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Covalent bond approach to the glass-transition temperature of chalcogenide glasses
J O U R N A L OF
NON-Cm]SOI ELSEVIER
Journal of Non-CrystallineSolids 189 (1995) 141-146
Covalent bond approach to the glass-transition temperature of chalcogenide glasses L. Tich~
a,*
H. Tichfi b
a Joint Laboratory of Solid State Chemistry of the Academy of Sciences of Czech Republic and University of Pardubice, 53009 Pardubice, Czech Republic b University of Pardubice, 532 10 Pardubice, Czech Republic
Received 29 August 1994; revised manuscript received 13 January 1995
Abstract
Good correlation between the glass-transition temperature, Tg, and the overall mean bond energy, (E), of a covalent network of a glass was found for 186 binary and ternary chalcogenide glasses. This correlation satisfies the Arrhenius relation for viscosity where the apparent activation energy of viscosity, E,, is empirically related to the overall mean bond energy. The chemical bond arrangement is probably the main factor influencing Tg in chalcogenide glasses.
I. Introduction
The glass-transition temperature, Tg, or the softening temperature, T~, is one of the most important parameters for characterization of the glassy state. Considerable attention has been devoted to the explanation of the origin of Tg and to the correlation of Tg with other physical or chemical properties of glasses. In chalcogenide glasses these correlations are mostly based on the explicit or implicit assumption that, to reach Tg (or Ts, or the onset of fluidity), one must overcome the cohesive forces responsible for the solid behavior of a glass. One should at least partly destroy the network of a given glassy matrix in such a way that entities of the network are macroscopically movable at T---Tg. Consequently, Tg is
* Corresponding author. Tel: 42-40 47 461. Telefax: +42-40 48 400. E-mail: [email protected].
related to the rigidity of the network which is usually associated with the mean coordination number, ( C N ) , a n d / o r it is related to some typical bond energy or cohesive energy between the atoms or entities of a glass. It was shown (for example by Dembovskii [1]) that for certain chalcogenide glasses Tg-~ 433 ( C N ) , where ( C N ) = . ~ i x i C N i , x i is the atomic fraction of the ith component of a glass and CN i is the coordination number of the ith atom. Another relation has been reported by De Neufville and Rockstad [2] for the correlation between the optical gap, Eg, Tg and (CN). This correlation is based on the Vogel-Fulcher relation for viscosity:
]-L - 1
exp
.
(1)
It is assumed that Tg can be taken as a measure of the onset of diffusion motion and thus it corresponds to a fixed value of viscosity (/X(Tg) = 1013 Poise). In this case Tg can be related to ( C N ) which reflects
0022-3093/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0022-3093(95)00202-2
= 1£0 1
L. Tich~, H. Tichd / Journal of Non-Crystalline Solids 189 (1995) 141-146
142
the number of bonds/atom (i.e., it reflects the connectedness of the network) and to Eg, a measure of the average bond energy. Hence Eq. (1) takes the form Tg = To +
6((CN) -2)Eg 32.2k, '
(2)
where 32.2 = In(IX(Tg)/IX0). The quantities To and 6 are the system parameters for a given class of glasses [2]. Sarrach et al. [3] showed that for G e S e - T e glasses the compositional dependence of Tg can be understood according to models which deal with local covalent bond configurations. It was found [3] that the compositional dependence of Tg can be related to the degree of polymerization and to the optical gap. Both of these together provide a measure of the dissociation energy of the nearest covalent bond in the system [3]. In selenide glasses, or in Se-rich glasses, Se, chains determine Tg and the modifications caused by the addition of impurities are responsible for a change in Tg according to Berkes [4]. Hence it is assumed that the average bond coordination in the system is directly related to Tg such that Tg = ( T g ( S e ) / Z ) ( C N ) .
(3)
For the A s - S e and G e - S e systems, the transform which gives the fraction A s z S e 3 / 2 ( x ' ) or GeSe4/z(X') in terms of As(x) or Ge(x)(x' = x / ( 1 - 3 x / 2 ) for the AsxSe l_x system and x' = x / ( 1 2 x ) for the GexSel_ x system) should be used for calculation of (CN) [4]. Linke [5] assumed that the mean atomization enthalpy, ( H a), reflects the cohesive forces which must be overcome to reach Tg and consequently Tg should be related to ( H a ) . For certain classes of glasses, Linke found Tg at ( H a) [5]. Tanaka [6], assuming an Arrhenius relation for the viscosity
Ix = Ixo exp( E / k s T ) ; Ixo = 10°-10 5 Poise,
Ix(Tg)
= 1013
Poise, (4)
derived the relation ln(Tg) = 1.6(CN) + 2.3.
