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 #


I. INTRODUCTION
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 ͑3D͒ transport can be described by the transport equation where N imp is the density of impurities, ⑀(p) is the energy spectrum of particles with the distribution function n(p), and the transition ͑scattering͒ probability W between the states p and pЈ is proportional to the impurity cross section ͑throughout the paper បϭ1). This allows to express the relaxation time operator Ϫ1 via the transport cross section tr . In quasi-2D systems with small spacing L between the walls, such as ultrathin films, the motion in x direction perpendicular to the walls is quantized and one deals with a set of 2D energy minibands ⑀ j (q) instead of the 3D spectrum ⑀(p). If the motion along the walls remains quasiclassical and the concentration of impurities is large, N imp 1/3 Lӷ1, the transport Eq. ͑1͒ can often be rewritten as a set of coupled equations for minibands ⑀ j (q) where the collision integral includes all impurity scattering processes with and without interband transitions j↔ jЈ. The related relaxation time operator is a matrix, j j Ј Ϫ1 . A similar matrix equation in kinetic theory of gases of particles with discrete internal states is sometimes called the Waldmann-Snider transport equation. Since the ͑weak͒ localization length is exponentially large in 2D systems, the quantum interference and localization problem can be approached af-ter the ''classical'' transport problem ͑2͒ is solved and the relaxation and diffusion parameters are found. The form of the collision operators in Eqs. ͑1͒ and ͑2͒ is universal for bulk impurities and does not depend on the geometry of the system. The main conclusion of this paper is that for a wide range of wall scattering problems one can still use the impuritylike Eq. ͑2͒ in which the transition probability W ͓i.e., the impurity cross-section j j Ј (q؊qЈ)͔ should be simply replaced by the correlation function of surface inhomogeneities j j Ј (q؊qЈ) with trivial constant factors. The form of the resulting transport equation is system independent in the same sense as Eq. ͑2͒ is independent of a particular realization of the random impurity system. We solve both the diffusion and localization problems and give a compact expression for the wall-related relaxation time operator j j Ј Ϫ1 .
In certain situations, the quantum transport equation is solved analytically. 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. Almost always, this quantum transport equation with surface or impurity scattering reduces to the form ͑2͒. The exception is a narrow quantum resonance region in which the gaps between the quantized energy levels ⍀ are comparable to the effective corrugation-induced perturbation. Since the energy gaps ⍀ increase with decreasing spacing L between the walls as 1/L 2 , this anomalous quantum resonance domain is unavoidable with miniaturization of the system. In the quantum resonance regime, the description of weak impurity and surface scattering becomes almost intractable. In this regime, transport processes are coupled to off-diagonal ͑mixed͒ quantum states and cannot be approximated by Eq. ͑2͒, which accounts only for the diagonal ͑pure͒ states. Outside of this quantum resonance region, the contribution of the off-diagonal mixed states is small and the quantum transport equation acquires the standard Boltzmann-Waldmann-Snider form ͑2͒. Below we study only the ''normal'' regime ͑2͒ and show that the surface-induced transition probability W is system independent for a wide range of systems and physical problems.
The wall roughness affects physical processes via the correlation ͑or lack thereof͒ of classical and quantum multiple scattering by the walls. The magnitude of the effect is a complicated function of four spatial scales-the particle wavelength , distance L between the walls, bulk mean-free path L b , and the correlation radii R of intra and interwall correlations of surface inhomogeneities. Another spatial scale, namely, the amplitude l of the wall corrugation, determines the ''strength'' of the surface scattering and, without the bulk scattering and outside of the resonance region, enters the results as a simple perturbative factor l 2 ͑see below͒. The main restriction on our results is that l is small, while the wavelength remains arbitrary. This restriction is not overly severe. ͓For example, the only nontrivial transport problem for narrow channels is that for slight roughness. If the roughness is strong, the particles are dephased by each wall collision, and the transport problem becomes trivial with the mean-free path LϳL͔. 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. [14][15][16] The diffusion results, with minor modifications, can be extended to quantized narrow quasi-1D channels as long as we do not consider localization effects, which are much stronger in quasi-1D channels than in quasi-2D films.

