Kinetic Roughening Research Articles

Machine learning method for roughness prediction

This work aims to employ machine-learning models, specifically neural networks, to predict the time evolution of the global surface roughness in a lattice model that represents a film growing on a d-dimensional substrate. We analyze the well-known ballistic deposition (BD) model for d=1, 2 since it presents strong corrections to the scaling, making it difficult to observe directly, via effective scaling exponents, its correspondence with the Kardar-Parisi-Zhang (KPZ) universality class. As an alternative to overcome this difficulty, we first intend to learn the time evolution of the global roughness for substrate sizes that are computationally viable to simulate. To test the learning, we apply two different methodologies for d = 1: the first one learns the Family-Vicsek scaling relation, and by doing the reverse transformation, we get the global roughness as a function of the time, and the second one learns the kinetic roughening directly from the time series data. For growth in d = 2 where applications arise and no exact KPZ scaling exponents are known, we apply the second methodology. However, we employ a more resilient learning model tailored for time series problems. Hence, the time required to generate the same amount of data, showing the evolution of global roughness, is reduced dramatically. Importantly, machine learning techniques capture the scaling corrections of the BD model, predicting an effective global roughness exponent, α, calculated from the learned data extracted from very large lateral sizes and times that cannot be simulated using lattice models. Our prediction is consistent with accurate estimates of the KPZ roughness exponent reported in the literature for d = 2.

Surface Evolution of Polymer Films Grown by Vapor Deposition: Growth of Local and Global Slopes of Interfaces.

The kinetic roughening of polymer films grown by vapor deposition polymerization was analyzed using the widely accepted classification framework of "generic scaling ansatz" given for the structure factor. Over the past two decades, this method has played a pivotal role in classifying diverse forms of dynamic scaling and understanding the mechanisms driving interface roughening. The roughness exponents of the polymer films were consistently determined as α=1.25±0.09, αloc=0.73±0.02, and αs=0.99±0.06. However, the inability to unambiguously assign these roughness exponent values to a specific scaling subclass prompts the proposal of a practical alternative. This report illustrates how all potential dynamic scaling can be consistently identified and classified based on the relationship between two temporal scaling exponents measured in real space: the average local slope and the global slope of the interface. The intrinsic anomalous roughening class is conclusively assigned to polymer film growth characterized by anomalous "native (background slope-removed) local height fluctuations". Moreover, the new analysis reveals that interfaces exhibiting anomalous scaling, previously classified as intrinsic anomalous roughening, could potentially belong to the super-rough class, particularly when the spectral roughness exponent αs is equal to 1.

Marangoni spreading on liquid substrates in new media art.

With the advent of new media art, artists have harnessed fluid dynamics to create captivating visual narratives. A striking technique known as dendritic painting employs mixtures of ink and isopropanol atop paint, yielding intricate tree-like patterns. To unravel the intricacies of that technique, we examine the spread of ink/alcohol droplets over liquid substrates with diverse rheological properties. On Newtonian substrates, the droplet size evolution exhibits two power laws, suggesting an underlying interplay between viscous and Marangoni forces. The leading edge of the droplet spreads as a precursor film with an exponent of 3/8, while its main body spreads with an exponent of 1/4. For a weakly shear-thinning acrylic resin substrate, the same power laws persist, but dendritic structures emerge, and the texture of the precursor film roughens. The observed roughness and growth exponents (3/4 and 3/5) suggest a connection to the quenched Kardar-Parisi-Zhang universality class, hinting at the existence of quenched disorder in the liquid substrate. Mixing the resin with acrylic paint renders it more viscous and shear-thinning, refining the dendrite edges and further roughening the precursor film. At larger paint concentrations, the substrate becomes a power-law fluid. The roughness and growth exponents then approach 1/2 and 3/4, respectively, deviating from known universality classes. The ensuing structures have a fractal dimension of 1.68, characteristic of diffusion-limited aggregation. These findings underscore how the nonlinear rheological properties of the liquid substrate, coupled with the Laplacian nature of Marangoni spreading, can overshadow the local kinetic roughening of the droplet interface.

