4.4.1 Validation of Simulation Approach
The experimental results presented in the previous section demonstrated that nonlinear contributions in the measuredJSC are detectable. A com-parison of the relative trend of the measuredJSCwith linear and nonlinear absorption in Si pointed out that the detected nonlinear contributions can be attributed to TPA.
However, the presented experimental approach is rather limited re-garding its temporal variations induced to the ultrashort pulse train (Fig. 4.6b). E.g. it does not allow for generation of continuous radiation and, thus, lacks comparability of pulsed versus continuous illumination, which is the most prominent comparison regarding applicability of ultra-short laser pulses in solar cell characterization. Therefore, a simulation approach based on the theoretical model derived in Section 4.2 will be presented in this section. Modeling the experimental conditions from Section 4.3 yields a validation of the simulation approach and justifies its application for a comparison of pulsed to continuous illumination.
The experimental conditions in Section 4.3 allow for application of the mostly reduced differential equation given in Section 4.2. In Section 4.2.3 it was demonstrated that injection dependence ofτand saturation in car-rier generation can be readily neglected for Si and the given spectral range (see Fig. 4.2b). Furthermore, the assumed illumination intensities allow for application of the QSS-approximation detailed in Section 4.2.4. Thus, Eq. (4.15) can be applied for a simulation of the experimental conditions.
Modeling the short circuit current requires incorporation of an ex-tremely high front surface recombination velocity S0 for simulating the carrier extraction (see Eq. (4.1)). As discussed e.g. by Giesecke in [20, pp. 244-245], this yields a crucial bottleneck in finite element approaches for solving Eq. (4.15), since a strongly confined spatial modification of carrier densities, by e.g. absorption or recombination, requires a fine spa-tial discretization ∆z. In turn, such narrow spaspa-tial discretization involves a fine temporal discretization for accurate finite element modeling as well (∆t∆z2/Da), resulting in tremendous computational effort.
One way to circumvent this bottleneck is the integration of a Green’s function that allows for an analytical description of the diffusion process.
Taking into account the less explicit but powerful approximation, that αtot ≈ αbb +σFCAn (λ/µm)xnND (which is only valid for low-injection conditions ∆n ND),13 the excess carrier density does not affect any of the pulses incident onto the solar cell. Thus, the generation term is unaltered for all pulses and can be written as
G(z, t) = I(t)αtotexp (−αtotz)
13This low-injection approximation is much weaker than the similar QSS-approximation (see Section 4.2.4) as not only the changes induced by a single pulse are considered, but the contribution of FCA in the steady-state condition of radiation-semiconductor interaction. Especially for wavelengths below 1150 nm the low-injection condition might be violated in Si.
Figure 4.9: Ratios of measuredJSCs for aligned and misaligned ring cavities (black squares, filled and open symbols denote measurements at different days) and simulated ratios (red line) versus wavelengths (after [33]). The error bars indicate the standard deviations of the measurement. The reddish area surrounding the simulated values denotes the standard deviation of the simulation accounting for the limited precision of temporal ring cavity alignment.
With this approximation Eq. (4.15) can be written as
∂∆n(z, t)
∂t =Da∂2∆n(z, t)
∂z2 +G(z, t)−∆n(z, t)
τ (4.17)
and the Green’s function approach from [20, pp. 56-60] can be applied.14 In Fig. 4.9 measured and simulated ratios of JSC for aligned versus misaligned ring cavities are plotted over excitation wavelength. The surements are given as black symbols (open and filled squares denote mea-surements on different days), including error bars as standard deviations.
The simulation result is given as red line. The generation term G(z, t) (Eq. (4.16)) has been modeled from temporal pulse train shapes I(t) for the aligned and misaligned setup configuration (compare Fig. 4.6b) and material parameters from [55, 58, 61]. The remaining parameters in Eq. (4.17) are given as follows: the ambipolar diffusion coefficient has
14In Appendix D the approach is sketched along the lines of [20, pp. 56-60].
been set to Da = 11.6 cm2s−1,15 the pulse duration to τFWHM = 160 fs (see Section 4.3.2.1) and the minority carrier lifetime toτ = 10µs.16
The reddish area surrounding the simulation result denotes a standard deviation of the simulation. This results from the assumed normal distri-bution of translation stage alignment with 10 µm standard deviation. It considers the limited accuracy of mechanical stage alignment and align-ment control by the autocorrelation signal. The red line represents the mean value after 1000 randomized simulations.
Simulation and experiment demonstrate a very good agreement up to 1350 nm. Especially the rising edge at shorter wavelengths is imitated very well. For longer wavelengths the measurements suffer from noise due to comparably low signals. Although some experimental data points are not covered, the general tendency is reproduced by the simulation. In con-clusion, the presented results demonstrate that the simulation approach is capable of a comparison of JSCs under different temporal characteristics of the illuminating radiation.
4.4.2 Pulsed Versus Continuous Illumination
As the simulation results presented in Fig. 4.9 demonstrate a good agree-ment to the measureagree-ments, theoretical model and simulation approach ap-pear to reflect the interaction of ultrashort laser pulses and short-circuited Si solar cells quite well. From this conclusion simulations covering the most prominent comparison of pulsed versus continuous illumination can be conducted.
For a comparison ofJSCs from pulsed and continuous illumination ul-trashort laser pulses with τFWHM = 100 fs and frep = 80 MHz are as-sumed. The average intensity Iav is varied in order to reflect various
15Da = DDnDp(n+p)
nn+Dpp ≈ Dp with electron and hole diffusion coefficients Dn,p and
∆nND [20, pp. 40-41]. With hole mobility bp ≈ 450 cm2V−1s−1 [65] the hole diffusion coefficient is given byDp=bpkBT/q≈11.6 cm2s−1.
16The exact value of minority carrier lifetime is of minor importance as carrier extraction rates are much higher than recombination rates in that lifetime range.
Figure 4.10: Deviation of theJSC for pulsed versus continuous illu-mination at various average illuillu-mination intensities for a Si solar cell (published in [33]). Ultrashort laser pulses withτFWHM= 100 fs and frep= 80 MHz are assumed for the simulations.
illumination conditions (e.g. high average intensities for localized charac-terizations and concentrator applications or low intensities for large area differential current response measurements).
In Fig. 4.10 deviations of simulatedJSCs for pulsed versus continuous illumination are plotted over photon energy (normalized by band gap en-ergy Eg = 1.12 eV) and wavelength. The presented results emphasize the impact of TPA that is increasing with wavelengths and illumina-tion intensities. It can be deduced that for standard measurement ap-plications (monochromatic illumination intensity Iav < 0.1 Wcm−2 and λ <1200 nm) the contribution of TPA is negligible. However, for highly concentrating measurement applications, as e.g. in local characterizations or for concentrator cell measurements, TPA might significantly affect the measurement outcome.