Evolution of the domain topology in a ferroelectric

Hexagonal manganites: Strong coupling of ferroelectricity and magnetic orders

Hexagonal manganites belong to an exciting class of materials exhibiting strong interactions between a highly frustrated magnetic system, the ferroelectric polarization, and the lattice. The existence and mutual interaction of different magnetic ions (Mn and rare earth) results in complex magnetic phase diagrams and novel physical phenomena. A summary and discussion of the various properties, underlying physical mechanisms, the role of the rare earth ions, and the complex interactions in multiferroic hexagonal manganites are presented in this review.

Characteristics and controllability of vortices in ferromagnetics, ferroelectrics, and multiferroics

Topological defects in condensed matter are attracting e significant attention due to their important role in phase transition and their fascinating characteristics. Among the various types of matter, ferroics which possess a switchable physical characteristic and form domain structure are ideal systems to form topological defects. In particular, a special class of topological defects-vortices-have been found to commonly exist in ferroics. They often manifest themselves as singular regions where domains merge in large systems, or stabilize as novel order states instead of forming domain structures in small enough systems. Understanding the characteristics and controllability of vortices in ferroics can provide us with deeper insight into the phase transition of condensed matter and also exciting opportunities in designing novel functional devices such as nano-memories, sensors, and transducers based on topological defects. In this review, we summarize the recent experimental and theoretical progress in ferroic vortices, with emphasis on those spin/dipole vortices formed in nanoscale ferromagnetics and ferroelectrics, and those structural domain vortices formed in multiferroic hexagonal manganites. We begin with an overview of this field. The fundamental concepts of ferroic vortices, followed by the theoretical simulation and experimental methods to explore ferroic vortices, are then introduced. The various characteristics of vortices (e.g. formation mechanisms, static/dynamic features, and electronic properties) and their controllability (e.g. by size, geometry, external thermal, electrical, magnetic, or mechanical fields) in ferromagnetics, ferroelectrics, and multiferroics are discussed in detail in individual sections. Finally, we conclude this review with an outlook on this rapidly developing field.

Low-energy structural dynamics of ferroelectric domain walls in hexagonal rare-earth manganites

Domain walls (DWs) in ferroic materials, across which the order parameter abruptly changes its orientation, can host emergent properties that are absent in the bulk domains. Using a broadband (10⁶ to 10¹⁰ Hz) scanning impedance microscope, we show that the electrical response of the interlocked antiphase boundaries and ferroelectric DWs in hexagonal rare-earth manganites (h-RMnO₃) is dominated by the bound-charge oscillation rather than free-carrier conduction at the DWs. As a measure of the rate of energy dissipation, the effective conductivity of DWs on the (001) surfaces of h-RMnO₃ at gigahertz frequencies is drastically higher than that at dc, whereas the effect is absent on surfaces with in-plane polarized domains. First-principles and model calculations indicate that the frequency range and selection rules are consistent with the periodic sliding of the DW around its equilibrium position. This acoustic wave-like mode, which is associated with the synchronized oscillation of local polarization and apical oxygen atoms, is localized perpendicular to the DW but free to propagate along the DW plane. Our results break the ground to understand structural DW dynamics and exploit new interfacial phenomena for novel devices.

Hexagonal RMnO3: a model system for two-dimensional triangular lattice antiferromagnets

The hexagonal RMnO3(h-RMnO3) are multiferroic materials, which exhibit the coexistence of a magnetic order and ferroelectricity. Their distinction is in their geometry that both results in an unusual mechanism to break inversion symmetry and also produces a two-dimensional triangular lattice of Mn spins, which is subject to geometrical magnetic frustration due to the antiferromagnetic interactions between nearest-neighbor Mn ions. This unique combination makes the h-RMnO3 a model system to test ideas of spin-lattice coupling, particularly when both the improper ferroelectricity and the Mn trimerization that appears to determine the symmetry of the magnetic structure arise from the same structure distortion. In this review we demonstrate how the use of both neutron and X-ray diffraction and inelastic neutron scattering techniques have been essential to paint this comprehensive and coherent picture of h-RMnO3.

