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Ionic conduction and dynamical regimes in silver phosphate glasses
Journal of Non-Crystalline Solids 307–310 (2002) 963–970 www.elsevier.com/locate/jnoncrysol
Ionic conduction and dynamical regimes in silver phosphate glasses M. Cutroni
a,*
, A. Mandanici a, P. Mustarelli b, C. Tomasi b, M. Federico
a
a
b
Dipartimento di Fisica, Universit a degli Studi di Messina and INFM, Unit a di Ricerca di Messina, ctr. Papardo salita Sperone, 31, 98166 Messina, Italy Dipartimento di Chimica-Fisica, Universit a di Pavia, INFM and CSTE-CNR via Taramelli 16, 27100 Pavia, Italy
Abstract Dielectric measurements on ion conducting silver metaphosphate glasses, pure and doped with silver sulphide or silver sulphate, have been performed in the microwave frequency region up to 60 GHz in a wide temperature range between 77 and 413 K. With the current extension of previous measurements, a frequency range larger than 10 decades has been investigated and several distinct conductivity regimes have been observed. Above the low frequency dc conductivity value, the progressive increase of conductivity with frequency has been found to obey first a power law with exponent p 2=3 and later another power law with exponent q 4=3, nearly independent of temperature and of composition. In the low temperature region a nearly constant loss contribution to the conductivity has been also detected. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction The use of glassy and polymer ionic conductors has allowed important progresses in the technological field, giving powerful, portable and durable energy sources. Most questions concerning structure of and dynamics of ions in these materials are still open [1]. Dynamics of mobile ions in ionic glasses can be explored by dielectric spectroscopy techniques, measuring ionic conductivity as a function of frequency. In fact, according to the linear response theory, the frequency dependent
*
Corresponding author. Tel.: +39-90 676 5013; fax +39-90 395 004. E-mail address: [email protected] (M. Cutroni).
(complex) conductivity is related to the velocity correlation function of the mobile species [2]. When experimental data are available over a large frequency range, extending from a few Hz to tens of GHz, as in the present study, the observation timescales cover the range from a second to about 20 ps. In the limit of long times the conductivity tends to assume a frequency independent value rdc , which is usually expressed by the following equation rdc ¼
Nq2 x20 C ; 6VkB T
ð1Þ
where N is the number of mobile ions, q the charge of each mobile ion, x0 the elementary jump distance of the ions, V the volume of the sample, kB the Boltzmann constant, T the temperature, C the
0022-3093/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 2 ) 0 1 5 6 1 - 2
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hopping rate, for a random ionic hopping [3]. The problem of relating the macroscopic dc conductivity (rdc ) to the microscopic ionic movement has been recently proposed by some authors [4]. At higher frequencies, crossing over a characteristic frequency fh , which depends on the material and on the temperature, the conductivity progressively increases as a function of frequency [3]. This behaviour, discussed by Jonscher [5] is common to a wide class of materials, and can be described by a simple power law rðf Þ ¼ r0 þ Af p
ð2Þ
or by the equivalent expression, p
rðf Þ ¼ r0 b1 þ ðf =fh Þ c:
ð3Þ
The hopping frequency fh marking the onset of the dispersive region is then the frequency at which the conductivity becomes the double of the plateau value r0 . The value of the power law exponent p seems to depend on the material investigated [6]. Also, its determination can be conditioned by the width of the frequency range in which experimental spectra are available [7–9]. As suggested recently, the value of the power law exponent, which characterizes the ac conductivity dispersion for some decades above the dc plateau, could be influenced by the dimensionality of the local cation conduction space [10]. Experiments at microwave frequencies [7,8,11–14] have revealed that in some glasses the conductivity is higher than that predicted from Eq. (2) and a proper description of the dispersive behaviour on the broad frequency range between Hz to GHz is given by a double power law, rðf Þ ¼ r0 þ Af p þ Bf q ;
ð4Þ
where 1 < q < 2. In this work the considerable width of the frequency range investigated has been exploited to evaluate the power law exponents of conductivity, which have been found nearly constant as a function of temperature for different glass compositions. Even at the end of the microwave frequency region available, conductivity did not show any levelling off, so an eventual high frequency plateau of ionic conductivity, like that experimentally ob-
served in the ionic crystal RbAg4 I5 [15] and predicted by all hopping models [3,16], is probably hidden because of vibrational contributions to the conductivity. 2. Experimental Phosphate glasses AgPO3 doped with Ag2 S or Ag2 SO4 have been prepared by melt quenching technique [17,18] and the existence of crystalline phases was ruled out by the results of X-ray diffraction measurements. The glassy samples with flat and parallel faces were accurately polished to fit exactly into the waveguide sections corresponding to the conventional frequency bands in the range between 8.0 and 60 GHz. Dielectric measurements have been performed detecting the reflection and transmission parameters of the waveguide line in which the sample was inserted. The actual temperature of the sample into the waveguide was checked locally by a thermocouple. A home made thermalization system has been used for measurements at constant temperature within 0.1 K in the range 77–413 K. Scalar transmission and reflection coefficients have been measured in the range 18.0– 60.0 GHz by a scalar network analyser HP8757A for samples of different thickness, while, in the frequency range 8.0–18.0 GHz, a vector network analyser HP8720C was used, giving forward and backward complex scattering coefficients of the waveguide line for one sample of a given thickness. An optimization procedure to minimize the difference between experimental curves and simulated response of the line was followed to obtain the best estimate for the real and the imaginary part of permittivity as a function of frequency. Complex permittivity in the frequency range 1 Hz– 13 MHz between 20 and 373 K was previously determined by impedance analyser measurements [7,8,14,19]. 3. Results The behaviour of frequency dependent conductivity in silver metaphosphate glass doped with silver sulphate ðAg2 SO4 Þx ðAgPO3 Þ1x is shown in
M. Cutroni et al. / Journal of Non-Crystalline Solids 307–310 (2002) 963–970
965
Fig. 1. Frequency dependent conductivity of ðAg2 SO4 Þx ðAgPO3 Þ1x glasses with x ¼ 0, 0.1, 0.3 at 296 K. Solid lines have been obtained by fitting experimental data to a Jonscher power law plus a new power law contribution, with exponent greater than one (Eq. (4)). The power law exponent corresponds to the mean slope of the power law curve in a log–log representation. For a comparison a straight line with slope 1 has been traced.