Unpredicted Scaling of the One-Dimensional Kardar-Parisi-Zhang Equation.

The celebrated Kardar-Parisi-Zhang (KPZ) equation describes the kinetic roughening of stochastically growing interfaces. In one dimension, the KPZ equation is exactly solvable and its statistical properties are known to an exquisite degree. Yet recent numerical simulations in the tensionless (or inviscid) limit of the KPZ equation [C. Cartes et al., The Galerkin-truncated Burgers equation: Crossover from inviscid-thermalized to Kardar-Parisi-Zhang scaling, Phil. Trans. R. Soc. A 380, 20210090 (2022).PTRMAD1364-503X10.1098/rsta.2021.0090; E. Rodríguez-Fernández et al., Anomalous ballistic scaling in the tensionless or inviscid Kardar-Parisi-Zhang equation, Phys. Rev. E 106, 024802 (2022).PRESCM2470-004510.1103/PhysRevE.106.024802] unveiled a new scaling, with a critical dynamical exponent z=1 different from the KPZ one z=3/2. In this Letter, we show that this scaling is controlled by a fixed point which had been missed so far and which corresponds to an infinite nonlinear coupling. Using the functional renormalization group (FRG), we demonstrate the existence of this fixed point and show that it yields z=1. We calculate the correlation function and associated scaling function at this fixed point, providing both a numerical solution of the FRG equations within a reliable approximation, and an exact asymptotic form obtained in the limit of large wave numbers. Both scaling functions accurately match the one from the numerical simulations.

Scaling of surface roughness in film deposition with height-dependent step edge barriers.

We perform kinetic Monte Carlo simulations of film growth in simple cubic lattices with solid-on-solid conditions, Ehrlich-Schwoebel (ES) barriers at step edges, and a kinetic barrier related to the hidden off-plane diffusion at multilayer steps. Broad ranges of the diffusion-to-deposition ratio R, detachment probability per lateral neighbor, ε, and monolayer step crossing probability P=exp[-E_{ES}/(k_{B}T)] are studied. Without the ES barrier, four possible scaling regimes are shown as the coverage θ increases: nearly layer-by-layer growth with damped roughness oscillations; kinetic roughening in the Villain-Lai-Das Sarma (VLDS) universality class when the roughness is W∼1 (in lattice units); unstable roughening with mound nucleation and growth, where slopes of logW×logθ plots reach values larger than 0.5; and asymptotic statistical growth with W=θ^{1/2} solely due to the kinetic barrier at multilayer steps. If the ES barrier is present, the layer-by-layer growth crosses over directly to the unstable regime, with no transient VLDS scaling. However, in simulations up to θ=10^{4} (typical of films with a few micrometers), low temperatures (small R,ε, or P) may suppress the two or three initial regimes, while high temperatures and P∼1 produce smooth surfaces at all thicknesses. These crossovers help to explain proposals of nonuniversal exponents in previous works. We define a smooth film thickness θ_{c} where W=1 and show that VLDS scaling at that point indicates negligible ES barriers, while rapidly increasing roughness indicates a small ES barrier (E_{ES}∼k_{B}T). θ_{c} scales as ∼exp(const×P^{2/3}) if the other parameters are kept fixed, which represents a high sensitivity on the ES barrier. The analysis of recent experimental data in the light of our results distinguishes cases where E_{ES}/(k_{B}T) is negligible, ∼1, or ≪1.

