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Origin and properties of the nearly constant loss in crystalline and glassy ionic conductors
Journal of Non-Crystalline Solids 307–310 (2002) 1024–1030 www.elsevier.com/locate/jnoncrysol
Origin and properties of the nearly constant loss in crystalline and glassy ionic conductors n a,*, J. Sanz b, C.P.E. Varsamis c, A. Rivera a, J. Santamarıa a, C. Leo G.D. Chryssikos c, K.L. Ngai d a
c
GFMC, Dpto. de Fisica Aplicada III, Universidad Complutense Madrid, 28040 Madrid, Spain b Instituto de Ciencia de Materiales de Madrid, CSIC, 28049 Cantoblanco, Spain Theoretical and Physical Chemistry Institute, National Hellenic Research Foundation, 48, Vass. Constantinou Avenue, Athens 11635, Greece d Naval Research Laboratory, Washington, DC 20375-5320, USA
Abstract We report on the temperature and mass dependence of the nearly constant loss (NCL) in several ionically conducting materials, both crystalline (Li0:18 La0:61 TiO3 and (Y2 O3 )0:16 (ZrO2 )0:84 ) and glassy (M2 O 3B2 O3 , M ¼ Li, Na, K and Rb). The magnitude A of the NCL is found to be larger for those materials showing higher dc conductivity values at room temperature, and shows weak temperature dependence of the form A / expðbTÞ in all cases. We have also found that A systematically decreases upon increasing the mass of the mobile alkali ion in the borate glasses. All these results point to the relaxation of ions in a very slowly decaying cage as the origin of the NCL. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 66.30.Hs
1. Introduction The ac conductivity of ionically conducting materials shows a linear frequency dependent term as the dominant contribution, i.e. r0 ðxÞ Ax, at sufficiently high frequencies and/or low temperatures [1,2]. Since this term results in a dielectric loss nearly independent of frequency, e00 ðxÞ ¼ r0 ðxÞ= x Axd , with d almost zero, such a behavior is
*
Corresponding author. Tel.: +34-913 944 435; fax: +34-913 945 196. E-mail address: carletas@fis.ucm.es (C. Leon).
usually known as nearly constant loss (NCL) [3–5]. There are other materials showing NCL behavior at low temperatures [6], but it seems to be a universal feature in the case of ionic conductors [1,4]. However, experimental conductivity data of ionic conductors showing a NCL as the dominant contribution in r0 ðxÞ are scarce in the literature, and systematic studies of these materials are desirable in order to investigate the origin and properties of the NCL. Researchers in the field mostly have focused their interest on the long-range ion motion at high temperatures, where the frequency dependence of the ac conductivity can be described by the Jonscher law [6],
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 8 - 5
A. Rivera et al. / Journal of Non-Crystalline Solids 307–310 (2002) 1024–1030
r0 ðxÞ r0 ½1 þ ðx=x0 Þn :
ð1Þ
In this temperature range, a dc conductivity r0 is found in the limit of low frequency, and a power law behavior is observed with a fractional exponent n above a characteristic frequency x0 . Both r0 and x0 are found to be thermally activated with about the same activation energy Er . This has been interpreted in different theoretical models [7–12] as indicating that the ac conductivity given by Eq. (1) originates from migration of ions by hopping between neighboring potential wells, which eventually gives rise to a dc conductivity at the lowest frequencies. Since ion hopping is a thermally activated process, it is at sufficiently low temperatures that the contribution given by Eq. (1) becomes negligible and the NCL becomes the dominant contribution to the ac conductivity in the experimental frequency window. For the usual measurement range of frequencies up to a few megahertz, and in typical ionic conductors, it is then necessary to perform complex admittance measurements well below room temperature to obtain a dominant contribution from the NCL. Accordingly, in order to observe a NCL behavior in the ac conductivity at room temperature it is usually necessary to extend the frequency range up to the gigahertz. We want to emphasize at this point that, although it is usually assumed in the literature [1–3,13–15] that the Jonscher conductivity and the linear frequency term (or NCL) are additive contributions, it has been recently shown that the total ac conductivity in ionic conductors can not be expressed by a simple addition of both contributions in the frequency domain [16,17]. The NCL behavior has been often explained in the past in terms of asymmetric double-well potential configurations and attributed to the motion of groups of ions [14,18,19]. Recently, the origin of the NCL has been proposed to be in the vibrational relaxation of ions within their cages, before they start hopping and contributing to the longrange ion transport [4,16]. More recently, NCL is identified with the very slow decay of the cage in the short time regime, when the probability of successful hop of the ions out of their potential wells is small [20]. There are also different theo-
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retical models [21,22] which propose there is no formal separation between the Jonscher law and the NCL, and the latter is just the consequence of ionic hopping but at a very short time scale. In order to further understand the NCL behavior and its origin, it is necessary to perform new and systematic studies of the ac conductivity of different ionic conductors at low temperatures and/or high frequencies. By investigating the properties of the NCL we shall get new insights about its nature and origin. In this paper we report experimental results of NCL in various ionic conductors and investigate the temperature dependence of the NCL and also the dependence of the NCL upon changes in the mass of the mobile ions. We have chosen materials with crystalline and glassy structures, with different atomic species as mobile ions, and showing different values of the dc conductivity at room temperature: a crystalline superionic conductor such as Li3x La2=3 x TiO3 , which shows a high dc conductivity value at room temperature due to the high mobility of lithium ions through the perovskite structure [23,24]; yttria stabilized zirconia (YSZ), where the mobile charge carriers are oxygen ions through the oxygen vacancies created in the cubic structure of zirconia by yttria doping [25,26]; and the family of alkali triborate glasses, M2 O 3B2 O3 , where M stands for Li, Na, K and Rb, which are the mobile ions [27,28]. Our experimental results point to the relaxation of ions in a very slowly decaying cage as the origin of the NCL.
