Theory of spin waves in magnetic overlayers and sandwiches

,h)urnal of Magnetism and Magnetic Materials 104-107 (19921 1876-1878 North-I lolland

Theory

of spin

waves

in m a g n e t i c

overlayers

M.S. Phan ", J. Mathon ", D.M. Edwards t, and R.B. Muniz

and

sandwiches

t'

" Department o.f Mathematics, ('ity Unicersity, London E C I V OHB, UK t, Dcpartment oJ"Mathematics, Imperial College, London SI¥7 2BZ, UK

Microscopic calculations of the energy arid lifetime of Spit) Waves (SW) in a ferromagnetic (FM) overlayer and of the exchange coupling between two FM layers separated by a Nonmagnetic (NM) spacer layer are r;.ported. The coupling is determined fl'om the energy gap between the in-phase and out-of-phase SW modes. It is found that the SW lifetime in an overlayer is very short because of tunneling of holes from the substrate to the minority hole band in the overlayer. The coupling between two FM layers is shown to oscillate as a function of the spacer thickness with a long period when the Fermi surface is close to a zone boundary. We report our microscopic calculations of the energy and lifetime of Spin Waves (SW) in a magnetic ovcrh|yer and of the exchange coupling between two magnetic h|yers separated by a nonmagnetic spacer layer. The coupling is determined from the energy gap between the in-phase and out-of-phase SW modes. We model the ferromagnetic ( F M ) a n d nonmagnctic (NM) layers in a multilayer by a sc tight-binding Hamiltonian

=

t;it:,,(:,,+ ~U~,z~),l,~ Iii,r

(!)

i

where U; is the intta-atomic Coulomb interaction which is non-zero only in the magnetic layers. To discuss spin waves wc require the transverse dynamic susceptibility in the mixed representation X,,+ (to, qlt), where qll is the wave vector parallel to the layers and i, j label atomic planes. It satis~cs for any layer system [I] a matrix equation k" )(~o, qll) = [ l (Uxtl~((°" qlf))] i
R~ x,!/'"(,o, q,,) -- - ( ~ u , , )

' E J'"[ dE{Ira ~;,~( E. I<,,) K" H

.-
~,i,11 ( ~o, ql!)

=(rrNl,)

'Ef/"'dE

× {1,:, (;,', ( e, ~,,) In, (;,', ( ~- + ,o, ~-) -lm¢;,~,(E, KH) ~m(;,I(E-,o,~)}.

(2)

The summation in eq. (2) is over the two-dimensional Brillouin Zone (BZ), E v is Fermi energy and K = KII + qtl"

Spin wave energies arc obtained from poles of X +, i.e., from the secular equation Dct[ I-

U R e x H V ( t o , qtl)l = 0 ,

(3)

and the lifetime of spin waves is determined from the imaginary part of X Wc consider a monolaycr of a strong fcrromagnct on (100) surface of ,, scmi-iafinitc nonmagnetic r e c t a ! lic substratc. The first step in computing SW cncrgics is to solve the sclf-tx,t;~i:;tcnt HF ground state problem. The HF one-electron Green t'unctioo.s G~,r wcrc determined by the method of adlaycrs [2]. The spin-independent potentials V, wcrc set equal to zero in the substratc and I<~) in the magnetic layer was chosen to give lhc layer an occupancy of n 1 + n ~ = (I.12 holes per atom, which is appropriate to nickel in this single-band model. An intra-atomic interaction U = 1.75W was chosen to make the ovcrlayer strongly ferromagnetic n r >> n j ( W is thc band width). The summations ovcr thc BZ required in the calculation of (no_,,) and in cq. (2) wcrc evaluated using the Cunningham points [3]. Finally, SW cncrgics o> wcrc dctcrmincd from cq. (3) for (I <-qll < av/a. Wc then rcpcatcd thc above calculation for an uns),pportcd FM monolaycr to compare the two systems. The SW spectra for the ovcrlaycr and unsupported monolaycr arc shown in fig. 1. There is very iittly difference between the two spectra at the bottom of the SW band. However, spin waves in the ovcrlaycr run very rapidly into the Stoner continuum and become critically damped for w = 0.08W. A very strong damping of spin waves in the ovcrlaycr is illustrated in fig. 2 where lm X + is shown for several values of qll" In contrast, Im X ~ for the unsupported monolaycr is infinitely narrow in the whole BZ. The reason tk~r this bchaviour is rather interesting. Although n r >> "~ inaplies that the ovcrlaycr is essentially a strong fcrromag-

0312-S853/L)2/$()5.(H) ,,, 1992 - Elsevier Science Publishers B.V. All rights reserved

M.S. Phan et al. / Theoo' of SI,V in FM overho,ers and sandwiches 1.5,

ix77

x10-5 15

vl

,-,10

=

D =

oo:

1'-4

(a)/

Q ee tll

tu = tll

til

0.=

.

