Download Electromagnetic Waves in Media with Ferromagnetic Losses

Electromagnetic Waves in
Media with Ferromagnetic
Losses
Jörgen Ramprecht
Doctoral Thesis
Electromagnetic Theory
Royal Institute of Technology
Stockholm, Sweden, 2008
TRITA-EE 2008:029
ISSN 1653-5146
ISBN 978-91-7415-011-7
KTH Electromagnetic Engineering
SE-100 44 Stockholm
SWEDEN
Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges
till offentlig granskning för avläggande av teknologie doktorsexamen torsdagen den 5
juni 2008 kl 13.15 i sal D3, Kungl Tekniska högskolan, Lindstedtsvägen 5, Stockholm.
c 2008 Jörgen Ramprecht
Copyright °
Tryck: Universitetsservice US AB
Abstract
The operation of a wide variety of applications in today’s modern society are heavily
dependent on the magnetic properties of ferromagnetic materials and their interaction with electromagnetic fields. The understanding of these interactions and the
associated loss mechanisms is therefore crucial for the improvement and future development of such applications.
This thesis is concerned with electromagnetic waves in media with ferromagnetic
losses. We model the dynamics of the magnetization of a ferromagnetic material
with the nonlinear Landau-Lifshitz-Gilbert (LLG) equation and study stability conditions on static solutions. Furthermore, with the aid of a small signal analysis this
equation is linearized around a stable static solution. From this analysis we obtain
a small signal permeability, which shows that ferromagnetic material in general are
gyrotropic with a resonant frequency behavior similar to that of a Lorentz material. In difference to dielectric Lorentz material, this resonance frequency can be
shifted with the aid of a bias field. For a specific bias field we obtain a frequency
behavior that mimics that of a material with electric conductivity losses. In terms
of losses per unit volume it is then possible to define a magnetic conductivity which
is independent of frequency.
We treat composite materials built from ferromagnetic inclusions in a nonmagnetic and nonconductinig background material. The composite material inherits the
gyrotropic structure and resonant behavior of the single particle. The resonance frequency of the composite material is found to be independent of the volume fraction,
unlike dielectric composite materials. For small enough particles, typically around
100 nm, it becomes energetically favorable to form a single domain in the particle,
where disturbances in the magnetization can propagate in the form of spin waves.
We study the possibility of exciting spin waves and derive a susceptibility that takes
spin waves into account. It is found that spin wave resonances are excited in the
gigahertz range and this could offer a way to increase the losses in a composite material. We also discuss some concerns regarding stability and causality of effective
material parameters for biased ferromagnetic materials.
Finally, we discuss the possibility of using magnetic materials in absorbing applications. We analyze the scattering of electromagnetic waves from a metal surface
covered with a thin magnetic lossy sheet. It is found that very thin magnetic layers
can provide substantial specular absorption over a wide frequency band. However,
magnetic specular absorbers, where the waves propagates just a fraction of the wavelength in the material, seem to require a certain amount of ferromagnetic material
which make them quite heavy and thereby limit its practical use. On the other
hand, for nonspecular absorbers where the waves propagates several wavelengths
in the material, the amount of magnetic material required for efficient absorption
seems to be substantially less than for specular absorbers. Thus, as nonspecular
absorbers, magnetic lossy materials could offer very thin and light designs.
iii
iv
List of papers
This thesis consists of a General Introduction and the following scientific papers:
I. Hans Steyskal, Jörgen Ramprecht and Henrik Holter. Spiral Elements for
Broad-Band Phased Arrays. IEEE Transactions on Antennas and Propagation, Vol. 53, No. 8, pp 2558-2562, August 2005.
II. Jörgen Ramprecht and Daniel Sjöberg. Biased magnetic materials in RAM
applications. Progress in Electromagnetics Research, vol. 75, pp. 85-117, 2007.
III. Jörgen Ramprecht, Martin Norgren and Daniel Sjöberg. Scattering from
a thin magnetic layer with a periodic lateral magnetization: application
to electromagnetic absorbers. Submitted to Progress in Electromagnetics
Research.
IV. Jörgen Ramprecht and Daniel Sjöberg. Magnetic losses in composite materials. Accepted for publication in Journal of Physics D: Applied Physics.
V. Jörgen Ramprecht and Daniel Sjöberg. On the amount of magnetic material
necessary in broadband magnetic absorbers. IEEE International Symposium
on Antennas and Propagation (AP-S 2008), San Diego, U.S., July 5-12, 2008.
VI. Daniel Sjöberg, Jörgen Ramprecht and Niklas Wellander. Stability and causality of effective material parameters for biased ferromagnetic materials. URSI
General Assembly (GA 2008), Chicago, U.S., August 7-16, 2008.
v
vi
The author’s contribution to the included papers
I. In this paper I contributed to all of the analysis and carried out all the
numerical calculations.
II. In this paper I did most of the analysis and derivations. I wrote the codes
and did the numerical examples. I wrote most of the bulk text.
III. In this paper I contributed to most of the analysis and derivations. I did the
main part of the derivation of the Fundamental equation, wrote the numerical
code and did the numerical examples. I also wrote most of the bulk text.
IV. In this paper I contributed to most of the analysis and derivations. I did the
main part of the analysis in the spin wave section. Both authors contributed
to the bulk text.
V. In this paper I did most of the analysis and derivations. I wrote all of the
bulk text.
VI. In this paper I contributed to the analysis and derivations.
vii
viii
Acknowledgements
I wish to express my deepest gratitude to my co-author Daniel Sjöberg for invaluable
guidance and support. Daniel, with his great knowledge and physical intuition, has
been a true source of inspiration and I am very grateful for our pleasant collaboration. I also wish to thank my supervisor Martin Norgren for guiding me through
courses, proofreading my manuscripts, sharing his insights and introducing me to
the field of classical electrodynamics.
During my time in the Electromagnetic Theory group at KTH I have also had the
privilege to work together with Hans Steyskal, which has been a delightful experience. Hans provided excellent guidance sharing his great experience and knowledge.
I thank you for all of this and also all the pleasant conversations.
Furthermore, I thank all present and former colleagues at the department for
contributing to the nice atmosphere. I have really enjoyed the discussions at fika
bordet, the pubs and all the activities. In addition, during my collaboration with
Daniel Sjöberg I have also had the opportunity to spend time with the Electromagnetic Theory group at Lund University. I thank Gerhard Kristensson and all
his colleagues for always making me feel welcome and for providing an inspiring
atmosphere.
Finally and most of all, I wish to thank my family and friends. My fantastic
friends for making sure that I don’t work too much and that I have a great time
outside the office, my sister Marita for all support and housing over the years, Hannes
and Viktor for letting me sleep in their bunk bed and not beating me too bad playing
playstation, Per Granberg for being a good friend and introducing me to the world
of physics, my brother Johan and my dear parents, Franz and Ragnhild, for always
supporting me. You all mean the world to me.
ix
x
Contents
Abstract . . . . . . . . . . .
List of papers . . . . . . . .
The candidate’s contribution
Acknowledgements . . . . .
. . . . . . . . .
. . . . . . . . .
to the included
. . . . . . . . .
. . . .
. . . .
papers
. . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. iii
. v
. vii
. ix
1 Introduction
1.1 A first approach towards magnetism and magnetic losses . . . . . . .
1
2
2 Ferromagnetism
2.1 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5
3 Applications
3.1 Magnetic recording . . . . . . . . .
3.2 Magnetic resonance imaging (MRI)
3.3 Transformers . . . . . . . . . . . .
3.4 Electrical motors and generators . .
3.5 Ferromagnetic nanoparticles . . . .
3.6 Absorbers . . . . . . . . . . . . . .
.
.
.
.
.
.
7
7
8
8
8
9
9
.
.
.
.
11
11
12
12
16
.
.
.
.
.
.
.
19
19
20
20
22
22
23
24
.
.
.
.
.
.
25
25
25
25
26
26
27
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
4 Magnetic absorbers
4.1 Specular absorbers . . . . . . . . . . . . . . .
4.2 Nonspecular absorbers . . . . . . . . . . . . .
4.2.1 Reduction of surface waves . . . . . . .
4.2.2 Surface wave reduction in phased array
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. . . . . .
. . . . . .
