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: 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 . We consider the dynamics of semiclassical wavepacket asymptotic (in the limit ↓ 0) solutions which are spectrally localized near to a crossing of two Bloch band dispersion functions of the periodic operator − 1 2 ∂ 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. arXiv:1709.05575v2 [math-ph] 1 Jun 2018

Propagation of Wave Packets for Systems Presenting Codimension One Crossings

“…The choice of δ = ε 2/9 optimizes the combined estimate and yields convergence of order ε m with m ≥ 5/9. We also want to emphasize that the method of proof we propose here allows to systematically avoid the impressive computations, which appear in [13] pages 65 to 72, and are also present in [39] via the reference [46] to which the authors refer therein.…”

“…More recently, in [39], A. Watson and M. Weinstein studied models arising in solid state physics in the context of Bloch band decompositions,…”

“…Some results on the subject focus on the evolution at leading order in ε of quadratic quantities of the wave function for initial data which are not necessarily wave packets (see [9,20] and the references therein). The main results devoted to wave packet propagation through smooth eigenvalue crossings are the references [13] and [39] mentioned above. There, for the specific Hamiltonian operators (4) and ( 5), respectively, the authors gave rather explicit descriptions of the propagated wave packet, exhibiting non-adiabatic transitions that occur at the crossing between the two eigenvalues that are of order √ ε.…”

Multiband effects in equations of motion of observables beyond the semiclassical approach

“…For multiband wave packets, especially when there are crossing points in the band structure, this issue is currently under investigation. Recent studies have shown that the collapse time of a packet initially localized in a single band whose evolution allows interband transitions after encountering a crossing point is also relatively long and it is determined by the 1/h scale [23,24]. Also, it has been shown that a multiband description of a wave packet on a generic honeycomb potential with Dirac points accurately approximates the full solution to the Schroedinger equation for large times even when the packet is centered at the crossing point [25].…”

“…Also, in a transport calculation, which includes scattering (not considered here), the time scale for spreading of the packet in real space must be larger than the scattering time. The collapse of the packet in reciprocal space is currently under investigation as noted earlier, although it has been shown that even for multiband wave packets with crossing points in the band structure, the collapse times are relatively long [22][23][24][25]32]. Although this issue needs further separate investigations, the EOM for the pair of linear bands can be obtained as…”