Schematic view of the studied structure for different boundary conditions. (a) The GaN slab is sandwiched between two electrodes connected to a voltage source. (b) The GaN slab is sandwiched between two electrodes in open-circuit conditions. (c) The GaN slab is sandwiched between two charged electrodes. The sign convention with respect to the  direction of GaN is shown in the legend for all cases. Red up arrows apply for positive bias, whereas blue down arrows apply for negative bias [see panels (a) and (c)]. Thus, the sign of the spontaneous polarization with respect to z is negative for and positive for (note that is opposite to  in the latter case), whereas for pointing down in the figure.
Dependence of strain on thickness and applied voltage under constant-E conditions for a GaN slab. (a) In-plane strain without including the electrostatic force correction . (b) Out-of-plane strain for . (c) including the electrostatic force correction . (d) for . (e) comparison between and as a function of the electric field intensity. (f) Dependence of the Poisson’s ratio and the effective piezoelectric coefficient on the electric field intensity.
Dependence of the out-of-plane strain on thickness and applied voltage under constant-E conditions for an in-plane GaN slab constrained to (a) (compressive) and (b) to (tensile).
Capacitance of a GaN slab as a function of applied voltage at different film thicknesses. The upper panel shows the capacitance for a 1 nm-thick GaN slab, whereas the lower panel shows the capacitance for a 10 nm-thick GaN slab. In both cases, the capacitance is obtained for unconstrained and compressive constrainted conditions. Black (red) lines correspond to the calculations (without) including the electrostatic force correction, i.e., . The maximum relative variation in capacitance in the displayed range between unconstrained and constrained results is also shown for comparison. Red color online corresponds to gray in the printed version.
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