  In this work, density functional computations and Monte Carlo simulations are
performed to investigate structural, electronic, magnetic, and thermodynamic
properties of Co/WS$_2$ junction, a semiconductor WS$_2$ monolayer covered by a
ferromagnetic cobalt monolayer. In addition to a conventional semilocal
exchange-correlation functional, three nonlocal functional, including the novel
ACBN0 scheme, are applied to obtain reliable electronic and magnetic
properties. It is argued that the ACBN0 scheme, is very efficient for first
principles description of the Co/WS$_2$ junction. The obtained electronic
structures evidence a trustworthy half-metallic gap in the majority spin
channel of the lowest energy configuration of the junction, promising for
spintronic applications. The obtained magnetic thermodynamics properties from
Monte Carlo simulations predict a Curie temperature of about 110\,K, which is
far small for device applications of this junction. The electronic and magnetic
properties of the system are calculated under various compressive and tensile
strains and it is shown that a tensile strain of about 4\% may effectively
improve thermal stability of half-metallic ferromagnetism in the Co/WS$_2$
junction.
