Surface Effects on Spinwave Resonance in Thin Magnetic Films - CaltechTHESIS

The roots of this deter-

minant give the allowed values of k1 and k2 .

The relations between

-28-

k1 and k2 and the elements of the 4 x 4 determinantal equation for
determining k1 are given in Appendix II. Schematically
det [aij] = 0

(3-9a)

At perpendicular resonance, the symmetry is such that the positive
and negative precession spinwave branches uncouple; and Eq. (3-9a)
becomes

II

{(k.

i=l

k.d
k.d
K0 cot -i-)(ki + K0 tan -i-> + 6K } = 0

where K0 = (Ksl + Ks 2 )/2A and 6K = (Ksl - Ks 2 )/2A.

(3-9b)

The (i=l) factor

gives the allowed positive precession spinwave wave vectors.

If Ksl

and Ks 2 are large and negative, the (i =2 ) term can give only two
allowed negative precession spinwave wave vectors; however, values
of the surface anisotropy of this magnitude are believed unrealistic.
Therefore, for values of surface anisotropy normally required to
match experimental data the spinwave excitations associated with this
model have purely sinusoidal or hyperbolic excitations corresponding
to the allowed values of k1 .
For each allowed k1 and k2 the required applied field can be
calculated from Eq. (3-1) and the equilibrium condition for the
static magnetization.

Solutions to Eq. (3-9) for a symmetric film

at perpendicular and parallel resonance are plotted in Fig. (3-3)
and (3-4).

For positive Ks at parallel resonance and negative Ks

at perpendicular resonance there is always one and sometimes two
surface modes (i.e.,k 1 is negative}; for the other sign of Ks there

-29-

55

Perpendicular Resonance
10
4Trt·1 = 10

y = 1.8xlo 7

=d'

A = 10- 6

f(GHz) = 9

d(Jh = 103

3i

2i

1i

-1

-10

-55
Fig. 3-3 Solutions (k 1d/n) to Eq. (3-9) for a symmetric film at
perpendicular resonance versus Ksd/2A.

-30-

3i

2i

1i

-1

Parallel Resonance
41TM = 10

y = 1.8xl0
-10

A = 10- 6

f(GHz) = 9
d(A) = 10

Fig. 3-4 Solutions (k 1d/n) to Eq. (3-9) for a symmetric film at
parallel resonance versus Ksd/2A.

-31-

are no surface modes.

It can be shown that at perpendicular reson-

ance if there are two surface modes then there is no allowed value of
k1 in the ran9e (O < k1d/n < l); this may not be true at parallel resonance as can be seen in Fig. (3-4).
Ratios of the coefficients in Eq. (3-8) can be determined for
each allowed value of k1 and k2 • A particular coefficient can be assumed unity or related to the applied field through Eq. (3-6a). For a
symmetric film at the parallel orientation, plots of m¢ normalized to
unity at the film center are shown in Fig. (3-5). Note that the k2 component (i.e., -R*m21 cos k2 z) is concentrated at the surfaces; this is
typically the case since k 2 is usually a large imaginary number.
Since the k2 components are concentrated at the surfaces and
have a small amplitude only a small error is made if these components
are neglected when calculating the power absorbed from (Eq. (3-6b).
With this approximation lvl 2 = IRI 2 and the power absorbed per unit
area of film is

= ho 2xMd
2et ( 1+ IR12)

+ sin

cos 2~)

(3-lOa)
where
(3-lOb)

-32-

ml2
mll

all

a31

a41

al2

a32

a42

al3

a33

a43

a21

a31

a41

a22

a32

a42

a23

a33

a43

and the aij's are given in Appendix II.
(3-9) factors into two parts.

(3-lOc)

For a symmetric film Eq.

One factor gives wave vectors which

correspond to mode shapes that are symmetric around the film center
(even modes); the other factor gives wave vectors which correspond
to antisymmetric mode shapes (odd modes).

For the antisymmetric mode

shapes cos~= 0; therefore, these modes are not excited (i.e., Pabs
= 0).

For the symmetric mode shapes cos~= 1, and the power

absorbed by the symmetric modes normalized by the k = 0 absorption
is plotted versus k1d/2TI in Fig. (3-6).

If a highly localized sur-

face mode exists (i .e., k1d/TI is a large imaginary number) the next
even mode has a wave vector in the range (i- and (3-4)); therefore, it is possible to have a surface mode (highest
field mode) with a power absorption smaller than the first body mode.
Eq. (3-9) has been expanded for the parallel resonance orientation; the result of this expansion is given in Appendix III along
with other results for some permalloy films at both the perpendicular
and parallel resonance orientations.

m11 cos(kz)

TRT

'-. -I R* 1 m21 cos ( kz)
4lTM = 1735
y = 1.767xlo 7
A =3.593xlo- 7
mcp

Fio. (3-5) The first three spin wave excitations at parallel resonance for a film
w1th YIG parameters, Ks=.05 ergs/em , and d=.4 microns

8i
6i

4i

2i

Fig. (3-6) Normalized power absorbed for a symmetric film with uniaxial anisotropy.
Material constants are for YIG, d=.846 microns.

1Oi

y = 1 .767x10 7
A = 3.593x1o- 7

4nM = 1735

10 k1d/2n

-35-

3.3.2 Tensorial Anisotropy
The tensorial anisotropy energy proposed by Yu (1975) gives
the following boundary condition on m (see Appendix I)
(n is the coordinate along the outward film normal)
Adm + (Klcos 2e + Kll sin 2 e)m = o
dn

( 3-11)

or
Adm + Kr(e)m = o
dn
For spinwave excitations given by

m=

I m . cos k.z
+ m . sin k .z
21
i=l 1 1

(3-12)

and asymmetric boundary conditions, the secular equation for ki is
k.d

k.d

II {(k.- K c o t1 -) (k. + K tan - 1 ) + (LlK)} = 0
i=l

( 3-13)

where d is the film thickness, KT 1 (e) and KT2 (e) the anisotropies at
the two surfaces, K0 = (KT 1 (e) + KT2 (e))/2A, and ~K = (KT 1( e)
- KT 2 (e))/2A.
Equation (3-13) is identical to Eq. (3-9b) i f KTl(e) = Ksl
KT (e) = Ks 2 ; therefore, the comments after Eq. (3-9b)
applying to uniaxial perpendicular anisotropy at perpendicular

and

resonance apply at all angles here.

For a symmetric film and mat

an angle e the allowed values of k1 versus Ks = KT(e) are plotted
in Fig. {3-3).

Finally, based upon an approximation for the uniaxial

anisotropy developed in chapter 4 the tensorial and uniaxial models
have similar properties if

-36-

'1

= Ks

-K IRI2
K II - ____;s:;...__><'"
- 1 + IRI2

(3-14)

The power absorbed for a film with this boundary condition is
given by Eq. (3-lOa) where

-t.K cos k1d/2
Recall that for a symmetric film and symmetric modes this expression
normalized to the k1 = 0 absorption is plotted in Fig. (3-6).
3.3.3 Surface Layer Model
A physically plausible model is that the surface regions of
the film have a chemical composition and structure different than
the bulk of the film.

This can be due to diffusion of elements

into the film or chemical reaction. As a first step in understanding
the properties of this model, the film may be assumed to have surface
layers of uniform magnetic properties which are different from the
bulk properties.

As required for a clearer understanding, the model

can be later extended to one in which variation in properties is represented by adjacent layers with graded properties or by an explicit
functional dependence.

The greater part of this thesis is restricted

to simple layers at each surface. The properties given these layers
represent a kind of average of the properties of the actual regions. It
will be shown in Chapter 6 that this model can quantitatively or in

-37some cases only qualitatively match most behavior observed in YIG
films.

Some of the failures are believed to be due to the variation

of the magnetic properties near the surface; that is, the assumption
of uniform layers i s s imply not accurate enouqh.
A schematic repre sentation of the model is given in Fig. (3-7};
the figure is schematic for a VIr, film grown on a GGG substrate.

The

two surface layers are obviously different and the exact properties are
unknown; but experimental and theoretical data suggest the properties
which should be ascribed to each.
In each of the three uniform regions Eq. {3-1} applies when the
appropriate material constants are used.

It is assumed that the

direction of the static magnetization in the surface region is the
same as the static magnetization of the bulk; at all angles of m
other than parallel and perpendicular to the film plane this is an
approximation not only i n the surface region but also in the bulk
near the interface.

Because of the exchange interaction, there is a

continuous smooth transition between the angles of equilibrium;
the transition region extends from the interface into both the bulk
and surface layers.

At 9 GHz the maximum variation for a half

magnetization surface layer is about 6° for YIG and 20° for Permalloy.
With the above assumptions the required boundary conditions at the
interface of the magnetic regions are easily derived from torque considerations.

They are
mb - ms
Mb - Ms
(3-15)
Ab amb
Mb Tz

As
Ms

ams

-38-

z~L

frPP c:ni nc; at

surface --.
nonstoichiometric YIG
Ms > Mb~ surface mode at perpendicular resonance

r-------~~~~~~~uu~~k--------------------1

z=Q 1-~~b~ surface mode at parallel resonance

BOUNDARY CONDITIONS at z=±¥

Y3 Fe 5o12
BULK

or

YIG

z=O

z=-¥0-t-------------------------------------------~

SUBSTRATE

Fig. (3-7)

Schematic representation of the surface layer model
for YIG. It is assumed that the static magnetization
in each region is in the same direction as in the
bulk of the film.

-39-

where mb, ms are the rf magnetization vectors;

Mb and Ms

are the

saturation magnetizations; and Ab, As are the exchange constants in
the bulk and surface regions, respectively.

At all surfaces between

magnetic and nonmagnetic materials the spins are taken to be free;

am

this requires that az = 0.
Spinwave excitations for a symmetric film with free spins at
z =

+(% + L) are given by
meb = m1b cos k1bz + m2bcos k2bz
COS klbz - 1\,* m2b cos k bz
m¢b ___l.Q_
Rb

(3-16)

mes = mls cos kls (z±%tL)

+ m2s cos k2s ( z±%tL)

_ mls
m¢s -~ cos kls (z±%tU

- Rs* m2s cos k2s (z±%tL)

These spinwave excitations and the boundary conditions in Eq. (3-15)
give a 4 x 4 determinant for determing the k-values.

The following

secular equation is obtained for klb'
klb Ab tan (k 1bD/2) =
Tls[(!Rbi2+1}(1Rsl2+l}T2s+l(l+RbR;}I2T2b]+IRb-Rsi2T2sT2bl
- [ l(l+f\,R;}I2T2s+(!Rbi2+1}(1Rsl2+l)T2b+lf\,-Rsi2T2s
where
Tls

klsAs tan(k 1sL}

T2s = k2SAS tan(k 2sL}
T2b = k2bAb tan ( k2b D/2)

(3-17)

-40-

2s

This analysis for an asymmetric film gives a secular equation in
the form of an 8 x 8 determinant; this determinant is given in
Appendix II.

Given the frequency, the angle of the static magneti-

zation, and the magnetic properties of the bulk and surface regions,
the secular equation can be solved for the spinwave wave vectors
that satisfY

the boundary conditions.

For each allowed wave

vector the required applied field Ha can be determined from Eq. 3-1
and the equilibrium conditions for the static magnetization in the
bulk region (Appendix I). (Note that (Eq. 3-17) can be applied to
a film having only one surface layer by replacing D/2 by D.

The other

surface would naturally have free spins.)
For YIG material constants and e = 0°, 30°, 60° and 90° the two
sides of Eq. (3-17) are plotted versus k1D/2n in Fig. (3-8). Note that
in the range e = n/3 to n/2 there is a root giving a surface mode (i.e.,
kl is imaginary). The angle where k1 = 0 satisfies Eq . (3-17) has been
call ed the critical angle; a more in depth discussion of the critical
angle will be given later.
Since garnets have widely varying losses depending on their compos ition and preparation, it is plaus ibl e that the damping parameter, a.

, I

Fig. (3-8)

II

,zj1'

"/

J "oc;;::::: --, ~ 0/2n

Plot of the right and left hand sides of Eq. (3-17) versus k D/2n . The material
constants are for YIG with a 200A half magnetization surface 1layer. The bulk
thickness is .42 microns.

rr==---

yb = 1.767xlo 7
A = 3.593xlo- 7

4nMb = 1735

......

-42could be different in bulk and s urface layers.

The effects of this on

the relative intensity of the modes can be calculated as follows. Using
Eq. (3-5) and ignoring the contribution of the k components the power
absorbed i s
(3-18)

abs

1(l +I R~ I ) ~(D+
Yb b

s in(2k 1bD/2)
'k

lb

c2 a
)+

sin(2k

M (1+ 1Rsl
)(2L+ k

Ys s

ls

L)
ls

)t

This expression normalized fork = 0 and ~/os = 1 has been plotted
in Fig. (3-9) for ~las =

1,

.3, and .1.

The increase in absorption

at kD/2n = 7 or 8 is due to the surface layer going through its
uniform resonance (i.e., kls ~ 0).

Note that the increased surface

a has more effect on the surface mode intensity than on the body

modes (except where k1sz 0).
By making approximations in equation (3-17) one can deduce some
of the physics of the surface layer model .

By assuming that RbzRs

and that klsl is sufficiently small to approximate tan(k 1sL),
Eq. (3-17) becomes

This equation is in the form of the secular equation for a symmetric
film with an ani sotropy energy, Km' and easy direction along M0 .
this case
( 3-19)

In

g;

7i

5i

3i

Fig. (3-9) Normal ized power absorbed versus
k D/2~ for YIG material constants
1 a half magnetization surface
and
layer 470A . Thickness of bulk
film is .423 microns. (yb=ys)

1i

""!'

"""

v v \../'\,
.,. . . . . ._ r> -f!">
f ~
kl D/2~
4 5
10

yb = 1.767xl07
Ab = 3.593xl o- 7

4nMb = 1735

-44-

An anisotropy field can be defined
2'

Hm =

2As

'f\L = - Ms kl s

(3-20)

This is approximately the exchange field necessary to satisfy the
resonance condition in the surface region.

If Hm is positive the

highest field mode will have a sinusoidal excitation in the bulk
and an exponential excitation in the surface region.

If H is
negative the highest field mode will have an exponential excitation
in the bulk (surface mode) and a sinusoidal excitation in the surface
region.

Fig. (3-10) shows several spinwave mode shapes at perpendi-

cular resonance.

The angle at which Hm = 0 is approximately the cri-

tical angle (see next section), and corresponds to the angle at
which both regions resonate with a "uniform excitation".
Finally the following interesting correspondence between the surface layer (yb=ys) and the perpendicular uniaxial anisotropy is worthy
of note. It was shown by Bajorek and Wilts (1971) that for thin surface
layers on permalloy at perpendicular resonance these two models have mode
positions which are in very close agreement no matter how thin the central bulk layer if the uniaxial model has the following properties. First
the film is uniform with bulk properties and the same magnetization as
the layered film. Secondly, the value of Ks is given by Eq. (3-19) evaluated at perpendicular resonance with k1b=O .

At parallel resonance

Wilts and Ramer showed that the same close agreement existed
(unpublished).

For YIG material constants (Yb =Ys)

Fig. (3-11)

is a comparison of the two models at all angles of the applied field.
It will be pointed out in Chapter 6 that the agreement is not as
remarkable for thicker surface Jayers on YIG or when yb ! Ys

yb= 1.767xl07

n=2

Ab= 3.593xlo- 7

n=3

n=4

Fig. {3-10) Perpendicular resonance mode shapes for the surface layer model and one
free . surface. The modes are numbered
with the0 highest field mode n=l .
The pa·rameters were for YIG, D=3700A and L=470A.

t4b/Ms =. 965

f\

41Tf·1b = 1735

Mb/Ms=2

Mb>Ms

n=l

-1=:>

()1

-46-

Surface layer
450
400
350
300
4TTM = 1735
= 1 .767xlo 7
A = 3.593xlo- 7

250

~ 200

4-

:::I
:z:::

Surface layer
150

0..100
0..
10

:z:::

50
Surface layer

-25
-50
Fig. (3-11)

Comparison of the surface layer model and the uniaxial
model using a value of Ks deduced for very thin surface
layers. The surface is a half magnetization 200A
layer, D=4550A . For the surface anisotropy d=4650A
and Ks=.061 ergs/cm2.

-473.4 Critical Phenomena
The typical spinwave spectrum has one large power absorption peak
and several smaller ones; the mode with the largest absorption usually
has the highest field position.

However, in some films one and some-

times two modes have a higher field position than the mode with the
largest absorption; these modes are called surface modes, and it will
be shown that they have properties similar to the surface modes introduced mathematically in the previous sections. It has been observed
that if the surface modes exist at one of the limiting orientations
(i.e., applied field parallel or perpendicular to the film plane) they
do not exist at the other. When films with surface modes are rotated
with respect to the applied field from the one limiting orientation,
the highest field mode increases in absorption intensity while the
largest absorption peak decreases in intensity. This behavior continues
until the highest field mode is observed to have the largest absorption
and some of the modes that were prominent actually vanish. In some
films all modes (except the highest field mode} vanish at about the same
angle.

In other films there are two angles where some of the modes

are observed to vanish, but a particular mode does not vanish at more
than one angle.

Beyond the angle where a particular mode has dis-

appeared it reappears and grows in intensity; but the highest field
mode remains the largest.

Even in films where the highest field

mode at both limiting orientations is dominant there are angles where
some of the lower field modes vanish.
In films with symmetric surface conditions or films where the
air film interface has been treated to ensure that the spins at this

-48-

surface are free (~~ = 0), all modes except the highest field mode
vanish at or very near one angle of the applied field.

Since all

modes do not vanish at exactly the same angle, this critical phenomenon
is characterized by that angle at which the second spinwave mode
vanishes, hereafter called the critical angle Be.

The temperature

dependence of this critical angle is believed important in determining the particular mechanism producing the surface pinning.
In terms of the surface models the observed small amplitude high
field modes are the surface modes introduced in the previous sections.
For the tensorial model and a symmetric film, there is one allowed
surface mode for O>KT(e)d/A>-2 and two allowed surface modes for
KT(e)d/A<-2; the second mode, however, is antisymmetric and is not
excited (i.e . , cos~ in Eq. (3-10) is zero).

Further, no other

antisymmetric mode is excited.
For the tensorial model and an asymmetric film, corresponding
values of KT( e ) at the two surfaces (KT 1 ( e ) and KT 2 (e)) required
for 0, 1, and 2 surface modes are plotted in Fig. (3-12).

