Patterns of Competitive Crystal Growth
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JUAN M. GARCIA-RUIZ AND A. RODRIGUEZ-NAVARRO Laboratorio de Estudios Cristalogrnificos. IAGM. CSIC-Universidad de Granada. Av. Fuentenueva s/n. Granada 18002. Spain
ABSTRACT
We present a computer simulation of the problem of competitive crystal growth of onedimensional and two-dimensional single crystals restricted to a two-dimensional space. Two similar competition indexes • and x accounting for the degree of preferred orientation developped during the growth process are defined and we show that they varies linearly with the film thickness. The existence of a clear formal similarity between • and x indexes defined in the direct space with a textural index r defined in the reciprocal space permits the identification of a competitive growth process by X-ray diffraction.
INTRODUCTION
The problem of interpreting the morphological output of a set of single crystals competing for space, has been a subject of long interest. The structures arising from this competition process affect directly to the physical properties of the materials containing them and in particular, they have been claimed to occurs during thin film formation1 . The problem of competitive growth was raised long ago by Lemmlein 2 who performed laboratory experiments and observed that only those crystals whose fastest growth directions are close to the normal to the substrate survive. This is the so called "law of survival of the faster". The same law was independently found by van der Drift3 when studying PbO thin films precipitated from vapour phase on a two-dimensional planar substrate. The experimental work performed by Lemmlein suggested that geometric selection by competitive growth leads to a linear population density of surviving crystals of n(h) proportional to 1/h, where h is the distance from the substrate. There is a probabilistic theory proposed by Kolmogorov 4 that found a similar result: i.e., n(h) proportional to 1/h'A for two-dimensional models and this potential law has been also recently found by Thijsen et al5 . Thin films can be described as layered structures in the sense that they form by an ill-defined number of crystals growing out from a surface/substrate by acretion of growth units overlying on the previously crystal faces. The very geometry of the thin film production provokes therefore texturation phenomena that occurs in the normal to the substrate meanwhile the
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Mat. Res. Soc. Symp. Proc. Vol. 355 ©01995 Materials Research Society
orientation of the crystals in all directions contained in the plane parallel to the substrate can be considered (in a first approach) unrelevant to the problem and in many cases it displays a homogeneous distribution. In these cases, interesting information can be obtained by using Xray powder diffractometer and comparing the intensities of the diffracted beam corresponding to the textured sample with those arising from a sample made of a ramdon distribution of crystallites. We discuss below the results of a computer simulation of the competitive crystal growth problem and the relation
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