CONCENTRATION EFFECT IN THE KONDO RESISTIVITY

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CONCENTRATION EFFECT IN THE KONDO RESISTIVITY

K. Matho, M. Beal-Monod

To cite this version:

K. Matho, M. Beal-Monod. CONCENTRATION EFFECT IN THE KONDO RESISTIVITY. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-213-C1-215. �10.1051/jphyscol:1971167�. �jpa-00214494�

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JOURNAL DE PHYSIQUE Colloque C I , supplkment au no 2-3, Tome 32, Fkvrier-Mars 1971, page C 1

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CONCENTRATION EFFECT IN THE KONDO RESISTIVITY

K. M A T H 0 (*)

C. R. T. B. T., C . N. R. S., C6dex 166, 38, Grenoble (Gare), France and M. T. BEAL-MONOD

Physique des Solides Facult6 des Sciences, 91, Orsay, France

Rbum6. - On calcule, en Born au deuxikme ordre, le temps de relaxation tlectronique dii a des paires d'impuretks magnktiques coupltes, dans une matrice non magnktique. Pour des concentrations croissantes en impuretks la contribution des paires la rbsistivitk Blectrique se traduit par une suppression du terme Kondo en InT a basse temptrature.

Pour le Mn dissous dans des rnktaux nobles on montre que l'accord avec l'exptrience est quantitativement satisfai- sant jusqu'a des tempkratures en dessous de celle du maximum de rksistivitk.

Abstract. - We have calculated, in the second Born approximation, the conduction electron scattering rate due to impurity pairs with long range magnetic coupling in a non magnetic host. The pair contribution to the electrical resistivity at non vanishing concentrations amounts to a suppression of the Kondo In T term at low temperature.

For Mn in noble metals quantitative agreement with experiments is observed down to temperatures below theresistivity maximum.

1. Introduction. - The low temperature resistance minimum in dilute magnetic alloys is described, in the limit of vanishing impurity concentration, by Kondo's one-impurity model and approximate solu- tions thereof [I], all giving resistances linear in concentration.

Experimentally, the range of validity of the one- impurity approach is limited by the onset of interaction effects often resulting in a concentration dependent resistance maximum below the minimum [2].

Theoretically, these effects are related to the RKKY- interaction [3] which has been taken into account by several authors [4, 5, 61 in an average way by means of a distribution of internal magnetic fields acting on each spin. This approach is certainly the best below the spin ordering temperature where the impurities are in some collective magnetic state. Above that temperature it is more suitable to consider directly the RKKY-interaction between pairs, triplets,

...

etc, of spins, thus making an expansion of the resistivity in powers of the concentration.

In second Born approximation A(T) is given by Kondo's formula [7], more generally by one of the resistance formulae referred to in [I].

We have calculated B(T) in second Born approximation and shall present evidence that this pair correction is sufficient to describe the experimental p(c, T) down to temperatures below the maximum.

2. Conduction electron scattering rate. - One of the authors has treated the problem of electron scattering on an impurity pair with magnetic energy El, = - WS,.S2 for spin S = 3 [8]. We keep the model described there and generalize the calculation to arbitrary spin S. The analytical result for the scat- tering rate l/z(E, T, W) per impurity as a function of electron energy E (counted from the Fermi level),

(*) Supported by the french-german Institute Max von Laue- Paul Langevin, Grenoble (France).

temperature T and coupling strength W will be published elsewhere [9].

In figure 1 the s-d exchange scattering part of l/z is shown as a function of

I

E 1 on a logaritmic scale,yat

T

+ for S = Z "?

FIG. 1. - Anomalous energy dependence of the conduction electron scattering rate.

kT = kTx = 10-2 EB in all four plots.

fixed temperature T, for several values of W. The curves marked f and af are for positive and negative sign of W, i. e. ferromagnetic or antiferromagnetic pairs, respectively. The dotted curve is the zero coupling scattering rate as given by Kondo. All curves are normalized to 117, the constant first Born result at

W = 0.

For the resistivity, an E-integration weighted with the derivative of the Ferrni function has to be perfor- med on z(E, T, W). As this weight function collects only the scattering within

I

E 1 < nkT of the Fermi surface it is evident from figure 1 that the effective scattering is always reduced as compared to the zero coupling limit.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971167

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²¹⁴ ^K. MATH0 AND M. T. BEAL-MONOD

3. The pair approximation of the resistivity. - For the calculation of B(T) in [I] a statistical average over the possible couplings W occuring in an alloy of impurity concentration c is carried out on the E-inte- grated scattering rate.

