Electric field controlled spin- and valley-polarized edge states in silicene with extrinsic Rashba effect

Zhiming Yu, Hui Pan, and Yugui Yao

Phys. Rev. B 92, 155419 – Published 14 October 2015

Abstract

In the presence of extrinsic Rashba spin-orbit coupling, we find that silicene can host a quantum anomalous Hall state with spin- and valley-polarized edge states, which can be effectively controlled by the exchange field and electric field. In this state, a pair of nontrivial edge states reside in one specific valley and have a strong but opposite spin polarization. A distinctive feature of this state is that both of the spin and valley indexes of the edge states can be switched by reversing the electric field. We also present a microscopic mechanism for the origin of this state. Our findings provide an efficient way to control the topologically protected spin- and valley-polarized edge states, which is crucial for spintronics and valleytronics.

Received 9 June 2015

DOI:https://doi.org/10.1103/PhysRevB.92.155419

Authors & Affiliations

Zhiming Yu¹, Hui Pan², and Yugui Yao^1,*

¹School of Physics, Beijing Institute of Technology, Beijing 100081, China
²Department of Physics, Beihang University, Beijing 100191, China

^*ygyao@bit.edu.cn

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Issue

Vol. 92, Iss. 15 — 15 October 2015

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Images

Figure 1
Phase diagram of silicene in the $(M, Δ)$ space with extrinsic Rashba SOC $λ = 0.2 t$ , calculated from the TB Hamiltonian (1). The solid lines separate the phases, which are indexed by the Chern number and the valley index of the edge state $(C, V_{edge})$ . We define $V_{edge} \equiv 1 / - 1$ when the gapless edge states are located around the $K / K^{'}$ valley, and $V_{edge} \equiv 0$ if there is no gapless edge state or the gapless edge states are not located around one valley. $M$ and $Δ$ are in units of $t$ .
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Figure 2
Band structures of the zigzag-terminated nanoribbon at marked points in Fig. 1, calculated from the TB Hamiltonian (1). The topological indices $(C, V_{edge})$ are (a) SV-QAHI1: $(1, 1)$ ( $Δ = 0.38 t$ and $M = 0.4 t$ ), (b) QAHI: $(2, 0)$ ( $Δ = 0 t$ and $M = 0.4 t$ ), and (c) SV-QAHI2: $(1, - 1)$ ( $Δ = - 0.38 t$ and $M = 0.4 t$ ). Blue (red) curves represent the edge states propagating along the top (bottom) edge of the nanoribbon. Energy is in units of $t$ and momentum is in units of $1 / a$ .
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Figure 3
Enlarged edge states of zigzag-terminated nanoribbons of SV-QAHI in the four quadrants of $(M, Δ)$ space. Strong spin polarization edge currents propagate along the edges of the nanoribbons. Blue (red) curves represent edge states propagating along the top (bottom) edge of the nanoribbon. By reversing the external electric field ( $Δ$ ) or exchange field ( $M$ ), the polarization direction of the spin and valley of the edge states can be switched.
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Figure 4
Skyrmion spin texture needs $s_{z}$ ranges from 1 to $- 1$ and the azimuth angle of spin $φ_{s} = arctan [s_{y} (k) / s_{x} (k)]$ changes from 0 to $2 π$ with any $s_{z}$ . Left: The contour map of $s_{z}$ around the $K$ valley of SV-QAHI for $M > 0$ and $Δ > 0$ , calculated from the effective Hamiltonian $H_{eff} (k)$ (4), which shows the reversal of spin and the trigonal warping effect around the valley. Right: Along the isoline of $s_{z} (k)$ , the azimuth angle of spin $φ_{s}$ changes from 0 to $2 π$ . Here, we present the behavior of $φ_{s}$ as a function of $θ_{k} = arctan (k_{y} / k_{x})$ with $s_{z} (k) = - 0.6$ . The isoline of $s_{z} (k) = - 0.6$ is marked by a red curve in the left figure. With other $s_{z} (k), φ_{s}$ also presents similar behavior (not shown). Hence, these two figures directly indicate that $H_{eff} (k)$ generates a skyrmion spin texture.
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Figure 5
The probability of $| A ↓ 〉$ and $| B ↑ 〉$ in the edge states of a zigzag-terminated SV-QAHI nanoribbon ( $M > 0$ and $Δ > 0$ ), obtained from the TB Hamiltonian (1). The solid lines intuitively show that the edge states are almost composed by the basis of $H_{eff} (k)$ : ${(A ↓, B ↑)}^{T}$ . The dashed lines show that the edge current propagating along the top and bottom edges have strong spin polarization, mainly composed of $| A ↓ 〉$ and $| B ↑ 〉$ , respectively.
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