## 2 Model Description
Ambipolar energy flow is the sum of diffusion and thermal energy currents inside the carrier subsystem and can be written as  
\[\vec{W}=\left\{E_{g}+2k_{B}T_{e}\left[H_{0}^{1}\left(\eta_{e}\right)+H_{0}^{1 }\left(\eta_{h}\right)\right]\right\}\vec{J}-\left(\kappa_{e}+\kappa_{h} \right)\
abla T_{e}. \tag{6}\]  
The dynamics of semiconductors under the irradiation of ultrashort laser pulses can be modeled with the system of three continuum equations [17; 28]: continuity equation for free carrier density and two coupled energy balance equations, one for the carriers and one for atoms:  
\[\frac{\partial n}{\partial t}+\
abla\cdot\vec{J}=S_{n}-\gamma n^{3}+\delta \left(T_{e}\right)n, \tag{7}\]  
\[C_{e-h} \frac{\partial T_{e}}{\partial t}=S_{u}-\
abla\cdot\vec{W}-\frac{C _{e-h}}{\tau_{e-h}}\left(T_{e}-T_{a}\right)\] \[-\frac{3}{2}k_{B}T_{e}n\frac{\partial n}{\partial t}\left\{\left[ 1-H_{\frac{1}{2}}^{\frac{3}{2}}\left(\eta_{e}\right)H_{\frac{1}{2}}^{\frac{3}{ 2}}\left(\eta_{e}\right)\right]\frac{\partial\eta_{e}}{\partial n}+\left[1-H_ {\frac{1}{2}}^{\frac{3}{2}}\left(\eta_{h}\right)H_{\frac{1}{2}}^{-\frac{1}{2} }\left(\eta_{h}\right)\right]\frac{\partial\eta_{h}}{\partial n}\right\},\] \[C_{a}\frac{\partial T_{a}}{\partial t}=\
abla\cdot\left(k_{a} \
abla T_{a}\right)+\frac{C_{e-h}}{\tau_{ep}}\left(T_{e}-T_{a}\right). \tag{9}\]  
The meanings of symbols in Equations (7) to (9) are the following, \(S_{n}\) is the source of new carriers (excitation rate of new carriers by the laser), \(S_{u}\) describes the energy source (rate of laser energy absorption), and \(T_{a}\) is atomic temperature. The terms on the right hand side of Equation (7) are responsible for the carrier generation, Auger recombination, and impact ionization, consequently. Equation (8) describes the energy balance in the photoexcited electron-hole pairs and is a nonlinear diffusion equation. The last Equation (9) describes the energy balance in the atomic subsystem. \(C_{e-h}\) is specific heat capacity of the electron-hole pairs:  
\[ \begin{split} C_{e-h}=&\frac{3}{2}nk_{B}\left[H_{ \frac{1}{2}}^{\frac{3}{2}}\left(\eta_{e}\right)+H_{\frac{1}{2}}^{\frac{3}{2}} \left(\eta_{h}\right)\right.\\ &+\left.T_{e}\frac{\partial\eta_{e}}{\partial T_{e}}\left[1-H_{ \frac{1}{2}}^{\frac{3}{2}}\left(\eta_{e}\right)H_{\frac{1}{2}}^{-\frac{1}{2}} \left(\eta_{e}\right)\right]+T_{e}\frac{\partial\eta_{h}}{\partial T_{e}} \left[1-H_{\frac{1}{2}}^{\frac{3}{2}}\left(\eta_{h}\right)H_{\frac{1}{2}}^{- \frac{1}{2}}\left(\eta_{h}\right)\right]\right].\end{split}  \tag{10}\]  
The total energy of electron-hole pairs consists of the energy gap and the kinetic energy of electrons and holes (taking into account the Fermi statistics),  
\[u=nE_{g}\left(n,T_{e}\right)+\frac{3}{2}nk_{B}T_{e}\left[H_{\frac{1}{2}}^{ \frac{3}{2}}\left(\eta_{e}\right)+H_{\frac{1}{2}}^{\frac{3}{2}}\left(\eta_{h }\right)\right]. \tag{11}\]  
The parameters used in the calculations as well as the meanings of other symbols are presented in Table 1.  
To present an example of the model application, we use the following source terms. The rate of free carrier density growth, \(S_{n}\), and the corresponding rate of their energy increase, \(S_{u}\), are given by  
\[S_{n}=\frac{\alpha I_{abs}\left(\vec{r},t\right)}{\hbar\omega}+\frac{\beta I_{ abs}^{2}\left(\vec{r},t\right)}{2\hbar\omega}, \tag{12}\]  
\[S_{u}=\alpha I_{abs}\left(\vec{r},t\right)+\beta I_{abs}^{2}\left(\vec{r},t \right)+\Theta nI_{abs}\left(\vec{r},t\right). \tag{13}\]  
In the last two equations, the first and second terms on the right hand side represent the influence of one- and two-photon absorption, respectively, and the third term in the second equation represents the laser energy absorption by the excited free carriers. One-dimensional (1D) laser heating problem is analyzed in this work. The laser is focused on the material surface.