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Characterization and Study of Ferromagnetic Resonance of Micro and Nano Ferrites at Microwave and Millimeter waves

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Characterization and Study of Ferromagnetic
Resonance of Micro and Nano Ferrites at
Microwave and Millimeter waves
A dissertation
submitted by
Anjali Sharma
In partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
in
Electrical Engineering
TUFTS UNIVERSITY
February, 2014
ADVISER: Dr. M.N. Afsar
UMI Number: 3612824
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UMI 3612824
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Abstract
Ferrites have been used in magnetic recording devices, isolators, circulators and as
permanent magnets for a long time. However, the recent developments in semiconductor
technology and nanoelectronics have led to a significant growth in high frequency applications.
As the device size becomes smaller and applications shift towards millimeter wavelengths, the
need for novel materials capable of handling high frequencies has increased. This has generated
interest in the study of materials in the nanoscale domain. Nanoferrites are being extensively
studied for their potential application as high frequency absorbers, information storage media,
circulators, isolators, etc. Other areas of growing interest for these materials include biomedical
engineering, alternate energy, aerospace engineering and defense applications.
Nanoferrites consist of metal substituted iron oxide nanopowders that have average
particle size below 100 nm. At these dimensions the domain wall resonance can be avoided since
materials can exist in single domain state and thus such materials prove useful for high frequency
applications. Nano-size materials have unique mechanical, electrical and magnetic properties.
The unique properties of nanomaterials could be attributed to their structure which is close to
that of an isolated atom or molecule. The properties of nanomaterials may not necessarily be
predicted from those observed at larger scales. In fact, the electromagnetic properties of
materials are known to change as particle sizes are reduced to the nanoscale. Consequently, it is
necessary to characterize these materials in order to understand their behavior and better predict
their potential use in high frequency applications.
A waveguide based set-up has been used here to perform transmission and reflection
measurements on powdered nano-ferrite samples at microwave frequencies using a vector
network analyzer. This measurement set-up is capable of accurately measuring the material
1
properties in terms of s-parameters in the frequency range from 2 GHz to 40 GHz. The
electromagnetic properties, namely, magnetic permeability and electric permittivity are derived
from these parameters. The algorithm has been specifically written to calculate the real and
imaginary parts of permittivity and permeability of the powdered nano-ferrite samples.
The measurements were also performed on micro-size samples to understand the
dependence of material properties on particle dimensions. In order to verify the observed
difference in the micro- and nano-size samples, the same powders were also analyzed by another
technique at higher frequencies. Transmittance measurements were performed in the millimeter
wave frequency range from 40 GHz to 120 GHz using a free-space quasi-optical millimeter wave
spectrometer. The set-up is equipped with high-power backward wave oscillators. The complex
permittivity for both micro- and nano-ferrites has been calculated from the measured
transmittance spectrum. It was observed that the constitutive material properties, namely
permittivity and permeability, as well as the ferromagnetic resonance frequency of the samples
vary with the change in particle dimensions. Based on the results of these measurements, a
model for calculating the ferromagnetic resonance frequency of ferrite powders has been
derived, which takes into account the size and shape of the particles in the sample. It can be
concluded from the size-dependent absorption properties observed in this study that these
materials show promise as tunable millimeter wave absorbers.
2
Acknowledgement
I am grateful to my advisor, Dr. Mohammed Nurul Afsar for his continuous guidance and
support during my research. He provided me with the great opportunity to work in his High
Frequency Material Measurement and Information Center Laboratory at Tufts University. His
advice and ideas have been a source of great motivation for my work.
I am also thankful to the committee members, Prof D. Preis, Prof. X and Prof. X for
serving in my defense committee and providing useful review of my work.
I would like to thank Dr. Mahmut Obol and Dr. Konstantin A. Korolev for their
assistance with the instruments in the laboratory and fruitful discussions about the experimental
results. I am also thankful to other members of Prof. Afsar’s research group. I am extremely
grateful to Dr. John McCloy at Pacific Northwest National Laboratory for his support with this
research.
I would especially like to thank my parents and siblings for their constant love and care.
This thesis would not have been possible without their support. I am also grateful to all my
friends at Tufts University.
3
Table of Contents
Abstract ........................................................................................................................................... 0
Acknowledgement .......................................................................................................................... 3
1. Introduction ............................................................................................................................... 14
1.1 Background ......................................................................................................................... 14
1.2 Motivation for the study of nanoferrites ............................................................................. 18
1.3 Problem Statement .............................................................................................................. 22
1.4 Thesis Objective.................................................................................................................. 23
1.5 Thesis Organization ............................................................................................................ 25
2. Theoretical Background ............................................................................................................ 26
2.1 Origin of magnetism in materials ...................................................................................... 26
2.1.1 Diamagnetic materials ............................................................................................. 27
2.1.2 Paramagnetic materials ............................................................................................ 28
2.1.3 Ferromagnetic materials........................................................................................... 29
2.1.4 Anti-ferromagnetic materials ................................................................................... 30
2.1.5 Ferrimagnetic materials ........................................................................................... 30
2.2 Neel’s model of ferrimagnetism ........................................................................................ 34
2.3 Interaction between electromagnetic fields and materials ................................................. 35
2.3.1 Constitutive material properties ................................................................................ 37
2.3.2 Absorption of electromagnetic energy ...................................................................... 39
3. Microwave Measurement Techniques ...................................................................................... 45
3.1 Introduction ......................................................................................................................... 45
3.2 Brief Review of Microwave Techniques ............................................................................ 46
3.3 Two port network ................................................................................................................ 49
3.4 Wave Propagation in a Rectangular Waveguide ................................................................ 52
3.5 Experimental Set up ............................................................................................................ 57
3.5.1 Transmission Unit ...................................................................................................... 57
3.5.2 Measurement Unit ...................................................................................................... 58
3.6 Reflection and Transmission Coefficient............................................................................ 63
3.7 Determination of Permittivity and Permeability ................................................................. 67
4. Measurement Results at Microwave Frequencies..................................................................... 74
4.1 Introduction ......................................................................................................................... 74
4
4.2 Teflon Results ..................................................................................................................... 76
4.3 Nano-size hexagonal ferrites............................................................................................... 79
4.4 Nano-size spinel ferrites ..................................................................................................... 89
4.5 Nano-size Rare Earth Ferrites ............................................................................................. 97
5. Measurement results at millimeter frequencies ...................................................................... 103
5.1 Introduction ....................................................................................................................... 103
5.2 Theoretical Background .................................................................................................... 104
5.3 Experimental Details......................................................................................................... 105
5.4 Results ............................................................................................................................... 110
6. Size effect on ferromagnetic resonance in ferrites .................................................................. 121
6.1 Introduction ....................................................................................................................... 121
6.2 Properties of nano- and micro-size ferrite powders .......................................................... 122
6.3 Structural analysis of ferrite powders ............................................................................... 127
6.4 Ferromagnetic resonance .................................................................................................. 136
6.5 FMR Calculations ............................................................................................................. 142
6.6 Quality Control ................................................................................................................. 146
7. Error Analysis ......................................................................................................................... 150
7.1 Introduction ....................................................................................................................... 150
7.2 Systematic errors in microwave measurements ................................................................ 151
7.3 Random errors in microwave measurements .................................................................... 159
7.4 Uncertainty in millimeter wave measurements................................................................. 166
Conclusion .................................................................................................................................. 170
List of publications ..................................................................................................................... 172
References ................................................................................................................................... 175
5
List of Figures
Figure 2.1 Dipole orientation in (a) diamagnetic and (b) paramagnetic materials with
and without an external field………………………………………………….......... 29
Figure 2.2 (a) Atomic magnetic dipoles align in parallel directions in a ferromagnetic
material, (b) adjacent dipoles align in opposite directions with no net dipole
moment in anti-ferromagnetic materials, (c) in ferrimagnetic materials,
adjacent dipoles align in opposite directions but with a net dipole moment. ..............31
Figure 2.3 Dipoles are aligned in certain regions of the material called domains. A
material may consist of several domains, oriented in mutually random directions. ... 32
Figure 2.4 The electric and magnetic fields are perpendicular to each other and to the
direction of wave propagation. Such a wave is incident on the front end of a
material with intrinsic properties ε and µ. .................................................................. 40
Figure 2.5 The incident wave (I) is partly reflected (R) at the air-material interface. The
wave propagating inside the material attenuates depending in the absorption
properties of the material. The remaining energy is transmitted out of the
material (T). ............................................................................................................... 41
Figure 2.6 A multilayer structure formed by stacking two materials in medium 1, and
(b) the equivalent transmission line model, where the impedance of each
component is equal to the equivalent impedance seen at the interface of the
corresponding layer. .................................................................................................... 43
Figure 3.1 Schematic representation of a two-port network showing the incident and
reflected wave at each port. Hollow rectangular metallic waveguides were
used as transmission lines that guide the EM wave between the sample and
6
network analyzer. ........................................................................................................ 49
Figure 3.2 Geometry of a rectangular waveguide ......................................................................... 52
Figure 3.3 Block diagram of the measurement setup ................................................................... 58
Figure 3.4 The energy incident on the material under test is partially reflected at the first
material interface and the rest of the energy is transmitted through the material. ...... 59
Figure 3.5 Schematic diagram of the internal components of the network analyzer. ................... 60
Figure 3.6 Measurement set-up for measurement in X band. The network analyzer, coaxial
cables, waveguide flanges and adapters are shown. ................................................... 61
Figure 3.7 Standard waveguides used for each band are shown. The largest waveguide is
for R band and the smallest one is for Ka band. ........................................................ 62
Figure 3.8 Electromagnetic waves transmitting through and reflected from a sample in a
transmission line. ........................................................................................................ 63
Figure 4.1 Empty sample holders used for the measurement of G, C and Ku frequency
bands. .......................................................................................................................... 76
Figure 4.2 Variation of permeability and permittivity of barium hexaferrite nano-powder
with frequency. The average values of the imaginary permittivity and
permeability are 0.0516 and 0.0706, respectively. ..................................................... 81
Figure 4.3 Variation of front-end impedance of barium hexaferrite nano-powder
with frequency ............................................................................................................ 83
Figure 4.4 Variation of reflectance of 2cm thick sample of barium hexaferrite nanopowder with frequency. .............................................................................................. 84
Figure 4.5 Variations of permittivity and permeability of strontium hexaferrite nanopowder with frequency. .............................................................................................. 85
7
Figure 4.6 Variation of front-end impedance of strontium hexaferrite nano-powder with
frequency..................................................................................................................... 86
Figure 4.7 Variation of reflectance of 2cm thick sample of barium hexaferrite nanopowder with frequency .............................................................................................. 87
Figure 4.8 Material properties of Barium ferrite micro-powders are shown. The average
value of real and imaginary parts of permittivity is observed to be 3.065 and
0.0559 whereas real and imaginary parts of permeability are 1.1069 and
0.0536, respectively. ................................................................................................... 88
Figure 4.9 Variations of permittivity and permeability of Copper Iron Oxide nanopowder
with frequency. ........................................................................................................... 90
Figure 4.10 Variation of front-end impedance of Copper Iron Oxide nano-powder with
frequency.................................................................................................................. 91
Figure 4.11 Variation of reflectance of 2 cm thick sample of Copper Iron oide nanopowder with frequency ............................................................................................ 91
Figure 4.12 Variation of permittivity and permeability of Copper Iron Oxide nanopowder with frequency ............................................................................................ 92
Figure 4.13 Variation of front-end impedance of Copper Zinc Iron Oxide nanopowder with frequency ............................................................................................ 93
Figure 4.14 Variation of reflectance of 2 cm thick sample of Copper Iron oxide nanopowder with frequency ............................................................................................ 93
Figure 4.15 Variation of permittivity and permeability of Nickel Zinc Iron Oxide nanopowder with frequency bands .................................................................................. 94
Figure 4.16 Variation of front-end impedance of Copper Zinc Iron Oxide nano-powder
8
with frequency ......................................................................................................... 95
Figure 4.17 Variations of reflectance of 2 cm thick sample of Copper Iron oxide nanopowder with frequency ............................................................................................ 96
Figure 4.18 (a) Variations of permittivity and permeability of LSCF nanopowder with
frequency................................................................................................................. 98
Figure 4.18 (b) Variation of permittivity and permeability of LSM nanopowder with
frequency.................................................................................................................. 99
Figure 4.18 (c) Variation of permittivity and permeability of YSZ nanopowder with
frequency. ............................................................................................................ 99
Figure 4.19 (a) Variations of impedance of YSZ nanopowder with frequency.......................... 100
Figure 4.19 (b) Variations of impedance of LSM nanopowder with frequency ......................... 101
Figure 4.19 (c) Variations of impedance of LSCF nanopowder with frequency........................ 101
Figure 5.1 Horn antennas used for transmitting the incident signal at Q, V and W bands. ........ 106
Figure 5.2 Experimental set-up for measurement at millimeter wave frequencies .................... 107
Figure 5.3 Experimental set-up for millimeter wave measurements. ........................................ 108
Figure 5.4 Specially designed sample holders for BWO based spectrometer. The
thickness of the sample holder used here is 12mm. .................................................. 109
Figure 5.5 Transmitance variations of micro- and nano-size strontium ferrite powders
in the millimeter wave range..................................................................................... 113
Figure 5.6 Real (a) and imaginary (b) parts of permeability for Strontium nano-ferrite
powder measured in the millimeter wave frequency range. ..................................... 116
Figure 5.7 Real (a) and imaginary (b) parts of permeability for Strontium micro-ferrite
powder measured in the millimeter wave frequency range. ..................................... 116
9
Figure 5.8 Real (a) and imaginary (b) parts of permeability for Barium nano-ferrite
powder measured in the millimeter wave frequency range. ..................................... 118
Figure 5.9 Real (a) and imaginary (b) parts of permeability for Barium micro-ferrite
powder measured in the millimeter wave frequency range. .................................. 119
Fig. 6.1 The transmittance spectrum for micro- and nano-size strontium ferrite powders
measured using the BWO technique at Q, V and W bands. ........................................ 124
Figure 6.2 Transmittance spectrum recorded for two slightly different densities of
strontium hexaferrite ............................................................................................... 126
Figure 6.3 X-ray diffraction spectra of (a) strontium hexaferrite and (b) barium
hexaferrite. ................................................................................................................ 128
Figure 6.4 Scanning electron micrographs of barium hexaferrite nano-powder for
different resolutions can be seen. The magnification level for the images
shown here are (a) 5,000, (b) 10,000, (c) 20,000, (d) 40,000. The particle
size is much smaller than the 1 µm mark indicated in the third inset. ...................... 132
Figure 6.5 The micro-size powder of barium hexaferrite consisted of agglomerates
in the size range from 3 to 15 µm. It can be seen in inset C that the primary
particle size lies between 0.5 to 1 µm. .................................................................... 135
Figure 6.6 Several lots of samples contained incomplete phases or impurities such as
Fe2O3. X-ray studies of a few samples are shown here. ........................................... 147
Figure 6.7 Transmittance spectrum for different lots of barium hexaferrite samples. The
hexaferrite phase ....................................................................................................... 149
was missing in lots 2 and 3. This can also be concluded from this data due
to the missing resonance. .......................................................................................... 149
10
Figure 7.1. Network analyzer measurement set-up is shown with the sources of errors
marked in the red boxes........................................................................................... 153
Figure 7.2 Block diagram of the waveguide set-up. Port1 and port 2 indicate the
location of the network analyzer ports while the desired measurement
port is at the interface of the material under test. .................................................... 153
Figure 7.3 Thru connection for calibration procedure. ............................................................... 155
Figure 7.4 Block diagram of the reflect connection. .................................................................. 157
Figure 7.5 Block diagram of the line connection. ....................................................................... 158
Figure 7.6 Experimental uncertainty and standard deviations for the transmittance spectra. ... 167
Figure 7.7 Experimental uncertainty and standard deviations for the reflectance spectra. ....... 168
11
List of Tables
TABLE 3.1 Cut-off frequency for different modes inside the waveguide ................................... 56
TABLE 3.2 Cut-off frequency for higher order modes for X band .............................................. 57
TABLE 3.3 Properties of rectangular waveguides used in material characterization
measurements ........................................................................................................... 62
TABLE 4.1 List of nano-size powders investigated ..................................................................... 75
TABLE 4.2 Dielectric properties of solid Teflon samples ........................................................... 78
TABLE 4.3 Dielectric and magnetic properties of hexaferrite nanopowder ................................ 87
TABLE 4.4 Dielectric ad magnetic properties of spinel ferrites .................................................. 96
TABLE 4.5 Dielectric and magnetic properties of rare earth perovskite nanopowders ............. 100
TABLE 5.1 Investigated millimeter wave frequency bands ....................................................... 106
TABLE 5.2 Best fit value of permittivity for hexaferrite samples ............................................. 114
TABLE 5.3 Resonance frequency of ferrite powders at different sizes ..................................... 120
TABLE 6.1 Electro-Magnetic Properties Measured At Microwave Frequencies ...................... 122
TABLE 6.2 Electro-Magnetic Properties Measured At Millimeter wave Frequencies .............. 123
TABLE 6.3 Particle size determination of barium and strontium ferrite ................................... 129
TABLE 6.4 Calculated values of resonance frequencies of Barium and Strontium hexaferrite
nanopowders without demagnetization effects. ..................................................... 143
TABLE 6.5 Calculation of demagnetization factor for different particle shapes in barium
hexaferrite nano-powder....................................................................................... 144
TABLE 6.6 Calculated value of resonance requency in the millimeter wave range .................. 145
TABLE 6.7 XRD study of micro and nano hexaferrite powders ............................................... 148
TABLE 7.1 Standard Deviation in the Derived Parameter Values for barium hexaferrite ........ 159
12
TABLE 7.2 Standard deviation in the measurement of s-parameters at each frequency band
for barium hexaferrite nanopowder ............................................................................ 162
TABLE 7.3 Average error percentage in dielectric properties of solid and powdered samples 164
TABLE 7.4 Average error percentage in dielectric properties of solid and powdered samples 165
TABLE 7.5 Results of repeated reference scans at each frequency band .................................. 166
TABLE 7.6 Standard deviation in the measured transmittance and reflectance spectrum for
millimeter wave frequency bands ............................................................................... 168
13
Chapter 1
Introduction
1.1 Background
With rapid developments occurring in the field of wireless communication and high frequency
devices, the use of magnetic and dielectric materials finds wider applicability, not necessarily
restricted to military and stealth applications only. The ever increasing growth of commercial
applications in the higher frequency range of the radio wave spectrum has augmented the need to
find materials that can support these applications. This radio frequency (RF) range is viewed as
an important area of development due to broader bandwidths and reduction in device size at
these frequencies. In addition, the high frequency range is of great interest due to lower
propagation losses and reduced fabrication costs of the devices at these frequencies [1].
In the electromagnetic spectrum, the radio frequency spectrum covers frequencies from a
few megahertz to several hundreds of gigahertz. The higher frequency range of this spectrum can
be further divided into the microwave and millimeter wave frequency bands. The term
microwave refers to the frequency region from 300 MHz to 30 GHz. This corresponds to the
wavelengths from 1 meter to 1 centimeter. Numerous applications in this frequency range
14
include, for example, cell phones, air traffic control radars and global positioning systems, etc.
The higher end of the radio frequency region from 30 GHz to 300 GHz is often referred to as
millimeter waves since these frequencies correspond to wavelengths in the millimeter range.
This frequency range supports a wide variety of applications such as bio-medical applications,
remote sensing and as absorbers for high frequency devices [1-2].
During the last four decades, the smallest feature of a transistor has shrunk from 10 um
down to 30 nm. Rapid developments in the semiconductor industry has led to the shrinking of
device size which in turn has triggered the development of material preparation techniques in the
micrometer to nanometer domains as well as material characterization techniques for these
particle sizes and frequency domains. As an example, consider the development of
electromagnetic absorbers. Radar absorbing materials have been in use for a long time now to
reduce the radar cross section (RCS) in military applications. The first electromagnetic wave
absorber came into light in the mid 1930s to improve the front to back ratio of a 4GHz antenna
[3]. But nowadays there is broader area of applications for absorbing materials. The extensive
use of wireless technology in commercial products has made electromagnetic interference a
serious concern for the designers. Absorbing materials are used to eliminate the unwanted
electromagnetic leakages and to reduce interference among wireless devices. Since many
different types of wireless devices are being used currently, application specific absorbers are
desired. Depending on the application, the absorbers can be extremely light weight, narrowband
or wideband and thin or multi-layered. As a result, several types of absorbers have been
developed over the years [4]. The design of absorbers not only requires the ability to fabricate
special materials but also the ability to study their constitutive electromagnetic properties and
15
tailor them to provide custom solutions for various applications [5]. This is true for other
applications of electromagnetic materials as well.
Several materials with desirable magnetic and dielectric properties have been developed
for the high frequency applications. The class of materials consisting of oxides and
semiconductors doped with transition metal elements or rare earth metals have been used for the
design of devices in the microwave and millimeter wave frequency ranges. Oxides of iron have
substantial technological value largely because they possess the combined properties of a
magnetic material and an electric insulator. Materials formed with iron (III) oxide, Fe2O3, as the
principal constituent are referred to as ferrites. The basic formula for ferrites was identified by
Hilppert in 1909 as MeO.Fe2O3, where Me is a divalent metal ion [6]. These are compounds of
metals with iron oxide. However, in contrast to metals, ferrites are non-conducting oxides which
allow total penetration of electromagnetic fields. Ferrites were initially used as magnetic cores.
Over the years, ferrites have proved to be versatile magnetic materials since they are relatively
inexpensive, stable and have a wide range of applications [6]. Ferrite materials have been under
intense research for several years due to their favorable electromagnetic properties [7]. They are
in use in many industries such as automobile, telecommunication, data processing, electronics
and instrumentation [8]. This can be attributed to suitable properties of ferrites such as high
saturation magnetization and electrical resistivity, good chemical stability and low electrical
losses. Ferrites have been used in information storage media such as magnetic tapes and floppy
disks, in transformer cores and high frequency circuits [9]. More recently, ferrites are also used
in millimeter wave ICs and power handling devices. Lower RF loss makes these materials useful
in the design of microwave devices such as isolators and phase shifters. Resonance isolators can
require up to 30-35 kOe magnetic fields, which require more space and increased cost. By
16
utilizing the high internal anisotropy of hexaferrites, isolation levels of up to 20 dB can be
achieved by using external field as small as 500 Oe [10]. Ferrites are also used as inductive
components in low noise amplifiers, voltage-controlled oscillators and impedance matching
networks [11].
The performance of these materials in their bulk form is limited upto a few megahertz
due to their higher electrical conductivity and domain wall resonance [12]. However, the recent
technological advances in electronics industry demand ever more compact devices for work at
higher frequencies [14-15]. One way to solve this problem is by synthesizing the ferrite particles
in the nanometric scale. When the size of the magnetic particle is smaller than the critical size,
the particle is in a single domain state, thus avoiding domain wall resonances. Such materials can
work at higher frequencies. The recent developments in fabrication techniques have opened the
possibility to manufacture ferrites in nano-scale domain. This has given way to new research
areas and fields of application for the ferrites.
The term, nano, is commonly used to refer to one-billionth (10-9) of a unit. In the context
of nanomaterials, this refers to materials that are composed of particles with at least one
dimension in the nanoscale range. Dimensions up to 100 nm, sometimes even 1000 nm, fall in
the nanometer range [16]. Nanoferrites are ferrite compounds consisting of particles with the
smallest dimension in the nanoscale region. These materials are of great interest since their
dimensions approach that of an individual atom or molecule. As a result, the properties of
nanoferrites are significantly different from those of the materials in bulk [17]. While bulk
ferrites still remain important magnetic materials, the nanomaterials have emerged as strong
candidates for electronics as well as medical applications [18, 19]. These are discussed in the
next section.
17
1.2 Motivation for the study of nanoferrites
In the past, the applications of magnetic and dielectric materials in permanent magnets for
frequency tuning of microwave devices or in electromagnetic absorbers employed materials in
the bulk form. However, as the device size is shrinking, smaller magnets and absorbers are
needed. Study of nanoparticles is currently an area of intense scientific research not only to
support electronic applications with modern IC revolution but also to help develop potential
applications in the field of biomedical engineering. In the context of biomedical engineering
nanoparticles play the role of a bridge between bulk materials and atomic structures.
The properties of a material change as its physical size shrinks, due to the increased
percentage of atoms on the surface of the material. For a given amount of material, the surface
area compared to the volume increases as the particle size decreases. Thus the proportion of
constituent atoms on or near the surface increases. For instance, consider a microparticle with a
radius of 30 µm and surface area of 0.01 mm2. The same mass of this material in nanoparticulate
form, with each particle having a radius of 30 nm will consist of 1 billion nanoparticles, with a
total surface area of 11.3 mm2. Thus compared to microparticles, nanoparticles have a very large
surface area and higher number of particles per unit mass [16]. This feature is important because
many reactions occur on the surfaces of materials. The small particle form of a material may
have properties quite different from its properties in bulk form. For example, zinc oxide particles
have been found to have superior UV blocking properties compared to the bulk substitute. This is
one of the reasons why zinc oxide particles are often used in the preparation of sunscreen lotions
[20]. Gold nanoparticles melt at much lower temperature (~ 3000C) than gold slabs (10640C)
[21]. Absorption of solar radiation in photovoltaic cells is much higher in nanoparticles than it is
in thin films of continuous sheets of a bulk material because of the enhanced effective surface
18
area in the former case. Particles in smaller size domain extend the range of applications of their
bulk counterparts.
Microwave ferrite devices are used to control energy propagation by a static or
switchable dc magnetic field [9]. Another application of ferrites is in frequency tuning of
electronic components, which has traditionally required external magnets. However, at high
frequencies, the magnetic bias required increases the size and cost of the components. This can
be avoided by using self-biased hexaferrites that have high effective internal magnetic anisotropy
that can provide the permanent magnetic bias [22], thus eliminating the need for bulky external
magnets.
Electromagnetic interference (EMI) suppression is needed for radar targets and in
electronic devices that are densely packed. Factors such as thickness, weight, mechanical
strength and environmental conditions limit the choice of available absorbing materials. As the
clock speeds of electronic devices approach microwave frequencies, it has become essential to
study the high frequency properties of the materials used in electronics devices and components.
Reduction in component size would result in faster speed and higher density of logic on
processors and circuit boards. Further, due to the significant increase in the density of
components in electronic systems such as high-speed digital interfaces in computers, digital
cameras, scanners, and so forth electromagnetic interference becomes a serious issue. Special
materials are used in the design of circuits and packaging applications to ensure electromagnetic
compatibility in the smaller designs [23]. For the case of narrow-band absorbers, an optimum
solution would require a compromise between absorber thickness and absorption characteristics.
Material parameters can be optimized once the intrinsic material properties are known. In the
case of wide band absorbers the number of parameters increases and optimization becomes a
19
complex task. Such a design can be greatly simplified by using electromagnetic simulation tools.
But the accuracy of the results from such simulations is determined by the accuracy with which
the material properties are known. In the development of computer aided design circuit
simulation programs, both magnitude and phase data are required for accurate models. Therefore
if the exact dependence of material properties on frequency is known, application specific
absorbing materials can be designed. The intrinsic material properties have been used to study
the microwave power absorption of ferrites [24] and magnetic loss in microwave heating [25].
Microwave remote sensing is another field where material characterization plays an
important role. In remote sensing, materials are identified by studying the reflection and
scattering of microwave signals by the target specimens [26], which in turn depend on the
constitutive properties of the target material. Thus by characterizing materials and creating a
database, identification of materials by remote sensing can be done more efficiently.
Nanomaterials are also proving useful in the field of biomedical engineering. Due to their
small size, nanopoarticles are valuable in investigating cells since their size is comparable to the
size of important biological constituents of the body. For instance, viruses are 20-400nm, cell 10100 um, proteins 5-50 nm, antibodies 10 nm, bacteria 100nm-10um and red blood cells are 10
um. The small size allows the nanoparticles to penetrate physiological barriers and travel within
the circulatory system to perform functions such as the magnetic resonance enhancement, iron
delivery for the production of RBCs and drug delivery to improve the availability of seruminsoluble drugs. Nanomaterials with particle size upto 50nm can travel the circulatory system of
the body and thus provide drugs to desired cells. These nanoparticles can be manipulated by an
external magnetic field. This means that the particles can be guided using an external field for
targeted drug delivery inside the body. They can also be coated with biological molecules to bind
20
them to a biological entity [16]. They can also resonantly respond to time varying magnetic field.
Thus energy can be transferred from external field to the biological systems by tagging them
with magnetic nanoparticles. Magnetic nanoparticles have recently been used to treat bone
diseases by using surface modified magnetic nanoparticles [27]. These are also being used in
cancer treatment. Unlike the conventional cancer treatment methods, the affected cells can be
targeted exclusively and the healthy cells close to an infected one can be saved by using
magnetic nanoparticles. Magnetic nanoparticles can also be used in hyperthermia treatment. Due
to small size, the particles can produce a more localized and effective solution. Ferrite
nanoparticles have wide range of sizes, diversity and chemical stability as compared to metal
nanoparticles [28]. The nanoparticles have beneficial properties such as magnetic guidance and
induction heating in AC magnetic fields.
Another novel application of nanoferrites is in magnetic resonance imaging. In MRI, the
person under examination is exposed to a RF pulse in the presence of an external magnetic field.
The resulting changes in the spin of the protons in water molecules are used to construct a three
dimensional image of the subject. Use of the magnetic nanoparticles in biological systems can
enhance the detection sensitivity [29]. Nanomaterial tubes are ideal probe tips for scanning
microscopy due to their small diameter which maximizes the resolution and high aspect ratio
[30]. With further improvement in imaging technology, magnetic nanoparticles exhibiting strong
resonances can be used to perform localized imaging. The scope of applications for nanopowders
has driven the need for extensive research in this field.
A thorough review of the applications of nano-ferrites in RF and microwave applications
has been presented [31]. In order to use materials in applications listed above and beyond,
precise characterization of materials is a necessary requirement. Thus the knowledge of intrinsic
21
material properties is essential to understand and predict the behavior of magnetic and dielectric
materials and thus utilize them to their full potential in electromagnetic applications.
1.3 Problem Statement
Over the last 50 years, many devices have been developed across the microwave spectrum.
However, many challenges remain to be addressed to meet the demands for miniaturization,
broader bandwidths, higher operating frequencies and reduced costs [9]. From the discussion in
the previous section it is evident that the knowledge of material properties is indispensable for
the development of novel applications in the fields of electronic and biomedical engineering.
Characterization of constitutive material properties allows scientists to understand, analyze and
modify the behavior of the materials. This also provides the flexibility to tailor the material
properties according to the application. However, higher frequencies of operation and smaller
sizes of the samples make characterization challenging. The measurement methodology is
limited by the size of components needed to make measurements at such high frequencies and
also by the sensitivity of the measurement set-up to interference from other sources. Any
additional length of cables or connectors adds significant delay and phase change at high
frequencies.
The properties of materials depend on the frequency of the application, size and
composition of the sample. Controlled growth of powdered ferrites is difficult and as a result the
final size and shape of the particles may be different from what was intended. Physical properties
of nanoparticles are influenced significantly by the manufacturing technique [32]. The
preparation conditions such as chemical composition, sintering temperature and amount of
substitutions have been known to affect the physical properties of the fabricated material.
22
Therefore, in materials research and engineering, microwave techniques for the characterization
of material properties can be used as nondestructive testing tools to check if the final products
match the specifications that they were designed for. Distribution of particle sizes and inter
particle spacing affect the magnetic properties. Therefore, it is important to know the effect of
particle size on the material properties. In order to do so, it is necessary to collect data for
nanopowders of different sizes and analyze the relationship between their size, composition,
morphology and their electromagnetic responses. One of the critical challenges faced currently
by researchers in the field of nanotechnology is the lack of instruments to characterize and
manipulate the materials prepared at the nanometer level. Needless to say that characterization of
nanopowders requires extreme sensitivity and accuracy.
Furthermore, the particles in the powdered samples tend to form aggregates. These
aggregates can vary in size from a few nanometers to micrometer. It may not always be possible
to avoid aggregate formation. Therefore it is essential to check the state of the sample during
measurement and perform controlled measurements.
1.4 Thesis Objective
