Roughness effects on spin waves

~H ELSEVIER

Journalof magnetism and magnetic materials

Journal of Magnetism and Magnetic Materials 163 (1996) 207-215

Roughness effects on spin waves Daniel Mercier *, Jean-Claude Serge L~vy ixtboratoire de Magn~tisme des Su(f~ces, Unil:ersit~ Paris 7

Denis Diderot, 75251 Paris cedex 05, France

Received 26 October 1995

Abstract

A simple modelling of surface roughness in magnetic thin films shows that roughness tends to unpin spin waves at external surfaces. This unpinning effect increases with the roughness height as long as this height remains small. In spin wave resonance, the spin wave intensity spectrum is shown to be quite sensitive to roughness, and new surface spin waves appear because of this roughness. Keywords: Spin waves; Spin wave resonance; Thin films; Surface roughness

1. Introduction Recently, in investigations of giant magnetoresistance in multilayered materials with alternate magnetic and non-magnetic layers [1-3], many experiments have provided evidence for long range effective ferromagnetic or antiferromagnetic couplings through a non-magnetic layer. These effective ferromagnetic or antiferromagnetic couplings strongly depend upon the thickness t of the non-magnetic layer, often with an oscillatory behaviour. As a result of this sharp dependence, magnetic effects are quite sensitive to the surface and interface roughnesses which modulate the interlayer thickness. Experimentally, roughness can be varied by using convenient substrata and buffers [4,5] as well as by thermal treatments as carried out for F e / C r and for C o / C u [6,7]. Different roughnesses are observed and anaCorresponding author. Fax: + 33-4633-9401.

lyzed by means of grazing incidence X-ray scattering with evidence for an average height and a transverse correlation distance as well as a fractal profile [4-7]. From a theoretical point of view, Monte Carlo pair potential simulations of layer growth on a substratum, with or without mismatch, also indicate the general occurrence of roughness during early growth [8], i.e. a surface roughness extended over several atomic layers. This roughness depends on the pseudo-pair potential shape, i.e. on the surface tension. During the process of simulated growth, the mismatch and elastic difference between adatoms and the substratum play central roles in island pattern formation [8]. Thus from experimental and theoretical evidence for roughness in magnetic multilayers, the question arises of observing roughness by means of a purely magnetic measurement such as spin wave resonance. Spin wave resonance is a standard magnetic technique for studying surfaces and surface pinning [9,10], which is expected to be

0304-8853/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PII S 0 3 0 4 - 8 8 5 3 ( 9 6 ) 0 0 3 0 0 - 9

208

D. Mercier, J.-C.S. L~'ty / Yournal qf Magnetism antiMagnetic Materials 163 (1996) 207 215

quite sensitive to the magnetic properties near the surface and thus to the local structure. This defines the goal of the present paper. In order to deal with the influence of roughness on spin wave propagation, a discrete, local approach of the propagation is useful. In fact, a transfer matrix [11-15] which easily distinguishes between surface pinnings and unpinnings is used [11,12,15]. In this description, the main local parameter is the layer-tolayer exchange coupling J,,.,, + ~, which is usually the same over the whole atomic layer. The roughness is then understood as a sharp variation of this parameter J, ..... ~ in the vicinity of the sample surface, either because of a lower concentration of magnetic sites and spins or because of a local exchange variation. In the simple effective medium model developed below, a linear variation of the effective layerto-layer exchange coupling J,,.,,_ j between a null value out of the sample and its bulk value, is assumed. In other words, the only parameter of this model is the average roughness height. The roughness transverse correlation length is neglected as well as the fi'actal profile since spin wave resonance involves, in practice, an integration on the rf wave plane. Since the spin wave correlation length is larger than the usual roughness transverse correlation length [4-7] then, at a given height, there is only an averaged effect of the roughness. There are obvious similarities between this discrete treatment of roughness and the continuous treatments of a non-uniform distribution of magnetization or exchange integrals as performed by Portis [16] and more recently by Watts [17]. However, the present discrete treatment enables us to account for the rapid roughness variation over just a few atomic planes of the roughness height. In this paper, only a single film with symmetric roughness conditions at its surface is studied. The case of roughness in trilayers and more generally in multilayers is left for future consideration. The parameters of this study are the bulk exchange parameter J, the height d of the roughness and the thickness N of the film, since we are interested in a soft material where bulk anisotropy can be neglected, while surface anisotropy defines the spin pinning at the surface [9,10]. An introduction of the roughness model in the spin wave transfer matrix is given in Section 2, while Section 3 deals with practi-

cal results. Concluding remarks are noted in Section 4.

