Valley-splitting and valley-dependent inter-Landau-level optical transitions in monolayer ${\mathrm{MoS}}_2$ quantum Hall systems

The valley-dependent optical selection rules in recently discovered monolayer group-VI transition-metal dichalcogenides (TMDs) make possible optical control of valley polarization, a crucial step towards valleytronic applications. However, in the presence of Landau-level (LL) quantization such selection rules are taken over by selection rules between the LLs, which are not necessarily valley contrasting. Using ${\mathrm{MoS}}_2$ as an example we show that the spatial inversion-symmetry breaking results in unusual valley-dependent inter-LL selection rules, which is controlled by the sign of the magnetic field and directly locks polarization to valley. We find a systematic valley splitting for all LLs in the quantum Hall regime, whose magnitude is linearly proportional to the magnetic field and is comparable with the LL spacing. Consequently, unique plateau structures are found in the optical Hall conductivity, which can be measured by the magneto-optical Faraday rotations.

Figure 1

(a) Schematic of the spin-valley coupled band structure of TMDs. Red (blue) represents spin up (down), respectively. (b) and (c) Conduction and valence band LLs for ${\mathrm{MoS}}_2$ under $B_{\perp }$ = 20 T. $K$ $\left(K^{\prime }\right)$ valley is on the left (right). The crossing-LL states in the conduction band are from the dangling bonds on the zigzag edges [34], which do not affect the LLs. The dashed line is a guide to eye for valley splitting and also marks filling level $\nu \left(K\right)=4$, $\nu \left(K^{\prime }\right)=2$. (d) ${\Delta }_c^{01}$ $({\Delta }_v^{01})$ is the absolute energy difference between the LL 0 $\left(-1\right)$ in $K$ valley and 1 (0) in $K^{\prime }$ valley in conduction (valence) band. $\hslash {\omega }_0^c$ $(\hslash {\omega }_0^v$ ) is the LL spacing in conduction (valence) band. (e) same as (d) calculated from orbital magnetic moment and effective mass approximation. $a$ is the lattice constant.

Figure 2

Optical absorption spectrum in the quantum Hall regime. The solid (dashed) lines represent $K$ $\left(K^{\prime }\right)$ valleys, respectively. Red (blue) represents spin up (down), respectively. Spin-valley polarized transitions between $(n,n^{\prime })$ are labeled. (a) and (d) $\nu \left(K\right)=4$, $\nu \left(K^{\prime }\right)=0$. (b) and (e) $\nu \left(K\right)=4$, $\nu \left(K^{\prime }\right)=2$. (c) and (f) $\nu \left(K\right)=6$, $\nu \left(K^{\prime }\right)=2$. $\Gamma =0.1\hslash {\omega }_0$. (g) and (h) Schematic of the inter-LL transitions corresponding to (b) and (e), and (c) and (f). Red (solid) crosses indicate Pauli blocking. Green (dashed) crosses indicate vanishing probability.

Figure 3

Optical Hall conductivity in units of ${\sigma }_0=e^2/h$ with filling factors $\nu \left(K\right)=6$, $\nu \left(K^{\prime }\right)=2$, corresponding to Fig. 2. (a) Optical conductivity ${\sigma }_{xy}$ and ${\sigma }_{xx}$ for spin up, $K^{\prime }$ valley. (b) Spin down, $K$ valley. (c) Total ${\sigma }_{xy}$; the dashed line schematically shows the canceling out situation when ${\Delta }_c^{01}={\Delta }_v^{01}$. (d) Total ${\sigma }_{+/-}$. Only the real part is plotted.

Figure 4

Plateau structure for the optical Hall conductivity ${\sigma }_{xy}$ (black solid line) at $\omega =1.786$ eV and the static quantum Hall conductivity (green solid line) in the electron-doped regime. The red (blue) dashed line represents the spin up (down) or $K^{\prime }$ $\left(K\right)$ valley component of ${\sigma }_{xy}$, respectively. Numbers on the solid black line indicate the corresponding quantum Hall conductivity.