Measurement of topological Berry phase in highly disordered graphene

We have observed the quantum Hall effect (QHE) and Shubnikov–de Haas (SdH) oscillations in highly disordered graphene at magnetic fields up to 65 T. Disorder was introduced by hydrogenation of graphene up to a ratio H/C $\approx 0.1\%$. The analysis of SdH oscillations and QHE indicates that the topological part of the Berry phase, proportional to the pseudospin winding number, is robust against introduction of disorder by hydrogenation in large-scale graphene.

Figure 1

(a) Raman spectrum of graphene with 4 minutes of hydrogenation exposure (red) and pristine graphene (blue). (b) Raman map of the peak intensity ratio $I_D/I_G$ of the same region of a graphene sheet on ${\mathrm{SiO}}_2/\mathrm{Si}$ before and after an 8 minute hydrogenation exposure. Macroscopic inhomogeneity is observed, including striations associated with the Cu growth substrate. (c) Conductivity ${\sigma }_{xx}$ of pristine graphene and hydrogenated graphene versus carrier density $n$ shows a trend of decreasing conductivity upon hydrogenation. The neutrality point, not observable in all samples, was identified from magnetic frequency analysis of Shubikov–de Haas oscillations.

Figure 2

The measured longitudinal resistance $R_{xx}$ and Hall resistance $R_{xy}$ of a large-area CVD graphene sample G79 versus back-gate voltage $V_G$ at temperature $T=0.3$ K and constant magnetic field $B=35$ T. The observed QHE filling factor sequence $\nu =-6,-2,+2$ corresponds to the anomalous Hall sequence for massless Dirac fermions in monolayer graphene.

Figure 3

(a) The measured $R_{2pt}$ of hydrogenated graphene sample HGTO2 on a logarithmic scale versus applied magnetic field $B$ and back-gate voltage $V_G$, at a sample temperature $T=1.5\mathrm{K}$. SdH oscillations and a strong positive magnetoresistance near neutrality are observed. (b) The measured $R_{xy}$ of hydrogenated graphene sample HG78 on a logarithmic scale versus applied magnetic field $B$ and back-gate voltage $V_G$, at a sample temperature $T=0.3\mathrm{K}$. A $\nu =-2$ plateau at $R_{xy}=12550\mathrm{\Omega }$ is observed above a field of $20\mathrm{T}$. (c) The measured $R_{xx}$ of hydrogenated graphene sample HG78 (log scale) and pristine graphene sample G79 (linear scale) versus applied gate voltage, over a temperature range $T=0.3–83\mathrm{K}$ at a constant field of $B=35\mathrm{T}$. The minima in $R_{xx}$ are labeled with corresponding filling factors.

Figure 4

(a) The measured $R_{2pt}$ versus $1/B$ of HG78 at a gate voltage $V_G=-90\mathrm{V}$. The local minima and maxima of $R_{2pt}$ versus $1/B$ are identified. (b) A Landau fan diagram of LL index versus $1/B$ for sample HG78 at different gate voltages. The SdH frequency $B_F$ at each gate voltage is extracted from the slope of the LL fan (linear fit indicated by dashed lines). The intercept $\beta /2\pi =-0.53\pm 0.14$ is in good agreement with $\beta =\pi$.

Figure 5

The Landau fan diagrams of samples (a) HG75 at $T=0.3\mathrm{K}$, (b) HGTO2 at $T=1.5\mathrm{K}$, and (c) HGTO1 at $T=1.5\mathrm{K}$. The topological Berry phase $\beta$ is extracted for each sample from the intercept at $1/B=0$ of the best-fit lines to at least two LL's.

Figure 6

The scattering time $\tau$ extracted by a Lifshitz-Kosevich best fit to SdH oscillations with a Dingle damping factor $R(B,T)=R_{0}exp(-\pi \hslash k_F/eB{v}_F\tau )$, plotted versus carrier density $n$, for six graphene samples.

Figure 7

The extracted SdH frequency $B_F$ versus carrier density scaled by the flux quantum $n{\mathrm{\Phi }}_0$ for six graphene samples. The best-fit slope (black dashed line) on the hole side for hydrogenated graphene samples is $\delta B_F/\left|\delta n{\mathrm{\Phi }}_0\right|=0.228\pm 0.022$, in agreement with 4-fold LL degeneracy. Extension of the best-fit slope to the electron side is shown with a gray dashed line.