Advanced solar cell materials and solar cells ...

Advanced solar cell materials and solar cells analytical modeling Abdelhalim Zekry Ain Shams University Abstract In this paper, a review of the advanced solar cell material will be presented. These materials will be compared with metallic semiconductor materials. Then analytical models are developed for the organic and perovskite solar cells. Estimated performance parameters of the solar cells show satisfactory agreement with the measured values published in the literature. Anomalies in the behavior of some of such cells are accounted for and a model is proposed for it. 1 organic versus metallic semiconductors Most advanced solar cells are made of either organic or hybrid organo metallic materials. So, in this section, it is required to compare the two classes of semiconductors the well-known metallic and the to-know organic. What are the similarities and differences between them? [1] The first criterion of the comparison is the chemical bonding among the constituting units of the material: The metallic semiconductors are characterized by covalent and ionic bonds between their constituting atoms such as silicon and thereby they crystalize in diamond or zinc blende structure. The organic semiconductors are molecular in nature and thereby form van der wal forces between them. Such bond is much weaker than the covalent bod. Both type of materials can be crystalline, polycrystalline or amorphous. Because of the molecular nature of the organic material and their weak bond, the energy level splitting of the molecular orbitals will be small and therefore the bands will be narrow. The valence band in the organic semiconductor is termed homo band and the conduction band is termed lumo band. The density of state in every band is assumed Gaussian [2]. In the metallic semiconductor crystals the splitting in every band is much larger leading to wide energy bands. The density of states is parabolic near the band edges. The band gap in the organic material is direct as it preserves the molecular nature while the band gap can be direct like gallium arsenide or indirect like silicon. The energy band structure affects the electrical and the optical performance of the material. So, the organic materials have much larger effective mass than the metallic semiconductor and therefore much less mobility [3] as the mobility is inversely proportional to the effective mass.

If any material is polycrystalline the mobility will be further reduced as the grain size decreases. In the limit if the material becomes amorphous, not only the mobility will be degraded to its minimum value but also the effective energy gap increases. It is so that the organic materials are prepared using low temperature processes [4] such as solution based method resulting in either amorphous or at the best polycrystalline materials compared to crystalline or semi crystalline metallic materials. This contributes to further lowering of mobility of the organic material. However one has not to forget that this material production method runs at low temperature and with much simpler apparatus rendering them very cost effective. The other important property of the organic semiconductor that relay on their unique energy band structure is that they have a very large absorption coefficient of light [5] such that only a fraction of a micrometer of the material is sufficient to fully absorb the radiation in their absorption range. This makes them a strong alternative as a solar cell material. Now we continue the discussion; this time about the mobile carrier concentrations and their sources in the materials. Generally in any semiconductor there are two types of mobile charges, the electrons in the conduction band and the holes in the valence band. If the material is pure and intrinsic the source of the electrons and holes will be the thermal generation of electron hole pairs. The concentration of the electrons will be equal to that of the holes n0=p0=ni. According to the mass action law, no po = ni2. The intrinsic concentration ni2= Nc Nv exp Eg/kT, where Nv, Nc are the effective density of states, Eg is energy gap and T is the absolute temperature. This law is a general low and applicable for all semiconductors. Organic semiconductor materials have relatively high energy gaps compared to the most common semiconductor metallic material, the silicon Si. The effective density of states according to the data in the literature [6] are not much different from those of silicon, therefore the intrinsic concentration of the organic materials is very small. Accordingly, organic materials approach the insulators rather than the semiconductors. In the sense, practically in their intrinsic state they behave as insulator. As a matter of purity of the material, metallic semiconductor are produced with high purity but the organic semiconductors are produced with less purity such they contain impurities leading to make them either p-type or n-type conducting. Once doped either intentionally or unintentionally, one can define for them a Fermi level as the in the metallic material. Otherwise one describes them as insulators in the sense

