# On the Melting Thresholds of Semiconductors under Nanosecond Pulse Laser Irradiation
## 3 Results and Discussion
### Silicon  
Some ambiguity exists in the reported melting point of crystalline Si, ranging from 1683 to 1690 K [7; 29]. For the temperature dependence of \(c_{\text{p}}\), we took the data from Ref. [41] with a stronger variation over the range of solid-state temperatures than the dependence used for \(c\)-Si in Ref. [5]. The \(c\)-Si thermal conductivity was approximated by the expression from the measured data reported in [42]. For the liquid state, we use \(c_{\text{p}}=910\) J/(kg-K) and \(\kappa=50.8+0.029(T-T_{\text{m}})\) W/(m\(\cdot\)K) [5]. The reflectivity and the absorption coefficient for \(c\)-Si are temperature dependent and given by the relations presented in [43]. The optical properties of molten silicon are described according to the calculated data for 694 nm [44] and measured data for 352 nm [45]. The data on the properties for solid and liquid silicon used in the present modeling are summarized in Tables A-A4 of Appendix A.  
Interestingly, our simulation data for Si (Table 1) somewhat overestimate the melting threshold fluence measured in [37; 38], while they are systematically lower than the simulated \(F_{\text{th}}\) values presented in [6]. The experimental investigations reported in [37] for 532 nm ns laser irradiation give an interval of increasing reflectivity between 330 and 380 mJ/cm\({}^{2}\). The value of 380 mJ/cm\({}^{2}\) was identified as a threshold for reaching a high reflectivity (-70%), probably indicating melting to a depth of approximately one optical skin layer (-10 nm), and the maximum reflectivity of 73% was observed at 450 mJ/cm\({}^{2}\). A similar value of around 400 mJ/cm\({}^{2}\) was determined as a threshold for melting based on time-resolved reflectivity measurements [38]. The theoretical calculations [6] give higher values for the melting thresholds than those calculated in this work and measured in [37; 38]. The main reason for this can be seen in the difference in the absorption coefficient change with temperature. For instance, the absorption coefficient of \(c\)-Si at a 694 nm wavelength at temperatures close to the melting point is approximately five times larger in our case (taken from [43]) than in calculations presented in [6]. Note that the analytical theory [27] predicts an even higher threshold fluence for Si under the same irradiation  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & \(\lambda\), nm & \(\tau\), ns & \(F_{\text{th}}\), mJ/cm\({}^{2}\) & \(F_{\text{th}}\), mJ/cm\({}^{2}\) \\  & & & **This Work** & **Literate Data** \\ \hline CdTe & 248 & 20 & 46 & 50, 50 \({}^{*}\)[39] \\  & 694 & 15 & 68 & 78 [28] \\  & & 30 & 80 & 98 [28] \\  & & 70 & 103 & 130 [28] \\ \hline InP & 532 & 7 & 106 & 97 [30] \\  & 694 & 15 & 165 & \\  & & 30 & 211 & \\  & & 70 & 296 & \\ \hline \hline \end{tabular}
\end{table}
Table 1: _Cont._conditions (694 nm, 30 ns, see Table 1). As a whole, our modeling data for Si are in reasonable agreement with most of the published data, thus demonstrating that our model approach can be used for other semiconducting materials. The calculations with various pulse durations, \(\tau_{\mathrm{L}}\), show that the melting threshold increases with \(\tau_{\mathrm{L}}\) proportionally to app \(\tau_{\mathrm{L}}\)\({}^{0.2}\) (Table 1), i.e., the dependence is considerably weaker than the \(\propto\tau_{\mathrm{L}}\)\({}^{0.5}\) dependence predicted for the _evaporation_ threshold for fairly long (nanosecond and longer) laser pulses [46].