## M.A. Makeev, A.-L. Barabási

# Effect of surface morphology on the sputtering yields: II. Ion sputtering from rippled surfaces

#### Nuclear Instruments & Methods In Physics Research Section B 222, 335-354 (2004)

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Off-normal ion bombardment of solid targets with energetic particles often leads to development of periodically modulated structures on the surfaces of eroded materials. Ion-induced surface roughening, in its turn, causes sputtering yield changes. We report on a comprehensive theoretical study of the effect of rippled surface morphology on the sputtering yields. The yield is computed as a function of the parameters characterizing the surface morphology and the incident ion beam, using the Sigmund’s theory of ion sputtering. We find that the surface morphology development may cause substantial variations in the sputter yields, depending on a complex interplay between the parameters characterizing the ripple structure and the incident ion beam. For certain realizations of the ripple structure, the surface morphology is found to induce enhanced, relative to the flat surface value, sputtering yields. On the other hand, there exist regimes in which the sputtering yield is suppressed by the surface roughness below the flat surface result. We confront the obtained theoretical results with available experimental data and find that our model provides an excellent qualitative and, in some cases, quantitative agreement with the results of experimental studies.

## M. A. Makeev, A.-L. Barabási

# Effect of surface morphology on the sputtering yields: I. Ion sputtering from self-affine surfaces

#### Nuclear Instruments & Methods In Physics Research Section B 222, 316-334 (2004)

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As extensive experimental studies have shown, under certain conditions, ion bombardment of solid targets induces a random (self-affine) morphology on the ion-eroded surfaces. The rough morphology development is known to cause substantial variations in the sputtering yields. In this article, we present a theoretical model describing the sputter yields from random, self-affine surfaces subject to energetic ion bombardment. We employ the Sigmund’s theory of ion sputtering, modified for the case of self-affine surfaces, to compute the sputter yields. We find that the changes in the sputtering yield, associated with the non-planar surface morphology, are strongly dependent on the parameters characterizing the surface roughness (such as the saturation width and the correlation length) and the incident ion beam (such as the incident ion energy and the deposited energy widths). It is shown that, for certain ranges of the parameters variations, surface roughness leads to substantial enhancements in the yield, with magnitude of the effect being more than 100%, as compared to the flat surface value. Furthermore, we find that, depending on the interplay between these parameters, the surface roughness can both enhance and suppress the sputter yields.

## M. Makeev, R. Cuerno, A.-L. Barabási

# Morphology of ion-sputtered surfaces

#### Nuclear Instruments & Methods In Physics Research Section B 197, 185-227 (2002)

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We derive a stochastic nonlinear continuum equation to describe the morphological evolution of amorphous surfaces eroded by ion bombardment. Starting from Sigmunds theory of sputter erosion, we calculate the coefficients appearing in the continuum equation in terms of the physical parameters characterizing the sputtering process. We analyze the morphological features predicted by the continuum theory, comparing them with the experimentally reported morphologies. We show that for short time scales, where the effect of nonlinear terms is negligible, the continuum theory predicts ripple formation. We demonstrate that in addition to relaxation by thermal surface diffusion, the sputtering process can also contribute to the smoothing mechanisms shaping the surface morphology. We explicitly calculate an effective surface diffusion constant characterizing this smoothing effect and show that it is responsible for the low temperature ripple formation observed in various experiments. At long time scales the nonlinear terms dominate the evolution of the surface morphology. The nonlinear terms lead to the stabilization of the ripple wavelength and we show that, depending on the experimental parameters, such as angle of incidence and ion energy, different morphologies can be observed: asymptotically, sputter eroded surfaces could undergo kinetic roughening, or can display novel ordered structures with rotated ripples. Finally, we discuss in detail the existing experimental support for the proposed theory and uncover novel features of the surface morphology and evolution, that could be directly tested experimentally.

