# On the Melting Thresholds of Semiconductors under Nanosecond Pulse Laser Irradiation
## 1 Introduction  
The material processing of semiconductors using short and ultrashort laser pulses is one of the key technologies in various fields, including microelectronics, photonics, photovoltaics, sensor devices. It has been employed for enhancing dopant diffusion [1], crystallization [2] and selective modification of multilayer structures [3], and in studies of kinetics of structural changes in materials [4]. Its main advantages are the high level of controllability and variety of wavelengths that can be selected to fit a particular material and an application. One of the basic parameters for laser material processing that involves surface patterning, modification, crystallization and ablation, is the melting (damage) threshold. This key parameter is usually determined in the experimental studies of laser material processing as a reference point for controlling the laser modification process and for testing theoretical models developed with the aim of better understanding the fundamental processes at laser-matter interaction [5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20]. For detection of the phase change, several approaches are used, such as time-resolved reflectivity (TRR) [9; 11; 19; 20; 21; 22], acoustic [23] and electrical conductivity [24] measurements and pump-probe microscopy [25], or combined techniques as in [13], where time-of-flight velocity distributions together with the evaporation rate and reflectivity were measured and analyzed. In some cases, the experiments identify a transient state where only a minor or instable change of the studied parameter is observed and thus the obtained melting threshold, i.e., minimal laser energy density (laser fluence), needed to reach this state corresponds to the melting onset. In other cases, a stable change of the measured value is detected, corresponding to the established molten phase on the irradiated surface at higher fluences. Thus, in TRR experiments, the obtained data on the reflectivity of the molten phase and melting threshold are strongly dependent on the sensitivity and time resolution of themeasurements [19] and on the wavelength of the probe laser beam [20]. For instance, a UV beam, which probes a thinner surface layer, results in a lower melting threshold of silicon as compared to a visible probe beam with a higher absorption depth under identical conditions [20]. We believe, therefore, that the threshold values obtained with the TRR technique are generally overestimated if we assume the onset of melting as a threshold. It should also be noted that the fluence interval with a transient melting is dependent on the material purity and surface quality such as roughness, which can lead to a locally changed absorption, as well as on the presence of hotspots in the laser beam profile. In modeling, it is typically assumed that the melting threshold corresponds to the energy density needed for the temperature of the surface of the material to reach the melting point, \(T_{\text{m}}\)[5; 6; 26; 27; 28]. A comprehensive analytical theory has previously been developed for optical heating of semiconductors using the coupled diffusion equations for heat transfer and excited carrier density and taking into account a variety of non-linear processes involved, such as free-carrier and two-photon absorption, radiative and Auger recombination and temperature shift of the band gap [26]. The theory was applied to evaluate the melting thresholds of a number of semiconductors irradiated by laser pulses in a range of pulse durations from nanoseconds to a millisecond, and in some cases a reasonable agreement with available experimental data was obtained [27; 28]. However, the practical use of this theory is rather complicated as it requires data on a large number of physical properties of the material, including those related to free carrier dynamics and their dependencies on the temperature and carrier density, which have to be analytically integrated over laser-heating time.