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THE COMBINING OF DIELECTRIC AND
ANELASTIC RELAXATION MEASUREMENTS IN
THE STUDY OF POINT DEFECTS IN INSULATING
CRYSTALS
A. Nowick
To cite this version:
THE COMBINING OF DIELECTRIC AND ANELASTIC RELAXATION MEASUREMENTS IN THE STUDY OF POINT DEFECTS IN INSULATING CRYSTALS
A.S. NOWICK
H e n r y Krumb S c h o o l o f Mines, C o l u m b i a U n i v e r s i t y , New Y o r k , NY 1 0 0 2 7 , U.S.A.
Résumé
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En ce qui concerne l ' é t u d e dés défauts ponctuels dans l e s c r i s t a u x i s o l a n t s , nous montrons q u ' i l e s t possible d ' o b t e n i r plus d'informations sur l a cinétique de ces defauts en u t i l i s a n t simultanément des mesures de relax- ation diélectriques e t de relaxation anglastiques, qu'en u t i l i s a n t ces derniéres uniquement. Cela e s t dû au f a i t que ces deux types de mesures con- cernent d i f f é r e n t s modes exci t é s de relaxation. Plusieurs exampl es i l 1 ustrant ce propos sont donnés, en p a r t i c u l i e r , nous avons gtudié: de cristaux de quartz contenant des défauts Al-Na,des c r i s t a u x de Céria contenant des defautç Y V O (Vo représentant une lacune d10xyg3ne ionise) e t de Céria contenant des ions Sc3+ qui sont i s o l é s e t vont dans une configuration non-centrée.Abstract
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In t h e study of point defects in insulating c r y s t a l s , i t i s shown t h a t more information about defect kinetics i s obtainable by combininr! measurements of d i e l e c t r i c relaxation with those of a n e l a s t i c relaxation than from a n e l a s t i c i t y alone. This i s t r u e because the two types of experiments usually excite d i f f e r e n t relaxational modes. Examples of such studies are given, including: quartz c r y s t a l s with Al-Na defects, c e r i a with YVo defects (where Vo i s an oxygen-ion vacancy), and ceria with isolated S c 3 + ions, which go i n t o an off-center configuration.1
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INTRODUCTIONAnelastic relaxation due t o point defects i n c r y s t a l s i s usuallydescribable i n terms of one or more Debye peaks, of the form
6 s U T
tan
9
=--
1
+
~ 2 . ~ 2where 4 i s the angle of lag of s t r a i n behind the s t r e s s ( t h e " l o s s angle"), Ç i s
the appropriate compliance and 6s t h e maçnitude of i t s relaxation, cd the angular frequency and T t h e relaxation time. Because T-'generally obeys an Arrhenius r e l a -
t i o n , t a n 4 y i e l d s a symmetric peak e i t h e r as a function of 1/T (where T i s absolute temperature) o r of log w
.
Such Debye peaks a r e observed whenever the symmetry of point defects i s lower than t h a t of the crystal in which they reside. In such a case the defect hasn
equivalent orientations anioung which reorientation may occur. The detailad theory/l,2/ shows t h a tn
orientations give r i s e t on
relaxational modes, each of which has i t s own unique relaxation time,r
.
For each mode,^-lmayb e p r e s s e d in terms of al 1 the possible reorientation frequencies. Detai led expressions have been worked out f o r every crystal and defect symmetry. /1/
Some relaxational modes are excited by various s t r e s s systenis. Others may be exci- ted by an e l e c t r i c f i e l d ( i n d i e l e c t r i c materials) giving r i s e t o a d i e l e c t r i c loss of the form
2 2
t a n s = ( 6 & / ~ ) W T / ( ~ + w T ) (2)
CIO-508 JOURNAL
DE
PHYSIQUEwhere 6 i s t h e d i e l e c t r i c l o s s angle, E t h e a p p r o p r i a t e d i e l e c t r i c constant, and 6 s
t h e r e l a x a t i o n o f E which depends on t h e c o n c e n t r q t i o n o f t h e d e f e c t s and t h e i r e l e c t r i c d i p o l e s t r e n g t h .