Fig 1. A similar trend was found at room temperature in ðAg2 SÞx ðAgPO3 Þ1x glasses [20]. Conductivity values can increase of about five orders of magnitude when frequency is increased from some Hz up to 60 GHz. At low frequencies a plateau region is observed, while at higher frequencies a dispersive behaviour sets in, with a continuous increase up to microwave frequencies. Enhancement of conductivity can be obtained by increasing the dopant salt concentration, but the difference due to the composition is minimized in the region of highest available frequencies. In that region the slope of conductivity in the log–log plots clearly exceeds one. Also increasing temperature (Fig. 2) has the effect of enhancing the conductivity and shifting the onset of dispersive region up to higher frequencies. Temperature variations of conductivity are much more pronounced at low than at high frequencies. The existence of an eventual high frequency conductivity plateau cannot be observed explicitly within the frequency range available. Analysis of experimental data is now presented in separate steps, to study the different dynamical regimes observable in conductivity spectra.
3.1. Low frequency dispersion Fitting the experimental data in the low frequency part of the spectrum to the simple power law equation (3) the plateau value of conductivity and the hopping frequency fh can be estimated. The temperature dependence of dc conductivity is nicely described by an Arrhenius equation (5) on a wide temperature range between 250 and 373 K for the pure silver phosphate glass and on a similar range for the doped compounds. DEdc rdc T ¼ K0 exp : ð5Þ kB T Also the typical frequency fh , which marks the onset of the dispersive region, is found to be thermally activated. The hopping rate, C ¼ 2pfh [3], will then be described by the equation DEh C ¼ C0 exp : ð6Þ kB T Energy values DEh obtained are nearly equal to those, DEdc , deduced from dc conductivity values. Referring to Eq. (1), this would suggest that the concentration of mobile ions and the elementary hopping distance x0 remain nearly independent of
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Fig. 2. Conductivity of ðAg2 SO4 Þ0:3 ðAgPO3 Þ0:7 glass at selected temperatures between 173 and 273 K at low and high frequencies up to 60 GHz. Solid lines have been obtained by fitting experimental data to Eq. (4).
Table 1 Prefactors and activation energies related to the temperature dependence of the conductivity and of the hopping rate for AgPO3 glass pure or doped/modified with Ag2 S or Ag2 SO4 , obtained by fitting experimental data of conductivity and hopping rate to Arrhenius equations (5) and (6) respectively Glass
Log[K0 (X1 cm1 K)]
DEdc =kB (K)
Log[C0 (rad/s)]
DEh =kB (K)
AgPO3 (Ag2 S)0:3 (AgPO3 )0:7 (Ag2 SO4 )0:3 (AgPO3 )0:7
4.97 0.14 4.23 0.14 3.5 0.1
6123 93 4270 75 4156 55
15.1 0.2 13.9 0.2 13.1 0.2
6142 114 4100 113 3924 127
temperature, across the temperature range studied. Fitting parameters concerning temperature dependence of dc conductivity and hopping rate are listed in Table 1. A problem encountered when fitting low frequency dispersion data is that, depending on the temperature and on the composition, some parts of the dynamic conductivity shift out of the experimental frequency range, so rarely one curve has both a well-defined rdc plateau and a complete dispersive part. In particular, hiding a part of the dispersion region has the effect of changing the value of the power law exponent. This value seems to vary as a function of temperature [7], but the
same apparent variation can be simply simulated by ignoring some low frequency or high frequency regions of experimental spectra [9]. Several scaling approaches have been recently proposed, which in principle could allow one to study the spectral shape of ac conductivity by analysing a resulting master curve [21]. Following the procedure indicated by Roling [22], frequencies are scaled by a factor rdc T , while conductivities are divided by rdc . In Fig. 3 wideband experimental data on AgPO3 glass at room temperature up to 60 GHz are compared with a master curve obtained from conductivity curves in the range 1 Hz– 13 MHz between 210 and 290 K. It is evident
M. Cutroni et al. / Journal of Non-Crystalline Solids 307–310 (2002) 963–970
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Fig. 3. Master curve of frequency dependent conductivity for AgPO3 glass obtained from conductivity data in the frequency range 1 Hz–13 MHz at selected temperatures between 210 and 290 K. Frequencies are scaled by a factor rdc ðT ÞT , while conductivities are divided by rdc ðT Þ. For comparison the same scaling is applied on conductivity data of the same glass at room temperature up to the frequency of 60 GHz.
