The AC Conductivity of Polycrystalline LISICON, Li2+ ...

662

J. Electrochem. Soc.: S O L I D - S T A T E S C I E N C E A N D T E C H N O L O G Y

March 1983

in these regions. The elongated and irregular island shapes argue against a liquid-like coalescence due to surface migration of molecules as discussed previously for GaAs (1).

A n y discussion of this paper will appear i n a Discussion Section to be published in the D e c e m b e r 1983 JOURNAL. All discussions for the December 1983 Discussion Section should be submitted b y Aug. 1, 1983.

Summary

Publication costs of this article were assisted by the Colorado State University.

The TEM micrographs show that the anodic oxidation of I n P is initiated b y n u c l e a t i o n and island growth. I n contrast to GaAs, the islands on I n P are smooth without rough edges. However the islands on I n P are a p p r o x i m a t e l y the same height and area as those observed on GaAs. The same basic island shape of the I n P islands was observed for all growth conditions tested. Some variation from r u n to r u n was observed but this was not consistent and could not be a t t r i b u t e d to the electrolyte, electrolyte pH, or i n t e n sity of illumination. It is believed that the observed variations were due to the quality of the substrate polish and not the changes in the growth conditions. On m a n y of the samples, the islands were in rows separated by ~0.2-0.6~. These are thought to be due to surface steps which cause a heterogeneous nucleation. The coalescence of the islands appeared to be caused by rapid growth in the neck region between the two touching islands similar to that observed for GaAs.

Acknowledgments We wish to t h a n k K e n t Geib for his help with sample p r e p a r a t i o n and Dr. Morita for his assistance with the electron microscope. This work was supported by ARO. Manuscript submitted J u n e 15, 1982.

REFERENCES 1. W. H. Makky, F. Cabrera, K. M. Geib, a n d C. W. Wilmsen, J. Vac. Sci Technol., 20, 417 (1982). 2. D. L. Lile and D. A. Collins, Appl. Phys. Lett., 28, 554 (1976). 3. C. W. Wilmsen and R. W. Kee, J. Vac. Sci. TechnoL, 14, 953 (1977). 4. K. P. P a n d e and G. G. Roberts, ibid., 16, 1470 (1979). 5. K. M. Geib and C. W. Wilmsen, ibid., 17, 952 (1980). 6. D. H. L a u g h l i n and C. W. Wilmsen, Appl. Phys. Lett., 37, 915 (1980). 7. D. A. Baglee, D. H. Laughlin, C. W. Wilmsen, and D. K. Ferry, "The Physics of MOS Insulators," G. Luscovsky, S. T. Pantelides, and F. L. Galeener, Editors, Pergamon, New York (1980). 8. A. A. S t u d n a and G. J. Gualitieri, Appl. Phys. Lett., 39, 965 (1981). 9. A. Yamamoto and C. Uemura, Electron. Lett., 18, 63 (1982). 10. J. F. Wager, W. H. Makky, and C. W. Wilmsen, Thin Solid Films, 95, 343 (1982). 11. J. F. Wager and C. W. Wilmsen, J. Appl. Phys., To be published. 12. W. H. Makky, Unpublished.

