Kinetic growth with surface relaxation: Continuum versus atomistic models

SURFACE ROUGHNESS IN MOLECULAR BEAM EPITAXY

This paper discusses the roughness of surfaces described by nonlinear stochastic partial differential equations on bounded domains. Roughness is an important characteristic for processes arising in molecular beam epitaxy, and is usually described by the mean interface width of the surface, i.e. the expected value of the squared Lebesgue norm. By employing results on the mean interface width for linear stochastic partial differential equations perturbed by colored noise, which have been previously obtained, we describe the evolution of the surface roughness for two classes of nonlinear equations, asymptotically both for small and large times.

Exponent inequalities in dynamical systems

In this Letter, we derive exponent inequalities relating the dynamic exponent z to the steady state exponent Γ for a general class of stochastically driven dynamical systems. We begin by deriving a general exact inequality, relating the response function and the correlation function, from which the various exponent inequalities emanate. We then distinguish between two classes of dynamical systems and obtain different and complementary inequalities relating z and Γ. The consequences of those inequalities for a wide set of dynamical problems, including critical dynamics and Kardar-Parisi-Zhang-like problems, are discussed.

Statistics of interfacial fluctuations of radially growing clusters

The dynamics of fluctuating radially growing interfaces is approached using the formalism of stochastic growth equations on growing domains. This framework reveals a number of dynamic features arising during surface growth. For fast growth, dilution, which spatially reorders the incoming matter, is responsible for the transmission of correlations. Its effects include the erasing of memory with respect to the initial condition, a partial attenuation of geometrically originated instabilities, and the restoration of universality in some special cases in which the critical exponents depend on the parameters of the equation of motion. In this sense, dilution rends the dynamics more similar to the usual one of planar systems. This fast growth regime is also characterized by the spatial decorrelation of the interface, which, in the case of radially growing interfaces, naturally originates rapid roughening and scale-dependent fractality, and suggests the advent of a self-similar fractal dimension. The center-of-mass fluctuations of growing clusters are also studied, and our analysis suggests the possible nonapplicability of usual scalings to the long-range surface fluctuations of the radial Eden model. In fact, our study points to the fact that this model belongs to a dilution-free universality class.

Modelling of epitaxial film growth with an Ehrlich-Schwoebel barrier dependent on the step height

The formation of mounded surfaces in epitaxial growth is attributed to the presence of barriers against interlayer diffusion in the terrace edges, known as Ehrlich-Schwoebel (ES) barriers. We investigate a model for epitaxial growth using an ES barrier explicitly dependent on the step height. Our model has an intrinsic topological step barrier even in the absence of an explicit ES barrier. We show that mounded morphologies can be obtained even for a small barrier while a self-affine growth, consistent with the Villain-Lai-Das Sarma equation, is observed in the absence of an explicit step barrier. The mounded surfaces are described by a super-roughness dynamical scaling characterized by locally smooth (facetted) surfaces and a global roughness exponent α > 1. The thin film limit is featured by surfaces with self-assembled three-dimensional structures having an aspect ratio (height/width) that may increase or decrease with temperature depending on the strength of the step barrier.

Roughness exponents and grain shapes

In surfaces with grainy features, the local roughness w shows a crossover at a characteristic length r(c), with roughness exponent changing from α(1)≈1 to a smaller α(2). The grain shape, the choice of w or height-height correlation function (HHCF) C, and the procedure to calculate root-mean-square averages are shown to have remarkable effects on α(1). With grains of pyramidal shape, α(1) can be as low as 0.71, which is much lower than the previous prediction 0.85 for rounded grains. The same crossover is observed in the HHCF, but with initial exponent χ(1)≈0.5 for flat grains, while for some conical grains it may increase to χ(1)≈0.7. The universality class of the growth process determines the exponents α(2)=χ(2) after the crossover, but has no effect on the initial exponents α(1) and χ(1), supporting the geometric interpretation of their values. For all grain shapes and different definitions of surface roughness or HHCF, we still observe that the crossover length r(c) is an accurate estimate of the grain size. The exponents obtained in several recent experimental works on different materials are explained by those models, with some surface images qualitatively similar to our model films.

Interface Roughness: What is it and How is it Measured?

A panel discussion on interface roughness was held at the Fall 1992 Materials Research Society meeting. We present a summary of the results presented by the invited speakers on the application and interpretation of X-ray reflectivity, atomic force microscopy (AFM), scanning tunneling microscopy (STM), photoluminescence and transmission electron microscopy. A transcript of the moderated discussion is provided in the final section.

Epitaxial Growth and Recovery: an Analytic Approach

Most detailed studies of morphological evolution during epitaxial growth and recovery make use of computer-based simulation techniques. In this paper, we discuss an alternative, analytic approach to this problem which takes explicit account of the atomistically random processes of deposition and surface diffusion. Beginning with a master equation representation of the dynamics of a solid-on-solid model of epitaxial growth, we derive a discrete, stochastic equation of motion for the surface profile. This Langevin equation is appropriate for growth studies. In particular, we are able to provide a microscopic justification for a non-linear continuum equation of motion proposed for this problem by others on the basis of heuristic arguments. During recovery, the deposition flux and its associated shot noise are absent. We analyze this process with a completely deterministic equation of motion obtained by performing a statistical average of the original stochastic equation. Results using the latter compare favorably with full Monte Carlo simulations of the original model for the case of the decay of sinusoidally modulated initial surfaces.

A Continuum Approach to Two-Component Thin Film Growth

Theoretical effort so far in understanding epitaxial growth has focused mainly on the one-component growth, i.e. growth that can be fully characterized by a surface (or height) profile. The predictions are also quite limited to the height-height correlation functions as a function of substrate size and the amount of deposition. In this paper, we consider the case of a two-component growth which is quite common in metallic thin films. Instead of using large-scale simulation, we first write down the appropriate two-component growth equations in continuum form. These equations are carefully designed such that in the limit of one-component growth the corresponding equation is recovered. Analytical and numerical analysis of the proposed equations allow us to study the long-range physics associated with these growth processes. Comparison with computer growth experiments is also mentioned.

