5.3.8 Analytical treatment of domain wall energy

Ferroelectric domains form as a result of instabilities due to alignment of dipoles in a crystal. The free energy change involved in the formation of a domain is given as

(5.29)

where
U_c : effect of applied field on the domain energy
U_p and U_x : bulk electrical and elastic energies
U_d : depolarization energy and
U_w : domain energy
U_d is the energy related to the internal field set up in the crystal by the polarization and not compensated. The internal field opposes the applied field E and hence is called as depolarizing fieldThis is dependent on the domain size (d) and is expressed as

(5.30)

Here ε^* is a constant determined by the dielectric constant of the material, t is crystal thickness, P₀ is the polarization at the center of the domain.

U_w is the domain energy and can be expressed in terms of surface energy of the wall (γ), domain width (d) and crystal volume (V) and is given as

(5.31)

To get a stable domain structure, this ΔG has to be minimized. In (5.29), U_p and U_x are the same in each domain, irrespective of domain thickness and so are independent of domain size and hence are treated as constant. Therefore one only needs to consider U_w and U_d when ΔG is minimized w.r.t. domain wall thickness d i.e. dΔG/dd = 0 resulting in

(5.32)

Here, one can see that domain size is dependent on surface energy, crystal thickness and polarization, showing a competition between the surface energy of the wall and polarization (governing the depolarizing field). This shows that larger the surface energy is, larger the domain size is which makes sense because of large energy requirement, the interface area needs to be smaller. Secondly, the larger the polarization is, the larger the depolarizing field will be and large would be the driving force for the domains to form and hence smaller domains will be preferred.