shifted from the spectral absorption range towards the red (Stokes shift). Because of the spectral shift, a reabsorption event of the light emitted by the dye is much less probable than absorption of the incident (and un-shifted) light. Therefore, the internally emitted light travels more or less undisturbed through the concentrator. The angular emission characteristic of the dye is omnidirectional. For this reason, a fraction of the internally emitted light is transported to the edges of the fluorescent concentrator via total internal reflection (TIR). This transport mechanism is used also e.g. in optical fibres. Spectrally matched solar cells are placed at the edges of the fluorescent concentrator and utilize the transported light. As the front surface of the fluorescent concentrator is larger than the side surface, the light impinging on the solar cell is concentrated.
47
Spectrally selective photonic crystals
Figure 4.1: Working principle of the fluorescent concentrator. The fluorescent concentrator consists of a transparent matrix material (e.g. PMMA) in which a fluorescent dye is introduced. Light in a certain spectral range is absorbed by the dye and reemitted at a higher wavelength (Stokes shift). Because of total internal reflection, a fraction of the light is internally transported to the edges of the concentrator, where it is utilized by solar cells. The main loss mechanism The reasons for the moderate efficiencies of the fluorescent concentrators are found in several loss mechanisms that reduce the internal light guiding efficiency of the system and therefore the amount of light reaching the solar cell. The most important loss mechanism is the loss through the escape cone of TIR (Figure 4.2). Only light emitted into an angular range with angles greater than a critical angle θc is totally internally reflected. The critical angle is given by
⎛1⎞ ⎝n⎠
θ c = arcsin⎜ ⎟
(4.1)
In this equation, n denotes the refractive index of the matrix material. All light emitted into angles smaller than θc only experiences Fresnel reflections but no TIR. In a first assumption, this light is considered lost 6. This assumption is founded on the fact that light emitted into the escape cone has to be reflected many times, each time with a reflectance considerably lower than one. Common materials for fluorescent concentrators (like PMMA) have a refractive index close to n = 1.5. 7 The critical angle is therefore θc=41.8°. The fraction of light lost inside this cone depends on the emission characteristics of the dye. This characteristic is unknown and only measurable by indirect methods. Ray tracing simulations performed by M. Bendig [Ben08] indicate, however, that the assumption of an isotropic emission is expedient. Assuming an isotropic emission, the fraction lost through the escape cone is given by 2π
θc
∫ ∫ φ= π π ∫ ∫ 0 2
0
0
0
sin ϕ dθ dϕ
/2
sin ϕ dθ dϕ
= 1 − cosθ c = 0.26
(4.2)
Due to the escape cone alone, 26% of all the light inside the concentrator [Zas81] is lost. This number is a lower boundary for the fraction lost, as several other mechanisms accrue. The lost fraction is increased e.g. if reabsorption events or absorption in the matrix material are considered. An elaborated discussion about other loss mechanisms and more realistic values of the fraction of light lost can be found in [Slo08]. 6
I have performed a simple consideration taking into account Fresnel reflections. The difference in the fraction of light transported to the edges, with and without Fresnel reflections for a A = 2x2cm² fluorescent concentrator was below 5% relative. For larger fluorescent concentrators, it becomes even smaller. 7 In literature, a value of n = 1.49 is also found frequently. The change in the results caused by this small difference in the refractive index is negligible.
48
4.1 Photovoltaic concepts The losses caused by the escape cone of the total internal reflection represent, therefore, the most important loss mechanism in the fluorescent concentrator. Other problems that occur are low quantum efficiencies of the dye, unwanted absorptions in the matrix material, and an insufficient ultraviolet (UV) stability of the dyes.
Figure 4.2: The main loss mechanism of the fluorescent concentrator. For the light emitted by the dye, there are two possibilities. (I) the light is emitted into an angle greater than the cut off angle of total internal reflection. In this case, it is transported to the edges of the concentrator. (II) the emitted light has an angle smaller than the cut off angle of total internal reflection. In this case, the light leaves the concentrator and is lost. This loss mechanism causes losses of 26% of the light in the concentrator. The photonic concept The goal when implementing photonic structures is to reduce the escape cone losses. The basic idea is to use a structure which is transparent for the absorbed light and mirroring for the emitted light. In that way, the light emitted into the escape cone is reflected and therefore also guided to the edges (Figure 4.3). A premise for this concept is that the spectral ranges of absorption and photoluminescence are separated. This premise is only satisfied within limits, as will be discussed in the next passage. The attributes of the used filter that need to be examined are •
The reflectance and transmittance of the filter in the spectral range of absorption and emission
•
The position of the reflection edge of the filter
Figure 4.3: Illustration of the photonic concept. The fluorescent concentrator is equipped with photonic structures. These structures are transparent for the absorbable light (blue arrow) but reflect the emitted light (red arrow). A filter with this characteristic will trap the light in the escape cone of TIR (grey underlaid).
49
Spectrally selective photonic crystals
This concept has already been proposed and investigated with different photonic structures by several groups [Ric04][Rau06]. In these works, mainly the performance of the fluorescent concentrator and attached solar cells were examined. The focus of this work shall rest more on the photonic structures. For this purpose, photonic structures were optimized especially with respect to their characteristics of certain fluorescent concentrators (section 4.3). The effect on the light guiding efficiency was simulated in detail (section 4.4) and exemplary fluorescent concentrator systems with spectrally selective optical elements were realized and characterized experimentally (section 4.5). Investigated fluorescent concentrators As already mentioned, the fluorescent concentrator plates consisted of PMMA and typically had a size of A = 2x2cm², although larger samples with sizes up to A = 50cm² have also been used. PMMA shows absorption in the spectral range below λ = 400nm. This matrix absorption has not been considered in the theoretical examination. To ensure that the error of this neglect is controllable, only wavelengths above λ = 400nm have been considered. Currently, a number of different fluorescent concentrators in a large variety of colors exist. In this work, three different fluorescent concentrators that seemed most promising to yield high efficiency were primarily used. The dyes in these fluorescent concentrators were developed some 20 years ago by different companies (e.g. BASF) and are non commercial. Also the fluorescent concentrator plates used were of comparable age. The dyes are denoted after the names given by the companies; the examples used in this work are denoted JMC4, BA2411 and BA856 and for the sake of brevity, with the fluorescent concentrators denoted using the same convention. The interpretation of the absorption- and photoluminescence spectra of the fluorescent concentrators measured with an integrating sphere is non trivial. One reason is that the spectral shift deployed by the dye is not detected by the integrating sphere. Another reason is that the emission of the dye is omnidirectional. Therefore when measuring the transmission or reflection, always also radiation emitted by the dye and leaving the concentrator through the corresponding escape cone is detected. For details about the determination of spectral data from a fluorescent concentrator see [Gol09]. The relevant information for the design of the spectrally selective filter is the spectral position of absorption and emission. How this information is obtained is described in the following. The spectral position of the absorption was measured using a photo spectrometer and an integrating sphere (or “Ulbricht sphere”). In the integrating sphere, the fluorescent concentrator is illuminated with light of a certain wavelength λ0, which is varied. All light entering the integrating sphere is detected independently of its wavelength. In that way, the transmission T and the reflection R of the fluorescent concentrator were measured (see Figure 4.4a for the setup). However, when measuring the transmission, not only the transmitted light is measured, but also the light reemitted into the forward direction of the escape cone with another wavelength λ1. This light enters the sphere, is detected and recorded as transmitted light at λ0 8. The same holds for the reflection measurement. Not only the reflected light is measured, but also the light reemitted into the backward direction of the escape cone. The absorption shown in Figure 4.4b is calculated as Abs = 1-T-R. The magnitude of this measurement therefore gives the absorption reduced by the reemission of the dye in the escape cone.
8
This effect may be used to determine the effectiveness of the filter on the light guiding efficiency, as I will show in section 4.5.
50
4.1 Photovoltaic concepts
b)
a)
absorption [%]
100
absorption JMC4 absorption BA2411 absorption BA856
80 60 40 20 0
400
500
600
700
wavelength [nm]
Figure 4.4: Absorption characteristics of different dyes used. In Figure 4.4a the measurement method to obtain reflection and transmission data is shown. When measuring the characteristics of a fluorescent concentrator, the light emitted by it must be considered. The emitted light is measured inclusively both for transmission and reflection. The calculated absorption is therefore distorted by the emitted light into forward and backward direction. The absorption obtained by this method is shown in Figure 4.4b (measurement by J.C. Goldschmidt). The second piece of information relevant for the filter design is the spectral position of the emission. This characteristic is measured by detecting the light emitted at the edge of a concentrator. The light leaving the concentrator here is the light relevant for an attached solar cell. This characteristic is measured by using a photo spectrometer. The setup for this measurement is show in Figure 4.5a. The results of the measurement of the spectral emission characteristics of the different fluorescent concentrators are shown in Figure 4.5b. At this point a potential error occurs, because the path from a dye to the edge of the fluorescent concentrator is typically much longer than a path to the filter. On the longer path, more reabsorption events occur which will influence the spectrum. It is therefore possible that the spectrum used for the optimization of the filter is not exactly the one which actually arrives at the filters. This point has not been investigated further and certainly deserves some attention in future approaches. a)
b)
100
emission [%]
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JMC4 BA2411 BA856
60 40 20 0
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500
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700
wavelength [nm]
Figure 4.5: The setup for the measurement is sketched in Figure 4.5a. The result of the measurement of the spectral emission characteristics of different fluorescent concentrators is shown in Figure 4.5b. The obtained emission characteristic is the one relevant for the solar cell. The spectral region of maximum emission has to be regarded for the filter (measurement by J.C. Goldschmidt).
51
Spectrally selective photonic crystals
For the time being I assumed that it is most disadvantageous to have a filter that reflects in the absorption range of the dye, because the unwanted reflection affects all impinging light, while the wanted reflection affects only the fraction of light in the escape cone. All light lost because of unwanted reflections must be overcompensated by gains through the desired reflection, and is therefore bought dearly. On the other hand, the emission characteristic obtained by this measurement is the one relevant for the solar cell, and the spectral position of maximum emission is the one with the most relevance for the filter. It proved to be a good approach to choose the spectral position where the reflectance of the filter changes from low to high. Typically, the wavelength that was chosen was the one for which absorption and dye emission were equal.
4.1.2 Spectrum Splitting At this point I want to abandon the example of the fluorescent concentrator and turn towards the second example, spectrum splitting. Some attention was paid recently to the topic of spectrum splitting [Ime06]. This strategy is used for several high efficiency concepts [Bar07][Gom08][Ell87] that aim towards efficiencies of more than η = 40%. The basic idea is to use an optical element to separate the solar radiation and direct it to solar cells with different band-gaps. The advantage of this strategy is that typical problems of stacked multijunction solar cells, such as lattice or current matching, are avoided. Additionally, by spatially separating the solar cells, freedom of interconnection is gained. In the scheme of this work, one concept was investigated [Gro09] in some more detail. In this concept the spectrum splitting is achieved by a special geometry and different spectral selective mirrors. The concept is sketched in Figure 4.6.
Figure 4.6: Schematic sketch of the concept for spectrum splitting investigated. The spectral distribution on the different solar cells is achieved by geometric arrangement and equipping every solar cell with a spectrally selective filter. The geometric arrangement assures that the solar cell with the highest band gap receives the solar radiation first and the solar cell with the lowest band gap last. The spectrally selective filters reflect the light not used by each solar cell to the next solar cell. This attempts to illuminate every solar cell with the part of the spectrum that yields the highest benefit. The geometry of the concept is such that the solar cells receive solar irradiation in a sequence that corresponds to their band gap. The solar cell with the highest band gap is illuminated first, the solar cell with the lowest band gap last. Each solar cell is equipped with a spectrally selective filter that reflects all light that is more beneficially used by a
52
4.2 Considerations of principles subsequent solar cell. In that way each solar cell may be illuminated in the most beneficial manner. The optimum characteristics for the spectrally selective filters in this concept are: each filter shall be transparent for the light that is indicated for the solar cell upon which the filter is placed. Each filter must reflect the light indicated for a subsequent solar cell (for this reason the last solar cell needs no filter). The solar cells are tilted by θ = 45° towards the normal; the filters therefore need to be optimized for a θ =45° incidence.
4.2 Considerations of principles In this section I will discuss principal considerations about the demands on spectrally selective filters for different PV concepts. An elaborated treatment of the problem for the fluorescent concentrator is given, as well as a short discussion on the optical characteristics of the spectrum splitting approaches, in [Pet09].
4.2.1 Considerations for the fluorescent concentrator concept In this subsection demands in order for a filter to be effective on a fluorescent concentrator shall be estimated. Two questions must here be answered: •
How much reflectance can be tolerated in the absorption range of the dye?
•
How high must the reflectance be in the photoluminescence range of the dye?
The filter will have a positive effect on the fluorescent concentrator if the gain due to increased light guiding efficiency is higher than the loss attributed to unwanted reflection in the absorption range. In this treatment, reabsorption is not taken into account. The amount lost because of the reflection of the filter Rabs(λ) in the absorption range can be calculated as
φ loss =
∫R
abs
(λ ) ⋅ α (λ ) ⋅ N am15 (λ )dλ
∫ α (λ ) ⋅ N am15 (λ )dλ
(4.3)
In this equation, a(λ) is the spectrally dependent absorption of the dye and Nam15(λ) is the solar photon flux. For the further considerations two different reflections need to be distinguished. The first is the initial reflection before the light enters the filter. This reflection causes losses, the corresponding reflectance is denoted Rabs. The second is the reflection after the emission in the fluorescent concentrator and is denoted R. This reflection induces the gain. Both reflections are caused by the filter but differ in several aspects. The important aspect here is that Rabs and R concern different spectral regions. Rabs concerns the spectral range where the dye absorbs and R concerns the photoluminescence range. In the further considerations, the reflections are each restricted to these spectral regions. Gains and losses will now be considered separately. I will start with an estimation of the gained light. The approach here will be to estimate the average length of a photon path inside the concentrator and to calculate the losses on this path. Subsequently the balance will be drawn between gained light and lost light. An investigation about gains and losses for different filters is given in section 4.4. To determine the gain induced by the filter, first the fraction of light transported to the edges for a reflectance R will be calculated. The reflectance is dependent on the angle of incidence and the wavelength R(θ,λ). The polar angle of the incident wave vector θ is defined against the normal of the filter. The path length q for a photon emitted under an angle θ until it reaches a lateral distance s to the place where it has been emitted is given by
q=
s sin(θ ) 53
(4.4)
Spectrally selective photonic crystals
The number of reflections k on this path is given by
k=
s t ⋅ tan(θ )
(4.5)
In this equation, t is the thickness of the fluorescent concentrator. The number of reflections a single photon experiences is an integer. However, this number k will be treated as a rational number. This is justified by the fact that many photons that are absorbed at different depths of the fluorescent concentrators are considered. The number of photons from a light beam with an initial number of photons I0 that have a wavelength λ0 and are emitted under an angle θ0 is reduced with every reflection. After k reflections, the number of photons Ik left is given by
I k (λ 0 , θ 0 ) = I 0 (λ 0 , θ 0 ) ⋅ R ( λ 0 , θ 0 ) k
(4.6)
For I0=1 this number gives the fraction of photons still in the fluorescent concentrator at a lateral distance s from the point of emission, if the emission occurred under an angle θ0. Now the average path length d ( x 0 , y 0 ) of a photon inside a fluorescent concentrator needs to be calculated. In the beginning the geometry of the fluorescent concentrator has to be defined. The fluorescent concentrator shall be of quadratic shape with a homogenous distribution of the dye and shall have a side length L. The expansion of the fluorescent concentrator is from -0.5 to +0.5 in the x and y direction in terms of a normalized length 1/L; distances given are therefore without units.
Figure 4.7: Geometric considerations to determine the average distance from a point (x0,y0) to the sides of a fluorescent concentrator. Each point sees the four sides of the concentrator under four different fields of view (δ1-δ4). The distance to a certain side must be determined for each side individually and in the corresponding angular range. The example shows the needed parameters for the right side of the fluorescent concentrator. First the distance d(δ) in dependence of the azimuth angle δ from a point (x0, y0) in the fluorescent concentrator to a side of the concentrator has to be calculated. This normalized distance is given by
54
4.2 Considerations of principles d (δ , x0 , y 0 ) =
x1 ( x 0 , y 0 ) cos(δ )
(4.7)
The geometric considerations are sketched in Figure 4.7. As the emission of the dye is angularly isotropic, the normalized distance d(δ) as shown is only valid in the angular range (δ1,δ2,δ3,δ4), under which a certain side of the fluorescent concentrator is seen from the point (x0, y0) and must be calculated for all four sides accordingly. The average normalized distance from the point(x0, y0) to all sides is given by: d ( x0 , y 0 ) =
1 ⎛⎜ d (δ , x 0 , y 0 ) dδ + ∫ d (δ , x 0 , y 0 ) dδ + 2 ⋅ π ⎜ δ∫1 δ2 ⎝ ⎞ ⎟ + d ( δ , x , y ) d δ d ( δ ) d δ , x , y 0 0 0 0⎟ ∫δ ∫δ 3 4 ⎠
(4.8)
The normalized distance d(δ), as well as the integral boundaries, are dependent on x0 and y0. To give some examples for the result of this function: d (0, 0) = 0.561 ; this is the average distance for a photon emitted directly in the center of the fluorescent concentrator. At the side, it is d (0, 0.5) = 0.383 and at the edge of the square d (0.5, 0.5) = 0.281 . The average normalized distance for all points on the fluorescent concentrator is given in Figure 4.8.
Figure 4.8: Map of the average distance for a photon to reach the edge of the fluorescent concentrator. The side length of the concentrator is normalized to 1. The average path length is therefore in terms of the side length. To calculate the number of photons actually reaching the side, the geometric ratio side length to thickness w = L/t of the fluorescent concentrator has to be taken into account. Fluorescent concentrators used for experiments have a side length of L=2.0cm and a thickness of t = 3mm. The geometric ratio in this case is close to w = 7. The fraction of photons remaining at a normalized distance d(δ) from the position from which they were emitted for a fluorescent concentrator with the geometric ratio w as a function of the polar angle θ into which the photons are emitted, is given by
55
Spectrally selective photonic crystals
⎡ d (δ , x0 , y 0 ) ⋅ w ⋅ ln( R(λ0 ,θ )) ⎤ n(δ ,θ , x0 , y 0 , R(λ ,θ ), w,θ c ) = Exp ⎢ ⎥ Tan(θ ) ⎣ ⎦
(4.9)
The average number of photons emitted at the position(x0, y0) that reach the edge of the fluorescent concentrator is given by n ( x0 , y0 , R(λ ,θ ), w,θ c ) =
⎛ 4 ⎡ θc ⎤⎞ ⎜∑ ⎢ n(δ ,θ , x0 , y0 , R(λ ,θ ), w,θ c ) dθ dδ ⎥ ⎟ ∫ ∫ 2 ⋅ π ⋅θ c ⎜⎝ j =1 ⎢⎣ δ j 0 ⎥⎦ ⎟⎠ 1
(4.10)
The integration here is given for all azimuth angles and for a polar angular cone defined by the critical angle θ = θc. The critical angle depends on the material of the fluorescent concentrator and on the conditions for a special system. Typically the critical angle is given by the cutoff angle for TIR reflection (which depends on the material), θc=Arcsin(1/n). However, the situation may be changed completely, if e.g. a 3D photonic crystal, like the opal, is deposited directly on top of the fluorescent concentrator. In an extreme case here TIR is completely destructed and the critical angle is θc = 90°. Until now the reflectance R(λ,θ) has been given as a function of the wavelength and the angle of incidence. For the further considerations of principles no dependence on the wavelength λ shall be assumed and the dependence on the angle of incidence θ is neglected throughout the rest of this chapter. In this section it therefore holds R(λ,θ) = R. To illustrate equation (4.10), some examples for the fraction of photons reaching the side are calculated. The parameters for these examples are: R =0.9, a critical angle θc = 41.8° (corresponding to the escape cone in PMMA) and a geometric ratio w = 7. The fraction n ( x0 , y 0 ) is then only dependent on the position on the fluorescent concentrator. The fraction of photons reaching the edges at the center is n (0,0) = 0.315 at the side n (0, 0.5) = 0.623 , and at the edge n (0.5, 0.5) = 0.786 . The fraction of photons reaching the sides for all positions on the concentrator is given in Figure 4.9.
Figure 4.9: Map of the fraction of photons in the escape cone reaching the side of the concentrator when a reflective filter is used. The reflectance of the filter is R=0.9, the geometric ratio is w = 7, and the critical angle is θc = 41.8° (corresponding to PMMA with n = 1.5). Depending on the position of the emission, between 30% and 80% of the photons are potentially gained by the filter.
56
4.2 Considerations of principles To calculate the average fraction of photons reaching the side of the fluorescent concentrator, function (4.10) has to be integrated over the lateral expansion 0.5 0.5
N ( R(λ ,θ ), w,θ c ) =
∫ ∫ n (x , y 0
0
, R(λ ,θ ), w,θ c ) dx dy
(4.11)
− 0.5 − 0.5
In Figure 4.10 the fraction of photons in the cone defined by the critical angle θc reaching the side of the concentrator, depending on the reflectance of the filter R and the critical angle θc is given.
Figure 4.10: Fraction of photons in the escape cone that reach the side of the concentrator depending on the reflectance of the filter and the critical angle θc .The geometric ratio is w = 7, corresponding to our experimental setup. The critical angle is defined through the refractive index contrast between the matrix material and air. It can be altered by destroying the conditions for total internal reflection. For a critical angle of θc=90, only the reflection of the filter is used for the transport of the photons to the edges. Now the question shall be answered which magnitude of reflectance is needed to obtain a significant increase of photons reaching the solar cell. A significant increase shall be achieved, if a fraction of 1/e of the photons in the escape cone reaches the solar cell (independent of the fraction of photons in the escape cone). The condition for this is
N ( R(λ ,θ ), w,θ c ) =
0.5 0.5
∫
!
∫ n ( x0 , y0 , R(λ ,θ ), w,θ c ) dx dy =
− 0.5 − 0.5
1 e
(4.12)
The result of this calculation is given in Figure 4.11. In case of an uncoupled filter on PMMA with a refractive index of n = 1.5, a critical angle of θc = 41.8° corresponding to 26% escape cone losses, and a geometric ratio of w = 7, the reflectance of the filter needed for a significant increase in the number of photons guided to the edges is R = 86.9%.
57
Spectrally selective photonic crystals
Figure 4.11: Reflectance of the filter required for a fraction 1/e of the photons to be guided to the sides in dependence of the critical angle θc. The geometric ratio is w = 7. Now the influence of the geometric ratio on the fraction of photons reaching the side in dependence of the filter reflectance needs to be examined. The result of this examination is given in Figure 4.12. For larger ratios, and therefore for larger fluorescent concentrator areas, the demands on the filters increase.
Figure 4.12: Fraction of photons in the escape cone that are guided to the sides in dependence of the reflectance of the filter and the geometric ratio. The geometric ratio is given logarithmically to the basis of 10. The critical angle is θc =41.8° corresponding to PMMA. Assuming a constant thickness of the fluorescent concentrator, the demands on the reflectance of the filter increase with increasing area. The system for testing had a geometric ratio of w = 7 corresponding to a value of 0.85 in this figure. Commercial use would require a factor of around w = 100, corresponding to a value of 2 in this figure. Much less light is transported to the edges in that case, meaning that the demands on the filter are much higher.
58
4.2 Considerations of principles Finally some considerations about gains and losses shall be given. Gains and losses are both related to the reflection of the filter. The losses are related to the unwanted reflections in the spectral absorption range of the dye. The gain is related to the filter reflection in the photoluminescence range. For a filter to have a positive effect more light must be gained than is lost. Assuming a fraction of 26% of the light in the escape cone, the filter can only have a positive effect if less than 26% are lost. The relation, how high the reflectance in the PL range has to be to compensate a certain loss is calculated with equation (4.12) by entering the lost fraction as target value. The resulting break even relation is shown in Figure 4.13.
100
Rpeak [%]
80 60 40
break even relation 20 0
0
5
10
15
20
25
Runwanted [%] Figure 4.13: Break even relation for the filter characteristic for a filter applied to a fluorescent concentrator with a geometric of w = 7 and 26% losses in the escape cone. The x-axis gives the amount of light lost due to reflections in the absorption range of the dye. The y-axis is the average reflection in the PL range of the dye. The blue line marks the reflection that is at least needed to compensate occurring losses.
4.2.2 Considerations for the spectrum splitting concept In this subsection I will discuss the influence of the filter characteristics on the performance of the spectrum splitting concept introduced in section 4.1.2. The concept describes a set of solar cells that are each equipped with a photonic filter and are illuminated subsequently by the light reflected from the filter on the previous solar cell. It is sensible here to start with the solar cell with the highest band gap und arrange the solar cells according to decreasing band gap. The number of photons N(R,T) which reaches the nth solar cell is given by n −1
N ( R, T ) = ∫ Tn (λ ) ⋅ ∏ Ri (λ ) ⋅ N AM 15 (λ ) dλ λ
(4.13)
i =1
In this equation NAM15(λ) is the number of photons in the AM15 spectrum. The fraction of photons L(R,T) reaching this solar cell is accordingly given by dividing equation (4.13) through the number of photons in the spectrum. The function n−1
ηmax (λ ) = Tn (λ ) ⋅ ∏ Ri (λ ) dλ
(4.14)
i =1
constitutes an upper limit to the quantum efficiency that may be reached with such a system and shall be addressed to as the optical limit. In the setup considered in this work, three solar cells were used. The optimum filter characteristics for this configuration are shown in Figure 4.14.
59
1.0
100
0.8
80
0.6
EQE cell 1 EQE cell 2 EQE cell 3
0.4
filter 1 filter 2 filter 3
0.2 0.0
60 40
transmission [%]
EQE [%]
Spectrally selective photonic crystals
20 0
400
600
800 1000 1200 1400 1600 1800
wavelength [nm]
Figure 4.14: Optimum filter characteristics for a setup with three solar cells. The filters themselves shall not absorb light so that the filter reflection R is given by R = 1-T. Also given is an exemplary external quantum efficiency (EQE) for a triple solar cell setup. To estimate the optical limit some simplifications shall be made. Each filter has a spectral range of high reflectance in which the light is reflected to the subsequent solar cells. For this spectral range (which is different for each filter) an effective reflectance Rn shall be assumed which is independent of the wavelength. For all other wavelengths the filter shall transmit the light with an effective transmission Tn . The number of photons used by each solar cell in the optical limit is than given by n −1
N ( R, T ) = ∑ Tn ⋅ ∏ Ri ∫ N AM 15 (λ ) dλ m
i =1
(4.15)
λm
The integration in this equation is confined to each spectral range m in which Rm and Tm are constant. Each solar cell now shall be intended for a third of the photons in the spectrum. The filter on each solar cell shall be transparent with T for the third of the spectrum intended for the subjacent solar cell and reflective with R for the rest of the spectrum. The total fraction of photons used by the system is then given by 1 1 1 1 1 L( R, T ) = T + R T + R 2T + (1 − T ) R 2T + (1 − T ) R 3T + O( R ) 3 3 3 3 3
(4.16)
In the considered setup the solar cells are arranged in a triangle. In that way part of the light reflected by the filter in front of a solar cell is reflected and reaches the solar cell again on the way back. This is represented by the last two terms. O(R) represents terms of higher order that are neglected. Function (4.16) is shown in Figure 4.15. The figure shows that, especially for a high performance of the filters with transmission and reflection higher than 90%, the effect of the reflection on the efficiency is more sensible than the one of the transmission. This is an advantage when it comes to the design of the filters. Generally a reflectance close to 100% can be obtained in the intended spectral range while in the other spectral range(s) the transmission will be reduced by a couple of percent. To finally give a guiding value for the estimation of the performance of the filters: for every percent reflection and transmission are decreased 1.4% less photons reach the solar cells.
60
4.3 Optimized spectrally selective photonic structures
Figure 4.15: Fraction of photons reaching one of the three solar cells in the described setup. The figure is given in dependence of R , the effective reflectance of the filter in the spectral range not intended for the subjacent solar cell and T , the effective transmission in the spectral range intended for the solar cell. It is assumed that the performance for all filters is equal.
4.3 Optimized spectrally selective photonic structures For application in solar cells, the demands on the photonic structures are quite high. For a certain PV system, the reflection must be as low as possible in one spectral range λ1 and as high as possible in another spectral range λ2 that typically is located next to λ1. Therefore, a steep edge at the transition high reflection- high transmission is also required. Furthermore, the position of this edge must be accurately definable. Concerning 1D photonic structures, the demands on the reflection and transmission characteristic may be met by using a sufficiently large number of layers. The demands on the position of the edge depend on the accuracy with which the single layers may be deposited. For 3D photonic structures the characteristic is defined by the quality with which such a structure is realized. In section 4.4, it will be highlighted what is theoretically possible with photonic structures and what characteristics may be expected optimally. The optimization given in this section is performed with regard to an application on a fluorescent concentrator system. The spectrum splitting approach is not mentioned explicitly. The reason for this is that the optimization for both concepts is quite similar, however, the performance of the filter for the spectrum splitting concept is so high that in practice only the application of a band stop filter is sensible. All simulations and optimizations performed for this concept were therefore similar to the ones given in the part for the band stop filter.
4.3.1 1D crystals The optimized Rugate filter The rugate filer was introduced in section 2.3. With the rugate filter, harmonic reflections are attenuated. However, for the application on the fluorescent concentrator, a further suppression of the harmonic reflections is needed and additionally the still occurring sidelobe reflections have to be eliminated. A strategy to obtain the desired characteristic with rugate filters has been suggested by Southwell [Sou88][Sou89]. The optimization is performed in two steps:
61
Spectrally selective photonic crystals
1. The envelope function of the refractive index is modulated. This step aims towards a suppression of the sidelobe reflections 2. The refractive index function is matched to the adjoined materials. With this step, the average reflection that occurs due to the refractive index contrast of filter and adjoined materials is reduced. For the modulation of the envelope and the matching, several functions have been proposed and showed an effective suppression of sidelobe reflections. I worked with a Gaussian modulation of the envelope function was used and a seventh order reciprocal function was applied for the matching. The resulting equation of the refractive index is given by 2 ⎛ ⎛ Λ ⎞ 1 ⎞ ⎛ sΛ ⎞ ⎛ 2π ⎞ ⎟ ⋅ z ⎟ Exp⎜ − ⎜ s − z ⎟ / ⎜ ⎟ n( z ) = H ( z ) + δn ⋅ sin ⎜ ⎜ 2 ⎝ Λ ⎠ ⎠ ⎝ 4 ⎠ 2 ⋅ Ln 2 ⎟⎠ ⎝ ⎝
(4.17)
In this equation, s is the number of periods for the filter and H(z) is the seventh order polynomial, which is given by
⎧ δn ⎪ n+ 2 ⋅ (1 + 2Λ ⋅ z ) 7 ⎪ δn ⎪ H ( z ) = ⎨n + 2 ⋅ (1 + 2Λ ⋅ ( sΛ − z )) 7 ⎪ ⎪ n ⎪ ⎩
for z < Λ / 2 for z > sΛ − Λ / 2
(4.18)
else
The characteristic of the resulting filter is shown in Figure 4.16. a)
b) wavelength [nm] 100
200
300
400
600 1000
1.9
reflectance
refractive index
2.0
1.8 1.7 1.6 1.5 1.4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
z[µm]
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0 15
3.0x10
15
2.5x10
15
2.0x10
15
1.5x10
1.0x10
15
5.0x10
14
frequency [Hz]
Figure 4.16: Characteristic of the optimized rugate filter. The envelope function is modulated with a Gaussian form and the average refractive index of the filter is matched to the fluorescent concentrator material with a polynomial function. Oscillations are attenuated and vanish, especially at higher frequencies. Because of the lowered overall refractive index contrast, the reflectance is weakened. For the filter 10 periods have been used. For comparison the characteristics of the normal rugate filter are given in light grey. The average refractive index contrast of this filter is lower, resulting in reduced reflectance in the peak region. This may be compensated by using more periods. By using more periods, the reflectance of the main peak becomes stronger, as well as the peak at the double
62
4.3 Optimized spectrally selective photonic structures frequency. Oscillations between the peaks are attenuated. Unfortunately, even for optimized rugate filters, it is very difficult to suppress unwanted reflections completely. Especially the reflection at the half design frequency is difficult to eliminate. For fluorescent concentrators with an emission range of the dye in the range of λ = 600nm, the optimization is sufficient because the reflection at the half design frequency is below the absorption range of the dye. The characteristic of an optimized rugate filter designed for application on a fluorescent concentrator with the dye BA2411 is shown in Figure 4.17. b)
2.0
100
1.9
80
reflectance [%]
refractive index
a)
1.8 1.7 1.6
absorption JMC4 PL emission JMC4 reflection filter
60 40 20
1.5 0
2
4
z [μm]
6
8
10
0 350
400
450
500
550
600
650
700
wavelength [nm]
Figure 4.17: Optimized Rugate Filter designed for application on a fluorescent concentrator with the dye BA2411. Note that the reflectance characteristic is plotted against the wavelength and not against the frequency as in the other graphs. Additionally, only the spectral range relevant for the application is included. For the filter, 60 periods were used with a period of 160nm (PL emission measured by J.C. Goldschmidt). . The band stop filter The rugate filter has some disadvantages. A principle disadvantage is that for a rugate filter the reflection and transmission characteristics are defined for the whole spectrum. This is a disadvantage because the reflection peak, for example, has a fixed width and is often narrower than required. For the applications however, only a certain spectral range is of interest. Restricting the definition of the optical characteristics to this rang of interest increases the freedom of design. A practical disadvantage is that a rugate filter requires a continuous refractive index profile. This is a disadvantage because a continuous refractive index profile makes high demands on the production. The technology to produce these profiles is not commonly available and less accurate than would be needed. These problems may be solved by using another design approach. In the approach of the band stop filter, only two layers with different refractive indices are used and the thickness of the layers is varied. A lot of different designs are possible to obtain the desired reflectance characteristic. One way to design a band stop filter [The81] with a suitable characteristic is the needle optimization method [Sul96]. I used an evolutionary model to optimize the filter 9. The result of this optimization is shown in Figure 4.18.
9
The evolutionary algorithm is straightforward and a great deal of literature exits in the field [Mac01]. A stack of thin films is preset with a fix number of layers. The thickness of the single layers is independently and randomly varied until a specified goal is reached or a break condition is fulfilled.
63
Spectrally selective photonic crystals
b)
2.0
100
1.9
80
R, A, PL [%]
refractive index
a)
1.8 1.7 1.6
absorption BA2411 PL emission BA2411 reflection filter
60 40 20
1.5 0
1
2
3
4
5
z[µm]
0 350
400
450
500
550
600
650
700
750
wavelength [nm]
Figure 4.18: Refractive index profile (Figure 4.18a) and reflectance characteristic (Figure 4.18b, blue line) of a band stop filter. The filter consists of 50 layers with the refractive indices n1 = 1.5 and n2 = 2. It was designed using an evolutionary algorithm. Shown are also the absorption (red triangles) and the photoluminescence spectrum (black dots) of the fluorescent concentrator BA2411 (PL emission and absorption measured by J.C. Goldschmidt). One problem occurring especially, but not only, for periodic filters is that the reflectance characteristic is also dependent on the angle of incidence. The angular dependence of the filters used in this subsection is described by the Bragg effect. The Bragg effect results in a blue shift of the reflection peak for increasing angles of incidence. A more pronounced treatment of this effect is found in chapter 6. The angular dependence is disadvantageous because the reflectance is shifted into the absorption range of the dye. The filter, therefore, no longer reflects the emitted light, as well as reflecting light in the absorption range of the dye. This is especially disadvantageous for diffuse light. It is possible to design filters with a reduced angular-dependent reflectance by using the needle optimization technique. The resulting filters are non-periodic and the Bragg effect does not occur. Imenes et al. demonstrated the design of such a filter, which would also be suitable for applications regarding fluorescent concentrators [Ime06].
4.3.2 3D crystals The opal Opals were introduced in section 2.3. Concerning opals, one question was whether to use normal opals or inverter ones. From the simulation point of view, comparable results may be achieved with both kinds of crystals, therefore both kinds have been implemented into the RCWA method. From the band structures (Figure 2.13), the inverter opal should show a broader reflection peak and is therefore initially preferable. Necessary for the simulation are the material parameters. Here, different considerations hold for the normal opal and for the inverted opal. The production of normal opals demands a very high monodispersity of the used spheres; otherwise, staple errors occur that are the source of scattering [All04]. Disorder in this fashion will severely influence the optical properties of the opal [Gal04]. The demands on the material quality limit the supply of materials suitable to produce opals. Common materials are PMMA, SiO2, and polystyrene. All of these materials have a refractive index of close to n = 1.5. The opals used for the applications on fluorescent concentrators were made of PMMA.
64
4.4 Results of the simulations of the enhanced light guiding efficiency inside the fluorescent concentrator This constraint doesn’t exist for the inverted opal; here a number of different materials are suitable for the inversion process. A common material is TiO2. The refractive index of TiO2 is in the range of n = 2.8. However, as the filling process is incomplete and the TiO2 in the crystal is nanoporous, the effective refractive index for the filling medium is much lower. This refractive index was identified experimentally to be close to n = 1.7 for some inverted opals. For the simulation of the inverted opal therefore this refractive index was used. Simulations of the spectral reflection characteristic for both kinds of opals, designed for the application on a fluorescent concentrator, were performed using the RCWA method. The results of this calculation are shown in Figure 4.19. The obtained characteristics are such that a beneficial effect for an application on fluorescent concentrators should be possible. b) a)
80
100
absorption JMC4 PL emission JMC4 reflection opal reflection inverted opal
60 40 20 0 350
60 40 20 0
400
450
500
opal 22 layers opal 10 layers
80
reflectance [%]
R, A, PL [%]
100
550
600
650
700
400
wavelength [nm]
500
600
700
wavelength[nm]
Figure 4.19: Simulated reflection characteristic of an opal and an inverted opal designed for an application on a fluorescent concentrator with the dye denoted JMC4. The opal consists of spheres with a refractive index n = 1.5 and a diameter of D = 254.7nm. 22 sphere layers were assumed. The inverted opal consists of air spheres in a matrix with a refractive index of n = 1.7. 19 sphere layers were assumed. (PL emission & absorption measured by J.C. Goldschmidt). Also given is a comparison of the reflection characteristics for opals of different thickness. Also given in Figure 4.19 are two opals of different thickness. One of the major problems when producing opals is that the opal becomes the more imperfect the thicker it gets. In very thick opals only a certain number of layers contributes to the photonic effect so that the effective number of layers is lower than the actual number. For this reason the reflectance required has never been reached experimentally and an effective ten layer opal fits the measured characteristics better than a perfect 20 layer opal.
4.4 Results of the simulations of the enhanced light guiding efficiency inside the fluorescent concentrator Now the amount of light gained by applying a photonic structure to the fluorescent concentrator can be calculated. The application of the filter has two consequences for the fluorescent concentrator. On the one hand, a fraction of the light in the escape cone is transported to the edges and therefore gained. A fraction of the incident light is reflected by the filter and therefore is not absorbed by the concentrator and must be considered lost. The fraction of light lost because of the filter is (compare equation (4.3)).
φ loss =
∫
Abs
Rabs (λ ) ⋅ α (λ ) ⋅ N am15 (λ )dλ
∫
Abs
α (λ ) ⋅ N am15 (λ )dλ
(4.19)
The integration is over the absorption range of the dye. The fraction of light gained because of the filter is
65
Spectrally selective photonic crystals
0.5 0.5
φ gain =
∫ ∫ ∫ n (x PL
0
, y 0 , R, w,θ c ) dx dy ⋅ N PL (λ ) dλ
(4.20)
− 0.5 − 0.5
∫
PL
N PL (λ ) dλ
In this calculation, the angular dependence of the reflectance of the filter is ignored. The total amount of light gained or lost through the filter is given by
φ = φ gain ⋅
N esc ⋅ (1 − φ loss ) − φ loss N0
(4.21)
The fraction of light in the escape cone that would be lost without the filter is denoted N esc / N 0 here. With the data given in section 4.3, this method may be used to analyze the filters presented in section 4.2. Each filter is used on a fluorescent concentrator with the dye for which it was designed. Two cases are analyzed. In the first case, the filter is not optically coupled to the fluorescent concentrator. In this case, the light is partly transported via total internal reflection and partly because of the reflection of the filter. The reflection of the filter only affects light in the escape cone of total internal reflection. In our experimental setup with a fluorescent concentrator consisting of a material with a refractive index n = 1.5, the escape cone consists of all light that is emitted between θ = 0° and θ = 41.8°. Assuming an isotropic emission, the fraction of light in the escape cone is N esc / N 0 = 26%. The filter acts therefore only on this 26% of the light. The main disadvantage of not coupling the filter to the fluorescent concentrator is that the reflection of the filter, including the reflections at the backside of the substrate, fully adds to the reflection at the surface of the fluorescent concentrator. The filter reflection in the absorption range of the dye must therefore be considered a loss. The filter will have a positive effect if the amount of light gained by the filter exceeds the one lost by this reflection. The analysis of the filters is given in Table 4.1.For the sake of comparability, the simulation results here are given for one dye (in this case JMC4), though due to the different photoluminescence properties, the filters are differently apt for the different dyes.
