[PDF] Electromagnetic Waves in Media with Ferromagnetic Losses

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Electromagnetic Waves in

Media with Ferromagnetic

Losses

JÄorgen 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 tillstºand av Kungl Tekniska hÄogskolan framlÄagges till o®entlig granskning fÄor avlÄaggande av teknologie doktorsexamen torsdagen den 5 juni 2008 kl 13.15 i sal D3, Kungl Tekniska hÄogskolan, LindstedtsvÄagen 5, Stockholm.

Copyright

c°2008 JÄorgen Ramprecht

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 interac- tion with electromagnetic ¯elds. The understanding of these interactions and the associated loss mechanisms is therefore crucial for the improvement and future de- velopment 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 con- ditions 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 mate- rial. In di®erence to dielectric Lorentz material, this resonance frequency can be shifted with the aid of a bias ¯eld. For a speci¯c bias ¯eld 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 de¯ne a magnetic conductivity which is independent of frequency. We treat composite materials built from ferromagnetic inclusions in a nonmag- netic and nonconductinig background material. The composite material inherits the gyrotropic structure and resonant behavior of the single particle. The resonance fre- quency 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 o®er a way to increase the losses in a composite ma- terial. We also discuss some concerns regarding stability and causality of e®ective material parameters for biased ferromagnetic materials. Finally, we discuss the possibility of using magnetic materials in absorbing ap- plications. 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 wave- length 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 e±cient absorption seems to be substantially less than for specular absorbers. Thus, as nonspecular absorbers, magnetic lossy materials could o®er very thin and light designs. iii iv

List of papers

This thesis consists of a General Introduction and the following scienti¯c papers: I. Hans Steyskal, JÄorgen Ramprecht and Henrik Holter. Spiral Elements for Broad-Band Phased Arrays.IEEE Transactions on Antennas and Propaga- tion, Vol. 53, No. 8, pp 2558-2562, August 2005. II. JÄorgen Ramprecht and Daniel SjÄoberg. Biased magnetic materials in RAM applications.Progress in Electromagnetics Research, vol. 75, pp. 85-117, 2007. III. JÄorgen Ramprecht, Martin Norgren and Daniel SjÄoberg. Scattering from a thin magnetic layer with a periodic lateral magnetization: application to electromagnetic absorbers. Submitted toProgress in Electromagnetics

Research.

IV. JÄorgen Ramprecht and Daniel SjÄoberg. Magnetic losses in composite materi- als. Accepted for publication inJournal of Physics D: Applied Physics. V. JÄorgen Ramprecht and Daniel SjÄoberg. 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 SjÄoberg, JÄorgen Ramprecht and Niklas Wellander. Stability and causal- ity of e®ective 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 SjÄoberg 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 collabora- tion. 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 ¯eld 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 experi- ence. 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 ¯ka bordet, the pubs and all the activities. In addition, during my collaboration with Daniel SjÄoberg I have also had the opportunity to spend time with the Electro- magnetic 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 o±ce, 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v The candidate's contribution to the included papers . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1 Introduction 1

1.1 A ¯rst approach towards magnetism and magnetic losses . . . . . . . 2

2 Ferromagnetism 5

2.1 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Applications 7

3.1 Magnetic recording . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Magnetic resonance imaging (MRI) . . . . . . . . . . . . . . . . . . . 8

3.3 Transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.4 Electrical motors and generators . . . . . . . . . . . . . . . . . . . . . 8

3.5 Ferromagnetic nanoparticles . . . . . . . . . . . . . . . . . . . . . . . 9

3.6 Absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Magnetic absorbers 11

4.1 Specular absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.2 Nonspecular absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.2.1 Reduction of surface waves . . . . . . . . . . . . . . . . . . . . 12

4.2.2 Surface wave reduction in phased array antennas . . . . . . . 16

5 Mathematical models of ferromagnetic materials 19

5.1 The Landau-Lifshitz-Gilbert model . . . . . . . . . . . . . . . . . . . 19

5.2 Small signal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.2.1 The static solution . . . . . . . . . . . . . . . . . . . . . . . . 20

