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The determination of hopping rates and carrier concentrations in ionic conductors by a new analysis of ac conductivity
Solid State Ionies 8 (1983) 159-164 North-Holland PublishingCompany
THE DETERMINATION OF HOPPING RATES AND CARRIER CONCENTRATIONS IN IONIC CONDUCTORS BY A NEW ANALYSIS OF ac CONDUCTIVITY D.P. ALMOND School of Materials Science, University of Bath, Claverton Down, Bath BA2 7,4 Y, UK
and G.K. DUNCAN, A.R. WEST Department of Chemistry, University of Aberdeen, Meston Walk, Old Aberdeen AB9 2UE, Scotland, UK
Received 7 October 1982 ABSTRACT The a.c. conductivity o(m) of ionic materials takes the form, o(m) = o(o) + Amn. The c a r r i e r hopping rate, Up, is obtained from the new expression o(o) = Aconn and the c a r r i e r concentration is estimated from o(o). The contribution of creation and migratio~ terms to the activation energy for conduction may be determined from the thermal activation of o(o) and ~D and the corresponding entropy terms quantified. D a t a have been analyzed for four widely d i f f e r e n t ibnic materials: single crystal Na B-alumina, p o l y c r y s t a l l i n e Li4SiO~, AgTl~AsO~ glass and Ca(NO3)2/KN03 glass and melt. For each, the c a r r i e r concentration and hopping rates have been obtained.
The conductivity of an ionic conductor is determined by both the concentration of the mobile ions and the rate at which they are able to hop from site to site in the material. Experimentally, the separation of these two terms has been found, in general, to be difficult. Methodswhich have been used with some success include: the Hall effect [ l ] , the doping of ionic crystals with aliovalent impurities [2] and the electrochemical measurement of a c t i v i t i e s in glasses [ 3 ] . None of these methods appear, however, to be readily applicable to a l l ionic conductors. In this paper we demonstrate that c a r r i e r concentrations and hopping rates can be separated by a new treatment of a.c. conductivity. The potential scope of the method is i l l u s t r a t e d by the analysis of four very d i f f e r e n t materials: a single crystal superionic conductor (Na B-alumina), a p o l y c r y s t a l l i n e i o n i c conductor ( L i . S i O , ) , a vitreous e l e c t r o l y t e (AgTI.AsO. glass) and a poorly conducting n i t r a t e glass (40 mol % Ca(NOa)2 - 60 mol % KN03) both above and below the glass transformation temperature. The d.c. c o n d u c t i v i t y of a c r y s t a l l i n e i o n i c conductor is determined by the expression obtained from the theory of random walk [ 4 ] : a(o) = Kmp = (Ne2aZ/kT) yc(l -C)mp
(l)
where y is a geometrical factor that may include a correlation factor, c is the concentration of mobile ions on N equivalent l a t t i c e sites per unit volume, a is the hopping distance, e is the electronic charge, k is Boltzmann's
constant, T is absolute temperature and Up is the hopping rate of the ions. The hopping rate is thermally activated and is given by: Up = moeXp(-AGa/kT) = moexp(ASa/k)exp(-Ea/kT) = meexp(-Ea/kT)
(2)
where &Ga, Ea and ASa are the free energy, activation energy and entropy of the hopping process, mQ is the vibrational frequency of ions in t h e l r l a t t i c e sites and me is an e f f e c t ive attempt frequency which includes the entropy term. The concentration c of mobile ions may or may not be temperature dependent. In the former case,
c
=
CoeXp(-AGc/kT)
=
CoeXp(ASc/k)exp(-Ec/kT)
= CeeXp(-Ec/kT)
(3)
where AGc, Ec and ASc are the free energy, activation energy and entropy for the creation of mobile carriers and ce is the effective i n f i n i t e temperature c a r r i e r concentration. Measurements of d.c. conductivity, over a range of temperatures, provide numerical values for the net activation energy and pre-exponential factor for the overall conduction process. The d i f f i c u l t i e s which often arise in c o r r e l a t -
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D.P. A l m o n d et al. / Hopping rates and carrier concentrations in ionic conductors
160
ing such values w i t h the t h e o r e t i c a l expressions, ( I ) , ( 2 ) , ( 3 ) , are : a) i t is not possible t o determine, unambiguously, whether or not the c a r r i e r c o n c e n t r a t i o n is t h e r m a l l y a c t i v a t e d , i . e . whether the measured a c t i v a t i o n energy of o ( o ) is (E a + Ec) or Ea alone. b) i t is not p o s s i b l e t o determine the magnitudes of me and c e since t h e y are present as a product in the p r e - e x p o n e n t i a l f a c t o r . Up to now the main reasons f o r using a . c . techniques to measure i o n i c c o n d u c t i v i t y have been t o avoid the n e c e s s i t y of non-blocking e l e c t r o d e s and t o f a c i l i t a t e the s e p a r a t i o n of g r a i n boundary and bulk phenomena in p o l y crystalline materials. I d e a l l y , a frequency independent c o n d u c t i v i t y , equal to the d.c. c o n d u c t i v i t y , should be obtained a t h i g h e r frequencies where e l e c t r o d e and grain boundary phenomena are i n s i g n i f i c a n t . In p r a c t i c e , such a frequency independent p l a t e a u is observed, often over several decades of frequency, but is l i m i t e d at higher frequencies by an a d d i t i o n a l bulk phenomenon. This takes the form of a power law increase in c o n d u c t i v i t y w i t h frequency. The a . c . c o n d u c t i v i t y a(m) is found t o v a r y w i t h angular frequency co as ~(co)
=
o(o)
+
A~n
(4)
