Coherent wave-packet evolution in coupled bands

Abstract

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 include a pseudo-spin 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 pseudo-spin. 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 up and down 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.Comment: 8 pages, 1 figur

Noncommutative geometry and non-Abelian Berry phase in the wave-packet dynamics of Bloch electrons

“…[40] The second term of Eq. (37) gives, when inserted into the commutator, ∂ǫ loc (k,x, t) ∂x νx ν , ∇ kµ mn = − ∂ 2 ǫ loc ∂k µ ∂x νx ν .…”

“…(43) are indeed SU (N ) gauge invariant. In order to obtain a complete set of EOM, however, we still need to know an EOM forz(t) defined in (40). The details of its derivation is given in Appendix B, and the result is,…”

Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium

“…It is worth noticing that an alternative but related approach, considering the electromagnetic wave evolution in a gravitational field within the Bargmann−Wigner equations, has been offered recently in [13]. Our approach is in essence equivalent to that developed in [14] for electron wave-packet evolution in coupled bands. In our case non-Abelian evolution appears due to anisotropic correction in the Hamiltonian in the presence of Abelian Berry gauge field.…”

Ultrafast Control of Strong-Field Electron Dynamics in Solids

“…This expression can only be applied as long as ∆E i j (k) in the denominator is not equal to zero. The case of degenerate bands presents additional mathematical challenges [27,35,36]; in particular, the transition matrix elements ξ ξ ξ i j (k) are singular at degeneracies [36]. For reference, we also give the relation between ξ ξ ξ i j (k) and the matrix elements of the position operator between the Bloch functions [15,37,38]:…”

“…In the tunneling regime (γ K 1), charge carriers are predominantly created at the extrema of the electric field, each of which launches an electron wave packet. For a wave packet launched at a time t 0 , it is convenient to introduce a semiclassical displacement: 35) where the group velocity v(k) should correspond to the most probable quantum path of the wave packet in reciprocal space. As long as k(t) is not limited to the first Brillouin zone, this approach is most useful if the probabilities of Bragg scattering at the edges of the Brillouin zone are either negligibly small or close to 100%.…”