## J. Kim, B. Kahng, A.-L. Barabási

# Nanoscale wire formation on sputter eroded surface

#### Applied Physics Letters 81, 3654-3656 (2002)

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Rotated ripple structures (RRS) on sputter-eroded surfaces are potential candidates for nanoscale wire fabrication. We show that the RRS can form when the width of the collision cascade in the longitudinal direction is larger than that in the transverse direction and the incident angle of ion beam is chosen in a specific window. By calculating the structure factor for the RRS, we find that they are more regular and their amplitude is more enhanced compared to the much studied ripple structure forming in the linear regime of sputter erosion.

## H. Jeong, B. Kahng, S. Lee, C.Y. Kwak, A.-L. Barabási, J.K. Furdyna

# Monte Carlo simulation of sinusoidally modulated superlattice growth

#### Physical Review E 65, 031602 (2002)

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The fabrication of ZnSe/ZnTe superlattices grown by the process of rotating the substrate in the presence of an inhomogeneous flux distribution instead of the successively closing and opening of source shutters is studied via Monte Carlo simulations. It is found that the concentration of each compound is sinusoidally modulated along the growth direction, caused by the uneven arrival of Se and Te atoms at a given point of the sample, and by the variation of the Te/Se ratio at that point due to the rotation of the substrate. In this way we obtain a ZnSe12xTex alloy in which the composition x varies sinusoidally along the growth direction. The period of the modulation is directly controlled by the rate of the substrate rotation. The amplitude of the compositional modulation is monotonic for small angular velocities of the substrate rotation, but is itself modulated for large angular velocities. The average amplitude of the modulation pattern decreases as the angular velocity of substrate rotation increases and the measurement position approaches the center of rotation. The simulation results are in good agreement with previously published experimental measurements on superlattices fabricated in this manner.

## B. Kahng, H. Jeong, A.-L. Barabási

# Nanoscale structure formation on sputter eroded surface

#### Journal of the Korean Physical Society 39, 421-424 (2001)

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We investigate the morphological features of sputter eroded surfaces, demonstrating that while at short times ripple formation is described by the linear theory, after a characteristic time, the nonlinear terms determine the surface morphology, by monitoring the surface width and the erosion velocity. Furthermore, we show that sputtering under normal incidence leads to the formation of spatially ordered uniform nanoscale islands or holes. We nd that while the size of these nanostructures is independent of flux and temperature, it can be controlled by ion beam energy.

## B. Kahng, H. Jeong, A.-L. Barabási

# Spatial ordering of stacked quantum dots

#### Applied Physics Letters 78, 805–807 (2001)

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We investigate the growth conditions necessary to form an ordered quantum dot crystal by capping spatially ordered quantum dots and growing a new layer of dots on top of the capping layer. Performing Monte Carlo simulations and developing analytic arguments based on the stress energy function, we demonstrate the existence of an optimal capping layer thickness, external flux, and temperature for the formation of quantum dot crystals.

## B. Kahng, H. Jeong, A.-L. Barabási

# Quantum dot and hole formation in sputter erosion

#### Applied Physics Letters 78, 805–807 (2001)

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Recently, it was experimentally demonstrated that sputtering under normal incidence leads to the formation of spatially ordered uniform nanoscale islands or holes. Here, we show that these nanostructures have inherently nonlinear origin, first appearing when the nonlinear terms start to dominate the surface dynamics. Depending on the sign of the nonlinear terms, determined by the shape of the collision cascade, the surface can develop regular islands or holes with identical dynamical features, and while the size of these nanostructures is independent of flux and temperature, it can be modified by tuning the ion energy.

## A.-L. Barabási

# Thermodynamic and kinetic mechanisms in self-assembled quantum dot formation

#### Materials Science and Engineering B 67, 23–30 (1999)

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Heteroepitaxial growth of highly strained structures offers the possibility to fabricate islands with very narrow size distribution, coined self-assembling quantum dots (SAQD). In spite of the high experimental interest, the mechanism of SAQD formation is not well understood. We will show that equilibrium theories can successfully predict the island sizes and densities, the nature and the magnitude of the critical thickness needed to be deposited for SAQD formation, as well as the onset of ripening. Furthermore, the flux and temperature dependence of the SAQDs is described using kinetic Monte Carlo simulations.