φgain
Simulated Rugate filter (Figure 4.17) with JMC4 45.59%
Simulated band stop filter (Figure 4.18) with JMC4 83.71%
Simulated Opal (Figure 4.19) with JMC4 32.20%
φloss
4.58%
4.60%
8.64%
φ
6.73%
16.17%
-0.99%
Table 4.1: Gains and losses induced by the optically uncoupled filter. The total gain Φ is calculated with equation (4.21) and a fraction of light in the escape cone N esc / N 0 = 26% . Only the simulated rugate filter and band stop filter have a positive effect on the light guiding efficiency. Opal and inverted opal yield comparable results. Only the result for the normal opal is given here. The transport mechanism based on TIR inside a fluorescent concentrator depends on the flatness of its surface. If the flatness is violated several mechanisms may occur that account for losses. At an uneven surface, for example, light has multiple chances of hitting the surface under an angle inside the escape cone. At a structured surface modes of diffraction may occur that couple out light. This second example deserves special attention, as it shows that TIR is not guaranteed anymore, if a 2D or 3D photonic crystal is deposited onto the fluorescent concentrator. How much TIR is influenced is hard to tell, but it is expedient to assume that in such a case, less light is transported via TIR and more light is transported via the filter. To estimate, how
66
4.5 Experimental results the performance of the fluorescent concentrator is influenced if the filter has a bigger share in the transport, the extreme case is investigate in which all light is transported via filter reflection. The critical angle in this case is θc =90° (this would also be the case, if no refractive index contrast occurs between fluorescent concentrator and surrounding, e.g. if the concentrator was encapsulated). The fraction of the light in the escape cone is N esc / N 0 = 100% . This case is compared to the one where no filter is attached to the fluorescent concentrator and the light is transported via TIR. This is given in the last line of Table 4.2 under the assumption that in the escape cone 26% of the light are situated. Simulated band stop filter (Figure 4.18) with JMC4
Simulated opal (Figure 4.19) with JMC4
Simulated Rugate filter (Figure 4.17) with JMC4
Simulated Rugate filter (Figure 4.17) with BA856
φgain
93.65%
52.89%
56.60%
82.88%
φloss
3.11%
8.64%
4.58%
25.47%
φ
87.63%
39.68%
49.43%
36.31%
φ − 74%
13.63%
-34.32%
-24.57%
-37.69%
Table 4.2: Gains and losses induced by the filters if no total internal reflection occurs and all light is transported to the edges by filter reflection. The total gain Φ is calculated with equation (4.21) and a fraction of light in the escape cone N loss / N 0 = 100% . If the total internal reflection were perfect, 74% of the light would be guided to the sides. The last line gives the light guiding efficiency of the filter compared to this 74%. The results given in Table 4.2 show how important TIR as transport mechanism in the fluorescent concentrator is. If the light is only transported by the filter reflection, the performance of all systems is much worse than if TIR was the main transport mechanism. Besides the band stop filter, no filter induces a positive effect any more. It is therefore not advisable to attach filters to a fluorescent concentrator, if TIR is affected. The considerations shown here have been made for different filters in regard of one selected dye (JMC4) but also for a different dye (BA856) and the rugate filter.
4.5 Experimental results 4.5.1 Fluorescent concentrator system efficiency The preparation of the measured samples shown in this subsection was performed by Jan Christoph Goldschmidt. He also performed the characterization of fluorescent concentrators, i.e. measurement of the absorption and the PL spectra, measurements of the efficiencies and the quantum efficiencies of the solar cells attached to the fluorescent concentrators as well as the light beam induced current (LBIC) measurements shown in this work. His thesis [Gol09] is strongly recommended for details about the fluorescent concentrator systems. The measurements are used to illustrate the effect of the photonic structures. I performed the characterization of the photonic structures, the calculation of gains and losses, as well as reflection and transmission measurements of advanced fluorescent concentrator systems and the considerations about the parasitic reflection. Rugate Filter Several rugate filters were purchased commercially from Fraunhofer IST. These filters were specially fabricated for the application on different fluorescent concentrators (Figure 4.20). The first filter was optimized for the application on a fluorescent concentrator with the dye
67
Spectrally selective photonic crystals
JMC4. The reflectance of this filter is close to one between 550nm < λ0 < 650nm. The second filter was optimized for the application on a fluorescent concentrator with the dye BA856. The reflectance of this filter is close to one between 600nm < λ0 < 720nm. Both filters show unwanted reflections in the absorption range of the dyes, causing considerable losses. a) b) 100
absorption JMC4 PL emission JMC4 reflection filter
100 80
R, A, PL [%]
R, A, PL [%]
80 60 40 20 0 350
absorption BA856 PL emission BA856 reflection filter
60 40 20
400
450
500
550
600
650
700
750
wavelength [nm]
0
400
450
500
550
600
650
700
750
wavelength [nm]
Figure 4.20: Reflection characteristic of two rugate filters purchased from Fraunhofer IST (blue line). The filters were optimized for application on fluorescent concentrators with the dyes BA856 and JMC 4. Also shown are the absorption (red triangles) and the photoluminescence (black dots) of the fluorescent concentrator. The filters show a very high reflectance in the photoluminescence range of the dye. Unfortunately, a lot of unwanted reflections occur in the absorption range of the dye, causing losses. Using the simulation method presented in section 4.2, I calculated the estimated effect of the filters on the light guiding efficiency of a fluorescent concentrator with a geometric factor of w = 7. Simulated rugate filter (Figure 4.17) with JMC 4
Real rugate filter (Figure 4.20a) with JMC 4
Simulated rugate filter (not shown) with BA856
Real rugate filter (Figure 4.20b) with BA856
φgain
45.59%
67.44%
52.18%
76.95%
φloss
4.58%
14.46%
7.67%
22.68%
φ
6.73%
0.54%
4.86%
-7.21%
Table 4.3: Predicted effect of the rugate filter on the performance of a fluorescent concentrator. For comparison, the performance for the optimized simulated rugate filter is also given. Comparing the simulated to the real rugate filter, a broader reflection range is noticeable. With the real rugate filter more than Φgain = 75% of all light in the escape cone is guided to the edges, whereas in the simulated case this fraction is only ca. Φgain = 50%. However, for the real filter the unwanted reflections cause losses of more than Φloss = 20% compared to Φloss = 5% for the optimized case. For both systems measurements of the current / voltage (IV)-characteristic for a fluorescent concentrator with and without the photonic structure were performed. The results of these measurements are shown in Table 4.4. The measurements were performed on fluorescent concentrators with the particular dye, a size of A = 2x2 cm² and a thickness of t = 3mm. A gallium arsenide (GaAs) solar cell was optically coupled to one side of the fluorescent concentrator. The other sides remained untouched. Within the precision of the measurement
68
4.5 Experimental results method, no effect of the filters on the efficiency of the solar cell was detectable. Within reasonable considerations about the occurring errors, the theoretical predictions agree with the measured values.
ηwithout
Real rugate filter (Figure 4.20a) with JMC 4 1.41%
Real rugate filter (Figure 4.20b) with BA 856 1.49%
1.39%
1.51%
ηwith
-1.4% + 1.3% δηrel Table 4.4: Measured efficiency of fluorescent concentrator system with and without a photonic structure. Within the precision of the measurement method, no effect from the structure was detectable [Gol06]. To conclude, my interpretation of the presented results is that with the rugate filter the light guiding efficiency of fluorescent concentrators is increased, but the losses in the absorption range of the dye lead to a negligible or even negative effect. Reducing these losses is a technological question and is associated with the control over the refractive index profile and AR coatings. At the time of this work, no filters with a sufficiently low reflectance characteristic in the absorption range of the dye were available; nevertheless, if unwanted reflection can be reduced, rugate filters are an interesting option, especially for fluorescent concentrators that work further in the infrared. Band stop filter A band stop filter was purchased commercially from mso Jena. This filter was produced specially for the application on a fluorescent concentrator with the dye BA2411 (Figure 4.21). This filter consisted of two materials, though the exact parameters are unknown to me. The reflectance of this filter is close to R = 1 in the spectral range from 550nm < λ0 < 700nm. The unwanted reflections in the absorption range of the dye are very low (R = 2% 3% 375nm < λ0 < 550nm). This filter showed the best characteristic of all the filters tested.
R, A, PL [%]
100
absorption BA2411 PL emission BA2411 reflection filter
80 60 40 20 0 350
400
450
500
550
600
650
700
750
wavelength [nm]
Figure 4.21: Reflection characteristic of a band stop filter purchased from mso Jena (blue line). The filter was optimized for the application on fluorescent concentrator with the dye BA2411. Also shown are the absorption (red triangles) and the photoluminescence (black dots) of the fluorescent concentrator. The filters show a very high reflectance in the photoluminescence range of the dye and little reflectance in the absorption range.
69
Spectrally selective photonic crystals
Using again the simulation method presented in section 4.2, I calculated the estimated effect of the filters on the light guiding efficiency of a fluorescent concentrator with a geometric factor of w = 7. Simulated edge filter (Figure 4.18) with BA 2411
Real edge filter (Figure 4.21) with BA 2411 83.85% 92.03% φgain 4.40% 3.11% φloss 16.44% 20.09% φ Table 4.5: Predicted effect of the edge filter on the performance of a fluorescent concentrator. For comparison, the performance for the optimized simulated edge filter is also given. In the case of the band stop filter, the predicted performance of the filter is even better than that of the simulated. Following the simulations, the high reflectance of the filter in the photoluminescence range of the dye guides ca. 90% of the photons to the edges [Pet07]. The transmission in the absorption range of the dye causes losses that are lower than the ones that would appear for a glass on top of the system. For the purchased filter, measurements of the IV-characteristic for a fluorescent concentrator with and without the photonic structure were performed. The results of these measurements are shown in Table 4.6. The measured efficiency enhancement is in good accordance with the predicted enhancement of the light guiding efficiency. Real edge filter (Figure 4.21) with BA 2411 3.31% ηwithout 3.89% ηwith 18.7% δηrel Table 4.6: Measured efficiency of fluorescent concentrator system with and without a photonic structure. A relative increase in efficiency of more than 18% was measured. Hence the band stop filter shows the desired effects on the efficiency of a solar cell attached to the fluorescent concentrator. On this example, I analyzed the effect on the light guiding efficiency in some more detail. One possibility to characterize the effect of a photonic structure on a fluorescent concentrator system is to measure the transmission of the system fluorescent concentrator / filter (Figure 4.22a) with an integrating sphere. In the arrangement shown in this figure, the photonic filter will reflect all light emitted by the dye in the backward direction. It is important to bear in mind that the integrating sphere detects all light with the excitation wavelength and ignores the Stoke shift. Consequently the integrating sphere will detect the effect of the filter as an increase in the transmission in the absorption range of the dye. This effect is shown in Figure 4.22b where the transmission of the fluorescent concentrator with and without photonic structure is shown. The difference between the transmission with and without the filter gives directly the fraction of light in terms of the incident light reflected by the photonic structure.
N R ,inc = Twith − Twithout
(4.22)
Setting this fraction of light in relation to the absorption in the fluorescent concentrator and assuming a quantum efficiency of QE = 95%, the fraction of the light emitted by the dye which is reflected by the photonic structure is obtained.
N R ,int =
N R ,inc QE ⋅ ADye 70
(4.23)
4.5 Experimental results
b)
a)
100
transmission [%]
80
edge filter cover glass
60 40 20 0 350 400 450 500 550 600 650 700 750 800
wavelength [nm]
Figure 4.22: Transmission characteristic of an edge filter applied to a fluorescent concentrator with the dye BA 2411. The filter was designed to reflect all light between λ = 550nm and λ = 700nm. The increased transmission of the system of fluorescent concentrator / filter is a measure of how much light the filter traps in the concentrator. Without the filter this light is emitted into the hemisphere opposite to the integrating sphere. Assuming an equal emission into both hemispheres, the fraction of light additionally detected corresponds to half of the light in the escape cone and should therefore be at least LR,int = 13%. The result of this calculation is shown in Figure 4.23. 450
475
500
525
550
575 100
0.20
80
0.15
60 40
0.10 0.05 0.00 425
20
1/2 light in escape cone dye absorption 450
475
500
525
absorption [%]
half fraction of light in the escape cone
425 0.25
0 550
575
wavelength [nm]
Figure 4.23: Fraction of light in the escape cone derived from transmission measurements. The difference between the transmission measurements with and without filter is generated by the light in the escape cone reflected by the filter. The difference should therefore be equal to half of the light in the escape cone. Assuming isotropic emission, this fraction should be at least 13% (red line). This method indicates escape cone losses between 30% and 35%. As this figure shows, the assumed fraction of light in the absorption range of the dye is between LR,int = 15% and LR,int = 18%, indicating a fraction of light in the escape cone of ncone = 30% to ncone = 35%. Close to λ = 550nm this fraction drops because of the inset of the reflection peak of the filter at this wavelength. A possible explanation for the wavelength dependence of this characteristic is that for light which is absorbed at smaller wavelengths more reabsorption events occur (note that in Figure 4.23 the characteristic is given in dependence of the wavelength of absorption). With more reabsorption events the probability for an emission into the escape cone
71
Spectrally selective photonic crystals
increases. This results in a bigger fraction of light in the escape cone for smaller wavelengths. Another method to characterize the effect of the photonic structure is given by the LBIC measurement [Mar84]. Using the method for a fluorescent concentrator, every spot on the fluorescent concentrator is illuminated separately and the current of the attached solar cell is measured [Gol08]. The result of this measurement is shown in Figure 4.24. Especially interesting about this method is that spatial information about the effects of the photonic structure is gained. b) a) without photonic structure with photonic structure
signal [a.u.]
600
500
400
300 0
2
4
6
8
10
distance from solar cell [cm]
Figure 4.24: LBIC scan of the effect of a fluorescent concentrator with the dimensions 10cm x 5cm x 3mm.and a photonic structure on top of it. Figure 4.24a shows the .complete scan and Figure 4.24b a cross section of the measurements, with and one time without the band stop filter. The light guiding efficiency is increased over the full length of the concentrator. The high efficiency for the system without photonic structure close to the solar cell is caused by scattering of the substrate. The photonic structure initially reflects the scattered light, so that close to the solar cell the efficiency is decreased. Finally the external quantum efficiency (EQE) was measured for a fluorescent concentrator system with and without the band stop filter. The results of this measurement are shown in Figure 4.25. Where Figure 4.24 provided the spatial information of the effect of the photonic structure, Figure 4.25 gives the spectrally resolved effect of the photonic structure directly on the solar cell.
15
BA241, white bottom reflector
100 80
with Filter without Filter
60 10 40 5
reflection [%]
EQE [%]
20
20 Reflection Filter
0 300
400
500
0 600
wavelength [nm]
Figure 4.25: EQE measurement of the solar cell at the edge of the fluorescent concentrator with and without photonic structure. In the spectral ranges where the band stop filter reflects (below λ = 400nm, above λ = 550nm), a reduced EQE is perceptible. A clear increase of the EQE appears, however, for all wavelengths with a considerably low reflectance. This increase is caused by the improved light guiding efficiency. (EQE measured by J.C. Goldschmidt)
72
4.5 Experimental results It has to be said that the EQE is given with respect to the incident wavelength, which is different from the wavelength that the solar cell actually receives. The internal wavelength is shifted into the photoluminescence (PL) - range of the dye and therefore is located in the spectral range where the filter shows a high reflection. Opal
reflectance / tranmission / absorption in %
The opals used in the experiments were produced at the University of Mainz and the Fraunhofer IWM in Halle. In a first series of measurements, opals and inverted opals were compared. The result of this comparison is shown in Figure 4.26. 70
normal opal inverted opal
60 50 40 30 20 10 0 350
400
450
500
550
600
650
700
750
wavelength in nm
Figure 4.26: Measured spectral reflection characteristic of an opal (black dots) and an inverter opal (green triangles). The normal opal is clearly preferable to the inverted opal. The reflection peak of the opal is narrower, corresponding to the narrower pseudogap, but much less unwanted reflections occur and the reflection in the peak is higher. (Opals produced by L. Steidl and B. Lange, University of Mainz). The main disadvantage of the inverted opal is the very high reflectance in the absorption range of the dye. This reflection is mainly due to scattering that is caused by numerous imperfections produced during the inversion process of the opal. With a reflectance here as high as R = 30% there exists no chance that an inverted opal may have a beneficial effect on a fluorescent concentrator. Inverted opals have therefore not been used further for an application on a fluorescent concentrator as tremendous efforts to improve the opal quality are required. The tested normal opal structures used for the subsequent experiments were designed for the application on a fluorescent concentrator with the dye JMC4. The opal shown in Figure 4.27 was one with the lowest ratio of maximum reflectance to unwanted reflection achieved during the entire period of this work for this specific application. This opal reaches a maximum reflection of ca. R = 70% at λ = 600nm. It is experimentally very hard to exceed this value, as for thicker opals layers are in considerable disorder and contribute mostly to scattering and not to the photonic properties. The reflection peak is much narrower than the photoluminescence range, so that the enhanced light guiding efficiency only occurs for a fraction of the emitted photons. Unwanted reflections in the absorption range of the dye are in the range of R = 15% to R = 20%, so considerable losses have to be expected here.
73
Spectrally selective photonic crystals
R, A, PL [%]
100
absorption JMC4 PL emission JMC4 reflection filter
80 60 40 20 0
400
500
600
700
wavelength [nm] Figure 4.27: Reflection characteristic of an opal produced at the University of Mainz. The crystal was optimized for application on a fluorescent concentrator with the dye JMC4. Also shown are the absorption (red triangles) and the photoluminescence (black dots) of the fluorescent concentrator. The crystal shows a comparably low reflectance of maximally 70% in the photoluminescence range of the dye and high unwanted reflection in its absorption range. (Dye reflection and absorption measured by J.C. Goldschmidt) Using once more the simulation method presented in section 4.2, I calculated the estimated effect of the filters on the light guiding efficiency of a fluorescent concentrator with a geometric factor of w = 7. Simulated opal 22 layers (Figure 4.19a) with JMC 4
Simulated opal 10 layers (Figure 4.19b) with JMC 4
opal ca. 10 layers (Figure 4.27) with JMC 4
φgain
52.89%
20.48%
25.61%
φloss
8.64%
7.57%
11.54%
φ
3.92%
-5.45%
-5.65%
Table 4.7: Predicted effect of the opal on the performance of a fluorescent concentrator. Given are the analytically obtained value (first two columns) and the measured value (last column). Two opals were simulated; one opal with a reflection such that a considerable amount of photons is transported to the edges (left column) and one opal with a reflectance comparable to the real one (middle column). Neither the simulated nor the measured opals yield a positive result. However, the simulated opal was designed so that an appreciable fraction of light is transported to the edges. For the simulated opal, gain and loss are balanced and the effect of the opal is negligible. However, a thicker opal will theoretically produce a positive effect. Looking at the measured opal, the reflectance in the reflection peak is considerably lower and more unwanted reflections are produced. The total yield is therefore negative. For real opals to provide a positive effect, efforts to improve quality have to be made. Finally, the efficiency of a fluorescent concentrator system with the opal shown in Figure 4.27 attached to the front surface (Table 4.8) was measured. Here as well the effect is negative.
74
4.5 Experimental results Real opal (Figure 4.27) with BA 2411 3.31%
ηwithout ηwith
3.03%
-8.46% δηrel Table 4.8: Measured efficiency of fluorescent concentrator system with and without an opaline photonic structure. The efficiency is decreased by 8.5%. During this work, some effort was put towards an improvement of opals. Especially, one aim was to increase the magnitude of the reflection peak. For this reason, several batches of opals were produced. The endeavor to increase this magnitude was successful; however the problem of the unwanted, parasitic reflections was not solved. To analyze the development of the opals, the maximum reflectance in the peak as well as the parasitic reflection was measured. The result shows that an increasing peak reflection is connected to an increasing parasitic reflection (Figure 4.28). With the opals currently available it seems not possible to achieve an improvement of fluorescent concentrators.
parasitic reflection [%]
40 batch I batch II batch III
(july 07) (sept. 07) (oct. 08)
30
26%
20
10
0
0
20
40
60
80
maximum reflectance [%]
Figure 4.28: Dependence of parasitic reflectance on the maximum reflectance for normal opals. The red line gives the theoretical limit, above which no positive effect is possible by the application of a filter. The blue line gives the performance required, if the escape cone losses are ncone = 26% (break even relation).
4.5.2 Spectrum splitting system The spectrum splitting concept was the topic of a diploma thesis by Bernhard Gross [Gro092]. Bernhard Gross performed the characterization of the filters and the measurements of the efficiencies and the quantum efficiencies. For details about this concept, I recommend his diploma thesis. The measurements are used to illustrate how a very efficient PV device may be constructed using spectrally selective photonic structures. My part in this work was in the development of the concept and in the specification of the filter characteristics, as well as in the simulation of the filters. For the device described in section 4.1.2, two spectrally selective filters were used in front of the first two solar cells. No filter was used for the last solar cell as in this particular realization all remaining light was intended to be used by this solar cell. The filters were produced at mso Jena. Special about these filters is that they are optimized for an incidence of θ =45° corresponding to the device geometry. The spectral transmission characteristic of these filters is shown in Figure 4.29.
75
Spectrally selective photonic crystals
a)
b) optimum reflection simulated single measured single simulated system
80 60 40 20 0
400
600
800
1000 1200 1400 1600 1800
wavelength [nm]
100
transmission [%]
transmission [%]
100
optimum reflection simulated single measured single simulated system
80 60 40 20 0
400
600
800 1000 1200 1400 1600 1800 2000
wavelength [nm]
Figure 4.29: Spectral transmission characteristics for two spectrally selective filters produced at mso Jena for the spectrum splitting device. The filters are band stop filters similar to the one shown in Figure 4.18. The first filter was produced for a GaInP/GaAs dual junction solar cell absorbing light with a wavelength below λ < 850nm. The second filter was produced for a silicon solar cell that absorbed light with a wavelength below λ < 1080nm. As the device consisted of three solar cells, two filters were needed. The filters were designed for optical coupling directly to the solar cell. As it is not possible to measure the transmission characteristic if the filter is coupled to the solar cell, this characteristic was measured initially without solar cell (“measured single”). The measurements here were in good accordance with the simulation for the same case (“simulated single”). Following that, the performance of the filter when coupled to the solar cell was also simulated to obtain the expected transmission of the filter in the system (“simulated system”). For a PV system with the filters shown in Figure 4.29, the external quantum efficiency under one sun was measured. The used solar cells were a GaInP/GaAs (gallium indium phosphide / gallium arsenide) dual junctions solar cell for light with a wavelength λ < 850nm, a silicon solar cell for light with a wavelength between 850nm < λ < 1080nm, and a gallium stibnite (GaSb) solar cell for light with a wavelength λ > 1080nm. The result of this measurement is shown in Figure 4.30a. Also measured was the efficiency of the system under several conditions. One measurement was performed calibrated indoors and one measurement outdoors. The measured efficiency indoors resulted in a value of η=30.7%.Two points must be adverted to for this value. First: the light impinged on the device with a large angular spreading. The device was not laid out for that condition. The angular spreading has a considerable effect on the filter performance which was optimized for a θ =45° incidence. This reduces the efficiency as well as the second point: the indoor measurement was performed with an aperture. The intensity of the incident beam was calibrated to have an intensity of I = 1000W/m² at the aperture. As the solar cell area is larger than the aperture area, the intensity at the solar cells was considerably lower than 1000W/m². This comparably low intensity at the solar cells also had a share in reducing the solar cell efficiency. Both of these points play a much smaller role for the outdoor measurement. The outdoor measurement yielded with η=34.1% a much higher efficiency. For this measurement, the solar radiation in Freiburg was used directly. As sunlight has only a small angular spreading, this efficiency is probably a better measure for what may be achieved by this system than the indoor measurement. An even higher efficiency can be expected if the whole system is measured under concentration. However, the angular spreading of the incident light should be kept as low as possible. The optimization to concentrating conditions is a future task here. The measured system efficiency is shown in Figure 4.30b.
76
4.5 Experimental results
b)
a)
device, series connection
ISE CalLab (13.03.2009) - 25°C
1.0
25
0.8
st.
outdoor measurement march 31 2009, 16:44 2 DNI = 706 W/m 2 GNI = 806 W/m
0.6
System GaInP GaAs Si GaSb
0.4 0.2 0.0
current [mA]
EQE [%]
20 15
T = 20.7 °C 2 Aaperture = 2.51 cm
10
FF = 84.9% 2 2 Isc = 25.1 mA (Isc = 12.5 mA/cm for 1000 W/m )
5
Voc = 3.24 V ηGNI = 34.1%
400
600
800 1000 1200 1400 1600 1800
0 0.0
wavelength [nm]
0.5
1.0
1.5
2.0
2.5
3.0
3.5
voltage [V]
Figure 4.30: Measured external quantum efficiency for the spectrum splitting concept. The external quantum efficiency for each solar cell (GaInP/GaAs, Si, GaSb) is shown as well as the quantum efficiency for the system (thick grey line) (a). Also given is the IV characteristic of an outdoor measurement. Following the considerations given in subsection 4.2.2 the maximum quantum efficiency possible for the solar cells can be calculated. The results of these considerations are shown in Figure 4.31. Also given are the measured quantum efficiencies of the three solar cells used. The optical efficiency of the system is very high. The solar cell performance is not limited by the performance of the filters in each intended spectral range. Outside the spectral region for which each solar cell is intended, the optical limit is a good description for the solar cell performance. A small spectral inaccuracy exists for both measurements (transmission and EQE) which explains the spectral shift between the measured EQE and the optical limit for the second solar cell. 1.0
EQE [%]
0.8 0.6
EQE cell 1 EQE cell 2 EQE cell 3 EQE max 1 EQE max 2 EQE max 3
0.4 0.2 0.0
400
600
800 1000 1200 1400 1600 1800
wavelength [nm]
Figure 4.31: Comparison between the measured quantum efficiency and the maximum possible optically limited quantum efficiency, regarding the characteristics of the filter and the considerations given in subsection 4.2.2.
77
Spectrally selective photonic crystals
4.6 Summary of the chapter & perspective In this chapter, I have investigated photovoltaic concepts that use spectrally selective photonic structures. The photonic structures here are used in different ways. Either they serve to prevent loss mechanisms or they are used for spectral selection purposes. Concerning spectral selectivity, two very different PV concepts were considered that were introduced in section 4.1. The first concept was the advanced fluorescent concentrator. In this concept spectrally selective photonic structures are used to reduce escape cone losses that account for at least ncone = 26% of the incident light. The topic of the second concept is spectral splitting. The approach of this concept is to distribute light onto separated solar cells with different band gaps. Spectrum splitting concepts are intended for very high efficiencies exceeding η = 40%. In section 4.2, considerations of principles on the demands of the spectrally selective filters for the different PV concepts are given. For the advanced fluorescent concentrator concept, I have established an analytical model to calculate the amount of photons gained by the application of a spectrally selective structure. One result of this model was that the reflectance of the filter in the photoluminescence range of the dye should exceed 90% for fluorescent concentrators with dimensions comparable to the ones used. For the spectrum splitting concept I have calculated the influence on the fraction of photons utilized by a solar cell in dependence of the filter performance. One result here was, that for a good filter the influence of the transmission characteristics are more sensitive than the influence of the reflection characteristics. A guiding value for this concept is that if the filter performance is reduced by 1% (i.e. reducing reflectance and transmittance by 1% in the corresponding spectral ranges) the efficiency of the solar cell is reduced by 1.4%. In section 4.3 spectrally selective photonic crystals were introduced and an optimization towards their application on fluorescent concentrators is given. The three exemplary systems that were chosen are the rugate filter, the band stop filter and the normal and the inverted opal. No explicit example for an optimization for the spectrum splitting concept is given. Filters used here are similar to the presented band stop filter. In section 4.4, the analytical model developed in section 4.2 for the fluorescent concentrator was applied to the filters introduced in section 4.3. For the theoretically optimized filters, a positive effect was predicted for the rugate filter and the band stop filter. Due to an insufficient reflectance in the peak, the predicted effects of opal and inverted opal were negative. In section 4.5 experimental results of the filter application are presented. For the application on the fluorescent concentrators, results using the analytical model from section 4.2 and measurements for each the same photonic structure are compared, demonstrating the accuracy of the established model. Using the purchased filters, an enhancement of the light guiding efficiency of δη = 20% was predicted and an enhancement of δη = 19% in solar cell efficiency was measured for the band stop filter. This corresponds to ca. ¾ of the escape cone losses and shows that spectrally selective filters can be used to efficienctly reduce these losses. For the other filters no enhanced light guiding efficiency could be obtained due to too high unwanted reflections.First experimental measurements of a fabricated spectrum splitting device resulted in an efficiency of η = 34.1% (measured by B. Groß) of the attached solar cells nunder outdoor conditions. To my knowledge, this is the highest efficiency hitherto measured under one sun illumination. At this point I want to give a short evaluation of the results given in this chapter. Concerning the fluorescent concentrator concept, a positive aspect was that the application of a spectrally selective filter was successful. A high percentage of the escape cone losses were regained; the light guiding efficiency and the efficiency of an attached solar cell could be enhanced. Another positive aspect is that the established model resulted in an accurate prediction of the enhancement of the light guiding efficiency.
78
4.6 Summary of the chapter & perspective One demerit is that the band stop filter with which the best results were obtained is a complex thin film structure. The production of such structures is too expensive for a commercially sensible application. Other filters investigated did not reach sufficient characteristics to induce an enhancement of the fluorescent concentrator system efficiency. Especially for the opals neither sufficient reflection nor transmission in the corresponding spectral ranges could be achieved. A positive aspect of the spectrum splitting concept is that already in a first experiment a very high efficiency was measured for the investigated setup. The positive results of both concepts show the potential of spectrally selective filters. A disadvantage of this concept was also that the applied filters were complex and expensive thin film structures. However, as this concept is intended for an application under concentrated illumination, the application of advanced optical devices is possible. A future task to further increase the efficiency of fluorescent concentrators with spectrally selective filters concerns the emission characteristic of the fluorescent dye. The spectral emission characteristic has only been measured at the edges of the concentrators and not at the top or the bottom. Therefore the measured photons had a much longer path to the spectrometer than they had to the filter. On this longer path the chance of reabsorption is much higher, which influences the spectrum reaching the filter. The angular emission has been assumed to be isotropic. Though this is probably qualitatively a good assumption experimental results exits that indicate spectral dependence of the angular emission characteristic. For a further optimization of the filters, these effects should be taken into account. Another task concerns the quality of opals. Opals comprehend the possibility of a cheap and large scale deposition, however, as was shown, opals produced today have too many defects and show too high unwanted reflections. In principle a positive effect with opals should be possible if more than 20 sphere layers could be assembled with no or little imperfection. Therefore deposition methods should be investigated that lead to an improved performance of opals. A possibility of doing so would be to structure the fluorescent concentrators before depositing the opal. A deeper consideration could include thermodynamical considerations with the aim to investigate the conditions that result in opal defects. A future task concerning the spectrum splitting concept is the optimization of this concept to concentrating conditions. A major point here will be the angular spreading of the incident light. This will present a problem both for the geometric setup and the performance of the filters. Considerations given in chapter 6 will have to be included here. As a perspective I want to discuss two concepts in short which have been developed and patented during the scheme of this work but haven not yet been closely investigated yet. The first concept concerns upconversion and the second is an advanced version of the fluorescent concentrator that includes photonic effects. In the example of the fluorescent concentrator, it was shown how spectral selectivity may be used in a system with frequency conversion of the incident light. The fluorescent concentrator is an example for a system with downconversion (i.e. the frequency is lowered). The opposite example would be upconversion, and here as well an improvement with spectrally selective filters is possible. An exemplary system combining up- and downconversion is shown in Figure 4.32. Concerning upconversion, photonic crystals exhibit another useful property. Including the upconverting material into a photonic crystal, the emission of the upconverting material may be increased if the transition is in resonance with a mode of the photonic crystal [Gal062]. This allows using upconverting materials already for much lower light concentrations which may help to include upconverting materials into solar cells.
79
Spectrally selective photonic crystals
Figure 4.32: Design of an advanced upconversion system. The fluorescent concentrator absorbs a broad spectral region, and emits in the narrow absorption region of the upconverter. By this means, the incident solar photon flux is concentrated spectrally and geometrically, both increasing the efficiency of the upconverter. a) A photonic structure (left inset) serves as a back reflector for the solar cell and ensures light trapping in the fluorescent concentrator. Another photonic structure (right inset) prevents the upconverted light from being reabsorbed in the fluorescent concentrator. Similar considerations may also be used for a concept that could considerably increase the light guiding efficiencies of fluorescent concentrators. The basic idea here is to include a fluorescent material in such a way into a photonic crystal, that emission is enhanced in some directions and suppressed in other directions. This could be achieved by including a thin layer with a thickness smaller than the wavelength of the considered light into a photonic crystal or to build a fluorescent concentrator out of luminescent materials. This concept, which was given the name “Nanofluko”, is shown in Figure 4.33. One task here concerns the inclusion of photoluminescence inside a photonic structure. This is not possible with the RCWA method and other simulation methods like FDTD need to be evaluated here. a)
b)
Figure 4.33: Schematic sketch of the “Nanofluko” concept. A thin layer of fluorescent material is included into a photonic crystal. In the figure, the photonic crystal is a Bragg stack (a). Another possibility is to include the luminescent material directly into the photonic crystal. As an example the dye is included into the spheres of which the opal is made (b). In both cases, the intention is the same, to suppress the emission of the dye in one direction and to force all emission into the desired directions.
80
5 Diffractive structures The topic of this chapter are structures that perform a specific change of the direction of light inside a solar cell. The aim of this approach is to increase the internal path length of light. Such a specific change of direction is obtained by adopting diffractive structures i.e. gratings. By now, several approaches have been accomplished on this topic by various groups. First results for gratings were presented by Gale [Gal90] and Kies [Kie91]. The probably bestnoticed work regarding submicrometer gratings was performed by Heine & Morf [Hei95] in 1995. In this work, binary gratings with a period of Λ = 620nm are studied. For this period, the 2nd order of diffraction travels parallel to the surface. Also investigated are blazed gratings with a period of 660nm. Heine & Morf report a path length enhancement of approximately a factor of 5 for the blazed grating in a silicon wafer with a grating depth of d = 70nm. In a subsequent work [Mor97] from 1997, Morf, Kies & Heine report an increase in quantum efficiency in the infrared for a flat silicon solar cell with a thickness of t = 40µm and no antireflection coating. One important point in their work is the suggestion that the surface recombination is not increased by the incorporation of the grating. Zaidi, Gee and Ruby [Zai00] in 2000 extended the concept to other grating designs, including 1D triangular patterns and 2D hole patterns. Increased quantum efficiency was shown for the triangular grating. As light trapping concepts are especially beneficial for thin film technologies, the approach of gratings has also been used in this field. One example here is the study of gratings for thin amorphous silicon solar cells by Morf in 2002 [Mor02], in which the grating was used to reduce the reflection of a solar cell made of amorphous silicon with a thickness of t = 310nm. Another example is the application of gratings for light trapping purposes in organic solar cells, published by Niggemann et al. [Nig02]. A recently published work by Bermel et al. suggests the combination of Bragg reflectors and diffractive photonic structures [Ber07]. In the last few years, interest in the topic of diffractive gratings has increased, as shown by the increasing number of publications [Fen07][Isa08]. However, despite all efforts, no commercialized product of a solar cell with diffractive grating has emerged yet. In this chapter, I will mainly investigate diffractive gratings in the context of crystalline silicon solar cells. This kind of solar cell is well established and highly optimized. Improving a crystalline silicon solar cell would show the high potential for a diffractive structure also on other kinds of solar cells. Another reason to use crystalline silicon solar cell is that they are well commercialized and the potential of developing a concept fit for the market is highest here. One goal of this chapter shall therefore be to investigate the potential of back side gratings for this particular application. The models and considerations discussed were all developed with regard to this goal. The presented results form the basis of experiments currently performed. The considerations given in this chapter are adaptable to other types of thick solar cells and materials and partially also to thin film solar cells as well. To expand the theoretical methods and results to other thick solar cells only the material parameters would need to be changed. Beside these considerations, I have also tried to take the general view on gratings in PV concepts. Considerations of principles, e.g. are not restricted to a special kind of solar cell. In section 5.1, the concept of the back side grating in a solar cell is introduced and the special circumstances for textured solar cells are outlined. The relative orientation of front surface texture and grating and, consequently, the direction under which light impinges on the grating, plays a special role here. In section 5.2, considerations of principles are given for the maximum path length enhancement of grating structures. The potential current that maximally can be gained by the application of a grating is calculated. The models and methods that were used to investigate the gratings theoretically are presented. Section 5.3 introduces the different types of gratings that have been investigated. In section 5.4 the results of the theoretical investigations with the different methods are given, optimum parameters for the gratings are defined, and the effect on solar cells is analyzed. In section 5.5 first experimental results for gratings that were introduced in solar cells are presented.
81
Diffractive structures
5.1 Photovoltaic concepts In this section I will introduce the PV concepts investigated in this work. Two concepts were considered each with the aim of increasing the path length of light inside the solar cells for the weakly absorbed light. The first concept is a back side grating on a silicon solar cell with a flat front surface. This concept will be the topic of subsection 5.1.1. In subsection 5.1.2 I will discuss the case for a back side grating on a crystalline silicon solar cell with a pyramidal front surface texture.
5.1.1 Back side gratings for solar cells with a flat front surface The introduction of gratings into solar cells with a flat front surface is a promising concept. In such a solar cell, the direction of light inside the solar cell is always close to normal, or at least very steep. Changing the direction of the internal light will therefore almost certainly have a positive effect on the internal path length, and consequently on the absorption (Figure 5.1). In this context especially, gratings are an interesting option for thin film solar cells and most concepts demonstrated in the past few years aim for a path length enhancement in such solar cells [Mor02][Nig02][Ber07]. An advantage for thin solar cells is that structuring the active medium is not necessarily needed if a structured substrate is used. In that way losses induced by damages at the rear surface are reduced. Thin film solar cells typically have thicknesses of a few µm. However, as was shown already in the introduction, also thick crystalline silicon solar cells benefit from a path length enhancement in the near infrared region (Figure 1.1). The approach in this work is to show, what can be achieved with a grating in a crystalline silicon solar cell. As a testing device, a solar cell with a thickness of d = 40µm is examined, but also for a solar cell with a thickness of d = 200µm results are shown.
Figure 5.1: Sketch of the basic idea of the diffractive structure in a solar cell. In a solar cell without grating, the light is just reflected at the rear surface. The grating adds several orders of diffraction that induce a path length enhancement and consequently an increased absorption. The increased absorption increases the generation of charge carriers and, eventually, the solar cell efficiency.
5.1.2 Back side gratings for solar cells with a pyramidally textured front surface One task of this chapter shall be to answer the question of whether it is possible to maintain a positive effect from diffractive structures in textured solar cells as well. A positive effect emerges, as a rule of thumb, if the direction of light after diffraction is more complanate than before. To examine the effects of a grating, one possibility is therefore to compare the light paths before and after diffraction.
82
5.1 Photovoltaic concepts The workhorse for front surface textured monocrystalline silicon solar cells is the pyramidal texture. Especially for a regular texture with inverted pyramids [Cam87], the main paths of the internal light may be obtained analytically [Sch94] (Figure 5.2). For a first analytical estimation of the effect of inverted pyramids, the model shown in this figure has been adopted.
Figure 5.2: Paths of the light in a pyramidally textured solar cell for the case of a double reflection on opposite surfaces. The figures given represent the case for light with a wavelength of λ = 1040nm and a solar cell with an antireflection coating of silicon oxide with a thickness of t = 105nm. In the silicon two main paths result. The most important transports 76.4% of the incident light under an angle of θ =41.4°. The second transports 19.4% of the light under an angle of θ =59°. Following this models’ assumptions, for normal incident light two main paths exist inside the solar cell after entering through the front surface. The most important path transports 76.4% of the incident light (it has a transport efficiency ηθ of 76.4%) and has an angle of θ=41.4° against normal. The second path transports 19.4% of the incident light and has an angle of θ=59°. Assuming a perfect mirror, these paths are a correct description for the travel of the light from the front surface to the rear side and back to the front surface again. As no directional change of the light occurs inside the solar cell, most of the light will couple out through the front surface again and only a small fraction will be reflected due to Fresnel reflections. However, all light experiencing more than one reflection has been ignored in the further considerations. The contemplation discussed until here has been made for inverted pyramids. Similar results are though obtained for random pyramids [Cam90, Kin91], too and the general idea also holds. A problem that must be discussed at this point concerns the relative orientation of grating and surface structure. This problem will be referred to as the alignment problem. The origin of the alignment problem is that the pyramidal front surface texture and the back side grating have different symmetries. The front surface texture has a 90° rotational symmetry, assuming inverted pyramids, while the back side structure only has a 180° rotational symmetry. For this reason, light coming from the front surface texture, has two possible azimuthal components when impinging on the grating, depending on what relative orientation the facet of the texture the light impinged on had. This phenomenon is shown in Figure 5.3.