A. Systems and problems
The aim of this paper is to describe the effect of scattering by random surface inhomogeneities on physical processes in ballistic systems with random rough surfaces. The results describe the following classes of problems: • diffusion flows along random rough walls in thin films, wires, layers, quantum wells, waveguides, etc.; • quantum and classical bouncing-ball problems with a random rough wall and an arbitrary holding potential U(x), which returns the particle ͑the ''bouncing ball''͒ back to the wall; • motion and localization of adsorbed or weakly bound particles on rough substrates; • localization of particles in rough channels; • transport and localization in multilayer systems with rough transparent interfaces between the layers.
The confinement in the x direction perpendicular to the walls is necessary to ensure the repeated scattering by the walls. This confined motion can be quantized; the motion along the walls is quasiclassical. The particles can have arbitrary spectrum ⑀(p)ϭ⑀(Ϫp), such as p 2 /2m, ''relativistic'' spectrum cp, spectra with gaps, etc. Specific physical examples and experimental applications are discussed in Sec. III. Geometrically, the systems can be split into four groups: • systems with a single random rough wall x ϭ(y,z), ͗͘ϭ0; the particles are bound to it by some attractive potential U(x), U(x→ϱ)ϭ0; Uϭϱ on the wall; • single-wall systems, xϭ(y,z), with particles confined near the wall by a holding potential, such as gravity or electric field, with U(x→ϱ)ϭϱ; • system with two rough external walls, xϭϮL/2 ϯ 1,2 (y,z), ͗ 1,2 ͘ϭ0, with an arbitrary potential U(x) in between; Uϭϱ on the walls; • multilayer systems with corrugated transparent interfaces between the layers, xϭx ␣ ϩ ␣ (y,z), and an arbitrary bulk potential U(x) in between ͑the potential changes abruptly by ͓U͔ ␣ on the interface͒; the confinement in x direction is ensured either by impenetrable external walls or by a holding potential.
Theoretical description of systems with random boundaries is often hindered by a common difficulty. It can be illustrated in the calculation of the corrugation-induced correction to the matrix element ͗ 1 ͉Ĥ ͉ 2 ͘ for particles with the Hamiltonian Ĥ ϭĤ 0 ϩV and the orthonormalized wave functions ϭ (0) ϩ␦ (Ĥ 0 and (0) describe the same system but with flat walls; V and ␦ are the corrugationinduced changes͒, The wave functions (0) for the flat walls are known and the first term in Eq. ͑4͒ is the same as in all perturbative schemes. Analytical or computational calculations of the corrugation-induced ␦ require a set of wave functions that can be used as a basis. The use of the basis set assumes that all wave functions are defined in the same space. For randomly corrugated walls, especially with the boundary condition ϭ0, the domain of existence of the wave functions is not the same as for the flat-geometry functions (0) and the proper basis set cannot be introduced explicitly. Thus, ␦ cannot be easily calculated using the functions (0) as an expansion basis. As a result, the second term in Eq. ͑4͒ is much more complicated than the first. This issue arises in almost any theory of systems with random rough walls. The energy factor in this second term indicates that this term does not contribute to the Waldmann-Snider Eq. ͑2͒. As it is shown in Ref. 6, the contribution of this term is large only in the resonance situations when Eq. ͑2͒ breaks down. In this paper we do not consider the quantum resonance domain and assume that Eq. ͑2͒ is valid. Then the only issue is the calculation of the scattering probability W j j Ј in Eq. ͑2͒, i.e., of the matrix elements ͗ j The real question is, of course, how to define this ''corrugation-induced perturbation V '' mathematically. After that, the calculation of the matrix elements becomes straightforward.

B. Theoretical approaches
There are numerous theoretical approaches to scattering by surface corrugation. We will mention only those that are best suited for transport and dissipative phenomena and ignore the ones aimed at description of wave fronts, diffraction patterns, spectral shifts, mean field corrections, etc. These approaches, though not equally convenient and general, lead to identical results outside of the quantum resonance region. Inside the quantum resonance region, the cumbersome calculations are system specific, while most of the existing approaches fail.

S-matrix approach
One of the options is to start from the single-wall S matrix. 5,17 Because of the interwall interference, the two-wall scattering S matrix does not factorize into a product of two single-wall S matrices, and the exact expressions for the single-wall S matrix are not sufficient for solving the multiwall problems in the ultraquantum regime. This approach has not been used beyond the simplest two-wall system without interwall interference. 18

Adiabatic approach
Kawabata's ''adiabatic'' approximation for the wave function 19 can circumvent the difficulties caused by the lack of explicit basis wave functions in systems with randomly corrugated boundaries. Suppose one deals with a flow cannel with two random walls, If the change of the wave function along the walls is slow, one can start from the ''adiabatic'' wave function This wave function assumes a slow variation of the wall shape along the channel, qRӷ j, jl/L(R is the ''size'' of inhomogeneities, i.e., the correlation radius of surface corruga-tion͒. This condition differs from the perturbative condition ͑3͒ and is sufficient for the calculation of the matrix elements and the reflection coefficient. However, the transport calculations use the perturbative Boltzmann equation, which requires an extra condition of the smallness of the wall corrugation. As a result, the transport restrictions in Ref. 19 are equivalent to supplementing of Eq. ͑3͒, lӶR,L, by a strong extra condition qRӷS(S is the number of occupied and/or accessible minibands͒. Taken together, these restrictions are stronger than the ones used in other calculations. On the positive side, the adiabatic method, when applicable, makes the physics transparent. The adiabatic method fails in the quantum resonance regime.

Mapping transformation
One of two approaches used in this paper is the exact mapping of the problem with the corrugated boundaries onto an equivalent problem with flat boundaries and distorted bulk. This approach to transport was suggested first by Tesanovic et al. 20 and Trivedi and Ashcroft 21 without an explicit expression for the coordinate transformation. Independently, one of the authors and S. Stepaniants [22][23][24]6 and, later, Bratkovsky and Rashkeev 25 introduced the explicit Migdallike mapping transformation for transport and localization calculations. In a different context, the mapping transformation approach to systems with nonuniform walls has been used for electromagnetic and acoustic wave scattering, diffraction patterns, wave guides, etc. for many years ͑see Refs. 2,4,8,9 and 26-34 and references therein͒.
This method provides an unambiguous definition of the corrugation-induced perturbation V . For a single random rough wall, xϭ(y,z), the flattening transformation is a simple coordinate shift, XϭxϪ͑ y,z ͒.

͑7͒
The conjugate momentum transformation,

͑8͒
transforms the Hamiltonian Ĥ 0 (p,r) into Ĥ 0 (P,R) ϩV (P,R,ˆ‰) with a random bulk part V that depends on the wall corrugation (y,z). For example, the quadratic Hamil- Two walls ͑5͒ with the average spacing L can be flattened simultaneously by stretching the film, while the coordinates in the plane of the wall may be left unchanged, The conjugate momentum transformation identifies the effective random bulk distortion Vˆ 1,2 ͖. In the case of quadratic Hamiltonian, [22][23][24] Ϫy,z Ј ͪ P x P y,z ϩ P y,z ͩ X L ϩy,z Ј Ϫ

