Scattering Theory in an N-Pole Semiconductor Quantum Device: The Unitarity of the Current S-Matrix and Current Conservation

In a number of previous publications, scattering theory for N-pole semiconductor quantum devices was developed. In the framework of the Landauer-Büttiker formalism, an S-matrix was constructed with the aid of an R-matrix providing a mapping of the in-going waves onto the out-going waves. These waves include propagating waves and evanescent waves, the latter of which decay exponentially in the leads which are connected to the active region of the N-pole device. In order to formulate the current conservation in the N-pole device, it is necessary to define the current S-matrix schematically as S˜=k1/2Sk-1/2, where <i>k</i> contains the information about the k-vectors of the mentioned in- and out-going waves. In this paper, we show how the complete current S-matrix is calculated including the coupling between the propagating and evanescent components and coupling to the bound states in the active device region. One then finds a sub-matrix of S˜ which is unitary and which is restricted to the space of the propagating components. We demonstrate that current conservation is associated with the unitarity just of this sub-matrix.