83
Diffractive structures
Figure 5.3: Relative alignment of grating and front surface texture. Depending on the orientation of the facet the light impinges on, relative to the grating, two azimuthal components emerge with a difference of 90°. The consequence of this is that 50% of the light impinges on the grating with another direction than the other 50%. Because of the alignment problem each 50% of the light takes another direction as the other 50% when impinging on the grating. This may cause problems, because the grating needs to be optimized for each orientation independently. To elude the alignment problem, several possibilities exist. One is to tilt the grating by δ = 45°. If that is done, the azimuthal component for light impinging on the grating is always |δ| = 45°, regardless of the pyramidal flank on which it impinges. The second possibility is to use 2D gratings with the same orientation as the front surface structure. In this case the grating has the same rotational symmetry as the front surface texture and all light impinges on the grating with the same relative orientation (Figure 5.4). a)
b)
Figure 5.4: Possibilities to elude the alignment problem. In Figure 5.4a the 1D grating is tilted by δ = 45° towards the front surface texture. This results in a single azimuthal component of |δ| = 45° for light impinging on the grating. In Figure 5.4b a 2D grating is adopted with the same orientation as the front surface texture. In that case, the azimuthal component for light from both pyramidal flanks is equivalent and is denoted as δ = 0° 10.
Also the 2D structure may be tilted towards the front surface orientation by δ = 45° to obtain a single azimuthal component, in that case of δ = 45°.
10
84
5.2 Considerations of principles
5.2 Considerations of principles In this section I will discuss some considerations of principles concerning the potential of grating structures for a path length enhancement. Subsequently I will introduce the methods used to calculate the effects of a grating on a solar cell. For this purpose two methods were employed: a semi-analytical method to estimate the gain in current density produced by a grating and a rigorous model to estimate the generation profile of charge carriers in a solar cell. The first method was used to determine the optimum grating parameters. The second method is used to simulate rigorously the electromagnetic field inside the solar cell. This method is far more accurate but also far more complex and time consuming than the first.
5.2.1 Potential of gratings in a silicon solar cell In this subsection I want to give a short estimation of the potential for gratings in a crystalline silicon solar cell. The approach made here aims for a path length enhancement and consequently an increased efficiency for the part of the solar spectrum which is close to the band edge of silicon at λ = 1100nm. Therefore a spectral range between λ = 900nm and λ = 1127nm is considered (the reason to choose this truncated wavelength of λ = 1127nm lies in the available material data of silicon). The total number of photons in the AM1.5 spectrum in this spectral range is ca. N = 6.44 ⋅ 10 20 cm −2 s −1 .The total current density that would be produced if all these photons were absorbed and would generate an exciton is j = 10.33mA/cm². To estimate the maximum current density produced by a solar cell made of crystalline silicon solar cell with a flat surface, I calculated the number of photons absorbed in the silicon bulk and scaled the result with the elementary charge. The current density calculated in that way shall be addressed to as the potential current density. λ =1127 nm
j pot = e0
∫N
AM 1.5
(λ ) ⋅ (1 − exp[−2 ⋅ t ⋅ α (λ )])
(5.1)
λ =900 nm
The factor 2 in this equation emerges from the presence of a perfect back side mirror. The potential current density produced by a solar cell with a thickness of t =40µm is jpot = 3.79mA/cm². For a solar cell with a thickness of t = 200µm the potential current density is jpot = 7.10mA/cm². This shows the potential gain in current density that may be achieved by a grating in the considered spectrum 11. Especially in thin solar cells a large fraction of the photons in the red part of the spectrum are not used. I have also estimated the potential current density for a pyramidally textured crystalline silicon solar cell for the spectral range between λ = 900nm and λ = 1127nm. In that case, the contributions of the different paths due to reflection and diffraction at the structured surface need to be summed up
j mir = ∑ηθ ⋅ e0 ⋅ θ
λ =1127 nm
∫N
λ =900 nm
AM 1.5
2 ⎡ ⎤ ⋅ t ⋅ α ( λ ) ⎥ ) dλ (1 − exp ⎢− ⎣ cos θ ⎦
(5.2)
The transport efficiencies ηθ and directions of the different paths are given in Figure 5.2. The potential current density generated by a solar cell with a thickness of t = 40µm is jpot = 4.57mA/cm². For a solar cell with a thickness of t = 200µm, the potential current density is jpot = 7.71mA/cm². To compare this rough estimation with the actual situation in a solar cell, the current density generated by a textured solar cell with a thickness of t = 200µm was also calculated from EQE measurements. These calculations resulted in a value of j = 7.9mA/cm² 11
Typically crystalline silicon solar cells are very good absorbers for light with a wavelength below λ = 900nm. The potential therefore only changes little when considering also lower wavelengths
85
Diffractive structures
which is slightly larger than the current density calculated from the absorption. This may be explained as the simple estimation of the potential current density underestimates the path length of the light inside the solar cell. However, the comparison shows that even this very rough prediction yields acceptable results. Knowing the current density generated by a solar cell under the specified conditions, the potentially gained current density can be defined as the current density generated by the unused photons. This potentially gained current density constitutes an upper limit for what can be achieved by a light trapping device. potentially gained current density (900nm-1127nm) flat crystalline silicon solar cell, 6.54mA/cm² t = 40µm with rear side reflector flat crystalline silicon solar cell, 3.23mA/cm² t = 200µm with rear side reflector textured crystalline silicon solar cell, 5.76mA/cm² t = 40µm with rear side reflector textured crystalline silicon solar cell, 2.62mA/cm² t = 200µm with rear side reflector calculated from EQE measurement on a 2.43mA/cm² t = 200µm solar cell with pyramidal texture Table 5.1: Potentially gained current density for a textured and an untextured solar cell with thicknesses of t = 40µm and t = 200µm. The potentially gained current density is obtained by counting all unabsorbed photons per second and cm² in the spectral range λ = 900nm – λ = 1127nm and multiplying them with the elementary charge e0. For comparison, the potentially gained current density from the unused photons calculated from an EQE measurement of a solar cell with a thickness of t = 200µm is shown. Given, that a very good crystalline silicon solar cell generates a current density of ca. 40mA/cm² [Gre99], the potential in this part of the spectrum is considerable.
5.2.2 Maximum path length enhancement for concepts with a discrete change of the light direction inside the PV device In this subsection I want to discuss the potential for a path length enhancement with a structure that induces a discrete change in the internal direction of light. An optimum path for the light in a solar cell would be achieved if all light travelled parallel to the (flat) solar cells’ surface. Assuming normal incidence, this would mean a directional change of 90°. The thermodynamic limits for a structure that performs such a directional change will be estimated. This approach follows an argument originally given by Thomas Kirchartz [Bad09]. Initially, the small angular divergence of sunlight of θs = 4.7 mrad has to be taken into account. Considering light with an angular divergence instead of parallel light, conservation of étendue must be taken into account. Without any further specifications, the following assumptions for the structure can be made: 1. 2. 3.
Rays with a direction perpendicular to the back surface are directionally changed so that they move parallel to the surface. Every other ray is diffracted in a way that the étendue of the incoming light beams is conserved. The directional change shall have a rotational symmetry, which is probably impossible to realize in practice but constitutes an upper limit for any structure.
86
5.2 Considerations of principles The structure shall be realized within a solar cell consisting of a material with the refractive index n surrounded by air (n=1). Incoming radiation shall have an étendue [Wel78, Bor99]
dE = n 2 dΩdA = dΩdA
(5.3)
After refraction at the front surface, étendue conservation leads to the same result as Snell’s law
sin θ i = n ⋅ sin θ d
(5.4)
This is the relation between the angle θi of the incoming light beam and the angle θd of the refracted light beam. The light beam impinging on the back surface is then reflected so that the angle of each ray is as high as possible without changing the étendue. Since the illuminated area and the refractive index stays the same, étendue conservation requires θd
π /2
0
π / 2 −θ c
∫ sin θ cos θ dθ =
∫ sin θ cos θ dθ
(5.5)
2⎛ π
⎞ ⇔ sin 2 (θ d ) = cos ⎜ − θ d ⎟ ⎝2 ⎠
The situation described by equation (5.5) is the one for a cone in which every single direction is turned by 90°. Looking at it in that way, a unique path in the solar cell from entry to egress may be allocated for each incident direction. These paths may be described with the following steps: The sunlight enters the solar cell and traverses it (1). At the back side it is reflected and changes the direction because of the structure (2). The reflected light takes its path back to the front surface where it is possibly totally internally reflected (3). The light, reflected once more, impinges on the structure, is reflected and directionally changed. Because of the light path reversibility, it now has the original direction again (4). The light then impinges on the front surface and leaves the solar cell (5). The described path is shown in Figure 5.5. Considering this path, the maximum path length enhancement for the light in the solar cell is given by θs
⎞ ⎛ 1 1 ⎜ 2 ⋅ + ∫0 ∫ ⎜⎝ cosθ 2 (θ1 ) sin θ 2 (θ1 ) ⎟⎟⎠ sin θ1 dϕ dθ1 ⇔ θs
∫∫
sin θ dϕ dθ
0
⎞ ⎛ ⎟ ⎜ 1 1 ⎟ ⎜ 2⋅ ∫⎜ + ⎟ sin θ 1 dθ 1 ⎡ ⎤ ⎡ ⎤ 1 1 ⎛ ⎞ ⎛ ⎞ 0⎜ ⎟ ⎜ cos ⎢ Arc sin ⎜⎝ n sin θ 1 ⎟⎠⎥ sin ⎢ Arc sin ⎜⎝ n sin θ 1 ⎟⎠⎥ ⎟ ⎣ ⎦ ⎣ ⎦⎠ ⎝ θs
θs
∫
⇔
sin θ dθ
0
θs
2⋅
∫ 0
sin θ dθ 1 − sin 2 θ / n 2 θ2
∫ sin θ dθ
θs
+ n ∫ dθ 0
⎡ θs ⎤ ≈ 2n ⋅ ⎢ ⎥ for θ s 〈〈 1 ⎣1 − cosθ s ⎦
(5.6)
0
The maximum path length enhancement for the divergence of sunlight with a structure that selectively changes the direction of light in a solar cell is resultantly given by L = 853 ⋅ n . For silicon with n = 3.5 this results in a factor of L = 2980.
87
Diffractive structures
Figure 5.5: Path of the light inside a solar cell with a structure that changes the internal direction by exactly 90°.
5.2.3 The binary grating To perform the directional change, grating structures shall be employed. The well-known characteristic of a grating is that it distributes light into several orders of diffraction that are emitted into specific angles. The relation between the angle θ under which the mth order is emitted, the angle of incidence θ0, the period Λ of the grating and the wavelength λ is given by
sin θ + sin θ 0 =
m ⋅ λ0 , m = (K,−1, 0, 1, K) n⋅Λ
(5.7)
From equation (5.7) it is clear that the desired directional change with a grating is only obtained for a single design wavelength λ0. The first order of diffraction for the design wavelength is diffracted into θ = 90° if λ0/(n Λ) = 1 (assuming normal incidence). The first demand, the directional change of the light parallel to the solar cell, is therefore controlled by choosing the grating period accordingly. For a flat solar cell, i.e. normal incidence on the grating, the period is given by Λ = λ 0 / n . For crystalline silicon with λ0 = 1100nm and n = 3.5, this corresponds to an initial value of Λ = 314nm. For a textured solar cell, the situation is not so easy anymore because of the multiple directions form which light impinges on the grating. For an incidence of θ0 =41.4°, the optimum period is Λ = 261nm, for θ0 =59°, the optimum period is Λ = 243nm. The initial value must be chosen somewhere in between. Considering the alignment problem, the predictions become even more difficult. No predictions for an azimuthal value other than δ = 0° are given in this section. One possibility to suppress the 0th order diffraction is the application of a λ/4 effect in a binary grating. A similar effect is used e.g. for antireflection coatings. For a binary grating, λ/4 effect is obtained if the grating depth is defined by d = λ0 / (4 ⋅ neff ) , neff being the effective refractive index in the grating region. This consideration uses the approach of effective media. Using effective media is not justified in this case, because the period of the grating is too large. Especially, differences must be expected for TE and TM polarization. The approach has been used nevertheless to find an initial value for the grating depth that can be used as a starting point for the optimization. I want to emphasize, that finding such initial values for the parameters is the whole point of the considerations given here. The considerations are illustrated in Figure 5.6.
88
5.2 Considerations of principles
Figure 5.6: Illustration of the concept of a back side reflective grating that changes the direction of a normal incident beam with the design wavelength λ0 by 90°. The period of the grating Λ determines that the first order diffraction has a direction along the solar cell. The grating depth d assures destructive interference for light diffracted into the zeroth order, so that all light for the design wavelength is diffracted into the first order.
efficiency diffracted st into 1 order [%]
80
direction
60
90 80 70 60
40
50
20
40
0
30 600
700
800
900
1000
1100
st
TE polarization TM Polarization
100
direction of 1 order [deg]
The initial value for the grating depth, in this approach, is defined by the effective refractive index in the grating region. Assuming that each of the materials fills half of the volume, a simple approach is to take the average of both refractive indices. One material typically used in this work to fill the grating was SiO2 with a refractive index of n2 = 1.5. The effective refractive index for this configuration is neff = 2.5. The initial value for the grating depth is consequently d = 110nm. This consideration has been given for normal incidence. For non normal incidence the thickness scales with a factor cos(θ) where θ is the direction in the effective medium. As neither the effective index is known exactly nor the direction of incidence is unique, finding an initial value is pure guesswork and the optimization has been started with the same value as for normal incidence. Having retrieved initial values for the binary grating, a first estimation of the path length enhancement that may be achieved with such a grating shall be given and the wavelength dependence of the effect shall be explored. For this reason, a binary grating has been implemented into the RCWA method (see section 3.1) and direction and diffraction efficiencies of the diffracted orders have been calculated for normal incidence. The path length enhancement is determined by these two factors, diffraction efficiency and diffraction angle (see part 5.2.5) Implementation and simulation results are shown in Figure 5.7 b) a)
wavelength [nm]
Figure 5.7: Implementation of an exemplary binary grating. The parameters used for this grating were the initial values derived in the previous paragraph for normal incidence. Figure 5.7b shows the efficiency that is diffracted into the first order for the different polarizations.
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Diffractive structures
Figure 5.7 demands further interpretation. The first issue is the difference in the characteristics for the different polarizations. This effect can be explained from the approach of effective media. The grating has different properties along all three axis which results in different effective refractive indices for TE and TM polarization. Additionally, the indices will depend on the wavelength because of the change in the ratio Λ/λ. Using the positions of the different maxima and a thin film approach, for TE polarization a refractive index of n = 2.41 for λ = 1019nm and n = 2.31 for λ = 678nm is obtained. Another issue is that a lot of effects were neglected in the simple scalar approach. Wave optical effects make the scalar prediction inaccurate, especially so close to the design wavelength. The characteristics of the grating depend on the exact realisation, including issues like the thickness of the SiO2 layer and the optical properties of the reflector (e.g. surface plasmons play a role here.). All these issues make an analytical prediction of grating parameters and performance very difficult. Finally, when looking at solar cells one is not so much interested in the path length enhancement for a certain wavelength but how the grating has to be designed to induce a maximum efficiency enhancement. What these points show is: the optimum parameters for a certain grating depend on the exact context of the grating and cannot be predicted with a simple analytical approach. The context depends on the particular concept and on the technical possibilities. In silicon solar cells, for example, a technique typically used to implement a structure is reactive ion etching. The technique limits the shape of the grating. A crucial issue is that a back side grating bears the risk of an increased surface recombination. This issue limits the materials adoptable. For this reason, I have set up several models to optimize gratings in their context. Before I discuss these models, I briefly want to discuss another grating concept.
5.2.4 The blazed grating Apart from the λ/4 effect, another well-known way exists to trigger high diffraction efficiencies into a single order of diffraction. This ways is the use of blazed gratings. Blazed gratings feature a saw tooth profile. The normal of a facet of this saw tooth has an angle θb against the grating normal. The concept of the blazed grating is depicted in Figure 5.8.
Figure 5.8: Concept of the blazed grating. Light incident with an angle θ0 against the grating normal is diffracted. The diffracted orders are defined by the angles θi. The facet normal is tilted towards the grating normal by an angle θb High diffraction efficiency is achieved if for incident light and one diffracted order (denoted ith order) mirroring conditions are satisfied with respect to the facet normal. The blaze condition with the blaze angle θb is therefore given by
90
5.2 Considerations of principles θ 0 − θ b = θ i + θ b , (i = …-1,0,1,…)
(5.8)
Conventionally a grating of this type is called a blazed grating, if the blaze condition is satisfied for a low order of diffraction (i = 0, 1 or 2) and an echelette grating, if the condition is satisfied for a higher order of diffraction. The wavelength λb to which this condition corresponds is called the blaze wavelength and is given by
i ⋅ λb = 2 ⋅ Λ ⋅ [sin θ b ⋅ cos(θ 0 − θ b )]
(5.9)
The blaze conditions are depicted in Figure 5.9
Figure 5.9: Illustration of the blazing conditions. The incident light is diffracted such that the facet is a mirror for incident and diffracted light. In equations (5.9) no dependence on the refractive index was included. If the light is incident from a medium other than air, the blaze wavelength must be scaled accordingly. Now the initial values for the blazed grating shall be determined. For normal incidence, the desired effect is obtained for θb = 45° and i = 1. Inserting this into equation (5.9), the relation Λ = λb/n is obtained. Considering the situation for a crystalline silicon solar cell, this results again in an initial value of Λ = 314 nm for the grating period. For non normal incidence, the situation becomes again complicated because of multiple directions. For an angle of incidence of θ0 = 41.4°, the blaze angle is θb = 65.7° and the period is Λ = 189nm. For an angle of incidence of θ0 = 59°, the blaze angle is θb = 74.5° and the period is Λ = 169nm.Here also the alignment problem was not considered. To compare the binary grating to the blazed grating, the intensities in the 1st order for the different gratings with the initial parameter values derived in this paragraph for normal incidence are shown in Figure 5.10. In this figure, the blaze conditions is satisfied for λ = 1100nm. Seeing how the characteristic of the blazed grating is similar to the one of the binary grating, the same objections to the analytical approach hold as for the binary grating. Also the blazed grating has to be optimized in each single case.
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Diffractive structures
TE binary TM binary TE blazed TM blazed
efficiency diffracted st into 1 order [%]
100 80 60 40 20 0 600
700
800
900
1000
1100
wavelength [nm]
Figure 5.10: Comparison of the binary grating and the blazed grating by means of the efficiency diffracted into the 1st order. For both gratings a similar characteristic is obtained, though the one for the blazed grating is shifted by ca. 70nm from the one of the binary grating.
5.2.5 Semi-analytical model to calculate the effect of a grating In this subsection I will introduce the semi analytical model that has been used to estimate the effect of a grating on a solar cell. The basic idea of this model is to calculate whether a grating increases or decreases the exciton generation inside a solar cell, compared to the case that no grating is introduced. For a single wavelength, the effect of the grating is positive, if the path length l of the light inside the solar cell is increased. For light that has a direction defined by a polar angle θ against normal inside a solar cell with the thickness t the path length for crossing the solar cell once is given by
l=
1 ⋅t cosθ
(5.10)
Assuming a specular back side reflection, this path length is doubled. Now, the path is restricted from the entrance through the front surface until the second impinging of the beam on the same surface. Fresnel reflections or total internal reflection at the front surface are neglected. This restriction will hold for all calculations performed with this method. The total path length will therefore be underestimated, and consequently also the fraction of light absorbed and the excitons generated will be too low. After introducing a grating into the solar cell, several paths emerge as the light is diffracted into several orders. The total path length of the light is now given as the sum over all orders of the paths weighted with the diffraction efficiency ηm diffracted into the mth order
l=
1 1 ⋅ t + ∑η m ⋅t cosθ cosθ m m
(5.11)
In this equation, θ defines the direction of the incident light inside the solar cell, θm defines the direction of the diffracted light and ηm are the diffracted efficiencies. The diffracted directions and efficiencies are calculated using the RCWA method (see section 3.1). The decision whether the introduction of a grating has a positive effect or not depends only on the angles of diffraction θm. The method will therefore not predict the absorption correctly, but it will predict, whether the grating induces a path length enhancement or not and it will
92
5.2 Considerations of principles allow for a comparison between different gratings. The paths of light beams in a solar cell with and without grating considered with this model are shown in Figure 5.11. b)
a)
Figure 5.11: Path of light inside a solar cell with and without a grating. To calculate the absorption, the path length is determined from entrance through the front surface to egress, when a beam impinges on the same surface for the second time.Figure 5.11a shows the case for a solar cell with a back side mirror. Only one path exists for inward and the backward direction. Figure 5.11b shows the case for a solar cell equipped with a back side grating. The inward path is unchanged but the backward path is split up. Let the initial number of photons per unit time, unit area and wavelength in the solar cell be N0(λ). The number of photons absorbed under these conditions is given by inserting equation (5.11) into Lambert-Beers law
⎡ ⎤ 1 1 ⋅ t ⋅ α (λ ) + ∑ η m ⋅ t ⋅ α (λ ) ⎥ ) Abs (λ ) = N 0 (λ ) ⋅ (1 − exp ⎢− cos θ m m ⎣ cos θ ⎦
(5.12)
In this equation, α(λ) is the wavelength-dependent absorption coefficient of the solar cell material. To calculate a current density from the absorption, I assumed that every absorbed photon creates an electron-hole pair (i.e. a quantum efficiency of 1). From equation (5.12), a current density is derived by integrating over the AM1.5 spectrum and multiplying by the elementary charge
j gra = e0 ⋅ ∫
AM 1.5
⎡ ⎤ 1 1 ⋅ t ⋅ α (λ ) + ∑ η m ⋅ t ⋅ α (λ )⎥ )dλ N 0 (λ ) ⋅ (1 − exp ⎢− cos θ m m ⎣ cos θ ⎦
(5.13)
This value is compared to the one derived for the case that no grating is introduced in the solar cell but a perfect back side reflector is located at the bottom of the solar cell. This value is given by
2 ⎡ ⎤ N 0 (λ )(1 − exp ⎢− ⋅ t ⋅ α (λ )⎥ )dλ AM 1.5 ⎣ cosθ ⎦
j mir = e0 ⋅ ∫
(5.14)
The gain in current density by the grating is consequently given by jgra-jmir. It is therefore possible to reduce the current density generated by a solar cell with a grating if the diffraction results into paths with a steeper direction than the one the light impinges with on the grating. This is in some situations the case e.g. for textured solar cells.
5.2.6 Rigorous model to calculate the effects of a grating The model presented in the last subsection was used to optimize the gratings to the most advantageous period and grating depth. However, the model does not describe the
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Diffractive structures
distribution of the EM field inside the solar cell correctly and apparently underestimates the effects of the grating on a solar cell (see section 5.5). Important points causing an inaccuracy of the last model are that light is mostly treated as a ray and not as a wave, and that the path of the light inside the solar cell is tracked until the second internal reflection. For this reason a second, rigorous model was set up to calculate the absorption inside the solar cell correctly. This model uses the RCWA method to calculate the distribution of the electric field inside the solar cell. A big advantage of the RCWA is that the complete structure, including front side texture and back side structure, may be implemented at once. However, this simulation is very time-consuming and therefore does not lend itself feasibly to optimization. It may be used, though, to calculate the correct absorption profile for an optimized setup. The task to solve with this model is now to calculate a 1D generation profile of charge r carriers inside a solar cell in which the 3D distribution of the electric field E ( x, y , z ) is known. In a first step this problem is reduced to a 2D problem by considering cross sections throughout the solar cell in the x-z-plane for a constant y. For each cross section a generation profile is calculated and the average over all cross sections is taken to obtain the total profile. The result of the RCWA simulation is the electric field in the planes that are defined by the cross sections. The electric field is given in the form of a matrix Eij. For each point in the matrix, the absorption at position (i, j) is given by
Aij =
σ 2
⋅ | Eij | 2
(5.15)
where σ is the electrical conductivity of the material [Nig02]. A typical value for a silicon solar cell is σ = 1Ωcm. From the matrix containing the 2D absorption information an absorption profile along the zaxis has to be calculated which is then used for solar cell simulation e.g. in PC1D [Bas90]. A problem arising here is that, because of the front-side texture it is not straightforward to define a z-axis. I defined the vertical position (position on the z-axis) of a point in the solar cell as the shortest distance from this point to the surface of the solar cell. The total absorption in an interval [z0, z0+dz] is given as the integral over all points with a shortest distance to the surface between [z0, z0+dz].
A[ z 0 , z 0 + d z ] =
z0 + d z
∫ A ( z) ij
(5.16)
z0
Regions of equal distance to the front surface are shown in Figure 5.12. The absorption profile is directly related to the generation profile of the solar cell over
g[λ1 , λ1 + dλ ] ~ n[λ1 , λ1 + dλ ] ⋅ A[λ1 , λ1 + dλ ]
(5.17)
where n[λ1, λ1+dλ] is the number of photons in the illuminating spectrum in the interval [λ1, λ1+dλ]. As this calculation is performed under the assumption of coherent light, oscillations occur in the resulting absorption profile. These coherence effects can be suppressed by two strategies. Either the calculation is performed for several wavelengths and the average is calculated or, in the calculation, the interval dz is chosen such that over a sufficiently large number of oscillations is integrated. The best results are obtained, if both strategies are combined. However, as a single calculation of the profile is extremely time-consuming, typically only the integration over the interval dz has been performed.
94
5.3 Different types of gratings
Figure 5.12: Sketch of the absorption distribution inside a solar cell calculated with the RCWA method. To calculate the 1D absorption profile in the z direction, the absorption within a layer bordered by two surfaces that are defined by a distinct distance to the front surface is integrated. Each integral defines one point of the absorption profile. The figure shows one cross section for a constant y-value. The complete 3D problem is solved by averaging over several cross sections.
5.3 Different types of gratings In this section I will introduce the different kinds of gratings investigated in the scheme of this work. The simplest version and first grating considered is the binary grating. Subsequently the binary grating is refined to a more realistic, trapezoidal version the geometry of which was derived from SEM (scanning electron microscope) pictures. Following that the blazed grating is introduced. The reason to broach the issue of blazed gratings is the demonstrated efficacy of such gratings. Because of the alignment problem introduced in section 0, a 2D grating with a pyramidal shape was investigated as well. As a reminder; the aim of this investigation was to examine the effect of the different gratings on crystalline silicon solar cells. The choice of this particular type of solar cell reflects on the implementation in many aspects. The materials and material parameters implemented are the ones for crystalline silicon solar cells. One process to integrate the grating into the solar cell is to structure the bulk material directly. The structuring process sets limits to the grating shape and possibly also the grating orientation, depending on what kind of etching process is used. Great attention must be paid to avoid surface damages and to effectively passivate the solar cell. These demands limit the choice of materials adoptable to this system. The issues mentioned here have been taken into account for the design of the following gratings.
5.3.1 The binary grating The first example of a grating which has been investigated is the binary grating. The binary grating emerged from the idea to suppress the 0th order diffraction with a λ/4 effect. To introduce a binary grating into a crystalline silicon solar cell, first the grating is processed directly into the silicon and afterwards the structured silicon is passivated by a thermal oxidation step which produces SiO2. The thermalization step also drives the grating into the silicon. With such a process, a binary shape is probably not feasible. The binary grating therefore serves here as a reference to compare the path length enhancement induced by
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Diffractive structures
other gratings to this ideal one. Initially I want to introduce some terms and definitions that play a role in the discussion of gratings: •
Period Λ: smallest length of a repetitive element constituting the grating. The period is decisive for the number and direction of diffractive orders
•
Grating depth d: height difference between bar and ditch. The depth is decisive for the efficiency distribution into the different orders
•
Bar width b: Gauge of the bar
•
Ditch width a: Gauge of the ditch
•
Aspect ratio s: Ratio of depth and period d/p
•
Filling factor f: Ratio of ditch width and period a/p
Some of these terms are included in Figure 5.13, which illustrates the binary grating. A last point that should be mentioned is that typical crystalline silicon solar cells possess a back side reflector made of aluminium. This reflector was also implemented as a perfect mirror (extinction coefficient k = 100). The reflection of the light causes the light to impinge on the grating twice, provided the SiO2 layer is sufficiently thick. Once, on the way from the front surface and once on the way back from different directions. This double impact of the grating is another point that makes an analytical treatment extremely difficult. In the simulations, the period and the depth of the grating were varied. The filling factor was kept constant at f = 50%.
Figure 5.13: Schematic sketch of the implementation of the binary grating into the RCWA method. The grating has a binary profile. It is located at the surface Si (n=3.5) and SiO2 (n= 1.5). Behind the grating, a perfect mirror is located.
5.3.2 The trapezoidal shape First experiments were performed to introduce a binary grating with the parameters from the scalar approach into a silicon solar cell. Structuring and thermal oxidation were performed both at IAO in Jena and at Fraunhofer ISE in Freiburg. A cross section of this structure was used to make SEM pictures to define the actual shape achieved with the thermal oxidation process. The result of the SEM pictures has been used to implement a more realistic shape of the grating. This more realistic shape was approximated by a trapezoidal form. The implementation of this shape is shown in Figure 5.14.
96
5.3 Different types of gratings
Figure 5.14: Schematic sketch of the implementation of a grating with a trapezoidal shape into the RCWA method. It is located at the surface Si (n=3.5) and SiO2 (n= 1.5). Behind the grating, a perfect mirror has been affixed. This grating is obtained from the binary grating by reducing the bar width without changing anything else. The bar width is reduced symmetrically so the angle α is equal on the right and the left edge. The most important difference between the binary grating and the trapezoidal grating is that the edges of the trapezoidal grating are not perpendicular, but rather have an angle α against the normal. This angle is equal at both sides of the bar, which results in a symmetrical shape of the grating. The angle α is given by
tan α =
Λ−a−b 2⋅d
(5.18)
One possibility to proceed for an optimization of the grating depth is to hold the angle α constant and vary ditch width and bar width implicitly with the grating depth. Instead of doing so, I kept ditch width and bar width constant and implicitly varied α. The disadvantage of this procedure is that constancy of α is probably a better approximation of the etching process. However, the implementation of the variation performed was much easier and much faster. The rectangular shape was implemented to get an idea of the effects a deviation of the optimum, binary shape induces. For a future implementation of the grating with a shape obtained from an established process, contour accuracy is, of course, essential. The parameters derived from the SEM measurement were: period p = 307nm, bar width b = 88nm, ditch width a = 177nm, depth d = 62 nm. For this grating, the parameters varied were period Λ and depth (for constant a and b).
5.3.3 The blazed grating Results reported by Heine & Morf [Hei95] indicate that, at least for a solar cell with a flat front surface, the blazed grating showed better results than the binary shaped grating. For this reason I have also implemented a blazed grating for which the effects on a crystalline silicon solar cell with and without a front surface texture were examined. The advantage of the blazed grating is that a high efficiency may be diffracted in one order of diffraction. This is expected especially if the blaze condition is satisfied A straightforward procedure for the optimization of the blazed grating would be to vary the period and the blaze angle. For a better comparability between the different gratings I chose not to vary the blaze angle directly but the grating depth. The optimization was therefore similar to the one of the binary and the trapezoidal grating. With the grating depth the blaze angle is varied implicitly. To define the shape of the grating one more parameter needs to be
97
Diffractive structures
set. This parameter is the filling factor (here: complanate width / period). In all simulations this factor was f = 0.75. The optimization approach departs from the idea of a blazed grating somehow. It is possible that an optimization on the basis of the blaze angle is more target – oriented. Against this argument can be said that the optimization given here is incomplete for all gratings in the sense, that not all occurring parameters were varied. A set of parameters had to be chosen somewhere and I chose comparability over target – oriented. I motivate this decision on the fact that the diffraction characteristics given in Figure 5.6and Figure 5.10 for both gratings are similar. A sketch of the blazed grating and the important parameters is shown in Figure 5.15.
Figure 5.15: Schematic sketch of the implementation of the blazed grating into the RCWA method. The grating is located at the surface Si (n=3.5) and SiO2 (n= 1.5). Behind the grating, a perfect mirror is located.
5.3.4 The pyramidal grating (2D) The idea to also implement a 2D grating emerged from the alignment problem. The basic idea here was to repeat the process of the front surface texture on a much smaller scale for the back side. Taking into account the crystalline structure of silicon, this results in pyramids of a specific form with a fixed ratio Λ/d (it is given by Λ / d = 2 ). A sketch of the resulting structure is given in Figure 5.16. As the depth of the pyramids is defined over the period, both values were varied simultaneously.
Figure 5.16: Implementation of the pyramidal structure into the RCWA method. Like at the front surface, the pyramids are defined by the crystalline structure of silicon. As the structures on front and rear side are aligned automatically, no problems emerge from relative alignment. Also the pyramidal grating was equipped with a rear side reflector.
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5.4 Simulation results
5.4 Simulation results For the simulation of the gratings, I proceeded as follows: in a first step an optimization of the grating parameters using the semi-analytical model was performed. In a second step, these gratings were implemented into the RCWA method and the absorption profile was calculated rigorously. Using the semi-analytical model, first the grating period was varied and the produced current was examined. This optimization of the period was performed both for a flat surface and for a textured surface. For the textured surface, an aligned as well as a δ = 45° tilted orientation were implemented. To this approach it has to be said that the global maximum depends on more than the two varied parameters. Even on the hyper surface spanned by period and depth, the global maximum may be missed by the procedure. The optimization was performed in awareness of these constraints. The idea was to first vary the period Λ to determine which part of the spectrum should be affected most. Subsequently the grating depth was varied to maximize the effect. As the grating was intended to enhance the solar cell absorption in the spectral range of low absorption close to λ = 1100nm, the investigation was confined to a spectral range between λ = 900nm and λ = 1127 nm. This spectral range was chosen also with regard to the assumed thickness of the silicon wafer of t = 40µm. A simple slab of this material absorbs almost all light with λ = 900nm, so no, or only little, light with a smaller wavelength arrives at the grating. Having assessed the parameters for an optimum grating performance, each optimized grating was implemented into the RCWA method to simulate the generation profile of excitons within the solar cell. For these simulations, again a wafer with a thickness of t = 40µm was assumed. The different simulation methods yield different measures for the evaluation of the gratings. In the semi analytical model, the measure is the gain in current density jgain given in mA/cm². The gain in current density is the difference between the current density calculated for a solar cell with and without a grating. A negative gain in current density means that the grating induces a loss. The calculation of the gain in current density systematically underestimated the effect of the grating; it is only used to estimate relative effects which can be used to find the optimum grating parameters or to compare different gratings. The measure in the rigorous model is the absorption profile at a certain wavelength given as fraction of the incident light absorbed per depth. The rigorous model gives a better estimation of the effect of the grating. However, as the calculation is extremely time consuming, only few wavelengths were considered.
5.4.1 Optimization of the grating period For the three 1D gratings, this optimization of the grating period was performed for a fixed grating depth. For the 2D pyramidal grating, the period defines the depth as well, so both parameters were varied simultaneously. The optimization has first been applied to the case of a solar cell with a flat front surface, and subsequently also to a solar cell with a textured surface, considering different relative orientations of grating and front surface texture. The solar cell thickness in all cases was t = 40µm. Flat surface solar cell The results of the optimization of the grating period for the different gratings integrated into a solar cell with a flat front surface are shown in Figure 5.17. Light impinges on the cell perpendicularly, under 0°. Without further light trapping, the path length of the light inside the solar cell is minimal under these conditions. A grating will always have a positive effect because every deviation from the 0° direction results in a path length enhancement.
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Diffractive structures
To optimize the grating period of the 1D gratings, first a fix grating depth needs to be assumed. For all three gratings the same value of d = 215nm has been used. This value was obtained from a first series of simulations. For the 2D grating, the depth is defined by the period. Comparing the binary and the trapezoidal grating, it seems that a deviation from the binary shape (represented by the trapezoidal grating) results in a reduction of the grating performance. The performance of the binary shaped grating seems to be an upper limit for deviating grating shapes. The gain in current density for the binary grating shows a plateau between Λ = 310nm and Λ = 380nm. The maximum value for the gain for the binary grating is obtained for a period of Λ = 310nm. This value is close to the one obtained from the theoretical considerations of Λ = 314nm. For the trapezoidal grating the performance is reduced more for smaller periods than it is for larger ones. The maximum performance is obtained for a period of Λ = 370nm. For the blazed grating, the optimum period is Λ = 300nm. This period corresponds to a blaze angle of θb = 43.7° and a blaze wavelength of λb = 1049nm. The obtained period is close to the one obtained from theoretical considerations. The 2D pyramidal grating shows only a small enhancement of the current density. A maximum emerges for a grating period of Λ = 370nm.
2
gain in current density [mA/cm ]
gain (900nm-1127nm) binary trapezoidal blazed pyramidal
0.4 0.3 0.2 0.1 0.0 250
baseline = 3.79mA/cm 300
350 period [nm]
400
2
450
Figure 5.17: Optimization of the grating period for the different gratings integrated into a crystalline silicon solar cell with a flat front surface. The three 1D gratings all had a depth of d = 215nm. The grating period has been varied between Λ=250nm and Λ=400nm. Textured solar cell, aligned orientation Most important for the simulation of the performance of a 1D grating integrated into a solar cell with a pyramidally textured front surface is the existence of two components of the light incident on the grating with two different azimuthal components (see Figure 5.2). These two components emerge from the two possibilities of relative orientation of pyramidal facet and grating. As both different facets appear with equal commonness, both components are of equal importance. In a first step, an aligned orientation was assumed, i.e. the grating has in one direction the same orientation as the pyramidal facets. The two occurring components for this orientation is either δ = 0° or δ = 90°. For the 2D pyramidal grating the situation is different; here, the geometry of the grating is equal to that of the front surface texture. Consequently, only one relative orientation occurs. The results of the optimization for the different gratings are shown in Figure 5.18. The most important result of this simulation is that it possible to induce a positive effect with a grating on the path length in a pyramidally textured crystalline silicon solar cell. However, unlike for the flat front surface, the grating may now have also a negative effect if the parameters are chosen unwisely. This is especially visible for the δ=0° orientation. For this orientation a small maximum occurs for all gratings for a period of 175nm < Λ < 180nm. Apart from
100
5.4 Simulation results the 2D pyramidal grating, the effect of the gratings becomes negative with increasing periods. The δ=0° orientation corresponds to the case for which theoretical considerations were given for non normal incidence. The obtained period for the binary grating disagrees with the theoretical predictions, even worse; a grating with the predicted period produces a loss. For the blazed grating the period is in accordance with the theoretical prediction. The blaze angle is θb = 57° and therefore differs from the theoretical prediction. For the δ = 90° orientations the period for which the maximum performance is obtained changes for the 1D gratings to a larger value of ca. Λ = 400nm. Also, the gain in current density is much stronger emphasized for the δ = 90° orientations. Comparing the binary to the trapezoidal grating, the binary again seems to be limiting the performance of the trapezoidally shaped grating. The blazed grating shows the strongest effects, positive and negative. The performance of the 2D grating is comparable to the performance of the 1D gratings for the δ = 0° orientation. It is worse for the δ = 90° orientation. b - 90°)
a - 0°)
gain (900nm-1127nm)
0.10 0.05 0.00 -0.05 -0.10 -0.15 -0.20 100
binary trapezoidal blazed pyramidal 150
200
250
2
gain in current density [mA/cm ]
2
baseline = 4.57mA/cm
2
gain in current density [mA/cm ]
gain (900nm-1127nm) 0.15
0.5
0.3 0.2 0.1 0.0
2
-0.1 100
300
350
400
450
500
binary trapezoidal blazed pyramidal
0.4
baseline = 4.57mA/cm 150
200
250
300
350
400
450
500
period [nm]
period [nm]
Figure 5.18: Optimization of the grating period for the different gratings integrated into a crystalline silicon solar cell with a textured front surface. The three 1D gratings all had a depth of d = 215nm. For the aligned orientation, two components for the impinging light occur with the azimuthal components δ = 0° and δ = 90° for the 1D gratings. For each of these components the optimization has been performed separately. As the 2D pyramidal grating has the same geometry as the front surface texture, both components are equivalent. Figure 5.19 shows the average of the gain in current density for both orientations. This averaged result has to be considered for the definition of the optimum period. As was shown in Figure 5.18, for the 1D gratings, the maximum performance emerges for different periods for the different orientations. Furthermore, the δ = 0° component causes a negative contribution for the period, where the δ = 90° orientation shows a maximum. This reduces the effect of the grating on the gain in current density. The most important result of this examination is that the major contribution to the current is made by the fraction of light with the δ = 90° orientation. The period is chosen accordingly and is given by Λ = 410nm for the binary and for the trapezoidal grating and Λ = 390nm for the blazed grating. As light with this orientation has not been considered in the theoretical approach, the obtained periods differ strongly from the ones predicted theoretically. For the 2D pyramidal grating, the period for which the maximum gain in current density is obtained is Λ = 180nm.
