Hybrid resonant phenomenon in a metamaterial structure with integrated resonant magnetic material

Abstract

We explore the hybridization of fundamental material resonances with the artificial resonances of metamaterials. A hybrid structure is presented in the waveguide environment that consists of a resonant magnetic material with a characteristic tuneable gyromagnetic response that is integrated into a complementary split ring resonator (CSRR) metamaterial structure. The combined structure exhibits a distinct hybrid resonance in which each natural resonance of the CSRR is split into a lower and upper resonance that straddle the frequency for which the magnetic material's permeability is zero. We provide an analytical understanding of this hybrid resonance and define an effective medium theory for the combined structure that demonstrates good agreement with numerical electromagnetic simulations. The designed structure demonstrates the potential for using a ferrimagnetic or ferromagnetic material as a means of creating a tunable metamaterial structure.

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arXiv:0810.4871v1 [cond-mat.mtrl-sci] 27 Oct 2008
Hybrid resonant phenomenon in a metamaterial structure with integrated resonant
magnetic material
Jonah N. Gollub1, David R. Smith2, and Juan D. Baena3
1Department of Physics, University of California, San Diego, CA, 92037 USA
2Department of Electrical and Computer Engineering, Duke University, Durham, NC, 27708 USA and
3Department of Physics, National University of Colombia, Bogot´ a, Colombia
(Dated: October 27, 2008)
We explore the hybridization of fundamental material resonances with the artificial resonances
of metamaterials. A hybrid structure is presented in the waveguide environment that consists of
a resonant magnetic material with a characteristic tuneable gyromagnetic response that is inte-
grated into a complementary split ring resonator (CSRR) metamaterial structure. The combined
structure exhibits a distinct hybrid resonance in which each natural resonance of the CSRR is split
into a lower and upper resonance that straddle the frequency for which the magnetic material’s
permeability is zero. We provide an analytical understanding of this hybrid resonance and define
an effective medium theory for the combined structure that demonstrates good agreement with nu-
merical electromagnetic simulations. The designed structure demonstrates the potential for using a
ferrimagnetic or ferromagnetic material as a means of creating a tunable metamaterial structure.
INTRODUCTION
Metamaterials encompass a class of artificial electro-
magnetic media that provide electromagnetic properties
beyond those available from natural materials [1, 2, 3].
Their unique electromagnetic properties are obtained by
harnessing the resonant behavior of many periodic and
subwavelength composite structures. Unfortunately, the
resonant nature of metamaterials also ensures that they
are frequency dispersive and limited to a narrow fre-
quency band of operation.
this constraint is to design tunable metamaterials, which
though still dispersive, can be tuned to have the desired
properties over the frequency range of interest. A tunable
metamaterial can be made by placing a material with a
tunable electromagnetic parameter, permittivity or per-
meability, in a region of the metamaterial cell where the
local fields are concentrated. Tuning the local fields will
in turn affect the bulk electromagnetic response of the
metamaterial. This has been demonstrated by tuning the
capacitance in the gap region of metamaterials composed
of split-ring resonators (SRRs) using ferroelectric mate-
rials [4, 5]. In this paper we investigate a tunable meta-
material structure that incorporates a resonant magnetic
material, that in contrast to ferroelectric material, is it-
self frequency dispersive over the range of operation. The
two dispersive systems, metamaterial structure and mag-
netic material, combine to exhibit a distinct hybrid res-
onance for which we provide an analytical model and
demonstrate its good agreement with numerical results.
Gyromagnetic materials have a permeability tensor of
the form [6],
One method of bypassing
µ = µ0


