Kinetic growth with surface relaxation: Continuum versus atomistic models

Self-consistent expansion results for the nonlocal Kardar-Parisi-Zhang equation

In this paper various predictions for the scaling exponents of the nonlocal Kardar-Parisi-Zhang (NKPZ) equation are discussed. I use the self-consistent expansion (SCE), and obtain results that are quite different from the result obtained in the past, using dynamic renormalization-group analysis, a scaling approach, and a self-consistent mode-coupling approach. It is shown that the results obtained using SCE recover an exact result for a subfamily of the NKPZ models in one dimension, while all the other methods fail to do so. It is also shown that the SCE result is the only one that is compatible with simple observations on the dependence of the dynamic exponent z in the NKPZ model on the exponent rho characterizing the decay of the nonlinear interaction. The reasons for the failure of other methods to deal with NKPZ are also discussed.

Surface kinetics and generation of different terms in a conservative growth equation

A method based on the kinetics of adatoms on a growing surface under epitaxial growth at low temperature in (1+1) dimensions is proposed to obtain a closed form of the local growth equation. It can be generalized to any growth problem where surface morphology is governed by adatom diffusion. The method can be easily extended to higher dimensions. The kinetic processes contributing to various terms in the growth equation are identified from the analysis of in-plane and downward hops. In particular, processes corresponding to the term that breaks h-->-h symmetry and the curvature dependent term are discussed. Effects of these terms on the stable to unstable transition in (1+1) dimension are analyzed. In (2+1) dimensions, it is shown that an additional asymmetric term is generated due to the in-plane curvature associated with mound-like structures. This term is independent of any diffusion barrier differences between in-plane and out-of-plane migration. It is shown that terms generated in the presence of downward hops are the relevant terms in a growth equation. A growth equation in closed form is obtained for various growth models introduced to capture most of the processes in experimental molecular beam epitaxial growth. The effect of dissociation is also considered and is seen to have a stabilizing effect on growth. It is shown that for uphill current the growth equation approach fails to describe the growth since a given single equation does not apply over the entire substrate.

Crossover and universality in the Wolf-Villain model

The transition rules of the Wolf-Villain model for the deposition and instantaneous relaxation of particles on a lattice are expressed as a Langevin equation for the height fluctuations at each site. A coarse-graining transformation of this equation shows directly that this model belongs to the Edwards-Wilkinson universality class, in agreement with kinetic Monte Carlo simulations. The crossover from the Mullins-Herring equation is explained by the transformation under coarse graining of the coefficients in the equation of motion.

Time-reversed dielectric-breakdown model for erosion phenomena

A time-reversed dielectric-breakdown model in which the annihilating probability of a particle on the surface site (x,h) depends on Laplacian field phi(x,h,t) as P(x,h,t)=| inverted Delta phi(x,h,t)|(kappa)/ Sigma(x,h)| inverted Delta phi(x,h,t)|(kappa) is suggested. This model is shown to be a theoretical model that covers a variety of eroding surfaces from the linear phenomena with dynamic exponent z=1 to those showing nonlinear behavior. phi(x,t) is defined to satisfy the Laplace equation inverted Delta (2)phi=0 with the boundary condition phi=0 on the material and phi=1 far from the material. The model with 0.5 < or = kappa < or = 2 is found to follow the linear growth equation with z=1 as the diffusion-limited erosion, which is also a time-reversed version of diffusion-limited deposition. For small kappa, the dynamical scaling property of the eroding surface belongs to the Kardar-Parisi-Zhang universal class as the time-reversed Eden model. The model with kappa >2.5 does not show any surface roughening behavior.

