An exactly soluble model for the combined direct and indirect exchange interaction between two magnetic moments in an electron gas

Physica 54 (1971) 425-441 0 Nwth-Hollaad Publishing Co.

AN EXACTLY

SOLUBLE

AND INDIRECT BETWEEN

MODEL

FOR THE COMBINED

EXCHANGE

TWO MAGNETIC

MOMENTS

DIRECT

INTERACTION IN AN ELECTRON

GAS

W. A. SMIT Laboratorium voor Vaste-Stof-Fysica,

Universiteit van Groningen, Nederland and

G. VERTOGEN Ilzstituut VOOYTheoretische Natuurkunde, Universiteit van Groningen, Nederland Received 20 January 1971

Synopsis An exactly soluble model is presented for the case of two localized magnetic moments with spin S = 4, which are directly coupled by the Heisenberg interaction of strength G and indirectly by the s-f or s-d interaction of strength J via the conduction electrons. This Zener model displays important differences among the four existing cases corresponding with the possible signs of G and J, e.g., both magnetic moments are only correlated in an antiferromagnetic way in case the direct interaction is antiferromagnetic. It appears that the most interesting case viz. a ferromagnetic s-f interaction and an antiferromagnetic Heisenberg interaction displays a very unexpected feature, namely the correlation function describing the relative direction of both magnetic moments in the ground state exhibits a discontinuity as a function of J/G when the spin clouds, which are built up by the electron gas around the moments, overlap.

1. Introduction.

The idea of an indirect

interaction

between

localized

magnetic moments via the s-f interaction of these moments with the conduction electrons is basic to an understanding of the magnetic behaviour of the rare-earth metals. This type of interaction is known as the RudermanKittel-Kasuya-Yosida (RKKY) interactionl+3*3). Because of a number of difficulties associated with the derivation of the RKKY interaction an exactly soluble model concerning this type of interaction was developed4). This RKKY model treats the case of two magnetic moments with spin S = & in s-f interaction with an electron gas. The purpose of this paper is to show that this RKKY model can be extended to an exactly soluble model which contains the aspect of an indirect interaction between the localized magnetic moments via the s-f (or s-d) interaction with the conduction electrons as well as the aspect of a direct 425

426

interaction interaction

W. A. SMIT AND

G. VERTOGEN

between the localized magnetic moments, where the direct is assumed to be a Heisenberg one. The most interesting feature

of this model is that it gives a complete

description

of the interplay

be-

tween the direct and the indirect interaction, each of which separately can give rise to a completely different correlation between both magnetic moments. Clearly there is some analogy between this model and the situation in crystals of metals and alloys of the iron group where both the direct exchange interaction between the spins due to the overlap of the d clouds, and the indirect interaction produced through the s-d exchange mechanism are important. Like most exactly soluble models this model also exhibits a serious defect, namely, the kinetic energy is assumed to be the same for all one-particle levels, which are perturbed by the s-f interaction. This assumption is closely related with the one concerning the nature of the s-f (or s-d) exchange integral J(&‘, k). B ecause the exact nature of this integral is unknown the relevance of this model to the situation in metals is unclear. Although it might be argued with good reason that the relevance of exactly soluble models just lies in the fact that they are exactly soluble and as such more than make up for their deficiencies, it has to be noted that in fact the underlying model represents a physical situation. In the case of a narrow conduction band the assumption of the constancy of the kinetic energy is very reasonable. This model, therefore, represents the situation of two magnetic impurities built into a solid with a narrow conduction band, where only the Heisenberg interaction between both magnetic moments and the s-f (or s-d) interaction of the moments with the conduction electrons is taken into account. The organization of this paper is as follows. In section 2 the model is presented. In section 3 the eigenvalues, eigenstates and relevant correlation functions are given, while in section 4 the results are discussed. 2. The model. The model consists of two directly interacting localized magnetic moments with spin S = 4 built into a free-electron gas. This direct interaction has the form -GSr.Ss. The interaction between the localized spins and the electron gas is assumed to have an s-f (or s-d) character. Now the following assumption is made concerning the nature of the exchange integral J(k’, k). After dividing the momentum space into three different regions separated by two spherical surfaces centred on the origin it is assumed that 1. In region I and III the exchange integral J(k’, k) equals zero; 2. In region II J(k’, k) equals a constant J/Q, where Q denotes the volume of the system. Region II is defined by the wavenumbers kF + D, where D
EXCHANGE