5.2.2 The small signal solution . . . . . . . . . . . . . . . . . . . . . 22

5.3 Stability of the static magnetization . . . . . . . . . . . . . . . . . . . 22

5.3.1 Thin plate with an applied ¯eld in the normal direction . . . . 23

5.3.2 Spherical particle with an applied ¯eld in arbitrary direction . 24

6 Summary of papers 25

6.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6.5 Paper V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6.6 Paper VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

xi xii

1 Introduction 1

N S Figure 1: A compass needle aligning itself with Earth's magnetic ¯eld.

1 Introduction

Magnetism is one of mankind's earliest scienti¯c 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 ¯rst 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 de- veloped. This has resulted in magnetism nowadays being a huge industry with enormous ¯nancial 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 mag- net 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 ¯gure the direction of the compass needles are randomly distributed in the absence of a magnetic ¯eld. When a magnetic ¯eld is applied the compass needles align themselves along the ¯eld, as illustrated in the right ¯gure.

1.1 A ¯rst approach towards magnetism and magnetic losses

One of the ¯rst 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 ¯eld. On the Earth's surface this magnetic ¯eld is everywhere approximately pointing towards north, see Figure 1. Thus, we conclude thatthe magnetic compass needle aligns itself with the direction of the magnetic ¯eld. At this point we don't care what the magnetic ¯elds 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 calledmagnetic momentthat may be viewed as a tiny compass needle; see Figure 2. In general these compass needles assume random di- rections and when added up, their contributions cancel each other. But if a magnetic ¯eld is applied, all of the compass needles start to align along the magnetic ¯eld and together they may be represented by a total compass needle for the material. Materials that responds to magnetic ¯elds in this way are said to bemagneticand when the atoms are aligned the material ismagnetized. The total compass needle then represents thedirection of the magnetizationof the material. As the magnetic ¯eld 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 ¯eld. The most familiar among such ma- terials are probably those classi¯ed asferromagneticor more commonly referred to as permanent magnets. They way a material responds to an electromagnetic ¯eld 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 havemagnetic 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 °uid, where the °uid represents the surrounding media. As the compass needle turns in the °uid 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 ¯eld. In this sense the magnetic ¯eld loses energy to the magnetic material, hence the terminology magnetic losses. The simpli¯ed illustration given here hardly provides us with the answers on what magnetism really is, but it may give some physical insight into how a mag- netic material is magnetized and how energy is dissipated in the material. From this simple model we have learned that a magnetic ¯eld interacts with a magnetic material in such fashion that the atoms aligns in a certain way to produce a mag- netization, 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 ¯elds and ferromagnetic materials. We also investigate the possibil- ity 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 di®erent domains are not parallel. When applying an external magnetic ¯eld, domains which are already closely aligned to the ¯eld grow on the expense of others. As a result a net magnetization is achieved. This magnetization remains even when the external magnetic ¯eld 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 ¯eld. Such materials are typically classi¯ed as ferromagnetic or ferrimagnetic materials [2]. In this section we brie°y 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. Ferromag- netic e®ects 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 ¯eld. Thus, ferromagnetic materials seem to have a spontaneous magnetization where the alignment of the magnetic moments are \frozen in". At ¯rst, 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 interac- tion is due to an internal magnetic ¯eld,the Weiss ¯eld, that aligns the magnetic moments. Heisenberg showed that this ¯eld is the result of the quantum mechani-

