Ferroelectricity
The ferroelectricity phenomenon is 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. This is similar to permanent magnetism in ferromagnetics, which explains the name used for such materials. Application of an electric field can rearrange the domains resulting into the net polarization in the material. At very large electric fields, the polarization 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.
All domains are assumed to have polarization of the same magnitude Ps, but the polarization can have different orientations. The applied magnetic field changes the domain orientation, and the resulting net polarization in the ferroelectric material is found from the following nonlinear implicit relation:
(2-8)
The effective field in the material is given by
where the matrix α characterizes the interdomain coupling.
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:
Hystersis modeling
The Jiles–Atherton hysteresis model for ferroelectric materials is available COMSOL Multiphysics. The model assumes that the total polarization can be represented as a sum of reversible and irreversible parts
The polarization change can be computed from the following incremental equation:
where the reversibility is characterized by the matrix cr, and the anhysteretic polarization is found from a relation similar to Equation 2-8:
The change of the irreversible polarization can be computed from
where the pinning loss is characterized by the matrix kp.
Reference
6. R.C. Smith and C.L. Hom, “A Domain Wall Model for Ferroelectric Hysteresis,” Journal of Intelligent Material Systems and Structures, vol. 10, no. 3, pp. 195–213, 1999.