Temperature dependent tunability data and modelling in the paraelectric Ba 0 . 7 Sr 0 . 3 TiO 3 solid solutions

The fi eld dependence of the dielectric constant (dc-tunability) as a function of temperature in (Ba,Sr)TiO3 ceramics was investigated. Dense and homogeneous BaxSr1-xTiO3 ceramics prepared by Pechini method show low losses, high dielectric constant of ~4000 and typical ferroelectric properties at room temperature. The dctunability at various temperatures, including the Curie range, was obtained. A single polarization mechanism was used in the polar state of the ceramics to describe the tunability data. A good correlation between theory and experiment was found. In the paraelectric phase, a model of random uncorrelated non-interacting dipolar units in a double well potential was considered.


I. Introduction
(Ba,Sr)TiO 3 (BST) solid solutions are basic ceramic materials for microwave devices in wireless communications (fi lters, phase shifters, voltage-controlled oscillators, etc.) [1].Their functional properties have been intensively studied in relationship with their preparation technique and processing parameters, composition and microstructural properties, in single-crystal form, ceramics, epitaxial, polycrystalline fi lms or heterostructures [2][3][4][5].In spite of extensive studies dedicated with the BST solid solutions, the basic physics associated to the functional properties is still of large interest and insuffi ciently understood.For applications in microelectronics it is necessary to have low losses and a strong variation of permittivity with the applied fi eld ("tunability").High permittivity and tunability are obtained in the ferroelectric phase, because these properties are related to the ferroelectric non-linearity.The main disadvantage of using the ferroelectric phase is the hysteretic behaviour that is not long-expected in such applications.For this reason, it was suggested to use the paraelectric phase around the phase transition, where permittivity is high, losses are low and tunabi lity is high.The study of a system around its phase transition is also interesting because many non-linear components will contribute to the total polarization.
Taking into account these considerations, Ba x Sr 1-x TiO 3 ceramics were investigated in this work.An optimum of the desired properties were obtained for the composition x = 0.30.For this concentration, the temperature dependence of the dc-tunability was investigated.The experimental data were analysed using a model of random uncorrelated non-interacting dipolar units in a double well potential.

II. Experimental
The ceramic samples were prepared by pressing the pure perovskite powders followed by sintering in air at 1350°C/3h.The powders were prepared by a polymerized complex method based on the Pechini-type reaction route, as described in detail in a previous work [6].The phase purity of the perovskite powders and sintered ceramics was checked by X-ray diffraction measurements performed with a Shimadzu XRD 6000 diffractometer using Ni-fi ltered CuKα radiation (λ = 1.5418Å) with a scan step increment of 0.02° and with a counting time of 1 s/step, for 2θ ranging between 20-80°.
The electrical measurements were performed on the parallel-plate capacitor confi guration.Gold electrodes were applied by rf-sputtering on the polished surfaces of the sintered ceramic disks.The complex impedance in the frequency domain 1-10 6 Hz and for temperatures below 200°C was determined by using an impedance analyzer (Solartron, SI 1260).For obtaining accurate tunability data, a circuit was designed and realized [7], in which the high voltage was obtained from a dc-function generator coupled with a TREK 30/20A-H-CE amplifier.The ceramic pellets were placed in a cell containing transformer oil to avoid breakdown at high voltages.

III. Results and discussion
X-ray diffraction patterns obtained for the BST ceramics show single-phase composition with the cubic symmetry at room temperature [8].The SEM image shows a good densifi cation, with a bimodal grain size distribution consisting in exaggerated grown grains with size of ~ 40-50 μm and smaller grains of ~ 3-5 μm (Fig. 1).The temperature dependence of the relative permittivity and the dielectric losses at a few frequencies are presented in the Fig. 2. BST ceramics have excellent dielectric properties, with high values of the permittivity at room temperature (~ 4000) and low loss-es in the range 25-200°C (tan δ < 6%).As expected from the literature data, the ferroelectric BST with the composition x = 0.30 has a Curie temperature around T C = 20°C and is in its paraelectric state at room temperature [9,10].The present dielectric data also confi rm these properties (Fig. 2).
The dc-tunability was determined at several temperatures around the ferroelectric-paraelectric phase transition and shown in the Fig. 3.A strong nonlinearity of the ε(E) dependence is observed, with a tendency towards saturation for fi elds of ~ 20 kV/cm.At room temperature, the sample shows high tunability, n = ε(0)/ε(E) of ~ 1.60 for E ~ 20 kV/cm as it can be seen in Fig. 3.For observing the infl uence of temperature, the dc-tunablity data in range of temperature from 20°C to 70°C were determined (Fig. 4).With increasing temperature, the tunability of BST decreases almost linear from n = 2.6 at 20°C to n = 1.2 at 70°C.
In spite of its paraelectric character, the BST ceramics still show high tunability.One might ask what the origin of this strong non-linear response in the nonpolar state of the BST system is.In the ferroelectric state only one polarization mechanism contribution can be considered, while for the paraelectric state, an overlapping of polarization mechanisms are rather expected, related to: (i) the existence of the dipolar nanodomains (versus individual dipoles or spins) and (ii) the presence of some degree of correlations among domains and co-operative freezing of the orientation degrees of freedom below a critical temperature.
For describing the complex polarization process in such system, a simple model considering random noninteracting dipolar units in a double-well potential is considered [11].It was assumed that N = N 1 +N 2 is the total number of dipolar units, where N 1 and N 2 are the number of the occupied wells 1 and 2, respectively, and the relative population of the double wells under the fi eld E was calculated.The resulted polarization P of the system of volume V depends on the number of not (1) Taking into account this polarization-fi eld dependence, the permittivity and tunability at a fi xed temperature are calculated as: (2) from which a logarithmic dependence is obtained: (3) It is worth to mention that this model is valid in the situation when the correlations between the neighbouring dipolar units are negligible.If the model well describes the permittivity-fi eld data for the paraelectric state, a linear dependence of the function should be obtained.The tunability data for the BST ceramics were arranged and represented according to the equation (3).Linear fi ts according to this function were performed for different tempe ratures, as shown in the Fig 5 .For fi elds below 20 kV/cm, the agreement between the equation (3) and the experimental data is very good (Fig. 5), allowing to determine the temperature dependence of the average value of the dipole moment p 0 by linear regression.Using this model, the sizes of fi eld-orientable nanopolar regions can be estimated to be in the range of 7.6-8.8nm for temperatures of 20-70°C.A more detailed study in a large range of temperatures in the paraelectric state is expected to give information on the temperature range of stability of the nanopolar domains in Ba 0.7 Sr 0.3 TiO 3 ceramic.

IV. Conclusions
The tunability of Ba 0.7 Sr 0.3 TiO 3 ceramics in the paraelectric state was investigated as a function of tempera-ture.The experimental data were analyzed considering a simple model with fi eld re-orientable non-interacting dipolar units.Using this model, the temperature dependent average dipole moment can be determined and the size of nanopolar regions was estimated in the range of ~7.6-8.8 nm for temperatures of 20-70°C.