The A-C Conductivity of Polycrystalline LISICON, Li2+ Zn, xGe04, and a Model for Intergranular Constriction Resistances P. G. Bruce~ and A. R. West University of Aberdeen, Ddpartment of Chemistry, Meston Walk, Old Aberdeen AB9 2UE, Scotland ABSTRACT The a-c conductivity ofa polycrystalline, Li + ion conducting solid electrolyte, LISICON, Li2+2xZnl-xGeO4,has been measured over the frequency range 10-3-107Hz and attention focused on the methods of data analysis. The data show intragranular (bulk) and grain boundary effects in which, for a given sample, both effects are characterized by a similar activation energy; a constriction resistance model for the latter is proposed based on air gaps existing in polycrystalline sinters of less than theoretical density. The data only crudely fit a Debye-like equivalent circuit, composed of frequency independent resistors and capacitors, but fit well an equivalent circuit that contains Jonscher elements, i.e., frequency dependent admittances, in addition to normal R and C elements. Two such Jonscher elements are needed, one for the grain boundary and one for the intragranular contribution. With this circuit, departures from ideality in, e.g., the cpmplex impedance, admittance, and electric modulus planes are accounted for. The a-c conductivity of solid electrolytes is generally analyzed in terms of equivalent circuits composed of f r e q u e n c y i n d e p e n d e n t resistors and capacitors. Very often, however, the a-c response does not correspon~l to that predicted by simple, ideal circuits; for instance, semicircles in the complex impedance Z* plane are often broadened and distorted into a s y m m e t r i c arcs. Various attempts have been made to deal with this problem: empirical functions based on the Cole-Cole (1-3) or Cole-Davidson (4) expressions have been used to fit the e x p e r i m e n t a l results; a d i s t r i b u t i o n of hopping processes leading to a distribution of relaxation times has also b e e n postulated (5). Such n o n ideal response is only observed in electrolytes with a high concentration of charge carriers. These are just the systems i n which cooperative interactions are m a n ifest. ~Present address: Department of Inorganic Chemistry, University of Oxford, Oxford OXl 3QR, England.

A recent approach which is g a i n i n g increasing acceptance is based on 5onscher's "universal law of dielectric response" (6, 7), which, u n l i k e the above approaches, can be derived from theories of cooperative migration (8-11). When this theory is applied to a-c conductivity, frequency d e p e n d e n t elements must be introduced into the e q u i v a l e n t circuit. This approach, utilizing one frequency d e p e n d e n t element, has been used b y Jonscher (7, 12), Huggins and others (13-15) to model the a-c response of polycrystalIine materials. We have s h o w n (16) that the b u l k a-c response of single crystal E-alumina can be treated in the same way, and that the observed large departures from Debyelike behavior can be fitted to a relatively simple e q u i v alent circuit. I n this paper, we discuss the application of Jonscher circuits to a polycrystaUine Li + ion conducting electrolyte LISICON, and show how both the g r a i n b o u n d a r y a n d b u l k regions m a y be satisfactorily described, e x t e n d i n g previous approaches b y utilizing

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V o l I30, No. 3

A-C CONDUCTIVITY

663

two f r e q u e n c y d e p e n d e n t elements. W e also show h o w the d e p a r t u r e s f r o m D e b y e - l i k e response are m a n i fested in all four complex formalisms, t h e a d m i t t a n c e (Y*), i m p e d a n c e ( Z * ) , p e r m i t t i v i t y (,*), a n d m o d ulus (M*), which a r e i n t e r r e l a t e d (17) Y* -- C Z * ) - I = j~,Co(M*)-1 = j~Co,*

[1]

w h e r e ~ is t h e a n g u l a r frequency, 2~f, and Co is t h e v a c u u m capacitance of the c o n d u c t i v i t y ceiL In a d d i t i o n to the a b o v e - m e n t i o n e d r e p r e s e n t a t i o n of g r a i n b o u n d a r y response, the origin of such g r a i n b o u n d a r i e s in L I S I C O N is discussed a n d a m o d e l for t h e m presented.

