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Cooperative enhancement of the Kondo effect in heavy-fermion systems
~/~
Solid State Conmunications, Vol.62,No.9,
pp.627-631,
1987.
0038-1098/87 $3.00 + .00
Printed in Great Britain.
Pergamon Journals Ltd.
COOPERATIVE ENtIANCI~EY~ OF TIE KONI)O EFFECT IN IF_~VY-FEI~ION SYSTI~S C . - I . Kim, Y. Kuramoto+ and T. Kasuya Department o f Physics, Tohoku U n i v e r s i t y , Sendal 980, Japan +Department of A p p l i e d Physics, Tohoku U n i v e r s i t y , Sendal 980, J a p a n
( R e c e i v e d 19 J a n u a r y 1987,
in r e v i s e d form 20 F e b r u a r y 1987, by T. Tsuzuki}
I n f l u e n c e o f h i g h c o n c e n t r a t i o n of f e l e c t r o n s on t h e Kondo e f f e c t i s s t u d i e d by u s e o f t h e self-consistent perturbation theory with respect to hybridization. I t i s shown t h a t p a r t i a l d e l o c a l i z a t i o n o f f e l e c t r o n s l e a d s t o c o o p e r a t i v e enhancement o f t h e Kondo r e s o n a n c e i n t h e f-electron density of states.
S t r o n g c o r r e l a t i o n s among f e l e c t r o n s in some of r a r e - e a r t h and a c t i n i d e compounds l e a d t o e x t r a o r d i n a r y m e t a l l i c b e h a v i o r s u c h as g i g a n t i c s p e c i f i c h e a t apd s u p e r c o n d u c t i v i t y l w i t h heavy Cooper p a i r s . These c o m p ( x a ~ a r e c a l l e d Kondo l a t t i c e o r h e a v y - f e r m i o n s y s t e m s and p o s e c h a l l g r ~ [ i n g p r o b l e m s t o t h e c o n d e n s e d matter theory.~'°' A complicating feature is t h a t s e v e r a l c o m p e t i n g e f f e c t s a r e i n v o l v e d in the system. For example, t h e Kondo s c r e e n i n g e f f e c t i s i n t e r f e r e d by t h e [ ~ Y i n t e r a c t i o n . In a d d i t i o n , t h e number o f c o n d u c t i o n e l e c t r o n s a v a i l a b l e f o r t h e Kondo s c r e e n i n g i s t o o s m a l l t o c o n f i n e a l l t h e f - e l e c t r o n l o c a l moments. The f e l e c t r o n s , on t h e o t h e r hand, can be p a r t i a l l y d e l o c a l i z e d by h y b r i d i z a t i o n w i t h conduction-band states. For p r o p e r u n d e r s t a n d i n g of the system it is necessary to consider
with infinitely strong local f-electron repulsion. Taking an a r b i t r a r y f - e l e c t r o n s i t e , we i n t r o d u c e t h e r e s o l v e n t s w i t h ommplex e n e r g y z such as
Ro(Z) = [ Z - ~ o l Z ) ] - l , f o r fO and f l t h e e n e r g y of Fermi l e v e l . include both
RIIZ} = [ z - ~ f - ~ l ( Z ) ] - I
configurations. Here ~f d e n o t e s the bare f level relative to the The s e l f - e n e r g i e s Z 1 } I z } a n d ~ l ( z ) i n t r a - and i n t e r - s i t e e f f e c t s of
hybridization. They are determined by the integral equations
~l}(Zl : n
these competing effects on an equal footing. In this paper we argue that the partial delocalization of f electrons is the dominant intersite effect at temperatures comparable to or higher than the Kondo temperature T K. As a result the Kondo effect is enhanced, rather than suppressed as might be anticipated by a naive consideration, by intersite interactions. This conclusion is obtained bYsuse of the self-consistent perturbation theory which includes intersite interactions by a moan-field approximation. The use of the self-consistent perturbation theory fgr heavy-fermion systems was initiated by Grewe who, however, neglected the effect of intersite interactions back on the Kondo effect. Our basic observation to account for the intersite interaction is that the number z n of
~ WI~lf(~IRIIZ+~I,
El(Z} = ~8"~(~)[l=f(~llRo(Z=~)
,
where f ( ~ l iS the Fermi distribution function and W(~I is the renormalized hybridization intensity with inclusion of intersite interactions. ~ This way of incorporating the intersite interaction can be regarded as a kind of moanfield approximation which is accurate up to Otl/z }- We note that the factor I/z is assoclated with each cycle o4 conductionelectron propagation which starts from a site and, after hybridization at different sites, returns to the original site. For simplicity we take a model where conduction bands are also nfold degenerate and hybridize with only sNch f states that have the same quantum number.1 Then W(~) is g i v e n by
effective neighbors around a given f-electron s i t e i s much l a r g e r t h a n u n i t y i n a c t u a l systems. I t s h o u l d t h e n be more a d v a n t a g e o u s t o s t a r t from t h e l i m i t of I / Z n - - ~ O , r a t h e r t h a n t o increase the number of interacting neighbors
W(~) ~"
one by one. On the other hand, for the singlesite Kondo system, the reciprocal of the degeneracy n of f-electron states has turned out to be a proper exp~n,qion parameter in the whole temperature range. ' We adopt t~e extended J non-crossing approximation /XNCA} which is the simplest self-consistent scheme with I/z n and I/n being regarded as small expansion
=
-
(V21~}ImG'c(t ÷ i81
Here V i s t h e h y b r i d i z a t i o n is positive infinitesimal.
.
(I}
m a t r i x e l e m e n t and 8 G" r e p r e s e n t s t h e
site-diagonal component of the conductionelectron propagator of a fictitious system in which hybridization with t~e concerned felectron site is excluded. ~ In the single-site NCA th~ bare propagator has been used instead of G'~.~ On the other hand the propagator Gc(k,z) of the real system is given by
parameters. The model we u s e i s t h e Anderson l a t t i c e 627
628
THE KONDO EFFECT IN HEAVY-FERI41OE SYSTEMS Z(.(Z) T(z} = Z c l z ) l l l - - ~ - k ] ~ l .