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Diffractive structures
binary trapezoidal blazed pyramidal
2
gain in current density [mA/cm ]
gain (900nm-1127nm) 0.20 0.15 0.10 0.05 0.00 -0.05
2
baseline = 4.57mA/cm -0.10 100
150
200
250
300
350
400
450
500
period [nm]
Figure 5.19: Average of the calculated gain in current density for both optimizations. As for the 1D gratings, the greatest contribution is attributed to the δ = 90° component, all of which show a maximum for periods at ca. Λ = 400nm. For the 2D pyramidal grating the maximum gain in current density is located at a period of Λ = 180nm Textured solar cell, tilted orientation As was shown in the last part, it is possible to generate a positive effect on the path length of light inside a solar cell by the integration of a grating. However, the presence of the two different orientations and the corresponding averaging reduces the potential benefit. For that reason I have also simulated the performance of the grating for a δ = 45° tilted orientation. For this orientation only a single azimuthal component of |δ| = 45° for all gratings exists. The result of this simulation is given in Figure 5.20. For this orientation, the period for which the maximum gain in current density is expected is located at Λ = 220nm (for the 1D gratings) resp. Λ = 230nm (2D pyramidal grating) and is therefore similar for all gratings. For these configurations, no theoretical perditions were made. An interesting aspect of the results is that the best performance is provided by the 2D pyramidal grating. Another interesting point is that the characteristic of the binary and the trapezoidal grating are very similar. Again it seems that a variation of the binary shape reduces the grating performance, however the reduction is apparently much smaller than for the other orientations.
2
gain in current density [mA/cm ]
gain (900nm-1127nm) 0.3
binary trapezoidal blazed pyramidal
0.2 0.1 0.0 -0.1 -0.2
baseline = 4.57mA/cm 200
250
300
350
2
400
450
500
period [nm]
Figure 5.20: Optimization of the grating period for the different gratings integrated into a crystalline silicon solar cell with a textured front surface. The three 1D gratings all had a depth of d = 215nm. For the tilted orientation, only one component for the impinging light remains with the azimuthal components δ = 45° for all gratings. For all gratings, the maximum is located at.ca. Λ = 225nm.
102
5.4 Simulation results 5.4.2 Optimization of the grating depth In the second step of the optimization, the grating depth is optimized for the three 1D gratings (binary, trapezoidal and blazed grating). With the grating depth, the diffraction efficiencies in the different orders are varied. The optimum depth shall be obtained for a maximum gain in current density. An evaluation of the grating performance is given after the optimization of both, period and grating depth. The solar cell thickness in all cases was t = 40µm. Flat surface solar cell The result of the optimization of the grating depth for the three 1D gratings, each with the optimum period, is shown in Figure 5.21. The obtained gain in current density is a measure that can be used to compare the performance of the different gratings. The optimum depth of the binary grating is d =130nm. The gain in current density with the optimized parameters is jgain = 0.4 mA/cm². The parameters obtained from the optimization are close to the ones coming from the theoretical considerations. However, the differences are noticeable and the performance of the optimized grating is better than the one of a grating with the initial values. For the trapezoidal grating two maxima occur. These are a global maximum of jgain = 0.3mA/cm² for a grating depth of d = 250nm and a local maximum of jgain = 0.26mA/cm² for a grating depth of d = 130nm. For both maxima, the gain in current density is lower than for the binary grating. For the given conditions, a deviation of the binary shape seems to reduce the grating performance. For the trapezoidal grating, the grating parameters cannot be predicted from theoretical considerations. This shows that an accurate implementation of the actual, experimentally obtained, grating shape is necessary for an optimization. For the blazed grating the optimum grating depth is d = 285nm. Under the given conditions, the blazed grating shows the best performance with a gain in current density of jgain = 0.52mA/cm². This result agrees with one given by Heine & Morf [Hei95]. The obtained optimum parameters correspond to a blaze angle of θb = 51.71° and a blaze wavelength of λb = 1021nm. These conditions differ considerably from the ones obtained by theoretical considerations and the grating performance of the optimized grating is better than the one for a grating with the initial parameters (a depth of d = 225nm corresponds to a blaze angle of θb = 45° and a blaze wavelength of λb = 1050nm).
2
gain in current density [mA/cm ]
gain (900nm-1127nm) 0.5 0.4 0.3 0.2 0.1 0.0
baseline = 3.79mA/cm 0
50
2
binary trapezoidal blazed
100 150 200 250 300 350 400
depth [nm]
Figure 5.21: Optimization of the grating depth for the three 1D gratings integrated into a crystalline silicon solar cell. The periods used were the periods for which the maximum gain in current density was calculated in the optimization in the last subsection. For the binary grating this was Λ = 310nm, for the trapezoidal grating Λ = 370nm, for the blazed grating Λ = 300nm.
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Diffractive structures
Textured solar cell, aligned orientation The optimization of the grating period indicated already that the δ=90° component provides the major contribution to the gain in current density of a grating introduced into a textured solar cell and in aligned orientation. To ascertain this result, the grating depth was optimized for both components, each with the optimum period. The result of this calculation is shown in Figure 5.22. a - 0°)
b - 90°) gain (900nm-1127nm)
gain (900nm-1127nm)
0.25 0.20 0.15 0.10 0.05 0.00 -0.05
2
baseline = 4.57mA/cm 0
50
100 150 200 250 300 350 400 450
depth [nm]
0.5 2
gain in current density [mA/cm ]
binary trapezoidal blazed
2
gain in current density [mA/cm ]
0.30
0.4 0.3 0.2 0.1 0.0
baseline = 4.57mA/cm 0
50
2
binary trapezoidal blazed
100 150 200 250 300 350 400
depth [nm]
Figure 5.22: Optimization of the grating depth for the three 1D gratings integrated into a crystalline silicon solar cell with a textured front surface. For the aligned orientation, two components for the impinging light occurs with the azimuthal components δ=0° and δ=90, For each of these components, the optimization has been performed separately. The periods used were the periods for which the maximum gain in current density was calculated in the optimization in the last subsection. For the δ=0° component this was Λ=180nm for all three gratings, for the δ=90° component this was Λ = 410nm for the binary and the trapezoidal grating and Λ = 390nm for the blazed grating. The simulation results given show that the assumption of the importance of the components was correct. The δ=90° component provides the major contribution. Therefore this contribution is used to define the optimum grating depth for the three 1D gratings. A comparison with the theoretically derived values makes no sense, because the different orientations were not considered. The maximum gain in current density for the binary and the trapezoidal grating is obtained for a grating depth of d = 130nm. The performance of the trapezoidal grating in this case is better than the performance of the binary grating. This shows that a deviation of the binary shape does not influence the grating performance in a textured solar cell. This is an encouraging result considering the grating fabrication. The optimum grating depth for the blazed grating is d = 250nm. The gain in current density provided by the grating is defined by the contributions of both orientations, therefore the contribution of the δ=0° component for the optimized period has to be considered and the average must be calculated. After doing so, a maximum gain in current density of jgain = 0.20mA/cm² is obtained for the binary and the blazed grating and jgain = 0.21mA/cm² for the trapezoidal grating. This shows that it is possible to induce positive effect on the path length of light inside a pyramidally textured solar cell by a grating. The positive effect is smaller, however, than the one in a flat solar cell. Textured solar cell, tilted orientation The results of the optimization of the grating depth for the tilted orientation are shown in Figure 5.23. The depth has only been varied up to a value of d=450nm. This value corresponds, for the given optimum period, to an aspect ratio of ca. s = 2. As gratings with
104
5.4 Simulation results the corresponding aspect ratios are probably not feasible for production, the simulation was halted there. Looking at the optimization of the grating depth, maxima appear for very large grating depths or none at all in the range for which the optimization was performed. However, for the binary and the trapezoidal gratings, a local maximum of jgain = 0.17mA/cm², and jgain = 0.16mA/cm² respectively appear for a grating depth of d = 150nm. Confining the aspect ratios to values below 1, this seems a reasonable result for an optimization of the grating depth. For the blazed grating, however, no such maximum occurs and though comparable values to those of the other gratings are obtained, they are obtained only for much larger grating depths. Compared to the results of the aligned orientation, it has to be said that the tilted orientation shows a worse performance than the aligned one, even though the alignment problem is avoided.
2
gain in current density [mA/cm ]
gain (900nm-1127nm) 0.30
binary trapezoidal blazed
0.25 0.20 0.15 0.10 0.05 0.00
2
baseline = 4.57mA/cm
-0.05 0
50 100 150 200 250 300 350 400 450
depth [nm]
Figure 5.23: Optimization of the grating depth for the three 1D gratings integrated into a crystalline silicon solar cell with a textured front surface. For the δ = 45° tilted orientation only one relative orientation is left. The periods used were the periods for which the maximum gain in current density was calculated in the optimization in the last subsection. For all 1D gratings this period was Λ = 220nm. The maximum gain in current density appears for gratings with very high aspect rations or not at all in the range of the simulation. However very high aspect ratios were not investigated as they are probably not feasible (at least with the applied methods).
5.4.3 Calculated absorption profile To calculate the absorption profile, the method presented in subsection 5.2.6 was used. This method uses the RCWA to simulate the distribution of the EM field inside a solar cell, and to calculate the absorption profile. For this, the electrical conductivity is needed. In all calculations a value of 1 Ωcm was used. Unlike in the semi-analytical method used in the previous part, only a 1D implementation of this problem was possible, so no dependence on the azimuth angle could have been surveyed. That means that for the textured solar cell only the δ = 0° orientation was implemented. The results given for the textured surface therefore only serve for principle. The complete 2D implementation is currently being performed and will be the subject of future work. When calculating the absorption profile, it has to be considered that the calculation is performed assuming perfect coherence. The coherence manifests itself in several aspects: On the one hand the calculated absorption oscillates with the considered wavelength of the incident light. On the other hand the profile oscillates strongly throughout the solar cell. To eliminate these oscillations caused by coherence, the profile was integrated over a small
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Diffractive structures
wavelength range and a small spatial range. However, as the calculations were extremely time consuming, the number of different wavelengths that were taken into account were limited so that some coherence effects remain in the shape of the absorption profile. Flat surface solar cell From a first series of simulations I calculated the wavelength-dependent absorption profile in an untextured silicon solar cell with a thickness of t = 40µm for four different cases (no grating, binary grating, trapezoidal grating and blazed grating 12) and five different wavelengths (λ = 900nm to λ = 1100nm in steps of 50nm). The resulting absorption profiles are shown in Figure 5.24. The results are scaled such that an integration over the thickness in µm yields the total absorptance in percent. In a first simulation the parameters of the gratings were chosen according to the results of the semi – analytical simulations. This resulted in very little effect of the grating, calculated with the rigorous model. To achieve acceptable results with the rigorous model, the grating depth had to be adapted. This contradiction between the semi – analytical model and the rigorous model concerning the grating depth has not been solved yet. The parameters chosen were: a period of Λ = 310nm for all gratings and a grating depth of d = 130nm for the binary grating and d = 70nm for the real and the blazed grating. For a wavelength of λ = 900nm the effect of the grating on the absorption profile is very small. Within the uncertainty of the calculation method no significant effect on the form of the absorption profile is perceptible. However, the total absorptance is increased a little (the absorptance may be calculated with much higher accuracy than the profile as the calculation of this value is much faster and many calculations may be performed to eliminate coherence effects). The effect of the grating on the absorption profile is best visible for a wavelength of λ = 950nm. More light is absorbed at the rear side of the solar cell because here the path is increased. As at the rear side more light is absorbed, less light reaches the front surface after the reflection. Therefore the grating causes a balancing of the depth dependence of the absorption. Additionally the increased path causes an increased total absorptance. For λ = 1000nm and λ = 1050nm the absorption at the rear side is increased so much that now in the solar cell with grating more light is absorbed at the rear side than at the front. This effect diminishes when increasing the wavelength furthermore. For a wavelength of λ = 1100nm the absorption profile is flat for all cases. Yet the increased path length still results in an increased absorption. Comparing the three gratings, all of them show a comparable performance with a tendency towards a weaker effect for the blazed grating.
12 The 2D pyramidal grating could not have been considered, as this simulation only takes into account 1D structures.
106
5.4 Simulation results
λ=900nm
λ=950nm
without binary trapez. blazed
3.5
3.5
90.6% 92.0% 91.9% 91.8%
absorption / depth [% of incident light / μm]
absorption / depth [% of incident light / μm]
4.0
3.0 2.5 2.0 1.5
0
10
20
30
2.5
1.5 1.0
depth [µm]
0
absorption / depth [% of incident light / μm]
absorption / depth [% of incident light / μm]
30
40
30
40
0.5
1.0
0.5
without: binary: trapez.: blazed:
38.5% 42.2% 42.4% 41.2%
10
20
0
30
40
depth [µm]
0.10 0.08 0.06 without: binary: trapez.: blazed:
0.04 0.02
0
10
2.70% 3.36% 3.29% 3.18%
20
0.4 0.3 0.2
without: binary: trapez.: blazed:
0.1 0.0
0
10
11.6% 13.8% 13.6% 13.2%
20
depth [µm]
λ=1100nm
absorption / depth [% of incident light / μm]
20
λ=1050nm
1.5
0.00
10
depth [µm]
λ=1000nm
0.0
70.8% 74.0% 73.8% 73.3%
2.0
0.5
40
without: binary: trapez.: blazed:
3.0
30
40
depth [µm]
Figure 5.24: Absorption profile in a flat solar cell with different grating implemented at the rear side and for different wavelengths. Also given is the integrated absorption over the entire thickness of the solar cell. With increasing wavelengths the absorption profile becomes flatter because of the decreasing absorptivity of silicon. The grating causes an additional flattening of the absorption profile as it increases the absorption, especially at the rear side of the solar cell. This effect manifests itself formidably at a wavelength of λ = 950nm. For larger wavelengths the grating increases the absorption at the rear side so much that it becomes larger than the absorption at the front.
Textured solar cell The same calculation as for the flat front surface was performed for a wafer with a pyramidal texture and a thickness of t = 40µm. Again the considered wavelengths were λ = 900nm to λ = 1100nm in steps of 50nm. The result of this calculation is given in Figure 5.25.
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Diffractive structures
λ=900nm
λ=950nm 4.0
without binary real blazed
5
94.2% 94.3% 94.3 % 94.3 %
absorption / depth [% of incident light / μm]
absorption / depth [% of incident light / μm]
6
4 3 2 1
0
10
20
30
1.5 0
10
20
30
λ=1000nm
λ=1050nm
40
1.50 without binary real blazed
2.25 2.00
absorption / depth [% of incident light / μm]
absorption / depth [% of incident light / μm]
2.0
depth [µm]
67.9% 70.8% 70.8 % 69 %
1.75 1.50 1.25 0
10
20
30
40
λ=1100nm 0.6
0.5
without binary trapez. blazed
10.0% 13.7 % 13.9 % 12.2 %
20
30
0.4
0.3
0
10
without binary trapez. blazed
1.25
33.0% 38.1 % 38.1 % 35.6 %
1.00
0.75
0.50
0
10
20
30
40
depth [µm]
depth [µm]
absorption / depth [% of incident light / μm]
87.2% 88.3% 88.4 % 87.9 %
2.5
depth [µm]
2.50
0.2
3.0
1.0
40
without binary trapez. blazed
3.5
40
depth [µm]
Figure 5.25: Absorption profile in a textured solar cell with different gratings implemented at the rear side and for different wavelengths. Also given is the integrated absorption over the entire depth. The absorption is the highest at the top of the solar cell because of the exponential decay and the increased surface because of the texture. Especially for large wavelengths the exponential decay loses importance and the main effect comes from the increased surface area. Where the grating shows a positive effect, it does so over the complete depth of the solar cell and not only at the rear side.
With this simulation only the δ = 0° component could have been displayed and therefore the component with the minor contribution. Also in this simulation the grating depth chosen is not the one obtained for the semi analytical model. The parameters used for all gratings were: a period of Λ = 180nm and the depth is d = 187nm. The form of the absorption profile for the textured wafer is influenced by two effects. On the one hand, it is influenced by the exponential decay due to Lambert-Beers law, and on the other hand, by the increased surface due to the texture. The texture influences the absorption profile because a volume defined by a certain distance interval to the front surface is much larger close to the surface than far from it (see Figure 5.12). For λ = 900nm most of the light is absorbed before reaching the grating and no significant difference in the profiles for all cases is perceptible. With increasing wavelengths (and
108
5.4 Simulation results consequently decreasing absorption), the absorption for the system with gratings becomes noticeably larger than for the system without gratings. For all systems, the largest effect is produced by the binary and the trapezoidally shaped grating. Nearly no difference in the integrated absorption occurs for theses systems. These results confirm the ones from the semi-analytical model. For a textured front surface, a deviation from the binary shape has little effect. Also, the blazed grating shows a positive effect, but the performance of the binary shaped grating and the trapezoidal grating are better. Because of the already flattened path for the light inside the textured solar cell, the enhancement of the absorption at the back side is comparably weaker and the effect of the flattening of the absorption profile is not visible for this case.
5.4.4 Summary The results of the different optimizations are summarized in Table 5.2 - Table 5.4 for the flat front surface, the textures front surface with aligned orientations and the textures front surface with tilted orientation towards the ´grating. optimum period
optimum depth
gain in current density
binary grating
310nm
130nm
0.40mA/cm²
trapezoidal grating
370nm
250nm
0.30mA/cm²
blazed grating
300nm
285nm
0.52mA/cm²
pyramidal grating
370nm
-
0.13mA/cm²
flat front surface
Table 5.2: Summary of the results for different gratings applied to a solar cell with a flat front surface. Note that the given gain in current density underestimates the actual effect of the grating and serves mainly for the purpose of comparison. For a solar cell with a flat front surface, the simulation indicates that the best result is obtained with a blazed grating. This result agrees with the one given by Heine & Morf [Hei95]. A good result is also obtained with the binary grating. A deviation from the binary shape (represented by a trapezoidal shape) reduces the grating performance. The performance of the 2D pyramidal grating is not good and provides no real alternative to the 1D gratings. The parameters for the binary grating differ only little from the ones predicted theoretically. Nevertheless, the performance of the optimized grating was better than the one of a grating with the predicted parameters. For all other gratings, the parameters differ considerably from the ones predicted by theory. The results given in Table 5.3 indicate that it is possible to induce a gain in current density with a grating also for a pyramidally textured crystalline silicon solar cell. The results in this table correspond to a grating aligned to the front surface texture. Compared to the flat front surface, the gain is reduced. The gain in current density is comparable for all 1D gratings (binary, trapezoidally shaped and blazed) and lower for the 2D grating. The major part to the current is contributed by light with an azimuthal component of δ = 90°. This contribution also defined the grating period. As this configuration was not considered in the theoretical predictions, the obtained grating period is larger for all 1D gratings than predicted theoretically. As a consequence, also the grating depth differs from the theoretical predictions.
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Diffractive structures
optimum period
optimum depth
gain in current density
binary grating
410nm
130nm
0.20mA/cm²
trapezoidal grating
410nm
130nm
0.21mA/cm²
blazed grating
390nm
250nm
0.20mA/cm²
pyramidal grating
180nm
-
0.12mA/cm²
textured front surface / aligned orientation
Table 5.3: Summary of the results for different gratings applied to a solar cell with a textured front surface. Grating and texture are oriented aligned. Note that the given gain in current density underestimates the actual effect of the grating and serves mainly for the purpose of comparison. A third investigation was performed for a grating that was tilted by δ = 45° towards the front side texture. The gain in current density for the 1D gratings is comparable, but tends to be lower than for the aligned orientation (especially for the trapezoidal grating). More important is that for the tilted orientation, the required grating period is much smaller than for the aligned orientation and the required grating depths tend to be greater. As this results in increased difficulties in fabrication, the aligned orientation is probably preferable. For the titled orientation, the highest gain in current density is calculated for the 2D pyramidal grating. However, also this grating is probably very difficult to fabricate.
optimum period
optimum depth
gain in current density
binary grating
220nm
150nm
0.17mA/cm²
trapezoidal grating
220nm
150nm
0.16mA/cm²
blazed grating
220nm
430nm
0.20mA/cm²
pyramidal grating
230nm
-
0.26mA/cm²
textured front surface / 45° tilted orientation
Table 5.4: Summary of the results for different gratings applied to a solar cell with a textured front surface. Grating and texture are oriented with a tilt of δ = 45°. Note that the given gain in current density underestimates the actual effect of the grating and serves mainly for the purpose of comparison. To calculate the absorption profile, a 1D implementation of a solar cell with a grating was performed in the RCWA method (1D here is in the sense of a 1D implementation into the RCWA code, that means the structure has a periodicity only along one axis.) for a flat and for a textured solar cell. This implementation is comparable to the cases described in Figure 5.17 and Figure 5.18a. From this implementation, the absorption and the absorption profile were calculated directly. The obtained effect was smaller than expected for the semi analytical model but still a positive effect on the absorption for the textured and the untextured case was found. This is however not surprising, as in a 1D model the relative orientation of texture and grating cannot be included and therefore only the minor contribution of the δ = 0° component is considered. A future task here will be to establish a complete 2D model of the problem. The result of the evaluation of the relative absorption enhancement, calculated with the 1D RCWA method, is shown in Figure 5.26. Once more
110
5.5 Experimental results it shall be emphasized that the complete process of absorption with the interplay of front and back surface needs to be considered to obtain the absorption profile. This is the big advantage of the applied method. b)
a) binary grating real grating blazed grating
25 20 15 10 5 0 900
950
1000
binary grating trapezoidal grating blazed grating
40
relative absorption enhancement [%]
relative absorption enhancemet [%]
30
1050
1100
wavelength [nm]
30 20 10 0 900
950
1000
1050
1100
wavelength [nm]
Figure 5.26:Relative absorption enhancement calculated with the RCWA method for a flat wafer (left) and a textured wafer (right).
5.5 Experimental results The approach to realize a solar cell with a rear side grating has been started roughly at the same time as the simulations. However, simulation results are naturally produced much faster than experimental ones. Therefore at the time being, no solar cell with a rear side grating has been realized. The experimental situation at this moment is that first gratings were introduced into silicon and were characterized by optical and electrical methods. Therefore, in this section an overview of the results obtained at present will be given. The experimental results shown here were performed by Christian Helgert (University Jena, ebeam lithography structuring of the wafers), Pauline Voisin and Hubert Hauser (Fraunhofer ISE, preparation of the wafers, edging and characterisation). My part in this work was the simulation and determination of the grating parameters.
5.5.1 Grating shape A first important result concerns the actual shape of the grating etched into silicon. The procedure here includes a two step process. Initially, an etchings mask is produced on the silicon wafer. As the desired periods of ca. Λ = 300nm are very small, they are hard to fabricated by conventional photolithography. A first mask has therefore been produced with electron beam lithography [Cor00] at Fraunhofer IOF in Jena. With this mask, a first grating was etched into silicon by reactive ion etching (RIE) (Figure 5.27a). The grating obtained by this method has very steep edges and its shape is close to binary. However, this grating is not suitable for a solar cell process. The reason for this is that the rear surface is badly damaged by the ion bombardment within the RIE process. This damage is the source of surface recombination that drastically reduces the carrier lifetime and therefore spoils the solar cell performance. The approach to prevent these losses was therefore to heal the surface damage by growing the grating into the silicon in a thermal oxidation step. This thermal oxidation is the second step in producing the grating. In this step SiO2 layer is created and the grating is transferred from the interface air/Si to the interface Si/SiO2. An SEM picture of a grating driven into silicon by that procedure is given in Figure 5.27b.
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Diffractive structures
a) Nano-grating fabricated by electron beam lithography and etched into silicon.
b) Nano-grating after the growth of a 105nm thermal oxide.
Protective resist SiO2 Si
Figure 5.27: Scanning Electron Microscope (SEM) images of the nano-grating (a) etched into the back side of the silicon wafer and (b) after a thermal oxidation (oxide thickness: 105nm).In Figure 5.27b a two step etching process was used to separate and uncover the silicon structure from the SiO2 structure (SEM picture by Pauline Voisin). A grating similar to the one shown in Figure 5.27b. was used to obtain the parameters used for the trapezoidally shaped grating.
5.5.2 Comparison of measured and simulated results A first characterization of the grating on a textured wafer was performed making reflection R and transmission T measurements of a textured wafer [Voi09]. For the illustration of the comparison, a new measure was chosen. This measure is the relative absorption enhancement. The relative absorption enhancement is the absorption gained by the application of the grating divided by the absorption of the solar cell without grating at a certain wavelength. This measure was chosen, because a comparison of absolute values was difficult. The difficulties were cause by initial reflection on the measured wafers’ surface. The relative absorption enhancement is not affected by this initial reflection. Provided, the used absorption coefficient is correct, therefore, the relative absorption enhancement is a good measure to compare simulation and experiment. The comparison is shown in Figure 5.28.
relative absorption enhancement [%]
30 25
RCWA simulation measurement
20 15 10 5 0 950
975
1000
1025
1050
1075
1100
wavelength [nm]
Figure 5.28: Relative absorption enhancement for a grating in a textured wafer simulated with the RCWA method and calculated from reflection- and transmission measurements. The measured wafer had a thickness of t = 200µm. Simulation and measurements yield comparable results (measurements by Pauline Voisin).
112
5.6 Summary of the chapter & perspective
Comparing RCWA simulation and measurement shows that the spectral range in which the absorption enhancement is predicted is correct. Also it shows that the predicted relative absorption enhancement has the right magnitude. By trend, the RCWA simulation even underestimates the achievable absorption enhancement. In the simulation, the grating shows no more effect for wavelengths of λ = 1100nm and above. This is because no diffraction occurs here due to the chosen grating period. In the measurement an absorption enhancement is observed as well as for wavelengths above λ = 1100nm. The absorption enhancement here is probably caused by scattering effects and not by diffraction. Scattering may also be the reason why the measured absorption enhancement exceeds the predicted one below λ = 1100nm. The accordance between measured and simulated result is encouraging as it is a strong hint that the other predictions are also correct.
5.6 Summary of the chapter & perspective In this chapter I have investigated the effect of diffractive structures on the optical properties of solar cells. Especially back side gratings integrated into textured and untextured crystalline silicon solar cells were considered. The aim of this approach was to increase the path length of light inside the solar cell by discretely changing the direction of the internal radiation. An optimum is achieved, if the direction of radiation inside the solar cell is changed to propagating alongside the solar cell. Such a discrete change in the direction is obtained by diffractive structures. In section 5.1 the photovoltaic concepts considered in this chapter were introduced. The concepts are distinguished by their surface properties. The first approach features a flat front surface. Flat front surfaces are realized e.g. for thin film solar cells on flat substrates. The second approach features a textured front surface. As an example, a pyramidal texture was chosen. Pyramidal textures are realized for contemporary crystalline silicon solar cells. For textured solar cells, it has to be considered, that light will impinge on a back side grating under non normal incidence and with different orientations. The presence of different orientations gives rise to difficulties. These difficulties haven been subsumed under the term “alignment problem” Strategies have been outlined to avoid the alignment problem. In section 5.2 considerations of principles have been presented. The considerations concerned the potential of gratings in solar cell. First, an estimation of what can be achieved with light trapping was given by calculating the number of unused photons in the red part of the spectrum (900nm < λ < 1127nm). These unused photons could maximally generate a current density between j = 2.4mA/cm² and j = 6.5mA/cm² (compared to jSC ≈ 40mA/cm² for a good contemporary crystalline silicon solar cell) depending on the thickness of the solar cell and whether the front surface is textured or not. Subsequently, the maximum possible path length enhancement L with a structure that induces a discrete directional change was calculated. Considering the conservation of étendue, this consideration resulted in a factor of L = 853n with n the refractive index of the solar cell material. Following that two grating concepts were discussed in detail that comprehend the potential of a large path length enhancement. These concepts are the binary grating and the blazed grating. Finally in this section, two models were introduced to estimate the effects of the gratings. A semi-analytical method to optimize the grating parameters and a rigorous method to calculate the absorption profile for a solar cell with a grating rigorously. In section 5.3 the four different types of gratings implemented into the simulation method were discussed. The first grating was the already mentioned binary shaped grating. Using SEM pictures, the binary shape was refined to fit the experimentally conditions. This refinement resulted in a grating with a trapezoidal shape. This shape was used to estimate the effects of a deviation from the optimum binary shape. An advantage over the binary grating was predicted for the blazed grating, which was considered as a third example. The
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Diffractive structures
final example was a 2D pyramidal grating which was chosen as a concept to elude the alignment problem. In section 5.4 the simulation methods introduced in section 5.2 were applied to the gratings introduced in section 5.3. First the semi-analytical was used to optimized the grating period and the grating depth subsequently. This optimization has been performed both for a crystalline silicon solar cell with a flat and with a pyramidally textured front surface. For the textured front surface different relative orientations of grating and texture were considered. The optimization for the flat front surface confirmed the result, that the blazed grating yielded the best results. A gain in current density of jgain = 0.52mA/cm² was calculated for a blazed grating with a period of Λ = 300nm and a depth of d = 285nm. It has to be said, that the given gain in current density systematically underestimates the effect of the grating and mainly serves for the purpose of comparison. Also the binary grating showed a good performance. Deviations from the binary shape seem to reduce the grating performance. The considered 2D grating yielded the least beneficial results. For the textured front surface, all 1D gratings yielded comparable results. For the trapezoidal grating, a gain in current density of jgain = 0.21mA/cm² was calculated for a grating with a period of Λ = 410nm and a depth of d = 130nm. It has to be said, that the model probably underestimated the gain in current density. Interesting about these results is that the major contribution is provided by the radiation that impinges on the grating with an azimuthal component of δ = 90°. The outlined strategies to elude the alignment problem yielded comparably bad results and provide no options. The rigorous model confirmed the positive effect that is induced by the grating on the absorption for the different PV concepts. Furthermore it shows that the absorption profile is flattened by the application of the gratings. Several differences occurred between the results provided by the two simulation methods. The most important of these differences is that the methods disagree on the optimum grating depths. In section 5.5 finally, first experimental results of gratings introduced in silicon solar cells are presented. Unfortunately, no solar cells with rear side gratings have yet been produced and so only limited comparison between simulation and experiments was possible. However, first test structures were introduced into silicon wafers and the shape of the grating could be investigated with SEM. Also first measurements of the absorption in a wafer with a grating were performed and showed similarities to the simulated one. At this point I want to give an evaluation of the results of this chapter. One goal of the presented approach was to answer the question whether it is possible to induce a positive effect with a grating in a front side pyramidally textured crystalline silicon solar cell. This question deserves an affirmative answer. Positive effects were found for all considered gratings and every considered PV concept. This shows that gratings are a promising concept for the application even on thick solar cells. Another promising result is that the found grating parameters for this concept are convenient i.e. the optimum period is not too small and the depth is not too large. This convenience is caused by the fact that the major contribution to the gained current density is provided by radiation that has an azimuthal component of δ = 90° when impinging on the grating. This component even yields results that are comparable to the ones for a solar cell with a flat front surface. Encouraging is also that first experimental results about the effect of the grating are in agreement with simulated results. A demerit about the presented results is that the two simulation methods disagree on several points. The most important point is the prediction about the optimum grating depth. No solution to this contradiction has been found yet. Another unpleasant point is that with the presented methods provide no resilient predictions about the actual gain in current density that is generated. The semi-analytical method features a systematic underestimation of the effect while the rigorous method is so time consuming that hardly enough points can be calculated. Both methods consider only optical effects and electrical issues are neglected. The estimation of what is possible with a grating therefore remains very rough. A third demerit concerns the concepts to avoid the alignment problem. All of these concepts were
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5.6 Summary of the chapter & perspective unsuccessful and no solution was found yet. However, this demerit can be recaptured if a 1D groove texture is used instead of a 2D pyramidal texture. The most important future task concerning diffractive structures will be the realization of an actual solar cell with integrated grating. Only when a solar cell is realized and characterized the simulation can be validated and adapted to the actual resulting structure. An important point in the realization will be to prevent losses at the interface grating / solar cell. A future task concerning the simulation will be to accommodate the two simulation methods to one another. Here several additional simulations will be necessary. Another task will be the complete implementation of the problem into the 2D RCWA code. Only then the absorption profile of the solar cell in its full complexity and with all occurring orientations is accessible. As a perspective, I want to discuss briefly a concepts that has developed from the alignment problem only recently but has not been investigated in detail yet. This alignment problem concerns the existence of different orientations of light incident on the back side grating for a pyramidal front surface texture. Especially the presence of two different azimuthal components that are tilted by 90° severely reduced the performance of a grating. Several strategies to solve this problem have already been outline, yet none of them was successful. On the other hand, one result of the semi-analytical model was, that the highest contribution for a solar cell with a textured surface is made by the light with an azimuthal component of δ = 90°. Using not a pyramid but a line pattern as front surface texture allows to direct all light on the back side structure with this orientation. This reduction to one azimuthal component increases the path length enhancement of the grating, even more so, as the resulting geometry supports TIR of the light impinging on the front surface the second time. The setup of this concept is sketched in Figure 5.29.
Figure 5.29: Combination of a line pattern as front surface texture and a back side grating. Texture and structure are tilted by 90° towards each other. This reduces the number of azimuthal components to one and increases the effect of the grating. Additionally, the setup supports TIR so that both light trapping concepts, front surface texture and back side structure, are optimally combined.
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6 Angularly selective photonic crystals Angular selectivity in PV is used to affect one attribute of radiation; its angular spreading. One other prominent concept exists that also affects this attribute: concentration. Concentrating concepts are well established and have been realized in a large variety. An obvious advantage of concentration is the reduction of the illuminated solar cell area which allows for the use of complex and expensive materials. Yet, concentration also affects the thermodynamical working conditions of the solar cell which results in increased efficiency. The disadvantages of concentration are that a tracking system is required and that the concentration achieved practically is well below the thermodynamical limit. Apart from concentration, only few concepts are known that work with angular spreading. One that could be mentioned is the design of solar thermal devices with a distinct angular emission characteristic. In this chapter I will investigate the application of angularly selective optical elements to PV devices. These elements are only little investigated hitherto, especially are no filters established with a distinct angularly selective characteristic. One goal of this chapter will therefore be an estimation of what is possible with angularly selective filters in principle. Another goal will be the search for effects that result in an adequate characteristic, The effects induced by the application of angularly selective optical elements are related to concentration and feature similar advantages and disadvantages. A part of the description given in this chapter will therefore be borrowed from concentration. The mode of operation for angular selectivity in PV resembles the effects induced by total internal reflection. Another part of the description is therefore related to total internal reflection. Section 6.1 introduces the concept of angular selectivity and angular confinement. The basic idea of how angularly selective optical elements may be used for a beneficial effect in a PV converter is discussed here as well as additional preliminaries needed. Subsequently I introduce an exemplary system in which an angularly selective photonic crystal is used for light trapping. Section 6.2 gives considerations about principles. I show how angular selectivity may be used for highly efficient light trapping and how the Shockley-Queisser limit for PV devices is influenced by angular confinement. Subsequently, I discuss the Bragg effect as one possibility to create angular selectivity. To conclude this section, I broach possibilities to create an angularly dependent reflection characteristic different from that of the Bragg reflection. Section 6.3 presents different photonic structures that I have investigated theoretically and experimentally with regard to their angular-dependent reflection characteristics. An interpretation of these characteristics follows. The investigated structures were: the rugate filter and the band stop filter (1D photonic structures), the checkerboard structure (a 2D photonic structures) and the opal (a 3D photonic structure). Section 6.4 shows experimental results of systems to which the angularly selective photonic structures introduced in section 6.3 were attached. In a first measurement series, the absorption enhancement produced by the light trapping effects of the filters in different absorbers was examined. In a second measurement also the increase in quantum efficiency for a solar cell with a band stop filter was measured.
6.1 Angularly selective optical elements in PV In this section I will introduce the concept of how angularly selective optical elements can be used for PV applications. In subsection 6.1.1, I will define what I mean by angular selectivity and introduce model characteristics for an angularly selective optical element. The most important effects are light trapping and suppression of radiative recombination. These mechanisms are discussed in detail. Another important issue are mechanisms that result in a broadening of the angular emission characteristics of the PV converter. I will introduce such mechanisms in subsection 6.1.2. In subsection 6.1.3, exemplary realisation of PV concept using angular selectivity for light trapping purposes are discussed.
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6.1 Angularly selective optical elements in PV 6.1.1 Basic idea An angularly selective optical element is an element which is transparent for radiation impinging on it within a solid angle Ω1 and mirroring for radiation within a solid angle Ω2.For a PV application the element should be transparent for incident sunlight. For a tracked system the solid angle of transparency is defined by the solar disc. The solar disc covers an angular range that is described by a polar angle of θS = 4.7mrad [Wür05] for the sun only, and θS = 44mrad when considering circumsolar radiation as well [Bui03]. The ideal optical element is therefore transparent for sunlight impinging on it between perpendicular incidence (θ = 0°) and θS and it is mirroring for light coming from all other directions. The angular range of transparency is called the “angular acceptance range” (Figure 6.1). In many ways the mode of operation of such an element can be considered similar to total internal reflection.
Figure 6.1: Schematic sketch of the characteristics of an angularly selective optical element. The angular acceptance range, in this case, corresponds to a polar angle of θs = 35°. All radiation impinging under angles smaller than θs passes the element undisturbed. All other radiation is reflected. The basic idea how an angularly selective optical element creates a beneficial effect for a PV system is shown in Figure 6.2. Two processes define the radiation balance of a PV element. These processes are absorption and emission of radiation. A requisition needed for a beneficial application of an angularly selective optical element is that the solid angle under which radiation leaves the PV system Ωemit is larger than the one from which radiation is absorbed Ωabs. In the figure, Ωabs/emit are defined by the polar angles θS and θE. A typical situation for a solar cell without further optical elements is that Ωabs equals the solid angle defined by the solar disc while Ωemit is the complete hemisphere. The beneficial effect is induced by restricting Ωemit. 13. The benefit manifests itself in several aspects. These aspects shall be clarified here by means of two examples. In the first example a PV device shall be assumed in which the light direction is randomized by a certain process (e.g. scattering at a rough surface). Furthermore the solar cell shall not absorb the internal light completely. Consequently radiation will leave the PV device with a great angular spreading, i.e. Ωemit is large compared to Ωabs. In this example, all radiation that leaves the PV device and outside the device is not in the angular acceptance range of the angularly selective filter will be reflected back into the PV device. Consequently, the path length for radiation inside the PV device is enhanced which results in an increased quantum efficiency. This effect, an increased quantum efficiency induced by a path length enhancement, is the first beneficial aspect of the application of angularly selective filters. 13
PV elements are typically equipped with a rear side reflector so that absorption and emission of radiation only occur at one surface. Emission is thus only considered in one hemisphere. The presence of a back side reflector is typically assumed, even if not explicitly mentioned in the text.
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Angularly selective photonic crystals
Considerations of principles concerning this effect are given in subsection 6.2.1. An experimental investigation is given in section 6.4. In the second example, a PV device shall be assumed in which the most important recombination mechanism is radiative recombination. Radiative recombination can be assumed to be an isotropic process and radiation leaves the PV device with a great angular spreading (θE = π ). In that way, the situation in this example is similar to the one in the first example, however, the consequence is different. Like in the first example, some radiation coming from the PV device is reflected back. In that way, losses due to radiative recombination are reduced. This affects the radiative balance of the PV device and consequently its theoretical efficiency limit. This effect, an increase in the efficiency limit due to a confinement of radiative recombination, is the second beneficial aspect of the application of angularly selective filters. Considerations on this effect are given in subsection 6.2.2. b) a)
Figure 6.2: Sketch of the basic concept for angularly selective filters. The initial situation is sketched in Figure 6.2a. The sun emits radiation on a PV system from a solid angle defined by θS. The PV system emits radiation into the complete hemisphere. This emission is reduced by introducing an angularly selective optical element. This element will only allow radiation to be transmitted if it has a polar component smaller that the one of the angular acceptance range. In order to simplify the further text, from now on all radiation coming from the PV device regardless of the source shall be referred to as “emitted radiation”. Especially radiation that is entering the device, is internally reflected and leaves again falls in this category. All radiation entering the PV device shall be referred to as “absorbed radiation” 14.