## S. Park, B. Kahng, H. Jeong, A.-L. Barabási

# Dynamics of ripple formation in sputter erosion: nonlinear phenomena

#### Physical Review Letters 83, 3486–3489 (1999)

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Many morphological features of sputter eroded surfaces are determined by the balance between ion-induced linear instability and surface diffusion. However, the impact of the nonlinear terms on the morphology is less understood. We demonstrate that, while at short times ripple formation is described by the linear theory, after a characteristic time the nonlinear terms determine the surface morphology by either destroying the ripples or generating a new rotated ripple structure. We show that the morphological transitions induced by the nonlinear effects can be detected by monitoring the surface width and the erosion velocity.

## I. Daruka, A.-L. Barabási, S. J. Zhou, T. C. Germann, P. S. Lomdahl, A. R. Bishop

# Molecular-dynamics investigation of the surface stress distribution in a GeSi quantum dot superlattice

#### Physical Review B 60, R2150-R2153 (1999)

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The surface stress distribution in an ordered quantum dot superlattice is investigated using classical molecular dynamics simulations. We find that the surface stress field induced by various numbers ~from 1 to 9! of Ge islands embedded in a Si~001! substrate is in good agreement with analytical expressions based on pointlike embedded force dipoles, explaining the tendency of layered arrays to form vertically aligned columns. The short-ranged nature of this stress field implies that only the uppermost layers affect the surface growth and that their influence decreases rapidly with layer depth.

## S. Lee, I. Daruka, C.S. Kim, A.-L. Barabási, J.K. Furdyna, J.L. Merz

# Lee et al. reply to Dynamics of ripening of self-assembled II-VI semiconductor quantum dots

#### Physical Review Letters 83, 240 (1999)

#### Read the abstract

Despite extensive investigation, little is still known about the physical mechanisms responsible for quantum dot (QD) formation in II-VI semiconductor systems, especially when compared to their group-IV or III-V counterparts. However, the distinct chemical and microscopic features characteristic of the various materials make these diverse systems rather exciting to study and compare. We therefore welcome the Comment by Kratzert et al. [1] that sheds new light on the CdSe island formation on ZnSe. The method used by them—in situ ultrahigh vacuum atomic force microscopy (AFM)—provides valuable information that was not accessible before: it allows one to probe the dynamics of QD formation without external influences (such as the influence of the atmosphere), and it offers minimum delay between QD formations and their characterization. Specifically, and in contrast with our findings [2], these new results do not manifest room temperature ripening of CdSe islands. These new observations, combined with a number of other results recently reported (see below), suggest the existence of three distinct island types:

## I. Daruka, J. Tersoff, A.-L. Barabási

# Shape transition in growth of strained islands

#### Physical Review Letters 82, 2753–2756 (1999)

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Strained islands formed in heteroepitaxy sometimes change shape during growth. Here we show that there is typically a first-order shape transition with island size, with the discontinuous introduction of steeper facets at the island edge. We present a phase diagram for island shape as a function of volume and surface energy, showing how surface energy controls the sequence of island shapes with increasing volume. The discontinuous chemical potential at the shape transition drastically affects island coarsening and size distributions.

## C. Lee, A.-L. Barabási

# Spatial ordering of islands grown on patterned surfaces

#### Applied Physics Letters 73, 2651-2653 (1998)

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We demonstrate that growth on a sample patterned with an ordered defect array can lead to islands with rather narrow size distribution. However, improvement in the size distribution is achieved only if the growth conditions ~flux and temperature! have optimal values, determined by the patterning length scale. Since the scanning tunelling and the atomic force microscopes are capable of inducing surface perturbations that act as potential preferential nucleation sites, our work demonstrates that nanoscale surface patterning can improve the ordering of platelets and self-assembled quantum dots.

