Quantum Hall effect and general narrow-wire circuits

A theoretical study is made of charged-particle motion in planar circuits made up of narrow wires, in the presence of a magnetic field of arbitrary strength, in the ballistic approximation. The basic element is a four-terminal junction of narrow wires, and its detailed properties are calculated. Its scattering probabilities strongly reflect the influence of quantum effects of the junction, including subband thresholds and virtual, resonant states, and the Hall resistance calculated from them may depart considerably from the classic wide-wire result. Physical features are related to the emergence of pinned Landau levels as the field strength increases. By suitably modifying the four-terminal method, it is extended to include elbow (‘‘ssL’’) and tee (‘‘ssT’’) junctions. The results are then used to construct linear six- and eight-terminal junctions, whose resistive properties are discussed. The Hall resistance is predicted to depend on which arms are used and on the stub spacing. The application of the method to the general linear series of junctions is then outlined. The four-terminal results are also applied to a square eight-terminal junction, to show the presence and the consequences of the Aharonov-Bohm effect in circuits with closed loops.