Coarsening in inhomogeneous systems

A class of simple, (2+1)-dimensional, discrete models is reviewed, which allow us to study the evolution of surface patterns on solid substrates during ion beam sputtering (IBS). The models are based on the same assumptions about the erosion process as the existing continuum theories. Several distinct physical mechanisms of surface diffusion are added, which allow us to study the interplay of erosion-driven and diffusion-driven pattern formation. We present results from our own work on evolution scenarios of ripple patterns, especially for longer timescales, where nonlinear effects become important. Furthermore we review kinetic phase diagrams, both with and without sample rotation, which depict the systematic dependence of surface patterns on the shape of energy depositing collision cascades after ion impact. Finally, we discuss some results from more recent work on surface diffusion with Ehrlich-Schwoebel barriers as the driving force for pattern formation during IBS and on Monte Carlo simulations of IBS with codeposition of surfactant atoms.

Simulating discrete models of pattern formation by ion beam sputtering.

We study dynamic scaling in a model with collective diffusion (CD) of isolated atoms in terraces and irreversible aggregation at step edges. Simulations are performed in two-dimensional substrates with several diffusion to deposition ratios R identical with D/F. Data collapse of scaled roughness distributions confirms that this model is in the class of the fourth-order nonlinear growth equation by Villain, Lai, and Das Sarma (VLDS) with negligible finite-size effects, while estimates of scaling exponents show some discrepancies. This result is consistent with the prediction of a recent renormalization group approach and improves previous numerical works on related models. The roughness follows dynamic scaling as W=Lalpha/R1/2f(xi/L), with correlation length xi=(Rt)1/z, where z is the dynamic exponent. We also propose a limited mobility (LM) model where the incident atom executes up to G steps before a new atom is adsorbed, and irreversibly aggregates at step edges. This model is also shown to belong to the VLDS class. The size of the plateaus in the film surface increases as G1/2 and the lateral correlation scales as G1/2t1/z. The time evolution of the roughness reproduces that of the CD model if an equivalent parameter G approximately R2/z is chosen. This suggests the possibility of using LM models with tunable diffusion length to simulate processes with simultaneous diffusion of many atoms. A scaling approach is used to justify exponent values and dynamic relations for both models, including the significant decrease of surface roughness as R or G increases.

Dynamic scaling in thin-film growth with irreversible step-edge attachment.

In this paper, we study many geometrical properties of contour loops to characterize the morphology of synthetic multifractal rough surfaces, which are generated by multiplicative hierarchical cascading processes. To this end, two different classes of multifractal rough surfaces are numerically simulated. As the first group, singular measure multifractal rough surfaces are generated by using the $p$ model. The smoothened multifractal rough surface then is simulated by convolving the first group with a so-called Hurst exponent, ${H}^{*}$. The generalized multifractal dimension of isoheight lines (contours), $D(q)$, correlation exponent of contours, ${x}_{l}$, cumulative distributions of areas, $\ensuremath{\xi}$, and perimeters, $\ensuremath{\eta}$, are calculated for both synthetic multifractal rough surfaces. Our results show that for both mentioned classes, hyperscaling relations for contour loops are the same as that of monofractal systems. In contrast to singular measure multifractal rough surfaces, ${H}^{*}$ plays a leading role in smoothened multifractal rough surfaces. All computed geometrical exponents for the first class depend not only on its Hurst exponent but also on the set of $p$ values. But in spite of multifractal nature of smoothened surfaces (second class), the corresponding geometrical exponents are controlled by ${H}^{*}$, the same as what happens for monofractal rough surfaces.

Geometrical exponents of contour loops on synthetic multifractal rough surfaces: multiplicative hierarchical cascade p model.