2. Experimental Samples of YSZ were (1 0 0) oriented single crystals supplied by ESCETE with composition ZrO2 –16 mol% Y2 O3 and 10 5 1 mm sized. Ceramic samples of composition Li0:18 La0:61 TiO3 were prepared by heating a stoichiometric mixture of high purity LiCO3 , La2 O3 and TiO2 reagents at 1350 °C in air for several hours (5–11 h) and then quenched to room temperature. Sintered cylindrical pellets 5 mm in diameter and 1 mm thick, with evaporated silver electrodes, were used for electrical measurements. Li content was checked by
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ICP analysis. Alkali borate glasses with the chemical formula M2 O 3B2 O3 (where M stands for Li, Na, K and Rb) were prepared from stoichiometric ground mixtures of B2 O3 and alkali carbonates by melting in platinum crucibles in an electric furnace at 1100 °C for 20 min. The liquids were quenched by pouring into 10 mm diameter, cylindrical brass molds, followed by annealing at 10 °C below Tg . The resulting glass cylinders were cut with a low speed diamond saw into 1 mm thick disks. Silver electrodes were evaporated on the flat surfaces of the samples prior to electrical measurements. Complex admittance was determined in the audio frequency range (100 Hz to 100 kHz) at low temperatures (8–300 K) by using a closed cycle helium cryostat and precision LCR meter HP4284. High frequency complex admittance measurements from 1 MHz to 1 GHz were made at room temperature with an HP4291A impedance analyzer.
Fig. 1. Temperature dependence of the real part of the conductivity, r0 ðxÞ, at fixed frequencies for Li0:18 La0:61 TiO3 . Conductivity data are shown for different frequencies (300 Hz, 1, 3, 10, 30 and 100 kHz, from bottom to top) in a linear temperature scale. At low temperatures, data at the different frequencies are spaced according to a linear frequency dependence. This is emphasized in the inset, where we have plotted the imaginary part of the permittivity, or dielectric loss, e00 ðxÞ ¼ r0 ðxÞ=e0 x, for the same frequencies, showing that these losses are nearly frequency independent, e00 ðxÞ A. Only at the highest temperatures, for which ionic hopping contributes to the diffusion, any frequency dependence becomes noticeable. The errors of the data are smaller than the size of the symbols.
3. Results The temperature dependence of the ac conductivity of Li0:18 La0:61 TiO3 is shown in Fig. 1 for several fixed frequencies from 300 Hz to 100 kHz and in the temperature range from 300 down to 8 K. It can be observed that, at low temperatures, experimental conductivity data sets are spaced according to a linear frequency term, r0 ðxÞ ¼ xe0 e00 ðxÞ Ae0 x. While at higher temperatures, the conductivity data change to assume a sublinear frequency dependence (see Eq. (1)). This can be seen more clearly in the inset where e00 ðxÞ is plotted against temperature. The data from the six different frequencies collapse onto a single curve at low temperature, indicating that e00 ðxÞ is almost independent of frequency and therefore a genuine NCL. At high temperatures the experimental data e00 ðxÞ for the different frequencies in the inset can be seen to diverge, due to the appearance of a different contribution to the ac conductivity described by the Jonscher law. In a similar fashion, Fig. 2 shows the temperature dependence of the ac conductivity of YSZ with a 16% molar yttria content. Each data curve was obtained by varying
Fig. 2. Temperature dependence of the real part of the conductivity, r0 ðxÞ, at fixed frequencies for (Y2 O3 )0:16 (ZrO2 )0:84 . Conductivity data are shown for different frequencies (1, 3, 10, 30 and 100 kHz, from bottom to top) in a linear temperature scale. To emphasize the linear frequency dependence of the conductivity, the imaginary part of the permittivity, or dielectric loss, e00 ðxÞ ¼ r0 ðxÞ=e0 x, is plotted in the inset for the same frequencies. Note that only at the highest temperatures, where ionic hopping contributes to diffusion, data at different frequencies separate from each other. The errors of the data are smaller than the size of the symbols.
