Noncoarsening Origin of Logarithmic-Normal Size Distributions during Crystallization of Amorphous Thin Films

Abstract

It is demonstrated that the formation of logarithmic-normal crystallite size distributions during crystallization of amorphous thin films can be a consequence of the dynamics of random nucleation and growth without the involvement of coarsening. Our analytical result is supported by experimentally observed crystallite size distributions during the crystallization of amorphous Si thin films obtained from transmission electron microscopy. [S0031-9007(97)05141-7] The phase transformation from the amorphous and liquid to the crystalline state, leading to the formation of polycrystalline materials, is of fundamental scientific and technological importance. As an example, the transition from an amorphous to a polycrystalline semiconducting film upon annealing is essential for processing electronic materials, such as thin film transistors [1,2] and solar energy converters [3]. Other examples are the formation of polycrystalline bulk Si material solidified via supercooling of a Si melt [4] or the deposition of polycrystalline Si films from the vapor phase [5]. All of these processes establish nonmonotonic crystallite size distributions that can be well approximated by a logarithmic-normal (log-normal) distribution [4]. The size distribution within a polycrystalline semiconductor material can have a significant impact on device performance. This is especially true for minority carrier devices with highly recombination-active grain boundaries [6].A fundamental understanding of the transition from an amorphous phase to the crystalline state is still far from complete. Previous studies concerned themselves merely with the understanding of the earliest stages of nucleation and growth. In this case, the amorphous phase can be treated to be infinite; size distributions thus monotonically decrease with grain size [7]. The depletion of the amorphous phase, however, significantly influences the dynamics of nucleation and growth and determines the subsequent development of the size distribution.In this Letter, we describe the time evolution of the cluster size distribution beyond the early stages of nucleation and growth. We demonstrate that the interplay between the dynamics of nucleation and growth results in a transition from monotonic size distributions in the early stages to nonmonotonic size distributions as a function of the grain radius r in the later stages similar to the log-normal distribution L͑ln r͒~exp͓2͑ln r 2 m͒ 2 ͞2s 2 ͔, where m and s represent the median and width of the distribution, respectively [8]. Our results thus reveal an alternative origin for the development of lognormal size distributions. This recognition is essential, since the formation of log-normal size distributions have generally been attributed to the coarsening process [9,10]. The physical concept described here has recently been proposed by the authors on an intuitive basis [11], and is here analytically established for the first time.The dynamic evolution of the crystallite size distribution in the early stages of random nucleation and gro...

On the origin of logarithmic-normal distributions: An analytical derivation, and its application to nucleation and growth processes

“…Lognormal size distributions in phase transformations have conventionally been attributed to coarsening [6,7]. Our earlier work [8], however, pointed out that such distributions might occur due to time dependent kinetics of nucleation and growth, without the involvement of coarsening. Literature on nucleation describes the initial or early stages of nucleation [9,10] and there are a number of derivations of nucleation rates for these early stages, see eg.…”

“…The corresponding parameter v 0 = 1.72 ± 0.34 µm/h is taken from Ref. [8], which also states the crystallization parameters t 0 = 1.5 ± 0.5h and t c = 4 ± 0.5h, necessary for Eq. (8), using d = 2 for amorphous Si films solid phase crystallized at 600…”

Alternative Origin of Log-Normal Size Distributions of Crystallites in Controlled Solid-Phase Crystallization of Amorphous Si Thin Films

“…The intrinsic depletion of the nucleation sites during the nucleation and growth leads naturally to a nonlinear effect in the subsequent evolution of the CSD. Although it was suggested that the interplay between the dynamics of nucleation and growth could give rise to a log-normal CSD [8,9], no definite experimental evidence has been obtained excluding involvement of coarsening of the crystallites in the formation process.…”

Comment on “Noncoarsening Origin of Logarithmic-Normal Size Distributions during Crystallization of Amorphous Thin Films”

“…Bergmann, Shi, and Krinke [1] (BSK) use an analytical description of nucleation and growth to explain the observed grain size distribution in an amorphous Si film. It is shown below, however, that the proposed description does not produce the known limit of transient nucleation which arises for a negligible crystallization fraction, X͑t͒ (and which corresponds to t c !`in Eq.…”

“…(4) of BSK should have a different form [5(b)] (again, Bergmann, Shi, and Krinke [1] miss a large logarithmic term). On the other hand, the reported experimental depletion of 2.8% is remarkably small.…”