OPTIMAL DESIGN OF FERROELECTRIC CERAMICS MICROSTRUCTURE

OPTIMAL DESIGN OF FERROELECTRIC CERAMICS MICROSTRUCTURE K. P. Jayachandran, jaya@dem.ist.utl.pt J. M. Guedes, jmguedes@ist.utl.pt IDMEC, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal H. C. Rodrigues, hcr@ist.utl.pt IDMEC, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal Abstract. Piezoelectricity (or electric-field-induced strain) in ferroelectrics has found extensive applications in sensors and actuators and offers great potential for next generation high density storage devices such as NvRAM. Ceramic ferroelectrics which can be manufactured at a fraction of the cost of single crystals, which are prone to deficiencies such as depolarization and chemical inhomogeneity, are attracting keen interest lately. Recent reports on ceramic BaTiO 3 suggest that grain-orientation in a flux of random grain boundaries could greatly enhance the piezoelectricity. These findings underscore previous notion of the role of grain boundaries in the control and design of ceramic FE materials. Piezoelectric properties in ceramics can be optimized by a proper choice of the parameters which control the distribution of grain orientations although this choice is complicated and it is impossible to analyze all possible combinations. In this work we have implemented a finite element based computational homogenization model, characterizing ferroelectric properties, together with a stochastic optimization technique of simulated annealing to solve the optimization problem. We show that there could be an optimum choice of distribution parameters available at which the ceramic material shows better piezoelectric performance than its oriented single crystal counterpart. Keywords: Ferroelectrics, Ceramics, Optimization, Computational 1. INTRODUCTION The spatial configuration of crystallographic grains and their orientation distribution (texture) has a vital role in the piezoelectric anisotropy exhibited by conventional as well as new generation ferroelectric (FE) polycrystals and thin films (Scott, 2007). Ferroelectrics display extraordinary physical behaviors that make them crucial for many devices such as sensors and actuators and have been extensively studied for their applications of nonvolatile and high-speed random access memories (Uchino, 2000; Eerenstein et al., 2006). It is well established that some FEs in single crystal form display enhanced piezoelectricity when poled along a nonpolar direction (Park and Shrout, 1997; Wada et al., 2005). Nonpolar in the sense that a direction other than the spontaneous polarization direction. An as-grown polycrystalline FE is an aggregate of grains with randomly oriented polarizations. This randomness in polarizationvector orientation renders the resultant piezoelectricity of the material to be marginal or zero. Albeit the resultant polarization is zero for as-grown ceramic, overall piezoelectricity can be enabled by the application of an external electric field called poling field. Nevertheless, if we make use of this randomness judiciously in the design of FEs, it generates polycrystals (ceramics) with tailor-made configurations of grains useful for applications (e.g., Jayachandran et al. 2008). Recent reports (Wada et al, 2007) showing enhancement of macroscopic piezoelectricity in ferroelectric polycrystals by the introduction of randomness in texture offer vast prospect in material optimization. As stated above, the aggregate texture (orientation distribution) of an unpoled polycrystal can be assumed to have a uniform random distribution. With the strength of the poling field increases, the nature of the distribution becomes Gaussian or normal (Ruglovsky et al., 2006; Uetsuji et al., 2004). To arrive at an optimum texture of the ferroelectric polycrystal at which the material exhibits maximum piezoelectric performance, a global optimization method has to be employed. Stochastic optimization techniques like simulated annealing (SA) are quite suitable in this respect as the objective function is not sensitive to the starting point of the iterative process (Kirkpatrick et al., 1983; Sonmez, 2007). Besides being insensitive to the starting point, SA can search a large solution space and they can escape local optimum points thanks to the freedom for occasional uphill moves. Kirkpatrick et al. (1983) first proposed simulated annealing as a powerful stochastic optimization technique. The complex structure of configuration space is treated analogous to the state of material controllable by an adjustable parameter, the temperature, in simulated annealing. In other words, annealing is a strategy by which an optimum state can be approached by controlling the temperature. Annealing involves heating the material matrix to high temperature and then let it be cooled slowly so that at each step a near thermal equilibrium is achieved and finally render the material to a stable minimum energy crystalline (ordered) state. The idea to explore analogy of the annealing used in solid state physics with the optimization problems gives rise to simulated annealing technique. A control parameter similar to the temperature in physical annealing is introduced in optimization which will