. . . . . .
antennas
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5 Mathematical models of ferromagnetic materials
5.1 The Landau-Lifshitz-Gilbert model . . . . . . . . . . . . . . . . . .
5.2 Small signal model . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 The static solution . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 The small signal solution . . . . . . . . . . . . . . . . . . . .
5.3 Stability of the static magnetization . . . . . . . . . . . . . . . . . .
5.3.1 Thin plate with an applied field in the normal direction . . .
5.3.2 Spherical particle with an applied field in arbitrary direction
6 Summary
6.1 Paper
6.2 Paper
6.3 Paper
6.4 Paper
6.5 Paper
6.6 Paper
of papers
I. . . . . .
II . . . . .
III . . . . .
IV . . . . .
V . . . . .
VI . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
xi
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
xii
1 Introduction
1
N
S
Figure 1: A compass needle aligning itself with Earth’s magnetic field.
1
Introduction
Magnetism is one of mankind’s earliest scientific discovery. It has been claimed that
the Chinese used the compass sometime before 2500 B.C., and at around 800 years
B.C. it was known, both in Europe and China, that certain stones could attract
iron. One of the first to document observations on magnetic phenomenon was the
Greek philosopher Thales of Miletus, who about 600 B.C. reported that Loadstone
attracts Iron. Another early documentation was made by Plato in his dialogue Ion,
where the discovery of the magnet, a stone that attracts iron by an invisible force, is
mentioned [1]. At this point however, the Greek philosophers were more interested in
placing the phenomenon in a divine context than to explain and predict the wonders
of nature. For many centuries magnetism remained a mysterious phenomenon whose
true nature was yet to be revealed.
Extensive research, especially during the twentieth century, has unveiled many
secrets associated with magnetism and a quite good understanding has been developed. This has resulted in magnetism nowadays being a huge industry with
enormous financial impact and with numerous applications where the properties
of magnetic materials are crucial. Today magnetism is a concept and phenomenon
which most people are aware of and take for granted. We have all used a magnet, for
instance when attaching a piece of paper to our refrigerator with a refrigerator magnet or navigating with the aid of a compass. However, little has probably changed
since the early discoveries in terms of the common knowledge on magnetism. To
most people magnetism is still some obscure force that attracts objects of certain
metals and most of the time when using applications where magnetism is exploited
we are not even aware that magnetism is the underlying cause. One of the main
reasons for this is simply that magnetism is a very intricate subject.
2
General Introduction
Figure 2: A magnetic material where the atoms are represented by compass needles.
In the left figure the direction of the compass needles are randomly distributed in the
absence of a magnetic field. When a magnetic field is applied the compass needles
align themselves along the field, as illustrated in the right figure.
1.1
A first approach towards magnetism and magnetic losses
One of the first questions concerning magnetism that comes to mind might be: What
do we mean by a magnetic material? For a person without a degree in engineering
or similar, the complex theories on magnetism are usually not of much assistance
when searching for answers. But perhaps we can shed some light on the subject
by turning to the well known compass mentioned earlier. We are all familiar with
the fact that the needle of a compass always points in the direction of what we call
north. This means that everywhere, no matter where we go, there is a force that
makes sure that the needle points in this direction. As it turns out, the needle itself
is a magnet and the force exerted on it is due to the Earth’s magnetic field. On the
Earth’s surface this magnetic field is everywhere approximately pointing towards
north, see Figure 1. Thus, we conclude that the magnetic compass needle aligns
itself with the direction of the magnetic field. At this point we don’t care what the
magnetic fields is nor how it is created but just treat it as an invisible direction in
space with which a compass needle aligns itself.
All materials consist of very small elements called atoms. For some materials
these atoms posses a so called magnetic moment that may be viewed as a tiny
compass needle; see Figure 2. In general these compass needles assume random directions and when added up, their contributions cancel each other. But if a magnetic
field is applied, all of the compass needles start to align along the magnetic field
and together they may be represented by a total compass needle for the material.
Materials that responds to magnetic fields in this way are said to be magnetic and
when the atoms are aligned the material is magnetized. The total compass needle
then represents the direction of the magnetization of the material. As the magnetic
field is removed the atoms return to a state of random orientation and the magne-
1 Introduction
3
tization disappears. However, there are materials where the magnetization remains
even after removing the applied magnetic field. The most familiar among such materials are probably those classified as ferromagnetic or more commonly referred to
as permanent magnets.
They way a material responds to an electromagnetic field is determined by the
so called electromagnetic properties of the material. Being a magnetic material is
one such property. Another important electromagnetic property is the losses inside
the material. In the case of a magnetic material, we have magnetic losses. This
naturally raises the question, what do we mean by losses? Once again the analogy
with a compass needle may be instructive. The aligning atom may be pictured by
a compass needle immersed in a fluid, where the fluid represents the surrounding
media. As the compass needle turns in the fluid there will be friction between the
two, resulting in heat being generated in the material. The energy that heats the
material comes from the energy in the magnetic field. In this sense the magnetic
field loses energy to the magnetic material, hence the terminology magnetic losses.
The simplified illustration given here hardly provides us with the answers on
what magnetism really is, but it may give some physical insight into how a magnetic material is magnetized and how energy is dissipated in the material. From
this simple model we have learned that a magnetic field interacts with a magnetic
material in such fashion that the atoms aligns in a certain way to produce a magnetization, and during this interaction energy is dissipated in the material. One of
the main purposes of this thesis is to describe these phenomenons in a more detailed
way and thereby we hope to increase our understanding of the interaction between
electromagnetic fields and ferromagnetic materials. We also investigate the possibility of using ferromagnetic materials in absorbing applications where the magnetic
losses are utilized to absorb electromagnetic energy.
4
General Introduction
2 Ferromagnetism
5
Figure 3: Domains in a ferromagnetic material. To the left an unmagnetized
material where the magnetizations in the different domains are not parallel. When
applying an external magnetic field, domains which are already closely aligned to
the field grow on the expense of others. As a result a net magnetization is achieved.
This magnetization remains even when the external magnetic field is removed.
2
Ferromagnetism
As mentioned in the previous section, there are certain magnetic materials that
remain magnetized even in the absence of an applied magnetic field. Such materials
are typically classified as ferromagnetic or ferrimagnetic materials [2]. In this section
we briefly review some of the properties associated with ferromagnetic materials.
2.1
General properties
The name ferromagnetic originates from the latin word ferrum. Ferrum means iron,
which is one of the most common examples of ferromagnetic materials. Ferromagnetic effects are typically quite strong and therefore easy to detect, and this is
probably the reason why observations on magnetism can be dated back as early
as to the ancient Greeks. In ferromagnetic materials there is a strong interaction
between the neighbouring magnetic moments (the compass needles) of each atom.
This interaction may produce a net magnetization, a permanent magnet, even in
the absence of an external magnetic field. Thus, ferromagnetic materials seem to
have a spontaneous magnetization where the alignment of the magnetic moments
are “frozen in”.
At first, one might think the reason to this phenomenon is the classical dipolar
interaction between two magnetic moments. However, this dipolar interaction is
rather weak and is, even for quite low temperatures, overcome by thermal agitation.
In further attempts to describe ferromagnetism, Weiss postulated that the interaction is due to an internal magnetic field, the Weiss field, that aligns the magnetic
moments. Heisenberg showed that this field is the result of the quantum mechani-
6
General Introduction
cal exchange interaction, which has no classical analog [3–5]. This Wiess field, also
named the exchange field, may be viewed as a magnetic field aligning the magnetic
moments in the material and thus producing the spontaneous magnetization. Nevertheless, despite the existence of permanent magnets, our everyday experience tells us
that some iron objects do not appear to have any magnetic effects. To explain this,
the domain theory was proposed by Weiss in 1907 [6]. The concept of this theory
is that the alignment occurs in relatively small regions, called domains; see Figure
3. Each domain, though small, contains a huge amount of magnetic moments, all
lined up. However, the direction of different domains need not be parallel. The
domains may be arranged in such fashion that the resultant magnetic moment is
approximately zero, which explains that ferromagnetic materials may be unmagnetized. The direction of the magnetization in a domain in this case is determined
by the magnetocrystalline anisotropy field. This field is due to the atomic lattice
and directs the magnetization along certain crystallographic axes called directions
of easy magnetization [2, p. 471]. In terms of alignment of magnetic moments both
the exchange field and the anisotropy field acts as if classical magnetic fields. But
both of them are of quantum mechanical origin and do therefore not enter into the
Maxwell equations. The effect of the exchange field and the anisotropy field is thus
to align the moments in certain easy directions. What is then responsible for setting
up the domain structure? To understand this we also have to include the dipolar interaction between the magnetic moments. Landau and Lifshitz showed that domain
structure is a consequence of the various interactions - exchange, magnetocrystalline
anisotropy and magnetic dipole - of a ferromagnetic material [7].
