The Scaling Behavior of Classical Wave Transport in Mesoscopic Media at the Localization Transition

TL;DR

This paper investigates how classical waves propagate in disordered media at the Anderson localization transition, revealing that their scaling behavior differs from electrons and depends on sample geometry, with implications for understanding wave transport.

Contribution

It demonstrates that classical wave transport exhibits unique scaling laws influenced by sample geometry, contrasting with electron behavior, and develops a self-consistent approach to analyze this phenomenon.

Findings

Weak localization is geometry-dependent, weaker in cubic and slab geometries.

The diffusion constant scales as ln(L)/L in non-spherical geometries.

Transmission scales as ln L / L^2, differing from electron behavior.

Abstract

The propagation of classical wave in disordered media at the Anderson localization transition is studied. Our results show that the classical waves may follow a different scaling behavior from that for electrons. For electrons, the effect of weak localization due to interference of recurrent scattering paths is limited within a spherical volume because of electron-electron or electron-phonon scattering, while for classical waves, it is the sample geometry that determine the amount of recurrent scattering paths that contribute. It is found that the weak localization effect is weaker in both cubic and slab geometry than in spherical geometry. As a result, the averaged static diffusion constant D(L) scales like ln(L)/L in cubic or slab geometry and the corresponding transmission follows <T(L)>~ln L/L^2. This is in contrast to the behavior of D(L)~1/L and <T(L)>~1/L^2 obtained previously for electrons or spherical samples. For wave dynamics, we solve the Bethe-Salpeter equation in a disordered slab with the recurrent scattering incorporated in a self-consistent manner. All of the static and dynamic transport quantities studied are found to follow the scaling behavior of D(L). We have also considered position-dependent weak localization effects by using a plausible form of position-dependent diffusion constant D(z). The same scaling behavior is found, i.e., <T(L)>~ln L/L^2.