where a(o) is the d.c. c o n d u c t i v i t y , A is a temperature dependent parameter and n is found to take values between 0 and I . Jonsche~ [ 5 ] has suggested t h a t t h i s power law is a ' u n i v e r s a l ' p r o p e r t y of m a t e r i a l s t h a t is r e l a t e d t o the dynamics of hopping conduction. R e c e n t l y , we have shown [ 6 ] t h a t t h e r e is a simple r e l a t i o n s h i p between a(o) and A. The c o n d u c t i v i t y o(co) can then be expressed as • o(co)
,
l-n
n
= K~p + ~CJp co
(5)
where K was defined in equation (1). By combining equations (4) and (5) the hopping rate cop is given by co = ( ~ ( o ) / A ) I / n (6) P This provides a p a r t i c u l a r l y sir.lple way of quantifying m . The t h e o r e t i c a l temperature dependences of p o(o) and A may be obtained by s u b s t i t u t i n g f o r K and cod from ( I ) , (2) and (3) in ( 5 ) . For m a t e r i a l s 'in which the c a r r i e r c o n c e n t r a t i o n is independent of temperature, o n l y mo is t h e r m a l l y a c t i v a t e d and the a c t i v a t i o n energies of a(o) and A are Ea and (I - n ) E a, r e s p e c t i v e l y . For m a t e r i a l s in which the c a r r l e r c o n c e n t r a t i o n is a l s o t h e r m a l l y a c t i v a t e d , K is t h e r m a l l y a c t i v a t e d and consequently, the a c t i v a t i o n energies of ~(o) and A become Ec + Ea and E + (I - n ) E a, respectively, c The above r e l a t i o n s h i p s have been used to analyse a.c. c o n d u c t i v i t y data from f o u r d i f f e r e n t i o n i c conductors. The f i r s t three : s i n g l e c r y s t a l Na B-alumina, p o l y c r y s t a l l i n e L i , S i 0 4 and AgTI,AsO, g l a s s , appear t o have temperature independent c a r r i e r c o n c e n t r a t i o n s
whereas the f o u r t h , K/Ca n i t r a t e g l a s s , appears to have a temperature dependent c a r r i e r c o n c e n t r a t i o n above the glass t r a n s f o r m a t i o n temperature, Tg. C o n d u c t i v i t y data at several temperatures f o r the f i r s t t h r e e m a t e r i a l s are shown in Fig. I . The s o l i d l i n e s through the data are f i t s to equation (4). The values of o(o) and A used f o r these f i t s are shown p l o t t e d in conventional Arrhenius format as Fig. 2. In each case, the a c t i v a t i o n energies of a(o) and A are in the r a t i o 1 : (I - n ) as p r e d i c t e d . The hopping rates up c a l c u l a t e d from the same data using equation (6) are shown in Arrhenius format as Fig. 3. The data were f i t t e d by the s t r a i g h t l i n e s shown using l i n e a r regression and values of the a c t i v a t i o n energies and e f f e c t i v e attempt frequencies were obtained from the slopes and i n f i n i t e temperature intercepts, respectively. The a c t i v a t i o n energies f o r each are w i t h i n experimental e r r o r the same as those found f o r a(o) in Fig. 2. This i n d i c a t e s t h a t the thermal a c t i v a t i o n s of these c o n d u c t i v i t i e s are caused by the thermal a c t i v a t i o n s of the hopping rates alone. The e f f e c t i v e attempt frequency me of B-alumina obtained from Fig. 3 is v e r y s i m i l a r to the value 2 x 1012 Hz associated with the v i b r a t i o n al frequency mo obtained d i r e c t l y from i n f r a red a b s o r p t i o n measurements [ I 0 ] . This shows t h a t the e n t r o p y term ASa in equation (2) is small in t h i s p a r t i c u l a r m a t e r i a l . Several authors [4, I I ] have iTnd,~ed suggested t h a t these e n t r o p y terms should be small in f a s t ion conductors such as B-alumina. For L i , S i O , and the Ag+ g l a s s , the e f f e c t ive attempt f r e q u e n c i e s , I0 zsHz Fig. 3, are c o n s i d e r a b l y higher than l i k e l y values of corresponding v i b r a t i o n a l f r e q u e n c i e s , ~1013Hz. This r a i s e s the question o f the magnitude o f the e n t r o p y of a c t i v a t i o n ASa in these, and o t h e r , i o n i c conductors. Some i n s i g h t may be obtained by using the formula f o r ~Sa which was d e r i v e d by Wert and Zener [12] in t h e i r t h e o r y of i n t e r s t i t i a l d i f f u s i o n in m e t a l s : AS BEa (7) a T m where Tm is the m e l t i n g temperature and B is a dimensionless number, ~ 0 . 4 , which is r e l a t e d to the temperature dependence of the shear modulus of the m a t e r i a l . Estimates of the magnitude of the term exp(ASa/k) in me, obtained using equation (7) are 40, I I and 1.3 f o r the Ag+ g l a s s , Li4SiO, and B-alumina respectively. These e n t r o p y values match, to w i t h i n a f a c t o r of f i v e or b e t t e r , those determined using equation (2) and Fig. 3 and assuming wo ~ 1013 Hz f o r L i , S i 0 4 and the Ag+ glass. In view of the e r r o r s in e s t i m a t i n g me, 8 and mo ( e s p e c i a l l y f o r Li4Si04 and the Ag+ glass) t h i s is a l l t h a t can be expected at t h i s stage. A f t e r the hopping rates have been determined, i t is p o s s i b l e to estimate the carrier concentrations. For B-alumina, however
161
D.P. A l m o n d et al. / Hopping rates and carrier concentrations in ionic conductors
m
Li&SiO4
n : 0.68
Z
AgTILAs~
n :0.68
f184K
E u
..(Z 0
~ 4 8 0 K
b
2
3
A
5
2
6
3
1 4 0 K ~
~
4 5 6 log f(Hz)
2
3
4
5
6
7
FIG. 1 Conductivity data for single crystal Na 6-alumina (ref. 6,7), polycrystalline Li,SiO, (ref. 8) and Ag~l,AsO4 glass (ref. 9) at a selection of temperatures. Solid curves are f i t s of equation (4) to the experimental data. Errors in n are estimated as ±0.03. The experimental data for 6-alumina and AgTI,AsO, at the highest temperatures and low frequencies show a conductivity dispersion associated with electrode polarisation effects.
