Search

We report on the investigation of height distributions (HDs) and spatial covariances of two-dimensional surfaces obtained from extensive numerical simulations of the celebrated Clarke-Vvedensky (CV) model for homoepitaxial thin film growth. In this model, the effect of temperature, deposition flux, and strengths of atom-atom interactions are encoded in two parameters: the diffusion to deposition ratio $R=D/F$ and $\varepsilon$, which is related to the probability of an adatom "breaking" a lateral bond. We demonstrate that the HDs present a strong dependence on both $R$ and $\varepsilon$, and even after the deposition of $10^5$ monolayers (MLs) they are still far from the asymptotics in some cases. For instance, the temporal evolution of the HDs' skewness (kurtosis) displays a pronounced minimum (maximum), for small $R$ and $\varepsilon$, and only at long times it passes to increase (decrease) toward its asymptotic value. However, it is hard to determine whether they converge to a single value or different nonuniversal ones. For large $R$ and/or $\varepsilon$, on the other hand, these quantities clearly converge to the values expected for the Villain-Lai-Das Sarma (VLDS) universality class. A similar behavior is observed in the spatial covariances, but with weaker finite-time effects, so that rescaled curves of them collapse quite well with the one for the VLDS class at long times. Simulations of a model with limited mobility of particles, which captures some essential features of the CV model in the limit of irreversible aggregation ($\varepsilon=0$), reveal a similar scenario. Overall, these results point out that the study of fluctuations in homoepitaxial thin films' surfaces can be a very difficult task and shall be performed very carefully, once typical experimental films have $\lesssim 10^4$ MLs, so that their HDs and covariances can be in the realm of transient regimes.

The effect of geometry in the statistics of \textit{nonlinear} universality classes for interface growth has been widely investigated in recent years and it is well known to yield a split of them into subclasses. In this work, we investigate this for the \textit{linear} classes of Edwards-Wilkinson (EW) and of Mullins-Herring (MH) in one- and two-dimensions. From comparison of analytical results with extensive numerical simulations of several discrete models belonging to these classes, as well as numerical integrations of the growth equations on substrates of fixed size (flat geometry) or expanding linearly in time (radial geometry), we verify that the height distributions (HDs), the spatial and the temporal covariances are universal, but geometry-dependent. In fact, the HDs are always Gaussian and, when defined in terms of the so-called "KPZ ansatz" $[h \simeq v_{\infty} t + (\Gamma t)^{\beta} \chi]$, their probability density functions $P(\chi)$ have mean null, so that all their cumulants are null, except by their variances, which assume different values in the flat and radial cases. The shape of the (rescaled) covariance curves is analyzed in detail and compared with some existing analytical results for them. Overall, these results demonstrate that the splitting of such university classes is quite general, being not restricted to the nonlinear ones.

Circular KPZ interfaces spreading radially in the plane have GUE Tracy-Widom (TW) height distribution (HD) and Airy$_2$ spatial covariance, but what are their statistics if they evolve on the surface of a different background space, such as a bowl, a cup, or any surface of revolution? To give an answer to this, we report here extensive numerical analyses of several one-dimensional KPZ models on substrates whose size enlarges as $\langle L(t) \rangle = L_0+\omega t^{\gamma}$, while their mean height $\langle h \rangle$ increases as usual [$\langle h \rangle\sim t$]. We show that the competition between the $L$ enlargement and the correlation length ($\xi \simeq c t^{1/z}$) plays a key role in the asymptotic statistics of the interfaces. While systems with $\gamma>1/z$ have HDs given by GUE and the interface width increasing as $w \sim t^{\beta}$, for $\gamma<1/z$ the HDs are Gaussian, in a correlated regime where $w \sim t^{\alpha \gamma}$. For the special case $\gamma=1/z$, a continuous class of distributions exists, which interpolate between Gaussian (for small $\omega/c$) and GUE (for $\omega/c \gg 1$). Interestingly, the HD seems to agree with the Gaussian symplectic ensemble (GSE) TW distribution for $\omega/c \approx 10$. Despite the GUE HDs for $\gamma>1/z$, the spatial covariances present a strong dependence on the parameters $\omega$ and $\gamma$, agreeing with Airy$_2$ only for $\omega \gg 1$, for a given $\gamma$, or when $\gamma=1$, for a fixed $\omega$. These results considerably generalize our knowledge on the 1D KPZ systems, unveiling the importance of the background space in their statistics., Comment: 8 pages and 8 figures, including a Suppl. Material

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., Comment: 5 pages, 4 figures and 2 tables

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 $d_s = 2$. We show that their cumulants follow a Family-Vicsek type scaling, and, at early times, when $\xi \ll l$ ($\xi$ is the correlation length), the rescaled SLRDs are given by log-normal distributions, with their $n$th cumulant scaling as $(\xi/l)^{(n-1)d_s}$. This give rise to an interesting temporal scaling for such cumulants $\left\langle w_n \right\rangle_c \sim t^{\gamma_n}$, with $\gamma_n = 2 n \beta + {(n-1)d_s}/{z} = \left[ 2 n + {(n-1)d_s}/{\alpha} \right] \beta$. 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 $\gamma_n$'s (and, consequently, $\alpha$, $\beta$ and $z$) in nice agreement with their respective universality class. Thus, it is an useful framework for numerical and experimental investigations, where it is, usually, hard to estimate the dynamic $z$ and mainly the (global) roughness $\alpha$ exponents. The stationary (for $\xi \gg l$) SLRDs and LEHDs of Kardar-Parisi-Zhang (KPZ) class are also investigated and, for some models, strong finite-size corrections are found. However, we demonstrate that good evidences of their universality can be obtained through successive extrapolations of their cumulant ratios for long times and large $l$'s. We also show that SLRDs and LEHDs are the same for flat and curved KPZ interfaces., Comment: 11 pages, 10 figures, 4 tables

We investigate solid-on-solid models that belong to the Kardar-Parisi-Zhang (KPZ) universality class on substrates that expand laterally at a constant rate by duplication of columns. Despite the null global curvature, we show that all investigated models have asymptotic height distributions and spatial covariances in agreement with those expected for the KPZ subclass for curved surfaces. In $1+1$ dimensions, the height distribution and covariance are given by the GUE Tracy-Widom distribution and the Airy$_2$ process, instead of the GOE and Airy$_1$ foreseen for flat interfaces. These results imply that, when the KPZ class splits into the curved and flat subclasses, as conventionally considered, the expanding substrate may play a role equivalent to, or perhaps more important than the global curvature. Moreover, the translational invariance of the interfaces evolving on growing domains allowed us to accurately determine, in $2+1$ dimensions, the analogue of the GUE Tracy-Widom distribution for height distribution and that of the Airy$_2$ process for spatial covariance. Temporal covariance is also calculated and shown to be universal in each dimension and in each of the two subclasses. A logarithmic correction associated to the duplication of column is observed and theoretically elucidated. Finally, crossover between regimes with fixed-size and enlarging substrates is also investigated., Comment: 11 pages, 9 figures