We investigate numerically the homogenized permittivities of composites made of low-index dielectric inclusions in a ferroelectric matrix under a static electric field. A refined model is used to take into account the coupling between the electrostatic problem and the electric field dependent permittivity of the ferroelectric material, leading to a local field enhancement and permittivity change in the ferroelectric. Periodic and pseudorandom structures in two dimensions are investigated, and we compute the effective permittivity, losses, electrically induced anisotropy, and tunability of these metamaterials. We show that the tunability of such composites might be substantially enhanced in the periodic case, whereas introducing disorder in the microstructure weakens the effect of the enhanced local permittivity change. Our results may be useful to guide the synthesis of novel composite ceramics with improved characteristics for controllable microwave devices.

Ferroelectric materials play a crucial role in reconfigurable microwave devices with typical applications including antenna beam steering, phase shifters, tunable power splitters, filters, voltage controlled oscillators, and matching networks.1 Both bulk ceramics and thin films have been employed to design frequency agile components2–4 and metamaterials.5,6 The main reason for using ferroelectric materials is their strong dependence of their permittivity ε on an applied electric field E, which is measured by their tunability defined as n=ε(0)/ε(E), along with a nonhysteresis behavior when used in their paraelectric state. The key requirements for antenna and microwave applications are large tunability and low losses. These two characteristics are correlated and one has to find a trade-off for optimal device performance, which can be quantified by the so-called commutation quality factor K=(n1)2/(ntanδ(0)tanδ(E)), where tanδ is the loss tangent. These materials have usually high permittivity values even at microwave frequencies, often leading to slow response time and impedance mismatch, which can be an issue in some practical applications. Thus, it has been considered to mix ferroelectric ceramics with low-index and low-loss nontunable dielectrics in order to reduce both permittivity values and losses or to use porous ceramics to achieve the same goals without unwanted chemical reactions at the boundaries between dissimilar materials. In particular, the addition of magnesium oxide in barium strontium titanate (BST) ceramics has been shown to decrease the losses while keeping good tunability.7,8 Ceramics such as Pb(Zr,Ti)O3 (PZT) and BaTiO3 (BT) have been used as fillers in polymer-based composites with high dielectric constant.9 Other mixtures include metal–polymer composites10 and electroactive polymers such as poly(vinylidene fluoride) (PVDF) with high index dielectric inclusions.11 

The effective parameters of those composites have been investigated12–15 and it has been found that the permittivity can be greatly reduced while losses are much less sensitive to the dielectric phase addition and, in some situations, can lead to a small increase of the tunability of the mixtures. Analytical models based on the Bruggeman effective medium approach for low concentration of dielectrics were derived for different configurations (columnar, layered, and spherical inclusions models) and have been successfully compared with numerical simulations and experiments.12 In the context of porous ferroelectrics, the homogenized properties strongly depend on the size and morphology of the pores.16,17 Recently, the concept of tailoring the nonlinear properties of ferroelectric and dielectric structures by local field engineering has been introduced.18–20 It was shown through finite element calculation including the nonlinear coupling that, by employing composite materials made of linear dielectric inclusions into a ferroelectric matrix, one can lower the permittivity while maintaining high tunability, due to the local field in the ferroelectric phase that is tuned by the linear dielectric phase. Moreover, the effect of grain sizes in ferroelectric ceramics was studied using a model taking the field enhancement into account at the grain boundaries, and the predicted behavior was successfully compared to experimental data.19 Generally, there is a need for refined theoretical and numerical models to explain and design tunable materials and composites with tailored nonlinear properties. Note that the general method followed by our coupled model could be applied to other types of tunable system where local field enhancement and amplification is relevant, including ferromagnetic metamaterials,21 liquid crystals based devices,22 or field-enhanced carrier dynamics in doped semiconductors at other frequency ranges, particularly in the terahertz and near-infrared.23,24

