First-principles study of polarization and piezoelectric properties of Zn$_{1-x}$Mg$_x$O

Wurtzite ZnO can be substituted with up to $\sim$30\% MgO to form a metastable Zn$_{1-x}$Mg$_x$O alloy while still retaining the wurtzite structure. Because this alloy has a larger band gap than pure ZnO, Zn$_{1-x}$Mg$_x$O/ZnO quantum wells and superlattices have been much studied recently as promising candidates for applications in optoelectronic and electronic devices. Here, we report the results of an {\it ab-initio} study of the spontaneous polarization of Zn$_{1-x} $Mg$_x$O alloys as a function of their composition. We perform calculations of the crystal structure based on density-functional theory in the local-density approximation, and the polarization is calculated using the Berry-phase approach. We decompose the changes in polarization into purely electronic, lattice-displacement mediated, and strain mediated components, and quantify the relative importance of these contributions. We consider both free-stress and epitaxial-strain elastic boundary conditions, and show that our results can be fairly well reproduced by a simple model in which the piezoelectric response of pure ZnO is used to estimate the polarization change of the Zn$_{1-x}$Mg$_x$O alloy induced by epitaxial strain.