Hexagonal rare-earth manganites and ferrites: a review of improper ferroelectricity, magnetoelectric coupling, and unusual domain walls

Menglei Li *a, Hengxin Tan b and Wenhui Duan cd
aDepartment of Physics, Capital Normal University, Beijing 100048, China. E-mail: limenglei@cnu.edu.cn
bMax Planck Institute of Microstructure Physics, Weinberg 2, 06120 Halle (Saale), Germany
cState Key Laboratory of Low-Dimensional Quantum Physics and Collaborative Innovation Center of Quantum Matter, Department of Physics, Tsinghua University, Beijing 100084, China
dInstitute for Advanced Study, Tsinghua University, Beijing 100084, China

Received 24th April 2020 , Accepted 4th June 2020

First published on 25th June 2020

Hexagonal rare-earth manganites and ferrites are well-known improper ferroelectrics with low-temperature antiferromagnetism/weak ferromagnetism. In recent decades, new multi-functional device concepts and applications have provoked the exploration for multiferroics which simultaneously possess ferroelectric and magnetic orders. As a promising platform for multiferroicity, hexagonal manganites and ferrites are attracting great research interest among the fundamental scientific and technological communities. Moreover, the novel type of vortex-like ferroelectric domain walls are locked to the antiphase structural domain walls, providing an extra degree of freedom to tune the magnetoelectric coupling and other properties such as conductance. Here, we summarize the main experimental achievements and up-to-date theoretical understanding of the ferroelectric, magnetic, and magnetoelectric properties, as well as the intriguing domain patterns in hexagonal rare-earth manganites and ferrites. Recent work on non-stoichiometric compounds will also be briefly introduced.

image file: d0cp02195d-p1.tif

Menglei Li

Menglei Li is currently a lecturer in the Department of Physics at Capital Normal University, China. She received her BS degree from Beijing Normal University in 2010 and PhD degree in condensed matter physics from Tsinghua University in 2015. She did her postdoctoral research at the Institute of Applied Physics and Computational Mathematics from 2015 to 2017. Her research interests are focused on the first-principles calculations of ferroelectric and multiferroic materials.

image file: d0cp02195d-p2.tif

Hengxin Tan

Hengxin Tan is now a postdoctoral research fellow at the Max-Planck Institute for Microstructure Physics in Germany. He received his BS in applied physics from Dalian University of Technology, China, in 2013 and PhD in physics from Tsinghua University, China, in 2019 under the supervision of Prof. Wenhui Duan. One of his current research focuses is on multiferroic materials, especially identifying and designing new multiferroic materials in low dimensions.

image file: d0cp02195d-p3.tif

Wenhui Duan

Wenhui Duan is currently a professor in the Department of Physics at Tsinghua University. He received his PhD degree in 1992 from Tsinghua University. After two years of postdoctoral research at the Central Iron & Steel Research Institute, he joined Tsinghua University as a faculty in 1994 and became a full professor in 2000. He is a member of the Chinese Academy of Sciences and a fellow of the American Physical Society. His research interests focus on quantum mechanical studies of condensed matter and materials, including solids, low-dimensional materials, quantum materials, and ferroelectrics. He has contributed over 350 peer-reviewed journal articles.

1 Introduction

Ferroelectrics are materials which have spontaneous electric polarization that can be reversed by an external electric field. Ferroelectric materials are similar to ferromagnetic materials which possess magnetic field switching magnetization, but are less well-known as systematic studies did not begin until 1920 when American scientist Valasek discovered the unique ferroelectricity in Rochelle salt.1 Ferroelectric and ferromagnetic materials both have wide applications in various energy transform devices, transducers, integrated circuits, and information storage devices.2–4 To seek versatile functional devices with more degrees of freedom, and more importantly, to realize the cross control of magnetization with an electric field and vice versa, exploration into materials which simultaneously have two ferroic orders – mainly referring to the ferroelectricity and (ferro)magnetism – were started in the late 1950s. Now, such materials with multiple orders are called multiferroics.5 Russian scientists attempted to achieve magnetic perovskite ferroelectrics by doping dn ions at the non-magnetic B site, but the resultant diluted magnetism renders low magnetic transition temperatures.6,7 The first intensely investigated single-phase multiferroic was nickel–iodine boracite (Ni3B2O13I), which was reported in 1966.8 Although several multiferroic materials have been discovered in nature or artificially synthesized,9 the obtained compounds are rather complex, with performance far from satisfactory. As shown in ref. 10, the search for multiferroic materials had eased by the 1970s.

In 2000, Spaldin pointed out that the fabrication of magnetic ferroelectrics by substituting the B-site cation with some magnetic cations in perovskites would not work well.11 In a typical ferroelectric perovskite with ABO3 formula, polar displacements are favored when the B cation has an empty d orbital, which is unlikely to introduce magnetism due to a lack of partly occupied d or f orbitals. When the B-site cation has d or f electrons, however, the ferroelectric distortion will be suppressed, which is supported by the phonon dispersion curves from density functional calculations.11 This work shed light on a brand-new approach for the search for multiferroics: The d0vs. dn problem in traditional displacive ferroelectrics should be circumvented, and new mechanisms for ferroelectricity have to be established. Soon after, several breakthroughs were made that encouraged studies into multiferroics. Revisiting the BiFeO3 thin film in 2003 showed that at room temperature the antiferromagnetic order coexists with the ferroelectricity driven by the asymmetry of the 6s2 lone-pair electrons of Bi.12 In the same year, Kimura and colleagues discovered a spin-driven ferroelectricity in perovskite TbMnO3, where the spiralling magnetic order induces weak ferroelectricity.13 In 2004, the ferroelectric mechanism of hexagonal YMnO3 (h-YMnO3), where a noncollinear antiferromagnetic order also exists at low temperature, was revealed to be a geometric effect from the structural instabilities instead of the bond mode.14 Since the electric polarization in TbMnO3 and h-YMnO3 are the byproducts of other orders rather than the primary order parameter, the ferroelectricity in them is called improper ferroelectricity.15 Other known improper ferroelectric mechanisms include the ferroelectricity induced by charge ordering in oxides such as Fe3O416 and LuFe2O417,18 (however, the ferroelectricity in LuFe2O4 is still under debate19). The above improper ferroelectrics are multiferroics as well. They are further divided into the so-called type-I multiferroics (h-YMnO3 and LuFe2O4) where ferroelectricity and magnetism have different origins; and type-II multiferroics (TbMnO3) where the spin ordering induces ferroelectricity.20 Typically, the type-I multiferroics have large polarization and high transition temperatures but very weak magnetoelectric coupling, while the type-II multiferroics have relatively small polarization and low transition temperatures but very strong magnetoelectric coupling.

