Nanoscale ferroelectrics are expected to exhibit various exotic domain configurations, such as the full flux-closure pattern that is well known in ferromagnetic materials. Here we observe not only the atomic morphology of the flux-closure quadrant but also a periodic array of flux closures in ferroelectric PbTiO3 films, mediated by tensile strain on a GdScO3 substrate. Using aberration-corrected scanning transmission electron microscopy, we directly visualize an alternating array of clockwise and counterclockwise flux closures, whose periodicity depends on the PbTiO3 film thickness. In the vicinity of the core, the strain is sufficient to rupture the lattice, with strain gradients up to 10(9) per meter. Engineering strain at the nanoscale may facilitate the development of nanoscale ferroelectric devices.

https://science.sciencemag.org/content/sci/348/6234/547.full.pdf

Observation of a periodic array of flux-closure quadrants in strained ferroelectric PbTiO3 films

The first self-consistent computations of piezoelectricity in a ferroelectric are presented. Bulk spontaneous polarization, dynamical charges ( ${Z}^{*}$), and the full piezoelectric stress tensor of ${\mathrm{PbTiO}}_{3}$ were determined from ground-state Berry's phase calculations using the all-electron linearized augmented plane wave method. Both the proper and total piezoelectric moduli were computed and were found to be significantly different. The theoretical piezoelectric stress moduli, ${e}_{15}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3.15\mathrm{C}/{\mathrm{m}}^{2}$, ${e}_{31}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}0.93\mathrm{C}/{\mathrm{m}}^{2}$, and ${e}_{33}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3.23\mathrm{C}/{\mathrm{m}}^{2}$, agree well with single crystal experimental data.

First-Principles Study of Piezoelectricity in PbTiO 3

We develop a general and unified first-principles theory of piezoelectric and flexoelectric tensor, formulated in such a way that the tensor elements can be computed directly in the context of density-functional calculations, including electronic and lattice contributions. We introduce practical supercell-based methods for calculating the flexoelectric coefficients from first principles, and demonstrate them by computing the coefficients for a variety of cubic insulating materials including C, Si, MgO, NaCl, CsCl, BaZrO${}_{3}$, BaTiO${}_{3}$, PbTiO${}_{3}$, and SrTiO${}_{3}$.

First-principles theory and calculation of flexoelectricity

First-principles supercell calculations and the modern theory of polarization are used to compute the ${e}_{33}$ piezoelectric coefficients of $\mathrm{Pb}({\mathrm{Zr}}_{0.5}{\mathrm{Ti}}_{0.5}{\mathrm{O}}_{3})$ (PZT) and $0.60\mathrm{Pb}({\mathrm{Mg}}_{1/3}{\mathrm{Nb}}_{2/3}){\mathrm{O}}_{3}+0.40{\mathrm{PbTiO}}_{3}$ (PMN-PT) alloys. A drastic enhancement by a factor of 2.7 is found for ${e}_{33}$ when going from PZT to PMN-PT. The huge value of ${e}_{33}$ in PMN-PT comes from the large response of the internal coordinates of Pb, Ti, Nb, and O atoms to a macroscopic strain. On the other hand, the Mg atoms contribute little to the piezoelectricity for dielectric and elastic reasons.

/pdf/intrinsic-piezoelectric-response-in-perovskite-alloys-pmn-pt-4df3mwnimj.pdf

Intrinsic piezoelectric response in perovskite alloys : pmn-pt versus pzt

We report on first-principles density functional perturbation theory calculations and inelastic neutron scattering measurements of the phonon density of states, dispersion relations, and electromechanical response of $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_{3}$, $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_{3}$, and $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_{3}$. The phonon density of states of the quantum paraelectric $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_{3}$ is found to be fundamentally distinct from that of ferroelectric $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_{3}$ and $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_{3}$ with a large, $70--90\phantom{\rule{0.3em}{0ex}}\mathrm{meV}$, phonon band gap. The phonon dispersion and electromechanical response of $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_{3}$ reveal giant anisotropies. The interplay of covalent bonding and ferroelectricity strongly modulates the electromechanical response and gives rise to spectacular signatures in the phonon spectra. The computed charge densities have been used to study the bonding in these perovskites. Distinct bonding characteristics in the ferroelectric and paraelectric phases give rise to spectacular vibrational signatures. While a large phonon band gap in $A\mathrm{Ti}{\mathrm{O}}_{3}$ perovskites seems to be a characteristic of quantum paraelectrics, anisotropy of the phonon spectra correlates well with ferroelectric strength. These correlations between the phonon spectra and ferroelectricity can guide future efforts at custom designing still more effective piezoelectrics for applications. These results suggest that vibrational spectroscopy can help design novel materials.

Large phonon band gap in Sr Ti O 3 and the vibrational signatures of ferroelectricity in A Ti O 3 perovskites: First-principles lattice dynamics and inelastic neutron scattering

Brillouin light scattering was used to obtain the elastic and piezoelectric constants for tetragonal PbTiO3 single crystals at room temperature. Approximately 170 measurements of longitudinal and transverse acoustic wave velocities were inverted using literature dielectric constant values to obtain the full set of elastic and piezoelectric constants. Our data indicate a greater electromechanical anisotropy in the polar c direction than previously reported, but are otherwise in general agreement with previous studies. We discuss briefly the degree to which individual elastic and piezoelectric constants are resolved.

Elastic properties of tetragonal pbtio3 single crystals by brillouin scattering

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Elastic properties of tetragonal pbtio3 single crystals by brillouin scattering