Quantized systems with randomly corrugated walls and interfaces

Effect of scattering by random surface inhomogeneities on transport along the walls and localization in ultrathin systems is analyzed. A simple universal surface collision operator is derived outside of the quantum resonance domain. This operator contains all relevant information on statistical and geometrical characteristics of weak roughness and can be used as a general boundary condition on the corrugated surfaces. In effect, the boundary problem for the three-dimensional ~3D! transport equation is replaced by the explicit matrix collision operator coupling a set of 2D transport equations. This operator is applied to a variety of systems including ultrathin films and channels with rough walls, particles adsorbed on or bound to rough substrates, multilayer systems with randomly corrugated interfaces, etc. The main emphasis is on quantization of motion between the walls, though the quasiclassical limit is considered as well. The diffusion and mobility coefficients, localization length, and other parameters are expressed analytically or semianalytically via the intrawall and interwall correlation functions of surface corrugation. @S0163-1829~99!00935-2# Recent progress in micro- and nanofabrication, multilayer systems, pure materials, vacuum technology, etc., made the study of particle and wave interaction with system boundaries vital for almost all branches of physics. Below, we concentrate on some universal features of wall and interface scattering. More precisely, we consider effects of scattering by random surface corrugation without energy accommodation. Scattering of particles by random surface inhomogeneities contributes to the randomization of momentum, formation of the mean-free path, quantum interference, and, often, localization. Though this effect of surface scattering looks transparent, it is not easy to express it in terms of geometrical and statistical properties of surface inhomogeneities, especially for quantized systems ~see Refs. 1‐5 and, for recent references, our preceding publication 6 !. At first glance, the effects of scattering by random surface inhomogeneities should not be qualitatively different from scattering by bulk impurities. However, while the basic effects of impurity scattering are described in textbooks, a similar simple account of surface scattering is missing. One of the reasons is technical: the range of impurity interaction is usually short while the corresponding parameter for surface scattering, namely the correlation radius of surface inhomogeneities, can be large. The second reason is more fundamental. It is intuitively clear that the role of surface scattering is higher in ultraquantum small-size systems of longwave particles. In such quantized systems even the impurity scattering is not well understood and is much more complicated than in standard quasiclassical problems. If one disregards quantum interference and localization, the effect of impurity scattering on quasiclassical threedimensional ~3D! transport can be described by the transport equation dn~ p! dt 52pNimp E W~ p2p8!@n~ p8!2n~ p!# 3d~e p2e p8 ! d 3 p8 ~ 2p! 3 , ~1!