Buildup dynamics of transmission resonances in superlattices

Abstract

A closed analytical expression for the time-dependent probability density of transmitted particles in superlattices is derived from a formal solution of the time-dependent Schr\"odinger equation. Such an expression consists of Breit-Wigner type resonance terms and interference contributions with explicit time dependence, which is applied to different superlattices to describe step-by-step how the transmission resonances are constructed as a function of time. In particular, it is found that for incidence at a resonance of position ${\ensuremath{\varepsilon}}_{n}$ and width ${\ensuremath{\Gamma}}_{n},$ the buildup of the transmission peaks is governed by a simple exponential law ${\mathcal{T}}_{n}({\ensuremath{\varepsilon}}_{n}{,t)=T}_{n}^{\mathrm{peak}}[1\ensuremath{-}\mathrm{exp}(\ensuremath{-}{\ensuremath{\Gamma}}_{n}t/2\ensuremath{\Elzxh}){]}^{2},$ where ${\mathcal{T}}_{n}({\ensuremath{\varepsilon}}_{n},t)$ is the probability density at the right edge of the superlattice and ${T}_{n}^{\mathrm{peak}}$ the height of the corresponding transmission peak. We show that our results are valid for periodic superlattices as well as for asymmetrical or even disordered potential profiles.