LOCAL ENVIRONMENT EFFECT IN ALLOY MAGNETIZATION

(1)

HAL Id: jpa-00214093

https://hal.archives-ouvertes.fr/jpa-00214093

Submitted on 1 Jan 1971

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

LOCAL ENVIRONMENT EFFECT IN ALLOY MAGNETIZATION

D. Kim

To cite this version:

D. Kim. LOCAL ENVIRONMENT EFFECT IN ALLOY MAGNETIZATION. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-755-C1-756. �10.1051/jphyscol:19711262�. �jpa-00214093�

(2)

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

-

755

LOCAL ENVIRONMENT EFFECT IN ALLOY MAGNETIZATION

D. J. KIM

Francis Bitter National Magnet Laboratory (*)

Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139

Rbum6. - L'effet de l'environnement local sur le comportement magnetique d'une impuretk dans un metal, suggere pour la premi6re fois par Jaccarino et Walker est etudie par un modele &Anderson etendu au cas d'impuretks multiples.

On explique cet effet par le fait que la susceptibilite de I'impurete depend de la distribution des autres impuretes dans son entourage.

Abstract. - The local environment effect in the magnetic behavior of an impurity in a metal first suggested by Jacca- rino and Walker is studied on the basis of the Anderson model extended to the many impurity case. The local environment effect is understood from the fact that the magnetic susceptibility of an impurity depends on the distribution of other impurities in its neighborhood.

Since the work of Friedel, Anderson and Wolff the occurrence or non-occurrence of a local magnetic moment on an impurity in a metal was understood in close correlation with the bulk properties of the host metal such as the density of states at the Fermi surface.

Later in 1965, however, Jaccarino and Walker [l, 21 presented a counter viewpoint based on the analysis of Co Knight shift in RhPd host. Jaccarino and Walker suggested that the occurrence or non-occurrence of a localized moment on an impurity is determined mainly by the number of other surrounding impurities in its immediate vicinity rather than the bulk properties of the host metals. This local environment effect has been seen in a number of other alloys [3].

In this paper we present a simple discussion on the possible mechanisms for the local environment effect based on the Anderson model extended to the many impurity case [4]. By using the two-time Green's function we calculate the self-energy of an impurity due to its interaction with the other surrounding impurities, The real and imaginary parts of the self-energy give rise, respectively, to the shift and the broadening (or narrowing) of the impurity state. The local environment effects originate from the fact that the self-energy of an impurity depends on the distribution of other impurities in its immediate neighborhood. We show this by calculating the magnetic susceptibility of the impurity.

We assume the impurity levels are not orbitally degenerate and the host metal is represented by a free electron band. Then the Green's function of the ith impurity Gi,(o) is calculated in the Hartree-Fock approximation :

G ~ * ( u ) = [ ~ - E ~ ~ ' + U ~ N ~ ~ - & ~ Fo+(w)- z~+(w)]-'

In eq. (I), E:+ = E:

k

pH7 where E? is the unperturbed energy level of the ith impurity and f p H is the Zeeman energy due to the external magnetic field H along the z axis, Ui is the Coulomb repulsion at the

(*) Supported by the U. S. Air Force Oflice of Scientific Research.

ith impurity site, N,, is the expectation value of the number of localized electrons at the ith impurity, V i is the matrix element of the s-d mixing of the ith impurity with the host metal conduction band which we assume to be real and

where ^E,, ^{= E,}

+

pH is the energy of a conduction electron with wave number k including the Zeeman energy. p:(w) which is obtained from the imaginary part of Fo(w) is the unperturbed density of states of the host metal conduction electrons. Notice if the species of all impurities are the same, we can drop the suffices i's from E:, Ui and Vi. Also when we refer to a quantity without the external magnetic field (and at the paramagnetic state) we can drop the spin suffices f . The self-energy due to the interaction with other impurities, Zik(u), is given by eq. (2). The.difference between Gi+(w) and the single impurity: limit Green's function G:,(U), is just this self-energy term.

In eq. (21, T i j is the direct transfer energy betweell the ith and jth impurities and

For a parabolic band

where D is the volume of the system, m is the mass of a conduction electron and ^E, is the bottom of the conduction band.

A Green's function of the form of eq. (1) was used earlier to discuss the effective exchange interaction between localized moments [5]. In these earlier works[5], however, the occurrence of a local moment was assumed from the beginning. In this paper the magnetic susceptibility of the individual impurity states is calculated from Gi(w) by assuming the impurity state does not form a local moment.

The magnetic susceptibility of the ith impurity, xi, is defined in terms of the expectation value of the number of electrons at the impurity

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711262

(3)

C 1 - ⁷⁵⁶ D. J. KIM

where Ni, is obtained from the imaginary part of

and f (o) is the Fermi distribution function.

