Circuit type simulations of magneto-transport in the quantum Hall effect regime

Abstract

Localization in the bulk is one of the most important ingredients for the theory of thequantum Hall effect and much attention has been paid to this topic for more than twodecades. However, less effort has been made to model the current transport itself. Networkmodels are frequently used in this context and an answer should be given as to whetherthese are also suitable for modelling the lateral distribution of experimentally excitedcurrents and voltages in the quantum Hall effect (QHE) regime. The term ‘network model’is of more general meaning and therefore the term ‘circuit type simulations’ should be usedinstead for expressing this kind of modelling. In preceding papers a Landauer–Büttikertype representation of bulk current transport has been successfully used for thenumerical simulation of the magneto-transport of two-dimensional electron systems inthe high magnetic field regime. This approach allows us to build up a networkmodel, which describes correctly the effect of non-equilibrium currents injected viametallic contacts as in real experiments. In this context we suggest a networkmodel, which serves as a circuit type representation of magneto-transport. It isdemonstrated that it is in full agreement with a treatment of bulk current transport asa quantum tunnelling process between magnetic bound states, which exist inthe high magnetic field regime. Additionally, we find a striking correspondencebetween our network representation and the bulk current picture in terms ofmixed phases mapped on a chequerboard: at half filled Landau level (LL) coupleddroplets of a quantum Hall (QH) liquid phase and coupled droplets of an insulatorphase exist at the same time, with each of them occupying half of the bulk area.Removing a single electron from such a QH liquid droplet at half filling completesthe QH plateau transition to the next higher QH plateau, while adding a singleelectron to such a droplet at half filling completes the QH plateau transitionto the previous lower QH plateau. As a consequence, the sharpness of the QHplateau transitions on the magnetic field axis depends on the typical size of thedroplets, which can be understood as a measure of the disorder in the sample. Wefurthermore demonstrate that our model can also be used as a theoretical concept fordiscretization of a long range random potential on a regular grid. We also find acorrespondence between our model and the so called localization picture of theQH plateau transitions. Alternative approaches like the Chalker–Coddington(CC) network as the coherent counterpart and a resistor network as the classicalcounterpart are discussed in comparison with our network approach. On the onehand the CC model was originally developed to treat the localization problem;on the other hand it also assigns currents to channels, and in this context theCC model will be discussed in the context of circuit type simulations as well.Finally, we compare our simulation results to results obtained on the basis of anadapted lattice model by Ando (1998 Surf. Sci. 361/362 270) for the QHE regimeand find our results in full agreement for the case of strong inelastic scattering.