THE INFLUENCE OF SURFACE ENERGY ON INCOMMENSURATE SUPERSTRUCTURES

Abstract

Phase portraits of the incommensurate superstructures described by the Livshits invariant were studies. The construction of phase portraits executed in the Python software environment using the scipy library. Using the set_integrator method, the integrator "vode" was chosen, which the usual solver of the differential equation is.

The influence of parameters, the stability of the initial phase (T) and the anisotropic interaction (K), which is determined by the Dzyaloshinsky invariant on the phase portraits of the investigated system of differential equations, was considered. We have established that the anisotropic interaction described by the Dzyaloshinsky invariant leads to a violation of the spatial periodicity of the superstructure with the appearance of the amplitude and frequency modulation of the superstructure.

Investigations of the stability of the system on the magnitude of surface energy were carried out. As a result, the superficial energy reduces the number of bifurcations of the system, indicating the transition of the system to a steady state. It is characterized by two mutually symmetric attractors.

The system goes into a metastable state with an increase of the electric field intensity. It is characterized by a constant value of the magnitude of the wave vector of the superstructure, but not a zero mean value of the spatially modulated spontaneous polarization value. The increase of the value of the electric field, unlike to the influence of the surface energy, is accompanied by a decrease in the number of existing metastable states, and in the final stage the system passes to a state described by the existence of a single metastable state. Consequently, in contrast to the influence of surface energy, the action of an electric field does not lead to the removal of the degeneration of the investigated system.

In the surface layer, an incommensurate superstructure is absent because its energy is less than the surface energy. The bulk part of the crystal is characterized by greater energy of the incommensurate superstructure in relation to the surface energy. In a thin-layer crystal, the transition from the inhomogeneous state to a homogeneous state occurs when the energy of the incommensurate superstructure becomes equal to the surface energy.

Key words: incommensurate superstructures, phase portraits, surface energy.