CHAPTER II

ELECTROMAGNETIC WAVE PROPAGATION IN

FERRITES AND FERRITE/DIELECTRIC SYSTEMS

2.1 Introduction

2.2 Ferrite Permeability Tensor

2.3 Propagation in an Unbounded Ferrite Medium

2.3.1 Propagation Parallel to the Direction of Magnetization

2.3.2 Propagation Perpendicular to the Direction of Magnetization

2.3.3 The Effect of Damping

2.4 Regimes of Wave Propagation

2.5 Propagation in Magnetized Ferrimagnetic Films

2.6 Method of Analysis: An Overview

2.7 Propagation in A YIG Film: Theory

2.7.1 Results and Discussion

2.8 Propagation in Ferrite/Dielectric/Ferrite Structure

2.8.1 Derivation of Dispersion Relation

2.8.2 Results and Discussion

2.9 Summary

References

2.1 INTRODUCTION

Since ferrites are highly insulating and have significant amount of

anisotropy at microwave frequencies, useful interaction between the

magnetic properties of the material and electromagnetic waves can be

expected. The anisotropic properties of ferrites can be understood by

treating the spinning electron as a gyroscope as mentioned in the first

chapter. If a static magnetic field is applied to the ferrite medium, an

electromagnetic wave will propagate in it differently in different directions

or the wave propagation in magnetized ferrites are nonreciprocal in nature.

This effect is utilized to fabricate directional devices such as isolators,

circulators and gyrators. Also the interaction of ferrimagnetic material with

the propagating fields can be controlled by adjusting the strength of the

bias field. This effect leads to variety of control devices such as phase

shifters, switches and tunable resonators and filters [2-4].

This chapter is meant to systematically represent the interaction

between electromagnetic wave and magnetized ferrimagnetic materials. In

the following section, the electromagnetic modelling of a ferrite medium

under magnetization through its permeability tensor (Polder tensor) is

undertaken. In section 3, propagation in an unbounded ferrite medium is

discussed. Different regimes of wave propagation in ferrites are introduced

in section 4. Propagation in magnetized ferrimagnetic films is discussed in

23

section 5. In section 6, an outline of the method of analysis employed in

this study is given. Detailed discussions of wave propagation in a

standalone ferrite film and in a ferrite/dielectric/ferrite hybrid structure are

presented in sections 7and 8 respectively.

2.2 FERRITE PERMEABILITY TENSOR

It was D. Polder who very cleverly accommodated all the specialities of

magnetized ferrites into a tensor called the Polder tensor through his paper,

“On the theory of ferromagnetic resonance”, in 1948 [1]

Consider the spinning of an electron in a ferrite sample which is

subjected to a static magnetic field B0 in the z) direction and to a time

varying field H of time dependence , where ‘tje ω ω ’ is the signal frequency

and 1−=j .

The field zBo) will generate a saturation magnetization zM )

0 in the

sample. The time varying field H will also generate a magnetization Mac.

Therefore the total magnetization in the sample is

Mt = zM )0 + Mac (2.1)

The total applied field intensity is

Ht = zH )0 + Mt ( 2.2)

Hence the total magnetic field induction in the sample is

B = 0μ (Ht + Mt), (2.3)

24

where, 0μ is the permeability of free space.

The torque exerting on the spinning electron by the applied sinusoidal

field is

τ = m × B (2.4)

where, m is the magnetic moment of the electron.

If p is the angular momentum of the electron and γ = 1.759 × 1011

C/Kg, is the gyromagnetic ratio, then equation (2.4) becomes

τ = γ− p×B (2.5)

But, τ = dp/dt (2.6)

From equations (2.2), (2.3), (2.4), (2.5) and (2.6)

dMt /dt = -γ 0μ Mt × Ht (2.7)

The total magnetization is also varying with time dependence .

Also the components of AC field are very small when compared with the

static field. Considering these factors into account, the component form of

equation (2. 7) is

tje ω

ωj Mx = - γ 0μ [Mx H0 – Hy M0]

ωj My = γ 0μ [Mx H0 – Hx M0]

ωj Mz = 0.

On rearrangement, one may write

25

ωj Mx + γ 0μ H0 Mx = γ 0μ M0 Hy

- γ 0μ H0 Mx + ωj My = - γ 0μ M0 Hx

ωj Mz = 0. (2.8)

The first two equations of the above set are simultaneous equations in Mx

and My and on solving them one shall arrive at

Mx = χ Hx – j κ Hy

My = j κ Hx + χ Hy

Mz = 0 (2.9)

where,

22 ωωωω

χ−

=o

mo , κ = 22 ωωωω−o

m ,

with 0ω = γ 0μ H0 and =mω γ 0μ M0. (2.9a)

On using equations (2.2) and (2.9), equation (3) can be modified into

B = oμ rμ , ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+ 0HHHH

z

y

x

where, the relative permeability called the Polder tensor is given by

rμ = , (2.10) ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+−+

1000101

χκκχ

jj

26

It is now possible to represent the interaction of an electromagnetic

field with a magnetized ferrite by using the Polder tensor (2.10) as the

material permeability in a natural way.

