Parametric amplification of microwave magnetoelastic and magnetostatic waves in yttrium iron garnet

il H

BIBLIOTHEEK TU Delft P 1968 7251

C 662104 3110

MICROWAVE MAGNETOELASTIC AND

MAGNETOSTATIC WAVES IN

YTTRIUM IRON GARNET

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR EN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL D E L F T OP GEZAG VAN DE RECTOR MAGNIFICUS DR. IR. C. J. D. M. VERHAGEN, HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUURKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VER-D E VER-D I G E N OP WOENSVER-DAG 23 APRIL 1969

TE 14 UUR

DOOR

HERMAN van de VAART

I. INTRODUCTION 1 A. Ferromagnetic Resonance 1 B. Spin Waves 1 C. Magnetoelastic Waves 3 D. Magnetostatic Waves 4 E. Relaxation 5 F. Nonlinear Effects 6

G. Parametric Interaction with Magnetoelastic Waves 8

H. Scope of this Thesis 9 II. LINEAR CHARACTERISTICS 12

A. Free Energy 12 B. Spin Waves 15 C. Magnetoelastic Waves 18

p. Magnetostatic Waves 41 III. NONLINEAR CHARACTERISTICS 55

A. Spin Wave Instabilities 55 B. Magnetoelastic Wave Instabilities 69

C. Magnetoelastic Wave Instabilities, Including

Magnetoelastic Anisotropy 94 IV. EXPERIMENTS 111 A. Experimental Arrangement 111 B. Linear Experiments 117 C. Nonlinear Experiments 123 V. DISCUSSION 137 A. Magnetoelastic Wave Amplification 137

B. Geometrical Ray Theory 141 C. Threshold for Upper Branch Magnetoelastic Rays 147

D. Magnetostatic Wave Amplification 153

A. Yttrium Iron Garnet 161

B. Relation Between the Saturation Magnetostriction

Constants and the Magnetoelastic Constants 166

C. Derivation of the Instability Threshold 170

D. Ray Equations 172

ACKNOWLEDGMENTS 178

REFERENCES 179

ABSTRACT 185

SAMENVATTING 187

A. Ferromagnetic Resonance

Landau and Lifshitz first described the basic phenomena of ferromagnetic resonance in a paper which appeared in 1935. The first experimental observation of ferromagnetic resonance at micro-wave frequencies was made more than a decade later, however, and

2

was reported by Griffiths in 1946. These data were interpreted by Kittel in 1947, by using classical equations of motion for the magnetic dipoles and taking into account the surface demagnetizing factors and magnetic anisotropy. ' This understanding of ferro-magnetic resonance was developed on the basis of a collection of coupled spins, precessing in phase throughout the sample. However, higher order modes of oscillation, in which the precession has a different phase in different zones of the sample, were observed by White and Solt in 1956, by using an inhomogeneous rf exciting field. The explanation was given by Walker, vrfio assumed that the characteristic distance for phase reversal of these modes was suf-ficiently long so that the exchange coupling was very small com-pared to the wavelength for electromagnetic propagation in the medium. In this case, the modes of oscillation were found to be due to magnetostatic forces arising from the externally applied field and the dipolar fields of the sample magnetization only; hence, the name magnetostatic modes. Since then, these modes have been observed in samples of various shapes.

B. Spin Waves

The concept of spin waves, in which there is a progres-sive variation of phase or amplitude of precession from dipole to

o

dipole, was first introduced by Bloch in 1930. He derived the 3/2

T ' law describing the variation of the magnetization with tem-perature T. The mechanism for sustaining the wave motion in a ferromagnetic crystal is found in the internal exchange field, which tends to align the dipoles. An rf magnetic driving field,

applied at a particular location In a ferromagnetic specimen, will cause the spins to precess with a larger precession angle than their neighbors. The exchange force between the dipoles will cause a neighboring spin to Increase its precession angle, thus providing a mechanism for the propagation of this precessional disturbance.

9

Herring and Kittel showed that a spin wave could be described by a modified equation of motion for the uniform precession, the modification being a restoring term containing the exchange field,

The generation of spin waves in small ferrite spheres was the basic concept in explaining the high power resonance experiments by Bloerabergen, Damon and Wang. ~ They observed that at a high rf driving power the main resonance declined, and that an additional absorption was observed at a lower applied magnetic field. It was pointed out by Suhl that these nonlinear effects could be attributed to the parametric coupling between the uniform precession and the spin wave modes, and that beyond the instability threshold the spin waves are excited to a level much

13 14 higher than that corresponding to the thermal equilibrium. ' These high power effects will be treated in detail in a later sec-tion of this thesis, since they form the basis of the parsunetric amplification of magnetoelastic waves, the central topic of this investigation.

The direct generation and detection of spin waves in 15

ferrites made a great stride forward when Schlomann in 1961 proposed to employ the nonuniformity of the internal dc magnetic

field in samples of nonellipsoidal shape, specifically long rods with plane end faces. He showed that if the magnetic field is

larger at the surface than Inside the material (which is the case if the rod is magnetized perpendicular to its axis) the wavelength of the spin waves will become larger as they approach the surface.

If the inhomogeneity is large enough to change the character of the wave function from periodic to exponential near the surface, a net dipole moment exists, where the spin waves can be excited by means of a uniform rf field.

A similar condition exists in a normally magnetized disk, and it was with such a sample of yttrium iron garnet that Eshbach in 1962 performed the first experiments showing direct excitation and detection of short wavelength spin waves.

C. Magnetoelastic Waves

At this point in the discussion of spin waves, it is necessary to introduce the concept of magnetoelastic coupling between the magnetic and the elastic modes in ferrimagnetic crys-tals, since practically all of the experiments relating to the propagation of spin waves involve a combination of these modes with elastic waves, the so-called magnetoelastic waves. In fact,

in the above-mentioned experiments by Eshbach, it was possible to tune the propagating mode smoothly through the interaction region from pure spin waves to pure elastic waves, by varying the applied dc magnetic field.

The magnetoelastic coupling between the strain in a ferromagnetic crystal and the direction of magnetization, resulting from the anisotropy energy, produces an interaction between the spin waves and elastic vibrations. It was shown by Kittel that this interaction is particularly large when the wavelengths and

17

frequencies of the two modes are equal. If this condition is nearly satisfied, the normal modes of the system are no longer purely magnetic or purely elastic but a mixture of the two kinds of excitation. Schlomann and Joseph showed that for spin waves traveling through a region of nonuniform magnetic field the con-version into elastic waves, aid vice versa, depends on the gradient of the internal magnetic field at the crossover point, the point within the sample at which the two wavelengths are the same, and on the strength of the magnetoelastic interaction. They con-cluded that for small samples of yttrium iron garnet the field gradients encountered are usually such that it is justified to assume that substantially all the spin wave energy is converted

19 into elastic waves.