Experimental The synthesis of l i t h i u m zinc g e r m a n a t e phases and t h e i r analysis b y x - r a y diffraction has been discussed p r e v i o u s l y (18). Pellets for c o n d u c t i v i t y m e a s u r e m e n t s w e r e p r e p a r e d b y cold-pressing p o w d e r s of the -vn-type solid solutions at 45,000 psi, followed b y sintering at t e m p e r a t u r e s b e t w e e n 1000 ~ and ll00~ for 1-2 hr. A few pellets w e r e c o v e r e d with p o w d e r of t h e same composition to reduce lithia loss a n d sintered at 1200~ for 5-10 hr. P e l l e t densities w e r e t y p i c a l l y 75-80% of the t h e o r e t i c a l values. In some cases, u n s i n t e r e d p e l lets w e r e p a r t i a l l y m e l t e d b y h e a t i n g in closed P t e n velopes for 10-15 rain at t e m p e r a t u r e s above the solidus. No significant distortion of the pellets occurred. This gave densities ~-, 85% of the theoretical value. Gold electrodes were a p p l i e d to opposite pellet faces, g e n e r a l l y b y v a p o r deposition. Care w a s t a k e n to m a s k the edges of the p e l l e t to p r e v e n t p e n e t r a t i o n of t h e gold onto t h e sides of the pellet. In o t h e r cases, gold electrodes w e r e a p p l i e d w i t h E n g l e h a r d paste, h e a t e d at 150~176 in successive steps to r e m o v e the organic solvent, decompose the o r g a n o - g o l d complex, and h a r d e n t h e metallic residue. In some instances, before the electrodes w e r e applied, the p e l l e t surfaces w e r e polished w i t h successively finer grades of silicon carb i d e paper, using e t h a n o l as lubricant, and finally w i t h a r o t a r y polisher using 1 ~m d i a m o n d paste. The i n t e r n a l m i c r o s t r u c t u r e and composition of some pellets w e r e e x a m i n e d b y scanning electron microscopy combined w i t h m i c r o p r o b e analysis. The conductivity jig took the f o r m of a stainless steel G - c l a m p , Fig. 1. The cell was held b e t w e e n high p u r i t y silica glass disks and p r e s s u r e m a i n t a i n e d b y a spring l o a d e d p l u n g e r ( e v a p o r a t e d e l e c t r o d e s ) . Connection was m a d e to the m e a s u r i n g a p p a r a t u s b y i n serting gold foil strips b e t w e e n each silica disk and celt face. Each gold foil s t r i p was w r a p p e d a r o u n d a p l a t i n u m w i r e m o u n t e d w i t h i n a stainless steel cylinder, f r o m w h i c h it was i n s u l a t e d b y a concentric silica glass tube. I n t e r n a l surfaces of the jig h e a d w e r e lined w i t h an insulating m a t e r i a l ( p y r o p h y l l i t e ) . The t e m p e r a t u r e was m e a s u r e d w i t h a C h r o m e l / A l u m e l thermocouple, the tip of which was m a i n t a i n e d in close p r o x i m i t y to t h e cell for m a x i m u m accuracy. The jig was m o u n t e d in a silica glass tube, thus p e r m i t ting m e a s u r e m e n t s to be m a d e in v a c u u m or different atmospheres. The e n t i r e a s s e m b l y was c l a m p e d cent r a l l y inside a n o n i n d u c t i v e l y wound, h o r i z o n t a l tube furnace a t t a c h e d to a E u r o t h e r m P.D.I. controller. T e m p e r a t u r e s w e r e controlled and m e a s u r e d to _2~ Two t e r m i n a l a - c m e a s u r e m e n t s w e r e conducted o v e r the f r e q u e n c y r a n g e 10-3-107 Hz, using a c o m b i nation of b r i d g e and a u t o m a t e d phase sensitive d e t e c tion techniques. F o r the range 100 Hz-70 kHz, a W a y n e K e r r B224 b r i d g e was used with a B r o o k d e a l 9472 signal source and 9654 null detector. The signal g e n e r a t o r o u t p u t was m a i n t a i n e d at 25 m V rms throughout; h o w ever, the voltage o u t p u t f r o m the b r i d g e d e p e n d e d on the p a r t i c u l a r a d m i t t a n c e r a n g e selected. A n accuracy, in t h e m e a s u r e d a d m i t t a n c e values, of b e t t e r t h a n 3% was t y p i c a l l y obtained. F o r the range 70 kHz-7 MHz, a B602 rf b r i d g e and c o m b i n e d s o u r c e / d e t e c t o r unit, SR268L, w e r e used a l -

