Nonlinear elasticity in III-N compounds: Ab initio calculations

We have studied the nonlinear elasticity effects in zinc-blende and wurtzite crystallographic phases of III-N compounds. Particularly, we have determined the pressure dependences of elastic constants in InN, GaN, and AlN by performing ab initio calculations in the framework of plane-wave pseudopotential implementation of the density-functional theory. The calculations have been performed employing two exchange-correlation functionals, one within the local density approximation and the other within the generalized gradient approximation. We have found that ${\mathrm{C}}_{11}$, ${\mathrm{C}}_{12}$ in zinc-blende nitrides and ${\mathrm{C}}_{11}$, ${\mathrm{C}}_{12}$, ${\mathrm{C}}_{13}$, ${\mathrm{C}}_{33}$ in wurtzite nitrides depend significantly on hydrostatic pressure. Much weaker dependence on pressure has been observed for ${\mathrm{C}}_{44}$ elastic constant in both zinc-blende and wurtzite phases. Further, we have examined the influence of pressure dependence of elastic constants on the pressure coefficient of light emission, $d{E}_E∕dP$, in wurtzite $\mathrm{In}\mathrm{Ga}\mathrm{N}∕\mathrm{Ga}\mathrm{N}$ and $\mathrm{Ga}\mathrm{N}∕\mathrm{Al}\mathrm{Ga}\mathrm{N}$ quantum wells. We have shown that the pressure dependence of elastic constants leads to a significant reduction of $d{E}_E∕dP$ in nitride quantum wells. Finally, we have considered the influence of nonlinear elasticity of III-N compounds on the properties of hexagonal nitride quantum dots (QDs). For typical wurtzite $\mathrm{Ga}\mathrm{N}∕\mathrm{Al}\mathrm{N}$ QDs, we have shown that taking into account pressure dependence of elastic constants results in the decrease of volumetric strain in the QD region by about 7%. Simultaneously, the average $z$ component of the piezoelectric polarization in the QDs increases by $0.1\mathrm{MV}∕\mathrm{cm}$ compared to the case when linear elastic theory is used. Both effects, i.e., decrease of volumetric strain as well as increase of piezoelectric field, decrease the band-to-band transition energies in the QDs.

Figure 1

(a) The axial ratio, $c∕a$, and (b) the internal parameter, $u$, of the wurtzite structure as a function of the hydrostatic pressure for GaN, InN, and AlN. Solid lines are added to guide the eye.

Figure 2

Elastic constants of wurtzite structure as a function of the hydrostatic pressure for GaN (a), InN (b), and AlN (c). Full and empty symbols represent values obtained using LDA-DFT and GGA-DFT methods, respectively. Solid and dashed lines are added to guide the eye.

Figure 3

Elastic constants of zinc-blende structure as a function of the hydrostatic pressure for GaN (a), InN (b), and AlN (c). Full and empty symbols represent values obtained using LDA-DFT and GGA-DFT methods, respectively. Solid and dashed lines are added to guide the eye.

Figure 4

Pressure coefficients of light emission energies $(d{E}_E∕dP)$ for wurtzite ${\mathrm{In}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{N}∕\mathrm{Ga}\mathrm{N}$ QWs as a function of quantum well width. Stars represent the experimental values of $d{E}_E∕dP$ taken from Ref. 21, squares correspond to theoretical values of $d{E}_E∕dP$ obtained using linear elastic theory, and circles correspond to $d{E}_E∕dP$ calculated using nonlinear elastic theory. Full and empty symbols represent values of $d{E}_E∕dP$ obtained using elastic constants calculated within LDA-DFT and GGA-DFT approaches, respectively. Solid and dashed lines are added to guide the eye.

Figure 5

Pressure coefficients of emission energies $(d{E}_E∕dP)$ for wurtzite $\mathrm{Ga}\mathrm{N}∕{\mathrm{Al}}_{0.8}{\mathrm{Ga}}_{0.2}\mathrm{N}$ QWs as a function of QW width. Stars represent the experimental values of $d{E}_E∕dP$, taken from Ref. 6. Squares correspond to theoretical values of $d{E}_E∕dP$ obtained using linear elastic theory. Circles correspond to $d{E}_E∕dP$ calculated using nonlinear elastic theory. Full and empty symbols represent values of $d{E}_E∕dP$ obtained using elastic constants calculated within LDA-DFT and GGA-DFT approaches, respectively. Solid and dashed lines are added to guide the eye.

Figure 6

Profiles of the built-in hydrostatic pressure (a) and volumetric strain (b) for a $\mathrm{Ga}\mathrm{N}∕\mathrm{Al}\mathrm{N}$ QD, taken from the center of the QD along the $z$ axis, i.e., parallel to the [0001] growth direction. Dotted and dashed lines show profiles obtained using linear elastic theory while dashed-dotted and solid lines correspond to results obtained using nonlinear elastic theory.