dictate the number of states to be accessed in going through the successive steps of the optimization algorithm before being settled in the minimum energy state (the optimum configuration). 2. PROBLEM STATEMENT AND MODEL SETUP As remarked above, the crystallographic grains (crystallites) in an as-grown polycrystal are randomly oriented and require three angles to describe its orientation with reference to a fixed coordinate system. Euler angles (φ, θ, ψ) can completely specify the orientation of the crystallographic coordinate system embedded in crystallites and thereby the orientation of crystallites (Goldstein, 1978). 2.1. Design variables and objective function The orientation distribution of a poled polycrystalline FE follows a Gaussian distribution with the probability distribution given by 2 2 f (, ) (1/ 2 )exp ( ) / 2. (1) Here and are the parameters of the distribution viz., the mean and the standard deviation respectively. α stands for the Euler angle (φ, θ, ψ). In a 3D case as in the present work, parameters and perform the role of the control parameters which will decide the scatter of the orientations (Euler angles) and hence be critical to the piezoelectric response of the polycrystalline ferroelectric material. Hence and are the design variables of the optimization problem. Thus we are aiming to find an optimum set of these parameters from a solution space controlled by the laws of coordinate transformations from a crystallographic coordinate system embedded in the grains to a local coordinate system which coincides with the global frame of reference. Also, the solution space is bounded by distribution parameters and ranging from those of uniform (in the case of random polycrystal) to those of Gaussian distribution (in the case of poled polycrystal). A fairly uniform kind of distribution can be achieved by putting standard deviation () equals 5 and for a poled ceramic ferroelectric the is set near zero. The piezoelectric strain coefficient d 33 is the most significant and widely used figure of merit of ferroelectric materials in piezoelectric applications like actuators (e.g., Park and Shrout, 1997). In single crystalline materials like BaTiO 3 the piezoelectric coefficients shows a maximum when they are poled along a nonpolar axis. However, our objective is to search possible ways of enhancing the piezoelectricity in ceramic ferroelectric materials. Given the difficulties in synthesizing good quality single crystals of fairly large size for integration and also the non-reliability of reproduction, polycrystals are often preferred to single crystals in device applications. In polycrystalline piezoelectric materials, the state of strain is inhomogeneous. Understanding the local and global ferroelectric response of these topologically complex materials by combining mathematical modeling and simulation could help effectively engineer material configurations. Keeping in mind the goal of getting the maximum piezoelectric efficiency from polycrystalline material, the present objective is to maximize the piezoelectric coefficients d 33 by the optimum choice of grain distribution. The grain distribution parameters chosen by the SA algorithm will prompt a normal random generator thereby create a set of Euler angles (φ, θ, ψ). These Euler angles will dictate the coordinate transformation in the electromechanical property tensors appearing in the homogenization equations. 2.1. Homogenization Effective material properties are calculated using the mathematical homogenization method. The numerical solution of the coupled piezoelectric problems is sought using the finite element method (FEM) to eventually compute the homogenized piezoelectric coefficients. The finite element method used for this study correlates each randomly oriented grain in a polycrystalline material with each element of the finite element mesh. Each grain in a polycrystalline material is assumed to be made of a single, pinned, chemically homogeneous ferroelectric domain. The asymptotic analysis and homogenization of the piezoelectric medium (Galka et al., 1992; Nelli Silva et al., 1999) has resulted in the macroscopic piezoelectric coefficients e 33 and thereby d 33. Full integration (2-point Gaussian integration rule in each direction) is used for the evaluation of the stiffness, piezoelectric and dielectric matrices and for the homogenization. As the representative unit-cell is expected to capture the response of the entire piezoelectric system, particular care is taken to ensure that the deformation across the boundaries of the cell is compatible with the deformation of adjacent cells. Hence all the load cases are solved by enforcing periodic boundary conditions in the unit-cell for the displacements and electrical potentials. The numerical simulation of ceramic BaTiO 3 is done using the parameters of single crystal data from Zgonic et al. (1194) using the present homogenization model computationally implemented in Fortran. 3. SIMULATED ANNEALING