Phase composition of polycrystalline HfNx (0.45 ≤ x ≤ 1.60) and effects of low-energy ion irradiation on microstructure, texture, and physical properties

We have investigated the phase composition of HfNx as a function of x and the effects of low-energy ion irradiation on the microstructure and physical properties of polycrystalline layers grown on SiO2 at 350 °C by ultrahigh vacuum reactive dc magnetron sputtering of Hf in mixed N2/Ar discharges. X-ray diffraction and Rutherford backscattering spectrometry results show that the phases obtained in polycrystalline HfNx layers with increasing x are hcp-structure α-Hf:N (x ≲ 0.6); multiphase mixtures consisting of α-Hf, NaCl-structure δ-HfN, rhombohedral ɛ-Hf3N2, and/or ζ-Hf4N3 (0.6 ≲ x ≲ 0.9); δ-HfN single phase (0.9 ≲ x ≲ 1.3); and mixtures of δ-HfN and higher nitrides (x ≳ 1.3). HfNx layers with 0.9 ≲ x ≲ 1.2 grown under mild ion irradiation (incident ion energy Ei ≃ 7 eV and ion-to-Hf flux ratios Ji/JHf = 1−3) are underdense with mixed orientation, low in-plane stress, and rough surface morphology due to limited adatom mobilities resulting in kinetic roughening and atomic shadowing during film growth. However, the use of intense ion irradiation (Ei = 25 eV and Ji/JHf = 4−20) results in HfNx layers, which are fully dense with strongly 111-oriented texture, compressive in-plane stress, and smooth surfaces due to ion irradiation enhanced adatom surface mobilities. In addition, the latter films have lower resistivity and higher hardness. For stoichiometric δ-HfN layers, ρ decreases from 69.7 to 35.2 μΩ cm and H increases from 22.1 to 27.4 GPa, with increasing ion-irradiation intensity. However, for HfNx layers with 1.2 ≲ x ≲ 1.6, the correspondingly higher steady state atomic N surface coverages during deposition alter growth kinetics in favor of 001 texture with a fully dense structure and compressive in-plane stress.

Effect of Random Environment on Kinetic Roughening: Kardar–Parisi–Zhang Model with a Static Noise Coupled to the Navier–Stokes Equation

Effect of Random Environment on Kinetic Roughening: Kardar–Parisi–Zhang Model with a Static Noise Coupled to the Navier–Stokes Equation

Microscopic ordering of supercooled water on the ice basal face

Growth of ice crystals is ubiquitous around us, but we still do not know what is occurring at the forefront of crystallization. In general, the interfacial structure is inseparably involved in the microscopic ordering during crystal growth. However, despite its importance in nature and technology, the intrinsic role of the interfacial structure in the melt growth of ice remains to be elucidated. Here, using extensive molecular dynamics simulations, we comprehensively explore how supercooled water molecules are incorporated into the ice basal face. Structural and dynamic characterizations of the ice-water interface demonstrate that the ice basal face is sharp at the molecular level and its growth proceeds layer-by-layer through two-dimensional nucleation without any intermediate structures. We further quantify the crossover from layerwise to adhesive growth, called kinetic roughening, with the height difference correlation and the normal growth rate analysis. Moreover, we identify the presence of an ultra-low density water layer in contact with the structural interface, which assists two-dimensional nucleation at a small amount of supercooling without involving any triggers, such as dislocations.

Kinetic roughening and magnetic study of Co electrodeposited thin films

Kinetic roughening and magnetic study of Co electrodeposited thin films

Nonequilibrium criticality driven by Kardar-Parisi-Zhang fluctuations in the synchronization of oscillator lattices

The synchronization of oscillator ensembles is pervasive throughout nonlinear science, from classical or quantum mechanics to biology, to human assemblies. Traditionally, the main focus has been the identification of threshold parameter values for the transition to synchronization as well as the nature of such transition. Here, we show that considering an oscillator lattice as a discrete growing interface provides unique insights into the dynamical process whereby the lattice reaches synchronization for long times. Working on a generalization of the Kuramoto model that allows for odd or non-odd couplings, we elucidate synchronization of oscillator lattices as an instance of generic scale invariance, whereby the system displays space-time criticality, largely irrespective of parameter values. The critical properties of the system (like scaling exponent values and the dynamic scaling Ansatz which is satisfied) fall into universality classes of kinetically rough interfaces with columnar disorder, namely, those of the Edwards-Wilkinson (equivalently, the Larkin model of an elastic interface in a random medium) or the Kardar-Parisi-Zhang (KPZ) equations, for Kuramoto (odd) coupling and generic (non-odd) couplings, respectively. From the point of view of kinetic roughening, the critical properties we find turn out to be quite innovative, especially concerning the statistics of the fluctuations as characterized by their probability distribution function (PDF) and covariance. While the latter happens to be that of the Larkin model irrespective of the symmetry of the coupling, in the generic non-odd coupling case the PDF turns out to be the Tracy-Widom distribution associated with the KPZ nonlinearity. This brings the synchronization of oscillator lattices into a remarkably large class of strongly-correlated, low-dimensional (classical and quantum) systems with strong universal fluctuations.