Coincidentally, most of the landmark findings in improper ferroelectrics and multiferroics are closely related to the manganite and ferrite families, therefore it is no exaggeration to say that the rediscovery of manganites and ferrites greatly promoted the renaissance of multiferroics. Since there are already quite a few reviews on BiFeO321,22 and the type-II multiferroics of perovskite-type rare-earth manganites,23,24 in this review we focus on the hexagonal rare-earth manganites and ferrites including h-RMnO3, h-RFeO3 and RFe2O4 (R = rare-earth ion). Hexagonal rare-earth manganites and ferrites are important not only because they include two types of typical improper ferroelectrics – namely h-RMnO3 and h-RFeO3 with geometric ferroelectricity, and RFe2O4 with charge-order induced ferroelectricity – but also because these systems have an abundance of fascinating physical phenomena. The 120° noncollinear spin ordering of Mn/Fe implies a non-ignorable role of the spin–orbit coupling term, and thus the nature of magnetoelectric coupling is complicated, which makes the cross control between magnetism and ferroelectricity much more interesting. Moreover, the hexagonal structure and trimerizing instability give rise to the novel cloverleaf domain patterns that cannot be completely polarised by the external electric field, and have never been observed in traditional ferroelectrics. The unique domain patterns contain rich physics since structural trimerization, ferroelectricity, magnetism and charge conduction are all coupled at the domain walls.

This review is organized as follows. We summarize both experimental findings and theoretical investigations on the structural and ferroelectric properties of h-RMnO3 and h-RFeO3 and the charge ordering in RFe2O4, followed by the magnetic phase diagram and magnetoelectric coupling in these families. In Section 5, we provide an overview of the characterization and theoretical understanding of the interlocked structural-ferroelectric domain walls. Before a summary of this review is made in the final section, a short introduction of the non-stoichiometric effect is presented in Section 6.

2 Ferroelectricity

Rare-earth manganites and ferrites can crystallize in perovskite and hexagonal forms, as shown in Fig. 1. The perovskite phase is similar to BaTiO3 and PZT, featured by the compactly arranged corner-sharing oxygen octahedra. Before moving on to the ferroelectric properties, we first provide an overview of the empirical rule for the formation of hexagonal rare-earth manganites and ferrites.
image file: d0cp02195d-f1.tif
Fig. 1 (a) Orthorhombic phase of RMnO3 or RFeO3 with the space group Pbnm. This figure shows a GdFeO3-type distorted perovskite structure. (b) Hexagonal RMnO3 (h-RMnO3) or h-RFeO3 with the space group P63cm. (c) RFe2O4 structure. (d) Tolerance factor and stable structure against the ionic size of rare-earth ions in RMnO3. In the figure, the unlabelled 4 dots in the deep pink area next to Ho correspond to Er, Tm, Yb, and Lu (from right to left), meaning that RMnO3 with R = Ho–Lu crystallize in the P63cm hexagonal phase. YMnO3 and ScMnO3 also stabilize in the hexagonal phase, but they are not shown here. In the light pink area, the three unlabelled dots between Dy and Sm correspond to Tb, Gd, and Eu. Image (d) is reprinted/modified with permission from ref. 25. Copyright (2019) Frontiers.

In compounds with formula ABO3 (A and B are both metallic ions), Goldschmidt defined a tolerance factor image file: d0cp02195d-t1.tif to assess the stability of the structure on the assumption that a cubic perovskite-type crystal is formed.26 Here, rA, rB and rO are the ionic radii of the A-site cation, B-site cation, and oxygen anion, respectively. In a conventional cell of the cubic perovskite, A cations occupy the corners of the cubic, B cations occupy the body-center sites, and O anions occupy the face-centered sites. For most ABO3 compounds that form in a cubic perovskite structure, the values of t are in the range 0.8–0.9. A value of t larger than 0.9 can lead to ferroelectric distortion with the elongation of the cubic cell such as in BaTiO3, PbTiO3 and KNbO3.27 While in most RMnO3 and RFeO3 compounds, t is smaller than 0.9. One structure family RMnO3 and RFeO3 could form is the GdFeO3-type distorted perovskite in the orthorhombic phase with rotating oxygen octahedra [Fig. 1(a)], and the other structure family is the layered hexagonal phase of the space group P63cm or P63/mmc [Fig. 1(b)]. Due to the different electronic structures of Mn and Fe ions, the ground state structures of RMnO3 and RFeO3 are different.28 For R = Ho–Lu, Y, and Sc, RMnO3 is stable in the hexagonal structure, as shown in Fig. 1(d). While bulk RFeO3 all stabilize in the orthorhombic phase. Nevertheless, hexagonal RFeO3 (h-RFeO3) can still be stabilized via various methods, e.g., non-aqueous solvothermal,29 coprecipitation from an amorphous precursor,30 epitaxial growth on the trigonal substrate,31 and chemical doping in the Fe site or R site in bulk crystals.32 As for the RFe2O4 family, with R being Y, Ho, Er, Tm, Yb, Lu, or In, they all crystallize in a R[3 with combining macron]m structure,33–35 a phase related to h-RFeO3, as shown in Fig. 1(c).

2.1 Geometric ferroelectricity

In traditional ferroelectrics such as perovskite oxides BaTiO3, ferroelectricity is led by the softening of the polar distortion mode, meaning that the energy will be lowered when the ions shift along one of the polarization directions. In this case, the polarization is the primary order parameter to drive the paraelectric to ferroelectric phase transition. But ferroelectricity in h-RMnO3/h-RFeO3 is driven by a totally different geometric force, as we will see in this section.

