Growth in a restricted solid-on-solid model

Anisotropic ballistic deposition model with links to the Ulam problem and the Tracy-Widom distribution

We compute exactly the asymptotic distribution of scaled height in a (1+1)-dimensional anisotropic ballistic deposition model by mapping it to the Ulam problem of finding the longest nondecreasing subsequence in a random sequence of integers. Using the known results for the Ulam problem, we show that the scaled height in our model has the Tracy-Widom distribution appearing in the theory of random matrices near the edges of the spectrum. Our result supports the hypothesis that various growth models in (1+1) dimensions that belong to the Kardar-Parisi-Zhang universality class perhaps all share the same universal Tracy-Widom distribution for the suitably scaled height variables.

Partition functions and metropolis-type evolution rules for surface growth models with constraints

We study dynamical scaling properties of the surface growth model with the Metropolis-type evolution rule from a partition function Z= sum ([h(r)])II (h(max))(h=h(min))1/2(1+z(n(h))), where z is a fugacity-like quantity and n(h) is the number of sites with height h in a surface configuration [h(r)]. The partition function describes a 2-particle correlated growth model when z=-1 and a self-flattening growth model when z=0. For one-dimensional equilibrium surfaces, the scaling properties for z>or=-1 except z=1 are all one phase with roughness exponent alpha=1/3 and growth exponent beta approximately equal 0.22. For the growing (eroding) surfaces, there exists a phase transition at z=0 from the grooved phase (alpha=1) for -1<or=z<0 to the ordinary Kardar-Parisi-Zhang phase (alpha=1/2) for z>0.

Scaling function for surface width for free boundary conditions

We study the restricted curvature model with both periodic and free boundary conditions and show that the scaling function of the surface width depends on the type of boundary conditions. When the free boundary condition is applied, the surface width shows a new dynamic scaling whose asymptotic behavior is different from the usual scaling behavior of the self-affine surfaces. We propose a generalized scaling function for the surface width for free boundary conditions and introduce a normalized surface width to clarify the origin of the superrough phenomena of the model.

Dynamical scaling of surface growth in simple lattice models

We present extensive simulations of the atomistic Edwards-Wilkinson (EW) and Restricted Edwards-Wilkinson (REW) models in 2+1 dimensions. Dynamic finite-size scaling analyses of the interfacial width and structure factor provide the estimates for the dynamic exponent z=1.65+/-0.05 for the EW model and z=2.0+/-0.1 for the REW model. The stochastic contribution to the interface velocity U due to the deposition and diffusion of particles is characterized for both the models using a blocking procedure. For the EW model the time-displaced temporal correlations in U show nonexponential decay, while the temporal correlations decay exponentially for the REW model. Dynamical scaling of the temporal correlation function for the EW model yields a value of z, which is consistent with the estimate obtained from finite-size scaling of the interfacial width and structure factor.

Spontaneous formation of dynamical patterns with fractal fronts in the cyclic lattice Lotka-Volterra model

Dynamical patterns, in the form of consecutive moving stripes or rings, are shown to develop spontaneously in the cyclic lattice Lotka-Volterra model, when realized on square lattice, at the reaction limited regime. Each stripe consists of different particles (species) and the borderlines between consecutive stripes are fractal. The interface width w between the different species scales as w(L,t) approximately L(alpha)f(t/L(z)), where L is the linear size of the interface, t is the time, and alpha and z are the static and dynamical critical exponents, respectively. The critical exponents were computed as alpha=0.49+/-0.03 and z=1.53+/-0.13 and the propagating fronts show dynamical characteristics similar to those of the Eden growth models.

Anomaly of the height-height correlation functions in self-flattening surface growth

By Monte Carlo simulations and scaling theories, we consider the height-height correlation function G(r,t;L) of the one-dimensional equilibrium self-flattening surface growths, where the deposition (evaporation) attempt only at the globally highest (lowest) site is suppressed. G(r,t:L) is shown to satisfy the anomalous scaling behavior G(r,t;L)=L(2alpha)g(1)(r/L(delta),t/L(z)) or G(r,t;L)=t(2beta)g(2)(r/t(1/z(')),L/t(1/z)). Here alpha, beta, and z are the roughness, growth, and dynamic exponents, respectively, for the surface width, with alpha=1/3 and z=alpha/beta=3/2. Anomalous exponents z(') and delta are found to satisfy z(')=9/4 and delta=z/z('). We also show that anomalous behavior of G(r,t;L) can be understood from a scaling theory based on the competition between local random-walk-like behavior and the global-length-scale suppression.