INTERACTION

BETWEEN

TWO MAGNETIC MOMENTS

427

4. All one-particle levels in region II have the same kinetic energy E. It is obvious that the hamiltonian of the system breaks up into three commuting pieces. Only that part of the hamiltonian concerned with region II is of interest, because the hamiltonian for regions I and III is simply the free-electron one. The hamiltonian for region II reads N

HII =

E

N

J

C c&,ckn, ?L=l,CT

__ 252

IZ n,m=l

=

Z exp[i(km-h2) p=l

-&I

x [(c$,,+c,~+ - c;~_c~,_)S; + ck+,+ck,-S; + c;/~+SP+]

- G&*&t.

(2.1) In formula (2.1) R, denotes the position of magnetic moment 6 with spin S,, while the total number of states in region II is given by 2N, N spin-up and N spin-down states, where N is approximately given by N = QkiD/x2. In order to investigate the properties of the hamiltonian HII it is of advantage to change to a new representation. This representation appears to bed) 3

N

C d$d,,l=l,o

HII=&

where the operators 01 =

g

(1 +

i=l

Oi (i = 1, 2, 3) are defined by Cl -

fvqq

x [p=l i (d,+,d,+02 = $-{l

d&d,-)

03 =

G-+d2+

-

+ (d:+d,-

The operators

S; + d:+d,_S,

+ d;+d,+ - d;-d,_

+ d$+dl-)(S1

+ (d,f_d,+ + d;-d,+)(St

f(R) =

+ dp+_d,+S;],

(2.3a)

function

- d;-dl-)

(2.3b)

(S,z + S;)

+ S,) + !%)I.

(2.3~)

f(R) is given by

cos kpR sin DR k;DR3

+ d,+_d,+S,+

d,+_d,_) S; + dz++d2_S, + d,+_d,+S,+],

sf(R)[(d:,d,+

The appearing

S; + d,+,d,_S,

- [l - f2(R)]‘}

x [(dl++dr+ - d;_d,_) +

(2.2)

C 02-G&.&,

sin kpR sin DR +

kpDR2

cos kFR cos DR -

d& and d,, simply satisfy the well-known

k;R2

.

(2.4)

anticommutation

W. A. SMIT AND G. VERTOGEN

428

relations. Evidently

it is sufficient to study the hamiltonian

(2.5) because the mutual influence determined

of both magnetic

by this hamiltonian.

moments

In the next section

on each other is

the eigenvalues

and the corresponding eigenfunctions IET, k> of the hamiltonian presented, while also the correlation functions

kI&~&IE~,

Er

H;,

are

k>

are given. The correlation functions $ have been calculated in order to obtain information about the relative direction of the magnetic moments. The eigenfunctions IET, k> are expressed as linear combinations of the functions

appearing in ref. 4.

3. Gelteral solution.

With the aid of a number of simple group-theoretical

arguments the following

results are obtained

1. E’;“l = -_tG,

(3.1 a)

IE’;Zl, 1) = (El, I), 1"1: =

IE’;“l, 2) = (El, 3,

~;

=

G%*S2>~.:

=

IEr, 3), (3. lb) (3.lc)

4.

(3.2a)

2. Ey2 = PG,

3.

IEY2> = IEI, 4>,

(3.2b)

~2

(3.2~)

= -2.

ET = E - g

[I + f(R)] -

IEF, I> = IE2, I), (ET,

4.

=



ET = E -

$

IE?,l>

IE3, I>,

IET,

IEF,2>

=

&=

[I - f(R)] -

=

<&*Sz>

lE2,2>,

(3.3b)

T&=

(3.3c)

&=&

(3.4a)

$G, IEP, 2> = IE3,2>,

(3.4b)

IE?, 4> = IE3,4),

3> = IE3,3),

&

=

IEY, 4) = [Ez, 4>,

3) = IE2,3),


(3.3a)

&G,

T2 =

Gl's2g3

=

<&*S2>7,*

=

t.

