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LOW-TEMPERATURE SPECIFIC HEAT ANOMALY OF THE ONE-DIMENSIONAL IONIC CONDUCTOR
HOLLANDITE
H.V. Löhneysen, H. Schink, W. Arnold, H. Beyeler, L. Pietronero, S. Strässler
To cite this version:
H.V. Löhneysen, H. Schink, W. Arnold, H. Beyeler, L. Pietronero, et al.. LOW-TEMPERATURE SPECIFIC HEAT ANOMALY OF THE ONE-DIMENSIONAL IONIC CONDUCTOR HOLLAN- DITE. Journal de Physique Colloques, 1981, 42 (C6), pp.C6-193-C6-195. �10.1051/jphyscol:1981657�.
�jpa-00221593�
JOURNAL DE PHYSIQUE
CoZZoque C6, suppZdment au n012, Tome 42, &cembre 1981 page C6- 193
LOW-TEMPERATURE S P E C I F I C HEAT ANOMALY OF THE ONE-DIMENSIONAL I O N I C CONDUCTOR HOLLANDITE
*H
*
+%H . v . LGhneysen, H.J. Schink, W. A r n o l d
,
H.U. Beyeler, L. P i e t r o n e r o andS. ~ t r g s s l e r *
2. PhysikaZisches InsCitut, RWTH Aachen, F. R. G.
"BBC-~esearoh Centre, Baden, Switzerknd
Most amorphous m a t e r i a l s c o n t a i n l o c a l i z e d e x c i t a t i o n s which s t r o n g l y i n f l u - ence t h e i r low temperature p r o p e r t i e s ; f o r example, f o r temperatures below a few K the s p e c i f i c heat v a r i e s almost l i n e a r 1 y w i t h temperature, and t h e thermal conductiv- i t y i s d r a s t i c a l l y reduced compared t o c r y s t a l 1 in e m a t e r i a l s and g e n e r a l l y e x h i b i t s a T dependence 2 /I/. I t i s now accepted t h a t these e x c i t a t i o n s a r i s e from t u n n e l l i n g
of c e r t a i n p a r t s o f the disordered network between two p o t e n t i a l energy minima /2/.
However, the question o f t h e microscopic o r i g i n o f t h e t u n n e l l i n g systems has n o t been solved y e t . S i m i l a r p r o p e r t i e s have been observed i n superionic conductors /3/, and t h i s seems t o be p a r t i c u l a r l y promising i n order t o e l u c i d a t e t h e microscopic o r i g i n o f these e x c i t a t i o n s because i n superionic conductors the d i s o r d e r can be re- duced t o two dimensions o r even t o one dimension. We have t h e r e f o r e c a r r i e d o u t spe- c i f i c - h e a t measurements i n the one-dimensional superionic conductor h o l l a n d i t e and we e x p l a i n i t s s p e c i f i c heat i n terms o f a microscopic model.
I n h o l l a n d i t e (KpxMgxTi8-xO16), t h e m o b i l e K i o n s r e s i d e i n a l i n e a r p e r i o d i c poten-
+
t i a l which i s substoichiometrica11 y occupied. The channel s c o n t a i n i n g the K i o n s t.
