Engineered phase competition in $A$-site-ordered manganites $R$BaMn$_2$O$_6$ ($R$=Y and rare earth elements) from first principles

($A/A^\prime$)MnO$_3$ manganites for which the $A$-site cations order in layers, e.g., $R$BaMn$_2$O$_6$ ($R$=Y and rare earth elements) show higher charge and orbital ordering temperatures as compared with $A$-site disordered manganites. The degree of MnO$_6$ octahedra rotation, and therefore the Mn-O-Mn angle and Mn-O bond length, in $R$BaMn$_2$O$_6$ varies strongly with the ionic size of the rare earth ion. In fact $R$BaMn$_2$O$_6$ spans from a ferromagnetic metal ($R$=La) to an $A$-type antiferromagnetic metal ($R$=Pr and Nd), to a CE-type charge/orbital-ordered insulator ($R$=Sm-Y). The tuning of the electronic and magnetic ground states coincides with changes in the rotation patters and structural transitions from tetragonal ($R$=La-Nd), to orthorhombic ($R$=Sm-Gd), to monoclinic ($R$=Tb-Y) as the radius of $R$ decreasing, reflecting the competition among charge, orbital, magnetic, Jahn-Teller, and lattice degrees of freedom. In this talk we present the epitaxial phase diagram calculated from first-principles for these $A/A^\prime$ layered manganites and discuss the possibility of using an electric-field to control the competition among these phases via octahedral rotation induced ferroelectricity.