The effects of crystallographic texture and microstructure are analyzed for polycrystalline tetragonal BaTiO 3 , pseudotetragonal PZN-PT, and cubic BaTiO 3 . For tetragonal BaTiO 3 and pseudotetragonal PZN-PT, we demonstrate that a high anisotropy of the single-crystal properties induces an apparent enhancement in the macroscopic piezoelectric response. For tetragonal BaTiO 3 , the predicted macroscopic piezoelectric constants d 31 and d 33 are enhanced with respect to its single-crystal value at the expense of the spatial contributions from d 15 . For samples possessing fiber texture, an optimal response is predicted for samples that are not perfectly textured. Similarly, an apparent enhancement of the macroscopic value of d 15 is predicted for PZN-PT. For cubic BaTiO 3 , the low anisotropy of the underlying crystal properties induces a uniform decrease of the macroscopic electrostrictive constant, Q 11 , with decreasing texture. A completely random polycrystal provides 0.8570.05 times its single-crystal response.
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A model and numerical framework is developed for piezoelectric materials. The model treats the piezoelectric and electrostrictive effects by incorporating orientation-dependent, single-crystal properties. The method is implemented in Object Oriented Finite Element program, a public domain finite element code, so it can be applied to arbitrary two-dimensional microstructures with crystallographic anisotropy. The model is validated against analytic solutions. Consistency of the method for known cases permits application of the technique to more complicated twodimensional systems. The piezoelectric and electrostrictive response is determined for a few simple device geometries and provides insight for design and convergence criteria.
J ournalThe polarization vector is related to the applied electric fieldẼ and total polarizationD by:where E 0 is the permittivity of vacuum. z Because this is an expansion about a stable state, it is implicit that hysteretic effects, domain wall motion, charge separation, mass diffusion, and other irreversible kinetic effects cannot be operative, thus these equations apply. In a ferroelectric, there may be irreversible phase transitions that include switching events and local motion of domains.
We propose a new type of ordered colloid, the "ionic colloidal crystal" (ICC), which is stabilized by attractive electrostatic interactions analogous to those in atomic ionic materials. The rapid self-organization of colloids via this method should result in a diversity of orderings that are analogous to ionic compounds. Most of these complex structures would be difficult to produce by other methods. We use a Madelung summation approach to evaluate the conditions where ICC's are thermodynamically stable. Using this model, we compare the relative electrostatic energies of various structures showing that the regions of ICC stability are determined by two dimensionless parameters representing charge balance and the spatial extent of the electrostatic interactions. Parallels and distinctions between ICC's and classical ionic crystals are discussed. Monte Carlo simulations are utilized to examine the glass transition and melting temperatures, between which crystallization can occur, of a model system having the rocksalt structure. These tools allow us to make a first-order prediction of the experimentally accessible regions of surface charge, particle size, ionic strength, and temperature where ICC formation is probable.
As discussed previously, interfacial roughness in one-dimensional photonic crystals (1DPCs) can have a significant effect on their normal reflectivity at the quarter-wave tuned wavelength. We report additional finite-difference time-domain (FDTD) simulations that reveal the effect of interfacial roughness on the normal-incidence reflectivity at several other wavelengths within the photonic bandgaps of various 1DPC quarter-wave stacks. The results predict that both a narrowing and red-shifting of the bandgaps will occur due to the roughness features. These FDTD results are compared to results obtained when the homogenization approximation is applied to the same structures. The homogenization approximation reproduces the FDTD results, revealing that this approximation is applicable to roughened 1DPCs within the parameter range tested (rms roughnesses < 20% and rms wavelengths < 50% of the photonic crystal periodicity) across the entire normal incidence bandgap.
Dielectric reflectors that are periodic in one dimension, also known as one-dimensional photonic crystals (1DPCs), have become extremely useful tools in the optics industry due to the presence of wavelength-tunable photonic bandgaps. However, little is known about the practical effects of manufacturing defects, such as interfacial roughness, on this technologically useful property of 1DPCs. We employ a finite-difference time-domain code to gain further insight into the effect of interfacial roughness on the reflectivity of quarter-wave-tuned 1DPCs in the center of the bandgap at normal incidence. This provides an estimate of the magnitude of the effect of the roughness for even the most-robust incidence conditions.
As previously reported [Opt. Lett. 29, 2791 (2004)], one-dimensional photonic crystals exhibit a decrease in their normal reflectivity if their interfaces are not flat. We show that the homogenization approximation accurately predicts this diminished optical response by comparing results with finite-difference time-domain (FDTD) simulations applied to the same roughened structures. Within the parameter range tested (rms roughness < 20% and rms wavelengths < 100% of the photonic crystal periodicity), the homogenization approximation accurately reproduces the reflectivities obtained by the FDTD simulations, which are much more computationally expensive.
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