Flexoelectricity, incommensurate phases and the Lifshitz point

Abstract

The solutions for the minimizers of the energy density f (q, p) = Aq² + Bq⁴ + p² + gA,B + β(q'p - p'q)+ |q'|² +κ|p'|²] describe the flexoelectric effect with a flexoelectric coupling coefficient β. The order parameters q and p can be visualized as strain and polarisation, respectively. The parameter κ denotes the ratio of intrinsic length scales for q and p. We show that the structural ground-states include 3 phases, namely the paraelastic state q = p = 0, the ferroelastic state where polarization exists inside and near twin boundaries, and the incommensurate (modulated) phases with a very rich array of structural modulations ranging from nearly pure sine waves to kink arrays and ripple states. The phases coincide in the multicritical Lifshitz point. Linear flexoelectricity p~q' is encountered only approximately inside the ferroelastic phase and near the phase boundary between the paraelastic phase and the incommensurate phase. The relationship between the polarisation and the strain gradient is highly non-linear in all other states. The spatial profiles and energy distributions are discussed in detail.

Polarization curling and flux closures in multiferroic tunnel junctions

“…The high degree of disorder with a certain remanent periodicity observed here suggests that the system actually enters into an incommensurate phase. It has been recently predicted that general ferroelastic systems in a certain range of dimensionality no longer undergo a simple para-ferro phase transition, instead entering, across the tri-critical Lifshitz point, an incommensurate phase18. The driving force for this is the flexoelectric effect.…”

“…We obtain local dipole maps by considering the B-site ion displacement with respect to the centre of the O 2− octahedra (Δ B , see the ‘Methods' section). As the film thickness is reduced the ferroelectric domains undergo a transition between 3 regimes: from Kittel-like domains, as seen in thick films17, through complex curling resembling an incommensurate phase18 to monodomain material in the very thinnest layers. We find that, in contrast to thicker films where the domain walls are thin and relatively smooth19, the domain walls in ultrathin ferroelectrics develop into topological entities with a flourishing domain structure involving polarization curling, spanning from Bloch(Néel)-Ising structures to flux closure and vortices.…”

Hydrodynamics of domain walls in ferroelectrics and multiferroics: Impact on memory devices

“…And this year Salje et al [12][13][14] found that ferroelastics are described well by hydrodynamics. The modest breakthrough in the present case follows the recent paper by Scott,3 which suggests that if indeed domain walls follow hydrodynamic flow, they should exhibit hydrodynamic instabilities also.…”

“…5,[12][13][14][15] Salje's Model: The basic assumption in Salje's model is that unlike ferroelectric switching, the hysteresis in ferroelastic switching is dominated by continuum fluid mechanics and not the lattice symmetry. He points out that for the strain reversal step from -S to +S, under a positive (reversing) stress to an initially negative strain, the ferroelastic hysteresis is dominated by viscous flow, with a complex domain structure sometimes p.6 describable as a "domain glass.…”

Ferroelastic Domain Boundary-Based Multiferroicity

“…Theoretical arguments lead to the conclusion that vortex structures are expected to exist [68,69] although no direct domain observations were yet reported. Nevertheless, the results of Zhao et al [69] were directly inspired by the simulations in SrTiO 3 [18] and may yet be the strongest evidence that such flicker vortex states may be observable in SrTiO 3 .…”