The revisiting of h-RMnO3 started from the prototypical h-YMnO3 in 2004.14,36 The established geometric ferroelectricity and the two-step phase transition are well accepted now. Early in the 1960s, the room-temperature ferroelectric phase and high-temperature paraelectric phase were found unambiguously,37 but the ferroelectric transition path was under much debate. h-YMnO3 in the ferroelectric phase is in space group P63cm with six formula units per structural unit cell, and the high-temperature paraelectric phase is in space group P63/mmc with two formula units per unit cell. Each Mn ion occupies a 5-fold coordinated site and forms a bipyramid with surrounding oxygen ions (three in-plane oxygen ions and two apical oxygen ions), and each in-plane oxygen ion links three bipyramids as the shared corner, resulting in a sole close-packed MnO layer. Meanwhile, the triangular planes formed by R cations separate the MnO layers. Ismailzade and Kizhaev once reported a phase transition at 930 K in which YMnO3 transforms from P63cm ferroelectric phase to a nonpolar phase with space group P63/mcm,38 however, this intermediate nonpolar phase has not been observed in later experiments.39,40

According to group theory analysis, the high-symmetry space group P63/mmc has four subgroups including the low-symmetry group P63cm of the ferroelectric phase.41 The transition from the P63/mmc to P63cm group has three possible paths as shown in Fig. 2(a).42 The middle path through an intermediate P63/mcm phase corresponds to the scenario proposed in ref. 38, which has been excluded from the real phase transition in h-YMnO3 according to experimental observations. The first path goes through a ferroelectric distortion first, and then a polyhedral tilting which leads to the trimerization of the unit cell. The third path directly links the P63cm structure to the high-symmetry P63/mmc phase, and also happens in two steps. In the first step, although the K3 mode tilting triples the original unit cell, polarization has not emerged yet. It is the displacive mode at zone center in the next step that is responsible for the ferroelectricity. Based on single-crystal X-ray diffraction experiments and density functional calculations, Spaldin and co-workers revealed that the correct ferroelectric transition path in h-YMnO3 is the last one, where the distortions from the high-symmetry structure include the buckling of the MnO5 bipyramids and the imbalanced out-of-plane displacements of Y cations that give rise to net polarization, as shown in Fig. 2(b).14

image file: d0cp02195d-f2.tif
Fig. 2 (a) Allowed phase transition paths connecting the high-temperature paraelectric phase P63/mmc to the ferroelectric phase P63cm. Each path is named by the irreducible representation (irrep) of the P63/mmc phase. These irreps are associated with certain structural distortions that determine the possible intermediate phases. Z is the number of the formula units in a structural unit cell. The upmost path first goes through a ferroelectric distortion which does not triple the unit cell, and then goes to polyhedral tilting, leading to the trimerization of the bipyramids. On the contrary, the second path involves a unit-cell tripling distortion, which induces a nonferroelectric intermediate phase, followed by a ferroelectric distortion. The last path directly points to the polar structure of P63cm via the K3 mode. Note that the K3 mode itself does not introduce polarization. With another displacive Γ mode the final ferroelectric phase is obtained. Reprinted with permission from ref. 36. Copyright (2005), the American Physical Society. (b) Schematic plot of ferroelectric h-YMnO3 in the two enantiomorphous polarized states. The left one corresponds to a polarization-down state and the right one is with an upward polarization. The atomic displacements from the high-symmetry paraelectric phase are indicated by arrows. Reprinted with permission from ref. 14. Copyright (2004), Macmillan Publishers Ltd: Nature Materials.

Similar to h-YMnO3, the paraelectric to ferroelectric phase transition for other h-RMnO3 compounds takes two steps, therefore, the experimental data shows two transition temperatures. Cooling from the high-temperature structure, the structural transition from the P63/mmc to P63cm space group first arises above 1000 K. At lower temperature, a nonferroelectric-to-ferroelectric phase transition happens while the space group remains unchanged. Hence the phase transition of the h-RMnO3 series follows the third path in Fig. 2(a). The transition temperatures for the two transitions, i.e. a higher temperature for the structural transition and a lower one for the nonferroelectric-to-ferroelectric transition, are listed in Table 1 for h-RMnO3 single crystals.

Table 1 Transition temperatures and polarization (from experiments and theoretical calculations) for h-RMnO3. Here TFE is the transition temperature of nonferroelectric-to-ferroelectric phase transitions and Tnpl is the transition temperature for the transition from the P63/mmc phase to P63cm phase. The extremely low polarization measured in h-TmMnO3 and h-LuMnO3 are ascribed to the leakage current
Compound YMnO3 HoMnO3 ErMnO3 TmMnO3 YbMnO3 LuMnO3
T FE (K) 92038 87337 83337 62144 99338 75045
T npt (K) ≈122046 125047 ≈131046 ≈127046 ≈129046
P exp. (μC cm−2) 5.548 5.648 5.648 0.144 5.648 0.09645
P theory (μC cm−2) 6.214 8.049 8.149 8.349 8.549

As mentioned, only RMnO3, where R = Y, Ho–Lu, naturally stabilize in the hexagonal phase in the bulk state. Other rare-earth manganites and ferrites are normally stable in the orthorhombic phase. It is noteworthy that in the past two decades, not only h-RMnO3 (R = Sm–Dy) were obtained in a thin-film state by epitaxial growth on a certain substrate,50 but also h-RFeO3, where R = Y, Dy–Lu, were synthesized. A comprehensive review on the hexagonal ferrites from an experimental perspective can be found in ref. 28. The practical substrates include Al2O3(0001), Y-stabilized zirconium oxide (YSZ)(111) and Al2O3(0001) buffered with Pt(111) layers.31,51–55 For example, on ZrO2(Y2O3)(111) substrates, Bossak and co-workers epitaxially stabilized h-RFeO3 (R = Eu–Lu) thin films at 900 °C.31 Akbashev and co-workers reported the successful fabrication of h-RFeO3 (R = Er–Lu) using metal–organic chemical vapor deposition.51 All the thin-films of h-RMnO3 and h-RFeO3 were grown along the c-axis.

