Ferroelectric Vortices and Related Configurations

22Ferroelectric Vortices and Related

Configurations

Sergei Prosandeev1,2, Ivan I. Naumov3, Huaxiang Fu1, Laurent Bellaiche1,Michael P.D. Campbell4, Raymond G.P. McQuaid4, Li-Wu Chang4,

Alina Schilling4, Leo J. McGilly5, Amit Kumar4, and J. Marty Gregg4

1Physics Department, University of Arkansas, USA2Physics Department and Research Institute of Physics, Southern Federal University, Russia

3Carnegie Institution of Washington, USA4School of Maths and Physics, Queen’s University Belfast, Northern Ireland, United Kingdom

5Ceramics Laboratory, Ecole Polytechnique Federale de Lausanne (EPFL), Switzerland

22.1 Insights from Simulations and Theory

22.1.1 Introduction

Phase transition in systems of reduced dimension has been a subject of long-standinginterest both fundamentally and technologically [1–4]. Fundamentally, interactions in low-dimensional systems are truncated along certain directions or are screened differently thanin bulk, which will significantly alter collective behaviors and relative stabilities of differ-ent structural phases. Meanwhile, the thermal fluctuations in low-dimensional systems aremore destructive to the formation of ordered phases than in bulks [3]. As a result, phasetransformations in reduced dimensions can be profoundly different from what they are inbulk counterparts.

Moreover, when systems are near the critical points of phase transitions, their propertiesare often very sensitive to small modifications of temperature, pressure, composition, or

Nanoscale Ferroelectrics and Multiferroics: Key Processing and Characterization Issues, and Nanoscale Effects,First Edition. Edited by Miguel Alguero, J. Marty Gregg, and Liliana Mitoseriu.© 2016 John Wiley & Sons, Ltd. Published 2016 by John Wiley & Sons, Ltd.

Ferroelectric Vortices and Related Configurations 701

external strain, which usually leads to exceptionally large responses that have technologi-cal applications. This is particularly true in ferroelectric (FE) systems. For instance, whenundergoing ferroelectric transitions, these FEs give rise to large piezoelectric, dielectric, orelectromechanical responses [5]. The increasing demands of smaller but efficient nanoscaledevices require that these large responses continue to remain as devices are miniaturized.

Furthermore, bulk ferroelectrics, which develop non-zero polarization after undergoingthe paraelectric-to-ferroelectric phase transformation at the Curie temperature, give rise tobistable or multistable degenerate states. These multistable states can be utilized for storingdata in non-volatile ferroelectric random access memories (NFERAMs) [6].

Compared to bulk ferroelectrics, zero-dimensional FE dots promise to increase the stor-age density of NFERAMs thousandsfold. Obviously, this anticipated benefit would dependon whether phase transitions and multistable states continue to exist in zero-dimensional FEstructures. This further explains why phase transitions in ferroelectrics of reduced dimen-sion are technologically relevant, in addition to the fundamental interest described above.Interestingly, the recent intense investigation of FE nanostructures have resulted in the sur-prising and exciting discoveries of dipolar patterns that do not exist in the bulk counter-part, because of electrical and mechanical boundary conditions as well as size effects thatare inherent to low-dimensional objects. Examples of such patterns are electric vorticesor related flux-closure configurations, which were first predicted by simulations [7–9] andthen experimentally confirmed (see [10] to [20]).

The aims of this article are fourfold: (1) to first briefly describe numerical approachesthat have been used to discover and further simulate FE vortices and related configurations;(2) to report and discuss findings about such patterns and their original order parameters inFE nanostructures; (3) to emphasize novel possibilities of switching due to these patterns;and (4) to provide examples of other unusual phenomena and effects that originate from FEvortices and other complex dipolar organizations.

22.1.2 Methods

Let us first (briefly) review several different computational and theoretical schemes thathave been employed to mimic and study properties of ferroelectric vortices.

22.1.2.1 Effective Hamiltonian Approach

One common scheme is the so-called effective Hamiltonian approach, for which the totalenergy of ferroelectric nanostructures is generally written as:

𝜀tot(ui, vi, 𝜂H) = 𝜀Heff(ui, vi, 𝜂H) + 12𝛽

∑𝜄

⟨Edep

⟩⋅ Z∗ui −

∑i

E ⋅ Z∗ui − a3∑

j

𝜎j𝜂H,j

(22.1)

where the degrees of freedom are: (1) the local soft mode in the unit cell i, ui, whose prod-uct with the effective charge Z∗ yields the local electrical dipole, {pi}, in this cell; (2)the homogeneous strain tensor, 𝜂H; and (3) inhomogeneous strain-related (acoustic-type)displacements in unit cell i, vi [21]. Note that there are effective Hamiltonians that alsoinclude oxygen octahedral tiltings and magnetic dipoles as other degrees of freedom (see,for example, [22] and [23]).

702 Nanoscale Ferroelectrics and Multiferroics

In Equation (22.1) 𝜀Heff represents the intrinsic (effective Hamiltonian) energy of thestudied ferroelectrics, with its parameters having been generally extracted by performingfirst-principles calculations on small cells. Analytical expressions of 𝜀Heff are typicallythose of bulk systems (see, for example, [21], [24], and [25]), except for two main modifica-tions. The first change consists of adding energetic terms that are associated with the directinteraction between the vacuum surrounding the nanoferroelectric and both the dipolesand inhomogeneous strain near the surface [7, 26]. The second modification consists inreplacing the (reciprocal-space-based) matrix associated with long-range dipole–dipoleinteractions in the bulk [21] by the corresponding matrices characterizing dipole–dipoleinteractions in the zero-dimensional, one-dimensional, or two-dimensional ferroelectric –implying that no supercell periodic boundary conditions are used to simulate ferroelectricnanostructures. Such matrices are given by Naumov and co-workers [27,28] and correspondto ideal open-circuit (OC) conditions, for which no possible screening of the polarization-induced surface charges can exist. These electrical boundary conditions naturally leadto the existence of a maximum depolarizing field (denoted by <Edep> and determinedfrom the atomistic approach of [28]) inside the nanoferroelectric for a non-vanishingpolarization.

The second term of Equation (22.1) mimics a screening of <Edep> via the 𝛽 parameter.More precisely, the residual depolarizing field resulting from the combination of the firstand second term of Equation (22.1) has a magnitude equal to (1 − b)|<Edep>|. Therefore𝛽 = 0 corresponds to ideal OC conditions, while an increase in 𝛽 lowers the magnitude ofthe resulting depolarizing field; 𝛽 = 1 corresponds to ideal short-circuit (SC) conditions forwhich the depolarizing field has vanished (i.e. for which all polarization-induced surfacecharges are screened).

The third and fourth terms of Equation (22.1) represent the effect of an electric field,E, and applied stress, 𝜎, on properties of the investigated system, respectively [29], with abeing the five-atom lattice constant at 0 K and the sum over j in the fourth term runningfrom 1 to 6 for the components of the strain tensor (as consistent with Voigt notations).

Effective Hamiltonians can be implemented within both Monte Carlo (MC) and molecu-lar dynamics (MD) simulations. In the MC scheme, a large number of MC sweeps (typicallyof the order of 40 000) is used to first equilibrate the system and then compute statisticalaverages – at each temperature, electric field, and/or stress values. In order to find the groundstate, the temperature is usually decreased in small steps, starting from high temperatures(for which any long-range dipolar order is absent). Within the MD technique, statisticalquantities can be computed as a function of time and an alternating AC electric field canalso be applied.

