Effect of temperature fluctuation on Debye-Waller factors and Debye temperatures of diamond-structure Si and Ge

Abstract

The statistical moment method in statistical mechanics has been applied to investigate temperature effects on the thermodynamic properties of diamond-structure Si and Ge compounds. The analytical expressions of thermal-induced atomic displacement and the mean-square displacement (MSD) of these semiconductors have been derived. The advance of this method is that it has included the anharmonic effects of thermal lattice vibrations by taking into account the higher-order atomic displacement terms. Numerical calculations have been performed for Si and Ge semiconductors up to 800 K. Our theoretical calculations are compared with previous experiments and computations showing reasonable agreement. The obtained MSDs have also been used to derive the Debye temperatures of these two compounds. We have shown that while the MSD function increases rapidly, the Debye-Waller factor decreases promptly as temperature rises. The slopes of MSD curves of Si and Ge are, respectively, $5.4688\times {10}^{-5}$ Ų/K and $6.9194\times {10}^{-5}$ Ų/K at temperature beyond 150 K. The calculations of the present work can be used as an appropriate reference for the future experiments.