Universality classification of restricted solid-on-solid type surface growth models

We consider the restricted solid-on-solid (RSOS) type surface growth models and classify them into dynamic universality classes according to their symmetry and conservation law. Four groups of RSOS-type microscopic models--asymmetric (A), asymmetric-conserved (AC), symmetric (S), and symmetric-conserved (SC) groups--are introduced and the corresponding stochastic differential equations (SDEs) are derived. Analyzing these SDEs using dynamic renormalization group theory, we confirm the previous results that A-RSOS, AC-RSOS, and S-RSOS groups belong to the Kardar-Parisi-Zhang class, the Villain-Lai-Das Sarma class, and the Edwards-Wilkinson class, respectively. We also find that SC-RSOS group belongs to a new universality class featuring the conserved-cubic nonlinearity.

Mounding instability and incoherent surface kinetics

Mounding instability in a conserved growth from vapor is analyzed within the framework of adatom kinetics on the growing surface. The analysis shows that depending on the local structure of the surface, kinetics of adatoms may vary, leading to disjoint regions in the sense of a continuum description. This is manifested particularly under the conditions of instability. Mounds grow on these disjoint regions and their lateral growth is governed by the flux of adatoms hopping across the steps in the downward direction. Asymptotically ln t dependence is expected in (1+1) dimensions. Simulation results confirm the prediction. Growth in (2+1) dimensions is also discussed.

Dynamical surface structures in multiparticle-correlated surface growths

We investigate the scaling properties of the interface fluctuation width for the Q-mer and Q-particle-correlated deposition-evaporation models. These models are constrained with a global conservation law that the particle number at each height is conserved modulo Q. In equilibrium, the stationary roughness is anomalous but universal with the roughness exponent alpha=1/3, while the early time evolution shows nonuniversal behavior with the growth exponent beta varying with models and Q. Nonequilibrium surfaces display diverse growing and stationary behaviors. The Q-mer model shows a faceted structure, while the Q-particle-correlated model shows a macroscopically grooved structure.

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.

Universality class of discrete solid-on-solid limited mobility nonequilibrium growth models for kinetic surface roughening

We investigate, using the noise reduction technique, the asymptotic universality class of the well-studied nonequilibrium limited mobility atomistic solid-on-solid surface growth models introduced by Wolf and Villain (WV) and Das Sarma and Tamborenea (DT) in the context of kinetic surface roughening in ideal molecular beam epitaxy. We find essentially all the earlier conclusions regarding the universality class of DT and WV models to be severely hampered by slow crossover and extremely long-lived transient effects. We identify the correct asymptotic universality class(es) that differs from earlier conclusions in several instances.

Self-consistent expansion for the molecular beam epitaxy equation

Motivated by a controversy over the correct results derived from the dynamic renormalization group (DRG) analysis of the nonlinear molecular beam epitaxy (MBE) equation, a self-consistent expansion for the nonlinear MBE theory is considered. The scaling exponents are obtained for spatially correlated noise of the general form D(r-r('),t-t('))=2D(0)[r-->-r(')](2rho-d)delta(t-t(')). I find a lower critical dimension d(c)(rho)=4+2rho, above which the linear MBE solution appears. Below the lower critical dimension a rho-dependent strong-coupling solution is found. These results help to resolve the controversy over the correct exponents that describe nonlinear MBE, using a reliable method that proved itself in the past by giving reasonable results for the strong-coupling regime of the Kardar-Parisi-Zhang system (for d>1), where DRG failed to do so.

Universality class of the restricted solid-on-solid model with hopping

We study the restricted solid-on-solid (RSOS) model with finite hopping distance l(0), using both analytical and numerical methods. Analytically, we use the hard-core bosonic field theory developed by the authors [Phys. Rev. E 62, 7642 (2000)] and derive the Villain-Lai-Das Sarma (VLD) equation for the l(0)=infinity case, which corresponds to the conserved RSOS (CRSOS) model and the Kardar-Parisi-Zhang (KPZ) equation for all finite values of l(0). Consequently, we find that the CRSOS model belongs to the VLD universality class and that the RSOS models with any finite hopping distance belong to the KPZ universality class. There is no phase transition at a certain finite hopping distance contrary to the previous result. We confirm the analytic results using the Monte Carlo simulations for several values of the finite hopping distance.

