In this paper, we report on the growth of highly uniform MoS2 films, mostly consisting of monolayers, on SiC surfaces with different doping levels (n- SiC epitaxy, ~1016 cm-3, and n+ SiC substrate, ~1019 cm-3) by sulfurization of a pre-deposited ultra-thin MoOx films. MoS2 layers are lowly strained (~0.12% tensile strain) and highly p-type doped (<Nh>≈4×1019 cm−3), due to MoO3 residues still present after the sulfurization process. Nanoscale resolution I-V analyses by conductive atomic force microscopy (C-AFM) show a strongly rectifying behavior for MoS2 junction with n- SiC, whereas the p+ MoS2/n+ SiC junction exhibits an enhanced reverse current and a negative differential behavior under forward bias. This latter observation, indicating the occurrence of band-to-band-tunneling from the occupied states of n+ SiC conduction band to the empty states of p+ MoS2 valence band, is a confirmation of the very sharp hetero-interface between the two materials. These results pave the way to the fabrication of ultra-fast switching Esaki diodes on 4H-SiC.