  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. ....