(3.4c)

EXCHANGE

INTERACTION

JN

5. E’4” = E + -=

/ET,

BETWEEN

TWO

MAGNETIC

MOMENTS

429

[l + f(R)] + &G + 4

x [1 + f(R)1 - G

(3.5a)

1) = ~1 IE4, 1) + bllE5, I),

(3.5b)

IEy,2)

=

alIE4,2)


+

(3.5c)

blIE5,2),

<&*S2>~2

2[1-fz(R)1~+[4--6f(R)+2f"(K)lb

1

a' 2(8-22f2(R)+[4--f(R)l[4+2f(R) -2f2(R)lb)*

4 --[

1

2[1-fa(R)]a-[4-6f(R)+2f2(R)]+ + ”

2{8-2fz(R)

- [4-f(R)][4+2f(R)

2

-2fz(R)]t}*

(3.5d)

’

@I = ~J2G[sgnf(R)1[3--3f(R)lt[4-2f(R)l-6

,~4-2fvw+m1 -

-G ~P+ml+

- @4-2f(R)l-l ~4-qq1~[1

G2

+f(ql"

7 4522

[4--2f(R)l[l+f(R)l -G;

$-

G[4-2fwi-l

[l+f(R)l+ G2

[4--2fvw~

-‘>

(3.5e)

+fw*

(3.5f) with sgnf(R)

= 1

sgn f(R) = - 1

6.

if

f(R)2 0,

if

f(R) < 0.

(3.W

EY=F+$[I+~(R)]+;G

f

‘g

I-&?, 1) =

a2IE4,

--

[4--2fPW 1) +

+f(R)l

b2 IE5, I>,

- G$

Cl+f(R)l

+ G2jt

(3.6a) (3.6b)

430

W. A. SMIT

Iq?,2i

=

azIE4,2)

'j;1=

+

ANL) G. VERTOGEN

(3.6~)

WE5,2>,

c%*s2g2

2L.l -f2(R)l*+[4-6f(W+2fV)l*

2(8-22f2(~)+[4-~(~)l[4+2f(~)-22f2(~)l~}t

1

2[1 -/2(R)]“-[4-6f(R)+2f2(R)]~

2

+ b2 2{8-2f2(R)-[4--f(R)][4+2f(R)-2f2(Z2)]a}t a2 = gJ2G[sgnf(R)1[3--3f(R)l,

[4--2f(m -

twi

[4--2t(H)l[l

- G $51

+f(w]

[4-2f(R)lh

[1 +f(qlk

- Gz

[1 +/WI

+@)I

+ Gz

+ Gz

- G[4-2f(R)]-l[4--2f(R)l*[l +/@)I”

7.

(3.6d)

[4--2f(R)i-r

- q4-2rpq-1

4522

’

T

[4--2fuw

+ f(R)1 - G $

T

[4--2twiv

+mi

- G$

[1 +W)l

[1+@)I

+ G2 + Gt}-i.

-‘I

(3.6e)

1 (3.6f)

Er=E+$[l-f(R)]+& ,4+2f(R)][l IE’g”, 1) = a3 I&,

IEr,2>= &

- G$

[l-f(K)]+G2 ', (3.7a) (3.7b)

1> + b3 IE7, I>,

asIEs,2> =

-f(R)]

+

(3.7c)

b3IE7,2),


2[1 -f2(W-[4+6f(R)+2f2(R)lb - [4+f(R)][4--2/(R) -2f2(R)lt)*

+ b3 2(8-2/2(R)

1’ 2

a3= -~J2G[sgnf(R)1[3+3f(R)lt [4+2f(R)l-* w[4+2f(w[l

-f(K)1 - G $

[1 -f(R)1

+ G2

(3.7d)

