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Low-energy vibrations in superionic glasses
Solid State Ionics 105 (1998) 97–102
Low-energy vibrations in superionic glasses b, a a a A. Bartolotta *, G. Carini , G. D’Angelo , G. Tripodo a
Dipartimento di Fisica and Istituto Nazionale di Fisica della Materia, Universita’ di Messina, Contrada Papardo, Salita Sperone 31, I-98166 S. Agata ( ME), Italy b Istituto di Tecniche Spettroscopiche del C.N.R. Contrada Papardo, Salita Sperone 31, I-98166 S. Agata ( ME), Italy
Abstract Measurements of low temperature specific heat Cp (1.5 K–25 K) have been carried out in AgI–Ag 2 O–B 2 O 3 superionic glasses. It has been revealed that modifications of the network coherence change the magnitude of the hump in the specific heat, which is observed when it is plotted as Cp /T 3 : the increasing connectivity (defined as the number of bridging bonds per network forming ion) reduces the deviations of Cp from a T 3 behavior. Finally comparative measurements of low temperature Cp and low frequency Raman scattering (below 100 cm 21 ) in the same glass permit us to assess the frequency dependence of the photon-vibration coupling coefficient C(v ) and the spectral density of low-energy vibrational modes g(v ). Keywords: Superionic glasses; Specific heat; Raman scattering; Borate glasses PACS: 63.50. 1 x; 65.40. 1 g; 78.30.Ly
1. Introduction The low temperature specific heat Cp of amorphous solids shows marked deviations from the prediction of the Debye model [1]: in the temperature region above 1 K, it exhibits a broad maximum when plotted as Cp /T 3 and is appreciably greater than that evaluated by the sound velocities. These anomalies, which are universal characteristics of the amorphous state, arise from a density of low-energy vibrational states (DOS), given by the contributions of extended states, the usual phonons, and quasilocal harmonic modes. The harmonic nature of the additional modes, corresponding to a temperature independence of their spectral density, has been proved by a number of inelastic neutron scattering *Corresponding author.
experiments [2–4]. It is believed that they arise from the collective motion of groups of atoms which can be considered as ‘‘quasimolecules’’ frozen in to the glass. Different theoretical models [5–7] have been suggested to explain the nature of low-energy vibrational states which give rise to the excess specific heat, even though none of them accounts satisfactorily for the mechanisms driving the formation of the quasiparticles. Recent measurements of low temperature Cp (0.1– 3 K) in As x Se 12x over a wide range of concentrations revealed a significant dependence on the average coordination number of the contributions arising from both the two level systems (TLS, below 1 K) and the Debye-like vibrations (above 1 K) [8]. Now we believe that a consistent picture of the correlation between low energy excitations and the rigidity of the network could require a more signifi-
0167-2738 / 98 / $19.00 1998 Elsevier Science B.V. All rights reserved. PII S0167-2738( 97 )00454-2
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cant parameter than the coordination number, such as the connectivity defined as the number of bridging bonds per network forming ions (NFI). This is because the coordination number does not permit to distinguish between bridging and non bridging bonds, the latter leading to the network coherence breakdown. Consequently our study concerning the possible origin for the additional low energy vibrations will adopt the connectivity as the driving parameter. The model systems chosen for this analysis are the silver borate superionic glasses, a class of materials where the connectivity can be varied by changing the Ag 2 O content [9,10]. The additional vibrations over the Debye-like phonons account for both the excess of specific heat in the region between 1 and 20 K and also the Boson peak (BP) in the low frequency region (below 100 cm 21 ) of the Raman spectrum. The heat capacity is insensitive to fine detail in the phonon spectrum, but more direct information can be obtained by Raman scattering measurements. It has been proved, in fact, that, in consequence of the absence of translation symmetry, the low frequency Raman intensity reflects the product between the light-vibration coupling coefficient C(v ) and the DOS g(v ) [11]: C(v )g(v )[n(v, T ) 1 1] I expt (v, T )~]]]]]]], v
(1)
where n(v )51 /(e " v / K B T 2 1) is the Bose–Einstein population factor and 1 /v the harmonic propagator. The complexity of the vibrational characteristics of disordered systems can affect both the functions C(v ) and g(v ). In particular it has been shown in a number of glasses [12–14] that, in the low frequency region, the DOS exhibits a broad maximum (the BP), while C(v ) shows a v a frequency dependence where a is usually lower than 1 [15,16]. Very recently we proved that comparative measurements of low frequency Raman spectroscopy and low temperature heat capacity in the same glass permit us to assess the frequency dependence of C(v ) and the absolute magnitude of the DOS [17]. By applying the same procedure we were able to determine the total and the excess low-energy g(v ) in one of the superionic glasses investigated.
