  We study how strain affects orbital ordering and magnetism at the interface
between SrMnO$_3$ and LaMnO$_3$ from density-functional calculations and
interpret the basic results in terms of a three-site Mn-O-Mn model. Magnetic
interaction between the Mn atoms is governed by a competition between the
antiferromagnetic superexchange of the Mn t$_{2g}$ core spins and the
ferromagnetic double exchange of the itinerant e$_g$ electrons. While the core
electrons are relatively unaffected by the strain, the orbital character of the
itinerant electron is strongly affected, which in turn causes a large change in
the strength of the ferromagnetic double exchange. The epitaxial strain
produces the tetragonal distortion of the MnO$_6$ octahedron, splitting the
Mn-e$_g$ states into x$^2$-y$^2$ and 3z$^2$-1 states, with the former being
lower in energy, if the strain is tensile in the plane, and opposite if the
strain is compressive. For the case of the tensile strain, the resulting higher
occupancy of the x$^2$-y$^2$ orbital enhances the in-plane ferromagnetic double
exchange owing to the larger electron hopping in the plane, causing at the same
time a reduction of the out-of-plane double exchange. This reduction is large
enough to be overcome by antiferromagnetic superexchange, which wins to produce
a net antiferromagnetic interaction between the out-of-plane Mn atoms. For the
case of the in-plane compressive strain, the reverse happens, viz., that the
higher occupancy of the 3z$^2$-1 orbital results in the out-of-plane
ferromagnetic interaction, while the in-plane magnetic interaction remains
antiferromagnetic. Concrete density-functional results are presented for the
(LaMnO$_3$)$_1$/(SrMnO$_3$)$_1$ and (LaMnO$_3$)$_1$/(SrMnO$_3$)$_3$
superlattices for various strain conditions.