1 2 warn VECTOn ZN UNZT~ 11a

0

,¢

3

Fig. i. Spin wave dispersion in an overlayer (a) and in an unsupported monolayer (b) for qll in [100] direction. (W is the band width.)

net, holes from the susbstrate can tunnel into the down spin 'O-d l -l t J- ' of """ I, I 1 ~ . ~-ver!a:,'cr at all o,-,,.,-,,i,.,: --.~ F ~ i } < .E. . < . . .E r. where E b is the bottom of the up spin hole band. Therefore, there is no well-defined Stoner gap for an overlayer and spin waves decay into electron-hole pairs even for to-= O. This may have important implications * ; . .

-

;

aqx=O. S

3

7 Z2

~'Ib4 c E-

~"

.

=

°

,,<

aqx=Z'

C 0

"----2

I SPI.~I W A V E

ENERGY

IN U N I T S

W/6

Fig. 2. Spin wave line widths in an overlayer for aqi! = 0.5. 1.o and rr.

.

.

.

.

.

11

L

L

i

I

I~

18

25

N Fig. 3. Energy of the out-of-phase spin wave mode w. in a sandwich as a function of the number of atomic planes in the spacer layer N.

for the temperature dependence of the ovcrlayer magn,..uization in !he SW rc~im,~. The model dcscribcd abovc was also uscd to calculate SW cncrgics of two FM monolaycrs cmbcddcd in u NM metal. Thc mo ivation is to dctcrminc thc indirect cxchangc couplir g J bctwccn the FM laycrs from thc cncrgy diffcrcncc bctwccn thc in-phase and outof-phase SW modes of thc multilaycr system. This is prcciscly what is measured in lighl scattering cxpcriments on F e / C r sandwiches [4]. Such cxperiments show oscillations of J as a function of the spaccr thickness. Since exchange coupling between the layers is much weaker than that in the layers, transvcrsc SW modes with n o n z e r o qll arc not excited and thcrc arc only two SW modes to bc considcrea: the modc w(qj~ = 0 ) = 0 when the two total laycr momcnts prccess in phase and the mode to(qll = 0) = w. when they preccss out of phasc. The cncrgy w. is proportional to thc intcrlaycr coupling J. Thc cncrgics of both SW modcs wcrc d c t e r m i n c d from the pcaks of lm X,+x, whcrc N is the n u m b c r of at. planes in thc spaccr. Wc used E~ = - 0 . ! 7 5 W (tb.e Fermi st_,rfacc i~ c!(~c m BZ boundary), U = 20W (very strong FM) and chose the spin-independent potentials in the FM layers so that the n u m b e r of majority holes in these layers was equal to the n u m b e r of holes of either spin in thc NM layer. This is cxactly equivalent to the modcl uscd in rcf. [5] to calculate J from the total energy diffcrcncc. Wc imposed an FM ground state for all N, which means that we expcct to obtain the to = 0 peak (Gold-

1878

M.S. Phan et al. / Theory of S W bl FM ot,er/ayers and sandwiches

stone mode) in all cases but the out-of-phase mode 0% only when J is ferromagnetic (the correct ground state). This is born out by our calculation and 0% ot J is shown in fig. 3 as a function of the spacer thickness N. We find oscillations of J with a period ---- 10 at. layers in perfect agreement with the results of the total energy calculation [5].

References [1] [2] [3] [4]

J. Mathon, Phys. Rev. B 24 (1981) 6588. J. Mathon, J. Phys.: Condens. Matter ! (1989) 2505. S.L. Cunningham Phys. Rev. B 10 (1974) 4988. P. Griinberg, S. Demokritov, A. Fuss, M. Vohl and J.A. Wolf, J. Appl. Phys. (in press). [5] D.M. Edwards and J. Mathon, J. Magn. Magn. Mater. 93 (1991) 85.