Extensive studies on linear growth processes with spatiotemporally correlated noise in arbitrary substrate dimensions

An extensive analytical and numerical study on a class of growth processes with spatiotemporally correlated noise in arbitrary dimension is undertaken. In addition to the conventional investigation on the interface morphology and interfacial widths, we pay special attention to exploring the characteristics of the slope-slope correlation function S(r,t) and the [Q]-th degree residual local interfacial width w[Q](l,t), whose importance has been somewhat overlooked in the literature. Based on the above analysis, we give a plausible theoretical explanation about the various experimental observations of kinetically and thermodynamically unstable surface growth. Furthermore, through explicit examples, we show that the statistical methods of calculating the exponents (including the dynamic exponent z, the global roughness exponent α, and the local roughness exponent α(loc)), based on the scaling of S(r,t) and w[Q](l,t), are very reliable and rarely influenced by the finite time and/or finite-size effects. Another important issue we focus on in this paper is related to numerical calculation. For the specific class of growth processes discussed in this paper, we develop a very efficient and accurate algorithm for numerical calculation of the dynamics of interface configuration, the structure factor, the various correlation functions, the interfacial width and its variants in arbitrary dimensions, even with very large system size and very late time. The proposed systematical algorithm can be easily generalized to other linear processes and some special nonlinear processes.

Dynamic scaling in thin-film growth with irreversible step-edge attachment

We study dynamic scaling in a model with collective diffusion (CD) of isolated atoms in terraces and irreversible aggregation at step edges. Simulations are performed in two-dimensional substrates with several diffusion to deposition ratios R identical with D/F. Data collapse of scaled roughness distributions confirms that this model is in the class of the fourth-order nonlinear growth equation by Villain, Lai, and Das Sarma (VLDS) with negligible finite-size effects, while estimates of scaling exponents show some discrepancies. This result is consistent with the prediction of a recent renormalization group approach and improves previous numerical works on related models. The roughness follows dynamic scaling as W=Lalpha/R1/2f(xi/L), with correlation length xi=(Rt)1/z, where z is the dynamic exponent. We also propose a limited mobility (LM) model where the incident atom executes up to G steps before a new atom is adsorbed, and irreversibly aggregates at step edges. This model is also shown to belong to the VLDS class. The size of the plateaus in the film surface increases as G1/2 and the lateral correlation scales as G1/2t1/z. The time evolution of the roughness reproduces that of the CD model if an equivalent parameter G approximately R2/z is chosen. This suggests the possibility of using LM models with tunable diffusion length to simulate processes with simultaneous diffusion of many atoms. A scaling approach is used to justify exponent values and dynamic relations for both models, including the significant decrease of surface roughness as R or G increases.

Transient regimes and crossover for epitaxial surfaces

We apply a formalism for deriving stochastic continuum equations associated with lattice models to obtain equations governing the transient regimes of epitaxial growth for various experimental scenarios and growth conditions. The first step of our methodology is the systematic transformation of the lattice model into a regularized stochastic equation of motion that provides initial conditions for differential renormalization-group (RG) equations for the coefficients in the regularized equation. The solutions of the RG equations then yield trajectories that describe the original model from the transient regimes, which are of primary experimental interest, to the eventual crossover to the asymptotically stable fixed point. We first consider regimes defined by the relative magnitude of deposition noise and diffusion noise. If the diffusion noise dominates, then the early stages of growth are described by the Mullins-Herring (MH) equation with conservative noise. This is the classic regime of molecular-beam epitaxy. If the diffusion and deposition noise are of comparable magnitude, the transient equation is the MH equation with nonconservative noise. This behavior has been observed in a recent report on the growth of aluminum on silicone oil surfaces [Z.-N. Fang, Thin Solid Films 517, 3408 (2009)]. Finally, the regime where deposition noise dominates over diffusion noise has been observed in computer simulations, but does not appear to have any direct experimental relevance. For initial conditions that consist of a flat surface, the Villain-Lai-Das Sarma (VLDS) equation with nonconservative noise is not appropriate for any transient regime. If, however, the initial surface is corrugated, the relative magnitudes of terms can be altered to the point where the VLDS equation with conservative noise does indeed describe transient growth. This is consistent with the experimental analysis of growth on patterned surfaces [H.-C. Kan, Phys. Rev. Lett. 92, 146101 (2004); T. Tadayyon-Eslami, Phys. Rev. Lett. 97, 126101 (2006)].

Dynamic scaling study of vapor deposition polymerization: a Monte Carlo approach

The morphological scaling properties of linear polymer films grown by vapor deposition polymerization are studied by 1+1D Monte Carlo simulations. The model implements the basic processes of random angle ballistic deposition (F) , free-monomer diffusion (D) and monomer adsorption along with the dynamical processes of polymer chain initiation, extension, and merger. The ratio G=D/F is found to have a strong influence on the polymer film morphology. Spatial and temporal behavior of kinetic roughening has been extensively studied using finite-length scaling and height-height correlations H(r,t). The scaling analysis has been performed within the no-overhang approximation and the scaling behaviors at local and global length scales were found to be very different. The global and local scaling exponents for morphological evolution have been evaluated for varying free-monomer diffusion by growing the films at G=10 , 10(2), 10(3), and 10(4) and fixing the deposition flux F. With an increase in G from 10 to 10(4), the average growth exponent beta approximately 0.50 was found to be invariant, whereas the global roughness exponent alpha(g) decreased from 0.87 (1) to 0.73 (1) along with a corresponding decrease in the global dynamic exponent z(g) from 1.71(1) to 1.38(2). The global scaling exponents were observed to follow the dynamic scaling hypothesis, z(g)=alpha(g)/beta. With a similar increase in G however, the average local roughness exponent alpha(l) remained close to 0.46 and the anomalous growth exponent beta(*) decreased from 0.23(4) to 0.18(8). The interfaces display anomalous scaling and multiscaling in the relevant height-height correlations. The variation in H(r,t) with deposition time t indicates nonstationary growth. A comparison has been made between the simulational findings and the experiments wherever applicable.