This study investigates numerically the effective permittivity of composites made of dielectric inclusions in a ferroelectric matrix by using a two-scale convergence method.25,26 The originality lies in the fact that a fully coupling model is employed to calculate the electrostatic field distribution when a uniform biasing field is applied on the structures, which will result in a local modification of the permittivity in the ferroelectric phase due to the microstructure. As compared to a simple uncoupled model where the ferroelectric phase is only modified through the biasing field, the resulting effective permittivity, dielectric losses, tunability, and anisotropy significantly differ. In contrast with earlier studies in the literature,18,19 we account for the nonlinear coupling beyond the first iteration and use a two-scale convergence homogenization analysis to obtain the effective parameters at higher frequencies, instead of a capacitance-based model valid in the static regime. This is an important point as, contrary to most homogenization procedures that are based on a quasistatic approximation, the two-scale convergence method fixes the frequency and lets the characteristic size of the system (the periodicity of the composites) tend to zero.26 This asymptotic analysis allows one to study the frequency dependence of the effective parameters. In addition, analytical models for the effective permittivity routinely employed in the literature, such as Maxwell-Garnett or Bruggeman theories, are limited to a few canonical shapes of the inclusions and cannot handle arbitrary geometries and media with spatially varying properties. This last point is of particular importance in the context of this study since we have to account for the field induced local permittivity change.

The model we developed has been implemented with the finite element method (FEM), and we realize a systematic computational study of ferroelecric-dielectric mixtures. First, we consider metamaterials consisting of a square array of parallel dielectric rods with circular cross section in a ferroelectric host and then investigate the effect of random distribution of those rods within the unit cell.

We consider a composite made of a ferroelectric material with anisotropic permittivity εf(E) that is dependent on an applied electric field E and a nontunable dielectric of permittivity εd, which are both nonmagnetic. The structures under study are invariant along the z direction, which leads to the standard decomposition of the wave equation in the transverse electric case (TE, electric field parallel to the direction of invariance) and the transverse magnetic case (TM, magnetic field parallel to the direction of invariance). A uniform biasing field is applied in order to be able to tune the effective permittivity. Modeling homogenized properties of this type of mixtures can be done by assuming that the electric field distribution is uniform throughout the sample so that the study of the tunability is essentially achieved by changing the value of the properties in the ferroelectric phase and computing the effective permittivity of the composite. We refer to this approach as the uncoupled model in the following. However, a more accurate description is to take into account the change of the electric field by the microstructure, if any. We, therefore, need to solve an electrostatic equation to find the field distribution within the material, but its solution depends on the permittivities of both materials, and the permittivity in the ferroelectric phase depends on this induced electric field: this leads to a strongly coupled problem.

We use barium strontium titanate (BST) as our ferroelectric material.BaxSr1xTiO3 samples were fabricated using the conventional sintering method with a barium ratio of x=0.6 to obtain a dielectrically tunable material as reported in the literature.11,27 The tunability was measured using an impedance analyzer up to 100 MHz and at 3.8 GHz using a loaded microstrip split ring resonator.28 The measured tunability of the in-house BST samples of 27% under 1 kV/mm DC bias was in agreement with those reported elsewhere.11,27 The method presented is, however, general and relies only on the gradient of the dielectric tunability vs electric field and could be applied to any tunable host material. The normalized permittivity value as a function of biasing field is reported in Fig. 1.

FIG. 1.

Variation of the ferroelectric permittivity as a function of the applied electric field [dots: measurements; dashed lines: fit to formula (1)], for the static case (green) and at microwave frequencies (orange, f=3.8GHz). The fitting parameters are given in Table I.

FIG. 1.

Variation of the ferroelectric permittivity as a function of the applied electric field [dots: measurements; dashed lines: fit to formula (1)], for the static case (green) and at microwave frequencies (orange, f=3.8GHz). The fitting parameters are given in Table I.

Close modal
TABLE I.

Fitting parameters to model (1) for the measured permittivity values as a function of applied electric field shown in Fig. 1.

Caseεf(0)α (μm2/V2)β (μm4/V4)
Static 3050 0.120 0.024 
f=3.8GHz 165 0.240 0.079 
Caseεf(0)α (μm2/V2)β (μm4/V4)
Static 3050 0.120 0.024 
f=3.8GHz 165 0.240 0.079 

To describe the permittivity, we make use of the Landau potential given by F(P,E)=F0+aP2/2+bP4/4+cP6/6EP, where E is the applied electric field and P is the polarization.29,30 Variations of the permittivity with the temperature can be taken into account through the coefficients a, b, and c, but we assume that we are working at a constant room temperature. We further assume that the material is not subject to any stress so that the variation of permittivity due to mechanical constraints is irrelevant. The equation of state

