Materials Transactions, Vol. 45, No. 5 (2004) pp. 1469 to 1472 Special Issue on Advances in Computational Materials Science and Engineering III #2004 The Japan Institute of Metals

Ab initio Modeling of the Stress-Strain Response of SiAlON (Si6zAlzOzN8z, z ¼ 0:5 and 1)

Cenk Kocer1;2, Naoto Hirosaki1 and Shigenobu Ogata3;4

1Advanced Materials Laboratory, National Institute for Materials Science, Tsukuba 305-0044, Japan 2University of Sydney, School of Physics, Sydney NSW 2006, Australia 3Department of Mechanical Engineering and Systems, Graduate School of Osaka University, Suita 565-0871, Japan 4Handai Frontier Research Center, Graduate School of Osaka University, Suita 565-0871, Japan

A derivative of the Si3N4 is the quaternary SiAlON solid solution. In this paper the characteristic stress-strain response of the - and c-SiAlON phases is investigated using an ab initio computational procedure, for the 11 strain component, where different substitutions of the atomic pairs, Al-O, were performed. From the modeled data the ‘ideal’ strengths and other material constants were estimated for the two polymorphs. Estimates of the elastic constants were found to be in reasonable agreement with existing data.

(Received November 10, 2003; Accepted February 3, 2004) Keywords: SiAlON, mechanical properties, strength, stress-strain curve

1. Introduction z > 4 a supersaturated SiAlON system is obtained. There are two well-known polymorphs of nitride (Si3N4) is a non- ceramic that exhibits , which are the - and -Si3N4 phases. The lattice desirable high temperature properties.1) However, the non- configuration of these two polymorphs have been outlined by oxide ceramic is inherently difficult to produce, since, in many workers,8) and various properties have been inves- general, high temperatures, high pressures and additives are tigated in detail.9–13) Both of the phases have an underlying required during the fabrication process. Nevertheless, the use atomic structure that is hexagonal and only differs along the of oxide additives does produce a variation of silicon nitride, z-axis, the stacking sequence. The -phase, (density of which is the quaternary SiAlON solid solution. The solid 3:183 gcm3) is generally synthesized at ambient pressure solution can be produced using reaction or hot (below 2000 K) and the -phase, the more stable of the two, pressing techniques, utilizing oxide, nitride and oxynitride (density of 3:200 gcm3) is obtained from a transformation powders. The SiAlON solid solution exhibits excellent of the -to-phase at high temperatures.8) Moreover, it is properties, such as excellent resistance to wear, generally believed that both polymorphs can be synthesized and thermal shock, yet can be formed at lower temperatures concurrently over a wide temperature range. Recent studies and exhibits greater thermodynamic stability, making it a have shown that there exists a cubic spinel phase of silicon suitable alternative to silicon nitride. Furthermore, variations nitride. Zerr et al.14) reported, in 1999, the synthesis of a in additional parameters (i.e. the physico-chemical proper- cubic spinel structure, c-Si3N4, with a density of 3:93 ties) in the SiAlON system could produce increased strength, 0:12 gcm3 (23% higher than the -or-phases). wear resistance and chemical inertness, along with other Likewise, studies to date have reported the synthesis of -, advantageous electronic properties.2) - and c-SiAlON. Naturally, the well known single crystal - It is well-known that the mechanical properties and and -SiAlON phases exhibit a hexagonal lattice structure fracture behavior of sintered silicon nitride composite that clearly is derived from the - and -Si3N4 phases. materials, at elevated temperatures above 1200 C, are Furthermore, the existence of c-Si3N4 has led to the directly related to the intergranular glassy phase, which is hypothesis that cubic spinel SiAlON can be produced by an mostly made up of , at triple junctions or between the appropriate process. Recently, the discovery of cubic spinel 3) single crystal Si3N4 grains (generally, -Si3N4 particles). SiAlON (c-Si6zAlzOzN8z, z ¼ 1:8 and 2.8) was reported, Additionally, it is known that the thickness and properties of the cubic phase was synthesized using a shock compression the intergranular phases (including other properties4)) can be method,15) with other workers reporting the use of other altered, using various sintering additives, where in most cases methods.16) rare-earth elements are used. The addition of the elements Various studies have been reported that have employed and into sintered materials effects the computational and experimental methods to understand growth of grains and the strength of the crystalline-glass better the properties of the SiAlON system.17–22) Using interface. In particular, it has been observed that at high various methods the atomic and electronic structure, bulk temperatures the addition of these additives produces a modulus, various mechanical properties and the lattice material, SiAlON, which is easier to densify and exhibits parameters of crystalline -, - and c-SiAlON have been greater ductility. Thus, the SiAlON system could offer further studied. The mechanical response of the SiAlON lattice advances, as a significant material in high strength, high structures to applied strains has not so far been reported. In temperature applications.5,6) The chemical formula that is this paper, an ab initio numerical scheme is employed to used to define the system is given as Si6zAlzOzN8z, where z determine the stress-strain response of the - and c-SiAlON is any value between the limits 0 to approximately 4,7) and for phases. The -phase has not been modeled, at this time, due 1470 C. Kocer, N. Hirosaki and S. Ogata to inherent difficulties in the complex lattice structure. The (a) determination in this case of the equilibrium structure with the stacking appropriate Al and O pair substitutions is of ongoing work. In sequence the following sections the method of calculation is outlined in Si N Al O detail, including an outline of the crystal structures employed A for the two -, -phases. Following this, the results obtained B from the simulation procedure are presented, and finally, these results are briefly discussed. (b) (c)