Magnetoelastic wave propagation has been studied exten-sively In the past few years, especially in single crystal rods of yttrium iron garnet. Strauss first observed magnetoelastic wave

20 21

propagation in an axially magnetized rod. ' In his experiments, the waves propagated along the field, as opposed to Eshbach's experiments, where the propagation occurred at right angles to the field. The unique feature of magnetoelastic wave propagation is that the group velocity of these waves can be varied from zero up to the velocity of pure elastic waves by merely changing the mag-netic field strength or the frequency. This property makes the application of these effects very attractive for dispersive delay lines in the microwave region, as required in pulse compression radar systems.

A complete analysis of the propagation characteristics of magnetoelastic waves forms an integral part of this thesis, and is given in Chap. II,

D. Magnetostatic Waves

The wavelengths of the magnetoelastic waves, studied by Eshbach and Strauss, are quite small compared to the sample dimensions. Therefore, the wave propagation is dominated by exchange forces, and dipolar forces can be neglected. However, when the wavelengths of the spin waves become comparable to a

sample dimension, dipolar forces are dominant, and exchange effects can be neglected. The propagation of these long wavelength spin waves, termed magnetostatic, was first suggested by Fletcher and

22

Kittel, and subsequently observed by Damon and Van de Vaart in 23

a normally magnetized disk. Olsen and Yaeger observed similar 24

propagation in axially magnetized YIG rods, and Damon and Van de 25

Vaart in transversely magnetized rods.

The propagation characteristics in all these cases can be easily derived by combining Maxwell's equations with the equa-tion of moequa-tion for the magnetizaequa-tion, including the relevant boundary conditions. ' The dispersive behavior of these waves

is closely related to the properties of magnetoelastic waves and has been part of this investigation. A more detailed description of the dispersion of magnetostatic waves is given in Sec. II.D. E. Relaxation

In ferromagnetic resonance experiments, there are always losses due to the interaction of the spins with the surrounding medium. The losses produce damping of the precessional motion, and the resonance linewidth is an indication of how fast the pre-cessing spins transfer their energy out of the uniform mode. A fast relaxation results in rapid damping of the precessional motion

OO and a broad resonance line. Yaeger, Gait, Merritt and Wood in 1950 observed a linewidth of 70 Oe in single crystal nickel fer-rite, resulting in a relaxation time which was at least four orders of magnitude shorter than could be accounted for theoretically at

29 that time.

Since the direct relaxation from the uniform precession to the lattice obviously was not the dominant mechanism, another energy transfer path was required. The logical choice was the reservoir of the spin wave modes. Kittel and Abrahams considered the dipolar coupling between spin wave modes and, although the basic concept of the spin waves as an intermediate step in the relaxation process later on was proved to be correct, the discrep-ancy between theory and experiment could only be reduced by two

30 orders of magnitude, but could not be removed completely, Anderson and Suhl in 1955 then found that if one considered a finite sample and included the appropriate demagnetizing factors, rather than an infinite sample, spin waves with the same, or lower, frequency as the uniform precession were permitted, thus providing an energy-conserving process of scattering of the uniform

preces-31

sion into the spin wave modes. The detailed coupling mechanism was identified one year later by Clogston, Suhl, Walker and Anderson as being a result of irregxilaiities in the magnetic

lat-32

linewidth increased when the frequency of the uniform precession 33 was equal to the frequency of a large number of spin waves,

Le Craw, Spencer and Porter showed that in yttrium iron garnet (YIG) the linewidth was proportional to the size of the irregularities created on the surface of the sample,

F, Nonlinear Effects

The successful explanation of the coupling mechanism between the uniform precession and the spin waves was a result of

31

a study by Anderson and Suhl of some resonance experiments on nickel ferrite at high microwave power, performed in 1952 by Bloembergen , Damon and Wang. ' ' These experiments were designed to measure the various relaxation times and to distin-guish between two mathematical models for damping, the so-called

1 ^6 S7 Landau-Lifshitz form and the Bloch-Bloembergen form, '

However, several novel effects were encountered i^ich could not be explained by the conventional theory of ferromagnetic resonance, It was found that the ordinary ferromagnetic resonance saturated at a level far below that predicted by the linearized equation of motion of the magnetization. Also, a secondary, rather broad, absorption was observed at a field strength below resonance. The

31 first effect was explained in 1955 by Anderson and Suhl; the

38 explanation for the second effect was given by Suhl in 1956. It was shown that both effects arose from a spontaneous transfer of motion from the uniform precession to certain spin waves. It was realized that the spin waves are coupled to the uniform pre-cession in second and higher orders through the dipolar and exchange fields that accompany the spin waves. ' At ordinary signal levels, this effect imposes a small additional loss on the uniform mode. Above a certain threshold level, however, the power transfer to the spin waves surpasses the normal losses. The spin waves then grow to large anplltude, absorbing energy from the uniform precession and leading to the effects observed. It will be shown in Chap, III that nonlinear terms of first order in

uniform precession amplitude lead to excitation of spin waves at half the driving frequency (subsidiary absorption or first order Suhl instability) and that nonlinear terms of second order in uniform precession amplitude lead to excitation of spin waves at the driving frequency (decline of the main resonance or second order Suhl instability).

The first nonlinear process involving acoustic modes was 39

reported by Spencer and Le Craw, Low frequency (a few MHz) acoustic oscillations in a polished q)here of YIG were observed above a certain threshold level of microwave power when the mag-netic field was adjusted such that the ferrimagmag-netic resonance occurred at the difference frequency between the applied microwave signal and the frequency of the elastic oscillations. In this case, the uniform precession parametrically excited the magnetostric-tively coupled acoustic and magnetostatic modes. This was called the magnetoacoustic resonance (MAR).

For all these instabilities, the dc and rf fields are at right angles to each other. They are therefore called transverse pumped instabilities to distinguish these from another nonlinear effect, suggested by Morgenthaler and Schlomann, known as the parallel pumped instability. The latter effect is produced by applying an rf magnetic field of sufficiently high amplitude parallel to the dc magnetic field. A coupling occurs between this

longitudinal rf field and the spin waves because certain spin waves precess on an elliptical rather than a circular cone, due to their dipolar interaction, and pairs of these modes can be excited by a sufficiently strong driving field. The mechanism is similar to that of the subsidiary absorption, and the unstable spin waves are at half the driving frequency. The parallel pump instability has been extremely useful for investigating relaxation processes in ferrimagnetic materials because the unstable spin wave is accurately defined by the strength of the applied magnetic field.

As can be expected, the instability thresholds are

strongly modified in the presence of magnetoelastic coupling. This 42

was first observed in 1960 by Turner in parallel pump experiments on YIG spheres. He noted two sharp increases in the threshold as a function of magnetic field, which he was able to relate to a direct interaction of spin waves with longitudinal and transverse elastic waves of the same frequency and wavevector. From these data, the magnetoelastic coupling constant could be determined and a direct measure of the exchange constant was obtained.

G. Parametric Interaction with Traveling Magnetoelastic Waves The nonlinear effects with either transverse or parallel pumping were observed on the driven mode only (except for the MAR); the existence of excited spin waves was inferred from the theo-retical model. It seemed logical to expect that, since a method was known to generate and detect spin waves in YIG samples, in principle it should be possible to detect the spin waves in a

non-linear process directly, and even to use these interactions for parametric anqjlification. The successful combination of nonlinear

instabilities and the linear generation and detection of spin waves and magnetoelastic waves, resulting in the observation of both these effects, form the basis of this thesis.

