close

Вход

Забыли?

вход по аккаунту

?

Dynamic Magnetics:Microwave Interactions with Magnetic Structures on both a Macroscopic and Microscopic Scale

код для вставкиСкачать
Dynamic Magnetics:
Microwave Interactions with Magnetic Structures
on both a Macroscopic and Microscopic Scale
by
Timothy J. Fal
B.S., University of Colorado at Colorado Springs, May 17, 2004
M. S., University of Colorado at Colorado Springs, December 16, 2006
A dissertation submitted to the Graduate Faculty of the
University of Colorado at Colorado Springs
in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Applied Science
Department of Physics
2011
UMI Number: 3450521
All rights reserved
INFORMATION TO ALL USERS
The quality of this reproduction is dependent upon the quality of the copy submitted.
In the unlikely event that the author did not send a complete manuscript
and there are missing pages, these will be noted. Also, if material had to be removed,
a note will indicate the deletion.
UMI 3450521
Copyright 2011 by ProQuest LLC.
All rights reserved. This edition of the work is protected against
unauthorized copying under Title 17, United States Code.
ProQuest LLC
789 East Eisenhower Parkway
P.O. Box 1346
Ann Arbor, MI 48106-1346
ii
© Copyright By Timothy J. Fal and Robert E. Camley 2011
All Rights Reserved
iii
This dissertation for Doctor of Philosophy in Applied Science degree by
Timothy J. Fal
has been approved for the
Department of Physics
by
___________________________________________
Robert E. Camley, Chair
___________________________________________
Zbigniew Celinski
___________________________________________
Marek Grabowski
___________________________________________
Anatoliy Glushchenko
___________________________________________
Anatoliy Pinchuk
________________________
Date
iv
Fal, Timothy J. (Doctor of Philosophy in Applied Science)
Dynamic Magnetics: Microwave Interactions with Magnetic Structures on both a
Macroscopic and Microscopic Scale
Dissertation directed by Professor Robert E. Camley
Microwave interactions with magnetic materials are an important topic with
research being done for signal processing devices and magnetic storage. There is an
interest in how magnetic materials affect electromagnetic signals and in turn, how the
signals affect the magnetic material. In this work, we present analytical and numerical
calculations for electromagnetic waves interacting with thin-film magnetic materials as
well as magnetic nano structures.
The first part of this dissertation focuses on how a microwave signal is affected by
a magnetic material. These results can be applied to signal processing devices such as
notch filters, band pass filters, phase shifters, and isolators. The materials considered in
the analysis are iron, M-type barium hexagonal ferrite, permalloy, and yttrium iron
garnet.
The second part of this dissertation focuses on how a magnetic material is
affected by a microwave signal. Micromagnetic simulations are used to track the
magnetization direction of a magnetic nano structures as it evolves with time in an
oscillating magnetic field. The results demonstrate the possibility of microwave assisted
magnetic reversal for magnetic nano squares and bilayer exchange spring structures.
I dedicate this work to my parents, Liz and John Fal, and to my wife Holly.
vi
AKNOWLEDGEMENTS
I would like to thank my graduate advisor Bob Camley for giving me so many
opportunities, a lot of insight, and his tireless work for this graduate program. I would
also like to thank Zbigniew Celinski for all of his support of the graduate program, fellow
students, and myself. Thank you to Karen Livesy and Veera Venugopal for collaborating
with me on projects and giving me sound, useful advice. Thank you to Sam Milazzo for a
significant part of my undergraduate physics education. Thank you as well to Marek
Grabowski who instructed me in a majority of my graduate level classes.
Much of this work was supported by the DOA Grant No. W911NF-04-1-0247, the
US Army Research Office Grant No. DAAD19-02-1-0174, and the NSF Grant No. DMR
0907063.
Finally, I would like to thank Evan McHugh for introducing me to physics many
years ago.
vii
TABLE OF CONTENTS
CHAPTER
I.
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
II.
BACKGROUND: Dynamic Magnetic Permeability . . . . . . . . . . . . . .
8
The Landau-Lifshitz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
Ferromagnetic Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
The Dynamic Permeability Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Permeability with Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Demagnetizing Fields and Thin Film FMR . . . . . . . . . . . . . . . . . . . 20
III.
Thin Films Structures and Boundary Condition Calculations . . . . . . 24
Calculation for a Small 3-Layer Ferromagnetic Notch filter . . . . . 24
Transmission Results for the small 3-Layer Notch Filter . . . . . . . . 30
Reflections and Insertion Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
IV.
Devices with Hexagonal-ferrites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
The Calculation for the Hexagonal Ferrite waveguide . . . . . . . . . . 43
Transmission Results for the Hexagonal Ferrite Waveguide . . . . . 50
Using the waveguide as a Phaseshifter . . . . . . . . . . . . . . . . . . . . . . . 60
Experimental Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
viii
V.
Non-reciprocal Ferromagentic Devices using
Attenuated Total Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Surface Polaritons and Three-layer ATR . . . . . . . . . . . . . . . . . . . . . 72
Four-layer ATR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
VI.
BACKGROUND: Micromagnetic Simulations . . . . . . . . . . . . . . . . . . . 95
The Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
The Anisotropy Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
The Effective Exchange Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
The Dipole Field and the FFT Method . . . . . . . . . . . . . . . . . . . . . . .100
VII.
Accessing Multiple States in Magnetic Squares with Microwaves . . .107
The Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108
The Multi-State Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .110
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .115
VIII. Microwave Assisted Switching in an Exchange Spring Bilayer. . . . . . .116
Theoretical Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .120
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .129
ix
IX.
SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .130
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132
APPENDICES
Appendix- A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .139
x
TABLES
Table
5.1
Magnetic Layer Thickness and Return Loss Differences . . . . . . . . 92
xi
FIGURES
Figure
1.1
Basic microstrip geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Oersted fields around a microstrip . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3
Schloemann notch filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.4
Simple bit patterened media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.1
Simple macrospin precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2
Small angle precession of magnetization . . . . . . . . . . . . . . . . . . . . . . 12
2.3
Small angle precession with an oscillating magnetic field . . . . . . . . 14
2.4
Macrospin precession with the direction of damping . . . . . . . . . . . 17
2.5
Real parts of the permeability tensor elements . . . . . . . . . . . . . . . . 19
2.6
Imaginary parts of the permeability tensor elements . . . . . . . . . . . . 20
2.7
Thin film geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1
Three layer notch filter geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2
Transmission loss vs. frequency for different
positions of an Fe layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3
Maximum transmission loss vs. magnetic film thickness . . . . . . . . 32
3.4
Rejection bandwidth vs. magnetic layer position . . . . . . . . . . . . . . . 34
3.5
Cross-section of a two region waveguide . . . . . . . . . . . . . . . . . . . . . . 36
3.6
Theoretical return loss due to insertion vs. frequency . . . . . . . . . . . 38
3.7
Experimental loss due to insertion vs. frequency . . . . . . . . . . . . . . . 39
4.1
Notch filter geometry with BaM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
xii
4.2
Transmission vs. frequency for 0.5 μm of
BaM with different fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3
Transmission vs. frequency for 4.0 μm of
BaM with different fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.4
Resonant frequency vs. BaM filling factor . . . . . . . . . . . . . . . . . . . . 54
4.5
Resonant frequency vs. linewidth . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.6
Transmission vs. frequency for 0.5 μm
of out-of-plane permalloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.7
Transmission vs. frequency for 0.5 μm of in-plane BaM . . . . . . . . . 60
4.8
Phase difference vs. applied field for 0.8 μm of BaM
for different frequencies
4.9
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Phase difference vs. applied field with highly attenuated
results removed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.10
Phase difference vs. applied field for 1.6 μm of BaM
with highly attenuated results removed . . . . . . . . . . . . . . . . . . . . . . 63
4.11
Phase difference vs. applied field for 0.8 μm of BaM
for higher frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.12
Phase difference vs. applied field for 1.6 μm of BaM
for higher frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.13
Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.14
Theoretical results for experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.1
Three-layer ATR geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2
Four-layer ATR geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
xiii
5.3
Polariton Dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.4
Return Loss vs. frequency for a three-layer
ATR geometry using YIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.5
Return Loss map of frequency vs. wavevector
for a three-layer ATR geometry using YIG . . . . . . . . . . . . . . . . . . . 83
5.6
Four-layer ATR geometry comparison
with the three-layer ATR geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.7
Return Loss map of frequency vs. wavevector
for a four-layer ATR geometry using BaM. . . . . . . . . . . . . . . . . . . . . 86
5.8
Return Loss vs. frequency for a four-layer
ATR geometry using 0.1 cm of BaM . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.9
Return Loss vs. frequency for a four-layer
ATR geometry, comparing different applied fields . . . . . . . . . . . . . 88
5.10
Return Loss map of frequency vs. wavevector
For a four-layer ATR geometry using 0.01 cm of BaM . . . . . . . . . . 89
5.11
Return Loss vs. frequency for a four-layer
ATR geometry using 0.01 cm of BaM . . . . . . . . . . . . . . . . . . . . . . . . 90
5.12
Return Loss vs. frequency for a four-layer
ATR geometry using 0.001 cm of BaM with bandpass behavior . . . 91
5.13
A typical ATR geometry and lateral ATR waveguide . . . . . . . . . . . 93
6.1
Magnetic structure divided into cells
for the finite difference method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
xiv
6.2
A row of spins with an effective
exchange length between two selected spins . . . . . . . . . . . . . . . . . . . 99
6.3
Average magnetic field in one volume
created by another volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101
6.4
Average dipolar field in one cell due to all other cells . . . . . . . . . . .102
6.5
Time comparison between the double iteration method
and the FFT method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .106
7.1
Nano-square geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108
7.2
The three states of the nano-square . . . . . . . . . . . . . . . . . . . . . . . . .110
7.3
Time evolution of the z-component of the magnetization . . . . . . . .112
7.4
Square state based on driving field vs. applied field . . . . . . . . . . . .114
8.1
The geometry of the exchange spring bilayer . . . . . . . . . . . . . . . . . .118
8.2
Hysteresis loops for a structure made of high anisotropy
material and an the exchange spring bilayer structure . . . . . . . . . .121
8.3
Frequency spectrum for standing spin wave modes
across the bilyaer structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .122
8.4
Spatial profile for standing spin wave modes . . . . . . . . . . . . . . . . . .123
8.5
Time evolution of the z-component of magnetization
in the exchange spring bilayer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .124
8.6
Final state of bilayer structure based on
the strength of the driving field . . . . . . . . . . . . . . . . . . . . . . . . . . . . .125
8.7
Hysteresis loops for the bilayer structure without a driving field
and with a driving field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .127
CHAPTER I
INTRODUCTION
Ferromagnetic materials have played a large role in developing solid-state
technology in the last 50 years. There continues to be a great deal of interest in
ferromagnetism for technology today, particularly in layered and patterned structures that
can be grown with thicknesses on the order of microns. Communication devices based on
these structures are being developed to work in the microwave range of 1-100 GHz and
promise to be more efficient and compact than the large, bulky devices currently
implemented.1-11 Growing layers of magnetic materials for potential devices can be time
consuming and requires costly equipment.12 To reduce the time and cost it is beneficial to
explore analytic and numerical results for the dynamics of these materials. In
this
dissertation we present analytical and numeric results for the interactions between
ferromagnetic materials and electromagnetic waves in the microwave frequency range.
We consider a variety of structures that may be implemented in devices. The focus of the
first part of this dissertation is on how magnetic materials change microwaves. The focus
for the second part is on how microwaves can change magnetic materials. We will
consider different microwave signal processing devices, such as notch filters, band pass
filters, phase shifters, and isolators, as well as how magnetic storage and bit reversals are
affected by microwaves. Chapters III, IV, V, VII, and VIII present new results on these
topics and have been submitted for review and accepted in scientific journals.
The signal processing devices discussed in Chapters III and IV cam be
constructed from a variety of geometries, including microstrips, striplines and co-planar
2
waveguides. The microstrip geometry is a thin strip of conductive material, typically
mounted on a dielectric with a ground plane below that. The top strip has an oscillatory
current running through it, leading to electromagnetic fields in the region near the strip.
The oscillating current, possibly being brought into the strip from some kind of receiver
like an antenna, creates the signal of interest. The signal comes in through one port and
leaves through a second port, seen in Fig. 1.1.
Figure 1.1: Basic microstrip geometry.
One can obtain an intuitive understanding of the microstrip in the following way:
the current in the strip (signal line) creates Oersted fields around it that oscillate at the
same frequency as the current, as shown in Fig. 1.2.
3
Figure 1.2: Oersted fields oscillating around a conductive strip.
The oscillating magnetic field, in turn, creates oscillating electric fields in an
electromagnetic wave. A magnetic material in close proximity to the strip can potentially
modify the signal.
Chapters III and IV explore the interaction of a magnetic material
with a microwave that could be generated by a microstrip. Schloeman et. al. created some
of the first microstrip filters with a magnetic layer next to the signal line.2
The
dimensions of that device were on the order of 3 cm in length and a few hundred microns
in thickness. The results of those experiments showed that the incident signals were
absorbed at frequencies matching the resonant frequencies of the magnetic material.
Since then the microstrip geometry using a magnetic thin film has been improved and
made more efficient.7-11
4
Figure 1.3: Notch filter device from Schloemann et at [2]. A microstrip of 2 μm of Al or Au with
0.1 μm of Fe directly below it. An applied magnetic field orients the magnetization of the
magnetic material and sets the magnetic resonance frequency.
Some of the types of signal processing devices that can be made from microstip
structures are notch filters, bandpass filters, and phase shifters. A notch filter, or bandstop
filter, is a device that blocks a small range of frequencies from passing through it. In
other words, it makes a notch in the transmission spectrum. A bandpass filter works in
the opposite way. The bandpass filter allows a small range, or band, of frequencies to
pass through it, blocking off all other frequencies. A phase shifter is a device that will
shift the phase of a signal. By modifying something about the device, for example an
applied H-field, the phase of the signal is altered. The last type of signal processing
device we will consider is the isolator. An isolator is a device that will allow a signal to
pass in one direction through the device, but not in the opposite direction.
5
In Chapter III we attempt to find a way to improve the notch filter. We look for an
optimal placement of the magnetic layer to get the greatest signal damping in the notch.
Chapter IV examines notch filters and phase shifters using M-type barium hexagonal
ferrite (BaM). Hexagonal ferrite allows for the processing of higher frequencies than the
materials previously used due to its high crystalline anisotropy. Chapter V proposes a
possible modified microstip device that would use attenuated total reflection to achieve
notch filter, bandpass filter, and isolator behavior.
Chapter V concludes the investigation of the topic on how a microwave is
changed by a magnetic material and the remainder of the dissertation considers the
inverse problem - whether a microwave signal can be used to alter the magnetic material.
These remaining chapters deal with magnetic reversals and patterned magnetic nanostructures.
Reversing magnetic orientations has been of great interest for many years13-15,
particularly in data storage. Because of new film growth and lithography techniques, bit
patterned media shows promise for being a method to create high density magnetic
storage media.21-26A bit patterned medium is one where every single bit in the device has
a well defined magnetic structure. Current techniques can make magnetic squares or dots
with side lengths or diameters on the order of 20-50 nm in size. With further reductions
in size, this could produce magnetic storage densities of around 2000 Giga-bits per square
inch, which is four times the areal densities of most hard drives made today. This
neglects the stacking of disks, which effectively increases areal density by a multiple of
how many disks are stacked, one on top of the other.
6
Figure 1.4: Diagram of a bit patterned media. An array of small magnetic squares.
There are thermal stability issues with a bit’s magnetization as it gets smaller,
particularly for structures below 50 nm in scale. In order to make these structures more
stable, materials with high crystalline magnetic anisotropy are being considered.
Unfortunately, the large anisotropy field means a very large magnetic field is normally
required to reverse the magnetization direction. Therefore, if such a material is used,
there is a need to find new ways to temporarily reduce the magnetic stability.
One
possible method to do this is to use oscillating magnetic fields in order to destabilize a
magnetic state to aid magnetization switching. This technique is called microwave
assisted magnetic reversal (MAMR).16-20 In order to evaluate such a process,
micromagnetic computer simulations can be used. Chapter VI outlines the principles used
in these simulations and Chapters VII and VIII explore the use of microwaves to aid in
the switching of magnetic orientations.
7
Chapter VII examines magnetic nano-sized squares with three possible stable
magnetic states. The square is subjected to microwave oscillating magnetic fields at
different strengths and frequencies, and this can determine the which of the states remain
stable. . Chapter VIII focuses on bi-layer structures. Each layer has a different crystalline
anisotropy, one with a strong anisotropy the other with a weak one. The simulations in
Chapter VIII are concerned with reducing the strength of magnetic fields used to switch
the magnetic structure by applying an additional oscillating magnetic field.
Before looking at applications and results, Chapter II reviews the physics
involved with in dynamic calculations of magnetic materials. The concepts in that chapter
will be used in all the following chapters.
8
CHAPTER II
BACKGROUND: Dynamic Magnetic Permeability
In order to make the reader more familiar with microwave interactions with
ferromagnetic materials, we present some of the theory involved.3,27-30 The magnetic
properties of a material depends on the magnetic moments of each individual atom in the
material. In general, there can be both spin and orbital angular momentum contributions
to the magnetic moment. However, in the common elements of Fe, Ni and Co, the spin
contribution is dominant.
In ferromagnetic materials the magnetic moments of the
individual atoms are nearly parallel to each other. This alignment is a quantum
mechanical effect produced by the overlapping electron wave functions of the atoms and
the Pauli Exclusion principle. This is called the exchange interaction and will become
important in the following chapters, but for now we will simply say that it aligns the
atomic magnetic moments in the material. The volumetric density of the magnetic
moments is the material’s magnetization M.
The Landau-Lifshitz Equation
The magnetic field, B, inside a magnetic material is the result of two terms. In cgs
units, the terms are:

B

H

4 M
(2.1).
9


There is a contribution from a field H and the magnetization of the material M . The

magnetization of the material is influenced by H and can, in some limits, be represented
as a linear function of the H-field through the magnetic susceptibility

M

H
In general, the susceptibility
(2.2).

is a tensor. When Eq. (2.2) is put into Eq. (2.1), we have
the expression with the permeability of the material:

B

H
(2.3)
where μ is the permeability of the material



1 4
(2.4).
We are particularly interested in how a magnetic material would respond to
an

oscillating field h . Such an interaction can be contained in Eq. (2.4).
We can find a dynamic susceptibility if we have a microscopic model of how an
individual magnetic moment moves in response to an H-field. We note that the torque

on an individual moment m is given by
 
m H

(2.5)
The torque, in turn, is given by the time derivative of the angular momentum. For a
single electron with angular momentum based on spin only, the angular momentum is

given by S . The magnetic moment can be related to the angular momentum through the
Bohr magneton
B
10

m
g
e 
S
2mc
g(

/

)
S
B
(2.6)
where the Bohr magneton μB, is given by
e
2m e c
B
(2.7)
and me is the mass of an electron, the speed of light is c, and the charge of an electron is
e.
The parameter g arises from relativistic effects and is typically about 2 for most transition
metal ferromagnets.
Using Eqs (2.5) – (2.7) we arrive at the Landau-Lifshitz equation (LL equation)
for the motion of a magnetic moment in an applied field

dm
dt
Here
g
B
 
m H
(2.8).
/  and the negative sign in the electron charge is made explicit in Eq. (2.8)
by using the absolute value and the minus sign. If the magnetization is a region of space


is uniform, then Eq. (2.8) can be written with M replacing m .
Ferromagnetic Resonance
The conditions that the magnetization behaves as one giant macrospin and that
angular momentum must be conserved, requires that
the magnitude of the total
magnetization is conserved. Because torque is defined as the change of angular
momentum with time, in order for the angular momentum to change it must do so by
11
changing its direction. This is what magnetic moments do in an applied H-field, instead
of reducing their magnitude, they change their direction and precess around the field, as
illustrated in Fig. 2.1.
Figure 2.1: The precession of a macrospin, represented by the magnetization M of a magnetic
structure, about the applied H-field H. The vector denoting the direction of the change of
magnetization with time is given by the LL equation.
For given |γ| and magnetization we can find the precession frequency in an
infinitely extended material.
In Fig. 2.2, we consider a small precession for the magnetization. The H-field is in
the z-direction with the magnetization mostly saturated in that direction. The angle of
precession is very small, much less than 1o, so we can say that Mz is approximately equal
to the material’s saturation magnetization Ms. The time dependent components of the
magnetization are the x and y components. We represent them here as lowercase mx and
my to remind us that the values of these components are very small. The LL equation then
becomes:
t
mx
mx
0
my
Mz
my
Mz
0
H
(2.9).
12
Figure 2.2: Simplified model of small angle precession of magnetization about applied field H.
Vector mr is the time dependant component of the magnetization. If the angle θ is small enough,
we can approximate Mz as Ms.
The components mx and my are assumed to have the same time dependence to
allow precession at particular frequency ωo, which gives them the form:
mi
m i0e
i
ot
(2.10).
Performing the time derivative and cross product gives us two equations.
i
o
mx
my H
(2.11)
i
o
my
mx H
(2.12)
Substituting one equation into the other we find the frequency is
o
H
(2.13).
This frequency is the ferromagnetic resonance (FMR) of an infinite ferromagnet.
The only material parameter in this case is the gyromagnetic ratio |γ|, which is around 2.9
GHz/kOe for many ferromagnetic materials. So, a ferromagnetic material with a 2 kOe
13
magnetic H-field applied to it would precess at about 5.8 GHz, which falls into the
microwave range. This is a simple result for a simple model. The FMR frequency for a
finite magnetic structure could differ from this form. H is not just an applied external
field, but the total effective field acting on the magnetization. The H-field could include
interactions unique to the material or a particular geometry.
The Dynamic Permeability Tensor
We have discussed how the magnetization of a material responds to a static Hfield. The devices we are interested in also make use of the dynamic H-fields from
electromagnetic waves,. So to account for a response to a dynamic field we add an
oscillating magnetic field to our simple model. Fig. 2.3 is just like Fig. 2.2, except we
have now added a driving magnetic field h. Just like the time dependant magnetization
components, the magnitude of the driving h-field is small (hence the lower case
representation).
The LL equation becomes:
t
mx
mx
hx
my
Mz
my
Mz
hy
H hz
(2.14)
We give the driving h-field a typical time dependant form:
hi
h i0e
i t
(2.15)
14
Figure 2.3: Simplified model of small angle precession about an applied field H with a small
oscillating magnetic field h. The precession will have the same frequency as that of the oscillating
driving field.
We assume that the time dependence of the magnetization and the driving field
will be the same.
mi
mi0e
i t
(2.16)
Performing the time derivative and cross product gives us:
i
mx
m y (H h z ) M s h y
my
0
M s h x m x (H h z )
mx h y myh x
(2.17)
We linearize this set of equations by setting any product of a small m component
and small h component to zero. This simplifies Eq. (2.17) into two equations:
mx
my
i
m yH Msh y
Msh x m x H
(2.18)
Substituting one equation into the other, we arrive at solutions for mx, and my.
15
mx
Ms
my
2
i hy
2
(
H2
2
2
i hx
Ms
2
(
hxH
H2
(2.19)
)
h yH
2
(2.20)
)
Separating out the h components and writing the equations in a matrix form, we
get:
2
mx
my
Ms
H
2
o
i
2
2
o
2
2
i
2
o
2
2
o
H
hx
hy
(2.21)
2
where ωo is given by Eq (2.13).
Eq. (2.21) is Eq. (2.19) and (2.20) rewritten in a matrix form. This is what we
were looking for, an expression that gives us the dynamic magnetization as it relates to an
oscillating h-field. The matrix that relates the two vectors (magnetization and driving
field) is the magnetic material’s susceptibility. According to Eq. (2.4), this also gives us
the permeability:

i
1
i
0
t
0
t
0
1
1
0
(2.22)
Where the matrix elements μ1 and μt are given by:
2
1
1 4 Ms
2
o
H
2
(2.23)
16
t
4 Ms
2
o
(2.24)
2
The most obvious problem with this expression is when the driving frequency ω
equals the FMR frequency ωo and both Eqs. (2.22) and (2.23) blow up. These results are
based on “perfect” materials where all the work done by the driving field is transferred to
the magnetic material. According to the second law of thermodynamics, not all of the
energy pumped into the system will be turned into work. There will be losses and they
need to be accounted for in the equations of motion.
Permeability with Damping
The complete Landau-Lifshitz Equation typically has a term added to handle the
damping

dM
dt
 
M H
M

 
M [ M H]
(2.25).
The term α is the damping factor, a real unitless number. The damping of a material is
often measured by the width of an observed absorption peak in a scan over either the
frequency of the signal or of an
external applied H-field. This provides either a
frequency linewidth or field linewidth. The damping factor α can be related back to these
linewidths.