(5)
In this derivation it is assumed that the activation energy of viscosity, E, is given by the equation E = E0n-1,
(6)
where E o = 0.18 eV is a typical cohesive energy of van der Waals bonding and the term n 1 (n = 5) gives the number of atoms belonging to an atomic unit. Further, it was assumed that fluidity is warranted, e.g., by slipping movements [7] of distorted layers. This means that the van der Waals cohesive force should be overcome for softening of a glass to be reached. However, for chalcogenide glasses with homopolar bonds of the type As-As; G e - G e Tanaka supposed [6] that the fact that Tg is nearly compositionally invariant reflects the fact that, for homopolar bond energy, Eh, the relation E h < Eo n(cN) 1 could hold and consequently the softening of the glass could be due to the breaking of these bonds. We note that values of Tg in A s - S e - T e glasses satisfy this assumption (Tg atE h) [8]. In a recent paper by Sreeram et al. [9], the modified Gibbs-DiMarzio equation in the form Tg = T0/[1 - / 3 ( ( C N ) - 2)]
(7)
has been successfully used to calculate values of Tg of some G e - S b - S e , G e - S b - S e - T e and G e - A s S b - S e - T e glasses. The term ((CN) - 2 ) in Eq. (7) stresses the role of cross-linking, TO is Tg of the non-cross-linked parent chain and /3 is a parameter of the system. If the glass-transition temperature is related to the network rigidity (here the influence of defect states created by broken bonds, VAPS and IVAPS as well as kinetic phenomena are neglected) then Tg should be related not only to the connectedness of the network (which is reflected in (CN)) but should also be related to the quality of connections, i.e., to some bond energy between atoms of the network (in the case of a covalent network) or to some interaction energy between entities (in the case of a molecular solid). Actually, only De Neufville and Rockstad [2], Sarrach et al. [3] and Tanaka [6] took into consideration both the connectedness (related to (CN)) and some energy (related to Eg, see Eq. (2) or related to E 0, see Eq. (6)). In this report we examine the correlation between the glass-transition temperature of 186 chalcogenide glasses and the overall bond energy of the covalent glassy network. Following the results of Sreeram et al. [9] and our recent communication [10], we assume that, since chemical ordering effects seem to be
L. Tich~, 11. Tich6 /Journal of Non-Crystalline Solids 189 (1995) 141-146
pronounced in Tg versus chemical composition dependences, the covalent bond approach can be taken as an acceptable first approximation for a network of chalcogenide glasses.
2. Data and data selection.
For comparative reasons we examine the Tg data of different glassy systems from the literature. Since the data are taken from different sources the different techniques of glass preparation, measurements and determination of Tg values can affect the comparability of the data. However, we suppose that the uncertainty in Tg values for such reasons cannot exceed the value = -T-0.1 Tg. In Table 1 are summarized systems whose Tg data are taken into consideration, the number of Tg values used, references, bonds and bond energies used for the calculation of the overall mean bond energy. In the data selection, two restrictions have been made. First, we take into consideration Tg values of glasses with chalcogen content < 90 at.% because in glasses with higher chalcogen content the matrix is composed by entities of the parent glass ( S e m , S n) and by entities arising from cross-linking. In this case the molecular character of the glassy matrix and its influence on the Tg values perhaps should not be neglected and the covalent bond approach might not be correct. It is in this region that Tanaka's [6] approach should probably be taken into consideration. Second, there are indications that for glasses in the G e - A s - S e system with the content of Se < 55 at.% the radial distribu-
143
tion function [17] gives an estimate of (CN) = 3.63.7. This indicates that the standard assumption C N ( G e ) = 4 , C N ( A s ) = 3 and C N ( S e ) = 2 is not valid. According to Krebs [18], this could be due to the low selenium content. In such cases, in the liquid state the network has to break apart because there are not sufficient Se atoms to form enough G e - S e - A s bridges, the mobility of the atoms increases and hence the atoms are forced to higher coordination numbers. For this reason we take into consideration only Tg values of G e - A s - S e glasses with Se > 55 at.%, where we believe: C N ( G e ) = 4, C N ( A s ) = 3, CN(Se) = 2.
3. Discussion
We assume that to reach Tg one should overcome some energy barrier. There are probably two contributions to this barrier. The first is accompanied by breaking of the network in such a way that mobile entities are created. In the second contribution, most probably a smaller one, some energy is necessary to reorient entities to move. We suppose that the first contribution is the most important and it is relevant to the covalent bond approach. Moreover we suppose that this contribution is proportional to some overall mean bond energy, ( E ) , which is a function of the mean coordination number ((CN)), the type of bonds, the degree of cross-linking and the bond energy, forming a network. Systems examined (Table 1) can be divided into two groups. In the first group (I), for all glasses discussed CN(Se)= 2 is
Table 1 The systems whose Tg data were used, the number of Tg values used, references, chemical bonds and bond energies a System
No. of Tg values
Ref.
Bond
Bond energy
Bond
(eV) As-Se As-S Ge-Se Ge-S Ge-As-Se Ge-Sb-Se Ge-Ga-Se Ge-ln-Se
13 13(10) 10 b 10 42 32 32 37
[4] [10] [10,12] [13] [10] [14] [15] [16]
As-Te As-Se Sb-Se As-S ln-Se Ge-Se Ga-Se Ge-S
1.41 1.8 1.86 2.0 2.09 2.12 2.32 2.4
a Heteropolar bond energies were calculated using Pauling's relation. b The value of Tg for Ge40Se60 of 600 K has been taken from Ref. [12].
