SUSCEPTIBILITY AND RESISTIVITY OF PARAMAGNETIC Ni-Cu ALLOYS

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SUSCEPTIBILITY AND RESISTIVITY OF PARAMAGNETIC Ni-Cu ALLOYS

M. Fibich, A. Ron

To cite this version:

M. Fibich, A. Ron. SUSCEPTIBILITY AND RESISTIVITY OF PARAMAGNETIC Ni-Cu ALLOYS.

Journal de Physique Colloques, 1971, 32 (C1), pp.C1-748-C1-749. �10.1051/jphyscol:19711259�. �jpa- 00214089�

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

-

748

SU SCEPTIBILITY AND RESISTIVITY OF PARAMAGNETIC Ni-Cu ALLOYS

M. FIBICH and A. RON

Department of Physics Technion, Israel Institute of Technology, Haifa, Israel

Resumk. - Nous proposons un modele pour les alliages paramagnetiques Ni-Cu oh des agrkgats magnktiques possk- dant un moment gkant ( w 1 0 ~ ~ ) interagissent via les electrons itinkrants. Nous montrons que l'interaction prksente des oscillations du type Ruderman-Kittel, avec une longuenr d'onde du mgme ordre de grandeur que la taille des agregats.

I1 en resulte une susceptibilite de Curie-Weiss avec 0 negative (interaction antiferromagnktique) B faible concentration de Ni. On a obtenu une rksistivitk du type de Kondo modifik, qui depend aussi de la taille des agregats.

Abstract. - A model for paramagnetic Ni-Cu alloys is proposed, wherein magnetic clusters with giant moment ( y 10p~), interact via the itinerant electrons. The interaction is shown to exhibit Ruderman-Kittel type oscillations wlth a wavelenght of the order of the cluster size. A Curie-Weiss susceptibility results in 0 becoming negative (antiferroma- gpetic interaction) at low Ni concentration. A modified Kondo resistivity is obtained which also depends on the cluster SIX.

I. Susceptibility. - Nix-Cu,

-,

alloys have been studied extensively over a wide range of x, the atomic Ni concentration. On the ferromagnetic side of the critical composition (x > 0.44) neutron scattering [I], high temperature susceptibility [2] and saturation moment [3, 41 measurements indicate that a magnetic clustering phenomenon occurs. The clusters are spin polarization clouds which are formed in local Ni-rich regions of the random alloy.

Recently Kouvel and Comly [5] extended these measurements in to the paramagnetic region of Ni concentration (0.32

<

x < 0.44). From their bulk magnetization measurements they find that zero-field susceptibility

x0

has the form

In eq. (I), X' is essentially independent of temperature and Ni concentration. In the Curie-Weiss term, B and 8 decrease with decreasing Ni concentration, and in particular 8 becomes negative (indicating antiferro- magnetic interaction between clusters) below x 0.39.

To account for the Curie-Weiss susceptibility and for the variation of 8 with x, we propose a model of localized magnetic clusters interacting via the itinerant electrons. The clusters are randomly distributed in the alloy, each one extending over a large number of neighbouring sites (12-20). The itinerant electrons interact with the magnetic moment of each cluster as a whole, rather than with its atomic constituents.

Thus, the interaction Hamiltonian is Hi,, =

2

^J(Rn

-

ri) S,. S(ri)

.

n i (2)

Here R, are the (random) coordinates of the (center of the) n-th cluster, R i the coordinates of the itinerant electrons. S, is the n-th cluster spin operator and s(ri) the spin operator of the itinerant electrons. J(R, - ri) is the interaction coupling and has a finite range in space, of the order of the cluster size.

The cluster spin susceptibility can be expressed by

where the sum extends over the cluster coordinates.

x,,, is the time integral of the correlation

<

Sn(t) Sm(0)

>

and is shown diagramaticcally in figure 1.

The shaded bubble represents x,,, the open bubble represents

x,,

0 (the paramagnetic spin susceptibility of non-interaction spins), and the wiggly line repre- sents V,,,, the effective cluster-cluster interaction (located at R , and Rmt respectively). The diagram in figure 1 corresponds to the integral equation

0

Xnm = Xnm -!-

C XL

^{Vn'm, Xm'm}

n'm'

FIG. 1. - Diagramatic representation of Eq. (4) for the suscep- tibility XnG. The indicesrefer to cluster positions, the wavy lines represents the effective interaction ^V,,',',. The open bubble and the shaded bubble represent x0,, (the susceptibility of

non-interacting spins) and x,, respectively.

where the non-interacting spin susceptibility is given by

M being the (average) cluster magnetic moment.