Evolution of the statistical distribution in a topological defect network

The complex networks of numerous topological defects in hexagonal manganites are highly relevant to vastly different phenomena from the birth of our cosmos to superfluidity transition. The topological defects in hexagonal manganites form two types of domain networks: type-I without and type-II with electric self-poling. A combined phase-field simulations and experimental study shows that the frequencies of domains with N-sides, i.e. of N-gons, in a type-I network are fitted by a lognormal distribution, whereas those in type-II display a scale-free power-law distribution with exponent ∼2. A preferential attachment process that N-gons with a larger N have higher probability of coalescence is responsible for the emergence of the scale-free networks. Since the domain networks can be observed, analyzed, and manipulated at room temperature, hexagonal manganites provide a unique opportunity to explore how the statistical distribution of a topological defect network evolves with an external electric field.

Duality of topological defects in hexagonal manganites

We show that the spontaneous symmetry breaking in multiferroic hexagonal manganites can be chemically manipulated to yield two complementary ground states: the well-known ferroelectric P6(3)cm and an antipolar P3c phase. Both symmetry breakings yield topologically protected vortex defects, with the antipolar vortices dual to those of the ferroelectric. This duality stems from the existence of 12 possible angles of MnO5 tilting, and broad strain-free walls with low energy spontaneously emerge through an intermediate P3c1 state, providing a complete unified symmetry description.

Theoretical Methods of Domain Structures in Ultrathin Ferroelectric Films: A Review

This review covers methods and recent developments of the theoretical study of domain structures in ultrathin ferroelectric films. The review begins with an introduction to some basic concepts and theories (e.g., polarization and its modern theory, ferroelectric phase transition, domain formation, and finite size effects, etc.) that are relevant to the study of domain structures in ultrathin ferroelectric films. Basic techniques and recent progress of a variety of important approaches for domain structure simulation, including first-principles calculation, molecular dynamics, Monte Carlo simulation, effective Hamiltonian approach and phase field modeling, as well as multiscale simulation are then elaborated. For each approach, its important features and relative merits over other approaches for modeling domain structures in ultrathin ferroelectric films are discussed. Finally, we review recent theoretical studies on some important issues of domain structures in ultrathin ferroelectric films, with an emphasis on the effects of interfacial electrostatics, boundary conditions and external loads.

Direct observation of multiferroic vortex domains in YMnO3

Topological vortices with swirling ferroelectric, magnetic and structural anti-phase relationship in hexagonal RMnO₃ (R = Ho to Lu, Y, and Sc) have attracted much attention because of their intriguing behaviors. Herein, we report the structure of multiferroic vortex domains in YMnO₃ at atomic scale using state-of-the-art aberration-corrected scanning transmission electron microscopy (STEM). Two types of displacements were identified among six domain walls (DWs); six translation-ferroelectric domains denoted by α+, γ−, β+, α−, γ+ and β−, respectively, were recognized, demonstrating the interlocking nature of the anti-vortex domain. We found that the anti-vortex core is about four unit cells wide. In addition, we reconstructed the vortex model with three swirling pairs of DWs along the [001] direction. These results are very critical for the understanding of topological behaviors and unusual properties of the multiferroic vortex.

Unfolding of vortices into topological stripes in a multiferroic material

Multiferroic hexagonal RMnO(3) (R=rare earths) crystals exhibit dense networks of vortex lines at which six domain walls merge. While the domain walls can be readily moved with an applied electric field, the vortex cores so far have been impossible to control. Our experiments demonstrate that shear strain induces a Magnus-type force pulling vortices and antivortices in opposite directions and unfolding them into a topological stripe domain state. We discuss the analogy between this effect and the current-driven dynamics of vortices in superconductors and superfluids.

Direct visualization of magnetoelectric domains

The coupling between the magnetic and electric dipoles in multiferroic and magnetoelectric materials holds promise for conceptually novel electronic devices. This calls for the development of local probes of the magnetoelectric response, which is strongly affected by defects in magnetic and ferroelectric ground states. For example, multiferroic hexagonal rare earth manganites exhibit a dense network of boundaries between six degenerate states of their crystal lattice, which are locked to both ferroelectric and magnetic domain walls. Here we present the application of a magnetoelectric force microscopy technique that combines magnetic force microscopy with in situ modulating high electric fields. This method allows us to image the magnetoelectric response of the domain patterns in hexagonal manganites directly. We find that this response changes sign at each structural domain wall, a result that is corroborated by symmetry analysis and phenomenological modelling, and provides compelling evidence for a lattice-mediated magnetoelectric coupling. The direct visualization of magnetoelectric domains at mesoscopic scales opens up explorations of emergent phenomena in multifunctional materials with multiple coupled orders.