Transport Equation and Diffusion in Ultrathin Channels and Films

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 (thick) 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

Interference of Bulk and Boundary Scattering in Films with Quantum Size Effect

The interference between boundary and bulk scattering processes is analyzed for ultrathin films with random rough walls. The effective collision and transport relaxation times for scattering by random bulk and surface inhomogeneities are calculated, when possible analytically, in quantum size effect conditions. The transport and localization results are expressed via the bulk transport parameters and statistical characteristics of the surface corrugation. The diagrammatic calculation includes second-order effects for boundary scattering and full summation for bulk processes. The interference contribution is large in systems with robust bulk scattering and can be comparable to, or even exceed, the pure wall contribution to the transport coefficients

Localization and Diffusion in Quasi-2D Helium and Hydrogen Systems with Corrugated Boundaries

Localization length and diffusion coefficient are calculated for quantized quasi-2D low-temperature systems with randomly corrugated boundaries or substrates. Applications include electrons on helium or hydrogen surfaces, quasiparticles in capillaries and ultrathin films, surface states, ultracold neutrons in gravitational traps, etc

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

Transport Phenomena at Rough Boundaries

We present a simple method describing transport processes near rough boundaries. By a proper coordinate transformation we reduce a transport problem with rough walls to an equivalent problem with smooth flat walls, but with some random bulk inhomogeneities. In many cases the last problem can be treated perturbatively, leading to simple expressions for relevant transport coefficients via the correlation function of surface inhomogeneities. We calculate diffusion and conductivity in films, phonon, and photon diffusion, quantum corrections to conductivity, and the single-particle diffusion coefficient

Transport in Channels and Films with Rough Surfaces

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

Boundary Effects and Spin Waves in Spin-Polarized Quantum Gases

Boundary conditions are derived for spin dynamics of spin-polarized quantum gases near nonmagnetic walls. We are interested mostly in boundary-induced line shifts and attenuation of spin waves, and in the possibility of having a macroscopic boundary condition for systems close to a Knudsen ballistic regime. We consider the effects caused by roughness of the wall and by surface adsorption. By a proper coordinate transformation, we reduce the problem of particle collisions with an inhomogeneous nonmagnetic wall to an equivalent problem with a specular homogeneous wall but with stochastic bulk imperfections. As a result, the boundary effects are described by some additional bulklike transverse spin-diffusion coefficient inversely proportional to the angular harmonics of the correlation function of surface inhomogeneities. This leads to an effective macroscopiclike boundary condition for transverse spin dynamics responsible for the boundary effects in spin-wave resonances. The situation changes drastically at low temperatures because of an appearance of an adsorbed boundary layer which renormalizes the molecular field near the wall, and leads to additional effective spin-exchange processes. The experimental implications for helium and hydrogen systems are discussed

Diffusion and Localization of Ultra-Cold Particles on Rough Substrates

Diffusion and localization of ultra-cold particles moving along randomly corrugated substrates is analyzed quasianalytically. The particles are either bound to the substrate or pressed to it by the external holding field. The localization length and diffusion coefficient are expressed explicitly via the correlation radius of surface inhomogeneities. This quantum bouncing hall problem with a random rough wall is solved analytically in three limiting cases of longwave particles, large gaps between bound states, and single-state occupancy. Elsewhere, the diffusion coefficient and localization length are evaluated numerically for Gaussian correlation of inhomogeneities. The results are applied to ultra-cold neutrons in the gravitational trap, electrons on helium and hydrogen surfaces, and hydrogen particles bound to helium surface. Experimental observation of weak 2D localization for neutrons and electrons requires further cooling and substrate preparation

Statistical Quasiparticles in Transverse Dynamics of Gases

We analyze the validity of the Fermi-liquid approach to transverse dynamics of spin-polarized gases at arbitrary temperatures. We demonstrate that the diagrammatic kinetic equation for transverse processes can be formulated as a simpler, but completely equivalent equation in terms of ‘‘statistical quasiparticles.’’ The equation includes all coherent and dephasing molecular-field terms as well as the dissipative collision integral up to the second order. Beyond the second order, the results become very complicated, and a quasiparticle approach loses its attraction. We give the expressions for the effective interaction function and collision integral for statistical quasiparticles, applicable at all temperatures, and discuss the implications of this concept at high temperatures. The interaction function contains anomalous pole terms which do not exist in equations for longitudinal dynamics. This provides a somewhat unexpected interpretation for zero-temperature dissipative processes, observed recently in spin dynamics, and for controversial molecular field terms (the so-called I2 terms) as imaginary (pole) and real (principal) parts of the quasiparticle interaction function. These molecular field terms with complicated analytical structure do not vanish completely, as was assumed earlier, in the Boltzmann region, but contribute to higher-order density terms. With an emphasis on quantum gases, we discuss how to reconcile various physical assumptions inherent to different kinetic approaches to dilute gases