Constitutive material properties of powdered ferrite samples are studied in this work. Materials
have been studied in solid form in the past. However, here the samples are studied as pure
powders, without the need to form solid samples by diluting them with epoxy or other forms of
conductive binder. Two different measurement techniques have been applied to characterize the
samples in the broad frequency range from 1.7 GHz to 120 GHz. For microwave measurements,
covering frequencies between 1.7 GHz to 40 GHz, a transmission/reflection based measurement
technique has been used to obtain the intrinsic material properties. A vector network analyzer is
23
employed to measure the s-parameters of the material under study. Rectangular waveguides are
used to connect the sample to the measurement ports of the network analyzer. The measured data
is used to calculate the complex dielectric permittivity and magnetic permeability of the samples.
For the millimeter wave measurements, a quasi-optical high frequency spectrometer is
used. Backward wave oscillators are used as the high frequency power sources. The signal
propagates through two channels. The optical path, consisting of horn antennas and lenses to
guide the Gaussian beam, is used for the measurement and the other channel made up of high
frequency components such as directional couplers and attenuators acts as the reference path.
Power is divided between these two arms by using a waveguide-based directional coupler. The
permittivity and permeability values are derived from the transmission data of the sample using
curve fitting functions. A thorough error analysis has been attempted to identify the possible
sources of errors.
The samples investigated in this work include M-type hexaferrites, spinels and perovskite
ferrites. Hexaferrite samples with identical material composition but different particle sizes are
used. The study revealed a dependence of material properties of hexagonal ferrites on the
morphology of the particles in the powdered sample. The morphology of the samples was
studied using scanning electron micrographs (SEM) and their compositions were verified using
X-ray diffraction. The possible reasons for the difference in behavior of nano-powder and micropowder samples are suggested. A calculation model has been defined to determine the resonance
frequency for such samples. The proposed model takes into account the structural properties of
the particles in the sample. The results indicate that it is possible to use the hexaferrites as high
frequency tunable absorbers.
24
1.5 Thesis Organization
The first part of the thesis is an introduction to ferrite materials and the measurement techniques
employed for measuring dielectric and magnetic properties at the microwave and millimeter
wave frequencies. The measurement results are presented in the later sections of the thesis. The
basic principles needed to understand the properties of ferrites are discussed in chapter 2. The
measurement technique used in the microwave measurements and the modified algorithm to
interpret the data are presented in chapter 3. The limitations of the existing material
characterization techniques are also discussed here. The results for microwave and millimeter
wave measurements of ferrite samples are analyzed in chapters 4 and 5, respectively. The
dependence of material properties on the size of the particles is discussed in chapter 6. Chapter 7
provides an analysis of the possible errors in the results. The relevance and future scope of this
work is discussed in the conclusion.
25
Chapter 2
Theoretical Background
Extensive use of the iron based compounds including the ferrites in traditional and innovative
technologies has been mentioned in chapter 1. Magnetically anisotropic materials such as the
hexagonal ferrites hold special promise to realize these and other applications. Micro- and nanosize particles of these materials further extend the scope of such applications, particularly so in
the microwave and millimeter spectral regions. In this chapter, a brief introduction to the
different types of magnetic materials and their interaction with the external electromagnetic
fields is provided. The role of permeability and permittivity of these materials in this process is
crucial. Their suitability or otherwise to various applications critically depend on these material
properties.
2.1 Origin of magnetism in materials
In order to comprehend the electromagnetic behavior of a material, it is important to understand
how the electromagnetic fields interact with its fundamental units, that is, the electrons present in
the atoms (molecules) and the associated magnetic moment of the atoms (molecules). An
26
electron is a charged body that exerts a force on other charged bodies. A stationary charge is
considered a source of electric field and the charges present in this field experience a force. On
the other hand, a moving charge (current) creates a magnetic field. Inside a material, the atoms
consist of electrons that orbit around the nucleus, generating current loops and thus the atom has
a magnetic dipole moment associated with it. This magnetic moment is along its axis of rotation.
Additionally, the electron also spins around its own axis. The net magnetic moment of an atom is
the vector sum of the magnetic moments of each electron. The magnetic properties of a material
are mainly determined by these magnetic moments which result from the orbiting and spinning
of electrons in the atom. Net magnetic moment present in unit volume of the material defines its
magnetization. In the presence of an external magnetic field, the force of the field exerts a torque
that tends to align these dipoles. The magnetic behavior of a material in the presence of an
external field can be understood by studying the interaction between the magnetization of the
material and the external field. Materials are classified based on the response of their magnetic
dipoles to an external field. The different types of magnetic materials are briefly described here.
2.1.1 Diamagnetic materials
In the absence of an external field, diamagnetic materials do not have a net magnetic moment
since the electrons in such a material are paired up with anti-parallel spins. When an external
field is applied, the orbits of the electron change such that the net induced magnetic moment is in
a direction opposite to the direction of the applied magnetic field. The magnitude of this induced
magnetic moment is extremely small.
27
2.1.2 Paramagnetic materials
Some materials possess an unpaired electron spin in the atom and due to the incomplete
cancelation of the electron spins the atoms possess a net magnetic moment. However, in the
absence of an external magnetic field, the atomic moments are randomly oriented and the
material does not exhibit any magnetization. In the presence of an external field, the magnetic
moments get aligned along the direction of the external field. As soon as the applied field is
removed, the alignment and thus the magnetization is lost.
The behavior of the dipoles in diamagnetic and paramagnetic materials, with and without an
external field is shown in figure 2.1. On the left side, the behavior of the dipoles is shown in the
absence of external field and on the right side an external field, field direction indicated by the
arrow, is applied. In case of figure 2.1 (a) diamagnetic materials, the atoms do not possess any
net moment because all the spins are paired in opposite directions. In the presence of an external
field, the induced dipoles point opposite to the applied field and the material possesses a weak
net magnetization. In case of figure 2.1 (b) paramagnetic materials the atoms possess a net
dipole moment but these are randomly oriented in the absence of external field.
(a) Diamagnetic material
28
(b) Paramagnetic material
Figure 2.1 Dipole orientation in (a) diamagnetic and (b) paramagnetic materials with and without an
external field.
2.1.3 Ferromagnetic materials
Both diamagnetic and paramagnetic materials are considered non-magnetic materials since they
do not possess any net magnetization in the absence of an external field. There are certain
metallic materials, like transition elements, which have permanent magnetic moment and exhibit
magnetism even without any external field.
In a ferromagnetic material the permanent magnetic moments result from unbalanced
electron spins in the atom. The coupling interaction, commonly called the exchange interaction
among atoms, tends to align the magnetic dipoles in neighboring atoms along the same direction.
In the process a ferromagnetic material attains what is called the spontaneous magnetization, i.e.,
the material gets magnetized without the help of an external magnetic field. Therefore these
materials possess net magnetic moment even in the absence of an external field. This
phenomenon is shown in figure 2.2 (a). Examples of ferromagnetic materials include iron, cobalt
and nickel.
29
2.1.4 Anti-ferromagnetic materials
There are certain materials other than ferromagnetic materials which show different kind of
coupling among the magnetic moments in adjacent atoms. Unlike ferromagnetic materials the
neighboring dipoles tend to align in an antiparallel manner. If the magnetic dipoles are arranged
such that half of the dipoles align themselves in the direction opposite to that of the rest of the
dipoles, then they cancel out each other’s effect and there is no net magnetic moment in the
material. It should be noted that the dipoles are assumed to be of the same strength in this case.
Such a material is called anti-ferromagnetic material. Examples of anti-ferromagnetic material
are chromium and manganese. Neel proposed the two sublattice model to explain
antiferromagnetism. This is discussed in section 2.2.
2.1.5 Ferrimagnetic materials
Ferrimagnetic materials have similar behavior as anti-ferromagnetic materials except that the
neighboring dipoles in these materials have unequal strengths and hence they do not completely
cancel each other’s effect. Thus the materials possess a net magnetization. These materials are
sometimes also called ferrites. The typical examples of ferrites are magnetite (Fe3O4) and barium
ferrite (BaFe12O19). Magnetite is a good example to understand the unequal moments. The
chemical formula for Fe3O4 can be re-written as, Fe2+O2- (Fe3+)2 (O2-)3. Thus Fe exists in both
Fe2+ and Fe3+ states. The net spin magnetic moment for each of these ions is different and they
are present in a 1:2 ratio in the compound. This results in a net magnetic moment in magnetite.
This example is discussed in more detail in section 2.2
30
(a)
(b)
(c)
Figure 2.2 (a) Atomic magnetic dipoles align in parallel directions in a ferromagnetic material, (b)
adjacent dipoles align in opposite directions with no net dipole moment in anti-ferromagnetic materials,
(c) in ferrimagnetic materials, adjacent dipoles align in opposite directions but with a net dipole moment.
Figure 2.2 shows the arrangement of dipoles in the three categories of magnetic materials
described in section 1.1.3 to 1.1.5. However, the arrangement of dipoles in various magnetic
materials can be more complicated than what is shown in this figure. The dipoles may not be
arranged in the same direction over the entire extent of the material.
The mutual spin alignment (parallel or anti-parallel) may exist over the entire volume of
the material or over certain regions of the crystal. In general, at a macroscopic level, a material is
composed of several regions of aligned dipoles but the alignment direction can vary from one
region to the other. Each small volume region consisting of mutually aligned dipoles is called a
domain. An example of a material containing several domains is shown in figure 2.3. The
boundaries of the domains are commonly called domain walls. Although these are just shown as
lines in figure 2.3, it should be realized that these basically represent areas along which the
dipole orientation direction changes gradually.
31
Figure 2.3 Dipoles are aligned in certain regions of the material called domains. A material may consist of
several domains, oriented in mutually random directions.
Thus a material can have several domains and each domain can have a different
orientation of the magnetic dipoles and hence magnetization. For a polycrystalline material, each
grain may consist of more than a single domain. The net magnetic field for such a material is the
vector sum of the magnetizations of all the domains. Of course, larger domain volumes would
have a more significant contribution towards the net magnetization. In case of single domain, all
the dipoles are aligned. Thus materials with single domain would have a higher magnetization as
compared to polycrystalline materials consisting of several randomly oriented domains. The
value of magnetization when all the magnetic dipoles present in the material are aligned with the
external field is referred to as the saturation magnetization.
Another important feature of polycrystalline materials is that they may have preferred
orientation directions for the grains. Even for a single crystal ferromagnetic material, the
magnetization depends on its crystallographic orientation relative to the direction of the applied
magnetic field. It is easier to achieve higher values of magnetization if the field is applied along
certain preferred directions. For instance, the effect of applied field direction on the
32
magnetization has been studied for nickel and cobalt single crystals [33-34]. It was observed that
in these materials there is one crystallographic direction in which magnetization is the easiest.
This direction is termed as the easy magnetization axis. If the field is applied along this direction,
then very small value of magnetic field is required to achieve saturation magnetization. This is
often the lowest energy state for the dipoles. The dipoles in a domain also tend to align along the
easy axis. The reason for a material to form multiple domains is to achieve a state of minimum
internal energy. The size of a domain depends on the overall balance of energy in the material.
In a single domain ferromagnetic material, the magnetization does not vary across the
material. The size range over which a ferromagnet exists as a single domain is generally quite
narrow. There is a critical radius for a material such that for particle or grain size less than the
critical radius, the ferromagnetic particles stay uniformly magnetized as single domain [35].
Ferromagnets tend to be in a state of uniform magnetization when the diameter of the particles is
between 10 nm and 100 nm [36].
The dependence of the magnetic behavior on crystallographic orientation is termed as
magnetic anisotropy. The origin of this anisotropy goes back to the spin-orbit interaction at the
atomic level. The distribution of the orbits of the electrons in a transition metal is anisotropic,
which in turn affects the distribution of spins and hence magnetization via the spin-orbit
interaction. Magnetic anisotropy follows the crystal symmetry. The advantage of high magnetic
anisotropy is that such materials do not easily demagnetize. Such materials find use as permanent
magnets. On the other hand, materials with low magnetic anisotropy can easily change their
magnetization and are used to make inductors and transformer cores.
33
2.2 Neel’s model of ferrimagnetism
L. Neel in 1948 proposed the existence of two magnetic sublattices in an antiferromagnetic
substance. The dipoles in each sublattice point in the same direction but the two sublattices have
magnetizations in exactly opposite directions. These dipolar orientations are caused by what is
called the superexchange interaction. Unlike in a ferromagnetic substance where adjacent dipoles
are in close neighborhood (with the resulting tendency to align parallel to each other), the dipoles
in an antiferromagnetic substance are rather far apart. The oxygen anions with zero magnetic
dipole moments lie between neighboring magnetic cations. It is the super exchange interaction
among the magnetic cations mediated by the intervening non-magnetic oxygen anions which
lead to dipolar alignment in each sublattice. The interaction among the dipoles belonging to
different sublattices is stronger than the interaction among the dipoles belonging to the same
sublattice so that the two sublattices can align in opposite directions.
The orientation of the dipoles in each sublattice competes with the disorientational
tendency due to the thermal energy of the overall lattice. The thermal energy increases with
increasing temperature. At a certain critical temperature called Neel temperature, individual
sublattices get demagnetized and the material is no longer antiferromagnetic and becomes
paramagnetic. Consequently the material is anti-erromagnetic below the Neel temperature and
paramagnetic above it.
In an antiferromagnetic material, the dipolar moments in the two sublattices are of equal
strength but oppositely oriented, leading to zero net magnetization. In ferrimagnetic materials
(ferrites), the dipoles on the two sublattices may have unequal strengths but oppositely oriented.
Thus the net magnetization in a ferromagnetic substance is small but not zero. However, not all
ferrimagnetic materials obey this scheme. It may happen, as in the case of magnetite (Fe 3O4),
34
that interactions within and between the sublattices may be comparable in strength but all cations
may not occupy identical crystallographic sites in the unit cell. For instance, for magnetites there
are 16 Fe3+ cations and 8 Fe2+ cations in the unit cell. The unit cell has two kinds of sites. Site A
has four oxygen anions near neighbors in a tetrahederal orientation and site B has eight oxygen
anions near neighbors in octahederal sites. There are 8 A sites and 16 B sites in the unit cell. In
normal spinel structure, the divalent cations occupy all 8 A sites and trivalent cations occupy all
16 B sites. But in magnetite (FeO.Fe2O3) with inverted spinel structure, 8 trivalent cations
occupy all 8 A sites and the remaining 8 trivalent cations and all 8 divalent cations occupy the 16
B sites. Sites A and B belong to different sublattices. All dipoles on A site are oriented
oppositely to dipoles on the B sites. The trivalent cations in the two sublattices yield zero
magnetic moment because of their oppositely oriented dipoles. The net reduced magnetization in
magnetite is due to the 8 divalent cations in the unit cell.
2.3 Interaction between electromagnetic fields and
materials
A TEM electromagnetic wave is composed of perpendicular electric and magnetic fields. The
time-varying electric and magnetic fields can mutually sustain each other. The direction of
propagation of the wave is perpendicular to the direction of either field. Maxwell’s equations
describe the propagation of the electromagnetic wave in a material. During propagation, the
electric and magnetic fields associated with the wave interact with the electric and magnetic
constituents of the material. The electric and magnetic phenomena have been known for a long
time but the connection between the two was established much later. James C. Maxwell was the
first to formalize the missing link between the electrical and magnetic effects. He gave a
35
mathematical formulation known as Maxwell’s equations which describe the interrelated
behavior of electric and magnetic fields inside a medium. Maxwell’s equations have the form,
⃗
(2.1)
⃗
⃗
⃗
⃗
⃗
where D is the electric displacement vector, E is the electric field strength, B is the magnetic flux
density, H is the magnetic field strength, and J is the current density. These are all vector
quantities, ρ is the charge density. These quantities are related by the following constituting
relations,
⃗
⃗
⃗
⃗
⃗
Here, σ is the conductivity of the material, and ε and µ are the permittivity and
permeability of the material. Permittivity and permeability are complex quantities and are
explained in more detail in the following section. These equations indicate that the response of a
material to electromagnetic fields is determined essentially by three constitutive parameters,
namely, permittivity, permeability and conductivity. These parameters also determine the spatial
extent to which the electromagnetic field can penetrate into the material at a given frequency.
36
Thus the knowledge of these parameters is crucial to develop and utilize these materials for
specific applications at a given frequency.
2.3.1 Constitutive material properties
As mentioned earlier permittivity and permeability are important material properties that
describe the behavior of the material in the presence of an external field and can be used to
predict the interaction of the material with electromagnetic waves. Being complex quantities, the
permittivity and permeability can be expressed as,
where ε’ and µ’ are the real parts of permittivity and permeability and ε” and µ” are the
imaginary parts of permittivity and permeability, respectively. However, the terms most
commonly used to define a material property are relative permittivity and relative permeability,
defined as
where
(
) is the relative permittivity (permeability),
(permeability) of a medium and
( ) is the complex permittivity
( ) is the permittivity (permeability) of free space.
Combining equations (2.10) and (2.11) gives
37
where we have used
.
The complex index of refraction N is defined as,
⁄
⁄
Defining,
And using equations (2.7) and (2.8) gives,
(
(
)
)
where n is the real part of the complex index of refractions and is usually referred to as index of
refraction or refractive index. K is called the absorption coefficient. It should be noted that the
index of refraction n and absorption coefficient K depend on the dielectric (ε) and magnetic (µ)
properties of the medium. Thus we have only two equations to derive the four unknowns. In the
past, two approaches have been used to extract the permittivity and permeability values. The first
method is used for dielectric samples. For such materials, the real part of permeability is taken as
unity and the imaginary part is neglected (assuming no magnetic losses). Now there are only two
unknowns and these can be easily evaluated using Eqs. (2.15) and (2.16). For more complicated
materials, such as ferrites, the evaluation of material properties is not as direct. In such materials,
the permeability values cannot be assumed to be unity since the permeability changes near the
resonance frequency. The calculations are performed in two steps. First, the dielectric properties
are determined at frequencies outside the ferromagnetic resonance region, assuming the real and
imaginary part of permeability to be unity and zero, respectively. This assumption may not
38
always hold as for example for some of the rare earth ferrites discussed in chapter 4. However,
ignoring such complications, we may use these values of the dielectric parameters to extract the
values of permeability in the resonance region. These methods may suffice is some situations for
estimating the material parameters but they are not very accurate. A more precise method to
calculate permeability in the ferromagnetic resonance region is based on Schloemann’s equation
[37],
[
⁄
⁄
]
The values of Ms, and Ha are also evaluated from the curve fitting program by using the
known values of real and imaginary parts of the permittivity.
2.3.2 Absorption of electromagnetic energy
An electromagnetic wave interacts with a material in two ways, which are, energy storage and
energy dissipation [38]. The real parts of these parameters determine the amount of energy from
an external field that can be stored in the material. The permittivity is associated with the energy
stored in terms of the electric field and permeability is associated with the energy stored in terms
of the magnetic field. The imaginary parts of these parameters are associated with the loss of
energy. They indicate how dissipative a material is to an external field. The conductivity of the
material is, in fact, related to the imaginary part of permittivity.
39
where σ is the electric conductivity and ω is the angular frequency. Another important property
of materials is the loss tangent. The dielectric loss tangent describes the energy dissipation of a
dielectric material and is defined as the ratio of the imaginary part of permittivity to real part of
permittivity.
The loss tangent is an indicator of the extent of energy being absorbed by the medium.
Thus for the medium to be an effective absorber, the loss tangent should be as large as possible.
In a similar way, the magnetic loss tangent can be defined as,
Now that all the important material parameters have been defined, consider an
electromagnetic wave propagating inside a material. This is shown in figure 2.4.
Figure 2.4 The electric and magnetic fields are perpendicular to each other and to the direction of wave
propagation. Such a wave is incident on the front end of a material with intrinsic properties ε and µ.
40
T
I
R
Figure 2.5 The incident wave (I) is partly reflected (R) at the air-material interface. The wave propagating
inside the material attenuates depending in the absorption properties of the material. The remaining
energy is transmitted out of the material (T).
Figure 2.5 shows the side view of a material as a wave is incident on it. The incident
wave (I) can be completely or partially reflected at the air-material interface, depending on the
constitutive material properties. As the wave propagates through the material it loses some of its
energy and finally the transmitted wave comes out of the material. Now the amount of signal
coming out of the material depends on losses suffered in the material in addition to the relection
losses at the interfaces. First consider the interaction of the wave with the material at the front
end. The amount of energy reflected from the front surface of the material depends on the
impedance seen by the incident wave. The reason for reflection of the wave is the impedance
mismatch. The impedance of a material is defined as,
√
The characteristic impedance of a material is defined in terms of its permittivity and
permeability. The characteristic impedance of a medium can be re-written as,
√
√
41
where Z0 is the impedance of free space which can be calculated by using the values of free
space permittivity and permeability,
√
Thus in order to minimize the front end reflection the impedance of the material, as seen
by the incident wave, should be as close to 377 ohm as possible. This impedance match and
hence minimization of reflection at the air-material interface is the first condition that has to be
satisfied in order to design an absorber. A good absorber would minimize the reflected energy
and make sure that the entire energy incident at the boundary enters the material. The transmitted
wave as it propagates in the medium continuously keeps storing the energy of the wave by
establishing electric and magnetic fields within the medium. At the same time the energy is
getting dissipated in the medium in the form of heat, which is desirable for absorbing materials.
The material should be lossy enough to completely terminate the wave inside it so that nothing is
transmitted. This is the second condition in the design of absorbers [39]. As discussed earlier, the
loss properties of a material are captured in the permittivity and permeability of the material.
For several applications, several layers of different materials are used to achieve strong
broadband absorption. A two-layer structure is shown in figure 2.6 (a). The equivalent
transmission line model used to analyze such structures is shown in figure 2.6 (b). Each layer in
this structure is characterized by its characteristics impedance and represented as the circuit
components of a transmission line.
42
1
Ei
2
Er
3
Z2
Et
Z3
Zin
d
(a)
(b)
Figure 2.6 A multilayer structure formed by stacking two materials in medium 1, and (b) the equivalent
transmission line model, where the impedance of each component is equal to the equivalent impedance
seen at the interface of the corresponding layer.
In this figure, Zin represents the input impedance seen into the transmission line which is
the impedance seen by the incident wave. For multilayered structures, Zin is determined by
taking all the successive layers into account. Using the transmission line analysis, the input
impedance can be calculated as,
where d is the thickness of the absorber and γ is the propagation constant of the wave given by,
√
This is the effective impedance as seen by the incident wave at the interface between
media 1 and 2. The overall power reflection coefficient is given by,
43
Multilayer absorbing material is often coated over a metallic body to reduce its
backscatter. It therefore becomes essential to include a metal backing when analyzing these
absorbers. In the above example if we consider medium 3 as metallic in nature, the power
reflection coefficient at the interface of medium 1 and 2 becomes,
Theoretically, the condition for zero reflection can be derived by substituting the right
hand side of equation (2.27) with zero to get,
In practice it is not possible to get a zero reflection off the front surface. In this work,
front-end reflection less than -15 dB is considered satisfactory.
44
Chapter 3
Microwave Measurement Techniques
3.1 Introduction
It was mentioned in the previous chapter that nanoferrites hold great potential for applications in
a wide variety of fields. This is more so for devices working in the higher frequency range.
However, the pre-requisite for all such developments is a detailed knowledge of the intrinsic
properties of these materials. Unfortunately, this knowledge is lacking, particularly so in the
higher frequency range. In fact most of the existing experimental techniques for the measurement
of intrinsic properties of the promising materials have limitations. Furthermore, for designing
materials for high frequency or wide bandwidth applications, it is necessary to measure material
properties over the entire operational frequency range. The high performance commercial
electromagnetic simulation tools also require accurate data of the material properties to replicate
the behavior of the material and provide results that match the actual behavior of the material
very closely. Needless to say that precise material characterization techniques are needed.
The two properties of interest here are the permittivity and permeability of the samples
under investigation. Complete information on the permittivity and permeability of a medium can
45
be obtained from the scattering parameters or s-parameters measured at the material boundaries.
We have used a transmission-reflection (T/R) based waveguide technique that employs a vector
network analyzer to measure the scattering parameters in the microwave frequency range from 2
GHz to 40GHz. Using this technique, one can generate both the reflection and transmission
spectra when the frequency of an electromagnetic wave traveling through the material under
study is changed in a continuous fashion. Based on the reflection and transmission spectra
obtained from these measurements, the frequency interval over which the sample exhibits strong
absorption can also be determined. The algorithm used to determine the permittivity and
permeability from the measured data has been modified over the years to improve accuracy.
Since the measurements are being carried out over a wide range of frequencies, the size of the
measurement components, which include rectangular waveguides and sample holder, must be
changed for each frequency band. Therefore multiple adapters and connectors have to be used.
This necessitates frequent instrument calibration which adds to the error involved in the
measurement process. The test methodology and calculation algorithms have to be modified to
compensate for these errors and provide accurate data on the properties of the materials. The
theoretical and experimental details of the technique employed in this study are discussed in this
chapter. We begin by reviewing the existing techniques available for these measurement and
their limitations.
3.2 Brief Review of Microwave Techniques
Several microwave techniques have been introduced to characterize the electrical and
magnetic properties of materials [40]. On a broad scale, these techniques can be divided into two
categories, namely, resonant and non-resonant methods. Resonant methods are based on the fact
46
that the resonance frequency and Q-factor of a dielectric resonator depend on the permittivity
and permeability values. The changes in the resonance frequency and Q-factor values with the
introduction of the sample in the resonator are used to determine the properties of the sample
[41]. The sample either forms the resonator or a part of it in the measurement circuit. Although
resonant methods provide accurate values of the material properties, their application is limited
by the fact that they can provide data only at a single frequency or at a few discrete frequency
points. Non-resonant methods, on the other hand, are applied to measure material properties over
a wider range of frequencies. The resonance method also restricts the sample size that can be
used for measurement. The resonance method is one of the several rigorous techniques used in
measuring magnetic properties of ferrites. Other proposed methods have been found to be
inefficient due to complicated experimental configurations or redundant measurement
repetitions.
Other measurement techniques include free-space methods, open-ended coaxial probe
techniques, cavity resonators, full body resonance techniques and transmission line techniques.
Although these measurements are quite reliable, the minimum diameter requirement for the
Gaussian beam in free space measurement limits the operational frequency range. The errors in
free-space measurement are primarily due to diffraction effects at the edges of the sample and
multiple reflections between the horn antennas [42]. The errors due to diffraction can be avoided
by keeping the target material diameter at least three times the diameter of the Gaussian beam.
The disadvantage of this is the large amount of sample needed at lower frequencies. This is
especially undesirable for the measurement of nano-ferrites since the fabrication is a complex
process and the samples are relatively more expensive.
47
The fundamental idea in non-resonant techniques is to let the incident wave propagate
towards the material and measure the reflected and transmitted wave amplitudes. Different types
of transmission lines can be used to direct the energy, such as, coaxial lines or metallic
waveguides. Non resonant methods are based on either reflection measurements alone or on
transmission and reflection measurements together. As the name suggests, in reflection method,
only the reflected wave is studied whereas in transmission/reflection (T/R) method, both
reflected and transmitted waves are studied. In the former case only one parameter is measured,
either the permittivity or permeability. For permittivity measurements, the material is assumed to
be non-magnetic and the thickness of the sample is chosen to be much larger than the aperture
diameter of the open-ended coaxial line used for the measurement. Shorted reflection method is
used for the measurement of permeability, assuming the permittivity to be same as the free space
permittivity. Thus this methodology is limited in its application.
The transmission/reflection method is used to measure both permittivity and permeability
values. In the T/R method, the sample is inserted inside a section of the transmission line and the
permittivity and permeability of the sample are derived from the analysis of the reflected and
transmitted waves from the two port network thus formed. Several algorithms exist to derive the
permittivity and permeability of the material from the measured transmission and reflection
coefficients [43-45]. These will be discussed in a subsequent section. The first step is to measure
the reflection and transmission coefficients of the two port system containing the sample. A
vector network analyzer is employed for this purpose. By measuring the amplitudes and phases
of the transmission and reflection coefficients of the system, a network analyzer reveals all the
network characteristics of that system. The transmission lines used for these measurements can
be in the form of hollow rectangular waveguides. One can also apply the free space measurement
48
technique instead. In free space measurement, a set of horn antennas is used to provide the
incident electromagnetic wave and detect the transmitted signal. The advantage of this technique
is that the sample can be of any convenient shape and the temperature of the sample can be
varied easily for low or high temperature measurements. However, multiple reflections between
the horn antennas and diffraction effects at sample edges lead to inaccurate results. The set-up
also becomes bulky when measuring over a wide frequency range. Here, a waveguide based T/R
measurement technique has been used to measure the parameters of the network.
3.3 Two port network
The measurement of current or voltage for rectangular waveguides is difficult at
microwave or higher frequencies [46]. At these frequencies, the network is characterized in
terms of the transmitted and reflected waves at each port. Consider a general two port network as
shown in figure 3.1.
Figure 3.1 Schematic representation of a two-port network showing the incident and reflected wave at
each port. Hollow rectangular metallic waveguides were used as transmission lines that guide the EM
wave between the sample and network analyzer.
49
An electromagnetic wave is assumed to be incident on the device under test (DUT) at
port 1. Here, the device under test can be a solid or powdered sample or a device such as
amplifier. The two ports represent the input to the DUT and the output from the DUT.
Depending on the properties of the DUT, a part of this wave is reflected at the first air-DUT
interface and the rest of the wave enters the DUT. The propagation of wave inside the DUT is
governed by the properties of the medium inside the DUT.
In the network shown here,
and
represent the incident waves and
and
represent the reflected waves at ports 1 and 2, respectively. These can also be thought of as input
and output ports of the network. The scattering parameters or s-parameters relate the incident and
reflected wave at each port. The scattering matrix (S) can be expressed as,
( )
(
)(
)
(3.1)
The equations relating the incident and reflected waves for the two port network are given by,
(3.2)
Where S11 = input reflection coefficient
S12 = reverse transmission coefficient
S21 = forward transmission coefficient
S22 = output reflection coefficient
and
are the square roots of the incident and reflected powers at port i.
The input power reflection coefficient is calculated as the ratio of the power of the
reflected wave at the input port to the power of the incident wave at the input port, with the
output incident wave set to be zero. It represents the amount of incident power that is reflected
50
back at the input port and does not reach the output port. Thus, in a well matched network, the
value of
would be close to zero.
|
(3.3)
The forward transmission coefficient is computed by taking the ratio of the output
reflected power and the input incident power with the output incident wave set to zero,
|
(3.4)
This parameter measures the amount of incident power that reaches the output port. It is
sometimes also referred to as the gain of the device under test.
In a similar manner, the rest of the parameters can be expressed as,
|
(3.5)
|
For a symmetric system,
and
are equal. As described in chapter 2, the reflected
power would be minimum when there is no impedance mismatch. Thus
and
would be
equal to zero for a well matched network.
Another important property of two port networks is reciprocity. A two-port system is said
to be reciprocal if the S matrix and its transpose are equal. In other words, the forward and
reverse transmission coefficients are equal for a reciprocal network. The condition for reciprocity
can be expressed as,
(3.7)
If in addition to the condition described above, the input and output power reflection
coefficients are also identical then the system is said to be reciprocal.
51
For the measurement of powdered samples, the objective is to measure the s-parameters
of the two port network formed by placing the sample under study between the two ports. A
vector network analyzer is used to measure the s-parameters. A network analyzer measures the
wave reflected from and transmitted through the material under test. The sample is loaded in a
waveguide between the ports of the network analyzer and an electromagnetic wave is incident on
this material. The reflected and transmitted waves are analyzed to calculate the intrinsic material
properties. Propagation of an EM wave in a rectangular waveguide is described in the next
section.
3.4 Wave Propagation in a Rectangular Waveguide
Waveguides, in general, have high power handling capability and low loss as compared
to other forms of transmission lines. Waveguides provide a restricted guided path for
electromagnetic waves to propagate. Coaxial cables, hollow or dielectric metal pipes and fiber
optical cables are all waveguides. For the measurements in this work, hollow rectangular
waveguides are used.
Figure 3.2 Geometry of a rectangular waveguide
52
Fig. 3.2 shows the geometry of a rectangular waveguide. For the purpose of the analysis,
the waveguide is considered to be homogeneous and infinitely long in the z direction. The wave,
is assumed to propagate along the z-direction, (α=0 for lossless line). Expanding the
curl equation,
⃗
⃗
(3.8)
as
̂
̂
̂
|
⃗
|
(3.9)
(3.10)
(
)
and
The spatial variation in z can be calculated,
(
)
(
Using this value to simplify the curl equations,
53
)
A
similar
expansion
can
also
be
⃗
performed
on
Ampere’s
equation,
⃗
To give,
These six equations can be manipulated to produce equations for the transverse components of
electric and magnetic fields yielding,
(
)
(
)
(
)
(
where
and
)
.
In these equations, kc is the cut-off wavenumber and k is the wavenumber in the material.
These equations are used to analyze the waves traveling inside a waveguide structure. For a
rectangular waveguide, the waves propagate in transverse electric or TE mode (
transverse magnetic or TM (
) or
) mode inside the waveguide [39]. In rectangular waveguides
we assume TE mode propagation only. For TE modes these equations reduce to,
54
(
)
(
)
(
)
(
)
The z-component of the magnetic field component must satisfy the Helmholtz wave
equation,
(
)
where ,
These equations can be solved by the method of separation of variables. The propagation
constant is,
√
where m, n = 0,1….(
(
)
(
)
) represent the mode of the propagating wave.
This equation would yield real results when
. This condition provides the
condition for wave propagation and defines the cutoff wavenumber as,
√(
)
(
)
The cutoff frequency for any mode is given by,
55
√
√
√(
)
(
)
Waves will propagate inside the waveguide when the frequency exceeds
. The mode
with the lowest cutoff frequency will therefore be the dominant mode. For TE mode, TE10 mode
has the lowest cutoff frequency given by,
√
For X-band waveguide, the cross-sectional dimensions are a = 2.286cm and b = 1.016cm.
The cut-off frequencies for different modes are calculated in Table 3.1.
TABLE 3.1
CUT-OFF FREQUENCY FOR DIFFERENT MODES INSIDE THE WAVEGUIDE
(GHz)
m
n
1
0
6.562
2
0
13.123
0
1
14.764
1
1
16.156
In the X-band region (8.2-12.4 GHz) only the TE10 mode can propagate in the waveguide
regardless of how it is excited. This is called single mode operation and is most often used as the
preferred mode for hollow waveguides. It can be shown that no TM mode can propagate in the X
band waveguide either. For TM modes, if either m or n is zero, then E field = H field = 0. This
means that no TM modes with m = 0 or n = 0 are allowable in a rectangular waveguide.
56
TABLE 3.2
CUT-OFF FREQUENCY FOR HIGHER ORDER MODES FOR X BAND
(GHz)
m
n
1
1
16.156
1
2
30.248
2
1
19.753
Rectangular, circular, elliptical and all hollow, metallic waveguides cannot support TEM
waves. Two separate conductors are required for TEM waves [39].
3.5 Experimental Set up
The block diagram in figure 3.3 is a simplified representation of the measurement set up,
excluding the internal intricacies of the devices or components, used by us to investigate the
dielectric and magnetic properties of ferrites at the microwave frequencies. It can be divided into
two main blocks, namely, the transmission unit and measurement unit. We now briefly discuss
the contents and roles of these blocks.
3.5.1 Transmission Unit
The transmission unit consists of the transmission medium used to carry signals between
the measurement planes and the sample. Rectangular waveguides are employed for this purpose.
The theory of the wave propagation in rectangular waveguides has been discussed in section 3.4.
57
Figure 3.3 Block diagram of the measurement setup
3.5.2 Measurement Unit
The heart of the measurement unit is the network analyzer used to measure the
amplitudes and phases of the transmitted and reflected waves at the two measurement ports. The
high frequency network analysis involves incident, reflected and transmitted waves traveling
along the transmission lines- waveguides in the present case. Network analysis is concerned with
the accurate measurement of the ratios of the reflected signal to the incident signal, and the
transmitted signal to the incident signal. Theory of the network analyzer has already been
discussed in section 3.3.
58
Figure 3.4 The energy incident on the material under test is partially reflected at the first material
interface and the rest of the energy is transmitted through the material.
The automatic vector network analyzer can characterize the sample material primarily in
the microwave range. It has limitation in the higher frequency range. The input signal is a
frequency swept sinusoid. A source supplies the stimulus for the test system with variable
frequency and power levels. The components of the network analyzer are discussed next.

