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For a multilayer solar cell composed of seven layers that facilitate the solar cell to use the seven colours of the white light for maximum efficiency, the expression ...
Indian Journal of Pure & Applied Physics Vol. 43, June 2005, pp. 432-438

Theory of multilayer solar cells K M Khanna, R Ekai, C K Ronno, S K Rotich & P K Torongey Moi University, Physics Department, P O Box 1125, Eldoret, Kenya Received 15 September 2004; revised 25 January 2005; accepted 18 April 2005 An expression has been derived for the absorption coefficient α of an incident radiation with energy ħω by a semiconductor with band-gap ΔE0. For a multilayer solar cell composed of seven layers that facilitate the solar cell to use the seven colours of the white light for maximum efficiency, the expression for the solar current ISC has been used to calculate the value of the charge carrier density n in each layer. ISC should be the same through all the layers for the layers to be connected in optical series. For such an arrangement, the efficiency of the multilayer solar cell turns out to be more than 42%. Keywords: Multilayer solar cell, Solar current, Charge carrier density IPC Code: F24J2/00

1

Introduction The present low efficiency of the solar cells is due to the incomplete use of the solar spectrum. Since different semiconductor materials have different bandgaps, they will separately respond to different parts of the solar spectrum and, therefore, it is conceptually possible to put several junctions in optical series. This may allow conversion efficiencies of 30-40 per cent or even more. Thus, different parts of the solar spectrum could be utilized selectively by directing frequencies of the spectrum on appropriate cell materials. This spectrum splitting capability is the key to higher efficiency of a solar cell. The main requirement is that maximum light absorption should take place along the cell thickness and the photogenerated currents may be of the order of 13mAcm-2 or more. For all stacked solar cells, proper current matching of the series connected cells is a fundamental requirement and, without this, the cell with the lowest current would limit the current of the complete device. Microcrystalline silicon (μc-Si:H) has gained a lot of interest as low bandgap absorber material in stacked thin-film solar cells during the last few years. Several groups have reported conversion efficiencies of more than 12% for stacked solar cells with μc-Si:H absorber layers1-4. For the simplest stacked cell configuration, the aSi:H/μc-Si:H tandem and a thick (300nm) a-Si:H top

cell are needed to meet the current matching requirements. Such thick a-Si:H cells are known to degrade strongly and thus the benefits of the μc-Si:H material are partly sacrificed. A concept to avoid this problem, is the triple cell configuration. For triple solar cells, the overall current will be split between three cells facilitating the use of a thin wide bandgap a-Si:H top cell with high open circuit voltage (VOC) and low degradation1. In this study, we have tried to solve some of the problems arising out of the multilayer or stacked solar cells. The problem of producing high efficiency solar cells is, therefore, to be solved in the following manner. The first step involves the preparation of crystals with band structure adjusted to the solar spectrum for maximum photon absorption leading to high efficiencies. The second step involves the understanding that there are more than two layers of materials having different optical and electrical properties. The layers will have different energy bandgaps, absorption coefficients, refractive indices, critical angles, etc. The absorption of light will depend on the values of these quantities. Generally, the absorption of light takes place in two ways, direct transitions and indirect transitions. The direct transition depends on the probability of an electron meeting a photon, whereas an indirect transition involves three particles: an electron, a photon, and a phonon. This means that the indirect

KHANNA et al.: MULTILAYER SOLAR CELLS

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transition is a less probable process than the direct transition process. Hence, light absorption coefficient should be greater for the direct transitions than for the indirect transitions. For longer efficiency solar cells, we shall assume the existence of direct transitions. In devising the theory of light absorption by electrons, we shall not take into account of the Coulomb interactions between holes and electrons created in the process of photon absorption. Every atomic system has an infinite set of discrete energy levels corresponding to a finite motion of the electron. When the potential energy of the interaction is normalized to zero at infinity, the total energy of the electrons is negative. For positive values of the energy, the electron is not bound to the ion and is moving freely. It is these freely moving electrons that will be responsible for the electric current in a solar cell. The electric current in any system depends on the number of free electrons that constitute the current and also on the velocity of the electrons since the electric current is I = neυA, where n is the number of free electrons per unit volume (carrier concentration), e the charge on the electron, υ the velocity of the electron and A is the cross-section area of the system through which the current flows. However, the magnitude of the current in a solar cell will depend on the intrinsic light absorption by the electron. Hence, it will depend on the probability of a photon being absorbed by an electron and, consequently, on the absorption coefficient α. If we represent the solar current by ISC, then we can write,

