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Raman-spectroscopy studies on rigidity percolation and fragility in Ge–(S,Se) glasses
Journal of Non-Crystalline Solids 266±269 (2000) 872±875
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Raman-spectroscopy studies on rigidity percolation and fragility in Ge±(S,Se) glasses Yong Wang a,*, Mitsutaka Nakamura a, Osamu Matsuda b, Kazuo Murase a a
Department of Physics, Graduate School of Science, Osaka University, 1-1 Machikaneyama,Toyonaka, Osaka 560-0043, Japan b Division of Applied Physics, Graduate School of Engineering, Hokkaido University, Sapporo 060-8628, Japan
Abstract We have investigated the Raman spectra in a range 3±100 cmÿ1 in chalcogenide glasses, Gex S1ÿx and Gex Se1ÿx , with 0 6 x 6 0:42 in the average coordination number, hri, from 2 to 2.84. The relation between the rigidity percolation, based on the prediction of the mean-®eld constraint theory, and a relative degree of fragility, representing the structural relaxation of the network, are demonstrated by varying the structures of the glasses. The obtained fragility of the glasses for x < 0:20 hri < 2:4 is larger than that for x > 0:20. A decrease of the fragility at about hri 2:4 is attributed to the predicted rigidity percolation threshold where the glass changes from ¯oppy to rigid, and mesoscopic structural changes beyond the mean-®eld theory. For hri > 2:4, fragility depends on chemical bonds between Ge±Se and Ge±S systems, since the local structure directly contributes to the properties of the overcoordinated network. Ó 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Several papers have been devoted to investigate the germanium chalcogenide glass system Ge± (S,Se) in terms of properties of network varied by changing Ge content [1±8]. The average coordination number, hri, of the glasses varies from 2.0 to 2.8 by adding Ge to Gex Se1ÿx or Gex S1ÿx from x 0 to 0.40. Numerous techniques have been used to investigate short- and medium-range orders in these glasses [1±7]. The medium-range order, although no common understanding is reached, holds the key to the solution of various problems of glasses. A mean-®eld constraint theoretical approach by Phillips and Thorpe [9,10]
* Corresponding author. Tel.: +81-6 6850 5375; fax: +81-6 6850 5376. E-mail address: [email protected] (Y. Wang).
predicts the rigidity percolation threshold at hri 2:4, corresponding to x 0:20 for Gex Se1ÿx and Gex S1ÿx . A qualitative change in the network glasses, from being easily deformable at hri < 2:4 to being rigid at hri > 2:4, has been demonstrated by many experimental results [1±8]. For discriminating relaxation properties of non-crystalline material, particularly the liquid, Angell [11] proposes a fragility index degree depending on the departure from the Arrhenius dependences of the average of the relaxation time. Sokolov et al. [12] proposed two methods to obtain the relative fragility of solid glasses by investigating the Raman scattering spectra at the glass-transition temperature Tg or the speci®c heat at temperatures much less than Tg . In the Ge±As±Se system, the rigidity percolation and fragility have been studied by measuring thermal properties [8]. We also have made an explication of the relation between the rigidity percolation and frigidity in Gex Se1ÿx
0022-3093/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 9 ) 0 0 8 5 8 - 3
Y. Wang et al. / Journal of Non-Crystalline Solids 266±269 (2000) 872±875
873
glasses by low-frequency Raman scattering measurement [7]. In this work, we study the low-frequency Raman spectra of Gex S1ÿx and Gex Se1ÿx glasses in a range 3±100 cmÿ1 . In these two glassy compounds, the bonding of the network is covalent. However, the network properties, such as spectral and actual network dimensionalities, between the Gex S1ÿx and Gex Se1ÿx glasses dier. The relation between the fragility and rigidity percolation in both Gex S1ÿx and Gex Se1ÿx glasses are discussed. The dierent properties in Gex S1ÿx and Gex Se1ÿx are explained from the point of view of medium-range order. 2. Experimental The Gex S1ÿx , Gex Se1ÿx samples, 0 6 x 6 0:40, were prepared by quenching liquids of Ge, S or Se elements in iced water. Raman scattering measurements were performed at room temperature by using an 800 nm excitation light of an argon-ion laser pumped Ti:sapphire laser (Spectra Physics 3900S). The excitation power was less than 10 mW to reduce structural changes. During accumulation of signals, the as-prepared samples were blown by Ar gas to remove the background due to the rotational modes of atmosphere molecules. The scattered light was analyzed with a triple grating polychromator (Jobin Yvon T64000) and detected by a charge-coupled-device (CCD) detector. 