4t
Na /B-alumina
Li/.Si 04
n:O-60
n :0.68
2
1 log o'(o)T 0 or - ~ 1 log AT }
o-(olT -~Eo = 17± 2 kJ moL-1 80±2
26±2
tl
§
0,
2
4
6
8 10 0 2 Z. 1000IT (K '1)
0
FIG. 2 Arrhenius plots of o(o)T and AT.
4
6
8 10
16 2
D.P. Almond et al. /Hopping rates and carrier concentrations in ionic conductors
161 i
15 i
I
Na /~-alumine
T
Liz.Si04
"
78-*4
I
Ag71 z.A sO4
14 13
li Iog~(secP)1(119~
=15-+2kJrnO[-1
/
\
25±2
8
7
•
6
1000/T ( K4 ) FIG.
3
Arrhenius plots of ion hopping rates, Up. equation ( I ) may be used only semiquantitatively because of uncertainties in y, a and N : the mechanism of conduction is an interst~t~alcy one involving several Na+ ions and two types of l a t t i c e s i t e , the Beevers-Ross and mid-oxygen, are involved in the conduction. Further + d i f f i c u l t i e s arise with Li,SiO, and the Ag glass because the conduction mechanisms are not known and in the Ag+ glass, the structure is completely unknown. We estimate the net c a r r i e r concentration c' using the approximation: c' = c ( l - c ) N (8) and substituting into equation (I) wish assumed average values for a and y of 3 A and 0.33 respectively. The net a l k a l i ion c a r r i e r concentrations, c ' , are then calculated to be 22%~(Na B-alumina) 0.5% (Li,SiO,) and I . I % (Ag" glass). The composition of the Na B-alumina crystal was ~ 1.2 Na20.IIA~203 and the high conductivity of B-alumina has been attributed to the 0.2 excess soda content. Our estimate of the c a r r i e r concentration of ~22% is in f a i r agreement with t h i s . The room temperature conducti v i t y of the Ag÷ glass is comparable to that of Na B-alumina despite i t s lower c e r r i e r concent-
ratio~. However, in the Ag+ glass, the density of Ag- ions is about 4 times greater than the density of Na+ ions in B-alumina and therefore, the ionic c a r r i e r density difference between the two materials is only a factor of ~5. This difference is offset by the higher (~7 fold) hopping rate in the Ag+ glass at room temperature as shown in Fig. 3. This enhanced hopping rate can be a t t r i b u t e d , almost e n t i r e l y , to the large entropy of activation of the Ag+ glass. The estimated room temperature conductivity of Li,SiO, is some I0 decades less than that of the other two materials. This is largely due to the much reduced hopping rate in Li,Si04, Fig. 3, which in turn is a consequence of i t s much higher activation energy. Conductivity data for the fourth material the K/Ca n i t r a t e glass, are shown in Fig. 4a. The data f a l l into two groups, above and below the glass t r a n s i t i o n temperature, Tg, ~60°C. These data were analysed in the same way and i t is evident that the q u a l i t y of some of the f i t s , solid lines Fig. 4a, is not so good as that for the other three materials, Fig. I. The reasons for this are not understood. Nevertheless, reasonable estimates of o(o) and A were obtained from the low and high frequency l i m i t i n g regions
D.P. A l m o n d et al. / Hopping rates and carrier concentrations in ionic conductors
0 4Ca{NO3}2 0.6KNO3 353.3K~
Tg~ 3 ~ 0 6 3
: :~ ~
_ . w ~ 0'63
"T E
b -o', 11 o
&3
(a}
T~_
0
1
2 log f
3 IHz)
L
5
6
(b)
~
log o-(o)T or
328_t 17 kJ mo[ 4
cr(o)T
IogAT
i5 ,
;g
, 36_.5,
T~ 2.8
3-0 3.2 1000IT (K -1)
3.4
7 (c) 6
57_.1/., kJ mot -1
5
log 0 P4 (sec -1) 3 2 1
Tg !