Fig. 1. A-C conductance jig though at t h e e x t r e m e s of this f r e q u e n c y range, the e x p e r i m e n t a l data points showed some scatter. The a p p l i e d v o l t a g e was 25 mV rms. F o r the f r e q u e n c y r a n g e 10-3-10 ~ Hz, a S c h l u m b e r g e r S o l a r t r o n F r e quency Response A n a l y z e r (F.R.A.) Model 1170 was used. N o r m a l l y 500 mV rms was a p p l i e d across a s t a n d a r d resistance and the cell, which w e r e connected in series; only a p r o p o r t i o n of this a p p l i e d voltage a p p e a r e d across the cell therefore, a n d this p r o p o r t i o n v a r i e d w i t h frequency, since the cell i m p e d a n c e was f r e q u e n c y dependent. Generally, the m a t c h i n g and continuity of results f r o m the F.R.A. and the two bridges was good.

Results Impedance data--general comments.--Three t y p i c a l sets of a-c data, in the f o r m of c o m p l e x i m p e d a n c e p l a n e plots, are given in Fig. 2 for A u / L I S I C O N / A u cells. T h e L I S I C O N solid e l e c t r o l y t e composition was Li2+2xZnl-zGeO4: x = 0.55; similar b e h a v i o r was o b s e r v e d for o t h e r x values. The d a t a s e p a r a t e clearly into a high f r e q u e n c y region, which contains two s e m i circles, and a low f r e q u e n c y region, w h i c h contains a s p i k e that was sometimes l i n e a r and sometimes curved. The spike was associated w i t h t h e e l e c t r o d e - e l e c t r o lyte interface since it was sensitive to polishing of the pellet surface p r i o r to application of the electrodes. The h i g h f r e q u e n c y region was assigned to the a - c response of the electrolyte itself, since it was i n d e p e n d e n t of the m a g n i t u d e of the a p p l i e d electric field, at least up to 9.0V rms. W i t h i n the high f r e q u e n c y region, Fig. 2, two slightly distorted semicircles are present, b u t t h e i r r e l a t i v e sizes are different in the t h r e e cases. The m a g n i t u d e of the low f r e q u e n c y semicircle v a r i e d c o n s i d e r a b l y f r o m cell to cell and at the same t e m p e r a t u r e . In some cases it was the l a r g e r of the two (c) a n d in others it was the s m a l l e r (a, b ) . This indicates that the two semicircles had a different origin. The high f r e q u e n c y semicircle, w h i c h passes t h r o u g h the origin, was associated with t h e i n t r a g r a n u l a r or b u l k response of the solid electrolyte. The intercept, Rg, on the Z' axis gives the resistance of the grains.

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J. Electrochem. Soc.: S O L I D - S T A T E S C I E N C E AND T E C H N O L O G Y

664

20

March I983

Toc

125Hz ~: 1 02

400

I

300

200

150

100

50

I

~

I

l

I

s xl0 -4 10

101 9 "~

"'.. 7J~Hz 10

I I [

2/

20 Rg g~ gb30 ;z' / 5h. ~

I

{ia )i

10o

\

o-T

/~0

I0-4

\\

ohrnlcr~ 1K j -1 10 12 SmH;?

50 3"/

~c~2 100KHz

30

9

9

3~Hz

(b) I

i

20

I

I

&O

I

i

Rg 50

I

I

i

l o -3

i

80 Rg + Rgb

120

140

16o

Z ' / ~ . x 10-z"

0y

120

Z"/o o D_• -4 /+ 0

0

S/

1 ~K~z 150KHz

20

/

Rg

100

1L0

1(~:'

I 15

20

2.5

30

1O00 K - I T

Fig. 3. Arrhenius plot of the intracrystalline conductivity (1/Rg)-Q and intercrystalline conductivity ( 1 / R g b ) - - I of an LiSICObl solid solution.

~

Rg + Rg b 250 Z' / ~- x 10-4

300

340

380

Fig. 2. Typical complex impedance plots for three Au/LISICON/ Au cells, at 59~ a) cell constant, k ---- 0.55 cm-1; b) cell constant, k ~ 0.47 cm-1; c) cell constant, k ---- 0.64 cm -1.