The

boundaries between the regions were determined from the condition
for a uniform precession mode.
from Eq. (3-13) by setting

This condition (easily obtained

k1 = 0) is given by

{3-21)
where

-5

TWO SURFACE MODES

\I.e

-10

ONE SURFACE MODE

NO SURFACE MODES

KT2{e)d/A

10

KTll(e)d/A

Fig. (3-12) Values of Kr(e) at the two surfaces required for 0, 1, and 2 surface modes.

-10

ONE SURFACE MODE

10

1.0

-50-

In the region for 2 surface modes and when KTl ~ KT 2 ' the second surface mode is excited (i.e.,

cos~~

antisymmetric surface mode.

The first or high field surface mode is

0); this mode is called a quasi-

called a quasi-symmetric surface mode.

It is easy to see from Fig.

(3-6) and Eq. (3-10) how these two surface modes could have a smaller
absorption than the third mode; mathematically this is due to their
hyperbolic decay away from the film surface versus the sinusoidal behavior of the third spinwave (first body) mode.
If KT 1(e/2) and KT2 (e/2) have values which will give the spinwave
spectrum two surface modes and KT 1 (o) and KT 2(o) are both positive
then as the magnetization is rotated from the parallel to perpendicular
resonance orientation, the surface modes become the two highest field
body modes.

Further, three conditions can exist which will cause mode

to vanish as observed experimentally,
KTl(e) = -KT2(8)
KTl ( 8) = KT 2 ( 8)
or
For the condition KT1 (e) = -KT 2 (e) the allowed wave vectors are
n = 1,2,3,4,···

(3-22)

From Eq. (3-10) the power absorbed for n even is zero and the power absorbed for n odd is not zero unless KT 1 (e) = KT 2 (e) = 0 .

The modes

corresponding ton odd and even are calledquasi-antisymmetric and quasisymmetric, respectively.

The mode corresponding ton = 0 (the quasi-

symmetric uniform precession mode) only occurs under this condition if

-51KTl = KT 2 = 0; under this condition all modes except the uniform precession mode vanish.

It was pointed out that for a symmetric film

the antisymmetric modes are not excited; therefore, under the condition
KT 1 ( o ) = KT 2 ( e) quasi-antisymmetric modes become antisymmetric modes
and vanish.
In the above description of the mathematical behavior of the
tensorial model the variable e is the angle of the magnetization.

In

the experimental situation the film is held fixed with respect to the
angle, S, of the applied field; however, the direction of the magnetization varies only slightly as the magnitude of the applied field is
swept over the range of interest.

The surface layer and perpendicu-

lar uniaxial anisotropy models are mathematically more complex.

It

can be s hown that they too have a mathematical behavior which can
explain the above experimental behavior; in fact, this is shown
roughly by the following argument .

The surface layer has properties

similar to the uniaxial anisotropy model (see the previous section);
the uniaxial anisotropy has properties similar to the tensorial model
if

Kl and Kll are given by Eq. (3-14).

Since the tensorial model can

represent the above experimental data the others will also.
The critical ang le, Sc' was defined as the angle where the
second spinwave mode in a symmetric film vanishes.

This angle can be

estimated for the various models by solving simultaneously Eq. (3-1),
the magnetization equilibrium conditions and the applicable equation
for the propagation constant k1 = 2n/O (e.g., Eq. (3-17) for the surface l ayer).

This is only approximate for the uniaxial

-52and surface layer models; the absorption amplitude is not zero because of the surface layer and the negative precession components of
the mode in the bulk contribute to the excitation.
The angle of the magnetization, ec, when B = Be, k1 = 2n/D
discussed for the three models below.

For the uniaxial perpendicu-

lar anisotropy, ec is plotted versus Ks in Fig. (3-13).

The range

of values that is reasonable for analyzing experimental data for YIG
films is (-.3

At the lower limit there is a highly

localized surface mode at perpendicular resonance; at the upper limit
there is a highly localized surface mode at parallel resonance.
Within this region of Ks, ec only varies a few degrees.
does not match the observed experimental data.

This behavior

For the tensorial

model ec is given by
(3-23)
Therefore, any variation of ec can be matched by the appropriate choice
of Kl and K , although this would not be physically meaningful unless
11
some understanding of the origins of Kl and K11 were established.
Finally for the surface layer model, the critical angle variation
depends on the assumed properties of the layers; the physics required
to match the observed variation is discussed in greater detail in later
chapters.

-53-

.6

.4

.2

10

20

30

-.2

-.4

-.6

-.8

Fig. (3-13)

The calculated variation of ec with K5 in the
perpendicular uniaxial anisotropy model.

-54-

Chapter 4
Absorption Calculations
4.1

Introduction
In this chapter, the techniques used in calculating the power

absorbed by ferromaqnetic films are presented.

The theory as devel-

oped in Chapter 2 and this chapter has been discussed by many
authors; in particular the magnetics group from Yale University has
been very active in this area (at the head of this group is Dr.
Barker).

The material in this chapter has been repeated because of

simplification, additions, and for completeness.

The simplifications

are apparent only if one is familiar with the previous work; therefore, they are not discussed.

The additions are the approximations

to the boundary conditions discussed in the final section; these approximations are useful because the computer computations are
simplified.

The calculated power absorption data presented for an

asymmetric film with surface layers were obtained using this method;
this method is believed to accurately represent the resonance process.
In the typical resonance experimental situation the magnetic
film is placed in a cavity or strip line at a position of no tangential electric field and a large tangential magnetic field.

Since

power is absorbed by the magnetic sampl e, a smal l tangential electric
field is required at the surface; therefore, the fields inside the
sample chamber are perturbed in order to meet this demand.

The

large tangential magnetic field, however, is little changed by this
perturbation.

Two methods for calculating the power absorbed by the

-55-

film have been used .

Both of these methods have field configurations

around the sample which approximate the experimental situation
described above.
below.

These field configurations are briefly described

In any case, it is implicitly assumed that the perturbation

in field structure is negligibly small and that the differences in
the calculated power absorption are less than experimental error or
resolution.
In the first method, the magnetic field is provided by incident
and anti-incident plane waves.

(Due to the film structure and the

possibility of a transmitted wave, it is not proper to use the term
reflected wave.)

The incident plane waves of amplitude h0 /2 are in

phase, linearly polarized with magnetic field along ax, the perpendicular to the film plane projection of the magnetization.

The solution

requires that the anti-incident plane waves be slightly elliptically
polarized with a small component of the magnetic field perpendicular
to the incident plane wave; this anti-incident wave is out of phase
with the incident wave and its magnitude is such that the amplitude
of the total field along

a is slightly different than h .

In the

second method the incident and anti-incident plane waves are nearly
equal in magnitude and are oppositely elliptically polarized such that
the magnetic fields at the film surfaces are in phase, exactly linearly polarized along aX with amplitude h0 .

In both methods the

resultant electric fields can have both aX and ay components of
arbitrary (small) amplitude and phase as demanded by the magnetic
medium.

-56-

4.2

Power Absorption
The general situation shown in Fig. (4-1) is described as follows:

From the space surrounding the magnetic film there are waves incident
upon both surfaces of the film (hi+' hi-); propagating into the
surrounding space away from the magnetic film are anti-incident waves
(ha~ ha -).

Inside the magnetic film the magnetic field is given by a

superposition of eight terms like Eq. (2-2);
(4-1)
From the second of Maxwell's Equations (Eq. (2-12)) the electric field
inside is given by
(4-2)
where cr' is given in Eq. (2-13).

The time varying magnetization is

related to the magnetic field h(z) by Eq. (2-13).

The ratio,

hyn/hxn = vn' can be obtained from the equation of motion of the
magnetization (Eq. (2-14)) and Eq. (2-13);
i 1T 1

vn = - -w cos e

(4-3)

At z = d/2 the continuity of tan~ential electric and magnetic
fields requires that
-+

hi

e.+ +

-+

+-

+-

hxax + hyay

(4-4a)

++e+a = exax
+ eyay

(4-4b)

+ ha

+ + +
where hx, hy, ex, and ey are the components of h(z) (Eq.(4-l)) and
e(z) (Eq. (4-2)) at z = d/2.

-57-

Thin Magnetic Film
-+ -+
hi , e.1
-+
h , -+
ea

- ic
e= 47ro' az
m,~.= E
'~'

n=l nx n

m =-~ h v Q ei(knz+wt)
e n=l" n n

/cos(e)

surface values
h+ h+ etc.
x' y'
z=-d/2: h~,h;, ex, etc.
z=d/2:

Fig. (4-1)

Schematic representation of the magnetic film and
mathematical fields.

-58By using

simple

relations

between electric and magnetic plane

waves in free space Eq. (4-4) can be written in component form as
h: + h+ = h+
1x
ax
h: + h+ay = h+y
1y

(4-Sa)

h+ + Zoh+ay = ex+
- 0 iy

(4-5c)

zoh+ix - Zoh+ax = ey+

(4-Sd)

(4-5b)

Finally, the following continuity equations at z = d/2 are easily
obtained from Eq. {4-5)
2Z0 hix = Z0 hx + ey

(4-6a)

2Z o h.lY = Z0 hy - e X

(4-6b)

At z = -d/2, a similar procedure to the above gives
(4-7a)
2Z 0 h:1 Y = Z0 hy + e X

(4-7b)

Similar expressions to Eqs. (4-6) and (4-7) can be obtained for the
anti-incident waves at each surface.

These expressions are useful

in obtaining an understanding of the required waves in free space.
From the Poynting theorem, the average power flow per unit
surface area into a region is given by
IT =
~ex fll . n dSdt
TS

f f

(4-8)

surf
where the integral is over the entire bounding surface with inward

unit normal n, and surface area S.
this can be written as

For sinusoidal time variations

-59-

IT = ~ Re [

f ~rr

(ex Ji"*)

· ndS]

(4-9)

surf
For the planar geometry considered here, the power absorbed through
both surfaces per unit area of one surface (or as is commonly called
per unit area of film) is
(4-10)
where -+
e and ,...+
h are the fields at z = 2
and -e- and r:n- those at z = -

c ( + +*
c ( - -*
- -* }
P = Re { Srr
ey hx - ex+ hy+*) + Srr
-ey hx + exhy

( 4-11)

The continuity equations Eqs . (4-6) and (4-7), the continuity of
tangential fields at all interfaces between magnetic material of
different properties, and the magnetic boundary conditions at all
surfaces and interfaces provide enough equations to solve for the
unknowns (h+,
x e+,
x hnx , etc.) provided the incident fields(h~1 h.-)
are specified . Therefore, the power absorbed can be calculated from
. t
Eq. ( 4-11). For
= hi- = 2 e 1w ax the above description constitutes the first method discussed in the introduction. This

fl/

method is easily adapted to the study of transmission of electromagnetic radiation through films, where the second method discussed
below does not contain this flexibility.
When the magnetic fields at the film surfaces are specified
(i.e., method number two in the introduction), the computation or
computer time required to calculate the power absorbed can be
sign1· ficant 1y re d uce d .

If h+ = h

= h0 an d h+Y = h-Y = 0 ' t hen t he

-60-

power absorbed per unit area of film is
ch
P = -8n0 Re ( ey - e-)

(4-12)

The equations necessary for the computation of (e; - e~) come from
the magnetic boundary conditions at all surfaces and interfaces
between magnetic materials of different properties, the continuity
of tangential fields at all interfaces, and the following surface
equations.
ik d/2
h+ = 1 = E
hnx e n
n=l
8 h
-ik d/2
nx e n
hx = 1 = E
n=l
e+iknd/2
h+ = 0 = E
hnx vn
n=l
hy = 0 = E hnx vn e-iknd/2
n=1
ey

ey = - 2Tio '

n=l

knd
knhnx sin 2

(4-13a)
(4-13b)
(4-13c)
(4-13d)
(4-13e)

This set of equations is one more in number than for method one,
but ey+ - ey- can be directly determined .
Due to the factoring of Eq . (2-16) at parallel and perpendicular
resonance orientations, the power absorption calculations at these
orientations are simplified .

At parallel resonance the tangential

fields associated with the six wave vectors from Eq. (2 - 19) and the
two wave vectors from Eq . (2-18) are linearly polarized perpendicular

-61, respectively.
and parallel to M

In terms of the equations this

gives
hnx = 0

n = 7,8

(4-14a)

n =0

n = 1,2,3,4,5,6

(4-14b)

'V

With these conditions, Eqs. (4-13) are reduced to
eiknd/2
h+ = 1 = E
hnx
n-1
eiknd/2
h; = 1 = E hnx
n=l
-c 6E
ey ex = 2na •
knhnx sin knd/2
n=l

(4-15a)
(4-15b)
(4-15c)

Since the summation extends only ton= 6, the parallel resonance
calculation is simplified.

For the ~erpendicular resonance orientation
Eq. (2-16) factors into two quadratics in K2 (Eq.(2-17)). Associated

with each quadratic are four field components with circular polarization; the sense of precession or rotation of this polarization is
positive (negative) for the fields with wave vectors from the
i =1 (i=2) equation.

Using method two, the linearly polarized inputs

(hx+ = 1, hy+ = 0, hx- = 1, hy- = 0) are resolved into two oppositely
polarized circular waves of half magnitude; these polarizations are
completely uncoupled, i.e., the response of the system associated
with one sense of precession is not affected by the other.

In

general, this is true at perpendicular resonance for any magnetic boundary condition that requires isotropic pinning of the magnetization.

-62-

The power absorbed can be calculated for each of the two circular
drive fields individually (see Appendix IV). The power absorbed due
to the negative precession drive is small, slowly varying and can be
neglected when compared to the resonant characteristics of the
positive precession response.
The equations necessary to solve for the power absorbed in the
following four cases are given in Appendix IV.
1.

A film with asymmetric perpendicular uniaxial anisotropies

and the magnetization at a general angle, e.
2.

A film with asymmetric tensorial anisotropies and the

magnetization at a general angle, e.
3.

A film with asymmetric tensorial or perpendicular uniaxial

anisotropies; the magnetization is in the perpendicular resonance
orientation.

The symmetries discussed above and method two are

utilized.
4.

A film with asymmetric surface layers with the magnetization

in the perpendicular resonance orientation.

The symmetries discussed

above and method two are utilized.
In the first two examples the equations for both method one and
two are presented.

-63-

4 .3 Approximate Absorption Calculations
The calculations discussed in the previous section are not too
unreasonable for the anisotropy models where the maximum number of
equations is 9; however, when the asymmetric surface layer problem
was considered it was found to have 24 unknowns.

The surface layer

model also required the roots of three equations like Eq. (2-16) to
be found.

Even the symmetric film calculation at perpendicular

resonance had 9 equations.
order.

An approximation was believed to be in

The calculated power absorption data presented in Chapter 6

for an asymmetric film with surface layers were obtained using this
approximation.

The positive and negative precession spin· wave vectors

were approximately factored from Eq. (2-16).

Secondly, the boundary

condition at the interface between the layers was approximated such
that the negative precession wave vectors were not required.

These

approximations were found to give very good results for symmetri c
films at parallel resonance where they are the least accurate .
The approximation to the secular equation will be presented here
and the approximation to the boundary condition is presented in
Appendi x V.

Results from the calculations are presented for the

perpendicular uniaxial anisotropy and the surface layer model at the
end of th i s section.
Equation (2 -14) written in matrix form is

-64TT2

-H"2'

io'

TTl

me

(4-16)

m

or

.._..
G m=

With £ = 0 the matrix G can be exactly diagonalized by a similarity
transformation U+G U (Kobayashi, 1973).

(1 + IRl2) l/2

(1 + IRI2)1/2
u =

( 4-17)

-R*
(1 + I Rl2) 1/2

(1 + IRI2)1/2

.._..
Upon applying this transformation to G, the following is obtained
All

.._..
AU+ m =

me

Al2
u+

A21

A22

=0

(4-18a)

m

e)

IRI + cos
K4
K2
-~
i2£
i2£
(-K~
All
1 + IRI2

(4-18b)

· e
-1·2 £ R s1n
Al2 = -A21 =
(1 + IRI )

(4-18c)

-65-

As before, the dispersion relation is given by det (A) = 0; however,
if A12 and A21 are negligible, it is approximately given by A =A =o .
11 22
The secular equation approximation (Yelon et ~ 1974) is to neglect
A12 and A21 (Note A12 and A21 are zero for perpendicular resonance) .
The ellipticity and sense of precession of the components of the
magnetization for A11 and A22 are found by examining each eigenvector
independently .

The eigenvectors obviously are
for

and

A11 = 0

(4-19a)

~2 ( ~) for A22 = 0

(4-19b)

where

u• (::)

u+ (:: ) =

~ 1 (~)

(4-19c)

~2( ~ )

(4-19d)

Multiplying the above by U gives

( ::}

~1

( 1/(1 + IRI2) 1/ 2 )

-R*/(1 +I Rl2)

1/

me

:9- = -1/R*
mcp

( 4-20)

and

(:J=

l-12

( R/(1 + I Rl2) 1/2)

2 l/2

1I ( 1 + I R1 >

9.J!. = R
mcp

(4-21)

-66-

Therefore, A11 = 0 gives two spinwaves with negative precession;
like the negative precession spinwaves of perpendicular resonance,
these will contribute little to the resonance phenomena.

The spin-

waves given by A22 = 0, however, have positive spin precession and
are the major contributors to the resonance behavior .

The spinwave

ellipticities here are the same as those associated with the K and
K2 wave vectors in Eq. (3-2).
If the boundary conditions are such that the positive and
negative precession spinwaves are not coupled, then one would proceed
as follows:

1) the linearly polarized input (h + of method two)

would be resolved into two oppositely polarized elliptical waves with
hxp = h;/(1 + IRI ), hxn = IRI h;/(1 + IR! ) and ellipticies (R cos e),

* respectively .
(-cos e/R),

(These can be matched by the film plane

projection of the positive and negative precession spinwave
respectively);

fields

2) The power absorbed for each of these polari za-

tions would be calculated as in the perpendicular resonance case
discussed earlier.
Of the three boundary conditions treated in this thesis only the
tensorial model falls into the class of uncoupled positive and
negative precession spinwaves .

Therefore, further approximations

had to be made to simplify the calculations involving the other two.
The uniaxial anisotropy model is approximately a tensorial model with
(see Appendix V)
(4-22a)

-67-

(4-22b)

By using the procedure outlined above i t was found that the 11 exact 11
spectra and those due only to the positive precession spinwaves
had almost exactly the same characteristics (i.e., mode position, mode
linewidth and relative mode amplitude); but, the power absorbed by
the positive precession mode, P+, was not in good agreement with the
.. exact .. calculation.