0
i
2.8
i
i~
i
i
3.0 3.2 10001T (K ~ )
i
i
3-4
FIG. 4 (a, b, c) Conductivity data f o r K/Ca n i t r a t e glass and supercooled melt a) c o n d u c t i v i t y data ( r e f . 13) and f i t t e d curves, b) Arrhenius p l o t s of o(o) T and AT, c) Arrhenius p l o t of Up.
163
of these data. Results are shown plotted in Arrhenius format in Fig. 4b. In the low temperature region, below Ta, the activation energies of o(o) and A are Tn the r a t i o l : (l - n ) as with the other three materials, Fig. 2. Above Tg, however, this relationship does not hold. The activation energies are in the r a t i o of I:0.5 whereas a r a t i o I:0.37 might be expected since in this region n ~ 0.63. The hopping rates, uD, calculated using equation (6) and the values of ~(o) and A shown in Fig. 4b, are plotted in Arrhenius format as Fig. 4c. In the low temperature region, below Ta, the activation energy for ~n is very s i m i l a r ~ t o t h a t of o ( o ) , Fig. 4b. Ab6ve Tn, however, the a c t i v a t i o n energy of ~(o) is s i g n i f i c a n t l y higher than t h a t of Up. This is i n t e r p r e t e d as being due to a temperature dependent c a r r i e r concentration above TO. The temperature dependence of o ( o ) , equati6ns ( I ) , ( 2 ) , ( 3 ) , is in g e n e r a l , caused by the thermal a c t i v a t i o n of both the c a r r i e r c o n c e n t r a t i o n , c, and the hopping rate u D. Thus the a c t i v a t i o n energy f o r o ( o ) , Fig. ~b, above TQ is taken to be Ea + Ec whereas t h a t of Up is ~ Ea alone, i.e. Ea = 257 kJ mol "I and Ec = 71 kJ mol " I , In the presence of a t h e r m a l l y a c t i v a t e d c a r r i e r concentration i t was shown e a r l i e r t h a t o(o) and A should have a c t i v a t i o n energies in the r a t i o (E c + Ea) _~c(EC + ( I - n ) E ~ ) . _ Using the above values OT and Ea with = 0.63 t h i s r a t i o becomes 1:0.5 which i.s e s s e n t i a l l y the same as the observed r a t i u , Fig. 4b. The c a r r i e r concentration in the n i t r a t e glass was estimated in the same way as f o r the other three systems. Assuming that only K+ ions c o n t r i b u t e to the c o n d u c t i v i t y , the concentration of mobile K+ tons is ~ 0.1% below Tg. Using the new method of analysis proposed here we are able to u t i l i s e the dynamical c h a r a c t e r i s t i c s of a.c. c o n d u c t i v i t y to separate the c o n t r i b u t i o n s made by the m o b i l i t y and concentration of ions to d.c. c o n d u c t i v i t y . The method has been used successfully on four m a t e r i a l s of widely d i f f e r i n g s t r u c t u r e s and conductivities and each with a d i f f e r e n t mobile c a t i o n : Li +, Na+, K+ and Ag+: In the evaluation of ion hopping r a t e s , the c o n t r i b u t ion of the associated entropy term has become e v i d e n t . The q u a n t i f i c a t i o n of t h i s term raises i n t e r e s t i n g questions as to i t s physical o r i g i n in i o n i c conductors. The a b i l i t y to separate the c o n t r i b u t i o n s of c a r r i e r c o n c e n t r a t i o n , with i t s possible temperature dependence, and hopping rates providesa powerful new way of studying ionic transport. ACKNOWLEDGEMENT We thank Dr. M.D. Ingram f o r providing o r i g i n a l c o n d u c t i v i t y data f o r the Ag+ glass.
D.P. Almond e# al. /Hopping rates and carrier concentrations in ionic conductors
164
REFERENCES (I)
D . S . Nevanan, C. Frank, R.W. Matlack, S. Twining and V. Krishnan, Electrochim. Acta 22 (1977) 811.
(8)
I.M. Hodge, M.D. Ingram and A.R. West, J. Amer. Ceram. Soc., 59 (1976) 360 and unpublished results.
(2)
L.W. Barr and A.B. Lidiard, in "Physical Chemistry, An Advanced Treatise", Volume I0 Solid S t a t e , Ed. W. Jost, Academic Press, 1970, p. 151.
(9)
R.J. Grant, M.D. Ingram, L.D.S. Turner and C.A. Vincent. J. Phys. Chem., 82 (.1978) 2838.
(3)
D. Ravaine and J.L. Souquet, Chem. Glasses 18 (1977) 27.
(io)
S.J. Allen Jr, and J.P. Remeika, Phys. Rev. Letters 33 (1976) 1478.
(4)
R.A. Huggins, in "Diffusion in Solids, Recent Development", Ed. A.S. Nowick and J.J. Burton, Academic Press, 1975, p. 445.
(11)
P. McGeehin and A. Hooper, 12 (1977) I.
(12)
C. Wert and C. Zener, Phys. Rev., 76 (1949) 1169.
Phys.
5)
A.K. Jonscher,
6)
D . P . Almond, A.R. West and R.J. Grant, Solid State Comms. in press.
Nature 267 (1977) 673.
7)
R.J. Grant, M.D. Ingram and A.R. West, unpublished.
J. Mat. Sci.,
13) F.S. Howell, R.A. Bose, P.B. Macedo and C.T. Moynihan, 78 (1974) 639.
