Spin wave resonance studies of inhomogeneous La,Ga:YIG epitaxial films

INHOMOGENEOUS La,Ga:YIG EPITAXIAL FILMS

B. HOEKSTRA

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t-SPIN WAVE RESONANCE STUDIES OF INHOMOGENEOUS La,Ga: YIG EPITAXIAL n L M S

BIBLIOTHEEK TU Delft P 1781 4355

INHOMOGENEOUS La,Ga : YIG EPITAXIAL FILMS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL TE DELFT,

OP GEZAG VAN DE RECTOR MAGNIFICUS PROF. DR. IR. F. J. KIEVITS, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN TE VERDEDIGEN OP WOENSDAG 15 NOVEMBER 1978

TE 13.30 UUR

DOOR

BOB HOEKSTRA

NATUURKUNDIG INGENIEUR

GEBOREN TE NUENEN, GERWEN EN NEDERWETTEN

1978 Gema - Eindhoven

Aan: Geraldine Marieke

General Introduction 1

References 3 Chapter I. THEORY OF SPIN WAVE RESONANCE IN THIN

FILMS 4 1.1. Introduction 4 1.2. The equations of motion 7

1.3. Normal modes of a homogeneous film in perpendicular resonance 13

1.4. Surface modes 15 1.5. Oblique resonance 16 1.6. The perpendicular resonance of a multilayer 20

1.7. Oblique resonance of a multilayer 31 1.8. The relation between the surface layer model and the surface

anisotropy model 33 1.9. Normal modes of films with linearly varying anisotropy . . . . 35

1.10. Determining the exchange stiffness constant 40

References 41 Chapter II. EPITAXIAL GROWTH OF GARNETS BY LPE . . . 43

II. 1. Some properties of garnets 43 11.2. The LPE growth process 44 11.3. Experimental procedures 48

References 49 Chapter III. THE RESONANCE EQUIPMENT 51

III. 1. The spectrometer 51 III.2. Magnetostriction measurements 54

References 58 Chapter IV. SPECTRA A N D STRUCTURE OF La,Ga:YIG FILMS 59

IV.l. Introduction 59 IV.2. The perpendicular resonance spectrum of film I grown at 850 °C 63

IV.3. The perpendicular resonance spectrum of film II grown at 935 °C 64

IV.4. The angular dependence of the spectrum of film II 70 IV. 5. The spectrum of film III, an example of a linearly varying

aniso-tropy 76 IV.6. Determining the exchange constant of films I, II and III . . . . 79

IV.7. Discussion 82 References 85

TIONS AND THE GROWTH PROCESS 86

V.l. Introduction 86 V.2. The effects of meltback and diffusion 87

V.3. The dependence of the properties of La,Ga: YIG films on growth

temperature and lattice mismatch 90 V.4. The origin of the parameter variations 94

V.5. Conclusions 99 References 101 Appendices 102

A. Equations of motion of a thin film with z-dependent anisotropy for

arbitrary field direction 102 B. Normal modes of a film with a linear gradient of the anisotropy 104

List of frequently used symbols and abbreviations 106

Summary 109 Samenvatting ,. 110

Curriculum vitae 112

Laboratories in Eindhoven The work has for the larger part been pub-lished before in separate articles The experiments were carried out in a fruitful cooperation with the co-authors of these articles G Bartels, F van Doveren, J M Robertson, W T Stacy and R P van Stapele Particularly the pleasant and close cooperation with J M Robertson, who grew most of the films, has made it possible to explore the La.Ga YIG system in a relatively short time Many of my colleagues have contributed to the work in the discussions, particularly P F Bongers The work would not have been complete without the chemical analyses performed by P Paans W van Erk has contributed by his comments on the manuscript

Finally I would like to thank the management of Philips Research Labor-atories for giving me the opportunity to prepare this thesis

GENERAL INTRODUCTION

A thin plate of a magnetic crystal can exhibit a domain structure with so-called magnetic bubbles. Bubbles are cylindrical domains (see fig. 1) in a film magnetized by a magnetic field perpendicular to the film plane. In the bubbles the magnetization direction is opposite to the applied field. Magnetic bubbles were first observed by Kooy and Enz ') and their device possibilities were demonstrated by Bobeck ^) in 1967. This has resulted in the development of magnetic bubble memory devices ^).

Fig. 1. A magnetic bubble can exist in a thin magnetic film if a magnetic field is applied normal to the film. Within the cylindrical bubble the magnetization is opposite to the applied field.

The thin films applied in the devices are single crystals with the garnet structure. They are grown epitaxially on nonmagnetic garnet substrates by liquid phase epitaxy (LPE). A necessary condition for the existence of magnetic bubbles is that the preferred direction of magnetization is perpendicular to the film. Despite their cubic structure the garnets exhibit a uniaxial anisotropy with the film normal as principal axis. The film normal is the preferred direction if this uniaxial anisotropy is positive. It is induced either by the crystal growth process or by the stress due to the mismatch with the substrate lattice.

At the time when we initiated the experiments described in this thesis it was realized that the epitaxial growth process does not yield a homogeneous film. When the dipping technique is used a substrate is dipped into an undercooled melt and initially the growth rate is fairly high. It drops however to a steady state growth rate which is often limited by the diffusion of fresh material

towards the growth face. Since the crystal composition depends on the growth rate it was suspected that this variation of the growth rate leads to a com-positional variation in the growth direction.

It was also known that a gradient of the magnetic parameters could affect the properties of bubbles in these layers. The most important effect is that particular gradients can suppress bubbles with complicated wall states (hard bubbles) which have lower velocities than normal bubbles for a given drive. These hard bubbles are undesired in devices. It is necessary to cover the surface of a uniform film with a surface layer which has its magnetization in the plane of the film in order to suppress these hard bubbles (this layer is often pro-duced by ion implantation of the surface). In our laboratory films had been obtained which, as grown, did not support hard bubbles. X-ray diffraction showed that these films sometimes had two sections parallel to the surface with different lattice constants.

Haisma et al.*) suggested that the double layer structure of these films might be as effective in suppressing hard bubbles as the ion-implantation method. This has prompted our studies of the inhomogeneities in these films. It soon became clear to us that these double layer films were very similar to implanted films. They had a bulk with positive uniaxial anisotropy (implying that the energy is lowest if the magnetization direction is normal to the film, a con-dition for the existence of bubbles). A thin surface layer had negative uniaxial anisotropy and accordingly its magnetization direction was in the plane. Our studies have been aimed at finding how this particular situation comes aboui. The thin surface layer is located near the substrate and we will demonstrate that it is due to the transient effects in the growth process.

The garnet materials used in magnetic bubble devices are derived from the magnetic yttrium-iron garnet (YjFejOij). Combinations of rare earth ions can be substituted for Y and diamagnetic Ga or Al can be substituted for Fe ions. Usually a combination of rare earth ions is required in order to obtain a sufficiently large, induced uniaxial anisotropy. The films in our laboratory which are hard-bubble-free as grown, are of approximate composition Y2.85Lao,,5Fe3.75Gai.250i2. They contain no magnetic rare earth ions which are known to damp the spin precession. As a consequence high domain wall velocities can be realised in these films. The resonance spectra of these films have lines of 10 Oe width. The spectra show many intense resonance lines due to different precession modes. These are spin wave resonance modes. It is well known that in thin films it is possible to excite standing spin wave reso-nance modes across the film thickness. The spacing and intensity of these modes is determined by the boundary condition at the surfaces and by the exchange stiffness. It is also well known that variations of the magnetic properties across the film thickness affect the spin wave modes. We have used this effect to deduce the profile of the magnetic parameters across the film

thickness from the spin wave resonance spectrum. Thus we have been the first to supplement data on the compositional gradient across the film thickness with data on the magnetic parameters. By varying the growth conditions we have been able to demonstrate a correlation between the profile of the magnetic parameters and the crystal growth kinetics. Thus we are not dealing with spurious effects but with parameter variations which are intrinsic proper-ties of films grown by the liquid phase epitaxial process. We will even use the spectra to demonstrate some aspects of the crystal growth.

