Wolf Villain Research Articles

Bifractality in the one-dimensional Wolf-Villain model.

We introduce a multifractal optimal detrended fluctuation analysis to study the scaling properties of the one-dimensional Wolf-Villain (WV) model for surface growth. This model produces coarsened surface morphologies for long timescales (up to 10^{9} monolayers) and its universality class remains an open problem. Our results for the multifractal exponent τ(q) reveal an effective local roughness exponent consistent with a transient given by the molecular beam epitaxy (MBE) growth regime and Edwards-Wilkinson (EW) universality class for negative and positive q values, respectively. Therefore, although the results corroborate that long-wavelength fluctuations belong to the EW class in the hydrodynamic limit, as conjectured in the recent literature, a bifractal signature of the WV model with an MBE regime at short wavelengths was observed.

Analyzing asymmetry in the Wolf-Villain model via the study of the persistence of various initial height fluctuations

The Wolf-Villain (WV) model, which, in some literature, has shown trait of up-down asymmetry, is investigated through the study of the persistence probability of height fluctuations in simulated film surfaces. The persistence probability is the probability that the height fluctuation of each site does not cross its initial value (h 0) along a time interval. When averaged over all possible values of h 0, the probability is known to exhibit a power law decay behavior with the exponent called persistence exponent. In this work, instead of averaging over all h 0, we consider a fixed value of h 0 and find that the persistence probability of the WV model still decreases with time as a power law if the fixed value meets a necessary condition, i.e. h 0 is roughly equal to or larger than the saturated roughness of the film. The persistence exponent for each h 0 is measured and found to decrease as the value of h 0 increases. Scaling form of the persistence probability at a specific h 0 is also studied. Notably, all results obtained here are in agreement with those of the up-down symmetric models appeared in literatures despite the fact that the WV model is an asymmetric model. These results support the idea from other studies that the asymmetry in the WV is very weak.

Statistical investigation on the dynamic behaviors of maximal spatial persistence in saturated surfaces of discrete growth models

The dynamic scaling behaviors of the maximal spatial persistence τmax in saturated surface of (1+1)-dimensional discrete growth models, including Wolf–Villain model, Family model, Ballistic Deposition model and Restricted Solid-on-Solid model, are investigated by means of Kinetic Monte Carlo method. Linear relationships are found for the mean value <τmax(L)>≈A⋅L, and the root-mean square deviation (i.e. the second moment) σ(L)≈B⋅L (where σ=(〈τmax2〉−〈τmax〉2)1∕2), respectively, versus the system size L. The distribution of the maximal spatial persistence is also analyzed, and data collapsing for different system sizes to a single curve implying good scaling behavior. It is shown that the distribution of maximal spatial persistence does not satisfy the commonly used extreme statistical distribution, namely Weibull and Gumbel distribution, but can better conform to Asym2Sig distribution.

Schramm–Loewner evolution theory of the asymptotic behaviors of (2+1)-dimensional Wolf–Villain model

Abstract Although extensive analytical and numerical work has focus on investigating the (2+1)-dimensional Wolf–Villain (WV) model, some problems concerning its asymptotical behaviors, such as the universality class to which it belongs remain controversial. The Schramm–Loewner evolution (SLE) theory is an attractive approach to describe the fluctuation phenomena and random processes, and has also been applied to the analysis of the stochastic growth of surfaces. In this work, we applied SLE theory to the analysis of the saturated surface to conduct in-depth research, with a new perspective, on the asymptotical behaviors of the (2+1)-dimensional WV model. On the basis of the analysis of the saturated surface contour lines, we determine that the diffusion coefficient calculated for (2+1)-dimensional WV model is κ = 2.91 ± 0.01 , and that for Family model is κ = 2.88 ± 0.01 . Accordingly we can conclude from the view of SLE theory, that the (2+1)-dimensional WV model, similar to the Family model, also belongs to the Edwards–Wilkinson universality class.

