Geometry dependence in linear interface growth

Abstract

The effect of geometry in the statistics of 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 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 v∞t + (Γt) β χ], their probability density functions P (χ) 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.

Height fluctuations in homoepitaxial thin film growth: A numerical study

“…From dynamic scaling theory, one expects that C S (r, t ) w 2 F (r/ξ ), where ξ is the correlation length parallel to the surface, with F (x) being a, in principle, universal and geometry-dependent spatial covariance. Once it is usually hard to obtain accurate estimates of ξ and the covariances obtained here for the CV model display a pronounced minimum at r = r min (due to a modulated oscillatory decreasing behavior, analogous to that found in other models with dynamics dominated by surface diffusion [42,43]), we will analyze the rescaled covariances by plotting C S (r, t )/w 2 against r/r min . In this way, all rescaled curves coincide at r/r min = 0 (where C S /w 2 = 1) and have minima at r/r min = 1, so that they shall collapse whenever they have the same shape.…”

“…Importantly, the (one-point) height distributions (HDs) and (two-point) spatial covariances, which are central quantities determining the statistics of fluctuating surfaces, are believed now to display universal behaviors, beyond an interesting dependence on geometry. In fact, this has been widely established for systems belonging to the Kardar-Parisi-Zhang (KPZ) [26] class in both 1D [27][28][29][30][31][32][33][34][35][36][37] and 2D [38][39][40][41] substrates, whereas a single study exists indicating the same for the VLDS class [42], as well as another one for linear universality classes [43]. In face of this, HDs and spatial covariances have been used in several recent works as a tool for determining the universality class of real thin film surfaces [44][45][46][47][48][49].…”

Height fluctuations in homoepitaxial thin film growth: A numerical study

“…From dynamic scaling theory, one expects that C S (r, t) w 2 F (r/ξ), where ξ is the correlation length parallel to surface, with F (x) being an, in principle, universal and geometry-dependent spatial covariance. Once it is usually hard to obtain accurate estimates of ξ and the covariances obtained here for the CV model display a pronounced minimum at r = r min (due to a modulated oscillatory decreasing behavior, analogous to that found in other models with dynamics dominated by surface diffusion [42,43]), we will analyze the rescaled covariances by plotting C S (r, t)/w 2 against r/r min . In this way, all rescaled curves coincide at r/r min = 0 (where C S /w 2 = 1) and have minima at r/r min = 1, so that they shall collapse whenever they have the same shape.…”

“…Importantly, the (1point) height distributions (HDs) and (2-point) spatial covariances, which are central quantities determining the statistics of fluctuating surfaces, are believed now to display universal behaviors, beyond an interesting dependence on geometry. In fact, this has been widely established for systems belonging to the Kardar-Parisi-Zhang (KPZ) [26] class in both 1D [27][28][29][30][31][32][33][34][35][36][37] and 2D [38][39][40][41] substrates, whereas a single study exists indicating the same for the VLDS class [42], as well as another one for linear universality classes [43]. In face of this, HDs and spatial covariances have been used in several recent works as a tool for determining the universality class of real thin film surfaces [44][45][46][47][48][49].…”

Transition between chaotic and stochastic universality classes of kinetic roughening

“…Equivalently, finite systems whose size decreases (increases) linearly with time display TW-GOE (GUE) statistics [19][20][21], while analogous transitions have been assessed for changes in the background topology [22] or in the rate of system-size change [23]. Furthermore, the existence of universality subclasses induced by similar changes in geometrical constraints carries over to the main linear [24] and nonlinear [25] universality classes of kinetic roughening other than KPZ, making this a robust trait of (this type of) criticality far from equilibrium.…”