F(P,E)P=aP0+bP03+cP05E=0

gives the dependence of the polarization on the applied electric field, with P0 being the equilibrium polarization. Along the direction of a uniform applied electric field, the relative permittivity is given by

εf(E)=[2F(P,E)P2]1=εf(0)1+αP02+βP04,
(1)

where εf(0)=1/a, α=3b/a, and β=5c/a. The fitting parameters are given in Table I. As the norm of the field increases, the permittivity decreases with a characteristic bell curve typical for a ferroelectric material in its paraelectric state. Furthermore, assuming that the crystalline principal axes of the ferroelectric material are oriented in the coordinate directions and that the diagonal components of the permittivity tensor are only function of the corresponding bias electric field components,31 we have

εf(E)=(εxxf(Ex)000εyyf(Ey)000εzzf(Ez)),
(2)

where each of the diagonal components has the functional form given by Eq. (1). Note that we will use the static values of permittivity for the electrostatic modeling, while we are interested in the homogenized values of permittivity at microwaves.

The composites under study are made of two materials, thus their permittivity is represented by a piecewise defined tensor ε(r,E), which is equal to εf(E(r)) in the ferroelectric phase and diag(εd) in the dielectric phase. In the following, we consider two different cases for the biasing field. Because of the form (2) assumed for the ferroelectric permittivity tensor, εzz will not be changing for a field in the plane orthogonal to the z axis. This is the only component being relevant for TE polarization, so we consider in this case a uniform biasing electric field applied along the direction of invariance E0=E0ez. On the other hand, the in-plane components of εf are tuned by Ex and Ey, therefore, without loss of generality, we consider a uniform applied electric field directed along the x axis E0=E0ex for the TM polarization case. To calculate the total electric field in the material, one has to solve for the potential V satisfying Gauss’ law

(εV)=0.
(3)

Note that for the TE case, the solution is trivial since the structures are invariant along z so that the electric field is equal to the uniform biasing field, and we will thus not study it in the following. However, in the TM case, the situation is much more complex: this is a coupled problem since the electric field E=V derived from the solution of Eq. (3) depends on the permittivity distribution, which itself depends on the electric field. The coupled system formed of Eqs. (2) and (3) is solved iteratively until there is convergence on the norm of the electric field. Here, we would like to emphasize that the permittivity in the ferroelectric material, although uniform initially, is spatially varying due to the nonuniform distribution of the total electric field.

When the period of the composite metamaterial is much smaller than the wavelength, one can describe the properties of the composite by a bulk medium with homogenized parameters. The effective permittivity for TM polarization is calculated using a two-scale convergence homogenization technique.25,26 For this purpose, one has to find the solutions ψj of two annex problems Pj, j={1,2},

[ξ(ψj+rj)]=0,
(4)

where r=(x,y)T is the position vector in the xy plane and ξ=εT/det(ε). The homogenized tensor ξ~ is obtained with

ξ~=ξ+ϕ,
(5)

where denotes the mean value over the unit cell. The elements of the matrix ϕ represent correction terms and are given by ϕij=ξψij. Finally, the effective permittivity tensor can be calculated using ε~=ξ~T/det(ξ~).

Note that the TE case, which we shall not study here as no coupling happens, is trivial since the homogenized permittivity is simply the average of the permittivity in the unit cell: ε~=ε.

In the following numerical results, the dielectric phase is supposed to be lossless and nondispersive with εd=3, while the ferroelectric material follows the permittivity described in Sec. II A and has a constant loss tangent, tanδf=102. Equations (3) and (4) are solved with a finite element method using the open source packages Gmsh32 and GetDP.33 In both cases, we use a square unit cell Ω of length d with periodic boundary conditions along x and y. Second order Lagrange elements are used, and the solution is computed with a direct solver (MUMPS34).

Let us now consider a periodic square array of infinitely long dielectric rods of a circular cross section of radius r embedded in a ferroelectric matrix.

We first study the convergence of the coupled problem on the particular case with dielectric filling fraction f=πr2/d2=0.5 and E0=2MV/m. Figures 2(a) and 2(b) show the convergence of the real part and loss tangent of the components of the homogenized permittivity tensor, respectively. The yy components converge quickly and are almost unaffected by the coupling process, whereas the xx components change substantially from the initial conditions. This is due to the effect of the redistribution of the electrostatic field within the unit cell [see Figs. 2(c) and 2(d)], where the x component of the electric field is still much stronger than the y component, even if spatially varying in the ferroelectric medium. At equilibrium, the electric field is concentrated close to the y axis in between two neighboring rods. This, in turn, affects the permittivity distribution [see Figs. 2(e) and 2(f)] and the homogenized properties of the composite.

