Nonuniversal behavior of finite quantum Hall systems as a result of weak macroscopic inhomogeneities

Long-range nontopological edge currents in charge-neutral graphene

Van der Waals heterostructures display numerous unique electronic properties. Nonlocal measurements, wherein a voltage is measured at contacts placed far away from the expected classical flow of charge carriers, have been widely used in the search for novel transport mechanisms, including dissipationless spin and valley transport^1-9, topological charge-neutral currents^10-12, hydrodynamic flows¹³ and helical edge modes^14-16. Monolayer^1-5,10,15-19, bilayer^9,11,14,20 and few-layer²¹ graphene, transition-metal dichalcogenides^6,7 and moiré superlattices^8,10,12 have been found to display pronounced nonlocal effects. However, the origin of these effects is hotly debated^{3,11,17,22-24}. Graphene, in particular, exhibits giant nonlocality at charge neutrality^1,15-19, a striking behaviour that has attracted competing explanations. Using a superconducting quantum interference device on a tip (SQUID-on-tip) for nanoscale thermal and scanning gate imaging²⁵, here we demonstrate that the commonly occurring charge accumulation at graphene edges^23,26-31 leads to giant nonlocality, producing narrow conductive channels that support long-range currents. Unexpectedly, although the edge conductance has little effect on the current flow in zero magnetic field, it leads to field-induced decoupling between edge and bulk transport at moderate fields. The resulting giant nonlocality at charge neutrality and away from it produces exotic flow patterns that are sensitive to edge disorder, in which charges can flow against the global electric field. The observed one-dimensional edge transport is generic and nontopological and is expected to support nonlocal transport in many electronic systems, offering insight into the numerous controversies and linking them to long-range guided electronic states at system edges.

Analyzing Longitudinal Magnetoresistance Asymmetry to Quantify Doping Gradients: Generalization of the van der Pauw Method

A longitudinal magnetoresistance asymmetry (LMA) between a positive and negative magnetic field is known to occur in both the extreme quantum limit and the classical Drude limit in samples with a nonuniform doping density. By analyzing the current stream function in van der Pauw measurement geometry, it is shown that the electron density gradient can be quantitatively deduced from this LMA in the Drude regime. Results agree with gradients interpolated from local densities calibrated across an entire wafer, establishing a generalization of the van der Pauw method to quantify density gradients.

Non-universality of scaling exponents in quantum Hall transitions

We have investigated experimentally the scaling behaviour of quantum Hall transitions in GaAs/AlGaAs heterostructures of a range of mobility, carrier concentration, and spacer layer width. All three critical scaling exponents γ, κ and p were determined independently for each sample. We measure the localization length exponent to be γ ≈ 2.3, in good agreement with expected predictions from scaling theory, but κ and p are found to possess non-universal values. Results obtained for κ range from κ = 0.16 ± 0.02 to κ = 0.67 ± 0.02, and are found to be Landau level (LL) dependent, whereas p is found to decrease with increasing sample mobility. Our results demonstrate the existence of two transport regimes in the LL conductivity peak; universality is found within the quantum coherent transport regime present in the tails of the conductivity peak, but is absent within the classical transport regime found close to the critical point at the centre of the conductivity peak. We explain these results using a percolation model and show that the critical scaling exponent depends on certain important length scales that correspond to the microscopic description of electron transport in the bulk of a two-dimensional electron system.

Localization of fractionally charged quasi-particles

An outstanding question pertaining to the microscopic properties of the fractional quantum Hall effect is understanding the nature of the particles that participate in the localization but that do not contribute to electronic transport. By using a scanning single electron transistor, we imaged the individual localized states in the fractional quantum Hall regime and determined the charge of the localizing particles. Highlighting the symmetry between filling factors 1/3 and 2/3, our measurements show that quasi-particles with fractional charge e* = e/3 localize in space to submicrometer dimensions, where e is the electron charge.

The microscopic nature of localization in the quantum Hall effect

The quantum Hall effect arises from the interplay between localized and extended states that form when electrons, confined to two dimensions, are subject to a perpendicular magnetic field. The effect involves exact quantization of all the electronic transport properties owing to particle localization. In the conventional theory of the quantum Hall effect, strong-field localization is associated with a single-particle drift motion of electrons along contours of constant disorder potential. Transport experiments that probe the extended states in the transition regions between quantum Hall phases have been used to test both the theory and its implications for quantum Hall phase transitions. Although several experiments on highly disordered samples have affirmed the validity of the single-particle picture, other experiments and some recent theories have found deviations from the predicted universal behaviour. Here we use a scanning single-electron transistor to probe the individual localized states, which we find to be strikingly different from the predictions of single-particle theory. The states are mainly determined by Coulomb interactions, and appear only when quantization of kinetic energy limits the screening ability of electrons. We conclude that the quantum Hall effect has a greater diversity of regimes and phase transitions than predicted by the single-particle framework. Our experiments suggest a unified picture of localization in which the single-particle model is valid only in the limit of strong disorder.

Dynamical scaling of the quantum Hall plateau transition

Using different experimental techniques, we examine the dynamical scaling of the quantum Hall plateau transition in a frequency range f=0.1-55 GHz. We present a scheme that allows for a simultaneous scaling analysis of these experiments and all other data in literature. We observe a universal scaling function with an exponent kappa=0.5+/-0.1, yielding a dynamical exponent z=0.9+/-0.2.

Universal scaling functions for bond percolation on planar-random and square lattices with multiple percolating clusters

Percolation models with multiple percolating clusters have attracted much attention in recent years. Here we use Monte Carlo simulations to study bond percolation on L1xL2 planar random lattices, duals of random lattices, and square lattices with free and periodic boundary conditions, in vertical and horizontal directions, respectively, and with various aspect ratios L(1)/L(2). We calculate the probability for the appearance of n percolating clusters, W(n); the percolating probabilities P; the average fraction of lattice bonds (sites) in the percolating clusters, <c(b)>(n) (<c(s)>(n)), and the probability distribution function for the fraction c of lattice bonds (sites), in percolating clusters of subgraphs with n percolating clusters, f(n)(c(b)) [f(n)(c(s))]. Using a small number of nonuniversal metric factors, we find that W(n), P, <c(b)>(n) (<c(s)>(n)), and f(n)(c(b)) [f(n)(c(s))] for random lattices, duals of random lattices, and square lattices have the same universal finite-size scaling functions. We also find that nonuniversal metric factors are independent of boundary conditions and aspect ratios.

High frequency conductivity in the quantum hall regime

We have measured the complex conductivity sigma(xx) of a two-dimensional electron system in the quantum Hall regime up to frequencies of 6 GHz at electron temperatures below 100 mK. Using both its imaginary and real part we show that sigma(xx) can be scaled to a single function for different frequencies and several transitions between plateaus in the quantum Hall effect. Additionally, the conductivity in the variable-range hopping regime is used for a direct evaluation of the localization length xi. Even for large filling factor distances deltanu from the critical point we find xi approximately equals deltanu(-gamma) with a scaling exponent gamma = 2.3.