Realization of Valley and Spin Pumps by Scattering at Nonmagnetic Disorders

Xing-Tao An, Jiang Xiao, M. W.-Y. Tu, Hongyi Yu, Vladimir I. Fal’ko, and Wang Yao

Phys. Rev. Lett. 118, 096602 – Published 1 March 2017

Abstract

The recent success in optical pumping of valley polarization in two-dimensional transition metal dichalcogenides (TMDs) has greatly promoted the concept of valley-based informatics and electronics. However, between the demonstrated valley polarization of transient electron-hole pair excitations and practical valleytronic operations, there exist obvious gaps to fill, among which is the valley pump of long-lived charge carriers. Here we discover that the quested valley pump of electrons or holes can be realized simply by scattering at the ubiquitous nonmagnetic disorders, not relying on any specific material property. The mechanism is rooted in the nature of the valley as a momentum space index: the intervalley backscattering in general has a valley contrasted rate due to the distinct momentum transfers, causing a net transfer of population from one valley to another. As examples, we numerically demonstrate the sizable valley pump effects driven by charge current in nanoribbons of monolayer TMDs, where the spin-orbit scattering by nonmagnetic disorders also realizes a spin pump for the spin-valley locked holes. Our finding points to a new opportunity towards valley spintronics, turning disorders from a deleterious factor to a resource of valley and spin polarization.

Received 10 September 2016

DOI:https://doi.org/10.1103/PhysRevLett.118.096602

Physics Subject Headings (PhySH)

Spintronics Valleytronics

Transition metal dichalcogenides

Green's function methods

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Xing-Tao An^1,2, Jiang Xiao^3,4,5, M. W.-Y. Tu¹, Hongyi Yu¹, Vladimir I. Fal’ko⁶, and Wang Yao^1,*

¹Department of Physics and Center of Theoretical and Computational Physics, University of Hong Kong, Hong Kong, China
²School of Science, Hebei University of Science and Technology, Shijiazhuang, Hebei 050018, China
³Department of Physics and State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China
⁴Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China
⁵Institute for Nanoelectronics Devices and Quantum Computing, Fudan University, Shanghai 200433, China
⁶National Graphene Institute, University of Manchester, Booth St E, Manchester M13 9PL, United Kingdom

^*wangyao@hku.hk

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Issue

Vol. 118, Iss. 9 — 3 March 2017

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Images

Figure 1
(a) Schematics of the momentum transfers (upper) of the scattering channels (lower) for the electron incident in valley $- K$ . (b) Electron incident in valley $K$ . The valley-flip reflection coefficients $R_{- K, K}$ in (a) and $R_{K, - K}$ in (b) can differ because of the distinct momentum transfers (solid curved arrows in the upper panels), whereas the valley-flip transmission coefficients $T_{K, - K}$ and $T_{- K, K}$ are always equal. For an incident flux with one electron per valley, the disorder scattering thus transfers a net population of $R_{- K, K} - R_{K, - K}$ from valley $- K$ to $K$ . (c) When a charge current ( $j_{c}$ , purple arrow) is driven through, the valley current ( $j_{V}$ , green arrows) is pumped out from both sides of the disorders. (d) Numerical demonstration of the valley pump uses monolayer ${MoS}_{2}$ nanoribbons with disorders randomly distributed over a rectangular scattering region, where the pumped valley current is examined at the boundaries of the disordered region.
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Figure 2
(a) Conduction sub-bands from the tight-binding model without spin-orbit coupling. $E_{c}$ denotes the band edge. (b) The coefficients of intra and intervalley transmission and reflection by the disordered region [cf., Fig. 1], as functions of the Fermi energy $E_{F}$ , plotting the numbers of electrons scattered into the corresponding channels out of $N_{τ}$ incident electrons in valley $τ$ , where $N_{τ}$ is the number of sub-bands crossed by $E_{F}$ in valley $τ$ . (c) The valley pump rate $P_{V} \equiv 2 (R_{- K, K} - R_{K, - K})$ , (d) the overall transmission $T_{sum} \equiv T_{K, K} + T_{- K, - K} + T_{K, - K} + T_{- K, K}$ , and (e) the valley depolarization rate $Γ_{V} \equiv 2 (T_{K, - K} + T_{- K, K} + R_{K, - K} + R_{- K, K})$ . The length of the disordered region $L = 90 a$ in (b)–(e), and $n_{i}$ is the disorder density. (f) The valley pump efficiency (i.e., per charge transmitted) $P_{V} / T_{sum}$ as a function of $n_{i}$ , at $L = 90 a$ . (g) $P_{V} / T_{sum}$ as a function of $L$ , at $n_{i} = 2 %$ . $E_{F} - E_{c} = 0.26 eV$ in (f) and (g). The width of the zigzag nanoribbon is $W = 27.7 a$ ( $a$ is lattice constant). In all plots, the dots represent calculated values averaged over 500 configurations of randomly generated disorder distributions, with the fluctuations shown as the error bar.
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Figure 3
(a) Valence band edges in monolayer TMDs with the valley-dependent spin splitting of magnitude $λ$ . Red and blue denote spin up and down, respectively. Green line cuts of the two-dimensional bands give the one-dimensional sub-bands of the nanoribbons (with periodic boundary condition). The two curved solid arrows denote the spin-flip reflections $R_{↑, ↓}$ and $R_{↓, ↑}$ , which correspond to distinct momentum transfers on a finite Fermi surface with the spin-valley locking. The inset is the schematic of a nonmagnetic disorder with spin-flip hopping between three nearest-neighbor Mo sites. (b) Spin-conserved and spin-flip reflection coefficients calculated for monolayer ${MoS}_{2}$ nanoribbons of zigzag, (4,1) and (2,1) orientations, respectively, plotting the numbers of holes scattered into the corresponding channels out of $N_{↑ (↓)}$ incident holes in the spin $↑ (↓)$ state, where $N_{↑ (↓)}$ is the number of spin $↑ (↓)$ sub-bands crossed by $E_{F}$ . The disorder density $n_{i} = 1 %$ . (c) The spin pump efficiency $P_{S} / T_{sum}$ at several disorder densities, where $P_{S} \equiv 2 (R_{↑, ↓} - R_{↓, ↑})$ . In (b) and (c), the widths of zigzag, (4,1) and (2,1) nanoribbons are 27.7, 27.5, and $29.1 a$ , respectively, and the lengths of the disordered regions are 90, 79.4, and $73.3 a$ , respectively. The dots represent averages over 500 configurations of randomly generated disorder distributions, with the fluctuations shown as the error bar.
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