The phenomenon of spontaneous formation of quasihexagonal arrays of electric bubbles in systems of electric dipoles subjected to an external electric field bias is studied via ab initio-based simulations of ultrathin ferroelectric films. We find that the formation of such topologically nontrivial superstructures is driven by development and evolution of soft-mode triplets with equal wavelengths and eigenfrequencies. The concurrent development of these soft-modes takes place above the transition point and the soft-modes become frozen-in near the transition temperature, where frequencies of the soft modes attain a minimum value. A further gradual cooling (or heating) of the superstructures leads to an increase in the soft-mode frequencies, following a square root law around the transition point. We quantify the thermal behavior of the degree of quasihexagonal ordering and find that the dipole structure factor intensities show the same critical behavior around the transition point as the soft-mode frequencies. The critical behavior, associated with the topological changes, is consistent with the Landau theory of phase transitions.

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• Ferroelectricity
• Topological materials
• Ferroelectrics
• Thin films
• Molecular dynamics