Derivation of continuum stochastic equations for discrete growth models

We present a formalism to derive the stochastic differential equations (SDEs) for several solid-on-solid growth models. Our formalism begins with a mapping of the microscopic dynamics of growth models onto the particle systems with reactions and diffusion. We then write the master equations for these corresponding particle systems and find the SDEs for the particle densities. Finally, by connecting the particle densities with the growth heights, we derive the SDEs for the height variables. Applying this formalism to discrete growth models, we find the Edwards-Wilkinson equation for the symmetric body-centered solid-on-solid (BCSOS) model, the Kardar-Parisi-Zhang equation for the asymmetric BCSOS model and the generalized restricted solid-on-solid (RSOS) model, and the Villain-Lai-Das Sarma equation for the conserved RSOS model. In addition to the consistent forms of equations for growth models, we also obtain the coefficients associated with the SDEs.

Dynamic renormalization group study of a generalized continuum model of crystalline surfaces

We apply the Nozières-Gallet dynamic renormalization group (RG) scheme to a continuum equilibrium model of a d-dimensional surface relaxing by linear surface tension and linear surface diffusion, and which is subject to a lattice potential favoring discrete values of the height variable. The model thus interpolates between the overdamped sine-Gordon model and a related continuum model of crystalline tensionless surfaces. The RG flow predicts the existence of an equilibrium roughening transition only for d=2 dimensional surfaces, between a flat low-temperature phase and a rough high-temperature phase in the Edwards-Wilkinson (EW) universality class. The surface is always in the flat phase for any other substrate dimensions d>2. For any value of d, the linear surface diffusion mechanism is an irrelevant perturbation of the linear surface tension mechanism, but may induce long crossovers within which the scaling properties of the linear molecular-beam epitaxy equation are observed, thus increasing the value of the sine-Gordon roughening temperature. This phenomenon originates in the nonlinear lattice potential, and is seen to occur even in the absence of a bare surface tension term. An important consequence of this is that a crystalline tensionless surface is asymptotically described at high temperatures by the EW universality class.

Roughness scaling in cyclical surface growth

The scaling behavior of cyclical growth (e.g., cycles of alternating deposition and desorption primary processes) is investigated theoretically and probed experimentally. The scaling approach to kinetic roughening is generalized to cyclical processes by substituting the number of cycles n for the time. The roughness is predicted to grow as n(beta) where beta is the cyclical growth exponent. The roughness saturates to a value that scales with the system size L as L(alpha), where alpha is the cyclical roughness exponent. The relations between the cyclical exponents and the corresponding exponents of the primary processes are studied. Exact relations are found for cycles composed of primary linear processes. An approximate renormalization group approach is introduced to analyze nonlinear effects in the primary processes. The analytical results are backed by extensive numerical simulations of different pairs of primary processes, both linear and nonlinear. Experimentally, silver surfaces are grown by a cyclical process composed of electrodeposition followed by 50% electrodissolution. The roughness is found to increase as a power law of n, consistent with the scaling behavior anticipated theoretically. Potential applications of cyclical scaling include accelerated testing of rechargeable batteries and improved chemotherapeutic treatment of cancerous tumors.

Fokker-Planck equation for lattice deposition models

An asymptotically exact Fokker-Planck equation for the height fluctuations of lattice deposition models is derived from a Van Kampen expansion of the master equation. Using an Edwards-Wilkinson-type model as an example, the solution of the equivalent Langevin equation reproduces the surface roughness and lateral height correlations obtained with kinetic Monte Carlo (KMC) simulations. Our discrete equations of motion thereby provide an exact analytic and computational alternative to KMC simulations of these models.

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.

Amorphous thin film growth: effects of density inhomogeneities

A nonlinear stochastic growth equation for the spatiotemporal evolution of the surface morphology of amorphous thin films in the presence of potential density variations is derived from the relevant physical symmetries and compared to recent experimental results. Numerical simulations of the growth equation exhibit a saturation of the surface morphology for large film thickness originating from the inclusion of the density inhomogeneities. Furthermore, we argue why moundlike surface structures observed on vapor deposited amorphous films are not the result of the Grinfeld instability.