For a given composition, x, and temperature, the v a l u e of Rg was the same, w i t h i n a factor of 2-3, in all samples studied [large changes in R~ occurred on prolonged aging of samples, however (24) : see later]. Using the relation ~RC = 1, which holds at the m a x i m u m of an ideal semicircle, the capacitance of the high frequency semicircle was evaluated as ~ 1 - 2 pf cm - I . This corresponds with that expected from the dielectric polarization of the crystals, assuming a typical b u l k dielectric c o n s t a n t of ~5-25. The second, low frequency semicircle is a t t r i b u t e d to a grain b o u n d a r y impedance in series with the i n t r a g r a n u l a r impedance. The value of its capacitance, C~D, calculated from the m a x i m u m of the second semicircle is in the range 10-400 pf cm -1 and represents charge polarization at the grain boundaries. These Cgb values are relatively small compared with grain b o u n d a r y capacitances observed in solid electrolytes, such as Na ~ - a l u m i n a [800 pf cm -1 (Ref. 19) 21, nf cm - i (Ref. 20)], and doped CeO2 [50 n f cm -1 (Ref. 21)]. The m a g n i t u d e of the grain b o u n d a r y resistance, Rgb, varied from sample to sample at the same composition and was u s u a l l y i n the range 0.2 Rg < Rgb ~ 8Rg. Although the m a g n i t u d e s of Rg and Rgb were n o r m a l l y different, an i m p o r t a n t result is that, for a given sample, they appeared always to have the same activation energy, in contrast to the results of Bayard (22). Typical results are shown in Fig. 3 for a pellet in which both Rg and Rgb could be d e t e r m i n e d over a range of temperatures. The two sets of data fall on straight lines that are parallel w i t h i n e x p e r i m e n t a l error. Similarly, parallel sets of data were obtained: (i) for a v a r i e t y of compositions, x; (it) for samples that had u n d e r g o n e a phase t r a n s i t i o n to give one of various ~v-derivative phases; (iii) for samples that had aged, i.e., had u n d e r g o n e a m a r k e d decrease in conductivity on a n n e a l i n g (24).

The s i m i l a r i t y i n the activation energies of R~ a n d Rgb indicates that charge migration in the i n t r a c r y s talline and grain b o u n d a r y regions occurred b y the same process. I n particular, the grain b o u n d a r y effect is believed to be associated with a constriction of c u r r e n t pathways at the necks b e t w e e n crystals. A model for this is presented in the third section. Various alternative explanations for th~ occurrence of more than one semicircle were considered but were rejected. These alternative explanations, with reasons for their rejection are as follows: (i) The presence of a surface layer on the electrolyte at the interface with the electrode, arising from, e.g., surface attack (23) or electrochemical reactions. This layer would have a different structure and composition to the bulk, and probably, therefore, a different activation energy for conduction, in contrast to the results. (it) If the crystal conductivity was m a r k e d l y anisotropic, the presence of crystals i n different orientations could possibly give rise to b r o a d e n i n g of, or resolution into, two complex impedance semicircles. Again, however, it is likely that the two associated resistances would have different activation energies. (iii) Compositional v a r i ations b e t w e e n or within the crystallites (i.e., a variation in x) could give rise to the appearance of more t h a n one complex impedance semicircle. However, the activation energy for conduction in these LISICON solid solutions is very composition d e p e n d e n t (24), and a n y additional semicircles arising from compositional inhomogeneities should have different activation energy from the bulk. (iv) Dielectric r e l a x a t i o n m a y occur due to the formation of dipoles between, e.g., substitutional and interstitial Li + ions. If this was the cause of the appearance of two semicircles, it can be shown that the dielectric r e l a x a t i o n time, which should be a f u n d a m e n t a l p r o p e r t y for each composition, in fact varied from sample to sample. This possibility is discounted, therefore. Details of the analysis and a n e q u i v a l e n t circuit which m a y be used to represent a solid electrolyte that contains both free ions and dipoles, is given in the Appendix.

