Application of the Intergranular Impedance Model in Correlating Microstructure and Electrical Properties of Doped BaTiO3

Microstructure properties of barium-titanate based materials, expressed in grain boundary contacts, are of basic importance for electric properties of this material. In this study, the model of intergranular impedance applied on a two-grain contact is considered. Globally, a BaTiO3-ceramics sample consists of a large number of mutually contacted grains, which form clusters. Such clusters can be presented as specific fractal formations. For each of them, it is possible to establish the equivalent electrical model and, for a defined set of input parameters, using symbolic analysis, obtain the frequency diagram. The influence of fractal structure is especially stressed. Realizing the totality of relations between cluster grain groups, their microelectrical schemes and corresponding frequency characteristics, on one hand, and the global equivalent electrical scheme and corresponding acquired frequency characteristics of BaTiO3-ceramics samples, on the other hand, we set a goal of correlating experimental results with the summing effect of microelectric equivalent schemes. The model is successfully tested on doped bariumtitanate ceramics.


Introduction
Doped barium-titanate ceramic is attracting much interest for its application as resistors with a positive temperature coefficient of resistivity (PTCR), multilayer ceramic capacitors (MLCC), thermal sensors etc [1,2].In the process of BaTiO 3 -ceramics consolidation, technological parameters like pressing pressure, initial sample's density, sintering temperature and time, as well as different dopants, essentially determine the final electrical properties of the ceramics.A slight change of a particular consolidation parameter, or the change of the dopant concentration can significantly change the microstructure, thus influencing electrical properties of the specimens.Since grain size and distribution considerably affect electrical properties of barium-titanate based materials, correlation of their microstructure and electrical properties has been investigated most extensively by numerous authors [3][4][5].It has been shown that electrical properties of non-doped and doped BaTiO 3ceramics are mainly controlled by the barrier structure, domain motion of domain boundaries and the effects of internal stress in the grains [6][7].Using variable-frequency measurements, and irrespective of the nature of the material under study, it is possible to separate the effects of electroactive components such as grain boundary and grain [8].As a result an impedance model can be used for probing the relaxational processes during charge transport within the electronic ceramics.The equivalent circuit can be envisaged as a series of grouped elements, i.e. several resistance and capacitance elements in parallel, each corresponding to a different process occurring at the electrode interface, at the grain boundaries or within the grains themselves [9].Therefore, microstructure properties of barium-titanate based materials, expressed in grain boundary contacts, are of basic importance for electric properties of this material.Usually, for electronic materials design, the microstructure of this type of ceramics can be considered to be as sketched in the cross section of  The model illustrates the polarization and the conduction contributions to the total impedance of the system, which can be represented by an electrical equivalent network consisting of three RC branches.The elements of the network may be attributed to the permittivities, conductivities, and geometrical extensions of the grain bulk (G), grain boundary regions (GB), and electrical interfaces (EI).It is well known that both the intergranular structure and electrical properties depend on the ceramics diffusion process.Therefore it is essential to have an equivalent circuit model that provides a realistic representation of the electrical properties.Having this in mind, the purpose of this article is to determine an intergranular impedance model based on intergranular capacity of BaTiO 3ceramics doped with different rare earth additives.

Experimental Procedure
The samples were prepared from high purity (>99.98)commercial BaTiO 3 powder (MURATA) with [Ba]/[Ti]=1,005 and reagent grade Er 2 O 3 and Yb 2 O 3 powders (Fluka chemika).Er 2 O 3 and Yb 2 O 3 dopants were used in the amounts to have 0.01, 0.1 and 0.5 wt% Er or Yb in BaTiO 3 .Starting powders were ball milled in ethyl alcohol for 24 h using a polypropylene bottle and zirconia balls.After drying at 200 o C for several hours, the powders were pressed into disks 7 mm in diameter and 3 mm in thickness under 120 MPa.The compacts were sintered 4 h in air up to 1380 o C. The microstructures of as sintered or chemically etched samples were observed by a scanning electron microscope (JEOL-JSM 5300) equipped with an energy dispersive X-ray analysis spectrometer (EDS-QX 2000S system).Prior to electrical measurements silver paste was applied on flat surfaces of specimens.Capacitance, dissipation factor and impedance measurements were done using an Agilent 4284A precision LCR meter in the frequency range of 20 to 10 6 Hz.The illustrations of the microstructure simulation were generated by Mathematica 6.0 software.