2. Model

Neglecting bulk anisotropy as is usual for magnetically soft samples, the spin Hamiltonian reduces to two parts, the exchange part //~× and the Zeeman part Hz~.cm~.1, with

H~,, = - 4 E Z , , , s , .

s~,,

( l)

and

Hz ....... = - ~, E n . s.t ,

(2)

]'

where S r is the spin vector of the electron located at site f, and Jt.~. is the exchange integral for sites f and g, and is non-zero only between nearest neighbours. The effective magnetic field H accounts for the magnetostatic correction and lies in the direction z normal to the film. In the central part of the film, the exchange parameter is constant and equal to J, while near the surface, because of the averaged roughness of height d, it varies linearly with a step of J / d at each layer. The equilibrium orientation of all St is collinear with the z-axis since the applied magnetic field in spin wave resonance for this perpendicular configuration is larger than the saturation field. We therefore change from spin ½ operators S t to Pauli operators bt, with S ~

i

t

S~

t

=

+

,

I + 7(bt -b,) I

s; =

+ bt)

(3)

+

-

bt.

From the commutation relations between Pauli operators, their equations of motion are derived when considering the restricted Hamiltonian HQ which is quadratic in Pauli operators. With this standard low temperature approximation, we have for a standing wave of time dependence e x p ( - i w t ) d 1 i-~t(b~,)=wb~,=hb,+~EJt~.(b~-b,). t

(4)

D. Mercier. J.-C.S. Ldry / Journal of Magnetism and Magnetic MateriaLs 163 (1996) 207-215

When summing over all spins g which belong to the same layer 1 parallel to the film surface, we have:

where an extra layer 0 has been formally introduced. At the surfaces, the spin pinning conditions result from surface anisotropies at the two external layers [10,18] and are easily introduced in all cases:

1

E (~,-h)b~

= 7

~,,~1

E

209

jr.~(t,~-t,t)

~,,~I ,f'~l- I

b I = albo,

1

+ 7

E

J,,(b.~ - b,).

bN ~

(5)

ON bN-

1"

gEl.f~l+ I

Together with Eq. (8), these define the characteristic equation by equating to zero a suitable linear combination of matrix elements of r [12,15]. Within the central part of t-he ferromagnetic film, away from the rough part, the transfer matrix does not depend on n and reads:

Taking advantage of the translational invariance for directions parallel to the layer and of the effective selection rule for spin wave resonance ( k , = k, = 0) which is consistent with the roughness model, this reduces to a one-dimensional problem: 2( w - h) b; = ( b, - b, , ) J,., + (b/-

1

_T= {

b,+,) J/./+,,

2a + 1 2a

1] 1 '

(9)

(6)

where the JI.I+L a r e effective layer-to-layer exchange parameters. Assuming that the film has a simple cubic structure with (1,0,0) surfaces then leads to a value of J for J~./+~ when both layers 1 and l +_ 1 belong to the central part. Otherwise, Eq. (6) involves two different exchange parameters. The general Eq. (6) links the spin deviation in one layer to the spin deviations in the two previous layers, and thus leads to a transfer matrix formalism with two-by-two matrices. In order to focus attention on the possible local pinning or unpinning of spin waves, i.e. nodes and maxima, a spin wave vector field with components ( b , , , b , , - b,,_ ~), i.e. the spin wave amplitude b,, on atomic layer n and the difference between two neighboring layers, is introduced. The spatial evolution of this field defines Poincar6's transfer matrices T,, which read:

with a = ( h - w ) / J . Near the surfaces, the averaged effective exchange parameters are taken into account, and define d other 2 × 2 transfer matrices for each surface. Finally, the whole transfer matrix r for a film of N layers involving two symmetric surfaces of d layers reads: r=

T I'T~_' " " " TT'V' -2IdT,

d "'"

--

b,,-b,,

i

d

n

2a n

L=

+1 d

11

2 a - --

2a d + 1

b,v- bx-I

,7-.