A. Rivera et al. / Journal of Non-Crystalline Solids 307–310 (2002) 1024–1030
Fig. 3. Temperature dependence of the real part of the conductivity, r0 ðxÞ, at fixed frequencies for Li2 O 3B2 O3 . Conductivity data are shown for different frequencies (1, 3, 10, 30 and 100 kHz, from bottom to top) in a linear temperature scale. The linear frequency dependence of the conductivity is better shown in the inset, where we have plotted the imaginary part of the permittivity, or dielectric loss, e00 ðxÞ ¼ r0 ðxÞ=e0 x at the same frequencies. At low temperatures the losses are nearly frequency independent, e00 ðxÞ A. Only at the highest temperatures, for which ionic hopping contributes to the diffusion, any frequency dependence becomes noticeable. The errors of the data are smaller than the size of the symbols.
the temperature from 100 to 400 K at constant frequency ranging from 1 to 100 kHz. The inset shows the temperature dependence of the imaginary part of the dielectric permittivity, e00 ðxÞ, which is clearly a NCL up to almost room temperature in this frequency range. Fig. 3 shows conductivity vs temperature plots, at five different fixed frequencies between 1 and 100 kHz, for the lithium triborate glass (Li2 O 3B2 O3 ). The NCL behavior at low temperatures is again more clearly observed in the inset, where the temperature dependence of the dielectric loss e00 ðxÞ is plotted. It can be observed that the magnitude A of the NCL is much higher in Li0:18 La0:61 TiO3 , (A ¼ 1:0 0:2 at T ¼ 100 K), than in (Y2 O3 )0:16 (ZrO2 )0:84 , (A ¼ 3:6 0:4 10 2 at T ¼ 100 K), or Li2 O 3B2 O3 , (A ¼ 2:4 0:2 10 2 at T ¼ 100 K). Interestingly, the magnitude A, for all the samples analyzed, was found to be only weakly dependent on temperature, especially in the case of the lithium triborate glass which shows a very slow increase of the NCL when increasing temperature in a wide range. In fact, although we found that the best fit to the
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weak temperature dependence of the NCL was an exponential function of temperature, it could be also well described by a power law or even a linear dependence on temperature. Solid lines in Figs. 1–3 are fits to an exponential temperature dependence of the form A / expðbT Þ, with values of 1:8 0:1 10 2 , 3:9 0:3 10 3 and 5 1 10 4 K 1 of the parameter b for Li0:18 La0:61 TiO3 , (Y2 O3 )0:16 (ZrO2 )0:84 , and Li2 O 3B2 O3 respectively. Fig. 4(a) shows conductivity vs frequency plots at 300 K, from 1 MHz up to 1 GHz, for the four alkali triborate glasses studied in this work. Note that the experimental conductivity data have been shifted vertically to separate them from each other for clarity. It can be readily observed that a linear frequency dependence of the ac conductivity is found for all samples over this frequency range. Best fits are obtained for a frequency dependence of the form, r0 ðxÞ ¼ Aeo xp , with an exponent p ¼ 0:99 0:03 for all samples. The magnitude A of the NCL at 300 K is thus obtained for each alkali borate from these fits. The dependence of the nearly constant loss on the mass of alkali ions is presented in Fig. 4(b). We have plotted data at two different temperatures, 300 and 60 K, and both data sets show a similar trend: the magnitude of the NCL decreases slightly but systematically as the alkali mass increases.
4. Discussion It has been recently proposed [4,16] that the NCL is due to vibrational relaxation of mobile ions within their cages before they start hopping. Within this picture the magnitude of the NCL would be related to the mean squared displacement of ions in the vibrational motion. In the case of a harmonic potential defining the cage the mean squared displacement of the ions is a constant. Since the NCL behavior implies a slow increase of the mean squared displacement with the logarithm of time, anharmonicity of the potential of the cages might play an important role. More recently [20], NCL has been associated with a very slow decay of the cages and consequently anharmonicity of the potential may become time dependent,
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Fig. 4. (a) Frequency dependence of the real part of the conductivity, r0 ðxÞ, at room temperature and frequencies in the range 1 MHz to 1 GHz, for the alkali triborate glasses M2 O 3B2 O3 , with M ¼ Li, Na, K and Rb. Data sets corresponding to M ¼ Li, Na and K have been multiplied by 103 , 102 and 10, respectively, for clarity. Solid lines represent best fits to a linear frequency dependence, i.e. a constant loss contribution to the conductivity. (b) Mass dependence of the dielectric loss, e00 ðxÞ A, at 60 and 300 K for the alkali triborate glasses shown in (a). The magnitude of the NCL increases by a factor of two in changing the alkali ion from rubidium to lithium. Dotted lines are fits to a power law dependence of the form m a with exponents a ¼ 0:35 0:03 and 0:31 0:05 at 60 and 300 K respectively. Solid lines represent a dependence of the form m 1=3 , which describes within error for the experimental data.
increasing slowly with the logarithm of time. Such time dependent anharmonicity could explain also why the mean squared displacement, and therefore the NCL, does not show a linear temperature dependence characteristic of harmonic vibration, but weak exponential temperature dependence A / expðbT Þ.