## S. Lee, I. Daruka, C.S. Kim, A.-L. Barabási, J.L. Merz, J.K. Furdyna

# Dynamics of ripening of self-assembled II-VI semiconductor quantum dots

#### Physical Review Letters 81, 3479-3482 (1998)

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We report the systematic investigation of ripening of CdSe self-assembled quantum dots (QDs) on ZnSe. We investigate the size and density of the QDs as a function of time after deposition of CdSe has stopped. The dynamics of the ripening process is interpreted in terms of the theory of Ostwald ripening. Furthermore, the experimental results allow us to identify the growth mode of the QD formation process.

## R. Albert, A.-L. Barabási, N Carle, A. Dougherty

# Driven interfaces in disordered media: determination of universality classes from experimental data

#### Physical Review Letters 81, 2926-2929 (1998)

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While there have been important theoretical advances in understanding the universality classes of interfaces moving in porous media, the developed tools cannot be directly applied to experiments. Here we introduce a method that can distinguish the isotropic and directed percolation universality classes from snapshots of the interface profile. We test the method on discrete models whose universality class is well known, and use it to identify the universality class of interfaces obtained in experiments on fluid flow in porous media.

## M. A. Makeev, A.-L. Barabási

# Effect of surface roughness on the secondary ion yield in ion sputtering

#### Applied Physics Letters 73, 1445–1447 (1998)

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There is extensive experimental evidence that, at low temperatures, surface erosion by ion bombardment roughens the sputtered substrate, leading to a self-affine surface. These changes in the surface morphology also modify the secondary ion yield. Here, we calculate analytically the secondary ion yield in terms of parameters characterizing the sputtering process and the interface roughness.

## I. Daruka, A.-L. Barabási

# Equilibrium phase diagrams for dislocation free self-assembled quantum dots

#### Applied Physics Letters 72, 2102–2104 (1998)

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The equilibrium theory of self-assembled quantum dot ~SAQD! formation can account for many of the experimentally observed growth modes. Here, we show that despite the large number of material constants entering the free energy of strained islands, there are only four topologically different phase diagrams describing the SAQD formation process. We derive each of these phase diagrams and discuss the physical properties of the predicted growth modes.

## M. A. Makeev, A.-L. Barabási

# Secondary ion yield changes on rippled interfaces

#### Applied Physics Letters 72, 906–908 (1998)

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Sputter erosion often leads to the development of surface ripples. Here we investigate the effect of the ripples on the secondary ion yield, by calculating the yield as a function of the microscopic parameters characterizing the ion cascade ~such as penetration depth, widths of the deposited energy distribution! and the ripples ~ripple amplitude, wavelength!. We find that ripples can trongly enhance the yield, with the magnitude of the effect depending on the interplay between the ion and ripple characteristics. Furthermore, we compare our predictions with existing experimental results.

## I. Derenyi, C.-S. Lee, A.-L. Barabási

# Ratchet effect in surface electromigration: smoothing surfaces by an ac field

#### Physical Review Letters 80, 1473–1476 (1998)

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We demonstrate that for surfaces that have a nonzero Schwoebel barrier the application of an ac field parallel to the surface induces a net electromigration current that points in the descending step direction. The magnitude of the current is calculated analytically and compared with Monte Carlo simulations. Since a downhill current smoothes the surface, our results imply that the application of ac fields can aid the smoothing process during annealing and can slow or eliminate the Schwoebel-barrier-induced mound formation during growth.

## M. A. Makeev, A.-L. Barabási

# Ion-induced surface diffusion in ion sputtering

#### Applied Physics Letters 71, 2800–2802 (1997)

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Ion bombardment is known to enhance surface diffusion and affect the surface morphology. Here we demonstrate that preferential erosion during ion sputtering can lead to a physical phenomenon reminiscent of surface diffusion, what we call effective surface diffusion ~ESD!, that does not imply mass transport along the surface and is independent of the temperature. We calculate the ion-induced ESD constant and its dependence on the ion energy, flux and angle of incidence, showing that sputtering can both enhance and suppress surface diffusion. The influence of ion-induced ESD on ripple formation and roughening of ion-sputtered surfaces is discussed and summarized in a morphological phase diagram.

