Unusual Polarization Patterns in Flat Epitaxial Ferroelectric Nanoparticles

We investigate the effects of a lattice misfit strain on a ground state and polarization patterns in flat perovskite nanoparticles (nanoislands of ${\mathrm{BaTiO}}_3$ and PZT) with the use of an ab initio derived effective Hamiltonian. We show that the strain strongly controls the balance between the depolarizing field and the polarization anizotropy in determining the equilibrium polarization patterns. Compressive strain favors 180° stripe or tweed domains while a tensile strain leads to in-plane vortex formation, with the unusual intermediate phase(s) where both ordering motifs coexist. The results may allow us to explain contradictions in recent experimental data for ferroelectric nanoparticles.

Figure 1

Low-temperature polarization distributions on the central $z$ plane in PZT (a)–(c) and BTO (d)–(f) nanoparticles in unstrained states. (a),(d)- $19\times 6$ nanodisks, (b),(e)- $39\times 10$ nanodisks and (c),(f)- $39\times 39\times 10$ particles with square footprints. Shown in red, blue, green, and black are the dipoles directed predominantly along [$\pm 100$] and [$0\pm 10$] in PZT and along [$\pm 1\pm 10$] in BTO systems.

Figure 2

The $z$ component of the toroidal moment as a function of strain in $19\times 6$ (a) and $39\times 10$ (b) nanoparticles. The strains are measured relative to the LDA-calculated lattice constants in bulk cubic structures. Note that in free-standing states the particles are characterized by the nonzero “residual” strains marked by vertical lines. The residual strain increases in $39\times 10$ compared to $19\times 6$ particle, especially in PZT.

Figure 3

Low-temperature three-dimensional polarization patterns in the free-standing (001) PZT (a),(b) and BTO (c),(d) nanoparticles under compressive strain of $-2\%$. (a),(c)- $39\times 10$ nanodisks, (b),(d)- $39\times 39\times 10$ particles with square footprints. The small rectangular insets show the cross sections of a central vertical plane marked by the yellow lines. The red and blue arrows show local dipoles with positive and negative $z$ component, respectively.

Figure 4

Vertical polarization component in the $19\times 6$ PZT disk at different misfit strains. While the values are radially averaged, they correspond to an instant moment of time and represent the fluctuating dipole structure. The lines represent 4th order polynomial fits to the instant polarizations.