Analytical and Numerical Modeling of Surface Morphologies in Thin Films
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Franqois Y. GENIN Lawrence Livermore National Laboratory, Chemistry and Materials Science Division, Livermore, California 94550. ABSTRACT Experimental studies have shown that strains due to thermal expansion mismatch between a film and its substrate can produce very large stresses in the film that can lead to the formation of holes and hillocks. Based on a phenomenological description of the evolution of a solid surface under both capillary and stress driving forces and for surface and grain boundary selfdiffusion, this article provides, for the first time, analytical and numerical solutions for surface profiles of model geometries in polycrystalline thin films. The results can explain a variety of surface morphologies commonly observed experimentally and are discussed to give some practical insights on how to control the growth of holes and hillocks in thin films. INTRODUCTION The long term stability of thin films under a variety of conditions has been extensively studied both theoretically" 9 and experimentally.' 0 26 Models have been developed to describe surface morphologies for different ideal geometries.27 -3 ' These models have provided solutions for the profiles of surfaces evolving under capillary and stress driving forces.
Excellent tutorials of the theoretical procedure used in such models can be found in a chapter by Mullins32 and an article by Srolovitz.33 In these treatments, grain boundary (GB) self-diffusion as a transport mechanism is not included since it was not relevant to the particular cases that were modeled (i.e. single crystal surface under stress or thermal grooving in stress-free sheets). In most practical cases, however, morphological changes in polycrystalline thin solid films are not only driven by both capillary forces and stresses but involve transport of atoms along GBs.
Only recently have models of thin films begun to include these two driving forces and transport mechanisms simultaneously,34'35 using earlier approaches
that are
crack growth.3 6' 37
more
common
in mechanical
metallurgy
to model
A similar trend is found in modeling the sintering
particles .38,39
83 Mat. Res. Soc. Symp. Proc. Vol. 389 © 1995 Materials Research Society
of
The theoretical description of the motion of atoms goes through the usual steps27'32 of assigning a chemical potential in the solid (here, on the surface of that solid), expressing the gradients of chemical potential, the flux of atoms they create, and by translating the divergence of these fluxes into a rate of atoms accumulation via the continuity equation. Finally, this rate of accumulation of atoms can be related to the speed of motion of an element of surface along its normal and converted into a surface profile for each time. In most studies that take a constitutive approach (as opposed to an atomistic one), Herring's expression for the chemical potential4"' 4" on the surface or in the GB is the starting point. The properties of the solid are usually assumed to be independent of crystallographic orientation.
Finally,
derivation
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