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Selective-catalyst formation for carbon nanotube growth by local indentation pressure
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Thin Solid Films 516 (2008) 859 – 862 www.elsevier.com/locate/tsf
Selective-catalyst formation for carbon nanotube growth by local indentation pressure T. Yasui ⁎, Y. Nakai, Y. Onozuka Nagaoka University of Technology, 1603-1 Kamitomioka, Nagaoka, Niigata 940-2188, Japan Available online 14 June 2007
Abstract We studied the selective formation of Co catalyst particles as a function of indentation pressure. We subjected a Co (8 nm thickness)/Si substrate pre-annealed at 600 °C to indentation processing. The catalytic function was confirmed in the indentations by the selective growth of carbon nanotubes (CNTs) at 800 °C. The number density of CNTs against the indentation pressure was investigated against indentation loads for two types of indenter: a Berkovich indenter with a ridge angle of 115° and a Berkovich indenter with a ridge angle of 90°. The pressures above 7 GPa applied by the former indenter enhanced Co atomization acting as a catalyst function for CNT growth (35 CNTs in one indentation). In contrast to this, the number of CNTs was markedly reduced when the latter indenter was used with pressures less than 3 GPa. The pop-out phenomenon was observed in unloading curves at pressures above 7 GPa. These results indicate that metastable Si promotes the self-aggregation of catalyst particles (Co) leading to the selective growth of CNTs within indentations at pressures above 7 GPa. © 2007 Elsevier B.V. All rights reserved. Keywords: Carbon; Catalysis; Phase transitions; Silicides
1. Introduction Since application of carbon nanotubes (CNTs) was first demonstrated in components such as switches and memories [1] in nanoscale electronic devices, research into the practical use of CNTs has been advancing. To date, several studies on processes [2–4] for integrating CNTs have been carried out. However, numerous difficulties still remain in aligning CNTs in arbitrary locations. The processes proposed thus far include the catalyst chemical vapor deposition (CVD) method by which CNT growth is started at catalytic metal particles such as particles of Co [3], Fe [4–6], or Ni [7]. Homma et al. attempted to form a single-wall CNT (SWNT) by forming a Fe catalyst over a Si pillar array formed in advance, and then carrying out catalyst CVD [4]. Ishida et al. formed nanoscale Fe dot arrays by lithographically anchored nanoparticle synthesis (LANS), which is an application of electron-beam (EB) lithography, and then grew CNTs using the Fe dot arrays as catalysts [5]. On the other hand, Chen et al. recently reported CNT selective growth at hand-made randomly positioned ⁎ Corresponding author. Tel./fax: +81 258 47 9706. E-mail address: [email protected] (T. Yasui). 0040-6090/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2007.06.034
grooves using the thermal decomposition of iron phthalocyanine on a silicon substrate [8]. However, simple and low-cost processes are required for practical CNT wiring processes. We have previously applied a nano-indentation method [9] to align catalyst particles and successfully demonstrated selective CNTs formation [10] in each indentation. In this study, we investigated selective catalyst particle formation as a function of local indentation pressure and investigated the mechanism of catalytic function on the Si surface induced by the formation of indentations. 2. Processes First, Co was deposited as a catalyst by ion sputtering forming an 8-nm-thick layer over the entire surface of the Si substrate. Indentations were formed in the surface of the Co/Si substrate using an nano-indentation tester (Elonix ENT-1100) which measures ultra-low displacements while applying a load (P–h curve) via the indenter tip (see Fig. 1); the nano-indentation tester also provides data on the hardness and the Young's modulus of ultra-thin films [11]. The local pressure caused by the indentations formed metastable Si and inhibited Co/Si combinations, causing selective Co aggregation within the indentations.
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T. Yasui et al. / Thin Solid Films 516 (2008) 859–862
Fig. 1. Schematic drawings of an indenter (left) from the direction of the ridge of the tip and a bird's-eye view of the indenter tip (inset). The indenter tip has a threesided pyramid shape with a ridge angle (θ) of 115° for a Berkovich indenter, and that of 90° for a modified Berkovich indenter.