An unmagnetized ferromagnetic specimen may be magnetized with the aid of
an external magnetic field. The magnetization process takes place by essentially
two independent processes. For weak applied fields domains favorable oriented with
respect to the field grow at the expense of unfavorably oriented domains; see Figure
3. This is also referred to as domain-wall motion and even for quite weak fields this
process becomes irreversible, resulting in some net magnetization even when the field
is removed. For strong enough fields the material will finally consist approximately
of one domain. At this point the domain magnetization rotates (domain rotation)
toward the direction of the field. When the alignment is complete the material is
said to be saturated and the net magnetization is called the saturation magnetization. Removal of the applied field from the saturated material will now result
in a strong remanent magnetization; i.e., a permanent magnet has been produced.
This remanent magnetization is an example of hysteresis. This basically means that
the magnetization depends on the complete history of the excitation and not just
the final value. Another typical property of ferromagnetic materials is that they
are nonlinear, i.e., the magnetization depends on the applied magnetic field in a
nonlinear way.
3 Applications
(a) Transformer with an iron
core
7
(b) Cassette tape
(c) Floppy disk
Figure 4: Applications of ferromagnetic materials.
3
Applications
From the brief introduction on ferromagnetism in the previous chapter we can at
least say one thing for sure: Magnetism is indeed a very difficult subject. There are
many obscure mechanisms associated with the phenomenon, which of some, even to
this day, are not well understood. Nevertheless, magnetism is a phenomenon that is
widely used in many applications in our world today. In this section we will review
some of them. Also note that, depending on the application, there are different
requirements on the losses in the ferromagnetic material. In some applications, high
losses are desirable (e.g., absorbers), while in others it is crucial to maintain low
losses (e.g., transformers, motors etc.).
3.1
Magnetic recording
For several decades, magnetic recording has been a common and popular way to
store information. The most well known examples are perhaps the cassette tape,
the floppy disk and the hard disk drive. Magnetic recording of the human voice was
carried out by the Danish engineer Valdemar Poulsen already in 1896 with aid of
a ferromagnetic wire [8]. Later, around 1920, the magnetic tape was invented and
in 1928 the German Fritz Pfleumer filed a patent for coating iron particles onto a
strip of paper as a recording medium. In 1957, IBM introduced the first hard disk
drive. In the beginning the storage capacity and the quality of the recording was
quite poor. Along with improved understanding of the behavior of ferromagnetic
materials the performance of magnetic recording media has improved substantially.
For instance, today a hard disk drive can store a huge amount of data in an area of
just a few square centimeters.
In magnetic recording devices the remanent magnetization is utilized. By applying a magnetic field we can control the direction of the magnetization in the recording
ferromagnetic material and when removing the field, the material “remembers” this
direction. In this fashion we can magnetize different parts of the material in different
directions and thus store information with the aid of the remanent magnetization.
For further details on magnetic recording, see [9, 10].
8
3.2
General Introduction
Magnetic resonance imaging (MRI)
Magnetic resonance imaging is a widely used diagnostic technique. In this application a conventional permanent magnet made from ferromagnetic material can be
used to create a quite strong magnetic field. This field is applied to the patient in
order to align the atoms in the body. Then, while maintaining this strong field, a
weaker magnetic field is applied. The weak magnetic field pushes the atoms out
of alignment and different parts of the body are affected in different ways. In this
way different structures of the body can be resolved. MRI is particularly useful for
evaluating tumors and showing abnormalities in the heart and blood vessels. This
diagnostics method also has the attractive property that it causes minimal damage
to cellular structures.
There are however drawbacks to using permanent magnets for this application.
For instance, they are extremely large and heavy, and produce a relatively weak field.
But MRIs using permanent magnets are very inexpensive to maintain. To achieve
larger field strengths, superconducting magnets is often used. Despite being very
costly, this is the most common type of magnet used in MRI today. An exhaustive
presentation on MRI can be found in for instances [11, 12].
3.3
Transformers
A transformer is a device that transfers energy from one electrical circuit to another
through electromagnetic induction. The inductive coupling between the circuits
is enhanced with the aid of an (ferromagnetic) iron core. A key application of
transformers is within the field of transmission and distribution of electrical power.
Here transformers are used to transform the electrical power to a high-voltage/lowcurrent form, providing economical transmission of power over long distances and
thus permitting generation to be located remotely from points of demand. Power
transformers have been in use since the late 1880’s and its working principle is wellknown. But still, they are subject to extensive research. One of the main concerns
is to understand and find ways to reduce the magnetic losses associated with the
iron core. Due to the vast amount of transformers in the electrical networks over
the world, this research is essential to the network owners looking to cut down on
their expenses. Some of the recent research can be found in [13], and for a detailed
description on power transformers see [14, 15].
3.4
Electrical motors and generators
Electrical motors convert electric energy into mechanical energy, while generators
operate in the reverse. Some electric motors rely upon a combination of an electromagnet and a permanent magnet, for instance the synchronous motor. Some typical
applications for such motors are washing machines, refrigerators, air conditioning,
fans and automotive applications. We refrain ourselves from going into the details
of the workings of such motors and refer the interested reader to some of the many
textbook on the subject, see for instances [16–20].
3 Applications
3.5
9
Ferromagnetic nanoparticles
The advanced manufacturing processes of today have made it possible to produce
ferromagnetic particles of dimensions in the nanometer (a billionth of a meter) range
with well-defined characteristics and narrow distribution of particle size. Commercial products are available on the web1 . Ferromagnetic particles seem to have an
enormous potential and there are numerous applications. In the field of medical applications, vesicles that contain ferromagnetic particles are used to deliver drugs. Via
an applied external magnetic field one can target drugs to specific locations inside
the human body in order to destroy cancer tumours. Ferromagnetic nanoparticles
are also used in magnetic hyperthermia, which is a recent non-invasive therapeutic alternative principally used as a complementary therapeutic tool in oncology.
In this application, heating of the ferromagnetic particles by an external magnetic
field is utilized. The heat generation results in damage weakening of the tumour
and also making it more sensitive to other therapeutic means. Ferrofluids is another
application of ferromagnetic nanoparticles, which is used in many areas such as sealing, bearing, sensing, and contrast agent in MRI etc. Ferromagnetic nanoparticles
are also an important component in magnetic recording devices and in composite
materials for electromagnetic absorbing applications. To learn more about these
applications see [21–27] and the references therein. Due to the many applications,
research on ferromagnetic nanoparticles has been of great interest over the years
and an understanding of the interaction between these particles and an externally
applied magnetic field is of great importance. In Paper IV we investigated the magnetic properties and losses of ferromagnetic nanoparticles in a composite material.
3.6
Absorbers
In certain situations it is of interest to absorb electromagnetic energy. The absorption can be accomplished by using materials with electric and/or magnetic lossmechanisms that convert electromagnetic energy into heat. Since the main purpose
of an absorber is to dissipate energy, we are now interested in materials with rather
high losses. Typical fields of application for absorbers are: electromagnetic measurements [28], military applications [29], and phased array antennas [30, 31].
Besides good absorption, there are usually additional requirements that need to
be considered when constructing absorbers. For instances, it is often desirable that
the absorber is thin, light, durable and that it performs well over a wide range of
frequencies. In the next section, with these requirements in mind, we will compare
magnetic with electric absorbers and discuss some of the pros and cons with the
different designs.
1
http://nanoprism.net/
10
General Introduction
4 Magnetic absorbers
11
λ/4
Figure 5: Thin magnetic lossy sheet attached to a PEC surface (left) and a thin
electric lossy sheet suspended by a quarter of a wavelength from the PEC surface
(right).
4
Magnetic absorbers
It is common practise to divide absorbers into two categories: Specular and nonspecular absorbers. A specular absorber is used to absorb incoming electromagnetic
waves and thus preventing them from being reflected. This is achieved by clothing
the scattering object with an absorbing material. On the other hand, a nonspecular absorber typically absorbs electromagnetic waves propagating along structures.
