Resistivity and thermopower for the Ce-Kondo system in the crystalline field

Journal of Magnetism and Magnetic Materials 76 & 77 (1988) 245-246 North-Holland, Amsterdam

245

RESISTIVITY A N D T H E R M O P O W E R FOR THE C e - K O N D O S Y S T E M

IN T H E CRYSTALLINE FIELD A. OKIJI, N. KAWAKAMI and A. N A K A M U R A Department of Applied Physics, Osaka Unic,ersi(v, Suita, Osaka 565, Japan The crystal-field effects on the resistivity and the thermopower for the Ce-Kondo system are investigated precisely at low temperatures. The enhancement of the thermopower is discussed for the hexagonal field. The anisotropic properties of above quantities are mentioned briefly.

The Bethe Ansatz method has been applied successfully to the precise evaluation of the transport and low-frequency dynamical quantities for the Kondo system at low temperatures [1,2]. In this paper we investigate the crystal-field effects on the resistivity and the thermopower for the Ce Kondo system at low temperatures. The method is based on the exact solution of the impurity system with the aid of the local Fermi liquid theory [2]. According to the straightforward analysis of the T-matrix, transport coefficients for the impurity system can be written in terms of the density of states for the localized f-state, O,,,(c). The resistivity R and the thermopower S at low temperatures are written as ~2/v(0~

0,,

S~,. = ( e r ) - ~r ( ' ) //. . . ~

= x, y, z),

(aa)

self-energy 2;f,,(c):~,,*, = 1 - d ~ f r n ( e ) / d c [ c = 0. With the use of the Bethe Ansatz method, O,,,(0) and Z'f,,,(0) can be calculated exactly through the Friedel sum rule. The evaluation of "y* of each f-state is done in the following manner. We consider the case where the J = 5 / 2 multiplet splits into three doublets, each doublet being specified by the index i = 1, 2, 3 for the physical quantities. Here we introduce the spin susceptibility X~i) (g~s = 1) for each doublet and the generalized charge susceptibility X~i-j). The latter quantity represents the susceptibility for the charge fluctuation between ith and j t h doublets. By extending the local Fermi liquid theory [3] to the case including the crystalline field, we obtain the exact relation among the quantities introduced above (detail for derivation will be published elsewhere),

,

(lb)

y,* = {(1/2)X(/) + ( 1 / 9 ) ( 2 X I j " + 2 x I J k ) + 2X~i))}

where

x,o,

__,~ = c

/O,(0),

f d~

k k.k~

g.."' = c( ~2T2/3) f de.

akmOo,(O) k.k.E

o~k,.O,,,(O) m

O~knOn

02(0) = (2

,v2/a )xf..(o)[

with akin= I ( k ° l r n ) I 2 , c is the numerical constant and I k o ) ( ]m)) represents the state for the conduction electron with wave vector k and spin o (f-electron state). Here A is the resonance width of the f-electron and the quantity "g* is the enhancement factor defined in terms of the f-electron

for j , k ~ i .

(2)

With this simple formula we can evaluate the enhancement factor Z* by means of the Bethe Ansatz solution. In the following calculation of R and S, we assume for simplicity that the first and second excited states lie energetically above A and 2A from the ground doublet. There exist still many choices of the level scheme even for the given values of the energy splitting. We discuss the characteristic properties of the above quantities with the use of a simple case for the hexagonal field: 1 + 5 / 2 ) is ground doublet and 1_+1/2) (I + 3 / 2 ) ) is first (second) excited state (the simplest case for cubic field has been discussed in ref. [2]). In fig. 1 the results for the resistivity and the thermopower are shown as a function of the

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246

d. Okiji et al. / Resistivity and thermopower for the Kondo .~3'stem 1.5--

(a)

I

i

R×x

1 / n,-

0.5 [

(b)

l

__1

~ / /

4

~2

0

1

A/T~61

2

Fig. 1. (a) Resistivity (normalized at A = 0) and (b) T-linear part of the thermopower: So =-~2/(3e) and the energy scale T~K 6~ is the Kondo temperature for A = 0. The suffices xx and zz mean the quantities parallel to the a and c-axis, respectively. crystal-field splitting A. In this figure the z-axis is chosen as the crystal c-axis. It is seen that the resistivity for the a-axis is increased and then decreased, while that for the c-axis is decreased monotonically. In the large crystalline field, both of the resistivities R~. and R : : vanish instead of reaching the unitality limit [4]. On the other hand the thermopower is seen to be enhanced in the presence of the crystalline field, which is due to the increase of y,* defined in (2). In the large field the thermopower is shown to be proportional to A 2 analytically.

While the t h e r m o p o w e r is enhanced generally in the crystalline field, the anisotropy of the resistivity and of the thermopower depends strongly on the level scheme employed. Furthermore it appears in different ways for the resistivity and the thermopower, the systematic classification of which seems to be difficult. Below we summarize results briefly for three cases of the hexagonal field in the same parameter regime shown in fig. 1 (anisotropy of the exchange coupling is not included here). (a) L ± 5 / 2 > ground doublet: the anisotropy is largest in three cases both for the resistivity and the thermopower (see fig. 1). (b) ] _+3/2> ground doublet: R , , > R : : and S,, > S:. but anisotropy is small (for instance S, , / S : : _< 1.2). (c) I ± 1 / 2 ) g r o u n d doublet: R : : > R , , while difference of S,, between S:: is vanishingly small. We have calculated the resistivity and the therm o p o w e r of the Ce K o n d o system precisely and discussed characteristic properties due to the crystal-field effects. The detail of the discussion including the comparison with experiments will be presented elsewhere.

References

[1] A. Okiji, Springer Series in Solid State Sciences 77 (1988) p. 63. [21 N, Kawakami and A. Okiji, Japan. J. Appl. Phys. 29 (1987) 449. [3] K. Yosida and K. Yamada, Prog. Theor. Phys. 53 (1975) 1286.

[4] K. Hanzawa, K. Yamada and K. Yosida, J. Phys. Soc. Japan 56 (1987) 678.