Bond energy
Bond
(eV) In-In Sb-Sb As-As Ga-Ga Ge-Ge Se-Se S-S Ge-ln
1.3 1.31 1.38 1.48 1.63 1.9 2.2 1.47
Bond energy
(eV) Ge-Sb Ge-As Ge-Ga
1.48 1.54 1.6
144
L. Tichf~,H. Tichf/Journal of Non-CrystallineSolids 189 (1995) 141-146
fulfilled. This is the case for binary systems A,C1-,(Asx(S, Se)a x and Gex(S, Se)l_ x) and ternary systems AxByC z where B = As or Sb. Since CN(Ge) = 4, CN(As) = CN(Sb) = 3, CN(Se) = CN(S) = 2, the mean coordination numbers are
of cross-linking/atom, Pr for R > 1 and Pp for R
(CN) = x C N ( A ) + (1 - x ) C N ( C )
Pp(II) = ( z C N ( C ) + y ) / ( x + y
(CN) = x C N ( A ) + y C N ( B ) + z C N ( C ) .
(8)
In the second group (II), AxByC z = Gex(Ga,In)ySez, the coordination number of B atoms is assumed to be four [15,19]: C N ( G a ) = C N ( I n ) = 4. Since the electron configuration of Ga and In atoms i s sZp 1, we assume that the fourfold coordination of these atoms is due to a dative bond with a p-lone pair on Se atoms. Consequently some Se atoms should be threefold coordinated. Hence, the mean coordination number for these glasses is
Pr(I, II) = ( x C N ( A ) + y C N ( B ) ) / ( x + y + z ) , Pp(I) = z C N ( C ) / ( x + y + z ) ,
ffSc= Pr Ehb,
R > 1,
ff]c = PpEhb,
R < 1,
(12)
where the average heteropolar bond energy is given by Ehb = [ xCN(A)EA_ c + yCN(B)EB_c] /(xCN(A) + yCN(B)),
+ ( z - y)CN(C),
(9)
where 3y means that there are y threefold-coordinated Se atoms and hence there are ( z - y) twofoldcoordinated Se atoms (CN(C) = 2). We note that Eq. (9) (for z > y, which in our case is fulfilled) agrees with ( C N ) = 8 - 4 x - 3 y - 6 z , (x+y+z=l) calculated on the basis of the formal valence shell model [20]. We showed [10] that, apart from GexAsySz, every system considered (or in the families of the GexSbyS % system) the quantity, R, defined as + yCN(B))
R(II) = ( z C N ( C ) - y ) / ( x C N ( A )
+z).
The symbols I, II stand for groups I and II of glasses discussed above. The mean bond energy of the average cross-linking/atom (/~c) is given by
(CN) = x C N ( A ) + y C N ( B ) + 3y
R(I) = z C N ( C ) / ( x C N ( A )
(11)
+ yCN(B)), (10)
divides a system (or a family of the system) into three parts. For R = 1, the system reaches the stoichiometric composition since only heteropolar bonds are present. It is a point of the chemical threshold where usually maximal values of Tg are observed. For R > 1, the system is chalcogen-rich (r). There are heteropolar bonds and chalcogen-chalcogen bonds present. For R < 1, the system is chalcogenpoor (p). There are only heteropolar bonds and 'metal-metal' bonds present 1. We define the degree 1 Since bulk glasses are considered, a chemical bond ordering model is assumed.
(13)
where EA_ c, EB_ c are the heteropolar bond energies of A - C and B - C heteropolar bonds, (see Table 1). We define the average bond energy per atom of the 'remaining matrix', Erm, as ffSrm= 2(½(CN) - P r ) E c _ c / ( C N ) ,
R>I,
ff~rm = 2(½(CN) - Pp)E< > / ( C N ) ,
R
In Eq. (14), Ec_ c is the homopolar bond energy of S - S or Se-Se bonds (see Table 1) and E<) = (EA_ a + EB_ B + EA_B)/3 is the average bond energy of a 'metal-metal' bond in the chalcogen-poor region. Finally the overall mean bond energy is given by ( E ) =/~c + ffSrm.
(15)
For the 186 glasses summarized in Table 1, we calculated the values of the overall mean bond energy, ( E ) . A plot of Tg versus ( E ) (actually versus ( E ) - 0.9) is shown in Fig. 1. Despite the scatter of Tg values, a good correlation between Tg and ( ( E ) 0.9) is evident. For most of the data the accuracy is not worse than _0.1 Tg. This correlation satisfies the Arrhenius relation for the viscosity in the form -
/X(Tg) = / z 0 exp(Eu/kBTg ).
