Time Periodic Behavior of Multiband Superlattices in Static Electric Fields

Abstract

We use an analytic perturbation expansion for the two-band system of tight binding electrons to discuss Bloch oscillations and Zener tunneling within this model. We make comparison with recent numerical results and predict analytically the frequency of radiation expected from Zener tunneling, including its disappearance, as a function of the system parameters. PACS numbers: 71.70.Ej, 73.40.Gk, 73.20.Dx Typeset using REVTEX 1 One of the earliest predictions of the theory of electrons in periodic crystals is of Bloch oscillations — time periodic motion at frequency ωB = Fd/h̄ under application of a uniform constant field F in a crystal potential of lattice period d, associated with Bragg reflection at Brillouin Zone boundaries. But in ordinary atomic crystals, subjected to accessible applied (electric) fields, scattering typically disrupts the coherence of electronic motion in times very short compared to the Bloch period TB = 2π/ωB, preventing observation of the phenomenon. However, the lattice constant d of semiconductor superlattices, typically two orders of magnitude larger, correspondingly reduces TB by the same factor. Esaki and Tsu 1 recognized 25 years ago, at the dawn of this new technology, that this greatly improved the prospects of experimental observation of Bloch oscillations and related phenomena. Moreover, the characteristic frequencies lie in the interesting far infrared region. Not surprisingly, then, there has been a great deal of subsequent activity, both experimental and theoretical, in the study of high quality semiconductor superlattices under the influence of static and time periodic electric fields. Transport and both linear and nonlinear optical response have been of special interest. In particular, Bloch oscillations have been shown to survive the addition of Coulomb interactions to the simple theory, and far infrared radiation from such oscillations has now been observed. The one-dimensional periodicity of a quantum well superlattice gives as the eigenstates for independent (noninteracting) electrons a set of “minibands” associated with motion in the growth direction, perpendicular to the wells. For obvious reasons of simplicity, much of the theoretical study has been limited to consideration of a single isolated miniband. But, as emphasized particularly in a recent Letter by Rotvig, Jauho and Smith (RJS), there are essential interesting phenomena — notably Zener tunneling — introduced by additional bands in the presence of a static electric field. These authors reported a numerical study of the density matrix of a two-band model, exhibiting a number of interesting types of periodic behavior of the electrons. It is the purpose of the present paper, (i) to make use of analytic results in the form of an exact perturbation expansion to explore the corresponding behavior throughout the space of the parameters characterizing the two-band superlattice, and (ii) to 2 make use of our analytically predicted behavior to compare with selected numerical results in order to test the convergence of the perturbation series so as to define its range of validity for application to other problems and properties. The structure of the series suggests (correctly, we will show) that the convergence is much more rapid than imposed by the obvious limits that can be set analytically. The two-band model is, of course, the simplest extension beyond the single band picture, and it does introduce those features characteristic of interband communication. But is it a reasonable approximation to any physical situation to focus on a single pair of minibands while neglecting the higher bands which inevitably also exist? As we have pointed out before, there is at least one realistic situation where this should capture the dominant physical behavior. A superlattice can be grown dimerized, with the unit cell consisting, e.g., of a pair of quantum wells, separated from the adjacent pair by a larger (wider) barrier than that which separates the two wells of the basis pair. Then the lowest level of an individual well is split within the “molecular” pair. These two levels form a pair of bands in the superlattice which are well separated energetically from those arising from higher levels in the well, and it is then reasonable to neglect all except that lowest pair of bands. We will use the standard two-band tight-binding Hamiltonian in a uniform static electric field E: H = ∑

Cite this paper

@inproceedings{Hone1995TimePB, title={Time Periodic Behavior of Multiband Superlattices in Static Electric Fields}, author={Daniel W. Hone and Xian-geng Zhao}, year={1995} }