2
Ϫy,z Ј ͪ P xͬ , Ϯ ϭ 2 Ϯ 1 , and the problem with the corrugated walls is mapped onto the equivalent bulk problem with flat walls, XϭϯL/2, and randomly distorted bulk, V . Similar mapping transformation can be used for cylindrical geometry. 25 If the particle spectrum ⑀(p) is nonquadratic, the mapping transformation ͑10͒ and ͑11͒ leads to a more complicated, but still treatable, expression for the effective bulk distortion V . If the potential field U(x) between the walls is not uniform, the distortion V in Eq. ͑12͒ should be supplemented by

͑13͒
One should be cautious when using the mapping transformation in the form ͑10͒ and ͑11͒: the Jacobian J of this transformation J 1. The standard diagrammatic or perturbative techniques implicitly assume that the Jacobian Jϭ1.
To avoid mistakes, one can restore the volume to Jϭ1 by supplementing the transformation ͑10͒ by an additional stretching the system in y,z directions instead of Eq. ͑11͒. This has been done in Ref. 6. The result shows that the corrections to V ͑12͒ related to this additional transformation in y,z plane are small and can be disregarded outside of the anomalous quantum resonance region ⍀ ϳ1. Thus, in nonresonance calculations, one can leave the coordinates y,z unchanged and disregard the Jacobian-generated terms in the integrals. By leaving these coordinates unchanged, one can, with the same accuracy, extend the two-wall mapping transformation to the multilayer geometry by applying the coordinate transformations ͑10͒ and ͑11͒ to each layer independently. Inside the resonance region, this cannot be done since the forced change of the y,z coordinates makes the momenta in each layer different from each other.
The calculation of the matrix elements of the perturbation ͑9͒, ͑12͒, and ͑13͒ is straightforward. Though the mapping transformation and the intermediate expressions for the matrix elements are different for each geometry, the final expression for the transition probabilities W j j Ј (q,qЈ) with ⑀ j (q)ϭ⑀ j Ј (qЈ) for the transport Eq. ͑2͒ is not systemspecific with the exception of the quantum resonance regime.
The mapping transformation approach seems indispensable because of its consistency and known accuracy on each step of the calculation. At present, this is the only viable approach for calculations in the anomalous quantum resonance regime.

Direct perturbation calculation
The simplest approach is to calculate directly the matrix elements of the potential using the wave functions for flat geometry as has been suggested by Fishman and Calecki 35 for two-wall systems without interwall correlation of inhomogeneities ͑see also Ref. 36͒.
In this subsection, this approach is extended to particles with an arbitrary spectrum ⑀(p), multilayer systems with corrugated external and interlayer walls, and to systems with possible interwall interference and with a nonuniform potential between the walls. The transparent interlayer walls separate different layers of a multilayer system from each other and are characterized by finite steplike jumps ͓U͔ ␣ of potential. The potential becomes infinite on external impenetrable walls.
In this approach, the walls are replaced by some infinitely narrow potential U(x). If the straight wall xϭx ␣ is described by where s is the coordinate in the plane y,z, q is the conjugate momentum, and the unperturbed wave functions for the flat geometry ⌿ i are chosen real.
On external walls, the potential U becomes infinite and should be excluded from the integrals using the Schrodinger equation for the motion in x direction, and the integration by parts yields Since we are interested only in the processes without changes in energy, Eq. ͑2͒, i.e., only in the first term in the right-hand side in Eq. ͑4͒, the first term in the right-hand side of Eq. ͑17͒ should be disregarded. The final expression for the matrix elements of the corrugation-induced ''perturbation'' of the external wall (␣) is ͓cf. Eq. ͑14͔͒

͑18͒
When the particle spectrum is not quadratic, the calculations should be slightly modified. Instead of Eq. ͑18͒, one gets The full matrix element V ik is the sum over ␣ of V ik (␣) for all interlayer interfaces ͑14͒ and external walls ͑18͒ ͓or ͑19͔͒.
This approach, by design, disregards the resonance contributions and the second term in the expression ͑4͒ for the matrix elements. The accuracy is not always clear; it can be evaluated from the comparison with the more consistent mapping transformation approach. We used both approaches outside of the quantum resonance domain. The transition probabilities W calculated using the matrix elements ͑14͒ and ͑18͒ turned out to be the same as the ones calculated for V ͑9͒, ͑12͒, and ͑13͒ under the condition E i ϭE k though the matrix elements V ik themselves were not identical.

C. Scattering probabilities and correlation functions of surface corrugation
The corrugation-induced bulklike matrix elements V ik ͑14͒ and ͑18͒ play the same role for boundary scattering as the scattering amplitude for scattering by bulk impurities. The role of the impurity cross section is played by the transition probability which should be averaged over random inhomogeneities

