Under various conditions of the growth process, when the presumably unconventional superconductor Sr$_2$RuO$_4$ (SRO) contains micro-inclusions of Ru metal, the superconducting critical temperature increases significantly. An STEM study shows a sharp interface geometry which allows crystals of SRO and of Ru-metal to grow side by side by forming a commensurate superlattice structure at the interface. In an attempt to shed light as to why this happens, we investigated the atomic structure and electronic properties of the interface between the oxide and the metal micro-inclusions using density functional theory (DFT) calculations. Our results support the observed structure indicating that it is energetically favored over other types of Ru-metal/SRO interfaces. We find that a $t_{2g}$-$e_g$ orbital mixing occurs at the interface with significantly enhanced magnetic moments. Based on our findings, we argue that an inclusion mediated interlayer coupling reduces phase fluctuations of the superconducting order parameter which could explain the observed enhancement of the superconducting critical temperature in SRO samples containing micro-inclusions.
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