It was found, however, that if hxp was changed

to
(4-23)

then even the power absorbed was in very good agreement at all angles
of the magnetization.

A comparison is given in Table {4-1) .

Here

the amplitude, linewidth, and peak positions of P+ is compared with
method two calculations for a symmetric permalloy film; the power
absorbed by the negative precession spinwaves was small and slowly
varying.
An approximate boundary condition for the positive precession
spin waves at the interface of two magnetic layers was deduced from
Eq. (3-17) (See Appendix V)

~2b

((1 + 1Rsl 2 ) {1 + 1Rbl 2 )) 1/ 2 Mb

(1 + ~Rs )

~2s

(4-24a)

-68-

A comparison of the "exact" and approximate, P+, calculations at
parallel resonance (the orientation at which the approximation should
be the least accurate) is given in Table (4-2) for a YIG film.
(Note:

Eq. (4-23) was used for hxp · )

The excellent agreement found

in this case and the uniaxial case presented earlier, lets one use
this approximation with confidence that the calculated data represents
the resonance process.

-69Table (4-1)

p+
"Exact"
Amplitude Amplitude
. 2
(er~/cm ) (erg/em )

"Exact'~

p+

p+

Position
(Oe)

"Exact"
Position l1H*
(Oe)
(Oe)

l1H*
(Oe)

3179.9

3179.9

43.1

43.1

6.884xlo 6 6.884xlo 6
4.503xlo 5 4.503xl0 5
6.005x10 4 6.005x10 4

2878.5

2878.5

21.1

21.1

2122.1

2122.1

18.4

18.4

1 .450x1o 4 1.45xlo4

876.8

876.8

17.6

17.6

1.038xl0 7 1.038xlo 7
3.8llxl05 3.853xl05
4.144xlo 4 4.378xlo 4

2082

2081.9

40.5

40.5

1799.4

1799.3

21.3

21.3

1047.2

1047.0

18.4

18.4

80

1.502xlo 7 1. 502xl0 7

882.4

882.4

38.1

38. l

90

1.513x1o 7 1.513xlo 7

860.9

860.6

37.9

38.0

30

Film Properties
r~

Rho

= 887.4 G
= 14.3 Micro-Ohm-em

d = 2023 A
Alpha = .00457

"' 1 . 143x 10- 6

= Ks 2 = .22 Erg/cm 2
= 9.44 GHz

l.8484xlo 7 inv Oe-sec

* l1H is the inflection point 1inewidth

-70Table (4-2)
Film Properties
Mb = 138 . 1 G
CL

b = CLs

.00105

Ab = As = 3.593xl0- Erg/em

= 9.16 GHz

yb = ys

= 1 . 767xlo 7 inv Oe-sec

Mb/Ms = 1.95

L = 470 A

D = 4230 A
FMR Position (Oe)

Li newi dth ( Oe)

FMR

Position

(P+)

"Exact"

(P+)

2596.6

2595.6

3.9

3.9

2494.0

2494.0

3.9

3.9

2426.6

2426.6

3.9

3.9

Mode NR

Linewi dth

AbsorEtion (er~/cm )

"Exact" (P+)

"Exact"

1.053xlo 7 1 .054xlo 7
2.018xl0 7 2.015xlo 7
1.426xl06 1. 426xlo 6

-71Chapter 5
Surface Layer Properties
5.1

Introduction
It has been asserted earlier that the surface layer model can

be made to match the experimental resonance spectra observed in YIG;
however, the required surface layers must represent a realistic
average of the properties believed to exist in the surface regions.
Some evidence for the existence of surface regions of different properties than the bulk properties is given below.

Etching experiments

on YIG films have shown that the source of surface modes is located
within 100-800~ of the surface.

An example of the mode field position

behavior during an etching experiment on a film with two surface modes
is provided in Fig. (5-l).

The lower surface mode made the transition

to a body mode in the first 200 or 300~ of etching.
mode was concentrated at the film-air interface.

Presumably this

The field position

of the other surface mode was invariant until the film was less than
1500~ thick.

The remarkable behavior below this thickness indicates

that there was an interface surface region of considerable thickness
with properties different from the bulk.

This second mode only ap-

peared if the film had been annealed in this case at 1200°C for 6
hours; this in itself is suggestive that a diffusion may occur between
substrate and film.
Surface modes were observed at perpendicular resonance by Henry
et ~ (1973) after overcoating with Si0 2 or argon implantation.
It is physically plausible that these processes would give a region

(])

_J

LL

.......

<.9

::;E

1.15 1.1

3000

,..

1.0

0.9
0.7

0.6

APPROXIMATE FILM THICKNESS (fLm)
0.8

YJG/GdGaG
Parallel Resonance
Room Temperature
v =9.16 GHz

• • • • • • • • ••• • • •

••

·"

-'\

••

• •••

• • • • • • ••• • • •
2sooJ!!
• •• • • •
•••
2300

Fig . (5-l) Effect of etching on the resonant-field position of the hig h-field spin wave
modes for a YIG film having two surface modes . After Yu et . al. (1975)

-73at the air-film interface with magnetic properties different than
the bulk.
In the following section some of the properties of bulk garnet
materials are described.

The final section presents some of the

properties to be ascribed to the two surface layers of a YIG film;
other properties will be presented in the next chapter as needed to
explain the experimental data.
5.2

Properties of Garnet Materials
The simplest chemical formula for garnet materials is R3P2Q3o12 .

The basic crystal structure is cubic with eight formula units per
unit cell, i.e., 160 atoms 96 of which are oxygen.

Each oxygen ion

lies at a vertex that is common to four polyhedra of oxygen, one
octahedron, one tetrahedron, and two dodecahedra, as indicated in
Fig. (5-2).

The orientations of the polyhedra vary throughout the

unit cell, although the type of symmetry for each is retained.
cations occupy the interstitial sites.

The

The cations denoted by P and

Q occupy the octahedral or [a] sites and the tetrahedial or [d] sites,
respectively.

The other metal ions, R, are surrounded by eight

oxygen ions located at the corners of a skewed cube, or, as it is
often called, a dodecahedron, [c] sites.
In the magnetic garnets, R is typically a trivalent combination
of rare-earth and yttrium ions; P and Q are trivalent combinations of
Fe +3 , Ga +3 and Al +3 .

An example is (Gd0 _7v1 _55 Yb 0 _75 ) Ga 0 _9Fe 4 . 1o12 .
In a magnetized state the net moment of the
P3+ ions in the (a)

sites and the net moment of the R3+ ions are in one direction and

-74-

F~)+ (a) AT (OO"fl

r~H(d)AT(otil

yH AT

tft rl AND AT (0 ~i)

oot• COMMON TO POI.YHE:OAA AT (ij, z.,

Fig (5-2)

j + It)

Structure of yttrium iron garnet (After Gilleo, et. ·•al.,
1958)

-75-

Q3+ ions in the (d) sites are in the opposite direction.
The alignment is due to superexchange interactions of the Q3+ ions
that of the

in (d) sites with those in (a) and (c) sites via
The moment of an Fe 3+ ion is 5 Bohr magnetons .

o2- intermediaries.

In rare earth substituted YIG the variation of the net moment of
the Fe 3+ system with temperature is similar to that of ferromagnetic
however,
Th e momen t con t r1.b ut·10n due t o magne t•1c R3+ 1ons,

meta 1s.

is quite different as shown in Fig. (5-3) for Gd 3Fe 5o12 .

The net

moment at absolute zero (that for R ions and (a) site Fe 3+ less that
for (d) site Fe 3+) can be fairly large. As the temperature increases
for garnets like Gd 3Fe 5o12 the net moment decreases to zero at a
temperature called the compensation temperature. Above the compensation temperature the moment for Fe 3+ in the (a) sites dominates and
does so up to the Neel temperature where the moment again drops
to zero.
Fig.(5-4)shows the net magnetization of several garnets as a
function of temperature. Introduction of Ga3+ and Al 3+ for Fe3+
is known to reduce the moment and the Neel temperature of the material.
"The rules for ionic site preference in the garnets may be
summarized as follows:
1.

The octahedral and tetrahedral sites appear to prefer exclu-

sively ions with spherical or pseudospherical electronic configuration.

The dodecahedral sites are not selective in this regard.

2.

Site preferences depend on relative ionic sizes: (a) If

an ion has a spherical electronic configuration in both octahedral
and tetrahedral crystal fields, the larger the ion, the greater

-76-

-"'

.....

41

:It

600

The magnetization of the GdiG sublattices per formula
un;t as a function of temperature ( After R. Pauthenet)

Fig. (5-3)

.<:

e-

..

:It

700

Fig. (5-4)

Experimental values of the spontaneous magnetization of
various simple garnets as a function of temperature.
The formula unit is P3Fe 5o12 where the P is indicated
for each curve ( After R. Pauthenet)

-77will be the preference for the octahedral site.

The dodecahedral

sites are usually occupied by the largest metal ions present.
(b) The substitution of one ion for another in a particular garnet
is limited by the relative sizes of all of the ions involved."
(Geller 1970)
The garnets are magnetostrictive and the dominant crystalline
anisotropy of the rare-earth garnets is cubic with easy axis along
the body diagonals; however, a number of techniques are known to add
large anisotropies to these films.

An example of these large

anisotropies is the growth induced anisotropy in bubble related
garnets.

This anisotropy results from site ordering of the two or more

rare-earth ions incorporated in the particular garnet; the magnitude of
this anisotropy is in part determined by the size differences of the R
ions involved.

(Rosencwaig et !l (1971), Gyorgy et !l (1973).

Another anisotropy observed at low temperatures (i.e., less than
100°K) is associated with Fe 2+ ions in octahedral sites. In the
situations where this anisotropyhas been most studied, the Fe 2+ was
generated by introducing Si 4+ into the lattice. The Si 4+ is believed
to be in the tetrahedral sites because of the small size and preference for a coordination number of four. The Fe 2+ is believed to
be in the octahedral sites because of the larger size and a preference
for a coordination number of six. The Fe 2+ ion is about the same size
as the sc 3+ ion which prefers octahedral sites exclusively in the
garnets .

There are four types of octahedral sites, distinguished

by a different local symmetry and characterized by the local trigonal
axis which lies in one of the four directions.

Electrons

-78-

(Fe2+ ions) in sites whose trigonal axis is nearest to the magnetization direction have a slightly lower energy than those in other
sites.

At low temperatures the excess electrons become trapped in

those sites and produce anisotropy.

The magneto-optical effects of

this anisotropy have been thoroughly studied.
with white

Irradiation

light leads to a redistribution of electrons that

essentially destroys the anisotropy {Gyorgy et ~ {1970)).
5.3 Surface Layers in Garnet Films
It was schematically pointed out in Fig. {3-7) that the properties
to be ascribed to the two surface layers are different.
erties of a particular layer depend

The prop-

upon the history associated

with the film; therefore, it is impossible for universal properties
to be determined; however, some properties that could be easily
associated with a particular layer are given below.

In general,

any magnetic material constant may be different; this includes the
magnetization, M, the gyromagnetic ratio, y , the exchange constant,
A , the damping parameter, a , and the anisotropies, Ka.
A and a

Of these

are expected to make little difference for very thin layers

but may play a role if the layers become thicker.

Variations of M,

Ka, and y however, have considerable effect on the field position of
the s pinwave modes.

Significant variation in

near compensation in ferrimagnets,

is unlikely except

therefore this effect

should only appear as a sensitive function of temperature. Variations
in

M or Ka

have effects that are not easily separated.

-795.3. 1 Film-Air Interface
The physical mechanism producing the surface anisotropy or
spin pinning at this surface is not clear; however. through etching
experiments. it has been isolated to a thin surface region. In any
orientation a mode localized at a surface will exist if the surface
layer tends to resonate at a larger field.

Two possible origins of

this are: (a) a layer with different anisotropy energies than the
bulk or (b) a layer of different magnetization.

In the latter case.

the surface mode appears in perpendicular resonance ( a = 0°) if the
s urface magnetization is increased. and in parallel resonance ( a = 90° )
if the surface magnetization is decreased.
One possible source of a larger surface magnetization is the
existence of oxygen vacancies in the surface region causing Fe 2+ ions
in the octahedral sites.

A reduction of the total moment in the

octahedral sites would increase the total magnetization. Further.
the Fe 2+ may have the effect of producing significant anisotropies
at lower temperatures. and these anisotropies could be sensitive to
irradiation by light.
5 . 3.2

Film-substrate Interface
This surface region is believed to be of variable chemical

composition v3 _YGdYFe 5 _xGaxo 12 where the thickness of the layer.
and y and x are dependent on prior annealing treatment.

The behavior

of the 6d moment in the partially substituted YIG should be little
different than that of stoichiometric GdiG shown in Fig. (6-2)
except for a reduced value of MGd.

If the Gd magnetization is suf-

-80-

ficiently high, the strong temperature dependence of the gadolinium
magnetization will produce a compensation temperature where the net
magnetization vanishes.

Further, it is expected that at temperatures

below compensation the resultant magnetization in the surface layer
will be anti-parallel to the applied field since the principal exchange coupling is through the iron sublattices and because the
surface layer is believed thin compared to the width of a typical
domain wall.

The gyromagnetic ratio for a ferrimagnet varies with

the sublattice magnetization in accordance with an effective g
factor (Wangsness 0953, 1954, 1956))

_ MFe - MGd
eff - MFe MGd
gFe

gGd

The temperature dependence of the critical angle associated with
this interface can be explained if gFe > gGd and if the surface layer
has a compensation temperature near ll0°K.

From effective g measure-

ments in GdlG, it was deduced by Calhoun et ~ (1958) that gGd is
slightly lower than gFe·

By varying the frequency and measuring the

perpendicular FMR field, gFe was determined to be 2.008±.002.

Based

on these considerations, the values of gFe and gGd were chosen to be
2.008 and 2.000, respectively.

In order to have a compensation

temperature near 110°K, it was estimated from experimental and
molecular field analysis data of Figures (5-3) and (5-4) that the
room temperature magnetizations in the surface layer should be in

-81-

the range .4 MFe < MGd < .3 MFe·

In the analysis, the room tempera-

ture ratio of MGd to MFe was varied within the above limits with MFe
chosen 15-25% lower than the bulk value due to the possibility of
gallium substitution.
Any atomic substitution in the surface layer may also give rise
to an in-plane strain since the layer is epitaxial with thick film
and substrate .

Through magnetostrictive interactions this can give

rise to a substantial perpendicular uniaxial anisotropy such as is
well known in bubble material garnets.

This anisotropy is to a large

extent indistinguishable from a change in magnetic moment. Therefore
in what follows a change in 4nM in the surface layers could be in
part a change in this anisotropy.

-82-

Chapter 6
Comparison of Experimental
and Calculated Data
6.1

Introduction
The previous chapters have introduced experimental phenomena

(Chapter 1}, theory (Chapters 2-4), garnet material properties (Chapter 5).

This chapter will address the thesis that observed surface

phenomena in thin YIG films can be explained by surface layers with
magnetic properties different from the bulk.

Calculations utilizing

the other models of surface pinning are also provided where instructive.
Experimental data from four films are compared with calculated
spinwave spectra.

Comparison of calculations with experimental data

from other than these four films are qualitative.

The material con-

stants for these four films are given in Table (6-1}; for convenience

the samples have been designated CIT 1, CIT 2, OSU 1, OSU 2.

The two

samples measured in this laboratory were cut from a single garnet film
grown by CVO process on a [111] oriented wafer.
was annealed in dry 02 for 6 hours at 900°C.

One sample (CIT 2)

The unannealed sample

(CIT 1) has a surface mode at perpendicular resonance while the
sample CIT 2 has one at parallel resonance.

The surface mode of the

CIT 1 sample is believed localized at the air-film surface since it
had been overcoated after gr~~th; the surface mode of the CIT 2 sample
was shown to be localized at the air-film interface by etching away
the outer surface.

The data for the other two films are taken from a

Ls
Mb/Ms

Ls
Mb/Ms
yb/ys
a.b/ a.s

yb

4nMb

(A)

(A)

(A)
(Oe)
( sec -1 0e-1)

Sample
Orientation
Total Thickness (~)
Annealing Temp. (°C)
Annealing Time (Hr)
Frequency
(GHz)
(erg/em)

20
1.1

200
.821
.3

4450
1701.2
1.767xlo 7

470
1.95

SUBSTRATE-FILM SURFACE LAYER
200
1.1

4200
1735
1.767xlo 7

BULK CONSTANTS
3800
1701.2
1.767xlo 7

[100]
.471
1200
9.16
3.593xl0- 7

[100]
.467
1000
9.16
3.593xlo- 7

800
1.4

290
1.45

3620
1735
1 .767xl0 7

osu 2

osu 1

CIT 2
[111 J
.47
900
5.966
3.593xlo- 7

AIR-FILM SURFACE LAYER
700
1.38
.54

5.966
3.593xlo- 7

CIT 1
[111 J
.467

TABLE (6-1)

co

(A)
(ergs/cm 2)
(ergs/cm 2)
(ergs/cm 2)
(ergs/cm 2)
4680
4712
-.0488
-.061
.0019
.0019

4680
-.049
.023
.0019
-.001

4280
.114
- .048
4270
4441
.1085
.14

UNIAXIAL ANISOTORPY ~10DEL
4300
4489
.1
. 161
.019
.019

.9977
1400
510

TENSORIAL ANISOTROPY MODEL
4300
.1
-.040
.019
- .01

1545
.9

3850
4391
.112
.195
.065
.075

3860
.118
-. 0505
.067
-.0265

.999
1500
261

@ not determined

fit to the experimental data (see the text)

* calculated from the surface layer model for comparison with other value obtained from a best

(A)
(A)
D calculated*
(ergs/cm2)
Ksl
Ksl calculated *
(ergs/cm 2)
Ks2
Ks 2 calculated*

Kl, 1
KII ,1
Kl,2
KII ,2

ab/as

yb/ys
4nMFe at 300°K (Oe)
4n~1Gd at 300°K (Oe)

TABLE (6-1) CONT.

co

-85-

paper by Yu, Tuck, and Wigen (1975).

Both films were cut from a

single garnet film grown by CVD on a [100] oriented wafer.

One of

the films (OSU 1) was annealed at 1000°C for 6 hours and has a single
surface mode at parallel resonance.

The other film (OSU 2) was an-

nealed at 1200°C for 6 hours and has two parallel resonance surface
modes.

In the latter case, two surface modes indicate that both sur-

faces of the film have been altered.