EXCHANGE

INTERACTION

JN

-

-

-

2Q

BETWEEN

G[4+2f(R)i-1

I

~[4+2f(w~(~)j

bg=&/2

1

-

g

[

-G~~w(~)I

- @4+2/(R)1’}

J2N2

+

I

MOMENTS

431

-WI

- G ~CI

g[4+2f(R)m

-f(R)1

- G$

(3.7e)

+ GUT]-‘,

[4+2f(Wh

4R2[4-(-2ww1

1 X

MAGNETIC

[4+2f(R)lt 11-f(W”

JzN2

x

TWO

Cl-/@)I+

-WI

+ ~“)i]

[ 1 -f(R)]

+ G2 -*.

(3.7f)

I

8. E”=E++(R),+&G - ;{~[4+2f(R),[l-j(R)] IEY, 1) =

a41&,

IEy,2)

a4IEs,2>

=

<&~sz>~l=

---

1) + +

- G+$f(R)]

b4lE7, I),

(3.8b)

b4IE7,2),

(3.8~)

4

2[1 -f2(~)1*+[4+W)+2f2(JW

u4

[

2{8-22f2(~)+[4+f(~)l[4-22f(~)-22f2W1t}t

2[1 -f”(W-[4+6f(~)+2fz(R)lt ’ b4 2{8-2f2(R)-[4+f(R)j[4-2f(R)-2f2(R)]*}~

~4 =

-~J2G[sgnf(R)1[3+3f(R)lt

[4+2f(R)l-h

X

- G;

$$4+2f(R)lP -

X

(3.8a)

<&*sz>y2

1

[

+ G2 ‘,

-f(R)1

- Q4+2f(R)l-l

$$+21(R)j[l -f(R)3

bq= -+J2

4522

g

-f(R)]

[I

J;

- G -_[I

- G[4+2f(R)l-l

[4+2f(R)l[l

II1-f(R)1

[4+2f(R)Y

1’ 2

(3.8d)

+ G”

-f(R)l’

t -f(R)1

4

+ G2

,

(3.8e)

II

[4+2f(R)l+ [l -f(R)]* - G +I

492 [4+2f(R)l[l--f(R)]-

Gg

-f(R)]

+ G2>‘l

[I -f(R)]

+ G2 -*.

(3.8f)

432

W. A. SMIT AND

9. p=2E_JN_ 8

‘G z’

252 I-Es,

I-q, 1) =

l>,

IE;;",2) =

IE;;“,4) = /Es, 4>, ~l= =

10.

<&.S2>;ll4=

I&?,

3) = IEs, 3>, (3.9b)

jE8,5>,

~3 (Sl.S2);115 =

(3.9c)

4.

E$nl = 2~ - $G,


l

s,>;,y

(3.1 Oa) (Er’, 2) = (Es, 2?,

= (S1. s,>$$ = (&!?I.s,>;;

IErI, 3) = IEg, 3>, = 2.

E”

10

(3.1 la)

lEr2, 2) = lEg, 5>,

1) = I&, 4>,

(S1* s,g;

=

= (S1 s,>T;

IG 4

(3.1 lc)

L FJ2N2 [l -f2W1

+

2

a5lE lo, 1) +

I&?,, 3 = a5lEl0,2>

+

(3.1 lb)

= -2.

l

xF +

IE';",, 1) =

+ G2 ',

(3.12a) (3.12b)

b5 lE11, I), b5

PII, 3,

(3.12c)

IEC, 3> = as IE 1093) + bs lE11,3>, G%*sz>y~,l

=

<&*Sz>;"u,2

1

G 2

--4 a5 =

[I

=

J2N2

F

G 2%,1-f”(R),

+ y

(3. lob) (3.1 Oc)

Er2=2c+$G, lEr2,

12.

IEs,B,

IE;(“,5> =

G%S2)72=

IE T’, 1) = IEg, l),

11.