͑20͒
Since the matrix elements V ik ͑14͒ and ͑18͒ are linear in , the transition probabilities W ͑20͒ are quadratic in . Their averages are expressed via the correlation functions of surface corrugation defined as The correlation functions ␣␣ describe the intrawall correlations of inhomogeneities, and ␣␤ (␣ ␤)ϭ ␤␣ the interwall correlations. Usually, the inhomogeneities on all surfaces are similar, ␣␣ ϭ ␤␤ , but are not correlated between different surfaces, ␣ ␤ ϭ0. Occasionally, the corrugations of different boundaries can be correlated with each other, ␣ ␤ 0. 37 Experimentally, the correlation function ͑21͒ is not necessarily Gaussian. 4,38 When possible, we avoid specifying the form of the correlation function and express the results via the angular harmonics of ik (s). In applications and numerical calculations, when the form of ik (s) has to be specified, we use the most common Gaussian correlation function To avoid the parameter clutter, we assume in numerical applications that all correlation radii R ␣␤ are the same, while the amplitudes of inhomogeneities may be different with some typical scale l ␣␤ ͑ s͒ϭa ␣␤ l 2 exp͑Ϫs 2 /2R 2 ͒. ͑23͒ The matrix elements V ik ͑14͒ and ͑18͒ allow direct calculation of the corrugation-induced transition probabilities W j j Ј (q,qЈ) ͑20͒ between the quantum states ( j,q) and ( jЈ,qЈ). For transparent interlayer walls with finite changes in potential ͓U͔ ␣ ͑14͒, the corrugation-induced transition probabilities ͑20͒ are where ⌿ j␣ ϭ⌿ j (x ␣ ). Diagonal terms W j j Ј ␣␣ (q؊qЈ) describe the transitions caused by repeated scattering from one interface ␣. The terms W ␣␤ with ␣ ␤ describe the effect of interwall interference in consecutive scattering from two dif-ferent corrugated surfaces. For external impenetrable walls, the probabilities W ͑18͒ and ͑20͒ are and the external wall (␣)-interlayer wall (␤) interference term in the transition probability is for internal interfaces can be used for particles with any spectrum ⑀(p)ϭ⑀(Ϫp), while Eqs. ͑25͒ and ͑26͒ assume ⑀ϭp 2 /2m. For particles with an arbitrary spectrum ⑀(p)ϭ⑀(Ϫp) ͑19͒ in a homogeneous two-wall system, Eq. ͑25͒ should be replaced by

͑27͒
The most prominent feature of the above scattering probabilities is that these expressions are not system specific and the form of the equations is universal. All system-specific information is hidden, in the form of simple constant factors, in the boundary values of the wave functions ͑or their de-rivatives͒ for ideal flat geometry.

D. Standard Waldmann-Snider transport equation and the relaxation time
The transport equation outside of the quantum resonance region ⍀ϳ1 has the standard Boltzmann-Waldmann-Snider form ͑2͒ and is determined by the sum of all corrugation-induced W (␣␤) The integration over dqЈ is done using the ␦ function, and v j j Ј ϭ(‫⑀ץ‬ j Ј q Ј /‫ץ‬qЈ) q Ј ϭq j jЈ for isotropic ⑀ jq . As always in the transport theory, the angular integration is eliminated by using the angular harmonics. The currents are given by the first harmonic of the distribution n j (1) ϵ j the equation for which involves only the zeroth and first harmonics (0,1) W j j Ј (q,q j j Љ ) of W(q؊q j j Ј ) over the angle qq j j Ј , This collision operator Ϫ1 can serve, outside of the anomalous quantum resonance regime, as a general boundary condition for a wide range of confined ballistic systems.
In effect, we replaced a boundary problem for the 3D transport equation by the matrix collision operator that couples a set of 2D transport Eqs. ͑29͒. This matrix-transport equation and the relaxation time operator closely resemble those for the bulk impurity problem. The angular harmonics of the surface correlation function, (0) (͉q؊q j j Ј ͉)Ϫ (1) (͉q؊q j j Ј ͉), play the role of the impurity transport cross section, tr (͉q؊q j j Ј ͉)ϭ (0) Ϫ (1) . Equations ͑29͒ for the distribution functions j (q) are still very complicated. The level of complexity depends on the number of equations S and by the range of q for which this set should be solved. The number of equations is given by the number S of occupied or energetically accessible minibands ⑀ j (q) and can be very large or even infinite. In each equation, the arguments of the distribution functions j and j Ј , q, and q j j Ј (q), are different reflecting the integral nature of the collision operator. Sometimes, the number of equations and the number of the relevant values of q become finite. For degenerate fermions, the values of q and q j j Ј in the equation for j (q) are the Fermi momenta q j and q j Ј for minibands j and jЈ, E F ϭ⑀ j (q j )ϭ⑀ j Ј (q j Ј ), while the number S of the occupied minibands "⑀ j (0)рE F … is restricted by the Fermi energy E F . For the single-particle problems, the values of q and q j j Ј are the momenta q j and q j Ј of the particle with the overall energy E in the minibands j and jЈ, Eϭ⑀ j (q j )ϭ⑀ j Ј (q j Ј ), while only a finite number of the minibands, ⑀ j (0)рE, are energetically accessible. In other situations, as, for example, for particles with the Boltzmann distribution function, the number of occupied minibands is technically infinite, while the equations should be solved for all values of q.
In three important situations the equations decouple from each other, the matrix j j Ј Ϫ1 becomes diagonal, j j Ј Ϫ1 Ӎ(1/ j )␦ j j Ј , and Eqs. ͑29͒ can be solved analytically. This happens when ͑i͒ only the first ͑lowest͒ quantum state j is occupied or is energetically accessible, Sϭ jϭ jЈϭ1; then the set ͑29͒ reduces to a single equation ͑ii͒ LӶR; the small clearance between the surfaces makes the energy gaps between the minibands ⍀ j j Ј ϰ1/L 2 large leading to a suppression of the interband transitions in comparison to intraband scattering, W j j Ј ӶW j j ͑iii͒ the particle wavelength is large, qRӶ1, and all the transition probabilities W are constant (0) WϷ2W(qϭ0) with the zero first harmonic (1) Wϭ0 ͑quantum reflection, cf.
In all other cases, Eqs. ͑29͒ remain coupled and should be solved numerically. The condition ͑ii͒ is purely geometrical and depends only on the preparation of the walls and the interwall spacing L. The other two situations require that the characteristic energy of particles should be small in comparison with the interwall gaps 1/mL 2 in the case ͑i͒ or in comparison with the quantum energy 1/mR 2 in the case of quantum reflection ͑iii͒.
Equations ͑29͒-͑32͒ represent the main general result of the paper. The next section contains the practical applications. The calculations can be done for any surface correlator (s). We will supplement the results by the commonly used Gaussian correlators ͑22͒ with the hypergeometric angular harmonics (0) ik Ϫ (1) ik ϭ4a ik l 2 R 1 2 F 1 ͑ 3/2,2,Ϫ2q 2 R 2 ͒. ͑33͒

