1558 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 6, JUNE 2004
Magneto-Dielectrics in Electromagnetics:
Concept and Applications
Hossein Mosallaei, Student Member, IEEE, and Kamal Sarabandi, Fellow, IEEE
Abstract—In this paper, the unique features of periodic mag-
neto-dielectric meta-materials in electromagnetics are addressed.
These materials, which are arranged in periodic configurations,
are applied for the design of novel EM structures with applications
in the VHF-UHF bands. The utility of theses materials are demon-
strated by considering two challenging problems, namely, design
of miniaturized electromagnetic band-gap (EBG) structures and
antennas in the VHF-UHF bands. A woodpile EBG made up
of magneto-dielectric material is proposed. It is shown that the
magneto-dielectric woodpile not only exhibits band-gap rejection
values much higher than the ordinary dielectric woodpile, but
also for the same physical dimensions it shows a rejection band
at a much lower frequency. The higher rejection is a result
of higher effective impedance contrasts between consecutive
layers of the magneto-dielectric woodpile structure. Composite
magneto-dielectrics are also shown to provide certain advantages
when used as substrates for planar antennas. These substrates are
used to miniaturize antennas while maintaining a relatively high
bandwidth and efficiency. An artificial anisotropic meta-substrate
having
, made up of layered magneto-dielectric and
dielectric materials is designed to maximize the bandwidth
of a miniaturized patch antenna. Analytical and numerical
approaches, based on the anisotropic effective medium theory
(AEMT) and the finite-difference time-domain (FDTD) technique,
are applied to carry out the analyzes and fully characterize the
performance of finite and infinite periodic magneto-dielectric
meta-materials integrated into the EBG and antenna designs.
Index Terms—Anisotropic effective medium theory (AEMT),
antenna miniaturization, electromagnetic band-gap (EBG)
materials, finite-difference time-domain (FDTD) technique,
magneto-dielectrics, meta-materials, periodic structures.
I. INTRODUCTION
T
HE USE OF mobile wireless communication for a wide
number of applications is on the rise. Depending on the
application (data rate, range, etc.), the frequency, bandwidth,
transmitted power, and modulation scheme of todays wireless
systems may vary widely. However, independent of the appli-
cation, three antenna features are always sought in a wireless
system. These include: 1) compactness; 2) power efficiency; and
3) affordability.
In recent years, considerable efforts have been devoted
toward miniaturization of RF electronics and development of
power efficient amplifiers. However, the current state of the art
in power efficient and miniaturized antennas, integrated filters,
and multiplexers leaves much to be desired. Most RF passive
Manuscript received July 11, 2003; revised August 18, 2003.
The authors are with the Department of Electrical Engineering and Computer
Science, University of Michigan, Ann Arbor, Ann Arbor, MI 48109-2122 USA
(e-mail: hosseinm@engin.umich.edu).
Digital Object Identifier 10.1109/TAP.2004.829413
components such as filters, lumped elements, and substrates
are made using high quality materials. Currently, high dielec-
tric, low loss materials are widely used in the fabrication of
miniaturized filters, diplexers, and antennas [1]–[4]. At VHF
through the lower UHF band where miniaturization of RF
components and antennas are highly desirable, the application
of magneto-dielectric composite meta-materials is proposed.
Engineering effective medium properties of a composite
material by proper arrangement of constituent dielectric and
magneto-dielectric materials provides more degrees of freedom
in achieving desired functionalities.
Most ferrite materials are highly lossy in the VHF range
and up. To achieve a low-loss ferrite with relatively high
permeability
capable of operating up to 500 MHz,
researchers at Trans-Tech Inc.
1
align Z-type hexaferrite ce-
ramic material that can be found in any desired shape. Basically
barium hexaferrite
is blended with cobalt oxide
and barium carbonate into a slurry which can be shaped using
a sacrificial mold and heat treatment.
In this paper, two challenging problems in RF engineering,
namely, design of EBG structures and miniaturized antennas are
addressed. Dielectric band-gap materials are a class of periodic
structures that can be used in the construction of high
filters
and isolators among circuit subsystems [1], [2], [5], [6]. Com-
pactness and improved band-gap rejection levels are two desired
features of band-gap materials. Utilizing the magneto-dielectric
material it is shown that not only can the size be reduced, but
also a much higher rejection level can be obtained when com-
pared to the dielectric-only EBG structures.