This thesis is divided in five chapters. The first chapter deals with the theory of resonance in inhomogeneous films. Gradients of the uniaxial anisotropy are considered. The separation of the modes into localized modes, which carry the information on the inhomogeneities, and volume modes, which can be used to determine the exchange stiffness constant, is illustrated. In the experi-mental work the localized modes are used to deduce the parameter profiles. Chapter II deals with some properties of garnets relevant to our work and describes the LPE growth process by the dipping technique. The growth kinetics are discussed since the relation between the inhomogeneities and the growth kinetics is the subject to chapter V. The resonance equipment and various ex-perimental procedures are discussed in chapter III, particularly the milliscope cavity which is a slightly modified standard cavity permitting local measure-ments on large plates. In chapter IV the profiles of the uniaxial anisotropy are determined of three films which are characteristic for La,Ga: YIG films grown at various temperatures. The limitations of the SWR method in determining the profiles are discussed.

Finally, in chapter V we will put things together. First the origin of the uniaxial anisotropy in these films is determined and next the parameter profiles are shown to be directly related to compositional gradients which in turn are related to the variations of the growth rate during the LPE dipping process, particularly during the initial transient period.

REFERENCES ^) C. Kooy and U. Enz, Philips Res. Repts. 15, 7 (1960). 2) A. H. Bobeck, Bell Syst. Tech. J. 48, 3287 (1969).

^) A. H. Bobeck and E. Delia Torra, Magnetic Bubbles, North Holland Publ. Comp, Amsterdam (1975).

CHAPTER I

THEORY OF SPIN WAVE RESONANCE IN THIN FILMS

I.l. Introduction

Spin waves are excitations of a system of exchange coupled spins. A traveling spin wave has no net instantaneous rf magnetic moment and thus does not couple to a spatially uniform rf field. In ferromagnetic resonance experiments the uniform rf field therefore excites only the uniform resonance (wavevector ^ = 0) in which the magnetization of the material precesses as a whole. For arbitrary sample shapes however inhomogeneous rf demagnetizing fields do give rise to excitation of long wavelength spin wave modes. The dipole fields due to the divergence of the rf magnetization at the sample surfaces cause a fc-dependence of their frequency (magnetostatic modes). These were first ob-served by White and Solt ') in 1956 and later studied experimentally by Dillon ^) and theoretically by Walker ^). For large k (wavelength < sample size) the sur-face dipole fields average to zero. However, for large k the exchange effects become significant. These are usually treated in a continuum approximation which applies if the wavelength is large compared to the ionic distances. In 1958 Kittel *) demonstrated theoretically that a uniform rf field can excite ex-change dominated spin waves directly in a thin film because of the boundary conditions at the surfaces (exchange dominated means that the A:-dependence of the frequency is mainly due to the exchange energy). In essence Kittel's argument was that the surface spins are in a special position. They have a resonance frequency different from bulk spins and are therefore pinned while the bulk resonates. In a thin slab with thickness L the boundary conditions on the transverse component m of M are m{z) = 0 at z = 0 and z = L. The normal modes are thus m{z) = sin (nnzIL) with « = I, 2, 3, . . . . For odd n these modes have an instantaneous net transverse magnetic moment and thus couple to a uniform exciting field. If the film thickness L is sufficiently small (i5a 1 fxm) the exchange energy of these modes is high enough so that the separation of the frequencies of the modes (mode spacing) exceeds the line-width. This was demonstrated by Seavey and Tannenwald ^) in thin permalloy films. Permalloy is the name for alloys of Ni and Fe. The composition may range from 70% Ni-30% Fe to 90% Ni-10% Fe. The films were of interest for their possible application in memory elements ^). Spin wave resonance could be used to measure the exchange constant. Furthermore, the dependence of the spectrum on the boundary conditions could be used to measure the magnetic surface state ' ) .

Kittel suggested that the surface spin pinning could be due to either the special situation of the surface spin (Neel surface anisotropy '')) or to the presence of

an antiferromagnetic surface layer In 1962 however, Wigen et al ^) demon-strated that a thin surface layer with a slightly lower magnetization can account for the spin pinning Such a surface layer is thin compared to the total film thick-ness, but its thickness is large compared to the crystal lattice constant In this model the mismatch between the frequencies of the bulk spins and surface layer spins causes a dynamic pinning due to the continuity condition at the interface This naturally explained the observation of a critical angle effect if the magnetic field is applied at an angle with the film normal For this critical angle the angular dependent frequencies of the surface layer and the bulk become equal so that the pinning disappears and only one mode, the uniform precession, is excited This also drew attention to an important problem which would charac-terize all subsequent work: the metallic films did not have uniform properties across their thickness and, worst of all, these non-umformities were not repro-ducible ' ) . This shows up in mode spacmgs and mode intensities that deviate from the simple quadratic laws that follow from the simple Kittel theory (see I 3) A variety of models has been proposed to account for the observed spectra. These can be divided roughly into two: I surface anisotropy mo-dels 10,11,12-) vvhich assign an additional anisotropy energy to a monolayer of surface spins only and II volume mhomogeneity models which assume that the magnetic properties, usually M, vary across the film cross section Naturally the latter model is the most flexible since a variety of functions M{z) can be used, such as parabolic '^), linear '*), trigonometric " ) or thin but finite sur-face layers ^.IS.IT^ Films with spectra showing a mode spacing quite close to the quadratic Kittel spacing have been prepared '^ ' ' ) . Even in these films it IS probable that the small surface anisotropy is not an intrinsic anisotropy of a monolayer of surface spins but represents the effect of a residual mhomogeneity, throughout the film thickness or in a thin surface layer.

In 1968 the first thin layers of yttrium-iron-garnet (YIG), grown epitaxially on non-magnetic substrates (usually GGG, gadolinium-gallium-garnet) became available ^°) These films permitted the first observation ^') of magnetostatic modes, exchange modes and admixtures of these (magneto-exchange modes) on one sample The modes are named according to the energy term which dominates the ^-dependence of the frequency Magnetostatic modes are long wavelength modes which have little exchange energy and mainly magnetostatic (dipolar) energy Exchange modes are short wavelength modes with dominating exchange energy In samples with finite dimensions the normal modes have wave-lengths such that the dimension in a given direction is a multiple of half the wavelength in that direction. In a thin garnet film with a large ratio of the lateral dimensions to the film thickness the wavelength in the plane is rather long so that modes with different in-plane wavevectors have different magneto-static energy Exchange energy is mainly associated with the wavevector normal to the film plane. The magnetoexchange modes ^^) have comparable

exchange and magnetostatic energies. A resonance mode particularly suitable for a study of the surface condition is the surface exchange mode. The name implies that it is a mode with mainly exchange energy and thus with a short wavelength. It exists only when the surface spins have a lower resonance fre-quency than the bulk spins. Its wavevector is parallel to the film normal and is imaginary so that it is an excitation which has maximum amplitude at the surface and decays exponentially into the bulk. The surface mode had been predicted theoretically by Puszkarski ^^) and independently by Sokolov ^'*) from a quantummechanical solution of the spin wave resonance problem in a film with surface anisotropy. Its existence also follows directly from the semiclassical models involving surface layers or surface anisotropy (see 1.4). It was first observed in YIG films in 1972 ^'). Often the name non-propagating surface mode is used. This is to stress that it does not have an in-plane propagation such as the propagating magnetostatic surface modes (Damon and Eshbach 1961 ^*)), These are long wavelength modes with an imaginary component of k in the direction of the film normal. The character-istic decay length is usually several times the film thickness.