3. Results and Discussion Fig. 1 shows the reduced Stokes-Raman intensity, I x I xexp =xn x 1, of the glassy Gex S1ÿx from 3 to 100 cmÿ1 . (Spectra of Gex Se1ÿx glasses can be found in our previous report [7].) The temperature factor, xn x 1, is x1 ÿ exp ÿhx=kT ÿ1 . The reduced intensity is determined by the vibrational density of states (VDOS) through the light-to-vibration coupling coecient, C x, which can be determined by comparison of inelastic neutron- and Ramanscattering spectra. In glassy materials, the low-frequency Raman scattering comes from two kinds of contributions
Fig. 1. Depolarized low-frequency Raman scattering spectra for dierent composition of Gex S1ÿx . The spectra normalized to the intensity at the boson peak maximum (pointed by up arrows) around 20 cmÿ1 . From top to bottom, the spectra are those of x 0:07, 0.10, 0.15, 0.20, 0.25, 0.30, 0.33, 0.35, 0.37, 0.40, 0.42. Down arrows point the minimum discussed in the text.
[13]: quasielastic scattering, which is usually ascribed to some kind of relaxational motion, and the boson peak, a vibrational motion. The boson peak (e.g. Fig. 1 at about 20 cmÿ1 ), observed in the spectra of most glasses, is an experimental demonstration of the properties of the VDOS and not of the frequency-dependence of the C x [12]. It has been recently shown that the quasielastic scattering, dominating spectra at frequencies less than the boson peak, has a relationship to the b process analyzed within a mode coupling theory [14]. In Fig. 1 the relaxational contribution decreases with x, while the vibrational contribution increases with x up to the maximum of the boson
874
Y. Wang et al. / Journal of Non-Crystalline Solids 266±269 (2000) 872±875
peak. Thus, a minimum, below the boson peak, appears due to the overlap of these two contributions. Sokolov et al. [15] used the ratio, Imin =Imax (de®ned as the ratios of the intensities at the minimum to that at the boson peak maximum), at the glass-transition temperature to describe the fragility (i.e., departure from the Arrhenius dependence of viscosity or relaxation time). We have succeeded in explaining the hri-dependence of the relative fragility, Imin =Imax , in Ge±Se system from the point of view of the medium-range order [7]. In this paper, the following discussion will be carried out with regard to the ratio, Imin =Imax , to a relative degree of the fragility of Gex S1ÿx and Gex Se1ÿx glasses. In an undercoordinated network glass hri < 2:4, the mean-®eld constraint theory predicts the existence of ¯oppy-modes, a low-frequency or `zero-frequency' vibrational excitations [9,10]. The inelastic neutron scattering measurements [6] directly detect the ¯oppy modes in composition with hri < 2:4 in Gex Se1ÿx glasses. The reported ¯oppy modes should aect the Raman spectra particularly in the low-frequency region through the light-to-vibrations coupling coecient, C x. In Gex Se1ÿx , as shown in Fig. 2(a), the decrease of fragility was observed at the rigidity percolation threshold of hri 2:4 where the glass changes from a ¯oppy to a rigid structure. The decrease of the low-frequency intensity ratio, Imin =Imax , can be explained by either a larger boson peak or a smaller relaxational distribution which reduces the intensity at the minimum between the Rayleigh line and the boson peak or both. The mechanism of light scattering at these frequencies is still controversial. Nevertheless, the intensity ratio, Imin =Imax , at the percolation threshold, hri 2:4, will be regarded by us as important evidence of the ¯oppy-mode in the undercoordinated and disappearing in the overcoordinated network glasses Gex Se1ÿx . Further, we could imagine beyond the mean-®eld picture that some structural phase transformation occurs even in mesoscopic or intermediate spatial extent [3±5]. In the meantime, the decrease of the fragility in the range 2:0 6 hri 6 2:2, corresponding to the Ge content 0 6 x 6 0:10, is related to an enlargement of
Fig. 2. hri dependence of the ratio, a relative degree of fragility, de®ned as the intensity at the minimum to that at the boson peak maximum. Upper axis shows the corresponding Ge content. The lines are drawn as a guide to the eye.
