Possible Twinning Operations during Directional Solidification of

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Possible Twinning Operations during Directional Solidification of Multi-Crystalline Silicon J.W. Jhang, T. Jain, H.K. Lin, and Chung-Wen Lan Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.8b00115 • Publication Date (Web): 05 Mar 2018 Downloaded from http://pubs.acs.org on March 8, 2018

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Crystal Growth & Design

Fig. 1

The schematic diagram to illustrate the concept of T matrix. In this case, the corresponding T matrix is 𝑇⟨11̅1̅⟩,60° which means the misorientation axis / angle pair is ⟨11̅1̅⟩ and 60°.

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Crystal Growth & Design 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Fig.3 The schematic diagrams to explain the difference between the crystalline orientations of the twin grain resulted from pure tilt (a) and pure twist (b) twining mechanisms. Green grain and yellow grain indicate the parent grain and the twin grain, respectively.


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Possible Twinning Operations During Directional Solidification of MultiCrystalline Silicon

J.W. Jhang, T. Jain, H.K. Lin, C.W. Lan*

Department of Chemical Engineering, National Taiwan University, Taipei, 10617, Taiwan

*Corresponding author: [email protected]; Tel.: 886-2-2363-3917

Fax: 886-2-2363-3917

1



Abstract

Twinning is an important phenomenon during crystal growth of multicrystalline silicon. During twinning, the nucleated new grain is in twin relation with the parent grain, and the twin relation mathematically could be operated through a transformation matrix (T matrix). We have derived the possible T matrices for twinning on the (111) facets, and some of them have not well recognized before. As applied to the available electron backscatter diffraction (EBSD) data, we found that in addition to the twist Σ3 operation, the well accepted mechanism for twin formation, there are four other possible twinning operations.

Keywords: Silicon, Twinning, Transformation matrix, EBSD

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1. Introduction Twinning is a frequently observed and important phenomenon during solidification of multi-crystalline silicon (mc-Si) used for solar cells. Twinning usually occurs on the (111) facet grooves [1-6], and is a major source for the formation of the Σ3 twin boundary. Because the Σ3 twin boundaries are the most electrically inactive CSL (Coincidence Site Lattice) grain boundaries (GBs), and they are often preferred for mc-Si solar cells. On the other hand, for the recently developed high-performance mc-Si technology [7-8], which requires the more random GBs for stress relaxation, the Σ3 twin boundaries are not preferred at the initial stage of the ingot growth. Therefore, the understanding and the control of twinning are crucial for the growth of mc-Si. Electron backscatter diffraction (EBSD) is a powerful tool for the analysis of the crystal orientation and GB types. The Euler angles are available in EBSD data to for each grain, which are useful information for understanding twinning and identifying GB types. Wong et al. [6] used this tool to study twinning during directional solidification of mc-Si, and they found that twinning often occurred at the three-grain tri-junction grooves. Recently, Jain et al. [9] further developed a 3D model to predict the twinning probability, and the preferred grain for twinning was correctly predicted. The Σ3 twin boundary is a coherent GB, and the atomic arrangement of the twin grain

can be formed by either 60° twist around < 111 > or 70° symmetric tilt from [110].

Because twinning occurs on the {111} plane, the structure of the two types of twining operations are structurally indistinguishable [10], and they have the same atomic arrangement [11]. Therefore, it is difficult to distinguish the twinning operation from

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the structures of the parent and the twin grains. Fortunately, the transformation matrix (T matrix) is a useful tool for the analysis, because it focuses on the orientation relationship instead of the structural one. The use of the mirror symmetric operation for T matrixes was reported recently [12-13], but still some experimental results could not be explained by the proposed T matrixes. Because the previous matrices were calculated based on the two grains being misoriented by 180°, they were not sufficient at all. In this study, we propose a method to calculate the T matrixes. Apart from the T matrices mentioned in the literatures [12-13], we derive more T matrixes for twining. Moreover, we use the T matrices to find the possible twinning operations from the experimental results [6], and the possible twinning operations may be useful to better understand the twin formation during crystal growth.