22.1.2.2 Phase Field Method

In contrast to the effective Hamiltonian method, which is atomistic in nature, the phasefield method employs a continuum approach, in which the polarization field, P(r), is theorder parameter. A variation functional F(P(r)) is developed, which consists of three mainenergies:

F = FGL + Felectrost + Felastic (22.2)


where FGL is the Ginzburg–Landau contribution:

FGL =∫

[f0 (P(r)) + 1

2𝛽ijkl

𝜕Pi

𝜕xj

𝜕Pk

𝜕xl

]d3r (22.3)

with

f0(P(r)) = 𝛼1

(P2

1 + P22 + P2

3

)+ 𝛼11

(P4

1 + P42 + P4

3

)+ 𝛼12

(P2

1P22 + P2

1P23 + P2

2P23

)+ 𝛼111

(P6

1 + P62 + P6

3

)+ 𝛼112

(P2

1

(P4

2 + P43

)+ P2

2

(P4

1 + P43

)+ P2

3

(P4

1 + P42

))+ 𝛼123P2

1P22P2

3 (22.4)

Here the subscripts 1, 2, and 3 correspond to the x, y, and z Cartesian components, respec-tively.

The coefficients in this expansion can be found from first-principles calculations or fromfitting experimental data. The homogeneous contribution to the free energy expanded intoTaylor series with respect to the polarization is f0(P(r)), up to the 6th order. The secondterm in Equation (22.3) gathers the gradient corrections of this expansion.

Moreover, the electrostatic interaction energy, Felectrost, appearing in Equation (22.2) iscalculated as [30]:

Felectrost =1

2𝜀0𝜀∞

3∑i=1

3∑j=1

∫

kikj

k2PikPjk

d3k(2π)3

− E ⋅ P0 (22.5)

where Pik is the ith component of the Fourier transform of the polarization field P(r), and kis the corresponding wave vector. P0 is the homogeneous part of the polarization, associatedwith the zero wave vector. Note that the zero wave vector contribution is excluded from thesummation associated with the first term and 𝜀0 is the dielectric constant of vacuum and𝜀∞ is the high-frequency dielectric permittivity.

In Equation (22.2), Felastic is an elastic energy, which is computed as:

Felastic =12

3∑i=1

3∑j=1

3∑k=1

3∑l=1

Cijkl∫

[𝜂ij(r)𝜂kl(r)

]d3r −

3∑i=1

3∑j=1

𝜎ij∫

[𝜂ij(r)

]d3r (22.6)

where Cijkl are elastic coefficients, 𝜂 is the strain, and 𝜎 is the external stress. Practically[30], the strain is related to the polarization field, via the electrostriction coefficients Qijkl:

𝜂ij(r) =3∑

k=1

3∑l=1

QijklPkPl (22.7)

A variation method is used to find the polarization field numerically [30], based on theGinzburg–Landau kinetic equation:

𝜕Pi

𝜕t= −L

𝜕F𝜕Pi

+ 𝜉i (22.8)

where 𝜉i is the Cartesian components of the Langevin noise [31].


22.1.2.3 Shell Model Calculations

The shell model is an atomistic approach, like the effective Hamiltonian method. However,unlike this latter scheme, the shell model incorporates all structural degrees of freedom butalso relies on rather simple analytical forms of interatomic potentials (which we will dis-cuss below). In the shell model, each ion is described as a massive core that is connectedto the outer shell by a spring. Notice that, in ferroelectrics, the core–shell interaction isrequired to be anisotropic, and this anisotropy is crucial for the description of phases hav-ing different symmetry (see, for example, [32] and references therein). These shells arealso connected to each other by additional springs. Among these springs, some are mim-icking long-range electrostatic interactions while others simulate short-range interactions.Two different types of short-range interatomic potentials are usually employed. Firstly, aRydberg potential, V(r) = (a + br) exp(r∕p), is used, for example for the Pb–Ti, Pb–O, andTi–O pairs in PbTiO3 systems. Secondly, the Buckingham potential V(r) = c exp(r∕𝜁 ) + fr6

is often used for O–O interactions, in, for example, PbTiO3 systems. The a, b, 𝜌, c, 𝜁 , andf coefficients in these potentials are usually found from least-square fitting of first princi-ples calculations or experimental phonon spectra. Then molecular dynamics calculationsare typically performed in order to find various properties, including the dipolar pattern.

22.1.2.4 First-Principles Approach

First-principles calculations are also atomistic in nature. They do not require any input fromexperiments and are based on the density functional theory [33]. This theory demonstratesthat the many-electron system ground-state properties can be uniquely determined from theelectron charge density, once knowing the energy functional (such determination is typi-cally done in a self-consistent way using Kohn–Sham equations). Unfortunately, there isan analytically unknown part of this functional, that is the so-called exchange and corre-lation term. Various solutions have been proposed and tested for that latter term, and theiraccuracies were basically found to depend on the type of systems to be investigated [34].

22.1.3 Discovery of Unusual Patterns in Ferroelectric Nanostructures and TheirAssociated Order Parameters

Properties of FE nanostructures are largely governed by the existence of a depolarizationfield, which is much stronger than the demagnetization field in magnetic nanosystems. Con-sequently, polarization and monodomain may vanish in low-dimensional FEs in order toavoid the drastic increase in energy caused by the depolarization field. Even in systemswith incomplete screening such as in SrRuO3/BaTiO3/SrRuO3 [35], where the depolariza-tion field is significantly weakened as compared to systems under open-circuit electricalboundary condition [36, 37], there is still a critical thickness below which an FE mon-odomain could disappear.

Ferroelectricity in zero-dimensional dots is even more interesting. A pioneer studyrevealed that there are large atomic off-center displacements in barium titanate nanodotshaving less than 5 nm for their lateral size, despite the fact that these dots are under open-circuit electrical boundary condition [7]. These displacements were shown to be robustas different short-range interactions were used to mimic different capping matrix materials.Although the net total polarization vanishes, the existence of large off-center displacements


is nevertheless important for the following reason. If we assume that atomic off-center dis-placements do not exist in dots, very large electric fields will then be required to generateindividual dipoles (much like in, for example, semiconductors), which will be detrimentalfor technological applications. In contrast, with large off-center displacements in FE dots,one may only need a small electric field to generate ferroelectric polarization by aligningthe already-existing dipoles, a conclusion which was indeed borne out by theoretical sim-ulations when the electric field was applied in BaTiO3 dots [7].

The vanishing net polarization at all temperatures and the lack of a ferroelectric phasein dots being under open-circuit electrical boundary conditions reveal that there are nopolarization-induced multistable states in these dots. One therefore cannot use the polar-ization itself for the purpose of making non-volatile ferroelectric memories. This initiallyappeared to be a drawback at the early stage of the FE dot study.

However, the situation quickly changed, mostly because a new type of phase transitionwas discovered in Pb(Zr,Ti)O3 (PZT) nanosystems [8]. Instead of looking at the net polar-ization P =

∑i pi, Naumov, Bellaiche, and Fu found and examined a new order parameter,

namely the toroidal moment G =∑

i ri × pi, where pi is the local dipole moment of site ithat is located at ri. By performing finite-temperature simulations using a first-principles-derived effective Hamiltonian for Pb(ZrTi)O3 nanodisks and nanorods, they showed thatthis new order parameter vanishes at high temperature due to the fact that dipoles becomerandomly oriented. However, this order parameter is very large when the temperature iswell below a certain critical value, demonstrating clearly the existence of a phase transi-tion [8]. This phase transition originates from the spontaneous formation of the toroidalmoment.

