Ferroelectricity enhancement in confined nanorods: Direct variational method

Ferroelectric nanoparticles of different shapes and their nanocomposites are actively studied by modern physics. Because of their applications in many fields of nanotechnology, the size effects and the possible disappearance of ferroelectricity at a critical particle volume attract a growing scientific interest. In this paper we study the size effects of the cylindrical nanoparticle phase diagrams allowing for effective surface tension and depolarization field influence. The Euler-Lagrange equations were solved by the direct variational method. The approximate analytical expression for the paraelectric-ferroelectric transition temperature dependence on nanoparticle sizes, polarization gradient coefficient, extrapolation length, effective surface, tension and electrostriction coefficient was derived. It was shown that the transition temperature could be higher than the one of the bulk material for nanorods and nanowires in contrast to nanodisks, where the decrease appears. We show that among all cylindrical shapes a nanobar reveals the minimal critical volume. Therefore we predict conservation and enhancement of ferroelectric properties in nanorods and nanowires. Obtained results explain the observed ferroelectricity enhancement in nanorods and could be very useful for the elaboration of modern nanocomposites with perfect polar properties.