Waves propagating along structures are sometimes referred to as surface waves and
they can, like the specular case, be eliminated by covering the structure with an
absorbing material.
4.1
Specular absorbers
In terms of thickness and bandwidth, magnetic media have some features making
them more attractive than their electric counterparts. For example, when reducing
the reflection from a perfect electric conductor (PEC) surface with the aid of a thin
isotropic single layer absorber, it can be shown that a thin magnetically lossy sheet
can be placed directly onto the surface whereas the corresponding electrically lossy
sheet must be suspended a quarter of a wavelength from the surface by using an
additional dielectric layer [29, 32]; see Figure 5. These designs are often referred to as
Salisbury screens. This is a result of the fact that the magnetic and electric field has
their maximum at the PEC surface and at a distance of a quarter of a wavelength
from the PEC, respectively. Therefore, the most efficient placement of a lossy sheet
for absorption is at the maximum of its corresponding field, e.g., a magnetic lossy
sheet should be placed at the maximum of the magnetic field. Provided that the
proper frequency behavior and large enough losses can be achieved, a magnetic
absorber can be made very thin with, in theory, an infinite bandwidth, whereas the
thickness and bandwidth of the electric absorber is limited by the quarter of a wave
length requirement. Hence, magnetic media has the possibility to offer designs with
larger bandwidth and less space occupancy than electric media. However, finding
materials that meet the conditions required of a magnetic Salisbury screen is very
difficult. The required frequency behavior and losses can at most be achieved within
a finite frequency range, resulting in a finite bandwidth.
Based only on the assumption of linear materials and the principle of causality,
12
General Introduction
Rozanov [33] has derived an upper bound for the bandwidth at a specified thickness
and reflection level, for a physically realizable absorber. In this paper he shows the
following relation
Z ∞ ³
1 ´
ln
dλ ≤ 2π 2 µs d
(4.1)
|Γ(λ)|
0
where Γ is the reflection coefficient, λ the free-space wavelength, µs the static permeability of the layer and d the thickness of the layer. This relation shows that
we can obtain low reflection only in a limited band. The bandwidth of a reflection
below Γ0 is then bounded by
2π 2 µs d
f1 − f2
<
f1 f2
c ln |Γ10 |
(4.2)
where we used λ = c/f . This bound also tells us that magnetic absorbers have the
potential of being more broadband than electric absorbers.
Some of the drawbacks of magnetic materials, compared to electric, are that
they usually are heavy and have a large conductivity. A large conductivity leads
to a very small penetration depth of the electromagnetic wave into the material,
resulting in a nearly total reflection of the wave. One way to circumvent this is to use
composite materials with ferromagnetic particles. In this way it might be possible
to reduce both weight and conductivity. Thus, in reality a magnetic absorber will
most likely consist of a composite material with ferromagnetic particles rather than a
bulk material. Further disadvantages with magnetic media are that they usually are
anisotropic and rather difficult to analyze due to the complex nature of ferromagnetic
materials. In Paper II-VI we discuss these problems and study the behavior of
ferromagnetic media in detail, and apply the results to specular absorbers.
4.2
Nonspecular absorbers
The excitation of waves along surfaces sometimes results in unwanted effects. For
instance in phased array antennas surface waves are known to cause scan blindness
[34, 35], and in radar cross section reduction applications surface waves sometimes
produce scattering in undesirable directions [29, p. 227]. In such applications it
is therefore of interest to eliminate these effects and we therefore investigate the
possibility of reducing surface waves by using a thin lossy sheet attached to the
surface, i.e., a nonspecular absorber.
4.2.1
Reduction of surface waves
Surface waves along planar interfaces with isotropic lossy materials placed on top
have been thoroughly investigated, see for instance [36] for a nice introduction.
In [37, 38] surface waves on slabs with magnetic losses are treated and for an analysis
including lossless anisotropic materials see [39]. In this section we briefly review some
of the results for the case of a lossy isotropic media.
4 Magnetic absorbers
13
Figure 6: A lossy slab with thickness d placed on a PEC.
The geometry is depicted in Figure 6 and we study a TM-type surface wave
propagating along the z axis. The transverse field components in the different
regions are: Free space region (y ≥ d):
Hx = eik1 (y−d) eiβz
k1
Ez = Z0 Hx = Z1 Hx
k0
(4.3)
(4.4)
and inside the layer (0 ≤ y ≤ d):
cos(k2 y) iβz
e
(4.5)
cos(k2 d)
k2
Ez = i
Z0 Hx tan(k2 y) = Z2 Hx
(4.6)
k0 ε
q
√
In these equations, k0 = ω ε0 µ0 and Z0 = µε00 are the wave number and impedance
of the free space. The quantities k1 and k2 are the transverse wave numbers of
the wave field outside and inside the layer, respectively. The quantity β is the
longitudinal wave number. The Helmholtz equation yields the following relations
between the wavenumbers
Hx =
k12 + β 2 = k02 ,
k22 + β 2 = k02 εµ
(4.7)
which gives
k22 − k12 = k02 (εµ − 1)
(4.8)
where ε and µ are the relative permittivity and the relative permeability, respectively. Enforcing the boundary conditions for the field components at the interfaces
y = d and y = 0 results in the eigenvalue equation
ik2 tan(k2 d) = εk1
which together with (4.7) turns into the following dispersion relation
q
q
q
2
2
2
2
D(k0 , β) = k0 εµ − β tan(d k0 εµ − β ) + iε k02 − β 2 = 0
(4.9)
(4.10)
14
General Introduction
This result can also be obtained from the equivalent transmission-line circuit for
propagation along the y axis. Replacing the region y ≤ d by a surface impedance
Zs = Z2 at y = d and the free space region by Z1 , and using the zero reflection
condition Zs = Z1 , results in (4.10).
For the slab to support a surface wave it is required that Im(k1 )> 0, which
means that the surface impedance Zs must be inductive. It can be shown that, for
TE surface waves, the surface impedance must be capacitive. Thus, for thin slabs,
where Zs is inductive, there can only exist TM surface waves. In general there are
several solutions to (4.10) corresponding to different surface wave modes. Except
for the first TM mode, all modes, both TM and TE, are associated with a cutoff
frequency below which no propagation occurs. For sufficiently thin slabs, only the
first TM mode exists. Furthermore, in difference to surface waves on lossless slabs,
modes on a lossy slabs also have an upper cutoff frequency [37]. This upper cutoff
frequency for TM waves occurs when the surface impedance changes sign, i.e., going
from inductive to capacitive.
The most efficient damping of TM surface waves is achieved by placing a magnetic lossy sheet in close contact with the PEC where the magnetic field has its
highest value. It is also found in [38] that increasing the magnetic losses is more
efficient than increasing the electric losses if one wishes to achieve large surface
wave attenuation. The damping of the surface wave along the slab is determined by
the imaginary part of the longitudinal wave number. Inserting (4.9) into (4.7) and
solving for β gives
p
β = k0 1 − Zs /Z0
(4.11)
The value of β may be estimated by setting k22 = k02 εµ in Zs . This is approximately
valid for materials with large ε and µ, since the direction of the refracted wave in the
material will be very close to normal for all angles of incidence. This approximation
is used in the following numerical examples.
A plot of the imaginary part of β in dB per free-space wavelength for a surface
wave at 3 GHz is given in Figure 7. The material parameters at this frequency
were set to µ = 2.1 + i1.0 and ε = 20.45 + i0.73, as given in [37]. From this
figure we see that the attenuation rapidly becomes larger as the layer thickness
increase. After reaching a peak value there is a sudden decrease and for sufficiently
large thickness the imaginary part of β becomes negative, showing that the surface
impedance becomes capacitive and the TM surface wave can no longer exist. The
peak value of the attenuation is about 18 dB/λ0 and occurs for a layer thickness
of about 0.032λ0 , which corresponds to 3.2 mm at 3 GHz. Solving (4.10), without
the previous approximation, using the method described in [37], gives a value of the
attenuation of 17.90 dB/λ0 at a layer thickness of 0.032λ0 . Hence, the approximation
made for the data in Figure 7 seems to be legitimate in this case.
Furthermore, in Figure 8 we compare a pure magnetic absorber with a pure
electric absorber. From this result we see that for thin slabs, magnetic losses provide
more efficient damping of surface waves. Thus, magnetic nonspecular absorbers have
the potential of offering designs with less thickness than absorbers with only electric
losses.
