Zhang Model Research Articles

Dynamical properties of the Zhang model of self-organized criticality

Critical exponents of the infinitely slowly driven Zhang model of self-organized criticality are computed for $d=2,3$ with particular emphasis devoted to the various roughening exponents. Besides confirming recent estimates of some exponents, new quantities are monitored and their critical exponents computed. Among other results, it is shown that the three dimensional exponents do not coincide with the Bak, Tang, and Wiesenfeld (abelian) model and that the dynamical exponent as computed from the correlation length and from the roughness of the energy profile do not necessarily coincide as it is usually implicitly assumed. An explanation for this is provided. The possibility of comparing these results with those obtained from Renormalization Group arguments is also briefly addressed.

Fe/Au multilayer growth fronts: Comparison of X-ray diffuse scattering results with atomic-force microscopy profiles

The interfacial roughness morphology of sputtered Fe/Au multilayers was derived by combining the results from X-ray specular and non-specular (diffuse) scattering. From the non-specular intensity collected by a two-dimensional image plate we determined the in-plane height–height correlation function 〈 H i( R) H i(0)〉 and the growth correlation length as a function of feature size. From these parameters we determined both the static and dynamic roughness exponents α and z, which agree with the Kardar–Parisi–Zhang (KPZ) model of surface growth. The height difference function derived from the diffuse scattering is compared with several model functions that incorporate the asymptotic behavior of 〈 H( R) 2〉 for the cases of R→0 and R→∞. We show that only one particular function is appropriate. The evolution of the interfacial slope is determined from the roughness exponents, and it is shown that it decreases as a function of time. This asymptotic smoothing is analyzed using the noiseless and stochastic KPZ equations. The noiseless KPZ equation predicts that the growth front is composed of paraboloids, which increase in lateral size and coalesce. This result is supported by an Atomic-force Microscopy measurement of the surface of one of the Fe/Au Multilayers.

Morphological study of kinetic roughening on amorphous and microcrystalline silicon surface

An ex situ study of surface morphology of amorphous (a-Si:H) and microcrystalline silicon ( μc-Si:H) has been carried out to understand their growth mechanism using an atomic force microscope. For a-Si:H, a standard Karder, Parisi and Zhang (KPZ) model well describes the kinetics of the roughness evolution with increasing film thickness except at T s∼360°C, where a significant roughness enhancement is observed. The morphology of μc-Si:H on Si(001) substrate shows a similar roughness formation at much lower temperatures of 150 to 200°C. This roughness formation is found to be correlated with the surface hydrogen coverage.

A dynamical system approach to SOC models of Zhang's type

We discuss Zhang's model of SOC in the framework of hyperbolic dynamical systems with singularities. The fractal structure of the invariant energy distribution, correlation decay-like phenomena, and symbolic coding are discussed.

Symmetries and fixed point stability of stochastic differential equations modeling self-organized criticality

A stochastic nonlinear partial differential equation is built for two different models exhibiting self-organized criticality, the Bak, Tang, and Wiesenfeld (BTW) sandpile model and the Zhang's model. The dynamic renormalization group (DRG) enables to compute the critical exponents. However, the nontrivial stable fixed point of the DRG transformation is unreachable for the original parameters of the models. We introduce an alternative regularization of the step function involved in the threshold condition, which breaks the symmetry of the BTW model. Although the symmetry properties of the two models are different, it is shown that they both belong to the same universality class. In this case the DRG procedure leads to a symmetric behavior for both models, restoring the broken symmetry, and makes accessible the nontrivial fixed point. This technique could also be applied to other problems with threshold dynamics.

Exact scaling in surface growth with power-law noise.

We investigate a continuum formulation of surface growth following the Kardar-Parisi-Zhang equation [Phys. Rev. Lett. 56, 889 (1986)] with a power-law distribution of the magnitudes of regional advances. This formulation describes Zhang's ballistic-deposition model [J. Phys. (Paris) 51, 2129 (1990)] with power-law noise and possibly recent fluid-displacement experiments. Our exact theory predicts a transition of the scaling behavior from power-law-noise domination to a Gaussian-noise regime as the power increases. An apparent contradiction with previous simulations is due to a logarithmic correction to the scaling at the transition and to anomalous-growth effects. Analogous scaling behaviors are derived for the Edwards-Wilkinson model [Proc. R. Soc. London Ser. A 381, 17 (1982)] with power-law noise

Kinetic roughening with power-law waiting time distribution

The authors introduce a surface growth model where the elementary events are characterized by a waiting time distribution P( tau ). Exact relations to directed polymer statistics and to continuous time random walk problems are established. For P( tau ) approximately 1/ tau mu +1 the behaviour is similar to that of the Zhang model where rare-event-dominated kinetic roughening occurs due to a power-law noise in the surface increments. A careful correction to scaling analysis of the numerical results in 1+1 dimensions indicates universality with the Zhang model for fixed values of mu .

SIMULATIONS OF KINETIC ROUGHENING WITH POWER-LAW NOISE ON THE CONNECTION MACHINE

We present results of large scale simulations on the Connection Machine (CM) on the scaling behavior of the Zhang model and its variants for the kinetics of self-affine interfaces with power-law noise. Details on implementing this problem on a massively parallel computer such as the CM are given. Our calculations for the case when the amplitude η of the noise has a distribution P(η)~η−1−µ are in good agreement with earlier findings of non-universality for µ<7. We present data which suggest that for µ≥7 the model is in the universality class of Gaussian noise.

Probability distribution of the interface width in surface roughening: analogy with a Levy flight

The authors present a closed-form expression for the probability distribution of the height fluctuations in the Zhang model of anomalous surface roughening. The result-which includes both the steady state behavior and the time evolution to the steady state-is based on an analogy between the (d=1+1)-dimensional 'surface' problem and a d=1 Levy flight. In the limit case of conventional ballistic deposition they obtain a Gaussian distribution for the height fluctuations. Their results are corroborated by detailed numerical simulations.

Ballistic deposition with power-law noise: A variant of the Zhang model.

We study a variant of the Zhang model [Y.-C. Zhang, J. Phys. (Paris) 51, 2113 (1990)], ballistic deposition of rods with the length l of the rods being chosen from a power-law distribution P(l)\ensuremath{\sim}${\mathit{l}}^{\mathrm{\ensuremath{-}}1\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\mu}}}$. Unlike in the Zhang model, the site at which each rod is dropped is chosen randomly. We confirm that the growth of the rms surface width w with length scale L and time t is described by the scaling relation w(L,t)=${\mathit{L}}^{\mathrm{\ensuremath{\alpha}}}$w(t/${\mathit{L}}^{\mathrm{\ensuremath{\alpha}}/\mathrm{\ensuremath{\beta}}}$), and we calculate the values of the surface-roughening exponents \ensuremath{\alpha} and \ensuremath{\beta}. We find evidence supporting the possibility of a critical value ${\mathrm{\ensuremath{\mu}}}_{\mathit{c}}$\ensuremath{\approxeq}5 for d=1 with \ensuremath{\alpha}=1/2 and \ensuremath{\beta}=1/3 for \ensuremath{\mu}>${\mathrm{\ensuremath{\mu}}}_{\mathit{c}}$, while for \ensuremath{\mu}${\mathrm{\ensuremath{\mu}}}_{\mathit{c}}$, \ensuremath{\alpha} and \ensuremath{\beta} vary smoothly, attaining the values \ensuremath{\alpha}=\ensuremath{\beta}=1 for \ensuremath{\mu}=2.