The Co particles in the indentations have been observed to successfully act as a catalyst for selective high-quality CNT growth. This has already been reported in detail in our previous study [10]. Reproducibility of the number density of CNTs within indentations was confirmed using a 5 × 5 array of indentations formed by scanning an X–Y sample stage of the nano-
indentation tester. We used two types of indenter: a Berkovich indenter with a ridge angle (θ in Fig. 1) of 115° (115° indenter) [11] and a Berkovich indenter with a ridge angle of 90° (90° indenter). The maximum indentation loads were in the range of 30 to 50 mN for the 115° indenter and in the range of 10 to 40 mN for
Fig. 2. SEM images of grown CNTs are shown against maximum loads of indentation. CNTs within the indentations were observed for 115° indenter (upper row). In contrast to this, the occurrence of CNTs was much decreased by the decrease in the pressures due to pile-ups and to the presence of cracks for the 90° indenter (lower column).
T. Yasui et al. / Thin Solid Films 516 (2008) 859–862
Fig. 3. P–h curves for the Co/Si substrate using two types of indenter: one with a ridge angle of 90° and another with a ridge angle of 115°. The maximum load was 30 mN. A pop-out phenomenon was observed in the unloading process for 115° indenter.
the 90° indenter. The indentation depth was about 1 μm or less, sufficient to penetrate the Co layer and leave an impression on the Si surface. Next, the sample surface was annealed at 600 °C for 5 min under a He atmosphere (6.2 sccm) to form fine Co particles. An infrared radiation heater (Thermal Riko IVF 298Q) was used for annealing and to facilitate the growth of CNTs [10]. The Co atomization was enhanced at the indented sections, while the combining of Si and Co was enhanced in the nonindented sections by annealing through CVD [10]. Catalyticthermal CVD was performed at 800 °C for 10 min, in which an alcohol vapor of 5 × 102 Pa was supplied. The alcohol vapor was produced by a vaporization system (LINTEC LV-2100), heated to 52 °C, and supplied to the CVD chamber. The number of CNTs within the indentations was observed with a secondary electron microscope (SEM; Hitachi S-4100). 3. Results and discussion Fig. 2 shows SEM images of indentations formed on the Co/ Si surface where catalytic alcohol CVD was performed at 800 °C for 10 min. Images of indentations made with a 115° indenter (upper row) and a 90° indenter (lower column) are shown in relation to maximum indentation load. CNTs grown within the indentation made by the 115° indenter can be clearly observed and the Co catalytic function succeeded in promoting CNT growth. Selective growth in an indented area has been previously confirmed on the basis of radial breathing mode peaks in micro-Raman spectra within a resolution of a few microns [10]. In contrast to this, none, or very few, CNTs were observed within indentations formed with the 90° indenter for indentation loads up to 40 mN. On the basis of the results of our previous study, indent-induced pressures for the latter case are considered to be insufficient to cause the emergence of meta-stable Si phases [12] and facilitate Co self-aggregation [10]. The differences are supported by the discontinuity in the P–h curve, the so-called “pop-out” phenomenon (solid curve in Fig. 3) [13], for the 115° indenter. The pop-out while unloading is characteristically observed for Si [14] when the meta-stable Si
861
phases emerged in the indentations [12]. This phase transformation strongly depends on the indentation pressure. These meta-stable Si phases inhibit Co–silicide formation and enhance Co aggregation [10]. In contrast to this, no distinct anomaly was observed in the P–h curve obtained for the 90° indenter (dotted curve in Fig. 3). Thus, in the latter case, the pressures inside the indentations are considered to be insufficient to cause the unstable Si crystal phases to occur. Fig. 4 shows the number of CNTs formed within indentations against pressure. Pressures for the 115° indenter (closed squares) were obtained from the maximum loads divided by the indentation area. The pressures for the 90° indenter, however, should be corrected because the applied loads were reduced by cracks propagating along an indent (see Fig. 2). Therefore, the pressures (closed circles in Fig. 4) were corrected using a classical cracking model [15]; K r ¼ pffiffiffiffiffiffi pa; where σ is the fracture stress per crack, a is the crack length, and K isthe stress intensity factor having a value of 0.9 1 MPadm2 for Si [16]. We obtained a mean value of “a” and then estimated the fracture stress (3σ) for each indentation. The “pile-up” area (see Fig. 2) as well as the abovementioned cracks causes the pressures to be reduced. Thus, correction of the pressures (corrected values denoted by closed circles in Fig. 4) was performed by subtracting 3σ from the pressure which took into account the pile-up area. The pressures above 7 GPa obtained for a 115° indenter agree with the values where Si phase transformation emerged [12]. This situation promoted Co self-aggregation, and thus CNT growth was enhanced inside the indentations by the Co catalytic functions [10]. On the other hand, the number of CNTs is low or zero in the indentations formed with the 90° indenter (closed circles in Fig. 4). This is because the pressures below 3 GPa (see Fig. 4) were insufficient to cause the Si phase transition reported by Ge
Fig. 4. The number of CNTs within an indentation as a function of pressure for two types of indenter. The pressures are the maximum loads divided by each indentation area for the 115° indenter. The pressures for the 90° indenter (⁎) were corrected for the occurrence of cracks and pile-ups. The errors denote the variation of a 5 × 5 iteration for each condition.