Gclk,z} = [Z-Ck-Zc(Z)] - I ' where c k is t h e e n e r g y o f a c o n d u c t i o n e l e c t r o n w i t h momentum k, and the s e l f - e n e r g y Zc(Z) is independent of k up t o O ( I / z _ ) . The r e l a t i o n between Gc(Z) and the s i t e - d i a g o n a l component Gc(Z) of tbe r e a l p r o p a g a t o r is Gc(Z } ~ I ~
G c ( k , z } = ~ - c ( z } [ l ÷ Z c ( Z ) G c ( Z } ] ' (2}
where N i s t h e m m b e r of t h e l a t t i c e s i t e s , Here we have n e g l e c t e d p o s s i b l e d e v i a t i o n of Z_(z) in the fictitious system c a u s e d by the c~ange in the local environment. This change does not affect the O(l/z n} accuracy in determnin9 the resolvents. Given the self-energy E~(z), we can thus derive the resolvents. In order to close the self-consistent chain of equations, we must derive X_(z} back from the resolvents. As in the sing)e-site NCA we obtain the t-matrix T(i~ n} of the conduction-electron propagator with tlatsubara frequency icn by V2 Ttic n, =~-f IC ~-~-e-~ZRotZ)RltZ+i~n ,. ZI = ~C ~dz
We d e f i n e t h e f - e l e c t r o n by pf(c)
'31
e -Dz [Ro(Z) ÷ n R l ( Z ) l ,
~clZ) l_E((z)/(Z_~k):
(4)
l
Gc(k,z} e x p l i k - ( R i - R j •} ]
= gij(z}
.
d e n s i t y of s t a t e s
pf(c}
= -Imn~-k~Gf(k,c÷iS}.
The set of s e l f - c o n s i s t e n t equations is solved by numerical i t e r a t i o n . As in the case of the CPA the c r y s t a l s t r u c t u r e enters o n l y i m p l i c i t l y in the form of the d e n s i t y of conduction-band s t a t e s . We take a s i m p l e model s u c h as -(V2/n)
The scheme which u s e s t h i s r e l a t i o n i s r e f e r r e d t o as t h e 0 scheme. T h e r e i s a p o s s i b i l i t y in E q . ( 4 ) t h a t t h e s o l u t i o n f o r Zc(Z) does n o t e x i s t i f some i n a c c u r a c y i s i n v o l v e d in T ( z ) . S i n c e t h e XNCA d e r i v e s T(z) w i t h a c c u r a c y of O ( I / n ) and O i l / z _ ) , t h i s p o s s i b i l i t y c a n n o t be elimnated. We ~ h e r e f o r e a l s o t r y an a l t e r n a t i v e scheme in which t h e r e l a t i o n between T(z) and Z c ( z ) i s a p p r o x i m a t e d b u t t h e e x i s t e n c e of Zc(Z) i s g u a r a n t e e d . In t h e l a t t e r scheme, h e r e a f t e r r e f e r r e d t o as t h e A scheme, T(z) i s i n t e r p r e t e d as t h e o n - s i t e t - m a t r i x which a p p e a r s i n t h e e x p a n s i o n o f t h e p r o p a g a t o r Gc such as
-~
The 0 scheme and the A scheme are complementary t o each o t h e r ; the former may be too s t r i c t on the r e s t r i c t i o n of the s i t e summation in view of the O ( I / z n) accuracy in T ( z ) , w h i l e the l a t t e r is too loose. The optimum r e s t r i c t i o n w i l l be between these two extremes. Thus i f the r e s u l t s d e r i v e d by 0 and A schemes are close to each o t h e r , one can judge t h a t they do not depend on the a p p r o x i = a t i o n s in r e l a t i n g T(z) and Z c ( Z } . In both 0 and A schemes the f e l e c t r o n propagator w i t h momentum k is 9iven by Y.cIZI/V 2 G f ( k , z ) = 1.2(.(z}/lZ_Ck)
where ~ is the r e c i p r o c a l of the temperature T and the i n t e q r a t i o n contour C e n c i r c l e s a l l s i n m a l a r i t i e s of the inteqrand in the c o u n t e r clockwise direction, in the o r i g i n a l XNCA, T(i~ n ) is r e g a r d e d as t h e site-diagonal e l e m e n t of t~e t-matrix of the whole system. Then T(z) and Zc(Z) is related to each other by the exact relation T(z) = I ~
Vol. 62, No. 9
Im g i i l c + i 6 l
= WoO(D-loll,
where O(x) is the Heaviside step f u n c t i o n and D is the h a l f - w i d t h of the conduction band. In o r d e r to see the importance of the i n t e r s i t e i n t e r a c t i o n back on the Kondo e f f e c t we have a l s o Performed c a l c u l a t i o n f o r the s i D g l e - s i t e 0 model and f o r the Grewe approximation which is s i m i l a r to the A scheme but w i t h replacement of W'(c} by WoOID-I~I). F i g u r e I shows t h e f - e l e c t r o n d e n s i t y of s t a t e s f o r t h e n = 2 c a s e w i t h t h e foil, owing s y s t e m p a r a m e t e r s : nW0 -- 500 K, D = 101 K, ~:f = -1500 K. The Kondo t e m p e r a t u r e d e f i n e d by TK = D(n~____OOll/n
cf exp (~0-0 },
(6 }
is equal to 16 K. We have set T = 150 K where the Kondo resonance a l r e a o y appears near tl~e Fermi l e v e l . The appearance of the Kondo resonance at T ~ I0 TK has been o ~ e r v e d a l s o in t h e s i n g l e - s i t e model w i t h n = 6. A remarkable f e a t u r e in F i g . I i s t h a t t h e i n t e r s i t e i n t e r a c t i o n e n h a n c e s t h e Kondo r e s o n a n c e . The d i f f e r e n c e between t h e 0 and A schemes i s insi~ificant as compared w i t h t h a t between t h e
+ ~ qiQ(z)T(z)gQj(z}
+ ~Q m ~~ g i ~ ( z } T l z ) g j ~ m ( z ) T ( z } [ g m j ( Z ) + p#m ~ g mv~ ( z ) T ( z ) g~~ ( z ) ] where 9 i 3 i z ) is t h e s i t e r e p r e s e n t a t i o n of t h e bare propagator. In E q . ( 5 ) p can be equal t o ~. I n c l u s i o n of t h i s term, which i n t r o d u c e s an e r r o r of O ( I / Z n ) , l e a d s t o a s i m p l e r e l a t i o n s h i p
