Nonlinear Quantum Waves in Semiconductor Superlattices

Semiconductor superlattices constitute an ideal laboratory for investigating coherent electron dynamics and nonlinear charge transport in periodic nanostructures. Their miniband electronic structure gives rise to Bloch oscillations, Wannier-Stark localization, resonant tunneling, electric-field domain formation, and self-sustained current oscillations over a wide range of temporal and spatial scales. This article reviews the multiscale theoretical framework developed to describe these phenomena, from semiclassical Boltzmann-Poisson-BGK kinetic equations to fully quantum Wigner-Poisson formulations. It shows how systematic asymptotic analysis bridges microscopic quantum kinetics and macroscopic nonlinear transport. Systematic hydrodynamic reductions yield generalized and quantum drift-diffusion equations that retain the essential effects of scattering, self-consistent electrostatics, coherent tunneling, and quantum interference while remaining amenable to analytical investigation and efficient numerical simulation. These reduced descriptions account for Gunn-type oscillations mediated by traveling waves, the coexistence of Bloch and Gunn oscillations under appropriate conditions, and nonlinear transport involving multiple minibands. The review concludes by discussing how the methodology developed for semiconductor superlattices may be extended to emerging quantum materials, including moire superlattices, twisted bilayer graphene, Floquet-engineered systems, and topological materials, where Berry curvature, quantum geometry, orbital magnetic moments, and strong electronic correlations enrich the dynamics of coherent quantum waves.