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.

Growth model with restricted surface relaxation

We simulate a growth model with restricted surface relaxation process in d=1 and d=2, where d is the dimensionality of a flat substrate. In this model, each particle can relax on the surface to a local minimum, as the discrete surface relaxation model, but only within a distance s. If the local minimum is out from this distance, the particle evaporates through a refuse mechanism similar to the Kim-Kosterlitz nonlinear model. In d=1, the growth exponent beta, measured from the temporal behavior of roughness, indicates that in the coarse-grained limit, the linear term of the Kardar-Parisi-Zhang equation dominates in short times (low-roughness) and, in asymptotic times, the nonlinear term prevails. The crossover between linear and nonlinear behaviors occurs in a characteristic time t(c) which only depends on the magnitude of the parameter s, related to the nonlinear term. In d=2, we find indications of a similar crossover, that is, logarithmic temporal behavior of roughness in short times and power law behavior in asymptotic times.

Restricted curvature model with suppression of extremal height

A discrete growth model with a restricted curvature constraint is investigated by measuring both the surface width and the height difference correlation function. In our model, where an extremal height is suppressed, the surface width W shows the roughness exponent alpha approximately 0.561 and the dynamics exponent z approximately 1.69 in one substrate dimension. However the correlation function has an unusual scaling behavior and produces different wandering exponent alpha(') approximately 1.33 and its dynamic exponent z(') approximately 4. The discrepancy is due to the fact that the correlation length increases with a power law t(1/z(')) until it reaches the value proportional to Ldelta at time t(s) approximately L(z), where L is the system size and delta is the "window exponent" satisfying the relation delta=z/z(')=alpha/alpha('). delta is a new exponent to characterize the window size of the system.

Fluctuations of self-flattening surfaces

We study the scaling properties of self-flattening surfaces under global suppression on surface fluctuations. Evolution of self-flattening surfaces is described by restricted solid-on-solid type monomer deposition-evaporation model with reduced deposition (evaporation) at the globally highest (lowest) site. We find numerically that equilibrium surface fluctuations are anomalous with roughness exponent alpha approximately equal to 1/3 and dynamic exponent z(W) approximately equal to 3/2 in one dimension (1D) and alpha=0 (log) and z(W) approximately 5/2 in 2D. Stationary roughness can be understood analytically by relating our model to the static self-attracting random walk model and the dissociative dimer-type deposition-evaporation model. In case of nonequilibrium growing-eroding surfaces, self-flattening dynamics turns out to be irrelevant and the normal Kardar-Parisi-Zhang universality is recovered in all dimensions.

Effect of spatial bias on the nonequilibrium phase transition in a system of coagulating and fragmenting particles

We examine the effect of spatial bias on a nonequilibrium system in which masses on a lattice evolve through the elementary moves of diffusion, coagulation, and fragmentation. When there is no preferred directionality in the motion of the masses, the model is known to exhibit a nonequilibrium phase transition between two different types of steady state, in all dimensions. We show analytically that introducing a preferred direction in the motion of the masses inhibits the occurrence of the phase transition in one dimension, in the thermodynamic limit. A finite-size system, however, continues to show a signature of the original transition, and we characterize the finite-size scaling implications of this. Our analysis is supported by numerical simulations. In two dimensions, bias is shown to be irrelevant.

Passive random walkers and riverlike networks on growing surfaces

Passive random walker dynamics is introduced on a growing surface. The walker is designed to drift upward or downward and then follow specific topological features, such as hill tops or valley bottoms, of the fluctuating surface. The passive random walker can thus be used to directly explore scaling properties of otherwise somewhat hidden topological features. For example, the walker allows us to directly measure the dynamical exponent of the underlying growth dynamics. We use the Kardar-Parisi-Zhang (KPZ) -type surface growth as an example. The world lines of a set of merging passive walkers show nontrivial coalescence behaviors and display the riverlike network structures of surface ridges in space-time. In other dynamics, such as Edwards-Wilkinson growth, this does not happen. The passive random walkers in KPZ-type surface growth are closely related to the shock waves in the noiseless Burgers equation. We also briefly discuss their relations to the passive scalar dynamics in turbulence.

