**5. Charge carrier transport studied by electron-beam induced current 43**

**5.2. Modeling: Cross section electron-beam induced current**

**5.2.1. Low injection: Comparison to analytical model**

In the following, a summary of the influence of parameters such as the electron diffusion
length Le,CISe, absorber doping density Na,CISe and back contact recombination velocity
of electrons S_{e,BC} on simulated cross section EBIC profiles is presented for low injection
conditions. A comparison of the numerically simulated profiles to curves obtained from
the analytical model for the collection function of a CISe solar cell, which is introduced
in Sections 2.2 and 5.1, makes it possible to draw conclusions about limitations of the
analytical description and the range of validity.

In Figure 5.5, a comparison of numerically and analytically simulated EBIC profiles for
different electron beam energies E_{b} is shown. For the numerical simulations, the
param-eters listed in Table A.1 were used, and for the analytical calculations, the corresponding
parameters were assumed (see Figure 5.4). The agreement between the corresponding
curves is good and there are only minor deviations in the region of the edges of the space
charge region.

Figure 5.5: Comparison of analytically and numerically simulated cross-section EBIC profiles
assuming an electron beam currentI_{b}of 10 pA and different electron beam energies
E_{b}. The standard set of parameters was used (see Figure 5.4).

5.2 Modeling: Cross section electron-beam induced current 51

Influence of the electron diffusion length

In order to investigate the influence of the electron diffusion length in the CISe layer,
the electron capture cross section σ_{e} of the midgap acceptor state in the CISe layer was
changed for the numerical simulations yielding electron diffusion lengths as listed in Table
5.1.

Table 5.1: Electron capture cross sections σe and resulting electron life times τe,CISe and
dif-fusion lengths L_{e,CISe} used for the numerical and analytical simulations shown in
Figure 5.6.

Profile σ_{e} (cm^{2}) τ_{e,CISe} (ns) L_{e,CISe} (µm)

(i) 10^{−13} 0.5 0.23

(ii) 10^{−14} 5 0.72

(iii) 10^{−15} 50 2.3

(iv) 10^{−16} 500 7.2

### (a) (b)

Figure 5.6: (a) Simulation of EBIC profiles assuming different electron diffusion lengthsLe,CISe

(b) Comparison of simulated and analytically derived profiles for electron diffusion lengths of 230 nm and 2.3µm.

In Figure 5.6, the corresponding numerically simulated EBIC profiles are shown (a) and compared to analytically derived profiles. The profiles are normalized to the maximum current value of the profile simulated by use of the standard set of parameters stated in Table A.1. If the electron diffusion length is larger than 2 µm, the profiles show only minor differences (curves (iii) and (iv)). For small electron diffusion lengths (≤ 1µm), the profile in the quasi neutral region of the absorber layer as well as the maximum cur-rent value and the width of the maximum depend on the electron diffusion length. A comparison of the numerically simulated to the analytically derived profiles shows that the simplified assumption of a collection probability of one in the complete space charge region is not a valid assumption for small electron diffusion lengths. In this case, recom-bination within the space charge region cannot be neglected and influences the shape of

52 Chapter 5. Charge carrier transport studied by electron-beam induced current the profile significantly. Thus, an evaluation of cross section EBIC data based on the simplified assumptions of the analytical description of the collection function introduced in Section 2.2 can lead to misinterpretations concerning the width of the space charge region of the solar cell and the electron diffusion length in the absorber layer.

In the following, the error of a fit based on the analytical description of the collection
function by neglecting recombination in the space charge region in case of a low
elec-tron diffusion length shall be quantified exemplarily. In Figure 5.7, the best fit of the
numerically simulated EBIC profile for an electron diffusion length of 230 nm with an
analytically derived profile is shown. The fitting parameters are: L_{e,CISe}= 260 nm, w_{SCR}

= 420 nm and S_{e,BC} =10^{7} cm/s. This corresponds to a deviation of 13% in diffusion
length and 36% in width of space charge region.

Figure 5.7: Numerically simulated EBIC profile forL_{e}of 230 nm andw_{SCR}(profile (i) in Figure
5.6) and best fit using the analytical description of the collection function. The
used fitting parameters are: Le,CISe = 260 nm, wSCR = 420 nm and Se,BC = 10^{7}
cm/s.

Influence of the absorber layer doping density

In the following, the influence of the CISe doping density on cross section EBIC profiles is considered in more detail. The smaller the effective doping density, the larger is the extension of the space charge region and the smaller is the capacitance of the device.

SCAPS simulations of the high frequency capacitanceC_{HF}of the solar cell were performed
and the width of the space charge region for different shallow doping densitiesN_{a,CISe} was
calculated via Equation 5.12. The results are shown in Table 5.2.

In Figure 5.8, the simulated EBIC profiles are displayed. While the agreement between the corresponding curves is good for a width of the space charge region of less than 700 nm, the deviations are larger for the widest assumed space charge region of 1.3 µm.

Influence of the back-contact recombination velocity

In Figure 5.9, the influence of the back-contact recombination velocity for electrons S_{e,BC}
on EBIC profiles is displayed (a) and simulated profiles are compared to analytically

de-5.2 Modeling: Cross section electron-beam induced current 53

Table 5.2: Assumed variation in absorber doping density N_{a,CISe} and corresponding
capaci-tanceC and width of the space charge region w_{SCR}for the simulation of the EBIC
profiles shown in Figure 5.8.

Curve Na,CISe (cm^{−3}) CHF (nF/cm^{2}) wSCR (µm)

(i) 2×10^{16} 44.2 0.273

(ii) 5×10^{15} 26.3 0.458

(iii) 2×10^{15} 18.3 0.658

(iv) 2×10^{14} 8.8 1.368

### (a) (b)

Figure 5.8: (a) Simulated EBIC profiles for different shallow doping densities Na,CISe of the
absorber layer and correspondingly widths of the space charge region: (i) 2×
10^{16}cm^{−3} (ii) 2×10^{15}cm^{−3} (standard) (iii) 5×10^{15}cm^{−3} (iv) 2×10^{14}cm^{−3} (b)
Comparison of the numerically and analytically simulated profiles for different
doping densitiesN_{a,CISe}.

rived profiles (b). For S_{e,BC} > 10^{5} cm/s, the collection probability in the back contact
region is low even for a large electron diffusion length in the range of the layer thickness.

A significant part of the electrons generated close to the back contact recombine there and do not reach the pn-junction. For lower recombination velocities (curves (iii) and (iv)), recombination at the back contact is less important. In this case, the EBIC profile is rather influenced by the electron diffusion length, i.e. recombination within the quasi neutral region than recombination at the back contact. The agreement between numerical and analytical simulations is good (Figure 5.8 (b)).