that the master curve does not match to experimental data at microwave frequencies, even though a master curve is obtained for some decades below and close to the onset of the dispersive region. Thus, only in a suitable frequency/temperature range, a time–temperature equivalence seems to be valid. To account for the real shape of the conductivity dispersion curves a parameter K which affects the decay of the mismatch function gðtÞ has been recently introduced in the concept of mismatch and relaxation (CMR) model [23]. 3.2. Analysis of broadband conductivity data A fit on the whole frequency range available requires a new power law contribution, to describe the microwave region, in which the slope of conductivity in a log–log scale exceeds one. Using Eq. (4) a good fit of experimental data is obtained with exponents p ffi 2=3 and q ffi 4=3 nearly independent of composition (see Fig. 1 and Ref. [20]) and also independent of temperature, for fixed compositions (see Figs. 2 and 4). These values are closely similar to those recently reported for other
superionic glasses [24]. An apparent temperature dependence of the power law exponents would be obtained if the low and high frequency regions were analyzed separately. Coefficients A and B of Eq. (4) are found to be thermally activated (Table 2), with activation energies lower than that associated with rdc As it can be seen from Fig. 4, lowering the temperature, the dc conductivity plateau progressively disappears from the available experimental frequency window, while the remaining conductivity spectrum in the MHz region is linear (in the log–log plot) with a slope approaching 1 as the lowest temperature, 20 K, is reached. Actually, disagreement between the fitting curve and the experimental spectra in the MHz region already appears at temperatures below about 200 K. The data can be reproduced only if a further power law contribution, with exponent one is considered. As usual, the suitable use of more fitting parameters gives an improved agreement between the curve fit and experimental data. The extra conductivity term nearly linear in frequency corresponds to a nearly constant loss
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Fig. 4. Frequency dependent conductivity of ðAg2 SÞ0:3 ðAgPO3 Þ0:7 glass at selected temperatures between 133 and 273 K in the frequency range 5 Hz–60 GHz. Conductivity data between 173 and 273 K have been fitted to a double power law equation. Experimental conductivity values at 20 K in the range from 105 Hz up to about 107 Hz (from Ref. [7]) are also reported and compared with a straight line with a slope of one (dotted line).
Table 2 Prefactors and activation energies obtained by fitting experimental conductivity data of AgPO3 glass, pure or doped/modified with Ag2 S or Ag2 SO4 , to a double power law in frequency with thermally activated coefficients: K0 DEdc DEA DEB exp þ A0 exp f p þ B0 exp fq rðf ; T Þ ¼ T kB T kB T kB T Glass
K0 (X1 cm1 K)
Edc =kB (K)
A0 (X1 cm1 K)
EA =kB (K)
p
B0 (X1 cm1 K)
EB =kB (K)
q
AgPO3 (Ag2 S)0:3 (AgPO3 )0:7 (Ag2 SO4 )0:3 (AgPO3 )0:7
1:55 105 1:71 104 4:09 103
6272 4271 4234
3:95 108 2:53 108 4:17 108
1803 1184 1395
0.67 0.67 0.66
2:07 1016 1:17 1016 1:79 1016
301 118 195
1.33 1.33 1.33
[25–27], being e00 ðf Þ ¼ rðf Þ=ð2pf e0 Þ, where e0 is the permittivity of vacuum and e00 ðf Þ is the imaginary part of the frequency dependent dielectric function. This term is almost independent of temperature, so at sufficiently low temperatures it becomes the dominant contribution than the low frequency power law contribution with exponent p. It is worthwhile to point out that the choice of a fitting function containing a nearly constant loss contribution, rðf ; T Þ / f , in addition to the Jon-
scher power law (2) would cause a slight decrease of the numerical value of the exponent p, e.g. from 0.66 to 0.58 in the case of AgPO3 glass.
4. Discussion The conductivity dispersion in frequency can be considered as a probe of ion dynamics on different time and length scales. At sufficiently long times ionic contribution to the conductivity is well ac-
M. Cutroni et al. / Journal of Non-Crystalline Solids 307–310 (2002) 963–970
counted for by a random hopping model. This gives a frequency independent conductivity, which is thermally activated. Activation energy DEdc and hopping frequency are remarkably similar to the corresponding values, which characterize a sub-Tg mechanical absorption peak in acoustic attenuation at ultrasonic frequencies [17–19]. Evidently, even in a nearly frozen glassy matrix, a fraction of atomic units is still able to hop at random, giving rise to electric and mechanical relaxation processes. By inspection of the dielectric spectra as a function of frequency, a smooth transition from random hopping to a different dynamical regime is marked by an onset frequency. In the power law regime, the response function associated with ionic motion is a stretched exponential in time [3,4]; motions develop in a limited spatial extent, which does not exceeds a threshold length n [6,28]. In a relatively short time an ion will be allowed to visit only a limited set of spatial configurations, covering shorter distances and crossing over lower potential barriers. The power law behaviour of conductivity with exponent q > 1, observed at higher frequencies, in the microwave frequency range, could be related to localized hopping processes in a time dependent effective potential for the ion [3,29]. Other authors indicate the ionic vibrational motion as the origin of the f q conductivity behaviour [30]. At sufficiently low temperatures the fingerprint of dynamical processes at even lower energy is found. It appears as a linear dependence of conductivity on frequency (nearly constant loss regime), which is not very sensitive to temperature changes. Regarding this aspect, the response of the system can be modelled considering a distribution of asymmetric double well potentials [31]. Recent investigations suggest that the near constant loss response could be caused by a slow decay of the cage that confines the ion within its potential well [27]. Their contributions should be more evident especially in ionic glasses, where mobile ions are decoupled from the frozen glassy matrix. The existence of several distinct dynamical regimes in the frequency dependent conductivity of the ionic glasses investigated also corresponds to the different relaxational contributions which
969
characterize the mechanical response of the same materials at ultrasonic frequencies [7,8,17,18]. The activation energies associated with the different terms concurring in the total mechanical absorption aðT Þ, in a large temperature range between a few K and 350 K, are close to those characterizing the separate contributions to the electrical conductivity (Table 2).
