Wavepackets in inhomogeneous periodic media: propagation through a one-dimensional band crossing

TL;DR

This paper studies how semiclassical wavepackets behave when passing through a band crossing in a one-dimensional periodic medium, revealing the excitation of a second wavepacket with opposite velocity, modeled by a Landau-Zener type process.

Contribution

It provides a detailed analysis of wavepacket dynamics at band crossings in inhomogeneous periodic media, highlighting the emergence of a second wavepacket with opposite group velocity.

Findings

A second wavepacket is excited at the band crossing with opposite velocity.

Wavepacket dynamics are qualitatively different from well-separated bands.

Results align with a Landau-Zener-type model.

Abstract

We consider a model of an electron in a crystal moving under the influence of an external electric field: Schroedinger'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 $\epsilon$. We consider the dynamics of $\textit{semiclassical wavepacket}$ asymptotic (in the limit $\epsilon \downarrow 0$) solutions which are spectrally localized near to a $\textit{crossing}$ of two Bloch band dispersion functions of the periodic operator $- \frac{1}{2} \partial_z^2 + V(z)$. 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 $\textit{opposite}$ group velocity to the incident wavepacket. We then show that our result is consistent with the solution of a `Landau-Zener'-type model.