The stabilized h-LuFeO3 is the most investigated h-RFeO3 compound. It is very similar to h-YMnO3 in its ferroelectric structure with the P63cm group. The P63cm phase is maintained up to 1050 K in h-LuFeO3 films (20–60 nm) grown on α-Al2O3(001) substrates and h-LuFeO3 films grown on Pt/Al2O3 substrates.53 The room-temperature ferroelectricity is also verified by the polarization–electric field (PE) hysteresis loop which shows a remnant polarization of ∼6.5 μC cm−2. Also, similar to h-RMnO3 possessing a two-step phase transition, h-LuFeO3 goes through a nonferroelectric-to-ferroelectric phase transition at about 563 K after the first structural phase transition.55 In epitaxially grown h-TmFeO3 films, the ferroelectric transition happens at 430 K and the spontaneous polarization is 8.7 μC cm−2.56 While the h-ErFeO3 thin film has a measured spontaneous polarization of 0.2 μC cm−2, much smaller than the predicted value, yet the origin of this discrepancy is not clear.57 Besides, h-YbFeO3 is quite fascinating since the two-step phase transition follows the first path in Fig. 2(a), different from most hexagonal manganites and ferrites. The first transition occurs at ∼470 K, yielding an intermediate ferroelectric phase P63mc with remnant P = 4.0 μC cm−2 (measured at 300 K). The onset of the second phase transition is at about 225 K with the enhanced polarization of 10.0 μC cm−2 (15 K).52 Using first-principles calculations, the bulk properties of h-RFeO3 can be predicted with no limitation.58 Most of the experimental data of synthesized h-RFeO3 in a thin-film state are comparable to the calculated properties of the bulk state. Besides, the bulk h-RFeO3 can be stabilized by doping with other cations at the R site or Fe site, which will be elaborated on in Section 6.

Next we introduce the controversial ferroelectricity in hexagonal InMnO3. Although indium does not belong to the rare-earth series, h-InMnO3 is a close relation to h-RMnO3. Some researchers have reported that h-InMnO3 is also a ferroelectric material with P63cm space group at room temperature,59,60 while others found that the room-temperature structure of h-InMnO3 is nonpolar.61,62 Kumagai et al. proposed that room-temperature h-InMnO3 has a centrosymmetric structure (P[3 with combining macron]c1),62 in which tilting of the MnO5 bipyramids also leads to the tripling of the unit cell as in the ferroelectric phase of h-YMnO3, but the “up-none-down” displacements of In ions cancel out, as shown in Fig. 3(a). With the help of density functional calculations, Kumagai et al. attributed the nonferroelectricity in h-InMnO3 to the strong hybridization between In-d and h-R states, which resists the ferroelectric displacements of In.62 This conclusion is extended to the hexagonal manganites as the absence of R–O hybridization is essential to the geometric ferroelectricity. However, using transmission electron microscopy (TEM) and high-angle annular dark-field (HAADF) scanning electron microscopy (STEM), Huang et al. found that the ground state of h-InMnO3 is the ferroelectric P63cm phase while the P[3 with combining macron]c1 phase is stable near 950 °C. Interestingly, the P[3 with combining macron]c1 phase can be quenched at room temperature and coexist with the ferroelectric phase.63

image file: d0cp02195d-f3.tif
Fig. 3 (a) Side views of the ferroelectric P63cm phase and centrosymmetric P[3 with combining macron]c1 phase for the h-RMnO3 compounds. The indications of the spheres are the same as those in Fig. 1. The arrows on the R ions indicate displacements from the high-temperature P63/mmc phase. (b) Landau free energy landscape of the h-YMnO3 as a function of K3 order parameter (Q, Φ) with the P63/mmc phase at the top in the middle. The polar (P), antipolar (AP), and intermediate (IM) phases occur at the minima (red), maxima (blue), and intermediate (grey) regions at the brim of the Mexican hat shape, respectively. (c) Relationship between the high-symmetry nonpolar phase and the three low-symmetry phases. (d) Displacement of R ions along the c-axis in the polar, intermediate, and antipolar phases. Images (b–d) are reprinted/modified with permission from ref. 43. Copyright (2019), the American Physical Society.

A Landau theory study suggested that the nonferroelectric-to-ferroelectric phase transition in h-YMnO3 at around 920 K relates to a P63cmP3c1 ↔ P[3 with combining macron]c1 phase diagram.64 The order parameter of the K3 mode for the structural phase transition at 1250 K contains two components: one describes the tilting amplitude of the bipyramids (Q) and the other describes the azimuth of the tilting with the angle projected on the ab-plane (Φ).65 The relationship between polarization P and the K3 order parameter is PzQK3[thin space (1/6-em)]cos[thin space (1/6-em)]3Φ. In the ferroelectric P63cm phase, the tilting always points to the Y cation, corresponding to image file: d0cp02195d-t2.tif, where n represents an integer. When Φ is equal to image file: d0cp02195d-t3.tif, the structure retains its nonpolar P[3 with combining macron]c1 phase, as is the situation in h-InMnO3. Therefore, the P[3 with combining macron]c1 phase can be called an antipolar phase in contrast to the polar phase P63cm. When Φ is between these two special values, the structure is in an intermediate state with the P3c1 space group which allows for a non-zero polarization no larger than that of the P63cm phase. The Mexican hat-like energy landscape of h-YMnO3 connects the three lower-symmetry phases (P63cm, P[3 with combining macron]c1 and P3c1 phases) to the high-temperature phase P63/mmc, and the symmetry relationship between the space groups and the Y displacements in these phases are shown in Fig. 3(b)–(d). Combining neutron scattering techniques with first-principles-based effective Hamiltonian, Skjærvø et al. established the local structure at the transition temperature and revealed the unconventional nature of the phase transition at around 920 K, namely the mixed displacive order–disorder type of phase transition, which can also be anticipated in h-RMnO3 materials other than h-YMnO3.43