It is quite clear from the permeability matrix that when the ferrite

material is magnetized to saturation with a static field, the magnetic

permeability of the ferrite becomes anisotropic with rotational symmetry

about the direction of the DC magnetic field. Also it may be noted that all

the magnetic properties of a ferrite material is contained in the permeability

matrix and hence the properties of ferrites can be manipulated as and when

required by manipulating the values of the static field, the frequency of the

wave field, direction of application of the static field etc. Using the Polder

tensor, the propagation of electromagnetic wave in an un bounded ferrite

medium can be studied as has been done in the following section.

2.3 PROPAGATION IN AN UNBOUNDED FERRITE MEDIUM

In this section, the wave propagation in an unbounded ferrite medium which

is under an applied magnetic field is attempted. Three cases are separately

studied: (1) propagation parallel to the biasing field, (2) propagation

perpendicular to the biasing field and (3) effect of damping.

27

2.3.1 PROPAGATION PARALLEL TO THE DIRECTION OF MAGNETIZATION

An electromagnetic wave of angular frequency ω is supposed to be

propagating in the Z direction with field dependence, , in an

unbounded, Z direction magnetized ferrite medium.

)( kztje −ω

In this context, 0=∂∂

=∂∂

yx, and using equations (2.9) and (2.10),

components of Maxwell’s curl equation for electric field E can be written as

β Ey = - ω 1μ Hx + jω 2μ Hy

β Ex = jω 2μ Hx + ω 1μ Hx

0 = Hz (2.11)

where, 1μ = )1(0 χμ + and κμμ 02 = .

Components of Maxwell’s curl equation for magnetic field, H are

β Hy = ω ε Ex

β Hx = - ω ε Ey

0 = Ez (2.12)

where ε is the permittivity of the ferrite medium.

From equations (2.11) and (2.12) we get,

± ( β2 _ ω 2 1μ ε )2 = ω 4 ε 2 22μ

Or β2 = ω 2 ε ( 1μ ± 2μ )

For a wave propagating in the positive Z direction,

28

β = ω )( 21 μμε ± (2.13)

Let the two values of β are

βL = ω )( 21 μμε + and βR = ω )( 21 μμε −

when β = βL, Ex = - j Ey and

when β = βR, Ex = j Ey

Therefore βL and βR correspond to two circularly polarized waves

whose E phasors are moving in the clockwise and anticlockwise directions

respectively. These two circularly polarized waves are considered as

fundamental modes of propagation in an infinite ferrite medium. In short,

the transmission of a linearly polarized wave through a ferrite medium can

be represented in terms of the two fundamental modes characterized by βL

and βR.

The respective effective permeabilities are

= + = +μ 1μ 2μ oμ [1 + ωm /(ω0 - ω )] (2.14)

and = - = −μ 1μ 2μ oμ [1 + ωm /(ω0 + ω )] (2.15)

Variation of μ+ and μ- with frequency is shown in Fig.2.1 It is quite

clear that μ+ alone is having a resonance at ω = ω0. Existence of different

propagation characteristics for two circularly polarized waves of the same

frequency having opposite sense of rotation of E vectors is a significant

feature of wave propagation through ferrites.

29

0 4 8 12 16 20

-80

-40

0

40

80

1

2

Perm

eabi

lity

Frequency, G Hz

Fig.2.1. Variation of μ+ (curve 1) and μ - (curve 2) with frequency.

Since these circularly polarized waves have different values of

propagation constants ( and ), the two electric field vectors rotate at

different rates. The electric field vectors of these circularly polarized waves

combine to produce a maximum, when sum of phase shifts experienced by

the two waves is 4

Lβ Rβ

π radians. The corresponding distance defines the

wavelength ‘λ ’ for the linearly polarized wave. One may write

30

Lβ λ + Rβ λ = 4π

Or λ = RL ββπ+

4 = )/2()/2(

4RL λπλπ

π+

(2.16)

where,

= and = . Lλ )/2( Lβπ Rλ )/2( Rβπ

When the wave travels through one wavelength, electric field vector,

E of the linearly polarized wave rotates through an angle φΔ given by

φΔ = 2

λβλβ RL − (2.17)

The rotation of the direction of electric field, E of a linearly polarized

wave passing through a magnetized ferrite medium is known as Faraday

rotation. This phenomenon is analogous to the rotation of plane of

polarization of light when travelled through paramagnetic liquid [5].

The property of Faraday rotation in ferrites is the basis of

nonreciprocal propagation in magnetized ferrite and their applications.

31

2.3.2 PROPAGATION PERPENDICULAR TO THE DIRECTION OF MAGNETIZATION

Consider the propagation of the plane electromagnetic wave through an

infinite ferrite medium magnetized in the X direction. Let the wave

propagates in the Z direction.

