Not Found - Wolfram Cloud

Ferroelectricity in cubic symmetry	Ferroelectricity in hexagonal symmetry
Ferroelectricity in tetragonal symmetry	Ferroelectricity in trigonal symmetry

Ferroelectricity is characterized by a finite polarization in the sample. We discuss the phenomenological theory of ferroelectricity in the framework of the Landau theory.

GTLandauExpansion

gives a symmetry-adapted free energy expansion of generalized order parameters.

Ferroelectricity in cubic symmetry

Install the point group O_h and compute the character table.

The polarization transforms as a vector, i.e., as the irreducible representation T_1u. We compute the representation matrices:

The free energy expansion up to fourth order is given by

Ferroelectricity in tetragonal symmetry

The corresponding point group is D_4h.

The in-plane components of the polarization, P_x and P_y transform as E_u. In contrast, P_z transforms as A_2u.

The free energy expansion up to fourth order for the in-plane polarization is given by

The free energy expansion up to fourth order for the z-component is given by

The coupled free energy expansion is given by

Ferroelectricity in hexagonal symmetry

The corresponding point group is D_6h.

The in-plane components of the polarization, P_x and P_y transform as E_1u. In contrast, P_z transforms as A_2u.

The free energy expansion up to fourth order for the in-plane polarization is given by

The free energy expansion up to fourth order for the z-component is given by

The coupled free energy expansion is given by

Ferroelectricity in trigonal symmetry

The corresponding point group is D_3h.

The in-plane components of the polarization, P_x and P_y transform as E' In contrast, P_z transforms as A₂''.

The free energy expansion up to fourth order for the in-plane polarization is given by

The free energy expansion up to fourth order for the z-component is given by

The coupled free energy expansion is given by

Landau theory for ferroelectricity