Different behavior of threading edge dislocation conversion during the solution growth of 4H–SiC depending on the Burgers vector

Different behavior of threading edge dislocation conversion during the solution growth of 4H–SiC depending on the Burgers vector

Available online at www.sciencedirect.com ScienceDirect Acta Materialia 81 (2014) 284–290 www.elsevier.com/locate/actamat Different behavior of threa...

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Available online at www.sciencedirect.com

ScienceDirect Acta Materialia 81 (2014) 284–290 www.elsevier.com/locate/actamat

Different behavior of threading edge dislocation conversion during the solution growth of 4H–SiC depending on the Burgers vector Shunta Harada a,⇑, Yuji Yamamoto a, Kazuaki Seki b, Atsushi Horio a, Miho Tagawa a, Toru Ujihara a a

Department of Materials Science and Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan b Department of Crystalline Materials Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan Received 9 July 2014; received in revised form 15 August 2014; accepted 16 August 2014 Available online 18 September 2014

Abstract Threading edge dislocation (TED) conversion during the solution growth of SiC was investigated by synchrotron X-ray topography. TEDs were converted to basal plane dislocations by lateral growth with the advance of macrosteps on the growth surface during the solution growth. TEDs with the Burgers vector parallel to the step-flow direction were converted to basal plane dislocations with a high probability. On the other hand, the conversion ratio of TEDs whose Burgers vector is not parallel to the step-flow direction was much lower. The variation in elastic energy of the basal plane dislocations after the conversion depending on the Burgers vectors led to the different behavior of TED conversion. Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Silicon carbide; Dislocations; X-ray topography; Crystal growth

1. Introduction Silicon carbide (SiC) has received a great deal of interest in recent years due to its potential application as a nextgeneration power device because of its excellent power device properties [1]. For the achievement of high-performance SiC power devices, high-quality SiC crystal is necessary. Defects in SiC are known to affect the performance and/or long term reliability of power devices [2–6]. However, SiC crystal contains various defects such as micropipes (MPs), threading screw dislocations (TSDs), basal plane dislocations (BPDs) and threading edge dislocations (TEDs) [3]. Commercial SiC wafers are usually produced by the physical vapor transport method. Although the crystal quality of the SiC wafer has been improved and ⇑ Corresponding author.

E-mail addresses: [email protected], [email protected]. ac.jp (S. Harada).

the density of MPs which causes the degradation of breakdown voltage has been decreased below 0.1 cm1 [7], thousands of dislocations still exist in the wafers. Solution growth is one of the methods to achieve high-quality crystal growth because the condition of solution growth is close to thermal equilibrium [8]. Indeed, some positive effects on the crystal quality of SiC during the solution growth have been reported by many researchers [9–16]. During the solution growth, MPs existing in the seed crystal have been reported to be terminated [9]. Kamei et al. have reported the BPD reduction during the solution growth [10]. Recently, we have revealed that most of the TSDs in seed crystal were converted to the extended defects on the basal planes by macrosteps formed on the growth surface due to the step bunching [17–20]. Furthermore, the TSD conversion by the macrosteps can be drastically enhanced by the step flow growth on a vicinal seed crystal [17,18]. The TSD conversion leads to the drastic improvement of the crystal quality because the defects would laterally

http://dx.doi.org/10.1016/j.actamat.2014.08.027 1359-6454/Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

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propagate on the basal plane and finally get out of the crystal [15,16]. Although TEDs were assumed to be less harmful than TSDs, recently TEDs were reported to evidently reduce time-dependent dielectric breakdown lifetime [5]. In this study, for the complete elimination of dislocations in SiC crystal, we investigated the behaviors of TEDs during the solution growth of SiC on a vicinal seed crystal. 2. Experimental procedure A single crystal of 4H–SiC was grown in an inductionheating furnace (Nisshin-Giken Co., Ltd) by the topseeded solution growth method. Details of the growth configuration are given in Ref. [21]. High-purity silicon (11 N) was placed in a graphite crucible and heated to the growth temperature under a high-purity Ar gas flow. The seed crystals were 4H–SiC (0 0 0 1) Si face with 4° offcut towards [1 1  2 0] (10  10 mm2). Growth was carried out at 1903 K for 1 h under a temperature gradient of 36 K cm1. The bottom of the crucible is higher temperature than the surface of the solvent. After the growth, the residual solvent on the crystal was removed by a mixture of HNO3 and HF. In the present study, we obtained a grown layer with a thickness of 10 lm. Grazing incidence synchrotron reflection X-ray topography was performed using a monochromatic X-ray beam (k = 0.150 nm) at BL15C in the Photon Factory at the High-Energy Accelerator Research Organization, Japan. The applied g vector was 1 1  2 8. Surface morphology of the grown crystal was observed by a differential interference contrast (DIC) microscope (Leica DM4000M) using a Nomarski-type prism.

