Possible realization and protection of valley-polarized quantum Hall effect in $\mathrm{Mn}/\mathrm{W}{\mathrm{S}}_2$

By using the first-principles calculations and model analyses, we found that the combination of defected tungsten disulfide monolayer and manganese adsorption may give a $K{K}^{\prime }$ valley spin splitting up to 210 meV. This system also has a tunable magnetic anisotropy energy and its Fermi level sits right in a large band gap. Therefore it can be used for the realization of the valley-polarized anomalous Hall effect and for the exploration of other valley related physics without using optical methods. A protective environment can be formed by covering it with a hexagonal BN layer, without much disturbance to the benign properties of $\mathrm{Mn}/\mathrm{W}{\mathrm{S}}_2$.

Figure 1

(a) Three diffusion pathways for Mn atoms (inset) and the corresponding energy profiles. (b) Side and top views of charge density difference for $\mathrm{Mn}/\mathrm{W}{\mathrm{S}}_2$, blue and red colors represent charge accumulation and depletion, respectively. (Inset) Side and top views of spin density of $\mathrm{Mn}/\mathrm{W}{\mathrm{S}}_2$, red and blue colors represent the minority and majority spin parts, respectively.

Figure 2

(a) The band structure of $\mathrm{Mn}/\mathrm{W}{\mathrm{S}}_2$ with SOC included. The red (blue) line represents the spin up (down) components. (The inset is the effect of dopants concentration on the valley splitting of $\mathrm{Mn}/\mathrm{W}{\mathrm{S}}_2$. These four dopants concentration are corresponding to the case of one Mn atom periodic doping in 7 × 7, 6 × 6, 5 × 5, and 4 × 4 defected $\mathrm{W}{\mathrm{S}}_2$, respectively.) (b) The corresponding projected density of states of all atoms, where dash-red lines and solid-blue lines are for the spin-up and spin-down channels, respectively.

Figure 3

(a) Left and right panels are the distributions of Berry curvature in the whole Brillouin zone of $\mathrm{Mn}/\mathrm{W}{\mathrm{S}}_2$ with the Fermi level lying in the gap and intervalley, respectively. (b) The anomalous Hall conductivity ${\sigma }_{xy}$ as a function of Fermi level. (c) The variation of valley splitting $\mathrm{\Delta }K{K}^{\prime }$ as the magnetization of $\mathrm{Mn}/\mathrm{W}{\mathrm{S}}_2$ rotates. The insets show the schematics of the anomalous valley Hall effect (insets).

Figure 4

(a) The Fermi level dependent total and spin channel decomposed MAEs of $\mathrm{Mn}/\mathrm{W}{\mathrm{S}}_2$ from the rigid band model analyses. (b) The MAE and valley splitting of $\mathrm{Mn}/\mathrm{W}{\mathrm{S}}_2$ as a function of external electric field (ε). The inset show the charge density difference of $\mathrm{Mn}/\mathrm{W}{\mathrm{S}}_2$ caused by the external electric field of −0.2 and 0.2 V/Å, and blue and red colors represent charge accumulation and depletion, respectively.