Third-order elastic constants and biaxial relaxation coefficient in wurtzite group-III nitrides by hybrid-density functional theory calculations

We determine the second-order elastic constants (SOECs) and the third-order elastic constants (TOECs) for wurtzite AlN, GaN, and InN using the hybrid-density functional theory calculations with the plane wave basis sets. We apply the analytical formulas for the deformation gradient tensors as functions of the Lagrangian strain in order to eliminate the truncation errors in the Taylor expansion series of the deformation gradients and to facilitate the calculation of the Lagrangian stress. We show that the convergence criteria for the calculation of the TOECs with respect to the k-points density and the plane wave cutoff energy are similar for the strain-energy method and the strain-stress approach. The strain-energy method turns out to be more stable against the numerical errors than the strain-stress approach, which requires smaller tolerance for the precision of the self-consistent calculations. The SOECs, extracted by the method of least squares, are consistent with the experimental data and the previous ab initio calculations. Then, we investigate the biaxial relaxation coefficient for AlN, GaN, and InN, subjected to biaxial stress in the plane perpendicular to the c axis of the wurtzite structure. This coefficient determines the relationship between the in-plane and out-of-plane strain components in thin films and quantum wells grown on c-plane substrates. We demonstrate that for InN and AlN, the biaxial relaxation coefficient increases significantly with the in-plane strain, whereas it shows the opposite behavior in GaN. These results are well described by the third-order elasticity theory and they cannot be modeled by the linear theory of elasticity, which predicts no dependence of the biaxial relaxation coefficient on the in-plane strain. Therefore, the obtained TOECs should prove very useful for the modelling of strain-related phenomena in heterostructures, nanostructures and devices made of the group-III nitride semiconductors.