2. Experimental details Samples of the (AgI) x ((Ag 2 O) y (B 2 O 3 ) 12y ) 12x system, x and y being the molar fractions, were prepared in the binary (x50) and ternary form using the same procedure as described elsewhere [18]. The sound velocity measurements of longitudinal (vl ) and shear (vt ) waves were performed at 5 MHz and at room temperature by a pulse echo technique. The bulk (B) and shear (G) moduli have been determined by the usual expressions: B5 r v 2l 24 / 3G; G5 r v 2t . The specific heat was measured in the range between 1.5 and 30 K using an automated calorimeter which operated by the thermal relaxation method, using a silicon chip as the sample holder on to which a sample of about 15–30 mg was bonded by Apiezon grease. The random error is apparent from the figures and any systematic errors are believed to be less than 3–4%.
3. Results and discussion The experimental results of the specific heat, obtained between 1.5 K and 30 K for (AgI) x ((Ag 2 O) y (B 2 O 3 ) 12y ) 12x glasses and plotted as Cp (T ) /T 3 , are compared to those reported for pure B 2 O 3 glass [19] in Fig. 1a. The Debye contributions have been evaluated using the Debye temperatures QD , as determined by the ultrasonic waves velocities at room temperature: QD 5269 K for B 2 O 3 [20]; QD 5335 K for (Ag 2 O) 0.2 (B 2 O 3 ) 0.8 ; QD 5331 K for (Ag 2 O) 0.33 (B 2 O 3 ) 0.67 and QD 5237 K for (AgI) 0.4 ((Ag 2 O) 0.2 (B 2 O 3 ) 0.8 ) 0.6 . They are reported as continuous lines in Fig. 1a and evidence in all the samples the existence of an excess specific heat having the characteristic shape (for a glass) of a broad peak. The maximum of the hump is located at about 4 K in B 2 O 3 glass and shifts to about 8 K in silver borate glasses. More importantly, its magnitude shows an unexpected dependence on Ag 2 O concentration: the excess specific heat, which in the glass with y50.2 is lower than in B 2 O 3 , exhibits a substantial increase in the glass with y50.33. The addition of AgI to the silver borate glass with y50.2 gives rise to an enhancement of the excess specific heat and to a lowering of the maximum temperature.
A. Bartolotta et al. / Solid State Ionics 105 (1998) 97 – 102
Fig. 1. (a) The temperature dependence of Cp (T ) /T 3 for (AgI) x ((Ag 2 O) y (B 2 O 3 ) ( 12y) ) ( 12x) glasses: (s) x50.0, y50.0; (m) x50.0, y50.2; (n) x50.0, y50.33; (1) x50.4, y50.2. The horizontal lines show the Debye values calculated from sound velocity data. (b) The temperature dependence of Cp (T ) /T 3 for (Na 2 O) y (B 2 O 3 ) ( 12y) glasses: (s) y50.0; (n) y50.06; (,) y5 0.16; (y) y50.25; (data from Refs. [19] and [26]).
Significant differences between the behaviors of Cp (T ) /T 3 are also observed below 2.5 K and arise from the contribution of the ‘two level systems (TLS)’. This term, associated to the tunneling motions of groups of atoms (or single atoms), is approximately linear in temperature and becomes dominant below 1 K [1]. The differences between the upturns of the pure B 2 O 3 and silver borate glasses could be associated to variations in the TLS density, as pointed out in a low temperature study of the ultrasonic characteristics of these systems [21,22]. Unfortunately, the limited temperature range explored (down to 1.5 K) prevents any significant evaluation of the magnitude of this linear term. An explanation for the observed behaviors can be
99
given by considering the role of silver as modifier ion of the borate network. In fact the random network of B 2 O 3 is essentially built up on planar triangles BO 3 [23]. It has been proved by NMR data [9] and MD calculations [10] that the addition of silver oxide up to a molar fraction of about y50.25 assists the formation of BO 4 tetrahedric groups by crosslinks between the planar units and gives rise to an increase of the network coherence. The existence of such groups is directly shown by the analysis of the Raman bands at about 770 cm 21 [24]. The concentration of BO 4 groups (as well as the related stiffness) increases up to y50.25 with a rate equal to y /(12y), which corresponds to the formation of two BO 4 groups for each oxygen introduced by the silver oxide. For y.0.25 the BO 4 formation rate decreases and non-bridging oxygens (NBO) appear, whose number is insignificant at lower concentration. The NBO are predominant in BO 3 groups [10], near which the metallic ions should be placed in order to preserve the electrical neutrality. The consequent result is a decrease of the glassy network coherence and a softening of the structure. In the case of the ternary glass the analysis of the vibrational dynamics [24,25] led to the belief that both the AgI and the borate matrix preserve their local structures. This implies that AgI tetrahedra, weakly bonded to the BO 3 or BO 4 groups of the borate matrix, must be coordinated in order to allow the fast ionic diffusion typical of these kinds of glasses. The above considerations on the structure show that the connectivity of the borate network critically depends on the silver oxide or halide concentrations: it increases up to y¯0.25 and decreases for higher values of y or for x.0 because of the NBO formation or the inclusion of less tied AgI polyhedra. By considering the bridging oxygens per NFI, the connectivity changes from 3 in pure B 2 O 3 to 3.25 in the glass with y50.2 and to about 3.04 in that with y50.33. The latter value was roughly evaluated by assuming that 70% of the BO 3 groups present have one NBO [10]. For the sake of completeness we like to emphasize that, besides the connectivity, a further element must be considered for an accurate determination of the strengthening of the network: the coordination of the cations which occupy sites in the existing interstices. The variations of the connectivity regulate the
