Intervalley Scattering and Localization Behaviors of Spin-Valley Coupled Dirac Fermions

We study the quantum diffusive transport of multivalley massive Dirac cones, where time-reversal symmetry requires opposite spin orientations in inequivalent valleys. We show that the intervalley scattering and intravalley scattering can be distinguished from the quantum conductivity that corrects the semiclassical Drude conductivity, due to their distinct symmetries and localization trends. In immediate practice, it allows transport measurements to estimate the intervalley scattering rate in hole-doped monolayers of group-VI transition metal dichalcogenides (e.g., molybdenum dichalcogenides and tungsten dichalcogenides), an ideal class of materials for valleytronics applications. The results can be generalized to a large class of multivalley massive Dirac systems with spin-valley coupling and time-reversal symmetry.

Figure 1

Low-energy effective band structures of the $M{X}_2$ monolayer at $K$ and $K^{\prime }$ valleys. The Fermi surface (at the top edges of the shadowed areas) intersects the highest two valence bands of opposite spin orientations (marked by $\uparrow$ and $\downarrow$). Horizontal dotted arrows indicate the intervalley spin-flip scattering.

Figure 2

(a) Sketch of band structure. $V$ measures the Fermi energy from the valence band top ($V\in [0,2\lambda ]\mathrm{eV}$). (b)–(d) Magnetoconductivity $\Delta \sigma$ when the Fermi energy intersects the highest valence bands. The positive (negative) logarithmic magnetoconductivity is the signature for the WL (WAL), which indicates that intravalley (intervalley) scattering is stronger than intervalley (intravalley) scattering. (b) Only WL in the absence of intervalley scattering (${\tau }_0/{\tau }_{\mathrm{I}}=0$) for different $V$. (c) The crossover between WL and WAL at $V=0.1\mathrm{eV}$ and for different ${\tau }_0/{\tau }_{\mathrm{I}}$. A larger ${\tau }_0/{\tau }_{\mathrm{I}}$ means stronger intervalley scattering. (d) Zoom-in of (c) for ${\tau }_0/{\tau }_{\mathrm{I}}=0.1$, 1, 10. Parameters: $\Delta =1.66\mathrm{eV}$ and $\lambda =0.075\mathrm{eV}$ [15], mean free path $\ell =10\mathrm{nm}$, and phase coherence length ${\ell }_{\varphi }=300\mathrm{nm}$.

Figure 3

The quantum conductivity ${\sigma }^F$ as a function of temperature $T$ for different ${\tau }_0/{\tau }_{\mathrm{I}}$. Positive (negative) ${\sigma }^F$ corresponds to WAL (WL). Other parameters are the same as those in Figs. 2c, 2d. The signature that intervalley (intravalley) scattering is stronger than intravalley (intervalley) scattering is given by a positive (negative) ${\sigma }^F$, as well as by the fact that ${\sigma }^F$ increases (decreases) with decreasing $T$.

Figure 4

WL and WAL in the conduction bands. Spin-up ($\uparrow$) and spin-down ($\downarrow$) are offset for clarity. The solid and dotted arrows represent spin-flip and spin-conserved scattering, respectively.