Transport equation and diffusion in ultrathin channels and films

Abstract

A rigorous perturbative transport equation for ballistic particles in thin films with random rough walls is derived by the diagrammatic Keldysh technique for both quasiclassical and quantized motion across the film. The derivation is based on canonical Migdal transformation that replaces a transport problem with random rough walls by an equivalent problem with flat boundaries and randomly distorted bulk. The rigorous derivation requires a modification of our previously used transformation to avoid non-Hermitian perturbations. The unusual nondiagonal structure of the effective scattering operator makes the transport equation different from the standard Waldmann-Snider equation when the distance between quantized levels for the motion across the film is comparable to the wall-induced perturbation. Outside of this anomalous quantum resonance region, the transport equation is similar to that for scattering by bulk impurities. The magnitude of the anomaly is calculated for degenerate particles and Gaussian correlations of surface inhomogeneities. The transport problem is solved analytically for the single-band occupancy and in the limiting cases of very large and very small correlation radii of inhomogeneities for an arbitrary correlation function of surface roughness. Elsewhere, the transport equation is analyzed numerically for the Gaussian correlation function. Transport coefficients are expressed explicitly via the angular harmonics of the surface correlation function; in the anomalous region, the results contain certain supplemental correlators. The results reveal various effects of interwall correlations on transport including an oscillatory dependence on the number of occupied minibands. The transition from quantum to quasiclassical description of ballistic motion across the ͑thick͒ film can be hindered by residual interwall interference effects similar to those in classical optic problems for thick films without bulk attenuation. Erroneous matrix elements in our previous calculations have been corrected. ͓S0163-1829͑98͒05943-8͔

Quantized systems with randomly corrugated walls and interfaces

“…Two additional conditions that ensure the quasiclassical motion along the walls and the absence of the quantum resonance are also not very restrictive. 6 The results below are obtained for quasi-2D systems with impenetrable external walls and/or transparent interlayer boundaries in multilayer systems. Since the localization length in weakly inhomogeneous 2D systems is exponentially large, [8][9][10][11][12][13] one can start from ''usual'' transport and diffusion and study the localization effects after or in the frame of the diffusion problem.…”

“…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 ͒.…”

“…To illustrate the versatility of the results, we present a wide spectrum of applications such as conductivity of ultrathin metal films and channels, multilayer systems, single-particle diffusion of quasiparticles in helium systems, quantum bouncing ball problem for trapped ultracold neutrons or electrons on helium surface, weakly bound states on corrugated substrates, etc. Our preceding paper 6 contains a rigorous diagrammatic derivation of the quantum transport equation for quasi-2D systems with weak scattering by random rough walls ͑or random impurities͒. The essential difference from the standard Keldysh technique in combination with impurity averaging 7 is the quantization of motion in x direction perpendicular to the walls resulting in the quantum ͑matrix͒ form of the transport equation; the motion along the walls remains quasiclassical.…”

Quantum size effect and the two types of interference between bulk and boundary scattering in ultrathin films

“…The mapping transformation approach allows one not only to develop a 165432-3 mathematically rigorous derivation for the bulk quantum transport equation and the collision operator, which reflects the boundary roughness in the initial problem, but also to understand the limitations and accuracy of alternative approaches to the problem. 24 Most of the other perturbative approaches have an accuracy similar to that of the simplified nonunitary mapping transformation. However, in contrast to the mapping transformation, there is no clear way to improve the accuracy of these approaches.…”

“…The wall-induced transition probabilities W jj (q,q ) between the states j (q) and j (q ) are determined by the correlation functions of surface inhomogeneities on both walls, ζ 11 and ζ 22 , and by the interwall correlation of surface inhomogeneities ζ 12 . 24 When the metal film can be treated as a 2D square well, the equations for these transition probabilities are quite simple:…”