5. Conclusions Useful hints about ion dynamics in the glass network are obtained by studying the dynamical response of ionic vitreous materials in terms of electrical conductivity in a large range of frequencies and temperatures. It seems that decreasing the temperature and probing the system at sufficiently high frequencies some dynamical processes involving progressively lower activation energies are called into play. Energy values associated with the different contributions to the conductivity are closely similar to those previously deduced from the analysis of acoustic attenuation data at ultrasonic frequencies on the same glasses [8]. Looking at the frequency dependence of conductivity, the localization of hopping processes can be monitored by the numerical value of the power law exponent associated with each conductivity regime. At higher frequencies the localization character increases and a considerable number of elementary local relaxations greatly enhances the conductivity in respect of its low frequency plateau value, which is due instead to hopping processes successful on a macroscopic diffusion scale. The physical reason to have power law exponents with the same numerical value, independent of temperature, in glassy ionic conductors of different composition, remains unknown. The finding of a high frequency power law exponent q 1:3, lower than the value of 2 expected for a Debye-like hopping in a rigid potential, could indicate that the effective potential experienced by the ion is time dependent [3]. This agrees with the possibility that ion hopping is accompanied by local adjustments of the glass structure, like variations of bond angles or distances of non-bridging oxygens or
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terminal units of atomic chains. On the other hand, the occurrence of both the f 1 and f q terms in the real part of the conductivity rðf ; T Þ could be accounted for by considering that, in some frequency/temperature ranges, the ionic mean square displacement, hr2 ðtÞi, is entirely due to the vibrational part of the ion motion, rather than to the diffusive part [30]. Further investigations at microwave frequencies and in the far infrared spectroscopic region are foreseen in order to achieve a better characterization of the superlinear frequency dependence of conductivity observed in the present study.
Acknowledgements The authors are grateful to Professor K. Funke and Dr Cornelia Cramer for their help and support concerning high-frequency measurements in the range 18–60 GHz and for interesting discussions.
References [1] A. Bunde, K. Funke, M.D. Ingram, Solid State Ionics 105 (1998) 1. [2] R. Kubo, J. Phys. Soc. Jpn. 12 (1957) 570. [3] K. Funke, Progr. Solid State Chem. 22 (1993) 111. [4] K.L. Ngai, C. Le on, Phys. Rev. B 60 (1999) 9396. [5] A.K. Jonscher, Nature (London) 267 (1977) 673. [6] D.L. Sidebottom, Phys. Rev. Lett. 83 (1999) 983. [7] M. Cutroni, A. Mandanici, A. Piccolo, C. Fanggao, G. Saunders, P. Mustarelli, Solid State Ionics 90 (1996) 167. [8] M. Cutroni, A. Mandanici, Solid State Ionics 105 (1998) 149. [9] D.L. Sidebottom, J. Non-Cryst. Solids 244 (1999) 223. [10] D.L. Sidebottom, Phys. Rev. B 61 (2000) 14507. [11] C. Cramer, K. Funke, T. Saatkamp, Phil. Mag. B 71 (1995) 701. [12] A. Pradel, M. Ribes, J. Non-Cryst. Solids 172–174 (1994) 1222. [13] S.R. Elliott, Solid State Ionics 70&71 (1994) 27. [14] M. Cutroni, A. Mandanici, A. Piccolo, C. Fanggao, G.A. Saunders, P. Mustarelli, Philos. Mag. B 73 (1996) 349. [15] K. Funke, T. Kloidt, D. Wilmer, C.J. Carlile, Solid State Ionics 53–56 (1992) 947. [16] K.L. Ngai, Comments Solid State Phys. 9 (1980) 141; K. Funke, J. Non-Cryst. Solids 172–174 (1994) 1215;
