# Influence of Hot Carrier and Thermal Components on Photovoltage Formation across the p-n Junction
## 3 Results
The calculated \(U_{gen}(t)\) is presented in Figure 3a in green. Additionally, the experimental trace of the total photovoltage is depicted here. As expected, good agreement was seen only at the end of the picture. Figure 3: Sequence of photovoltage pulse simulation: (a) carrier generation-induced component (green); (b) hot carrier photovoltage (red) and sum of the two components (blue) and (c) thermal lattice heating-caused photovoltage (violet) and sum of the three components (orange). For comparison, the experimental photovoltage trace is shown in black. Hot carrier energy relaxation time in GaAs is of the order of one picosecond [26]. It means that the free carriers are hot as long as the laser pulse is present:  
\[\overline{U}_{hc}(t)\ =\ K_{hc}\,I(t), \tag{5}\]  
and the time constant is \(\tau_{hc}=\tau_{p}\). The value of coefficient \(K_{hc}\) was tailored to fit \(U_{hc}(t)\) with the negative peak of the experimental trace. The hot carrier component \(U_{hc}(t)\) and the sum of the two calculated components, \(U_{hc}(t)+U_{gen}(t)\), are presented in Figure 3b. Obviously, to reach better congruence with the measured photovoltage, one needs a slow negative component to be added to the calculated binary trace. Heating effects, no matter whether they are of the free carriers or of a semiconductor lattice, are linear phenomena. Thus, the forcing function of the thermalization-caused component is expressed as  
\[\overline{U}_{th}(t)\ =\ K_{th}\,I(t). \tag{6}\]  
For calculation of \(U_{th}(t)\), the value of \(\tau_{th}\) = 160 ns was estimated by measuring the time-decay of the transient thermoelectric (TTE) voltage [27] between two contacts on the sample's substrate exposed to the laser radiation. The magnitude of \(K_{th}\) was tailored for the best fitting of the calculated and experimental results. As Figure 3c shows, addition of the third, thermal, component (violet trace) improved the proposed model and gave good agreement with the experimental data. Thus, the model is provided with an ability to separate all the components and to evaluate the contribution of each of them to the total photovoltage. The dependence of the peak values of all three derived components, \(U_{gen}(t)\), \(U_{hc}(t)\) and \(U_{th}(t)\), on laser intensity are depicted in Figure 4. Each point was calculated by applying the above model for each photovoltage trace experimentally fixed at the appropriate laser intensity level. As the laser intensity increased and the two-photon absorption started to be detected (at approximately about 20% of the maximum intensity value), several notices should be pointed out. First, the magnitude of the measured \(U_{gen}\) was lower than the modeled one (compare the values in Figures 2b and 4). It confirms that the classical photovoltage across the p-n junction was lowered by the opposing heating effects. Second, the dependence of the \(U_{hc}\) component passed into the sublinear character, that is, its input into the total signal seemed to diminish. This must be caused by the relative lack of light photons directly used for the carrier heating since part of them now was used for the carrier generation. Third, the interplay between the measured and calculated \(U_{hc}\) component peak values  
Figure 4: Simulated photovoltage components versus light intensity: the hot carrier \(U_{hc}\) in red circles, the generation-caused \(U_{gen}\) in blue squares and the thermal one \(U_{th}\) in green triangles. The calculated ratio \(U_{hc}/U_{th}\) is shown in black diamonds. The lines are guides for the eye of linear (green) and square law (blue) dependence. also changed: the experimental one led at low laser intensities while it gave place to the calculated one at the higher excitation level. Influence of the other two components was responsible for these changes: \(U_{lh}\) supported the experimental \(U_{lhc}\) at low intensities (see cases b and c in Figure 3), and \(U_{gon}\) cut down \(U_{hc}\) as it started manifesting itself.