Physics of Ferroic and Multiferroic Domain Walls

It is with great pleasure that we present this special issue focused on topological structures in ferroic materials. It is composed of invited contributions from the International Workshop on Topological Structures in Ferroic Materials held at the University of South Wales, Sydney, Australia from 19-21 st May 2015. The aim of the workshop was to capture current research interests regarding interfaces, domains, domain walls, vortices and skyrmions in ferroics as an emerging and highly active fi eld in solid state physics and materials science. Signifi cant emphasis was placed on understanding the fundamental commonalities that bridge topological defects across various types of ferroic materials, and therefore lessons learned on how new and groundbreaking functionalities afforded by the aforementioned concepts could be translated into emerging technologies.

BiFeO3 simultaneously shows antiferromagnetic and ferroelectric order with high transition temperatures, i.e. T N ∼ 370°C and T C ∼ 830°C, respectively. Naturally, it has been inferred that coupling exists between the magnetic and ferroelectric order parameters like in the multiferroic manganites with low transition temperatures. A thorough investigation of the ferroelectric properties of BiFeO3 is therefore in line with the understanding of its multiferroic behaviour. Here, we review the ferroelectric properties of epitaxial (001) oriented BiFeO3 films grown by different techniques on several substrates. Structural characterization along with ferroelectric quantitative analysis point at the high quality of the films. Emphasis is put on identifying the various polarization variants and domain dynamics under an applied bias. In these studies, to unravel the intricate ferroelectric domain structure, piezo-force microscopy scans have been taken along the principal crystallographic directions. Two cases have been analysed. First, a 600 nm thick film grown on SrTiO3 (001) with a thin SrRuO3 underlayer exhibits a mosaic domain pattern due to the presence of both up and down polarization domains. Mainly four polarization domains have been identified in this case, which correspond to two structural domains. Second, epitaxial BiFeO3 films grown on DyScO3 (110) and miscut SrTiO3 (001) with a thin SrRuO3 underlayer show stripe patterns, with mainly two down polarization domains. A single structural domain of orthorhombic SrRuO3 epitaxial underlayer induces this changes in the domain structure of BiFeO3. The suppression of up domains by changing the substrate conditions prove the possibility of ferroelectric domain engineering. The three possible polarization switching mechanisms, namely 71 and 109° rotations, as well as 180° rotation, have been identified by following the domain dynamics in a two-domain epitaxial BiFeO3 film. Interestingly, 180° polarization reversal seems to be the most favorable switching mechanism in epitaxial films under an applied bias along [001]. The observation of both ferroelastic and ferroelectric switching processes open exciting possibilities for the optimization of BiFeO3’s ferroelectric properties and investigation of magnetoelectric coupling in epitaxial films. A recent photoemission study using linearly polarized X-rays proved the coupling between the ferroelectric and antiferromagnetic domain structures.

Ferroelectric domain walls are atomically sharp topological defects that separate regions of uniform polarization. The discovery of electrical conductivity in specific types of walls gave rise to "domain wall nanoelectronics," a technology in which the wall (rather than the domain) stores information. This paradigm shift critically hinges on precise nanoengineering of reconfigurable domain walls. Using specially designed nanofabricated electrodes and scanning probe techniques, we demonstrate a prototype nonvolatile ferroelectric domain wall memory, scalable to below 100 nm, whose binary state is defined by the existence or absence of conductive walls. The device can be read out nondestructively at moderate voltages (<3 V), exhibits relatively high OFF-ON ratios (~103) with excellent endurance and retention characteristics, and has multilevel data storage capacity. Our work thus constitutes an important step toward integrated nanoscale ferroelectric domain wall memory devices.

Magnetic domains in well annealed [001] whisker crystals have been studied by X-ray topography. The three-dimensional shape of the rectangular domain observed by the powder pattern technique was found to be a cuboid. Qualitative explanation is given to the image contrast of 90° domain walls by the dynamical theory of X-ray diffraction in distorted crystals. The lattice distortion around the line of intersection of 90° and 180° domain walls was determined by the analysis of butterfly images. It was also found that the 180° domain wall is made visible by the application of magnetic field. Fringe patterns appeared when the whiskers took the closure domain structure across the axis and also were elastically bent to produce the serrated domain wall. Dynamical calculation proves that they are the moiré fringes produced by the double diffraction in the adjacent domains. Some observations about whisker tips and kinked whiskers are also described.