A. Application of the general equations
The form of the general equations for the effective relaxation time ͑29͒-͑32͒ is universal. All system-specific information-the energy spectrum ⑀ j (q) and the unperturbed wave functions ⌿(x ␣ ) is hidden in the expressions for the transition probabilities W ͑24͒-͑27͒.
The practical application of these equations is straightforward. One should start from the Schrodinger equation for the ''flat'' geometry with uncorrugated surfaces and find the energy spectrum ⑀ j (q) and the values of unperturbed wave functions ⌿ j (x ␣ ) on interlayer interfaces and ⌿ j Ј(x ␣ ) on external walls. These data, together with the correlation function of surface corrugation, provide the transition probabilities W j j Ј (␣␤) (q,qЈ) ͑24͒-͑27͒ and their angular harmonics (0,1) W j j Ј (␣␤) . Then one should solve the transport Eq. ͑29͒ or, in simpler situations, use the expressions for the kinetic coefficients via the relaxation time ͑30͒-͑32͒. Below this procedure is illustrated for diverse physical systems with various geometries.

B. Conductivity and mobility in ultrathin films: degenerate particles
The most direct application is the calculation of diffusion, conductivity, and mobility coefficients for particles in films and channels with an average clearance L between the corrugated walls in the absence of external field U(x). This was done in detail in our previous paper 6 for degenerate fermi-ons. The paper contains the analytical expressions for conductivity , graphs of in different ranges of p F R, R/L, and S, the analysis of interwall correlations for this geometry, etc. Though the approach was less general than the one discussed above, the results are the same and should not be repeated.

C. Conductivity and mobility in ultrathin films: Boltzmann particles
The transport Eq. ͑29͒ for particles with the Boltzmann distribution function, in contrast to the degenerate fermions, involves an infinite number of minibands and should be solved for all values of q, and not only for a set of Fermi momenta q j .
The convenient representation for the distributions j in Eq. ͑29͒ is (F is the driving force͒. Then the conductivity ͑mobility͒ acquires the form ϭϪ e 2 L 3 while the transport Eq. ͑29͒ becomes The relaxation times j j Ј (q) are defined by Eq. ͑29͒.
To avoid cumbersome equations, we give the analytical results directly for the Gaussian distribution of inhomogeneities and evaluate the dimensionless conductivity ⌸(R/L,x) defined as ϭ 2e 2 L 2 4 l 2 Z⌸͑R/L,x ͒, ͑37͒ Zϭ2NL 2 /, xϭq T Rϵͱ2mTR. When the clearance between the walls is small, q T LӶ1, the thermal energy is small in comparison to the interband gaps and only the lowest miniband ⑀ 1q has a noticeable occupation. Then the dimensionless conductivity ͑37͒ becomes ⌸ϭ L 2 8R 2 1 a 11 ϩa 22 ϩ2a 12 . ͑38͒ Interband transitions are suppressed in comparison with intraband scattering if R/Lӷ1 ͓this is a sufficient condition for q T Lտ1; for small q T LӶ1 one returns to the single-band situation ͑38͔͒. Then the dimensionless conductivity ⌸(R/L,q T R) decouples into the product of two singleparameter functions ⌽ 1 (q T L) and ⌽ 2 (q T R) Functions ⌽ 1 (x) and ⌽ 2 (x) are plotted in Figs. 1 and 2. The third analytical case is the case of long-wave particles Function ⌽ 3 (x) is plotted in Fig. 3. A numerical example for ⌸ ͑37͒ in an intermediate range of parameters is given in Fig. 4 for two values of the interwall correlation amplitude a 12 ϭ0 ͑solid curve͒ and a 12 ϭ0.8 ͑dotted line͒. As expected, the gradual filling of minibands for Boltzmann particles results in much smoother curves than for degenerate fermions in Ref. 6 for which the singularities in reflect abrupt changes in the number of occupied minibands with changing Fermi energy. The effect of interwall correlations is also less dramatic than in the degenerate case.

D. Single-particle diffusion in thin films and narrow channels
The single-particle diffusion in channels and films with corrugated walls is similar to the diffusion problem for degenerate fermions. In equilibrium, a particle with energy E can be in any of S accessible minibands ⑀ j (q)ϭ(1/ 2m)͓( j/L) 2 ϩq 2 ͔ for which ⑀ j (0)рE. The equilibrium distribution function n (0) is Equations ͑29͒ reduce to a set of S coupled linear equations for distributions j (q j ) with the values of momenta q j ϭͱ2mEϪ( j/L) 2 , 1 Sm where ٌ is the density gradient that causes the diffusion, D is the single-particle diffusion coefficient, and is still defined by Eq. ͑29͒. The single-particle mobility coefficient b is related to D by the Einstein equation DϭbE. The set of equations Eq. ͑42͒ is easily solved numerically. Analytical solutions are available in the same three cases. For low-energy particles ͓only one band is accessible, Eq.

͑30͔͒, ͑43͒
Dϭ where ⑀ ϭ2mE(L/) 2 , z j ϭq j R. For ultrathin channels ͑31͒, while for a long-wave particle 2mER 2 Ӷ1, For Gaussian correlations of inhomogeneities, these equations reduce to for ultranarrow channels R/Lӷ1, and Dϭ ͑ Ϫ1 ͒ jϩS a 12 a 11 ϩa 22 , for long-wave particles ͓here z j ϭq j Rϭ(2mE Ϫ 2 j 2 /L 2 ) 1/2 R͔. An example of the single-particle diffusion coefficient, parameterized as is given in Fig. 5 for R/Lϭ0.003, a 11 ϭa 22 ϭ1, and two values of interwall correlations, a 12 ϭ0 ͑solid line͒ and a 12 ϭ0.7 ͑dotted line͒. The contribution of interwall correlation is an oscillating function of the number S of minibands ⑀ jq accessible to a particle with energy E and changes from destructive to constructive depending on whether S is even or odd. This is a general feature that can be observed for any energy spectrum at RӶрL for not very large values of S. At larger S, especially in the quasiclassical regime Sӷ1, the contribution of interwall correlations vanishes.