Fig. 1. A Nomarski image taken from the grown crystal on the 4° off-axis Si face seed crystal.

3. Results Surface morphology of the grown crystal is shown in Fig. 1. A train of macrosteps due to step bunching was observed on the whole grown surface. The step-flow direction was [1 1  2 0], which corresponds to the off-oriented direction of the seed crystal. Fig. 2 shows X-ray topography images taken from the seed crystal and the grown crystal at the same position. The large circular contrasts like A and small circular contrasts like B and C in Fig. 2a correspond to TSDs and TEDs in the seed crystal. In the topography image of the grown crystal (Fig. 2b), the asymmetric knife-shaped line contrasts of the extended defect on the basal plane like A0 and line contrasts of BPDs like B0 were observed. By comparing two topography images, TSDs in the seed crystal were converted to the defects on the basal planes extending to the step-flow direction, as we have previously reported [17]. The TSD conversion took place with a high probability and most of TSDs were converted to the extended defect on the basal planes. Furthermore, some TEDs were converted to BPDs. The TED B was converted to the BPDs extending to the step-flow direction as B0

Fig. 2. X-ray topography image taken from (a) the seed crystal and (b) the grown crystal at the same position.

during the solution growth, while the other TED as C was propagated in the grown crystal. To further investigate the TED/BPD conversion behavior, we focus on the Burgers vector of the TEDs. TEDs having six different Burgers vectors of 1/3[1 1  2 0], 1/ 3[1 2 1 0], 1/3[2 1 1 0], 1/3[11 2 0], 1/3[1 2 1 0] and 1/3[2 11 0] are possible. Although the magnitudes of the Burgers vectors are all the same, the directions of the Burgers vectors are different. Fig. 3 shows the magnified X-ray topography images of the TEDs in the seed crystal with the different Burgers vectors. The shape of a bright contrast at the center and the position of dark contrasts are different from

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Fig. 3. Magnified X-ray topography images of the TEDs with the different Burgers vectors (a–f) in seed crystal. The images correspond to (a) 0°-TED, (b) 60°-TED, (c) 120°-TED, (d) 180°-TED, (e) 240°-TED and (f) 300°-TED.

each other. Because of the different strain field, the Burgers vector of TEDs before solution growth can be identified [22]. Angles between the step-flow direction and the Burgers vectors of TEDs are 0°, 60°, 120°, 180°, 240° and 300°. Thus, here we refer to these six types of TEDs as “0°-TED”, “60°-TED”, “120°-TED”, “180°-TED”, “240°-TED” and “300°-TED”, respectively. The behaviors of the TED/BPD conversion during the solution growth depend on the direction of the Burgers vectors of the TEDs. Table 1 summarizes the TED/BPD conversion with the different Burgers vectors. Most of the TEDs with the Burgers vector parallel to the step-flow direction (0°- and 180°-TED) were converted to the BPDs. In that case, these BPDs were pure screw dislocation because the Burgers vector and the line vector were parallel. On the other hand, the conversion ratio of TEDs with the Burgers vector not parallel to the step-flow direction Table 1 TED/BPD conversion behavior with the different Burgers vector of TEDs. Variation of TEDs

Quantity

TED/BPD conversion

Conversion ratio (%)