Since 0=∂∂

=∂∂

yx, then from Maxwell’s curl equation of electric field

E,

β Ey = - oμ ω Hx

-j β Ex = -j ω (μ Hy – j κ Hz)

ω κ Hy = j ω μ Hz (2.18)

From Maxwell’s curl equation of Magnetic field H,

j β Hy = j ω ε Ex

-j β Hx = j ω ε Ey

Ez = 0 (2.19)

From equations (2.18) and (2.19), one shall get

β2 Ey = ω2 μoε Ey

μ( β2 –ω2 μ ε )Ex = - ω2 κ2ε Ex (2.20)

One solution to (2.20) occurs for

βo = ω εμo (2.21)

32

with Ex = 0 and hence Hy = 0. Then the complete fields are

oE = zjo

oeEy β−)

oH)

= zjoo

oeYEx β−− ) (2.22)

The admittance is

Yo = ωε /β = oμε (2.23)

This wave is called the ordinary wave, because it is unaffected by the

magnetization of the ferrite. This happens whenever the magnetic field

components transverse to the bias direction are zero ( Hy = Hz = 0 ).

Another solution to (2.20) occurs for

eβ = ω εμv , (2.24)

with Ey = 0, where μv is called the effective permeability of ferrite, given

by

vμ = μκμ 22 − (2.25)

This wave is called the extraordinary wave and is affected by the

magnetization of the ferrite.

The complete field is given by

oe ExE ))= e- j βe z

33

eH)

= EoYe ( μκjzy )) + ) e- j βe z , (2.26)

where, Ye = eβ

ωε = eμε

These fields constitute a linearly polarized wave, but note that the

magnetic field has a component in the direction of propagation. In addition

to an Z component magnetic field, the extraordinary wave has electric and

magnetic fields of the ordinary wave also. Thus, a wave polarized in the Y

direction will have a propagation constant βo (ordinary wave), but a wave

polarized in the X direction will have a propagation constant βe (extra

ordinary wave). This effect, where the propagation constant depends on the

polarization direction, is called birefringence. Birefringence often occurs in

optics, where the index of refraction can have different values depending

on the polarization.

2.3.3 THE EFFECT OF DAMPING

Gyromagnetic resonance occurs when the forced precession frequency, ω

is equal to the free precession frequency, 0ω = γ 0μ H0. In the absence of

loss, the response may be unbounded in the same way as in the case of an

inductance-capacitance resonant circuit. But all ferrite materials have

various loss mechanisms that damp out such singularities. If we include the

34

damping aspect into the picture, then χ and κ appearing in equation

(2.10) take complex forms like [5]

χχχ ′′−′= j (2.27)

κκκ ′′−′= j (2.28)

with = 'χ( )( )( ) 222

02222

0

220

2200

41 αωωαωω

αωωωωωωω

++−

+− mm (2.29)

= ''χ( )( )

( )( ) 2220

22220

2220

41

1

αωωαωω

αωωαωω

++−

++m (2.30)

' =κ( )( )

( )( ) 2220

22220

2220

41

1

αωωαωω

αωωωω

++−

+−m (2.31)

''κ = ( )( ) 222

02222

0

20

41

2

αωωαωω

αωωω

++−m (2.32)

where, 0ω and mω are defined in equation (2.9a) and α is a dimensionless

damping constant.

Most ferrites used for microwave applications have a low loss at

microwave frequencies. In that case α <<1 and the resonance frequency is

again at ω0.

The imaginary components of χ and κ give a measure of the power

absorbed in the ferrite due to resonance. The resonant line width is the

width of the resonance curve where the magnitude of χ ′′ and κ ′′ is half it’s

35

peak value. Variation of real susceptibility with frequency is plotted in Fig.

2.2, for α = 0.003. From equation (2.30) or (2.32), one may get

ωΔ = γ HΔ = 2 0ω α , (2.33)

so that damping constant can be obtained from measured quantities.

Typical linewidths range from less than 100 Oe (for YIG) to 100-500 Oe

(for ferrites); single crystal YIG can have a linewidth as low as 0.3 Oe.

0.0 0.4 0.8 1.2 1.6 2.0

-20

0

20

Normalized Frequency, 0/ωω

Nor

mal

ized

real

susc

eptib

ility

,

Fig.2.2 Variation of real susceptibility with frequency

36

2.4 REGIMES OF WAVE PROPAGATIN

Eventhough, the precession of magnetization in any small region of ferrite

is assumed to be uniform and any variation of the microwave field is

negligible, there are other modes of motion of the magnetization which

vary with very short wavelengths within a small region of the ferrite. They

are called spinwaves [7]. They can be excited in the ferrite when the

microwave magnetic field intensity exceeds a certain critical field value.

They contribute to the attenuation in the ferrite and to nonlinear effects at

high peak power [8]. For a given frequency, the critical magnetic field

varies with static bias magnetic field. The minimum microwave critical

field is given by [9]

Hc = 2/12 ])2/(1[)2/(1)]/(1)[/(2

ωωωωωωωω

mm

omkH++−−Δ , (2.34)

where, is the spinwave linewidth and other symbols are already

defined in section 1 of this chapter.

kHΔ

The wavelength of spinwaves is primarily determined by the effective

exchange field which arises from the exchange energy and aligns the

electron spins in a magnetic material. Under the influence of an external

disturbing magnetic field, the electron spins precess as a single unit and it

is this precession which gives the ferrites their useful magnetic properties.

