A resource efficient design strategy to optimise the temperature coefficient of capacitance of BaTiO3-based ceramics using finite element modelling

An in-house finite element modelling package is used to simulate the electrical response of core–shell microstructures of BaTiO3-based (BT) dielectric materials as a design strategy for multilayer ceramic capacitors. By combining this method with experimental material properties, both the temperature coefficient of capacitance (TCC) and the volume ratio of undoped BT ceramic (core phase) to chemically doped BT ceramic (shell phase) to optimise temperature stability can be predicted. This is a faster, more reliable and efficient design strategy than current techniques which are based on iterative experimental protocols. This methodology is illustrated using rare earth-free NaNbO3-doped BT (NNBT) ceramics as an example for which we predict and experimentally confirm an optimised volume fraction ratio of 0.66 undoped BT core to 0.34 of a 2.5% NNBT shell.


Introduction
Driven by the rapid growth in the number and type of commercial electronic devices, the need for multilayer ceramic capacitors (MLCC) is increasing and at present exceeds 2 trillion units per year. 1 Ferroelectric BaTiO 3 (BT) is the material of choice for such devices due to its high permittivity ($1700 at room temperature, rising to a sharp maximum of $10 000 at the Curie temperature, T c of $120 C). However, the strong temperature dependent permittivity needs to be modied to meet industrial requirements. For example, a AE 15% temperature variation in capacitance (and therefore permittivity) is required to meet X7R (À55 to 125 C) and X8R (À55 to 150 C) standards. 2 To achieve such standards, BT is chemically doped to improve the temperature stability of the permittivity of the MLCC, thereby decreasing the magnitude of the Temperature Coefficient of Capacitance (TCC). A wide range of additives and dopants to BT have been considered with a particular focus on intermediate sized trivalent ions such as Y 3,4 and rare earths. [5][6][7][8] One rare earth used frequently is Dy due to its ability to dope onto either or both the Ba and Ti sites. 9 Ceramic processing is designed to create limited diffusion of these elements into BT grains and this heterogeneous distribution creates a 'coreshell'-type ceramic microstructure. 3,4,[10][11][12][13][14][15] In its simplest form, the sub-grain microstructure consists of a grain-core based on un-doped BT with T c $ 120-130 C surrounded by an outer-shell grain region with a distribution of dopants that suppress and smear out T c to lower temperatures. Core-shell microstructures therefore, create a much broader permittivity-temperature prole as compared to un-doped BaTiO 3 ceramics. The overall permittivity-temperature prole depends on many parameters, including; grain size, dopant concentration and distribution and volume fraction of core and shell regions. For a given grain size and dopant concentration, controlling dopant distribution by ceramic processing (e.g. varying heating/cooling rates, sintering temperatures and dwell times) are employed to create different volume fractions of core and shell regions to systematically modify the permittivity-temperature prole of BT-based ceramics. 16 Thus, TCC can be controlled over a sufficiently wide temperature range to satisfy industrial standards via optimisation of appropriate core-shell volume fractions. 13,16 As TCC is controlled by a combination of the material properties and physical microstructure, this poses signicant challenges in both the fabrication of devices and in their characterisation post fabrication. For example, state-of-the-art MLCCs based on BT ceramics oen consist of more than 500 dielectric layers 17 that range from 0.8-2.0 mm thick (corresponding to only $5-15 BT grains per layer) interleaved by parallel layers of nickel electrodes.