4 Magnetic absorbers
15
20
15
10
β′′ (dB/λ0)
5
0
−5
−10
−15
−20
0
0.01
0.02
0.03
d/λ
0.04
0.05
0.06
0
Figure 7: Surface wave attenuation per free space wavelength of propagation as a
function of layer thickness for a thin slab with magnetic losses.
20
β′′ (dB/λ0)
15
10
5
0
0
0.005
0.01
0.015
d/λ0
0.02
0.025
0.03
Figure 8: Comparison of surface wave attenuation between a slab with pure magnetic losses and a slab with pure electric losses.. The solid line shows the attenuation
for a slab with µ = 2.1 and ε = 20.45 + i0.73. The dashed line shows the attenuation
for a slab with µ = 2.1 + i0.73 and ε = 20.45.
16
4.2.2
General Introduction
Surface wave reduction in phased array antennas
As mentioned earlier, the presence of surface waves in phased array antennas may
cause scan blindness. The surface wave is bound to the array surface which means
that no real power enters or leaves the array; the surface wave stores energy and
this typically results in a total reflection at the feed of the antenna.
For certain phased array antennas, scan blindness, i.e., excitation of a surface
wave, is possible whenever the longitudinal wavenumber βSW of the surface wave
coincides with an evanescent Floquet mode propagation constant βFL [34, 35]. Since
2
βSW
= k02 − k12 , we have that βSW /k0 ≥ 1 for lossless slabs. This means that it is
possible to excite surface waves even when the element spacing is small enough to
avoid grating lobes.
The best way to avoid scan blindness is probably to design the antenna so that no
surface waves are excited. This can be done by choosing the proper element spacing,
layer thickness, and material parameters. However, in some situations this might
not be possible. One might not have access to the proper material for instance. In
this case, one may use an absorber to eliminate the surface waves. It is, however,
important to understand that the absorber might affect the radiation properties of
the antenna. Even though the surface waves are suppressed, it might happen that
a substantial amount of power is absorbed rather than radiated. It is therefore
important to investigate how much power that is actually absorbed and how much
that is radiated.
In Paper I, an infinite phased array with spiral elements is investigated. It is
found that when scanning of broadside, resonances in the active reflection coefficient appear. Here we investigate the possibility of eliminating these resonances by
covering the entire antenna with a thin absorbing layer. The numerical calculations
were done using PB-FDTD [40]. Plots of the active reflection coefficient for exactly
the same geometry as given in Figure 1 in Paper I, are shown in Figure 9. It is
seen that a lossy magnetic layer with thickness of 0.6 mm efficiently eliminates the
resonances. The magnetic losses were modeled by a magnetic conductivity, i.e.,
µ = µ0 + iσm /µ0 ω, with σm = 1.0 · 104 Ω/m. This result in a variation of the imaginary part of the permeability between 0.4-4.0 in the frequency range 2-18 GHz,
which is reasonable for ferromagnetic media.
To avoid that the magnetic absorber short circuits the antenna we use a composite material in order to reduce the electric conductivity of the ferromagnetic
material. The amount of magnetic material needed in order to achieve the results in
Figure 9 is estimated by studying the magnetic conductivity in a composite material
with ferromagnetic spherical particles in a non magnetic background material. In
Paper IV an expression for the magnetic conductivity for such a composite material
is found to be
α
(4.12)
σm = µ0 ωS f1
1 + α2
where f1 is the volume fraction of the magnetic material, ωS is the intrinsic precession
angular frequency, and α the Landau-Lifshitz-Gilbert damping constant. Using the
material parameters for iron as given in Paper IV with σm = 1.0 · 104 and α = 0.2,
4 Magnetic absorbers
17
Active Reflection Coefficient
1
0.8
0.6
0.4
0.2
0
2
4
6
8
10
12
Frequency (GHz)
14
16
18
Figure 9: Active reflection coefficient for a phased array with spiral elements, 45
degrees scan in the xz-plane. Solid curve is without absorbing layer and dashed
with a magnetic absorbing layer. Layer thickness is 0.6 mm, µ = 1 + iσm /µ0 ω with
σm = 1.0 · 104 Ω/m, and ε = 1.
solving for f1 in (4.12) gives a volume fraction of
f1 ≈
σm
1.0 · 104
=
≈ 0.11
µ0 ω S α
4π · 10−7 · 2π · 60 · 109 · 0.2
(4.13)
The product f1 d together with the mass density, %m , of the inclusions give the
required amount of magnetic material per surface area. For a 0.6 mm thick absorber
with iron particles, where %m = 7870 kg/m3 [2, p. 24], we obtain
f1 d%m = 0.11 · 6 · 10−4 · 7870 ≈ 0.5 kg/m2
(4.14)
From this we conclude that quite thin magnetic composite materials of low weight
have the potential of providing efficient elimination of surface waves.
In this investigation we covered the whole antenna with a thin absorbing layer.
As mentioned earlier, besides suppressing surface waves this might also result in
some power being absorbed rather than radiated. For real finite antennas it might
be possible to avoid this and eliminate the surface waves by using an absorbing layer
only at the edges of the antenna as suggested in [30].
18
General Introduction
5 Mathematical models of ferromagnetic materials
5
5.1
19
Mathematical models of ferromagnetic materials
The Landau-Lifshitz-Gilbert model
The precise mechanisms of interaction between the magnetic moments in a ferromagnetic material are very complex and still remain obscure. A review of these
mechanisms can be found in [41]. Awaiting a better understanding one often turns
to a phenomenological description of the different effects in a ferromagnetic material. In 1935 Landau and Lifshitz proposed a phenomenological model that describes
the time evolution of the magnetic moment per unit volume [7]. This model has
subsequently been modified. In 1955 Gilbert proposed a refined phenomenological
model of the losses [42]. Both of these models are based on the observation that
the magnetic moment of a charged particle is proportional to its mechanical angular
momentum. Since the time derivative of angular momentum equals the torque, we
end up with the following model of the magnetization
∂M
M
∂M
= −γµ0 M × H eff + α
×
∂t
MS
∂t
(5.1)
where the first term on the right hand side represents a non-dissipative precession
of the magnetization about the effective magnetic field H eff , and the second term,
proportional to α, represents a phenomenological dissipative aligning process of the
magnetization with the effective magnetic field. Furthermore, it is seen that the
right hand side is orthogonal to M , which results in that the magnitude of the
magnetization is preserved, |M | = MS , where MS is the saturation magnetization.
On a mesoscopic scale this means that the individual atomic magnetic moments at
each point in space are aligned so as to produce a locally uniform magnetization
corresponding to the saturation magnetization. On the other hand, on a macroscopic scale, the magnetization can point in different directions, making possible a
macroscopic magnetization of arbitrary strength from zero to the saturation magnetization. In real materials, however, relaxation processes, which do not preserve
|M |, are possible and for this purpose Bloch and Bloembergen introduced yet another form of the damping term [24, 43, 44]. In this thesis however, we a are only
concerned with magnitude preserving processes, i.e., we use the model described by
(5.1).
The constant γ = ge/2me = 1.759 · 1011 C/kg is the gyromagnetic ratio for the
material, where g ≈ 2 is the spectroscopic splitting factor, and e and me are the
charge and mass of the electron, respectively. The dimensionless constant α represents the losses, and is a purely phenomenological constant, i.e., it is not necessarily
associated with a particular loss mechanism.
The effective field H eff is the local field producing the torque on the magnetic
moment. It has several contributions, of which some are of quite different origin
than that of the classical magnetic field described by the Maxwell equations. Besides from the classical magnetic field, the effective field also includes effects like
exchange interactions and magnetocrystalline anisotropy, in order to account for
20
General Introduction
the possibility of domain formation. For a detailed presentation of the different
components included in the effective field we refer to [5, 45–48].
5.2
Small signal model
In many applications it is often of interest to study situations when the magnetic
specimen is subjected to a weak time-varying magnetic field. For this purpose it
is therefore motivated to perform a small signal analysis of the nonlinear LandauLifshitz-Gilbert model [47–50]. It is then assumed that the magnetization only
slightly deviates from a static equilibrium state, i.e., we treat these deviations as
small perturbations. We therefore assume that the classical magnetic field has one
static bias part and one signal part (time convention e−iωt ), with the resulting splitting of the magnetization and the effective field
H = H 0 + H 1 e−iωt ,
M = M 0 + M 1 e−iωt ,
H eff = H eff,0 + H eff,1 e−iωt
(5.2)
where index 0 corresponds to fields constant in time, and time harmonic fields are
indexed by 1.