2. Calculation Method & Results

The equilibrium structure, elastic constants and other properties of the SiAlON polymorphs were determined using the Vienna Ab-initio Simulation Package (VASP).23–25) The VASP package provides for the core region and valence electrons of the atoms in the supercell to be described by the (d) Vanderbilt ultrasoft pseudopotential.26) In addition, the electron-electron exchange interaction was described using Silicon (tetrahedral bond) the generalized gradient approximation (GGA) and the local Silicon (octahedral bond) density approximation (LDA). The GGA employs a Perdew- 27) 28) 91 (PW91) functional form, and a Ceperley-Alder form Atom substitution site is employed in the LDA. In many cases, the two potential functions provide comparable results.29,30) Therefore, in this study, as a matter of convenience, only the LDA results of the x stress-strain data are presented. The numerical integration of the Brillouin zone was performed using a discrete 4 4 4 z y and 6 6 6, for - and c-SiAlON, respectively, Mon- 31) Fig. 1 An illustration of the - and c-Si6zAlzOzN8z single crystal lattice khorst-Pack k-point sampling, and the plane wave cutoff structure. Figures a, b, and c, are the -single crystal lattice for the z ¼ 0:5 17 was chosen as 7:9 10 J. (case 1 and 2) and z ¼ 1 (case 3), respectively. Figure d, is the c-single During the simulation procedure at each step, the applied crystal lattice for z ¼ 1. The pair substitution positions, of Al-O, are strain was increased by a uniform 1%. The strain definition highlighted. used in this study is equivalent to the ‘engineering strain’ definition used by Morris et al.32) The relaxation process, using the conjugate gradient method, was performed for a orthorhombic lattice, made up of two unit cells stacked along peak force at each atomic site of 1:6 1011 J/m, and a peak the c-axis, consisting of a total of 28 atoms. The initial lattice 8 2 10 stress in the supercell of 1 10 kg/m.s . At each subsequent parameters, ao and co, were defined as 7:595 10 and step, the supercell configuration of the previous step was 2:902 1010 m, respectively, from experimental data.35) employed after relaxation of the supercell.33) It is important For the -phase the modeling was restricted to three simple to note that at each step the conjugate gradient method was cases of atom substitution in Si6zAlzOzN8z, where z would performed only after a finite temperature of 1 K was applied be 0.5 and 1. In the first case, z ¼ 0:5, where the first and last to the supercell structure for 1 1013 s. In this case, the atomic positions in the 14 atom Wyckoff unit cell are favored predefined temperature value was selected to provide a for the pair substitution,17) and thus, the substitution is made sufficient amount of energy to the supercell to displace the such that in position 1, N ! O, and in position 14, Si ! Al, 10 atomic configuration by a small amount. All subsequent data see Fig. 1(a), (with an optimized lattice, ao ¼ 7:590 10 10 were calculated for a temperature of 0 K. and co ¼ 5:805 10 m). In the second case z ¼ 0:5 as Clearly, the -SiAlON lattice structure is obtained from well, where the last and fourth atomic positions in the the -Si3N4 lattice structure. As a matter of convenience, in Wyckoff unit cell are favored for the pair substitution, and the following a detailed discussion of the - and c-Si3N4 thus, the substitution is made such that in position 4, N ! O, lattice structures are not presented, the reader is directed to and in position 14, Si ! Al, see Fig. 1(b), (with an optimized 11,12) 10 10 the literature for a detailed discussion. In this work, the lattice, ao ¼ 7:569 10 and co ¼ 5:789 10 m). In underlying -Si3N4 unit cell structure was obtained from the the third case, z ¼ 1, the substitution is made at both the sites hexagonal P63 configuration. It is generally accepted that used in case 1 and 2, and thus, the substitution is made such between the P63=m and P63 configurations, both of which that in position 1 and 4, N ! O, and at position 10 and 14, have been used to describe the -Si3N4 unit cell structure, the Si ! Al, see Fig. 1(c) (with an optimized lattice, ao ¼ 10 10 difference is insignificant and results in similar data of the 7:584 10 and co ¼ 5:822 10 m). structural properties of the single crystal.3,34) In this study the In the case of the cubic spinel structure, the underlying c- relaxed supercell structure was found to resemble the P63=m Si3N4 unit cell structure was obtained from the cubic Fd-3m symmetry configuration. As mentioned, the SiAlON structure space group configuration. In the standard unit cell config- is obtained by a substitution of Si and N pairs with Al and O, uration there are 56 atoms, using the Wyckoff notation, there respectively. The supercell structure was configured as an is one group of Si atoms (octahedral bonds) in the 8a [1/8, ab initio Modeling of the Stress-Strain Response of SiAlON (Si6zAlzOzN8z, z ¼ 0:5 and 1) 1471

