Finite Element Implementation of a Thermodynamic Description of Piezoelectric Microstructures

Abstract

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.

Modeling 180° Domain Switching Population Dynamics in Polycrystalline Ferroelectrics

“…Similarly, if the c ‐axis of the grains makes a large misorientation angle with the applied field, those grains will remain unswitched. The volume fraction of the simple switching population is highly dependent on grain size, degree of tetragonality, and crystallographic texture of a polycrystal, as discussed in previous publications …”

“…The mechanical equilibrium equation and Coulomb's law in its differential form are given by,where is the spatially dependent stress vector and is the spatially dependent total polarization vector. The local stress and polarization are coupled through the set of constitutive relationsin which Є ij is the dielectric permittivity, the e ijk is the piezoelectric tensor in its e ‐form, C ijkl is the elastic stiffness tensor, and α kl is the kl ‐th component of the thermal expansion tensor.…”

“…include the contribution of spontaneous polarization to the dielectric displacement as readily described by Nye, and formulated by García et al . in the case of piezoelectric materials . The thermal expansion strain term, α kl Δ T , includes the non‐elastic strain contributions that result from cooling down the processed sample from the Curie temperature.…”

“…where M is the polarization domain mobility. The mechanical equilibrium equation and Coulomb's law in its differential form are given by, 46,47…”

“…in the case of piezoelectric materials. 46 The thermal expansion strain term, a kl DT, includes the non-elastic strain contributions that result from cooling down the processed sample from the Curie temperature. The stresses that result from this additional contribution to strain will lead to built-in processing fields that will bias the response of the solid.…”