Atomic Structure and Diffusion in Amorphous Si B C N by Molecular Dynamics Simulation

Full text

(1)Materials Transactions, Vol. 43, No. 7 (2002) pp. 1506 to 1511 Special Issue on Grain Boundaries, Interfaces, Defects and Localized Quantum Structures in Ceramics c 2002 The Japan Institute of Metals. Atomic Structure and Diffusion in Amorphous Si–B–C–N by Molecular Dynamics Simulation Katsuyuki Matsunaga1 , Yuji Iwamoto2 and Yuichi Ikuhara1 1 2. Engineering Research Institute, School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan Japan Fine Ceramics Center, Nagoya 456-8587, Japan. We carried out molecular dynamics simulation of amorphous silicon nitride containing boron and carbon, in order to investigate the short-range atomic arrangement and diffusion behavior. In amorphous Si–B–N, boron atoms are in a nearly threefold coordinated state with nitrogen atoms, while boron atoms in amorphous Si–B–C–N have bonding with both carbon and nitrogen atoms. Carbon atoms in Si–B–C–N are also bonded to silicon atoms. The self-diffusion constant of nitrogen in Si–B–N becomes much smaller than that in amorphous Si3 N4 . Also, amorphous Si–B–C–N exhibits smaller self-diffusion constants of constituent atoms, even compared to Si–B–N. Addition of boron and carbon is important in decreasing atomic mobility in amorphous Si–B–C–N. This may explain the increased thermal stability of the amorphous state observed experimentally. (Received January 28, 2002; Accepted February 28, 2002) Keywords: molecular dynamics, short-range structure, atomic diffusion, thermal stability. 1. Introduction It is well known that amorphous Si-based nonoxide materials are good candidates for high temperature applications due to their excellent thermal and mechanical properties.1–8) Amorphous Si3 N4 -based materials, such as Si–C–N and Si– B–N, can be prepared via polymer pyrolysis using suitable precursors. One of their interesting properties is that these materials are able to maintain the amorphous state at more than 1500◦ C. It has also been reported that they exhibit excellent oxidation and creep resistance at high temperatures, keeping their amorphous state.9–12) The high temperature stability of amorphous Si3 N4 -based materials strongly depends on additional elements such as boron and carbon. Baldus et al. reported that amorphous Si–B–N and Si–B–C–N materials retain amorphous up to 1700◦ C and 1900◦ C, respectively.5, 6) Riedel et al. showed that amorphous Si–B–C–N remains in the amorphous state up to 2000◦ C.4) Very recently, Wang et al. also reported that amorphous Si–B–C–N keeps its amorphous state up to 2200◦ C.8) These Si–B–N and Si–B–C–N systems exhibit much higher thermal stability, as compared to binary amorphous Si–N.2) Moreover, it is likely that thermal stability of Si–B–C–N is higher than that of Si–B–N. We can say, therefore, that addition of both boron and carbon into the amorphous state is able to improve the thermal stability significantly. Since atomic structures in amorphous Si–B–N and Si–B– C–N should be closely related to the excellent thermal stability, a number of experimental studies on the atomic structures were performed using nuclear magnetic resonance, X-ray diffraction and neutron diffraction techniques.13–15) These experiments indicated that the amorphous network of Si–B–N is composed of SiN4 and BN3 structural units and has a homogeneous distribution of elements through the amorphous state. Heinemann et al. also studied short-range atomic ordering and elemental distribution in amorphous Si–B–N and Si–B–C–N, from their pair distribution functions and high-. resolution elemental mapping using energy-filtering transmission electron spectroscopy.16) They suggested that amorphous Si–B–C–N may contain Si–C, B–C and C–C bonds in addition to Si–N and B–N bonds, although their experimental data did not clearly reveal the presence of these additional bonds in the amorphous network. Due to such a complicated atomic structure of Si–B–C–N, however, characteristics of the amorphous network have not been fully