Faceted patterns and anomalous surface roughening driven by long-term correlated noise

TL;DR

This paper studies how long-term correlated noise affects surface growth in the KPZ model, revealing the emergence of faceted patterns and anomalous roughening not explained by traditional theories.

Contribution

It demonstrates that long-term correlated noise induces faceted surface patterns and anomalous roughening in KPZ growth, challenging existing theoretical frameworks.

Findings

Faceted patterns dominate surface morphology with increased noise correlation.

Anomalous kinetic roughening occurs due to long-term noise correlations.

Conventional renormalization group methods do not capture these phenomena.

Abstract

We investigate Kardar-Parisi-Zhang (KPZ) surface growth in the presence of long-term correlated noise. By means of extensive numerical simulations of models in the KPZ universality class we find that, as the noise correlator range increases, the surface develops a pattern of macroscopic facets that completely dominate the dynamics and induce anomalous kinetic roughening. This novel phenomenon is not described by the conventional dynamic renormalisation group calculations and can explain the singular behavior observed in the analytical treatment of this problem in the seminal paper of Medina {\it et al} [Phys. Rev. A {\bf 39}, 3053 (1989)].

Equations

$${\partial }_{t}h(\mathbf{x},t)=\nu {\nabla }^{2}h+\lambda (\nabla h{)}^2+\eta (\mathbf{x},t),$$

$$⟨\eta (\mathbf{x},t)\eta ({\mathbf{x}}^{\prime },t^{\prime })⟩=2D\delta (\mathbf{x}-{\mathbf{x}}^{\prime })\delta (t-t^{\prime }).$$

$$⟨\eta (x,t)\eta (x^{\prime },t^{\prime })⟩=2D{\delta }_{x,x^{\prime }}|t-t^{\prime }{|}^{2\theta -1},$$

$$h(x,t+1)=\mathrm{Max}[h(x,t)+\eta (x,t),h(x-1,t),h(x+1,t)]$$

$$S(k,t)=k^{-(2\alpha +d)}s(k{t}^{1/z}),$$

$$s(u)\sim\left\{\begin{array}[]{lcl}u^{2(\alpha-\alpha_{s}})&{\rm if}&u\gg 1\\ u^{2\alpha+d}&{\rm if}&u\ll 1\end{array}\right.,$$

$${\partial }_{t}h(\mathbf{x},t)={\nabla }^{2}h+\lambda (\nabla h{)}^2+\eta (\mathbf{x}),$$

$$⟨\eta (\mathbf{x})\eta ({\mathbf{x}}^{\prime })⟩=\delta (\mathbf{x}-{\mathbf{x}}^{\prime }).$$

$${\partial }_t\varphi ={\nabla }^2\varphi +\eta (\mathbf{x})\varphi (\mathbf{x},t).$$

Full text

The dynamics of surfaces and interfaces driven by random fluctuations has many applications in modern condensed matter science and statistical physics including the description of surfaces formed by particle deposition processes in thin-film growth (e.g. molecular-beam epitaxy, sputtering, electrodeposition, and chemical-vapor deposition) Barabási and Stanley (1995); Krug (1997), advancing fracture cracks in disordered materials Alava et al. (2006), and fluid-flow depinning in disordered media Alava et al. (2004). The dynamics of scale-invariant interfaces is also relevant to understand many classical problems in statistical mechanics including directed polymers in random media, minimal energy paths, and localization in disordered media Kardar (1998); Halpin-Healy and Zhang (1995). Remarkably, it has also been recently found that there exists a deep connection of interface kinetic roughening with the evolution of perturbations, the so-called Lyapunov vectors, in chaotic spatially extended dynamical systems Pazó et al. (2013, 2016); Pikovsky and Politi (1998); Pazó et al. (2014, 2008); Pikovsky and Politi (2001); Szendro et al. (2007a); Pazó et al. (2009). All these theoretical interconnections between apparently distant problems make the understanding of all aspects of kinetic surface roughening a central theme in modern statistical physics.

The time evolution, Langevin-type, equation for the surface height $h(\mathbf{x},t)$ in most of the above mentioned systems satisfies a set of fundamental symmetries (rotational, translational in $\mathbf{x}$, time invariance, etc), including the fundamental shift symmetry $h\to h+c$, where $c$ is a constant Barabási and Stanley (1995). The latter immediately leads to scale-invariant surfaces/interfaces without fine tuning of any external parameters or couplings, which implies generic power-law decay of the spatio-temporal surface height correlations Hentschel (1994). In a nutshell, a surface is said to be scale invariant if its statistical properties remain unchanged after re-scaling of space and time according to the transformation $h(\mathbf{x},t)\to b^{\alpha}h(b\,\mathbf{x},b^{1/z}\,t)$, for any scaling factor $b>1$ and a certain combination of critical exponents $\alpha$ and $z$ Barabási and Stanley (1995); Krug (1997).