The observed change in the magnitude of A of different samples at a given temperature is much smaller than that found for the dc conductivity. For example, in the family of alkali triborates, A varies by about a factor of two at 300 K, while the dc conductivity decreases by more than three orders of magnitude in changing the alkali ion from lithium to rubidium. However, changes in both quantities might be related, since samples with a higher dc conductivity show also a higher value of A. The same trend is found in the case of Li0:18 La0:61 TiO3 , which shows a high dc conductivity at room temperature (rdc 3 10 4 S cm 1 ) and has also the highest value of A, almost two orders of magnitude higher than that of the other samples analyzed. This correlation between the magnitude of A and the dc conductivity value is consistent with the previously published observation [4] that ionic conductors showing high values of the NCL are all superionic conductors, having very high dc conductivities even at room temperature. This fact, together with the contrast between the weak temperature dependence of the NCL and the thermally activated behavior of the dc conductivity, suggest that the NCL is related to the ions, but has a different origin than the dc conductivity. Another interesting property of the NCL is its dependence on the mass of the mobile ions, which has not been investigated in the past. We have found at constant temperature a systematic decrease in the magnitude of A upon increasing the mass of the mobile ions in the family of alkali triborates. Since the room temperature far-infrared spectra of these glasses show that anharmonicity decreases with the increase in the mass of the alkali ions [27], the observed mass dependence of NCL is consistent with the proposed connection of NCL to anharmonicity of the cage potential [4,16] and the origin of NCL from time dependent anharmonicity caused by very slow decay of the cage. Thus the moderate decrease of NCL with increasing alkali ion mass and with decreasing temperature according to A / expðbT Þ may stem from the same source. At this time it is too early to identify the true source of NCL. In passing we mention other possible connections. In a loosely related problem
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of lattice vibration with anharmonic restoring forces, it has been shown that the on-site anharmonic potential modeled by either a hard /4 potential or a Morse potential can slow down dramatically the vibrational energy relaxation [29– 31]. Interestingly, the dependence of the magnitude of the NCL on the alkali ion mass found in the present work may also be explained within this picture. In the case of an ion vibrating in a cage defined by a harmonic potential the characteristic mean-squared displacement is independent of the mass of the ion [32]. However, in the case of an anharmonic potential of the /4 type, it is found [33,34] that the mass dependence is proportional to m a with an exponent a ¼ 1=3. We have plotted as dashed lines in Fig. 4(b) the best fits to a powerlaw dependence on the alkali mass of the value of A obtained for the alkali triborates. The value obtained for the exponent a is 0:35 0:03 and 0:31 0:05 at 60 and 300 K respectively. In fact, solid lines in Fig. 4(b) represent a mass dependence of the form m 1=3 , which accounts within experimental error, and at both temperatures, for the observed change in the magnitude of the NCL with the mass of the alkali ion. However, we cannot say for sure that these models bear any relation to NCL.
5. Conclusions We have determined the temperature dependence of the nearly constant loss by ac conductivity measurements in the audio frequency range and varying temperature from 300 down to 8 K. The magnitude A of the nearly constant loss is found to be weakly dependent on temperature. Good fits are obtained in a wide temperature range to an expression A / expðbT Þ for all the samples analyzed. We have also found, in the family of alkali triborate glasses, that the magnitude of the nearly constant loss at constant temperature decreases systematically upon decreasing the mass of the mobile ions. The observed mass dependence can be described by a power law of the form A / m 1=3 within experimental error. These properties of the nearly constant loss suggest that it may arise from the mean-squared displacement of
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the ions increasing slowly with time due to the very slow decay with time of the cages confining the ions.
Acknowledgements Financial support from Spanish CICYT grant MAT98-1053-C04 is acknowledged. The work performed at NRL was supported by ONR.
References [1] W.K. Lee, J.F. Liu, A.S. Nowick, Phys. Rev. Lett. 67 (1991) 1559. [2] K.L. Ngai, H. Jain, O. Kanert, J. Non-Cryst. Solids 222 (1997) 383. [3] A.S. Nowick, A.V. Vaysleb, W. Liu, Solid State Ionics 105 (1998) 121. [4] K.L. Ngai, J. Chem. Phys. 110 (1999) 10576. [5] J.R. Macdonald, J. Chem. Phys. 115 (2001) 6192. [6] A.K. Jonscher, Dielectric Relaxation in Solids, Chelsea Dielectric, London, 1983. [7] F.S. Howell, R.A. Bose, P.B. Macedo, C.T. Moynihan, J. Phys. Chem. 78 (1974) 639. [8] K.L. Ngai, Comments Solid State Phys. 9 (1979) 127. [9] K.L. Ngai, Phys. Rev. B 48 (1993) 13481. [10] K. Funke, Prog. Solid State Chem. 22 (1993) 111. [11] J.C. Dyre, J. Appl. Phys. 64 (1988) 2456. [12] J.R. Macdonald, J. Appl. Phys 90 (2001) 153, and references therein. [13] C. Cramer, K. Funke, T. Saatkamp, Philos. Mag. B 71 (1995) 701. [14] H. Jain, X. Lu, J. Non-Cryst. Solids 196 (1996) 285. [15] C. Le on, M.L. Lucia, J. Santamaria, Phys. Rev. B 55 (1997) 882. [16] C. Le on, A. Rivera, A. Varez, J. Sanz, J. Santamarıa, K.L. Ngai, Phys. Rev. Lett. 86 (2001) 1279. [17] A. Rivera, C. Le on, J. Sanz, J. Santamarıa, C.T. Moynihan, K.L. Ngai, Phys. Rev. B, in press. [18] H. Jain, S. Krishnaswami, Solid State Ion. 105 (1998) 129. [19] A.S. Nowick, B.S. Lim, A.V. Vaysleb, J. Non-Cryst. Solids 172–174 (1994) 1243. [20] K.L. Ngai, R.W. Rendell, C. Le on, these Proceedings, p. 1039. [21] K. Funke, D. Wilmer, Solid State Ion. 136&137 (2000) 1329. [22] J.C. Dyre, T.B. Schroder, Rev. Mod. Phys. 72 (2000) 873. [23] C. Le on, M.L. Lucia, J. Santamaria, M.A. Paris, J. Sanz, A. Varez, Phys. Rev. B 54 (1996) 184; C. Le on, J. Santamaria, M.A. Paris, J. Sanz, J. Ibarra, L.M. Torres, Phys. Rev. B 56 (1997) 5302.