-

bl -- bo

~

To •

,= I - -

bl - bo

(8)

,

'

,, +

n+l

t

Starting from layer 1 and ending at layer N defines the complete transfer matrix v.

j

+ 1

--

(7)

"

(10)

and the extension to asymmetric films and multilayers is easily understood. Two conjugate basic nonunitary rough surface matrices read:

,, + i

b,,+l-b,,

T~Tt,_

(ll) 2a d ti

n + 1 17

In a continuous treatment of the equation of motion (Eq. (6)), this means that the bulk equation b" + 2ab = 0

(6')

210

of Magnetism and Magnetic Materials 163 (1996) 207-215

D. Mercier, J.- C.S. Ldl'y / Journal

includes a damping term b' and that the stiffness term a is larger near the surface than in the bulk. Thus the standard Fourier treatment of the completed equation of motion (Eq. (6')) involves both Hermite polynomials and Airy functions [19]. In practice, the local wavevector k both increases and becomes

slightly complex when the spin wave is considered near the surfaces. There is thus a strong difference from harmonic waves. However, a quite similar discretized Schr/Sdinger equation occurs for the Stark effect in quantum wells [20]. Finally, the characteristic equation is deduced

pln-pln 400 O 10000

1 I

2 I

3 I

4 I

5 I

6 t

7 I

8 I

9 I

10 10000

-8800

9900-

-9800

98000

Q

-8700

9700-

9600-

~9600

O

-8800

9500-

-9400

94000 I

=

9300-

-9300

9200-

-9200

-9100

9100-

I e

9000-

-9000

8900-

-8900

8800-

-8800 0

8700-

-8700

8600-

-8600

8500-

-8500

8400-

-8400

8300-

-8300

8200-

-8200

0100-

-8100

8000

I

I

I

I

I

I

I

I

1

2

3

4

5

6

7

8

I g

8000 10

d Fig. 1. The resonant magnetic fields H, of the first modes of a film with 400 + 2 d layers and pinned surfaces.

D. Mercier, J.-C.S, L[t,y / Journal of Magnetism and Magnetic Materials 163 (1996) 207-215

bulk wavevectors k i and a set of resonant fields h i follow from this solution. Calculation of the partial transfer matrices starting from the first layer enables us to determine the spin wave propagation through

from the number N of layers, and the number d of layers which ensure the roughness property and the pinning conditions at the external surfaces. Thus a set of numerical roots a i is obtained, and a set of

'

1h

2]1

2 ;.

3OO '

4OO '

II 2-400-2 pin-pin

i 0

I 100

I 200

I 3o0

15-400-15 pin-pin

I 400

n

Fig. 2. The spin wave behaviour of the first modes of a film with 400 + 2 d layers and pinned surfaces when (a) d = 2 and (b) d = 15.

D. Mercier, J.-C.S. L&y / Journal of Magnetism and Magnetic Material.s 163 (1996) 207 215

212

F

I. 4o~

ao4

s-

1

, i

o

82100

U~00

t

e6I. . . .

100

90~0

92t00

94t00

~0O

I

9o~J

looo~

I-I

6-400-6 pin-pin I

I. 40

[

i

4 I i

J

ol

i

i

iri,i 15-400-15 pin-pin

i 98oo

10000

H

Fig. 3. The spin wave intensities and resonant fields for the first modes of a film with 4 0 0 + 2 d layers and pinned surfaces when (a) d = 6 and ( b ) d = 15.

D. Mercier, J.-C.S. L[t'y / Jot r u I qf M~ ~,,neti~m ~nd Magnetic Materials 163 (1996) 207-215

100

213

I

Z

9-400-9 unpin-unpin

45-

40-

35-

30-

25-

20-

15-

10-

5-

P 0

r

r

9-400-9 unpin-unpin

H

Fig. 4. (a) The spin wave b e h a v i o u r of the first modes of a film with 4 0 0 + 2 d layers and unpinned surfaces when d = 6. (b) The spin wave intensities and resonant fields for the same film.

