Unconventional Quantum Hall Effect and Tunable Spin Hall Effect in Dirac Materials: Application to an Isolated ${\mathrm{MoS}}_2$ Trilayer

We analyze the Landau level (LL) structure in a ${\mathrm{MoS}}_2$ trilayer and find a field-dependent unconventional Hall plateau sequence $\nu =\dots ,-2M-6,-2M-4,-2M-2,-2M-1,\dots ,-5,-3,-1,0,2,4,\dots$. Because of orbital asymmetry, the low-energy Dirac fermions become heavily massive and the LL energies grow linearly with $B$, rather than with $\sqrt{B}$. Spin-orbital couplings break spin and valley degenerate LL’s into two groups, with LL crossing effects present in the valence bands. In a $p\mathrm{\text{−}}n$ junction, spin-resolved fractionally quantized conductance appears in two-terminal measurements with a controllable spin-polarized current that can be probed at the interface. We also show the tunability of zero-field spin Hall conductivity.

Figure 1

(a) and (b) Electronic band structure near valley $K$ and $K^{\prime }$. (c) LL’s with $n=0,10,\dots ,80$ orbitals. (d)–(f) Enlarged view of the LL’s in Group I, II, and III in (c). The $n\ne 0$ LL’s are broken into ${\tau }_z{s}_z=1$ doublets and ${\tau }_z{s}_z=-1$ doublets, due to the SOC and inversion asymmetry. LL crossing occurs between group II and III. The $n_{\mathrm{I}}=0$ LL is spin degenerate and only appears at valley $K$. The $n_{\mathrm{II}}$, $n_{\mathrm{III}}=0$ LL’s are spin-filtered and appear only at valley $K^{\prime }$.

Figure 2

The example of valence band LL crossing effect described in the text. All LL’s are doubly degenerate except that the $n_{\mathrm{III}}=0$ LL is nondegenerate. The negative number denotes the filling factor of the region bounded by the surrounding LL’s. The crossing points have degeneracy $g=3$ at $n_{\mathrm{III}}=0$ and $g=4$ otherwise.

Figure 3

Left panel: The solid (dashed) curves represent spin $\uparrow$ ($\downarrow$) bands and the parallel lines denote their LL’s. Regions $A\mathrm{\text{−}}D$ are energy windows separated by the three lifted energies of the four $n=0$ LL’s that are depicted by red (bold) lines. Right panel: a schematic $p\mathrm{\text{−}}n$ junction with two different filling factors ${\nu }_1$ and ${\nu }_2$ at two regions.

Figure 4

(a) Valence band spin Hall conductivity ${\sigma }_{\mathrm{SH}}(e^2/h)$ as a function of ${ϵ}_f$ and $u$ that reduces inversion asymmetry. The yellow (orange) line denotes the top of the ${\tau }_z{s}_z=+1(-1)$ valence band. (b) Quantum phase transitions between the QVH and QSH states at $u=\Delta \pm \lambda$.