J. Phys. Chem.,
THE DETERMINATION OF HOPPING RATES AND CARRIER CONCENTRATIONS IN IONIC CONDUCTORS BY A NEW ANALYSIS OF ac CONDUCTIVITY D.P. ALMOND School of Materials Science, University of Bath, Claverton Down, Bath BA2 7,4 Y, UK
and G.K. DUNCAN, A.R. WEST Department of Chemistry, University of Aberdeen, Meston Walk, Old Aberdeen AB9 2UE, Scotland, UK
Received 7 October 1982 ABSTRACT The a.c. conductivity o(m) of ionic materials takes the form, o(m) = o(o) + Amn. The c a r r i e r hopping rate, Up, is obtained from the new expression o(o) = Aconn and the c a r r i e r concentration is estimated from o(o). The contribution of creation and migratio~ terms to the activation energy for conduction may be determined from the thermal activation of o(o) and ~D and the corresponding entropy terms quantified. D a t a have been analyzed for four widely d i f f e r e n t ibnic materials: single crystal Na B-alumina, p o l y c r y s t a l l i n e Li4SiO~, AgTl~AsO~ glass and Ca(NO3)2/KN03 glass and melt. For each, the c a r r i e r concentration and hopping rates have been obtained.
The conductivity of an ionic conductor is determined by both the concentration of the mobile ions and the rate at which they are able to hop from site to site in the material. Experimentally, the separation of these two terms has been found, in general, to be difficult. Methodswhich have been used with some success include: the Hall effect [ l ] , the doping of ionic crystals with aliovalent impurities [2] and the electrochemical measurement of a c t i v i t i e s in glasses [ 3 ] . None of these methods appear, however, to be readily applicable to a l l ionic conductors. In this paper we demonstrate that c a r r i e r concentrations and hopping rates can be separated by a new treatment of a.c. conductivity. The potential scope of the method is i l l u s t r a t e d by the analysis of four very d i f f e r e n t materials: a single crystal superionic conductor (Na B-alumina), a p o l y c r y s t a l l i n e i o n i c conductor ( L i . S i O , ) , a vitreous e l e c t r o l y t e (AgTI.AsO. glass) and a poorly conducting n i t r a t e glass (40 mol % Ca(NOa)2 - 60 mol % KN03) both above and below the glass transformation temperature. The d.c. c o n d u c t i v i t y of a c r y s t a l l i n e i o n i c conductor is determined by the expression obtained from the theory of random walk [ 4 ] : a(o) = Kmp = (Ne2aZ/kT) yc(l -C)mp
(l)
where y is a geometrical factor that may include a correlation factor, c is the concentration of mobile ions on N equivalent l a t t i c e sites per unit volume, a is the hopping distance, e is the electronic charge, k is Boltzmann's
constant, T is absolute temperature and Up is the hopping rate of the ions. The hopping rate is thermally activated and is given by: Up = moeXp(-AGa/kT) = moexp(ASa/k)exp(-Ea/kT) = meexp(-Ea/kT)
(2)
where &Ga, Ea and ASa are the free energy, activation energy and entropy of the hopping process, mQ is the vibrational frequency of ions in t h e l r l a t t i c e sites and me is an e f f e c t ive attempt frequency which includes the entropy term. The concentration c of mobile ions may or may not be temperature dependent. In the former case,
c
=
CoeXp(-AGc/kT)
=
CoeXp(ASc/k)exp(-Ec/kT)
= CeeXp(-Ec/kT)
(3)
where AGc, Ec and ASc are the free energy, activation energy and entropy for the creation of mobile carriers and ce is the effective i n f i n i t e temperature c a r r i e r concentration. Measurements of d.c. conductivity, over a range of temperatures, provide numerical values for the net activation energy and pre-exponential factor for the overall conduction process. The d i f f i c u l t i e s which often arise in c o r r e l a t -
0 167-2738/83/0000-0000/$ 03.00 © 1983 North-Holland
D.P. A l m o n d et al. / Hopping rates and carrier concentrations in ionic conductors
160
ing such values w i t h the t h e o r e t i c a l expressions, ( I ) , ( 2 ) , ( 3 ) , are : a) i t is not possible t o determine, unambiguously, whether or not the c a r r i e r c o n c e n t r a t i o n is t h e r m a l l y a c t i v a t e d , i . e . whether the measured a c t i v a t i o n energy of o ( o ) is (E a + Ec) or Ea alone. b) i t is not p o s s i b l e t o determine the magnitudes of me and c e since t h e y are present as a product in the p r e - e x p o n e n t i a l f a c t o r . Up to now the main reasons f o r using a . c . techniques to measure i o n i c c o n d u c t i v i t y have been t o avoid the n e c e s s i t y of non-blocking e l e c t r o d e s and t o f a c i l i t a t e the s e p a r a t i o n of g r a i n boundary and bulk phenomena in p o l y crystalline materials. I d e a l l y , a frequency independent c o n d u c t i v i t y , equal to the d.c. c o n d u c t i v i t y , should be obtained a t h i g h e r frequencies where e l e c t r o d e and grain boundary phenomena are i n s i g n i f i c a n t . In p r a c t i c e , such a frequency independent p l a t e a u is observed, often over several decades of frequency, but is l i m i t e d at higher frequencies by an a d d i t i o n a l bulk phenomenon. This takes the form of a power law increase in c o n d u c t i v i t y w i t h frequency. The a . c . c o n d u c t i v i t y a(m) is found t o v a r y w i t h angular frequency co as ~(co)
=
o(o)
+
A~n
(4)
