Ab initio piezoelectric properties of wurtzite ZnO-based alloys: Impact of the c/a cell ratio

The piezoelectric (PE) and stiffness tensors of 32-atom supercells of ZnO-based alloys have been obtained from ab initio simulations using density functional perturbation theory in the local density approximation. Low concentration for substituents to Zn, O, or both were considered in unstrained and biaxially strained supercells. The ${d}_{33}$ coefficient for unstrained ${\mathrm{Zn}}_{15}{\mathrm{YO}}_{15}\mathrm{N}$ and ${\mathrm{Zn}}_{15}{\mathrm{LaO}}_{15}\mathrm{N}$ alloys are, respectively, 17.5 and $18\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.28em}{0ex}}\mathrm{pC}\phantom{\rule{0.28em}{0ex}}{\mathrm{N}}^{\ensuremath{-}1}$, whereas ${e}_{33}$ is $1.7\phantom{\rule{0.28em}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{C}\phantom{\rule{0.28em}{0ex}}{\mathrm{m}}^{\ensuremath{-}2}$ for both alloys. These values are significantly improved compared to simulated values for pristine ZnO (${d}_{33}=11.4\phantom{\rule{0.28em}{0ex}}\mathrm{pC}\phantom{\rule{0.28em}{0ex}}{\mathrm{N}}^{\ensuremath{-}1}$ and ${e}_{33}=1.3\phantom{\rule{0.28em}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{C}\phantom{\rule{0.28em}{0ex}}{\mathrm{m}}^{\ensuremath{-}2}$). Applying 2% tensile strain on ${\mathrm{Zn}}_{15}{\mathrm{YO}}_{15}\mathrm{N}$ results in an increase of the ${\mathrm{e}}_{33}$ coefficient to 2.$1\phantom{\rule{0.28em}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{C}\phantom{\rule{0.28em}{0ex}}{\mathrm{m}}^{\ensuremath{-}2}$, a 62% increase over the value calculated for pristine ZnO. We confirm for a variety of ternary and quaternary ZnO-based alloys that a linear relation is verified between the ${\mathrm{e}}_{33}$ coefficient and the cell ratio $c/a$, described by a slope $\ensuremath{\approx}\ensuremath{-}9\phantom{\rule{0.28em}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{C}\phantom{\rule{0.28em}{0ex}}{\mathrm{m}}^{\ensuremath{-}2}$. Our results also indicate that the PE coefficients follow the same trends with respect to changes in $c/a$ caused by variations in chemical composition or by applying biaxial strain. Based on this correlation, we propose a simple method to identify promising candidates among piezoelectric alloys in the wurtzite family, effectively reducing the intensive computational resources needed to obtain optimal PE performance for applications compatible with the many requirements of thin film growth and processing.