Thermodynamics of nanodomain formation and breakdown in scanning probe microscopy: Landau-Ginzburg-Devonshire approach

Thermodynamics of tip-induced nanodomain formation in scanning probe microscopy of ferroelectric films and crystals is studied using the analytical Landau-Ginzburg-Devonshire approach and phase-field modeling. The local redistribution of polarization induced by the biased probe apex is analyzed including the effects of polarization gradients, field dependence of dielectric properties, intrinsic domain-wall width, and film thickness. The polarization distribution inside a ``subcritical'' nucleus of the domain preceding the nucleation event is shown to be ``soft'' (i.e., smooth without domain walls) and localized below the probe, and the electrostatic field distribution is dominated by the tip. In contrast, polarization distribution inside a stable domain is ``hard'' (i.e., sharp contrast with delineated domain walls) and the spontaneous polarization reorientation takes place inside a localized spatial region, where the absolute value of the resulting electric field is larger than the thermodynamic coercive field. The calculated coercive biases corresponding to formation of switched domains are in a good agreement with available experimental results for typical ferroelectric materials. The microscopic origin of the observed domain-tip elongation in the region where the probe electric field is much smaller than the intrinsic coercive field is the positive depolarization field in front of the moving-counter domain wall. For infinitely thin domain wall the depolarization field outside the semiellipsoidal domain tip is always higher than the intrinsic coercive field that must initiate the local domain breakdown through the sample depth while the domain length is finite in the energetic approach evolved by Landauer and Molotskii (we refer the phenomenon as Landauer-Molotskii paradox). Our approach provides the solution of the paradox: the domain vertical growth should be accompanied by the increase in the charged domain-wall width.