Energy gap of the even-denominator fractional quantum Hall state in bilayer graphene

TL;DR

The study addresses even-denominator fractional quantum Hall states in Bernal bilayer graphene and their potential non-Abelian anyons. It combines thermally activated transport in Corbino devices with direct chemical potential sensing to extract both activation and thermodynamic gaps, illustrating gap suppression by disorder via the relation $n_{ ext{qp}} \\propto \exp(-\Delta_{ ext{qp}}/2k_B T)$. State-of-the-art iDMRG calculations for BLG, including inter-Landau-level screening, yield $\Delta_{ ext{qp}}^{\mathrm{DMRG}} = 0.011 E_C$ (with $E_C = e^2/(4\pi \epsilon_0 \epsilon_{\mathrm{hBN}} \ell_B)$), corresponding to about $5.6\,\mathrm{K}$ at $B=12\,\mathrm{T}$. A slow-disorder Wigner-crystal model reconciles measured gaps, giving $\Delta_{ ext{qp}}^{\mathrm{fit}} \approx 7\,\mathrm{K}$ for the $1/2$ state and $11.6\,\mathrm{K}$ for the $2/3$ state, within ~20% of the DMRG value, and establishing bilayer graphene as a robust platform for probing non-Abelian anyons.

Abstract

Bernal bilayer graphene hosts even denominator fractional quantum Hall states thought to be described by a Pfaffian wave function with nonabelian quasiparticle excitations. Here we report the quantitative determination of fractional quantum Hall energy gaps in bilayer graphene using both thermally activated transport and by direct measurement of the chemical potential. We find a transport activation gap of 5.1K at B = 12T for a half-filled N=1 Landau level, consistent with density matrix renormalization group calculations for the Pfaffian state. However, the measured thermodynamic gap of 11.6K is smaller than theoretical expectations for the clean limit by approximately a factor of two. We analyze the chemical potential data near fractional filling within a simplified model of a Wigner crystal of fractional quasiparticles with long-wavelength disorder, explaining this discrepancy. Our results quantitatively establish bilayer graphene as a robust platform for probing the non-Abelian anyons expected to arise as the elementary excitations of the even-denominator state.

Figures (10)

• Figure 1: Chemical potential and inverse compressibility of bilayer graphene fractional quantum Hall states.(A) Device schematic showing the hBN layers (blue), top and bottom graphite gates (dark grey), monolayer graphene detector layer connected to Corbino contacts, and bilayer graphene sample layer. (B) Optical image of the Corbino contacts to the monolayer graphene detector. White dashed lines show the trajectory of a chiral edge state along trenches etched through the device, which ensures contact between the metal and dual gated sample bulk. (C) The top panel shows the measured $\mu$ at $B=13.8T$ and $T=50mK$ in the partially filled N=0 level spanning $0<\nu<1$. The bottom panel shows the inverse compressibility, $d\mu/d\nu$, calculated by numerically differentiating the data in the top panel. (D) The same as C, but for the partially filled N=1 orbital Landau level spanning $-1<\nu<0$.
• Figure 2: Comparison of activation and thermodynamic gap sin a partially filled $N=1$ Landau level.(A) Two terminal conductance measured in a Corbino geometry as a function of filling factor at $B=12T$ for different temperatures. The temperature spacing is 5mK. (B) Activation gap from the Arrhenius fit for $\nu=-3+1/2$ (red) and $\nu=-3+2/3$ (blue). (C) Chemical potential measurement near $\nu_0=-1+1/2$ (red dots) at $B=13.8T$. Theory fit using the Wigner crystal model in the clean limit (light red line) and in the disordered limit (red line). (D) Chemical potential measurement near $\nu_0=-1+2/3$ (blue dots) at $B=13.8T$. Theory fit using the Wigner crystal model in the clean limit (light blue line) and in the disordered limit (blue line).
• Figure 3: Temperature dependent $\mu$ near fractional filling at B=13.8T.(A) Chemical potential near half filling of an $N=1$ Landau level at several different temperatures. (B) Chemical potential jump across the incompressible states as a function of temperature for different filling factors in an $N=1$ LL (dots). The solid lines are a low temperature fit, $\Delta \mu(T) =\Delta_0-bT^2$. (C) Chemical potential jump $\Delta_0$ extracted from the fit for different fractional states in the N=0 ($\tilde{\nu}=\nu$, red dots) and (N=0 orbital) and $N=1$ ($\tilde{\nu}=\nu+1$, orange dots) orbital Landau levels. (D) Temperature decay parameter $b$ extracted from the fit same states.
• Figure S1: Sample fabrication chronological steps.(A) Optical microscope image of the final heterostructures with the layer borders highlighted with colors (B) Holes are defined in the top gates. (C) Smaller hole in a Pac-Man shape to the monolayer are etched within the top gate holes. (D) Contact trenches are defined together with openings to contact all the layers. (E) Metal deposition to make contact to the different layers. (F) Overdose PMMA is used to define bridges on top of the exposed top gate edges. (G) Final image of the chemical potential device. (H) Final image of the Corbino transport device.
• Figure S2: Measurement principle.(A) Device schematic showing the dual graphite gated sample with a graphene monolayer detector capacitively coupled to the bilayer graphene sample of interest. (B) Monolayer detector conductance as a function of top gate voltage at B=13.5T. The four flux fractional state conductance minima $\nu=-2+7/9$, highlighted with the red arrow, is used as a sharp detector of the chemical potential change in the bilayer graphene sample.