The epitaxially grown single crystal YIG films have uniform properties ex-cept for narrow layers at the surfaces. The first films had surface modes located at the substrate-film interface presumably due to a diffusion layer ^'^). These were grown by chemical vapour deposition (CVD) at a high temperature (1300 °C) where diffusion between substrate and film during the growth process is appreciable. Surface anisotropy models have been successful in explaining the spectra of these films 27.28,29^ y j Q grown by liquid phase epitaxy at a-bout 900 °C had an effective surface anisotropy two orders of magnitude lower than that of CVD films, presumably because of the smaller diffusion rates at the lower growth temperature ^°). For YIG films one can conclude that inhomogeneous surface layers cause the effective surface pinning condition. In this thesis we report on substituted YIG films which, due to the growth kinetics of the LPE process, have considerable parameter variations across the film thickness ^'~^*). These films are definitely dirtier than the pure YIG films, they are very inhomogeneous. As stated in the general introduction the in-homogeneities have interesting consequences for the application of these films in bubble devices. In addition the inhomogeneities are closely related to the growth process and can be used to investigate the growth process ^^-^i). in this chapter we will present the theory of spin wave resonance in thin films, largely based on the original theories of the early sixties: Kittel *), Pincus *'), Portis '^), Schlomann '*) and Wigen and coworkers *•'). We will consider the case where the effects of the parameter gradients and the exchange stiffness on the resonance modes are of comparable magnitude, which represents the situation in our films.

magnetic field is applied normal to the film and next if the magnetic field is applied at an angle with the normal (oblique resonance). Only the uniaxial anisotropy and the magnetization are assumed to vary from layer to layer. The particular case of a linear gradient of the uniaxial anisotropy across the film thickness is treated in 1.9. For comparison with previous work we will discuss also surface anisotropy models and the relation between those and the multilayer theory. Finally the determination of the exchange stiffness constant is discussed.

1.2. The equations of motion

In a thin plate of material with thickness L a coordinate system is defined as drawn in fig. 1.2.1. We will assume throughout this thesis that the material is

Fig. 1.2.1. Definition of the coordinate axes with respect to the thin film sample.

saturated, so that there are no domains. Due to the sample shape there is an anisotropic demagnetizing energy. A thin film can be considered as a flat ellipsoid for which the anisotropic part of the demagnetizing energy (£'dem) 's given by

(1.2.1)

^dem = -InM^SVCl^ 0.

The angle 0 is the angle between the magnetization M and the film normal ( = z-direction). The magnetization M is the magnetic moment per volume and the energies v.'ill also be given per volume throughout this thesis. Independent of the sample shape there is also an anisotropic energy, the magnetocrystalline

anisotropy energy (see II. 1). Thin films often have an induced uniaxial aniso-tropy energy £"„ with the film normal as the principal axis:

£„ = K^ sin^ 0 + higher order terms. (1.2.2) Ku is the first uniaxial anisotropy constant. Higher order constants are usually

small at room temperature. Since the equations 1.2.1 and 1.2.2 have the same dependence on 0 we can define an effective anisotropy constant K^'" as

KJ" = /:„ - 27zM\ (1.2.3) A thin film can be ascribed a uniaxial anisotropy energy £„ of the form of

Eq. 1.2.2 with the anisotropy constant AT/" which represents the induced uni-axial anisotropy and the demagnetizing energy. These two terms cannot be separately determined unless one studies magnetostatic modes which have different demagnetizing energy depending on the wavevector. We consider only exchange modes and thus cannot separate the two terms in the uniaxial aniso-tropy energy. The single crystalline garnet films have cubic symmetry and therefore have an anisotropy energy with cubic symmetry. It is usually written a s " ) :

£cub = ^ i C a i ' a a ' + oi^^^z^ + « 3 ' « i ' ) . (1-2.4) Ki is the first cubic anisotropy constant. The a,'s are the direction cosines of

the magnetization with respect to the cubic [100] axes. A cubic [111] axis is normal to the film plane in our films.

In order to saturate the film a field H has to be applied. Resonance with H applied normal to the film will be called perpendicular resonance while parallel resonance refers to resonance with H applied in the film plane. For the present we will consider perpendicular resonance only. The anisotropy energies given by Eqs. 1.2.2 and 1.2.4 can then be represented by effective fields. From Eqs. 1.2.2 and 1.2.4 it is found that for small angular deviations 0 of M from the film normal in a [111] oriented film the anisotropy energy is given by

£ ^ - H I ^1 - 2 ^ / " ) 0'. (1.2.5) Comparing this with the magnetostatic energy if a magnetic field H is applied

along the film normal:

E = - MH cos 0 ^ ^MH0^ + constant, (1.2.6) it is seen that an anisotropy field H^ parallel to the film normal can be defined

as ^ ' ) :

The equation of motion for the magnetic moment M of the material under the influence of the fields H and H^ is given by the familiar equation ^*)

dM

= -yMx(H + Hfd, (1.2.8) dt

where the gyromagnetic ratio y equals

y = g\e\l2m,c. (1.2.9) y is positive, g is the g-factor, e the charge of the electron, m^ the electron mass

and c the speed of light. We note (see fig. 1.2.2) that dM/dt = M is

perpen-Fig. 1.2.2. The precession of the magnetization vector M about the applied field H.

dicular to both M and H + H^ so that the moment M will precess around the field H + H^ on a cone. Substitution of

m = (cos mt • Cx + sin cot • Cy) (1.2.10) for the transverse component of Af (m < M) yields the familiar condition

for resonance

(o = y{H + HK). (1.2.11) If the rf magnetization is non-uniform there is a torque on M due to the

ex-change interaction. In the continuum approximation, when the wavelength of the spatial variation of /n is large compared to the spin-spin distance, it can be represented by " ) :

2A

^ „ = - - V W , (1.2.12) M^

where A is the exchange stiffness constant. Finally the damping is often given in the Gilbert f o r m " ) :

a M

"damping = — . (1.2.13) y M

This relation is similar to the often used Landau-Lifshitz ^*) form for small damping. In reference 36 it is demonstrated that for higher values of the damp-ing constant a the Landau-Lifshitz dampdamp-ing gives in addition to the term 1.2.13 an eflFectively higher y-value:

yeff = (1 + a") Y- (1.2.14) We will neglect this correction to y in the following. In the final results y can

be replaced by y^ft for high values of damping. The equation of motion now becomes

2A a .

M=-yMx(H+ HK + V^M M + h). (1.2.15) M^ yM

We have added a driving rf field h. To facilitate the solution of Eq. 1.2.15 we take a new coordinate system e||,ej^ which rotates around the z-axis with angular frequency m, with the same sense of rotation as M rotates around the effective field (compare fig. 1.2.2):

Cii = cos (ot • Cjc + sin (ot • Cy

e^ = —sin cot • Cx + cos wt • e,. (1.2.16) We choose the exciting field

A = /!C|„ (1.2.17) which is stationary in the rotating frame. A linearly polarized field (amplitude

2h), usual in experimental situations, can be considered as the resultant of two oppositely rotating fields of amplitude h. The field which rotates with the same sense of direction as the precession of the magnetization (Eq. 1.2.10, fig. 1.2.2) is given by Eq. 1.2.17. The effect of the component with opposite sense of rotation can be neglected ^*).

The precessing magnetization will look like

M = WiiCii + m_^e^ + M^e^, (1.2.18) where the components of M are time independent.

We will consider only variations of the rf magnetization and the magnetic material properties along the z-direction. So neither spin waves in the plane of the film nor variations of the parameters in the plane will be considered.

W e will assume t h r o u g h o u t this thesis that only t h e uniaxial anisotropy con-stant K„ z-dependent.

We now solve Eq. 1.2.15 by substituting h a n d M from Eqs. 1.2.17 a n d 1.2.18, C|| = -l-cocj^, e^ = —£0^11 from Eq. 1.2.16 a n d linearizing the equations in the transverse components of M a n d the exciting field h.

This yields the equations:

2A d^/W|| ft) OLco m^^{H + HK.{Z)) = »i|| - Mh m^ M dz^ y y (1.2.19) 2A d^/Mj^ (o CLU) -— ——- - mAH + HK_{Z)) = m^ + m,,. M dz'^ y y

Before actually solving these equations for t h e normal modes we will derive a relation for t h e coupling t o the driving field. Therefore we assume that we know the solutions m„(z) in case t h e driving field a n d the damping are zero. In that case E q . 1.2.19 reduces t o :

/2A d^ \ (o„ \ M dz^ / y

The normal modes m„(z), which we choose real, form a complete set of orthonormal functions, so that

L

fm,iz)mj{z)dz = dij. (1.2.2!) o

T h e modes excited by t h e rf field can be written as f"\\i^) = 'Za„m„(z), Wi(z) = X b,/n„(z).

n

Substitution in Eq. 1.2.19 yields:

'2/1 d^ ft)\ aw (1.2.22)

(

(

ft) — w„ aw L «» '"'.(2) = - Mh X Km^z), V Y n (o — co„ ao) I , bn niniz) = + ^ a„w„(z). Y Y n (1.2.23) (1.2.24)

Multiplication with /Mp(z) and integration over the film thickness gives, using Eq. 1.2.21: w — w„ aft) '-•a,, -\ b^ = - Mh fmp(z) dz Y Y o a w CO — ojp a„ + b„ = 0. (1.2.25) Y Y

Finally, we find for a^, and bp-.

