Joseph B. Krieger

Quantum transport and solid-state dynamics for Bloch electrons in an electric field

Abstract A novel formalism for treating Bloch electron dynamics and quantum transport in electric fields of arbitrary strength and time dependence is presented. In this formalism, the electric field is described through the use of the vector potential; this choice of gauge leads to a natural set of basis functions for describing Bloch electron dynamics in an electric field, even if the field is inhomogeneous . Quantum transport results for Bloch electrons in a spatial homogeneous but arbitrarily time dependent electric field undergoing elastic scattering from randomly distributed impurities (Kohn-Luttinger revisited) are presented; that is, a non-linear “Boltzmann equation” is derived with collision integrals involving not only memory and intracollisional field effects, but also including explicit band-mixing transients such as effective mass dressing and Zener tunneling. The application of this formalism to quantum transport in inhomogeneous electric fields is also addressed; specific applications to problems involving tunneling through “band-engineered” tunneling barriers and impurity scattering in electric fields of arbitrary strength and time dependence are discussed.

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

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.

Dynamics of a Bloch electron in a uniform electric field; Bloch oscillations

Abstract The dynamics of a Bloch electron in a spatially uniform electric field is investigated by employing a vector potential to represent the effect of the field. This method has the advantage of maintaining the translational periodicity of the Hamiltonian in the presence of the field, and avoids the difficulties associated with the use of a scalar potential. We find that both the tunneling probability to other bands and the ladder-like structure in the optical absorption are obtained without any assumption concerning the existence of Wannier-Stark energy levels.

Quantum Transport For Bloch Electrons In Homogeneous Time Dependent Electric Fields

Quantum Transport Equations for Bloch electrons interacting with randomly distributed impurities in the presence of a homogeneous electric field of arbitrary strength and time dependence are derived. The equations account for all possible quantum effects to lowest nonzero order in the scattering strength, including intra and interband scattering, interband Zener tunneling and non-linear transient transport, and contain effects previously not anticipated, such as coherent impurity scattering, and field and time dependent scattering matrix elements.