Dimensional crossover in Kardar-Parisi-Zhang growth.

Two-dimensional (2D) Kardar-Parisi-Zhang (KPZ) growth is usually investigated on substrates of lateral sizes L_{x}=L_{y}, so that L_{x} and the correlation length (ξ) are the only relevant lengths determining the scaling behavior. However, in cylindrical geometry, as well as in flat rectangular substrates L_{x}≠L_{y} and, thus, the surfaces can become correlated in a single direction, when ξ∼L_{x}≪L_{y}. From extensive simulations of several KPZ models, we demonstrate that this yields a dimensional crossover in their dynamics, with the roughness scaling as W∼t^{ß_{2D}} for t≪t_{c} and W∼t^{ß_{1D}} for t≫t_{c}, where t_{c}∼L_{x}^{1/z_{2D}}. The height distributions (HDs) also cross over from the 2D flat (cylindrical) HD to the asymptotic Tracy-Widom Gaussian orthogonal ensemble (Gaussian unitary ensemble) distribution. Moreover, 2D to one-dimensional (1D) crossovers are found also in the asymptotic growth velocity and in the steady-state regime of flat systems, where a family of universal HDs exists, interpolating between the 2D and 1D ones as L_{y}/L_{x} increases. Importantly, the crossover scalings are fully determined and indicate a possible way to solve 2D KPZ models.

One-point height fluctuations and two-point correlators of (2+1) cylindrical KPZ systems.

While the one-point height distributions (HDs) and two-point covariances of (2+1) Kardar-Parisi-Zhang (KPZ) systems have been investigated in several recent works for flat and spherical geometries, for the cylindrical one the HD was analyzed for few models and nothing is known about the spatial and temporal covariances. Here, we report results for these quantities, obtained from extensive numerical simulations of discrete KPZ models, for three different setups yielding cylindrical growth. Beyond demonstrating the universality of the HD and covariances, our results reveal other interesting features of this geometry. For example, the spatial covariances measured along the longitudinal and azimuthal directions are different, with the former being quite similar to the curve for flat (2+1) KPZ systems, while the latter resembles the Airy_{2} covariance of circular (1+1) KPZ interfaces. We also argue (and present numerical evidence) that, in general, the rescaled temporal covariance A(t/t_{0}) decays asymptotically as A(x)∼x^{-λ[over ¯]} with an exponent λ[over ¯]=ß+d^{*}/z, where d^{*} is the number of interface sides kept fixed during the growth (being d^{*}=1 for the systems analyzed here). Overall, these results complete the picture of the main statistics for the (2+1) KPZ class.

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, Îµ, 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.

An electrodeposition model with surface relaxation predicts temperature and current effects in compact and dendritic film morphologies.

We study a thin-film electrodeposition model that represents the relaxation of the deposited material by adatom diffusion on quenched crystal topographies and considers simple mechanisms of cation flux in the electrolyte. The results of numerical simulations with collimated flux and a rapid cation reduction in contact with the deposit relate the surface roughness and the adatom hop numbers with two model parameters. A comparison with the results of a collective diffusion model for vapor deposition shows differences in the surface morphologies but similarities in scaling relations, which suggest thermally activated (Arrhenius) forms for the parameters of the electrodeposition model and relate one of them to the applied current. Simulations with purely diffusive cation flux and possible pore formation in simple cubic lattices show the growth of self-organized structures with leaf shapes (dendrites) above a compact layer that covers the flat electrode. The thickness of this layer and the average dendrite size also obey scaling relations in terms of the model parameters, which predict that both sizes decrease with the applied current, in agreement with recent experimental studies. Under all flux conditions, an increase in adatom diffusivity with temperature implies an increase in the average sizes of low-energy surface configurations, independently of their particular shapes. Finally, we note that a previously proposed model for electrodeposition produced similar morphologies, but the quantitative relations for the characteristic sizes differ from those of the present model, which also advances with a consistent interpretation of temperature effects.

Kardar-Parisi-Zhang growth on square domains that enlarge nonlinearly in time.

