Inverse transition of labyrinthine domain patterns in ferroelectric thin films.

Phase separation is a cooperative process, the kinetics of which underpin the orderly morphogenesis of domain patterns on mesoscopic scales1,2. Systems of highly degenerate frozen states may exhibit the rare and counterintuitive inverse-symmetry-breaking phenomenon3. Proposed a century ago4, inverse transitions have been found experimentally in disparate materials, ranging from polymeric and colloidal compounds to high-transition-temperature superconductors, proteins, ultrathin magnetic films, liquid crystals and metallic alloys5,6, with the notable exception of ferroelectric oxides, despite extensive theoretical and experimental work on the latter. Here we show that following a subcritical quench, the non-equilibrium self-assembly of ferroelectric domains in ultrathin films of Pb(Zr0.4Ti0.6)O3 results in a maze, or labyrinthine pattern, featuring meandering stripe domains. Furthermore, upon increasing the temperature, this highly degenerate labyrinthine phase undergoes an inverse transition whereby it transforms into the less-symmetric parallel-stripe domain structure, before the onset of paraelectricity at higher temperatures. We find that this phase sequence can be ascribed to an enhanced entropic contribution of domain walls, and that domain straightening and coarsening is predominantly driven by the relaxation and diffusion of topological defects. Computational modelling and experimental observation of the inverse dipolar transition in BiFeO3 suggest the universality of the phenomenon in ferroelectric oxides. The multitude of self-patterned states and the various topological defects that they embody may be used beyond current domain and domain-wall-based7 technologies by enabling fundamentally new design principles and topologically enhanced functionalities within ferroelectric films.

The stabilization of a single domain in free-standing ferroelectric nanocrystals.

High resolution electron microscopy, electron diffraction and electron holography were used to study individual free-standing ∼ 30 nm barium titanate nanocrystals. Large unidirectional variations in the tetragonal distortion were mapped across the smaller nanocrystals, peaking to anomalously large values of up to 4% at the centers of the nanocrystals. This indicated that the nanocrystals consist of highly strained single ferroelectric domains. Simulations using an effective Hamiltonian for modeling a nanocrystal under a small depolarizing field and negative pressure qualitatively confirm this picture. These simulations, along with the development of a phenomenological model, show that the tetragonal distortion variation is a combined effect of: (i) electrostrictive coupling between the spontaneous polarization and strain inside the nanocrystal, and (ii) a surface-induced effective stress existing inside the nanodot. As a result, a 'strain skin layer', having a smaller tetragonal distortion relative to the core of the nanocrystal, is created.

Control of ferroelectricity and magnetism in multi-ferroic BiFeO3 by epitaxial strain.

Recently, strain engineering has been shown to be a powerful and flexible means of tailoring the properties of ABO3 perovskite thin films. The effect of epitaxial strain on the structure of the perovskite unit cell can induce a host of interesting effects, these arising from either polar cation shifts or rotation of the oxygen octahedra, or both. In the multi-ferroic perovskite bismuth ferrite (BiFeO3-BFO), both degrees of freedom exist, and thus a complex behaviour may be expected as one plays with epitaxial strain. In this paper, we review our results on the role of strain on the ferroic transition temperatures and ferroic order parameters. We find that, while the Néel temperature is almost unchanged by strain, the ferroelectric Curie temperature strongly decreases as strain increases in both the tensile and compressive ranges. Also unexpected is the very weak influence of strain on the ferroelectric polarization value. Using effective Hamiltonian calculations, we show that these peculiar behaviours arise from the competition between antiferrodistortive and polar instabilities. Finally, we present results on the magnetic order: while the cycloidal spin modulation present in the bulk survives in weakly strained films, it is destroyed at large strain and replaced by pseudo-collinear antiferromagnetic ordering. We discuss the origin of this effect and give perspectives for devices based on strain-engineered BiFeO3.

Properties of epitaxial films made of relaxor ferroelectrics.

Finite-temperature properties of epitaxial films made of Ba(Zr,Ti)O3 relaxor ferroelectrics are determined as a function of misfit strain, via the use of a first-principles-based effective Hamiltonian. These films are macroscopically paraelectric at any temperature, for any strain ranging between ≃-3% and ≃+3%. However, original temperature-versus-misfit strain phase diagrams are obtained for the Burns temperature (Tb) and for the critical temperatures (Tm,z and Tm,IP) at which the out-of-plane and in-plane dielectric response peak, respectively, which allow the identification of three different regions. These latter differ from their evolution of Tb, Tm,z, and/or Tm,IP with strain, which are the fingerprints of a remarkable strain-induced microscopic change: each of these regions is associated with its own characteristic behavior of polar nanoregions at low temperature, such as strain-induced rotation or strain-driven elongation of their dipoles or even increase in the average size of the polar nanoregions when the strength of the strain grows.

Field-induced percolation of polar nanoregions in relaxor ferroelectrics.

