  Using direct variational method with 2-parametric trial function and
Landau-Ginzburg-Devonshire thermodynamical approach, we derived analytical
expressions for polarization spatial redistribution in the ferroelectrics
caused by the biased Scanning Probe Microscope probe. We demonstrate that the
shape of nanodomain induced by the probe electric field can be either oblate or
prolate depending on the ferroelectric nonlinearity strength. For typical
ferroelectric material parameters and probe apex geometry the domain nucleus
aspect ratio is close to dielectric anisotropy factor. Corresponding coercive
biases of a stable domain formation are in reasonable agreement with available
experimental results. Spike-like domains typical for Landauer-Molotskii rigid
approach appear in the considered case only when depolarization field energy
contribution strongly dominates over the nonlinear correlation and field
effects and domain wall energy.