The double cross product term M [M

H] supplies a new vector that moves M
toward Ms, as seen in Fig. 2.4. The resulting motion is a spiral decay into smaller and
17
smaller precession angles until the magnetization comes to rest in the direction of the
applied H-field.
Figure 2.4: Macrospin precession with the direction of the damping vector.
If we were to add a driving field, as we did in the previous section, the resulting
equation of motion would be:
mx
my
t
Mz
mx
my
Mz
hx
hy
H hz
Mz
mx
my
Mz
mx
my
Mz
hx
hy
H hz
(2.26).
We follow the same procedure used to find the permeability of an undamped
magnetic material, we must linearize the equations. The permeability with the damping
terms in them can be found as30

i
1
i
0
t
t
1
0
0
0
1
(2.27).
18
where μ1, and μt are:
2
1
1 4 Ms
t
4 Ms
2
2
H 2 (1
H 2 (1
2
H(1
) 2i
H
2
(2.28)
) 2i
H
2
(2.29).
2
2
) i
Using the permeability elements, we can gain information about the material's
resonance, and we can express the material’s damping in terms of linewidth. If we were
to keep ω fixed frequency and plot μ1 and μt as functions of the applied field H, there is a
signature near an applied field of H
/ | | . How the signature spreads out near that
field is determined by the damping constant α. This is the connection between the
experimental parameter of a material’s linewidth and the damping factor α. Fig. 2.5
shows the real parts of both μ1 and μt as a function of applied field using Fe as the
magnetic material with a Ms = 1.7 kG, |γ| = 2.9 GHz/kOe, the damping set to α = 0.03,
and the frequency f = 8 GHz. For both the real parts, there is a maximum and minimum
centered around about 2.76 GHz. For one of the elements we measure the difference
between one minimum and maximum, we get a value of 0.165 kOe.
19
linewidth = 0.165 kOe
80
Realative
Permeability
60
Real part of
40
20
0
-20
-40
Real part of
-60
-80
2.0
2.5
3.0
3.5
Applied Field (kOe)
t
4.0
Figure 2.5: Real parts of the permeability tensor elements μ1 and μt. The linewidth between the
minimums and maximums is 0.165 kOe.
Fig. 2.6 is the plot of the imaginary parts of μ1 and μt. The signature is now a peak
or dip centered around 2.76 GHz. Measuring the width of the peak across half of its
maximum value (full width at half max), we again come up with 0.165 kOe. Again, this
spreading of the function is due to the damping constant α, which is related to the
experimental linewidth. In this case, the damping constant of α = 0.03 is related to a
linewidth of 0.165 kOe. The value of the of the field-linewidth is given by
H 2
In the next chapters, we will use these forms for the permeability.
(2.30).
20
Realative
Permeability
150
linewidth = 0.165 kOe
Imaginary part of
100
50
0
-50
-100
-150
2.0
Imaginary part of
2.5
t
3.0
3.5
4.0
Applied Field (kOe)
Figure 2.6: Imaginary parts of the permeability tensor elements μ1 and μt. The linewidth between
the minimums and maximums is 0.165 kOe, typical full width at half max measurement.
Demagnetizing Fields and Thin Film FMR
We now consider the ferromagnetic resonance for a thin film. The film is infinite
in the y and z plane, but has a thickness in the x-plane.
Figure 2.7: Geometry of a thin film. The y and z directions are infinite, and there is a finite
thickness in the x-direction.
21
As we will see, the most significant change in the calculation comes from the static and
dynamic demagnetizing fields introduced by effective magnetic surface charges at the
boundaries. We start with H-field being represented as the divergence of a scalar
magnetic potential

H
(2.31).
M
The potential’s value at a point in space (r) is determined by the surrounding
magnetization (the magnetization at all points r'). The potential is then given by the
integral

M(r )
M (r )
1
  dV
r r
V
(2.32).
Making an approximation that the magnetization is uniform everywhere in the
volume, we can write an expression for the average dipolar H-field in the volume, V,
surrounding the observation point r:

Hd
1
V

dV M
V
V

r
1
 dV
r
 
N M
(2.33).
The tensor N is a strictly geometric relationship between volume V and V’. The elements
of this tensor are given by these integrals
N ij
1
V
dV
V
V
xi
1
  dV
xj r r
(2.34).
These volume integrals are generally changed to surface integrals using a version
of Guass’s theorem
22

N
1
V
dV
V
V
1
  dV
r r


1
dA
dA  
VA
r r
A
(2.35).
This is a general demagnetization tensor. When the observation point lies inside

the magnetic volume the elements of the tensor N are known as demagnetizing factors.
The general tensor relationship between a small volume centered around r' and an
observation point at r is known as the Newell tensor.29
Using the results obtained from the Newell tensor for an infinitely extended film
in the yz plane but with a finite thickness in the x-direction and a uniform magnetization,
the average H-field from the dipolar interactions is

Hd
4 m x x̂
(2.36).
This is the demagnetizing field in a thin film with uniform magnetization. Assuming that
the static magnetization is in the plane of the film, along the z direction, one notes that
the demagnetizing field is dependent on a component of magnetization that precesses.
Hence, this is a dynamic demagnetizing field. It might be a little disturbing that the field
has no apparent dependence on the thickness of the film, but this is because of the infinite
components in the y and z directions. Essentially the result is the same as that for the
electric field inside an infinitely extended capacitor. If the thin film or the capacitor were
not infinite in the y and z directions, the results would depend on thickness.
This method for finding the demagnetizing field can also be applied to dipolar
fields from non-local volumes of magnetic materials and becomes critical in the
micromagnetic simulations.
23
To observe the effects of the demagnetizing fields, we consider the usual
geometry: a thin film with the z-component of magnetization Mz in-plane and an applied
static magnetic field also along the z-axis. Assuming there is small angle precession
around the applied field, the Mz component is taken to be the saturation magnetization,
and the demagnetizing field depends on the component of magnetization normal to the
film. With the damping set to zero for simplicity, we get the following LL equation:
t
mx
mx
4 mx
my
Mz
my
Mz
0
H
(2.37).
Using the same procedure that gave us Eq. (2.14), we get an expression for the FMR of a
thin film
FMR
H( H 4 M s )
(2.38).
This is more complicated compared to the bulk FMR case given by Eq. (2.14). The most
notable change is that it now includes a dependence on the magnetization of the material.
So, if we used the 2.9 GHz/kOe gyromagnetic ratio |γ| and applied a 2 kOe field to thin
film of Fe with a Ms of 1700 G, we would get a FMR of 19.8 GHz.
The goal of this chapter was to give the reader an understanding of the magnetic
permeability tensor which will be used extensively in the following work. In the
following chapters, there may be differences in the explicit forms of the presented
expressions, but this is due to different choices of coordinate systems and definitions of
damping. These choices were each made on an individual basis and the work is presented
in its original form.
CHAPTER III
Thin Films Structures and Boundary Condition Evaluation
The first device presented in this chapter is the notch filter. We give results predicting
the performance of a notch filter device with different magnetic components. We model
an electromagnetic wave forced into a planar waveguide. The device is composed of two
conducting plates with one or more dielectric spacer along with a magnetic thin film
sandwiched between the conducting plates. The magnetic film interacts with the wave
and changes it. We evaluate what happens to the wave in the guide by applying boundary
conditions set by Maxwell’s electrodynamic equations.
We investigate a different structure from the Schloemann geometry mentioned in
Chapter I.2 The ferromagnetic film is placed in an arbitrary position between the outer
conductive layers of the waveguide as is shown in Fig. 3.1 and is surrounded by dielectric
material on both sides. This geometry produces a significant improvement in attenuation
compared to the filter where the ferromagnet is directly next to the conductive layers. In
addition, the width of the frequencies absorbed by the filter is reduced slightly, giving
one better frequency resolution.
Calculation for a Small 3-Layer Ferromagnetic Notch filter
The geometry of the waveguide is shown in Fig. 3.1. The conductive layers are
silver (Ag). The silver sandwiches two dielectric spacer layers, which we model as
silicon dioxide (SiO2). The magnetic component is in-between the spacers, which we
25
have chosen to be iron (Fe). A transverse magnetic (TM) mode propagates down the z
axis parallel to the external applied field Ho. For the TM mode the oscillating magnetic
field is directed along the x axis only. The electric field has a longitudinal component,
Ez, and a transverse component, Ex.
Figure 3.1: Geometry of the three layer notch filter. The wave propagates along the z axis,
parallel to the applied field. The outer metallic layers are assumed to be a highly conductive
material such as Ag.
The behavior of the magnetic material is described by a permeability tensor given
by30-32
1

( )
4 M( H 0 i
( H0 i )2
4 M
i
( H0 i )2
0
)
2
2
4 M
( H0 i )2
4 M( H 0 i
1
( H0 i )2
0
i
0
2
)
2
i
1
0
1
i
0
t
t
1
0
0
0
1
(3.1).
26
H0 is the applied field, M is the saturation magnetization, γ is the gyromagnetic ratio, ω is
the frequency of the electromagnetic (EM) wave, and Γ is the damping factor. The
damping factor for a set frequency can be related to the measured full width at half
maximum ferromagnetic resonance (FMR) line width (ΔH) of a material. In the range of
small damping, Γ can be taken to be the measured full width at half maximum linewidth.
We will assume that Γ is linearly dependant on ω and has this form:
( H0
)
(3.2)
Here ΔH0 is the FMR linewidth at zero frequency and α is a parameter that determines
how much the FMR linewidth changes with frequency.
The use of the complete tensor would make the problem complicated and has
been shown to be unnecessary.3 One can instead use a scalar permeability given by
(
xx
yy
2
1
2
t
v
)
,
(3.3)
1
(3.4)
1.
zz
Using the two curl equations from Maxwell's equations we find the following wave
equation for the system:
(

H)
2
c2 t 2
v
Hx
Hy
Hz
v
(3.5)
The equation has the same form for all regions but the permeability μ and permittivity ε
change from region to region. Because we are dealing with a TM mode only Hx is nonzero.30,31 Furthermore we assume the width of the guide is sufficiently large that there is
no variation in the x direction. These equations then reduce to one equation
27
2
2
Hx
y2
2
Hx
z2
v
2
c
t2
(3.6)
Hx .
Because Hx is a wave propagating along the z axis it has the form
Hx
H x ei ( k z z
t ) ik y y
e
(3.7)
.
When we substitute this equation into the wave equation we find two solutions for k y
which are the negative of each other.
2
ky
i k
2
z
v
(3.8)
c2
We expect three independent values of ky, one for each region. These values are
denoted by ky1, ky2, and ky3, each with respect to the region in which they are found. Eq.
(3.10) then applies in all regions with the appropriate choices for μ and ε.
In each of the regions, the x-component of the H field can be represented as the
sum of two terms with different ky values, each with an unknown amplitude:
Hx
(A e
ik y1y
A e
ik y1y
)e i ( k z z
t)
.
(3.9)
Using the curl of the H-field, we can represent the z component of the E field by
the same unknown amplitudes:
Ez
(A e
ik y1y
A e
ik y1y
)e i ( k z z
t)
.
(3.10)
Where η is a constant defined by the permittivity of the region, the k y of the
region, the frequency of the electromagnetic wave (ω), and the speed of light (c).
ck y
(3.11)
28
The constraints on what waves can propagate in this geometry are determined by
the boundaries where the layers meet. We can relate the values of Hx and Ez at the first
boundary of a region (y = yo) to the second boundary of the same region (y = y1). We
evaluate Eqs. (3.11) and (3.12) at the second boundary, y1, and solve for the constants A+
and A- in terms of Hx(y1) and Ez(y1). We then evaluate Eqs. (3.11) and (3.12) at the first
boundary, y0, and using the results for A+ and A- we obtain a relationship that can be
described with a matrix transferring the values Hx and Ez at one boundary into terms of
Hx and Ez at the next boundary.
H x (y y0 )
E z (y y0 )
i sin( k y y)
cos(k y y)
i sin( k y y)
cos(k y y)
H x (y
y1 )
E z (y
y1 )
(3.12)
Eq (3.14) is a well known relationship.33,34 Where Δy is y1-yo. This supplies us
with the form of the transfer matrix relating Hx and Ez at the boundaries on either side of
a single region.35

mn
cos(k yn y n )
i sin( k yn y n )
n
i sin( k yn y n )
(3.13)
cos(k yn y n )
Because the tangential components of E and H are continuous, the transfer matrix
in Eq. (3.15) also relates Hx and Ez at the first boundary of region to the first boundary of
region two. With appropriate transfer matrices, we can make a connection from the
region one boundaries to the region two boundaries and then to the region three
boundaries. This can be represented nicely in a progression of transfer matrices. To relate
the fields at first boundary in region one to those at the last boundary in region three, we
must include the transfer associated with each layer:
29
H x (y y0 )
E z (y y0 )
   H (y y3 )
m1 m 2 m 3 x
E z (y y3 )
(3.14)
The resulting transfer matrix is a multiplication of all three transfer matrices.

M
  
m1 m 2 m 3
M11
M 21
M12
M 22
(3.15)
At the outer boundaries, y0 and y3, the boundary conditions are that Ez must be
zero because the Ag is considered as highly conductive. Thus
H x (y
y0 )
0
 H (y y3 )
M x
0
(3.16)
This gives the relation that the element M21 of the transfer matrix must be zero.
Expanding out M21, we get:
iM 21
1
sin( k y1 y1 ) cos(k y 2 y 2 ) cos(k y 3 y 3 )
2
cos(k y1 y1 ) sin( k y 2 y 2 ) cos(k y 3 y 3 )
3
cos(k y1 y1 ) cos(k y 2 y 2 ) sin( k y 3 y 3 )
1
3
(3.17)
sin( k y1 y1 ) sin( k y 2 y 2 ) sin( k y 3 y 3 )
2
0
One can make additional approximations, e.g. k y1 y1 and k y3 y 3 are both small and
then expand out the sine terms in Eq. (3.19). However, this does not substantially
simplify the resulting equation.
We find a solution to Eq. (3.19) numerically for kz as a function of . One picks a
real frequency
and a complex wavevector kz. This generates all the ky values in each
region. These values are then plugged into the matrix element M21. If the result is zero,
our guess for kz was good and we can pick a different frequency and start over. If not, we
30
guess a new value for kz. The guessing is done efficiently by using a root finding method
which searches the complex kz space.
Transmission Results for the small 3-Layer Notch Filter
In all the following graphs the same parameters were used, unless otherwise
stated. The permittivity of SiO2 is 4.0 (ε1 = ε3 = 4.0) the total thickness of the two
dielectric layers adds up to 4.5 microns, the applied field is 1000 Oe, the saturation
magnetization for Fe is 1700 G, the gyromagnetic ratio is γ
1.803 x 107 rad/G-sec, and
the permittivity for the Fe layer changes as a function of frequency and conductivity ζm =
107 /Ωm; ε2 = 1 + i ζm/ ε0ω. The damping factor Γ has been set by the parameters ΔH0 =
120 Oe and α=0.016.
The numerical method, outlined in the previous section, provides a complex
propagation wavevector kz for every frequency ω. The real part of kz gives information
on the wavelength and the imaginary part of kz determines the attenuation of the wave as
it propagates. In Fig. 3.2 we plot the transmission loss of the wave as a function of
frequency for a set of filters where the magnetic element is placed in different positions.
If the value is zero, then there are no transmission losses, and the EM wave gets
through the waveguide at the same intensity it went in with. By examining the dips of the
transmission, we notice that the largest attenuation occurs at the ferromagnetic resonance
frequency (ωres).
res
H 0 (H 0
4 M)
(3.18)
31
Rejection Band at -10 dB/cm
Transmission Loss (dB/cm)
0
-15
Fe at the edge.
-30
(.75 microns
from the edge)
-45
-60
-75
(1.5 microns
from the edge)
9
12
15
Frequency (GHz)
Fe in the middle.
(2.25 microns
from the edge)
18
Figure 3.2: Transmission as a function of frequency for different positions of the magnetic Fe
layer. The Fe film is 0.1 microns thick. There are four positions graphed, each with the Fe film
moved a certain distance away from one edge of the waveguide: 0 microns away, 0.75 microns,
1.5 microns, and 2.25 microns (right in the middle). The attenuation at resonance is greatest for
when the Fe film is in the center of the waveguide, and the dip width is the narrowest at the same
position.
We also notice that the largest attenuation occurs when the film is positioned
directly in the middle of the waveguide with equal amounts of dielectric on either side of
the film.
32
Fig. 3.3 shows transmission loss at the resonant frequency for varying thicknesses
of the magnetic material while keeping the amount of dielectric constant (4.5 microns
total thickness).
Transmission Loss (dB/cm)
0
-20
-40
Difference between the
middle and the edge results
Fe at the edge
of the waveguide
-60
-80
Fe in the middle
of the waveguide
0.0
0.1
0.2
0.3
0.4
0.5
Thickness of Magnetic Film (microns)
Figure 3.3: Transmission loss at resonance as a function of the thickness of the Fe film. The sum
of the total thickness for all dielectric layers is 4.5 microns. For the magnetic material at the edge,
the attenuation reaches a maximum at 0.10 microns. In the middle, the attenuation reaches a
maximum at 0.20 microns.
The results show that for extremely thin films (between 0.01-0.05 microns) the
position of the magnetic film doesn't make much difference, both the middle and the edge
positions have the same attenuation. For thicker films, the difference between attenuation
in the two positions increases. After 0.2 microns we find that the attenuation for both the
33
middle and the edge positions have reached their maximum values, and there is about a
constant 30 dB/cm difference between the two.
The saturation value for attenuation when the Fe film is at the edge of the
waveguide is about -55 dB/cm, and the saturation value for the Fe in the middle is about
-86 dB/cm. This could be the result of the skin depth in the Fe.11,31 A skin depth
calculation for Fe (which is frequency dependent),
2
(3.19)
gives about 0.1 microns at the FMR frequency. The EM wave must exist inside the Fe
film to be absorbed, but due to the skin depth it can only penetrate a limited distance. The
amount of the Fe the EM wave "sees" determines how much power is lost to the film. If
the film is on the edge of the waveguide, then the EM wave only sees one side of the
film. If the Fe film is in the center of the waveguide, then the EM wave penetrates both
sides of the film. From Fig. 3.3 we see that when the Fe film is at the edge of the
waveguide the attenuation reaches a constant value at a thickness of about 0.1 microns.
This distance is a measure of the skin depth in the Fe at resonance. In contrast, when the
film is in the center of the waveguide the attenuation becomes constant at a thickness of
about 0.2 microns. This further demonstrates the fact that the EM wave penetrates from
both sides of the Fe film.
Another important issue is how narrow or wide the transmission dips are. The
width of the dip will characterize the rejection band, which is the range of frequencies
that a notch filter effectively blocks out. This chapter uses the criterion that the rejection
band spans the region where the transmission loss is below -10 dB/cm.
34
As seen in Fig. 3.2, the transmission dip for the Fe film in the middle of the waveguide is
narrower than the dip for the film at the edge of the waveguide. Fig. 3.4 illustrates this
characteristic by plotting the width of the rejection band as a function of the position of
the magnetic film. The graph shows that the width reaches a minimum when the Fe film
Rejection Band at -10 dB/cm
(GHz)
is at the center of the waveguide.
6.0
Fe at the edges
of the waveguide
5.9
5.8
5.7
Fe in the Center
of the Waveguide
5.6
5.5
5.4
5.3
0
1
2
3
4
Position of the Fe film from
an edge of the waveguide (microns)
Figure 3.4: Rejection bandwidth as a function of the magnetic film's distance from the edge of the
waveguide. The bandwidth decreases as the magnetic film approaches the center of the
waveguide. The total sum of the dielectric layers is 4.5 microns. The Fe film is 0.1 microns thick.
We have seen that increasing the magnetic film thickness (up to the skin depth)
increases attenuation. The increase in film thickness also increases the width of the
rejection band. In Fig. 3.5 we plot transmission as a function of frequency with the
35
magnetic film in the middle of the waveguide for four different thicknesses of the Fe
film: 0.075 microns, 0.100 microns, and 0.150 microns. Clearly the smallest thickness
has the smallest rejection band. Fig. 3.6 summarizes this behavior and displays the
rejection bandwidth as a function of the Fe film thickness. The rejection bandwidth
increases almost linearly as a function of thickness 0.05 microns.
Reflections and Insertion Loss
Using the filter described above in an application, the wave would likely be
guided to the filter using a standard microstrip waveguide. This raises the issue of
reflection of the electromagnetic waves as they move from one structure to another. As
seen in Fig. 3.5, we again apply boundary conditions to find an approximation of the
reflection properties at the interface between the two waveguides (one with the magnetic
material, one without). As a quick approximation, let us assume the wave-guide passage
consists primarily of dielectric. In this case, there is only one allowed traveling wave of
TM character. For thin waveguides, this mode is nearly TEM in nature, and we consider
a case where there are no components of the H or E fields that are parallel with the
direction of propagation. The wave number k1 in the fully dielectric waveguide is:
k1
c
(3.20)
We consider the case where the wave starts in Region 1 of the waveguide (all dielectric)
and then encounters the structure with the magnetic layer, Region 2. An incident
electromagnetic wave in the first section traveling in the +z direction encounters the
interface at z = 0 and is partially reflected. For the nearly TEM mode we assume that
36
there is only a y component of the electric field. This y component can be used to find the
components of the H field. From the equation,

