  Vortices consisting of $90^\circ$ quadrant domains are rarely observed in
ferroelectrics. Although experiments show polarization flux closures with
stripe domains, it is as yet unclear why pure single vortices are not commonly
observed. Here we model and explore the energy of polarization patterns with
vortex and stripe domains, formed on the square cross-section of a barium
titanate nanowire. Using phase-field simulations, we calculate the associated
energy of polarization patterns as a function of nanowire width. Further, we
demonstrate the effects of surface energy and electrical boundary conditions on
equilibrium polarization patterns. The minimum energy equilibrium polarization
pattern for each combination of surface energy and nanowire width is mapped for
both open-circuit and short-circuit boundary conditions. The results indicate a
narrow range of conditions where single vortices are energetically favorable:
nanowire widths less than about 30nm, open-circuit boundary condition, and
surface energy of less than 4N/m. Short-circuit boundary conditions tend to
favor the formation of a monodomain, while surface energy greater than 4N/m can
lead to the formation of complex domain patterns or loss of ferroelectricity.
The length scale at which a polarization vortex is energetically favorable is
smaller than the typical size of nanoparticle in recent experimental studies.
The present work provides insight into the effects of scaling, surface energy
and electrical boundary conditions on the formation of polarization patterns.

Materials Science