Dynamic scaling of stochastic thermodynamic observables for chemical reactions at and away from equilibrium.

Physical kinetic roughening processes are well-known to exhibit universal scaling of observables that fluctuate in space and time. Are there analogous dynamic scaling laws that are unique to the chemical reaction mechanisms available synthetically and occurring naturally? Here, we formulate an approach to the dynamic scaling of stochastic fluctuations in thermodynamic observables at and away from equilibrium. Both analytical expressions and numerical simulations confirm our dynamic scaling ansatz with associated scaling exponents, function, and law. A survey of common chemical mechanisms reveals classes that organize according to the molecularity of the reactions involved, the nature of the reaction vessel and external reservoirs, (non)equilibrium conditions, and the extent of autocatalysis in the reaction network. Varying experimental parameters, such as temperature, can cause coupled reactions capable of chemical feedback to transition between these classes. While path observables, such as the dynamical activity, have scaling exponents that are time-independent, the variance in the entropy production and flow can have time-dependent scaling exponents and self-averaging properties as a result of temporal correlations that emerge during thermodynamically irreversible processes. Altogether, these results establish dynamic universality classes in the nonequilibrium fluctuations of thermodynamic observables for well-mixed chemical reactions.

The transition of growth behaviors of moderate Sn fraction Ge1-xSnx (8 % < x < 15 %) epilayers with low temperature molecular beam epitaxy

Epitaxial breakdown phenomenon for Ge1-xSnx thin film with Sn fraction of less than 6 % during low-temperature molecular beam epitaxy has been reported [J. Appl. Phys. 97, 044904, 2005]. However more Sn incorporation is required for direct bandgap transition of GeSn alloy, which has much attractive for GeSn optoelectronic devices. In this work, moderate Sn fraction Ge1-xSnx (8 %∼15.3 %), as well as composition-stepped multiple GeSn layers, grown on Ge (001) substrate by molecular beam epitaxy at 120–150 ℃ are investigated. Epitaxy breakdown occurs for the Ge1-xSnx epilayer with Sn fraction of about 8 % and 10 %, even with low Sn fraction buffer layers, in which the film delaminates into a fully compressively strained bottom layer with dislocation free and a fully relaxed top layer spontaneously. Once the epitaxy breakdown occurs, the amorphous columnar structures extend into the following deposited GeSn epilayer, regardless of Sn fraction. While, the strain in the whole Ge1-xSnx epilayer with Sn fraction of 15.3 % is directly released during growth by generation of misfit and threading dislocations or agglomeration of islands. Those results indicate that there are three growth modes including fully compressive strained thin GeSn epilayer, epitaxial breakdown induced self-delamination of GeSn layer with moderate Sn fraction, and strain relaxed Ge1-xSnx epilayer with high Sn fraction of above 15 %. The complex behavior can be attributed to the competition between kinetic roughening and strain- driven GeSn agglomeration or generation of dislocations. The understanding of the growth behavior is helpful for controlling crystal quality of GeSn thin films for their application in optoelectronic devices.

The nonequilibrium potential today: A short review

A brief review is made of the birth and evolution of the “nonequilibrium potential” (NEP) concept. As if providing a landscape for qualitative reasoning were not helpful enough, the NEP adds a quantitative dimension to the qualitative theory of differential equations and provides a global Lyapunov function for the deterministic dynamics. Here we illustrate the usefulness of the NEP to draw results on stochastic thermodynamics: the Jarzynski equality in the Wilson–Cowan model (a population-competition model of the neocortex) and a “thermodynamic uncertainty relation” (TUR) in the KPZ equation (the stochastic field theory of kinetic interface roughening). Additionally, we discuss system-size stochastic resonance in the Wilson–Cowan model and relevant aspects of KPZ phenomenology like the EW–KPZ crossover and the memory of initial conditions.