43 The first such observation was made by Matthews in a nearly axially magnetized rod of yttrium iron garnet and involved pure elastic waves of 500 to 800 MHz, generated by means of a quartz transducer, A pump at twice the signal frequency was applied at the other end of the rod, resulting in a growth of the elastic waves. The effects could not be explained in detail, al-though it was al-thought to be due to the coupling between spin waves and elastic waves.

The parametric excitation and amplification of propa-gating magnetoelastic and magnetostatic waves were first observed by Damon and Van de Vaart at X-band frequencies at a temperature

44 45

46

reported. These effects were observed in YIG rods, magnetized either axially or transversely, and in normally magnetized YIG disks, at frequencies from 1 to 10 GHz, and at temperatures from 1.5 K to room temperature.

The experiments use a pulse-echo technique. The sample is placed in a microwave cavity, located in a magnetic field. Linear wave packets are excited by a pulsed microwave oscillator, and the echo pattern is displayed on an oscilloscope. For para-metric amplification, a second microwave source at twice the signal frequency is used.

H. Scope of this Thesis

In this thesis, we are concerned with the combination of the linear and nonlinear processes involving propagating microwave magnetoelastic and magnetostatic waves in single crystal YIG rods and disks, resulting in the parametric generation and amplification of these waves. The experimental results will be described, and an analysis of the various instability processes will be given.

The linear characteristics of the magnetoelastic and magnetostatic waves are extensively reviewed in Chap. II. Starting from the expression for the free energy, the equations of motion for spin waves, elastic waves and magnetoelastic waves are derived. From these, the dispersion relation is obtained, giving the rela-tionship between the angular frequency and the wavenumber. The dispersion relation is then combined with the internal field in a rod and disk to give the propagation time as a function of mag-netic field strength. For spin waves with wavelengths comparable to a sample dimension, the dispersion relation is modified by magnetostatic boundary conditions. In this case, a new dispersion relation is obtained. Combining this dispersion relation with the Internal field gives the magnetostatic wave delay as a function of field,

The nonlinear characteristics are described in Chap, III, The first and second order Instability thresholds for pure spin

waves are derived first, resulting in the familiar Suhl

instabili-ties: the spin waves most likely to go unstable are those

propa-gating at 45 degrees with the internal field for the first order <

instability, while spin waves propagating parallel to the internal

field are most likely to go unstable in the second order instability.

The magnetoelastic wave instability is treated next,

first for the case urtiere the internal field is aligned with one of

the cubic axes of YIG. It is found that the lowest threshold is

obtained for pure spin waves and that the threshold increases when

magnetoelastic crossover is approached. These results, however,

do not explain the observed behavior. A satisfactory explanation

of the experimental results can be arrived at by relaxing the

restriction that the applied field be aligned with the crystal axes

of YIG and by taking the magnetoelastic anisotropy into account. t

A complete description of the experimental arrangement

and the linear and nonlinear experiments is given in Chap. IV.

Pulse-echo techniques were used to observe the magnetoelastic and

magnetostatic echoes. The experiments were performed at frequencies

ranging from 1 GHz at room tenperature to 10 GHz at liquid helium

tenperatures. In the case of magnetoelastic waves, echoes could

be amplified to a level which in all cases exceeded the input

signal. The time delay of the echo is a function of applied

mag-netic field strength, and amplification is observed only if the pump

is applied at a particular time. The amplification in a rod is

largest when the field is directed ~ 15 degrees off the rod axis,

and in this case the pump must be applied approximately when the

signal is in the magnetoelastic crossover region. For

magneto-static waves in an axially magnetized rod, the experiments resulted

in the so-called "pulse recall with variable time delay"; a

micro-wave signal. Injected into the YIG rod, will not be detected unless

it is recalled by the application of the pump signal,

In order to obtain agreement between theory and

experi-ments, it is necessary to show that the angle between the

wave-vector and the local magnetic field is very small for the

propagating magnetoelastic waves in an axially magnetized rod. That this is indeed the case can be shown by using the recently developed ray theory analysis for magnetoelastic waves. The results of such an analysis are described in Chap. V, together with a comparison between the threshold data and the experiments. A brief description of the nonlinear effects involving magnetostatic waves concludes Chap. V.

The results are summarized in Chap. VI.

The new developments to be found in this thesis are: 1. The first observation of parametric generation and amplification of microwave magnetoelastic and magnetostatic waves in YIG.

2. The derivation of the theoretical instability thresholds for several processes involving transversely pumped, magnetoelastic shear waves, including magnetoelastic anisotropy.

3. The combination of the threshold data with the geometrical ray theory to obtain a minimum in the degenerate mag-netoelastic wave threshold near crossover, as required to explain the experimental results.

II. LINEAR CHARACTERISTICS

A. Free Energy

A convenient s t a r t i n g point for the derivation of the

equations of motion i s the expression for the free energy of a

47

ferrimagnetic cubic c r y s t a l :

E= -M-Hj ( 2 . 1 . a )

+71 a® M-V* M - 11 M-M ( 2 . 1 . b )

+K^ (.m^^+m\%mV) + K^(m V m ^ ) ( 2 . 1 . c)

+ ^ ^ V x x • ^ ^ ' V V z z ^ + ^b^^m^m^e^y+mym^e^^+m^m^e^^) ( 2 . 1 . d )

Term (2,1.a) is the interaction of the magnetization S with the internal magnetic field Hj, which differs from the applied magnetic field H^j, by the demagnetizing field resulting from the magnetic poles on the surface of the san^jle. This is sometimes referred to as the Zeeman energy. Term (2.1.b) represents the exchange energy, where Tl is a molecular field coefficient and a is the distance between neighboring spins. The term TIM, the Weiss field term, is very large and is responsible for the existence of ferromagnetism. It can nevertheless be omitted from further con-sideration since it provides no restoring torque for small dis-placements of U from equilibrium and thus does not affect the

—4

dynamic problem of the motion of M.

The term ( 2 . 1 . c ) i s the magnetocrystalline energy, which

acts in such a way that the magnetization tends to be directed

along certain preferred crystallographic axes, vrfiich are called the

d i r e c t i o n s of easy magnetization. Energy i s required to turn the

magnetization from an "easy" axis to a "hard" a x i s , which i s the

magnetocrystalline or anisotropy energy. It can be described by the first two or three terms of an infinite power series in the direc-tion cosines of the magnetizadirec-tion with respect to the crystal axes, and it must have the symmetry of the crystal lattice. Thus for a cubic crystal:

E „ = K (m m+m m +m m ) + K (m, m m ) + (2 2)

mc 1 x y x z y z s x y z ' ""••'••'

where K and K are the first and second order anisotropy constants and m , m and m are the direction cosines of the magnetization

X y z

with respect to the crystal axes:

- M

»s

•" ~

"" ^ "x + J "y •*" """z • (2.3)

with

•"y^M,

«z

•"2 = 5 ; •

For YIG, the anisotropy energy generally can be neglected, since K /Mg Ri 40 Oe (K is negligible), which is small compared to the internal magnetic field (~3500 Oe) for resonance at X-band fre-quencies.