0°-TED 60°-TED 120°-TED 180°-TED 240°-TED 300°-TED

57 71 110 46 28 40

54 3 2 42 6 2

95 4 2 91 21 5

(60°-, 120°-, 240°- and 300°-TED) was much lower than 0°- and 180°-TED. The BPDs converted from 60°-, 120°-, 240°- and 300°-TED were mixed dislocation. Fig. 4 shows the X-ray topography images of the BPDs converted from the TEDs with the different Burgers vector. It should be noted that the line vector of the BPDs are tilted from the step-flow direction. The BPDs converted from the 0°- and 180°-TED were slightly tilted from the step-flow direction to the right and left, as Fig. 4a and d. The BPDs converted from the 60°- and 240°-TED were tilted from the step-flow direction to the left and the BPDs converted from the 120°- and 300°-TED were tilted to the right at an angle of 10°. 4. Discussion Growth dislocations usually propagate in the directions close to, and frequently parallel to, the growth direction [23,24]. On the other hand, in the present results, the threading dislocations parallel to the growth directions were converted to the basal plane defects perpendicular to the growth direction. This phenomenon is closely related to the macrosteps due to step bunching on the growth surface [17]. During the solution growth on a vicinal seed crystal, lateral growth with the advance of macrosteps to the [1 1 2 0] direction proceeds. Thus the reorientation of dislocations is caused by the step flow of the macrosteps. Similar dislocation reorientations by the advance of macrosteps

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Table 2 Energy factor K for BPDs extending to the [1 1 2 0] direction with the Burgers vector parallel to the step-flow direction and not parallel to the step-flow direction.

Fig. 4. X-ray topography images of BPD converted from (a) 0°-TED, (b) 60°-TED, (c) 120°-TED, (d) 180°-TED, (e) 240°-TED and (f) 300°-TED after the solution growth.

were also reported in the epitaxial growth of GaN and AlN [25,26]. The conversion behaviors of TEDs during the solution growth of 4H–SiC depend on the Burgers vector. One possible reason for the different behavior of the TED/BPD conversion is the different strain energy of the BPD after the conversion. Here we calculate the elastic energy of these two types of BPDs. The elastic strain energy per unit length of a straight dislocation is expressed as follows [23,27,28]:   Kb2 R E¼ ln ð1Þ r0 4p where K is so-called energy factor, b is the modulus of the Burgers vector b, R is the outer cutoff radius and r0 is the inner cutoff radius. The energy factor K depends on the Burgers vector b, the line vector of the dislocation l and elastic constant cij. The variation of elastic energy of a dislocation is often considered as proportional to the energy factor K assuming that the logarithm term is independent of the line vector l [23]. In the Cartesian coordinate (X1, X2, X3), in which the X3 axis is parallel to the dislocation line and the X2 axis lies in the (0 0 0 1) plane, the energy factor K of a dislocation in a hexagonal crystal is expressed as follows [29,30]:  1 ð2Þ K ¼ 2 K 11 b21 þ K 22 b22 þ K 33 b23 þ 2K 13 b1 b3 b in which b0 þ q ½S 1 þ S 2 dðd3  dÞ d b0 þ q ¼ ½d3 S 1 þ S 2 dðdd3  d1 Þ d   b0 þ q d S 5 þ S 6 dðd  d3 Þ þ ¼ d d3

K 11 ¼ K 22 K 33

K 13 ¼

b0 þ q ½S 3 þ S 4 dðd  d3 Þ d

Direction of Burgers vector

Energy factor K (GPa)

Parallel to step flow (pure screw BPD) Not parallel to step flow (mixed BPD)

178 202

and b1, b2, b3 are the components of the Burgers vector in the Cartesian coordinate (X1, X2, X3). Other symbols are elucidated in the Appendix A. The coefficients Klm depend only on the elastic constant cij and the angle a between the [0 0 0 1] axis and the dislocation line vector [29,30]. By using the reported elastic constant of 4H–SiC [31], we can evaluate the energy factor K for the pure screw BPD and the mixed BPD. Table 2 summarizes the energy factor K. The energy factor K for the pure screw BPD is lower than that for the mixed BPD. From Eq. (1), the elastic strain energy E is proportional to the energy factor K. Thus, the elastic strain energy E of the pure screw BPD is expected to be smaller than that of the mixed BPD. This indicates that the pure screw BPDs converted from the 0°- and 180°-TEDs are more stable than the mixed BPDs converted from the 60°-, 120°-, 240°- and 300°-TEDs. Thus the 0°- and 180°-TEDs are more easily converted to the BPDs than the other types of TEDs. Generally speaking, the elastic energy of a pure screw dislocation with the Burgers vector parallel to the line vector is smaller than that of an edge dislocation or a mixed dislocation [32]. The same is true of the BPDs in 4H–SiC and this fact would affect the selective TED conversion during the solution growth. The mixed BPDs converted from the 60°- and 240°TEDs were tilted from the step-flow direction to the left, and the mixed BPDs converted from 120°- and 300°-TED were tilted to the right. This phenomenon can be also understood by considering the elastic strain energy of the BPDs. The elastic strain energy of the mixed BPD is higher than that of the pure screw BPD. In order to reduce the elastic strain energy per unit length, the mixed BPDs would be inclined from the step-flow direction to the Burgers vector. By the incline from the step-flow direction, the length of the dislocation increases. Generally, a dislocation line l tends to adopt a direction for which its strain energy per unit growth thickness is a minimum [23]. Here we consider the straight BPD propagating to the direction tilting at h to the [1 1 2 0] step-flow direction on a vicinal seed crystal with the off-angle of u (in this case u = 4°), as schematically illustrated in Fig. 5. The elastic strain energy W of the BPD per unit growth thickness can be expressed as W ¼