Signal separator: It measures a portion of the incident signal to provide a reference.
This is done by using splitters or directional couplers. The second function of the signal splitting
hardware is to separate the incident and reflected waves at the input of the DUT. Couplers and
bridges are used for this purpose. The reference channel provides the reference signal against
which test channel readings are compared for making phase measurements. This two channel
scheme serves another purpose. Taking the ratio of test channel to reference channel results in
considerable cancellation of amplitude and phase noise variation that might be present on the
microwave source or receiver LO.

Detector: Scalar network analyzers use diode detectors. For both magnitude and phase
detection, the tuned receiver uses a local oscillator to mix the RF signal down to a lower
intermediate frequency. The LO is either locked to the RF or the IF signal so that the receivers in
the network analyzer are always tuned to the RF signal present at the input. The IF signal is
bandpass filtered, which narrows the receiver bandwidth and greatly improves sensitivity and
59
dynamic range. Modern analyzers use ADC and DSP to extract magnitude and phase information
from the IF signal. Tuned receivers provide the best sensitivity and dynamic range, and also
provide harmonic and spurious signal rejection. The narrow IF filter produces a considerably
lower noise floor, resulting in a significant sensitivity improvement.

Processor/Display: This is where the reflection and transmission data is formatted in
ways that make it easy to interpret the measurement results. Most network analyzers have similar
features such as linear and logarithmic sweeps, linear and log formats, Smith charts, etc.
A simplified block diagram of the network analyzer is shown in figure 3.5.
Figure 3.5 Schematic diagram of the internal components of the network analyzer.
A photograph of the set-up used to measure the powdered samples is shown in figure 3.6.
The sample is placed between the two waveguide flanges. These are connected to the ports of the
60
network analyzer via connectors and cables. A 2.4mm coaxial cable is used to connect the
network analyzer ports to waveguide ports. A 2.4mm connector is good for measurements up to
50 GHz. Since the measurements were carried out across eight different frequency bands, the
inner dimensions of the waveguides used varied between 109.22 x 54.61 mm for R band to 7.112
x 3.556 mm for Ka band. Figure 3.7 shows all the waveguides that were used in the study. In
order to match the different waveguide adapters to the coaxial cable on the network analyzer,
additional connectors had to be used.
Figure 3.6 Measurement set-up for measurement in X band. The network analyzer, coaxial cables,
waveguide flanges and adapters are shown.
61
Figure 3.7 Standard waveguides used for each band are shown. The largest waveguide is for R band and
the smallest one is for Ka band.
The dimensions of the waveguide and sample holder used for each frequency band are
summarized in table 3.3.
TABLE 3.3
PROPERTIES OF RECTANGULAR WAVEGUIDES USED IN MATERIAL CHARACTERIZATION
MEASUREMENTS
Frequency
Frequency
Cut-off
Dimensions
Sample holder
Band
(GHz)
Frequency
(inches)
thickness
(GHz)
(cm)
R band
1.70-2.60
1.37
4.30x2.15
4.52
S band
2.60-3.95
2.08
2.84x1.34
2.82
G band
3.95-5.85
3.15
1.87x0.87
1.27
C band
5.85-8.20
4.37
1.37x0.62
1.25
X band
8.20-12.4
6.52
0.90x0.40
0.50
Ku band
12.4-18.0
9.49
0.62x0.31
0.64
K band
18.0-26.5
14.06
0.42x0.17
0.45
62
Ka band
26.5-40.0
21.09
0.28x0.14
0.30
3.6 Reflection and Transmission Coefficient
The s-parameters are calculated by applying boundary conditions to the two-port network
formed with the sample. Consider a schematic representation of the two-port network as shown
in figure 3.8. Here, the material under test is placed between two transmission lines. The
thickness of the sample is L2 and the length of the transmission lines on either side of the sample
are L1 and L3. Ports 1 and 2 represent the measurement ports for the network analyzer. It is clear
from this figure that the actual measurement ports are not at the DUT interfaces. This should be
taken into account when analyzing the measured parameters. The desired measurement would be
at the sample holder walls but instead the measurement system consists of sections of
transmission lines filled with air and the material under test. To calculate the precise sample
properties, the set-up shown in figure 3.8 can therefore be divided into three regions.
Figure 3.8 Electromagnetic waves transmitting through and reflected from a sample in a transmission line.
63
The first region consists of an empty transmission line (air-filled/free space) of length
.
The electromagnetic wave is assumed to be incident at this port. This is followed by a piece of
transmission line filled with the material under study (
) and another piece of air-filled
transmission line ( ). Region 1 consists of the wave incident on the material and the wave
reflected from the air-material interface. The electric field in this region can be expressed as,
where
represents the net electric field in the transmission line region 1,
is the propagation
constant in the transmission line filled with free space, x represents the location of a point in
region 1 and
and
are the constants to be determined by applying boundary conditions on
the fields. In this expression, the first term represents the incident wave and the second term is
for the reflected wave.
Proceeding in a similar manner, the equations for regions 2 and 3 can also be written. The
second region represents the waves travelling inside the material. The third region consists of the
transmitted wave which is measured at port 2. The electric fields for these regions (
and
)
can be written as,
where, γ is the propagation constant inside the sample and the rest of the quantities are as defined
above. The wave gets reflected inside the material as well and therefore the expression for
electric field in region 2 represents two waves travelling in the opposite directions. As can be
seen in the figure, the third region consists of the transmitted wave only. The propagation
constant inside the material (γ) and in free space ( ) are,
64
√
(
√( )
(
)
)
where ω is the angular frequency, c is the speed of light in vacuum and
is the cutoff
wavelength of the transmission line. The constants Ci mentioned in equations (3.35) – (3.37) can
be determined by applying the boundary conditions at the interface. The continuity of the
tangential components of the electric field at the two interfaces give,
|
|
|
|
The lengths L1 and L2 are indicated in figure 3.8. The boundary condition on the magnetic field
requires an additional assumption that no surface currents are generated so that the tangential
component of magnetic field is continuous across the interface giving,
|
|
|
|
Applying these boundary conditions the solution for the s-parameters of the two-port
network for the case of the symmetric scattering matrix, S12 = S21 gives,
65
where R1 and R2 are the reference plane transformations at the two ports, given by:
here, L1 and L3 are the lengths of the air line between the ports and sample interface.
Here, z represents the reference plane transformation for the wave being transmitted in the
sample of length
.
Additionally, S21 for the empty sample holder is given as,
The reflection coefficient can also be expressed in terms of permeability (µ) and
propagation constant (γ),
The next step is to calculate the values of permittivity and permeability from the data
collected for s-parameters. For the case of non-magnetic materials, the unknown quantities are
the real and imaginary parts of permittivity, length of material (L2) and the reference plane
transformations, R1 and R2. The length of sample is considered unknown since the position of the
sample inside the transmission line may not always be known precisely. However, in some cases
it can be a known quantity. Thus the system consists of nine real equations for five unknown
quantities. The four complex equations (3.44, 3.45, 3.46, and 3.49) are actually eight real
equations. The ninth equation is for the total length of the transmission line,
66
For the case of magnetic materials, the real and imaginary parts of permeability are
unknown, making the number of unknown quantities to be seven. Thus the system of equations
is over determined and several methods can be used to calculate the unknown parameters.
3.7 Determination of Permittivity and Permeability
Several algorithms exist for determining the permittivity and permeability of the sample
from the above results. The Millimeter and Sub-Millimeter Waves Laboratory at Tufts
University has an improved version to increase the accuracy of the measurements [47].
Nicolson and Ross [43] developed a broadband simultaneous measurement technique
using coaxial transmission line. In their algorithm, they combined the equations for S11 and S21
and derived explicit formulas for the calculation of permittivity and permeability. Nicolson and
Ross derived S21 and S11 from time domain measurements using Fourier transform. This method
had two major shortcomings. First, the determination of permeability and permittivity is bandlimited, depending on the time response of the pulse and its repetition frequency. Secondly, in
using discrete Fourier transform, errors arise due to truncation and aliasing. According to their
algorithm, the power reflection coefficient was calculated as:
√
where,
The reflection coefficient is defined as the ratio of the reflected signal to the incident
signal. Therefore, the value of reflection coefficient is always less than unity. It is equal to unity
67
in case of completely reflective surfaces. The positive or negative sign in equation (3.51) is
chosen such that the condition,| |
Here,
, is satisfied. The transmission parameter is calculated as:
is the free space permeability and µ is the permeability of the sample. Using
these values the permittivity and permeability are calculated as,
√( )
[( )
(
[
( )]
( )
)]
where
It should be noted here that the argument of the natural logarithm function in Eq. (3.56)
consists of a complex number. The phase factor of 1/z in Eq. (3.56) repeats its value after a phase
change of 2πn. Therefore the phase of the transmission coefficient does not change when the
sample length increases by multiple of wavelength.
To overcome this ambiguity, Weir
introduced the use of group delay to accurately determine permeability and permittivity. The
measured value of group delay is compared to the calculated value in order to find the correct
root. Group delay through the material is strictly a function of the total length of the material.
Therefore phase ambiguity can be resolved by calculating the value of group delay using the
derived permittivity and permeability values,
68
√
The value of group delay thus computed is compared with the measured value of group
delay, which is determined from the slope of the phase of the transmission coefficient (  ) versus
frequency using the following equation,
The correct root should satisfy,  g ,n
 g  0 .
Thus phase ambiguity can be resolved by matching the calculated and measured group
delay. But this is not a very consistent method.
Another limitation of the existing methods is that the solution becomes divergent at
frequencies corresponding to half-wavelength in the sample [45]. At these frequency points, the
magnitude of S11 becomes very small and the uncertainty in the phase becomes very large. Since
the solution is proportional to
, the equations become algebraically unstable as S11 0. Many
researchers use samples that have a length less than nλ/2 at the highest measurement frequency
to resolve this issue. But the use of thin samples lowers the measurement sensitivity due to
uncertainty in reference plain positions [48]. Therefore, a major limitation of Nicolson-RossWeir algorithm is that the results are unstable at certain frequencies.
James Baker-Jarvis developed an iterative method to calculate the value of the
permittivity. This proposed method minimizes the instability of the equations used by NicolsonRoss-Weir and allows measurements to be taken on samples of arbitrary length. Sample length
and air length are treated as unknowns in this system of equations. The solution is therefore
69
independent of reference plane position and sample length. For example, one useful combination
that was proposed is shown below,
{[
]
[
]}
here, β is a function of sample length, uncertainty in s-parameter values and loss characteristics
of material. For low loss materials, S21 is strong and β is zero whereas for high loss materials S11
dominates, so large value of β is appropriate. In general, β is given by ratio of the uncertainties in
S21 and S11. Since this is an iterative procedure, an initial value is required to start the
calculations. The solution for permittivity obtained from Nicolson-Ross-Weir algorithm can be
used as a starting point.
All the modifications suggested so far either need an initial guess parameter or choice of
appropriate integer values of phase. A novel technique for the measurement of samples with
arbitrary length was developed at the Tufts Millimeter and Sub-Millimeter Waves laboratory.
This technique allows the simultaneous calculation of the permittivity and permeability of the
sample by using a modified value of the transmission coefficient. This method does not require
initial guess parameters, thus making the measurement set up more accurate. The first advantage
of this algorithm is that it allows the measurement of different thickness values of the sample. In
previous work, the sample is required to completely fill the sample holder. However, the
proposed algorithm corrects for the presence of air between the sample and waveguide
interfaces, in case the sample does not occupy the entire volume of the sample. It should be
noted that this still requires that the surface of the powder inside the sample holder is smooth and
flat. This is especially helpful when measuring across various frequency bands since it allows to
minimize the variation in the density of the sample. In other methods, the air gap between the
sample and sample holder edges is corrected by filling it up with a conducting material [49]. But
70
this affects the calculated parameter values as well and the sample dilution must be accounted
for. The modified s-parameters used in the measurements in our work are given below,
(
̃
√
(
̃
)
√
)
where, l is the quarter wavelength difference between thru and line in air, d is the thickness of
the sample inside the waveguide, k0 is the wavenumber in the sample and kc is the cutoff
wavenumber. These equations take into account the effect of using samples with thickness (d)
values that are smaller than the waveguide shim used in the experimental setup.
In the measurements performed in this study, a phase unwrapping technique was used to
resolve this phase ambiguity. Whenever the jump in the value of the phase from one
measurement frequency to the next is more than π, all subsequent phases are shifted by 2π in the
opposite direction.
It was also noticed that the permeability and permittivity of the loaded sample affect the
cut-off frequency for the waveguide band. This was accounted for in the calculations by
including the cut-off frequency for each band in the derivation of permeability and permittivity
from the data for s-parameters. The propagation constant inside the waveguide can be calculated
as,
| |
Considering the effect of loaded material’s magnetic and dielectric properties in the calculation
of the cutoff wavelength of the propagation constant, we get
71
(√( )
√
(
) )
( )
Using this, we can write the equations for the determination of permittivity and permeability as
[50],
(
)(
| |
)
√( )
(
( ) (
)(
)(
(
)
) (√( )
| |
)
(
) )
where, Г is the reflection coefficient, T is the transmission coefficient,
propagation constant for the
and
are
mode with and without the material inserted in the waveguide,
a is the longer dimension of the rectangular waveguide,
is the phase of the transmission
coefficient and d is the material thickness.
In case of rectangular waveguide measurements, the permittivity and permeability can
also be expressed in terms of the equivalent impedance and refractive index as [51],
⁄
(
)
⁄
(
where
)
(
)
is the wave impedance and N is the complex refractive index, already
defined in chapter 2. The permittivity and permeability can now be defined in terms of the
equivalent impedance (
) and equivalent refractive index (
72
) as,
The experiments carried out in the lab suggest that these modifications are necessary and
known materials were measured to confirm the accuracy of the measurement technique. The
derived permeability and permittivity data is very reliable and not effected by the scattering
voltage ratios of the vector network analyzer. This method is also applicable to non-reciprocal
circuits. Previous methods depend on the values of all four S parameters and required that S11 =
S22 and S21 = S12. Since the proposed method uses only the values of parameters S11 and S21, it
can be used to calculate the permittivity and permeability for non-reciprocal networks as well.
Finally the relative permittivity of the material is calculated as,
(
here,
)
, a is the longer dimensions of the rectangular waveguide.
73
Chapter 4
Measurement Results at Microwave
Frequencies
4.1 Introduction
The results of the measurements described in the previous chapter are presented and discussed in
this chapter. A considerable amount of useful information can be derived from these results. In
addition to the permittivity and permeability values of a number of ferrite materials, the loss
factor and reflection coefficient of the samples are also obtained. These characteristics help
predict the possible use of these materials in different applications such as the radar absorbing
materials and microwave filters. Wherever published data is available in literature, the properties
of the nano-powders have been compared with the properties for the same composition of
samples in bulk form. A few deviations from the expected results have been observed and these
have been discussed in the following chapters. By controlling the material type (dielectric or
magnetic) and its thickness, loss factor, impedance and internal design, the performance of a
microwave absorbing material can be optimized for a single narrow band frequency, multiple
frequencies, or over a broad frequency spectrum.
74
A brief description of the nanopowders analyzed in the microwave and millimeter wave
frequency ranges is enumerated in Table 4.1. The particle sizes mentioned in this table are as per
the manufacturer’s data sheets. In addition to these samples, a solid sample of Teflon was also
studied using the waveguide technique in each frequency band. This is used as a reference
material to validate the ability of the proposed algorithm for determining the permittivity and
permeability values from the measured s-parameters. Powders of barium ferrite and strontium
ferrite consisting of micro-size particles were also studied to observe if the differences in the
properties of the nano- and micro-size ferrites can be detected using the proposed measurement
technique.
TABLE 4.1
Sample
LIST OF NANO-SIZE POWDERS INVESTIGATED
Chemical Formula
Sample Description
Barium nano-ferrite
BaFe12O19
Avg. particle size 50-100 nm
Barium micro-ferrite
BaFe12O19
Avg. particle size 5-50 µm
Strontium nano-ferrite
SrFe12O19
Avg. particle size 50-100 nm
Strontium micro-ferrite
SrFe12O19
Avg. particle size 5-50 µm
Copper Iron Oxide
CuFe2O4
Avg. particle size 50-100 nm
Copper Zinc Iron Oxide
CuFe2O4Zn
Avg. particle size 50-100 nm
Nickel Zinc Iron Oxide
Fe2NiO3Zn
Avg. particle size 50-100 nm
La0.6Sr0.4Co0.2 Fe0.8O3-δ
Avg. granular size 30 – 40 µm
La0.8Sr0.2MnO3
Avg. granular size 40 µm
(ZrO2)92(Y2O3)8
Avg. granular size 0.5 µm
Lanthanum Strontium Cobalt
Ferrite (LSCF)
Lanthanum Strontium
Manganite (LSM)
Yttrium Stabilized Zirconia
75
Lanthanum Strontium Cobalt Ferrite, Lanthanum Strontium Manganite, Yttrium
Stabilized Zirconia and micro-size Barium and Strontium ferrite samples were purchased from
Trans-tech, Inc. and the rest of the samples were obtained from Sigma Aldrich.
The powders are filled inside the sample holder that is placed between the waveguides. A
transparent tape is used on one side of the sample holder to prevent the powdered sample from
falling off. The use of tape does not affect the measurement results. This was verified on the
solid Teflon sample. The sample holders used in the measurement process are shown in figure
4.1. Sample holders for G, C and Ku frequency bands are shown here.
Figure 4.1 Empty sample holders used for the measurement of G, C and Ku frequency bands.
4.2 Teflon Results
Teflon has long been used for various household and industrial applications. The small value of
the loss tangent at high frequencies makes it useful for applications in electrical engineering.
The dielectric constant of Teflon is known to be around 2.1.
76
The complex values of the
permittivity and permeability of Teflon have been reported to be ε = 2.045 - j0.00045, and µ = 1
.00 – j0.00, respectively [49]. Since the dielectric constant of Teflon is almost twice the dielectric
constant of air, the waves inside this material would cover larger electric lengths than the waves
travelling in free space of the same physical extent. Teflon is often used as a substrate in
electrical industry, as insulators in the form of spacer or tubing and in the production of discrete
electrical components, such as the capacitors.
Teflon has been reported to have a stable permittivity value of about 2.03 in X band and
loss tangent value of less than 0.0004 [44]. Since it is non-magnetic, the expected value of real
permeability is close to unity. The s-parameters of the Teflon samples were measured in each of
the eight microwave frequency bands between 2 to 40 GHz. The algorithm presented in chapter 3
was used to derive the permittivity and permeability values from the measured data.
Solid samples of teflon that tightly fit the sample holder for each frequency band were
used for these measurements. The average values of the real and imaginary parts of the teflon
permittivity obtained for each frequency band are listed in table 4.2. These values are in close
agreement with the known values for Teflon. The permeability of Teflon samples was also
measured but did not show much variation with frequency. The average permeability across the
measurement frequency range from 1.77 GHz to 40 GHz was close to 1.032. The close match
between the expected and observed permittivity and permeability values of Teflon validates the
measurement technique and proposed algorithm.
77
TABLE 4.2
Frequency
Band
DIELECTRIC PROPERTIES OF SOLID TEFLON SAMPLES
Average value
Sample
Imaginary
Thickness (cm)
Real Permittivity
Permittivity
Dielectric Loss
Tangent
R
10.9
2.0628
0.000571
0.000277
S
7.241
2.0577
0.000519
0.000252
G
1.27
2.0497
0.000484
0.000236
C
1.25
2.0423
0.000416
0.000204
X
0.50
2.0375
0.000402
0.000198
Ku
0.64
2.0321
0.000387
0.000191
K
0.45
2.0252
0.000344
0.000169
Ka
0.30
2.0181
0.000315
0.000156
We now present the results for different ferrite powders investigated in this work. The
permittivity and permeability values of these samples were obtained using waveguide based
transmission/reflection technique described in chapter 3. In preparing samples for measurement,
the powders were filled inside the waveguide shim with one end taped to keep the powdered
samples in the sample holder. The effect of using an adhesive tape has been discussed in more
detail in the error analysis (chapter 7). Sufficient care was taken while loading the samples in the
waveguide shim to make sure that the powder is tightly packed, without any air gaps. The
powder is added in steps and pressed at every stage to prevent the formation of agglomerates and
air gaps inside the sample holder. We expect the samples to maintain uniform density during
measurement.
78
4.3 Nano-size hexagonal ferrites
The generic formula for M-type ferrites is MO.6Fe2O3, where M can be any divalent ion. The
two hexaferrite samples studied here are barium (BaO.6 Fe2O3 or BaFe12O19) and strontium
hexaferrites (SrFe12O19). Compared to ferrites with inverted spinel or garnet structures,
hexagonal ferrites have larger intrinsic magnetic fields. Due to the hexagonal symmetry of the
structure, such ferrites have a major preferred axis, i.e., there are certain preferred directions
along which it is easier to magnetize the material. This makes them highly anisotropic. As a
consequence of the effectively large internal magnetic anisotropy, these materials remain
magnetized even after the external applied field is removed. This diminishes the need for an
external magnetic field and makes M-type hexagonal ferrites very useful in the design of high
frequency devices. Bulk hexagonal ferrite samples have been extensively used in microwave
applications such as transformers and circulators. Due to the growing interest in these materials,
the basic characteristics of nano-sized M-type hexaferrite powders have been extensively studied
[52].
Recently these materials are also being investigated for use as microwave absorbers [5356] due to their large magnetic losses in the microwave range. The hexagonal ferrites are suitable
as radar absorbing materials due to their large permeability values and favorable dielectric
properties at microwave frequencies. Several substituted barium hexaferrite samples are being
designed for such applications [57-60]. Substitution for the Fe3+ and Ba2+ ions is an effective
method to vary the magnetic properties of barium hexaferrite. Common substitutions include ZnTi, Ni-Zr or Ti-Cu. It has been found that the absorption properties can be manipulated by
varying the amount of substitution [61].
79
The permittivity and permeability of these materials have been extensively studied in
bulk form [62]. When materials are produced in powder form, depending on the particle size, the
microwave behavior may change. However, not much has been reported on these materials in
powder form. This could be due to the lack of accurate high frequency characterization
techniques suitable for the powder samples.
The variation of the real and imaginary parts of the permittivity and permeability of
barium hexaferrite nanopowders at microwave frequencies are shown in figure 4.2. It should be
noted that the plots shown here are generated from average values over certain frequency points.
The data is collected at 201 frequency points for each frequency band. This would result in a
total of 1608 frequency points for the entire data. Since it is difficult to show a plot with such a
large number of data points, only certain selective data points are marked in the plots. It is
immediately clear that the measured values of the real part of the permittivity for nano-powders
are much smaller compared to the known values of real part of permittivity of bulk barium
hexaferrite, which lie between 13 and 16 [63, 64]. These numbers refer to the real part of the
permittivity. The real part of the permeability of barium hexaferrite in bulk form has been
reported to be as high as 6 [63]. It should be noted that the bulk properties are reported for solid
samples whereas the samples studied here are being measured in powder forms. The
permittivities for M-type barium and strontium ferrites in bulk form have been reported in
literature to lie between 14 and 16 [64, 65]. These are significantly different from the values
measured here for their powder samples. In another study, a different measurement technique,
namely, cavity resonator technique was applied to study powder samples of barium and
strontium hexaferrites [62]. The particle sizes were much bigger than the samples studied here.
The important point to note, however, is that the parameter values reported by them differ
80
significantly from the values for bulk samples. For an approximate powder size of 100µm, the
average real parts of the permittivity for barium and strontium ferrite at microwave frequencies
were reported to be 2.5 and 2.6, respectively. The magnetic and dielectric properties of the
sample are affected by the state of the sample, i.e., whether bulk or particulate form. The reasons
for the difference between bulk properties and micro- or nano-size powder samples is further
investigated in chapter 6.
1.6
1.4
Parameter Value
1.2
1
0.8
Real Permeability
Imaginary Permeability
Real Permittivity
Imaginary Permittivity
0.6
0.4
0.2
0
2
5
8
11
14
17
20
23
26
29
32
35
38
Frequency (GHz)
Figure 4.2 Variation of permeability and permittivity of barium hexaferrite nano-powder with frequency.
The average values of the imaginary permittivity and permeability are 0.0516 and 0.0706, respectively.
For
sample density of 0.4004 g/cm3, the average values of the real and imaginary parts of
the permeability of barium ferrite are 1.0642 and 0.0706 while the average permittivity values
are 1.4647 and 0.0516, respectively. The value of real part of permittivity shows slight variation
with frequency. The real part of permeability is fairly constant over the measured frequency
range, except for the drop at the end of Ka band. For some of the bands, a trivial variation in the
81
parameter value at the band extremities is observed during measurements. This manifests into a
discontinuity in the data plots at frequencies that correspond to the end of one band and
beginning of the successive band. As can be seen in figure 4.2, this difference in values from one
frequency band to the next is very small. Since the same density of the sample was used for
measurement at each frequency band the data continuity is good.
The non-zero values of the imaginary parts of permittivity and permeability suggest that
the ferrites are lossy in nature. This means that an electromagnetic wave propagating through
these materials would lose energy. This is an essential property for absorbing materials. The
dielectric and magnetic loss tangent values for each sample are listed in Table 4.3. Another
important observation here is that the permittivity and permeability values are very close to each
other. This means that the impedance seen at the front end of the material would be close to the
free space impedance (377 ohm) and hence there will be very little reflection at the air-ferrite
interface. This satisfies the first condition for the absorbing materials described in chapter 2 and
thus these powders may be useful as non-reflecting coatings for microwaves. The impedance
values shown here have been calculated from the permittivity and permeability values using the
following equation,
√
√
The frequency dependence of impedance thus obtained is shown in figure 4.3. It can be
observed that the front end impedance of barium ferrite nanopowder approaches the free space
impedance value of 377 ohm towards the high end of microwave frequencies.
82
340
Impedance
335
330
325
320
315
2
6
10
14
18
22
26
30
34
38
Frequency (GHz)
Figure 4.3 Variation of front-end impedance of barium hexaferrite nano-powder with frequency
Bulk barium hexaferrite is known to have resonance frequency around 48 – 50 GHz [64].
Results of our measurement on ferromagnetic resonance frequency of Barium hexaferrite powder
are presented in chapter 5. It suffices here to state that resonance reqeuncy of barium hexaferrite
powder also falls within the above range. The resonance corresponds to the maximum absorption
frequency of the sample. From the data presented here, it can be predicted that nano size-barium
ferrite should be an excellent microwave absorbing material at relatively lower frequencies, near
the higher end of Ka band. In order to better understand the performance of the sample as an
absorber, we calculate the front end reflectance. In chapter 2 the front end reflectance has been
shown to have the form,
Here, medium1 is air and medium 2 is nano-size barium ferrite powder. The power
reflection coefficient Γ, when multiplied by the incident energy gives the amount of energy that
gets reflected from the air-sample interface due to impedance mismatch between the two media.
83
Figure 4.4 gives the frequency dependence of Γ. This behavior is as expected. As the front end
impedance becomes close to 377 ohm, the reflected energy decreases considerably. At
frequencies higher than 34 GHz, the reflectance is less than -10dB, which is a practical value for
an absorber. The performance improves significantly beyond 38 GHz with power reflection
coefficient touching -25 dB at 38 GHz. Plots 4.3 and 4.4 are mutually consistent and suggest that
Barium hexaferrite nanopowder should have excellent non-reflecting properties at higher
frequencies. The magnetic loss shows an increase. The real part of permittivity decreases in
magnitude, becoming closer to the permeability value near the frequency of resonance. The
reflectivity results shown below are for a sample of thickness of 2 cm.
Frequency (GHz)
2
6
10
14
18
22
26
30
34
38
Reflection Coeeficient (dB)
0
-5
-10
-15
-20
-25
-30
Figure 4.4 Variation of reflectance of 2cm thick sample of barium hexaferrite nano-powder with
frequency.
The properties of strontium hexaferrite nanopowder are discussed next. Figure 4.5 shows
the variation of permeability and permittivity of strontium hexaferrite nanopowder with
frequency. The overall variation pattern in this case is quite similar to the one found for the
84
corresponding barium based nanopowder (figure 4.2). The values of the permittivity and
permeability are somewhat higher for the strontium nanopowder than for the barium
nanopowder. The average values of real parts of permittivity and permeability obtained for
strontium hexaferrite are 1.689 and 1.088, respectively. Further, the real part of the permeability
of strontium hexaferrite nanopowder does not vary much with frequency except perhaps towards
the higher end of frequencies used in our measurements. The magnetic loss factor for the
strontium nanopowder shows a minor increase towards the higher end of the frequency range.
The imaginary part of permittivity is very small and would not contribute much towards
absorption in this case for the strontium nanopowder. However, the magnetic loss is considerably
higher. We can therefore expect significant energy attenuation during propagation in strontium
hexaferrite nanopowder.
2
1.8
Parameter Value
1.6
1.4
1.2
1
0.8
Real Permeability
Imaginary Permeability
Real Permittivity
Imaginary Permittivity
0.6
0.4
0.2
0
2
.
6
10
14
18
22
26
30
34
38
Frequnecy (GHz)
Figure 4.5 Variations of permittivity and permeability of strontium hexaferrite nano-powder with
frequency.
85
Characteristic Impedance
312
310
308
306
304
302
300
298
296
2
6
10
14
18
22
26
30
34
38
Frequency (GHz)
Figure 4.6 Variation of front-end impedance of strontium hexaferrite nano-powder with frequency.
The difference between real parts of permittivity and permeability of Strontium
hexaferrite nanopowder is small but somewhat larger than what was observed for barium
hexaferrite nano-powder. As a result, its characteristic impedance in this frequency range is not
as close to the free space impedance value of 377 ohm as it was in the case of barium hexaferrite
nanopowder. The variation of impedance of strontium hexaferrite nano-powder with frequency is
shown in figure 4.6. In this frequency range, the highest impedance value is 310 ohm at about 30
GHz. The fluctuation in values after that frequency could be due to measurement inaccuracies.
But it can be concluded that the impedance rises towards the end of the frequency range
investigated here, just as in the case of Barium hexaferrite nanopowder.
The variation of power reflection coefficient for Strontium hexaferrite nanopowder
shown in figure 4.7 is in general agreement with its impedance values. The impedance was
observed to be best matched at 30 GHz and this corresponds to the sharp -16 dB minimum in
reflectance observed in figure 4.7.
86
Frequency (GHz)
2
6
10
14
18
22
26
30
34
38
Reflection Coefficient (dB)
0
-4
-8
-12
-16
-20
Figure 4.7 Variation of reflectance of 2cm thick sample of barium hexaferrite nano-powder with
frequency
The results of measurement for the two nano-size hexaferrites are summarized in table
4.3. The data in this table provide only the average value of each parameter across the wide
frequency range from 2 to 40 GHz.
Material
TABLE 4.3
DIELECTRIC AND MAGNETIC PROPERTIES OF HEXAFERRITE NANOPOWDER
Dielectric Permittivity
Magnetic Permeability
Density
Real
Imaginary
Loss
Real ImaginLoss
(g/cc)
Part
Part
Tangent
Part ary Part Tangent
Bulk
Permitti
vity
Barium
1.464
0.052
0.016
1.064
0.071
0.041
0.4004
16 [64]
1.689
0.030
0.018
1.088
0.0658
0.060
0.3531
14[65]
Hexaferrite
Strontium
Hexaferrite
87
It is clear from figures 4.2 and 4.5 that the parameter values are somewhat frequency
dependent. For instance, the dielectric loss tangent value is very small at lower frequencies but
increases steadily with frequency. This behavioral phenomenon cannot be captured in average
values and thus the frequency dependent analysis is essential. A single average value for any
parameter undermines the fact that the value is high at certain frequencies and falls at others.
For comparison, micro-size samples of Barium hexaferrite were also studied. The
frequency dependence of real and imaginary parts of permittivity and permeability for this
powder is shown in fig 4.8. The average value of real parts of permittivity and permeability for
the micro-size sample were found to be 3.065 and 1.1069, respectively. An interesting
observation here is that the values for the micro-size powders are different from those measured
for the nano-size powder. This is explored in more detail in chapter 6.
3.5
Measured Value
3
2.5
Real Permeability
Imaginary Permeability
Real Permittivity
Imaginary Permittivity
2
1.5
1
0.5
0
2
8
14
20
26
Frequency (GHz)
32
38
Figure 4.8 Material properties of Barium ferrite micro-powders are shown. The average value of real and
imaginary parts of permittivity is observed to be 3.065 and 0.0559 whereas real and imaginary parts of
permeability are 1.1069 and 0.0536, respectively.
88
4.4 Nano-size spinel ferrites
Spinel ferrites generally have a cubic crystal structure. Their general formula is MFe2O4, where
M is a divalent cation. Common examples of divalent cations include manganese, zinc, nickel
and copper. Spinel ferrites have been utilized as absorbing materials in various forms for many
years due to their large magnetic losses and large resistivity [66-67]. Nickel-zinc spinel ferrites
have been prepared with different amounts of substitution for use as absorbers in the ISM band
(at a frequency of 2.4 GHz) for mobile devices [68]. As was seen in the case of hexaferrites,
within the GHz range, the permittivity of a ferrite is almost constant with its imaginary part close
to 0. Therefore, the dielectric loss is negligible and its absorbing performance mainly depends on
the magnetic loss. However, in the microwave region, the application of spinel ferrites are
limited at the lower end of the microwave frequency 1-3 GHz because of their lower natural
resonance frequencies in comparison with those of other types of ferrites, such as hexaferrites
[66-67]. One of the frequently used methods to tune the complex permeability of spinel ferrites is
to dope the ferrites with many metallic ions, such as copper, zinc, cobalt or manganese. In order
to do this, it is necessary to know the precise values of permittivity and permeability and the
relation between material properties and microwave absorption. For instance, Co-doped NiZn
spinel ferrite or multilayer copper substituted nickel-zinc spinel ferrite have been used for
absorption in the microwave region [69]. Thus the performance of spinel ferrites can be extended
to microwave region by substituting them with metallic ions such as copper, zinc, cobalt, lithium
or manganese. In order to do this in an efficient way and get desired performance, it is important
to know the constitutive properties of the spinel material first.
A remarkable characteristic of spinel structure is that it is able to form an extremely wide
variety of solid solutions [70]. This means that the composition of a given ferrite can be easily
89
modified, without altering the basic crystalline structure. In this work, three spinel ferrite powder
samples were studied, each consisting of particles with sizes in the nanometer domain. The
results of our measurements for each sample are presented here.
Figure 4.9 shows the permittivity and permeability of copper iron oxide. The average
values of the real parts of the permittivity and permeability of this nano-ferrite are obtained to be
1.507 and 1.03, respectively. Real part of the permeability shows only marginal changes with
frequency but real part of permittivity undergoes considerable changes over the frequency range
