Solid-on-solid rules and models for nonequilibrium growth in 2+1 dimensions

Abstract

On the basis of a detailed stochastic simulation we find that nonequilibrium growth in 2+1 dimensions, within the simple solid-on-solid approximation, is surprisingly rich with its dynamical universality depending sensitively on the local atomistic relaxation rules of the growth model. We establish connections between our computed dynamical growth exponents for various physically plausible local growth models with those given by a recently proposed fourth-order nonlinear continuum diflferential equation.PACS numbers: 61.50. Cj, 05.40.+J, 05.70.Ln, 68.55.Bd In a solid-on-solid (SOS) growth model, incident atoms stick only to the tops of the already existing substrate atoms which are arranged in a lattice [1]. The resultant growing structure is, therefore, a lattice of columns whose heights increase as matter is added from outside. The SOS model has been extensively used in both equilibrium and nonequilibrium crystal growth studies. While being conceptually simple, it also describes well many real situations where vacancies and overhangs are relatively rare. In this Letter, we present detailed simulation results which demonstrate that nonequilibrium growth, even within the simple SOS approximation, is extremely rich in 2+1 dimensions (i.e., two-dimensional substrate, ^'=2, with growth in the other direction), leading to many different growth universality classes depending upon the precise nature of the local growth rules. We find that the local atomistic relaxation rule and, in particular, its dependence on the atomic coordination number which defines the model determine the universality class of the growth model. This surprising richness in the SOS nonequilibrium growth is totally unanticipated, because the simplicity of the model makes one expect some simple universality independent of the details of the local growth rules. In fact, it is interesting to note that more complicated nonequilibrium growth models involving ballistic deposition (where impinging atoms may stick to the sides of the existing columns, thus allowing overhangs and vacancies) are all believed to belong to a single universality class, namely, the KPZ universality [2], whereas the simpler SOS growth models cannot be characterized by a single universality class.We impose two additional restrictions on the SOS model, motivated both by practical relevancy and computational tractability, namely, neglect of desorption and, in some cases [3] as discussed below, of any upward atomic relaxation [4]. The continuum equation governing the dynamics of growth under these conservative (i.e., no desorption and SOS restrictions) conditions has the general form [5] bh/bt^dSl^h-bS^h + (fourth-order nonlinear terms) +where rjixj) is the Gaussian white noise satisfying the correlation {T](xj)7]{xj'))^D8 (x-x)d(t-t') with x and x' the lateral coordinates. Since there is no evaporation [6], vacancies [7], or a tilted substrate [8], the (Vh)^ term of the KPZ equation does not appear, as the growth process is manifestly current conserving. In the presence of the ...

Dynamics of a Toom interface in three dimensions

“…Our model is distinguished from that of BAS in that the four neighboring spins are of equidistance, while they are not in the BAS model. Interestingly, this difference of the local dynamic rules leads to different universality classes, which is in accordance with a recent claim that dynamic universality class depends on local dynamic rules in (2+1) dimensions [6]. We studied the model for both unbiased and biased cases, and found that our model belongs to the Edwards-Wilkinson (EW) [7] universality for the unbiased case and the AKPZ universality in the wea/c-coupling limit [4] for the biased case.…”

Nonuniversality in models of epitaxial growth

“…10,11 We will briefly survey some results obtained for models with modified rules and in higher dimensions in Sec. III.…”

“…Apart from the intrinsic theoretical interest, the motivation was a hope to better understand growth processes in molecular beam epitaxy ͑MBE͒ and other epitaxial growth techniques. Different variants of simplified ͑''toy''͒ discrete models with relaxation only after deposition, [5][6][7][8][9][10][11][12][13][14][15] as well as more realistic full diffusion ͑FD͒ models, 7, 16 -18 have been employed, and the question of whether they belong to a new universality class has been addressed. Early numerical calculations seemed to suggest that this is actually the case 6 -8,10,16 but more recent results 12,14,18,19 show that in many growth models with surface diffusion the true asymptotic behavior is either of the well-known Edwards-Wilkinson type ͑which is only logarithmically rough in dϭ3) or that growth is unstable.…”

Set of measures to analyze the dynamics of nonequilibrium structures

“…Patterns growth in the Cahn-Hilliard Equation (used to model spinodal decomposition) are shown to exhibit three distinct stages. Two models of surface growth, namely the continuous Kardar-Parisi-Zhang (KPZ) model and the discrete Restricted-Solid-On-Solid (RSOS) model are shown to have different saturation exponents.PACS numbers: 47.54.+r,47.20.Hw,05.70.Np,64.75.+g,89.75.Kd,05.50.+q, In recent years, there has been a considerable amount of effort to analyze non-equilibrium interfacial growth and pattern formation in experimental and model systems [1,2,3,4,5,6,7,8,9,10]. The phenomena studied include spinodal decomposition [1,5,11,12], chemical pattern formation [4], surface growth [2,3,7,8] and epitaxial growth [9,10].…”

“…The phenomena studied include spinodal decomposition [1,5,11,12], chemical pattern formation [4], surface growth [2,3,7,8] and epitaxial growth [9,10]. Microscopic modeling of these phenomena is highly complex, and most microscopic details of such structures depend on initial conditions and stochastic effects.…”