L (ft) — Wp) Mh f m^iz) dz 0 (w — WpY + a^w^ L (1.2.26) OLOjMh f nipiz) dz n b„ = (ft) — Wp)^ + a^w^

The power absorption is obtained from jh • dm, where we substitute dm = (dmjdt) dt and integrate over the film thickness, to obtain

2 = 0 0 '

We substitute m and A from Eqs. 1.2.18 and 1.2.17:

m = WiiCii -I- m^e^, h = he^^, (1.2.28) and use Eq. 1.2.16 to obtain

hoi f

P = / w ^ d z (1.2.29) z=0

With Eqs. 1.2.22 and 1.2.26 we obtain finally

L \ 2 / m„{z) dz ) .

P = -Y.^^ —• (1.2.30) L „(«()- co„y +

a?-oP-For small damping such that a^w^ < (w„ - w„+i)^ the excited mode is al-most purely one of the functions m„{z) (see Eq. 1.2.26) and the power ab-sorbed for w sa w„ is simply " ) :

We thus see that the absorption is simply proportional to the square of the integral of the instantaneous transverse magnetic moment.

1.3. Normal modes of a homogeneous film in perpendicular resonance

Let us first neglect the damping (a <c 1), and take the exciting field /j = 0 and solve Eq. 1.2.19 for the normal modes of the film. Eq. 1.2.19 then becomes

2A d^ni CO

— —--{H+H^{z))m = - - m , (1.3.1) M dz-' y

where m is the transverse magnetization. We further simplify the problem by taking Hg^ constant throughout the film thickness. Eq. 1.3.1 then becomes

2A d^m ( / M dz^ \ The general solution of Eq. 1.3.2 is

m = a- e"^ + b • e''"^. (1.3.3) Substitution of Eq. 1.3.3 into Eq. 1.3.2 yields

w 2A

— = H + Hg + k\ (1.3.4) y M

This is the well-known quadratic dispersion-relation for spin waves. Traveling waves occur only for w/y > H + H^ %o that the wavevector k is real. Alternativ-ely one can say that at constant frequency these occur only if / / < //„„ where Hu„ is the field for uniform resonance (/c = 0):

ft)

7/„„ = Hg. (1.3.5) Y

In the case of our thin film there are boundary conditions to be considered at the surfaces at z = 0 and z = L. Due to these boundary conditions only solutions of the form of Eq. 1.3.3 are allowed with specific wavevectors quan-tized in units 1/Z,. These form standing spin waves across the film thickness with either nodes or maximum amplitude (zero derivative) at the surface, depending on the boundary conditions. The possible normal modes in those cases where the spins are either completely pinned {m = 0) or completely unpinned (dm/dz = 0) at the surfaces have been listed in table 1.3.1. The relative intensities have been calculated from Eq. 1.2.31. The mode shapes m{z) have been drawn in fig. 1.3.1. Most of these results were first derived by Kittel *) and Pincus " ) . It is to be noted that in a film with no surface pinning only the uniform mode is excited by a uniform field. Of all higher order modes m{z) integrates to zero over the film thickness. If the spins are pinned on one side only, the spacing of the modes with respect to //„„ increases quadratically with

TABLE 1.3.1

Normal modes of a homogeneous film depending on the boundary condition at the surfaces. The field for resonance is given hy H = cojy — H^^- {2AIM)k^

= H,„-{2AIM)k' pinning condition at z = 0 z = L wavevector k = nnIL m{z) intensity quantizaton unpinned unpinned n = 0 , 1 , 2 , . . . cos {rmzjL) ^0 for n = 0 = 0 f o r « 7^0 pinned unpinned « = 1 3 J. " 2 ' 2 ' 2 ' • • • sin (nnzjL) ~lln^ pinned pinned n = 1 , 2 , 3 , . . . sin {nnzjL) ~ l/«^ n odd = 0 ,n even ^ ^ : z = L z = 0

^ 7

Fig. 1.3.1. The first few normal modes of a thin film The dotted lines mark the fields where the modes are excited (at constant frequency) and m(z) has been drawn Top unpinned spins on both surfaces. Middle: pinned spins at z = 0, unpinned spins at z = 1. Bottom: pinned spins at z = 0 and z = L. Compare table 1.3.1.

n and their intensity drops with the square of the modenumber n. If the spins are pinned on both sides half of the modes have zero intensity and the intensity of the others drops again with the square of the modenumber n.

1.4. Surface modes

Thus far we have considered simple boundary conditions at the surfaces. A frequently used more general phenomenological boundary condition is in terms of a surface anisotropy constant '^) K^:

{dmldz)lm = ± KJA. (1.4.1) The — sign applies to the top surface, the + sign to the bottom surface of the

film. Ks in essence acts to shift the resonance frequency of the monolayer of surface spins with respect to the resonance of the bulk. If K^ is very large the surface spins are far off resonance and cause a pinned condition at the surface (jn = 0). For small K^ Eq. 1.4.1 gives dmjdz ^ 0 (unpinned surface spins).

A very interesting mode appears in the case that the surface spins are freeer than the bulk, e.g. Aj < 0 in perpendicular resonance. Then a so-called sur-face mode 23,24,27^ vj'\Xh purely imaginary wavevector can be excited which has maximum amplitude at the surface and decays exponentially into the bulk. To demonstrate this, let us consider a thin slab with surfaces at z = 0 and z = L and assume KJi^L) = 0 and KXO) = K^. If the rf magnetization in the slab has the shape m = a- cos {kz -\- cp) then the boundary conditions at z = 0 and z = L lead to

tan kL = KJAk. (1.4.2) In the case that k = ik' is purely imaginary Eq. 1.4.2 reads

-Xa.nhk'L = KJAk'. (1.4.3) Both the lefthand side and righthand side of Eqs. 1.4.2 and 1.4.3 have been

plotted in fig. 1.4.1, where k'L is plotted to the left and kL to the right. The function tanA:L together with the functions — tanh ArX form a set of curves labeled 0, 1, 2 , . . . , each of which is intersected once by the function KJAk or KJAk'. The allowed values of k are found at these intersections. In the case ATj > 0 there are only solutions in the righthand side of the graph, with real values of k.

The surface mode with imaginary k is found only in the case K^ < 0 a.t the intersection in the third quadrant. The z-dependence of this mode is simply

m{z) ~ coshA:'(z- L), (1-4.4) which satisfies the boundary condition at z = L. If K^ is very large and negative

the surface mode is highly localized {k'L > 1). Notice in fig. 1.4.1 that the first few modes with real k (volume or bulk modes) have A:L «» (« -f- \)7i {n = 0,

Fig I 4 1 Mode diagram for a film with a surface anisotropy K, on the z = 0 surface k is the spin wave wavevector and L is the film thickness The function KJAk intersects every curve I (i = 0, 1, 2, 3, . ) once The curves i represent the function tan ArZ. Every intersection represents a normal mode In the case A'j < 0 there is a solution with imaginary wavevector k = ik' which represents a surface mode.

1 , 2 , . . . ) like in the case of pinning on one side (see table 1.3.1 and fig. 1.3 1). The number of these modes increases with KJA. High order modes are closer to kL s» nn and thus are eflFectively unpinned. In the case K^ > 0 when the surface spins are less free than the bulk, there is no surface mode. Again the first few modes are pinned at z = 0, while high-order modes are unpinned. In all cases the number n of the mode (« = 0, 1, 2, . . .) equals the number of zero-crossings of /M„(Z). Since there are two surfaces the number of surface modes is either 0, 1 or 2 in this model.