The cluster-cluster effective interaction mediated by the itinerant electrons is given by

where ~ , ( q ) is the itinerant electron spin density sus- ceptibility. Since J(r) has a finite extent in space, of the order of a mean cluster size, J2(q) is very small outside the range

I

q

I

,< Q, where l/Q is of the order of the interaction range (cluster size). Since Q < k,, k, being the Fermi momentum, and ~ , ( q ) is characteris- tically fairly constant over a range of 2 k,, we may replace x,(q) in eq. (6) by ~ ~ ( 0 ) . From eq. (3), (4) and (6) we find, with these approximations

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711259

SUSCEPTIBILITY AND RESISTIVITY OF PARAMAGNETIC Ni-Cu ALLOYS C 1 - 749

where N, is the number of clusters in the system, F is given by

sin qrn

F(r,) = - (8)

and r, is the distance from a given cluster to the n-th one.

Since the clusters are randomly distributed in the alloy, the measured susceptibility per atom is the configurational average of eq. (7) divided by the num- ber of atoms. One can show that the variance of the quantity F(r,) is of the order N,'. We further approximite

<

F(rn)

>

by CF

(<

r

>),

where

<

^r

>

is the mean nearest neighbour distance. For a n

random distribution the probability to find a nearest neighour at r is [6] W(r) = n, enp

(

^- 4 n ^{n, r}

')

^where

is the density of clusters, no being the atomic density of the alloy. The average

<

^r

>

is thus found to be

<

^r

>

^{= 0.554 n, 'I3. With}

x0

given in eq. (5), the average susceptibility per atom assumes the Curie- Weiss form (see Eq. (I)), where

where

and

Xe

is the electronic susceptibility per particle.

This will lead to F(r) of the form sin Qr - Qr cos Qr F(r) = J ;

( Q d 3 (11)

As seen from eq. (10) 8 varies with the cluster concentration C and thus is a function of Ni concen- tration. In particular since F oscillates, 8 becomes negative at low Ni concentration. In addition, compa- ring eq. (10) with Kouvel and Comly's data for 6 and B (C,, in reference 5), we can estimate Jo and Q- the strength and the inverse range of the interaction.

If we assume that M and Q are concentration independent and fit the data of ref. [5] at 8 = 00

[I] HICKS (T. J.), RAINFORD (B.), KOWEL (J. S.), LOW (G. G.) and COMLY (J. B.), Phys. Rev. Letters, 1969, 22, 531.

[2] RYAN (F. M.), PUGH (E. W.) and SMOUCHOWSKI (R.), Phvs. Rev.. 1959. 116. 1106.

[3] AHERN

(s.

A.), MARTIN (M. J.

c.)

and SUCKSMITH (W.), Proc. Roy. Soc. (London), 1958, 248, 145.

[".

^{m.;. OK] 113}

and 8 = 240, we obtain

P

⁼ 0,28 and a = 0,8 x lo6 -

.

Taking M = 10 pB, we

(e. m. u.)

find Q = 0,94 n;l3 and 5, pB2 no 1 eV. We find, however, that with these assumption, eq. (10) does not fit the data too well at other points, giving some- what higher values for 8. It seems likely that a better fit can be obtained assuming Q to increase slowly with decreasing the Ni concentration (notice the Q3 dependence of a).

11. Resistivity. - Within the framework of the model proposed in Sec. I, one should expect to find in the paramagnetic Ni-Cu alloy, a resistance mini- mum just as it is found in dilute magnetic alloys.

We follow Kondo's [7] original treatment of the resistivity and express the relaxation time , z, which enters the Boltzmann equation, by

where the transition probability, Wk,, is given by Wkk' = 2 zS(S+I)

N

x ( ~ ' ( k - k') + 4 J(k-k') g(k, k', ek)

)

b(ck - E ~ ) (13) and

Here J(k - k') is a constant Jo, only for

I

k - k' )

<

^Q

and zero otherwise. Carrying out the calculation of the resistivity p, in the standard way, we find

where p, = [3 n nS(S

+

^{1)/2 eZ AE,]} ( V / N ) J:, as in Kondo's paper.

Expressing p = C(Q/2 kF)4 [po f (Q/2 k,)' p1 log T]

and adding the phonon contribution aT5, to the resis- tivity, we find

and

If we take C, the cluster concentration from the sus- ceptibility data, the measurement of Tmin and p,, - pmin for different alloys, could provide infor- mation on the dependence of

e3

M on concentration.

We wish to thank Dr. J. S. Kouvel for sending us his data prior to publication. One of us (A. R.) wishes to thank Dr. K. H. Bennemannfor stimulatingdiscussions.

ences

[4] ROBBINS (C. G.), CLAUSS (H.) and BECK (P. A.), J.

Appl. Phys., 1969, 40, 2269.

[5] KOWEL (3. S.) and COMLY (J. B.), Phys. Rev. Letters, 1970. 24. 598.

[6] CHAUDRASEKGAR-(S.), Rev. Mod. Phys., 1943, 15, 1.

[7] KONDO (J.), Prog. Theor. Phys., 1964, 32, 37.