their energy levels are aligned to the vacuum level. If the material is doped, the energy level through a system of materials is aligned to Fermi level at thermal equilibrium. This is a very important point and it is investigated intensively in the literature [7] The doping of the metallic semiconductors is by atomic substitution while in the organic material it is by adding doping molecules. There are donor molecules and acceptor ones. Both materials react to the incident light with the proper wavelength by generating excitons in the materials. The excitons dissociate directly after their generation because of the larger screening between their electrons and holes as the dielectric constant of the metallic semiconductors are much larger than those of organic materials. So, one can speak of photogeneration of electron hole pairs. In contrary, the excitons are strongly bound in organic materials and need assistance to be dissociated before they recombine again. This assistance can be accomplished by an electric field or by donor acceptor interfacing. Electrons and holes can be injected by forming junctions with different materials, this is valid for both type of materials. This concerns the mobile charge carriers and their main sources in the both classes of materials. Now, we will try to compare the current conduction in the two classes of materials. The mobile charge carriers can drift under the influence of the applied electric field and can also diffuse under the concentration gradients. So, there will be the drift current and the diffusion current with their known formulations: Jdrift= sigma E, where sigma is the electric conductivity and E is the intensity of the electric field, and Jdiff= - qDp dp/dx where q is the electron charge, D the diffusion constant and dp/dx is the concentration gradient for holes. Similar equation can be written for electrons In organic semiconductors, assuming they are free of charge carriers, they behave as insulators where the current in them will be either limited by injection, injection limited current, or by building up space charge in the bulk of the organic material. This is because the mobility of organic materials is very low. The mobility of the charge carriers in amorphous organic semiconductors is modeled by hopping motion between energy sites separated by potential barriers with help of the applied electric field and the thermal energy. Allowed sites are irregularly spaced from each other with an average distance (a) and disorder factor gamma. The energy distribution of the allowed sites follows a Gaussian distribution with a standard deviation sigma.

According to this hopping model the mobility is given by u = uinf exp[ -( 3sigma/5kBT)^2+ 0.78((sigma/kBT)^3/2 - gamma) sq. root(qaF/sigmma), where uinf is the mobility in the limit T tends to infinity, sigma gives the width of the Gaussian distribution of the density of states, kB is Boltzmann's constant and T the temperature, gamma gives the positional disorder of transport sites, a is the intersite spacing, and q and F are the elementary charge and the field strength, respectively.One sees from the given expression that the mobility increases by increasing the electric field and temperature [8], [9] recombination The recombination is the disappearance of a mobile electron in a hole. This process leads to the loss of mobile charge carriers and thereby affects much the electrical characteristics of the semiconductor devices including solar cells and light emitting diodes. The recombination mechanisms were studied intensively in metallic semiconductors and to less extent in the organic semiconductors. From the conceptual point of view, the recombination mechanisms occurring in the metallic semiconductor also occur in the organic semiconductors. These mechanisms can be classified into radiative and nonradative types. Radiative recombination is a consequence of the direct fall of electrons from the conduction band to the valence band while the nonradiative one when the fall of the electrons occurs through trap levels in the bandgap. These trap levels are called recombination centers. Every recombination mechanism has its specific dependence on the hole and electron concentrations and its rate constants.There is other recombination mechanism which exists in the organic semiconductors. It is the Langevin mechanism which occurs as direct consequence of an electron and holes comes in the field of the other. They get this chance when they meet while moving with low mobility. Its recombination rate r lv can be expressed by the expression: r lv=q (mun+mup) pn/ epsilon, Where mun, mup are the electron and hole nobilities, p,n are the hole and electron concentrations, epsilon is the permittivity and q is the electron charge. It is found that this mechanism is applicable in case of the organic LEDs, while it must be reduced by an appreciable factor when applied to organic solar cells.Generally the rate can be expressed by r lv= kr pn, where kr is the recombination rate constant that can be determined experimentally. On contacting metals with organic material MOrS, The first challenge for determining the current of such junction is the energy alignment at the metal organic semiconductor interface. When the semiconductor material

is doped then one can define for it a Fermi-level and accordingly the Fermi level will be constant at the interface similar to the metalinorganic semiconductor. In case of intrinsic organic semiconductor it behaves as an insulator because of the relatively large bandgap. In this case, the alignment will be according to the vacuum energy level. That is at equilibrium, the energy levels align themselves to the vacuum level Evac. Consequently, the electron transfer from one material to the other is governed by the energy difference between the two intended levels and their probability of occupation. Such simplifying assumption can help get mathematical expressions for the currents. However, experimental measurements showed that there is a potential step at the metal organic semiconductor that may be due to unintentional doping of the material. But here we will assume that the material is free of doping. Inspecting the literature concerned with the currents in the MOrS diodes with one charge type it is found that the current can be either injection limited from the M-S contact or it can be space charge limited. The space charge limited current exists when it is much less than injection current. The injection current is limited by the height of the injection barrier at the contact which is more or less independent of the applied voltage except the image force barrier lowering and equal to the contact difference of potential for the ideal interface. When the contact is injection rich, it can act as an Ohmic contact. So, in case of the Ohmic contacts the current will be space charge limited. Organic semiconductors are significantly sensible for the humidity as well as oxygen. Therefore they are needed to protect them permanently by proper encapsulation. In addition, they are sensitive to temperature and they can get unstable with severe environmental variations. 2 The mixed donor-acceptor bulk heterojunction solar cell The most prevalent organic solar cell is the bulk heterojunction solar cells. It has the highest achieved conversion efficiency till now and it is nominated for commercialization [10]. The physical model of such solar cells is shown in Fig 1.

