High-resolution tunnelling spectroscopy of a graphene quartet

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Electrons in a single sheet of graphene behave quite differently from those in traditional two-dimensional electron systems. Like massless relativistic particles, they have linear dispersion and chiral eigenstates. Furthermore, two sets of electrons centred at different points in reciprocal space (‘valleys’) have this dispersion, giving rise to valley degeneracy. The symmetry between valleys, together with spin symmetry, leads to a fourfold quartet degeneracy of the Landau levels, observed as peaks in the density of states produced by an applied magnetic field. Recent electron transport measurements have observed the lifting of the fourfold degeneracy in very large applied magnetic fields, separating the quartet into integer1, 2, 3, 4 and, more recently, fractional5, 6 levels. The exact nature of the broken-symmetry states that form within the Landau levels and lift these degeneracies is unclear at present and is a topic of intense theoretical debate7, 8, 9, 10, 11. Here we study the detailed features of the four quantum states that make up a degenerate graphene Landau level. We use high-resolution scanning tunnelling spectroscopy at temperatures as low as 10mK in an applied magnetic field to study the top layer of multilayer epitaxial graphene. When the Fermi level lies inside the fourfold Landau manifold, significant electron correlation effects result in an enhanced valley splitting for even filling factors, and an enhanced electron spin splitting for odd filling factors. Most unexpectedly, we observe states with Landau level filling factors of 7/2, 9/2 and 11/2, suggestive of new many-body states in graphene.

At a glance


  1. Landau level spectroscopy of epitaxial graphene on SiC.
    Figure 1: Landau level spectroscopy of epitaxial graphene on SiC.

    Tunnelling spectroscopy of the Landau level states in a plot of differential conductivity versus sample bias. Tunnelling parameters are as follows: set-point current, 400pA; sample bias, −300mV; modulation voltage, 1mV. Left inset, high-resolution scanning tunnelling microscopy image of the graphene honeycomb lattice. Tunnelling parameters: set-point current, 100pA; sample bias, −250mV; T = 13mK. Right inset, Landau level peak energy position (relative to the N = 0 level) versus the square root of NB. Excellent scaling is observed in the linear relationship, yielding a carrier velocity of (1.08±0.03)×106ms−1 (1σ).

  2. Landau levels of epitaxial graphene on SiC as a function of magnetic field.
    Figure 2: Landau levels of epitaxial graphene on SiC as a function of magnetic field.

    A series of dI/dV line scans, taken vertically through the moiré region in Supplementary Fig. 1, as a function of magnetic field. Each panel shows the dI/dV intensity in a colour scale (from −1 to 12nS). The vertical axis within each panel is distance, D, from 0 to 40nm. A splitting of the N = 0, 1 and 2 Landau levels can been seen in different field ranges. Tunnelling parameters: set-point current, 200pA; sample bias, −250mV; modulation voltage, 250μV; T = 13mK.

  3. Electron density and filling-factor variation as a function of magnetic field in epitaxial graphene.
    Figure 3: Electron density and filling-factor variation as a function of magnetic field in epitaxial graphene.

    a, The electron density, n = νB/Φ0, determined from the filling of the Landau levels (red symbols), as a function of B, where ν is the filling factor and Φ0 is the flux quantum. The dashed blue lines correspond to densities at constant filling factors ranging from ν = 3 to 14. b, The integral of the dI/dV spectra of the filled K′ valley (rightmost peaks in Fig. 4), from the middle of the N = 1 Landau level to zero sample bias, EF, as a function of field (Fig. 4). The plateaux correspond to stable filling factors. c, Electron density versus B in the region around 11.5T, showing the transitions between the half-filled states at ν = 9/2 and ν = 11/2 between ν = 4 and 5 and, respectively, ν = 5 and 6. Filling factors were determined by taking the ratio of the integrated areas in b and dividing by the area of a single Landau level (area at ν = 5) and adding four (two for N = 0 and two for the K valley (leftmost peaks in Fig. 4)). The calculated filling factors are 5.46±0.06, 4.54±0.04 and 3.52±0.05 (1σ). Error bars, 1σ.

