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Within the Landau-Ginsburg-Devonshire phenomenological approach we study the ferroic nanosystems properties changes caused by the flexo-effect (flexoelectric, flexomagnetic, flexoelastic) existing spontaneously due to the inhomogeneity of order parameters. Exact solution for the spatially inhomogeneous mechanical displacement vector allowing for flexocoupling contribution was found for nanowires and thin pills. Strong influence of flexo-effect in 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 flat geometry in radial direction transforms into the saucer-like one. The new phenomena can be considered as true manifestation of the spontaneous flexo-effect existence. It was shown that flexo-effect leads to (a) the appearance of new linear and nonlinear contribution 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, (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 is 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. The divergences of dielectric permittivity and correlation radius at some critical value of flexoelectric coefficient originate from the critical radius dependence on the coefficient. This peculiar behavior shows a new way to govern material properties. The effect of the correlation radius renormalization by the flexoelectric effect leads to the renormalization of the intrinsic width of ferroelectric domain walls.

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General approach for consideration of primary ferroic (ferroelectric, ferromagnetic, ferroelastic) nanoparticles phase transitions was proposed in phenomenological theory framework. The surface stress, order parameter gradient, striction as well as depolarization and demagnetization effects were included into the free energy. The strong intrinsic surface stress under the curved nanoparticle surface was shown to play the important role in the shift of transition temperature (if any) up to the appearance of new ordered phase absent in the bulk ferroic. The approximate analytical expression for the size-induced ferroelectric transition temperature dependence on cylindrical or spherical nanoparticle sizes, polarization gradient coefficient, correlation radius, intrinsic surface stress and electrostriction coefficient was derived. It was shown that the transition temperature of nanoparticle could be higher than the one of the bulk material. The best conditions of ferroelectric properties conservation and enhancement in nanowires correspond to the radius 5-50nm and compressive surface stress. Under the favorable conditions size effects (spatial confinement) induces ferroelectric phase in incipient ferroelectrics nanowires and nanospheres. The prediction of size-induced ferroelectricity in KTaO3 nanorods with radius less then 5-20 nm at room temperatures could be important for the next step of device miniaturization based on 3D nanostructures.Comment: 35 pages, 11 figures, 2 appendices, to be submitted to Physical Review