Fig. 1 The physical model of organic bulk heterojunction solar cells

It consists of the absorber donor acceptor mixture layer sandwiched between the etl flour doped tin oxide acting as a transparent anode. The htl must be heavily doped p-type material to form ohmic contact with the FTO layer and at the same time its valence band edge must lie equal or above the conduction band edge of the mixture while its conduction band edge must lie much above the conduction band edge to reflect the electrons back to the absorber and accept the holes coming from the absorber. The etl on the other side of the absorber must be heavily doped n-type

Fig. 2a,b The energy level diagram before and after contacting material to build an ohmic contact with the cathode metal and its conduction band edge must be aligned or under the conduction band edge of the mixture to accept the electrons coming from the absorber and have a valence band edge higher than that of the absorber to reject holes back to absorber. The energy band diagram before and after contacting the solar cell materials are shown in Fig 2a and 2b. When the solar cell is illuminated, the material then absorbs all the incident solar radiation in its absorption wavelength range. Also assuming the mixing is complete, every donor molecule is connected to an acceptor molecule which means that the excitons will be dissociated once formed as they find directly a dissociation acceptor agent. Therefore, the excitons transport and dissociation efficiencies are close to one. Then, the solar conversion efficiency will be limited by the collection of the generated electron holes and the dark current as in the metallic solar cells. We will derive expressions for the collected photocurrent IL and the dark current in such cells. The collected photocurrent IL Here it is assumed that solar cell absorber thickness dd makes out the whole active layer thickness and the absorption coefficient alpha is then equal for the mixed material of double the thickness of the donor material. So, a thickness of active material dt of 200 nm will generate

electron hole density of Iph= q alpha F dd =16 mA/cm^2 with absorption coefficient alpha= 5x10^4 /cm, the incident photon flux at AM1 F= 1017 photons/cm2 and q the electron charge = 1.6x10-19 As. This current must be collected by the electric field associated with the built in voltage phi. The incident light will generate excitons which will dissociate at the donor acceptor interface and build polarons. The electrons of the polarons will be dispatched by the electric field to the cathode while the holes will be swept to the anode. While the generation of the polarons are homogeneous in the volume the collected hole current will change linearly from zero at the side of the etl, it will be maximum at the htl side as depicted in Fig 3.

Fig. 3 The photoelectric effect and the distribution of currents and charge carriers The distribution of collected electron current will be reversed where it will be zero at the side of the htl and maximum at the etl as demonstrated in fig 3. The hole distribution follows the hole current while the electron distribution follows the electron current as shown fig 6.3. The hole concentration p(y) is then given by p(y)= Jph (dt-y)/ (qdt up E), with the parameters have their usual meaning, Likewise, the electron concentration can be written as n(y)= Jph y/ (qdt un E), The recombination current will be now a volume recombination following the Shockley Reed Hole relation as: Jrec= Integral q CNt [(p(y) n(y) – nie^2)/ (p(y) +n(y))] dy, Where C is the recombination coefficient and Nt is the volume trap density. Assuming un=up, the electron and hole concentration distribution will be linear and symmetrical around the horizontal line y= dt/2 as depicted in Fig 3. Substituting p(y) and n(y) in the current equation and integrating over the whole thickness dt, we get a final expression

