Research ArticleCONDENSED MATTER PHYSICS

Observation of the quantum valley Hall state in ballistic graphene superlattices

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Science Advances  18 May 2018:
Vol. 4, no. 5, eaaq0194
DOI: 10.1126/sciadv.aaq0194
  • Fig. 1 Device structure and characterization of our hBN/graphene/hBN superlattices.

    (A) Microscope image of our typical hBN/graphene/hBN superlattice device. The Hall bar geometry is such that width W = 1.0 μm and length L = 2.5 μm. (B) Schematic cross section of the device with Cr/Au contacts on the edge yielding one-dimensional contacts. We applied the backgate voltage (Vg) through the 90-nm thickness of SiO2 and the 20-nm thickness of hBN. (C) (red curve) Measured nonlocal resistance Rnl (I: 62, V: 53) and (blue curve) longitudinal resistivity ρxx (I: 14, V: 65) as a function of Vg without magnetic fields at 1.5 K. Sharp increases of the ρxx at the Vg of approximately 0 and −21 V correspond to a DP and a SDP, respectively. Inset shows schematic pictures of the measurement setup. (D) A logarithmic-scale plot of the longitudinal conductivity (σxx) as a function of Vg and magnetic fields (B) applied perpendicular to the substrate at 6 K. (E) Transverse resistance (Rtr) oscillation at 6 K, which is Rtr as a function of Vg and B. The terminal configurations are the same for the Rnl in (C). (F) Experimental estimation of the energy gaps derived by the Arrhenius plots of the resistivity of the DP and SDP as a function of measured temperature T. We estimated the gaps to be Eg = 2Δ = 32 and 14 meV at the DP and SDP, respectively.

  • Fig. 2 Longitudinal (local) resistance Rxx and nonlocal resistance Rnl with different terminal configurations on the six-terminal device measured without magnetic fields at 1.5 K.

    (A) Schematic showing the terminal number. (B) Rxx for (I: 14, V: 65); (C) Rnl for (I: 61, V: 53), (I: 61, V: 54), and (I: 61, V: 43); and (D) Rnl for (I: 62, V: 43), (I: 62, V: 53), as a function of Vg. Dotted lines show theoretical values of the resistance based on the minimal edge-state model described in the text. Both Rxx and Rnl are consistent with the theoretical value apart from fluctuations. The ratio of experimental values of Rnl(I: 61, V: 53)/Rnl(I: 61, V: 54), Rnl(I: 61, V: 53)/Rnl(I: 61, V: 43), and Rnl(I: 62, V: 53)/Rnl(I: 62, V: 43) are approximately 2.

  • Fig. 3 Nonlocal magnetoresistance in graphene superlattices.

    (A) Schematics of the measurement setup (left) and the energy band structure in magnetic fields (right), which show how the band structure is reconstructed when the magnetic field is included. In the band structure, we do not take into account the role of orbital magnetism for simplicity, which leads to broadened Landau bands overlapped due to disorder and finite-temperature effects. (B) Rnl(I: 62, V:53) is mapped as a function of Vg and B at 1.5 K. Dashed white lines correspond to the plateau denoted by the black arrows in (C). (C) Rnl versus B for five Vg’s near the SDP, marked with the same color as arrows on the top of (B). Black arrows show the regime where the energy gap with hot spots is kept, linking to the white dashed lines in (B).

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/5/eaaq0194/DC1

    Supplementary Text

    fig. S1. False-color AFM image of our hBN/graphene/hBN stack.

    fig. S2. Influence of thermal cycles and charge impurities on our device.

    fig. S3. Shubnikov-de Haas oscillation and estimation of the Fermi velocity.

    fig. S4. Magnetotransport in our ballistic hBN/graphene/hBN sample.

    fig. S5. Temperature dependence of the nonlocal resistance Rnl.

    fig. S6. Magnetic field dependence of the Rnl.

    References (3242)

  • Supplementary Materials

    This PDF file includes:

    • Supplementary Text
    • fig. S1. False-color AFM image of our hBN/graphene/hBN stack.
    • fig. S2. Influence of thermal cycles and charge impurities on our device.
    • fig. S3. Shubnikov-de Haas oscillation and estimation of the Fermi velocity.
    • fig. S4. Magnetotransport in our ballistic hBN/graphene/hBN sample.
    • fig. S5. Temperature dependence of the nonlocal resistance Rnl.
    • fig. S6. Magnetic field dependence of the Rnl.
    • References (32–42).

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