What are ferroelectric materials? Explain with the example of BaTiO3. How are they different from piezoelectric materials?

Ferroelectric materials are a class of materials that exhibit spontaneous polarization, which means that they have a permanent electric dipole moment even in the absence of an external electric field. This polarization can be reversed by applying an external electric field, making these materials fundamentally distinct from other dielectric materials, where the polarization is induced only under an applied electric field. The phenomenon of ferroelectricity is analogous to ferromagnetism in magnetic materials, where spontaneous magnetization occurs. Ferroelectric materials are typically crystalline and exhibit non-linear dielectric properties, with their behavior depending on factors such as temperature, external electric fields, and crystal symmetry.

A quintessential example of a ferroelectric material is barium titanate (BaTiO₃). BaTiO₃ is a perovskite-structured compound that undergoes a phase transition at a critical temperature known as the Curie temperature (Tc), which is around 120°C for BaTiO₃. Below Tc, BaTiO₃ adopts a tetragonal structure that exhibits spontaneous polarization, while above Tc, it transforms into a cubic structure where the polarization disappears, and the material becomes paraelectric. This transition is driven by changes in the symmetry of the crystal lattice.

At low temperatures (below Tc), BaTiO₃ crystallizes in a tetragonal unit cell, where the titanium ion (Ti⁴⁺) is displaced _____ ______ ____ _______ _______ _________.
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