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

摘要

The theory of Bloch electron dynamics for carriers in homogeneous electricand magnetic fields of arbitrary time dependence is developed in the frameworkof the Liouville equation. The Wigner distribution function (WDF) is determinedfrom the single particle density matrix in the ballistic regime, i.e.,collision effects are excluded. The single particle transport equation isestablished 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, includingmultiband 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 invariantand is expressed explicitly in terms of the position, kinetic momentum, andtime. The methodology for developing the WDF is illustrated by deriving the exactWDF equation for free electrons in homogeneous electric and magnetic fields.The methodology is then extended to the case of electrons described by aneffective Hamiltonian corresponding to an arbitrary energy band function. Intreating the problem of Bloch electrons in a periodic potential, themethodology for deriving the WDF reveals a multiband character due to theinherent nature of the Bloch states. In examining the single-band WDF, it isfound that the collisionless WDF equation matches the equivalent Boltzmanntransport equation to first order in the magnetic field. These results arenecessarily extended to second order in the magnetic field by employing aunitary transformation that diagonalizes the Hamiltonian using the ABR tosecond order. The work includes a discussion of the multiband WDF transportanalysis and the identification of the combined Zener-magnetic field inducedtunneling.