Another interesting application of magneto-dielectrics is in
the area of antenna miniaturization. Antenna miniaturization
using high permittivity materials as substrates has been at-
tempted in the past [4], [7]. Although miniaturization can be
achieved using high dielectric materials, there are two draw-
backs. One problem stems from the fact that the field remains
highly concentrated around the high permittivity region (field
confinement), which results in low antenna efficiency and
narrowband characteristics. The second drawback pertains to
the fact that the characteristic impedance in a high permittivity
medium is rather low which creates difficulties in impedance
matching of the antenna. These aforementioned problems can
be effectively circumvented if one uses a magneto-dielectric
material. Magneto-dielectric materials can also miniaturize the
antenna by the same factor however using moderate values of
permittivity and permeability
. Thus, the issue
of strong field confinement is minimized and the medium is
1
Trans-Tech, Inc., Adamstown, MD 21710 USA is a subsidiary of Skyworks
Solutions, Inc., Woburn, MA 01801.
0018-926X/04$20.00 © 2004 IEEE
MOSALLAEI AND SARABANDI: MAGNETO-DIELECTRICS IN ELECTROMAGNETICS: CONCEPT AND APPLICATIONS 1559
Fig. 1. 1-D periodic magneto-dielectric: (a) Plane wave excitation and its corresponding coordinate system, (b) N-layer periodic structure in
y
or
z
directions,
(c) unit cell of
i
th layer; it is assumed that this layer is periodic in
y
and infinite in
z
directions.
far less capacitive when compared to the dielectric-only high
permittivity material. Furthermore, since the characteristic
impedance of magneto-dielectric medium
is
close to that of the surrounding medium
it allows for ease
of impedance matching over a much wider bandwidth.
To obtain the interactions of electromagnetic waves with
complex finite and infinite periodic magneto-dielectric
meta-materials, analytical and numerical approaches based
on anisotropic effective medium theory (AEMT) and the
finite-difference time-domain (FDTD) technique are used. In
the AEMT method the periodic magneto-dielectric material is
represented approximately as an anisotropic effective medium
with permittivity and permeability tensors
and , and the
wave propagation characteristics are determined analytically.
The FDTD full wave analysis used here allows for arbitrary
placement of the FDTD boundary using a periodic boundary
condition (PBC) or a perfectly matched layer (PML) or a
combination of these depending on the problem at hand.
II. P
ROBLEM’S FORMULATION
In this section, the formulations necessary for characterizing
composite magneto-dielectric meta-materials are developed.
Depending on the complexity of the structure under study,
analytical or numerical methods will be used. The analytical
model has the advantage of providing insight into the nature of
EM waves interacting with the medium, whereas, the numerical
method will be used for complicated structures for which an
analytical solution does not exist.
A. AEMT
In construction of magneto-dielectric structures one may con-
sider one-dimensional (1-D) periodic layers made up of dielec-
tric and magnetic materials. The layers can be stacked on top
of each other with an arbitrary orientation direction. One can
use a quasistatic mixing model [8], [9] to obtain the effective
medium properties of periodic magneto-dielectrics. However,
this approach is only applicable when the period is much smaller
than a wavelength (quasistatic region). In what follows, a gen-
eral formulation for periodic magneto-dielectric media is pre-
sented that provides accurate results for period values as high
as
.
Fig. 2. Anisotropic effective material model for the 1-D periodic
magneto-dielectric: (a) A unit cell is viewed as a slice of slab that is
infinite in
x
and
z
, and the fact that a medium of periodic slabs can be modeled
by an effective magneto-dielectric material and (b) N-layer anisotropic effective
"
0
media representing the original layered 1-D periodic magneto-dielectric
[Fig. 1(b)].
AEMT is a very capable method for analyzing periodic struc-
tures with relatively large periods. It is shown that 1-D periodic
dielectric layers can be viewed as a uniaxial anisotropic medium
with the optical axis along the periodic direction. The relation-
ship between the effective constitutive parameters and the geo-
metrical and electrical parameters of the periodic medium com-
posed of only dielectric materials was described in [10]. To
obtain the performance of a stack of magneto-dielectric periodic
layers, AEMT must be extended to include magnetic materials
as well.
The geometry of an N-layer 1-D periodic structure stacked
along the
direction and illuminated by an arbitrary incident
plane wave is depicted in Fig. 1. Each periodic layer is assumed
to be made of both dielectric and magnetic materials with peri-
odicity
(layers are periodic in the or directions), where
the subscript
represents the th region . The
direction and polarization of the incident wave and its corre-
sponding coordinate system are defined in Fig. 1(a). Note that a
linear polarization is denoted by the angle
between the elec-
tric field and a reference direction defined by
, where
is the incident wave propagation direction.
To obtain the effective constitutive parameters for each layer
the medium is assumed to be infinite in the
direction as il-
lustrated in Fig. 2(a). This assumption is applied only where
1560 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 6, JUNE 2004
the effective parameters of the layers are obtained and in the
remainder of the analysis the actual thickness of layers is con-
sidered. This appears to be a very good approximation even
when the layers are relatively thin. The 1-D periodic medium is
then analyzed by considering the possible solution of Maxwell’s
equations in the absence of any source. This is accomplished by
invoking the Floquet periodic boundary conditions in addition
to the traditional electric and magnetic field boundary condi-
tions [11], [12]. Examining the possible modes of propagation
it is shown that the medium acts as a uniaxial anisotropic ma-
terial. The original structure [Fig. 1(b)] can then be reduced to
an
N-layer medium composed of anisotropic layers [Fig. 2(b)]
with effective constitutive parameters given by the terms
and , whose response to any arbitrary source can be ana-
lytically determined.