Characterisation of core-shell microstructures have been attempted using several methods. In a recent study, Jeon et al. 16 showed an optimised ratio for X7R existed in their system for a shell thickness (Y-doped) of approximately one third of the core (BT) radius. Microstructural analysis of the core and shell volume fraction was performed using analytical transmission and scanning electron microscopy. Core and shell volume fractions have also been measured in commercial positive temperature coefficient of resistance (PTCR) thermistors based on BaTiO 3 . 18 Analysis of impedance spectroscopy data using complex electric modulus, M*, plots and spectroscopic plots of the imaginary component of the electric modulus, M 00 , suggested shell regions are approximately one h of the thickness of the core regions. This estimate was conrmed by conductive atomic force microscopy that revealed Schottky barriers $500 nm wide for an average grain size of $5 mm. 19 As the demand for the number of BaTiO 3 -based MLCC increases, reducing rare-earth dopant concentration is becoming a necessity based on scarcity, geopolitical uncertainty and cost. The trial and error experimental approach to coreshell volume faction optimisation based on various compositional formulations and ceramic-processing conditions is time and resource intensive, as are the analyses required to establish core-shell volume ratios in ceramics. Here, we demonstrate a new strategy of functional materials design using a combination of nite element modelling (FEM) and experimental data. First, we use experimentally measured permittivity-temperature proles from a range of BT-based ceramics to provide input to an FEM model. This is used to assist and guide the design process by identifying the optimised volume ratio of un-doped BT ceramics (core phase) to doped ceramic (shell phase) to obtain the lowest magnitude of TCC. Finally, ceramics based on predicted volume fractions are fabricated to verify the FEM results. This methodology is faster, more reliable and resource efficient than current techniques which are based purely on an iterative experimental protocol. Novel rare earth-free NaNbO 3doped BaTiO 3 (NNBT) ceramics are used to illustrate this new methodology.

Modelling methodology
The modelling uses an in-house developed nite element package, ElCer, 20,21 to simulate the response of an electroceramic material. This incorporates a time domain nite element method (TDFEM) to solve Maxwell's equations in space and time. A powerful aspect of this approach is that it permits the electronic response to be calculated from a combination of material properties (conductivity, permittivity) and the ceramic microstructure of the system, including grain boundaries, grain cores and grain shells of various shape and size. The system is meshed using a combination of tetrahedra and prisms, solved and analysed for the electrical response. Each electrically active region, such as a grain or grain boundary phase, can be assigned its own unique time constant (i.e. conductivity and permittivity) allowing the ability to create both microstructural and electrical heterogeneity within ceramics. This model can then be solved for the electrical response and the impedance data simulated using ElCer without the requirement of an equivalent circuit consisting of some combination of resistors, capacitors and/or constant phase elements.
To use the FEM method for optimising the TCC for multilayer capacitor materials, we propose a novel resource efficient strategy. We rst design a microstructure to mimic that found experimentally, namely a volume fraction of core and shell phases. We assign a permittivity to each phase based on experimental data obtained from ceramics of individually fabricated phases. FEM is then used to simulate the total capacitance of the system and converted, using the geometry of the model, to obtain the effective permittivity. This is repeated, using the temperature-permittivity prole for the individual core and shell phases, replicating the response of the system over a dened temperature range and converted to TCC by normalising the permittivity data using eqn (1) where the change in temperature DT is relative to 25 C and D3 is the change of relative permittivity 3 r again relative to the value obtained at 25 C and denoted by 3 25 . Finally, this is repeated for different volume fractions as well as various shell proles to identify the ideal combination of phases to optimise TCC. To accomplish this requires two key features. The rst is a microstructure that represents a core and shell granular system with grain boundaries. The second is the material properties, specically (in this case) the permittivity, as a function of temperature.

Experimental methodology
A series of samples based on The dried sample was then sieved through a 200 mm mesh and the resulting powder reacted in an Al 2 O 3 crucible for 6 h at 1140 C. The powder was then ball-milled, dried, sieved again and pressed into 10 mm diameter pellets in a uniaxial press at $0.3 tonne for $1 min. Green pellets were then placed on Pt-foil in an Al 2 O 3 crucible and sintered in air for 8 h at 1400 C. Ceramics were coated with Au-paste electrodes and placed in an in-house designed rig that was inserted into a programmable tube furnace. The permittivity-temperature proles were obtained using an LCR meter (Agilent E4980 Precision LCR Meter, Agilent Technologies) and the results shown in Fig. 1 for a xed frequency of 100 kHz with an applied ac voltage of 100 mV. Data points were collected every 60 s from room temperature to 200 C using a non-inductively wound tube furnace at a rate of 1 C min À1 . All data were corrected for sample geometry.

Modelling generation and simulation
Previously we have shown that grain shape can inuence the impedance response of ceramics. 21 To accommodate such microstructural effects, we incorporate into the design a realistic grain size and distribution that includes a unique irregular shape for each grain but with a well-dened core-shell volume fraction, each with specied material properties (permittivity and conductivity).