## I. Daruka, A.-L. Barabási

# Dislocation free island formation in heteroepitaxial growth: a study at equilibrium

#### Physical Review Letters 79, 3708–3711 (1997)

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We investigate the equilibrium properties of strained heteroepitaxial systems, incorporating the formation and the growth of a wetting film, dislocation-free island formation, and ripening. The derived phase diagram provides a detailed characterization of the possible growth modes in terms of the island density, equilibrium island size, and wetting layer thickness. Comparing our predictions with experimental results we discuss the growth conditions that can lead to stable islands as well as ripening.

## A.-L. Barabási

# Self-assembled island formation in heteroepitaxial growth

#### Applied Physics Letters 70, 2565-2567 (1997)

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We investigate island formation during heteroepitaxial growth using an atomistic model that incorporates deposition, activated diffusion, and stress relaxation. For high misfit the system naturally evolves into a state characterized by a narrow island size distribution. The simulations indicate the existence of a strain assisted kinetic mechanism responsible for the self-assembling process, involving enhanced detachment of atoms from the edge of large islands and biased adatom diffusion.

## I. Daruka, A.-L. Barabási

# Island formation and critical thickness in heteroepitaxy

#### Physical Review Letters 78, 3027 (1997)

#### Read the abstract

In a recent Letter Chen and Washburn [1] proposed a mechanism for island nucleation in large-mismatch heteroepitaxy. The predicted coverage sQd dependence of the 3D island density ri sQd reproduces the fast increase in the island density near the critical coverage Qc ø 1.6 ML [2]. Here we show that the critical coverage predicted by Ref. [1] depends strongly on the growth rate, thus contradicting, among others, the experimental results of Refs. [2,3].

## A.-L. Barabási

# Self-organized superlattice formation in II-VI and III-V semiconductors

#### Applied Physics Letters 70, 764–767 (1997)

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There is extensive recent experimental evidence of spontaneous superlattice ~SL! formation in various II–VI and III–V semiconductors. Here we propose an atomistic mechanism responsible for SL formation, and derive a relation predicting the temperature, flux, and miscut dependence of the SL layer thickness. Moreover, the model explains the existence of a critical miscut angle below which no SL is formed, in agreement with results on ZnSeTe, and predicts the formation of a platelet structure for deposition onto high symmetry surfaces, similar to that observed in InAsSb.

## A.-L. Barabási, E. Kaxiras

# Dynamic scaling in conserved systems with coupled fields: application to surfactant-mediated growth

#### Europhysics Letters 36, 129-134 (1996)

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We present an analytical study of the interaction of two nonequilibrium conservative fields. Due to the conservative character of the relaxation mechanism, the scaling exponents can be obtained exactly using dynamic renormalization group. We apply our results to surfactant-mediated growth of semiconductors. We find that the coupling between the surfactant thickness and the interface height cannot account for the experimentally observed layered growth, implying that reduced diffusion of the embedded atoms is a key mechanism in surfactant-mediated growth.

## P. Jensen, A.-L. Barabási, H. Larralde, S. Havlin, H.E. Stanley

# A fractal model for the first stages of thin film growth

#### Fractals 4, 321–329 (1996)

#### Read the abstract

In this paper, we briefly review a model that describes the diffusion-controlled aggregation exhibited by particles as they are deposited on a surface. This model allows us to understand many experiments of thin film deposition. In the Sec. 1, we describe the model, which incorporates deposition, particle and cluster diffusion, and aggregation. In Sec. 2, we study the dynamical evolution of the model. Finally, we analyze the effects of small cluster mobility and show that the introduction of cluster diffusion dramatically affects the dynamics of film growth. Some of these effects can be tested experimaentally.

## S. V. Buldyrev, L.A.N. Amaral, A.-L. Barabási, S.T. Harrington, S. Havlin, R. Sadr-Lahijani, H.E. Stanley

# Avalanches in the directed percolation depinning and self-organized depinning models of interface roughening

#### Fractals 4, 307–319 (1996)

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We review the recently introduced Directed Percolation Depinning (DPD) and Self-Organized Depinning (SOD) models for interface roughening with quenched disorder. The difference in the dynamics of the invasion process in these two models are discussed and different avalanche definitions are presented. The scaling properties of the avalanche size distribution and the properties of active cells are discussed.

