Solidification behavior of highly supercooled polycrystalline silicon droplets

Materials Science in Semiconductor Processing 15 (2012) 722–730

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Solidification behavior of highly supercooled polycrystalline silicon droplets Sudesna Roy n, Teiichi Ando Department of Mechanical & Industrial Engineering, Northeastern University, Boston, MA 02115, United States

a r t i c l e in f o

abstract

Available online 17 June 2012

Polycrystalline silicon balls are popularly used for solar cells to lower cost and improve their light collection capability. This article investigates on the microstructure evolution during solidification of polycrystalline silicon. The uniform droplet spray process, a controlled capillary jet break-up process, which enables stringent control of the nucleation and microstructure evolution during solidification of alloy droplets in a thermal spray process, was used to produced mono-sized silicon droplets. The experimental parameters for production of silicon droplets were established and the solidification behavior of silicon droplets was investigated using the modification of the free dendritic growth model and the dendritic fragmentation model. This enabled us to correctly establish the transition supercooling for transformation from lateral growth mode to continuous growth mode to be between 81 K and 172 K. The model was used to predict the microstructure of polycrystalline silicon droplets for solar cells produced by a droplet based manufacturing method, enabling greater control over process parameters and its relation to the final microstructure. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Poly/Multi-crystalline silicon Rapid solidification processing Nucleation kinetics Microstructure evolution Materials characterization

1. Introduction Recently silicon is used in solar cells in the form of polycrystalline spherical balls. While these balls reduce the overall efficiency as compared to single crystal silicon, they increase the surface area of absorption and the light trapping efficiency of the modules due to its spherical shape and moreover, reduce the cost associated with producing single-crystalline wafers [1]. A common problem with production of spherical silicon balls is the deformation of the droplets into a non-spherical and irregular shape due to the volume expansion and formation of twinned and faceted dendrites associated with solidification in silicon [2]. However, it is understood that with the change in the solidification behavior, the n Correspondence to: Department of Mechanical & Industrial Engineering, Northeastern University, 360 Huntington Avenue, Boston, MA 02115, United States. Tel.: þ 1 617 373 2215. E-mail address: [email protected] (S. Roy).

1369-8001/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mssp.2012.05.006

droplets become more spherical and smooth [3,4]. This is mainly attributed to the change in the growth mechanism from lateral, at low supercooling, to continuous, at high supercoolings. The supercooling for the transition has been reported to be  100 K by Nagashio et al. [5] for boron-doped silicon and Aoyama and Kuribayashi [3] and Aoyama et al. [6] for pure silicon. These studies have been explicitly reported on the microstructural evolution in silicon performed by electromagnetic levitation experiments [3–6]. Since they are designed to get a broader picture of the solidification behavior of silicon, there are no studies to correlate the solidification behavior of pure silicon to the actual microstructure obtained in thermal spray processes commonly used to produce polycrystalline silicon balls. Some studies by Zeng et al. [7] and Miller et al. [8] have established the process parameters for obtaining silicon droplets about 1 mm in diameter, but fail to characterize the droplet solidification behavior. The present investigation addresses the conditions for dendritic growth in silicon