A first-principles-based effective Hamiltonian is used to investigate low-temperature properties of Ba(Zr,Ti)O(3) relaxor ferroelectrics under an increasing dc electric field. This system progressively develops an electric polarization that is highly nonlinear with the dc field. This development leads to a maximum of the static dielectric response at a critical field, E(th), and involves four different field regimes. Each of these regimes is associated with its own behavior of polar nanoregions, such as shrinking, flipping, and elongation of dipoles or change in morphology. The clusters propagating inside the whole sample, with dipoles being parallel to the field direction, begin to form at precisely the E(th) critical field. Such a result, and further analysis we perform, therefore, reveal that field-induced percolation of polar nanoregions is the driving mechanism for the transition from the relaxor to ferroelectric state.

Broken local symmetry in paraelectric BaTiO3 proved by second harmonic generation.

Precursor dynamics of a cubic to tetragonal ferroelectric phase transition in BaTiO3 is studied by the accurate measurement of the second harmonic generation (SHG) integral intensities. A finite signal holds for the SHG integrated intensity above the ferroelectric Curie temperature T(c)=403 K. Above the Burn's temperature T(d)≈580 K, the power law with the exponent γ=1 shows normal SHG nature originating from the hyper-Raman scattering by dynamical polar excitations, while, below T(d), a SHG signal from polar nanoregions becomes dominant with the larger exponent γ=2. Such a crossover of the power law exponent near T(d) is discussed on the basis of the effective Hamiltonian method and Monte Carlo simulation.

Finite-temperature properties of Ba(Zr,Ti)O3 relaxors from first principles.

A first-principles-based technique is developed to investigate the properties of Ba(Zr,Ti)O(3) relaxor ferroelectrics as a function of temperature. The use of this scheme provides answers to important, unresolved and/or controversial questions such as the following. What do the different critical temperatures usually found in relaxors correspond to? Do polar nanoregions really exist in relaxors? If yes, do they only form inside chemically ordered regions? Is it necessary that antiferroelectricity develop in order for the relaxor behavior to occur? Are random fields and random strains really the mechanisms responsible for relaxor behavior? If not, what are these mechanisms? These ab initio based calculations also lead to deep microscopic insight into relaxors.

Strain dependence of polarization and piezoelectric response in epitaxial BiFeO3 thin films.

Epitaxial strain has recently emerged as a powerful means to engineer the properties of ferroelectric thin films, for instance to enhance the ferroelectric Curie temperature (T(C)) in BaTiO(3). However, in multiferroic BiFeO(3) thin films an unanticipated strain-driven decrease of T(C) was reported and ascribed to the peculiar competition between polar and antiferrodistortive instabilities. Here, we report a systematic characterization of the room-temperature ferroelectric and piezoelectric properties for strain levels ranging between -2.5% and +1%. We find that polarization and the piezoelectric coefficient increase by about 20% and 250%, respectively, in this strain range. These trends are well reproduced by first-principles-based techniques.

Thickness-dependent polarization of strained BiFeO3 films with constant tetragonality.

We measure the ferroelectric polarization of BiFeO3 films down to 3.6 nm using low energy electron and photoelectron emission microscopy. The measured polarization decays strongly below a critical thickness of 5-7 nm predicted by continuous medium theory whereas the tetragonal distortion does not change. We resolve this apparent contradiction using first-principles-based effective Hamiltonian calculations. In ultrathin films, the energetics of near open circuit electrical boundary conditions, i.e., an unscreened depolarizing field, drive the system through a phase transition from single out-of-plane polarization to nanoscale stripe domains. It gives rise to an average polarization close to zero as measured by the electron microscopy while maintaining the relatively large tetragonal distortion imposed by the nonzero polarization state of each individual domain.

Phase transitions in epitaxial (-110) BiFeO3 films from first principles.

The effect of misfit strain on properties of epitaxial BiFeO3 films that are grown along the pseudocubic [110] direction, rather than along the usual [001] direction, is predicted from density-functional theory. These films adopt the monoclinic Cc space group for compressive misfit strains smaller in magnitude than ≃1.6% and for any investigated tensile strain. In this Cc phase, both polarization and the axis about which antiphase oxygen octahedra tilt rotate within the epitaxial plane as the strain varies. Surprisingly and unlike in (001) films, for compressive strain larger in magnitude than ≃1.6%, the polarization vanishes and two orthorhombic phases of Pnma and P2(1)2(1)2(1) symmetry successively emerge via strain-induced transitions. The Pnma-to-P2(1)2(1)2(1) transition is a rare example of a so-called pure gyrotropic phase transition, and the P2(1)2(1)2(1) phase exhibits original interpenetrated arrays of ferroelectric vortices and antivortices.

BiFeO3 films under tensile epitaxial strain from first principles.

Density-functional calculations are performed to predict structural and magnetic properties of (001) BiFeO(3) films under tensile epitaxial strain. These films remain monoclinic (Cc space group) for misfit strains between 0% and ≈8%, with the polarization, tilt axis and magnetization all rotating when varying the strain. At a tensile strain ≈8%, these films undergo a first-order phase transition towards an orthorhombic phase (Ima2 space group). In this novel phase, the polarization and tilt axis lie in the epitaxial plane, while the magnetization is along the out-of-plane direction and the direction of the antiferromagnetic vector is unchanged by the phase transition. An unexpected additional degree of freedom, namely, an antiphase arrangement of Bi atoms, is also found for all tensile strains.