µ1
iµ2 0
−iµ2 µ1 0
001

, (1)
where µ1and µ2are resonant functions of frequency. In
Magnetic Layer
Dielectric Layer
E
H
kz
I/2I/2
H
60 7080 90100110 120
-2
2
4
6
8
Frequency (Ghz)
Relative Parameters
Real(ε) Imag(ε)
Real(μ) Imag(μ)
1st Resonance2nd Resonance
y
x
z
FIG. 1: A diagram of a unit cell of the CSRR parallel plate
waveguide is shown. At resonance, current flows along the
edges of the CSRR structure and produces a magnetic field
that is approximately perpendicular in the gap region. The
extracted permeability and permittivity are shown for the fre-
quency range that includes the first two resonances of the
CSRR waveguide (without the influence of the magnetic ma-
terial).
this study we consider a simplified “generic” resonant
magnetic material which is isotropic and does not in-
clude the off-diagonal permeability component, µ2, in or-
der to elucidate the underlying physics of the hybridiza-
tion. Never-the-less, this structure is suggestive of how
ferrimagnetic or ferromagnetic materials might be used
to make a tunable metamaterial structure. At the end
of this letter we briefly discuss the potential for integrat-

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ing a real gyromagnetic material into the metamaterial
structure.
Metamaterials with artificial magnetic properties have
been constructed using SRR structures and exhibit an
approximate magnetic Drude-Lorentz response that has
been well documented in the literature [2]. A related
resonant structure is called the complimentary split-ring
resonator (CSRR) and has the “complementary” meta-
surface of a SRR (as shown in Fig. 1). The dual CSRR
and SRR structures obey Babinet’s principle of equiva-
lence with equal resonant responses [7, 8], but the CSRR
is excited by a perpendicular electric field while the SRR
structure is excited by a perpendicular magnetic field.
The CSRR structure is well suited for implementation
of the metamaterial concept in a waveguide geometry be-
cause the CSRR structures can be etched into the ground
plane of a waveguide and excited by an incident TEM
wave. Just as for bulk metamaterial structures, it is pos-
sible to define effective bulk parameters for a waveguide
metamaterial structure [9]. By Babinet’s equivalence, the
CSRRs permittivity is the dual of the SRR structure and
is given by
ǫ(ω) = 1 −
Fω2
0+ iωΓc,
ω2− ω2
(2)
where F is a constant, ω0 is the resonant angular fre-
quency, Γcis the dissipation factor, and ω is the angular
frequency [3]. The resonant frequency ω0 is related to
the inductance, L, and capacitance, C, of the structure
by
ω0=
1
√LC.
(3)
The inclusion of a thin layer of magnetic material below
the CSRR structure as shown in Fig. 1 provides a means
to influence the the local fields of the CSRR structure.
We consider a magnetic material with relative permeabil-
ity of the form,
µr(ω) =
(ξMs+ ξH0− iωΓm)2− ω2
(ξH0− iωΓm)(ξMs+ ξH0− iωΓm) − ω2
where Msis the magnetic saturation of the material, H0
is the magnetic bias field, Γmis the damping constant,
and ξ=γµ0is a constant with γ defined as the gyromag-
netic ratio [10]. The CSRR resonance and magnetic ma-
terial resonance interact to produce a hybrid resonance.
