Controlled instability and multiscaling in models of epitaxial growth

Fragility of Kardar-Parisi-Zhang universality class in the presence of temporally correlated noise

We study numerically a family of surface growth models that are known to be in the universality class of the Kardar-Parisi-Zhang equation when driven by uncorrelated noise. We find that, in the presence of noise with power-law temporal correlations with exponent θ, these models exhibit critical exponents that differ both quantitatively and qualitatively from model to model. The existence of a threshold value for θ below which the uncorrelated fixed point is dominant occurs for some models but not for others. In some models the dynamic exponent z(θ) is a smooth decreasing function, while it has a maximum in other cases. Despite all models sharing the same symmetries, critical exponents turn out to be strongly model dependent. Our results clearly show the fragility of the universality class concept in the presence of long-range temporally correlated noise.

Solution of the 1D KPZ Equation by Explicit Methods

The Kardar–Parisi-Zhang (KPZ) equation is examined using the recently published leapfrog–hopscotch (LH) method as well as the most standard forward time centered space (FTCS) scheme and the Heun method. The methods are verified by reproducing an analytical solution. The performance of each method is then compared by calculating the average and the maximum differences among the results and displaying the runtimes. Numerical tests show that due to the special symmetry in the time–space discretisation, the new LH method clearly outperforms the other two methods. In addition, we discuss the effect of different parameters on the solutions.

From cellular automata to growth dynamics: The Kardar-Parisi-Zhang universality class

We demonstrate that in the continuous limit the etching mechanism yields the Kardar-Parisi-Zhang (KPZ) equation in a (d+1)-dimensional space. We show that the parameters ν, associated with the surface tension, and λ, associated with the nonlinear term of the KPZ equation, are not phenomenological, but rather they stem from a new probability distribution function. The Galilean invariance is recovered independently of d, and we illustrate this via very precise numerical simulations. We obtain firsthand the coupling parameter as a function of the probabilities. In addition, we strengthen the argument that there is no upper critical limit for the KPZ equation.

Faceted patterns and anomalous surface roughening driven by long-range temporally correlated noise

We investigate Kardar-Parisi-Zhang (KPZ) surface growth in the presence of power-law temporally correlated noise. By means of extensive numerical simulations of models in the KPZ universality class we find that, as the noise correlator index increases above some threshold value, the surface exhibits anomalous kinetic roughening of the type described by the generic scaling theory of Ramasco et al. [Phys. Rev. Lett. 84, 2199 (2000)PRLTAO0031-900710.1103/PhysRevLett.84.2199]. Remarkably, as the driving noise temporal correlations increase, the surface develops a characteristic pattern of macroscopic facets that completely dominates the dynamics in the long time limit. We argue that standard scaling fails to capture the behavior of KPZ subject to long-range temporally correlated noise. These phenomena are not not described by the existing theoretical approaches, including renormalization group and self-consistent approaches.

Local roughness exponent in the nonlinear molecular-beam-epitaxy universality class in one dimension

We report local roughness exponents, α_{loc}, for three interface growth models in one dimension which are believed to belong to the nonlinear molecular-beam-epitaxy (nMBE) universality class represented by the Villain-Lais-Das Sarma (VLDS) stochastic equation. We applied an optimum detrended fluctuation analysis (ODFA) [Luis et al., Phys. Rev. E 95, 042801 (2017)2470-004510.1103/PhysRevE.95.042801] and compared the outcomes with standard detrending methods. We observe in all investigated models that ODFA outperforms the standard methods providing exponents in the narrow interval α_{loc}^{}∈[0.96,0.98] quantitatively consistent with two-loop renormalization group predictions for the VLDS equation. In particular, these exponent values are calculated for the Clarke-Vvdensky and Das Sarma-Tamborenea models characterized by very strong corrections to the scaling, for which large deviations of these values had been reported. Our results strongly support the absence of anomalous scaling in the nMBE universality class and the existence of corrections in the form α_{loc}^{}=1-ε of the one-loop renormalization group analysis of the VLDS equation.