where a(o) is the d.c. c o n d u c t i v i t y , A is a temperature dependent parameter and n is found to take values between 0 and I . Jonsche~ [ 5 ] has suggested t h a t t h i s power law is a ' u n i v e r s a l ' p r o p e r t y of m a t e r i a l s t h a t is r e l a t e d t o the dynamics of hopping conduction. R e c e n t l y , we have shown [ 6 ] t h a t t h e r e is a simple r e l a t i o n s h i p between a(o) and A. The c o n d u c t i v i t y o(co) can then be expressed as • o(co)
,
l-n
n
= K~p + ~CJp co
(5)
where K was defined in equation (1). By combining equations (4) and (5) the hopping rate cop is given by co = ( ~ ( o ) / A ) I / n (6) P This provides a p a r t i c u l a r l y sir.lple way of quantifying m . The t h e o r e t i c a l temperature dependences of p o(o) and A may be obtained by s u b s t i t u t i n g f o r K and cod from ( I ) , (2) and (3) in ( 5 ) . For m a t e r i a l s 'in which the c a r r i e r c o n c e n t r a t i o n is independent of temperature, o n l y mo is t h e r m a l l y a c t i v a t e d and the a c t i v a t i o n energies of a(o) and A are Ea and (I - n ) E a, r e s p e c t i v e l y . For m a t e r i a l s in which the c a r r l e r c o n c e n t r a t i o n is a l s o t h e r m a l l y a c t i v a t e d , K is t h e r m a l l y a c t i v a t e d and consequently, the a c t i v a t i o n energies of ~(o) and A become Ec + Ea and E + (I - n ) E a, respectively, c The above r e l a t i o n s h i p s have been used to analyse a.c. c o n d u c t i v i t y data from f o u r d i f f e r e n t i o n i c conductors. The f i r s t three : s i n g l e c r y s t a l Na B-alumina, p o l y c r y s t a l l i n e L i , S i 0 4 and AgTI,AsO, g l a s s , appear t o have temperature independent c a r r i e r c o n c e n t r a t i o n s
whereas the f o u r t h , K/Ca n i t r a t e g l a s s , appears to have a temperature dependent c a r r i e r c o n c e n t r a t i o n above the glass t r a n s f o r m a t i o n temperature, Tg. C o n d u c t i v i t y data at several temperatures f o r the f i r s t t h r e e m a t e r i a l s are shown in Fig. I . The s o l i d l i n e s through the data are f i t s to equation (4). The values of o(o) and A used f o r these f i t s are shown p l o t t e d in conventional Arrhenius format as Fig. 2. In each case, the a c t i v a t i o n energies of a(o) and A are in the r a t i o 1 : (I - n ) as p r e d i c t e d . The hopping rates up c a l c u l a t e d from the same data using equation (6) are shown in Arrhenius format as Fig. 3. The data were f i t t e d by the s t r a i g h t l i n e s shown using l i n e a r regression and values of the a c t i v a t i o n energies and e f f e c t i v e attempt frequencies were obtained from the slopes and i n f i n i t e temperature intercepts, respectively. The a c t i v a t i o n energies f o r each are w i t h i n experimental e r r o r the same as those found f o r a(o) in Fig. 2. This i n d i c a t e s t h a t the thermal a c t i v a t i o n s of these c o n d u c t i v i t i e s are caused by the thermal a c t i v a t i o n s of the hopping rates alone. The e f f e c t i v e attempt frequency me of B-alumina obtained from Fig. 3 is v e r y s i m i l a r to the value 2 x 1012 Hz associated with the v i b r a t i o n al frequency mo obtained d i r e c t l y from i n f r a red a b s o r p t i o n measurements [ I 0 ] . This shows t h a t the e n t r o p y term ASa in equation (2) is small in t h i s p a r t i c u l a r m a t e r i a l . Several authors [4, I I ] have iTnd,~ed suggested t h a t these e n t r o p y terms should be small in f a s t ion conductors such as B-alumina. For L i , S i O , and the Ag+ g l a s s , the e f f e c t ive attempt f r e q u e n c i e s , I0 zsHz Fig. 3, are c o n s i d e r a b l y higher than l i k e l y values of corresponding v i b r a t i o n a l f r e q u e n c i e s , ~1013Hz. This r a i s e s the question o f the magnitude o f the e n t r o p y of a c t i v a t i o n ASa in these, and o t h e r , i o n i c conductors. Some i n s i g h t may be obtained by using the formula f o r ~Sa which was d e r i v e d by Wert and Zener [12] in t h e i r t h e o r y of i n t e r s t i t i a l d i f f u s i o n in m e t a l s : AS BEa (7) a T m where Tm is the m e l t i n g temperature and B is a dimensionless number, ~ 0 . 4 , which is r e l a t e d to the temperature dependence of the shear modulus of the m a t e r i a l . Estimates of the magnitude of the term exp(ASa/k) in me, obtained using equation (7) are 40, I I and 1.3 f o r the Ag+ g l a s s , Li4SiO, and B-alumina respectively. These e n t r o p y values match, to w i t h i n a f a c t o r of f i v e or b e t t e r , those determined using equation (2) and Fig. 3 and assuming wo ~ 1013 Hz f o r L i , S i 0 4 and the Ag+ glass. In view of the e r r o r s in e s t i m a t i n g me, 8 and mo ( e s p e c i a l l y f o r Li4Si04 and the Ag+ glass) t h i s is a l l t h a t can be expected at t h i s stage. A f t e r the hopping rates have been determined, i t is p o s s i b l e to estimate the carrier concentrations. For B-alumina, however
161
D.P. A l m o n d et al. / Hopping rates and carrier concentrations in ionic conductors
m
Li&SiO4
n : 0.68
Z
AgTILAs~
n :0.68
f184K
E u
..(Z 0
~ 4 8 0 K
b
2
3
A
5
2
6
3
1 4 0 K ~
~
4 5 6 log f(Hz)
2
3
4
5
6
7
FIG. 1 Conductivity data for single crystal Na 6-alumina (ref. 6,7), polycrystalline Li,SiO, (ref. 8) and Ag~l,AsO4 glass (ref. 9) at a selection of temperatures. Solid curves are f i t s of equation (4) to the experimental data. Errors in n are estimated as ±0.03. The experimental data for 6-alumina and AgTI,AsO, at the highest temperatures and low frequencies show a conductivity dispersion associated with electrode polarisation effects.