investigated here. Unlike the hexaferrite samples, the loss values are observed to be higher at the
lower frequency end and decrease in the higher frequency range.
1.8
1.6
Parameter Value
1.4
1.2
1
0.8
0.6
0.4
Real Permeability
Imaginary Permeability
Real Permittivity
Imaginary Permittivity
0.2
0
2
6
10
14
18
22
26
Frequency (GHz)
30
34
38
Figure 4.9 Variations of permittivity and permeability of Copper Iron Oxide nanopowder with frequency.
The impedance values for copper iron oxide are shown in figure 4.10. The characteristic
impedance is closest (though not very close) to free space impedance at low frequencies and then
seems to fall with frequency but not so smoothly. This is generally consistent with the
90
reflectance variations obtained for the air-sample interface (Fig. 4.11). The reflectance
characteristic reaches -20 dB at low frequencies and subsequently worsens as the frequency
increases.
Characteristic Impedance
320
316
312
308
304
300
296
2
6
10
14
18
22
26
30
34
38
Frequency (GHz)
Figure 4.10 Variation of front-end impedance of Copper Iron Oxide nano-powder with frequency
Frequency (GHz)
2
6
10
14
18
22
26
30
34
38
0
Reflectance (dB)
-5
-10
-15
-20
-25
Figure 4.11 Variation of reflectance of 2 cm thick sample of Copper Iron oide nano-powder with
frequency
91
Next, the results for Copper Zinc Iron Oxide are presented. Fig. 4.12 shows that the
addition of zinc results in a higher value of the real part of the permittivity and substantially
higher magnetic loss. The real part of the permeability is still close to unity but since the real part
of the permittivity is considerably higher than that of copper iron oxide, the characteristic
impedance of copper zinc iron oxide sample is farther removed from free space value. As a
result, it shows poor front end reflection performance. The front end impedance and reflectance
variations of copper zinc iron oxide are shown in figures 4.13 and 4.14, respectively. The
reflectance value does not reach the -10 dB limit.
2
Parameter Value
1.6
1.2
0.8
0.4
Real Permeability
Imaginary Permeability
Real Permittivity
Imaginary Permittivity
0
2
6
10
14
18
22
26
30
34
38
Frequency (GHz)
Figure 4.12 Variation of permittivity and permeability of Copper Iron Oxide nanopowder with frequency
92
310
Characteristic Impedance
308
306
304
302
300
298
296
294
292
2
6
10
14
18
22
26
30
34
38
Frequency (GHz)
Figure 4.13 Variation of front-end impedance of Copper Zinc Iron Oxide nano-powder with frequency
Frequency (GHz)
2
6
10
14
18
22
26
30
34
38
0
Parameter Value
-1
-2
-3
-4
-5
-6
-7
-8
Figure 4.14 Variation of reflectance of 2 cm thick sample of Copper Iron oxide nano-powder with
frequency
93
The third spinel ferrite sample studied by us is the nickel zinc iron oxide. The variations
of permittivity and permeability with frequency for this sample are shown in figure 4.15.
1.6
Parameter Value
1.4
1.2
1
0.8
0.6
0.4
Real Permeability
Imaginary Permeability
Real Permittivity
Imaginary Permittivity
0.2
0
2
6
10
14
18
22
26
Frequency (GHz)
30
34
38
Figure 4.15 Variation of permittivity and permeability of Nickel Zinc Iron Oxide nanopowder with
frequency bands
The material properties of nickel zinc iron oxide have been studied in bulk form at lower
microwave frequencies in the S band [71]. The permittivity and permeability plots of figure 4.15
show that the nickel zinc iron oxide is expected to perform well at high frequencies due to the
close proximity in the values of the real parts of these parameters. The lower value of
permittivity results in good impedance matching at the air-sample interface. The loss factors are
not very high but compare favorably with barium hexaferrite loss factors. The impedance
variations are shown in Fig. 4.16. At higher frequencies, the impedance is fairly close to the free
space value. This is reflected in the high reflectance value (nearly -25 dB) at these frequencies
(Fig. 4.17).
94
The average values of the real parts of the permeability and permittivity of Copper Zinc
Iron Oxide nano-powder in the frequency range explored by us are 1.058 and 1.313, respectively.
These are much lower than the values reported for bulk samples. For bulk samples of these
ferrite, the real part of the permeability has been reported to be around 18.42 at 2.44 GHz and
15.12 at 3.98 GHz and the imaginary part of permeability has been reported to be 0.825 and
0.615 at 2.44 GHz and 3.98 GHz, respectively.
Characteristic Impedance
355
350
345
340
335
330
325
320
2
6
10
14
18
22
26
30
34
38
Frequency (GHz)
Figure 4.16 Variation of front-end impedance of Copper Zinc Iron Oxide nano-powder with frequency
95
2
6
10
14
Frequency (GHz)
18
22
26
30
34
38
Reflection Coefficient (dB)
0
-5
-10
-15
-20
-25
-30
Figure 4.17 Variations of reflectance of 2 cm thick sample of Copper Iron oxide nano-powder with
frequency
Results of measurement on spinel ferrites are summarized in table 4.4.
TABLE 4.4
Material
DIELECTRIC AD MAGNETIC PROPERTIES OF SPINEL FERRITES
Dielectric Permittivity
Magnetic Permeability
Density
Bulk
(g/cc)
Properties
0.085
0.5204
-
0.150
0.5454
-
Real
Imaginary
Loss
Real
Imaginary
Loss
Part
Part
Tangent
Part
Part
Tangent
CIO
1.507
0.075
0.0499
1.03
0.087
CZIO
1.637
0.0618
0.0377
1.037
0.156
µ= 15.12
NZIO
1.313
0.037
0.028
1.058
0.091
0.086
0.2754
[71]
96
4.5 Nano-size Rare Earth Ferrites
Oxides of rare earth metals are often classified as orthoferrites. These ferrites, in general, have
perovskite crystal structure, with formula MFeO3, where M represents a rare earth trivalent
cation such as Yttrium [72]. These materials are classified as ferrites as different cations occupy
different sub-lattices. This results in a net magnetic moment, making these materials behave as
ferromagnetic materials.
Current problems of microelectronics strongly call for materials with large
magnetoresistance at room temperature [73]. From this standpoint manganite perovskites with a
general formula Ln1-xAxMnO3, where Ln is a trivalent cation like La and A is a divalent alkalineearth cation like Ca, Sr, Ba, are very promising compounds demonstrating colossal
magnetoresistance (CMR) effect, by many orders of magnitude exceeding the giant
magtoresistance of multilayered films and granulated systems [74]. In addition to CMR,
manganite perovskites show a variety of other interesting magnetic and electrical properties,
including ferromagnetic and antiferromagnetic ordering of manganese magnetic moments,
charge ordering of the Mn3+ and Mn4+ cations and orbital ordering of the manganese 3d orbitals
[74-76]. These magnetic nanoparticles have great potential for use in a wide range of
applications, including magnetic recording media, sensors, catalysts, magnetic refrigeration,
medicine, etc. Lanthanum Strontium Manganite (LSM) nanoparticles have been used as
mediators for hyperthermia applications [77]. Thus these materials have generated a lot of
interest in the past years. We therefore decided to investigate the magnetic and dielectric
properties of nanoparticles of some of manganite perovskites at microwave frequencies. The
materials studied by us are Lanthanum Strontium Manganite (LSM), Yttrium stabilized Zirconia
(YSZ) and Lanthanum Strontium Cobalt Ferrite (LSCF).
97
Making measurements on the samples of these materials was most challenging. The
samples for these powders tend to exist as agglomerates. Thus even after pressing the powder
well, the distribution of particles was not uniform in the sample holder. As a result, the powder
shifted very easily inside the sample holder. The experiment was very sensitive to any movement
of the sample holder. So in order to carry out these measurements, the sample was never
completely filled in the sample holder and the sample holder was always kept horizontal once the
powder was filled in.
The rare earth perovskites investigated by us have already been listed in Table 4.1. They
were acquired from Trans-Tech. The variations of the permittivity and permeability of these
materials are shown in figure 4.18 (a), (b), (c).
6
Parameter Value
5
4
3
Real Permeability
Imaginary Permeability
Real Permittivity
Imaginary Permittivity
2
1
0
2
6
10
14
18
22
26
30
34
38
Frequency (GHz)
Figure 4.18 (a) Variations of permittivity and permeability of LSCF nanopowder with frequency
98
5
4.5
Parameter Value
4
3.5
3
2.5
Real Permeability
Imaginary Permeability
Real Permittivity
Imaginary Permittivity
2
1.5
1
0.5
0
2
6
10
14
18
22
26
30
34
38
Frequency (GHz)
Figure 4.18 (b) Variation of permittivity and permeability of LSM nanopowder with frequency.
3.5
Parameter Value
3
2.5
2
Real Permeability
Imaginary Permeability
Real Permittivity
Imaginary Permittivity
1.5
1
0.5
0
2
6
10
14
18
22
26
Frequency (GHz)
30
34
38
Figure 4.18 (c) Variation of permittivity and permeability of YSZ nanopowder with frequency.
Except for the YSZ nanopowder (Fig. 4.18c), other rare earth perovskites show only
marginal variations in permeability and permittivity in the microwave frequency range. The real
99
parts of the permittivity and permeability of these rare earth perovskites are considerably higher
than the values for other materials studied here. The imaginary parts of the permittivity and
permeability of these materials are almost zero, implying that these materials are not lossy. The
average parameter values are summarized in table 4.6.
TABLE 4.5
DIELECTRIC AND MAGNETIC PROPERTIES OF RARE EARTH PEROVSKITE NANOPOWDERS
Material
Dielectric Permittivity
Magnetic Permeability
Real
Imaginary
Loss
Real
Imaginary
Loss
Part
Part
Tangent
Part
Part
Tangent
LSCF
4.7704
0.0513
0.01
2.162
0.021
0.0097
LSM
4.639
0.079
0.017
2.17
0.0176
0.008
YSZ
3.133
0.0089
0.003
1.308
0.0024
0.002
The characteristic impedances of these samples are plotted in figure 4.19 (a)-(c). As
expected, the characteristic impedance of these samples substantially differs from the free space
impedance value.
Characteristic Impedane
250
248
246
244
242
240
238
236
2
6
10
14
18
22
26
30
34
38
Frequency (GHz)
Figure 4.19 (a) Variations of impedance of YSZ nanopowder with frequency
100
Characteristic Impedance
261
260
259
258
257
256
255
2
6
10
14
18
22
26
30
34
38
Frequency (GHz)
Figure 4.19 (b) Variations of impedance of LSM nanopowder with frequency
Characteristic Impedance
268
264
260
256
252
248
244
2
6
10
14
18
22
26
30
34
38
Frequency (GHz)
Figure 4.19 (c) Variations of impedance of LSCF nanopowder with frequency
101
Due to the poor impedance matching, it is evident that these samples are not suitable for
absorbing applications. However, their loss factors would be a major consideration in devices
where higher losses cannot be tolerated. The rare earth perovskites, LSCF and LSM with
relatively high permeability of about 2 may find use in other magnetic applications.
Frequency (GHz)
0
2
6
10
14
18
22
26
30
34
38
Reflectance (dB)
-1
-2
-3
-4
-5
-6
-7
Fig.4.20 Variation of reflectance of LSCF nanopowder with frequency
102
Chapter 5
Measurement results at millimeter
frequencies
5.1 Introduction
It was mentioned in chapter 4 that the intrinsic material properties of micro- and nano-sized
barium and strontium hexaferrite powders are different at microwave frequencies. To further
explore the size dependent behavior of these materials we decided to measure their dielectric and
magnetic properties at still higher frequencies using a different measurement technique. A quasioptical backward wave oscillator was employed to measure the variation of the power
transmittance of Barium and Strontium hexaferrite micro- and nano- powders. These materials
show strong ferromagnetic resonances in the millimeter wave frequency range. The permittivity
and permeability values are obtained by comparing the measured power transmittance spectra of
these samples with the theoretically obtained ones. The measurement results for micro- and
nano-sized barium and strontium ferrite powders at the millimeter wave frequency range are
presented in this chapter. The results show that the magnetic properties such as the ferromagnetic
resonance frequency and effective magnetic anisotropy vary with the particle size in powders.
103
The spectrometer designed here reduces the measurement errors that occur in spectrometers
employing pure optical technology by using a combination of guided wave technology and
optical technology [78].
5.2 Theoretical Background
By using appropriate BWO source and waveguide elements such as directional couplers, phase
shifters and antennas, the power transmittance spectrum of the sample is measured in three
frequency bands, namely, Q (34-56 GHz), V (44-76 GHz) and W (70-117 GHz) bands. The
wave propagating through the sample under test may undergo several reflections at the
⁄
boundaries of the sample. The sample transmittance T
which takes into account the
interference among the multiple reflected waves coming out of the sample takes the form [79],
where the power reflection coefficient for normal incidence at either surface of the sample is,
(
(
)
)
where r is the amplitude transmission coefficient of the sample, d is the sample thickness, n and
K are the real and imaginary parts of the complex index of refraction of the sample and
intensity of radiation incident on the sample.
104
is the
The expression for α in equation 5.5 suggests that this parameter depends on the
wavelength inside the medium, which in turn suggests that α would vary with frequency. As a
result, the expression for evaluating T (Eq. 5.1) also oscillates as a function of the frequency.
Thus the amplitude of the transmittance spectrum is expected to oscillate with frequency. The
maxima and minima of the oscillating signal respectively correspond to the constructive and
destructive interference among the multiply reflected waves exiting the sample. The location of
maxima and minima in the transmittance as a function of the frequency depends on the argument
of the sine function in the denominator of Eq. 5.1 and their strengths depend on the reflection
coefficient R (Eq. 5.2) of the sample. The maxima in the transmittance spectrum occur when the
argument of the sine function in the denominator of Eq. (5.1) is a multiple of π, so that the
sinusoidal term is at its minimum value. The reduction in transmittance within the sample
described by Eq. 5.3 takes into account the magnetic and dielectric losses suffered by the wave
inside the sample. It should be noted that phase φ (Eq. 5.4) vanishes in a lossless material (k = 0).
5.3 Experimental Details
The barium and strontium ferrite samples were studied using a quasi-optical millimeter wave
technique developed at the High Frequency Materials Measurement and Information Center at
Tufts University. The quasi-optical spectrometer has been in use for some time now and the
technology has been continually improved over the years [80]. The measurement system
employs backward wave oscillators (BWO) as high power sources to observe the transmission of
energy through the sample. The BWOs can give about 30 mW of power at each frequency point.
The frequency range from 34 to 117 GHz is divided into three bands, shown in Table 5.1. Three
different sets of sources, detectors and antennas were used to cover these frequency bands.
105
TABLE 5.1
INVESTIGATED MILLIMETER WAVE FREQUENCY BANDS
Frequency band
Frequency Range
Q
34 – 56 GHz
V
44 – 76 GHz
W
70 – 117 GHz
The horn antennas used for each frequency band in the experimental set-up are shown in
figure 5.1. As expected, the size of the antenna depends on the operational frequency band.
Figure 5.1 Horn antennas used for transmitting the incident signal at Q, V and W bands.
The block diagram of the measurement set-up is shown in figure 5.2. The high power
backward wave oscillator source is used to generate the millimeter waves. As seen in the block
diagram, the modulated signal is divided into two channels. Instead of using a pure optical
technique, which has been reported to have high measurement error [78], the measurement
system developed here uses a guided wave technology combined with a quasi-optical millimeter
106
wave system for precision measurements. One of the channels is made up of the waveguide
components to serve as a reference for phase information. The other channel consists of an
optical transmit and receive system. Horn antennas are used to radiate and receive energy.
Figure 5.2 Experimental set-up for measurement at millimeter wave frequencies
Waveguide based isolators and directional couplers are used to divide the signal into the
two different propagation paths. Isolators are used to prevent the radiated signal from entering
the source. The channel made out of the waveguide components is used to provide reference
signal for phase measurements. In the optical channel, the transmitted signal propagates along a
path consisting of four lenses, placed as shown in the figure. The first two lenses help to
effectively expose the sample to the signal radiated by the transmitter antenna. The remaining
lenses focus the transmitted Gaussian beam exiting the sample on to the receiver horn. The
quasi-optical channel is placed inside the pole pieces of a 1 Tesla magnet to attain partial
saturation of the magnetization in the samples. The material under study is placed in the common
107
focal planes of the lenses on its two sides. The signal from the quasi-optical channel, after
passing the receiver horn antenna is combined with the signal from the reference channel by the
directional coupler similar to the one used for the division of the input signal into the two
channels. The two signals are combined at the detector and the interference of the combined
waves is recorded with a detector. Specially written Lab View based software is used to
automate the spectrometers and process the measured data.
Figure 5.3 Experimental set-up for millimeter wave measurements.
108
A photograph of the measurement set-up used in our lab is shown in figure 5.3. The
sample is placed between the poles of the electromagnet. First, the reference data is collected by
running the set-up without placing any sample in the quasi-optical path. Next, the same
procedure is repeated with the sample located in the propagation path of the transmitted wave in
the optical channel.
The powders were carefully packed inside the sample holder. The volume of the sample
used for millimeter wave measurements is much larger than what was used for microwave
measurements. The sample containers designed for the experiment are shown in figure 5.4. The
sample holder is made out of aluminum frame with mylar films fixed through Teflon strips on
either side. The container is about 6 cm high. The thickness of the frame and hence of the sample
is 12 mm. The thickness of mylar sheet is 0.05 mm. Prior to filling the sample holder, the
powdered samples were manually milled to make sure that the sample does not form
agglomerates. The powders were carefully packed inside the sample holder in gradual amounts
to make sure that there are no air gaps within the sample.
Figure 5.4 Specially designed sample holders for BWO based spectrometer. The thickness of the sample
holder used here is 12mm.
109
The computer controlled quasi-optical measurement system records the power
transmission spectrum of the sample under test and the permeability and permittivity of the
sample are extracted from this spectrum by fitting the theoretical curves to the experimental
ones, as explained in the next section. As mentioned earlier, the reference data is collected by
running the set-up without placing any sample between the lenses. Next, the same procedure is
repeated with the sample located in its position in the optical channel. The power transmittance
spectrum is then obtained as the ratio of the measured point to point voltages at the detector with
and without the sample.
5.4 Results
The power transmittance spectra of micro- and nano-powders of Barium and Strontium
hexaferrites were obtained in the millimeter range using the quasi-optical technique described in
the previous section. The results in the form of the transmittance plots for the strontium
hexaferrite are shown in figure 5.5. The strong dips in the transmittance plots between 50-60
GHz correspond to the ferromagnetic resonances of micro- and nano-powders of Strontium
hexaferrite. The ferromagnetic resonance for nano-powders is much sharper than the one for
micro-powder. The flattening of ferromagnetic resonance of micropowder is due to power
broadening. Ferromagnetic resonance in bulk samples of strontium hexaferrite under similar
experimental conditions are known to be even more broad.
Next we note that the dips due to ferromagnetic resonances in the transmittance plots are
followed by nearly sinusoidal transmittance variations of relatively smaller amplitudes for each
of the samples. The small amplitude sinusoidal variations correspond to the oscillations
discussed in section 5.2 in the context of Eq. 5.1. The appearance of interference among the
110
multiply reflected waves within the samples is evident. The small amplitude of these oscillations
reflects the fact that the sample power reflection coefficient R (Eq. 5.2) is not very large (R<<1)
at the millimeter wave frequencies.
We now describe how complete information on the permeability and permittivity of the
samples can be obtained from the measured variations in the power transmittance of Barium and
Strontium hexaferrite powders. At the outset, we note that the transmittance is not significantly
affected by the magnetic interaction of the external field with the magnetic dipoles present in the
material at frequencies sufficiently away from the natural ferromagnetic resonance of the
material. Further, the dielectric behavior of these ceramic type materials has only weak linear
dependence on frequency within the millimeter wave frequencies. The dielectric resonances
(absorption) take place at much higher frequencies.
We shall identify the spectral portions of the transmittance plots where magnetic effects
are dominant and where they are weak. The real and imaginary parts of permeability are
extracted from the spectral regions where magnetic effects are overwhelmingly high. The
permittivity values are extracted where magnetic effects are rather weak.
As mentioned earlier, sample transmittance described by Eq. 5.1 oscillates with
frequency. The frequency difference of two successive peaks of transmittance when magnetic
effects can be ignored is given by
, where c is the velocity of light in vacuum, n is the real
part of the index of refraction and d is the thickness of the sample. This frequency difference
may change somewhat from peak to peak. One can then take the mean value of this frequency
difference. The refractive index n of the material can be obtained from the mean frequency
difference measured among the transmittance oscillations far removed from the ferromagnetic
resonance. The absorption coefficient k can be obtained from the overall decay of the
111
transmittance in the spectral region where the magnetic effects are rather weak. This happens in
the spectral range around 90-110 GHz. Similarly the absorption coefficient calculated from the
decay of the transmittance in the 90-110 GHz essentially corresponds to the dielectric losses ( ).
A curve fitting function is used to match the measured transmittance spectra to the
transmittance spectrum calculated theoretically from equation. The transmittance spectrum of the
sample is measured by taking the ratio of the detector output with and without the sample placed
between the lenses. The values of n and k are optimized such that the experimental and
theoretical graphs can be overlaid on each other as closely as possible. The goal is to get the
peaks in the two waveforms to line up in position and overlap in magnitude by changing the
values of n and k. These parameters are recorded for all peaks and the average value of the real
and imaginary parts of the refractive index are determined. An alternative method is to express
these parameters in terms of first or higher order equations in frequency and then use frequency
dependent parameter values for further calculations.
Schloemann’s equation (Eq. 2.17) can be used to obtain the best fit to the transmittance
variations in the neighborhood of the ferromagnetic resonances. The permittivity coefficients
and
obtained from the estimated values of n and K in the high frequency range of the
millimeter wave frequencies can be used in this fitting procedure. The transmittance spectra
measured for two strontium hexaferrite samples are shown in figure 5.5. These samples consist
of different sizes of the particles in the material. The marked dips in the transmittance spectra
(figure 5.5) observed around the resonance frequencies indicate that strontium ferrite powders
can be used as efficient absorbers in the frequency range of their resonances.
112
Sr Ferrite, Micro, 2.13 g/cc
1
Sr Ferrite, Nano, 0.57 g/cc
Transmittance
0.8
0.6
0.4
0.2
0
30
40
50
60
70
80
90
100
110
Frequency (GHz)
Figure 5.5 Transmitance variations of micro- and nano-size strontium ferrite powders in the millimeter
wave range.
Next, the real and imaginary parts of permittivity were determined by the procedure
mentioned earlier. The values obtained here are slightly higher than the permittivity values
obtained at microwave frequencies using the network analyzer based technique discussed in
chapter 4. Since the value of permittivity obtained from this technique is based on curve fitting, it
may not be as accurate as the one obtained using the waveguide technique at microwave
frequencies. This could be due to different densities of the samples used in the two measurement
techniques and accuracy of the algorithms. However, as at microwave frequencies, we find that
the permittivity of nano-powder is lower than the permittivity of micro-powders in the millimeter
wave range as well. The values of permittivity for nano- and micro-sized powders at millimeter
wave frequencies are summarized in table 5.2.
113
TABLE 5.2
BEST FIT VALUE OF PERMITTIVITY FOR HEXAFERRITE SAMPLES
Sample
Density
(g/cc)
Millimeter wave
Measurements
Ɛ’
Ɛ’’
Barium Ferrite
Micro
1.83
4.41
0.029
Barium Ferrite
Nano
0.59
1.88
0.01
Strontium
Ferrite Micro
2.13
5.8
0.057
Strontium
Ferrite Nano
0.57
2.15
0.012
Using these values of permittivity, the real and imaginary parts of permeability were
calculated by a similar curve fitting procedure. The best fit results for nano- and micro-sized
strontium ferrite samples in the millimeter wave frequency range are shown in figure 5.6 and 5.7,
respectively. The phenomena of ferromagnetic resonance can be seen more clearly in these plots.
It should be noted here that Schloemann’s equation is useful in calculations close to the
ferromagnetic resonance frequency. However, as we move away from the resonance frequency,
the accuracy of this method decreases. In fact, for frequencies that are distant from the resonance
frequency, more accurate calculations can be performed by using Polder’s theory [87].
Therefore, the permeability data that is presented in the following plots is not accurate at higher
frequency points.
114
Real Part of Permeability
1.09
Strontium Ferrite, Nano
ρ = 0.57 g/cm3
Fres = 48.2 GHz
1.05
1.01
0.97
0.93
0.89
30
40
50
60
70
80
90
100
110
Frequency (GHz)
(a)
Imaginary part of permeability
0.24
Strontium Ferrite, Nano
ρ = 0.57 g/cm3
Fres = 48.2 GHz
0.2
0.16
0.12
0.08
0.04
0
30
40
50
60
70
80
Frequency (GHz)
(b)
115
90
100
110
Real part of permeability
Figure 5.6 Real (a) and imaginary (b) parts of permeability for Strontium nano-ferrite powder measured in
the millimeter wave frequency range.
2
Strontium Ferrite, Micro
ρ = 2.13 g/cm3
Fres = 53.1 GHz
1.6
1.2
0.8
0.4
0
30
40
50
60
70
80
90
100
110
Frequency(GHz)
Imaginary part of permeability
(a)
2
Strontium Ferrite, Micro
ρ = 2.13 g/cm3
Fres = 53.1 GHz
1.6
1.2
0.8
0.4
0
30
40
50
60
70
80
90
100
110
Fequency (GHz)
(b)
Figure 5.7 Real (a) and imaginary (b) parts of permeability for Strontium micro-ferrite powder measured
in the millimeter wave frequency range.
116
A careful inspection of the plots in figure 5.6 and 5.7 lead us to conclude the best fit
permeability values obtained in the lower frequency range (30 – 40GHz) are in good agreement
with the values obtained from the microwave measurements. It is also observed that the microsized powders show higher values of intrinsic material properties as compared to the nano-sized
powders in both microwave and millimeter wave frequency ranges.
An important observation from these plots concerns the resonance phenomena in these
materials. It can be observed that the resonance frequencies are different for the two sizes of the
powders. The resonance frequency seems to shift to lower values for nano-powders as compared
to micro-sized powders. Resonance is the phenomena by which absorption occurs in ferrite
samples. This means that the absorption frequency for a given composition of the ferrite sample
can be tuned by changing the size of the particles in the sample. The best fit permeability plots
for barium ferrite samples are shown in figures 5.8 and 5.9. The best fit results or barium ferrite
powder samples are qualitatively similar to those for strontium ferrite powder.
Real part of permeability
1.08
Barium Ferrite, Nano
ρ = 0.59 g/cm3
Fres = 42.5 GHz
1.04
1
0.96
0.92
30
40
50
60
70
80
Frequency (GHz)
(a)
117
90
100
110
Imaginary part of permeability
0.16
Barium Ferrite, Nano
ρ = 0.59 g/cm3
Fres = 42.5 GHz
0.12
0.08
0.04
0
30
40
50
60
70
80
90
100
110
Frequency (GHz)
(b)
Figure 5.8 Real (a) and imaginary (b) parts of permeability for Barium nano-ferrite powder measured in
the millimeter wave frequency range.
Real part of permeability
2
Barium Ferrite, Micro
ρ = 1.83 g/cc
Fres = 49.2 GHz
1.6
1.2
0.8
0.4
0
30
40
50
60
70
80
Frequency (GHz)
(a)
118
90
100
110
Imaginary part of permeability
2
1.6
Barium Ferrite, Micro
ρ = 1.83 g/cm3
Fres = 49.2 GHz
1.2
0.8
0.4
0
30
40
50
60
70
80
90
100
110
Frequency (GHz)
(b)
Figure 5.9 Real (a) and imaginary (b) parts of permeability for Barium micro-ferrite powder measured in
the millimeter wave frequency range.
The best fit values of the resonance frequencies of micro- and nano-powders of strontium
and barium hexaferrites are given in table 5.3. From the data shown above, it is clear that the
material properties for solid ferrite are different from those of the powdered ferrites. Further,
these properties depend on the size of the particles in the powder. The data is also compared to
values reported in literature using different techniques. There is no data available to compare the
nano-ferrite properties but considerable study has been done on bulk and powdered ferrites with
particle size in micrometer or greater size region. For example, the ferromagnetic resonance of
barium ferrite pressed-powder sample was studied using optically switched spectrometer [64].
119
TABLE 5.3
RESONANCE FREQUENCY OF FERRITE POWDERS AT DIFFERENT SIZES
Resonance Frequency
Material
Particle size
(GHz)
Bulk
50-53 [64]
Micro
49.2
Nano
42.5
Bulk
57.0 [81]
Micro
53.1
Nano
48.15
Barium
Hexaferrite
Strontium
Hexaferrite
It is clear from the measurement procedure described above that the parameter estimation
for the BWO based millimeter wave measurement technique is based on trial and error and curve
fitting of the expected and measured values. Thus this technique is not as accurate for
permittivity and permeability measurements as the transmission/reflection based waveguide
technique. However, if the measured transmittance spectrum is accurate, the information derived
from these procedures provides a good estimation of the material properties [82].
The material properties for micro- and nano-ferrites at microwave and millimeter wave
frequencies have been discussed here. The values from the two measurement techniques are not
exactly equal but they both reveal similar trend of material properties in case of micro- and nanopowders. This size dependence of these material properties is further analyzed in the following
chapter and the causes for these differences are identified.
120
Chapter 6
Size effect on ferromagnetic resonance in
ferrites
6.1 Introduction
The intrinsic properties of different sizes of barium and strontium hexaferrite powders at
microwave and millimeter wave frequencies were presented in chapters 4 and 5, respectively.
Significant differences were observed in the properties of the micro- and nano-size powders of
these ferrites at these frequencies. A deeper analysis is needed to understand the origin of these
differences. In this chapter, the dependence of the material properties on the size of the particles
is investigated. To make further progress, it is necessary to understand the role of the
composition and internal morphology of the samples. Therefore, in addition to microwave and
millimeter wave characterization, structural analysis of the samples was also performed. The
results indicate that properties such as the ferromagnetic resonance frequency and effective
magnetic anisotropy vary with the particle size. Barium and strontium hexaferrite show similar
behavior with changing particle size.
121
6.2 Properties of nano- and micro-size ferrite powders
The frequency dependence of the real and imaginary parts of the permittivity and permeability of
barium hexaferrite nano- and micro-powders were discussed in chapter 4 (fig 4.2 and 4.8,
respectively). The permittivity of both the nano and micro powders of this ferrite was found to be
much smaller compared to the permittivity of bulk barium hexaferrite, which lies between 13 and
16 [63, 64]. It was further observed that the measured permittivity of the micro powder was
slightly larger than that of the nano powder. Similar behavior was observed for strontium microand nano-ferrite powders as well. The average values of the permittivity and permeability of the
nano and micro-sized powders for each sample, measured in the microwave frequency range are
summarized in Table 6.1.
TABLE 6.1
ELECTRO-MAGNETIC PROPERTIES MEASURED AT MICROWAVE FREQUENCIES
Barium Hexaferrite
Frequency
(GHz)
Nano-sized
Powder
Strontium Hexaferrite
Micron-sized
Powder
Nano-sized
Powder
Micron-sized
Powder
4
μ'
1.11
ε'
1.43
μ'
1.06
ε'
3.07
μ'
1.09
ε'
1.68
μ'
1.02
ε'
3.83
8
1.14
1.50
1.06
3.05
1.09
1.72
1.03
3.77
12
1.14
1.50
1.10
3.04
1.08
1.74
1.03
3.82
16
1.11
1.45
1.12
3.09
1.09
1.73
1.04
3.91
20
1.06
1.44
1.13
3.06
1.07
1.69
1.06
3.95
24
1.04
1.49
1.13
3.11
1.06
1.68
1.04
3.89
28
1.08
1.48
1.12
3.08
1.11
1.66
1.05
3.74
32
1.10
1.46
1.03
3.06
1.1
1.65
1.08
3.85
36
1.14
1.44
1.08
3.01
1.09
1.69
1.06
3.93
40
1.12
1.42
1.07
3.06
1.07
1.63
1.04
3.98
122
Barium and strontium micro-powders (particle size 50-100 µm) have been studied in the
past in the frequency range from 8 GHz to 26.5 GHz using a cavity resonator [62]. Our results
for micro-powders of these materials are in close agreement with their published results reported
in [62]. The slight differences in the two results could be attributed to the difference in particle
size and sample density used in the two measurements. However, only limited literature exists on
these materials in the small size range. Therefore, it is difficult to say if these results are indeed
typical. But we have observed similar results in the millimeter wave frequency measurements
also.
At this point, we feel that intrinsic material properties depend on the size of the particles
in the powder. For reference, the millimeter wave results are also summarized in Table 6.2. Since
the saturation magnetization will be higher for the single domain nano-powders, resonance
frequency is expected to be lower for nano-powders. This is in agreement with what was
observed in the measurements. The detailed calculations are presented in the next section.
TABLE 6.2
ELECTRO-MAGNETIC PROPERTIES MEASURED AT MILLIMETER WAVE FREQUENCIES
Sample
Density
(g/cc)
Millimeter wave
Measurements
Ɛ’
Ɛ’’
Ferromagnetic
resonance frequency
(GHz)
Barium
Ferrite Micro
1.83
4.41
0.029
49.2
Barium
Ferrite Nano
0.59
1.88
0.01
42.5
Strontium
Ferrite Micro
2.13
5.8
0.057
53.1
Strontium
Ferrite Nano
0.57
2.15
0.012
48.2
123
As discussed in chapter 5, the values of permittivity and permeability of the samples at
millimeter wave frequency range are derived from the observed power transmittance spectrum.
The transmittance spectra for strontium ferrite samples are reproduced in figure 6.1. The
resonance phenomenon can be clearly seen in the transmittance spectrum.
Sr Ferrite, Micro, 2.13 g/cc
1
Sr Ferrite, Nano, 0.57 g/cc
Transmittance
0.8
0.6
0.4
0.2
0
30
40
50
60
70
80
Frequency (GHz)
90
100
110
Fig. 6.1 The transmittance spectrum for micro- and nano-size strontium ferrite powders measured using
the BWO technique at Q, V and W bands.
Under similar conditions, strontium hexaferrite in bulk form is known to have a
resonance frequency between 50 – 65 GHz [81], implying strong absorption of electromagnetic
energy entering the sample in this frequency range and the material can be used as an absorber.
From the transmittance spectrum in figure 6.1, it can be concluded that strontium hexaferrite
samples consisting of nano- and micro-size particles exhibit resonance within the frequency
range of resonance of bulk strontium hexaferrite. The resonance in the case of micro powder is
spread out and therefore the exact value of the resonance frequency cannot be determined from
124
this data alone. The absorption width seen in figure 6.1 for micro-size strontium ferrite may not
be the actual width of the ferromagnetic resonance. The visual inspection of the micro-sized
samples showed that the particles form aggregates inside the sample holder. Thus the
measurement surface was not smooth and formation of cracks and pores was visible. The
different crystallites present in the sample tend to orient randomly and form local grain
boundaries. All these factors contribute to the broadening of the observed linewidth [83].
However, in the case of nano-size sample, the powder was packed much more smoothly and
showed no discernible cracks or discontinuity inside the sample holder.
It should also be noted here that the level of transmittance observed in this plot does not
represent the maximum amount of absorption possible through the sample. Further study reveals
that the strength of absorption depends on the density of the sample packed inside the sample
holder. For instance, it cannot be concluded from figure 6.1 that micro-size strontium ferrite
powder shows stronger absorption than nano-size strontium ferrite powder. This is because the
densities of the samples differ considerably. For reference, the specific gravity of solid ferrite
ceramics (for both barium and strontium) is about 4.7 -5.2 g/cm3. However, when used in
powdered form, the specific gravity is much lower. The density for micro-powder is about 2
g/cm3, which is about 2.5 times lower than the usual solid ferrite samples. The density for nanopowder is about 0.57 g/cm3, which is almost 9 times lower. In order to better understand the
effect of density on the measured transmittance levels two different densities of the same powder
were used for the measurement. The difference in density was achieved by gradually increasing
the amount of pressure applied during the packing of the sample. Figure 6.2 shows the
transmittance spectrum recorded for two different densities of the nano-size strontium hexaferrite
sample. In one case, the powder was filled in the sample holder without applying any extra
125
pressure. Next, external pressure was applied in order to tightly pack more powder in the same
sample holder. The sample can be packed even more tightly inside the sample holder by using
special milling techniques to achieve higher density values that are comparable to solid samples.
It can be concluded from the results shown in figure 6.2 that higher density of the sample results
in stronger absorption at resonance frequency. However, the sample density does not affect the
frequency at which the resonance occurs. The weaker absorption observed for lower sample
density is due to air dilution in the scarcely packed sample.
1
Transmittance
0.8
0.6
Sr-Fe Nano, 0.52 g/cc
Sr-Fe Nano, 0.57 g/cc
0.4
0.2
34
44
54
64
74
84
94
104
114
Frequency (GHz)
Figure 6.2 Transmittance spectrum recorded for two slightly different densities of strontium hexaferrite
Quantitative differences in the transmittance of micro- and nano-powder samples of
strontium hexaferrite shown in figure 6.1 can be summarized as,
a. The strength of absorption is different for the two types of powder,
b. Wider absorption width is observed for micro-powders as compared to nano-powders,
126
c. Resonance frequency is lower for nano-powders as compared to micro-powders.
The first two observations in this list have already been explained. In order to understand
the dependence of resonance frequency on the particle size, structural studies were carried out on
the samples. The results are presented in the following section.
6.3 Structural analysis of ferrite powders
The powders were characterized for their composition and size using the following
techniques,