2. Methodology To get the T matrices is very straightforward. First, we transform the

misorientation axis/angle pair (axis= < >, angle= ) into the quaternion [14]. Then, the quaternion is converted into the corresponding T matrix [15]. The detail calculation is shown as the following:

= √

, = √

= cos !,

, = √

= sin !,

= sin !,

4


,

(1)

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= sin !,

(2)

where , , and are the components of the quaternion. Then, the T matrix could be found as:

+ − −

$%&, = ' 2+ − , 2+ + ,

2+ + ,

+ − − 2+ − ,

2+ − ,

2+ + , -.

+ − −

(3)

If we take the twist twining as an example, the schematic diagram is illustrated in

Fig. 1, where the corresponding T matrix is $%11.1.&,60° , which means that the misorientation axis/angle pair is %11.1.& and 60°, respectively. Because the operations

could be performed on different {111} planes, we need to derive all the T matrices for the twist twining, as well as the tilt twinning, and they are summarized in Table 1. As shown in Table 1(a), there are twelve T matrices for the pure twist Σ3 relationship. On the other hand, as shown in Table 1(b), there are twenty four T matrices for the pure tilt Σ3 relationship. An interesting thing is that there exists a “122” numerical relationship in each row and column of the T matrices corresponding to the Σ3 relationship. An important concept needed to be mentioned is that the rows or columns of a T matrix cannot be arbitrarily exchanged with each other. When any two rows or columns are exchanged, the sign of one row or column has to be reversed to keep the determinant positive. In addition, even though the determinant can remain positive, it may still lead to a different twin orientation. Hence, the T matrix applied to satisfy the orientation relationship between two neighboring grains should be unique. Again, taking the case in Fig. 1 as an example, the corresponding T matrix could be

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calculated as follows:

=

/

√0

, =

1/ √0

= cos

, =

23°

= sin

23°

= sin

= sin

0.666 $%//4/4&,23° = $%/4//&,123° = 5 0.333 −0.666

!,

√0

(4)

,

!,

23°

!,

!, and

23°

1/

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−0.666 0.666 −0.333

(5)

2. 2 1.

0.333 2 / 0.6668 = 0 51 0.666 2.

1 28. 2

(6)

Table 1 All the T matrixes correspond to twist (a) and tilt (b) Σ3 relationships. There are twelve T matrices for the twist Σ3 relationship and twenty-four T matrices for the tilt Σ3 relationship. (a) Angle Axis

%111& %1.11& %11.1& %111.&

180°/−180° 1 1. 2 52 1. 3 2 2

2 28 1.

1 2 51. 3 2

1 1. 2. 52. 1. 3 2 2.

2 2.8 1.

1 2 52. 3 . 1

1 1. 2. 52. 1. 3 . 2 2 1 1. 2 52 1. 3 . . 2 2

2. 28 1. 2. 2.8 1.

1 2 52. 3 1 1 2 52 3 1

60°

−60°

2 2 1.

1. 28 2

1 2 52 3 . 1

1. 2 2

2 1.8 2

1 2 2.

2 18 2

1 2 51 3 2

2. 2 1

1. 2.8 2

1 2 2 1. 2 2.

2. 1.8 2 2. 18 2

6


1 2 51 3 . 2 1 2 51. 3 . 2

2. 2 1. 2 2 1

1 28 2 1 2.8 2

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70°

(b) Angle Axis

%011& %01.1&

%011.& %01.1.& %101& %1.01&

%101.& %1.01.& %110& %1.10& %11.0& %1.1.0&

1 1 52. 3 2 1 1 52. 3 . 2 1 1 52 3 2 1 1 52 3 . 2 1 2 52. 3 1 1 2 52. 3 . 1 1 2 52 3 . 1 1 2 52 3 1 1 2 51 3 2 1 2 51. 3 2 1 2 51. 3 . 2 1 2 51 3 . 2

