Boundary Modes from Periodic Magnetic and Pseudomagnetic Fields in Graphene

Võ Tiến Phong and E. J. Mele

Phys. Rev. Lett. 128, 176406 – Published 29 April 2022

Abstract

Single-layer graphene subject to periodic lateral strains is an artificial crystal that can support boundary spectra with an intrinsic polarity. This is analyzed by comparing the effects of periodic magnetic fields and strain-induced pseudomagnetic fields that, respectively, break and preserve time-reversal symmetry. In the former case, a Chern classification of the superlattice minibands with zero total magnetic flux enforces single counterpropagating modes traversing each bulk gap on opposite boundaries of a nanoribbon. For the pseudomagnetic field, pairs of counterpropagating modes migrate to the same boundary where they provide well-developed valley-helical transport channels on a single zigzag edge. We discuss possible schemes for implementing this situation and their experimental signatures.

Received 13 August 2021
Revised 10 December 2021
Accepted 20 March 2022

DOI:https://doi.org/10.1103/PhysRevLett.128.176406

Physics Subject Headings (PhySH)

Edge states Surface states Topological phases of matter Valleytronics

Graphene Topological insulators Topological materials

Band structure methods

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Võ Tiến Phong and E. J. Mele ^*

Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

^*mele@physics.upenn.edu

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Issue

Vol. 128, Iss. 17 — 29 April 2022

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Images

Figure 1
Graphene in a periodically modulated magnetic field. (a) Real-space and (b) reciprocal-space representations of graphene in a periodic magnetic field. (c) Band structure for $N = 14$ and $δ t = 0.3$ . The energy eigenstates are color coded by their valley expectation value $⟨ V ⟩$ as in Ref. [11]. We observe that the bands close to $E = 0$ are exceptionally well valley polarized. Within each valley, there is a doublet band at $E = 0$ , and singlet bands elsewhere. Band structures on a finite zigzag (d) and armchair (e) nanoribbon showing protected topological edge states at every insulating gap shown. In the zigzag configuration, we see that the edge states are highly valley polarized. Electron density distributions for some representation edge states are shown below each band structure. Here, each ribbon has 175 carbon atoms across its width.
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Figure 2
Graphene in the presence of a periodic strain. (a) Lattice realization of the gauge field defined in the text with the bond strength indicated by the thickness of the bonds. In this gauge, $C_{3 z}$ and $M_{x}$ are evidently preserved. Band structures on a finite zigzag (b) nanoribbon showing edge states at every insulating gap. These edge states sometimes meet at $T$ -invariant momenta and form small avoided crossings. There are no corresponding edge modes in the armchair configuration (c). The bands are colored according to their valley character, as detailed in Fig. 1. Here, $N = 14$ , $δ t = 0.3$ , and each ribbon has 175 carbon atoms across its width. (d) A four-terminal setup to measure the existence of the one-sided boundary states when the chemical potential resides within one of the bulk gaps. When a potential difference is applied across $V_{1}$ and $V_{2}$ , a current flows across the two terminals. On the other hand, when a potential difference is applied across $V_{3}$ and $V_{4}$ , no current is detected. (e) Local density of states as a function of energy $E$ and position $y$ . The midgap states shown in blue are localized to only one edge. The bulk states shown in yellow are extended throughout the sample. For this simulation, the ribbon has 150 carbon atoms across its width.
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Figure 3
Designing strain fields. (a) $h (r)$ and (b) its Gaussian curvature with $ϕ = - π / 4$ as defined in the text. Following Ref. [10], we show (c) the estimated bandwidth of the energy manifold at charge neutrality and (d) the estimated magnitude of the first insulating gap above charge neutrality. The yellow and cyan lines show the Coulomb energy scale $E \sim 14.4 Å eV / ε L$ for $ε = 4$ and $ε = 10$ , respectively. The color maps have units $\log_{10} (eV)$ .
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