(16)
Taking /x(Tg) -~ 1 0 1 3 Poise, /z o = 1 0 - 3 Poise and E ~ = ( E ) - 0.9, we obtain Tg--314 ( ( E ) - 0 . 9 ) . The linearization of Tg versus ( E ) data using a
L. Tich~, H. Tich6 /Journal of Non-Crystalline Solids 189 (1995) 141-146
least-square fit gives: Tg = 311 ( ( E ) - 0.9) (see full line in Fig. 1). For comparison, in Fig. 2 the plot of Tg versus (CN) is shown indicating a higher scatter in Tg values. We believe that our result can be taken as an acceptable demonstration that the bonding arrangement mainly determines the value of Tg in chalcogenide glasses. This does not mean that intermolecular interactions have no influence on Tg. There are experimental indications (e.g. from pressure experiments, see, for example, Refs. [21,22]), that intermolecular interactions influence Tg. Moreover, it is perhaps this interaction which plays a role in relaxation processes in the glass-transition region. We believe that this interaction enhances the changes of some properties in the vicinity of Tanaka's threshold [23,24]. However, the overall compositional trend in Tg seems to be influenced mainly by the chemical bond arrangement.
145
GeS 2
,-o 0
750
0
O
650
GeSe 2
O O
~o
OO 0 O
^00;0
0 o
oo o o ~ , ~ 8 q g~oo o
o_
550
oOO
~v
~D
~ 0
0
o o'_
~ 450
0
o
000
As 2 $3 - - , - - i : ~ O o As2
3 q~
q5 o U
350
o6)°n~ ~e
I
2.2 GeS2
-
750
-~ o;
Ge Se 2 ---~o°
0
...~e.~As2S 3
oo 0.6
o
>
The authors are indebted to Professor D.L. Kinser and to the referees for kind comments. This work was supported by the Grant Agency of the Czech Republic under Grant No. 2 0 3 / 9 3 / 0 1 5 5 .
O
I
I
2.8
For 186 glasses, with Tg ranging from ~ 320 to 760 K, a good correlation between Tg and ( E ) was observed in the form: Tg-~311 ( ( E ) - 0.9). This correlation satisfies the Arrhenius relation for viscosity.
~0
350
I
2.6
4. Conclusion
550
450
I
2,4
Fig. 2. The variation of Tg with the mean coordination number, (CN). For data sources, see Table l.
v
650
o o°
stope ~ 311 [K/eV] I
1
1.6 /..E) - 0.9
References
2.6 [eV]
Fig. 1. The variations of Tg with the 'overall' mean bond energy ( ( E ) - 0 . 9 ) . Full line indicates a least-squares fit. The slope of the full line is approximately 311 K / e V . For data sources, see Table 1.
[1] S.A. Dembovskii, Phys. Chem. Glasses 10 (1969) 73. [2] J.P. De Neufville and H.K. Rockstad, in: Amorphous and Liquid Semiconductors, Vol. 1, ed. J. Stuke and W. Brenig (Taylor and Francis, London, 1974) p. 419. [3] D.J. Sarrach, J.P. De Neufville, and W.L. Hawarth, J. NonCryst. Solids 22 (1976) 245.
146
L. Tich~, H. Tichd /Journal of Non-Crystalline Solids 189 (1995) 141-146
[4] J.S. Berkes, in: Proc. 4th Int. Conf. on Physics of Non-Crystalline Solids, 1976, ed. G.H. Frischat (TransTech Publications, Aedermannsdorf, Switzerland, 1977) p. 405. [5] D. Linke, Proc. llth Int. Congr. on Glass, Vol. 1, ed. J. G6tz, Prague, 1977, p. 149. [6] Ke. Tanaka, Solid State Commun. 54 (1985) 807. [7] L.E. Busse, Phys. Rev. B29 (1984) 3639. [8] L. Tich~), H. Tich~ and V. Smrcka, Mater. Lett. 15 (1992) 202. [9] A.N. Sreeram, D.R. Swiler and A.K. Varshneya, J. NonCryst. Solids 127 (1991) 287. [10] L. Tich~ and H. Tichfi, Mater. Lett. 21 (1994) 313. [11] Z.U. Borisova, Glassy Semiconductors (Plenum, New York, 1981). [12] M. Vlfiek, L. Tich#, J. Klikorka and A. Tfiska, J. Mater. Sci. Lett. 7 (1988) 335. [13] J. M~ilek, J. Non-Cryst. Solids 107 (1989) 323.
[14] S. Mahadevan and A. Giridhar, J. Non-Cryst. Solids 143 (1992) 52. [15] A. Giridhar and S. Mahadevan, J. Non-Cryst. Solids 152 (1993) 42. [16] A. Giridhar and S. Mahadevan, J. Non-Cryst. Solids 151 (1992) 245. [17] Yu. G. Pozdniakov and V.M. Pozdniakov, Izv. Acad. Nauk, USSR, Inorg. Mater. 10 (1974) 1932 (in Russian). [18] H. Krebs, Fundamentals of Inorganic Crystal Chemistry (McGraw-Hill, London, 1968) p. 354. [19] J.Z. Liu and P.C. Taylor, J. Non-Cryst. Solids 114 (1989) 25. [20] J.Z. Liu and P.C. Taylor, Solid State Commun. 70 (1989) 81. [21] Ke. Tanaka, Phys. Rev. B30 (1984) 4549. [22] M.I. Klinger, Solid State Commun. 65 (1988) 949. [23] Ke. Tanaka, Phys. Rev. B39 (1989) 1270. [24] E. Vateva, E. Skordeva and D. Arsova, Philos. Mag. B67 (1993) 225.
