TC/96/:

United Nations Educational Scientific and Cultural Organizationand

international Atomic Energy Agency

INTERNATIONAL CENTRE EOR THEORETICAL PHYSTCS

INTERLAYER COUPLINGIN METALLIC MAGNETIC MULTILAYERS

Cheng-chung Lee (Z. Z. Li)1

International Centre for Theoretical Physics. Trieste, Italy

Rui STien a.Tid Ming-wen XiaoDepartment, of Physics. Nanjing University,

Nanjing. 210093, People's Republic of China.

ABSTRACT

We obtain the exact Creen's functions of the Anderson s-d mixing model for magnetic

mull.ibwers within the mean-field l.Tieory of t.Tic: on-site Coulomb repulsion. Tl. IN HIIOWTI

that the coupling oscillates in the experimental range of the spacer thickness only when

t.Tic N-d mixing is strong enough, and t.Tiat. the polarization energies of s find d <;l<;c.tronN

weaken the interlayer coupling remarkably. We also find that the thermal dependence is

determined by bot.Ti t.Tic propcrticN of t.Tic Fermi Nurfacc of t}i<; Hpacer and the exchange

splitting between the two spin subbands in the ferromagnetic layers.

MIP AM ARE - T1UESTE

September "1996

'Permanent address: Department of Physics, Nanjing University. Nanjing 210093, Peo-ple's Republic of China.

Oscillatory exchange; coupling between (wo ferraTnagn<;l.ic (F'M) la.yers separated by

nonmagnetic (KM) spacer has been well observedW. For the spacers of 3d transition metals

l.he OKtilUriion period is about. 10 A (except. Cr), and I.IK; coupling sl.rength measured at l.he

first antiferromagnetic (AT) peak is 0.1—OAmJ/m2^. A linear thermal dependence with

a. decrea.se of about. 20% bel.vveen 4.2K and 400K lias also been observed^. Furlhermore,

the experiment verifies that the polarization energy plays an important role in the coupling

strengl.h^. There have already been several l.heories l.o understand 1.1K; phenomena., <;.g.,

the ab initio numerical calculations ^ , the quantum well theory^ and the Huderman-

Kil.tel-Ka.suya-Yosida. (RKKY) approach^. T'}i<; RKKY approach is more transparent, in

physics, however, its approximation of a contact interaction between the spins and the

conduction electrons is ral.her crude. Moreover .bot.h I.IK; RKKY theory^ and I.IK; quanf.inn

well theory^ attributed the thermal dependence only to the properties of l.he Fermi

surface (FS) of the JNM spacer so as to yield a temperature dependence of the coupling

as (T/T0)/a\](T/T0) where the chara.cterist.ic l.emperal.ure is defined as To = hvpI'lirksTj

with L the spacer thickness and •[>>,•• the Fermi velocity of the conduction electrons in the

spacer layer. With t.his result., Ref. [9] analyzed the experimenl.a.l data, and gave a. vjr an

order lower in magnitude. For these reasons, there is still a need to improve the theoretical

situation.

In this paper, we report an approach based on the Anderson s-d mixing model. Such an

investigation has been previously atl.empl.ed by Bruno^1"]. However, l.he coupling strengl.h

in Hef. [10] is a result of the 1th order perturbation of the mixing strength so that it is

only valid for weak s-d mixing limit and cannot, show the influence of l.he mixing strengl.h

on the interlayer coupling. In fact, the s-d mixing strength should be quite large in the

F'M layered sysl.ems consisting of l.ra.nsil.ion metals, and l.he polarization energies of s and

d electrons have a strong influence on the coupling^. Therefore, a microscopic theory

for l.hose systems should be able l.o treat the ca.se with large; mixing sl.rength and include;

the effect of the polarization energies of both s and d electrons. This is the aim of our

paper. Within l.he mean-field l.heory, we have; obl.ained l.he exact. Green's functions of

the Anderson model for the layered system. With those Green's functions, the coupling

strengl.h has been derived, it. includes the effect, of l.he pola.riza.l.ion energies of both s and

d electrons, and can be applied to the case with arbitrary mixing strength. It is shown

t.hat. the int.erla.yer coupling oscillates in the experimental range of l.he spacer thickness

only when the s-d mixing strength is strong enough, that the polarization energies of s and

<] electrons weaken t.Tic: interla.yer coupling remarkably and that the l.hernial dependence

is determined by both the properties of the iermi surface of the spacer and the exchange

splil.ting between the two spin snbba.nds in t.Tic: rerroma.gnel.ic layers.