The simulated annealing (SA) algorithm is based on the Metropolis algorithm (Metropolis et al., 1953) for simulating the behavior of an ensemble of atoms that are cooled slowly from their melted state to the minimum energy ground state. The ground state or minimum energy state corresponds to the global optimum we are seeking in material optimization. In order to apply SA to a piezoelectric material, we must first introduce the notion of system energy. In the present setup the piezoelectric coefficient d 33 acts as system energy and we are seeking a maximization of d 33. Following the discussion given in this paper, d33( ) d33(, ). Here [0, 5] and [0, / 2].The distribution parameters are selected randomly. The main goal of SA is to find the ground state(s), i.e., the minimum energy configuration(s), with a relatively small amount of computation. Minimum energy states are those that have a high likelihood of existence at low temperature. The likelihood that a configuration, R i, is allowed to exist is equal to the Boltzmann probability factor, ER ( ) i PR ( i ) exp kt B (2), where T is the temperature and k B is the Boltzmann constant. For computational convenience k B is often treated as unity. It is obvious from the above relation that as the temperature decreases, the Boltzmann distribution concentrate on the states with lowest energy and finally, when the temperature approaches zero, only the minimum energy states have the non-zero probability of occurrence. To simulate the evolution of thermal equilibrium of a solid for a fixed T, Metropolis et al. (1953) proposed a Monte Carlo method, which generates a sequence of states of the solid. Given the current state of energy E 1, another set of design variables are randomly generated which will eventually calculate the another energy E 2. If the difference in energy E, between the current state and the new state (E 2 -E 1 ) is positive (negative in the case of minimization) then the process is continued with the new state. If E 0 then the probability of acceptance of the new state is given by exp( E / k B T) for maximization problems as in the present work. Following this criterion, the system eventually evolves into thermal equilibrium. Once thermodynamic balance is reached at a given temperature the temperature is lowered slightly and new chain of iterations will be executed before the system finally ends up in equilibrium. 4. RESULTS AND DISCUSSION As stated above, we have six design variables, viz.,,,, and which corresponds to the standard deviations and means of the orientation [Euler angles (φ, θ, ψ)] distributions expressed in radians. The temperature T is set to fall by 20% from each of the previous step, i.e., T k+1 = 0.8T k. Ideally we must start the iteration with an initial guess of the design variables randomly picked up from [0, 5] and [0, / 2]. To verify correctness of the algorithm, first we have "Energy", Piezoelectric coefficient d 33 (pc/n) 240 200 160 120 80 12 10 8 6 4 2 0 "Temperature" Figure 1. Energy (Piezoelectric coefficient d 33 ) as a function of temperature in single crystal BaTiO 3 applied this optimization procedure to the case of single crystal BaTiO 3. Thus we started with Euler angles (φ, θ, ψ) alone without going to the assumption of distribution of grain orientations since a single crystal has no grain structure.

All the three angles are allowed values between limits (,, ). The evolution of the objective function d 33 with the temperature is shown in Fig. 1. The piezoelectric coefficient d 33 obtained after optimization is 223.7 pc/n which compares exactly with our homogenization results reported recently (Jayachandran et al., 2009). The solution (φ, θ, ψ) is (-2.182, 0.873, -0.175). This corresponds to one of the {111} planes of the BaTiO 3 single crystal along which the maximum piezoelectric coefficient of d 33 = 203 pc/n is measured by Wada et al. (1999). The optimization of polycrystal BaTiO 3 is treated next. We analyze the most general case with (,, ) [0, 5] and (,, ) [0, /2]. The results are shown in Fig. 2. The solution (,,,,, ) obtained is (4.7, 0.873, 0, 0.698, 1.8, 1.223). The objective function converges with a value d 33 = 273.7 pc/n which is much higher than both [001] poled and [111] poled single crystals. This supplements our suggestion (Jayachandran et al., 2008) that randomness in the orientation of grains, if utilized judiciously, might be useful for manufacturing piezoelectric ceramics which outperform single crystals. The solution suggests one should keep the Euler angle (φ and ψ) related to the orientation of ab-plane of the crystallites to be in random rather than keeping their value at zero while the orientation θ, of c-axes is kept at 0.698 radians (θ = 40). This would further point out that c-axes of the crystallites should be constrained to have a specific orientation while the ab-plane need not be kept at a specific orientation. This condition will deliver a better piezoceramic. 280 "Energy", piezoelectric coefficient, d 33 pc/n 260 240 220 200 180 160 140 120 100 80 1 0.9 0.8 0.7 "Temperature" 0.6 0.5 0.4 Figure 2: Energy (Piezoelectric coefficient d 33 ) as a function of temperature in FE polycrystal BaTiO 3 5. ACKNOWLEDGEMENTS KPJ acknowledges the award of Ciência 2007 by the FCT, Portugal. Partial support from the project PTDC/EME- PME/67658/2006 is also acknowledged. 6. REFERENCES Eerenstein, W, Mathur, N.D. and Scott, J.F., 2006. Multiferroic and magnetoelectric materials, Nature, Vol. 442, pp. 759-765. Galka, A., Telega J.J., and Wojner, R., 1992. Homogenization and thermo piezoelectricity, Mech. Res. Commun., Vol. 19, pp. 315-324. Goldstein, H., 1978. Classical Mechanics, Addison-Wesley, Reading, MA. Jayachandran, K.P., Guedes, J.M., and Rodrigues, H.C., 2008. Piezoelectricity enhancement in ferroelectric ceramics due to orientation, Appl. Phys. Lett., Vol. 92, 232901. Jayachandran, K.P., Guedes, J.M., and Rodrigues, H.C., 2009. Enhancement of the electromechanical response in ferroelectric ceramics by design, J. Appl. Phys., Vol. 105, 084103. Kirkpatrick, S., Gelatt, Jr., C.D. and Vechi, M.P., 1983. Optimization by simulated annealing, Science, Vol. 220, pp. 671-680. Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A., and Teller, E., 1953. Equation of state calculations by fast computing machines, J. Chem. Phys., Vol. 21, pp. 1087-1092.

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