Results and Disscussion
The necessary step in designing BaTiO 3 -ceramics is research of the correlation between the material's microstructure, consolidation parameters and final properties.Structure research based on the application of basic methods strongly provides better understanding of dielectric properties, especially for pure and doped BaTiO 3 ceramics.These properties are mainly influenced by the region near the grain boundary and in the grain contact.This is the reason for the great importance of modeling grain geometry, grain boundary surfaces and the geometry of grain contacts.In order to establish the model of intergranular impedance for doped barium titanate, it is important to notice that microstructure properties of BaTiO 3 based materials, expressed in their grain boundary contacts, are of basic importance for electric properties of these materials.The barrier character of the grain boundaries is especially pronounced for doped BaTiO 3 materials which are used as PTC resistors.Basically two types of dopants can be introduced into BaTiO 3 : large ions of valence 3+ and higher, can be incorporated into Ba 2+ positions, while the small ions of valence 5+ and higher, can be incorporated into the Ti 4+ sublattice [11,12].Usually, the extent of the solid solution of a dopant ion in a host structure depends on the site where the dopant ion is incorporated into the host structure, the compensation mechanism and the solid solubility limit [14].For rare-earth-ion incorporation into the BaTiO 3 lattice, the BaTiO 3 defect chemistry mainly depends on the lattice site where the ion is incorporated [7].It has been shown that the three-valent ions incorporated at the Ba 2+ -sites act as donors, where the extra donor charge is compensated by ionized Ti vacancies (V " " Ti ), the three-valent ions incorporated at the Ti 4+ -sites act as acceptors where the extra charge is compensated by ionized oxygen vacancies (V .. O ), while the ions from the middle of the rareearth series show amphoteric behavior and can occupy both cationic lattice sites in the BaTiO 3 structure [13].It has been established that the Ba/Ti ratio also influences the incorporation of rare earth dopants into the barium titanate lattice [14].Investigations of the influence of Er on BaTiO 3 defect chemistry pointed out that for Ba/Ti>1 Er 3+ enters on the titanium site, and for Ba/Ti < 1, it enters on the barium site.Moreover, erbium incorporation on barium or titanium sites introduces a charge mismatch with the lattice (an aliovalent dopant) that must be charge compensated to achieve overall neutrality [15].As a result of rare-earth addition, abnormal grain growth and the formation of deep and shallow traps at grain boundaries influenced by the presence of an acceptor-donor dopant can be observed.
Our investigations showed that the microstructure of the samples doped with Er 2 O 3 or Yb 2 O 3 exhibit similar microstructure characteristics with the existence of an intergranular capacity.The samples sintered with Er 2 O 3 showed that the grains were irregularly polygonal shaped (Fig. 2a), although in Yb doped BaTiO 3 the grains are more spherical in shape (Fig. 2