understood. The amorphous state of Si–B–C–N finally crystallizes into thermodynamically stable phases such as Si3 N4 , BN and SiC.17) At high temperatures before crystallization, the amorphous network will undergo rearrangement so as to bring about phase separation into Si3 N4 -rich, BN-rich and SiCrich regions. Since constituent atoms are homogeneously distributed in the amorphous state, diffusion processes are necessary for the crystallization behavior. Riedel et al. revealed that crystallization of amorphous Si3 N4 is rate-controlled by diffusion processes.18) It can be expected, therefore, that addition of boron and carbon will have a great influence on diffusion properties of amorphous Si–B–C–N systems. Bill et al. also suggested that boron has a retarding effect on atomic mobility in amorphous Si–B–C–N, from their experimental observations of the microstructures after crystallization.17) To clarify the physical origin of the unexpected thermal stability of amorphous Si–B–C–N, however, it is necessary to examine the amorphous network and effects of boron and carbon atoms on diffusion processes when these atoms are homogeneously distributed in the amorphous state. In this study, we performed molecular dynamics (MD) simulations of amorphous Si–B–C–N, in order to investigate the atomic structure and diffusion behavior. By comparing with the amorphous systems of Si–N and Si–B–N, effects of addition of boron and carbon on the thermal stability of amorphous Si–B–C–N will be discussed in terms of atomic diffusion at a high temperature..

(2) Atomic Structure and Diffusion in Amorphous Si–B–C–N by Molecular Dynamics Simulation. 2. Computational Procedure We conducted MD simulations of amorphous Si–B–C–N using the program MASPHYC (Fujitsu). Amorphous systems were calculated in the constant-volume and constanttemperature ensemble. We employed systems with chemical compositions of Si3 N4 , Si3 B3 N7 and Si3 B3 C3 N4 . The system of Si3 B3 C3 N4 corresponds to the one where a part of N atoms in Si3 B3 N7 are replaced by carbon atoms. To create an initial structure for MD, a random mixture with a composition of Si3 N4 , Si3 B3 N7 or Si3 B3 C3 N4 in a cubic simulation box was generated, where the density was 2.5 g/cm3 . This is because several experimental studies of precursor-derived amorphous Si3 N4 -based materials showed a density of around 2.5 g/cm3 . The system of Si3 N4 contains 896 atoms, while those of Si3 B3 N7 and Si3 B3 C3 N4 have 832 atoms. To generate amorphous structures from the initial random mixtures, simulations at 8000 K for 4.0 × 10−12 s (4.0 ps) were carried out. Subsequently, systems were equilibrated at 1400 K for 30.0 ps. After equilibration, further MD runs were performed at 1400 K for 120.0 ps. Three-dimensional periodic boundary conditions were imposed on the systems. A time step of 0.2 × 10−15 s (0.2 fs) was employed for all runs. In this study, the Tersoff potential was used to describe atomic interactions in Si–B–C–N systems.19–22) Detailed expressions of the Tersoff potential are given elsewhere.19) This potential model includes three-body interactions between atoms, which represent bond-bending and bond-stretching effects on atomic interactions due to the covalent nature of bonding. This was originally developed for one-component systems such as Si and C, and was later extended to multicomponent systems such as SiC and Si3 N4 .21, 23) In this study, we used available potential parameters in Ref. 24) for Si, B and N, and those in Ref. 25) for C. Although the original definition of the Tersoff potential contains bonding interactions of all possible atomic pairs in a simulated system,21) we took account of bonding interactions of Si–N, Si–C, Si–Si, B–N and B–C in this study. This is because these are expected to dominate in the network structure of amorphous Si–N, Si–B– N and Si–B–C–N according to previous studies.4, 6, 16) Amorphous structures calculated by MD were analyzed in terms of pair distribution functions (PDFs). A total PDF G(r) is calculated as G(r) = ci c j gi − j (r), (1). 