2 2 1 2 2 1. 2. 2 1. 2. 2 1 2 1 2. 2 1 2 2. 1 2. 2. 1 2 1 2 2. 1. 2 2 1. 2 2. 1 2 2

2. 18 2 2 1.8 2 2. 1.8 2 2 18 2 1 28 2 1. 2.8 2 1. 28 2 1 2.8 2 2. 28 1 2. 2.8 1 2 28 1 2 2.8 1

110°

1 1. 52. 3 2 1 1. 52. 3 . 2 1 1. 52 3 2 1 1. 52 3 . 2 1 1 52. 3 2 1 1 52. 3 . 2 1 1 52 3 . 2 1 1 52 3 2 1 1 52 3 2 1 1 52. 3 2 1 1 52. 3 . 2 1 1 52 3 . 2

2 1 2 2 1 2. 2. 1 2. 2. 1 2 2 1. 2. 2 1. 2 2. 1. 2. 2. 1. 2 2 1 2. 2. 1 2 2. 1 2. 2 1 2

2. 28 1 2 2.8 1 2. 2.8 1 2 28 1 2 28 1 2. 2.8 1 2. 28 1 2 2.8 1 2. 28 1. 2. 2.8 1. 2 28 1. 2 2.8 1.

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3. Comparison with experiments We used the electron backscatter diffraction (EBSD) data from Wong et al. [6] (referred as Wong’s experiments) for the analyses. Besides the twist Σ3 mechanism, a public accepted mechanism for twin formation, four other kinds of twinning operations were found. We classified the observed twinning operations into types A, B, C, D, and E. We will illustrate three cases for types B, C and D, and two cases for types A and E. An interesting observation is that there were three kinds of twinning operations that were neither publicly accepted pure tilt Σ3 nor pure twist Σ3 twinning from our analysis. However, the original EBSD data (Horiba Nordlys F+) only indicated the twist Σ3 twin boundaries between the parent grain and the twin grain. The Euler angles of the parent grain and the twin grain of these cases were obtained from the EBSD data. Moreover, the EBSD orientation map was considered to be the ;< − =< plane and the grain growth direction was considered to be the >̂ direction, i.e.

the 〈001〉 direction towards the reader. The projection of GBs on the ;< − =< plane is also obtained by EBSD.

First, we need to transform the Euler angles into the corresponding rotation

matrix for both parent grain and twin grain. The definition of Euler angles (B/ , Φ, B )

of EBSD is according to Bunge’s convention, and rotation matrix (R) corresponding to the Euler angles [16] is given as follows: cosFB/ G cosFB G − cosFΦG sinFB/ G sinFB G cosFB G sinFB/ G + cosFB/ G cosFΦG sinFB G sinFΦG sinFB G D = E−cosFΦG cosFB G sinFB/ G − cosFB/ G sinFB G cosFB/ G cosFΦG cosFB G − sinFB/ G sinFB G cosFB G sinFΦGH. sinFB/ G sinFΦG − cosFB/ G sinFΦG cosFΦG

9


(7)


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With this rotation matrices of both grains, there exists a transformation matrix (T matrix) between two rotation matrixes. The corresponding equations are shown as the following:

DI = $ DJ or

(8)

$ = DI DJ1/ ,

(9)

where DJ and DI are the rotation matrices of the parent grain and the twin grain,

respectively, and DJ1/ is the inverse of the rotation matrix of the parent grain. Again,

for the T matrix, we could also find the corresponding misorientation axis/angle pair asfollows [17]:

= cos 1/[

],

IF/,/GIF , GIF0,0G1/

MN = MR = MS =

IF ,0G1IF0, G OPQFG

(10)

,

IF0,/G1IF/,0G OPQFG

,

IF/, G1IF ,/G OPQFG

.