NON-Cm]SOI ELSEVIER
Journal of Non-CrystallineSolids 189 (1995) 141-146
Covalent bond approach to the glass-transition temperature of chalcogenide glasses L. Tich~
a,*
H. Tichfi b
a Joint Laboratory of Solid State Chemistry of the Academy of Sciences of Czech Republic and University of Pardubice, 53009 Pardubice, Czech Republic b University of Pardubice, 532 10 Pardubice, Czech Republic
Received 29 August 1994; revised manuscript received 13 January 1995
Abstract
Good correlation between the glass-transition temperature, Tg, and the overall mean bond energy, (E), of a covalent network of a glass was found for 186 binary and ternary chalcogenide glasses. This correlation satisfies the Arrhenius relation for viscosity where the apparent activation energy of viscosity, E,, is empirically related to the overall mean bond energy. The chemical bond arrangement is probably the main factor influencing Tg in chalcogenide glasses.
I. Introduction
The glass-transition temperature, Tg, or the softening temperature, T~, is one of the most important parameters for characterization of the glassy state. Considerable attention has been devoted to the explanation of the origin of Tg and to the correlation of Tg with other physical or chemical properties of glasses. In chalcogenide glasses these correlations are mostly based on the explicit or implicit assumption that, to reach Tg (or Ts, or the onset of fluidity), one must overcome the cohesive forces responsible for the solid behavior of a glass. One should at least partly destroy the network of a given glassy matrix in such a way that entities of the network are macroscopically movable at T---Tg. Consequently, Tg is
* Corresponding author. Tel: 42-40 47 461. Telefax: +42-40 48 400. E-mail: [email protected].
related to the rigidity of the network which is usually associated with the mean coordination number, ( C N ) , a n d / o r it is related to some typical bond energy or cohesive energy between the atoms or entities of a glass. It was shown (for example by Dembovskii [1]) that for certain chalcogenide glasses Tg-~ 433 ( C N ) , where ( C N ) = . ~ i x i C N i , x i is the atomic fraction of the ith component of a glass and CN i is the coordination number of the ith atom. Another relation has been reported by De Neufville and Rockstad [2] for the correlation between the optical gap, Eg, Tg and (CN). This correlation is based on the Vogel-Fulcher relation for viscosity:
]-L - 1
exp
.
(1)
It is assumed that Tg can be taken as a measure of the onset of diffusion motion and thus it corresponds to a fixed value of viscosity (/X(Tg) = 1013 Poise). In this case Tg can be related to ( C N ) which reflects
0022-3093/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0022-3093(95)00202-2
= 1£0 1
L. Tich~, H. Tichd / Journal of Non-Crystalline Solids 189 (1995) 141-146
142
the number of bonds/atom (i.e., it reflects the connectedness of the network) and to Eg, a measure of the average bond energy. Hence Eq. (1) takes the form Tg = To +
6((CN) -2)Eg 32.2k, '
(2)
where 32.2 = In(IX(Tg)/IX0). The quantities To and 6 are the system parameters for a given class of glasses [2]. Sarrach et al. [3] showed that for G e S e - T e glasses the compositional dependence of Tg can be understood according to models which deal with local covalent bond configurations. It was found [3] that the compositional dependence of Tg can be related to the degree of polymerization and to the optical gap. Both of these together provide a measure of the dissociation energy of the nearest covalent bond in the system [3]. In selenide glasses, or in Se-rich glasses, Se, chains determine Tg and the modifications caused by the addition of impurities are responsible for a change in Tg according to Berkes [4]. Hence it is assumed that the average bond coordination in the system is directly related to Tg such that Tg = ( T g ( S e ) / Z ) ( C N ) .
(3)
For the A s - S e and G e - S e systems, the transform which gives the fraction A s z S e 3 / 2 ( x ' ) or GeSe4/z(X') in terms of As(x) or Ge(x)(x' = x / ( 1 - 3 x / 2 ) for the AsxSe l_x system and x' = x / ( 1 2 x ) for the GexSel_ x system) should be used for calculation of (CN) [4]. Linke [5] assumed that the mean atomization enthalpy, ( H a), reflects the cohesive forces which must be overcome to reach Tg and consequently Tg should be related to ( H a ) . For certain classes of glasses, Linke found Tg at ( H a) [5]. Tanaka [6], assuming an Arrhenius relation for the viscosity
Ix = Ixo exp( E / k s T ) ; Ixo = 10°-10 5 Poise,
Ix(Tg)
= 1013
Poise, (4)
derived the relation ln(Tg) = 1.6(CN) + 2.3.