6.1.2 Scattering processes in PV converters In this subsection first the term scattering needs to be clarified. Like the term absorption, scattering in this chapter is used in two senses. The first, closer sense is the classical one and scattering here means a random change of the trajectory of a photon that interacts with a medium. This interaction results in angular spreading of an incident light ray. In the broader sense, in which the term is used in this chapter, scattering describes all processes that result in an angular spreading of radiation, regardless of the interaction. All that is considered for this meaning of scattering is the angular characteristic. Other effects e.g. on the wavelength are ignored. In this sense, scattering describes the relevant process 14
The absorbed radiation in this context is the radiation that would be absorbed by a solar cell with a quantum efficiency of 1. Having considered ideal conditions first, this is where the term comes from. For a device with a realistic absorption, not all of the “absorbed radiation” is actually absorbed. This is the case, e.g. i.e. if an absorption or extinction coefficient is given or if a value for the absorption is given It is important not to confuse these two meanings of absorption throughout the text. However, I think the meanings are clear from the context. Where they might not be, I added comments to the text.
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6.1 Angularly selective optical elements in PV for the application of angularly selective optical elements. In that sense, classical scattering is a part of scattering as it is described here. The examples given in this subsection are meant to illustrate which effects can be subsumed under the term scattering as it is used here. An important term in this context is the one of a Lambertian characteristic. The Lambertian characteristic is the angular emission characteristic of a planar light source and is defined by a cosine dependence on the polar angle. If the light distribution inside a device with a planar surface is isotropic, a Lambertian characteristic is obtained. Stepping a bit further, a combination of scattering and angular selectivity may also be used for concentrating light in different PV concepts. Such concepts that combine angular and spectral selectivity are described e.g. in [Ort04] or in [Goe08]. Classical scattering The most straightforward example of a process resulting in an angular spreading of radiation is classical scattering. Different mechanisms exist that result in classical scattering and a classical scattering mechanism may be applied easily to almost every device. For different types of solar cells, the path length enhancement is achieved by different techniques. Thin film solar cells of amorphous silicon, for example, may be deposited on a roughened superstrate [Tie83] [Bec03] [Ter06][Roc08], or a diffuse back reflector may be introduced [Goe81]. Diffuse back reflectors may also be introduced into thick silicon wafer solar cells [Kra08]. An additional example is the inclusion of scattering centres. Even if a single scattering event does not result in a Lambertian characteristic, the desired characteristics are rapidly obtained from multiple scattering events [Yab81]. Classical scattering, in difference to the other mentioned mechanism, only affects the trajectory of a photon an not its frequency Radiative recombination An effect occurring in indirect, yet more significantly in direct semiconductors is radiative recombination. Radiative recombination is the exact opposite of absorption. An electron hole pair annihilates to a photon. As the emission characteristic of this process is isotropic, radiative recombination inside a PV converter results in a Lambertian characteristic. Radiative recombination is one of three principle recombination mechanisms in solar cells (the others are Auger recombination and Shockley-Read-Hall recombination) [Pra83][Pra86]. For an estimation of the solar cell efficiency, Shockley and Queisser (SQ) [Sho61] considered radiative recombination the only permissible recombination mechanism. Under these conditions, angularly selective optical elements induce a strong effect on the efficiency of a solar cells. If other recombination mechanisms are allowed, the importance of radiative recombination is reduced and the effect of the optical elements on the efficiency is weakened. Radiative recombination is a very important recombination mechanism [Mac01] for direct semiconductors such as GaAs [Var67].For indirect semiconductors like silicon, radiative recombination is the least dominant recombination mechanism [Sch74]. A special aspect of radiative recombination is the process of photon recycling in multijunction solar cells. Photon recycling is the reabsorption of a photon emitted by radiative recombination [Dum57]. In single junction solar cells, photon recycling is the process supported by angularly selective optical elements. As the photons generated by radiative recombination have energies close to the band gap of the semiconductor, they are typically absorbed only weakly (α ≈ 1/m). Angularly selective filters produce a considerable path length enhancement in the PV converter and therefore increase the probability of photon recycling. The situation is slightly different for multijunction solar cells. Here most of the radiatively emitted photons are absorbed by a subsequent solar cell with a lower band gap [Let03]. These photons therefore are not lost, as they were in a single junction solar cell, but they generate an electron hole pair with less energy than they otherwise could. In that way a
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Angularly selective photonic crystals
larger fraction of the photon energy is turned into heat and the potential efficiency is reduced. The path of a photon to a subsequent solar cell is blocked by the application of an angularly selective filter. The photon therefore is trapped in the junction where it has been created increasing the probability of a reabsorption there. The effect of angularly selective filters on III-V solar cells is currently under investigation and will not be discussed further in this work. First simulations showed, however, that angular confinement may have a measurable effect on the solar cell efficiency, but that this effect is small. Dye emission In the fluorescent concentrator, radiative processes are used for the transport of radiation. This system is described in detail in chapter 4. A fluorescent concentrator consists of a polymer plate, into which a fluorescent dye is included [Goetz77]. The dye absorbs radiation and reemits it in a random direction. Several experimental results indicate that the assumption of an isotropic emission characteristic of the dye is justified [Ben08]. The emitted radiation is transported to the edges of the concentrator via total internal reflection (TIR). However, not all light is trapped in the concentrator: a fraction of the light is emitted into the escape cone of TIR and leaves the concentrator [Zas81]. This radiation will have a Lambertian characteristic outside the concentrator. As the concentrator consists of a polymer with a typically refractive index of ca. n = 1.5, ca. 26% of the internal radiation are lost. The approach of using an angularly selective filter to reduce escape cone losses resembles the concept described in chapter 4 of using a spectrally selective one. The angularly selective filter mimics TIR for the escaping light and traps most of it in the concentrator. In that way the light guiding efficiency is increased.
6.1.3 PV concepts with angular confinement The basic concept of a PV system with an angularly selective optical element is sketched in Figure 6.3. Solar radiation (blue arrow) impinges on an angularly selective optical element. The set up shall be arranged so that the solar radiation is always within the angular acceptance range. For an angular acceptance range as it was defined in Figure 6.1, this requires tracking. 15 In the PV system the light is scattered by an appropriate mechanism. In the figure this mechanism is depicted as scattering centers, though every other mechanism described in the last subsection is possible, too. For the theoretical functionality of a PV concept with an angularly selective filter, all that matters is that the radiation expands its angular spreading after it has passed the optical element. Some radiation is now absorbed in the PV system. The radiation which is not absorbed or which is reemitted at some point in the PV system has a broad angular range (red stars). This light impinges on the optical element again (red arrows). All photons that are emitted into the angular acceptance range of the optical element pass the optical element and leave the system. All other photons are reflected by the optical element and therefore are trapped in the PV system where they are absorbed [Ulb08]. The description given until now has been kept very general to preferably cover every possible system. Additionally, I want to give a description of one concept which has been realized to test the principle. Experimental results obtained with the corresponding set up are given in section 6.4. The concept realized consisted of a thin film solar cell made of amorphous silicon on a roughened superstrate and an angularly selective photonic structure. The experimental set up is illustrated in Figure 6.4a. The aim of this approach was to increase the absorption in the long wavelength regime of this particular solar cell (λ > 760nm). This solar cell had a thickness of ca. d = 2µm and the quantum efficiency in the 15 One idea for a concept that would require no tracking is the application of an angularly selective filter, the angular acceptance range of which mimics the sun’s orbit
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6.1 Angularly selective optical elements in PV considered spectral range is limited by absorption. The roughened superstrate generally only serves for light trapping purposes. The roughening results in a (classical) scattering of radiation and consequently in a path length enhancement. The quantum efficiency of this solar cell is given in Figure 6.4b. In the long wavelength regime, despite of light trapping, the quantum efficiency is still low due to an incomplete absorption.
Figure 6.3: Basic concept for a PV system with angular selectivity. Light enters the PV system by passing the angularly selective optical element (blue arrow). Subsequently, the light changes its angular characteristic (red stars) and impinges on the optical element again (red arrows). If the light has a direction within the angular acceptance range of the filter, it leaves the system. All other light is reflected by the optical element and is trapped within the system. In the discussed concept the roughened superstrate not only induces a path length enhancement inside the solar cell but also guarantees an angular characteristic of the radiation coming from the solar cell that is appropriate for he concept. The expected effect of the angularly selective filter is an increased light trapping of the radiation in the solar cell, and consequently an increased absorption and an increased quantum efficiency. Generally, increasing the path length increases the amount of absorbed photons. This leads to a higher short circuit current density jSC or it allows decreasing the thickness of the solar cell while maintaining the same jSC. For solar cells made from materials with low minority carrier lifetimes, such as multicrystalline silicon, the bulk recombination is reduced with decreasing thickness. With good surface passivation, this results in increased open circuit voltage VOC. Light trapping improves the solar cell performance in both cases. b)
a)
100
EQE α-Si solar cell
EQE [%]
80 60 40 20 0 300
400
500
600
700
800
λ [nm]
Figure 6.4: Schematic sketch of the light trapping effect generated by an angularly selective filter applied to a thin film silicon solar cell on a roughened superstrate. In the given example, the aim was to increase the quantum efficiency in the spectral regime where it is limited by absorption. The scattering is achieved by a roughened surface of the superstrate. Also given is the quantum efficiency of the used solar cell without the optical element.
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Angularly selective photonic crystals
6.2 Considerations of principles In this section I will discuss how angular confinement affects the efficiency of a solar cell. The topic of subsection 6.2.1 is the light trapping induced by angular confinement. Here, I will calculate the path length enhancement induced by the application of an angularly selective filter. In subsection 6.2.2 I will give considerations about the thermodynamical effect of angular confinement. Here, I will investigate how the SQ limit is affected and discuss the close relationship between angular confinement and concentration. In subsection 6.2.3 I will investigate the angular-dependent reflection characteristic of Bragg-like photonic crystals. I will show that the Bragg description is valid for many different photonic crystals and how this effect may be used to create angular confinement. In subsection 6.2.4, I will discuss effects other than the Bragg effect that lead to a distinct angular-dependent reflection characteristic. Here I will present simulations of more complex photonic crystals and investigate their respective characteristics.
6.2.1 Path length enhancement Path length enhancement induced by the optical element For the considerations of principles, I assume an angularly selective optical element with optimum characteristics (see Figure 6.1). The characteristics of the element are completely defined by the angular acceptance range, which is characterized by the polar angle θs. 16 Within the acceptance range the optical element has a transmission of T = 1, outside this range it is a perfect mirror with R = 1. I will now show that the path length enhancement for such an element is equivalent to the reciprocal of the fraction of light in the angular acceptance range. I will also point out under which conditions this argument holds. Concluding, I will calculate the path length enhancement. The considerations given follow an argument by Yablonovitch [Yab81]. According to Landau and Lifshitz, the internal light intensity Iint inside a medium in bb equilibrium with external blackbody radiation I ext is proportional to the external black body radiation with a proportional factor equivalent to the square of the refractive index of the medium n² [Lan69] bb I int (ν , x) = n 2 (ν , x) I ext (ν )
(6.1)
In this equation ν denotes the frequency of the internal light and x denotes the position. Equation (6.1) remains valid for collimated illumination, as long as the light behaviour inside the medium is ergodic 17. Adequate ergodic behaviour is given if a randomization of the light inside the system occurs quickly enough, that is to say before the light may leave the system. In our examples this is achieved by the processes described in 6.1.2, resulting in a Lambertian characteristic 18. By virtue of the backside reflection the internal light intensity has been doubled, so that now it holds
16
Initially θs is undefined and the results are given as a function of θs. However, as the notation indicates, θs is closely related to the solid angle of the solar disc and if not said other wise and a concrete value is assigned, it always holds θs = 4.7mrad. 17
The term ergodic is related to the quasi-ergodic hypothesis. The hypothesis states that a steady-state, temporally averaged, light-intensity distribution is identical to a statistical phase-space intensity distribution. An illustrative example for an ergodic system is the expectation of the characteristics of a die. Its characteristic may be determined by throwing one die repetitively or by throwing many dice simultaneously. 18
Realistic reflectors will show non-Lambertian characteristics. Adequate randomization for non -Lambertian but diffuse reflectors is achieved if multiple reflections occur. At least after the second or third reflection the light will have almost Lambertian characteristics and most light will be trapped.
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6.2 Considerations of principles I int (ν , x) = 2 ⋅ n 2 (ν , x) I ext (ν )
(6.2)
with Iinc(ν,x) being the intensity of the incident light. (Note that these equations describe idealized conditions i.e. Fresnel reflections at the surface are completely neglected.) The equilibrium condition described by equation (6.2) holds for a solar cell with light incident in a beam, but with radiation exchange with the complete hemisphere. With angular confinement, the solid angle into which the solar cell radiates is restricted, but the incident intensity remains unchanged. If the fraction of internal radiation that is in radiation exchange with the surroundings is reduced by a factor L, the internal intensity has to be increased by the same factor to ensure that the equilibrium condition (6.2) is met. At this point, it is important to note that this consideration is only valid in the limit of low absorption. Low absorption plays a role here at two points: first, the internal intensity of radiation has to be reached. If the absorption is too high, the internal intensity will never reach the equilibrium value. Second, the radiation needs to be distributed equally throughout the entire volume. This is only possible for a low absorption; otherwise an exponential dependence of light intensity and thickness will occur. The absorption Abs in the volume under the given conditions is proportional to the internal intensity
Abs ~ ∫ α ⋅ I int dV ~ α ⋅ l ⋅ I int ⋅ Ainc
(6.3)
In this equation dV is a volume element, α is the absorption coefficient, Ainc is the surface area upon which light is incident and l is the path length of the light inside the medium. This path length can be used as a definition of the effective thickness of the solar cell. An increased internal intensity therefore is equivalent to an increased path length of the light inside the medium. In fact, the conclusion shown in equation (6.2) is also obtained from geometric- optical considerations only taking into account the path length. This proves the initial assumption, namely that the path length enhancement is equivalent to the internal intensity of radiation which is the reciprocal of the fraction of light in the angular acceptance range of the optical element. The assumptions needed up until here are equilibrium of internal and external radiation and low absorption. Now I want to investigate the path length enhancement that is obtained by an optical element with the characteristics given in Figure 6.1. The effect that I want to investigate here is the path length enhancement induced by the filter only. To described this effect all refractive indices are set to n = 1. This procedure is justified because typically an air gap exists between PV element and optical element. Still, the refractive index of the PV element influences the light trapping of the entire system and effects induced by a refractive index other than n = 1 will be investigated later. The light coming from the PV element and impinging on the filter shall now have a Lambertian characteristic.
dI int (θ ) = I int, 0 cosθ dθ
(6.4)
For this characteristic, and a restriction of the radiation exchange to a solid angle defined by
θs (acceptance range of the optical element), the fraction of radiation in this steric range L is
given by [Goe81]
2π π / 2
L=
∫ ∫I
int, 0
0 0 2π θ s
∫∫I
int, 0
cosθ sin θ dθ dϕ = cosθ sin θ dθ dϕ
1 sin 2 θ s
(6.5)
0 0
The maximum possible restriction of the solid angle is given by the projection of the solar disc and is defined by the angle θs = 4.7mrad. For this value, a maximum path length
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enhancement of L = 46200 is obtained 19. If the circumsolar radiation is also considered (θs = 44mrad), the path length enhancement still reaches a value of L = 525. Path length enhancement in a PV element with the condition of TIR In the last part, the path length enhancement induced by the angularly selective optical element only was investigated. In this part, effects shall be considered that are induced by a refractive index other than n = 1 of the PV element. More precisely, the situation that shall now be given is: radiation is scattered inside a PV element consisting of a material with the refractive index n2. The PV element is surrounded by a medium with a refractive index n1 < n2. At the interface between the PV element and the surrounding, TIR occurs. This situation is illustrated in Figure 6.5. Because of TIR only a fraction of the internal light leaves the PV element, all other light is trapped. This process, the combination of TIR and scattering, adds an additional contribution to light trapping in a PV device.
Figure 6.5: Illustration of the light trapping effect discussed in this part of the subsection. Inside the PV element a scattering mechanism exists indicated here by the red scattering centers (other mechanisms are possible as well). A consequence of scattering is now that for a fraction of the light TIR occurs. TIR traps the light inside the PV element. This adds a further contribution to light trapping. I will now calculate the path length enhancement generated by this effect. The derivation of the path length enhancement follows an approach by Yablonovitch [Yab81]. A similar approaches is found by Green [Gre02]. The approach given here is based on a detailed balance of the light incident upon a small area element dA and the light in the escape cone of TIR emitted through the same element. The intensity that escapes Iesc (still assuming Lambertian distribution and already having performed the integral over θ) is given by θc
I esc = ∫ I int cosθ sin θ dθ = 0
1 I int sin 2 θ c 2
(6.6)
Using the definition of the critical angle of TIR θc = arcsin 1/n, the escaping intensity in terms of to the internal intensity is given by
I esc =
I int 2n 2
⇔
I int = 2n 2 ⋅ I esc
(6.7)
Assuming equilibrium conditions, only a fraction of the internal radiation is in radiation exchange with the surroundings. As was shown in the last part of this subsection, and as follows from the assumption of detailed balance, a decreased escaping intensity results in an increased internal intensity compared to the incident intensity Iinc coming from the sun 19 As the fraction of light in the considered cone is equivalent to the path length enhancement, no differentiation is made in the symbols.
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6.2 Considerations of principles (always under the assumption of low absorption). Therefore, in equation (6.7) Iesc can be exchanged for Iinc.
I int = 2n 2 ⋅ I inc
(6.8)
This is the same result as equation (6.2), but now obtained explicitly for a medium in which TIR occurs. Finally, the effect of the increased intensity on the absorption needs to be calculated to obtain the path length enhancement. To do this it is beneficial to introduce another factor Bint, the internal intensity per unit angle. Bint and Iint are related through the equation
I int = ∫ Bint dΩ
(6.9)
dΩ being the integration over the sphere. For a uniform distribution of the internal light, it holds that
I int = 2π ⋅ Bint
(6.10)
The absorption Abs is now given by π
Abs = ∫ α ⋅ Bint dV dΩ = 2π ⋅ α ⋅ Bint ⋅ Ainc ⋅ t ⋅ ∫ sin θ dθ 0
= 2 ⋅ α ⋅ t ⋅ I int ⋅ Ainc = 4 ⋅ n² ⋅ α ⋅ t ⋅ I inc ⋅ Ainc
(6.11)
In this equation dV is a volume element in the bulk, α is the absorption coefficient and t is the thickness of the solar cell. To obtain the absorption enhancement and consequently the increased path length, this case needs to be compared to the case for which no scattering and no rear side reflection of light occurs. In that case Bint is a δ − distribution at the direction of incidence θinc. The relation of Bint and Iint is then given by
Bint = I int ⋅ δ (θ inc ) = I inc ⋅ δ (θ inc )
(6.12)
A comment needs to be made upon the claim of the equality of Iint and Iinc in equation (6.12). Up until now, I have always considered the presence of a back side reflector, which produces a factor of two for Iint. The mirror reflection was already introduced in equation (6.2). Equation (6.12) now, defines the case to which all other cases are compared. This comparable case is kept as simple as possible. The means that equation (6.12) describes the case for light passing through a simple slab without any reflection and especially without a backside mirror. After this short recapitulation, the absorption for the comparable case Abs0 is calculated.
Abs 0 = ∫ α ⋅ Bint dV dΩ = α ⋅ I inc ∫ dV = α ⋅ I inc ⋅Ainc ⋅ t
(6.13)
Comparing this result to the result of equation (6.11), angular randomization and TIR result in a path length enhancement of a factor of 4n². Both effects, light trapping by TIR and angular confinement are completely independent, though both processes have similar working principles. Light is scattered inside a medium and a fraction of the light with certain polar angles is trapped. Both effects may be combined, because a Lambertian characteristic inside a medium stays Lambertian outside, despite TIR. The maximum path length enhancement possible for such a system in which both processes are combined is L = 4n²/sin²θs [Min90][Gre95]. For a silicon solar cell with a refractive index of n = 3.5, the factor 4n² produces a value of ca. L = 50. Combined with the maximum path length enhancement of L = 46200 for a restriction to the solid angle of the solar disc, this light trapping system provides a maximum path length enhancement of L = 2.2 E6.
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Angularly selective photonic crystals
6.2.2 Efficiency limit for systems with angular confinement In this subsection I investigate how the Shockley-Queisser limit for a PV converter is influenced by angular confinement. This process is very closely related to concentration, and therefore both phenomena will be considered in the following paragraph. In the first part, I investigate the generation of entropy due to the change of the spreading of incident and emitted radiation. Here I show that the generation of entropy is prevented by changing this spreading. Generation of entropy affects the Shockley-Queisser limit, i.e. the efficiency limit of a PV converter that is completely defined by the incident and emitted photon flux. The approach given here builds on several publications. A first formulation of a similar idea, was presented by Araujo [Ara94]. Badescu [Bad05] investigated angularly and spectrally selective emitting surfaces for PV systems under one sun illumination. The thermodynamic limit for the maximum voltage has been given by Markvart [Mar07]. The potential of concentrating systems is well known and described e.g. by de Vos [Vos82] and Minano [Min90]. Étendue and entropy In a photovoltaic conversion process, the irreversible generation of entropy leads to a reduction of the theoretical efficiency limit. One of these losses is due to the étendue expansion between absorbed and emitted radiation [Mar08]. The étendue ε(τ,t) describes the phase space volume of light. In this context it is beneficial to consider an ensemble of photons. For such an ensemble, Liouville’s equation holds.
∂ ε (τ , t ) = {H , ε (τ , t )} ∂t
(6.14)
In this equation τ describes the ensemble of the phase space coordinates, H is the Hamilton operator of the light and {} are Poisson brackets. From Liouville’s equation, it follows that the phase space volume for conservative systems 20 remains constant (Liouville’s theorem). The étendue dε of a light beam in air being received from a solid angle dΩ by an infinitesimal surface element dA can be defined as
dε = cosθ dΩ dA
(6.15)
with θ being the polar angle between the surface normal of dA and the incident light. To calculate the étendue ε for an extended receiving system, an integration must be performed over the aperture area Ainc and over the solid angle Ωinc from which radiation is received.
ε=
∫ ∫ cosθ dΩdA
Ainc Ω inc
(6.16)
The étendue can also be calculated for an emitting system. In this case, the integration has to be performed over the emitting area Aemit and over the solid angle Ωemit in which radiation is emitted. For light incident from a cone, with θinc being half of the opening angle, and a flat, not tilted illuminated area Ainc (see Figure 6.6), the étendue is given bye
ε = π sin 2 θ inc Ainc
(6.17)
This equation describes the étendue for a PV converter under the assumption that the light is received and emitted in a cone. From the definition (equation (6.15)) and this result, the étendue can be understood as a measure of how spread out a light beam is in terms of 20
Conservative systems in this context are imaging systems like lenses or mirrors. The conservation of étendue may be used as a definition for imaging systems. For non-imaging systems the étendue is not conserved but always increased. Systems are non-imaging if e.g. scattering occur
126
6.2 Considerations of principles angular divergence and illuminated or emitting area. From equation (6.17) it is also obvious that constancy of étendue means that if the aperture area is decreased, the angular spreading has to be increased and vice versa. A conservative optical element is shown in Figure 6.6.
Figure 6.6: Illustration of the conservation of étendue. The product of aperture area A and angular spreading θ is constant. If an optical element changes e.g. the aperture area form Ainc to Aext, the angular spreading is changed consequently for θinc to θext. The étendue expansion of a PV converter is defined by two angles. The first angle θinc describes the cone of radiation incident on the PV converter. The second angle θext describes the cone into which the PV converter emits radiation. This cone is nothing more than the angular acceptance range. The initial situation is that the PV converter receives radiation from the narrow cone into which solar radiation impinges on the earth, defined by the already introduced polar angle θs. The angular acceptance range in a first consideration is the complete hemisphere corresponding to a polar angle of θext = π. Because of this disparity of the cones of received and emitted radiation, the PV converter initially is a source of entropy. The entropy per photon σ generated by the enlargement of the angular spreading is defined by the étendue of the incident εinc and the emitted εext radiation . It is given by [Mar08]
σ = k B ln
ε ext ε inc
(6.18)
In this equation kB is the Boltzmann factor. This equation already shows that the production of entropy is influenced equally from both cones, as it only depends on the ratio of the étendues. Furthermore, no entropy is generated if the étendues for incident and emitted radiation are equal. For a solar cell with the aperture area Ainc, the étendue of the incident sunlight is given by εinc = 1/46200 Ainc. The étendue of the emitted radiation for a solar cell with the area Ainc emitting into the complete hemisphere is given by εem = 1 Ainc. Concentration of light is equivalent to an enhancement of angular spreading of the incident light. As the étendue cannot increase, the illuminated solar cell area needs to be decreased simultaneously which is the desired effect for concentration. A maximum concentration is obtained if the spreading of the concentrated light is θ = 90°. The condition for maximum concentration is therefore given by
Ainc sin ²θ S = Aext ⇒ C max =
Ainc 1 1 = = Aext sin ²θ S 46200
(6.19)
That means that the solar cell area is decreased by a factor of 46200 compared to the aperture area upon which the solar radiation impinges (i.e. the lens or mirror). As the solar cell area is decreased, the étendue of the emitted radiation is also decreased and is now given by εem =1/46200 Ainc (note that Ainc here is the aperture area and not the solar cell
127
Angularly selective photonic crystals
area). Étendue of the incident and emitted radiation are equal, consequently maximum concentration turns a solar cell into a conservative system. A reduction of the angular acceptance range by angularly selective optical elements reduces the étendue of the emitted radiation. A maximum confinement is achieved if the angular acceptance range of the solar cell is restricted to the solid angle of the sun. Under these conditions, the étendue of the emitted radiation is given by εem =1/46200 Ainc. Again étendue of received and emitted radiation are equal and therefore maximum angular confinement also turns a solar cell into a conservative system. Inserting equation (6.17) into equation (6.18), the entropy generated by conical incidence and conical emission of a PV converter is given by
σ = k B ln
sin 2 θ ext sin 2 θ inc
(6.20)
With the values given above, this equation results in a total entropy generation per photon of σ = 9.26 10-4eV/K for a PV converter without any further optical elements. The entropy generation is reduced to zero if the cones for absorbed and emitted radiation are equal. An additional consequence that is obtained from equation (6.17) as well as equation (6.20) is that the cone for emitted radiation cannot be smaller than that for incident radiation. Voltage, current and efficiency in the SQ limit Now the effect of a change of the angular divergence on the Shockley-Queisser limit is investigated. According to Würfel [Wür05], the efficiency η of a solar cell is given by
η=
j mpVmp ∞
∫ hϖ djγ
, sun
(hω )
(6.21)
0
In this equation jmp and Vmp are the current density and voltage at the maximum power point. The integral in the denominator is the total power in the solar radiation ( djγ , sun describes the photon flux per energy). The assumptions for the calculation of the Shockley-Queisser limit are now that all photons with energy higher than the band gap energy are absorbed with a probability of one. All other photons are not absorbed. Each absorbed photon creates an electron-hole pair with an energy defined by the difference of the quasi Fermi levels. Furthermore, the only allowed recombination mechanism is radiative recombination. That means that the charge current density is defined completely by the absorbed and emitted photon current density
j c = e0 jγ ,emit (V ) − e0 jγ ,abs
(6.22)
In this equation e0 is the elementary charge. The photon current densities are defined by [Wür05]
djγ ,emit (hω ) =
Ω ext 4π 3 h 3 c 2
djγ , abs (hω ) =
(hω ) 2 dhω ⎡ hω − ( E FC − E FV ) ⎤ exp ⎢ ⎥ −1 kT0 ⎦ ⎣
(hω ) 2 dhω Ωinc 4π 3h 3c 2 ⎡ hω ⎤ exp ⎢ ⎥ −1 ⎣ kTS ⎦
128
(6.23)
(6.24)
6.2 Considerations of principles Equation (6.24) is the emitted photon current density derived from the generalized Planck law. Equation (6.25) is the absorbed photon current density and is derived directly from the blackbody radiation. In these equations EFC and EFV are the quasi Fermi energies of the conduction and valence bands, T0 is the temperature of the PV converter, and TS is the temperature of the sun. The one important point about equations (6.24) and (6.25) in this consideration is, however, only that the photon current densities contain the solid angles for absorption and emission Ωabs/emit as prerequisites. These angles are given by
Ω ext / inc = 2π
θ ext / inc
∫ sin θ dθ = π
sin (θ ext / inc )
2
(6.25)
0
The IV characteristic of a PV converter under the described conditions is given by ∞ ⎡ ⎛e V ⎞ ⎤∞ j c = e0 ⎢exp⎜ 0 ⎟ − 1⎥ ∫ djγ ,emit (hω ) − e0 ∫ djγ ,abs (hω ) ⎣ ⎝ kT ⎠ ⎦ ε G εG
(6.26)
As was shown earlier, the photon current densities depend linearly on the solid angles Ωabs/Ωemit. Equation (6.26) can therefore be rewritten as ⎞ ⎛ ⎞⎟ ⎛ ⎜ ∞ ⎟ ⎜ ⎛ ∞ ⎞ e jc (Ω ext , Ω inc , V ) = Ω ext ⎜⎜ e0 ∫ djγΩ,emit (hω ) exp⎜ 0 V − 1⎟ ⎟⎟ − Ω inc ⎜ e0 ∫ djγΩ,abs (hω ) ⎟ ⎜ ε ⎟ kT ⎟⎟ ⎜{ G ⎝14G42443⎠ ⎜⎜ 1ε4 42443 c2 ⎟ ⎠ ⎝ c1 c3 ⎠ ⎝
(6.27)
In equation (6.27) jγΩ describes the photon current density per solid angle. For a given PV converter with a defined level of the quasi Fermi energies, the terms marked by the brackets are constants (c1, c2, c3) and the current density is only a function of the two solid angles and the voltage. Equation (6.27) may therefore be simplified to
j c (Ω ext , Ω inc , V ) = Ω ext ⋅ c1 [exp(c 2 ⋅ V ) − 1] − Ω inc ⋅ c3
(6.28)
The condition for the voltage at the maximum power point is given by
Pmax = j mp ⋅ Vmp
⇔ ∂ V ( j c ⋅ V ) | mp = 0 ⇔
exp(c 2 ⋅ Vmp ) ⋅ (1 + c 2 ⋅ Vmp ) = 1 +
⎛ sin θ inc Ω inc c3 = 1 + ⎜⎜ Ω ext c1 ⎝ sin θ ext
2
⎞ c3 ⎟⎟ ⎠ c1
(6.29)
Vmp therefore is a function of the ratio of the solid angles Ωabs/Ωemit. The dependence of Vmp on this ratio is given in Figure 6.7. As this function has a strictly positive slope, a maximum of Vmp is reached at the maximum value of Ω abs / Ω emit . As the solid angle into which radiation is emitted cannot be smaller than the one from which radiation is received, the maximum is reached if both angles are equal.
129
Vmp [a.u.]
Angularly selective photonic crystals
0 Ωinc/Ωext [a.u.]
Figure 6.7: Dependence of Vmp on the ratio of the two solid angles of absorbed and emitted light Ωinc/Ωext. Notable is that the voltage at the maximum power point only depends on the ratio of the solid angles. Whether the ratio is defined via concentration or via angular confinement is of no importance. This result is somewhat counterintuitive. For concentration an increased electron density occurs due to the decreased solar cell area, which for angular confinement does not occur. In case of concentration, the increased electron density is the cause for the increased voltage. For angular confinement the amount of radiatively recombining electronhole pairs is reduced so that the emitted photon current decreases. As the equilibrium situation is always defined by gains and losses, the same equilibrium situation is obtained either by increasing the incident photon current density or by decreasing the emitted photon current density. For this reason, angular confinement results in an equivalent voltage increase to that achieved via concentration. Having examined the voltage at the maximum power point, now the current at the same point needs to be examined. Equation (6.28) gives the current density. However, if concentrating and non-concentrating systems shall be compared, the current needs to be related to the actual solar cell area Acell.
[
]
j mp (Ω ext , Ω inc ) ⋅ Acell = Ω ext ⋅ Acell ⋅ c1 exp(c 2 ⋅ Vmp ) − 1 − Ω inc ⋅ Acell ⋅ c3
(6.30)
From the conservation of étendue, it is necessary that changing the solid angle for the incident radiation is equivalent to changing the solar cell area such that Acell Ωinc = const. Inserting this into equation (6.30) results in
j mp (Ω ext , Ω inc ) ⋅ Acell =
[
]
1 ′ ′ ⋅ c1 exp(c 2 ⋅ Vmp ) − 1 − c3 Ω inc Ω ext
(6.31)
Therefore the total current at maximum power point (or any point) also depends only on the ratio Ω inc / Ω ext . Bearing in mind that the current is defined as negative, a maximum occurs for the maximum ratio of Ω inc / Ω ext . This means that the current also shows a maximum for conservative systems. As c1, being the dark current per solid angle, is much smaller than c3, the current that is produced due to illumination per solid angle, the effect on the total current is much smaller than the effect on the voltage. In other words: the total current at maximum power point undergoes only minor changes with the two solid angles. As current and voltage only depend on the ratio of both solid angles, the efficiency also only depends on this ratio (Figure 6.8).
130
6.2 Considerations of principles
Figure 6.8: Shockley-Queisser limit for photovoltaic systems depending on the solid angle of absorption and emission. If both angles are equal, the system is conservative. A possible realization of a conservative system is maximum concentration, another one maximum angular confinement. For all conservative systems, the same Shockley-Queisser limit holds. In a conservative systems, the solid angles for absorption and emission are equal. For thermodynamical reasons, Ωext cannot be smaller than Ωinc.(these cases are represented by the yellow area). Typically the opposite holds. A standard solar cell receives radiation only from the solid angle of the solar disc but emits into the complete hemisphere. No values for the efficiencies are given in the graph as the efficiency depends on the band gap energy of the semiconductor used. For silicon, the efficiency limit for the initial situation is 31% whereas the limit for the conservative system is 41%
6.2.3 Angular-dependent characteristic of Bragg-like systems In the last two subsections I have discussed how angular confinement affects solar cells. In this subsection I want to give a first example for a mechanism resulting in an angularly dependent reflection of photonic crystals. This characteristic concerns the angular dependence of the Bragg condition. The Bragg condition is very important, as the reflection of many photonic crystals may be considered a Bragg reflection. For example, the Bragglike reflection of opals has been shown by Gajiev [Gaj05]. Considering the Bragg reflection adds a new aspect to the consideration of angular confinement; the wavelength dependence. I will determine the wavelength-dependent path length enhancement that can theoretically be achieved with Bragg-like systems. Additionally I investigate the spectral range in which angular selectivity will occur. For the Bragg reflection, a system is considered that consists of several equidistant surfaces. The condition for a Bragg reflection is satisfied if the waves reflected at each surface interfere constructively (Figure 6.9). All system that can be described in that way will be referred to as Bragg-like.
131
Angularly selective photonic crystals
Figure 6.9: Illustration of the Bragg reflection. Light is incident on two surfaces under the angle α. At each surface a fraction of the light is reflected. Because of the distance d of the two surfaces, the reflected light at each surface has a phase difference δ. The Bragg condition is fulfilled if the phase difference is equal to a multiple of the wavelength of the light. Let the angle under which the light is incident be α. At each surface, a fraction of the incident light is reflected. If the surfaces have a distance d, the phase difference δ of the light reflected at each surface is given by
δ = 2 ⋅ d ⋅ cos α
(6.32)
For constructive interference, the phase difference must be a multiple of the wavelength of the incident light. This demand results in the Bragg condition
2 ⋅ d ⋅ cos α = m ⋅ λ
(m = 1,2,3,…)
(6.33)
Until now, no refractive index was assumed between the surfaces. If a refractive index 21 n is assumed between the surfaces the wavelength in the medium changes and also the angle according to Snell’s law. The Bragg condition then changes to
λ ⎡ ⎞⎤ ⎛1 2 ⋅ d ⋅ cos ⎢arcsin⎜ sin α ⎟⎥ = m ⋅ 0 n ⎠⎦ ⎝n ⎣
⇔ 2 ⋅ d ⋅ (n² − sin ²α ) = m ⋅ λ0
(6.34)
In this equation λ0 is the vacuum wavelength of the incident light and n is the average refractive index of the element. Equation (6.34) describes the wavelength dependence of the reflection peak of a Bragg-like system. If to Braggs’ equation is referred, equation (6.34) is what is meant. Equation (6.34) describes the spectral as well as the angular dependence of Bragg like systems. A connection between spectral and angular dependence is typical for most photonic crystals. The angular confinement of a give crystal must therefore always be related to the spectral range within which it is valid. As a consequence of equation (6.34), the reflection peak which is caused by the Bragg effect is shifted towards lower wavelengths for increasing angles of incidence.
21
The refractive index in this case describes the average refractive index of the Bragg-like filter. A simple approach to calculate the average refractive index is adding up the refractive indices of the single components weighted with the filling fraction.
132
6.2 Considerations of principles This blue shift of the reflection peak allows to use a Bragg like reflector as an angularly selective filter. Equation (6.34) describes the position of the reflection peak of a Bragg like optical element. This peak has a certain width and the equation describes the angular characteristic of the centre position of this peak. For the moment, the reflection peak shall be assumed to be a step function. with reflection R = 1 in the spectral range of the peak and R = 0 outside the peak. The centre position shall be denoted as λ0, and the spectral range of the peak shall be defined by the interval [λ-, λ+]. Now the assumption can be made, that the angular characteristics of λ-(α) and λ+(α) are the same as the one for λ0(α) (equation (6.34)). This assumption is justified by experimental results (see e.g. Figure 6.14). The situation for a certain wavelength λ-(α0) is that it experiences a reflection of R = 0 for angles smaller than α0 and a reflection R = 1 for angles greater than α0. This characteristic exactly describes an angularly selective optical element for a wavelength λ-(α0) with an angular acceptance range defined by α0. This situation is shown in Figure 6.10.
Figure 6.10: Illustration of how a Bragg like optical element can be used as an angularly selective filter. λ-(α) describes the angular characteristic of the edge of the reflection peak. For the wavelength λ-(α0) = 525nm, this element constitutes an angularly selective filter with an angular acceptance range defined by the angle α0.≈ 45°. The distance of the lattice planes for the Bragg filter is d = 200nm, the average refractive index n = 1.5. The angular characteristic given in Figure 6.10 is typical for all Bragg-like reflectors. One last comment about this characteristic is necessary. In Figure 6.10. it was tacitly assumed that the reflection peak is so broad, that once the blue line is crossed, the reflection stays R = 1 for all greater angles. This is not the case if the reflection peak is narrow. For a narrow reflection peak, it is possible that for higher angles the peak is crossed and the wavelength again reaches an area with R = 0. This case is shown in Figure 6.18. From equation (6.5) the path length enhancement associated with a certain acceptance angle is known. From this the path length enhancement created by a Bragg-like reflector can be calculated. The wavelength-dependent path length enhancement associated with the exemplary Bragg-like filter shown in Figure 6.10 is given in Figure 6.11.
133
Angularly selective photonic crystals
Figure 6.11: Wavelength-dependent path length enhancement for the exemplary Bragg-like reflector presented in Figure 6.10. Only when close to the reflection peak for normal incidence(λ = 600nm) is the path length enhancement considerably larger than a factor of ten (red line). This shows that Bragg-like systems are not very effective angularly selective filters. Equation (6.34) defines not only the form of the angular characteristic, but also the spectral range for which angular selectivity is obtained. This spectral range is defined by the wavelengths for normal incidence (θ = 0°) and for parallel incidence (θ = 90°). It depends on the effective refractive index of the Bragg-like reflector. The lower this effective index is, the more pronounced the blue shift of the reflection peak with increasing angles turns out to be. This dependence is shown in Figure 6.12.
ratio lH90°LêlH0°L
0.8 0.6 0.4 0.2 0 1
1.5
2 2.5 3 refractive index
3.5
4
Figure 6.12: Illustration of the ratio s of the Bragg peak at θ =90° to the Bragg peak at θ =0°. The ratio gives the maximum spectral range which may be covered by angular confinement. A ratio of 0.5 means that a range from one frequency to the double frequency is maximally covered by the angular selectivity. The ratio increases with increasing refractive indices, therefore when using the Bragg effect, a low refractive index of the filter is beneficial.
134
6.2 Considerations of principles Typically, dielectric filters like conventional DBRs or rugate filters have effective refractive indices between n = 1.5 and n = 2. For a refractive index of n = 1.5 and a reflection peak for normal incidence positioned at λ = 600nm angular confinement occurs between λ = 447nm and λ = 600nm. Lower average refractive indices and therefore broader spectral ranges for the angular confinement are possible for 3D photonic crystals consisting of spherical particles in air. This marks a theoretical advantage of 3D photonic crystals.
