**5. Measurement of the lifetime at the surface (τs) and in the bulk (τb)**

By measuring the temporal decay of the PC illuminating the sample with highly (*α*(ℏ*ω*)=105 cm-1) and less absorbed light (*α*(ℏ*ω*)<<102 cm-1) it should be possible the measure τs and τb. We made the temporal PC decay visible with a 500 MHz scope by employing chopped (18 Hz) continuous wave (cw) laser beams at 488.0 nm (2.54 eV) and 632.8 nm (1.96 eV), by us‐ ing an Ar-Kr and He-Ne laser, respectively. Both laser beams were unfocused resulting in the rather moderate intensity of about 0.6 W/cm2 and, as for the PC measurements above, the driving electric field was again 200 V/cm.

**Figure 3.** Fit (broken line) of the measured photocurrent spectrum in Fig. 1 (symbols) using Eqs. (36) and (37). The fit

**Figure 4.** Comparison of the absorption coefficient (a) after Dutton and (b) modeled with Eqs. (36) and (37).

hardly differs from the one in Fig. 2 (a).

202 Optoelectronics - Advanced Materials and Devices

**Figure 5.** Photocurrent decay vs. time measured under the illumination of a laser emitting at (a) 488.0 nm and (b) 632.8 nm. The symbols represent the measurements, while solid and broken lines represent the fits done with Eq. (38).

Figure 5 shows the experimental results (symbols) and fits (solid and broken lines) of the decay. The fits were performed with the Kohlrausch function [19], which is an extension of the exponential function with one additional parameter γ that can range between 0 and 1,

$$I\_{\rm ph}(t) = I\_{\rm ph}^0 \exp[-(t \mid \tau\_{\rm s,b})^\vee] \tag{38}$$

The following parameters resulted in the best fits for surface and bulk: τs=1.2 ms and γ=0.53, and τb=7.6 ms and γ=0.82, respectively, resulting in τs/τb=0.16. Despite the straightforward‐ ness of the experiment, which did not consider electric charging effects, the number is only approximately a factor 2.4 and 1.6 off from the predicted value using the data of Dutton and the modeled absorption edge, respectively. It is worthwhile to note that the Kohlrausch de‐ cay can be expressed as linear superposition of simple exponential decays, i.e.,

$$\mathbf{\dot{\varepsilon}} \exp \mathbf{\tilde{f}} - (t \,/ \tau\_{s,b})^\gamma \mathbf{\tilde{J}} = \left[ p(\mathbf{u}, \, \mathbf{y}) \exp \mathbf{\tilde{f}} - t \,/ (u \, \tau\_{s,b}) \mathbf{\tilde{J}} du, \,\text{where } \mathbf{p(u, \gamma)} \text{ is the weight function for the decay} \right]$$

4 Department of Physics and Astronomy, Bowling Green State University, USA

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**References**

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86-91.

time τ=uτs,b. The mean relaxation time is<*τ* > = *∫* 0 *∞* exp −(*t* / *τs*,*b*)*<sup>γ</sup> dt* = *τs*,*<sup>b</sup> γ Γ*( 1 *<sup>γ</sup>* ), where Γ is the

gamma function. For a simple exponential decay, <*τ* > = *τs*,*b*, and, therefore, the necessity of the Kohlrausch function to fit the temporal PC decay confirms the involvement of various time constants and is strong supportive evidence for the multi-time scale relaxation model introduced by Eq. (29).

### **6. Conclusion**

We have derived a spectral PC formula based on general principles for the standard set‐ up used in experiments. By explicitly including the spatial variation of the recombina‐ tion rate along the light propagation - i.e., at the surface and in the bulk region – we were able to demonstrate that PC spectra can be accurately described by the BU model. Equivalently good agreements between theory and experiment were found by using *α*(ℏ*ω*) values either experimentally determined or straightforwardly modeled by the density of states and the modified Urbach rule. Furthermore, we have shown that the detailed the‐ oretical model of the spatial variation of τ(z) is not critical for the generation of the PC near the band gap. However, the presented detailed theory correctly fit and explains the experimental observations. It reveals the firm physical key mechanism for understand‐ ing of the PC peak near the gap energy: The peak takes place due to the different recom‐ bination rates of the excited carries near the surface and bulk. We also showed that the τs/τb ratio found by temporally resolved PC measurements reasonably agrees with the results from the PC fits, accommodating the commonly used concept in optoelectronics, i.e., the use of carrier lifetimes.