I n t h e present paper, 1 s h a l l c o n f i n e m y s e l f e n t i r e l y t o t h e study o f i o n i c m a t e r i a l s ( i n s u l a t o r s ) i n which d i e l e c t r i c as w e l l as a n e l a s t i c r e l a x a t i o n can be measured. i t i s intended t o show t h a t g r e a t e r i n f o r m a t i o n about d e f e c t m i g r a t i o n processes may be obtained by combining these two types o f measurements than from a n e l a s t i c i t y alone. Some o f t h e p o s s i b i l i t i e s t o expected, i n o r d e r o f i n c r e a s i n g complexity, a r e t h e f o l l o w i n g :
1 ) The r e l a x a t i o n r a t e , r - l , f o r d i e l e c t r i c and a n e l a s t i c r e l a x a t i o n a r e t h e same. 2) The two r e l a x a t i o n r a t e s a r e c o n t r o l l e d by t h e same atomic jump b u t i n v o l v e d i f f e r e n t geometrical f a c t o r s . ( I n t h i s case, t h e a c t i v a t i o n energy i s t h e same f o r both r e l a x a t i o n s b u t t h e preexponentials d i f f e r by a f a c t o r t h a t i s c h a r a c t e r i s t i c o f t h e d e f e c t i nvolved)
.
3 ) The two r e l a x a t i o n r a t e s may be c o n t r o l l e d by d i f f e r e n t atomic jumps, so t h a t t h e i r a c t i v a t i o n energies a l s o d i f f e r .
These v a r i o u s p o s s i b i l i t i e s w i l l be i l l u s t r a t e d h e r e i n by t h r e e examples taken from r e c e n t s t u d i e s of. i n t e r e s t i n g i o n i c c r y s t a l s .
II. THE CASE OF
a
-
QUARTZ WITH Al-Na DEFECTSC r y s t a l l i n e quartz (a-quartz)
,
whether n a t u r a l l y o c c u r r i n g o r s y n t h e t i c a l l y grown, g e n e r a l l y contains A?*
ions s u b s t i t u t i n g f o r Si4+. / 3 / . Because o f t h e charge d i f f e r e n c e , t h e A l'+
i o n i s compensated by an i n t e r s t i t i a l monovalent i o n , e.g. Na+, L i + o r H+. The A l '4- thus s i t s a t t h e c e n t e r o f a d i s t o r t e d A10 t e t r a h e d r o n w i t h t h e monovalent i n t e r s t i a l i o n nearby. A mode1 of t h e much-studted Al-Na d e f e c t i s shown i n Fig.1. The o n l y symmetry o f t h e t e t r a h e d r o n i s a s i n g l e 2 - f o l d (C2) a x i s , and s i n c e t h e Nai s i t s o f f o f t h i s axis, t h e r e a r e two e q u i v a l e n t p o s i t i o n s f o r t h e Na+ on t h e tetrahedron. The c r y s t a l , however i s t r i g o n a l (D3 symmetry) w i t h a 3 - f o l d a x i s along t h e z d i r e c t i o n ( p e r p e n d i c u l a r t o F i g . l ) , so t h a t t h e r e a r e t h r e e d i f o f e r e n t l y o r i e n t e d t e t r a h e d r a on which t h e d e f e c t may r e s i d e , each r o t a t e d by 120 from t h e others. I n t h i s case, t h e r e f o r e , t h e defect i s t r i c l i n i c (no symmetry) and n = 6. For such a d e f e c t i t has been shown /4/ t h a t t h e r e a r e two types o f r e l a x a t i o n a l modes. The type designated by l e t t e r A 2 ( i n group t h e o r e t i c a l n o t a t i o n ) i s e x c i t e d o n l y by an e l e c t r i c f i e l d p a r a l l e l t o t h e 3 - f o l d screw a x i s (z a x i s ) . On t h e o t h e r hand, t h e t y p e designated by l e t t e r E i s e x c i t e d e i t h e r by an e l e c t r i c f i e l d p e r p e n d i c u l a r t o t h e z axis, o r by any s t r e s s system t h a t c o n t a i n s shear components. The E modes a c t u a l l y g i v e r i s e t o two r e l a x a t i o n times, b u t one o f these r e q u i r e s m i g r a t i o n o f t h e A1 3+ ion, a processFig.1
