DFT study of itinerant ferromagnetism in $p$-doped monolayers of MoS$_2$

We use density functional theory to explore the possibility of making the semiconducting transition-metal dichalcogenide MoS$_2$ ferromagnetic by introducing holes into the narrow Mo $d$ band that forms the top of the valence band. In the single impurity limit, the repulsive Coulomb potential of an acceptor atom and intervalley scattering lead to a twofold orbitally degenerate effective-mass like $e'$ state being formed from Mo $d_{x^2-y^2}$ and $d_{xy}$ states, bound to the K and K$'$ valence band maxima. It also leads to a singly degenerate $a'_1$ state with Mo $d_{3z^2-r^2}$ character bound to the slightly lower lying valence band maximum at $\Gamma$. Within the accuracy of our calculations, these $e'$ and $a'_1$ states are degenerate for MoS$_2$ and accommodate the hole that polarizes fully in the local spin density approximation in the impurity limit. With spin-orbit coupling included, we find a single ion magnetic anisotropy of $\sim 5\,$meV favouring out-of-plane orientation of the magnetic moment. Pairs of such hole states introduced by V, Nb or Ta doping are found to couple ferromagnetically unless the dopant atoms are too close in which case the magnetic moments are quenched by the formation of spin singlets. Combining these exchange interactions with Monte Carlo calculations allows us to estimate ordering temperatures as a function of the dopant concentration $x$. For $x \sim 9\%$, Curie temperatures as high as 100K for Nb and Ta and in excess of 160K for V doping are predicted. Factors limiting the ordering temperature are identified and suggestions made to circumvent these limitations.