  Polar skyrmions are topologically stable, swirling polarization textures with
particle-like characteristics, which hold promise for next-generation,
nanoscale logic and memory. While understanding of how to create ordered polar
skyrmion lattice structures and how such structure respond to applied electric
fields, temperature, and film thickness remains elusive. Here, using
phase-field simulations, the evolution of polar topology and the emergence of a
phase transition to a hexagonal close-packed skyrmion lattice is explored
through the construction of a temperature-electric field phase diagram for
ultrathin ferroelectric PbTiO3 films. The hexagonal-lattice skyrmion crystal
can be stabilized under application of an external, out-of-plane electric field
which carefully adjusts the delicate interplay of elastic, electrostatic, and
gradient energies. In addition, the lattice constants of the polar skyrmion
crystals are found to increase with film thickness, consistent with expectation
from Kittel law. Our studies pave the way for the development of novel ordered
condensed matter phases assembled from topological polar textures and related
emergent properties in nanoscale ferroelectrics.