scanning electron microscopy (SEM)

laser particle size distribution (PSD)

X-ray diffraction (XRD)
The XRD spectra for the four samples (micro- and nano- size barium and strontium
hexaferrite) are shown in figure 6.3 (a)-(b). Single phases without any impurity are observed for
each sample.
127
(a)
(b)
Figure 6.3 X-ray diffraction spectra of strontium hexaferrite (a) and barium hexaferrite (b).
128
The primary particle and agglomerate sizes of barium and strontium hexaferrite were
determined using the scanning electron microscopy technique (SEM) and also by the laser
particle size distribution technique (PSD). Crystallite sizes in the sample were determined from
the x-ray study. There is a good correlation between the data obtained from the different
techniques. These measurements were performed at Pacific Northwest National Laboratory
(PNNL, WA). The size distribution and average particle as well as agglomerate sizes are
presented below, in Table 6.3.
TABLE 6.3
PARTICLE SIZE DETERMINATION OF BARIUM AND STRONTIUM FERRITE
XRD
Particle Diameter (nm)
Agglomerate Diameter (nm)
Powder
Crystallite
Size
PSD
SEM
PSD
SEM
size (nm)
Micro
161
1000-2000
500-1000
20000-50000
3000-15000
Nano
60
100-200
20-50
500-600
500-1000
The particle sizes obtained using particle size distribution (PSD) method are higher than
the values obtained using other methods. However, it should be noted that the resolution of this
technique is limited to these values and so it cannot measure smaller sizes [90]. Thus the SEM
results are more reliable.
Based on these measurements, it can be concluded that the nano-particles are about 20 50 nm in size whereas the micro-particles are about 0.5 - 2.0 µm. These sizes are lower than
what was mentioned in the sample data sheet supplied by the manufacturer. From the average
particle size calculated above and the crystallite dimensions obtained from X-ray diffraction, it
can be concluded that particles exist as single crystallites in the nano-powder samples. However,
129
in the micro-powder samples, the particles exist in polycrystalline state. This is in agreement
with published results [84]. The critical size for particles to exist as single crystallites calculated
for these powders using the theory due to Kittel is about 100 nm, which is smaller than the
particle size in micro-powders but larger than the particles in the nano-powder [85]. The direct
observation of the domain structure of barium hexaferrite small particles at room temperature by
Goto, Ito and Sakurai also confirms that the critical size is in the region of 1 um [86]. Therefore,
it can be concluded that the nanoparticles definitely exist as single domains whereas the micro
particles may exist in polycrystalline state. When the size of the particle is smaller than the
critical size for multi-domain formation, it exists in a single-domain, where the domain-wall
resonance is avoided, and therefore, the material could be used at higher frequencies [22]. By
definition, the magnetic moments in a single domain particle are aligned in the same direction.
Therefore single domain material is expected to possess the maximum saturation magnetization
for that particular size and composition. However, in reality the alignment may not be
completely uniform and the materials are assumed to be in a quasi-uniform magnetization state,
when their diameter is a few nanometers.
Next, the morphology of the samples was studied using scanning electron micrograph.
The images from the high resolution microscope are shown in figures 6.4 and 6.5. The SEM
pictures show that the powders consist of agglomerates ranging in size from 3 to 15 um. The
high resolution pictures show that individual particles have sizes much below 1 um. It is difficult
to ascertain the exact shapes of individual particles from these images. These shapes appear to be
in between spherical and hexagonal. Figure 6.4 shows the micrographs obtained for the nanosize barium hexaferrite powder.
130
(a)
(b)
131
(c)
(d)
Figure 6.4 Scanning electron micrographs of barium hexaferrite nano-powder for different resolutions can
be seen. The magnification level for the images shown here are (a) 5,000, (b) 10,000, (c) 20,000, (d)
40,000. The particle size is much smaller than the 1 µm mark indicated in the third inset.
132
Each graph (figure 6.4a to 6.4d) shows the sample morphology at a different magnification level.
In order to get a visual estimate of the particle size, the scale has been shown in each plot. For
instance, in figure 6.4 (a), it can be seen that the particle size is much smaller than the 5um
dimension indicated in the image. The individual particle size can be observed to be much
smaller than the 0.5um mark indicated in figure 6.4 (d). However, in some cases the agglomerate
size can be close to this dimension.
Figure 6.5 shows the SEM data obtained for the micro-size powders. The individual
particle size was observed to be about 0.5um to 1 um whereas the agglomerate sizes range from
3 um to 15 um.
(a)
133
(b)
(c)
134
(d)
Figure 6.5 The micro-size powder of barium hexaferrite consisted of agglomerates in the size range from
3 to 15 µm. It can be seen in inset C that the primary particle size lies between 0.5 to 1 µm.
Based on these images, it can be concluded that the shape of the particles is near
spherical in case of nano-powders. For the case, of micro-powders, the shape of the individual
particles is not clearly distinguishable but the shape of the agglomerates in each case seems to be
close to hexagonal. The resonance frequency is known to depend on the crystallite size and shape
[72]. In the next section, we discuss the phenomenon of ferromagnetic resonance and calculate
resonance frequencies of hexaferrite nanoparticles using the information gathered so far on the
crystal structure and morphology of hexaferrites and see if the calculated results agree with our
observations.
135
6.4 Ferromagnetic resonance
The phenomenon of ferromagnetic resonance occurs when a ferromagnetic substance is
simultaneously subjected to a dc magnetic field and a relatively weak alternating magnetic field,
perpendicular to each other, provided the frequency of the alternating magnetic field satisfies
what is called the resonance condition. The dc magnetic field includes the externally applied dc
field and any fields existing in the medium due to the demagnetization and crystallographic
anisotropic effects. The alternating magnetic fields in ferromagnetic resonance are in the radio
and microwave frequency regimes. Kittel’s theory of ferromagnetic resonance (1947) was
extended by Polder (1949) [87]. Below, we describe Polder’s phenomenological theory of
ferromagnetic resonance.
6.4.1 Polder’s Theory of Ferromagnetic Resonance
To begin with, ferromagnetic material of arbitrary shape and size is considered. The shape and
size of the ferromagnetic material become relevant because of the long range character of the
dipolar interaction among its atoms and molecules possessing non-zero magnetic moments. The
crystal anisotropy favors orientation of magnetic moments in preferred crystallographic
directions. Further, energy dissipation also occurs during the ferromagnetic resonance. These
effects, neglected in Polder’s theory, can be subsequently incorporated in a phenomenological
manner. A ferromagnetic material acquires a net magnetization in the presence of the external dc
magnetic field. The motion of an elementary dipole under these conditions is quite like the
motion of a spinning top. The net magnetic field seen by the elementary dipole is,
(⃗ )
⃗
⃗⃗
136
where ⃗ is the macroscopic field and ⃗⃗ is the magnetization (dipole moment per unit volume)
produced in the ferromagnetic substance. The equation of motion of an elementary dipole of
magnetic moment
is,
(⃗
where gyromagnetic ratio
⃗⃗ )
is the ratio of the dipole moment of the atom and its angular
momentum. The magnetization ⃗⃗ is given by
⃗⃗
where
is the number of atoms per cc. Combining eq. (6.2) and eq. (6.3) gives
⃗⃗
⃗⃗
⃗
In the absence of the alternating magnetic field, the magnetization ⃗⃗⃗⃗⃗ and the dc field ⃗⃗⃗⃗ are
parallel. The net magnetic field and magnetization in the presence of the alternating magnetic
field are
⃗
⃗⃗⃗⃗
⃗
⃗⃗
⃗⃗⃗⃗⃗
⃗⃗
where ⃗⃗⃗⃗⃗ is the saturation magnetization produced by the dc field ⃗
and ⃗⃗ is the ac
magnetization produced in the material. On substituting equations (6.5) in eq. (6.4), we obtain,
⃗⃗
(⃗⃗⃗⃗⃗
⃗⃗⃗⃗
⃗
⃗⃗⃗⃗⃗
⃗⃗
⃗⃗⃗⃗
⃗⃗
⃗)
Last term on the RHS being a product of two small quantities may be neglected. Further
⃗⃗⃗⃗⃗
⃗⃗⃗⃗
⃗⃗⃗⃗ and ⃗⃗⃗⃗⃗ may be taken along the z-axis and ⃗ to lie in the xy plane. Eq. (6.6), then yields,
(
)
137
Rearranging terms, we obtain the components of the ac magnetization as,
where
Resonance occurs when the frequency of the alternating magnetic field approaches
leading to enhancement in the induced alternating magnetization ⃗⃗ .
The magnetic susceptibility is defined by the tensor equation,
[
The total magnetization ⃗⃗
,
[
⃗)
[ ](⃗⃗⃗⃗
[
where
]
]
][
]
[ ] ⃗ can now be expressed as,
[
][
,
Recalling the relationship,
138
]
,
⃗
we can write, ⃗
⃗
⃗⃗⃗⃗
⃗
⃗⃗
⃗⃗⃗⃗⃗
⃗⃗
⃗⃗⃗⃗
⃗
⃗⃗
⃗
We have for the alternating magnetic field, ⃗
⃗
⃗
⃗⃗
With eq. (6.9), eq. (6.15) yields,
where
Eq. (6.15) can be written in the tensor notation as
⃗
where [ ]
[
[ ]⃗
]
Eq. (6.11) or equivalently eq. (6.16) can easily be extended to incorporate the effects due to
crystal anisotropy and energy dissipation. We can require the permeability tensor to be consistent
with crystal symmetry. Further, allowing
and
in the permeability tensor to be complex
quantities can account for the energy dissipation in ferromagnetic resonance.
139
From the equations defining the permeability tensor, it is clear that the permeability
tensor has a singularity at
. This value of ω corresponds to the ferromagnetic resonance.
The physical meaning of this expression is that when the operating RF frequency is equal to
,
the energy from the microwave field is transferred most efficiently to the system of spins. The
closer the frequency of the microwave field to the natural precession frequency (
), the greater
the energy absorbed by the spins [22].
In the discussion so far, the medium is assumed to be infinite and the losses associated
with dipole motion are not explicitly considered. However, the resonance frequency is affected
by the loss mechanisms associated with demagnetization, magnetic anisotropy and crystalline
imperfections in polycrystal structures. To consider the finite size of the material, the magnetic
field inside the sample should be evaluated by applying electromagnetic boundary conditions.
Demagnetization factors can be easily obtained for uniformly shaped samples like sphere,
cylinder or a slab. The modification in the magnetic field due to demagnetization can be
expressed as,
and
where N is the demagnetization vector, Hi and hi are the reduced dc and ac fields due to finite
size of the sample. The complex susceptibility can be obtained by substituting Eq. (6.18) in Eq.
(6.4) and proceeding in a similar way as described above for the case of infinite materials. The
final values of the susceptibility tensor thus obtained are,
́
[ ́
]
[ ́
140
(
)
]
[ ́
́
́
]
[ ́
(
)
́
]
(
[ ́
)
]
where,
́
][
{[
where
,
,
(
)
]}
⁄
are the demagnetizing factors applicable for the x, y, z directions. In
the presence of demagnetization and anisotropic effects, this expression gets modified. For a
⁄ . For other simple shapes, these factors can be calculated. The
sphere,
demagnetizing factors so defined satisfy the condition
. A method to simplify
these equations for further analysis is often used [88]. The solution to the boundary problem to
evaluate the magnetic fields inside a ferrite is very difficult. However, these solutions have been
worked for certain specific shapes of the ferrite sample. Here the results for hexagonal and
spherical ferrites are used to calculate the expected resonance frequency for the micro- and nanoferrites studied here.
We next consider the effect of anisotropy on the ferromagnetic resonance frequency.
Following Kittel [89], the anisotropic field Hanis can be taken to be in the xy plane of a single
crystal. In that case the effect of anisotropy can be described by additional demagnetizing factor
and
such that
141
so that
changes to
due to the anisotropic field,
and
to
in Eq. (6.21). For small perturbation
takes the value
for a uniaxial crystal, so that
Values of
are tabulated. Since nano-particles generally exist as single domains with
larger values of the magnetization
as compared to the micro-particles, the anisotropic effects
are less prominent for nano-particles in comparison to the micro-powders.
6.5 FMR Calculations
Using the theory and equations presented in section 6.4, one can calculate the ferromagnetic
resonance (FMR) frequency for the samples. Taking a simpler approach, if the demagnetization
and anisotropic effects are neglected, the resonance frequency is given by,
or,
where,
here
is the external field (1 kOe in the present case), g is the spectroscopic splitting factor
with a value of 2, e is the electron charge (1.6 x 10-19 C) and m is the mass of electron (9.109 x
10-31 kg). Using these values, γ is evaluated to be 1.759 x 108 C/g.
142
Equation (6.26) needs to be modified to incorporate demagnetization and anisotropic
effects. First, the ferromagnetic resonance frequency is calculated by neglecting the
demagnetizing effects but including the anisotropic effects. The resonance frequency takes the
form,
where
The values for saturation magnetization and anisotropy constant for single crystals of
barium and strontium hexaferrites have been calculated by Shirt and Buessen [92]. These are
listed in Table 6.4, along with the calculated values of the resonance frequency using Eq. (6.27).
TABLE 6.4
CALCULATED VALUES OF RESONANCE FREQUENCIES OF BARIUM AND STRONTIUM HEXAFERRITE
NANOPOWDERS WITHOUT DEMAGNETIZATION EFFECTS.
Sample
K1
(erg/cm2)
Ms (G)
Hanis
H0
Calculated
Measured
(Oe)
(kOe)
FMR (GHz)
FMR (GHz)
Ba-ferrite
3250000
380
17,092
1
50.60
42.50
Sr-ferrite
3570000
380
18,775
1
55.37
48.15
The calculated values of the resonance frequency are much higher than those observed in
the measurements. Thus it is clear that the demagnetizing effects cannot be neglected here.
We now incorporate the demagnetization effects by modifying Eq. (6.27) to
(
)
143
where,
We have used
. The value of the demagnetization factor (N) calculated by Glass
is about 4 for spherical particles and 8 for hexagonal particles [91]. The effective anisotropy has
been measured to be approximately 15.5 kOe for spherical particles and 12.9 kOe for plate-like
shape of the particles [84]. This data can be used to calculate the values of N for different shapes
of the particles by using the following relationship,
where
is the effective anisotropy field. The derived values of the demagnetization factor are
shown in table 6.5.
TABLE 6.5
CALCULATION OF DEMAGNETIZATION FACTOR FOR DIFFERENT PARTICLE SHAPES IN BARIUM
HEXAFERRITE NANO-POWDER
Particle
K1
Demag.
Ms (G)
shape
(erg/cm2)
(kOe) Factor (N)
Spherical
3250000
380
15.5
8.1
Hexagonal
3250000
380
14
4.2
Plate-like
3250000
380
12.9
11
It was observed earlier in the morphological study of the samples that the nano-powders
exist nearly as spherical particles. Assuming the particles to be spherical with N = 4.2 and using
144
(
)
the calculated resonance frequency for barium hexaferrite nano-particles is 46.14 GHz which is
on the higher side of the measured value. However, if we use
applicable for the hexagonal
shape, the calculated value for the resonance frequency for barium hexaferrite nano-particles is
42.15 GHz, which is much closer to the experimental value. As mentioned earlier, the actual
particle shape may lie in between spherical and hexagonal, perhaps more towards hexagonal
shape. Table 6.6 gives the final results for Barium and Strontium ferrites. The agreement
between the calculated and experimental values is quite good. Therefore this model can be used
to calculate the ferromagnetic resonance frequency of different sizes of hexaferrite powders.
TABLE 6.6
CALCULATED VALUE OF RESONANCE REQUENCY IN THE MILLIMETER WAVE RANGE
K1
Hanis
Demag.
Calculated
Measured
Sample
Ms (G)
(erg/cm2)
(Oe) (kOe) Factor (N) FMR (GHz) FMR (GHz)
Barium
3250000
380
17,092
1
4.2
46.14
42.50
Nano-ferrite
3250000
380
17,092
1
8
42.15
Strontium
3570000
380
18,775
1
4.2
50.90
Nano-ferrite
3570000
380
18,775
1
8
46.89
48.15
It is clear from these measurements that the resonance frequency can be tuned by
changing the size and shape of the particles. Nano-ferrites can be used to achieve sharper
resonance and hence absorption over specific frequencies in the millimeter wave range. It is not
145
easy to fabricate absorbers at these frequencies but barium and strontium nano-ferrite show
promise.
6.6 Quality Control
To ensure that the differences observed in the properties of ferrite samples are not due to
measurement errors, such as non-uniform packing of powder inside the sample holder, several
lots of the same material were studied and measurements were repeated for each sample. An Xray diffraction pattern was taken to confirm the completion of the phase transition to the
hexagonal structure of BaFe12O19.
The diffractogram indicates that the sample is almost
completely transformed to BaFe12O19 and no other phases were detectable.
Some interesting results were observed during the measurement of the powdered
samples. As mentioned earlier the samples studied here are commercially available powders. The
structural results shown in previous sections reveal that the actual particle dimensions may vary
from the dimensions reported in the datasheet. The preparation of powders with nano-size
particles is challenging. Various techniques have been developed to prepare ultrafine ferrite
powders. Some of the commonly used methods are chemical co-precipitation method [93-94],
glass crystallization method [95-97], organic resin method [98], and hydrolysis method metalorganic complexes [99]. The advantage of developing small-sized particles is that phase
transformations can be achieved at lower temperatures. It has been seen here and other similar
studies that the crystallite size, particle size distribution and inter particle spacing significantly
influence the behavior of the material [100]. In recent years, significant scientific effort has gone
into developing techniques for the controlled growth of nano-powders. The synthesis techniques
used for the preparation of nano-powders over the years have been reviewed [101].
146
Several different lots of barium and strontium hexaferrite nano-powders were purchased
to carry out the measurements. However, during millimeter wave characterization, some barium
hexaferrite samples did not show any resonance. In order to understand this, all samples were
studied with X-ray diffraction. The samples presented here are in addition to the ones that were
used for measurement and discussed in this work. X-ray results for these new samples are shown
in Fig. 6.5.
Figure 6.6 Several lots of samples contained incomplete phases or impurities such as Fe2O3. X-ray studies
of a few samples are shown here.
The barium hexaferrite nanopowders shown in this figure contained at least 50 volume
percent of ε-Fe2O3. These samples are labeled as lot 1 to lot 3. Lot 1 shows 50% barium
hexaferrite by volume but lot 2 and lot 3 show almost no traces of barium hexaferrite. However,
barium ferrite micro-power shows single phase only. This means that the preparation method is
147
correct but it is not reliable. For the three lots of barium hexaferrite, the percentage of each phase
was determined by Rietheld method. These measurements were carried out at PNNL. The results
are shown in Table 6.6.
TABLE 6.7
XRD STUDY OF MICRO AND NANO HEXAFERRITE POWDERS
Sample
Nano- Ba ferrite
Micro- Ba ferrite
Lot Number
Phases (vol %)
Barium
hexaferrite
Iron oxides
Lot 1
56.2
43.8
Lot 2
-
100
Lot 3
-
100
-
100
-
As expected, only the barium hexaferrite micro-powder shows a single phase. The nanosized barium hexaferrite samples were either partially or completely composed of iron oxides.
This characterization method does not completely distinguish between γ- Fe2O3 and ε- Fe2O3. So
the compositions shown in Table 6.6 might not be exact but the indicated percentage of barium
hexaferrite phase is correct. Presumably these iron oxide phases resulted from an incomplete
synthesis of barium hexaferrite during the preparation process. Another iron oxide, α- Fe2O3 is
known to be used as a precursor for the production of barium hexaferrite.
These samples were also characterized at millimeter wave frequency range. Out of the
three barium ferrite samples mentioned here, only lot# 1 showed resonance phenomena
(millimeter wave absorption) during the characterization process, which could be due to the
148
partial presence of barium hexaferrite phase in the sample. The other two samples did not show
any resonance. The transmittance spectra for the three nano-ferrite samples are shown in figure
6.7.
Figure 6.7 Transmittance spectrum for different lots of barium hexaferrite samples. The hexaferrite phase
was missing in lots 2 and 3. This can also be concluded from this data due to the missing resonance.
These results suggest that the BWO based measurement technique used here is sensitive
enough to imply the absence of a certain phase and thus millimeter wave absorption could be
used as a quality control technique in the manufacture of ferrite nano-powders. The advantage of
this technique is that it is non-destructive and quick. It can be concluded from the results shown
in section 6.5 that the performance of the samples depends on the morphology of the particles in
the sample. But controlled fabrication of powders in this size is not possible. The particle size
and shape gets deviated from the intended design. As can be seen in table 6.6, even though the
same procedure is used to manufacture several samples of these powders the outcomes can be
different. The microwave or millimeter wave properties can be used as an additional check to
confirm if the sample constituents and dimensions are correct.
149
Chapter 7
Error Analysis
7.1 Introduction
In order to interpret the measured parameter values correctly, it is essential that we identify the
possible sources of error and uncertainties in the measurement process and apply the corrective
action, wherever possible. The occurrence of experimental uncertainties is inevitable but a good
estimate of these uncertainties can help in validating the accuracy of the experiment.
The error in any measurement is classified as either random or systematic error.
Experimental uncertainties that can be revealed by repeated measurements are called random
errors [102]. But all types of experimental uncertainties cannot be assessed by repeated
measurements. The repeatable and stable uncertainties in measurement are called systematic
errors. Systematic errors are due to imperfections in the network analyzer and the test set up.
They are repeatable (therefore predictable) and are assumed to be time invariant. Systematic
150
errors are characterized during the calibration process and mathematically removed during
measurements. Random errors, on the other hand, are unpredictable since they vary with time in
a random fashion. Therefore, they cannot be removed by calibration. The main sources of
random errors are instrument noise.
In addition to these errors, the system may also suffer from drift error. Drift errors occur
due to change in system performance after calibration. Drift is primarily caused by temperature
variation and it can be removed by further calibration. The timeframe over which a calibration
remains accurate is dependent on the rate of drift that the test system undergoes in the user’s test
environment. Drift can be minimized by providing a stable ambient temperature. The
measurements were performed under constant temperature and humidity conditions and so no
significant drift errors were observed. The systematic and random errors in the microwave
measurement procedure are discussed in the following sections.
7.2 Systematic errors in microwave measurements
Systematic errors are caused by imperfections in the test equipment and test set-up. These errors
do not vary over time and so they can be characterized through calibration and mathematically
removed during the measurement process. The process of removing systematic errors from the
network analyzer s-parameter measurement is called measurement calibration.
In network analyzer based measurements, the need for calibration arises because of the
presence of several physical components between the sample under study and the measurement
equipment. These introduce losses and phase delays in the measurement. In our experiments,
these components include waveguide adapters, connectors and cables. The measurement set-up
described in chapter 3 is shown again in figure 7.1. All the additional components added to
connect the sample with the network analyzer have been highlighted in this figure. It should be
151
noted here that the frequency range covered in the frequency band defines the size of the
waveguide that can be used for the measurements. Each of the eight frequency bands studied
here had a specific rectangular slot dimension associated with them. As a result, in this
measurement eight different waveguides were connected to the network analyzer ports. The
network analyzer cables are 2.4 mm coaxial cable. But depending on the band, the connectors on
the waveguide varied from 2.4 mm (Ka band) to N-type SMA (R band). Thus a variety of
adapters were used in the measurement process and each frequency band was calibrated before
carrying out the measurements.
152
Figure 7.1. Network analyzer measurement set-up is shown with the sources of errors marked in the red
boxes.
As seen in figure 7.1, there are several components between the measurement plane of
the network analyzer and the desired measurement plane, which is at the sample interface. All
these components, including the black coaxial cable shown in the figure, become part of the error
box. The presence of special components in the measurement path leads to a difference between
the measurement port of the network analyzer and the sample interfaces. These are marked as
VNA measurement port and desired measurement port in figure 7.2.
VNA Measurement
Plane
Port 1
VNA Measurement
Plane
Desired
Measurement
Plane
Material
under
Test
Error Box I