1507. to atomic pair i − j can be obtained from R n i − j (r)dr. z i − j = (1/n i ). (4). 0. z i − j means the number of atoms j around atom i in a sphere of radius R. Dynamic properties of atoms were examined by analyzing mean square displacements (msd). The msd of atom i is given by 2 2 [rm (t + t0 ) − rm (t0 )] . (5) msd = δr i = (1/n i ) m. A plot of msd versus time is a measure of the vibrational motion and self-diffusion of atom i in a simulated system. The self-diffusion constant of atomic species i, Di , was calculated from the gradient of the msd plot using the Einstein relation: Di = (1/6)dδr 2 i /dt.. (6). 3. Results and Discussion 3.1 Local atomic coordinates in amorphous Si–B–C–N As a typical example, an atomic structure of amorphous Si3 B3 C3 N4 obtained by the present MD simulation is displayed in Fig. 1. It can be seen that constituent atoms are homogeneously distributed, and three-dimensional periodicity of atomic arrangement as would be found in crystalline material is absent. Structural information about calculated amorphous networks can be obtained by analyzing PDFs, G(r). Calculated G(r) curves of amorphous Si3 N4 , Si3 B3 N7 and Si3 B3 C3 N4 are plotted in Fig. 2. In our previous study, we found that the overall profiles and peak positions of the calculated G(r) curves for amorphous Si3 N4 and Si3 B3 N7 were in a good agreement with experimental data by X-ray diffraction and neutron diffraction measurements.24) The first peak at 0.172 nm in the G(r) curve for amorphous Si3 N4 (Fig.. i, j. where ci = n i /N .. (2). n i and N indicate the number of atomic spieces i and the total number of atoms in a system, respectively. gi − j (r) is the partial PDF for an atomic pair i − j expressed as gi − j (r) = (V /n i n j )(n i − j (r)/4πr 2 ∆r),. (3). where n i − j (r) indicates the number of atomic species in a spherical shell between r − ∆r/2 and r + ∆r/2 centered at atom i. The average coordination number z i − j with respect Fig. 1 Atomic structure of amorphous Si–B–C–N generated by MD..

(3) 1508. K. Matsunaga, Y. Iwamoto and Y. Ikuhara. Fig. 2 Total PDFs of (a) amorphous Si3 N4 , (b) Si3 B3 N7 and (c) Si3 B3 C3 N4 .. 2(a)) corresponds to nearest neighboring Si–N pairs. The Si– N bond length is very close to that reported by experiment (0.173 nm).26) In contrast, a broader peak is located at around 0.30 nm, which is due to Si–Si and N–N atomic pairs. In the case of crystalline Si3 N4 , the interatomic distance between Si atoms at centers of neighboring SiN4 tetrahedra is in the range of 0.28–0.32 nm. The distance of N–N at corners of a SiN4 tetrahedron is also in the range of 0.27–0.31 nm. Since angular distortion of linkages between SiN4 units is possibly included in the amorphous network, the distances of Si– Si and N–N should exhibit wider variation than those in the crystalline state. Thus, these two atomic correlations overlap, forming a broad peak at around 0.30 nm. In the case of amorphous Si3 B3 N7 (Fig. 2(b)), the G(r) curve has an extra peak located at 0.148 nm, as compared to that of amorphous Si3 N4 . This is due to B–N bonding pairs in the amorphous state. Such a peak was also observed by experiment of Hagenmayer et al. (at 0.145 nm).15) The B–N bond length is close to those in hexagonal BN (0.145 nm) and in amorphous BN (0.150 nm).27) In contrast, it is found that the G(r) curve of amorphous Si3 B3 C3 N4 (Fig. 2(c)) is very similar to that of Si3 B3 N7 . Therefore, characteristics of the G(r) curve for amorphous Si3 B3 C3 N4 due to carbon addition cannot be easily understood by a simple comparison with that for amorphous Si3 B3 N7 . In order to examine the G(r) curve of amorphous Si3 B3 C3 N4 in detail, partial PDF curves gi − j (r) for individual atomic pairs are plotted in Fig. 3. The most prominent peak of gB–N (r) is located at 0.150 nm. The strongest peaks of gB–C (r) and gC–C (r) are situated at 0.147 nm. These atomic correlations contribute to the first peak at around 0.15 nm in the G(r) curve of Fig. 2(c). The gSi–N (r) and gSi–C (r) curves have the