The result of this decomposition is that the dynamics split in two equations
0 = −γµ0 MS m0 × H eff,0
−iωM 1 = −γµ0 MS [M 1 × H eff,0 /MS + m0 × H eff,1 ] − iωαm0 × M 1
(5.3)
(5.4)
where M 0 = MS m0 , and |m0 | = 1.
5.2.1
The static solution
The first of these equations is part of Brown’s equations in micromagnetics [46, p.
27] and together with the static Maxwell equations and the appropriate boundary conditions it yields the static magnetization M 0 and the static magnetic field
H 0 solutions. The combined equations are nonlinear and difficult to solve even
numerically; this is the field of computational micromagnetics. In general, the resulting magnetization direction m0 will vary within the magnetic particle, see for
instances [51, 52].
In many applications it is possible to simplify the problem by using a model
consisting of a spheroidal particle immersed in a homogeneous external field H e0 .
For this special case, the particle is uniformly magnetized, and the total classical
field within the particle can be shown to be
H 0 = H e0 − MS Nd m0
(5.5)
where Nd is the demagnetization tensor for the particle. A table of demagnetization
tensors for different extremes of spheroidal particles is found in Table 1. Brown’s
equation (5.3) require that the effective field is either parallel to m0 , that is, H eff,0 =
βMS m0 for some scalar β, or H eff,0 = 0.
5 Mathematical models of ferromagnetic materials
Shape

Spherical
Circular needle
Plate
1/3
 0
0
1/2
 0
0
0
0
0
21
N

0
0
1/3 0 
0 1/3
0 0
1/2 0
0 0
0 0
0 0
0 1
Table 1: Different demagnetization tensors for different shapes.
For the case H eff,0 = βMS m0 it is then possible to show that the static magnetization for this particular geometry is given by
m0 = (βI + N)−1 H e0 /MS
(5.6)
where N = Nd + Nc and Nc is the magnetocrystalline anisotropy tensor as defined
in [47, 48]. The scalar β is determined from the normalization requirement |m0 | = 1
and it is seen from (5.6) that it depends on H e0 , MS and the shape of the specimen
via Nd . For an applied field H e0 along the principal axis of the spheroidal particle
the scalar becomes (ignoring crystalline anisotropy)
β=±
|H e0 |
−N
MS
(5.7)
where N is the demagnetization factor corresponding to the principal axis of the
particle coinciding with the applied field. As an example, for the special case
of an
|H e |
applied field along the normal direction of a thin plate we obtain β = ± MS0 − 1. As
seen from this, the static solution M 0 is ambiguous.
Furthermore, the condition H eff,0 = 0 may produce additional solutions not given
by (5.6). For instances, in the case of an applied field along the normal direction
of a thin plate, the condition H eff,0 = 0 yields |H e0 | = ẑ · M 0 = Mz . Since the
magnetization in general has both a normal component Mz and a lateral component
M⊥ with Mz2 + M⊥2 = MS2 , this gives
s
|H e0 |2
e
M 0 = Mz ẑ + M⊥ n̂ = |H 0 |ẑ + MS 1 −
n̂
(5.8)
MS2
where n̂ and ẑ are unit vectors in the lateral and normal direction, respectively.
Though being acceptable, not all solutions of (5.3) are stable (equilibrium) solutions.
The stability of the static solutions will be discussed further shortly, but first we
will briefly review the small signal solution of (5.4).
22
5.2.2
General Introduction
The small signal solution
Having found the static solution, equation (5.4) can be solved to give the following
small signal relation [48]
M 1 = χH 1
(5.9)
Here χ represent a small signal susceptibility
defined by µ = I + χ is given by

µ1 −iµg

µ = iµg µ2
0
0
and the small signal permeability,

0
0
1
(5.10)
where the z axis corresponds to the direction of the zeroth order magnetization M 0 .
The components are
β + Nc,22 − iαω/ωS
2
− (ω/ωS )2
(β + Nc,22 − iαω/ωS )(β + Nc,11 − iαω/ωS ) − Nc,12
β + Nc,11 − iαω/ωS
µ2 (ω) = 1 +
2
(β + Nc,22 − iαω/ωS )(β + Nc,11 − iαω/ωS ) − Nc,12
− (ω/ωS )2
ω/ωS − iNc,12
µg (ω) =
2
(β + Nc,22 − iαω/ωS )(β + Nc,11 − iαω/ωS ) − Nc,12
− (ω/ωS )2
µ1 (ω) = 1 +
(5.11)
(5.12)
(5.13)
where we used that the tensor Nc is symmetric, i.e., Nc,12 = Nc,21 and defined
ωS = γµ0 MS .
For small uniformly magnetized spheroidal particles, where the total small signal
field within the particle is H 1 = H e1 − Nd M 1 , a relation between the small signal
magnetization M 1 and the external field H e1 , i.e., M 1 = γH e1 , can easily be obtained. A short calculation shows that the tensor γ is obtained by the substitution
Nc → Nc + Nd in (5.11)-(5.13) and in terms of stability and resonance frequency,
it is in Paper IV and VI argued that the tensor γ is the appropriate representation
rather than µ.
5.3
Stability of the static magnetization
In order for the small signal solution M 1 to make sense it is important that the
static solution M 0 , which (5.1) is linearized about, represents a stable solution.
Otherwise it is no longer guaranteed that the application of a small signal field H 1
will result in small deviations M 1 from the static solution.
One way to determine the stability of the static solution is to examine the eigenvalues of the γ tensor, and this method is described in Paper VI. In this section we
choose a different approach.
As mentioned before, the static solution is obtained from Brown’s equation (5.3).
However, this equation does not tell us anything about the stability of the solution.
In order to investigate stability, we study the total free energy of the particle and
5 Mathematical models of ferromagnetic materials
23
again we consider the special case of a spheroidal particle. For a uniformly magnetized spheroidal particle with volume V , the total free energy is the sum of the
anisotropy energy Uan and the magnetostatic energy of the particle [45, 46, 53]
Z
µ0
1
Ftot = −
M 0 · (2H e0 − NM 0 ) dV = µ0 MS2 m0 · Nm0 V − µ0 MS m0 · H e0 V
2
2
(5.14)
which should be minimal for a stable solution. From now on we will work with
dimensionless quantities and divide both sides by µ0 MS2 V . For simplicity we ignore
crystalline anisotropy, i.e., Nc = 0 and express Nd with respect to its principal
axes, i.e., Nd is diagonal. We also assume that the particle is rotation symmetric
about the z-direction, i.e., Nx = Ny = N⊥ . With these assumption and using
|m0 |2 = m2x + m2y + m2z = 1, the normalized free energy can be written (neglecting
constant terms which disappear in the minimization procedure) [46]
1
1
Fnorm.,tot = − (N⊥ − Nz ) m2z −m0 ·he0 = − (N⊥ − Nz ) cos2 θ−he0,z cos θ−he0,⊥ sin θ
2
2
(5.15)
e
e
where θ is the angle between the magnetization m0 and the z axis, h0,z and h0,⊥ are
the normalized components of the applied field parallel and perpendicular to the z
axis, respectively. We also observe that, due to the symmetry of the problem, the
equilibrium magnetization lies in the plane defined by the z axis and the applied
field.
