Interband coherence response to electric fields in crystals: Berry-phase contributions and disorder effects

Abstract

In solid state conductors, linear response to a steady electric field is normally dominated by Bloch state occupation number changes that are correlated with group velocity and lead to a steady state current. Recently it has been realized that, for a number of important physical observables, the most important response even in conductors can be electric-field induced coherence between Bloch states in different bands, such as that responsible for screening in dielectrics. Examples include the anomalous and spin-Hall effects, spin torques in magnetic conductors, and the minimum conductivity and chiral anomaly in Weyl and Dirac semimetals. In this paper we present a general quantum kinetic theory of linear response to an electric field which can be applied to solids with arbitrarily complicated band structures and includes the inter-band coherence response and the Bloch-state repopulation responses on an equal footing. One of the principal aims of our work is to enable extensive transport theory applications using computational packages constructed in terms of maximally localized Wannier functions. To this end we provide a complete correspondence between the Bloch and Wannier formulations of our theory. The formalism is based on densitymatrix equations of motion, on a Born approximation treatment of disorder, and on an expansion in scattering rate to leading non-trivial order. Our use of a Born approximation omits some physical effects and represents a compromise between comprehensiveness and practicality. The quasiparticle bands are treated in a completely general manner that allows for arbitrary forms of the spin-orbit interaction and for the broken time reversal symmetry of magnetic conductors. We demonstrate that the inter-band response in conductors consists primarily of two terms: an intrinsic contribution due to the entire Fermi sea that captures, among other effects, the Berry curvature contribution to wave-packet dynamics, and an anomalous contribution caused by scattering that is sensitive to the presence of the Fermi surface. To demonstrate the rich physics captured by our theory, we explicitly solve for some electric-field response properties of simple model systems that are known to be dominated by interband coherence contributions. At the same time we discuss an extensive list of complicated problems that cannot be solved analytically. Our goal is to stimulate progress in computational transport theory for electrons in crystals.

Quantum kinetic theory of thermoelectric and thermal transport in a magnetic field

“…(15). We can follow the procedure that was done for the case of transport induced solely by an electric field [19,27]. In linear response, we write the electron density matrix ρ as ρ = ρ 0 + ρ T , where ρ 0 is the equilibrium density matrix and ρ T is the correction to ρ 0 in linear order in a temperature gradient −∇T/T .…”

“…As one can see from its form, the off-diagonal part of the thermal driving term is responsible for the Berry phase contribution to transport coefficients such as the Nernst conductivity of systems with broken time-reversal symmetry in the absence of a magnetic field. The solution to this equation is [27]…”

“…We treat disorder effects on transport coefficients within the Born approximation. Our starting point is the quantum kinetic equation in the presence of disorder [27] ∂ ρ ∂t…”

“…Focusing on the elastic-scattering case and using the notation of Ref. [27] we find that the band-diagonal part of…”

“…The full expression for K( ρ ), including the contributions from the off-diagonal density matrix S , can be found in Ref. [27].…”