Roughness distribution of multiple hit and long surface diffusion length noise reduced discrete growth models

Abstract Conventionally, the universality class of a discrete growth model is identified via the scaling of interface width. This method requires large-scale simulations to minimize finite-size effects on the results. The multiple hit noise reduction techniques ( m > 1 NRT) and the long surface diffusion length noise reduction techniques ( l > 1 NRT) have been used to promote the asymptotic behaviors of the growth models. Lately, an alternative method involving comparison of roughness distribution in the steady state has been proposed. In this work, the roughness distribution of the (2 + 1)-dimensional Das Sarma–Tamborenea (DT), Wolf–Villain (WV), and Larger Curvature (LC) models, with and without NRTs, are calculated in order to investigate effects of the NRTs on the roughness distribution. Additionally, effective growth exponents of the noise reduced (2 + 1)-dimensional DT, WV and LC models are also calculated. Our results indicate that the NRTs affect the interface width both in the growth and the saturation regimes. In the steady state, the NRTs do not seem to have any impact on the roughness distribution of the DT model, but it significantly changes the roughness distribution of the WV and LC models to the normal distribution curves.

Vorticity Generation by Rough Walls in 2D Decaying Turbulence

In this work we present Lattice Boltzmann simulations of a decaying vortex array in a 2D rectangular domain, which is bounded by a random rough wall from one side. In order to separate the effects of the collisions with the rough wall, the opposite (smooth) rigid wall is placed at a larger distance from the center of the vortex array. Periodic boundary condition is imposed in the perpendicular direction. Well defined random roughness is generated by the widely studied Wolf–Villain surface growth algorithm. The main finding is that collisions with a rough wall generate excess vorticity compared with a smooth boundary, while the kinetic energy decreases monotonously. A proper measure is the integrated excess enstrophy, which exhibits an apparent maximum at an “optimal” roughness range. Numerical values of the excess enstrophy are very sensitive to a particular configuration (wall shape and vortex lattice randomization), however the “optimal” roughness exhibits surface features of similar characteristic sizes than that of the decaying vortices.

Conformal Invariance of Isoheight Lines of the (2+1)-Dimensional Wolf-Villain Surfaces

In order to learn more about the statistical properties of the surface fluctuations of the Wolf-Villain (WV) model and confirm the applicability of the Schramm-Loewner evolution theory (SLEκ) to WV surfaces, the contour lines of the (2+1)-dimensional saturated surfaces of the WV model are investigated numerically. The results obtained show that the isoheight lines of the WV surfaces are conformally invariant and can also be described in the frame of the SLEκ theory with the diffusivity κ=4. The calculated fractal dimension of the contours is df=1.31±0.02.

Transitions in a probabilistic interface growth model

We study a generalization of the Wolf–Villain (WV) interface growth model based on aprobabilistic growth rule. In the WV model, particles are randomly deposited onto asubstrate and subsequently move to a position nearby where the binding is strongest. Weintroduce a growth probability which is proportional to a power of the numberni of bindingsof the site i: . Through extensive simulations, in(1 + 1) dimensions, we find three behaviors depending on theν value: (i) ifν is small, acrossover from the Mullins–Herring to the Edwards–Wilkinson (EW) universality class; (ii) for intermediatevalues of ν, a crossover from the EW to the Kardar–Parisi–Zhang (KPZ) universality class; and, finally, (iii) forlarge ν values, the system is always in the KPZ class. In(2 + 1) dimensions, we obtain three different behaviors: (i) a crossover fromthe Villain–Lai–Das Sarma to the EW universality class for smallν values; (ii) the EW class is always present for intermediateν values; and (iii) a deviation from the EW class is observed for largeν values.

Mound morphology of the 2+1 -dimensional Wolf–Villain model caused by the step-edge diffusion effect

The mound morphology of the 2+1-dimensional Wolf–Villain model is studied by numerical simulation. The diffusion rule of this model has an intrinsic mechanism, i.e., the step-edge diffusion, to create a local uphill particle current, which leads to the formation of the mound. In the simulation, a noise reduction technique is employed to enhance the local uphill particle current. Our results for the dynamic exponent 1 / z and the roughness exponent α obtained from the surface width show a dependence on the strength of the step-edge diffusion. On the other hand, λ ( t ) , which describes the separation of the mounds, grows as a function of time in a power-law form in the regime where the coalescence of mounds occurs, λ ( t ) ∼ t n , with n ≈ 0.23 – 0.25 for a wide range of the deposition conditions under the step-edge diffusion effect. For m = 1 , a noise reduction factor of unity, the behavior of λ ( t ) in the saturated regime is also simulated. We find that the evolution behavior of λ ( t ) in the whole process can be described by the standard Family–Vicsek scaling.

