Ferroelectroelasticity
The ferroelectroelasticity and ferroelectricity phenomena are related to phase transitions in materials. In its ferroelectric phase, the material exhibits spontaneous polarization, so that it is constituted of domains with nonzero polarization even at zero applied field. Electrostriction in ferroelectroelastic materials can be related to the domain rotation. Thus, the applied electric field can both rearrange the domains resulting into the net polarization and rotate the domains mechanically. Thus, the material extends in the direction of the electric field and contracts in the direction perpendicular to the field. The domain rotation can be affected by an applied mechanical stress, which also results into the effective polarization. At very large electric fields, the electrostrictive effect saturates, as all ferroelectric domains in the material are aligned along the direction of the applied field. Domain wall interactions can also lead to a significant hysteresis in the polarization and strain.
The direct electrostrictive effect for a material of arbitrary symmetry can be represented as the following additive contribution to the strain (Ref. 1):
which is quadratic in polarization P. Due to the symmetry, the fourth order tensor Q can be effectively represented by a 6-by-6 coupling matrix. The number of independent components in the matrix depends on the material symmetry. For example, for piezoelectric ceramics, the matrix can be characterized by three independent components: Q11, Q12 and Q44.
For ferroelectroelastic materials, the polarization vector is nonlinear function of the electric field and possible mechanical stress in the material.
If hysteresis in the material can be neglected, the polarization is computed from the implicit relation
where the anhysteretic polarization function is assumed to have the following special form:
(3-84)
where Ps is the saturation polarization.
The polarization shape is characterized by the function L with the following properties:
For weak effective fields, the polarization is nearly linear
and can be characterized by the initial electric susceptibility matrix χ0.
For strong fields, the polarization magnitude approaches the saturation value
Two possible choices are the Langevin function
and a hyperbolic tangent (Ising spin model):
The effective electric field is given by
(3-85)
where E is the applied electric field, α is a material parameter called the inter-domain coupling, and the mechanics stress is computed assuming mechanically linear material as
where C is the fourth order elasticity tensor. The last term in Equation 3-85 represents the inverse electrostrictive effect.
The effective tangent piezoelectric coupling tensor can be computed as
where ε0,vac is the electric permittivity of free space, and
is the tangent electric susceptibility matrix. The piezoelectric coupling tensor d is a third order tensor. Due to the symmetry, it can be conventionally represented by a 3-by-6 matrix with only few nonzero components. An important observation from the above formula is that the piezoelectric coefficients should reach their maximum (or minimum) at certain strength of the applied bias field. This is because P is zero at zero applied field, while χe tends to zero at large applied field magnitudes because of saturation.
If the mechanical deformation due to the electrostriction is assumed to be volume preserving, the following form can be used for the electrostrictive strain in case of isotropic material:
where λs is the saturation electrostriction, and the deviatoric part can be computed for any matrix A as
The stress in Equation 3-85 is replaced then by
and tangent piezoelectric coupling tensor becomes
Hysteresis modeling
The Jiles–Atherton model for ferroelectric hysteresis assumes that the total polarization can be represented as a sum of reversible and irreversible parts. The polarization change is computed from the following incremental equation:
(3-86)
where the anhysteretic polarization is found using Equation 3-84, and the irreversible polarization change is computed as
where the pinning loss is characterized by the parameter kp.
Equation 3-86 can be solved using either a time-dependent analysis or a stationary parametric sweep.
References
1. F. Li, , Jin, Z.Xu, and S. Zhang, “Electrostrictive effect in ferroelectrics: An alternative approach to improve piezoelectricity,” Appl. Phys. Reviews, vol. 1, no. 1, pp. 011103-1–011103-21, 2014.
2. C. L. Horn and N. Shankar, “A finite element method for electrostrictive ceramic devices,” Int. J. Solids Structures, vol. 33, pp. 1757–1779, 1995.
3. R.C. Smith and Z. Ounaies. “A Domain Wall Model for Hysteresis in Piezoelectric Materials,” J. Int. Mat. Sys. Struct., vol. 11, no. 1, pp. 62–79, 2000.