The system under our consideration is the two FM monolayers i'L and FH separated

by N nonma.gnel.ic monolayers. The disl.a.nce between FL and FR is R^ = ( N-\-~\ )ri, where

d is the spacing between two atomic planes. The system can be described theoretically

by t.Tic: Anderson model H = H, + Hd + ^£ ; [ l l ] where

1L = y £i. ,. c\ , ci. L „ . (la)KII .kz jd

11,!= y EkdL d.ik +- Y SR diRi^.R diRi „ , (lb)

lL,d= 4i

H<;re. Hc represents l.he HamiHoriiriii of l.he conduction electrons (K eleclrons) in OK; spri.c<;r

with energy band e^ *,. Hi represents the iiamiltonian of d electrons in the FM layers

with cTKjrgy band Ek U l.he on-sik; Coulomb repulsion, i = =p'l dcmol.ing the left, or right,

FM layer, and ti\\ the lattice vector in the plane. llx,i is the hybridization term and V is

l.he mixing strengl.h. which is proportional to \/\/N. Now we treat l.he on-sil.e Coulomb

repulsion with the mean-field approximation according to E^, ^ = Kj. + U{n_l). The

exchange; split.l.ing b<;l.w<;<;n two spin subbands reads Urn wh<;re rn is the average; local

magnetic moment, in the mean-field approximation, the iiamiltonian of d electrons in

l.he FM layers, i.e. Hd, reduces to

!' Z—I

and the total Iiamiltonian becomes

The (jreen's function can be solved exactly because the total iiamiltonian is quadratic.

For arbitrary V, we obl.ain l.hose exact GreeiVs [inicl.ions in l.he layered system as follows:

II 7 '- * ' l |

| | | | f ( \-\ _ P2 ' (3C)

where

fi'1>(fc||,W) = ( W - J f e ^ - i ' ) - 1 , (3d)

! , (3e)

1/ 'i(,ikzl

(30

'The a b o v e Gr<;<;n:K finid.ioTis c a n Tx; coiiKidcrcjd as t.Tic: gcTKjraliz.al.ion of vvlial, (.•a.roli l ias

obtained for two magnetic impurities^12]. (]:,i(T represents the (Green's function of the d

dccl.roTis in OK; i pUnic when the l.vvo F'M UWCJTK ar<; infinitely apart, i.e. there is no

coupling. Due to the s-d mixing, the localized state E\! a is broadened into a virtual

bound slate wilh an energy Kliift V and a vvidlh A. The dependence; of th<; inl<;rla.yer

coupling on the distance R:\ is entirely contained in the function 1(\

Evaluating ihe iherinodynaTnic av<;rage of ihe K-d hybridizalion <;nergy {H^;} 1'>V

fluctuation-dissipation theorem, one obtains

i,/l||,fc-;,<7

ik** f d^)l(( j4 JU +C.C.i\ J — rx,

where; /(it?) is tli<; Fermi [iniction. TIK; t}i<;rTnodynaTnic potenlial Q of ihe Nysleni can be

obtained by the ieynman's theorem as

where the integral over Feynman's adjustable parameter A means that all the functions

under ihe integral are ihe KolulioiiK corresponding to ihe Kysl<;in whose HamilIonian is

The inlerlayer coupling conKtanl p<;r unit. area. J (bilinear lerm) can Tx; defined as

J = (QF — QAF)/2A. where A is the area of the plane and the superscripts indicate

respectively t.Tic: FM or AF arrangement bet.vveen t.Tic: l.vvo FM layers. Afl.er some: alg<;T)raic:.

calculations one obtains

(6a)

wlic:re we: assume: that the axis of the spin quantization is along the dirc:cl.ion of t.Tie

magnetization of the right plane (i = +1) . Eq. (6a) can be written in a more symmetric

form

(6b)

by assuming that the axis of the spin quantization is along the direction of the magne-

t.izal.ioTi of cil.licr plane; r<;spc<:l.ivdy. Tf GTK; n<;gl<;d.K I.IK; conlribulioiiH of t.Tic: polariza.Uon

energies of both s and d electrons, i.e.. attributes the coupling strength only to the hy-

bridization energy ffS!;. t.Tic: coupling slrengtli reduces to

J' = [(U%M)-{11-Zn]/(2A)

