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.
We investigate the phase transitions and the properties of the topological insulator in InGaN/GaN and InN/InGaN double quantum wells grown along the [0001] direction. We apply a realistic model based on the nonlinear theory of elasticity and piezoelectricity and the eight-band k·p method with relativistic and nonrelativistic linear-wave-vector terms. In this approach, the effective spin‒orbit interaction in InN is negative, which represents the worst-case scenario for obtaining the topological insulator in InGaN-based structures. Despite this rigorous assumption, we demonstrate that the topological insulator can occur in InGaN/GaN and InN/InGaN double quantum wells when the widths of individual quantum wells are two and three monolayers (MLs), and three and three MLs. In these structures, when the interwell barrier is sufficiently thin, we can observe the topological phase transition from the normal insulator to the topological insulator via the Weyl semimetal, and the nontopological phase transition from the topological insulator to the nonlocal topological semimetal. We find that in InGaN/GaN double quantum wells, the bulk energy gap in the topological insulator phase is much smaller for the structures with both quantum well widths of 3 MLs than in the case when the quantum well widths are two and three MLs, whereas in InN/InGaN double quantum wells, the opposite is true. In InN/InGaN structures with both quantum wells being three MLs and a two ML interwell barrier, the bulk energy gap for the topological insulator can reach about . We also show that the topological insulator phase rapidly deteriorates with increasing width of the interwell barrier due to a decrease in the bulk energy gap and reduction in the window of In content between the normal insulator and the nonlocal topological semimetal. For InN/InGaN double quantum wells with the width of the interwell barrier above five or six MLs, the topological insulator phase does not appear. In these structures, we find two novel phase transitions, namely the nontopological phase transition from the normal insulator to the nonlocal normal semimetal and the topological phase transition from the nonlocal normal semimetal to the nonlocal topological semimetal via the buried Weyl semimetal. These results can guide future investigations towards achieving a topological insulator in InGaN-based nanostructures.
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