Static phase and dynamic scaling in a deposition model with an inactive species

We extend a previously proposed deposition model with two kinds of particle, considering the restricted solid-on-solid condition. The probability of incidence of particle C (A) is p (1-p). Aggregation is possible if the top of the column of incidence has a nearest neighbor A and if the difference in the heights of neighboring columns does not exceed 1. For any value of p>0, the deposit attains some static configuration, in which no deposition attempt is accepted. In 1+1 dimensions, the interface width has a limiting value W(s) approximately p(-eta), with eta=3/2, which is confirmed by numerical simulations. The dynamic scaling relation W(s)=p(-eta)f(tp(z)) is obtained in very large substrates, with z=eta.

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.

Pseudospectral method for the Kardar-Parisi-Zhang equation

We discuss a numerical scheme to solve the continuum Kardar-Parisi-Zhang equation in generic spatial dimensions. It is based on a momentum-space discretization of the continuum equation and on a pseudospectral approximation of the nonlinear term. The method is tested in (1+1) and (2+1) dimensions, where it is shown to reproduce the current most reliable estimates of the critical exponents based on restricted solid-on-solid simulations. In particular, it allows the computations of various correlation and structure functions with high degree of numerical accuracy. Some deficiencies that are common to all previously used finite-difference schemes are pointed out and the usefulness of the present approach in this respect is discussed.

Width distributions and the upper critical dimension of Kardar-Parisi-Zhang interfaces

Simulations of restricted solid-on-solid growth models are used to build the width distributions of d=2-5 dimensional Kardar-Parisi-Zhang (KPZ) interfaces. We find that the universal scaling function associated with the steady-state width distribution changes smoothly as d is increased, thus strongly suggesting that d=4 is not an upper critical dimension for the KPZ equation. The dimensional trends observed in the scaling functions indicate that the upper critical dimension is at infinity.

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.

Slow crossover to Kardar-Parisi-Zhang scaling

The Kardar-Parisi-Zhang (KPZ) equation is accepted as a generic description of interfacial growth. In several recent studies, however, values of the roughness exponent alpha have been reported that are significantly less than that associated with the KPZ equation. A feature common to these studies is the presence of holes (bubbles and overhangs) in the bulk and an interface that is smeared out. We study a model of this type in which the density of the bulk and sharpness of the interface can be adjusted by a single parameter. Through theoretical considerations and the study of a simplified model we determine that the presence of holes does not affect the asymptotic KPZ scaling. Moreover, based on our numerics, we propose a simple form for the crossover to the KPZ regime.

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.

Anomalous crossover behavior at finite temperature

We introduce a stochastic growth model where the growth is controlled by a temperaturelike parameter T. The model shows various types of dynamical behavior as T changes from 0 to infinity. For T=0 the growth process belongs to the quenched Kardar-Parisi-Zhang (KPZ) universality class, whereas it belongs to the Edwards-Wilkinson (EW) universality class for T=infinity. In the intermediate range 0<T<infinity, the model shows an anomalous crossover behavior from the quenched KPZ to the thermal KPZ class. The KPZ nonlinearity is generated by an anisotropic effect of the quenched noise which exists only for T<infinity in our model. We also study crossovers between different types of scaling behavior of the interface width for various T's.

Reconstructed rough growing interfaces: ridge-line trapping of domain walls

We investigate whether surface reconstruction order exists in stationary growing states at all length scales or only below a crossover length l(rec). The latter behavior would be similar to surface roughness in growing crystal surfaces; below the equilibrium roughening temperature they evolve in a layer-by-layer mode within a crossover length scale l(R), but are always rough at large length scales. We investigate this issue in the context of Kardar-Parisi-Zhang (KPZ) type dynamics and a checkerboard type reconstruction, using the restricted solid-on-solid model with negative monatomic step energies. This is a topology where surface reconstruction order is compatible with surface roughness and where a so-called reconstructed rough phase exists in equilibrium. We find that during growth reconstruction order is absent in the thermodynamic limit, but exists below a crossover length l(rec)>l(R), and that this local order fluctuates critically. Domain walls become trapped at the ridge lines of the rough surface, and thus the reconstruction order fluctuations are slaved to the KPZ dynamics.

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.