Kittel law in BiFeO3 ultrathin films: a first-principles-based study.

A first-principles-based effective Hamiltonian is used to investigate the thickness dependency of the size of straight-walled domains in ultrathin films made of the multiferroic BiFeO3 (BFO) material. It is found that the Kittel law is followed, as in ferroelectric or ferromagnetic films. However, an original real-space decomposition of the different energetic terms of this effective Hamiltonian allows the discovery that the microscopic origins of such a law in BFO films dramatically differ from those in ferroelectric or ferromagnetic films. In particular, interactions between tilting of oxygen octahedra around the domain walls and magnetoelectric couplings near the surface (and away from the domain walls) play an important role in the observance of the Kittel law in the studied BFO films.

Discovery of incipient ferrotoroidics from atomistic simulations.

An effective Hamiltonian technique is used to investigate the effect of quantum vibrations on properties of stress-free KTaO3 nanodots under open-circuit electrical boundary conditions. We discover that these vibrations suppress the paraelectric-to-ferrotoroidic transition, or, equivalently, wash out the formation of vortex states. Such suppression leads to the saturation of the so-called ferrotoroidic susceptibility at low temperature, and to a peculiar local structure that exhibits short-range, needlelike correlations of the individual toroidal moments.

Electric field induced by dynamical change of dipolar configurations in ferromagnets.

An analytical expression for the electric field, Eint, induced by any dynamical change of dipolar configuration is derived for ferromagnets. Effective Hamiltonian simulations are further conducted to realistically compute such field in an asymmetric permalloy ring. It is found that Eint mostly consists of short pulses that are correlated with the rapid temporal change of the magnetic toroidal moment in this low-dimensional ferromagnet, thus providing macroscopic information about the dynamical change of magnetic vortices. Discussion about the connection between Eint and some electric fields recently mentioned in the literature is also provided.

Controlling double vortex states in low-dimensional dipolar systems.

The reversal process of the chirality of each opposite vortex belonging to a double vortex state in ferromagnetic hysterons, via the application of in-plane magnetic fields, is reported. Simulations reveal that such a process involves the formation of four intermediate states, including original ones. Hysteresis loops can occur only in a counterclockwise fashion because of one of these intermediate states. Double vortex states can also be controlled by electric fields in ferroelectric nanostructures of different shapes, but with some key differences with respect to the ferromagnetic case.

Control of vortices by homogeneous fields in asymmetric ferroelectric and ferromagnetic rings.

Effective Hamiltonians have been used (i) to demonstrate that the shape asymmetry of ferromagnetic rings is essential to the recently discovered switching of the chirality of their vortices by homogeneous magnetic fields, via a transition into onion states; (ii) to reveal that an electric vortex can also be controlled by a homogeneous electric field in asymmetric ferroelectric nanorings, via the formation of antiferrotoroidic pair states rather than onion states; and (iii) to provide the fundamental reason that allows such control, namely, two new interaction energies involving a vector characterizing the asymmetry, the applied field, and the toroidal moment.

Quantum paraelectricity coexisting with a ferroelectric metastable state in single crystals of NaNbO(3): a new quantum effect.

We experimentally show that, in contrast to the data having been collected so far, some single crystals of NaNbO(3) exhibit a dielectric permittivity of several thousand, at low T, and this value is saturated when approaching 0 K on cooling. Other sodium niobate crystals (having larger dielectric losses) present a first-order phase transition to a ferroelectric phase on cooling (at 80-200 K). The width of the thermal hysteresis in these crystals increases when the temperature of the phase transition obtained on heating decreases. The dielectric permittivity at the phase transition obtained on cooling shows a tendency to increase and saturate, when the thermal hysteresis increases. We identify the ground state of the sodium niobate crystal exhibiting the smallest dielectric losses (in the studied set of crystals) as a novel quantum paraelectric state coexisting with a metastable ferroelectric state. In principle, the crystal presenting the state of quantum paraelectricity can be considered as having the largest (among the crystals studied) thermal hysteresis, for which the low boundary is below 0 K.

Properties of ferroelectric nanodots embedded in a polarizable medium: atomistic simulations.

An atomistic approach is used to investigate finite-temperature properties of ferroelectric nanodots that are embedded in a polarizable medium. Different phases are predicted, depending on the ferroelectric strengths of the material constituting the dot and of the system forming the medium. In particular, novel states, exhibiting a coexistence between two kinds of order parameters or possessing a peculiar order between dipole vortices of adjacent dots, are discovered. We also discuss the origins of these phases, e.g., depolarizing fields and medium-driven interactions between dots.

Controlling toroidal moment by means of an inhomogeneous static field: an ab initio study.

A first-principles-based approach is used to show (i) that stress-free ferroelectric nanodots under open-circuit-like electrical boundary conditions maintain a vortex structure for their local dipoles when subject to a transverse inhomogeneous static electric field, and, more importantly, (ii) that such a field leads to the solution of a fundamental and technological challenge: namely, the efficient control of the direction of the macroscopic toroidal moment. The effects responsible for such striking features are revealed and discussed.