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D. Kn€ odler, P. Pendzig, W. Dieterich, Solid State Ionics 86–88 (1996) 29; K. Funke, Z. Phys. Chem. 154 (1987) 251. M. Cutroni, A. Magistris, M. Villa, Solid State Ionics 53–56 (1992) 1232. S. Scotti, M. Villa, P. Mustarelli, M. Cutroni, Solid State Ionics 53–56 (1992) 1237. C. Fanggao, G.A. Saunders, Z. Wei, D.P. Almond, M. Cutroni, A. Mandanici, Solid State Ionics 109 (1998) 89. A. Mandanici, M. Cutroni, C. Cramer, K. Funke, P. Mustarelli, C. Tomasi, in: A. Messina (Ed.), Nuclear and Condensed Matter Physics-AIP, vol. 513, 1999, p. 150. B. Roling, A. Happe, K. Funke, M.D. Ingram, Phys. Rev. Lett. 78 (1997) 2160; D.L. Sidebottom, Phys. Rev. Lett. 82 (1999) 3653; T.B. Schrøder, J.C. Dyre, Phys. Rev. Lett. 84 (2000) 310; A. Ghosh, A. Pan, Phys. Rev. Lett. 84 (2000) 2188. B. Roling, Solid State Ionics 105 (1998) 185. K. Funke, B. Roling, M. Lange, Solid State Ionics 105 (1998) 195; K. Funke, R.D. Banhatti, S. Brueckner, C. Cramer, C. Krieger, A. Mandanaci, C. Martiny, I. Ross, Phys. Chem. Chem. Phys. (2002), in press. C. Cramer, K. Funke, T. Saatkamp, D. Wilmer, M.D. Ingram, Z. Naturforsch 50a (1995) 613; C. Cramer, K. Funke, B. Roling, T. Saatkamp, D. Wilmer, M.D. Ingram, A. Pradel, M. Ribes, G. Taillades, Solid State Ionics 86–88 (1996) 481. W.K. Lee, J.F. Liu, A.S. Nowick, Phys. Rev. Lett. 67 (1991) 1559; A.S. Nowick, B.S. Lim, A.V. Vaysleyb, J. Non-Cryst. Solids 172–174 (1994) 1243; A.S. Nowick, A.V. Vaysleyb, Wu. Liu, Solid State Ionics 105 (1998) 121. D.L. Sidebottom, P.F. Green, R.K. Brow, Phys. Rev. Lett. 74 (1995) 5068. K.L. Ngai, J. Chem. Phys. 110 (1999) 10576, and private communication. B. Roling, C. Martiny, K. Funke, J. Non-Cryst. Solids 249 (1999) 201; B. Roling, J. Non-Cryst. Solids 244 (1999) 34. A. Bunde, K. Funke, M.D. Ingram, Solid State Ionics 86–88 (1996) 1311; C. Cramer, K. Funke, T. Saatkamp, D. Wilmer, M.D. Ingram, Z. Naturforsch. 50a (1995) 613; K. Funke, B. Roling, M. Lange, Solid State Ionics 105 (1998) 195. K.L. Ngai, H. Jain, O. Kanert, J. Non-Cryst. Solids 222 (1997) 383. K.S. Gilroy, W.A. Phillips, Philos. Mag. B 43 (1981) 735; A. Pradel, G. Taillades, C. Cramer, M. Ribes, Solid State Ionics 105 (1998) 139; X. Lu, H. Jain, O. Kanert, J. Non-Cryst. Solids 172–174 (1994) 1436.
Ionic conduction and dynamical regimes in silver phosphate glasses M. Cutroni
a,*
, A. Mandanici a, P. Mustarelli b, C. Tomasi b, M. Federico
a
a
b
Dipartimento di Fisica, Universit a degli Studi di Messina and INFM, Unit a di Ricerca di Messina, ctr. Papardo salita Sperone, 31, 98166 Messina, Italy Dipartimento di Chimica-Fisica, Universit a di Pavia, INFM and CSTE-CNR via Taramelli 16, 27100 Pavia, Italy
Abstract Dielectric measurements on ion conducting silver metaphosphate glasses, pure and doped with silver sulphide or silver sulphate, have been performed in the microwave frequency region up to 60 GHz in a wide temperature range between 77 and 413 K. With the current extension of previous measurements, a frequency range larger than 10 decades has been investigated and several distinct conductivity regimes have been observed. Above the low frequency dc conductivity value, the progressive increase of conductivity with frequency has been found to obey first a power law with exponent p 2=3 and later another power law with exponent q 4=3, nearly independent of temperature and of composition. In the low temperature region a nearly constant loss contribution to the conductivity has been also detected. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction The use of glassy and polymer ionic conductors has allowed important progresses in the technological field, giving powerful, portable and durable energy sources. Most questions concerning structure of and dynamics of ions in these materials are still open [1]. Dynamics of mobile ions in ionic glasses can be explored by dielectric spectroscopy techniques, measuring ionic conductivity as a function of frequency. In fact, according to the linear response theory, the frequency dependent
*
Corresponding author. Tel.: +39-90 676 5013; fax +39-90 395 004. E-mail address: [email protected] (M. Cutroni).
