Marginally Self-Averaging One-Dimensional Localization in Bilayer Graphene

The combination of a field-tunable band gap, topological edge states, and valleys in the band structure makes insulating bilayer graphene a unique localized system, where the scaling laws of dimensionless conductance $g$ remain largely unexplored. Here we show that the relative fluctuations in $\ln g$ with the varying chemical potential, in strongly insulating bilayer graphene (BLG), decay nearly logarithmically for a channel length up to $L/\xi \approx 20$, where $\xi$ is the localization length. This “marginal” self-averaging, and the corresponding dependence of $⟨\ln g⟩$ on $L$, suggests that transport in strongly gapped BLG occurs along strictly one-dimensional channels, where $\xi \approx 0.5\pm 0.1\mu \mathrm{m}$ was found to be much longer than that expected from the bulk band gap. Our experiment reveals a nontrivial localization mechanism in gapped BLG, governed by transport along robust edge modes.

Figure 1

Device characteristics and conductance fluctuations. (a) Device schematic. (b) Transfer characteristics ($g-V_{\mathrm{tg}}$) for a few $V_{\mathrm{bg}}$ at 275 mK. (c) Mean conductance $⟨g_{\mathrm{CNP}}⟩$ at the charge neutrality point (CNP) as a function of $D$ for several devices with different channel lengths. Inset shows extrapolated values of $⟨g_{\mathrm{CNP}}⟩$ to $D=0\mathrm{V}/\mathrm{nm}$. (d) Carrier density ($n$) dependence of $⟨g⟩$ for $|D|=0.75\mathrm{V}/\mathrm{nm}$. (e) Conductance fluctuations as a function of small variations in $n$ induced by the top gate at three different points marked in (d). (f) Variance of conductance fluctuations $⟨(\mathrm{\Delta }g{)}^2⟩$ as a function of $n$ for several $D$.

Figure 2

Conductance distribution. Probability distribution function of conductance [$P(g_{\mathrm{CNP}})$ and $P(\ln g_{\mathrm{CNP}})$] around the CNP for different $L$ and $D$, with a fixed length $L=3.68\mu \mathrm{m}$ at (a) $D=0\mathrm{V}/\mathrm{nm}$ and (b) $|D|=0.75\mathrm{V}/\mathrm{nm}$; (c),(d) for a fixed $|D|=0.75\mathrm{V}/\mathrm{nm}$ at (c) $L=0.67\mu \mathrm{m}$ and (d) $L=10.48\mu \mathrm{m}$. Insets in (b)–(d) show the corresponding probability distributions of $\ln g_{\mathrm{CNP}}$. The dashed and solid lines represent normal and log-normal distributions, respectively. (e) Average logarithm of conductance around the CNP ($⟨\ln g_{\mathrm{CNP}}⟩$) as a function of $L$ for $|D|=0.75\mathrm{V}/\mathrm{nm}$. The expected variations of $⟨\ln g_{\mathrm{CNP}}⟩$ from SKL [20] (black solid line), RR [21] (red solid line), and Anderson model [17] (green dashed line) are also indicated. The error bars indicate the standard deviation of $⟨\ln g_{\mathrm{CNP}}⟩$ over ensemble variations. (f) The variance of $\ln g_{\mathrm{CNP}}$, $\mathrm{var}(\ln g_{\mathrm{CNP}})$, as a function of channel length $L$ for $|D|=0.75\mathrm{V}/\mathrm{nm}$. The $L$ dependence of variance expected from SKL, RR, and Anderson model, with $\xi \approx 0.5\mu \mathrm{m}$, are shown. The error bars represent the uncertainty in log-normal fits.

Figure 3

Self-averaging of $\ln g_{\mathrm{CNP}}$ in gapped bilayer graphene. (a) Relative fluctuations $R_{\ln g}=\mathrm{var}(\ln g_{\mathrm{CNP}})/⟨\ln g_{\mathrm{CNP}}{⟩}^2$ as a function of $L$ for high $|D|$ ($>0.5\mathrm{V}/\mathrm{nm}$). The solid lines represent a logarithmic decay $R_{\ln g}\propto {[\ln (2L/\xi )]}^{-2}$ expected from the SKL and RR models with $\xi \approx 0.5\mu \mathrm{m}$. Inset shows $R_{\ln g}$ versus $L$ for $|D|=0.3$, 0.45, and $0.5\mathrm{V}/\mathrm{nm}$. Here, strong self-averaging is indicated by the decay in $R_{\ln g}$ as $R_{\ln g}\propto L^{-2}$ (solid line). The error bars have been computed as the net error from uncorrelated relative errors in $⟨\ln g_{\mathrm{CNP}}⟩$ and $\mathrm{var}(\ln g_{\mathrm{CNP}})$. (b) Schematic depicting hopping transport along disordered 1D edge of the bilayer graphene with gapped bulk. (c) $⟨g_{\mathrm{CNP}}⟩$ as a function of $T^{-0.5}$ for three devices at a fixed $|D|=0.75\mathrm{V}/\mathrm{nm}$, where the fits $⟨g⟩\sim \exp -(T_0/T{)}^{1/2}$ indicate the validity of 1D hopping transport.