Jrec= q C Nt nm dt/3, with nm is the maximum electron current at y=dt as illustrated in Fig 3. Substituting nm in the above expression of the recombination current we get finally Jrec= Jphdt/3 un E Tau, where Tau is the lifetime= 1/CNt. The above equation can be further reduced to meaningful form Jrec= (dt/ Lef) Jph, with Lef= (3 un E Tau) is the effective drift length where it is equal to three times the drift length of the electrons in the their lifetime. To make Jrec much smaller than the photocurrent one has to construct the cell such that the effective drift length be much greater than the thickness of the mixed absorber material. This is a very important design criterion. We see from Fig 3 of the electron and hole distribution that the center of the electrons are displaced from that of holes leading to a debiasing electric field in the opposite direction of the built in electric field. This results in enhanced recombination current losses. However, this effect will be negligible as the structure criterion mentioned above is satisfied. The effect of biasing is straight forward where the net potential barrier will be phi-V, with V the forward bias voltage. The equation of Jrec will be modified only by reducing phi to a phi-V leading to the increase of Jrec and the collected photocurrent JL becomes smaller. The dark current and the illuminated current Basically the current is a recombination current in the volume of the active layer. When a forward bias is applied on the cell, the hole concentration in the donor layer varies exponentially with the potential barrier according to the Boltzmann relation, then the hole concentration can be expressed by the relation p= phtl exp (-phi+V)(dt-y)/Vtdt, And the electron concentration by n=netl exp (-phi + V) y/ dt Vt, The current density due to the volume recombination in the active layer is given by Jd= integral over y [q (np – nie^2)/ (n + p)] dy / Tau, This current will be maximum at n=p=nie exp V/2, It follows that Jd as a maximum possible value, can be written in the form, Jd= q nie(exp V/2 -1) dt/ Tau = Js( exp V/2 -1), with Js= q nie dt/Tau With Tau= 1 ms, nie=10^10/cm^3, and dt=0.2 µm the reverse saturation current Js= 3.2x10^-11 A/cm^2, The open circuit voltage is achieved when the dark current = the photocurrent, then Voc= 2Vt ln (Jph/Js) = 1 V, which is greater than that of the finger structure. If Tau= 0.1 ms, then Voc becomes 0.88 V. The effect of the illumination on the dark characteristics is that nL and pL will be higher than that of the dark only. In case of relatively high nL and

pL compared to nd and pd, the da concentrations, The combined net recombination current Jrecdl cannot be expressed as a superposition dark current and recombination photocurrent, but it can be calculated directly using the recombination current formula above. Then the terminal current J of the solar cell = Jph – Jrecdl. With a fill factor of 0.8, this cell has a conversion efficiency of about 11 percent in agreement with the achieved organic solar cell efficiency now [11].

3 The perovskite solar cells The perovskite solar cells are the most recent photovoltaic devices. Perovskites are composed of organic metal halide. The best material till now is CH3NH3PbI3 [12]. It has good optoelectronic properties for solar cell applications. In addition, It has relatively high dielectric constant, a direct bandgap near the optimum value for the highest conversion efficiency, a high absorption coefficient as Ga AS and better in some wavelengths, high mobility and small effective masses with the electron and hole mobility in the same range enhancing the bipolar conduction, relatively high diffusion length in the order of one micrometer, highest stability compared to the other perovskites, easy to synthesize and produce in thin films at low temperatures. It gave till now photoconvertion efficiency PCE, of more than 20 percent approaching that of single crystal silicon solar cells [13]. However, they are still having some obstacles for commercialization such as instability and long term stability against environmental stresses like temperature and incident solar radiation. The inclusion of toxic Pb is also undesirable. There are research efforts to substitute Pb or to mitigate its toxicity by tight encapsulation of the solar cells and end of life recycling [14]. We are going to develop a generic model of such solar cells as we did with the organic solar cells. The physical model of such solar cells is shown in Fig 4.

Fig. 4 The physical model of perovskite solar cells It consists of the absorber perovskite layer sandwiched between the htl and the metal anode from one side and the etl and the glass substrate coated with flour doped tin oxide acting as a transparent cathode. It will be assumed here that the perovskite layer will be intrinsic and as the perovskite has wide bangap, it will be

almost initially free of charge carriers and behaves much like the absorber in the bulk heterojunction organic solar cells. The function and properties of the transport layers are the same as in the organic solar cells. The etl must be heavily doped n-type material to form ohmic contact with the fto layer and at the same time its conduction band edge must lie equal or under the conduction band edge of the perovskite while its valence band edge must lie much under the valence band edge to reflect the holes back to the absorber and accept the electrons coming from the absorber. The htl on the side of the absorber must be heavily doped p-type material to build an ohmic contact with the anode metal and its valence edge must be aligned or under the valence band edge of the

Fig. 5a,b The energy level diagram of perovskite solar cell before and after contacting pervoskite to accept the holes coming from the absorber and have a conduction band edge higher than that of the absorber to reject the electrons back to absorber. The energy band diagram before and after contacting the solar cell materials are shown in Fig 6.5a,b. The contact difference of potential phi= phih-phie. If one positions the Fermi levels at the conduction band edge and the valence band edge in the htl and etl, respectively , one gets the highest possible work function difference equal to energy gap of the perovskite. If the conduction in such perovskite solar cells is analogous to the conduction in the hybrid bulk solar cells, then all derived current voltage characteristics will be applicable and one can adapt them for the description of the electrical characteristics. This is the case if the perovskite behaves as an insulator

with low motilities such as in the donor acceptor blend. Assuming that the photogeneration rate is G and the material absorbs uniformly through its thickness dp, then the photo generation current will be Iph= qG dp A/cm2. With G= 5x1021/ cm3, dp= 0.2 um, Iph= 16 mA/cm^2. Practical devices have double this thickness, so the photocurrent may amount to 32 mA/cm2. The generated excitons will be dissociated thermally once generated because the dielectric constant of the perovskite material is high enough to screen the binding force in the excitons. This is one of the merits of the perovskites compared to organic semiconductors. The built in electric field in the perovskite will separate the electron hole pairs and convey them to the transport layers and then to the metallic electrodes. The electrons to the cathode and the holes to the anode. It follows that an electron charge and hole charge will be accumulated in the perovskite as shown in Fig 6.