To perform this average we assume :

1. An fcc crystal structure as host matrix with a random distribution of impurities.

2. An explicit expression for the space dependence of the RKKY-interaction, W = W , F(2 k, r

+

O), with a normalized oscillating function F such that W , > 0 is the amplitude of the first oscillation outside the origin. r is the impurity distance within the pair, k, the Fermi wave number and O a phase shift that may vary within 0 < 0 < ^.naccording to different theories [lo].

Figure 2 shows - (B(T)/A(T)) as a function of W , / k T on a log-log plot. A(T) is Kondo's expression[7].

slope 1

/

FIG. 2. - The ratio of the coefficients from expression (1).

The solid line is the result of simplified statistics : the positive and negative oscillations of F are averaged over separately to give two monotonic branches of WRKKy, one positive or ferromagnetic, the other negative or anti-ferromagnetic, both decreasing like r - 3 . At the same time the probabilities to find a pair with distance r are taken in a continuum model. The vertical bars indicate possible deviations calculated numerically by performing lattice sums with the real oscillating P and with various values of 0. All the possible curves merge into the solid line at values of W,/kT where the slope in the log-log plot begins to be unity. The horizontal line for - A/B ⁼100 is drawn to point out the following : As B is the coefficient of the c2 term and c < lo-', the value of - (B/A) must be of the order of lo2 for the pair correction to become important.

Thus, in this region of W,/kT the one impurity resistivity is modified by a multiplicative factor contai- ning the effect of the interactions :

The constant a depends on the crystal structure and _- the spin value S. For fcc lattices, a = 3.4 v ' ~ .

The relation (2) between the coefficients A and 3 has been derived in second Born approximation. In order to use it in the experimental range of temperatures one has to make the tacit assumption that the same relation holds also in a non perturbative calculation.

Theoretical considerations as to the upper limit in concentration of the validity of our approach will be published elsewhere [9]. Here we present some experimental evidence that the pair approximation may be sufficient down to temperatures well below the resistivity maximum.

4. Cornparaison with experiment. - The experi- mental points in figure 3 are data of Jericho et al. Ill]

on the AgMn system. The continuous curves corres- pond t o t h e fitted expression :

where the last term is an additional normal residual resistivity proportional to c and independent of T. ( 3 )

FIG. 3. - p(c, T) of A g - Mn alloys.

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CONCENTRATION EFFECT IN THE KONDO RESISTIMTY C 1

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also describes the low concentration data [ll], where Mn in other noble metals gives equally good fits no maximum is observed in the experimental tempe- while alloys with other 3 d impurities do not seem to

rature range. follow the simple law (2).

Nominal concentrations given by the authors are The existence of a long-lived magnetic nmment on

c, = 3 350, c, = 1 250, c, = 558, in ppm. ~h~ concen- the isolated impurity, in contrary to a spin fluctuation, is necessary for the s-d exchange model combined trations chosen for the exact fit are c; = 3 380 ;

with the term = - W S 1 . S, to be

c; = 1 230, ck = 550, which is within the error bars. applicable. hi^ is most likely the case for M ~ . References

[I] KONDO (J.), Review in : Solid State Physics, Aca- [6] HARRISON (R. J.) and K L E ~ N (IM. W.), Phys. Rev., demic Press 1969, Vol. 23, chapters 12 ff. 1970, B 1, 940 ; Phys. Rev., 1967, 154, 540 and [2] Review articles cited in [8], ref. 2. Phys. Rev., 1968, 167, 878 (E).

[7] See expression (14-13) [I].

131 K1rrEL demic Press, (Ch.1, Review 1968, in Vol. : 22. State Aca- [8] [9] B B A ~ - M ~ ~ ~ ~ BBAL-MONOD (M. (M. T.) T.), and MATHO PhyS. Rev., 1969, (K.) : to be 178, pub11- 874.

141 BEAL (M. T.), J. Phys. Chem. Solids, 1964, 25, 543. shed.

[5] SILVERSTEM (S. D.), Phys. Rev. Lett., 1966, 11, 466. [lo] See ref. 25-25e in [3].

[ l l ] JHA (D.) and JERICHO (M. H.) : to be published.