Quantum scattering in one-dimensional periodic structures: A Green's function approach solved through continued fractions

In the present contribution, we discuss quantum scattering in 1D periodic finite lattices of $N$ localized potentials by means of an exact Green's function approach. By considering continued fraction techniques, we solve the resulting recurrence relations, thus being able to derive the full structure reflections ${R}_{N}$ and transmission ${T}_{N}$ amplitudes in a closed analytic form. The framework allows for dealing with extremely large arrays, in some examples for $N$ up to ${10}^{10}$ cells (or building blocks). For so great $N$'s, in practice the protocol can unveil most of the basic features of the energy band structures of the corresponding infinite systems, demanding relatively little computational effort. We further investigate general scattering properties of distinct lattices, e.g., when their cells are spatially asymmetric or composed by two or more elementary shapes, each shape commonly modeled in the literature in terms of Dirac's delta, rectangular, trapezoidal, and triangular barriers. As concrete applications, we address the problem of parameter optimization of heterostructures used to build solar cells and the identification of some transmission resonance modes, relevant in the study of band-pass transmission in superlattices.