To demonstrate the procedure outlined above, consider an ar-
bitrary incident plane wave illuminating a periodic layer with
periodicity in the
direction and infinite in the and direc-
tions. The direction of propagation of the incident wave is given
by
(1)
where
is a transverse vector in - plane. Since the structure
is symmetric in the
- plane, the electromagnetic fields can be
separated into
and polarizations, where . The
EM waves for the electric and magnetic polarizations in one unit
cell of the
th layer must satisfy the scalar wave equation
(2a)
(2b)
where , depending on the polarization, can be
or . The solution to (2) is found to be
(3a)
(3b)
where
(4a)
(4b)
The continuity and periodicity (Floquet theorem) boundary
conditions are next applied for determining the natural modes
in the periodic structure for both polarizations. For
polar-
ization the application of these boundary conditions results in
the following transcendental equation:
(5)
The application of duality to (5) determines the following tran-
scendental equation for the
polarization case:
(6)
From (4a) and (4b), it can be shown that
, which together with (5) or (6) provide
the solutions for
, , and for the -or -
polarization cases, respectively.
The electromagnetic behavior of the
th periodic medium for
the dominant mode can be modeled with a uniaxial effective
magneto-dielectric slab. In a uniaxial anisotropic material with
the optical axis along the
direction and tensors take the
following forms:
(7a)
(7b)
and the dispersion relations for
or polarizations are given
by
(8a)
(8b)
where subscripts “
” and “ ” denote the extraordinary and ordi-
nary waves, respectively [13]. It should be pointed out that in an
electric and magnetic uniaxial medium both ordinary and extra-
ordinary waves are independent of incident wave polarizations.
The
th periodic layer can be thus presented as an anisotropic
material if
, , and satisfy (8) for both polarizations.
The parameters
, , , and of the th effective medium
can be obtained from
(9a)
(9b)
(10a)
(10b)
for a given value of
. Although effective and appear to be a
function of the angle of incidence, the numerical solution shows
MOSALLAEI AND SARABANDI: MAGNETO-DIELECTRICS IN ELECTROMAGNETICS: CONCEPT AND APPLICATIONS 1561
this dependence on incident angle is very small even when is
as large as
[10]. In the special case where ( is
the wavelength in
th layer) the above formulation simplifies to
the well-known quasistatic solution
(11a)
(11b)
(12a)
(12b)
It is also interesting to note that the expressions for the effective
permittivity and permeability are separable, as expected. That
is
and are only functions of and and and are
only functions of
and .
Therefore, utilizing the generalized AEMT, the layers of the
1-D periodic magneto-dielectric can be modeled with as an
anisotropic media with the
and tensors [Fig. 2(b)]
and the stack of periodic medium can be analytically charac-
terized (see Appendix A).
B. FDTD Technique
In Section II-A the AEMT was applied to provide an ana-
lytic formulation that describes the behavior of 1-D periodic
layers. However, in order to accurately characterize the per-
formance of electromagnetic waves in more complex periodic
magneto-dielectric meta-materials a more general technique is
needed. Here, a powerful method based on the FDTD approach
is applied [14]. The FDTD in conjunction with the PBC and
PML walls [15], [16] allows for characterizing plane wave prop-
agation within an arbitrary 3-D periodic structure. The FDTD
method is appropriate for this application as it can provide the
EM behavior of the magneto-dielectrics over a wide frequency
range. Prony’s extrapolation scheme is also integrated to effi-
ciently expedite the computational time of the FDTD formula-
tion. The FDTD code used here is capable of analyzing both
finite and infinite periodic magneto-dielectrics.
III. N
OVEL APPLICATIONS OF
MAGNETO-DIELECTRICS
In this section, the feasibility of designing periodic mag-
neto-dielectric meta-materials useful for RF and wireless
applications is demonstrated. Two specific examples are con-
sidered: (a) electromagnetic band-gap (EBG) structures with
superior properties, and (b) substrate design for miniaturized
wideband planar antennas with enhanced performance.
