Electrical and acoustical relaxation in fast ionic conductors

Solid State lonics 40/41 (1990) 922-925 North-Holland

ELECTRICAL AND ACOUSTICAL RELAXATION IN FAST I O N I C C O N D U C T O R S R. VAITKUS, A. KEZIONIS, V. SAMULION1S, A. ORLIUKAS and V. SKRITSKIJ Physics Department Vilnius UniversiO,, 232054 l~7lmus. Lithuania. USSR

The ultrasonic velocity, attenuation coefficient, electric conductivity and dielectric perminivity have been measured in wide temperature and frequency ranges in various crystalline, polycrystalline and glassy solid electrolytes. The high relaxational ultrasonic attenuation m a x i m a and corresponding velocity dispersion have been observed. Such behaviour is supposed to be determined by acoustoionic interactions of two types: (i) the piezoelectric interaction in the piezoelectric superionic; (ii) interaction caused by acoustic wave modulation of chemical potential felt by mobile ions in the centrosymmetric materials. In polycrystalline and glassy superionics the non-Debye behaviour for ultrasonic relaxation has been established. In such materials the characteristic frequency dependencies of ionic conductivity and dielectric permittivity were obtained. A simple theoretical model describing both frequency and temperature dependencies of electrical and acoustical properties of solid electrolytes is proposed. This model is based on continuous distribution of activation energies in a limited interval and describes the experimental data fairly well.

1. Introduction In recent years considerable attention has been given to the subject of relaxational phenomena in solid electrolytes. For these purposes simultaneous electric and ultrasonic investigations can be very valuable [ 1,2 ]. Usually the relaxation of fast ionic subsystem manifests itself as frequency dispersion of electric conductivity as well as high relaxational temperature peaks of ultrasonic attenuation and corresponding velocity dispersion [3,4]. Such behaviour has been observed in glassy [ 5 ], polycrystalline [ 6 ] and crystalline [7] solid electrolytes. The ultrasonic relaxations appear due to acoustoionic interactions of different types [8,9]. The relaxation time obtained from ultrasonic measurements or by means of low frequency internal friction techniques is very close to the Maxwell relaxation time, which can be deduced from electric measurements. Usually experimental ultrasonic and electrical relaxation peaks are much broader than an ideal Debye peak. Two different approaches have been proposed to characterise such non-Debye behaviour. One way is to describe this relaxation by Jonsher's universal response function [ 10]. The validity of this treatment has recently been questioned by Elliott [11]. Another traditional approach is to describe electrical and 0167-2738/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )

acoustical response in solid electrolytes using the distribution of relaxation times. It was demonstrated that such description requires simple and more physically defined distribution functions [ 5,12-14 ]. To our mind namely this approach could be applied to explain experimental results in various solid electrolytes (including polycrystalline samples and glasses) in wide frequency and temperature ranges. The purpose of this paper is to report the investigation results of electrical and ultrasonic relaxations in wide frequency and temperature ranges including phase transition points in crystalline, polycrystalline and glassy solid electrolytes. A simple theoretical model similar to that of ref. [12] describing both frequency and temperature dependencies of electrical and ultrasonic properties is presented. The continuous distribution of activation energies in some limited interval was introduced in theoretical calculations and correspondence between experimental and theoretical data is obtained.

2. Theory The complex electric impedance for superionic material with high electric conductivity can be written according to [ 13 ]

R. Vaitkuset al. / Fast ionic conductors

923

oo Re #(T, og)=a'

2 = I rg(z)dz 0

--

(1)

1 +ogz

E m a x - - Emin

kT

COCoc

× (arctg ogZma,- arctg ogrma.) I "q-O) 2"max

where r is the relaxational strength, T= TO exp(AE/ kT) the relaxation time, AE the activation energy, g ( r ) the distribution function of r, and og=2nv the angular frequency. If we suppose that r=r%coo (Coo is the dielectric permittivity when cot>> 1, % is the electric constant) we get ~=

1 i To e x p ( A E / k T ) f ( A E ) d(AE) 1 + iogz

(3)

Now we make the presumption that the activation energy is distributed in the interval Emm-Ema~ with equal probability' (i.e. f ( A E ) = const within this energy interval) and To is independent of AE. We will call this the continuous distribution of activation energies in a limited interval. After some mathematical calculations follows Re Z _ kT(arctg o91"max- - arctg (,OTmin ) C0%oo9 (Emax-gmin) ~

kT

Emax - Emirt

kT

C0 C~

= go exp ( A E / k T ) .