do n o t communicate w i t h neighbouring channels. By a n a l y s i s o f d i f f u s e X-ray s c a t t e r - i n g a good d e s c r i p t i o n o f the s t a t e o f order has been obtained /4/. Each channel c o n t a i n s one s i t e per u n i t c e l l along the c-axis and the f r a c t i o n a l occupancy of these s i t e s w i t h K
+
ions i s equal t o x. Very important i s the i o n - i o n i n t e r a c t i o n w i t h i n each channel : i t removes the degeneracy o f the energies o f d i f f e r e n t c o n f i g - u r a t i o n s and causes the e q u i l i b r i u m p o s i t i o n o f t h e i o n s t o s h i f t from t h e l o c a l minima o f the backgroond p o t e n t i a l . The question i s : what are the c o n f i g u r a t i o n a l s t a t e s i n such a system and how do they c o n t r i b u t e t o t h e s p e c i f i c heat /5/? Specif- i c - h e a t measurements have been c a r r i e d o u t i n two samples w i t h x = 0.77 and x = 0.78 a t temperatures between 70 mK and 3 K. The measurements were made i n a d i l u t i o n re- f r i g e r a t o r using samples weighing about 100 mg each, and the r e s u l t s are p l o t t e d i n Fig. 1 as a f u n c t i o n of temperature. The s p e c i f i c heat f i r s t increases r a p i d l y w i t h temperature and then e x h i b i t s a maximum a t .8 K f o r x = .78 and a t 1 K x = .77. The o v e r a l l behaviour d i f f e r s markedly from t h a t found i n glasses /2/ and i n B-alumina o r LiN3 /3/. Furthermore, i t i s very s t r i k i n g t h a t a d i f f e r e n c e o f o n l y 1% in x causes the s p e c i f i c heat t o change by a f a c t o r o f f o u r . Such a d r a s t i c e f f e c t c a l l s f o r special a t t e n t i o n .*
present address: F r a u n h o f e r - I n s t i t u t , 6600 Saarbriicken, FRGArticle published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981657
C6- 194 JOURNAL DE PHYSIQUE
Let us s t a r t the discussion of the specific heat by defining an array of length n as a s e r i e s of n consecutive wells occupied by ions which are surrounded by empty wells. For s u f f i c i e n t l y high density of ions (x
2.75) we can neglect the oc- currence of two or more empty wells adjacent t o -
Yeach other.
Aconfiguration i s then specified by % -
the s e t of array lengths [n] , e.g. [ n ] = 4;3;3;. . . , d6
indicates t h a t we have a vacancy, then four occu- pied wells, a vacancy, three occupied wells e t c . I t has been shown t h a t the t o t a l potential energy of a given configuration including the contribu-
d7tion of the ionic relaxation in each well due t o
the ion-ion interaction can be written in terms
a05 0.1 1 5of a spin type Hamiltonian acting only between
T (K1empty s i t e s /4/: Fig.1: Specific heat of hollandite
Here
Jand
C1 (51)are parameters related t o the b a r r i e r height of the background potential and t o the strength of the ion-ion interaction. The variable O;. assumes the value 0 or
1depending on whether the j - t h s i t e i s occupied or empty, and n* i s the total number of occupied s i t e s between the s i t e s
jand
j ' .The sum extends t o a l l pairs of empty s i t e s . With
x=.77and using parameters appropriate t o hollandite, a contribution t o the specific heat T240 K i s expected due t o t r a n s i t i o n s of type
. . ; 4 ; 4 ; . . . -. . ,3;5,. . . The energy d i f f e r e n c e d , between these two configurations can be estimated from Eq.
(1)limiting the interaction t o nearest holes /4,6/:
However, f o r pairs of configurations t h a t are degenerate
w i t hrespect t o the i n t e r - action between nearest holes l i k e ... ;3;4; ..-. . ;4;3.. . the interactions up t o second nearest empty wells must be included. Transitions 1 ike .. . . .4;3;4;3;. . . -
.;4;4;3;3;. .. might occur which have a c h a r a c t e r i s t i c g a p d 2 /6/:
A' = ~ ( 6 ) + CI8) -2C(71= a3d1 (3)
Because a = . 3 , 4 i s about 40 times smaller t h a n d , and gives r i s e t o a s p e c i f i c 2
heat peak a t T = 1-2 K which agrees nicely w i t h the experimental findings.
