Edge states in silicene nanodisks

Ko Kikutake, Motohiko Ezawa, and Naoto Nagaosa

Phys. Rev. B 88, 115432 – Published 25 September 2013

Abstract

Silicene is a honeycomb structure of silicon atoms which shares many intriguing properties with graphene. We investigate the electronic properties of a silicene nanodisk, which is silicene with a closed edge. In the case of a graphene nanodisk, the number of the zero-energy modes is given by the lower bound of the inequality in a bipartite system, $ξ_{0} \geq 2 | N_{A} - N_{B} |$ , where $ξ_{0}$ is the number of zero-energy modes and $N_{A / B}$ is the number of sites in a $A / B$ sublattice. They are standing wave states. In the case of a silicene nanodisk, they propagate around a nanodisk as a helical current, because silicene is a topological insulator due to the spin-orbit interaction. We have found the counting rule of the zero-energy states and constructed the low-energy theory of zigzag triangular silicene. We also show the validity of the bulk-edge correspondence in a nanodisk with a rough edge by calculating the local probability amplitude and current. Finally, the signatures of the topological phase transition induced by an external electric field are discussed. Our results will be observable by means of scanning tunneling microscopy experiments.

Received 1 August 2013

DOI:https://doi.org/10.1103/PhysRevB.88.115432

Authors & Affiliations

Ko Kikutake¹, Motohiko Ezawa¹, and Naoto Nagaosa^2,1

¹Department of Applied Physics, University of Tokyo, Hongo 7-3-1, 113-8656, Japan
²RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan

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Vol. 88, Iss. 11 — 15 September 2013

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Images

Figure 1
(a1), (a2) Local probability amplitude and current of the up-spin $π$ mode, when $N$ is odd. We have taken $λ_{SO} = 0.1 t, λ_{R} = 0$ for illustration. (b–d) Energy spectra with parameters shown within figures. The vertical axis is $E / t$ , and the horizontal axis is the index of the eigenstates. We have taken $N_{A} = 18, N_{B} = 15$ . There are six edge mode (green) in the graphene nanodisk and two $π$ modes (red) in the silicene nanodisk.Reuse & Permissions
Figure 2
See the caption of Fig. 1. Here $N$ is even, and we have taken $N_{A} = 25, N_{B} = 21$ . There are eight edge mode (green) in the graphene nanodisk and two $π$ modes (red) in the silicene nanodisk.Reuse & Permissions
Figure 3
Illustration of the correspondence between the edge modes of a zigzag silicene nanoribbon (solid lines) and a zigzag triangular silicene nanodisk (dots) for $N = 9$ and $N = 10$ . The edge modes represent almost straight lines connecting the tips of the Dirac cones in the conduction and valence bands in a nanoribbon. The green dots represent the energy spectrum of a nanodisk. We focus on one edge of a nanodisk. Since the length is finite, the momentum is quantized. When $N$ is odd, there exist two states at the $π$ point on the Fermi level, which we call the $π$ modes. When $N$ is even, there exist four states near the $π$ point in the valence bands, which we also call the $π$ modes.Reuse & Permissions
Figure 4
(a) Local density of the $π$ mode in an triangular nanodisk with a defect. The blue circle represents the magnitude of the local density. The defect of the edge is colored in red. (b) Local current of the $π$ mode. Red (blue) arrows indicate flows between the nearest-neighbor (second-nearest-neighbor) sites, respectively. We have taken $λ_{SO} = 0.1 t$ for illustration.Reuse & Permissions
Figure 5
Local probability amplitude (a1), (b1) and local current (a2), (b2) of the up-spin $π$ mode. The local probability current circulates in the clockwise direction. Panels (a3) and (b3) show energy spectra. The vertical axis is $E / t$ , and the horizontal axis is the index of the eigenstates. We have taken $N_{A} = 1087$ and $N_{B} = 1091$ . There exist eight edge modes.Reuse & Permissions
Figure 6
(a) Energy spectrum as a function of $λ_{V}$ for the system with $λ_{SO}$ as indicated. The system size is $N = 40$ . The vertical axis is the energy in unit of $t$ . The two lines are linear extrapolations from the band spectrum, which cross each other at the phase transition point of the bulk system. (b) Schematic illustration of the energy spectrum. The energy levels between the two lines for $λ_{V} < λ_{SO}$ represent the edge modes. The red curve is the extremum $\pm E (λ_{V})$ in the energy spectrum given by (6.1), while the thin lines are $E = \pm (λ_{V} - λ_{SO})$ . (c) Numerical fitting of $Δ$ by the curve given by (6.2).Reuse & Permissions

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