1/8,1/8] position, the second group of Si atoms (tetrahedral 70 z = 0.5 [case 1] bonds) in the 16d [1/2,1/2,1/2] position and the N group of 60 z = 0.5 [case 2] z = 1 [case 3] atoms in the 32eð½x; x; x: where x ¼ 0:25 þ ) position. In β 50 - Si3N4

this work a supercell of 14 atoms was employed, and the c- GPa] SiAlON structure is obtained by a substitution of a Si and N σ/ 40

pair with Al and O, respectively. The simulation procedure Stress [ 30 was applied to the simple case of single pair substitution in c- 20 Si6zAlzOzN8z, where z would be 1. In this case the sixth 10 and seventh atomic positions in the 14 atom unit cell are 0 favored for pair substitution (this configuration of the cubic 0 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 structure was shown to be the most stable compared to Strain several other configurations,36) and thus, the substitution is Fig. 2 The 11 induced stresses (in units of GPa) in -Si6zAlzOzN8z, for made such that in position 7, N ! O, and in position 6, Si ! z ¼ 0:5 (case 1 and 2) and z ¼ 1 (case 3), as a function of applied strain Al (Fig. 1(d)). The lattice constants, determined experimen- deformation (for a temperature of 0K). The plot is of the LDA tally, and the optimized value, were originally reported by pseudopotential results only. In addition, data of -Si3N4 single crystal 38) Zerr et al.14) as a ¼ 7:80 0:08 [1010 m] and ¼ 0:074. are also presented. However, in this work the values obtained by Jiang et al.37) were employed for the initial dimensions of the supercell, 60 a ¼ 7:73391 1010 m and ¼ 0:084. The initial supercell c - Si5A lON7 lattice configuration, using the dimensions given by Jiang et 50 c - Si3N4 al.37) was relaxed and thus, for all subsequent calculations the 40 relaxed structure was used as the zero strained equilibrium GPa] 10 σ/ structure (a ¼ 7:7141 10 m). 30

