JOURNAL OF APPLIED PHYSICS 107, 013511 共2010兲
Relationship between grain boundary structures in Si multicrystals and generation of dislocations during crystal growth Noritaka Usami,a兲 Ryusuke Yokoyama, Isao Takahashi, Kentaro Kutsukake, Kozo Fujiwara, and Kazuo Nakajima Institute for Materials Research (IMR), Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan
共Received 14 September 2009; accepted 19 November 2009; published online 7 January 2010兲 We attempted to clarify relationship between grain boundary structures in Si multicrystals and generation of dislocations during crystal growth. Systematic variation of grain boundary structures was realized by employing dendritic nucleation at the initial stage of crystal growth. Etch-pit observation revealed that the contact angle of adjacent dendrite crystals to form a grain boundary affects generation of dislocations. Experimentally observed dislocation density was found to be well correlated with shear stress around the grain boundary calculated by finite element analysis. © 2010 American Institute of Physics. 关doi:10.1063/1.3276219兴 I. INTRODUCTION
Si multicrystals are widely used as a starting material for solar cells due to the relatively low production cost and high performance.1 However, further reduction in costs per watt peak is mandatory for global deployment of photovoltaic technology. “Defect engineering” in Si multicrystals2–8 has been regarded as one of the promising approaches to reach the goal. Especially, crystal growth technique to manipulate defects has been intensively pursued for realization of highquality multicrystalline Si ingot.9–12 Recently, Fujiwara et al. developed “dentritic casting method,” which includes dendritic nucleation of controlled crystallographic orientation at the bottom of the crucible and subsequent directional solidification.9 As a result, Si multicrystals with large crystal grain size and controlled crystallographic orientation have been realized. If such a sophisticated control has not been made, the melt growth of Si multicrystals is started with inhomogeneous nucleation at the bottom of the crucible. Resultantly, randomly oriented multicrystals with random grain boundaries are formed at the initial stage. During subsequent directional solidification, dislocations are generated from random grain boundaries, and the dislocation density is dependent on microstructures in multicrystals.13–16 Although various mechanisms of generation of dislocations exist including condensation of point defects, this implies that the dislocation density could be altered by controlling grain boundary structures formed at the initial stage of crystal growth. In order to verify the concept to manipulate dislocations by controlling grain boundary structures and develop a crystal growth technology to realize high-quality Si multicrystals, relationship between grain boundary structures and dislocation density must be understood. For this purpose, systematic change in grain boundary structures is necessary to simplify the problem. It would be ideal if the method is compatible with practical crystal growth technology to faciliAuthor to whom correspondence should be addressed. Tel.: ⫹81 22 215 2014. Electronic mail:
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a兲
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tate implementation of the fundamental knowledge. Therefore, we attempted to utilize dendritic nucleation in order to systematically change grain boundary structures in Si multicrystals formed at the initial stage of the crystal growth. The dendrite crystals contain parallel twins with 兵111其 grain boundary planes in the primary arm, and their preferential orientation is limited to either 具110典 or 具112典 by the amount of supercooling.17–20 As a consequence, the upper plane of the dendrite crystal is controlled to be either 兵112其 or 兵110其. Therefore, the contact angle between adjacent dendrite crystals can be employed as a structural parameter to describe variation in the grain boundary structures while controlling the upper planes, as illustrated in Fig. 1. In this research, we investigated the relationship between the grain boundary structures and the dislocation density. Etch-pit observation revealed that the contact angle between adjacent dendrite crystals affects generation of dislocations. Physics behind will be discussed based on comparison between experimentally observed dislocation density and calculated shear stress around the grain boundary. II. EXPERIMENTS
Si multicrystals were grown using a Bridgman-type vertical furnace. Quartz crucibles with an inner diameter of 70–
FIG. 1. Definition of the contact angle, , between two adjacent dendrite crystals.
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FIG. 2. 共Color兲 A typical result of orientation distribution in Si multicrystals cut perpendicular to the growth direction. The multicrystalline structure is formed mainly by four dendrite crystals, and their main arms are shown in dotted lines.
100 mm were utilized. To prevent the ingot from sticking to the crucible, the inner wall of the crucible was coated with a Si3N4 layer. The electronic grade feedstock was utilized to minimize the impact of impurities. The amount of the feedstock was chosen to be 350–700 g depending on the size of the crucible. After forming the Si melt, vertical temperature gradient was prepared to permit directional solidification, and crystal growth was started with dendritic nucleation by establishing supercooling larger than 10 ° C.21 Subsequently, temperature was cooled down at a typical rate of −0.5 ° C / min. The total growth time before all the melt are solidified was within 200 min. The grown crystal was cut either parallel or perpendicular to the growth direction. Distribution of crystal orientations and grain boundary characters was investigated using the electron backscattering diffraction pattern 共EBSP兲 analysis. Dislocation density was investigated by counting etch pits formed by the Sopori etching after mechanical and chemical polishing.22 III. RESULTS A. Dislocation density around grain boundaries
Figure 2 shows a typical result of orientation distribution in Si multicrystals cut perpendicular to the growth direction at the height of 6.3 mm. The sample contains four large crystal grains. The dotted lines are added to show the main arm of dendrite crystals as a guide to the eyes. The existence of parallel twins helped us to assign that they originate from dendrite crystals. As shown in Table I, crystal orientations of the upper planes in these crystal grains were found to be close to either 具110典 or 具112典. The existence of parallel twins and orientations of these crystal grains manifest that these originate from dendrite crystals. Although small deviations
FIG. 3. 共Color兲 Pictures of Si multicrystals to have a grain boundary with different contact angles of 共a兲 2°, 共b兲 63°, and 共c兲 73°. It is seen that the increase in the contact angle results in the increase in etch pits.
