Strain-Induced Pseudomagnetic Fields in Twisted Graphene Nanoribbons

Dong-Bo Zhang, Gotthard Seifert, and Kai Chang

Phys. Rev. Lett. 112, 096805 – Published 6 March 2014

Abstract

We present, for the first time, an atomic-level and quantitative study of a strain-induced pseudomagnetic field in graphene nanoribbons with widths of hundreds of nanometers. We show that twisting strongly affects the band structures of graphene nanoribbons with arbitrary chirality and generates well-defined pseudo-Landau levels, which mimics the quantization of massive Dirac fermions in a magnetic field up to 160 T. Electrons are localized either at ribbon edges forming the edge current or at the ribbon center forming the snake orbit current, both being valley polarized. Our result paves the way for the design of new graphene-based nanoelectronics.

Received 29 May 2013

DOI:https://doi.org/10.1103/PhysRevLett.112.096805

Authors & Affiliations

Dong-Bo Zhang^1,4, Gotthard Seifert², and Kai Chang^3,*

¹Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Avenue SE, Minneapolis, Minnesota 55455, USA
²Physikalische Chemie, Technische Universität Dresden, D-01062 Dresden, Germany
³SKLSM, Institute of Semiconductors, Chinese Academy of Sciences, P.O. Box 912, Beijing 100083, China
⁴Beijing Computational Science Research Center, Beijing 100084, China

^*Corresponding author. kchang@semi.ac.cn

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Vol. 112, Iss. 9 — 7 March 2014

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Images

Figure 1
(a) A twisted GNR (upper panel) and its horizontal projection (lower panel). The coordinate system is defined as $x$ along the GNR width dimension and $y$ along the GNR axis. Twisting is around the GNR axis. The twist-induced pseudomagnetic field of $K$ valley-specified electronic states, $B_{K}$ directs reversely on the two sides of GNR axis. The pseudomagnetic field of $K^{'}$ valley electronic states $B_{K^{'}} = - B_{K}$ by time reversal symmetry. The edge currents $J_{K}$ and $J_{K^{'}}$ arising from $K$ valley and $K^{'}$ valley, respectively, are counterpropagating. The current $S_{K}$ and $S_{K^{'}}$ associated with the snake orbit are also counterpropagating but in a serpentine manner around the GNR axis. (b) Strain distribution $ε_{y y}$ along the GNR width, $x$ at twist rate $γ = 0.61 ° / nm$ and (c) the corresponding pseudomagnetic field $B_{K}$ for 170 nm wide GNRs with different chiralities $χ$ according to Eq. (1) with $a_{0} = 1.42 Å$ , $β = 2$ , $c = 1$ , and $t_{0} = 2.7 eV$ . (d) A near-field scanning optical microscope setup including contacts $I$ (injector) and $C$ (collector) for the detection of localized edge states (solid curve) and snake states (dashed curve).
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Figure 2
Electronic band structures (upper panel) and density of states (lower panel) of a 170 nm-wide zigzag GNR at twist rate (a) $γ = 0 ° / nm$ and (b) $γ = 0.61 ° / nm$ . The down arrows indicate the extra electronic states arising from zigzag edges. In (b), the solid dot (upper panel) indicates the snake orbit states, and the vertical dashed lines (lower panel) indicate the Landau levels with $n = 0$ , $\pm 1$ , $\pm 2$ , $\pm 3$ , $\pm 4$ according to Eq. (2) with $B_{eff} = 85 T$ . (c) The first Landau level energy, $E_{1}$ , and (d) the effective magnetic field, $B_{eff}$ , versus GNR width $W_{0}$ .
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Figure 3
Electronic band structures (upper panel) and density of states (lower panel) of a 176 nm-wide armchair GNR at twist rate (a) $γ = 0 ° / nm$ and (b) $γ = 0.66 ° / nm$ . In (b), the solid dot (upper panel) indicates the snake states, and the vertical dashed lines (lower panel) indicate the LLs $n = \pm 1$ , $\pm 2$ , $\pm 3$ , $\pm 4$ according to Eq. (2) with $B_{eff} = 136 T$ . The down arrows indicate the zero LL.
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Figure 4
Wave function spatial distributions ( $| Ψ |^{2}$ ) along the zigzag GNR width $x$ (a) for the first four LLs $n = 0$ , $\pm 1$ , $\pm 2$ , $\pm 3$ as shown in Fig. 2 at $κ = 0$ and (b) for the snake orbit whose location in the energy spectrum is indicated in Fig. 2 at $κ = 0.7 π$ . In (a), the electronic states represented by the red dashed line and blue solid line are degenerate. The vertical dashed lines indicate the maxima of the wave functions for the different LLs.
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Figure 5
Wave function spatial distributions ( $| Ψ |^{2}$ ) along armchair GNR width $x$ (a) for the first four LLs $n = \pm 0 (zero LL)$ , $\pm 1$ , $\pm 2$ , $\pm 3$ as shown in Fig. 3 at $κ = π$ and (b) for the snake orbit whose location in the energy spectrum is indicated in Fig. 3 at $κ = 0.2 π$ . In (a), the electronic states represented by the red dashed line and blue solid line are degenerate. The vertical dashed lines indicate the maxima of the wave functions for the different LLs.
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