Marginally self-averaging one-dimensional localization in bilayer graphene

In a disordered system, a macroscopic variable X is spatially ergodic, or self-averaging, when its relative fluctuations R_X =〈(ΔX)²〉/〈X²〉 → 0 as L → ∞, where L is a linear dimension and 〈...〉 represents averaging over different realizations of disorder. For strongly localized noninteracting carriers the electrical conductance g does not self-average, but its logarithm ln g does, in a manner that is determined by the dimensionality and the scaling properties of Anderson localization for L » ξ, the localization length. In this work, we show that in the strongly insulating bilayer graphene (BLG), the relative fluctuations in ln g with chemical potential decay nearly logarithmically for channel length up to L ≈ 20ξ. This 'marginal' self-averaging along with its associated dependence of 〈ln g〉 on L, suggest that transport in strongly gapped BLG takes place via strictly one-dimensional channels, with the ξ ≈ 0.5 ± 0.1 μm much longer than that expected from the bulk bandgap. Our experiment not only reveals a nontrivial localization mechanism in BLG based on robust edge modes, but is also the first demonstration of the marginal self-averaging nature of strong localization in one dimension.