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T. Yasui et al. / Thin Solid Films 516 (2008) 859–862
et al. [12] and to cause Co aggregation [10]. Thermal effects promoted interfacial diffusion of Co and Si and then Co-silicides emerged through the CNT growth process. Thus, Co lost the catalytic function under the lower indentation pressures. 4. Conclusion We observed the occurrence of a catalytic (Co) function induced by indentation pressures above 7 GPa for CNTs growth within the indentations made in a Co thin film/Si system. Indentation pressures below 3 GPa were insufficient to cause the Si phase transition and occurrence of Co self-aggregation, and thus the catalytic function for CNTs growth was inhibited. The latter case was applicable to pressures for a 90° indenter which took into account cracks and pile-ups. The pressure is a controllable core factor in the indent-selective and catalytic growth of CNTs in a Co film/Si system. Acknowledgements We would like to acknowledge the contributions of T. Kasahara and co-workers at Niigata Sanyo Electronics Co., Ltd. for providing high-quality Co–Si substrates. This work is supported by Core Research for Evolutional Science and Technology (CREST), Japan Science and Technology (JST).
References [1] A. Bachtold, P. Hadley, T. Nakanishi, C. Dekker, Science 294 (2001) 1317. [2] W.Z. Li, S.S. Xie, L.X. Qian, B.H. Chang, B.S. Zou, W.Y. Zhou, R.A. Zhao, G. Wang, Science 274 (1996) 1701. [3] M. Terrones, N. Grobert, J.P. Zhang, H. Terrones, J. Olivares, W.K. Hsu, J.P. Hare, A.K. Cheetham, H.W. Kroto, D.R.M. Walton, Chem. Phys. Lett. 285 (1998) 299. [4] Y. Homma, Y. Kobayashi, T. Ogino, T. Yamashita, Appl. Phys. Lett. 81 (2002) 2261. [5] M. Ishida, H. Hongo, F. Nihey, Y. Ochiai, Jpn. J. Appl. Phys. 43 (2004) L1356. [6] S. Fan, M.G. Chapline, N.R. Franklin, T.W. Tombler, A.M. Cassell, H. Dai, Science 283 (1999) 512. [7] J.I. Sohn, C.-J. Chois, S. Lee, T.-Y. Seong, Appl. Phys. Lett. 78 (2001) 3130. [8] Y. Chen, J. Yu, Appl. Phys. Lett. 87 (2005) 033103. [9] T. Yasui, M. Yasuda, D. Nezaki, M. Takata, B.P. Zhang, Y. Segawa, Thin Solid Films 464–465 (2004) 273. [10] T. Yasui, S. Nishimura, Jpn. J. Appl. Phys. 45 (2006) L800. [11] “Metallic materials—instrumented indentation test for hardness and materials parameters—Part 1: test method”, ISO 14577 Part 1: 2002. [12] D. Ge, V. Domnich, Y. Gogotsi, J. Appl. Phys. 95 (2004) 2725. [13] W.C. Oliver, G.M. Pherr, J. Mater. Res. 7 (1992) 1564. [14] H. Saka, A. Shimatani, M. Suganuma, Suprijadi, Philos. Mag., A 82 (2002) 1971. [15] I.N. Sneddon, M. Lowengrub, in: R.F. Drenick, H. Hochstadt, D. Gillette (Eds.), Crack Problems in the Classical Theory of Elasticity, The SIAM Series in Applied Mathematics, John Wiley and Sons, Inc., 1969, p. 1. [16] C.P. Chen, M.H. Leipold, Am. Ceram. Soc. Bull. 59 (1980) 469.