1 B
c t

E
(3.21)
we find that only an x component of the H field exists.
Hx
Ey
ic
v
z
(3.22)
In this case we are not using the complete tensor for the permeability. Instead we use the
scalar permeability from Eq (2).
Figure 3.5: Cross-section of the waveguide divided into two regions. Region 1 does not have a
magnetic layer. Region 2 has a magnetic layer. The wavevector in region 1 will be different than it
is in region 2.
The wave in region one is a superposition of both the incident and reflected parts.
However, after the interface, the wave will continue on with a different wavevector. So in
the two regions, denoted by subscripts 1 and 2, we can represent our E and H fields as:
37
E y1
H x1
E y2
A I eik1z
ic
v
ARe
(k1A I e ik1z
ik1z
(3.23)
k 1A R e
ik 1z
)
A T eik 2z
ic
H x2
v
(3.24)
(3.25)
(k 2 A T e ik 2z )
(3.26)
Because the parallel components of E and H must be continuous over the
interface we are able to solve for the amplitudes of the fields in terms of the wave
numbers. The reflection coefficient is ratio of the intensity of the reflected portion of the
E field and of the incident E field. We find
R
IR
II
k1 k 2
k1 k 2
2
(3.27)
The wavevector k1 is found from Eq (15) and k2 comes from the results of our
calculations in the first part of this paper. Note that k2 is complex and both the real and
imaginary parts are necessary for an appropriate result.
We compare the results of the approximation in Fig. 3.6 with an experiment in
Fig. 3.7. The experiment was performed using a microstrip geometry where a 0.1 micron
layer of Fe and 4.5 microns layer of SiO2 were used. The Fe film was at the top of the
dielectric, just under the metallic signal line.
38
The experiment measured the return loss of the signal entering the microstrip as a
function of the frequency for different applied fields. Using the boundary condition
method we find slightly different results from the experiment.
Return Loss (dB)
-4
0.715 kOe
-6
1.057 kOe
-8
1.745 kOe
-10
-12
-14
-16
0.385 kOe
0
Fe 0.1 microns
SiO2 4.5 microns
10
20
30
Frequency GHz
40
Figure 3.6: Return Loss vs. Frequency for an EM wave in a SiO2 waveguide (providing a k1 wave
number) interfacing with a waveguide geometry consisting of a 0.1 microns thick layer of Fe and a
4.5 microns thick layer of SiO2 (which provides the parameters for the calculation of a k2 wave
number).
The results for the boundary condition method in Fig 3.6 show a rise in the
reflected signal and a peak right around the FMR frequency. The maximum reflection is
between –8 dB and –5 dB. The maximum return loss again decreases as the applied field
is increased. After the peak, the return loss drops by about 7-8 dB’s in a matter of 3-4
39
GHz, reaches a minimum and then begins to increase again. The return loss at higher
frequencies converges for all the curves at a value of around -11 dB.
-4
Return Loss (dB)
-6
0.385 kOe
-8
-10
0.715 kOe
1.057 kOe
-12
1.745 kOe
-14
-16
Fe 0.1 microns
SiO2 4.5 microns
-18
0
5
10 15 20 25 30 35 40
Frequency GHz
Figure 3.7: Experimental Data of Return Loss vs. Frequency for a microstrip geometry using a
0.1 microns thick layer of Fe and a 4.5 microns thick layer of SiO 2.
The experimental results in Fig. 3.7 display similar trends. The return loss peaks
in the experiment occur slightly after the predicted FMR frequency. The maximum
reflected signal decreases as the field is increased, this is in agreement with the boundary
condition method. At frequencies beyond the peak, the return loss drops sharply by about
6-9 dB. After the drop, the return loss rises again, converging on a value of near –8 dB,
also agreeing with the calculation. Qualitatively, the results agree on the most important
points: 1) The maximum reflected signal occurs near the FMR frequency and 2) The
maximum reflected signal becomes smaller as the external field is increased. The values
40
for the maximum reflected signal ranged from -5 to -7 dB for the boundary condition
method, and from -7 to -11 dB in the experiment. Given the approximate nature of the
calculations, this level of agreement seems appropriate.
These results validate the
assumption that the signal can be inserted into the waveguide and the results of our
calculations within the structure are appropriate.
Conclusions
In conclusion, for a fixed thickness of the magnetic film, the attenuation at
resonance is at a maximum when the magnetic layer is at the center of the waveguide. In
addition, when the magnetic film is at the center of the structure, the rejection bandwidth
is minimized.
These features can be understood in terms of the skin depth of the magnetic film.
The attenuation of the EM wave is determined, in part, by how far it can penetrate the
magnetic material. If a film is thinner than the material’s skin depth, then it does not
matter where the film is positioned in the waveguide geometry. However, if a thicker
layer is used, it must be positioned away from the conductive edge in order for the
additional material to effectively contribute to the absorption of the wave.
We have also calculated the reflection of a wave entering from a non-magnetic
waveguide into a waveguide with a magnetic layer. We find that the reflection is largest
slightly above the FMR frequency. The experimental results show the reflection is
always below
-7 dB. The theoretical results also show a maximum reflected signal of
about -5 dB, but this could be reduced if the damping in the calculations was increased.
CHAPTER IV
Devices with Hexagonal-ferrites
We also investigated a notch filter using a hexagonal-ferrite as the ferromagnetic
layer in the waveguide. M-type Barium hexagonal ferrites are of interest because of their
high crystalline anisotropy and low losses. The strong anisotropy of the Barium
hexagonal ferrite (BaM) allows for the orientation of its magnetization in a thin film to
point out of the plane of the film. We investigated a waveguide with a film of a
hexagonal ferrite and a film of a dielectric material (SiO2) put between two conductive
layers in a microstrip-like geometry as shown in Fig. 4.1. The hexagonal ferrite is
assumed to be grown with its c axis out-of-plane. In this case, the magnetization and a
large anisotropy field will also point out-of-plane. Furthermore, in contrast to most of the
previous work using ferromagnetic metals, a magnetic field is applied perpendicular to
the surface of the waveguide.
There are several advantages to using a hexagonal ferrite: 1) The conductivity is
low, therefore, there are no issues with nonmagnetic losses. Because of this, the insertion
loss in the device is generally smaller compared to a filter using ferromagnetic metals. 2)
Hexagonal ferrite film thicknesses comparable to metallic films produce transmission
loss dips with greater symmetry than their metallic counterparts. 3) The internal
anisotropy fields produce an operational frequency close to 47 GHz at low fields (at least
1 kOe), compare this to the original Schloemann’s filter that required a large field (10
kOe) to reach the same operating frequency.2
42
For comparison, we also present results for a few different waveguides. In one
addition case the hexagon ferrite film has an in-plane anisotropy and magnetization. The
other addition case uses a metallic magnetic layer that has its magnetization forced out of
plane by an external field. We also evaluate devices made from hexagonal ferrites as
phase shifters for the magnetization oriented both in-plane and out-of-plane.
The main results are the following: 1) The structure outlined here can be used as a
tunable notch filter, with an attenuation of over 50 dB/cm at the notch. This is true even
with fairly thin hexagonal ferrite films around one micron thin and relatively poor
linewidths of 400 Oe. 2) There is a small shift in the frequency of the notch as the
thickness of the hexagonal ferrite film is changed. The thicker the film, the greater the
shift in frequency. 3) It is possible in the same device to get substantial phase shifts, on
the order of 360o/cm with relatively small losses, less than 2dB/cm. 4) A geometry where
the magnetization and anisotropy field lie in-plane requires a smaller external field to
obtain the same operational frequency as the out-of-plane structure.
We point out that the structure considered here is small, with thicknesses of the
hexagonal ferrite on the order of one micron. There has been a lot of work designed to
grow high quality hexagonal ferrite films with thicknesses of several microns or more
using a variety of different methods.36-38 In addition there have been some phase shifters
built with hexagonal ferrite films which are .2 mm thick.
39-40
This theoretical work
shows that thicker films may not be necessary, at least for some devices.
43
The Calculation for the Hexagonal Ferrite waveguide
The geometry of the waveguide is shown in Fig. 4.1. The magnetic field is
directed along the z axis, as is the magnetization. To be specific, the propagation of the
electromagnetic wave is along the y axis, although by symmetry all directions in the
plane are equivalent.
Figure 4.1: Geometry of the notch filter. The wave propagates along the y axis, perpendicular to
the applied field, Ho. The outer metallic layers are assumed to be a highly conductive material
such as Ag.
The behavior of the magnetic material is described by a dynamic permeability
tensor given by3

i
1
i
0
t
t
1
0
0
0
1
(4.1).
44
2
1
1 4 M
t
4 M
H
2
2
i
2
i
i
2
H
2
H H
H H
(4.2)
(4.3)
M is the saturation magnetization. H is the sum of the applied external field (H0), the
internal anisotropy field (Ha), and the static demagnetizing field (-4πM). |γ| is the absolute
value of the gyromagnetic ratio, ΔH is the measured full width half maximum
ferromagnetic resonance (FMR) linewidth for the material, and ω is the frequency of the
interacting electromagnetic wave.
There is a resonance in the permeability elements
when Δω2 = 0. Δω2 is given by
2
2
H2
2
(4.4).
Here the factor δ is close to 1 if the linewidth is small and is given by
1
2
H2
4
2
(4.5),
The FMR linewidth is generally frequency dependent. Based on experiments36,41,
we can assume that the FMR linewidth for barium hexagonal ferrite is linearly dependent
on frequency. We will assume it has the form:
H
H0
| |
(4.6)
where ΔH0 is the FMR linewidth at zero frequency and ψ is a unitless parameter that
determines how the linewidth changes with frequency.
45
If there is no damping, then ΔH = 0 and δ = 1. If this is the case, the denominator
for both μ1 and μt goes to zero when ω = |γ|H. This gives us a resonance frequency for our
material,
| |H
res
(4.7).
The final results of these calculations will show significant absorption near this
frequency.
In order to find the behavior of an electromagnetic wave in our device, we must
solve the wave equations which arise from Maxwell's equations using the boundary
conditions at the interfaces between materials. The wave equations of our system are
found from the two curl equations of Maxwell's equations.
hy
y
z
x
x
hz
y
hx
z
hx
y
hy
z
hz
x
z
x
y
hx
z
hy
x
hz
y
hz
x
hx
y
hy
2
c2 t 2
1
hx
i thy
i thx
hz
1
hy
(4.8)
z
The equation has the same form for all regions but the permittivity ε, and the
permeability components μ1 and μt change from region to region. The main difference
between this calculation and the calculation in the previous chapter is that we use the full
tensor.
We assume that the h-field for our EM wave can be represented by this form:

h
 i ( k r
h 0e
t)
(4.9)
where the vector h0 represents the amplitude components of the h-field. Because we are
dealing with an infinite planar geometry and the permeability is oriented out-of-plane, the
46
direction of propagation in the plane is arbitrary. Therefore, we can choose the wave to
propagate in the y-direction. This constraint makes the x-component of the wave vector
zero (kx=0) and the wave equation from the double curl equations can be rewritten for
each region
2
k
2
y
k
2
z
2
i
1
c2
2
i
0
t
c2
hx
2
k
t
c2
2
z
k yk z
c2
hy
hz
2
0
k yk z
k
2
y
c2
0
(4.10).
1
This matrix containing the permittivity, the permeability, frequency, the k y’s and kz’s will
be referred to as matrix Q. The kz’s are the wavevectors for the standing wave modes in
the different layers and ky is the wavevector describing propagation. The relationship can
be simply written as:

Qh
0
(4.11)
For a non-trivial solution to this equation the determinant of matrix Q is equal to
zero. This gives us a dispersion relationship in each region:
2
kz
4
2
ky (
1
1) 2
c
2
1
2
kz
2
ky
2
2
c
ky
2
2
1
c
2
(
2
1
2
t
)
0
(4.12)
We define the following two quantities with wavevector units:
2
q1
ky
2
1
ky
2
2
(4.13)
c2
2
q2
ky
2
c2
1/ 4
2
ky
2
1
c2
(
2
1
2
t
)
(4.14)
47
Eq. (4.12) becomes:
kz
4
q12 k z
2
q 42
0
(4.15)
This gives four solutions for kz in each region.
q12
kz
q14
2
4q 42
(4.16)
We can find relationships between the components of the h-field. In other words,
for a given kz we can find hy and hz in terms of hx. For example, we can write this as
hy
(k z )h x
hz
(k z )h x
(4.17)
(4.18)
where α and β are coefficients with a dependence on permeability, permittivity,
frequency and ky. These are found from the Q-matrix acting on the h-field in Eq. (4.10).
We let hx be a superposition of the four waves associated with the four kz vectors
4
hx
Ane
ik z n z ik y y
e
(4.19)
n 1
Then we can represent the solution of the components hy and hz as they relate to the
solution for hx:
4
hy
(k zn )A n e
ik z n z ik y y
e
(4.20)
n 1
and
4
hz
(k z n )A n e
n 1
ik z n z ik y y
e
(4.21).
48
The h-field can now be represented in each region of our waveguide in terms of the xcomponent of the field, hx, with four unknowns A1 to A4
One now must consider the transverse magnetic wave in the dielectric layer.
Examining Eq. (4.12) with
1
1 and
t
0 we find the usual dispersion relationship
2
kz
2
k
2
y
c2
0 . This leads to two solutions for kz:
2
kz
i k 2y
(4.22)
c2
In addition, there is partial diagonalization of the matrix in Eq. (4.10) which shows that hy
and hz have no dependence on hx. This means that there are two independent modes that
can exist. One mode has only an hx component, and the other mode has only hy and hz
components. Because of these modes are possibile, the h-fields for the dielectric must be
considered a superposition of these two modes with the weight of their importance
determined by coefficients similar to those we have used in Eq. (4.19). The h-field
components in a dielectric region can be written as:
hx
[A5eik z z
hy
[A7 eik zz
hz
[ ( k z )A 7eik zz
A 6e
A8e
ik z z
ik z z
]e
]e
ik y y
(4.23)
ik y y
( k z ) A8e
(4.24)
ik z z
]e
ik y y
(4.25)
where η is the coefficient relating hz to hy, found from Eq. (4.10).
Because there are two regions (hexagonal ferrite, and the SiO2), there are four
different kz values and four different A coefficients for each region. This gives a total of
49
eight A coefficients. We will see that there are eight boundary conditions which will
allow us to obtain a complete set of equations for the unknown A values.
The E and h fields in each region need to satisfy the boundary conditions at the
interfaces between materials. These conditions are that the tangential components of E
and h are continuous. We can find the E fields for each region from the h-field by using
Maxwell's equations and substituting in Eqs. (19-21) or (23-25), based on the region.
We assume the tangential E field is zero at the interfaces between the dielectric
and the top, highly conductive, layer and between the hexagonal ferrite and the bottom,
highly conductive, layer. This condition will give us four equations: two for the Ex and Ey
components at the top of the waveguide and two for the Ex and Ey components at the
bottom of the wave guide. Looking at the boundary of the dielectric layer and the
hexagonal ferrite, we have four additional equations: two equations for the Ex and Ey
components and two for the hx and hy components. Thus there are a total of 8 equations.

We can write this set of equations in a matrix form, i.e. GA
0 where the elements of

the vector A are defined in Eqs. (19-21) and Eqs. (23-25). The elements of G are given
in Appendix A.
Every element of G is a function of frequency, ω, and wavevector, ky, (all kz’s are
functions of ky). For a non-trivial solution, the determinant of the matrix G must be zero:

det G( , k y )
0
(4.26)
This condition provides us with the dispersion relation, the relationship between a wave’s
frequency and wavevector.
The solution is found through the following numerical
method. We choose a particular frequency ω and guess a value for ky. From the wave
equations in the different regions, we can find a solution for kz in terms of ω and ky.
50
These values are substituted into the matrix G, and the determinant of G is calculated. If
it is zero, the guess for ky was good and we can pick a different frequency and start over.
If it is not zero, we guess a new value for ky. The guessing is done efficiently by using a
root finding method which searches the complex ky space.
Transmission Results for the Hexagonal Ferrite Waveguide
In all the graphs we have used the following parameters. The reported permittivity
of barium hexagonal ferrite varies according to different sources.42,43 Based on this we
chose a permittivity of 10 for M-type barium hexagonal ferrite. The saturation
magnetization of the M-type barium hexagonal ferrite is 0.334 kG, and the internal
anisotropy field is 16.4 kOe. We do not consider variations in the magnetization or
anisotropy field due to changes in temperature. However, it has been shown in the
cataloged properties of hexaferrites that the Curie temperatures for these materials are
around 700-750 K. In a range of 100 K centered on room temperature, the anisotropy
field increases by about 3% and the magnetization decreases by about 23% as the
temperature increases.44 This would lead to a small shift of about 4-5 GHz in the resonant
frequency over the whole range. The permittivity of SiO2 is 4.0. The SiO2 could be
replaced by a variety of dielectrics with different permitivities. The point of this layer is
to provide a portion of the waveguide that facilitates propagation of the electromagnetic
wave. In some previously grown devices the dielectric layer doubled as the substrate on
which the magnetic material was grown2 but this is not always the case, as seen in more
recent thin film devices.8 We have chosen SiO2 for our results because it is commonly
used as the filler layer in microstrip devices.
51
We explore the transmission of the filter as a function of frequency for changes in
the thicknesses of the dielectric layer, the thickness of the hexagonal ferrite thin film, the
external applied field (H0), and the damping in the hexagonal ferrite (determined by ΔH).
The material’s FMR linewidth (ΔH) is set by the parameters ΔH0 and ψ from Eq. (4.6).
The numerical method, outlined in the previous section, provides a complex
propagation wavevector ky for every frequency ω. The real part of ky gives information
on the EM wave’s phase at a given position y, and the imaginary part of k y determines the
transmission loss of the wave as it propagates.
For a notch filter device, the interest is the transmission as a function of
frequency. Fig. 4.2 gives the results for a filter with 4.0 microns of dielectric and 0.5
microns of barium hexagonal ferrite and plots the transmission at different applied fields.
The linewidth at 40 GHz is 400 Oe (ψ = 200 Oe |γ|/(40GHz), ΔH0 = 200 Oe). The results
show a substantial symmetric dip or notch in the transmission, with the position of the
notch tunable with an external field. For this geometry the maximum transmission loss is
about 80 dB/cm.
The largest attenuation occurs near the ferromagnetic resonance
frequency (ωres ). A closer examination shows that the greatest transmission loss for each
dip occurs about 0.8 GHz past the calculated FMR frequency from Eq.(4.4).
52
Transmission (dB/cm)
0
-20
4.0 microns SiO2
-40
0.5 microns BaM
200 Oe linewidth
-60
1 kOe
-80
5 kOe
3 kOe
-100
30
35
40
45
50
Frequency (GHz)
55
60
Figure 4.2: Transmission as a function of frequency for three different values of the applied field.
The thickness of the BaM is 0.5 μm. The thickness of the SiO2 film is 4.0 μm. The linewidth of
the BaM is set at 400 Oe at 40 GHz by the parameters ΔH0 = 200 Oe and ψ/|γ| = 200 Oe /40 GHz.
Fig. 4.3 shows results for a waveguide where the thicknesses of the dielectric
layer and the hexagonal ferrite layer are reversed compared to those used in Fig. 4.2. The
SiO2 layer is 0.5 microns thick and the hexagonal ferrite layer is 4.0 microns thick. The
results show the same features as in Fig. 4.2, except the losses are all much greater, from
about 80 dB/cm in Fig. 4.2 to about 600 dB/cm in Fig. 4.3, and the shift in frequency of
the maximum transmission loss above the FMR frequency is also greater. In Fig. 4.3 the
position of the notch is about 5 GHz higher than the calculated FMR frequency. Fig. 4.3
also shows that the notch is no longer symmetric in the case of the thicker hexagonal
ferrite films.
53
Transmission (dB/cm)
0
-100
-200
-300
0.5 microns SiO2
-400
4.0 microns BaM
200 Oe linewidth
-500
1 kOe
-600
-700
35
3 kOe
40
5 kOe
45
50
55
60
Frequency (GHz)
65
Figure 4.3: Transmission as a function of frequency for three different values of the applied field.
The thicknesses of the BaM and SiO2 are reversed from Fig. 4.2. The thickness of the BaM is 4.0
μm. This thickness of the SiO2 film is 0.5 μm. The linewidth of the BaM is the same as in Fig.
4.2. Note the attenuation at the notch is significantly larger than that seen in Fig. 4.2.
The expected FMR behavior for ferromagnets is that the resonant frequency of the
material shifts linearly with the applied field. However, in comparing the results from
Figs. 4.2 and 4.3, a surprising feature emerges - the notch position depends not only on
the applied field, but also on the thickness of the hexagonal ferrite. Fig. 4.4 plots the
frequency of the maximum transmission loss versus the ratio of thickness of the BaM to
the total thickness of the waveguide in a 3 kOe applied magnetic field. The graph shows a
significant shift in the frequency of maximum attenuation away from the calculated
ferromagnetic resonance frequency, Eq. (4.4). The shift is nearly linear as a function of
the filling fraction. The maximum attenuation frequency of around 48.5 GHz occurs
54
when the hexagonal ferrite fills the entire waveguide. This is 6.0 GHz above the
calculated 42.5 GHz FMR for a 3 kOe applied field.
Frequency of Maximum
Transmission Loss (GHz)
50
48
Ho = 3 kOe
46
44
42
0.0
0.2
0.4
0.6
0.8
1.0
BaM filling fraction
Figure 4.4: Frequency of the notch as a function of the filling fraction ratio of the thickness of the
BaM divided by the entire thickness between the metallic layers in the notch filter. The applied
field is 3kOe, and the linewidth is the same as in Fig. 2 and Fig. 4.3.
This shift in the notch position is a result obtained from the full boundary
condition calculation. In an FMR experiment the effective wavevector is zero, i.e. one
has a uniform oscillating magnetic field, and one expects the maximum absorption at the
resonant frequency. However, in this experiment k ≠ 0. The propagation wavevector k y
is dependent on all the kz’s (standing waves) in each region and on the thicknesses of the
55
various films through the boundary conditions. This relationship between the changes in
thickness and the changes in standing waves influences the frequency of the transmission
loss of the wave.
We can obtain some idea of why the frequency of the maximum transmission loss
is not simply at the resonance frequency given by Eq. (4.4). If one considers the
propagation of magnetostatic waves in an infinite material, then Maxwell's equations may
be reduced to an equation for the magnetic scalar potential φ which obeys an anisotropic
Laplace equation:
1
2
2
2
x2
y2
z2
0
(4.27)
A wave in this material will have the form
Ae
i(k y y k zz)
e
i t
(4.28)
and substitution of this form into Eq. (4.27) gives a connection between the propagation
wavevectors and the frequency through the permeability.
1(
)
k 2z
(4.29)
k 2y
If we use Eq. (2) and let the damping go to zero, we obtain
( / )2
H2
4 MH
k 2z
1
k 2y
(4.30)
One should note two a special limits: 1) if propagation is parallel to the magnetic field,
then
ω = |γ|H and 2) for propagation perpendicular to the magnetic field ω2 = |γ|2H ( H
+ 4πM ). The point is that the frequency of the wave is not necessarily just |γ|H , but
56
depends on the direction of propagation. A similar analysis can be found in Xu Zuo et
al’s work.39,40
The effective direction of propagation in the dynamic problem depends on the full
use of Maxwell's equations. For an out-of-plane hexagonal ferrite filter with the
dimensions used in Fig. 2, our results show that the ratio of k z2/ky2 is typically in the 1-13
range. According to the magnetostatic case given by Eq. (4.30), this would produce
resonance at a frequency shifted about 0.5-2.5 GHz above the FMR predicted by Eq.
(4.7). According to Fig. 4.2 and Fig. 4.4 the frequency of maximum attenuation is shifted
by 0.8 GHz above the predicted frequency, which falls into the 0.5-2.5 GHz range. For
the parameters in Fig. 4.3, our results kz2/ky2 show magnitudes between 0.2 and 5.0.
According to Eq. (4.30), this would make the shift range between 1-5 GHz. The shift of
6.0 GHz shown in Figs. 4.3 and 4.4 is just out of this range. Because we are not
examining a magnetostatic case, the results are not expected to match perfectly.
However, Eq. (4.30) does give results for wavevectors in the range of our calculation
results that are shifted off the Eq. (4.7) results by the same order of magnitude. These
results give an intuition that the frequency of maximum absorption is close to ω = |γ|H
but increases slightly from that value depending on the values of kz and ky, which are
needed to solve the boundary conditions. This implies that the resonant frequency is
indeed related to the thickness of the waveguide.
There is also a shift in the frequency of the maximum transmission loss from the
damping. The solid line in Fig. 4.5 is a graph of the frequency of the notch versus the
FMR linewidth. The linewidth is for a frequency of 40 GHz for a 3 kOe field. The dashed
line in this figure is derived from Eq. (4.4) with the damping included. There is
57
approximately 0.8 GHz difference between them at a 1000 Oe linewidth. That difference
increases to about 1.6 GHz difference at a linewidth of 5000 Oe. At low linewidths, the
difference between these two values can be attributed to the shift in frequency as a
function of thickness as discussed earlier. As the linewidth increases it also affects the
Frquency of Maximum
Transmission Loss (GHz)
wavevector, making the thickness shift greater.
44.5
H0= 3 kOe
44.0
4.0 SiO2
43.5
0.5 BaM
43.0
42.5
42.0
Predicted Resonance from
Equation 4
1000
2000
3000
4000
5000
FMR linewidth at 40.0GHz (Oe)
Figure 4.5: Frequency of the maximum attenuation as a function of linewidth. The thickness of
the BaM and SiO2 layers is the same as in Fig 4.2, as are the linewidth parameters. The solid line
is the frequency of maximum attenuation obtained from the full boundary condition calculation.
The dashed line is the calculated FMR frequency from Eq. (4.7).
In addition to the results for the out-of-plane hexagonal ferrite, we also consider
results for a magnetic layer that is metallic with an out-of-plane magnetization. Such an
58
orientation could be useful, even for the metallic ferromagnet, because this geometry
allows one to use a large out-of-plane magnetic field over a short distance, i.e. the
thickness of the film and substrate.
In contrast, the in-plane geometry requires a
magnetic field extending over several millimeters.
These results are based on the same two-layer geometry shown in Fig. 4.1, except
that the hexagonal ferrite is replaced by Permalloy. The parameters used for Permalloy
are M = 0.850 kG, ζ = 1.67 x 105 C2/Nm2-sec for its conductivity, and |γ| = 1.80 x 107
rad/G-sec. Fig. 4.6 shows a similar behavior for transmission as that seen in Figs. 4.2 and
4.3. However, the operational frequencies for devices made out of Permalloy are
considerably lower than for the BaM devices, even when much larger external fields are
used. There are two reasons for this. Permalloy does not have the large anisotropy field
of BaM and the larger M in Permalloy also means a larger demagnetizing field which
must be overcome by the external field.
Barium hexagonal ferrite can also be grown to have its anisotropy field in-plane.
The theory for this geometry has been published earlier, for more discussion on this
consider references.3,30,45 For the purpose of this paper, we only present the results of this
theory. Fig. 4.7 shows the transmission loss results for the same film thicknesses and
damping parameters as used in Fig. 4.2; the difference is that the permeability used for
the calculation is for an in-plane magnetization with an in-plane anisotropy field. The
most obvious feature is that the in-plane geometry results in a higher operating frequency
for the same applied field. For the in-plane case, the static demagnetizing field does not
reduce the effect of the external field, and the dynamic demagnetizing fields also increase
the frequency. The magnitude of the maximum transmission loss is about 20 dB more for
59
the out-of-plane case (Fig. 4.2) compared to the in-plane case in Fig. 4.7. We note that in
Fig. 4.7 the magnitude for maximum attenuation decreases as the external field is
increased, this is in contrast with the results in Fig. 4.2. The behavior of the linewidth
and the maximum attenuation can be complicated. It can either increase or decrease as a
function of field or frequency. This effect is discussed by Kuanr et al.41
Transmission (dB/cm)
0
-15
-30
4.0 m SiO2
-45
0.5 m permalloy
400 Oe linewidth
-60 13 kOe
14 kOe
-75
8
10
12
15 kOe
14
16
18
20
Frequency (GHz)
Figure 4.6: Transmission as a function of frequency for three different fields for a notch filter
with Permalloy as the active element. The external field is directed out-of-plane as shown in Fig.
4.1. Note that the frequencies are all considerably lower for the Permalloy film compared to the
BaM film in Fig. 4.2.
60
Transmission (dB/cm)
0
-10
-20
4.0 microns SiO2
-30
0.5 microns BaM
400 Oe linewidth
-40
-50
1 kOe
3 kOe
2 kOe
-60
45
50
55
60
65
Frequency (GHz)
Figure 4.7: Transmission as a function of frequency for a BaM filter with three different values of
the applied field. The anisotropy axis and the applied field are now directed in-plane, and parallel
to the propagation direction. The thickness of the BaM is 0.5 μm. The thickness of the SiO2 film
is 4.0 μm. Note that the frequencies are all higher than those found for the BaM film with the outof-plane geometry as shown in Fig. 4.2.
Using the waveguide as a Phaseshifter
The planar waveguides with metallic ferromagnetic films can also be used as
phase shifters, however the performance in the typical microstrip filter shows significant
damping. In this section will explore the phase shifting properties of a microstrip filter
using BaM as the active material.
61
Fig. 4.8 shows, for fixed frequencies, the phase difference in the real part of the
propagation wavevector, ky, as a function of a change in the external magnetic field. To
be more specific, Fig. 4.8 plots the phase difference of ky(Ho = 2kOe) – ky(Ho).
Phase difference
(rad/cm)
20
15
10
5
0
-5
-10
-15
48 GHz
45 GHz
51 GHz
4.0 microns SiO2
0.8 microns BaM
200 Oe linewidth
0
2
4
6
Field Difference from 2 kOe
Figure 4.8: Phase shift at different frequencies as a function of the change in the magnetic field
from 2 kOe. The thickness of the BaM is 0.8 μm. The thickness of the SiO2 film is 4.0 μm. The
linewidth of the BaM is set at 200 Oe at 40 GHz by the parameters ΔH0 = 100 Oe and ψ= 100 Oe
|γ| /40 GHz. These results are for the out-of-plane geometry.
The results shown are found with the boundary condition and root finding method
outlined in the theory section. The BaM layer in Fig. 4.8 is thicker than the one used in
Fig. 4.2 (0.8 μm compared to 0.5 μm) and the damping is lower (ΔH = 200 Oe, with ψ =
100 Oe |γ|/(40 Ghz), and ΔH0 = 100 Oe). One sees that substantial phase shifts are
possible; however the largest values all occur near the ferromagnetic resonance
62
frequency. This means that there is a lot of attenuation, which makes most of the shifted
signal unusable.
For practical devices, a transmission loss of no more than -2 dB/cm is a
reasonable cut off. Fig. 4.9 is Fig. 4.8 redrawn with all phase shifts that operate at a field
with transmission losses below -2 dB/cm removed. The phase shifting range has been
substantially limited. For the 45 GHz signal, the phase can be shifted from 0 rad/cm to 0.3 rad/cm at 0.16 kOe away from 2 kOe. For larger field differences the phase shift
ranges from 5 rad/cm to 3 rad/cm. For higher frequencies, the general trends of the
Phase difference
(rad/cm)
behavior are the same but, the phase shifts become smaller.
5
4
3
2
1
0
-1
-2
-2 dB/cm cut off
45 GHz
45 GHz
48 GHz
51 GHz
4.0 microns SiO2
0.8 microns BaM
200 Oe linewidth
48 GHz
51 GHz
0
2
4
6
Field Difference from 2 kOe
Figure 4.9: Phase shift at different frequencies as a function of the change in the magnetic field
from 2 kOe. This figure uses the same data as Fig. 4.8, except that it omits frequencies that
produce transmissions less than -2 dB/cm. These results are for the out-of-plane geometry.
63
To achieve larger phase shifts, one approach might be to make the layers of the
BaM thicker. In addition, to have a wider range of usable applied fields, the damping
could be decreased, thereby decreasing the linewidths of the transmission loss dips. These
two strategies are used in Fig. 4.10 which shows the same type of graph as Fig. 4.9 but,
with a BaM layer that is twice as thick (BaM = 1.6 μm), and has half the damping (ΔH =
100 Oe, with ψ = 50 Oe |γ|/(40GHz), and ΔH0 = 50 Oe). Furthermore, the investigated
frequencies are increased by +3 GHz each. The phase shifts are noticeably larger, and the
width of the lower field ranges has also been increased. For example, the 48 GHz phase
shift now ranges from 8.0 rad/cm to 4.4 rad/cm as the field difference changes from +3.5
kOe to +8 kOe. Again the phase shifts at higher frequencies are somewhat smaller.
Phase difference
(rad/cm)
-2 dB/cm cut off
8
6
4
2
0
-2
-4
51 GHz
48 GHz
54 GHz
4.0 microns SiO2
54 GHz
48 GHz
51 GHz
1.6 microns BaM
100 Oe linewidth
0
2
4
6
8
Field Difference from 2 kOe
Figure 4.10: Phase shift at different frequencies as a function of the change in the magnetic field
from 2 kOe. Although this figure is the same type of graph as Fig. 9, the parameters in this figure
differ from Fig. 9. The thickness of the BaM is 1.6 μm. This thickness of the SiO2 film is 4.0 μm.
The linewidth of the BaM is set at 100 Oe at 40 GHz by the parameters ΔH 0 = 50 Oe and ψ = 50
Oe |γ| /40 GHz. These results are for the out-of-plane geometry.
64
We have also studied the phase shifting behavior for the geometry where the
magnetization and anisotropy are in-plane. Because the demagnetizing field is no longer
opposing the anisotropy field, results are readily obtained using lower applied fields. Fig.
4.11 compares differences in phase between an original zero-applied-field state and one
at a higher field of Ho. This figure assumes that the magnetization has already been
established and is being maintained by the anisotropy field. For 57 GHz, the high field
phase shift varies from 5.5 rad/cm to 4.0 rad/cm as the field changes from 3.4 kOe to 7
Phase Difference
(rad/cm)
kOe.
7
6
5
4
3
2
1
0
-1
-2
-2 dB/cm cut off
57 GHz
60 GHz
63 GHz
57 GHz
4.0 m SiO2
0.8 m BaM
200 Oe linewidth
60 GHz
0
63 GHz
1 2 3 4 5 6
Applied H field (kOe)
7
Figure 4.11: Phase shift at different frequencies as a function of the change in the magnetic field
from 0 kOe. The parameters are the same as in Fig. 4.9, but the geometry is for an in-plane
magnetization.
65
Fig. 4.12 graphs the phase difference versus the field difference for a thicker, less
damped BaM layer in the in-plane geometry. The new parameters are 1.6 μm of BaM
with a damping of ΔH = 100 Oe (ψ = 50 Oe |γ|/(40GHz ), and ΔH0 = 50 Oe). As
expected, one finds larger phase shifts and a larger field range compared to the results of
Fig. 4.11. For example, for 57 GHz, the high field shifts now range from 10.5 rad/cm to
Phase Difference
(rad/cm)
7.0 rad/cm in a range of fields from 3.5 kOe to 7.0 kOe.
12
10
8
6
4
2
0
-2
-4
-2 dB/cm cut off
57 GHz
57 GHz
60 GHz
63 GHz
4.0 microns SiO2
1.6 microns BAM
100 Oe linewidth
60 GHz 63 GHz
0
1 2 3 4 5 6
Applied H field (kOe)
7
Figure 4.12: Phase shift at different frequencies as a function of the change in the magnetic field
from 0 kOe. The parameters in this graph are the same as in Fig. 4.10, but the geometry is for an
in-plane magnetization.
66
Experimental Comparison
A notch filter experiment with BaM components was performed by Z. Wang et al
46
. The experiment transmitted a signal down a co-planar wave guide with a 5 μm thick
film of BaM placed on top of it, followed by a thick layer of sapphire (>150 μm). There
was also be a small air gap in between the co-planar waveguide and the BaM. Fig. 4.13
shows a) the transmission of that signal as a function of frequency, and b) the phase of
the signal as a function of frequency. The experiment was performed with four different
applied fields: 4.62 kOe, 5.17 kOe, 5.62 kOe, and 6.31 kOe. We compare these
experimental results with Fig. 4.14, theoretical results we calculated for an infinite planar
wave guide with three layers: a 5 μm thick layer of BaM , 150 μm thick layer of
sapphire, and a 5 μm air gap.
Figure 4.13: Experimental results for a coplanar waveguide with 5 μm of BaM on top of it.
Figure 4.14: Theoretical results for a infinite planar wave guide with a 5 μm of BaM magnetic
layer.
67
The theoretic results match quite well with the experimental results. The
transmission dips have an error of about ± 1 GHz in frequency. The magnitude of the
transmission dips have an error less than ± 1 dB. The magnitudes of the phases don’t
match but, this was expected due to the experiment having sections of the signal line
without BaM present and the theory only considered the regions with BaM. Even though
the magnitudes of phase seems not to match, they are still qualitatively close in frequency
behavior.
Conclusions
We have theoretically studied the performance of tunable notch filters and phase
shifters which use BaM hexagonal ferrites as the active element in a small planar device.
The devices operate at frequencies in the 40-60 GHz range with relatively low applied
magnetic fields. We have considered anisotropy fields both out-of-plane and in-plane. As
a notch filter, the device has significant attenuation at the notch, typically in the 60
dB/cm range, even for BaM films which are less than a micron in thickness.
Because hexagonal ferrite is not a metal, electromagnetic waves propagating in
the filters do not have the significant eddy current losses which can be found in filters
using ferromagnetic metals. This allows devices to be made with thicker hexagonal
ferrite layers and can lead to larger attenuations at the notch and to larger phase shifts.
As a phase shifting device, the out-of-plane hexagonal ferrite geometry requires
strong fields, between 6 kOe and 8 kOe. However phase shifts on the order of 360o/cm
are possible for materials which are about one micron in thickness and which have an
FMR linewidth less than 200 Oe at 40 GHz. An in-plane hexagonal ferrite phase shifting
68
device with the same thickness and damping parameters can also show a significant
tunable phase shift at lower fields between 3 kOe and 6 kOe.
CHAPTER V
Non-reciprocal Ferromagentic Devices using Attenuated Total Reflection
Polaritons are electromagnetic oscillations in a material strongly coupled with
magnetic or electric dipole excitations. There has been a good deal of research done on
surface polaritons.47-51 A well known and widely used technique for studying the
properties of both bulk and surface polaritons is attenuated total reflection (ATR). ATR
is also used in material analysis and biological analysis.52-55 Additional work has shown
the connection of polaritons and plasmons with ATR.56-59 There have been a number of
experimental studies of antiferromagnetic materials using ATR or reflectivity as a probe
of both bulk and surface polaritons.60-62
Previous work for ATR considered an electromagnetic wave interacting with a
three layered planar structure as shown in Fig. 5.1.56 For the three layer case, the first
layer is generally a dielectric prism guiding electromagnetic waves. The waves encounter
an interface with a second layer, which we will call the gap layer (in previous work, the
gap layer was often taken to be a vacuum). If the incident angle that the incoming wave
makes with the interface is adequately large, and the permittivity of the prisim is greater
than the permittivity of the gap, then the electromagnetic wave totally internally reflects
off the interface, and the reflected power of the wave is equal to the incident power.
However, there is also an evanescent wave in the gap which exponentially decays with
distance from the upper interface. If the gap layer is thin enough, a third layer beneath it
70
could interact with the evanescent wave. If the third layer is a semi-infinite magnetic
material, the evanescent wave can excite a magnetic polariton in this layer. This takes
energy away from the reflected wave, typically in a limited frequency range.
Figure 5.1: Three-layer ATR geometry. The prism layer is treated as semi-infinite. The gap layer
has a finite thickness d. The magnetic layer is also semi-infinite. M is the magnetization of the
magnetic layer and Ho is an applied magnetic field.
This chapter explores the physics of attenuated total reflection (ATR) in a
multiple layered heterostructure in order to find possible alternative methods for creating
non-reciprocal devices. A non-reciprocal device is a system where the reflection
coefficient is different if the directions of the incident and reflected waves are reversed.
The origin of non-reciprocal ATR comes from nonreciprocal polaritons traveling in
opposite directions on the surface of a magnetic material. These polaritons attenuate the
reflection of an incoming electromagnetic wave, resulting in a possible isolator or
71
circulator device application. In fact, we propose a device with an effective ATR-like
structure which can be built on a planar geometry using microstrip-like structures.
In addition to the Fig. 5.1 geometry, this chapter examines a different geometry.
This different case, illustrated in Fig. 5.2, involves a fourth layer, as also investigated by
Ruppin.61,62 The fourth layer is a semi-infinite dielectric layer. This four-layer geometry
is necessary for our proposed device.
Figure 5.2: Four-layer ATR geometry. The incoming electromagnetic wave reflects off the
interface between the prism layer and the gap layer at an incident angle (θ). Permitivities ε 1, ε2,
and ε4 are all for dielectric materials. The magnetic layer has permittivity and permeability given
by ε3, μ1, and μt . The gap thickness and magnetic layer thickness are given by d 1 and d2. There is
an applied static magnetic field in the z-direction (H0).
The key result of our calculations is that the ATR geometry exhibits strong nonreciprocal behavior even for thin magnetic films. For relatively thick gap and magnetic
layers (0.05 cm and higher), the non-reciprocity follows the results for surface polaritons
defined by the gap material and the magnetic material. For thinner magnetic films, the
72
non-reciprocal behavior is complex. Results are presented for yttrium iron garnet (YIG)
and M type barium hexagonal ferrites (BaM). The frequency range of non-reciprocal
behavior can be quite large and is tunable both by an external field and by changing the
thicknesses of the gap layer, the magnetic film, or both. For example, nonreciprocal
behavior is found in BaM from 45–80 GHz. ATR geometries display a wide range of
possible applications, such as notchfilters, bandpass fileter, and isolators. The type of
application possible depends not only on the materials, but also on choosing the incident
wave angle, the gap thicknesss, and the magnetic layer thickness.
Surface Polaritons and Three-layer ATR
We examine solutions for electromagnetic waves with the electric field in the z
direction, transverse to the planar structure, and parallel to a static magnetic field Ho.
These transverse electric (TE) modes propagate is the xy plane.
Before we calculate the ATR results, we obtain the dispersion relation for a
surface polariton on a magnetic medium. We start with the permeability tensor of a given
magnetic material:

i
1
i
0
t
t
1
0
0
0
1
(5.1).
This particular tensor is for a ferromagnetic material with an external magnetic field
applied in the z-direction (see Fig. 5.1 or Fig. 5.2). The factors μ1 and μt are well-known
functions of the applied field in the z-direction, Ho, the frequency of the interacting
electromagnetic wave, ω, the magnetization of the material, M, and the linewidth of that
material, ΔH.60 By using Maxwell’s equations, a wave equation can be made and
73
evaluated at the boundary between two different materials (one of which is the magnetic
material defined by permeability). From this evaluation the dispersion relationship of a
magnetic spin wave confined to the interface between a semi-infinite dielectric and a
semi-infinite magnetic medium has been found to be :
2
kx
2
1
c
(
v
1)(
1
v d
m
2
) 2
2
t
2
d
(
v
t
1) 2
1
1
( 1(
2
d
2
m
)
d
(
m
v
1
1))
2
4
t
(5.2).
The dielectric material’s permittivity is εd and the magnetic material’s is εm. The
components of the magnetic material’s permeability tensor are μ1 and μt. The propagation
wavevector is kx (which is parallel to the surface), c is the speed of light, and ω is the
wave’s frequency. The factor μv is the voight permeability and is defined as
2
2
1
t
(5.3).
v
1
Eq (5.2) is an extension to the result found in Harstien et al.48 In that work they
considered only the case where εd is 1. In our device, we will need to consider different
values of εd.
The dispersion relationship for a surface mode is nonreciprocal. Notice that there
are four possible solutions for Eq. (5.2). From the initial conditions used for this equation
we find that only two of the four solutions are physical. The two physical solutions are:
kx
kx
1
(
v
1)(
1
v d
m
) 2
c
2
t
(
1
c
(
v
1
1)(
v d
m
) 2
2
d
v
(
1
2
d
v
1
1) 2
2
t
t
1
( 1(
2
2
d
m
)
d
m
(
v
1
1))
2
4
(5.4),
t
( 1(
t
1
1) 2
4
2
t
2
d
2
m
)
d
m
(
v
1
1))
(5.5).
74
Eqs. (5.4) and (5.5) show that there is a different dispersion relationship
depending on which direction the wave is propagating (notice the different sign at the
beginning of Eq. (5.5) and in its radical). Surface spin waves are non-reciprocal and
therefore we expect non-reciprocal behavior from interactions with them. Eq. (5.4) and
(5.5) establish what polaritons can be excited on the surface of a magnetic material.
In addition to the dispersion relationship for spin waves on the surface of a
magnetic material, we also consider spin waves in the bulk of the material. This
dispersion relationship is much simpler,
kx
2
ky
2
2
c2
v
(5.6).
Where kx and ky are the components of the propagation wavevector k in the
magnetic material. The point of this is that the bulk mode spin waves do not exhibit any
directional preference. Solutions for k in one direction are the same as for k in the
opposite direction.
Now we consider the ATR of the three layer case (Fig. 5.1). The prism contains a
TE electromagnetic wave traveling within it. Both the prism and magnetic layers are
semi-infinite. The prism layer is defined by the permittivity ε1 and has a scalar
permeability of 1. The gap layer has a permittivity of ε2, where ε2 < ε1, and a
permeability of 1. The third layer has a permittivity of ε3, and a permeability tensor
defined by Eq (5.1). The angle incidence for the incoming electromagnetic wave in the
prism is defined by θ. For θ greater than or equal to the critical angle established by ε 1
and ε2, an evanescent surface wave travels along the interface of the prism and the gap
layers. Depending on the gap’s thickness, the evanescent wave will interact with the
semi-infinite magnetic layer and create polaritons. The energy required to make the
75
polaritons must come from the incident electromagnetic wave, therefore the reflected
wave will have energy less than the incident. The solution for the reflection coefficient in
the prism has been found in previous work56 by finding wave-like solutions using the
boundary conditions of the layers. We present the results for the reflection amplitude
here
R3
A 1
A 1
layer
(5.7).
Where,
A
Fe
Fe
k y2
2ik y 2 d
2ik y 2 d
1
k y3
v
F
k y2
1
v
1 i 1
sin
c
1 k y2
t
1
sin
1
k y3
t
1
1
sin
c
(5.8)
(5.9)
c
The subscripts in Eqs. (5.8) and (5.9) refer to the layer which each value is associated
with. The quantities ky2, and ky3 are wavevectors in the y-direction in layers 2 and 3 as
shown in Fig. 5.2. These values will be defined explicitly later on in this paper; for now
we shall say that the ky’s are functions of the electromagnetic wave’s frequency,
permeability, and permittivity of the layer being considered. The thickness of the second
layer is d (this notation will change in the four-layer case). The results for the three-layer
case will provide a check for the solutions to the four layer case, which we will now
discuss.
76
Four-layer ATR
We present solutions for the reflection coefficients of the four-layer case in more
detail. The four-layer geometry is shown in Fig. 5.2.We again consider a TE
electromagnetic wave in each layer of the geometry. If we construct a wave solution in
each layer that is a superposition of an incident wave and a reflected wave. Using the
coordinates in Fig 5.2, we propose a position and time dependant solution for the zcomponent of the E field (since we are only considering a TE mode) in each of the four
layers to be a summation of two waves:
E zn
ik y n y
A 2n 1e
A 2n e
ik y n y
eik x x e
i t
(5.10).
In Eq. (5.10), n is the layer index, the A’s are the coefficients for individual waves
being summed, kx is the propagation wavevector parallel to the interfaces, ω is the
angular frequency of the wave, and kyn is the y-component of the wavevector in layer n.
The z-component of the wave vector has been set to zero.
For the next few steps, we will need to consider Maxwell’s two curl equations:

E

H
1
c

H
t
(5.11)

E
t
1
c
(5.12)
The H-fields in each layer can be derived from the E-field using Eq. (5.11).
H xn
c
vn
H yn
c
vn
E zn
y
E zn
x
tn
i
1n
i
tn
1n
E zn
x
E zn
y
(5.13)
(5.14)
77
The solutions for the kyn components of the wave vector in each layer can be solved in
terms of the kx component, the angular velocity ω, the material’s permittivity εn, and the
material’s permeability. To do this, first take a curl of Eq. (5.12). Then substitute the left
side of Eq. (5.11) into the right side of the double curl of H in order to arrive at a wave
equation


H

H
2
2
c2 t
(5.15).
Using Eqs. (5.9) and Eqs (5.13)-(5.15), we obtain a connection between the ky values, the
parallel wavevector kx, the wave frequency, and the material’s properties:
k yn
kx
2
2
c2
n vn
(5.16).
Given an incident wave, which specifies ω, ε, kx and μv , the ky values in each
layer can be uniquely determined..We set kx based on the angle of incidence (θ) and the
wave vector magnitude (k) of the electromagnetic wave in the prism layer.
The
magnitude of the wave vector in the prism would be:
k
c
1
(5.17).
This gives the x-component of the wave vector in all layers:
kx
k sin
(5.18).
We can now represent the E-field and H-field of an electromagnetic wave in
every layer of the geometry with the only unknown quantities being the incident and
reflected coefficients in each layer. In the first layer, we have two unknown coefficients,
A1 and A2. The second and third layers each have another two unknowns. The fourth
layer only has one unknown coefficient because the wave will continue on infinitely in
78
the y-direction and therefore does not have a reflected part. This gives us a total of seven
unknowns.
The boundary conditions are used to solve for these unknowns. The z-component
of the E-field must be continuous across the boundaries. From this we produce 3
equations, one for each boundary. Three more equations can be produced from the xcomponent of the H-field also being continuous at the boundaries.
We now have six equations with seven unknowns. In order to reduce the number
of unknowns, we divide each equation by A1. This will give us solutions for six unknown
ratios. The first ratio, (A2/A1) tells us how much of the original incoming wave is
reflected from the layered structure. This is the reflection amplitude.
All of the boundary condition equations at the interfaces of y = 0, d1, and d1+d2
can now be written in a 6
1
k y1
1
k y2
ik y 2 d1
0
e
0
k y2 e
0
0
ik y 2 d1
6 matrix form:
1
k y2
e
ik y 2 d1
k y2 e
0
0
0
ik y 2 d1
e
k y3
0
0
0
e
k y3
i
ik y 2 d1
t
i
0
0
e
e
kx
1
ik y3 ( d1 d 2 )
t
1
kx
e
0
0
ik y 2 d1
ik y 3d1
t
k y3 i
v3
1
e
k y3 i
ik y 3d1
0
v3
ik y3 ( d1 d 2 )
ik y 3 ( d1 d 2 )
v3
0
e
kx
t
kx
e
1
e
ik y 4 ( d1 d 2 )
ik y 3 ( d1 d 2 )
k y4 e
v3
ik y 4 ( d1 d 2 )
A2
A1
A3
A1
A4
A1
A5
A1
A6
A1
A7
A1
1
k y1
0
0
0
0
(5.19).
This set of linear equations can be solved numerically. The reflection coefficient
we’ve defined as the ratio of the coefficients A2 and A1.
R
A2
A1
(5.20).
79
However, in the results section we will be examining the return loss of the geometry,
which is the log of the modulus squared of the reflection coefficient:
*
A A
L 10 log( R ) 10 log 2 * 2
A1 A1
2
(5.21).
Results and Discussion
In our calculations we considered two different magnetic materials as examples,
yittrium iron garnet (YIG) and M-type barium hexagonal ferrite (BaM). The permittivity
used for YIG was ε3 = 15.0, the saturation magnetization was M = 141.6 G and the
gyromagnetic ratio was |γ| = 2.801 GHz/kOe. The linewidth used for the YIG was about
100 Oe at 10 GHz. The linewidth was assumed to change linearly with frequency at a rate
of 5 Oe per GHz. For the BaM, the permittivity used was ε3 = 10.0, the magnetization
was M = 334 G and the gyromagnetic ratio was also |γ| = 2.801 GHz/kOe. The BaM also
had a strong crystalline anisotropy constant of K = 2.74x106 egr/cm3, which results in an
anisotropy field of 16.4 kOe. The damping used for BaM was 200 Oe at 40 GHz and
changed with frequency at a rate of 2.5 Oe per GHz. All figures have an applied field of 3
kOe in the z-direction, unless specifically stated otherwise. Other unique parameters will
be given for each particular example.
Fig. 5.3 shows frequency versus propagation wavevector (kx) for surface and bulk
polariton solutions for a two layer, semi-infinite geometry. The solid blue lines are the
solutions for the surface polaritons, given by Eqs. (5.4) and (5.5). The bulk polariton
modes are the blue hatched regions, given by Eq. (5.6). The permittivity for the semiinfinite dielectric layer was εd = 1.5, and the semi-infinite magnetic layer was YIG. This
80
demonstrates the non-reciprocal nature of the surface polaritons. The surface modes for +
kx are clearly different than the surface modes for – kx.
scan lines
Frequency (GHz)
20
18
16
14
12
surface
modes
bulk modes
10
-15
-10 -5
0
5
10 15
Propagation Vector kx (1/cm)
Figure 5.3: Dispersion relationship for allowed bulk polaritons in YIG and surface polariton
modes for two semi-infinite materials, one dielectric and one magnetic, based on equations 4, 5,
and 6. The dielectric material has a permittivity of εd = 1.5, and the magnetic layer uses YIG
parameters. Also included are scan lines, k x
/c
1
sin , based on an incident EM wave in
an ATR geometry for a prism with a dielectric constant of ε 1 = 11.6. The incident angle is θ = ±
70o.
The black lines in Fig. 5.3 maps out scan line indicating the dispersion relation for
a light wave in the prism. This is given by k x
/c
1
sin . One expects a transfer of
energy from the incident beam to the magnetic polariton when the scan line overlaps with
the surface or bulk modes. In Fig. 5.3 the black scan lines show results for a prism with
permittivity of ε1 = 11.6 and incident angles of θ = ±70o. The positive scan line intersects
81
the surface polariton mode at about 17.9 GHz. On the other side of the graph, the
negative scan line intersects the surface mode at about 10.4 GHz. We expect that the
lowest reflectivity from an ATR geometry with these incident angles would be at these
frequencies.
Fig. 5.4 shows the ATR plot of the return loss versus frequency for incident
angles θ=±70o in the three layer geometry, with a prism permittivity of ε =11.6, gap
permittivity of ε2=1.5, and YIG as the semi-infinite magnetic material. The gap
permittivity matches the value for the dielectric in Fig. 5.3. The thickness of the gap is
0.05 cm. The black line is for θ = +70o and the red line is for the opposite direction of
propagation, θ = -70o. The lowest return loss values occur about where we would
expect. For the positive propagation vector, θ = +70o, the lowest return loss is at 18.5
GHz. For the negative propagation vector, θ = -70o, there is a sharp decrease in return
loss at around 10.4 GHz. We note that the lower frequency region of nonreciprocity is
near the ferromagnetic resonance frequency (near 9 GHz).
The nonreciprocity for the
higher frequencies (near 17 GHz) is related to the surface mode and the upper bulk band.
The ATR values do not match exactly with the results for the surface polariton,
18.5 GHz compared to 17.9 GHz, but this should also be expected. The Fig. 5.3 values
are based on a different geometry than the ATR geometry, therefore the surface polariton
modes excited should not be exactly the same.
82
-kx direction
0
Return Loss (dB)
-5
-10
+kx direction
-15
-20
-25
-30
5
10
15
20
25
Frequency (GHz)
Figure 5.4: A plot of the return loss verses frequency for a three-layer ATR case with YIG as the
magnetic material. The prism has a permittivity of ε1 = 11.6. The gap is 0.05 cm thick and has a
permittivity of ε2 = 1.5. The incident angles are θ = ±70o and the applied field is Ho = 3 kOe.
There are two regions of strong nonreciprocity, one near the usual ferromagnetic resonance
frequency (near 9 GHz) and one substantially higher in frequency related to the upper bulk
polariton band.
To get a more complete picture, Fig. 5.5 shows a return loss map, which is the
return losses for all possible frequencies and propagation wavevectors in a three-layer
ATR geometry. The layer materials and their thicknesses are the same as in Fig. 5.4. In
this figure, we only include physical results. Regions for non-physical results are shown
with a hatched pattern. The outer hatched region represents incident angles that are
greater than 90o
kx
c
1
(5.22).
83
The inner hatched region represents propagation phase velocities greater than the speed
of light,
kx
c
(5.23).
White Lines = surface modes from Fig. 5.3
20
Return
18
Frequency (GHz)
Loss (dB)
0
16
-3.000
-6.000
14
-9.000
-12.00
12
-15.00
10
-15
-10
-5
0
5
10
Propagation Vector kx (1/cm)
15
Figure 5.5: A return loss map for frequency versus propagation wavevector kx for a three-layer
ATR geometry with YIG as the magnetic material. In addition, the dispersion relationships for
surface polaritons between a semi-infinite YIG layer and a semi-infinite dielectric layer with εd =
1.5 are shown as white lines. We see strong nonreciprocity occurs near the intersection of the
surface polariton with the upper bulk polariton band. The prism has a permittivity of ε 1 = 11.6, the
gap is 0.05 cm thick and has a permittivity of ε2 = 1.5. The applied field is Ho = 3 kOe.
The return loss map of Fig. 5.5 outlines both the bulk and surface polariton
regions seen in Fig. 5.3. The lowest return loss features in Fig. 5.5 seem to correspond to
84
where the surface modes and bulk modes intersect. The white lines in Fig. 5.5 show the
dispersion relation for surface modes between a semi-infinite dielectric with permittivity
εd = 1.5 and semi-infinite YIG from Fig. 5.3. The white lines are there to draw attention
to the correlation between the surface modes and points of low reflectivity. The range of
nonreciprocal behavior is immediately obvious in Fig. 5.5.
Using these types of plots, we were able to see that as the gap became thicker the
structure became more reflective over all due to the weaker interaction of the evanescent
wave with the magnetic layer. The regions of lowest reflection were closely related to the
surface polariton between the gap material and the magnetic layer. With a thinner gap,
the regions of lowest reflection are due the surface polariton between the prism and the
magnetic layer rather than the gap and the magnetic layer.
We now consider the four layer geometry, where the magnetic layer has a finite
thickness, followed by a semi-infinite dielectric with a relatively high permittivity of ε4 =
15.0. In Fig. 5.6, we show the evolution of the reflected signal as the thickness of the
magnetic film is increased from 0.05 cm to 3 cm. Each graph in Fig. 5.6 contains the
semi-infinite YIG results, as dotted lines and the forward and reverse propagation return
loss for the finite YIG film as solid lines. As the thickness is increased, we see a gradual
convergence with the semi-infinite case except with oscillatory deviations that could be
attributed to Fabry-Perot interference. Even for the thinnest films, however, one can find
substantial nonreciprocal behavior.
85
0
-10
-kx
-20
Return Loss (dB)
-30
-40
0
Three-layer case
+kx
+kx
a)
b)
YIG = 0.05 cm
-kx
YIG = 0.1 cm
-10
-kx
-20
-30
-40
0
c)
+kx
YIG = 0.3 cm
-kx
d)
+kx
YIG = 0.5 cm
-10
-kx
-20
-30
-40
e)
5
+kx
YIG = 1 cm
10
15
20
Frequency (GHz)
-kx
f)
25 5
+kx
YIG = 3 cm
10
15
20
25
Frequency (GHz)
Figure 5.6: A series of four-layer ATR return loss plots, each with a different thickness for the
YIG magnetic layer. The dashed lines, which are the same in a) through f), show the results for a
infinitely thick YIG film. The incident angles are θ = ± 70o for all cases. The gap thickness is 0.05
cm.
We now look at results involving barium hexagonal ferrite (BaM) as the magnetic
material. This is motivated by the fact that BaM has a large internal anisotropy and the
resulting excitations are therefore at a much higher frequency than what is found for YIG.
Fig. 5.7 shows a return loss map of frequency vs. wavevector for a four-layer structure.
The prism permittivity is ε1 = 11.6, the dielectric gap is 0.03 cm thick with a permittivity
of ε2 =
1.5, the BaM layer is 0.1 cm thick, and the semi-infinite 4th layer has a
86
permittivity of ε4 = 15. The two white lines show the surface modes for a structure based
on a semi-infinite BaM and semi-infinite gap dielectric according to Eqs. (5.4) and (5.5).
White lines are the results for surface modes
Frequency (GHz)
85
Return
Loss (dB)
80
75
70
0
-3.000
65
-9.000
60
55
-15.00
-40
-20
0
20
40
Propagation Vector kx (1/cm)
Figure 5.7: A return loss map for frequency versus propagation wavevector kx for a four-layer
ATR geometry with BaM as the magnetic material. There is strong nonreciprocal behavior in the
57-59 GHz and in the 71-75 GHz ranges. The dielectric gap is 0.03 cm thick, the BaM layer is 0.1
cm thick. The applied field is Ho = 3 kOe. The two white lines show the surface modes for a
structure based on semi-infinite BaM and semi-infinite dielectric εd = 1.5.
The overall results for Fig. 5.7 display clear non-reciprocal behavior. On the
positive kx side of the graph, there is a relatively strong signature along the path of the
surface polariton, in the 70-75 GHz range. On the negative side, there are regions of low
reflection where we would expect the negative propagating surface polariton in the 57 to
59 GHz frequency range.
87
Fig. 5.8 shows results for return loss as a function of frequency for
40 o
with the same geometry and parameters as Fig. 5.7. The black line shows the results for
forward propagation, and the red line is for the reversed signal. We can observe the acute
non-reciprocity at 73 GHz, where there is over a 45 dB return loss difference between the
positive and negative propagation. Around the dip, ranging between 72 to 75 GHz, there
is a band of frequencies where the positive propagation has return losses below -10 dB
and the negative propagation has return losses above -1 dB. This region behaves as an
isolator, allowing propagation in the negative direction, but not the positive direction.
There is another frequency region of 53 to 60 GHz where propagation in the negative
direction is strongly attenuated while propagation in the forward direction is allowed.
-kx direction
Return Loss (dB)
0
-10
+kx direction
-20
-30
-40
50
60
70
80
Frequency (GHz)
90
Figure 5.8: A plot of the return loss verses frequency for a four-layer ATR case with BaM as the
magnetic material. There are two distinct frequency regions that exhibit nonreciprocal behavior.
The dielectric gap is 0.03 cm thick and the BaM layer is 0.1 cm thick. The incident angles are θ =
± 40o and the external field is H0 = 3 kOe.
88
The properties of these surface polaritons can be modified and tuned by the
applied magnetic field Ho. Fig. 5.9 uses the same parameters of Fig. 5.8, except the
applied field has been reduced from Ho = 3 kOe to Ho = 2 kOe. The 3 kOe case is
represented by dotted blue lines, while the 2 kOe case uses the black and red scheme of
previous graphs. With the reduction of field, the isolator bands have shifted down in
frequency.
Ho = 3 kOe
Return Loss (dB)
0
-10
-20
-30
-40
50
-kx
Ho = 2 kOe
+kx
Ho = 2 kOe
60
70
80
Frequency (GHz)
90
Figure 5.9: A plot of the return loss verses frequency for two 4-layer ATR cases with BaM as the
magnetic material but with different magnetic fields. A change of 1 kOe in the magnetic field
essentially shifts all features of the reflectivity, including the nonreciprocity, by a 2.8 GHz. The
BaM layer is 0.1 cm thick. The dashed lines represent the results where the applied field is H 0 = 3
kOe. The solid lines are for an applied field of H0 = 2 kOe. The incident angles are θ = ± 40o.
The previous plots using BaM have used a relatively thick gap and magnetic
layers. When these parameters are reduced, the return loss landscape becomes modified
and looks different from the bulk and surface polariton dispersion curves implied in Fig.
89
5.7. This can be seen in Fig. 5.10. For Fig. 5.10, the gap thickness is now 0.02 cm and the
BaM layer is 0.01 cm thick.
White lines are the results for surface modes.
85
Frequency (GHz)
80
Return
Loss (dB)
75
0
-3.000
70
65
-9.000
60
55
-15.00
-40
-20
0
20
40
Propagation Vector kx (1/cm)
Figure 5.10: A return loss map for frequency versus propagation wavevector k x for a 4-layer ATR
geometry with BaM as the magnetic material. The dielectric gap is 0.02 cm and the BaM layer is
0.01 cm thick. The applied field is H0 = 3 kOe. The two white lines show the surface modes for a
structure based on a semi-infinite BaM and semi-infinite dielectric εd = 1.5. The strongest
nonreciprocal behavior occurs near the intersections of the surface modes with the bulk bands.
Despite the thin BaM film, Fig. 5.10 still displays significant nonreciprocity. We
see this more clearly for a particular example in Fig. 5.11 which uses all the parameters
of Fig. 5.10 but only shows the return loss for incident angles
75 o . There is a large
dip in return loss in the negative direction at about 58 GHz, but very high reflection in the
90
positive direction. As we saw earlier, the position of the isolation frequency can be tuned
by changing the applied field Ho.
Return Loss (dB)
0
+kx
-5
-10
-15
-20
-kx
-25
-30
50
60
70
80
Frequency (GHz)
90
Figure 5.11: A plot of the return loss verses frequency for a four-layer ATR case with a BaM
layer of 0.01 cm in thickness. We note a distinct nonreciprocity even for this thin BaM film. The
incident angles are θ = ± 75o and the applied field is H0 = 3 kOe.
Depending on the parameters chosen, the importance of the surface modes can be
lessened and one may even be able to take advantage of the just the bulk modes of the
ATR geometry. Fig. 12 give results for a possible band pass filter. The gap thickness was
0.001 cm, the thickness of the BaM was 0.1 cm, and the angles of incidence were θ = ±
30o. There was non-reciprocal behavior, but it was restricted to below 60 GHz and above
70 GHz. Between those frequencies the device has high reflectivity and outside of that
region the return loss did not go over -5 dB. In fact, outside of 50 GHz and 75 GHz, the
91
return loss didn’t go over -10 dB. Essentially, the region between 60 and 70 GHz has
high reflectivity because there are no surface or bulk magnetic polaritons in this region.
0
Return Loss (dB)
-5
+kx
-kx
-10
-15
-20
-25
-30
40
50
60
70
80
Frequency (GHz)
90
Figure 5.12: A plot of the return loss verses frequency for a four-layer ATR case with BaM as the
magnetic material. In this case the dielectric gap is very thin (0.001 cm thick) and the BaM layer is
0.1 cm thick. With a small gap, the incident wave couples well to the bulk waves in the magnetic
material, and the resulting reflection looks like a band-pass filter with limited nonreciprocal
behavior. The incident angles are θ = ± 75o and the applied H-field is H0 = 3 kOe.
Two factors that play big roles in the dramatic changes of the reflection landscape
are the thickness of the gap and the thickness of the magnetic layer. As the gap becomes
thinner, the overall reflection is reduced because the evanescent wave is much stronger in
the magnetic material encouraging a transfer of energy from the electromagnetic wave in
the prism to that in the magnetic material.
A reduction in the thickness of the magnetic layer generally reduces the
nonreciprocity. For example, starting with the angle of incidence, gap thickness, and
92
BaM thickness used in Fig. 5.11, we present Table 5.1 for additional calculations for
different BaM thicknesses, where the first column gives the thickness of the BaM, the
second column gives the frequency where the greatest difference occurs between positive
and negative propagation, and ΔL is the difference in return loss between the two
propagation directions at that frequency. When the BaM has reached a micron in
thickness, the signals only differ by about ½ a dB at the frequency of greatest difference.
Magnetic Layer Thickness and Return Loss Differences
BaM (cm)
0.01
0.005
0.001
0.0005
0.0001
Frequency (GHz)
58.7
59.3
59.8
59.8
59.8
ΔL (dB)
25 .9
15.1
4.1
2.3
0.5
Table 5.1
It may be possible to create an effective ATR like geometry in a planar microstrip
structure. This is illustrated in Fig. 5.13. We propose that it may be possible to use
lithography to create planar devices that uses a v-shaped signal line to guide an
electromagnetic signal close to a multiple layer interface. Fig. 5.13 b) shows a possible
device where a microstrip imports a signal to a gap layer near a magnetic layer. For a
device like this to produce non-reciprocal ATR results, it would need the magnetic
material to have its magnetization oriented out of the plane. This is one of the reasons we
have presented results using BaM, which has a high crystalline anisotropy creating the
out of plane orientation. In both geometries the oscillating E field is parallel to the static
magnetization, and the oscillating H field is perpendicular to the magnetization.
93
We note that the planar device is different from the structure we have considered
here, and we do not believe it will exactly reproduce our results. However, based on the
physical principles we have explored, it should exhibit non-reciprocal behavior which
could be adjusted by many of the parameters we have discussed.
Figure 5.13: Figure a) is a side view of the typical ATR geometry - a prism separated from a
magnetic layer by a gap. The incident wave creates an evanescent wave in the gap which can
interact with the magnetic material. Figure b) is a top view of a planar geometry proposed which
would utilize the physics of ATR. A microstrip waveguide directs an EM wave to close proximity
of a magnetic material, with the magnetization directed into the page, parallel with the applied
magnetic field.
Conclusions
The four-layer ATR geometries using YIG and BaM magnetic layers exhibit
strong non-reciprocal behavior for magnetic films of thicknesses between 0.001 cm and 3
cm. The range of non-reciprocity can be tuned by changing the applied magnetic field,
94
gap layer thickness or magnetic layer thickness. For a YIG magnetic layer, the range of
non-reciprocal frequencies can be adjusted between 5 and 20 GHz. For a BaM magnetic
layer, the non-reciprocal range is between about 50 and 80 GHz.
With the parameters
used here, we find the isolation in terms of return loss can be 30 to 40 dB, with the
minimum loss being only -3 dB. We also calculate the properties of a reciprocal band
pass filter based on BaM in a similar geometry. The band pass filter used a very thin gap
layer compared to the nonreciprocal devices discusser earlier.
We also propose a possible planar device, designed to utilize the physics involved
in the ATR geometry. This device could be adjusted to accommodate different reciprocal
and non-reciprocal devices, such as notch filters, band pass filters, isolators, or
circulators.
CHAPTER VI
Background: Micromagnetic Simulations
The true magnetization of a magnetic structure, particularly in nanosized objects,
can be quite complicated. In real materials, local regions of magnetization can change
orientation throughout the structure. It is difficult to analytically model such a landscape.
Computer simulations are a way to evaluate these complicated structures. There are two
main techniques currently used in dynamic computer modeling of magnetic structures: 1)
The Finite Difference Method, and 2) the Finite Element Method.64 In this chapter, we
will talk about the principles involved in the Finite Difference Method. Then in Chapters
VII and VIII we will use it to study the effects oscillating magnetic fields have on
nanosized magnetic squares and on magnetic bilayer structures.
The Finite Difference Method
This method relies on dividing a magnetic structure up into individual cells, as illustrated
in Fig 6.1. The main approximation is that each cell’s magnetization is uniform. The LL
equation is then used to evaluate the motion for each cell

