Theory of superconductivity in hole-doped monolayer ${\mathrm{MoS}}_2$

We theoretically investigate the Cooper-pair symmetry that can be realized in hole-doped monolayer ${\mathrm{MoS}}_2$ by solving linearized BCS gap equations in the three-orbital attractive Hubbard-like model in the presence of the atomic spin-orbit coupling. In hole-doped monolayer ${\mathrm{MoS}}_2$, both spin-orbit coupling and the multiorbital effects are more prominent than those of an electron-doped system. Near the valence band edge, the Fermi surfaces are composed of three different types of hole pockets; namely, one consists mainly of the almost spin degenerate $|d_{z^2}\rangle$ orbital near the $\mathrm{\Gamma }$ point, and the others are the spin-split upper and lower bands near the $\mathrm{K}$ and ${\mathrm{K}}^{\prime }$ points arising from the $|d_{x^2-y^2}\rangle$ and $|d_{xy}\rangle$ orbitals. The number of relevant Fermi pockets increases with the 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 $T_{\mathrm{c}}$ due to the small density of states of the split bands. For further doping, the conventional spin-singlet state (SS) appears in the $\mathrm{\Gamma }$ pocket, which has a mixture of the spin-triplet, orbital-singlet (ST-OS) and spin-singlet, orbital-triplet (SS-OT) states in the $\mathrm{K}$ and ${\mathrm{K}}^{\prime }$ 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 $T_{\mathrm{c}}$ emerges at high doping that involves all three Fermi pockets.

Figure 1

Crystalline structure of ${\mathrm{MoS}}_2$ drawn by vesta [48] and the first Brillouin zone. (a) The unit cell of the bulk ${\mathrm{MoS}}_2$ with the trigonal prismatic coordination. The red point represents an inversion center. (b) The top view of monolayer ${\mathrm{MoS}}_2$. The blue and green spheres represent Mo and S atoms, respectively. ${\mathit{R}}_{1-3}$ and $-{\mathit{R}}_{1-3}$ stand for the Mo-Mo nearest-neighboring vectors, and ${\mathit{a}}_1$ and ${\mathit{a}}_2$ are the unit vectors as the unit length $a\equiv 1$. The diamond-shaped area indicates the two-dimensional unit cell. (c) The two-dimensional Brillouin zone. ${\mathit{b}}_1$ and ${\mathit{b}}_2$ are the reciprocal unit vectors.

Figure 2

Band structure, the DOS, and the Fermi surfaces of monolayer ${\mathrm{MoS}}_2$. (a) The spin dependence. The red and blue lines represent up- and down-spin bands, respectively. (b) The orbital dependence, where the red, green, and blue lines represent $|0\rangle , |+2\rangle$, and $|-2\rangle$, respectively. (c) The doping dependence of the DOS at the Fermi level. The doubly stepwise behavior appears when the Fermi level goes across the top of the split bands as shown in (d) the enlarged view of (a). (e)–(g) The Fermi surfaces at the representative doping rates in regions A–C.

Figure 3

Doping dependences of $T_{\mathrm{c}}$ and the Cooper-pair components for $U=-0.5$ eV. (a) $J=-U/5$ (ferromagnetic), (b) $J=-U/2$, (c) $J=U/5$ (antiferromagnetic), and (d) $J=U/2$. The black line (left axis) represents $T_{\mathrm{c}}$, and the red, blue, and green lines (right axis) represent the components, $\psi$ (SS), $D_z$ (SS-OT), and $d_z$ (ST-OS), respectively. The pairing states $(d_x,d_y)$ and $(D_x,D_y)$ have much lower $T_{\mathrm{c}}$ (not shown). The vertical dotted lines indicate the doping level at which the Fermi-surface topology changes. Note that $d_z$ changes sign at $x\sim 0.06$ in B for (a) and at the boundary between B and C for (b).

Figure 4

Schematic illustrations of the representative pairing states for $U=-0.5$ eV and $J=-U/2$ corresponding to the case in Fig. 3 (a) in region A, (b) in region B, and (c) in region C. The red, green, and blue colored areas indicate the spin-split hole pockets of the $|0\rangle , |+2\rangle$, and $|-2\rangle$ orbitals, respectively. The size of the arrows roughly indicates the magnitude of the pairing components.