Relative stabilities of Si polytypes under biaxial stress: A first-principles study

We explore the possibility of the production of Si polytypes via the total energies and free energies obtained in the density-functional theory and the density-functional perturbation theory. We first calculate the total energies of three polytypes of the tetrahedrally coordinated sp(3) Si atoms having the two-layer, three-layer, and four-layer stacking periods in the hexagonal unit cell ("2H", "3H", and "4H") as a function of the in-plane lattice constant a using the local-density approximation (LDA) and generalized gradient approximation (GGA) comparatively. In the LDA calculation, the 4H phase is energetically the most favorable among three phases in the range of 3.55 angstrom <= a <= 3.73 angstrom. While the 4H phase is found to be more stable than the 2H phase at a relatively wide range of the a value we studied, the 2H phase is found to be more stable than the 3H (3C) phase in the range of 3.57 angstrom <= a <= 3.61 angstrom. In the GGA calculation, on the other hand, the 4H phase becomes the most favorable phase for almost the same range as the LDA case, 3.57 angstrom <= a <= 3.73 angstrom. The 2H phase also becomes as stable as the 3H phase as in the case of the LDA around the small a value of 3.62 angstrom, although the 3H phase energy is always lower than that of the 2H phase in the GGA calculation. Considering the chemical vapor deposition (CVD) growth process, we calculate the biaxial stress values to reduce the lattice constant of the substrate Si crystal to realize the small a values, which can lead to the homoepitaxial growth of the 4H phase. The values obtained are 4.69 GPa and 8.14 GPa for the LDA and GGA, respectively. Next, we study the thermal effect on the relative stabilities of the 3 phases. We calculate the phonon dispersion, bulk modulus and thermal expansion including the vibrational and thermal effect. From the comparison with experimental values of the 3C phase, it is found that the LDA results show much better agreement with experiment than the GGA, indicating that the LDA calculation is reliable to predict thermal properties of the real systems. Finally, we derive the free energy as a function of a under several designated temperatures. The relative stabilities of the 2H and 4H phases with respect to the 3H phase are found to be enhanced. At around a = 3.65 A, stabilities of 2H and 4H phases relative to the 3H phase become most prominent. Although stabilities of the 2H and 4H phases relative to the 3H phase are reduced with increasing temperature at around a = 3.60 A, 2H phase is found to be as stable as 3H and 4H phases at higher temperatures, indicating the possibility of the production of the 2H phase via high-temperature CVD process with the biaxial stress.