Tiago J. Oliveira

Effects of grains’ features in surface roughness scaling

We study the local and global roughness scaling in growth models with grains at the film surfaces. The local roughness, measured as a function of window size r, shows a crossover at a characteristic length rc, from a rapid increase with exponent α1 to a slower increase with exponent α2. The result α1≈1 is explained by the large height differences in the borders of the grains when compared to intragrain roughness, and must not be interpreted as a consequence of a diffusion dominated intragrain dynamics. This exponent shows a weak dependence on the shape and size distribution of the grains, and typically ranges from 0.85 for rounded grain surfaces to one for the sharpest ones. The scaling corrections of exactly solvable models suggest the possibility of slightly smaller values due to other smoothing effects of the surface images. The crossover length rc provides a reasonable estimate of the average grain size in all model systems, including the cases of wide grain size distributions. In Kardar-Parisi-Zhang ...

Kardar-Parisi-Zhang universality class in (2+1) dimensions: universal geometry-dependent distributions and finite-time corrections.

The dynamical regimes of models belonging to the Kardar-Parisi-Zhang (KPZ) universality class are investigated in d=2+1 by extensive simulations considering flat and curved geometries. Geometry-dependent universal distributions, different from their Tracy-Widom counterpart in one dimension, were found. Distributions exhibit finite-time corrections hallmarked by a shift in the mean decaying as t(-β), where β is the growth exponent. Our results support a generalization of the ansatz h=v(∞)t+(Γt)(β)χ+η+ζt(-β) to higher dimensions, where v(∞), Γ, ζ, and η are nonuniversal quantities whereas β and χ are universal and the last one depends on the surface geometry. Generalized Gumbel distributions provide very good fits of the distributions in at least four orders of magnitude around the peak, which can be used for comparisons with experiments. Our numerical results call for analytical approaches and experimental realizations of the KPZ class in two-dimensional systems.

Universal fluctuations in Kardar-Parisi-Zhang growth on one-dimensional flat substrates.

We present a numerical study of the evolution of height distributions (HDs) obtained in interface growth models belonging to the Kardar-Parisi-Zhang (KPZ) universality class. The growth is done on an initially flat substrate. The HDs obtained for all investigated models are very well fitted by the theoretically predicted Gaussian orthogonal ensemble (GOE) distribution. The first cumulant has a shift that vanishes as t(-1/3), while the cumulants of order 2≤n≤4 converge to GOE as t(-2/3) or faster, behaviors previously observed in other KPZ systems. These results yield evidences for the universality of the GOE distribution in KPZ growth on flat substrates. Finally, we further show that the surfaces are described by the Airy(1) process.

Solution of an associating lattice-gas model with density anomaly on a Husimi lattice.

We study a model of a lattice gas with orientational degrees of freedom which resemble the formation of hydrogen bonds between the molecules. In this model, which is the simplified version of the Henriques-Barbosa model, no distinction is made between donors and acceptors in the bonding arms. We solve the model in the grand-canonical ensemble on a Husimi lattice built with hexagonal plaquettes with a central site. The ground state of the model, which was originally defined on the triangular lattice, is exactly reproduced by the solution on this Husimi lattice. In the phase diagram, one gas and two liquid [high density liquid (HDL) and low density liquid (LDL)] phases are present. All phase transitions (GAS-LDL, GAS-HDL, and LDL-HDL) are discontinuous, and the three phases coexist at a triple point. A line of temperatures of maximum density in the isobars is found in the metastable GAS phase, as well as another line of temperatures of minimum density appears in the LDL phase, part of it in the stable region and another in the metastable region of this phase. These findings are at variance with simulational results for the same model on the triangular lattice, which suggested a phase diagram with two critical points. However, our results show very good quantitative agreement with the simulations, both for the coexistence loci and the densities of particles and of hydrogen bonds. We discuss the comparison of the simulations with our results.

Maximal- and minimal-height distributions of fluctuating interfaces.