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Mode1 o f d i s t o r t e d tetrahedron, t h e b a s i c s t r u c t u r a l u n i t o f a-quartz,showing A1 3+ r e p l a c i n g S i 4 + w i t h an associated i n t e r s t i t i a l Nai ion. The l i n e C2 i s
a x i s o f t h e t e t r a h e d r o n !Sée F i g . 1 ) . The r e l a x a t i o n r a t e s have been shown t o be
/4/
-
1-
1r (A2) = T (E) = 2 LINa
i.e., t h e same f o r b o t h modes. Since t h e a c t i v a t i o n energy f o r t h i s i n t e r s t i t i a l jump i s v e r y low, t h e r e l a x a t i o n peaks a r e observed a t cryogenic temperatures (near 30 K f o r 1 kHz). D i e l e c t r i c r e l a x a t i o n u s u a l l y covers t h e frequency range o f . 1 - . I O kHz w h i l e t h e a n e l a s t i c r e l a x a t i o n i s observed i n AT-cut a c o u s t i c reasonators v i b - r a t i n g a t a f i x e d frequency o f 5 MHz.
I n view o f Eq. ( 3 ) i t i s p e r m i s s i b l e t o combine t h e d a t a f r o m b o t h sources t o o b t a i n an A r r h e n i u s p l o t which covers t h e e n t i r e range from 1 kHz t o 5 MHz. I n t h i s way,precis; values o f t h e a c t i v a t i o n energy (0.050 eV) and frequency f a c t o r ( 1 x
10 sec
-
) a r e obtained. /5/It i s noteworthy t h a t b o t h techniques can d e t e c t Al-Na d e f e c t s when t h e y a r e p r e s e n t a t a c o n c e n t r a t i o n as l o w as 1 ppm.
III- CASE OF Ce07 WITH Y V o DEFECTS
C e r i a (Ce02) i s a ceramic m a t e r i a l that,gossesses t h e f l u o r i t e s t r u c t u r e . When doped w i t h l o w e r v a l e n t c a t i o n s (e.g.Ca o r Y *), charqe compensation occurs t h r o u g h t h e f o r m a t i o n o f oxygen-ion vacancies ( V O ) . The m a t e r i a l t h e n becomes a good oxygen-ion conductor a t h i g h temperatures an& i s o f i n t e r e s t f o r f u e l ce11 a p p l i c a t i o n s . /6/ The t r i v a l e n t dopants (e.g. Y ) a r e o f s p e c i a l i n t e r e s t . Here, f o r e v e r y two Y 3+ i o n , we expect t o o b t a i n one V O . A t l o w e r temperatures, t h e s e
should a s s o c i a t e t o f o r m a Y 2 V o t r i p l e t . Since m o b i l i t y on t h e c a t i o n i c s u b l a t t i c e i s f r o z e n i n a t about 1000°C, however, t h e f o r m a t i o n o f such t r i p l e t s m i g h t n o t be k i n e t i c a l l y p o s s i b l e a t l o w e r temperatures, whecf o n l y t h e V o i s mobile. Accord- i n g l y , we were l e d t o expect t h a t h a l f o f t h e Y i o n s were a s s o c i a t e d t o form Y V O p a i r s (which c a r r y a n e t p o s i t i v e charge), and t h e o t h e r h a l f reinain unassociated ( c a r r y i n g a n e t n e q a t i v e charge). Thus an a r r a y o f charged d e f e c t g i s c r e a t e d . But how c o u l d we prove o r d i s p r o v e t h i s hypothesis?