Universality and corrections to scaling in the ballistic deposition model

In order to analyze some controversies on the equivalence between ballistic deposition (BD) and the Kardar-Parisi-Zhang (KPZ) theory, we simulated the BD model in one and two dimensions. Effective exponents betaL were obtained in the growth regions, which were rigorously determined for various lengths L. Effective exponents alphaL were obtained from saturation widths in the steady-state regimes. In d=1 we found betaL=beta+AL(-lambda) and alphaL=alpha+BL(-delta), with asymptotic exponents consistent with the KPZ values beta=1/3 and alpha=1/2, and correction-to-scaling exponents 0.2 < or approximately = lambda < or approximately = 0.4 and 0.6 < or approximately = delta < or approximately = 0.8. These strong finite-size corrections explain the previous discrepancies between numerical estimates for BD and the exact KPZ results. In d=2 we could only obtain reliable estimates of alphaL, which are consistent with KPZ values if finite-size corrections with delta approximately 0.4 are considered.

Kinetic roughening model with opposite Kardar-Parisi-Zhang nonlinearities

We introduce a model that simulates a kinetic roughening process with two kinds of particle: one follows ballistic deposition (BD) kinetics and the other restricted solid-on-solid Kim-Kosterlitz (KK) kinetics. Both of these kinetics are in the universality class of the nonlinear Kardar-Parisi-Zhang equation, but the BD kinetics has a positive nonlinear constant while the KK kinetics has a negative one. In our model, called the BD-KK model, we assign the probabilities p and (1-p) to the KK and BD kinetics, respectively. For a specific value of p, the system behaves as a quasilinear model and the up-down symmetry is restored. We show that nonlinearities of odd order are relevant in this low nonlinear limit.

Kinetic roughening with anisotropic growth rules

Inspired by the chemical etching processes, where experiments show that growth rates depending on the local environment might play a fundamental role in determining the properties of the etched surfaces, we study here a model for kinetic roughening that includes explicitly an anisotropic effect in the growth rules. Our model introduces a dependence of the growth rules on the local environment conditions, i.e., on the local curvature of the surface. Variables with different local curvatures of the surface, in fact, present different quenched disorder and a parameter p (which could represent different experimental conditions) is introduced to account for different time scales for the different classes of variables. We show that the introduction of this time scale separation in the model leads to a crossover effect on the roughness properties. This effect could explain the scattering in the experimental measurements available in the literature. The interplay between anisotropy and the crossover effect and the dependence of critical properties on parameter p is investigated as well as the relationship with the known universality classes.

Entropy fluctuations for directed polymers in 2+1 dimensions

We find numerically that the sample to sample fluctuation of the entropy DeltaS is a more sensitive tool in distinguishing low from high temperature behaviors than the common corresponding fluctuation in the free energy. In 1+1 dimensions we find a single phase for all temperatures, since (DeltaS)(2) is always extensive. In 2+1 dimensions we find a behavior that at first sight might appear to be a transition from a low temperature phase where (DeltaS)(2) is extensive to a high temperature phase where it is subextensive. This is observed in spite of the relatively large system we use. The observed behavior is explained not as a phase transition but as a strong crossover behavior. We use an analytical argument to obtain (DeltaS)(2) for high temperature, and find that while it is always extensive it is also extremely small, and that the leading extensive part decays very quickly to zero as the temperature is increased.

Dynamical self-affinity of damage spreading in surface growth models

The dynamical anisotropic scaling properties of the surface growth models are restudied by use of the damage spreading concept. For that the vertical damage spreading distance d( perpendicular) of a damaged column as well as the lateral damage spreading distance d(||) is introduced. The scaling Ansatze for &dmacr;( perpendicular)(d(||),t), D(||) identical with<d(||)> and D( perpendicular) identical with<&dmacr;( perpendicular)> are suggested. The critical property of the probability distribution P(d(||),t) for the survived damages is also suggested. The suggested scaling relations are tested by simulating various growth models with substrate dimension d=1. From these results it can be concluded that the critical property or dynamical self-affinity of a surface growth model can also be determined by investigating the damage spreading.

River networks on the slope-correlated landscape

We study the morphologies of river networks on various landscapes. In general, the probability density distribution of drainage area a of the river network scales as P(a) approximately a(-tau). We consider a slope-slope correlation function G(r) and define the persistent length R where G(r=R) becomes zero. In our restricted solid on solid network model, R is independent of the system size L and tau is close to 4/3, which is the value of the Scheidegger's river network model with random walk process. We also consider an avalanche model, where R is proportional to L. There is a large slope-slope correlation length and the river network does not follow the directed random walk process with tau approximately 1.42.

