Theory of hot-electron quantum diffusion in semiconductors

In the linear response regime close to equilibrium, the fluctuation-dissipation theorem relates linear transport coefficients via the well-known Green–Kubo or Einstein relation. The latter embodies a deep connection between fluctuations causing diffusion and dissipation, which are responsible for a finite mobility. Far from equilibrium, however, the Einstein relation is no longer valid so that both the mobility and diffusivity gain their own physical integrity. Consequently, beyond a linear response, both quantities have to be described by different approaches. Unfortunately, there is a strong imbalance of research activities devoted to the study of both transport mechanisms in semiconductors. On one hand, the rich physics of high-field quantum drift in semiconducting structures has a long history and has reached a high level of sophistication. On the other hand, there are only comparatively few and unsystematic studies that cover quantum diffusion of carriers under high-field conditions. This review aims at reducing this gap by presenting a unified approach to quantum drift and quantum diffusion. Starting from a semi-phenomenological basis, a quantum theory of transport coefficients is developed for one- as well as multi-band models. Physical implications are illustrated by selected applications whereby the quantum character of the approach is emphasized. Furthermore, the basic unified treatment of transport coefficients is extended by accounting for the two-time dependence of one-particle correlation functions in quantum statistics. As an application, a phononless transport mechanism is identified, which solely originates from the double-time nature of the evolution. Finally, additional examples are presented that illustrate the important role played by quantum diffusion in semiconductor physics.