On modeling quantum point contacts in quantum Hall systems

Quantum point contacts (QPC) are a key instrument in investigating the physics of edge excitations in the quantum Hall effect. However, at not-so-high bias voltage values, the predictions of the conventional point QPC model often deviate from the experimental data both in the integer and (more prominently) in the fractional quantum Hall regime. One of the possible explanations for such behaviors is the dependence of the tunneling between the edges on energy, an effect not present in the conventional model. Here we introduce two models that take QPC spatial extension into account: wide-QPC model that accounts for the distance along which the edges are in contact; long-QPC model accounts for the fact that the tunneling amplitude originates from a finite bulk gap and a finite distance between the two edges. We investigate the predictions of these two models in the integer quantum Hall regime for the energy dependence of the tunneling amplitude. We find that these two models predict opposite dependences: the amplitude decreasing or increasing away from the Fermi level. We thus elucidate the effect of the QPC geometry on the energy dependence of the tunneling amplitude and investigate its implications for transport observables.

💡 Research Summary

The paper addresses a long‑standing discrepancy between the conventional point‑contact model of quantum point contacts (QPCs) in quantum Hall (QH) systems and experimental observations at moderate bias voltages. In the standard model the tunnelling amplitude is taken to be a single, energy‑independent parameter located at a single spatial point. This works well for very low bias (eV ≪ kBT) but fails when the bias exceeds the thermal energy yet remains well below the bulk gap, a regime where many experiments on both integer and especially fractional QH states report deviations in the current‑voltage characteristics and in the low‑frequency shot noise.

To explain these deviations the authors propose two phenomenological extensions that explicitly incorporate the finite spatial extent of a real QPC. Both models are formulated for the integer QH regime (ν = 1) and are analyzed within the Landauer‑Büttiker scattering framework, where the central object is the energy‑dependent transmission probability T(E).

1. Wide‑QPC model
In this model the two counter‑propagating chiral edge modes overlap over a finite width W (of order a few magnetic lengths) along the direction of propagation. The tunnelling Hamiltonian contains a uniform amplitude ζ acting over a length L. Solving the scattering problem yields

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