The term (2.1.d) describes the interaction of the mag-netic and elastic systems. It is observed that, if a single crystal of a ferro- or ferrimagnetic crystal is magnetized, the crystal becomes strained. This is known as magnetostriction. It occurs because the anisotropy energy depends on the strain in such a way that the crystal will deform spontaneously, if to do so will lower the anisotropy energy. To express the dependence of the anisotropy energy on the strain, the energy is expanded in a Taylor series:

BE

^mc = C - ^ 1^^ a ë ^ «ij -^ (2.4)

«diere E _ refers to the unstrained state of the crystal. The _mc second term in Eq. (2.4) is referred to as the magnetostriction or magnetoelastic energy. It is a function of both the direction cosines and the components of the strain tensor e.., and thus represents the coupling between the magnetic and elastic systems:

dE dE dE rSE dE dE n. rac^ ^ mc. ^ mc. ^„' - . / . " - . " t = -5 e + T — — e + T e +2 me oe xx oe yy de zz XX yy •'•' zz

or

rBE SE 9E -1

J -aCe ^ « ! £ e 4^-S!^ (2.5)

E_^ = b, (m.,e +m*e +m^e ) + 2b (m m e +m m e +m m e ) , (2,6) me 1 X XX y yy z zz a x y xy y z yz x z xz • ^ • where

9E OE

^ ™r =

BT:

-"<^

\

"-i-j

=

a i j

.

(2.T)

b and b are known as the magnetoelastic coupling ^ ^ 48

constants. The values for YIG are:

e a b = 3,48 X 10 ergs/cm ; b = 6.96 X 10 ergs/cm

They are directly related to the saturation magnetostriction

constants \^^, \^^ by

where c,, , c, _, c^^ are the elastic stiffness constants (see

Appendix B ) .

Finally, term (e) in Eq. (2.1) describes the elastic

energy density, which is the energy stored in the lattice as a

result of the elastic stiffness.

The equation of motion can now easily be determined by

equating the time rate of change of the angular momentum -rr to the

torque m x V _ E . The ratio of magnetic moment to angular momentum

m

IS the gyromagnetic ratio Y chosen positive for electrons. Thus:

i | f = m x V E . (2.9)

B. Spin Waves

Applying Eq. (2.9) to the first two terms of Eq. (2,1),

the equation of motion for exchange-dominated spin waves is found

f =-Y(Mxifi)

-yo^^^ ,

(2.10)

where D = exchange constant. For YIG, D = 5,17 x lO"* Oe/cm^,

In what follows, it will be assumed that the biasing

magnetic field is in the z-direction, parallel to a (100) direction

of the cubic unit cell of YIG (Fig, 1). It is also assumed that Hj

is parallel to z. The transverse components of the magnetization

are M^ and M , the saturation magnetization Mg ?» M^, and

H + M^ « M?. The transverse components of the magnetic field

X y a

are H and H • these consist of the driving field and the dipolar

field.

Equation (2.10) can be written in component form:

ST

= - Y [ H i - D v ^ M y + Y M 3 H y

1

Z

u

"z

r^-II^

L s i ^ r f

|(

Fig. 1. Spin wave propagation d i r e c t i o n and precession of the

magnetization about Hj.

We now consider a plane wave solution to Eq. (2.11) so t h a t M^ and

M are proportional to e^ "^ ~ ''^ . The dipolar f i e l d i s determined

from Maxwell's equations:

V X iïjj = 0

V • ff. = - 4nV-ii} ; hence

d Jc2

(2.12)

Clearly a l l spin waves propagating under the same angle with Hj

have the same frequency, so t h a t we can r e s t r i c t the discussion to

the x-z plane without loss in g e n e r a l i t y , (Fig. 1). Thus

and

i(d«x + Y(Hi+Dk^)My = 0

(Y(Hi+Dk=) + Oj, sin^ej M^-iuJUy = 0 ,

s

Y(Hj+Dk^) [Y(Hj+Dk^) + % sin^e]

(2.13)

(2.14)

where f^ =

and t h e notation cu, i s adopted when r e f e r r i n g to

the precession frequency of a spin wave. For spin waves

propa-gating along the f i e l d 9 = 0 , the d i p o l a r f i e l d vanishes and

u)^(O) = Y(Hi+Dk^)

(2.15)

For spin waves propagating at right angles to the field, 6 = n/2

and we find:

(Ü,J(TT/2)

=

QHj+Dk^)(Bj+Dk^)] , (2,16)

where B. = H.+4nM . A plot of

w./y

is shown in Fig. 2. These

relations are only valid for wavelengths much smaller than the

sample size. When the wavelength becomes comparable to a sample

dimension, magnetostatic boundary conditions have to be satisfied,

and the extrapolation of the spin wave branches to k=0 is invalid.

t

Y

/•HjBi

C, Magnetoelastic Waves

Applying Eq, (2,9) to the magnetoelastic energy term in

Eq. ( 2 . 1 ) , we obtain:

T l2m„^ m„e +b ( m e +m„e^^A- 2m /b m e +b (m„e +m e )'\|

I y V1 z zz s y yz X xz y z \ i y yy s x xy z yz y j

[

2m {b m e +b (m e +m e )> -2m„«) m„e ^+b„(m e +m e )^l

_{zv 1 X XX s y xy z xz-* x V i z zz s y yz x x z y j}

ic |2m^/b, m e +b (m e +m e )"\ - 2m /"b m e +b (m e +m e )'\|.

I x V i y yy s x xy z y z y yV i x xx s y xy z xz y j

(2.17)

As in Sec. II,B, the discussion w i l l be restricted to small

pre-cession angles; m and m « 1 and m t^ 1. Equation (2.17) Ihen

reduces t o :

or

^ f =-ïp^''sV3^^C2(b,e^^)] (2.18)

E

a^ + l f j

\

urtiere the strain component e . . i s replaced by an e l a s t i c

displace-ment vector R. which i s related to e . . by:

The vector R represents the displacement of a p a r t i c l e from i t s

p o s i t i o n in the unstrained solid to i t s new position in the

d i s t o r t e d s o l i d . The z-component of the equation of motion i s

redundant. Equations (2,11) and (2,19) then give the complete

equation of motion for the magnetization, including magnetoelastic

coupling:

dH r = /&R

3 R " \

"I

ÖM fa

/SR 3 R ' \

1

- r f = Y (H.-DV ) H +b k ^ + -3-^ - M H , (2.22)

ot 1 X s i oz dx J s X

The equation of motion of the displacement vector R can be found

in the standard way by considering the forces acting on an element

49

of volume in the c r y s t a l ; we then find:

a^R ÖX ax ax

PT-T = T^+ T^+ T^ (2.23)

OX ox oy oz

with similar equations for the y- and z-directions, where p is the density and

X = the force applied in the x-direction to a unit area of a plane perpendicular to the x-axis.

X = the force applied in the x-direction to a unit area of plane perpendicular to the y-axis, etc.

