The Transfer Matrix Method and the Theory of Finite Periodic Systems. From Heterostructures to Superlattices

Abstract

Historically, the physics, energy spectrum, and wave functions of superlattices have had different levels of understanding. Unlike the standard approaches, based on theorems for infinite systems, the theory of finite periodic systems (TFPS) constitutes a genuine quantum approach, with closed analytical results. In this theory the finite number of unit‐cells is an essential condition, while the number of propagating modes, as well as, the potential profiles (or refractive indices) are arbitrary. In this work, the transfer matrix definitions, symmetry properties, group representations, and relations with the scattering amplitudes are reviewed. The derivation of multichannel matrix polynomials (which reduce to Chebyshev polynomials in the one‐propagating mode limit), analytical formulas for resonant states, energy eigenvalues, eigenfunctions, parity symmetries, and discrete dispersion relations for superlattices with different confinement characteristics are reviewed. After showing the inconsistencies and limitations of hybrid approaches that combine the transfer‐matrix method with Floquet's theorem, some applications of the TFPS to multichannel negative resistance, ballistic transistors, coupled channels transport, spintronics, superluminal, and optical antimatter effects are revisited. Two high‐resolution superlattice related experiments are reviewed: the tunneling time in photonic band‐gap and the optical response of blue‐emitting diodes. It is shown that the TFPS accurately predicts the experimental results.