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Ionic and polaronic glassy conductors: conductivity spectra and implications for ionic hopping in glass
SOLID
STATE Solid State Ionics 85 (1996) 293-303
Ionic and polaronic glassy conductors: conductivity spectra and implications for ionic hopping in glass K. Funke”‘“, C. Cramera, B. Roling”, T. Saatkamp”, “Institut fiir
Physikaiische
D. Wilmer”, M.D. Ingramb
Chemie, Schlosspiatz 417, 48149 Miinster,
hDepartment of Chemistry, University
Germany
of Aberdeen, Meston Walk, Aberdeen, AB9 TUE. UK
Abstract spectra have been taken of a lithium ion and a polaron conducting glass. The former exhibits plateaux in its hopping conductivities and two different power-law exponents in the dispersive regime. This is readily explained if ions, contrary to polarons, have a pronounced site preference. At the same time, this view also explains the well known differences in the dependence of carrier mobility on carrier concentration in the two types of glasses. Complete
conductivity
high-frequency
Keywords:
Glassy electrolytes;
Polaronic
glassy conductors;
Conductivity
1. Introduction Frequency-dependent ionic conductivities of solid electrolytes, &(o), contain valuable information on the dynamics of the elementary steps of hopping motion of mobile ions in these materials. The same holds true for polaronic conductors. Measurement of complete conductivity spectra extending from d.c. to the far- or mid-infrared does, however, require application of various experimental techniques which are not normally available in one laboratory. This may be the reason why complete spectra of o-((w) are still extremely rare. Two well-known examples are shown in Figs. la and lb. In Na,O. 3Si0, glass [1,2], there is a continuous increase of log u vs. log w, and even of the slope of this function, until the vibrational regime is attained at about 1 THz, cf. Fig. la. The situation *Corresponding
author. Email: [email protected]
0167.2738/96/$15.00 Copyright PII SOl67-2738(96)00073-2
01996
Elsevier
spectra;
Power laws; High-frequency
plateau;
Site preference
is different in RbAgJ,, see Fig. lb, where the increase of the hopping conductivity is restricted to a regime denoted by II in the figure. This regime is followed by the so-called high-frequency plateau, which is observed in regime I [3,4]. Note also that in regime II the slope, p, is always smaller than one, while in Fig. la the slope is found to exceed the value of one. As pointed out earlier [3-51, the conductivity spectra of Fig. lb are in general agreement with the predictions of the jump relaxation model [6]. By contrast, the spectra of Fig. la do not show the high-frequency plateaux predicted by this model. Moreover, power-law exponents larger than one cannot be explained in terms of jump relaxation, if only hops via equivalent sites are being considered
[61. In this paper we present complete conductivity spectra of a lithium ion conducting glass of composition B,O, * 0.56 Li,O * 0.45 LiBr and of a
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294
K. Funke et al. I Solid State Ionics 85 (1996) 293-303
employ the concept of non-equivalent sites when interpreting the polaronic conductivity spectra. For polarons the effect of site preference does not seem to exist. As the polaron moves from site to site, there is apparently no “memory” left behind in the glass structure which can influence the movements of other polarons.
2. Experimental spectra
b)
I
aaoooo00 0000000 e
129K 0000000' ' I 8
10
12
I
bl,#~l Fig. 1. (a) absorptivity/conductivity spectra of Na,O-350, glass at various temperatures, after Refs. [ 1,2]. 1 Neper cm-’ correC’cm-‘. (b) Dynamic conductivity of the sponds to 3.10-’ crystalline solid electrolyte RbAg,l, at temperatures between 129 K and 298 K, after Ref. [3].
polaron conducting glass of composition P205 . 4(V,O,, V204). In both cases, the low and high frequency parts of the spectra are well explained by jump relaxation [6] and vibrational contributions, respectively. In the ion conducting glass, it is possible for the first time to separate the hopping and vibrational contributions to the conductivity and to identify high frequency plateaux of the hopping conductivity. However, the shapes of the spectra of the ion conducting glass call for an extension of the basic version of the jump relaxation model [6], to include the existence of non-equivalent sites, i.e. of preferred and less favoured ones. This is done by combining the jump relaxation and dynamic structure models [7]. On the other hand, there is no need to
Complete ionic conductivity spectra of glassy B,O, .0.56 Li,O - 0.45 LiBr have been taken at twelve temperatures, ranging from 173 K to 573 K. Examples are given in Fig. 2a and b. Evidently, the overall shape of the spectra is similar to Fig. la. In particular, a high-frequency plateau is not visible. In Fig. 3 we present complete conductivity spectra of the polaron conducting glass P,Os .4(V,O,, V,O,). Again, the overall shape of the spectra is similar to Fig. la. However, it has to be borne in mind that in this example the low and high frequency parts of the spectra are due to the motion of entirely different charge carriers. Polaronic hopping is dominant below about 100 GHz while the far-infrared conductivity maximum is due to vibrational motion of the ions. In spite of the apparent similarity of the spectra of Figs. 2 and 3, closer inspection reveals significant differences arising from different dynamic properties of the respective charge carriers. This will be discussed in the following.
3. Ton conducting B,O, * 0.56 Li,O. 0.45 LiBr In the far infrared, the conductivity spectra of Figs. 2 and 3 are found to vary with frequency as w2, the temperature dependence being rather small. This feature can be regarded as the low-frequency flank of the lowest-lying vibrational mode. The vibrational contribution to the conductivity is now easily extrapolated to lower frequencies and taken out of the total spectra. The resulting spectra are then interpreted in terms of the hopping motion of the mobile charged species. In the following, this procedure is applied to the lithium ion conducting glass B,O, .0.56 Li,O * 0.45 LiBr.
K. Funke et al. I Solid State lonics
573 K 08
/
I:: 6
7
6
9
slope - p
10
11
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2 1 0 $
-1
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85 (1996)
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glasses, because they predict a high-frequency plateau which has never been observed. This argument is now invalidated. The reason for high-frequency plateaux not being directly visible in experimental conductivity spectra of glass is the existence of a broad vibrational contribution which swamps the high-frequency part of the hopping conductivity. In the high-frequency-plateau regime, the sign of the applied electric field changes so fast that every hop or displacement of an ion is recorded and contributes to the conductivity. At slightly higher FIR frequencies, the period of the field is already comparable with the duration of an individual hop, and this implies the breakdown of the very concept of hopping conductivity. At the same time, the subtraction described above is no longer feasible, if the frequency exceeds about 400 GHz. Considering the experimental hopping conductivities of Fig. 4 ( + ) we find that they are well in line with the predictions of the jump relaxation model as long as the frequency is not too high, whereas characteristic deviations are observed in the microwave and millimetre-wave regimes. Let us start our discussion at comparatively low frequencies. The d.c. conductivity, g(O), obeys an Arrhenius law,
-6 -7'
t 2
I 4
6
6
10
12
conductivity
spectra of B,0;0.56
(1)
14
h4,,ww Fig. 2. Complete
a(O) . T m exp( - E,, lkT),
Li,O.O.45
LiBr glass at (a) 573 K and (b) 323 K. The error bars indicate the
with E,,=O.49 eV-k.5670 K. Fig. 6 is an Arrhenius plot of U(0). At all temperatures, the low-frequency part of . . qo,(w) 1s nicely fitted by
largest possible error over the entire frequency range.
