  We have studied the influence of epitaxial strain on magnetic orderings and
the couplings between the spin and polar phonons in the 1:1 SrMnO$_3$/LaMnO$_3$
superlattice from first principles. Magnetic phase transitions of the
superlattice induced by epitaxial strain are observed, consistent with previous
reports. We find that oxygen octahedral rotations lower the ground state energy
but do not destroy the magnetic phase transitions induced by strain. We compute
zone center phonon frequencies and eigenvectors as functions of epitaxial
strain and magnetic ordering. A substantial increase of the coupling strength
between the spin and the lowest-frequency polar mode is observed for tensile
strains. This increase can be attributed to a change of character of the lowest
mode resulting from different relative couplings of the various polar modes to
epitaxial strain. Finally, spin-phonon coupling strengths are computed in a
Heisenberg formalism. This analysis directly reveals the changes in exchange
couplings due to specific atomic displacements or phonon modes, as well as the
nonequivalence of the out-of-plane exchange couplings across LaO layers and
across SrO layers, the latter being the result of the artificial structuring in
the superlattice.
