Minibands in twisted bilayer graphene probed by magnetic focusing

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Science Advances  17 Apr 2020:
Vol. 6, no. 16, eaay7838
DOI: 10.1126/sciadv.aay7838
  • Fig. 1 Moiré minibands and TMF measurements.

    (A) Schematics of the moiré superlattice induced by the twist of graphene layers. Here, two graphene sheets are rotated by an angle θ relative to each other, which creates an additional spatial periodicity λS = a/[2 sin (θ/2)] (a is graphene’s lattice constant) with the unit cell area of AS=3/2λS2. (B) Band structure of TBG graphene in the K valley of the Brillouin zone calculated for the twist angle θ = 1.87°, as discussed in section S3. (C) Optical image of TBG device D1 with θ = 1.87°. Scale bar, 4 μm. (D) Two examples of TMF signals measured in device D2 (D = 0 V nm−1) at 5 K for the carrier density 3.7 × 1012 cm−2 (left) and 9.3 × 1012 cm−2 (right) at a distance of 4.9 μm from the injector. The latter is close to the main and secondary neutrality points, respectively, as illustrated in (B). The insets are examples of focusing caustics near the main (left) and secondary (right) neutrality points (see more examples in fig. S4). Arrows highlight the focal points for caustics, red star marks the current injection point, and red lines show typical trajectories that extend from the injector to the first focal point.

  • Fig. 2 Transverse magnetic focusing map.

    (A) Focusing signal Rf as a function of the magnetic field and carrier density measured at 2 K for device D1 in zero displacement field, D = 0 V nm−1. Color scale: blue to red, ±3 ohms. (B) TMF map calculated from the energy spectrum shown in Fig. 1B using a numerical method described in section S4. The angle between the zigzag edge of one of the monolayers and the sample boundary is taken as 45° to avoid any spurious effects of crystallographic alignment. As demonstrated in section S4, the calculated TMF map is only very weakly sensitive to the mutual orientation between graphene and the sample edge, confirming the generality of our results. (C) Contour plot of the first conduction miniband shown for the K valley of the Brillouin zone for zero (left) and nonzero (right) displacement fields. Black and red dashed lines outline the shape of the Fermi surfaces for carrier densities marked by black and red dashed lines in (A); the latter corresponds to equivalent doping levels relative to the main (black) and secondary (red) neutrality points. The color scale is from 0 to 154 meV. (D) Rf as a function of magnetic field and carrier density for device D2 measured at T = 2 K and D = 0.75 V nm−1 at a distance of 8.5 μm from the injector (more data are shown in fig. S5). Color scale: blue to red, ±0.2 ohm. (E) TMF map calculated numerically for device D2 in a displacement field (see sections S3 to S5 for details), which shows the splitting of the focusing peaks originating from the different miniband dispersion at κ and κ′.

  • Fig. 3 Temperature dependence of magnetic focusing.

    (A) Temperature dependence of the TMF signal measured at two characteristic carrier densities for device D1 (see legends). T was varied from 2 to 30 K (blue to red). (B) T dependence of the relative scattering length (see text) extracted from experimental data for consecutive focusing peaks. Absolute scattering lengths for several relative orientations of the crystallographic axes and the sample edge are shown in fig. S6. Dashed line shows T−2 dependence. The inset shows the ratio of the areas under the first and second focusing peaks in (A) as a function of T. Arrows correspond to A2/A1 = 0.8 and 0.65 (see text). Error bars indicate the accuracy of determining A2/A1; large errors at T > 20 K are due to the relatively large background signal as the focusing peaks become strongly suppressed.

  • Fig. 4 Bulk transport properties of TBG.

    (A) Resistivity as a function of carrier density measured at 5 K for device D1. The inset shows the same data on a logarithmic scale. (B) Hall resistivity as a function of the carrier density for D1. Black arrows in (A) and (B) mark neutrality points, and red arrows mark vHS. (C) Resistivity versus magnetic field measured at different temperatures for device D1 at n = 3.1 × 1012 cm−2. Red dots on a 40-K curve highlight the positions of Brown-Zak oscillations.

Supplementary Materials

  • Supplementary Materials

    Minibands in twisted bilayer graphene probed by magnetic focusing

    A. I. Berdyugin, B. Tsim, P. Kumaravadivel, S. G. Xu, A. Ceferino, A. Knothe, R. Krishna Kumar, T. Taniguchi, K. Watanabe, A. K. Geim, I. V. Grigorieva, V. I. Fal’ko

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    • Sections S1 to S6
    • Figs. S1 to S6
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