Electron scattering in quantum waveguides with sources and absorbers. I. Theoretical formalism

We develop a novel method based on sources and absorbers to examine quantum scattering in finite, nanoscale systems. We show that the Cauchy (mixed) boundary conditions (BCs) are required to put the scattering theory into an action integral formulation. These complex BCs are reduced to simpler Dirichlet BCs by introducing totally absorbing “stealth regions.” Material properties of these enclosing regions are optimized to give decaying solutions so that the scattering amplitudes vanish at the finite boundaries. With the active scattering region now surrounded by absorbers, we construct an “electron antenna” to provide incident waves. The method retains all the physical aspects of the conventional theory while providing new insights into “near-field” scattering effects. The action integral is discretized and evaluated to derive the local wavefunction everywhere. In two-dimensional quantum waveguides, we obtain the scattered wavefunction for geometrically complex scattering centers, showing the flexibility of our method. The modal decomposition of reflected and transmitted waves allows us to obtain transmission coefficients for both propagating and evanescent modes. Using group theory, we develop selection rules for the scattered modes depending on the symmetry of the potential. Our method outperforms the limitations of traditional perturbative estimates, transfer-matrix, S-matrix discretizations, and other schemes to provide a complete nonasymptotic variational description for electron transport in quantum waveguides.