FIG. 2.

Convergence of the coupled problem. Real part (a) and loss tangent (b) of the components of the homogenized permittivity tensor as a function of iteration step i. The values are normalized to the corresponding quantities for the bulk ferroelectric material. The distribution of the normalized electric field (color map: magnitude in the logarithmic scale; lines: equipotential contours) and of the xx component of the permittivity tensor is shown for i=1 [(c) and (d)] and i=12 [(e) and (f)].

FIG. 2.

Convergence of the coupled problem. Real part (a) and loss tangent (b) of the components of the homogenized permittivity tensor as a function of iteration step i. The values are normalized to the corresponding quantities for the bulk ferroelectric material. The distribution of the normalized electric field (color map: magnitude in the logarithmic scale; lines: equipotential contours) and of the xx component of the permittivity tensor is shown for i=1 [(c) and (d)] and i=12 [(e) and (f)].

Close modal

We computed the effective parameters of these metamaterial structures for different radii of the rods and studied their behavior when subjected to an external electrostatic field (see Fig. 3). The results of our coupled model differ significantly from the uncoupled one. Increasing the dielectric fraction lowers the effective permittivity, while the losses are slightly reduced but much less sensitive. Due to the inhomogeneous redistribution of the permittivity over the ferroelectric domain, the overall tunability changes. For the periodic composites studied in this section, taking into account the coupling leads to an effective tunability increase with higher dielectric concentration, and that is larger than the tunability of bulk ferroelectric. This can be seen in Fig. 3(c), where we plot the tunability of the composites along the x axis, n~(E)=ε~xx(E)/ε~xx(0), normalized to the tunability of the bulk ferroelectric n(E)=εxxf(E)/εxxf(0). Two concurrent effects are at stake here: on the one hand, the dilution of ferroelectric makes the composite less tunable, but on the other hand, the rearrangement of the electrostatic field surrounding the inclusion and its concentration in some region will cause a higher permittivity change locally. The relative strength of those phenomena is governed by the shape of the inclusion and its permittivity, and so, it is envisioned that the performance of the composites might be enhanced by engineering their microstructure. Those observations are consistent with previously published numerical and experimental results18 where the local field enhancement in porous ferroelectrics has been shown to possibly increase tunability with reducing permittivity for small porosity levels. Our approach also agrees with an analytical spherical inclusion model predicting an increase of the tunability with the dilution of the ferroelectric.12 

FIG. 3.

Effective parameters of the 2D metamaterials as a function of the applied electric field for various filling fraction of dielectric. (a) Normalized permittivity, (b) normalized loss tangent, (c) normalized tunability, and (d) anisotropy factor. The solid lines correspond to the coupled model, and the dashed lines to the uncoupled model. The values are normalized to the corresponding quantities for the bulk ferroelectric material.

FIG. 3.

Effective parameters of the 2D metamaterials as a function of the applied electric field for various filling fraction of dielectric. (a) Normalized permittivity, (b) normalized loss tangent, (c) normalized tunability, and (d) anisotropy factor. The solid lines correspond to the coupled model, and the dashed lines to the uncoupled model. The values are normalized to the corresponding quantities for the bulk ferroelectric material.

Close modal

The geometry of the unit cell is symmetric so the homogenized material is isotropic when no field is applied. But when the sample is biased, the permittivity distribution becomes asymmetric due to the inhomogeneity of the electric field, thus making the effective material properties anisotropic. This geometric effect is added to the anisotropy arising from the material properties of the ferroelectric phase itself and depending on the topology and permittivity of the rods, one effect would be predominant. In the case studied here, the equilibrium permittivity distribution varies strongly along the bias direction and much less orthogonally to it, which adds anisotropy by diminishing the effective permittivity in the x direction. This local field induced effect is what makes the anisotropy stronger in our coupled model compared to the uncoupled one [cf. Fig. 3(d), where we plot the anisotropy factor α=εxx/εyy]. Those subtle phenomena can only be rigorously taken into account by employing a coupling formalism and are responsible for the difference observed when compared to a simple uncoupled model.