E cos h  cos u

ð3Þ

where E is elastic strain energy per unit length defined in Eq. (1). Considering that E can be assumed to be proportional to the energy factor K and the energy factor K can be expressed as a function of h for a given Burgers vector b, one can get

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Fig. 5. Schematic illustration of the BPD in 4H–SiC on a vicinal seed crystal with the off-angle of u propagating to the direction tilting at h to the step-flow direction. When the crystal grows with the thickness of d, the length of the BPD increases by the length of AB.

W ðhÞ /

KðhÞ cos h

ð4Þ

Thus the BPD would tend to adopt the tilting angle h when K(h)/cos h is a minimum. The Burgers vector of the BPDs converted from 0°-, 60°-, 120°-, 180°-, 240°- and 300°-TEDs in the Cartesian coordinate (X1, X2, X3) can be expressed as follows: b0 ¼ ð0; b sin h; b cos hÞ b

60

b

120



ð5Þ 

¼ ð0; b sinðh  60 Þ; b cosðh  60 ÞÞ ¼ ð0; b sinðh þ 60 Þ; b cosðh þ 60 ÞÞ

b180 ¼ ð0; b sin h; b cos hÞ b240 ¼ ð0; b sinðh  60 Þ; b cosðh  60 ÞÞ b300 ¼ ð0; b sinðh þ 60 ; b cosðh þ 60 ÞÞ By substituting equation (5) into Eq. (2), one can get K(h) for the BPDs with each Burgers vector: 1 1 K 0 ðhÞ ¼ ðK 22 þ K 33 Þ  ðK 22  K 33 Þ cosð2hÞ 2 2 1 1 K 60 ðhÞ ¼ ðK 22 þ K 33 Þ  ðK 22  K 33 Þ cosð2h  60 Þ 2 2 1 1 K 120 ðhÞ ¼ ðK 22 þ K 33 Þ  ðK 22  K 33 Þ cosð2h þ 60 Þ 2 2 K 180 ðhÞ ¼ K 0 ðhÞ

ð6Þ

K 240 ðhÞ ¼ K 60 ðhÞ K 300 ðhÞ ¼ K 120 ðhÞ Fig. 6 shows the calculated curves of K(h) and K(h)/cos h with the different Burgers vector of the BPDs. The values of K0°(h), K60°(h) and K120°(h) are minimum at h = 0°, 60° and 60°, respectively, which corresponds to the stability of a pure screw dislocation with the Burgers vector parallel to the line vector. The values of K0°(h)/cos h, K60°(h)/ cos h and K120°(h)/cos h are minimum at h = 0°, 9.1° and 9.1°, respectively. Thus, the BPDs converted from TEDs are classified in three types according to the Burgers vector. One is the pure screw BPDs converted from the 0°- and 180°-TEDs; the second is the mixed BPDs converted from the 60°- and 240°-TEDs and the third is the mixed BPDs converted from the 120°- and 300°-TEDs. From the

Fig. 6. Calculated curves of (a) K(h) and (b) K(h)/cos h with the different Burgers vector of the BPDs.

calculation of K(h)/cos h, the pure screw BPDs converted from the 0°- and 180°-TEDs would tend to propagate in the step-flow direction. On the other hand, the mixed BPDs converted from the 60°- and 240°-TEDs tend to propagate in the direction tilted from the step-flow direction to the left, and the mixed BPDs converted from the 120°- and 300°-TEDs tend to propagate in the direction tilted from the step-flow direction to the right. These results are generally consistent with the present X-ray topography observation. However, these calculations of the elastic energy cannot provide an account for the slight tilt from

