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We use \textit{ab initio} calculations to systematically study the phasediagram of multiferroic Sr$_{1-x}$Ba$_x$MnO$_3$ ($0 \leq x \leq 1$) as afunction of chemical doping, epitaxial strain and external pressure. We findthat 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 betweentwo multiferroic phases, one antiferromagnetic with a smaller polarization andone ferromagnetic with a larger polarization. A giant effectivemagneto-electric coupling and cross-field control (electric field control ofmagnetism or magnetic field control of polarization) can be achieved in thevicinity of the transition. The dependence of the theoretically computedtransition point on the choice of exchange correlation functionals isdetermined and is found to be non-negligible. We also show that the perovskitestructure of BaMnO$_3$ can be stabilized relative to its hexagonal polymorphsat pressures larger than 20 GPa.

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