G. VERTOGEN

[1-/2(R)]”

(3.12d)

t )1

;;,J

[1-f2(R)1 + G”

-1

,

(3.12e)

+ 2G”

g

[1-/2(R),

+ G”

[1

,

(3.12f)

-f2(W1 + G2

+ +-/2(R)]*{

‘g

[1

-f2(R)1+ G2

s

(3.1%)

EXCHANGE

INTERACTION

13. E;‘“, = 2~ + $G lE;n,,1) =

BETWEEN

z’

a6IEl0,

g

1) +

TWO

[I -f(K)]

MAGNETIC

MOMENTS

+ G”)‘,

(3.13a) (3.13b)

b61'511, I>,

IE’iZ,,2> = ~6 IElo, 2) + b6 IEm

(3.13c)

3,

(3.13d)

IEE, 3) = @6 IElo, 3> + b6 IEII, 3>, <&*Sz&

=

<&*Sz>‘;“l,2

z

=

a6 = G 2 g

y~,3

[1-/2(R)]

$

[ 1-/2(R)]

433

+ G”

,

(3.13e)

+ 2G2 b 4

[l-/“(WI

+

G”

,

(3.13f)

11 &I =

-

;

[1 -f”(R)]

+ {g

[l -f2(R)]

+ G”H

+ 2G2

[l -f”(R)]

[I-f"(R)] + w G2

14.

E;n2 = 2~ + g IEE

1) =

c%s2>;,1

15.

16.

-

g

IE12, I), =

[1+8fz(K)]i I&%

2)

‘;“z,2

=

lE’$, l> = 2~ + $

+ z

I%

lE;,2>

1) =

lE13,

I),

<&-S2&

=

Erd=2&+-

JN m +iG

+ i 2 IE;>

=

y3,2

J2N2 F [4-33fz(R)]

~7 IE14> +

b7 IE15>,

=

-

tG,

(S1*S2);n2,3

= =

IE13,2),

(s1*s2>y3,3

-

2G +

(3.1

(3.14a) IE;, 3) = IEm 3>,

IE12,2>,

[l +8f2(R)1*

.

=

-

&.

(3.14c)

iG,

(3.15a)

I&$,3) =

(3.14b)

$.

+ G2 ‘,

= IE13,3>,

(3.15b) (3.15c)

(3.16a) (3.16b)

434

W. A. SMIT

<&

l

AND

G. VEKTOGEN

sz>li;

1

2[1-p(K)]*+~4-77f”(K)+3f4(K)]~

4 --L

u7 {32-32/2(R)+6/4(R)+[16-

10/2(R)][4-33f2(R)]~}~

1‘,

2[1--f~(R)]~-[4--7fz(R)$3f4(R)]~

(3.16~)

+ b7 {32-32~2(R)+6f4(R)-[16-10f2(R)J[4-3f2(R)]~)) a7 = &/2G[3--3fa(R)]* x

[

J2N2 [4--3fz(K)] szz

x [4-3f”(R)J4

-

2G $

g[4-3/“(R)]

z

17.

[4-3/“(K)]-? + G” -

-

- G[4-3f’L(R)]-1

{g

2G;

+ G”

G[4-3/z(K),-‘}

-‘.

(3.16d)

[4--3/“(R)]*

[4--3/d(R)]

-

2G ;

+ G”

’

[4--3f2(K)]

-

2G z

+ G”

.

E’i”5 = 2~ + g

-

(3.16e)

+ *G

-

2G ;

+ G” ‘,

(3.17a)

lE;IL5>= as lE14> + 6s lE15),

--LU8 41 + b8

{32-32f~(R)+6f4(R)+[162[1-f~(2?)]~+[4--7f”(R)+3f4(K)]~ 10f2(R)][4-33f2(R)]~}*m

12,

1 2[ -p(R)]*-[4-7p(K) +3f4(R)]h {32-32f2(R)+6f4(R)-[16-lOOf2(R)][4-33f2(K)]*}t

u8 = &/2G[3--3fz(R)]k

[4--3f”(R)],

(3.17c)

[4--3/z(R)]-h

[4--3fz(R)]

x

(3.17b)

-

2G g

JzN2 522 [4-3f2(R)]

+ Gz +

-

2G g

JN Q

- G[4-3f”(R)]-1

+ G2

b-1 II ,

(3.17d)

EXCHANGE

INTERACTION

bg = --gJ2

BETWEEN

TWO

MAGNETIC

18. E’;” = 3.5 - $

[I +f(R)]

- 2G g

+ G2

- 2G g

+ G2 -‘.

