Finite-difference approach to edge-state transport in quantum wires and multiterminal devices

Abstract

We investigate multimode electronic transport in a quantum wire in the presence of a uniform magnetic field. Using a finite-difference scheme for the evaluation of the free-electron Schr\"odinger equation, we obtain Harper's equation, which is transformed into an eigenvalue problem. The resulting scattering states of the wire are used to describe the magnetoconductance for free electrons as well as for a periodic lattice potential. In the latter case we obtain a conductance pattern that is related to the Hofstadter butterfly. The calculated current distribution in a disordered quantum Hall device confirms the edge-state picture of the transport in the quantum Hall regime.