6.2.4 Other mechanisms to control angular selectivity The Bragg condition is a very strong mechanism controlling the angular characteristic of photonic crystals. However, there are certain possibilities to tune the angular characteristic of photonic crystals to characteristics beyond Bragg. This subsection is only meant to give a short overview of these possibilities, but a detailed consideration has not been performed and is the subject of ongoing work. Periodic structures In periodic structures, the Bragg condition is strictly valid. It is therefore impossible to create a 1D periodic photonic structure which doesn’t show a characteristic governed by Braggs’ condition. This is also true for 2D and 3D structures. However, the lattice constant relevant for the position of the Bragg reflection may change, so that for different directions different conditions apply. This idea is sketched in Figure 6.13 for the example of the checkerboard structure.
Figure 6.13: Sketch of the checkerboard structure. Depending on the direction of incidence, different lattice distances are relevant for the calculation of the position of the Bragg peak. For normal incidence, the relevant distance of the layers is d1, while for an incidence of θ=45°, the relevant distance is d2. Figure 6.13 shows that a simple consideration for a Bragg reflection cannot be correct for such a crystal. It is not straightforward anymore to determine what the angular-dependent reflection characteristics of such a crystal looks like. The reflection characteristics of the checkerboard structure are shown in Figure 6.17 in subsection 6.3.2. A future task will be to find and investigate crystals with non-Bragg-like angular-dependent characteristics to create angular selectivity.
135
Angularly selective photonic crystals
Non-periodic structures The limitation for the angular characteristics of periodic structures does not apply to nonperiodic structures. It is therefore possible to tune the angular selectivity by using nonperiodicity. However, this process is also currently insufficiently investigated. In this work, only 1D non-periodic structures were considered, and these structures were also mainly governed by Braggs’ condition (see subsection 6.3.1). It is however possible to influence the angular selectivity in a guided way by this approach. Imenes et al. used a special optimization algorithm called the “needle optimization method” to create a non-periodic layer system which showed no angular selectivity over a broad angular range [Ime06]. This approach could also be used to create layer systems with the goal of increasing the angular dependence of the reflection. However, this approach has not been carried out in the scheme of this work, making it an interesting option for future investigations.
6.3 Angularly selective photonic structures In this section I will introduce different photonic crystals and investigate their angulardependent reflection characteristics. 1D, 2D and 3D photonic structures were investigated theoretically and, for available crystals, experimentally as well. For the theoretical examination the available tools (RCWA simulation and characteristic matrices) were used. Some photonic structures were realized for different purposes and characterized with a gonio-reflectometer. The results presented here are chosen examples that show relevant effects. The examples are ordered after their dimensionality.
6.3.1 1D crystals From the 1D crystals two examples where chosen, the rugate filter and the band stop filter. Specimen of both kinds of filters were purchased so that a theoretical as well as an experimental characterization was possible. The rugate filter is a periodic filter with a sinusoidal function of the refractive index. For this filter the Bragg condition is valid. The example serves as an illustration of how such a filter is used to create angular selectivity. The band stop filter is a non-periodic system. This filter has been used in the experiments to demonstrate the effect of an angularly selective filter on the light trapping in a solar cell. Even though this filter is non-periodic, Braggs’ equation still provides a satisfactory description of the reflection characteristic. Optimized rugate filter Rugate filters were introduced in section 2.3. The optimization of the rugate filter was performed in section 4.3. According to equations (4.17) and (4.18) (compare Figure 4.17), a rugate filter was designed which showed a reflection peak at λ = 650nm. The results of the simulation of the angular-dependent reflection characteristic are shown in Figure 6.14a. To describe the angular-dependent reflection characteristic, Braggs’ equation (6.34) has been used with an average refractive index of n = 1.75 and a period of Λ = 160nm. Figure 6.14a shows that Braggs’ equation describes the peak center position as well as the edges of the reflection peak. A rugate filter similar to the one simulated was fabricated at Fraunhofer IST. The reflection characteristic of this filter is shown in Figure 6.14b. Again Braggs’ equation was used to describe the blue shift of the reflection peak. The parameters in this case were an average refractive index of n = 1.75 and a period of Λ = 170nm (the period was slightly larger than in the simulations). The simulated reflection characteristic differs in some points from that of the real filter. The reflection peak in the measurement is narrower, and narrows further with increasing angles of incidence. The difference between simulation
136
6.3 Angularly selective photonic structures and experiment is caused by several effects. For one, dispersion and absorption were neglected in the simulation. However, this point plays a minor role as materials used typically for the production of rugate filters are SiN and SiO2. In the investigated spectral range, these materials show neither much dispersion nor considerable absorption. Crucial for the characteristics of the rugate filter is that in the production, the refractive index profile is approximated within limits. The refractive index and the thickness of the single layers both can only be controlled up to a certain point. The visible differences are therefore most likely effects of this approximation. The light trapping effect of a rugate filter was examined theoretically by Fahr [Fah08]. b) a)
Figure 6.14: Simulated (a) and measured (b) angular-dependent reflection characteristic of a rugate filter with a refractive index profile specified in Figure 4.17. The filter was designed for a reflection at λ = 650nm. The measured filter was fabricated at Fraunhofer IST. Also shown is the blue shift of the reflection peak predicted with Braggs’ equation. The parameters used were n = 1.75 for both filters and a period of d=160nm for the simulated filter and d = 170nm for the real filter. The band stop filter The band stop filter is a layer system similar to the Bragg filter, though the thickness of the layers is not periodically arranged but rather optimized to fulfil certain demands on the reflection characteristic as are a very high reflection and transmission in the particular spectral ranges. As the band stop filter is non-periodic, the Bragg condition does not strictly apply here. However, when designing a band stop filter, a typical procedure is to start with a Bragg filter that shows a reflection in the designated spectral region, and optimize this filter towards the desired characteristics.. Usually, after the optimization enough periodicity remains so that Braggs’ equation may still be used to describe the angular dependence of the band stop filter. Saying “enough periodicity remains”, I mean that the distribution of the thicknesses of the single layers the filter consists of, varies in a band around the periodic value. The distribution of the thickness of the single layers for an edge filter is shown in Figure 6.15. It has to be said that the blue shift, described by Braggs’ equation, is the regular case for thin film assemblies. However, combination of thin films exist that show a characteristic that differs considerably from Bragg. In fact, strategies exist, to use certain assemblies to create such differing characteristics. It is therefore not straightforward, that a band stop filter obeys the Bragg condition.
137
Angularly selective photonic crystals
layer thickness [nm]
180
160
140
120
100
n = 1.5 n=2
80
Figure 6.15: Distribution of the thicknesses of the single layers for a band stop filter consisting of two materials with the refractive indices n1 = 1.5 and n2 = 2.The red lines give the thickness the layers would have in an ordinary Bragg reflector. The edge filter was developed starting with such a Bragg filter and using an evolutionary algorithm. The figure shows clearly that the distribution around these lines only occurs in a certain range so that the resulting system is still close to a periodic system. The angular-dependent reflection characteristic of the edge filter with the corresponding layer thicknesses was simulated using the approach of characteristic matrices (see section 3.1). The results of this simulation are shown in Figure 6.16a. Clearly observable is that the angular characteristic still follows Braggs’ equation even though the filter is non-periodic. The parameters used for the fit were a refractive index n = 1.75 and a period of Λ = 237.5nm. A band stop filter with a similar characteristic was purchased commercially from mso Jena. The refractive index profile of this filter is not known. Neither was information available about the used materials, refractive indices, dispersion relation or absorption. The refractive index profile of the purchased filter is therefore probably different from the one obtained from the simulation. The angular-dependent reflection characteristic of this filter was measured using a gonio-reflectometer and a Fourier spectrometer [Kem09]. Only angles smaller than θ = 75° were considered for the gonio-reflectometer measurements because data for higher angles could not be recorded dependably 22. The results of this measurement are shown in Figure 6.16b. Also shown is the blue shift predicted by Braggs’ equation using the same parameters that were used for the simulated filter. Braggs’ equation describes the reflection characteristic with constraints. For angles smaller than θ = 30° the characteristics of simulated and real filter are in accordance and are well described by Braggs’ equation. For angles greater than θ = 30°, the characteristics of the simulated band stop filter still is well described by Braggs’ equation. The angular characteristic of the real filter splits and is better described by two overlapping characteristics (blue edge and red edge). Yet, this particularity has not been observed in more detail but is probably caused by non-periodicity. Also observable are a suppression of reflections for wavelengths smaller than the ones in the peak region and a high reflection of wavelengths in the spectral range above the peak.
The projected area of the sample on the detector decreases fast (for θ = 75° it is ca. 0.25 of the real sample area) and the reflection increases rapidly due to Fresnel effects. The measurement of the gonio reflectometer results in a reflectance close to one for all angles above θ = 75° which can be seen in Figure 6.14. In that measurement all angles up to θ = 90° where considered.
22
138
6.3 Angularly selective photonic structures
b)
a)
Figure 6.16: Simulated (a) and measured (b) angular-dependent reflection characteristic of a band stop filter with a reflection edge at λ=760nm. The filter shows the expected blue shift and the desired angular-dependent reflection for wavelengths below 760nm. The simulated filter consists of two materials with n1 = 1.5 and n2 = 2.0 and was optimized using an evolutionary algorithm. The real filter was purchased from mso Jena. Also shown is the blue shift of the reflection peak predicted with Braggs’ equation. The used parameters were n = 1.75 and Λ = 237.5nm.
6.3.2 2D crystals In this subsection the example of the checkerboard structure is investigated. This example is used to illustrate the peculiarities of 2D structures and to demonstrate that also periodic structures with a very strict periodicity are not necessarily described by Braggs’ equation. This is important when aiming towards angularly selective optical elements with characteristics different from Bragg. The checkerboard structure The checkerboard structure was introduced in subsection 6.2.4 as an exemplary structure for which the Bragg description cannot be valid. To simulate the angular-dependent reflection characteristic of the checkerboard structure, the RCWA method was used (see section 3.1). The results of this simulation are shown in Figure 6.17 It is important to note that for 2D structures, the characteristics for TE and TM polarization are decoupled and may therefore be different. Especially for TE polarization a red shift of the reflection peak is observable that cannot be explained with Braggs’ equation (Bragg always results in a blue shift). An idea how the angular dependent characteristics of these structures look like can be obtained from the band structure (see section 2.2). The shift of a certain band towards lower or higher frequencies is shown there. Care needs to be taken when converting the crystallographic directions into angles of incidence, especially because the band structures contains more directions than the polar – wavelength plot. The band structure plot does not show how distinct a certain reflection is and how the characteristic is outside the band gaps.
139
Angularly selective photonic crystals
b) TM polarization
a) TE polarization
Figure 6.17: Angular-dependent reflection characteristic of the checkerboard structure with a period of Λ = 1 µm. The structure was described in Figure 2.10. The parameters used were n1 = 1 and n2 = 2. The characteristics of this crystal are different for TE- and TM polarization. The angular dependant characteristic cannot be described with Braggs’ equation.
6.3.3 3D crystals From the 3D crystals, I chose the opal as an exemplary system. This example was chosen because it illustrates how the Bragg effect influences the reflection characteristics of a 3D photonic structure. Also it illustrates the limitations of the Bragg description. As opals were realized for different potential application during this work, experiments were performed which made it possible to compare simulations and experiments. Being a 3D crystal, for the opal the polar as well as the azimuth dependence of the reflection characteristics need to be considered. I will show, however, that the azimuthal dependence for this particular crystal plays a minor role. The opal The angular-dependent reflection characteristics of an opal depend on the wavelength, the polar- and the azimuth angle. The parameter space is therefore larger than for a 1D crystal and the illustration of the characteristics needs to be displayed in a set of plots rather than in a single one. I start with a discussion of the dependence of the reflection characteristic on the polar angle and the wavelength for a fixed azimuth angle. Here I also show in how far the Bragg description is applicable to an opal. I continue with a discussion of the reflection characteristics in dependence of the polar and the azimuth angles for a fixed wavelength. The azimuth dependence turns out to be comparatively weak for this particular crystal, so that the general picture of the angular-dependent characteristics of the opal is already derived from the variation of the polar angle. An opal consists of a discrete number of stacked sphere layers. These layers form an equidistant lattice with the distance d111 23, which describes the distance of the planes in which the centers of the spheres are located considering L direction, and is given by
23
111 here indicate the Miller indices. The L direction is identical to the 111 –direction where are the Miller indices.
140
6.3 Angularly selective photonic structures d111 =
2 ⋅r 3
(6.35)
In this equation r is the radius of the spheres. For normal incidence, the distance d111 is the lattice distance valid for the Bragg description. The approach is now to maintain this lattice distance and adopt the angular dependence from equation (6.34). This description is valid is long as d111 can be considered as the valid lattice distance. The angular dependence assumed for the opal is consequently given by
λ0 (α ) =
8 ⋅ r ⋅ (n ² − sin ²α ) 3
(6.36)
In this equation n is the average refractive index of the opal [Gaj05]. This average refractive index is lower than that of the 1D crystal, the reason being the opal consists partly of air. As the spheres in the opal are closely packed, they have a volume-filling fraction of 74%. Assuming a refractive index of n = 1.49 for the spheres, an effective refractive index of n = 1.36 is obtained. From Figure 6.12, it is known that this low average refractive index results in a strong pronouncement of the blue shift (wavelength change between θ = 0° and θ = 90° of ca. 0.7). To simulate the angular-dependent reflection characteristics of an opal, the RCWA method was used. To keep the results close to the measurements, the same parameters for the simulation were used as for the produced opals. In the given example these parameters were: sphere diameter D = 255nm, refractive index of the spheres n = 1.49 (corresponding to PMMA). The results of the simulated reflection characteristic are given in Figure 6.18a. The opal samples were fabricated at the university of Mainz. The angular-dependent reflection characteristics of these crystals were measured using a gonio-reflectometer and a Fourier spectrometer. Again, only angles smaller than θ = 75° were considered. The results of these measurements are given in Figure 6.18b. Also shown is the blue shift predicted with Braggs’ equation for both cases Simulation and measurement are in good accordance. The differences between simulation and measurement are due to disorder in the fabricated crystal. This disorder results in a narrowing of the reflection peak and a suppression of sidelobes. a) b)
Figure 6.18: Simulated a) and measured b) angular-dependent reflection characteristic of an opal (opal produced by Lorenz Steidl, University of Mainz). The sphere diameter was d=255nm and the spheres were made of PMMA, for which a refractive index of n = 1.49 was assumed in the simulation. Also shown is the blue shift predicted with Braggs’ equation (6.36) with d111 = 208.1nm and n = 1.36. This low average refractive index is the reason for the comparably strong blue shift. The black line marks the cross section for which the dependence on the azimuth angle was calculated. The black line in Figure 6.18b corresponds to a wavelength of λ = 540nm and marks the cross section that is also shown in Figure 6.20 For the calculation of the path length enhancement, I assumed that the reflection peak is so wide that once a certain wavelength at a certain angle has crossed the edge into the area of
141
Angularly selective photonic crystals
high reflectance it stays there for all greater angles. This is no longer true for the opal. In fact, looking at the cross section at λ = 540nm in Figure 6.18b, it is clear that for the opal angular selectivity occurs only between ca. θ ≈ 25° and θ ≈ 40°. Still this characteristic results in a certain path length enhancement because some light is reflected by the filter. The exact value in this example depends on the geometry of the PV system. I will not go into further detail here. The small width of the reflection peak is a serious limitation for the use of opals as angularly selective filters, and a further examination should concentrate on crystals with a broader reflection peak as are e.g. opals with a higher index contrast or the diamond structure. Looking at the Bragg fit in Figure 6.18, it yields a good description of the characteristics of the reflection peak for small angles of incidence. Yet for the opal, changing the direction of incidence means changing the relative orientation of incident light and crystal and therefore moving towards another point of high symmetry on the Brillouin zone. Reaching another symmetry point, another lattice distance is relevant and the Bragg fit becomes incorrect 24. In the opal, the next point of high symmetry is the U point, which, relative to the L point, appears under an angle of θ = 54.7°. The Brillouin zone now describes only the geometry of the crystal orientation, for the real opal effects of finite thickness also need to be considered. As the L direction is the growth direction of the opal, i.e. the lattice planes have a large expansion, it is much more dominant than the U direction. To investigate the area of validity of the Bragg description, the result of the corresponding fit has been compared to the predictions for the form of the reflection peak from the band structure and from the RCWA simulation. The result of this consideration is shown in Figure 6.19. 600
wavelength [nm]
580 560 540 520 500
gap band structure gap RCWA gap position (Bragg theory)
480 460 440 0
10
20
30
40
50
60
angle [deg]
Figure 6.19: Comparison of the calculation of the angular dependence of the stop gap between the 2nd and the 3rd band for different methods. Equation (6.36) was used to calculate the gap position (green triangles). Figure 6.19 shows that for angles smaller than θ < 40°, all three descriptions are in good accordance but for greater angles differences occur. The RCWA describes the reflection of the opal in the peak region with high accuracy as can be seen in comparison to the measurements. The difference of RCWA simulation and band structure calculation can be explained with the finite thickness of the first and the infinite thickness of the latter. The differences between Bragg theory and RCWA can be explained by the limitations of the Bragg theory for the appearance of another lattice distance. From the comparison of the methods, the area of validity of the Bragg equation can be assumed to be until ca. θ = 50°. Finally I investigated the dependence on the azimuth angle. For this purpose, the reflection characteristics were simulated in dependence of polar and azimuth angle for a fixed 24
A possibility to proceed here would be to calculate the effective lattice distance under the assumption that the Bragg description is valid also for higher angle and compare the results to the lattice distances for the different orientations. I haven’t tried that but it could be an interesting option for future considerations.
142
6.4 Experimental results wavelength. As an example for such a simulation, the characteristic for a wavelength of λ = 540nm is shown in Figure 6.20. In the figure, the r-component corresponds to the polar angle θ and the ϕ - component to the azimuth angle δ. A cross section at an azimuthal component of δ =0° in Figure 6.20 (black line) corresponds to a cross section at λ = 540nm in Figure 6.18b (black line). The figure shows the angular selectivity (red, resp. orange circle) and also the sixfold symmetry of the crystal. However, the major point which I meant to illustrate with this figure is that the azimuthal dependence on the reflection characteristic is considerably weak. This has also been found also for simulations at other wavelengths which are not displayed here.
Figure 6.20: Simulated reflection depending on the polar and the azimuth angle for the opal described in Figure 6.18 given separately for both polarizations. The polar angle was varied between θ = 0° and θ = 90°, the azimuth angle between δ = 0° and δ = 360°. The chosen wavelength was λ = 540nm. An azimuth angle of 0° (black line) corresponds to the situation in Figure 6.18.The general picture is the same for both polarizations. The reflection peak appears as a circle..
6.4 Experimental results In this section I will present measurements that show how the light trapping effect discussed in subsection 6.2.1 is used to enhance the absorption and also the quantum efficiency of a PV converter. For this purpose, different photonic structures were applied to different absorbers and a thin film solar cell on a roughened superstrate. To demonstrate the effect on the absorption, reflection- and transmission measurements were performed. The effect on the quantum efficiency was measured directly for a solar cell. It was possible to show indications of an increased absorption for all used photonic structures and also an increase in quantum efficiency for the used solar cell was demonstrated.
6.4.1 Absorption enhancement The absorption enhancement was investigated using reflection and transmission measurements using a Varian Cary 500 spectrometer. The system that was created always consisted of three components: 1. The angularly selective photonic structure 2. The absorber (a thin layer of an absorbing material or a solar cell) 3. A scattering device (separate or already included in the absorber) From all of these components, measurements of the reflection and, in some cases, the transmission were performed. This allowed for the obtainment of absorption of the single components by assuming
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Angularly selective photonic crystals
A = 1−T − R
(6.37)
The procedure to obtain the difference in the absorption, with and without the angularly selective photonic structure, now differs for the different photonic structures used and the procedure applied will be mentioned separately. The biggest problem that occurred for the measurement of the absorption enhancement for opal and rugate filters was the presence of multiple reflections. For this reason a system was realized in which absorption and scattering were already combined. This system was used to investigate the induced path length enhancement of the band stop filter. Opal To measure the effect of the opal on the absorption, a special absorber system consisting of a thin layer of non-stoichiometric SiC deposited on a glass substrate was used. This material is very well known at Fraunhofer ISE and can be tuned to have a band edge at a desired wavelength. To add the scattering mechanism, a diffuse backside reflector of Barium Sulphate (BaS) on aluminium was added to the system. An illustration of this setup is shown in Figure 6.21.
Figure 6.21: Setup for the measurement of the absorption enhancement in a thin absorber deposited on a glass substrate with an angularly selective filter applied to it. The light (black arrow) passes the filter, the absorber on a substrate, and is scattered by a scattering back side reflector. A fraction of this light (red arrow, path I) leaves the system again through the angular acceptance range θc of the filter. Another fraction is reflected by the filter and impinges on the absorber again, thus increasing the absorbed fraction (red arrow, path II). The aim was to compare the absorption in the absorbing material for a system with and without the angularly selective filter. In a first step, therefore, reflection and transmission of the single components were measured and the absorption calculated. In a second step, the absorption within the absorbing layer was calculated for the system absorber – scattering device. Surface reflection and multiple reflections were taken into account here.
144
6.4 Experimental results 1 − Rsys1 − Tabs Aabs1 =
∞ ⎞ ⎛ ⋅ Aref ⋅ ⎜1 + ∑ ( Rref ⋅ Rabs ) 2 n ⎟ ⎠ ⎝ n =1 1 − Rabs
1 − Rsys1 − Tabs ⋅ Aref ⋅
= (6.38)
1 1 − ( Rref ⋅ Rabs ) 2
1 − Rabs In this equation Rsys1 denotes the reflection of the system absorber – scattering device. Tabs and Rabs denote the transmission and the reflection of the absorber. Rref and Aref denote the reflection and the absorption of the scattering device. In a third step, the absorption in the absorbing layer for the system of angularly selective filter – absorber – scattering device was calculated. Again multiple reflections were considered.
Aabs 2 =
1 − Rsys − f ( Aref ) − f ( A fil ) ∞ ⎛ ⎛ ⎞⎞ 2 1 − ⎜⎜ ( R fil + A fil ) + T fil ⋅ Rabs ⋅ ⎜1 + ∑ ( Rabs ⋅ R fil ) 2 n ⎟ ⎟⎟ ⎝ n =1 ⎠⎠ ⎝
=
(6.39)
1 − Rsys − f ( Aref ) − f ( A fil ) ⎛ ⎞ 1 2 ⎟ 1 − ⎜ ( R fil + A fil ) + T fil ⋅ Rabs ⋅ 2 ⎜ ⎟ − ⋅ 1 ( R R ) abs fil ⎝ ⎠ In this equation Rfil, Tfil and Afil denote reflection, transmission and absorption of the filter. In this approach it is assumed that the total absorptions in the angularly selective filter f ( A fil ) and in the scattering device f ( Aref ) , each with regard to multiple light passages, are small so that they can be treated as corrections to the absorption in the absorber. This approach is confirmed by the measurements of the components. However, a noticeable error occurs if they are not taken into account. The absorption in both components is obtained by summing up the absorption occurring every time, light passes the element. Because of the numerous possible paths this results in complicated sums that fortunately converge very fast 2
2
f ( Aref ) = T fil ⋅ Tabs ⋅ Aref ⋅ (1 + Rref ⋅ ( Rabs + Tabs ⋅ R fil + Rabs ⋅ Rref + ...))
(6.40)
and 2
f ( A fil ) = A fil + T fil ⋅ Tabs ⋅ Rref ⋅ A fil ⋅ (1 + Rabs ⋅ R fil + ...) The result of these calculations is given in Figure 6.22.
145
(6.41)
Angularly selective photonic crystals
absorption [%]
50
80
opal reflectance abs. absorber only abs. absprber + opal
70
40
60
30
50
20
40
10
30
0 450
reflectance [%]
60
20
475
500
525
550
wavelength [nm]
Figure 6.22: In this figure the absorption in a thin absorber layer, once in a system of absorber – scattering device (black squares) and once for the systems opal - absorber – scattering device (green squares) is shown. Also given is the reflection of the opal for normal incidence (red dots). The blue shift of the opal reflectance (not shown) induces a path length enhancement which results in an increased absorption. (opal produced by L. Steidl, University of Mainz). The presented calculation to obtain the absorption in the absorber is quite complicated and therefore susceptible to errors. A good indicator for the correctness of the approach is to compare the calculated absorption for wavelengths of ca. λ = 450nm. In this wavelength range the filter induces no more angular confinement and the absorption in the absorber should be the same with and without filter. The figure shows that the calculated absorption in this region for both cases is very close, which confirms the approach. Close to the reflection peak of the filter, angular selectivity is expected because of the blue shift, of the opal reflection and the calculation results in an absorption which is up to 30% relative higher with the opal than without. This result is a strong indication that the predicted path length enhancement really occurs. Rugate Filter To obtain the absorption enhancement induced by a rugate filter, the same method was applied as described in the previous part for the opal. The setup for the measurement was similar, though another absorber was used. Already from the measurement of the systems, without actually calculating the absorption in the absorber, an enhancement was plainly visible. To calculate the absorption in the absorber, again equations (6.38) and (6.39) were applied. The corrections from the correction terms (equations (6.40) and (6.41)) were much smaller for the rugate filter than they were for the opal, which is an indication that the errors in the result for the rugate filter are smaller. The obtained absorption enhancement is shown in Figure 6.23. In this example, the effect on the absorption is distinct. For wavelengths below and close to the reflection peak, the absorption in the absorber was more than doubled by the application of the filter 25 which is a strong indication towards the expected path length enhancement. Again the approach must be tested by comparing the calculated absorption for the systems with and without filter for low wavelengths. The difference in this case was below 5% and the accordance therefore smaller than for the opal.
25 As the filter reflects all light in the reflection peak, nearly no light is absorbed there and the calculation yields no dependable results. The concept is only sensible if the induced losses by the filter reflection for normal incidence are negligible.
146
6.4 Experimental results
100
absorption [%]
80
80
60 40
60
abs. absorber only abs. absorber + filter reflectance rugate filter
40
20 0 450
reflectance [%]
100
20
500
550
600
0
wavelength [nm]
Figure 6.23: Absorption in an absorber, once in a system of absorber – scattering device (black squares) and once for the systems rugate filter - absorber – scattering device (green squares). Also given is the reflection of the rugate filter for normal incidence (red dots). The blue shift of the rugate filter reflectance (not shown) induces a path length enhancement which causes an increased absorption. (rugate filter produced at Fraunhofer IST). Band Stop filter To examine the absorption enhancement, a band stop filter with the characteristic shown in Figure 6.16b was used. The filter was designed specially for application on an amorphous silicon thin film solar cell and had a reflection peak that started at λ = 760nm for normal incidence. The used solar cell was a solar cell of amorphous silicon on a roughened glass substrate [Bec03], specially fabricated for the given purpose at Forschungszentrum Jülich. Because of the rough substrate, the light is diffused directly by solar cell and substrate. Additionally, an aluminium reflector was deposited on the backside of the solar cell to increase the backside reflection. The region of low absorption for this solar cell starts at λ ≈ 750nm (see Figure 6.4b). As solar cell and mechanism of diffusion are realized in a single system, the procedure described in the previous parts cannot be applied to the band stop filter. To obtain the absorption in the solar cell, it was assumed that no absorption other than the one inside the solar cell occurs. Under this assumption, the absorption is easily obtained by reflection and transmission measurements. The procedure here was as described in the following: the spectral reflectance Rcell and transmission Tcell of the solar cell system (solar cell on substrate + aluminium reflector) alone was measured. As the solar cell system showed no transmission at all in the spectral region of interest, the reflectance is a direct measure of the amount of light that is absorbed in the solar cell system Acell
Acell = 1 − Rcell
(6.42)
The results of this measurement are shown in Figure 6.24. Subsequently, the reflection of the solar cell system with the band stop filter placed on top of the solar cell system was measured. The filter had been deposited on BA7 glass substrates. Independently from the characteristics of the filters, this causes additional reflection losses from the air-glass interface. In this example, this reflection was not corrected by a calculation but compared to a measurement with a covering glass placed on top of the solar cell. This procedure shows directly the effects of the additional reflection. Also the covering glass accommodates the fact that in an outdoor application the filter may be deposited directly on the covering glass of a module. All of these systems showed no transmission, so the reflection is again a direct measure for the absorption in the entire system. As the absorption of filter and covering glass are
147
Angularly selective photonic crystals
100
100
90
75
80
50
70 60 550
reflection filter abs. cell only abs. cell + filter abs. cell + covering glass 600
650
700
25
750
reflection [%]
absorption [%]
negligible in the spectral region of interest, the obtained absorption is that of the solar cell. The results of these measurements are also shown in Figure 6.24.
0
wavelength [nm]
Figure 6.24: Obtained absorption for the solar cell system only (black squares), the solar cell systems with the photonic filter placed on top (green triangles) and with the covering glass (blue triangles). Additionally shown is the reflection of the filter (red dots, filter produced by mso Jena). In Figure 6.24 no corrections were performed on the calculation and the absorption enhancement is directly obtained from the reflection measurement. Therefore Figure 6.24 is a clear demonstration of an increased absorption due to angular confinement. Additionally, the figure shows that the absorption in the low wavelength regime is only minorly lowered compared to the solar cell alone. This is because of an antireflection effect resulting in low reflection for the filter system (i.e. the filter was deposited on one side of the glass and an antireflection coating was deposited on the other side). The covering glass lowers the performance of the solar cell for smaller wavelengths because of reflection losses. For the solar cell, measurements of the quantum efficiency were also performed. To be able to compare the absorption enhancement with the results from the measurement of the quantum efficiency, the relative increase in absorption was also calculated. The results of this consideration are shown in Figure 6.25.
abs1 / abs2 [%]
130
abs1 = cell + filter
120
abs2 = cell only
110
abs2 = cell only
+25%
abs1 = cell + covering glass
100 90 80 550
600
650
700
750
wavelength [nm] Figure 6.25: The relative change in absorption. The relative absorptions are shown for a system with filter relative to the solar cell only (red dots) and a system with covering glass (blue squares). The relative absorption is increased up to 25%.
148
6.4 Experimental results 6.4.2 Enhancement of the quantum efficiency In a second set of measurements, the external quantum efficiency (EQE) was measured for the same systems. The results of these measurements are shown in Figure 6.26 This figure shows the external quantum efficiencies for the systems consisting of solar cell only, solar cell + filter, and solar cell + covering glass. Also given is the reflection of the filter. Because of reflection losses, the system with covering glass shows the lowest EQE for all wavelengths. The filter shows less reflection losses and the EQE for the system with filter is higher than that for the system with covering glass for all wavelengths. For wavelengths between 640nm < λ < 740nm, the EQE for the system with filter is even higher than that for the solar cell only. This corresponds to the results of the absorption measurements and also to the result of equation (6.36). It must be emphasized that the filter still produces reflection losses in this spectral region. However, the increased quantum efficiency shows that the positive effects of the photonic light trap overcome the negative ones. The ratios of the EQEs between the systems of solar cell + filter / solar cell only and solar cell + covering glass / solar cell only are given in Figure 6.26b. The form of the curves is similar to those for the absorption enhancement in Figure 6.25, and the maximum value of an enhancement of 25% is also similar. This shows that the absorption in a thin solar cell of amorphous silicon is increased by up to and exceeding 25% by the photonic light trap, and that this increased absorption is transferred to an increase in quantum efficiency. b) 100
60
75
40
50
20
0 550
cell only cell + filter2 cell + cover glass reflection filter 2 600
650
25
700
750
130
EQE1/EQE2 [%]
80
reflection [%]
EQE [%]
a)
120 110
EQE1 = cell + filter 2
+25%
EQE2= cell only EQE1 = cell + cover glass EQE2 = cell only
100
0
90 550
wavelength [nm]
600
650
700
750
wavelength [nm]
Figure 6.26: Figure 6.26a shows the measured quantum efficiency for the solar cell system only (black squares), the solar cell system with one of the filters (green triangles) and the solar cell system with a covering glass (blue triangles). Also given is the reflection of the filter (red dots). For wavelengths greater than λ = 650nm, the system with filter shows a quantum efficiency that is higher than for the system without, even though the filter produces reflection losses here. In Figure 6.26b, the ratio between the quantum efficiencies between the system with filter and without filter (red dots) and between the system with filter and with cover glass are shown (blue triangles). The quantum efficiency is increased up to 25%. One interesting point about Figure 6.26b is that for wavelengths greater than λ = 700nm, the system solar cell + covering glass shows a higher quantum efficiency than the solar cell only. A similar characteristic may also be observed in Figure 6.25 for the absorption, though in an attenuated form. The increased absorption and quantum efficiency in this spectral range are caused by Fresnel reflections that also show a certain angular dependence. It must be said that, looking at the whole spectrum, the solar cell alone showed the best performance when compared to a system of solar cell + filter or solar cell + covering glass. This is because a filter or covering glass induces reflection losses. These losses affect all wavelengths and are currently not overcompensated by the gain produced by the angularly
149
Angularly selective photonic crystals
selective filter. However, the performance of the system consisting of solar cell + filter was improved when compared to the system of solar cell + covering glass solar cell + covering glass
solar cell + covering glass weighted with AM 1.5
compared to solar cell + covering glass
100.0%
100.0%
compared to solar cell + filter
104.4%
105.5%
compared to solar cell 107.9% 108.0% only Table 6.1: Comparison of the integrated quantum efficiencies for the systems solar cell, solar cell + filter and solar cell + covering glass.. Band stop filter and covering glass induce reflection losses compared to the case of the solar cell alone. However, assuming that a covering glass is always needed in a module and comparing the band stop filter to that case, the filter induces a positive effect. In the first column, the relative efficiency is given with respect to the EQE directly. In the second column, the values are weighted with the AM1.5 spectrum 26.
6.5 Summary of the chapter & perspective In this chapter I have investigated concepts which use angularly selective photonic structures for PV applications. In the investigation I have concentrated on two aspects of such concepts. One aspect is that angular confinement can be used to create a light trapping effect. The other aspect concerns the suppression of radiative recombination which increases the theoretical efficiency limit. In section 6.1, the concept of angular selectivity was introduced including a definition for angularly selective optical elements. Concepts were introduced and discussed that use angular confinement in PV device. A refined discussion for one concept was given in which light trapping induced by angular confinement was realized. Special attention has been paid on mechanisms that cause angular spreading of the radiation coming from the solar cell. A large angular spreading is a necessary condition for the concept. Another necessary condition is tracking. In section 6.2 considerations of principles were given. First the light trapping effect was investigated. The path length enhancement induced by an angularly selective optical element is given by a factor sin-2(θc), θc defining the angular acceptance range of the photonic structure. The theoretically maximum possible path length enhancement is given for a confinement to the solid angle of the solar disc. Under that condition the path length enhancement is given by a factor of L = 46200 Considering TIR, the path length is enhanced by an additional 4n², with n the refractive index of the PV converter which results in a maximum possible enhancement of L = 2.23E6 for a silicon solar cell with n = 3.5. Angular confinement reduces the entropy generation due to a change of the angular spreading. Reduced entropy generation results in an increased efficiency limit. This effect is related to concentration; both, angular confinement and concentration change the ratio of the solid angles for incident and emitted radiation of a PV device. For both the same efficiency limit holds which, for silicon solar cells, is about 41% efficiency. One effect generating angular selectivity is the Bragg effect, which has been investigated and
26 An objection that cannot be dismissed, when looking at these numbers is that the comparison between uncoated covering glass and the filter system is not entirely fair, because of the antireflection coating of the filter. However, a comparing system had to be chosen, and a first, unsophisticated choice was the simple glass slab. Another possible choice would have been a glass with an antireflection coating. However, also the comparison with this system is not perfect.
150
6.5 Summary of the chapter & perspective evaluated for different photonic structures. Alternative effects to create angular selectivity were briefly discussed. In section 6.3 different exemplary photonic structures are investigated for their angularly dependent reflection characteristics. Exemplary 1D photonic structures are the rugate filter and the band stop filter which were investigated theoretically and experimentally. An exemplary 2D photonic structure is the checkerboard structure. This structure was investigated only theoretically. The opal is a 3D photonic structure which was investigated theoretically and experimentally In section 6.4, experimental results concerning the path length enhancement induced by an angularly selective optical element are presented. For all investigated photonic structures, an increased absorption was obtained in different absorber systems in the expected spectral range. In case of the rugate filter, the absorption was more than doubled in a thin absorbing layer. For the band stop filter the absorption in the thin film solar cell was increased by up to 25%, in the investigated spectral region. The increased absorption induced an increase in quantum efficiency of also 25%. This confirms experimentally that angular confinement can be used to increase the absorption and the efficiency of a PV converter. From the EQE the absolute efficiency of the system was calculated. This calculation resulted in an increase of efficiency of 5.5% when compared to a solar cell with a covering glass, At this point I want to give an evaluation of the results presented in this chapter. A positive aspect about angularly selective optical elements is their high potential to improve PV systems. I know of no concept with a higher possible path length enhancement. Additionally, the Shockley Queisser limit is influenced in the same way as with concentration. Both aspects make angular confinement highly interesting for photon management. A positive aspect of the theoretical considerations was that the close relation between angular confinement and concentration could be shown. This is best visible in Figure 6.8 where the dependence of the Shockley Queisser limit on the solid angle for incident and emitted radiation are plotted. A positive aspect about the experiments was the proof of an increased absorption and an increased quantum efficiency induced by angularly selective filters. This is a good result especially when considering that the angular confinement of the used filter was not very effective. Encouraging was that with the Bragg effect one effect creating angular confinement was identified and put into action. The spectral range of the angular confinement achieved with this effect is limited narrowly, though and the obtained path length enhancement is mostly not very high. Other effects to achieve angular confinement have been investigated but none of the corresponding concepts could have been realized. The lack of a promising concept to create an efficient angular confinement is a negative point. Another negative point concerns the realization of the applied band stop filter that yielded the best results. This filter is a highly complex device and too expensive for a commercial application. At this point, it is proven that angularly selective concepts can be realized to improve the PV efficiency but other filters have to be found for an application. The main future task concerning angularly selective filters therefore will be the development of filters with a more pronounced angular-dependent characteristic. I have already tried to outline strategies for this task. On the one hand an investigation of more complex 3D photonic structures is possible. If the relevant lattice planes change fast with the direction of incidence, the reflection characteristic may not be governed by the Bragg effect anymore. A promising structure here is e.g. the diamond structure or similar structures. On the other hand, aperiodic structures seem promising. Techniques like the needle optimization method could be used to design 1D structures with an optimized angular selectivity. This method could especial used to increase the spectral range over which angular selectivity occurs. As a perspective for angularly selective photonic structures I want to briefly discuss two concepts that have been developed at the end of this work and have therefore not yet been
151
Angularly selective photonic crystals
realized. The first concept is very simple and can be outlined in a few sentences. As was shown in the experiment, already Fresnel reflections at the glass surface have a certain angularly selective effect. An even stronger effect can be expected by antireflection coatings. The idea for a multilayer antireflection coating on a solar cell or on a covering glass is therefore to not only optimize the coating for the antireflection effect but also for the reflection, so that a certain reflection is created above the band edge of the material, which is shifted to lower wavelengths for increasing angles. The second concept is an integration of angularly selective filters into concentrating devices. as a secondary concentrator 27. The idea of this concept, the combination of angular selectivity and concentration, is sketched in Figure 6.27. This concept has several advantages. On the one hand, concentrating devices would need to get to very high concentrations to actually increase the angular spreading to angles close to Ω = 90°. Such high concentrations are not practical at the present. The angular spreading of a concentration in the range of 1000fold is theoretically ca. Ω = 10°. On the other hand, it will be very hard to realize angularly selective systems with very small angular acceptance ranges. Therefore each element is active in the angular area that cannot be served by each other element; i.e. the elements are complementary. If the angular acceptance range of the selective filter and the spreading on the light impinging on the solar cell (in the figure this would be the light after the lens) could be matched, the system would not increase the étendue and therefore entropy production would be reduced. Another advantage is that the angularly selective element could be integrated in an existing concentrating device. Here even expensive filters could be used, as they only need to be active on the small area of the solar cell on which the light is concentrated. Apart from the thermodynamical improvement, also still the light trapping would also still occur for such a system.
Figure 6.27: Schematic sketch of the combination of angular selectivity and concentration. The concentrating device (here: lens) concentrates the light so that it changes its angular spreading from θS = 4.7mrad to a higher value θL> θS. The angularly selective element on the other hand restricts the emission of the PV system to exactly the same cone as the incident light θE = θL.
27
Note that the angularly selective structure will not induce an area concentration but an intensity concentration.
152
7 Summary & perspective The thesis so far… The goal of this work was to explore the possibilities of using photonic concepts for PV applications. These concepts aim towards an improved utilization of solar photons by either increasing the amount of absorbed photons or reducing the amount of those photons leaving the solar cell. To accomplish these functions, which can be subsumed into the term “light trapping”, different photonic structures have been investigated. These structures are united by a typical length scale. The typical structure size is comparable to the wavelength of the considered light. A consequence of this comparability of scales is that the effects induced by these structures are based on interference. In this work, I have concentrated on three such effects: The first is spectral selectivity, the second is diffraction and the third effect is angular selectivity. To investigate the presented concepts simulations and experiments have been performed.