To generate the microstructure, a Voronoi tessellation method is employed. This creates a set of surfaces from a collection of seed points that can be used to describe a granular structure. Previously, we have distributed these points onto a cubic array to generate brick-shaped grains and included random uctuations to their position to form irregular shaped grains. We now extend this moving towards a more realistic granular shape by placing the seed points on a body centred cubic structure. When the Voronoi tessellation algorithm is run, these points form a collection of tessellated truncated octahedra. A variable is included to add a distribution to the grain shape and volume by modifying the position of the points by a small amount. Using this method, we generate 341 points, ensuring the average spacing creates grains approximately 500 nm in diameter with a realistic Gaussian distribution. This is consistent with state-of-the-art MLCCs and the particle size of the starting BT powder. To generate the core-shell volume fraction, the seed point placement is used as the centroid of the associated grain. The associated surface of the grain is then copied and shrunk to create an inner surface that represents the core. This shrinking can then be controlled to obtain the desired volume fraction of the two (i.e. core and shell) phases. Finally, we add between the grains a thin region of 10 nm assigning it the properties of a grain boundary.
Once designed, GMSH 22 is used to mesh the structure. A convergence study was performed and established that using over 30 000 nodes and 150 000 individual tetrahedral elements provided convergence through all systems presented here. The full microstructure required is shown in Fig. 2. These structural parameters allow the simulation to be generated in <1 h on a desktop computer with optimised convergence and reliability of the solution. To test the optimisation methodology, we simplify the material properties by studying the effect of combining a core of BT with a shell of doped NNBT on the system's capacitance as a function of volume fraction and temperature. The temperature window over which to optimise was chosen as 25 to 150 C however, it should be noted that this can easily be modied to identify any TCC temperature window. The core and shell are assigned a temperature dependent permittivity extracted from Fig. 1. We set the values of conductivity for undoped BT and the doped NNBT at an arbitrary value of s BT ¼ 100 mS m À1 and the grain boundary to be s gb ¼ 0.1 mS m À1 . These values ensure the current ows through the grains via the grain boundaries (i.e. brick work layer model) as opposed to percolating down the grain boundaries. These values are assumed to be independent of doping level and temperature. We also choose the grain boundary permittivity to be xed at 3 r ¼ 100 and also constant as a function of temperature consistent with previous simulations. 20,21 As ceramics contain thousands of individual grains sintered together, the electrical measurement of capacitance that is experimentally determined represents the average response of all grain shapes and sizes. To ensure the simulation results are independent of any statistical anomalies generated by the relatively limited number of irregular shaped grains, each volume ratio was generated ten times and the results averaged.

Optimising TCC
To optimise the TCC of an MLCC, we rst generate different volume fractions combining each of the differently doped materials with a core material based on un-doped BT. We dene a core volume fraction V f as the ratio of the core volume (undoped BT) to that of the total volume of the grain. For example, V f ¼ 1 indicates a system that is made only of un-doped BT. As the core is shrunk, maintaining the total volume of the grain, we begin to surround the core by a doped material (shell phase). When the volume of core and shell phases are equal this corresponds to V f ¼ 0.5. This continues to the point where V f ¼ 0, which denotes the system is made entirely of doped material.
For each system, the electrical response for the collection of grains is simulated and the DC capacitance (i.e. at a frequency ¼ 0 Hz) calculated. This is converted to a relative permittivity, 3 r , of the system, using the geometrical factor (thickness/electrode area) of the sample. As shown in Fig. 3, for each of the Fig. 2 The complete microstructure of the system incorporating a two-phase core shell granular structure with associated grain boundary. Here, we show equal volumes of core and shell. (a) Representation of an individual meshed grain highlighting the core and shell regions and diameters required to generate equal volume fractions whereas (b) and (c) show a slice plot through the full microstructure highlighting the 341 individual grains each with a core and shell defined by a user defined volume fraction.