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droplets in a droplet based manufacturing process and also evaluates the fragmentation behavior of the dendrites in silicon droplets with interest in finding conditions for the production of spherical silicon balls. The uniform-droplet spray process is used to produce the spherical droplets. The uniform-droplet spray (UDS) process is a dropletbased manufacturing process where mono-sized droplets are generated by the controlled breakup of a laminar jet of molten metal ejected through an orifice while applying a regulated perturbation. Details of this process can be found elsewhere in the original patent filed at MIT by Chun and Passow [9]. The droplets produced have identical size ( 73%) and similar thermal history. Thus thermal state of the traveling droplets can be computed using the Newtonian cooling conditions, namely the in-flight solidification model, and would represent the actual state of the droplets as reported by DiVenuti and Ando [10]. Moreover, experimental parameters in this process are decoupled which makes it accurate and easy to correlate the experimental parameters in the process to the solidification behavior of the droplets. 2. Models The solidification behavior of silicon is quite different and unusual in comparison to that of metals and alloys. The growth mode is reported to change with increasing supercooling, from lateral mode (advancement of interface by formation of ledges, commonly found in solidification of non-metallic materials) to continuous mode (continuous movement of interface, commonly observed in metallic materials) [11]. Direct photographic evidence indeed shows the solidification behavior in moderate and high supercoolings to be of copious nucleation and growth, similar to that of metals [4]. Thus in spite of such anomalous behavior in silicon, the solidification behavior in silicon can be assumed to be similar to that of metals. 2.1. Free dendritic growth model The free dendritic growth model, developed by DiVenuti and Ando [10], is used to compute the dendrite growth characteristics. For the case of pure materials the model, as used in this work, is similar to the BCT model developed by Boettinger et al. [12]. The total supercooling here has three components, which are given as

DT ¼ DT r þ DT t þ DT k

ð1Þ

where DTr is the curvature supercooling, DTt the thermal supercooling and DTk is the kinetic supercooling. These components of supercooling for pure metals are expressed as

DT r ¼ DT k ¼

2G R Vi

m

ð2Þ

The interface velocity is expressed as   DG V i ¼ V o 0 R Ti

723

ð4Þ

where Vo is the maximum crystallization velocity, DG is the free energy across the solid–liquid interface, R0 is the gas constant and Ti is the temperature of the interface. The tip radius of the dendrite is given by the smallest stable perturbation, or the marginally stable wavelength, of the solid–liquid interface. The tip radius for pure metals can be expressed as

G=sn  R¼  P t DH=C p xt

ð5Þ

where sn is the stability constant equal to 1=4p2 , Pt is the thermal Peclet number, DH is the enthalpy and Cp is the specific heat capacity and xt is the thermal stability function given by 1

xt ¼ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð6Þ

1þ 1=sn P2t

2.2. Nucleation kinetics model The initial liquid cooling temperature at which nucleation takes place in the droplet was simulated with the droplet nucleation model by Wu and Ando [13] with the aid of an in-flight droplet cooling model, which is a part of the in-flight solidification model [10]. The droplet nucleation model, with the aid of a few sets of reference data, is used to predict nucleation temperature under any cooling conditions, producing continuous cooling transformation (CCT) diagrams of internal nucleation of droplets. The dendrite growth behavior during the recalescence was predicted by the free dendritic growth model and the calculated growth kinetics were incorporated in the inflight droplet solidification simulation. The expression for the droplet nucleation kinetics for internal heterogeneous nucleation is given as !  Z TN pD3 Mn ðhT þ 1Þ Q N n ðhT þ1Þ3 dt exp  dT ¼ 1 ð7Þ 6 ðT L TÞ2 RT dT TðT l TÞ2 TL Mn ¼

Nn ¼

8pð1cos yÞg2i T 2L Do C s a4o DH2v ð16=3Þpg3 f ðyÞ kB ðDHv =T L Þ2

ð8Þ

ð9Þ

where D is the droplet diameter, t is the time, TN is the nucleation temperature, TL is the liquidus temperature, Q is the activation energy and h is factor that relates the temperature dependence of the solid–liquid interfacial energy, gi, where gi ¼cTþ d and h¼c/d. M* and N* are material specific constants, given by Eqs. (8) and (9), that depend on the alloy properties; such as wetting angle, y, lattice parameter, ao and enthalpy of fusion per unit volume, DHv.

ð3Þ

where R is the radius of the dendritic tip, G is the Gibbs– Thompson coefficient, Vi is the tip velocity and m is the interface kinetic coefficient.