The intracrystalline (bulk) a-c response.--In the above analysis, it was assumed that the b u l k properties could be approximated b y a parallel combination

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VoL 130, No. 3

A-C CONDUCTIVITY

of a resistance a n d a capacitance. In fact, departures from this ideal b e h a v i o r occurred, as shown in Fig. 4a, where data are presented as spectroscopic plots of M" and Z" against frequency. I n the ideal case, the M", Z" peaks should be similar-shaped, Debye peaks (Y2a = 1.14 decades) with their m a x i m a occurring at the same frequency (17). In fact, the peak m a x i m a occur at different frequencies i n Fig. 4a, and the M" peak is noticeably broadened on the high frequency side. The Z" peak shows a spur on the low f r e q u e n c y side which is due to the onset of polarization associated with the grain b o u n d a r y effect. I n order to analyze f u r t h e r these n o n i d e a l data, use is made of a n e q u i v a l e n t circuit that contaifis a frequency d e p e n d e n t admittance, Y* (~), in parallel with the n o r m a l R and C elements, Fig. 5b. This circuit was also used to analyze data for single crystal beta a l u m i n a (16). The method of data analysis is as follows: First, the impedance data are replotted in the admittance formalism, on logarithmic scales. This is done i n Fig. 4b for one data set. The Y' plot shows two steps, or f r e q u e n c y i n d e p e n d e n t plateaus, one below 300 Hz and one b e t w e e n --~ 104 and 105 Hz, and a high f r e q u e n c y dispersion, above -~ 105 Hz. The low f r e q u e n c y plateau represents the overall electrolyte conductivity a n d after correction for the cell constant, is given by (Rg 4- Rgb) -1. The high frequency plateau represents the bulk, i n t r a c r y s t a l l i n e conductivity, ( R g ) - L The Y" plot shows a n a p p a r e n t plateau bet w e e n l0 s a n d 104 Hz, with a steep r a m p at higher frequencies and a n o t h e r dispersion at lower frequencies. At this stage we are concerned with the b u l k response only, which is given b y the data above , - 104 Hz.

70

(a)

30

//

Z"/r,. x 1 ~4 40

',X,

Y* : Rg -1 + j~C| Jr (A~ n -}- jB~ n)

[2]

The term in brackets is Y* (~), where A, B, and n are constants, B : A t a n ( n n / 2 ) and ~ ---- 2nf: t h i s - d e scribes the n o n i d e a l features of the response. Therefore Y' : Rg-1 + Awn,

Y" = B,~n + ~C|

[3]

since Y* = Y' + jY"

[4]

The equation for Y' contains two terms; Rg -1 is the value of the b u l k conductivity and gives rise to the plateau in Y' b e t w e e n ,~ 10~-105 Hz; A~ n represents the conductance ramp above -,-10 5 Hz. The equation for Y" also contains two terms; o~C~ gives the limiting slope of u n i t y observed above ,~ 104 Hz. The t e r m B~n should give a low frequency slope o] n, b u t below ,~ 10~ Hz, grain b o u n d a r y p h e n o m e n a influence the Y" data a n d the region c o r r e s p o n d i n g to B~ n is not clearly distinguished. Values of the various p a r a m e ters, extracted from Fig. 4b by curve fitting, are given in Table I. A test of the q u a l i t y with which Eq. [2]

Rg -,NvWV~

Ca) Rg ~AAAA/VV~

M"x103 15

#

30

10

20

5

10 I

I

103

*

104

10"410 -5 (b) y', y"/ 5z-I ~

The b u l k a d m i t t a n c e datal above ~ 104 Hz, can be represented by circuit (b), Fig. 5 whose complex a d m i t t a n c e is

25

50

102

665

105 f/Hz

I

106

I 10 7

/ Y "

y

,

13 10

I 104

(b) Rg

io 6

Rgb ANV~A

10-7 108 I 102

I 105 flHz

I 106

II cob

I 107

Fig. 4. (a) Z"-Iog I and M"-log f plots for an LISICON solid solution, x = 0.55, cell d, k = 0.55 cm -1. The solid line represents an ideal Debye peak. Temperature: 39~ (b) Log Y' and Ioq Y"-Iog f plots of the data in (a). So!id circles are experimental points. Solid lines are calculated admittances.