Continuous and discrete modeling of the decay of two-dimensional nanostructures

In this work we review some recent research on the surface diffusion-mediated decay of two-dimensional nanostructures. These results include both a continuous, vectorial model and a discrete kinetic Monte Carlo approach. Predictions from the standard linear continuous theory of surface-diffusion-driven interface decay are contrasted with simulational results both from kinetic and morphological points of view. In particular, we focused our attention on high-aspect-ratio nanostructures, where strong deviations from linear theory take place, including nonexponential amplitude decay and the emergence of several interesting nanostructures such as overhangs developing, nanoislands and nanovoids formation, loss of convexity, nanostructures-pinch off and nanostructures-break off, etc.

Random deposition of particles of different sizes

We study the surface growth generated by the random deposition of particles of different sizes. A model is proposed where the particles are aggregated on an initially flat surface, giving rise to a rough interface and a porous bulk. By using Monte Carlo simulations, a surface has grown by adding particles of different sizes, as well as identical particles on the substrate in (1+1) dimensions. In the case of deposition of particles of different sizes, they are selected from a Poisson distribution, where the particle sizes may vary by 1 order of magnitude. For the deposition of identical particles, only particles which are larger than one lattice parameter of the substrate are considered. We calculate the usual scaling exponents: the roughness, growth, and dynamic exponents alpha, beta, and z, respectively, as well as, the porosity in the bulk, determining the porosity as a function of the particle size. The results of our simulations show that the roughness evolves in time following three different behaviors. The roughness in the initial times behaves as in the random deposition model. At intermediate times, the surface roughness grows slowly and finally, at long times, it enters into the saturation regime. The bulk formed by depositing large particles reveals a porosity that increases very fast at the initial times and also reaches a saturation value. Excepting the case where particles have the size of one lattice spacing, we always find that the surface roughness and porosity reach limiting values at long times. Surprisingly, we find that the scaling exponents are the same as those predicted by the Villain-Lai-Das Sarma equation.

Atomic Force Microscopy Study of the Kinetic Roughening in Nanostructured Gold Films on SiO2

Dynamic scaling behavior has been observed during the room-temperature growth of sputtered Au films on SiO2using the atomic force microscopy technique. By the analyses of the dependence of the roughness, σ, of the surface roughness power,P(f), and of the correlation length,ξ, on the film thickness,h, the roughness exponent,α = 0.9 ± 0.1, the growth exponent,β = 0.3 ± 0.1, and the dynamic scaling exponent,z = 3.0 ± 0.1 were independently obtained. These values suggest that the sputtering deposition of Au on SiO2at room temperature belongs to a conservative growth process in which the Au grain boundary diffusion plays a dominant role.

Geometric principles of surface growth

I introduce a new equation describing epitaxial growth processes. This equation is derived from a simple variational geometric principle and it has a straightforward interpretation in terms of continuum and microscopic physics. It is also able to reproduce the critical behavior already observed, mound formation and mass conservation, but however does not fit a divergence form as the most commonly spread models of conserved surface growth. This formulation allows me to connect the results of the dynamic renormalization group analysis with intuitive geometric principles, whose generic character may well allow them to describe surface growth and other phenomena in different areas of physics.

Renormalization of stochastic lattice models: epitaxial surfaces

We present the application of a method [C. A. Haselwandter and D. D. Vvedensky, Phys. Rev. E 76, 041115 (2007)] for deriving stochastic partial differential equations from atomistic processes to the morphological evolution of epitaxial surfaces driven by the deposition of new material. Although formally identical to the one-dimensional (1D) systems considered previously, our methodology presents substantial additional technical issues when applied to two-dimensional (2D) surfaces. Once these are addressed, subsequent coarse-graining is accomplished as before by calculating renormalization-group (RG) trajectories from initial conditions determined by the regularized atomistic models. Our applications are to the Edwards-Wilkinson (EW) model [S. F. Edwards and D. R. Wilkinson, Proc. R. Soc. London, Ser. A 381, 17 (1982)], the Wolf-Villain (WV) model [D. E. Wolf and J. Villain, Europhys. Lett. 13, 389 (1990)], and a model with concurrent random deposition and surface diffusion. With our rules for the EW model no appreciable crossover is obtained for either 1D or 2D substrates. For the 1D WV model, discussed previously, our analysis reproduces the crossover sequence known from kinetic Monte Carlo (KMC) simulations, but for the 2D WV model, we find a transition from smooth to unstable growth under repeated coarse-graining. Concurrent surface diffusion does not change this behavior, but can lead to extended transient regimes with kinetic roughening. This provides an explanation of recent experiments on Ge(001) with the intriguing conclusion that the same relaxation mechanism responsible for ordered structures during the early stages of growth also produces an instability at longer times that leads to epitaxial breakdown. The RG trajectories calculated for concurrent random deposition and surface diffusion reproduce the crossover sequences observed with KMC simulations for all values of the model parameters, and asymptotically always approach the fixed point corresponding to the equation proposed by Villain [J. Phys. I 1, 19 (1991)] and by Lai and Das Sarma [Phys. Rev. Lett. 66, 2899 (1991)]. We conclude with a discussion of the application of our methodology to other growth settings.

Multifractal behavior of the surfaces evolved with surface relaxation

A discrete model exhibiting conserved dynamics with nonconserved noise involving particles of different nature, termed as linear and nonlinear, is proposed here. The morphology of the surface has been studied with different abundances of these particles. The saturated surface, slowly evolved from a lower contribution of nonlinear particles to a higher contribution of nonlinear particles, splits into four distinct scaling regimes with three crossover lengths. Each regime is characterized by different scaling property. It is shown that when the contribution of the nonlinear particles crosses a critical value, the surface morphology shows a linear-nonlinear "phase transition." The roughness exponent in a nonlinear regime is well compared with that of the continuum nonlinear equation in a molecular beam epitaxy (MBE) class as well as a MBE motivated discrete model.