In kinetic surface roughening the Kardar-Parisi-Zhang (KPZ) Kardar et al. (1986) equation plays a central role as the simplest, nonlinear, out-of-equilibrium model in the continuum exhibiting scale-invariant solutions. KPZ describes a universality class of surface growth that brings together many different surface dynamics that share the same symmetries including Eden growth, ballistic deposition, and Kim-Kosterlitz growth models among many others Barabási and Stanley (1995); Krug (1997). The KPZ equation describes the evolution of the interface height $h(\mathbf{x},t)$ at time $t$ and substrate position $\mathbf{x}$ in $d+1$ dimensions and is given by Kardar et al. (1986)

[TABLE]

where $\eta(\mathbf{x},t)$ is an uncorrelated noise

[TABLE]

The KPZ equation with uncorrelated noise has scale-invariant solutions as can be rigorously proven by calculating height-height correlation functions by means of dynamic renormalization group (RG) techniques Kardar et al. (1986). In $1+1$ dimensions one can obtain the exact critical exponents $\alpha=1/2$ and $z=3/2$ almost straightforwardly (see for instance Barabási and Stanley (1995)) after realizing that, on the one hand, the stationary solution of the Fokker-Planck equation associated with the Langevin dynamics, Eq. (1), in $d=1$ is identical to that of the linear ($\lambda=0$) case, implying $\alpha=1/2$ and, on the other hand, that Eq. (1) satisfies Galilean invariance (in any dimension), which implies the hyperscaling relation $\alpha+z=2$. In contrast, exact exponents are not known in higher dimensions, due to the existence of a strong-coupling fixed point for $d>1$ that cannot be approached with pertubative RG calculations, and have been estimated from numerical simulations Kim and Kosterlitz (1989); Kim et al. (1991); Moser et al. (1991); Ala-Nissila et al. (1992, 1993); Forrest and Tang (1990); Miranda and Aarão Reis (2008); Alves et al. (2014); Alves and Ferreira (2016) or analytical approximations of several sorts Colaiori and Moore (2001); Canet et al. (2010); Schwartz and Edwards (1992); Fogedby (2005, 2006).

In many applications the noise is actually correlated. If correlations are short-ranged one expects that, in the long wavelength limit, the critical exponents should remain exactly the same as in the case of uncorrelated noise. Indeed, this result can be rigorously proven by, for instance, RG arguments. However, in systems where noise correlations are long-ranged or long-termed the critical exponents do depend on the noise correlator decay exponents, as was early shown by Medina et al. Medina et al. (1989) using perturbative RG.

While the effect of spatially correlated noise in KPZ has been extensively studied in the literature Halpin-Healy (1989); Meakin and Jullien (1989); Zhang (1990); Margolina and Warriner (1990); Amar et al. (1991); Peng et al. (1991); Wu et al. (1995); Pang et al. (1995); Jeong et al. (1995); Li (1997); Janssen et al. (1999); Kuittu et al. (1999); Katzav and Schwartz (1999); Verma (2000); Kloss et al. (2014), the case of algebraic temporal correlations has remained virtually unexplored so far Lam et al. (1992); Katzav and Schwartz (2004); Strack (2015); Song and Xia (2016). Interestingly, the effect of long-term correlated noise in kinetic roughening has earned renewed interest as it has been very recently shown to play a crucial, and not yet completely understood, role in the interface picture of infinitesimal perturbations (Lyapunov vectors) Pazó et al. (2009); Romero-Bastida et al. (2010); Pazó et al. (2014) and the scaling of the Lyapunov exponent fluctuations in spatio-temporal chaos Pazó et al. (2016).

In this Letter we find that KPZ growth in the presence of temporally correlated noise gives rise to surfaces with a faceted structure and the accompanying anomalous kinetic roughening. The origin of the faceted pattern can be traced back to the strong localization properties of the field $\phi(\mathbf{x},t)=\exp h(\mathbf{x},t)$ and the same mechanism is expected to be relevant for other surface growth models. Our conclusions are based upon extensive numerical simulations of two models in the same universality class. We studied ballistic deposition as an example of a discrete growth model with KPZ symmetries. We also performed a direct numerical integration of the KPZ equation (1) with a long-term correlated noise.

