  We investigate electron transport across a complex oxide heterointerface of
La$_{0.67}$Sr$_{0.33}$MnO$_3$ (LSMO) on Nb:SrTiO$_3$ (Nb:STO) at different
temperatures. For this, we employ the conventional current-voltage method as
well as the technique of Ballistic Electron Emission Microscopy (BEEM), which
can probe lateral inhomogeneities in transport at the nanometer scale. From
current-voltage measurements, we find that the Schottky Barrier height (SBH) at
the LSMO/Nb:STO interface decreases at low temperatures accompanied by a larger
than unity ideality factor. This is ascribed to the tunneling dominated
transport caused by the narrowing of the depletion width at the interface.
However, BEEM studies of such unbiased interfaces, do not exhibit SBH lowering
at low temperatures, implying that this is triggered by the modification of the
interface due to an applied bias and is not an intrinsic property of the
interface. Interestingly, the SBH at the nanoscale, as extracted from BEEM
studies, at different locations in the device is found to be spatially
homogeneous and similar both at room temperature and at low temperatures. Our
results highlight the application of BEEM in characterizing electron transport
and their homogeneity at such unbiased complex oxide interfaces and yields new
insights into the origin of the temperature dependence of the SBH at biased
interfaces.
