Multifractal Conductance Fluctuations in High-Mobility Graphene in the Integer Quantum Hall Regime

We present the first experimental evidence for the multifractality of a transport property at a topological phase transition. In particular, we show that conductance fluctuations display multifractality at the integer quantum Hall plateau-to-plateau transitions in high-mobility mesoscopic graphene devices. The multifractality gets rapidly suppressed as the chemical potential moves away from these critical points. Our combination of experimental study and multifractal analysis provides a novel method for probing the criticality of wave functions at phase transitions in mesoscopic systems, and quantum criticality in several condensed-matter systems.

Figure 1

(a) A schematic diagram illustrating the dependence of the DOS of 2DEG on energy in the IQH regime. The extended states (brown regions) at the centers of the disorder-broadened Landau levels are separated by the localized regions (light blue). The quantum phase transitions between the localized and extended states occur at the mobility edge around the critical energies $E_{C1}$ and $E_{C2}$. (b) A false-color SEM image of device 1DC8. (c) Plots showing the dependence of $R$ on $V_G$ at different temperatures for 1DC8. The measurements were carried out at $B=0\mathrm{T}$. The left inset shows an enlarged plot near the Dirac point [see Supplemental Material [7], S2]. The inset at the right indicates the contacts used in the measurement. (d) Plots of $G_{XY}$ versus the filling factor $\nu$ measured at different temperatures and magnetic field $B=16\mathrm{T}$ for 1DC8. The arrows mark the positions of IQH critical points at which the localization-delocalization transitions occur. Inset: the contacts used in the measurement. The arrow indicates the chirality of electron transport at $B=16\mathrm{T}$. The corresponding number density $n$ is given on the top axis of (c) and (d).

Figure 2

Plots of $G_{XY}$ versus $\nu$ measured during $\nu =1\leftrightarrow \nu =2$ IQH transition at (a) a fixed value of the magnetic field $B=16\mathrm{T}$ and (b) a fixed number density $n=4.75\times {10}^{11}{\mathrm{cm}}^{-2}$ for the device 1DC8. The corresponding values of $n$ are shown on the top axes. (c) Plots of two different traces of $G_{XY}$ versus $B$, measured between $\nu =1.64$ and $\nu =1.54$, showing the reproducibility of the mesoscopic conductance fluctuations. The datasets have been shifted vertically for clarity. (d) Plots of segments of data of $G_{XY}^{\prime }$ versus $B$, for the $\nu =1\leftrightarrow \nu =2$ transition, measured at different temperatures $T$ and at a fixed carrier density $n=4.75\times {10}^{11}{\mathrm{cm}}^{-2}$. The data have been vertically offset for clarity. The data presented in (c) are from a different cooldown cycle than those in (a), (b), and (d).

Figure 3

(a) Plots of $\log [F_q(s)]$ versus $\log [s]$ [see Eq. (2)], for $q=-4$ (green circles) and $q=4$ (blue circles) for a typical data segment of $G_{XY}^{\prime }$. The thick lines are the linear fits to the data points. (b) Plot of $h(q)$ versus $q$ for the data segment shown in (a). (c) Plot of the singularity spectrum $f(\alpha )$ versus $\alpha$ obtained from (b). The maximum value of this spectrum is $f_{\max }(\alpha )=1$ located at ${\alpha }_0\simeq 2.21$ (marked by an arrow). The data were obtained at $\nu =1.47$ and 20 mK for the device 1DC8.

Figure 4

(a) Plot of the spectral width $\mathrm{\Delta }\alpha$ versus $\nu -{\nu }_C$ for the devices 1DC8 and B15D4. The vertical line marks the $\nu =1\leftrightarrow \nu =2$ critical point. The gray-shaded curve is a guide to the eye. The error bars for the $\nu -{\nu }_C$ axis are calculated from the intrinsic impurity level in the device; the error bars in the $\mathrm{\Delta }\alpha$ axis are derived from the error in calculating the slopes of the plots of $\log [F_q(s)]$ versus $\log [s]$ (Supplemental Material, S5). (b) Plot of $\mathrm{\Delta }\alpha$ versus $T$ for the device 1DC8, measured at $\nu =1.6$. The red-shaded curve is a guide to the eye.