Density functional theory study of the structural and electronic properties of single and double acceptor dopants in MX2 monolayers

Density functional theory calculations are used to systematically investigate the structural and electronic properties of MX$_2$ transition metal dichalcogenide monolayers with M = Cr, Mo, W and X = S, Se, Te that are doped with single (V, Nb, Ta) and double (Ti, Zr, Hf) acceptor dopants on the M site with local $D_{3h}$ symmetry in the dilute limit. Three impurity levels that arise from intervalley scattering are found above the valence band maxima (VBM): an orbitally doubly degenerate $e'$ level bound to the $K/K'$ VBM and a singly degenerate $a'_1$ level bound to the $\Gamma$-point VBM. Replacing S with Se or Te lowers the $\Gamma$ point VBM substantially with respect to the $K/K'$ VBM bringing the $a'_1$ level down with it. The relative positions of the impurity levels that determine the different structural and electronic properties of the impurities in $p$-doped MX$_2$ monolayers can thus be tuned by replacing S with Se or Te. Single acceptors introduce a magnetic moment of 1$\, \mu_{\rm B}$ in all MX$_2$ monolayers. Out-of-plane magnetic anisotropy energies as large as 10 meV/dopant atom are found thereby satisfying an essential condition for long-range ferromagnetic ordering in two dimensions. For double acceptors in MS$_2$ monolayers, both holes occupy the high-lying $a'_1$ level with opposite spins so there is no magnetic moment; in MSe$_2$ and MTe$_2$ monolayers the holes occupy the $e'$ level, a Jahn-Teller (JT) distortion wins the competition with exchange splitting resulting in the quenching of the magnetic moments. Even when the JT distortion is disallowed, magnetic double acceptors have a large in-plane magnetic anisotropy energy that is incompatible with long-range magnetic ordering in two dimensions. ....