Quantum-Hall Activation Gaps in Graphene

We have measured the quantum-Hall activation gaps in graphene at filling factors $\nu =2$ and $\nu =6$ for magnetic fields up to 32 T and temperatures from 4 to 300 K. The $\nu =6$ gap can be described by thermal excitation to broadened Landau levels with a width of 400 K. In contrast, the gap measured at $\nu =2$ is strongly temperature and field dependent and approaches the expected value for sharp Landau levels for fields $B>20\mathrm{T}$ and temperatures $T>100\mathrm{K}$. We explain this surprising behavior by a narrowing of the lowest Landau level.

Figure 1

(a) Schematic Landau-level structure in graphene. The white areas are extended states in the center of the Landau levels, with the localization radius of order of the sample size; the gray areas represent localized states in between. The arrows indicate the position of the chemical potential for the corresponding filling factor. (b) Hall resistivity ${\rho }_{xy}$ for holes in a single-layer graphene device as a function of the magnetic field. The traces were recorded at 4.2 K (blue, solid line) and RT (red, dashed line) at a fixed gate voltage corresponding to a carrier concentration $n=-1.0\times {10}^{12}{\mathrm{cm}}^{-2}$. (c) Scanning electron micrograph of the graphene multiterminal device. Hall conductivity ${\sigma }_{xy}$ (d) and conductivity ${\sigma }_{xx}$ (e) at 4.2 K (blue, solid line) and at RT (red, dashed line) as a function of the gate voltage at $B=30\mathrm{T}$.

Figure 2

Resistivity ${\rho }_{xx}$ as a function of carrier concentration at $B=15\mathrm{T}$ for different temperatures. The arrows mark the $\nu =\pm 2$ and the $\nu =6$ minima.

Figure 3

Arrhenius plots of ${\rho }_{xx}$ in the high temperature range for the gaps at $\nu =2$ (a) and at $\nu =6$ (b) for different magnetic fields. (c) Energy gaps $2{\Delta }_a$ between two Landau levels as a function of magnetic field for $\nu =+2$ (full red triangles), $\nu =-2$ (open black circles), and $\nu =6$ (full blue squares) as deduced from the Arrhenius plots. The dashed (red) and dotted (blue) lines are the theoretically expected energy gaps for sharp Landau levels. The inset shows schematically the density of states for a sharp zeroth Landau level and broadened higher Landau levels for electrons and holes at 30 T. The form and width of the higher Landau levels were extracted from experimental data. Extended states are represented by the white areas, localized states by the dashed areas.