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 the quantum Hall effect and much attention has been paid to this topic for more than two decades. However, less effort has been made to model the current transport itself. Network models are frequently used in this context and an answer should be given as to whether these are also suitable for modelling the lateral distribution of experimentally excited currents 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 used instead for expressing this kind of modelling. In preceding papers a Landauer–Büttiker type representation of bulk current transport has been successfully used for the numerical simulation of the magneto-transport of two-dimensional electron systems in the high magnetic field regime. This approach allows us to build up a network model, which describes correctly the effect of non-equilibrium currents injected via metallic contacts as in real experiments. In this context we suggest a network model, which serves as a circuit type representation of magneto-transport. It is demonstrated that it is in full agreement with a treatment of bulk current transport as a quantum tunnelling process between magnetic bound states, which exist in the high magnetic field regime. Additionally, we find a striking correspondence between our network representation and the bulk current picture in terms of mixed phases mapped on a chequerboard: at half filled Landau level (LL) coupled droplets of a quantum Hall (QH) liquid phase and coupled droplets of an insulator phase 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 completes the QH plateau transition to the next higher QH plateau, while adding a single electron to such a droplet at half filling completes the QH plateau transition to the previous lower QH plateau. As a consequence, the sharpness of the QH plateau transitions on the magnetic field axis depends on the typical size of the droplets, which can be understood as a measure of the disorder in the sample. We furthermore demonstrate that our model can also be used as a theoretical concept for discretization of a long range random potential on a regular grid. We also find a correspondence between our model and the so called localization picture of the QH plateau transitions. Alternative approaches like the Chalker–Coddington (CC) network as the coherent counterpart and a resistor network as the classical counterpart are discussed in comparison with our network approach. On the one hand 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 the CC 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 an adapted lattice model by Ando (1...

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