The mechanism of interaction can be understood by con-
sidering the dynamics of the current flow when the CSRR
is resonating. An incident TEM wave in the waveguide
drives current symmetrically in and out of the CSRR
structure along the edges of the gap region as shown in
Fig. 1. The current is concentrated on either side of the
gap of the CSRR structure but flows in opposite direc-
tions creating a nearly perpendicular magnetic field in
the gap and the regions directly above and below the
(4)
gap. Consequently, determining the inductance of the
CSRR structure is analogous to analyzing the magnetic
capacitor model of parallel current sheets filled with some
volume fraction q (to be determined) of frequency depen-
dent magnetic material. Exploiting this simple model,
the inductance is found to be,
L = µ0
?
µr(ω)
µr(ω)(1 − q) + q
?
ggeom, (5)
where µr(ω) is the relative permeability of the magnetic
material and ggeom is a constant with units of length
that is determined by the geometry of the CSRR struc-
ture. If the parallel current approximation were exact
then ggeom= (1/2)wd/h with w the length of the gap, d
the width of the gap, and h the height of the gap (the fac-
tor of 1/2 follows from the parallel current flow into the
CSRR structure). In general, the geometrical function
is more complex but it can be extracted from numerical
simulations of the empty CSRR waveguide as we demon-
strate below. The capacitance of the CSRR structure
follows from the capacitance across the gap (above and
below the metal-surface). It is given by, C = ǫ0fgeom,
where again fgeomis a constant with units of length that
is determined by the geometry of the CSRR structure
and can be extracted from numerical simulations.
Inserting Eq. 5 into Eq. 3 provides an equation that
can be solved to determine the resonant frequency of the
hybrid CSRR/magnetic structure,
ω′
0=
1
?
µr(ω′
0)(1−q)+q
0)
µr(ω′
ω0.(6)
where, ω0= 1/?ǫ0µ0fgeomggeom. We note that Eq. 6 re-
duces to the resonant frequency of the empty structure in
the case µr(ω) is of unity. As previously mentioned, the
magnetic field generated in the gap region of the CSRR at
resonanceis predominately perpendicular (see Fig. 1) and
interacts principally with the magnetic material through
the x-component of the permeability. In fact, simulations
(not shown here) have shown that variation of any of the
other diagonal permeability components has no effect on
the response of the structure. We can insert the perme-
ability Eq. 4 into Eq. 6 and solve to get the new resonant
frequency of the hybrid structure. Eq. 6 is transcendental
in nature, as a result of the magnetic materials frequency
dependence, but it can be solved straightforwardly using
numerical methods.
In order to understand the dynamics of Eq. 6 it is in-
structive to plot the left and right side of the equation as
a function of frequency, as shown in Fig. 2(a). Note that
the solution to Eq. 6 is found at the intersection of these
two lines, i.e. where the function is self-consistent. The
characteristic phenomenon of the hybridization is seen
to be a splitting of the CSRR’s resonance into a lower,
1a, and upper, 1b, hybrid resonance which straddle the