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the step-flow direction of the pure screw BPDs converted from the 0°- and 180°-TEDs. In the present elastic energy calculation, the core structure of the dislocation is not taken into consideration. The BPDs with Burgers vector b = 1/3h1 1  2 0i forms an extended dislocation consists of two partial dislocations with the Burgers vectors b1 = 1/ 3h1 0  1 0i and b2 = 1/3h0 1  1 0i accommodating with the Shockley type stacking fault between them [33]. The structures of extended dislocations for the BPDs converted from the 0°- and 180°-TEDs may be different from each other in terms of the partial dislocations and the stacking fault between them. Thus the BPDs converted from the 0°- and 180°-TEDs might be tilted to the different directions although the energy factor K for the both BPDs is identical. Since 0°- and 180°-TEDs are converted to BPDs by [1 1 2 0] step flow growth, all types of TEDs can be converted to BPDs by at least three different step flows with the direction of [1 1  2 0], [ 12 1 0] and [ 2 1 1 0], in principle. Alternatively, all types of TEDs would be converted by the increase of the growth thickness because some TEDs with the Burgers vector not parallel to the step-flow direction were definitely converted to the BPDs, although the conversion rate was lower than 0°- and 180°-TEDs. Thus the TEDs would be completely eliminated by using the TED/BPD conversion during the solution growth because the converted BPDs would be laterally propagated and get out of the crystal. 5. Summary TED conversion during solution growth of SiC was investigated by synchrotron X-ray topography. The results obtained are summarized as follows.

increase of the growth thickness because some TEDs with the Burgers vector not parallel to the step-flow direction are converted to BPDs although the conversion rate is low. Acknowledgements The authors are grateful to Dr H. Yamaguchi and Dr K. Hirano for the X-ray topography measurements. The Xray topography was performed under the approval of the Photon Factory Program Advisory Committee (Proposal No. 2011G247). This study was partly supported by a Grant-in-Aid for Scientific Research (23246004 and 24686078). Appendix A The symbols appearing in Eq. (2) are expressed by the elastic constant cij and the angle a between the [0 0 0 1] axis and the dislocation line vector as follows [29,30]:   b0 2 g cos4 a þ d13 d43 S1 ¼ 2 2kðb0 þ qÞ S2 ¼ S3 ¼ S4 ¼ S5 ¼

(1) During the solution growth of 4H–SiC on an off-axis Si face seed crystal, TEDs are converted to BPDs by lateral growth with the advance of macrosteps. (2) The TED/BPD conversion behaviors vary according to the direction of the Burgers vector. The TEDs with the Burgers vector parallel to the step-flow direction (0° and 180°-TEDs) are converted to the pure screw BPDs at a high rate. On the other hand, the conversion rate is low for the mixed TEDs with the Burgers vector not parallel to the step-flow direction (60°, 120°, 240° and 300°-TEDs). The elastic energy of the pure screw BPDs is smaller than that of the mixed BPDs, which would lead to the different dislocation conversion behavior during the solution growth. (3) The BPDs converted from the 120°- and 300°-TEDs are tilted from the step-flow direction to the right and the BPDs converted from the 60°- and 240°TEDs are tilted to the left because a dislocation tends to adopt a direction minimizing the elastic energy. (4) All types of TEDs can be converted to BPDs by at least three different step flows with the direction of [1 1  2 0], [ 12 1 0] and [ 2 1 1 0], in principle. Alternatively, all types of TEDs would be converted by the

289

S6 ¼

4b0 2 d23 cot2 a ðb0 þ qÞ2   b0 2 sin 2a d13 qd23  gb0 cos2 a 4kðb0 þ qÞ

2

2b0 cot aðb0  q cos 2aÞ 2

ðb0 þ qÞ  2  gb0 þ d13 q2 sin2 2a 8kðb0 þ qÞ

2

4qb0 cos2 a 2

ðb0 þ qÞ

d21 ¼

k þ l sin2 a  m sin4 a k

d2 ¼

2k þ l sin2 a k

b0 þ q sin2 a b0 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 d¼ d2 þ 2d1 2 b0 ¼ c66 d23 ¼

q ¼ c44  c66 k ¼ c11 c44   l ¼ c11 c33  c213  2c44 ðc11 þ c13 Þ   m ¼ c11 c33  c213  c44 ðc11 þ c33 þ 2c13 Þ d13 ¼ c11 c33  c213 g ¼ c11 c33  ðc13 þ 2c44 Þ