.b).
For the lowest concentration, the size of the grains was large (up to 60 μm), but by increasing the dopant concentration the grain size decreased.As a result, for 0.1 wt% of dopant the average grain size was from 20 to 30 μm, and for the samples doped with 0.5 wt% of dopant grain size decreased to the value up to 10 μm as can be seen from the cumulative grain size distribution curves for doped BaTiO 3 , given in Fig 3. Spiral concentric grain growth has been noticed for the samples sintered with a low concentration of Er 2 O 3 or Yb 2 O 3. For these samples the formation of a "glassy phase" indicated that sintering occurred in a liquid phase.
EDS analysis has shown that for the small concentrations of Er and Yb, a uniform distribution can be noticed (Fig. 4a), while the increase of dopant concentration led to coprecipitation between grains (Fig. 4b).As a result, the region between the grains can be represented by an electrical equivalent network consisting of three RC branches as noted in the introduction.
All this allowed us to consider the BaTiO 3 -ceramics sample as a system with a large number of mutually contacted grains which form clusters.For each of them, it is possible to establish an equivalent electrical model and, for a defined set of input parameters, using symbolic analysis, obtain the frequency diagram.Realizing the totality of relations between clusters grains groups, their microelectrical schemes and corresponding frequency characteristics, on one hand, and a global equivalent electrical scheme and corresponding acquired frequency characteristics of BaTiO 3 -ceramics samples, on the other hand, we set a goal of correlating experimental results with the summing effect of microelectric equivalent schemes.According to the microstructures we obtained for BaTiO 3 doped with Er 2 O 3 and Yb 2 O 3 it can be concluded that the global impedance of a barium-titanate ceramics sample, which contains both the resistor and capacity component, can be presented as a "sum" of many clusters of micro-resistors and micro-capacitors connected in tetrahedral lattice.Our microstructure analysis showed that at the beginning the microstructure can be simulated as cluster of spheres or ellipsoids which later deform as a result of the densification process and contacts formation.Therefore, for the general model of the stereological configuration of BaTiO 3 the density of the sample can be defined by mutual positions of neighboring grains (Fig. 5).The developed model gives the distribution of grain contacts through the sample volume.As it can be seen from the typical spatial situation of a four grain cluster (Fig. 6), the positions of the neighboring grains can be: 1. in touching contact; 2. slightly immerging one into another and 3. not touching each other.This configuration logically leads to the tetrahedral scheme of mutual electrical influence of BaTiO 3 grains.The impedances are at each edge of the tetrahedron, as shown on Fig. 7.The vertices (in Fig. 7 displayed as small spheres) are stylized grains, while impedances contain resistance and capacity between two grains.Our new approach includes fractal geometry in describing the complexity of the spatial distribution of BaTiO 3 grains.The best fractal model is a sponge model, or and it is more correct to say, a kind of three -dimensional lacunary set (a set with voids).The structure of tetrahedral influence may be established in each spatial sense, which means that one has a tetrahedral lattice that fills the space.
This kind of fractals is sometimes identified with a deterministic construction like a Cantor set (in Ρ), Sierpinski triangle or Sierpinski square (in Ρ 2 ), with a Sierpinski pyramid or Menger sponge (in Ρ 3 ), and so on.In our case of electric impedance property, the Sierpinski pyramid (Fig. 8) might be an adequate paradigm for the first instance inquiry.The starting pyramid T 0 and the first two iterations, shown in Fig. 7, give the initial part of the orbit of the so called Hutchinson operator W, that is T 1 = W(T 0 ), and T 2 = W(T 1 ) = W 2 (T 0 ).The boundary case, T = W ∞ (T 0 ) is an exact fractal set with a Hausdorff dimension D H = 2.Some of the impedances in this lattice are missing following the lacunary fractal pattern.According to [16,17], the fractal dimension of a typical ceramics sample is estimated to be about 2.087.So in this case, the fractal dimension of BaTiO 3 -ceramics is slightly below 3.The higher values of fractal dimensions can be obtained for a denser BaTiO 3 -ceramics.The dimension close to 3 means that not many impedances are lacking in the tetrahedral lattice.On the other hand, dimensions closer to 2.5 alert to a high presence of pores which means that many impedances are to be swept in the model of the tetrahedral impedance lattice.What is sure is that BaTiO 3 sintered ceramics can be considered as a lacunar fractal rather than a percolation fractal or succolar fractal [18].
In order to calculate equivalent impedance for a wide frequency range, the equivalent electrical circuit for a ceramic material can be introduced as a impedance containing two capacitances C and C p , an inductance L and a resistance R.
The dominant electrical parameter of our model is the capacitance C. The connection between C and geometrical and/or structural properties can be established by an assumption that the contact region can be viewed as a planar micro-capacitance.Fig. 10.shows a geometric model of two spheres in contact, where r c is the radius of the spherical particles, and x is the neck radius.Since the spherical shape and the circular neck profile are assumed, the contact surface is the circle of the area πx 2 .Because of center-to center approach, the dielectric thickness 2h is a function of d (the distance between centers of particles) according to the relation 2h=2r c -d.Therefore, the capacitance can be written as: where ε o and ε r are the dielectric constants of vacuum and the ceramic material, respectively and α is a correction factor obtained by a constructive approach to the fractal structure.Taking into account that by the fractal theory α can be presented as: where D≈2.08744 is the fractal (Hausdorff) dimension of the intergrain contact surface and D T =2 is the topological dimension of the surface, it can be concluded that for BaTiO 3 doped ceramics contact surfaces are of low-irregularity which is characterized by the small difference D -D T ≈0.08744.Taking this into account, calculations of micro-capacitance generated in grains contacts of BaTiO 3 doped with Er 2 O 3 and Yb 2 O 3 have been performed (fig.11).In order to calculate equivalent impedance of the sample, conductance as another dominant electrical parameter should be taken into account.It is a parasitic parameter that is given in terms of capacitance with tanδ as a measure of losses, i.e., 1/R=G=ωC⋅tanδ.The intergranular impedance model also contains two additional parameters; inductance L, and capacitance C p .Their nature cannot be correlated with geometrical parameters of grains in general way.
In order to determine an algebraic equation describing equivalent intergranular impedance in terms of circuit parameters the following equation can be used: Based on the proposed equivalent circuit and the theory of impedance analysis for the model of three aggregate spheres the equivalent impedance can be defined by: ( ) where Z 12 , Z 13 , Z 23 are the intergranular impedances between two adjacent particles.Then, this model can be inserted for any contact region inside the multi-particle model system during its microstructure development.Thus, electrical properties are determined in general by a series combination of such impedances.