5.3.1
Thin plate with an applied field in the normal direction
In Figure 10, plots of the normalized free energy for the special case of a thin plate
with an applied field in the normal direction (+z-direction) are shown. All extremum points are solutions of Brown’s equation (5.3), minimum extremum points
corresponds to stable solutions and maximum extremum points corresponds to unstable solutions. From the figure we see that in the absence of an applied field, the
free energy has its minimum at θ = ±π/2 and maximum at θ = 0, ±π. Thus, in this
case the stable solutions (equilibrium state of the magnetization) are in the lateral
direction of the plate and the unstable solutions are in the normal direction. As
the applied field is increased, the stable solutions move towards θ = 0, which means
that the magnetization corresponding to these solutions starts to pick up a normal
component. Finally, at |H e0 | = MS , the two stable solutions merge with the unstable
solution at θ = 0 to form one stable solution. Also note that there are always two
unstable solutions at θ = ±π. Thus, for field strengths corresponding to |H e0 | ≥ MS ,
the stable solution is always at θ = 0, i.e., in the normal direction and parallel with
the applied field, and the unstable is always anti-parallel with the magnetic field. At
intermediate field strengths 0 ≤ |H e0 | ≤ MS , the solutions in the normal directions
are unstable, while the remaining stable solutions corresponds to a magnetization
with both normal and lateral components. Furthermore, the solutions at θ = 0, ±π
are those corresponding to solutions of (5.3) where H eff,0 = βMS m0 with β given
by (5.7). On the other hand, a static magnetization with a lateral component corresponds to solutions of (5.3) with H eff,0 = 0. From this we conclude that in order
24
-π
-π/2
General Introduction
π/2
π
θ
Figure 10: Normalized free energy of a thin plate (Nz = 1 and N⊥ = 0) with an
applied field in the normal direction. The different curves corresponds to: he0,z = 0
(solid line), he0,z = 0.5 (dashed line), he0,z = 1 (dotted line), and he0,z = 1.5 dashdotted line).
to obtain a stable magnetization in the normal direction of the plate, a very strong
field is required, and the precise conditions for stability is
β=
|H e0 |
−1>0
MS
(5.16)
Note that this condition discards the minus sign in (5.7).
5.3.2
Spherical particle with an applied field in arbitrary direction
For a spherical particle Nz = N⊥ = 1/3, and in this case we see from (5.14) that
in the absences of an applied field, any direction of M 0 corresponds to a minimum
and thus a stable solution. Applying a magnetic field in an arbitrary direction will
result in a stable solution parallel with the applied field and an unstable solution
in the anti-parallel direction. Using (5.6) with N = 1/3 we obtain the following
stability condition
|H e0 |
− 1/3 > −1/3
(5.17)
β=
MS
6 Summary of papers
6
6.1
25
Summary of papers
Paper I
In this paper, we present a numerical analysis of an infinite periodic array of planar
spiral elements with octave bandwidth. For off-broadside scan the array is found to
exhibit very narrow resonances, which are independent of scan angle; see Figure 3
in this paper. They occur when the spiral arms are multiples of half a wavelength,
in which case the current forms a high amplitude standing wave along the spiral
arms as shown in Figure 5. An efficient way to eliminate these resonance is to spoil
the symmetry of the spirals. This is achieved by making the arms unequally long,
resulting in a power reflection < −10 dB for scanning up to 45o in both planes
over the range 6.5-13.0 GHz. Finally, we also discuss the equivalent 3-port for this
nonsymmetrical array element and evaluate the element polarization performance.
It is found that the element is approximately circularly polarized.
6.2
Paper II
In this paper we give a short review on electric and magnetic absorbers. It is argued
that magnetic absorber have the potential of offering more broadband and thinner
designs than their electrical counterpart.
In our analysis of magnetic absorbers we model the magnetization of a ferromagnetic material with the Landau-Lifshitz-Gilbert (LLG) equation. From a small
signal approximation we obtain a small signal permeability which shows that the
material in this approximation behaves as if gyrotropic with a resonant frequency
dependence which can be controlled by a bias field. Furthermore, the reflection
coefficient for normally impinging waves on a PEC covered with a ferromagnetic
material, biased in the normal direction, is calculated. In this case the eigenmodes
in the material are circularly polarized where the eigenvalues of the small signal permeability describes an effective permeability. We found that in general there will be
two distinct resonance frequencies in the reflection coefficient, one associated with
the precession frequency of the magnetization and one associated with the thickness
of the layer. The former of these resonance frequencies can be controlled by the
bias field and for a bias field strength close to the saturation magnetization one can
achieve low reflection (around -20 dB) for a quite large bandwidth (more than two
decades).
6.3
Paper III
In paper II we examined scattering from a PEC covered by a thin magnetic lossy
layer magnetized in the normal direction. It was found that this design required a
very strong bias field, which might be hard to achieve in a practical realization. It
was also found that good absorption could only be achieved for one polarization. In
this paper we extend the analysis to include the case where the magnetic layer now
is magnetized in the lateral direction.
26
General Introduction
A magnetized thin layer mounted on a PEC surface is considered as an alternative
for an absorbing layer. The magnetic material is again modeled with the LandauLifshitz-Gilbert equation, but with a lateral static magnetization that also may have
a periodic variation along one lateral direction. The scattering problem is solved
by means of an expansion into Floquet-modes, a propagator formalism and wavesplitting. Numerical results are presented, and for parameter values close to the
typical values for ferromagnetic materials, reflection coefficients below -20 dB can
be achieved for the fundamental mode over the frequency range 1-4 GHz, for both
polarizations. It is found that the periodicity of the medium makes the reflection
properties for the fundamental mode almost independent of the azimuthal direction
of incidence, for both normally and obliquely incident waves.
6.4
Paper IV
The disadvantages with absorbers of magnetic material are that they tend to be
quite heavy and that they posses a very high electric conductivity which prevents
electromagnetic waves from being transmitted into the material, resulting in a total reflection of the wave. We therefore discuss some of the problems involved in
homogenization of a composite material built from ferromagnetic inclusions in a
nonmagnetic background material.
The small signal permeability for a ferromagnetic spherical particle is combined
with a homogenization formula to give an effective permeability for a biased composite material. The composite material inherits the gyrotropic structure and resonant
behavior of the single particle. The resonance frequency of the composite material is found to be independent of the volume fraction, unlike dielectric composite
materials. The magnetic losses are described by a magnetic conductivity which
can be made independent of frequency and proportional to the volume fraction by
choosing a certain bias. Finally, some concerns regarding particles of small size,
i.e., nanoparticles, are treated and the possibility of exciting exchange modes are
discussed. It is found that it is possible to excite exchange resonance modes in the
gigahertz range, a result that also agrees with experiments. These exchange modes
may be an interesting way to increase losses in composite materials.
6.5
Paper V
In this paper we use the results in Paper IV to analyze a specular absorber made of
a biased magnetic composite material. The composite material consists of ferromagnetic spherical particles embedded in a nonmagnetic and nonconducting background
material. Unfortunately the results in this paper shows that the weight of the RAM
cannot be reduced with the aid of a composite material. Thus, in order to obtain good absorption for a thin magnetic absorber, a certain amount of magnetic
material is required which makes such an absorber quite heavy. However, a lower
bound for the thickness of a composite RAM is obtained, which could be used as a
rule of thumb for engineers designing absorbers. This bound shows that magnetic
absorbers, though heavy, can be made very thin.
References
6.6
27
Paper VI
Finally, in the last paper we discuss stability and causality of effective material
parameters for biased ferromagnetic materials. We demonstrate that the stability and causality of the small signal material parameters describing single domain
ferromagnetic spherical particles depend on which field is taken as the input. The
small signal permeability derived from a linearization of the Landau-Lifshitz-Gilbert
equation describing a ferromagnetic particle is shown to be unstable from which one
might incorrectly infer that causality is violated. Though surprising at first sight,
the explanation is due to the fact that the internal field is not independent of the
magnetization, and stability and causality is recovered when taking the external
field as input. Thus, the small signal susceptibility can not be deemed unphysical
just on account of its poles, which simply reflects the choice of input signal.
When the effective material permeability of a composite material is calculated
with a standard mixing formula, the result is shown to be stable and causal, in spite
of the fact that the calculation was formally made using unstable susceptibilities
describing the particles. This due to the fact that the external field to the particle,
which is the proper input signal, in this case corresponds to the field inside the
composite. The field inside the composites is independent of the magnetization and
thus makes the effective permittivity stable.
References
[1] Plato, Ion. 380 B.C.
[2] C. Kittel, Introduction to Solid State Physics. New York: John Wiley & Sons,
7 ed., 1996.
[3] W. Heisenberg Z. Physik, vol. 49, 1928.
[4] J. H. V. Vleck, “A survey of the theory of ferromagnetism,” Reviews of modern
Physics, vol. 17, no. 1, pp. 27–47, 1945.
[5] C. Kittel, “Physical theory of ferromagnetic domains,” Reviews of modern
Physics, vol. 21, no. 4, pp. 541–583, 1949.
