# Influence of Hot Carrier and Thermal Components on Photovoltage Formation across the p-n Junction
## 3 Results  
Figure 2a presents oscilloscope trace of the photovoltage across the GaAs p-n junction induced by the laser pulse at 20% of its maximum intensity value. It is seen that the response consisted of at least two components of opposite polarity. The positive one corresponded to electron-hole pair generation and their separation by the electric field of the junction (process 1 in Figure 1a). Additionally, the negative part obviously agreed with the carrier flow above the potential barrier (red arrows in Figure 1a), i.e., it could be attributed to the hot carrier photovoltage. The laser photon energy was less than the forbidden energy gap of GaAs, \(Inv=1.17\) eV and \(E_{g}=1.42\) eV, respectively. Thus, the intraband light absorption could be the reason of the hot carrier photovoltage formation as it took place in the case of CO\({}_{2}\) laser excitation [18]. On the other hand, GaAs is known for its relatively expressed two-photon absorption [24]. This feature can be assumed to be the reason of the carrier pair generation-caused photovoltage formation. Really, the measured peak value of the negative component \(U_{\textit{neg}}\) shows linear dependence on the laser intensity (red circles in Figure 2b); this is an inherent feature of the heating-caused effects. While the positive component \(U_{\textit{pos}}\) followed the square law thus proving its origin to be related with the mechanism of two-photon absorption (blue squares in Figure 2b). Figure 2: (a) Experimental photovoltage signal (green) across GaAs p–n junction at the moderate light intensity level; laser pulse (blue) is added for time-dependent comparison. (b) Dependence of measured peak values of the negative \(U_{\textit{neg}}\) (red circles) and positive \(U_{\textit{pos}}\) (blue squares) photovoltage components on laser light intensity; the lines are guides for the eye of linear (red) and square law (blue) dependence. Moreover, the presence of hot carriers was unavoidably followed by their fast thermalization and resulting rise of the junction temperature. Therefore, the total photoresponse across the GaAs p-n junction should be composed of three components,  
\[U_{tot}(t)\ =\ U_{luc}(t)+U_{th}(t)+U_{gen}(t), \tag{1}\]  
where \(U_{luc}(t)\) is the hot carrier photovoltage, \(U_{th}(t)\) is the thermal component resulting from thermalization of hot carriers and subsequent heating of the lattice and \(U_{gen}(t)\) is the electron-hole pair generation-based component arising due to two-photon absorption. The polarities of \(U_{hc}(t)\) and \(U_{th}(t)\) were the same and were opposite to the sign of \(U_{gen}(t)\). Our model assumed the p-n junction as a first-order linear time-invariant (LTI) system [25], and the time-response of it in a general case was characterized by the differential equation  
\[\tau\frac{dU}{dt}+U\ =\ \overline{U}(t), \tag{2}\]  
where \(\tau\) represents the exponential decay constant, and \(U=U(t)\) is a photoresponse function of time. The forcing function \(\overline{U}(t)\) depends on the laser pulse and on the physical phenomenon giving rise to a particular photoresponse component. In our case, the laser pulse can be properly approximated as  
\[I(t)\ =\ I_{p}\bigg{(}\frac{t}{\tau_{p}}\bigg{)}^{m}\textit{exp}\bigg{[}m \bigg{(}1-\frac{t}{\tau_{p}}\bigg{)}\bigg{]}, \tag{3}\]  
where \(I_{p}\) is the peak intensity, \(I_{p}=I(\tau_{p})\) and \(\tau_{p}\) is the laser pulse rise time. The best fit with the experimental laser pulse shape was achieved with index \(m=10\). As it was mentioned earlier, generation of electron-hole pairs was caused by a nonlinear phenomenon of two-photon absorption. Therefore, the forcing function has a square dependence on the laser intensity:  
\[\overline{U}_{gen}(t)\ =\ K_{gen}\,I^{2}(t). \tag{4}\]  
To solve Equation (2) in respect to \(U_{gen}(t)\), the time constant \(\tau_{gen}\) was measured as the exponential decay of the positive component, and the coefficient \(K_{gen}\) was chosen to fit the experimental photovoltage magnitude far after the end of the laser pulse.