cr,,,(low Figs. 4a and 4b show the resulting hopping-conductivity spectra ( + ), as well as the total conductivity ones. The most exciting feature is the appearance of a high-frequency plateau. This is more clearly seen when the data are looked at more closely between 1O’O and 10” Hz, see Figs. 5a and 5b. At about IO” Hz the data merge into their plateau values. With regard to their overall shape, the spectra ( + ) of Fig. 4 do, indeed, resemble those of crystalline electrolytes, cf., Fig. lb. They are also in broad agreement with the predictions of the coupling [8-lo] and jump relaxation [ 1 l] models. It has often been claimed that these models do not apply for
w) = d(o).
[ 1 + (&*)“I with p = 0.6.
(2)
In this equation, the angular frequency 1 lt, marks the onset of the power-law dispersion. According to the jump relaxation model, the continuation of Eq. (2) towards higher frequencies would have to be %,.,Ul(@)
= a(0) + [a(m) - V(O)] . [ 1 + (wtJ’]
-p.
(3)
Here, the angular frequency 1 lt, marks the transition into the high-frequency plateau, where the hopping conductivity attains the value g(m). Evi-
296
K. Eunke et al. I Solid State lonics 85 (1996) 293-303
2
-
1
-
~523
K
+4?3
K
q
423
K
x 373 K
a
0
-
-1
-
-2
-
0 323
K
e e B
A
abps.0.7
-4 -3
// I
-51 0 Fig. 3. Complete
conductivity
dently, Eq. (2) is the low-frequency of Eq. (3), valid at w< l/t,, with (T(O). t; = [CT(m)- CT(O)]. t’;.
’ 6
I
2
4
w) = a(m).
with q > 1.
approximation
(4)
[ 1 + (~t,)-‘]-~ (5)
In the fits presented in this paper we have used somewhat larger values of q also
q = 1.3. However,
I
’
10
12
spectra of P,O,.4(V,O,,
The continuation of the low-frequency part of the experimental hopping spectra as expected from Eq. (3) is presented in Fig. 7. The three regimes I, II, III already encountered in Fig. lb reappear in Fig. 7. As will be shown later, the spectra of Fig. 7 would correspond to a situation where the mobile lithium ions perform hops via preferred sites (called A) exclusively. The parameter values used in Fig. 7 will later be employed for fitting a modified version of the jump relaxation model to the actual spectra. For further details concerning the construction of Fig. 7, see Ref. [12]. Comparison of the actual hopping-conductivity spectra with the spectra of Fig. 7 reveals systematic differences at microwave frequencies. In fact, the low-frequency behaviour is well described by Eq. (2), while the high-frequency part is close to q,,(high
I
6
V,O,)
L
J
14
at various temperatures.
fit the data (see below), and it seems realistic to characterize the range of possible q values by 1.35 qs2.0. In Fig. 8, a superposition of q,,(low w) from Eq. (2) and a,,,(high w) from Eq. (5) is shown to fit the experimental hopping conductivities very well. Eq. (5) differs from Eq. (3) by q#p and even q> 1. The latter inequality is impossible in the original jump relaxation model, in which all sites and all hops are geometrically alike [6]. Instead, according to the dynamic structure model [7], we have to distinguish between at least two kinds of site in a single-alkali glass. These are the so-called A sites which are optimally configured to meet the requirements of each mobile At ion (here: Li+), and the c sites which provide less room and are less favourable. If an A+ ion leaves an A site, this site is supposed to stay A for some time and then to transform into a c site. Correspondingly, if an A+ ion hops into a c site and manages to stay, then the r? site is supposed to relax into an A site. There are thus two kinds of site relaxation: (i) the shifting of the Coulomb cage as described by the jump relaxation model and (ii) the A to c and c to A relaxation as described by the dynamic structure model. Effect (i) is achieved by the hopping of neigh-
K. Funke et al. I Solid State Ionics 85 (1996) 293-303
291
2 0.5 i t
a)
r
o-
0
573 K
OC
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00
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12
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11.2 11.4 11.6 11.6 12
12.2
vib
-7 2
’
10.2 10.4 10.6 10.6 11
14
ba,&vW
Fig. 4. Total experimental obtained after subtracting (+), of glassy B,O;0.56 323 K.
conductivity spectra (0) and spectra vibrational contributions out of them Liz0.0.45 LiBr at (a) 573 K and (b)
bouring mobile ions, while (ii) necessitates a restructuring of the glassy matrix. Therefore, (i) is assumed to be much faster than (ii). In this paper we will show that the low-frequency behaviour described by Eq. (2) and its continuation according to Eq. (3) are caused by hops of the mobile lithium ions via their preferred A sites, whereas the high-frequency behaviour described by ECq. (5) is mainly due to hops from A sites into adjacent c sites, immediately followed by backward hops. Regarding site relaxation, the spectra clearly
Fig. 5. Representations
of the high-frequency
plateau
regions
Fig. 4 at the same temperatures.
-7
1
1.5
2
2.5
3
3.5
4
4.5
5
1000 K/-r
Fig. 6. Arrhenius plot of the d.c. conductivity Li,O.O.45 LiBr glass.
of B,O,.O.56
of
K. Funke et al. I Solid State lonics
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85 (1996)
3.0,
293-303
,
2.0 1 .o
500
K
333
K
250
K
200
K
0.0
0
-1.0 -2.0
d 5 I<
-3.0
s I< -;
-4.0
rp -e
-5.0
H
-6.0 -7.0
6.0
6.0
toa,.@/H~) Fig. 7. Log-log representation L&0.0.45 LiBr, cf. Eq. (3).
of the calculated
-6.0 % 4.0
A to A hopping contribution
to the frequency-dependent
conductivity
of glassy B,O, .0.56
_1
5.0
6.0
7.0
6.0
0.0
la.0
11.0
12.0
ba,,WW Fig. 8. Hopping conductivity spectrum of B,O;0.56 L&0.0.45 LiBr glass at 373 K (0). regime, p=O.6) and Eq. (5) (high-frequency regime, q= 1.3).
reflect effect (i), while effect (ii) is not readily observed, cf. Section 4. In the original jump relaxation model, the hopping into a neighbouring site is activated with an energy A. Being smaller than the activation energy for successful hops, E,, of Eq. (l), A determines the
Solid lines calculated
via Eq. (2) (low-frequency
temperature dependence of o-(m) of Eq. (3), i.e. the spacing of the high-frequency plateau conductivities in the plot of Fig. 7. On the other hand, the corresponding spacing of the actual high-frequency plateau values, U(W) of Eq. (S), would be given by the activation energy required for a hop into a c site.
