Domains of electrically induced valley polarization in two-dimensional Dirac semiconductors

V. A. Kochelap, V. N. Sokolov, and K. W. Kim

Phys. Rev. B 104, 075403 – Published 3 August 2021

Abstract

Electrically induced formation of valley-polarized free-carrier domains is theoretically investigated in two-dimensional (2D) honeycomb lattice systems exhibiting topological valley transport and the valley Hall effect. The results show that under a strong electric field $E$ applied along a nanostrip, the domains can be formed across the strip when this transverse dimension is comparable to the intervalley diffusion length $L_{i v}$ . Further, these domains are found to be characterized by two distinct length scales dependent on $E$ and $L_{i v}$ , i.e., the extended diffusion length $L_{ext}$ ( $\propto L_{i v} E$ ) and the compressed diffusion length $L_{com}$ ( $\propto L_{i v} / E$ ). The former determines the extension of the domain plateaus, whereas the latter specifies the width of the domain wall. Within each of the domains (excluding a narrow region of the domain wall; $L_{ext} ≫ L_{com}$ ), the charge carriers are fully polarized belonging to only one of the valleys, providing a pure bulk valley current inside the considered domain region. The domain properties including the polarization amplitude and domain wall position are analyzed as a function of the applied electric field and intervalley scattering rates at the edges. The position of the domain wall, which determines the relative extension of the plateaus, can be controlled by a transverse current flowing between the Hall-type contacts. The current-voltage characteristic demonstrates a superlinear behavior in the range of electric fields corresponding to the valley-polarization domain formation. This feature is a distinctive signature of the anomalous transport arising from Bloch-band Berry curvature, which enhances the longitudinal conductance and diminishes the channel resistance. By reversing the applied electric field, the valley polarization and localization of the valley-polarized currents can be abruptly switchable. We suggest that the studied scheme of electrically induced domains of valley polarization in the 2D honeycomb lattice systems can be used in novel applications with all-electrical control and manipulation of the valley degree of freedom.

Received 26 March 2021
Revised 15 July 2021
Accepted 15 July 2021

DOI:https://doi.org/10.1103/PhysRevB.104.075403

Physics Subject Headings (PhySH)

Honeycomb lattice Transition metal dichalcogenides Two-dimensional electron system

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

V. A. Kochelap¹, V. N. Sokolov ^1,*, and K. W. Kim^2,3

¹Department of Theoretical Physics, Institute of Semiconductor Physics, NASU, Pr. Nauki 41, Kiev 03028, Ukraine
²Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, North Carolina 27695, USA
³Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA

^*sokolov@isp.kiev.ua

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Issue

Vol. 104, Iss. 7 — 15 August 2021

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Images

Figure 1
Schematic illustration of the sample geometry for the electrically induced valley-polarization domains. When the external electric field $E$ increases, the valleys $K$ and $K^{'}$ are asymmetrically occupied due to the size effect on the intervalley diffusion length $(d_{y} \sim L_{i v})$ . The skewed arrows $K$ and $K^{'}$ depict the electron flows to the sample edges under the valley Hall effect.
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Figure 2
Phase portrait of Eq. (9a). Integral curves are shown for different values of integration constant $\bar{C}$ : curves 1, $1^{'}, \bar{C} = - 0.2$ ; 2, $2^{'}, \bar{C} = 0$ ; 3, $\bar{C} = 1$ ; 4, $\bar{C} = 2$ . Dashed line 5 is for the BCs with $S^{\pm} = 0$ ( $E_{x}^{2} = 1.25$ ). Arrows indicate the direction of motion along the integral curves with increasing $ζ$ . Two portions of curve 2 $(\bar{P} ≷ 0)$ correspond to wide (semi-infinite) samples.
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Figure 3
Valley polarization $P (ζ)$ calculated by numerical solution of Eqs. (8a) and (8b) for different values of $E_{x}$ ( $δ = 2.5$ ). (a) $S^{\pm} = 0$ and (b) $S^{+} = 0, S^{-} = \infty$ . The curves are 1 (black), $E_{x} = 1$ ; 2 (red), 5; 3 (green), 10.
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Figure 4
Transverse distributions of dimensionless current densities calculated using the valley polarization of curve 2 (red) in Fig. 3 ( $S^{\pm} = 0; E_{x} = 5, δ = 2.5$ ). (a) Line 1 (black) is the total longitudinal current density $J_{x} = J_{x}^{K} + J_{x}^{K^{'}}$ ; lines 2 (blue) and 3 (green) are for $J_{x}^{K}$ and $J_{x}^{K^{'}}$ , respectively. Lines 4 (blue) and 5 (green) are for $J_{y}^{K}$ and $J_{y}^{K^{'}}$ , respectively. (b) An arbitrary-scale map of vector field given by the valley current densities $J^{K}$ (blue) and $J^{K^{'}}$ (green). The direction of arrows is defined by angle $φ_{K (K^{'})}$ , where $tan (φ_{K (K^{'})}) = J_{y}^{K (K^{'})} / J_{x}^{K (K^{'})}$ , and the arrow length is defined by total valley current density $J^{K (K^{'})} = J_{x}^{K (K^{'})} \hat{i} + J_{y}^{K (K^{'})} \hat{j}$ .
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Figure 5
Transverse distributions of dimensionless current densities calculated using the valley polarization of curve 2 (red) in Fig. 3 ( $S^{+} = 0, S^{-} = \infty; E_{x} = 5, δ = 2.5$ ). (a) Line 1 (black) is the total longitudinal current density $J_{x} = J_{x}^{K} + J_{x}^{K^{'}}$ ; lines 2 (blue) and 3 (green) are for $J_{x}^{K}$ and $J_{x}^{K^{'}}$ , respectively. Lines 4 (blue) and 5 (green) are for $J_{y}^{K}$ and $J_{y}^{K^{'}}$ , respectively. (b) Vector field map given by the valley current densities $J^{K}$ (blue) and $J^{K^{'}}$ (green).
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Figure 6
Normalized electric charge density $ϱ (ζ) / ϱ_{0}$ [ $= e_{0} N δ κ (ζ) / 2 e_{0} N L_{D} E_{x} π^{- 1} L_{i v}^{- 1}$ ] calculated for three different values of dimensionless field, i.e., $E_{x} = 1$ (curve 1, black), $E_{x} = 5$ (curve 2, red), and $E_{x} = 10$ (curve 3, green). The heights of curves 2 and 3 are reduced by a factor of 0.2 and 0.1, respectively, for ease of presentation. In all three cases, $S^{\pm} = 0$ and $δ = 2.5$ are assumed.
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