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Complex permittivity in ionically conducting solids: a hopping model
IOURNA
Journal of Non-Crystalline Solids 136 (1991)60-66 North-Holland
L OF
I~'~R~III~
~0IlI~
Complex permittivity in ionically conducting solids: a hopping model F. H e n n a, S.R. Elliott b a n d J.C. G i u n t i n i a Laboratoire de Physicochimie des Matdriaux, URA DO407CNRS, Equipe de Chimie-Physique, Universit~ de Montpellier 11, Place Eugene Bataillon, 34095 Montpellier Cddex, France b Department of Chemistry, University of Cambridge, Lensfield Rd, Cambridge CB2 1EW, UK Received 8 April 1991 Revised manuscript received 8 July 1991
Using both the correlated barrier hopping model accountingfor the dielectric losses and the Anderson and Stuart model describing the potential barrier in ionically conducting glasses, the calculation of the dielectric processes in ionic solids is extended to the complex perminivity and the results are compared with the behaviour described by the empirical Cole-Davidson function. Experimental results for the ac conductivity obtained on Na20-3SiO 2 glass are fitted by the theoretical behaviour predicted by this model.
1. Introduction In a recent paper [1], we developed a model, based on a hopping mechanism, accounting for the high frequency part (namely, above the dielectric relaxation peak) of the polarisation conductivity, o,'(w), or the dielectric loss, E"(o~) (~r'(w) =wE"(oJ)), in some ionically conducting solids. A theoretical expression was obtained which corresponds to the experimental behaviour: ~r'(w) =AoJ s where the temperature behaviour of the exponent, s, seems to be a characteristic parameter of a material. Although, some authors [2-4] have pointed out that s varies with frequency and tends to unity, we may reasonably assume that s is constant throughout the frequency range [10 ~ 10 7 Hz]. As a consequence, the value of s can be determined by a classical least-squares method. The theoretical behaviour resulting from this investigation is similar to some experimental data observed in ionically conducting systems such as ionic glasses. In this paper, we extend the calculation of o"(~o) or e"(co) via a numerical method on both
sides of the dielectric relaxation peak and, using the Kramers-Kr6nig transformation, we determine the theoretical behaviour of E'(w). As a consequence, we show that the complex permittivity, E*, diagram calculated for this model is similar to those generally obtained by different approaches such as the Cole-Davidson empirical function [5] or to the Kohlrausch-Williams-Watts (KWW) relaxation function [6] which can be related to the Cole-Davidson model [7] or the Elliott-Owens diffusion controlled relaxation (DCR) model [8]. Nevertheless, our approach is an alternative model to the Elliott-Owens D C R model, based on a series relaxation model, since we use a parallel relaxation process with a distribution of relaxation times. However, at this point, we remark that our work is strictly restricted to the dielectric or polarisation processes, as well as the previous cited functions. Therefore, we may keep in mind that this approach differs form a commonly used method, based on measurement of the /3 KWW parameter [9,10] which does not clearly distinguish fundamental differences between the electrical, ~rdc, and dielectric, o"(w), properties observed in classical bridge measure-
0022-3093/91/$03.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved
F. Henn et al. / Complex permittiuity in ionically conducting solids"
ments within the same range of frequency. In our case, we always refer to e"(w) or ~r'(co) as corresponding only to the dielectric or polarisation process. Finally, the paucity of experimental data has led us to compare the theoretical predictions of our model with experimental results obtained by us for N a 2 0 - 3 S i O 2 glass.
61
kT), to calculate the potential barrier, W, which must be surmounted by the charge carrier. In the case of ionic conductors, we adopt a model first proposed by Anderson and Stuart [14] to account for the activation energy of the dc conductivity, which assumes that the activation energy, W, is the sum of two energy terms: W = E~ + E h.
