CONTRIBUTION OF VACANCY MAGNETIC POLARONS TO THE SPECIFIC HEAT OF SOLID bcc 3He

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CONTRIBUTION OF VACANCY MAGNETIC

POLARONS TO THE SPECIFIC HEAT OF SOLID bcc 3He

M. Héritier, P. Lederer

To cite this version:

M. Héritier, P. Lederer. CONTRIBUTION OF VACANCY MAGNETIC POLARONS TO THE

SPECIFIC HEAT OF SOLID bcc 3He. Journal de Physique Colloques, 1978, 39 (C6), pp.C6-130-C6-

132. �10.1051/jphyscol:1978659�. �jpa-00218082�

JOURNAL DE PHYSIQUE

Colloque C6, suppliment au no

8,

Tome

39,

aotit

1978,

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_{C6- 130}

C O N T R I B U T I O N O F VACANCY M A G N E T I C POLARONS TO T H E S P E C I F I C H E A T OF S O L I D b c c 3 He

M. H 6 r i t i e r and P. Lederer

Laboratoire de Physique des s o l i d e s , Universitd Paris-Sud, Centre drOrsay, 91405 Orsay, France

RBsum6.- Nous montrons qu'un modsle de p o l a r o n magngtique de lacunes p e u t r e n d r e compte de l a c h a l e u r s p g c i f i q u e observde de 1 ' 3 ~ e s o l i d e cubique c e n t r d e n t r e I e t 5 0 mK. I1 f a u t une c o n c e n t r a t i o n de lacunes de 7 x lo-' pour un volume m o l a i r e d e 23,8 cm3/mole.

Abstract.- The c o n t r i b u t i o n of vacancy magnetic p o l a r o n s t o t h e s p e c i f i c h e a t of s o l i d b c c 3 ~ e i n t h e 50 mK

-

I mK temperature range i s shown t o account f o r t h e experimental d a t a provided a concen- t r a t i o n of 7 x low4 v a c a n c i e s i s p r e s e n t i n t h i s temperature range f o r a sample w i t h 23.8 cm3/mole.

Following two r e c e n t p a p e r s / l , 2 / , v a c a n c i e s i n bcc s o l i d 3 ~ e should give r i s e t o g i a n t magnetic polarons a t low enough temperatures. The r e a s o n i s t h a t t h e vacancy lowers i t s k i n e t i c energy by li- ning up surrounding n u c l e a r s p i n s and p r o p a g a t i n g c o h e r e n t l y w i t h i n t h e r e s u l t i n g p o l a r i z a t i o n cloud.

On t h e o t h e r hand, t h i s process c o s t s e n t r o p y and exchange energy, a f a c t which l i m i t s t h e p o l a r o n s i - s i z e . I n t h e s i m p l e s t model, above t h e s p i n orde- r i n g temperature T 131, t h e p o l a r o n r a d i u s R(T) va-

T t N

r i e s a s ( + 2k Ln2) .,where t i s a charac-

B N B

t e r i s t i c vacancy hopping energy, k i s Boltzmann's B

c o n s t a n t and a is t h e l a t t i c e parameter. The tempe- r a t u r e below which t h e polaron forms, was e s t i m a t e d / I / t o be about TI = 0.6 t / k B

"

30 mK. Although t h e Hubbard model was used t o i l l u s t r a t e t h e pola- ron c o n c e p t / l / i t i s c l e a r t h a t t h e p h y s i c a l con- c l u s i o n h o l d under very g e n e r a l c o n d i t i o n s / 2 / , ir- r e s p e c t i v e of models which have been used t o account f o r bcc 3 ~ e p r o p e r t i e s / 4 , 5 , 61 ; those c o n d i t i o n s a r e t h e e x i s t e n c e of an a l t e r n a t e l a t t i c e of l o c a l i - sed fermions, t h e e x i s t e n c e of a hopping mechanism c h a r a c t e r i z e d by an energy t , s u f f i c i e n t l y l a r g e compared w i t h exchange e n e r g i e s / l / . The q u e s t i o n a r i s e s a s t o t h e p h y s i c a l consequences of t h e e x i s - tence of vacancy magnetic p o l a r o n s . A l a r g e enough c o n c e n t r a t i o n of v a c a n c i e s w i l l have measurable e f f e c t s on v a r i o u s p r o p e r t i e s such a s magnetic sus- c e p t i b i l i t y , s p e c i f i c h e a t , e t c . See r e f e r e n c e /I 6/

about t h e low temperature ordered phase.