Growth with surface curvature on quenched potentials

A discrete growth model driven by the Laplacian of the surface curvature in quenched random media is discussed. The interface width W at the saturated regime obeys scaling W approximately Lalpha with alpha approximately 2.3, where L is the system size. Starting from an initial sine wave condition of a selected wavelength, we measure an autocorrelation function, and obtain the dynamic critical exponent z approximately 3.1. The model is expected to be described by the quenched Mullins-Herring equations.

Evolution of networks with aging of sites

We study the growth of a network with aging of sites. Each new site of the network is connected to some old site with probability proportional (i) to the connectivity of the old site as in the Barabasi-Albert's model and (ii) to tau(-alpha), where tau is the age of the old site. We find both from simulation and analytically that the network shows scaling behavior only in the region alpha<1. When alpha increases from -infinity to 0, the exponent gamma of the distribution of connectivities [P(k) approximately k(-gamma) for large k] grows from 2 to the value for the network without aging. The ensuing increase of alpha to 1 causes gamma to grow to infinity. For alpha>1, the distribution P(k) is exponentional.

Scaling behavior of cyclical surface growth

The scaling behavior of cyclical surface growth (e.g., deposition/desorption), with the number of cycles, n, is investigated. The roughness of surfaces grown by two linear primary processes follows a scaling behavior with asymptotic exponents inherited from the dominant process while the effective amplitudes are determined by both. Relevant nonlinear effects in the primary processes may remain so or be rendered irrelevant. Numerical simulations for several pairs of generic primary processes confirm these conclusions. Experimental results for the surface roughness during cyclical electrodeposition/dissolution of silver show a power-law dependence on n, consistent with the scaling description.

Self-organized growth model for the quenched herring-mullin equation

We introduce a simple self-organized growth model mimicking the dynamics of a driven tensionless interface in a random medium near the depinning threshold. The roughness and growth exponents for the model are obtained as zeta approximately 1.93 and beta approximately 0.96, respectively. We discuss the possible continuum equation describing the motion of a driven interface in our model.

Statistical models for carbon-nitrogen film growth

We studied models of deposition and erosion, with two species of particles, that represent quantitatively many features of amorphous carbon-nitrogen film grown under plasma enhanced chemical vapor deposition. In the original model, the columns of the deposit are independent, and particles C and N are released with probabilities p and 1-p, respectively. An incident C particle always aggregates upon contact with the surface. An N particle annihilates with a top C particle with probability q and aggregates with probability 1-q. An N particle always annihilates with a top N. A critical line separates the regimes of growth (p>q/2) and erosion (p<q/2). For fixed q, when p decreases towards the critical value p(c)=q/2, the bulk concentration of N (x(N)) increases, and the growth rate r decreases. The rxx(N) curve for q=0.25 agrees with data from films grown in acetylene-nitrogen atmospheres. In order to represent the blocking of surface bonds by hydrogen atoms, we considered a second model in which any aggregation process is accepted with probability alpha, otherwise it is rejected. For q=0.25 and alpha=0.3, the rxx(N) curve agrees with data from films grown in methane-nitrogen and methane-ammonia atmospheres. The fitting values of q and alpha were inferred from related experiments. In order to test the influence of lattice structure and spatial correlations, we also studied those models in simple cubic lattices, considering that the aggregation must satisfy the restricted solid-on-solid model conditions for the difference of heights in neighboring columns, while the erosion is random. We obtained similar results for rxx(N) curves, confirming the validity of those models to represent the kinetics of amorphous films growth. It was also observed that the surface roughness increases with x(N), which agrees qualitatively with several experiments on carbon-nitrogen films growth with ion bombardment.

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.

Nonlinear measures for characterizing rough surface morphologies

We develop an approach for characterizing the morphology of rough surfaces based on the analysis of the scaling properties of contour loops, i.e., loops of constant height. Given a height profile of the surface we perform independent measurements of the fractal dimension of contour loops, and the exponent that characterizes their size distribution. Scaling formulas are derived, and used to relate these two geometrical exponents to the roughness exponent of a self-affine surface, thus providing independent measurements of this important quantity. Furthermore, we define the scale-dependent curvature, and demonstrate that by measuring its third moment departures of the height fluctuations from Gaussian behavior can be ascertained. These nonlinear measures are used to characterize the morphology of computer generated Gaussian rough surfaces, surfaces obtained in numerical simulations of a simple growth model, and surfaces observed by scanning-tunneling microscopes. For experimentally realized surfaces the self-affine scaling is cut off by a correlation length, and we generalize our theory of contour loops to take this into account.