As a matter of brevity the single crystal structure of the - Stress [ and c-SiAlON phases are illustrated in a concise form in Figs. 20 1(a), (b), (c), and (d), with the atomic substitutions clearly 10 illustrated. The crystal structures are, however, well docu- 2,8,14,15) 0 mented in the literature and discussed in detail. For the 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -structure, Figs. 1(a), (b), and (c), the lattice is obtained Strain z through the stacking of the planes ABABAB..., along the - Fig. 3 The 11 induced stresses (in units of GPa) in c-Si6zAlzOzN8z, for axis. The c-structure is only illustrated for the [001] direction z ¼ 1, as a function of applied strain deformation (for a temperature of as a matter of convenience. 0 K). The plot is of the LDA pseudopotential results only. In addition, data 39) For - and c-SiAlON the induced stresses as a function of of c-Si3N4 single crystal are also presented. applied strain deformation in the 11 directions where determined (see Figs. 2 and 3, respectively). It is well known that beyond the point of maximum strain deformation the and 3, respectively (including data calculated for single 38,39) relaxed structure may show increased bond lengths at certain crystal - and c-Si3N4. Beyond the point of maximum bonds, suggestive of fracture. Clearly, beyond the point of tensile stress, in most cases the curve monotonically maximum induced stress, the curve would be expected to decreases indicating the failure of the lattice. The maximum monotonically decrease, indicating failure. However, in some stress data from Figs. 2 and 3, suggests that the introduction cases in this work a clear decrease in the induced stress is not of the Al and O elements does result in a decrease of the observed. It is found that in certain cases the change in the ‘ideal’ strength of the single crystal silicon nitride lattice, in structural configuration is sudden and drastic, and thus, an all cases. It should be noted, previously reported work17) accurate value for the stress at this point is not calculable. indicates that the introduction of a single Al impurity results Nevertheless, this is an indication that the structure has failed in the destabilization of the lattice due to interactions of the and the last calculated stress value is the maximum stress Al element with adjacent N atoms. Thus, to prevent this level for the applied strain component. outcome pair substitution is required, of mutually compen- The elastic constants, bulk and shear moduli, were sating impurities.17) It is also accepted that the Al-O bond is estimated and are given in Table 1: where possible the weaker than the Si-N bond. Therefore, for the SiAlON results are compared to values found in the literature. The polymorphs investigated, it is reasonable to accept, in all 2,3,17) relevant elastic constants are C11, C12, C13, C33 and C44, and cases, a decrease in the ‘ideal’ strength of the lattice. the bulk modulus was defined as [ðC11 þ 2C12Þ=3]. The shear The decrease in the ‘ideal’ strength of the SiAlON modulus was defined as the average of the tetragonal and polymorphs is, to a degree, expected and reasonable. What rhombohedral shear moduli, [ðC11 C12Þ=2] and C44, re- is even more interesting is the characteristic behavior, in the spectively. case of -SiAlON, of the stress-strain curve below the maximum induced stress limit. In Fig. 2, for the 11 applied 3. Discussion strain, the double pair substitution, case 3, exhibits a much lower maximum stress. In cases 1 and 2, the stress-strain As mentioned previously, for - and c-SiAlON the stress response is similar and the maximum stress difference is response as a function of applied strain is presented in Figs. 2 small. This result simply indicates what is expected, a greater 1472 C. Kocer, N. Hirosaki and S. Ogata

Table 1 The elastic constants, shear and bulk moduli estimated (for a temperature of 0 K) for -Si6zAlzOzN8z,(z ¼ 0:5 and 1) and 38Þ c-Si6zAlzOzN8z,(z ¼ 1). Where possible data is compared to values obtained from the literature of silicon nitride (- and 39Þ c- Si3N4).

-Si6zAlzOzN8z 38Þ 39Þ -Si3N4 c-Si5AlON7 c-Si3N4 Case 1 z ¼ 0:5 Case 2 z ¼ 0:5 Case 3 z ¼ 1 Elastic constants (GPa)