population of the randomly distributed GeSe4=2 tetrahedra, since such a glass structure is more rigid than that of the Se±Se chains. The increase of the fragility hri > 2:67 x > 0:33 indicates an increase of some kinds of structural disorder. It is probable that in the Ge-rich region with Ge±Ge bonds the structural disorder will also increase [16]. In Gex S1ÿx , a qualitative change of the hri dependence of fragility occurs around hri 2:4, as shown in Fig. 2(b). The turning point near the predicted rigid percolation transition threshold is also assumed to be the phase transition of the network from ¯oppy to rigid. Again, the fragility of the rigid Gex S1ÿx glasses is smaller than that of
Y. Wang et al. / Journal of Non-Crystalline Solids 266±269 (2000) 872±875
the ¯oppy ones. In the S-rich glasses, S8 -ring coexists with the network constructed by GeS4=2 tetrahedra and S±S chains. An expected distribution of relaxation motion, covering the lowfrequency range of the Raman spectra, results from the coexistence of S8 -ring structure which makes the network more fragile. For x < 0:07 in Gex S1ÿx , the relative fragility cannot be obtained by the method used here. The fragility minimum at the stoichiometric composition GeS2 hri 2:67, x 0:33 is assumed to be a result of the chemical eect, in which the stablest structure of the network occurs rather than the mechanical eect. In Gex Se1ÿx the fragility in the range of 2:2 6 hri 6 2:36, where the glass is ¯oppy, remains at about 0.55, and in the range of 2:4 6 hri 6 2:7, where the glass is rigid, is about 0.4. Based on these facts we suggest that the network of Gex Se1ÿx with the hri in these two ranges is an ideal system for demonstrating the Phillips±Thorpe prediction. In Gex S1ÿx near the rigidity percolation threshold, rigid glasses depend on composition. Although in these network systems, Gex S1ÿx and Gex Se1ÿx , the bonding between atoms are mainly covalent, the masses of Ge and Se atoms are almost the same and the mass of S atom is only one third. Based on the dierent properties of the Gex S1ÿx and Gex Se1ÿx glasses in the rigid regions, we suggest that chemical dierences should more directly contribute to the properties of an overcoordinated network [8]. 4. Conclusion We have investigated the low-frequency Raman spectra in a range 3±100 cmÿ1 in glassy Gex S1ÿx and Gex Se1ÿx x 0 ÿ 0:42. A relative degree of the fragility, usually described by the departure of viscosity or relaxation time from an Arrhenius function, has been obtained by an investigation of the line shape of low-frequency Raman spectra. The qualitative changes of the fragility at the average coordination number hri 2:4 x 2:0 is assumed to be evidence for the rigidity threshold predicted by the mean-®eld constraint theory
875
[9,10]. In the rigid region, the fragility has a chemical bond dependence, since the chemical dierences more directly contribute to the properties of overcoordinated network. Acknowledgements This work is partially supported by Grant-inAid for Scienti®c Research No. 09440117, No. 11740173 and Scienti®c Research on Priority Area `Cooperative Phenomena in Complex Liquids' from the Ministry of Education, Science, and Culture, Japan. One of us (Y.W.) acknowledges the support of the Inamori foundation for the Promotion of Science.