E. Quasiparticles with nonquadratic energy spectrum
So far, we were discussing the effect of weak surface roughness on particles with quadratic energy spectrum, ⑀ j (q)ϭ͓( j/L) 2 ϩq 2 ͔/2m. In this subsection we analyze the single-particle diffusion coefficient for particles with a nonquadratic spectrum such as ballistic solid-state electrons or quasiparticles in low-temperature helium systems with a low probability of inelastic processes. The latter include phonons ⑀ jq ϭc͓q 2 ϩ( j/L) 2 ͔ 1/2 , 3 He quasiparticles in He II, and ballistic quasiparticles in low-temperature 3 He. The scatter-ing by wall inhomogeneities defines the helium flow through capillaries or superleaks at ultralow temperatures and the thermomechanical effect. In superfluid 3 He, the results are affected by Andreev reflection. To account for these processes, the transport Eq. ͑29͒ should be re-written as two coupled sets of equations for quasiparticles and quasiholes.
The form of the transport Eq. ͑29͒ formally does not change irrespective of the particle spectrum while the transition probabilities acquire the form ͑27͒ If only one energy miniband is occupied, Eϭ⑀ 1 (q 1 ) Ͻ⑀ 2 (0), the single-particle diffusion coefficient ͑30͒ is (1) where the angular harmonics are defined as

͑52͒
In the limit of ultranarrow channels, LӶR, the interband transitions are negligible in comparison with intraband scattering and the diffusion coefficient is ͑31͒ In the case of quantum reflection q j RӶ1, all the correlators ik (q؊qЈ) in the kernel of the integral equation can be replaced by the constants ik (0), and the diffusion coefficient of the quasiparticle with the energy Eϭ⑀ jq j is equal to FIG. 5. Function ⌸(R/L,x), xϭR/ ϭR(2mE) 1/2 for the single-particle diffusion coefficient ͑49͒ Dϭ(L 2 /ml 2 )⌸(R/L,x) at R/L ϭ0.003, a 11 ϭa 22 ϭ1, a 12 ϭ0 ͑solid line͒ and a 12 ϭ0.7 ͑dotted line͒.
For Gaussian correlations ͑22͒, ⌼ ͑52͒ is expressed via the hypergeometric function 1 F 1 The frequency dependence of the diffusion coefficient for ''phonons'' ⑀ jq ϭc͓q 2 ϩ( j/L) 2 ͔ 1/2 at RӷL, Dϭ 2 a 11 ϩa 22 ϩ2a 12 is illustrated in Fig. 6 for Gaussian correlation of inhomogeneities. Sharp singularities in the curves appear in the points where the number S of minibands ⑀ jq accessible to the ''phonon'' with frequency increases by 1 with increasing .

F. Multilayer systems with corrugated interlayer walls
The effect of scattering by corrugated interlayer walls is similar to that for scattering by a corrugated external wall. The simplest illustration with the smallest number of parameters is a two-layer system with flat walls xϭ0;L and a rough interface xϭx 0 ϩ(y,z) between the layers, 0Ͻx 0 ϽL. The potential changes on the interface by U 0 . The transition probability ͑24͒ contains the correlation function of the interface inhomogeneities and the value of the unperturbed wave function ͉⌿ j (x 0 )͉ 2 on the interface, The unperturbed spectrum ⑀ j (q)ϭe j ϩq 2 /2m, and the wave function on the interface ⌿ j (x 0 ) should be found from the Schrodinger equation, as a function of the wall position ␦ϭx 0 /L and the potential difference between the layers ϭ2mU 0 (L/) 2 . This information on dimensionless parameters ẽ j (␦,) and ␥ j j Ј (␦,) ϭL 2 ͉⌿ j (x 0 )͉ 2 ͉⌿ j Ј (x 0 )͉ 2 is sufficient to find W ͑56͒, solve the transport Eq. ͑29͒, and find the conductivity and diffusion coefficients. The kinetic coefficients for a layered system depend not only on the characteristics of the corrugation, R/L and /R as in the previous sections, but also on the position and ''strength'' of the interface ␦ and .
In some sense, the effect of the transition from a single layer to a multilayer system is similar to the result of change in spectrum discussed in the previous subsection. The main difference between the two-layer metal film with a corrugated interface and a single-layer film with rough walls, apart from the obvious factor ␥, is the replacement of the quantum numbers j 2 by ẽ j ͓for example, the expressions for the Fermi momenta q j in the harmonics (0,1) (q j ) become q j 2 L 2 / 2 ϭϪẽ j instead of Ϫ j 2 ͔. If only the first miniband e 1 ϩq 2 /2m is occupied, then In ultrathin systems R/Lӷ1, For the long-wave particles, for the interface with the Gaussian correlation of inhomogeneities at R/Lϭ0.01 and the potential strength ϭ2m(L/) 2 U 0 ϭ90 for three positions of the interface, ␦ ϵx 0 /Lϭ 1 3 ; 1 2 ; 4 5 . The shape of the curves is less regular and is noticeably different from those for the conductivity of a single-layer film with rough walls.