’

(3.17e)

- fG,

(3.18a)

IE’L 1) = IEm

I>,

IE;ns, 2) =

IEE;Zs, 3)

3>,

IJ%;“,,4) = IEm

IEm

2>,

(3.18b)

4)s

?6,1 = ~6,~ = ~6,s = Y~,~ = ). 19. E;“7 = 3~ - g

[ 1--f(R)1 > -

fG,

PET;, 3 =

lE17,2>,

P&3> =

I&$7,4>=

lE17,4>,

lE17,3>, =

y,,2

20. E”18 = 3.5+ E [ 1+f(R)] 49 + ;

IETs”,,1) =

I

=

~,,3

=

(3.19b) y74

=

$.

(3.19c)

+ z‘G

[4--2fWlP +fWl -

g

(3.18~) (3.19a)

FE”,, l> = IE17, I>,


435

- G[4-3f2(R)1-‘}[4-Y”(R)1r

;

= IEm

MOMENTS

G;

[l +/@)I +

(3.20a) G2}i,

1) + bl IEm l),

(3.20b)

IE& 3 = al IEm 3 + bl IEm 3,

(3.20~)

y*J

(3.20d)

~1

=

IEm

y*,2

21. &,’ = 3~ + $

[I +/@)I

=

<&*Sz>&

+ $G

t (3.21a) IE&“,,1) = az IE18, 1) + bzIEm I>,

(3.21 b)

I&% 3 =

(3.21~)

<&*s2>;,l

a2 IEm =

2) +

bz IEm

yg9,2 =

2>, ~l.

(3.21d)

436

22.

W. A. SMIT

E;

= 3~ + g

+ ;

[l -f(R)]

g

AND

G. VERTOGEN

+ iG

[4+2f(K)][l

-f(R)]

-

G;

[I-f(R),+ Gz}',(3.22a) I

I-G,,

1) =

IGG,“,, 2)

=

a3 F20, a3 IE20,B


=

I&l,,

1)

g =

IE;;,2) =

+

b3 F21,

+

b3 IE21,2),

$k,2

23. E!$ = 3~ + g - ;

1)

[l -f(R)]

=

a4 lE20,2)

1) + +

(3.22~) (3.22d)


+ tG

[4+2f(R),[l

~4 lE20,

(3.2213)

I),

-f(R),

b4 IE21,

- Gg

[*

(3.23a)

-f(R)1

(3.23b)

I),

(3.23~)

b4lE21,2),

(3.23d) (3.24a)

24. EI/L,’= 4~ - $G, IE22,3>,

(3.2413) (3.24~) (3.25a) (3.25b)

(3.25~) 4. Discussion. In the following it is assumed that Y states are occupied in region II, where 2 I: Y I 2N - 2. Otherwise the number of eigenvalues of the hamiltonian Hi, will be reduced, which simplifies the problem in a rather drastic way. In this model both magnetic moments are subjected to two kinds of interaction. They are coupled by a direct exchange interaction of strength G and both interact with the conduction electrons via the so-called s-f interaction of strength J, These two types of interaction were first suggested by Zeners) in order to explain the ferromagnetism in the transition metals. Zener pointed out that the interaction of the magnetic moments with the conduction electrons would result in a ferromagnetic coupling between the magnetic moments, while the direct exchange coupling always possessed an antiferromagnetic character.

