Scaling behavior of classical wave transport in mesoscopic media at the localization transition

Abstract

The propagation of the 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 determines 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 D͑L͒ scales like ln͑L͒ / L in cubic or slab geometry and the corresponding transmission follows ͗T͑L͒͘ ϰ ln L / L 2 . This is in contrast to the behavior of D͑L͒ ϰ 1/L and ͗T͑L͒͘ ϰ 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 D͑L͒. We have also considered position-dependent weak localization effects by using a plausible form of position-dependent diffusion constant D͑z͒. The same scaling behavior is found, i.e., ͗T͑L͒͘ ϰ ln L / L 2 .

Reply to “Comment on ‘Scaling behavior of classical wave transport in mesoscopic media at the localization transition’ ”

“…͑1͒ gives rise to T ϰ ͑l / L͒ 2 behavior at large L's, they conclude that the T ϰ ͑l / L͒ 2 ln͑L / l͒ behavior we obtained in Ref. 2 was an artifact of replacing D͑z͒ with its harmonic mean. We would like to state here that Fig.…”

“…We believe, however, that there is a physical reason behind the deviations and this physical reason is precisely the one that gives rise to the T ϰ ͑l / L͒ 2 ln͑L / l͒ behavior found in Ref. 2.…”

Comment on “Scaling behavior of classical wave transport in mesoscopic media at the localization transition”

“…Appearance of this scaling in Ref. [1] should then be an artifact of replacing D(r) by its harmonic mean.…”

“…The correct scaling T ∝ 1/L 2 is obtained by properly treating the position dependence of D(r). In a recent paper [1] Cheung and Zhang (CZ) apply the self-consistent (SC) theory of localization to study the transmission of waves through a slab of disordered medium at the Anderson localization transition. The SC theory is a powerful tool to deal with the phenomenon of Anderson localization, but its application to disordered media of finite size requires some care.…”

“…CZ propose a way of overcoming technical difficulties caused by the position dependence of D(r) [1] (see also [9]). They average the equation for 1/D(r), their Eq.…”