1.5. Oblique resonance

In the previous calculations we have considered perpendicular resonance since then the precession is circular. This greatly simplifies the mathematics It IS also the usual experimental situation because the analysis of the spectrum IS easy. In our experiments however we have obtained many data from the dependence of the resonance modes on the field direction so that we have to discuss oblique resonance in detail For arbitrary field direction the anisotropy energy is not symmetric for small deviations of M from its equilibrium orien-tation as It was in Eq. 1.2.5. Therefore the torques in two orthogonal directions

are not equal and the precession becomes elliptical '*) so that we have to deal with two orthogonal transverse magnetization components. An additional complication is that the static magnetization is not parallel to H so that the equilibrium direction of M has to be determined as well. In appendix A we derive the equations of motion for a homogeneous film (the anisotropy is still assumed to be z-independent):

1 . 2A d^OT, — nif = (H cos & + g{0)) m„ —

-Y (1.5.1) 1 . + - W , = ( / / c o s ? ? -H/(6>))/«4 Y M dz^ 2A d^nii. M dz^

lUf and m„ are transverse components of M out of the equilibrium direction. •& is the tilt angle between H and M: & = 0^— 0. The coordinate systems have been drawn in fig. 1.5.1. The functions/(0) and g{0) depend on the anisotropy. For a film with cubic anisotropy and induced uniaxial anisotropy and the field H applied in a cubic (110)-plane/and g are:

K K /(6») = 2 —COS20F — ( 2

-M -M 13 sin^ ij) -\- \2 sin*y) K K

g{0) = 2 — cos^ 6> -h — (2 - 7 sin^ ip + 2 sin'^ xp). M M

(1.5.2)

Fig. 1.5.1. Coordinate systems and angle definitions for oblique resonance, mj and m„ are transverse components of M. e„ is normal to the film (compare fig. 1.2.1.).

y) is the angle between M and the cubic [001] axis in the (110)-plane. Higher order anisotropy constants ^') are neglected. The equilibrium angle 0 is found from the condition derived in appendix A (Eq. A16):

MH sin {0„ - © ) - / : „ sin 20 - Ki{sm 2rp) (1 - fsin^ ip) = 0. (1.5.3) The solutions of Eq. 1.5.1 are of the form

m^ = a cos wt • cos {kz + cp), „ , , m^ = b sin wr • cos {kz + <p),

where bja is the ellipticity.

Substitution of these solutions in Eq. 1.5.1 gives the resonance condition *): {(ojyf = (//cos ^ + g{0) + {2AIM)k^) (//cos & +f{0) + {2AIM)k^). (1.5.5) Solving Eq. 1.5.5 for the values of k we find

{2AlM)k^ = -Hcos&- K / ( 0 ) + g{0)) ± {coly) j / l - { ( / ( 6 > ) - g ( 0 ) ) / ( 2 w / y ) p . (1.5.6) The fields required to saturate a sample are larger than f{0) and g{0). Ex-perimentally the frequency is usually chosen such that co/y > f,g so that the argument of the square root is always positive. The values of k can then be either real (trigonometric solutions) or imaginary (hyperbolic solutions). For the solution k^ with the -I- sign in Eq. 1.5.6 the ellipticity is easily calcu-lated to be

{alby ={g- fWcoly) + l / l - { ( g - / ) / ( 2 w / y ) F . (1-5.7) For ease of notation we drop the argument 0 f r o m / a n d g. Eq. 1.5.7 shows that if g a n d / a r e equal the precession is circular, but for g ^ / t h e precession is elliptical. The sense of the precession is as drawn in fig. 1.2.2. The second type of solution ^^) k~ with the - sign in Eq. 1.5.6 has always an imaginary wavevector. Its ellipticity is

{alb)- = ( g - / ) / ( 2 w / y ) - yi-{{g-f)l{2coly)y (1.5.8) and the sense of precession is opposite to that of the k'^ solutions. From Eq.

1.5.6 it is clear that \k-\ is always very large compared to |^"''| since the third and the first term on the righthand side are both negative.

In an experiment the solutions k' usually have a very small amplitude com-pared to the k'^ solutions. However, the k~ solutions are needed to match the boundary conditions at the surfaces and cannot be neglected.

In the solutions 1.5.5-1.5.7 the angle «? = ©« — 0 is still unknown. In order to see the effect of the angle § we will apply a perturbation procedure in writing the solutions of the equations 1.5.1. Again, we take the solutions Eq. 1.5.4 and substitute in Eq. 1.5.1 to find

w a 4 yH w 6 4 yH COS ?? 4 — 4 ^ H MH ( f 2A cos J? 4 — 4 V H MH '\b = Q, (1.5.9) 0.

We neglect the usually relatively small cubic anisotropy and the parameter 2KJMH = A is taken as the small perturbation. Then we write:

Q = cojyH = 13«" 4 Ai3<i> 4 PQ^^^ + ... a = a'°' 4 Aa*i> 4 A^a<^> 4 . . .

b = i*"' 4 AM'' 4 A^ft") 4 . . .

^ere g " ' ^ ( 2 , / < " / ( 2 , ^ ( i ) ^(2) ^ ( 0 ) The zero ^ = ^<°> 4 A ^ < " 4 A^^<2> 4 . . g = g*"' 4 ?ig''' 4 A V ' ' + . . • / = / ( 0 ) + A/(i> 4 Ay<2> 4 . . = COS^ 0n, = ^'i>sin20H, = COS 2 0 H . = 2t?<i' Sin 20„, = i sin 20„, = - i s i n 4 0 „ , = /<"> = g(0) = 0.

-order part of Eq. 1.5.9 yields

/ 2A ^ « ' - 4 1 4 k^ (1.5.10) (1.5.11) \ MH • and a(0) = + ^,(0) (1.5.12) In the first order equation the angle •& does not yet enter, b u t / a n d g do. It is

not necessary to determine the first order mode in order to determine the frequency to first order. Knowing the zero order a'°' and i " " one finds directly:

^(1) = + ^(y(i) + ^(D).

Thus, using zero-order modes we find the resonance frequency to first order: CO [ 2A K^ 1

— = ± 1 4 k^ 4 ( / ' " 4 g " > ) . (L5.13)

yH I MH MH J

This result will be used in paragraph 1.7 when we discuss the oblique resonance of a multilayer. In each layer the parameters are z independent except for the tilt angle •& (due to the transition regions (walls) which form between successive layers with diflFerent tilt angles). If this is taken into account a and b in the solutions 1.5.4 must be z-dependent. However, this § does not show up in

the coeflflcients of a and b in Eq. 1.5.9 in zero'th and first order. Thus the resonance frequency to first order is still given by Eq. 1.5.13 in this case.

In the analysis of the experimental data we will need the equations for the perpendicular resonance (//j^ to film plane) and parallel resonance (// in the plane). The filmnormal is a [111] and the inplane direction is a [112] (the field rotates in a (llO)-plane). From Eqs. 1.5.6 and 1.5.2 we find:

-'^perp and //, w Y

IK

M

+ 1

1.6. The perpendicular resonance of a multilayer

From this section on we will consider gradients of the uniaxial anisotropy within the film thickness. We assume that at the surfaces the boundary con-dition is dmjdz = 0, that is no pinning. In first instance we assume that the uniaxial anisotropy constant changes stepwise. This results in a stepwise change of the field for uniform resonance since //„„ = cojy — H^ (Eq. 1.3.5). In fig. 1.6.1 we have drawn a double layer with thickness L = Ly + L2 on a.

L-/////'/ ^ ' substrate , / ' / / / / / / ' ^ u n l un2

Fig. 1.6.1. A double layer on a nonmagnetic substrate. The uniaxial anisotropy and the field for uniform resonance (//„„) are different for the two layers.

nonmagnetic substrate. The layers have thicknesses L^ and Lj and are uniform with fields for uniform resonance//„„i and//„„2. In the case of perpendicular resonance the solutions in each layer are of the form of Eq. 1.3.3. For the layers sketched in fig. 1.6.1, m has the form:

/Ml = fll • c o s fcjZ,

/M2 = ^2 ' cos k2 {z — Ly — L2). (1.6.1) These have been chosen such that they already fit the boundary condition

dw/dz = 0 at the outer faces. The solutions are matched at the interface by the requirement that '*'^^)

Wi(L,) = W2(L,), dm I

(L,) dm-, {L,). (1.6.2)

dz dz

From these continuity conditions we can determine the ratio ^2/^1 ^i^d a relation between ky and ^2:

^2 cos kyLy

(1.6.3) fli cos /:2^2

and '^)

A:i tan k^L^ 4 ATJ tan ^2^2 = 0- (1.6.4) There is a second relation between k^ and ATJ which simply states that there

is just one applied field H so that k^ and ATJ are related to the difiFerencc of the fields for uniform resonance of the two layers:

Fig. I 6.2 Graphical solution of the resonance equation (Eq. 1.2.3) for the double layer of fig. 1.6 1 for Z-i = L2- The set of curves A„ represents all combinations of ki and A2 that satisfy the condition 1.6.4. The lines B, C, D and E represent the condition 16 5 for either

Hum = Huni (f'). Hu„i > Hu„2 (C, D) or //„„, < //„„2 (E). At the intersections of these lines

With the curves A„ we find ^ i and k2 of the normal modes of the double layer The horizontal scale is linear m the negative of the applied field.

