  The transition metal dichalcogenide ($MX_{2}$, where $M$=Mo, W and $X$=S, Se,
Te) monolayers are of high interest for semiconducting applications at the
nanoscale level; this interest is due to both their direct band gaps and high
charge mobilities. In this regard, an in-depth understating of the related
Schottky barrier heights, associated with the incorporation of $MX_{2}$ sheets
into novel low-dimensional metal-semiconductor junctions, is of crucial
importance. Herein, we generate and provide analysis of the Schottky barrier
heights behavior to account for the metal-induced gap states concept as its
explanation. In particular, the present investigations concentrate on the
estimation of the charge neutrality levels directly by employing the primary
theoretical model, $i.e.$ the cell-averaged Green's function formalism combined
the complex band structure technique. The results presented herein place charge
neutrality levels in the vicinity of the mid-gap; this is in agreement with
previous reports and analogous to the behavior of three-dimensional
semiconductors. The calculated canonical Schottky barrier heights are also
found to be in agreement with other computational and experimental values in
cases where the difference between electronegativities of the semiconductor and
metal contact is small. Moreover, the influence of the spin-orbit effects is
herein considered and supports that Schottky barrier heights have metal-induced
gap state-derived character, regardless whether spin-orbit coupling
interactions are considered. The results presented within this report
constitute a direct and vital verification of the importance of metal-induced
gap states in explaining the behavior of observed Schottky barrier heights at
$MX_{2}$-metal junctions.
