Tunable Berry curvature and valley and spin Hall effect in bilayer ${\mathrm{MoS}}_2$

The chirality of electronic Bloch bands is responsible for many intriguing properties of layered two-dimensional materials. We show that in bilayers of transition metal dichalcogenides (TMDCs), unlike in few-layer graphene and monolayer TMDCs, both intralayer and interlayer couplings give important contributions to the Berry curvature in the $K$ and $-K$ valleys of the Brillouin zone. The interlayer contribution leads to the stacking dependence of the Berry curvature and we point out the differences between the commonly available 3R type and 2H type bilayers. Due to the interlayer contribution, the Berry curvature becomes highly tunable in double gated devices. We study the dependence of the valley Hall and spin Hall effects on the stacking type and external electric field. Although the valley and spin Hall conductivities are not quantized, in ${\mathrm{MoS}}_{2}2\text{H}$ bilayers, they may change sign as a function of the external electric field, which is reminiscent of the behavior of lattice Chern insulators.

Figure 1

(a) Band structure showing the CB and the VB of 3R stacked bilayer ${\mathrm{MoS}}_2$ along the $\mathrm{\Gamma }\text{−}K\text{−}M\text{−}\mathrm{\Gamma }$ direction of the BZ from DFT calculations. (b) The same for 2H stacked bilayer ${\mathrm{MoS}}_2$. The spin-orbit coupling is neglected in (a) and (b). (c) Schematic crystal structure of 3R stacked bilayer TMDCs in side and in top views. (d) Schematic crystal structure of 2H stacked bilayer TMDCs in side and in top views. In (c) and (d), the monolayers are shown as simple hexagonal lattices with two inequivalent sites. ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ denote the lattice vectors.

Figure 2

Comparison of numerical and analytical calculation of ${\mathrm{\Omega }}_z$ around the $K$ point for (a) 3R stacked and (b) 2H stacked bilayer ${\mathrm{MoS}}_2$. $\square$ show the results for CBs, $◯$ for VBs. In (a), brown color corresponds to bands in the bottom layer, purple to bands in the top layer, solid lines show the results of Eq. (4). Dashed lines indicate the Berry curvature ${\mathrm{\Omega }}_z^{\left(0\right)}$ of a monolayer, given by Eq. (4a). In (b), brown color corresponds to the layer at $-U_g$, purple to the layer at $+U_g$ potential, solid lines show the results of Eqs. (5a), (5b), and (6), dashed lines indicate ${\mathrm{\Omega }}_{z,cb}^{\left(0\right)}$ for the interlayer contribution given by Eq. (5a). For material parameters of the $\mathbf{k}·\mathbf{p}$ models see Table 1. In (b), we used $U_g=10\mathrm{meV}$. The plotted range corresponds to around 10% of the $\mathrm{\Gamma }\text{−}K$ distance in the BZ. (c) and (d) Schematics of the valley Hall conductivity contributions in the CB of 3R and 2H bilayers, respectively, when an in-plane electric field $\mathbf{E}$ is applied.

Figure 3

SOC effects in the band structure of bilayer ${\mathrm{MoS}}_2$. (a) DFT band-structure calculations along the $\mathrm{\Gamma }\text{−}K\text{−}M\text{−}\mathrm{\Gamma }$ line of the BZ for 3R bilayer. (b) The same as in (a) for 2H bilayer. (c) For 3R bilayers, close to the $\pm K$ points the layer index is an approximately good quantum number for each of the bands both in the CB and the VB. Neglecting the SOC (noSOC) the lowest CB is mostly localized to the top layer (dashed line), while the next CB band (solid line) to the bottom layer. The opposite is true for the two highest energy VBs. The bands are shifted in energy due to the interlayer band edge energy difference $2\delta E_{cc}$ and $2\delta E_{vv}$. (d) When SOC is taken into account for 3R bilayers, each of the bands become spin-split and spin polarized. Red corresponds to $\uparrow$, blue to $\downarrow$ spin polarization. The spin-splitting ${\mathrm{\Delta }}_{cb}$ of the two lowest CBs is much smaller than the interlayer splitting $\delta E_{cc}$. The situation is different for the VBs: here ${\mathrm{\Delta }}_{vb}\gtrsim \delta E_{vv}$ and therefore the spin-polarized bands have an alternating layer index. (e) For 2H bilayers, if SOC is neglected (noSOC), the two lowest energy CB are degenerate at the $\pm K$ points and weakly split due to the interlayer coupling away from the $\pm K$ points. The energy splitting of the two highest energy VBs is $2t_{\perp }$. Both layers contribute with equal weight to each of the bands. (f) When SOC is taken into account for 2H bilayers, in the CB there are two twofold degenerate and spin-unpolarized bands separated by an energy $2{\mathrm{\Delta }}_{cb}$ at the $\pm K$ point. A combined layer and spin index can be assigned to each of the four CB bands at the $\pm K$ point, away from the $\pm K$ points both layers contribute to each of the bands, but with different weights. In the VB, both layers contribute to each of the bands, even at the $\pm K$ points. Only if ${\mathrm{\Delta }}_{vb}\gg t_{\perp }$ do the bands become approximately layer polarized [26]. In (d) an (f), the spin polarization of the bands in the $-K$ valley can be obtained by taking the time-reversed states.

Figure 4

Schematic evolution of the four low-energy CB bands as a function of the interlayer potential $U_g$ at the $K$ point of the BZ. Spin-degenerate bands are shown with purple, $\uparrow$ polarized with red and $\downarrow$ polarized with blue. Solid line corresponds to bands at $+U_g$, dashed line to bands at $-U_g$ potential.

Figure 5

Schematic top view of the crystal lattice of (a) 3R and (b) 2H bilayer TMDC. For 3R bilayers in (a), the positions of the metal atoms in the unit cell are given by the vectors ${\mathbf{t}}_{\mathrm{Mo}}^t=\frac{a}{2}{(1,-1/\sqrt{3})}^T$ and ${\mathbf{t}}_{\mathrm{Mo}}^b={(0,0)}^T$. Metal atoms in different layers are shown by different colors. For 2H bilayers, in (b), only atoms in the top layer are visible. The position of the metal atoms in the unit cell are given by the vectors ${\mathbf{t}}_{\mathrm{Mo}}^t=\frac{a}{2}{(1,-1/\sqrt{3})}^T$ and ${\mathbf{t}}_{\mathrm{Mo}}^b=\frac{a}{2}{(1,1/\sqrt{3})}^T$. Here, $a$ is the lattice constant.

Figure 6

The Brillouin zone with the six $Q$ points and the $\pm K$ valleys.