Spin-orbit-mediated spin relaxation in monolayer MoS${}_2$

We study the intravalley spin-orbit-mediated spin relaxation in monolayers of MoS${}_2$ within a two bands effective Hamiltonian. The intrinsic spin splitting of the valence band as well as a Rashba-like coupling due to the breaking of the out-of-plane inversion symmetry are considered. We show that, in the hole doped regime, the out-of-plane spin relaxation is not very efficient since the spin splitting of the valence band tends to stabilize the spin polarization in this direction. We obtain spin lifetimes larger than nanoseconds, in agreement with recent valley polarization experiments.

Figure 1

(a) Top view of the lattice in real space. Red and blue dots represent the Mo and S atoms, respectively. (b) Brillouin zone, with the corresponding reciprocal-lattice wave vectors. The inequivalent $+\mathbf{K}$ and $-\mathbf{K}$ points are shown. (c) Conduction and valence bands at $\pm \mathbf{K}$ points within the effective model discussed in the text. The SO splitting is not considered in this figure. The numbers inside the kets which label the bands express the phases picked up the Bloch wave function under symmetry operations of the crystal structure; see Tables and .

Figure 2

Sketch of the microscopic processes which lead to the effective SO coupling terms discussed in the text. (a) First-order processes which lead to the splitting of the valence band. (b) Second-order processes associated to the splitting of the conduction band. (c) Second-order processes which lead to a Bychkov-Rashba coupling when ${\sigma }_h$ symmetry is broken.

Figure 3

Spin susceptibility $\chi$ as a function of the chemical potential $\mu$, for different values of $\Gamma$, calculated from Eq. (15). Solid black line corresponds to $\Gamma =0.001$ eV, blue dashed line to $\Gamma =0.01$ eV, and red dotted line to $\Gamma =0.1$ eV.

Figure 4

In-plane spin lifetimes as a function of the carrier concentration. Top: electron doping. Bottom: hole doping. Dashed black line corresponds to $\Gamma =0.001$ eV, dotted blue to $\Gamma =0.01$ eV, and solid red line to $\Gamma =0.1$ eV. Inset: In-plane spin lifetimes for electron concentrations in double logarithmic scale. Notice the different time scale in the top and bottom panels.

Figure 5

Out-of-plane spin lifetimes as a function of the carrier concentration. In black (dashed) $\Gamma =0.001$ eV, in blue (dotted) $\Gamma =0.01$ eV, in red $\Gamma =0.1$ eV. In all the cases ${\lambda }_{\mathrm{ext}}={10}^{-2}{\lambda }_{\mathrm{int}}$. Inset: Spin lifetime for hole concentrations with the correction given by Eq. (29).