6 General Introduction

cal exchange interaction, which has no classical analog [3{5]. This Wiess ¯eld, also namedthe exchange ¯eld, may be viewed as a magnetic ¯eld aligning the magnetic moments in the material and thus producing the spontaneous magnetization. Never- theless, despite the existence of permanent magnets, our everyday experience tells us that some iron objects donotappear to have any magnetic e®ects. To explain this, the domain theorywas 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 di®erent 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 unmagne- tized. The direction of the magnetization in a domain in this case is determined bythe magnetocrystalline anisotropy ¯eld. This ¯eld 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 ¯eld and the anisotropy ¯eld acts as if classical magnetic ¯elds. But both of them are of quantum mechanical origin and do therefore not enter into the Maxwell equations. The e®ect of the exchange ¯eld and the anisotropy ¯eld 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 in- teraction 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 ¯eld. The magnetization process takes place by essentially two independent processes. For weak applied ¯elds domains favorable oriented with respect to the ¯eld 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 ¯elds this

process becomes irreversible, resulting in some net magnetization even when the ¯eld is removed. For strong enough ¯elds the material will ¯nally consist approximately of one domain. At this point the domain magnetization rotates (domain rotation) toward the direction of the ¯eld. When the alignment is complete the material is said to be saturated and the net magnetization is calledthe saturation magneti- zation. Removal of the applied ¯eld 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 ofhysteresis. This basically means that the magnetization depends on the complete history of the excitation and not just the ¯nal value. Another typical property of ferromagnetic materials is that they arenonlinear,i.e., the magnetization depends on the applied magnetic ¯eld in a nonlinear way.

3 Applications 7

(a) Transformer with an iron core (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 di±cult 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 di®erent 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 °oppy 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 P°eumer ¯led a patent for coating iron particles onto a strip of paper as a recording medium. In 1957, IBM introduced the ¯rst 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 apply- ing a magnetic ¯eld we can control the direction of the magnetization in the recording ferromagnetic material and when removing the ¯eld, the material \remembers" this direction. In this fashion we can magnetize di®erent parts of the material in di®erent directions and thus store information with the aid of the remanent magnetization. For further details on magnetic recording, see [9,10].

8 General Introduction

3.2 Magnetic resonance imaging (MRI)

Magnetic resonance imaging is a widely used diagnostic technique. In this appli- cation a conventional permanent magnet made from ferromagnetic material can be used to create a quite strong magnetic ¯eld. This ¯eld is applied to the patient in order to align the atoms in the body. Then, while maintaining this strong ¯eld, a weaker magnetic ¯eld is applied. The weak magnetic ¯eld pushes the atoms out of alignment and di®erent parts of the body are a®ected in di®erent ways. In this way di®erent 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 ¯eld. But MRIs using permanent magnets are very inexpensive to maintain. To achieve larger ¯eld 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 ¯eld of transmission and distribution of electrical power. Here transformers are used to transform the electrical power to a high-voltage/low- current 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 well- known. But still, they are subject to extensive research. One of the main concerns is to understand and ¯nd 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 electro- magnet 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 9

3.5 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-de¯ned characteristics and narrow distribution of particle size. Commer- cial products are available on the web

1. Ferromagnetic particles seem to have an

enormous potential and there are numerous applications. In the ¯eld of medical ap- plications, vesicles that contain ferromagnetic particles are used to deliver drugs. Via an applied external magnetic ¯eld one can target drugs to speci¯c 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 therapeu- tic alternative principally used as a complementary therapeutic tool in oncology. In this application, heating of the ferromagnetic particles by an external magnetic ¯eld is utilized. The heat generation results in damage weakening of the tumour and also making it more sensitive to other therapeutic means. Ferro°uids is another application of ferromagnetic nanoparticles, which is used in many areas such as seal- ing, 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 ¯eld is of great importance. In Paper IV we investigated the mag- netic 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 absorp- tion can be accomplished by using materials with electric and/or magnetic loss- mechanisms 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 highlosses. Typical ¯elds of application for absorbers are: electromagnetic mea- surements [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 di®erent 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:Specularandnon- specularabsorbers. A specular absorber is used to absorb incoming electromagnetic waves and thus preventing them from being re°ected. This is achieved by clothing the scattering object with an absorbing material. On the other hand, a nonspecu- lar absorber typically absorbs electromagnetic waves propagating along structures. Waves propagating along structures are sometimes referred to assurface wavesand 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 re°ection 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 ¯eld has their maximum at the PEC surface and at a distance of a quarter of a wavelength from the PEC, respectively. Therefore, the most e±cient placement of a lossy sheet for absorption is at the maximum of its corresponding ¯eld,e.g., a magnetic lossy sheet should be placed at the maximum of the magnetic ¯eld. 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 in¯nite 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 o®er designs with larger bandwidth and less space occupancy than electric media. However, ¯nding materials that meet the conditions required of a magnetic Salisbury screen is very di±cult. The required frequency behavior and losses can at most be achieved within a ¯nite frequency range, resulting in a ¯nite 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 speci¯ed thickness and re°ection level, for a physically realizable absorber. In this paper he shows the following relationZ1 0 ln³1 j¡(¸)j´ d¸·2¼2¹sd(4.1) where ¡ is the re°ection coe±cient,¸the free-space wavelength,¹sthe static per- meability of the layer anddthe thickness of the layer. This relation shows that we can obtain low re°ection only in a limited band. The bandwidth of a re°ection below ¡

