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

Abstract

The equations of motion (EOM) for the position and gauge invariant crystal momentum are considered for multiband wave packets of Bloch electrons. For a localized packet in a subset of bands well-separated from the rest of the band structure of the crystal, one can construct an effective electromagnetic Hamiltonian with respect to the center of the packet. We show that the EOM can be obtained via a projected operator procedure, which is derived from the adiabatic approximation within perturbation theory. These relations explicitly contain information from each band captured in the expansion coefficients and energy band structure of the Bloch states as well as non-Abelian features originating from interband Berry phase properties. This general and transparent Hamiltonian-based approach is applied to a wave packet spread over a single band, a set of degenerate bands, and two linear crossing bands. The generalized EOM hold promise for novel effects in transport currents and Hall effect phenomena.

Transport theory within a generalized Boltzmann equation for multiband wave packets

“…The justification for a well-localized packet in real space comes from the semiclassical approximation used in transport for which the wavelengths of external fields are much larger than the spread of the packet in real space. We further note that multiband packets that are well localized in reciprocal and real spaces, one deals with degenerate or nearly degenerate Bloch states comprising the packets [13,15].…”

“…The anomalous velocity can give rise to a variety of Hall effects associated with charge carriers and photons as well as unusual thermoelectricity [7][8][9][10][11]. On the other hand, several researchers have considered Bloch electron wave packets extending over multiple energy bands [12][13][14][15][16]. In fact, it has been shown that the equations of motion for the centers of such multiband wave packets are different than those in the single band wave packet case, and one can obtain new features in the dynamics of spin, parity polarization currents, and non-Abelian transport [12][13][14][15][16].…”

“…On the other hand, several researchers have considered Bloch electron wave packets extending over multiple energy bands [12][13][14][15][16]. In fact, it has been shown that the equations of motion for the centers of such multiband wave packets are different than those in the single band wave packet case, and one can obtain new features in the dynamics of spin, parity polarization currents, and non-Abelian transport [12][13][14][15][16]. The discoveries of Dirac, Weyl, and other materials with nontrivial topology have elevated the importance of interband transitions and coherence of Bloch electrons, which undoubtedly are important for transport [17][18][19].…”

“…To this end, research findings strongly suggest that the single band semiclassical transport theory may need to be expanded to capture multiband effects from the materials electronic structure [8,[12][13][14][15][16][17]. This problem has been addressed in recent reports [8,12,14], where a quantum kinetic equation through a Wigner transformation of the Liouville equation for the usual density operator of energy eigenstates is derived.…”

“…For our approach, we define a density operator associated with a distribution of wave packets based on the quantum mechanical degrees of freedom in the wave packets. This is a new quantity that uses generalized equations of motion for wave packets containing multiband effects through a previously derived projection procedure [15]. Transport coefficients and related thermoelectric properties are derived within leading orders of the electrochemical potential and temperature gradients to linear order of electromagnetic fields in the system.…”