Date of Original Version
We present a simple and versatile description of transport of almost ballistic particles near rough boundaries with an emphasis on thin films and narrow channels. The main effects are associated with chaotization of motion as a result of repeated scattering from random walls. We show that the problem contains an additional mesoscopic length scale which is expressed explicitly via the amplitude and correlation radius (or the correlation function) of surface inhomogeneities, and the ratio of the particle wavelength to the correlation radius. The calculations are performed with the help of a canonical coordinate transformation which reduces a transport problem with rough random walls to a completely equivalent problem with ideal flat walls, but with some random bulk distortions. This problem is treated on the basis of a kinetic equation with a perturbative collision integral. In addition to the application of the Boltzmann transport equation for (quasi)particles with an arbitrary degree of degeneracy of the distribution function, we also include the results for a single-particle diffusion on the basis of the Focker-Plank equation. We calculate different transport coefficients for (quasi)particles with an arbitrary spectrum ε(p) with a bulk of calculations for particles with quadratic, p2/2m, and linear, cp, spectra. The calculations are made in classic and WKB regimes as well as in the case of quantized motion across the film.
All the transport coefficients are expressed via the first two angular harmonics of the correlation function of surface inhomogeneities which play the role of an effective transport cross section. The results include the effects of bulk impurities and changes in potential relief near the walls. We also calculate the quantum interference corrections to conductivity and localization and mesoscopic effects associated with reflections from random surface inhomogeneities, and the density of states in low-dimensional films. The mesoscopic properties are especially simple in the case of strong quantization of motion across the d-dimensional films when the problem becomes effectively equivalent to localization of d-1-dimensional motion in weak random potential. We discuss possible future applications of our method such as for porous media, boundary slip, etc.
Meyerovich, A. E., & Stepaniants, S. (1995). Transport in Channels and Films with Rough Surfaces. Phys.Rev. B., 51(23), 17116-17130. doi: 10.1103/PhysRevB.51.17116
Available at: http://dx.doi.org/10.1103/PhysRevB.51.17116