Effect of long-range interactions on the scaling of the noisy Kuramoto-Sivashinsky equation

The effects of long-range interactions on the scaling properties of the noisy Kuramoto-Sivashinsky (KS) equation are studied by the dynamic renormalization-group technique. It is found that the presence of long-range nonlinearity in the KS equation can produce new stable fixed points with varying critical exponents that depend on both the long-range interaction parameter rho and the substrate dimension d.

Extremal-point densities of interface fluctuations in a quenched random medium

We give a number of exact, analytical results for the stochastic dynamics of the density of local extrema (minima and maxima) of linear Langevin equations and solid-on-solid lattice growth models driven by spatially quenched random noise. Such models can describe nonequilibrium surface fluctuations in a spatially quenched random medium, diffusion in a random catalytic environment, and polymers in a random medium. In spite of the nonuniversal character for the quantities studied, their behavior against the variation of the microscopic length scale can present generic features, characteristic of the macroscopic observables of the system.

Bulk dynamics for interfacial growth models

We study the influence of the bulk dynamics of a growing cluster of particles on the properties of its interface. First, we define a general bulk growth model by means of a continuum Master equation for the evolution of the bulk density field. This general model just considers an arbitrary addition of particles (though it can be easily generalized to consider subtraction) with no other physical restriction. The corresponding Langevin equation for this bulk density field is derived where the influence of the bulk dynamics is explicitly shown. Finally, when a well-defined interface is assumed for the growing cluster, the Langevin equation for the height field of this interface for some particular bulk dynamics is written. In particular, we obtain the celebrated Kardar-Parisi-Zhang equation. A Monte Carlo simulation illustrates the theoretical results.

Amorphous thin film growth: minimal deposition equation

A nonlinear stochastic growth equation is derived from (i) the symmetry principles relevant for the growth of vapor deposited amorphous films, (ii) no excess velocity, and (iii) a low-order expansion in the gradients of the surface profile. A growth instability in the equation is attributed to the deflection of the initially perpendicular incident particles due to attractive forces between the surface atoms and the incident particles. The stationary solutions of the deterministic limit of the equation and their stability are analyzed. The growth of the surface roughness and the correlation length of the moundlike surface structure arising from the stochastic growth equation is investigated.

Renormalization group analysis of the anisotropic nonlocal kardar-parisi-zhang equation with spatially correlated noise

We study an anisotropic nonlocal Kardar-Parisi-Zhang (KPZ) equation with spatially correlated noise by using the dynamic renormalization group method. When the signs of nonlinear terms in parallel and perpendicular directions are opposite, the correlated noise coupled with the long ranged nature of interaction produces a stable non-KPZ fixed point for d<d(c). For the uncorrelated noise, the roughness and dynamic exponents associated with the stable fixed point are different from those of the isotropic nonlocal KPZ equation, while for the correlated noise the exponents are the same as those of the isotropic case.

Nonlocal effects in the conserved kardar-parisi-zhang equation

By using the dynamic renormalization group approach, we analyze a nonlocal conserved Kardar-Parisi-Zhang equation with spatially correlated conservative noise in order to study the effect of the long-range nature of interactions coupled with spatially correlated noise on the dynamics of a volume conserving surface. The roughness of the surface depends on both the long-range interaction strength and the spatial correlation parameter. The surface becomes less rough by the long-range interaction, while it becomes more rough by the spatial correlation of noise. We also study the nonlocal conserved Kardar-Parisi-Zhang equation with spatially correlated nonconservative noise.