[6] P. Weiss, “L’hypothèse du champ moléculaire et la propriété ferromagnétique,”
J. Physique, vol. 6, pp. 661–690, 1907.
[7] L. D. Landau and E. M. Lifshitz, “On the theory of the dispersion of magnetic
permeability in ferromagnetic bodies,” Physik. Z. Sowjetunion, vol. 8, pp. 153–
169, 1935. Reprinted by Gordon and Breach, Science Publishers, “Collected
Papers of L. D. Landau”, D. Ter Haar, editor, 1965, pp. 101–114.
[8] B. Bhushan, Mechanics and Reliability of Flexible Magnetic Media. New York:
Springer, 2000.
28
General Introduction
[9] H. N. Bertram, Theory of Magnetic Recording. Cambridge, U.K.: Cambridge
University Press, 1994.
[10] S. X. Wang and A. M. Taratorin, Magnetic Information Storage Technology.
Academic Press, 1999.
[11] R. B. Buxton, Introduction to Functional Magnetic Resonance Imaging: Principles and Techniques. Cambridge, U.K.: Cambridge University Press, 2002.
[12] C. Westbrook, C. K. Roth, and J. Talbot, MRI in Practice. Blackwell publishing, 3 ed.
[13] D. Ribbenfjärd, A lumped element transformer model including core losses and
winding impedances. Licentiate thesis, Royal Institute of Technology, Division of
Electromagnetic Engineering, School of Electrical Engineering, Royal Institute
of Technology
Teknikringen 33, SE-100 44 Stockholm, Sweden, 2007.
[14] S. Kulkarni and S. Khaparde, Transformer Engineering: Design and Practice.
New York and Basel: Marcel Dekker Inc., 2004.
[15] M. J. Heathcote, A. C. Franklin, and D. P. Franklin, The J & P Transformer
Book: A Practical Technology of the Power Transformer. Newnes, 13 ed., 2007.
[16] I. M. Gottlieb, Electric Motors & Control Techniques. McGraw-Hill, 2 ed.,
1994.
[17] J. F. Gieras and M. Wing, Permanent Magnet Motor Technology Revised. New
York and Basel: Marcel Dekker Inc., 2 ed., 2002.
[18] H. A. Toliyat and G. B. Kliman, Handbook of Electric Motors. CRC Press,
2004.
[19] R. Miller and E. P. Anderson, Audel Electric Motors. Indianapolis: Wiley pub.,
6 ed., 2004.
[20] A. Hughes, Electric Motors & Drives. Newnes, 3 ed., 2006.
[21] K. Raj and R. Moskowitz, “Commercial applications of ferrofluids,” J. Magn.
Magn. Mater., vol. 85, pp. 233–245, 1990.
[22] S. Odenbach, Magnetoviscous Effects in Ferrofluids. Springer, 2002.
[23] V. B. Bregar, “Advantages of ferromagnetic nanoparticle composites in microwave absorbers,” IEEE Trans. Magnetics, vol. 40, pp. 1679–1684, May 2004.
[24] K. H. J. Buschow, Handbook of Magnetic Materials, vol. 16. Amsterdam: Elsevier Science Publishers, 2006.
[25] V. P. Torchilin, Nanoparticulates as Drug Carriers. Imperial College Press,
2006.
References
29
[26] W. Andrä and H. Nowak, Magnetism in Medicine. Wiley-VHC., 2 ed., 2007.
[27] V. Rotello, Nanoparticles: Building Blocks for Nanotechnology. Springer, 2004.
[28] L. H. Hemming, Electromagnetic Anechoic Chambers: A Fundamental Design
and Specification Guide. New York: Wiley Interscience, 2002.
[29] E. F. Knott, J. F. Shaeffer, and M. T. Tuley, Radar Cross Section. 5601 N.
Hawthorne Way, Raleigh, NC 27613: SciTech Publishing Inc., 2004.
[30] M. Gustafsson, “RCS reduction of integrated antenna arrays with resistive
sheets,” J. Electro. Waves Applic., vol. 20, no. 1, pp. 27–40, 2006.
[31] M. Gustafsson, “Surface integrated dipole arrays with tapered resistive edge
sheets,” J. Electro. Waves Applic., vol. 21, no. 6, pp. 713–718, 2007.
[32] D. Sjöberg, “Circuit analogs for stratified structures,” Tech. Rep.
LUTEDX/(TEAT-7159)/1–18/(2007), Lund University, Department of Electrical and Information Technology, P.O. Box 118, S-221 00 Lund, Sweden, 2007.
http://www.eit.lth.se.
[33] K. N. Rozanov, “Ultimate thickness to bandwidth ratio of radar absorbers,”
IEEE Trans. Antennas Propagat., vol. 48, pp. 1230–1234, Aug. 2000.
[34] D. M. Pozar and D. H. Schaubert, “Scan blindness in infinite phased arrays of
printed dipoles,” IEEE Trans. Antennas Propagat., vol. 32, no. 6, pp. 602–610,
1984.
[35] D. M. Pozar and D. H. Schaubert, “Analysis of an infinite array of rectangular microstrip patches with idealized probe feeds,” IEEE Trans. Antennas
Propagat., vol. 32, no. 10, pp. 1101–1107, 1984.
[36] R. E. Collin, Field Theory of Guided Waves. New York: IEEE Press, second ed.,
1991.
[37] R. T. Ling, J. D. Scholler, and P. Y. Ufimtsev, “The propagation and excitation
of surface waves in an absorbing layer,” Progress In Electromagnetics Research,
vol. 19, pp. 49–91, 1998.
[38] J. H. Richmond, L. Peters, and R. A. Hill, “Surface waves on a lossy planar
ferrite slab,” IEEE Trans. Antennas Propagat., vol. 35, no. 7, pp. 802–808,
1987.
[39] G. Kristensson, “On the generation of surface waves in frequency selective
structures,” Tech. Rep. LUTEDX/(TEAT-7118)/1–38/(2003), Lund Institute
of Technology, Department of Electroscience, P.O. Box 118, S-221 00 Lund,
Sweden, 2003. http://www.es.lth.se.
[40] H. Holter, “Pb-fdtd, version 4.0,” 2008.
30
General Introduction
[41] J. B. Goodenough, “Summary of losses in magnetic materials,” IEEE Trans.
Magnetics, vol. 38, pp. 3398–3408, Sept. 2002.
[42] T. L. Gilbert, “A phenomenological theory of damping in ferromagnetic materials,” IEEE Trans. Magnetics, vol. 40, pp. 3443–3449, Nov. 2004.
[43] F. Bloch, “Nuclear induction,” Physical review, vol. 70, pp. 460–474, 1946.
[44] N. Bloembergen, “Nuclear induction,” Physical review, vol. 78, no. 5, pp. 572–
580, 1950.
[45] W. F. Brown, Micromagnetics. New York: Interscience Publishers, 1963.
[46] M. d’Aquino, Nonlinear Magnetization Dynamics in Thin-films and Nanoparticles. PhD thesis, Universita degli studi di Napoli “Federico II”, Facolta di
Ingegneria, Dec. 2004.
[47] J. Ramprecht and D. Sjöberg, “Biased magnetic materials in RAM applications,” Progress in Electromagnetics Research, vol. 75, pp. 85–117, 2007.
[48] J. Ramprecht and D. Sjöberg, “Magnetic losses in composite materials,” Tech.
Rep. LUTEDX/(TEAT-7169)/1–28/(2008), Lund University, Department of
Electrical and Information Technology, P.O. Box 118, S-221 00 Lund, Sweden,
2008. http://www.eit.lth.se.
[49] R. E. Collin, Foundations for Microwave Engineering. New York: McGraw-Hill,
second ed., 1992.
[50] D. M. Pozar, Microwave Engineering. New York: John Wiley & Sons, third ed.,
2005.
[51] S. Huo, J. Bishop, J. Tucker, W. Rainforth, and H. Davies, “Simulation of
3-D micromagnetic structures in thin iron platelet,” IEEE Trans. Magnetics,
vol. 33, pp. 4170–4172, 1997.
[52] A. Bagneres, “Three-dimensional numerical simulations of stripe chopping in a
magnetic perpendicular material,” IEEE Trans. Magnetics, vol. 35, pp. 4318–
4325, 1999.
[53] L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskiı̆, Electrodynamics of Continuous Media. Oxford: Pergamon, second ed., 1984.