The stress components are determined by:

X = — X ae XX V ^ oe •' xy etc. (2.24)

The energy density E is the sum of the elastic and magnetoelastic

energy densities, the last two terms in Eq. (2.1):

E = E + E . (2,25)

e me

Combining Eqs, ( 2 , 1 ) , ( 2 , 2 0 ) , ( 2 . 2 3 ) , (2,24) and ( 2 . 2 5 ) , we obtain

for the equation of motion of t h e x-component of the displacement

v e c t o r , retaining only f i r s t - o r d e r terms in the magnetization:

a^R a b aM

P T-r = '^.yK + (c, +c , ) r-(V-R) + — - r ^ , (2.26)

>' a^^ 44 X 18 44 3x Mg az

where we have used the fact that for cubic crystals with elastic

isotropy

c,, - c, = 2c ^ . (2.27)

For YIG,^°

c^^ = 26,9 X 10 d/cm^ ,

c = 10.77 X 10*^ d/cm* ,

c = 7.64 X 10^^ d/cm^ ;

hence, the assumption of e l a s t i c isotropy i s j u s t i f i e d in f i r s t

approximation. Similarly, we find for R and R :

y z a'R„ a b aM

p ^ - r ^ = c V ^ R „ + ( c +c ) — ( v - R ) + — — ^ (2.28)

The five coupled linear differential equations (2.21), (2.22), (2.26) (2.28) and (2.29)now describe the equations of motion for magnetoelastic waves for small amplitudes. As in Sec. II.B, we consider a plane wave solution, so that R , R and R are

propor-i(ut-i?-?) '^ y 2 tional to e , where u is the angular frequency of the magnetoelastic wave and k is in the x-z plane, making an angle 9 with the z-axis. Fig. 3,

k,,

/

z

f

F i g . 3 . Relationship between x and ycomponents and the l o n g i -t u d i n a l and -t r a n s v e r s e componen-ts of -the displacemen-t vector R.

Resolving the displacement into components along and t r a n s v e r s e to t h e wavevector: R = R cos9 + R, sine X t j ^ R = -R^ sine + R cose _{z t I} R = R/ , y t ' (2.30)

one o b t a i n s :

luM + YH.M - iYb,k R' cos9 = 0 ,

X K y 3 L

_s

(•Vfl. + a , sin e)M^ - iuM - lYb k(R. cos2e+R.sin2e) = 0 ,

b k

- i - | r M sin2e + (u^ - u)^)R, = 0 , (2.31)

b k

- i TÏT K cos2e + (13 - w^n. = 0 ,

b k

- 1 - 3 - M, cose + (.V - u)f)R/ = 0 ,

pM y t t

where H. = H, + Dk^ ,

_{k 1 '}

/c x'^

c = ( ~ ^ J ~ v e l o c i t y of longitudinal e l a s t i c waves,

0) = c k = angular frequency of longitudinal e l a s t i c

^ ^ waves,

'. - M

= v e l o c i t y of transverse e l a s t i c waves,

u) = c k = angular frequency of transverse e l a s t i c

waves.

The secular determinant of the set of equations (2.31) i s :

0^ = (''-«)^']r(''-'^tj fc^-o)^] - cT^oo* cos'e cos=2e

-CTU)^ hJ^-u^t] l(YHk+nMsin^eJ cos^e + YHj^ cos^26>

- L^-wnaw^ sin^2e I p -"^^JYH^ + (^^l cos^e) , (2.32)

Yb^

where a = - ^

44 s

The cases of interest here are waves traveling along the field:

9 = 0; and perpendicular to the field: e = TT/2.

For 9 = 0 , Eq. (2.32) reduces to:

Dj^(O) = (v^-u)p l(T5^-ci)^'')(w-uj^)+CTU)f| r(u^-u)^)(i;-u)^)-o<jü^1 = 0 .

( 2 , 3 3 )

It is clear that for 6 = 0 the longitudinal elastic waves are

com-pletely decoupled from the spin system. We will find this to be

true also in the case 9 = n/2. The second and third term in

Eq. (2.33) can be identified with negatively and positively

rotating circularly polarized waves. This can be verified by

sub-stituting M = M^ ± IM and R = R ± IR into Eq. (2.31). One

then obtains two secular equations, one describing positive

circu-lar pocircu-larization equal to the third term in Eq. (2.33), and one

describing negative circular polarization equal to the second term

in Eq. (2.33). The spin waves and elastic waves are coupled by

the term a = (Yb^c, M - ) . Equation (2.33) yields three positive

roots for transverse waves: two for (+) polarization and one for

(-) polarization. For cr = 0, corresponding to no magnetoelastic

coupling, these are:

13 = ui (+) spin wave term

" = "'t ^ • ' ^ , ,.

) elastic

13 = U)^ (-)J

wave terms

For a ^ 0, no closed form solutions can be obtained. An

approxi-mate positive root of the second term of Eq. (2.33) for (-)

polarization is:

For YIG, b

6.96 X 10^ erg cm"^ (ref 49), c

X itf^ dyne cm"^ (ref 50), M = 191 Oe (at 1,5°K) and Y = 1.76 x 10

sec" Oe~^

This gives CT = 6 x 10 sec ; and, for uj. in the

order of 6 x 10 sec~^ (X band), the second term in Eq. (2.34) can

be neglected. This means that for all practical purposes this

branch can be considered uncoupled from the spin system. Of

inter-est here are the two positive roots resulting from the third term

in Eq. (2.33):

(13-' U)^) (v-^

_a

_u)^

(2.35)

The largest interaction takes place when the wavelengths and the

frequencies of spin waves and elastic waves are equal. This is

called the crossover region.

point. Then,

v + m

Let 13 = 00 and k = k at that

cr cr

2ou and Eq. (2.35) reduces to:

cr

(13 _{<"t> ( ^}

-y

OU)

cr

(2.36)

The two branches are shown in Fig. 4. The minimum separation is

Fig. 4. Dispersion diagram for magnetoelastic waves propagating

in the direction of the internal magnetic field (9=0).

(Au) . = {2am Y^ and occurs at u = uu , For YIG, we had

min 8 *''^ a a = 6 X 10 sec"S so that (A,,) . « 8,7 x 10 sec" at 10 GHz.

min

This i s equivalent to 138 MHz, which i s small compared to the s i g

-nal frequency.

For 9 = n / 2 , Eq. (2.32) reduces t o :

D^(n/2) = (i^-u)p(i3^-u)^^) 1(13^-u)^) (13=-u)^) - G^l Y H J

(2.37)

Again i t i s found t h a t the longitudinal waves (propagation and d i s

-placement both in x - d i r e c t i o n ) are decoupled from the spin system,

But now also a shear wave branch e x i s t s (propagation in the

x-d i r e c t l o n , x-displacement in the y - x-d i r e c t i o n ) x-decouplex-d from the spin

system. The t h i r d term in Eq. (2.37) represents magnetoelastic

waves, propagating in the x - d i r e c t i o n , with displacement in the

z - d i r e c t i o n . They are shown in Fig. 5. The minimum.separation in

min

K Cr roughly^2 smaller than in the case of 9 = 0.