299
K. Funke et al. I Solid State lonics 85 (1996) 29_?-303
Before further discussing the frequency dependencies of Eqs. (3) and (5) let us first consider the activation energies involved. This is done in the Arrhenius-type representation of Fig. 9. The broken line marks the high-frequency conductivities according to Eq. (3) and Fig. 7. Experimental data points are included for both the high-frequency plateau and 40 GHz. Evidently, the experimental high-frequency data do not obey a simple Arrhenius law. However, the situation becomes quite simple in the limits of both high and low temperatures. At high temperatures, the limiting slope is very similar to the one of the straight line, the experimental conductivities being larger than those derived from the simple jump relaxation model. Therefore, we have been using identical activation energies A for both generic processes, i.e.,
exactly as WI-O. Although explanations have been suggested /14,15], it is felt that the “new universaliunsolved ty” can be regarded as a challenging problem. The solid lines fitting the two experimental spectra of Fig. 10 have been constructed by starting from the “new universality” baseline and superimposing the thermally activated contributions that have been identified. The latter include (i) the jump-relaxation-type spectra of Eq. (3) and Fig. 7, with p =0.6, and (ii) the high-frequency contribution described by Eq. (5), with q= 1.3. The solid lines in Fig. 9 have been obtained from the same model construction.
hops from A to A (cf. Fig. 7) and hops from A to c, cf. the high-frequency part of Fig. 8. At low temperatures, the conductivity appears to approach a limiting value. This is observed at all frequencies up to the high-frequency plateau. In fact, the experimental data seem to converge to the temperature-independent baseline included in Fig. 10. The existence of this low-temperature baseline has been termed “new universality” [ 131. Except for the high-frequency plateau regime, it is found to vary
4. Model conception
for ionic hopping
in glass
This section is devoted to a dynamical interpretation of a,,,,(high w) as described by Eq. (5). As stated earlier, the inequality q> 1 necessitates a modification of the original jump relaxation model to include the existence of different sites. According to the dynamic structure model, these are supposed to be the “good” A sites and the “bad” c sites. As a mobile A’ ion (here Li+) spends most of its time at A sites, the hops we have to consider are mainly A to
hf pkteau
1
40 GHr
3.0
5.0
7.0
1000 KIT
Li,O-0.45 LiBr at different frequencies. Fig. 9. Arrhenius type representation of hopping conductivities a,,,, (0) of glassy B,0;0.56 lines: model construction. The broken line marks the high frequency plateau values of chhclPr from Eq. (3) and Fig. 7.
Solid
300
K. Funke et al. I Solid State Ionics 85 (19%)
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7 d ‘i b a s”
-4.0
-6.O4.0
8
5.0
I 6.0
7.0
/ L
/
/
/’
b 5.0
0 0.0
0 10.0
’ 11.0
112.O
bn,,W/Hd
Fig. 10. Hopping conductivity line is the “new universality”
spectra of glassy B,0;0.56 L&0.0.45 LiBr (0). Solid lines result from our model construction. baseline attained in the low-temperature limit.
A and A to c. After an A to c hop, we expect the ion to “roll back” to its initial A site in a vast majority of cases. Fig. 11 shows the difference between an A to A hop and an A to (? hop, in terms of the singleparticle potential felt by the mobile ion. Fig. 12 is applicable to both situations displayed in Fig. 11. This means that Y may be A or c while X is always A.
&q.k
The broken
,,-lr
\
1’
i
\
6
I f
site X
A
?
site Y
Fig. 12. Effective single-particle potential of the central ion at the time of its initial hop from site X to site Y.
Once the ion is at Y, a competition sets in between two ways of relaxation. The ion may hop back to X or the neighbourhood may relax with respect to Y. Either one of these competing ways of relaxation can be characterized by a time constant or by its inverse, the respective rate. The rates are denoted by
r=o
Fig. 11. Typical single-particle potentials encountered by hopping ions (at t=O) in (a) A to ,& and (b) A to C hops. Each of the effective potentials constructed in (a) and (b) consists of two contributions: a site-sensitive potential (on top) and a Coulomb cage potential.
= initial back-hop rate,
= initial site-relaxation
rate.
In Eqs. (6a) and (6b) W(t) is the probability
(6a)
(6b) for the
301
K. Funke et al. I Solid State Ionics 8-5 (1996) 29.7-303
correlated backward hop not to have occurred until time t, and Ejback(t) is the time-dependent barrier height for that hop. It is now easy to show that the ratio t, lt* determines the exponent (p or q) in the frequencydependent hopping conductivity. This is seen by considering the correlated back-hop rate, - t&‘(r), and the rate of change of &&t), $ack(t), at sufficiently short times, when the site-relaxation process at Y is still far from being completed. Denoting the attempt frequency for a backward hop by vor, we have - W(r) =
W(r) -var. exp[-&,,(t)lkT].
(7a)
On the other hand, numerical simulations [ 161 as well as the jump relaxation model show that
1. The power-law of Eq. (8) now transforms into a power-law of a,,,(w): (i) If the ratio t, lt* is less than two, a,,,(w) is well approximated by uxx.v(w>= U,,,(w’)‘[l
+ (Wt,)-‘J-rl”*.
We thus reproduce Eq. (5) as well as the highfrequency part of Eq. (3), where the exponent is less than unity. The important result is power-law
exponent =
initial back-hop rate initial site-relaxation
C’b) holds in the short-time regime. With the definition Eq. (6a) and after a little algebra, we find at short times.
of
(8)
(u(O). w,,,
= (x;/z).
r,. [S(l) + I@(r)].
(9)
In Eq. (9), s(t) is the delta function, while x,, and r, are the X-Y distance and the forward hopping rate, respectively: *exp[-A/U]. r;, = VOX
(IO)
Here vex denotes the attempt frequency at site X. To retain simplicity, we suppose that the influence of cross terms, (v,(0)*vj(t)), where i and j, with i#j, denote different hopping ions, can be neglected. The contribution to the high-frequency hopping conductivity that is due to the hopping between sites X and Y is then proportional to the Fourier transform of the velocity autocorrelation function of Eq. (9):
rate ’ (13)
(ii) If the ratio t, lt* tends to infinity, i.e., in the absence of any site relaxation, W(t) becomes a simple exponential,
W(c)=,l$= (1 + t/tl))““’ ,
In the following, Eq. (8) will be used to derive Eq. (5). In particular, the exponent in Eq. (5), which may be q 11 or p< 1, turns out to be determined by the exponent in Eq. (8), i.e., by the ratio t,lt*. Let us first relate the back-hop rate to the (shorttime part of the) velocity autocorrelation function describing hops between X and Y:
(12)
implying a;(Jo)
= : exp(-t/r*),
(14)
Debye relaxation: = +J”)*
[l + (w$?]-‘.
