Theory of superconductivity in hole-doped monolayer MoS$_{2}$

We theoretically investigate the Cooper-pair symmetry to be realized in hole-doped monolayer MoS$_2$ by solving linearized BCS gap equations on the three-orbital attractive Hubbard-like model in the presence of the atomic spin-orbit coupling. In hole-doped monolayer MoS$_2$, both spin-orbit coupling and the multi-orbital effects are more prominent than those of electron-doped system. Near the valence band edge, the Fermi surfaces are composed of three different types of hole pockets, namely, one mainly consisting of the almost spin-degenerate $|d_{z^{2}}\rangle$ orbital near $\Gamma$ point, and the others of the spin-split upper and lower bands near ${\rm K}$ and ${\rm K}'$ points arising from the $|d_{x^{2}-y^{2}}\rangle$ and $|d_{xy}\rangle$ orbitals. The number of relevant Fermi pockets increases with increase of the doping. At very low doping, the upper split bands of $|d_{x^{2}-y^{2}}\rangle$ and $|d_{xy}\rangle$ are concerned, yielding extremely low $T_{\rm c}$ due to small density of states of the split bands. For further doping, the conventional spin-singlet state (SS) appears in the $\Gamma$ pocket, which has a mixture of the spin-triplet (orbital-singlet) (ST-OS) and spin-singlet (orbital-triplet) (SS-OT) states in the K and K$'$ pockets. The ratio of the mixture depends on the relative strength of the interactions, and the sign of the exchange interactions. Moderately strong ferromagnetic exchange interactions even lead to the pairing state with the dominant ST-OS state over the conventional SS one. With these observations, we expect that the fascinating pairing with relatively high $T_{\rm c}$ emerges at high doping that involves all the three Fermi pockets.