From Kardar-Parisi-Zhang scaling to explosive desynchronization in arrays of limit-cycle oscillators

Phase oscillator lattices subject to noise are one of the most fundamental systems in nonequilibrium physics. We have discovered a dynamical transition which has a significant impact on the synchronization dynamics in such lattices, as it leads to an explosive increase of the phase diffusion rate by orders of magnitude. Our analysis is based on the widely applicable Kuramoto-Sakaguchi model, with local couplings between oscillators. For one-dimensional lattices, we observe the universal evolution of the phase spread that is suggested by a connection to the theory of surface growth, as described by the Kardar-Parisi-Zhang (KPZ) model. Moreover, we are able to explain the dynamical transition both in one and two dimensions by connecting it to an apparent finite-time singularity in a related KPZ lattice model. Our findings have direct consequences for the frequency stability of coupled oscillator lattices.

Extreme fluctuations in stochastic network coordination with time delays

We study the effects of uniform time delays on the extreme fluctuations in stochastic synchronization and coordination problems with linear couplings in complex networks. We obtain the average size of the fluctuations at the nodes from the behavior of the underlying modes of the network. We then obtain the scaling behavior of the extreme fluctuations with system size, as well as the distribution of the extremes on complex networks, and compare them to those on regular one-dimensional lattices. For large complex networks, when the delay is not too close to the critical one, fluctuations at the nodes effectively decouple, and the limit distributions converge to the Fisher-Tippett-Gumbel density. In contrast, fluctuations in low-dimensional spatial graphs are strongly correlated, and the limit distribution of the extremes is the Airy density. Finally, we also explore the effects of nonlinear couplings on the stability and on the extremes of the synchronization landscapes.

Normal dynamic scaling in the class of the nonlinear molecular-beam-epitaxy equation

The scaling of local height fluctuations is studied numerically in lattice growth models of the class of the nonlinear stochastic equation of Villain-Lai-Das Sarma (VLDS) in substrate dimensions d=1 and 2. In d=1, the average local slopes of the conserved restricted solid-on-solid (CRSOS) models converge to a finite value in the long-time limit, with power-law corrections in time whose exponents are close to 0.1. Other VLDS models in d=1, such as that of Das Sarma and Tamborenea, show a divergence of local slopes up to 10(6) monolayers, typical of anomalous roughening, but a comparison of roughness distributions shows that they scale as the linear fourth-order growth equation in those time scales. Normal scaling is also obtained in a modified VLDS equation with instability suppression, in contrast to recent numerical works. In d=2, a CRSOS model and a model with lateral aggregation of diffusing particles show normal scaling of the local slopes, also with small correction exponents. These results consistently show that the VLDS class has normal dynamic scaling in d=1 and 2, in agreement with the theoretical predictions of Phys. Rev. Lett. 94, 166103 (2005), and they show that the apparently anomalous features observed in previous works are effects of large scaling correction terms or crossover effects.

Universal fluctuations in Kardar-Parisi-Zhang growth on one-dimensional flat substrates

We present a numerical study of the evolution of height distributions (HDs) obtained in interface growth models belonging to the Kardar-Parisi-Zhang (KPZ) universality class. The growth is done on an initially flat substrate. The HDs obtained for all investigated models are very well fitted by the theoretically predicted Gaussian orthogonal ensemble (GOE) distribution. The first cumulant has a shift that vanishes as t(-1/3), while the cumulants of order 2≤n≤4 converge to GOE as t(-2/3) or faster, behaviors previously observed in other KPZ systems. These results yield evidences for the universality of the GOE distribution in KPZ growth on flat substrates. Finally, we further show that the surfaces are described by the Airy(1) process.