(15)
(iii) The power-law exponent is, therefore, two for both limiting cases, t,/t”-+2 and t, it*-+“, the straight lines in a log u vs. log o plot being parallel to each other. As might have been expected, finite values of t, lt* larger than two yield results in between, again with an exponent of two. Summarizing (i) to (iii) one can say that Eq. ( 13) holds if t,lt* does not exceed two. If it does, the exponent stays at two. The two thermally activated contributions to the experimental spectra, featuring power-law exponents p < 1 and q > 1, are now easily explained. Hops via A sites may be treated by the original jump relaxation model. The model does, indeed, apply as only one kind of site is involved, the Coulomb relaxation of the neighbourhood being essentially due to successful x to A hops of other ions. In this case, the t, lt* ratio has a value of p < 1 and we obtain Eq. (3) and Fig. 7. Compared to an A to A hop, an A to c hop
302
K. Funke et al. / Solid State lonics 85 (1996) 293-303
has a smaller backward barrier height. Therefore, the back-hop rate, l/t *, is enhanced. As a consequence, the t, It* ratio is now found to be larger, exceeding one. Neighbouring c sites being quite common,
centration of mobile cations in a simple binary glass by c and their mobility by U, then the model predicts a power-law behaviour,
there is frequent A to c hopping, virtually always followed by a backward hop. This process is well described by Eq. (8) with t, lt* > 1, resulting in a power-law exponent q > 1. Only in a few cases does the mobile ion manage to stay at the c site which then transforms into an A site. However, these rare events do not appreciably contribute to the low-frequency conductivity. We should like to emphasize that there is no doubt about the existence of the q > 1 term in the spectra. Therefore, if Y is a c site, t, lt* is clearly larger than unity. On the other hand, the superposition of the and q> 1 terms causes some “new universality” arbitrariness in the choice of the best numerical value of q. The fit of Fig. 10 has been made using t, it* = 1.3, but the ratio may easily be larger than that. There is also some arbitrariness concerning the thermal activation of the q> 1 term at relatively low temperatures when it is not easily separated from the “new universality” baseline of Fig. 10. At relatively high temperatures, however, the activation energies of the p and q terms are indistinguishable. This implies roughly identical barrier heights for A to A and A to c hops. The different success of A to A and A to i; hops does, therefore, not result from different
UKCY,
hopping rates A to A and A to r?, but rather from different A to A and c to A back-hop barrier heights implying different backward-hopping rates.
5. Comparison of ionic and polaronic conduction in glass In the ion conducting glass, two power-law regions have been observed, with exponents p< 1 and q > 1. The interpretation given in the preceding section is based on the existence of non-equivalent sites A and c’, with a strong preference of the mobile cations for the optimally configured A sites. This concept is of the essence of the dynamic structure model [7]. One of the merits of this model is the successful explanation of the way ionic mobilities depend on ionic concentrations. Denoting the con-
y>o.
(16)
Experimentally, this power-law is well established. Interestingly, varying the degree of reduction in a polaronic glass like P205.4(V205,VZ04), i.e. varying the concentration of mobile polarons, is known to have little effect on the polaronic mobility, unless extremely large concentrations are attained [ 171. It is, therefore, evident that the dynamic structure model does not apply to polarons. One may further conclude that there should be no pronounced preference of polarons for certain sites and that the concept of favoured and less favoured sites should not play a role for polarons. As a consequence, there would be no need for modifying the original jump relaxation model to include the effect of different sites. Therefore, polaronic conductivity spectra would be expected to conform with Eq. (3), and there would be no justification for a second power law with an exponent q>l. This prediction can be tested by studying the spectra of the polaron conducting glass P205. 4(V,O,, V204). As in the ion conducting glass, the low-frequency data are well described by Eq. (2), the numerical value of p being now 0.7. The d.c. activation energy obeys the Arrhenius law of Eq. ( 1), with E,=O.41 eV In the dispersive regime, the activation energy is found to be (1 -p)*E,, in agreement with the jump relaxation model. Again, as in the ion conducting glass, the low-frequency flank of the vibrational conductivity varies as w2, and a temperature dependence of the position of this flank is not detected. Taking the vibrational conductivity out of the spectra does not yield a detectable high frequency plateau of the hopping conductivity. Evidently, it is entirely swamped by the broad vibrational contribution. Superposition of the hopping-conductivity component of Eq. (2) and the vibrational component does, in fact, suffice for an excellent fit to the experimental data. This is shown in Fig. 13 where emphasis is put on our results obtained in the microwave and millimetre-wave regimes.
K. Funke et al. I Solid State Ionics 8.5 (1996) 29.3-303
b 1.5
0.5
A 523 K +475K n 423 K
$
-0.5
c
x 373 K 0 923 K
303
analysis of the frequency dependent ionic conductivity was funded by the Deutsche Forschungsgemeinschaft. Financial aid for our British-German collaboration is provided by the British Council and the DAAD in the framework of the ARC programme. Financial help by the Fonds der Chemischen Industrie is also gratefully acknowledged.
I)
z7
-1.5
References -2.5
-3.5 B
10
11
12
13
h,,WW
Fig. 13. High-frequency conductivities of the polaron conducting glass P,0;4(Vz0,, V,O,). The solid lines result from the fit described in the text.
The tit of Fig. 13 proves the complete absence of a g> 1 term in the polaron conducting glass. Evidently, in glassy conductors, the effects of different sites and site preference are important for ions but of no significance for polarons. It is remarkable that it is the existence of such effects in ionic glasses which is responsible for anomalies seen at both d.c. and microwave frequencies.
Acknowledgments It is a pleasure to thank A. Bunde for fruitful discussions. He contributed essential views to our present paper. R. Holtwick helped with the preparation of the figures. The experimental and theoretical
[I] J. Wong and C.A. Angell, Glass, Structure by Spectroscopy (Dekker, New York, 1976). [2] A. Bums, G.D. Chryssikos, E. Tombari, R.H. Cole and W.M. Risen Jr., Phys. Chem. Glasses 30 (1989) 264. [3] R. Hoppe, T. Kloidt and K. Funke, Ber. Bunsenges. Phys. Chem. 95 (1991) 1025. [4] K. Funke, T. Kloidt, D. Wilmer and C.J. Carlile, Solid State Ionics 53-56 (I 992) 947. [5] K. Funke, Mat. Res. Sot. Symp. Proc. 210 (1991) 97. [6] K. Funke, Prog. Solid State Chem. 22 ( 1993) 1 I I. [7] A. Bunde, M.D. Ingram and P. Maass, J. Non-Cryst. Solids 172-174 (1994) 1222. [8] K.L. Ngai, Comm. Solid State Phys. 9 (1979) 127. [9] K.L. Ngai, Comm. Solid State Phys. 9 (1980) 141. [IO] K.L. Ngai, S.L. Peng and K.Y. Tseng, Physica A I91 (1992) 523. [II] K. Funke, J. Non-Cryst. Solids 172-174 (1994) 1215. [I21 C. Cramer, K. Funke, T. Saatkamp, D. Wilmer and M.D. Ingram, Z. Naturforsch. 50a (1995) in press, [I31 W.-K. Lee, J.F. Liu and A.S. Nowick, Phys. Rev. Lett. 67 (1991) 1559. [I41 M. Pollak and GE. Pike, Phys. Rev. Lett. 28 (1972) 1449. [I51 A.K. Jonscher, Dielectric relations in solids (Chelsea Dielectric Press, London, 1983). [I61 P. Maass, M. Meyer and A. Bunde. Phys. Rev. B. in press (1995). [I71 G.S. Linsley, A.E. Owen and F.M. Hayatee. J. Non-Cryst. Solids 4 (1970) 208.
