Valley Zeeman effect and spin-valley polarized conductance in monolayer ${\text{MoS}}_2$ in a perpendicular magnetic field

We study the effect of a perpendicular magnetic field on the electronic structure and charge transport of a monolayer ${\text{MoS}}_2$ nanoribbon at zero temperature. We particularly explore the induced valley Zeeman effect through the coupling between the magnetic field $B$ and the orbital magnetic moment. We show that the effective two-band Hamiltonian provides a mismatch between the valley Zeeman coupling in the conduction and valence bands due to the effective mass asymmetry and it is proportional to $B^2$ similar to the diamagnetic shift of exciton binding energies. However, the dominant term which evolves with $B$ linearly, originates from the multiorbital and multiband structures of the system. Besides, we investigate the transport properties of the system by calculating the spin-valley resolved conductance and show that, in a low-hole doped case, the transport channels at the edges are chiral for one of the spin components. This leads to a localization of the nonchiral spin component in the presence of disorder and thus provides a spin-valley polarized transport induced by disorder.

Figure 1

A top view schematic of a monolayer ${\text{MoS}}_2$ lattice structure in a two-terminal setup. Blue (orange) circles indicate the Mo (S) atoms. The nearest neighbor (${\delta }_i$) and the next nearest neighbor ($a_i$) vector are shown in the figure. Ribbon width and scattering region length are $W/a_0=3N/2-1,L/a_0=\sqrt{3}M$, respectively.

Figure 2

Contour plot of the orbital magnetic moment as function of the momenta along the $x$ axis at the conduction (top panel) band and the valence (below panel) band. $M$ is in units of $e^{2}V{a}_0^2/\hslash$ and the spin-orbit coupling is neglected in this figure.

Figure 3

Orbital magnetic moment as a function of the momentum along the $x$ axis for both the spin-up and -down components calculated by the six-band and the two-band models. Up (below) panel corresponds to the spin-up (-down) component and $M$ is in units of $e^{2}V{a}_0^2/\hslash$.

Figure 4

(Top panel) Landau levels as a function of the momentum in units of eV calculated by a tight-binding approach on a zigzag ribbon where $B=100$ T. (Bottom panel) Valley Zeeman splitting in units of meV as a function of the magnetic field in units of tesla for both the conduction and valence bands. In the inset: The mismatch between the valley Zeeman effect of the conduction and valence bands which is the splitting in PL spectrum for right- and left-handed polarized light as a function of the magnetic field in units of tesla. Note that blue (red) lines indicate spin-up (-down) states. We set $N=100$ as the ribbon width and the real Zeeman effect is not included in this figure.

Figure 5

Unipolar conductance according to the Landau level spectrum of the low-energy model. The Zeeman interaction corresponding to the real spin is not taken into account in this figure.

Figure 6

(a) and (b) Projected local density of states $\rho (y,E_k)$ for spin-up edge modes at $E_K=-0.89$ eV. The left- and right-going modes are localized on opposite edges. (c) and (d) The same as before for spin-down edge modes but the left- and right-going states are localized on the same edges. The edge modes are mostly constructed by $d_{xy}$ and $d_{x^2-y^2}$ orbitals of the molybdenum atoms.

Figure 7

(a) Unipolar conductance for a zigzag ribbon as a function of the Fermi energy in the presence of the perpendicular magnetic field and random on-site energy. (b) Spin polarization in the presence of the perpendicular magnetic field and random on-site energy. The Zeeman interaction corresponding to the real spin is not taken into account in this figure. We set $N=50$, $M=10$, and $B=150\mathrm{T}$.