The finding of a toroidal phase transition is of fundamental significance since, priorto this, scientists widely believed that there may not be any phase transition in zero-dimensional ferroelectric nanoparticles. The toroidal phase transition thus offers a new anddifferent thinking. Meanwhile, it is useful to point out that the toroidal phase transitionfound in a single dot does not contradict Landau’s phase transition theory. The latter isformulated for the thermodynamic limit, which applies to an array of dots with weak inter-actions between dots. In [8], an ordered vortex phase occurs in a single dot. When dots forman array as in the thermodynamic limit, the vortices in different dots may become randomwhen the coupling between dots is weak and when thermal fluctuation is sufficiently strong.

The existence of a toroidal phase transition is also technologically important. The toroidalmoment G may point up or down, therefore representing two degenerate states. Thesebistable states of the toroidal moment phase can thus be utilized to make ferroelectric mem-ory devices. More specifically, scientists can use one dot to store one data bit. The minimumdiameter of the disks that display low-temperature structural bistability was determined tobe 3.2 nm [8], enabling an ultimate NFERAM density of 60 x 1012 bits per square inch. Fur-thermore, since the interaction between toroid moments in different dots is generally weak,the problem of cross-talking between different memory units can thus be largely avoided,which is another advantage.

After the discovery of the toroidal phase transition in PZT nanodisks, ferroelectricvortex structures were subsequently confirmed by other studies using various differentmethods, such as phase field models, Ginzburg–Landau theory, atomistic simulations, andfirst-principles density functional calculations. The details of these important studies aredescribed in the following.


Using phase field simulations, Slutsker, Artemev, and Roytburd investigated domainstructures in confined nanoferroelectrics of different shapes and sizes [30]. They found anelectrostatically compatible circuit of 180 and 90◦ domains at equilibrium. The formationof 90◦ domains is very similar to the FE vortex, and the trend to minimize the residual stressand the stray field was found to be responsible for the crater-shaped sets of closed circuitsof 90◦ domains.

Munch and Huber examined the FE closure structures in tetragonal ferroelectric crystalsemploying a phase-field model, and their study shows a stable polarization vortex of sixdomains over a narrow range of aspect ratios in freestanding cuboidal nanodots [38]. Forother aspect ratios, the authors found that square polarization vortices of four domains arestable. In both cases, the stable FE vortex domains were shown to exist.

By self-consistently solving the Ginzburg–Landau equations and electrostatic equations,Lahoche, Lukyanchuk, and Pascoli found the multivortex toroidal states appearing in nano-metric ferroelectric cylinders [39]. The geometrical textures, critical temperatures, and sta-bility regions for these states were calculated, which supports the finding that depolarizationenergy is the driving force for vortex formation.

Stachiotti and Sepliarsky reported toroidal ferroelectricity in PbTiO3 nanoparticles [40]by using atomistic simulations based on first-principles-derived interatomic potentials.Their simulations revealed the existence of a ferroelectric bubble by the alignment of vor-tex cores along a closed path. The aspect ratio of the nanostructure was found to play animportant role in affecting the topology of the dipole configurations. For certain flat nanos-tructures, a multibubble state was shown to act as a bridge between the dipole patterns in0D nanodots and in 2D ultra-thin films.

Other convincing studies that support the existence of a ferroelectric vortex come fromdirect first-principles calculations. Using density functional calculations, Durgun et al. pre-dicted that polarization vortices appear in germanium telluride crystalline nanoplatelets andnanoparticles when the size is above a diameter of 2.7 nm [41]. The ferrotoroidic orderingis spontaneous and reversible, with the toroidal moment being the order parameter. Theauthors further emphasized the important role of inhomogeneous strain in stabilizing fer-roelectric vortices.

Pilania and Ramprasad reported complex curling vortex configurations of electric dipolesin PbTiO3 nanowires [42] by density functional theory calculations. The critical size for thevortex polarization instability was found to be 16 A. The authors also showed that strainand surface terminations may cause phase transitions between the curled vortex phase andthe conventional rectilinear axial polarization phase.

Furthermore, Shimada et al. demonstrated, also using direct first-principles calculations,a possible coexistence of linear polarization and vortex polarization at twist boundariesin PbTiO3 [43]. Near the twist boundary, in addition to the linear spontaneous polariza-tion along the normal direction, the authors also found a coexisting curled (or toroidal)polarization. The linear polarization is caused by the locally strengthened covalent Pb–Obond, while the curled polarization is caused by rotational in-plane displacements at thetwist boundary. It was also revealed that these two types of polarization can be consider-ably influenced by applying tensile strains. A tensile strain was found to enhance the linearpolarization, while it suppresses the vortex polarization.

The ferroelectric vortex also exists in thin films, in addition to FE dots and wires. Forinstance, first-principles-based calculations conducted on BiFeO3 thin films [44] revealed


an array of FE vortices, each having a core located close to the film surface and interface.Interestingly, such a peculiar closed-flux structure was then experimentally confirmed [16].Similarly, using a first-principles-based effective Hamiltonian approach, Wu et al. deter-mined the ferroelectric pattern in PbZr0.5Ti0.5O3 ultra-thin film [45]. Vortex stripes werereported to form in the FE thin films, with some relation to the 180◦ domains. The authorsalso studied how this structure responds to external electric fields. They found that whena local external field is applied, the vortex stripe transforms into the vortex loop structure[45]. In another study, Roy, Sarkar, and Dattagupta investigated the evolution of 180◦, 90◦,and vortex domains in ferroelectric films, using a Landau-like theory and its time-dependentgeneralization [46]. It was shown that the formation of the above domains (including FEvortex domains) in thin films depends largely on the choice of electrical boundary condi-tions and does not require strain fields, which is consistent with the result that FE vortexexists in free-standing FE dots under zero external strain [7].

For the purpose of technological applications, it is always desirable to be able to tune ormodify the system properties according to what is needed. This is also true for the ferro-electric vortex structure phase, and scientists have explored different approaches to modifythe FE vortex structure. Here we list two interesting possibilities. One was reported byProsandeev and Bellaiche who investigated properties of ferroelectric nanodots that areembedded in a polarizable medium [47]. The ferroelectric strength of the medium mate-rial was found to be important in determining the structural phase of the whole system.Some novel states, such as a state that exhibits a coexistence of two kinds of order param-eters, or a state possessing a peculiar order between dipole vortices of adjacent dots, werepredicted. Another interesting possibility is by lattice misfit strain, as reported by Naumovand Bratkovsky [48]. Using the first-principles-derived effective Hamiltonian, these authorsinvestigated the effects of external strains on the dipole structures in flat nanoparticles ofBaTiO3 and PZT. These authors revealed that strain strongly affects the structural phase bycontrolling the competition between the depolarizing field and the polarization anisotropy.Compressive strain was shown to favor the formation of 180◦ stripe or tweed domains, andtensile strain favors the formation of an in-plane vortex. An unusual intermediate phasewith the coexistence of both stripe domain and FE vortex was also predicted to be possible.Note that the effect of strain (as well as electrical boundary conditions) on properties of FEnanostructures was also revealed in computational work of [49].

