Quantum transport and the Wigner distribution function for Bloch electrons in spatially homogeneous electric and magnetic fields

TL;DR

This paper develops a gauge-invariant Wigner distribution function framework for Bloch electrons in homogeneous electric and magnetic fields, capturing multiband effects and Zener tunneling in a collisionless regime.

Contribution

It introduces an exact, gauge-invariant WDF formulation employing the accelerated Bloch state representation for arbitrary time-dependent fields.

Findings

WDF equation for free electrons in homogeneous fields derived

Single-band WDF matches Boltzmann equation to first order in magnetic field

Methodology extended to include multiband effects and Zener tunneling

Abstract

The theory of Bloch electron dynamics for carriers in homogeneous electric and magnetic fields of arbitrary time dependence is developed in the framework of the Liouville equation. The Wigner distribution function (WDF) is determined from the single particle density matrix in the ballistic regime, i.e., collision effects are excluded. The single particle transport equation is established with the electric field described in the vector potential gauge, and the magnetic field is treated in the symmetric gauge. The general approach is to employ the accelerated Bloch state representation (ABR) as a basis so that the dependence upon the electric field, including multiband Zener tunneling, is treated exactly. In the formulation of the WDF, we transform to a new set of variables so that the final WDF is gauge invariant and is expressed explicitly in terms of the position, kinetic momentum, and time. The methodology for developing the WDF is illustrated by deriving the exact WDF equation for free electrons in homogeneous electric and magnetic fields. The methodology is then extended to the case of electrons described by an effective Hamiltonian corresponding to an arbitrary energy band function. In treating the problem of Bloch electrons in a periodic potential, the methodology for deriving the WDF reveals a multiband character due to the inherent nature of the Bloch states. In examining the single-band WDF, it is found that the collisionless WDF equation matches the equivalent Boltzmann transport equation to first order in the magnetic field. These results are necessarily extended to second order in the magnetic field by employing a unitary transformation that diagonalizes the Hamiltonian using the ABR to second order. The work includes a discussion of the multiband WDF transport analysis and the identification of the combined Zener-magnetic field induced tunneling.