4t
Na /B-alumina
Li/.Si 04
n:O-60
n :0.68
2
1 log o'(o)T 0 or - ~ 1 log AT }
o-(olT -~Eo = 17± 2 kJ moL-1 80±2
26±2
tl
§
0,
2
4
6
8 10 0 2 Z. 1000IT (K '1)
0
FIG. 2 Arrhenius plots of o(o)T and AT.
4
6
8 10
16 2
D.P. Almond et al. /Hopping rates and carrier concentrations in ionic conductors
161 i
15 i
I
Na /~-alumine
T
Liz.Si04
"
78-*4
I
Ag71 z.A sO4
14 13
li Iog~(secP)1(119~
=15-+2kJrnO[-1
/
\
25±2
8
7
•
6
1000/T ( K4 ) FIG.
3
Arrhenius plots of ion hopping rates, Up. equation ( I ) may be used only semiquantitatively because of uncertainties in y, a and N : the mechanism of conduction is an interst~t~alcy one involving several Na+ ions and two types of l a t t i c e s i t e , the Beevers-Ross and mid-oxygen, are involved in the conduction. Further + d i f f i c u l t i e s arise with Li,SiO, and the Ag glass because the conduction mechanisms are not known and in the Ag+ glass, the structure is completely unknown. We estimate the net c a r r i e r concentration c' using the approximation: c' = c ( l - c ) N (8) and substituting into equation (I) wish assumed average values for a and y of 3 A and 0.33 respectively. The net a l k a l i ion c a r r i e r concentrations, c ' , are then calculated to be 22%~(Na B-alumina) 0.5% (Li,SiO,) and I . I % (Ag" glass). The composition of the Na B-alumina crystal was ~ 1.2 Na20.IIA~203 and the high conductivity of B-alumina has been attributed to the 0.2 excess soda content. Our estimate of the c a r r i e r concentration of ~22% is in f a i r agreement with t h i s . The room temperature conducti v i t y of the Ag÷ glass is comparable to that of Na B-alumina despite i t s lower c e r r i e r concent-
ratio~. However, in the Ag+ glass, the density of Ag- ions is about 4 times greater than the density of Na+ ions in B-alumina and therefore, the ionic c a r r i e r density difference between the two materials is only a factor of ~5. This difference is offset by the higher (~7 fold) hopping rate in the Ag+ glass at room temperature as shown in Fig. 3. This enhanced hopping rate can be a t t r i b u t e d , almost e n t i r e l y , to the large entropy of activation of the Ag+ glass. The estimated room temperature conductivity of Li,SiO, is some I0 decades less than that of the other two materials. This is largely due to the much reduced hopping rate in Li,Si04, Fig. 3, which in turn is a consequence of i t s much higher activation energy. Conductivity data for the fourth material the K/Ca n i t r a t e glass, are shown in Fig. 4a. The data f a l l into two groups, above and below the glass t r a n s i t i o n temperature, Tg, ~60°C. These data were analysed in the same way and i t is evident that the q u a l i t y of some of the f i t s , solid lines Fig. 4a, is not so good as that for the other three materials, Fig. I. The reasons for this are not understood. Nevertheless, reasonable estimates of o(o) and A were obtained from the low and high frequency l i m i t i n g regions
D.P. A l m o n d et al. / Hopping rates and carrier concentrations in ionic conductors
0 4Ca{NO3}2 0.6KNO3 353.3K~
Tg~ 3 ~ 0 6 3
: :~ ~
_ . w ~ 0'63
"T E
b -o', 11 o
&3
(a}
T~_
0
1
2 log f
3 IHz)
L
5
6
(b)
~
log o-(o)T or
328_t 17 kJ mo[ 4
cr(o)T
IogAT
i5 ,
;g
, 36_.5,
T~ 2.8
3-0 3.2 1000IT (K -1)
3.4
7 (c) 6
57_.1/., kJ mot -1
5
log 0 P4 (sec -1) 3 2 1
Tg !