first peaks at 0.175 nm and at 0.185 nm, respectively.. Fig. 3 Partial PDF curves for amorphous Si3 B3 C3 N4 .. These distances are close to bond lengths of nearest neighboring Si–N pairs in crystalline Si3 N4 (0.174 nm) and nearest neighboring Si–C in crystalline SiC (0.189 nm). Therefore, these two correlations overlap with each other, which results in the second peak at around 0.18 nm in the G(r) curve of Si3 B3 C3 N4 . Beyond 0.20 nm, possible kinds of atomic correlations in the amorphous state merge to form a broad peak. It is noted that the gSi–Si (r) curve in Fig. 3 exhibits the first peak at around 0.24 nm. This indicates that Si–Si bonding pairs about 0.24 nm in length are present in the amorphous network of Si3 B3 C3 N4 . Since Si–Si bonding pairs were also experimentally found in amorphous Si3 N4 ,28) it is plausible that amorphous Si3 B3 C3 N4 contains Si–Si bonds in the network structure. The first peaks of gi − j (r) curves correspond to nearest neighboring i − j pairs. Thus, average coordination numbers z i − j can be obtained by integrating gi − j (r) curves under first peaks according to eq. (4). Table 1 lists average coordination numbers in each amorphous system. The values of z Si–N in Si3 N4 and Si3 B3 N7 were smaller than those by experiment. As stated in the case of Si3 B3 C3 N4 , however, our simulated amorphous structures contain Si–Si bonds. Such Si–Si bonds for Si3 B3 N7 were not considered in the previous experimental study.15) The sum of z Si–N and z Si–Si is 4.00 for Si3 N4 and 3.83 for Si3 B3 N7 , which is close to the value of four as expected if Si atoms form tetrahedral units. As compared to amorphous Si3 N4 and Si3 B3 N7 , the value of z Si–N in amorphous Si3 B3 C3 N4 is smaller. However, Si atoms in Si3 B3 C3 N4 are bonded to C atoms with z Si–C = 1.32. In addition, amorphous Si3 B3 C3 N4 includes Si–Si bonds of.

(4) Atomic Structure and Diffusion in Amorphous Si–B–C–N by Molecular Dynamics Simulation Table 1 Calculated average coordination numbers for atomic pairs in the amorphous systems. Available experimental data15, 27) are also listed in parentheses. Average coordination number, z i − j Atomic Pair, i − j Si–N N–Si Si–Si B–N N–B Si–C B–C C–Si C–B C–C. Si3 N4. Si3 B3 N7. Si3 B3 C3 N4. 2.90 (∼ 3.3) 2.18 (∼ 2.4) 1.10 (∼ 0.6). 2.42 (3.4–3.7) 1.04 (1.4–1.6) 1.41 3.15 (2.8–2.9) 1.35 (1.2). 1.07 0.79 1.44 2.44 1.80 1.32 0.76 1.32 0.76 0.85. length 0.24 nm with z Si–Si = 1.44. This gives a total coordination number of Si of 3.83 (= z Si–N + z Si–C + z Si–Si ), which is a similar value to those in the cases of Si3 N4 and Si3 B3 N7 . Boron in Si3 B3 N7 has a nearly threefold coordinated state with nitrogen (z B–N = 3.15), whereas that in Si3 B3 C3 N4 exhibits a smaller coordination number with nitrogen (z B–N = 2.44). By incorporation of C atoms, however, B–C bonds are present with z B–C = 0.77, so that a total coordination number of B (= z B–N + z B–C ) is 3.21. This indicates that B atoms mainly form threefold coordinated units represented as BNx C y where x + y = 3. Such a local compositional mixture around boron in amorphous Si–B–C–N was not clearly detected by previous experiments. Heinemann et al., however, suggested the presence of B–C bonds in the amorphous network.16) In addition, carbonenriched amorphous Si–B–C–N, which was synthesized from a precursor including a number of B–C bonds, showed that the amorphous state brings about phase separation into Si3 N4 , SiC and BN at high temperatures.4) It is plausible that a local structure around boron changes from BC3 to BN3 at high temperatures. It can be assumed, therefore, that amorphous Si–B–C–N contains a number of BNx C3−x units in an intermediate state before crystallization. 