Fig. 5. Dispersion diagram for magnetoelastic waves propagating at right angles to the direction of the internal magnetic

The propagation velocity of magnetoelastic waves can now

be determined from Eqs. (2.35) and (2.37) by taking the derivative

of the frequency with respect to k: ^ . Since the splitting

between the upper and the lower branch is small, the magnetoelastic

wave along the upper branch can be considered as a pure spin wave

for 0 < k < k , and as an elastic wave for k > k ; a

magneto-cr magneto-cr

elastic wave along the lower branch can be considered as an elastic

wave for 0 < k < k , and as a spin wave for k > k .

cr '^ cr

The excitation of plane spin waves in a nonuniform

mag-15 51

netic field has been extensively described by Schlomann. ' He

showed that a net dipole moment exists for long wavelength spin

waves with k « 0. The propagating nature of the solution of the

dispersion relations can easily be shown by combining Eqs. (2.11)

and (2.15) for 9 = 0 and using M = M + IM . This results in:

X y

V^M + k^M = 0 . (2.38)

For waves propagating in the z-direction in a nonuniform internal

magnetic f i e l d , t h i s can be rewritten as:

dz

d lm\ ^ ^S(2)M = 0 , (2.39)

_{( ^ )}

^ u)^-YH.(z)

with k (z) = -Q

For (u. -

> 0, k^ > 0 and the rate of change of the slope

-rr

is negative for positive M. This means that eventually M must go

through zero. This changes the sign of the rate of change of the

slope, and the process will repeat, leading to an oscillating

function. If, for some region, k is positive and constant, then

the solution to Eq. (2.39) has the form e^ \ ^ , which shows

the propagating nature. If, however, we choose the same initial

s

at |z| = ", M never goes through zero, due to the fact that the signs of M and — are the same. The solution is then an exponen-tially decaying standing wave. Thus, the character of the wave changes at w. = YH.(z) from a propagating wave to a standing vibra-tion. The region where k = 0 is called the "turning point". In reality, this point is never reached. Both the introduction of spin wave losses and magnetostatic effects for low k spin waves will shift the turning point away from k = 0. However, the model describes the excitation of magnetoelastic waves quite well.

From the foregoing, it can be concluded that magneto-elastic waves can be generated magnetically only on the upper branch of the dispersion relation. On the lower branch, the propagating wave has a spin wave character only for k > k , and no turning point exists.

We now consider the specific case of a rod, magnetized along its axis (z-direction). The internal magnetic field is:

H. = H - H. , (2.40) 1 0 d

where H = applied field , H. = demagnetizing field .

It has the shape shown in Fig. 6. The field is stronger near the center of the rod than at the ends, due to the demagne-tizing effect of the end surfaces. The difference depends on the diameter-to-length ratio q; for small q, it approaches 2TrM. The exact formulation for the demagnetizing field H. along the axis is

52 given by Sommerfeld.

^

1

.

_

_

_

_

i

.

~r

Fig. 6. Internal field in an axially magnetized rod used for

delay time calculation of magnetoelastic waves.

where x

L

R

q

= normalized distance = —

= 21 = rod length ,

= radius of the rod ,

= 7 = aspect ratio .

Co

nsider now a value of H. for which the turning point is at x^.

A uniform rf field at frequency w couples into the net dipole

moment existing at the turning point, and, since u) - Y H ^ must be

positive, the generated spin wave propagates in the direction of decreasing internal field with increasing k. As the wave propagates toward lower internal field, the admixture of elastic component grows, until beyond the crossover point x„ the wave becomes pre-dominantly elastic. The elastic wave is reflected from the polished end surface of the rod and returns in a similar fashion to the turning point, where a fraction of the energy is detected, and the remainder reflected for another trip. With decreasing field strength, the turning point moves away from the end of the rod; in addition, the wavepacket spends a larger fraction of its travel in the spin wave portion of the spectrum. Both these effects contribute to a longer delay at lower field.

The round-trip time is given by:

T = T + T , (2.42) s e ' ^C with T^ •^- ' ^

^ ^ ^ ^ (^)

ff-^e = 2^ J -f = r (1 -''C^

Here T represents the time spent as a spin wave between turning point and crossover point, and T represents the time spent as an elastic wave between crossover point and the end of the rod. The frequency of z-directed spin waves is given in Eq, (2,15):

u^ = Y(H. + Dk ) .

Spin wave losses are introduced in the dispersion relation by re-AHjj

placing cu. with cu. + lY ~2~t where AH, is the spin wave linewidth, and replacing k with k + ik'. The spin wave dispersion relation then becomes:

AH,

\ + ^y 2 "^ "^"i "^ YD[k^- (k') ] + 2iYDkk' . (2.43)

By equating real and imaginary parts and eliminating k', a new

dispersion relation is obtained:

' \

[. Ml

K ~ . a a B

A-^\?\^

The spin wave group v e l o c i t y i s then:

(2.44)

YD

_2Y*D^k'

(v^I

From Eq. ( 2 . 4 4 ) :

" ^ 2YD j

A

1 + -\ / I +

( \ - ^ i ) '

(2.45)

(2.46)

Substituting Eq. (2,46) in Eq. (2.45) then yields the group

v e l o c i t y as a function of l-r- - H.j ;

—^=Y(2D)^

dk

ni

For AH, = O : k

dk

= 2YD^

i-y

- H.J .. (2.48)

The introduction of the spin wave loss term AH. results in a non-zero group velocity at the point where cu. = YH.:

[-^J = 2YD^

(AH|^)^

. (2.49)

In the ab ov e calculation of—n— , it is assumed that 53 54 AH is a constant, even though AH. is linearly dependent on k. However, this is justified since in the delay time calculation — - H. varies from 0 up to ~ 1100 Oe, and AH «^0.1 Oe; thus the

only contribution to T arising from the loss term is for -— - H.

s u*f Y 1

of the order of AH. . We must now express — - H. as a function of K T 1

the normalized position in the rod, x. We define (see Fig. 6 ) : H.(0) is H. at the center x = 0 ;

H.„ is H. ïAich satisfies ./ = H. (turning point) ; 10 1 Y 10 a K '

"^-0

H. (0) is H. which satisfies - ^ = H. at x = 0 10 1 Y 10

(turning point at x = 0) ;

H,(0) = H - H.(0) (strength of the demagnetizing field at x=0); AH.(O) = H.(0) - H.„ .