dM
dt
 
M H
M

 
M [ M H]
(6.1) .


M is the magnetization of a single cell and H is the average effective magnetic field in
the cell. This equation of motion is solved in a time integration scheme to find how the
96
magnetization of each individual cell evolves in time. Our results are based on either a 2nd
order Runga-Kutta method or an Adams-Bashforth method.65
Figure 6.1: Magnetic structure divided into cells. Each cell has a uniform magnetization.
In order to evaluate the equation for each individual cell we must be able to
account for the average effective field acting on each cell. The contributions of the
effective field are given by
 



H H 0 H A H ex H dip
(6.2).



H 0 is an applied field, H A is the local anisotropy field of the material, H ex is the

effective exchange field produced by neighboring cells, and H dip is the dipolar field

created from all the cells in the structure. The applied field, H 0 is chosen. The anisotropy
97



field H A , exchange field H ex , and the dipole field H dip must be calculated based on the
magnetization and geometry of the cells.
The Anisotropy Field

H A depends only on the magnetization of the cell that is being considered. It is
often based on the spin-orbit coupling and the interaction of neighboring atoms. Because
of this, the details of the lattice structure are important, and this interaction can make
particular directions of magnetization preferred. We can talk about it in terms of the
energy density due to the orientation of the magnetization. For different types of
crystalline anisotropy, the total energy density has different forms. For a single preferred
axis, or uniaxial anisotropy, it has the form
U
K cos2
(6.3).
K is the anisotropy constant, which is determined experimentally for a material.27 The
angle θ is the angle between the local magnetization and the easy axis (the preferred axis)
of the material. The H-field for interaction can be found by considering how the energy
density changes with the magnetization

H
U

M
(6.4).
The vector derivative is

M
x̂
mx
ŷ
my
ẑ
mz
(6.5).
98
If the easy axis is the z-axis, then the field along that axis would be
HA
HA
mz
[K cos
2
]
mz
m
K z
M
2K
cos
M
2
2K m z
M M
(6.6)
In the calculations done in later sections, this is the form we use for the average uniaxial
anisotropy field of a single cell.
The Effective Exchange Field

H ex depends on the relative orientation between the cell under consideration and
that of its neighboring cells. This short-range field accounts for the quantum mechanical
effect that allows for the existence of permanent magnets. This is due to overlapping
wave functions of the electrons orbiting atoms. Because it is short range, only the nearest
atomic neighbors are usually considered. In its most basic interaction form, the exchange
energy for two spins is given by
E
Here
 
J S1 S2
(6.7)


S1 and S 2 are the spins of the two electrons, and J is the exchange constant. J is
dependent on what the two atoms are and how far apart they are spaced. Because this
interaction depends on electron orbital overlap, J becomes very weak over large spacing.
It is because of this that when evaluating the exchange energy associated with a single
spin, one often uses only nearest neighbor atoms.
99
In the finite difference method, we are not considering interactions between single
atoms, we are dealing with volumes of spins. Even though the interaction is short ranged,
it still has an effect at much greater distances. From one spin, there is an effective
exchange on a spin located at a far distance (L) through the exchange interactions of all
the spins in between, see Fig. 6.2. Because our method does not deal with just one spin,
we consider the effective interaction between the average spins of two neighboring
regions and the length between the middle of each region (the exchange length). From
this idea comes the effective exchange constant. Instead of J, which is how the energy
scales with the difference between two spins, we have the exchange stiffness constant A.
The constant A describes how the energy scales with the change of magnetization over a
length (cgs units of erg/cm).
Figure 6.2: A row of spins. The circled spins influence each other through the spins in between
over a length L, giving rise to an effective exchange between the two.
We can represent the energy density due to exchange as:
U
A
mx
M
2
my
M
2
mz
M
2
(6.8).
The exchange stiffness constant A, for a simple cubic lattice, relates to J and S by the
atomic lattice spacing a:
100
A
JS2
2a
JM 2 a 5
2
(6.9).
With the energy density, we follow the same method used to find the field due to the
anisotropy by calculating the variation in energy density with respect to the
magnetization

H ex
U

M
2A
M
2

M
M
(6.10).
In the computer simulation, this derivative is turned into a difference equation. A
single component of the exchange field can be written as
H ex x ( x, y, z)
2A M x ( x 1)

x 2 M( x 1) 2
M x ( x 1)

2
M( x 1)
(6.11).
Where Hex-x(x,y,z) indicates the x-component the exchange field in some cell defined by
its location with x, y, and z. The Mx(x+1) term is the x-component of magnetization for
the nearest neighbor cell in the positive x-direction and Mx(x-1) term is the x-component
of the nearest neighbor cell in the negative x-direction. The denominators for the
respective terms are the modulus squared of the total magnetization in those cells.
The Dipole Field and the FFT method
The dipolar field in a cell being considered is dependent on all cells in the
structure. It is for this reason that this field is the most time consuming calculation in the
simulation. In Chapter II we established how to calculate the demagnetizing field for a
single cell.29 The demagnetizing field of a volume is its dipolar interaction with itself.
101
This method can also be used to evaluate the average dipolar field produced by one
volume acting on another. Recall from Eq. (2.33):

Hd
1
V

dV M
V
V

r
1
 dV
r
 
N M
(6.12) .
This double integration evaluates the dipole field at a point in volume V created from
every point in volume V’, and then taking an average of all field values across the entire
volume V, see Fig. 6.3. The Newell tensor is then dependant on the vectors locating the
 
x
two volume ( x
). Recalling Chapter II, we rewrite Eq. (2.35) as
  
N( x x )


1
dA
dA  
VA A r r
(6.13).
Where A and A’ are the surface areas of the respective volumes. The vectors x and x’ are
the relative locations of the volumes.
Figure 6.3: Average magnetic field inside volume V generated by volume V’. x and x’ are vectors
indicating the locations of the two volumes. r and r’ are vectors are vectors indicating specific
instances contained in the integral from Eq. (6.12).
102
This allows us to write the dipolar field in a given cell as summation of the
products of its relative position to all cells (including itself) and their magnetization.
Considering Fig 6.4, if we index the cells and ask for the average dipolar field in cell “j”,


we would have to sum up the relative products of N and M for all cells
 
H j (x j )
  

N ij ( x j x i ) M i ( x i )
(6.14).
i
Figure 6.4: Magnetic structure divided into cells. The average effective field Hj(xj) is due to the
contribution of the magnetization from all other cells, M i(xi).
The time cost in the simulation comes from the double iteration that must be
performed. The average dipolar field must be found for each individual cell, and that
calculation requires summing through each individual cell, shown in Eq. (6.14). The
amount of time for a computer to perform these operations increases rapidly with the
number of cells being considered. It is for this reason that an alternate way of evaluating
Eq. (6.14) is used.
103
A more efficient method of performing the calculation of Eq. (6.14) is done by
utilizing the convolution theorem to transform the summation into something that can be
evaluated faster and then transform the result back.66 As an example, we will consider a
simpler case. We will switch from summations of discrete cells to an integration over a
continuum, and we reduce the problem down to one dimension. Consider the H-field in
one dimension at a single point due to the products of the relative Newell tensor and
position dependant magnetization
H(x)
N(x
x ' ) M ( x ' )dx '
(6.15).
x'
The Newell tensor can be rewritten as an inverse Fourier transformation from some
function n(q)
1
2
N(x x' )
n(q )e
iq ( x x ')
dq
(6.16).
q
Plugging Eq. (6.16) back into Eq. (6.15) and regrouping some terms gives us
H(x )
q
1
2
M( x ' )e iqx ' dx ' n(q )e
iqx
dq
(6.17).
x'
We now recognize that the function in the brackets is only of x’ and it is a Fourier
transformation of another function m(q)
1
2
M( x ' )e iqx 'dx ' m(q )
x'
(6.18).
104
Substituting m(q) into Eq. (6.17), we get
H(x )
m( q ) n ( q )e
iqx
dq
(6.19).
q
It might be easier to identify the product of m(q) and n(q) simply as some function F(q)
H(x)
F(q ) e
iqx
dq
(6.20).
q
We again recognize this as an inverse Fourier Transformation.
1
2
F(q ) e
iqx
dq
h( x )
(6.21).
q
So the result is
H(x )
2 h( x )
(6.22).
This says that we can find the H-field at a particular point by taking the inverse Fourier
transformation of F(q), which is a product of the Fourier transformation of the Newell
tensors and the magnetization of the entire structure.
We will now change back to a discrete format and include three dimensions.
Using a different notation, we define T(f) as the discrete finite Fourier transformation of a
function and T-1(g) is a discrete finite inverse Fourier transformation. We can now write
a component of the average H-field for all cells as

Hi


T ( T( N ii ) T (M i ) )


1
T ( T( N ij ) T (M j ) )


T 1 ( T( N ik ) T(M k ) )
1
(6.23).
105

H i is a matrix containing all of the average i-components of the H-fields in every cell
of the structure. The subscripts i, j, and k denote vector components and tensor elements

(if i = x, then j = y and k = z). The Newel tensors N nm indicate the geometric relations for all

the cells surrounding a single cell. The magnetization matricies M n represent a single
directional component of magnetization for every cell in the structure. Please note that
the
2
factor from Eq. (6.21) is not obvious in Eq. (6.22). That constant is based on
the choice of how the discrete Fourier transformations are defined. The transformation
can be defined to do away with it.
Performing these transformations and inverse transformations may seem
complicated and one would question the benefit of performing the calculation in such a
way. The advantage comes in two ways:
1) The Newell tensor transformations are constant throughout the simulation.
The standard practice is to perform them once before the simulation begins,
store them, and then call them when needed.
2) Taking advantage of symmetries in the transformed space allows for a discrete
Fourier transformation to include redundant calculations. This reduces the
number of unique calculations that need to be done. A Discrete Fast Fourier
Transformation (FFT) scheme reduces the number of calculations to be done
from C2 (C = total number of cells) to approximately C ln(C). 66
106
These factors result in a significant run-time improvement depending on how many cells
are being included in the simulation. Fig. 6.5 shows the time taken to perform dipole
calculations on a particular computer using the two different methods. The double
iteration method quickly increases in time cost as the number of cells increases; the FFT
method increases at a much slower rate. From Fig. 6.5, for a 1024 cell structure the
double iteration method took 1459 seconds to perform 1000 cycles while the FFT method
only took 39 seconds.
Figure 6.5: Time to complete 1000 iterations of dipole calculations versus the number of cells
being considered. This graph compares the double-iteration, Eq. 6.12, and the FFT method, Eq.
6.21, used to calculate dipole fields. For large numbers of cells, the FFT method is clearly
advantageous.
These methods of calculating the effective anisotropy, exchange, and dipolar
fields will be used in the next two chapters to evaluate the time-dependant behavior of
magnetic nano-structures in microwave fields.
CHAPTER VII
Accessing Multiple States in Magnetic Squares with Microwaves
Magnetic reversals13-15, microwave assisted magnetic reversal (MAMR)16-20, and
nano-scale bit patterned media21-26 have been investigated recently because of their
potential use in high density magnetic storage systems. MAMR uses microwaves to
induce large angle precessions in the magnetization and dislodge it from a stable state.
To do this efficiently, one generally employs a frequency that is in resonance with one of
the normal modes of the magnetic structure.
A particularly interesting example that combines the ideas of MAMR bit
patterned media has recently been reported.
S. Li et al67 proposed a scheme with two
films made of different materials. Each film has a different resonant frequency so it is
possible to flip the magnetization of one structure but not the other with an oscillating
field that has a frequency that matches the resonance of only one of the layers. The
frequency-matched layer will experience large angle precessions and will switch, while
the other layer will not.
This paper demonstrates both a multistate micromagnetic structure68-72 and the use
of MAMR to change an initial magnetic state into one of two other states, or to let it
remain in its current state. We consider a thin, square magnetic structure with an applied
static magnetic field directed nearly opposite to the magnetization as shown in Fig 7.1.
There are three important magnetic orientations for this structure that are stable,
for both negative fields and zero fields: an initial state, an intermediate state and a
108
reversed state. We find that the strength of the oscillating microwave field can control
which one of the three allowed states emerges as the final configuration. As we will see,
the transition from one state to another occurs at a negative field, sometimes with the
addition of a microwave signal. After the orientational transition has occurred, the
microwave field can be removed and the static field reduced to zero, and all three states
remain stable at zero field.
Figure 7.1: A 10 nm x 160 nm x 160 nm structure of Fe-Ti-N with an in-plane average
magnetization (M). An external magnetic field (H0) is applied in the negative z-direction. An
oscillating magnetic field (hd) is applied in the y-direction.
The Calculation
We consider a thin, square-shaped magnetic element with an in-plane uniaxial
anisotropy. It is 10 nm thick and its sides are 160 nm long. The material, one form of
Fe-Ti-N 73-74, will be described by the following parameters: the saturation magnetization
(Ms = 1.106 kG), the uniaxial anisotropy constant for the z-axis (K = 11500 erg/cm3), the
109
gyromagnetic ratio (|γ| = 2.897 GHz/kOe), the exchange stiffness parameter (A = 5e-7
erg/cm) and the damping of the material (α = 0.04).
For the micromagnetic calculation, the structure is divided into 4096 cells. The
cells are square in the y and z plane, with dimensions of 10 nm thick and 2.5 nm sides.
The dynamic micromagnetic calculation applies the Landau-Lifshitz equation (LL
equation) to the magnetization of each of the cells to evolve them through time. The LL
equation is:

M
t
 
(M H eff )
Ms
  
(M M H eff )
(7.1)
where M is the magnetization of the cell and Heff is the averaged effective magnetic field
present in the cell.
The effective magnetic field for each cell (Heff) is made up of an external applied
field (H0), the effective exchange fields between nearest neighbor cells (Hex), dipole
fields created from all cells including the single cell’s own demagnetizing field ( Hdip),
the driving field from the microwave field when it is present (hdcos(ωt)), and the field
due to the uniaxial anisotropy (HA).