2. The model: theoretical approach The high-frequency part of the polarisation conductivity that is often observed experimentally in most dielectric solids (electronic or ionic) [11] may be explained on the assumption that the charge carriers move via a hopping mechanism between localised states. A model, called the correlated barrier hopping model, was developed several years ago [12] and is based on the assumption that the charge carriers hop between sites by means of thermal activation over the potential barrier separating the two sites, whose height is correlated with the intersite separation. This allows o"G0) or e"(w) to be expressed in a general way as a sum of individual Debye responses for each intersite separation, each assumed to contribute independently:
In the expression, E b is the electrostatic and E~ the mechanical strain contribution [15] involved in the hopping process, namely, e2
E b = Wm
a~
or
r~
q'(w) = f
E~= ~GR(r-rd)
2,
Rp
o92
- - k T r,, B + CR 2
1 -{- O)2T2
dr,
(3)
with
O)T
e"(w)=Lg(R)l+wzr2dr,
7r
and
VeOer R
where Wm is the energy required to remove a singly charged ion form its site to infinity, e o and e r are, respectively, the dielectric constant of vacuum and of the material, e is the charge of the hopping ion, G is the shear modulus and (r - ra) is the enlargement of the 'doorway' of radius r d between two neighbouring sites by an ion of radius r. Using these expressions, an analytical calculation of o-'(~o) leads to [1]
O)2T
o"(o~) = f l ~ g ( R ) 1 + w 2 r 2 d r
(2)
(1)
where g(R) is the distribution function of R, the hopping distance between a pair of sites, and r is the relaxation time related to that hop. The function g(R) can be written as a product of three terms: N 2
g( R) = -~--ot( R)p( R),
- W i n ( r / t o ) t~ + ~ / W 2 ( r / % ) 2t~ + 4BC R= and =kT/W
m•
In this relation, the parameters B and C are related to electrostatic and mechanical energies E~ and E b within the solid: e2
where N is the density of sites, a ( R ) is the polarisability of the dipole, that is the pairs, and p(R) is the spatial distribution of these pairs. The calculation of eq. (1) requires a relation between the variables R and r. It is necessary, when one assumes the hopping process to be thermally activated [13], namely, r = r 0 e x p ( W /
(4)
2C
B
'IT
and
C=
G ( r - - r d ) 2.
Wff(lff r
The integer p can take values between 2 and 6 depending on the power law exponents of R for a ( R ) and p(R). % is the inverse frequency of the vibrational motion of the particle in its site. Equation (3) can be solved either by approxima-
a
F. Henn et al. / Complex permittivity in ionically conducting solids ~
1.0"
-
-
-
-
-
-
-
_
_
_
-0.0
-0.2
0,8
• -0.4 m o t~
0,6 "o
,
-0.6
J~ 0.4 •
-0.8
0,2
'
'-1.0
0.0 .4
i
i
~
-2
0
2
log
F
-1.2
i
4
6
(freq)
Fig. 1. T h e o r e t i c a l v a r i a t i o n s o f t h e n o r m a l i s e d f u n c t i o n e ' ( t o ) in n o r m a l a n d log s c a l e as a f u n c t i o n o f f r e q u e n c y . T h e f o l l o w i n g r e p r e s e n t a t i v e v a l u e s w e r e u s e d : r~ = 10 - 2 s; w i = 10 - 3 H z ; tof = 107 H z ; Wm = 0.5 eV; G = 109 N m 2; er = 10; r - r d = 10 - t ° m; T = 300 K, a n d p = 6.
0.3
.?
0.2
-4
0.1
'-5
0,0
.
-4
-6
-2
0
2
4
6
8
log (fr©q) Fig. 2. T h e o r e t i c a l v a r i a t i o n o f t h e n o r m a l i s e d f u n c t i o n e"(w) as a f u n c t i o n o f f r e q u e n c y . T h e p a r a m e t e r v a l u e s u s e d w e r e t h e s a m e as f o r fig. 1.
63
F. Henn et al. / Complex permittivity in ionically conducting solids
range 10-2-10 - ' s. Finally, the frequency varies from oJ~ to wf such that
tion, as already done [1], or by numerical calculation. Here, we choose the latter way in order to calculate cr'(~o) and E"(~o) on either side of the relaxation peak. Then, using the Kramers-Kr6nig transformation, it is possible to calculate the real part of the permittivity E'(o~) from the expression for E"(o~). One can write from expression (3): = (T~
E"(w)
Rp
to0-~ << 1
+w2T 2d~"
(5)
and ~
Rp
E'(oJ) - E~ = o B- -+kCTR 2
1 1+
d~-
~o2~.2 7
oJf~-=>> 1.