37

If the uniformity of the motion is slightly disturbed, as always occurs

owing to thermal agitation, strong demagnetizing and exchange fields are

generated. Under certain conditions, these local disturbances can grow at

the expense of the external disturbing magnetic field and will then

propagate through the ferrite as spinwaves. They have a short wavelength

so that they may be analyzed as plane waves even within a small ferrite

sample.

If the amplitude of the applied ac field is greater than the critical

value, spinwaves are excited and they absorb power from the field and heat

up the ferrite material. The frequency of oscillation of these modes depends

on the size and shape of the sample. Their wavelength is much smaller than

the wavelength of any electromagnetic wave. They are called

magnetostatic modes [MSW], because the rate of change of magnetic field

with regard to time is almost nil [4, 7, 10] and hence they are characterized

by the magnetostatic form of Maxwell’s equations.

∇ . B = 0

∇ B = 0 (2.35) ×

where, B and H are the magnetic vectors of the disturbing electromagnetic

field.

In table 2.1, given below, the different kind of propagations of

electromagnetic wave in ferrimagnetic medium is given.

38

Table.2.1: Comparison of different kinds of waves in a ferrite medium

Type of waves Specialities

Electromagnetic waves

Both electric and magnetic dipolar

interactions are important but exchange

interactions are negligible. No frequency

degeneracy.

Spin waves

No frequency degeneracy. Wave length

much less than the free space wavelength.

Only the exchange interactions are important

Magnetostatic waves

Shortest wavelength of all the three kind.

Magnetic dipolar interactions dominates both

electric and exchange interactions.

Frequency degeneracy is there [7].

If the wavelength of the signal is not very much altered in the

medium, there will not be any change in the nature of the entering wave.

But if the propagation is such that the wave length is much less than the

free space wave length, then the exchange interactions will control the

propagation and the propagating wave is called spin waves as we have

seen. But if the propagation is such that the propagation constant is very

large and hence the wave length is very small comparable with the

dimensions of the ferrimagnetic medium and in that case magnetic dipole

interaction dominates both electric and exchange interactions and the mode

39

of propagation is called magnetostatic. If ‘k’ is the propagation constant in

the ferrite medium, then for the magnetostatic mode, ×∇ H varies with 1/k

[7] and since ‘k’ is very large, ×∇ H 0, and that is why this mode is

called magnetostatic. It was L. R. Walker who first investigated this kind

of wave propagation [11].

→

2.5 PROPAGATION IN MAGNETIZED FERRIMAGNETIC

FILMS

Study of MSW propagation in an unbounded ferrimagnetic film is well

covered in the literature [10-17]. Three pure MSW modes exist depending

on the orientation of the bias magnetic field relative to the YIG film and the

propagation direction. These modes are: magnetostatic surface waves

(MSSWs), magnetostatic forward volume waves (MSFVWs) and

magnetostatic backward volume waves (MSBVWs). For MSSWs, the film is

tangentially magnetized and the propagation vector k is perpendicular to the

biasing magnetic field. If the propagation vector is parallel to the bias field

in the case of a tangentially magnetized film, the mode is MSBVWs. But if

the field is normally magnetized, irrespective of the propagation direction,

the mode will be MSFVWs. All the three modes are dispersive and the

dispersion can be controlled by controlling the boundary conditions [4].

Between the structures supporting magnetostatic volume waves and

magnetostatic surface waves, there is an important difference. In films

40

supporting magnetostatic volume waves, the potential function is periodic

through the thickness of the film and hence volume waves exhibit multiple

thickness modes, where as in films supporting magnetostatic surface

waves, the potential function is not periodic through the thickness of the

film and hence surface waves do not exhibit multiple thickness modes.

Another important point is that for the surface mode, wave amplitude

decays exponentially from the surface [7].

2.6 METHOD OF ANALYSIS: AN OVERVIEW

In all the structures studied, magnetization is tangential to the ferrimagnetic

(YIG) films and propagation is considered perpendicular to the direction of

magnetization and tangential to the plane of magnetization.

In this work altogether ten structures are analyzed. Derivation of

analytical dispersion relation corresponding to transverse electric wave

(TE) propagation in each structure, starting from Maxwell’s equations and

finding its numerical solution is the principal procedure of analysis

adopted. Each dispersion relation is analyzed in detail to study the

influence of structural parameters on propagation in the respective

structures. A number of curves displaying various aspects of propagation

are plotted for each structure.

41

In the coming sections of this chapter, propagation of transverse

electrical wave in a direction perpendicular to the direction of

magnetization in a tangentially magnetized ferrite film and that in a hybrid

structure consisting sandwiched dielectric film in between ferrite films is

undertaken. The influence of structural parameters on the propagation is

studied in detail and presented.

In the third chapter, four ferrite-superconductor hybrid structures are

studied and in each case the dispersion relation is derived. In addition to

the dispersion curves a number of curves exhibiting different aspects of

wave propagation in the structures are displayed.

In the fourth chapter, nonlinear wave propagation in three ferrite –

nonlinear dielectric structures are undertaken. Here also from Maxwell’s

equations, dispersion relations are obtained and computation is done.