Simulation method… To perform simulations of photonic structures, and especially more dimensional photonic structures, one important tool was the rigorous coupled wave analysis (RCWA). An implementation of this method had been purchased and one task of this work was to check in how far this implementation was applicable to the photonic structures under consideration. For this reason I have performed several convergence considerations and comparisons to other simulation methods and to experimental data. The performed considerations resulted in the evaluation that the RCWA is a convenient method for the simulation of dielectric structures with periods in the sub-µm regime. It is versatile in the sense that almost arbitrary shapes can be considered and it is capable of simulating the optical near field. Results are obtained in a reasonable time for a reasonable choice of parameters. Among these parameters, the most important ones are those that define the approximation of the structure shape and the number of Fourier components. These parameters have to be chosen with great care. The method becomes very slow if many Fourier coefficients need to be considered. This is the case for structures with large periods (several µm), high refractive indices or considerable absorption. The RCWA yields results that are in accordance with those provided by other simulation methods and by experiments. Especially in the region of photonic band gaps the accordance is very good. Outside this region, even small variations have a strong influence on the results for all considered simulation methods. Comparisons between simulations and experiments need to consider this.
Spectral selectivity… Spectrally selective photonic structures are used in concepts in which the solar spectrum is divided into several parts. One concept featuring spectral selectivity is the fluorescent concentrator. The considered spectra here are defined by absorption and emission of a dye molecule. As absorption and emission are mostly separated due to the Stokes shift, they may be considered separately. The fluorescent concentrator transports light to the edges via TIR. This transport mechanism loses at least 26% of the internal radiation through the escape cone. The advanced fluorescent concentrator concept envisages regaining escape cone losses by the application of a spectrally selective photonic structure. To evaluate the application of spectrally selective photonic structures on fluorescent concentrators I have developed a theoretical model allowing for the formulation of minimum requirements of a filter and for the prediction of the effect a certain filter has on the light guiding ability of the concentrator. Subsequently I have theoretically designed and
153
Summary & perspective
optimized different photonic structures for the application on different fluorescent concentrators. These filters were the rugate filter, the band stop filter, and the opal. From all of these filters, samples that were realized following the specifications of the optimization were obtained. The filters were evaluated using the theoretical model and also experimentally. The best result was obtained with a band stop filter. With this filter an increase of 20% in light guiding efficiency was predicted and an increase of 19% in solar cell efficiency was measured, corresponding to almost ¾ of the escape cone losses. This result was refined by measuring the spectral (EQE measurement) and the lateral (LBIC measurement) characteristics of the same fluorescent concentrator system. The results of these measurements are shown in Figure 7.1. b)
a)
100 80
with Filter without Filter
60 10 40 5
500
400
20 Reflection Filter
0 300
without photonic structure with photonic structure
600
signal [a.u.]
15
BA241, white bottom reflector
reflection [%]
EQE [%]
20
400
500
300 0 600
0
wavelength [nm]
2
4
6
8
10
distance from solar cell [cm]
Figure 7.1: Effect of a band stop filter on a fluorescent concentrator. Figure 7.1a shows the spectrally resolved efficiency for a fluorescent concentrator with and without the filter. Figure 7.1b shows the laterally resolved collection efficiency of the same system. Altogether, the efficiency of the system was increased by 19%. Another concept concerning spectral selectivity is the spectrum splitting concept. Here the solar spectrum is divided into several parts that are each guided to a solar cell which generates the maximum benefit of this part of spectrum. Spectrally selective filters here undertake the task of dividing and guiding the solar spectrum. One concept that was realized using such filters reached an efficiency of η = 34.1% under outdoor illumination.
Diffraction… Diffractive structures are used to induce a specific change of the direction of radiation inside a solar cell. The aim of inducing such a directional change is to enhance the pathlength of light in the solar cell and consequently increase absorption and quantum efficiency. In this work the application of back side gratings in textured and untextured crystalline silicon solar cells was specifically investigated. Considering a textured front surface gives rise to several specific challenges that were subsumed under the term “alignment problem”. In a first theoretical consideration, the maximum possible path length enhancement of a structure inducing a specific directional change was calculated considering conservation of étendue. This consideration resulted in a value of L = 853n, with n being the refractive index of the solar cell. For silicon, this consideration results in a maximum value of ca. L = 3000. Using gratings to induce the directional change, the dependence on wavelength and polarization needs to be considered also. A consideration has been performed theoretically for the binary and the blazed grating with the aim to find initial parameters for a grating optimization. To optimize the gratings, two models were developed; a semi-analytical
154
Summary & perspective
model to estimate the path length enhancement and optimize the grating parameters and a rigorous model to calculate the effect of a grating on the absorption profile. Altogether four gratings were theoretically investigated using the two models. These gratings were: the ideal, binary grating, a grating with a trapezoidal shape based on SEM pictures, the blazed grating and a 2D pyramidal grating. One result of this optimization was that it was possible to induce a positive effect on the pathlength of light not only inside a flat crystalline solar cell but also inside a crystalline solar cell with a pyramidal front surface texture. An absorption enhancement of more than 20% relative was predicted in the long wavelength regime. Comparable results were also obtained in first measurements of prototype gratings introduced into silicon wafers. A first experimentally realized grating and the simulated absorption enhancement are shown in Figure 7.2. b)
a)
SiO2 Si
relative absorption enhancement [%]
40
Protective resist
binary grating real grating blazed grtaing
30 20 10 0 900
950
1000
1050
1100
wavelength [nm]
Figure 7.2: Grating introduced into a silicon wafer (a) and relative absorption enhancement for different gratings introduced into a silicon solar cell with a textured front surface calculated with the RCWA method (b).
Angular selectivity… Angularly selective optical elements have hitherto not been used in PV in the way they were considered in this work. For this reason, one goal was to investigate the potential of such structures. In a first theoretical consideration two effects induced by angular selectivity were investigated. The first effect is a path length enhancement generated by angular selectivity in combination with an appropriate scattering mechanism that results in increased quantum efficiency. The second effect is a suppression of radiative recombination, which results in an increased thermodynamical efficiency limit. Both effects depend on how far the angular acceptance range of the filter may be restricted. A maximum restriction is reached if only light coming from the solid angle that is occupied by the solar disc may pass the filter and all other light is reflected. Under these circumstances, the path length enhancement reaches a maximum of L = 46200. This path length enhancement can be expanded even further when combined with conventional light trapping with TIR. For silicon, a maximum path length enhancement of L = 2.23 106 was calculated, which is the highest value for a path length enhancement I know of. The suppression of radiative recombination results in the same efficiency limit as maximum concentration. For a silicon solar cell this limit is ca. η = 41%. Generally, a maximum efficiency is obtained if the angular spreading of radiation incident on a PV device is equal to the angular spreading of the radiation coming from the PV device (see Figure 7.3a). This equality can be achieved either by angular confinement or by concentration or by a combination of both. Following this theoretical consideration, I have investigated effects that result in an angular selective characteristic of photonic structures. One effect considered here especially is the
155
Summary & perspective
Bragg effect. This effect results in a blue shift of the spectral filter characteristics and can be used to create angular selectivity in a certain spectral range. A Bragg-like characteristic has been found theoretically and experimentally for most investigated photonic structures. Finally experiments were performed to investigate the path length enhancement induced by different filters by measuring absorption and quantum efficiency. An absorption enhancement was found for all investigated filters. With a band stop filter, the absorption enhancement was shown for a thin film solar cell of amorphous silicon. An increase of 25% was obtained in a certain spectral range. In the same spectral range, an increase of 25% of QE was also measured for the same system (see Figure 7.3b), proving the principle of the presented approach. a)
b)
EQE1/EQE2 [%]
130 120 110
EQE1 = cell + filter 2
+25%
EQE2= cell only EQE1 = cell + cover glass EQE2 = cell only
100 90 550
600
650
700
750
wavelength [nm]
Figure 7.3: Shockley-Queisser limit of the efficiency in dependence of the angular spreading of incident radiation (Ωinc) and radiation emitted by the solar cell (Ωext). A maximum efficiency is reached if the corresponding angles are equal, which holds e.g. for a system with maximum concentration or maximum angular confinement (a). Measured enhancement of the quantum efficiency for a system with angular confinement is shown in (b). In that example, the QE of a thin film solar cell of amorphous silicon was increased by 25%.
Evaluation of the results… To take a step backward, one main goal of the investigation of photonic concepts was to find ways to prevent photons from leaving PV systems in order to make solar cells thinner and more efficient. With all approaches discussed in this work, this goal was accomplished in one way or another. For some concepts, also additional effects have been found. Spectrally selective photonic structures were used successfully to reduce escape cone losses in fluorescent concentrators and generate very high efficiencies in a spectrum spitting concept. For diffractive structures it was shown that it is possible to induce an absorption enhancement in a crystalline silicon solar cell and especially in a crystalline silicon solar cell with a pyramidal front side texture. Several aspects of the novel concept of angular selectivity were investigated. One aspect found was a very high potential for a path length enhancement. The existence of a path length enhancement induced in a solar cell was also demonstrated experimentally. Another aspect concerns a similarity to concentration in principle that results in a correspondingly high efficiency limit. Altogether I believe I have shown the high potential that rests within photonic concepts for application in PV. Disadvantages include that many of the applied optical elements, and especially the ones that showed the best results were highly complex and, consequently, too expensive for a commercial product. The application of opals on fluorescent concentrators failed because of the insufficient quality of these photonic structures. In the simulations of diffractive
156
Summary & perspective
structures, inconsistencies concerning the optimum grating depth occurred that at time of publication have not yet been solved. Concerning angular selectivity, the Bragg effect generates angular selectivity only with limited efficiency. The search for structures with a more pronounced angular dependent characteristic has not resulted in structures with a satisfying characteristic at present.
Perspectives… At this point, an outlook shall be given concerning future tasks and open questions on the topic of photonic concepts for solar cells. Some tasks and challenges result from this work. Partly they were already mentioned in the evaluation and partly they emerged form the development of novel, more advanced concepts. Yet the field of photonic concepts is a wide one and only a small fraction of it was considered in this work. Therefore, interesting topics from other sources shall also be mentioned here. The RCWA method has proven to be a useful tool for the design and optimization of photonic structures. Investigations here mainly concerned optical properties. To some extent, the method has also been used to investigate the absorption within a solar cell. Concerning the optimization of photonic structures, one task will be to find a way to account for scattering processes. These processes could be investigated with the RCWA directly by increasing the considered unit cells, or they could be examined by adding other methods like algorithms based on Mie- and Rayleigh theory to the consideration. Concerning the investigation of gratings, an important issue will be to clear the inconsistencies that occurred for the optimization of the grating depth. A very important task concerns the coupling of electrical and optical approaches and the coordination of the corresponding simulation methods. An integrated electro – optical approach could be useful in many respects. Such an approach would allow the investigation of the absorption profile of a solar cell with an arbitrary photonic structure. Photonic processes inside solar cells could be examined directly and the understanding of underlying functions could be deepened. A possible application would be the optimization of diffractive structure in a complete 3D setup. Another possibility would be the development of novel PV concepts with e.g. a defined spatial absorption profile. Furthermore, the RCWA method could be used also to investigate other optical effects that were not considered in this work. Examples would be effects connected to plasmons [Roc082] or to photon jets. These effects could e.g. be used for local concentration. Several tasks and perspectives arise from the issue of spectrally selective optical elements. The spectrum splitting concept, for example, needs to be adapted to concentration conditions. This task mainly concerns two parts: I
An optimization of the geometry of the setup for an optimum illumination of the different solar cell and
II An optimization of the filters for light incident in a broad cone. This optimization of filters needs to consider the close relationship between angular- and spectral dependence of photonic structures. A possible task here could be the development of an optimization algorithm for non - periodic structures to control the angular characteristics of the filter. An optimized concentrating spectrum splitting device could achieve efficiencies above η = 40%. For the advanced fluorescent concentrator concept, one task concerns the application of spectrally selective filters. At the moment, these filters are too expensive to be interesting for a commercial product. For this reason methods have to be found to make filters cheap and to deposit them on large areas. One possibility here would be an a new approach with opals. To improve the quality of opals, they could be deposited on a textured fluorescent concentrator.
157
Summary & perspective
Another issue is the concept of the “Nanofluko” and several tasks arise here. A theoretical examination of this concept is needed to investigate its predicted increased light guiding ability. This examination could include an FDTD simulation of emission processes inside photonic structures. A method for the simulation of such emission processes could also be expanded to other concepts that combine photonic structures with luminescent materials. Examples here would be up- and downconversion concepts that aim for an increased luminescence induced by the photonic structure. Another task is the realization and experimental characterization of a “Nanofluko” as an advanced fluorescent concentrator system but also as an exemplary system for luminescent processes in photonic structures. The next step for diffractive structures will be the realization of a solar cell with a back side grating. This realization is in itself a challenging task, as the corresponding processes in the solar cell fabrication need to be developed. A decisive question for the application of diffractive structures in solar cells will be how much recombination occurs at the altered back surface. Answering this question will eventually also answer the question whether a grating can increase the efficiency of a solar cell. Several tasks are connected to these questions: •
One task concerns avoiding an increased back surface recombination. Here concepts are needed for an efficient electrical passivation of the diffractive structure without a change in the optical properties.
•
The semi-analytical model needs to be expanded towards a consideration of electrical loss mechanisms. One way of doing this would be to include experimental data. Furthermore, the parameters used for the implementation of the grating need to be adapted to the parameters of the fabricated grating in the solar cell. An extensive experimental characterization is needed for this.
•
The rigorous model needs to be advanced towards a full implementation of the 3D problem. Especially the question needs to be turned to how a 3D distribution of the EM field inside a solar cell can be transformed into a 1D or 2D absorption profile. The inconsistencies between the semi-analytical model and the rigorous model need to be cleared.
•
Especially advantageous for the theoretical investigation of diffractive structures would be the already mentioned integrated electro–optical approach. This method would allow e.g. considering electrical loss mechanisms directly and therefore a more accurate optimization of gratings would be possible. An electro – optical approach would also allow for investigating the impact of the changed absorption profile on the performance of the solar cell.
Another aspect of diffractive structures emerged from the alignment problem and concerns the relative orientation of diffractive structure and front side texture. A very good light trapping was predicted if the front surface texture features a line pattern and is tilted by 90° towards a diffractive structure that also features a line pattern. In these concepts the benefit of front surface texture and back side structure are combined in a very advantageous way. The discussed approaches are all meant for an application on already very efficient crystalline silicon solar cells but may be used, maybe even more advantageously, for other solar cell concepts as well. Photonic concepts featuring angular selectivity show a high potential for improving solar cells, yet they are still in their infancy. Accordingly, a lot remains to be done. One task in this field will be the design and fabrication of angularly selective photonic structures with a strongly pronounced angular selectivity. Especially structures with a narrow confinement of the angular acceptance range promise beneficial effects. To find structures with adequate properties, non-periodic and partially periodic structures could be
158
Summary & perspective
an interesting option. To design 1D photonic structures with controlled angular dependent characteristics, special algorithms are needed, as already mentioned. In the field of 2D and 3D photonic structures, fabrication methods often limit the possibilities in the structures’ design. An expedient procedure could therefore be to first investigate known structures for their angular characteristics and then optimize the most promising looking ones. Another task concerns the verification and optimization of the impacts of angular confinement on solar cells. A first confirmation of the light trapping effect was given in this thesis; however, the effect still needs to be improved greatly for a commercial application. The suppression of radiative recombination and the associated increased efficiency has not been confirmed yet experimentally, which will be a future task. An interesting concept could be the combination of concentration and angular confinement. This concept could improve concentrating solar cells by approaching the thermodynamic efficiency limit. An angularly selective structure could simply be added to existing concentrating setups. A fabrication of a prototype with adequate components and the characterization of the occurring effects could be a future task here. Another possible application could be the design of optimized antireflection coatings. These coatings could be optimized to show angular confinement in the weakly absorbed spectral region close to the band edge of the solar cell. This angular confinement would induce lighttrapping and therefore an enhanced absorption. An additional example for a concept that hitherto has not been mentioned in this work is the intermediate reflector in a stacked multijunction solar cell. The photonic structure here is included into a solar cell, which presents special demands on it. The photonic structure needs to be electrically conductive to allow a current transport from one solar cell to the other. As a consequence, the filter will also show considerable absorption in certain spectral ranges. Both aspects, electrical conductivity and absorption, have not been considered yet but present interesting issues.
Closing words… I believe that photonic concepts will play an increasingly important role on the way to improving PV devices. However, only few concepts have been investigated yet, let alone established in production. In this work new concepts were investigated and evaluated. Theoretical considerations showed the huge potential for increasing the absorption or reducing loss mechanisms. Experiments proved the principles of some concepts and showed where future work is needed. For other concepts, experimental work is still in progress and first results are expected soon. Photonic structures need to be developed further to increase their efficiency and reduce their costs. Yet, as solar cells get thinner and thinner, photonic concepts gain more and more in importance. With this work, I hope to have given an idea of the wide range of possibilities that exist in the field of photonic concepts.
159
Appendix A
Simulation of optical thin films
An optical thin film is described by a film of a constant thickness d and a constant, complex refractive Index N. A film is thin if the path difference between two partial waves propagating in opposite directions is less than the coherence length of the considered light. Incident on this film is a plane wave of light with the wavelength λ and the angle θin. This setup is sketched in Figure A.1.
Figure A.1: Plane wave incident on a thin film To describe this problem I will start with light impinging on a simple boundary, and then I will proceed to a thin film and finally transfer the result to an assembly of thin films The refractive index
r r
The answer of a medium affected by an electric field E (r , t ) may be written as a development of the electric moments (dipole moment, quadruple moment…)
r r r r ∇ ⋅ (ε 0 E (r , t ) + P(r , t ) + ...) = ρ
(A.1)
r
In this equation, P is called the macroscopic polarization. This approach will be restricted to isotropic media, which means that the macroscopic polarization shall depend linearly on the electric field.
r r r r r P (r , t ) = ε 0 χ (r ) E (r , t )
(A.2)
r
The proportionality constant χ (r ) is called the susceptibility of the medium which shall be constant in time. Similar considerations hold for the magnetic field. These considerations lead directly to the material equations (2.5) - (2.7). For a space-charge free medium, the following equation is derived from Maxwell’s equations
r r r r r r r ∂ ² E (r , t ) r ∂E (r , t ) ∇ ² E (r , t ) = ε 0ε (r ) + σ (r ) ∂t ² ∂t μ0 μ 1
(A.3)
Furthermore, all material parameters ε,μ,σ shall be scalar. Equation (A.3) may then be solved by the function
160
Appendix A
Simulation of optical thin films
r r r 1r E (r , t ) = E 0 exp(iω (t − r )) v
(A.4)
For equation (A.4) to be a solution of equation (A.3), it is necessary that
ω² v²
= ω ²ε 0 εμ 0 μ − iωσμ 0 μ
(A.5)
And with the equivalence c ² = 1 / μ 0 ε 0 it holds
μσ c² = εμ − i ωε 0 v²
(A.6)
with the dimensionless parameter c/v, which is denoted the complex refractive index N. N is of the form
N = n − ik
(A.7)
In this equation n is the refractive index and k the extinction coefficient. (The minus in equation (A.7) is a convention. Equation (A.7) is often found with a “+” and a negative value for k). In this work, when using N the complex refractive index is meant, while n is a real number and denotes the (real) refractive index. The simple boundary The case of the simple boundary is illustrated in Figure A.1. A plane wave with the angular frequency ω and the wavelength λ0 is incident from a non-absorbing medium with the refractive index N0 on a medium with the refractive index N1. In the incident medium the wave shall have an angle θ0 against the z-axis.
r r ⎛ ⎞ 2πN 0 ( x sin θ in + z cos θ in ) ⎟⎟ E inc ( x, y, z , t ) = E 0 exp⎜⎜ iωt − λ0 ⎝ ⎠
(A.8)
r r ⎛ ⎞ 2πN 0 ( x sin θ in + z cos θ in ) ⎟⎟ H inc ( x, y, z , t ) = H 0 exp⎜⎜ iωt − λ0 ⎝ ⎠
(A.9)
r
Generally the direction of propagation of the wave is given by the vector s .
r r r r s = α ⋅ ex + β ⋅ e y + γ ⋅ ez r
r
r
r
(A.10)
r
The vectors s , E and H ( E and H shall here only be the spatial vectors) are for every moment mutually perpendicular and form a right handed set. These properties permit the formation of an equation that is very useful for this approach.
H=
ε0 v r N 0 (s × E ) μ0
161
(A.11)
Appendix A
Simulation of optical thin films
The tangential components of the electric and magnetic field E(x, y, z, t) and H(x, y, z, t) are continuous across the boundary. The boundary is located at z = 0 and the tangential components must be continuous for all values of x, y and t. When the wave impinges on the boundary it will split into a reflected and a transmitted wave. Without knowing the exact form of these waves, it may be assumed that they can be described by an amplitude- and a phase term.
r r ⎛ ⎞ 2πN 0 E ref ( x, y, z , t ) = E 0,ref exp⎜⎜ iωt − ( x sin θ in − z cos θ in ) ⎟⎟ λ0 ⎝ ⎠
(A.12)
r r ⎛ ⎞ 2πN 0 ( x sin θ in − z cos θ in ) ⎟⎟ H ref ( x, y, z , t ) = H 0,ref exp⎜⎜ iωt − λ0 ⎝ ⎠
(A.13)
r r ⎛ ⎞ 2πN 1 Etra ( x, y, z , t ) = E 0,tra exp⎜⎜ iωt − (α t xt + β t y t + γ t z t ) ⎟⎟ λ0 ⎝ ⎠
(A.14)
r r ⎛ ⎞ 2πN 1 (α t xt + β t y t + γ t z t ) ⎟⎟ H tra ( x, y, z , t ) = H 0,tra exp⎜⎜ iωt − λ0 ⎝ ⎠
(A.15)
From the boundary conditions it holds that the angular frequency ω has to be equal for all terms at z = 0. The direction of the reflected and transmitted beam is given by Snell’s law
N 0 sin θ in = N 1 sin θ 1
(A.16)
The calculation gets very complicated for arbitrary polarizations of the incident wave. However, a sensible restriction here is to the cases of TE (transversal electric) and TM (transversal magnetic) polarization. From this set every other polarization may be generated by linear combination. Before the actual calculation, the positive directions of the electric and magnetic vectors for the different polarizations have to be clarified, because a certain freedom of choice exists here. The conventions used in this approach are given in Figure A.2.
Figure A.2: Convention defining the positive directions of the electric and magnetic waves for TE and TM polarization.
162
Appendix A
Simulation of optical thin films
TM polarization The components of the electric field parallel to the boundary are continuous across it
r r r E 0 cos θ in + E 0, ref cos θ in = E 0,tra cos θ 1
(A.17)
For the sake of simplicity this equation is rewritten to
Ein + E ref = Etra
(A.18)
The components of the magnetic field parallel to the boundary are continuous across it. With equation (A.11) it therefore holds
r r r ε0 ε0 ε0 N 0 E0 + N 0 E 0,ref = N 1 E 0,tra μ0 μ0 μ0
(A.19)
Or, using another simplification,
ε0 N0 ε0 N0 ε 0 N1 H in − H ref = H tra μ 0 sinθ in μ 0 sinθ in μ 0 sinθ 1 With that and introducing the optical admittance y =
(A.20)
ε0 N the so-called Fresnel μ0
amplitudes for reflection and transmission may be defined.
ρ TM =
τ TM =
E ref E in
y0 y1 − cos θ in cos θ 1 = y0 y1 + cos θ in cos θ 1
E tra = E in
2 y0 cos θ in y0 y1 + cos θ in cos θ 1
(A.21)
(A.22)
From the Fresnel amplitudes, the reflectance and transmittance may be calculated with
R = ρ² ,
T=
y1 τ² y0
(A.23)
TE polarization The components of the electric field parallel to the boundary are continuous across it
r r r E 0 + E 0,ref = E 0,tra 163
(A.24)
Appendix A
Simulation of optical thin films
For the sake of simplicity this may be rewritten as
Ein + E ref = Etra
(A.25)
The components of the magnetic field parallel to the boundary are continuous across it. With equation (A.11) and the used simplification it holds that
y 0 E in cos θ in − y 0 E ref cos θ in = y1 E tra cos θ 0
(A.26)
Which gives the Fresnel amplitudes
ρ TE =
y 0 cos θ in − y1 cos θ 1 y 0 cos θ in + y1 cos θ 1
(A.27)
τ TE =
2 y 0 cos θ in y 0 cos θ in + y1 cos θ 1
(A.28)
This term may be further simplified by using
η 0 = y cos θ in
η1 = y cos θ 1
and
(A.29)
Consequently, the Fresnel amplitudes are given by
ρ=
η 0 − η1 η 0 + η1
τ=
and
2η 0 η 0 + η1
(A.30)
Reflectance and transmittance, finally, are given by
η − η1 R= 0 η 0 + η1
2
and
T=
4 Re(η 0 ) Re(η1 )
η 0 + η1
2
(A.31)
Extension to a thin film When more than one surface exists, a number of partial waves will be produced by successive reflections. The properties of the film will be determined by the summation of these beams. Beams in the direction of incidence are denoted a+ and waves in the opposite direction are denoted a-. The interface film-substrate b (see Figure A.1) may be treated like the simple boundary. The waves in the film can be summed into one effective positive going and one effective negative going wave. The tangential components at this interface may be written as
Eb = E1+b + E1−b
(A.32)
H b = η1 E1+b − η1 E1−b
(A.33)
164
Appendix A
Simulation of optical thin films
The field at the other interface a at the same time and position is determined by changing the phase factor of the wave. If the thickness of the thin film is d, then the phase factor for the positive going wave is given by
exp(
2πiN 1 d cos θ 1
λ
) = exp(iδ )
(A.34)
In this equation θ1 may be complex. For the negative going wave the phase factor is exp(iδ). With this the electric field may be obtained
E a = E1+a + E1−a = E1+b exp(iδ ) + E1−b exp(−iδ ) i sin δ ⎛ exp(iδ ) + exp( −iδ ) ⎞ ⎛ exp(iδ ) + exp(−iδ ) ⎞ = Eb ⎜ ⎟ = E b cos δ + H b ⎟ + Hb ⎜ 2η1 2 2 ⎝ ⎠ ⎝ ⎠
(A.35)
and for the magnetic field it holds
H a = Eb iη1 sin δ + H b cos δ
(A.36)
These equations may be combined in the form of a matrix
i sin δ / η1 ⎤ ⎡ Eb ⎤ ⎡ E a ⎤ ⎡cos δ ⎢ H ⎥ = ⎢iη sin δ cos δ ⎥⎢H ⎥ ⎦⎣ b ⎦ ⎣ a⎦ ⎣ 1
(A.37)
This matrix is the characteristic matrix of a thin film. With the optical admittance Y=Ha/Ea the Fresnel amplitude for the reflection of a simple interface between an incident medium with the optical admittance η0 and a medium of the admittance Y are obtained.
ρ=
η0 − Y η0 + Y
and
τ=
2η 0 η0 + Y
(A.38)
This result is used to further modify equation (A.37).
Ea Eb
i sin δ / η1 ⎤ ⎡1 ⎤ ⎡ B ⎤ ⎡1 ⎤ ⎡cos δ ⎥ ⎢η ⎥ = ⎢C ⎥ ⎢Y ⎥ = ⎢iη sin δ cos δ ⎣ ⎦ ⎣ 1 ⎦⎣ 2 ⎦ ⎣ ⎦
(A.39)
An assembly of thin films With the tools now derived, the calculation of the optical properties of an assembly of thin films is relatively simple. Following the argumentation from equations (A.32) to (A.35), it can be stated that the characteristic matrix of an assembly of thin films is given by the product of the matrices for the single films.
i sin δ r / η r ⎤ ⎞⎟ ⎡1 ⎤ ⎡ B ⎤ ⎛⎜ r = q ⎡cos δ r = ⎥ ⎟ ⎢η ⎥ ⎢C ⎥ ⎜ ∏ ⎢iη sin δ cos δ ⎣ ⎦ ⎝ r =1 ⎣ r r r ⎦ ⎠⎣ m ⎦ The matrix entries are given by
165
(A.40)
Appendix A
δr =
Simulation of optical thin films
2πN r d r cosθ r
λ
,
η r ,TE =
ε0 ε0 1 N r cosθ r , η r ,TM = Nr μ0 μ0 cosθ r
(A.41)
Reflectance, transmittance and absorptance are calculated from this formula by 2
η B−C , R= 0 η0 B + C
T=
4η 0 Re(η m )
η0 B + C
2
,
A = 1− R −T
(A.42)
Equations (A.40) -(A.42) are also a good summary of this method, as these equations are all that is needed to implement the method.
166
Appendix B
The RCWA method
In the following subsection the RCWA method is described. First, a sketch of the method for arbitrary 1D gratings is given that follows a description by Turunen [Tur97]. Then a generalization for the 2D case is given that follows a description by Li [Li97]. Finally, some remarks about the formulation of the used code are given. General Formulation of the 1D Problem The periodicity of a structure is generated by a modulation of the refractive index. By virtue of this periodicity, discrete orders of diffraction appear in the far field. If the dimension of the periodic structure is much larger than the wavelength of the considered light, scalar approximations yield adequate results. For this purpose, ray tracing simulations may be used. As the structure dimensions become smaller, the scalar approximations no longer hold, and the wavelike nature of light must be taken into account. The origin of a rigorous method capable of doing so is formed once more by Maxwell’s equations (equations (2.1) (2.4)). In this section, all materials shall have a permittivity of μ ≈ 1 , which is true for most semiconductors and all materials used for practical purposes in this work. To calculate the diffraction efficiencies, the considered structure is divided into three regions as shown in Figure B.1Figure B.. Region I describes the superstrate, region II the modulated region (grating) which shall be extended infinitely in the x and y directions, and region III the substrate. All regions shall be isotropic; region I and region III shall additionally be homogenous. Without restriction to generality we may assume nI and nIII to be real. As the problem is formulated for the 1D case, all regions are invariant in the ydirection.
Figure B.1: Sketch of an arbitrary 1D grating and nomenclature. Let the incident light be plane waves E yinc ( x, z , t ) and H yinc ( x, z , t ) that are normal to the plane of incidence. The amplitudes are given by E0y, H0y and the frequency is ω. The wave is incident from region I, which shall be infinitely extended and invariant in the y-direction. E yinc ( x, z , t ) = E0 y exp(ik x x + ik z z − iωt ) = E0 y exp(ik0 nI ( x ⋅ sin θ in + z ⋅ cosθ in ) − iωt )
(B.1)
H yinc ( x, z, t ) = H 0 y exp(ik x x + ik z z − iωt ) = H 0 y exp(ik0 nI ( x ⋅ sin θ in + z ⋅ cosθ in ) − iωt )
(B.2)
with k 0 = 2π / λ and λ the wavelength of the light in vacuum. kx and kz are the components of k0 in x and z directions. They are given by
167
Appendix B
The RCWA method
k x = k0 nI sin θ in ,
k z = k0 nI cos θ in
(B.3)
In the following the factor exp(iωt) will be neglected because only the stationary case and not the transient oscillation is of interest. As the excitation happens with a harmonic field we may assume, that all fields will oscillate with the frequency ω after a sufficient time. Temporal deviations yield a factor iω. Because of the invariance in the y-direction and the isotropy, light incident to the x-z plane will split into two independent groups according to Maxwell’s equation. One group contains only the field components Ey, Hx and Hz (TE-polarization, the vector of the electric field strength is parallel to y)
∂E y
=0
∂y
∂H x ∂H z + =0 ∂x ∂z
∂E y ∂z ∂E y ∂x
(B.4)
(B.5)
+ iμ 0 ω H x = 0
(B.6)
− iμ 0 ω H z = 0
(B.7)
∂H x ∂H z − − σE y + iεε 0ωE y = 0 ∂x ∂z
(B.8)
The other group only contains Ex, Ez and Hy (TM-polarization, the vector of the magnetic field strength is parallel to y)
∂E x ∂E z + =0 ∂x ∂z ∂H y
=0
(B.10)
+ iμ 0 ω H x = 0
(B.11)
∂y
∂E y ∂z −
∂H y ∂z
∂H y ∂x
(B.9)
− σE x + iεε 0ωE x = 0
(B.12)
− σE z + iεε 0ωE z = 0
(B.13)
168
Appendix B
The RCWA method
TE polarization By inserting equations (B.6) and (B.7) into equation (B.8) the characteristic scalar Helmholtz equation for Ey is obtained.
1 ∂² 1 ∂² Ey + E y + (iσω + εε 0ω ²) E y = 0 μ 0 ∂x ² μ 0 ∂z ²
(B.14)
The according H-field may be determined from equations (B.5) and (B.6). We are now looking for solutions of Ey that satisfy the transition conditions at the boundaries (the components of the field and their deviations must be continuous) and the scalar Helmholtz equation (B.14) . The general approach is to describe the field in region I and III as a linear combination of plane waves. The components are identified with the discrete orders of diffraction. Afterwards a solution is searched for region II that may be written as a linear combination of modes, which are the solutions of the Helmholtz equation. The final solution is obtained by matching the amplitudes of the modes in the three regions with regard to the boundary conditions. The incident plane wave interacts with the infinite periodic grating. Every period is “excited” harmonically and interacts with every other period. After the transient oscillation, the only difference between the periods is the phase for which the wave interacts with the grating. For all periods the same temporal evaluation of the electromagnetic field occurs, but with a delayed excitation. It has to be mentioned that a period not only covers a part of region II but has an infinite elongation in the y- and z-direction and a length Λ along the xaxis. The path difference of the incident light wave between two periods is given by l = Λ sin(ϕ in ) , the time difference by l/c. l ⎞ ⎛ E y ( x + Λ, z ) exp(−iωt ) = E y ( x, z ) exp⎜ − iω (t − ) ⎟ = E y ( x, z ) exp(−iωt ) exp(ik x Λ) c ⎠ ⎝
(B.15)
Ey therefore is a pseudoperiodic function with
E y ( x + Λ, z ) =E y ( x, z ) exp(ik x Λ )
(B.16)
A new function may now be defined by
Fy ( x, z ) = E y ( x, z ) exp(−ik x x)
(B.17)
This function Fy(x, z) is periodic with the period Λ and may therefore be expressed in a Fourier series
⎛ 2πimx ⎞ Fy ( x, z ) = ∑ Fm ( z ) exp⎜ ⎟ ⎝ Λ ⎠ m = −∞ m =+ ∞
(B.18)
Using this new function Ey may be written as
E y ( x, z ) =
m =+ ∞
∑F
m = −∞
m
⎛ 2πimx ⎞ ( z ) exp⎜ ⎟ exp(ik x x) ⎝ Λ ⎠
169
(B.19)
Appendix B
The RCWA method
Every term of this sum is identified with one mode. The incident wave defines kx and the periodicity defines the different discrete x-components of all further possible k-vectors of the single modes. These k-vectors will in the following be denoted as k xm = k x + 2πm / Λ . Region I and region III The purpose of our approach was to find solutions of equation (B.14) . Inserting equation (B.19) into equation (B.14) , and using the orthogonality of the single modes the following equation is obtained
Fm ( z )(iσμ 0ω + εε 0 μ 0ω ² − k xm ² ) + Fm′′ ( z ) = 0
(B.20)
The single modes are obtained by inserting equation (B.20) into equation (B.19) and solving the obtained function. For region I and region III the solution is straightforward. With σI = σIII=0 and εε0µ0 =n²/c² equation (B.20) is solved by functions of the form
Fm ( z ) = Fm+ exp(ik zm z ) + Fm− exp(−ik zm z )
(B.21)
with k zm =
n ²ω ² n ²ω ² 2 2 , k xm − k xm ≤ c² c²
(B.22)
n ²ω ² n ²ω ² 2 , k xm > c² c²
(B.23)
2 k zm = i k xm −
Inserting equation (B.21) into equation (B.19) results in
E yIII ( x, z ) =
m =+ ∞
∑ (F
m = −∞
+ m
exp(ik zm z ) + Fm− exp(−ik zm z )) exp(ik xm x)
(B.24)
which is the description for the field in region III. The field in region I is similarly given by
E yI ( x, z ) = E yinc ( x, z ) +
m =+ ∞
∑ (F
m = −∞
+ m
exp(ik zm z ) + Fm− exp(−ik zm z )) exp(ik xm x)
(B.25)
where Eyinc is the corresponding component of the incident field. If kzm is real, this solution describes plane waves, or evanescent waves, if kzm is imaginary. In region I the Fm+ term corresponds to incident plane waves if kzm is real. If kzm is imaginary, the Fm+ term described external decreasing evanescent fields. These fields have vanishing amplitudes for physical reasons; FmI+ = 0 . Remaining are only the reflected orders with the amplitudes Rm = FmI− . With an analogous argumentation for region III, we obtain the transmitted orders with an + − amplitude Tm = FmIII and FmIII = 0 . As solutions for equation (B.19) in region I and region III remain the reflected and transmitted plane waves and the external decreasing evanescent fields. As a result of the Fourier expansion, the single plane waves occurring in equation (B.24) and (B.25) have discrete directions that depend on the angle of incidence and the period, but not on the exact geometry of the surface. For the case of radiant modes, the
170
Appendix B
The RCWA method
grating equation is obtained, in which the period Λ, the wavelength λ, the angle of incidence θin and the emission angle of the mth order θm are correlated.
sin θ m = sin θ in +
mλ , in region I Λn I
n III sin θ m = n I sin θ in +
mλ , in region III Λ
(B.26)
(B.27)
For a complete characterization, the directions and the diffraction of the efficiencies need to be known. The direction are given in equations (B.26) and (B.27), the diffraction efficiency may be calculated from the z-component of the Poynting vector as follows
η Rm =
k zmI Rm ² k z0I
(B.28)
η Rm =
k zmI Rm ² k z0I
(B.29)
Region II The periodic modulated region II may be separated into N sufficiently thin layers that are parallel to the x-y plane. Sufficiently thin here means that σn(x ,z) and εn(x, z) in each layer n may be regarded as invariant in the z-direction. To solve equation (B.14) the separation approach is used
E yn ( x, z ) = X n ( x) Z n ( z )
(B.30)
Inserting equation (B.30) into equation (B.14) , we obtain for the nth layer:
Z n′′ ( z ) + g n2 Z n ( z ) = 0
(B.31)
X n′′ ( x) + (iσ n ( x) μ 0ω + μ 0 ε 0 ε n ( x)ω ² − g n2 ) X n ( x) = 0
(B.32)
with the separation constant gn. The general solution for Zn(z) is straightforward and is given by
Z n ( z ) = a n exp(ig n z ) + bn exp(−ig n z )
(B.33)
Similar to the discussion for region I and region III Xn(x) has to be pseudoperiodic also in region II and may therefore be expanded into a Fourier series
X n ( x) =
m =∞
∑X
m = −∞
nm
171
exp(ik xm x)
(B.34)
Appendix B
The RCWA method
To determine Xn(x) we now need to also expand iσ n ( x) μ 0ω + μ 0 ε 0 ε n ( x)ω ² into a Fourier series. This is possible because σ n ( x) and ε n ( x) are by definition periodic functions with the period Λ.
iσ n ( x) μ 0ω + μ 0 ε 0 ε n ( x)ω ² =
p =∞
∑a
p = −∞
np
⎛ 2πipx ⎞ exp⎜ ⎟ ⎝ Λ ⎠
(B.35)
To obtain an equation for each Xnm, equations (B.35) and (B.34) have to be inserted into (B.32) and terms with the same exponent must be combined.
g X nm = 2 n
p =∞
∑ (a
p = −∞
nm − p
− k xp2 δ mp )X np
(B.36)
The set of equations for each layer may be written in the form of an infinite quadratic matrix
Mˆ n with the entries a nm − p − k xp2 δ mp r r Mˆ n X n = g n2 X n
(B.37)
2 This is a typical Eigenvalue problem and the qth Eigenvalue g nq and the corresponding
r
Eigenvector X nq can be calculated with numerical methods integrated e.g. into MATLAB.
r
By using equations (B.33) and (B.34), every Eigenvector X nq may be used to form a solution to the homogenous differential equation (B.14)
E ynq ( x, z ) =
m =∞
∑X
m = −∞
nqm
exp(ik xm x)(a nq exp(ig nq z ) + bnq exp(−ig nq z ))
(B.38)
The complete field in each layer is than given as the sum over all of these functions
E yn ( x, z ) =
q, m =∞
∑X
q , m = −∞
nqm
exp(ik xm x)(anq exp(ig nq z ) + bnq exp(−ig nq z ))
(B.39)
The method to obtain equation (B.39) shown here relates to the Fourier modal method. In the rigorous coupled wave analysis the same equation (B.39) obtained but with a slightly different method. In the RCWA the same separation into N layers is performed. The approach then is to choose Fnm(z) in equation (B.19) in the form
Fnm ( z ) = S nm ( z ) exp(ik nm z )
(B.40)
Again the Fourier expansion of the material parameters has to be performed (equation (B.35)). Inserting equations (B.40) and (B.35) into (B.14) , one obtains N systems of coupled 2nd order differential equations which may be transformed into 2N systems of 1st order differential equations of the form
r r S n′ ( z ) = Bˆ n S n ( z ) With the general approach
172
(B.41)
Appendix B
The RCWA method
S nm ( z ) =
q =∞
∑A
q = −∞
nq
S nmq exp(λ nq z )
(B.42)
whereas λ nq are the Eigenvalues of Bˆ n and S nmq are the components of the corresponding Eigenvector. This may again be used to form a solution of the homogenous differential equation (B.14) , and thus equation (B.39) is retrieved. The difference between the different methods lies in the interpretation of the single terms. From the Fourier modal method, for every layer a set of plane waves is obtained that each see different effective refractive indices. In the RCWA, on the other hand, in every layer modes propagate and the z-dependence of the mth mode is traced which depends on the amplitudes of the (m+1)th and (m-1)th modes. This means that only neighbouring modes couple. To determine the intensities of the different modes from equation (B.39), the single amplitudes Xnqm have to be calculated. This is done by choosing Ey and Hy in such a way that they are continuous at the borders of the single layers. Because of the linear independence of the Fourier modes, this is also true for every single Fourier term. As there are N+1 intersections between the layers and every Fourier component gives two contributions (Ey and Hy), there are 2(N+1) equations with 2(N+1) unknowns (N-times anq, N-times bnq, Rm and Tm), and as there are an equal number of Rm and Tm values and Eigenvalues gq, there are always as many equations as unknowns, and the system of equations may be solved e.g. with the Gauss algorithm. Up to this point, equation (B.39) is rigorous. In the numerical implementation, the infinity of the sums and the matrices limits the accuracy of the method. The problem of infinite sizes is solved by cutting the matrices. With this procedure information is lost, and it has to be assumed that a marginal amount of energy is transported in the modes that are cut away. The calculation time scales with the size of the matrix in the third power, which is a problem if many modes have to be considered. A rule of thumb is to use a number of modes that is twice as high as the number of propagable modes. In summary, the discussed method is appropriate to the calculation of periodic structures with a period in the range of the considered light wavelength. The larger the period, and the higher the refractive index in the structure, the more modes have to be considered for an accurate result. This leads to a rapidly growing time needed for the calculation or a decreased accuracy of the result. In this context, metallic gratings with a period larger than the considered light wavelength are especially problematic to calculate.. TM polarization For the TM polarization, it is more convenient to determine the vectorial components Hy than to look for solution of the coupled differential equations of Ex and Ez. This is possible by inserting equations (B.11) and (B.12) into (B.13).