also give us the percentage increase in the efficiency of the solar cell. Since the layers are put in optical series, the same current must flow through all the layers. Looking at Eq. (1), since e and A are constant, the following must hold, i.e.,

I SC = αI = αneυA

e ⎞ ⎛ ⎜Ρ − Α⎟ 2 c ⎠ = − h Δ + ihe ( Α∇) Η=⎝ 2m∗ 2m∗ m∗c

… (1)

In a multilayer solar cell, with each semiconductor layer having its own bandwidth, the absorption coefficient will be different for different layers. Since the layers are put in optical series, ISC should then be the same throughout each layer, whereas α will be different for different layers. For seven layers, the current flowing through the different layers may be given by the relation

I SC = α i ni eυi A ,

where i = 1, 2, 3, …7

… (2)

We have, therefore, reduced the problem for calculating the values of α1, α2, α3, α4, α5, α6 and α7 to obtain the value of ISC. This value of ISC will then be compared with the values of known currents I0 in the existing solar cells. The percentage increase in current will be [(ISC - I0)/ISC] × 100 A, and this will

α1n1υ1 = α2 n2υ2 = α3 n3υ3 = α4 n4υ4 = α5 n5υ5 = α6 n6υ6 = α7 n7υ7

… (3)

Hence, in manufacturing the multilayer solar cells, the carrier concentration in each layer will have to be adjusted in such a manner that Eq. (3) is satisfied. 2

Theoretical Derivations The intrinsic light absorption by electrons will be studied using the perturbation theory. The energy of the electron in a light wave described by its electric field intensity E and magnetic field induction B is used as a perturbation. Consequently, an electron having absorbed a photon goes over from a point in the Brillouin valence zone to the equivalent point in the Brillouin conduction zone. The vector potential A(r,t) can be introduced to describe E and B as (in the Gauss system), Ε=

1 ∂Α and Β = ∇ × Α c ∂t

… (4)

The Hamiltonian of the electron (system) in a crystal acted upon by radiation in the effective mass approximation can be written as, 2

+

ihe e2 Α2 Α + div 2m∗c 2m∗c2

… (5)

The term containing A2 will be of importance only if the semiconductors are illuminated by highintensity light produced by lasers. For weak light intensity radiation obtained from conventional light sources, the term containing A2 in Eq. (5) can be neglected. Since the vector potential satisfies the Lorentz condition, we can write,

divΑ = 0

… (6)

and, then, the perturbation operator κ can be written from Eq. (5) as,

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1⎡ i he e⎡ ih∇ ⎤ i h∇ ⎤ ( Α∇ ) = − ⎢ Α( − ) ⎥ = − ⎢ Α( −e ) m*c c⎣ m* ⎦ c⎣ m * ⎥⎦

κ=

=−

1 1 [ Α(−eυ )] = − (Αj ) c c

… (7)

where, j, is the current density operator, i.e.,

j = eυ

… (8)

Hence, the Hamiltonian operator for the light field acting on electrons in a semiconductor is of the form,

Η = Η0 + κ = −

h2 Δ +κ 2m *

… (9)

It is well understood that the perturbation leads to transitions from state to state. In a semiconductor, the transition takes place from the valence band to the conduction band. Consider a state in the valence band with the energy E1(k1) and the associated wave function Ψ1k1 given as, −i 1 Ψ1k1 (r , t ) = 3 eik1r e L2

E1 ( k1 ) t h

… (10)

Similarly, in the conduction band, consider a state with the energy E2(k2) and the associated wave function Ψ1k 2 given as, −i 1 Ψ2 k2 (r , t ) = 3 eik2r e L2

