Matrix method for tunneling in heterostructures: Resonant tunneling in multilayer systems

We present a scattering-matrix formalism applied for the first time to study electron transmission in extended nonperiodic semiconductor heterostructures. In contrast to transfer-matrix methods, it has the advantage of being stable for all systems without requiring the truncation of the growing exponential states. The method is applied to the study of resonant tunneling in GaAs/${\mathrm{Al}}_{\mathrm{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$As multilayer systems, with the influence of the higher band-structure minima included. The results show that the \ensuremath{\Gamma} and the X states give rise to resonances in the transmitted current. Although the positions of the \ensuremath{\Gamma} resonances are in qualitative agreement with an analysis using only the \ensuremath{\Gamma} minima, the wave function contains significant contributions from the X minima. The X resonances are associated with the AlAs layers. It is concluded that for energies below the conduction-band edge of the barrier material, a single-\ensuremath{\Gamma}-state analysis can give qualitative agreement with the many-band analysis, but above the conduction-band edge of an indirect-band-gap barrier, a full many-band model is needed.

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