This was confirmed by an etching

experiment (see Fig. (5-l)).
Typical experimental data consist of a set of spinwave spectra
obtained at different angles of applied field.

Field locations of

the three highest field modes are shown for a case in Fig. (6-1). The
most important feature in this figure is the separation of the modes
from the (calculated) uniform mode location.

Because of the large

variation in the uniform mode location, there is a great loss of
detail unless this separation is plotted instead of the actual mode
location.

All subsequent figures will show only the separations from

the calculated uniform mode location.

Comparison of experimental data

with the calculated uniform mode locat ion has one inherent difficulty;
Fig. (6-1) shows that a small error in alignment will affect the uniform mode position negligibly at perpendicular and parallel resonance,
but a significant error may res ult at other ang les (e.g., a .1 ° error
in alignment changes the uniform field by about 5 Oe at

B = 30°). This

may be the source of some of the difference between the calculated and
experimental data presented later.

-86-

380Qr---~--~--~----~--~--~--~----~~

2800

(1)

c:
......

"'C

(1)

.-c..
c..
<(

1800

1300 0

Fig. (6-1)

10

20

40
50
30
MAGNET ANGLE

60

70

80

90

Angle dependence of the resonant-field position of the
highest three field modes of sample CIT2. The solid
curve i s the calculated field position for the
uniform precession mode.

-87-

In attempting to duplicate the experimental spectra, the surface layer thickness, magnetization and in the case of substrate-film
layer the ratio of MGd to MFe were varied to give a best fit to the
mode locations at all angles.

The material constants that were de-

termined for the four films are given in Table (6-1).

Film thickness

for YIG is usually measured by an optical interference method.

In

the thickness range of .5 ~m, this appears to give an accuracy only
of the order of 10%.

Since the spinwave spectra are very sensitive

to thickness, it was necessary to vary the film thickness from the
optically measured value.
The orientation of the magnetization depends on the orientation
of the applied field with respect to the crystallographic axes.

For

simplicity of calculation, the experimental data were taken with the
applied field in the orientation described below.

For the [100]

oriented films the applied field was in a (100) plane at an angle
from the film normal.

13

For the [111] oriented films the applied field

was at an angle 13 from the film normal in a plane defined by the
normal and a line in the film plane 30° from the [TT2] axis.
6.2

Comparison of the Angular Spinwave Mode Field Position Data
Surface layers can force the bulk material to support surface or

body modes

as the highest field mode depending on whether the surface

layers tend to resonate at a higher or lower field than the bulk.
A layer with a reduced magnetization will resonate at a higher field
at the parallel orientation and a lower field at the perpendicular.
Therefore, a film with a reduced magnetization layer will have a
surface mode at parallel and not at perpendicular.

A layer with

increased magnetization produces the opposite effect.

-88Figures (6-2) and (6-3) show the angular dependence of the
resonance fields for the observed and calculated spinwave spectra
in the samples CIT 1 and CIT 2, respectively.

For both films, the

best fit to the experimental data was obtained by using two surface
layers, and a total film thickness of about .47 ~m.

The calculated

and experimental data for the two [100] oriented films (OSU 1 and
OSU 2) are shown in Figures (6-4) and

(6-5).

The best fit for

sample OSU 1 was obtained by using a single surface layer.
surfaces were obviously required for sample OSU 2.

Two

The best fit to

the data for both films was obtained using a total thickness of about
.47 ~m.

The thickness reported by Yu et al (1975) was .56 ~m; this

reported thickness is clearly inconsistent with the experimental
perpendicular resonance mode spacings and must be in error.

The above

mode position calculations utilized Eq. (3-17) for the sample with
one free surface and the 8 x 8 determinant in Appendix II for the
films with 2 layers.

The respective g•s and magnetization equilib-

rium relations for the [100] and [111] oriented films are given in
Appendix I.
One important observation can be made from the layer thickness
data given in Table (6-l).The total thickness required to match the
mode position data in an annealed film is slightly larger than that
required for an unannealed film or film annealed at a lower temperature.

The effect, however, is small.

It is instructive to compare the above with the results from the
tensorial and uniaxial surface anisotropy models.

The best fit to

-89-

+ CALCULATE~ ~ATA

0 EXPERIMENTAL ~ATA

8+ 'l
++++++It+

W~-8~8~·-8_·-------,~~&-~--~~~_w~~~

El

li
:J

El

El

+a-

MAGNET ANGLE
Fig. (6-2)

The angle dependence of the magnetic field separation
of the observed and calculated positions of the spinwave modes, HA, from the ca lculated position of the
uniform mode, flU, for sampl e CIT 1.

-90-

:ant

.a...++ +

,_.f. '&:.1

111+ftTB

+ CALaJLRTEI> I>ATR

a +

D EXPERIMENTAL I>ATA
a +
a +

a +
a a+
a a +a

/\

W i +B- ..at- -1=1+ a + +

a +

a +
I I +ril A- .fl + ~ iS+
It + a a r/e 1;, r:t"'r.t
[[

I"
:J

...

-Jm

MAGNET ANGLE
Fig . (6-3)

The angle dependence of the magnetic field separation of
the observed and calculated positions of the spinwave
modes, HA, from the calculated positions of the uniform
mode, HU, for sample CIT 2.

-91-

+ CALCULATED DATA
+at- tJ

++ + +
El

El

El

D EXPERIMENTAL DATA

El

El

ft

/\

[[
I~

+_ + + +
I " +at- -e +at- ±
""'"~Bee
:J

.&.

El

+EJ

-.1 (21,

MR5NET RN5LE
Fig. (6-4)

< DE5 > 9121

The angle dependence of the magnetic field separation
of the observed and calculated positions of the
spinwave modes, HA, from the calculated position of
the uniform mode,HU, for sample OSU 1.

-92 -

+ CALCULATED DATA

D EXPERIMENTAL DATA

/\

w ~ +at. fJ + Elt- ~ + +

[[

B B

B B B B

a +
~ +at- t +a+ + + +
a +
a + +

+B ...
l rilq--------------M-ra.-a~.~~+~-~-8--------------~
.&.

....J

r:l+

a+ a '

MAGNET ANGLE

Fig. (6-5)

The angle dependence of the magnetic field separation
of the observed and calcu lated positions of the spinwave modes, HA, from the calculated position of the
uniform mode, HU, for sample OSU 2.

-93-

the experimental data from sample OSU 1 is shown for all models in
Fig. (6-6).

A first conclusion would be that nothing has been proven

since all the models can be made to predict the same behavior;
however, closer observation of Table (6-1) shows the following contradictory result in the cases of the tensorial and uniaxial models.
The thickness required for two films with different annealing histories
but initially from the same wafer are significantly different.

Fur-

ther, the thickness required to match the mode data in an annealed
film is smaller (not larger) than that required for an unannealed
film or film annealed at a lower temperature.

Since this is such

good evidence that the pinning cannot be due to a surface interaction
of the type postulated, it is considered in greater detail below.
Basically it is to be shown that the mode position data for samples
OSU 1 and OSU 2 cannot be matched with reasonable accuracy if OSU 2
is required to be of equal or greater thickness than OSU 1.

The

mode spacings at perpendicular resonance simply will not allow it.
Similar arguments can be made for the two CIT films.

The perpen-

dicular uniaxial model will be used for this discussion; however,
since Kl and Kll are related via the critical angle similar statements can be made for the tensorial model.

For sample OSU 2 the two

surface energies were chosen to match the field position of the two
parallel resonance surface modes; this match actually has only a
slight dependence upon the thickness of the film.

The film thickness

was then chosen to match the separation of the perpendicular
resonance modes.

Is it possible to match the OSU 1 data with this

-94-

T.

h 51K 511< 5 K 5

5 T K

Tl

01 5TK 51l< 5TK 5TK 5T

[[

5 T K

I v

:J

STK STK STK STK 5 TK

UNIAXIAL KS
SURFACE LAYER
TENSORIAL KS

T K

Tl

TJ

< DEG >

9~

r:;T

5 T
-J~I1

MAGNET ANGLE
Fig. (6-6)

A comparison of the calculated angle dependence of the
magnetic field separation of spinwave modes from the
uniform mode for the surface layer, uniaxial
anisotropy, and the tensorial anisotropy models for
sample OSU 1.

-95-

thickness?

For sample OSU 1 the value of Ksl at one surface was de-

termined by matching the position of the surface mode at parallel
resonance (again this is nearly independent of film thickness). Using
this value and

Ks 2= 0 at the other surface, and the above thickness,

the separation of the perpendicular resonance modes from the uniform
mode is 7.4, 66.6 and 186.8 Oe.

Note that these are not near the

experimental values of 9, 56 and 154.5 Oe. If Ks 2 is greater than
zero the separation at the higher order modes is greater than above.
If Ks 2 less than zero the higher order mode separations can be reduced but a perpendicular resonance surface mode is produced; this
was not experimentally observed.

The only way to obtain complete

agreement is to increase the thickness.

Conversely, if the film with

two surface modes had the thickness which gives a good fit for OSU 1,
then the mode spacings for OSU 2 at perpendicular resonance are 19.6,
80.3, 186.0, which is not in agreement with the experimental values
of 25 . 1, 100. 1, 223. 1.

Considering the accuracy of the experimental

measurements, these differences are very large.
The above suggests that the tensorial and uniaxial models can
at best represent some sort of averaging of the surface layer properties; this was initially proposed by Bajorek and Wilts (1971).

In

section (3.3.3) it was pointed out that the surface layer and uniaxial
anisotropy have similar properties if Ks is determined from Eq. (3-19)
and the film thickness, d, is determined by requiring the total magnetization in the two models to be the same.

Table (6-1) gives a com-

pari son of this Ks and thickness with the Ks and thickness required to

-96-

match the experimental data.

Note that the agreement is good for

the films with thin layers while for the films with thicker layers
the agreement is not necessarily good at all.
6.3

Comparison of Spinwave Mode Intensity and Linewidth Data
Experimentally it is observed that the linewidth of the surface

modes is typically wider than the other resonance modes.

For sample

CIT l the surface mode linewidth is larger by as much as a factor of
two.

Using the theory presented in Chapter 4 the observed spinwave

mode intensity and linewidth variation can be explained if the damping parameter, as, of the s urface layers is assumed larger than the
damping parameter in the bulk of the material.
from earlier work support this assumption.

Two experimental facts

First, the resonance line-

width of rare-earth substituted garnets ("bubble materials") can be
many times (20-100) larger than that observed in good YIG films.
Secondly, the resonance linewidth of films irradiated with He 4 ions
(Stakelon et ~ (1975)) is wider than the linewidth of non-irradiated
films.

Therefore, disordering of the lattice (ion implantation) and

impurity substitution ("bubble materials") both apparently increase
the losses.

Si nce the surface layers are believed caused by either a

diffusion (impurity substitution) process or by l attice disordering
(ion implantation) the assumption that ab/as < 1 is plausible.
For sample CIT 1 the ratio ab/as = .3 was required to match the
linewidth variation observed at perpendicular resonance.

With this

ratio the theory also gives reasonable quantitative agreement (Table
6-2) for the intensity and linewidth variation in the two observed
modes at all angles where the mode position is accurately matched
(see Fig. 6-2).

If the experimental and calculated data are compared

-97based upon mode separation from the uniform precession mode, then
there is reasonable quantitative agreement for all angles.

For

comparison, calculated data using the uniaxial model are also provided
in Table (6-2).

Note, this model does not qualitatively match the

experimental data and cannot unless an additional mechanism is postulated at the surfaces; the same comments are true of the tensorial
model.
For the annealed [111] oriented film (sample CIT 2) even better
results were obtained as shown in Table (6-3).

At angles where the

calculated and experimental field positions match (Fig. (6-3)), the
linewidth and intensity data (calculated and experimental) are again

if ab/as = .54 in the 700 A layer and ab/as = .9

in good agreement

in the 200 A layer.
for comparison.

Data from the uniaxial model are also provided

The uniaxial model shows no difference in linewidth

between the surface and body modes and the mode intensities are in
poor agreement with the experimental results.

Note that at perpendic-

ular resonance orientation the second mode is smaller than the third
for both models; this is expected since the second mode is quasiantisymmetric and should be smaller than the quasi-symmetric third
mode.

For this sample (CIT 2), absorption derivative curves were

shown earlier in Fig. (1-2) for eight angular orientations.
6.4

Comparison of Temperature Dependence Data
The temperature dependence of the critical angle and the parallel

resonance spectrum have been reported by Yu et ~ (1975).

Measurements

were made with films that showed both one and two parallel resonance
surface modes at room temperature; these cases will be discussed

-98TABLE (6-2)
For film properties see Table {6-1) sample CIT 1.
All amplitudes given below are normalized to 100% for the largest
amplitude mode.

&H

Experimental

% Amp

% Amp

6H

6H

% Amp

11 . 5 5.7

25.6 100.

5.7

5.7

50.2

100.

11.6 5.7

15.4

100.

10

11.9

6.2

31.. 8 100.

6.1

6 .1 62.1

100 .

11.1 5.8

22.0

100.

15

8.1

5.8 90.9

100.

20

12.0

7.4

74.6 100.

7.1

7.0

100.

71.9

6.9

6.9

100.

11.

30

9.2

7.7

100. 7.4

7.5

6.9

100.

6.7

7.0

100.

40

8.0

100 . . 03

7.2

100.

7.5

100.

50

7. 1 7.5 100. . 31

6.7

100.

.3

7.2

7.2

100.

.7

60

6.5

6.5

100. 2.5

6.3

100.

1.1

7.0 6 . 9 100.

1.3

70

6.1

6.3

100. 3.0

5.9

5.3

100.

2.7

6.3

6.3

100.

1.7

80

5.9

6 . 2 100 . 3.3

5.7

5.2

100.

3.0

5.4

5 .4 100.

2.1

90

5.8 6.2

100. 3.8

5.6

5.2

100.

3.0

5.4

5.4

2.1

@ not determined

Uniaxial Anisotropy

Surface Layer

Angle

not observed

100.

58.6 100. 8.2
52.2 100. 8.8
50,5 100. 9.0

6.4

9. 0 7.1

9.1

8. 9 6.1

8.8 5,9 6.1

8.7 5.8 6.0

50.

60

70

80

90

77. 1 100 . 7 . 1

59.4 3.5

8.9 5.8 6.6 45.0 100.

5.9 5.9 5.9 94.2 100. 8.8

5.6 5.6 5.6 79.5 100. l 0.

5.7 5.7 5.7 82.3 l 00. 9.5

7.6

7.6

8.8 5.8 5.8 42.0 100. 8.7

8.8 5.8 6.0 42.0 100. 9.1

8.3 6.6 6.6 62.3 100.

78.4 6.0

6 .l

6.2 6.2 100.

91.7 5.5

8.0 6.9 6.6 100.

100.

3.0 2.0

1.0 2.0

24.0 1.4

100.

100.

100.

100.

6.9 6.9

6.9 a

1.0 3.0

34.3 2.3

7. 1 100. 1.0 .22

llH

7.3 7.6

5.9

5.7

Experimental
%Amp

6.6 6.6 6.7 100.

1.6 .44

2.9 2.5

For film properties see Table (6-1) sample CIT 2
All amplitudes are normalized to 100% for the largest amplitude mode

a not observed

@ not determined

6.5 6.8

7.4 100.

.04

0.

8.4 100.

8.1

40

7. 1

7.4 6.9 7.5 100.

1.2

4.1

7.6 8.1 9.0 100.

30

7.0 6.9 7.2 100.

7. 1 7.3 8.0 100. 4. 9 4.5

100. 2.3 4.0

20

6.0 6.1

6.1

6.1 6.2 6.5 100.

10

2.3 5.9

Uniaxial Model
llH
% Amp
5.7 5.6 5.6 100. 2.0 4.5

Angle

Surface Layer Model
llH
% Amp
5.7 5.7 5.9 100. 2.9 6.3

TABLE (6-3)

1.0
1.0

-100individually below.

The sample (OSU 3) with one parallel resonance

surface mode was [100] oriented, . 37 ~m thick, and annealed at 1000°
for 6 hours; this is not one of the films listed in Table (6-1). Upon
lowering the temperature below 300 °K, the parallel resonance surface
mode increased in intensity while the body modes decreased;

at a

critical temperature the once surface mode presumably became a uniform precession mode and the other body modes were not excited.

The

position of the critical angle was observed to shift toward the parallel orientation (Fig. (6-9)) such that at the above critical temperature
the critical angle was in the plane of the film.

Below the critical

temperature, Yu observed no critical angle or surface mode (i.e., only
body modes were observed).
It has been pointed out in Chapter 3 that the uniaxial model is
incapable of explaining these experimental results.

The tensorial

model can be made to match almost any variation but physical explanation of the variation in Kll and Kl is not convincing.

The degree to

which the surface layer model predicts the above behavior is explored
below.

For the following reasons the temperature dependence calculation

was made using the material constants associated with OSU 1:
1) Only the temperature dependence data were given for sample
OSU 3, so that accurate material constants are not known.
2) The computer analysis showed that the temperature variation
of the critical angle depends almost entirely on the magnetic characteri stics of the film-substrate s urface layer so
that the differen ce in total thickness i s unimportant .

-1013) The two samples had identical annealing histories: there-

fore, it is reasonable that the surface properties should
be approximately the same.
4) The detailed temperature dependence of the interface layer
magnetization can only be estimated in an approximate way.
Using the properties for the substrate-film layer given in Chapter 5
the temperature dependence was calculated roughly by holding MFe constant and increasing MGd linearly with decreasing temperature.

In

view of the largely qualitative nature of the comparison sought, a
more accurate treatment of the temperature variation of MGd and MFe
was not warranted.
Figures (6-7) and (6-8) show the calculated temperature dependence of the two highest field modes at parallel and perpendicular
resonance; Fig. (6-9) shows the calculated temperature dependence of
the critical angle.

Note that the calculated and experimental data are

in qualitative agreement down to the critical temperature where the
critical angle is observed in the film plane, and the uniform precession mode is excited at parallel resonance.

However, at a lower

temperature or higher MGd the model predicts a phenomenon that was not
observed by Yu, that is, a low temperature perpendicular resonance surface mode and associated critical angle (below 100 ° in Figs. (6-9) and
(6-8)).

It was speculated that this disagreement in experimental and

predicted behavior was due to over-simplification of the model.

In

any real system, diffusion will not produce a uniform layer but rather
an inhomogeneous region with a compensation layer that moves through

K>

SURFACE MODES

temperature

BODY MODES

~critical

TEMPERRTURE

::a~m

..