+ ..-,
(5)
s i n g l e - s i t e model and t h e p e r i o d i c one. If one n e q l e c t s t h e f e e d - b a c k e f f e c t o1 i n t e r s l t e i n t e r a c t i o n s as in t h e Grewe a p p r o x i m a t i o n , 6 t h e d e n s i t y of s t a t e s i s a l m o s t t h e same as t h a t of
Vol. 62, No. 9
0.40
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8,00
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0,20
t
B,O0
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T=I5K
I I I I
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El
n=2
T= 150Kt
n=2
0.30
629
THE KONDO EFFECT IN HEAVY-FERMION SYSTEMS
W
20
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t h e s i n q l e - s i t e model. The f - e l e c t r o n o c c u p a t i o n ntmber n f p e r s i t e i s 0.85 in b o t h t h e 0 and A schemes and i s s l i q h t l y s m a l l e r t h a n 0.86 of the sinqle-stte ledel. The r e a s o n why t h e i n t e r s i t e i n t e r a c t i o n e n h a n c e s t h e Kondo r e s o n a n c e becomes c l e a r by i n s p e c t i o n o f W(~). At t e = p e r a t u r e s s u f f i c i e n t l y h i g h e r t h a n T K t h e d i f f e r e n c e between W(~) and WO(~} ~ negliqible. Therefore t h e Kondo t e ~ e r a t u r e d e f i n e d from t h e most dominant l o g a r i t h m c term in t h e p e r t u r b a t i o n expau7~ion i s t h e samN~ in b o t h t h e s i r ~ l e - s i t e s y s t e m and t h e p e r i o d i c o n e . F i g u r e 2 s b o ~ s t h e r e s u l t s o f W~'(c) c a l c u l a t e d in t h e A s c h e l e . As t e m p e r a t u r e d e c r e a s e s , ~(~) is enhanced for 1 near the renormalized f level. This enhancement i s due t o t h e s p e c t r a l i n t e n s i t y t r a n s f e r r e d from f e l e c t r o n s , and i s p r o p e r l y c a l l e d the delocalization effect. The e f f e c t i s a l r e a d y s e e n a t T = 150 K. With f u r t h e r d e c r e a s e o f t e m p e r a t u r e , s a y . a t T = 15 K, W(~) e x h i b i t s a sharp maximum as w e l l as a d i p . The d i p r e p r e s e n t s a co=petincj e f f e c t t h a t t h e mmber o ( a v a i l a b l e c o n d o c t i o n - b a n d s t a t e s is limited. T h i s e f f e c t i s r e f e r r e d t o as t h e s a t u r a t i o n e f f e c t and i s d i s c u s s e d in more d e t a i l helow. The s a t u r a t i o n e f f e c t is i n t e r p r e t e d in F i g . 3 ( a } as t h e e x c l u s i o n o f a c o n d u c t i o n - b a n d s t a t e a p p e a r i n g wore t h a n o n c e a t a g i v e n instant. I t i s more c o n v e n i e n t t o a c c o t m t f o r t h e e x c l u s i ~ by a l l o w t n 9 t h e p r o c e s s in F i q . 3 ( a ) and a t t h e s a l e t i l e i n c l o d i n 9 a n o t h e r c o m p e ~ s a t i n 9 p r o c e s s which a l s o v i o l a t e s t h e Pauli Principle. F i g u r e 3 ( b } r e p r e s e n t s such a process. In t h e Grewe a p p r o x i m a t i o n , t h e r e s t r i c t i o n shoan i n F l g . 3 ( a ) i s s i B p l y n e g l e c t e d and t h e s a t u r a t i o n e f f e c t i s missed. The X ~ A i n c l u d e s t h e c o ~ n s a t i n 9 process i n
0
50
1000
F i q . I . The f - e l e c t r o n d e n s i t y o f s t a t e s a t T = 150 K f o r t h e s i n c j l e - s i t e s y s t e = ( d o t t e d l i n e } and f o r t h e p e r i o d i c s y s t e = c a l c u l a t e d by t h e t h r e e d i f f e r e n t s c h e l e s : Greue a p p r o x i m a t i o n ( d a s h - d o t t e d l i n e } , t h e 0 scheme ( d a s h e d l i n e } , and t h e A s c h e = e ( s o l i d l i n e } .
i
0
F i g . 2 . The r e n o r m a l i z e d h y b r i d i z a t i o n vs e n e r g y a t v a r i o u s t e ~ p e r a t u r e s .
intensity
F i g . 3 ( b ) as t h e s e l f - e n e r g y c o r r e c t i o n t o W(g}. Namely t h e XNCA c o u n t s h y b r i d i z a t i o n p r o c e s s a t R: and s u c c e s i v e o n e s a t many d i f f e r e n t s i t e s u~ich v i o l a t e the Pauli p r i n c i p l e p a r t l y . The saturation effect thus included tends to reduce t h e h y b r i d i z a t i o n i n t e n s i t y a t RO. On t h e o t h e r hand, as we have s e e n in F i g . 2 , t h e s e l f - e n e r g y correction also incorporates the delocalization e f f e c t of f e l e c t r o n s . In t h e Kondo l i m i t where nf i s v e r y c l o s e to u n i t y , the k s t a t e s r e l e v a n t to the s a t u r a t i o n e f f e c t a r e t h o s e n e a r t h e Fermi l e v e l . If nf i s much s m a l l e r than u n i t y , t h e r e l e v a n t k s t a t e s in h y b r i d i z a t i o n have e n e r g i e s n e a r t h e f - e l e c t r o n l e v e l which i s above t h e Fermi l e v e l . In t h e l i m i t o f s m a l l n f , e l e c t r o n c o r r e l a t i o n s
k R0
-
-
~
k R i
....
. ~ ~ _
_
_
(a)
R O
Ri
~
---~~___ (b)
Fiq.3. Exalples of h y b r i d i z a t i o n p r o c e s s e s v i o l a t i n g the Pauli p r i n c i p l e . T~e d a s h e d and wavy l i n e s r e p r e s e n t t h e f and f s t a t e s , r e s p e c t i v e l y , and k i s t h e m ~ e n t t m o f a conduction-band state. The s i t e s R0 and R i a r e d i f f e r e n t from each o t h e r .