## A.-L. Barabási

# Roughening of growing surfaces: kinetic models and continuum theories

#### Computational Materials Science 6, 127-134 (1996)

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The use of scaling concepts in understanding growth by molecular beam epitaxy (MBE) is increasingly important these days. Here we present a critical discussion on the advantages and disadvantages of kinetic theories and continuum models, two main methods frequently used to study the roughening and scaling of surfaces grown by MBE. Finally, some open problems faced by these approaches are also discussed.

## H. A. Makse, A.-L. Barabási, H.E. Stanley

# Elastic string in a random medium

#### Physical Review E 53, 6573–6576 (1996)

#### Read the abstract

We consider a one-dimensional elastic string as a set of massless beads interacting through springs characterized by anisotropic elastic constants. The string, driven by an external force, moves in a medium with quenched disorder. We find that longitudinal fluctuations lead to nonlinear behavior in the equation of motion that is kinematically generated by the motion of the string. The strength of the nonlinear effects depends on the anisotropy of the medium and the distance from the depinning transition. On the other hand, the consideration of restricted solid-on-solid conditions imposed on the string leads to a nonlinear term with a diverging coefficient at the depinning transition.

## A.-L. Barabási, G. Grinstein, M.A. Munoz

# Directed surfaces in disordered media

#### Physical Review Letters 76, 1481–1484 (1996)

## L.A.N. Amaral, A.-L. Barabási, H.A. Makse, H.E. Stanley

# Scaling properties of driven interfaces in disordered media

#### Physical Review E 52, 4087–5005 (1995)

## R. Cuerno, A.-L. Barabási

# Dynamic scaling of ion-sputtered surfaces

#### Physical Review Letters 74, 4746–4749 (1995)

#### Read the abstract

We derive a stochastic nonlinear equation to describe the evolution and scaling properties of surfaces eroded by ion bombardment. The coefficients appearing in the equation can be calculated explicitly in terms of the physical parameters characterizing the sputtering process. We find that transitions may take place between various scaling behaviors when experimental parameters, such as the angle of incidence of the incoming ions or their average penetration depth, are varied.

## L.A.N. Amaral, A.-L. Barabási, S.V. Buldyrev, S.T. Harrington, S. Havlin, R. Sadr-Lahijani, H.E. Stanley

# Avalanches and the directed percolation depinning model: experiments, simulations and theory

#### Physical Review E 51, 4655–4673 (1995)

## P. Jensen, A.-L. Barabási, H. Larralde, S. Havlin, H.E. Stanley

# Growth and percolation of thin films: a model incorporating deposition, diffusion and aggregation

#### Chaos, Solitons and Fractals 6, 227–232 (1995)

#### Read the abstract

We propose a model for describing diffusion-controlled aggregation of particles that are continually deposited on a surface. The model, which incorporates deposition, diffusion and aggregation, is motivated by recent thin film deposition experiments. We find, that the diffusion and aggregation of randomly deposited particles “builds” a wide variety of fractal structures, all characterized by a common length scale L1. This length LI scales as the ratio of the diffusion constant over the particle flux to the power l/4. We compare our msults with several recent experiments on two-dimensional nanostructures formed by diffusion-controlled aggregation on surfaces.