Recent N& measurements/7,8/ e x h i b i t an i n - c r e a s e of t h e magnetic s u s c e p t i b i l i t y below 5 mK with r e s p e c t t o t h e p r e d i c t i o n s of a simple n e a r e s t neighbour Heisenberg exchange ~ a m i l t o n i n a n . The l a t - t e r i s s a t i s f a c t o r y t o d e s c r i b e h i g h e r temperature

p r o p e r t i e s (T > 50 mK). Also t h e o r d e r i n g tempera- t u r e i s lower by a f a c t o r two than t h e expected c r i t i c a l temperature w i t h i n t h e same model/9/, and t h e t r a n s i t i o n i s f i r s t o r d e r . We have observed t h a t a c o n c e n t r a t i o n of about 4 x independent vacan- c i e s would account f o r t h e observed magnetic suscep- t i b i l i t y . Other mechanisms proposed t o account f o r t h e l a t t e r i n c l u d e four-spin exchange mechanism/5/, o r a s e l f c o n s i s t e n t f r e e z i n g of magnetic degrees of freedom/6/. Those two models a l s o account f o r t h e o r d e r of t h e t r a n s i t i o n .

The experimental d a t a on t h e vacancy concen- t r a t i o n a t low temperature i s s c a r c e . A s p e c i f i c h e a t term l i n e a r i n temperature which h a s been in-

t e r p r e t e d i n terms of a vacancy Fermi gas i n t h e 100 mK r e g i o n / l O / h a s n o t been confirmed by r e c e n t experiments/l I / .

I t i s c l e a r t h a t g i a n t magnetic polarons above T N w i l l c o n t r i b u t e an excees s p e c i f i c h e a t s i n c e a l a r g e number of n u c l e a r s p i n s a r e s t i l l f r o z e n above TN and a r e g r a d u a l l y l i b e r a t e d up t o T I . A t temperature s l i g h t l y above t h e o r d e r i n g tem- p e r a t u r e , a n a n a l y t i c a l e x p r e s s i o n of A C may be o b t a i n e d : f o r a c o n c e n t r a t i o n x of independent va- c a n c i e s :

kTLnZ

-t 815

A C = xk Ln ₂

-

_{t ( k ~ +2kTLn2}

₁

N

This e x p r e s s i o n assumes t h a t t h e vacancy i s com- p l e t e l y l o c a l i z e d w i t h i n t h e f e r r o m a g n e t i c volume and t h a t t h e p o l a r o n magnetization i s f u l l y s a t u r a - t e d . A t h i g h e r t e m p e r a t u r e s , t h e s p r e a d i n g of t h e vacancy wave f u n c t i o n o u t s i d e of t h e p o l a r o n beco- mes important and t h e n u c l e a r s p i n i n s i d e t h e pola- r o n a r e p a r t i a l l y d i s o r d e r e d / l / . Because of t h e s e two f a c t s , a simple e x p r e s s i o n of t h e vacancy f r e e energy i s n o t a v a i l a b l e / l 2 / . We have c a l c u l a t e d i t

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1978659

n u m e r i c a l l y , up t o T I , and computed t h e s p e c i f i c h e a t by g r a p h i c a l d e r i v a t i o n s . However, i n t h e v i c i - n i t y of T I , t h i s procedure l e a d s t o l a r g e u n c e r t a i n - t i e s on C. Above T I , t h e magnetic p o l a r o n v a n i s h e s , and s o does AC.

The vacancy c o n t r i b u t i o n t o t h e s p e c i f i c h e a t which i s n e g l i g i b l e o u t s i d e of t h e 1 mK

-

30 mK temperature range, must b e added t o t h e o t h e r con- t r i b u t i o n s of t h e c r y s t a l c

.

^It h a s been shown t h a t above 50 mK, a f t e r removing of t h e phonon term ( n e g l i g i b l e below 100 mK) co i s e s s e n t i a l l y t h e s p e c i f i c h e a t of n u c l e a r s p i n s coupled by a n e a r e s t neighbour Heisenberg exchange

/

1 1

/ .