It is worthy of noting that within the mean-field approximation to the on-site Coulomb

repulsion, the exchange coupling constant, .] in Eq. (6a. 6b) contains all the many-body

effects including the s-d hybridization energy, the polarization energies of s and d elec-

t.roTis. and all the perlurbalion effects of V up t.o infinite: order. Tf we: lake a first order

perturbation of F2 which corresponds to a kind of 4th order perturbation of V, our result

reduces t.o " [ ]

8-7T ./ J-oc

Now, we calculate: J from Eq. (6a). For the sake: of simplicity, we: assiniK; t.Tic: s

electrons to be the free electron gas with energy band .-i^ ... = (k\\2 + ki)/'2m* where

rn" is the e:ffe:ct.ive: mass, and t.Tic: d e:le:ct.rons t.o Tx; a two-enc:rgy-levc:l syste:m witli an

exchange splitting Urn. Additionally, we adopt, in our numerical calculations, the typical

value:s (e.g. Cu) d = "I.SA, t.Tic: Fernii energy tjr = T.OcV", and t.Tic: F<;rmi wave: v<;c:-tor

kF = 0.827(-/(f)[4]. Our results are shown in Figs. 1 to 3.

TTI Fig. 1, we show t.Tic: influence of t.Tic: mixing strength GTI the oscillal.ory bc:havior of

the interlayer coupling. We find in the experimental range of the spacer thickness (e.g.

15 <--30 nionolayers^ ) that t.Tic: inl.erlayc:r coupling does not oscillate: in t.Tic: vvc:ak s-d

mixing limit as shown in the inset of Fig. 1. When V gets larger, the coupling begins to

oscillate: at. long distance (sec: t.Tic: dol.ted line). Only when V is large; c:Tiough the cr-onpling

shows a good oscillatory behavior in the whole range. To interpret this phenomenon, we

introduce the formula of the lifc:t.iin<; of virtual bound slate at th<; FS. T(/; | | . tj-), in fr<;<;

electron approximation to the s electrons as

(9)

It is dependent on the 21) momentum ty. The lifetime increases when k\\ decreases.

Physically, t.hc: principal contributions t.o the; iTia.gnel.ic coupling must cronie from t.hc:

long-life state, if the mixing strength is strong enough then the denominator in Eq. (9)

is very large. To ensure that t.hc: lifelinie is long, t.hc: numerator should also be: large;. Tl

means that the 21) momentum k\\ must be restricted in its vicinity of the origin. Now

t.Tial fe|| varies only in a small vicinily of t.Tic: origin, t.hc: e;onpling stre;nglh in Fq. (6a.)

is solely determined by F(k\\,u;) with k\\ ^ 0, which means that the interference effect

from F(k\\,uj) wit.Ti dif['e:re;nt k\\ is of le;ss importance;. Tn this erase:, the; int.e;rla.yc:r c:xe;ha.nge;

coupling oscillates naturally. Otherwise, if the mixing strength is very small, the condition

of the; long-life: virtual bennid slate: doe;s ne)t give: e;f[ecrtive; re;slrie;lie)ns to t.hc: k\\. Under

this condition, the interference arising from F(k\\,u;) with different k\\ comes into effect

sue;h that t.Tic: e;_xcha.nge eronpling be;cr.emie;s noTie>se:illat.e>ry. We; ca.ulie)n t.Tia.l, the; a.syniplofie:-

behaviour of the coupling strength in long distance is always oscillatory10 irrespective as

t.e> vvhelher t.hc: mixing strength V is large or ne>t. This is beera.nse the; coupling stre;nglh is

only attributed to one of the oscillation functions F(k\\,io) with the momentum determined

by the; sta.liema.ry point.10 when R^r —> oc. Howe;vc:r, in t.hc: e;_xpe;riTnenla.l range; of t.hc:

spacer thickness which is not large enough, whether the coupling strength oscillates or

ne)t. is dependent, on the; magnitude of the; mixing strength. Thus the; sta.liema.ry pe)int

approximation is unsuitable for the experimental spacer thickness. According to the above

analysis, only when the; mixing stre;nglh V is strong e;nongh t.hc: eronpling stre;nglh oscrillatc:s

in the experimental range of the spacer thickness, in fact, the s-d mixing strength V ought

t.e> be: quit.e; large: in the; la.yere;d systc:ms eronsisting of t.vM.Tisit.ion nie;ta.ls, t.ha.l is why t.hc:

interlayer exchange couplings observed in this sort of systems oscillate.