(5)
In this derivation it is assumed that the activation energy of viscosity, E, is given by the equation E = E0n
(6)
where E o = 0.18 eV is a typical cohesive energy of van der Waals bonding and the term n
(7)
has been successfully used to calculate values of Tg of some G e - S b - S e , G e - S b - S e - T e and G e - A s S b - S e - T e glasses. The term ((CN) - 2 ) in Eq. (7) stresses the role of cross-linking, TO is Tg of the non-cross-linked parent chain and /3 is a parameter of the system. If the glass-transition temperature is related to the network rigidity (here the influence of defect states created by broken bonds, VAPS and IVAPS as well as kinetic phenomena are neglected) then Tg should be related not only to the connectedness of the network (which is reflected in (CN)) but should also be related to the quality of connections, i.e., to some bond energy between atoms of the network (in the case of a covalent network) or to some interaction energy between entities (in the case of a molecular solid). Actually, only De Neufville and Rockstad [2], Sarrach et al. [3] and Tanaka [6] took into consideration both the connectedness (related to (CN)) and some energy (related to Eg, see Eq. (2) or related to E 0, see Eq. (6)). In this report we examine the correlation between the glass-transition temperature of 186 chalcogenide glasses and the overall bond energy of the covalent glassy network. Following the results of Sreeram et al. [9] and our recent communication [10], we assume that, since chemical ordering effects seem to be
L. Tich~, 11. Tich6 /Journal of Non-Crystalline Solids 189 (1995) 141-146
pronounced in Tg versus chemical composition dependences, the covalent bond approach can be taken as an acceptable first approximation for a network of chalcogenide glasses.
2. Data and data selection.
For comparative reasons we examine the Tg data of different glassy systems from the literature. Since the data are taken from different sources the different techniques of glass preparation, measurements and determination of Tg values can affect the comparability of the data. However, we suppose that the uncertainty in Tg values for such reasons cannot exceed the value = -T-0.1 Tg. In Table 1 are summarized systems whose Tg data are taken into consideration, the number of Tg values used, references, bonds and bond energies used for the calculation of the overall mean bond energy. In the data selection, two restrictions have been made. First, we take into consideration Tg values of glasses with chalcogen content < 90 at.% because in glasses with higher chalcogen content the matrix is composed by entities of the parent glass ( S e m , S n) and by entities arising from cross-linking. In this case the molecular character of the glassy matrix and its influence on the Tg values perhaps should not be neglected and the covalent bond approach might not be correct. It is in this region that Tanaka's [6] approach should probably be taken into consideration. Second, there are indications that for glasses in the G e - A s - S e system with the content of Se < 55 at.% the radial distribu-
143
tion function [17] gives an estimate of (CN) = 3.63.7. This indicates that the standard assumption C N ( G e ) = 4 , C N ( A s ) = 3 and C N ( S e ) = 2 is not valid. According to Krebs [18], this could be due to the low selenium content. In such cases, in the liquid state the network has to break apart because there are not sufficient Se atoms to form enough G e - S e - A s bridges, the mobility of the atoms increases and hence the atoms are forced to higher coordination numbers. For this reason we take into consideration only Tg values of G e - A s - S e glasses with Se > 55 at.%, where we believe: C N ( G e ) = 4, C N ( A s ) = 3, CN(Se) = 2.
3. Discussion
We assume that to reach Tg one should overcome some energy barrier. There are probably two contributions to this barrier. The first is accompanied by breaking of the network in such a way that mobile entities are created. In the second contribution, most probably a smaller one, some energy is necessary to reorient entities to move. We suppose that the first contribution is the most important and it is relevant to the covalent bond approach. Moreover we suppose that this contribution is proportional to some overall mean bond energy, ( E ) , which is a function of the mean coordination number ((CN)), the type of bonds, the degree of cross-linking and the bond energy, forming a network. Systems examined (Table 1) can be divided into two groups. In the first group (I), for all glasses discussed CN(Se)= 2 is
Table 1 The systems whose Tg data were used, the number of Tg values used, references, chemical bonds and bond energies a System
No. of Tg values
Ref.
Bond
Bond energy
Bond
(eV) As-Se As-S Ge-Se Ge-S Ge-As-Se Ge-Sb-Se Ge-Ga-Se Ge-ln-Se
13 13(10) 10 b 10 42 32 32 37
[4] [10] [10,12] [13] [10] [14] [15] [16]
As-Te As-Se Sb-Se As-S ln-Se Ge-Se Ga-Se Ge-S
1.41 1.8 1.86 2.0 2.09 2.12 2.32 2.4
a Heteropolar bond energies were calculated using Pauling's relation. b The value of Tg for Ge40Se60 of 600 K has been taken from Ref. [12].