H eff

H0

H ex

H dip
h d cos( t ) ŷ

HA
(7.2)
The Multi-State Results
We are interested in a structure that holds a stable magnetic state even if an
applied static magnetic field is opposite to the magnetization of the structure. To find
such a state, we started with Ho positive and gradually brought it to a negative value. The
resulting equilibrium state was an S shaped state, as seen in Fig. 7.2a. This state was then
110
used as the starting point for all following simulations. The external magnetic field was
applied opposite to the magnetization, with a slight angle off the z-axis (0.01o). The slight
angle is included to avoid any non-physical symmetry, and we find that the rotation of the
S state is in the direction of this small angle.
For a negative field up to about -320 Oe the initial state configuration was stable.
We will simply refer to this as the initial state, Fig. 7.2a. For fields stronger than -320
Oe, the state flipped to an S shaped state, with magnetization almost anti-parallel to the zaxis (about 128o off the positive z-axis). We will refer to this as the reversed state, seen in
Fig. 7.2c.
Figure 7.2: Three graphical representations of the magnetization in the y-z plane of the structure
for the three distinct states (with an applied field of H0 = -220 Oe). The large arrow represents the
average magnetization and the small arrows indicate locally averaged magnetization. (a) the initial
state, (b) the intermediate state, and (c) the reversed state.
In order to study the microwave assisted switching, we used fields in the initial
state’s stability range (0 to-320 Oe) and added an oscillating magnetic field oriented
along the y-axis. To choose the oscillation frequency, we slightly disturbed the
equilibrium state, recorded My(t), and did a Fourier transform which resulted in a set of
111
peaks at different frequencies representing bulk and localized modes75-78. The frequency
for the most prominent mode and set it to the driving frequency for our oscillating field.
We found an approximately linear relationship for the driving microwave frequencies as
a function of the applied field Ho.
f
7.97
GHz
H0
kOe
4.025 GHz
(7.3)
This linear relationship is not valid for fields within 25 Oe of the switching field (-320
Oe).
In the MAMR study, we found a third stable state, with the magnetization oriented in
an S shape close to 900 relative to the z-axis. We refer to this as the intermediate state,
and it is shown in Fig. 7.2b. An interesting point is that this state may or may not be seen
if transitions are induced by a static applied field, e.g. in a standard hysteresis curve
measurement. For the chosen parameters in the hysteresis calculation, the intermediate
state did not appear. However, for cases where the saturation magnetization was lower
we did in fact see stable intermediate states for certain applied fields. For example, using
a saturation magnetization of 850 G (a value close to Permalloy), and applying an
opposing field of -80 Oe an intermediate state was produced.
To study the behavior of the system, we applied oscillating driving fields of
varying amplitudes (hd) for a fixed static field, Ho = -220 Oe (opposing the initial
magnetization). The frequency of the driving field was determined by Eq. (7.3) and was
held constant. Fig. 7.3 shows the results for the normalized time evolution of the average
z-component of magnetization for different values of hd.
The driving field was turned on at 3 ns in the simulation and removed at 7 ns. For
driving fields of hd = 30 Oe, the normalized mz component remains at about 0.82 after the
112
driving field is removed and the structure remains in the initial state. For driving fields of
hd = 50 Oe, the average mz component changes rapidly and then oscillates around mz = 0.14, indicating the structure is in the intermediate state. The highest driving field of hd =
85 Oe finishes with the mz component ending at -0.95, this means the structure is in the
reversed state.
Figure 7.3: The average normalized mz component of magnetization for the entire structure versus
time. There is an applied external field (H0) of -220 Oe. The different lines represent the results for
different amplitudes of the driving field (hd). The oscillating driving field starts at a time of 3 ns
and ends at 7 ns.
An interesting question is - why does adding a microwave of a given strength
cause the magnetic configuration to leave the initial state, but possibly become trapped in
the intermediate state? The answer is that the switching depends on the orientation and
frequency of the microwave field. The microwave is set to pump the y-component of
113
magnetization. In the initial state, this is effectively the transverse component of
magnetization. In the intermediate state the y direction is close to that of the longitudinal
component and the pumping is less effective. Furthermore, the driving frequency is set to
that of the principal resonance of the initial state. At this frequency, enough energy is
added so that the structure can get out of its local energy minimum and have a transition
to the intermediate state. The driving frequency is then no longer in resonance with the
normal modes of the new state. Evidence for this can be seen in the amplitude of the
oscillations in Fig. 7.3. Even when the initial state is stable, the oscillations in <mz> are
fairly large in the time range of 4 – 6 ns. In contrast, when the intermediate state is stable
the oscillations are considerably smaller even though hd is larger. Of course if hd is
sufficiently large, enough energy can be added to cause a transition to the reversed state.
We collected data for a range of applied fields and driving fields. Fig. 7.4 shows
the ending state results as a function of driving field amplitude (hd) at different static
applied fields, H0. In general larger driving fields lead to transitions to the intermediate or
reversed state. As Ho is made more negative, smaller hd values are needed to change the
initial state.
One parameter we found to be an issue was the angle of the applied field.
Changing the angle to 1o altered the range where the intermediate state occurred. The
intermediate state is now found for oscillating fields between 60 Oe to 65 Oe, in contrast
to the values of hd = 45 – 75 Oe for the 0.01o case. Nonetheless, all the main features of
the microwave driven switching remain true.
114
Figure 7.4: Phase diagram showing the final magnetic state of the structure after applying a
driving field (hd) for 4 ns in an external field (H0). The final state is shown as a function of the
driving field versus the applied field.
We also performed trials on a smaller square structure with a thickness of 10 nm
and side lengths of 80 nm. The results again were similar. The smaller structure flipped to
the reversed state at a lower static field (at about -280 Oe). The resonance frequencies in
the small structure were higher. For the smaller structure and with a static field of -160
Oe directed at a 1o angle away from the –z axis, the initial state was found to be stable up
to oscillating fields of 47 Oe. The intermediate state occurred for 48 Oe < hd < 56 Oe,
and the reversed state is found if hd is larger than 57 Oe.
115
Conclusions
In summary, we have explored a structure with several stable magnetic
configurations at zero magnetic field. We found that an added microwave field can
change the initial state to one of two other states. The final state is determined by the
frequency and strength of the driving field.
CHAPTER VIII
Microwave Assisted Switching in an Exchange Spring Bilayer
In recent years exchange spring structures79-81, typically layered structures
comprising a hard magnetic material and a soft magnetic material, have been
proposed as a method to increase magnetic data storage densities82-83.
In these
materials, a small external static magnetic field can reverse the magnetization in the
soft material. Because of the high anisotropy, however, the magnetization in the hard
material does not change, and a domain wall is created near the interface between the
two materials. Increasing the static magnetic field can then move the domain wall
through the hard material and lead to its reversal. This field is smaller than the field
normally required to switch the hard material in isolation and the hard layer reversal
is therefore referred to as “domain wall assisted.” Composite recording media may be
designed in this way with a large anisotropy (high thermal stability) but with a lower
switching field than is normally associated with hard materials.
Another way proposed to lower the energy required for magnetization reversal
is through so-called “microwave assisted reversal” .84-85 The main idea of microwave
assisted reversal is that applied microwave fields can resonantly transfer energy to a
magnetic system thereby inducing large amplitude dynamics. This may allow the
spin system to escape a metastable energy minimum. In contrast, a static applied field
can cause a reversal by altering the energy landscape and destroying an energy
minimum. A comparison of the switching time of single phase media to that for
117
exchange spring media can be found in Ref. [86].
Since the domain wall in the exchange spring represents a metastable state, it
therefore seems plausible that microwave magnetic fields may be used to help move
the domain wall and switch the hard material. In fact, microwave fields have been
found to move domain walls in a past experiment.87 In that experiment, the resonant
frequency of the domain wall was very different from the resonant frequency of the
domains. The application of a driving field with frequency that matched the domain
wall resonance was seen to most efficiently move the domain walls.
In this chapter we examine whether domain wall and microwave assisted reversal
can work together to lower the energy required for magnetization switching. We consider
a bilayer containing a magnetically hard and a soft thin film. A similar work has been
carried out by Li et al67 but for a small bit with magnetization perpendicular to the
interface. Here we consider in-plane magnetization due to the thin film geometry. Also,
the microwave fields used in Ref. [67] are too large (on the order of several kOe) to be
useful for applications so we limit our study to lower microwave powers.
In our results, we find that there are a number of different eigenmodes for the
exchange spring structure, each with a different frequency. We study the microwave
driven switching for each of these frequencies and find that a driving field set to the
frequency of the lowest eigenmode is most effective at switching the hard material. The
efficiency of switching is correlated with the profile of the eigenmode and the microwave
orientation.
118
Theoretical Calculations
The geometry of the exchange spring is illustrated in Fig. 8.1. Both layers have an
in-plane uniaxial anisotropy in the z direction, but the hard layer’s anisotropy constant is
32 times larger than that of the soft layer. Both layers start in a state of mutual alignment
in the +z direction. A static external field is then applied to both layers in the –z direction.
As the external field increases in strength, the magnetization in the soft layer turns to
align with the field while the hard layer remains pinned in the +z direction. Away from
the interface, the turn is complete, but near the interface the magnetization in each atomic
layer is at a different angle with respect to the –z axis. Thus a domain wall is formed
near the interface between the layers. The domain wall produces an additional torque on
the hard layer, encouraging it to switch. This allows the hard layer to switch and align
with the field at a lower static field strength than it would without the presence of the soft
layer.
Figure 8.1: The geometry of the exchange spring bilayer. The magnetization is initially oriented
in the +z direction in both materials. A negative static applied field is large enough to reorient
most of the magnetization in the soft layer but the hard layer remains pinned in the positive z
direction. An oscillating microwave field hd is added parallel to the y-axis.
119
For the simulations in this chapter we have chosen a set of material parameters.
For the saturation magnetization of both materials, 4πMs = 18.1 kOe. The uniaxial
anisotropy constant for the hard material is Kh = 5.3x106 erg/cm3, making the constant for
the soft material Ks = 0.166x106 ergs/cm3. The effective exchange constant for both
materials is A = 1.03x10-6 erg/cm. This is also taken to be the exchange constant across
the hard/soft interface.
For the micromagnetic calculation, the structure is divided into 64 cells in the x
direction. The cells are square in the y and z plane. The side lengths of the cells are 100
microns, which is quite large compared to the cell thickness of 1 nm. This is done so that
the calculation is consistent with earlier models which assumed a single spin in each layer
could represent the entire layer.89, 90, 80
The dynamic micromagnetic calculation applies the Landau-Lifshitz (LL)
equation to the magnetization of each of the cells to evolve it through time. The LL
equation is as follows:

M
t
 
(M H eff )
Ms
  
(M M H eff )
(8.1)


where M is the magnetization of the cell and H eff is the effective magnetic field present
in the cell and is given by.

H eff

H0

H ex

H dip

h d cos( t ) ŷ H A

Here H 0 is the external static magnetic field,
(8.2)

H ex is the effective exchange field

between nearest neighbor cells, H dip includes the dipole fields created from all cells
including the single cell’s own demagnetizing field , h d cos( t ) ŷ is the driving field
120
from the microwave field when it is present, and the field due to the uniaxial anisotropy

is H A . The dipole field is calculated efficiently through a standard FFT method.
To find the exchange spring state, with the magnetization in the soft film nearly
opposite to that in the hard material, we started with H0 positive and gradually brought it
to a negative value. The final magnetic field orientation was close to the –z direction,
with a slight angle off the negative z-axis (0.01°). The slight angle is included to avoid
any nonphysical symmetry.
Results and Discussion
In Fig. 8.2 we show two hysteresis curves. One curve is for an 8 nm film of the
hard material. The second curve is for an exchange spring bilayer with a 56 nm thick film
of the soft material, exchange coupled to the 8 nm thick film of the hard material. The
dashed loop shows the result for the hard material by itself, the solid line loop shows the
behavior of the exchange spring system. The addition of the soft layer allows the
structure to completely switch at an applied field of 1.12 kOe as opposed to an applied
field of 7.37 kOe. This is a reduction of the required switching field by 85%. The
question is whether the application of a microwave field may further reduce the switching
field.
The hysteresis loop maps out a region of static applied field strength where the
structure is in an exchange spring state. This range runs from about -0.38 kOe to -1.12
kOe. To explore how this structure is affected by a microwave field, we choose an
applied field in the middle of this range, H0 = -0.7 kOe. In order to drive the structure
121
effectively, the applied microwave frequency should match that of an eigenmode of the
bilayer.91
1.0
Mz (a.u.)
0.5
0.0
-0.5
-1.0
-10 -8 -6 -4 -2 0 2 4 6
Applied Field H0 (kOe)
8 10
Figure 8.2: Hysteresis loops for an 8 nm thick film of hard material (red dashed line) and a 64 nm
bilayer structure (blue solid line) with 56 nm of soft material and 8 nm of the hard material. Both
materials have 4πMs = 18.1 kOe. The hard material has an anisotropy constant of Kh = 5.3x106
erg/cm3. The soft material has an anisotropy constant of Ks = 0.166x106 ergs/cm3.
To find the frequencies of the eigenmodes, we numerically started the system in
an equilibrium state and added a small magnetic field pulse in the y-direction for 0.0005

ns. At the end of the pulse, we recorded m( t ) for each cell. We then did a Fourier
transform for the time evolution of magnetization in each cell, and summed the resulting
power spectrum over all cells. This results in a set of peaks at different frequencies
representing bulk and localized modes.75-
78
Because the chosen pulse was spatially
uniform, it is expected that the mode with the strongest response will be a mode with a
spatial profile that is also close to uniform. We will see this is in fact the case below.
122
Fig. 8.3 shows the frequency spectrum of the oscillations after the pulse is
applied. The peaks correspond to the resonant frequencies of the structure. In Fig. 8.4(a),
(b), (c) and (d) the spatial profiles of the first four modes are drawn as a function of
thickness through the bilayer.
Figure 8.3: Frequency spectrum of the magnetic eigenmodes in the 64 nm bilayer structure with
an applied field of -700 Oe.
The eigenmode profiles are constructed by first taking the Fourier transform of
mx(t) in each cell. From this one finds the amplitude A(i) and phase
a function of frequency.
i) in each cell i as
One can find a spatial profile of a mode at a particular
frequency by plotting A(i) cos( i)) as a function of the cell index i. This is equivalent to
an instantaneous “snapshot” of the out-of-plane precession for a particular mode at
frequency
75
123
Figure 8.4: Mode profiles for the four lowest frequency modes in the 64 nm bilayer structure. The
frequencies are taken from the frequency spectrum in Fig. 8.3. The darker shading shows the
location of the 8 nm hard film at the bottom of the 64nm thick bilayer.
It can be seen that the lowest resonant frequency corresponds to an excitation
which is mainly localized in the soft layer, with a maximum near the domain wall at x
=15 nm. As discussed earlier, the profile of the lowest frequency mode is excited with
the largest amplitude since it is the closest to a spatially uniform mode. In contrast, the
standing wave mode at 29.17 GHz is only weakly excited by the spatially uniform pulse
(see Fig. 8.3). Our calculations were compared to those found using a linear spin wave
theory89 with the dipolar contributions on each layer estimated as 4πMs and the
frequencies and mode profiles agreed quite well.
Next a spatially-uniform oscillating magnetic field (hd) is applied to the structure,
as shown in Fig. 8.1, with the frequency matching one of the resonant modes. The driving
field is oriented along the y axis, which is in-plane and perpendicular to the static applied
124
field. Fig. 8.5 shows an example of the time evolution of Mz (the average z component of
the magnetization of the whole structure) when the driving frequency is 10.74 GHz and
hd = 120 Oe. The microwave field is off from 0 ns to 1 ns and M z is initially near -0.5,
indicating that the structure is in the exchange spring mode. At 1 ns the driving field is
turned on. At this point, the magnetization undergoes a cycle and a half of oscillation
before the hard material flips over at around 1.25 ns. After the switching point, the
magnetization in the entire structure is almost completely aligned with the applied field
along the –z axis. This state no longer has a resonance at 10.74 GHz, which is made
apparent by the decay of the oscillation amplitude, even though the driving field is still
pumping the structure all the way up to 2 ns.
0.00
Mz (a.u.)
-0.25
-0.50
-0.75
-1.00
0.5
1.0
1.5
2.0
time (ns)
Figure 8.5: Time evolution of the average z-component of magnetization for the bilayer structure
in a -700 Oe applied field and a 120 Oe driving field. The driving field has a frequency of 10.74
GHz, matching the lowest magnon mode depicted in Fig. 4. The driving field is started at 1 ns.
125
We drove the structure at several resonant frequencies with different strengths of
the driving fields. Fig. 8.6 is a plot of the switching results at each of the resonant
frequencies. Depending on the strength of the driving field, the structure either switched
so that both layers lined up with the static field (solid), or it stayed in the exchange spring
state (hatched). The primary observation from Fig. 8.6 is that the microwave induced
switching occurs at lower hd values for lower frequencies. The lowest driving field
required for switching was 105 Oe when the frequency was 10.74 GHz.
Exchange
Spring
State
10.74 GHz
14.28 GHz
21.12 GHz
29.17 GHz
Switched
State
37.96 GHz
47.85 GHz
0.00
0.25
0.50
0.75
Driving field strength (kOe)
1.00
Figure 8.6: Final spin configuration as a function of the magnitude of the driving field. The
exchange spring state is illustrated by the hatched area and the solid area shows for what values of
the driving field the hard layer has switched (uniform state). Results for driving fields with
frequencies corresponding to that of different eigenmodes are shown. The static field is -700 Oe.
126
One can imagine at least two possible factors governing the switching of the hard
layer.
1) If the amplitude of the mode is largest near the domain wall then the switching should
occur at lower field strengths. The basic idea is that the larger amplitude precession helps
the magnetization jump over an energy barrier.
2) If the spatial profile of the eigenmode is close to that of the driving field, the energy in
the driving field will be more effectively transferred to that of the eigenmode, leading to a
larger amplitude precession and switching at a lower hd value.
From Fig. 8.4, it is apparent that, in this case, all the modes have a large
amplitude near the domain wall. In contrast, only the lowest frequency mode is close to
being spatially uniform, and this is why pumping this mode leads to switching at the
lowest power level.
Mathematically, the aerial density of power absorbed in the structure should be given by
x d
Aerial Power Density =
h d (x)
x 0
dm y ( x )
dt
x d
dx
h d ( x )m y ( x )dx .
(8.3)
x 0
Generally as the power absorbed is increased, the amplitude of the mode also increases.
x d
If hd is uniform, the expression for the power absorption reduces to
hd
m y ( x )dx and
x 0
thus the mode with frequency 10.74 GHz will absorb significantly more power than the
mode at f = 29.17 GHz because its profile has a larger value of the integral (see Fig. 8.4).
We also examined the case where the microwave strength was held constant, at
120 Oe, and the static field, Ho, was changed. The frequency of the driving field at each
Ho was set at the frequency of the lowest eigenmode for that value of Ho.
The
127
frequencies ranged from almost 9 GHz at -400 Oe to almost 16 GHz at -1000 Oe. Fig.
8.7 is a hysteresis loop constructed from these driven results (solid line) and compared to
the undriven hysteresis loop (dashed line). The static switching field in Fig. 8.6 is -550
Oe. If we sum the magnitude of the static switching field and the magnitude of the
driving field, we create an effective switching field (HES).
H ES
Hs
hd
(8.4)
Thus for the driven case, the effective switching field is 0.67 kOe. In the undriven case,
the switching field is 1.12 kOe. This gives a reduction in the effective switching field of
about 40%.
1.0
Mz (a.u.)
0.5
0.0
-0.5
-1.0
-2
-1
0
1
2
Applied Field H0 (kOe)
Figure 8.7: Hysteresis loops for the undriven bilayer structure (dashed line) and the bilayer
structure driven with a 120 Oe oscillating field (solid line). The frequency of the driving field is
the frequency of the lowest magnon mode in the structure, which varies as a function of static
applied field.
128
We also explored driving the structure with the microwave field oriented along
the x-axis and z-axis. The results were similar to those for hd along the y-axis in that
driving at the lowest frequency eigenmode caused switching at the lowest strength of
driving field. However, in both cases (hd along x, and hd along z) the amplitude of the
driving field required to switch the structure was almost twice what was required for the
y-axis oriented microwave. This again can be understood by considering the aerial power
density absorbed by the structure. In Eq. (8.3) the integrand is actually a dot product


