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Phase diagram of Sr$_{1-x}$Ba$_x$MnO$_3$ as a function of chemical doping, epitaxial strain and external pressure

We use \textit{ab initio} calculations to systematically study the phase diagram of multiferroic Sr$_{1-x}$Ba$_x$MnO$_3$ ($0 \leq x \leq 1$) as a function of chemical doping, epitaxial strain and external pressure. We find that by replacing Sr with Ba in cubic SrMnO$_3$ and imposing epitaxial strain, the material can be tuned to the vicinity of a first order transition between two multiferroic phases, one antiferromagnetic with a smaller polarization and one ferromagnetic with a larger polarization. A giant effective magneto-electric coupling and cross-field control (electric field control of magnetism or magnetic field control of polarization) can be achieved in the vicinity of the transition. The dependence of the theoretically computed transition point on the choice of exchange correlation functionals is determined and is found to be non-negligible. We also show that the perovskite structure of BaMnO$_3$ can be stabilized relative to its hexagonal polymorphs at pressures larger than 20 GPa.