630
Vol. 62, No. 9
THE KONDO EFFECT IN HEAVY-FERMION SYSTEMS
Play no role and the hybridization gap appears in the spectrum of electrons. Since the Kondo limit is continuously reached from the small nf limit as far as the system remains paramagnetic, it is natural to associate a dip in W(~} near the Fermi level with a kind of the hybridization gap. In order to understand why not only the gap but also a peak appear in W(~}, it is helpful to consider the spinless model which can be solved exactly. Using Eq.(2} with exact results for GclZ} andE(:{z)~ one can e a s i l y d e r i v e t h e q u a n t i t y Gc(Z). The r e s u l t a n t W'(~} has a gap around t h e f - e l e c r o n l e v e l and, in t h e c e n t e r of t h e _ g a p , a d e l t a - f u n c t i o n s p i k e w i t h a w e i g h t of O{VE/DZ}. The s p i k e i s due t o t h e f - c o m p o n e n t transferred by hybridization and appears because the fictitious system, for which ~ is defined, lacks the translational symmetry. The peak and the gap in W(e) are modified in the presence of strong electron correlations, but are still recognizable in Fi9.2. Unfortunately the self-consistent solution disappears in the 0 scheme beloe T = 126 K with the parameters for Fig. l. It is probable that the disappearance is caused by the inaccuracy in Trio_l, given by Eq.(3}, which is maximum for the ~egeneracy n = 2. The density of states calculated in the 0 scheme continues to show the enhancement of the Kondo resonance down to 126 ), and there is no indication of the significant RKKY interaction. In view of the breakdown of the 0 scheme, the results for W(e) at T < 126 K should be regarded as tentative. However, one can be sure about the dominance of the delocalization effect over the saturation effect shown in Fig. l since the qualitative aspect of the results does not depend on the choice of the scheme. Similar calculation has been performed for the case of n = 6. The bare hybridization intensity nW 0 is no~ chosen to be 135 K so that TK given by Eq.(6) remains the same as that of the n = 2 case. Other parameters are the same as those in Fi~s.l and 2. Figure 4 shows the results at T = 20 K together with the singlesite result and that of the Grewe approximation. The f-electron occupation number nf is 0.93 in all cases. It is found that the intersite interaction appears much weaker in the present case than in the n = 2 case. Since the t-matrix given by Eq.{3) is more accurate for n = 6 than for n = 2, the 0 scheme still has the selfconsistent solution at 20 K. The smaller effect of enhancement is consistent with the ohservation7that the intersite interaction is of order I/n. Since both the A and the 0 scheme give results very similar to each other, one can conclude that the cooperative enhancement persists down to temperatures comparable to T K, at least in the case of n = 6. From calculations at several temperatures we have confirmed that the effect of intersite interactions on the Kondo resonance becomes smaller as the temperature increases. At T = lO0 K, for example, there is still a prominent Rondo resonance in pf(e) but the difference between the single-site system and the periedic one is slight. We notice that the competition between the delocalization and saturation effects should
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F i g . 4 . The f - e l e c t r o n d e n s i t y of s t a t e s i n t h e c a s e of n = 6. The meaning of e a c h l i n e i s t h e same as t h a t i n F i g . i .
also be present in partially disordered systems such as substitutioDal allo~s. ExpePimentally, enhancement of the Kondo effect with i n o r e s s i ~ f-electron concentrati~ has been o b s e r v e d in several Ce-based heavy-fermion systems such as 9 -IO IV Ce La I Cu2Si~, CexLa 1 xCU6, Ce~..a I B6,
Ce h,_ Sg.12 A=on.-thesesyst
dramatic example is Ce-'Lalx x Sb which is a semimetal and does not s l ~ the Kondo effect in the dilute limit. The enhancement is most moderate in Cex iLa'-xB6 which has the crystalline-electric-field ground state with n = 4, in contrast to other systems with n = 2. Since Ce has a smaller ionic radius than that of t.a, the increase of T K could partly be ~ by stronger hybridization with hiqnher Ce concentration. It should be possible to discriminate between the cooperative enhancelmt and the increase of hybridization since the former becomes insignificant at te~))~eratures sufficiently higher than T K. For example, the magnetic susceptibility per Ce i ~ in CexLa I x B is independent of x for T > 100 K - 6 wherelthe, Kondo effect is already significant. Hence the Kondo temperature (~ l K) relevant to the high-tel)eratu~e regime hardly depends on x. however with decreasin~ temperature the Kondo screening effect on the susceptibilty becomes stronger for larger x. This is likely due to the cooperative enhancement demonstrated in this paper. On the other hand, Y has a smaller ionic radius than Ce. Without the cooperative enhancement, the Kondo effect is expected to become smaller with increasincj Ce concentration. This te~Jency has I0 been found for example in CexY l xAl2 . However, the opposite tendency has been reported for Cex -Yl Rh2" A possible interpretation for the la~terXis that the cooperative enhancement dominates over the decrease of hybridization.
Vol. 62, No. 9
THE KONDO EFFECT IN HEAVY-FERMION
In c o n c l u s i o n , t h e s e l f - c o n s i s t e n t c a l c u l a t i o n b a s e d on t h e ~ h a s shown t h e enhancement o f t h e Kondo e f f e c t i n h e a w f e r m i o n system5 a t t e m p e r a t u r e s h i g h e r t h a n TK.
SYSTEMS
631
The s a l e l i n e o f c a l c u l a t i o n s h a s been p e r f o r m a l s o a t t e m P e r a t u r e s lower t h a n TK~ D e t a i l e d result for the low-te~erature re
References I . For a r e v i e w o f e x p e r i m e n t s e e , e . g . , G.R. S t e w a r t , Rev. Mcxl. Phys. 56, 755 ( 1 9 8 4 ) . 2. P.A. Lee, T.M. Rice, J.W. S e r e n e , L . J . Sham, and J.W. W i l k i n s , Comments Condensed M a t t e r Phys. 12, 99 (19861. 3. V a r i o u s t h e o r e t i c a l a p p r o a c h e s a r e p r e s e n t e d in Theory of Heavy F e r m o n s and V a l e n c e F l u c t u a t i o n s , e d i t e d by T. Kasuya and T. Saso ( S p r i n c J e r - V e r l a g , B e r l i n , 1985). 4. P. N o z i e r e s , Ann. Phys. F r . l__q, 19 119851. 5. Y. Kuramoto, in R e f . 3 , p.152. 6. N. Greue, S o l i d S t a t e Commn. 50, 19 (19841. 7. P. Coleman, J . Maim. Ma~n. M a t e r . 47&48, 323 (19841. 8. H. Kojima, Y. Kuramoto, and M. T a c h i k i , Z. Phys. B54, 293 (19841. See a l s o N. Grewe, Z. P h y s i k B53, 271 (19831; P. Coleman, Phys. Rev. 1329, 3035 (19841; F.C. 7_~hancj and T.K. Lee, Phys. Rev. B3_~, 1556 119841;
9. lO. 11. 12.
13. 14.
N.E. B i c k e r s , D.L. Cox, and J.W. W i l k i n s , Phys. Rev. L e t t . 54, 230 119851. F.G. A l i e v , N.B. Bran(it, V.V. Moshchalkov, and S.M. Chuclinov, J. Low Temp. Phys. 57, 61 119841. A. Suaiyama, Y. Oda, H. Nagano, ¥. Onuki, and T. Komatsubara, J. Phys. Soc. Jpn. 54, 877 (19851. N. Sato, A. Stmiyama, S. K u n i i , H. Naga,o, and T. Kasuya, J. Phys. Soc. Jpn. 54, 1923 (19851. M. Sera, T. S u z u k i , and T. Kasuya, J. Magn. Magn. Mater. 31-34, 385 119831. R. S c h e f z y k , J . H e i b e l , F. S t e c j l i c h , R. F e l t e r , and G. Weber, J . Maqn. Magn. Mater. 47&48, 83 (19851 and r e f e r e n c e s t h e r e i n . A. H a r r u s , T. M i h a l i s i n , and E. Kemly, J . Magn. Maim. Mater. 47&4~, 93 (1985).