2

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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

Matsunami H, Kimoto T. Mater Sci Eng R 1997;20:125. Neudeck PG. IEEE Trans Electron Devices 1999;46:478. Neudeck PG. Mater Sci Forum 2000;338–342:1161. Skowronski M, Ha S. J Appl Phys 2006;99:011101. Yamamoto K, Nagaya M, Watanabe H, Okuno E, Yamamoto T, Onda S. Mater Sci Forum 2012;717–720:477. Senzaki J, Kojima K, Kato T, Shimozato A, Fukuda K. Appl Phys Lett 2006;89:022909. Mu¨ller SG, Sanchez EK, Hansen DM, Drachev RD, Chung G, Thomas B, et al. J Cryst Growth 2012;352:39. Dost S, Lent B. Single crystal growth of semiconductors from metallic solutions. Amsterdam: Elsevier; 2007. Yakimova R, Janze´n E. Diamond Relat Mater 2000;9:432. Kamei K, Kusunoki K, Yashiro N, Okada N, Takana T, Yauchi A. J Cryst Growth 2009;311:855. Yashiro N, Kusunoki K, Kamei K, Yauchi A. Mater Sci Forum 2010;645–648:33. Soueidan M, Ferro G. Adv. Funct. Mater. 2006;16:975. Ujihara T, Maekawa R, Tanaka R, Sasaki K, Kuroda K, Takeda Y. J. Cryst. Growth 2008;310:1438. Ujihara T, Seki K, Tanaka R, Kozawa S, Alexander, Morimoto K, et al. J Cryst Growth 2011;318:389. Harada S, Yamamoto Y, Seki K, Ujihara T. Mater Sci Forum 2013;740–742:189. Yamamoto Y, Harada S, Seki K, Horio A, Mitsuhashi T, Koike D, et al. Appl Phys Express 2014;7:065501. Yamamoto Y, Harada S, Seki K, Horio A, Mitsuhashi T, Ujihara T. Appl Phys Express 2012;5:115501.

[18] Ujihara T, Kozawa S, Seki K, Alexander, Yamamoto Y, Harada S. Mater Sci Forum 2012;717–720:351. [19] Harada S, Yamamoto Y, Seki K, Horio A, Mitsuhashi T, Tagawa M, et al. APL Mater 2013;1:022109. [20] Harada S, Yamamoto Y, Xiao SY, Tagawa M, Ujihara T. Mater Sci Forum 2014;778–780:67. [21] Harada S, Alexander, Kozawa S, Seki K, Yamamoto Y, Zhu C, et al. Cryst Growth Des 2012;12:3209. [22] Kamata I, Nagano M, Tsuchida H, Chen Y, Dudley M. J Cryst Growth 2009;311:1416. [23] Klapper H. Generation and propagation of defect during crystal growth. In: Dhanaraj G, Byrappa K, Prasad V, Dudley M, editors. Springer handbook of crystal growth. New York: Springer; 2010. [24] Klapper H. Mater Chem Phys 2000;66:101. [25] Shen XQ, Matsuhata H, Okumura H. Appl Phys Lett 2005;86:021912. [26] Bai J, Dudley M, Sun WH, Wang HM, Asif Khan M. Appl Phys Lett 2006;88:051903. [27] Hirth JP, Lothe J. Theory of dislocations. New York: McGraw-Hill; 1968. [28] Steed JW. Introduction to anisotropic elastic theory of dislocations. Oxford: Clarendon Press; 1973. [29] Teutonico LJ. Mater Sci Eng 1970;6:2747. [30] Savin MM, Chernov VM, Strokova AM. Phys Status Solidi (a) 1976;35:747. [31] Kamitani K, Grimsditch M, Nipko JC, Loong CK, Okada M, Kimura I. J Appl Phys 1997;82:3152. [32] Hull D. Introduction to dislocations. Oxford: Pergamon Press; 1975. [33] Hong MH, Samant AV, Pirouz P. Philos Mag A 2000;80:919.