Dipole structures that are more complex than the FE vortex can also occur in ferroelectricsystems. In such cases, one needs to define other order parameters. For instance, Prosandeevand Bellaiche introduced another interesting order parameter that involves a double crossproduct of the local dipoles with their positions [50, 51]. The authors have used this orderparameter to describe the helicity of the two domains in the onion states, the curvatureof the dipole configuration in flower states, and the simultaneous existence of vorticesof opposite chirality. Another example is the formation of the quadrant domain patternthat has been observed and simulated in free-standing single-crystal platelets of BaTiO3[52]. These quadrant domains self-organize themselves into an antivortex-like configura-tion, with the relative position of the quadrant core (with respect to the center of the platelet)being adjustable by varying the length-to-width ratio of the platelet sidewalls. These intrigu-ing features were explained by developing a Landau-type phenomenology in which theorder parameter is the amount of off-centering of the quadrant core and the thermodynamicdriver is the shape anisotropy of the platelet [52]. First-principles calculations conducted


in BaTiO3 nanowires [53, 54] also revealed other complex chiral patterns of the electricdipoles and topological defects that can be associated with a large winding number.

Interestingly, electric dipoles can also locally couple with other degrees of freedom,which can result in original phenomena when these electric dipoles organize themselvesinto vortices. For instance, the coupling between tiltings of oxygen octahedral (also termedantiferrodistortive motions) and ferroelectric vortices was recently predicted to give rise tonovel exotic chiral patterns for the tilting of oxygen octahedra in zero-dimensional ferro-electric and multiferroic dots [55]. Describing such chiral patterns requires the introductionand use of a novel order parameter, which can be thought of as an extension of the toroidalmoment through (i) the replacement of electric dipoles by vectors quantifying antiferrodis-tortive (AFD) motions and (ii) the introduction of negatives signs for some AFD domainsexisting inside these chiral patterns [55].

22.1.4 Controlling and Switching of Single, Double, and Multiple Vortex States

The possibility of controlling and, especially, switching ferroelectric vortex or vor-tices opens exciting opportunities for nanomemory devices, nanomotors, nanotransducers,nanoswitchers, nanosensors, etc. For example, switching between opposite chiralities of(bistable) vortices promises to increase the density of FE non-volatile random access mem-ories by five orders of magnitude [8]. As a new kind of dipolar ordering, vortex FE statesexhibit novel coupling with external fields. In fact, the curling dipolar pattern in ferroelec-tric nanostructures can be affected by homogeneous [56–58], inhomogeneous [59], curled[60–62], or combined (homogeneous plus curled) [60] electric fields. Even a mechanicalload can be used as an effective tool to control the vortex structures [63].

Though the homogeneous electric field does not directly interact with the toroidalmoment G, it interacts with the local dipoles pi at cell i via −

∑i E ⋅ pi, and, therefore,

can change the initial dipolar pattern. Naumov and Fu [56] studied vortex-to-polarizationtransformation in Pb(Zr0.5Ti0.5)O3 disk-shaped nanoparticles, caused by a homogeneouselectric field normal to the vortex plane and parallel to the initial G. By performing first-principles effective Hamiltonian simulations, they found that such a transformation followsan unusual path via a bridging phase, with the toroidal moment rotating by 90◦ with respectto the initial Gz (Figure 22.1). Later such a path was presumably observed by Gruvermanet al. [19] in their experimental study of switching of vertical polarization in small circularPZT capacitors. They discovered that a small domain remains unswitched in the center ofthe capacitors after switching; this effect was attributed to the formation of a vortex stateand to the transformation mechanism predicted by Naumov and Fu [56].

Surprisingly, the homogeneous electric field not only can unwind the curling dipolar pat-tern but can also switch the toroidal moment in ferroelectric disks or rings if the latter areasymmetric in shape; such an asymmetry can be induced, for example, by a cylindrical holeshifted from the central axis. This interesting possibility was demonstrated by Prosandeevet al. [57] in their effective Hamiltonian study of both ferromagnetic (Ni80Fe20) and fer-roelectric (PZT) asymmetrical nanorings. They found that the homogeneous electric fieldswitches the chirality of the ferroelectric vortices via the formation of antiferrotoroidic pairstates where two, large and small, vortices coexist but do not compensate each other, asshown in Figure 22.2. This intermediate phase differs from that in magnetic nanoparti-cles, where switching of the magnetic toroidal moment T (by a homogeneous magnetic


Figure 22.1 Toroidal moment G (using the left vertical axis) and net polarization Pz (using theright vertical axis) versus the magnitude of the electric field, E, in a nanodisk with a diameter of19a, where a is the lattice parameter. As the electric field exceeds ∼1.5 V nm−1, an intermediatephase with non-zero Gy occurs. This phase disappears as the field is further increased. Adaptedfrom [56]

Figure 22.2 Calculated hysteresis loops in asymmetric ferromagnetic rings (panels (a) and(b)) and in asymmetric ferroelectric rings (panels (c) and (d)). Panels (a) and (b) display thebehavior of the magnetization and magnetic toroidal moment, respectively, as a function ofthe applied homogeneous AC magnetic field. Panels (c) and (d) show the evolution of thepolarization and electric toroidal moment, respectively, versus the applied homogeneous ACelectric field. Insets schematize the rings’ geometry and the dipole arrangement in the (x, y)plane for different states around the hysteresis loops. Reprinted from [57]


field) involves onion states rather than antiferrotoroidic configurations (cf. Figure 22.2). Itis worth noticing that in the context of magnetic nanoparticles, the results of [57] are fullyconsistent with previous experimental studies (see, for example, [64]).

Some ferroelectric nanoparticles, especially flat ones shaped like the number 8, canposses a double vortex state for which the coexisting two vortices have opposite chiral-ities. Though the total toroidal moment vanishes here, it is possible to introduce a neworder parameter, the hypertoroidal moment h, that adequately describes such a peculiarstate [50, 51]. Since the moment h is different in symmetry from the toroidal moment, thedouble vortex state responses to the static homogeneous electric field are different thanthose to a single vortex state. Using an approach similar to that of [57], Prosandeev andBellaiche [58] showed for their model systems of two ferroelectric disks “welded” togetherthat the moment h can be reversed by static electric fields. Such a reversal can be viewedas a simultaneous switch of the sense of dipolar rotation (local G) in each of two vortices.

As mentioned above, homogeneous electric fields can control and even switch the direc-tion of the toroidal moment in asymmetrical nanoparticles. Analogously, inhomogeneouselectric fields are able to interact with the toroidal moment in symmetrical nanoparticles.Such an interaction was studied [59] for a 12 × 12 × 12 PZT nanodot with the help ofa first-principles-based Monte Carlo scheme. The inhomogeneous field was mimicked byplacing several point charges around the nanoparticle whose net charge is zero. Figure 22.3displays the temperature dependency of G in the absence and presence of the external fieldgenerated by two, + and −, point charges in the (x, y) plane. Though the field is rather weak,it nevertheless fixes the direction of the forming toroidal moment along the z axis; in theisolated dot this direction could be along any of the six <100> directions.