We then have:

H.(x) = H„ - H.(x)

and H„ = H.(0) + H.(0)

Hence:

and

^ - H.(x) = Hj^ - H.(0) + [H^(x) - Hj(0)] ,

y

or -f - H.(x) = Ql^(x) - H^(0)] - AH.(O) . (2.50)

H^(x) - H^(0) i s found from Eq. ( 2 . 4 1 ) :

H^(x) - H^(0) = 2TTM

1 - x

d+q')'^ [(l-x)%q=l^ [(1+x)

1 + X 1

(2.51)

The turning point x_, i s determined by setting k=0 in Eq, ( 2 , 1 5 ) ,

so that from Eq, ( 2 , 5 0 ) :

H^(x.j.) - H^(0) - AH.(O) = 0

""t ""o

The crossover point x„ i s determined by setting k = — = — in

Eq, ( 2 , 1 5 ) , so that * ^

H .(x„) - H.(0) - AH.(O) = D - ^ ,

d C d 1 c^ ( 2 , 5 3 )

Combining Eqs, (2,42), (2,47), ( 2 . 5 0 ) — (2.53) yields the round-trip time T as a function of AH.(O). The reason for using AH.(O) as a parameter is that the condition AH.(O) = 0, the field for which the turning point is at x = 0, is experimentally easily observed. It is the field for which the magnetostatic mode of propagation becomes allowed through the while rod (see Sec. II.D), which results in a distinct change in absorption.

The transit time and the locations of turning point and crossover point for a magnetoelastic wave have been computed at 9.3 GHz for a rod with L = 1 cm, R = 0.15 cm. The results are

shown in Fig. 7 where the total delay time T and the delay of the

0 K» 200 300 400 500 600 700 800 900 1000

A H| (o) IN Oe •

Fig. 7. The position of the turning point, the crossover point and the total delay time of magnetoelastic wave and the delay time of the elastic wave portion as a function of magnetic field strength in an axially magnetized rod.

elastic part T are plotted as a function of AH.(O), As was indi-cated before, the introduction of the loss term Y ( ^ I ^ / 2 ) in the dispersion relation only gives a significant contribution at small values of AH.(O); for instance, at AH.(O) = 1 Oe, the difference inTwith or without loss is 3 usee for T = 65 usee; & r AHi(0) = 100 0e, where T sa 13 ^isec, the difference is already reduced to 0.2 g.sec. For all practical purposes, one can use Eq. (2.48), since magneto-elastic wave echoes in general are attenuated too much to be

observed for T > ~ 30 iisec. As can clearly be seen in this graph, the elastic waves contribute only ~ 10% of the total delay; the major portion of the transit time is caused when the magnetoelastic wave is in the spin wave state. On the same graph are indicated the positions of the crossover point x-, and the turning point x™ as functions of AH.(O). The crossover point x- Is at x = 0.55, nearly one-quarter of the length down the rod, «rtien the turning point is in the middle of the rod, at x = 0. At the high field end, the turning point and crossover points are very close together, due to the high field gradient near the end surfaces.

The case of a disk or slab, magnetized perpendicular to its plane, differs in two respects from the axially magnetized rod. First, the spin waves travel at right angles to the dc field, and the spin wave dispersion relation for 9 = n/2, i.e., Eq. (2,16), must be used. Second, the field in the sample is weaker in the center than at the periphery of the disk. The difference in field strength is again dependent on the aspect ratio q = ^ , where s is the thickness and R the radius of the disk. For a small aspect ratio, it approaches 2nM.

The expression for the internal magnetic field in the central plane is given by:

H.(r) = H„ - H.(r)

where

H^(r) = 4TTM

^[jJ^lf^

r' . m

-r»

Vn^+r^ - 2rR cos

cos9 I

(2.54)

I t has the shape shown in Fig. 8. A spin wave generated at the

turning point r™ propagates in the d i r e c t i o n of decreasing f i e l d

toward the center of the disk. At the crossover point r - , i t

T

s

.i

H H ( 0 )

Fig. 8, Internal field in a normally magnetized disk used for

delay time calculation of magnetoelastic waves.

changes i t s character from spin wave to e l a s t i c wave. I t propagates

through the center as an e l a s t i c wave, changes back to a spin wave

at r - and i s detected at r „ , . At low f i e l d , both turning point and

crossover point are close to the disk edge and most of the t r a n s i t

occurs as an e l a s t i c wave. With increasing f i e l d , the crossover

point moves closer to the disk center and a larger fraction of the

t r a n s i t occurs as a spin wave, with longer delay, u n t i l the c r o s s

-over point i s at the disk center and the propagation i s purely spin

wave. I t can be shown t h a t in t h i s case the delay approaches a

constant value. Independent of f i e l d . With s t i l l further increase

in f i e l d , magnetostatic effects take over, and the t r a n s i t time

decreases rapidly (see Sec. I I . D ) .

The t r a n s i t time i s obtained from:

T = T + T^ , (2.55) s e '

r

T

with T = 2

_s

f

j ^

The spin wave dispersion r e l a t i o n i s given by Eq. ( 2 . 1 6 ) :

y

(Hj + Dk^) (Bj + Dk^) 1

dcu. H. + 2TTM + Dk

- T = 2YDk ^-— —-j7 . (2.56)

From Eq, (2.16) we obtain:

^[#

(2.57)

S u b s t i t u t i n g Eq. (2.57) in Eq. (2.56) y i e l d s :

^=?#^[-v^

1^

(2.58)

This r e l a t i o n can be g r e a t l y simplified if the approximation

YBjHj' = H. + 2nM i s used. For H- « 4TTM, the e r r o r introduced t h i s

way i s ~ T%. I t should be pointed out t h a t t h i s i s only v a l i d for

X-band frequencies and higher. For lower frequencies, the complete

expression must be used. With t h i s approximation, the group

v e l o c i t y becomes:

d r

1*^

(2.59)

By defining (see Fig. 8 ) :

H.(0) is H. at the center of the disk, r=0 ;

\

Hj^ is H^ for which ^ = V ^ i ^ ^ i ^ (turning point) ;

\

CU,

- / B T H ? can be replaced by AHi(O) - [ H ^ ( 0 ) - H^(r3, «liere

Y

A H . ( O )

= H.^(O) - Hj(0). H^(r) i s given by Eq. ( 2 . 5 4 ) , and from

t h i s :

H.(0) = 4nM d

['^] '

The turning point r» is determined by:

AH.(O) - | H ^ ( 0 ) - H^(r^)] = 0 . (2,61)

The crossover point r- is determined by:

s

U)

AHj(O) - [H^(0) - H^(r(,)] = D - | , (2.62)

•^t

where again the approximation H. + 2nM RS Y B J H J is used. Combining Eqs, (2,54), (2.55), (2.58), (2.60) — (2.62) yields the delay time, and the positions of the turning point and crossover point for a magnetoelastic wave in a disk. They have been computed for a wave at 9.3 GHz for a disk with s = 2,5 x lO"^ cm and

R = 1.875 x 10" cm. The results are shown in Fig. 9. For field strengths where H.(rp) < H.(0), the propagation is purely magnetic

1 w 1

and most of the delay takes place near the middle of the disk. H,(r) can then be expressed in a power series:

H.(t) = H.(0) + h r= + .... (2.63) U U s with h = ^ s

&

'm

r=0

120

A Hi(o)jn

Oe-Fig. 9. The position of the turning point, the crossover point,

and the total delay time of a magnetoelastic wave and the

delay time of the elastic wave portion as a function of

magnetic field strength in a normally magnetized disk.

From Eq. (2.54) we find:

h = 4nM I

i£

P ^

ïïï t' ^

^)f.

