Quasihexagonal arrays of electric-skyrmion bubbles in thin-film ferroelectrics: Pattern formation and structure

Topologically nontrivial patterns, such as quasiordered arrays of magnetic skyrmion bubbles and their electric analogs, have recently received the wide attention of researchers across disciplines. These structures and/or superstructures are considered as potential candidates for technological application, e.g., as the basis for new generations of memory devices. In this case, a manipulation of electric bubbles by external stimuli is of great essence. In the present work, we report ab initio-based simulations of ferroelectric $\mathrm{PbZ}{\mathrm{r}}_{0.4}{\mathrm{Ti}}_{0.6}{\mathrm{O}}_{3}$ ultrathin films, wherein quasiregimented arrays of electric bubbles in the form of quasihexagonal superstructures are formed under external electric-field bias at low temperatures. The emergence of these superstructures takes place in a range of external electric-field magnitudes and temperatures. The other structures, obtained by varying external electric-field magnitude, include vortex tubes and monodomains. An underlying mechanism of the pattern formation and their properties are discussed, with the primary focus put on the quasihexagonal superstructures. In particular, we find that the formation of these superstructures is associated with three ``polarization waves'' having reciprocal wave vectors of the same magnitude, and discuss mechanism of the superstructure formation. We also find that the external electric field modifies the average size of individual bubbles and the orientational order of quasihexagonal arrangement, and causes superstructure rotation with three distinct rotation angles identified.