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arXiv · 1709.05575

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

Alexander B. Watson·Michael I. Weinstein
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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.

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Alexander B. Watson, Michael I. Weinstein. 2017-09-16. Wavepackets in inhomogeneous periodic media: propagation through a one-dimensional band crossing. https://doi.org/10.1007/s00220-018-3213-x

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