## P. Jensen, A.-L. Barabási, H. Larralde, S. Havlin, H.E. Stanley

# Deposition, diffusion and aggregation of atoms on surfaces: a model for nanostructure growth

#### Physical Review B 50, 15316–15329 (1994)

#### Read the abstract

We propose a model that describes the diffusion-controlled aggregation exhibited by particles as they deposited on a surface.The model, which incorporates deposition, particle and cluster diffusion, and aggregation, is inspired by recent thin-film-deposition experiments. We find that as randomly deposited particles diffuse and aggregate they configure themselves into a wide variety of fractal structures characterized by a length scale L1. We introduce an exponent y that tunes the way the diffusion coefficient changes with cluster size: if the values of y are very large, only single particles can move, if they are smaller, all clusters can move. The introduction of cluster diffusion dramatically affects the dynamics of film growth. We compare our results with those of several recent experiments on two-dimensional nanostructures formed by diffusion-controlled aggregation on surfaces, and we propose several experimental tests of the model. We also investigate the spanning properties of this model and find another characteristic length scale L2 (L>>L1) above which the system behaves as a bond percolation network of the fractal structures each of length scale L1. Below L2, the system shows similarities with diffusion-limited aggregation. we find that L1 scales as the ratio of the diffusion constant over the particle flux to the power 1/4, whereas L2 scales with another exponent close to 0.9.

## L.A.N. Amaral, A.-L. Barabási, H.E. Stanley

# Universality classes for interface growth with quenched disorder

#### Physical Review Letters 73, 62–65 (1994)

## L.A.N. Amaral, A.-L. Barabási, S.V. Buldyrev, S. Havlin, H.E. Stanley

# New exponent characterizing the effect of evaporation on imbibition experiments

#### Physical Review Letters 72, 641–644 (1994)

#### Read the abstract

We report imbibition experiments investigating the effect of evaporation on the interface roughness and mean interface height. We observe a new exponent characterizing the scaling of the saturated surface width. Further, we argue that evaporation can be usefully modeled by introducing a gradient in the strength of the disorder, in analogy with the gradient percolation model of Sapoval et al. By incorporating this gradient we predict a new critical exponent and a novel scaling relation for the interface width. Both the exponent value and the form of the scaling agree with the experimental results.

## L.A.N. Amaral, A.-L. Barabási, S.V. Buldyrev, S. Havlin, H.E. Stanley

# Anomalous interface roughening: the role of a gradient in the density of pinning sites

#### Fractals 1, 818–826 (1993)

## A.-L. Barabási

# Surfactant-mediated surface growth: nonequilibrium theory

#### Fractals 1, 846–859 (1993)

#### Read the abstract

A number of recent experiments have shown that surfactants can modifiy the growth mode of an epitaxial film, suppressing islanding and promoting lyer-by-layer growth. hee, a set of coupled equations are introduced to describe the coupling between a growing interface and a thin surfactant layer deposited on the top of the nonequilibrium surface. The equations are derived using the main experimentally backed characteristics of the system and basic symmetry principles. The system is studied using dynamic-normalization-group scheme, which provides scaling relations between the roughness exponents. It is found that the surfactant may drive the system nto a novel phase, in which the surface roughness is negative, corresponding to a flat surface.

## A.-L. Barabási

# Surfactant-mediated growth of nonequilibrium interfaces

#### Physical Review Letters 70, 4102–4105 (1993)

#### Read the abstract

A number of recent experiments have shown that surfactants can modify the growth mode of an epitaxial film, suppressing islanding and promoting layer-by-layer growth. Here I introduce a set of coupled equations to describe the nonequilibrium roughening of an interface covered with a thin surfactant layer. The surfactant may drive the system into a novel phase, in which the surface roughness is negative, corresponding to a flat surface.

## A.-L. Barabási

# Dynamic scaling of coupled nonequilibrium interfaces

#### Physical Review A 46, R2977– R2980 (1992)

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We propose a simple discrete model to study the nonequilibrium fluctuations of two locally coupled (1+1)-dimensional systems (interfaces). Measuring numerically the tilt-dependent velocity we construct a set of stochastic continuum equations describing the fluctuations in the model. The scaling predicted by the equations is studied analitically using dynamic-renormalization-group theory and compared with simulation results.

## A.-L. Barabási, M. Araujo, H.E. Stanley

# Three-dimensional Toom model: connection to the Anisotropic Kardar-Parisi-Zhang Equation

#### Physical Review Letters 68, 3729–3732 (1992)

#### Read the abstract

A three-dimensional Toom model is defined and the properties of the interface separeting the two stable phases are investigated. Using symmetry arguments we show that in the zero-noise limit the model has only nonequilibrium fluctuations and that the scaling is decribed by the anisotropic Kardar-Parisi-Zhang equation. The scaling exponents are determined numerically and good agreement with the theoretical predictions is found.