2A

~{k^-k^) = H^„,-H,„2. (1.6.5) M

A convenient way to solve the equations 1.6.5 and 1.6.4 for k^ and k2 is by a graphical method. In fig. 1.6.2 we have plotted the relations 1.6.5 and 1.6.4 for the case Li = L^ versus the squares of the wave vectors in units of {n/LiY. The scale on the axes is thus linear in the applied field since from Eq. 1.3.4 we find

( ' i ) ' = ( « . . - « ) ( ^ y ( ^ ) . (1.6.6) The field increases from right to left and from top to bottom.

The set of curves A„ in fig. 1.6.2 represents the combinations of fci and /TJ that obey Eq. 1.6.4 and thus specifies all solutions that meet the boundary condi-tions. The second relation between k^ and /cj (Eq. 1.6.5) can be represented by a straight line in fig. 1.6.2, line B, for example, in the case //„„, = //„„2. The solutions of the spinwave equation are found as the intersections of line B with the set of curves A„. For this case we find ki = k2 = nnjlL^ = mij2L2 (« = 0, 1, 2, . . .) thus k = nnjL, as expected for a uniform film with no surface spin pinning (compare table 1.3.1). Curve C applies to a case //„„! — H^2 > 0. Note that every curve A„ is intersected once by lines B and C, so that the number of modes remains unchanged, they shift only. We will therefore always number the modes n = 0, 1,2, . . ., as for the uniform film. This mode number equals the number of zero crossings of m{z). In case C the modes n = 0 and 1 have a real ki, but an imaginary Aij. Hence they have a sinusoidal character in layer 1 and decay exponentially into layer 2. Mode n = 0 has ki close to nj2Li, as the first mode of layer 1 would have if there were pinning on one side. Apparently the continuity condition at the interface with layer 2 has the eflFect that would be produced if the spins in layer 1 were pinned at this interface.

The asymptotes in the second quadrant all have k^ = (2n 4 1)7T/2L, and describe waves in layer 1 only, fully pinned at the interface with layer 2. These are the solutions of Eq. 1.6.4 for (A:2L2)^ - » - - oo (compare table 1.3.1). Similarly, the modes in layer 2 pinned by layer 1 are found in the fourth quadrant. No solutions are found in the third quadrant, where none of the layers can support a spin wave. In the first quadrant the solutions are sinusoidal in both layers. From fig. 1.6.2 it can be seen that the modes with a high number M (n ^ 4) have all shifted by the same amount to a smaller value of ^2^-^2^/^^ on going from case B to case C. In paragraph 1.9 we will show that the high order modes are spaced as if the film is homogeneous with the weighted mean of the aniso-tropy of the two layers. Curve D is for a larger value of //„„, - //„„2. The number of modes localized in layer 1 is seen to increase. On reversing the sign

of //„„! — //„„2 (curve E) the first few modes become localized in layer 2 instead of layer 1. We will call modes which have an imaginary wavevector in one of the layers localized modes because the sinusoidal wave is localized in one layer only and the wave decays exponentially in the other layer. Modes with real wavevector throughout the film will be called volume or bulk modes, as usual in the literature.

The double layer with Ly = L2 discussed above gives an idea of the modes of a film with variations of //„„ throughout the film thickness. We will en-counter also films with a uniform bulk and a surface region where //„„ deviates. This can be approximated by a double layer with Lj <C L2. Figure 1.6.3 has

Fig. 1.6.3. Same as fig. 1.6.2 but for the case Z-i = L^jlQ. The dotted curve -4j corresponds to the case L] = Z.2/5.

been calculated for the case L, = Lj/lO. The set of curves. A„ have a slightly different shape from those of fig. 1.6.2. Line B has a slope (LJLj)^. If //„„i exceeds /^„„2 (curve C in fig. 1.6.3), mode « = 0 shifts strongly upwards to higher fields with respect to the volume modes. The z-dependence of m of this mode has been sketched in fig. 1.6.4 (top left). The mode has k^ real and k2 imaginary so that it is sinusoidal in layer 1 and decays exponentially in layer 2. Since layer 1 is only a thin surface layer the mode resembles the sur-face mode in the sursur-face anisotropy model which has an imaginary wavevector throughout the film thickness 23,24,27^ jj^g surface anisotropy model repre-sents the limiting case Li ^- 0 of the surface layer model. Modes 1, 2, etc., are volume modes, in this terminology, since these have a real wavevector through-out the film. With //„„i - //„„2 increasing from zero, mode n = 0 moves

H Q i Li ,0 ^1-2 3 U 5 6 - - H y 1 L1+L2 un

Fig. 1.6.4. Top part: m(z) of the first three normal modes (n = 0, 1, 2) of a double layer in the cases corresponding to the cases represented by lines C and E in fig. 1.6.3. At the surface the spins are taken free ((dm/dz = 0). The bottom part shows the variation of //„„(z) in the two cases and the field values of the normal modes. The parameters are L^ = L2IIO, MLi^ X (//„„i - //„„2)/(27t^A) = 40.2 (C), or -0.2 (E). H„„ is the mean field for uniforms resonance.

along the AQ curve and changes gradually from the uniform resonance mode {ki = /fj = 0) into a mode having ky = nj2Li, which is eflFectively pinned at the interface with layer 2. The wave has then been changed from a volume mode into a localized mode. The same can happen with higher-order modes. With increasing //„„i — //„„2, mode 1 shifts along curve A^ and can finally become the second one-side-pinned mode in layer 1 (/c, = InjlL^). The number of localized or surface modes therefore increases with increasing Hmi — Hu„2. Note that the first volume modes in fig. 1.6.4 have w «i 0 at the interface between the layers. This is consistent with the observation from fig. 1.6.3, that these modes have fcj ^ (2« 4 \)7tj2L2 as expected for one-side-pinned modes in layer 2 (compare fig. 1.3.1).

in fig. 1.6.3). As in cases C and D we will find localized modes, but these have k2 real. Since layer 2 is relatively thick this implies that the modes are sinusoidal in the largest part of the film and decay exponentially into the thin surface layer 1. The first modes lie close to /TJ = nnjL2 (« = i , f, . . .) showing that the modes are eflFectively pinned at the interface with layer 1 because of the rapid exponential decay into layer 1. This can be seen from the z-dependence of m of these modes, drawn in fig. 1.6.4. The pinning becomes less effective when |A^,| decreases, the decay becomes slower and A:, is finally real for mode 5 so that this is the first volume mode. In fig. 1.6.3 we have also drawn the curve A 5 for the case Li = LJS. This serves to illustrate that the curves A„ depend only slightly on the ratio of Lj and Lj so that the eflFect of a variation of this ratio on the mode spectrum can be studied qualitatively from figs. 1.6.2 and 1.6.3 by only changing the slope of the curves B-E, etc.

In fig. 1.6.5 we have plotted the field values of the modes versus the square of 500

_ 0 O X I I X -500 -1000 -1500 -2000 \ Fig. I.6.5b

\

k

A

-1,

i¥

i¥

Hf ^

Fig. 1.6.5. Field values of the normal modes of a double layer in the cases C, D (a) and E (b) of fig. 1.6.3, versus the modenumber. Parameters as in fig. 1.6.4 with L = 1.1 [xm and 2^/Af = 3xlO-*Oe/cm^ (in cases C and E //„„i - / / „ „ 2 = ± 592 Oe, in case D fluni — Hui\2 = 3552 Oe). Solid lines have been drawn through the field values for the modes of a uniform layer (case B, m fig. 1.6.3) according to Eq. 1.6.7. In all cases the higher order modes are little affected by the mhomogeneity while the low order modes are shifted The inset in (b) shows the field values of the low order modes in case E plotted versus modenumber using the numbering scheme n = V2. ' / j , '/2. as applicable to a uniform layer with pinned spins on one side (compare table 1.3.1.).

the modenumber « = 0, 1, 2, ... . The spacing of the modes is given with re-spect to the average field for uniform resonance Hy,„ = (//„„i • L-^ 4 //„„2 • L2)/ (Li 4 L2). The solid lines represent the relation

H^H,„-{2AIM){nnlLY, (1.6.7)

so that all field values would lie on those lines if the film were homogeneous. In all cases we note that the low order modes (which are strongly excited, see fig. 1.6.6) are shifted while the high order modes (with low intensity) are only little aflFected. In case C there is one surface mode {n = 0) at about 300 Oe above H^„ (fig. I.6.5a, see page 25).