3.2 Diffusion behavior As shown in the previous section, Si and B atoms in amorphous Si3 B3 C3 N4 have bonding with both C and N atoms in the amorphous network. Such a local compositional mixture of atoms is expected to be disadvantageous to transformation of the amorphous state into thermodynamically stable crystalline phases such as Si3 N4 , SiC and BN.17) Phase separation of the amorphous state, which would take place before crystallization, requires rearrangement of the amorphous network and atomic diffusion. In this section, we examine selfdiffusion behavior in amorphous Si–B–C–N. Before going to detailed discussion of atomic diffusion in amorphous Si3 B3 C3 N4 , as a simple example, we begin examining dynamic behavior of atoms in amorphous Si3 N4 . Figure 4 displays atomic trajectories of Si and N at 1400 K for 10.0 ps in crystalline (β-Si3 N4 ) and amorphous Si3 N4 . It can be seen in Figs. 4(a) and (b) that trajectories of Si and N atoms in the. 1509. crystalline state are much localized. This means that these atoms thermally vibrate around their mean atomic positions, and no diffusion can be observed in the perfect Si3 N4 lattice. As compared to this, Si and N atoms in the amorphous state exhibit broader trajectotries (Figs. 4(c) and (d)), indicating that these atoms undergo significant thermal vibration in the amorphous network. Since greater thermal vibration increases probabilities of atoms diffusing through the amorphous network, Si and N atoms have higher atomic mobility than those in the crystalline state. Self-diffusion behavior in the amorphous state can be analyzed from msd curves. Figure 5(a) shows msd curves of Si and N atoms in crystalline and amorphous Si3 N4 . It can be seen that the msd curves of Si and N in the crystalline state exhibit constant values with time. This corresponds to the localized trajectories of these atoms at around their mean atomic positions as shown in Fig. 4(a). In contrast, the msd curves in the amorphous state increase with increasing time, indicating that atomic diffusion takes place in the amorphous state. In particular, N atoms exhibit larger amplitude of the msd curve than that of Si. It can be said, therefore, that N atoms are able to move more considerably than Si atoms through the amorphous state. Due to the presence of B atoms in amorphous Si3 B3 N7 , msd curves of Si and N show different behavior from those in amorphous Si3 N4 . Figure 5(b) displays msd curves of Si, B and N in amorphous Si3 B3 N7 . The msd curves for Si and N increase with time in a similar manner to those for amorphous Si3 N4 . However, Si and N atoms in Si3 B3 N7 show smaller msd values than those in amorphous Si3 N4 . B atoms also show lower msd values, as compared to Si and N atoms. Atomic motion of constituent atoms in Si3 B3 N7 is much restricted due to the presence of boron. In addition, it should be noted in Fig. 5(b) that the msd curve of N becomes smaller than that of Si. It can be said that atomic motion of nitrogen is more reduced than that of Si by incorporation of B atoms. In the case of amorphous Si3 B3 C3 N4 (Fig. 5(c)), the msd curves of constituent atoms show smaller values, as compared to amorphous Si3 N4 and Si3 B3 N7 . C atoms also have as small msd values as B atoms. Thus, thermal displacements of atoms in amorphous Si3 B3 C3 N4 are further reduced by the presence of both boron and carbon. Self-diffusion constants can be obtained from gradients of the msd plots in Fig. 5. Calculated self-diffusion constants Di for each atomic species i are plotted in Fig. 6. As compared to amorphous Si3 N4 , the DN and DSi values are lowered in the case of Si3 B3 N7 . The variation of D N is more significant than that of DSi . It is noted that DB is also smaller than DSi and DN . Moreover, amorphous Si3 B3 C3 N4 exhibits smaller selfdiffusion constants of atoms, even compared to Si3 B3 N7 . The degree of decrease of DSi is as large as that of DN . The overall atomic diffusivity of the amorphous state is much lowered by addition of both boron and carbon. The decrease in DSi and DN because of boron and carbon addition can be attributed to the presence of B–N and Si–C linkages in the amorphous network. As shown in Table 1, B atoms in amorphous Si3 B3 N7 have bonding with N atoms in a threefold coordinated state. In our previous study,24) we found that atomic diffusivity of nitrogen in amorphous Si–B–.

