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.
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