0is then bounded by

f

1¡f2

f

1f2<2¼2¹sd

cln1 j¡0j(4.2) where we used¸=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 re°ection 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 di±cult 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 e®ects. 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 e®ects 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 brie°y review some of the results for the case of a lossy isotropic media.

4 Magnetic absorbers 13?

???? ? ? ????? ?  Figure 6: A lossy slab with thicknessdplaced on a PEC. The geometry is depicted in Figure 6 and we study a TM-type surface wave propagating along thezaxis. The transverse ¯eld components in the di®erent regions are: Free space region (y¸d): H x= eik1(y¡d)ei¯z(4.3) E z=k1 k 0Z

0Hx=Z1Hx(4.4)

and inside the layer (0·y·d): H x=cos(k2y) cos(k2d)ei¯z(4.5) E z= ik2 k

0"Z0Hxtan(k2y) =Z2Hx(4.6)

In these equations,k0=!p

"

0¹0andZ0=q

¹ 0 "

0are the wave number and impedance

of the free space. The quantitiesk1andk2are the transverse wave numbers of the wave ¯eld outside and inside the layer, respectively. The quantity¯is the longitudinal wave number. The Helmholtz equation yields the following relations between the wavenumbers k

21+¯2=k20; k22+¯2=k20"¹(4.7)

which gives k

22¡k21=k20("¹¡1) (4.8)

where"and¹are the relative permittivity and the relative permeability, respec- tively. Enforcing the boundary conditions for the ¯eld components at the interfaces y=dandy= 0 results in the eigenvalue equation ik2tan(k2d) ="k1(4.9) which together with (4.7) turns into the following dispersion relation

D(k0;¯) =q

k

20"¹¡¯2tan(dq

k

20"¹¡¯2) + i"q

k

20¡¯2= 0 (4.10)

14 General Introduction

This result can also be obtained from the equivalent transmission-line circuit for propagation along theyaxis. Replacing the regiony·dby a surface impedance Z s=Z2aty=dand the free space region byZ1, and using the zero re°ection conditionZs=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 impedanceZsmust be inductive. It can be shown that, for TE surface waves, the surface impedance must be capacitive. Thus, for thin slabs, whereZsis inductive, there can only exist TM surface waves. In general there are several solutions to (4.10) corresponding to di®erent surface wave modes. Except for the ¯rst TM mode, all modes, both TM and TE, are associated with a cuto® frequency below which no propagation occurs. For su±ciently thin slabs, only the ¯rst TM mode exists. Furthermore, in di®erence to surface waves on lossless slabs, modes on a lossy slabs also have an upper cuto® frequency [37]. This upper cuto® frequency for TM waves occurs when the surface impedance changes sign,i.e., going from inductive to capacitive. The most e±cient damping of TM surface waves is achieved by placing a mag- netic lossy sheet in close contact with the PEC where the magnetic ¯eld has its highest value. It is also found in [38] that increasing the magnetic losses is more e±cient 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¯gives