References [1] Y. Wang, O. Matsuda, K. Inoue, O. Yamamuro, T. Matsuo, K. Murase, J. Non-Cryst. Solids 232±234 (1998) 702. [2] P. Boolchand, W. Bresser, M. Zhang, Y. Wu, J. Wells, R.N. Enzweiler, J. Non-Cryst. Solids 182 (1995) 143. [3] X. Feng, W.J. Bresser, P. Boolchand, Phys. Rev. Lett. 78 (1997) 4422. [4] M. Taniguchi, T. Kouchi, I. Ono, S. Hosokawa, M. Nakatake, H. Namatame, K. Murase, J. Electron Spectrosc. Related Phenomena 78 (1996) 507. [5] M. Nakamura, O. Matsuda, K. Murase, Phys. Rev. B 57 (1998) 10228. [6] W.A. Kamitakahara, R.L. Cappelletti, P. Boolchand, B. Halfpap, F. Gompf, D.A. Neumann, H. Mutka, Phys. Rev. B 44 (1991) 94. [7] Y. Wang, M. Nakamura, O. Matsuda, K. Murase, Physica B 263&264 (1999) 313. [8] M. Tatsumisago, B.L. Halfpap, J.L. Green, S.M. Lindsay, C.A. Angell, Phys. Rev. Lett. 64 (1990) 1549. [9] J.C. Phillips, J. Non-Cryst. Solids 43 (1981) 37. [10] M.F. Thorpe, J. Non-Cryst. Solids 57 (1983) 355. [11] C.A. Angell, Science 267 (1995) 1924. [12] A.P. Sokolov, A. Kisliuk, D. Quitmann, E. Duval, Phys. Rev. B 48 (1993) 7692. [13] J. J ackle, in: W.A. Phillips (Ed.), Amorphous Solids: LowTemperature Properties, Springer, Berlin, 1981 pp. 135. [14] G. Li, W.M. Du, X.K. Chen, H.Z. Cummins, N.J. Tao, Phys. Rev. A 45 (1992) 3867. [15] A.P. Sokolov, E. R ossler, A. Kisliuk, D. Quitmann, Phys. Rev. Lett. 71 (1993) 2062. [16] Y. Wang, T. Komamine, T. Nakaoka, O. Matsuda, K. Inoue, K. Murase, these Proceedings, p. 904.
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Raman-spectroscopy studies on rigidity percolation and fragility in Ge±(S,Se) glasses Yong Wang a,*, Mitsutaka Nakamura a, Osamu Matsuda b, Kazuo Murase a a
Department of Physics, Graduate School of Science, Osaka University, 1-1 Machikaneyama,Toyonaka, Osaka 560-0043, Japan b Division of Applied Physics, Graduate School of Engineering, Hokkaido University, Sapporo 060-8628, Japan
Abstract We have investigated the Raman spectra in a range 3±100 cmÿ1 in chalcogenide glasses, Gex S1ÿx and Gex Se1ÿx , with 0 6 x 6 0:42 in the average coordination number, hri, from 2 to 2.84. The relation between the rigidity percolation, based on the prediction of the mean-®eld constraint theory, and a relative degree of fragility, representing the structural relaxation of the network, are demonstrated by varying the structures of the glasses. The obtained fragility of the glasses for x < 0:20 hri < 2:4 is larger than that for x > 0:20. A decrease of the fragility at about hri 2:4 is attributed to the predicted rigidity percolation threshold where the glass changes from ¯oppy to rigid, and mesoscopic structural changes beyond the mean-®eld theory. For hri > 2:4, fragility depends on chemical bonds between Ge±Se and Ge±S systems, since the local structure directly contributes to the properties of the overcoordinated network. Ó 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Several papers have been devoted to investigate the germanium chalcogenide glass system Ge± (S,Se) in terms of properties of network varied by changing Ge content [1±8]. The average coordination number, hri, of the glasses varies from 2.0 to 2.8 by adding Ge to Gex Se1ÿx or Gex S1ÿx from x 0 to 0.40. Numerous techniques have been used to investigate short- and medium-range orders in these glasses [1±7]. The medium-range order, although no common understanding is reached, holds the key to the solution of various problems of glasses. A mean-®eld constraint theoretical approach by Phillips and Thorpe [9,10]
* Corresponding author. Tel.: +81-6 6850 5375; fax: +81-6 6850 5376. E-mail address: [email protected] (Y. Wang).
