The specific-heat coefficient and the coherence temperature for the Kondo lattice system

~'~"'~ Solid State Communications, Vol.65,No.4, pp.257-260, ~i~ Printed in Great Britain.

1988.

0038-1098/88 $3.00 + .00 Pergamon Journals Ltd.

THE SPECIFIC-HEAT COEFFICIENT AND THE COHERENCE TEMPERATURE FOR THE KONDO LATTICE SYSTEM Hiroyuki Kaga and Haruhiko Kubo Department of Physics, Niigata University (Received 9 S e p t e m b e r 1987

by

Niigata

950-21, Japan

B. MUhlschlegel

)

By a finite-temperature approach of the Green-function decoupling approximation the appearance of a peak Y in the Kondo-lattice ma specific-heat coefficient is shown to result from the stabilization into the coherent state by the formation of a pseudogap at the Fermi level. This takes place at the coherence temperature T x given by Y T ~ 0.6 J/mole K, which becomes ~ 0.i T. for t ~ Ce compounds .max max . K in agreement wlth experlment (T K the Kondo temperature).

There appears a characteristic peak Yraax in the low-temperature specific-heat coefflcients y(T), enhanced by the Kondo effect, of the Ce Kondo lattice compounds CeA£~ [1,2], J CeCu2Si 2 [2], CeCu~ [3]. Because such a peak b is not observed in the alloys of these dilute and dense Kondo systems [2,3], it is generally ascribed to the coherence effect between Kondo lattice centers [2,4,5]. Similar coherence phenomena have been observed in the lowtemperature behaviors of resistivity [5,6], Hall coefficient [7,5], etc., below the socalled coherence temperatures T O which agree more or less in the different measurements. The purpose of this Letter is to demonstrate by explicit calculations the appearance of y in the Kondo lattice systems ax contrary to t~e impurity Kondo systems, and to clarify the nature of y -temperature T as max max the coherence temperature T . We are par0 ticularly interested in how T is defined in an expression, where those forma~he above Ce compounds, T = 0.2 ~ 0.5 K [1-3], experimentally obtainm~Xas an order of magnitude smaller than the Kondo temperatures T K = 3 a, 5 K [I3,5-7], can be understood. We study the non-degenerate Anderson lattice model,

proximations [8-12], in the perturbative method [4], in the variational method [13] and in the functional-integral methods [14,15] for the Kondo lattice states of the Anderson lattice model and the Kondo lattice Hamiltonian. Grewe [4] has first shown that the lattice Kondo resonance splits and develops a pseudogap near ~[ below T K. Based upon this result Bredl et [2] have interpreted their observations of y(T)-peaks ~ in the Ce compounds to be associated w ~ the pseudogap formation. More recently, Lacroix [15] has shown qualitatively that y can arise around the coherence temmax perature T O due to the temperature variation of the quasiparticle density of states at E FWe have recently made detailed quantitative studies on the lattice Kondo resonance with the forraation of a pseudogap and the felectron self-energy in the coherent Kondo lattice states as functions of temperature by numerically solving the self-consistent Greenfunction integral equations [12]. On the basis of the obtained f-electron Green functions G~(~) we construct explicitly the quasiparticle Green functions gT~,(~) using the standard procedure. The self-e~ergy Z(~) = ZR(~)+iZI(~) is defined from G~(~) =

ko +

1 ~-Ef-Z(w)

+

(2) V2 W-E~

+ { U fi~fi+fi+fi@

+ ~

v

ikR ~ (e kO

~d~ fio

+ H.C.)

,

Although this lattice self-energy Z(~) is ~independent, it has the coherent character contrary to the impurity self-energy zimp(~); the imaginary port ZI(~) becomes vanishingly small at low tempera£ures below T.K in the wide energy range near E F. The quasiparticle energy dispersion ~ = ~ is obtained from the poles of

(i)

and its impurity model in the finitetemperature Green-function decoupling method. The conduction band ~÷ is assumed to have the constant density of s~ates PO = I/2D in -D < E÷ < D (D=I) with the Fermi level E F at the - k origin, EF=0.