2.2 Electronic ferroelectricity

Different from RFeO3, the rare-earth ferrites with formula RFe2O4 (R = Dy–Lu, Y, In) crystallize in a rhombohedral R[3 with combining macron]m structure as shown in Fig. 1(c), where the FeO bilayers which form plane-sharing double bipyramids are separated by the triangular planes of R ions. LuFe2O4 is very intriguing as a prototype of charge-ordered systems. As is well known, space inversion symmetry breaking is the prerequisite for the emergence of ferroelectricity. Charge ordering of Fe ions, which also gives rise to spin ordering, breaks the inversion symmetry and it is possible to introduce multiferroicity, as shown in Fig. 4(a). In 2005, based on the dielectric spectroscopy and pyroelectric current measurements, Ikeda and co-workers concluded that LuFe2O4 has ferroelectricity arising from the charge ordering of Fe2+/Fe3+ ions with a polarization larger than that of BaTiO3.18 With the help of resonant X-ray scattering measurements, they also discovered that the charge ordering yields a image file: d0cp02195d-t4.tif superstructure with a propagation vector (1/3, 1/3, 0). Specifically, in an FeO bilayer, one sublayer has a surplus of Fe2+ ions and the other sublayer has a surplus of Fe3+ ions, thus a dipole moment emerges between the sublayers due to the disproportionation of the electron distribution, as shown in Fig. 4(b). Using first-principles calculations and Monte Carlo simulations, Xiang and Whangbo investigated three candidate charge orderings and found that the ones allowing ferroelectricity are energetically favored.66 The calculated ground state is a ferrielectric state with an average polarization of 26.3 μC cm−2, which is in excellent agreement with experimental results.
image file: d0cp02195d-f4.tif
Fig. 4 (a) Schematic plot of ferroelectricity emerging in a charge-ordered system as a result of inequivalent charge sites and unequal bonds. (b–e) Four types of proposed charge ordering in LuFe2O4. The dark green spheres and pink spheres represent Fe2+ and Fe3+ ions, respectively. The light green large arrows indicate the polarization of the FeO bilayers. (b) Charge ordering which leads to the parallel polarization in each FeO bilayer. (c) Charge ordering which leads to the antipolar FeO bilayers and hence antiferroelectricity. (d) Charge ordering which is nonpolar with charged FeO bilayers. (e) Charge ordering theoretically proposed by Xiang and Whangbo,66 which is the charge ordering found in YMn2O4.67

Subsequent experiments partly support the connection between ferroelectricity and charge ordering with an observed charge-order induced antiferroelectric state [Fig. 4(c)],68 but it is difficult to reproduce the pyrocurrent results.19,69–71 Based on the crystal structure refinement, de Groot and co-workers suggested a charge ordering with the stacking of charged FeO bilayers, in contrast to the polar bilayers previously suggested.72 This charge ordering, shown in Fig. 4(d), cannot lead to ferroelectricity. The energy of the nonpolar charge-ordered state was compared with those of the polar charge-ordered states by density functional theory investigations.73 However, the latter ones are more stable than the former one and the ground-state charge ordering is actually calculated to be the one proposed in ref. 68, in which the neighboring bilayers have opposite charge-order induced polarization, bringing about the antiferroelectricity in LuFe2O4. The discrepancy between theory and experiments has yet to be resolved.

There is less work regarding the charge ordering of other RFe2O4 compounds. The charge ordering in YFe2O4 is found to be a lot different from that in LuFe2O4.74 Nagata et al. observed a clear intrinsic ferroelectric PE loop in YbFe2O4, thus claiming the existence of ferroelectricity from charge ordering.75 They further showed that the charge ordering structure in YbFe2O4 can be modulated by the magnetic order from X-ray diffraction experiments and magnetic measurements.76

Although the ferroelectricity from charge ordering in RFe2O4 is still in doubt, investigations on h-LuFeO3/LuFe2O4 superlattice and h-YMnO3/YMn2O4 superlattice both reveal the existence of multiferroicity.67,73 Cheng and co-workers discovered that MnO bilayers form at the interface between the ferroelectric h-YMnO3 thin film and the substrate Al2O3.67 The MnO bilayer corresponds to the FeO bilayers in LuFe2O4, thus forming a YMn2O4 layer with different valences of Mn ions (Mn3+ and Mn2+), similar to the Fe ions in LuFe2O4. According to the hybridization between Mn 3d and O 2p states, Mn2+ favors longer MnO bonds, while Mn3+ favors shorter MnO bonds as shown in Fig. 5, leading to simple charge ordering: one sublayer only includes Mn3+ ions and the other sublayer only includes Mn2+ ions [Fig. 4(e)]. Both the charge-order induced polarization and the geometric polarization in h-YMnO3 layers contribute to the ferroelectricity of this h-YMnO3/YMn2O4/Al2O3 system. Since Fe2+ and Fe3+ ions form FeO bonds of similar length, such planar charge ordering cannot be stabilized in LuFe2O4.66 Mundy et al. introduced unit-thick LuFe2O4 into a h-LuFeO3 matrix to grow (LuFeO3)m/(LuFe2O4)1 superlattices.73 They found that the neighboring h-LuFeO3 layers drive LuFe2O4 into a ferroelectric state with the rumpling of Lu triangular planes, while simultaneously reducing the spin frustration in LuFe2O4. Those works only focused on the h-RFeO3 or h-RMnO3 matrix with mixing of unit-cell thick RFe2O4 or RMn2O4 layers, therefore, further studies are needed on superlattices with various components of RFe2O4 or RMn2O4.

image file: d0cp02195d-f5.tif
Fig. 5 (a and b) Schematic plots of occupation of the 3d levels for Mn ions with different valences. (c and d) Bond and anti-bond states between O2 − and Mn (Fe) cations. (e and f) Local structures of FeO and MnO bilayers. Mn ions with different valences result in a noncentrosymmetric MnO bilayer, while the structural symmetry of the FeO bilayer seems not to break down with different Fe ions. Reprinted with permission from ref. 67. Copyright (2018) the American Association for the Advancement of Science.

3 Magnetism

In this section, we summarize the work on magnetism in hexagonal rare-earth manganites and ferrites.

Unlike perovskite-type systems which usually yield quasi-one-dimensional magnetic order, h-RMnO3/h-RFeO3 systems consist of triangular Mn/Fe planes with frustrated magnetic order, and R planes which also contribute to the low-temperature magnetism on the condition that the R ions have partially filled f-shells, namely R = Ho, Er, Tm, and Yb. The ab-plane is an easy plane for magnetization. The intraplane superexchange interaction via the Mn(Fe)–O–Mn(Fe) path is much stronger than the interplane super–superexchange between adjacent Mn/Fe planes via the Mn(Fe)–OR–O–Mn(Fe) path. The magnetic moment of Mn/Fe can rotate within the ab-plane, creating noncollinear two-dimensional magnetic structures. Because of the triangular lattice and path of the intraplane superexchange, the magnetic moments of the trimerized Mn/Fe ions in the same layer are locked and arrange at 120° with each other below the Néel temperature (TN).77 The magnetic structure thus can be manifested by any one of the magnetic rotation angles Ψ measured from the local a-axis (Fig. 6). Besides, there are two types of coupling between adjacent Mn/Fe layers. If the magnetic moments in adjacent Mn/Fe layers couple antiferromagnetically, the magnetic structure is denoted with A (magnetic space group P63); if the interplane interaction favors the parallel arrangement between adjacent layers, the notation of the magnetic structure is B (magnetic space group P[6 with combining low line]3). In four principal cases with high symmetry where Ψ = 0° and 90°, the magnetic structures are denoted with A1(P63cm), A2(P63[c with combining low line][m with combining low line]), B1(P[6 with combining low line]3c[m with combining low line]) and B2(P[6 with combining low line]3[c with combining low line]m) as shown in Fig. 6.78 These symbols correspond to the four one-dimensional irreducible representations of the point group 6mm.79 From the perspective of symmetry, only A2 allows for a nonzero net magnetic moment from the canting of the in-plane magnetic moments.