STATE Solid State Ionics 85 (1996) 293-303
Ionic and polaronic glassy conductors: conductivity spectra and implications for ionic hopping in glass K. Funke”‘“, C. Cramera, B. Roling”, T. Saatkamp”, “Institut fiir
Physikaiische
D. Wilmer”, M.D. Ingramb
Chemie, Schlosspiatz 417, 48149 Miinster,
hDepartment of Chemistry, University
Germany
of Aberdeen, Meston Walk, Aberdeen, AB9 TUE. UK
Abstract spectra have been taken of a lithium ion and a polaron conducting glass. The former exhibits plateaux in its hopping conductivities and two different power-law exponents in the dispersive regime. This is readily explained if ions, contrary to polarons, have a pronounced site preference. At the same time, this view also explains the well known differences in the dependence of carrier mobility on carrier concentration in the two types of glasses. Complete
conductivity
high-frequency
Keywords:
Glassy electrolytes;
Polaronic
glassy conductors;
Conductivity
1. Introduction Frequency-dependent ionic conductivities of solid electrolytes, &(o), contain valuable information on the dynamics of the elementary steps of hopping motion of mobile ions in these materials. The same holds true for polaronic conductors. Measurement of complete conductivity spectra extending from d.c. to the far- or mid-infrared does, however, require application of various experimental techniques which are not normally available in one laboratory. This may be the reason why complete spectra of o-((w) are still extremely rare. Two well-known examples are shown in Figs. la and lb. In Na,O. 3Si0, glass [1,2], there is a continuous increase of log u vs. log w, and even of the slope of this function, until the vibrational regime is attained at about 1 THz, cf. Fig. la. The situation *Corresponding
author. Email: [email protected]
0167.2738/96/$15.00 Copyright PII SOl67-2738(96)00073-2
01996
Elsevier
spectra;
Power laws; High-frequency
plateau;
Site preference
is different in RbAgJ,, see Fig. lb, where the increase of the hopping conductivity is restricted to a regime denoted by II in the figure. This regime is followed by the so-called high-frequency plateau, which is observed in regime I [3,4]. Note also that in regime II the slope, p, is always smaller than one, while in Fig. la the slope is found to exceed the value of one. As pointed out earlier [3-51, the conductivity spectra of Fig. lb are in general agreement with the predictions of the jump relaxation model [6]. By contrast, the spectra of Fig. la do not show the high-frequency plateaux predicted by this model. Moreover, power-law exponents larger than one cannot be explained in terms of jump relaxation, if only hops via equivalent sites are being considered
[61. In this paper we present complete conductivity spectra of a lithium ion conducting glass of composition B,O, * 0.56 Li,O * 0.45 LiBr and of a
Science B.V. All rights reserved
294
K. Funke et al. I Solid State Ionics 85 (1996) 293-303
employ the concept of non-equivalent sites when interpreting the polaronic conductivity spectra. For polarons the effect of site preference does not seem to exist. As the polaron moves from site to site, there is apparently no “memory” left behind in the glass structure which can influence the movements of other polarons.
2. Experimental spectra
b)
I
aaoooo00 0000000 e
129K 0000000' ' I 8
10
12
I
bl,#~l Fig. 1. (a) absorptivity/conductivity spectra of Na,O-350, glass at various temperatures, after Refs. [ 1,2]. 1 Neper cm-’ correC’cm-‘. (b) Dynamic conductivity of the sponds to 3.10-’ crystalline solid electrolyte RbAg,l, at temperatures between 129 K and 298 K, after Ref. [3].
polaron conducting glass of composition P205 . 4(V,O,, V204). In both cases, the low and high frequency parts of the spectra are well explained by jump relaxation [6] and vibrational contributions, respectively. In the ion conducting glass, it is possible for the first time to separate the hopping and vibrational contributions to the conductivity and to identify high frequency plateaux of the hopping conductivity. However, the shapes of the spectra of the ion conducting glass call for an extension of the basic version of the jump relaxation model [6], to include the existence of non-equivalent sites, i.e. of preferred and less favoured ones. This is done by combining the jump relaxation and dynamic structure models [7]. On the other hand, there is no need to
Complete ionic conductivity spectra of glassy B,O, .0.56 Li,O - 0.45 LiBr have been taken at twelve temperatures, ranging from 173 K to 573 K. Examples are given in Fig. 2a and b. Evidently, the overall shape of the spectra is similar to Fig. la. In particular, a high-frequency plateau is not visible. In Fig. 3 we present complete conductivity spectra of the polaron conducting glass P,Os .4(V,O,, V,O,). Again, the overall shape of the spectra is similar to Fig. la. However, it has to be borne in mind that in this example the low and high frequency parts of the spectra are due to the motion of entirely different charge carriers. Polaronic hopping is dominant below about 100 GHz while the far-infrared conductivity maximum is due to vibrational motion of the ions. In spite of the apparent similarity of the spectra of Figs. 2 and 3, closer inspection reveals significant differences arising from different dynamic properties of the respective charge carriers. This will be discussed in the following.
3. Ton conducting B,O, * 0.56 Li,O. 0.45 LiBr In the far infrared, the conductivity spectra of Figs. 2 and 3 are found to vary with frequency as w2, the temperature dependence being rather small. This feature can be regarded as the low-frequency flank of the lowest-lying vibrational mode. The vibrational contribution to the conductivity is now easily extrapolated to lower frequencies and taken out of the total spectra. The resulting spectra are then interpreted in terms of the hopping motion of the mobile charged species. In the following, this procedure is applied to the lithium ion conducting glass B,O, .0.56 Li,O * 0.45 LiBr.