The s t r u c t u r e o f t h e Y V o defect i s shown i n Fig.2. T h i s i s a d e f e c t o f t r i g o n a l symmetry i n a c u b i c c r y s t a l , f o r which n = 8. Such a d e f e c t s h o u l d g i v e r i s e t o a n e l a s t i c r e l a x a t i o n o n l y under a shear s t r e s s o f t h e t y p e S I + $ ( o r u n i a x i a l s t r e s s a l o n g < I l l > and n o t under shear s t r e s s o f t y p e sll
-
s l e ( o r <100> u n i a x i a l s t r e s s ) ,VACANCY
HOST CATION I4+)
@
DOPANT CATION (2+ or 3+)C10-510 JOURNAL
DE
PHYSIQUEThus, t h e proposed model could be checked by a n e l a s t i c s t u d i e s i f s i n g l e c r y s t a l s were a v a i l a b l e . Unfortunately, c e r i a i s a ceramic o f very h i g h m e l t i n g p o i n t , and s i n g l e c r y s t a l s a r e n o t a v a i l a b l e . L e t us then consider u t i l i z i n g d i e l e c t r i c r e l a x - a t i o n . The modes e x c i t e d d i e l e c t r i c a l l y a r e those denoted by l e t t e r T w h i l e those e x c i t e d a n e l a s t i c a l l y a r e o f type T
.
Since t h e nearest-neighbor jump18f vacancy provides access t o a l 1 8 o r i e n t a t i o i q o f t h e Y V o p a i r (see Fig.21, i t i s c l e a r t h a t a l 1 r e l a x a t i o n r a t e s a r e c o n t r o l l e d by t h e s p e c i f i c jump r a t e o f the vacancy, which we c a l 1 W. It has been shown /2/ t h a t) = 4w 2 g (4) T-'(T 1 ) = 2w u ( 5 ) I n o t h e r words ( a n e l a s t i c ) / r l ( d i e l e c t r i c ) = 2. This r a t i o i s c h a r a c t e r i s t i c o f t h i s p a r t i c u l a r d e f e c t .
Experimentally, i n an a n e l a s t i c study r e p o r t e d i n ICIFUAS 7 /7/, we showed t h a t f o r Y
*
concentrations below 2. 1% a s i n g l e Debye peal: i s obtained. (The broadening t h a t develops a t higher concentrations shows t h e onset o f d e f e c t i n t e r a c t i o n s and o f higher c l u s t e r s . ) More r e c e n t l y , c a r e f u l d i e l e c t r i c s t u d i e s c a r r i e d o u t a t low Y concentrations, showed t h a t a r a t i o ( a n e l a s t i c ) / r - l ( d i e l e c t r i c ) = 1.94 was obtained. /8/ The closeness o f t h i s r e s u l t t o t h e t h e o r e t i c a l r a t i o o f 2 leaves l i t t l e doubt as t o t h e v a l i d i t y o f the proposed model.I V
-
THE LOW-
TEMPERATURE SC 3+ PEAK I N CERIAst- 3+
.