Figure 22.3 (a) Temperature dependency of the toroidal moment (squares) and polarization(dots) in a 12 x 12 x 12 PZT nanodot, and (b to d) resulting dipole pattern at 625, 300, and 1 K,respectively. The filled symbols of panel (a) correspond to simulations in which the inhomo-geneous field is turned on while open symbols show results for an isolated dot. The long andthick arrows of panels (b- to d) are guides for the eyes to show the tendency of some dipolesto align along some specific directions. Reprinted from [59]


Perhaps the most straightforward way to influence toroidal moment is to use a curledelectric field, ∇ × E, which directly interacts with G; since G and ∇ × E are thermodynam-ically conjugated parameters, the corresponding term in the thermodynamical potential issimply (∇ × E) ⋅ G. It should be stressed that the curled field cannot be generated by staticcharges, but only by the time-dependent magnetic field via∇ × E = −𝜕B∕𝜕t, as in a focusedlaser beam with a varying B-field component. In the simulations, it is convenient to presentthe curled fields in the form E = 1

2S × r, where S characterizes the strength and direction

of the vorticity. Using first-principles-based Monte Carlo simulations, Naumov and Fu [60]investigated the response of an FE vortex to S in PZT nanoparticles. They found that thedomain coexistence mechanism, when the −G domain grows while the original G domainshrinks under external fields, is not valid for the switching of G. Instead dipoles displayunusual collective behavior by forming a new intermediate vortex phase with a perpen-dicular (not opposite) toroid moment, as displayed in Figure 22.4(a), where this phase isdenoted as phase II and characterized by a finite Gy moment. The reversal of the vortex

30

(b)

(a)

phase I

Gx

GyGz

|G|

II III

15

0

–15

–30

0

10

–10

0 10 20

Monte Carlo sweeps (×200)

U (

meV

)U

(m

eV)

Gx,

y,z

(eÅ

2 )G

x,y,

z (e

Å2 )

30 40 50

–34

–32

–30

–28

–34

–32

–30

–28

–26

Figure 22.4 Evolution of the toroidal moment G and its magnitude |G| (using the left verticalaxis) and internal U energy per five-atom cell (solid black dots, using the right vertical axis) in ad = 19a nanodisk: (a) under an S = 0.25 mV A−2 curled electric field; (b) under the combinedaction of an E = 1.9 V nm−1 homogeneous field and an S = 0.04 mVA−2 curled field. For clarityof display, Gz is plotted after multiplying it by −1. Adapted from [60]


occurs by undergoing an evolution sequence of Gz → Gy → (−Gz). Along this path, theamplitude of the moment |G| (shown in red) changes only slightly, suggesting that the col-lective hydrodynamical mode for the vortex reversal is simply the rotation of G. Amongthe interesting findings of Naumov and Fu [60] is also the fact that the combined actionof homogeneous and curled electric fields can significantly reduce the switching energybarrier and drastically (by 600%) decrease the magnitude of the switching curled field (seeFigure 22.4(b)). This is due to the fact that both fields can give rise to a lateral vortex,which serves as a bridging phase between the two end states corresponding to −G and G,respectively.

Using phase field simulations, Wang [61] also studied the switching behavior of an FEvortex subjected to a curled electric field in a single-crystal ferroelectric tetragonal nanodot,in contrast to the nanodisks considered by Naumov and Fu [60]. These simulations revealedthat the vortex switching in such dots does not begin from the vortex core, as reported [60]but rather from the dot corners (which have the lowest elastic energy density). This discrep-ancy can be due to a geometry effect or attributed the fact that the direction of polarizationin [61] was restricted to the (x, y) plane. Within the same approach, Wang and Kamlah[62] investigated the intrinsic switching of a defect-free vortex in ferroelectric PbTiO3 nan-otubes under a curled electric field. It was found that the vortex switching goes through thenucleation of local vortices, which grow at the expense of the original one. In contrast to theresults obtained for nanodisks [60], they also found that during the switching a new vortex−G can coexist with the initial vortex G. The difference was attributed to the absence oftopological defects in nanotubes.

Furthermore, and as demonstrated recently by using phase field simulations, the vortexdomain structure in ferroelectric nanoplatelets can be controlled by a simple mechanicalload [63]. Vortex states with more/fewer vortices can be obtained by application of com-pressive/tensile strains to the nanoplatelet. Such transformations between different vortexstates allow one to effectively control the size and the number of vortices; they also lead toa variety of different temperature–stress phase diagrams.

22.1.5 Other Phenomena Related to Vortices and Complex Dipolar Patterns

FE vortices also exhibit several other distinct features. For instance, a nanosystem exhibit-ing an electric vortex in the (x, y) plane, and thus possessing an electrical toroidal momentalong the z axis, is more elongated along the x or y direction than along the z direction. Inother words, and as shown by Prosandeev and Bellaiche [65], the 𝜂H homogeneous strainhas Voigt components for which 𝜂H,3 is smaller than 𝜂H,2 = 𝜂H,1. The axial ratio c/a is there-fore smaller than 1 for electric vortices, which contrasts with the cases of systems exhibitinga spontaneous polarization [66] or antiferrodistortive motions [67] along the z axis. It wasdemonstrated [65] that the inhomogeneous strain, 𝜂I, is also quite peculiar when an FE vor-tex state forms. In particular, the 𝜂I,1 and 𝜂I,2 (Voigt) components of the inhomogeneousstrain have significant magnitude near the center of the FE vortex. This is because of theelastic elongation (that is induced by the coupling between the strain and electric dipoles)of all the domains forming the vortex. Other unusual signatures of FE vortices are the typeof strongly inhomogeneous electric field produced by these vortices outside the nanostruc-ture and the very weak interaction between symmetric vortices located at two different dots[65]. This latter interaction varies as 1/R9, with R being the distance between the centers ofthe two dots [65].


Moreover, the facts that the electric toroidal moment (1) is an order parameter and(2) involves the electric dipoles in its definition automatically imply that novel physicalresponses and tensors can be defined. As an example and as shown in [68], the magnitude,and sometimes even the direction, of the electric toroidal moment can be altered by apply-ing a strain or its conjugate field (i.e. a curled electric field). These effects can be termed aspiezotoroidicity and electric toroidal susceptibility, respectively, in analogy with piezoelec-tricity and electric susceptibility (that relate the change of the electrical polarization withstrain and homogeneous electric fields, respectively) in “normal” ferroelectrics.

Several other novel and promising phenomena can also emerge from the existence of fer-roelectric vortices. For instance, the combined experimental and theoretical works of [17]reported an enhanced electric conductivity at the FE vortex cores in multiferroic BiFeO3(BFO). More precisely, the formation of one-dimensional conductive channels that are acti-vated at voltages as low as 1 V (rather than at ∼ 3 V, as in the bulk counterpart or in domainwalls) was evidenced. Another example of original phenomena was also discovered in BFO:using a first-principles-derived effective Hamiltonian, Ren and Bellaiche [69] demonstratedthat the direction and magnitude of the magnetization can be controlled by curled electricfields, because of the inherent interactions between ferroelectric vortices, oxygen octahedraltilts, and magnetic dipoles. This novel magnetoelectric coupling was termed the magneto-toroidic effect and was also found to involve complex intermediate microscopic states, suchas electric vortices coexisting with antivortices.