Extensive numerical study of the anomalous dynamic scaling of the Wolf–Villain model

In order to study the microscopic physical mechanisms of roughness surfaces exhibiting the anomalous scaling behavior, the Wolf–Villain model in 1 + 1 and 2 + 1 dimensions is investigated by the kinetic Monte-Carlo simulation on long time and large length scale (the growth time and the system size are respectively extended to t = 2 29 , L = 400 000 for 1 + 1 dimensions, and t = 2 21 , L × L = 512 × 512 for 2 + 1 dimensions). In the 2+1-dimensional simulations, the noise reduction technique is employed so as to eliminate the crossover effects in the growth process. Our calculations show that the Wolf–Villain model in 1 + 1 dimensions very probably exhibits intrinsic anomalous scaling behavior in the time and length simulation range of this paper, and the 2 + 1 -dimensional Wolf–Villain model leads to a pyramidal mounded morphology. Some properties of the mounded pattern in the 2 + 1 -dimensional Wolf–Villain model are discussed in the final part of this presentation.

Scaling law analysis of paraffin thin films on different surfaces

The dynamics of paraffin deposit formation on different surfaces was analyzed based on scaling laws. Carbon-based films were deposited onto silicon (Si) and stainless steel substrates from methane (CH4) gas using radio frequency plasma enhanced chemical vapor deposition. The different substrates were characterized with respect to their surface energy by contact angle measurements, surface roughness, and morphology. Paraffin thin films were obtained by the casting technique and were subsequently characterized by an atomic force microscope in noncontact mode. The results indicate that the morphology of paraffin deposits is strongly influenced by substrates used. Scaling laws analysis for coated substrates present two distinct dynamics: a local roughness exponent (αlocal) associated to short-range surface correlations and a global roughness exponent (αglobal) associated to long-range surface correlations. The local dynamics is described by the Wolf–Villain model, and a global dynamics is described by the Kardar–Parisi–Zhang model. A local correlation length (Llocal) defines the transition between the local and global dynamics with Llocal approximately 700nm in accordance with the spacing of planes measured from atomic force micrographs. For uncoated substrates, the growth dynamics is related to Edwards–Wilkinson model.

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.

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 AND SELF-ORGANIZATION

A general procedure is presented for the multiscale analysis of atomistic models of fluctuating interfaces driven by the deposition of new material. Beginning with a lattice Langevin formulation of kinetic Monte Carlo simulations, we derive stochastic partial differential equations by regularizing the atomistic transition rules. Subsequent coarse-graining is accomplished by calculating renormalization-group trajectories from initial conditions determined by the regularized equations. We apply this methodology to the Wolf–Villain model of relaxation upon deposition and to a model for the self-organization of heteroepitaxial nanostructures.

Effects of the Ehrlich–Schwoebel potential barrier on the Wolf–Villain model simulations for thin film growth

Wolf–Villain (WV) model is a simple model used to study ideal molecular beam epitaxy (MBE) growth by using computer simulations. In this model, an adatom diffuses instantaneously within a finite diffusion length to maximize its coordination number. We study statistical properties of thin films grown by this model. The morphology of the WV model is found to be kinetically rough with a downhill particle diffusion current. In real MBE growth, however, there are additional factors such as the existence of a potential barrier that is known as the Ehrlich–Schwoebel (ES) barrier. The ES barrier is an additional barrier for an adatom that diffuses over a step edge from the upper to a lower terrace which is known to induce an uphill particle current. We found that with the addition of the ES barrier, the WV–ES model morphology is rough with mound formation on the surface when the barrier is strong enough. To confirm these results, the correlation function is also studied. We find no oscillation in the correlation function in the WV model. For the WV–ES model, the correlation function oscillates. These results confirm that a strong enough ES barrier can cause mound formation on the WV surface in our study.

Scaling of local interface width of statistical growth models

We discuss the methods to calculate the roughness exponent α and the dynamic exponent z from the scaling properties of the local roughness, which is frequently used in the analysis of experimental data. Through numerical simulations, we studied the Family, the restricted solid-on-solid, the Das Sarma–Tamborenea (DT) and the Wolf–Villain (WV) models in one- and two-dimensional substrates, in order to compare different methods to obtain those exponents. The scaling at small length scales do not give reliable estimates of α, suggesting that the usual methods to estimate that exponent from experimental data may provide misleading conclusions concerning the universality classes of the growth processes. On the other hand, we propose a more efficient method to calculate the dynamic exponent z, based on the scaling of characteristic correlation lengths, which gives estimates in good agreement with the expected universality classes and indicates expected crossover behavior. Our results also provide evidence of Edwards–Wilkinson asymptotic behavior for the DT and the WV models in two-dimensional substrates.