Maximal- and minimal-height distributions of fluctuating interfaces

Maximal- and minimal-height distributions (MAHD, MIHD) of two-dimensional interfaces grown with the nonlinear equations of Kardar-Parisi-Zhang (KPZ, second order) and of Villain-Lai-Das Sarma (VLDS, fourth order) are shown to be different. Two universal curves may be MAHD or MIHD of each class depending on the sign of the relevant nonlinear term, which is confirmed by results of several lattice models in the KPZ and VLDS classes. The difference between MAHD and MIDH is connected with the asymmetry of the local height distribution. A simple, exactly solvable deposition-erosion model is introduced to illustrate this feature. The average extremal heights scale with the same exponent of the average roughness. In contrast to other correlated systems, generalized Gumbel distributions do not fit those MAHD and MIHD, nor those of Edwards-Wilkinson growth.

Position-dependent expression of GADD45alpha in rat brain tumours

Although the complex and multifactorial process of tumour growth has been extensively studied for decades, our understanding of the fundamental relationship between tumour growth dynamics and genetic expression profile remains incomplete. Recent studies of tumour dynamics indicate that gene expression in solid tumours would depend on the distance from the centre of the tumour. Since tumour proliferative activity is mainly localised to its external zone, and taking into account that generation and expansion of genetic mutations depend on the number of cell divisions, important differences in gene expression between central and peripheral sections of the same tumour are to be expected. Here, we have studied variations in the genetic expression profile between peripheral and internal samples of the same brain tumour. We have carried out microarray analysis of mRNA expression, and found a differential profile of genetic expression between the two cell subsets. In particular, one major nuclear protein that regulates cell responses to DNA-damaging and stress signals, GADD45alpha, was expressed at much lower levels in the peripheral zone, as compared to tumour core samples. These differences in GADD45alpha mRNA transcription levels have been confirmed by quantitative analysis via real time PCR, and protein levels of GADD45alpha also exhibit the same pattern of differential expression. Our findings suggest that GADD45alpha might play a major role in the regulation of brain tumour invasive potential.

Finite-size effects in roughness distribution scaling

We study numerically finite-size corrections in scaling relations for roughness distributions of various interface growth models. The most common relation, which considers the average roughness <w(2)> as a scaling factor, is not obeyed in the steady states of a group of ballisticlike models in 2+1 dimensions, even when very large system sizes are considered. On the other hand, good collapse of the same data is obtained with a scaling relation that involves the root mean square fluctuation of the roughness, which can be explained by finite-size effects on second moments of the scaling functions. We also obtain data collapse with an alternative scaling relation that accounts for the effect of the intrinsic width, which is a constant correction term previously proposed for the scaling of <w(2)> . This illustrates how finite-size corrections can be obtained from roughness distributions scaling. However, we discard the usual interpretation that the intrinsic width is a consequence of high surface steps by analyzing data of restricted solid-on-solid models with various maximal height differences between neighboring columns. We also observe that large finite-size corrections in the roughness distributions are usually accompanied by huge corrections in height distributions and average local slopes, as well as in estimates of scaling exponents. The molecular-beam epitaxy model of Das Sarma and Tamborenea in 1+1 dimensions is a case example in which none of the proposed scaling relations work properly, while the other measured quantities do not converge to the expected asymptotic values. Thus although roughness distributions are clearly better than other quantities to determine the universality class of a growing system, it is not the final solution for this task.

Pseudospectral versus finite-difference schemes in the numerical integration of stochastic models of surface growth

We present a comparison between finite differences schemes and a pseudospectral method applied to the numerical integration of stochastic partial differential equations that model surface growth. We have studied, in 1+1 dimensions, the Kardar, Parisi, and Zhang model (KPZ) and the Lai, Das Sarma, and Villain model (LDV). The pseudospectral method appears to be the most stable for a given time step for both models. This means that the time up to which we can follow the temporal evolution of a given system is larger for the pseudospectral method. Moreover, for the KPZ model, a pseudospectral scheme gives results closer to the predictions of the continuum model than those obtained through finite difference methods. On the other hand, some numerical instabilities appearing with finite difference methods for the LDV model are absent when a pseudospectral integration is performed. These numerical instabilities give rise to an approximate multiscaling observed in earlier numerical simulations. With the pseudospectral approach no multiscaling is seen in agreement with the continuum model.

Renormalization of stochastic lattice models: basic formulation

We describe a general method for the multiscale analysis of stochastic lattice models. Beginning with a lattice Langevin formulation of site fluctuations, we derive stochastic partial differential equations by regularizing the transition rules of the model. Subsequent coarse graining is accomplished by calculating renormalization-group (RG) trajectories from initial conditions determined by the regularized atomistic models. The RG trajectories correspond to hierarchies of continuum equations describing lattice models over expanding length and time scales. These continuum equations retain a quantitative connection over different scales, as well as to the underlying atomistic dynamics. This provides a systematic method for the derivation of continuum equations from the transition rules of lattice models for any length and time scales. As an illustration we consider the one-dimensional (1D) Wolf-Villain (WV) model [Europhys. Lett. 13, 389 (1990)]. The RG analysis of this model, which we develop in detail, is generic and can be applied to a wide range of conservative lattice models. The RG trajectory of the 1D WV model shows a complex crossover sequence of linear and nonlinear stochastic differential equations, which is in excellent agreement with kinetic Monte Carlo simulations of this model. We conclude by discussing possible applications of the multiscale method described here to other nonequilibrium systems.

Multiscale theory of fluctuating interfaces: renormalization of atomistic models

We describe a framework for the multiscale analysis of atomistic surface processes which we apply to a model of homoepitaxial growth with deposition according to the Wolf-Villain model and concurrent surface diffusion. Coarse graining is accomplished by calculating renormalization-group (RG) trajectories from initial conditions determined by the regularized atomistic theory. All of the crossover and asymptotic scaling regimes known from computer simulations are obtained, but we also find that two-dimensional substrates show an intriguing transition from smooth to mounded morphologies along the RG trajectory.