Maximal- and minimal-height distributions (MAHD, MIHD) of two-dimensional interfaces grown with the nonlinear equations of Kardar-Parisi-Zhang (KPZ, second order) and of Villain-Lai-Das Sarma (VLDS, fourth order) are shown to be different. Two universal curves may be MAHD or MIHD of each class depending on the sign of the relevant nonlinear term, which is confirmed by results of several lattice models in the KPZ and VLDS classes. The difference between MAHD and MIDH is connected with the asymmetry of the local height distribution. A simple, exactly solvable deposition-erosion model is introduced to illustrate this feature. The average extremal heights scale with the same exponent of the average roughness. In contrast to other correlated systems, generalized Gumbel distributions do not fit those MAHD and MIHD, nor those of Edwards-Wilkinson growth.

Universal and nonuniversal features in the crossover from linear to nonlinear interface growth.

We study a restricted solid-on-solid model involving deposition and evaporation with probabilities p and 1 - p, respectively, in one-dimensional substrates. It presents a crossover from Edwards-Wilkinson (EW) to Kardar-Parisi-Zhang (KPZ) scaling for p approximately 0.5. The associated KPZ equation is analytically derived, exhibiting a coefficient lambda of the nonlinear term proportional to q identical with p - 1/2, which is confirmed numerically by calculation of tilt-dependent growth velocities for several values of p. This linear lambda - q relation contrasts to the apparently universal parabolic law obtained in competitive models mixing EW and KPZ components. The regions where the interface roughness shows pure EW and KPZ scaling are identified for 0.55< or =p< or =0.8, which provides numerical estimates of the crossover times tc. They scale as tc approximately lambda -phi with phi=4.1+/-0.1, which is in excellent agreement with the theoretically predicted universal value phi=4 and improves previous numerical estimates, which suggested phi approximately 3.

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.

Universality of fluctuations in the Kardar-Parisi-Zhang class in high dimensions and its upper critical dimension.

We show that the theoretical machinery developed for the Kardar-Parisi-Zhang (KPZ) class in low dimensions is obeyed by the restricted solid-on-solid model for substrates with dimensions up to d=6. Analyzing different restriction conditions, we show that the height distributions of the interface are universal for all investigated dimensions. It means that fluctuations are not negligible and, consequently, the system is still below the upper critical dimension at d=6. The extrapolation of the data to dimensions d≥7 predicts that the upper critical dimension of the KPZ class is infinite.

Finite-size effects in roughness distribution scaling

We study numerically finite-size corrections in scaling relations for roughness distributions of various interface growth models. The most common relation, which considers the average roughness as a scaling factor, is not obeyed in the steady states of a group of ballisticlike models in 2+1 dimensions, even when very large system sizes are considered. On the other hand, good collapse of the same data is obtained with a scaling relation that involves the root mean square fluctuation of the roughness, which can be explained by finite-size effects on second moments of the scaling functions. We also obtain data collapse with an alternative scaling relation that accounts for the effect of the intrinsic width, which is a constant correction term previously proposed for the scaling of . This illustrates how finite-size corrections can be obtained from roughness distributions scaling. However, we discard the usual interpretation that the intrinsic width is a consequence of high surface steps by analyzing data of restricted solid-on-solid models with various maximal height differences between neighboring columns. We also observe that large finite-size corrections in the roughness distributions are usually accompanied by huge corrections in height distributions and average local slopes, as well as in estimates of scaling exponents. The molecular-beam epitaxy model of Das Sarma and Tamborenea in 1+1 dimensions is a case example in which none of the proposed scaling relations work properly, while the other measured quantities do not converge to the expected asymptotic values. Thus although roughness distributions are clearly better than other quantities to determine the universality class of a growing system, it is not the final solution for this task.

Monte Carlo simulations of polymers with nearest- and next nearest-neighbor interactions on square and cubic lattices

We study a generalized interacting self-avoiding walk (ISAW) model with nearest- and next nearest-neighbor (NN and NNN) interactions on the square and cubic lattices. In both dimensions, the phase diagrams show coil and globule phases separated by continuous transition lines. Along these lines, we calculate the metric