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Fig. 2. The concentration behavior of the bulk (B) and shear (G) moduli in (Ag 2 O) y (B 2 O 3 ) ( 12y) glasses.
elastic moduli, whose concentration dependence is reported in Fig. 2, and also the low temperature anomalies of the heat capacity of borate glasses. Increasing connectivity leads to a relevant growth of both B and G and to a decrease of Cp (T ) /T 3 which corresponds to a smaller magnitude of the low energy DOS. A similar trend is also observed in sodium borate glasses for both the elastic moduli [20] and the low temperature specific heat [26], whose behaviors for a number of sodium oxide concentrations are reported for a comparison in Fig. 1b. The decrease of the connectivity, obtained by further addition of Ag 2 O from y50.2 to y50.33, causes a reduction in the elevation rate of the moduli and an increase of Cp /T 3 which becomes comparable to that of pure B 2 O 3 . It is believed that the network breakdown due to NBO works in competition with the network stiffening due to BO 4 groups, resulting in a larger excess of low energy vibrations over the contribution of the ordinary elastic waves. These peculiarities give evidence of the fact that the source of the anomalous specific heat lies in additional low-energy vibrational states whose density decreases with increasing connectivity. The relevant increase of Cp /T 3 in the ternary glass, which has a lower network coherence than the glass without AgI, appears to be in agreement with this conclusion (see Fig. 1a). The revealed features are consistent with the supposed nature of the low energy vibrations, which are ascribed to quasi-local modes in quasiparticles frozen in the glass. In this context the
coherence breakdown due to NBOs or to the inclusion of AgI polyhedra corresponds to the formation of less tied clusters of atoms in the network, which could be the source for the additional modes. The frequency dependence of C(v ) and the magnitude of g(v ) in the (AgI) 0.4 ((Ag 2 O) 0.2 (B 2 O 3 ) 0.8 ) 0.6 glass have been evaluated by fitting the low temperature heat capacity through the 35 K low-frequency Raman spectrum published by Carini et al. [27,28]. In fact it is expected [17] that a temperature of 35 K is low enough to depress all the relaxation mechanisms causing the light scattering excess, whose contribution, usually observed below 25 cm 21 , increases with increasing temperature much faster than the Bose population factor. By inserting C(v )~v a in the low temperature reduced Raman intensity (IR 5 Iexp v / [n(v, T )11]), see Eq. (1), it has been possible to fit Cp as expressed by the following equation: Cp ¯ Cv v0
5 3Nk B
"v E g(v) S]] k TD 0
2
B
exp("v /k B T ) 3 ]]]]]]2 dv, [exp("v /k B T ) 2 1]
(2)
where N is the number density and v0 the highest vibrational frequency. In addition to the exponent a, also the magnitude of g(v ) represents a fitting parameter. In Fig. 3a the experimental results of Cp (T ) /T 3 are compared to the theoretical fit and the resulting value for the exponent is a 50.25. This value of a is quite different from those usually found in glasses [4,27,28] and appears to be quite close to that (a 50.29) found in a sample of silica xerogel with a lower degree of connectivity than vitreous SiO 2 (connectivity equal to 4) [17]. It has been suggested that the reason for that anomalous value could be found in a low coherence of the network, which introduces relevant elasto-optical and elastic local inhomogeneities (the expected sources for the C(v ) behavior [1]). Similar conclusions can be also extended to the glass analyzed, where the inclusion of AgI polyhedra in the borate matrix causes local density fluctuations, leading to a connectivity which is surely lower than that characterizing the (Ag 2 O) 0.2 (B 2 O 3 ) 0.8 glass. This is clearly supported
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4. Conclusions By measurements of low temperature specific heats in silver borate glasses we have shown that the connectivity of the network affects strongly the spectral density of low energy vibrations which cause the hump in Cp /T 3 and the BP in the low frequency region of the Raman spectrum. In particular it has been found that an increasing connectivity leads to a significant reduction of the excess density of vibrational states which cause the excess specific heat at low temperatures. Furthermore by specific heat and Raman scattering measurements in the same superionic glass it has been possible to evaluate the frequency dependence of the coupling function C(v ) and the excess DOS. The comparison of the present results with those obtained in silica glass shows that the degree of the local homogeneity of the glassy network strongly affects both the C(v ) and g(v ).