In case D there are two surface modes (« = 0, 1) of which mode 0 is oflF scale (at //„„ 4 2488 Oe). In case D, it can be seen that a large number ( > 10) of resolved spin wave modes is needed in a experiment in order to obtain a measurement of A accurate to within 10%. When drawing a straight line through a plot as in fig. 1.6.5 its slope will be smaller than expected from Eq. 1.6.7. We have discussed that in case E the low order modes are eflFectively pinned at the interface with layer 1 (fig. 1.6.4). Thus the low order field values should he H = H^„ - {lAjM) {nn/LY with n = }, | , | , . . • and Z, = Lj (compare table 1.3.1 for pinning on one side). These values are represented by the solid line in the inset of fig. 1.6.5b and the first two modes fit to this line. On shifting line E further downward in fig. 1.6.3, representing an increase of ffuni ~ H^„y, the number of modes that fits to this line would increase.

We have also calculated the intensities of the modes. The intensity is pro-portional to the integrated power absorption of the absorption line: ( / P dH). The relative intensities can thus be calculated from Eq. 1.2.31. These have been plotted in fig. 1.6.6 for the case //„„2 = 3000 Oe, L2 = 1 [J.m, L, = Z-2/IO, and 2AIM = 3 X 10"^ Oe cm^. The values of //„„i corresponding to the cases C, D and E of fig. 1.6.3 have been indicated. If//„„i exceeds//„„2, mode« = 0 becomes a surface mode and loses intensity because of its increasing localization. If//„„i is lowered below //„„2 the intensity of the « = 0 mode decreases only slightly be-cause it becomes effectively pinned at the interface, while modes n= 1,2, etc., gain intensity, go through a maximum, and decrease asymptotically to the intensity of the higher-order modes of layer 2, pinned at the interface. These asymptotic intensities for the first three modes are given by the thin dashed lines in fig. 1.6.6. The maxima occur because the wave in the thin layer 1 also contributes to the intensity. This eflFect is relatively stronger for the high-order modes than for the low-high-order ones. In fact, for the higher modes, near the maximum, the spin wave is mainly driven by its tail in the thin layer 1. Note that for //„„, — //„„2 < —1000 Oe the intensity does not fall oflF mono-tonically with mode number. After an initial sharp decrease the intensity increases again, goes through a maximum for those modes which occur at a field close to //„„i and then falls oflF again. It is clear that the uniform mode {n = 0) in a uniform film (//„„i = //„„2) has nonzero intensity, whereas we have seen in table 1.3.1 that all other modes have vanishing intensity. Mode 0 has always finite intensity whereas for mode n there are n situations where its intensity vanishes. In addition to the case //„„, = //„„2 zero's occur whenever mode n has kiLJn = integer. From Eq. 1.6.4 it can be seen that in this case the continuity condition at the interface leads to /cj-^^/^ = integer. Unpinned modes (table 1.3.1) in each layer, each with / w,- dz = 0, incidentally match so that the intensity vanishes. The additional zero's for //„„i - //„„2 > 0 in fig. 1.6.6 all have ATIZ-I/TI = 1 and the modes then have a z-dependence as drawn in fig. 1.6.7 for mode n = 5.

j::2:i: ' : t ' i 1 LJ ^ \ J

-2000 E 0 C 2000

^

Hun1-Hun2 (Oe)

Fig. 1.6.6. Intensities of the normal modes (fig. 1.6.3.) of a double layer having Z-i = Z,2/10

and 2 / 4 / M = 3 x 1 0 " ' Oe cm^. L j = 1 (xm. If//„„i equals //„„2 only the uniform mode (n = 0) IS excited. If //„„i exceeds //„„2 mode n = 0 becomes a surface mode and looses intensity. The asymptotic intensities (thin dotted lines) are those of the normal modes of layer 2 pinned at the interface with layer 1 All intensities have been normalized with respect to the intensity of the uniform mode. Odd modes (dashed lines) and even modes (solid lines) have similar intensities (compare fig. 1.6.8).

In films with more than two layers m„{z) in layer n has the form

m„{z) = a„ cos k„z 4 b„ sin k„z. (1.6.8)

From the continuity of m and dmjdz at the interfaces one calculates the recursion formula

m

Fig. 1.6.7. »?(z)of moden = 5 at the dip of the intensity in fig. 1.6.6. For this particular external field *ii.i/;t = 1 and the instantaneous transverse moment vanishes.

bn+i

where

cosk„z„jcosk„+iZ„

k„{l 4 tan2/:„+,z„) {«n {k„+1 4 k„ tan k„z„ • tan k„+ iZ„) 4 4 b„ {k„+1 tan k„z„ - k„ tan /:„+ j z j } cosk„zJcosk„+iZ„

fc„+i (1 4 tan^ A:„+iZ„) {a„ {kn+1 tan k„+12„ - k„ tan k„z„) 4 4 b„ {k„+1 tan k„+ iZ„ • tan /:„z„ 4 k„)},

(1.6.9)

2« = Z-^"

and L; is the thickness of layer ;. If the film extends from ZQ = 0 to z^y = L the boundary conditions (no pinning) give the conditions:

br = 0 ,

aff sin ksL = bff cos k^L.

. (1.6.10) (1.6.11) Using Eq. 1.6.10 we can calculate a^ and t^ by applying Eq. 1.6.9 N — 1 times. Substitution of these into Eq. 1.6.11 yields an equation equivalent to Eq. 1.6.4 from which the normal mode frequencies are to be determined. For a triple layer one finds

ki tan^iLi 4A;2tanA:2/-2 + ^3tanA:3L3 = (A:i/:3//:2)tanA:,Li -taufciLj -tan/fsLa. (i.6.12) The mode diagrams become more complicated but the normal modes are still easily understood from the examples we have given for the double layer. The intensities however deserve some special attention. We will demonstrate some eflFects for a triple layer. In our example we consider a bulk layer (Z.2 = 1 l^^^ Hum = 3000 Oe) with one surface layer on each side (Lj = 0.02 [im, //„„i = //„„2 ± 500 Oe; L3 == 0.02 [im, //,„3 = variable).

0001 OOOCi -000001 2000 , 3000 , 4000 Oe 5000 S A S Hu„3 • 1500 2500 Xs 3500 Oe 4500 S Hun3 i i Hun

3500

3000

2500

®

Ls'

®

Fig. 1.6.8. Intensities of the normal modes of a film with two surface layers. The film param-eters are as drawn with L2 = 1 (im, Li = L3 = 0.02 y.m, lAjM = 3 x 10"* Oe cm^. Notice the different behaviour of the even and odd numbered modes. AS = antisymmetric film. S = symmetric film.

The intensities of the normal modes have been plotted in fig. 1.6.8. The even numbered modes have been drawn solid while the dashed curves are for the odd numbered modes. The two sets of modes behave distinctly diflFerent. Both sets have a monotonic decrease of the intensity with modenumber. The envelopes of the intensity versus n are equal only in the case of a double layer {Huni = ff^ni)- In the particular case of a symmetric layer (//„„, = //„„3) the uneven numbered modes have zero intensity. These modes are then exactly antisymmetric and thus do not couple to the uniform driving field.

Analogously the even numbered modes have zero intensity if the pinning is antisymmetric since then the deviation of the mode from the symmetric m{z) in the case //„„, = //„„2 = //„„3 is exactly antisymmetric. Similar results are found for the surface anisotropy model (see 1.8).