Models.-

In our numerical study we needed to generate very long time series of random numbers with long temporal correlations at every spatial position $x$. In particular, one must generate a spatially uncorrelated time series $\eta(x,t)$ at every lattice site $x$ and be certain that the power spectrum $\langle|\hat{\eta}(x,\omega)|^{2}\rangle$, where $\langle\centerdot\rangle$ denotes noise average, exhibits excellent scaling at low frequencies so that, at leading order, $\langle|\hat{\eta}(x,\omega)|^{2}\rangle\sim\omega^{-2\theta}$ for $\omega\to 0$ at every lattice site $x$ and the noise correlator scales as

[TABLE]

for time differences $|t-t^{\prime}|>L^{z}$. In this way, we can be certain that we have spatially uncorrelated noise with power-law scaling of the temporal correlations for times differences up to, at least, the saturation time in a system of size $L$. The exponent $\theta\in[0,1/2)$ characterizes the temporal correlation range of the noise that becomes more long-term correlated as $\theta$ is increased from zero.

For each lattice site $x$ we generated a noise sequence parametrized by $t$ using the Mandelbrot’s fast fractional Gaussian noise generator Mandelbrot and Wallis (1969); Mandelbrot (1971). This algorithm produces a random sequence of Gaussian distributed numbers $Z(t)$ which can be used in the numerical integration of the KPZ equation, $\eta(x,t)\equiv Z_{x}(t)$, by generating $L$ independent Mandelbrot sequences $Z_{1}(t),Z_{2}(t)\cdots,Z_{L}(t)$. In the case of the simulations of particle deposition by ballistic deposition we map $\eta(x,t)=1$ if $Z_{x}(t)>0$ and $\eta(x,t)=0$ otherwise. This digitalization of the noise enhances the statistics of the simulations by avoiding the formation of overhangs on the surface Lam et al. (1992). We checked that the noise generated has a correlator with the correct scaling at very long times with the desired decay exponent $\theta$ (see Supplemental Material SM ).

We simulated ballistic deposition by implementing the following discrete time evolution for the surface:

[TABLE]

where the height $h(x,t)$ is an integer and the noise $\eta\in\{0,1\}$ is temporally correlated as in Eq. (3) with an exponent $0\leq\theta<1/2$. Periodic boundary conditions were used and the algorithm is updated in parallel so that growth is attempted at all even (odd) sites at even (odd) time steps.

We also carried out a numerical integration of the KPZ equation with temporally correlated noise. For reasons that will become clear later the noise correlator (3) yields surfaces that develop a faceted pattern with an increasing slope, $\langle|\nabla h|\rangle$, as the correlation exponent $\theta$ is increased above certain threshold. This leads to a numerical instability in finite time for any discretization of Eq. (1). To avoid this we replace the nonlinear term $\lambda(\nabla h)^{2}$ by an arbitrary function $\lambda f[(\nabla h)^{2}]$ that saturates for large values of the argument. We have checked several choices for the control function $f(y)$ with similar results. For definiteness, the numerical results presented below correspond to the choice $f(y)=(1-e^{-cy})/c$, where $c>0$ is a constant. We used $c=0.1$ in our simulations. This is equivalent to include the infinite series of nonlinear terms $\lambda(\nabla h)^{2}[1+\sum_{n=1}^{\infty}(-c)^{n}(\nabla h)^{2n}/(n+1)!]$ in the evolution equation, Eq. (1), while respecting all KPZ symmetries. This trick stabilizes the numerical scheme for any $0<c<1$, as occurs in other growth models in which the average local slope, $\langle|\nabla h|\rangle$, becomes large Dasgupta et al. (1996, 1997). We discretized Eq. (1), with $f[(\nabla h)^{2}]$ replacing $(\nabla h)^{2}$, with an Euler finite-differences scheme and the noise correlator given in Eq. (3) (see Supplemental Material SM ).

Numerical Results.-

In all our simulations the surface is started from a initially flat profile $h(x,0)=0$ and periodic boundary conditions are used. As time evolves the surface becomes progressively rough and height fluctuations grow under the action of the random noise. Surface correlations are measured by means of height-height correlations, surface width, and the structure factor at different times in the evolution. Surface fluctuations saturate and become stationary at a characteristic time that scales with the system size, $t_{\times}\sim L^{z}$.

In Fig. 1 we plot the surface height for the ballistic deposition simulation in a system of size $L=8192$ and correlation exponents $\theta=0.15$ and $\theta=0.47$. It becomes apparent the spontaneous formation of a faceted pattern as the correlation index is increased. We found similar pattern formation in the numerical integration of the KPZ equation (see Supplemental Material SM ).