References Fig. 3. (a) Theoretical fit by Eq. (2) to the experimental Cp data in (AgI) 0.4 ((Ag 2 O) 0.2 (B 2 O 3 ) 0.8 ) 0.6 glass. The used g(v ) has been determined by the low frequency Raman spectrum at 35 K (see Eq. (1)). (b) Density of vibrational modes versus frequency as deduced from the fit shown in (a); the dotted line represents the Debye density of states. (c) Comparison between the densities of excess modes in (AgI) 0.4 ((Ag 2 O) 0.2 (B 2 O 3 ) 0.8 ) 0.6 and a-SiO 2 .
by the decrease of the moduli, which become B5 30.32 GPa and G513.53 GPa in the ternary glass. The behavior of g(v ) in the low-energy region, as determined by the comparison of the Raman intensity to the specific heat data through C(v ), is shown in Fig. 3b where the corresponding Debye DOS gD (v ) is also reported as a dotted line. Assuming that the excess modes are different from phonons and coexist with them, we can obtain the excess DOS by subtracting gD (v ) from the total g(v ). In Fig. 3c the resulting excess density of states is compared to that obtained for amorphous silica by the same procedure [17]. It results that a decreasing connectivity softens both the Debye and the excess DOS, enhancing the spectral density of vibrational modes at lower frequencies.
[1] For a review see, W.A. Phillips (Ed.), Amorphous Solids: Low-Temperature properties, Springer, Berlin, 1981. [2] U. Buchenau, M. Prager, N. Nucker, A.J. Dianoux, N. Ahmad, W.A. Phillips, Phys. Rev. B 34 (1986) 5665. ` J.L. [3] F.J. Bermejo, J. Alonso, A. Criado, F.J. Mompean, ` ` ` Martınez, M. Garcıa-Hernandez, A. Chahid, Phys. Rev. B 46 (1992) 6173. [4] G. Carini, G. D’Angelo, G. Tripodo, A. Fontana, A. Leonardi, G.A. Saunders, A. Brodin, Phys. Rev. B 52 (1995) 9342. [5] V.G. Karpov, M.I. Klinger, F.N. Ignatiev, Zh. Expt. Teor. Fiz. 84 (1983) 760 [Sov Phys. JEPT 57, 439 (1983)]. [6] R. Orbach, Science 231 (1986) 814 and Refs. therein. [7] C.C. Yu, J.J. Freeman, Phys. Rev. B 36 (1987) 7620. ¨ [8] O. Brand, H.V. Lohneysen, Europhys. Lett. 16 (1991) 455. [9] K.S. Kim, P.J. Bray, J. Non-Metals 2 (1974) 95. [10] M.C. Abramo, G. Carini, G. Pizzimenti, J. Phys. C 21 (1988) 527. [11] R. Shuker, R.W. Gammon, Phys. Rev. Lett. 4 (1970) 222. [12] U. Buchenau, H.M. Zhou, N. Nucker, K.S. Gilroy, W.A. Phillips, Phys. Rev. Lett. 60 (1988) 1318. [13] V.K. Malinovsky, V.N. Novikov, P.P. Parshin, A.P. Sokolov, M.G. Zemlyanov, Europhys. Lett. 11 (1990) 43. [14] A.P. Sokolov, U. Buchenau, W. Steffen, B. Frick, A. Wischmewsski, Phys. Rev. B 52 (1995-II) R9815. [15] A. Fontana, F. Rocca, M.P. Fontana, Phys. Rev. Lett. 58 (1987) 503. [16] A. Fontana, F. Rocca, M.P. Fontana, B. Rosi, A.J. Dianoux, Phys. Rev. B 41 (1990) 3778.
