A novel mesoscale state comprising of an ordered polar vortex lattice has been demonstrated in ferroelectric superlattices of PbTiO3/SrTiO3. Here, we employ phase-field simulations, analytical theory, and experimental observations to evaluate thermodynamic conditions and geometric length scales that are critical for the formation of such exotic vortex states. We show that the stability of these vortex lattices involves an intimate competition between long-range electrostatic, long-range elastic, and short-range polarization gradient-related interactions leading to both an upper and a lower bound to the length scale at which these states can be observed. We found that the critical length is related to the intrinsic domain wall width, which could serve as a simple intuitive design rule for the discovery of novel ultrafine topological structures in ferroic systems.
Bibliographical noteFunding Information:
The work is supported by U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award FG02-07ER46417 (L.-Q.C., F.X. and J.B.). Z.H. acknowledges the support by NSF-MRSEC Grant DMR-1420620 and NSF-MWN Grant DMR-1210588. A.R.D. acknowledges support from the Army Research Office under Grant W911NF-14-1-0104. L.W.M. acknowledges support from the National Science Foundation under Grant DMR-1451219. A.K.Y., C.T.N., and R.R. acknowledge support from the Office of Basic Energy Sciences, U.S. Department of Energy under contract no. DE-AC02-05CH11231. L.W.M. and R.R. acknowledge support from the Gordon and Betty Moore Foundations EPiQS Initiative, Grant GBMF5307.
- Ferroelectric superlattices
- geometric length scale
- phase-field simulations
- topological structures by design
- ultrafine polar vortex