Vortex-antivortex structures in PbTiO$_{3}$/SrTiO$_{3}$ superlattices*

When ultrathin ferroelectric layers of PbTiO$_{3}$ are embedded in superlattices with an incipient ferroelectric, such as SrTiO$_{3}$, the interplay between elastic, electrostatic, and gradient energies produce complex patterns of the electrical polarization. In particular, nanometer scale of vortex-antivortex arrays have been recently detected,\footnote{Yadav {\it et al.}, Nature {\bf 530}, 198 (2016)} and exotic properties such as the emergence of a negative capacitance have been measured.\footnote{\label{Zubko}P. Zubko {\it et al.}, Nature {\bf 534}, 524 (2016)} A realistic atomic simulation of these structures is difficult due to the large number of atoms required and the small differences in energies between the relevant phases. Here we use a recently developed second-principles method~$^{3,}$ \footnote{J. Wojdel {\it et al.}, J. Phys.: Condens. Matter {\bf 25}, 305401 (2013)} that treats all the lattice degrees of freedom with high accuracy at a modest computational cost. The effect of the periodicity, strain, temperature, and external electric fields in the formation of vortex-antivortex pairs is explored. We predict that some of the structures are chiral, that would make them optically active, supporting x-ray circular dichroism.