Tunneling calculations for systems with singular coupling matrices: Results for a simple model

The tight-binding approach has become one of the most common and useful methods for incorporating band-structure effects into the calculation of the tunneling resonances of resonant tunneling diodes, the energy levels oi quantum wells,and other heterostructure properties. For several years now, numerical stabilization methods have;allowed the use of light-binding models for even very long structures (similar to 3000 Angstrom). These methods still, however, suffer from a common deficiency: the reliance on a transfer-matrix calculation to determine the boundary conditions, The difficulty is rooted in the fact that the mere generation of a transfer matrix requires a matrix inversion which may not always be possible. Recently, we have shown how to obtain the complex band structure in the case of singular coupling matrices, for which a transfer matrix does not exist. Here we study a simple model, deliberately constructed in such a way that a transfer matrix does not exist, and demonstrate that its funneling properties are exactly what one anticipates from the bulk bands.