from exact 具110典 or 具112典 exist, we express orientations of upper planes of dendrite crystals as 具110典 or 具112典 in the following discussions. Figure 3 shows the representative pictures to show the impact of the contact angle on the dislocation density. It is noted that the samples were obtained from different ingots but almost the same height between 5.0 and 7.6 mm. In addition, straight grain boundaries were chosen to exclude impact of the shape of the grain boundary plane. It is obvious that the decrease in the contact angle leads to the decrease in the dislocation density. Results of 13 grain boundaries formed by dendrite crystals using 6 different ingots are summarized in Fig. 4. Three combinations of the upper planes, 具110典 versus 具110典, 具110典 versus 具112典, and 具112典 versus 具112典 are plotted in the same figure. Two symbols plotted on the top axis mean that dislocation density is too large and the exact evaluation was not possible due to the overlap of plural pits. Figure 4 suggests that parallel contact of dendrite crystals is preferable to decrease dislocations irrespective of the orientation of the upper plane. Furthermore, we investigated change in the dislocation density around grain boundaries with different contact angles of 2°, 8°, and 73° during directional solidification. For this purpose, samples sliced parallel to the growth direction were used. Figure 5 shows the spatial distribution of dislocation density along the growth direction. When the contact angle is 73°, dislocation density increases as crystal growth proceeds. On the other hand, dislocation density was kept low when
TABLE I. Results of orientation analysis of crystal grains in Si multicrystals shown in Fig. 2.
1 2 3 4
Orientation of the upper plane
具110典 or 具112典
Deviation 共deg兲
具13, 2 , 17典 具9 , 6 , 23典 具13, 7 , 26典 具19, 1 , 21典
具110典 具112典 具112典 具110典
9.3 11.5 10.6 3.5
FIG. 4. Relationship between the dislocation density and the contact angle of two adjacent dendrite crystals to form a grain boundary.
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FIG. 5. Spatial distribution of dislocation density along the growth direction. It is seen that the dislocation density is kept low in the entire range when the contact angle is small.
the contact angle is 2°. This suggests that the reduction in the contact angle is useful to suppress generation of dislocations during directional solidification. B. Finite element analysis of shear stress
The experimental results supported that dislocation density in Si multicrystals strongly depends on the grain boundary structures. In this section, we attempt to interpret relationship between them. Several mechanisms are known to exist to explain generation of dislocations. One is based on modification of local grain boundary structures during crystal growth to minimize grain boundary energy, which accompanies generation of dislocations. The other is the introduction of shear stress to slip planes, leading to plastic deformation. In the following discussions, we focus the latter since the introduction of relatively large shear stress could be expected owing to the large volume expansion of Si on crystallization as well as some possible temperature variations. The local shear stress around a grain boundary is considered to be dependent on the relative orientations to construct the grain boundary owing to the anisotropic elastic constants in Si.23 We speculated that this would explain inhomogeneous distribution of dislocations in Si multicrystals. To confirm the hypothesis, we started with three-dimensional finite element analysis of stress distribution in bicrystals shown in Fig. 6 as a simple model for multicrystals formed by two dendrite crystals. The model bicrystal is a cylinder with a diameter of 10 cm and a height of 1 cm to consist of two crystal grains. It is noted that this model corresponds to the Si crystal, which grows in a crucible along the z direc-
FIG. 6. 共Color兲 Illustration of a simple model bicrystal for finite element analysis.
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FIG. 7. 共Color兲 Calculated maximum shear stress around grain boundaries with 共a兲 具110典 vs 具110典, 共b兲 具110典 vs 具112典, and 共c兲 具112典 vs 具112典 as a function of the rotation angles, ␣ and , to define the model bicrystal.
