Atomic Vibrational Effect on Vacancy Concentration of Gray Tin (α-Sn): Computation Based on Density Functional Theory

The study of formation energy and the vibrational effect on the vacancy concentration in the gray tin (α-Sn) using density functional theory has been successfully carried out. The vacancy is modeled by using a supercell consisting of 64 atoms. The vibrational effect is obtained by calculating the phonon density of states (P-DOS) of perfect and vacancy gray tin through the Gaussian-function approach. It is found that the formation energy of Sn-vacancy is 1.89 eV, and the vacancy concentration at the melting point is 3.68 × 103 cm-3. However, the vacancy concentration increases significantly to 6.48 × 109 cm-3 when the vibrational effect is considered. It is expected due to the softening of P-DOS on the vacancy case.