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[17] A. Fontana, F. Rossi, G. Carini, G. D’Angelo, G. Tripodo, A. Bartolotta, Phys. Rev. Lett. 78 (1997) 1078. [18] G. Carini, M. Cutroni, M. Federico, G. Galli, G. Tripodo, Phys. Rev. B 30 (1984) 7219. [19] M.A. Ramos, R. Villar, S. Vieira, J.M. Calleja, in: J. Colmenero, A. Alegria (Eds.), Proc. of the Second Int. Workshop on Non-Cryst. Solids, World Scientific, Singapore 1990, p. 514. [20] J.T. Krause, C.R. Kurkjian, in: L.D. Pye, V.D. Frechette, N.J. Kreidl (Eds.), Borate Glasses, Vol. 12, Plenum, New York, 1978, p. 577. [21] G. Carini, M. Cutroni, M. Federico, G. Tripodo, Phys. Rev. B 37 (1988) 7021. [22] G. Carini, M. Cutroni, M. Federico, G. Galli, G. Tripodo, Solid State Comm. 74 (1990) 661.
[23] J.L. Hopkins, C.R. Kurkjian, in: W.P. Mason (Ed.), Physical Acoustics, Vol. IIB, Academic, New York, 1965, p. 91 and refs. cited therein. [24] G. Carini, M. Cutroni, A. Fontana, G. Mariotto, F. Rocca, Phys. Rev. B 29 (1984) 3567. [25] M.P. Fontana, B. Rosi, A. Fontana, F. Rocca, Phil. Mag. B 65 (1992) 143. [26] E.S. Pinango, M.A. Ramos, R. Villar, S. Vieira, in: J. Colmenero, A. Alegria (Eds.), Proc. of the Second Int. Workshop on Non-Cryst. Solids, World Scientific, Singapore 1990, p. 509. [27] A.P. Sokolov, A. Kislink, D. Quitmann, E. Duval, Phys. Rev. B 48 (1993) 7692. [28] A. Brodin, A. Fontana, L. Borjesson, G. Carini, L.M. Torell, Phys. Rev. Lett. 73 (1994) 2067.
Low-energy vibrations in superionic glasses b, a a a A. Bartolotta *, G. Carini , G. D’Angelo , G. Tripodo a
Dipartimento di Fisica and Istituto Nazionale di Fisica della Materia, Universita’ di Messina, Contrada Papardo, Salita Sperone 31, I-98166 S. Agata ( ME), Italy b Istituto di Tecniche Spettroscopiche del C.N.R. Contrada Papardo, Salita Sperone 31, I-98166 S. Agata ( ME), Italy
Abstract Measurements of low temperature specific heat Cp (1.5 K–25 K) have been carried out in AgI–Ag 2 O–B 2 O 3 superionic glasses. It has been revealed that modifications of the network coherence change the magnitude of the hump in the specific heat, which is observed when it is plotted as Cp /T 3 : the increasing connectivity (defined as the number of bridging bonds per network forming ion) reduces the deviations of Cp from a T 3 behavior. Finally comparative measurements of low temperature Cp and low frequency Raman scattering (below 100 cm 21 ) in the same glass permit us to assess the frequency dependence of the photon-vibration coupling coefficient C(v ) and the spectral density of low-energy vibrational modes g(v ). Keywords: Superionic glasses; Specific heat; Raman scattering; Borate glasses PACS: 63.50. 1 x; 65.40. 1 g; 78.30.Ly
1. Introduction The low temperature specific heat Cp of amorphous solids shows marked deviations from the prediction of the Debye model [1]: in the temperature region above 1 K, it exhibits a broad maximum when plotted as Cp /T 3 and is appreciably greater than that evaluated by the sound velocities. These anomalies, which are universal characteristics of the amorphous state, arise from a density of low-energy vibrational states (DOS), given by the contributions of extended states, the usual phonons, and quasilocal harmonic modes. The harmonic nature of the additional modes, corresponding to a temperature independence of their spectral density, has been proved by a number of inelastic neutron scattering *Corresponding author.