(complex) conductivity is related to the velocity correlation function of the mobile species [2]. When experimental data are available over a large frequency range, extending from a few Hz to tens of GHz, as in the present study, the observation timescales cover the range from a second to about 20 ps. In the limit of long times the conductivity tends to assume a frequency independent value rdc , which is usually expressed by the following equation rdc ¼
Nq2 x20 C ; 6VkB T
ð1Þ
where N is the number of mobile ions, q the charge of each mobile ion, x0 the elementary jump distance of the ions, V the volume of the sample, kB the Boltzmann constant, T the temperature, C the
0022-3093/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 2 ) 0 1 5 6 1 - 2
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hopping rate, for a random ionic hopping [3]. The problem of relating the macroscopic dc conductivity (rdc ) to the microscopic ionic movement has been recently proposed by some authors [4]. At higher frequencies, crossing over a characteristic frequency fh , which depends on the material and on the temperature, the conductivity progressively increases as a function of frequency [3]. This behaviour, discussed by Jonscher [5] is common to a wide class of materials, and can be described by a simple power law rðf Þ ¼ r0 þ Af p
ð2Þ
or by the equivalent expression, p
rðf Þ ¼ r0 b1 þ ðf =fh Þ c:
ð3Þ
The hopping frequency fh marking the onset of the dispersive region is then the frequency at which the conductivity becomes the double of the plateau value r0 . The value of the power law exponent p seems to depend on the material investigated [6]. Also, its determination can be conditioned by the width of the frequency range in which experimental spectra are available [7–9]. As suggested recently, the value of the power law exponent, which characterizes the ac conductivity dispersion for some decades above the dc plateau, could be influenced by the dimensionality of the local cation conduction space [10]. Experiments at microwave frequencies [7,8,11–14] have revealed that in some glasses the conductivity is higher than that predicted from Eq. (2) and a proper description of the dispersive behaviour on the broad frequency range between Hz to GHz is given by a double power law, rðf Þ ¼ r0 þ Af p þ Bf q ;
ð4Þ
where 1 < q < 2. In this work the considerable width of the frequency range investigated has been exploited to evaluate the power law exponents of conductivity, which have been found nearly constant as a function of temperature for different glass compositions. Even at the end of the microwave frequency region available, conductivity did not show any levelling off, so an eventual high frequency plateau of ionic conductivity, like that experimentally ob-
served in the ionic crystal RbAg4 I5 [15] and predicted by all hopping models [3,16], is probably hidden because of vibrational contributions to the conductivity. 2. Experimental Phosphate glasses AgPO3 doped with Ag2 S or Ag2 SO4 have been prepared by melt quenching technique [17,18] and the existence of crystalline phases was ruled out by the results of X-ray diffraction measurements. The glassy samples with flat and parallel faces were accurately polished to fit exactly into the waveguide sections corresponding to the conventional frequency bands in the range between 8.0 and 60 GHz. Dielectric measurements have been performed detecting the reflection and transmission parameters of the waveguide line in which the sample was inserted. The actual temperature of the sample into the waveguide was checked locally by a thermocouple. A home made thermalization system has been used for measurements at constant temperature within 0.1 K in the range 77–413 K. Scalar transmission and reflection coefficients have been measured in the range 18.0– 60.0 GHz by a scalar network analyser HP8757A for samples of different thickness, while, in the frequency range 8.0–18.0 GHz, a vector network analyser HP8720C was used, giving forward and backward complex scattering coefficients of the waveguide line for one sample of a given thickness. An optimization procedure to minimize the difference between experimental curves and simulated response of the line was followed to obtain the best estimate for the real and the imaginary part of permittivity as a function of frequency. Complex permittivity in the frequency range 1 Hz– 13 MHz between 20 and 373 K was previously determined by impedance analyser measurements [7,8,14,19]. 3. Results The behaviour of frequency dependent conductivity in silver metaphosphate glass doped with silver sulphate ðAg2 SO4 Þx ðAgPO3 Þ1x is shown in
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Fig. 1. Frequency dependent conductivity of ðAg2 SO4 Þx ðAgPO3 Þ1x glasses with x ¼ 0, 0.1, 0.3 at 296 K. Solid lines have been obtained by fitting experimental data to a Jonscher power law plus a new power law contribution, with exponent greater than one (Eq. (4)). The power law exponent corresponds to the mean slope of the power law curve in a log–log representation. For a comparison a straight line with slope 1 has been traced.
Fig 1. A similar trend was found at room temperature in ðAg2 SÞx ðAgPO3 Þ1x glasses [20]. Conductivity values can increase of about five orders of magnitude when frequency is increased from some Hz up to 60 GHz. At low frequencies a plateau region is observed, while at higher frequencies a dispersive behaviour sets in, with a continuous increase up to microwave frequencies. Enhancement of conductivity can be obtained by increasing the dopant salt concentration, but the difference due to the composition is minimized in the region of highest available frequencies. In that region the slope of conductivity in the log–log plots clearly exceeds one. Also increasing temperature (Fig. 2) has the effect of enhancing the conductivity and shifting the onset of dispersive region up to higher frequencies. Temperature variations of conductivity are much more pronounced at low than at high frequencies. The existence of an eventual high frequency conductivity plateau cannot be observed explicitly within the frequency range available. Analysis of experimental data is now presented in separate steps, to study the different dynamical regimes observable in conductivity spectra.
3.1. Low frequency dispersion Fitting the experimental data in the low frequency part of the spectrum to the simple power law equation (3) the plateau value of conductivity and the hopping frequency fh can be estimated. The temperature dependence of dc conductivity is nicely described by an Arrhenius equation (5) on a wide temperature range between 250 and 373 K for the pure silver phosphate glass and on a similar range for the doped compounds. DEdc rdc T ¼ K0 exp : ð5Þ kB T Also the typical frequency fh , which marks the onset of the dispersive region, is found to be thermally activated. The hopping rate, C ¼ 2pfh [3], will then be described by the equation DEh C ¼ C0 exp : ð6Þ kB T Energy values DEh obtained are nearly equal to those, DEdc , deduced from dc conductivity values. Referring to Eq. (1), this would suggest that the concentration of mobile ions and the elementary hopping distance x0 remain nearly independent of
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Fig. 2. Conductivity of ðAg2 SO4 Þ0:3 ðAgPO3 Þ0:7 glass at selected temperatures between 173 and 273 K at low and high frequencies up to 60 GHz. Solid lines have been obtained by fitting experimental data to Eq. (4).
