Development of the Grain Size Distribution During the Crystallization of an Amorphous Solid
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Development of the Grain Size Distribution During the Crystallization of an Amorphous Solid Andreas Bill1 and Ralf B. Bergmann2 1 California State University Long Beach, Department of Physics & Astronomy, 1250 Bellflower Blvd., Long Beach, CA 90840, U.S.A. 2 Bremen Institute for Applied Beam Technology (BIAS), Klagenfurter Str. 2, 28359 Bremen, Germany. ABSTRACT We present an overview of the theory developed over the last few years to describe the crystallization of amorphous solids. The microstructure of the crystallizing solid is described in terms of the grain size distribution (GSD). We propose a partial differential equation that captures the physics of crystallization in random nucleation and growth processes. The analytic description is derived for isotropic and anisotropic growth rates and allows for the analysis of different stages of crystallization, from early to full crystallization. We show how the timedependence of effective nucleation and growth rates affect the final distribution. In particular, we demonstrate that for cases described by the Kolmogorov-Avrami-Mehl-Johnson (KAMJ) model applicable to a large class of crystallization processes a lognormal type distribution is obtained at full crystallization. The application of the theory to the crystallization of silicon thin films is discussed. INTRODUCTION Crystallization of a solid is rarely homogeneous in a material. Instead, surfaces or local inhomogeneities often serve for the formation of nuclei and growth into crystalline grains. Depending on the material and the crystallization process, grains over various shapes and sizes form. The resulting microstructure is of importance in applications since it determines to a great extent the physical properties of the material. The magnetism of hard drives, the response of optically active materials, the resistivity of conductors, the stress response to external forces, all depend on the degree to which the material has crystallized and how many grains and grain boundaries per unit volume are found in a sample. To fabricate reliably components with specific functionality thus requires choosing the microstructure of the material. To reach this goal it is of interest to take distance from an empirical determination of the microstructure and rely on a more fundamental understanding of the crystallization process. In many processes the final product is obtained by first growing an amorphous sample, followed by a treatment such as annealing to crystallize the sample. The theory described below and detailed in Refs.[1-4] addresses the second part of sample production (see Fig. 1). The theory provides quantitative information on the crystallization of a solid by determining the timedependent grain size distribution N(g,t) . The distribution gives the number of grains per unit volume of given size found at a certain time during the crystallization of the amorphous solid. In this model g is a quantity that characterizes the size of the grain as, for example, the diameter or radius of the grain, the number of atom
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