Vortex lines in type-II superconductors display complicated relaxation processes due to the intricate competition between their mutual repulsive interactions and pinning to attractive point or extended defects. We perform extensive Monte Carlo simulations for an interacting elastic line model with either point-like or columnar pinning centers. From measurements of the space- and time-dependent height-height correlation function for lateral flux line fluctuations, we extract a characteristic correlation length that we use to investigate different non-equilibrium relaxation regimes. The specific time dependence of this correlation length for different disorder configurations displays characteristic features that provide a novel diagnostic tool to distinguish between point-like pinning centers and extended columnar defects.

Characterization of relaxation processes in interacting vortex matter through a time-dependent correlation length

A discrete model exhibiting conserved dynamics with nonconserved noise involving particles of different nature, termed as linear and nonlinear, is proposed here. The morphology of the surface has been studied with different abundances of these particles. The saturated surface, slowly evolved from a lower contribution of nonlinear particles to a higher contribution of nonlinear particles, splits into four distinct scaling regimes with three crossover lengths. Each regime is characterized by different scaling property. It is shown that when the contribution of the nonlinear particles crosses a critical value, the surface morphology shows a linear-nonlinear "phase transition." The roughness exponent in a nonlinear regime is well compared with that of the continuum nonlinear equation in a molecular beam epitaxy (MBE) class as well as a MBE motivated discrete model.

Multifractal behavior of the surfaces evolved with surface relaxation.

Extensive Monte Carlo simulations are performed on a two-dimensional random field Ising model. The purpose of the present work is to study the disorder-induced changes in the properties of disordered spin systems. The time evolution of the domain growth, the order parameter, and the spin-spin correlation functions are studied in the nonequilibrium regime. The dynamical evolution of the order parameter and the domain growth shows a power law scaling with disorder-dependent exponents. It is observed that for weak random fields, the two-dimensional random field Ising model possesses long-range order. Except for weak disorder, exchange interaction never wins over pinning interaction to establish long-range order in the system.

Dynamical properties of random-field Ising model.

We demonstrate the non-universal behavior of finite size scaling in (1+1) dimension of a nonlinear discrete growth model involving extended particles in generalized point of view. In particular, we show the violation of the universal nature of the scaling function corresponding to the height fluctuation in (1+1) dimension. The 2nd order moment of the height fluctuation shows three distinct crossover regions separated by two crossover time scales namely, tx1 and tx2. Each regime has different scaling property. The overall scaling behavior is postulated with a new scaling relation represented as the linear sum of two scaling functions valid for each scaling regime. Besides, we notice the dependence of the roughness exponents on the finite size of the system. The roughness exponents corresponding to the rough surface is compared with the growth rate or the velocity of the surface.

Non-universal finite size scaling of rough surfaces

We report a study of nonequilibrium relaxation in a two-dimensional random field Ising model at a nonzero temperature. We attempt to observe the coarsening from a different perspective with a particular focus on three dynamical quantities that characterize the kinetic coarsening. We provide a simple generalized scaling relation of coarsening supported by numerical results. The excellent data collapse of the dynamical quantities justifies our proposition. The scaling relation corroborates the recent observation that the average linear domain size satisfies different scaling behavior in different time regimes.

Characterization of kinetic coarsening in a random-field Ising model.

We investigate a novel (d + 1)-dimensional discrete erosion model for d = 1, 2 and 3. The dynamics of the model is controlled by the physically motivated erosion mechanism. The coarse grained nature of this erosion process has been well compared with the Kardar–Parisi–Zhang (KPZ) equation. The kinetic roughening of the discrete model shows the same scaling behavior as that of the KPZ equation in the dimensions d = 1, 2. Moreover, in this present discrete model in (3 + 1)-dimension almost smooth interface has been obtained with vanishingly small roughness exponent, indicating the model belongs to the weak coupling regime of KPZ universality class.

Pradipta Kumar Mandal

Papers

Multifractal behavior of the surfaces evolved with surface relaxation.

Dynamical properties of random-field Ising model.

Non-universal finite size scaling of rough surfaces

Characterization of kinetic coarsening in a random-field Ising model.

Kardar-Parisi-Zhang universality class of a discrete erosion model