E2 ( k2 ) t h

… (11)

To calculate the matrix element of the perturbation κ, we can define the vector potential in the form of a plane wave as Α (r , t ) = Α0 ei (ω t − gr )

… (12)

where g is a measure of the photon momentum such that the matrix element of the perturbation κ is given by,

∫Ψ

* 1k1

( r , t ) κ (r , t )Ψ 2 k2 ( r , t ) dτ

3

L

i ⎡ eh [ E1 ( k1 )− E2 ( k2 )+hω ]t ⎤ h ( Α0 k2 ) e = −⎢ ⎥ δ k1 + g , k2 ⎣ m*c ⎦

… (13)

Eq. (13) shows that the matrix element will be nonzero only if,

k1 + g = k2 ,

… (14)

p2 = p1 + hg

… (15)

i.e., the quasi-momentum conservation law should be satisfied in the process of light absorption. The quasimomentum of the final state (p2) is equal to the vector sum of the quasi-momentum of the initial state (p1) and the momentum of the photon (g). The transitions for which g « k1, give

p2 = p1 and k2 = k1

… (16)

That is, the electrons in going from the valence band to the conduction band retain the wave vector of the electron, and such transitions are called direct or vertical. In this study, we shall use the theory of direct transitions, although as we use the wavelengths from red to violet, g will have a finite variation. Of course, the electron having absorbed a photon will go over from a point in the Brillouin valence zone to the equivalent point in the Brillouin conduction zone. The electron absorbs energy to go over from the valence band to the conduction band will appear in the expression for the energy. Eq. (13), when used to calculate transition probability per unit time, will lead to a δ-function,

δ [ Ε1 (k1 ) − Ε2 (k2 ) + hω ]

… (17)

which provides for the energy conservation law to be satisfied in the process of light absorption, i.e.,

Ε2(k2) = Ε1(k1) + hω

… (18)

Now, the probability of electron transition from a unit volume of the k1 space into a unit volume of the k2 space per unit time is equal to,

Φ ( k1 , k2 ) =

2π e2 h2 ( Α0 k2 )2 2 2 h m* c

× δ [ E1 (k1 ) − E2 ( k2 ) + hω ]δ k1 + g ,k2

… (19)

The δ-functions in Eq. (19) emphasize the observance of energy and quasi-momentum conservation laws. In the equations that follow, we can drop the terms in δ-function since conservation laws are implied. The transition probability given in Eq. (19) can now be expressed in terms of the number of photons passing through the semiconductor. To do this

KHANNA et al.: MULTILAYER SOLAR CELLS

calculation, we make use of the fact that the average density of light energy is εE02/8π, and the energy flux is cεE02/8πµ, where ε is the permittivity of the medium, E0 electric field associated with the electromagnetic radiation, µ the refractive index of the medium, and c/µ is the velocity of light in the medium that absorbs photons. The number of photons q is obtained by dividing the energy flux by the energy ħω of one quantum, i.e.,

q=

1 c ε Ε02 hω 8π μ

… (20)

Now, the photon flux q can be expressed in terms of A02 instead of E02, and this can be done by correlating A0 and E0 and, then, by expressing the probability of electron transition in terms of the photon flux. To do this we write, using Eq. (12), ⎡

Ε=−

π⎤

1 ∂Α ωΑ0 ⎢⎣ωt − ( gr ) − 2 ⎥⎦ = e c ∂t c

… (21)

and, hence,

Ε0 =

ωΑ0 c

= gΑ0

… (22)

Substituting E0 in Eq. (20) we get,

Α 02 =

8πhcμ

εω

q

The transition probability given by Eq. (19) can now be written as,

16π 2h 2e 2 k22 μ cos 2 θ q Φ (k1 , k2 ) = cε (m* ) 2 ω

… (24)

Φ(k1 , k2 ) 16π 2h 2e 2 k22 μ cos 2 θ = q cε (m* ) 2 ω

Φ ph = σ q

c

=

μ

16π 2h 2e 2 k22 cos 2 θ ε ( m* ) 2 ω

… (25)