tion. The orientations of upper planes are chosen as 具110典 versus 具110典, 具110典 versus 具112典, and 具112典 versus 具112典, which corresponds to experiments. The rotation angles of each dendrite crystal, ␣ and , define the character of the grain boundary, which is related to the experimentally observed contact angle as will be discussed later. Generally, five parameters are needed to describe the geometry of a grain boundary.24 The three are to describe relative orientations of adjacent grains and the rest are to describe the grain boundary plane. It is noted that the grain boundary plane was assumed to be flat in the model. The model bicrystal is treated as continuous elastic material and the change in orientations of each crystal grain was represented by change in the elastic modulus. Therefore, the grain boundary is defined by discontinuous change in the elastic modulus, and atomistic bonds are not apparently considered. As a boundary condition, 0.01% displacement to the center was given to all the nodes located at the edge. This assumes external compressive force given to the crystal. Calculation was done using a commercial software ANSYS. After calculation, the maximum shear stress around the grain boundary was adopted among 12 values originating from eight equivalent 兵111其 slip planes and three equivalent 具110典 orientations. Figure 7 compares contour plots of the calculated maximum shear stress for 共a兲 具110典 versus 具110典, 共b兲 具110典 versus 具112典, and 共c兲 具112典 versus 具112典, respectively. When the orientation of the upper plane was chosen as 具110典, 180° rotation results in the same stress, reflecting the symmetry. When the upper planes are the same, the amount of the shear stress is seen to become small for ␣ = . This corresponds to single crystal where no discontinuous change in elastic constants at the grain boundary exists. When the orientation of the upper plane is 具112典, ␣- = 180 or ⫺180 shows relatively small value. This corresponds to ⌺3 grain boundaries with low grain boundary energy. In the case of 具110典 versus 具112典, any combinations of ␣ and  result in realization of single crystal or ⌺3 grain boundary. In spite of this fact, calculated
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FIG. 8. Two types of relative orientations of dendrite crystals with given upper planes and growth orientations.
shear stress is strongly dependent on ␣ and , which confirms that shear stress around the grain boundary is strongly dependent on structures of grain boundaries. IV. DISCUSSIONS
We attempt to compare experimental and calculated results. For this purpose, we assume that the grain boundary is formed to satisfy ␣ = − due to the almost equal lateral growth rate of adjacent dendrite crystals owing to the same amount of supercooling, and the grain boundary plane is formed perpendicular to the growth direction. In addition, we should pay attention to the symmetry of dendrite crystals. As shown in Fig. 8, two types of relative orientations of dendrite crystals can be considered for the given upper planes and growth orientations. We refer these two types as type A and type B in the following discussions. When the orientation in the upper plane is 具110典, clockwise 120° rotation of type A coincides with type B. When the orientation in the upper plane is 具112典, 180° rotation of type A coincides with type B. Since the appearance of type A and type B is uncontrollable, experimental results are supposed to contain these two types. By considering the existence of two types, we plot shear stress as a function of 2␣, which corresponds to the contact angle in experiments. Figure 9 shows the calculated shear stress as a function of the contact angle in the case of 具112典
FIG. 10. Summary of maximum calculated shear stress as a function of the contact angle for experimentally observed types of dendrite crystals.
versus 具112典. Four combinations, typeA/typeA, typeB/typeA, typeA/typeB, and typeB/typeB, are plotted. The line for typeA/typeA corresponds to the cross section with  = 360 − ␣ in Fig. 7共c兲. Since type B can be obtained by 180° rotation, the line for typeA/typeB corresponds to the cross section with  = 180− ␣ in Fig. 6共c兲, and so on. Similar plots are made for 具110典 versus 具110典 and 具110典 versus 具112典. EBSP clarified that grain boundaries with 具112典 versus 具112典 contained all four possible combinations. On the other hand, grain boundaries with 具110典 versus 具110典 contained only type A/typeA. For 具110典 versus 具112典, typeA/typeB and typeB/typeA were found. In order to compare calculated shear stress with experimentally obtained dislocation density 共Fig. 4兲, Fig. 10 was prepared by extracting only experimentally observed types. Obviously, a good correlation was found between calculated shear stress and dislocation density. These confirm that the amount of shear stress around the grain boundary is an important parameter to control generation of dislocations. When the contact angle of dendrite crystals is minimized, discontinuous change in elastic constants at the grain boundary is also decreased. As a consequence, the amount of shear stress around the grain boundary is decreased, which could suppress generation of dislocations. The obtained knowledge would be utilized for further precise control of dendritic nucleation so that we could realize highquality Si multicrystals with low dislocation density. V. SUMMARY
FIG. 9. Calculated shear stress as a function of the contact angle in the case of 具112典 vs 具112典.
We utilized dendritic nucleation to systematically control grain boundary structures and investigated dislocation density around the grain boundary as a function of the contact angle with adjacent dendrite crystals. In addition, threedimensional finite element analysis was done to investigate how shear stress depends on the structures of grain boundaries. Comparison of dislocation density and calculated shear stress was made by considering various types of dendrite crystals. Reasonably good agreement was found between them, indicating that the reduction in the shear stress around the grain boundary is one of the keys to reduce dislocation density in Si multicrystals.
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ACKNOWLEDGMENTS
This work was supported by a Grant-in-Aid for Scientific Research共S兲 共Grant No. 20226001兲 from the Ministry of Education, Culture, Sports, Science and Technology of Japan, and the New Energy and Industrial Technology Development Organization 共NEDO兲. IT acknowledges the support of Global COE program “Materials Integration” of Tohoku University 1
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