First-Principles Calculations of Elastic Properties of Cu $_{3}$ Sn and Cu $_{6}$ Sn

Elastic properties of a solid are closely related to many fundamental solid-state properties. Theoretical calculations on elastic constants are well motivated by the advance in computational technologies, especially when mechanical testing on submicron components is extremely difficult. Elastic constants of a number of anisotropic lattice systems have been calculated based on the density functional theory, and good agreements between computational and experimental results have been found. In this study, we report elastic properties of the Cu-Sn crystalline phases, the epsiv-Cu₃Sn and eta-Cu₆Sn₅, using first-principles calculations. The polycrystalline moduli obtained using the Voigt-Reuss scheme are 134.16 GPa for Cu₃Sn and 125.98 GPa for Cu₆Sn₅ . Calculation results show that these Cu-Sn crystalline phases have the greatest stiffness along the c -direction. In particular, the results reveal the unique anisotropic feature along a - and b -directions within the Cu₃Sn superstructure, which can hardly be resolved from experiments. Our results also suggest that the most compliant stiffness in the long-period direction is associated with the lattice modulation within the Cu₃Sn superstructure.