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Derivation of Open Circuit Voltage
APPENDIX
F
Derivation of Open Circuit Voltage step 1 Electrons and holes radiatively recombine in a semiconductor layer of an illuminated solar photovoltaic cell via interband transitions (i.e., conduction to valence band transitions separated by the discrete energy bandgap Eg). These emitted photons are known as luminescent photons. The emitted luminescent photon flux Φ outgoing from the solar photovoltaic cell is given by:
∞
π /2
Eg
0
Φ = exp( ∆µ / kBT ) ∫ a ( E )ϕ ( E ) dE ∫
2π
cos θ sin θ dθ ∫ d ξ , (F.1) 0
where ∆m is change in chemical potential (i.e., difference in the electron and hole gas chemical potentials or quasi-Fermi levels in a perturbed semiconductor such as a solar photovoltaic cell under sunlight illumination; we will assume that the separation of quasi-Fermi levels is constant), kB is Boltzmann’s constant, T is cell temperature, E is photon energy, a(E) is absorptivity, (E) is spectral photon flux, u is polar angle, and ξ is azimuthal angle. step 2 Now, to eventually find (E) – the emitted spectral photon flux – shown in the integrand of Equation F.1, we first need to determine the photon density of states. Here, we will follow, in part, the analysis from H. C. Casey, Jr. and M. B. Panish, Heterostructure Lasers Part A: Fundamental Principles, Academic Press, 1978. We consider a real-space cube of volume V = L3. Invoking periodic boundary conditions, we find kx = ky = kz = 2π / L. Therefore, the k-space volume is (2π / L)3. The incremental number of photon states is given by:
dN ( k ) = 2
4 π k 2 dk , (2π / L )3
Solar Photovoltaic Cells: Photons to Electricity. http://dx.doi.org/10.1016/B978-0-12-802329-7.00014-6 Copyright © 2015 Elsevier Inc. All rights reserved
(F.2)
114
Appendix F
where the two different photon polarization states are represented by the value of 2, the volume of a spherical shell is represented by 4π k2dk, the wave vector is represented by k = 2π nrv / c where nr is index of refraction, v is photon frequency, c is photon velocity in vacuum, and finally dk = (2π nr / c)dv. We make the substitutions and rewrite Equation F.2 as:
dN ( v ) = ( L3 )8π nr3v 2 c −3 dv.
(F.3)
Upon substituting v = E / h and dv = dE / h , where h is Planck’s constant, into Equation F.3, the photon density of states (i.e., number of states per unit volume per unit energy) is given by:
g ph ( E ) = dN ( E ) / VdE = 8π nr3 E 2 h −3c −3 .
(F.4)
Now, we can determine the photon density of states per unit solid angle. Photons are emitted in an isotropic manner, or over a full solid angle of Ω = 4π. Therefore, the photon density of states per solid angle is found by dividing Equation F.4 by 4π, which results in:
g ph , Ω ( E ) = 2 nr3 E 2 h −3c −3 .
(F.5)
step 3 The Bose–Einstein distribution function expresses the probability that photons (bosons) will be in a given energy state as:
f ( E ) = [exp( E / kBT ) − 1]−1
(F.6)
By invoking the “classical” regime, we may assume that exp(E / kBT)≫1, and therefore, Equation F.6 is rewritten as:
f ( E ) ≈ exp( − E / kBT ).
(F.7)
The spectral photon density is found by multiplying Equation F.5 by Equation F.7, resulting in:
D( E ) = 2 nr3 E 2 h −3c −3 exp( −E / kBT ).
(F.8)
The spectral photon density D(E) is converted to spectral photon flux (E) by multiplying Equation F.8 by the photon velocity c / nr.
Appendix F
115
Therefore, the spectral photon flux (that is shown in the integrand of Equation F.1) is given by:
ϕ ( E ) = 2 nr2 E 2 h −3c −2 exp( − E / kBT ).
(F.9)
step 4 Next, we will determine the radiative current density Jrad. To do this, we need to revert to Equation F.1 and multiply the emitted luminescent photon flux Φ by the electronic charge, e, which gives: 2π ∞ π /2 J rad = e exp( ∆µ / kBT ) ∫ E a ( E )ϕ ( E ) dE ∫ 0 cos θ sin θ d θ ∫ 0 d ξ . (F.10) g
We shall consider the ideal case for a solar photovoltaic cell in which a(E) = 1, which means unity absorptivity for photons with energy equal to or greater than the bandgap energy (E ≥ Eg). We consider the presence of a perfect backside mirror so that the emitted luminescent photons only escape out of the front surface of the cell, noting that nr = 1 for air. Now, solving the integrals in Equation F.10, we find:
∫
π /2 0
cos θ sin θ d θ ∫ 0 d ξ = π , 2π
(F.11)
and
∫
∞ Eg
2 E 2 h −3c −2 exp( − E / k BT ) dE
= 2 kBTh −3c −2 exp( − E g / kBT )[ E g2 + 2( kBT ) E g + 2( k BT )2 ].
