Kondo-lattice: Kondo-effect?

Zeitschrift fL r P h y s i k B

Z. Physik B 34, 37-44 (1979)

© by Springer-Verlag 1979

Kondo-Lattice: Kondo-Effect ?* B. Schuh Institut fiir Theoretische Physik der Universit~it zu K61n, K/Sln, West Germany Received February 16, 1979 The periodic s-d-model is investigated by Green function methods. We find that the in Tdivergence appearing in third order perturbation theory for the electron selfenergy is smoothed out but still shows up in a characteristic decrease (exchange coupling 7 0) of lifetime of electrons at the fermi level as T ~ 0 and a corresponding structure in the specific heat. However, in contrast to the single impurity case, the corresponding characteristic temperature does not vary like exp [-1/171] but always is of order 72. From a mean field theory of the model, one finds a critical temperature for a transition into a magnetically ordered state T~o which varies precisely like ?2. Thus an interesting competition between Kondo-effect and long-range magnetic order can be expected in this model.

I. Introductory Remarks By Kondo-lattice (or periodic s-d-model) we mean the periodic analog of the single-impurity s-d-exchange model which thus involves one "impurity"spin per cell. Probably the most interesting aspect in the Kondo-lattice problem is the possible interplay of long range magnetic order and the Kondo-effect. A preliminary study of critical properties of such kinds of systems was initiated by Doniach, Jullien and Fields on a 1D-analog of the Kondo-lattice, the "Kondo-necklace" [1]. Clearly, since these investigations center on general properties of the model system they are far from explaining quantitatively the low temperature anomalies of some rare-earth compounds such as CeA12 and others [2] which are partially attributed to the single-impurity Kondoeffect and have been explained rather successfully on the basis of a different model [3]. It is one aim of this work to develop an approach which may lead to a more quantitative analysis of the Kondo-lattice problem. Another reason for the interest in the periodic s-dmodel is its close resemblance to the periodic Ander* Work performed within the research program of the SFB 125 Aachen/Jiilich/K/51n

son-model which has been successfully applied to rare-earth compounds exhibiting the mixed-valence (MV) phenomenon [4]. Thus the Kondo lattice may be thought of as relevant for MV-systems in certain limiting cases. To illuminate this point, let us recall that in the one-impurity case the usual s-d-model results from a Schrieffer-Wolff transformation (SWT) performed on the Anderson-Hamiltonian [5], provided the Coulomb interaction energy U between two electrons localized at the same site is large enough (and the exchange interaction is restricted to the neighbourhood of the fermi-level and becomes constant there). An analogous transformation can be carried out for the periodic Anderson model H p A = 2 gkadkadka + 2 ( \ Eofz.+ fi. ÷ k,a i,a

u

+

+

]

+.

+h.c.)

[6] which involves two localized f-states with energy E o per cell. In this situation two cases have to be distinguished which are schematically depicted in Fig. 1. We state without proof that as long as the narrow band of localized states lies well below the

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38

B. Schuh: Kondo-Lattice: Kondo-Effect?

HpA I

Eo

EF=O

D

Eo+U

SWT U>>D

occur one may look for homogeneous ferromagnetic solutions: (Si)MF = :Se, (Ti)MF = : Te where e is a unit vector in the direction of the external field. The selfconsistency equations for vanishing field can be written down after a moment's thought:

H KL

T= ½sign (JS) tanh fl [JSI/2

HpA SWT

II

I

Eo 8F=O

D

HKL+@

U>>D

de p(a) [f(a -IJTI/2) - f ( a + IJrl/2)]

Eo+U

Fig. 1. Effect of the Schrieffer Wolff transformation on the periodic Anderson model in two typical jumping-off positions

conduction band, and U is large enough, U>D, the transformation leads to the Kondo lattice problem:

=E k,a

dL

J Z Ti. s,

S = ½sign (J T) N O

(1)

i

where T~,Si denotes the f,d-spin density at lattice site Ri(Ti=½ ~ %~,f/+f/~,, r: Pauli-matrices), respectively, and J is roughly given by J=21VIZ/Eo (The shift of the conduction band energies gk ~ % is O(J)). In case II of Fig. 1, however, i.e. if E o is of the order of the fermi-energy, even at low temperature exitations of f-electrons into the conduction band are possible and the hopping terms, schematically depicted by fi+fj in Fig. 1, which are generated by the transformation should be taken seriously. So in the case which is probably the most interesting for the mixed-valence phenomenon, the analogue of the SWT does not lead to the Kondo-lattice. Nevertheless /-/ICL is relevant even for MV-systems provided they are driven by external forces into a parameterregime where the localized moments dominate the behaviour of the system. In what follows we forget about the possible origin of the Kondo-lattice problem from the periodic Anderson-model and take (1) as a model for a fermi gas interacting via an exchange interaction (J) with fixed spins T~. Since our investigation centers on dynamic properties of the d-electrons and we leave out of consideration the possible long-range magnetic behaviour of the model, it is convenient to state some properties of the model concerning its magnetic behaviour. However, even a mean-field-theory of the magnetic behaviour becomes rather complex if one searches for spatially inhomogeneous solutions for the spontaneous magnetizations ( S I ) M F , ( T / ) M F (unfortunately this seems to be the case of interest if one believes that the Kondo-lattice may serve as an explanation model for compounds such as CeA12). But to get a rough idea whether a magnetic ordering can

where fi is the inverse temperature, No p(8) the density of d-states per atom and spin direction, f the fermi function and T12=3/4 has been taken for simplicity; the chemical potential is fixed at ~=0. For quantitative calculations a Lorentzian density of states will be used throughout this paper: p(~)=D2/(e2+D2); then the equation for S reads: S = 1 sign(JT)Im 0(½+ flD/2n +iflIJTl/4n) rc ~: Digamma function). Inserting T(S) in S (or vice versa) one sees that a nonzero spontaneous magnetization exists, provided the temperature is lower than fi~-i which has to be determined from

(ticJ /4 rc)2 . ~, ( {

.j_ ticD/2 n) =

1.

(2)

For J small or D large the critical temperature is explicitly given by

(D flc)--1

..~ (j/D)2~8 TO.

(3)

Equation (3) (or (2)) gives the mean field transition temperature at which the Kondo-lattice undergoes a transition fi'om an unordered paramagnetic/Paulistate (for T- and S-spins respectively) into a spatially homogeneous ferromagnetically ordered state. The transition occurs irrespective of the sign of J, the only difference being that the local d-electron spin density S and the local moment T are parallel (J>0) or antiparallel (J