Scaling of the quantum Hall plateau-plateau transition in graphene

We show that the width of the longitudinal magnetoconductivity peaks in graphene related to the $N=1$ Landau level displays a power-law type temperature dependence, $\Delta \nu \propto T^{\kappa }$, with $\kappa =0.37\pm 0.05$. Similarly, the derivative of the Hall conductivity at the plateau transition, $\left(d{\sigma }_{xy}/d\nu \right)$, scales as $T^{-\kappa }$ with $\kappa =0.41\pm 0.04$ for both the first and second Landau levels of electrons and holes. These results confirm the universality of a critical quantum Hall scaling in the higher Landau levels of graphene. In the zeroth Landau level, however, $\Delta \nu$ and $d{\sigma }_{xy}/d\nu$ are essentially temperature independent, pointing toward a different type of scaling that is possibly governed by a temperature independent intrinsic length.

Figure 1

(Color online) Longitudinal conductivity ${\sigma }_{xx}$ and Hall conductivity ${\sigma }_{xy}$ as a function of concentration $n$ (bottom axis) and voltage $V$ (top axis) at $B=20\text{T}$ for different temperatures. The conductivities were calculated by tensor inversion from the measured symmetrized resistivities for both magnetic field orientations. The inset shows the Landau level spectrum in graphene with the dashed regions indicating the localized states between two Landau levels.

Figure 2

(Color online) (a) $\Delta \nu$, distance between two extrema in the derivative $d{\sigma }_{xx}/d\nu$ of the first electron Landau level (solid symbols) and the first hole (open symbols) Landau level for different magnetic fields. (b) The derivative ${\left(d{\sigma }_{xy}/d\nu \right)}^{\text{max}}$ as a function of temperature for the same levels. (c) The derivative ${\left(d{\sigma }_{xy}/d\nu \right)}^{\text{max}}$ as a function of temperature for the second electron and hole level.

Figure 3

(Color online) Scaling behavior in the tails of the first electron Landau level for various magnetic fields as a function of relative filling factor $\left|{\nu }_c-\nu \right|/4$ with $\nu =nh/eB$. The factor 1/4 in the $x$ axis accounts for the fourfold Landau level degeneracy. The localization length on the right axis was calculated by Eq. (4) with $C=1$.

Figure 4

(Color online) (a) $\Delta \nu$, distance between two extrema in the derivative $d{\sigma }_{xx}/d\nu$ as a function of temperature for the zeroth Landau level. (b) The derivative ${\left(d{\sigma }_{xy}/d\nu \right)}^{\text{max}}$ as a function of temperature for the zeroth Landau level.