Experimental Observation of Quantum Hall Effect and Berry's Phase in Graphene

TL;DR

An experimental investigation of magneto-transport in a high-mobility single layer of graphene observes an unusual half-integer quantum Hall effect for both electron and hole carriers in graphene.

Abstract

When electrons are confined in two-dimensional (2D) materials, quantum mechanically enhanced transport phenomena, as exemplified by the quantum Hall effects (QHE), can be observed. Graphene, an isolated single atomic layer of graphite, is an ideal realization of such a 2D system. Here, we report an experimental investigation of magneto transport in a high mobility single layer of graphene. Adjusting the chemical potential using the electric field effect, we observe an unusual half integer QHE for both electron and hole carriers in graphene. Vanishing effective carrier masses is observed at Dirac point in the temperature dependent Shubnikov de Haas oscillations, which probe the 'relativistic' Dirac particle-like dispersion. The relevance of Berry's phase to these experiments is confirmed by the phase shift of magneto-oscillations, related to the exceptional topology of the graphene band structure.

Figures (3)

• Figure 1: Resistance, carrier density, and mobility of graphene measured at 1.7 K at different gate voltages. a, Resistance changes as a function of gate voltage in a graphene device shown in the optical microscope image in the right inset. The position of the resistance peaks varies from device to device, but the peak values are always of the order of$\sim 4 \mathrm{k} \Omega$, suggesting a potential quantum mechanical origin. The left inset shows a schematic diagram for the low energy dispersion relation near the Dirac points in the graphene Brillouin zone. Only two Dirac cones are inequivalent to each other, producing a two-fold valley degeneracy in the band structure. b, Charge carrier density (open circle) and mobility (filled circle) of graphene as a function of gate voltage. The solid line corresponds to the estimated charge induced by the gate voltage, $n_{s}=C_{g} V_{g} / e$, assuming a gate capacitance $C_{g}=115 \mathrm{aF} / \mu \mathrm{m}^{2}$ obtained from geometrical consideration.
• Figure 2: Quantized magnetoresistance and Hall resistance of a graphene device. a, Hall resistance (black) and magnetoresistance (red) measured in the device in Fig. 1 at$T=30 \mathrm{mK}$ and $V_{g}=15 \mathrm{~V}$. The vertical arrows and the numbers on them indicate the values of $B$ and the corresponding filling factor $v$ of the quantum Hall states. The horizontal lines correspond to $h / e^{2} v$ values. The QHE in the electron gas is demonstrated by at least two quantized plateaus in $R_{x y}$ with vanishing $R_{x x}$ in the corresponding magnetic field regime. The inset shows the QHE for a hole gas at $V_{g}=-4 \mathrm{~V}$, measured at 1.6 K . The quantized plateau for filling factor $v=2$ is well-defined and the second and the third plateau with $v=6$ and 10 are also resolved. b, The Hall resistance (black) and magnetoresistance (orange) as a function of gate voltage at fixed magnetic field $B=9 \mathrm{~T}$, measured at 1.6 K . The same convention as in $\mathbf{a}$ is used here. The upper inset shows a detailed view of high filling factor plateaus measured at 30 mK . c, A schematic diagram of the Landau level density of states (DOS) and corresponding quantum Hall conductance ( $\sigma_{x y}$ ) as a function of energy. Note that in the quantum Hall sates, $\sigma_{x y}=-R_{x y}{ }^{-1}$. The LL index $n$ is shown next to the DOS peak. In our experiment, the Fermi energy $E_{F}$ can be adjusted by the gate voltage, and $R_{x y}{ }^{-1}$ changes by an amount of $g_{s} e^{2} / h$ as $E_{F}$ crosses a LL.
• Figure 3: Temperature dependence and gate voltage dependence of the Shubnikov de Haas oscillations in graphene. a, Temperature dependence of the SdH oscillations at$V_{g}=-2.5 \mathrm{~V}$. Each curve represents $R_{x x}(B)$ normalized to $R_{x x}(0)$ at a fixed temperature. The curves are in order of decreasing temperature starting from the top as indicated by the vertical arrow. The corresponding temperatures are listed in the left inset. The left inset represents the SdH oscillation amplitude divided by temperature measured at a fixed magnetic field. The standard SdH fit yields the effective mass. The right inset is a plot of the effective mass obtained at different gate voltages. The broken line is a fit to the single parameter model described in the text, which yields $v_{F}=1.1 \times 10^{6} \mathrm{~m} / \mathrm{s}$, in reasonable agreement with the literature values. $\mathbf{b}$, A fan diagram for SdH oscillations at different gate voltages. The location of $1 / B$ for the $n_{\text{th }}$ minimum (maximum) of $R_{x x}$ counting from $B=B_{\mathrm{F}}$ is plotted against $n$ ( $n+1 / 2$ ). The lines correspond to a linear fit, where the slope (lower inset) indicates $B_{\mathrm{F}}$ and the $n$-axis intercept (upper inset) provides a direct probe of Berry's phase in the magneto-oscillation in graphene.