Under these conditions, we can observe a power law variation of e'(~o) (fig. 1) and of e"(w) (fig. 2) as a function of frequency in a log-log plot. As expected from theoretical considerations, e"(w) exhibits a loss peak at the frequency wp where ~OpT~---1. The low frequency part of the log-log variation of e"(~o) is linear with a slope equal to 1, whereas the high frequency side has a slope equal to s - 1. It is then possible to plot the complex permittivity diagram e " ( ~ o ) = f ( e ' ( w ) ) ( f i g . 3). The curve predicted by our model is very close to the curve described by the empirical Cole-Davidson equation characterised by the parameter /3, and the parameter s, used in our theoretical and experimental approach is related by/3 = 1 - s . This result is significant since an account of this type of complex-plane diagram, which is often observed, can be given by a microscopic model for transport in a ionic conducting material, such
(.o
Jr(, B + c R z k T I
and
(6)
3. Numerical calculation
A numerical calculation [16] was used to evaluate e'(w) and E"(~o). The relaxation time, ~'0, was taken to be equal to 10-'3 s and the upper limit, ~-~, corresponding to the inverse of the cut-off frequency, can be arbitrary fixed to be in the
6" 0.8 6°
- 6~
[] Cole-Davidson Elliott-Henn
[3=0,275 s=0,725
0.6
0.4
0.2'
//~,~,,~'i~
~N
~
6' 6~ 6 0 - 6~
0.0
,
0.0
0.2
0.4
0.6
0.8
1.0
Fig, 3. Normalised complex permittivity E* diagram: comparison of our model (calculated with the same values as for fig. 1) with the Cole-Davidson function.
F. Henn et al. / Complex permitticity in ionically conducting solids
64
as the Anderson-Stuart model applied to the CBH theory. Nevertheless, it is important to note here that the relaxation peak calculated here does not correspond to the electrical relaxation peak often observed when one plots the imaginary part, M", of the modulus [17] as a function of frequency. For the latter, the relaxation peak, %, corresponds to the ratio e0er/%c (i.e. the product RC) since the dc conductivity is not substracted from the ac conductivity measurements. Therefore, the parameter/3 obtained by fitting M " = f(to) cannot be rigourously compared with s via the relaxation, s = 1-/3 [18].
dictions and experimental data somewhat ambiguous. In the light of this ambiquity, we decided to apply our theoretical approach to the well known ionic glass N a 2 0 - 3 S i O 2 since the fitting of the experimental data requires knowledge of some physical constants, namely G (the shear modulus), e r (dielectric constant) and r - r d (the deformation factor) which are difficult to compile for each ionic solid, but which are known for this material.
4.1. Experimental results The N a 2 0 - 3 S i O 2 glass was prepared by the usual method [19], then cut and polished in order to obtain a pellet whose parallel faces were coated with a thin gold film. Conductivity measurements were carried out using a Gen-Rad frequency bridge from 10 to 105 Hz at different temperatures. Typical experimental results for a temperature of 300 K are shown in fig. 4. As usual, it can be observed that the total conductivity, trac, can be written as
4. Comparison with experimental data In this section, we limit our investigation to the high-frequency part of the polarisation conductivity and to the parameter s which can be easily measured from experiment. However, as already explained in the introduction, the number of experimental studies relating to the parameter s for different ionic conductors is rather small and this makes any comparisons between theoretical pre-
-5.0
trac = trdc + ~r'(to) = ~rdc+Ato s,
log (Crac) (f~-l.m-l)
-5.2
O
D
-5,4 [] D
-5.6 O
-5.8 O D
-6.0 []
[]
-6,9_
0
O
0
[]
O
log (freq) -6,4 0
I
i
i
2
4
6
Fig. 4. Log-log plot of the ac conductivity of N a 2 0 - 3 S i O 2 glass versus frequency at 300 K obtained experimentally.
F. Henn et aL / Complex permittil,ity in ionically conducting solids
65
0.80
• Na20
0.75
-
-
3SIO2
Theory
0.70
}I t tt t ttt tt t t
0.65
t
0.60
0.55
0.50 290
,
,
~
)
)
)
300
310
320
330
340
350
360
T(K) Fig. 5. Theoretical fit (line) and experimental data (points) for the ac conductivityfrequency exponent s obtained at different temperatures for Na20-3SiO 2 glass, usingvalues for the relevant parameters given in the text.
where ~rdc corresponds to the dc conductivity. The values of ~rdc and s can be measured for different temperatures (fig. 5) and their behaviour compared with previous results.
4.2. DC conductivity Although this form of the conductivity has been extensively studied, it is of no great interest in the present case. Nevertheless, we supply the results to demonstrate that the behaviour of our material corresponds to that reported previously. The dc conductivity obeys an Arrhenius law, namely, ~rdc= ~r0 e x p ( - E a / k T ) , with E a = 0.71 eV and ~r0 = 1.4 × 1 0 4 [ ' ~ - 1 m - 1. These values are very close to those previously measured by numerous authors [3,19-21].
