# On the Melting Thresholds of Semiconductors under Nanosecond Pulse Laser Irradiation
## 3 Results and Discussion
### Generalization of the Damage Threshold Data into a Predictive Dependence  
A wide set of data on the damage thresholds of five semiconductors under various ns-laser irradiation conditions are obtained in our calculations in the frames of a unified thermal model, and all the thresholds are in good agreement with available literature data, both experimental and theoretical ones. The obtained threshold values vary in a wide range depending on material, from \(\sim\)50 mJ/cm\({}^{2}\) for CdTe to almost 1 J/cm\({}^{2}\) for Si (Table 1). The irradiation conditions also affect the threshold values, which are generally smaller for shorter laser wavelengths and pulse durations. It is very attractive to generalize the obtained results in terms of a unified parameter combining the basic material properties (thermophysical and optical) in order to be able to predict the ns-laser-induced melting thresholds, at least approximately, without performing detailed simulations.  
[PERSON] [29] considered \"optimal\" melting conditions during ns-laser-induced thermal surface melting, when minimal laser energy is required for a certain melt depth. Assuming that such conditions are fulfilled when the melt depth is equal to the heat-diffusion length, he estimated the optimal laser fluence as:  
\[P_{\text{B}}=\frac{2\rho\Delta H}{1-R}\left(\frac{D}{\tau_{\text{L}}}\right)^ {\frac{1}{2}}\tau_{\text{L}} \tag{8}\]  
where \(D=\kappa/\rho c_{\text{p}}\) is the thermal diffusivity and \(\Delta H=L_{\text{m}}+c_{\text{p}}(T_{\text{m}}-300)\) is the total energy needed to heat the sample to the complete melting state from room temperature, A similar parameter was introduced in [46] as an evaporation threshold under ns-laser ablation (assuming naturally by \(\Delta H\) in Equation (8) the specific heat for evaporation instead of that for melting and omitting the \(2/(1-R)\) factor).  
Figure 4 shows the calculated melting threshold values plotted as a function of the \(P_{\text{B}}\) parameter, Equation (8), evaluated for all the studied materials using their room-temperature properties. All the data are nicely grouped around a straight line in the logarithmic plot. This clear correlation is rather surprising for such a simplified generalization approach when the material absorption coefficient and temperature dependencies of thermophysical properties are not taken into account. The least square fitting line in Figure 4 is described by a power law, \(F_{\text{th}}\approx 0.05\)\(P_{\text{B}}\).[16], which can be used for a rough estimation of the melting threshold of semiconductors based on their basic room-temperature properties.  
The parameter \(P_{\text{B}}\) predicts a growth of the melting threshold with the laser pulse duration as \(\tau_{\text{L}}\)[0.5]. However, as was noticed above, this is not always the case according to our simulations. Some semiconductors (Ge, CdTe, InP) follow closely the \(\tau_{\text{L}}\)[0.5] dependence, while others (Si, GaAs) demonstrate weaker dependencies (Table 1 and Figure 4). This finding agrees with calculations based on the theory in [26], which also predicts a nearly \(\tau_{\text{L}}\)[0.5] dependence for CdTe over a wide range of pulse durations [28] and a much weaker dependence for Si [27]. This is probably mainly due to a difference in the thermal diffusivity, \(D\), of the materials. Thus, at room temperature, \(D\approx 0.8\) cm\({}^{2}\)/s for Si, and it is around 0.35 cm\({}^{2}\)/s for Ge, InP and CdTe. A higher thermal diffusivity results in a higher heat diffusion length and smaller in-depth temperature gradients and thus in a lower heat flow from the surface at an increased pulse duration. The temperature dependencies of material parameters (included in our model simulations) can additionally affect the pulse duration dependence of the melting threshold.