Symmetry-based determination of space-time functions in nonequilibrium growth processes

We study the space-time correlation and response functions in nonequilibrium growth processes described by linear stochastic Langevin equations. Exploiting exclusively the existence of space- and time-dependent symmetries of the noiseless part of these equations, we derive expressions for the universal scaling functions of two-time quantities which are found to agree with the exact expressions obtained from the stochastic equations of motion. The usefulness of the space-time functions is illustrated through the investigation of two atomistic growth models, the Family model and the restricted Family model, which are shown to belong to a unique universality class in 1+1 and 2+1 space dimensions. This corrects earlier studies which claimed that in 2+1 dimensions the two models belong to different universality classes.

Temperature-driven change in the unstable growth mode on patterned GaAs(001)

We observe a dramatic change in the unstable growth mode during GaAs molecular beam epitaxy on patterned GaAs(001) as the temperature is lowered through approximately 540 degrees C, roughly coincident with the preroughening temperature. Observations of the As2 flux dependence, however, rule out thermodynamic preroughening as driving the growth mode change. Similar observations rule out the change in surface reconstruction as the cause. Instead, we find evidence that the change in the unstable growth mode can be explained by a competition between the decreased adatom collection rate on small terraces and a small anisotropic barrier to adatom diffusion downward across step bunches.

Morphology characterization of layer-by-layer films from PAH/MA-co-DR13: the role of film thickness

We report on the use of dynamic scale theory and fractal analyses in the study of distinct growth stages of layer-by-layer (LBL) films of poly(allylamine hydrochloride) (PAH) and a side-chain-substituted azobenzene copolymer (Ma-co-DR13). The LBL films were adsorbed on glass substrates and characterized with atomic force microscopy with the Ma-co-DR13 at the top layer. The granular morphology exhibited by the films allowed the observation of the growth process inside and outside the grains. The growth outside the grains was found to follow the Kardar-Parisi-Zhang model, with fractal dimensions of ca. 2.6. One could expect that inside the grains the morphology would be close to a Euclidian surface with fractal dimension of ca. 2 for any growth stage. The latter, however, was observed only for thicker films containing more than 10 bilayers. For thinner films the morphology was well described by a self-affine fractal. Such dependence of the growth behavior with the film thickness is associated with a more complete coverage of adsorption sites in thicker films due to diffusion of polymer molecules.

Strongly anisotropic roughness in surfaces driven by an oblique particle flux

Using field theoretic renormalization, an MBE-type growth process with an obliquely incident influx of atoms is examined. The projection of the beam on the substrate plane selects a "parallel" direction, with rotational invariance restricted to the transverse directions. Depending on the behavior of an effective anisotropic surface tension, a line of second-order transitions is identified, as well as a line of potentially first-order transitions, joined by a multicritical point. Near the second-order transitions and the multicritical point, the surface roughness is strongly anisotropic. Four different roughness exponents are introduced and computed, describing the surface in different directions, in real or momentum space. The results presented challenge an earlier study of the multicritical point.

Dynamic properties in a family of competitive growing models

The properties of a wide variety of growing models, generically called X-RD, involving the deposition of particles according to competitive processes, such that a particle is attached to the aggregate with probability p following the mechanisms of a generic model X that provides the correlations and at random [random deposition (RD)] with probability (1-p), are studied by means of numerical simulations and analytic developments. The study comprises the following X models: Ballistic deposition, random deposition with surface relaxation, Das Sarma-Tamboronea, Kim-Kosterlitz, Lai-Das Sarma, Wolf-Villain, large curvature, and three additional models that are variants of the ballistic deposition model. It is shown that after a growing regime, the interface width becomes saturated at a crossover time (tx2) that, by fixing the sample size, scales with p according to tx2(p) proportional variant p-y (P>0), where is an exponent. Also, the interface width at saturation (Wsat) scales as Wsat(p) proportional variant p-delta (p>0), where delta is another exponent. It is proved that, in any dimension, the exponents delta and y obey the following relationship: delta=y beta RD, where beta RD=1/2 is the growing exponent for RD. Furthermore, both exponents exhibit universality in the p --> 0 limit. By mapping the behavior of the average height difference of two neighboring sites in discrete models of type X-RD and two kinds of random walks, we have determined the exact value of the exponent delta. When the height difference between two neighbouring sites corresponds to a random walk that after walking <n> steps returns to a distance from its initial position that is proportional to the maximum distance reached (random walk of type A), one has delta=1/2. On the other hand, when the height difference between two neighboring sites corresponds to a random walk that after <n> steps moves <l> steps towards the initial position (random walk of type B), one has delta=1. Finally, by linking four well-established universality classes (namely Edwards-Wilkinson, Kardar-Parisi-Zhang, linear [molecular beam epitaxy (MBE)] and nonlinear MBE) with the properties of type A and B of random walks, eight different stochastic equations for all the competitive models studied are derived.

Scaling in the crossover from random to correlated growth

In systems where deposition rates are high compared to diffusion, desorption, and other mechanisms that generate correlations, a crossover from random to correlated growth of surface roughness is expected at a characteristic time t0. This crossover is analyzed in lattice models via scaling arguments, with support from simulation results presented here and in other works. We argue that the amplitudes of the saturation roughness and of the saturation time t(x) scale as t0(1/2) and t0, respectively. For models with lateral aggregation, which typically are in the Kardar-Parisi-Zhang (KPZ) class, we show that t0 approximately p(-1), where p is the probability of the correlated aggregation mechanism to take place. However, t0 approximately p(-2) is obtained in solid-on-solid models with single-particle deposition attempts. This group includes models in various universality classes, with numerical examples being provided in the Edwards-Wilkinson (EW), KPZ, and Villain-Lai-Das Sarma (nonlinear molecular-beam epitaxy) classes. Most applications are for two-component models in which random deposition, with probability 1-p, competes with a correlated aggregation process with probability p. However, our approach can be extended to other systems with the same crossover, such as the generalized restricted solid-on-solid model with maximum height difference S, for large S. Moreover, the scaling approach applies to all dimensions. In the particular case of one-dimensional KPZ processes with this crossover, we show that t0 approximately nu(-1) and nu approximately lambda(2/3), where nu and lambda are the coefficients of the linear and nonlinear terms of the associated KPZ equations. The applicability of previous results to models in the EW and KPZ classes is discussed.