0
i
2.8
i
i~
i
i
3.0 3.2 10001T (K ~ )
i
i
3-4
FIG. 4 (a, b, c) Conductivity data f o r K/Ca n i t r a t e glass and supercooled melt a) c o n d u c t i v i t y data ( r e f . 13) and f i t t e d curves, b) Arrhenius p l o t s of o(o) T and AT, c) Arrhenius p l o t of Up.
163
of these data. Results are shown plotted in Arrhenius format in Fig. 4b. In the low temperature region, below Ta, the activation energies of o(o) and A are Tn the r a t i o l : (l - n ) as with the other three materials, Fig. 2. Above Tg, however, this relationship does not hold. The activation energies are in the r a t i o of I:0.5 whereas a r a t i o I:0.37 might be expected since in this region n ~ 0.63. The hopping rates, uD, calculated using equation (6) and the values of ~(o) and A shown in Fig. 4b, are plotted in Arrhenius format as Fig. 4c. In the low temperature region, below Ta, the activation energy for ~n is very s i m i l a r ~ t o t h a t of o ( o ) , Fig. 4b. Ab6ve Tn, however, the a c t i v a t i o n energy of ~(o) is s i g n i f i c a n t l y higher than t h a t of Up. This is i n t e r p r e t e d as being due to a temperature dependent c a r r i e r concentration above TO. The temperature dependence of o ( o ) , equati6ns ( I ) , ( 2 ) , ( 3 ) , is in g e n e r a l , caused by the thermal a c t i v a t i o n of both the c a r r i e r c o n c e n t r a t i o n , c, and the hopping rate u D. Thus the a c t i v a t i o n energy f o r o ( o ) , Fig. ~b, above TQ is taken to be Ea + Ec whereas t h a t of Up is ~ Ea alone, i.e. Ea = 257 kJ mol "I and Ec = 71 kJ mol " I , In the presence of a t h e r m a l l y a c t i v a t e d c a r r i e r concentration i t was shown e a r l i e r t h a t o(o) and A should have a c t i v a t i o n energies in the r a t i o (E c + Ea) _~c(EC + ( I - n ) E ~ ) . _ Using the above values OT and Ea with = 0.63 t h i s r a t i o becomes 1:0.5 which i.s e s s e n t i a l l y the same as the observed r a t i u , Fig. 4b. The c a r r i e r concentration in the n i t r a t e glass was estimated in the same way as f o r the other three systems. Assuming that only K+ ions c o n t r i b u t e to the c o n d u c t i v i t y , the concentration of mobile K+ tons is ~ 0.1% below Tg. Using the new method of analysis proposed here we are able to u t i l i s e the dynamical c h a r a c t e r i s t i c s of a.c. c o n d u c t i v i t y to separate the c o n t r i b u t i o n s made by the m o b i l i t y and concentration of ions to d.c. c o n d u c t i v i t y . The method has been used successfully on four m a t e r i a l s of widely d i f f e r i n g s t r u c t u r e s and conductivities and each with a d i f f e r e n t mobile c a t i o n : Li +, Na+, K+ and Ag+: In the evaluation of ion hopping r a t e s , the c o n t r i b u t ion of the associated entropy term has become e v i d e n t . The q u a n t i f i c a t i o n of t h i s term raises i n t e r e s t i n g questions as to i t s physical o r i g i n in i o n i c conductors. The a b i l i t y to separate the c o n t r i b u t i o n s of c a r r i e r c o n c e n t r a t i o n , with i t s possible temperature dependence, and hopping rates providesa powerful new way of studying ionic transport. ACKNOWLEDGEMENT We thank Dr. M.D. Ingram f o r providing o r i g i n a l c o n d u c t i v i t y data f o r the Ag+ glass.
D.P. Almond e# al. /Hopping rates and carrier concentrations in ionic conductors
164
REFERENCES (I)
D . S . Nevanan, C. Frank, R.W. Matlack, S. Twining and V. Krishnan, Electrochim. Acta 22 (1977) 811.
(8)
I.M. Hodge, M.D. Ingram and A.R. West, J. Amer. Ceram. Soc., 59 (1976) 360 and unpublished results.
(2)
L.W. Barr and A.B. Lidiard, in "Physical Chemistry, An Advanced Treatise", Volume I0 Solid S t a t e , Ed. W. Jost, Academic Press, 1970, p. 151.
(9)
R.J. Grant, M.D. Ingram, L.D.S. Turner and C.A. Vincent. J. Phys. Chem., 82 (.1978) 2838.
(3)
D. Ravaine and J.L. Souquet, Chem. Glasses 18 (1977) 27.
(io)
S.J. Allen Jr, and J.P. Remeika, Phys. Rev. Letters 33 (1976) 1478.
(4)
R.A. Huggins, in "Diffusion in Solids, Recent Development", Ed. A.S. Nowick and J.J. Burton, Academic Press, 1975, p. 445.
(11)
P. McGeehin and A. Hooper, 12 (1977) I.
(12)
C. Wert and C. Zener, Phys. Rev., 76 (1949) 1169.
Phys.
5)
A.K. Jonscher,
6)
D . P . Almond, A.R. West and R.J. Grant, Solid State Comms. in press.
Nature 267 (1977) 673.
7)
R.J. Grant, M.D. Ingram and A.R. West, unpublished.
J. Mat. Sci.,
13) F.S. Howell, R.A. Bose, P.B. Macedo and C.T. Moynihan, 78 (1974) 639.
J. Phys. Chem.,