2.3. Fragmentation model The fragmentation behavior of the silicon dendrites was investigated using the fragmentation model by

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Karma [14]. The model is based on the assumption that the dendrites which are formed in the recalescence period fragment in the post-recalescence plateau period if the time for break-up, tbu, is greater than the plateau period, tpl. The dendrite break-up time, tbu, was computed according to the expression [14]

Dt bu ðDTÞ ¼

1 3 RðDTÞ3  oðkmax ; RðDTÞÞ 2 aG

ð10Þ

where R(DT) is the dendrite radius which depends on the supercooling, DT, a is the thermal diffusivity and G is the Gibb’s Thompson parameter. A geometrical parameter, zn, relates the trunk radius to the tip radius of the dendrites by a simple ratio.

The morphology of the droplets was then examined to estimate the nucleation distance. The fully molten droplets were completely deformed due to the rotation of the blades whereas the semi-solid droplets retained a hump like structure indicating an already nucleated droplet. This effectively provides a range where nucleation is probable. The solidified droplets and splats were ultrasonically cleaned in acetone and sectioned and polished and metalographically examined. Their microstructures were revealed by etching with a mixture of HNO3, HF and CH3COOH in the ratio (volume) 2:1:1 by swabbing it with a cotton tip for 5–7 s. 4. Results

3. Materials and method

4.1. Dendrite growth characteristics

Fig. 1 shows the schematic representation of the droplet generator which was used to produce the droplets. High purity silicon wafers (99.9999%) were melted in an alumina crucible lined with a layer of zirconia. Table 1 provides the typical experimental conditions used to produce the droplets. The nucleation distances of the droplets were determined by the splat quenching technique where the droplets were impinged on rotating blades positioned below the orifice. This method for determining the nucleation distance was first used by Bialiauskaya and Ando [15] and later adapted by Roy and Ando [16]. Details of the experimental set-up are provided in Table 1.

Fig. 2 shows the micrographs of the 550 mm and 390 mm droplets. The droplets have uneven and irregular surfaces commonly found in silicon droplets formed by similar rapid quenching methods, due to its volume expansion of 10% on solidification. The internal microstructure of droplets, however, differs with formation of large grained structure with /110S dendrites that often nucleate as twin boundaries in the 550 mm droplet whereas the 390 mm droplet had sharply defined orthogonal /100S dendrites as reported in the levitation studies by Aoyama et al. [6] and later by Nagashio et al. [2]. The /110S dendrites are reported to form as H shaped, where the major growth direction is the direction of the arms [11]. This occurs due to insufficient time for development of tertiary arms while the faceted secondary arms grow rapidly in the lateral direction. The microstructure of the droplets depends widely on their supercooling and hence on the size of the droplets for a given cooling media. It is generally accepted that smaller droplets cool faster and can be supercooled to a higher degree than larger droplets. Thus the smaller 390 mm droplet would supercool to a higher degree than the 550 mm droplet. With increasing supercooling the growth mode changed from lateral to continuous, which resulted in the formation of twinned /110S dendrites in 550 mm but orthogonal /100S dendrites in 390 mm droplets. From the micrographs it is also evident that the grain size increases drastically from 270 mm for 550 mm droplets to 90 mm for 390 mm droplets. The microstructure of 390 mm droplets is microcrystalline which indicates fragmentation that might have occurred in the droplets. The fragmentation behavior of the droplets is thus examined by the fragmentation model and the nucleation temperature of the droplets is experimentally determined by the splat quenching technique and used to compute the nucleation kinetics of the droplets. The data from these two droplet diameters were used to compute the dendrite growth characteristics as explained in detail in Section 2.1. Fig. 3 shows the interface velocity and dendrite tip radius calculated by the free dendritic growth model. As typical observed for pure metals the tip radius decreases monotonically whereas the tip velocity increases with increase in the supercooling.

Piezoelectric transducer Thermocouple Inert gas supply Vibration transducer rod -

Crucible Induction heater

+ Orifice

Droplet charging plates

Fig. 1. Schematic of the UDS droplet generator.

