3 N ov 2 00 7 A continuous non - linear shadowing model of columnar growth

We propose the first continuous model with long range screening (shadowing) that described columnar growth in one space dimension, as observed in plasma sputter deposition. It is based on a new continuous partial derivative equation with non-linear diffusion and where the shadowing effects apply on all the different processes. PACS numbers: 81.15.Aa, 68.35.Ct, 05.40.-a Submitted to: J. Phys. D: Appl. Phys. Fast Track Communications § Corresponding author: Fax +33(0)2 38 41 71 54, e-mail: [email protected] Non-linear growth model 2 Plasma sputtering is a common process for film growth which often exhibits wide columns more or less close packed separated by thin deep grooves [1, 2, 3, 4]. This columnar growth mainly results from a shadowing instability[5, 6, 7], where the elevated parts of the surface are more exposed to the sputtering while they shadow the incoming particles to the lower parts. The modelization of this shadowing instability has been well described by probabilistic Monte-Carlo methods (MC)[8, 9, 10] and also with continuous models based on partial derivative equations (PDE)[6, 7, 11, 12, 13, 14, 15, 16] including the seminal work of Bales and Zangwill [1]. However, both approach fail to describe at long times the strongly nonlinear columnar microstructures observed recently (see [2] for instance). In fact, although the continuous models gives tall and well separated columns at early time, only few sharp peaks remain later on[6, 14, 16]. Columnar structure using PDE has already been obtained by Gillet et al.[17] but in that case no shadowing effect was taken into account! On the other hand, discrete approaches using MC methods including shadowing have been developped and showed a fair description of the columnar structure, particularly through the formation of sharp column sides. However, these models cannot avoid the coarsening of the columnar structures showing larger and larger plateau as time increases, in contrast with experimental observations. The goal of this paper is to present a new continuous non-local model which includes both non-linear shadowing and diffusion effects to simulate columnar-like growth. We consider a two dimensional model where the one dimensional (1D) surface described by h(x, t) is subjected to receive particles from all directions not shadowed by the surface itself. Our starting point is deduced from the models developped initially by Bales and Zangwill[1] and by Karunasiri et al [6]: ∂h ∂t = RΩ(x, {h}) √ 1 + (∇h) + ν∇h + η (1) where the deterministic deposition term R is multiplied by the solid angle Ω(x, {h}) which modelizes the shadowing effect as a long range screening (see figure 1). ν is the diffusion/relaxation coefficient while η is the usual noise with zero mean < η >= 0 and its correlation given by < η(x, t)η(x, t) >= 2Dδ(x, x)δ(t, t). For small surface angles, we retrieve a KPZ-like equation [18] with shadowing effects (defining λ = πR): ∂h ∂t = ν∇h+ λ 2 (∇h) +RΩ(x, {h}) + η. (2) A complete study of these equations has shown that it is unable to reproduce columnar shapes corresponding to experiments and MC simulations [7, 14]. Indeed, in the most favourable situation, only broad peaks emerge instead of flat columns. Experiments suggest thus that the diffusion should be enhanced in the region more exposed to the flux. Moreover, we will assume that the flux also increases (greater than for normal shadowing) on the top of the columns compared to the grooves. Although we have no strong argument for it, we expect some point effect near the sharp edge Non-linear growth model 3