¯=k0p

1¡Zs=Z0(4.11)

The value of¯may be estimated by settingk22=k20"¹inZs. This is approximately valid for materials with large"and¹, 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¯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 to¹= 2:1 + i1:0 and"= 20:45 + i0:73, as given in [37]. From this ¯gure 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 su±ciently large thickness the imaginary part of¯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=¸0and occurs for a layer thickness of about 0:032¸0, 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=¸0at a layer thickness of 0:032¸0. 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 e±cient damping of surface waves. Thus, magnetic nonspecular absorbers have the potential of o®ering designs with less thickness than absorbers with only electric losses.

4 Magnetic absorbers 1500.010.020.030.040.050.06-20

-15 -10 -5 0 5 10 15 20 d/λ0

β′′ (dB/λ0)

Figure 7: Surface wave attenuation per free space wavelength of propagation as a function of layer thickness for a thin slab with magnetic losses.

00.0050.010.0150.020.0250.030

5 10 15 20 d/λ0

β′′ (dB/λ0)

Figure 8: Comparison of surface wave attenuation between a slab with pure mag- netic losses and a slab with pure electric losses.. The solid line shows the attenuation for a slab with¹= 2:1 and"= 20:45+i0:73. The dashed line shows the attenuation for a slab with¹= 2:1 + i0:73 and"= 20:45.

16 General Introduction

4.2.2 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 re°ection 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¯SWof the surface wave coincides with an evanescent Floquet mode propagation constant¯FL[34,35]. Since ¯

2SW=k20¡k21, we have that¯SW=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 a®ect 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 in¯nite phased array with spiral elements is investigated. It is found that when scanning of broadside, resonances in the active re°ection coe±- cient 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 re°ection coe±cient 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 e±ciently eliminates the resonances. The magnetic losses were modeled by a magnetic conductivity,i.e., ¹=¹0+ i¾m=¹0!, with¾m= 1:0¢104=m. This result in a variation of the imag- inary 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 com- posite 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 ¾ m=¹0!Sf1®

1 +®2(4.12)

wheref1is the volume fraction of the magnetic material,!Sis the intrinsic precession angular frequency, and®the Landau-Lifshitz-Gilbert damping constant. Using the material parameters for iron as given in Paper IV with¾m= 1:0¢104and®= 0:2,

4 Magnetic absorbers 17246810121416180

0.2 0.4 0.6 0.8 1

Active Reflection Coefficient

Frequency (GHz)

Figure 9: Active re°ection coe±cient 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,¹= 1 + i¾m=¹0!with ¾ m= 1:0¢104=m, and"= 1. solving forf1in (4.12) gives a volume fraction of f

1¼¾m

¹

0!S®=1:0¢104

4¼¢10¡7¢2¼¢60¢109¢0:2¼0:11 (4.13)

The productf1dtogether 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= 7870kg=m3[2, p. 24], we obtain f

1d%m= 0:11¢6¢10¡4¢7870¼0:5kg=m2(4.14)

From this we conclude that quite thin magnetic composite materials of low weight have the potential of providing e±cient 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 ¯nite 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 19

5 Mathematical models of ferromagnetic materi-

als

5.1 The Landau-Lifshitz-Gilbert model

The precise mechanisms of interaction between the magnetic moments in a ferro- magnetic 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 di®erent e®ects in a ferromagnetic mater- ial. 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 modi¯ed. In 1955 Gilbert proposed a re¯ned 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 @t =¡°¹0M£He®+®M M

