AFM/SEM Study of Thermally Induced Hillock Coalescence
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and mass scaling of the equations describing the interactions between those entities.
First, we briefly review the model and its application to data obtained independently by Vook and coworkers3 ,4 . The questions we hope to address are articulated in the succeeding section. We then present the results of atomic force microscopy (AFM) and scanning electron microscopy (SEM) experiments on aluminum films which provide some answers or insights into these questions. The Model The model has been described in detail in the literature 2 ,5 and is based on the Smoluchowski equations (1) with Jullien's asymptotic solution 6 (2) giving the time-dependent concentration, nk, of a cluster, hillock or some other association or entity having k monomer units, i.e. a k-mer. The model assumes that k-mers are produced only by coalescence of smaller clusters and are destroyed only when they coalesce with another cluster to form a still larger cluster. If there was a population of monomers at t = 0, say metal atoms impinging on a substrate from the gas phase, the formation of a film would commence by the formation of associations between the atoms. These "associations" might be thought of as islands, hillocks or by some other name.
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Mat. Res. Soc. Symp. Proc. Vol. 356 0 1995 Materials Research Society
dt nk
dk1 -_I-1Io Kj,k-jnjnk-j 2 j=1
- nk
Y,Kj,knj (j1 )
C ka e-bk
(2)
Although this model contains much of the hillock/void formation and growth process, there are processes which could lead to fragmentation of some islands or clusters. Although both unimolecular 7 and bimolecular processes 8 can be imagined, we will group them all together under the heading of "evaporation". There is no explicit term for evaporation in (1), but the rate constants implicitly relate the fact that not all encounters between i-mers and j-mers result in coalescence. The asymptotic solution, (2), can be interpreted in terms of the scaling of the bimolecular rate constants, Kij, relating the interaction of individual agglomerating entities, e.g. i-
mers and j-mers, with i and j. Under certain situations 9 , it is thought that the rate constants could
scale smoothly as in (3), and if so, they are termed "homogeneous".
Kxi,xj = .2°Kij
(3)
Then the rate constants Kij can be written explicitly in terms of i and j. This allows analysis of the equations and the possibility of discovering rigorous closed form mathematical connections between the parameters defining the Kij and hillock or cluster size distributions. Although we shall choose not to do so in this work, it is possible 10 to factor the Kij into a product of two factors. The first involves only the probability of a hard sphere(or other generic shape) encounter, while the second gives the likelihood of an encounter ending in coalescence. The results of the model, e.g. a distribution of cluster sizes produced in the long time limit, show what would happen if the distribution function were determined purely by kinetics, assuming no preferred pathways for the formation or destruction of any
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