Percolation transitions in bilayer graphene encapsulated by hexagonal boron nitride

We studied the plateau-plateau transitions that characterize the electrical transport in the quantum Hall regime in a high mobility bilayer graphene flake encapsulated by hexagonal boron nitride at magnetic fields up to 9 T and temperatures above 300 mK. We measured independently the exponent $\kappa$ of the temperature-induced transition broadening, the critical exponent $\gamma$ of the localization length, and the exponent $p$ ruling the temperature scaling of the coherence length, finding consistency with the relation $\gamma =p/2\kappa$. The observed value of $\kappa =0.30(0.28,0.32)$ deviates from that of the quantum Hall critical point. The obtained $\gamma =1.25(0.96,1.54)$ questions the validity of a pure Anderson transition, and reveals percolation as the underlying driving mechanism.

Figure 1

Longitudinal resistivity ${\rho }_{xx}$ vs back gate voltage $V_g$ at $T=5$ K in the absence of magnetic field. The CNP is found to be at 0.4 V. The right axis (red) shows ${\rho }^{-1}$ as a function of the carrier density $n$ (upper abscissa axis) at $T\sim 0.3$ K. Dashed lines highlight the linear dependence of ${\rho }^{-1}$ on $n$ away from the CNP.

Figure 2

(a)–(f) show the isotherms at a few selected temperatures of the Hall resistivity ${\rho }_{xy}$ as a function of the magnetic field at different densities. (a)–(c) show the QHE in the electronlike regime at densities $n=14.6\times {10}^{11} {\mathrm{cm}}^{-2},n=10.2\times {10}^{11} {\mathrm{cm}}^{-2}$, and $n=6.07\times {10}^{11} {\mathrm{cm}}^{-2}$, respectively. (d)–(f) show the QHE in the holelike regime at densities $n=(-)4.74\times {10}^{11} {\mathrm{cm}}^{-2},n=(-)6.84\times {10}^{11} {\mathrm{cm}}^{-2}$, and $n=(-)9.03\times {10}^{11} {\mathrm{cm}}^{-2}$, respectively. The insets show the isotherms of the longitudinal resistivity ${\rho }_{xx}$ as functions of the magnetic field. For clarity, gray horizontal lines indicating the values $h/j{e}^2$ are included. (g) shows ${(d{\rho }_{xy}/dB)}_{\text{max}}$ as a function of the temperature in log-log scale for the PPTs displayed in (a)–(f). The estimated experimental error bars are smaller than the size of the symbols. Dashed lines correspond to linear fits of the data according to ${(d{\rho }_{xy}/dB)}_{\text{max}}\propto T^{-\kappa }$.

Figure 3

Coherence length $L_{\varphi }$ as a function of temperature obtained by WL measurements at $n=2.7\times {10}^{11} {\mathrm{cm}}^{-2}$. The best fit of the data to $L_{\varphi }\propto T^{-p/2}$ (black line) yields $p=0.90\pm 0.02$, considering data for $T\ge 3$ K. The inset shows the WL peak as a function of the magnetic field (at a set of selected different temperatures), from whose fits (red solid lines) the coherence length at each temperature is obtained [28].

Figure 4

Log-log plot of the characteristic temperature $T_0$ of the conductivity in the tails of the Landau levels as a function of the relative filling factor for the analyzed PPTs. The factor $1/4$ in the abscissa axis is due to the fourfold degeneracy of the Landau levels. Dashed lines correspond to linear fits. The legend indicates the measured value of ${\nu }_c$ for the different transitions considered.