A large number of devices based upon man-made structures (multiple quantum wells and superlattices) derive their unique transport properties as a result of the physical structure of the device and the resulting energy levels of the structure. At present, the properties of these structures are generally analyzed using "single-well" energy levels and ignore the strong coupling between che wells. Whereas this yields qualitative understanding of the physics involved, the properties of these devices are derived from the strong coupling between the wells such as in resonant tunneling. In order to quantitatively understand and design new devices, an efficient algorithm for determining the bound and resonant spectra is required. We show that an algorithm based upon R-matrix propagation techniques is capable of determining the required spectra and is numerically stable. The adaptation of the R-matrix algorithm to one-dimensional potentials is discussed and the expressions that determine the bound and virtual energy spectra for single quantum wells are obtained to illustrate the method. Results for the bound and virtual energy spectra of various structures in the presence of an applied bias are then presented. In general, the new algorithm has several advantages in that it is fast, numerically stable, efficient, incorporates the boundary condition on the wave function and its derivative, and can be adapted to any one-dimensional potential.
"An Efficient Algorithm For Calculating Bound- And Resonant-Energy Spectra", Proc. SPIE 0792, Quantum Well and Superlattice Physics, (11 August 1987); doi: 10.1117/12.940849; https://doi.org/10.1117/12.940849