S£@M

@t (5.1) where the ¯rst term on the right hand side represents a non-dissipative precession of the magnetization about the e®ective magnetic ¯eldHe®, and the second term, proportional to®, represents a phenomenological dissipative aligning process of the magnetization with the e®ective magnetic ¯eld. Furthermore, it is seen that the right hand side is orthogonal toM, which results in that the magnitude of the magnetization is preserved,jMj=MS, whereMSis 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 macro- scopic scale, the magnetization can point in di®erent directions, making possible a macroscopic magnetization of arbitrary strength from zero to the saturation mag- netization. In real materials, however, relaxation processes, which do not preserve jMj, are possible and for this purpose Bloch and Bloembergen introduced yet an- other 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°=ge=2me= 1:759¢1011C=kg is the gyromagnetic ratio for the material, whereg¼2 is the spectroscopic splitting factor, andeandmeare the charge and mass of the electron, respectively. The dimensionless constant®repre- sents the losses, and is a purely phenomenological constant,i.e., it is not necessarily associated with a particular loss mechanism. The e®ective ¯eldHe®is the local ¯eld producing the torque on the magnetic moment. It has several contributions, of which some are of quite di®erent origin than that of the classical magnetic ¯eld described by the Maxwell equations. Be- sides from the classical magnetic ¯eld, the e®ective ¯eld also includes e®ects 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 di®erent components included in the e®ective ¯eld 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 ¯eld. For this purpose it is therefore motivated to perform a small signal analysis of the nonlinear Landau- Lifshitz-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 ¯eld has one static bias part and one signal part (time convention e

¡i!t), with the resulting split-

ting of the magnetization and the e®ective ¯eld H=H0+H1e¡i!t;M=M0+M1e¡i!t;He®=He®;0+He®;1e¡i!t(5.2) where index 0 corresponds to ¯elds constant in time, and time harmonic ¯elds are indexed by 1. The result of this decomposition is that the dynamics split in two equations

0=¡°¹0MSm0£He®;0(5.3)

¡i!M1=¡°¹0MS[M1£He®;0=MS+m0£He®;1]¡i!®m0£M1(5.4) whereM0=MSm0;andjm0j= 1.

5.2.1 The static solution

The ¯rst of these equations is part ofBrown's equationsin micromagnetics [46, p.

27] and together with the static Maxwell equations and the appropriate bound-

ary conditions it yields the static magnetizationM0and the static magnetic ¯eld H

0solutions. The combined equations are nonlinear and di±cult to solve even

numerically; this is the ¯eld of computational micromagnetics. In general, the re- sulting magnetization directionm0will 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 ¯eldHe 0. For this special case, the particle is uniformly magnetized, and the total classical

¯eld within the particle can be shown to be

H 0=He

0¡MSNdm0(5.5)

whereNdis the demagnetization tensor for the particle. A table of demagnetization tensors for di®erent extremes of spheroidal particles is found in Table 1. Brown's equation (5.3) require that the e®ective ¯eld is either parallel tom0, that is,He®;0= ¯M

Sm0for some scalar¯, orHe®;0= 0.

5 Mathematical models of ferromagnetic materials 21

Shape N

Spherical

0 @1=3 0 0

0 1=3 0

0 0 1=31

A

Circular needle

0 @1=2 0 0

0 1=2 0

0 0 01

A Plate 0 @0 0 0 0 0 0

0 0 11

A Table 1: Di®erent demagnetization tensors for di®erent shapes. For the caseHe®;0=¯MSm0it is then possible to show that the static magne- tization for this particular geometry is given by m

0= (¯I+N)¡1He

0=MS(5.6)

whereN=Nd+NcandNcis the magnetocrystalline anisotropy tensor as de¯ned in [47,48]. The scalar¯is determined from the normalization requirementjm0j= 1 and it is seen from (5.6) that it depends onHe

0,MSand the shape of the specimen

viaNd. For an applied ¯eldHe

0along the principal axis of the spheroidal particle

the scalar becomes (ignoring crystalline anisotropy)

¯=§jHe

0j M

S¡N(5.7)

whereNis the demagnetization factor corresponding to the principal axis of the particle coinciding with the applied ¯eld. As an example, for the special case of an applied ¯eld along the normal direction of a thin plate we obtain¯=§jHe0j M