(5) 1510. K. Matsunaga, Y. Iwamoto and Y. Ikuhara. Fig. 4 Trajectories of silicon and nitrogen at 1400 K for 10.0 ps. (a) and (b) correspond to trajectories in crystalline Si3 N4 , while (c) and (d) in amorphous Si3 N4 .. Fig. 5 Mean square displacement (msd) curves of constituent atoms in (a) amorphous Si3 N4 , (b) Si3 B3 N7 and (c) Si3 B3 C3 N4 . In (a), msd curves of silicon and nitrogen in crystalline Si3 N4 (denoted by “Si∗ ” and “N∗ ”) are also plotted for comparison.. N decreases as boron content increases. B–N linkages prevent nitrogen from diffusing through the amorphous state. Since the self-diffusion constant of boron is also smaller than that of nitrogen (see Fig. 6), they act as rigid linkages in the amorphous network. Then, the diffusivity of Si atoms bonded to. B–N linkages also decreases slightly, as shown in Fig. 6. In the case of amorphous Si3 B3 C3 N4 , the coordination number of z B–N is smaller than that in Si3 B3 N7 . Alternatively, carbon is bonded to boron and silicon (see Table 1). Since carbon atoms have bonding mainly with Si, such Si–.

(6) Atomic Structure and Diffusion in Amorphous Si–B–C–N by Molecular Dynamics Simulation. 1511. Si3 B3 N7 is much smaller than that in amorphous Si3 N4 , due to the presence of B–N linkages in the amorphous network. B–N bonds act as rigid linkages, preventing nitrogen atoms from diffusing through the amorphous state. (3) Our simulation shows that amorphous Si3 B3 C3 N4 has smaller self-diffusion constants of constituent atoms, even compared to Si3 B3 N7 . The atomic diffusivities of nitrogen and silicon decrease due to B–N and Si–C linkages in the network, respectively. The combined effect of B–N and Si–C on lowering atomic diffusivity explains the enhanced thermal stability of amorphous Si–B–C–N relative to amorphous Si– N and Si–B–N. REFERENCES. Fig. 6 Self-diffusion constants of atoms in each amorphous system calculated from the present MD simulation.. C bonds can also contribute to reducing atomic diffusivity in the amorphous state. In fact, our previous simulation of amorphous Si–C-N demonstrated that carbon addition leads to decrease in Si diffusivity due to covalent Si–C linkages.29) Thus, Si–C bonds in Si3 B3 C3 N4 also become rigid linkages in the network structure, in addition to B–N bonds. As a result, the values of DSi and DN in Si3 B3 C3 N7 become smaller, even compared to Si3 B3 N7 . When these rigid linkages of B–N and Si–C are homogeneously distributed throughout the amorphous state, rearrangement of the atomic structure, which would occur somewhere around a crystallization temperature, is inhibited. The combined effect of B–N and Si–C on lowering atomic diffusivity results in the enhanced thermal stability of amorphous Si–B–C–N relative to amorphous Si–N and Si–B–N. 4. Conclusions We performed MD simulation of amorphous Si–B–C–N, using the Tersoff potential. We calculated the atomic structures of the amorphous state, and analyzed self-diffusion constants of constituent atoms at 1400 K. The excellent