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Physica Scripta

Bloch electron in an electric field: a comparison between the exact wave function and the Houston approximation

John N. Churchill, F.E. Holmstrom

Published Nov 1, 1994• 1 citation
Publisher

Abstract

The time-evolution of a wave packet in a perfectly periodic lattice potential is considered for an electron that is in a zero-field Bloch energy eigenstate prior to the adiabatic application of a uniform d.c. electric field. Power series expansions in time are constructed for both the exact wave function and the Houston wave function. A comparison of the coefficients in these two expansions shows that the Houston function diverges instantly and drastically from the exact wave function. A special-case example is presented for which the exact wave function is pure-real for all values of time whereas the corresponding Houston function is not.

DOI
10.1088/0031-8949/50/5/021

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Journal of the Physical Society of Japan•Mar 15, 1995•4 citations
The time-evolution of a wave packet in a perfectly periodic lattice potential is considered for an electron that is in a zero-field Bloch energy eigenstate prior to the adiabatic application of a uniform d.c. electric field. Power series expansions in time are constructed for both the exact wave function and the Houston wave function. A comparison of the coefficients in these two expansions shows that the Houston function diverges instantly and drastically from the exact wave function. A special-case example is presented for which the exact wave function is pure-real for all values of time whereas the corresponding Houston function is not.
Energy eigenstates are superimposed, using the Stark ladder energy spectrum, to construct exact, closed-form wave packets for an electron in a uniform dc electric field and a virtual lattice. We find that (1) perfectly periodic temporal variation at the Bloch frequency does not necessarily imply that there is a spatial oscillation and (2) perfectly periodic variation in both spatial and temporal behavior at the Bloch frequency does not imply that the motion is related in any way to the equation, d k /d t = − eE/h .
The transient picture for a Bloch electron accelerating in an arbitrarily time-dependent homogeneous electric field is developed. The temporal sequence for the analysis includes the instant after electron injection, followed by the time required for a small change in electron wavenumber away from initial injection, leading to the final time evolution over many Bloch periods. The time-dependent behavior is studied using the properties of the Schrödinger equation. The electric field is described through the vector potential gauge, and the instantaneous eigenstates of the Bloch, electric-field-dependent Hamiltonian are used as basis states in describing the Bloch dynamics in the electric field. For each temporal sequence considered, the solution to the Schrödinger equation is established and comparatively discussed. The expectation value of the momentum is obtained for the special case of first order in a constant electric field; the resulting velocity derived is a field-dependent generalization of the natural Zitterbewegung-like behavior discussed in the recent literature. The early-time and long-time limits of the momentum expectation value and its time derivative demonstrate that the resistance to Bloch acceleration after initial band injection varies from real mass to effective mass dynamics as the electron accelerates through the band under the influence of electric field. This changing inertia from early injection of a free-mass electron is the result of the {\it real mass} electron {\it dressing-up} into the states of the crystal to become {\it an effective mass} electron. The ramifications of this temporal {\it dressing} behavior are discussed in considering the general dynamics of Bloch electrons subject to ultrastrong electric fields.
The propagation of one dimensional electron wave function in periodic potential have been simulated. Gaussian wave function was chose to simulate the wave packet for the electron. The simulation or periodic potential shows that it follows Bloch theorem. Many problems related with the shape of potential are simulated also. Initial energy chose as tested parameter, which correspond to the speed of wave packet and related with the energy of the electron, and its show that for greater energy the spreading of wave packet become slower. The dynamical parameter makes possibilities to evaluate the propagation every time we need.
The early quantum theory of electrical conductivity in crystal lattices by Bloch and Zener [1,2] led to the striking prediction that a homogeneous static electric field induces an oscillatory rather than uniform motion of the electrons. These so-called Bloch oscillations have never been observed in natural crystals because the scattering time of the electrons by the lattice defects is much shorter than the Bloch period. In semiconductor superlattices the larger spatial period leads to a much shorter Bloch period (,600 fs) and Bloch oscillations have recently been observed through the emission of THz radiation by the electrons [3]. Here we present Bloch oscillations of atoms in the fundamental energy band of a periodic optical potential. We directly measure the atomic momentum distribution evolving in time under the influence of a constant inertial force for various potential depths. We experimentally observe oscillation periods in the millisecond range as well as positive and negative effective masses. Bloch oscillations are a pure quantum effect which can be explained in a simple one-dimensional model. The periodicity of the lattice (period d) leads to a band structure (Fig. 1) of the energy spectrum of the particle and the corresponding eigenenergies Ensqd and eigenstates jn, ql (Bloch states) are labeled by the discrete band index n and the continuous quasimomentum q; Ensqd and jn, ql are periodic functions of q with period 2pyd and q is conventionally restricted to the first Brillouin zone g2pyd, pydg [4]. Under the influence of a constant external force F, weak enough not to induce interband transitions, a given Bloch state jn, qs0dl evolves (up to a phase factor) into the state jn, qstdl according to qstd ­ qs0d 1 Ftyh . (1) This evolution is periodic with a period tB ­ hyjFjd corresponding to the time required for the quasimomentum to scan a full Brillouin zone. The mean velocity in jn, qstdl
The time evolution of a Gaussian wave packet of an electron, which is confined in a quantum dot subjected to uniform magnetic fields, is analytically studied in terms of the exact Feynman kernel. The central position of the wave packet follows the classical trajectory. Although the motion of the central position is aperiodic due to its dual period, the variance of the probability density changes periodically. If the shape of the initial wave packet is the same as that of the wave function of the ground state, its shape never alters. For more general systems in which the periodic full revival of electron wave packets is realized, the characteristic of energy eigenvalues is discussed.