Spontaneous flexoelectric/flexomagnetic effect in nanoferroics

Within the Landau-Ginsburg-Devonshire phenomenological approach we study the ferroic nanosystem properties changes caused by the flexoeffect (flexoelectric and flexomagnetic) existing spontaneously due to the inhomogeneity of order parameters. An exact solution for the spatially inhomogeneous mechanical displacement vector allowing for flexocoupling contribution was found for nanowires and thin pills. Strong influence of the flexoeffect on nanorods and thin pills leads to the displacements of the atoms resulting into the unit-cell symmetry changes, which lead to the phase-transition temperature shift, as well as the nanoparticle shape distortions; in particular, flat cross sections transform into saucerlike ones. The phenomena can be considered as true manifestation of the spontaneous flexoeffect existence. It was shown that flexoeffect leads to (a) the appearance of linear and nonlinear contributions and renormalization of coefficients before the order-parameter gradient, (b) essentially influences the transition temperature, piezoelectric response, and the spatial distribution of the order parameter, and (c) results in renormalization of extrapolation length in the boundary conditions. These effects cannot be neglected for ferroelectrics, the renormalization being important for nanoparticles of arbitrary shape, while the linear and nonlinear terms are essential for the thin pills only. They are absent for nanowires with the order parameter directed along the wire axis. We demonstrated that the flexoelectric coupling decreases the polarization gradient self-consistently and so makes polarization more homogeneous. It was shown that revealed effect of the correlation radius renormalization by the flexoelectric effect leads to the renormalization of the intrinsic width of ferroelectric domain walls.