between the driving field h d and the dynamic magnetization m . Therefore, the relative
orientation between the two vectors is important. The dynamic magnetization in the
bilayer has its largest component in the y direction due to thin film demagnetizing effects
(see Fig. 8.1) and so the largest power absorption occurs when the microwave field is
applied in this direction to maximize the dot product in Equation (8.3).
We also investigated using a spatially non-uniform driving field in an attempt to
excite the non-uniform modes seen in Fig. 8.4b, 8.4c, and 8.4d more efficiently and
thereby cause switching at a lower value of hd. The results for a driving field, that
changes linearly through the thickness of the structure and has a sign change at the
midpoint, did not show an advantage for any mode. The driving field required to switch
the structure was increased for all the modes. In particular, the lowest frequency mode,
the most uniform eigenmode, became much more difficult to excite as opposed to being
the easiest when driven with a uniform microwave field. This is consistent with Eq. (8.3)
because now hd(x) is odd about the midplane of the film while my(x) is close to even
about the midplane and the resulting integral is small.
In order for there to be an
129
advantage to driving the higher frequency modes, the non-uniform field profile must
closely match the profile of the eigenmode that it is trying to excite. There are methods to
adjust the spatial profile of the driving field92 but adjusting the field profile to match the
profiles in Fig 8.4 (b-d) would be difficult.
Conclusions
We find that domain wall assisted and microwave assisted switching can be used
together to decrease the power required to reverse the magnetization in a hard material.
The applied field required to switch a hard thin film is decreased by 86% (for our
material parameters) when it is coupled to a soft material and an exchange spring is
formed. By adding a spatially-uniform, oscillating magnetic field (hd), and setting its
frequency to match the lowest frequency magnon mode in the structure, we show that the
switching field is reduced further by 40%. The construction of an exchange spring and
the application of a driving microwave field together lead to a reduction in the total
effective switching field of 91%.
The reduction in the total field required for switching, requires the application of
both a static and microwave field. The additional complexity that this creates might be
mitigated by several factors.
For example, one could saturate a larger region with an
oscillating field and localize a static applied field to switch particular bits. The inverse
situation, where the static field extends over a larger region and the microwave field is
localized could also be considered.
CHAPTER IX
Summary
This dissertation deals with issues that arise in signal processing and magnetic
storage. The majority of Chapters III, IV, VII, and VIII have been adapted from our
publications in scientific journals.93-96 The work is theoretical in nature, but the results
depend on a range of parameters that can be used in experiments. Indeed, some of the
predictions made in the theoretical calculations were later confirmed by experiments.46 In
general, the dissertation is concerned with how a microwave signal is changed by a
magnetic material and the inverse problem of how a magnetic material can be changed by
a microwave field.
The advent of nanosized structures makes the second topic
particularly interesting.
In Chapters III, IV, and V we dealt with how a microwave signal is altered as it
passes through a layered magnetic material. The calculations were done using Maxwell's
equations and the appropriate boundary conditions. This gave results for microwave
absorption and phase shifts in a magnetic material as a function of frequency and applied
magnetic field. The main results are the following:
1) Positioning a magnetic film directly in the middle of the dielectric spacer in a
microstrip geometry increases the signal attenuation at the resonant frequency and also
narrows the width of the absorption, optimizing notch filter results.
2) Using hexagonal ferrites as the magnetic element in notch filters or phase shifter
devices accommodates operational frequencies in the 50-70 GHz range for relatively low
131
static magnetic fields. Relatively thin hexagonal ferrite films (less than 1 micron) can
produce usable device results.
3) Using attenuated total reflection, a variety of non-reciprocal devices can be made from
multi-layered structures which include a magnetic layer.
The methods used in these sections do not account for non-linear effects and thus may
not be appropriate for high power microwaves. Despite this, the theoretical results often
come quite close to experimental results, as seen in Chapter IV.
Chapters VII and VIII dealt with how magnetic nano-structures are influenced
and even changed by an applied microwave field. The calculations were done with a
locally-developed micromagnetics program that is appropriate for both linear and
nonlinear effects. The key results in this portion of the dissertation were:
1) The geometry of a nano-structure can influence which magnetic states (C state, S state,
flower state, etc.) are stable. Driving the structure with an oscillating magnetic field at a
resonant frequency can influence which of these initially stable states remain stable and
effectively cause a transition from one magnetic state to another.
2) When trying to switch the orientation of a bilayer magnetic structure, the total
switching field can be substantially reduced by adding an oscillating magnetic field
matching the appropriate resonant frequency.
In future work, the micromagnetic simulations could also be used to evaluate the
microstrip devices and include non-linear effects. The boundary condition calculation
could be used to enhance the simulations by approximating how the oscillating field
would decay across the structure. Together, these methods could provide insight into a
wide range of experiments involving microwave interaction with magnetic structures.
132
REFERENCES
1
G.A. Prinz and J . J . Krebs , Appl. Phys. Letts. 2, 397 (1981).
2
E.Schloemann, R. Tustison, J. Weissman, H. J. Van Hook, and T. Varitimos, J. Appl.
Phys., 63, 3140 (1988).
3
V. S. Liau, T. Wong, W. Stacey, S. Ali, and E. Schloemann IEEE MTT-S, 3, 9560
(1991).
4
C. S. Tsai, Jun Su, and C. C. Lee, IEEE Trans. Magn., 35, 3178 (1999).
5
Salahun, P. Quėffėlec, G. Tannė, A.-L Adenot, A.-L., and O. Acher, J. Appl. Phys., 91,
5449 (2002).
6
I. Huynen, G. Goglio, D. Vanhoenacker, and A. Vander Vorst, IEEE Microwave and
Guided Wave Letters, 9, 401 (1999).
7
N. Cramer, D. Lucic, R. E. Camley, and Z. Celinski, J. Appl. Phys., 87, 6911 (2000).
8
N. Cramer, D. Lucic, D. K. Walker, R. E. Camley, and Z. Celinski, IEEE Trans. on
Magnetics, 37, 2392 (2002).
9
Bijoy Kuanr, L. Malkinski, R. E. Camley, Z. Celinski, and P. Kabos, J. Appl. Phys, 93,
8591 (2003).
10
Y. Zhuang, B. Rejaei, E. Boellaard, M. Vroubel, and J. N. Burghartz, IEEE Microwave
and Wireless Components Letters, 12, 1531 (2002).
11
Bijoy Kuanr, Z. Celinski, R.E. Camley, Appl. Phys. Lett., 83, 3969 (2003).
12
J. George, Preparation of Thin Films, (Marcel Dekker, INC., New York, 1992).
13
Plumer M L, van Ek J and Weller D, The Physics of UltraHigh Density Magnetic
Recording (Springer, New York , 2001).
133
14
H. F. Hamann, Y. C. Martin, and H. K. Wickramasinghe, Appl. Phys. Lett. 84(5), 810
(2004).
15
R. Hertel, S. Gliga, M. Fahnle, and C. M. Schneider, Phys. Rev. Lett. PRL 98, 117201
(2007).
16
Y. Nozaki, M. Ohta, S. Taharazako, K. Tateishi, S. Yoshimura, and K. Matsuyama,
Appl. Phys. Lett., 91, 082510 (2007).
17
J. Zhu, X. Zhu, and Y. Tang, IEEE Trans. on Magnetics, 44, 1 (2008).
18
P. Li, X. Yang, and X. Cheng, Proceedings of the SPIE, 7125, 6 (2008).
19
Z. Wang, and M. Wu, J. Appl. Phys., 105, 093903 (2009).
20
C. Nistor, K. Sun, Z. Wang, M. Wu, C. Mathieu, and M. Hadley, Appl. Phys. Lett., 95,
012504 (2009).
21
M. Todorovic, S. Schultz, J. Wong, and A. Scherer, Appl. Phys. Lett., 74 2516 (1999).
22
C. A. Ross, Henry I. Smith, T. Savas, M. Schattenburg, M. Farhoud, M. Hwang, M.
Walsh, M. C. Abraham, and R. J. Ram, J. Vac. Sci. Technol. B, 17, 3168 (1999).
23
J. Lohau, A. Moser, C. T. Rettner, M. E. Best, and B. D. Terris, Appl. Phys. Lett., 78,
990 (2001).
24
B. D. Terris, T. Thomson, G. Hu, Microsyst. Technol., 13, 189 (2006).
25
H. J. Richter, A. Y. Dobin, R. T. Lynch, D. Weller, R. M. Brockie, O. Heinonen, K. Z.
Gao, J. Xue, R. J. M. v. d. Veerdonk, P. Asselin, and M. F. Erden, Appl. Phys. Lett. 88,
222512 (2006).
26
X.Yang, S. Xiao, W. Wu, Y. Xu, K. Mountfield, R. Rottmayer, K. Lee, D. Kuo, and
D.Weller, J. Vac. Sci. Technol. B, 25, 2202 (2007).
27
C. Kittel , Introduction to Solid State Physics, (John Wiley & Sons, Inc., U. S., 2005).
134
28
J. D. Patterson, B. C. Baily, Solid-State Physics Introduction to the Theory, (Springer,
Verlag, ,2007).
29
A. J. Newell, W. Williams, D. J. Dunlop, J. Geophys. Res., 98, B6, 9551 (1993).
30
R. E. Camley and D. L. Mills, J. Appl. Phys., 82, 3058 (1997).
31
R. J. Astalos and R. E. Camley, J. Appl. Phys., 83, 3744 (1998).
32
R. J. Astalos and R. E. Camley, Phys. Rev. B, 58, 8648 (1988).
33
F.L. Pedrottti and L.S. Pedrotti, Intoduction to Optics, (Prentice Hall , NJ: Upper Saddle
River, 1993).
34
S. Treryokov, Analytical Modeling in Applied Electromagnetics, (Artech House, MA:
Norwood, 2003).
35
B. Wood and J. B. Pendry, Phys. Rev. B, 74, 115116 (2006).
36
Y. Song, S. Kalarickal, and C. E. Patton, J. Appl. Phys. 94, 5103 (2003).
37
S. D. Yoon and C. Vittoria, J. Appl. Phys., 93, 8597 (2003).
38
V. G. Harris et al, J. of Appl. Phys., 99, 08M911-08M911-5 (2006).
39
Xu Zuo, H. How, P. Shi, S.A. Oliver and C. Vittoria, IEEE Trans. Magn., 38, 3493
(2002).
40
Xu Zuo, P. Shi, S.A. Oliver, and C. Vittoria, J. Appl. Phys., 91, 7622 (2002).
41
Bijoy Kuanr, R. E. Camley, and Z. Celinski, Appl. Phys. Lett., 87, 012502 (2005).
42
S. B. Narang, I. S. Hudiara, J. of Ceramic Processing Research, 2(2), 113 (2006).
43
J. Qui, Q. Zhang, and M. Gu, J. Appl. Phys., 98, 103905 (2005).
44
J. Smit and H. P. J. Wijn, Ferrites: Physical Properties of Ferrimagnetic Oxides in
Relation to their Technical Application, (John Wiley & sons, NY: New York, 1959).
45
R. J. Astolas and R. E. Camley, J. Appl. Phys., 82, 3744 (1998).
135
46
Z. Wang,Y. Song,Y. Sun, J. Bevivino,M. Wu,V. Veerakumar, T. J. Fal, and R. E.
Camley, Appl. Phys. Lett., 97, 072509 (2010).
47 1
J. Schoenwald, E. Burstein, and J. M. Elson, Solid State Communications 88(11/12),
1067 (1993, accepted 1972).
48 2
A. Hartstein, E. Burnstein, A. A. Maradudin, R. Brewer, and R. F. Wallis, J. Physics
C: Solid State Physics, 6, 1266 (1973).
49
J. J. Burk, G.I. Stegeman, and T. Tamir, Phys. Review B 33(8), 5156 (1986).
50
K. Torrii, T. Koga, T. Sota, T. Azuhata, S. F. Chichiba, and S. Nakamura, J Physics:
Condensed Matter 12, 7041 (2000).
51
C. Arnold, Y. Zhang, and J. G. Rivas, Appl. Phys. Lett., 96, 113107 (2010).
52
K.G. Carr-Brion and J. A. Gadsden, J. Physics E series 2, 2, 155 (1969).
53
F. Kimura, J. Umemura, and T. Takenaka, Langmuir, 2, 96, (1986).
54
K. K. Chittur, Biomaterials, 19, 357 (1998).
55
Gregor Kos, Hans, Lohninger, and Rudolf Krska, Vibrational Spectroscopy, 29, 115
(2002).
56
R. E. Camley, and D. L. Mills, Physical Review B, 26(3), 1266 (1982).
57
M. R. F. Jensen, T. J. Parker, Kamsul Abraha, and D. R. Tilley, Phys. Rev. Lett., 75,
3756 (1995).
58
M. R. F. Jensen, S A Feiven, T J Parker and R E Camley, J. Phys: Condens. Matter, 9,
7233, (1997).
59
L. Remer, B. Lüthi, H. Sauer, R. Geick, and R. E. Camley, Phys. Rev. Lett., 56, 2752
(1986)
60
Y. J. Chen and G. M. Carter, Solid State Communications, 45(3), 277 (1983).
136
61
R. Ruppin, Phys. Letters A, 277, 61 (2000).
62
R. Ruppin, J. Phys: Condens. Matter, 13,1811 (2001).
63
T. Fal, and R. Camley, Journal of Applied Physics, 104, 2, 23910 (2008).
64
J. Fidler, and T. Schrefl, J. Phys. D: Appl. Phs., 33, R135, (2000).
65
U. M. Ascher, L. R. Petzold, Computer Methods for Ordinary Differential Equations
and Differential-Algebraic Equations, (Society for Industrial & Applied Mathematics,
U.S., 1998).
66
G. B. Arfken, and H. J. Weber, Mathematical Methods for Physicists, (Academic Press,
UK: London ,2001).
67
S.Li, B. Livshitz, H. N. Bertram, M. Schabes, T. Schrefl, E. E. Fullerton, and V.
Lomakin, Appl. Phys. Lett., 94, 202509 (2009).
68
Wolfgang Rave and Alex Hubert, IEEE Trans., Magn. 36, 3886 (2000).
69
D. Goll, G. Schutz, and H. Kronmuller, Phys. Rev. B, 67, 094414 (2003).
70
J. I. Martín, J. Nogués, Kai Liu, J. L. Vicent and Ivan K. Schuller , J. Magn. Magn.
Mater., 256, 449 (2003).
71
J. Li, and C. Rau, Nucl. Instrum. Meth B, 230, 518, (2005).
72
H.Wang, H. Hu, M. R. McCartney, and D. J. Smith, J. Magn. and Magn. Mater., 303,
237 (2006).
73
S.S. Kalarickal, P. Krivosik, J.Das, K. S. Kim, and C. E. Patton, Phys. Rev. B, 77,
054427 (2008).
74
J. Das, S. S. Kalarickal, K. S. Kim, and C. E. Patton, Phys. Rev. B, 75, 093902 (2009).
75
M. Grimsditch, G. K. Leaf, H. G. Kaper, D. A. Karpeev, and R. E. Camley, Phys. Rev.
B, 69, 174428 (2004).
137
76
R. D. McMichael, and M. D. Stiles, J. Appl.Phys., 97, 10J901 (2005).
77
R. E. Camley, B. V. McGrath, Y. Khivintsev, Z. Celinski, Roman Adam, Claus M.
Schneider and M. Grimsditch, Phys. Rev. B, 78, 024425 (2008).
78
Z. K. Wang, H. S. Lim, H. Y. Liu, S. C. Ng, M. Kuok, L. L. Tay, D. J. Lockwood, M.
G. Cottam, K. L. Hobbs, P. R. Larson, J. C. Keay, G. D. Lian, and M. B. Johnson, Phys.
Rev. Lett. PRL, 94, 137208 (2005).
79
E. F. Kneller and R. Hawig, IEEE Trans. Magn., 27, 3588 (1991).
80
E. E. Fullerton, J. S. Jiang, M. Grimsditch, C. H. Sowers, and S. D. Bader, Physical
Review B, 58, 12193 (1998).
81
E. E. Fullerton, J. S. Jiang, and S. D. Bader, J. Magn. Magn. Mater., 200, 392 (1999).
82
A. Yu. Dobin and H. J. Richter, J. Appl. Phys., 101, 09K108 (2007).
83
D. Suess, J. Lee, J. Fidler and T. Schrefl, J. Magn. Magn. Mater., 321, 545 (2009).
84
Y. Nozaki, M. Ohta, S. Taharazako, K. Tateishi, S. Yoshimura and K. Matsuyama,
Appl. Phys. Lett., 91, 082510 (2007).
85
J. Zhu, X. Zhu and Y. Tang, IEEE Trans. Magn., 44, 1 (2008).
86
M. A. Bashir, T. Schrefl, J. Dean, A. Goncharov, G. Hrkac, D. Allwood, and D. Suess,
IEEE Trans. Magn., 44, 3519 (2008).
87
J. Grollier, M. V. Costache, C. H. van der Wal and B. J. van Wees, J. Appl. Phys., 100,
024316 (2006).
89
K. L. Livesey, D. C. Crew and R. L. Stamps, Phys. Rev. B, 73, 184432 (2006).
90
M. Grimsditch, R. Camley, E. E. Fullerton, S. Jiang, S. D. Bader, and C.
H. Sowers, J. Appl. Phys., 85, 5901 (1999).
91
T. J. Fal and R. E. Camley, Appl. Phys. Lett., 97, 122506 (2010).
138
92
Y. V. Khivintsev, L. Reisman, J. Lovejoy, R. Adam, C. M. Schneider, R. E. Camley,
and Z. J. Celinski, J. Appl. Phys., 108 023907 (2010).
93
T. J. Fal, V. Veerakumar, B. Kuanr, Y. V. Khivintsev, Z. Celinski, R. E. Camley, J.
Appl. Phys., 102, 063907 (2007).
94
T. J. Fal, R. E. Camley, J. Appl. Phys., 104, 23910 (2008).
95
T. J. Fal, R. E. Camley, Appl. Phys. Lett. 97, 122506 (2010).
96
T. J. Fal, K. L. Livesy, R. E. Camley, J. Appl. Phys., (Accepted 3/08/2011, publication
pending).
139
Appendix- A
From Chapter IV, we give complete expressions for the elements of the G matrix
which represent the equations of the boundary conditions for the waveguide in Chapter

GA
IV. We had
0
G 11
which is explicitly given by
G 12
G 13
G 14
0
0
0
0
A1
0
G 21 G 22
G 23
G 24
0
0
0
0
A2
0
G 31
G 32
G 33
G 34
0
0
G 37
G 38
A3
0
G 41 G 42
G 43
G 44
G 45
G 46
0
0
A4
0
G 51
G 52
G 53
G 54
G 55
G 56
0
0
A5
0
G 61 G 62
G 63
G 64
0
0
G 67
G 68 A 6
0
G 78 A 7
0
0
0
0
0
0
0
G 77
0
0
0
0
G 85
G 86
0
0
A8
0
(A1)
The nonzero elements of G1n are associated with Ex at z = 0 and are given by
G1n
c
[
n
ky
n
k zn ]
(A2)
1
The nonzero elements of G2n are associated with Ey at z = 0 and are given by
G 2n
c
k zn
(A3)
1
The nonzero elements of G3n are associated with Ex at z = d1 and are given by
G 3n
c
eik zn d1 [
n
ky
n
k zn ]
for n = 1, 2, 3, 4
(A4)
1
G 37
c
e
2
ik z d1
[ ( k z )k y
( kz ) ]
(A5)
140
c
G 38
e
ik z d1
[ ( k z )k y
( kz ) ]
(A6)
2
The nonzero elements of G4n are associated with Ey at z = d1 and are given by
c
G 4n
eik zn d1 k zn
for n = 1, 2, 3, 4
(A7)
1
c
G 45
e
ik z d1
( kz )
(A8)
e
ik z d1
( kz )
(A9)
2
c
G 46
2
The nonzero elements of G5n are associated with hx at z = d1 and are given by
G 5n
eik zn d1
G 55
e
G 56
e
for n = 1, 2, 3, 4
ik z d1
(A10)
(A11)
ik z d1
(A12)
The nonzero elements of G6n are associated with hy at z = d1 and are given by
G 6n
ne
ik zn d1
G 67
e
ik z d1
G 68
e
ik z d1
for n = 1, 2, 3, 4
(A13)
(A14)
(A15)
The nonzero elements of G7n are associated with Ex at z = d1+d2 and are given by
G 77
c
e
2
ik z ( d1 d 2 )
[ ( k z )k y
( kz ) ]
(A16)
141
G 78
c
e
ik z ( d1 d 2 )
[ ( k z )k y
( kz ) ]
(A17)
2
The nonzero elements of G8n are associated with Ey at z = d1+d2 and are given by
G 85
c
e
ik z (d1 d 2 )
e
ik z ( d1 d 2 )
( kz )
(A18)
2
G 86
c
2
( kz )
(A19)
Документ
Категория
Без категории
Просмотров
0
Размер файла
6 623 Кб
Теги
sdewsdweddes
1/--страниц
Пожаловаться на содержимое документа