Solid State Conmunications, Vol.62,No.9,
pp.627-631,
1987.
0038-1098/87 $3.00 + .00
Printed in Great Britain.
Pergamon Journals Ltd.
COOPERATIVE ENtIANCI~EY~ OF TIE KONI)O EFFECT IN IF_~VY-FEI~ION SYSTI~S C . - I . Kim, Y. Kuramoto+ and T. Kasuya Department o f Physics, Tohoku U n i v e r s i t y , Sendal 980, Japan +Department of A p p l i e d Physics, Tohoku U n i v e r s i t y , Sendal 980, J a p a n
( R e c e i v e d 19 J a n u a r y 1987,
in r e v i s e d form 20 F e b r u a r y 1987, by T. Tsuzuki}
I n f l u e n c e o f h i g h c o n c e n t r a t i o n of f e l e c t r o n s on t h e Kondo e f f e c t i s s t u d i e d by u s e o f t h e self-consistent perturbation theory with respect to hybridization. I t i s shown t h a t p a r t i a l d e l o c a l i z a t i o n o f f e l e c t r o n s l e a d s t o c o o p e r a t i v e enhancement o f t h e Kondo r e s o n a n c e i n t h e f-electron density of states.
S t r o n g c o r r e l a t i o n s among f e l e c t r o n s in some of r a r e - e a r t h and a c t i n i d e compounds l e a d t o e x t r a o r d i n a r y m e t a l l i c b e h a v i o r s u c h as g i g a n t i c s p e c i f i c h e a t apd s u p e r c o n d u c t i v i t y l w i t h heavy Cooper p a i r s . These c o m p ( x a ~ a r e c a l l e d Kondo l a t t i c e o r h e a v y - f e r m i o n s y s t e m s and p o s e c h a l l g r ~ [ i n g p r o b l e m s t o t h e c o n d e n s e d matter theory.~'°' A complicating feature is t h a t s e v e r a l c o m p e t i n g e f f e c t s a r e i n v o l v e d in the system. For example, t h e Kondo s c r e e n i n g e f f e c t i s i n t e r f e r e d by t h e [ ~ Y i n t e r a c t i o n . In a d d i t i o n , t h e number o f c o n d u c t i o n e l e c t r o n s a v a i l a b l e f o r t h e Kondo s c r e e n i n g i s t o o s m a l l t o c o n f i n e a l l t h e f - e l e c t r o n l o c a l moments. The f e l e c t r o n s , on t h e o t h e r hand, can be p a r t i a l l y d e l o c a l i z e d by h y b r i d i z a t i o n w i t h conduction-band states. For p r o p e r u n d e r s t a n d i n g of the system it is necessary to consider
with infinitely strong local f-electron repulsion. Taking an a r b i t r a r y f - e l e c t r o n s i t e , we i n t r o d u c e t h e r e s o l v e n t s w i t h ommplex e n e r g y z such as
Ro(Z) = [ Z - ~ o l Z ) ] - l , f o r fO and f l t h e e n e r g y of Fermi l e v e l . include both
RIIZ} = [ z - ~ f - ~ l ( Z ) ] - I
configurations. Here ~f d e n o t e s the bare f level relative to the The s e l f - e n e r g i e s Z 1 } I z } a n d ~ l ( z ) i n t r a - and i n t e r - s i t e e f f e c t s of
hybridization. They are determined by the integral equations
~l}(Zl : n
these competing effects on an equal footing. In this paper we argue that the partial delocalization of f electrons is the dominant intersite effect at temperatures comparable to or higher than the Kondo temperature T K. As a result the Kondo effect is enhanced, rather than suppressed as might be anticipated by a naive consideration, by intersite interactions. This conclusion is obtained bYsuse of the self-consistent perturbation theory which includes intersite interactions by a moan-field approximation. The use of the self-consistent perturbation theory fgr heavy-fermion systems was initiated by Grewe who, however, neglected the effect of intersite interactions back on the Kondo effect. Our basic observation to account for the intersite interaction is that the number z n of
~ WI~lf(~IRIIZ+~I,
El(Z} = ~8"~(~)[l=f(~llRo(Z=~)
,
where f ( ~ l iS the Fermi distribution function and W(~I is the renormalized hybridization intensity with inclusion of intersite interactions. ~ This way of incorporating the intersite interaction can be regarded as a kind of moanfield approximation which is accurate up to Otl/z }- We note that the factor I/z is assoclated with each cycle o4 conductionelectron propagation which starts from a site and, after hybridization at different sites, returns to the original site. For simplicity we take a model where conduction bands are also nfold degenerate and hybridize with only sNch f states that have the same quantum number.1 Then W(~) is g i v e n by
effective neighbors around a given f-electron s i t e i s much l a r g e r t h a n u n i t y i n a c t u a l systems. I t s h o u l d t h e n be more a d v a n t a g e o u s t o s t a r t from t h e l i m i t of I / Z n - - ~ O , r a t h e r t h a n t o increase the number of interacting neighbors
W(~) ~"
one by one. On the other hand, for the singlesite Kondo system, the reciprocal of the degeneracy n of f-electron states has turned out to be a proper exp~n,qion parameter in the whole temperature range. ' We adopt t~e extended J non-crossing approximation /XNCA} which is the simplest self-consistent scheme with I/z n and I/n being regarded as small expansion
=
-
(V21~}ImG'c(t ÷ i81
Here V i s t h e h y b r i d i z a t i o n is positive infinitesimal.
.
(I}
m a t r i x e l e m e n t and 8 G" r e p r e s e n t s t h e
site-diagonal component of the conductionelectron propagator of a fictitious system in which hybridization with t~e concerned felectron site is excluded. ~ In the single-site NCA th~ bare propagator has been used instead of G'~.~ On the other hand the propagator Gc(k,z) of the real system is given by
parameters. The model we u s e i s t h e Anderson l a t t i c e 627
628
THE KONDO EFFECT IN HEAVY-FERI41OE SYSTEMS Z(.(Z) T(z} = Z c l z ) l l l - - ~ - k ] ~ l .