For disks with a small aspect r a t i o

h = 4nM -%•

3 BR

and H^(r) - H^(0)

h. r^ = 4 n M - ^ r ^

= 8R

Equation (2,59) then becomes:

- ^ = 2YD* | A H . ( 0 ) - h r® dk I I 1 s

[AH.(O) - h^rj ,

m

_ L - ƒ dr

After i n t e g r a t i o n :

T = — r ^ • sinh ^

m

[

AH. ( 0 ) 1

(2,67)

(2.68)

(2.69)

and

T = r,—TT which is independent of AH.(O) . (2.70)

>YD?^h ^

The fact t h a t the delay time approaches a f i n i t e value instead of

i n f i n i t y for AH.(O) - 0 i s due to the parabolic shape of the

demagnetizing f i e l d near the center of the disk. The motion of

the wavepacket i s a simple harmonic motion of a p a r t i c l e in a

para-bolic p o t e n t i a l well, and thus has a constant period Independent

of amplitude.

D. Magnetostatic Waves

The dispersion r e l a t i o n for 9=0 spin waves, as given in

Eq. ( 2 . 1 5 ) , approaches y = H. as the spin wavenumber k approaches

zero. However, In an a x i a l l y magnetized rod the uniform mode i s

at ^ = H. + 2nM, r a t h e r than at -^ = H.. Similarly, the dispersion

r e l a t i o n for e=n/2 spin waves approaches -^ ="^BjHj' as k -• 0, but

Y

for a normally magnetized disk the uniform mode i s at -^ = H^.

Obviously, the extrapolation of plane spin waves to k=0 i s i n v a l i d .

The reason i s that for spin waves with wavelengths comparable to a

sample dimension the spin wave spectrum i s strongly modified by

magnetic d i p o l a r e f f e c t s , in order to s a t i s f y the magnetostatic

boundary conditions. These long wavelength spin waves are t h e r e

-fore called magnetostatic waves because t h e i r behavior depends on

q u a s i - s t a t i c magnetic dipolar forces, and the dependence on

exchange forces can be neglected. Under t h e s e circumstances, the

spin wave dispersion r e l a t i o n i s influenced by the sample geometry,

and the plane spin wave approximation i s invalid. In t h i s t h e s i s

we w i l l not give a complete derivation of the magnetostatic d i s

-persion r e l a t i o n , since t h i s has been the subject of two extensive

papers. ' We s h a l l only show the most s a l i e n t points and

describe the modified behavior expected from the magnetostatic

spin wave dispersion r e l a t i o n s , as much as i s required for the

parametric a n p l i f i c a t i o n experiments to be described in the

follow-ing chapters. I

To obtain the magnetostatic mode spectrum, exchange

effects are i n i t i a l l y neglected, so t h a t the forces are purely

magnetostatic, a r i s i n g only from the applied magnetic f i e l d and

the dipolar f i e l d s of the sample magnetization. Combining

Maxwell's equations with the equation of motion for the

magnetiza-t i o n and inmagnetiza-troducing a magnemagnetiza-tic p o magnetiza-t e n magnetiza-t i a l magnetiza-t magnetiza-t a modified Laplace

equation in c y l i n d r i c a l coordinates i s obtained of the form:

(1 +

^a%. 1 hi/. 1 a%A a%

\^ar r a r r^ ^ ^ / ^^

= 0

(2.71.a)

inside the sample.

a t 1 aijt 1 a%

r or T op

a te

IF

= 0

(2.71.b)

outside the sample,

vrtiere

B i H l - f-^)

1 + K = ^

To obtain the boundary conditions, we have to specify the shape

of the sample.

For a disk of thickness s in the z - d i r e c t i o n , the boundary

conditions a r e :

a*

az

at

e

az

(2.72.a)

z = ± s/2

± s/2

for the continuity of the normal component of B and

*i = *,

z = ± s/2

(2.72.b) z = ± s/2

for the continuity of the tangential component of H. After separating the variables by writing • = R(r)Z(z)$(tp), the solution for the radial part of t. is a Bessel function of (kr):

R(r) = J (kr). By using the asymptotic expression for (kr) large compared to n, and combining this with Laplace's equations and the boundary conditions, we obtain after separating out the radial dependence:

['

id

+ K)^

+

\i^ +

_{"2] =}

where k is the transverse spin wave number, and successive values of n correspond to increasing values of k .

It is easily seen that, for k -> 0, -:r—• H., while, for k ^ - , ^-.YBTHT',

22

For a rod of radius R, the boundary conditions are:

(2.74.a)

a*.

+

iu at. ay

\

<

- (^)

Solutions of the radial part of Eq. (2.71) which are regular at r=0 and vanish at r=<» are a Bessel function of argument

in the interior of the sanple and a Hankel function of argument (ikr) outside the sample. Combining this with the boundary condi-tions yields: i d + H)*^ k j^ + -g-n H' • I 1 i k ^ - , n (2.75)

where the prime denotes differentiation with respect to the argu-ment. This can be rewritten in the form:

n(u - H ) = IkR -Sfp-i- 1(1+ K ) ' ^ k R ^ i j ^

n n (2.76)

The first term on the right-hand side approaches zero for k - 0; the second term approaches -2n(l + K ) ; hence

n(u - K ) = -2nd + K ) for k - 0 ,

or u + K = - 2 , from which we obtain cu. = Y(H. + 2nM) at k=0. For (kR) l a r g e , the Hankel function may be approximated by ( k r ) ^ e Equation (2.75) then becomes for n = 1:

i d + K)^-~

- 1 - k R (2.77)

which for (kR) large reduces to

K

IkR (2.78)

or

This y i e l d s :

\

UJ '

(2,80)

where Y. are the roots of J (Y) = 0. The magnetostatic mode

spectrum of a normally magnetized slab and an a x i a l l y magnetized

rod are shown in F i g s . 10a and 10b, r e s p e c t i v e l y . The slope of the

curves in both cases approaches zero for k -• °=. I t i s c l e a r ,

Fig, 10 a, Magnetostatic mode spectrum of a normally magnetized slab,

b, Magnetostatic mode spectrum of an a x i a l l y magnetized rod,

however, t h a t , for large values of k, exchange effects must be

i n c l u d e d / This means t h a t in the case of the normally magnetized

disk the magnetoelastic wave dispersion curve as shown in Fig, 5,

which approaches ca =

for k -• 0, and the magnetostatic

wave dispersion as shown in Fig, 10, which approaches the same

value of w. for k -• " , must be joined together, r e s u l t i n g in a

non-zero group v e l o c i t y for a l l values of k. Fig. 11a. I t i s here t h a t

a big difference a r i s e s with the a x i a l l y magnetized rod. If the

magnetostatic and the magnetoelastic dispersion curves for an

/ H I •• EXCHANGE DOMINATED SPIN WAVES MAGNETOSTATIC WAVES y(Hi+2TrM)-Y»i MAGNETOSTATIC WAVES ^TURNING POINT,-j^ =0 EXCHANGE DOMINATED SPIN WAVES

Fig. 11 a. Magnetoelastic and magnetostatic wave dispersion

joined together in the case of a normally magnetized slab.

b. Magnetoelastic and magnetostatic wave dispersion