Dispersion curves corresponding to different structural situations are

plotted. The main thrust given in the fourth chapter is for tunability (the

dependence of power propagation on the applied magnetic field and on the

frequency of the propagating signal) and nonreciprocal effect of power

flow in nonlinear structures. For this, expressions for power of the

structures are formulated and computation of power is done and curves

displaying different aspects of power flow in the structures are given.

42

In the fifth chapter, a microstrip line and a slot line on magnetized

ferrite substrates have been analyzed using spectral domain approach.

Dispersion has been examined for forward and reverse propagations and it

is seen that the propagation is nonreciprocal in both the structures

2.7 PROPAGATION IN A YIG FILM: THEORY

An yttrium iron garnet film of thickness ‘ ’ magnetized in the X direction

with magnetic field B is considered. Let a TE wave of angular frequency

1f

ω is propagating in it, in the Z direction with field dependence, . )( kztj −ω

e

Y

Air

X

B

Fig.2.3. Tangentially magnetized ferrite film.

With the above configuration, the electric and magnetic field

components of TE wave are ( )0,0,xE and ( )zy HH ,,0 respectively and x∂∂ = 0.

The Maxwell’s curl equations corresponding to wave propagation in the

YIG film can be written as

YIG Film

Air

y =f1

43

E = ×∇ 0ωμj− [ ]fμ H (2.36)

H = ×∇ fj εωε 0 E (2.37)

where, [ ]fμ = , (2.38) ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

− ηκκη

jj

00

001

is the polder tensor of the YIG film with χη += 1 . All the terms in the

polder tensor are defined in equation (2.9a). fε is the relative permittivity

of YIG and 0ε is the permittivity of free space. Therefore,

[ = ] 1−rμ 22

1κη − ⎟

⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−ηκκη

jj

00

001, (2.39)

For the configuration in Fig. 2.3, one can write

= ×∇

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

∂∂

−

∂∂

∂∂

∂∂

−

00

00

0

y

z

yz

(2.40)

From equations 2.36 and 2.37, one can have,

×∇ [ ] 1−rμ ×∇ E = E (2.41) rεεμω 00

2

On using equations 2.39, 2.40 and E = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

00

xE( )kztje −ω , one shall arrive at

44

×∇ [ ] 1−rμ ×∇ E= 22

1κη −

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛∂∂

−∂∂

∂+

∂∂∂

−∂∂

−

00

2

222

2

2

yE

zyE

jyz

Ej

zE xxxx ηκκη

(2.42)

= 22

1κη −

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛∂∂

−

00

2

22

yE

Ek xx ηη

(2.43)

since, z

Ex

∂∂ = - (2.44) xjkE

Therefore equation 2.41 becomes

( xvrx Ek

yE

μεεμω 0022

2

2

−−∂∂ ) = 0 (2.45)

where, ηκημ

22 −=v (2.46)

and it is called the effective permeability of the ferrimagnetic medium.

Let, = fk vrk μεεμω 0022 − (2.47)

Now equation 2.45 becomes,

xfx Ek

yE 2

2

2

−∂∂ = 0, (2.48)

Equation 2.48 is quite a general one and it can be used to represent the TE

wave propagation in any medium, of course with appropriate change in .

For example, in air medium, equation 2.48 becomes

fk

45

xxx Ek

yE 2

2

2

−∂∂ = 0, (2.49)

with, = xk 0022 εμω−k , (2.50)

since ,1=η 0=κ and 1=rε for air. In any isotropic medium with relative

permeability, dμ ( dμη = and 0=κ ), and relative permittivity dε , equation

2.48 becomes

xdx Ek

yE 2

2

2

−∂∂ = 0, (2.51)

with, = dk ddk μεεμω 0022 − (2.52)

From equation 2.36,

H = 0ωμ

j [ ] 1−rμ ×∇ E (2.53)

But H = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

z

y

HH0

( )kztje −ω , (2.54)

From equations 2.39, 2.40, 2.53 and 2.54, for the ferrimagnetic medium,

= ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

z

y

HH0

( kztje −ω )

( )220 κηωμ −

j

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

∂∂

−∂∂

∂∂

+∂∂

yE

zE

j

yE

jz

E

xx

xx

ηκ

κη0

(2.55)

46

Therefore, = yH ⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−y

EkE x

fv

x

μμωμ11

0

(2.56)

where, κκημ

22 −=f (2.57)

and Hz = ⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−y

EkEj x

vf

x

μμωμ1

0

(2.58)

Equation 2.55 can also be treated as a general one. For any isotropic

medium, with relative permeability dμ , equation 2.55 reduces to

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

z

y

HH0

( )kztje −ω = d

jμωμ0

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

∂∂

−

∂∂

yEz

E

x

x

0 (2.59)

Of the field components, ( )0,0,xE and ( )zy HH ,,0 in the structure

given in Fig. 2.3, and are the tangential field components at the

interfaces. In the light of the discussions above, the solutions to equation

2.48 for the three regions in Fig. 2.3 can be written as

xE zH

0<yxE = 1A )exp( ykx( )kztje −ω (2.60)

10 fyxE << = ( ))exp()exp( 32 ykAykA ff −+ ( kztje −ω ) (2.61)

1fyxE > = 4A )exp[( ykx− ( )kztje −ω (2.62)

47

where, and are defined in equations 2.47 and 2.50 respectively and

A

fk xk

1, A2 etc are constants.