(c) Fig. 5. Traditional (a) and new (b) equivalent circuits for the intracrystalline a-c response of a polycrystalline electrolyte. (c) Equivalent circuit for the entire a-c response of polycrystalline

LISICON.

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666

J. E I e c t r o c h e m . S o c . : S O L I D - S T A T E S C I E N C E A N D T E C H N O L O G Y

Table I. Equivalent circuit parameters extracted from Fig. 4b Bulk region R-i A B n C

3.9 x 4.3 x 5.9 x 0.60 1.3 x

k

0 . 5 5 c m -x

Grain boundary region

10-7 ohm-~ cm-1 10-n ohm-i cm-trad -a I0-~1ohm-i cm-~ rad -~ 10-i-~F cm-1

1.2 x 10-~ ohm-i em-~ 8.7 X 10-it o h m -~ crn-~ rad-. 2.3 x 10-~0olim-1 cm-~ rad-~ 0.77 1.9 x 10-'l F cm-1 0.55 cm -i

models the e x p e r i m e n t a l data is given by the closeness of the agreement b e t w e e n calculated and e x p e r i m e n t a l admittance data. This is described in section below, in which the total admittance of the electrolyte, i n c l u d ing both i n t r a g r a n u l a r a n d grain boundary-effects, is calculated a n d compared with the e x p e r i m e n t a l results. The solid line in Fig. 4b represents calculated values a n d it can be seen that the a g r e e m e n t with the e x p e r i m e n t a l data is good. The complex a d m i t t a n c e of circuit [Fig. 5 ( b ) ] m a y be converted into the three other complex formalisms using Eq. [1] and hence, the response of the e q u i v a lent circuit in all four formalisms m a y be calculated. The limiting high and low frequency expressions for each of the four formalisms are given in Table II. Equations [1] and [2] p e r m i t a better u n d e r s t a n d i n g of the different expressions of n o n i d e a l i t y seen i n the various formalisms and plots. For instance, the M " peak, on a logarithmic scale, has a limiting slope of 1 at frequencies below the peak m a x i m u m , b u t a l i m i t i n g slope of n -- 1 (0 < n < 1) at frequencies above the peak m a x i m u m . In contrast, the Z " peak has l i m i t i n g slopes of n and --1 at low and high frequencies, respectively, Table II. This accounts for the m a r k e d tendency, noted i n various ionic conductors a n d Fig. 4a, for the M " peak to be broader t h a n the Z " peak at frequencies above the peak m a x i m u m . Further, the simulations show that the m a x i m a of the M" and Z" peaks are not coincident, but that the M " s p e c t r u m always peaks at higher frequencies t h a n the Z " spectrum. The present results on LISICON show smaller departures from Debye response than observed in /~-alumina (16). This m a y be because the mobile ion concentration in LISICON is considerably less t h a n expected, in keeping with a model of ion t r a p p i n g to be discussed elsewhere (25). G r a i n b o u n d a r y e f ] e c t s . - - P o l y c r y s t a l l i n e LISICON sinters gave complex impedance plane plots that showed two semicircles, Fig. 2; the high frequency semicircle is a t t r i b u t e d to the b u l k response and the low frequency one to an i n t e r g r a n u l a r or grain b o u n d a r y effect. Since the resistances d e t e r m i n e d from the intercepts of the two semicircles on the Z' axis have similar activation energies in all samples studied, the origin of the low frequency semicircle is believed to be in the constriction of the c u r r e n t pathways at the necks which connect adjacent grains. These necks are clearly seen in a scanning electron micrograph, Fig. 6, a n d the porous n a t u r e of the sinter is also apparent. While, i n some systems, a n i n t e r g r a n u l a r blocking phase of different composition to the b u l k crystals m a y give rise to a constriction and a second semicircle in the a-c data (26), there is no evidence for such a second phase here: on extended exposure, G u i n i e r x - r a y powder diffraction photographs of powdered pellets, there was no evidence of a second phase. I n addition, electron microprobe analysis of materials such as those shown in Fig. 6a, showed no difference i n Zn,Ge content b e t w e e n the b u l k and i n t e r g r a n u l a r