G~(~) with ZI(E~) % 0 : V2 -

The f-electron Green function G~([0) and the Kondo-resonance phenomenon in the felectron density of state pf(~0) have been recently investigated in the decoupling ap-

~f

-

Z[E~)

_

0

.

(31

Then, the Green function G~(W), Eq. (2), can be written by expanding the denominator ~ around the quasiparticle energy ~ : 257

THE COHERENCE TEMPERATURE FOR THE KONDO LATTICE SYSTEM

258

Gk~(~)

400

zf

(4)

(a)

i

•( b )

i

L0 - E~ where

Vol. 65, No. 4

i

T=0.5K

200

nir~(u)

n(u)

,/0.5 K

(5) f Z~ is the renormalization

5K

20 K

factor for f electron

~ 5K

given by

4°

(E~ - E~) 21-I ' a~R(~0)

f(w)

0.5K -,,.,

(6)

10

-i

5K

The renormalization factor z~ for conduction electron is also obtained as

-

~T~0.5 K

.-

~5K

.

2%

z~ = z~k

V

The quasiparticle

,

0

2

-0.02 .

Green function

(8)

g~k(to) is

obtained from Eq.(4) by removing z~ so that the resultant g~k(to)+satisfies the spectral sum-rule for each state k with the very sma~l lifetime broadening; the imaginary part of ~ is much f more reduced than ~I(E~) because z~ << i: I %

Z~k Zk I Z~k ,~ + ~

associated with the Kondo resonance are shown in Fig.l(a) below the impurity Kondo temperature T K = 4.4 K, defined by T K = 1.14 D x exp(~leflA), A -= ~p V 2 Also plotted in 0 " imp Fig.l(b] are the correspondlng results n (tO), OfmP(~0) for the impurity system for comparison. The important difference is noticed that the sharp quasiparticle spike n(0~) which develops at lower temperatures than T K shifts its peak slightly below E F, corresponding to that of the Of(to), with a less-pronounced pseudogap; the impurity quasiparticle peak in nlmP(to) stays just on the Fermi level E The quasiparticle F" densities of states have such a large energy dependence near EF, besides the strong temperature dependence for T < T , that the specificK heat coefficents cannot be simply represented by the Fermi-level densities, n(E F) and imp n (E_). We have therefore computed the specific heats C(T) and CImP(T) by taking into account the full energy and temperature dependences of n(~) and nlmP(to), and obtained their coefficients by ~(T) = C(T)/T and ~mmP(T) = CImP(T)/T. The low-temperature behaviors of the specific-heat coefficients y(T) for the various Kondo lattice systems are presented in Figs.2(a) and (b) for the changes of the parameters I/~A (the approximate magnitude of

,

0

0.02J t_0.02

0

0.02

Fig.l. The Kondo-resonance densities of states imp of quasiparticle n(to), n (~) and f electron pf(~), p~mp(to) for (a) the lattice an~ (b) the impurity Kondo systems with T K = 4.4 K, i / ~ = 20, Ef = -0.04. 1

2.C

(9)

NOW, the densities of states, n(to) for quasiparticles and Of(to) for bare f electrons,

,

W

,

9~k(to)

~ EF Y . . . .

i./?,

,

,

81

,

~'

,

,

,

,

,

....... TK=4.4 K /

--

1

~A

=20

TK=1.6K

-5 E -~ 1.2 ~

F-

~.

0

. . . . . . . .

20 40 60 80 crrg)"l

l, U II

O.8 ~ "

t--

,,56

~- 0.4

0

100

T(K)

T(K)

(a)

(b)

Fig.2. Dependences of the lattice y(T) behaviors (a) on i / ~ (T < i0 K) and (b) on T. iT < 5 K) imp together with the impurlty y (T) for I/~A = 56 (broken line). The inset shows Ymax vs i/~£. •

~

the bare f-electron Kondo-resonance peak) and the Kondo temperature T K. Also shown in Fig.2(a) is the impurity coefficient ylmP(T) for comparison. First, we note that while the impurity ~lmP(T)

increases monotonously

with

THE COHERENCE TEMPERATURE FOR THE KONDO LATTICE SYSTEM

Vol. 65, No. 4

a e c r e a s i n g temperature, the lattice coefficients y(T) do e x h i b i t a peak ~ x" For the K o n d o t e m p e r a t u r e T K = 4.4 K clmae to the e x p e r i m e n t a l values [2,3] (3 % 5 K) of the Ce K o n d o lattice compounds and I/~A = 56, the Y m a x - t e m p e r a t u r e Tma x is o b t a i n e d as Tmax ~ 0.4

5 ,

,

•

.

