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

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Abstract

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 \({\epsilon}\). We consider the dynamics of semiclassical wavepacket asymptotic (in the limit \({\epsilon\downarrow 0)}\) solutions which are spectrally localized near to a crossing of two Bloch band dispersion functions of the periodic operator \({-\frac{1}{2} \partial^{2}_ {z} +V(z)}\). 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 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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Acknowledgements

The authors wish to thank George Hagedorn, Jianfeng Lu, and Christof Sparber for stimulating discussions. This research was supported in part by National Science Foundation Grant Nos. DMS-1412560, DMS-1620418 and Simons Foundation Math + X Investigator Award #376319 (Michael I. Weinstein).

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  1. Department of Mathematics, Duke University, Durham, NC, 27708, USA

    Alexander Watson

  2. Columbia University, New York, NY, USA

    Michael I. Weinstein

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  1. Alexander Watson
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Correspondence to Michael I. Weinstein.

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Communicated by H. Spohn

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Watson, A., Weinstein, M.I. Wavepackets in Inhomogeneous Periodic Media: Propagation Through a One-Dimensional Band Crossing. Commun. Math. Phys. 363, 655–698 (2018). https://doi.org/10.1007/s00220-018-3213-x

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