  The combination of ferromagnets with topological superconductors or
insulators allows for new phases of matter that support excitations such as
chiral edge modes and Majorana fermions. EuS, a wide-band-gap ferromagnetic
insulator with a Curie temperature around 16 K, and SrTiO$_3$ (STO), an
important substrate for engineering heterostructures, may support these phases.
We present scanning superconducting quantum interference device (SQUID)
measurements of EuS grown epitaxially on STO that reveal micron-scale
variations in ferromagnetism and paramagnetism. These variations are oriented
along the STO crystal axes and only change their configuration upon thermal
cycling above the STO cubic-to-tetragonal structural transition temperature at
105 K, indicating that the observed magnetic features are due to coupling
between EuS and the STO tetragonal structure. We speculate that the STO
tetragonal distortions may strain the EuS, altering the magnetic anisotropy on
a micron-scale. This result demonstrates that local variation in the induced
magnetic order from EuS grown on STO needs to be considered when engineering
new phases of matter that require spatially homogeneous exchange.