M a r c h 1983

regions. It is believed, therefore, that the necks or i n t e r g r a n u l a r regions are composed of m a t e r i a l of the same composition and s t r u c t u r e as the b u l k of the crystals. I n the following, a model is proposed to show how constriction of c u r r e n t p a t h w a y s b y the presence of air gaps b e t w e e n grains can lead to an e x t r a semicircle in the complex impedance plane. A schematic, idealized constriction b e t w e e n two grains is s h o w n i n Fig. 7a and a model e q u i v a l e n t circuit i n Fig. 7b. The essential difference b e t w e e n grain and constriction regions lies in their relative cross-section.al area. It is assumed that there are no extra resistances at the contact points b e t w e e n grains, and that the crystal structure passes c o n t i n u o u s l y from one grain to the next via the constricted region; it is also assumed that conduction is isotropic, a n d that differences in g r a i n orientation do not affect the behavior. The g r a i n a n d constriction resistances are set equal to R1 and Ra (the resistivities, Pi a n d ~ , are equal, b u t the two regions have a different effective cell constant). The capacitances of the t w o regions are C1 and C2; Ci corresponds to the b u l k capacitance of the grains, C~ represents the combination of the capacitance of the constriction region, Cc and the capacitance of the air gap, Ca. The Maxwell r e l a x a t i o n times, 9, of the grain and constriction regions are given, assuming simple geometries, b y "~i - " R 1 C 1

:

pc'Co

T~ : R~C2 where ,' is the b u l k dielectric constant and eo the p e r m i t t i v i t y of free space. But

c~ : cr + c~ Ac

(Ag

-

Ac)

-

where Ac, Ag, and lc are defined in Fig. 7 (a). A s s u m ing that Ag > ~ A~ Ae Ag

+

9

o lo

Hence Ag T~ = ~i + n - - ' - A~

I

[5]

,'

As an order of m a g n i t u d e calculation, if ,' = 10 and Ag ----- 90 Ac, then T2 ----- 10~i. I n complex impedance plots, semicircle m a x i m a occur w h e n ~ _-- I, and therefore, two semicircles separated by one decade in frequency would be expected. I n order to t r y and modify the constriction resistances, some pellets were p a r t i a l l y melted and their a-c conductivity measured. SEM photographs are shown in Fig. 6b, c for a pellet of composition x ---- 0.55, which was partially melted at ,,-1350~ and subseq u e n t l y annealed at 1150~ for 1 hr. Electrodes were then applied and the a-c response of this cell determined. After conductivity measurements, the electrodes were removed, the pellet crushed to a fine powder, repelleted, and sintered at 1150~ for 1 hr; an SEM micrograph is given in Fig. 6d. The two pellets should differ only in their texture, therefore. It can be seen that the partially melted pellet, Fig. 6b, c, had a much larger average grain size than the r e g r o u n d

Table II.. Limiting high and low frequency expressions calculated for the Jonscher circuit, Fig. 5(b) Y'

-> 0

-~ ~

1/R

Awn

Y"

Ben

~C|

e"

(B~"-l)/Co

C|

e"

1/~RCo

(A~ "~i) I Co

Z'

R (A~-~) IC|

Z" R~B~~

11~C~

M'

R~CoB~~+1 Co/C|

M"

RCo~ (CoA~--i) IC|

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Vol. 130, No. 3

667

A-C CONDUCTIVITY

/~/Oro/m / drea~Ag /

air~ Ca

area, Ac