  In ultra-thin two-dimensional (2-D) materials, the formation of ohmic
contacts with top metallic layers is a challenging task that involves different
processes than in bulk-like structures. Besides the Schottky barrier height,
the transfer length of electrons between metals and 2-D monolayers is a highly
relevant parameter. For MoS$_2$, both short ($\le$30 nm) and long ($\ge$0.5
$\mu$m) values have been reported, corresponding to either an abrupt carrier
injection at the contact edge or a more gradual transfer of electrons over a
large contact area. Here we use \textit{ab initio} quantum transport
simulations to demonstrate that the presence of an oxide layer between a
metallic contact and a MoS$_2$ monolayer, for example TiO$_2$ in case of
titanium electrodes, favors an area-dependent process with a long transfer
length, while a perfectly clean metal-semiconductor interface would lead to an
edge process. These findings reconcile several theories that have been
postulated about the physics of metal/MoS$_2$ interfaces and provide a
framework to design future devices with lower contact resistances.