4.3. Polarisation conductic, ity We report in fig. 5 the variation of s as a function of temperature for glassy N a z O - 3 S i O 2. In view of the experimental errors, s can be considered to be almost temperature-independent [21]. Although this behaviour has been pre-
viously reported in various studies on ionically conducting glasses [9,10], it cannot be assumed to be a general fact. In most amorphous semiconductors [2] and protonic conductors [22,23], s is strongly temperature-dependent. However, our model can account for both types of behaviour [1]. It is thus possible to compare the experimental results presented here with the theoretical behaviour of s. In both experimental and theoretical points, the parameter s is determined from a least-squares method fitting the polarisation conductivity c r ' ( t o ) = A t o s. Therefore, if one uses in the numerical calculation of eq. (3) the parameter values pertaining to glassy N a 2 0 - 3 S i O 2, namely G = 2.6 × 10 l° N m -2 and e r = 6.8 [14], one can adjust the theoretical values of s to agree with experiment-by taking Wm = 0.35 eV and r - r d = 0.315 A. These values are physically quite reasonable and support the validity of our approach. A value of Wm, i.e., 0.4 eV, has been calculated from another model [24]. Of course, this single comparison is, by itself, not sufficient and general verification of our theoretical approach necessitates comparison with the experimental results for other ionically conducting glasses.
66
F. Henn et al. / Complex permittivity in ionically conducting solids
5. Conclusion
We show in this paper how a model, based on a microscopic hopping mechanism for ionic transport in ionic solids, can be used to interpret experimental data for the polarisation conductivity. First, the complex dielectric function diagram, predicted theoretically, exhibits a shape similar to those commonly observed experimentally and described by the empirical function of Cole-Davidson. Further, the relation between the parameters s and /3, the frequency exponent of the polarisation conductivity and the Cole-Davidson parameter, respectively, namely, s = 1 -/3, is verified in this case. Second, it is possible to fit the experimental ac conductivity results obtained on Na20-3SiO 2 glass using this model. This fitting process gives values for the two parameters: Wm the electrostatic binding energy of the charge carrier and r - r d , the mechanical strain associated with the hopping process. F.H. is grateful to the European Community for provision of a grant.
References [1] S.R. Elliott and F. Henn, J. Non-Cryst. Solids 116 (1990) 179. [2] S.R. Elliott, Adv. Phys. 36 (1987) 135. [3] A. Burns, F.D. Chryssikos, E. Tombari, R.H. Cole and W.M. Risen Jr., Phys. Chem. Glasses 30 (1989) 264.
[4] J. Wong and C.A. Angell, Glass Structure by Spectroscopy (Dekker, New York, 1976). [5] D.W. Davidson and R.H. Cole, J. Chem. Phys. 19 (1951) 1484. [6] G. Williams and D.C. Watts, Trans. Faraday Soc. 66 (1971) 1484. [7] C.P. Lindsey and G.D. Patterson, J. Chem. Phys. 19 (1980) 3348. [8] S.R. Elliott and A.P. Owens, Philos. Mag. 60 (1989) 777. [9] G. Batzer-J611enbeck, O. Kanert, H. Jain and K.L. Ngai, Phys. Rev. B39 (1989) 6071. [10] K.L. Ngai and S.W. Martin, Phys. Rev. B40 (1989) 10550. [11] A.K. Jonscher, Dielectric Relaxation in Solids (Chelsea Dielectric, London 1983). [12] S.R. Elliott, Philos. Mag. 36 (1977) 673. [13] M. Pollak and G.E. Pike, Phys. Rev. Lett. 28 (1972) 1449. [14] O.L. Anderson and D.A. Stuart, J. Am. Ceram. Soc. 37 (1954) 573. [15] D.K. Mc Elfresh and D.G. Howitt, J. Am. Ceram. Soc. 69 (1986) C237. [16] J.C. Giuntini, B. Deroide, P. Belougne and J.V. Zanchena, Solid State Commun. 62 (1987) 739. [17] F.S. Howell, R.A. Bose, P.B. Macedo and C.T. Moynihan, J. Phys. Chem. 78 (1974) 639. [18] F. Henn, J.C. Giuntini and J.V. Zanchetta, J. Non-Cryst. Solids 131-133 (1991) 1084. [19] V. Provenzano, L.P. Boesch, V. Volterra, C.T. Moynihan and P.B. Macedo, J. Am. Ceram. Soc. 88 (1974) 492. [20] D. Ravaine, J.P. Diard and J.L. Souquet, J. Chem. Soc. Faraday Trans. II 71 (1975) 1935. [21] L.P. Boesch and C.T. Moynihan, J. Non-Cryst. Solids 17 (1975) 44. [22] J.C. Giuntini, J.V. Zanchetta and F. Henn, Solid State Ionics 28-30 (1988) 142. [23] F. Henn, J.C. Giuntini, J.V. Zanchetta, W. Granier and A. Taha, Solid State Ionics 42 (1990) 29. [24] F. Buet, J.C. Giuntini, F. Henn and J.V. Zanchetta, submitted to Philos. Mag.