Table 1 Experimental conditions for production of silicon droplets. Parameters Droplet size (mm) Orifice diameter (mm) Melt temperature (K) Frequency of perturbation (kHz) Oxygen content (ppm) Differential pressure (kPa)

Values 550 252 1853 2.89 9.60 193

390 168 1913 3.55 10.50 227

S. Roy, T. Ando / Materials Science in Semiconductor Processing 15 (2012) 722–730

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Fig. 2. Micrographs of silicon droplets of diameters (a) 550 mm and (b) 390 mm, showing their external (top row) and internal features (bottom row). Note differences in scale.

1.00E+03 Tip Velocity (m/s)

Tip Radius (m)

1.00E-04 1.00E-05 1.00E-06 1.00E-07

1.00E+01 1.00E-01 1.00E-03 1.00E-05 1.00E-07

1.00E-08 0

100

200 300 Supercooling (K)

400

0

100

200 300 Supercooling (K)

400

Fig. 3. Computed dendrite characteristics, i.e. (a) tip velocity and (b) tip radius, for silicon.

4.2. Droplet nucleation kinetics Fig. 4 shows the cross-sections of the splats obtained at the flight distances, z of 0.27 m and 0.24 for 550 mm and 390 mm droplets, respectively. The splat morphology indicates that the droplets were in a semi-solid or mushy state when they impinged on the rotating blades, thus indicating that nucleation probably occurred at the given flight distances. The nucleation distance was thus deduced to be 0.2670.01 m for the 550 mm droplets and 0.2370.01 m for the 390 mm droplets. It follows that these droplets nucleate within the respective ranges of flight distances. Kinetics for internal nucleation of molten silicon droplets was investigated using the nucleation model explained in Section 2.3. The nucleation temperatures of the 550 mm and 390 mm droplets were estimated as

TN ¼160678 K for 550 mm droplets and TN ¼1515717 K, for 390 mm droplets. Thus the larger 550 mm droplets have supercooling of 81 K, whereas the smaller 390 mm droplets have a higher supercooling of 172 K. Thus, it is evident that the transition from lateral growth mode to continuous growth model occurs between a supercooling of 81 K and 172 K in agreement with previous publication that reports the transition supercooling to be 100 K [6]. With the experimental nucleation data for 550 mm and 390 mm droplets and their simulated cooling schedules, the material specific constants, Mn and Nn, were calculated using Eq. (7). The values of Mn and Nn represent the potency of nucleant catalysts present in the melt. The potency may depend on the initial melt temperature, if the catalysts are likely to dissolve into the melt at high temperatures. However, the calculation of Mn and Nn values assumes that the catalysts are stable and independent

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Z = 0.27 m

Z = 0.24 m

Fig. 4. Splats obtained by quenching (a) 550 mm droplets and (b) 390 mm droplets nucleating at the given flight distances (z). Table 2 Thermo-physical constants of ASTM F75 alloy used in calculation of tbu. Parameters

Values

Ref.

Heat of fusion (DHf) (J/mole) Specific heat capacity (Cp) (J/mol K) Liquidus temperature (Tl) (K) Thermal diffusivity (a) (m2/s) Gibbs–Thompson coefficient (G) (mK)

50208 25.39 1687 8.5  10  5 1.96  10  7

TCWv2s TCWv2s TCWv2s [17] [17]

Table 3 Parameters used for computing the fragmentation model.

DT (K) Fig. 5. Computed CCT curves for silicon droplets.

of the initial melt temperature. The calculated value of Mn is 3.858  1018 K2 m  3 s  1 and that of Nn is 3.326  107 K3. With these values of Mn and Nn the nucleation temperatures of droplets of any size under any cooling conditions can be calculated using Eq. (7) and used to construct the CCT curves for nucleation in silicon droplets. Fig. 5 shows the CCT curves calculated with the model for 550 mm and 390 mm silicon droplets cooling under various conditions. The droplet nucleation kinetics model was used to predict nucleation temperature for droplets of other diameters. Although the present work did not produce mono-sized droplets finer than 390 mm, some satellite droplets (a 200 mm droplet and a 275 mm droplet) and a triplet of 550 mm droplet (850 mm droplet) were characterized to verify the predictions by the model. This is discussed in detail in Section 5. 4.3. Dendrite fragmentation The observation of multi-grained microstructure in the droplets suggests the possibility of fragmentation of the silicon dendrites, which is commonly observed in alloys and reported earlier [16]. The fragmentation behavior of dendrites of pure materials is not expected to occur in the same manner as that for alloys. This is expected as the crystals growing in the plateau period have very low supercooling of only a few degrees. Hence, the growing crystals would not have the typical dendritic morphology, but rather a mostly planar front. Hence, surface tension of the interface may not act on the same manner as it would