Thin Solid Films 516 (2008) 859 – 862 www.elsevier.com/locate/tsf
Selective-catalyst formation for carbon nanotube growth by local indentation pressure T. Yasui ⁎, Y. Nakai, Y. Onozuka Nagaoka University of Technology, 1603-1 Kamitomioka, Nagaoka, Niigata 940-2188, Japan Available online 14 June 2007
Abstract We studied the selective formation of Co catalyst particles as a function of indentation pressure. We subjected a Co (8 nm thickness)/Si substrate pre-annealed at 600 °C to indentation processing. The catalytic function was confirmed in the indentations by the selective growth of carbon nanotubes (CNTs) at 800 °C. The number density of CNTs against the indentation pressure was investigated against indentation loads for two types of indenter: a Berkovich indenter with a ridge angle of 115° and a Berkovich indenter with a ridge angle of 90°. The pressures above 7 GPa applied by the former indenter enhanced Co atomization acting as a catalyst function for CNT growth (35 CNTs in one indentation). In contrast to this, the number of CNTs was markedly reduced when the latter indenter was used with pressures less than 3 GPa. The pop-out phenomenon was observed in unloading curves at pressures above 7 GPa. These results indicate that metastable Si promotes the self-aggregation of catalyst particles (Co) leading to the selective growth of CNTs within indentations at pressures above 7 GPa. © 2007 Elsevier B.V. All rights reserved. Keywords: Carbon; Catalysis; Phase transitions; Silicides
1. Introduction Since application of carbon nanotubes (CNTs) was first demonstrated in components such as switches and memories [1] in nanoscale electronic devices, research into the practical use of CNTs has been advancing. To date, several studies on processes [2–4] for integrating CNTs have been carried out. However, numerous difficulties still remain in aligning CNTs in arbitrary locations. The processes proposed thus far include the catalyst chemical vapor deposition (CVD) method by which CNT growth is started at catalytic metal particles such as particles of Co [3], Fe [4–6], or Ni [7]. Homma et al. attempted to form a single-wall CNT (SWNT) by forming a Fe catalyst over a Si pillar array formed in advance, and then carrying out catalyst CVD [4]. Ishida et al. formed nanoscale Fe dot arrays by lithographically anchored nanoparticle synthesis (LANS), which is an application of electron-beam (EB) lithography, and then grew CNTs using the Fe dot arrays as catalysts [5]. On the other hand, Chen et al. recently reported CNT selective growth at hand-made randomly positioned ⁎ Corresponding author. Tel./fax: +81 258 47 9706. E-mail address: [email protected] (T. Yasui). 0040-6090/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2007.06.034
grooves using the thermal decomposition of iron phthalocyanine on a silicon substrate [8]. However, simple and low-cost processes are required for practical CNT wiring processes. We have previously applied a nano-indentation method [9] to align catalyst particles and successfully demonstrated selective CNTs formation [10] in each indentation. In this study, we investigated selective catalyst particle formation as a function of local indentation pressure and investigated the mechanism of catalytic function on the Si surface induced by the formation of indentations. 2. Processes First, Co was deposited as a catalyst by ion sputtering forming an 8-nm-thick layer over the entire surface of the Si substrate. Indentations were formed in the surface of the Co/Si substrate using an nano-indentation tester (Elonix ENT-1100) which measures ultra-low displacements while applying a load (P–h curve) via the indenter tip (see Fig. 1); the nano-indentation tester also provides data on the hardness and the Young's modulus of ultra-thin films [11]. The local pressure caused by the indentations formed metastable Si and inhibited Co/Si combinations, causing selective Co aggregation within the indentations.
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T. Yasui et al. / Thin Solid Films 516 (2008) 859–862
Fig. 1. Schematic drawings of an indenter (left) from the direction of the ridge of the tip and a bird's-eye view of the indenter tip (inset). The indenter tip has a threesided pyramid shape with a ridge angle (θ) of 115° for a Berkovich indenter, and that of 90° for a modified Berkovich indenter.