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0.00.51.0
Biasing Component H0 (kG)
1.52.02.53.0
60
65
70
Frequency (Ghz)
HFSS Simulations
Analytic Solution
Empty CSRR Resonance
0 2040 60 80100 120
20
40
60
80
100
120
Frequency (Ghz)
Frequency (Ghz)
1st Resonance (CSRR)
2nd Resonance (CSRR)
Line of equality
RHS for 1st Resonance
RHS for 2nd Resonance
1a
1b
2a
2b
(a)
(b)
FIG. 2: (a) The left hand side (line of equality) and right
hand side (RHS) of Eq. 6 are plotted for the 1st/2nd CSRR
resonance with H0 = 1.5 kG and characteristic parameters
stated in the text. (b) The dispersion curve for the 1st or-
der hybrid mode of the CSRR/magnetic material waveguide
is calculated both analytically and through numerical electro-
magnetic simulations as a function of the biasing field, H0.
frequency at which the magnetic material’s permeabil-
ity is zero (above resonance—not the zero point at reso-
nance). The hybridization is strongest when the natural
resonance of the CSRR structure is near the zero perme-
ability point of the magnetic material. If one increase the
biasing field of the magnetic material, such that its zero
permeability frequency advances through the CSRR’s
natural resonance, a characteristic growth and shift of
the 1aresonance is observed and then a subsequent decay
and shift of the 1bresonance is observed. Hence, tuning
the magnetic material effectively tunes the response of
the metamaterial structure. Because a CSRR structure
inherently exhibits multiple resonance (the first two res-
onances of the CSRR structure are shown in Fig. 1) it
is not only the fundamental resonance that is hybridized
but also the higher order resonances.
higher order resonances are highly damped if they are
far away from the magnetic material’s zero permeability
frequency. In practice this means that if the zero per-
meability frequency of the magnetic material is aligned
with the fundamental CSRR structure resonance, then
depending on losses in the system, residual higher order
hybrid resonances 2a, ..., na might be found near the
fundamental, 1a, hybrid resonance as shown in Fig. 2(a).
In contrast, the 1b, 2b, ..., naresonances remain largely
separated. This suggests that the 1b resonance—which
is strongly resonant, isolated from the other hybrid reso-
nances, and strongly tunable—maybe the most useful for
application.
However, these
In order to investigate the hybridization of the CSRR
and magnetic material we used Ansoft’s commercially
available electromagnetic finite element solver (HFSS).
HFSS calculates the scattering matrix for a simulated
structure via a full wave analysis. It is sufficient to sim-
ulate one CSRR unit cell, as shown in Fig. 1, and apply
appropriate boundary conditions to reproduce the char-
acteristic response of a parallel plate waveguide struc-
ture. The simulated structure consisted of two parallel
perfectly conducting plates and adjoining perpendicular
perfect magnetic boundaries. The structure was excited
through waveports on either sides of the waveguide with
a fundamental TEM mode. The CSRR structure was
cut into the top waveguide plate and an exterior volume
above the structure was defined in order to accommo-
date electromagnetic fields emanating above the struc-
ture. Perfect magnetic boundaries were defined on the
exterior volume’s surfaces perpendicular to the propa-
gation mode while perfect electric boundaries were de-
fined on the other surfaces including the top of the ex-
tra boundary volume (note that the height of the extra
volume is chosen such that on the boundary the electro-
magnetic fields decay to near zero—and hence its speci-
fication is irrelevant, but numerical convergence is faster
by defining an electric boundary here).
The geometrical parameters of the CSRR were chosen
to have a resonance near 64 Ghz. The unit cell was 1
mm, the ring radius was 0.4 mm, the neck was 0.15 mm
wide, and the notch width was 0.035 mm. The spac-
ing between the plates, which determines the strength
of the CSRR resonance, was set to 0.4 mm. The top
and bottom waveguide plates were 1 µm thick and had
the conductivity of copper (σ = 5.6 × 107). For sim-
plicity, the dielectric between the plates was set to have
a dielectric constant of ǫr = 1. The structure was first
simulated without the magnetic layer to determine the
Drude-Lorentz fit parameters in Eq. 2. A parameter ex-
traction was performed on the S-parameters [11] of the
simulated structure to determine the effective permit-
tivity and then a least square fit of the Drude-Lorentz
function, Eq. 2, was used to determine the resonant fre-
quency, ω0= 1/?ǫ0µ0ggeomfgeom; the constant, F; and
the loss tangent, Γc. In fact, CSRR structures (and
metamaterials in general) are not perfect Drude-Lorentz
resonators as they exhibit spatial dispersion effects near
resonance. These spatial dispersion effects were incor-
porated into our fitting procedure [12] to determine the
resonant frequency more accurately but then the simpler
Drude-lorentz model was used in our analytic analysis
with good accuracy.
Once the empty CSRR structure was characterized,
numerical simulations incorporating the magnetic ma-
terial were performed. In the interest of exploring the
potential of incorporating high frequency (40-70 Ghz)
thin film magnetic materials, such as hexagonal fer-
rites [13], we considered a 1 µm thick layer of magnetic