K. Funke et al. I Solid State lonics
573 K 08
/
I:: 6
7
6
9
slope - p
10
11
12
13
14
2 1 0 $
-1
-5 -
85 (1996)
2%
293-303
glasses, because they predict a high-frequency plateau which has never been observed. This argument is now invalidated. The reason for high-frequency plateaux not being directly visible in experimental conductivity spectra of glass is the existence of a broad vibrational contribution which swamps the high-frequency part of the hopping conductivity. In the high-frequency-plateau regime, the sign of the applied electric field changes so fast that every hop or displacement of an ion is recorded and contributes to the conductivity. At slightly higher FIR frequencies, the period of the field is already comparable with the duration of an individual hop, and this implies the breakdown of the very concept of hopping conductivity. At the same time, the subtraction described above is no longer feasible, if the frequency exceeds about 400 GHz. Considering the experimental hopping conductivities of Fig. 4 ( + ) we find that they are well in line with the predictions of the jump relaxation model as long as the frequency is not too high, whereas characteristic deviations are observed in the microwave and millimetre-wave regimes. Let us start our discussion at comparatively low frequencies. The d.c. conductivity, g(O), obeys an Arrhenius law,
-6 -7'
t 2
I 4
6
6
10
12
conductivity
spectra of B,0;0.56
(1)
14
h4,,ww Fig. 2. Complete
a(O) . T m exp( - E,, lkT),
Li,O.O.45
LiBr glass at (a) 573 K and (b) 323 K. The error bars indicate the
with E,,=O.49 eV-k.5670 K. Fig. 6 is an Arrhenius plot of U(0). At all temperatures, the low-frequency part of . . qo,(w) 1s nicely fitted by
largest possible error over the entire frequency range.
cr,,,(low Figs. 4a and 4b show the resulting hopping-conductivity spectra ( + ), as well as the total conductivity ones. The most exciting feature is the appearance of a high-frequency plateau. This is more clearly seen when the data are looked at more closely between 1O’O and 10” Hz, see Figs. 5a and 5b. At about IO” Hz the data merge into their plateau values. With regard to their overall shape, the spectra ( + ) of Fig. 4 do, indeed, resemble those of crystalline electrolytes, cf., Fig. lb. They are also in broad agreement with the predictions of the coupling [8-lo] and jump relaxation [ 1 l] models. It has often been claimed that these models do not apply for
w) = d(o).
[ 1 + (&*)“I with p = 0.6.
(2)
In this equation, the angular frequency 1 lt, marks the onset of the power-law dispersion. According to the jump relaxation model, the continuation of Eq. (2) towards higher frequencies would have to be %,.,Ul(@)
= a(0) + [a(m) - V(O)] . [ 1 + (wtJ’]
-p.
(3)
Here, the angular frequency 1 lt, marks the transition into the high-frequency plateau, where the hopping conductivity attains the value g(m). Evi-
296
K. Eunke et al. I Solid State lonics 85 (1996) 293-303
2
-
1
-
~523
K
+4?3
K
q
423
K
x 373 K
a
0
-
-1
-
-2
-
0 323
K
e e B
A
abps.0.7
-4 -3
// I
-51 0 Fig. 3. Complete
conductivity
dently, Eq. (2) is the low-frequency of Eq. (3), valid at w< l/t,, with (T(O). t; = [CT(m)- CT(O)]. t’;.
’ 6
I
2
4
w) = a(m).
with q > 1.
approximation
(4)
[ 1 + (~t,)-‘]-~ (5)
In the fits presented in this paper we have used somewhat larger values of q also
q = 1.3. However,
I
’
10
12
spectra of P,O,.4(V,O,,
The continuation of the low-frequency part of the experimental hopping spectra as expected from Eq. (3) is presented in Fig. 7. The three regimes I, II, III already encountered in Fig. lb reappear in Fig. 7. As will be shown later, the spectra of Fig. 7 would correspond to a situation where the mobile lithium ions perform hops via preferred sites (called A) exclusively. The parameter values used in Fig. 7 will later be employed for fitting a modified version of the jump relaxation model to the actual spectra. For further details concerning the construction of Fig. 7, see Ref. [12]. Comparison of the actual hopping-conductivity spectra with the spectra of Fig. 7 reveals systematic differences at microwave frequencies. In fact, the low-frequency behaviour is well described by Eq. (2), while the high-frequency part is close to q,,(high
I
6
V,O,)
L
J
14
at various temperatures.
fit the data (see below), and it seems realistic to characterize the range of possible q values by 1.35 qs2.0. In Fig. 8, a superposition of q,,(low w) from Eq. (2) and a,,,(high w) from Eq. (5) is shown to fit the experimental hopping conductivities very well. Eq. (5) differs from Eq. (3) by q#p and even q> 1. The latter inequality is impossible in the original jump relaxation model, in which all sites and all hops are geometrically alike [6]. Instead, according to the dynamic structure model [7], we have to distinguish between at least two kinds of site in a single-alkali glass. These are the so-called A sites which are optimally configured to meet the requirements of each mobile At ion (here: Li+), and the c sites which provide less room and are less favourable. If an A+ ion leaves an A site, this site is supposed to stay A for some time and then to transform into a c site. Correspondingly, if an A+ ion hops into a c site and manages to stay, then the r? site is supposed to relax into an A site. There are thus two kinds of site relaxation: (i) the shifting of the Coulomb cage as described by the jump relaxation model and (ii) the A to c and c to A relaxation as described by the dynamic structure model. Effect (i) is achieved by the hopping of neigh-
K. Funke et al. I Solid State Ionics 85 (1996) 293-303
291
2 0.5 i t
a)
r
o-
0
573 K
OC
0 0
6
0 0
p-o.5 o? 9.
00
-1
-
Q
Oo”O
y++++++
+
-3 vib
hop
-21
-4 6
7
E
0
-6
10
11
hop
12
:
13
14
4
6
6
IO
12
k
’
’
a
’
8
I
’
’
11.2 11.4 11.6 11.6 12
12.2
10.2 10.4 10.6 10.6 11
11.2 11.4 11.6 11.6 12
12.2
vib
-7 2
’
10.2 10.4 10.6 10.6 11
14
ba,&vW
Fig. 4. Total experimental obtained after subtracting (+), of glassy B,O;0.56 323 K.
conductivity spectra (0) and spectra vibrational contributions out of them Liz0.0.45 LiBr at (a) 573 K and (b)
bouring mobile ions, while (ii) necessitates a restructuring of the glassy matrix. Therefore, (i) is assumed to be much faster than (ii). In this paper we will show that the low-frequency behaviour described by Eq. (2) and its continuation according to Eq. (3) are caused by hops of the mobile lithium ions via their preferred A sites, whereas the high-frequency behaviour described by ECq. (5) is mainly due to hops from A sites into adjacent c sites, immediately followed by backward hops. Regarding site relaxation, the spectra clearly
Fig. 5. Representations
of the high-frequency
plateau
regions
Fig. 4 at the same temperatures.