Bond energy
Bond
(eV) In-In Sb-Sb As-As Ga-Ga Ge-Ge Se-Se S-S Ge-ln
1.3 1.31 1.38 1.48 1.63 1.9 2.2 1.47
Bond energy
(eV) Ge-Sb Ge-As Ge-Ga
1.48 1.54 1.6
144
L. Tichf~,H. Tichf/Journal of Non-CrystallineSolids 189 (1995) 141-146
fulfilled. This is the case for binary systems A,C1-,(Asx(S, Se)a x and Gex(S, Se)l_ x) and ternary systems AxByC z where B = As or Sb. Since CN(Ge) = 4, CN(As) = CN(Sb) = 3, CN(Se) = CN(S) = 2, the mean coordination numbers are
of cross-linking/atom, Pr for R > 1 and Pp for R
(CN) = x C N ( A ) + (1 - x ) C N ( C )
Pp(II) = ( z C N ( C ) + y ) / ( x + y
(CN) = x C N ( A ) + y C N ( B ) + z C N ( C ) .
(8)
In the second group (II), AxByC z = Gex(Ga,In)ySez, the coordination number of B atoms is assumed to be four [15,19]: C N ( G a ) = C N ( I n ) = 4. Since the electron configuration of Ga and In atoms i s sZp 1, we assume that the fourfold coordination of these atoms is due to a dative bond with a p-lone pair on Se atoms. Consequently some Se atoms should be threefold coordinated. Hence, the mean coordination number for these glasses is
Pr(I, II) = ( x C N ( A ) + y C N ( B ) ) / ( x + y + z ) , Pp(I) = z C N ( C ) / ( x + y + z ) ,
ffSc= Pr Ehb,
R > 1,
ff]c = PpEhb,
R < 1,
(12)
where the average heteropolar bond energy is given by Ehb = [ xCN(A)EA_ c + yCN(B)EB_c] /(xCN(A) + yCN(B)),
+ ( z - y)CN(C),
(9)
where 3y means that there are y threefold-coordinated Se atoms and hence there are ( z - y) twofoldcoordinated Se atoms (CN(C) = 2). We note that Eq. (9) (for z > y, which in our case is fulfilled) agrees with ( C N ) = 8 - 4 x - 3 y - 6 z , (x+y+z=l) calculated on the basis of the formal valence shell model [20]. We showed [10] that, apart from GexAsySz, every system considered (or in the families of the GexSbyS % system) the quantity, R, defined as + yCN(B))
R(II) = ( z C N ( C ) - y ) / ( x C N ( A )
+z).
The symbols I, II stand for groups I and II of glasses discussed above. The mean bond energy of the average cross-linking/atom (/~c) is given by
(CN) = x C N ( A ) + y C N ( B ) + 3y
R(I) = z C N ( C ) / ( x C N ( A )
(11)
+ yCN(B)), (10)
divides a system (or a family of the system) into three parts. For R = 1, the system reaches the stoichiometric composition since only heteropolar bonds are present. It is a point of the chemical threshold where usually maximal values of Tg are observed. For R > 1, the system is chalcogen-rich (r). There are heteropolar bonds and chalcogen-chalcogen bonds present. For R < 1, the system is chalcogenpoor (p). There are only heteropolar bonds and 'metal-metal' bonds present 1. We define the degree 1 Since bulk glasses are considered, a chemical bond ordering model is assumed.
(13)
where EA_ c, EB_ c are the heteropolar bond energies of A - C and B - C heteropolar bonds, (see Table 1). We define the average bond energy per atom of the 'remaining matrix', Erm, as ffSrm= 2(½(CN) - P r ) E c _ c / ( C N ) ,
R>I,
ff~rm = 2(½(CN) - Pp)E< > / ( C N ) ,
R
In Eq. (14), Ec_ c is the homopolar bond energy of S - S or Se-Se bonds (see Table 1) and E<) = (EA_ a + EB_ B + EA_B)/3 is the average bond energy of a 'metal-metal' bond in the chalcogen-poor region. Finally the overall mean bond energy is given by ( E ) =/~c + ffSrm.
(15)
For the 186 glasses summarized in Table 1, we calculated the values of the overall mean bond energy, ( E ) . A plot of Tg versus ( E ) (actually versus ( E ) - 0.9) is shown in Fig. 1. Despite the scatter of Tg values, a good correlation between Tg and ( ( E ) 0.9) is evident. For most of the data the accuracy is not worse than _0.1 Tg. This correlation satisfies the Arrhenius relation for the viscosity in the form -
/X(Tg) = / z 0 exp(Eu/kBTg ).
(16)
Taking /x(Tg) -~ 1 0 1 3 Poise, /z o = 1 0 - 3 Poise and E ~ = ( E ) - 0.9, we obtain Tg--314 ( ( E ) - 0 . 9 ) . The linearization of Tg versus ( E ) data using a
L. Tich~, H. Tich6 /Journal of Non-Crystalline Solids 189 (1995) 141-146
least-square fit gives: Tg = 311 ( ( E ) - 0.9) (see full line in Fig. 1). For comparison, in Fig. 2 the plot of Tg versus (CN) is shown indicating a higher scatter in Tg values. We believe that our result can be taken as an acceptable demonstration that the bonding arrangement mainly determines the value of Tg in chalcogenide glasses. This does not mean that intermolecular interactions have no influence on Tg. There are experimental indications (e.g. from pressure experiments, see, for example, Refs. [21,22]), that intermolecular interactions influence Tg. Moreover, it is perhaps this interaction which plays a role in relaxation processes in the glass-transition region. We believe that this interaction enhances the changes of some properties in the vicinity of Tanaka's threshold [23,24]. However, the overall compositional trend in Tg seems to be influenced mainly by the chemical bond arrangement.