6

The e:ont.ribiit.ie>n of the; polarization energy is sketched in Fig. 2. The solid line shows

our numerical results based on Eq. (6a) including both the s-d hybridization energy and

l.he pe>lariz.at.ie>n e;nergie;s of s an el el elee:t.rons. The dasheel line; represents the exchange

coupling obtained from Eq. (7) where only the s-d hybridization energy has been taken

ink) account. Comparing l.he solid line and the dashed line, one finds l.hal. l.he pola.riz.af.ion

energy can weaken the coupling strength remarkably.

We now t.urn l.o discuss l.he effect, of l.he exchange split.l.ing Urn on l.he thermal depen-

dence of the interlayer coupling. This temperature dependence can be approximated by

t.he formula below (see Appendix)

7 ' f+ c x ' ,-/•>. s ' n -r-xco^T -six

where the characteristic temperature T{> = tiv^/2iTkHR:\ is determined by the properties

of l.he FS of the; space;r. which is of l.he order of 10'̂ K, and T,, = hljrn/^irks is determined

by the exchange splitting Urn. The convergent factor i] = 2A(0. £y)fl.Jm in Eq. (10)

is determined by l.he width of l.he virt.ual bound slate at. l.he FS. The numerical results

of the approximate formula. Eq. (10), are shown in Fig. 3 as curve a through i for

various Urn. Eq. (10) indie;ale;s l.hal. l.he thermal ele;pe;nele;ne;e of l.he int.erlayer coupling

is controlled by both T{> and Tg. When T <^i rjTfn Eq. (10) reduces to the power law

J{T)/J(0) = 1 - a'T\ wlicn: a is a. constant, det.erTnined by To, Ta and q. WIK^TI T > To,

Eq. (10) reduces to J(T)/J(Q) = {T/T0)/sh(T/Tu), which has the same form as the usual

fbrTnulat7'̂ . They differ from each other by that in our result, t.his form is only valid for

the high temperature range but in the usual theory it is thought as suitable for the whole

t.emperat.ure range;. Tf l.he exchange splitting is small, l.he thermal dependence; described

with the usual formula controlled only by Tu gives a much smaller decrease of the coupling

sl.re;nglh when compare;d with our re;snll.s, whie:h e:an e;asily be se;e;n from curve S in Fig.

3. Therefore, at least in the Anderson s-d mixing model, the exchange splitting between

l.he two spin snbbanels in l.he; FM la.ye;rs e:an be; e:onsie]e;ree] as ane)l.he;r pe>SKiT)le re;ason for

the thermal dependence of the interlayer coupling.

Tn summary, within l.he frame;we>rk of l.he; me;an-fie;ld l.heory of l.he; on-sil.e; Conle)mb

repulsion, we have obtained the exact Cireen's functions of the Anderson model for the

layere;d system e;onsisl.ing of t.ransit.ie>n rnelals. which are suitable; for arbitrary s-d mixing

strength. Our result indicates that the interlayer coupling oscillates only when the mixing

7

sl.rengl.il is large; enough, l.lial. t.Tic: polarization energies of s and d electrons much 1GW<;TN

the coupling strength, and that the temperature dependence is determined by both prop-

erties of t.Tic Fermi Niirfacc of 1.1i<; spacer and I.IK; exchange splitting between l.Tie two spin

subbands in the EM layers.

Acknowledgements

One of the authors (C.C.L.) would like to thank the international Centre for Theoretical

Physics, Trieste, for TioHpit.aiil.y. This work was Hiipport.<;d by the Doet.oral Program

Foundation of High Education of China.

Appendix

Although Eq. (8) is only a 4th order pertubation of Eq. (6a), both of them give nearly

t.Tie same thermal dependence, which can be; verified by direct numerical calculations.