This expression is equivalent to that for the impedance of a sequence of parallel RC chains connected in series (R is assumed to follow an Arrhenius law). After replacing in (2) the distribution of relaxation times by the distribution of activation energies we can rewrite it as

0

In the low-frequency limit o9~0 we obtain from (7)

{2)

0

Coc~

(7)

a(T,0) =%c -

1 i T g ( r ) dr Co~ 1 +iogz "

2=

In 1 +o9 2T r2a i t t t- (arctg ogZmax - arctg ogrm~,)2

1 +o9

(4) P m2 a x

ImZ=2co~ooog(Emax-E~i~) ln 1 "~o9 2 Tmi 2n

(5)

(8)

If we suppose rmax >> rrnin and E m a x - - E m i n is of the order of k T, then ro ~ %/~oCo~ and E m a x = AE. ~oo can be measured at microwave frequencies. Then the parameters Emax and ro can be estimated from the measurements of the direct current % ~ = f ( T ) . Therefore only one parameter Emin must be chosen to obtain the best fit of theoretical and experimental data. The same approach was used for the description of ultrasonic attenuation and velocity. In centrosymmetric (nonpiezoelectric) materials the relaxation arises from the elastic wave modulation of the chemical potential felt by mobile ions [15]. The ultrasonic attenuation peak height (i.e. relaxation strength) depends on the square of the "deformational" potential constant A and concentration of mobile ions no. When the distribution of relaxation times is postulated, the ultrasonic attenuation coefficient a and relative velocity change A V~ Voo can be described by the following expressions:

noA 2 2

Tmax -- 1"min

oo

/" g(r)o92r dr

c~=2p--V~kT j

~ +--~-~r2 ,

0

(9)

oo

where rmax=To e x p ( E .... / k T ) , l"min=T0 exp(Emi./ kT). NOW it is easy to calculate the dielectric permittivity g(T, og) and the electric conductivity

0(T, og): Re

g(

T, og)=c: I:'max-Emin {oo1 In1"~-O9~~'2ax kT 2 1 + (2.) Tmin

× ( 1 1 In1 +O92Z~"~+..~.On 92T2i(arctgogZmax-arctgogrmin)2) - t (6)

AV noA2 f g ( r ) dr l/£~-2p~-~kT.I 1 +o92r2'

(10)

0

where o9 is the angular frequency of the ultrasonic wave, p is the density of the material, V~ is the ultrasonic velocity at low temperatures (ogr>> 1 ). For ultrasonic relaxation we use the same approach as mentioned above, r = ro exp ( A E / k T ) and f ( A E ) = const in a certain energy interval Er~i.- Er, ax. After simple calculations we get from (9) and (10)

924

0£=

R. V a i t k u s et al. / F a s t ionic c o n d u c t o r s

noA 3 2p V~ (Emax -- Emmin )

X o9(arctg OJTmax arctg torture), -

AV Wo° --

-

oC,

(11)

cm: 72

~ .

~ =IOMHz

.

I

'~YlO 2 v~

noA 2 1 1 + ~02Z2m=x In 2 2 n 2pV~(Emax-Emi,) 2 1 +o) 22"mi

Ema x - Emi n

kT

(12)

Expressions ( 11 ) and (12) describe only attenuation and velocity dispersion which are connected only with this specific aeoustoionic interaction mechanism described above. The a = f ( T ) and AV/ V~ = f ( T ) caused by other mechanisms, e.g. phonon contribution, must be eliminated when making the comparison of theoretical and experimental data.

140

180

220 T, K

Fig. 1. Temperature dependencies of ultrasonic velocity change and attenuation coefficient in Agl6Jl2P207. Parameters: Er, i,--0.22 eV, Em~=0.29 eV, V~= 1800 m / s (experimental value), To= 10 -15 s, noA2=2.7X 1025 eV2/m 3.

3. Experimental 5

The ultrasonic velocity and attenuation were measured by the pulse-echo method using a Z-cut LiN b O 3 transducers to excite and to detect ultrasonic signals [ 16 ]. The four-electrode method for electrical conductivity measurements was applied at low frequencies (v < 106 Hz). In the frequency range 106 to 4X 10 9 Hz the coaxial line method was used. At microwave frequencies ( v > 4X 10 9 H z ) the electric conductivity and permittivity was estimated by means of the microwave technique. The electric equipment for all electric measurements and also the procedures for sample preparation have been described earlier [ 17 ]. The polycrystalline samples were prepared from the corresponding powders pressing them into pellets.

