Finite-size effects in roughness distribution scaling

Height distributions in interface growth: The role of the averaging process

Height distributions (HDs) are key quantities to uncover universality and geometry-dependence in evolving interfaces. To quantitatively characterize HDs, one uses adimensional ratios of their first central moments (m_{n}) or cumulants (κ_{n}), especially the skewness S and kurtosis K, whose accurate estimate demands an averaging over all L^{d} points of the height profile at a given time, in translation-invariant interfaces, and over N independent samples. One way of doing this is by calculating m_{n}(t) [or κ_{n}(t)] for each sample and then carrying out an average of them for the N interfaces, with S and K being calculated only at the end. Another approach consists in directly calculating the ratios for each interface and, then, averaging the N values. It turns out, however, that S and K for the growth regime HDs display strong finite-size and -time effects when estimated from these "interface statistics," as already observed in some previous works and clearly shown here, through extensive simulations of several discrete growth models belonging to the EW and KPZ classes on one- and two-dimensional substrates of sizes L=const. and L∼t. Importantly, I demonstrate that with "1-point statistics," i.e., by calculating m_{n}(t) [or κ_{n}(t)] once for all NL^{d} heights together, these corrections become very weak, so that S and K attain values very close to the asymptotic ones already at short times and for small L's. However, I find that this "1-point" (1-pt) approach fails in uncovering the universality of the HDs in the steady-state regime (SSR) of systems whose average height, h[over ¯], is a fluctuating variable. In fact, as demonstrated here, in this regime the 1-pt height evolves as h(t)=h[over ¯](t)+s_{λ}A^{1/2}L^{α}ζ+⋯-where P(ζ) is the underlying SSR HD-and the fluctuations in h[over ¯] yield S_{1-pt}∼t^{-1/2} and K_{1-pt}∼t^{-1}. Nonetheless, by analyzing P(h-h[over ¯]), the cumulants of P(ζ) can be accurately determined. I also show that different, but universal, asymptotic values for S and K (related, so, to different HDs) can be found from the "interface statistics" in the SSR. This reveals the importance of employing the various complementary approaches to reliably determine the universality class of a given system through its different HDs.

Universality and dependence on initial conditions in the class of the nonlinear molecular beam epitaxy equation

We report extensive numerical simulations of growth models belonging to the nonlinear molecular beam epitaxy (nMBE) class, on flat (fixed-size) and expanding substrates (ES). In both d=1+1 and 2+1, we find that growth regime height distributions (HDs), and spatial and temporal covariances are universal, but are dependent on the initial conditions, while the critical exponents are the same for flat and ES systems. Thus, the nMBE class does split into subclasses, as does the Kardar-Parisi-Zhang (KPZ) class. Applying the "KPZ ansatz" to nMBE models, we estimate the cumulants of the 1+1 HDs. Spatial covariance for the flat subclass is hallmarked by a minimum, which is not present in the ES one. Temporal correlations are shown to decay following well-known conjectures.

Scaling, cumulant ratios, and height distribution of ballistic deposition in 3+1 and 4+1 dimensions

We investigate the origin of the scaling corrections in ballistic deposition models in high dimensions using the method proposed by Alves et al. [Phys. Rev. E 90, 052405 (2014)PLEEE81539-375510.1103/PhysRevE.90.052405] in d=2+1 dimensions, where the intrinsic width associated with the fluctuations of the height increments during the deposition processes is explicitly taken into account. In the present work, we show that this concept holds for d=3+1 and 4+1 dimensions. We have found that growth and roughness exponents and dimensionless cumulant ratios are in agreement with other models, presenting small finite-time corrections to the scaling, that in principle belong to the Kardar-Parisi-Zhang (KPZ) universality class in both d=3+1 and 4+1. Our results constitute further evidence that the upper critical dimension of the KPZ class, if it exists, is larger than 4.

Width and extremal height distributions of fluctuating interfaces with window boundary conditions

We present a detailed study of squared local roughness (SLRDs) and local extremal height distributions (LEHDs), calculated in windows of lateral size l, for interfaces in several universality classes, in substrate dimensions d_{s}=1 and 2. We show that their cumulants follow a Family-Vicsek-type scaling, and, at early times, when ξ≪l (ξ is the correlation length), the rescaled SLRDs are given by log-normal distributions, with their nth cumulant scaling as (ξ/l)^{(n-1)d_{s}}. This gives rise to an interesting temporal scaling for such cumulants as 〈w_{n}〉_{c}∼t^{γ_{n}}, with γ_{n}=2nβ+(n-1)d_{s}/z=[2n+(n-1)d_{s}/α]β. This scaling is analytically proved for the Edwards-Wilkinson (EW) and random deposition interfaces and numerically confirmed for other classes. In general, it is featured by small corrections, and, thus, it yields exponents γ_{n} (and, consequently, α,β and z) in good agreement with their respective universality class. Thus, it is a useful framework for numerical and experimental investigations, where it is usually hard to estimate the dynamic z and mainly the (global) roughness α exponents. The stationary (for ξ≫l) SLRDs and LEHDs of the Kardar-Parisi-Zhang (KPZ) class are also investigated, and, for some models, strong finite-size corrections are found. However, we demonstrate that good evidence of their universality can be obtained through successive extrapolations of their cumulant ratios for long times and large l. We also show that SLRDs and LEHDs are the same for flat and curved KPZ interfaces.