experiments [2–4]. It is believed that they arise from the collective motion of groups of atoms which can be considered as ‘‘quasimolecules’’ frozen in to the glass. Different theoretical models [5–7] have been suggested to explain the nature of low-energy vibrational states which give rise to the excess specific heat, even though none of them accounts satisfactorily for the mechanisms driving the formation of the quasiparticles. Recent measurements of low temperature Cp (0.1– 3 K) in As x Se 12x over a wide range of concentrations revealed a significant dependence on the average coordination number of the contributions arising from both the two level systems (TLS, below 1 K) and the Debye-like vibrations (above 1 K) [8]. Now we believe that a consistent picture of the correlation between low energy excitations and the rigidity of the network could require a more signifi-
0167-2738 / 98 / $19.00 1998 Elsevier Science B.V. All rights reserved. PII S0167-2738( 97 )00454-2
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cant parameter than the coordination number, such as the connectivity defined as the number of bridging bonds per network forming ions (NFI). This is because the coordination number does not permit to distinguish between bridging and non bridging bonds, the latter leading to the network coherence breakdown. Consequently our study concerning the possible origin for the additional low energy vibrations will adopt the connectivity as the driving parameter. The model systems chosen for this analysis are the silver borate superionic glasses, a class of materials where the connectivity can be varied by changing the Ag 2 O content [9,10]. The additional vibrations over the Debye-like phonons account for both the excess of specific heat in the region between 1 and 20 K and also the Boson peak (BP) in the low frequency region (below 100 cm 21 ) of the Raman spectrum. The heat capacity is insensitive to fine detail in the phonon spectrum, but more direct information can be obtained by Raman scattering measurements. It has been proved, in fact, that, in consequence of the absence of translation symmetry, the low frequency Raman intensity reflects the product between the light-vibration coupling coefficient C(v ) and the DOS g(v ) [11]: C(v )g(v )[n(v, T ) 1 1] I expt (v, T )~]]]]]]], v
(1)
where n(v )51 /(e " v / K B T 2 1) is the Bose–Einstein population factor and 1 /v the harmonic propagator. The complexity of the vibrational characteristics of disordered systems can affect both the functions C(v ) and g(v ). In particular it has been shown in a number of glasses [12–14] that, in the low frequency region, the DOS exhibits a broad maximum (the BP), while C(v ) shows a v a frequency dependence where a is usually lower than 1 [15,16]. Very recently we proved that comparative measurements of low frequency Raman spectroscopy and low temperature heat capacity in the same glass permit us to assess the frequency dependence of C(v ) and the absolute magnitude of the DOS [17]. By applying the same procedure we were able to determine the total and the excess low-energy g(v ) in one of the superionic glasses investigated.
2. Experimental details Samples of the (AgI) x ((Ag 2 O) y (B 2 O 3 ) 12y ) 12x system, x and y being the molar fractions, were prepared in the binary (x50) and ternary form using the same procedure as described elsewhere [18]. The sound velocity measurements of longitudinal (vl ) and shear (vt ) waves were performed at 5 MHz and at room temperature by a pulse echo technique. The bulk (B) and shear (G) moduli have been determined by the usual expressions: B5 r v 2l 24 / 3G; G5 r v 2t . The specific heat was measured in the range between 1.5 and 30 K using an automated calorimeter which operated by the thermal relaxation method, using a silicon chip as the sample holder on to which a sample of about 15–30 mg was bonded by Apiezon grease. The random error is apparent from the figures and any systematic errors are believed to be less than 3–4%.
3. Results and discussion The experimental results of the specific heat, obtained between 1.5 K and 30 K for (AgI) x ((Ag 2 O) y (B 2 O 3 ) 12y ) 12x glasses and plotted as Cp (T ) /T 3 , are compared to those reported for pure B 2 O 3 glass [19] in Fig. 1a. The Debye contributions have been evaluated using the Debye temperatures QD , as determined by the ultrasonic waves velocities at room temperature: QD 5269 K for B 2 O 3 [20]; QD 5335 K for (Ag 2 O) 0.2 (B 2 O 3 ) 0.8 ; QD 5331 K for (Ag 2 O) 0.33 (B 2 O 3 ) 0.67 and QD 5237 K for (AgI) 0.4 ((Ag 2 O) 0.2 (B 2 O 3 ) 0.8 ) 0.6 . They are reported as continuous lines in Fig. 1a and evidence in all the samples the existence of an excess specific heat having the characteristic shape (for a glass) of a broad peak. The maximum of the hump is located at about 4 K in B 2 O 3 glass and shifts to about 8 K in silver borate glasses. More importantly, its magnitude shows an unexpected dependence on Ag 2 O concentration: the excess specific heat, which in the glass with y50.2 is lower than in B 2 O 3 , exhibits a substantial increase in the glass with y50.33. The addition of AgI to the silver borate glass with y50.2 gives rise to an enhancement of the excess specific heat and to a lowering of the maximum temperature.