4. Results and discussion The large relaxational ultrasonic attenuation maxima and velocity dispersion have been observed for an ultrasonic wave propagating in piezoactive crystallographic directions in single crystals Ag3SbS3 and Ag3AsS3 [ 18 ]. This has been attributed to the acoustoionic interaction of piezoelectric type [ 9,19 ]. Now we would like to focus our attention to nonpiezoelectric (polycrystalline Ag16J12P203 samples, xAgJ-

6

7

8

9

~0 ~ O ~ K '

q2 l rr; ~''1

70 e 70-~ -

""

~

0 = iMHz

70 < 103

705

10z

109 ~,Hz

Fig. 2. Frequency and temperature dependencies of ionic conductivity in Agt 6J 12P207 • Parameters: Emin = 0.13 eV, Emax= 0.27 eV, % = 5 X 10-t6 s, e ~ = 12.

( 1 - x ) A g P O 3 glasses) solid electrolytes in which non-Debye behaviour is expressed very clearly. The longitudinal ultrasonic attenuation and velocity temperature dependencies at a fixed frequency ( 10 MHz) in polycrystalline Ag16J12P207 solid electrolyte are illustrated in fig. 1. The same distribution describes also the temperature and frequency dependencies of ionic conductivity in this material. The dependencies r r = f ( T ) and rr=f(v) are shown in fig. 2. The inadequacy in electrical and ultrasonic relaxation parameters could be associated with different geometry for ionic displacements in elastic or electric field. A similar ultrasonic relaxation has been established also in the glassy solid electrolyte xAgJ( 1 - x ) A g P O 3 x = 0 . 5 5 ) . Fig. 3 shows the tempera-

R. Vaitkus et aL / Fast ionic conductors

cn.fI 12

=

t,V

2

v-£10 0 -4 -8

925

c o n n e c t e d with the ionic c o n d u c t i v i t y , changes rapidly. T h i s m u s t affect the u l t r a s o n i c properties. Ult r a s o n i c a n o m a l i e s can be d e s c r i b e d by piezoelectric or " d e f o r m a t i o n " p o t e n t i a l a c o u s t o i o n i c i n t e r a c t i o n m e c h a n i s m a n d have b e e n o b s e r v e d n e a r s u p e r i o n i c phase t r a n s i t i o n s in AgJ [ 8 ] , Ag2HgJ4 [21] a n d CsDSeO4 [22 ].

-12 -16 °° ~o

%o

2~o

36o

Fig. 3. Temperature dependencies of ultrasonic velocity change and attenuation coefficient in 0.55AgJ-0.45AgPO3 superionic glass. Parameters: E=i,=0.17 eV, E ~ = 0 . 2 6 eV, noA2= 3.5 × 1025 eVZ/m a, V~= 2410 m/s, ro= 1.5× 10-13 s.

40

10 0 10-1

~ 790K_

130

20

lO-e

10

1~

References

7-,x

lb'°~,Hz

Fig. 4. Frequency dependencies of electric conductivity and dielectric permittivity in 0.55AgJ-0.45AgPO3 glass. Parameters: Er.i, = 0, E=,x=0.25 eV, r0= 1.5× 10-13 s, ~ = 11. ture d e p e n d e n c e o f the l o n g i t u d i n a l u l t r a s o n i c velocity a n d a t t e n u a t i o n coefficient. T h e m a i n feature is seen as a very large loss peak whose m a x i m u m is p o s i t i o n e d at a t e m p e r a t u r e o f a b o u t 230 K. T h e inflection p o i n t in the slope o f the s o u n d velocity appears at a b o u t the s a m e t e m p e r a t u r e . T h e m i n i m u m o f A V / V o o = f ( T ) n e a r T = 340 K is associated with a glass phase t r a n s i t i o n . T h e m a x i m u m value o f a a n d its p o s i t i o n in the t e m p e r a t u r e scale is in very good a g r e e m e n t with those w h i c h c a n be o b t a i n e d f r o m e x t r a p o l a t i n g the results [ 1 , 2 0 ] to o u r frequency and composition, Considerable deviation a p p e a r s o n l y in the high t e m p e r a t u r e region a n d it is d e t e r m i n e d by critical p h e n o m e n a w h e n Tg is approached. T h e c o m p a r i s o n o f e x p e r i m e n t a l a n d theoretical d a t a for the electric r e l a x a t i o n in 0 . 5 5 A g J 0.45AgPO3 glass is s h o w n in fig. 4. N e a r the superionic phase t r a n s i t i o n the r e l a x a t i o n time, w h i c h is

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