Ab initio calculations of third-order elastic constants and related properties for selected semiconductors

We present theoretical studies for the third-order elastic constants $C_{ijk}$ in zinc-blende nitrides AlN, GaN, and InN. Our predictions for these compounds are based on detailed ab initio calculations of strain-energy and strain-stress relations in the framework of the density functional theory. To judge the computational accuracy, we compare the ab initio calculated results for $C_{ijk}$ with experimental data available for Si and GaAs. We also underline the relation of the third-order elastic constants to other quantities characterizing anharmonic behavior of materials, such as pressure derivatives of the second-order elastic constants $c_{ij}^{\prime }$ and the mode Grüneisen constants for long-wavelength acoustic modes $\gamma (\mathbf{q},\mathbf{j})$.

Figure 1

Sample convergence tests for the third-order elastic constants in zinc-blende GaN. (a) illustrates the dependence of $C_{111}$ and $C_{144}$ on energy cutoff (Monkhorst-Pack sampling $11\times 11\times 11$ was applied for all points). (b) shows the analogous dependence on the density of $k$-points mesh (energy cutoff of $800\mathrm{eV}$ was used for all points). Note different scales for $C_{111}$ and $C_{144}$. Later on, all calculations for GaN have been performed with the energy cutoff of $800\mathrm{eV}$ and the Monkhorst-Pack sampling $11\times 11\times 11$.

Figure 2

Plots of strain-energy relations for GaN. Squares denote results of DFT computations; solid line represents a polynomial fit. See main text for details and Table for definitions of $f_A\left(\eta \right),f_B\left(\eta \right),\dots ,f_F\left(\eta \right)$.

Figure 3

Plots of strain-stress relations for GaN. Circles denote results of DFT computations; solid line represents a polynomial fit. See main text for details and Table for definitions of $g_{A1}\left(\eta \right),g_{A2}\left(\eta \right),\dots ,g_F\left(\eta \right)$. Curves for $g_C\left(\eta \right)$ and $g_F\left(\eta \right)$ coincide because of very similar values of $C_{144}$ and $C_{456}$ in GaN.

Figure 4

(a) Energy and (b) stress component $t_3$ as a function of linear strain parameter $ϵ$ for particular deformation ${\eta }_B$ for GaN cubic crystal. Full points denote results of DFT computations; solid and dashed lines indicate the curves obtained from nonlinear and linear elasticity theory, respectively.