An approach to the Klein-Gordon equation for a dynamic study in ferroelectric materials

Ferroelectric materials such as lithium niobate and lithium tantalate show a non-linear hysteresis behaviour, which may be explained by dynamical system analysis. The behaviour of these ferroelectrics is usually explained by domains and domain wall movements. So, the spatial variation of the domain wall was studied previously in order to see its effect on the domain wall width in the context of the Landau-Ginzburg functional. In the present work, both temporal and spatial variations of polarization are considered, and by using the Euler-Lagrange dynamical equation of motion, a Klein-Gordon equation is derived by taking the ferroelectrics as a Hamiltonian system. An interaction has been considered between the nearest neighbour domains, which are stacked sideways in a parallel array with uniform polarization. This interaction term is associated with the spatial term and when this interaction is assumed to be zero, the spatial term vanishes, giving rise to a Duffing oscillator differential equation, which can be also studied by a dynamic system analysis.