Formation of polycrystalline silicon with log-normal grain size distribution

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Applied Surface Science 123/124 (1998) 376 380

Formation of polycrystalline silicon with log-normal grain size distribution Ralf B. Bergmann a,*, Frank O. Shi a,1, Hans J. Queisser a, JiSrg Krinke b ~ Max-Planck-lnstitut fiir Festki~rperforschung, Heisenbergstr. l, D-70569 Stuttgart, Germany " lnstitutftir Werkstoffwissenschaften VII, Unicersitiit Erlangen, Cauerstr. 6, D-91058 Erlangen, Germany

Abstract Polycrystalline silicon films, prepared by annealing from amorphous precursors, are analyzed by transmission electron microscopy and reveal a logarithmic-normal distribution of grain sizes. Such size distributions also result from various other crystallization processes from non-crystalline phases. The cessation of nucleation due to the finite amorphous reservoir leads to these logarithmic-normal size distributions. The origin of these observed distributions is a result of nucleation and growth, rather than coarsening of crystallites. © 1998 Elsevier Science B.V. PACS: 64.60.Qb; 64.70.Kb; 81.10Jt; 81.30.Hd

The phase transformation from non-crystalline to crystalline state, leading to polycrystalline semiconductors, is of fundamental scientific importance. The phase transition from an amorphous to a polycrystalline semiconducting film, stimulated by annealing, e.g., is essential for electronic materials, such as thin film transistors [1] or solar energy converters [2,3]. Also, the formation of polycrystalline bulk Si material solidified via supercooling of a Si melt [4,5] or the deposition of polycrystalline Si films from the vapor [6] all lead to the formation of l o g a r i t h m i c normal ( l o g - n o r m a l ) grain size distributions. Optimal performance requires maximized grain sizes [ 1,2] especially for those semiconducting materials where grain boundaries enhance recombination and trap* Corresponding author. Tel.: +49-711-6891606: fax: +49711-6891010" e-mail: [email protected]. f On leave from the University of California, School of Engineering, Irvine, CA 92697-2575, USA.

ping, hence reduce carrier lifetimes and mobilities. Silicon is in this category. Understanding of transitions from an amorphous reservoir to the polycrystalline state is incomplete. Most studies concern themselves just with optimization of the average grain size (for recent reviews, see, Refs. [7,8]) [9]. We showed that this average depends on the free-energy barrier to nucleation, which itself is determined by the ratio of the growth rate of crystallites and the random nucleation rate [10,11]. Size distributions, however, received hardly any attention [12]. Such neglect is serious: First, the electronic properties, such as carrier diffusion lengths, directly depend on the size distribution [6]. Secondly, size distributions in precipitation reactions and many other phase transformations are frequently assessed as being asymptotically time-invariant and l o g - n o r m a l l y distributed [13]. There seem to be no a priori theories to establish conditions for the appearance of l o g - n o r m a l distributions.

0169-4332/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0169-4332(97)00494-7

R.B. Bergmann et al. / Applied Surface Science 123 / 124 (1998) 376-380

We present, for the first time, both transmission electron microscopy (TEM) data and principles for the development of log-normally distributed poly-Si. The reason for such log-normal behavior is the cessation to nucleate crystallization, caused by a depletion of nucleation sites. Our concept explains the development of log-normal size distributions that is not limited to the formation of ceramics [14]. We propose a route to log-normal distributions, which is generally valid for many phase transitions in nature and technology. Solid phase crystallization of amorphous Si (a-Si) films allows one to monitor the development of the size distribution by TEM at various stages of crystallization. We crystallize 1 ~ m thick a-Si films at 600°C; experimental details in Ref. [15]. Computer aided image analysis determines the grain size distribution by measuring the individual areas of about 1000 grains using TEM plan-view micrographs [3]. The equivalent diameter g of a grain relates to grain area A via g = 2~A-/~'. Fig. 1 gives the grain size distribution of a fully crystallized Si film. This distribution is well approximated by the normalized lognormal distribution [ 13]

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Fig. 1. Normalized grain size distribution f ( g , t ) / n ( t ) (bars) of a fully crystallized Si film after crystallization at 600°C for t - 9 h obtained from n(t) - 1050 grains. Solid line shows fit of l o g - n o r mal distribution A ( g ) to experimental data.

