Quantum Theory of Spontaneous Polarization in a Ferroelectric Materials

This paper presents a quantum-mechanical model of rotating dipoles in perovskite crystals that exhibit ferroelectric properties. The symmetric rotator model accounts for both the displacement of a crystal lattice site—which lowers its symmetry—and the deformation of electron density within the unit cell. Matrices for the projection operators of dipole moments are calculated, incorporating a transformation from a rotating "molecular" coordinate system to the laboratory frame. In the limit of minimal rotation, these matrices—up to a numerical factor—correspond to the projection operator matrices for angular momentum (for j=1) in both spherical (cyclic) and Cartesian coordinates. Furthermore, by accounting for the coupling with the intrinsic spin of the electron associated with the dipole, the total angular momentum of the dipole is found to correspond to j=1/2. It is shown that two rotating dipoles, subject to dipole-dipole interaction, can "pair up"—either antiparallel (resulting in a minimal total dipole moment) or parallel (increasing the total dipole moment). In the latter case, a system of N dipoles can generate a macroscopic dipole moment within a domain. A variational model is proposed to explain the possibility of a second-order phase transition characterized by the onset of an ordered state. A general estimate is provided for the critical temperature at which the system undergoes a phase transition into a polarized state.