Journal of Non-Crystalline Solids 136 (1991)60-66 North-Holland
L OF
I~'~R~III~
~0IlI~
Complex permittivity in ionically conducting solids: a hopping model F. H e n n a, S.R. Elliott b a n d J.C. G i u n t i n i a Laboratoire de Physicochimie des Matdriaux, URA DO407CNRS, Equipe de Chimie-Physique, Universit~ de Montpellier 11, Place Eugene Bataillon, 34095 Montpellier Cddex, France b Department of Chemistry, University of Cambridge, Lensfield Rd, Cambridge CB2 1EW, UK Received 8 April 1991 Revised manuscript received 8 July 1991
Using both the correlated barrier hopping model accountingfor the dielectric losses and the Anderson and Stuart model describing the potential barrier in ionically conducting glasses, the calculation of the dielectric processes in ionic solids is extended to the complex perminivity and the results are compared with the behaviour described by the empirical Cole-Davidson function. Experimental results for the ac conductivity obtained on Na20-3SiO 2 glass are fitted by the theoretical behaviour predicted by this model.
1. Introduction In a recent paper [1], we developed a model, based on a hopping mechanism, accounting for the high frequency part (namely, above the dielectric relaxation peak) of the polarisation conductivity, o,'(w), or the dielectric loss, E"(o~) (~r'(w) =wE"(oJ)), in some ionically conducting solids. A theoretical expression was obtained which corresponds to the experimental behaviour: ~r'(w) =AoJ s where the temperature behaviour of the exponent, s, seems to be a characteristic parameter of a material. Although, some authors [2-4] have pointed out that s varies with frequency and tends to unity, we may reasonably assume that s is constant throughout the frequency range [10 ~ 10 7 Hz]. As a consequence, the value of s can be determined by a classical least-squares method. The theoretical behaviour resulting from this investigation is similar to some experimental data observed in ionically conducting systems such as ionic glasses. In this paper, we extend the calculation of o"(~o) or e"(co) via a numerical method on both
sides of the dielectric relaxation peak and, using the Kramers-Kr6nig transformation, we determine the theoretical behaviour of E'(w). As a consequence, we show that the complex permittivity, E*, diagram calculated for this model is similar to those generally obtained by different approaches such as the Cole-Davidson empirical function [5] or to the Kohlrausch-Williams-Watts (KWW) relaxation function [6] which can be related to the Cole-Davidson model [7] or the Elliott-Owens diffusion controlled relaxation (DCR) model [8]. Nevertheless, our approach is an alternative model to the Elliott-Owens D C R model, based on a series relaxation model, since we use a parallel relaxation process with a distribution of relaxation times. However, at this point, we remark that our work is strictly restricted to the dielectric or polarisation processes, as well as the previous cited functions. Therefore, we may keep in mind that this approach differs form a commonly used method, based on measurement of the /3 KWW parameter [9,10] which does not clearly distinguish fundamental differences between the electrical, ~rdc, and dielectric, o"(w), properties observed in classical bridge measure-
0022-3093/91/$03.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved
F. Henn et al. / Complex permittiuity in ionically conducting solids"
ments within the same range of frequency. In our case, we always refer to e"(w) or ~r'(co) as corresponding only to the dielectric or polarisation process. Finally, the paucity of experimental data has led us to compare the theoretical predictions of our model with experimental results obtained by us for N a 2 0 - 3 S i O 2 glass.
61
kT), to calculate the potential barrier, W, which must be surmounted by the charge carrier. In the case of ionic conductors, we adopt a model first proposed by Anderson and Stuart [14] to account for the activation energy of the dc conductivity, which assumes that the activation energy, W, is the sum of two energy terms: W = E~ + E h.