Fig. (6-7) The calculated te~peratur2 dependerce of the magnetic field separation of
the positions of two parallel resonance spinwave modes,HA, from th~ position
of the uniform mode,HU, for sample OSU 1.

-lc;tl2!

~t

[[

7S:

__.

(i)

a .,

a +

+ +

SURF"'RCE MODES

60DY MODES

TEMPERRTURE

+++

K>

32Sii!l

Fig. (6-8) The calculated temperature dependence (+) of the magnetic field separation
of the positions of two perpendicular resonance spinwave modes, HA, from
the position of the uniform mode, HU, for sample OSU 1. Rough positions
of the surface mode observed with sample CIT 3 are also given (e).

Fl

-t2s:

:::J

a:

7S:

......

Fig. (6- 9)

121

r:t:

r--

v .1.+

a:

_j

a:

l.!J

_j

9121

K>

::U211i!1

with the experimental data for sample OSU 3 (see text for properties of OSU 3)

The calculated temperature dependence of the critical angle for sample OSU 1,

SURF'RCE MODE

BODY MODES ONLY

TEMPERATURE

SURF'RCE MODE +

__,

-105it as the temperature changes.

In order to try to understand the

effects of nonuni fonnity, the interface region was represented by
three adjacent surface layers of different thicknesses and different
properties.

Here the results were not entirely straightforward. Some

geometries produced results similar to the above and others predicted
completely different behaviors; however, in all cases, there was a low
temperature perpendicular resonance surface mode.

A final mathemati-

cal attempt involved integration of the equations of motion through
the thickness of an inhomogeneous film at perpendicular resonance;
this also predicted a low temperature perpendicular resonance surface
mode.
In view of these results one therefore would expect a low
temperature perpendicular resonance surface mode if the above assumptions are valid.

Experimentally Yu and Wigen did not see such a mode;

sample CIT 2 was carefully examined and showed no such mode.

However,

the expected mode was observed by Ramer and Wigen on a narrow linewidth, [111], LPE film annealed in a dry 02 atmosphere for 6 hours
(sample CIT 3).

For this sample, the perpendicular resonance absorp-

tion derivative curves at six temperatures between 90° and 50°K are
shown in Fig. (6-10); note the clear indication of the surface mode
below 80°K.

It was confirmed by etching away the outer surface that

this mode was associated with the film substrate interface.

In spite

of this apparent agreement, an inconsistency between the surface layer
model and the data was noted.

This is shown in Fig. (6-8) where

the observed surface mode resonant field for sample CIT 3 is roughly
plotted versus temperature for comparison with the predicted behavior

-106-

surface
mode

Fig. (6-10)

increasing field ~

Oerivative absorption curves at six temperatures taken at
perpendicular resonance for sample CIT3. The curves
show the formation of the perpendicular resonance
surface mode and that it has almost vanished at

T=50°K.

-107for sample OSU 1.

The experimentally observed mode does not con-

tinue to shift in field position as predicted by the model. Further
from Fig. (6-10) it can be seen that with decreasing temperature the
mode decreases in intensity; it was not detected at temperatures below
43°K.

This rapid decrease in intensity would be expected if the mode

continued to move away from the uniform mode as predicted by the
model, but it would not be expected if the mode remained roughly
stationary as indicated by the experimental data.
If the surface layer model is to represent the experimental data
at low temperatures, some other cause must be found to account for
the behavior observed.

There is one mechanism at low temperatures

that has not yet been considered.

In the analysis the net magneti-

zation on the surface side of the compensated region was assumed
aligned anti-parallel to the magnetization in the bulk of the film,
this alignment being due to exchange interaction between neighboring
Fe sites.

Since exchange is not the only torque acting on the

magnetization, complete alignment may not be achieved and some sort
of quasi-domain wall may be generated; the effects of such a quasiwall on the resonance boundary conditions are unknown and not easily
calculated.
The temperature dependence of the parallel resonance mode
spaci ng s for a film with two surface modes as measured by Yu is
shown if Fig. (6-11).

Since there is some question about the

interpretation of this data, the following quotation is extracted
from their paper.

"Without exception, i t is found that the high-

field surface mode, the quasi-symmetric surface mode, is observed

-108-

150

••

Q)

--

I (I)

•• •••••• • •
YIG/GdGaG
v=9.16 GHz
Annealed at !200°C

100

•••

200

••

300

Temperature ( K )
Fig. (6-11)

Separation of the first body mode and the first and
second surface modes, respectively, at parallel
resonance as a function of temperature for a YIG
film annealed at 1200°C. After Yu et. al. (1975)

-109to shift downward and becomes degenerate with the low-field surface
mode .. The data points are the resonance-field separations between
the two surface modes and the firs t body mode.

At a temperature

near l00°K, these two s urfa ce modes become degenerate in their resonance-field positions .

Below this temperature, this surface mode is

observed to appear at a nearly constant field separation above the
first body mode.

11

Physical data for this film is not given, but it is believed
similar to OSU 2 except for somewhat greater thickness.

For the same

reasons that OSU l was us ed in the calculations for temperature dependence, OSU 2 is used here; this dependence is shown in Fig. (6-12). In
this figure the mode positions with respect to the uniform mode position have been computed and plotted assuming that the properties
of the air-film interface are constant.

Note that the behavior

for the higher temperatures is in qualitative agreement with
the above;

that is, the quasi-symmetric mode increases in field

position away from the body modes then

11

Shifts downward

11

However,

at the lower temperatures the two surface modes are not degenerate.
It has been pointed out earlier that the surface layer model does
not predict the low temperature behavior accurately.

This is be-

lieved due to the s upposed invalid approximation of anti-parallel
spins in the diffusion region.

If this is indeed true for the dif-

fusion region produced by annealing at l000 °K then it probably has
a l arger effect on films anneal ed at l200°K; that is, th e diffusion
region i s thi cke r and close r to the thi ckness of a typi cal domain wall
(~ 1500~ in YIG) . In any e vent, the following may explain some of the

- 110-

SURFACE MODES

* zero intensity
0 r-~--~~~-+--~~~-+--~~~-+--~~~~

4n(MFe-MGd)
intensity mode

13 0

BODY MODES

-100
Fig. (6-12)

The calculated temperature dependence of the magnetic
field separation of the positions of three parallel
resonance spin wave modes , Hap , from the position of the
uniform mode,Hu, for sample
P OSU2.

-111discrepancy between the above calculations and the description of the
experimental data; however, it is speculation!

In Figs. (6-11) and

(6-12) the second mode is a quasi-antisymmetric mode.

It can be

shown that at the position marked by a * this mode is nearly
anti symmetric and has at most a very small excitation.

Within the

temperature range *- * in .Fig. (6-12) this quasi-antisymmetric mode

grows in intensity and becomes a large absorption mode.
a similar

Therefore,

transition from surface mode to nearly vanishing

antisymmetric surface mode to large body mode in a small temperature
range may have resulted in a misinterpretation of the experimental
data.

If the air-film interface also has properties dependent on

temperature as has been observed by Omaggio (1974)
resonance

at perpendicular

the behavior might be even more complicated and difficult

to interpret.
6.5 Comparison of Frequency Dependence Data
The frequency dependence of the experimental mode positions is
dependent on the orientation angle of the applied field.
ular resonance there is little dependence.

At perpendic-

For ion implanted films at

perpendicular resonance Omaggio and Wigen (1974) reported no frequency
dependence at room temperature .

The uniaxial and tensorial models pre-

dict no frequency dependence at perpendicular resonance.

The surface

layer model contains a frequency dependence at perpendicular resonance if
the gyromagnetic ratio of the bulk and surface regions are different;
however, this effect is smaller than the experimental resolution.

-112At any other angle the experimental data show a frequency dependence.

The parallel resonance configuration was chosen for comparison

because the equilibrium position of M0 is not a function of the
applied field strength.

The largest observed effect was in the posi-

tion of the parallel resonance surface mode.

The measured mode sep-

arations at 6 and 25 GHz for sample CIT 2 are presented in Table (6-4).
The calculated separations for the surface layer and uniaxial models
are also given.

Note that both models predict the experimental be-

havior; this is not surprising considering the agreement shown in Fig.
(3-11) between Ks and the surface layer with

yb= Ys·

The tensorial

model has no frequency dependence. unless Kll or Kl has a frequency or
field dependence.
6-6

Discussion and Conclusions
The data presented in this thesis are believed to show the

following:
1.

The observed phenomena cannot be explained by the uniaxial
and tensorial models.

2.

The observed phenomena are explained by surface regions with
magnetic properties different from the bulk properties.
These regions were approximated by uniform surface layers.
In the case of low temperatures where the model predicts
behavior which is not observed, it is believed that other
assumptions made to facilitate computations are not valid.

The extent to whi ch each of the three model s predicts the experimental data i s s ummari zed below.

Also discussed i s the microscopic

71.9

1st body mode-2nd body mode

160.0

165.2

1st body mode-3rd body mode

SAMPLE CIT 2

53.3

54.3

70.7

101.3

1st body mode-2nd body mode

P.erpendicu1ar
Resonance

69 .

Experimental
5.966 GHz 25 GHz

surface mode-1st body mode

Parallel Resonance

TABLE (6-4)

164.5

57.2

72.5

68.3

165 .0

57. 3

70.7

100. 1

Calculated Surface Layer
5.966 GHz
25 GHz

151.2

51. 7

79.5

69. 2

151.2

51.7

77.3

101 . 6

Calculated Un iaxi al
5.966 GHz 25 GHz

-'
.....

-114model for the tensorial surface anisotropy field at the filmsubstrate interface proposed by Wigen and Puszkarski (1976).
The perpendicular uniaxial anisotropy has been used by many
workers to match experimental data in metal films.

In these films

this model and the surface layer model cannot be distinguished. In
permalloy a 22~ half magnetization surface layer is equivalent to a
Ks = .22 ergs/cm 2 ; this represents a significant anisotropy and an
insignificant

(11~)

change in the total film thickness.

However,

YIG samples cut from the same wafer but with different annealing
histories must have significantly different thicknesses when this
model is used to match the experimental data; for example, the thickness required for an annealed sample is as much as 400~ thinner than
that required for an unannealed sample or sample annealed at a lower
temperature (see Table (6-1}}.

This thickness difference cannot be

understood in terms of a surface interaction alone.

In addition to

the above, this model cannot explain the following phenomena.
1) Temperature dependence of the critical angle (see Fig.
(3-13)).
2) The observed linewidth and intensity variation with mode
number and orientation of the applied field (see Tables
(6-2) and (6-3)).
This model does, however, predict the observed room temperature frequency dependence if the values of Ks and film thickness are chosen
to match the spectra at one frequency (see Table (6-4)).

-115-

The tensorial model was proposed by Yu et 2l (1975); in effect
this model is a generalization of the Puszkarski (1970) model which
assumes the surface spins are affected by a surface field that is
independent of the magnetization.

The tensorial model assumes a

surface anisotropy field dependent on the mean orientation of the
magnetization, but not on the instantaneous orientation.

As in the

case of the uniaxial anisotropy this model cannot explain why the
required thickness for an annealed sample is as much as 400R thinner
than that required for an unannealed sample or sample annealed at a
lower temperature (see Table (6-1)).

In atldition to the above, this

model does not explain the observed linewidth and intensity variation
with mode number and orientation of the applied field (see section
(6-3)).

This model can be made to match the observed frequency

dependence and the variation of the critical angle with temperature;
however, these are not physically meaningful unless some understanding of the origin of K_t and Kll is established.

Wigen and Puszkarski

(1976) proposed a microscopic model forKland Kllthat combines two
independent mechanisms.

The first mechanism involves an isotropic
static mean field interaction between the Gd 3+ and Fe 3+ cations in a
diffusion region at the film substrate interface; this field is dependent on the temperature and applied field.

The second field

arises from a uniaxial energy in the Hamiltonian which is proportional to <(S·n) 2> ; it is proposed that this term is due to Fe 2+
interacting with strong crystal field gradients at the interface.
The 1atter ani sot ropy is considered independent of the temperature and

-116-

applied field.

The isotropic term increases in magnitude with de-

creasing temperature.

Therefore, the desired temperature dependence

of Kl and Kll is qualitatively generated.

To explain the effects of

changing the frequency on the spectra, it was proposed that the isotropic term was field dependent. The hypothesis that Fe 2+ is present
at the surfaces was tested (Wigen et ~ (1976)) by observing a photoinduced change in the spectrum of a film at low temperatures (less
than 100°K); in view of the effects observed in Si 4+ doped YIG (Gyorgy
et ~ (1970)) this is considered good evidence for the presence of
Fe

2+

Assuming that the effects of the surface regions are lumped

into Kl and Kll , I feel that this model cannot be correct for the
following reasons:
1)

Only the static effects of the above mechanisms are included.

The dynamic effects are not second order and

unimportant .

For example, if the assumed tensorial field

depends on the instantaneous position of the magnetization
the boundary conditions on

m are changed and are in fact

identical to that of the perpendicular uniaxial anisotropy
with
2)

Ks = Kll- Kl (see Appendix I).
If an interaction between the Fe 3+ and Gd 3+ exists it has

to be in a finite region; the material in this region will

3)

be ferrimagnetic and s hould be treated dynamically as such.
The Fe 2+ is probably distributed throughout the surface region; therefore, the plausible effect is an anisotropy like
that observed in bulk materials.

A temperature dependence

is stro ngly suggested by the photo-induced effects (i.e., if

-117white light can change the effect then thermal agitation
probably can).
The effects of inhomogeneous films have been considered by many
workers at perpendicular resonance (Portis (1964), Sparks (1970),
Bajorek and Wilts (1971)); however, due to the mathematical complications, work at the other angles has been limited.

In this thesis the

effects of inhomogeneous surface regions are considered by assuming
uniform surface layers with properties that are averages of the actual
properties.
1)

The results from this model are summarized below:

The film thickness required for an annealed sample is
slightly greater than the thickness required for an unannealed sample or sample annealed at a lower temperature
(samples cut from the same wafer).

This is consistent with

a diffusion process (see Table (6-1)).
2)

The linewidth and intensity vari at ion with mode number and
orientation can be explained by making the plausible assumption that the surface layer has a larger damping constant
than the bulk (see Tables (6-2) and (6-3)).

3)

The room temperature frequency dependence of this model is
the same as experimentally observed (see Table (6-4)) .

4)

The temperature dependence of the critical angle is qualitatively explained down to the critical temperature (see
Fig. (6-10)).

The temperature dependence of the surface

mode spacings are qualitatively explained down to the critical temperature (see Figs. (6-11) and (6-12)).

Below the

-118-

critical temperature the assumptions that the magnetization in the surface regions is anti-parallel to the magnetization in the bulk is believed invalid.
5)

Although not addressed explicitly, the effects of Fe 2+ can
easily be incorporated into the surface layer model by an
additional anisotropy like that

observed in bulk materials.

Below the critical temperature a significant portion of the diffusion region is believed to have a magnetization that has passed
through ·compensation.

If exchange was the only torque exerted on the

spins the above magnetization would be anti-parallel to the magnetization
in the bulk.

However, a variation in direction is believed to exist

producing a quasi-domain wall; the effects of this variation are not
known.

It is therefore apparent that more work is necessary before

all resonance phenomena in YIG films are fully understood .

This thesis

has introduced a model which may explain the origin of many of these
phenomena; the phenomena not explained are believed to be due to
mechanisms (like the variation in the direction of

M mentioned above)

which are not easily incorporated into the computations.

-119-

REFERENCES

Akhiezer, A. I., Bar'yakhtar, V. G., and Kaganov, M. I., Soviet Phys.
Uspekhi

3, 567 (1961)

Ament,W. S., and Rado, G. T., Phys. Rev.

97, 1558 (1955)

Bailey, G. C., and Vittoria, C., Phys. Rev. B

~.

3247 (1973)

Bajorek, C. H., and Wilts, C. H., J. Appl. Phys. 42, 4324 . (1971)
Brown, S. D., Henry, R. D., Wigen , P. E., and Besser, P. J., Solid
State Comm. }l, 1179 (1972)
Calhoun, B. A., Smith, W. V. and Overmeyer, J., J. Appl. Phys. 29, 427
(1958)
Geller, S., J. Appl. Phys. ll• 30S (1970)
Gilleo, t·1. A., Geller, S., Phys Rev. llO, 73 (1958)
Gyorgy, E. M., Sturge, M. D., Van Uitert, L. G., Heilner, E. J., and
Grodkiewicz, W. H., J. Appl. Phys. 44, 438 (1973)
Gyorgy, E. M., Dillon, Jr., J. F., Remeika, J. P., I.B.M. J. of
Research }i, 321 (1970)
Henry, R. D., Besser, P. J., Heinz, D. M., and Mee, J. E., IEEE
Transactions.~,

535 (1973)

Kittel, C., Phys. Rev. 110, 1295 (1958)
Kobayashi, T., Barker, R. C., Bleustein, J. L., and Yelon, A., Phys.
Rev B I· 2373 (1973)
Landau, L. and Lifshitz, E., Physik Zeits. Sowjetunion ~. 153 (1935)
Liu, Y. J., Ph. D. Thesis Yale U. (1974)
Macdonald, J. R., Ph. D. Thesis, Oxford U. (1950)

-120Morrish, A. H., The Physical Principles of Magnetism, John Wiley and
Sons, New York (1965)
Omaggio, J. and Wigen, P. E., AlP Conf. Proc. 24, 125 (1974)
Pauthenet, R., Ann Phys. (Paris)!· (1958)
Portis, A. M., Appl . Phys. Letters£, 69 (1963)
Puzkarski, H., Acta Phys. Pol A 38, 217 (1970);38 , 899
Ramer, 0. G., and Wilts, C. H., Phys. Stat. Sol. (b) 73, (1976)
Rosencwaig, A.,Tabor, W. J., and Pierce, R. D., Phys. Rev. Lett. 26,
779 (1971)
Sparks, M., Phys. Rev. B l• 3831 (1970); l• 3856 (1970); l· 3869 (1970)
Stakelon, T. S., Yen, P., Puszkarski, H., Wigen, P. E., AlP Proceedings
of the 21st Conference on Magnetism and Magnetic Materials (1976)
Stakelon, T. S., Annual Report, Ohio State Univ. NSF Report #4, RF project
3457 (1975)
Vittoria, C.,Ph. D. Thesis, Yale U. (1970)
Wangsness, R. K., Phys . Rev.~. 1085 (1953); 93, 68 (1954); Am. J.