Ceria may a l s o be doped w i t h Sc which, l i k e Y 1s a l s o t r i v a l e n t b u t i s a much smaller ion. From i o n i c c o n d u c t i v i t y data, i t has been shown /9/ t h a t t h e ScV p a i r has a much higher b i n d i n g energy than t h e YV, p a i r (0.67 eV vs. 0.2 eV). A study o f t h e a n e l a s t i c behavior o f Sc
-
doped c e r i a shows a s u r p r i s i n g r e s u l t . I n a d d i t i o n t o t h e ScV, p a i r peak, which i s observed near 300 K ( f o r a frequency o f 8 kHz), an e x t r a Peak i s observed a t 125 K, as shown i n Fig.3. This e x t r a peak does n o t occur f o r any o f t h e l a r g e r t r i v a l e n t cations, e.g., Y, Gd o r La. D i e l e c t r i c l o s 3 a l s o shows a peak i n t h e same low-temperature range. A study o f double doped samples-
-
i.e., doped w i t h Y as w e l l as Sc-
-
provides convincing evidence t h a t t h i s new ~ e a k i s n o t due t o t h e ScV, p a i r s , b u t t o i s o l a t e d Sc -ioiis. / I O / Howeveb, t h ei s o l a t e d s u b s t i t u t i o n a l Sc has cubic surroundings and c o u l d n o t g i v e r i s e t o r e - l a x a t i o n , unless i t goes o f f - c e n t e r . The low a c t i v a t i o n energy i s a l s o c o n s i s t e n t w i t h motions i n v o l v i n g l e s s than one atomic distance. Our d i e l e c t r i c l o s s measure-
s i s t e n t w i t h t h e d i e l e c t r i c Arrhenius p l o t , and, i n f a c t , f a l l s h i g h e r i n T-' by a f a c t o r o f 25. Such a l a r g e f a c t o r does n o t appear t o be e x p l i c a b l e as a d i f f e r e n c e i n geometrical f a c t o r s f o r t h e d i f f e r e n t r e l a x a t i o n a l modes. Rather, i t suggests t h a t t h e two modes a r e c o n t r o l l e d by d i f f e r e n t atomic jumps. Such a r e s u l t i s con- s i s t e n t w i t h a very low symmetry f o r t h e o f f - c e n t e r c o n f i g u r a t i o n .
I n t e r e s t i n g l y enough, computer s i m u l a t i o n c a l c u l a t i o n s / I O / support o f f - c e n t e r be- h a v i o r o f t h i s d e f e c t , n o t so much o f t h e Sc i o n i t s e l f , b u t r a t h e r o f t h e cage o f 8 surrounding oxygen ions. The c a l c u l a t i o n y i e l d s a d e f e c t c o n f i g u r a t i o n which has no symmetry ( i .e. t r i c l i n i c ) . See F i g u r e 4.
CONCLUSIONS
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It i s hoped t h a t these examples show t h a t , i n each case, t h e oppor- t u n i t y t o study d i e l e c t r i c and a n e l a s t i c r e l a x a t i o n t o g e t h e r has given more u s e f u l i n f o r m a t i o n than i f a n e l a s t i c techniques alone were u t i l i z e d .ACKNOWLEDGMENTS
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This work was sponsored b o t h by t h e U.S. Department o f Energy and t h e U.S. A i r Force, RADC. The author a l s o wishes t o thank h i s present and former students: M.P. Anderson, R. Gerhardt, W-K Lee, J. Toulouse and D-Y Wang, whose c o n t r i b u t i o n s a r e quoted h e r e i n .REFERENCES
/1/ Nowick, A.S., Adv. Phys. 16 (1967) 1.
/2/ Nowick, A.S., i n P o i n t Defects i n S o l i d s , Vol.1, ed. Crawford, J.H. and S l i f k i n L.M., Plenum Press, N.Y. (1972 ) ,Chapter 3.
/3/ Fraser, D.B., i n Physical Acoustics, Vo1.5, ed. Mason, W.P., Academic Press,
N.Y. (1968), Chapter 2.
/4/ Nowick, A.S. and Stanley, M.W., i n Physics o f t h e S o l i d State, ed. Balakrishna, S., Academic Press, N.Y. (1969), Chapter 11.
/5/ Toulouse, J. and Nowick, A.S., J. Phys. Chem. S o l i d s , i n press.
/6/
Steele, B.C.H., S o l i d S t a t e I o n i c s1 2
(1984) 391./7/ Anderson, M.P. and Nowick, A.S., J. de Physique 10 (1981), Colloque C5
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823. /8/ Wang, D.Y. and Nowick, A.S., J . Phys. Chem. S o l i * 44 (1983) 639./9/ Gerhardt-Anderson, R. and Nowick, A.S., Sol i d ~ t a t e Ï o n i c s