Surface pattern formation and scaling described by conserved lattice gases

We extend our 2+1 -dimensional discrete growth model [Odor, Phys. Rev. E 79, 021125 (2009)] with conserved, local exchange dynamics of octahedra, describing surface diffusion. A roughening process was realized by uphill diffusion and curvature dependence. By mapping the slopes onto particles, two-dimensional nonequilibrium binary lattice model emerges, in which the (smoothing or roughening) surface diffusion can be described by attracting or repelling motion of oriented dimers. The binary representation allows simulations on very large size and time scales. We provide numerical evidence for Mullins-Herring or molecular-beam epitaxy class scaling of the surface width. The competition of inverse Mullins-Herring diffusion with a smoothing deposition, which corresponds to a Kardar-Parisi-Zhang (KPZ) process, generates different patterns: dots or ripples. We analyze numerically the scaling and wavelength growth behavior in these models. In particular, we confirm by large size simulations that the KPZ type of scaling is stable against the addition of this surface diffusion, hence this is the asymptotic behavior of the Kuramoto-Sivashinsky equation as conjectured by field theory in two dimensions, but has been debated numerically. If very strong, normal surface diffusion is added to a KPZ process, we observe smooth surfaces with logarithmic growth, which can describe the mean-field behavior of the strong-coupling KPZ class. We show that ripple coarsening occurs if parallel surface currents are present, otherwise logarithmic behavior emerges.

Conformal invariance of isoheight lines in a two-dimensional Kardar-Parisi-Zhang surface

The statistics of isoheight lines in the (2+1) -dimensional Kardar-Parisi-Zhang (KPZ) model is shown to be conformally invariant and equivalent to those of self-avoiding random walks. This leads to a rich variety of exact analytical results for the KPZ dynamics. We present direct evidence that the isoheight lines can be described by the family of conformally invariant curves called Schramm-Loewner evolution (or SLE_{kappa} ) with diffusivity kappa=8/3 . It is shown that the absence of the nonlinear term in the KPZ equation will change the diffusivity kappa from 8/3 to 4, indicating that the isoheight lines of the Edwards-Wilkinson surface are also conformally invariant and belong to the universality class of domain walls in the O(2) spin model.

Numerical study of the Kardar-Parisi-Zhang equation

We integrate numerically the Kardar-Parisi-Zhang (KPZ) equation in 1+1 and 2+1 dimensions using a Euler discretization scheme and the replacement of (nablah)(2) by exponentially decreasing functions of that quantity to suppress instabilities. When applied to the equation in 1+1 dimensions, the method of instability control provides values of scaling amplitudes consistent with exactly known results, in contrast to the deviations generated by the original scheme. In 2+1 dimensions, we spanned a range of the model parameters where transients with Edwards-Wilkinson or random growth are not observed, in box sizes 8< or =L< or =128 . We obtain a roughness exponent of 0.37< or =alpha< or =0.40 and steady state height distributions with skewness S=0.25+/-0.01 and kurtosis Q=0.15+/-0.1 . These estimates are obtained after extrapolations to the large L limit, which is necessary due to significant finite-size effects in the estimates of effective exponents and height distributions. On the other hand, the steady state roughness distributions show weak scaling corrections and evidence of stretched exponential tails. These results confirm previous estimates from lattice models, showing their reliability as representatives of the KPZ class.

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.

Localization in disordered media, anomalous roughening, and coarsening dynamics of faceted surfaces

We study a surface growth model related to the Kardar-Parisi-Zhang equation for nonequilibrium kinetic roughening, but where the thermal noise is replaced by a static columnar disorder eta(x) . This model is one of the many representations of the problem of particle diffusion in trapping or amplifying disordered media. We find that probability localization in the latter translates into facet formation in the equivalent surface growth problem. Coarsening of the pattern can therefore be identified with the diffusion of the localization center. The emergent faceted structure gives rise to nontrivial scaling properties, including anomalous surface roughening in excellent agreement with an existing conjecture for kinetic roughening of faceted surfaces. In a wider context, our study sheds light onto the scaling properties in other systems displaying this kind of patterned surface.