The discovery of a significant interfacial Dzyaloshinskii-Moriya Interaction (iDMI) in multi-layer thin films launched a fervent research effort into its impact on the behavior of chiral magnetic objects including domain walls (DWs).[1] The field driven creep of these Dzyaloshinskii domain walls as observed by Kerr microscopy remains a popular tool for evaluating the strength of the iDMI, but results are notoriously challenging to interpret.In this talk, we focus on experimental and theoretical investigations into the asymmetric field-driven expansion of magnetic domains in thin films with iDMI, which provide a fascinating landscape to understand creep of 1D elastic manifolds with an internal order parameter.[2] We will review early models of creep based on the DW energy, which matched well with certain experimental observations – namely, that an in-plane field (Hx) coupled with a perpendicular driving field (Hz) leads to asymmetric expansion of the domain as well as minima in the plot of x-velocity vs Hx. However, other common observations were not explained by this model including that, at large values of Hx, the high energy wall orientations reached or exceeded the velocity of low energy orientations. To explain this, Pellegren at el considered the more general elastic energy of the DW, given by the stiffness [σ(θ) + σ''(θ)].[3] By introducing a dispersive stiffness, which accounts for the finite lengthscale over which the wall deforms, it was indeed shown that the elastic energy of opposite extrema on the domain converge as Hx → ∞. This more general model explains some of the features that have also been explained by an attempt frequency that depends on wall chirality (via a chiral damping parameter).[4] The possible role of chiral damping will be discussed briefly.In more recent experiments, Brock et al showed that, in some cases – especially for weak DMI, the domain adopts a highly preferred growth direction with a dendritic structure that is neither parallel or anti-parallel to Hx.[5] The results cannot be explained by a chiral Bloch component to the DW, which would violate the symmetry of the system. Instead, we identify a steady-state DW substructure that results from the z-field torque on the internal DW magnetization (see figure 1c). This breaks the energetic mirror symmetry along the x-axis and leads to orientations of the DW with vanishing elastic energy (i.e. rapidly increased creep velocity). Remarkable agreement between an augmented form of the dispersive stiffness model (now accounting for steady-state configurations) and the experimental measurements is found. The material parameters that give rise to this seemingly anomalous behavior will be discussed. Other important factors such as internal DW excitations, field-dependent pinning, and a field-dependent DW width will be considered in this presentation as well. **

A model is discussed for magnetoelectric (ME) interactions originating from the motion of magnetic domain walls (DWs) in a multiferroic composite of orthoferrites $R{\mathrm{FeO}}_{3}$ $(R\text{FO}$) with magnetic stripe domains and a piezoelectric such as lead magnesium niobate-lead titanate (PMN-PT). The DWs in $R\text{FO}$ can be set in motion with an ac magnetic field up to a critical speed of 20 km/s, the highest for any magnetic system, leading to the excitation of bulk and shear magnetoacoustic waves. Thus, the ME coupling will arise from flexural deformation associated with DW motion (rather than the Joule magnetostriction mediated coupling under a static or quasistatic condition). A $c$ plane orthoferrite with a single N\'eel-type DW in the $bc$ plane and an ac magnetic field $H$ along the $c$ axis is assumed. The deflection in the bilayer due to DW motion is obtained when the DW velocity is a linear function $H$ and the resulting induced voltage across PMN-PT is estimated. It is shown that a combination of spatial and time harmonics of the bending deformation leads to (i) a linear ME coefficient defined by ${\ensuremath{\alpha}}_{E}=E/H$ and (ii) a quadratic ME coefficient ${\ensuremath{\alpha}}_{\mathrm{EQ}}=E/{H}^{2}$. The model is applied to yttrium orthoferrites (YFO) and a PMN-PT bilayer since YFO has one of the highest DW mobility amongst the orthoferrites. The coefficient ${\ensuremath{\alpha}}_{E}$ is dependent on the DW position, and it is maximum when the DW equilibrium position is at the center of the sample. In YFO/PMN-PT the estimated low-frequency ${\ensuremath{\alpha}}_{E}$ \ensuremath{\sim} 30 mV/cm Oe and resonance value is 1.5 V/(cm Oe). Since orthoferrites (and PMN-PT) are transparent in the visible region and have a large Faraday rotation, the DW dynamics and the ME coupling could be studied simultaneously. The theory discussed here is of interest for studies on ME coupling and for applications such as magnetically controlled electro-optic devices.

