Wavepackets in Inhomogeneous Periodic Media: Propagation Through a One-Dimensional Band Crossing

We consider a model of an electron in a crystal moving under the influence of an external electric field: Schrödinger’s equation in one spatial dimension with a potential which is the sum of a periodic function V and a smooth function W. We assume that the period of V is much shorter than the scale of variation of W and denote the ratio of these scales by ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\epsilon}$$\end{document}. We consider the dynamics of semiclassical wavepacket asymptotic (in the limit ϵ↓0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\epsilon\downarrow 0)}$$\end{document} solutions which are spectrally localized near to a crossing of two Bloch band dispersion functions of the periodic operator -12∂z2+V(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${-\frac{1}{2} \partial^{2}_ {z} +V(z)}$$\end{document}. We show that the dynamics is qualitatively different from the case where bands are well-separated: at the time the wavepacket is incident on the band crossing, a second wavepacket is ‘excited’ which has opposite group velocity to the incident wavepacket. We then show that our result is consistent with the solution of a ‘Landau–Zener’-type model.