Probing Rashba spin-orbit coupling by subcycle lightwave control of valley polarization

We perform nonperturbative calculations of light-field driven valley-polarization process in monolayer ${\mathrm{MoS}}_2$ which has additional Rashba spin-orbit couplings (SOCs). The ultrafast electron dynamics is simulated within the independent particle picture by solving density-matrix equations in the basis of linear combination of atomic orbitals, where tight-binding (TB) models including both intrinsic atomic and Rashba SOCs are used to calculate relevant matrix elements. We demonstrate that the Rashba-type SOCs can be manifested by suboptical-cycle control of valley selectivity excitations, in particular necessary via using few-cycle linearly polarized pulse with controlled carrier-envelope phase (CEP). This procedure will lead to a CEP-dependent valley Hall conductivity (VHC), which exhibits an important phase shift among different Rashba coupling strengths. The additional analysis shows that this phase shift is mainly determined by the $d_z^2$-orbital TB Rashba parameter from Mo atom and originates from contribution of conduction bands to VHC, where the Berry curvature modified by Rashba SOC plays a crucial role. Moreover, we also provide a qualitative interpretation on the Rashba-dependent VHC in terms of suboptical-cycle Landau-Zener-Stückelberg interference. Our results suggest a feasible approach for probing Rashba SOCs in hexagonal two-dimensional materials, and might pave the way of achieving more controls in the future valleytronics application.

Figure 1

(a) Top view of monolayer ${\mathrm{MoS}}_2$ lattice, with each sublattice occupied by a Mo (blue) and two S (yellow) atoms. (b) First Brillouin zone in the reciprocal lattice. ${\mathbf{G}}_1$ and ${\mathbf{G}}_2$ are two reciprocal lattice primitive vectors.

Figure 2

Total CB population distribution in the reciprocal space after the interaction with driving pulses with CEP (a) ${\varphi }_{\mathrm{CEP}}=0$, (b) ${\varphi }_{\mathrm{CEP}}=\pi /2$, and (c) ${\varphi }_{\mathrm{CEP}}=3\pi /2$. For ${\varphi }_{\mathrm{CEP}}=\pi /2$ and ${\varphi }_{\mathrm{CEP}}=3\pi /2$, the population distribution is plotted by the difference compared with panel (a).The driving light has a Gaussian envelope with the central wavelength $\lambda =8\mu \mathrm{m}$, duration 27 fs, and the peak intensity $I_0=0.3\mathrm{TW}/\mathrm{c}{\mathrm{m}}^2$. The dashed hexagon represents the first BZ, and the vertical dashed line guides the eyes to observe the population asymmetry between $K$ and $K^{\prime }$ valleys.

Figure 3

Comparison of Hall conductivity calculated as a function of CEP for four different Hamiltonian systems: the intrinsic SOC Hamiltonian (green dashed), spinless Hamiltonian (black-dashed-dotted), and full Hamiltonian with ${\alpha }_v=0.5\mathrm{eV}\AA$ and ${\alpha }_c=0$ (blue dotted), together with ${\alpha }_v=0$ and ${\alpha }_c=0.5\mathrm{eV}\AA$ (red-solid). The laser parameters are the same as Fig. 2.

Figure 4

(a) Maximum conductivity ${\sigma }_{\max }$ vs the laser intensity at a fixed ${\alpha }_c=0.5\mathrm{eV}\AA$. (b) Phase shift $\delta \left({\alpha }_c\right)$ as a function of ${\alpha }_c$, calculated from driving pulses with peak intensity $I_0=0.1\mathrm{TW}/\mathrm{c}{\mathrm{m}}^2$ (black squares), $I_0=0.3\mathrm{TW}/\mathrm{c}{\mathrm{m}}^2$ (red circles), $I_0=0.5\mathrm{TW}/\mathrm{c}{\mathrm{m}}^2$ (blue-up triangle), and $I_0=2.0\mathrm{TW}/\mathrm{c}{\mathrm{m}}^2$ (green-down triangle). Four symbols on the curve of panel (a) denote the selected intensity position corresponding to panel (b). The other laser parameters are the same as Fig. 2.

Figure 5

(a) Comparison of different CEP-dependent Hall conductivity, obtained from VB contribution at ${\alpha }_c=0$ (blue dashed), from CB contribution at ${\alpha }_{\mathrm{c}}=0$ (blue solid), from VB contribution at ${\alpha }_c=0.5\mathrm{eV}\AA$, and from CB contribution at ${\alpha }_c=0.5\mathrm{eV}\AA$. (b) CEP-dependent Hall conductivity calculated at ${\alpha }_c=0.0\mathrm{eV}\AA$ (blue solid), ${\alpha }_c=0.5\mathrm{eV}\AA$ (red-dashed), and ${\alpha }_c=1.0\mathrm{eV}\AA$ (green dot). For the three cases, Berry curvature is always chosen as the unmodified one corresponding to ${\alpha }_c=0.0\mathrm{eV}\AA$. The laser parameters are the same as Fig. 2.

Figure 6

(a), (b) The $\mathit{k}$-resolved difference $\mathrm{\Delta }{\rho }_c\left({\mathbf{k}}_0\right)$ between the residual CBs population resulting from excitation with a pair of CEP ${\varphi }_1$ and ${\varphi }_2$, calculated at two different Rashba coupling strength (a) ${\alpha }_c=0.3\mathrm{eV}\AA$ and (b) ${\alpha }_c=0.7\mathrm{eV}\AA$. (c) Extracted $\mathrm{\Delta }{\rho }_c\left({\mathbf{k}}_0\right)$ along $k_y=0$ changing with ${\alpha }_{\mathrm{c}}$. The red-dashed rectangle represents the transition region across which the sign of $\mathrm{\Delta }{\rho }_c\left({\mathbf{k}}_0\right)$ reverses. The laser parameters are the same as Fig. 2.

Figure 7

(a) Two-dimensional electron trajectories in $\mathit{k}$ space for CRB driving field. The electron starts from the initial ${\mathbf{k}}_0$, and the transition probability reaches the maximum (green-shaded circles) when electron passes near the $K$ valley. The electron undergoes excitation with large transition probabilities only once per optical cycle. (b), (d) Valley population asymmetry as a function of (b) two-color relative phase $\mathrm{\Delta }\phi$ for the CRB field, and (d) CEP of control pulse for the two linearly polarized superposition fields, calculated at ${\alpha }_c=0.0\mathrm{eV}\AA$ (red solid) and ${\alpha }_c=0.5\mathrm{eV}\AA$ (blue dashed). (c), (e) VHC ${\sigma }_{xy}$ as a function of (c) two-color relative phase $\mathrm{\Delta }\phi$ for the CRB field, and (e) CEP of control pulse for the two linearly polarized superposition fields, calculated at ${\alpha }_c=0.0\mathrm{eV}\AA$ (red solid) and ${\alpha }_c=0.5\mathrm{eV}\AA$ (blue dashed).

Figure 8

Schematics of the device for probing Rashba effect.

Figure 9

Comparison of CEP-dependent Hall conductivity, calculated for four different external electric fields perpendicular to monolayer surface: $E_z=0\mathrm{V}/\mathrm{nm}$ (blue dot), $E_z=10\mathrm{V}/\mathrm{nm}$ (green dashed), $E_z=25\mathrm{V}/\mathrm{nm}$ (red solid) and $E_z=50\mathrm{V}/\mathrm{nm}$ (black-dashed-dotted). The driving light has the central wavelength $\lambda =8\mu \mathrm{m}$, pulse duration 27 fs, and the peak intensity $I_0=0.1\mathrm{T}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$.