Multiparticle biased diffusion-limited aggregation with surface diffusion: A comprehensive model of electrodeposition

We present a complete study of the multiparticle biased diffusion-limited aggregation (MBDLA) model supplemented with surface diffusion (SD), focusing on the relevance and effects of the latter transport mechanism. By comparing different algorithms, we show that MBDLA + SD is a very good qualitative model for electrodeposition in essentially the whole range of current intensities provided one introduces SD in the model in the proper fashion. We have found that the correct procedure involves simultaneous bulk diffusion and SD, introducing a time scale arising from the ratio of the rates of the two processes. We discuss in detail the different morphologies obtained and compare them to the available experimental data with very satisfactory results. We also characterize the aggregates thus obtained by means of the dynamic scaling exponents of the interface height, allowing us to distinguish several regimes in the mentioned interface growth. Our asymptotic scaling exponents are again in good agreement with recent experiments. We conclude by discussing a global picture of the influence and consequences of SD in electrodeposition.

Extremal-point densities of interface fluctuations

We introduce and investigate the stochastic dynamics of the density of local extrema (minima and maxima) of nonequilibrium surface fluctuations. We give a number of analytic results for interface fluctuations described by linear Langevin equations, and for on-lattice, solid-on-solid surface-growth models. We show that, in spite of the nonuniversal character of the quantities studied, their behavior against the variation of the microscopic length scales can present generic features, characteristic of the macroscopic observables of the system. The quantities investigated here provide us with tools that give an unorthodox approach to the dynamics of surface morphologies: a statistical analysis from the short-wavelength end of the Fourier decomposition spectrum. In addition to surface-growth applications, our results can be used to solve the asymptotic scalability problem of massively parallel algorithms for discrete-event simulations, which are extensively used in Monte Carlo simulations on parallel architectures.

Interfacial coarsening dynamics in epitaxial growth with slope selection

We investigate interfacial dynamics of molecular-beam epitaxy (MBE) growth in the presence of instabilities inducing formation of pyramids. We introduce a kinetic scaling theory which provides an analytic understanding of the coarsening dynamics laws observed in numerous experiments and simulations of the MBE. We address MBE growth on crystalline surfaces with different symmetries in order to explain experimentally observed differences between the growth on (111) and (001) surfaces and understand the coarsening exponents measured on these surfaces. We supplement our kinetic scaling theory by numerical simulations which document that the edges of the pyramids, forming a network across the growing interface, are essential for qualitative understanding of the coarsening dynamics of molecular-beam epitaxy.

Reentrant mound formation in GaAs(001) homoepitaxy observed by ex situ atomic force microscopy

A study of the surface morphology of homoepitaxial GaAs(001) by means of ex situ atomic force microscopy in air reveals the reentrance of mounding behavior at low growth temperatures. A transition from statistical roughening to organized mound formation is observed as the growth temperature is reduced. We show by means of growth simulations that the observed morphology is compatible with anisotropic adatom diffusion in the presence of an Ehrlich-Schwoebel barrier. The mechanism leading to this kind of adatom kinetics at low temperatures is interpreted in terms of surfactant acting arsenic condensing on the surface.

Dynamics of rough interfaces in chemical vapor deposition: experiments and a model for silica films

We study the surface dynamics of silica films grown by low pressure chemical vapor deposition. Atomic force microscopy measurements show that the surface reaches a scale invariant stationary state compatible with the Kardar-Parisi-Zhang (KPZ) equation in three dimensions. At intermediate times the surface undergoes an unstable transient due to shadowing effects. By varying growth conditions and using spectroscopic techniques, we determine the physical origin of KPZ scaling to be a low value of the surface sticking probability, related to the surface concentration of reactive groups. We propose a stochastic equation that describes the qualitative behavior of our experimental system.

Dynamical universality classes of vapor deposition models with evaporation

Growth models for vapor depositions in which evaporation and deposition can occur, both at a randomly chosen column and at its nearest neighbor columns (NNCs), are studied by Monte Carlo simulations. The growth processes in these models are determined by comparing local chemical potentials of the chosen column and its NNCs to the chemical potential of vapor. The universality classes and the characteristics of the models are studied by the scaling ansatz of kinetic roughening and by measurements of tilt-dependent currents and height step widths. Through measurements of the ratio of number of growth processes at NNCs to those at the chosen column, the key processes which are relevant to the determination of the universality class are also studied.

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.