The effect of atomic relaxations on the temperature-dependent elastic constants (TDECs) is usually taken into account at zero temperature by the minimization of the total energy at each strain. In this paper, we investigate the order of magnitude of this approximation on a paradigmatic example: the C44 elastic constant of diamond and zincblende materials. We estimate the effect of finite-temperature atomic relaxations within the quasi-harmonic approximation by computing ab initio the internal strain tensor from the second derivatives of the Helmholtz free-energy with respect to strain and atomic displacements. We apply our approach to Si and BAs and find a visible difference between the softening of the TDECs computed with the zero-temperature and finite-temperature atomic relaxations. In Si, the softening of C44 passes from 8.6% to 4.5%, between T = 0 K and T = 1200 K. In BAs, it passes from 8% to 7%, in the same range of temperatures. Finally, from the computed elastic constant corrections, we derive the temperature-dependent Kleinman parameter, which is usually measured in experiments.
In principle the sum is extended aver all modes, including the zero-frequency acoustic ones for which Eq. (6) diverges. We do not treat this divergence problem because the acoustic modes will not contribute in the rest of the formulation.
We compute the derivative with respect to the bond length , : hence, a factor is introduced. The other factor, introduced in the derivative with respect to strain, is given by the energy-strain expansion ϵF: .
We used pseudopotentials Si.pz-nl-kjpaw_psl.1.0.0.UPF, C.pz-n-kjpaw_psl.1.0.0.UPF, Ga.pz-dnl-kjpaw_psl.1.0.0.UPF, As.pz-n-kjpaw_psl.1.0.0.UPF, and B.pz-n-kjpaw_psl.1.0.0.UPF.