EXCHANGE

INTERACTION

BETWEEN

TWO

MAGNETIC

MOMENTS

437

The underlying model contains both features of the Zener argument, namely the indirect interaction tends to align the moments in a ferromagnetic way as has been shown in ref. 4, while the antiferromagnetic character of the direct interaction can be taken into account by choosing the appropriate sign for G. A third effect on the correlation between the magnetic moments in the Zener theory, which is neglected in this model, is the effect of a variable kinetic energy of the conduction electrons. Because of statistics this effect counteracts a polarization of the conduction electrons and therefore influences the indirect interaction between both magnetic moments in a negative way. It is clear that this effect can be neglected in the case of a narrow conduction band. Corresponding with the possible signs of G and .J four different cases have to be considered. Bearing in mind that If(R) / < 1 the following groundstate energies and there with corresponding correlation functions are obtained. A. Both the s-f and Heisenberg coupling are ferromagnetic (J > 0, G > 0). In this case the ground state energy of Hrr reads

JN

-

EA=YE-

2LJ

-

while the corresponding <&*S2)A

=

a,

tG =

(r -

2) E +

correlation

E;I”,

function

(4.1)

is given by R # 0.

for all distances

(4.2)

The degree of degeneracy of the ground state equals 5. B. The s-f coupling is antiferromagnetic, while the Heisenberg coupling is ferromagnetic (J < 0, G > 0). In this case the ground-state energy of HII reads J2N2 [4-3f2(R)] =

2G JN + Ga ’ sz

(4.3)

(r-2)~+E;.

This state is a singlet. The correlation B

-

=

(see

<&*sZ>;

C. Both the s-f and Heisenberg G < 0). In this case the ground-state

&=ye+~+.@-~ 2.0

= (r-2)E+E&

function

2 t

fig. 1).

(4.4)

coupling are antiferromagnetic energy of HII reads

J2N2 [4-3f2(R)]

n2

is given by

-

2G +

(J < 0,

+ G2 * (4.5)

W. A. SMIT AND G. VERTOGEN

438

Fig. 1. The correlation function (S~-&)B as a function of 1JN/4GRI plotted on a logarithmic scale (J < 0, G > 0). Curve “a” denotes the case f(R) = 0, curve ‘lb” If(R) 1 = 0.4, and curve “c” If(R) 1= 0.9.

“4

0.2 0.1 0 -0.1

1

-0.2

-0.3 /d-J my -0.4 5

-0.5 -0.6 -0.7 -0.8

0.01

0.1

1

10

100

Fig. 2. The correlation function (Sl-S&c as a function of IJN/4GQj plotted on a logarithmic scale (J < 0, G < 0). Curve “a” denotes the case f(A) = 0, curve “b” If(R) I = 0.4, and curve “c” If(R) I = 0.9.

This state is a singlet and the correlation G1*Sz>c

=

<&*Sz>‘T*5

(see

function is given by fig. 2).

(4.6)

D. The s-f coupling is ferromagnetic, while the Heisenberg coupling is antiferromagnetic (J > 0, G < 0). In this Zener case three different situations occur according to the magnitude of /2(R) : (i) In case -2QG/JN < /2(R) < 1 the ground-state energy reads -

*G = (r -

2) E + ET,

(4.7)

EXCHANGE

INTERACTION

while the corresponding
=

BETWEEN

correlation

TWO MAGNETIC

function

is given by

( 4.8)

a.

The degree of degeneracy of this state equals 5. (ii) In case /e(R) < -252G/JN the ground-state ED

=

rF

?! + 1G

+

2J-2 = (r - 2) E while the correlation <&*sZ>D

=

J2N2 [4-3fe(R)]

_

4

-

2G p

+ G2 ’

sz (4.9)

function

equals (4.10)

~.

ED==YE--g

-

= (r -

energy is given by

Ey5,

+

(iii) In case f2(R) = -2QG/JN tG = (r -

the ground-state

energy reads

2) E + ET

2) E + EyI = (r - 2) E + E;,

while the corresponding D

439

MOMENTS

correlation

= +[5<&42);;l,

+

function 3<&42>~,l

(4.11) is given by +


-fa(R)[lO-3f2(R)] =

12[2-f2(R)][4-f”(R)]

The degree of degeneracy

(4.12)

.

of this state equals 9. (See also fig. 3.)

