Polycrystalline cells consist of grains with different sizes. Figure 2.31 on p. 49 shows the measured log-normal grain size distribution that is well described by the function [200,201]
( Ґ — 2^
1 1 [ ln(G/ G) cr’
(^exphl—-% which that describes a Gaussian distribution of the logarithm of the grain size G. The average grain size is G, and cr is the width parameter of this distribution. For simulations we would not want to simulate many grains with various grain sizes. The question we address here is: what is an appropriate choice for an effective grain size Geff that we choose to represent an ensemble of grains with a log-normal size distribution? To answer this question we proceed in two steps. First, we calculate the effective quantum efficiency diffusion length LQfiogn for a log-normal grain size distribution. We tactically assume that all grains have the same intra-grain diffusion length L, diffusion coefficient Dw and grain boundary recombination velocity Sgrb. In a second step, we calculate the effective grain sizes Geff that yields a quantum efficiency diffusion length L^Geff) = LQti0gn• As a result, we express Geff (cr, G) as a function of the parameters of the lognormal distribution.
To calculate LQjogm we first study the log-normal distribution. A variable substitution for ln(G / G) yields the nth moment
(G") = ]g”P(g) dG = G" expf—(и2 – и)
of the distribution P(G). The symbol < > denotes an average over the log-normal distribution. The first moment is the average grain size is <G> = G. The area-weighted average grain size is
С. 5 Effective grain size for log-normal grain size distribution
The log-normal-averaged diffusion length is
і (<-/Ф))
LQJogn (G2)
We average the area-weighted 1 !LQtiogm since the diode saturation current jQ = (q nQ Dn)/LQtiogn is also proportional to /LQjogn and since the factor G2 ensures that the diode saturation currents from various grains add up to the total saturation current. We assume all grains to be independent of each other. Carrier injection from one grain across a grain boundary into a neighboring grain is neglected.
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The effective grain size G^we are looking for fulfills the condition LQ{Gef) = LQ>iogm with Lq{Gejj) from Eq. (C.22) and L<Qiogn from Eq. (C.41). In Eq. (C.22), the grain size enters via the second term sgrb only. We therefore first consider a grain without volume recombination (/ = oo). We find from Eq. (C.22)
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From Eqs. (C.41) and (C.42) we calculate the effective diffusion length
which we may also express as
Geff = G3/,Gmv5/i (C.45)
We can thence deduce Geff from the average grain size G and the area-weighted average size Gaw of an experimental grain size distribution.
We checked Eq. (C.44) numerically for many cases (а є [0.1,2], L є [0.1 pm, 103 pm], sgrbfDn є [0, 103 pm-1], G = 1 pm,) including cases of nonnegligible volume recombination. In all these cases we find that a cell with periodic grain structure of grain size G^has the same diode saturation current as a cell with noninteracting grains that have a log-normal grain size distribution that is characterized by G and <x
ation 94
[1] 8IQE;'(x, Y,a,)
3a.
We first consider local properties and then turn to the global relationship of Lq and Lj.
Equality of local effective diffusion lengths Lqi and L Jt
We use properties of Laplace transforms to prove that the local effective diffusion length always fulfills the relation
[2] = OH, F