combinations of a BT core surrounded by either a 1, 2.5, 5 or 10% NNBT doped shell material a different response is generated. Some combinations of core and shell phases do not lead to at 3 r -temperature proles, Fig. 3(a) and (d) while others are more promising, Fig. 3(b) and (c). An example of a doped material (shell phase) not assisting in a smearing of 3 r versus temperature is highlighted when combining a 1% NNBT doped material surrounding a BT core, Fig. 3(a). The doped material has a large peak in permittivity at 100 C due to a decrease in T c associated with Na-and Nb-doping in contrast to BT with T c of $125 C. As single-phase materials, corresponding to V f of 1 (all core phase) and 0 (all shell phase), the high permittivity associated with each phase dominates, leading to a poor permittivity prole for any potential MLCC composition. Combining these two compositions does not enhance the prospects for any potential MLCC usage because at intermediate V f the increased permittivity of both phases blend together but maintain signicant peaks in 3 r at their respective Tc's. This is in contrast to using a 2.5% NNBT material where a well-dened T c does not exist, instead there is a broad 3 r -temperature prole with a maximum at $60 C, Fig. 3(b). When mixed with a BT core such that V f ¼ 0.50, a much atter 3 r -temperature prole is obtained, e.g. triangle symbols in Fig. 3(b).
To optimise V f of the two phases, 3 r is converted to TCC and the TCC-temperature proles shown in Fig. 4. The desired standard TCC criteria, e.g. X7R can then be overlaid to see if any of the material combinations meet the required performance. This optimisation methodology follows the experimental route to identify the best combination of materials however, it is limited since it depends on the number of volume fractions created and measured. Using FEM, we can quickly and systematically simulate many material (phase) combinations and volume fractions thus going beyond the conventional time and resource intensive experimental methodology required to optimise the TCC response in BT-based ceramics.
The ideal combination of core and shell phases would lead to a perfectly at response with the integral of TCC being as close to zero as possible over the measured temperature range. Realistically TCC will not be either positive or negative but will uctuate around TCC ¼ 0. To correctly identify the optimised V f , we integrate the absolute value of TCC from 25 to 150 C and plot it as a function of V f of un-doped BT, Fig. 5. This new optimisation route indicates that for a 1% doped NNBT material the response is an almost linear decrease between the two different phases. There is no performance enhancement to be obtained by combining undoped BT with a 1% doped NNBT material and even un-doped BT out performs any combination of these two phases.
For 5% and 10% NNBT doped materials, a minimum in the integrated TCC is observed and can be identied as V f $ 0.75 of un-doped BT however, the best candidate to produce the attest   integrated absolute value of TCC is combining 2.5% doped NNBT with un-doped BT with V f $ 0.66. Based on an average grain size of 500 nm, this corresponds to a $31 nm thick dopedshell surrounding an un-doped BT core of diameter $437 nm.
We use a combination of both modelling and experimental studies to verify this prediction. Firstly, using the same methodology as described above, we simulated a multi-granular model, where each grain has a core-shell structure of BT surrounded by a doped 2.5% NNBT shell set at V f ¼ 0.66. As shown in Fig. 6, this combination leads to a atter TCC response. Although the simulation predicts a higher 3 r of $3500 compared to that of $3000 from the experimental data obtained from the ceramic, there is excellent agreement in the temperature prole of the response. The observed change in absolute permittivity here is associated with extrinsic (experimental) parameters such as porosity, surface roughness, electrode contact problems and grain boundaries as opposed to any inaccuracies of our simplied model used to assess TCC.
More signicantly, the experimental fabrication at this V f shows there is only a maximum change of AE10% in TCC from 25 to 125 C, highlighting this system is a possible candidate (based solely on TCC) for the upper operating temperature of X7R devices. However, it would be unsuitable for X8R devices due to the large negative deviation in TCC (>À50%) obtained between 125 and 150 C. Sub-ambient experimental 3 r data (down to À55 C) for the ceramics would need to be collected to test if the variation in TCC is within the AE15% limit required for X7R specication. Moreover, TCC is only one of several materials requirements for BT-based MLCCs and other parameters need also to be screened, e.g. dielectric loss, breakdown, degradation/ fatigue and compatibility with base metal electrodes such as Ni.
Nevertheless, and in conclusion, the combined FEM-experimental approach described above represents a new and resource efficient design strategy to optimise the TCC of BaTiO 3based ceramics. Other dopant systems are now under investigation and are being tested to verify this design strategy with the longer term aim of producing new, rare-earth free formulations with suitable TCC behaviour for future generations of BT-based MLCCs. Fig. 6 Overlaid FEM simulation models and experimental data of temperature profiles for (a) 3 r and (b) TCC. Excellent agreement is obtained showing the prediction of combining a BT core with 2.5% doped NNBT with V f ¼ 0.66 provides potentially useful TCC behaviour.