S¡1. As

seen from this, the static solutionM0is ambiguous. Furthermore, the conditionHe®;0= 0 may produce additional solutions not given by (5.6). For instances, in the case of an applied ¯eld along the normal direction of a thin plate, the conditionHe®;0= 0 yieldsjHe

0j=^z¢M0=Mz. Since the

magnetization in general has both a normal componentMzand a lateral component M ?withM2z+M2?=M2S, this gives M

0=Mz^z+M?^n=jHe

0j^z+MSs

1¡jHe

0j2 M

2S^n(5.8)

where ^nand^zare 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 ¯rst we will brie°y review the small signal solution of (5.4).

22 General Introduction

5.2.2 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=ÂH1(5.9)

HereÂrepresent a small signal susceptibility and the small signal permeability, de¯ned by¹=I+Âis given by

¹=0

@¹

1¡i¹g0

i¹g¹20

0 0 11

A (5.10) where thezaxis corresponds to the direction of the zeroth order magnetizationM0.

The components are

¹

1(!) = 1 +¯+Nc;22¡i®!=!S

(¯+Nc;22¡i®!=!S)(¯+Nc;11¡i®!=!S)¡N2c;12¡(!=!S)2(5.11) ¹

2(!) = 1 +¯+Nc;11¡i®!=!S

(¯+Nc;22¡i®!=!S)(¯+Nc;11¡i®!=!S)¡N2c;12¡(!=!S)2(5.12) ¹ g(!) =!=!S¡iNc;12 (¯+Nc;22¡i®!=!S)(¯+Nc;11¡i®!=!S)¡N2c;12¡(!=!S)2(5.13) where we used that the tensorNcis symmetric,i.e.,Nc;12=Nc;21and de¯ned !

S=°¹0MS.

For small uniformly magnetized spheroidal particles, where the total small signal

¯eld within the particle isH1=He

1¡NdM1, a relation between the small signal

magnetizationM1and the external ¯eldHe

1,i.e.,M1=°He

1, can easily be ob-

tained. A short calculation shows that the tensor°is obtained by the substitution N c!Nc+Ndin (5.11)-(5.13) and in terms of stability and resonance frequency, it is in Paper IV and VI argued that the tensor°is the appropriate representation rather than¹.

5.3 Stability of the static magnetization

In order for the small signal solutionM1to make sense it is important that the static solutionM0, which (5.1) is linearized about, represents a stable solution. Otherwise it is no longer guaranteed that the application of a small signal ¯eldH1 will result in small deviationsM1from the static solution. One way to determine the stability of the static solution is to examine the eigen- values of the°tensor, and this method is described in Paper VI. In this section we choose a di®erent 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 mag- netized spheroidal particle with volumeV, the total free energy is the sum of the anisotropy energyUanand the magnetostatic energy of the particle [45,46,53] F tot=¡¹0 2 Z M

0¢(2He

0¡NM0)dV=1

2

¹0M2Sm0¢Nm0V¡¹0MSm0¢He

0V (5.14) which should be minimal for a stable solution. From now on we will work with dimensionless quantities and divide both sides by¹0M2SV. For simplicity we ignore crystalline anisotropy,i.e.,Nc=0and expressNdwith respect to its principal axes,i.e.,Ndis diagonal. We also assume that the particle is rotation symmetric about thez-direction,i.e.,Nx=Ny=N?. With these assumption and using jm0j2=m2x+m2y+m2z= 1, the normalized free energy can be written (neglecting constant terms which disappear in the minimization procedure) [46] F norm:;tot=¡1 2 (N?¡Nz)m2z¡m0¢he

0=¡1

2 (N?¡Nz)cos2µ¡he0;zcosµ¡he0;?sinµ (5.15) whereµis the angle between the magnetizationm0and thezaxis,he0;zandhe0;?are the normalized components of the applied ¯eld parallel and perpendicular to thez axis, respectively. We also observe that, due to the symmetry of the problem, the equilibrium magnetization lies in the plane de¯ned by thezaxis and the applied

¯eld.