2. The model: theoretical approach The high-frequency part of the polarisation conductivity that is often observed experimentally in most dielectric solids (electronic or ionic) [11] may be explained on the assumption that the charge carriers move via a hopping mechanism between localised states. A model, called the correlated barrier hopping model, was developed several years ago [12] and is based on the assumption that the charge carriers hop between sites by means of thermal activation over the potential barrier separating the two sites, whose height is correlated with the intersite separation. This allows o"G0) or e"(w) to be expressed in a general way as a sum of individual Debye responses for each intersite separation, each assumed to contribute independently:
In the expression, E b is the electrostatic and E~ the mechanical strain contribution [15] involved in the hopping process, namely, e2
E b = Wm
a~
or
r~
q'(w) = f
E~= ~GR(r-rd)
2,
Rp
o92
- - k T r,, B + CR 2
1 -{- O)2T2
dr,
(3)
with
O)T
e"(w)=Lg(R)l+wzr2dr,
7r
and
VeOer R
where Wm is the energy required to remove a singly charged ion form its site to infinity, e o and e r are, respectively, the dielectric constant of vacuum and of the material, e is the charge of the hopping ion, G is the shear modulus and (r - ra) is the enlargement of the 'doorway' of radius r d between two neighbouring sites by an ion of radius r. Using these expressions, an analytical calculation of o-'(~o) leads to [1]
O)2T
o"(o~) = f l ~ g ( R ) 1 + w 2 r 2 d r
(2)
(1)
where g(R) is the distribution function of R, the hopping distance between a pair of sites, and r is the relaxation time related to that hop. The function g(R) can be written as a product of three terms: N 2
g( R) = -~--ot( R)p( R),
- W i n ( r / t o ) t~ + ~ / W 2 ( r / % ) 2t~ + 4BC R= and =kT/W
m•
In this relation, the parameters B and C are related to electrostatic and mechanical energies E~ and E b within the solid: e2
where N is the density of sites, a ( R ) is the polarisability of the dipole, that is the pairs, and p(R) is the spatial distribution of these pairs. The calculation of eq. (1) requires a relation between the variables R and r. It is necessary, when one assumes the hopping process to be thermally activated [13], namely, r = r 0 e x p ( W /
(4)
2C
B
'IT
and
C=
G ( r - - r d ) 2.
Wff(lff r
The integer p can take values between 2 and 6 depending on the power law exponents of R for a ( R ) and p(R). % is the inverse frequency of the vibrational motion of the particle in its site. Equation (3) can be solved either by approxima-
a
F. Henn et al. / Complex permittivity in ionically conducting solids ~
1.0"
-
-
-
-
-
-
-
_
_
_
-0.0
-0.2
0,8
• -0.4 m o t~
0,6 "o
,
-0.6
J~ 0.4 •
-0.8
0,2
'
'-1.0
0.0 .4
i
i
~
-2
0
2
log
F
-1.2
i
4
6
(freq)
Fig. 1. T h e o r e t i c a l v a r i a t i o n s o f t h e n o r m a l i s e d f u n c t i o n e ' ( t o ) in n o r m a l a n d log s c a l e as a f u n c t i o n o f f r e q u e n c y . T h e f o l l o w i n g r e p r e s e n t a t i v e v a l u e s w e r e u s e d : r~ = 10 - 2 s; w i = 10 - 3 H z ; tof = 107 H z ; Wm = 0.5 eV; G = 109 N m 2; er = 10; r - r d = 10 - t ° m; T = 300 K, a n d p = 6.
0.3
.?
0.2
-4
0.1
'-5
0,0
.
-4
-6
-2
0
2
4
6
8
log (fr©q) Fig. 2. T h e o r e t i c a l v a r i a t i o n o f t h e n o r m a l i s e d f u n c t i o n e"(w) as a f u n c t i o n o f f r e q u e n c y . T h e p a r a m e t e r v a l u e s u s e d w e r e t h e s a m e as f o r fig. 1.
63
F. Henn et al. / Complex permittivity in ionically conducting solids
range 10-2-10 - ' s. Finally, the frequency varies from oJ~ to wf such that
tion, as already done [1], or by numerical calculation. Here, we choose the latter way in order to calculate cr'(~o) and E"(~o) on either side of the relaxation peak. Then, using the Kramers-Kr6nig transformation, it is possible to calculate the real part of the permittivity E'(o~) from the expression for E"(o~). One can write from expression (3): = (T~
E"(w)
Rp
to0-~ << 1
+w2T 2d~"
(5)
and ~
Rp
E'(oJ) - E~ = o B- -+kCTR 2
1 1+
d~-
~o2~.2 7
oJf~-=>> 1.
Under these conditions, we can observe a power law variation of e'(~o) (fig. 1) and of e"(w) (fig. 2) as a function of frequency in a log-log plot. As expected from theoretical considerations, e"(w) exhibits a loss peak at the frequency wp where ~OpT~---1. The low frequency part of the log-log variation of e"(~o) is linear with a slope equal to 1, whereas the high frequency side has a slope equal to s - 1. It is then possible to plot the complex permittivity diagram e " ( ~ o ) = f ( e ' ( w ) ) ( f i g . 3). The curve predicted by our model is very close to the curve described by the empirical Cole-Davidson equation characterised by the parameter /3, and the parameter s, used in our theoretical and experimental approach is related by/3 = 1 - s . This result is significant since an account of this type of complex-plane diagram, which is often observed, can be given by a microscopic model for transport in a ionic conducting material, such
(.o
Jr(, B + c R z k T I
and
(6)
3. Numerical calculation
A numerical calculation [16] was used to evaluate e'(w) and E"(~o). The relaxation time, ~'0, was taken to be equal to 10-'3 s and the upper limit, ~-~, corresponding to the inverse of the cut-off frequency, can be arbitrary fixed to be in the
6" 0.8 6°
- 6~
[] Cole-Davidson Elliott-Henn
[3=0,275 s=0,725
0.6
0.4
0.2'
//~,~,,~'i~
~N
~
6' 6~ 6 0 - 6~
0.0
,
0.0
0.2
0.4
0.6
0.8
1.0
Fig, 3. Normalised complex permittivity E* diagram: comparison of our model (calculated with the same values as for fig. 1) with the Cole-Davidson function.
