Anderson Localization in Anisotropically Random Media
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ANDERSON LOCALIZATION IN ANISOTROPICALLY RANDOM MEDIA PING SHENG*, WEIGE XUE*, ZHAO-QING ZHANG** AND Q. J. CHU** *Exxon Research & Engineering Co., Route 22 East, Annandale, NJ 08801 **Academia Sinica, Beijing, China ABSTRACT Dimensional cross-over behavior of Anderson localization is presented. By delineating the physical basis of localization, it is shown that the localization phenomenon is sensitive to the spatial dimension of the randomness. Analytic and numerical results demonstrate that in an anisotropically random medium there is a critical amount of anisotropy which separates the system behavior into a ID-like regime and a 3D-like regime. Dimensional crossover is proposed as a viable experimental approach to observe the mobility edge. I.
INTRODUCTION
The concept that a wave can be "localized" through random scatterings was first proposed by P. W. Anderson [1] in 1958 in the context of electron diffusion in random potentials. While the initial concern of the localization theory was mainly in the area of electronic properties of disordered materials, in the past decade there has been an increasing interest in the implications of the localization phenomenon for classical wave transport in random medium [2]. However, despite numerous studies that have clarified the physical basis of localization, it remains today as a difficult phenomenon to observe experimentally. For electronic systems, metal-insulator transition has indeed been studied in many materials by varying the charge carrier concentration. Here the inevitable presence of electron interaction, especially in the low-concentration range where the metal-insulator transition occurs, makes it nearly impossible to have a clean experimental verification of the Anderson localization. On the other hand, classical waves are much more difficult to localize then electrons. In particular, it was predicted that a critical ratio of indices of refraction between the scatterers and the surrounding medium has to be exceeded before classical wave localization becomes possible [3]. One of the primary motivations of our study on the dimensional cross-over behavior of Anderson localization is to propose it as a viable experimental approach for observing the mobility edge. Since localization is induced by varying the spatial dimension of the randomness, for electronic systems a high charge carrier'density can be maintained so as to minimize electron-electron interaction effects. For classical waves the use of lower-dimensional randomness would insure localization even for small ratio of indices of refraction. Thus in both cases there can be a clear-cut test of the Anderson localization concept. In what follows, we give a physical description of the localization phenomenon and its behavior in anisotropic random media. The most common characterization of wave localization is the exponential decay of the wave envelope. In contrast to the evanescent wave (which can result for the electromagnetic wave if the real part of the dielectric constant is negative, such as in metals)
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