Using equations 2.58 and 2.59, the tangential component, of the

magnetic field for the three regions in Fig.2.3 can be written as

zH

0<yzH = 0ωμxjk

− 1A )exp( ykx( )kztje −ω (2.63)

10 fyzH << = ( ))exp()exp( 32

0

ykkAykkAjfbfc −+

ωμ( kztje −ω ) (2.64)

1fyzH > =

0ωμxjk

4A )exp( ykx( )kztje −ω (2.65)

where, v

f

fb

kkkμμ

+= and = ckv

f

f

kkμμ

− . (2.66)

The boundary condition to be satisfied at the interfaces of the layered

structure is that the tangential components of the fields must be continuous.

On applying this boundary condition to the configuration in Fig.2.3, one

shall get

=0 (2.67)

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−−−−−−

−−−

)exp()exp()exp(00

)exp()exp()exp(0

111

111

fkkfkkfkkkkk

fkfkfk

xxfbfc

bcx

xff

−− 0111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

4

3

2

1

AAAA

48

The dispersion relation obtained from equation (2.67) is

( )( ))( )( cxbx

cxbx

kkkk −+kkkk +− (2.68) = )2exp( 1fk f

Equation 2.68 governs the propagation of electromagnetic wave in a

tangentially magnetized YIG film.

2.7.1 RESULTS AND DISCUSSION

The dispersion relation is solved numerically. In computation, the biasing

magnetic field is fixed at 0.057 Tesla and the saturation magnetization of

YIG is taken as 0.175 Tesla. The relative permittivity of YIG is fixed at 15.

The gyromagnetic ratio is taken as 1.759 × 10 C/Kg. 11

It has been found that the propagation is reciprocal. Propagation is

considered in a symmetrically bounded ferrite film with air on both sides.

A symmetrically bounded ferrite film can exhibit only the field

displacement nonreciprocity [7]. It means that when the direction of

propagation is reversed, the mode energy shift from one surface to the

other. But in the structures studied in the coming chapters, the ferrite film

is asymmetrically bounded and it has been found that the ferrite film is

showing very good nonreciprocal effect.

In Fig.2.4, the dispersion equation 2.68 is plotted for a film thickness

10 micrometer.

49

0E+0 4E+4 8E+4 1E+5 2E+5

2.0

2.5

3.0

3.5

4.0

Freq

uenc

y, G

Hz

Propagation constant, m -1

Fig.2.4. The dispersion in ferrite film.

In Fig.2.5, the effect of biasing magnetic field on the propagation is

established. The magnetic tuning is also found to be reciprocal for this

symmetrically air bounded ferrite film. Thus it is found that a

symmetrically bounded ferrimagnetic thin film can support surface waves

and that its propagation can be tuned by biasing magnetic field at a given

50

frequency. At a fixed biasing field, frequency tunability is also there. But a

symmetrically bounded thin film doesn’t exhibit nonreciprocal effect

0.052 0.056 0.060 0.064 0.068

0E+0

5E+5

1E+6

2E+6

2E+6

3E+6

P

ropa

gatio

n co

nsta

nt, m

-1

Biasing magnetic field, Tesla

Fig.2.5. Propagation dependence on biasing magnetic filed, plotted for a YIG film of thickness 10 micrometer. The frequency is set at 3.5 G Hz.

In Fig.2.6, the dependence of propagation on ferrite film thickness is

established. The biasing magnetic field is set at 0.057 Tesla. It is found that

the product of propagation constant and the corresponding thickness of the

51

ferrite film is a constant for a particular frequency and biasing magnetic

field.

0 20 40 60 80 100

0E+0

4E+5

8E+5

1E+6

2E+6

2E+6

Prop

agat

ion

cons

tant

, m-1

Thickness of the ferrite film, 10 -7 m

Fig.2.6. The dependence of propagation on ferrite film thickness plotted for a frequency 3.5 G Hz. The biasing magnetic field is set at 0. 057 Tesla.

It is already mentioned in section 2.3.3 that all ferrite materials have

various loss mechanisms and there is the chance of getting the propagating

signal damped. Fig. 2.7 is a plot between the ferrite damping factor α and

52

propagation constant. It is seen from the figure that the wave gets retarded

more and more as the damping factor is increasing.

0.00 0.01 0.02 0.03

2E+4

2E+4

2E+4

2E+4

2E+4

Prop

agat

ion

cons

tant

, m-1

Damping factor, α

Fig.2.7. Effect of damping on wave propagation in ferrite film. The ferrite film thickness is 10 μ m and the frequency is set at 3.5 G Hz

The TE wave propagation alone is considered because as far as

perpendicular propagation in tangentially magnetized ferrite film loaded

layered structures are concerned, the transverse magnetic wave propagation

53

is not tunable and not nonreciprocal. This aspect of the TM wave is

discussed in detail in the coming chapter.

2.8 PROPAGATION IN FERRITE/DIELECTRIC/FERRITE

STRUCTURE

The propagation of magnetostatic surface waves through the structure

shown in Fig.2.8 is undertaken in this section. The structure consists of a

dielectric film of thickness ‘d’ sandwiched between two ferrite films of

thicknesses ‘f1’ and ‘f2’. The two ferrite films are magnetized with fields of

the same strength B but in opposite directions along X axis.

