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

Abstract

Long-period systems and superlattices, with additional periodicity, have new effects on the energy spectrum and wave functions. Most of the theoretical approaches adjust theories that are acceptable for systems with large number of unit-cells n, but questionable for small n. Lately, a theory of finite periodic systems based entirely on transfer matrices and their mathematical and physical properties, valid for any number of propagation modes, any n, and arbitrary potential profile or refractive indices, was developed. In this review, we focus on this approach where finiteness is an essential condition. We also comment on the widely used hybrid approach that combines transfer matrices with Floquet's theorem, and show that Floquet's theorem implies transfer matrices of the compact subgroup. We review the transfer matrix definitions, symmetry properties, group representations, and relations with the scattering amplitudes. We summarize the derivation of the recurrence relation of matrix polynomials, which in the scalar limit are the Chebyshev polynomials, the analytical formulas for resonant states, energy eigenvalues, eigenfunctions, parity symmetries, and discrete dispersion relations, for superlattices with different confinement characteristics. 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.