(11)

It can be seen in Table 2 that the twinning operation of the two cases (A-1, and A2) for type A is a pure twist twinning, because the orientation relationship between the

parent grain and the twin grain is through 60° rotation about < 111 >. The twinning

operation for type B in Table 3 is a pure tilt, because the orientation relationship

between the parent grain and the twin grain is though 70° rotation about < 110 >.

The twinning operations for cases belonged to types C, D, and E in Tables 4, 5, and 6,

respectively, are neither pure tilt nor pure twist twinning. The orientation relationship

between the parent grain and the twin grain for the cases in Table 4 is through 180° 10


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rotation about < 112 >, the one for the cases in Table 5 is 146° rotation about < 113 >, and the one for the cases in Table 6 is 131° rotation about < 102 >. The

EBSD mappings at the tri-junction before and after the twinning of B-2 and C-2 and their corresponding Euler angles of parent and twin grains are shown in Figs. 2(a) and (b). These four kinds of twinning operations (types B, C, D, and E) are not the well accepted twist twinning.

Table 2 The corresponding Euler angles of the parent and twin grains and the T matrix calculated according to Eq. (9) for type A, where the misorientation axis/angle is [111]/60°. Case

Components

Parent grain A-1

Twin grain Parent grain

A-2

Twin grain

B/

Φ

Euler angles

324.6°

45.0°

154.2°

31.9°

238.7° 38.9°

B

8.2°

25.4°

46.5°

25.2°

76.5°

5.1°

T matrix

1 2 $%///&,123° = 52 3 . 1 $%///&,123°

1 2 = 52 3 . 1

1. 2 2 1.8 2 2 1. 2 2 1.8 2 2

Table 3 The corresponding Euler angles of the parent and twin grains and the T matrix calculated according to Eq. (9) for type B, where the misorientation axis/angle is [110]/70°. Case

Components

Parent grain B-1


B-2

Twin grain

B/

40.2°

Φ

Euler angles 8.8°

40.0°

46.9°

44.7°

330.9°

51.4°

197.2°

12.8°

288.6°

B

52.1° 76.4°

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T matrix

1 1 $%3/44/&,W3° = 52 3 . 2 $%/43/4&,W3°

1 2 = 52 3 1

2. 2 1 2. 1 2

2 18 2 1 2.8 2


Parent grain B-3

Twin grain

0.9°

282.7°

19.0° 48.9°

21.0° 37.8°

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$%3/4/4&,W3°

1 1 = 52 3 . 2

2. 2 1

2 18 2

Table 4 The corresponding Euler angles of the parent and twin grains and the T matrix calculated according to Eq. (9) for type C, where the misorientation axis/angle is [112]/180°. Case

Components

Parent grain C-1


C-2


C-3

Twin grain

B/

Φ

Euler angles

B

354.3°

42.5°

28.5°

247.3°

49.9°

45.7°

141.4° 65.9°

352.0° 137.8°

31.1° 21.1° 27.4° 46.8°

68.3° 47.2° 19.3° 61.1°

T matrix

$%//

&,/X3°

$%//

&,/X3°

$%//

&,1/X3°

1 2. = 51 3 2 1 2. = 51 3 2 =

1 2. 51 3 2

1 2 2. 28 2 1 1 2 2. 28 2 1 1 2. 2

2 28 1

Table 5 The corresponding Euler angles of the parent and twin grains and the T matrix calculated according to Eq. (9) for type D, where the misorientation axis/angle is [113]/146°. Case

Components

Parent grain D-1


D-2


D-3

Twin grain

B/

Φ

Euler angles

B

293.3°

44.2°

27.5°

182.4°

44.3°

44.9°

106.8°

14.0°

121.2° 78.0°

223.8°

4.0°

14.0° 40.9°

55.7° 0.3°

12.7° 43.7°

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T matrix

$%//0&,1/Y2°

1 2. = 52 3 1

1. 2. 2

1 2. = 51. 3 2

2 1 2. 28 1 2

$%//0&,1/Y2° = $%//0&,/Y2°

1 2. 52 3 1

1. 2. 2

2 18 2 2 18 2

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Table 6 The corresponding Euler angles of the parent and twin grains and the T matrix calculated according to Eq. (9) for type E, where the misorientation axis/angle is [102]/131°. Case

