Laplace transform of quantum efficiency spectra

The local carrier collection probability rfc(Z) is defined as the probability that a carrier generated at the depth Z is collected at the junction of a short-circuited cell. In this sec­tion, we assume a planar cell of thickness ^ with one-dimensional transport and the cell surface located at Z = 0. In contrast to the previous section, the thickness of the emitter We and the space charge region Wscr are not neglected and Wf=We + Wscr + Wbas – For a zero back surface reflectance of a cell with thickness Wf, the internal quantum efficiency

wj

IQE = J r]c(z) as exp(-as Z)dZ =as L{j7c(z)} (C.30)

Z=0

is the Laplace transform l{t]c(z) } of the local carrier collection efficiency rjc(Z) [209]. Sinkkonen et al. suggested that the local carrier collection efficiency

could be determined from the inverse Laplace transform of experimental data IQE(a^) and as. For the inverse Laplace transformation the quantum efficiency IQE(as(X)) is interpreted as a function of the absorption coefficient as. Unfortunately, the inverse Laplace transformation is numerically unstable. Therefore, the experimental data

IQEja,) P(a) as ek)

are first fitted by the ratio of two polynomials P(as) and Q(as) with the inverse Laplace transform

ri(z)=Y^jexp(Pz)

calculated by the Heaviside expansion theorem [362, 413]. The Д are the roots of the polynomials Q(as).

Figure C.9 shows the measured internal quantum efficiency of a crystalline Si solar cell with a film thickness Wf – 150 pm. The experimental quantum efficiency data (cir­cles) as well as the fit to the experimental data with the test function as P(as)/Q(as) are taken from Ref. [413]. The inverse Laplace transform of the test function yields the local carrier collection efficiency that is shown in Figure C.10. The maximum of the collec­tion efficiency rc(Z) indicates the depth of the junction, which is around Z = 0.4 pm here.

С. 4 Laplace transform of quantum efficiency spectra

For the extraction of recombination parameters in the emitter, fitting of the local car­rier collection efficiency in Figure C.10 was suggested with the analytical expression for the collection efficiency

Tha latter results from the solution of the diffusion equation in the emitter [209]. Within the space charge region of thickness Wscn unity collection efficiency

rjc(z)= 1 for We<Z<We+Wscr (C.35)

is assumed. The recombination parameters in the base are deduced from fitting the col­lection efficiency

, v Kh exp((Wf-Z)/Lb )+ exp(- {w – z)/L. )

pXz) = r ■ (w, T f – – T W 77"T – forw’+ w~^z^wf (C.36)

Kbas exp(fVbaa / Lhas)+ exp(- Wbas / Lhas)

depend on the minority carrier diffusion length De>baS9 the surface recombination velocity Se, bas, and the minority carrier diffusion length Le>bas in the emitter and the base, respec­tively. The result of the fit for our example is given in Ref. [413].

The authors of Refs. [209, 362, 413] claim that fitting the local carrier collection effi­ciency has advantages over the direct fitting of the internal quantum efficiency. Their Laplace technique does, however, include two fitting procedures: fitting the IQE data with a test function and fitting the theoretical collection efficiency rjc to the Laplace transform of the test function. The ambiguity in choosing an appropriate test function is fully avoided by directly fitting the IQE data with an analytical solution of the diffusion equation. In addition, for thin cells with light trapping, the carrier generation profile is not a single exponential function, and the basic Eq. (C.30) underlying this Laplace trans­form analysis is no longer valid [414].

In this work, we therefore prefer a direct fitting of the internal quantum efficiency IQE with appropriate analytical models to determine the recombination parameters. The local carrier collection efficiency is then given by Eqs. (C.34) to (C.36).