.

-

=

~I K

/,' /'

/I 2

075R [ a 16K Y~'~fx L o 4.4K

/

T 0.074 R "max= ~ ~

/,

./."

2

~

,~,

~-

ru

[-',attire T K

1

~"

"y"

T % y m m p T % 0 . 6 8 R which was o b t a i n e d by the K -ma$ K exact so±utlon for the impurity K o n d o p r o b l e m [16]. These results show the q u a l i t a t i v e agreements with the e x p e r i m e n t a l o b s e r v a t i o n s [2] for the Ce compounds [1-3] and alloys [2], imp lncludlng the values of T , y and y . m x m x max However, these e x p e r l m e n t a ~ rela~lons y T max 0.68R and T 0.i T are not the u n l v e r s a ~ m x K ones in the ~ a t t i c e case. In Fig.2(a) it is seen that even for the same TK(=4.4 K) both the Y m a x and Tma x values c a n change w i t h I/~A, •

,

TK /

K w h i c h is an order of magnitude smaller than T K ( T m a x ~ 0.i T K). Moreover, the Y m a x is a p p r o x i m a t e l y equal to the z e r o - t e m p e r a t u r e imp imp impurlty coefficlent y (0) , • mp max Ymax Y m a x ' and they satisfy the r e l a t i o n Y m a x

?

259

.

/

/

, ~(c), 0

I

, 1

,, 1.6K

,

"~

,

2

3

l/Yma x ( m o [ e K 2 / j )

.

w h i l e the impurity ~ i m p remains e s s e n t i a l l y the • m same, thus v l o l a t l n g ~ e above relations in the lattice case. The inset of Fig.2(a) shows that the Y m is roughly proportional to I/~A for ax the constant T. = 4.4 K. In Fig.2(b) no d e f i n i t e r e l a t l o n is found between T K and Y m a x or T K and Tmax t h o u g h Ymax decreases and Tmax increases with T KInstead of the relation y T ~ 0.68R max K w h i c h is absent for the lattice case, we have found an i n t e r e s t i n g r e l a t i o n s h i p b e t w e e n y ax and Tmax, i r r e s p e c t i v e of the K o n d o t e m p e r a t u r e T K and I/~A, w h i c h is shown in Fig.3. Here, various i/Ymax values are plotted against Tma x values by d a r k symbols, including the d i f f e r e n t K o n d o t e m p e r a t u r e s T K and h y b r i d i z a t i o n s A. It's a rather s u r p r l s e to see that there exists an a p p r o x i m a t e linear r e l a t i o n s h i p T = 0 . 0 7 4 R / Y m a x b e t w e e n Tma x and i/Ymax T ax the K o n d o lattice systems. (The d e v i a t i o n s of the c a l c u l a t e d values from the straight line can be i m p r o v e d by i n c r e a s i n g the number of the s p e c i f i c - h e a t data points.) Our impurity data fall on the line T = 0.75R/y Imp, w h i c h is close to the a f o r e m e n t i o n e d oma,x T = 0 . 6 8 R / ~ Imp, d e n o t e d by the broken {inc. The ° b s e r v e ~ a ~ a and T values [1-3] for CeAi~, ma j CeCu Si_, ~e~u_ are p ~ o t t e d by the double dotcirc{e symbols, (a),(b),(c), respectively, and are close to our lattice data point d e n o t e d by (x) (I/~A = 56, T K = 4.4 K). On the other hand, the a p p e a r a n c e of ~ m a x in the lattice case is related to the b e h a v l o r of n(E F) at t e m p e r a t u r e s below TK, w h i c h m a k e s a m a x l m u m at T by increasing and, at the • x same time, shiT~ing its peak below E F wlth d e c r e a s e in temperature. This m e a n s that T max is also the t e m p e r a t u r e at which the p s e u d o g a p on pf(L0) at ~ = EF b e c o m e s conspicuous. The o p e n i n g of a p s e u d o g a p towards a real gap occurs b e c a u s e [ i ( E F) tends to vanish b e l o w