The structures are constructed utilizing a typical magneto-di-
electric (hexaferrite) recently manufactured by Trans-Tech. A
sample of high performance Z-type hexaferrite is measured
using an Agilent E4991A impedance analyzer to obtain the
material permeability. However, establishing the correct di-
electric constant for the medium requires an unambiguous
measurement. Dielectric measurement methods using coaxial
transmission lines and the impedance analyzer with relatively
Fig. 3. Measured complex permeability of an aligned hexaferrite material
manufactured by Trans-Tech Inc.
thick specimens give high numbers because of the influence
of permeability and are not very reliable. The best approach is
to construct a quarter wave coaxial resonator and then obtain
the resonant frequency that is proportional to the square root
of permeability and permittivity. From that, and substituting
for the permeability, one can evaluate the permittivity. Also, by
measuring the total
of the material and given the magnetic
loss tangent, the dielectric loss tangent can be determined.
Fig. 3 shows the magnetic behavior
of a typical mag-
neto-dielectric material manufactured by Trans-Tech. Note that
the measured results are valid up to 1 GHz and the observed
peak around this frequency is basically not due to a material
resonance but rather to the device deficiency and measurement
equipment. The minimum material resonance of hexaferrite
should occur no lower than about 3 GHz. In the VHF range
(up to 300 MHz) the relative permeability is found to be ap-
proximately
and the material has a low magnetic loss
tangent of about 0.02. The dielectric constant for this material
is about
and the dielectric loss tangent is around 0.002.
The Trans-Tech material is used in the construction of designed
magneto-dielectrics presented in Sections III-A and B.
A. EBG Structures
EBG structures are abundant in nature and show interesting
and useful phenomena. In particular, characteristics such as fre-
quency stop-band, pass-band and band-gap can be identified [5],
[6]. Generally speaking, EBG materials are 3-D periodic struc-
tures that prevent the propagation of electromagnetic waves in
a specified band of frequency for all angles and for all polariza-
tion states.
A woodpile EBG, proposed by Ho et al. [17], is a periodic
structure capable of producing complete transmission band-gap
regions. In the design presented by Ho et al. only dielectric ma-
terials are used to obtain the band-gap structure. In this work, a
woodpile structure composed of magneto-dielectric material is
considered which shows superior band-gap characteristics.
Fig. 4 depicts the geometry of an eight-layer woodpile struc-
ture of dielectric and magneto-dielectric materials. The even-
1562 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 6, JUNE 2004
Fig. 4. Eight-layer woodpile EBG structure (periodic in
y
-
z
directions); even layers are rotated 90 relative to the odd layers: (a) Dielectric woodpile (
"
material),
(b) Magneto-dielectric woodpile (
"
0
material).
Fig. 5. Transmission coefficients of the dielectric and magneto-dielectric woodpile EBG for an arbitrary incident plane wave with
=90
,
=140
, and
=60
: (a) AEMT analytical results, (b) FDTD numerical results. The magneto-dielectric woodpile shows the lower band reject frequency (reduced size) and
enhanced rejection level.
numbered layers are similar to the odd-numbered layers except
that they are rotated by 90
. The AEMT analytical technique
and FDTD full wave analysis are applied to characterize the
structure by examining the transmission coefficient through the
woodpile for an arbitrary incident plane wave with
,
and a linear polarization specified by angle
, where is the angle between the electric field and refer-
ence direction
.
Fig. 5(a) and (b) show the transmission coefficients for two
different woodpiles, namely, dielectric and magneto-dielectric
in a free space background as a function of normalized fre-
quency
( is the periodicity in the direction)
using the AEMT and FDTD approaches, respectively. In these
simulations the filling factor
is 0.50 and .
The dielectric woodpile is constructed from dielectric material
with
(loss tangent ) and for the mag-
neto-dielectric
(Z-type hexaferrite) is used. All
layers have equal thicknesses of
. As observed, for the dielec-
tric material the AEMT and FDTD results have excellent agree-
ment. For the magneto-dielectric since
and
(material wavelength is ), the AEMT has a good
agreement with FDTD only up to the normalized frequencies of
about
where . For periodicity
more than one Bragg mode exists and the more general full wave
analysis must be applied. The results however shows the useful-
ness of AEMT in analyzing periodic structures having relatively
large periods, of up to
.
As observed in Fig. 5, the dielectric woodpile EBG gener-
ates a band-gap region at the normalized frequency
with in-band rejection of about . In this design
the rectangular grains of the odd-numbered layers (#1, 3, 5, 7)
are parallel to the electric field and hence produce a larger ef-
fective permittivity
than the even-numbered layers .
Therefore, a wave impedance contrast exists between the layers
that is responsible for the rejection level at
the band-gap region.
![Fig. 2. Anisotropic effective material model for the 1-D periodic magneto-dielectric: (a) A unit cell is viewed as a slice of slab that is infinite in x and z, and the fact that a medium of periodic slabs can be modeled by an effective magneto-dielectric material and (b) N-layer anisotropic effective " media representing the original layered 1-D periodic magneto-dielectric [Fig. 1(b)].](/figures/fig-2-anisotropic-effective-material-model-for-the-1-d-t893yo2k.webp)