[


]

[

[ 


]




]

Error Box II

[

Port 2

]

Figure 7.2 Block diagram of the waveguide set-up. Port1 and port 2 indicate the location of the network
analyzer ports while the desired measurement port is at the interface of the material under test.
The additional cables and adapters used to connect the sample holder to the network
analyzer ports introduce uncertainty in the measurement. These components are considered to be
153
error networks and their impact must be removed from the measured values in order to obtain the
actual characteristics of the sample being studied. This is done by calibrating the experimental
set-up before each measurement. The idea is to first characterize the error boxes over the entire
measurement frequency range and apply this knowledge to obtain actual error-corrected sample
measurements. This way the network analyzer’s measurement port is transformed to the desired
measurement port.
For s-parameter measurements, the reference planes are defined by means of a standard
calibration technique such as Short-Open-Load-Thru (SOLT), Line-Reflect-Reflect-Match
(LRRM) or Thru-Reflect-Line (TRL). These are all cascade-based two port techniques. These
eliminate the need to know the internal structure of the components forming the error network.
Here, a TRL calibration technique was applied. TRL technique does not rely on known standard
loads, such as matched loads or opens, and so it does not suffer from the errors occurring due to
load imperfections at higher frequencies. Each frequency band has a specific calibration kit. The
calculations involved with the calibration procedure are implemented via a software package.
The TRL calibration procedure consists of three steps.
Consider the block diagram of the set-up is shown in figure 7.2. As mentioned earlier, the
error box represents all the connectors, cables waveguides and waveguide adapters that are used
in the measurement path. Instead of standard loads, the calibration procedure uses three different
types of connections to characterize the error boxes. Since the system is considered to be a
cascade of three sub-systems, it is better to use ABCD matrix. The overall parameters of a
cascade system can be written as the product of the constituent networks, thus simplifying the
calculations [46]. For the system shown in figure 7.2, the network equation can be written as,
[
]
[
] [
154
] [
]
[
or simply,
]
[
] [
] [
]
here, [Am] represents the ABCD matrix measured for the overall two port network, [A1] and [A2]
are the ABCD matrices of the individual error boxes and [As] corresponds to the ABCD matrix
of the material under test. The goal is to determine the accurate parameters for the sample, that
is, the [As] matrix. Once [As] is determined, the ABCD matrix can be converted to the sparameter matrix. The first step is to determine the parameters for the error box. The three
scenarios used for this purpose are described below. It is assumed that the error network II is the
mirror image of error network I and that both networks are uncoupled.
The TRL calibration method employs known loads in the form of thru, reflect and line to
characterize the network. For thru connection, the two waveguide sections are connected
together. This set-up is shown in figure 7.3.
VNA Measurement
Plane
VNA Measurement
Plane
Port 1
Error Box I
Error Box II
Figure 7.3 Thru connection for calibration procedure.
The s-parameters of error network I can be converted into T-parameters using the wellknown conversion equations [ 39],
[
]
155
For the mirrored network II, the T-parameters can be obtained by simply swapping the
ports,
[
]
The T-matrix for the THRU set-up is given by the product of TI and TII,
[
]
This matrix can be converted back to s-matrix and compared with the measured sparameters for the set-up shown in figure 7.2. Let STHRU represent the overall s-parameters for
Thru connection.
[
]
[
]
Due to symmetry and reciprocity of the circuit, it is expected to observe
and
=
. By applying these conditions on the symmetric error box
matrices, the following relation between the known (
,
and
=
) quantities for the thru connection is obtained,
156
and
) and unknown (
,
It should be noted that for the case of thru connection the system is reciprocal (
). Next, the reflect measurement is carried out. For reflection measurement, a metallic short is
placed at the end of each waveguide. Identical equations are obtained for either port. The block
diagram and calculations have only been shown for port 1.
VNA Measurement
Plane
Port 1
Error Box 1
Figure 7.4 Block diagram of the reflect connection.
The measured s-parameter matrix is denoted by
and by following the similar
calculation procedure as above the following equation can be obtained,
[
here,
]
is the reflection coefficient at the short circuited end of the waveguide.
The last connection in the calibration procedure is line. This can be basically seen as
inserting a known length of transmission line between the waveguides. It can, however, not be
zero since that would be the case for thru connection. In this study, the length used was the same
as the thickness of the sample holder (l).
157
Figure 7.5 Block diagram of the line connection.
Following the same procedure of calculation as in step 1, the following relations are
obtained,
[
]
[
]
Equations (7.6) to (7.10) are used to characterize the error boxes. These form a set of five
equations with five unknowns, namely,
,
,
, γ and Γ. The derivation of these quantities
is complicated [39]. The final form of the s-parameter matrix for the error box is given here in
terms of the known measured quantities,
√
[
√
]
158
where,
Thus the s-parameters for the error boxes are now determined. These can be converted to
ABCD matrix and used in equation x to determine the ABCD matrix for the sample under study,
which is converted back to s-parameters for further calculations. We were able to maintain return
losses less than -50 dB for all bands. This allows us to neglect any unwanted reflections from the
inner walls of the waveguide.
7.3 Random errors in microwave measurements
Random error is defined as the result of a measurement minus the mean over a very large
number of measurements of the same measurand carried out repeatedly. These can be treated
statistically to find the variation in the values and specify the confidence level for the data. For
each sample, the measurements were repeated ten times and the result was averaged. The error
percentage and standard deviation observed in the measurements at 40 GHz is shown in table
7.1.
TABLE 7.1
STANDARD DEVIATION IN THE DERIVED PARAMETER VALUES FOR BARIUM HEXAFERRITE
Average Standard
Error
Material Property
Value
Deviation Percentage
Real Permeability
1.0642
0.0042
5.17
Imaginary Permeability
0.0706
0.0018
2.37
Real Permittivity
1.4647
0.0058
1.56
159
Imaginary Permittivity
0.0516
0.0027
3.21
The reasons for random errors in these measurements can be identified as uncertainty in
the experimental setup, including sample preparation and the errors due to the ambient
conditions. The network analyzer uses the same coaxial cable for all frequency bands. The size
of the waveguides for these bands varies from 4.32 inch to as small as 0.28inch across the 2 to 40
GHz frequency band. Thus several connectors have to be used to connect the different
waveguides to the cables of the network analyzer which makes the system more sensitive to
noise and drift. The adapters and connectors added to the waveguide were accounted for in the
calibration process. In order to minimize these errors, an effort was made to keep the whole setup as stable as possible. Extra care was taken to ensure that the measurement path included
minimum number of twists and turns and that the system remains unchanged between calibration
and measurement steps.
Another major source of random errors is the packing of the sample. The measurement
becomes more challenging since the samples are not in solid form but powder form. The sample
was manually milled right before filling in the sample holder to prevent the particles from
forming agglomerates. The sample surface must be flat and there should not be any air gaps
inside the sample holder. Since the sample is in powder form, the particles in the sample holder
get displaced easily even with a slight movement and the density becomes non-uniform. It is
important to make sure that the same uniform density is maintained when loading the sample as
well as at the time of measurement. Proper alignment tools must be used to make sure that the
sample holder and waveguides are completely aligned with each other and there is no air gap
between the sample holder and the waveguide surfaces.
160
All attempts were made to eliminate these sources of random errors in the measurements.
These were minimized by repeating the measurement ten times for each sample. The standard
deviation in the values of real part of permittivity and permeability was observed to be +0.0058
and +0.0042, respectively. To keep the density fixed across all frequency bands, different
samples had to be repacked for each measurement. A scale was used to weigh the mass of the
sample. The variation in values read from the scale is about +0.02%, which can be neglected.
The mean value and standard deviation in the measured values of real and imaginary
parts of the transmission and reflection s-parameter are shown in table 7.2. Since the reflection
and transmission
coefficients are used in the calculations, only these parameters have
been tabulated. The data presented in this table is the averaged result of ten repeated
measurements. The standard deviation (σ) is calculated as,
√ ∑
̅
where, N is the number of experiments (which is 10 in this case),
experiment and
is the data value at the
̅ is the mean of all the data points. Standard deviation shows how much the
measured data values deviate from the average value for that parameter. Sometimes the average
value is also referred to as the expected value. Thus calculating the standard deviation is a way of
identifying the variation in the measured values.
161
TABLE 7.2
STANDARD DEVIATION IN THE MEASUREMENT OF S-PARAMETERS AT EACH FREQUENCY BAND FOR
BARIUM HEXAFERRITE NANOPOWDER
Frequency
Real
Mean
Real
-0.108+0.0002
-0.365+0.0004
Imaginary
Imaginary
-0.031+0.0003
-0.693+0.0003
R
Correlation C
Mean
-0.931
-0.119+0.0003
-0.413+0.004
-0.871
-0.026+0.0007
-0.678+0.0005
S
Correlation C
Mean
-0.956
-0.096+0.0005
-0.397+0.0007
-0.755
-0.037+0.0007
-0.763+0.0006
G
Correlation C
Mean
-0.891
-0.081+0.0005
-0.371+0.0008
-0.639
-0.029+0.0006
-0.740+0.0009
C
Correlation C
Mean
-0.902
-0.158+0.0006
-0.427+0.0009
-0.726
0.018+0.0008
-0.825+0.0008
X
Correlation C
Mean
-0.849
-0.132+0.001
-0.459+0.0011
-0.802
-0.042+0.0011
-0.873+0.0013
Ku
Correlation C
Mean
-0.873
-0.162+0.0006
-0.319+0.0009
-0.634
-0.033+0.0007
-0.719+0.0009
K
Correlation C
Mean
-0.981
-0.131+0.009
-0.368+0.013
0.588
0.056+0.016
-0.678+0.012
Ka
Correlation C
-0.988
-0.419
It can be observed here that the standard deviation increases with frequency. For instance,
the standard deviation for all four parameters is observed to be the highest for Ka band. This is
because since the size of the waveguides and sample holder becomes too small at this frequency
162
range. The system becomes extremely sensitive to even the slightest variation in the set-up. Even
a tiny air gap becomes significant at this frequency. The calibration becomes tougher for the
higher frequency bands and the calibration procedure had to be repeated several times before
favorable results were achieved.
Correlation is another statistical tool that represents the relationship involving
dependence in the measured set of data. Correlations are useful since they provide an indication
on whether the values of two parameters are related in some way. If this relationship can be
predicted, then additional steps can be taken to improve the measurement results.
∑
∑
(
̅
̅ ∑
It can be seen in table 7.2 that the real parts of
̅ )
̅
and
are strongly correlated. This
means that the value of one is dependent on the other. The data also indicates that these
parameters are negatively correlated. This is expected since
energy that is reflected at port 1 or at the input to the sample and
is a measure of the amount of
is a measure of the amount
of energy that reaches port 2 after transmission through the material. Therefore, if
is large, it
means that less energy will be available for transmission and thus lower amount of energy will
reach port 2 as compared to the case when
is small. Thus the results obtained here are in
agreement with the expectations.
The average error percentage in the measurement of samples existing in two different
forms is shown in table 7.3. These values are also averaged for ten repeated measurements. The
random error percentage is calculated as,
163
|
̅|
̅
In table 7.3, the error percentage has been compared for solid and powdered samples.
Measurement of solid samples is relatively easier when compared to the measurement of samples
that are in the form of powder. This is because the particles in the powder can shift inside the
sample holder and give rise to air gaps. This leads to unexpected measurement results. The
powdered samples therefore need extra caution when packing inside the sample holder. Another
common source of error is the agglomeration of the particles at the corners of the sample holder
which leads to the formation of discontinuity and gaps at the corners. Solid samples, on the other
hand, fit perfectly inside the entire sample holder volume, leaving no room for air gaps. The
challenge with these samples is to cut them in exact dimensions so that they are neither loose nor
tight inside the sample holder. Solid samples of Teflon were especially cut out to the size of each
frequency band. These were used for reference. In tables 7.3 and 7.4 the error percentage for
samples is compared to that of nano-size barium hexaferrite sample.
TABLE7.3
AVERAGE ERROR PERCENTAGE IN DIELECTRIC PROPERTIES OF SOLID AND POWDERED SAMPLES
Real
Imaginary
Dielectric Loss
Sample
Permittivity
Permittivity
Tangent
Teflon
2.03 + 0.034
0.0005 + 0.041
0.0002 + 0.062
1.47 + 0.053
0.053 + 0.069
0.016 + 0.86
(solid)
Ba-ferrite
(powder)
164
The error percentage for powdered samples is higher than that for the solid samples. This
could be attributed to the challenges in measuring powdered samples that were described above.
However, the error percentage is still very small and within acceptable limits. Therefore, it can
be concluded that these measurements are reliable. The error percentage for dielectric loss
tangent is higher than that for the real and imaginary parts of permittivity. This is because the
loss tangent values were derived from the real and imaginary permittivity values and so the error
propagates along in the calculations. The error percentage in the calculation of magnetic
properties of the samples are summarized in table 7.4
TABLE7.4
AVERAGE ERROR PERCENTAGE IN MAGNETIC PROPERTIES OF SOLID AND POWDERED SAMPLES
Real
Imaginary
Magnetic Loss
Sample
Permeability
Permeability
Tangent
Teflon
0.26
0.78
1.56
1.064 + 0.015
0.071 + 0.019
0.041 + 0.023
(solid)
Ba-ferrite
(powder)
The major sources of errors in the microwave measurements of the samples have been
identified in this section. The error analysis shows that the results are within acceptable
tolerances and that this technique can reliably determine the real and imaginary parts of electric
permittivity and magnetic permeability for powdered samples. Next, the uncertainties in the
measurement of millimeter wave properties are analyzed.
165
7.4 Uncertainty in millimeter wave measurements
The experimental set-up for measurements at millimeter wave frequencies is more stable than the
waveguide based microwave measurement set-up. This set-up does not have a pre-defined
calibration procedure. The uncertainties in the BWO based measurement technique are
determined by performing several reference scans before taking the actual measurement. Both
transmittance and reflectance modes were scanned for each band while keeping the other
conditions unchanged. These scans were performed without any sample in the transmission path.
Repeated measurement of the transmittance and reflectance spectra, under constant conditions,
should yield identical results. Therefore the ratio of any two measurements of either
transmittance spectrum or reflectance spectrum should be unity. The results of the reference
measurements are summarized in table 7.5. These values were obtained as the average over
several measurements. In each case, the ratio was calculated for consecutive measurements
performed under identical conditions.
TABLE7.5
RESULTS OF REPEATED REFERENCE SCANS AT EACH FREQUENCY BAND
Frequency band
Q
V
W
Transmittance spectrum ratio
1.015
1.012
1.01
Reflectance spectrum ratio
1.02
1.018
1.015
The results of the reference scan seem quite good. Although the ratio is not exactly unity,
it is still very close. The reason for this variation observed in the test runs is the instability of the
electronic devices that constitute the measurement set-up [103]. These include the BWO tubes
166
and high power supply. From the values shown in table 7.5, it can be concluded that the
uncertainty in the measurements is about 2%. These measurements have been repeated ten times.
As mentioned in the case of microwave measurements, repeated measurements reduce the
random uncertainties in the experiment. The average value of these measurements and the
variation in the measured transmittance and reflectance spectra are plotted together in figure 7.6.
Since the standard deviation is not too large the plots appear to overlap in the figure. The inset in
figure 7.6 provides a magnified view of the plots. The slight variation is evident from this plot. It
can be concluded from these results that the measured values do not vary significantly. The
stability of several transmittance scans show that the systematic errors are also negligible in this
case.
Figure 7.6 Experimental uncertainty and standard deviations for the transmittance spectra.
167
Figure 7.7 Experimental uncertainty and standard deviations for the reflectance spectra.
The standard deviation for each case is also summarized in table 7.6. As discussed in chapter 5,
in case of millimeter wave measurements the material properties are derived by curve fitting the
calculated transmittance to the measured transmittance spectra. Therefore, it makes sense to
identify the uncertainties in the measurement of the transmittance plots alone. The variation in
the derived material properties would reflect the same uncertainties. It is not necessary to
characterize the curve fitting procedure since it is a manual step and depends on the precision of
the operator.
TABLE7.6
STANDARD DEVIATION IN THE MEASURED TRANSMITTANCE AND REFLECTANCE SPECTRUM FOR
MILLIMETER WAVE FREQUENCY BANDS
Frequency band
Standard Deviation
168
Transmittance Spectrum
Reflectance Spectrum
Q
+ 0.00177
+ 0.001
V
+ 0.00214
+ 0.0008
W
+ 0.00171
+ 0.0019
169
Conclusion
Microwave and millimeter wave properties of micro- and nano-size ferrite powders have
been studied. Ferrites with different crystal structure and particle sizes have been studied.
Measurement of samples in powder form is challenging at such high frequencies. Material
characterization techniques were developed to accurately determine the intrinsic material
properties, namely, permittivity and permeability of the powder samples. The network analyzer
based microwave measurement technique has been modified to obtain precise measurements of
material properties of powdered samples. Millimeter wave measurements have been performed
by a quasi-optical spectrometer. Since the properties have been found to be size dependent, it is
important that the measurement set-up is accurate enough to detect these differences.
The measurement of samples in powder form is a complex procedure. In the past, the
powder samples have been diluted with conductive materials or epoxy to fabricate a solid sample
for measurements. The procedure followed in this study does not use any such additive.
Therefore the properties of the samples can be determined precisely. This information is useful
to understand the behavior of the sample in the presence of electromagnetic energy. This
knowledge can also be applied to simulate the material more accurately. Designers can save time
and money by simulating their design before fabrication.
The results presented here reveal that the ferromagnetic resonance frequency of the ferrite
samples vary with the particle size. Therefore the material can be tuned to resonate at different
frequencies by changing the size of the particles. This is particularly useful in the design of high
frequency tunable absorbers. The resonance observed in the case of nanoparticles is much
sharper than that observed for bulk samples or samples containing micro-sized particles.
170
Absolutely no resonance was observed in case of samples that did not contain the proper phase
of barium hexaferrite. Preparation of ferrite powders in nano-size is challenging. The preparation
procedure and conditions affect the properties of the sample. The techniques developed in this
study can be used for non-destructive testing to check if the desired phase has been achieved or
not. Accurate characterization would allow scientists to utilize nanomaterials to their full
potential and achieve high performance in diverse applications.
171
List of publications
Journal Publications
1. A. Sharma, L. Chao, M. N. Afsar, “Microwave and millimeter wave ferromagnetic
absorption of nanoferrites,” IEEE Transactions on Magnetics, vol. 48, no. 11, November
2012.
2. Anjali Sharma, M. N. Afsar, “Effects of particle size on magnetic and dielectric properties of
ferrites at microwave and millimeterwave frequencies,” accepted at European Microwave
Conference’2013 and submitted to IEEE transactions on Magnetics (under review).
3. A.Sharma, M.N. Afsar, “ Measurement Technique for Characterizing Constitutive Material
Properties of Thin films,” accepted for oral presentation at International Instrumentation and
Measurement Technology Conference (I2MTC), 2013. Journal manuscript has been
submitted.
4. N.Rahman, A.Sharma, M.N.Afsar, “Characterization, design and optimization of low-profile
cavities for UWB spiral antennas,” International Journal of Electromangetics and
Applications, 2012.
5. A. Sharma, M. Obol, M.N. Afsar, “Microwave Complex Permeability and Permittivity
Measurements of Commercially Available Nano-ferrites,” IEEE Transactions on Magnetics,
vol. 47, No.2, Feb’2011.
6. A. Sharma, M.N. Afsar, “Microwave Complex Permeability and Permittivity of NanoFerrites,” Journal of Applied Physics, vol.109, Issue 7, April’2011.
172
7. Nahid Rahman, Mahumt Obol, Anjali Sharma, Mohammed Afsar, “Determination of
Negative Permeability and Permittivity of Metal Strip Coated Ferrite Disks Using the
Transmission and Reflection Technique”, Journal of Applied Physics,2010.
8. Zijing Li, Anjali Sharma, Ana M. Ayala, Mohammed Afsar, K.Korolev, “Broadband
Dielectric Measurements on Highly Scattering Materials”, IEEE Transactions on
Instrumentation and Measurement, Vol. 59, Issue 5, May 2010.
Conference Proceedings
1. A.Sharma, M.N.Afsar, “Improved waveguide based technique for measurement of magnetic
and electrical properties of powdered ferrite sample at microwave frequencies,” Joint MMM
Intermag Conference, 2013.
2. A. Sharma, L.Chao, M.N.Afsar, “Microwave and millimeter wave measurements of
electrical and magnetic properties of oxide based nanopowders” Joint MMM Intermag
Conference, 2013.
3. John McCloy, A.Sharma, M.N.Afsar, et.al. “Mangetic and Mossbauer studies of unirradiated
and gamma ray irradiated barium hexaferrite powders,” submitted to MMM 2010, revised
2012.
4. L.Chao, A.Sharma, B. Yu, M.N.Afsar, “Dielectric permittivity measurements of thin films at
microwave and terahertz frequencies,” European Microwave Conference, 2012.
5. A. Sharma,L.Chao, M.N.Afsar, O. Obi, Z. Zhou, N. Sun, “Permittivity and permeability
measurement of spin-spray deposited NiZn-ferrite thin film sample from 18 to 40 GHz,”
Intermag 2012 Conference.
173
6. A.Sharma, M.N.Afsar, “Accurate permittivity and permeability measurement of composite
broadband absorbers at microwave frequencies,” International Instrumentation and
Measurement Technology Conference, 2011.
7. Anjali Sharma, Nahid Rahman, Mahmut Obol, Mohammed Afsar, “Precise Characterization
and Design of Composite Absorbers for wideband Microwave Applications”, European
Microwave Conference’2010.
8. Anjali Sharma, Nahid Rahman, Mohammed Afsar, “Precise Permittivity and Permeability
Measurement of Absorbers over Broadband Microwave Frequency Range”, Conference of
Precision Electromagnetic Measurements’2010.
9. Nahid Rahman, Anjali Sharma, Mahmut Obol, Mohammed Afsar, Sandeep Palreddy, Rudolf
Cheung, “Broadband Absorbing Material Design and Optimization of Cavity-Backed, Twoarm Archimedean Spiral Antennas”, International Symposium on Antennas and
Propagation’2010.
10. Nahid Rahman, Anjali Sharma, Mahmut Obol, Mohammed Afsar, Sandeep Palreddy, Rudolf
Cheung, “Dielectric Characterization and Optimization of Wide-band, Cavity-backed Spiral
Antenna”, Applied Computational Electromagnetics Society’2010.
11. Anjali Sharma, Mahmut Obol, Mohammed Afsar, “Microwave Permittivity and Permeability
Properties and Microwave Reflections of Micro/Nano Ferrite Powders”, International
Instrumentation and Measurement Technology Conference’2009.
174
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Appendix
187
I2MTC 2009 - International Instrumentation and Measurement
Technology Conference
Singapore, 5-7 May 2009
Microwave Permittivity and Permeability Properties
and Microwave Reflections of Micro/Nano Ferrite
Powders
M. N. Afsar, IEEE Fellow, A. Sharma, and M. Obol, IEEE Member
High Frequency Materials Measurement and Information Center, Department of Electrical Engineering, Tufts University, Medford,
Massachusetts, USA
Abstract— The application of magnetic powders are becoming
attractive in electromagnetic shielding interferences in wireless
systems. Magnetic powders such as Fe3O4 are also very important
in magnetic resonance imaging as well as drug delivery applications
in biomedical researches. Also, some of the special catalyst nano
magnetic particles are increasingly useful in the fuel cell
applications. Hence, a good understanding of the electromagnetic
wave response such as permittivity and permeability of magnetic
powders may be important in modern microwave technological
applications. In this paper we present broad-band measurements
on the real and imaginary parts of permeability and permittivity of
micron sized commercial nickel ferrites of Ni1-x-yCoxCuyFe2-zdMnzO4 (TT2) and Ni1-xZnxFe2-yMnyO4 (TT86) in the frequency
range from 4 to 40 GHz. We also present broadband permeability
and permittivity of some other commercially available nano and
micron sized ferrite powders, so that this is a paper that explores
microwave properties of nine different nano and micron sized
magnetic powders in terms of permittivity and permeability
aspects.
Keywords-Components— Broadband, permeability, permittivity, TRL
calibration and important ferrite powders
I.
INTRODUCTION
It is obvious that the microwave permittivity and permeability of
these solid substances are very well known. The partial attempts
to measure the permeability and permittivity of these solid and
powdered materials were presented in the previous
investigations [1, 2, 3, 4, and 5]. The current technology adopts those
solid composites in the scheme of nano and micro sizes. The
precision measurements [1] showed that nano magnetic powders
microwave permittivity and permeability are very different
compared to their solid cases. It is our understanding that it may
be very useful to extend the work [1] into the broadband
microwave regime. As such, the objective of this work is to
present the magnetic characteristics of nine different powdered
ferrite composite materials. The three different commercially
available powdered magnetic composites, purchased from
Trans-Tech, are known as TT2, TT86 and Ba2Co2Fe12O22 (99%
pure phase hexaferrite). The average particle size of these
powders was between 3 and 6 micron. The other six different
nano ferrites (average particle sizes were less than 40 nm) used
978-1-4244-3353-7/09/$25.00 ©2009 IEEE
in the experiment are purchased from Sigma-Aldrich, Inc. To
carry out the powder measurements, the ferrite powders, whose
complex permittivity and permeability are to be determined, are
loaded inside calibrated waveguide shims and measured with a
network analyzer. The measurement procedure is basically
similar to the technique in [1]. The difference in this paper is to
apply the technique to six different bands. Here, the raw ferrite
powders were simply loaded inside waveguides without
applying any pressure so that the same density is maintained;
perhaps this approach will reduce particles interfering with each
other at some degree. The waveguides are different in physical
sizes for measurements at different frequency bands. In addition
to that this paper extrapolates the continuous scattering
parameter of broadband using microwave permittivity and
permeability in six different bands. This calculation approach
provides broadband reflectivity and transmittance of scattering
parameters which is often hard to obtain by using the free space
measurements [4]. The method applied here is the usual
calibration technique, known as TRL (through-reflect-line). The
calibrated parameters help to determine the complex
permeability and permittivity of the powdered ferrite
composites.
II.
THEORY OF WAVEGUIDE
The permeability and permittivity measurements of various
materials were reported in previous work [1], [2], [3], [4] and
[5]. The theory of this work is based on paper [1], and each
parameter in the equations well corresponds to the work of [1].
In order to expand this to the broadband measurements
purposes, in this work, several micro and nano powdered ferrite
composites were loaded in the waveguides of six different
rectangular waveguides of G (4 to 6 GHz), C (4 to 8 GHz), X (8
to 12.4 GHz), Ku (12.4 to 18 GHz), K (18 to 26.5 GHz) and Ka
bands (26.5 to 40 GHz), and were measured with the help of
TRL calibration. The Agilent’s 8510C Vector Network Analyzer
(VNA) was used to record the scattering parameters from
waveguides which were further used to perform transmission
and reflection measurements on the powdered ferrite
composites. The measurement method in a previous paper [1]
was used to obtain the permeability and permittivity of the
powders (equations 1, 2 and 3 are from paper [1]) and the BakerJarvis method [3] was used to calculate broadband scattering
parameters S11 and S21 (equations 4, 5, and 6 are from paper [3])
using permittivity and permeability in six different waveguides
in this paper. The purpose in doing so is to ensure that the
derived permeability and permittivity by the waveguide
technique [1] has the capability to reproduce the appropriate
scattering parameters as obtained by calculation method [3]. It is
also a reasonable idea to present broadband scattering
parameters for demonstrating purpose.
Γ=
T =e
μr
+1
εr
⎛ω ⎞
− j ⎜ ⎟ d μrε r
e ⎝c⎠
−γd
S11 (ω ) =
S 21 (ω ) =
Γ = K ± K 2 −1
~
~
S 2 − S 2 +1
K = 11 ~ 21
2S11
~
~
S +S −Γ
T = 11~ 21~
1 + ( S11 + S 21 )Γ
⎛
⎞⎜
1
⎛ 1 + Γ ⎞⎛ 1 ⎞⎛⎜
μ = − j⎜
⎟⎜
⎟⎜ ln( ) + j (2πn − ϕT ) ⎟⎟⎜
⎝ 1 − Γ ⎠⎝ 2πd ⎠⎝ T
⎠⎜
⎝
μr
−1
εr
III.
(1)
2
2⎞
⎛ 1 ⎞ ⎛ 1 ⎞ ⎟
⎜ ⎟ −⎜ ⎟ ⎟
⎜λ ⎟
⎝ 0 ⎠ ⎝ 2a ⎠ ⎟
⎠
=
(1 − T 2 )Γ
1 − T 2Γ 2
(1 − Γ 2 )T
(4)
(5)
(6)
1 − T 2Γ 2
EXPERIMENTAL RESULTS OF NICKEL FERRITES
Ni1-x-yCoxCuyFe2-z-dMnzO4 (TT2): Figure 1 represents the real
and imaginary parts of the permittivity and permeability of the