  4. High-resolution Landau level spectroscopy of the fourfold states that make up the N = 1 Landau level.
    Figure 4: High-resolution Landau level spectroscopy of the fourfold states that make up the N = 1 Landau level.

    a, A series of dI/dV line scans focusing on the Fermi level region of the N = 1 Landau level (LL), made in the same spatial location as in Fig. 2. At 11.125 and 11.5T, new stable half-filled Landau levels appear at filling factors of 11/2 and 9/2. b, Single dI/dV spectra from the middle regions of the panels in a for the indicated magnetic fields. The level separation energies are defined in the various panels. ΔEV, the lifting of the valley degeneracy (blue and red lines), is measured from the centres of the two spin-split states (see spectrum with B = 11.75T). The yellow lines indicate the position of the Fermi level at zero sample bias. We define three energy separations for the spin split peaks: ΔESL and ΔESR for the left and right spin-split levels, respectively (B = 11T); ΔESE measures the enhanced spin splitting when the Fermi level falls between the spin-split levels (B = 11.25T); and ΔESF measures the separation between the two half-filled Landau levels. c, Tunnelling dI/dV spectrum at a filling factor of 7/2 showing a similar 1/2-fractional state when EF is positioned between the left two spin split states (K valley) at a higher magnetic field of 14 T. Note change in horizontal scale and position of EF.Tunnelling parameters: set-point current, 200pA; sample bias, −250mV, modulation voltage, 50μV; T = 13mK.

  5. Energies in the epitaxial graphene N = 1 Landau level.
    Figure 5: Energies in the epitaxial graphene N = 1 Landau level.

    a, Valley splitting, ΔEV, measured for the N = 1 Landau level as a function of magnetic field. The solid blue line is a linear fit of the N = 1 data for fields less than 7T. The linear fit yields the effective g-factor gV(LL1) = 18.4±0.4 (1σ). The smooth cyan line is a guide to the eye and shows the N = 1 splitting peaking when EF is centred in the N = 1 Landau manifold at a filling factor of 4. Inset, schematic of the energy level structure of the N = 1 Landau level, showing the breaking of the valley symmetry, ΔEV, followed by the breaking of the spin symmetry, ΔES. b, Spin level energy separations as functions of magnetic field. The solid lines are linear fits yielding the g-factors gSL = 2.36±0.01 and gSR = 2.23±0.01 (1σ). The level separation energies are defined in Fig. 4. Error bars, 1σ.


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Author information


  1. Center for Nanoscale Science and Technology, NIST, Gaithersburg, Maryland 20899, USA

    • Young Jae Song,
    • Alexander F. Otte,
    • Hongki Min,
    • Shaffique Adam,
    • Mark D. Stiles &
    • Joseph A. Stroscio
  2. Maryland NanoCenter, University of Maryland, College Park, Maryland 20742, USA

    • Young Jae Song,
    • Alexander F. Otte &
    • Hongki Min
  3. Department of Physics and Astronomy, Seoul National University, Seoul, 151-7474, South Korea

    • Young Kuk
  4. School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA

    • Yike Hu,
    • David B. Torrance,
    • Phillip N. First &
    • Walt A. de Heer
  5. Department of Physics, University of Texas at Austin, Austin, Texas 78712, USA

    • Allan H. MacDonald


Y.J.S, A.F.O, Y.K. and J.A.S designed and constructed the millikelvin scanning probe microscopy system. The graphene scanning tunnelling microscopy/scanning tunnelling spectroscopy measurements were performed by Y.J.S, A.F.O and J.A.S. The graphene sample was grown by Y.H and W.A.d.H., and the surface was prepared and characterized by D.B.T and P.N.F. A theoretical analysis of the epitaxial graphene multilayer system was performed by H.M., S.A., M.D.S and A.H.M.

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