Coherent wave-packet evolution in coupled bands

We develop a formalism for treating coherent wave-packet dynamics of charge and spin carriers in degenerate and nearly degenerate bands. We consider the two-band case carefully in view of spintronics applications, where transitions between spin-split bands often occur even for relatively weak electromagnetic fields. We demonstrate that much of the semiclassical formalism developed for the single-band case can be generalized to multiple bands, and examine the nontrivial non-Abelian corrections arising from the additional degree of freedom. Along with the center of mass motion in crystal momentum and real space, one must also take into account the probability amplitudes to characterize the dynamics between the bands. We derive the wave packet energy up to the first order gradient correction and obtain the equations of motion for the real- and $k$-space center of the wave packet, as well as for the probability amplitudes. These equations include the non-Abelian Berry curvature terms and a non-Abelian correction to the group velocity. As an example, we apply our formalism to describe coherent wave packet evolution under the action of an electric field, demonstrating that it leads to electrical separation of spins. A sizable separation will be observed, with a large degree of tunability, making this mechanism a practical method of generating a spin polarization. We then turn our attention to a magnetic field, where we recover Larmor precession, which cannot be obtained from a single-band point of view. In this case, the gradient energy correction can be regarded as due to a magnetic moment from the self-rotation of the wave packet, and we calculate its value for the light holes in the spherical four-band Luttinger model.

Figure 1

Separation along the $y$-direction between light holes of opposite helicities as a function of $\tau$, the dimensionless time. The maximum separation occurs for $\tau =1$.