Quasi-bound states and resonances in heterostructures

The general theory for resonances of coherently propagating electron states in nanostructures is presented. Resonances in transmission as a function of electron energy, concomitant peaks in the density of states within the structure, and slowly escaping quasi-levels, are fundamentally related. Their connection depends on features of the continuation to complex energy of the energy dependence of system variables. Diode (parallel-plane) structures are first analyzed, with emphasis on fundamental formulas. Relations between the dwell times and phase (Wigner-Eisenbud) propagation delays are obtained. The relationship between transfer matrix, S-matrix, and a third computation-adapted matrix is shown. The general theory for multi-channel nanostructures is then developed in terms of the system S-matrix, generalizing diode results. The critical variances are the eigenvalues, and accompanying channel eigenvectors, of the S-matrix, as a function of energy. It is shown that the density of states within the structure when integrated over the energy range of a resonance equals just one electron, appropriate to a quasi-level.