Fig.3. The a p p r o x i m a t e relatiO~a ~ ma~ 0.074R b e t w e e n i/y and T for the m x x imp lattice (dark symbols) and T~e one y T % 0.75R b e t w e e n i/y Imp and T formate max impurity s y s t e m (open syr6bols) are seen to hold. The broken line is y l m p T = 0.68R, and the points (a),(b),m~),~c)~v and (x) d e n o t e the three e x p e r i m e n t a l data for CeA£ , CeCu Si , CeCu_ and the 2 one c a l c u l a t e ~ point wi~h I/~A b= 56, respectively.

T , thus T i m p l y i n g the c o h e r e n c e temperamax max ture T O . The o n s e t of the coherent state takes p l a c e at the c o n s t a n t specific h e a t y T % - m a x max 0 . 0 7 4 R % 0.6 J / m o l e K. If the e n t r o p y change in the i n c o h e r e n t K o n d o lattice states occurs as in the i m p u r i t y K o n d o states, then it follows that the total spin e n t r o p y R~n2 % 0 . 6 8 R is c o m p e n s a t e d for 90 % by the i n c o h e r e n t Kondo spin q u e n c h i n g above T and for the rest % I0 % by the ax c o h e r e n T - g a p f o r m a t i o n below Tmax, because Y m a x Tmax % 0.074R and y(T) ~ Ymax for T < Tmax.

If

the lattice K o n d o t e m p e r a t u r e T L is d e f i n e d by y T ~ R£n2 as in the impurity p i c t u r e for max L << the incoherent K o n d o states because Tma x TL, then T = ( imp/y ) T and T (% T~) ~ 0.i u me m x T L are obtalne~. ~or t~e Ce c max o m p o u n d su (~max) . imp and the alloys (ylmP)max' Y m a x ~ [ m a x and therefore T L ~ T K. The larger effective lattice K o n d o t e m p e r a t u r e s T L > T K are often obtained, w h i c h c o r r e s p o n d to I / ~ < 56, and are c o n s i s t e n t with the wider lattice incoherent Kondor e s o n a n c e widths (Fig.l), because the latter is g i v e n as % T K in the impurity picture. In conclusion, the a p p e a r a n c e of a peak y in the lattice s p e c i f i c - h e a t c o e f f i c i e n t ax stems from the s t a b i l i z a t i o n into the coherent state by the f o r m a t i o n of a gap on EF, w h i c h occurs at the c o h e r e n c e temperature T O close to Tma x given by Y m a x T m a x ~ ~ . I Rin2.

260

THE COHERENCE TEMPERATURE FOR THE KOND0 LATTICE SYSTEM

Vol. 65, No. 4

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9. i0. ii. 12. 13. 14. 15. 16.

Boppart (North-Holland, Amsterdam, 1982) p.371. H.G. Baumgartel and E. Muller-Hartmann, Ref.8, p.57. T. Costi, J. Magn. Magn. Mat. 47-48, 384 (1985). G. Czycholl, Phys. Rev. B31, 2867 (1985). H. Kaga, H. Kubo, and T. Fujiwara, to be published. T.M. Rice and K. Ueda, Phys. Rev. Lett, 55, 995 (1985). N. Read, D.M. Newns, and S. Doniach, Phys. Rev. B30, 3841 (1984). C. Lacroix, J. Magn. Magn. Mat. 60, 145 (1986). N. Andrei, K. Furuya, and J.H. Lowenstein, Rev. Mod. Phys. 55, 331 (1983).