Components

Parent grain E-1


E-2

Twin grain

B/

Φ

Euler angles

B

190.8°

44.5°

89.5°

85.2°

38.3°

79.1°

98.7°

316.6°

19.1° 14.0°

46.9° 76.0°

T matrix

$%/3

&,1/0/°

$%/3

&,1/0/°

1 1. = 52 3 2 1 1. = 52 3 2

2. 2. 1 2. 2. 1

2 1.8 2 2 1.8 2

4. Discussion We have shown how to utilize the derived T matrixes to find out the possible twinning operations in Wong’s experiments from their EBSD data. Similarly, we could also examine the experimental results by Kang et al. [13]. In their paper, the growth orientation (row vector) of the parent grain was multiplied by a T matrix (for

the < 111 >/180° pair) to get the orientation (row vector) of the twin grain. The

growth orientation 〈1. 1. 5〉 of the twin grain, which was observed in the experiment, could not be calculated using the T matrices for the < 111 >/180° pair. The correct T matrices

. 1 0〉/70.5°&, which was should be the ones for the misorientation axis/angle pair of 〈1

observed in the experiment [13]. On the other hand, in the method mentioned in [12], which we adopted, the growth orientation (column vector) of the twin grain was obtained by multiplying the corresponding T matrix by the growth orientation

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(column vector) of the parent grain as

Z[ = $ ∗ ][,

(12)

where ][ and Z[ are the growth orientations of the parent grain and the twin grain,

respectively. With this, the T matrix for the 〈1. 1 0〉/70.5°& pair could correctly predict the growth orientation (column vector) of the twin grain. Two cases in [13] had the

growth orientations of the parent/twin grains are %110& / %114& and %111& / %1.1.5& ,

respectively. The calculated T matrixes using Eq. (12) for both cases are $^/4/3_,W3° ,

which were not mentioned in the previous reports [12-13]. This indicates that the twinning operation for both cases was a pure tilt, and the corresponding misorientation axis/angle pair was also consistent with the Laue spots in [13] for both cases. The twist and tilt Σ3 twin boundaries can be expected to have cross-grid screw

dislocations and an array of edge dislocations parallel to [110], respectively [18].

They both lead to the same atomic arrangement, so that the structure of the F111G

twin plane would be identical. Although the grain structures of the twin grains would be the same resulted from the same atomic arrangement, their crystalline orientations would be different depending on different twinning operations. If we take the pure tilt Σ3 and pure twist Σ3 twinning as an example, the schematic diagrams is illustrated in Fig. 3 to clearly explain the difference between the crystalline orientations of the twin grains resulted from different twinning operations. The misorientation axis/angle pair for pure tilt and pure twist twinning operations are %1.01&/70° and %1.1.1&/60°, and the

corresponding schematic diagrams are illustrated in Figs. 3(a) and (b), respectively.

As shown, the crystalline orientations of the twin grains are different, even though the

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grain structures of the twin grain are the same. Interestingly, these five types of twinning operations have also been reported through theoretical analyses for a cubic bicrystal [19-20] for a long time, but we confirmed they indeed existed based on the EBSD data for mc-Si crystal growth. Moreover, types C, D, and E were not yet clearly classified into pure tilting, pure twisting or a combination of both operations. To identify this, we utilize the tilt/twist component (TTC) [21] as follows:

b[cd ∙ M b[J , TTC = M

(13)

b[cd and M b[J are the misorientation axis and the GB plane normal of the parent where M