To make the calculation of Ni+ feasible, we neglect the o dependence of the last two terms in the bracket of eq. (1) by replacing w with the chemical potential which we choose as the zero of measuring energy. In this approximation the density of states of an impurity becomes a Lorentzian :

Gi+(o) = [(o

-

E,,)

+

^iAilF1 ^(1')

with the width of the impurity state is given as Ai = A;

+

6Ai, where A: is the familiar single impurity limit width of the impurity state, Z V ; p:(0), and 6Ai is the modification of the width due to the pre- sence of other impurities :

6Ai = - ImZi(0). (3)

Notice that the magnetic field dependence of the width does not come into the calculation of the linear susceptibility and is therefore neglected. Similarly the energy level of the impurity state is given as Ei+ = Ei

+

AEi, where

where the coupling constant, cij, of the effective exchange interaction between impurities is given as

A more detailed discussion about 6Ai and cij is given in Ref. [4].

The possible origins of the local environment effect can be catalogued as being due to the impurity site dependence of the following effects. (I a) 6Ai : the broadening or narrowing of the impurity level, (I b) ReZi(0) : the shifting of the impurity level, (11)

C

cij Uj ANj : the exchange interaction between

j( 5 i).

impur~t~es. The effects of (I a) and (I b) modify the impurity density of states from pP(0) ⁼- l/n: Im G~(O) to pi(0) = - l/n Im Gi(0). The effect of (II) can be seen from the following coupled equations between the impurities susceptibilities

where

XP

⁼2 p2.pi(0)/1 - Uipi(0) and we used the [I] JACCARINO (V.), WALKER (L. R.), Phys. Rev. Letters,

1965, 15, 258.

[2] See also BROG (K. C.), JONES (Wm. H., Jr.), Phys.

Rev. Letters, 1970, 24, 58 ; NAGASAWA (H.), SAM (N.), J. Phys. Soc. Japan, 1969, 27, 1150 ; BLUM (N. A.), FRANKEL (R. B.), GUERTIN (R. P.), Bull. Am. Phys. Soc., 1970, 15, 762.

[3] For references see Ref. [4].

[4] KIM (D. J.), Phys. Rev., 1970, B 1, 3725.

low temperature limit approximation. The effects of (I aJ and (I b) are included in

xP.

In X: the modified impurity density of states pi(0) appears whereas in the single impurity limit susceptibility, XPO, pP(0) appears :

X:O = 2 ^p2p:(0)/1 - Ui pO(0). According to the Frie- del-Anderson-Wolff theory the occurrence of the local moment is associated with the divergence of the impurity susceptibility. In Ref. [4] we gave an illus- trative example of how it is possible that the impurities in the neighborhood of the ith impurity lead to a divergence in X; although

xPO

does not diverge. It is also possible that, contrary to the above situation, the surrounding other impurities make

Xo

smaller than leading to the disappearance of the local moment.

In CuNi alloys as the nickel concentration is increas- ed the density of states of nickel atoms seems to decrease [6] (therefore pi(0) < p;(~), or,

XP

^<

xPO)

although the system becomes more magnetic (X > XoO).

In this case the exchange interaction between impurities, effect (11), seems to be playing an important role.

We consider a cluster of nickel atoms, in which a central nickel atom (i

=

0) is surrounded by z other nickel atoms (i

=

1) at its nearest neighbor sites. We assume the exchange interaction is only between the central nickel atom and its nearest neighbor nickel atoms with coupling constant c,,. Then the coupled equation (7) for this cluster is solved to give

1 1

X" =

rn [x: +

z.01 -7 2 P

ud

X?]

1 1

XI =

c.0. [Xi +

c01, 2 P

uxl x;]

where the common denominator is given by

Notice the difference between the susceptibilities of a central nickel atom and the surrounding nickel atoms,

xo

and xl, is the numerator. The denominator is the same. Namely the ⁽⁽Curie point ^)>is the same but the

(( Curie constants ^)>which give the magnitudes of the local moments are different. As discussed in Ref. [4]

the coupling constant cot is on the order of 1 and (112 p2)

uX:

or (1/2p2)

u ~ ?

is on the order of the exchange enhancement factor of the impurity. In the present case of nickel in copper with the nickel enhancement factor bigger than 1 we can easily conceive of a situation where X, > X, >

xoO

even if

po(0)

<

Pl(0)

<

Po@) for ^z^N10.

'ences

[5] KIM @. J.), NAGAOKA (Y.), Pmgr. Theoret. Phys.

(Kyoto) 1963, 30, 743 ; ALEXANDER (S.), ANDER-

SON (P. W.), Phys Rev., 1964, 133, A 1594;

MORIYA (T.), Progr. Theoret. Phys. (Kyoto), 1965, 33, 157 ; CAROLI (B.), J. Phys. Chem.

Solids, 1967, 28, 1427 ; LIU (S. H.), Phys. Rev., 1967, 163, 472.

[6] SEIB @. H.), SPICER (W. E.), Phys. Rev., 1970, B 2, 1676, 1694.