Large Area Liquid Crystal Monodomain Field-Effect Transistors

Butyl, hexyl, and decyl derivatives of the liquid-crystalline organic semiconductor 5,5' '-bis(5-alkyl-2-thienylethynyl)-2,2':5',2' '-terthiophene were synthesized and studied with respect to their structural, optical, and electrical properties. By means of an optimized thermal annealing scheme the hexyl and decyl compounds could be processed into self-assembled monodomain films of up to 150 mm in diameter. These were investigated with X-ray diffractometry, which revealed a clearly single-crystalline monoclinic morphology with lamellae parallel to the substrate. Within the lamellae the molecules were found to arrange with a tilt of about 50 degrees with the rubbing direction of the polyimide alignment layer. The resulting, close side-to-side packing was confirmed by measurements of the UV/vis absorption, which showed a dichroic ratio of 19 and indicated H-aggregation. AFM analyses revealed self-affinity in the surface roughness of the monodomain. The compounds showed bipolar charge transport in TOF measurements, with hole mobilities reaching up to 0.02 cm(2)/Vs and maximum electron mobilities around 0.002 cm(2)/Vs. The hexyl derivative was processed into large-area monodomain top-gate field-effect transistors, which were stable for months and showed anisotropic hole mobilities of up to 0.02 cm(2)/Vs. Compared to multidomain bottom-gate transistors the monodomain formation allowed for a mobility increase by 1 order of magnitude.

Modeling growth from the vapor and thermal annealing on micro- and nanopatterned substrates

We propose a -dimensional mesoscopic model to describe the most relevant physical processes that take place while depositing and/or annealing micro- and nanopatterned solid substrates. The model assumes that a collimated incident beam impinges over the growing substrate; scattering effects in the vapor and reemission processes are introduced in a phenomenological way as an isotropic flow. Surface diffusion is included as the main relaxation process at the micro- or nanoscale. The stochastic model is built following population dynamics considerations; both stochastic simulations and the deterministic limit are analyzed. Numerical aspects regarding its implementation are also discussed. We study the shape-preserving growth mode, the coupling between shadowing effects and random fluctuations, and the spatial structure of noises using numerical simulations. We report important deviations from linear theories of surface diffusion when the interfaces are not compatible with the small slope approximation, including spontaneous formation of overhangs and nonexponential decay of pattern amplitudes. We discuss the dependence of stationary states with respect to the boundary conditions imposed on the system.

Langevin equations for fluctuating surfaces

Exact Langevin equations are derived for the height fluctuations of surfaces driven by the deposition of material from a molecular beam. We consider two types of model: deposition models, where growth proceeds by the deposition and instantaneous local relaxation of particles, with no subsequent movement, and models with concurrent random deposition and surface diffusion. Starting from a Chapman-Kolmogorov equation the deposition, relaxation, and hopping rules of these models are first expressed as transition rates within a master equation for the joint height probability density function. The Kramers-Moyal-van Kampen expansion of the master equation in terms of an appropriate "largeness" parameter yields, according to a limit theorem due to Kurtz [Stoch. Proc. Appl. 6, 223 (1978)], a Fokker-Planck equation that embodies the statistical properties of the original lattice model. The statistical equivalence of this Fokker-Planck equation, solved in terms of the associated Langevin equation, and solutions of the Chapman-Kolmogorov equation, as determined by kinetic Monte Carlo (KMC) simulations of the lattice transition rules, is demonstrated by comparing the surface roughness and the lateral height correlations obtained from the two formulations for the Edwards-Wilkinson [Proc. R. Soc. London Ser. A 381, 17 (1982)] and Wolf-Villain [Europhys. Lett. 13, 389 (1990)] deposition models, and for a model with random deposition and surface diffusion. In each case, as the largeness parameter is increased, the Langevin equation converges to the surface roughness and lateral height correlations produced by KMC simulations for all times, including the crossover between different scaling regimes. We conclude by examining some of the wider implications of these results, including applications to heteroepitaxial systems and the passage to the continuum limit.

Volatility, persistence, and survival in financial markets

We study the temporal fluctuations in time-dependent stock prices (both individual and composite) as a stochastic phenomenon using general techniques and methods of nonequilibrium statistical mechanics. In particular, we analyze stock price fluctuations as a non-Markovian stochastic process using the first-passage statistical concepts of persistence and survival. We report the results of empirical measurements of the normalized qth-order correlation functions fq(t), survival probability S(t), and persistence probability P(t) for several stock market dynamical sets. We analyze both minute-to-minute and higher-frequency stock market recordings (i.e., with the sampling time deltat of the order of days). We find that the fluctuating stock price is multifractal and the choice of deltat has no effect on the qualitative multifractal behavior displayed by the 1/q dependence of the generalized Hurst exponent Hq associated with the power-law evolution of the correlation function fq(t) approximately tHq. The probability S(t) of the stock price remaining above the average up to time t is very sensitive to the total measurement time tm and the sampling time. The probability P(t) of the stock not returning to the initial value within an interval t has a universal power-law behavior P(t) approximately t(-theta), with a persistence exponent theta close to 0.5 that agrees with the prediction theta=1-H2. The empirical financial stocks also present an interesting feature found in turbulent fluids, the extended self-similarity.