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is a fit of the log-normal distribution A(g) to the observed normalized grain size distribution f(g,t)/n(t) with n(t) being the number of grains that form the distribution after a crystallization time t=9h. To explain the occurrence of such log-normal grain size distributions, we divide nucleation and growth of crystallites is into three stages: (i) initial nucleation and growth of crystallites, (ii) depletion of nucleation sites with concurrent growth of crystallites until nucleation terminates, and (iii) asymptotic attainment of a steady-state distribution. A complete picture for the evolution of the size distribution in the early stages of crystallization (stage i), i.e., before the cessation of nucleation due to the depletion of nucleation sites, has recently been obtained [16]. Fig. 2a shows the evolution of the distribution f(g,t)/n(t) of crystallites in the early stages of nucleation and growth. Note that in the early stages, at times prior to the depletion of nucleation sites, the distribution is a monotonic function of crystallite size: The crystallite concentration f(g,t)/n(t) decreases with increasing size; this decrease is steeper for grains smaller than the critical size g , , as compared to grains exceeding g . . The evolution of Fig. 2a holds for early stages, i.e., before the depletion of nucleation sites; this behavior has been experimentally verified in the crystallization of a-Si thin films [11]. The distribution of grains with a size g >> g, already resembles a log-normal distribution at this stage. Once the fraction of crystalline Si is no more negligible compared to the amorphous phase (stage ii), the nucleation of Si crystallites in a-Si diminishes. Once the amorphous phase is essentially depleted, the size distribution of crystallites (as in Fig. 2a) will be essentially preserved for grains with a diameter g > g . + A. Here, A stands for the width of the nucleation barrier layer, (inset, Fig. 2b), defined as the cluster size region, in which the cluster formation energy W(g) deviates less or equal than kT from the maximum cluster size energy W *, i.e., ] W ( g ) - W*[ _< kT [16]. The smallest size then is determined by the size of critical nuclei. This critical size g , is given by the maximal W * of the cluster formation energy W(g). The examination of the evolution of the size distribution after the nucleation has ceased, requires

R.B. Bergmann et al. /Applied Surface Science 123/124 (1998) 376-380

378

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of critical nuclei. The nucleation barrier layer is critical for the formation of the small-sized part of the distribution because none of the clusters with g < g , - d will survive to grow into large crystallites, and half of crystalline clusters of size g, will eventually grow into large and stable sizes, while all of the clusters of the size g > g , + A are stable and will develop into large ones. Thus, a rapid transition results from almost zero concentration at g = g , - A to a high concentration at g = g , + A in the size range of 2A, which is very small. Hence, the size distribution of stable crystallites that will further grow into large ones by acquiring Si atoms can be represented as that in Fig. 2b. The following evolution is ruled by further growth of crystallites from the amorphous phase. If the growth rate is identical for all crystallites, the shape of the distribution, as depicted in Fig. 2b will be time-invariant until the complete depletion of the amorphous phase. The growth rate d g/dt, however, is size dependent according to [16]

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GRAIN SIZE g Fig. 2. (a) Development of the normalized time dependent grain size distribution f(g,t)/n(t) before the depletion of nucleation sites, g , represents the critical grain size. (b) Grain size distribution during depletion of nucleation sites. The inset qualitatively shows the cluster formation energy W(g) and the resulting critical grain size g . together with the width of the nucleation barrier layer A. (c) Change of the grain size distribution due to the size dependent growth rate of grains. Schematic drawings.

