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

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 $\text{semiclassical wavepacket}$ asymptotic (in the limit $\epsilon \downarrow 0$) solutions which are spectrally localized near to a $\text{crossing}$ 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 $\text{opposite}$ group velocity to the incident wavepacket. We then show that our result is consistent with the solution of a `Landau-Zener'-type model.

To appear in Communications in Mathematical Physics; 42 pages, 5 figures

References in corpus (4)

• A kinetic model for the transport of electrons in a graphene layer
• An Egorov Theorem for avoided crossings of eigenvalue surfaces
• Wavepackets in inhomogeneous periodic media: effective particle-field dynamics and Berry curvature
• Honeycomb Schroedinger operators in the strong binding regime

Cited by in corpus (2)

• Characterization of edge states in perturbed honeycomb structures
• Propagation of Wave Packets for Systems Presenting Codimension 1 Crossings