The Co particles in the indentations have been observed to successfully act as a catalyst for selective high-quality CNT growth. This has already been reported in detail in our previous study [10]. Reproducibility of the number density of CNTs within indentations was confirmed using a 5 × 5 array of indentations formed by scanning an X–Y sample stage of the nano-
indentation tester. We used two types of indenter: a Berkovich indenter with a ridge angle (θ in Fig. 1) of 115° (115° indenter) [11] and a Berkovich indenter with a ridge angle of 90° (90° indenter). The maximum indentation loads were in the range of 30 to 50 mN for the 115° indenter and in the range of 10 to 40 mN for
Fig. 2. SEM images of grown CNTs are shown against maximum loads of indentation. CNTs within the indentations were observed for 115° indenter (upper row). In contrast to this, the occurrence of CNTs was much decreased by the decrease in the pressures due to pile-ups and to the presence of cracks for the 90° indenter (lower column).
T. Yasui et al. / Thin Solid Films 516 (2008) 859–862
Fig. 3. P–h curves for the Co/Si substrate using two types of indenter: one with a ridge angle of 90° and another with a ridge angle of 115°. The maximum load was 30 mN. A pop-out phenomenon was observed in the unloading process for 115° indenter.
the 90° indenter. The indentation depth was about 1 μm or less, sufficient to penetrate the Co layer and leave an impression on the Si surface. Next, the sample surface was annealed at 600 °C for 5 min under a He atmosphere (6.2 sccm) to form fine Co particles. An infrared radiation heater (Thermal Riko IVF 298Q) was used for annealing and to facilitate the growth of CNTs [10]. The Co atomization was enhanced at the indented sections, while the combining of Si and Co was enhanced in the nonindented sections by annealing through CVD [10]. Catalyticthermal CVD was performed at 800 °C for 10 min, in which an alcohol vapor of 5 × 102 Pa was supplied. The alcohol vapor was produced by a vaporization system (LINTEC LV-2100), heated to 52 °C, and supplied to the CVD chamber. The number of CNTs within the indentations was observed with a secondary electron microscope (SEM; Hitachi S-4100). 3. Results and discussion Fig. 2 shows SEM images of indentations formed on the Co/ Si surface where catalytic alcohol CVD was performed at 800 °C for 10 min. Images of indentations made with a 115° indenter (upper row) and a 90° indenter (lower column) are shown in relation to maximum indentation load. CNTs grown within the indentation made by the 115° indenter can be clearly observed and the Co catalytic function succeeded in promoting CNT growth. Selective growth in an indented area has been previously confirmed on the basis of radial breathing mode peaks in micro-Raman spectra within a resolution of a few microns [10]. In contrast to this, none, or very few, CNTs were observed within indentations formed with the 90° indenter for indentation loads up to 40 mN. On the basis of the results of our previous study, indent-induced pressures for the latter case are considered to be insufficient to cause the emergence of meta-stable Si phases [12] and facilitate Co self-aggregation [10]. The differences are supported by the discontinuity in the P–h curve, the so-called “pop-out” phenomenon (solid curve in Fig. 3) [13], for the 115° indenter. The pop-out while unloading is characteristically observed for Si [14] when the meta-stable Si
861
phases emerged in the indentations [12]. This phase transformation strongly depends on the indentation pressure. These meta-stable Si phases inhibit Co–silicide formation and enhance Co aggregation [10]. In contrast to this, no distinct anomaly was observed in the P–h curve obtained for the 90° indenter (dotted curve in Fig. 3). Thus, in the latter case, the pressures inside the indentations are considered to be insufficient to cause the unstable Si crystal phases to occur. Fig. 4 shows the number of CNTs formed within indentations against pressure. Pressures for the 115° indenter (closed squares) were obtained from the maximum loads divided by the indentation area. The pressures for the 90° indenter, however, should be corrected because the applied loads were reduced by cracks propagating along an indent (see Fig. 2). Therefore, the pressures (closed circles in Fig. 4) were corrected using a classical cracking model [15]; K r ¼ pffiffiffiffiffiffi pa; where σ is the fracture stress per crack, a is the crack length, and K isthe stress intensity factor having a value of 0.9 1 MPadm2 for Si [16]. We obtained a mean value of “a” and then estimated the fracture stress (3σ) for each indentation. The “pile-up” area (see Fig. 2) as well as the abovementioned cracks causes the pressures to be reduced. Thus, correction of the pressures (corrected values denoted by closed circles in Fig. 4) was performed by subtracting 3σ from the pressure which took into account the pile-up area. The pressures above 7 GPa obtained for a 115° indenter agree with the values where Si phase transformation emerged [12]. This situation promoted Co self-aggregation, and thus CNT growth was enhanced inside the indentations by the Co catalytic functions [10]. On the other hand, the number of CNTs is low or zero in the indentations formed with the 90° indenter (closed circles in Fig. 4). This is because the pressures below 3 GPa (see Fig. 4) were insufficient to cause the Si phase transition reported by Ge
Fig. 4. The number of CNTs within an indentation as a function of pressure for two types of indenter. The pressures are the maximum loads divided by each indentation area for the 115° indenter. The pressures for the 90° indenter (⁎) were corrected for the occurrence of cracks and pile-ups. The errors denote the variation of a 5 × 5 iteration for each condition.
