Quasi-Steady-State Approximation

Owing to the previously discussed approximations and the accompa-nying simplifications resulting in the above set of differential equations (Eq. (4.8)), free carrier absorption (FCA) is the only term left that causes coupling of Eqs. (4.8a) and (4.8b). If significant changes in ∆n(z, t) take place at time scales of a pulse transit time through the sample, these changes affect the pulse propagation itself and require an alternating step-wise computation of Eqs. (4.8a) and (4.8b).

As this coupling involves a tremendous computational effort, a quasi-steady-state approximation (QSS-approximation) is introduced that as-sumes invariance of ∆n(z, t) for each single pulse, thereby yielding a de-coupling of Eqs. (4.8a) and (4.8b). The two fundamental constraints of this approximation can be stated as follows: firstly, carrier motion within a pulse transit time through the sample is negligible and, secondly, any variations in ∆n(z, t) that are induced by a single ultrashort pulse do not affect the intensity of the pulse itself. Hence, each pulse incident on the semiconductor sample interacts with a previously known excess car-rier distribution ∆n(z, t) that is invariant for the pulse itself.⁹ If this QSS-approximation holds,n(z, t) and, likewise, p(z, t) in Eq. (4.8b) be-come temporally invariant for the time the pulse propagates through the

9Naturally the pulse induces changes in ∆n, however, they are not seen by the pulse itself but by all subsequent pulses.

semiconductor. Consequently, Eq. (4.8b) can be directly solved and ap-plied to Eq. (4.8a) resulting in a single differential equation describing the interaction of ultrashort pulses and semiconductors.

For the QSS-approximation being a valid assumption, the following two conditions have to hold:

1. Carrier redistribution due to diffusion within a pulse transit time is negligible.

2. Either the relative change in FCA induced by a single pulse or total FCA itself, as compared to band-to-band absorption, are negligible.

The first condition is readily achieved when considering typical dif-fusion coefficients and solar cell thicknesses (note, that the discussion is limited to one spatial dimension). Charge carriers move randomly while a pulse propagates through a sample. From the pulse transit time τ_p through the device and the diffusion coefficient D the characteris-tic distance of this random motion can be determined as p

τ_pD (see Fig. 4.3a). For typical diffusion coefficients and device thicknesses this diffusion length is less than 0.1 µm, thus, even less than typical spatial discretizations in numerical approaches. Therefore, significant carrier re-distribution can be readily neglected for the time scale of a pulse transit through a semiconductor sample.

The second condition requires two effects to be considered: firstly, the variation in FCA induced by a single ultrashort pulse (denoted asδα_FCA) and, secondly, the impact of FCA relative to the band-to-band absorption processes. Starting withδαFCA, FCA itself is defined by [57, 58, 63]:

αFCA=σⁿ_FCA(λ/µm)^xnn+σ^p_FCA(λ/µm)^xpp, (4.9) with FCA cross sections σ_FCA^n,p and wavelength powers xn,p for electrons (n=ND+∆n) and holes (p=NA+∆n). Asnandpdepend on ∆n(z, t), knowledge of the change in ∆n(z, t) induced by a single pulse (denoted asδ∆n) is required for computation ofδα_FCA. From Eq. (4.8a),δ∆ncan

(a) (b)

Figure 4.3: (a) Diffusion length of charge carrier movement over device thickness for various diffusion coefficients. The considered time frame for the statistical movement is the time it takes for an ultrashort pulse to travel through a device of respective thickness. (b) Ratio of FCA to band-to-band absorption (αFCA/(αbb+βIpeak)) over change in FCA by a single pulse (δαFCA) for various average illumination intensities Iav, normalized photon energiesEph/Eg and initial excess carrier dis-tributions ∆niin then-type Si sample withND= 5·10¹⁵cm⁻³. The assumed ultrashort pulses have a pulse duration ofτFWHM = 100 fs and are emitted at a pulse period ofT = 12.5 ns.

be described as¹⁰

δ∆n≈ α_bbIpeak

hν +βI_peak² 2hν −∆ni

τ

!

τ_1/e2, (4.10) with I_peak =I_av/ f_repτ_1/e2

as a rectangular equivalent to a Gaussian-shaped ultrashort pulse peak intensity with pulse repetition ratef_rep = 1/T and pulse duration τ_1/e2 (discussed in Appendix C on page 219).

∆ni denotes the excess carrier density in the semiconductor immediately before the considered ultrashort pulse hits. With Eqs. (4.9) and (4.10)

10For simplicity, an infinitely thin semiconductor is assumed here. This makes any consideration of diffusion or depth-averaging of carrier generation redundant. Also, this represents a very conservative treatment asδαFCAis proportional to δ∆n(see Eq. (4.11)), which is most pronounced for non-depth-averaged excess carrier generation.

δα_FCAis given by

δα_FCA= (σ_FCAⁿ (λ/µm)^xn+σ_FCA^p (λ/µm)^xp)δ∆n

σ_FCAⁿ (λ/µm)^xn(ND+ ∆ni) +σ^p_FCA(λ/µm)^xp(NA+ ∆ni). (4.11) For FCA having a significant impact on the set of differential equa-tions given in Eq. (4.8), not only the induced change of FCA (δα_FCA) but also the FCA itself must be of significant magnitude. In Fig. 4.3b αFCA/(αbb+βIpeak) is plotted overδαFCAfor an idealizedn-type Si sam-ple (N_D = 5·10¹⁵cm⁻³) illuminated with ultrashort pulses of τ_FWHM = 100 fs pulse duration andf = 1/T = 80 MHz pulse repetition rate. In this idealized consideration no defect-states and, thus, no SRH-recombinations occur. Therefore, the minority carrier lifetime τ is purely intrinsically limited (including Auger and radiative recombination) and approximated from the parametrization published byRichter et al. in [22] using ∆ni.

In contrast to the marginal influence of the initial excess carrier density

∆ni(given as solid and dotted/dashed lines of same color) the average illu-mination intensityI_av affectsδα_FCAsignificantly, hence, shifting the lines to the right (see highlighted points in Fig. 4.3b, where identical symbols denote specific photon energies). Furthermore, there is a quite distinct dependence on the energy of the incident photons. Those photons, that are close to and below the band gap energy (E_ph/E_g≈1) are most likely to violate the QSS-approximation. This results from a continuous reduc-tion in band-to-band absorpreduc-tion with longer wavelengths, while there are still excess carriers being excited giving rise to significance of δα_FCA.¹¹ However, forIav ≤100 Wcm⁻² the presented approximation is working quite well as eitherδαFCAorαFCA/(αbb+βIpeak) is less than 1%.

If the QSS-approximation discussed above is considered to be valid, any incident pulse interacts with a sample of known excess carrier density

∆n(z, t) and changes induced by the pulse do not significantly affect the pulse itself. Thus, there is no relevant time-dependency in ∆n(z, t) when calculating the intensityI(z, t) of a single pulse and ∆n(z, t)→∆n(z).

11Please note that a reconsideration of this argumentation might be necessary for extreme pulse peak powersIpeakas nonlinear absorption might outperform the linear band-to-band absorption.

Consequently, Eq. (4.8b) can be solved directly by I(z, t) = I(t)αtot(z) exp (−αtot(z)z)

α_tot(z) +βI(t) (1−exp (−αtot(z)z)), (4.12) withαtot(z) =αbb+σⁿ_FCA(λ/µm)^xnn(z) +σ^p_FCA(λ/µm)^xpp(z).