GROUP THEORY TECH NOTE Landau theory for ferroelectricity 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
Oh and compute the character table.
The polarization transforms as a vector, i.e., as the irreducible representation
T1u . 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
D4h .
The in-plane components of the polarization,
Px and
Py transform as
Eu. In contrast,
Pz transforms as
A2u .
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
D6h .
The in-plane components of the polarization,
Px and
Py transform as
E1u. In contrast,
Pz transforms as
A2u .
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
D3h .
The in-plane components of the polarization,
Px and
Py transform as
E' In contrast,
Pz 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
