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

TL;DR

This paper reviews the transfer matrix method for finite periodic systems, detailing its mathematical foundations, properties, and applications in various advanced physical phenomena and experiments.

Contribution

It provides a comprehensive review of the transfer matrix theory for finite systems, including new analytical formulas and applications in modern physics.

Findings

Accurate predictions of tunneling time in photonic band-gap

Analytical formulas for energy eigenvalues and states

Applications to spintronics and optical effects

Abstract

Long-period systems and superlattices, with additional periodicity, have new effects on the energy spectrum and wave functions. Most approaches adjust theories for infinite systems, which is acceptable for large but not small number of unit cells $n$. In the past 30 years, a theory based entirely on transfer matrices was developed, where the finiteness of $n$ is an essential condition. The theory of finite periodic systems (TFPS) is also valid for any number of propagating modes, and arbitrary potential profiles (or refractive indices). We review this theory, the transfer matrix definition, symmetry properties, group representations, and relations with the scattering amplitudes. We summarize the derivation of multichannel matrix polynomials (which reduce to Chebyshev polynomials in the one-propagating mode limit), the analytical formulas for resonant states, energy eigenvalues, eigenfunctions, parity symmetries, and discrete dispersion relations, for superlattices with different confinement characteristics. After showing the inconsistencies and limitations of hybrid approaches that combine the transfer-matrix method with Floquet's theorem, we review some applications of the TFPS to multichannel negative resistance, ballistic transistors, channel coupling, spintronics, superluminal, and optical antimatter effects. We review two high-resolution experiments using superlattices: tunneling time in photonic band-gap and optical response of blue-emitting diodes, and show extremely accurate theoretical predictions.