Apart from the evident change in the visual aspect of the height profiles for $\theta>\theta_{\mathrm{th}}$, the coexistence of faceted patterns with scale invariant dynamics also leads to important effects in the scaling behavior of the surface height correlations. According to Ramasco et al. Ramasco et al. (2000) theory of generic kinetic roughening, scale-invariant faceted surfaces obey inherently different scaling functions (and exponents) arising from the patterned structure. Following Ramasco et al. Ramasco et al. (2000) the most general description of the scaling properties of a growing surface is best achieved by using the structure factor $S(k,t)=\langle\widehat{h}(\mathbf{k},t)\widehat{h}(-\mathbf{k},t)\rangle$, where $\widehat{h}(\mathbf{k},t)\equiv(1/L)^{d/2}\int d\mathbf{x}\;h(\mathbf{x},t)\exp(-i\mathbf{k}\cdot\mathbf{x})$ is the Fourier transform of the surface height $h(\mathbf{x},t)$, and $k=|\mathbf{k}|$. For kinetically roughening surfaces in $d+1$ dimensions we expect

[TABLE]

where the most general scaling function, consistent with scale-invariant dynamics, is given by Ramasco et al. (2000)

[TABLE]

with $\alpha$ being the global roughness exponent and $\alpha_{s}$ the so-called spectral roughness exponent Ramasco et al. (2000). Standard scaling corresponds to $\alpha_{s}=\alpha<1$. However, other situations may be described within the generic scaling framework, including super-roughening and intrinsic anomalous scaling, depending on the values of $\alpha_{s}$ and $\alpha$ Ramasco et al. (2000). For faceted surfaces, the case of interest for us here, one has $\alpha_{s}>1$ and $\alpha\neq\alpha_{s}$ so that two independent roughening exponents are actually needed to completely describe the scaling properties of the surface Ramasco et al. (2000).

It is important to remark that only the structure factor $S(k,t)$ allows one to obtain the distinctively characteristic spectral roughness exponent $\alpha_{s}$ typical of faceted growing surfaces. For instance, the usual global surface width $W(L,t)=\langle[h(x,t)-\overline{h}(t)]^{2}\rangle^{1/2}$, where $\overline{h}(t)$ is the spatial average height at time $t$ and the brackets $\langle\cdots\rangle$ denote average over noise, can be analytically calculated from Eqs. (4) and (5) using $W^{2}(L,t)=\int\frac{dk}{2\pi}S(k,t)$ to obtain $W(L,t)=t^{\alpha/z}{\cal F}(L/t^{1/z})$, with the standard scaling function is ${\cal F}(u)\sim u^{\alpha}$ for $u\ll 1$ and ${\cal F}(u)\sim\mathrm{const}$ for $u\gg 1$, as was shown in Ref. Ramasco et al. (2000). Hence, the anomalous spectral exponent $\alpha_{s}$ leaves no trace in the usual height-height correlation functions.

In our simulations we analyzed the structure factor $S(k,t)$ of surfaces produced with the ballistic deposition algorithm and the discretized KPZ equation for noise with correlation index $\theta$ varying in the interval $[0,1/2)$. In Figs. 2 and 3 we plot our results for ballistic growth with correlated noise exponent $\theta=0.15$ and $\theta=0.47$, respectively. These $\theta$ values correspond to those plotted in Fig. 1 for easy comparison. Data collapse analysis was used to obtain the scaling behavior of $S(k,t)$ (insets of Figs. 2 and 3). This analysis reveals that the structure factor indeed exhibits anomalous scaling, as corresponds to faceted scale-invariant surface roughening, in the case $\theta=0.47$ with a spectral roughness exponent $\alpha_{s}=1.23\pm 0.02$ and roughness exponent $\alpha=1.02\pm 0.01$, while $\alpha_{s}=\alpha=0.56\pm 0.02$ (i.e. standard scaling) for $\theta=0.15$.