Fig. 6 The current and charge distribution in illuminated perovskite solar cell They have the similar distributions of electrons and holes as in the organic solar cells. The major difference here is the much lower concentration due the much higher mobility of electrons and holes in perovskite. Consequently the backbiasing will be much less pronounced and negligible. Also, the recombination current Jrec has same expression as in the previous cells. Jrec= Jph dp/3 µn E Tau= Jph/ Lef, with Lef= 3 µn E Tau, the drift length in the perovskite. With the conventional parameters of µn= 1 cm2/vs, E= 1.5V/ 0.2 um= 7.5x10^4 V/cm, Tau= 0.1 us, Jrec= 0.0088 Jph, which means the photorecombination current is negligible. In fact the perovskite is much better performing than the donor acceptor organic absorber. If the device is forward biased at the same time by connecting a load resistance across it, the built in electric field will be smaller and the photo recombination current becomes larger as the

effective drift length decreases. However, this increase will be still have negligible impact on the collected photocurrent JL since JL= Jph- Jrecp. When the device is forward biased under dark condition, the perovskite will be injected by electrons from the etl and holes from the htl leading to excess of electrons and holes which may recombine with each other through traps in the volume of the material. The rate of the recombination process determines the dark current as in the bulk heterojunction organic solar cells, bhjosc. Therefore, one can apply the same dark current expression of the bhjosc to the perovskite solar cells. It follows that Jd at a maximum possible value of the recombination rate, can be written in the form, Jd= q nie(exp V/2Vt -1) dp/ Tau = Js( exp V/2 -1), with Js= q nie dp/Tau With Tau= 1 us, nie=10^7/cm^3, and dp =0.2 µm the reverse saturation current Js= 3.2x10^-11 A/cm^2. The effect of the illumination on the dark current is that it increases the excess concentration of both electrons and holes in the active region. This will lead to higher recombination rate consequently an increase in the dark current. But this extra recombination current is already considered previously as Jrecp. The question is whether the overall recombination current Jrect will be equal the superposition of the dark current and the recombined photocurrent. In this case then the superposition theorem will be applied. This will be valid if the recombination rate is proportional to the excess carrier concentration, which will be satisfied if the carrier lifetime is constant and independent of the carrier concentration. The superposition concept may be also applied when the excess due to photgeneration is much smaller than the dark concentration. If the effective drift length is much larger than the thickness of the absorber the excess concentration due to photogeneration will be very small and the superposition principle is prevailing. It follows that under this condition J= Jph- Jrecp- Jd = Jph- Jd In case of neutrality condition is satisfied in the absorber region, one can apply the classical pin diode model for the dark current. The recombination current is then given by: Jd= q ni e dp (exp V/2Vt -1)/Tau, It has the same form of the previous dark current except it is not an approximated formula. The open circuit voltage Voc= 2Vt ln (Jph/ Js), for the example solar cell here Voc= 1.07 V. The PCE is then amounts to 13.7 percent which may appreciably increase by increasing the thickness of the perovskite and approach the realized efficiency of more than 20 percent.

According to the foregoing analysis, both types of solar cells can be modeled by the classical one diode model developed for ordinary metallic solar cells. However, if the photo recombination Jrecp current is appreciable, one has to add this recombination current in parallel with the diode current. Also, one has to add the resistances of the etl and the htl. If these layers are organic they must be made thin and even they may be represented as a nonlinear resistor if the current is space charge limited in them. If the metal contacts are not ohmic, one has to add Leaky rectifying MS contacts in the equivalent circuit of the solar cell [15]. An equivalent circuit model containing all the above effects is depicted in Fig 7.

Fig, 7 The equivalent circuit of the nonideal advanced solar cells 4 Conclusions In this paper an analytical models are developed for the advanced solar structures. The models give performance parameter values which are in the range of the published measured ones. The causes of anomalies observed in practical solar cells are also discussed. A circuit model accounting for the behavior of practical solar cells is given. References

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