We finally study the effect of random distribution of the inclusions within the unit cell on the effective parameters of the composites. This is an important point as fabrication of randomly dispersed inclusions is much more easy from a technological perspective. For each filling fraction of the dielectric, we generated 21 numerical samples with inclusions of circular cross section of average radius r=d/20 that can vary by ±30%. Their center is chosen randomly and the rods are allowed to overlap. An example of distribution for f=0.5 is given in Fig. 4. The effective material properties are plotted in Fig. 5. Similar to the periodic case, the permittivity decreases with increasing dilution of ferroelectric, but for identical filling fraction, the permittivity is lower as compared to the periodic array, and the smaller the dielectric concentration, the larger is the difference. Losses decrease as well and the reduction is substantially larger than the periodic case, with higher variation from sample to sample as f increases. The effective tunability is on average smaller than that in the periodic case, and for low biasing fields and for some particular samples, it can be greater than the bulk tunability. However, at higher applied electric fields, normalized tunability becomes smaller than unity and is reduced as one adds more dielectric. For comparison, the homogenized parameters are plotted in Fig. 6 in the case where the coupling is neglected. One can see that the coupled and uncoupled models give similar results for the tunability, whereas the losses are still smaller for the coupled case at higher fields.

FIG. 4.

Permittivity distribution of the numerical samples used for f=0.5. The dark color indicates the ferroelectric material, while the light color represents the dielectric inclusions.

FIG. 4.

Permittivity distribution of the numerical samples used for f=0.5. The dark color indicates the ferroelectric material, while the light color represents the dielectric inclusions.

Close modal
FIG. 5.

Effective parameters of the pseudorandom composites as a function of the applied electric field for various filling fraction of dielectric, when the coupling is taken into account. (a) Normalized permittivity, (b) normalized loss tangent, (c) normalized tunability, and (d) anisotropy factor. The solid lines represent the average values over the 21 samples, and the lighter error bands show a confidence interval corresponding to the standard deviation. The values are normalized to the corresponding quantities for the bulk ferroelectric material.

FIG. 5.

Effective parameters of the pseudorandom composites as a function of the applied electric field for various filling fraction of dielectric, when the coupling is taken into account. (a) Normalized permittivity, (b) normalized loss tangent, (c) normalized tunability, and (d) anisotropy factor. The solid lines represent the average values over the 21 samples, and the lighter error bands show a confidence interval corresponding to the standard deviation. The values are normalized to the corresponding quantities for the bulk ferroelectric material.

Close modal
FIG. 6.

Effective parameters of the random 2D mixtures as a function of the applied electric field for various filling fraction of dielectric, when the coupling is neglected. (a) Normalized permittivity, (b) normalized loss tangent, (c) normalized tunability, and (d) anisotropy factor. The solid lines represent the average values over the 21 samples, and the lighter error bands show a confidence interval corresponding to the standard deviation. The values are normalized to the corresponding quantities for the bulk ferroelectric material.

FIG. 6.

Effective parameters of the random 2D mixtures as a function of the applied electric field for various filling fraction of dielectric, when the coupling is neglected. (a) Normalized permittivity, (b) normalized loss tangent, (c) normalized tunability, and (d) anisotropy factor. The solid lines represent the average values over the 21 samples, and the lighter error bands show a confidence interval corresponding to the standard deviation. The values are normalized to the corresponding quantities for the bulk ferroelectric material.

Close modal
FIG. 7.

Convergence of the coupled problem in the random case for one sample. Real part (a) and loss tangent (b) of the components of the homogenized permittivity tensor as a function of iteration step i. The values are normalized to the corresponding quantities for the bulk ferroelectric material. The distribution of the normalized electric field (color map: magnitude in the logarithmic scale, lines: equipotential contours) and of the xx component of the permittivity tensor is shown for i=1 [(c) and (d)] and i=12 [(e) and (f)].

FIG. 7.

Convergence of the coupled problem in the random case for one sample. Real part (a) and loss tangent (b) of the components of the homogenized permittivity tensor as a function of iteration step i. The values are normalized to the corresponding quantities for the bulk ferroelectric material. The distribution of the normalized electric field (color map: magnitude in the logarithmic scale, lines: equipotential contours) and of the xx component of the permittivity tensor is shown for i=1 [(c) and (d)] and i=12 [(e) and (f)].

