Dependence of the ferroelectric domain shape on the electric field of the microscope tip

A theory of an equilibrium shape of the domain formed in an electric field of a scanning force microscope (SFM) tip is proposed. We do not assume a priori that the domain has a fixed form. The shape of the domain is defined by the minimum of the free energy of the ferroelectric. This energy includes the energy of the depolarization field, the energy of the domain wall, and the energy of the interaction between the domain and the electric field of the SFM tip. The contributions of the apex and conical part of the tip are examined. Moreover, in the proposed approach, any narrow tip can be considered. The surface energy is determined on the basis of the Ginzburg-Landau-Devonshire theory and takes into account the curvature of the domain wall. The variation of the free energy with respect to the domain shape leads to an integro-differential equation, which must be solved numerically. Model results are illustrated for lithium tantalate ceramics.