Anisotropic thermal expansion of SnSe from first-principles calculations based on Grüneisen's theory

Based on Grüneisen's theory, the elastic properties and thermal expansion of bulk SnSe with the Pnma phase are investigated by using first-principles calculations. Our numerical results indicate that the linear thermal expansion coefficient along the a direction is smaller than the one along the b direction, while the one along the c direction shows a significant negative value, even at high temperature. The numerical results are in good accordance with experimental results. In addition, generalized and macroscopic Grüneisen parameters are also presented. It is also found that SnSe possesses negative Possion's ratio. The contributions of different phonon modes to NTE along the c direction are investigated, and it is found that the two modes which make the most important contributions to NTE are transverse vibrations perpendicular to the c direction. Finally, we analyze the relation of elastic constants to negative thermal expansion, and demonstrate that negative thermal expansion can also occur even with all positive macroscopic Grüneisen parameters.