… (26)

To obtain the absorption probability of one photon by one electron, Eq. (26) is to be divided by the flux produced by an electron, which is ħk2/m. Hence, the probability of a photon being absorbed by an electron is,

Φ ph 16π 2he 2 k2 cos 2 θ = hk2 εm*ω m*

Φ e, ph =

… (27)

We have now to calculate the absorption coefficient α. Since the probabilities of the direct and reverse transitions are equal, when calculating the light absorption co-efficient, both direct and reverse light-induced transitions should be taken into account. Spontaneous transitions or recombination transitions will be neglected. The number of photons absorbed per unit time will be,

δ q = ∫∫ [ f1 ( k1 ) f p (k2 )Φ (k1 , k2 ) 2

dτ k1 dτ k2 4π 3 4π 3

… (28)

The first term in Eq. (28) describes the number of absorbed photons and the second⎯the number of radiated photons. Under normal conditions, the energy level occupancy by electrons coincides with their equilibrium distribution over states, i.e., 1

f1 (k1 ) = e

where θ is the angle between A0 and k2. Since the transition of the electron from one state to the other is possible only as a result of photon absorption, the quantity Φ(k1,k2) is the photon absorption probability. Since it is proportional to the photon flux q, we can get the effective cross-section σq of the absorption of a single photon flux by one electron by dividing Φ(k1,k2) by the photon flux q, i.e.,

σq =

Now, if the probability per photon of the flux c/μ is represented as Φph, then,

− f2 (k2 ) f p1 (k1 )Φ (k1 , k2 )] × … (23)

435

E1 − F kT

≈ 1, f2 ( E2 ) ≈ 0, +1

1

f p1 ( k1 ) = e

F − E1 kT

+1

≈ 0, f p2 ( E2 ) ≈ 1

… (29)

That is, the valence band is more or less filled and the conduction band is practically free or empty. Since the conduction band is empty, the reverse transitions may be neglected. If, however, an inverse energy level occupancy be created, Eq. (28) would turn negative, and the semiconductor would amplify radiation instead of absorbing it. But such a situation is of no interest to us since we are interested in the

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absorption of radiation by a semiconductor so that we can calculate the increase in efficiency of a solar cell. Multiplying Eq. (28) by ħω, we shall get the energy absorbed per unit volume of the semiconductor per unit time. We are now equipped to calculate the absorption co-efficient α, which is defined as the probability of photon absorption over a unit length. α is also defined as the energy absorbed in a unit volume per unit time. If δq is the number of photons absorbed per unit time and q the photon flux, then, α can be written as,

α=

δ q ⎛ δ q ⎞ ⎛ hω ⎞ energy absorbed =⎜ ⎟⎜ ⎟= q ⎝ q ⎠ ⎝ hω ⎠ incident energy

e2 h2 μ π 4 cε (m*)2 ω

* = mred

… (30)

∫ cos θ 2

mn*m*p

… (34)

mn* + m*p

Using Eq. (33) in Eq. (31), we get an expression for the absorption coefficient α as,

α=

Integrating Eq. (28) using Eqs (19) and (24), we get an expression for α as,

α=

where, ΔE0 = EC – EV is the band-gap energy and * mred the reduced effective mass of the electron and the hole, and is given by,

∞ h2 k22 ⎞ 4π 4 ⎛ h2 e2 μ × − Δ − δ ω k h E dk2 ⎜ 2 0 * ⎟ π 4 cε ( m*p )2 ω 3 ∫0 2mred ⎝ ⎠ … (35)

Here, 4π 3 is due to integration over the angles in which the polar angle is measured from the direction of the vector potential. To calculate the integral in Eq. (35), we put,

Vk2

×k22δ [ E1 (k1 ) − E2 (k2 ) + hω ] dτ k2

… (31)

To exclude the dependence of α on E1, E2, and k22, we make use of the fact that E2 = E1 + ħω. It should be noted that the effective mass m* is the effective mass of the electron in the valence band or the hole mass. Writing k22 in terms of ω we get, h2 k22 = E2 − EC = E1 + hω − EC 2mn* h2 k 2 = EV − 1* + hω − EC 2mp