Rtip (m) R (m) zn

550 lm droplets

390 lm droplets

275 lm droplets

200 lm droplets

81 K (expt.) 7.2  10  6 15  10  6 2

172 K (expt.) 1.7  10  7 10  10  6 10

226 K (model) 5.0  10  8 10  10  6 50

320 K (model) 7.74  10  8 10  10  6 75

for alloys where the dendrites have a typical tree-like morphology with narrow dendrite trunks where fragmentation is expected. The fragmentation behavior of silicon was investigated with the help of a fragmentation model and compared with the metallographically obtained results. The thermo-physical constants used for the calculation are listed in Table 2. The driving force for fragmentation of pure silicon dendrites must necessarily be the capillary force of the solid–liquid interface. Table 3 shows the values of the geometrical parameter, zn calculated for each of the droplets. The ratio of the trunk radius, R(DT), estimated from the actual microstructures and tip radius, Rtip(DT) is calculated using the free dendritic growth model, as shown in Fig. 3(a). The expected coarsening of the dendrites in the last stages of solidification may produce some variation in the calculated value of zn. The coarsening is prominent in pure materials which produce nearly same trunk radius for all the solidified droplets. Fig. 6 shows the computed tbu vs. supercooling curves for 850 mm, 550 mm, 275 mm and 200 mm droplets along with their respective Dtpl vs. supercooling curves. The predictions made using the model are compared to the microstructures of the droplets, which are collectively shown in Table 4. The fragmentation model predicts no fragmentation for both 850 mm and 550 mm droplets at the determined supercoolings of 42 K and 81 K, respectively. This is verified by the microstructure of the

S. Roy, T. Ando / Materials Science in Semiconductor Processing 15 (2012) 722–730

Fig. 6. Computed tbu vs. supercooling curves for various droplets of diameter 850 mm to 200 mm.

droplets (Table 4) which essentially shows large grains with occurrence of elongated, twinned /110S dendrite. However, for 390 mm, 275 mm and 200 mm droplets with supercoolings of 172 K, 226 K and 320 K, respectively, the model predict fragmented dendrites. The respective micrographs, shown in Table 4, show microcrystalline microstructures indicative of fragmentation of dendrites. 5. Discussions Table 4 collectively shows the cross-sections of the droplets ranging in sizes from 850 mm to 200 mm. The 850 mm droplet with a supercooling of 42 K shows a twinned microstructure with /110S dendrites and stepped grain boundaries characteristic of lateral growth of the nucleating dendrites. However, the 200 mm and 275 mm droplets, both show a multi-grained microstructure with traces of /100S dendrites within the droplets. In these small droplets, probably a single event of nucleation produced subsequent /100S growth. As the /100S dendrites grew, some of the arms detached produced the multi-grained microstructure. The smaller 200 mm droplet, however, had fewer grains, indicating that more marginal conditions for fragmentation applied to this droplet. The supercoolings predicted by the nucleation kinetics model were 226 K for 275 mm droplet and 320 K for the 200 mm droplet. At these high supercoolings, previous studies [17,18] have also reported formation of /100S orthogonal dendrites. However, a more rigorous verification of the actual numerical values from the CCT curves was not possible due to the height and temperature limitations of the present UDS apparatus. The micrographs of the 390 mm, 275 mm and 200 mm droplets, in Table 4, shows growth striations in the grains which indicate direction of the migrating interface in the droplets [11,19]. This is, however, absent in both 550 mm and 850 mm droplets. The growth striations are usually found in Ge and Si crystals grown by the Czochralski method where the striations are believed to occur due to the presence of oxygen interstitials along the growth directions [19]. However, at lower interface velocities