Conclusion
Understanding the electrical properties of barium-titanate materials is important for modern devices applications and presents a challenge for simulation.In this study, the model of intergranular impedance is established using the equivalent electrical scheme characterized by a corresponding frequency characteristic.According to the microstructures we have obtained for BaTiO 3 doped with Er 2 O 3 and Yb 2 O 3 the global impedance of a barium-titanate ceramics sample contains both a resistor and capacity component.The resistor and capacity component was presented as a "sum" of many clusters of micro-resistors and micro-capacitors connected in the tetrahedral lattice.The positions of neighboring grains for the four grain cluster have been defined and according to them the tetrahedral scheme of mutual electrical influence of BaTiO 3 grains has been established.Fractal geometry has been used to describe the complexity of the spatial distribution of BaTiO 3 grains.The model of impedances between clusters of ceramics grains has been presented and calculations of micro-capacitance generated in grain contacts of BaTiO 3 doped with Er 2 O 3 and Yb 2 O 3 have been performed.By controlling shapes and numbers of contact surfaces on the level of the entire BaTiO 3 -ceramic sample, control over structural properties of this ceramics can be done, with the aim of correlation between material electronic properties and corresponding microstructure.

Fig. 1 .
Fig. 1.Sketched cross section of a ceramic sample with electrodes applied (a), corresponding simplified brick wall model (b), and the equivalent electrical network

Fig 5 .
Fig 5. Simulation of the 1000 grains of BaTiO 3 obtained from the general model of the stereological configuration.

Fig. 6 .
Fig. 6.Positions of neighboring grains for the four grains cluster.

Fig . 10
Fig .10Two-sphere contact model presented in the plane section.