Gclk,z} = [Z-Ck-Zc(Z)] - I ' where c k is t h e e n e r g y o f a c o n d u c t i o n e l e c t r o n w i t h momentum k, and the s e l f - e n e r g y Zc(Z) is independent of k up t o O ( I / z _ ) . The r e l a t i o n between Gc(Z) and the s i t e - d i a g o n a l component Gc(Z) of tbe r e a l p r o p a g a t o r is Gc(Z } ~ I ~
G c ( k , z } = ~ - c ( z } [ l ÷ Z c ( Z ) G c ( Z } ] ' (2}
where N i s t h e m m b e r of t h e l a t t i c e s i t e s , Here we have n e g l e c t e d p o s s i b l e d e v i a t i o n of Z_(z) in the fictitious system c a u s e d by the c~ange in the local environment. This change does not affect the O(l/z n} accuracy in determnin9 the resolvents. Given the self-energy E~(z), we can thus derive the resolvents. In order to close the self-consistent chain of equations, we must derive X_(z} back from the resolvents. As in the sing)e-site NCA we obtain the t-matrix T(i~ n} of the conduction-electron propagator with tlatsubara frequency icn by V2 Ttic n, =~-f IC ~-~-e-~ZRotZ)RltZ+i~n ,. ZI = ~C ~dz
We d e f i n e t h e f - e l e c t r o n by pf(c)
'31
e -Dz [Ro(Z) ÷ n R l ( Z ) l ,
~clZ) l_E((z)/(Z_~k):
(4)
l
Gc(k,z} e x p l i k - ( R i - R j •} ]
= gij(z}
.
d e n s i t y of s t a t e s
pf(c}
= -Imn~-k~Gf(k,c÷iS}.
The set of s e l f - c o n s i s t e n t equations is solved by numerical i t e r a t i o n . As in the case of the CPA the c r y s t a l s t r u c t u r e enters o n l y i m p l i c i t l y in the form of the d e n s i t y of conduction-band s t a t e s . We take a s i m p l e model s u c h as -(V2/n)
The scheme which u s e s t h i s r e l a t i o n i s r e f e r r e d t o as t h e 0 scheme. T h e r e i s a p o s s i b i l i t y in E q . ( 4 ) t h a t t h e s o l u t i o n f o r Zc(Z) does n o t e x i s t i f some i n a c c u r a c y i s i n v o l v e d in T ( z ) . S i n c e t h e XNCA d e r i v e s T(z) w i t h a c c u r a c y of O ( I / n ) and O i l / z _ ) , t h i s p o s s i b i l i t y c a n n o t be elimnated. We ~ h e r e f o r e a l s o t r y an a l t e r n a t i v e scheme in which t h e r e l a t i o n between T(z) and Z c ( z ) i s a p p r o x i m a t e d b u t t h e e x i s t e n c e of Zc(Z) i s g u a r a n t e e d . In t h e l a t t e r scheme, h e r e a f t e r r e f e r r e d t o as t h e A scheme, T(z) i s i n t e r p r e t e d as t h e o n - s i t e t - m a t r i x which a p p e a r s i n t h e e x p a n s i o n o f t h e p r o p a g a t o r Gc such as
-~
The 0 scheme and the A scheme are complementary t o each o t h e r ; the former may be too s t r i c t on the r e s t r i c t i o n of the s i t e summation in view of the O ( I / z n) accuracy in T ( z ) , w h i l e the l a t t e r is too loose. The optimum r e s t r i c t i o n w i l l be between these two extremes. Thus i f the r e s u l t s d e r i v e d by 0 and A schemes are close to each o t h e r , one can judge t h a t they do not depend on the a p p r o x i = a t i o n s in r e l a t i n g T(z) and Z c ( Z } . In both 0 and A schemes the f e l e c t r o n propagator w i t h momentum k is 9iven by Y.cIZI/V 2 G f ( k , z ) = 1.2(.(z}/lZ_Ck)
where ~ is the r e c i p r o c a l of the temperature T and the i n t e q r a t i o n contour C e n c i r c l e s a l l s i n m a l a r i t i e s of the inteqrand in the c o u n t e r clockwise direction, in the o r i g i n a l XNCA, T(i~ n ) is r e g a r d e d as t h e site-diagonal e l e m e n t of t~e t-matrix of the whole system. Then T(z) and Zc(Z) is related to each other by the exact relation T(z) = I ~
Vol. 62, No. 9
Im g i i l c + i 6 l
= WoO(D-loll,
where O(x) is the Heaviside step f u n c t i o n and D is the h a l f - w i d t h of the conduction band. In o r d e r to see the importance of the i n t e r s i t e i n t e r a c t i o n back on the Kondo e f f e c t we have a l s o Performed c a l c u l a t i o n f o r the s i D g l e - s i t e 0 model and f o r the Grewe approximation which is s i m i l a r to the A scheme but w i t h replacement of W'(c} by WoOID-I~I). F i g u r e I shows t h e f - e l e c t r o n d e n s i t y of s t a t e s f o r t h e n = 2 c a s e w i t h t h e foil, owing s y s t e m p a r a m e t e r s : nW0 -- 500 K, D = 101 K, ~:f = -1500 K. The Kondo t e m p e r a t u r e d e f i n e d by TK = D(n~____OOll/n
cf exp (~0-0 },
(6 }
is equal to 16 K. We have set T = 150 K where the Kondo resonance a l r e a o y appears near tl~e Fermi l e v e l . The appearance of the Kondo resonance at T ~ I0 TK has been o ~ e r v e d a l s o in t h e s i n g l e - s i t e model w i t h n = 6. A remarkable f e a t u r e in F i g . I i s t h a t t h e i n t e r s i t e i n t e r a c t i o n e n h a n c e s t h e Kondo r e s o n a n c e . The d i f f e r e n c e between t h e 0 and A schemes i s insi~ificant as compared w i t h t h a t between t h e
+ ~ qiQ(z)T(z)gQj(z}
+ ~Q m ~~ g i ~ ( z } T l z ) g j ~ m ( z ) T ( z } [ g m j ( Z ) + p#m ~ g mv~ ( z ) T ( z ) g~~ ( z ) ] where 9 i 3 i z ) is t h e s i t e r e p r e s e n t a t i o n of t h e bare propagator. In E q . ( 5 ) p can be equal t o ~. I n c l u s i o n of t h i s term, which i n t r o d u c e s an e r r o r of O ( I / Z n ) , l e a d s t o a s i m p l e r e l a t i o n s h i p
+ ..-,
(5)
s i n g l e - s i t e model and t h e p e r i o d i c one. If one n e q l e c t s t h e f e e d - b a c k e f f e c t o1 i n t e r s l t e i n t e r a c t i o n s as in t h e Grewe a p p r o x i m a t i o n , 6 t h e d e n s i t y of s t a t e s i s a l m o s t t h e same as t h a t of
Vol. 62, No. 9
0.40
XlO-3
8,00
I
0,20
t
B,O0
!
T=I5K
I I I I
/
!
, I 0
A
LU L#.
El
n=2
T= 150Kt
n=2
0.30
629
THE KONDO EFFECT IN HEAVY-FERMION SYSTEMS
W
20
1I ! ! I I I
4.00
f
I
2,00
0,10
150 I
0.00 0,00 -2000
i
!
i
-lO00
i
0
I!
....
. . . .
-50
,
,
,
i
. . . .