A microscopic view on acoustomigration

Stress-induced material transport in surface acoustic wave devices, so-called acoustomigration, is a prominent failure mechanism, especially in high-power applications. We used scanning probe microscopy techniques to study acoustomigration of metal structures in-situ, i.e., during the high-power loading of the device. Scanning acoustic force microscopy (SAFM) allows for the simultaneous measurement of the acoustic wavefield and the topography with submicron lateral resolution. High-resolution microscopy is essential as acoustomigration is a phenomenon that not only results in the formation of more macroscopic voids and hillocks but also affects the microscopic grain structure of the film. We present acoustic wavefield and topographic image sequences giving a clear insight into the nature of the film damage on a submicron scale. The 900 MHz test structures were fabricated on 36 degrees YX-lithium tantalate (YX-LiTaO3) and incorporated 420-nm thick aluminium (Al) electrodes. By correlating the acoustic wavefield mapping and the local changes in topography, we confirmed model calculations that predict the correspondence of damage and stress (i.e., hillocks and voids) are preferentially formed in areas of high stress. The way the film is damaged does not significantly depend on the applied power (for typical power levels used in this study). Furthermore, acoustomigration leads to smoother surfaces via lateral grain growth. Another contribution to the grain dynamics comes from the apparent grain rotation in the highly anisotropic stress field of an acoustic wave. Thus, through in-situ scanning probe microscopy techniques, one can observe the initial changes of the grain structure in order to obtain a more detailed picture of the phenomenon of acoustomigration.

Numerical study of roughness distributions in nonlinear models of interface growth

We analyze the shapes of roughness distributions of discrete models in the Kardar, Parisi, and Zhang (KPZ) and in the Villain, Lai, and Das Sarma (VLDS) classes of interface growth, in one and two dimensions. Three KPZ models in d=2 confirm the expected scaling of the distribution and show a stretched exponential tail approximately as exp(-x0.8), with a significant asymmetry near the maximum. Conserved restricted solid-on-solid models belonging to the VLDS class were simulated in d=1 and d=2. The tail in d=1 has the form exp(-x2) and, in d=2, has a simple exponential decay, but is quantitatively different from the distribution of the linear fourth-order (Mullins-Herring) theory. It is not possible to fit any of the above distributions to those of 1/f(alpha) noise interfaces, in contrast with recently studied models with depinning transitions.

Interplay of instabilities in mounded surface growth

We numerically study a one-dimensional conserved growth equation with competing linear (Ehrlich-Schwoebel) and nonlinear instabilities. As a control parameter is varied, this model exhibits a nonequilibrium phase transition between two mounded states, one of which exhibits slope selection and the other does not. The coarsening behavior of the mounds in these two phases is studied in detail. In the absence of noise, the steady-state configuration depends crucially on which of the two instabilities dominates the early time behavior.

Numerical study of discrete models in the class of the nonlinear molecular beam epitaxy equation

We study numerically some discrete growth models belonging to the class of the nonlinear molecular beam epitaxy equation, or the Villain-Lai-Das Sarma (VLDS) equation. The conserved restricted solid-on-solid model (CRSOS) with maximum height differences Delta H(max)=1 and Delta H(max)=2 was analyzed in substrate dimensions d=1 and d=2 . The Das Sarma and Tamborenea (DT) model and a competitive model involving random deposition and CRSOS deposition were studied in d=1. For the CRSOS model with Delta H(max)=1, we obtain the more accurate estimates of scaling exponents in d=1:roughness exponent alpha=0.94+/-0.02 and dynamical exponent z=2.88+/-0.04. These estimates are significantly below the values of one-loop renormalization for the VLDS theory, which confirms Janssen's proposal of the existence of higher-order corrections. The roughness exponent in d=2 is very near the one-loop result alpha=2/3, in agreement with previous works. The moments W(n) of orders n=2 , 3, 4 of the height distribution were calculated for all models, and the skewness S triple bond W3/W(3/2)(2) and the kurtosis Q triple bond W4/W(2)2-3 were estimated. At the steady states, the CRSOS models and the competitive model have nearly the same values of S and Q in d=1, which suggests that these amplitude ratios are universal in the VLDS class. The estimates for the DT model are different, possibly due to their typically long crossover to asymptotic values. Results for the CRSOS models in d=2 also suggest that those quantities are universal.

Initial-stage growth phenomena and distribution of local configurations of the restricted solid-on-solid model

We take a detailed study on the restricted solid-on-solid (RSOS) model with finite nearest-neighbor height difference S. We numerically show that, for all finite values of S, the system belongs to the random-deposition (RD) class in the early time stage and then crossovers to the Kardar-Parisi-Zhang (KPZ) class. We find that the crossover time scales as Szeta with the crossover exponent zeta=2.06. Besides, we analytically study the RSOS model by grouping consecutive sites into local configurations to obtain the Markov chain describing the time evolution of the probability distribution of these local configurations. For demonstration, we use the RSOS model with S=2 as an explicit example and calculate the correlation functions and even scaling exponents based on the obtained probability distribution of local configurations. The results are very consistent with those obtained from direct simulation of the RSOS model.

Kinetic roughening of ion-sputtered Pd(001) surface: beyond the Kuramoto-Sivashinsky model

We investigate the kinetic roughening of Ar+ ion-sputtered Pd(001) surface both experimentally and theoretically. In situ real-time x-ray reflectivity and in situ scanning tunneling microscopy show that nanoscale adatom islands form and grow with increasing sputter time t. Surface roughness W(t) and lateral correlation length xi(t) follow the scaling laws W(t) approximately t(beta) and xi(t) approximately t(1/z) with the exponents beta approximately 0.20 and 1/z approximately 0.20, for an ion beam energy epsilon=0.5 keV, which is inconsistent with the prediction of the Kuramoto-Sivashinsky (KS) model. We thereby extend the KS model by applying the coarse-grained continuum approach of the Sigmund theory to the order of O(inverted Delta(4),h(2)), where h is the surface height, and derive a new term of the form inverted Delta(2)(inverted Delta h)(2) which plays a decisive role in describing the observed morphological evolution of the sputtered surface.