grain, respectively. Both vectors are normalized into unity. The TTC value varies from 0 for pure tilt to 1 for pure twist, and the value between 0 and 1 indicates a mixed one. The first step for calculating the TTC value is to find out the misorientation axis in the initial Cartesian reference coordinate, because the calculated misorientation axis using Eq. (11) is in the coordinate system of the parent grain. The corresponding equation to calculate the misorientation axis in the initial fixed Cartesian coordinate is as the following:

b[cd = DJ ∗ M b[c , M

(14)

b[cd and M b[c are the misorientation axis in the initial fixed Cartesian coordinate where M system and in the coordinate system of the parent grain, respectively. Equations (1), (2), and (3) are then used to calculate the corresponding T matrix in the initial fixed Cartesian reference coordinate. The corresponding equations used to calculate eight < 111 > vectors of the parent grain and the twin grain are as the following:

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[111]J = DJ [111],

[111] I = $ d [111]J ,

Page 18 of 24

(15) (16)

where [111]J and [111] I are the eight normalized [111] vectors of the parent grain

and the twin grain at the initial fixed Cartesian reference system, respectively. In

addition, $ d is the T matrix in the initial fixed Cartesian reference system. As a result,

the calculated TTC value for types C, D, and E are 0, 0.52, and 0.77, respectively. Therefore, it is clearly that the twinning operations of type C is still a pure tilt, while and twinning operations of types D and E is a combination of tilt and twist Σ3

operations.

5. Conclusions In this study, we introduce a method to calculate all the T matrices for Σ3 twinning. An interesting observation is that there exists a “122” numerical relationship in each row and column of the T matrices corresponding to the Σ3 relationship. We further utilize the T matrices to explain the possible twinning operations for Wong’s experiments through their EBSD data. Besides the well-known twist Σ3 twinning, there are other four kinds of twinning operation from our analyses for mc-Si crystal growth. Two of them are pure tilt and the other two are the combination of tilt and twist Σ3 operations. These four kinds of operations are not the well accepted twinning for mc-Si, but they provide a reference for future study. We also apply our T matrices to identify the twinning operation in the previous report, and the results are consistent with the experimental data.

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Acknowledgement This work was supported by the Ministry of Science and Technology of Taiwan.

References [1] Noma, T.; Yonehara T.; Kumomi H. Crystal Forms by Solid-State Recrystallization of Amorphous Si-Films on SiO2. Appl. Phys. Lett. 1991, 59, 653655. [2] Trempa, M.; Reimann, C.; Friedrich, J.; Muller, G.; Oriwol, D. Mono-crystalline Growth in Directional Solidification of Silicon with Different Orientation and Splitting of Seed Crystals. J. Cryst. Growth 2012, 351, 131-140. [3] Ernst, F.; Pirouz, P. The Formation Mechanism of Planar Defects in Compound Semiconductors Grown Epitaxially on (100) Silicon Substrates. J. Mater Res. 1989, 4, 834-842. [4] Lin, H.K.; Wu, M.C.; Chen, C.C.; Lan, C.W. Evolution of Grain Structures during Directional Solidification of Silicon Wafers. J. Cryst Growth 2016, 439, 40-46. [5] Duffar, T.; Nadri, A. On the Twinning Occurrence in Bulk Semiconductor Crystal Growth. Scripta Mater 2010, 62, 955-960. [6] Wong, Y.T.; Hsu, C.; Lan, C.W. Development of Grain Structures of Multicrystalline Silicon from Randomly Orientated Seeds in Directional Solidification. J. Cryst. Growth 2014, 387, 10-15. [7] Stokkan, G. Twinning in Multicrystalline Silicon for Solar Cells. J. Cryst. Growth 2013, 384, 107-113. [8] Lan, C.W.; Yang, Y.M.; Yu, A.; Wu, Y.C.; Hsu, B.; Hsu, W.C.; Yang, A. Recent Progress of Crystal Growth Technology for Multi-crystalline Silicon Solar Ingot, Solid State Phenom. 2016, 242, 21-29. [9] Jain, T.; Lin, H. K.; Lan, C.W. Three Dimensional Modelling of Grain Boundary Interaction and Evolution during Directional Solidification of Multi-crystalline