(F.12)
Since the terms 2(kBT )Eg and 2(kBT )2 are much smaller than E g2 in Equation F.12, they are neglected in the interest of keeping only the dominant terms. Therefore, Equation F.10 for the radiative current density reduces to:
J rad = exp( ∆µ / k BT )2π eE g2 k BTh −3c −2 exp( − E g / k BT ).
(F.13)
step 5 In an ideal solar photovoltaic cell at steady state and open circuit and considering the absence of any nonradiative recombination, the photogenerated current density Jph (incoming absorbed solar photon
116
Appendix F
flux multiplied by the electronic charge) balances the radiative current density Jrad (outgoing emitted luminescent photon flux multiplied by the electronic charge). This condition, explained by the principle of detailed balance for constant quasi-Fermi level separation, is represented by:
J ph = J rad .
(F.14)
Therefore, substituting Equation F.13 into Equation F.14, we find:
J ph = exp( ∆µ / k BT )2π eE g2 k BTh −3c −2 exp( − E g / kBT ).
(F.15)
The change in chemical potential for an illuminated solar photovoltaic cell at the open circuit condition is expressed as ∆m = eV oc, and therefore, we rewrite Equation F.15 as:
J ph = exp( eVoc / k BT )2π eE g2 k BTh −3c −2 exp( − E g / k BT ).
(F.16)
Finally, following mathematical manipulation of Equation F.16, we arrive at the following expression for the detailed balance-limiting open circuit voltage Voc in an ideal solar photovoltaic cell with a perfect backside mirror,
−1 Voc = E g e −1 − k BTe −1 ln(2π eE g2 k BTh −3c −2 J ph ).
(F.17)
Written in this specific format, Equation F.17 offers the most intuitive understanding of the fundamental factors that govern Voc – the bandgap energy Eg, cell temperature T, and photogenerated current density Jph. Note that for a bifacial solar photovoltaic cell, the emitted luminescent radiation may escape from the front (top) surface of the cell into air or escape out of the rear (bottom) surface of the cell into air. Therefore, a factor of 2 would be included in the logarithm term in Equation F.17. If the solar photovoltaic cell instead contains a parasitically absorbing substrate (with no photovoltaic activity) on the rear, then the emitted luminescent radiation may escape from the front (top) surface of the cell into air or escape out of the rear (bottom) surface of the active region of the cell where it is absorbed in the parasitic substrate below. Therefore, in this case, a factor of (1 + nr2 ) would be included in the logarithm term in Equation F.17, where nr is the refractive index of the parasitically absorbing substrate. Ultimately, an ideal mirror on the back of the solar photovoltaic cell offers the optimal limiting Voc.
F
Derivation of Open Circuit Voltage step 1 Electrons and holes radiatively recombine in a semiconductor layer of an illuminated solar photovoltaic cell via interband transitions (i.e., conduction to valence band transitions separated by the discrete energy bandgap Eg). These emitted photons are known as luminescent photons. The emitted luminescent photon flux Φ outgoing from the solar photovoltaic cell is given by:
∞
π /2
Eg
0
Φ = exp( ∆µ / kBT ) ∫ a ( E )ϕ ( E ) dE ∫
2π
cos θ sin θ dθ ∫ d ξ , (F.1) 0
where ∆m is change in chemical potential (i.e., difference in the electron and hole gas chemical potentials or quasi-Fermi levels in a perturbed semiconductor such as a solar photovoltaic cell under sunlight illumination; we will assume that the separation of quasi-Fermi levels is constant), kB is Boltzmann’s constant, T is cell temperature, E is photon energy, a(E) is absorptivity, (E) is spectral photon flux, u is polar angle, and ξ is azimuthal angle. step 2 Now, to eventually find (E) – the emitted spectral photon flux – shown in the integrand of Equation F.1, we first need to determine the photon density of states. Here, we will follow, in part, the analysis from H. C. Casey, Jr. and M. B. Panish, Heterostructure Lasers Part A: Fundamental Principles, Academic Press, 1978. We consider a real-space cube of volume V = L3. Invoking periodic boundary conditions, we find kx = ky = kz = 2π / L. Therefore, the k-space volume is (2π / L)3. The incremental number of photon states is given by:
dN ( k ) = 2
4 π k 2 dk , (2π / L )3
Solar Photovoltaic Cells: Photons to Electricity. http://dx.doi.org/10.1016/B978-0-12-802329-7.00014-6 Copyright © 2015 Elsevier Inc. All rights reserved
(F.2)
114
Appendix F
where the two different photon polarization states are represented by the value of 2, the volume of a spherical shell is represented by 4π k2dk, the wave vector is represented by k = 2π nrv / c where nr is index of refraction, v is photon frequency, c is photon velocity in vacuum, and finally dk = (2π nr / c)dv. We make the substitutions and rewrite Equation F.2 as:
dN ( v ) = ( L3 )8π nr3v 2 c −3 dv.
(F.3)
Upon substituting v = E / h and dv = dE / h , where h is Planck’s constant, into Equation F.3, the photon density of states (i.e., number of states per unit volume per unit energy) is given by:
g ph ( E ) = dN ( E ) / VdE = 8π nr3 E 2 h −3c −3 .
