# On the Melting Thresholds of Semiconductors under Nanosecond Pulse Laser Irradiation
## 2 Model Description  
The thermal model is applicable for nanosecond laser pulse durations and longer pulses because the electron-lattice interaction phenomena, critical for shorter pulses, are not accounted for, and the coupling of laser energy to the material lattice is treated as an instantaneous local process [29]. This is justified by the fact that electron-lattice thermalization time is in the order of \(\sim\)10\({}^{-12}\) s for silicon and germanium [7], and similar time scales for the other materials under study. In addition, as we consider relatively low surface temperatures, below and near the melting point, we disregard evaporation phenomena, which, however, may slightly affect the melting process for compound semiconductors [30; 31]. For correct simulations of laser-matter interaction processes, material thermophysical and optical properties (and their temperature dependences) are of fundamental importance. The material parameters used in the presented calculations are summarized in Appendix A, Tables A1-A19. The sample is assumed to be flat, and the laser beam couples perpendicularly to the surface in the direction of the \(z\) axis. The simulations are considered as a one-dimensional (1D) problem that is a valid approximation, as long as the irradiation spot size (typically above 100 \(\upmu\)m for the considered experiments) is much larger than the absorption depth, as is our case. Indeed, for a ruby laser with the longest wavelength in this study, the absorption depth of the studied semiconductors is less than 0.4 \(\upmu\)m (see Appendix A). For shorter wavelengths, also investigated here, the absorption depths are even smaller. Therefore, the time-dependent temperature distribution in the irradiated target is governed by the heat-flow equation in its 1D form [32; 33]:  
\[(c_{\rm p}(T)\rho+L_{\rm m}\delta(T-T_{\rm m}))\frac{\partial T}{\partial t}= \frac{\partial}{\partial z}\left(\kappa(T)\frac{dT}{dz}\right)+S(z,t). \tag{1}\]  
Here, \(t\) is time, \(T\) is the temperature and \(c_{\rm p}\), \(\rho\), \(L_{\rm m}\), \(T_{\rm m}\) and \(\kappa\) are, respectively, the heat capacity, the density, the latent heat of fusion, the melting temperature and the thermal conductivity of the sample material. Energy supplied by the laser is represented by the source term \(S(z,t)\) as:  
\[S(z,t)=(1-R)I(t)\alpha\exp(-\alpha z) \tag{2}\]  
where \(R\) and \(\alpha\) are the surface reflectivity and the material absorption coefficient. The pulse intensity \(I(t)\) has a Gaussian temporal profile:  
\[I(t)=\frac{2F_{0}}{\tau_{\rm L}\sqrt{\frac{\ln 2}{\pi}}}\exp\left(-4\,\ln 2 \left(\frac{t}{\tau_{\rm L}}\right)^{2}\right), \tag{3}\]  
with \(F_{0}\) and \(\tau_{\rm L}\) being the peak fluence and the pulse duration. Equation (1) is solved numerically using the finite difference method and the implicit scheme that ensures a high numerical stability. For temperatures below the melting point, the finite difference form of Equation (1) is written on the numerical grid as:  
\[-\kappa_{\rm I}T_{\rm i-1}^{*}+\left(\frac{\Delta z^{2}}{\Delta t}\rho c_{\rm p }+\kappa_{\rm I}+\kappa_{\rm r}\right)T_{\rm i}^{*}-\kappa_{\rm r}T_{\rm i+1}^ {*}=\frac{\Delta z^{2}}{\Delta t}\rho c_{\rm p}T_{\rm i}+S(z,\ t), \tag{4}\]  
where  
\[\kappa_{\rm I}=\frac{\kappa_{\rm i-1}+\kappa_{\rm i}}{2},\ \kappa_{\rm r}=\frac{ \kappa_{\rm i+1}+\kappa_{\rm i}}{2} \tag{5}\]  
and index \(i\) refers to the numerical grid points. The temperature values, \(T^{*}\), are unknown at the time moment \(t_{\rm f}\), and \(T\) (without asterisk) corresponds to the known temperature at the time moment \(t_{\rm f-1}=t_{\rm f}-\Delta t\). One of the advantages of the implicit numerical scheme is that using large spatial and/or temporal steps (\(\Delta z\) and \(\Delta t\), respectively) does not affect its stability; however, too large \(\Delta t\) values can introduce truncation errors to the calculation results [34].