Self-consistent expansion for the Kardar-Parisi-Zhang equation with correlated noise

A minor modification of the self-consistent expansion (SCE) for the Kardar-Parisi-Zhang (KPZ) system with uncorrelated noise is used to obtain the exponents in systems where the noise has spatial long-range correlations. For d-dimensional systems with correlations of the form D((-->)r-(-->)r',t-t')=2D(0)/(-->)r-(-->)r'/2 rho-d)delta(t-t'), (rho>0), we find a lower critical dimension d(0)(rho)=2+2 rho, above which a perturbative Edwards-Wilkinson (EW) solution appears. Below the lower critical dimension two solutions exist, each in a different, distinct region of rho. For small rho's the solution of KPZ with uncorrelated noise is recovered. For large rho's a rho-dependent solution is found. The existence of only one solution in each region of rho is not a result of a competition between two solutions but a direct outcome of the SCE equation.

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.

Thermodynamically reversible generalization of diffusion limited aggregation

We introduce a lattice gas model of cluster growth via the diffusive aggregation of particles in a closed system obeying a local, deterministic, microscopically reversible dynamics. This model roughly corresponds to placing the irreversible diffusion limited aggregation model (DLA) in contact with a heat bath. Particles release latent heat when aggregating, while singly connected cluster members can absorb heat and evaporate. The heat bath is initially empty, hence we observe the flow of entropy from the aggregating gas of particles into the heat bath, which is being populated by diffusing heat tokens. Before the population of the heat bath stabilizes, the cluster morphology (quantified by the fractal dimension) is similar to a standard DLA cluster. The cluster then gradually anneals, becoming more tenuous, until reaching configurational equilibrium when the cluster morphology resembles a quenched branched random polymer. As the microscopic dynamics is invertible, we can reverse the evolution, observe the inverse flow of heat and entropy, and recover the initial condition. This simple system provides an explicit example of how macroscopic dissipation and self-organization can result from an underlying microscopically reversible dynamics. We present a detailed description of the dynamics for the model, discuss the macroscopic limit, and give predictions for the equilibrium particle densities obtained in the mean field limit. Empirical results for the growth are then presented, including the observed equilibrium particle densities, the temperature of the system, the fractal dimension of the growth clusters, scaling behavior, finite size effects, and the approach to equilibrium. We pay particular attention to the temporal behavior of the growth process and show that the relaxation to the maximum entropy state is initially a rapid nonequilibrium process, then subsequently it is a quasistatic process with a well defined temperature.

Nonequilibrium critical behavior in unidirectionally coupled stochastic processes

Phase transitions from an active into an absorbing, inactive state are generically described by the critical exponents of directed percolation (DP), with upper critical dimension d(c)=4. In the framework of single-species reaction-diffusion systems, this universality class is realized by the combined processes A-->A+A, A+A-->A, and A-->0. We study a hierarchy of such DP processes for particle species A,B,..., unidirectionally coupled via the reactions A-->B, ...(with rates mu(AB),...). When the DP critical points at all levels coincide, multicritical behavior emerges, with density exponents beta(i) which are markedly reduced at each hierarchy level i> or =2. This scenario can be understood on the basis of the mean-field rate equations, which yield beta(i)=1/2(i-1) at the multicritical point. Using field-theoretic renormalization-group techniques in d=4-epsilon dimensions, we identify a new crossover exponent phi, and compute phi=1+O(epsilon(2)) in the multicritical regime (for small mu(AB)) of the second hierarchy level. In the active phase, we calculate the fluctuation correction to the density exponent on the second hierarchy level, beta(2)=1/2-epsilon/8+O(epsilon(2)). Outside the multicritical region, we discuss the crossover to ordinary DP behavior, with the density exponent beta(1)=1-epsilon/6+O(epsilon(2)). Monte Carlo simulations are then employed to confirm the crossover scenario, and to determine the values for the new scaling exponents in dimensions d< or =3, including the critical initial slip exponent. Our theory is connected to specific classes of growth processes and to certain cellular automata, and the above ideas are also applied to unidirectionally coupled pair annihilation processes. We also discuss some technical as well as conceptual problems of the loop expansion, and suggest some possible interpretations of these difficulties.