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[24] J.A. Alonso, J. Sanz, J. Santamaria, C. Le on, A. Varez, M.T. Fern andez, Angew. Chem. Int. Ed. 39 (2000) 619. [25] A. Rivera, J. Santamaria, C. Le on, Appl. Phys Lett. 78 (2001) 610. [26] C. Le on, P. Lunkenheimer, K.L. Ngai, Phys. Rev. B 64 (2001) 184304. [27] G.D. Chryssikos, L. Liu, C.P. Varsamis, E.I. Kamitsos, J. Non-Cryst. Solids 235–237 (1998) 761. [28] A. Rivera, C. Le on, C.P.E. Varsamis, G.D. Chryssikos, K.L. Ngai, C.M. Roland, L.J. Buckley, Phys. Rev. Lett. 88 (2002) 125902.
[29] G.P. Tsironis, S. Aubry, Phys. Rev. Lett. 77 (1996) 5225. [30] A. Bikaki, N.K. Voulgarakis, S. Aubry, G.P. Tsironis, Phys. Rev. E 59 (1999) 1234. [31] S. Flach, G. Mutschke, Phys. Rev. E 49 (1994) 5018. [32] P.M. Chaikin, T.C. Lubensky, in: Principles of Condensed Matter Physics, Cambridge University, Cambridge, 1995, p. 387. [33] V.G. Karpov, M.I. Klinger, F.N. Ignat’ev, Sov. Phys. JETP 57 (1983) 439. [34] Y.M. Galperin, V.L. Gurevich, D.A. Parshin, Phys. Rev. B 32 (1985) 6873.
Origin and properties of the nearly constant loss in crystalline and glassy ionic conductors n a,*, J. Sanz b, C.P.E. Varsamis c, A. Rivera a, J. Santamarıa a, C. Leo G.D. Chryssikos c, K.L. Ngai d a
c
GFMC, Dpto. de Fisica Aplicada III, Universidad Complutense Madrid, 28040 Madrid, Spain b Instituto de Ciencia de Materiales de Madrid, CSIC, 28049 Cantoblanco, Spain Theoretical and Physical Chemistry Institute, National Hellenic Research Foundation, 48, Vass. Constantinou Avenue, Athens 11635, Greece d Naval Research Laboratory, Washington, DC 20375-5320, USA
Abstract We report on the temperature and mass dependence of the nearly constant loss (NCL) in several ionically conducting materials, both crystalline (Li0:18 La0:61 TiO3 and (Y2 O3 )0:16 (ZrO2 )0:84 ) and glassy (M2 O 3B2 O3 , M ¼ Li, Na, K and Rb). The magnitude A of the NCL is found to be larger for those materials showing higher dc conductivity values at room temperature, and shows weak temperature dependence of the form A / expðbTÞ in all cases. We have also found that A systematically decreases upon increasing the mass of the mobile alkali ion in the borate glasses. All these results point to the relaxation of ions in a very slowly decaying cage as the origin of the NCL. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 66.30.Hs
1. Introduction The ac conductivity of ionically conducting materials shows a linear frequency dependent term as the dominant contribution, i.e. r0 ðxÞ Ax, at sufficiently high frequencies and/or low temperatures [1,2]. Since this term results in a dielectric loss nearly independent of frequency, e00 ðxÞ ¼ r0 ðxÞ= x Axd , with d almost zero, such a behavior is
*
Corresponding author. Tel.: +34-913 944 435; fax: +34-913 945 196. E-mail address: carletas@fis.ucm.es (C. Leon).
