Reduction of the Set of Differential Equations

The high peak power of ultrashort laser pulses is associated with extreme optical intensities on short time scales followed by comparably long time durations of (almost) zero intensity. Considering several contributions that arise from this strong temporal confinement of the overall intensity, Eqs. (4.2) to (4.4) form a set of coupled differential equations that allows for an analysis of the interaction of ultrashort laser pulses and semiconduc-tors. Nevertheless, a closer examination of the individual contributions reveals that some of them can be neglected for the further discussion.

In this section several approximations will be presented that significantly simplify the set of differential equations.

Injection Dependence of Minority Carrier Lifetime τ

The strong temporal confinement of photons in an ultrashort laser pulse hitting a solar cell att=t0yields an intrinsic variation of ∆n(z, t) within a pulse period T =t₁−t₀ (see Fig. 4.1a). With an injection-dependent minority carrier lifetime τ(∆n(z, t)) this results in an effective lifetime, evaluated over an entire pulse period, that might differ significantly from the lifetime under continuous illumination. Consequently, deviations in the measured effective short circuit current may arise. However, if the deviation in ∆n(z, t), that is being caused by the injection dependence ofτ (denoted asδ1−2), is significantly smaller than the overall change of

∆n(z, t) within a pulse period (δ₀), the injection dependence of τ can be neglected. Thus, δ₁₋₂ δ0 has to be valid for negligence of injection dependence ofτ, which is equivalent to

Assuming thatτ1,2 T the exponential terms can be linearized and Eq. (4.5) can be reduced to

(a) (b)

Figure 4.1: (a) Illustration of the impact of different lifetimesτ1 and τ2on excess carrier density evolution within a pulse periodT=t1−t0. (b) (τ1−τ2)/τ2plotted over the ratioτ2/τ1. For Eq. (4.6) being valid a ratioτ2/τ1 close to unity is required.

Plotting (τ₁−τ₂)/τ₂ over the ratio τ₂/τ₁ (see Fig. 4.1b) emphasizes the requirement ofτ₁ ≈τ₂ for validity ofδ₁₋₂δ₀. With above’s basic assumption τ1,2 T, a generally very small change in δ0 is obtained.

As this is intrinsically tied to small variations in τ, τ1 ≈ τ2 is readily achieved.

Thus, if the formulated criteriaτ T holds, negligence of injection de-pendence ofτis valid even under open circuit conditions, where injection-dependent recombination is much more relevant than under short circuit conditions. For indirect semiconductors like Si with typical lifetimes in theµs or even ms regime [59], ultrashort pulse oscillators with repetition rates in the range of several tens of MHz readily fulfillτT.

In contrast to this, much shorter minority carrier lifetimes are present in direct semiconductors due to the significantly increased radiative decay (e.g. in the ns toµs regime for gallium arsenide (GaAs) [60]). Asτ6T in such cases, a weaker approximation has to be applied that takes into account the short circuit condition provided that substantial injection dependence of minority carrier lifetime cannot be ruled out. With this precondition the transit time τtrans can be introduced as a maximum average lifetime. τtrans describes the average time it takes for a charge

carrier to reach the opposite side of a device. Thus, after [20, pp. excited charge carriers will have passed the device junction and will have been electrically extracted from the device.

In Fig. 4.2a τ_trans is plotted over device thickness W for various diffusion coefficients. These transit times represent a most extreme assumption for the maximum average lifetime of minority carriers un-der short circuit conditions as an extremely shallow⁷ generation profile is assumed (α_bbW 1). As long as τ_trans τ, the carrier lifetime is virtually limited by the injection-independent transit time τtrans, thus, achieving injection-independence in the overall carrier lifetime⁸ (1/τ_sum= 1/τ+ 1/τ_trans≈1/τ_trans).

Saturation in Carrier Generation

Owing to the constant lifetime approximation reasoned above, iden-tical effective lifetimes for pulsed and continuous illumination can be assumed. However, the temporally averaged excess carrier density 1/TRT

0 ∆n(z, t) dt might still differ for the two illumination cases, if the virtually instantaneous excitation of charge carriers by ultrashort pulses (schematically shown in Fig. 4.1a) temporary violates the approximation

∆nN0.

For an estimation of such an impact the additional carriers induced within an ultrashort pulse period are assumed to be excited instanta-neously and non-depth averaged. The results of this most conservative treatment for a Si and GaAs sample are shown in Fig. 4.2b. In this first order estimation N₀ is approximated by the effective carrier density of states (DOS) in the conduction bands. From Fig. 4.2b it is apparent that

7Shallowdenotes an excitation of excess charge carriers close to the front surface of a device or wafer.

8Please note that this is a rather untypical notation asτtrans does not describe a recombination being associated withR= ∆n(z, t)/τ. Instead, the given equation rather illustrates the role ofτtransin comparison withτ’s resulting from recombination effects.

(a) (b)

Figure 4.2: (a) Excess carrier transit timesτtransover device thickness for various diffusion coefficientsD. The resulting transit times are a maximum assumption for the average lifetimes of minority carriers under short circuit conditions, as very shallow excitation conditions at the opposite side of the device junction are assumed. (b) Non-depth averaged linear carrier generation in Si and GaAs by ultrashort laser pulses. The plotted values correspond to the additional dose of excess charge carriers induced by a single ultrashort pulse with a given average intensity Iav at 12.5 ns pulse period. The energy of the exciting photons is normalized by the band gap energies of Si (Eg = 1.12 eV) and GaAs (Eg = 1.424 eV), respectively. Linear absorption data from [61] (Si) and [62] (GaAs).

the validity of approximatingN0−∆n(z, t)≈N0depends on both photon energy and illumination intensity, but holds true for broad spectral ranges and illumination intensities. Thus, the validity of this approximation de-pends on the applied radiation source and will be evaluated individually for the specific situations considered in the later part of this chapter.

Reduced Set of Differential Equations

If the previously discussed approximations regarding minority carrier life-time and saturation of carrier generation are valid, the set of differential

equations (Eqs. (4.2) and (4.4)) can be simplified to and constant minority carrier lifetimeτ.