## S.V. Buldyrev, A.-L. Barabási, F. Caserta, S. Havlin, H.E. Stanley, T. Vicsek

# Anomalous interface roughening in porous media: experiment and model

#### Physical Review A 45, R8313–R8316 (1992)

#### Read the abstract

We report measurements of the interface formed when a wet front propogates in paper by imbibition and we find anomalous roughening with exponent α=0.63±0.04. We also formulate an imbibition model that agrees with the experimental morphology. The main ingredient of the model is the propogation and pinning of a self-affine interface in the presence of quenched disorder, with erosion of overhangs. By relating our model to directed percolation, we find α~0.63.

## A.-L. Barabási, R. Bourbonnais, M. Jensen, J. Kertesz, T. Vicsek, Y.-C. Zhang

# Multifractality of growing surfaces

#### Physical Review A 45, R6951–R6954 (1992)

#### Read the abstract

We have carried out large-scale computer stimulation of experimentally motivated (1+1)- dimensional modes of kinetic surface roughening with power-law-distributed amplitudes of uncorrelated noise. The appropriately normalized qth-order correlation function of the height differences Cq(x)=<|h(x+x')-h(x')|q> shows strong multifractal scaling behavior up to a crossover length depending on the system size, i.e. Cq(x)~xqHq, where Hq is a continuously changing nontrivial function. Beyond the crossover length conventional scaling is found.

## A.-L. Barabási, P. Szepfalusy, T. Vicsek

# Multifractal spectra of multi-affine functions

#### Physica A 178, 17–28 (1991)

#### Read the abstract

Self-affine fiunctions F(x) with multiscaling height correlations Cq(x)~XqHq are described in terrms of the standard multifractal formalism with a modified assumption for the partition. The corresponding quantities and expressions are shown to exhibit some characteristic differences from the standard ones. According to our calculations the f(a) type spectra are not uniquely determined by the Hq spectrum, but, depend on the particular which is made for the dependence of N on x, where N is the number of points over which the average is taken. Our results are expected to be relevant in the analysis of signal type data obtained in experiments on systems which an underlying multiplicative process.

## A.-L. Barabási

# A model for the temporal fluctuations of the surface width: a stochastic one-dimensional map

#### Journal of Physics A 24, 17-28 (1991)

#### Read the abstract

A stochastic one-dimensional map is introduced to model the steady-state fluctuations of the surface width in far-from-equilibrium surface roughening. The dynamics of the map and the correlations in the time sequence are investigated. In particular, for power law distributed noise a non-trivial multi-affine behaviour is observed.

## A.-L. Barabási, T. Vicsek

# Multifractality of self-affine fractals

#### Physical Review A 44, 2730–2733 (1991)

#### Read the abstract

The concept of multifractality is extended to self-affine fractals in order to provide a more complete description of fractal surfaces. We show that for a class of iteratively constructed self-affine functions there exists an infinite hierarchy of exponents Hq describing the scaling of the qth order height-height correlationfunction Cq(x)~xqhq. Possible applications to random walks and turbulent flows are discussed. It is demonstratedon on the example of random walks along a chain that for stochastic lattice models leading to self-affine fractals Hq exhibits phase-transition-like behavior.

## A.-L. Barabási, T. Vicsek

# Tracing a diffusion-limited-aggregate: self-affine versus self-similar scaling

#### Physical Review A 41, 6881–6883 (1990)

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The geometry of diffusion-limited aggregation clusters is mapped into single-valued functions by tracing the surface of the aggregate and recording the X (or Y) coordinate of the position of a walker moving along perimeter of the cluster as a function of the arc length. Our numerical results and scaling arguments show that the related plots can be considered as self-affine functions whose scaling behavior is determined by the exponent H=1/D, where D is fractal dimension of the aggregates.