two different Trans-Techs powdered micron sized Nickel
ferrites.
−1
(2)
⎛c
⎝ f
ε = − j ⎜⎜
⎞
⎟⎟
⎠
2
⎞⎛⎜
1
⎛ 1 − Γ ⎞⎛ 1 ⎞⎛⎜
⎜
⎟⎜
⎟⎜ ln( ) + j (2πn − ϕT ) ⎟⎟⎜
⎝ 1 + Γ ⎠⎝ 2πd ⎠⎝ T
⎠⎜⎝
2 ⎞
⎛ 1 ⎞
⎟
1
⎜ ⎟ − ⎛⎜ ⎞⎟ ⎟
⎜λ ⎟
⎝ 2a ⎠ ⎟
⎝ 0⎠
⎠
2
(3)
According to the recorded S parameters from the VNA, the
complex permittivity and permeability of the powdered ferrite
composite materials were derived for each different band based
on the Weir algorithm [2], i.e., equation set (1) ,equation (2) and
equation (3). Although the permeability and permittivity could
be determined by the scattering parameters through the VNA,
the scattering parameters in six different bands are very
different. This is due to the geometrical dimensions of
calibration standards for each band and the thicknesses of the
appropriate loaded powder inside the waveguide as per the
recommended phase regime from Agilent. Although the group
delays of wave propagation approach would help to load the
powders until the shim is filled by powders [5], that group delay
approach would also need to know the initial guess to the
parameters. Sometime guessing is difficult, especially to the
magnetic materials when both permeability and permittivity are
unknown [3], for example. In order that this work keeps the
tedious and mechanical approach to the measurements, that is, to
load powders until we see the reasonable phases of scattering
parameters, to the manufacturer (Agilent). As long as we
determine the complex permeability and permittivity of
magnetic powder materials, we will always be able to regenerate
the scattering parameters for those powders in the entire
frequency range. The formulations for regenerating scattering
parameters may be as follows:
Fig. 1 Measured complex permeability and permittivity of TT2.
Fig. 2 Measured complex permeability and permittivity of TT86.
Figure 3 represents the calculated scattering parameters S11 and
S21 of these powders assuming the load (d) thicknesses of each
powder equal to 0.5mm for equation (5).
Calculated S-parameters of TT2, d = 0.5mm
Magnitude (dB)
0
magnitude of S11
-5
magnitude of S21
-10
phase of S11
-15
phase of S21
-20
-25
-30
-35
4
8
12
16
20
24
28
32
36
Phase (degree)
0
-20
-40
-60
-80
-100
-120
-140
-160
-180
5
40
Frequency in GHz
Fig. 3 Calculated scattering parameters of TT2, with thickness of loaded powder
as 0.5 mm.
Calculated S-param eters, TT86, d=0.5m m
Magnitudes
magnitude of S11
magnitude of S21
phase of S11
phase of S21
-10
-15
Phase (degree)
IV.
0
-20
-40
-60
-80
0
-5
-100
-120
-140
-160
-180
-20
-25
-30
-35
4
8
12
16 20 24 28
Frequency in GHz
32
reasonable data in this measurement process. It should be
understood that the presented permeability and permittivity at
the entire frequency region will be the average complex
permeability and permittivity. As for the TT2, that would be the
following, μ~r ≈ 1.15 − j 0.1 and ε~r ≈ 2.22 − j 0.2 .
Ni1-xZnxFe2-yMnyO4 (TT86): from Figure 2, one can easily note
the phenomena we had seen in Fig.1. The average complex
permeability and permittivity of TT86 is as follows,
μ~r ≈ 1.2 − j 0.1 and ε~r ≈ 2.3 − j 0.2 . One would note that the two
different microns sized powdered nickel spinel ferrites have
almost similar microwave properties in nature. Experimentally,
we were able to obtain complex permittivity and permeability of
the powders by using the scattering parameters for different
waveguides. The calculated S-parameters of TT86 are presented
in Fig. 4. Since we have not seen any unusual S-parameters by
inverse calculation it confirms that the measurement and
calculation methods used in this article are reliable for the
simultaneous permeability and permittivity measurements as
well as calculation of scattering parameters for other micron and
nano sized magnetic powders.
36
40
EXPERIMENTAL RESULTS OF OTHER FERRITES
In section 3, we succeeded to determine the broadband
permeability and permittivity of two different technologically
important micron sized powdered nickel ferrites. Therefore, in
this section, we are simply interested in determining broadband
permittivity and permeability of other technologically important
nano and micron sized ferrites. It may be important to present
the measurements of these powdered ferrites as a purpose of
ferrite data library. Figures 5, 6, 7, and 8 are for the available
micron sized powdered planar hexaferrite of Co2Y
(Ba2Co2Fe12O22) and powdered nano M-type hexaferrites of
Ba2M (BaFe12O19) and SrM (SrFe12O19).
Fig. 4 Calculated scattering parameters of TT86, with thickness of loaded
powder as 0.5mm.
We must point out here that the obtained permittivity and
permeability in each waveguide has some discrepancies with
other waveguides. This is reasonable, because ideally, according
to TRL standards of waveguides, the shim thickness should be
within quarter wavelengths of central operation frequencies and
that standards were usually in air; but this is very hard to
maintain for general cases unless each TRL standard shim
thickness was specifically reconfigure out to the unknown
loading materials. As such a mathematical averaging method
(least squares method of polynomials) was applied to show the
continuous data in entire frequency range for six different bands.
So the following broadband measurements presented in the
paper would be able to overwrite the boundaries of waveguides.
The estimated errors from the different bands of permittivity and
permeability, which are less than 5% would be considered a
Fig. 5 Measurements showed that the real permittivity of ferrites depend on the
sizes of particles of ferrites.
The broadband measurements showed that micron sized
hexaferrites have slightly higher permittivity compared to the
micron sized nickel ferrites in section 3. However, the nano
hexaferrites showed microwave properties that are close to that
of air, in showing low permittivity and low permeability with
low losses as well. Now, we present the Spinel nano ferrites
microwave complex permeability and permittivity starting from
Fig.9 to Fig.12 (see figures from 9 to 12).
Fig. 6 Measurements showed that the planar micron sized planar hexaferrites
dielectric loss is also slightly higher than nano sized M-type hexaferrites.
Fig. 9 Measurement showed the nano Spinel ferrites real permittivity,
permittivity of nano magnetite (Fe3O4) has exceptional permittivity.
Fig. 7 Measuremnts showed that micron sized planar hexaferrite has larger
permeability compare to the nano sized M-type hexaferrites.
Fig. 10 Measurements showed that Spinel ferrites imaginary permittivity, the
magnetite has strong imaginary component.
Fig. 8 Measurements showed that the nano and micron sized hexaferrite are in
the state of low magnetic losses, the differences probably describes the
systematic and human errors in measurements process.
Figures 9, 10, 11, and 12 are for the nano sized available
powdered spinel ferrites broadband permeability and
permittivity measurements. The measured data showed that
powdered nano spinel ferrite of Fe3O4 has exceptional properties
compared to the other powdered nano spinel ferrites. It may be
contributed either to the aggregate properties of magnetite, since
it has strong dipoles or to the semiconductor properties in
magnetite between Fe2+and Fe3+. Besides that, the other nano
ferrite powders showed pretty similar permittivity and
permeability behaviors which can be seen in the above figures.
Most notable phenomena is that the microwave permittivity of
the nano powders hover close to the permittivity of the air, and
as expected the microwave permeability of the nano powders are
similar to air. These properties of the nano powders may be
useful in the technology that looks for materials with properties
similar to air. It should also be very interesting that although the
microwave permittivity and permeability of those nano materials
are close to air, those powders are capable of generating
significant reflectivity at the higher frequency region.
Fig. 11 Measurements showed that nano Spinel ferrites real permeability is
approximately one which is reasonable.
matter of magnetic particles. For permittivity measurements
alone, we have not seen any difference between TT2 and TT86.
Calculating the reflection (S11) of both powders, it has been
shown that the powders are capable to generate more reflectivity
at higher frequencies compared to their lower frequency bands.
This is consistent with what we obtained for scattering
parameters using six different waveguides. It is our
understanding that it was not necessary to show the calculated
scattering parameters for other ferrite powders since the
calculation is straightforward now. According to the nano ferrite
powders measurements, one would realize that the powdered
nano hexaferrites in general would have smaller microwave
losses compared to the powdered spinel ferrites at comparable
frequencies. Also, relative permittivities for nano spinel ferrites
are little higher compared to the relative permittivities of nano
hexaferrites. Interestingly, nano ferrite of magnetite is much
different from the other nano spinel ferrites which are regarded
to their relative real permittivity and imaginary permittivity. In
summary, it is obvious now that the relative permittivity of the
nano and micron sized ferrites is very different as compared to
their solid state relative permittivity. Thus, this paper presents
following important concepts. The nano magnetite (Fe3O4)
powder is for MRI measurement that has a very high
permittivity compared to other nano powders. Perhaps, the
permittivity of the magnetite makes it very special in biomedical
applications. Besides that micron sized ferrite have generally
two to there order higher permittivity compare to the air. Lastly,
several other nano powders permittivity and permeability are
hovering close to air while they have the capability to generate
microwave reflectivity; this will turn out to be useful for non
destructive sensor or drug delivery application.
Acknowledgment
This research is supported by a contract from US Army National
Ground Intelligence Center.
References
Fig. 12 Measurements showed that nano Spinel ferrites imaginary permeability
is very small, the differences may describes the systematic and human errors in
the measurement process.
DISCUSSION AND CONCLUSIONS
The complex permittivity and permeability of nine different
composite ferrite powders were determined by using six
different rectangular waveguides. The obtained results are
reasonably good; the measurements showed that permittivity and
permeability depend on the density, size, and volume fraction
[1] N. N. Al-Moayed, M. N. Afsar, U. A. Khan, S. McCooey, M. Obol, “ Nano
Ferrites Microwave Complex Permeability and Permittivity Measurements
by T/R Technique in Waveguide”, IEEE, Transactions on Magnetics,
Vol.44, No.7, July 2008.
[2] W. B. Weir, “Automatic measurement of complex dielectric constant and
permeability at microwave frequencies,” Proceedings of the IEEE, Vol. 62,
No.1, pp. 33-36, January 1974.
[3] J. Baker-Jarvis, E. J. Venzura, and W. A. Kissick, “Improved technique for
determining complex permittivity with the transmission/reflection method,”
IEEE Trans. Microwave Theory Tech., vol. 38, No.8, pp. 1096-1103,
August 1990.
[4] R. Grignon, M. N. Afsar, Y. Wang and S., Butt, “Microwave broadband
free-space complex dielectric permittivity measurements on low loss
solids,” IMTC 2003-Instrumentation and Measurement Technology
Conference, Vail CO, USA, 20-22 May 2003.
[5] A. Bahadoor, Y. Wang and M. N. Afsar, “Complex permittivity and
permeability of barium and strontium ferrite powders in X, KU, and Kband frequency ranges”, JOURNAL OF APPLIED PHYSICS 97, 10F105,
2005.
308
IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 2, FEBRUARY 2011
Microwave Complex Permeability and Permittivity Measurements
of Commercially Available Nano-Ferrites
Anjali Sharma and Mohammed N. Afsar
High Frequency Material Measurement and Information Laboratory, Tufts University, Boston, MA 02155 USA
The dielectric permittivity and magnetic permeability of nano-sized magnetic powders are studied for a wide microwave frequency
range from 2 to 40 GHz. These powders are studied for their potential application as soft microwave absorbers in wireless systems as well
as for use in the manufacture of small magnetic recording tape, transformers and circulators. A newly developed and precise transmission-reflection based waveguide technique has been used to carry out measurements across the eight frequency bands. These materials
have been accurately characterized for the first time across this frequency range. In this paper, we present equations for simultaneous
calculation of permittivity and permeability across a wide frequency band using the vector network analyzer. To account for variable
sample thickness in the in-waveguide measurements, the algorithm for transmission and reflection calculations has been modified. The
results show that the dielectric permittivity depends on the particle size and density of the sample used for measurement.
Index Terms—Nano-ferrites, permeability, permittivity, TRL calibration, waveguide technique.
I. INTRODUCTION
ANO POWDERS are agglomerates of nanoparticles that
have an average size between 1 and 100 nm. Iron based
nano-powders are spherical metal nanostructure particles,
typically 20–40 nm in size. Electromagnetic properties of solid
forms of ferrite compounds have been studied in the past [1].
Various sizes of nano-ferrite powders are now commercially
available. Measurements carried out on nano-sized powders
have shown that the microwave permittivity and permeability
of these materials differ from those of the material in solid state
[2]. Therefore, there is a growing interest in studying these
powders to understand how the material properties are affected
by particle size.
Nano-powders of metal substituted iron oxide find extensive
application in biomedical and integrated circuit technology.
These materials prove very useful in biological drug delivery
systems, DNA-separators as well as in magnetic recording and
information storage [3]. Due to their low conductivity, these
materials have also been used in other microwave applications
such as transformers and circulators. Recently, these powders
are being investigated for their potential as microwave absorbing materials. Microwave absorbers have been in demand
for stealth applications. With the use of higher and broader
frequency bands for communication and military operations,
the need for materials that absorb electromagnetic radiation
at these frequencies has increased. To identify the use of
nano-ferrites in these applications, it is important to determine
their electromagnetic properties. The absorption capability
of a material depends on the intrinsic material properties,
namely, permittivity and permeability, which may be frequency
dependent [4]. The frequency spectra for the complex relative
permittivity and magnetic permeability of a material is therefore
N
Manuscript received July 09, 2010; revised August 10, 2010; accepted August
27, 2010. Date of current version January 26, 2011. Corresponding author: A.
Sharma (e-mail: Anjali.sharma@tufts.edu).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TMAG.2010.2073457
of prime importance to engineers and scientists who deal with
applications where large frequency variations are common.
Both permittivity and permeability are complex quantities.
The extent to which an electromagnetic wave is attenuated and
phase shifted depends on the permittivity and permeability of
the material at that frequency. The real part of permittivity (permeability) determines the amount of energy that is stored in the
material from the external electrical (magnetic) field while the
imaginary part determines the amount of energy lost by the wave
in traversing through the material [4]. Therefore, a large real permittivity implies that a large phase change occurs as the wave
traverses the material and a large imaginary part of permittivity
indicates that the magnitude of the wave will be heavily attenuated. This explains why the knowledge of the exact dependence
of these parameters on frequency is required to understand the
behavior of the material.
In this work, five nano-ferrite powders were characterized
to determine their electromagnetic properties over a wide
frequency range from 2 to 40 GHz. This microwave frequency
range is important because of lower propagation loss and
cheaper fabrication of microwave devices at these frequencies
as compared to the millimeter wave frequency range [5]. A
transmission-reflection (T/R) based waveguide technique that
employs a vector network analyzer was used to measure the
scattering parameters for each sample. Using this technique,
one can record both reflection and transmission spectra generated when an electromagnetic wave travels through the material
under study. Based on the reflection and transmission spectra
obtained from the measurements, the frequency at which the
sample exhibits strong absorption can be determined. The
algorithm used to determine the permittivity and permeability
from the measured data has been modified over the years to
improve accuracy [6]–[8]. The algorithm used in this study
takes into account the effect of the thickness of the loaded
material and its permittivity and permeability on the cutoff
wavelength inside the waveguide. Using this technique, precise
electromagnetic properties were determined for the commercially available nano-powders. A thorough error analysis is also
presented to account for the systematic and random errors in
the measurement process.
0018-9464/$26.00 © 2010 IEEE
SHARMA AND AFSAR: PERMEABILITY AND PERMITTIVITY MEASUREMENTS OF COMMERCIALLY AVAILABLE NANO-FERRITES
TABLE I
SAMPLE DESCRIPTION
309
waveguide delay in the vector network analyzer to further remove errors. The modified equations used in the measurements
are given below:
(1)
II. SAMPLE DESCRIPTION
The nano-powders studied in this work were purchased
from Sigma Aldrich. The powdered samples have a grain
size that varies between 40 and 100 nm for different samples.
The frequency range from 2 to 40 GHz is divided into eight
frequency bands, namely, R (1.70–2.60 GHz), S (2.60–3.95
Ghz), G (3.95–5.85 GHz), C (5.85–8.20 GHz), X (8.20–12.40
GHz), Ku (12.4–18.00 GHz), K (18.00 GHz–26.5 GHz) and
Ka (26.50–40.00 GHz). Depending on the covered frequency
range, each band corresponds to specific waveguide dimensions. To maintain uniformity in the measurements taken across
different bands, the density of each nano-powder was kept
constant across all bands. With density and volume fixed for
each band, the appropriate weight of the sample to be filled in
the waveguide shim was calculated. The weight of the sample
holder is measured with and without the powder to ensure
that the required density is maintained. To maintain uniform
distribution of the powder and avoid the presence of any air
gap in the sample holder, the sample was steadily filled in the
shim. The density of each sample used in the measurement is
summarized in Table I.
Strontium and barium ferrite are M-type hexaferrite whereas
the rest of the powders are spinel nano ferrites.
Lossless transparent tape was used on either sides of the
shim to hold the powder inside the shim. It has been shown
that the inclusion of tape has negligible effect on the measured
S-parameter [9].
where is the quarter wavelength difference between thru and
line in air, is the thickness of the sample inside the waveguide, is the wavenumber of the sample, and is the cutoff
wavenumber. These equations take into account the effect of
values that are smaller than
using samples with thickness
the waveguide shim used in the experimental setup.
Ideally, the shim thickness should be within quarter wavelengths of the central operation frequency. But this is hard to
maintain since the thickness used for each sample depends on
the density requirement. The difference between the material
thickness and sample holder thickness was taken in to account
when performing the calculations. Each frequency band was
sub-divided into 201 frequency points at which measurement
was taken, giving a total of 1206 frequency points. It should be
noted that the data at the end of one band may show some discontinuity with the data at the starting frequencies of the successive band. This may be due to experimental errors, depending on
how carefully the operator used the instruments. To remove the
experimental discrepancies in the data across different bands a
mathematical averaging method (least squares method of polynomials) was used. The derived data is very reliable and not effected by the scattering voltage ratios of the network analyzer.
The final form of the equations used to determine permittivity
and permeability is shown below:
(2)
(3)
III. EXTRACTION OF PERMITTIVITY AND PERMEABILITY
The transmission-reflection based waveguide technique has
been widely used to determine the properties of solids. It can be
further modified for the measurement of soft powders. The basic
technique has been explained in [10]. The vector network analyzer measures the scattering parameters of the 2-port network
formed by the waveguide shim filled with the sample under
study. The algorithm proposed by Baker-Jarvis was then used
to derive the permittivity and permeability values from this data
[8]. However, this requires an initial guess parameter to remove
phase ambiguities. We were able to eliminate this ambiguity by
using phase unwrapping technique [10]. Additionally, the cutoff
frequency for each frequency band was calculated and set as the
where is the reflection coefficient, is the transmission coefand
are propagation constant for the
ficient,
mode with and without the material inserted in the waveguide,
is the longer dimension of the rectangular waveguide,
is the phase of transmission coefficient, and is the material
thickness.
IV. RESULTS
The accuracy of waveguide based measurements has been
validated by measuring the electromagnetic properties of
a known material, namely, yttrium iron garnet (YIG) [10].
According to the Pacific Microwave Ceramics Brochure, the
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 2, FEBRUARY 2011
Fig. 3. The measurement result for S-parameters for the barium ferrite
nanopowder sample.
Fig. 1. For a sample thickness of 0.5 mm, the permittivity and permeability of
YIG has been measured in X band [10].
TABLE II
ELECTROMAGNETIC PROPERTIES MEASURED AT DIFFERENT PARTICLE SIZE
Fig. 2. At a density of 0.4004 g/cm , the average value of real and imaginary
parts of permeability for barium ferrite is 1.1046 and 0.0706 while for permittivity it is 1.4647 and 0.0516, respectively.
permittivity of YIG is around
. This value is in
close agreement with our results, shown in Fig. 1.
The same technique has now been applied to nano-powders
and the values of permittivity and permeability thus obtained for
barium ferrite nano-powder are shown in Fig. 2.
It can be observed that the value of real part of permeability
remains stable throughout the frequency range except for
a slight dip observed through K band. The real part of permitis close to 1.46 0.02 throughout the range, which is
tivity
a very small variation. It is also observed here that the imaginary
and permeability
are very close
parts of permittivity
to zero. Hence, the loss tangent for this sample is expected to be
very small. For example, at 20 GHz, the dielectric loss tangent
is 0.0095 whereas the magnetic loss tangent is 0.01. At this freis measured to be 11.61 dB. It can be
quency, the value of
inferred that reflection occurs from the air-sample interface at
this frequency. Thus, it behaves as a weak absorber at this frequency. But as we move towards higher frequencies, the value
improves. At 40 GHz, it is measured to be 31.12 dB.
of
This is expected since barium ferrite has been shown to exhibit
strong absorption between 47 and 60 GHz [11]. The measured
and
are shown in Fig. 3.
values of
To analyze the effect of particle size on measured electromagnetic property, another set of data was collected for the
micron-sized powder of barium ferrite. The grain size of this
sample varied from 50–100 m, with a density of 1.3 g/cm .
The data obtained for the two samples is compared in Table II.
The difference in the observed values of permittivity and
permeability can also be attributed to the difference in sample
density. There is not much difference in the permeability values,
but permittivity shows significant dependence on density of the
sample. The density dependence of these parameters has been
established at higher frequencies as well [11]. Higher value
of permittivity is observed if the particles are more densely
packed. It is therefore very important to keep the density
constant throughout the measurements.
The calculated permittivity and permeability for strontium
ferrite nano-powder are shown in Fig. 4. The real part of permittivity falls from 1.49 at 4 GHz to 1.24 at 40 GHz, whereas the
real part of permeability remains close to 1. An analysis of the
-parameters of strontium ferrite powder shows that the absorption properties improve at higher frequency. Strontium ferrite is
known to show strong absorption between 50 and 63 GHz [11].
As in the case of barium ferrite powder, high density strontium ferrite has a higher real part of permittivity. At a density of
1.3 g/cm , the real permittivity varies very little over the entire
frequency range and has an average value of 2.633. The real part
of permeability varies negligibly over the frequency bands with
an average value of 1.063. Based on the study of these two materials, it can be said that real part of permittivity increases with an
SHARMA AND AFSAR: PERMEABILITY AND PERMITTIVITY MEASUREMENTS OF COMMERCIALLY AVAILABLE NANO-FERRITES
Fig. 4. At a density of 0.3531 g/cm , the average value of real and imaginary
parts of permeability for strontium ferrite is 1.0941 and 0.0730 while for permittivity it is 1.4061 and 0.04465, respectively.
311
Fig. 7. At a density of 0.2754 g/cm , the average value of real and imaginary
parts of permeability for nickel zinc iron oxide is 1.0797 and 0.2012 while for
permittivity it is 1.3132 and 0.1232, respectively.
V. ERROR ANALYSIS
Fig. 5. At a density of 0.5204 g/cm , the average value of real and imaginary
parts of permeability for copper iron oxide is 1.029 and 0.1061 while for permittivity it is 1.5073 and 0.0906, respectively.
Fig. 6. At a density of 0.5454 g/cm , the average value of real and imaginary
parts of permeability for copper zinc ferrite is 1.0376 and 0.2092 while for permittivity it is 1.5951 and 0.0888, respectively.
increase in the density of the powder. A comparison of the properties of these M-type nano-sized hexaferrites to micron sized
planar hexaferrite shows that the planar structures have higher
dielectric loss [12].
Similar curves obtained for the constitutive properties of the
rest of the nano-powders are shown in the following figures.
The spinel nano-ferrites are relatively weak absorbers in this
for
is only 21.27 dB
frequency range. The
at 40 GHz. A study of these materials in the higher frequency
range would give a better picture of their absorption properties.
A comparison of the results obtained for the two types of
nano-powders shows that the spine-type ferrites have limited use
in this high frequency range. The ferrites with hexagonal structure show natural resonance in the GHz range and are therefore
more effective as high-frequency absorbers.
For the completeness of experimental data obtained from the
vector network analyzer, the possible sources of errors and uncertainties in measurement are identified in this section. Experimental uncertainties that can be revealed by repeated measurements are called random errors [13]. These were minimized by
repeating the measurement six times for each sample. The standard deviation in the values of real part of permittivity and permeability was observed to be 0.0058 and 0.0042, respectively. To keep the density fixed across all frequency bands, the
scale was used to weigh the mass of the sample. The uncertainty
in reading the scale correctly is 0.0002 g, which can be neglected. There are several factors that account for the random
errors in the measurements. The connectors used between the
network analyzer and waveguide make the system sensitive to
noise and drift. These errors can be minimized by keeping the
whole set-up clean and stable. Since the sample is in powder
form, the particles in the sample holder get displaced easily even
with a slight movement and the density becomes non-uniform.
It is important to make sure that the same uniform density is
maintained when loading the sample as well as at the time of
measurement.
All experimental uncertainties cannot be assessed by repeated
measurements. The repeatable and stable uncertainties in measurement are called systematic errors [13]. In this experiment,
the systematic errors caused by the imperfections of the system
were reduced by careful calibration of the whole measurement
system before the actual measurements were taken. TRL (thrureflect-load) calibration procedure was used here. It removes the
errors caused by the difference in the location of reference plane
and measurements plane arising due to cables and connectors.
In all measurements, we were able to maintain return loss less
than 50 dB. This allows us to neglect any reflections occurring
from the walls of the waveguide.
VI. CONCLUSION AND FUTURE WORK
The dielectric permittivity and magnetic permeability of several nano-sized ferrite powders were successfully determined
by using the waveguide technique across a wide range of frequency. Measurements show that the real part of permittivity
of ferrites depends on the size of the particles and the density
312
IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 2, FEBRUARY 2011
inside the sample holder. Future work may involve the determination of permittivity and permeability for the same powders
packed at different densities. The effect of an external magnetic
field should also be studied. The potential of these powders as
microwave absorbers can be better analyzed by measuring the
reflection spectra of a metal backed sample of the powders.
ACKNOWLEDGMENT
The authors would like to thank Dr. M. Obol for sharing technical knowledge on the waveguide technique and J. Qui for assistance in the use of vector network analyzer.
REFERENCES
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permeability of barium and strontium ferrite powders in X, Ku and
K-band frequency ranges,” J. Appl. Phys., vol. 97, no. 10, May 2005.
[3] J. M. Greneche, “Magnetic properties of nano-ferrites,” in Conf. Int.
Espectroscopia, France, 2009.
[4] K. Vinoy and R. Jha, Radar Absorbing Material. Norwell, MA:
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[6] A. M. Nicolson and G. F. Rossc, “Measurement of intrinsic properties of materials by time techniques,” IEEE Trans. Instrum. Meas., vol.
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[7] W. B. Weir, “Automatic measurement of complex dielectric constant
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33–36, Jan. 1974.
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pp. 1096–1103, Aug. 1990.
[9] Y. Wang, M. N. Afsar, and R. Grignon, “Complex permittivity and
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IEEE Antennas Propag. Soc. Int. Symp., 2003, vol. 4, pp. 619–622.
[10] N. Al-Moayed, M. N. Afsar, U. A. Khan, S. McCooey, and M. Obol,
“Nano ferrites microwave complex permeability and permittivity measurements by T/R technique in waveguide,” IEEE Trans. Magn., vol.
44, no. 7, pp. 1768–1772, Jul. 2008.
[11] K. A. Korolev, L. Subramanium, and M. N. Afsar, “Complex permittivity and permeability measurements of ferrite powders at millimeter
waves,” in Eur. Microwave Conf., Oct. 2005.