-7
1
1.5
2
2.5
3
3.5
4
4.5
5
1000 K/-r
Fig. 6. Arrhenius plot of the d.c. conductivity Li,O.O.45 LiBr glass.
of B,O,.O.56
of
K. Funke et al. I Solid State lonics
298
85 (1996)
3.0,
293-303
,
2.0 1 .o
500
K
333
K
250
K
200
K
0.0
0
-1.0 -2.0
d 5 I<
-3.0
s I< -;
-4.0
rp -e
-5.0
H
-6.0 -7.0
6.0
6.0
toa,.@/H~) Fig. 7. Log-log representation L&0.0.45 LiBr, cf. Eq. (3).
of the calculated
-6.0 % 4.0
A to A hopping contribution
to the frequency-dependent
conductivity
of glassy B,O, .0.56
_1
5.0
6.0
7.0
6.0
0.0
la.0
11.0
12.0
ba,,WW Fig. 8. Hopping conductivity spectrum of B,O;0.56 L&0.0.45 LiBr glass at 373 K (0). regime, p=O.6) and Eq. (5) (high-frequency regime, q= 1.3).
reflect effect (i), while effect (ii) is not readily observed, cf. Section 4. In the original jump relaxation model, the hopping into a neighbouring site is activated with an energy A. Being smaller than the activation energy for successful hops, E,, of Eq. (l), A determines the
Solid lines calculated
via Eq. (2) (low-frequency
temperature dependence of o-(m) of Eq. (3), i.e. the spacing of the high-frequency plateau conductivities in the plot of Fig. 7. On the other hand, the corresponding spacing of the actual high-frequency plateau values, U(W) of Eq. (S), would be given by the activation energy required for a hop into a c site.
299
K. Funke et al. I Solid State lonics 85 (1996) 29_?-303
Before further discussing the frequency dependencies of Eqs. (3) and (5) let us first consider the activation energies involved. This is done in the Arrhenius-type representation of Fig. 9. The broken line marks the high-frequency conductivities according to Eq. (3) and Fig. 7. Experimental data points are included for both the high-frequency plateau and 40 GHz. Evidently, the experimental high-frequency data do not obey a simple Arrhenius law. However, the situation becomes quite simple in the limits of both high and low temperatures. At high temperatures, the limiting slope is very similar to the one of the straight line, the experimental conductivities being larger than those derived from the simple jump relaxation model. Therefore, we have been using identical activation energies A for both generic processes, i.e.,
exactly as WI-O. Although explanations have been suggested /14,15], it is felt that the “new universaliunsolved ty” can be regarded as a challenging problem. The solid lines fitting the two experimental spectra of Fig. 10 have been constructed by starting from the “new universality” baseline and superimposing the thermally activated contributions that have been identified. The latter include (i) the jump-relaxation-type spectra of Eq. (3) and Fig. 7, with p =0.6, and (ii) the high-frequency contribution described by Eq. (5), with q= 1.3. The solid lines in Fig. 9 have been obtained from the same model construction.
hops from A to A (cf. Fig. 7) and hops from A to c, cf. the high-frequency part of Fig. 8. At low temperatures, the conductivity appears to approach a limiting value. This is observed at all frequencies up to the high-frequency plateau. In fact, the experimental data seem to converge to the temperature-independent baseline included in Fig. 10. The existence of this low-temperature baseline has been termed “new universality” [ 131. Except for the high-frequency plateau regime, it is found to vary
4. Model conception
for ionic hopping
in glass
This section is devoted to a dynamical interpretation of a,,,,(high w) as described by Eq. (5). As stated earlier, the inequality q> 1 necessitates a modification of the original jump relaxation model to include the existence of different sites. According to the dynamic structure model, these are supposed to be the “good” A sites and the “bad” c sites. As a mobile A’ ion (here Li+) spends most of its time at A sites, the hops we have to consider are mainly A to
hf pkteau
1
40 GHr
3.0
5.0
7.0
1000 KIT
Li,O-0.45 LiBr at different frequencies. Fig. 9. Arrhenius type representation of hopping conductivities a,,,, (0) of glassy B,0;0.56 lines: model construction. The broken line marks the high frequency plateau values of chhclPr from Eq. (3) and Fig. 7.
Solid
300
K. Funke et al. I Solid State Ionics 85 (19%)
293-303
7 d ‘i b a s”
-4.0
-6.O4.0
8
5.0
I 6.0
7.0
/ L
/
/
/’
b 5.0
0 0.0
0 10.0
’ 11.0
112.O
bn,,W/Hd
Fig. 10. Hopping conductivity line is the “new universality”
spectra of glassy B,0;0.56 L&0.0.45 LiBr (0). Solid lines result from our model construction. baseline attained in the low-temperature limit.
A and A to c. After an A to c hop, we expect the ion to “roll back” to its initial A site in a vast majority of cases. Fig. 11 shows the difference between an A to A hop and an A to (? hop, in terms of the singleparticle potential felt by the mobile ion. Fig. 12 is applicable to both situations displayed in Fig. 11. This means that Y may be A or c while X is always A.
&q.k
The broken
,,-lr
\
1’
i
\
6
I f
site X
A
?
site Y
Fig. 12. Effective single-particle potential of the central ion at the time of its initial hop from site X to site Y.
Once the ion is at Y, a competition sets in between two ways of relaxation. The ion may hop back to X or the neighbourhood may relax with respect to Y. Either one of these competing ways of relaxation can be characterized by a time constant or by its inverse, the respective rate. The rates are denoted by
r=o
Fig. 11. Typical single-particle potentials encountered by hopping ions (at t=O) in (a) A to ,& and (b) A to C hops. Each of the effective potentials constructed in (a) and (b) consists of two contributions: a site-sensitive potential (on top) and a Coulomb cage potential.
= initial back-hop rate,
= initial site-relaxation
rate.
In Eqs. (6a) and (6b) W(t) is the probability
(6a)
(6b) for the
301
K. Funke et al. I Solid State Ionics 8-5 (1996) 29.7-303
correlated backward hop not to have occurred until time t, and Ejback(t) is the time-dependent barrier height for that hop. It is now easy to show that the ratio t, lt* determines the exponent (p or q) in the frequencydependent hopping conductivity. This is seen by considering the correlated back-hop rate, - t&‘(r), and the rate of change of &&t), $ack(t), at sufficiently short times, when the site-relaxation process at Y is still far from being completed. Denoting the attempt frequency for a backward hop by vor, we have - W(r) =
W(r) -var. exp[-&,,(t)lkT].
(7a)
On the other hand, numerical simulations [ 161 as well as the jump relaxation model show that
1. The power-law of Eq. (8) now transforms into a power-law of a,,,(w): (i) If the ratio t, lt* is less than two, a,,,(w) is well approximated by uxx.v(w>= U,,,(w’)‘[l
+ (Wt,)-‘J-rl”*.
We thus reproduce Eq. (5) as well as the highfrequency part of Eq. (3), where the exponent is less than unity. The important result is power-law
exponent =
initial back-hop rate initial site-relaxation
C’b) holds in the short-time regime. With the definition Eq. (6a) and after a little algebra, we find at short times.
of
(8)
(u(O). w,,,
= (x;/z).
r,. [S(l) + I@(r)].
