  We investigate the effect of strain and film thickness on the orbital and
magnetic properties of LaSrCrO$_3$ (LSCO)/LaSrMnO$_3$ (LSMO) heterostructures
using bulk magnetometry, soft X-ray magnetic spectroscopy, first-principles
density functional theory, high-resolution electron microscopy and X-ray
diffraction. We observe an anti-parallel ordering of the magnetic moments
between the ferromagnetic LSMO layers and the LSCO spacers leading to a
strain-independent ferromagnetic ground state of the LSCO/LSMO heterostructures
for LSMO layers as thin as 2 unit cells. As the LSMO thickness is increased, a
net ferromagnetic state is maintained, however, the average magnetic moment per
Mn is found to be dependent on the magnitude of the substrate-induced strain.
The differences in the magnetic responses are related to preferential
occupation of the Mn $x^2-y^2$ (in-plane) d-orbitals for tensile strain and
$3z^2-r^2$ (out-of-plane) orbitals under compressive strain leading to
competing ferromagnetic and anti-ferromagnetic exchange interactions within the
LSMO layers. These results underscore the relative contributions of orbital,
structural and spin degree of freedom and their tunability in atomically-thin
crystalline complex oxide layers.
