Local Compressibility Measurements of Correlated States in Suspended Bilayer Graphene

Bilayer graphene has attracted considerable interest due to the important role played by many-body effects, particularly at low energies. Here we report local compressibility measurements of a suspended graphene bilayer. We find that the energy gaps at filling factors $\nu =\pm 4$ do not vanish at low fields, but instead merge into an incompressible region near the charge neutrality point at zero electric and magnetic field. These results indicate the existence of a zero-field ordered state and are consistent with the formation of either an anomalous quantum Hall state or a nematic phase with broken rotational symmetry. At higher fields, we measure the intrinsic energy gaps of broken-symmetry states at $\nu =0$, $\pm 1$, and $\pm 2$, and find that they scale linearly with magnetic field, yet another manifestation of the strong Coulomb interactions in bilayer graphene.

Figure 1

(a) Schematic illustration of the measurement setup. (b) Inverse compressibility as a function of carrier density at $B=2\mathrm{T}$. Incompressible peaks at $\nu =0$, $\pm 2$, and $\pm 4$ occur due to the decreased density of states between Landau levels. (c) Chemical potential as a function of carrier density at $B=2\mathrm{T}$, obtained by integrating the data in (b). Steps in chemical potential occur at $\nu =0$, $\pm 2$, and $\pm 4$ as electrons begin to occupy the next Landau level. In panels (b) and (c), experimental data are shown in blue and fits, based on a Lorentzian at each filling factor, are shown in red.

Figure 2

(a) Gap size, extracted from a Lorentzian fit, as a function of magnetic field for $\nu =0$ (blue), $\pm 1$ (magenta), $\pm 2$ (green), $\pm 4$ (red), and $\pm 8$ (black). Squares correspond to electrons, and circles to holes. Gap size at all filling factors is well described by linear scaling with magnetic field, with fits given by the solid lines. (b) Incompressible peak width as a function of magnetic field for $\nu =0$, $\pm 2$, $\pm 4$, and $\pm 8$ [same colors as in (a)]. The $\nu =0$ peak is significantly narrower than the others. Peak width does not strongly depend on magnetic field.

Figure 3

(a) Inverse compressibility as a function of carrier density and magnetic field. Incompressible peaks occur at $\nu =0$ and $\pm 4$. Below about 0.2 T, peak height along $\nu =\pm 4$ increases with decreasing field, culminating in an incompressible peak at zero field. (b) Zoom-in on the low-field behavior of the sample, taken at a different location on the flake from that in (a). Distinct peaks along $\nu =\pm 4$ persist all the way to zero field. (c) Line cuts of incompressibility along the red arrows shown in panel (b) for magnetic fields between $B=0$ and 0.175 T in steps of 0.035 T. Curves are offset for clarity. Data are shown in blue, and the red curves are two-Lorentzian fits, except for the zero-field fit, which is composed of only one Lorentzian. (d) Gap size at $|\nu |=4$ for electrons (green), holes (red), and their sum (cyan). Data for $|B|>0.03\mathrm{T}$ are based on a two-Lorentzian fit. The blue circles at low field describe the jump in chemical potential across the charge neutrality point, as modeled by a single Lorentzian fit. Solid black lines correspond to ${\Delta }_4=3.9(B[\mathrm{T}])\mathrm{meV}$, as derived from the data at high magnetic fields.

Figure 4

(a) Two-terminal conductance as a function of carrier density and magnetic field. Quantum Hall plateaus at $\nu =\pm 4$ emerge at very low field, and the onset of the highly resistive state at $\nu =0$ is also apparent. (b) Derivative of conductance with respect to carrier density. Local peaks and minima with the same slope as $\nu =\pm 4$ are visible, indicative of localized states. These localized states persist down to zero magnetic field. (c) Conductivity along filling factors $\nu =-4$ (red), 0 (blue), and 4 (green) as a function of magnetic field.