Origins of scaling corrections in ballistic growth models

We study the ballistic deposition and the grain deposition models on two-dimensional substrates. Using the Kardar-Parisi-Zhang (KPZ) ansatz for height fluctuations, we show that the main contribution to the intrinsic width, which causes strong corrections to the scaling, comes from the fluctuations in the height increments along deposition events. Accounting for this correction in the scaling analysis, we obtain scaling exponents in excellent agreement with the KPZ class. We also propose a method to suppress these corrections, which consists in dividing the surface in bins of size ɛ and using only the maximal height inside each bin to do the statistics. Again, scaling exponents in remarkable agreement with the KPZ class are found. The binning method allows the accurate determination of the height distributions of the ballistic models in both growth and steady-state regimes, providing the universal underlying fluctuations foreseen for KPZ class in 2 + 1 dimensions. Our results provide complete and conclusive evidences that the ballistic model belongs to the KPZ universality class in 2+1 dimensions. Potential applications of the methods developed here, in both numerics and experiments, are discussed.

Distribution of scaled height in one-dimensional competitive growth profiles

This work investigates the scaled height distribution, ρ(q), of irregular profiles that are grown based on two sets of local rules: those of the restricted solid on solid (RSOS) and ballistic deposition (BD) models. At each time step, these rules are respectively chosen with probability p and r=1-p. Large-scale Monte Carlo simulations indicate that the system behaves differently in three succeeding intervals of values of p: I(B) ≈ [0,0.75),I(T) ≈ (0.75,0.9), and I(R) ≈ (0.9,1.0]. In I(B), the ballistic character prevails: the growth velocity υ(∞) decreases with p in a linear way, and similar behavior is found for Γ(∞) (p), the amplitude of the t(1/3)-fluctuations, which is measured from the second-order height cumulant. The distribution of scaled height fluctuations follows the Gaussian orthogonal ensemble (GOE) Tracy-Widom (TW) distribution with resolution roughly close to 10(-4). The skewness and kurtosis of the computed distribution coincide with those for TW distribution. Similar results are observed in the interval I(R), with prevalent RSOS features. In this case, the skewness become negative. In the transition interval I(T), the system goes smoothly from one regime to the other: the height distribution becomes apparently Gaussian, which motivates us to identify this phenomenon as a transition from Kardar-Parisi-Zhang (KPZ) behavior to Edwards-Wilkinson (EW) behavior back to KPZ behavior.

Roughness exponents and grain shapes

In surfaces with grainy features, the local roughness w shows a crossover at a characteristic length r(c), with roughness exponent changing from α(1)≈1 to a smaller α(2). The grain shape, the choice of w or height-height correlation function (HHCF) C, and the procedure to calculate root-mean-square averages are shown to have remarkable effects on α(1). With grains of pyramidal shape, α(1) can be as low as 0.71, which is much lower than the previous prediction 0.85 for rounded grains. The same crossover is observed in the HHCF, but with initial exponent χ(1)≈0.5 for flat grains, while for some conical grains it may increase to χ(1)≈0.7. The universality class of the growth process determines the exponents α(2)=χ(2) after the crossover, but has no effect on the initial exponents α(1) and χ(1), supporting the geometric interpretation of their values. For all grain shapes and different definitions of surface roughness or HHCF, we still observe that the crossover length r(c) is an accurate estimate of the grain size. The exponents obtained in several recent experimental works on different materials are explained by those models, with some surface images qualitatively similar to our model films.

Growth instability due to lattice-induced topological currents in limited-mobility epitaxial growth models

The energetically driven Ehrlich-Schwoebel barrier had been generally accepted as the primary cause of the growth instability in the form of quasiregular moundlike structures observed on the surface of thin film grown via molecular-beam epitaxy (MBE) technique. Recently the second mechanism of mound formation was proposed in terms of a topologically induced flux of particles originating from the line tension of the step edges which form the contour lines around a mound. Through large-scale simulations of MBE growth on a variety of crystalline lattice planes using limited-mobility, solid-on-solid models introduced by Wolf-Villain and Das Sarma-Tamborenea in 2+1 dimensions, we show that there exists a topological uphill particle current with strong dependence on specific lattice crystalline structure. Without any energetically induced barriers, our simulations produce spectacular mounds very similar, in some cases, to what have been observed in many recent MBE experiments. On a lattice where these currents cease to exist, the surface appears to be scale invariant, statistically rough as predicted by the conventional continuum growth equation.

Scaling of surface roughness and polymer structure in a model for film growth and polymerization

We study a model of growth of polymer films using numerical simulations and scaling concepts. During the deposition, each new monomer flows in a direction perpendicular to the substrate, aggregates at the first contact with the deposit and executes up to G steps along the polymers, propagating an existing chain or nucleating of a new polymer. Some qualitative results agree with those of a previous model for vapor deposition polymerization (VDP) with collective diffusion, such as the roughness increase and density decrease with G . This supports the interpretation of G as a ratio between diffusion coefficient and monomer flux. We perform a systematic study of scaling properties of the outer surface roughness and of polymer size and shape. For large G , the polymers are stretched in the direction perpendicular to the substrate and have typical size increasing as G(1/2). This is explained by the solution of the problem of random walk trapping, which illustrates the connection of surface processes and bulk properties. The distributions of polymer sizes are monotonically decreasing for all G and very broad, thus a large number of small chains and of chains much larger than the average is found in typical samples. The outer surface roughness obeys Kardar-Parisi-Zhang scaling, in contrast to the apparent anomalous scaling of previous VDP models with oblique monomer flux. However, the calculation of reliable exponents requires accounting for huge finite-size corrections. Possible applications and extensions of this model are discussed.

Numerical study of the Kardar-Parisi-Zhang equation

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