0.3 .

r1

0.2 . 0.1 0. -0.1

I -0.3 -0.2 . -0.4 ,4 lJG U-G -0.5 -

I

I I I I I I : I

t I I I I I I I I 7 I I I I t I I

I, I I I I I I I I

-0.6

- 0.8 0.01

I

0.1

1 I&

10

10

-

Fig. 3. The correlation iunction (S1-Ss)n as a function of 1JN/4GR( plotted on a logarithmic scale (J > 0, G ( 0). Curve “a” denotes the case f(R) = 0, curve “b” If(R) 1= 0.4, and curve “c” If(R)1 = 0.9.

440

W. A. SMIT

AND

G. VERTOGEN

In all four cases the ground state is a state in which two electrons are coupled with the magnetic moments. Generally the correlation function depends on the distance R between the two magnetic moments as well as on the ratio between the interaction strengths J and G. In the figs. 1, 2 and 3 the correlation functions are given as functions of the ratio between IJW4Ql and IQ on a logarithmic scale for three different values of If(R namely If(R)) equals 0, 0.4 and 0.9. Actually G is dependent on the distance between the magnetic moments, meaning that both G and f(R) change when the distance between the moments is varied. The function f(R) has an oscillatory behaviour, while its amplitude decreases with increasing distance R. f(R) is the overlap between the two spin clouds gr(x) and gs(x) which are built up by the electron gas around the positions of the magnetic moments: f(R) = j dx g;(x) gs(x) (see ref. 4). In case f(R) = 0 and in the absence of a direct interaction the correlation between the two magnetic moments disappears, 0, G > 0) both the direct and indirect interactions lead to a purely ferromagnetic coupling resulting in a correlation function, which always equals a. In case B (J < 0, G > 0) the correlation function always has a ferromagnetic character. This is due to the fact that the indirect interaction never gives rise to a correlation with an antiferromagnetic character in the ground state. As is to be expected the correlation between the two magnetic moments can only be of an antiferromagnetic character in the ground state in the case of a direct antiferromagnetic interaction (cases C and D). This situation gives rise to a coupling, whose character depends on the value of f(R) and the ratio between G and J (figs. 2 and 3). For small values of 1JN/4GQl, i.e. the direct interaction dominates the s-f interaction, the antiferromagnetic character of the coupling becomes more pronounced with increasing overlap of the spin clouds. This is a remarkable situation, because one would expect the contrary. For f(R) = 0 means that the spin clouds, which are built up by the electron gas around the localized magnetic moments, do not overlap, while it is just this overlap that gives rise to the ferromagnetic correlation. One would, therefore, expect that an increasing overlap always diminishes the antiferromagnetic character of the correlation. This is, however, not the situation as pointed out by the calculations. In case C (J < 0, G < 0) the correlation function C = -&. This means that the correlation functions as functions of 1JN/4GOI for different values of f(R) cross each other, as shown in fig. 2.

EXCHANGE

INTERACTION

BETWEEN

In cases B, C and D the dependence ratio between

TWO

MAGNETIC

of the correlation

MOMENTS

function

441

on the

J and G is most pronounced in the region 0, 1 I JN/dGO I 2.

Finally the very peculiar nature of the correlation function has to be stressed in the case of a ferromagnetic s-f interaction and an antiferromagnetic direct interaction (see fig. 3). Here a discontinuity appears. With increasing direct interaction the correlation discontinuously jumps from a purely ferromagnetic one to one with a strongly antiferromagnetic character. Acknowledgements. The authors wish to thank Professor A. J. Dekker for his stimulating interest in this subject, J. J. Hallers for many valuable discussions and the University’s Computing Centre for the numerical computation of the correlation functions. This work is part of the research program of the Foundation for Fundamental Research on Matter (F.O.M.).

REFERENCES 1) Ruderman, 2)

Kasuya,

M. A. and Kittel, C., Phys. Rev. 96 (1954) 99.

T., Progr. theor. Phys. 16 (1956) 45.

3)

Yosida,

4)

Vertogen,

K., Phys. Rev. 106 (1957) 893.

5)

Zener, C., Phys. Rev. 81 (1951) 440.

G., Physica 48 (1970) 509.