image file: d0cp02195d-f6.tif
Fig. 6 Magnetic structures of h-RMnO3/h-RFeO3. In (a and b), only Mn/Fe ions are shown. Blue and orange spheres represent Mn/Fe in the lower layer (z = 0) and upper layer (z = 1/2), respectively. In (c) only R ions are shown. Green and purple spheres represent ions at Wyckoff positions 2a and 4b. Arrows indicate the direction of the magnetic moments. (a) A general magnetic structure where Ψ is the angle between the magnetic moment of a chosen Mn/Fe ion and the local a-axis. Left (right) panel is the case where the coupling between neighboring layers is antiferromagnetic (ferromagnetic). (b) Different magnetic structures for Fe/Mn ions. Here A1, A2, B1, and B2 magnetic structures are also denoted with the symbols Γ1, Γ2, Γ3, and Γ4, respectively, in some literature. (c) Collinear magnetic orders for R ions in A1, B1, and B2 magnetic structures according to ref. 79. The A2 order is not shown, where R ions at 2a sites and 4b sites align ferromagnetically.

Using second harmonic generation (SHG) in combination with linear magneto-optical techniques, Fiebig and co-workers determined the low temperature magnetic structures for h-RMnO3 (R = Sc, In, Y, Yb–Lu).78 Under zero magnetic field, below TN (ranging from 73 K to 124 K) all h-RMnO3 compounds, except h-YMnO3 which has a B1-type order, show the B2 magnetic structure first (Fig. 7). When the temperature decreases, three B-type magnetic phases coexist in a h-ScMnO3 single crystal. In h-HoMnO3 and h-LuMnO3, three B-type magnetic orders also emerge in a certain range of temperature. Below 5 K, h-HoMnO3 and h-ErMnO3 both have a spontaneous magnetic phase transition from B-type to A-type, which is supposed to be related to the R ions. Cooling under a nonzero magnetic field along the c-axis (Hz), h-ErMnO3, h-TmMnO3, and h-YbMnO3 show H-induced ferri- or ferromagnetic behavior manifested by the hysteresis loops, thus indicating a magnetic phase transition from B2 to A2 order. Munawar and Curnoe simulated the magnetic phase diagram of h-RMnO3 using group theory and the Landau model, and reproduced the main features observed in SHG experimental results.79 Yen et al. investigated the magnetic phase diagrams of h-RMnO3 from R = Ho to Yb via magnetic and dielectric measurements.80 But their results showed no indication of hysteretic behavior across the phase boundaries in h-ErMnO3, h-TmMnO3 and h-YbMnO3. Moreover, they also demonstrated that the stability of the B2 order is robust towards a much higher magnetic field. Further theoretical and experimental studies on the stable magnetic phase of h-RMnO3 are needed to uncover the reason for the discrepancies.

image file: d0cp02195d-f7.tif
Fig. 7 Magnetic phase diagrams for h-RMnO3. Reprinted with the permission from ref. 78. Copyright (2003) the American Institute of Physics.

In h-RMnO3 where R = Ho, Er, Tm, and Yb, the paramagnetic moments of the R-f electrons can screen the anomaly arising from the spin ordering associated with Mn3+ ions, and add difficulty to the measurement of TN.81 Experimental studies on the magnetic properties of these h-RMnO3 can be found in ref. 82–84 (ErMnO3), ref. 85 and 86 (HoMnO3), ref. 87 (Ho1−xDyxMnO3), ref. 88–91 (YbMnO3), ref. 92 (R = Er, Tm, Yb, Lu), and ref. 93 (R = Dy, Tb, Ho, and Yb). In h-HoMnO3, neutron powder diffraction measurements have demonstrated that below 25.4 K, the Ho ions at Wyckoff position 4b become magnetically ordered along the c-axis while those at Wyckoff position 2a remain paramagnetic down to the lowest experimental temperature of 1.7 K.85 In h-ErMnO3, the exchange interaction between Mn3+ ions dominates the magnetic order and triggers the 4f moments of the Er ions of the 4b site to arrange in the B2 order at 10 K < T < TN, whereas below 10 K the exchange between the Er ions of the 2a site governs the magnetic structure and renders a magnetic reorientation from B2 to A2 order.84 The magnetic phase reorientations in h-YbMnO3 below TN are similar to those in h-ErMnO3. The B2 magnetic order of Mn3+ ions below TN = 80 K polarizes the Yb3+ moments at the 4b site, and below 20–30 K the magnetic moments of the 4b-site Yb3+ ions strongly increase. The Yb ions of the 2a site begin to order at 5 K, which drags the Mn ions and other Yb ions into the magnetic structure of A2.91 There is also theoretical work on the magnetic ground states and phase diagrams of h-RMnO3 based on density functional theory,80,94,95 however, it is a challenge to take the 4f electrons of R ions into consideration for the less accurate description by the functionals available. Therefore, density functional study on the 4f magnetism from R ions is relatively scarce.

