## 2 Photoinduced Heat Transfer Across Thin Semiconductor: Theoretical Model  
The model of photoinduced heat propagation is derived under the following assumptions:* The sample is excited by an optical pulse of irradiance \(I(t)=I_{0}f(t)\) [W/m\({}^{2}\)], where \(f(t)\) describes the time dependence of the incident irradiance: \[f(t)=h(t)-h(t-T)\] (1) where \(h(t)\) is denoted the Heaviside step function and with \(T\) [s] is the pulse duration. * Before the excitation of optical radiation, the whole structure and its environment are at the same temperature--\(T_{amb}\) [K]. * The deexcitation-relaxation processes due to photon-phonon interactions are assumed to be much faster than the rate of change of the rising edge of the optical pulse. Thus, the heat sources formed by these processes follow the temporal shape of the optical pulse [37]. * We consider a semiconductor disk uniformly illuminated across its cross-sectional surface normal to the direction of light propagation (Figure 1), allowing the entire problem of optically generated heat propagation to be analyzed using a one-dimensional approximation. * It is assumed that the surrounding gas does not absorb the incident radiation. Heat sources are generated solely within the sample; however, the resulting thermal disturbance affects the surrounding area. * The sample is considered optically opaque, i.e., \(\beta l_{s}\gg 1\), where \(\beta\) is the optical absorption coefficient [m\({}^{-1}\)] and \(l_{s}\) [m] is the sample thickness. Thus, the excitation optical beam penetrates only a thin layer of the semiconductor near its illuminated surface. The optically generated heat due to photon-phonon interactions can be described as a surface heat source [38]: \[H_{T}(t)=(1-R)I_{0}\delta(x)f(t)=S_{0}\delta(x)f(t)\] (2) where \(R\) is the sample's reflectance, \(\delta\) is the Dirac delta function, and \(S_{0}=(1-R)I_{0}\). Consequently, the optically generated concentration flux of charge carriers can also be described as surface excitation [39]:  
\[g(t)=(1-R)I_{0}\frac{E_{G}}{E}\delta(x)f(t)=S_{0}\frac{E_{G}}{E}\delta(x)f(t) \tag{3}\]  
where \(E_{G}\) and \(E\) signify the energy gap of semiconductor and photon energy, respectively, and \(E\geq E_{G}\)[18]. * The slow heat sources generated by charge carrier recombinations, denoted as \(H_{g}^{SR}(t)\), \(H_{b}^{SR}(t)\), and \(H^{BR}(x,t)\), can be described by [18; 32; 33; 34]  
Figure 1: The geometry of the problem of optically excited temperature variations in semiconductors. \[H_{g}^{SR}(t)=E_{\rm{G}}s_{g}n(0,t) \tag{4}\]  
\[H_{b}^{SR}(t)=E_{\rm{G}}s_{b}n(l_{s},t) \tag{5}\]  
\[H^{BR}(x,t)=E_{\rm{G}}\frac{n(x,t)}{\tau} \tag{6}\]  
where \(s_{g}\) and \(s_{b}\) are parameters describing the surface recombination velocity at the illuminated (\(g\)) and non-illuminated (\(b\)) surfaces of the semiconductor sample. The terms \(n(x,t)\), \(n(0,t)\), and \(n(l_{s},t)\) represent the concentration of photoinduced minority carriers within the sample and at the sample surfaces, illuminated and unilluminated, respectively, obtained by solving the diffusion equation for minority charge carriers induced by a surface source, as described by by Equation (3). The surface recombination velocity depends strongly on the surface treatment of the semiconductor. Chemically etched surfaces typically exhibit low recombination velocities due to the removal of surface defects, while untreated or contaminated surfaces can have significantly higher recombination rates. Additional passivation techniques--such as thermal oxidation, hydrogenation, or dielectric layer deposition (e.g., SiO\({}_{2}\), Al\({}_{2}\)O\({}_{3}\))--are commonly employed to further reduce surface defect density and electronically stabilize the surface [40]. However, surface recombination velocities are generally higher than volumetric recombination rates, which are intrinsic to the semiconductor and primarily depend on the doping level.