The specific features of the antiferromagnetic domain structure, magnetization, and polarization induced by an inhomogeneous micromagnetic distribution in films of bismuth ferrite multiferroics have been investigated. It has been shown that the magnetic domain structure correlates with the ferroelectric domain structure, and the character of the rotation of the antiferromagnetic vector depends on the type of ferroelectric domain walls. An asymmetry in the distribution of the antiferromagnetic vector has been observed for the cases of 109° and 71° ferroelectric domain walls. It has been demonstrated that there are differences in the distributions of the polarization and magnetization in bismuth ferrite films with ferroelectric domains separated by 109° and 71° walls. The basic mechanisms responsible for the magnetization in domain walls in multiferroics have been considered.

The transverse anisotropy constant and the related D\\"oring mass density are\nkey parameters of the one-dimensional model to describe the motion of magnetic\ndomain walls. So far, no general framework is available to determine these\nquantities from static characterizations such as magnetometry measurements.\nHere, we derive a universal analytical expression to calculate the transverse\nanisotropy constant for the important class of perpendicular magnetic\nmultilayers. All the required input parameters of the model, such as the number\nof repeats, the thickness of a single magnetic layer, and the layer\nperiodicity, as well as the effective perpendicular anisotropy, the saturation\nmagnetization, and the static domain wall width are accessible by static sample\ncharacterizations. We apply our model to a widely used multilayer system and\nfind that the effective transverse anisotropy constant is a factor 7 different\nfrom the when using the conventional approximations, showing the importance of\nusing our analysis scheme.\n

Ferroelectric domain walls (DWs) are promising structures for assembling future nano-electronic circuit elements on a larger scale since reporting domain wall currents of up to 1 mA per single DW. One key requirement hereto is their reproducible manufacturing by gaining preparative control over domain size and domain wall conductivity (DWC). To date, most works on DWC have focused on exploring the fundamental electrical properties of individual DWs within single-shot experiments, with an emphasis on quantifying the origins of DWC. Very few reports exist when it comes to comparing the DWC properties between two separate DWs, and literally nothing exists where issues of reproducibility in DWC devices have been addressed. To fill this gap while facing the challenge of finding guidelines for achieving predictable DWC performance, we report on a procedure that allows us to reproducibly prepare single hexagonal domains of a predefined diameter into uniaxial ferroelectric lithium niobate single crystals of 200 and 300 μm thickness, respectively. We show that the domain diameter can be controlled with an uncertainty of a few percent. As-grown DWs are then subjected to a standard procedure of current-limited high-voltage DWC enhancement, and they repetitively reach a DWC increase of six orders of magnitude. While all resulting DWs show significantly enhanced DWC values, their individual current–voltage (I–V) characteristics exhibit different shapes, which can be explained by variations in their 3D real structure reflecting local heterogeneities by defects, DW pinning, and surface-near DW inclination.

We evaluated the thicknesses of domain walls (DWs) in rhombohedral BiFeO3 thin films epitaxially grown on SrRuO3-covered SrTiO3 (100), (110), and (111) single crystals by scanning nonlinear dielectric microscopy (SNDM). The SNDM phase signal revealed an abrupt change in the sign of polarization components normal to the surface at DWs within one or two unit cells. On the other hand, the SNDM amplitude signals gradually changed across DWs, corresponding to the change in the electrostatic potential. The minimum thicknesses estimated from the SNDM amplitude signals are 4, 1, and 2 nm for 71, 109, and 180° DWs, respectively. The relationship between these DW thicknesses and the nature and situation of DWs is discussed taking account of the polarization configuration in adjacent domains.

Domain wall (DW) excitation spectrum for a bubble-state element has been investigated using three-dimensional micromagnetic simulations within a wide frequency range (0.1--100 GHz). Multiple DW resonance lines have been identified; localized modes at the DW in the low-frequency part of the spectrum and a set of high-frequency DW excitations above the frequency gap of domains spreading into the adjacent domains. In the last case, various DW modes with and without rotational symmetry coupled to multinodal domain modes are revealed. Due to the geometrical confinement leading to a strongly nonuniform static effective field, high-frequency DW modes with the same number of nodes but different localizations and frequencies exist, which prevents the use of a mode indexation with independent quantization numbers. The effects of the size of the element and a longitudinal polarizing magnetic field on the DW spectrum are then analyzed. In particular, the softening of the fundamental DW mode is evidenced at the first-order phase transitions between the bubble state and the saturated states. These numerical findings are discussed in light of theoretical models.

