Resonant reflection and transmission in a conducting channel with a single impurity

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Abstract
We consider a narrow conducting channel in a two-dimensional electron gas with a single impurity in the bottleneck part of the channel. In contrast with previous works no specific assumptions concerning the shape of the channel and the impurity potential are made. It is shown that any attractive impurity potential generates sharp Breit-Wigner-type resonances—dips and peaks—in the graph of conductance versus Fermi energy. These correspond to resonant reflection and transmission due to quasibound states in the impurity potential. A simple formula for the conductance in terms of resonance energies and partial widths is found. The widths are directly related to the impurity and channel potentials. It is demonstrated that in the case of a uniform conducting channel any attractive impurity, no matter how weak, always produces zero in the conductance at a certain Fermi energy.