Expanding k\\ in the vicinity of zero and LO in the vicinity of £>•. one obtains from Eq. (8)

B r+K "1 "I , .xJ{T) = — 0 / diof(Lo)lm[ : —]-e' . (A.I)

A = 2RN/vF, p = Urn/2, 7 = A{0,eF) , (A.2)

where B is a constant which is independent of T an<i v^ is the Fermi velocity. By a

contour integral on the complex plane;, one; obtains

=o IA-w + f) + p J

where kg is t.Tie "Boltzmann constant.. W<; introduce; two auxiliary functions

(A.4)

(A.5)

it can easily be verified that f\ (1'. A. p, 7) satisfies the differential equation

x \ (A.6)

with t.Tic: condition

/,(A -> +oc) = 0 . |

The solution of Eq. (A6) with the condition Eq. (A7) can be found as

1 (A.S)

! (A.9)

y) S 1 L F \ " njy- f\ -,Q\

g t.Tie inl.egra.1 variables^ x = p(y — A) , o n e o b t a i n s

where

r)/Urn , (A.12)

Tg =p/~kH = Um/2nkH • (A.13)

2xknRN • (A.14)

Finally we get

J(T) _ t\T)

J(0) F(T = 0)

,—n-r;

(A.15)

TTI Fig. 3. c-nrvc A. R and C ar<; t}i<; <;xri.d. results evaluated from Eq. (6a.) for £/m = 0.3,

0.35 and 0.4el''. Curve a. b and <: are the corresponding ones obtained from the approxi-

mate formula. Eq. (10). Clearly, one can find that t.Tiey a.re quil.e good a.pproxima.t.ioriH.

References

[I] (Jltrathin Magnetic Structures 1. An Introduction to the Electronic, Magnetic and

Sini.clv.ral Properties. <;d. J. A. C. Bland and R. Heiririch. Springer-Verlag, Berlin

(1994).

[2] S. S. P. Parkin Phys. Rev. Lett. 67 (1991) 3598.

[3] S. S. P. Parkin, R. Bliadra, am] K. P. Roche, Phys. Rev. Lell. 66 (1991) 2152.

[4] J. E. (Mega and F. J. Binipsd, Pliys. Rev. Lcl.1.. 69 ( 1992) 844.

[5] P. Lang. L. Nordstrom, H. Zeller. and P. 11. Dederichs. Phys. llev. Lett. 71 (1993)

"1927.

[6] J. Mat lion. Murielle Villeret, R. B. Muniz. J. d'Albuquerque e ("astro, and L). M.

Edwards, Pliys. R<;v. Let,!.. 74 (1995) 3696.

[7] P. Bruno and C. Clmpperl: Pliys. R<;v. Lcl.L 67 (1991) "1602: Pliys. Rev. B 46

(1992) 261.

[8] D. M. Edwards, J. Mallion, R. B. Mnniz: and M. S. Plian, Pliys. Rev. Loll, 67

(1991) 193.

[9] Z. Zhang. L. Zhou. P. E. Wigen, and K. Ounadjela. Phys. Rev. Lett. 73 (1991)

336.

[10] P. Bruno. J. Magn. Magn. Mater. 116 (1992) L13.

[II] P. W. Anderson. Phys. Uev. 124 (1961) 41.

[12] B. Caroli, J. Pliys. Cheni. Solids 28 (1967) 1427.

10

0.6

20)

0.2

o.o

5-0.2=1o

-0,4

0.01

0.00

0.00

-0.02

-0.03

V=0.1eV

J I

0 5 10152025

V=0.3eVV=0.4eVV=0.5eV

0 5 10 15 20 25Number of spacer layer

Fig. 1. Exchange coupling J with different mixing strength versus the number of spacer

layers for Urn = QAeV and T = SQOK

11

1 -

0

- 1 -

ooCDGO

- 2 -

oXI

- 3

- 40 5 10 15 20

Number of spacer layer25

Fig. 2. Exchange coupling J versus the number of spacer layers for Um = OAeV, V —

Q.5eV, and T = SOOK. The solid line is evaluted by Eq. (6a). the dashed line is calculated

byEq.(7).

12

1.0

0.8 -

0.6 -

0.4 -

0 200 400 600Temperature (K)

800

Fig. 3. The temperature dependence of the interlayer coupling J(T)/J(0). Curve A, B

and C is calculated from Eq. (6a) for Um — 0.3, 0.35 and OAeV respectively. Curve a

through * is the result obtained from Eq. (10) for Um = 0.3, 0.35, 0.4, 0.5, 0.6, 0.7, 0.8,

0.9 and l.OeV. Curve S is evaluted from the usual formula J(T)/J(0) = (T/TQ)/sh(T/T0).

13