Ferroelectric domains in PbTiO_{3}/SrTiO_{3} superlattices are studied using synchrotron x-ray diffraction. Macroscopic measurements reveal a change in the preferential domain wall orientation from {100} to {110} crystallographic planes with increasing temperature. The temperature range of this reorientation depends on the ferroelectric layer thickness and domain period. Using a nanofocused beam, local changes in the domain wall orientation within the buried ferroelectric layers are imaged, both in structurally uniform regions of the sample and near defect sites and argon ion-etched patterns. Domain walls are found to exhibit a preferential alignment with the straight edges of the etched patterns as well as with structural features associated with defect sites. The distribution of out-of-plane lattice parameters is mapped around one such feature, showing that it is accompanied by inhomogeneous strain and large strain gradients.

Ferroelectric domain walls are a completely new type of functional interface, which have the potential to revolutionize nanotechnology. In addition to the emergent phenomena at domain walls, they are spatially mobile and can be injected, positioned, and deleted on demand, giving a new degree of flexibility that is not available at conventional interfaces. Progress in the field is closely linked to the development of modern microscopy methods, which are essential for studying their physical properties at the nanoscale. In this article, we discuss scanning electron microscopy (SEM) as a powerful and highly flexible imaging technique for scale-bridging studies on domain walls, continuously covering nano- to mesoscopic length scales. We review seminal SEM experiments on ferroelectric domains and domain walls, provide practical information on how to visualize them in modern SEMs, and provide a comprehensive overview of the models that have been proposed to explain the contrast formation in SEM. Going beyond basic imaging experiments, recent examples for nano-structuring and correlated microscopy work on ferroelectric domain walls are presented. Other techniques, such as 3D atom probe tomography, are particularly promising and may be combined with SEM in the future to investigate individual domain walls, providing new opportunities for tackling the complex nanoscale physics and defect chemistry at ferroelectric domain walls.

Mesoscopic polarization structures such as polar nanoregions and polarization domain walls are the key factors connecting microscopic fundamental polarization structures with macroscopic practical dielectric properties. A snapshot observation of the domain in the relaxor ferroelectrics $\mathrm{Pb}[{(\mathrm{M}{\mathrm{g}}_{1/3}\mathrm{N}{\mathrm{b}}_{2/3})}_{1\text{\ensuremath{-}}x}\mathrm{T}{\mathrm{i}}_{x}]{\mathrm{O}}_{3}$ just in the vicinity of morphotropic phase boundary region was performed by use of 7-ps single shot soft x-ray laser pulse. A self-assembled evolution of the oblique polarization domain was observed in $\mathrm{Pb}[{(\mathrm{M}{\mathrm{g}}_{1/3}\mathrm{N}{\mathrm{b}}_{2/3})}_{72.2}\mathrm{T}{\mathrm{i}}_{27.8}]{\mathrm{O}}_{3}$ under the sample temperature decreased with the thermal equilibrium condition. Based on energetic discussion, antiphase shift of domain wall pairs keeping flat boundaries was proposed for the dielectric response. A sharp enhancement in the dielectric response at the vicinity of morphotropic phase boundary region reported previously was recognized as evidence for the hierarchic nature of the present oblique polarization domain wall.

The possibility to control the charge, type, and density of domain walls allows properties of ferroelectric materials to be selectively enhanced or reduced. In ferroelectric-ferroelastic materials, two types of domain walls are possible: pure ferroelectric and ferroelastic-ferroelectric. In this paper, we demonstrated a strategy to control the selective ferroelectric or ferroelastic domain wall formation in the (111) single-domain rhombohedral PMN-PT single crystals at the nanoscale by varying the relative humidity level in a scanning probe microscopy chamber. The solution of the corresponding coupled electro-mechanical boundary problem allows explaining observed competition between ferroelastic and ferroelectric domain growth. The reduction in the ferroelastic domain density during local switching at elevated humidity has been attributed to changes in the electric field spatial distribution and screening effectiveness. The established mechanism is important because it reveals a kinetic nature of the final domain patterns in multiaxial materials and thus provides a general pathway to create desirable domain structure in ferroelectric materials for applications in piezoelectric and optical devices.