G. Diffusion of particles bound to rough substrates
Another application is single-particle diffusion of particles weakly bound to or adsorbed on rough substrates when the size L of the bound state is larger than the amplitude of the surface inhomogeneities, Lӷl. Such particles move along the substrate, ⑀(q)ϭ⑀ 1 ϩq 2 /2m*, and experience scattering by its inhomogeneities. We will define the size of the bound state L via the derivative of the unperturbed wave function on the wall, The coefficients in Eq. ͑63͒ are chosen in such a way so that for a channel with two impenetrable walls the value of L in ͑63͒ would coincide with the channel width. If there is only one bound state, the expression for the single-particle diffusion coefficient is similar to Eq. ͑43͒ where qϵ(2mEϪ⑀ 1 ) 1/2 . In the case of Gaussian correlation of inhomogeneities, Equation ͑64͒ reduces to Dϭ 2m*R 4 l 2 ͩ L ͪ 6

⌽͑qR͒, ͑65͒
where the function ⌽(x) is defined by Eq. ͑46͒. Possible applications are weakly bound electron states in solids, 3 He surface states inside He II, and hydrogen atoms on helium surface. 39,40 In the last two cases, ⑀ 1 ϳ1 K, Lϳ12 Å, while at Tϳ1 K the ripplon corrugation has parameters R ϳ20 Å, lϳ0.8 Å, and the coefficient in Eq. ͑65͒ is 0.1ប/m.

H. Bouncing ball problem: electrons on helium surface, neutrons in a gravitational trap, etc.
A similar class of problems is often referred to as a ''bouncing ball'' problem. In this problem, a particle bounces repeatedly from a wall after being returned to it by an external field. We are interested in a version of this problem in which the reflection of the quantum particle by the static rough wall xϭ(y,z) is accompanied by scattering by weak random surface inhomogeneities ͗͘ϭ0 and leads to the particle diffusion along the wall. Two typical examples are ultracold neutrons in a gravitational trap 41 and electrons pressed to helium or hydrogen surface by an electric field ͑see also proceedings 42 ͒. In both cases, the holding external fields-mgx and eEx, respectively-are linear functions of coordinates making the problems identical.
The transition probability for scattering by the wall inhomogeneities is given by Eq. ͑25͒, with the energy spectrum ⑀ jq ϭ⑀ j ϩq 2 /2m. The unperturbed wave functions for the flat wall and linear holding potential are the Airy functions where C j are the normalization coefficients. The energy eigenvalues ⑀ j ϭ2 1/3 2/3 ⑀ j /mgL are given by the zeroes of the Airy function, Ai(Ϫ⑀ j )ϭ0; ⑀ 1,2,3, . . . ϭ2.34; 4.09; 5.52, . . . . The spacial scale is determined by the size of the first, closest to the wall discrete state Lϭ( 2 /m 2 g) 1/3 , Eq. ͑63͒. Finally, the transition probabilities ͑66͒ reduce to The transport Eq. ͑29͒ with the transition probabilities ͑68͒ can be solved analytically in the same three situations. The single-particle diffusion coefficient in the first miniband where the last expression describes Gaussian correlations of wall inhomogeneities; q 1 2 ϭ2m(EϪ⑀ 1 ), and ⌽(x) is given by Eq. ͑46͒. Particles cannot access the second miniband as long as 4.13Ͻͱ2mELϽ5. 45. In the case of large interband spacing L 2 /R ϳͱ2mEL 2 /RӶ1, the interband transitions are suppressed and the transport Eqs. ͑29͒ decouple. Then the diffusion coefficient is where q j 2 (E)ϭ2m(EϪ⑀ j ). In the case of Gaussian correlations, the diffusion coefficient is is given in Fig. 8 (HϵE/mg is the amplitude of the particle jumps͒. One of the most interesting applications is the system of ultracold neutrons 41 in a gravitational trap with a macroscopically inhomogeneous ''floor.'' The typical neutron parameters are Lϭ( 2 /m 2 g) 1/3 ϭ1.58ϫ10 Ϫ3 cm, ͱ2mER ϭ1.6ϫ10 3 Rv ͑here R is measured in cm, and the neutron velocity vϭͱ2E/m-in cm/s͒. At present, the neutrons can be trapped with velocities down to vϭ100 cm/s ͑jump amplitudes Hϳ5 cm). 43,44 Parameters of the artificially created roughness in experiment 43 were l,Rϳ10 Ϫ2 cm. Experimentally, the velocity distribution around the average value of v is very narrow and the fraction of low-velocity neutrons is insignificant. Thus, Hӷl,RӷL meaning that Sӷ1 and that the quasiclassical expression in Eq. ͑71͒ provides a fairly accurate description of diffusion and relaxation parameters.
A similar system is the system of electrons above helium or hydrogen surface in weak electric field. Numerically, in fields Eϭ10 3 V/cm, mg in the above equations should be replaced by eEϭ1.6ϫ10 Ϫ9 erg/cm, while Lϭ( 2 /meE) 1/3 ϭ3.78ϫ10 Ϫ6 cm. The electron-helium system 45,46 differs from the trapped ultracold neutrons in two ways. First, the inhomogeneities of the surface of liquid helium or thick helium films are ripplons ͑at Tϳ1 K, Rϳ20 Å, l ϳ0.8 Å) and are not static. Though this does not necessarily change the results, a more direct application is the electron system above a thin helium film on the surface of inhomogeneous solid substrate in a setup similar to the quasi-1D electron-helium system of Ref. 47 or electrons on solid hydrogen. The scale of inhomogeneities in a setup of the type 47 is large, lϳRϳ1mӷL meaning that Sӷ1 and that one should use the quasiclassical results for diffusion coefficient. Another peculiarity is that the electron in strong electric field creates a dimple on the helium surface. This makes the effective mass dependent on the electric field and leads, in the limit of large fields, to selftrapping or auto-localization of electrons in heavy ripplonic polarons. As a result, our description can be used without modifications only in the relatively low electric fields. The most promising application of this type may be the system of electrons on the surface of solid hydrogen.