usually known as nearly constant loss (NCL) [3–5]. There are other materials showing NCL behavior at low temperatures [6], but it seems to be a universal feature in the case of ionic conductors [1,4]. However, experimental conductivity data of ionic conductors showing a NCL as the dominant contribution in r0 ðxÞ are scarce in the literature, and systematic studies of these materials are desirable in order to investigate the origin and properties of the NCL. Researchers in the field mostly have focused their interest on the long-range ion motion at high temperatures, where the frequency dependence of the ac conductivity can be described by the Jonscher law [6],
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 8 - 5
A. Rivera et al. / Journal of Non-Crystalline Solids 307–310 (2002) 1024–1030
r0 ðxÞ r0 ½1 þ ðx=x0 Þn :
ð1Þ
In this temperature range, a dc conductivity r0 is found in the limit of low frequency, and a power law behavior is observed with a fractional exponent n above a characteristic frequency x0 . Both r0 and x0 are found to be thermally activated with about the same activation energy Er . This has been interpreted in different theoretical models [7–12] as indicating that the ac conductivity given by Eq. (1) originates from migration of ions by hopping between neighboring potential wells, which eventually gives rise to a dc conductivity at the lowest frequencies. Since ion hopping is a thermally activated process, it is at sufficiently low temperatures that the contribution given by Eq. (1) becomes negligible and the NCL becomes the dominant contribution to the ac conductivity in the experimental frequency window. For the usual measurement range of frequencies up to a few megahertz, and in typical ionic conductors, it is then necessary to perform complex admittance measurements well below room temperature to obtain a dominant contribution from the NCL. Accordingly, in order to observe a NCL behavior in the ac conductivity at room temperature it is usually necessary to extend the frequency range up to the gigahertz. We want to emphasize at this point that, although it is usually assumed in the literature [1–3,13–15] that the Jonscher conductivity and the linear frequency term (or NCL) are additive contributions, it has been recently shown that the total ac conductivity in ionic conductors can not be expressed by a simple addition of both contributions in the frequency domain [16,17]. The NCL behavior has been often explained in the past in terms of asymmetric double-well potential configurations and attributed to the motion of groups of ions [14,18,19]. Recently, the origin of the NCL has been proposed to be in the vibrational relaxation of ions within their cages, before they start hopping and contributing to the longrange ion transport [4,16]. More recently, NCL is identified with the very slow decay of the cage in the short time regime, when the probability of successful hop of the ions out of their potential wells is small [20]. There are also different theo-
1025
retical models [21,22] which propose there is no formal separation between the Jonscher law and the NCL, and the latter is just the consequence of ionic hopping but at a very short time scale. In order to further understand the NCL behavior and its origin, it is necessary to perform new and systematic studies of the ac conductivity of different ionic conductors at low temperatures and/or high frequencies. By investigating the properties of the NCL we shall get new insights about its nature and origin. In this paper we report experimental results of NCL in various ionic conductors and investigate the temperature dependence of the NCL and also the dependence of the NCL upon changes in the mass of the mobile ions. We have chosen materials with crystalline and glassy structures, with different atomic species as mobile ions, and showing different values of the dc conductivity at room temperature: a crystalline superionic conductor such as Li3x La2=3 x TiO3 , which shows a high dc conductivity value at room temperature due to the high mobility of lithium ions through the perovskite structure [23,24]; yttria stabilized zirconia (YSZ), where the mobile charge carriers are oxygen ions through the oxygen vacancies created in the cubic structure of zirconia by yttria doping [25,26]; and the family of alkali triborate glasses, M2 O 3B2 O3 , where M stands for Li, Na, K and Rb, which are the mobile ions [27,28]. Our experimental results point to the relaxation of ions in a very slowly decaying cage as the origin of the NCL.
2. Experimental Samples of YSZ were (1 0 0) oriented single crystals supplied by ESCETE with composition ZrO2 –16 mol% Y2 O3 and 10 5 1 mm sized. Ceramic samples of composition Li0:18 La0:61 TiO3 were prepared by heating a stoichiometric mixture of high purity LiCO3 , La2 O3 and TiO2 reagents at 1350 °C in air for several hours (5–11 h) and then quenched to room temperature. Sintered cylindrical pellets 5 mm in diameter and 1 mm thick, with evaporated silver electrodes, were used for electrical measurements. Li content was checked by
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ICP analysis. Alkali borate glasses with the chemical formula M2 O 3B2 O3 (where M stands for Li, Na, K and Rb) were prepared from stoichiometric ground mixtures of B2 O3 and alkali carbonates by melting in platinum crucibles in an electric furnace at 1100 °C for 20 min. The liquids were quenched by pouring into 10 mm diameter, cylindrical brass molds, followed by annealing at 10 °C below Tg . The resulting glass cylinders were cut with a low speed diamond saw into 1 mm thick disks. Silver electrodes were evaporated on the flat surfaces of the samples prior to electrical measurements. Complex admittance was determined in the audio frequency range (100 Hz to 100 kHz) at low temperatures (8–300 K) by using a closed cycle helium cryostat and precision LCR meter HP4284. High frequency complex admittance measurements from 1 MHz to 1 GHz were made at room temperature with an HP4291A impedance analyzer.
Fig. 1. Temperature dependence of the real part of the conductivity, r0 ðxÞ, at fixed frequencies for Li0:18 La0:61 TiO3 . Conductivity data are shown for different frequencies (300 Hz, 1, 3, 10, 30 and 100 kHz, from bottom to top) in a linear temperature scale. At low temperatures, data at the different frequencies are spaced according to a linear frequency dependence. This is emphasized in the inset, where we have plotted the imaginary part of the permittivity, or dielectric loss, e00 ðxÞ ¼ r0 ðxÞ=e0 x, for the same frequencies, showing that these losses are nearly frequency independent, e00 ðxÞ A. Only at the highest temperatures, for which ionic hopping contributes to the diffusion, any frequency dependence becomes noticeable. The errors of the data are smaller than the size of the symbols.
