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Cited by 541Topological Materials and PhenomenaGraphene research and applicationsQuantum and electron transport phenomena

Zak phase and the existence of edge states in graphene

Open access
Delplace, P., Ullmo, D., Montambaux, G.·2011·Physical Review B
DOI 10.1103/physrevb.84.195452
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Abstract

We develop a method to predict the existence of edge states in graphene ribbons for a large class of boundaries. This approach is based on the bulk-edge correspondence between the quantized value of the Zak phase $\mathcal{Z}({k}_{\ensuremath{\parallel}})$, which is a Berry phase across an appropriately chosen one-dimensional Brillouin zone, and the existence of a localized state of momentum ${k}_{\ensuremath{\parallel}}$ at the boundary of the ribbon. This bulk-edge correspondence is rigorously demonstrated for a one-dimensional toy model as well as for graphene ribbons with zigzag edges. The range of ${k}_{\ensuremath{\parallel}}$ for which edge states exist in a graphene ribbon is then calculated for arbitrary orientations of the edges. Finally, we show that the introduction of an anisotropy leads to a topological transition in terms of the Zak phase, which modifies the localization properties at the edges. Our approach gives a new geometrical understanding of edge states, and it confirms and generalizes the results of several previous works.

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  • Introducing anisotropy along the vertical and diagonal directions into graphene leads to broken six-fold symmetry and level crossing with deformation of the bandstructure.

    “In contrast, introducing the anisotropy along the vertical direction and diagonal direction into the graphene leads to the broken six-fold symmetry and level crossing with the deformation of the bandstructure [29-31].”

    From Observing movement of Dirac cones from single-photon dynamics (2021) · cites this paper as [30]

    Section IV.A says that changing the hopping ratios moves the Dirac points, and Section IV.B says that increasing anisotropy can merge them and then open a band gap. The copy does not specifically describe anisotropy along vertical and diagonal directions or state that it breaks six-fold symmetry; the reported band touching and merging also do not establish a general level-crossing claim. Page numbers are unavailable in our copy. (LLM, conf=0.83)

    Experimental — accuracy not yet measured

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Research works printed but not linked: 34

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Authors
Delplace, P., Ullmo, D., & Montambaux, G.
Affiliations
  • University of Geneva (CH)
  • Université Paris-Sud (FR)
ORCID
Venue
Physical Review B
Publisher
American Physical Society (APS)
Published
2011-11-23
Volume
84
Issue
19
Pages
195452
ISSN
  • 1098-0121 (ISSN-L)
  • 1550-235X
DOI
10.1103/physrevb.84.195452
OpenAlex
W1979276597
Cited by
541
References
3
Citations by year
2026: 25 · 2025: 37 · 2024: 58 · 2023: 50 · 2022: 38 · 2021: 62 · 2020: 41 · 2019: 47 · 2018: 55 · 2017: 44 · 2016: 37 · 2015: 17 · 2014: 11 · 2013: 16 · 2012: 2
FWCI
5.04
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review B (publishedVersion)
  • HAL (submittedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Funders
Agence Nationale de la Recherche (ANR-09-NANO-0016, ANR-09-NANO-016-01)
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Topological Materials and Phenomena
  • Physical Sciences › Materials Science › Materials Chemistry › Graphene research and applications
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum and electron transport phenomena
Keywords
Zak phase, edge states, graphene nanoribbons, bulk-boundary correspondence, Berry phase, zigzag edge, anisotropy, topological phase transition, edge localization

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