Generalized discretization of the Kardar-Parisi-Zhang equation

We report the generalized spatial discretization of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions. We solve exactly the steady state probability density function for the discrete heights of the interface, for any discretization scheme. We show that the discretization prescription is a consequence of each particular model. We derive the discretization prescription of the KPZ equation for the ballistic deposition model.

Scaling of local slopes, conservation laws, and anomalous roughening in surface growth

We argue that symmetries and conservation laws greatly restrict the form of the terms entering the long wavelength description of growth models exhibiting anomalous roughening. This is exploited to show by dynamic renormalization group arguments that intrinsic anomalous roughening cannot occur in local growth models. However, some conserved dynamics may display superroughening if a given type of term is present.

Persistence of randomly coupled fluctuating interfaces

We study the persistence properties in a simple model of two coupled interfaces characterized by heights h(1) and h(2) , respectively, each growing over a d -dimensional substrate. The first interface evolves independently of the second and can correspond to any generic growing interface, e.g., of the Edwards-Wilkinson or of the Kardar-Parisi-Zhang variety. The evolution of h(2) , however, is coupled to h(1) via a quenched random velocity field. In the limit d-->0 , our model reduces to the Matheron-de Marsily model in two dimensions. For d=1, our model describes a Rouse polymer chain in two dimensions advected by a transverse velocity field. We show analytically that after a long waiting time t(0) -->infinity , the stochastic process h(2) , at a fixed point in space but as a function of time, becomes a fractional Brownian motion with a Hurst exponent, H2 =1- beta(1) /2 , where beta(1) is the growth exponent characterizing the first interface. The associated persistence exponent is shown to be theta(2)(s) =1- H2 = beta(1) /2 . These analytical results are verified by numerical simulations.

Roughening of the interfaces in (1+1) -dimensional two-component surface growth with an admixture of random deposition

We simulate competitive two-component growth on a one-dimensional substrate of L sites. One component is a Poisson-type deposition that generates Kardar-Parisi-Zhang (KPZ) correlations. The other is random deposition (RD). We derive the universal scaling function of the interface width for this model and show that the RD admixture acts as a dilatation mechanism to the fundamental time and height scales, but leaves the KPZ correlations intact. This observation is generalized to other growth models. It is shown that the flat-substrate initial condition is responsible for the existence of an early nonscaling phase in the interface evolution. The length of this initial phase is a nonuniversal parameter, but its presence is universal. We introduce a method to measure the length of this initial nonscaling phase. In application to parallel and distributed computations, the important consequence of the derived scaling is the existence of the upper bound for the desynchronization in a conservative update algorithm for parallel discrete-event simulations. It is shown that such algorithms are generally scalable in a ring communication topology.

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.

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.

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.

Anomalous roughening in experiments of interfaces in Hele-Shaw flows with strong quenched disorder

We report experimental evidence of anomalous kinetic roughening in the stable displacement of an oil-air interface in a Hele-Shaw cell with strong quenched disorder. The disorder consists of a random modulation of the gap spacing that is transverse to the growth direction (tracks). Experiments were performed by varying the average interface velocity and the gap spacing, and measuring the scaling exponents. The following values of the scaling exponents were obtained; beta approximately 0.50, beta* approximately 0.25, alpha approximately 1.0, alpha(loc) approximately 0.5, and z approximately 2. When there is no fluid injection, the interface is driven solely by capillary forces, and a higher value of beta of approximately beta=0.65 is measured. The presence of multiscaling and the particular morphology of the interfaces, characterized by large height differences that follow a Lévy distribution, confirms the existence of anomalous scaling. From a detailed study of the motion of the oil-air interface, we show that the anomaly is a consequence of different local velocities on the tracks plus the coupling in the motion between neighboring tracks. The anomaly disappears at high interface velocities, weak capillary forces, or when the disorder is not sufficiently persistent in the growth direction. We have also observed the absence of scaling when the disorder is very strong or when a regular modulation of the gap spacing is introduced.