5.3.1 Thin plate with an applied ¯eld in the normal direction

In Figure 10, plots of the normalized free energy for the special case of a thin plate with an applied ¯eld in the normal direction (+z-direction) are shown. All ex- tremum points are solutions of Brown's equation (5.3), minimum extremum points corresponds to stable solutions and maximum extremum points corresponds to un- stable solutions. From the ¯gure we see that in the absence of an applied ¯eld, the free energy has its minimum atµ=§¼=2 and maximum atµ= 0;§¼. 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 ¯eld is increased, the stable solutions move towardsµ= 0, which means that the magnetization corresponding to these solutions starts to pick up a normal component. Finally, atjHe

0j=MS, the two stable solutions merge with the unstable

solution atµ= 0 to form one stable solution. Also note that there are always two unstable solutions atµ=§¼. Thus, for ¯eld strengths corresponding tojHe

0j ¸MS,

the stable solution is always atµ= 0,i.e., in the normal direction and parallel with the applied ¯eld, and the unstable is always anti-parallel with the magnetic ¯eld. At intermediate ¯eld strengths 0· jHe

0j ·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 atµ= 0;§¼ are those corresponding to solutions of (5.3) whereHe®;0=¯MSm0with¯given by (5.7). On the other hand, a static magnetization with a lateral component cor- responds to solutions of (5.3) withHe®;0= 0. From this we conclude that in order

24 General Introduction-π-π/2

?π/2π ? ? ? ?? ? ? ?? ? ? ?? ? ? ?? ? θ ? ? ? ? ? Figure 10: Normalized free energy of a thin plate (Nz= 1 andN?= 0) with an applied ¯eld in the normal direction. The di®erent curves corresponds to:he0;z= 0 (solid line),he0;z= 0:5 (dashed line),he0;z= 1 (dotted line), andhe0;z= 1:5 dash- dotted line). to obtain a stable magnetization in the normal direction of the plate, a very strong ¯eld is required, and the precise conditions for stability is

¯=jHe

0j M

S¡1>0 (5.16)

Note that this condition discards the minus sign in (5.7).

5.3.2 Spherical particle with an applied ¯eld in arbitrary direction

For a spherical particleNz=N?= 1=3, and in this case we see from (5.14) that in the absences of an applied ¯eld, any direction ofM0corresponds to a minimum and thus a stable solution. Applying a magnetic ¯eld in an arbitrary direction will result in a stable solution parallel with the applied ¯eld and an unstable solution in the anti-parallel direction. Using (5.6) withN= 1=3 we obtain the following stability condition

¯=jHe

0j M

S¡1=3>¡1=3 (5.17)

6 Summary of papers 25

6 Summary of papers

6.1 Paper I

In this paper, we present a numerical analysis of an in¯nite periodic array of planar spiral elements with octave bandwidth. For o®-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 e±cient 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 re°ection<¡10 dB for scanning up to 45oin 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 o®ering more broadband and thinner designs than their electrical counterpart. In our analysis of magnetic absorbers we model the magnetization of a ferro- magnetic 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 ¯eld. Furthermore, the re°ection coe±cient 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 per- meability describes an e®ective permeability. We found that in general there will be two distinct resonance frequencies in the re°ection coe±cient, 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 ¯eld and for a bias ¯eld strength close to the saturation magnetization one can achieve low re°ection (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 ¯eld, 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 Landau- Lifshitz-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 wave- splitting. Numerical results are presented, and for parameter values close to the typical values for ferromagnetic materials, re°ection coe±cients 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 re°ection 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 to- tal re°ection 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 e®ective permeability for a biased compos- ite material. The composite material inherits the gyrotropic structure and resonant behavior of the single particle. The resonance frequency of the composite mater- ial 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 ferromag- netic spherical particles embedded in a nonmagnetic and nonconducting background material. Unfortunately the results in this paper shows that the weight of