Close modal

The redistribution of electric field, permittivity, and convergence of the effective parameters are displayed in Fig. 7. The effect of disorder plays an important role here: the electrostatic field gets concentrated in between neighboring inclusions and the smaller the gap, the higher the field, hence a greater local permittivity change. In addition, even if the distribution is random, one expects that the anisotropy due to geometry would cancel for a sufficiently large number of rods (which is the case as the mean anisotropy factor is close to 1 when no bias field is applied). However, the anisotropy due to ferroelectric properties is important in this case as well, as both the x and y components of the electrostatic field are playing a role. Because of the relative positions of the rods, both εxx and εyy are affected by the coupling so that the anisotropy factor for higher fields is reduced as compared to the periodic case. However, even if there is a substantial variability from sample to sample, on average, the anisotropy factor decreases with increasing dielectric concentration.

We have studied the homogenized properties of dielectric/ferroelectric mixtures using a rigorous model that takes into account the coupling between the electrostatic field distribution and the field dependent ferroelectric permittivity tensor. After convergence of the coupled problem, the effective permittivity tensor is calculated using a two-scale convergence homogenization theory. The results obtained by this model differ significantly from a simple assumption that the permittivity of the ferroelectric responds just to the uniform biasing field. We have considered both periodic and random arrays of dielectric rods in a ferroelectric matrix in 2D and studied their effective properties for TM polarization as a function of dielectric concentration and bias field. Importantly, adding more low index and low loss dielectric allows to decrease the overall permittivity significantly and slightly lower the losses. For the periodic case, the tunability is higher than the bulk due to local field enhancement, whereas this effect is strongly suppressed when the disorder is introduced. The asymmetric redistribution of the permittivity induces an effective anisotropy that is added to the one arising purely from the ferroelectric material. The properties of the composites are affected by multiple factors: geometry and the spatially dependent electric field that will induce locally a tunable, anisotropic response in the ferroelectric phase depending on its amplitude and direction. This suggests that the performances of the composites may be enhanced by distributing the two phases in an optimal way to get high tunability and low losses. Further work in that direction is needed as well as extending this study to 3D media. Finally, because the permittivity of the dielectric is much smaller than the ferroelectric one, it would be of great interest to use high contrast homogenization theory35,36 to study these kinds of mixtures. This would reveal the frequency dependent artificial magnetism due to “microresonances” in the high index phase and potentially lead to composites with tunable effective permeability.

This work was funded by the Engineering and Physical Sciences Research Council (EPSRC), UK, under a grant (No. EP/P005578/1) “Adaptive Tools for Electromagnetics and Materials Modelling to Bridge the Gap between Design and Manufacturing (AOTOMAT).”

The authors would like to thank Henry Giddens for performing the measurements of ferroelectric permittivity used in this paper.

The codes necessary to reproduce the results in this article are freely available online at this address: https://www.github.com/benvial/ferromtm.