1.7. Oblique resonance of a multilayer

If the external field is not applied along a symmetry direction such that the torque is isotropic the mathematics becomes rather complicated. In paragraph 1.5 we have seen that for a single isolated layer the precession is in general elliptical, the transverse magnetization components Wj and m„ are unequal and both have to be solved ^^•^'). In order to find the normal modes of a multilayer we have to solve the Eqs. 1.5.1 in each layer and require conti-nuity of ffi{,w„ and their derivatives at the interfaces. First however, we will have to determine the magnetization direction which will vary with z. So, instead of Eq. 1.5.3 we have to solve the tilt angle •& from the full equation derived in appendix A (Eq. A16):

-2A{d^&jdz^) 4 MH sin &- K^ sin 2{0„ - §)- K^ (sin 2{xpu - §)) x

X ( 1 - lsin^(v;H- ^)) = 0. (1.7.1) This is a rather awkward equation to solve. The value of & is of the order of

KjMH, where K is K^ or K^. In order to find the normal mode frequencies to first order in the parameter KjMH we need not know the angle &, as shown in the perturbation calculation in 1.5. It also suffices to know the functions m^ and W2„ to zero'th order. This is the circular precession approximation. We can simply calculate the first order fields for uniform resonance of every layer from Eqs. L5.2andL5.13:

CO Ku Ki . , ,

//„„ = — 4 — (1 - 3 COS^^^H) ( 2 - 1 0 sin^ VH + T" S'"* VH). (1-7.2) y M M

The problem has now been reduced to solving the resonance equations for a multilayer with varying //„„. Thus the modes are found using the theory of paragraph 1.6.

In a more accurate calculation it is necessary to calculate both the ellipticity of the precession and the non-alignment of H and M since both effects give rise to contributions of the order of {KjMHy. Only in the case when the field is applied along a principal direction of the sum of the uniaxial and the cubic anisotropy (implying zero derative to 0 of the anisotropy energy) is the tilt angle 1? zero so that only the ellipticity matters. In [111] films with induced uniaxial anisotropy the film normal is a principal direction of both the cubic and the uniaxial anisotropy. According to Eq. 1.2.5 the energy is symmetric around [111 ] for small angular deviations of the magnetization so that the precession is even circular. Since we usually rotate the applied field in a (110) plane the

in-plane direction is a [112] and then the tilt angle is nonzero and the pre-cession is elliptical.

First we have to solve the static equilibrium problem (Eq. 1.7.1). In order to facilitate the analytical solution of 1.7.1 we linearize in the angle &, which is of the order KjMH:

2A d^&

4 «?-)?''' = 0, MH dz^

where

&'" = —— sin2 0H 4 — ^ (sin2 ^H) (1 - f sin^ y}„) MH MH

??'" is the equilibrium & for an isolated layer for which d^i?/dz^ = 0. Eq. 1.7.3 is easily solved. The general solution has the form

& = Ci exp{-zjL„) 4 C2 exp (z/LJ 4 C3, (1.7.5) where c, are constants and L„ is a length:

L„ = y2AjMH. (1.7.6) It is illustrative to solve Eq. 1.7.3 at first for an infinite medium with a stepwise

change of K^dX z = 0, as sketched in fig. 1.7.1. The boundary conditions are hm «? = 1^2"'. lim ^ = ^1'" and d and d^/dz are continuous at z = 0.

Z - * -t- 00 Z~* — 00

Fig. 1.7.1. Varidtion of the magnetization direction & at the interface between two layers with equilibrium magnetization direction ^ i ' " . The width of the transition region (wall) is about

2Ly, where L„ is given by Eq. 1.7.6.

(1.7.3)

The resulting function & is:

& = ^^e, + ^ (^^e, _ ^^.o) exp (^/^J (^ < 0)

= ^ 2 ' " - i ('^2''' - ^1'") exp ( - Z / 0 (^ > 0) ^ • • '' and has been sketched in fig. 1.7.1. The thickness of the transition region (or wall) at z = 0 is of the order of 2L„,. Thus the thickness of this wall is not the same as that of a regular Blochwall which depends on the anisotropy and the exchange interaction ^'). The present wall width is the result of a competition between the exchange interaction which favors a wide wall and the magneto-static energy MH which favors 1} = •&"*. Thus it is less wide than a Blochwall and strongly field dependent.

Clearly, if a multilayer has layers thick compared to L^,, walls as given by 1.7.7 will form at every interface while in the bulk of each layer •& = •&"*. The order of magnitude of L„ for YIG is only 10~* cm. For a bubble garnet, which has a small M, it is slightly larger (up to 5 x 10~* cm). If the layer thicknesses are of the order of, or smaller than, Ly„ the walls are not well separated. For a triple layer the shape of ^ is:

7?j = ,?j=« -f- a cosh (z/L^), 0 < z <Li

^^ = ^ / i + acosh(z/LJ 4 ( i ^ i " - ^2"") cosh ((z - Li)jL„), Li < z < Li + L2

^3 = ^3«o -I- c cosh ( ( z - L)jL„), Li + L2 < z < L, (1.7.8) with sinh(L3/LJ sinh ((L2 ^-^3)/^ J a = (i^a'o - ^2'") 4 {'&2"' - ^1'") sinh {LjL^ sinh {LjL,J) sinh {{L, 4 L2)jL„) sinh {LJL^) sinh {LjL^) sinh {LjL„) and L = Z-i 4 L2 4 L3 is the sum of the layer thicknesses. The boundary condition at the surfaces is di?/dz = 0. This condition is similar to the con-tinuity conditions 1.6.2.

For a multilayer with i?(z) according to Eq. 1.7.8 the equations of motion contain an additional term in d&jdz. The full equations are given in Eq. A17 in appendix A. Moreover, the t e r m s / a n d g are now z-dependent even within every layer. Therefore the solutions have been obtained by numerical integra-tion only, which will be discussed in IV. 3.

1.8. The relation between the surface layer model and the surface anisotropy model

In 1.4 we have discussed the concept of a surface anisotropy constant and in 1.6 we have mentioned that some features of the normal modes are similar

in a surface layer model and a surface anisotropy model. In this paragraph we will demonstrate that for small surface anisotropy the models are identical, in which case the surface anisotropy constant has a simple physical meaning.

The equations governing the normal modes in both models are Eqs. 1.4.2 and 1.6.4:

ktankL = KJA, (1.8.1) ki tanfciLi = —/cj tan/CiZ-a.

We take layer 2 as the thin surface layer (fig. 1.6.1). In the surface layer model free spins are assumed at z = 0 and z = L {dmjdz = 0) and it is the con-tinuity condition at z = Lj which leads to the relation between k^ and /cj. In the surface anisotropy model the boundary condition is specified as the ratio {dmjdz)jm at the surface at z = L (see 1.4.1). If the surface layer is small, so that kiLi < 1 {i = 1, 2), then we can write the second equation of 1.8.1 to first order in fc.Lj (/ = 1, 2) as:

fci tan k^L = {k^^ - k2^) L2, (1.8.2) where L = Li 4 / . j - ^^ this case the surface layer model and surface

aniso-tropy model are the same if

KJA={k^-k2')L2. (1.8.3) Substitution of k^ from Eq. 1.3.4 yields

K, = -WL2 {Hu.2 - H^ni)- (1.8.4) If we consider uniaxial anisotropy only we can express 1.8.4 for perpendicular

resonance as:

K, = L2{K,2 - ^ui). (1.8.5) This equation clearly shows the physical meaning of the surface anisotropy

constant K^.

In general the surface layer model yields a frequency and field dependent boundary condition according to Eq. 1.6.4. The ratio {dmldz)jm depends on the external field (at constant frequency) and thus on the modenumber. This was apparent in figs. 1.6.4 and 1.6.5 where the various numbering schemes mentioned in fig. 1.3.1 and table 1.3.1 were applicable in successive field ranges (for example case E in fig. 1.6.5). Only in limiting cases is the approximation of a surface anisotropy constant appropriate. One of these is the case of small surface anisotropy discussed above when for all experimental fields kiL2 < 1 (/ = 1, 2) so that the surface anisotropy is small {KJAk < 1).

In the work on garnets a surface anisotropy model based on a quantum-mechanical calculation by Puszkarski has been successfully used by Yu et al.''^) to interpret spectra of YIG films. The final results of those calculations are