Y Air

y = f1 + d + f2 Ferrite B

y = f1 + d Dielectric y = f1 Ferrite B X

Air

Fig.2.8. Geometry of the ferrite/dielectric/ferrite structure

54

2.8.1 DERIVATION OF DISPERSION RELATION

Consider the propagation of transverse-electric (TE) wave along Z

direction, with field components ( ) ( )( )zyx HHE ,,0,0,0, through the structure

in Fig.3.1. On proceeding in the same line as done in section 2.5.1, from

Maxwell’s equations, the following wave equation for can be derived: xE

2

2

yEx

∂∂ - = 0 (2.69) xi Ek 2

where, for air; xi kk = fi kk = for the ferrite films and for the

dielectric film. , and are defined in equations 2.47, 2.50 and 2.51

respectively.

di kk =

fk xk dk

The solutions to equation (2.69) for the five regions in the structure

can be written as

0<yxE = (2.70) )exp(1 ykA x

10 fyxE << = )exp()exp( 32 ykAykA ff −+ (2.71)

dfyfxE +<< 11 = +)exp(4 ykA d )exp(5 ykA d− (2.72)

211 fdfydfxE ++<<+ = +)exp(6 ykA f )exp(7 ykA f− (2.73)

21 fdfyxE ++> = )exp(8 ykA x− , (2.74)

55

A1, A2 etc. are arbitrary constants. A dependence of all propagating field

components on time t and z through ))(exp( kztj −ω has been assumed

throughout and is not shown explicitly.

The tangential magnetic field component is calculated from the

Maxwell’s equation E =

zH

[ ]rj μ×∇ ωμ0 B and the following expressions

corresponding to the five regions are obtained,

0<yzH = 0

1

ωμAjkx− (2.75) )exp( ykx

10 fyzH<<

= ( ))exp()exp( 320

ykkAykkAjfqfp −+

ωμ (2.76)

dfyfzH+<< 11

= ( ))exp()exp( 540

ykAykAjk

ddd

d −−−

μωμ (2.77)

211 fdfydfzH++<<+

= ( ))exp()exp( 760

ykkAykkAjfnfm −+

ωμ] (2.80)

21 fdfyzH++>

= 0

8

ωμAjkx )exp( ykx− (2.81)

where,

pk = v

f

f

kkμμ

−′

, v

f

fq

kkkμμ

+′

= , = mkv

f

f

kkμμ

−′′

, v

f

fn

kkkμμ

+′′

= ,

κκημ −

=′f22

, κκημ −

−=′′f22

and ηκημ −

=v

22

. (2.82)

56

Now, on imposing the requirement of continuity of tangential

components of electric and magnetic fields at the interfaces, the dispersion

relation can be obtained as:

hkx

hkn

hkm

tkn

tkm

tk

d

dtk

d

d

fk

d

dfk

d

dfkp

fkp

qpx

hkhkhk

tktktktk

fkfkfkfk

xff

ffdd

ddff

xff

ffff

fdff

ekekek

ekekek

ek

ek

ek

ekek

kkkeee

eeeeeeee

−

−

−

−−

−−

−−

−

−−−

−

−−−−

−−−−

−−

00000

0000

0000

0000000000

0000000000000111

1

1111

1111

μμ

μμ

= 0

(2.83)

where, t = f1 + d and h = f1 + d + f2

2.8.2 RESULTS AND DISCUSSION

The dispersion relation (2.83) is numerically solved. Films of standard

ferrite, YIG is taken in the structure. In computation, the biasing magnetic

field is fixed at 0.057 Tesla and the saturation magnetization of YIG is

taken as 0.175 Tesla. The relative permittivity of YIG is fixed at 15 and

that of the dielectric film is taken as unity. The gyromagnetic ratio is taken

as 1.759 × 1011 C/Kg.

The dispersion relation is plotted in Fig.2.9. The ferrite film

thicknesses are 1000 nm for the bottom film and 100 nm for the upper film.

57

The propagation starts at around 3.2 G Hz only. The two ferrite films are

magnetized in opposite directions

0E+0 4E+4 8E+4 1E+5 2E+5

3.0

3.2

3.4

3.6

Fre

quen

cy, G

Hz

Propagation constant, m -1

Fig.2.9. Variation of propagation constant with frequency. The ferrite film thicknesses are 1000 nm and 100 nm respectively and that of the dielectric film is 100 nm.

It has been found in the last section that electromagnetic wave

propagation in an isolated ferrite film is nonreciprocal. But in the present

structure where a dielectric film separate two ferrite films, the propagation

58

is nonreciprocal provided the upper and lower ferrite films are magnetized

in opposite directions. The nonreciprocal effect increases when the two

ferrite films are of unequal thicknesses. In Fig. 2.10, the nonreciprocal

effect of wave propagation in the ferrite/dielectric/ferrite structure is

presented.

