Homogenization of periodic and random ferroelectric-dielectric composites

We investigate the homogenized permittivity of ferroelectric-dielectric mixtures 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. Periodic and random structures in two dimensions are investigated and we study the effective permittivity, losses, electrically induced anisotropy and tunability of those metamaterials.


I. INTRODUCTION
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 components 2-4 and metamaterials 5,6 . The main reason of 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). 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 = (n − 1) 2 /(n tan δ(0) tan δ(E)), where tan δ is the loss tangent.
These materials have usually high permittivity values, 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 to low-index and low-loss non-tunable dielectrics in order to reduce both permittivity values and losses. The effective parameters of those composites have been investigated 7-10 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 lead to a small increase of the tunability of the mixture. In the context of porous ferroelectrics, the homogenized properties strongly depends on the size and morphology of the pores 11,12 . In addition, the influence of local field enhancement in the tunability has been shown to be of particular importance 13-15 . This study investigates the effective permittivity of composites made of dielectric inclusions in a ferroelectric matrix by using a two-scale convergence method 16,17 . 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. As compared to earlier sudies in the litterature 13,14 , we account for the non-linear coupling beyond the first iteration and use two-scale convergence homogenization analysis to obtain the effective parameters at higher frequencies, instead of a capacitance-based model. We first study metamaterials consisting of a squarre 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.

II. THEORY AND NUMERICAL MODEL
We consider a composite made of a ferroelectric material with anisotropic permittivity ε f (E) that is dependent on an applied electric field E, and a non tunable dielectric of permittivity ε d , which are both non-magnetic. 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. Modelling 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 this approach as to 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.

A. Permittivity model
We use barium strontium titanate (BST) as our ferroelectric material. Measurements have been carried out at electrostatics and at microwave frequencies, and the normalized permittivity value as a function of biasing field are reported on Fig. (1).
To describe the permittivity, we make use of the Landau potential given by F (P, E) = F 0 +aP 2 /2+ bP 4 /4 + cP 6 /6 − EP , where E is the applied electric field and P is the polarization 18,19 . Variations of the permittivity with the temperature can be taken into account through the coefficients a, b and c, but we assume 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  Table I. gives the dependence of the polarization on the applied electric field, with P 0 being the equilibrium polarization. Along the direction of a uniform applied electric field, the relative permittivity is given by: where ε f (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 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 20 , we have: where each of the diagonal components have the functional form given by Eq.
(1). Note that we will use the static values of permittivity for the electrostatic modelling, while we are interrested in the homogenized values of permittivity at microwaves.  Fig. (1).
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 ε f (E(r)) in the ferroelectric phase and diag(ε d ) 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, ε zz 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 E 0 = E 0 e z . On the other hand, the in-plane components of ε f are tuned by E x and E y , therefore, without loss of generality, we consider a uniform applied electric field directed along the x axis E 0 = E 0 e x for the TM polarization case. To calculate the total electric field in the material, one has to solve the electrostatic equation for the potential V : 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 emphasise that the permittivity in the ferroelectric material, although uniform initially, is spatially varying due to the non-uniform distribution of the total electric field.

C. Homogenization
The effective permittivity for TM polarization is calculated using two scale homogenization technique 16,17 . For this purpose, one has to find the solutions ψ j of two annex problems P j , where r = (x, y) T is the position vector in the xy plane and ξ = ε T /det(ε). The homogenized tensorξ is obtained with:ξ where . denotes the mean value over the unit cell. The elements of the matrix φ represent correction terms and are given by φ ij = ξ∇ψ i j . Finally the effective permittivity tensor can be calculated usingε =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 .

III. NUMERICAL RESULTS
In the following numerical results, the dielectric phase is supposed to be lossless and non dispersive with ε d = 3 while the ferroelectric material follows the permittivity described in section II A and has a constant loss tangent tan δ f = 10 −2 .

A. Two dimensional periodic metamaterial
Lets us now consider a periodc square array of infinitely long dielectric rods of 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 We computed the effective parameters of these metamaterial structures for different radii of the rods and studied their behaviour 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. In the case studied here, 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. 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. 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 is local field induced effect is what makes the anisotropy stronger in our coupled model compared to the uncoupled one (cf. Fig. (3.d)). 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.

B. Random case
We finally study the effect of random particle distribution 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 centre is chosen randomly and the rods are allowed to overlap. An example of distribution for f = 0.5 is given on Fig. (6). The effective material properties are plotted on Fig. (4).
Similarly 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 can be greater than the bulk tunability.
However, at higher applied electric fields, normalized tunability becomes smaller than unity and is The redistribution of electric field, permittivity and convergence of the effective parameters are displayed in Fig. (5). The effect of disorder plays an important role here: the electrostatic field gets concentrated in between neighbouring inclusions and the smaller the gap the higher the field, hence a greater local permittivity change. In addition, even if the distribution of particle is random, one expects that the anisotropy due to geometry would cancel for a sufficiently large number of particles (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 ε xx and ε yy 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.

IV. CONCLUSION
We have studied the homogenized properties of dielectric/ferroelectric mixtures using a rigorous model that take into account the coupling between the electrostatic field distribution and the field dependant ferroelectric permittivity tensor. After convergence of the coupled problem, the effective permittivity tensor is calculated using two scale convergence homogenization theory. The results obtained by this model differ significantly from a simple assumption that the permittivity of the ferroelectric respond 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 disorder is introduced.
The assymetric redistribution of the permittivity induce en 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 suggest 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 theory 24,25 to study this kind of mixtures. This would reveal the frequency dependant artificial magnetism due to "micro-resonances" in the high index phase and potentially lead to composites with tunable effective permeability.