100
150
E[K]
E[K]
t h e s i n q l e - s i t e model. The f - e l e c t r o n o c c u p a t i o n ntmber n f p e r s i t e i s 0.85 in b o t h t h e 0 and A schemes and i s s l i q h t l y s m a l l e r t h a n 0.86 of the sinqle-stte ledel. The r e a s o n why t h e i n t e r s i t e i n t e r a c t i o n e n h a n c e s t h e Kondo r e s o n a n c e becomes c l e a r by i n s p e c t i o n o f W(~). At t e = p e r a t u r e s s u f f i c i e n t l y h i g h e r t h a n T K t h e d i f f e r e n c e between W(~) and WO(~} ~ negliqible. Therefore t h e Kondo t e ~ e r a t u r e d e f i n e d from t h e most dominant l o g a r i t h m c term in t h e p e r t u r b a t i o n expau7~ion i s t h e samN~ in b o t h t h e s i r ~ l e - s i t e s y s t e m and t h e p e r i o d i c o n e . F i g u r e 2 s b o ~ s t h e r e s u l t s o f W~'(c) c a l c u l a t e d in t h e A s c h e l e . As t e m p e r a t u r e d e c r e a s e s , ~(~) is enhanced for 1 near the renormalized f level. This enhancement i s due t o t h e s p e c t r a l i n t e n s i t y t r a n s f e r r e d from f e l e c t r o n s , and i s p r o p e r l y c a l l e d the delocalization effect. The e f f e c t i s a l r e a d y s e e n a t T = 150 K. With f u r t h e r d e c r e a s e o f t e m p e r a t u r e , s a y . a t T = 15 K, W(~) e x h i b i t s a sharp maximum as w e l l as a d i p . The d i p r e p r e s e n t s a co=petincj e f f e c t t h a t t h e mmber o ( a v a i l a b l e c o n d o c t i o n - b a n d s t a t e s is limited. T h i s e f f e c t i s r e f e r r e d t o as t h e s a t u r a t i o n e f f e c t and i s d i s c u s s e d in more d e t a i l helow. The s a t u r a t i o n e f f e c t is i n t e r p r e t e d in F i g . 3 ( a } as t h e e x c l u s i o n o f a c o n d u c t i o n - b a n d s t a t e a p p e a r i n g wore t h a n o n c e a t a g i v e n instant. I t i s more c o n v e n i e n t t o a c c o t m t f o r t h e e x c l u s i ~ by a l l o w t n 9 t h e p r o c e s s in F i q . 3 ( a ) and a t t h e s a l e t i l e i n c l o d i n 9 a n o t h e r c o m p e ~ s a t i n 9 p r o c e s s which a l s o v i o l a t e s t h e Pauli Principle. F i g u r e 3 ( b } r e p r e s e n t s such a process. In t h e Grewe a p p r o x i m a t i o n , t h e r e s t r i c t i o n shoan i n F l g . 3 ( a ) i s s i B p l y n e g l e c t e d and t h e s a t u r a t i o n e f f e c t i s missed. The X ~ A i n c l u d e s t h e c o ~ n s a t i n 9 process i n
0
50
1000
F i q . I . The f - e l e c t r o n d e n s i t y o f s t a t e s a t T = 150 K f o r t h e s i n c j l e - s i t e s y s t e = ( d o t t e d l i n e } and f o r t h e p e r i o d i c s y s t e = c a l c u l a t e d by t h e t h r e e d i f f e r e n t s c h e l e s : Greue a p p r o x i m a t i o n ( d a s h - d o t t e d l i n e } , t h e 0 scheme ( d a s h e d l i n e } , and t h e A s c h e = e ( s o l i d l i n e } .
i
0
F i g . 2 . The r e n o r m a l i z e d h y b r i d i z a t i o n vs e n e r g y a t v a r i o u s t e ~ p e r a t u r e s .
intensity
F i g . 3 ( b ) as t h e s e l f - e n e r g y c o r r e c t i o n t o W(g}. Namely t h e XNCA c o u n t s h y b r i d i z a t i o n p r o c e s s a t R: and s u c c e s i v e o n e s a t many d i f f e r e n t s i t e s u~ich v i o l a t e the Pauli p r i n c i p l e p a r t l y . The saturation effect thus included tends to reduce t h e h y b r i d i z a t i o n i n t e n s i t y a t RO. On t h e o t h e r hand, as we have s e e n in F i g . 2 , t h e s e l f - e n e r g y correction also incorporates the delocalization e f f e c t of f e l e c t r o n s . In t h e Kondo l i m i t where nf i s v e r y c l o s e to u n i t y , the k s t a t e s r e l e v a n t to the s a t u r a t i o n e f f e c t a r e t h o s e n e a r t h e Fermi l e v e l . If nf i s much s m a l l e r than u n i t y , t h e r e l e v a n t k s t a t e s in h y b r i d i z a t i o n have e n e r g i e s n e a r t h e f - e l e c t r o n l e v e l which i s above t h e Fermi l e v e l . In t h e l i m i t o f s m a l l n f , e l e c t r o n c o r r e l a t i o n s
k R0
-
-
~
k R i
....
. ~ ~ _
_
_
(a)
R O
Ri
~
---~~___ (b)
Fiq.3. Exalples of h y b r i d i z a t i o n p r o c e s s e s v i o l a t i n g the Pauli p r i n c i p l e . T~e d a s h e d and wavy l i n e s r e p r e s e n t t h e f and f s t a t e s , r e s p e c t i v e l y , and k i s t h e m ~ e n t t m o f a conduction-band state. The s i t e s R0 and R i a r e d i f f e r e n t from each o t h e r .