Scaling of the interface roughness in Fe-Cr superlattices: self-affine versus non-self-affine

We have analyzed kinetic roughening in Fe-Cr superlattices by energy-filtered transmission electron microscopy. The direct access to individual interfaces provides both static and dynamic roughness exponents. We find an anomalous non-self-affine scaling of the interface roughness with a time dependent local roughness at short length scales. While the deposition conditions affect strongly the long-range dynamics, the anomalous short-range exponent remains unchanged. The different short- and long-range dynamics outline the importance of long-range interactions in kinetic roughening.

Layer-by-layer epitaxy in limited mobility nonequilibrium models of surface growth

We study, using noise-reduction techniques, layer-by-layer epitaxial growth in limited mobility solid-on-solid nonequilibrium surface growth models, which have been introduced in the context of kinetic surface roughening in ideal molecular beam epitaxy. Multiple hit noise reduction and long surface diffusion length lead to qualitatively similar layer-by-layer epitaxy in (1+1)- and (2+1)-dimensional limited mobility growth simulations. We discuss the dynamic scaling characteristics connecting the transient layer-by-layer growth regime with the asymptotic kinetically rough growth regime.

Dynamic finite-size scaling of the normalized height distribution in kinetic surface roughening

Using well-known simple growth models, we have studied the dynamic finite-size scaling theory for the normalized height distribution of a growing surface. We find a simple functional form that explains size-dependent behavior of the skewness and kurtosis in the transient regime, and obtain the transient- and long-time values of the skewness and kurtosis for the models. Scaled distributions of the models are obtained, and the shape of each distribution is discussed in terms of the interfacial width, skewness, and kurtosis, and compared with those for other models. Exponents eta(+) and eta(-), which characterize the form of the distribution, are determined from an exponential fitting of scaling functions. Our detailed results reveal that eta(+)+eta(-) approximately 4 for a model obeying usual scaling in contrast to eta(+)+eta(-)<4 with eta(-)=1 for a model exhibiting anomalous scaling as well as multiscaling. Since we obtain eta(+)+eta(-) approximately 4 for a model exhibiting anomalous scaling but no multiscaling, we conclude that the deviation from eta(+)+eta(-) approximately 4 is due to the presence of multiscaling behavior in a model.

Generic dynamic scaling in kinetic roughening

We study the dynamic scaling hypothesis in invariant surface growth. We show that the existence of power-law scaling of the correlation functions (scale invariance) does not determine a unique dynamic scaling form of the correlation functions, which leads to the different anomalous forms of scaling recently observed in growth models. We derive all the existing forms of anomalous dynamic scaling from a new generic scaling ansatz. The different scaling forms are subclasses of this generic scaling ansatz associated with bounds on the roughness exponent values. The existence of a new class of anomalous dynamic scaling is predicted and compared with simulations.

Characteristics of driven polymer surfaces: growth and roughness

Using a Monte Carlo simulation, the growth and roughness characteristics of polymer surfaces are studied in 2+1 dimensions. Kink-jump and reptation dynamics are used to move polymer chains under a driving field where they deposit onto an impenetrable attractive wall. Effects of field (E), chain length (L(c)), and the substrate size (L) on the growing surfaces are studied. In low field, the interface width (W) shows a crossover from one power-law growth in time (W approximately t(beta(1))) to another (W approximately t(beta(2))), before reaching its asymptotic value (W(s)), with beta(1)( approximately 0.5+/-0.1)<beta(2)( approximately 0.6-1.0). For short chain lengths (L(c)=4), the saturated width (W(s)) is independent of the substrate length (L), while for long chain lengths, W(s) decays with L before becoming independent at large L. W(s) depends strongly on the magnitude of the field: for short chains, W(s) approximately E-delta with delta approximately 0.4, while for long chains, it varies nonmonotonically with E.