predicts the rigidity percolation threshold at hri 2:4, corresponding to x 0:20 for Gex Se1ÿx and Gex S1ÿx . A qualitative change in the network glasses, from being easily deformable at hri < 2:4 to being rigid at hri > 2:4, has been demonstrated by many experimental results [1±8]. For discriminating relaxation properties of non-crystalline material, particularly the liquid, Angell [11] proposes a fragility index degree depending on the departure from the Arrhenius dependences of the average of the relaxation time. Sokolov et al. [12] proposed two methods to obtain the relative fragility of solid glasses by investigating the Raman scattering spectra at the glass-transition temperature Tg or the speci®c heat at temperatures much less than Tg . In the Ge±As±Se system, the rigidity percolation and fragility have been studied by measuring thermal properties [8]. We also have made an explication of the relation between the rigidity percolation and frigidity in Gex Se1ÿx
0022-3093/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 9 ) 0 0 8 5 8 - 3
Y. Wang et al. / Journal of Non-Crystalline Solids 266±269 (2000) 872±875
873
glasses by low-frequency Raman scattering measurement [7]. In this work, we study the low-frequency Raman spectra of Gex S1ÿx and Gex Se1ÿx glasses in a range 3±100 cmÿ1 . In these two glassy compounds, the bonding of the network is covalent. However, the network properties, such as spectral and actual network dimensionalities, between the Gex S1ÿx and Gex Se1ÿx glasses dier. The relation between the fragility and rigidity percolation in both Gex S1ÿx and Gex Se1ÿx glasses are discussed. The dierent properties in Gex S1ÿx and Gex Se1ÿx are explained from the point of view of medium-range order. 2. Experimental The Gex S1ÿx , Gex Se1ÿx samples, 0 6 x 6 0:40, were prepared by quenching liquids of Ge, S or Se elements in iced water. Raman scattering measurements were performed at room temperature by using an 800 nm excitation light of an argon-ion laser pumped Ti:sapphire laser (Spectra Physics 3900S). The excitation power was less than 10 mW to reduce structural changes. During accumulation of signals, the as-prepared samples were blown by Ar gas to remove the background due to the rotational modes of atmosphere molecules. The scattered light was analyzed with a triple grating polychromator (Jobin Yvon T64000) and detected by a charge-coupled-device (CCD) detector. 3. Results and Discussion Fig. 1 shows the reduced Stokes-Raman intensity, I x I xexp =xn x 1, of the glassy Gex S1ÿx from 3 to 100 cmÿ1 . (Spectra of Gex Se1ÿx glasses can be found in our previous report [7].) The temperature factor, xn x 1, is x1 ÿ exp ÿhx=kT ÿ1 . The reduced intensity is determined by the vibrational density of states (VDOS) through the light-to-vibration coupling coecient, C x, which can be determined by comparison of inelastic neutron- and Ramanscattering spectra. In glassy materials, the low-frequency Raman scattering comes from two kinds of contributions
Fig. 1. Depolarized low-frequency Raman scattering spectra for dierent composition of Gex S1ÿx . The spectra normalized to the intensity at the boson peak maximum (pointed by up arrows) around 20 cmÿ1 . From top to bottom, the spectra are those of x 0:07, 0.10, 0.15, 0.20, 0.25, 0.30, 0.33, 0.35, 0.37, 0.40, 0.42. Down arrows point the minimum discussed in the text.