We have systematically analyzed the scaling behavior of the structure factor for the ballistic deposition model and the discretized KPZ equation as the correlation range of the noise, $\theta$, is varied in the interval $[0,1/2)$. We computed the scaling exponents $\alpha$ and $\alpha_{s}$ from data collapse analysis of $S(k,t)$ for system sizes $L=2048,4096$, and $8192$. Note that large system sizes are required for the surface to develop a faceted pattern before it saturates. Our main results are summarized in Fig, 4, where the global roughness exponent $\alpha$ and the spectral roughness exponent $\alpha_{s}$ are plotted as a function of the noise correlation index $\theta$ for ballistic deposition growth. Similar results were obtained for the numerical integration of the KPZ equation with correlated noise (see Supplemental Material), demonstrating that our findings are robust within the universality class. Figure 4 clearly shows that the roughness exponents split up at $\theta_{\mathrm{th}}=0.25\pm 0.03$ for the ballistic deposition model (a similar threshold value $\theta_{\mathrm{th}}\approx 0.23$ was found for the KPZ equation SM ). So that, below the threshold one finds standard scaling with $\alpha=\alpha_{s}$ and no facets. In contrast, as the noise correlation range is increased above the threshold, faceted surfaces are formed and $\alpha_{s}$ moves away from $\alpha$. For the sake of comparison, Fig. 4 also shows the main theoretical approximations to KPZ with temporally correlated noise. We can see that, while the global roughness exponent is nicely predicted by the dynamic RG calculations of Medina et al. Medina et al. (1989) above the threshold, it fails to describe the correlation effects below $\theta_{\mathrm{th}}$. Obviously, no present theory is capable of predicting the existence of facets and the associated spectral roughness exponent as the noise correlation range is increased.

Discussion.-

While we lack of a complete theory to explain the numerical findings reported above, we can put forward some arguments to rationalize the emergence of the faceted patterns and the associated anomalous scaling.

Let us start by considering the limit case of KPZ dynamics with the noise at site each site $\mathbf{x}$ fixed at all times, but uncorrelated from site to site– namely, the problem of KPZ with columnar noise:

[TABLE]

where $\eta(\mathbf{x)}$ is a spatially uncorrelated noise

[TABLE]

This problem corresponds to the limit $\theta=1/2$ of the long-term correlated noise case in Eq. (3). It is well known Szendro et al. (2007b) that the columnar KPZ equation (6) exhibits facet formation arising from the exponential localization of the field $\phi(\mathbf{x},t)\equiv\exp h(\mathbf{x},t)$ around some random centers $\mathbf{x}_{c}$. The auxiliary $\phi$ can be interpreted as the probability density of particles diffusing in a random potential $\eta(\mathbf{x})$:

[TABLE]

The multiplicative-noise term in Eq. (7) leads to sharply localized solutions around random localization centers. The stochastic field $\phi$ has an exponential profile, $\sim\exp(-|\mathbf{x}-\mathbf{x}_{c}|/\xi)$, around any typical center $\mathbf{x}_{c}$ with a certain localization extent $\xi$. The solutions of Eq. (7) are, therefore, a superposition of these exponentially localized functions. In turn, this leads to a surface $h$ formed by facets with their cusps at the localization centers, $h\sim\ln\phi\sim\pm|\mathbf{x}-\mathbf{x}_{c}|/\xi$, as shown by Szendro et al. Szendro et al. (2007b). There, the reported values of the global and spectral roughness exponents were $\alpha=1.07\pm 0.05$ and $\alpha_{s}=1.5\pm 0.05$, respectively, in $d=1$.

We conjecture that this localization picture can be essentially extended for $\theta<1/2$. Our numerical results indicate that the mechanism for the formation of facets based on localization remains valid while $\theta>\theta_{\mathrm{th}}$, where $\theta_{\mathrm{th}}\approx 1/4$. Remarkably, the value $\theta=1/4$ was already shown to play a special role in the perturbative RG approximation of Medina et al. Medina et al. (1989), as the point at which the renormalized noise amplitude $D^{*}(\omega)$ has a singular correction at leading $\omega$-order. Further singularities appear at larger values of $\theta$ making the RG treatment ill-constructed. Our results strongly suggest that the infinitely many singularities are a reflection of the failure of the RG treatment to describe the appearance of a new exponent $\alpha_{s}\neq\alpha$. Given these results, it becomes clear that a generalization of the RG theory would be required to describe the generic scaling form (5) of the spectral function. Such a generalization should be able not only to fix the nonphysical singularities but it would also provide a coherent mathematical picture of anomalous kinetic roughening as a whole.

Acknowledgements.

J. M. L. thanks D. Pazó for discussions and a critical reading of the manuscript. This work has been partially supported by the Program for Scientific Cooperation “I-COOP+” from Consejo Superior de Investigaciones Científicas (Spain) through project No. COOPA20187. A. A. is grateful for the financial support from Programa de Pasantías de la Universidad de Cantabria in 2017 and 2018 (Projects No. 70-ZCE3-226.90 and 62-VCES-648), and CONICET (Argentina). J. M. L is supported by Dirección General de Investigación Cientıífica y Técnica, MICINN (Spain), through the project No. FIS2016-74957-P.