… (32)

m *p is the effective mass of the hole, and EC and EV are the energies of charge carriers in the conduction and valence bands, respectively. For k2 = k1, and using Eq. (32), we can write

δ [ E1 ( k2 ) − E2 (k2 ) + hω ]

⎡ h2 k 2 ⎤ = δ ⎢(hω − ΔE0 ) − *2 ⎥ 2mred ⎦ ⎣

1

⎞ 2 12 ⎟⎟ x ⎠

… (36)

Using Eq. (36), we can write, 5

1 ⎛ 2m* ⎞ 2 3 k dk2 = ⎜ 2red ⎟ x 2 dx 2⎝ h ⎠ 4 2

… (37)

The integral in Eq. (35) can now be written as,

where mn* is the effective mass of the electron and

⎡ ⎤ h2 k22 h2 k22 E = δ ⎢ EV − − − + hω ⎥ C * * 2mn 2 mp ⎣⎢ ⎦⎥

* ⎛ 2mred h 2 k22 ⎜ x= k , hence, = 2 * ⎜ h2 2mred ⎝

1 2



∫ 0

5

* ⎛ 2mred ⎞ 2 32 ⎜ 2 ⎟ x δ (hω − ΔE0 − x) dx ⎝ h ⎠

5

3 1 ⎛ 2m* ⎞ 2 = ⎜ 2red ⎟ (hω − ΔE0 ) 2 2⎝ h ⎠

Substituting Eq. (38) into Eq. (35) we get,

α = A(hω − ΔE0 )

3 2

… (39)

where the constant A is given by,

… (33)

… (38)

5 * e 2 (2mred )2 A= 3 3 3π ch ω (m*p ) 2 μ

2

KHANNA et al.: MULTILAYER SOLAR CELLS

Eq. (39) shows that for vertical transitions in the small range hω − ΔE0 , the absorption coefficient α is 3

proportional to (hω − ΔE0 ) 2 . Substituting Eq. (39) into Eq. (2), we get an expression for the so-called matching current ISC in a multilayer solar cell, i.e., 3

I SC = A(hω − ΔE0 ) 2 eniυi

… (40)

where i = 1, 2, …7, for the seven layers of the solar cell. The velocity υ of the charge carrier will depend on the different values of ħω and ΔE0 since for each layer in the solar cell, ħω and ΔE0 will change. For a given material of the layer, ħω and ΔE0 will be known, and since ½meυi2 = (hω − ΔE0 ) , the value of υ will be known. Knowing the value of the refractive index μ for each layer, the value of A can also be calculated. The problem will be to fix the value of the carrier concentration ni for each layer such that ISC is the same in the solar cell since all the seven layers are in optical series. 3

Results and Discussion In the fabrication of multilayer solar cells, the range of carrier concentration that has been used so far2 is between 4 × 1016 and 5 × 1019 cm-3, and the thickness between 0.01 and 3.5 μm. To fabricate a multilayer solar cell, each layer will have to be grown with accurate carrier concentration and thickness. For a proper matching current ni may have to be adjusted such that ISC is the same through each layer.

437

For a given colour used in the multilayer solar cell,

ω will vary with the colour and so is the value of A. As a result, the value of α will be different for different layers constituting the multilayer solar cell. Once α is known, ISC can be calculated from Eq. (1). The value of ISC has to be fixed that will flow through each layer, and using Eq. (1), we can calculate the value of the carrier concentration n for each layer. This value of the carrier concentration can be used to fabricate the multilayer solar cell. In the spectrum range of λ between 4000 and 8000Å, the energies associated with each wavelength are given in Table 1. The values of the refractive indices of the materials used in this research have been taken from the Handbook of optical constants3. In a recently reported three-layer tandem solar cell2, the open circuit voltage, the short circuit current density and the average resistance per layer are 2.42V, 13.44mAcm-2, and 60Ω, respectively. If the short circuit current density for the seven-layer solar cell is 10mAcm-2, then, the open circuit voltage will be 4.2V. Consequently, the seven-layer solar cell will have a resistance of about 420Ω. Using the relations derived in this work together with appropriate tandem solar cell parameters, a 4.2V open circuit voltage of a seven-layer solar cell gives an efficiency of 42%. Figure 1 shows the carrier concentration as a function of the short circuit current and the wavelength of the incident radiation. As can be seen in Fig. 1, a multilayer solar cell with a short circuit current of 10mAcm-2 has a carrier concentration ni in the layer of InGaP(Ca2S) material of around 2.8 × 1017 charge carriers per cm3. Since the bandwidth of InGaP is 1.9eV, a current of 20mAcm-2 will, then, give a carrier concentration ni of about 5.6 × 1017 particles per cm3, which is within the limits suggested in Ref. 2.

