Modeling the Grain Size Distribution during Solid Phase Crystallization of Silicon
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Modeling the Grain Size Distribution during Solid Phase Crystallization of Silicon Andreas Bill1, Anthony V. Teran1 and Ralf B. Bergmann2 1 Department of Physics & Astronomy, California State University Long Beach, 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 analyze the grain size distribution during solid phase crystallization of Silicon thin films. We use a model developed recently that offers analytical expressions for the timeevolution of the grain size distribution during crystallization of a d-dimensional solid. Contrary to the usual fit of the experimental results with a lognormal distribution, the theory describes the data from basic physical principles such as nucleation and growth processes. The theory allows for a good description of the grain size distribution except for early stages of crystallization. The latter case is expected and discussed. An important outcome of the model is that the distribution at full crystallization is determined by the time-dependence of the nucleation and growth rates of grains. In the case under consideration, the theory leads to an analytical expression that has the form of a lognormal-type distribution for the fully crystallized sample. INTRODUCTION In applications one targets specific functionalities of a material, which are generally defined by the microstructure of the solid. More often than not one relies on empirical knowledge to determine the growth conditions for which the desired microstructure is obtained. This is for example the case in the crystallization of an amorphous solid such as Silicon. It is of interest to be able to determine the microstructure from a more fundamental point of view. One important tool to characterize the micromorphology of a solid during its crystallization is the time-dependent grain size distribution (GSD) N(g,t) , where g defines some physical quantity that measures the size of the grain (diameter of the grain, number of atoms, volume, etc.) [1-6]. N counts the number of grains that have a certain size g at time t during crystallization. Recently we developed a theory describing the evolution of the GSD during the crystallization of a d-dimensional solid ( d = 2 for a thin film, d = 3 for a bulk solid) [5,6]. The theory considers random nucleation and growth (RNG) processes, but the model might be applicable to a variety of growth phenomena both in and outside the realm of solid-state physics. We apply the theory [5,6] to solid phase crystallization of a Si thin film [1-4]. Within the RNG model nuclei are first formed at random in the sample. These nuclei grow while others are created. This leads to a time dependent GSD, which can for example be observed by Transmission Electron Microscopy (TEM). The procedure employed for the evaluation of the cross-sectional TEM samples is described in [3]. In the present application we compare the theoretically determined GSD N(g,t) to the measured one. Here g describ
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