(F.4)
Now, we can determine the photon density of states per unit solid angle. Photons are emitted in an isotropic manner, or over a full solid angle of Ω = 4π. Therefore, the photon density of states per solid angle is found by dividing Equation F.4 by 4π, which results in:
g ph , Ω ( E ) = 2 nr3 E 2 h −3c −3 .
(F.5)
step 3 The Bose–Einstein distribution function expresses the probability that photons (bosons) will be in a given energy state as:
f ( E ) = [exp( E / kBT ) − 1]−1
(F.6)
By invoking the “classical” regime, we may assume that exp(E / kBT)≫1, and therefore, Equation F.6 is rewritten as:
f ( E ) ≈ exp( − E / kBT ).
(F.7)
The spectral photon density is found by multiplying Equation F.5 by Equation F.7, resulting in:
D( E ) = 2 nr3 E 2 h −3c −3 exp( −E / kBT ).
(F.8)
The spectral photon density D(E) is converted to spectral photon flux (E) by multiplying Equation F.8 by the photon velocity c / nr.
Appendix F
115
Therefore, the spectral photon flux (that is shown in the integrand of Equation F.1) is given by:
ϕ ( E ) = 2 nr2 E 2 h −3c −2 exp( − E / kBT ).
(F.9)
step 4 Next, we will determine the radiative current density Jrad. To do this, we need to revert to Equation F.1 and multiply the emitted luminescent photon flux Φ by the electronic charge, e, which gives: 2π ∞ π /2 J rad = e exp( ∆µ / kBT ) ∫ E a ( E )ϕ ( E ) dE ∫ 0 cos θ sin θ d θ ∫ 0 d ξ . (F.10) g
We shall consider the ideal case for a solar photovoltaic cell in which a(E) = 1, which means unity absorptivity for photons with energy equal to or greater than the bandgap energy (E ≥ Eg). We consider the presence of a perfect backside mirror so that the emitted luminescent photons only escape out of the front surface of the cell, noting that nr = 1 for air. Now, solving the integrals in Equation F.10, we find:
∫
π /2 0
cos θ sin θ d θ ∫ 0 d ξ = π , 2π
(F.11)
and
∫
∞ Eg
2 E 2 h −3c −2 exp( − E / k BT ) dE
= 2 kBTh −3c −2 exp( − E g / kBT )[ E g2 + 2( kBT ) E g + 2( k BT )2 ].
(F.12)
Since the terms 2(kBT )Eg and 2(kBT )2 are much smaller than E g2 in Equation F.12, they are neglected in the interest of keeping only the dominant terms. Therefore, Equation F.10 for the radiative current density reduces to:
J rad = exp( ∆µ / k BT )2π eE g2 k BTh −3c −2 exp( − E g / k BT ).
(F.13)
step 5 In an ideal solar photovoltaic cell at steady state and open circuit and considering the absence of any nonradiative recombination, the photogenerated current density Jph (incoming absorbed solar photon
116
Appendix F
flux multiplied by the electronic charge) balances the radiative current density Jrad (outgoing emitted luminescent photon flux multiplied by the electronic charge). This condition, explained by the principle of detailed balance for constant quasi-Fermi level separation, is represented by:
J ph = J rad .
(F.14)
Therefore, substituting Equation F.13 into Equation F.14, we find:
J ph = exp( ∆µ / k BT )2π eE g2 k BTh −3c −2 exp( − E g / kBT ).
(F.15)
The change in chemical potential for an illuminated solar photovoltaic cell at the open circuit condition is expressed as ∆m = eV oc, and therefore, we rewrite Equation F.15 as:
J ph = exp( eVoc / k BT )2π eE g2 k BTh −3c −2 exp( − E g / k BT ).
(F.16)
Finally, following mathematical manipulation of Equation F.16, we arrive at the following expression for the detailed balance-limiting open circuit voltage Voc in an ideal solar photovoltaic cell with a perfect backside mirror,
−1 Voc = E g e −1 − k BTe −1 ln(2π eE g2 k BTh −3c −2 J ph ).
(F.17)
Written in this specific format, Equation F.17 offers the most intuitive understanding of the fundamental factors that govern Voc – the bandgap energy Eg, cell temperature T, and photogenerated current density Jph. Note that for a bifacial solar photovoltaic cell, the emitted luminescent radiation may escape from the front (top) surface of the cell into air or escape out of the rear (bottom) surface of the cell into air. Therefore, a factor of 2 would be included in the logarithm term in Equation F.17. If the solar photovoltaic cell instead contains a parasitically absorbing substrate (with no photovoltaic activity) on the rear, then the emitted luminescent radiation may escape from the front (top) surface of the cell into air or escape out of the rear (bottom) surface of the active region of the cell where it is absorbed in the parasitic substrate below. Therefore, in this case, a factor of (1 + nr2 ) would be included in the logarithm term in Equation F.17, where nr is the refractive index of the parasitically absorbing substrate. Ultimately, an ideal mirror on the back of the solar photovoltaic cell offers the optimal limiting Voc.