(9)
In Eq. (9), s(t) is the delta function, while x,, and r, are the X-Y distance and the forward hopping rate, respectively: *exp[-A/U]. r;, = VOX
(IO)
Here vex denotes the attempt frequency at site X. To retain simplicity, we suppose that the influence of cross terms, (v,(0)*vj(t)), where i and j, with i#j, denote different hopping ions, can be neglected. The contribution to the high-frequency hopping conductivity that is due to the hopping between sites X and Y is then proportional to the Fourier transform of the velocity autocorrelation function of Eq. (9):
rate ’ (13)
(ii) If the ratio t, lt* tends to infinity, i.e., in the absence of any site relaxation, W(t) becomes a simple exponential,
W(c)=,l$= (1 + t/tl))““’ ,
In the following, Eq. (8) will be used to derive Eq. (5). In particular, the exponent in Eq. (5), which may be q 11 or p< 1, turns out to be determined by the exponent in Eq. (8), i.e., by the ratio t,lt*. Let us first relate the back-hop rate to the (shorttime part of the) velocity autocorrelation function describing hops between X and Y:
(12)
implying a;(Jo)
= : exp(-t/r*),
(14)
Debye relaxation: = +J”)*
[l + (w$?]-‘.
(15)
(iii) The power-law exponent is, therefore, two for both limiting cases, t,/t”-+2 and t, it*-+“, the straight lines in a log u vs. log o plot being parallel to each other. As might have been expected, finite values of t, lt* larger than two yield results in between, again with an exponent of two. Summarizing (i) to (iii) one can say that Eq. ( 13) holds if t,lt* does not exceed two. If it does, the exponent stays at two. The two thermally activated contributions to the experimental spectra, featuring power-law exponents p < 1 and q > 1, are now easily explained. Hops via A sites may be treated by the original jump relaxation model. The model does, indeed, apply as only one kind of site is involved, the Coulomb relaxation of the neighbourhood being essentially due to successful x to A hops of other ions. In this case, the t, lt* ratio has a value of p < 1 and we obtain Eq. (3) and Fig. 7. Compared to an A to A hop, an A to c hop
302
K. Funke et al. / Solid State lonics 85 (1996) 293-303
has a smaller backward barrier height. Therefore, the back-hop rate, l/t *, is enhanced. As a consequence, the t, It* ratio is now found to be larger, exceeding one. Neighbouring c sites being quite common,
centration of mobile cations in a simple binary glass by c and their mobility by U, then the model predicts a power-law behaviour,
there is frequent A to c hopping, virtually always followed by a backward hop. This process is well described by Eq. (8) with t, lt* > 1, resulting in a power-law exponent q > 1. Only in a few cases does the mobile ion manage to stay at the c site which then transforms into an A site. However, these rare events do not appreciably contribute to the low-frequency conductivity. We should like to emphasize that there is no doubt about the existence of the q > 1 term in the spectra. Therefore, if Y is a c site, t, lt* is clearly larger than unity. On the other hand, the superposition of the and q> 1 terms causes some “new universality” arbitrariness in the choice of the best numerical value of q. The fit of Fig. 10 has been made using t, it* = 1.3, but the ratio may easily be larger than that. There is also some arbitrariness concerning the thermal activation of the q> 1 term at relatively low temperatures when it is not easily separated from the “new universality” baseline of Fig. 10. At relatively high temperatures, however, the activation energies of the p and q terms are indistinguishable. This implies roughly identical barrier heights for A to A and A to c hops. The different success of A to A and A to i; hops does, therefore, not result from different
UKCY,
hopping rates A to A and A to r?, but rather from different A to A and c to A back-hop barrier heights implying different backward-hopping rates.
5. Comparison of ionic and polaronic conduction in glass In the ion conducting glass, two power-law regions have been observed, with exponents p< 1 and q > 1. The interpretation given in the preceding section is based on the existence of non-equivalent sites A and c’, with a strong preference of the mobile cations for the optimally configured A sites. This concept is of the essence of the dynamic structure model [7]. One of the merits of this model is the successful explanation of the way ionic mobilities depend on ionic concentrations. Denoting the con-
y>o.
(16)
Experimentally, this power-law is well established. Interestingly, varying the degree of reduction in a polaronic glass like P205.4(V205,VZ04), i.e. varying the concentration of mobile polarons, is known to have little effect on the polaronic mobility, unless extremely large concentrations are attained [ 171. It is, therefore, evident that the dynamic structure model does not apply to polarons. One may further conclude that there should be no pronounced preference of polarons for certain sites and that the concept of favoured and less favoured sites should not play a role for polarons. As a consequence, there would be no need for modifying the original jump relaxation model to include the effect of different sites. Therefore, polaronic conductivity spectra would be expected to conform with Eq. (3), and there would be no justification for a second power law with an exponent q>l. This prediction can be tested by studying the spectra of the polaron conducting glass P205. 4(V,O,, V204). As in the ion conducting glass, the low-frequency data are well described by Eq. (2), the numerical value of p being now 0.7. The d.c. activation energy obeys the Arrhenius law of Eq. ( 1), with E,=O.41 eV In the dispersive regime, the activation energy is found to be (1 -p)*E,, in agreement with the jump relaxation model. Again, as in the ion conducting glass, the low-frequency flank of the vibrational conductivity varies as w2, and a temperature dependence of the position of this flank is not detected. Taking the vibrational conductivity out of the spectra does not yield a detectable high frequency plateau of the hopping conductivity. Evidently, it is entirely swamped by the broad vibrational contribution. Superposition of the hopping-conductivity component of Eq. (2) and the vibrational component does, in fact, suffice for an excellent fit to the experimental data. This is shown in Fig. 13 where emphasis is put on our results obtained in the microwave and millimetre-wave regimes.
K. Funke et al. I Solid State Ionics 8.5 (1996) 29.3-303
b 1.5
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A 523 K +475K n 423 K
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-0.5
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analysis of the frequency dependent ionic conductivity was funded by the Deutsche Forschungsgemeinschaft. Financial aid for our British-German collaboration is provided by the British Council and the DAAD in the framework of the ARC programme. Financial help by the Fonds der Chemischen Industrie is also gratefully acknowledged.
I)
z7
-1.5
References -2.5
-3.5 B
10
11
12
13
h,,WW
Fig. 13. High-frequency conductivities of the polaron conducting glass P,0;4(Vz0,, V,O,). The solid lines result from the fit described in the text.
The tit of Fig. 13 proves the complete absence of a g> 1 term in the polaron conducting glass. Evidently, in glassy conductors, the effects of different sites and site preference are important for ions but of no significance for polarons. It is remarkable that it is the existence of such effects in ionic glasses which is responsible for anomalies seen at both d.c. and microwave frequencies.
Acknowledgments It is a pleasure to thank A. Bunde for fruitful discussions. He contributed essential views to our present paper. R. Holtwick helped with the preparation of the figures. The experimental and theoretical
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