First principles prediction of a morphotropic phase boundary in the Bi(Zn$_{1/2}$Ti$_{1/2}$)O$_3$-(Bi$_{1/2}$Sr$_{1/2}$)(Zn$_{1/2}$Nb$_{1/2}$)O$_3$ alloy*

We present a density functional theory study on alloys of the tetragonally distorted Bi(Zn$_{1/2}$Ti$_{1/2}$)O$_3$ (BZT) and the rhombohedrally oriented (Bi$_{1/2}$Sr$_{1/2})($Zn$_{1/2} $Nb$_{1/2})$O$_3$ (BSZN). We find that compositions with $\ge$ 50\% BZT are tetragonally distorted with the polarization pointing mainly along the [001] direction. Conversely, for low concentrations of BZT the polarization is rhombohedrally oriented. Based on these results we propose a phase diagram with a possible monoclinc phase between 25\% and 50\% BZT where this material may have a useful piezoelectric response.