Resonant tunnelling between the chiral Landau states of twisted graphene lattices

Journal name:
Nature Physics
Volume:
11,
Pages:
1057–1062
Year published:
DOI:
doi:10.1038/nphys3507
Received
Accepted
Published online

Abstract

A class of multilayered functional materials has recently emerged in which the component atomic layers are held together by weak van der Waals forces that preserve the structural integrity and physical properties of each layer. An exemplar of such a structure is a transistor device in which relativistic Dirac fermions can resonantly tunnel through a boron nitride barrier, a few atomic layers thick, sandwiched between two graphene electrodes. An applied magnetic field quantizes graphenes gapless conduction and valence band states into discrete Landau levels, allowing us to resolve individual inter-Landau-level transitions and thereby demonstrate that the energy, momentum and chiral properties of the electrons are conserved in the tunnelling process. We also demonstrate that the change in the semiclassical cyclotron trajectories, following an inter-layer tunnelling event, is analogous to the case of intra-layer Klein tunnelling.

At a glance

Figures

  1. Device structure and misaligned Brillouin zones.
    Figure 1: Device structure and misaligned Brillouin zones.

    a, Schematic of the device showing the two misaligned graphene lattices (bottom, red and top, blue) separated by a boron nitride tunnel barrier (upper yellow region). An optical micrograph is shown in Supplementary Fig. 1. b, Dashed black lines show the Brillouin zone boundary for electrons in the bottom graphene layer. Red arrows show the vector positions of the Dirac points Kb± (red circles) relative to the Γ point. Blue arrows show the positions of the Dirac points in the top layer, Kt± (blue circles), misoriented at an angle θ to the bottom layer.

  2. Magnetic field-induced resonances in the conductance.
    Figure 2: Magnetic field-induced resonances in the conductance.

    aI(Vb) curves measured when Vg = 0 for B = 0 (black dashed) and 4T (red solid), the latter offset by 7.5μA for clarity. Insets i and ii show the relative energies of the Dirac cones, displaced by ΔK, in the bottom (red) and top (blue) electrodes at the voltages V1 and V2 marked by the labelled vertical arrows. The intersections of the cones are shown by the thick yellow curves. The Fermi circles of the two layers are shown in white. b, Differential conductance, G(Vb), measured at Vg = −40V (blue lower curve), Vg = 0V (red middle curve), and Vg = 40V (green upper curve) when B = 4T and temperature T = 4K. Red and green curves are offset by 250μS and 350μS, respectively (that is, dotted lines mark G = 0 for the three curves). Red horizontal bars in a,b mark the Vb > 0 region where conductance peaks can be observed when Vg = 0.

  3. Differential magnetoconductance maps: experiment and theory.
    Figure 3: Differential magnetoconductance maps: experiment and theory.

    Colour maps showing G(Vb, Vg) at T = 4K measured (a) and calculated (b) when B = 2T and measured (c) and calculated (d) when B = 4T. Colour scales for a,c are in microseconds, whereas for b,d they are normalized to the maximum conductance in the maps. Black and yellow dashed curves enclose regions around Vb = 0 within which only conduction–conduction band (region labelled c–c with Vg > 0), or only valence–valence band (region labelled v–v with Vg < 0) tunnelling occurs. White right angles in a and b mark upper corners of the colour map regions enlarged in Fig. 5a,b, respectively. Filled black circles running top left to bottom right (bottom left to top right) show loci along which the chemical potential the bottom (top) layer intersects with the Dirac point in that layer.