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Silicon. J. Cryst. Growth 2018, 485, 8-18. [10] Ratanaphan, S.; Yoon, Y.; Rohrer, G.S. The Five Parameter Grain Boundary Character Distribution of Polycrystalline Silicon. J Mater Sci 2014, 49, 4938-4945. [11] Hornstra, J. Models of Grain Boundaries in the Diamond Lattice .2. Tilt about (001) and Theory. Physica 1960, 26, 198-208. [12] Uccelli, E.; Arbiol, J.; Magen, C.; Krogstrup, P.; Russo-Averchi, E.; Heiss, M.; Mugny, G.; Morier-Genoud, F.; Nygard, J.; Morante, J.R.; Morral, A.F.I. Threedimensional Multiple-order Twinning of Self-Catalyzed GaAs Nanowires on Si Substrates Nano Lett 2011, 11, 3827-3832. [13] Kang, D.Z.; Liu, J.H.; Jiang, C.B.; Xu, H.B. Correlation between Growth Twinning and Crystalline Reorientation of Faceted Growth Materials during Directional Solidification. Cryst Growth Des 2015, 15, 3092-3095. [14] Pusztai, T.; Bortel, G.; Granasy, L. Phase Field Theory of Polycrystalline Solidification in Three Dimensions. Europhys Lett 2005, 71, 131-137. [15] Shoemake, K. Animating Rotation with Quaternion Curves. ACM 1985, 19, 245254. [16] Goldstein, H. In Classical Mechanics(3rd ed.), MA: Addison-Wesley, 1980, pp.153. [17] Lloyd, G.E.; Farmer, A.B.; Mainprice, D. Misorientation Analysis and the Formation and Orientation of Subgrain and Grain Boundaries. Tectonophysics 1997, 279, 55-78. [18] Hornstra, J. Models of Grain Boundaries in the Diamond Lattice .1. Tilt about . Physica 1959, 25, 409-422. [19] Pumphrey, P.H.; Bowkett, K.M. Axis/angle Pair Description of Coincidence Site Lattice Grain Boundaries. Scripta Metallurgica 1971, 5, 365-369. [20] Brandon, D.G.; Ralph, B.; Ranganathan, S.; Wald, M.S. A Field Ion Microscope Study of Atomic Configuration at Grain Boundaries. Acta Metallurgica 1964, 12, 813-821. [21] Amouyal, Y.; Rabkin, E.; Mishin, Y. Correlation between Grain Boundary Energy and Geometry in Ni-rich NiAl. Acta Mater 2005, 53, 3795-3805.

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manuscript title Possible Twinning Operations During Directional Solidification of Multi-Crystalline Silicon author list J.W. Jhang, T. Jain, H.K. Lin, C.W. Lan TOC graphic

Synopsis Through the analyses of the EBSD data of multi-crystalline silicon from directional solidification, we found five types of twinning operations from the parent grain orientation to get the twin grain orientations. The top shows the grain orientations before and after twinning. The bottom shows the corresponding geometrical operations (left) and the mathematical representation.

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Figure Captions: Fig. 1 The schematic diagram to illustrate the concept of T matrix. In this case, the corresponding T matrix is 𝑇⟨11̅1̅⟩,60° which means the misorientation axis/angle pair is ⟨11̅1̅⟩ and 60°. Fig. 2 EBSD mapping of the tri-junction before and after the twin nucleation: (a) case B-2 and the corresponding Euler angles of parent and twin grains; (b) case C2 and the corresponding Euler angles of parent and twin grains. G1 means parent grain. Fig. 3

The schematic diagrams to explain the difference between the crystalline orientations of the twin grain resulted from the pure tilt (a) and the pure twist (b) twining operations. The green grain and yellow grain indicate the parent grain and the twin grain, respectively.



Fig. 2


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Possible Twinning Operations during Directional Solidification of

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