Anomalous ballistic scaling in the tensionless or inviscid Kardar-Parisi-Zhang equation

The one-dimensional Kardar-Parisi-Zhang (KPZ) equationis becoming an overarching paradigm for the scaling of nonequilibrium, spatially extended, classical and quantum systems with strong correlations. Recent analytical solutions have uncovered a rich structure regarding its scaling exponents and fluctuation statistics. However, the zero surface tension or zero viscosity case eludes such analytical solutions and has remained ill-understood. Using numerical simulations, we elucidate a well-defined universality class for this case that differs from that of the viscous case, featuring intrinsically anomalous kinetic roughening (despite previous expectations for systems with local interactions and time-dependent noise) and ballistic dynamics. The latter may be relevant to recent quantum spin chain experiments which measure KPZ and ballistic relaxation under different conditions. We identify the ensuing set of scaling exponents in previous discrete interface growth models related with isotropic percolation, and show it to describe the fluctuations of additional continuum systems related with the noisy Korteweg-de Vries equation. Along this process, we additionally elucidate the universality class of the related inviscid stochastic Burgers equation.

Kinetic roughening in the nonlocal Kardar–Parisi–Zhang growth: Pseudospectral versus finite difference schemes

To investigate the effects of nonlocal interaction in kinetic roughening, we adopt two independent numerical methods including pseudospectral (PS) and finite difference (FD) schemes to simulate the nonlocal Kardar–Parisi–Zhang (NKPZ) equation in (1+1)-dimensions. We find that the values of growth exponent β increase gradually with the spatial decay exponent ρ, indicating that the scaling properties are dependence on nonlocal interactions within the effective spatial decay region. However, our simulations also imply that both the local roughness exponent αloc and the spectral roughness exponent αs decrease slightly with ρ. Our results could be consistent with each other using these two numerical methods, and the comparisons and differences among the existing analytical predictions and numerical results are also discussed.

Spreading fronts of wetting liquid droplets: Microscopic simulations and universal fluctuations.

We have used kinetic Monte Carlo (kMC) simulations of a lattice gas to study front fluctuations in the spreading of a nonvolatile liquid droplet onto a solid substrate. Our results are consistent with a diffusive growth law for the radius of the precursor layer, R∼t^{δ}, with δ≈1/2 in all the conditions considered for temperature and substrate wettability, in good agreement with previous studies. The fluctuations of the front exhibit kinetic roughening properties with exponent values which depend on temperature T, but become T independent for sufficiently high T. Moreover, strong evidence of intrinsic anomalous scaling has been found, characterized by different values of the roughness exponent at short and large length scales. Although such a behavior differs from the scaling properties of the one-dimensional Kardar-Parisi-Zhang (KPZ) universality class, the front covariance and the probability distribution function of front fluctuations found in our kMC simulations do display KPZ behavior, agreeing with simulations of a continuum height equationproposed in this context. However, this equationdoes not feature intrinsic anomalous scaling, at variance with the discrete model.

Step barrier effects during early stages of the kinetic roughening of fcc(111) surfaces

The influence of the additional Ehrlich–Schwoebel step barrier and temperature on the early stages of kinetic roughening and mound formation of fcc(111) surfaces is studied by means of kinetic Monte Carlo simulations. Increasing the Ehrlich–Schwoebel barrier, the growth mode develops from nearly layer-by-layer growth to statistical (Poisson) growth mode with the formation of wedding-cake-like shaped mounds. The evolution of the growth morphology is characterized by scaling laws with effective critical exponents. On the one hand, coarsening for growth without step barrier follows exponents neff=0.35 and βeff=0.20 for the characteristic lateral distance (coarsening exponent) and the rms roughness, respectively. On the other hand, coarsening is strongly suppressed for large step barriers (neff=0.05, βeff=0.52, Poisson growth) in agreement with different experimental results, e.g., for the formation of mounds during the homoepitaxy of Ag(111) and Pt(111). The lateral roughness on short distances is governed by the low roughness exponent α=0.58 for all growth conditions independently of the growth mode.