Table 1 Prefactors and activation energies related to the temperature dependence of the conductivity and of the hopping rate for AgPO3 glass pure or doped/modified with Ag2 S or Ag2 SO4 , obtained by fitting experimental data of conductivity and hopping rate to Arrhenius equations (5) and (6) respectively Glass
Log[K0 (X1 cm1 K)]
DEdc =kB (K)
Log[C0 (rad/s)]
DEh =kB (K)
AgPO3 (Ag2 S)0:3 (AgPO3 )0:7 (Ag2 SO4 )0:3 (AgPO3 )0:7
4.97 0.14 4.23 0.14 3.5 0.1
6123 93 4270 75 4156 55
15.1 0.2 13.9 0.2 13.1 0.2
6142 114 4100 113 3924 127
temperature, across the temperature range studied. Fitting parameters concerning temperature dependence of dc conductivity and hopping rate are listed in Table 1. A problem encountered when fitting low frequency dispersion data is that, depending on the temperature and on the composition, some parts of the dynamic conductivity shift out of the experimental frequency range, so rarely one curve has both a well-defined rdc plateau and a complete dispersive part. In particular, hiding a part of the dispersion region has the effect of changing the value of the power law exponent. This value seems to vary as a function of temperature [7], but the
same apparent variation can be simply simulated by ignoring some low frequency or high frequency regions of experimental spectra [9]. Several scaling approaches have been recently proposed, which in principle could allow one to study the spectral shape of ac conductivity by analysing a resulting master curve [21]. Following the procedure indicated by Roling [22], frequencies are scaled by a factor rdc T , while conductivities are divided by rdc . In Fig. 3 wideband experimental data on AgPO3 glass at room temperature up to 60 GHz are compared with a master curve obtained from conductivity curves in the range 1 Hz– 13 MHz between 210 and 290 K. It is evident
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Fig. 3. Master curve of frequency dependent conductivity for AgPO3 glass obtained from conductivity data in the frequency range 1 Hz–13 MHz at selected temperatures between 210 and 290 K. Frequencies are scaled by a factor rdc ðT ÞT , while conductivities are divided by rdc ðT Þ. For comparison the same scaling is applied on conductivity data of the same glass at room temperature up to the frequency of 60 GHz.
that the master curve does not match to experimental data at microwave frequencies, even though a master curve is obtained for some decades below and close to the onset of the dispersive region. Thus, only in a suitable frequency/temperature range, a time–temperature equivalence seems to be valid. To account for the real shape of the conductivity dispersion curves a parameter K which affects the decay of the mismatch function gðtÞ has been recently introduced in the concept of mismatch and relaxation (CMR) model [23]. 3.2. Analysis of broadband conductivity data A fit on the whole frequency range available requires a new power law contribution, to describe the microwave region, in which the slope of conductivity in a log–log scale exceeds one. Using Eq. (4) a good fit of experimental data is obtained with exponents p ffi 2=3 and q ffi 4=3 nearly independent of composition (see Fig. 1 and Ref. [20]) and also independent of temperature, for fixed compositions (see Figs. 2 and 4). These values are closely similar to those recently reported for other
superionic glasses [24]. An apparent temperature dependence of the power law exponents would be obtained if the low and high frequency regions were analyzed separately. Coefficients A and B of Eq. (4) are found to be thermally activated (Table 2), with activation energies lower than that associated with rdc As it can be seen from Fig. 4, lowering the temperature, the dc conductivity plateau progressively disappears from the available experimental frequency window, while the remaining conductivity spectrum in the MHz region is linear (in the log–log plot) with a slope approaching 1 as the lowest temperature, 20 K, is reached. Actually, disagreement between the fitting curve and the experimental spectra in the MHz region already appears at temperatures below about 200 K. The data can be reproduced only if a further power law contribution, with exponent one is considered. As usual, the suitable use of more fitting parameters gives an improved agreement between the curve fit and experimental data. The extra conductivity term nearly linear in frequency corresponds to a nearly constant loss
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Fig. 4. Frequency dependent conductivity of ðAg2 SÞ0:3 ðAgPO3 Þ0:7 glass at selected temperatures between 133 and 273 K in the frequency range 5 Hz–60 GHz. Conductivity data between 173 and 273 K have been fitted to a double power law equation. Experimental conductivity values at 20 K in the range from 105 Hz up to about 107 Hz (from Ref. [7]) are also reported and compared with a straight line with a slope of one (dotted line).