This latter coexistence was also predicted to occur in the ground state of the nanocom-posite made of periodic arrays of BaTiO3 nanowires that are embedded in a matrix formedby another FE material [70]. In this ground state, the nanowires all possess FE vorticeshaving the same, clockwise or counterclockwise, rotation while antivortices (and also vor-tices with identical rotation than in the nanowires) can be found in the matrix. This stateis thus analogous to the so-called “phase-locked” phase that has been recently observedin magnetic systems [71]. Interestingly, a spontaneous polarization further exists along thenanowire direction, in addition to the toroidal moment that is associated with the FE vor-tices. Such a combination was shown to result in natural optical activity and its controlby electric fields [72]. In other words, the plane of polarization of linearly polarized lightshould rotate when passing through the nanocomposite, with the sense of this rotation beingswitchable by an electric field. Reasons behind this natural optical activity and its controlby electric field control are the linear response of the toroidal moment to a homogeneouselectric field (thanks to the coupling between electrical polarization and electric vortices)and the field-induced transition between the dextrorotatory and laevororotatory forms ofthe ground state [72].

Another computational study predicted that gyrotropy can also exist, but in a specificcrystallographic state of BFO films that are under compressive strain and that are grownalong the (unusual) [–110] direction [73]. This phase has the P212121 space group andpossesses interpenetrated arrays of FE vortices (having the same sense of rotation) and FEantivortices. Such arrays were found to originate from the simultaneous existence of twounstable off-center phonon modes, one at the M-point of the cubic first Brillouin zone andthe other one at the X-point of this Brillouin zone.

The nanocomposite studied by Louis et al. [70] was further predicted to exhibit anotherunusual phenomena, that is a temperature-driven vortex core transition (VCT) for whichthe vortex cores transforms from being axisymmetric to possessing a “broken symmetry”(that was further found to be correlated to inhomogeneous in-plane polar displacements)


[70]. VCT transitions have been previously known to exist in some complex systems, suchas superfluids [74], but never in ferroelectrics before the study of Louis et al. [70].

Furthermore, Prosandeer et al. [75] used an effective Hamiltonian approach withinmolecular dynamics to study electrocaloric effects in PZT nanodots exhibiting an FE vor-tex. It was found that the changes in temperature, ΔT(t), induced by a time-dependentchange in an AC electric field, ΔE(t), are related to each other via ΔT(t) = 𝛼ΔE(t) + 𝛾t,where 𝛼 and 𝛾 are time-independent positive coefficients. In other words, a variation lin-ear with time, 𝛾t, is superimposed on to the usual electrocaloric effect given by 𝛼ΔE(t).Such linear addition originates from the expansion/compression of the vortex pattern (viaflipping of dipoles) under the AC field and characterizes a constant heating of the nanodot.

Another original prediction was offered by Prosandeer et al. [76]: simulations using afirst-principles-based effective Hamiltonian (within the path-integral quantum Monte Carlomethod (PI-QMC) [24, 77, 78]) indicated that quantum, zero-point phonon vibrations leadto the suppression of the ferroelectric vortex, and thus to the annihilation of the electrictoroidal moment, in stress-free nanodots made of KTaO3 (KTO) and being under electricalboundary conditions. This suppression was found to be accompanied by the formation of apeculiar local structure that possesses short-range, needle-like correlations of the individualtoroidal moments. The annihilation of the global electric toroidal moment in KTO dots isreminiscent of the known fact that quantum vibrations also result in the disappearance ofthe spontaneous polarization in the so-called incipient ferroelectrics bulks (such as the KTObulk) [79, 80]. This analogy is at the origin of the term “incipient ferrotoroidics” given byProsandeer et al. [76] for systems such as KTO dots.

22.1.6 Summary from the Simulations and Theory Perspective

In summary, we have tried to provide a comprehensive review about theoretical and com-putational studies on FE vortices and related complex patterns of electric dipoles. We hopethat this review will be of benefits to expert of this fascinating field of research, but alsoto “newcomers” interested in learning more about this exciting topic. We also hope thatnumerous predictions resulting from simulations performed in the last nine years will beexperimentally confirmed soon. Finally, we apologize in advance for (involuntary) omis-sions of works in this rapidly developing and active research direction.

22.2 Insights from Experiments

22.2.1 Introduction

It should be evident from the preceding section of this chapter that theoretical studies, usingboth atomistic and phenomenological approaches, have made clear predictions about theexistence and behaviour of ferroelectric vortices [7, 8, 36, 59, 81]. Spurred on by thesepredictions, experimentalists have made significant efforts to try to find them in real mate-rials. The fact that magnetic vortex states had already been directly imaged in ferromagnets(whose dipole characteristics are often analogous to ferroelectrics) [82–84] gave genuineconfidence of success.

However, after a decade or more of searching, categorical and conclusive observationsof genuine ferroelectric vortices have not yet been made. That is not to say that there has


been no relevant research progress; on the contrary, a great deal has been learned from thesearch so far: flux-closure structures, often seen as the precursors for vortex formation, havebeen discovered to occur in response to depolarizing fields [10–13, 85,86] or as a result ofspecific switching experiments [14, 17, 18], and continuous rotations in dipole orientationsin ferroelectrics have been directly imaged for the first time [15, 16]. Such observationspoint strongly to the possibility of vortex formation and perhaps it is only a matter of timebefore suitable experimental conditions are found in which vortices could be stabilized. Forthe moment, however, they remain elusive.

It may therefore come as a surprise to find that experimental literature is commonly pop-ulated with matter-of-fact statements describing a specific dipole or domain arrangementas a ferroelectric vortex. To critically evaluate the validity, or otherwise, of such statements,let us first examine a number of the defining characteristics of a vortex configuration that,although perhaps not sufficient, are certainly necessary for its existence to be claimed:

(i) The topological winding number of a vortex is +1.(ii) The rotation of local dipoles in the vortex structure should be such that each is oriented

perpendicular to the radial vector from the vortex core to that dipole.(iii) The dipole pattern should only contain a single topological singularity, not a compound

set of singularities with a net winding number of +1.

22.2.2 Vortices and Topological Winding Numbers

The first characteristic listed above is that the winding number of a dipole pattern needsto be +1 for it to even be considered as a possible vortex state [87]. The winding numberclassification allows different topological patterns to be meaningfully grouped together. Todetermine the winding number associated with a region of dipoles, a closed path loop inthe ferroelectric needs to be considered (Figure 22.5(a)). One should imagine traversingthis loop in an anticlockwise sense and then, on a separate diagram, sequentially plottingthe orientation of each dipole encountered. In this separate diagram, each dipole consid-ered should be imagined to sit at the centre of a circle, and the point at which the vectordirection of the dipole intersects the circle circumference should then be plotted as a pole(Figure 22.5(b)). The number of times that the plotted poles undergo a full rotation, in ananticlockwise sense, is the winding number associated with the dipoles in the region of theferroelectric enclosed by the initially considered loop. Rotations of dipoles in a clockwisesense are designated as negative winding numbers. Interestingly, a vortex possesses a wind-ing number of +1 irrespective of the sense in which it acts (clockwise or anticlockwise; seeFigure 22.5(c)). Dipole patterns with winding numbers of −1 are distinctly non-vortex innature, illustrated by the quadrupole (or perhaps “antivortex”) pattern example given inFigure 22.6(a).

