Unveiling quantum Hall transport by Efros-Shklovskii to Mott variable-range hopping transition in graphene

The quantum Hall effect is universal and expected to occur in all two-dimensional electron systems in perpendicular high-magnetic field. We revisit quantum localization thanks to the high-energy scale of the quantum Hall effect in graphene, where the electron dynamics obey the Dirac equation. We solve a long debated question on the nature of electron transport in the transition region between Hall resistance plateaus. Is it of metallic or of variable-range hopping type as proposed, respectively, by Pruisken and Polyakov-Shklovskii? To the best of our knowledge, no experiment was able to discriminate between these models. Here, measurements of the conductance peak width scaling exponents with both temperature and current and determination of the localization length validates the variable-range hopping scenario. This shows that the usual assumption of a metallic behavior of the two-dimensional electron gas (2DEG) between Hall resistance plateaus is unnecessary for macroscopic samples.

Figure 1

Summary of the expected exponents for the FWHM: $\Delta \nu ={(T/T_1)}^{\mu }$ and $\Delta \nu ={(I/I_1)}^{\kappa }$; $T_1$ and $I_1$ are constant. $p$ is a nonuniversal parameter which was found to be equal to 2 in many previous experiments.

Figure 2

Color plot of the longitudinal resistivity versus filling factor and with temperature steps around 5 K at $B=16.5$ T for sample S3.

Figure 3

Color plot of the longitudinal resistivity versus filling factor and bias current ranging from 10 nA to 100 $\mu$A with variable step ranging from 10 nA to 10 $\mu$A on sample S3 at $B=16.5$ T and 4.2 K.

Figure 4

Plots of $T{\rho }_{xx}$ versus $1/\sqrt{T}$ at 14.8 T for various filling factors near $\nu =-6$ and $\xi$ above and below 600 nm on sample S3. The red solid and dashed lines are a E-S law's fit of the data while the green solid line uses Mott's law.

Figure 5

Effective temperature $T_{\text{eff}}\left(I\right)$ (see definition in text) versus bias current $I$ on plateau for different filling factors. Inset: Effective temperature for sample S3 at $B=16.5$ T on the plateau $\nu =-2$ away from the center of the plateau (at $\nu =1,3$). The VRH part shows a power 1 slope, whereas the activated part shows a power 0.4 slope.

Figure 6

Activation energies for S3 and S4 at different magnetic fields for $\nu =\pm 2$; $\pm 6;\pm 10$. Squares are for sample S3 and triangles for S4. The solid lines show the values expected from the Dirac equation. Green, blue, and red solid line curves are for $\nu =\pm 2$, $\pm 6$, and $\nu =\pm 10$, respectively.

Figure 7

(a) Top figure: localization length $\xi$ versus filling factor for S3 for different magnetic fields. The horizontal solid line $\xi =600$ nm shows the threshold above which screening of interactions by the gate plays a role. (b) Bottom figure: $T_0^{1/2.3}$ as a function of $\nu -{\nu }_C$ for $\nu =[\pm 2,\pm 6]$ and $\nu =[\pm 2,0]$ at different magnetic fields, the black curve is for S4 while the others are for S3. The solid lines are guides for the eyes. Red (green) circle corresponds to the curve at $\nu =4.6$ (5.61) of Fig. 4.

Figure 8

FWHM of the ${\rho }_{xx}$ peaks for temperature measurements (left) and for bias current measurement (right). Blue, green, and red curves are data for the peaks between $\nu =\pm 10$ and $\pm 6$, $\nu =\pm 6$ and $\pm 2$, and $\nu =-2$ and 2, respectively. Squares are for S3 and triangles are for S4. E-S scenario and Pruisken scenario (with $p=2$) are represented by dashed black lines, the Mott scenario is represented by the solid red line. $T_{*}$ and $I_{*}$ are scaling temperature and current in order that the value of $\text{FWMH}=1$ when $T/T_{*}=1$ and $I/I_{*}=1$.