Study of strain evolution mechanism in Ge1−xSnx materials grown by low temperature molecular beam epitaxy

Ge1−xSnx materials with constant and step-graded compositions have been successfully grown on Ge/Si(001) substrates by using low temperature molecular beam epitaxy (LT-MBE). It has been observed that both completely strained and partially relaxed GeSn materials with the same composition could be formed within the same sample, without adjusting any growth parameters. The residual in-plane strain in GeSn changes in a specific pattern from the GeSn/Ge interface towards the surface. The lower section of the GeSn material remains fully strained and free of dislocations, while most of the threading dislocations are located in the upper section of the layer, causing considerable strain relaxation within this section while maintaining the composition unchanged. This behavior could be explained by kinetic roughening and dislocation generation mechanisms at low temperatures and is remarkably different from GeSn materials grown by chemical vapor deposition (CVD), which exhibit a gradual strain relaxation process as growth continues. This work contributes to the fundamental understanding of the strain relaxation mechanisms of GeSn materials grown by MBE, which is instructive for improving the material quality in the future.

Simulations of Dissolution of Initially Flat Calcite Surfaces: Retreat Velocity Control and Surface Roughness Scaling

The far-from-equilibrium alkaline dissolution of initially flat and defect-free calcite surfaces is modeled with kinetic Monte Carlo simulations of site detachment from a Kossel crystal and with scaling approaches. The surface retreat velocity is strongly dependent on the detachment rate of the highest coordination sites, which represent molecules in the middle of (101̅4) terraces. The comparison with a recent velocity estimate from digital holographic microscopy predicts the removal rate of those molecules between 2 × 10–7 and 4 × 10–6 s–1 at room temperature, which improves a previous estimate from a grain dissolution model. The activation energy for this molecular-scale process is estimated as 92 ± 6 kJ/mol (assuming the same prefactor ∼3 × 1010 s–1 of kink and step sites). The areal density of nonterrace sites (mostly steps and kinks) is also related to that removal rate, so we propose that the measurement of this density may provide independent estimates of the above quantities. Arrhenius plots of the retreat velocities obtained in high-temperature simulations predict the (macroscopic) activation energy of 69 ± 4 kJ/mol for dissolution of smooth surfaces. We also show that the scaling of the surface roughness in time and size is the same as in the Kardar–Parisi–Zhang (KPZ) equation of kinetic roughening. However, room-temperature roughening of initially smooth calcite surfaces is so slow that it is unlikely to be observed in typical experimental times and the alkaline conditions modeled here. Since the rates in acidic media are larger and arguing that the microscopic symmetries of the molecule detachment processes should be preserved, we suggest the investigation of KPZ scaling under those conditions.

Kinetic roughening and nontrivial scaling in the Kardar–Parisi–Zhang growth with long-range temporal correlations

Long-range spatiotemporal correlations may play important roles in non-equilibrium surface growth processes. In order to explore the effects of long-range temporal correlation on dynamic scaling of growing surfaces, we carry out extensive numerical simulations of the (1 + 1)- and (2 + 1)-dimensional Kardar–Parisi–Zhang (KPZ) growth system in the presence of temporally correlated noise, and compare our results with previous theoretical predictions and numerical simulations. We find that surface morphologies are obviously affected with long-range temporal correlations, and as the temporal correlation exponent increases, the KPZ surfaces develop gradually faceted patterns in the saturated growth regimes. Our results show that the temporal correlated KPZ system displays evidently nontrivial dynamic properties when 0 < θ < 0.5, the characteristic roughness exponents satisfy α < α s, and α loc exhibits non-universal scaling within local window sizes, which differs with the existing dynamic scaling hypotheses, both in the (1 + 1)- and (2 + 1)-dimensions.