GROWTH EQUATIONS, DISCRETE MODELS AND MBE EXPERIMENTS

The question of the validity of the scaling ansatz in discrete deposition models and their connection with the scaling exponents of continuum differential equations is addressed. We specifically focus on the scaling properties of the Wolf–Villain type models and, as an extension of this model, on the influence of attractive and repulsive interactions up to second neighbors on the scaling relation. As an example of technological relevance, we present the evolution of steps in the vicinal (100) surface of Si during deposition at relatively low temperatures. We have found that, in general, one should not expect that discrete models, as well as real crystals, exhibit scaling.

Epitaxial mounding in limited-mobility models of surface growth

We study, through large-scale stochastic simulations using the noise reduction technique, surface growth via vapor deposition, e.g., molecular beam epitaxy (MBE), for simple nonequilibrium limited mobility solid-on-solid growth models, such as the Family model, the Das Sarma--Tamborenea model, the Wolf-Villain (WV) model, the larger-curvature (LC) model, and other related models. We find that $(d=2+1)$-dimensional surface growth in several noise reduced models (most notably the WV and LC models) exhibits spectacular quasiregular mound formation with slope selection in their dynamical surface morphology in contrast to the standard statistically scale-invariant kinetically rough surface growth expected (and earlier reported in the literature) for such growth models. The mounding instability in these epitaxial growth models does not involve the Ehrlich-Schwoebel step-edge diffusion barrier. The mounded morphology in these growth models arises from the interplay between the line tension along step edges in the plane parallel to the average surface and the suppression of noise and island nucleation. The line tension tends to stabilize some of the step orientations that coincide with in-plane high-symmetry crystalline directions, and thus the mounds that are formed assume quasiregular structures. The noise reduction technique developed originally for Eden-type models can be used to control the stochastic noise and enhance diffusion along the step edge, which ultimately leads to the formation of quasiregular mounds during growth. We show that by increasing the diffusion surface length together with supression of nucleation and deposition noise, one can obtain a self-organization of the pyramids in quasiregular patterns. The mounding instability in these simple epitaxial growth models is closely related to the cluster-edge diffusion (as opposed to step-edge barrier) driven mounding in MBE growth, which has been recently discussed in the literature. The epitaxial mound formation studied here is a kinetic-topological instability [which can happen only in $(d=2+1)$-dimensional, or higher dimensional, growth, but not in $(d=1+1)$-dimensional growth because no cluster diffusion around a closed surface loop is possible in ``one-dimensional'' surfaces], which is likely to be quite generic in real MBE-type surface growth. Our extensive numerical simulations produce mounded (and slope-selected) surface growth morphologies which are strikingly visually similar to many recently reported experimental MBE growth morphologies.

Numerical test of the damping time of layer-by-layer growth on stochastic models.

We perform Monte Carlo simulations on stochastic models such as the Wolf-Villain (WV) model and the Family model in a modified version to measure the mean separation l between islands in a submonolayer regime and the damping time t* of layer-by-layer growth oscillations in one dimension. The stochastic models are modified, allowing for diffusion within interval r upon deposition. It is found numerically that the mean separation and the damping time depend on the diffusion interval r, leading to the fact that the damping time is related to the mean separation as t* approximately l(4/3) for the WV model and t* approximately l(2) for the Family model. The numerical results are in excellent agreement with recent theoretical predictions.

Elementary local configurations, microscopic rules and Langevin equations for two models of molecular-beam epitaxy

Stochastic Langevin equations for the Wolf-Villain (WV) and Das Sarma-Tamborenea (DT) models are derived using the alternative method recently developed by Costanza (1997 Phys. Rev. E 55 6501) which avoids the complications arising in the calculation of the first moment of the transition rate required in the master equation approach. The calculations are compared with those recently published by Zhi-Feng Huang et al (1996 Phys. Rev. E 54 5935) obtaining the same results, as is expected. The microscopic rules are derived from the set of 243 elementary local configurations needed for the description of these two models and after using simple summation rules these were reduced to 16 for the DT model and to 25 for the WV model. The number of microscopic rules needed for the description of the present models are considerably larger than the previously solved models.