∂H y ⎤ ∂H y ⎤ ∂ ⎡ 1 1 ∂ ⎡ + ⎥ + i μ 0 ωH y = 0 ⎢ ⎥ ⎢ ∂x ⎣σ − iεε 0ω ∂x ⎦ ∂z ⎣σ − iεε 0ω ∂z ⎦
(B.43)
The corresponding E-field may be determined from equations (B.11) and (B.12) into (B.13). The approach to solve equation (B.43) is similar to that of solving (B.14) for the TE polarization. For region I and III we may therefore write directly and in analogy to equation (B.20)
J m ( z )(iσμ 0ω + εε 0 μ 0ω ² − k xm ² ) + J m′′ ( z ) = 0
173
(B.44)
Appendix B
The RCWA method
Where Jm(z) are the Fourier amplitudes in the expansion of the magnetic field. Furthermore, the field in region I and region III may, similar to equations (B.24) and (B.25), be written as
H yIII ( x, z ) =
m =+ ∞
∑ (J
m = −∞
H yI ( x, z ) = H yinc ( x, z ) +
+ m
exp(ik zm z ) + J m− exp(−ik zm z )) exp(ik xm x)
m =+ ∞
∑ (J
m = −∞
+ m
exp(ik zm z ) + J m− exp(−ik zm z )) exp(ik xm x )
(B.45)
(B.46)
In region II again the separation approach is used.
H yn ( x, z ) = Wn ( x)Yn ( z )
(B.47)
As with equation (B.35), the material parameters from equation (B.43) (which are periodic with period Λ) need to be expanded into a Fourier series
1 = σ ( x) − iε ( x)ε 0ω
p =∞
∑c
p = −∞
np
⎛ 2πipx ⎞ exp⎜ ⎟ ⎝ Λ ⎠
(B.48)
Also Wn(x) also has to be expanded into a Fourier series
Wn ( x ) =
m =∞
∑W
m = −∞
nm
exp(ik xm x)
(B.49)
Inserting equations (B.47), (B.48) and (B.49) into equation (B.43) a linear equation system is obtained. This may be written in the form of a matrix
r r Mˆ nWn = g n2 Nˆ nWn
(B.50)
Inversion of the matrix Nˆ n and matching to the boundary conditions yield the complete information about the field. Generalization to 2D structures The periodicity of the structure shall now be in x-direction with the period Λx and in ydirection with the period Λy. For this case no division into TE and TM polarization exists. In regions I and III a solution of Maxwell’s equation may be built from plane waves. For equation (B.19), a 2D equivalent is given through
E y ( x, z ) =
⎛ 2πimy ⎞ ⎛ 2πimx ⎞ ⎟ exp(ik y y ) ⎟⎟ exp(ik x x) exp⎜ Flm ( z ) exp⎜⎜ ⎜ ⎟ Λ Λ l , m = −∞ x y ⎝ ⎠ ⎝ ⎠ l , m =+ ∞
∑
(B.51)
Similar equations hold for Ex, Hy and Hx. The structure is again separated into layers with zinvariant material parameters ε(x, y), μ(x, y). A Fourier expansion of the material parameters and the electromagnetic field and consequently an insertion into the field equations leads to an Eigenvalue problem with now four field components and n × m Fourier components (if n is the number of Fourier components used for the x-direction and m is the number of Fourier components in the y-direction).
174
Appendix C
Implementation into “Reticolo Code 2D”
In this section I want to give a short introduction of how to implement a given photonic structure into the MATLAB-based RCWA program “Reticolo Code 2D”. The most important points on how to use this code are summarized, and comments have been made about special aspects for the implementation of 3D photonic structures. I will not comment on all parameters that are needed to define a structure, but I will mention the ones that have physical relevance. The remaining parameters will be briefly commented upon in the “sample program” section. Input parameters i.
ii.
iii. iv. v.
The first input parameter relevant for the simulation is the vacuum wavelength of the considered light λ. In the program it is denoted “LD” (longeur d’onde). One calculation is performed for one single wavelength. For light in a medium the wavelength is corrected automatically; the given wavelength is always the one in vacuum even if the refractive index for the incident medium differs from unity. The wavelength is also the reference length for the entire system. It has therefore no unit. All other dimensions are defined in the same unit of the wavelength. This is done because only relative values of spatial dilation and wavelength are relevant for the calculation of the optical properties. The second input parameter is the period length in the x and y directions lx, ly. In the program, this parameter is given as D = [lx, ly]. The period defines a rectangular unit cell. The period length is always given in the same dimension as the vacuum wavelength. Crystal systems implemented into the code always have to be designed as rectangular systems, which is problematic for arbitrary crystal systems and may result in very large unit cells. I will give the unit cell of an opaline system as an example later in the text. The polar angle θ which defines the polar component of the direction of the incident light and is given in degrees. In the code it is denoted as theta. The polar angle also defines the parallel component of the k-vector of the incident light kparallel = ninc sin(θ). The azimuth angle d defines the azimuthal component of the direction of the incident light and is given in degrees. In the code it is denoted as δ. Finally, one parameter is the number of Fourier components into each direction nfur. In the code, this parameter is denoted as nn = [nfurx, nfury]. The number of Fourier components may therefore be defined independently for each direction. However, typically the same number of Fourier components for each direction was used. Considered are always the Fourier components –nfur,…,nfur.
For a given rectangular representation of the unit cell, one proceeds by deconstructing the structure into different layers. The layers have a refractive index n1 and may contain different forms and different refractive indices nins. The definition of one layer is of the form textures{x} = {n1,[cx1,cy1,Lx1,Ly1,nins1,nd1],[cx2,cy2,Lx2,Ly2,nins2,nd2],…}
In this definition, x gives the number of the layer. This number is used by the program later to identify the layer. The symbols in the square brackets define the insertion: i. ii. iii. iv.
cx1,cy1 give the position of the center of the first inversion. The position 0,0 corresponds to the center of the rectangle with the period -lx/2, lx/2, -ly /2, ly/2. Lx1,Ly1 give the dimension of the first insertion along the x- and the y-axis. nins1 gives the refractive index of the first inversion nd1 defines the form of the first inversion. Possible are rectangular and elliptical shapes. nd1 gives the number of rectangles that are used to create the elliptical shape
175
Appendix C
Implementation into “Reticolo Code 2D”
(see convergence considerations, Figure 3.15). Lx1,Ly1 define the lengths of the halfaxis. If Lx1 = Ly1 the insertion has a spherical shape. It is important that the program inherently considers the periodicity of the structure. That means that for an inclusion which exceeds the unit cell, the exceeding fraction appears on the other side of the unit cell. Figure C.1 shows a sketch of how an inverted opal is deconstructed into layers. In the third layer, the effect of the periodicity is perceptible. Two circular inclusions are defined in this inclusion, located at each side center. The half circles exceeding the borders of the unit cell each appear on the other side of the unit cell.
Figure C.1: Two textures of the composition of an opal. The opal itself is deconstructed into layers with spherical inclusions which have different radii depending on the position of the layer. After the definition of the layers, the structure has to be assembled. This is done by the profile command. This command includes information about the order of the layers and their thickness. The profile command is of the form: profile =
{{[dx1,…,dxn],[x1,…xn],N1},{[dxn+1,…,dxm],[xn+1,…,xm] ,N2},…}
The information about a sequence of layers xi and their thickness dxi is combined in a curly bracket. Within a first square bracket the thicknesses of the layers dxi are given, and within a second square bracket the sequence of the layers xi is defined. It is possible to arrange different sequences separated by curly brackets. The order in which the layers are given in the profile command is the order in which the layers are assembled, whereas the first layer is the top layer. The last number Ni gives the number of repetitions for a certain sequence. Output parameters The information about the diffracted field is divided by the polarization and the direction of the incident light. The polarization is divided into TE and TM and marked by a corresponding suffix. For a given direction of incidence (defined by the angles θ and δ), the light may impinge on the structure from the top and from the bottom. Both cases are calculated and are divided by the suffix “top” or “bottom”. Also divided is between reflected and transmitted field so that all in all eight different fields in the result have to be distinguished. The information about the reflected field in TE polarization for incidence from the top is included in the data set named: result.TEinc_top_reflected
Corresponding data sets exist for the seven other cases. Each of these data sets contains the following numerical results
176
Appendix C
Implementation into “Reticolo Code 2D”
Order1:
The nth propagable order in the x-direction
Order2:
The nth propagable order in the y-direction
Polar angle (θ):
The polar component of the reflected and transmitted wave for each order
Azimuthal angle (δ):
The azimuthal component of the reflected and transmitted wave for each order
Efficiency:
The total efficiency in each order (efficiencies are defined as the ratio of the flux on the diffracted Poynting vector and the incident Poynting vector)
Efficiency_TE:
The efficiency in TE polarization in each order
Efficiency_TM:
The efficiency in TM polarization in each order
Amplitude_TE:
The complex amplitude in TE polarization in each order
Amplitude_TM:
The complex amplitude in TM polarization in each order
The data denoted result.TEinc_top_reflected.efficiency
is a vector containing the efficiencies of all diffracted orders in TE polarization for a wave incident from the top. This data contains all information about the optical far field. Exemplary program In the following table the most important commands are summarized in the form of a commented exemplary program. LD = 0.4;
vacuum wavelength of the incident light
D = [0.02,0.034641];
period in x- and y-direction
n_inc = 1;
refractive index of incident medium
n_sub = 1.5;
refractive index of substrate
n_1 = 1;
refractive index of layers
n_2 = 1.5;
refractive index of insertions
theta = 0;
polar angle of incident light
k_par=n_inc*sin(theta*pi/180);
parallel component of the k vector for incident light
delta = 0;
azimuthal component of incident light
parm=res0;
command to initialize the default values (required for computation). Default
177
Appendix C
Implementation into “Reticolo Code 2D”
values may be changed if special symmetries occur. parm.res1.champ=1;
command to control if the EM field is calculated accurately. A value of 1 means accurate computation, 0 means that only the far field is calculated.
nn = [3,3];
number of Fourier components considered in each direction
textures = cell(1,31);
defines the number of layers used. Required for computation
textures{1} = n_inc;
texture 1 defines the incident medium
textures{2} = n_sub;
texture 2 defines the substrate
textures{3}= {n_1,[0 ,-0.1,0.1,0.1,n_2,15], [-0.1,0.1,0.1,0.1 n_2,15],};
definition of the different layers. Number and positions of the inclusions depend on the implemented structure. The defined number of layers and the actual number of layers have to agree.
… textures{31}= {n_1,[0,-0.2,0.2,0.2,n_2,15], [0.1,0,0.2,0.2 ,n_2,5],}; aa=res1(LD,D, textures, nn, k_par, delta, arm);
command starting the calculation of the eigenmodes associated to all layers (first step of the computation)
profile={{1,1},{{[0.002 , 0.002 ,…,0.002], [7,8,9,10,11,12,13,…,31], 8},{0,2}};
definition of the profile. Three sequences are given. The first defines the incident medium, the second gives the opal structure with 8 repetitions and the third defines the substrate. Each sequence is separated by curly brackets.
result=res2(aa, profile);
command starting the computation of the diffracted waves (second step of computation)
Reflektiert_TE400 = sum(result.TEinc_top_reflected.eff iciency)
definition of the output variables for direct use or to write in an output file
Reflektiert_TM400 = sum(result.TMinc_top_reflected.eff iciency) Transmittiert_TE400=sum(result.TEi nc_top_transmitted.efficiency) Transmittiert_TM400=sum(result.TMi nc_top_transmitted.efficiency) …
command deleting all temporarily stored data (in practice a very important command, keeps your disk clean. If forgotten, calculation becomes slower)
retio
178
Appendix C
Implementation into “Reticolo Code 2D”
Simulation of the optical far field The total reflected / transmitted efficiency for one polarization is calculated by summing up the efficiencies of each reflected wave in every order with the corresponding command. For example, the reflectance for one wavelength and TE polarization for light incident from the top is given by sum(result.TEinc_top_reflected.efficiency)
Equivalent commands hold for the other polarization and the transmission. The absorption in the structure is calculated 1 - R - T. To obtain the total RTA, the average of the values obtained for both polarizations is used (TE+TM)/2. Depending on each command used and the dependency against which the information is plotted, the graphs shown in chapter 3 are obtained. Simulation of the optical near field Additionally to the optical far field also the optical near field is calculated with the RCWA method. The optical near field corresponds to solutions in region II (Figure B.1). In the program the calculation of the near field is controlled by the command AkkuratE = 1
If the value for this command is switched to 0 no calculation of the near field is performed. The information about the near field contains intensity of all six field components (Hx, Hy, Hz. Ex, Ey, Ez.). To extract this information, the intensity for all components is calculated for discrete points in a cross section through the structure and given in form of a matrix. For this a number of commands are needed. x = linspace(-D(1), D(1), 21);
defines a cross section along the y-axis with several points on the x-axis. In the given case 21 points are defined between a double period along the x-axis (fromD(1) to D(1))
y = 0;
defines the y-coordinate of the crosssection
einc = result.TEinc_top.PlaneWave_TE_Eu;
defines the conditions for incident light. In the given case, this is TE polarization incident from the top
parm.res3.npts=[0,30,30,30,30,…]
defines the resolution for each texture. Consequently the number of entries is equal to the number of layers the complete structure consists of. A value of 30 means that each layer is divided into 30 sub layers.
[e,z,o] = res3(x,y,aa,profile,einc,parm);
command to actually calculate the near field. The information is divided into three parameters. “e” contains the
179
Appendix C
Implementation into “Reticolo Code 2D”
electromagnetic field quantities, “z” contains the corresponding spatial information and “o” is the corresponding effective refractive index. All information about the electromagnetic field is therefore contained in the [e,z,o] file which depends on the x-coordinate, the y-coordinate, the eigenmodes in all textures (aa), the profile, the conditions for the incident light and a parameter file parm. The parameter file contains, for example, information about the resolution (parm.res3.npts) or which component of the EM field should be considered. This may be set by the command parm.res3.champs = [n]
where n may be a number between 1 and 6 with 1 = Ex, 2 = Ey, 3 = Ez, 4 = Hx, 5 = Hy, 6 = Hz. Additionally also the value 0 may also be chosen. In that case the information considered is not the EM field but the refractive index of the structure. To extract the information about the EM-field from the simulation the [e,z,o] file has to be saved and evaluated. Finally the near field may be plotted by using the command parm.res3.trace = 1
180
Appendix D
abbreviations, symbols, constants
List of abbreviations abbreviation
description
BaS
Barium Sulphate
DBR
Distributed Bragg Reflector
EM
Electro Magnetic
EQE
External Quantum Efficiency
FCC
Face Centered Cubic
FDTD
Finite Difference Time Domain
FMM
Fourier Modal Method
FWHM
Full Width at Half Maximum
GaAs
Gallium Arsenide
GaInP
Gallium Indium Phosphide
GaSb
Gallium Stibnite
HCP
Hexagonal Closest Package
IOF
Fraunhofer-Institut für Angewandte Optik und Feinmechanik
ISE
Fraunhofer-Institut für Solare Energiesysteme
IST
Fraunhofer-Institut für Schicht- und Oberflächentechnik
IV
Current / Voltage
IWM
Institut für Werkstoff- Mechanik
KKR
Korringa Kohn Rostoker
LBIC
Light Beam Induced Current
LD
Longeur d’Onde (wavelength)
MIT
Massachusetts Institute of Technology
MLU
Martin Luther Universtität
mso
Mikro Schicht Optik
NaCl
Sodium Chloride
181
Appendix D
abbreviations, symbols, constants
abbreviation
description
PBG
Photonic Band Gap
PC
Photonic Crystal
PL
Photo- Luminescence
PMMA
Poly- Methyl- Meth- Acrylate
PV
Photovoltaic
RCWA
Rigorous Coupled Wave Analysis
RTA
Reflection / Transmission / Absorption
SEM
Scanning Electron Microscope
Si
Silicon
SiC
Silicon Carbide
SiN
Silicon Nitride
SiO2
Silicon di- Oxide
SQ
Shockley Queisser
TE / TM
Transversal Electric / Transversal Magnetic
TiO2
Titanium di- Oxide
TIR
Total Internal Reflection
UV
Ultra Violet
Physical Constants quantity
symbol
value
unit
vacuum speed of light
c
2.998 ⋅ 10 8
ms-1
elementary charge
e
1.602 ⋅ 10 −19
As
Planck's constant
h
6.626 ⋅ 10 −34
Js
reduced Planck's constant
h
1.055 ⋅ 10 −34
Js
Boltzmann’s constant
k or kb
1.3806 ⋅ 10 −23
JK-1
permittivity of free space
ε0
8.8542 ⋅ 10 −12
Fm-1
permeability of free space
µ0
4π ⋅ 10 −7
Hm-1
182
Appendix D
abbreviations, symbols, constants
List of symbols symbol
unit
description
α
Deg
angle
α(λ)
m-1
absorption coefficient
δ
rad
phase difference
δ
deg
azimuth angle, also tilit between front side texture and back side structure
δη
relative change in efficiency
δ mp
Kronecker delta function
Δn
refractive index difference
ε
m2sterad
étendue
εext
m2sterad
étendue of emitted radiation
εinc
m2sterad
étendue of incident radiation
ε (r )
Fm-1
relative permittivity, if isotropic and homogenous only ε
η
%
efficiency
ηi
%
diffraction efficiency
ηθ
%
transport efficiency of ray in a trextured solar cell
η
S
tilted optical admittance, may be labeled
r
κρ
Fourier coefficient
λ
m
wavelength
λ0
m
design wavelength
λnq
eigenvalues of Bˆ n
Λ
m
period length
μ (r )
Hm-1
relative permeability, if isotropic and homogenous only μ
ν
s-1
frequency
r
r
ν
vector
183
Appendix D
abbreviations, symbols, constants
symbol
unit
description
Ω
sterad
solid angle
φ
fraction of photons in escape cone
φ
%
total balance of photons
φgain
%
fraction of photons gained
φloss
%
fraction of photons lost
r
ψ (r )
pseudoperiodic function
θ
deg
polar angle
θc
deg
cutoff angle for TIR / acceptance angle
Θ
Hermitian operator
ρ/τ
Fresnel amplitudes, reflection / transmission
r
ρ (r , t )
Cm-3
electric charge density
σ
eVK-1
entropy
Sm-1
electric conductivity, if isotropic and homogenous only σ
r
σ (r ) τ
Ensemble of phase space coordinates deg s-1
ω
r
χ (r )
angular frequency susceptibility
ζ
deg
zenith angle
a
m
ditch width of grating
a
m
lattice constant
an , bn
amplitudes of the separated electric field
anp , cnp
Fourier amplitudes of the expanded material parameters
r a
lattice vectors
A Aij
m²
area absorption in matrix representation
184
Appendix D
abbreviations, symbols, constants
symbol
unit
description
Aext
m2
area of emission
Ainc
m2
aperture area absorption, used to clarify the difference to area A
Abs b
m
bar width of grating
r b
reciprocal lattice vector
B, C
entries of the characteristic matrix of a thin film assembly
Bint
m-2sterad-1
internal intensity per unit angle
r r B(r , t )
T
magnetic flux density
r
matrix defining the relation between S n ( z ) and
Bˆ n
r S n' ( z )
d
m
grating depth / depth of a thin film may be labeled
d(δ)
m
distance to a side of the fluorescent concentrator
d ( x0 , y0 )
m
average distance to a side of the fluorescent concentrator
d111
m
lattice distance in the direction
D
m
sphere diameter
r r D(r , t )
Cm-2
electric induction density
EFC
eV
energy level of conduction band
EFV
eV
energy level of valence band
Eij
Vm-1
electric field in matrix representation
r r E (r , t )
Vm-1
strength of the electric field (components may be denoted Ex,y,z) filling factor of grating
f
Fy ( x, z )
Vm-1
periodic function, describing the electric field
Fm ( z )
Vm-1
diffracted modes of the electric field
185
Appendix D
symbol
abbreviations, symbols, constants
unit
description
g[λ1, λ1+dλ]
generation of charge carriers in a certain interval
gn
separation constants
H
Hamilton operator
H(z)
function for index matching
r r H (r , t )
Am-1
strength of the magnetic field (components may be denoted Hx,y,z) number of photons left in the fluorescent concentrator
I I
Wm-2
light intensity
Iinc(ν)
Wm-2
incident light intensity
Iint(ν,x)
Wm-2
internal light intensity
Iesc
Wm-2
escaping light intensity
Iext(ν)
Wm-2
external light intensity
j
Am-2
current density
jγ ,abs (hω )
m-2s-1
absorbed photon current density
jγ ,emit (hω )
m-2s-1
emitted photon current density
jγ , sun (hω )
m-2s-1
photon current density of solar photons
jc
Am-2
charge current density
j gra
A
current density for a device with grating
j mir
A
current density for a device with a perfect back side mirror
jmp
Am-2
current density at maximum power point
j pot
A
maximum possible electron generated current
jSC
Am-2
short circuit current density
r r j (r , t )
Am-2
current density
186
Appendix D
abbreviations, symbols, constants
symbol
unit
description
Jm(z)
Am-1
Fourier amplitudes in the expansion of the magnetic field
k
extinction coefficient
k
number of reflections of a photon in the fluorescent concentrator
r k
wave vector, if only one component (index x,y,z), without arrow
l
m
path length of light
l
m
path difference of light between two periods
L
fraction of photons reaching the solar cell
L
path length enhancement
L
m
side length of a fluorescent concentrator
m
index of the diffracted order
m
summation index
m
kg
mass
Mˆ n
Matrix describing one layer in the RCWA code
n
refractive index. If not denoted otherwise, real
ni
labeled refractive index
neff
effective refractive index
n
average refractive index
n ( x0 , y 0 )
fraction of photons reaching the edge of the fluorescent concentrator
n(δ ,θ , x0 , y0 , R(λ ,θ ), w,θ c )
fraction of photons remaining in the fluorescent concentrator
n ( x0 , y0 , R(λ ,θ ), w,θ c )
average fraction of photons remaining in the fluorescent concentrator
ncone
%
fraction of photons in the escape cone
nd
number of segments considered in the x-y plane
ndi
number of segments considered in the z-direction
187
Appendix D
symbol
abbreviations, symbols, constants
unit
description
nfur
number of Fourier components considered
N
refractive index. If not denoted otherwise, complex
N0
m-2s-1
total photon density, also N0(λ)
Nesc
m-2s-1
photons density in the escape cone
NR,inc
fraction of photons reflected by photonic structure related to incident light
NR,int
fraction of photons reflected by photonic structure related to internal light
N ( R(λ ,θ ), w,θ c )
average fraction of photons reaching the side of a fluorescent concentrator
N am15 (λ )
m-2s-1
photons density of the AM1.5 spectrum number of photons in the PL spectrum at a certain wavelength
N PL (λ )
r P
Cm-2
macroscopic polarization
Pmax
W
maximum power of a PV converter
q
m
lateral distance of a photon to the emitting dye molecule
QE
%
quantum efficiency
R, T, A
%
reflection, transmission, absorption, may be labeled
r
m
radius aspect ratio of grating
s s
m
distance of a photon to the emitting dye molecule
r s
propagation direction of a light wave
S nm ( z )
expansion of Fnm ( z )
t
m
thickness (of solar cell)
T
K
temperature
T0
K
temperature of PV converter
TS
K
temperature of sun
188
Appendix D
symbol
abbreviations, symbols, constants
unit
r u (r )
description
Bloch function
V
V
potential
Vmp
V
voltage at maximum power point geometrical ratio
w Wn(x)
Am-1
separation of magnetic field in x-direction
x1(x0,y0)
m
lateral distance to the side of a fluorescent concentrator
X, U, L, Γ…
crystallographic orientations
Xn(x)
Vm-1
separation of electric field in x-direction
y
S
optical susceptance
Y
S
optical admittance
Yn(z)
Am-1
separation of magnetic field in z-direction
Zn(z)
Vm-1
separation of electric field in z-direction
189
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196
Publications Publications in scientific journals related to this work
1. Light trapping, a new approach to spectrum splitting A. Goetzberger, J.C. Goldschmidt, M. Peters, P. Löper Solar Energy Materials & Solar Cells, 92 (2008), 1570-1578 2. Theoretical and experimental analysis of photonic structures for fluorescent concentrators with increased efficiencies J. C. Goldschmidt, M. Peters, L. Prönneke, L. Steidl, R. Zentel, B. Bläsi, A. Gombert, S. Glunz, G. Willeke, U. Rau. Phys. stat. sol. (a) 205, No. 12, 2811–2821 (2008) 3. Directional selectivity and ultra-lighttrapping in solar cells C. Ulbrich, S. Fahr, J. Üpping, M. Peters, T. Kirchartz , C. Rockstuhl, R. Wehrspohn, A. Gombert, F. Lederer and U. Rau Phys. stat. sol. (a) 205, No. 12, 2831–2843 (2008) 4. 3D photonic crystal intermediate reflector for micromorph thin-film tandem solar cell Andreas Bielawny, Johannes Üpping, Paul T. Miclea, Ralf B. Wehrspohn, Carsten Rockstuhl, Falk Lederer, Marius Peters, Lorenz Steidl, Rudolf Zentel, Seung-Mo Lee, Mato Knez, Andreas Lambertz, Reinhard Carius Phys. stat. sol. (a) 205, No. 12, 2796-2810 (2008) 5. Selective laser ablation of SiNX-layers on textured surfaces for low temperatur front side metallizations A. Knorz, M.Peters, A. Grohe, C.Harmel, R.Preu Prog. Photovolt: Res. Appl. 17, 127–136 (2009) 6. Increasing the efficiency of fluorescent concentrator systems Jan Christoph Goldschmidt , Marius Peters, Armin Bösch, Henning Helmers, Frank Dimroth, Stefan W. Glunz, Gerhard Willeke Solar Energy Materials & Solar Cells 93, 176–182 (2009) 7. The effect of photonic structures on the light guiding efficiency of fluorescent concentrators M. Peters, J. C. Goldschmidt, P. Löper, B. Bläsi, and A. Gombert Journal of Applied Physics 105, 1, (2009), 8. Novel methods to characterize the light guiding of fluorescent concentrators J. C. Goldschmidt, M. Peters, M. Hermle, S. W. Glunz Accepted for publication in JAP 9. The photonic light trap – improved light trapping in solar cells by angularly selective filters M. Peters, J. C. Goldschmidt, T. Kirchartz, B. Bläsi Accepted for publication in Solar Energy Materials & Solar Cells
197
Publications
Conference proceedings
1. Photonmanagement for full spectrum utilization with fluorescent materials J. C. Goldschmidt, M.Peters, P.Löper, S. W. Glunz, A.Gombert, G. Willeke Proceeding Quantsol meeting Bad Hofgastein 2007 2. Application of Photonic Structures on Fluorescent Concentrators, Marius Peters, Jan Christoph Goldschmidt, Philipp Loeper, Andreas Gombert, Gerhard Willeke Proceeding for the 22nd European Photovoltaic Solar Energy Conference and Exhibition, Milan 2007 3. Efficient Upconversion Systems for Silicon Solar Cells P. Löper, J. C. Goldschmidt, M. Peters, D. Biner, K. Krämer, O. Schultz, S. W. Glunz, J. Luther, Proceeding for the 22nd European Photovoltaic Solar Energy Conference and Exhibition, Milan 2007 4. Advanced Fluorescent Concentrator System Design J. C. Goldschmidt, M. Peters, P. Löper, O. Schultz, F. Dimroth, S. W. Glunz, A. Gombert, G. Willeke, Proceeding for the 22nd European Photovoltaic Solar Energy Conference and Exhibition, Milan 2007 5. Optical Confinement in Recrystallised Wafer Equivalent Thin Film Solar Cells S. Janz, M. Kuenle, M. Peters, S. Lindekugel, E. J. Mitchell, S. Reber, Proceeding for the 22nd European Photovoltaic Solar Energy Conference and Exhibition, Milan 2007 6. Photonic Crystals for the Efficiency Enhancement of Solar Cells M.Peters, J.C. Goldschmidt, P. Loeper B. Blaesi, A. Gombert, Proceeding for the EOS topical meeting on diffractive optics, Barcelona 2007 7. Photonic Structures for the Application on Solar Cells M. Peters, J. C. Goldschmidt, P. Löper, C.Ulbrich, T. Kirchartz, S. Fahr, B.Bläsi, S. W. Glunz, A. Gombert Proceeding for QUANTSOL winter workshop, Bad Gastein 2008 8. Progress in Photon Management for Full Spectrum Utilization with Luminescent Materials J. C. Goldschmidt, M. Peters, P. Löper, O.Schultz, F. Dimroth, A. Bett, A. Gombert, S. W. Glunz, G. Willeke Proceeding QUANTSOL winter workshop Bad Gastein 2008 9. Design of photonic structures for the enhancement of the light guiding efficiency of fluorescent concentrators M. Peters, J. C. Goldschmidt, P. Löper, L. Prönneke, B. Bläsi, A. Gombert Proceeding SPIE Strassbourg 08
198
Publications
10. Directional selectivity and light-trapping in solar cells Carolin Ulbrich, Stephan Fahr, Marius Peters, Johannes Üpping, Thomas Kirchartz, Carsten Rockstuhl, Jan Christoph Goldschmidt, Philipp Löper, Ralf Wehrspohn, Andreas Gombert, Falk Lederer and Uwe Rau Proceeding SPIE Strassbourg 08 11. Lighttrapping with Angular Selective Filters Marius Peters, Jan Christoph Goldschmidt, Philipp Loeper, Benedikt Bläsi, Gerhard Willeke Proceedings of the 23rd European Photovoltaic Solar Energy Conference and Exhibition, Valencia 2008 12. Upconversion for Silicon Solar Cells: Material and System Characterisation P. Löper, J. C. Goldschmidt, M. Peters, D. Biner, K. Krämer, S. W. Glunz Proceedings of the 23rd European Photovoltaic Solar Energy Conference and Exhibition, Valencia 2008 13. Efficiency Enhancement of Fluorescent Concentrators with Photonic Structures and Material Combinations J. C. Goldschmidt, M. Peters, F. Dimroth, S. W. Glunz, G. Willeke Proceedings of the 23rd European Photovoltaic Solar Energy Conference and Exhibition, Valencia 2008 14. Material Characterization for Advanced Upconverter Systems S. Fischer, J. C. Goldschmidt, P. Löper, S. Janz, M. Peters, S. W. Glunz, A. Kigel, E. Lifshitz, K. Krämer, D. Biner, G. H. Bauer, R.Brüggemann Proceedings of the 23rd European Photovoltaic Solar Energy Conference and Exhibition, Valencia 2008 15. Advanced Upconverter Systems with Spectral and Geometric Concentration for high Upconversion Efficiencies Jan Christoph Goldschmidt, Philipp Löper, Stefan Fischer, Stefan Janz, Marius Peters, Stefan W. Glunz, Gerhard Willeke, Efrat Lifshitz, Karl Krämer, Daniel Biner Presented at the IUMRS-ICEM 2008 International Conference on Electronic Materials, Sydney 2008 16. Simulation of Fluorescent Concentrators M. Bendig, J. Hanika, H. Dammertz, J. C. Goldschmidt, M. Peters, M. Weber Interactive Ray Tracing 2008 17. Photonic structures and solar cells M. Peters, J.C. Goldschmidt, P. Löper, B.Bläsi Proceeding for QUANTSOL winter workshop, Rauris 2009 18. Silicon quantum dot superstructures for all-silicon tandem solar cells P. Löper, M. Künle, A. Hartel, J. C. Goldschmidt, M. Peters, S. Janz, M. Hermle, S. W. Glunz, M. Zacharias Proceeding for QUANTSOL winter workshop, Rauris 2009 19. Photon Management with luminescent Materials J. C. Goldschmidt, S. Fischer, P. Löper, M. Peters, L. Steidl, M. Hermle, S. W. Glunz Proceeding for QUANTSOL winter workshop, Rauris 2009
199
Publications
Books
1. Photonic Concepts For Solar Cells M. Peters, A. Bielawny, B. Bläsi, R. Carius, S.W. Glunz, J.C. Goldschmidt, H. Hauser, M. Hermle, T. Kirchartz, P. Löper, J. Üpping, R. Wehrspohn, G. Willeke in Physics of nanostructured solar cells, V. Badescu and M. Paulescu (Editors) Nova Science Publishers (to be published 2009) Other publications
1. Incorporating a secondary electrode in the dye solar cell U. Würfel, M. Peters, A. Hinsch, R. Kern Proceeding for the 20th European Photovoltaic Solar Energy Conference and Exhibition, Barcelona 2005 2. Developments for Dye Solar Modules, Results from an Integrated Approach A.Hinsch, S. Behrens, H. Bönnemann, A. Drewitz, F. Einsele, D. Faßler, D. Gerhard, H. Gores, S. Himmler, G.Khelashvili, D. Koch, G. Nasmudinova, U. Opara-Krasovec, M. Peters, P. Putira, U. Rau, R. Sastrawan, T. Schauer, S. Sensfuß, C. Siegers, K. Skupien, J. Walter, P. Wasserscheid, U. Würfel, M. Zistler Proceeding for the 21st European Photovoltaic Solar Energy Conference and Exhibition, Dresden 2006 3. Zurück in die Zukunft - Mit neuen Konzepten erlebt eine alte Konzentratortechnologie ihre Renaissance J.C. Goldschmidt, M.Peters, F. Dimroth, S.W. Glunz Erneuerbare Energien 2008 4. Detailed Experimental and Theoretical Investigation of the Electron Transport in a Dye Solar Cell by Means of a Three-Electrode Configuration U. Würfel, M. Peters, and A. Hirsch J. Phys. Chem. C 10.1021 jp077016, (2008) Patents
1. “Solarelement mit gesteigerter Effizienz und Verfahren zur Effizienzsteigerung” Inventor: J.C. Goldschmidt, P. Löper, M. Peters, Patentnummer: DE 102007045546 A: 20070924 2. "Düseneinheit, Verfahren zur Herstellung der Düseneinheit sowie eine Vorrichtung mit einer Düseneinheit" Inventor: D. Kray, K. Priewasser, M. Peters, Patent pending 3. “Photovoltaik-Vorrichtung und deren Verwendung” Inventor: A. Götzberger, J.C. Goldschmidt, P. Löper, M. Peters, Patent pending
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Acknowledgments This thesis emerged from the work done for the projects “Nanosun” and “Nanovolt” at Fraunhofer ISE in the past three years. During this time I had the opprtunity of working with many inspiring characters. Quite a few of them contributed directly or indirectly to this work. At this point I want to thank everyone for the wonderful time, the discussions and the aid they gave me. I thank Prof. Eicke R. Weber for giving me the wonderful opportunity to do this work at Fraunhofer ISE. I thank Dr. Benedikt Bläsi for all the encouragement, the many discussions, the support he gave me and his indefatigable eagerness in correcting my numerous mistakes. I thank PD Dr. Andreas Gombert for his advice and I wish him always six inches of water under his keel I thank Jan Christoph Goldschmidt for the fellowship, the working atmosphere, buying ice cream, sharing loony ideas, brief the excellent time. I wish him all the best for his marriage. I thank Philipp Löper for the collaboration, the working atmosphere and the tea, for climbing up rooftops and rocks. I thank Hubert Hauser and Marcel Pfeiffer both for collaboration and fruitfull discussion, results as well as written and read texts. I thank everyone in the group MIO for the nice working atmosphere and for assistance with different measurements: Andreas Johannes Wolf, Michael Nitsche, Jörg Mick, Volker Kübler and all the HiWis. Also I thank all the people from the department MAO for the manifold support; especially Dr. Werner Platzer, Dr. Armin Zastrow, Franz Brucker and Christina Hildebrandt for organizing the ultimate Frisbee group. I thank the PhD students from the projects Nanosun and Nanovolt: Andreas Bielawny for cold showers, Liv Prönneke for the memory of Philipp Osram, Carolin Ulbrich for trading keys, Johannes Üpping for long walks through Jena, Thomas Kirchartz for shrewd ideas, Stephan Fahr for barbecue, Christian Helgert for electron beams and a night out in Barcelona and Lorenz Steidl for spheres and Blasenwürste. It was a pleasure working with you. I thank everyone else from the projects NANOSUN and NANOVOLT for making the excellent collaboration possible. Prof. Uwe Rau, Dr. Carsten Rockstuhl, Prof. R. Wehrspohn, Dr. R. Carius, Prof. F. Lederer, Prof. R. Zentel, PD Dr. Stefan Schweizer, Prof. Gero von Plessen, Prof. G. Bauer as well as everyone else in their working groups who took part in the projects. I thank Dr. Pauline Voisin, Dr. Martin Hermle, Dr. Stefan Janz, Dominik Suwito and Matthias Künle for the enthusiasm, SiC for all circumstances and beautiful stars. I thank Johannes Hanika especially for the Voronoi mapper and the people from University Ulm for the nice cooperation, especially Sabrina and Holger Dammertz, Marion Bendig und Christian Kempter. I thank Marcel Rothfelder for his great work with the elipsometer.
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Acknowledgments
I thank Tim Rist, Juliane Klatt and Wesley Dopkins for measurements, computer work and corrections. I thank my former bureau cohabitants Gerhard Peharz, Armin Bösch and Alexander Wekkeli especially for ice cream breaks in summer. I thank Elisabeth Schäfer for many measurements in the neatest lab I’ve ever seen. I thank Dr. Stefan Glunz, especially for the silicon forest workshops. I thank Prof. Adolf Goetzberger for the opportunity of working with him and the many interesting discussion and ideas Additionally I want to thank Bernhard Groß, Peter Kailuweit, Stefan Fischer and Simon Philipps as well as Lisbeth Rochlitz and Marc Hofmann for the ISE music night. I thank everyone from the F-floor at the solar info center, especially those who kept the coffee machine running. I want to thank everyone who had a part in this work and whom I did not mention here by name. I humbly ask for their indulgence. This includes all the people from ISE who kept up the most pleasant working atmosphere, all the people from FMF and the University of Freiburg. Especially, I thank all my friends and my family who accompanied me during my time here and gave me a great time in Freiburg. And finally I thank Astrid for everything.
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