ELECTROMAGNETIC WAVE PROPAGATION IN FERRITES AND...
Transcript of ELECTROMAGNETIC WAVE PROPAGATION IN FERRITES AND...
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