3. Results The temperature dependence of the ac conductivity of Li0:18 La0:61 TiO3 is shown in Fig. 1 for several fixed frequencies from 300 Hz to 100 kHz and in the temperature range from 300 down to 8 K. It can be observed that, at low temperatures, experimental conductivity data sets are spaced according to a linear frequency term, r0 ðxÞ ¼ xe0 e00 ðxÞ Ae0 x. While at higher temperatures, the conductivity data change to assume a sublinear frequency dependence (see Eq. (1)). This can be seen more clearly in the inset where e00 ðxÞ is plotted against temperature. The data from the six different frequencies collapse onto a single curve at low temperature, indicating that e00 ðxÞ is almost independent of frequency and therefore a genuine NCL. At high temperatures the experimental data e00 ðxÞ for the different frequencies in the inset can be seen to diverge, due to the appearance of a different contribution to the ac conductivity described by the Jonscher law. In a similar fashion, Fig. 2 shows the temperature dependence of the ac conductivity of YSZ with a 16% molar yttria content. Each data curve was obtained by varying
Fig. 2. Temperature dependence of the real part of the conductivity, r0 ðxÞ, at fixed frequencies for (Y2 O3 )0:16 (ZrO2 )0:84 . Conductivity data are shown for different frequencies (1, 3, 10, 30 and 100 kHz, from bottom to top) in a linear temperature scale. To emphasize the linear frequency dependence of the conductivity, the imaginary part of the permittivity, or dielectric loss, e00 ðxÞ ¼ r0 ðxÞ=e0 x, is plotted in the inset for the same frequencies. Note that only at the highest temperatures, where ionic hopping contributes to diffusion, data at different frequencies separate from each other. The errors of the data are smaller than the size of the symbols.
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Fig. 3. Temperature dependence of the real part of the conductivity, r0 ðxÞ, at fixed frequencies for Li2 O 3B2 O3 . Conductivity data are shown for different frequencies (1, 3, 10, 30 and 100 kHz, from bottom to top) in a linear temperature scale. The linear frequency dependence of the conductivity is better shown in the inset, where we have plotted the imaginary part of the permittivity, or dielectric loss, e00 ðxÞ ¼ r0 ðxÞ=e0 x at the same frequencies. At low temperatures the losses are nearly frequency independent, e00 ðxÞ A. Only at the highest temperatures, for which ionic hopping contributes to the diffusion, any frequency dependence becomes noticeable. The errors of the data are smaller than the size of the symbols.
the temperature from 100 to 400 K at constant frequency ranging from 1 to 100 kHz. The inset shows the temperature dependence of the imaginary part of the dielectric permittivity, e00 ðxÞ, which is clearly a NCL up to almost room temperature in this frequency range. Fig. 3 shows conductivity vs temperature plots, at five different fixed frequencies between 1 and 100 kHz, for the lithium triborate glass (Li2 O 3B2 O3 ). The NCL behavior at low temperatures is again more clearly observed in the inset, where the temperature dependence of the dielectric loss e00 ðxÞ is plotted. It can be observed that the magnitude A of the NCL is much higher in Li0:18 La0:61 TiO3 , (A ¼ 1:0 0:2 at T ¼ 100 K), than in (Y2 O3 )0:16 (ZrO2 )0:84 , (A ¼ 3:6 0:4 10 2 at T ¼ 100 K), or Li2 O 3B2 O3 , (A ¼ 2:4 0:2 10 2 at T ¼ 100 K). Interestingly, the magnitude A, for all the samples analyzed, was found to be only weakly dependent on temperature, especially in the case of the lithium triborate glass which shows a very slow increase of the NCL when increasing temperature in a wide range. In fact, although we found that the best fit to the
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weak temperature dependence of the NCL was an exponential function of temperature, it could be also well described by a power law or even a linear dependence on temperature. Solid lines in Figs. 1–3 are fits to an exponential temperature dependence of the form A / expðbT Þ, with values of 1:8 0:1 10 2 , 3:9 0:3 10 3 and 5 1 10 4 K 1 of the parameter b for Li0:18 La0:61 TiO3 , (Y2 O3 )0:16 (ZrO2 )0:84 , and Li2 O 3B2 O3 respectively. Fig. 4(a) shows conductivity vs frequency plots at 300 K, from 1 MHz up to 1 GHz, for the four alkali triborate glasses studied in this work. Note that the experimental conductivity data have been shifted vertically to separate them from each other for clarity. It can be readily observed that a linear frequency dependence of the ac conductivity is found for all samples over this frequency range. Best fits are obtained for a frequency dependence of the form, r0 ðxÞ ¼ Aeo xp , with an exponent p ¼ 0:99 0:03 for all samples. The magnitude A of the NCL at 300 K is thus obtained for each alkali borate from these fits. The dependence of the nearly constant loss on the mass of alkali ions is presented in Fig. 4(b). We have plotted data at two different temperatures, 300 and 60 K, and both data sets show a similar trend: the magnitude of the NCL decreases slightly but systematically as the alkali mass increases.