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T. Yasui et al. / Thin Solid Films 516 (2008) 859–862
et al. [12] and to cause Co aggregation [10]. Thermal effects promoted interfacial diffusion of Co and Si and then Co-silicides emerged through the CNT growth process. Thus, Co lost the catalytic function under the lower indentation pressures. 4. Conclusion We observed the occurrence of a catalytic (Co) function induced by indentation pressures above 7 GPa for CNTs growth within the indentations made in a Co thin film/Si system. Indentation pressures below 3 GPa were insufficient to cause the Si phase transition and occurrence of Co self-aggregation, and thus the catalytic function for CNTs growth was inhibited. The latter case was applicable to pressures for a 90° indenter which took into account cracks and pile-ups. The pressure is a controllable core factor in the indent-selective and catalytic growth of CNTs in a Co film/Si system. Acknowledgements We would like to acknowledge the contributions of T. Kasahara and co-workers at Niigata Sanyo Electronics Co., Ltd. for providing high-quality Co–Si substrates. This work is supported by Core Research for Evolutional Science and Technology (CREST), Japan Science and Technology (JST).
References [1] A. Bachtold, P. Hadley, T. Nakanishi, C. Dekker, Science 294 (2001) 1317. [2] W.Z. Li, S.S. Xie, L.X. Qian, B.H. Chang, B.S. Zou, W.Y. Zhou, R.A. Zhao, G. Wang, Science 274 (1996) 1701. [3] M. Terrones, N. Grobert, J.P. Zhang, H. Terrones, J. Olivares, W.K. Hsu, J.P. Hare, A.K. Cheetham, H.W. Kroto, D.R.M. Walton, Chem. Phys. Lett. 285 (1998) 299. [4] Y. Homma, Y. Kobayashi, T. Ogino, T. Yamashita, Appl. Phys. Lett. 81 (2002) 2261. [5] M. Ishida, H. Hongo, F. Nihey, Y. Ochiai, Jpn. J. Appl. Phys. 43 (2004) L1356. [6] S. Fan, M.G. Chapline, N.R. Franklin, T.W. Tombler, A.M. Cassell, H. Dai, Science 283 (1999) 512. [7] J.I. Sohn, C.-J. Chois, S. Lee, T.-Y. Seong, Appl. Phys. Lett. 78 (2001) 3130. [8] Y. Chen, J. Yu, Appl. Phys. Lett. 87 (2005) 033103. [9] T. Yasui, M. Yasuda, D. Nezaki, M. Takata, B.P. Zhang, Y. Segawa, Thin Solid Films 464–465 (2004) 273. [10] T. Yasui, S. Nishimura, Jpn. J. Appl. Phys. 45 (2006) L800. [11] “Metallic materials—instrumented indentation test for hardness and materials parameters—Part 1: test method”, ISO 14577 Part 1: 2002. [12] D. Ge, V. Domnich, Y. Gogotsi, J. Appl. Phys. 95 (2004) 2725. [13] W.C. Oliver, G.M. Pherr, J. Mater. Res. 7 (1992) 1564. [14] H. Saka, A. Shimatani, M. Suganuma, Suprijadi, Philos. Mag., A 82 (2002) 1971. [15] I.N. Sneddon, M. Lowengrub, in: R.F. Drenick, H. Hochstadt, D. Gillette (Eds.), Crack Problems in the Classical Theory of Elasticity, The SIAM Series in Applied Mathematics, John Wiley and Sons, Inc., 1969, p. 1. [16] C.P. Chen, M.H. Leipold, Am. Ceram. Soc. Bull. 59 (1980) 469.