[12] M. N. Afsar, A. Sharma, and M. Obol, “Microwave permittivity and
permeability properties and microwave reflections of micro/nano ferrite powders,” in IMTC’2009.
[13] J. R. Taylor, An Introduction to Error Analysis. South Orange, NJ:
University Science Books, 1982, pp. 81–93.
Microwave complex permeability and permittivity of nanoferrites
Anjali Sharma and Mohammed N. Afsar
Citation: Journal of Applied Physics 109, 07A503 (2011); doi: 10.1063/1.3536659
View online: http://dx.doi.org/10.1063/1.3536659
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/109/7?ver=pdfcov
Published by the AIP Publishing
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JOURNAL OF APPLIED PHYSICS 109, 07A503 (2011)
Microwave complex permeability and permittivity of nanoferrites
Anjali Sharmaa) and Mohammed N. Afsar
High Frequency Materials Measurement and Information Center, Department of Electrical Engineering,
Tufts University, Medford, Massachusetts 02155, USA
(Presented 18 November 2010; received 1 October 2010; accepted 29 October 2010; published
online 18 March 2011)
Complex permittivity and permeability of nanosized magnetic powders are studied for eight different
bands to cover the frequency range from 2–40 GHz. A precise transmission-reflection based
waveguide technique has been used to carry out measurements across the frequency bands. The
algorithm for transmission and reflection calculations has been modified to account for variable
sample thickness in the in-waveguide measurement setup. Diluted nanopowder sample has also been
studied. The ferromagnetic resonance in nanoferrites has been observed to be sharper compared to
microsized ferrites. Detailed analysis of measurement uncertainties and instrumentation errors has
C 2011 American Institute of Physics. [doi:10.1063/1.3536659]
been performed. V
I. INTRODUCTION
Iron based magnetic powders have been studied for several years now.1 The use of ferrites in waveguides for isolator applications is a well-established concept in the field of
microwave technology. In this paper, we are presenting the
study of nanosized magnetic powders at microwave frequency range. This frequency range is important because of
lower loss and cheaper fabrication of microwave devices as
compared to millimeter waves.2
Nanopowders consist of particles with an average size
between 1–100 nm. Iron based nanopowders are spherical
metal nanostructure particles, typically 20–40 nm in size.
Measurements carried out on nanosized powders have shown
that the microwave permittivity and permeability of these
materials differ from those of the material in solid state.3
These powders are useful in biomedical engineering as they
serve as good drug deliver solutions and DNA separators.
They also find extensive application in magnetic recording
and information storage.3 In our investigation, we have discovered that the ferromagnetic resonance for nanoferrite
powders becomes sharper compared to micro-sized ferrite
powders at microwave and millimeter wave frequencies.
Therefore, an important application of these nanoferrite powders is in the design of isolator, modulator, and phase rotator.
These ferrite components can be made better tunable by
using nanoferrite powders. To identify the use of nanoferrites
in common ferrite applications, for example, absorption of
electromagnetic energy, it is important to determine their
constitutive properties. The absorption capability of a material depends on the intrinsic material properties, namely, permittivity and permeability, which may be frequency
dependent.4 Therefore there is a growing interest in studying
the properties of these powders.
A transmission-reflection based waveguide technique
that employs a vector network analyzer was used to measure
a)
Author to whom correspondence should be addressed. Electronic mail:
Anjali.Sharma@tufts.edu.
0021-8979/2011/109(7)/07A503/3/$30.00
the scattering parameters for nanoferrite samples. Using this
technique, one can record both reflection and transmission
spectra generated when an electromagnetic wave travels
through the material under study. Based on the reflection
spectrum, the frequency at which the sample exhibits good
absorption can be determined. The algorithm used to
determine the permittivity and permeability from the measured data has been modified over the years to improve accuracy.5–7 However, the algorithm used in this study takes into
account the effect of the thickness of the loaded material and
its permittivity and permeability on the cutoff wavelength
inside the waveguide. A thorough error analysis is also presented to identify the systematic and random errors in the
measurement process.
II. SAMPLE DESCRIPTION
The three nanopowders studied in this work were purchased
from Sigma Aldrich, Milwaukee, WI. The powdered samples
have a grain size that varies between 40 and 100 nm for different
samples. The sample properties are summarized in Table I.
The frequency range from 2 to 40 GHz is divided into
eight frequency bands, namely, R (1.70–2.60 GHz), S (2.60–
3.95 GHz), G (3.95–5.85 GHz), C (5.85–8.20 GHz), X
(8.20–12.40 GHz), Ku (12.4–18.00 GHz), K (18.00–26.5
GHz), and Ka (26.50–40.00 GHz). Each band corresponds to
a different waveguide dimension and therefore the volume of
sample used was different in each band. To maintain constant density for each nanopowder, appropriate weight of the
sample to be filled in the waveguide shim was calculated.
The weight of the sample holder is measured with and without the powder to ensure that the required density is maintained. To load uniform distribution of the powder in the
waveguide and avoid the presence of any air gap in the sample holder, the sample was steadily filled in the shim. Lossless transparent tape was used on either sides of the shim to
hold the powder inside the shim. It has been shown that
the presence of such a tape does not affect the measured
s-parameter.8
109, 07A503-1
C 2011 American Institute of Physics
V
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07A503-2
Sharma and Afsar
J. Appl. Phys. 109, 07A503 (2011)
TABLE I. Sample description.
Name
Barium ferrite
Strontium ferrite
Iron (II, III) oxide
Chemical Formula
Density (g/cm3)
BaFe12O19
SrFe12O19
Fe3O4
0.4004
0.3531
1.291
III. EXPERIMENTAL SETUP
The transmission-reflection based waveguide technique
has been used for the measurement of soft powders. The
method is based on the transmission and reflection (T/R)
technique introduced by Nicolson-Ross5 and Weir, 6 and the
widely acclaimed formulations for transmission and reflection coefficients proposed by Baker-Jarvis.7 We were able to
reduce the phase ambiguities arising due to initial guess parameter. The details on the accuracy of the method can be
found in Ref. 9. Additionally, the cutoff frequency for each
frequency band was calculated and set as the waveguide
delay in the vector network analyzer to further remove
errors. Ideally, the shim thickness should be within quarter
wavelengths of the central operation frequency. But this is
hard to maintain since the thickness used for each sample
depends on the requirement to maintain constant density for
all frequency bands. The difference between the material
thickness and sample holder thickness was taken in to
account when performing the calculations. The modified
equations used in the measurements are given below
pffiffiffiffiffiffiffiffiffi
2
2
S~11 ¼ S11 ejð0 k0 kc Þ
pffiffiffiffiffiffiffiffiffi
2
2
S~21 ¼ S21 ejððldÞ k0 kc Þ ;
(1)
where l is the quarter wavelength difference between thru and
line in air, d is the thickness of the sample inside the waveguide, k0 is the wave number of the sample, and kc is the cutoff wave number. These equations take into account the effect
of using samples with thickness (d) values that are smaller
than the waveguide shim used in the experimental setup.
Each frequency band was subdivided into 201 frequency
points at which measurement was taken, giving a total of
1206 frequency points. It should be noted that the data at the
end of one band may show some deviation from the data at
the starting frequencies of the successive band. This may be
due to experimental errors, depending on how carefully the
operator used the instruments. To remove the experimental
discrepancies in the data across different bands, a mathematical averaging method (least squares method of polynomials)
was used. The derived data is very reliable and not affected
by the scattering voltage ratios of the network analyzer. The
final form of the equations used to determine permittivity
and permeability is shown below
2 c
1C
1
1
lnð Þ þ jð2pn uT Þ
e¼j
f
1 þ C 2pd
jT j
0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
2 2
1
1 A
;
(2)
@
k0
2a
l¼
gcTE10
jc0TE10
0
1
1
lnð T jÞ þ jð2pn uT ÞC
1þC
1 B
B jr
C
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ j
@
2 ffi A;
1 C 2pd
1
1 2
k0 2a
(3)
where C is the reflection coefficient, T is the transmission
coefficient, c0TE10 and cTE10 are propagation constant for the
TE10 mode with and without the material inserted in the
waveguide, a is the longer dimension of the rectangular
waveguide, /T is the phase of transmission coefficient and d
is the material thickness.
IV. RESULTS
To verify the accuracy of the measurement technique
we first measured a known sample, Teflon, in X-band. The
permittivity of Teflon was measured to be 2.1 throughout the
band with an average variation of 60.001. The results therefore agree with the known published value for permittivity
of Teflon.
Figure 1 shows the measured permeability and permittivity values for iron (II, III) oxide. Permeability of iron (II,
III) oxide nanopowder shows very slight variation with frequency whereas permittivity values show dependence on
frequency.
The permeability and permittivity results for rest of the
samples described in Sec. II are summarized in Table II.
It has been shown that barium ferrite and strontium ferrite exhibit strong absorption at frequencies between 47–60
GHz and 50–63 GHz, respectively.1 In the microwave frequency range studied here, it was observed that for these
samples the s-parameters, namely, S11 and S21, improve at
higher frequency. At 40 GHz, S11 for barium ferrite nanopowder was measured to be 31.12 dB. But at lower frequencies, these parameters show poor values, Fig. 2.
In order to see if we can improve the low frequency
behavior of barium ferrite nanopowder, we prepared a
diluted sample of barium ferrite. This sample was made by
adding epoxy solution to barium ferrite nanopowder. The
density of this sample was measured to be 1.31 g/cm3. The
reflection spectrum of this sample was studied in X-band and
FIG. 1. (Color online) The average value for the real and imaginary parts of
the measured permeability for iron (II, III) oxide are 0.9755 and 0.2266
while for permittivity it is 5.25 and 1.25, respectively.
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07A503-3
Sharma and Afsar
J. Appl. Phys. 109, 07A503 (2011)
TABLE II. Measured value of real and imaginary permeability and
permittivity.
Barium Ferrite
Frequency (GHz)
4
8
12
16
20
24
28
32
36
40
Strontium Ferrite
m0
e0
m0
1.11
1.14
1.14
1.11
1.06
1.04
1.08
1.10
1.14
1.12
1.43
1.50
1.50
1.45
1.44
1.49
1.48
1.46
1.44
1.42
1.10
1.09
1.09
1.09
1.08
1.05
1.12
1.12
1.12
1.09
e0
1.49
1.49
1.5
1.5
1.44
1.397
1.369
1.35
1.287
1.22
FIG. 3. The measurement result for s-parameters for the diluted barium ferrite nanopowder sample in X-band.
significant improvement was observed in the S11 and S21 parameter values. The results are shown in Fig. 3. Lower values of S11 are observed for the diluted sample. This is
because epoxy improves the absorption rate of the sample,
indicated by the lower S11 values.
the whole measurement system before the actual measurements
were taken. TRL (thru-reflect-load) calibration procedure was
used here. In all measurements, we were able to maintain return
loss less than 45 dB. This allows us to neglect any reflections
occurring from the walls of the waveguide.9
V. UNCERTAINTY AND ERROR ANALYSIS
VI. CONCLUSION
To completely define the accuracy of the experimental data
obtained from the vector network analyzer, the possible sources
of errors and uncertainties in measurement are identified. Any
experiment consists of two types of errors, namely, random error
and systematic error. Experimental uncertainties that can be
revealed by repeated measurements are called random errors.10
The main sources of random errors in this experiment were the
connectors used between the network analyzer and waveguide,
and the nonuniform nanopowder densities. Since the sample is
in powder form, the particles in the sample holder get displaced
easily and the density becomes nonuniform. It is important to
make sure that the same uniform density is maintained when
loading the sample as well as at the time of measurement. These
errors were accounted for in this study by repeating the measurement five times for each sample. The standard deviation in
the values of real part of permittivity and permeability was
observed to be 60.005 and 60.003, respectively.
The experimental uncertainties that cannot be assessed
by repeated measurements are called systematic errors.10 In
this experiment, the systematic errors caused by the imperfections of the system were reduced by careful calibration of
The magnetic permeability and dielectric permittivity of
nanosized ferrite powders were successfully determined by
using the transmission/reflection based waveguide technique
across a wide range of frequency. The values of S11 and S21
were studied to determine the potential of these materials as
absorbers. Barium and Strontium ferrites can be used for
making stronger magnets. The ferromagnetic resonance for
both barium and strontium nanoferrites appear between 40–
60 GHz, whereas iron oxide is better for microwave frequencies. Our finding reveals that the ferromagnetic resonance
for nanoferrite powders is very sharp compared to microsized ferrites. This will allow the use of nanoferrite powders
in isolator, modulator, phase rotator and circulator applications for achieving sharper tunability. It has been shown in
this study that the absorption properties of these samples can
be improved by diluting the sample with epoxy. Further
experimentation with different densities of the diluted sample can be carried out to find the most suitable concentrations
of the nanopowder and epoxy in the diluted sample.
FIG. 2. The measurement result for s-parameters for the barium ferrite
nanopowder sample in X-band.
1
K. A. Korolev, L. Subramanium, and M. N. Afsar, “Complex permittivity
and permeability measurements of ferrite powders at millimeter waves,”
in Proceedings of 35th European Microwave Conference, Paris, France,
3–5 October 2005.
2
C. Vittoria, J. Magn. Magn. Mater. 71, 109 (1980).
3
A. Bahadoor, Y. Wang, and M. N. Afsar, J. Appl. Phys. 97, 28 (2005).
4
K. Vinoy and R. Jha, Radar Absorbing Material (Kluwer Academic,
Boston, 1996).
5
A. M. Nicolson and G. F. Ross, IEEE Trans. Instrum. Measur. IM-19, 377
(1970).
6
W. B. Weir, “Proc. IEEE 62, 33 (1974).
7
J. Baker-Jarvis, E. J. Venzura, and W. A. Kissick, IEEE Trans. Microwave
Theory Tech. 38, 1096 (1990).
8
Y. Wang, M. N. Afsar, and R. Grignon, IEEE Trans. Antennas Prop. 4,
619 (2003).
9
N. Al-Moayed, M. N. Afsar, U. A. Khan, S. McCooey, M. Obol, IEEE
Trans. Magn. 44, 1768 (2008).
10
John R. Taylor, An Introduction to Error Analysis (University Science
Books, Sausalito, CA, 1982), pp. 81–93.
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 48, NO. 11, NOVEMBER 2012
2773
Microwave and Millimeter Wave Ferromagnetic Absorption of Nanoferrites
Liu Chao, Anjali Sharma, and Mohammed N. Afsar, Fellow, IEEE
Department of Electrical and Computer Engineering, Tufts University, Medford, MA 02155 USA
Complex dielectric permittivity and magnetic permeability of several commercially available nanoferrites have been studied over
a broad microwave and millimeter wave frequency range. Nano-sized barium, strontium, copper, zinc, nickel substituted iron oxide
powders with different lattice structures are investigated. A transmission-reflection based in-waveguide technique that employs a vector
network analyzer was used to determine the scattering parameters for each sample in two microwave bands (18–40 GHz). A free space
quasi-optical spectrometer energized by backward wave oscillators was used to acquire the transmittance spectra in the millimeter wave
frequency range (30–120 GHz). Relatively broad and sharp ferromagnetic resonance of hexagonal barium ferrite and strontium ferrite
are observed in millimeter wave frequency range. The ferromagnetic resonance peak for nano-sized hexagonal ferrite powder material
moves to lower frequencies compared to micro-sized and solid hexagonal ferrites. An X-ray diffraction measurement is performed on
these hexagonal ferrites to understand the magnetic behavior and the structure.
Index Terms—Dielectric permittivity, ferromagnetic resonance, magnetic domain, magnetic permeability, nanoferrite.
I. INTRODUCTION
N
ANOFERRITES consist of metal substituted iron oxide
nanopowders that have average particle size below 100
nm. These nanosize ferrites show different dielectric and magnetic properties in microwave and millimeter wave frequencies
compared to micro-size and solid ferrites. Therefore there is a
growing interest in studying these powders to understand how
the material properties are affected by the particle size.
The use of nanoferrites is significant in many aspects, such
as biological drug delivery systems, DNA-separators as well as
in magnetic recording and information storage [1]. These nanosize ferrites are also useful in other microwave applications such
as transformers, absorbers and circulators. It is necessary to investigate their electromagnetic properties and understand their
changing magnetic behavior with particle dimension.
In the microwave measurement, a vector network analyzer
together with waveguides was employed to determine the scattering parameters of the nanoferrites inside the waveguide in K
band and Ka band. From the S-parameters, complex permittivity
and permeability are evaluated by an improved algorithm. The
millimeter wave measurement is based on a free space quasioptical spectrometer powered by backward wave oscillators.
Initially precise transmittance spectra over a broad millimeter
wave frequency range from 40 GHz to 120 GHz are acquired.
Later the transmittance spectra are converted into complex permittivity and permeability spectra. These ferrite powder materials are further characterized by x-ray diffraction (XRD) to understand the crystalline structure relating to the strength and the
shift of the ferromagnetic resonance affected by the particle size.
vector network analyzer measures the scattering parameters of
the 2-port network formed by the waveguide shim filled with
the sample under study as shown in Fig. 1.
The nanopowders were filled in the sample holder that was
placed between the waveguides. It is important to ensure that the
sample must fill the entire area of the sample holder so that there
are no air gaps at the corners of the shim or between the powders.
The sample was packed tight enough such that changing the
orientation of the shim does not cause any shift in the particles.
The algorithm proposed by Baker-Jarvis was then used to
derive the permittivity and permeability values from this data
[2]. The phase unwrapping technique was employed to avoid
the use of initial guess parameter [3]. Additionally, the cut-off
frequency for each frequency band was calculated and set as
the waveguide delay in the vector network analyzer to remove
errors. The modified equations used in the measurements are
given below,
where is the quarter wavelength difference between thru and
line (in air), is the thickness of the sample inside the waveguide,
is the wavenumber of the sample and is the cutoff
wavenumber.
The final form of the equations used to determine permittivity
and permeability is shown below
II. MEASUREMENT TECHNIQUE
A. Transmission-Reflection Waveguide
The transmission-reflection based waveguide technique has
been widely used to determine the properties of solids. It can
be further modified for the measurement of soft powders. The
Manuscript received March 02, 2012; revised April 27, 2012; accepted May
15, 2012. Date of current version October 19, 2012. Corresponding author:
L. Chao (e-mail: stream.chao@gmail.com).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TMAG.2012.2200666
where, is the reflection coefficient, T is the transmission coefand
are propagation constant for the
ficient,
mode with and without the material inserted in the waveguide,
0018-9464/$31.00 © 2012 IEEE
2774
IEEE TRANSACTIONS ON MAGNETICS, VOL. 48, NO. 11, NOVEMBER 2012
Fig. 1. Schematic diagram of nanoferrites in waveguide.
Fig. 2. Schematic diagram of the free-space quasi-optical millimeter-wave
spectrometer in the transmittance mode with BWO as radiation source.
is the longer dimension of the rectangular waveguide,
is the phase of transmission coefficient and is the material
thickness.
Fig. 3. Complex permittivity and permeability in K and Ka bands of
nanopowder. Real permeability is around 1.05 and the real
permittivity is around 1.3. The imaginary part of permeability and permittivity
are 0 and 0.05, respectively.
TABLE I
SAMPLE PREPARATION
B. Quasi-Optical Spectrometer
Free space millimeter wave quasi-optical spectroscopy technique, including technical details and measurement uncertainties analysis, has been successfully employed and presented by
several researchers [4]–[7]. This study presents complex dielectric and magnetic measurements at millimeter waves performed
by the free space quasi-optical spectrometer in transmittance
mode [6], [7]. Three high vacuum, high power backward wave
oscillators (also called carcinotrons) (BWO) have been used as
sources of coherent radiation continuously tunable in the range
from 30 to 120 GHz. A couple of pyramidal horn antennas and
a set of polyethylene lenses along the propagation path from
the source antenna to the receiver antenna have been adjusted
to form a Gaussian beam as well as to focus the beam into the
sample. The diameter of the millimeter wave beam focused into
the sample has been found to be around a few millimeters. The
simplified schematic diagrams of the millimeter wave quasi-optical spectroscopic system are shown in Fig. 2.
The mathematical relationships between transmittance and
reflectance spectra, and refractive and absorption indexes are
presented below
where is the speed of light, is the refractive index of the
sample material, is the absorption index, is the complex
magnetic permeability of the sample material, is the complex dielectric permittivity, is the transmittance, is the reflectance, is the phase of the transmitted wave, and is the
phase of reflected wave.
III. SAMPLE PREPARATION
Five nanoferrites, namely barium ferrite (
,
, SrM), copper iron oxide
BaM), strontium ferrite (
, CAS-No.12018-79-0), copper zinc iron oxide
(
, CAS-No.66402-68-4), nickel zinc iron oxide
(
, CAS-No.12645-50-0) examined in this work
(
were purchased from Sigma Aldrich. The grain size of the powders varies from 40 nm to 100 nm for different kind of ferrites
[8]. Density of the ferrites in the sample holder is determined
by weighting from an accurate balance over the dimensions of
the sample holders. The density and purity are listed in Table I.
IV. RESULTS AND DISCUSSION
Complex dielectric permittivity and magnetic permeability
spectra obtained in microwave frequency range using transmission-reflection in-waveguide technique are shown below.
Transmittance spectra of hexagonal ferrites are acquired in
the millimeter wave frequency from 40 GHz to 120 GHz. The
angles from 5 to
X-ray diffraction data are collected from
70 in 0.015 step size on these hexagonal ferrite nanopowders.
The complex dielectric permittivity and magnetic permeability of nickel zinc iron oxide, copper zinc iron oxide and
copper iron oxide nanopowder in 18–40 GHz frequency range
are shown in Fig. 3, Fig. 4, Fig. 5, respectively. The real parts
of these copper, zinc, nickel substituted
of permittivity
nanopowders are much smaller than their solid form. One can
treat these nanopowders as diluted by the air between nano
particles. The real part of magnetic permeability
value is
almost the same as the value for micro size and solid form. The
values are close to zero for
imaginary parts of permittivity
specifying to have high electric resistivity. No ferromagnetic
resonance is observed in this frequency region for these samples and the imaginary parts of permeability
values are
very small (close to 0.01 to 0.05).
CHAO et al.: MICROWAVE AND MILLIMETER WAVE FERROMAGNETIC ABSORPTION OF NANOFERRITES
Fig. 4. Complex permittivity and permeability in K and Ka bands of
nanopowder. Real permeability is around 1.1 and the real
permittivity is around 1.5. The imaginary part of permeability and permittivity
are 0 and 0.05, respectively.
2775
Fig. 6. Millimeter wave transmittance spectra of barium and strontium
nanoferrites. Ferromagnetic resonance peaks are observed at 42.5 GHz and
48.2 GHz, respectively.
TABLE III
COMPLEX PERMITTIVITY AND RESONANT FREQUENCY
TABLE IV
ANISOTROPY FIELD AND SATURATION MAGNETIZATION
Fig. 5. Complex permittivity and permeability in K and Ka bands of
nanopowder. Real permeability is observed in 1.0–1.2 region. The real permittivity varies around 1.5. The imaginary part of permeability and permittivity are
0.1.
To evaluate the complex magnetic permeability spectra,
Schlömann’s equation [9] for partially magnetized ferrites has
been used:
TABLE II
SAMPLE AVERAGE PERMITTIVITY AND PERMEABILITY
Table II shows the average permittivity and permeability of
,
,
nanopowders in K and Ka
bands from 18 GHz to 40 GHz.
Transmittance spectra of hexagonal barium (BaM) and
strontium (SrM) nanoferrites measured by the quasi-optical
technique are shown in Fig. 6. A deep and sharp absorption
in transmittance spectra has been observed for both barium
and strontium nanoferrites in 40–60 GHz frequency range.
This deep absorption is the natural ferromagnetic resonance
that shifts to millimeter wave range due to the strong magnetic
anisotropy of barium and strontium ferrites. The periodic structure observed in all transmittance spectra at the frequencies
above zone of deep absorption represents channel fringes. The
analysis of channel fringes allows us to determine the complex
dielectric permittivity value of materials.
where
is the frequency,
is anisotropy field,
is
saturation magnetization, is the gyromagnetic ratio. Demagnetizing factors are determined by the theory of Schlömann’s
model for nonellipsoidal bodies. The complex permittivity and
permeability together with the center of the ferromagnetic resonance are shown in Table III.
From the ferromagnetic resonance, the hexagonal barium and
strontium nanoferrites show relatively strong anisotropy field of
and
and weak saturation magand
, respecnetization of
tively. However, these anisotropy fields and saturation magnetization are smaller comparing to the solid barium and strontium
and
ferrites which have anisotropy field of
, saturation magnetization of
and
, respectively. Comparison of anisotropy
field and saturation magnetization between nano-sized and solid
hexagonal ferrite is summarized in Table IV.
To understand the weak saturation magnetization is straightforward because the nanoferrites are actually diluted by the air
between each particle even though the layer was compressed.
The reduced anisotropy field is interesting for it is the intrinsic
characteristic affected by the crystal structure. But the physical
2776
IEEE TRANSACTIONS ON MAGNETICS, VOL. 48, NO. 11, NOVEMBER 2012
resonance completely at room temperature. The size of barium
and strontium nanoferrite powders measured in this paper is
right between the upper limit of single domain size and the
lower limit size of turning into superparamagnetism.
V. CONCLUSION
Fig. 7. XRD spectra for
nanoferrite and micro-ferrite.
Fig. 8. XRD spectra for
nanoferrite and microferrite.
change of the powder size does affect the anisotropy field of
these hexagonal ferrites. The X-ray diffraction was then performed on this nanoferrites and the diffraction pattern is compared to micro size barium and strontium ferrites in Fig. 7 and
Fig. 8.
The x-ray diffraction spectra show that both barium and
strontium keep the same crystalline structure in micropowder
and nanopowder particle size. This further demonstrates that the
shifting of ferromagnetic resonance (towards lower frequency)
and reduced anisotropy field are not caused by any crystal
structure change. The micro size particle of the hexagonal
ferrite has almost the same anisotropy field as the solid ferrite.
This is due to the domain size of the hexagonal ferrite. The
upper limit of single magnetic domain should have the size of
about 100 nanometer.
The nanoferrite powder with a physical dimension smaller
than this single magnetic domain size will lead to a lower
ferromagnetic resonance frequency. At the upper limit of single
domain size, all of the particle’s internal magnetization is
aligned to reduce the system energy to the lowest [10]. Therefore, at upper limit of single domain size, ferrite has the largest
anisotropy field which is the sum of all magnetic moment in the
particle. Below this physical upper limit of single domain size,
the anisotropy field of the ferrite is determined by the volume
of the particle until the dimension drops to a certain size. The
spins of the magnetic moment will no longer be aligned without
the application of an external magnetic field because of random
thermal flips. As the powder dimension turns to even smaller
size, the hexagonal ferrite is deduced to lose ferromagnetic
The complex dielectric permittivity and magnetic permeability are measured in microwave and millimeter wave
frequency range. The improved vector network analyzer based
transmission-reflection technique with waveguide is applied
to several nanoferrites in microwave frequency. Quasi-optical backward wave oscillator spectrometer was employed
to determine the transmittance of these nanoferrite samples.
Ferromagnetic resonances on hexagonal barium and strontium
nanoferrites are observed in the millimeter wave frequency
range by this millimeter wave spectrometer. The ferromagnetic
resonance is shifting to the lower frequency and the anisotropy
field reduces to lower strength. X-ray diffraction was performed on these hexagonal nanoferrites. The diffraction spectra
demonstrate that the crystal structure keeps the same as larger
size barium and strontium ferrites. The ferromagnetic resonance moving is caused by magnetic domain size limit. Further
research on the detail reasons leading to this phenomenon will
be carried through.
REFERENCES
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[6] K. N. Kocharyan, M. Afsar, and I. I. Tkachov, “Millimeter-wave magnetooptics: New method for characterization of ferrites the millimeterwave range,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp.
2636–2643, Nov. 1999.
[7] K. A. Korolev, C. Shu, L. Zijing, and M. N. Afsar, “Millimeter-wave
transmittance and reflectance measurement on pure and diluted
carbonyl iron,” IEEE Trans. Instrum. Meas., vol. 59, no. 11, pp.
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[8] A. Sharma and M. N. Afsar, “Microwave complex permeability and
permittivity measurements of commercially available nano-ferrites,”
IEEE Trans. Magn., vol. 47, pp. 308–312, 2011.
[9] E. Schlomann, “Microwave behavior of partially magnetized ferrites,”
J. Appl. Phys., vol. 41, pp. 1350–1350, 1970.
[10] E. C. Stoner and E. P. Wohlfarth, “A mechanism of magnetic hysteresis
heterogeneous alloys,” Phil. Trans. Roy. Soc. Lond.. A, Math. Phys.
Sci., vol. 240, pp. 599–642, 1948.
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