[13]: quasielastic scattering, which is usually ascribed to some kind of relaxational motion, and the boson peak, a vibrational motion. The boson peak (e.g. Fig. 1 at about 20 cmÿ1 ), observed in the spectra of most glasses, is an experimental demonstration of the properties of the VDOS and not of the frequency-dependence of the C x [12]. It has been recently shown that the quasielastic scattering, dominating spectra at frequencies less than the boson peak, has a relationship to the b process analyzed within a mode coupling theory [14]. In Fig. 1 the relaxational contribution decreases with x, while the vibrational contribution increases with x up to the maximum of the boson
874
Y. Wang et al. / Journal of Non-Crystalline Solids 266±269 (2000) 872±875
peak. Thus, a minimum, below the boson peak, appears due to the overlap of these two contributions. Sokolov et al. [15] used the ratio, Imin =Imax (de®ned as the ratios of the intensities at the minimum to that at the boson peak maximum), at the glass-transition temperature to describe the fragility (i.e., departure from the Arrhenius dependence of viscosity or relaxation time). We have succeeded in explaining the hri-dependence of the relative fragility, Imin =Imax , in Ge±Se system from the point of view of the medium-range order [7]. In this paper, the following discussion will be carried out with regard to the ratio, Imin =Imax , to a relative degree of the fragility of Gex S1ÿx and Gex Se1ÿx glasses. In an undercoordinated network glass hri < 2:4, the mean-®eld constraint theory predicts the existence of ¯oppy-modes, a low-frequency or `zero-frequency' vibrational excitations [9,10]. The inelastic neutron scattering measurements [6] directly detect the ¯oppy modes in composition with hri < 2:4 in Gex Se1ÿx glasses. The reported ¯oppy modes should aect the Raman spectra particularly in the low-frequency region through the light-to-vibrations coupling coecient, C x. In Gex Se1ÿx , as shown in Fig. 2(a), the decrease of fragility was observed at the rigidity percolation threshold of hri 2:4 where the glass changes from a ¯oppy to a rigid structure. The decrease of the low-frequency intensity ratio, Imin =Imax , can be explained by either a larger boson peak or a smaller relaxational distribution which reduces the intensity at the minimum between the Rayleigh line and the boson peak or both. The mechanism of light scattering at these frequencies is still controversial. Nevertheless, the intensity ratio, Imin =Imax , at the percolation threshold, hri 2:4, will be regarded by us as important evidence of the ¯oppy-mode in the undercoordinated and disappearing in the overcoordinated network glasses Gex Se1ÿx . Further, we could imagine beyond the mean-®eld picture that some structural phase transformation occurs even in mesoscopic or intermediate spatial extent [3±5]. In the meantime, the decrease of the fragility in the range 2:0 6 hri 6 2:2, corresponding to the Ge content 0 6 x 6 0:10, is related to an enlargement of
Fig. 2. hri dependence of the ratio, a relative degree of fragility, de®ned as the intensity at the minimum to that at the boson peak maximum. Upper axis shows the corresponding Ge content. The lines are drawn as a guide to the eye.
population of the randomly distributed GeSe4=2 tetrahedra, since such a glass structure is more rigid than that of the Se±Se chains. The increase of the fragility hri > 2:67 x > 0:33 indicates an increase of some kinds of structural disorder. It is probable that in the Ge-rich region with Ge±Ge bonds the structural disorder will also increase [16]. In Gex S1ÿx , a qualitative change of the hri dependence of fragility occurs around hri 2:4, as shown in Fig. 2(b). The turning point near the predicted rigid percolation transition threshold is also assumed to be the phase transition of the network from ¯oppy to rigid. Again, the fragility of the rigid Gex S1ÿx glasses is smaller than that of
Y. Wang et al. / Journal of Non-Crystalline Solids 266±269 (2000) 872±875