F. Henn et al. / Complex permitticity in ionically conducting solids
64
as the Anderson-Stuart model applied to the CBH theory. Nevertheless, it is important to note here that the relaxation peak calculated here does not correspond to the electrical relaxation peak often observed when one plots the imaginary part, M", of the modulus [17] as a function of frequency. For the latter, the relaxation peak, %, corresponds to the ratio e0er/%c (i.e. the product RC) since the dc conductivity is not substracted from the ac conductivity measurements. Therefore, the parameter/3 obtained by fitting M " = f(to) cannot be rigourously compared with s via the relaxation, s = 1-/3 [18].
dictions and experimental data somewhat ambiguous. In the light of this ambiquity, we decided to apply our theoretical approach to the well known ionic glass N a 2 0 - 3 S i O 2 since the fitting of the experimental data requires knowledge of some physical constants, namely G (the shear modulus), e r (dielectric constant) and r - r d (the deformation factor) which are difficult to compile for each ionic solid, but which are known for this material.
4.1. Experimental results The N a 2 0 - 3 S i O 2 glass was prepared by the usual method [19], then cut and polished in order to obtain a pellet whose parallel faces were coated with a thin gold film. Conductivity measurements were carried out using a Gen-Rad frequency bridge from 10 to 105 Hz at different temperatures. Typical experimental results for a temperature of 300 K are shown in fig. 4. As usual, it can be observed that the total conductivity, trac, can be written as
4. Comparison with experimental data In this section, we limit our investigation to the high-frequency part of the polarisation conductivity and to the parameter s which can be easily measured from experiment. However, as already explained in the introduction, the number of experimental studies relating to the parameter s for different ionic conductors is rather small and this makes any comparisons between theoretical pre-
-5.0
trac = trdc + ~r'(to) = ~rdc+Ato s,
log (Crac) (f~-l.m-l)
-5.2
O
D
-5,4 [] D
-5.6 O
-5.8 O D
-6.0 []
[]
-6,9_
0
O
0
[]
O
log (freq) -6,4 0
I
i
i
2
4
6
Fig. 4. Log-log plot of the ac conductivity of N a 2 0 - 3 S i O 2 glass versus frequency at 300 K obtained experimentally.
F. Henn et aL / Complex permittil,ity in ionically conducting solids
65
0.80
• Na20
0.75
-
-
3SIO2
Theory
0.70
}I t tt t ttt tt t t
0.65
t
0.60
0.55
0.50 290
,
,
~
)
)
)
300
310
320
330
340
350
360
T(K) Fig. 5. Theoretical fit (line) and experimental data (points) for the ac conductivityfrequency exponent s obtained at different temperatures for Na20-3SiO 2 glass, usingvalues for the relevant parameters given in the text.
where ~rdc corresponds to the dc conductivity. The values of ~rdc and s can be measured for different temperatures (fig. 5) and their behaviour compared with previous results.
4.2. DC conductivity Although this form of the conductivity has been extensively studied, it is of no great interest in the present case. Nevertheless, we supply the results to demonstrate that the behaviour of our material corresponds to that reported previously. The dc conductivity obeys an Arrhenius law, namely, ~rdc= ~r0 e x p ( - E a / k T ) , with E a = 0.71 eV and ~r0 = 1.4 × 1 0 4 [ ' ~ - 1 m - 1. These values are very close to those previously measured by numerous authors [3,19-21].
