Ferroelectric Crystals

Domains:
For ferromagnetic crystals, where the domain wall consists of a transition region in which the magnetization vector turns over gradually from a given direction to the opposite one. This is because the thickness of a ferromagnetic domain wall is a result of the compromise between the exchange energy and the anisotropy energy. The former would require a domain wall as wide as possible, the latter attempts to do just the contrary, in order to avoid the directions of "hard magnetization", but the exchange term is orders of magnitude larger than the anisotropy term and the result is a rather thick domain wall.
For ferroelectric crystals, there is no real counterpart of the magnetic exchange energy. Based on a dipole interaction calculations, the interaction energies of parallel and antiparallel dipole arrays do not differ much from each other, thus allowing a narrow domain wall. Most ferroelectrics, especially those with low symmetry, has high anisotropy, which prevents the spontaneous polarization from deviating from the ferroelectric axis. Hence, the polarization vector will not rotate within the wall, but rather will decrease in magnitude, passing through zero, and increase on the other side with opposite sign. Also, the elastic energy will also favor a rather thin domain wall.

For single domain ferroelectric crystal, a hysteresis loop will be observed. When the crystal consists many domains, the loop changes. Especially when there are equal number of positive and negative domains, no piezoelectric effect can be observed and the spontaneous polarization cannot be reversed by a mechanical stress. However, an electric field can reverse the spontaneous polarization and thus the piezoelectric polarity. The relationship between strain and field is characterized by a quadratic hysteresis loop. In actual crystals, different portions of the sample alter orientation at different parts of the cycle and a smooth curve results, which is called a "butterfly loop".


Domain wall energy:
Mert modified the Bloch Wall theory in ferromanetics for the case of BaTiO₃. The total wall energy was taken to be the sum of the energy due to dipole the interactions (corresponding to the exchange energy in the ferromagnectic case) and the anisotropy energy. Calling t the wall thickness, the wall energy E_wall can be written as:
E_wall=A/t+Bt,
A and B are functions of the lattice constant, spontaneous strain and the elastic constants.
Mert estimated to be of the order of one lattice constant, the corresponding energy was estimated to of the order of 7 ergs/cm².

The process of polarization reversal
It is quite evident that when the polarization of a ferroelectric crystal is reversed by means of an external electric field, the domain configuration must undergo radical changes.
The reversal process can be interpreted in terms of two mechanisms: nucleation of domains and domain wall movements.
Under low external field, the rate of nucleation is so low that the switching is primarily governed by the nucleation process. At low field, the probability of forming new domains depends exponentially upon the applied field.
p_n=p₀e^-α/E
It follows that:
1/t_s~1/t_n=1/t_∞e^-α/E
At high field, on the other hand, the nucleation rate is so large that the swithching time is primarily determined by the domain wall motions.
Assuming that the domain wall movement can be described by:
v=d/t_d=&muE
Where d is the distance covered by the wall in motion (thickness of the sample), and μ is the mobility,we obtain:
1/t_s~1/t_d=μE/d=μV/d²
This implies that, in the region of high fields, the switching time should depend quadratically upon the thickness of the sample.
At low field, on the other hand, the sidewise expansion of the nucleated domains may bring contributions to the switching transient.
At high field, the forward motion of the domain walls is very much faster than any sidewise motion.