727

the grains are reported to be free of striations [20] as observed for the 550 mm droplets. Using the experimentally determined supercooling of 81 K for 550 mm droplets and 172 K for 390 mm droplets, the transition interface velocities (from Fig. 3(b)) may be estimated to be between 2 m/s and 13 m/s, respectively. Similar striations have also been observed in electromagnetically levitated silicon splats supercooled to above 100 K, as explained in Section 1. The Dtpl vs. supercooling line, as shown on the plots in Fig. 6, are almost straight lines with a small slope. This is due to the high latent heat of fusion of which makes the amount of liquid left in the droplet at the end of recalescence very large, unless the supercooling is comparable to the hypercooling limit, DThyp (1977 K for silicon). Thus, a supercooling of 400 K is about 0.2  DThyp, which leaves 80% of the droplet volume still molten at the end of recalescence. Since the solidification rate in the post recalescence period is limited by the rate of heat extraction, droplets with low to moderate supercoolings would all have similar values of Dtpl. Although the micrographs of droplets diameter 390– 200 mm have microcrystalline microstructure indicative of fragmentation, the micrographs themselves show well developed orthogonal dendrites. The fragmentation is thus not clearly discernable in these micrographs probably due to the marginal conditions of fragmentation in these droplets. Fig. 7 shows another such 390 mm droplet and 200 mm droplet that more clearly shows fragmented dendrites. The cross-like features (highlighted) are observed to be fragments of orthogonal /100S dendrites. Nagashio et al. [17] have reported on fragmentation of dendrites occurring at a supercooling 261 K, where crosslike dendrite fragments were observed to detach in the initial post-recalescence plateau period. Thus, the microstructure does confirm the occurrence of fragmentation in 390 mm droplets as predicted by the model. The chance of finding a fragmented dendrite in these droplets depends on the direction of cross-sectioning the droplets and the orientation of the dendrites in the droplets. It is also expected that the number of fragmented dendrites are very few in the smaller 200 mm droplets, which makes it difficult to accurately pin-point the location of fragmented dendrites in the droplets. However, fragmentation itself is considered to have occurred in all droplets r390 mm regardless of the sectioning and orientation, as predicted by the model and metallographically verified. Although fragmentation is ascertained to occur in pure silicon droplets when sufficiently supercooled Z172 K, the fragmentation is not as extensive as is commonly observed in alloy droplets [16]. This may be due to the rapid growth of the planar dendritic front in the postrecalescence stage which does not allow for complete fragmentation with formation of fine equiaxed grains. The fragmentation of thermal dendrites in pure materials in the post-recalescence plateau period would only occur in the very initial stage of the plateau period when the dendrites are characterized by the typical dendritic morphology. This was in fact verified by Ngashio et al. where the dendrites in large droplets of 1.8 mm diameter fragmented in the initial 25 ms of the plateau period ( 2 s),

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Table 4 Comprehensive tabulation of characteristics of silicon droplets. Droplet size

Microstructure Low magnification

Supercooling Dendrite fragmentation High magnification

850 mm

42 K

Twinned /110S dendrites with no fragmentation

550 mm

81 K

Twinned /110S dendrites with no fragmentation

390 mm

172 K

Orthogonal /100S dendrites with fragmentation

275 mm

226 K

Orthogonal /100S dendrites with fragmentation

200 mm

320 K

Orthogonal /100S dendrites with fragmentation

i.e., the time for break-up was 1.25% of the plateau period [17]. Thus, when the remaining liquid solidifies, controlled mainly by the extraction of heat, the dendrites

would lose their morphology and grow with a planar front. This is in fact observed in the microstructures of the droplets where the grains with /100S orthogonal