1.
A. K.
Tagantsev
,
V. O.
Sherman
,
K. F.
Astafiev
,
J.
Venkatesh
, and
N.
Setter
, “
Ferroelectric materials for microwave tunable applications
,”
J. Electroceram.
11
,
5
66
(
2018
).
2.
O.
Vendik
,
E.
Hollmann
,
A.
Kozyrev
, and
A.
Prudan
, “
Ferroelectric tuning of planar and bulk microwave devices
,”
J. Supercond.
12
,
325
338
(
1999
).
3.
M.
Lancaster
,
J.
Powell
, and
A.
Porch
, “
Thin-film ferroelectric microwave devices
,”
Supercond. Sci. Technol.
11
,
1323
(
1998
).
4.
X.
Xi
,
H.-C.
Li
,
W.
Si
,
A.
Sirenko
,
I.
Akimov
,
J.
Fox
,
A.
Clark
, and
J.
Hao
, “
Oxide thin films for tunable microwave devices
,”
J. Electroceram.
4
,
393
405
(
2000
).
5.
T. H.
Hand
and
S. A.
Cummer
, “
Frequency tunable electromagnetic metamaterial using ferroelectric loaded split rings
,”
J. Appl. Phys.
103
,
066105
(
2008
).
6.
H.
Zhao
,
L.
Kang
,
J.
Zhou
,
Q.
Zhao
,
L.
Li
,
L.
Peng
, and
Y.
Bai
, “
Experimental demonstration of tunable negative phase velocity and negative refraction in a ferromagnetic/ferroelectric composite metamaterial
,”
Appl. Phys. Lett.
93
,
201106
(
2008
).
7.
P.
Irvin
,
J.
Levy
,
R.
Guo
, and
A.
Bhalla
, “
Three-dimensional polarization imaging of (Ba,Sr)TiO3:MgO composites
,”
Appl. Phys. Lett.
86
,
042903
(
2005
).
8.
U.-C.
Chung
,
C.
Elissalde
,
M.
Maglione
,
C.
Estournès
,
M.
Paté
, and
J. P.
Ganne
, “
Low-losses, highly tunable Ba0.6Sr0.4TiO3/MgO composite
,”
Appl. Phys. Lett.
92
,
042902
(
2008
).
9.
B.
Wang
,
L.
Luo
,
F.
Ni
,
P.
Du
,
W.
Li
, and
H.
Chen
, “
Piezoelectric and ferroelectric properties of (Bi1xNa0.8K0.2Lax)0.5TiO3 lead-free ceramics
,”
J. Alloys Compd.
526
,
79
84
(
2012
).
10.
X.
Li
,
Y.-F.
Lim
,
K.
Yao
,
F. E. H.
Tay
, and
K. H.
Seah
, “
Ferroelectric poly(vinylidene fluoride) homopolymer nanotubes derived from solution in anodic alumina membrane template
,”
Chem. Mater.
25
,
524
529
(
2013
).
11.
G.
Hu
,
F.
Gao
,
J.
Kong
,
S.
Yang
,
Q.
Zhang
,
Z.
Liu
,
Y.
Zhang
, and
H.
Sun
, “
Preparation and dielectric properties of poly(vinylidene fluoride)/Ba0.6Sr0.4TiO3 composites
,”
J. Alloys Compd.
619
,
686
692
(
2015
).
12.
V. O.
Sherman
,
A. K.
Tagantsev
,
N.
Setter
,
D.
Iddles
, and
T.
Price
, “
Ferroelectric-dielectric tunable composites
,”
J. Appl. Phys.
99
,
074104
(
2006
).
13.
L.
Jylha
and
A. H.
Sihvola
, “
Tunability of granular ferroelectric dielectric composites
,”
Prog. Electromagn. Res.
78
,
189
207
(
2008
).
14.
V. O.
Sherman
,
A. K.
Tagantsev
, and
N.
Setter
, “Tunability and loss of the ferroelectric-dielectric composites,” in 14th IEEE International Symposium on Applications of Ferroelectrics, ISAF-04, 2004 (
IEEE
,
2004
), pp. 33–38.
15.
K. F.
Astafiev
,
V. O.
Sherman
,
A. K.
Tagantsev
, and
N.
Setter
, “
Can the addition of a dielectric improve the figure of merit of a tunable material?
,”
J. Eur. Ceram. Soc.
23
,
2381
2386
(
2003
).
16.
K.
Okazaki
and
K.
Nagata
, “
Effects of grain size and porosity on electrical and optical properties of PLZT ceramics
,”
J. Am. Ceram. Soc.
56
,
82
86
(
1973
).
17.
R.
Stanculescu
,
C. E.
Ciomaga
,
L.
Padurariu
,
P.
Galizia
,
N.
Horchidan
,
C.
Capiani
,
C.
Galassi
, and
L.
Mitoseriu
, “
Study of the role of porosity on the functional properties of (Ba,Sr)TiO3 ceramics
,”
J. Alloys Compd.
643
,
79
87
(
2015
).
18.
L.
Padurariu
,
L.
Curecheriu
,
C.
Galassi
, and
L.
Mitoseriu
, “
Tailoring non-linear dielectric properties by local field engineering in anisotropic porous ferroelectric structures
,”