I. Scattering by surface inhomogeneities and localization
The above expressions for diffusion coefficient D(E) and mean-free path Lϭ2D/v determine the ͑weak͒ localization length R for particles with energy E ͑Refs. 14 and 8͒ where S is the number of minibands ⑀ jq accessible for a particle with energy E. The diffusion results from the previous sections provide an adequate description of the localization exponent ͑75͒ in various systems. Experimental observation of the weak 2D localization ͑75͒ is possible when the exponent is not very large, Շ20. In order to have a reasonable localization length, one should try to decrease the particle energy E, decrease the correlation radius R and the thickness L, and increase the amplitude of inhomogeneities l. Numerical estimates show that the 2D localization ͑75͒ can be observed almost exclusively for low-energy particles for which only the first miniband ⑀ 1q is accessible, S(E)ϭ1. At higher energies, the exponent ͑75͒ becomes too large.
Comparison of Eqs. ͑46͒, ͑65͒, and ͑69͒ for diffusion of the low-energy particles within the single miniband shows that the localization exponent can be written in a universal form as ͑E ͒ϭ 2 2R 4 l 2 ͩ L ͪ 6 ⌽͑qR͒ a 11 ϩa 22 ϩ2a 12 ͑76͒ ͓in single-wall systems, the factor 1/(a 11 ϩa 22 ϩ2a 12 ) should be replaced by 1͔. The plot of function ⌽(x), Eq. ͑46͒, is given in Fig. 9. This function grows very rapidly with increasing x, and one is unlikely to see localization for qR Ͼ1.5. The universality of the coefficient in Eq. ͑76͒ for all these diverse systems is explained by the choice of coefficients in the definition of the spatial scale L via the derivative of the normalized wave function on the wall ͑63͒.
After qR reaches a certain critical value x c ϭq c R, the second energy miniband becomes accessible and the localization exponent is not described by Eq. ͑76͒ and Fig. 9 any more. These critical values x c are different for different systems. For example, for channels with two impenetrable walls x c ϭq c Rϭͱ3R/L; for a bouncing ball, the critical value is smaller, x c ϭq c Rϵ2 1/3 2/3 (⑀ 2 Ϫ⑀ 1 ) 1/2 (R/L)ϭ3.57R/L. These critical values of x c ϭq c R are outside Fig. 9 for all LрR. The localization with more than one accessible minibands can be observed only for R/LӶ1. The multiband lo- calization exponent is not universal because the spectra and interband gaps depend on the system geometry though the typical energy dependence of the localization exponent resembles the sawlike curve in Fig. 6.

IV. DISCUSSION AND SUMMARY
In summary, we derived general expressions for the relaxation time operator for ballistic particles scattered by weak surface roughness in quantized systems. These expressions provide, often analytically, the values of transport and localization parameters. The results are illustrated for such diverse systems as classical or degenerate particles in ultrathin films and channels, multilayer systems, single-particle diffusion, diffusion of particles weakly bound to rough substrates, quantum bouncing balls with rough walls, etc.
Outside of the quantum resonance region, the transport equation and relaxation time operator ͑29͒, expressions ͑24͒-͑27͒ for the transition probabilities via the intra-and interwall correlation functions of surface corrugation, and the analytical expressions ͑30͒-͑32͒ for the corrugationinduced relaxation time, are system independent. All systemspecific information is hidden in the energy spectrum ⑀ j (q) and the values of unperturbed wave functions ⌿(x ␣ ) ͑or their derivatives͒ on the walls. Inside the resonance region described in Ref. 6, the situation is much more complicated even numerically. At present, we cannot provide any general expressions, including those for effective scattering probabilities, for this regime and can approach the problem only on a system-by-system basis. Needless to say, all the difficulties associated with the resonance region disappear in the quasiclassical regime, i.e., for large-scale systems.
Most of the problems discussed in the paper are either transport problems or transport-related ones. The natural question is whether it is possible to get a similar description for other physical problems such as the effect of weak sur-face roughness on particle spectra, mean field, diffraction patterns, etc. In short, the answer is no. A simple picture exists only for the effects that can be described with the help of collision integrals with the energy ␦ functions ␦(E 1 ϪE 2 ) in them ͑2͒. The wall-induced spectral changes involve parts of the collision operator not with the energy ␦ functions ␦(E 1 ϪE 2 ), but with the principal part integrals P(1/E 1 ϪE 2 ). As a result, such processes should take into account the terms (E 1 ϪE 2 )͗␦ 1 ͉ 2 (0) ͘ in the matrix elements ͑4͒. These terms, in turn, are not universal and cannot be easily incorporated into the equations.
Similar argument explains the difficulty of calculations in the quantum resonance regime. In this regime, there is a strong coupling between pure and mixed quantum states of particles ͑coupling between diagonal and off-diagonal Green's functions͒ that requires to include the principal part integrals along with the energy ␦ functions even in the parts of the collision operator responsible for transport. Even intuitively, it is clear that in resonance the matrix elements (E 1 ϪE 2 )͗␦ 1 ͉ 2 (0) ͘ do not disappear from the equations. We want to emphasize that all these difficulties associated with the resonance regime exist for quantized films with usual bulk impurities as well. As it was shown in Ref. 6, the quantum resonance regime ⍀ j j Ј ϳ1 corresponds to a moderately large number S of accessible minibands ⑀ jq . The quasiclassical domain of large quantum numbers Sӷ1 ͑thick films͒ and the ultraquantum case of small S ͑ultrathin films͒ are outside of the anomalous resonance region.
The next step should be the simultaneous study of boundary and bulk scattering in thin films including the interference terms beyond the Matthiessen's rule.