Spontaneous polarization in ferroelectric materials is the property that the polar crystals show electric polarity even without an external electric field. Ferroelectric materials show two or more spontaneous polarization states and can be changed from one state to another by the application of an electric field. Ferroelectric substances normally present separate regions called domains that have different spontaneous polarization directions. Note that in each domain the polarization state is uniform. In other words, each ferroelectric domain is a cluster of individual unit cells that are oriented in the same direction.
In fabrication of ferroelectric thin films, the films are generally grown at high temperatures above the Curie temperature (TC) holding for the bulk ferroelectrics. The strain relaxation in the films normally takes places during cooling. After the phase transition from the paraelectric to ferroelectric phase occurred, the symmetry is broken and differences in lattice parameters are introduced. That is, after cooling through TC, the symmetry is lowered due to the transition from cubic to tetragonal phases. This symmetry lowering can lead to a preferential formation of domains that is able to relieve the accumulated stress in the film.
In general, the stress in ferroelectric thin films is relieved by forming misfit dislocations and/or twins and by splitting into different ferroelastic domains [1, 2]. However, the term “domain” can not only refer to crystallographic domains but also to the domains where only the polarization reverses.
Figure 3535 presents the polydomain structures formed due to multiple misfit relaxations within an epitaxial tetragonal phase on a (001)-oriented cubic substrate. In this case, there are three possible domain configurations between the epitaxial film and substrate:
i) c-domains are oriented so that their c-axis is normal to the film/substrate interface.
ii) a1 and a2 domains are oriented so that their c-axis is parallel to the film/substrate interface and is aligned in  and  directions of the substrate.
If the lattice parameters of the thin film are slightly different from the lattice parameter of the substrate at room temperature, the mismatch will be relieved by introducing a small amount of domains with twinning mechanism.
Figure 3535. (a) Domain configurations within an epitaxial tetragonal phase on a (001)-oriented cubic substrate . (b) The schematic illustration of a domain wall between 90° domains (or a- domain) and c- domains in the tetragonal phase . c and a represents the lattice parameters of the tetragonal ferroelectric phase, while b represents the lattice parameter of the cubic substrate.
Table 3535 lists the orientations of the domains depending on the crystal structure and spontaneous polarization of the domains in ferroelectrics.
Table 3535. Orientation of the domains in all crystalline phases of ferroelectrics.
||Orientations of the domains
||Parallel to the cube
edge (6 possible directions in total)
||Parallel to the body diagonal (8 possible directions in total)
||Parallel to the face diagonal (12 possible directions in total)
At zero field (E0) the electric displacement within a single domain has two values (-Pr and +Pr), representing the opposite orientations of the spontaneous polarization. In a multiple-domain crystal the average zero-field displacement can have any value between these two extremes (-Pr < D < +Pr).
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