A. Bartolotta et al. / Solid State Ionics 105 (1998) 97 – 102
Fig. 1. (a) The temperature dependence of Cp (T ) /T 3 for (AgI) x ((Ag 2 O) y (B 2 O 3 ) ( 12y) ) ( 12x) glasses: (s) x50.0, y50.0; (m) x50.0, y50.2; (n) x50.0, y50.33; (1) x50.4, y50.2. The horizontal lines show the Debye values calculated from sound velocity data. (b) The temperature dependence of Cp (T ) /T 3 for (Na 2 O) y (B 2 O 3 ) ( 12y) glasses: (s) y50.0; (n) y50.06; (,) y5 0.16; (y) y50.25; (data from Refs. [19] and [26]).
Significant differences between the behaviors of Cp (T ) /T 3 are also observed below 2.5 K and arise from the contribution of the ‘two level systems (TLS)’. This term, associated to the tunneling motions of groups of atoms (or single atoms), is approximately linear in temperature and becomes dominant below 1 K [1]. The differences between the upturns of the pure B 2 O 3 and silver borate glasses could be associated to variations in the TLS density, as pointed out in a low temperature study of the ultrasonic characteristics of these systems [21,22]. Unfortunately, the limited temperature range explored (down to 1.5 K) prevents any significant evaluation of the magnitude of this linear term. An explanation for the observed behaviors can be
99
given by considering the role of silver as modifier ion of the borate network. In fact the random network of B 2 O 3 is essentially built up on planar triangles BO 3 [23]. It has been proved by NMR data [9] and MD calculations [10] that the addition of silver oxide up to a molar fraction of about y50.25 assists the formation of BO 4 tetrahedric groups by crosslinks between the planar units and gives rise to an increase of the network coherence. The existence of such groups is directly shown by the analysis of the Raman bands at about 770 cm 21 [24]. The concentration of BO 4 groups (as well as the related stiffness) increases up to y50.25 with a rate equal to y /(12y), which corresponds to the formation of two BO 4 groups for each oxygen introduced by the silver oxide. For y.0.25 the BO 4 formation rate decreases and non-bridging oxygens (NBO) appear, whose number is insignificant at lower concentration. The NBO are predominant in BO 3 groups [10], near which the metallic ions should be placed in order to preserve the electrical neutrality. The consequent result is a decrease of the glassy network coherence and a softening of the structure. In the case of the ternary glass the analysis of the vibrational dynamics [24,25] led to the belief that both the AgI and the borate matrix preserve their local structures. This implies that AgI tetrahedra, weakly bonded to the BO 3 or BO 4 groups of the borate matrix, must be coordinated in order to allow the fast ionic diffusion typical of these kinds of glasses. The above considerations on the structure show that the connectivity of the borate network critically depends on the silver oxide or halide concentrations: it increases up to y¯0.25 and decreases for higher values of y or for x.0 because of the NBO formation or the inclusion of less tied AgI polyhedra. By considering the bridging oxygens per NFI, the connectivity changes from 3 in pure B 2 O 3 to 3.25 in the glass with y50.2 and to about 3.04 in that with y50.33. The latter value was roughly evaluated by assuming that 70% of the BO 3 groups present have one NBO [10]. For the sake of completeness we like to emphasize that, besides the connectivity, a further element must be considered for an accurate determination of the strengthening of the network: the coordination of the cations which occupy sites in the existing interstices. The variations of the connectivity regulate the
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Fig. 2. The concentration behavior of the bulk (B) and shear (G) moduli in (Ag 2 O) y (B 2 O 3 ) ( 12y) glasses.