Pinning of tumoral growth by enhancement of the immune response

Tumor growth is a surface phenomenon of the molecular beam epitaxy universality class in which diffusion at the surface is the determining factor. This Letter reports experiments performed in mice showing that these dynamics can, however, be changed. By stimulating the immune response, we induced strong neutrophilia around the tumor. The neutrophils hindered cell surface diffusion so much that they induced new dynamics compatible with the slower quenched-disorder Edwards-Wilkinson universality class. Important clinical effects were also seen, including remarkably high tumor necrosis (around 80%-90% of the tumor), a general increase in survival time [the death ratio in the control group is 15.76 times higher than in the treated group (equivalent to a Cox's model hazard ratio of 0.85; 95% confidence interval 0.76-0.95, p=0.004)], and even the total elimination of some tumors.

Persistence in nonequilibrium surface growth

Persistence probabilities of the interface height in ( 1+1 ) - and ( 2+1 ) -dimensional atomistic, solid-on-solid, stochastic models of surface growth are studied using kinetic Monte Carlo simulations, with emphasis on models that belong to the molecular beam epitaxy (MBE) universality class. Both the initial transient and the long-time steady-state regimes are investigated. We show that for growth models in the MBE universality class, the nonlinearity of the underlying dynamical equation is clearly reflected in the difference between the measured values of the positive and negative persistence exponents in both transient and steady-state regimes. For the MBE universality class, the positive and negative persistence exponents in the steady-state are found to be theta(S)(+) =0.66+/-0.02 and theta(S)(-) =0.78+/-0.02, respectively, in ( 1+1 ) dimensions, and theta(S)(+) =0.76+/-0.02 and theta(S)(-) =0.85+/-0.02, respectively, in ( 2+1 ) dimensions. The noise reduction technique is applied on some of the ( 1+1 ) -dimensional models in order to obtain accurate values of the persistence exponents. We show analytically that a relation between the steady-state persistence exponent and the dynamic growth exponent, found earlier to be valid for linear models, should be satisfied by the smaller of the two steady-state persistence exponents in the nonlinear models. Our numerical results for the persistence exponents are consistent with this prediction. We also find that the steady-state persistence exponents can be obtained from simulations over times that are much shorter than that required for the interface to reach the steady state. The dependence of the persistence probability on the system size and the sampling time is shown to be described by a simple scaling form.

Transient evolution of surface roughness on patterned GaAs(001) during homoepitaxial growth

We have investigated the length scale dependence of the transient evolution of surface roughness during homoepitaxial growth on GaAs(100), patterning the surface lithographically with an array of cylindrical pits of systematically varied sizes and spacings. Our atomic force microscopy measurements show that the amplitude of the surface corrugation has nonmonotonic behavior in both the length scale dependence and time evolution. This behavior allows us to rule out a number of existing continuum models of growth.

Scaling properties of self-expanding surfaces

Scaling properties of self-expanding surfaces are studied with a comparison to those of self-flattening surfaces [Phys. Rev. E 66, 040602(R) (2002)]. The evolution of self-expanding surfaces is described by a restricted solid-on-solid type monomer deposition-evaporation model in which both deposition at the globally lowest site and evaporation at the globally highest site are suppressed. We find numerically that equilibrium surface fluctuation has a scaling behavior with a roughness exponent alpha approximately 1 in one dimension (1D). In contrast, 2D equilibrium surfaces show the same dynamical scaling behavior with alpha=0 (log) and dynamic exponent z approximately 5/2 as 2D self-flattening surfaces. Stationary roughness can be understood analytically by relating the self-expanding growth model to self-repelling random walks. In the case of nonequilibrium growing/eroding surfaces, self-expanding dynamics cause the fluctuation of surfaces to be characterized by alpha approximately 1 in both 1D and 2D.

Mound formation and coarsening from a nonlinear instability in surface growth

We study spatially discretized versions of a class of one-dimensional, nonequilibrium, conserved growth equations for both nonconserved and conserved noise using numerical integration. An atomistic version of these growth equations is also studied using stochastic simulation. The models with nonconserved noise are found to exhibit mound formation and power-law coarsening with slope selection for a range of values of the model parameters. Unlike previously proposed models of mound formation, the Ehrlich-Schwoebel step-edge barrier, usually modeled as a linear instability in growth equations, is absent in our models. Mound formation in our models occurs due to a nonlinear instability in which the height (depth) of spontaneously generated pillars (grooves) increases rapidly if the initial height (depth) is sufficiently large. When this instability is controlled by the introduction of a nonlinear control function, the system exhibits a first-order dynamical phase transition from a rough self-affine phase to a mounded one as the value of the parameter that measures the effectiveness of control is decreased. We define an "order parameter" that may be used to distinguish between these two phases. In the mounded phase, the system exhibits power-law coarsening of the mounds in which a selected slope is retained at all times. The coarsening exponents for the spatially discretized continuum equation and the atomistic model are found to be different. An explanation of this difference is proposed and verified by simulations. In the spatially discretized growth equation with conserved noise, we find the curious result that the kinetically rough and mounded phases are both locally stable in a region of parameter space. In this region, the initial configuration of the system determines its steady-state behavior.

Nonlinear stochastic equations with calculable steady states

We consider generalizations of the Kardar-Parisi-Zhang equation that accommodate spatial anisotropies and the coupled evolution of several fields, and focus on their symmetries and nonperturbative properties. In particular, we derive generalized fluctuation-dissipation conditions on the form of the (nonlinear) equations for the realization of a Gaussian probability density of the fields in the steady state. For the amorphous growth of a single height field in one dimension we give a general class of equations with exactly calculable (Gaussian and more complicated) steady states. In two dimensions, we show that any anisotropic system evolves in long time and length scales either to the usual isotropic strong coupling regime or to a linearlike fixed point associated with a hidden symmetry. Similar results are derived for textural growth equations that couple the height field with additional order parameters which fluctuate on the growing surface. In this context, we propose phenomenological equations for the growth of a crystalline material, where the height field interacts with lattice distortions, and identify two special cases that obtain Gaussian steady states. In the first case compression modes influence growth and are advected by height fluctuations, while in the second case it is the density of dislocations that couples with the height.