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_bof 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 (cm²) 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 =10⁷ 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_eof 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 = 10⁷ 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_HFof 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_SCRfor the simulation of the EBIC profiles shown in Figure 5.8.

Curve Na,CISe (cm⁻³) CHF (nF/cm²) wSCR (µm)

(i) 2×10¹⁶ 44.2 0.273

(ii) 5×10¹⁵ 26.3 0.458

(iii) 2×10¹⁵ 18.3 0.658

(iv) 2×10¹⁴ 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× 10¹⁶cm⁻³ (ii) 2×10¹⁵cm⁻³ (standard) (iii) 5×10¹⁵cm⁻³ (iv) 2×10¹⁴cm⁻³ (b) Comparison of the numerically and analytically simulated profiles for different doping densitiesN_a,CISe.

rived profiles (b). For S_e,BC > 10⁵ 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)).