Wave-packet dynamics in slowly perturbed crystals: Gradient corrections and Berry-phase effects

We present a unified theory for wave-packet dynamics of electrons in crystals subject to perturbations varying slowly in space and time. We derive the wave-packet energy up to first order in the gradient of perturbing potentials and different types of Berry-phase terms for the semiclassical dynamics and the quantization rule. For electromagnetic perturbations, we recover the corrections of orbital magnetization energy and the anomalous velocity purely within a single-band picture without invoking inter-band couplings. For deformations in crystals, besides a deformation potential, we obtain a Berry phase term in the Lagrangian due to lattice tracking, which gives rise to new terms in the expressions for the wave-packet velocity and the semiclassical force. For multiple-valued displacement field surrounding a dislocation, this term manifests as a phase analogous to Aharonov-Bohm phase, which we show to be proportional to the Burgers vector of the dislocation. To provide a concrete example of the corrections mentioned above, we study a bi-atomic two-dimensional hexagonal lattice in the tight-binding approximation that can be used to model some semiconducting thin films. We proceed to show that the orbital magnetization of a Bloch state can not be understood from its periodic current distribution, but from the current distribution of a wave packet peaked at the Bloch state. We thereby graphically demonstrate that the orbital magnetization corresponds to spin-like rotation of the wave packet. We then show that it can give rise to a macroscopic magnetization under non-equilibrium conditions, and that its contribution to paramagnetic susceptibility is of the same order of magnitude as the Landau-Peierls diamagnetic susceptibility. As for the Berry-phase term, we briefly discuss how it could modify the magneto-resistance.