Tunable skewed edges in puckered structures

Marko M. Grujić, Motohiko Ezawa, Milan Ž. Tadić, and François M. Peeters

Phys. Rev. B 93, 245413 – Published 15 June 2016

Abstract

We propose a type of edges arising due to the anisotropy inherent in the puckered structure of a honeycomb system such as in phosphorene. Skewed-zigzag and skewed-armchair nanoribbons are semiconducting and metallic, respectively, in contrast to their normal edge counterparts. Their band structures are tunable, and a metal-insulator transition is induced by an electric field. We predict a field-effect transistor based on the edge states in skewed-armchair nanoribbons, where the edge state is gapped by applying arbitrary small electric field $E_{z}$ . A topological argument is presented, revealing the condition for the emergence of such edge states.

Received 7 December 2015

DOI:https://doi.org/10.1103/PhysRevB.93.245413

Physics Subject Headings (PhySH)

Band gap

Nanoribbon Phosphorene

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Marko M. Grujić^1,2,*, Motohiko Ezawa^3,†, Milan Ž. Tadić¹, and François M. Peeters²

¹School of Electrical Engineering, University of Belgrade, P.O. Box 3554, 11120 Belgrade, Serbia
²Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium
³Department of Applied Physics, University of Tokyo, Bunkyo, Tokyo 113-8656, Japan

^*marko.grujic@etf.bg.ac.rs
^†ezawa@ap.t.u-tokyo.ac.jp

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Issue

Vol. 93, Iss. 24 — 15 June 2016

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Images

Figure 1
Illustrations of (a) sZZ nanoribbon with $N_{sZZ} = 5$ , (b) sAC nanoribbon with $N_{sAC} = 8$ , (c) nZZ nanoribbon with $N_{nZZ} = 5$ , and (d) nAC nanoribbon with $N_{nAC} = 8$ periodic along the parallel direction. Unit cells are outlined for each case by the gray area. The colors denote the ridge structure, with red sublayer atoms shifted vertically above the blue sublayer atoms.
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Figure 2
Band structure of (a) sZZ, (b) sAC, (c) nZZ, and (d) nAC nanoribbons with width $N = 40$ . There are QFBs (red curves) in the sAC and nZZ nanoribbons, but not in the sZZ and nAC nanoribbons. (I) and (II) depict the real-space localization of states belonging to QFBs at the crossed points in (b) and (c), respectively. The magenta disks (blue circles) denote the probability weight on the atoms in the upper (lower) sublayer. The probability weight between the left and right edges is identical.
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Figure 3
The lattice of (a) skewed-bearded nanoribbons and (b) skewed-dangled nanoribbons, while the corresponding band structure is shown in (c) and (d), respectively.
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Figure 4
Illustration of the (a1) skewed-zigzag, (b1) normal-zigzag, (c1) skewed-armchair, (d1) normal-armchair edges, (e1) skewed-beard, (f1) normal beard, (g1) skewed-dangled edge (type I), (h1) normal-dangled and (i1) skewed-dangled edge (type II). We note that there are two inequivalent types of the skewed-dangled edge depending on the type of the dangled atoms. (a2), (b2), (c2), (d2), (e2), (f2), (g2), (h2) and (i2): Plots of the loop for the topological number (6). The horizontal axis is $Re [F_{k} (k_{⊥})]$ , while the vertical axis is $Im [F_{k} (k_{⊥})]$ . The loop encircles the origin for all $k$ in the case of the skewed-armchair and normal-zigzag edges $(N_{wind} = 1)$ , while it does not for all $k$ in the case of the skewed-zigzag and normal-armchair edges $(N_{wind} = 0)$ . Color indicates the momentum $k$ . Red indicates $k = 0$ and cyan indicates $k = π$ . (a3), (b3), (c3), (d3), (e3), (f3), (g3), (h3) and (i3): The band structure of nanoribbons with the corresponding edges. The edge states are perfectly flat since they are calculated based on the simple anisotropic honeycomb model. We have set $t_{>} / t_{<} = 2.5$ .
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Figure 5
(a) Band structure of $N = 40$ sZZ nanoribbon for $E_{⊥} = 30$ mV/Å. (b) The impact of electric field along the sZZ nanoribbon width on the band gap for $N = 10, 20, 40$ , and 100. (c) Quasiflat bands of sAC nanoribbons without external field (red curves) and with side gating $E_{⊥} = 10$ mV/Å (blue dotted curves). The resulting conductance reveals the degeneracy and the band gap of the bands. (I) and (II) depict the real-space localization of states belonging to QFBs denoted by two cyan crosses in (c). The magenta disks (blue circles) denote the probability weight on the atoms in the upper (lower) sublayer. The wave functions are localized at one edge.
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Figure 6
Band structures of (a) even $(N = 40)$ and (b) odd $(N = 41)$ nZZ nanoribbons, and (c) even $(N = 40)$ and (d) odd $(N = 41)$ sAC nanoribbons, in the absence (solid red curves) and presence (dotted curves) of perpendicular electric field $E_{z}$ . The corresponding conductance is shown in the right panels for all cases. Note the lack of parity effect in skewed-armchair nanoribbons. (I) and (II) depict the real-space localization of states belonging to QFBs. The probability weight between the upper and lower sublayers is imbalanced due to the Stark effect induced by $E_{z}$ .
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