  Realizing an optimal Schottky interface of graphene on Si is challenging, as
the electrical transport strongly depends on the graphene quality and the
fabrication processes. Such interfaces are of increasing research interest for
integration in diverse electronic devices as they are thermally and chemically
stable in all environments, unlike standard metal/semiconductor interfaces. We
fabricate such interfaces with n-type Si at ambient conditions and find their
electrical characteristics to be highly rectifying, with minimal reverse
leakage current ($<$10$^{-10}$ A) and rectification of more than $10^6$. We
extract Schottky barrier height of 0.69 eV for the exfoliated graphene and 0.83
eV for the CVD graphene devices at room temperature. The temperature dependent
electrical characteristics suggest the influence of inhomogeneities at the
graphene/n-Si interface. A quantitative analysis of the inhomogeneity in
Schottky barrier heights is presented using the potential fluctuation model
proposed by Werner and G\"{u}ttler.
