Microphysical Model of Ferroelectricity

In this paragraph, we present one simple model, which could be nsed to explain the spontaneous polarization from the point of view of microphysical theoiy of ciystal lattice. Especially, the instability of the long-wave optical phonon will be discussed. Cochran and Anderson (1960) used the idea of so called soft-mode (Cochran 1959, 1960,1961,1969 Cochran and Cowley 1967) for the explanation of the ferroelectric phase transition natnre in perovskites in 1959/1960 for the first time. This idea has been further extended in numerous theoretical as well as experimental works since that time (see e.g. Agrawal and Peny 1971 Blinc and ZekS 1974 Mitsui et al. 1976 Nakamura 1966 Perry 1971 Petzelt and Fousek 1976 Scott 1974 Silverman 1969). 

Basic idea of the theory will be demonstrated in simplified form for cubic crystal with two atoms in the elementary nnit. Let ns suppose just one-dimensional lattice vibrations (i.e. all atoms in certain lattice plane move together) - either parallel, or perpendicnlar to the wave propagation direction. In such case, we could further simplify our model to one-dimensional chain of two regularly repeated atoms. 

Let the chain be composed of ion A with the mass ma and charge +e and ion B with the mass mb and charge -e regularly distributed in line on the distance Mia (Fig. 5.13). 

Period of the chain is equal to a. Let us suppose the linear relationship between the interaction force between the nearest neighbors and atomic displacement. Every internal motion of the lattice could be represented by the superposition of the mutually orthogonal waves as follows from the lattice dynamic theoiy (see e.g. Bom and Huang 1954 Leibfried 1955). Aiy lattice wave could be described by the wave vector K from the first Brillouin zone in the reciprocal space. Dispersion curve co K) has two separated branches (for 2 atoms in the primitive unit), which could be characterized as acoustic and optic phonons. If we suppose also the transversal waves (along with longimdinal ones), we can get three acoustic and three optical phonon branches. There is always one longitudinal (LA or LO) and two mutually perpendicular transversal (TA or TO) phonons. 

Let us consider the wave vectors K from the vicinity of the first Brillouin zone origin (i.e. K 0). It corresponds to the infrared active lattice vibrations with X a. The optical phonon branch has the highest vibration frequency possible for the atom chain in that case. The ions vibrate with the opposite phases and amplitudes inversely proportional to their masses. Dipole moments are effectively created in each elementary unit and therefore the crystal is polarized. Polarization of the crystal causes the internal periodical electric field E at the position of each atom. This field contributes to the additional electric force on each ion, either by - -eEi, or by —eE force. Let us further denote the stiffness of the nearest neighbor interaction (i.e. spring constant) by C and displacement of ions A and B by w.4, ub respectively. The ion displacements follow the differential equation of motion 

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