Fundamental properties of an interface evolving on a domain of size L, such as its height distribution (HD) and two-point covariances, are known to assume universal but different forms depending on whether L is fixed (flat geometry) or expands linearly in time (radial growth). The interesting situation where L varies nonlinearly, however, is far less explored and it has never been tackled for two-dimensional (2D) interfaces. Here, we study discrete Kardar-Parisi-Zhang (KPZ) growth models deposited on square lattice substrates, whose (average) lateral size enlarges as L=L_{0}+ωt^{γ}. Our numerical simulations reveal that the competition between the substrate expansion and the increase of the correlation length parallel to the substrate, ξ≃ct^{1/z}, gives rise to a number of interesting results. For instance, when γ<1/z the interface becomes fully correlated, but its squared roughness, W_{2}, keeps increasing as W_{2}∼t^{2αγ}, as previously observed for one-dimensional (1D) systems. A careful analysis of this scaling, accounting for an intrinsic width on it, allows us to estimate the roughness exponent of the 2D KPZ class as α=0.387(1), which is very accurate and robust, once it was obtained averaging the exponents for different models and growth conditions (i.e., for various γ^{'}s and ω^{'}s). In this correlated regime, the HDs and covariances are consistent with those expected for the steady-state regime of the 2D KPZ class for flat geometry. For γ≈1/z, we find a family of distributions and covariances continuously interpolating between those for the steady-state and the growth regime of radial KPZ interfaces, as the ratio ω/c augments. When γ>1/z the system stays forever in the growth regime and the HDs always converge to the same asymptotic distribution, which is the one for the radial case. The spatial covariances, on the other hand, are (γ,ω)-dependent, showing a trend towards the covariance of a random deposition in enlarging substrates as the expansion rate increases. These results considerably generalize our understanding of the height fluctuations in 2D KPZ systems, revealing a scenario very similar to the one previously found in the 1D case.

Time increasing rates of infiltration and reaction in porous media at the percolation thresholds.

The infiltration of a solute in a fractal porous medium is usually anomalous, but chemical reactions of the solute and that material may increase the porosity and affect the evolution of the infiltration. We study this problem in two- and three-dimensional lattices with randomly distributed porous sites at the critical percolation thresholds and with a border in contact with a reservoir of an aggressive solute. The solute infiltrates that medium by diffusion and the reactions with the impermeable sites produce new porous sites with a probability r, which is proportional to the ratio of reaction and diffusion rates at the scale of a lattice site. Numerical simulations for r≪1 show initial subdiffusive scaling and long time Fickean scaling of the infiltrated volumes or areas, but with an intermediate regime with time increasing rates of infiltration and reaction. The anomalous exponent of the initial regime agrees with a relation previously applied to infinitely ramified fractals. We develop a scaling approach that explains the subsequent time increase of the infiltration rate, the dependence of this rate on r, and the crossover to the Fickean regime. The exponents of the scaling relations depend on the fractal dimensions of the critical percolation clusters and on the dimensions of random walks in those clusters. The time increase of the reaction rate is also justified by that reasoning. As r decreases, there is an increase in the number of time decades of the intermediate regime, which suggests that the time increasing rates are more likely to be observed is slowly reacting systems.

Statistics of adatom diffusion in a model of thin film growth.

We study the statistics of the number of executed hops of adatoms at the surface of films grown with the Clarke-Vvedensky (CV) model in simple cubic lattices. The distributions of this number N are determined in films with average thicknesses close to 50 and 100 monolayers for a broad range of values of the diffusion-to-deposition ratio R and of the probability ε that lowers the diffusion coefficient for each lateral neighbor. The mobility of subsurface atoms and the energy barriers for crossing step edges are neglected. Simulations show that the adatoms execute uncorrelated diffusion during the time in which they move on the film surface. In a low temperature regime, typically with Rε≲1, the attachment to lateral neighbors is almost irreversible, the average number of hops scales as 〈N〉∼R^{0.38±0.01}, and the distribution of that number decays approximately as exp[-(N/〈N〉)^{0.80±0.07}]. Similar decay is observed in simulations of random walks in a plane with randomly distributed absorbing traps and the estimated relation between 〈N〉 and the density of terrace steps is similar to that observed in the trapping problem, which provides a conceptual explanation of that regime. As the temperature increases, 〈N〉 crosses over to another regime when Rε^{3.0±0.3}∼1, which indicates high mobility of all adatoms at terrace borders. The distributions P(N) change to simple exponential decays, due to the constant probability for an adatom to become immobile after being covered by a new deposited layer. At higher temperatures, the surfaces become very smooth and 〈N〉∼Rε^{1.85±0.15}, which is explained by an analogy with submonolayer growth. Thus, the statistics of adatom hops on growing film surfaces is related to universal and nonuniversal features of the growth model and with properties of trapping models if the hopping time is limited by the landscape and not by the deposition of other layers.