Phys. 24, 60 (1956)
Wigen, P. E., Stakelon, T. S., Puszkarski, H., Yen, P., AlP Proceedings
of the 21st Conference on Magnetism and Magnetic Materials (1976)
Yelon, A., Spronken, G., Bui-Thieu, T., Barker, R. C., Liu, Y. J., and
Kobayaski, T., Phy. Rev. B ~. 1070 (1974)
Yu, J. T., Turk, R. A. , Wigen, P. E., Phys. Rev. !l• 420 (1970)

-121Appendix I
1-1.

Discussion of YIG Anisotropies and the Equilibrium Conditions on M

The equations which describe the effects of anisotropies on the
resonance process are given by Eq. (2-10) in terms of the anisotropy
energy .

The dominant anisotropies in YIG fi l ms are the cubic crystalline

anisotropy and magnetostrictive anisotropy.

The effects of the crystal-

line anisotropy in YIG are well described in terms of the standard first
order expansion of the direction of M

(i.e. K1>> K2 ).

The crystalline

anisotropy energy is usually written in terms of the direction cosines
of M from the cubic axes

When written in terms of the spherical polar coordinates of the text,
the expression is different for different film orientations .

For the Kl

term and the [100] oriented films
Kl
. 4
. 22
. 22 ]
EA = 2 [ s1n a s1n ~ + s1n a

I-2

For the [111 J oriented films with· ; measured from a (1T2) axis
. 4
. 3
cos3~ + c~s4a ]
EA = Kl [ s1n4 a
3 s1n a cosa

_a

I-3

If the tension is along the film normal, it is shown below that
for the [100] and [111] oriented films the magnetostrictive anisotropy
is uniaxial with easy or hard axis normal to the film plane.
tensor components of the tension given by

a . . =cry.y .

lJ

1 J

With the

(the direction

cosines of the tension are yl, y2 , y3 ), the magnetostrictive energy
is (Morrish (1965))

-122-3

Er~ = 2

'-100°

2 2
2 2
2 2
a1 Y1 + a2 Y2 + a3 Y3

- 3"111 ° (a1a2y1y2 + a2a3y2y3 + a3a1y3y1
For the [100] oriented films and y3=1, y1=y2=o
-3
-3
EM = 2 "loooa3
= 2'-lOOo cos e

I-4

I-5

For the [111] oriented films and y1=y2=y3= l/13 the non-isotropic tenns
of the energy are
I-6

rt is easily shown that
cose = -1( a1 + a 2 + a 3 )
v3
therefore, for the [111] oriented film
I-7
for Eq. (I-2) are
The g.'s
g1 ( e .~ )

-Hk
. 2
= SnM [(3sln e

I-8

sin4e cose/2s ine ]
-3Hk
g (e.~) = lG nM [ sin e cose sin4$ ]
where Hk = 2K 1/M. The gi ' s for Eq. (I-3) are
g·1 ( e.~ ) = -(Hk/8nM)[ -8sin 4e + 7sin 2e - }:ose
1:2 cos3~ sin2e {1 - ~ sin 2e)]
. 2 26

g {e,q,) = -(Hk/8nM)[s~n
2 . 2 )J

~1n6

I -9

3 cos e + -r2 cos3q, sin2e (1 +

-123The gi 's for a uniaxial anisotropy E = -Ku cos 2 (e) (like EM in Eq. (I-5)
and (I-7)) are
gl{ e ) = -{Ku/2~M ) cos2e
g2{ 6) = -(K u/2~M ) cos a
g3{e) = 0

I-10

In general the equilibrium conditions can be determined from the
requirement given by Eq. (2-11) .

The equilibrium conditions which are

applicable to the experimental and calculated data presented in
chapter 6 are presented here.

For the [100] orientation with applied

magnetic field, Happ' in a (100) plane at an angle a from the normal,
the equilibrium condition is
0 = (4~M- 2Ku/M)sin2e - 2Happsina - (Hk/2)sin4a
where Ku=- 3A 100cr/2 from Eq. (I-5), and Hk=2 K1/M.

I-11

The equilibrium con-

dition for the [111] oriented film with the applied field, Happ' at an
angle a from the film normal in a plane defined by the normal and a line
in the plane 30° from the (Tf2) axis are
0 = (Hk/~sin e cose sin3~+ 1 (1:3Happsin~ sina )/2

I-12

- (Happ/2) cos~ s ine

0 = Happ ( -case sine + ~/2) case cos~ sine + .5cose sin~ sine)
+ (4~M - 2Ku/M)cose sine - (Hk/2)[sin 3e case - (4/3)cos 3es ine

-V2 COS3$( Sin 2e- (4/3) sin 4 a)]
where Ku=-3A111cr /2, and Hk=2 K1/M.

Note that in Eq. (I-ll) and (I-12)

that the uniaxial anisotropy field substracts from 4nM and has an
identical angular dependence .
eq uations.

Thi s i s also true in the resonance

-124-

I-2.

Surface Boundary Conditions
The surface boundary condition is generally a statement that the

surface anisotropy torque is balanced by a surface exchange torque.
Since we are at the boundary of the ferromagnet, the surface exchange
is related to the slope of the magnetization, dm/dn, rather than to its
second derivative.

The anisotropy torque can be defined in terms of an

equivalent surface field or a surface anisotropy energy.

In the first

case the surface torque requires the elementary calculation MxHs
the magnetization is perturbed from its equilibrium position.

when

In the

second case the torque can be obtained by taking appropriate angular
derivatives of the energy function.

If desired, calculation of these

derivatives can be interpreted as calculation of an equivalent
anisotropy surface field.

An unresolved question is whether this

surface field varies with the dynamic (small angle) motion of the
surface magnetization or depends only on the equilibrium position of the
magnetization or is completely independent of the orientation of M.
None of these assumptions complicate the analysis, they simply give
different boundary conditions.
If the anisotropy energy is expressed in terms of the angular
orientation of M (i.e.,e and~) then a satisfactory procedure is to
define an equivalent anisotropy field which is at all times perpendicular
toM.
Hs = Hso + 11s
where Hso = - vmEs /aM

a i s the lattice constant

-125-

- a
- a
vm = (eaaa + sine e~~ )

hs = (1/M) (m · vm) Hso
~ a
'(m . vm) = (rna · ae
+sine ~)

The field Hs is written as an expansion in terms of Hso the field when
M is in its equilibrium position and hs a small additional component
which arises from a small displacement of Mfrom its equilibrium
position (M = M0 + m8e8 + m~e~ ) .
In the equilibrium position, the surface torque per unit area (on
one atomic layer) is
Tso = aMo x Hso
and this is balanced by an equilibrium exchange torque between the
surface spins and ( for simple cubic lattice) the spins in the next
atomic layer.

With an arbitrary displacement m, the surface torque

becomes
Ts = a(M0 + m) x (Hso + hs)
+ a(M0 x Hso + M0 x hs + m x Hso )
where the second order term mx hs has been omitted.

For this same

displacement, the exchange torque per unit area becomes
rex= [ -aMO X Hso + a(-maHs~ + m~Hsa ) er
+ 2A a m~2A a rnaMan ea -M a n· e~ J
It will be noted that the first and second terms in Tex

are the

ne9ative of the fir st and third terms of fs' so the equilibrium
condition (Tex + Ts) = 0 becomes
t s + t ex = 0

-126-

~~hen

the anisotropy field is obtained in other ways which give a

radial component, then one must include the a and ~ components of

mx Hso which are not cancelled by the second term in the exchange
torque .
ts = a [(-Mhs~ + m~Hsor>ea + (Mhsa - maHsor)e~ ]
It should be noted that the partial derivatives in tex do not include
the static or equilibrium values of the derivatives which are required
to balance the equilibrium torque aM0 x Hso·

The equilibrium values

of the derivatives are much larger than the dynamic derivatives by a
factor of the order of M0 /m since this is roughly the ratio of the surface torques balanced by these two components of the exchange torque.

Example 1

Uniaxial anisotropy Es = Ks sin 2a
-Ks

sin 2a ea

-2K
h = _s [ cos 2a m e + cos 2a m,~. e,~. ]
a~1 2
a a
"' "'
2K
ts = ~ cos a m~ ea - cos 2a rna e~ ]
This gives the boundary condition given in equation (3-7).

-127Example 2

Tensorial field (Yu et al (1975)) (assumed constan~ with
respect to dynamic Mvariations.

Hso = [(Hlcos

e0 ) er +

- H1 ) sin e0 cos e0 ee ]

( HII

Let

e0 +HI! sin

2KI
Hl = __.~...
aM

This gives the boundary condition given in Eq. (3-11).
Example 3

Tensorial field but allowing it to vary with dynamic M.

Hso = [ Hl cos e + Hll si n e ) er + (HII - Hl) sine cose ee]

hs = M [ (Hl sin e + Hll cos e ) me ee + Hll m~ e~ ]

ts = ~(
Kl- Kll ) cos 2em~ -ee - ( Kl- Kll ) cos2e me -e~ ]

Note that this gives the boundary condition given in Eq. (3-7) with

-128Appendix II

The elements of the determinants given in Eqs . (3-9a) and (3-lOc)
are given below.

11 =~K cos(2a) cos(k 1d/2)

12 ~-k 1 sin(k 1d/2) + K0 cos(2e ) cos(k 1d/2)

a 13= ~K cos 2 (e) cos(k 1d/2)

sin(k 1d/2) + K0 cos 2 (e) cos(k 1d/2)
a 21 = k1 cos(k 1d/2) + K0 cos(2e ) sin(k 1d/2)
a 14= -k

a 22 = ~K cos(2e) sin(k 1d/2)
a 23 = k1 cos(k 1d/2) + K0 cos 2 (a) sin(k 1d/2)
a 24 = ~K cos 2 (e) sin(k 1d/2)
a ~ = ~K cos(2e) cos(k d/2)

sin(k 2d/2) + K0 cos(2e) cos(k d/2)
a 33= -RR* ~K cos(k 2d/2)
a =-RR* (-k 2 s in(k 2d/2) + K0 cos 2 (e) cos(k 2d/2)
34
a = k cos(k 2d/2) + K0 cos(2e) sin(k 2d/2)
41
a42 = ~K cos(2e ) sin(k 2d/2)
a = -RR* (k 2 cos(k 2d/2) + K0 cos 2{e) sin(k 2d/2))
43
a 44 = -RR* ~K cos 2(e) sin(k 2d/2)
32

= -k

where

Ksl ,K 52 are th e surface anisotropies at the two surfaces ; and
K0 = (Ks 1+K 52 )/2A, and ~K= (Ksl-Ks 2 )/2A

-129The elements of the determinant for the secular equation using
asymmetric surface layers are:

a 31 = cos(k 1bD/2)
a41 = sin(k 1bD/2)
a 51 = cos(k 2bD/2)
a61 = sin(k 2bD/2)
a 71 = -cos(k 1s 1L1 )
as,= -cos(k2slll)
a 12= -cos(k 1s 2L2 )
a22= -cos(k2s2L2)
a32= a31

a52= a51
a62= -a61
a33= a31/Rb
a43= a41/Rb
a53= -a51 Rb*
a63= -a61 Rb*
a73= a71 /Rsl

a83=-a81 Rsl
a14= a121Rs2

a24= -a22 R;2
a34=a33
a44=-a43

-130as4= aS3
a64= -a63
a3s= -klb a41
a4s= klb a31
ass=-k2ba61
a6s= k2baS1
a7s= -k, sl sin(k 151 L1 )
ass= -k2sl sin(k 251 L1 )
a16= kls2 sin(k 152 L2 )
a26= k2s2 sin(k2s2L2)
a36= -a3S
a46= a4S
as6= -ass
a66= a6S
a37= a35/Rb
a47= a45/Rb
as7= -aSS Rb*
a67= -a6S Rb*

an= a75/Rs 1
as7= -aSSRs* 1
a,s= a16/Rs2

a2s= -a26Rs2
a3s= -a37
a4s= a47
ass= -as7

-131a68= a67

The film characteristics are illustrated below .

kl b
k2b

Rb
=--M4----------- D------------~~

M-----~---------- d------------~--~~

-132 Appe nd ix I II

Amplitude of ferromagnetic spin-wave resonance in thin
films
C. H. Wilts and 0. G. Ramer
Co/iforolo ltUtll•t< of T(Received 27 September 1974; In fintl fonn 17 October 1975)
The effect of conductivity on the spin·wave spectrum of than Permalloy fenomillflelic films has been
ll\vestipted. J( conductivity cfrecta are Included and a SJmplc surface anisotropy is auu.med. it is known
that the calculated mode loc:ationJ and amplitudes for Permalloy films arc in excellent aareement with
IOtTle experimental data in the ranae 100-2700 A. in thac:kn- If oonductivity efrecll are omoued, a much
simpler calculation is possible, but the error in mode location and amplitude has been unk nown. For both
perpendicular and parallel resonance acomctria. detai led calculations reported here have s hown that mode
Jocation.s are not s•snificantly affected over the above thickness ranae. and that I he main mode amplitude is

in error by only 20% at 800 A thickness However, for 2000 A thickness, the main mode amplitude is in
error by a factor of 2 5
PACS numben 76 50., 75. 70.

INT RODUCTION

The existence of standing spin-wave modes In evaporated polycrystalllne ferromagnetic thin films was established many years ago and the approximately quadratic dispersion has been used by several workers for
measurement of the magnetic exchange constant. '•' At
a fixed frequency, the modes were spaced in applied
field approximately as the square of Integers which
describe (roughly) the number of hall-waves In the
standing-wave pattern. Using a semiclassical theory of
spin-wave dispersion in an insulating medium, the observed deviations from a square law were explained
qualitatively' by inhomogeneity In the film or by a surface anisotropy which provides partial pinning of the
spins at the surface. Attempts to explain the observed
amplitudes of the resonances have had only limited
success. 4 •' However It is uncertain whether disagreements were due to poor samples, due to Imposition of
Improper boundary conditions, due to the neglect of
conductivity in the film, or due to inadequacy of the
phenomenological model for magnetization dynamics.
Several papers In the last two decades flave given a
mathematical formulation for treatment of conducting
media utilizing Maxwell's equations and the LandauLifshltz equation . ._. None of these have applied this
formul ation to a theoretical comparison with experimental data. A recent treatment by Bailey and Vlttorta•
Is the first serious attempt to use this formulation to
match experimental data. Mode locations were matched
with very good accuracy, but due to an Invalid approximation In treating the magnetic losses, the predicted
amplitudes and lmewldths for the higher (shorter -wavelength) spin-wave modes deviated widely from the experiment. After correctton of this error,'" the theoretical predictions were In good agreement with experiment
for all modes observed In a set of four Permalloy films
ranging In thickness from 800 to 2700 A. Although the
!,.;elusion of conductivity effects greatly complicated the
calculations, no effort was made by Bailey and Vlttorla
to confirm the importance of including this effect. The
purpose of this paper is to compare the results of such
accurate calculations with simple approximations which
Ignore the effect of conductivity.

Ferromagnetic resonance Is observed with a static
magnetic field a pplied in any direction with respect to
the film. For simplicity of analysis, the experiments
are often done with the magnetic field parallel or
perpendicular to the film plane even though resonance
at an oblique angle Is a more powerful technique. Parallel and perpendicular resonance are the only cases considered In Ref. 9 and are therefore the only ones con side red In this paper.
MATHEMATICAL FORMULATION

Since a quantum-mechanical treatment of this system
Is Intractable, It Is customary to use Maxwell's equations coupled with the Landau-Lifshttz phenomenological
equation. In this equation M is treated as a vector of
fixed magnitude which moves In reaction to t he total
effective field and a small phenomenological dissipative
term provides an energy loss. These equations or motion have been amply discussed In the references cited
earlier. However due to dllferences In notation, the
Landau-LIIshltz equation is repeated here.
dM
dt
= - ')t4X(8 0 + hrt t h., + h• + • • •).