In order t o understand the strong concentration dependence of the s p e c i f i c heat we
need to take a closer look a t the t r a n s i t i o n s t h a t give r i s e t o the peak. The prob-
a b i l i t y of having a series of values l i k e the s t a r t i n g configuration of Eq. ( 3 ) in
the ground s t a t e i s maximum i f arrays with n-3 and n=4 appear with equal probabili-
t y . According t o t h i s argument t h e maximum s p e c i f i c heat i s obtained f o r n = 3.5 (x=<n>/( <n>+l)=.77 w h i l s t i t tends t o zero f o r (rl>=3(x=.75) and <r~=4(x=.80). For i n t e r m e d i a t e values o f x a more d e t a i l e d treatment i s necessary, which, i n f a c t , reproduces the s t r o n g v a r i a t i o n o f t h e s p e c i f i c heat by a f a c t o r o f 4 /6/.
COmparing the absolute magnitude between the measured s p e c i f i c heat and t h e t h e o r e t - i c a l expected one we f i n d t h a t the t h e o r e t i c a l curve i s much l a r g e r . r h i s i s due t o the f a c t t h a t the system has been t r e a t e d as an i n f i n i t e l y l o n g chain, w h i l e t h e actual samples c o n t a i n a h i g h d e n s i t y o f i m p u r i t i e s so t h a t t h e r e a l system i s a c o l l e c t i o n o f separate segments o f v a r i o u s lengths. The i n c l u s i o n o f f i n i t e c h a i n l e n g t h i n t o theory d r a s t i c a l l y reduces the magnitude o f the s p e c i f i c heat and we recover the experimental value i f we assume t h e average separation between b l o c k i n g i o n s t o be about 10 l a t t i c e s i t e s .
A f u r t h e r i n t e r e s t i n g question i s how the t r a n s i t i o n a c t u a l l y occurs. I n t h e model discussed above a c o n f i g u r a t i o n a l t r a n s i t i o n can be regarded as the hop o f an i o n i n t o an empty s i t e i n c l u d i n g the r e l a x a t i o n o f a l l o t h e r ions. The t u n n e l l i n g p a r t i - c l e i s t h e r e f o r e t h e jumping i o n accompanied by t h e o t h e r r e l a x i n g ions. Taking an e f f e c t i v e b a r r i e r o f about 0.04 eV between the c o n f i g u r a t i o n a l s t a t e s /4/ and t a k - i n g the formula f o r r e l a x a t i o n due t o t u n n e l l i n g , we estimate a r e l a x a t i o n t i m e r which i s a t l e a s t o f the order o f seconds a t T = 1 K. This value i s r a t h e r u n c e r t a i n owing t o the unknown parameters d e s c r i b i n g t h e degree o f overlap c f the wave- f u n c t i o n s needed t o c a l c u l a t e r
.
For comparison, t h e c l a s s i c a l r e l a x a t i o n t i m e (Arrhenius law) a t t h i s temperature would be o f t h e order o f 10 years. 3I n sumnary, we should l i k e t o p o i n t o u t t h a t the low-temperature s p e c i f i c heat ano- malies o f a disordered system have been analyzed i n terms o f a microscopic model.
Experiment and theory c o n f i r m t h a t the d i s o r d e r due t o d i f f u s i n g and i n t e r a c t i n g ions i n an otherwise p e r i o d i c p o t e n t i a l can g i v e r i s e t o very low energy e x c i t a t i o n s b u t n o t t o a continuous d i s t r i b u t i o n o f e x c i t a t i o n s . The t r a n s i t i o n between the s t a t e s occurs v i a t u n n e l l i n g .
References
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84,
2029 (1976)/2/ See c o l l e c t e d papers i n Amorphous Sol i d s , ed. W.A. P h i l 1 ip s , Springer,N.Y. (1981) /3/ P.J. Anthony and A.C. Anderson; Phys. Rev.
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B46, 3827 (1977)E. Gmelin and K. Guckelsberger; J. Phys. C,
14,
L21 (1981)/4/ H.U. Beyeler, L. P i e t r o n e r o and S. S t r a e s s l e r ; Phys. Rev.
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822, 2988 (1980) /5/ a more extensive r e p o r t of the r e s u l t presented here i s given by t h e p r e s e n tauthors i n Phys. Rev. L e t t . - 46, 1213 (1981)
/6/