  4. Energy alignment and tunnelling rates between the Landau levels of the two graphene electrodes.
    Figure 4: Energy alignment and tunnelling rates between the Landau levels of the two graphene electrodes.

    a,b, Dirac cones showing the energy–wavevector dispersion relation, E(k), for electrons in the bottom (red) and top (blue) graphene layers when B = 0 and Vb = 0.28V (a) and 0.58V (b). Rings of constant energy on the surface of the cones show the energies and semiclassical k-space radii of LLs with indices nb and nt. The black rings in a and b highlight nb = 1 to nt = 3 and nb = 2 to nt = 16 transitions, respectively. Occupied electron states in the bottom (top) layer are shaded dark red (blue) up to the Fermi level, μb, t, in that layer. c, Colour map showing tunnelling rates, W(nb, nt), normalized to the maximum rate in the plot, for scattering-assisted transitions (details in Supplementary Information) between LLs with indices nb and nt in the bottom and top electrodes. The dotted and solid curves show the loci calculated using equation (2). For all panels, B = 4T.

  5. Effect of chirality on the differential magnetoconductance: experiment and theory.
    Figure 5: Effect of chirality on the differential magnetoconductance: experiment and theory.

    ac, Colour maps showing G(Vb, Vg) for B = 2T. a,b are enlargements of the lower parts of the colour maps in Fig. 3a,b respectively (defined by white right angles). a, Experimental data (T = 4K). b, Calculated using the full model with chiral electrons. c, Calculated using non-chiral wavefunctions—that is, each comprising a single simple harmonic oscillator state. Colour bars in a are in microseconds and colour bars in b,c are in normalized units. d, The solid curves in ac enclose regions of the colour map where tunnelling is only v–v (labelled L) or a mixture of v–v and v–c (labelled R). e, Bar charts showing the ratio, left fenceGright fenceL/left fenceGright fenceR, of the mean conductance in regions L and R (see d) for the measured data (red), and calculated for chiral (yellow) and non-chiral (blue) electrons.

  6. Electron wavefunctions and semiclassical cyclotron orbits in the two graphene layers for figure-of-8 and nested tunnelling transitions.
    Figure 6: Electron wavefunctions and semiclassical cyclotron orbits in the two graphene layers for figure-of-8 and nested tunnelling transitions.

    a,b, Upper: vertical (horizontal) curves show the real (imaginary) parts of the real space electron wavefunction in the bottom (red curves) and top (blue curves) graphene electrodes, respectively, with B = 4T for nb = 1 (red) and nt = 3 (blue) (a) and nb = 2 (red) and nt = 16 (blue) (b). The x axis is scaled by lB2 for comparison with lower plots: circles show corresponding figure-of-8 and nested cyclotron orbits in k-space (kx, ky axes inset and direction of motion marked by arrows) with orbit centres separated by ΔK. The vertical black lines connecting upper and lower parts of the figure show the classical turning points.

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Author information

Affiliations

  1. School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK

    • M. T. Greenaway,
    • E. E. Vdovin,
    • O. Makarovsky,
    • A. Patanè,
    • T. M. Fromhold &
    • L. Eaves
  2. Institute of Microelectronics Technology and High Purity Materials, RAS, Chernogolovka 142432, Russia

    • E. E. Vdovin &
    • S. V. Morozov
  3. National University of Science and Technology ‘MISiS, 119049 Leninsky pr.4, Moscow, Russia

    • E. E. Vdovin &
    • S. V. Morozov
  4. School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK

    • A. Mishchenko,
    • M. J. Zhu,
    • K. S. Novoselov,
    • A. K. Geim &
    • L. Eaves
  5. Physics Department, Lancaster University, Lancaster LA1 4YB, UK

    • J. R. Wallbank &
    • V. I. Falko
  6. Centre for Mesoscience and Nanotechnology, University of Manchester, Manchester M13 9PL, UK

    • Y. Cao,
    • A. V. Kretinin &
    • A. K. Geim

Contributions

Y.C. and A.V.K., fabricated the devices. E.E.V., A.M., O.M., A.P., K.S.N., A.V.K., M.J.Z. and L.E., designed and/or carried out the experiments. M.T.G., T.M.F., L.E., E.E.V., A.M., S.V.M., J.R.W., V.I.F., K.S.N. and A.K.G., undertook the interpretation of the data. M.T.G. performed the calculations. M.T.G., T.M.F. and L.E. wrote the manuscript with contributions from the other authors.

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The authors declare no competing financial interests.

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