Unlike h-RMnO3, most h-RFeO3, though usually stabilized in the thin film form, show a canted weak ferromagnetism from the triangular antiferromagnetic order below TN = 60–150 K. The transition temperatures TN are also usually higher than those of h-RMnO3. The experimental TN for h-RFeO3 (R = Lu, Yb, Er, Ho, Dy, and Tb) are listed in Table 2 as extracted from published work. Akbashev and co-workers measured the magnetic properties of h-RFeO3 (R = Lu, Er–Tb) epitaxially grown on (111)ZrO2(Y2O3) substrates with a highly-sensitive vibrating sample magnetometer and revealed that a weak magnetic moment emerges in all h-RFeO3 thin films.51 They discussed the possible origins of the weak ferromagnetism and proposed that the extra d-electron from Fe3+ ions compared to Mn3+ ions might be the key factor. Wang et al. reported that h-LuFeO3 is a room-temperature multiferroic,53 however, later experiments on h-LuFeO3 thin films found that h-LuFeO3 transforms into a ferromagnetically canted antiferromagnetic state (A2) below 155 K, much lower than room temperature, irrelevant to the thickness of the thin film.96 Besides, labyrinth-like weak ferromagnetic domains below 147 K in a 200 nm thick h-LuFeO3 film are observed after zero field cooling.97 Another work published in 2018 reported the successful stabilization of single phase h-LuFeO3 in bulk form without doping by sol–gel method.98 The antiferromagnetic phase transition in the bulk state is confirmed to occur at 130 K, and another anomaly at 42 K appears in magnetization–temperature measurements, which, however, do not correspond to any spin reorientation from neutron experiments. Significantly, h-LuFeO3 thin films keep on attracting much research interest as a promising candidate for multiferroics at higher temperatures.

Table 2 Néel temperatures TN of h-RFeO3
LuFeO3 YbFeO3 ErFeO3 HoFeO3 DyFeO3 TbFeO3
T N (K) 12051 12052 10051 13051 70–8051 12051
13053,98 6099 12057

In h-RFeO3 with magnetic R ions, such as h-YbFeO3, the R ions can also be polarized by the effective magnetic field caused by the magnetic order of Fe3+ ions, further enhancing the saturation magnetization of h-RFeO3.52,99,100 In h-HoFeO3, element specific X-ray magnetic circular dichroism measurements also showed a weak ferromagnetic order of Fe ions and exchange coupling between Ho3+ and Fe3+ ions.101 Xu et al. performed first-principles calculations on the electronic and magnetic properties of h-RFeO3, where R includes all the Lanthanides, which are in accordance with the experimental results.58 But as with h-RMnO3, magnetism due to R ions still needs to be further explored.

There are few works on the magnetism of RFe2O4 materials. Interestingly, Wu et al. found that the collective freezing of nanoscale pancake-like ferrimagnetic domains with large uniaxial magnetic anisotropy is related to the promotion of the magnetic coercivity in LuFe2O4.102 The interactions between magnetism, charge ordering and ferroelectricity in these systems are undoubtedly very complicated, but can also be colourful and fascinating.

4 Magnetoelectric coupling

Magnetoelectric effect in multiferroic materials, especially in type-II multiferroics where spin ordering induces ferroelectricity, has been comprehensively reviewed by Dong et al.103,104 Here we briefly introduce related works on the hexagonal rare-earth manganites and ferrites which are type-I multiferroics.

Magnetoelectric coupling is the mutual coupling between magnetism and ferroelectricity. At the linear response level, the magnetoelectric effect, namely the phenomenon that the electric polarization is changed by an applied magnetic field or the magnetization changes upon applying an electric field, can be described by the linear relation image file: d0cp02195d-t5.tif or image file: d0cp02195d-t6.tif, where [small alpha, Greek, circumflex] is the second-rank magnetoelectric coupling tensor including the magnetic permeability and the dielectric constant of the medium.105 The renaissance of magnetoelectric materials and magnetoelectric effect was ignited by the discovery of the spin-induced ferroelectricity in TbMnO3, which is an orthorhombic manganite.13 On the other hand, hexagonal manganites and ferrites are also well-known multiferroic materials, thus enabling the cross coupling between ferroelectric and magnetic orders and providing a platform for novel magnetoelectric phenomena. Actually, anomalies in the dielectric constant at the magnetic transition temperature have long been observed in hexagonal manganites as evidence of magnetoelectric coupling.106–108 At high temperature, the dielectric constant drops upon the application of an external magnetic field in h-ErMnO3.109 Not only the dielectric constant, but also the electric polarization, undergoes a drastic change at the magnetic transition temperature in h-HoMnO3.110,111 The reverse effect that the magnetic phase is controlled by an external electric field in h-RMnO3 (R = Ho–Yb) has been reported.105,112 Similar phenomena have also been observed in h-RFeO3.52,113

Tracing the microscopic mechanism for the magnetoelectric coupling, one can dissect this effect into electronic, ionic, and strain-mediated responses.114 Most theoretical studies focus on the mechanisms based on spin–orbit coupling and spin–lattice coupling. The spin–orbit coupling is essential because it is the only term in the Hamiltonian that relates the magnetic degree of freedom to the electric degree of freedom.115 Nevertheless, the gigantic magneto-elastic coupling indirectly connects the (anti)ferromagnetic order to the ferroelectricity via atomic displacements, which has also been emphasized in h-RMnO3.103,104,116–118 Next, the two types of mechanisms are expounded separately.

4.1 Mechanism based on spin–orbit coupling

The Dzyaloshinskii–Moriya (DM) interaction, which was first proposed by Dzyaloshinskii to explain the weak ferromagnetism in antiferromagnetic crystals,121 and further elaborated on by Moriya as a consequence of the spin–orbit coupling,122,123 plays an indispensable role in the magnetoelectricity of many multiferroics, especially type-II multiferroics. The DM interaction has the form [D with combining right harpoon above (vector)]ij·([S with combining right harpoon above (vector)]i × [S with combining right harpoon above (vector)]j), where [D with combining right harpoon above (vector)]ij is the DM coefficient between the spin [S with combining right harpoon above (vector)]i and [S with combining right harpoon above (vector)]j; therefore, it only exists in systems with noncollinear spin structure. In multiferroic h-RMnO3 and h-RFeO3, the in-plane spins arrange in a 120° antiferromagnetic structure, thus the DM interaction is non-negligible. In hexagonal manganites, a symmetry analysis yields that Mz[Q with combining right harpoon above (vector)]·[L with combining right harpoon above (vector)], where [L with combining right harpoon above (vector)], is the order parameter describing magnetic order [L with combining right harpoon above (vector)] = Lxy(cos[thin space (1/6-em)]Ψ[x with combining circumflex] + sin[thin space (1/6-em)]Ψŷ) (the definition of Ψ is found in Fig. 6) and [Q with combining right harpoon above (vector)] is the order parameter describing the K3 mode of the bipyramid tilting [Q with combining right harpoon above (vector)] = Qk3(cos[thin space (1/6-em)]Φ[x with combining circumflex] + sin[thin space (1/6-em)]Φŷ) (the definition of Φ is in Section 2.1).119,120 According to the relationship: only when the magnetic order parameter has a parallel component with the K3 order parameter, magnetization along the z-axis arises. After analyzing the four principal magnetic patterns, only the A2 magnetic phase (or phases which contain A2) has nonzero linear magnetoelectric coupling and allows a weak ferromagnetic moment along the z-axis. (Here, the z direction is along the c-axis and xy-plane is the ab-plane.) In A1 and B2 magnetic patterns, spins are perpendicular to the K3 tilting vector, therefore [Q with combining right harpoon above (vector)]·[L with combining right harpoon above (vector)] = 0. While in B1 order, the canting-induces weak out-of-plane magnetic moments that cancel each other out between neighboring MnO/FeO layers.

