EXCHANGE ENHANCED SUSCEPTIBILITY OF A SUBSTITUTIONAL IMPURITY

(1)

HAL Id: jpa-00213990

https://hal.archives-ouvertes.fr/jpa-00213990

Submitted on 1 Jan 1971

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific 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.

EXCHANGE ENHANCED SUSCEPTIBILITY OF A SUBSTITUTIONAL IMPURITY

J. Appelbaum, D. Penn

To cite this version:

J. Appelbaum, D. Penn. EXCHANGE ENHANCED SUSCEPTIBILITY OF A SUBSTI- TUTIONAL IMPURITY. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-508-C1-509.

�10.1051/jphyscol:19711169�. �jpa-00213990�

(2)

JOURNAL DE PHYSIQUE Colloque C 1, supplément au n° 2-3, Tome 32, Février-Mars 1971, page C 1 - 508

EXCHANGE ENHANCED SUSCEPTIBILITY OF A SUBSTITUTIONAL IMPURITY

J. A. APPELBAUM

Bell Telephone Laboratories, Incorporated Murray Hill, New Jersey and D. R. PENN

Brown University Providence, Rhode Island

Résumé. — Le problème d'une impureté magnétique dans une bande de conduction étroite est étudié avec les fonc- tions de Green à deux temps.

Nous avons employé un modèle de Wolff où l'interaction Coulombienne répulsive, de valeur U, est limitée au site de l'impureté.

Nous avons résolu l'équation fonction de Green découplée dans la limite U infini en présence d'un champ magné- tique fini.

De cette solution un calcul de la susceptibilité magnétique dans un champ nul est fait et des résultats numériques sont obtenus.

Nous trouvons que pour un couplage suffisamment faible entre l'atome d'impureté et ses voisins, on obtient une loi de Curie pour % pour les quatre décades de température étudiées.

Abstract. — The problem of a magnetic impurity in a narrow conduction band is studied using double time Green's functions. We have used a Wolff model in which a repulsive Coulomb interaction of strength U is limited to the impurity site.

We have solved the decoupled Green's function equations in the infinite U limit in the presence of a finite magnetic field. From this solution a conserving calculation of the zero field magnetic susceptibility x is performed and numerical results obtained. We find that for sufficiently weak coupling between the impurity atom and its neighbors a Curie law behavior for x can be obtained over the four decades of temperature studied.

In this paper we report on a calculation of the exchange enhanced susceptibility of a substitutional impurity [1, 2].

The model we use is a modification of the Wolff model [3]. The Hamiltonian used is

(1) where d^ia creates an electron with spin c/2 = ± •£ at site i, U is the strength of the coulomb repulsion at site i = 0, V is the shift in the zero of energy at the impurity site, and A is the magnetic energy of the electrons due to the external field ; the g value of the electron at all sites is assumed equal.

The hopping integral Ttj is taken to have the form (2) where

(3) We study (1) by an equation of motion technique [4].

The equation of motion for the one-electron Green's function is

(4)

where

We now write an equation for the two particle Green's function

(5) We are interested in the infinite U case so we can ignore the last term on the right hand side of (5).

Equations of motion are written for the two new Green's functions on the r. h. s. of (5) and are then decoupled according to the prescription :

at most one of /, j , f = 0.

This results in our never separating equal-time operators referring to the impurity site and leads to a complicated self consistent integral equation for the localized Green's function Doa(w) which takes the form [5] :

(6) where

(7)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711169

(3)

EXCHANGE ENHANCED SUSCEPTIBILITY OF A SUBSTITUTIONAL IMPURITY C 1

-

⁵⁰⁹

We have specialized to a Lorentzian density of states for the conduction band with width D, and introduced a cut-off in the integrals in (6),

+

nD, because of the slow convergence of the Lorentzian. Equation (6), together with its complex conjugate spin flipped counterpart, constitute a pair of simultaneous non- linear integral equations which can be solved by the techniques of Mushkelishvilli [6].

With the substitutions

the two simultaneous equations we need to solve are

-

^b' ^-^i(o

+

^A

-

V)/d -I- X2(o

+

^{A -}ⁱ⁶⁾

1

+

^{i ( o}

+

^A

-

V)/d

+

^@,(o

+

^A

+

^id) ⁽¹⁰⁾

with

f ( o '

+

A) do'

(1 1)

nD f ( o t

-

A) Y-(o' - d

-

id) d o '

z -

^0' ⁽¹²⁾

where

b = I - 2 < n o i - > . f (14) The solution to (9) and (10) for fixed b* has been extensively discussed ; the solution has the form

1

SnD

In Hf(o') do' x exp - --

2 n i ^-nD Z - o' (1 5) where C is a constant and Ht(o') a function of X , ,,(o) and polynomials in o. There is a similar solution for @,(Z). Our aim is to use these formal solutions to calculate the susceptibility

x

from the expression

x

= lim (no-

-

^n,+)/A

.

A - t O

(16)

Now from (12) and (13) it is easily shown that

The problem of calculating

x

reduces then to making an asymptotic expansion of (15) as well as a similar expression for cB2 and inserting it into (17). 1n the weak field case this leads directly to an expression for

x

in terms of the moments of H(o). The problem is therebye reduced to a long numerical calculation.

In Table I we show

x

calculated for different tempe-

The susceptibility as a function of temperature for the case

(The second column represents the self consistent no, the third the adjusted no.)

ratures. Notice that while is exchange enhanced, especially at low temperatures it does not obey a Curie Law. The local impurity occupation number, no, has been determined self consistently. If we adjust no by less than. 1 % we find that

x

obeys rather well a Curie Law over four decades.

It is difficult to access the proper role that no serves as an adjustable parameter. It may be that since we have treated no approximately by introducing an energy cut-off it is not unreasonable to expect that the proper n, for

x

might differ from that obtained

(( self-consistently >>.

Another possibility is as follows. A major feature of decoupling schemes is that they introduce in a realistic fashion width to the impurity levels due to their interaction with the band electrons. This width tends to peg the susceptibility at a value more like that of the host band than an isolated spin. We know that the strong Coulomb repulsion at the impurity acts to overcome this tendency and suppresses this one particle width in so far as it enters X. What we may be observing is that a small part of the one par- ticle spectral width enters into the calculation of 2 due to the errors in the treatment of higher order non-singular terms by the decoupling proceedure.

By varying no we are compensating for this temperature independent piece and allowing the Fermi surface singularities to generate the Curie law.

References

[I] APPELBAUM (J. A.) and PENN (D. R.), Phys. Rev., [4] ZUBAREV (D. N.), Usp. Fiz. Navk, 1960, 71, 71.

1969, 188, 874, Phys. Rev. and references contain- [English trans1 : Soviet Phys., Vsp., 1960, 3 ,

ed therein. 320.1

123 A more detailed report of the work contained herein [S] Eq. (6) has the same form as derived by THEUMANN is being submitted to Physical Review. (A.), Phys. Rev., 1969, 178, 978.

[3] WOLFF (P. A.), Phys. Rev., 1961, 124, 1030. [6] MUSKHELISHVILI (N. I.), Singular Integral Equations, 1946, P. Noordhoff ltd. Groningen, Holland.