Table 2 Prefactors and activation energies obtained by fitting experimental conductivity data of AgPO3 glass, pure or doped/modified with Ag2 S or Ag2 SO4 , to a double power law in frequency with thermally activated coefficients: K0 DEdc DEA DEB exp þ A0 exp f p þ B0 exp fq rðf ; T Þ ¼ T kB T kB T kB T Glass
K0 (X1 cm1 K)
Edc =kB (K)
A0 (X1 cm1 K)
EA =kB (K)
p
B0 (X1 cm1 K)
EB =kB (K)
q
AgPO3 (Ag2 S)0:3 (AgPO3 )0:7 (Ag2 SO4 )0:3 (AgPO3 )0:7
1:55 105 1:71 104 4:09 103
6272 4271 4234
3:95 108 2:53 108 4:17 108
1803 1184 1395
0.67 0.67 0.66
2:07 1016 1:17 1016 1:79 1016
301 118 195
1.33 1.33 1.33
[25–27], being e00 ðf Þ ¼ rðf Þ=ð2pf e0 Þ, where e0 is the permittivity of vacuum and e00 ðf Þ is the imaginary part of the frequency dependent dielectric function. This term is almost independent of temperature, so at sufficiently low temperatures it becomes the dominant contribution than the low frequency power law contribution with exponent p. It is worthwhile to point out that the choice of a fitting function containing a nearly constant loss contribution, rðf ; T Þ / f , in addition to the Jon-
scher power law (2) would cause a slight decrease of the numerical value of the exponent p, e.g. from 0.66 to 0.58 in the case of AgPO3 glass.
4. Discussion The conductivity dispersion in frequency can be considered as a probe of ion dynamics on different time and length scales. At sufficiently long times ionic contribution to the conductivity is well ac-
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counted for by a random hopping model. This gives a frequency independent conductivity, which is thermally activated. Activation energy DEdc and hopping frequency are remarkably similar to the corresponding values, which characterize a sub-Tg mechanical absorption peak in acoustic attenuation at ultrasonic frequencies [17–19]. Evidently, even in a nearly frozen glassy matrix, a fraction of atomic units is still able to hop at random, giving rise to electric and mechanical relaxation processes. By inspection of the dielectric spectra as a function of frequency, a smooth transition from random hopping to a different dynamical regime is marked by an onset frequency. In the power law regime, the response function associated with ionic motion is a stretched exponential in time [3,4]; motions develop in a limited spatial extent, which does not exceeds a threshold length n [6,28]. In a relatively short time an ion will be allowed to visit only a limited set of spatial configurations, covering shorter distances and crossing over lower potential barriers. The power law behaviour of conductivity with exponent q > 1, observed at higher frequencies, in the microwave frequency range, could be related to localized hopping processes in a time dependent effective potential for the ion [3,29]. Other authors indicate the ionic vibrational motion as the origin of the f q conductivity behaviour [30]. At sufficiently low temperatures the fingerprint of dynamical processes at even lower energy is found. It appears as a linear dependence of conductivity on frequency (nearly constant loss regime), which is not very sensitive to temperature changes. Regarding this aspect, the response of the system can be modelled considering a distribution of asymmetric double well potentials [31]. Recent investigations suggest that the near constant loss response could be caused by a slow decay of the cage that confines the ion within its potential well [27]. Their contributions should be more evident especially in ionic glasses, where mobile ions are decoupled from the frozen glassy matrix. The existence of several distinct dynamical regimes in the frequency dependent conductivity of the ionic glasses investigated also corresponds to the different relaxational contributions which
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characterize the mechanical response of the same materials at ultrasonic frequencies [7,8,17,18]. The activation energies associated with the different terms concurring in the total mechanical absorption aðT Þ, in a large temperature range between a few K and 350 K, are close to those characterizing the separate contributions to the electrical conductivity (Table 2).
5. Conclusions Useful hints about ion dynamics in the glass network are obtained by studying the dynamical response of ionic vitreous materials in terms of electrical conductivity in a large range of frequencies and temperatures. It seems that decreasing the temperature and probing the system at sufficiently high frequencies some dynamical processes involving progressively lower activation energies are called into play. Energy values associated with the different contributions to the conductivity are closely similar to those previously deduced from the analysis of acoustic attenuation data at ultrasonic frequencies on the same glasses [8]. Looking at the frequency dependence of conductivity, the localization of hopping processes can be monitored by the numerical value of the power law exponent associated with each conductivity regime. At higher frequencies the localization character increases and a considerable number of elementary local relaxations greatly enhances the conductivity in respect of its low frequency plateau value, which is due instead to hopping processes successful on a macroscopic diffusion scale. The physical reason to have power law exponents with the same numerical value, independent of temperature, in glassy ionic conductors of different composition, remains unknown. The finding of a high frequency power law exponent q 1:3, lower than the value of 2 expected for a Debye-like hopping in a rigid potential, could indicate that the effective potential experienced by the ion is time dependent [3]. This agrees with the possibility that ion hopping is accompanied by local adjustments of the glass structure, like variations of bond angles or distances of non-bridging oxygens or
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terminal units of atomic chains. On the other hand, the occurrence of both the f 1 and f q terms in the real part of the conductivity rðf ; T Þ could be accounted for by considering that, in some frequency/temperature ranges, the ionic mean square displacement, hr2 ðtÞi, is entirely due to the vibrational part of the ion motion, rather than to the diffusive part [30]. Further investigations at microwave frequencies and in the far infrared spectroscopic region are foreseen in order to achieve a better characterization of the superlinear frequency dependence of conductivity observed in the present study.
Acknowledgements The authors are grateful to Professor K. Funke and Dr Cornelia Cramer for their help and support concerning high-frequency measurements in the range 18–60 GHz and for interesting discussions.
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