Giant magnetoresistance and perfect spin filter in silicene, germanene, and stanene

Stephan Rachel and Motohiko Ezawa

Phys. Rev. B 89, 195303 – Published 12 May 2014

Abstract

Silicene, germanene, and stanene are two-dimensional topological insulators exhibiting helical edge states. We investigate global and local manipulations at the edges by exposing them to (i) a charge-density-wave order, (ii) a superconductor, (iii) an out-of-plane antiferromagnetic, and (iv) an in-plane antiferromagnetic field. We show that these perturbations affect the helical edge states in a different fashion. As a consequence one can realize quantum spin-Hall effect without edge states. In addition, these edge manipulations lead to very promising applications: a giant magnetoresistance and a perfect spin filter. We also investigate the effect of manipulations on a very few edge sites of a topological insulator nanodisk.

Received 6 December 2013
Revised 19 March 2014

DOI:https://doi.org/10.1103/PhysRevB.89.195303

Authors & Affiliations

Stephan Rachel¹ and Motohiko Ezawa²

¹Institute for Theoretical Physics, TU Dresden, 01062 Dresden, Germany
²Department of Applied Physics, University of Tokyo, Hongo 7-3-1, 113-8656, Japan

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Issue

Vol. 89, Iss. 19 — 15 May 2014

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Images

Figure 1
QSH nanoribbons (with $λ = 0.2 t$ and a ribbon width $W = 128$ atoms) where one half of the system is exposed to one of the following perturbations: (a) CDW order leading to inner edge states at the boundary between the CDW and QSH regions. We have set $M = 2 λ$ . (b) OP-AF order leading to inner edge states at the boundary between the magnetic and QSH regions. We have set $m_{OP} = 2 λ$ . (c) IP-AF order not leading to inner edge states. We have set $m_{IP} = λ$ . (d) Superconducting order not leading to inner edge states. Magenta (cyan) solid [dotted] lines represent up-spin (down-spin) polarized modes localized at the outer [inner] edge. We have set $Δ = λ$ .
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Figure 2
Band structure of “asymmetric” hybrid ribbons, i.e., a QSH ribbon where region 1 with width $W_{1}$ is exposed to the OP-AF exchange field ( $m_{OP} = 2 λ$ ) while region 2 with width $W_{2}$ is unaffected ( $m_{OP} = 0$ ). The total width of the ribbon is $W = W_{1} + W_{2}$ . (a) Nanoribbon with $W = 128$ atoms where $W_{1} = 64$ and $W_{2} = 64$ . (b) Nanoribbon with $W = 126$ atoms where $W_{1} = 64$ and $W_{2} = 62$ . (c) Nanoribbon with $W = 128$ atoms where $W_{1} = 63$ and $W_{2} = 65$ .
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Figure 3
Nanoribbon with a width of $W = 128$ atoms and SO coupling $λ = 0.2 t$ . (a) IP-FM exchange field is applied only to the outermost sites of the upper edge. Only the upper edge modes are gapped. (b) IP-FM exchange field is applied to the outermost sites of both edges. All edge modes are gapped. We have set $m_{IP} = λ$ .
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Figure 4
Local manipulations on honeycomb nanodisks ( $λ = 0.5 t$ ) consisting of 800 lattice sites: the local density $| ψ_{σ} {(x, y) |}^{2}$ of the edge state is shown which is proportional to the radius of the dots. On the left side, $| ψ_{↑} |^{2}$ is plotted in magenta, on the right side, $| ψ_{↓} |^{2}$ is plotted in cyan. (a) OP-AF exchange field ( $m_{OP} = 2 t$ ) is applied to a small region at the edge. The helical edge modes detour the perturbation. Note that the edge state $ψ$ corresponding to the energy closest to zero is shown; it is a pure $↑$ -spin state. (b) IP-AF exchange field ( $m_{IP} = 2 t$ ) is applied to five adjacent sites at the edge. At the perturbation, no edge state is present. Instead, the spin is flipped and sent back. Again the edge state $ψ$ with energy closest to zero is shown; the state has equal weights for $↑$ - and $↓$ -spin components.
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Figure 5
Illustration of a giant magnetoresistance. We take a nanoribbon ( $W = 128$ ), to which the FM order is applied only at the outermost sites of both edges. We have set $λ = 0.2 t$ . (a) When the FM order is out of plane ( $θ = 0$ ), there are edge states yielding a quantized conductance. We have set $m_{OP} = λ$ . (b) When the FM order has finite in-plane component ( $θ > 0$ ), there are no metallic edge modes. We have set $m_{IP} = λ$ .
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Figure 6
Illustration of a perfect spin filter. A solid (dotted) boundary represents an edge where metallic edge modes (do not) emerge. (a) In general, helical modes circulate around a sample. (b) By introducing the IP-AF order to the lower half of the sample, helical modes only propagate along the upper edge. In this example, only $↑$ spins are transported from the left to the right lead.
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