R-matrix Formalism for Electron Scattering in Two Dimensions with Applications to Nanostructures with Quantum Dots

We investigate the scattering phenomena in two dimensions produced by a general finite-range nonseparable potential. This situation can appear either in a Cartesian geometry or in a heterostructure with cylindrical symmetry. Increasing the dimensionality of the scattering problem, new processes as the scattering between conducting channels and the scattering from conducting to evanescent channels are allowed. For certain values of the energy called resonance energy, the transmission through the scattering region changes dramatically in comparison with an one-dimensional problem. If the potential has an attractive character, even the evanescent channels can be seen as dips of the total transmission. The multi-channel current scattering matrix is determined using its representation in terms of the R-matrix. The resonant transmission peaks are characterized quantitatively through the poles of the current scattering matrix. Detailed maps of the localization probability density sustain the physical interpretation of the resonances. Our formalism is applied to a quantum dot in a two-dimensional electron gas and to a conical quantum dot embedded inside a cylindrical nanowire.