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Fig. 7. Micrographs showing metallographic evidence of fragmentation in silicon droplets. (a) Micrographs of 390 mm indicating fragmented dendrites. (b) Optical micrographs of 200 mm droplet showing dendrite fragments.

dendrites have a planar grain boundaries clearly formed in the post-recalescence plateau period. Moreover, solute segregation which is prominent in alloys and absent in pure materials also encourages rapid growth of dendrites formed in the recalescence stage, so that the dendrites lose their typical tree-like morphology. Thus as observed in the micrographs, the dendrite fragments, although oriented differently, remain in the same grain. This is clearly shown for the 200 mm droplet in Fig. 7(b), where fragmentation does not result in grain refinement and only 2–3 grains are found in each droplet. However, the maximum supercooling observed for silicon (DThyp ¼1977 K) droplets is only 0.16  DThyp. Thus, droplets with sufficiently higher supercooling would probably show more extensive fragmentation of dendrites. Similar studies by Devaud and Turnbull [21] and Lau and Kui [22] have observed equiaxed microcrystalline grains due to extensive fragmentation of dendrites in germanium droplets (300–700 mm diameter) only when supercooled above 300 K (DThyp  1200 K). 6. Conclusions The nucleation kinetics model with the help of a few experimentally determined nucleation data was used to construct a CCT curve that can be used to calculate the nucleation temperature for any droplet size with any cooling condition. The supercooling in 550 mm droplet with twinned /110S dendrites was found to be around

81 K whereas the supercooling in 390 mm droplet with orthogonal /100S dendrites was found to be  172 K. The transition supercooling for transformation from lateral growth mode to continuous growth mode occurs in between 81 K and 172 K. A droplet size of o400 mm diameter when quenched in helium gas would result in formation of orthogonal /100S dendrites with microcrystalline microstructure. Direct metallographic evidence and application of the fragmentation model confirms the fragmentation of dendrites occurring in a pure material. However, complete fragmentation of the dendrites, at low to moderate supercoolings, was not observed probably due to the rapid growth of the planar front in the post-recalescence plateau period which results in loss of the typical tree-like morphology of the dendrites. Results from the free dendritic growth model and the metallographic evidence both confirm the presence of skinny thermal dendrites that partially fragment in the initial stages of the post-recalescence period. The use of these droplets in a solar module has yet to be accomplished. Silicon is doped with either a pentavalent or trivalent element to form p-doped or n-doped silicon which is used in these solar cell modules. The doping concentration, however, when in the ppm level will not cause the microstructure to be different from that of pure silicon. It can thus be modeled as a pure element. However, if significant quantity of the doping element needs to be added, it then has be considered as an alloy and modeled accordingly. It is thus expected that silicon

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doped in the ppm level, which is usually the case, will follow the same solidification and fragmentation behavior as pure silicon and can be modeled using the same experimental parameters.

Acknowledgment The authors thank Fukuda Metal Foil & Powder Co. Ltd., Kyoto, Japan for funding and supporting this work. References [1] T. Maruyama, H. Minami, Solar Energy Materials and Solar Cells 79 (2) (2003) 113–124. [2] K. Nagashio, H. Murata, K. Kuribayashi, Acta Materialia 52 (2004) 5295–5301. [3] T. Aoyama, K. Kuribayashi, Materials Science & Engineering A 304–306 (2001) 231–234. [4] Z. Jian, K. Nagashio, K. Kuribayashi, Metallurgical & Materials Transactions A 33A (2002) 2947–2953. [5] K. Nagashio, H. Okamoto, K. Kuribayashi, I. Jimbo, Metallurgical & Materials Transactions A 36A (2005) 3407–3413. [6] T. Aoyama, Y. Takamura, K. Kuribayashi, Metallurgical & Materials Transactions A 30 (1999) 1333–1339. [7] X. Zeng, E.J. Lavernia, J.M. Schoenung, Scripta Metallurgica et Materialia 32 (8) (1995) 1203–1208.

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