Table 1⎯Bandgap energies of incident radiation as a function of wavelength λ, and values of bandgap energies of selected semiconductor materials in the wavelength range between 4000Å to 8000Å λ (Å)

Photon energy (eV)

4000 4500 5000 5500 6000 7000 8000

3.11 2.76 2.48 2.26 2.07 1.78 1.55

Material

SiC(Hex) Al2Te3 SiC(Cubic) Se(ZnTe) InGaP(Ca2S) CdSe GaAs

Band-gap energy (eV)

Index

2.9 2.5 2.3 2.1 1.9 1.7 1.4

2.765 3.010 2.664 3.111 3.352 2.812 3.679

μ

α × 10-7 (m-1)

Charge carriers n (1017 cm-3)

9.054 0.134 9.984 7.498 8.452 3.408 8.718

2.567 1.547 2.458 3.530 3.021 11.320 3.093

INDIAN J PURE & APPL PHYS, VOL 43, JUNE 2005

n (c m

-3

)

438 1 0

1 9

1 0

1 8

1 0

1 7

1 0

1 6

λ λ λ λ λ λ λ

0

5

= = = = = = =

1 0 1 5 2 0 2 5 -2 IS C ( m A c m )

0 0 0 0 0 0 0

.4 .4 .5 .5 .6 .7 .8

μm 5 μm μm 5 μm μm μm μm

3 0

Fig. 1⎯ Charge carrier concentration as a function of the short circuit current

4

Conclusions A comprehensive theory of multilayer solar cells that considers the perturbation of electrons and holes has been developed. The theory relates the short circuit current to the absorption coefficient, electronic charge, velocity of the electrons and the carrier concentration. Results show that a seven-layer solar cell that uses the seven colours of light has carrier concentrations in the range between 1.2 × 1016 and 5.0 × 1018 cm-3. Additional results show that the calculated efficiency of the seven-layer solar cell is about 42%. Incidentally, at present, we do not have the experimental facilities to manufacture such a solar

cell. However, laboratories with appropriate facilities may manufacture such solar cells with the information contained in this paper. It should be pointed out that the photoconductivity4 of a-Se85Te15 has been studied to understand the effect of Sn impurity on the photoconductive behaviour of binary Se85Te15. This is one of the materials used in our multilayer solar cell. Our future calculations will include the effects of energy barriers at each interface, series resistance in stacked solar cells since it will be large, and the reverse transitions from the conduction band. If laser light is used to illuminate the surface of the semiconductor, then the effect of the term containing A2 in Eq. (5) will also have to be considered. Acknowledgement The authors would like to thank members of the Department of Physics, Moi University for providing facilities that led to the completion of this work. References 1 Wieder S, Rech B, Beneking C, Siebeke F, Reetz W & Wagner H, Proceedings of the 13th Photovoltaic Solar Energy Conference, Nice, France (1995) 234. 2 Takahashi K, Yamada S & Unno T, High Efficiency AlGaAs/GaAs Tandenden Solar Cells, UDC. 621.385.51:523.9 - 7. 3 Palk E D, Handbook of Optical Constants of Solids II, [Academic Press Inc.], 1991. 4 Sharma V, Thakur A, Chandel P S, Madhok G, Goyal N & Tripathi S K, Indian J Pure & Appl Phys, 42 (2004) 845.