In literature, the topological patterns seen in rare earth manganites are often referredto as vortices [88–90] and yet this claim never seems to be explicitly justified. Withoutdoubt, very interesting domain patterns are present in these improper ferroelectrics; thepoints at which sets of domain walls intersect are particularly noteworthy as they clearlycreate some form of line defect around which the sense of the ferroelastic distortion andferroelectric dipole vector vary. However, the winding number associated with the dipolearrangement around this line defect is rarely discussed. In Figure 22.7, piezoresponse forcemicroscopy (PFM) images from a single crystal of YbMnO3 are shown. As the plane of


1

2

3

4

5

6

7

8

(a)

3

2

1

8

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6

4

5 n = +1

(b)

1

2

3

4

5

6

7

8

(c)

7

6

5

4

3

2

8

1 n = +1

Figure 22.5 The winding number can be determined by considering the way in which dipoleorientation varies while traversing around a closed loop (dashed ellipse) in an anticlockwisesense (a). At each point on the journey, encountered dipoles are considered as vectors origi-nating at the centre of a circle; these vectors are then represented, on a second diagram, bythe poles at which they intersect the circle circumference (b). The winding number is the num-ber of times sequentially encountered poles describe complete anticlockwise rotations on thissecond diagram. As can be seen, both clockwise and anticlockwise vortices generate windingnumbers of +1 (c)

the section imaged contains the polar axis and this axis is approximately perpendicular tothe PFM cantilever, the lateral PFM images completely capture the polar orientations in thevarious domains (sketched schematically in the same figure). Considering a loop pathwaythat encapsulates the point at which the domain walls intersect, we can readily generatean associated representation of the topology (Figure 22.7(b)). Each dipole vector is onlyassociated with one of two diametrically opposite poles. While it might be tempting toassign a winding number, it is, in fact, impossible: the sense of rotation, required to movebetween oppositely oriented domains, is undefined. The vortex terminology, widely usedin manganite literature, is therefore baffling and seems unjustified on the basis of the dipoleconfigurations at least.

A very important aspect of this winding number discussion is that, although a value of+1 is necessary for a topological pattern to be described as a vortex, it is not sufficient.Many dipole patterns that are clearly not vortices may also have +1 winding numbers: theradial pattern of dipoles in Figure 22.6(b) is a good example.


1

2

3

4

5

6

7

8

7

8

1

2

3

4

6

5 n = −1

(a)

1

2

3

4

5

6

7

81

8

7

6

5

4

2

3 n = +1

(b)

Figure 22.6 A quadrupole, or antivortex, dipole arrangement is an example of a pattern thatgenerates a winding number of −1 (a). A winding number of +1 is a necessary condition for avortex, but it is not a sufficient one, as can be seen by the radial dipole pattern in (b)

32 1 6

5

4

(a) (b)

1 μm

1,3,5

2,4,6

(c)

Figure 22.7 Piezoresponse force microscopy (PFM) images (amplitude (a) and phase (b))of an interesting region of a YbMnO3 single crystal; in this region, domain walls intersect toform a line defect, manifest in the 2D planar section imaged as a point defect. Traversing in aloop around this defect, in an anticlockwise sense, allows for an attempted analysis of windingnumber (c), as was previously illustrated in Figures 22.5 and 22.6. Here, however, the numberof times that the dipole vector rotates in traversing the loop is undefined; even the sense of therotation across successive 180◦ domain walls is unclear. A definitive winding number cannottherefore be assigned


2

1

3

4

(a) 1 μm

2

1

4

3 n = +1

(b)

Figure 22.8 (a) Flux-closure patterns generated by the net polarization in each of four packetsof ferroelastic domains (or superdomains). (b) A winding number of +1 can be associated withthese kinds of flux-closure arrangements, but the dipole orientation does not vary continuouslyaround the core; rather it changes direction in discrete jumps such that, despite the windingnumber value, quadrant flux closure cannot be considered as a vortex

22.2.3 Continuous versus Sudden Dipole rotation

By inspection, the form of a vortex structure (Figure 22.5) is such that there is a continuousrotation in the orientation of dipole vectors around the singularity at the centre of the vortex(the vortex core). Formally, this property might be described by a necessity for dipoles to beoriented perpendicular to the radial vector from the vortex core to the dipole (ri). In addition,the sense of all dipole vectors (clockwise or anticlockwise) around the core should be thesame. Another way to express this might be that, for a given magnitude of local dipole vector(pi), the magnitude of the ferrotoroidal moment, G =

∑i ri × pi (discussed extensively in

the first half of this chapter), should be at a maximum for the vortex state to be claimed.Flux-closure structures (Figure 22.8), such as those mapped by McQuaid et al. [12],

McGilly and Gregg [11], Ivry et al. [13], Chang et al. [86], and Balke et al. [14, 17], havewinding numbers of +1 and could therefore be considered as vortex contenders on the basisof winding number alone. However, it is clear that the rotation of dipole vectors around thecore is not continuous; rather its orientation changes abruptly, usually achieved through fourdistinct 90◦ rotations in the dipole, or net dipole, orientation. Arguments have been madethat, as one moves closer to the centre of a flux closure pattern, there may be a transitionfrom discrete polar rotation to continuous rotation and that, therefore, the existence of fluxclosure could be sufficient to claim the presence of a localized central vortex that extendsover a limited region of space, even if this is not directly observed. This is an interestingpoint of view. It is certainly topologically allowable for dipole patterns with the same wind-ing numbers to transform from one pattern to another with distance from the topologicaldefect core (see Figure 22.9 for an example of this) [87].

Direct high-resolution mapping of dipoles, using aberration-corrected transmission elec-tron microscopy [15, 16], has definitively shown that continuous rotations of dipoles in aferroelectric can occur. However, it is telling that, even in the landmark study by Jia andco-workers [15], full 360◦ polar rotation does not occur around a single core point (only


2

3

1

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3

4

56

7

8

92

4

(a)

4

3

2

1 n = +1

(b)

12

3

4

5 6

7

8

9

n = +1

(c)

Figure 22.9 Dipole patterns with the same winding number can transform continuously fromone to another as the distance from the topological defect at the center increases. In this exam-ple the interior of the pattern in (a) is a vortex, which transforms continuously into a radialpattern. As can be seen, successive dipoles encountered in the loop pathways marked by thedashed lines in (a) both indicate winding numbers of +1 (b and c)

180◦ rotation is observed) and thus the dipolar pattern is not associated with a true vortexcore singularity. This leads us to the third key feature of a true ferroelectric vortex discussedin detail below.

22.2.4 Unique Core Singularity

When ferromagnetic vortices are imaged, in permalloy disks for example [83,84], a singlevortex core is observed, at which the local magnetic dipole is forced to orient perpendicu-lar to the plane of the vortex structure and around which continuous dipole rotation in thevortex plane is found. In ferroelectrics, the structure of the vortex core may differ. Indeed,local polarization may disappear, rather than reorient, since, in ferroelectrics (unlike ferro-magnetics), the magnitude as well as the orientation of local dipoles can vary. Nevertheless,the existence of a single topological defect at the core is still expected, as a signature of thetrue vortex state.