3.0 3.2 3.4 3.6

0E+0

1E+5

2E+5

3E+5

4E+5

5E+5

k

(For

war

d) –

k (B

ackw

ard)

, m

-1

Frequency, G Hz

Fig.2.10. The nonreciprocal effect of propagation in the

ferrite/dielectric/ferrite structure. The ferrite film thicknesses are 1000 nm and 100 nm and the dielectric film thickness is 100 nm. The ferrite films are magnetized in the opposite directions.

59

The thickness of the dielectric film in the structure has profound

influence on propagation in the structure and it is displayed in Fig. 2.11.

0 20 40 60

38000

40000

42000

44000

46000

48000

Prop

agat

ion

cons

tant

, m -1

Dielectric film thickness, nm

Fig.2.11. The effect of dielectric film thickness on propagation in the structure. The ferrite films are of thickness 100nm each. The ferrite films are magnetized in opposite directions.

The nature of propagation in the structure is related with the

thicknesses of the ferrite films. This aspect is shown in Fig.2.12, where the

ratio of the ferrite film thicknesses is plotted against the propagation

60

constant. It can be seen that the propagation constant declines with increase

in ferrite film thickness ratio.

0 4 8 12

0

200000

400000

600000

Prop

agat

ion

cons

tant

, m -1

Ratio of ferrite film thicknesses, f1/f2

Fig.2.12. Dependence of propagation on thicknesses of ferrite films. The dielectric film is of thickness 100 nm.

The ferrite/dielectric/ferrite structure is magnetically tunable also. In

Fig.2.13, the tunability of the structure with magnetic field and the

associated nonreciprocal effect are displayed. The two ferrite films are

magnetized with equal fields but in the opposite directions.

61

0.05 0.06 0.07

-6E+5

-4E+5

-2E+5

0E+0

2E+5

4E+5

Forward

Backward

Prop

agat

ion

cons

tant

, m -1

Magnetic field, Tesla

Fig.2.13 The Magnetic tunability of the structure and the associated nonreciprocal effect. The ferrite film thicknesses are 100m and 1000nm and the dielectric film thickness is 100nm.

2.9 SUMMARY

In this chapter, different aspects of interaction between electromagnetic

waves and ferrimagnetic materials were brought out. Polder tensor, which

governs the interaction between ferrites and electromagnetic waves, is

derived in section 2. In section 3, propagation in an unbounded ferrite

62

medium is discussed. Different regimes of wave propagation in ferrites are

introduced in section 4. Different types of wave propagation in magnetized

ferrimagnetic films are discussed in section 5. In section 6, the method of

analysis used in this study is outlined. Detailed discussions of wave

propagation in a standalone ferrite film and in a ferrite/dielectric/ferrite

hybrid structure are presented in sections 7and 8 respectively.

REFERENCES

1. D. POLDER, “On the theory of ferromagnetic resonance,” Phil. Mag.

Vol.40, 1949

2. D. M. BOLLE and L. R. WHICKER, “Annotated literature survey of

microwave ferrite materials and devices,” IEEE Tans. Magn. Vol.11, 1975

3. D. WEBB, “Status of microwave technology in the United States,” IEEE

MTT-S. Int. Microwave Symp. Atlanta, 1993.

4. W. S. ISHAK, “Magnetostatic wave technology: a review”, IEEE Proce.

Vol.76, 1988.

5. D. M POZAR, Microwave engineering, Addison-Wesley, New York,

2000.

6. A. J. BADAN FULLER, Ferrites at microwave frequencies, Peter

Peregrinus Ltd, 1987.

7. D. D. STANCIL, Theory of magnetostatic waves, Springer-Verlag, 1993

8. SUHL (1956), “The nonlinear behaviour of ferrites at high microwave

signal levels,” Proc. IRE, 1956, Vol.44, 1270.

63

9. J. Nicolas (1980), “Microwave ferrites,” in Ferrimagnetic Materials, E. P.

Wohlfarth. Ed. New York: North Holland, Vol.2.

10. KABOS and V. S. STALMACHOV, Magnetostatic waves and their

applications, Chapman and Hall, 1994.

11. L. R. WALKER, “Mgnetostatic mode in ferromagnetic resonance,” Phys.

Rev, Vol.105, 1957.

12. R. W. DAMON and H. VAN DER VAART, “Propagation of

magnetostatic spin waves at microwave frequencies in a normally

magnetized disc,” J. Appl. Phys., Vol.36, 1965

13. W. L. BONGIANNI, “Magnetostatic propagation in a dielectric layered

structure,” J. Appl. Phys., Vol.43, 2541.

14. M. S. SODHA and N. C. SRIVASTAVA, Microwave propagation in

ferrimagnetics, New York, Plenum, 1981

15. J. P. PAEKH, K.W. CHANG, and H. S. Tuan , “Propagation

characteristics of magnetostatic waves,” Circuits Systems Signal

Processing, Vol.4, 1985

16. T. W. O KEEFFE and R. W. PATTERSON, “Magnetostatic surface wave

propagation in finite samples,” J. Appl. Phys., Vol.49, 1978

17. J. D. ADAM and S. N. BAJPAI, “Magnetostatic forward volume wave

propagation in YIG strips”, IEEE Trans. Magn., Vol.18, 1982.

64