(1)

The gyromagnetlc ratio y Is taken to be a positive number so t hat a negative sign Is explicitly used In Eq. (1);
Ro Is the static inter nal field Including the static demagnetizing field; h., Is the local rf magnetic field including both applled fields and rf demagnetizing fields.
The term h .. Is an effective rf field due to exchange
coupling between the adjacent nonparallel spins:
(2)

where A Is the exchange constant and k Is the wave
number of the spin wave m = m,exp{t(wl%ky)). The vector m Is the rf component of M, assumed small in
magnitude compared toM, and therefore (to first order)
perpendicular to the equilibrium position of M , I. e. , M
= M 0 + m where M0 Is parallel to 8 0 and m Is perpendicular to 8 0 • T he magnetic damping is treated phenomenologically by Introducing h,, an effective damping rf field
fiel d. It Is written here in the Gilbert form":
(3)

-133The magnitude of the damping Is described by the relaxation frequency X, or by the dimensionless damping
constant cr = 'A/YM. Additional fields to represent crystalline or uniaxial anlsotroples are readily Included.
Since they do not contribute to the effects studied In this
paper, they have been omitted.
The boundary condition on M Is an unsettled matter
although most workers Invoke one of four situations:
(1) spins unpinned, (2) spins completely pinned, (3) a
uniaxial anisotropy energy with easy or hard axis along
H0 , or (4) a surface anisotropy energy with easy or hard
axis perpendicular to the surface. The last of these
appears to be more consistent with experimental results
and Is use d both in Ref. 9 a nd here. The surface anisotropy energy Is assumed to have the form Ill,
= - K, (n• v)', where nIs the outward pointing unit vector
nor mal to the film surface and i•= M/ M Is a unit vector
In the direction of M. However a layer of reduced
magnetization or a surface layer of reduced demagnetlzlng field has an equivalent effect In "pinning" the magne tization at the surface if the anisotropy constant K. and
the surface layer thickness arc given appropriate
values. • In terms of the surface anisotropy, the boundary condition on M Is
(4a)

(4b)

where 0 = M.,/ M, and 9 and

directions with respect to the nor mal ii.
The solution to the above model Is desclbed below In
greater detail than In Refs. 7 and 8 In order to facilitate
comparison with the approximate solution developed
later. The excitation Is unUorm linearly polarized
e lectromagnetic radiation normal to the film surfaces.
Appropriate boundary conditions are satisfied and power
absorbed Is calculated from the Poynting vector at the
surface. Small-amplitude sinusoidal motion Is assumed
so that the equations are linearized. Calculations of the
power absorbed and the steady-state standing-wave
pattern are carried out by digital computer.
AI a given frequency, amplitude of external rf field,
static magnetic field, and Him orlenlatlon, the calculation predicts the amplitude of m and h throughout the
film and the power absorbed. The resonance condition
is determined by locating a maximum in the power absorbed while sweeping either field or frequency. There
are lour components o! the standing-wave pattern, each
with a characteristic polarization and complex propagation cons tant k. For the case o! perpendicular resonance, the response breaks up into circularly polarized
pairs. One pair has positive precession (in the sense
- mxH0 ), the other negative. Hence In the analysis, the
external rteld Is resolved Into components of opposite
circular polarization.
The following discussion relates to the positively
polarized components which are the only ones that partictpate significantly In the resonance process. The two
components are not In ph:~se with each other and since
even one component varies in both phase and magnitude

through the film thickness, all components of m and b
are described by complex numbers. In the discussion
below, the subscripts (r) and (im) refer to real and
Imaginary parts o! these complex numbers and
subscripts 1 and 2 refer to the components correspondIng to the two values of k, ordered so that lk 1 1 < lk2 1.
The coordinate system Is shown In Fig. 1, where the
y axis Is normal to the film. The basic normalization
Is to set the magnitude of the circularly polarized rf
field at both surfaces equal to ~11 0 , with phase chosen
so that at I = 0
h. = h 1 }~L)+ h.,_(~L)= }h0 ,

It, = - h 11 ., (iLl- h,,.,(iLl = 0.
For fields and frequencies normally used In the laboratory for spin-wave spectra of metal ferromagnetic
rums, the components have the following characteristics
at resonance: the component h 2 (y) Is nearly Independent of y and approximately equal to il•o: the components
h•,., ''~.r• and '"•• are all much smaller than 110 for all
values of y. The magnetization component m,(y) being
proportional to ll,(y) is also nearly Independent of y, but
Is small . The component m 1 (y) ls proportional to lt 1 (y ),
but the proportionality constant Is so large that m,(y)
» m 1 (y) even though h,(y) Is small. The significant component m 1 (y) Is largely Imaginary and var ies withy in a
manner governed by the spin-wave k value, k 1 • Since k 1
is nearly purely Imaginary , the variation of m 1 is nearly
sinusoidal. In otherwords, to a rough approximation
h.,_ Is equal to the external rf field and Its degree of independence on y coupled with the smallness of lt1 shows
the degree to which the magnetic field Is uniform In the
metal. The resonance variation of m Is described by
"''••• which Is roughly sinusoidal in v and 90" out o!
phase with the rf magnetic field, h,. All other components of m and h are small.
To summarize, the quantities of Interest are as
follows:
(1) applied field at resonance, H, = H 0 - 4~rM:

(2) spin-wave k value •(k 1 ) 1,.:

"'
--~

Fl G. 1, Field re lntlons at resonnnce in the perpencllcular reR onance conflgurntton.

-134(3) resonance amplitude of m • m,,.(O);

Is shown In Fig. 1. The entire pattern rotates about H 0
with angular velocity w. The vector dm/dt Is equal to
the first term on the right-hand side of Eq . (5), so that
the resonance condition Is

(4) relative surface amplitude of m (a measure of

surface pinning) Is approximated by the ratio

[m11 .V[m 11• (0)]:

1)
!!!.=(H
o +~k

(5) power absorbed per unit area equals the sum of

(6)

Poynting vectors at the two surfaces.
Although the power absorbed Is given directly, the
power absorption due to resonance requires subtraction
of the "background" power which Is a significant part
of the total for some modes. This Is done most simply
by plotting the power as a function of applied field, and
drawing a smooth curve under the resonances .
Similar considerations hold for parallel resonance,
except that the applied field Is linearly polarized perpendicular to M 0 and with amplitude h 0 • There are three
elliptically polarized components of m Instead of two
circular polarizations. In general one of these has positive polarization and describes the (roughly) uniform rf
field driving the magnetization, the other of positive
polarization Is the resonant spin-wave mode, and the
third of negative polarization Is a surface component of
negligible amplitude. The fourth component Is linearly
polarized but Is not exclled by the assumed external
field.

The last two terms of the equation balance In the sense
described earlier that the torque Mv0 Xhrl Integrated
over the thickness balances the dissipation torque ( a /
Y)v0 Xdm/dt also Integrated over the thickness.
Using a sinusoidal mode shape and the surface anisotropy boundary condition of Eq. (4) with
0 )= 1, It Is
not difficult to show that the secular equation for the
resonant spin-wave k values Is

(n•v

[k- K0 cotnkL)j[k + K 0 tan(ikL)] + (AK)1 =0,

(7a)

where L Is the film thickness, K, and K. the surface
anlsotroples at the two surfaces, 'K0 =(K, + K, )/2A,
and AK = (K, - K, )/ 2A. For our present purposes, It Is
adeq11ate to consl~er the symmetric case K •• = K ~2 , In
which case only half of the k values are taken (I. e. ,
those corresponding to symmetric rather than antisymmetric mode shapes). The symmetric modes are
obtained by setting the first factor of Eq. (7a) equal to
zero.
[k- K0 cot( ~kL))= 0.

AN APPROXIMATE SOLUTION
The simplest approximation for locating resonant
modes neglects both conductivity and >. losses. Maxwell's equations neglecting conductivity and displacement current include the rf component of demagnetizing
field due to the component of m perpendicular to the
surface, h.,= - 4~m,e,. Combining this with Eq. (1)
gives s imple relations fo r the mode locations. In the
ferromagnetic Insulator (<7= 0), If the >. losses are very
small (a« I), the amplitude of resonance and power
absorbed can be approximated by assuming the external
driving rf field to penetrate the medium without attenuation o r phase shift and the resulting magnetization variation to consist o f purely sinusoidal or hyperbolic components again without phase shift.

(7b)

Recognizing the H 0 Is the sum of the applied field H
and the demagnetizing field - 411M, the resonance
occurs at the field
(8)

where k Is a solution of Eq. (7). Assuming the magnetization m has an amplitude m 0 cos(ky), where y Is measured from the center of the fUm, the relative amplitude at the surface Is cos(}kL). From the discussion
following Eq. (6), It readily follows tflat the peak amplitude Is g iven by

!!! _ 2Ylt0
M -

sln(~kL)

(9)

aw kL + sin(kL)

If the driving torque Integrated over the thickness Is

balanced against the dissipation torque also Integrated
over the thickness, the resonance amplitude Is obtnlned.
Note that these torques do not b~lance locally. However,
the E'XChange interaction Is so stron~ thnl lnslgnlflc~nt
chan~es in mode shnpe a re able to provide the local
torque balance without significant change in amplitude.
As discussed earlier fot· the case of perpendicular
resonance, attention Is focused on a positive circularly
polarized magnetic field of amplitude ~/1 0 and frequency
w. For perpendicular resonance there is no rf demagnetizing field and the Landau-Ltfshltz equation becomes
simply
( 5)

For positive values of K • and for s ufficiently small
values of damping (a « I), all mode shapes are nearly
simple sine waves with negll~ible phase shift throug h
the film. AI reson:tnce, the relation between M and hrl

(10)
For the case or parallel resonance Vn Is parallel to
the z axis, 0 = e,. The field h.., outside the sample Is
linearly polarized, h0 e,, and the "component inside the
film Is assumed to have the same value. The boundary
condition on mIsgiven by Eq. (4) with (n•ii0 )=0. The
magnetization variation Is no longer a circularly polarIzed spin wave, but i s a linear combination or two e l liptically polarized standing waves. It Is easily shown
front Eqs. (4) and (5) with 0 = e, that these standing
waves are either sinusoidal or hyperbolic, and are
c haracterized by wave vectors k 1 e, and k 1 e,. If the wave
numbers are ordered so that lk 1 1 < lk1 1, then k 1 Is
usually Imaginary for all modes, while k 1 is real for all
modes except the first, in which case the sign of
depends on the sign or K,. The wave numbers k, and k 1 are
obtained from the roots of a dispersion relation which

k:

-135Is somewhat more complicated than Eq. (7a):

TABLE I. Aeeumed properties or magnetic fllma.

{+ Rk 2 (k 1 + K0 cot(ik 1L)] + R" 1k 1(k2 + K0 cot(}k 2L)]}

Thickness W
41'M (G)

{Rk,(k, - K 0 tan(ik 1 L)] + Jr'k,(k,- K 0 tan(tk,L>l}

2023
11151.4
1.l·f3><10..
1.8484><10'
o. 00457
6.3><1o'•
0.275
9.44

A (erg/e m)

'Y IOe aec)-1

x (Rk, tan(}k1 L) + R" 1k 1 tan(-ik,L))
x[Rk, cot(~k1 L) + R" 1k 1 cot(!k,L)](AK)' = 0,

"'a Cesu)

(11a)

K• lerg!em1 )

790
11216.7
1.143 x1o-•
1.8484 >< 10'
o. 00455
6.3 ><1011
0.200
9.44

k~ = k!- (21-M / A)(1 + 40')' 1',

I t:Hd

R = 20( 1 + (1 + 401 ) ' ' ' ] · • ,

a • 6.3><10~0em.

and
0=w/ 4l1M)'.

+ m 20 ), where

As In the case of perpendicular resonance, If the
boundary conditions are symmetric, then AK = O, and
the equation factors Into separate relations for the symmetric or antlsymmetrlc modes. The symmetric modes
are those obtained by setting the rtrst factor of Eq.
(Ita) to zero. With some simplification this becomes

~ _

m 10 -

(1 +(I+

:o•)ll•) cot(¥)+ 1 =0.

m.(tL>= m 10 cosUk 1 L) + m 10 cos(!k1 L).

Using Eq. (5) and the value of k 1 obtained from Eq.
(11), the resonance rteld H. Is found to be
(12)

111

The two components corresponding to k 1 and k 1 have
quite different characteristics. For K. "> 0, the principal
component corresponding to k 1 Is hyperbolic in y for
the main mode, and stnusoldal for all other modes. It
has a positive precession and an ellipticity given by

= + i/ R.

(O) = 41r 0 rM (

aw

1 )
sln(Jk,L)
1 + R 2 k 1L + sin(k 1L)'

( 16 )

and (17) should be used with k 1 replaced by the magnltude of k 1 and the sines replaced by hyperbolic sines.

The smaller component corresponding to k, Is hyperbolic in y for all modes. ll has negative precess ion and
elllpticlty.
(m,/m.)2

ll s hould be noted that when k 1 Is imaginary, Eqs. (16)

(13a)

(m/m) 1 = -iR.

(15)

The maximum value of "'•• Is lm20 1 cosh( I }k,l L). For
thin films of typical ferromagnetic metals, It can be
shown that this maximum value Is about two or three
orders or magnitude smaller than m 10 • In consequence
the amplitude or m and power absorbed can be approximated closely by Ignoring m,. With this approximation,
the amplitude becomes

(llb)

H. = 2.-Af((t + 40 2 ) 1 1' -1]- (2A/M)k!.

(14)

and at the surface the amplitude Is

~ (t-(1 + 4 ~1)1/i) cot ( ¥ )

+It

_ k 1 sln(!k 1L)
k 2 stnOk,L) •

rt should be emphasized that the results given in Eqs.
(6)-(17) are approximations based on assumptions or a
uniform driving rf field, negligible phase shift of m and
h through the film thickness, and sinusoidal or hyperbolic mode shapes. For Insulators, small values of a,
and for film thicknesses normally encountered, the er rors are negligible.

(13b)

The amplitude of "'• at the center of the film can be
written as the sum of the two components m.(O) = (m 10
TABLE 11. Predicted mode properties h'Om lnoulotor model.

79o A

2o23 A

Thickness

Mode No.

Pel"pendtcutar resonance

n. r>e>1

Eq. (8)
Eq, (7)
II (cm· ) xlo-'
Eq. (9)
mo/M
cos(jkL)
m !!Lll m (O)
Power (eJ'1t/cm1lx 10'1 ~'q, 00)

r; r:ower

14327.9
1.122
0. 04073
0.423
I . 732
IOCtl

14011.7
3.679
o. 0089:1
-0.11.17
0. 0700
4.04

13252.0
6.559
0. 00:137
0.939
0.0093:1
0. !\4

12005.3
9.561
o. 00168
-0.970
0.00226
0.13

141:14.1
I. 891
0. 037311
o. 734
0. 7107
100'1

Parallel resonance
Eq. (12)
Eq. (11)
~· (cm·1) >< I Q..
m(O) / M
l::q. (16)
cos(jk1 L)
m(~L) /m(O)
Power !erg/cm ) >< I ct- Eq. (17)

H. (()(!)

. power

Rfl2.211
0.4:1:1
0. 0616
1.0119
:1.:1!16
100' •

61!1.12
:1.0411
0. 00246
-0.990
0.0024K
0.07:1'.

Ar.l.:l:l

o. r.r.o
0. 06:14
I. 0 19
l.:l:tll
100'.

12588.6
11.469
0. 00391
- o. 979
0. 0049r.
0. 70

-136In the cases of perpendicular and parallel resonance
respectively, Eqs. (7)-(10) and (11 )-(17) can be used
to predict the location of the resonances, the spin -wave
k value, the amplitude of the magnetization precession,
the relative magnetization amplitude (effective pinning)
at the surface, and the power absorbed. Only the first
and last of these are experimentally observable. Equation (7) for dispersion, Eq. (8) for resonance field, and
Eq. (10) for power dissipated In resonance are also
obtained by the quantum-mechanical treatment of
Puzkarskl. 11 His Eqs. I 3. 24 and II 1. 8 reduce to Eqs.
(7) and (10) In the limit that the l attice constant approaches zero. Similar equations lor parallel resonance
have not been found In the literature.

In Ref. 9 have been used except that an average surface
anisotropy has been assumed at both surfaces, the only
effect being to eliminate excitation of the very small
anttsymmetric modes and to simplify the approximate
calculations. The specific values of the physical constants are given In Table I.
In the case of perpendicular resonance, the spin-wave
number k , mode location, magnetization rf amplitude,
surface pinning, and power absorbed were calculated for
the first four symmetric modes of the 2023-A film and
the first two symmetric modes of the 790-A film. The
results for the Insulator approximation [Eqs. (7)-{17) I
are given In Table II. For parallel resonance the same
data are tabulated for the first two symmetric modes
of the thick film and the first mode of the thin rum .
Only the larger in-plane component of m Is tabulated.

COMPARISON OF CALCULATION
For comparison of the approximations above with
calculations which accurately Include the effects of conductivity, two of the films tested experimentally and
theoretically by Bailey and Vlttoria 0 have been chosen.
In particular a thick film (2023 A) was selected where
the effect of conductivity Is expected to be significant,
and a thin film (790 A) where the effect of conductivity
Is expected to be small. The physical constants chosen

For the more accurate conductivity model, In addition to the basic case, four other cases were considered
.corresponding to conductivity reduced by a factor or 10,
100, and 1000 and finally conductivity reduced by a factor of 1000 and a reduced by a factor of 10. Results for
perpendicul ar resonance are given In Table m where
they are also compared with data for the Insulator ap-

TABLE 01. Predicted mode properties In perpendicular resonance.

LDaa parameter

Thickness

14327. 7
14327.9
14327.9
14:127.9
14327.9

14013.2
14011.9
14011.11
14011. 7
14011. 7

1 3252.3
1 3252.0
1 3252.0
n2s2. o
1 32fi2. 0

12005.4
12005.3
12005.:1
1 200r..3
1 2005.3

14334.1
14334. I
14334.1
14334 . I
14334.1

12588.7
12588.6
12588.6
12a8A.6

14:127.9

14011. 7

1 32fi2. 0

1200!t,:l

14334.1

1 25811. G

Mode No.

H. fOe)

6 x1o"
6 XIO"
6>< 1014
6 >< 1 0"
6 x 1o"

o. 0046
0.0046
0. 0046
0. 0046
o. 00046

ln!iiulator moctet

I. 4:19
1.190

Insulator model

'i me/ J\1

lnauL'ltor model
Relative surface
amplitude
m Cil l m (0)

lnsulntor model
'\ power
ab~orbed

tnsu lntor model

6.559

125RR.G

1.149
1.122

:1.67:1
3. 679
3.679
3.679
3.679

6.559

9. 561
9. !'ifll
9. !i61
9. 561
9. 561

1.122

3.679

6. :'"J59

9. 561

1.891

8.469

2 5.9
67. a
83.4
85.5
98.3

17.9
21.4
21.8
21.9
2 1.9

7.6r.
8. 21
8. 27
8. 211
8.28

3.96
4.11
4 . 12
4.1 2
4.1 2

71.0
94.6
97.4
97.7
99.R

10.0
10.4
10. ~.

100.0

21.9

8.29

4.1 2

100.0

lO. !i

0.071
0.277
0.322
0.327
0.422

- o. 840
-0.837
-0.837
- 0.8:17
-0.837

0. &40
0. &40
0. &40
0. &40
0. &40

-0.970
-0.970
-0.970
-0.970
- o. 970

0. 702
o. 726
0. 728
0. 728
0. 73 4

-0.979
-0.979
-0.979
- o. 979
- o. 979

0. 423

-0.837

o. 940

-0.970

o. 734

-0.979

:19.1
RO. !i
911.4
99. 11
911.4

3. 26
3. &4
4. 01
4. 01
4.03

0. 50

0.~

0.1:1
0.1:1
0.13
0 .1 3
0. 1 3

81.7
97.8
99.11
100.0
99.>1

0.66
0.69
o. 70
0. 70
o. 711

100.0

4. 04

o. ro4

o. 13

100.0

o. 70

t .tr,:l

k (cm· 1)xlo-'

790 ~

202:1 .\

6. 559
6.!i59

6.559

o. a:t

o. r..t

o. ro4

1.9fi6
1.901
I. 897
1. 897
1.891

11.469
8.469
8.469
8.4 69
11.469

10.!i
lO. !i

-137100%

-o.;...;..;.
--20001..

- - - eooA.

Nom•nal Conducflvlfy of Permalloy~

(D

~ 10•1.
(/)

(D

cr

2nd Mode

a..

3w
cr

TABI.E IV. Predicted mode properties In parallel resonance.
Loss par~metcr

Mode No.

H. fOe)

6 >< 1 0 11
6 >

o. 0046
0. 0046
0. 0046
o. 00046

862.04
862.28
862.28
862.28

602,3
615. u
6 1 7.9
618.t2

1161 , 34
1161.:1:1
1161.:13
861 . :1:1

fnftulator model

1162.28

6t8.12

AGI. 33

r. )(to"

lnsulatot· model

41n Mode
1016
CONDUCT lVI TY

FIG. 2. Variation of mode Intensities with conductivity for
case of perpendlcu tar re80nance.

