Ab initio quasi-harmonic thermoelasticity, piezoelectricity, and thermoelectricity of polar solids at finite temperature and pressure: Application to wurtzite ZnO

Abstract

We generalize a previously established ab initio approach-originally developed for hexagonal close-packed (hcp) metals-to accommodate solids with both internal and external degrees of freedom. This extension enables the thermodynamic and thermoelastic characterization of insulators, including those with non-vanishing piezoelectric and pyroelectric tensors. Utilizing Density Functional Theory (DFT) and Density Functional Perturbation Theory (DFPT) within the quasi-harmonic approximation, we derive the pressure and temperature dependence of these properties. Specifically, we investigate internal degrees of freedom using two distinct frameworks: the Zero Static Internal Stress Approximation (ZSISA) and Full Free Energy Minimization (FFEM). We then compare these approximations by computing internal and external thermal expansions, as well as temperature-dependent piezoelectric and pyroelectric tensors. Finally, we demonstrate the generalized formalism by calculating the thermodynamic properties of wurtzite ZnO across a broad range of pressures and temperatures.

Figures (17)

• Figure 1: The wurtzite crystal structure of ZnO, illustrating the internal and external parameters utilized in this study. The external lattice parameters are defined by the cell dimensions $a$ and $c$, while the internal parameter $u$ characterizes the relative position of the anion and cation sublattices along the $[0001]$ axis. (Rendered using VESTA 3 momma_vesta_2011)
• Figure 2: Energy contours as a function of the lattice parameter $a$ and the $c/a$ ratio (red lines). The intersection of the dashed blue lines identifies the global energy minimum. The yellow and blue curves represent the configurations where the stress corresponds to a uniform pressure at $0$ K and $700$ K, respectively. The yellow dots indicate the five specific geometries sampled to map the $0$ K stress-pressure curve. Green lines denote the isobars at $0$, $40$, and $80$ kbar. The thin dotted lines show the $5\times 5$ mesh of crystal parameters employed for the Helmholtz free energy calculations.
• Figure 3: PBEsol phonon dispersions of ZnO calculated at the $10$ K equilibrium geometry. The mode symmetries are indicated by color-coding corresponding to the irreducible representations of the point group of the ${\bf q}$ point indicated in the figure. Colored panels represent paths along the Brillouin zone borders; pink and yellow panels utilize projective representations for symmetry classification (refer to the thermo_pw documentation for detailed color convention definitions). For comparison, experimental inelastic neutron scattering data are included from Ref. hewat_lattice_1970 and Ref. thoma_lattice_1974 (blue dots), along with data from Ref. serrano_phonon_2010 (red dots).
• Figure 4: PBEsol thermal expansion tensor components, $\alpha_{11}$ and $\alpha_{33}$, of ZnO as a function of temperature. The red and dashed green lines represent our current results obtained using pseudo-dojo pseudopotentials within the FFEM and ZSISA frameworks, respectively. For comparison, we show the ZSISA results from Ref. rostami_anisotropic_2025 (dashed blue, also using Dojo PP) and the FFEM (solid yellow) and ZSISA (dashed yellow) results from Ref. masuki_full_2023 (calculated with PAW PPs). Experimental data points are taken from Ref. ibach_thermal_1969.
• Figure 5: PBEsol thermal expansion coefficient $\alpha_{xx}$ of ZnO as a function of temperature, calculated at pressures of $0$ kbar (red), $40$ kbar (green), and $80$ kbar (blue). Continuous lines represent the results obtained using the FFEM, while dashed lines correspond to the ZSISA approximation. The comparison illustrates the suppression of thermal expansion with increasing pressure and the divergence between the two methodological frameworks as temperature rises.