4. Discussion It has been recently proposed [4,16] that the NCL is due to vibrational relaxation of mobile ions within their cages before they start hopping. Within this picture the magnitude of the NCL would be related to the mean squared displacement of ions in the vibrational motion. In the case of a harmonic potential defining the cage the mean squared displacement of the ions is a constant. Since the NCL behavior implies a slow increase of the mean squared displacement with the logarithm of time, anharmonicity of the potential of the cages might play an important role. More recently [20], NCL has been associated with a very slow decay of the cages and consequently anharmonicity of the potential may become time dependent,
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Fig. 4. (a) Frequency dependence of the real part of the conductivity, r0 ðxÞ, at room temperature and frequencies in the range 1 MHz to 1 GHz, for the alkali triborate glasses M2 O 3B2 O3 , with M ¼ Li, Na, K and Rb. Data sets corresponding to M ¼ Li, Na and K have been multiplied by 103 , 102 and 10, respectively, for clarity. Solid lines represent best fits to a linear frequency dependence, i.e. a constant loss contribution to the conductivity. (b) Mass dependence of the dielectric loss, e00 ðxÞ A, at 60 and 300 K for the alkali triborate glasses shown in (a). The magnitude of the NCL increases by a factor of two in changing the alkali ion from rubidium to lithium. Dotted lines are fits to a power law dependence of the form m a with exponents a ¼ 0:35 0:03 and 0:31 0:05 at 60 and 300 K respectively. Solid lines represent a dependence of the form m 1=3 , which describes within error for the experimental data.
increasing slowly with the logarithm of time. Such time dependent anharmonicity could explain also why the mean squared displacement, and therefore the NCL, does not show a linear temperature dependence characteristic of harmonic vibration, but weak exponential temperature dependence A / expðbT Þ.
The observed change in the magnitude of A of different samples at a given temperature is much smaller than that found for the dc conductivity. For example, in the family of alkali triborates, A varies by about a factor of two at 300 K, while the dc conductivity decreases by more than three orders of magnitude in changing the alkali ion from lithium to rubidium. However, changes in both quantities might be related, since samples with a higher dc conductivity show also a higher value of A. The same trend is found in the case of Li0:18 La0:61 TiO3 , which shows a high dc conductivity at room temperature (rdc 3 10 4 S cm 1 ) and has also the highest value of A, almost two orders of magnitude higher than that of the other samples analyzed. This correlation between the magnitude of A and the dc conductivity value is consistent with the previously published observation [4] that ionic conductors showing high values of the NCL are all superionic conductors, having very high dc conductivities even at room temperature. This fact, together with the contrast between the weak temperature dependence of the NCL and the thermally activated behavior of the dc conductivity, suggest that the NCL is related to the ions, but has a different origin than the dc conductivity. Another interesting property of the NCL is its dependence on the mass of the mobile ions, which has not been investigated in the past. We have found at constant temperature a systematic decrease in the magnitude of A upon increasing the mass of the mobile ions in the family of alkali triborates. Since the room temperature far-infrared spectra of these glasses show that anharmonicity decreases with the increase in the mass of the alkali ions [27], the observed mass dependence of NCL is consistent with the proposed connection of NCL to anharmonicity of the cage potential [4,16] and the origin of NCL from time dependent anharmonicity caused by very slow decay of the cage. Thus the moderate decrease of NCL with increasing alkali ion mass and with decreasing temperature according to A / expðbT Þ may stem from the same source. At this time it is too early to identify the true source of NCL. In passing we mention other possible connections. In a loosely related problem
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of lattice vibration with anharmonic restoring forces, it has been shown that the on-site anharmonic potential modeled by either a hard /4 potential or a Morse potential can slow down dramatically the vibrational energy relaxation [29– 31]. Interestingly, the dependence of the magnitude of the NCL on the alkali ion mass found in the present work may also be explained within this picture. In the case of an ion vibrating in a cage defined by a harmonic potential the characteristic mean-squared displacement is independent of the mass of the ion [32]. However, in the case of an anharmonic potential of the /4 type, it is found [33,34] that the mass dependence is proportional to m a with an exponent a ¼ 1=3. We have plotted as dashed lines in Fig. 4(b) the best fits to a powerlaw dependence on the alkali mass of the value of A obtained for the alkali triborates. The value obtained for the exponent a is 0:35 0:03 and 0:31 0:05 at 60 and 300 K respectively. In fact, solid lines in Fig. 4(b) represent a mass dependence of the form m 1=3 , which accounts within experimental error, and at both temperatures, for the observed change in the magnitude of the NCL with the mass of the alkali ion. However, we cannot say for sure that these models bear any relation to NCL.
5. Conclusions We have determined the temperature dependence of the nearly constant loss by ac conductivity measurements in the audio frequency range and varying temperature from 300 down to 8 K. The magnitude A of the nearly constant loss is found to be weakly dependent on temperature. Good fits are obtained in a wide temperature range to an expression A / expðbT Þ for all the samples analyzed. We have also found, in the family of alkali triborate glasses, that the magnitude of the nearly constant loss at constant temperature decreases systematically upon decreasing the mass of the mobile ions. The observed mass dependence can be described by a power law of the form A / m 1=3 within experimental error. These properties of the nearly constant loss suggest that it may arise from the mean-squared displacement of
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the ions increasing slowly with time due to the very slow decay with time of the cages confining the ions.
Acknowledgements Financial support from Spanish CICYT grant MAT98-1053-C04 is acknowledged. The work performed at NRL was supported by ONR.
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