thermal stability of amorphous Si–B–C–N was investigated in terms of atomic diffusivity. The results obtained are summarized as follows. (1) In our simulated structure of amorphous Si3 B3 N7 , boron atoms are nearly threefold coordinated by nitrogen atoms, which is consistent with experiments previously reported. While boron atoms in amorphous Si3 B3 C3 N4 are bonded to both carbon and nitrogen atoms, indicating that a local compositional mixture of carbon and nitrogen is present around boron. Carbon atoms in Si3 B3 C3 N4 also have bonding with silicon atoms. (2) The calculated self-diffusion constant of nitrogen in. 1) D. Seyferth and G. Wiseman: J. Am. Ceram. Soc. 67 (1984) C132-33. 2) R. Riedel and W. Dressler: Ceram. Int. 22 (1996) 233–239. 3) R. Riedel, G. Passing, H. Schönfelder and R. J. Brook: Nature 355 (1992) 714–716. 4) R. Riedel, A. Kienzle, W. Dressler, L. Ruwisch, J. Bill and F. Aldinger: Nature 382 (1996) 796–798. 5) H.-P. Baldus, M. Jansen and O. Wagner: Key Eng. Mater. 89–91 (1994) 75–80. 6) H.-P. Baldus, O. Wagner and M. Jansen: Mat. Res. Soc. Symp. Proc. 271 (1992) pp. 821–826. 7) O. Funayama, H. Nakahara, A. Tezuka, T. Ishii and T. Isoda: J. Mater. Sci. 29 (1994) 2238–2244. 8) Z. C. Wang, F. Aldinger and R. Riedel: J. Am. Ceram. Soc. 84 (2001) 2179–2183. 9) D. Galusek, F. L. Riley and R. Riedel: J. Am. Ceram. Soc. 84 (2001) 1164–1166. 10) R. Riedel, H.-J Kleebe, H. Schönfelder and F. Aldinger: Nature 374 (1995) 526–528. 11) L. An, R. Riedel, C. Konetschny, H.-J. Kleebe and R. Raj: J. Am. Ceram. Soc. 81 (1998) 1349–1352. 12) R. Riedel, L. M. Ruswisch, L. An and R. Raj: J. Am. Ceram. Soc. 81 (1998) 3341–3344. 13) L. Wüllen, U. Müller and M. Jansen: Chem. Mater. 12 (2000) 2347– 2352. 14) G. Jeschke, M. Kroschel and M. Jansen: J. Non-Cryst. Solids 260 (1999) 216–227. 15) R. M. Hagenmayer, U. Müller, C. J. Benmore, J. Neuefeind and M. Jansen: J. Mater. Chem. 9 (1999) 2865–2870. 16) D. Heinemann, W. Assenmacher, W. Mader, K. Kroschel and M. Jansen: J. Mater. Res. 14 (1999) 3746–3753. 17) J. Bill and F. Aldinger: Adv. Mater. 7 (1995) 775–787. 18) R. Riedel and M. Seher: J. Eur. Cerm. Soc. 7 (1991) 21–25. 19) J. Tersoff: Phys. Rev. B 37 (1988) 6991–6999. 20) J. Tersoff: Phys. Rev. Lett. 61 (1988) 2879–2882. 21) J. Tersoff: Phys. Rev. B 38 (1988) 9902–9905. 22) J. Tersoff: Phys. Rev. B 39 (1989) 5566–5568. 23) F. de Brito Mota, J. F. Justo and A. Fazzio: Phys. Rev. B 58 (1998) 8323–8328. 24) K. Matsunaga and Y. Iwamoto: J. Am. Ceram. Soc. 84 (2001) 2213– 2219. 25) K. Matsunaga, C. Fisher and H. Matsubara: Jpn. J. Appl. Phys. 39 (2000) L48–L51. 26) T. Aiyama, T. Fukunaga, K. Niihara, T. Hirai and K. Suzuki: J. NonCryst. Solids 33 (1979) 131–139. 27) H. Grigoriew and J. Leciejewicz: Thin Solid Films 172 (1989) L75–79. 28) M. M. Guraya, H. Ascolani, G. Zampieri, J. I. Cisneros, J. H. Dias da Silva and M. P. Cantao: Phys. Rev. B 42 (1990) 5677–5684. 29) K. Matsunaga, Y. Iwamoto, C. A. J. Fisher and H. Matsubara: J. Ceram. Soc. Jpn. 107 (1999) 1025–1031..

(7)

No results found