4.3. Polarisation conductic, ity We report in fig. 5 the variation of s as a function of temperature for glassy N a z O - 3 S i O 2. In view of the experimental errors, s can be considered to be almost temperature-independent [21]. Although this behaviour has been pre-
viously reported in various studies on ionically conducting glasses [9,10], it cannot be assumed to be a general fact. In most amorphous semiconductors [2] and protonic conductors [22,23], s is strongly temperature-dependent. However, our model can account for both types of behaviour [1]. It is thus possible to compare the experimental results presented here with the theoretical behaviour of s. In both experimental and theoretical points, the parameter s is determined from a least-squares method fitting the polarisation conductivity c r ' ( t o ) = A t o s. Therefore, if one uses in the numerical calculation of eq. (3) the parameter values pertaining to glassy N a 2 0 - 3 S i O 2, namely G = 2.6 × 10 l° N m -2 and e r = 6.8 [14], one can adjust the theoretical values of s to agree with experiment-by taking Wm = 0.35 eV and r - r d = 0.315 A. These values are physically quite reasonable and support the validity of our approach. A value of Wm, i.e., 0.4 eV, has been calculated from another model [24]. Of course, this single comparison is, by itself, not sufficient and general verification of our theoretical approach necessitates comparison with the experimental results for other ionically conducting glasses.
66
F. Henn et al. / Complex permittivity in ionically conducting solids
5. Conclusion
We show in this paper how a model, based on a microscopic hopping mechanism for ionic transport in ionic solids, can be used to interpret experimental data for the polarisation conductivity. First, the complex dielectric function diagram, predicted theoretically, exhibits a shape similar to those commonly observed experimentally and described by the empirical function of Cole-Davidson. Further, the relation between the parameters s and /3, the frequency exponent of the polarisation conductivity and the Cole-Davidson parameter, respectively, namely, s = 1 -/3, is verified in this case. Second, it is possible to fit the experimental ac conductivity results obtained on Na20-3SiO 2 glass using this model. This fitting process gives values for the two parameters: Wm the electrostatic binding energy of the charge carrier and r - r d , the mechanical strain associated with the hopping process. F.H. is grateful to the European Community for provision of a grant.
References [1] S.R. Elliott and F. Henn, J. Non-Cryst. Solids 116 (1990) 179. [2] S.R. Elliott, Adv. Phys. 36 (1987) 135. [3] A. Burns, F.D. Chryssikos, E. Tombari, R.H. Cole and W.M. Risen Jr., Phys. Chem. Glasses 30 (1989) 264.
[4] J. Wong and C.A. Angell, Glass Structure by Spectroscopy (Dekker, New York, 1976). [5] D.W. Davidson and R.H. Cole, J. Chem. Phys. 19 (1951) 1484. [6] G. Williams and D.C. Watts, Trans. Faraday Soc. 66 (1971) 1484. [7] C.P. Lindsey and G.D. Patterson, J. Chem. Phys. 19 (1980) 3348. [8] S.R. Elliott and A.P. Owens, Philos. Mag. 60 (1989) 777. [9] G. Batzer-J611enbeck, O. Kanert, H. Jain and K.L. Ngai, Phys. Rev. B39 (1989) 6071. [10] K.L. Ngai and S.W. Martin, Phys. Rev. B40 (1989) 10550. [11] A.K. Jonscher, Dielectric Relaxation in Solids (Chelsea Dielectric, London 1983). [12] S.R. Elliott, Philos. Mag. 36 (1977) 673. [13] M. Pollak and G.E. Pike, Phys. Rev. Lett. 28 (1972) 1449. [14] O.L. Anderson and D.A. Stuart, J. Am. Ceram. Soc. 37 (1954) 573. [15] D.K. Mc Elfresh and D.G. Howitt, J. Am. Ceram. Soc. 69 (1986) C237. [16] J.C. Giuntini, B. Deroide, P. Belougne and J.V. Zanchena, Solid State Commun. 62 (1987) 739. [17] F.S. Howell, R.A. Bose, P.B. Macedo and C.T. Moynihan, J. Phys. Chem. 78 (1974) 639. [18] F. Henn, J.C. Giuntini and J.V. Zanchetta, J. Non-Cryst. Solids 131-133 (1991) 1084. [19] V. Provenzano, L.P. Boesch, V. Volterra, C.T. Moynihan and P.B. Macedo, J. Am. Ceram. Soc. 88 (1974) 492. [20] D. Ravaine, J.P. Diard and J.L. Souquet, J. Chem. Soc. Faraday Trans. II 71 (1975) 1935. [21] L.P. Boesch and C.T. Moynihan, J. Non-Cryst. Solids 17 (1975) 44. [22] J.C. Giuntini, J.V. Zanchetta and F. Henn, Solid State Ionics 28-30 (1988) 142. [23] F. Henn, J.C. Giuntini, J.V. Zanchetta, W. Granier and A. Taha, Solid State Ionics 42 (1990) 29. [24] F. Buet, J.C. Giuntini, F. Henn and J.V. Zanchetta, submitted to Philos. Mag.