further study of the shape of the distribution of the small sized grains. The transition from a very small number concentration to a large concentration arises in an exceedingly narrow size range, because the transition occurs in the nucleation barrier layer, which is very narrow, as seen in Fig. 2b. The width of the nucleation barrier layer is much smaller than the size

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Here, ~- stands for the characteristic time for the onset of nucleation, while /z is the driving force for the crystallization determined by the difference of the chemical potential between the amorphous and the crystalline phase. Eq. (2) indicates that the growth rate becomes size independent for crystallites larger than about 5 g , . Thus, depending on the duration of the nucleation, only the shape of the distribution for those crystallites with a size g < 5 g , , will be further altered, as depicted in Fig. 2c. Once the entire distribution moves beyond 5 g , , the shape of the size distribution becomes invariant until the complete depletion of the amorphous phase (stage iii). We thus present a framework to understand the evolution of size distributions beyond early crystallization stages. These results complement our early work on the dynamic evolution of the size distribution of crystallites in the early stages of nucleation and growth [16]. The development of the log-normal distribution starts as soon as the nucleation stops due to depletion of nucleation sites. This exhaustion may happen at a very early stage, when the crystallized volume fraction is still small. The cessation of nucle-

R.B. Bergmann et al. / Applied Surface Science 123/124 (1998) 376-380

ation can happen due to (1) the depletion of nucleation sites as discussded in detail in Ref. [16]; and (2) the dynamics of nucleation itself as discussed in Ref. [17]: A decrease in the number of nucleation sites can lead to a dramatic influence on the dynam-

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Fig. 3. Grain size distributions f ( g , t ) / n ( t ) of 1 /zm thick Si films crystallized at 600°C as determined from plan-view TEM. Crystallite size distribution after a crystallization time of: (a) t = 5 h, 20% of the film crystallized, size distribution of fully crystallized films after (b) t = 9 h (same distribution shown as linear plot in Fig. 1), and (c) t = 15 h.

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ics of nucleation. Depending on the rate of decrease in the number of nucleation sites, nucleation can be fully suppressed well before the complete exhaustion of nucleation sites. Our model is supported by our TEM investigations of the size distributions found in solid phase crystallized Si films. Fig. 3a-c show grain size distributions of partly and fully crystallized films. The use of a double-log scale reveals the developvisualizes the size increase of the large grains. Fig. 3a presents the distribution at a relatively early stage (t = 5 h). Only 20% of the film is crystallized at this stage, however, the population of the small sized grains is already decreasing. During further growth, small grains increase in size, and the distribution is depleted from small grains, see Fig. 3b (compare Fig. 2c). At this stage, after crystallization for t = 9 h, the film is fully crystallized. The consumption of small grains by larger grains [18], leads to a further depletion of small grains; the distribution finally reaches its steady state, see Fig. 3c. In conclusion, we present both evidence from transmission electron microscopy and theoretical arguments to demonstrate that the size distribution of crystallites in polycrystalline silicon films prepared from crystallization of their amorphous precursors are log-normal. This finding results from the depletion of the amorphous phase, and the continuing growth of individual crystallites. The distribution is always highly skewed as there is an almost abrupt change in the concentration for smaller crystallites, and a slow variation of the concentration for large crystallites. Our analysis gives further evidence of the generality of the log-normal distribution. Further investigations will address quantitative solutions of the time evolution up to the development of a stationary distribution. Such quantitative evaluations will establish a firm basis for the prediction and tailoring of size distributions in a large variety of material systems.

Acknowledgements F.G.S. thanks H.J. Queisser for providing a visiting fellowship and the hospitality he received from R.B. Bergmann and other friends at MPI during his

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R.B. Bergmann et al. / Applied Surjace Science 123/124 (1998) 376 380

visit. We thank H.P. Strunk and J.H. Werner for their support. Parts of this work were supported by the German Ministry for Education, Science, Research and Technology (BMBF).

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