C: m t/!tf
lnoulalor model
Relative
surfnee
am plitude
,.,(jL)/ m (O)

'f power
absorbed
Insulator model

proxlmation. Five quantities are compared: applied
field (mode location), wave number k, magnetlzatlon rf
amplitude, surface pinning, and power absorbed. As
remarked earlier, only the first and last of these are
exper imentally observable. It is Immediately obvious
that there Is no slgnl!lcant difference In mode location.
The main resonance for the thickest film is given within
0. 2 Oe by the Insulator theory. The second mode Is the
one which Is most affected by conductivity effects, since
it rides well up on the side or the main resonance for
thick films. Even so , it is within 0. 1 Oe for 790 A, and
the approximation deviates by only 1 . 5 Oe for 2000 A.
Reducing conductivity by a factor of tO brln~s the two
models In agreement within 0. 2 Oe even for this mode
of the thick film.
On the other hand the power absorbed by the 2000-A
film exhibits serious disagreement (factor of 2. 5) for
the first mode, moderate disagreement (20':; ) for the
second mode, and good agreement only for the higher
modes. In the case of the 790-A film, even though the
first mode is located by both models within 0. I Oe, the
amplitude of resonance Is in disagreement by nearly
201 . However the amplitude of higher modes Is In excellent ar:reement. Data for power absorbed by all
modes Is shown In Fig. 2.
Results for parallel resonance are shown In Table IV.
There are fewer modes than for perpendicular resonance, but the results are in similar agreement except
for two features: (t) the second (symmetric) mode of
the thick film is In significant disar:reement (t 6 Oe) for
basic conductivity; (2) except for the lowest loss cases,

0.433
0.433

:1.04 8

o. !i07
o. 5nG

44.6
89.0
99.9
98.7

2.8
3.11
4.0
4.1

83.7
98.0
99.9
99,7

4.0

t OO'T

tOO'l

0,696

-o. 997
-0.997
-o. 996
-o. 990

I, 019
t.019
1. 0 19
l. 019

1.089 -0.990

1. 0 19

1.097
1.089
l, 088
1.089

Insulator model

..

3.152
:1.066
3.058
3,048

• 0,625

k (cm• 1) >

l·t.

790 J..

6 >

-~f.------·-- _ _2~~0~
3rd Mode

Thickness
202:1 J..

44. 8
89.0
99. 8
98.7

o. oe•
o.o6•
o. oG•
0.074

10G'f

0. 07:J'l' 10Cff

83.6
97.9
99.8
99.6

"The two k va lue• are roughly equal In elze eo that It Is not
possible to assign the deelgnatlon "spin-wave" component to

either va lue.
bMode 3mplltude ext•·emely small , and shnpe o r resonance
highly aoymmetrlc so that amplitude can only be estimated.

the main mode in parallel resonance Is characterized by
three k values, two of which are roughly of equal size.
Neither can be c learly designated as belonging to the
"spin- wave" component. Nevertheless, the observable
quantities, mode location and power ai.Jsorbed, are obtained for thP main mode with essentially the same accu racy as an LhP. case or perp4'•ldlcul~r resonance. Althou~h the field location or this not obtaint>d with hil(h accur.wv, Its amplitude is less
than 0. 1';· . It ·~un l ikely that iconsidered in attempting an accurate measurement of
exchange constant .
Study of Tables I-IV and Fi~. 2 shows that the
conduchvltv must tw decreased by a factor of 10 before
the first mode amplitude is predicted with tO'.l' accuracy
at 2000 A. For ROO A, reduction or u by a factor of 2
will give tO'~ accuracy. In any case the excellent a~ree ­
ment at low conduclJv ity in all tabulated data of Tables
Ill and IV demonstrates the validity In the correlation
between the components of the two solutions discussed
earlier, and the validity or the approximations used in
the insulator model at least in the range of magnetic
losses up to those observed In Permalloy films.

-138•z. Fratt, Phye. Status Solidi 2, 141 7 09621.
'C.II. Wilts and S. Lal, IEEE Trans. Magn. MAG -8, 280
(1972).
sc. H. Bajorek and C. ll. Wilts, J. Appl. Phys. 42, 4324
(1971).
'P.E. Wigen, C . F. Koot, and M . R. Shanabarger, J. Appl.
Phys. 36, 3302 (1964).
'lit. SJ)3rke , Phys. Rev. Lett. 22, 1111 (19691.
"w.S. AmentandG.T. Rado, Phys. Rev. 97,1558 (19~5).
'J R. MacDonald, Phys. Rev. 103, 280 0956).
•c. Vlttorla, G.C. Dailey, R.C. Barker, and A, Ye1on,
Ph yo. Rev. B 7, 2112 (1973).

'o. C. Batley a nd C. Vlttorla, Phys. Rev. D 8, 3247 0973).

••c. Vlttorta (prtv:\te commu nication).

11The lAndau- Lifshitz (LIJ and Gilbert damping Ierma are

exactly equivalent If In Eq. 0) one usee y for the LL case
and replaces It by y(l + o-2) for the Gilbert case. For ferromagnetic metals this represents an Insignificant change In
gyromagnetlc ratio of the order of I part In 10 000. The
G ilbert form Is somewhat easter to manipulate In the present
context.
11 H. Puzkarakt, Acta Phys. Pol. A 38. 217 (1970); 38 . H99
(1970).

-139Appendix IV
IV-1.

Anisotropy Models
The perpendicular uniaxial anisotropy has been considered by

many workers; it has been shown by Bailey et al (1973) that different
or asymmetric surface anisotropies can be chosen in order to match
experimental data in a set of permalloy films.

When considering method

one (presented in the text) Eqs . (4-6) and (4-7) are given by

2Z h+ . =
0 Xl

n=l

hxn (Zo + Zn) eikndl2

2Z h+ . = E
o y1
n=l
2Z h-. =
0 Xl

hxn vn

n=l

2Z h-. = E
o y1
n=l

(Z + z ) eik dl2

IV-la
IV-lb

(Z _ z ) e-ik dl2
o n

IV-lc

hxn vn (Zo- Zn) e-ikndl2

IV-ld

hxn

Zn = ick n141To '
IV-le
For the resonance calculations it was pointed out in section (4.1) that
h+. = h- . = h/2
Xl

Xl

IV-1 f

h+. =

h~i =0

IV-lg

Yl

For transmission calculations (sometimes called antiresonance
calculations), the following conditions are imposed (Liu (1974))
h+. = h 12
Xl

IV-2a
IV-2b

The magnetic boundary conditions given by Eq. (3-7) can be written as
IV-3a
0 = E Q h (ik + (K 1A) cos 2e ) eikndl 2
xn
n=l
0 = E Q v ' h ( ik + (K 1A) cos2e ) eikndl 2 IV-3b
s1
n=l n n xn

-1408

0 = E Qn hxn ( -ik n + (K s2 /A) cos 2e) e-iknd/ 2
n=l

IV-3c

0 = E Qn vn hxn (-ik n + ( Ks2/A) cos2e ) e-iknd/ 2
n=l

IV-3d

v i = m /m~ = i-rrl/n l = - v/cos e
IV-3e
A computer program has been written that solves the above equations for
the eight unknowns (eight hxn), and calculates the power absorbed by
Eq. ( 4-4) .
If the tensorial anisotropy (Eq . (3-11) were to be considered
Eq . (IV-3) would be replaced by the following equations.

0 = E Qn hxn (ik n + ~ 1 (e )/A) eiknd/2
n=l

IV-4a

0 = E Qn vn hxn (ik n + KTl (e)/A) eiknd/ 2
n=l

IV-4b

0 = E Qn hxn(-ikn + KT2( e )/A) e-iknd/2
n=l

IV-4c

0 = E Qn v ~ hxn (-ikn + KT2( e )/A) e-iknd/2
n=l

IV-4d

If method two is to be used Eqs. (IV-1) are replaced by Eqs. (4-13) .
The magnetic boundary condition equations remain unchanged .

IV-2.

Perpendicular resonance with asymmetric anisotropies
In this orientation it has been shown that the polarization

of the fields associated with a specific wave vector have either
positive or negative circular polarization; and the linearly polarized
input can be resolved into two oppositel'y polarized circular waves of
half magnitude .

Further, the two polarizations are completely

uncoupled; one is made up of the positive precession wave vectors and
the other the negative precession wave vectors.

The wave vectors

-141are given by solutions to Eq. (2-18).

The development is the same for

both the positive and negative polarizations, therefore only the
positive precession is treated.

The Ks can be either the energy

associated with either the perpendicular uniaxial or tensorial
anisotropies.
The six boundary conditions come from the continuity of
tangential components of hand e and pinning conditions on m at each
surface.

Since the polarizations are circular only the equations for
h+ h;, ey+-ey- are used. The equations are':
x'
E X

n=l n
-E X

n=l n

eiknd/2

= h+X = .5

zn sin(iknd/2) = e;-e;

E Q

xn (ik n + K1/A) eiknd/

e -ik nd/2 = h-:x = .5

n=l n
l: X

n=l n

n: lon

IV-5a

(-ik

IV-5b
2 = 0

IV-5c
IV-5d

+ K /A) e-iknd/ 2 =0

IV-5e

The power absorbed per 2unit area is then given by
P = Re [ s_ _Q_ ( e + - e-) ]
8 2

IV-3.

Surface Layer Model
The surface layer model is a simple extension of the cases

already presented; it simply involves more unknowns and hence more
equations.

The mathematics are the same .

are 24 unknowns .

In the most general case there

The magnetic boundary conditions imposed on mat the

interfaces are given by Eq. (3-15).

The spins at the outer surface

-142of the film are assumed free (i.e. dm/dn=O).

In each region of the

film there are eight wave vectors and Eqs. (4-1)-(4-3) apply to each
region.

Since it is a trivial exercise to write down the equations

necessary to solve for the power absorbed, only the case of
perpendicular resonance with asymmetric surface layers is given below.
One surface layer has properties with the subscript f, the other has
properties with the subscript g .

The magnetic boundary conditions at

the z=D/2 interface are
IV-6a
Ab

--

Mb n=l

Qbn bn kbn eikbnD/2

= ~ E Qgn gn kgn
Mg n=l

IV-6b

The magnetic boundary conditions at the z=-0/2 interface are
IV-6c

Ab
Q b k e-ikbn°1 2 = Af
Q f k
Mb n=l bn n bn
Mf n=l fn n fn

IV-6d

The equations specifying the continuity of tangential hat the + and
-0/2 interfaces respectively are
E bn eikbn° 12 = E gn
IV-7a
n=l
n=l
IV-7b
E bn e-ikbn°12 = E fn
n=l
n=l
The equations s pecifying the continuity of tangential e at the + and
-0/2 interfaces respectively are

-1434
orb n=l

orb

n=l

1 4E
g n=l

k eikb D/2
bn bn

gn kgn

IV-7c

1 E4
=· --,.-fn kfn
f n=l

IV-7d

=~

k e-ikbnD/2
bn bn

The equations specifying the free magnetic spins at the z=D/2 + Lg
and z=-D/2 - Lf surfaces respectively are
0 = E Qgn gn kgn eik gn Lg
n=l
0 = E Qfn f n kfne-ikfnlf
n=l

IV-8a
IV-8b

The equations specifying the continuity of tangential hat the
z= D/2 + Lg and z= -D/2 - Lf surfaces respectively are (note that
method two of the text is used here)
. 5 = E gn e i kgn Lg
n=l
.5 = E f n e-ikfnlf
n=l
Finally the condition specifying ey+ - ey- is

ey

ik L
k_
[(l,
E gn kgn e gn g)
ey
41T
ag n=l
- ----.
af

n=l

fn kfn e-ikfnlf)]

IV-9a
IV-9b

IV-10

-144Appendix V

V-1.

Approximations to Boundary Conditions
It was pointed out in section 4.3 that if the boundary conditions

are such that the positive and negative spinwave branches uncouple then
the approximate positive and negative precession wave vectors can be
used to match the boundary conditions separately.

Unfortunately, of

the three models presented only the tensorial model falls into this
class.

Therefore, further approximations had to be made to simplify

the calculations involving the other two.

These approximations are

presented in the following sections.

V-1.1 Uniaxial Perpendicular Anisotropy Approximation
For the uniaxial perpendicular anisotropy model the approximation
was found by applying the similarity transformation U of Eq. (4-17}
to the boundary condition given in Eq. (3-7} .

The result of this

transformation is

KsR sin e

K;

dil+A

. 28
-KSR* Sln
A(l + IRI }
where

lJl

A(l + IRI2}

=0

Ks
dn +Ad

V-1

ll2

cos2e + cos e)/(l + IRI }
K~ = Ks(cos2e + IRI 2 cos 2e}/(l + IRI 2 }

K: = Ks(IRI

The approximation is to drop the off diagonal terms.

The boundary con-

dition for the positive precession spinwave branches is

Ks
dn +Adll2

=0

V-2

-145The approximate power absorbed can now be calculated as outlined in the
text of chapter 4.
V-1.2

Surface Layer Model Approximations
For the surface layer model there are two approximations to Eq.

{3-17) which will be considered.

The first approximation yields an

effective surface anisotropy with easy or hard axis along the
equilibrium direction of the magnetization.

In calculations involving

this effective anisotropy the power absorbed in the surfuces is
neglected; therefore, for thick layers a significant error exists in the
calculated spinwave mode intensities.

The second approximation gives

an approximate boundary condition between the bulk and surface layers
in which the positive and negative precession wave vectors are
uncoupled.

With this approximation the power absorbed due to the

positive precession spinwave

can be calculated.

This calculation

gives an accurate picture of the resonance process.
V-1.2.1

Surface Layer Effective Anisotropy

The first approximation to the surface layer problem is obtained
by dividing the numerator and denominator of the right hand side of
Eq. (3-17) by Abk 2b tan(k 2bD/2), and dropping all terms remaining
with Abk 2b tan(k 2bD/2) in the denominator. The result is
-(klsTls(l + R:Rb)2 + IRs - Rbl2 k2sT2s)
V-3
(JRsl + 1)(1Rbl + 1)
The secular equation for a symmetric film and boundary conditions given
by an anisotropy with easy or hard axis along M0 and energy KL is

-146V-4
Comparing Eq. (V-3) and (V-4) shows that the surface layer can be
approximated by an effective anisotropy energy, KL, given by the
right hand side of Eq. (V-3).

For this approximation of the surface

layer the power absorbed can be calculated in the same manner as
discussed for the other anisotropy models.
V-1.2.2

Surface Layer Approximate Boundary Condition

By making a further approximation to Eq. (V-3) an approximate
boundary condition between the bulk and surface layer can be deduced.
This approximation is to assume that IRs - Rbl 2 = 0. With this
approximation Eq. (V-3) becomes
A k tan(k D/2) =
b lb
lb

-(k

(1 + R R ) 2

s b
ls 1 s
( 1Rsl2 + 1)(1Rbl2 + 1)

V-5

An approximate boundary condition between the bulk and surface m~
which gives this secular equation and the C in Eq. (3-18) with the
same approximation is

( 1 + 1Rsl2)

Mb(l + RbRs)

Mb

dmpb
dz

(1 + RbRs)

V-6a

Ms

dm~s

dz

V-6b

Ms (1 + 1Rbl )
The corresponding expression for m6 is obtained by replacing m~ by m6 /R.
By using Eq. (4-21) the approximate boundary for ~ 2 is
given by Eq. (4-24).

-147-

Appendix

VI

Ferromagnetic Resonance Introduction
The theories of ferromagnetism propose that the magnetization is
due primarily to the magnetic moment of the electron.

Although the

origin of this moment is quantum mechanical in origin, most of the
phenomena involving ferromagnetism can be addressed classically.

In

this approach the ferromagnetic material is characterized by a magnetization, ~. which is associated with an opposite angular momentum
L=M/y

{y

is the gyromagnetic ratio).

The motion of the magnetization

is usually analyzed in terms of the Landau-Lifshitz equation.

This equa-

tion is easily obtained by equating the rate of change of angular
momentum and the torque (MxHeff)
dM
dt
In this equation

= -yMxHeff

Heff is the total effective field acting on

M.

The

sources of these fields are presented in the text of the thesis.
Ferromagnetic resonance is a phenomenon in which the magnetization
of a ferromagnetic sample exhibits a resonance when subjected to a harmonic magnetic field.

The magnetic resonance is manifested by a maximum

in the harmonic response of the magnetization or by a maximum in the
power absorbed from the driving system.

In a resonating elastic system,

the resonances are found at (or near) the eigenfrequencies of the normal
modes of the lossless elastic system.

These modes are strongly depen-

dent on the sample shape and boundary conditions; and the oscillations

-148-

can be treated by an analysis of the elementary excitations, phonon s.
Similarly in ferromagnetic bodies, the resonances are found at (or
near) the eigenfrequencies of the normal modes of the lossless magnetic system.

These modes are also strongly dependent on the sample

shape and boundary conditions.

The magnetic variations can be treated

in terms of elementary excitations; these excitations are magnons in
the quantum mechanical description and spinwaves in the classical
description. The spinwaves are described by functions of the form
m e i (l<·r +·wt) .

To eliminate the shape dependence in ferromagnetic resonance
the samples are usually made in the form of thin films.

These samples

are then driven by an approximately uniform magnetic field applied in
phase at both film surfaces.

In this case the excitations are stand-

ing spinwaves with K normal to the film plane.

Even with this simple

geometry, the mode locations and intensities are dependent upon the
magnetic boundary conditions at the surfaces.

The normal modes of the

system depend not only on the frequency, but on the static magnetic
field which is used to ensure that the magnetic system is not broken
up into magnetic domains and to establish the resonant frequency in a
range convenient for experimental observation.

The resonances can be

excited as "lines" of the spinwave spectrum by sweeping the frequency
at fixed magnetic field, or by sweeping the magnetic field at fixed
frequency.

For reasons of experimental convenience and accuracy the

latter scheme is almost invariably used.
Spinwave spectra have been investigated for a number of reasons.
In principle they provide one of the most accurate means of determining a number of the fundamental magnetic constants: saturation

-149-

magnetization, M , gyromagnetic ratio, y , exchange constant, A , to
name just a few.

They also provide a powerful method of studying

relaxation or loss processes in magnetic materials.

However, to

exploit the potential accuracy of this method in almost all of these
applications it is necessary to have an accurate analytic statement
of the boundary condition at the surface of a ferromagnet.

It is

surprising that this understanding remains elusive after 15 or 20
years of continuous research effort.