elastic moduli, whose concentration dependence is reported in Fig. 2, and also the low temperature anomalies of the heat capacity of borate glasses. Increasing connectivity leads to a relevant growth of both B and G and to a decrease of Cp (T ) /T 3 which corresponds to a smaller magnitude of the low energy DOS. A similar trend is also observed in sodium borate glasses for both the elastic moduli [20] and the low temperature specific heat [26], whose behaviors for a number of sodium oxide concentrations are reported for a comparison in Fig. 1b. The decrease of the connectivity, obtained by further addition of Ag 2 O from y50.2 to y50.33, causes a reduction in the elevation rate of the moduli and an increase of Cp /T 3 which becomes comparable to that of pure B 2 O 3 . It is believed that the network breakdown due to NBO works in competition with the network stiffening due to BO 4 groups, resulting in a larger excess of low energy vibrations over the contribution of the ordinary elastic waves. These peculiarities give evidence of the fact that the source of the anomalous specific heat lies in additional low-energy vibrational states whose density decreases with increasing connectivity. The relevant increase of Cp /T 3 in the ternary glass, which has a lower network coherence than the glass without AgI, appears to be in agreement with this conclusion (see Fig. 1a). The revealed features are consistent with the supposed nature of the low energy vibrations, which are ascribed to quasi-local modes in quasiparticles frozen in the glass. In this context the
coherence breakdown due to NBOs or to the inclusion of AgI polyhedra corresponds to the formation of less tied clusters of atoms in the network, which could be the source for the additional modes. The frequency dependence of C(v ) and the magnitude of g(v ) in the (AgI) 0.4 ((Ag 2 O) 0.2 (B 2 O 3 ) 0.8 ) 0.6 glass have been evaluated by fitting the low temperature heat capacity through the 35 K low-frequency Raman spectrum published by Carini et al. [27,28]. In fact it is expected [17] that a temperature of 35 K is low enough to depress all the relaxation mechanisms causing the light scattering excess, whose contribution, usually observed below 25 cm 21 , increases with increasing temperature much faster than the Bose population factor. By inserting C(v )~v a in the low temperature reduced Raman intensity (IR 5 Iexp v / [n(v, T )11]), see Eq. (1), it has been possible to fit Cp as expressed by the following equation: Cp ¯ Cv v0
5 3Nk B
"v E g(v) S]] k TD 0
2
B
exp("v /k B T ) 3 ]]]]]]2 dv, [exp("v /k B T ) 2 1]
(2)
where N is the number density and v0 the highest vibrational frequency. In addition to the exponent a, also the magnitude of g(v ) represents a fitting parameter. In Fig. 3a the experimental results of Cp (T ) /T 3 are compared to the theoretical fit and the resulting value for the exponent is a 50.25. This value of a is quite different from those usually found in glasses [4,27,28] and appears to be quite close to that (a 50.29) found in a sample of silica xerogel with a lower degree of connectivity than vitreous SiO 2 (connectivity equal to 4) [17]. It has been suggested that the reason for that anomalous value could be found in a low coherence of the network, which introduces relevant elasto-optical and elastic local inhomogeneities (the expected sources for the C(v ) behavior [1]). Similar conclusions can be also extended to the glass analyzed, where the inclusion of AgI polyhedra in the borate matrix causes local density fluctuations, leading to a connectivity which is surely lower than that characterizing the (Ag 2 O) 0.2 (B 2 O 3 ) 0.8 glass. This is clearly supported
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4. Conclusions By measurements of low temperature specific heats in silver borate glasses we have shown that the connectivity of the network affects strongly the spectral density of low energy vibrations which cause the hump in Cp /T 3 and the BP in the low frequency region of the Raman spectrum. In particular it has been found that an increasing connectivity leads to a significant reduction of the excess density of vibrational states which cause the excess specific heat at low temperatures. Furthermore by specific heat and Raman scattering measurements in the same superionic glass it has been possible to evaluate the frequency dependence of the coupling function C(v ) and the excess DOS. The comparison of the present results with those obtained in silica glass shows that the degree of the local homogeneity of the glassy network strongly affects both the C(v ) and g(v ).
References Fig. 3. (a) Theoretical fit by Eq. (2) to the experimental Cp data in (AgI) 0.4 ((Ag 2 O) 0.2 (B 2 O 3 ) 0.8 ) 0.6 glass. The used g(v ) has been determined by the low frequency Raman spectrum at 35 K (see Eq. (1)). (b) Density of vibrational modes versus frequency as deduced from the fit shown in (a); the dotted line represents the Debye density of states. (c) Comparison between the densities of excess modes in (AgI) 0.4 ((Ag 2 O) 0.2 (B 2 O 3 ) 0.8 ) 0.6 and a-SiO 2 .
by the decrease of the moduli, which become B5 30.32 GPa and G513.53 GPa in the ternary glass. The behavior of g(v ) in the low-energy region, as determined by the comparison of the Raman intensity to the specific heat data through C(v ), is shown in Fig. 3b where the corresponding Debye DOS gD (v ) is also reported as a dotted line. Assuming that the excess modes are different from phonons and coexist with them, we can obtain the excess DOS by subtracting gD (v ) from the total g(v ). In Fig. 3c the resulting excess density of states is compared to that obtained for amorphous silica by the same procedure [17]. It results that a decreasing connectivity softens both the Debye and the excess DOS, enhancing the spectral density of vibrational modes at lower frequencies.
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