the ¯oppy ones. In the S-rich glasses, S8 -ring coexists with the network constructed by GeS4=2 tetrahedra and S±S chains. An expected distribution of relaxation motion, covering the lowfrequency range of the Raman spectra, results from the coexistence of S8 -ring structure which makes the network more fragile. For x < 0:07 in Gex S1ÿx , the relative fragility cannot be obtained by the method used here. The fragility minimum at the stoichiometric composition GeS2 hri 2:67, x 0:33 is assumed to be a result of the chemical eect, in which the stablest structure of the network occurs rather than the mechanical eect. In Gex Se1ÿx the fragility in the range of 2:2 6 hri 6 2:36, where the glass is ¯oppy, remains at about 0.55, and in the range of 2:4 6 hri 6 2:7, where the glass is rigid, is about 0.4. Based on these facts we suggest that the network of Gex Se1ÿx with the hri in these two ranges is an ideal system for demonstrating the Phillips±Thorpe prediction. In Gex S1ÿx near the rigidity percolation threshold, rigid glasses depend on composition. Although in these network systems, Gex S1ÿx and Gex Se1ÿx , the bonding between atoms are mainly covalent, the masses of Ge and Se atoms are almost the same and the mass of S atom is only one third. Based on the dierent properties of the Gex S1ÿx and Gex Se1ÿx glasses in the rigid regions, we suggest that chemical dierences should more directly contribute to the properties of an overcoordinated network [8]. 4. Conclusion We have investigated the low-frequency Raman spectra in a range 3±100 cmÿ1 in glassy Gex S1ÿx and Gex Se1ÿx x 0 ÿ 0:42. A relative degree of the fragility, usually described by the departure of viscosity or relaxation time from an Arrhenius function, has been obtained by an investigation of the line shape of low-frequency Raman spectra. The qualitative changes of the fragility at the average coordination number hri 2:4 x 2:0 is assumed to be evidence for the rigidity threshold predicted by the mean-®eld constraint theory
875
[9,10]. In the rigid region, the fragility has a chemical bond dependence, since the chemical dierences more directly contribute to the properties of overcoordinated network. Acknowledgements This work is partially supported by Grant-inAid for Scienti®c Research No. 09440117, No. 11740173 and Scienti®c Research on Priority Area `Cooperative Phenomena in Complex Liquids' from the Ministry of Education, Science, and Culture, Japan. One of us (Y.W.) acknowledges the support of the Inamori foundation for the Promotion of Science.
References [1] Y. Wang, O. Matsuda, K. Inoue, O. Yamamuro, T. Matsuo, K. Murase, J. Non-Cryst. Solids 232±234 (1998) 702. [2] P. Boolchand, W. Bresser, M. Zhang, Y. Wu, J. Wells, R.N. Enzweiler, J. Non-Cryst. Solids 182 (1995) 143. [3] X. Feng, W.J. Bresser, P. Boolchand, Phys. Rev. Lett. 78 (1997) 4422. [4] M. Taniguchi, T. Kouchi, I. Ono, S. Hosokawa, M. Nakatake, H. Namatame, K. Murase, J. Electron Spectrosc. Related Phenomena 78 (1996) 507. [5] M. Nakamura, O. Matsuda, K. Murase, Phys. Rev. B 57 (1998) 10228. [6] W.A. Kamitakahara, R.L. Cappelletti, P. Boolchand, B. Halfpap, F. Gompf, D.A. Neumann, H. Mutka, Phys. Rev. B 44 (1991) 94. [7] Y. Wang, M. Nakamura, O. Matsuda, K. Murase, Physica B 263&264 (1999) 313. [8] M. Tatsumisago, B.L. Halfpap, J.L. Green, S.M. Lindsay, C.A. Angell, Phys. Rev. Lett. 64 (1990) 1549. [9] J.C. Phillips, J. Non-Cryst. Solids 43 (1981) 37. [10] M.F. Thorpe, J. Non-Cryst. Solids 57 (1983) 355. [11] C.A. Angell, Science 267 (1995) 1924. [12] A.P. Sokolov, A. Kisliuk, D. Quitmann, E. Duval, Phys. Rev. B 48 (1993) 7692. [13] J. J ackle, in: W.A. Phillips (Ed.), Amorphous Solids: LowTemperature Properties, Springer, Berlin, 1981 pp. 135. [14] G. Li, W.M. Du, X.K. Chen, H.Z. Cummins, N.J. Tao, Phys. Rev. A 45 (1992) 3867. [15] A.P. Sokolov, E. R ossler, A. Kisliuk, D. Quitmann, Phys. Rev. Lett. 71 (1993) 2062. [16] Y. Wang, T. Komamine, T. Nakaoka, O. Matsuda, K. Inoue, K. Murase, these Proceedings, p. 904.