Despite being necessary for ferroelectric vortices, winding numbers of +1 and contin-uous uniform dipole rotation do not guarantee the existence of a singular defect core. Aspointed out by Mermin [87], an aggregation of topological defects can act in concert togenerate apparent winding numbers, at a distance, that are given by the sum of their indi-vidual winding numbers, but are not necessarily representative of any one of the individualtopological defects within the aggregate. Figure 22.10 illustrates this point: two defects,each with a winding number of + 1∕2, are in close proximity. While pathway loops enclos-ing the defects separately clearly demonstrate that neither one has a winding number of +1,


(a)

23

4 1

41

2 3

2

1

8

7

6

5

4

3

3

21

4

n = +1/2

(b)

3

2

1

8

7

6

4

5 n = +1

(c)

Figure 22.10 The dipole pattern illustrated in (a) has two topological singularities, each witha winding number of + 1∕2 (b). If a loop pathway is chosen that encompasses both of thesesingularities, then the winding number observed will be the sum of their individual windingnumbers (c). Thus, something that might appear to have the properties of a vortex on a largescale could be composed of several topological defects, none of which is a true vortex state

loops that enclose both defects together do indeed generate +1 as the net winding number.One might suggest that the difference between a single vortex core and a compound one,made from potentially many non +1 topological defects, is immaterial. Yet, if a region ofdielectric contained an array of equal numbers of both vortex (+1) and antivortex (–1) struc-tures, we would not equate this to a perfectly undisturbed and uniform ferroelectric domain,despite the fact that, for large enough pathway loops, these two dipole patterns would pos-sess the same net zero winding number. Thus, in the limit, a compound set of topologicaldefects, which together generate a+1 winding number, should not be considered to be a truevortex core. Several experimental observations on ferroelectric dipole configurations withwinding numbers of +1 (flux-closure states) illustrate a tendency for compound, rather thansingle, topological core defects. These studies may suggest that a vortex core singularity isenergetically unfavorable in ferroelectrics.

McGilly et al. [10] examined the patterns formed at the boundaries between bundles of90◦ stripe domains (superdomains) in BaTiO3 single-crystal lamellae and nanostructures.While many of these boundaries were composed of conventional 180◦ domain walls, theyfound that some were more complex in form. By considering all possible self-consistentsets of dipole orientations in these more complex junctions (illustrated in Figure 22.11),McGilly et al. concluded, from symmetry arguments, that they had to be comprisedof chains of flux-closure and quadrupole pairs. The resolution of STEM allowed for


(a)

(b)

<11

0>

<110>

100 nm

50 nm

Figure 22.11 (a) Scanning transmission electron microscopy (STEM) images of a junctionbetween bundles of 90◦ ferroelastic domains (superdomain bundles), where the existence offlux-closure dipole structures is inferred. (b) Part of the boundary has been magnified and thedipole orientations thought to be present plotted, along with an inferred 180◦ domain wall notimaged in STEM (interpretation after McGilly et al. [10])

information to be obtained on the nature of the core structure in these flux-closure objects:in Figure 22.11(b), it is clear that the domains indicated by the light and dark STEM con-trast do not converge to a single point. Rather, the light contrast domains show needle-pointmorphology and stop short of the junction itself. The core region of the flux-closure struc-ture is entirely dark contrast. The implication is that, rather than form a single core at whichall four domains converge (a fourfold vertex), a distributed compound core forms, consist-ing of two threefold vertices, spatially separated by a short 180◦ domain wall, with little ifany contrast on STEM. The two topological defects then form a pair, with a total windingnumber of +1, but composed of two + 1∕2 objects, conceptually in direct analogy with theschematic given in Figure 22.10.

While the dipole orientations in McGilly et al.’s study were inferred, Chang et al. [86]were able to obtain more direct information, using piezoresponse force microscopy (PFM)on similar flux-closure objects, along the edges of single-crystal lamellae of Pb(Zn, Nb)O3–PbTiO3 (PZN–PT). Here superdomain bundles of 90◦ stripe domains were found to forminto flux-closure quadrants on a mesoscale (1–10 μm). Again the form of the flux-closurestructure was such that a single fourfold core was avoided and instead two threefold junc-tion structures with + 1∕2 winding number formed (see Figure 22.12(a) to (c)). This PZN–PT study was particularly interesting, as detailed mapping of the boundary between super-domains in the centre of the mesoscale flux-closure structure (Figure 22.12(d) and (e))


<110>

(a) (b) (c)

(e)(d) 200 nm

1 μm

<11

0>

Figure 22.12 Piezoresponse force microscopy (PFM) imaging of the edge of a thin single-crystal lamella of Pb(Zn,Nb)O3–PbTiO3 (PZN–PT). Piezoresponse amplitude (a) and phasemaps (b) revealed the existence of flux-closure structures on length scales between 1 and10 𝜇m, comprised of four superdomains (distinct bundles of stripe domains) arranged intoquadrants (c). Higher resolution mapping of the region enclosed by the rectangular box in(a) showed that, at the junction between bundles of 90◦ ferroelastic domains, flux closure wasagain present, but in this case on a length-scale of hundreds of nanometers (d). The two upperpanels in (d) show that one set of domains terminates somewhat short of the actual boundary,whereas the other set looks to meet and abut at the boundary. The detailed dipole structure atthe junction is hence given by the schematic in the lower panel in (d), a region that has beenblown up and rotated in (e). Comparison of the schematics in (c) and (e) clearly suggests thatthe nested flux-closure structures are self-similar, despite the quite different lengths of scales atwhich they exist (interpretation after Chang et al. [14])

revealed another set of flux-closure states in the nanoscale size regime (length scales oforder 100 nm). Detailed interpretation of vector PFM information led to a reconstruction ofthe small-scale flux-closure structures, illustrated schematically in Figure 22.12(e), whichwere remarkably similar to the mesoscale flux-closure structure in which they were embed-ded. Such nested self-similar morphology, generating flux closure with a compound rather


than a single core, across rather different length scales, strongly suggests that true vortexsingularities in ferroelectrics may be too energetically unstable to be generally observed.This may be one of the most serious barriers to ferroelectric vortex formation.

22.2.5 Summary and Conclusions from an Experimental Perspective

There is no doubt that predictions of ferroelectric vortices, resulting from computationalsimulations, prompted a great deal of excitement in the experimental ferroelectric commu-nity. Many researchers sought to be the first to discover ferroelectric vortex states in realmaterials. This resulted in either speculative interpretations of data in terms of the exis-tence of vortices [19, 20, 91, 92], when other interpretations were equally valid, or in thewidespread inappropriate use of the term “vortex” to describe dipole patterns that had eitherno, or only limited, similarities to genuine vortex patterns in terms of dipole distributions atleast [16,17, 88–90]. After around a decade of searching, we can now afford a frank evalua-tion of the observations made so far: no true ferroelectric vortices have been unequivocallyseen.

However, several phenomena consistent with the possibility of vortex existence havebeen established: ferroelectric flux-closure states (often seen as a precursor to the formationof vortices) have been found and studied [10–14, 17,18, 85,86] and gradual and continuousrotations of electrical dipoles, previously a point of contentious debate, have been directlyimaged [15, 16].

Acknowledgments

Work reviewed in Section 22.1 has been financially supported by ONR Grants N00014-11-1-0384 and N00014-12-1-1034. The authors also acknowledge discussion with scien-tists sponsored by NSF Grant DMR-1066158, ARO Grant W911NF-12-1-0085, and theDepartment of Energy, Office of Basic Energy Sciences, under Contract ER-46612. Workreviewed and discussed in Section 22.2 has been financially supported by the Engineer-ing and Physical Sciences Research Council (EPSRC) under grant awards: EP/F004869/1,EP/H047093/1, and EP/H04339X/1, by The Leverhulme Trust, under the International Net-work scheme (F/00 203/V), and by the Department of Employment and Learning (DEL) inNorthern Ireland.

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Ferroelectric Vortices and Related Configurations