A Numerical Study of Stress Controlled Surface Diffusion During Epitaxial Film Growth
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of this paper is on the S-K growth mode, which occurs when there is a lattice mismatch between film and substrate, and when the surface energy of the substrate is larger than the sum of those of the film and the interface. Important factors which are known to influence the film growth include surface diffusion, deposition, and the film/substrate interface. Special cases where film growth is dominated by surface diffusion have been extensively studied by first-order perturbation analyses [2-10]. It is now well understood that due to the mismatch-induced stress in the film, a flat film surface is unstable in that small undulations can increase exponentially under stresscontrolled surface diffusion. More recent analytical/experimental/numerical investigations [11-17] have further shown that the surface roughening can lead to formation of cusps which bear the same stress singularity as cleavage cracks. An overview of this subject, including discussions on three dimensional aspects of cusp formation, can be found in [18]. To fully understand surface roughening and defect nucleation during film growth, it is necessary to consider the combined influence due to surface diffusion, deposition, and the film/substrate interface. For example, it is known that the interface accounts for the layer-by-layer growth pattern during the initial stage of the S-K mode [19]. The rate of deposition may alter the rate of island formation and growth. In this paper we examine the role of the interface and that of film deposition rate in the presence of surface diffusion. Our investigation proceeds as follows: 33
Mat. Res. Soc. Symp. Proc. Vol. 356 0 1995 Materials Research Society
First the kinetic equation of film growth which involves surface diffusion and deposition will be discussed, following closely the derivation in [20]. We then propose a continuum boundary layer model to imitate the influence of the interface. This model is capable of explaining the S-K growth mode as well as the F-vdM and V-W modes. Perturbation and numerical methods based on the kinetic equation and the interface model are developed to demonstrate the film morphology under different deposition rates. PROBLEM STATEMENTS Consider a semi-infinite substrate covered by a layer of film under the plane strain condition. A set of Cartesian coordinate axes are attached to the film/substrate structure with the x and y-axis being placed parallel and normal to the interface, respectively. The surface of the film at time t is described by a function y = f(x, t) where we assume the profile is two-dimensional. Only symmetric and periodic functions will be considered, i.e. f(x, t) = f(-x, t) = f(x + A,t); A and k = 2w/A denote the surface wavelength and wavenumber. The film thickness h(t) is determined by
(1)
f(Xt)d
h(t) =
The pressure on the film surface exerted by a vapor phase above the film will be ignored. For simplicity, the film and the substrate are assumed to be isotropic in every aspect and have identical elastic properties with shear modulus y and Poisson ratio
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