145
GeS 2
,-o 0
750
0
O
650
GeSe 2
O O
~o
OO 0 O
^00;0
0 o
oo o o ~ , ~ 8 q g~oo o
o_
550
oOO
~v
~D
~ 0
0
o o'_
~ 450
0
o
000
As 2 $3 - - , - - i : ~ O o As2
3 q~
q5 o U
350
o6)°n~ ~e
I
2.2 GeS2
-
750
-~ o;
Ge Se 2 ---~o°
0
...~e.~As2S 3
oo 0.6
o
>
The authors are indebted to Professor D.L. Kinser and to the referees for kind comments. This work was supported by the Grant Agency of the Czech Republic under Grant No. 2 0 3 / 9 3 / 0 1 5 5 .
O
I
I
2.8
For 186 glasses, with Tg ranging from ~ 320 to 760 K, a good correlation between Tg and ( E ) was observed in the form: Tg-~311 ( ( E ) - 0.9). This correlation satisfies the Arrhenius relation for viscosity.
~0
350
I
2.6
4. Conclusion
550
450
I
2,4
Fig. 2. The variation of Tg with the mean coordination number, (CN). For data sources, see Table l.
v
650
o o°
stope ~ 311 [K/eV] I
1
1.6 /..E) - 0.9
References
2.6 [eV]
Fig. 1. The variations of Tg with the 'overall' mean bond energy ( ( E ) - 0 . 9 ) . Full line indicates a least-squares fit. The slope of the full line is approximately 311 K / e V . For data sources, see Table 1.
[1] S.A. Dembovskii, Phys. Chem. Glasses 10 (1969) 73. [2] J.P. De Neufville and H.K. Rockstad, in: Amorphous and Liquid Semiconductors, Vol. 1, ed. J. Stuke and W. Brenig (Taylor and Francis, London, 1974) p. 419. [3] D.J. Sarrach, J.P. De Neufville, and W.L. Hawarth, J. NonCryst. Solids 22 (1976) 245.
146
L. Tich~, H. Tichd /Journal of Non-Crystalline Solids 189 (1995) 141-146
[4] J.S. Berkes, in: Proc. 4th Int. Conf. on Physics of Non-Crystalline Solids, 1976, ed. G.H. Frischat (TransTech Publications, Aedermannsdorf, Switzerland, 1977) p. 405. [5] D. Linke, Proc. llth Int. Congr. on Glass, Vol. 1, ed. J. G6tz, Prague, 1977, p. 149. [6] Ke. Tanaka, Solid State Commun. 54 (1985) 807. [7] L.E. Busse, Phys. Rev. B29 (1984) 3639. [8] L. Tich~), H. Tich~ and V. Smrcka, Mater. Lett. 15 (1992) 202. [9] A.N. Sreeram, D.R. Swiler and A.K. Varshneya, J. NonCryst. Solids 127 (1991) 287. [10] L. Tich~ and H. Tichfi, Mater. Lett. 21 (1994) 313. [11] Z.U. Borisova, Glassy Semiconductors (Plenum, New York, 1981). [12] M. Vlfiek, L. Tich#, J. Klikorka and A. Tfiska, J. Mater. Sci. Lett. 7 (1988) 335. [13] J. M~ilek, J. Non-Cryst. Solids 107 (1989) 323.
[14] S. Mahadevan and A. Giridhar, J. Non-Cryst. Solids 143 (1992) 52. [15] A. Giridhar and S. Mahadevan, J. Non-Cryst. Solids 152 (1993) 42. [16] A. Giridhar and S. Mahadevan, J. Non-Cryst. Solids 151 (1992) 245. [17] Yu. G. Pozdniakov and V.M. Pozdniakov, Izv. Acad. Nauk, USSR, Inorg. Mater. 10 (1974) 1932 (in Russian). [18] H. Krebs, Fundamentals of Inorganic Crystal Chemistry (McGraw-Hill, London, 1968) p. 354. [19] J.Z. Liu and P.C. Taylor, J. Non-Cryst. Solids 114 (1989) 25. [20] J.Z. Liu and P.C. Taylor, Solid State Commun. 70 (1989) 81. [21] Ke. Tanaka, Phys. Rev. B30 (1984) 4549. [22] M.I. Klinger, Solid State Commun. 65 (1988) 949. [23] Ke. Tanaka, Phys. Rev. B39 (1989) 1270. [24] E. Vateva, E. Skordeva and D. Arsova, Philos. Mag. B67 (1993) 225.