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Transmission (dB)
5560657075
Frequency(Ghz)
-20
-15
-10
-5
5560 6570 75
-20
-15
-10
-5
55 6065 70 75
-20
-15
-10
-5
55 606570 75
-20
-15
-10
-5
Bias = 0 kG
Bias = 3 kG
Bias = 2 kG
Bias = 1 kG
HFSS Simulation
Analytic Solution
1a
1a
1a
1a
1b
1b
1b
1b
2a
2a
2a
2a
FIG. 3: The transmission results are shown for the HFSS
electromagnetic simulation and the analytic prediction using
effective medium theory.
material with parameters (Eq. 4): ξ = 18 GhzkG−1,
Γm= (0.70 Ghz)/ω and ξMs= 380 Ghz. The permeabil-
ity was calculated in MATLAB and imported into HFSS
as a data table of real values (Real[µ]) and loss tangent
values (Imag[µ]/Real[µ]). As previously mentioned, it is
only the x-component of µr(ω) that contributes to the in-
teraction. Using the fitted parameters extracted from the
empty structure we solved the analytic equation, Eq. 6,
for the first hybrid mode, 1a/1b, using a numerical root
finding method implemented in MATHEMATICA. The
magnetic filling fraction, q, was determined to have a
value of 0.013 through comparison to the HFSS simula-
tions results. A comparison of the analytic dispersion
curve calculated from Eq. 6 versus the numerical simu-
lations performed with HFSS is shown in Fig. 2(b) and
demonstrates excellent agreement with variation of the
bias field.
The transmission through the CSRR/magnetic mate-
rial waveguide can be calculated analytically by assigning
bulk electromagnetic parameters to the waveguide struc-
ture. The effective bulk permittivity of the waveguide
structure can be calculated by inputing Eq. 6 into Eq. 2
and then solving the Fresnel equations [11] to determine
the transmission and reflection of the structure. This
was done for the CSRR/magnetic waveguide structure
and compared to HFSS simulations as shown in Fig. 3.
In the HFSS simulations a splitting of the second order
CSRR resonance can also be seen. In the analytic ex-
pression we were able to reproduce this by applying our
6364 65
Frequency (Ghz)
Frequency (Ghz)
Relative Permeability
ξMs=380 Ghz
ξMs=189 Ghz
ξMs=38 Ghz
60 62 6466 6870
-5
5
10 2030 40 50 6070
-100
Hybrid Structure
Relative Permittivity
-50
50
100
Magnetic Material
zero point
(a)
(b)
FIG. 4: (a) The real part of the permeability is shown for sev-
eral magnetization values of the magnetic material with the
bias field chosen to have the same zero permeability frequency.
(b) The resulting hybrid permeability is shown.
technique to the first (electric) and second (magnetic)
resonance of the CSRR structure. The overall agreement
is seen to correlate well over a range of biasing values.
It is remarkable to note that the width and strength
of the hybrid resonances is primarily a function of the
CSRR’s resonant properties. The properties of the mag-
netic material’s resonance (and filling fraction q) predom-
inantly determines the bandwidth over which the hybrid
mode interaction exists. In Fig. 4 the permeability of the
magnetic material, Eq. 4, and the associated effective
permeability of the hybrid structure are plotted for vari-
ous value of magnetization Ms. When the biasing field is
chosen such that the zero point of the magnetic material
correlates to the empty CSRR structure resonance, we
see that the relative permeability strength of the hybrid
structure is the same but that the splitting of the res-
onance is smaller for smaller values of Ms values. This
suggest that it may be possible to harness narrowly reso-
nant magnetic materials, that cannot be utilized directly,
by using this metamaterial approach (though only over
a narrow bandwidth). These potential materials include
hexagonal ferrites and even antiferromagnetic materials
such as MnF2[14].
In summary, the hybrid resonance that results from
combining a resonant magnetic material and CSRR
structure results in a unique hybrid resonance which can
be harnessed to make tunable metamaterial structures.
Gyromagnetic materials have a resonant permeability
tensor form similar to that considered here but with the
added complexity of off-diagonal resonant components.
As a result, the resonant response of the combined struc-
ture has a more complicated dependance on the orienta-
tion of the biasing direction of the gyromagnetic mate-
rial with respect to the geometry of the CSRR structure
[15, 16]. Preliminary work suggest that if we limit our-

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selves to using thin layers we should expect to see an
analogous phenomenon as demonstrated here. Though
gyromagnetic materials have been used directly to make
tunable microwave devices [17, 18, 19], using them in-
directly in metamaterial structures has the potential ad-
vantage of increasing the range of effective material prop-
erties while at the same time reducing the amount of the
magnetic material needed in the structure and its asso-
ciated losses.
This research was supported by U.S. Army Research
Office DOA under Grant No. W911NF-04-1-0247. We
also acknowledgesupport from the Air Force Office of Sci-
entific Research through a Multiply University Research
Initiative under Contract No. FA9550-06-1-0279.
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