Nonlinear Valley and Spin Currents from Fermi Pocket Anisotropy in 2D Crystals

The controlled flow of spin and valley pseudospin is key to future electronics exploiting these internal degrees of freedom of carriers. Here, we discover a universal possibility for generating spin and valley currents by electric bias or temperature gradient only, which arises from the anisotropy of Fermi pockets in crystalline solids. We find spin and valley currents to the second order in the electric field as well as their thermoelectric counterparts, i.e., the nonlinear spin and valley Seebeck effects. These second-order nonlinear responses allow two unprecedented possibilities to generate pure spin and valley flows without net charge current: (i) by an ac bias or (ii) by an arbitrary inhomogeneous temperature distribution. As examples, we predict appreciable nonlinear spin and valley currents in two-dimensional (2D) crystals including graphene, monolayer and trilayer transition-metal dichalcogenides, and monolayer gallium selenide. Our finding points to a new route towards electrical and thermal generations of spin and valley currents for spintronic and valleytronic applications based on 2D quantum materials.

Figure 1

(a), (b) Carrier distributions of a spin-up Fermi pocket at valley $A$ and a spin-down pocket at valley $\overline{A}$, in an electric field along $+x$ (a) or $-x$ (b) direction. The anisotropy of the Fermi pocket results in a difference in the currents from $A$ and $\overline{A}$, giving rise to a valley (spin) current quadratic in the field. The horizontal red (blue) arrows correspond to the current from the Fermi pocket $A$ ($\overline{A}$), with the arrow thickness denoting the magnitude. (c) Such quadratic dependence in the field makes possible the generation of dc valley and spin currents by ac electric field, in the absence of net charge current.

Figure 2

(a) Hole pockets at $K$ valleys, (b) at $\mathrm{\Gamma }$ point, and (c) electron pockets at $\mathrm{\Lambda }$ valleys in monolayer TMDs. Red (blue) denotes spin up (down). (d) Displacement of $K$ pockets by an electric field in the armchair direction, where the thick red (blue) arrow corresponds to the current from the Fermi pocket $K$ ($\overline{K}$). The valley (spin) current flows perpendicular to the field. The thin red (blue) arrows illustrate the group velocity on the displaced $K$ ($\overline{K}$) valley Fermi surface. (e) Dependence of the spin valley current direction (orange arrow) on the relative angle $\theta$ between the field (green arrow) and the crystalline axis.

Figure 3

(a) Polarized EL from $p$-$n$ junction along armchair direction. The EL has an overall circular polarization $R\sim j_v/j_c$. Green (yellow) color denotes the hole (electron) doped region, and red (blue) color denotes the $\sigma -$ ($\sigma +$) circular polarization. The hole has larger anisotropy and, hence, larger nonlinear valley current, which determines the EL polarization. (b) Spatial pattern of EL polarization from $p$-$n$ junction along the zigzag direction.

Figure 4

(a) The dimensionless coefficient $\alpha$ that measures the ratio between the valley (spin) current and the charge current by a temperature gradient [see Eq. (6)]. (b) Spin and valley currents can be generated by an arbitrary inhomogeneous temperature distribution. The charge current vanishes as long as the temperatures at the two ends of the device equal.