We have assumed t h a t t h i s remains t r u e a t lower temperatures and t h a t c may b e given, down t o 5 mK by t h e h i g h tem- p e r a t u r e expansion :

357 35;

C =

The s p e c i f i c h e a t below 20 mK down t o 1 mK h a s been r e p o r t e d i n reference/l3/: It e x h i b i t s an anomaly a s shown on f i g u r e 1 . This anomaly i s i n excess

F i g . 1 : S p e c i f i c h e a t a t V = 23.8 cm3/mole. The e x p e r i m e n t a l p o i n t s a r e from r e f e r e n c e I l l / , ( d o t s ) and r e f e r e n c e /13/, ( t r i a n g l e s ) . The dashed l i n e i s t h e p r e d i c t i o n of a n e a r e s t neighbour Heisenberg Harniltonian. The s o l i d l i n e is our e s t i m a t i o n . The d o t t e d l i n e i s a smooth i n t e r p o l a t i o n between t h e two regimes.

w i t h comparison t o c and has n e g a t i v e s l o p e i n

0'

t h e C T' v s . 1 /T p l o t below 20 mK. Two independent v

measurements seem t o c o r r o b o r a t e t h i s o b s e r v a t i o n 1141. However, t h e r e a r e c a l l s f o r more a c c u r a t e measurements/l7/. The s o l i d l i n e i n f i g u r e 1 , i s a one parameter f i t of our t h e o r e t i c a l r e s u l t t o t h e experimental p o i n t s . The only parameter i s t h e c o n c e n t r a t i o n x of v a c a n c i e s . The f i t i s o b t a i n e d f o r x = 7 x lo-'. T h i s c o n c e n t r a t i o n i s assumed t o

b e c o n s t a n t throughout t h e temperature range 1 mK

-

20 mK. T h i s l a s t p o i n t i s i n l i n e w i t h our i n t e r p r e - t a t i o n of s u s c e p t i b i l i t y measurements ; we t h i n k of v a c a n c i e s a s b e i n g f r o z e n by t h e polaron formation below T / 1 2 / . The experimental p o i n t s i n f i g u r e 1

1

a r e o b t a i n e d f o r a molar volume V = 23.8 cm3/mole.

The molar volume i n t h e experiment d i s c u s s e d i n re- f e r e n c e / I / i s 24.2 cm3/mole, s o t h a t d i r e c t compa- r i s o n is n o t s t r a i g h t f o r w a r d . However, f o r e q u a l molar volume, t h e d i s c r e p a n c y between t h e two con- c e n t r a t i o n s needed t o e x p l a i n s p e c i f i c h e a t and s u s c e p t i b i l i t y can b e e s t i m a t e d t o b e a s most a fac- t o r 3.

The agreement between t h e o r y and experiment is s u r p r i s i n g l y good, i f one overlooks t h e r a t h e r l a r g e c o n c e n t r a t i o n of v a c a n c i e s needed. The tem- p e r a t u r e range and t h e shape of t h e curve a r e cor- r e c t l y given. The c e n t r a l d i f f i c u l t y w i t h t h i s sche- me is t h e r e q u i r e d o r d e r of magnitude f o r t h e vacan- cy c o n c e n t r a t i o n away from t h e m e l t i n g c u r v e / 8 / . It i s c l e a r t h a t t h e e q u i l i b r i u m c o n c e n t r a t i o n of a c t i - v a t e d v a c a n c i e s a t TI i s much t o o low. The presence of ground s t a t e vacancies h a s been s u g g e s t e d / l 3 / , b u t w i t h o u t any experimental evidence. I n conclu- s i o n we do n o t know what would b e t h e mechanism r e s p o n s i b l e f o r t h e r a t h e r l a r g e vacancy concentra- t i o n we need t o account f o r t h e s u s c e p t i b i l i t y and s p e c i f i c h e a t d a t a . Also, i t i s expected t h a t t h e whole s e t of thermodynamic anomalies is r e l a - t e d , s o t h a t a model capable of e x p l a i n i n g t h e s u s c e p t i b i l i t y and s p e c i f i c h e a t anomalies shodld probably a l s o e x p l a i n t h e o r d e r o f t h e t r a n s i t i o n and t h e low o r d e r i n g temperature.

T h i s is n o t t h e c a s e i n any obvious f a s h i o n f o r va- cancy models. The model assuming a very l a r g e f o u r s p i n exchange/4,5/ on t h e o t h e r hand, accounts f o r t h e o r d e r of t h e t r a n s i t i o n and f o r t h e s u s c e p t i b i - l i t y anomaly, b u t does not account f o r t h e s p e c i f i c h e a t one. Another e x p l a n a t i o n f o r the magnetic anomalies r e l i e s i n t h e mixed s p i n l a t t i c e n a t u r e of t h e s p i n exch'ange o p e r a t o r / 6 / . It i s n o t c l e a r t h a t t h i s i n t u i t i v e approach can account f o r t h e s p e c i f i c h e a t anomaly. I n any c a s e magnetic po- l a r o n s form around v a c a n c i e s i n those models a s w e l l . The q u e s t i o n a s t o what i s t h e dominant me-

chanism f o r t h e v a r i o u s observed anomalies i n t h e 20 mK- 1 mK temperature range s t a y s open.

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