D. Mercier, J.-C.S. Ldt,y / Journal qf Magnetism and Magnetic Materials 163 (1996) 207-215

214

the sample for each eigen mode ( b l ,~b 2~. . . . . b N) i therefore the spin wave intensity for each mode:

Ii = K

(

Evb]'

~/£v( bti,)+

and

(12)

Thus it is easy to compare fields and intensities with experimental results.

3. Numerical results The computations are performed for films with N = 400 + 2 d magnetic layers which exhibit several modes in the considered field range. The parameter of interest is the roughness thickness d which is varied from 2 to 15 atomic layers in order to examine different cases. We consider symmetric surfaces with either pinned or unpinned boundaries. The characteristic equation is obtained by putting to zero the convenient linear combination of matrix elements of r according to the pinning conditions. This characteristic equation is solved by means of a Newton method. In the calculations, the values of the parameters are: ~o= 104G and J = 10(' G. With these values, the resonant fields h i, the spin wave behaviour of the first spin wave modes and the spin wave mode intensities are computed. First, the influence of the roughness height d in the case of both pinned surfaces is shown in Fig. 1, where the resonant fields of the first modes are plotted. The effect is a general increase of the resonant field h, with increase of the roughness height d. This is well understood from the analysis of the spin wave behaviour shown in Fig. 2, where the limiting cases d = 2 and d = 15 are plotted. Quite obviously, there is a rapid variation of the spin wave amplitude near the rough surface because the local wavevector is larger there than in the bulk. Since the spin wave is simultaneously damped near the rough surface, as already seen, its amplitude is weak. As a result, when the roughness thickness d is increased, spin waves appear to be effectively more and more un-

bi,,,

pinned near the surfaces. Of course this rapid variation of amplitudes near the surface is stronger for modes of high rank than for the first modes, because of already larger wavevectors k, as clearly seen from Fig. 2. For perfectly unpinned surfaces, without any roughness, the uniform mode is a resonant mode, and the only one to be observed in spin wave resonance [18], thus the emergence of this effective case, because of surface roughness, can easily be seen from the plot of spin wave intensities shown in Fig. 3 for d = 6 and d = 15 when external surfaces are pinned. Comparison between the two results clearly shows that the decrease in intensity of the high rank modes when the roughness thickness increases is more effective than their shift towards higher field, shown in Fig. 1. In Fig. 3, unobserved resonant modes are noted by sharp lines of identical height. Finally, the case of magnetically unpinned surfaces, with roughness, must be considered. In this case, the spin wave damping near the surface leads to the appearance of a surface mode which lies quite close to the uniform mode and is the only observed mode, as shown in Fig. 4, where the same convention for unobserved modes as in Fig. 3 is used. In Fig. 4, both the simple intensity spectrum and the spin wave behaviour of the first modes are reported for d = 9. Unambiguously there is a surface mode since its resonant field is higher, even if only slightly, than that of the uniform mode: h u = 1 0 4 G.

4. Concluding r e m a r k s The above results demonstrate well that spin wave resonance must be quite sensitive to roughness effects even on single films, for different spin boundary conditions, and that for some boundary conditions roughness surface spin waves are obtained. The computations were performed on the basis of a simple model for the action of roughness on magnetic properties. Their simplicity warrants their validity. Experimentally it would be interesting to compare similar samples with different roughnesses, i.e. different non-magnetic substrata and different thermal treatments, and to correlate the spin wave resonance results with other measurements of roughness. Of

D. Mercier, J.-C.S. Ldcy/Journal c~fMagnetism and Magnetic Materials 163 (1996) 207-215

course, the consideration of roughness in trilayers, for instance, is also of interest for spin wave resonance experiments.

[5] [6]

Acknowledgements The authors are pleased to acknowledge very useful discussions with Dr. J.S.S. Whiting, Dr. R. Watts and M. Jackson from the University of York (UK). This work was carried out in the framework of contract CHRX-CT93-0320 of the European Economic Community.

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