Category Third Generation Photovoltaics

Solutions to Selected Problems Exercise 2.1





(a) Energy at earth’s mean distance/unit area/unit time = (rs/des)2 energy at sun’s surface/unit area/unit time

= (rs/deS)2 OTs4

Projected area of earth = nre2 Surface area of earth = 4nre2

Incoming energy = outgoing energy (rs/des)2 oTs4nre2 = oTe44nre2 Te = Ts (rs/des)1/2/21/2 = 289.38 K

(b) Trick Question!

The earth absorbs 0.7 of incident radiation but also only emits 0.7 as much. Hence the same radiation balance is maintained and:

Te = 289.38K

(c) The incoming radiation decreases to 0.7 but outgoing decreases to 0.6. Therefore the earth is hotter with such emission properties:

Te’ = (0.7 / 0.6 / 7 4 Te = 300.75 K

(d) This increase, 2°C, is roughly that predicted by a doubling of atmospheric CO2! The new value of emissivity is:

302.75 = (0.7 / є’ f74Te є = 0.584

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Quasi-Fermi Levels E.1 Introduction

The concept of a fermi level was introduced in Chap. 4 to describe the energy distribution of electrons and holes in thermal equilibrium. In non-equilibrium, quasi-fermi levels or imrefs provide a useful tool for semiconductor device analysis as outlined below. Formally, they correspond to the electrochemical potentials of non-equilibrium thermodynamics.

E.2 Thermal Equilibrium

A system in thermal equilibrium has a single, constant-valued fermi level. The case of p-n junction in thermal equilibrium is shown in Fig. E.1. For non­degenerate conditions (n « Nc, p « Nv):

n = NCexp [(Ef-EC)/kT] (E.1)

p = NV exp [(,Еу – Ef)/ kT (E.2)


Fig. E.1: P-N junction in thermal equilibrium.

E.3 Non-Equilibrium

When voltage is applied and/or the p-n junction is illuminated, the concept of a fer...

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. C.5 More General Integrals

Подпись: lj Ol,e) Подпись: І (+e)~kl±k (n-e)/(j - к)! k=0 Подпись: (C.21)

For integral j, this has a finite (J + 1) number of non-zero terms and the integral simplifies to:

For example,

в (п, є) = £3в) (п-є)/6 + є2Р1(ц-є)/2 + £/?2 (п-є) + в (п-є)

(C.22) where each term on the right can be evaluated using Eq. (C.16).

The inverse of such a function for integral j can be found by expanding к (п )

in a power series in 0 (п):

I±(п) = I± (п) + І а±г[ї0(п)]r (C.23)

r=2 ‘


a±:2 = (2k -1)/2k+1 (C.24)

ak 3 = (3k -1)/3k+1 – (2k -1)/2k+1 (C.25)

with terms for larger values of r becoming increasingly complex. This expression shows that, for negative x:

I+ (п)< I+ (п)< I+(п)+[I+ (п)]2(2k -1)/2k+1 (C.26)


/-(п) > I-(п) > I-(п)-[I-(п)]2(2k -1)/2k+1 (C.27)

Retaining only the first term in such an expansion, Eq. (C...

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C.4 Approximate Expressions

Considerable effort has been invested in deriving approximate expressions for Fermi-Dirac integrals as reviewed elsewhere (Blakemore 1982).

One compact approximation for negative arguments is based on the following expression:


where the constant Cj is chosen to minimise error over the desired range. This expression has a similar expansion to that of the integrals of interest with an identical first term. The second term is also identical if Cj is given the value 2-(j + ]). Cj chosen in the range:

Подпись: (C.14)

Подпись: (C.10)

2-(j+1) <> C± <>±(1II±(0)-1)

will give the best results for negative arguments, where the value on the right is ensures that the expression gives the correct value for n = 0. The value of the left
hand side will cause the expression to underestimate the integral while the va...

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Fermi-Dirac and Bose-Einstein Integrals

C.1 Functional Expressions

image202 Подпись: (C.1)

Fermi-Dirac integrals of the form

occur widely in semiconductor transport theory and carrier density calculations, where E is the energy above the edge of the respective band normalised by the thermal energy kT and n is a similarly normalised value of the carrier fermi – energy or electrochemical potential. Г(п) is the Gamma function equal to (n = 1)!

for positive integral values of n and 4n (2m – 1)!!/2m for n = (m + Ч2), where the!! sign represents the product 1.3 … (2m – 1).

image204 Подпись: (C.2)

In a similar way, the Bose-Einstein integral is defined:

his integral has similar application to the case of bosons and its properties have been explored elsewhere, although not as fully as the Fermi-Dirac integrals...

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Physical Constants



V alue*


electronic charge

1.602176462(63) x 10-19C


electron rest mass

9.10938188(72) x 10-28g 9.10938188(72) x 10-31 kg


circle circumference to diameter (calculable)



velocity of light in vacuum (exact value)

2.99792458 x 1010 cm/s

2.99792458 x 108 m/s


permittivity of vacuum (exact value)

8.854187817.. . x 10-14 F/cm

8.854187817.. . x 10-12 F/m


Planck constant

6.62606876(52) x 10-27 erg-s 6.62606876(52) x 10-34 J-s


reduced Planck constant (h/2n)

1.054571596(82) x 10-27 erg-s 1.054571596(82) x 10-34 J-s


Boltzmann constant

1.3806503(24) x 10-16 erg/K 1.3806503(24) x 10-23 J/K


Stefan-Boltzmann constant [2n5k4/(15h3c2)]

5.670400(40) x 10-12 Wcm-2K-4 5...

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From the analysis of this book, it appears that there are sufficient options for improving the performance of solar photovoltaic cells beyond the single junction limits, that greatly improved performance, at some stage in the future, is very likely.

The tandem cell approach of Chap. 5 already demonstrates that such enhanced performance is feasible. Cell technologies right at the top in terms of single junction cell performance and right at the bottom have already benefited from the tandem approach. Experimental gains to date have been in the 20-25% range, relative to single junction devices, compared to theoretically achievable boosts of over 100%. For tandems involving a large number of cells, a generic approach to tandem cell design would be desirable. An example is shown in Fig. 10...

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Without Filter

image199 Подпись: (9.27)

If the filter is removed, the analysis parallels that of the previous section with Eq. (9.18) remaining valid. However, the simplification made possible by Eqs. (9.19) and (9.20) no longer apply. Instead, the equivalent of Eq. (9.21) becomes:

Where E h/N h, N c/N h and Ec/E h are given by the same expressions as Eqs. (9.6), (9.7) and (9.8) but where TR is replaced by TH and the argument of the /(-functions equals – (EG – qVH/kTH), while that of the P* functions equals – ( EG – qVC)/kTC. In the most efficient mode of operation, these arguments approach zero, with the ratios approaching, for kTH << EG:

EH/NH “ EG[1 + (kTH/EG)P1 /P0 ] (9.28)

Nc/N h « (Tc/Th)( p0 / P0 )[1 + 2(kTc/Eg в / p0 – 2 kTH /Eg в / P0 ]


Ec/EH = (Tc/Th)(P0/P0)[ 1 + 3(kTc/Eg )p1 /P0 -3(kTH/Eg )P1 /P0...

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