Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                

Ballistic electron channels including weakly protected topological states in delaminated bilayer graphene

T. L. M. Lane, M. Anđelković, J. R. Wallbank, L. Covaci, F. M. Peeters, and V. I. Fal'ko
Phys. Rev. B 97, 045301 – Published 11 January 2018
PDFHTMLExport Citation
Abstract
Authors
Article Text
  • INTRODUCTION
  • ELECTRONIC PROPERTIES OF A SINGLE…
  • ELECTRONIC SPECTRUM OF A DELAMINATION IN…
  • DELAMINATION SPECTRUM IN A PERPENDICULAR…
  • DISCUSSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We show that delaminations in bilayer graphene (BLG) with electrostatically induced interlayer symmetry can provide one with ballistic channels for electrons with energies inside the electrostatically induced BLG gap. These channels are formed by a combination of valley-polarized evanescent states propagating along the delamination edges (which persist in the presence of a strong magnetic field) and standing waves bouncing between them inside the delaminated region (in a strong magnetic field, these transform into Landau levels in the monolayers). For inverted stackings in BLGs on the left and right of the delamination (AB-2ML-BA or BA-2ML-AB, where 2ML indicates two decoupled monolayers of graphene), the lowest-energy ballistic channels are gapless, have linear dispersion, and appear to be weakly topologically protected. When BLG stackings on both sides of the delamination are the same (AB-2ML-AB or BA-2ML-BA), the lowest-energy ballistic channels are gapped, with a gap ɛg scaling as ɛg∝W−1 with delamination width and ɛg∝δ−1 with the on-layer energy difference in the delaminated part of the structure. Depending on the width, delaminations may also support several “higher-energy” waveguide modes. Our results are based on both the analytical study of the wave matching of Dirac states and tight-binding model calculations, and we analyze in detail the dependence of the delamination spectrum on the electrostatic conditions in the structure, such as the vertical displacement field.

    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    2 More
    • Received 11 October 2017

    DOI:https://doi.org/10.1103/PhysRevB.97.045301

    ©2018 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Ballistic transportEdge states
    1. Physical Systems
    Graphene
    1. Techniques
    Tight-binding modelk dot p method
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    T. L. M. Lane1,2,*, M. Anđelković3, J. R. Wallbank1, L. Covaci3, F. M. Peeters1,3, and V. I. Fal'ko1,2

    • 1National Graphene Institute, University of Manchester, Manchester M13 9PL, United Kingdom
    • 2School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
    • 3Department Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium

    • *thomas.lane-3@postgrad.manchester.ac.uk

    Article Text (Subscription Required)

    Click to Expand

    References (Subscription Required)

    Click to Expand
    Issue

    Vol. 97, Iss. 4 — 15 January 2018

    Reuse & Permissions
    Access Options
    Author publication services for translation and copyediting assistance advertisement

    Authorization Required

    Log In
    Other Options
    ×

    Images

    • Figure 1
      Figure 1

      Sketch of the considered delamination. The decoupled monolayers lie in the region 0<y<W, connected to two bilayer graphene regions (y≤0 and y≥W). Corresponding energy dispersions within each distinct zone are shown with the electrostatic variables of the model indicated. These would be tuned by varying the vertical displacement field Ez across the system via top and bottom gates in each region. Below are images depicting the two different Bernal stacking arrangements discussed in the text.

      Reuse & Permissions
    • Figure 2
      Figure 2

      (a) Probability of reflection back on the same (upper) layer as the incoming wave for AB stacking within the bilayers. (b) Results for the opposite BLG stacking (BA) which exhibits identical reflection probability dependence under transformation kx→−kx. (c) and (d) Phase ϕ acquired upon reflection back on the same layer corresponding to (a) and (b).

      Reuse & Permissions
    • Figure 3
      Figure 3

      Interface states (solid red curves) and low energy BLG bands (dashed red curves) at y=0 for Δ=0.2eV and δ=U=0 in the K+ (top) and K− (bottom) valley. Blue curves depict the bands for nonzero skew interlayer coupling, γ3, for a delamination along the zig-zag direction, illustrating the negligible effect that this additional hopping term has on the spectrum at a delamination edge.

      Reuse & Permissions
    • Figure 4
      Figure 4

      (a)–(d) Wave function across the 2ML channel for specific energy and momenta [indicated on (e)] in the sublattice basis. (e)–(h) Permitted energy bands calculated from the continuum model within the bilayer gap and (i)–(k) the results of an equivalent TB calculation approach including continuum bands which extend above the bilayer band edge. Interface modes (red curves) are identified crossing between the low-energy bilayer band edges (orange shaded regions). Plotted with green (blue) dashed lines are the conical dispersions of the upper (lower) monolayer with energy difference δ between their Dirac points. (l) and (m) Top and side profiles of the LDOS for the same parameters as in (c) and (d). Calculations were performed for W=200 Å, Δ=0.2 eV, and U=0 around the K+ valley.

      Reuse & Permissions
    • Figure 5
      Figure 5

      (a)–(d) and (i)–(l) Electron wave functions across the delamination (gray shaded region) for specific energy and momenta indicated in (e) and (h) in sublattice basis (A1,B1,B2,A2)T. (e)–(h) Energy structure of the delamination for increasing interlayer energy gap δ within the monolayer region. Green (blue) dashed lines denote the upper (lower) monolayer Dirac cones in the 2ML region, and the orange region denotes the low-energy bulk bilayer band edge. Calculations are for W=200 Å, Δ=0.2 eV, and U=0 around the K+ valley.

      Reuse & Permissions
    • Figure 6
      Figure 6

      Interband states for different symmetry-breaking energy of-sets U for both (a) and (b) AB-2ML-AB and (c)–(f) AB-2ML-BA stacking with W=200 Å and Δ=δ=0.2 eV. In (a)–(d), calculated using the continuum model, the red dashed line indicates the 2ML region energy midpoint, and green and blue dashed lines indicate the positions of the bulk 2ML Dirac cones. (e) and (f) Density of states of the system calculated from the TB model.

      Reuse & Permissions
    • Figure 7
      Figure 7

      (a)–(d) Band structure and (e)–(l) LDOS maps at specific values of energy and wave vector for the nanoribbon in the presence of a perpendicular magnetic field. (e–h) and (i–l) depict the distribution of the states at the value of ɛ and kx marked on the band structure plots with yellow circles and red squares, respectively. Calculations are for W=200 Å, Δ=0.2 eV, and U=0 in the vicinity of the K+ valley.

      Reuse & Permissions
    • Figure 8
      Figure 8

      Energy dispersion for an AB-2ML-BA delamination with armchair edges, calculated using the TB model for Δ=0.2 eV, δ=U=0, and W=200Å.

      Reuse & Permissions
    • Figure 9
      Figure 9

      Dependence of energy gaps on delamination width. Open squares and dashed curves show data and fitting for the AB-2ML-AB system, respectively, while solid squares and solid curves show data for the AB-2ML-BA system. Black points represent data extracted from the TB model for the AB-2ML-BA system. (a)–(c) Band structures at W=200 Å, W=400 Å, and W=600 Å, respectively for the AB-2ML-AB system. (d)–(f) Band structures at W=200 Å, W=400 Å, and W=600 Å, respectively, for the AB-2ML-BA system. Plots are calculated for Δ=0.2 eV and δ=U=0 around the K+ valley.

      Reuse & Permissions
    ×

    Sign up to receive regular email alerts from Physical Review B

    Log In

    Cancel
    ×

    Search

    Article Lookup

    Paste a citation or DOI
    Enter a citation
    ×