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Even-denominator fractional quantum Hall states in bilayer graphene

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Science  03 Nov 2017:
Vol. 358, Issue 6363, pp. 648-652
DOI: 10.1126/science.aao2521
  • Fig. 1 FQH states in BLG.

    (A) Energy spectrum of the BLG LLs near the charge neutrality point (CNP). The lowest two LLs with Landau orbital index N = 0 and 1 are energetically degenerate, leading to the eightfold degeneracy of the LLL. The BLG wave function with Landau orbital N = 0 is identical to the lowest Landau orbital wave function for conventional (nonrelativistic) systems, whereas the orbital N = 1 wave function is a mixture of the conventional Landau orbitals 0 and 1. The schematic shows the wave function distribution on the four atomic sites of BLG, for the N = 0 and 1 Landau orbital states. The blue-shaded schematic wave function corresponds to conventional Landau orbital 0, and the red-shaded schematic wave function corresponds to conventional Landau orbital 1. (B) Schematic of the device geometry. (C) σxx as a function of filling factor v and magnetic field B at T = 20 mK and (left) D = –100 mV/nm and (right) D = 35 mV/nm for the LLL ( –4 ≤ v ≤ 4). (D and E) σxy and Rxx, acquired by sweeping B at T = 0.3 K and fixed carrier densities, n = 2.2 × 1011 cm–2 and 7.7 × 1011cm–2, corresponding to filling fractions spanning (D) 0 ≤ v ≤ 1 and (E) 1 ≤ v ≤ 2. Bottom axis labels the filling fraction v, with corresponding B values on the top axis.

  • Fig. 2 Behavior of the energy gap.

    (A) Δ for all even-denominator states at B = 18 T and D = 35 mV/nm. (B) Δ of the Embedded Image state as a function of magnetic field B, at D = 35 mV/nm and T = 0.3 K. (C) Longitudinal conductance σxx versus v at different B. Measurements are performed by changing carrier density n while keeping displacement field constant at D = 35 mV/nm. Traces are offset for clarity. (D) Energy gap of the Embedded Image state normalized to its B = 0 value, versus parallel field for two fixed values of perpendicular field.

  • Fig. 3 Tuning with displacement field.

    (A) σxx at T = 0.2 K and B = 14.7 T versus filling factor v and displacement field D for the LLL of BLG, 0 ≤ v ≤ 4. (B) Schematic phase diagram labeling the ground-state order for the same filling fraction range as shown in (A). The blue- and red-shaded areas are occupied by broken-symmetry states with orbital index 0 and 1, respectively, whereas the dark and light color tones denote respectively the two different valley-isospin and layer polarizations. Dashed and solid black lines correspond to phase transitions between broken-symmetry states with different valley-isospin and orbital index, respectively. Vertical solid lines represent incompressible states observed in transport measurements. (C) Δ as a function of D for Embedded Image and Embedded Image at B = 14.7 T. (D) Δ versus D for Embedded Image at various B.

  • Fig. 4 Phase diagram for the 3/2 state gap.

    Shown is the contour line for Embedded Image as a function of B and D. The valley polarization transition at nonzero D is shown as the black dotted line, and the orbital polarization transition is shown as the black dashed line. The even-denominator state disappears in the orbital 0 region (gray-shaded area).

Supplementary Materials

  • Even denominator fractional quantum Hall states in bilayer graphenen

    J. I. A. Li, C. Tan, S. Chen, Y. Zeng, T. Taniguchi, K. Watanabe, J. Hone, C. R. Dean

    Materials/Methods, Supplementary Text, Tables, Figures, and/or References

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    • Materials and Methods
    • Figs. S1 to S10
    • References

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