Appl. Phys. Lett.
100
,
252905
(
2012
).
19.
L.
Padurariu
,
L.
Curecheriu
,
V.
Buscaglia
, and
L.
Mitoseriu
, “
Field-dependent permittivity in nanostructured BaTiO3 ceramics: Modeling and experimental verification
,”
Phys. Rev. B
85
,
224111
(
2012
).
20.
A.
Cazacu
,
L.
Curecheriu
,
A.
Neagu
,
L.
Padurariu
,
A.
Cernescu
,
I.
Lisiecki
, and
L.
Mitoseriu
, “
Tunable gold-chitosan nanocomposites by local field engineering
,”
Appl. Phys. Lett.
102
,
222903
(
2013
).
21.
L.
Carignan
,
A.
Yelon
,
D.
Menard
, and
C.
Caloz
, “
Ferromagnetic nanowire metamaterials: Theory and applications
,”
IEEE Trans. Microw. Theory Techn.
59
,
2568
2586
(
2011
).
22.
D. H.
Werner
,
D.-H.
Kwon
,
I.-C.
Khoo
,
A. V.
Kildishev
, and
V. M.
Shalaev
, “
Liquid crystal clad near-infrared metamaterials with tunable negative-zero-positive refractive indices
,”
Opt. Express
15
,
3342
3347
(
2007
).
23.
G. R.
Keiser
and
P.
Klarskov
, “
Terahertz field confinement in nonlinear metamaterials and near-field imaging
,”
Photonics
6
,
22
(
2019
).
24.
K.
Fan
,
H. Y.
Hwang
,
M.
Liu
,
A. C.
Strikwerda
,
A.
Sternbach
,
J.
Zhang
,
X.
Zhao
,
X.
Zhang
,
K. A.
Nelson
, and
R. D.
Averitt
, “
Nonlinear terahertz metamaterials via field-enhanced carrier dynamics in GaAs
,”
Phys. Rev. Lett.
110
,
217404
(
2013
).
25.
G.
Allaire
, “
Homogenization and two-scale convergence
,”
SIAM J. Math. Anal.
23
,
1482
1518
(
1992
).
26.
S.
Guenneau
and
F.
Zolla
, “
Homogenization of three-dimensional finite photonic crystals
,”
J. Electromagnet. Waves Appl.
14
,
529
530
(
2000
).
27.
S.
Agrawal
,
R.
Guo
,
D.
Agrawal
, and
A. S.
Bhalla
, “
Tunable BST:MgO dielectric composite by microwave sintering
,”
Ferroelectrics
306
,
155
163
(
2004
).
28.
M. A. H.
Ansari
,
A. K.
Jha
, and
M. J.
Akhtar
, “
Design and application of the CSRR-based planar sensor for noninvasive measurement of complex permittivity
,”
IEEE Sens. J.
15
,
7181
7189
(
2015
).
29.
L. D.
Landau
,
J.
Bell
,
M.
Kearsley
,
L.
Pitaevskii
,
E.
Lifshitz
, and
J.
Sykes
,
Electrodynamics of Continuous Media
(
Elsevier
,
2013
), Vol. 8.
30.
K.
Zhou
,
S. A.
Boggs
,
R.
Ramprasad
,
M.
Aindow
,
C.
Erkey
, and
S. P.
Alpay
, “
Dielectric response and tunability of a dielectric-paraelectric composite
,”
Appl. Phys. Lett.
93
,
102908
(
2008
).
31.
C. M.
Krowne
,
M.
Daniel
,
S. W.
Kirchoefer
, and
J. A.
Pond
, “
Anisotropic permittivity and attenuation extraction from propagation constant measurements using an anisotropic full-wave Green’s function solver for coplanar ferroelectric thin-film devices
,”
IEEE Trans. Microw. Theory Techn.
50
,
537
548
(
2002
).
32.
C.
Geuzaine
and
J.-F.
Remacle
, “
Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities
,”
Int. J. Numer. Methods Eng.
79
,
1309
1331
(
2009
).
33.
P.
Dular
,
C.
Geuzaine
,
F.
Henrotte
, and
W.
Legros
, “
A general environment for the treatment of discrete problems and its application to the finite element method
,”
IEEE Trans. Magn.
34
,
3395
3398
(
1998
).
34.
P. R.
Amestoy
,
I. S.
Duff
,
J.
Koster
, and
J.-Y.
L’Excellent
, “
A fully asynchronous multifrontal solver using distributed dynamic scheduling
,”
SIAM J. Matrix Anal. Appl.
23
,
15
41
(
2001
).
35.
G.
Bouchitté
and
D.
Felbacq
, “
Homogenization near resonances and artificial magnetism from dielectrics
,”
C. R. Math.
339
,
377
382
(
2004
).
36.
K.
Cherednichenko
and
S.
Cooper
, “
Homogenization of the system of high-contrast Maxwell equations
,”
Mathematika
61
,
475
500
(
2015
).