630
Vol. 62, No. 9
THE KONDO EFFECT IN HEAVY-FERMION SYSTEMS
Play no role and the hybridization gap appears in the spectrum of electrons. Since the Kondo limit is continuously reached from the small nf limit as far as the system remains paramagnetic, it is natural to associate a dip in W(~} near the Fermi level with a kind of the hybridization gap. In order to understand why not only the gap but also a peak appear in W(~}, it is helpful to consider the spinless model which can be solved exactly. Using Eq.(2} with exact results for GclZ} andE(:{z)~ one can e a s i l y d e r i v e t h e q u a n t i t y Gc(Z). The r e s u l t a n t W'(~} has a gap around t h e f - e l e c r o n l e v e l and, in t h e c e n t e r of t h e _ g a p , a d e l t a - f u n c t i o n s p i k e w i t h a w e i g h t of O{VE/DZ}. The s p i k e i s due t o t h e f - c o m p o n e n t transferred by hybridization and appears because the fictitious system, for which ~ is defined, lacks the translational symmetry. The peak and the gap in W(e) are modified in the presence of strong electron correlations, but are still recognizable in Fi9.2. Unfortunately the self-consistent solution disappears in the 0 scheme beloe T = 126 K with the parameters for Fig. l. It is probable that the disappearance is caused by the inaccuracy in Trio_l, given by Eq.(3}, which is maximum for the ~egeneracy n = 2. The density of states calculated in the 0 scheme continues to show the enhancement of the Kondo resonance down to 126 ), and there is no indication of the significant RKKY interaction. In view of the breakdown of the 0 scheme, the results for W(e) at T < 126 K should be regarded as tentative. However, one can be sure about the dominance of the delocalization effect over the saturation effect shown in Fig. l since the qualitative aspect of the results does not depend on the choice of the scheme. Similar calculation has been performed for the case of n = 6. The bare hybridization intensity nW 0 is no~ chosen to be 135 K so that TK given by Eq.(6) remains the same as that of the n = 2 case. Other parameters are the same as those in Fi~s.l and 2. Figure 4 shows the results at T = 20 K together with the singlesite result and that of the Grewe approximation. The f-electron occupation number nf is 0.93 in all cases. It is found that the intersite interaction appears much weaker in the present case than in the n = 2 case. Since the t-matrix given by Eq.{3) is more accurate for n = 6 than for n = 2, the 0 scheme still has the selfconsistent solution at 20 K. The smaller effect of enhancement is consistent with the ohservation7that the intersite interaction is of order I/n. Since both the A and the 0 scheme give results very similar to each other, one can conclude that the cooperative enhancement persists down to temperatures comparable to T K, at least in the case of n = 6. From calculations at several temperatures we have confirmed that the effect of intersite interactions on the Kondo resonance becomes smaller as the temperature increases. At T = lO0 K, for example, there is still a prominent Rondo resonance in pf(e) but the difference between the single-site system and the periedic one is slight. We notice that the competition between the delocalization and saturation effects should
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also be present in partially disordered systems such as substitutioDal allo~s. ExpePimentally, enhancement of the Kondo effect with i n o r e s s i ~ f-electron concentrati~ has been o b s e r v e d in several Ce-based heavy-fermion systems such as 9 -IO IV Ce La I Cu2Si~, CexLa 1 xCU6, Ce~..a I B6,
Ce h,_ Sg.12 A=on.-thesesyst
dramatic example is Ce-'Lalx x Sb which is a semimetal and does not s l ~ the Kondo effect in the dilute limit. The enhancement is most moderate in Cex iLa'-xB6 which has the crystalline-electric-field ground state with n = 4, in contrast to other systems with n = 2. Since Ce has a smaller ionic radius than that of t.a, the increase of T K could partly be ~ by stronger hybridization with hiqnher Ce concentration. It should be possible to discriminate between the cooperative enhancelmt and the increase of hybridization since the former becomes insignificant at te~))~eratures sufficiently higher than T K. For example, the magnetic susceptibility per Ce i ~ in CexLa I x B is independent of x for T > 100 K - 6 wherelthe, Kondo effect is already significant. Hence the Kondo temperature (~ l K) relevant to the high-tel)eratu~e regime hardly depends on x. however with decreasin~ temperature the Kondo screening effect on the susceptibilty becomes stronger for larger x. This is likely due to the cooperative enhancement demonstrated in this paper. On the other hand, Y has a smaller ionic radius than Ce. Without the cooperative enhancement, the Kondo effect is expected to become smaller with increasincj Ce concentration. This te~Jency has I0 been found for example in CexY l xAl2 . However, the opposite tendency has been reported for Cex -Yl Rh2" A possible interpretation for the la~terXis that the cooperative enhancement dominates over the decrease of hybridization.
Vol. 62, No. 9
THE KONDO EFFECT IN HEAVY-FERMION
In c o n c l u s i o n , t h e s e l f - c o n s i s t e n t c a l c u l a t i o n b a s e d on t h e ~ h a s shown t h e enhancement o f t h e Kondo e f f e c t i n h e a w f e r m i o n system5 a t t e m p e r a t u r e s h i g h e r t h a n TK.
SYSTEMS
631
The s a l e l i n e o f c a l c u l a t i o n s h a s been p e r f o r m a l s o a t t e m P e r a t u r e s lower t h a n TK~ D e t a i l e d result for the low-te~erature re
References I . For a r e v i e w o f e x p e r i m e n t s e e , e . g . , G.R. S t e w a r t , Rev. Mcxl. Phys. 56, 755 ( 1 9 8 4 ) . 2. P.A. Lee, T.M. Rice, J.W. S e r e n e , L . J . Sham, and J.W. W i l k i n s , Comments Condensed M a t t e r Phys. 12, 99 (19861. 3. V a r i o u s t h e o r e t i c a l a p p r o a c h e s a r e p r e s e n t e d in Theory of Heavy F e r m o n s and V a l e n c e F l u c t u a t i o n s , e d i t e d by T. Kasuya and T. Saso ( S p r i n c J e r - V e r l a g , B e r l i n , 1985). 4. P. N o z i e r e s , Ann. Phys. F r . l__q, 19 119851. 5. Y. Kuramoto, in R e f . 3 , p.152. 6. N. Greue, S o l i d S t a t e Commn. 50, 19 (19841. 7. P. Coleman, J . Maim. Ma~n. M a t e r . 47&48, 323 (19841. 8. H. Kojima, Y. Kuramoto, and M. T a c h i k i , Z. Phys. B54, 293 (19841. See a l s o N. Grewe, Z. P h y s i k B53, 271 (19831; P. Coleman, Phys. Rev. 1329, 3035 (19841; F.C. 7_~hancj and T.K. Lee, Phys. Rev. B3_~, 1556 119841;
9. lO. 11. 12.
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N.E. B i c k e r s , D.L. Cox, and J.W. W i l k i n s , Phys. Rev. L e t t . 54, 230 119851. F.G. A l i e v , N.B. Bran(it, V.V. Moshchalkov, and S.M. Chuclinov, J. Low Temp. Phys. 57, 61 119841. A. Suaiyama, Y. Oda, H. Nagano, ¥. Onuki, and T. Komatsubara, J. Phys. Soc. Jpn. 54, 877 (19851. N. Sato, A. Stmiyama, S. K u n i i , H. Naga,o, and T. Kasuya, J. Phys. Soc. Jpn. 54, 1923 (19851. M. Sera, T. S u z u k i , and T. Kasuya, J. Magn. Magn. Mater. 31-34, 385 119831. R. S c h e f z y k , J . H e i b e l , F. S t e c j l i c h , R. F e l t e r , and G. Weber, J . Maqn. Magn. Mater. 47&48, 83 (19851 and r e f e r e n c e s t h e r e i n . A. H a r r u s , T. M i h a l i s i n , and E. Kemly, J . Magn. Maim. Mater. 47&4~, 93 (1985).