C11 405 395 379 423 534 533

C33 537 519 508 538

C12 180 181 179 196 198 191

C13 117 109 116 115

C44 99 88 91 104 309 341 Shear Modulus 105.8 97.5 95.5 111.8 238.5 258–340 (GPa) Bulk Modulus 201.0 190.3 187.0 257.0 310.0 280–411 (GPa) weakness due to the two pair substitution. Unfortunately, the 12) N. Hirosaki, S. Ogata, C. Kocer, H. Kitagawa and Y. Nakamura: Phys. data presented is limited, and thus, definite conclusions are Rev. B 65 (2002) 134110–134121. 13) W. Y. Ching, L. Ouyang and J. D. Gale: Phys. Rev. B 61 (2000) 8696– difficult to derive. In order to gain greater insight, various 8700. other pair substitutions should made, in particular for the c- 14) A. Zerr, G. Miehe, G. Serghiou, M. Schwarz, E. Kroke, R. Riedel, H. SiAlON case, and data of the bond lengths at each step of Fue, P. Kroll and R. Boehler: Nature 400 (1999) 340–342. strain deformation should be analyzed. The authors are 15) T. Sekine, H. He, T. Kobayashi, M. Tansho and K. Kimoto: Chem. currently performing this analysis. Phys. Lett. 344 (2001) 395–399. 16) M. Schwarz, A. Zerr, E. Kroke, G. Miehe, I.-W. Chen, M. Heck, B. Thybusch, B. Poe and R. Riedel: Angew. Chem. Int. Ed. Engl. 41 4. Summary (2002) 789–793. 17) S. V. Okatov and A. L. Ivanovskii: Int. J. Inorg. Mater. 3 (2001) 923– An ab initio density functional numerical technique was 930. employed to determine the induced stresses, in - and c- 18) F. K. Van Dijen, R. Metselaar and R. B. Helmholdt: J. Mater. Sci. Lett. 6 (1987) 1101–1102. SiAlON (Si6zAlzOzN8z, z set to 0.5 and 1), as a function of 19) L. Gillott, N. Cowlam and G. E. Bacon: J. Mater. Sci. 16 (1981) 2263– applied strain deformation, for the 11 strain component. For 2268. -SiAlON, Si6zAlzOzN8z, z set to 0.5 and 1 the ‘ideal’ 20) M. Haviar and Ø. Johannesen: Adv. Ceram. Mater. 3 (1988) 405–407. tensile strength, in the [100] direction, was found to be 43, 46, 21) I. Tanaka, S. Nasu, H. Adachi, Y. Miyamoto and K. Niihara: Acta. and 40 GPa, respectively. For the c-SiAlON phase, Metall. Mater. 40 (1992) 1995–2001. 22) K. Tatsumi, I. Tanaka, H. Adachi and M. Yoshiya: Phys. Rev. B 66 Si6zAlzOzN8z, z set to 1 the ‘ideal’ tensile strength, in the (2002) 165211–165218. [100] direction, was found to be 38 GPa. Furthermore, the 23) G. Kresse and J. Hafner: Phys. Rev. B 49 (1994) 14251–14269. elastic constants of the two polymorphs were also determined 24) G. Kresse and J. Furthmu¨ller: Comp. Mater. Sci. 6 (1996) 15–50. and compared with existing results, which demonstrated 25) G. Kresse and J. Furthmu¨ller: Phys. Rev. B 54 (1996) 11169–11186. reasonable agreement. 26) P. Vanderbilt: Phys. Rev. B 41 (1990) 7892–7895. 27) J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh and C. Fiolhais: Phys. Rev. B 46 (1992) 6671– REFERENCES 6687. 28) D. M. Ceperley and B. J. Alder: Phys. Rev. Lett. 45 (1980) 566–569. 1) F. de Brito Mota, J. F. Justo and A. Fazzio: Phys. Rev. B 58 (1998) 29) E. Soignard, M. Somayazulu, J. Dong, O. F. Sankey and P. F. 8323–8328. McMillan: J. Phys. Condens. Matter 13 (2001) 557–563. 2) T. Ekstro¨m and M. Nygren: J. Am. Ceram. Soc. 75 (1992) 259–276. 30) Y.-N. Xu and W. Y. Ching: Phys. Rev. B 51 (1995) 17379–17389. 3) W.-Y. Ching, M.-Z. Huang and S.-D. Mo: J. Am. Ceram. Soc. 83 31) H. J. Monkhorst and J. D. Pack: Phys. Rev. B 13 (1976) 5188–5192. (2000) 780–786. 32) J. W. Morris Jr., C. R. Krenn, D. Roundy and M. L. Cohen: Mater. Sci. 4) G. Pezzotti, K. Ota and H.-J. Kleebe: J. Am. Ceram. Soc. 79 (1996) Eng. A309–310 (2001) 121–124. 2237–2246. 33) D. Roundy and M. L. Cohen: Phys. Rev. B 64 (2001) 212103–212106. 5) H.-J. Kleebe, M. K. Cinibulk, R. M. Cannon and M. Ru¨hle: J. Am. 34) R. Belkada, M. Kohyama, T. Shibayanagi and M. Naka: Phys. Rev. B Ceram. Soc. 76 (1993) 1969–1977. 65 (2002) 092104–092108. 6) X. Jiang, Y.-K. Baek, S.-M. Lee and S.-J. L. Kang: J. Am. Ceram. Soc. 35) K. Kato, Z. Inoue, K. Kijima, J. Kawada, H. Tanaka and T. Yamane: J. 81 (1998) 1907–1919. Am. Ceram. Soc. 58 (1975) 90–91. 7) L. J. Gauckler, H. L. Lukas and G. Petzow: J. Am. Ceram. Soc. 58 36) L. Ouyang and W. Y. Ching: App. Phys. Lett. 81 (2002) 229–231. (1975) 346–347. 37) J. Z. Jiang, K. Stahl, R. W. Berg, D. J. Frost, T. J. Zhou and P. X. Shi: 8) R. Gru¨n: Acta Crystallogr. B35 (1979) 800–804. Europhys. Lett. 51 (2000) 62–67. 9) A. Y. Liu and M. L. Cohen: Phys. Rev. B 41 (1990) 10727–10734. 38) S. Ogata, N. Hirosaki, C. Kocer and Y. Shibutani: Acta Mater. In press 10) S. Ogata, H. Kitagawa and N. Hirosaki: Proceedings of the 10th (2003). International Congress on Fracture, (Elsevier Science CD-ROM, 39) C. Kocer, N. Hirosaki and S. Ogata: Phys. Rev. B 67 (2003) 035210– ICF100514OR, 2001). 035214. 11) S. Ogata, N. Hirosaki, C. Kocer and H. Kitagawa: Phys. Rev. B 64 (2001) 172102–172106.