The internal energy of ferroelectric domain structures characterized by pre-fractals of the pentad Cantor sets

Abstract

The laminated ferroelectric 180 degrees domain structure optically observed in uniaxial ferroelectric KDP is periodic along the normal to the domain walls. Each half-period of the domain structure is characterized by the first four pre-fractals of the pentad Cantor sets. In order to investigate the thermodynamic stability of these pre-fractal domain structures, the electrostatic energy U_E(n) and the wall energy U_W(n) of the nth pre-fractal domain structure have been formulated as functions of the period in the framework of the Kittel theory. It is shown that under a constant period the electrostatic energy U_E(3) of the third pre-fractal domain structure takes the least value while the wall energy U_W(n) increases monotonically with increasing ordinal number n of the pre-fractal domain structure. The equilibrium half-period minimizing the internal energy including U_E(n) and U_W(n) is proportional to the square root of the thickness of a crystal plate. The equilibrium half-period increases monotonically with increasing ordinal number n of the pre-fractal domain structure. The equilibrium internal energy takes the least value in the zeroth pre-fractal domain structure and the second-least value in the second prefractal domain structure. The ratio of the latter to the former is nearly equal to unity, with the value of 1.10.