When the linear magnetoelectric coupling is present, the out-of-plane magnetization can be modified by an electric field as shown in Fig. 8. In ferroelectric/antiphase boundary vortex-like domain structures (which will be elaborated on in Section 5), if the magnetoelectric coupling coefficient changes its sign in each domain, the magnetization across the vortex structure keeps the same direction as shown in Fig. 8(c).120 Such domain structures were observed in h-ErMnO3via magnetoelectric force microscopy, providing a direct visualization for magnetoelectric domains.119 If the net magnetization is clamped to the K3 order parameter, that is the sign of magnetoelectric coefficient α is unchanged, the magnetic domains will also couple to the ferroelectric domains. Das et al. argued that this domain configuration can be expected under some circumstances and thus an electric field can be used to reverse the magnetization by reversing the polarization, as illustrated in Fig. 8(d) and (e).120 Moreover, alternating net magnetic moments at ferroelectric domain walls in h-ErMnO3 were observed.124 In h-YMnO3, where the B1 magnetic order does not allow the linear magnetoelectric effect,78 the experimental observation manifests a magnetoelectric coupling as well as a weak magnetic component along the z-axis.125 A symmetry reduction in h-YMnO3 to the magnetic space group of P[6 with combining low line]3 which contains an A2 component might explain this phenomenon.116,125,126

image file: d0cp02195d-f8.tif
Fig. 8 (a) A diagram of effective magnetoelectric coupling via the K3 trimerization mode. (b) Schematics of how the application of an electric field alters the net magnetization by modifying the buckling angle of the MnO5 bipyramids. (a and b) are reprinted with permission from ref. 119. Copyright (2014), Macmillan Publishers Ltd: Nature Materials. (c–e) The cloverleaf-like ferroelectric/structural antiphase domain structure for the A2 magnetic phase. The antiferromagnetic spins rotate by (c) |π/3|, or (d) |2π/3|. In case (c), the canting magnetization remains unchanged across the ferroelectric domain walls while the sign of the magnetoelectric coefficient (αME) changes. Case (d), shows the contrary, the sign of αME is not changed while magnetization changes its direction across the ferroelectric domain walls which leads to coupled magnetic–ferroelectric domains, therefore, the magnetization and ferroelectric polarization can be simultaneously tuned by the electric field as shown in (e). (c–e) are reprinted with permission from ref. 120. Copyright (2014) Macmillan Publishers Ltd: Nature Communications.

4.2 Mechanism based on spin–lattice coupling

Indirect magnetoelectric coupling via exchange striction or electrostrictive/magnetostrictive effects also plays an important role in the cross control between magnetic and electric degrees of freedom. Lee et al. proposed that the giant magnetoelastic coupling leads to the magnetoelectric coupling since the large atomic displacements at the magnetic transition temperature TN connect the electric dipole moment with the magnetic phase transition.127 Other work on h-RMnO3 (R = Ho, Yb, Sc, Y) further pointed out that the positions of Mn ions influence the super–superexchange coupling between Mn in adjacent planes, and hence determines the magnetic order to be A-type or B-type.118 Nevertheless, a simple model merely based on the geometrical frustration effect via exchange interaction seems to hardly account for all the experimental facts. In h-HoMnO3, the two anomalies of the polarization at the spin-reorientation temperature and Ho ordering temperature are explained by the lattice shrinking and Ho off-center displacements respectively, both of which induce different interplane super–superexchange interactions associated with different spin orders.110 A theoretical study based on the normal double-exchange model instead of the spin–orbit coupling was established to explore the coupling between ferroelectric order and magnetic order in orthorhombic manganites, monoclinic BiMnO3 and hexagonal manganites, which works equally well for the h-HoMnO3 and orthorhombic TbMnO3.128 As to h-RFeO3, Wang et al. predicted theoretically that the structural distortion might mediate the coupling between the ferroelectric and magnetic orders in h-LuFeO3.113 Using first-principles calculations, Xu et al. predicted that the hydrostatic pressure in h-RFeO3 with small R ions, can cause the sudden disappearance of linear magnetoelectric coupling at some critical values.58 In a word, diverse models have been built to describe the spin–lattice coupling, which contributes significantly to the magnetoelectric coupling in h-RMnO3 and h-RFeO3.

5 Ferroelectric domain wall

In ferroic materials, domain structures inevitably emerge as a consequence of the minimization of free energy. During the annealing process, the ferroic materials tend to crystalize into different domains, and in each domain the order parameter freezes along the same orientation. In the film growth or surface modulation, various domain structures arise to minimize the surface energy related to the depolarization or demagnetization field. Domain walls (DWs) are the transition areas separating domains with distinct orientations, which have various widths and can be either neutral or charged.129–132 In multiferroics, domain structures and domain walls influence the response to the applied field, and result in anomalous properties distinguishable from the bulk structure, such as enhanced/decreased conductance, magnetoelectric effect and magnetization.120,133–135 Especially, the charged DWs attract carriers and charged defects, giving rise to exotic functionalities.136 In conventional ferroelectrics such as perovskites, ferroelectric DWs are as narrow as several unit-cells due to the strong interplay between lattice deformation and polarization.137 In hexagonal rare-earth manganites and ferrites, ferroelectric DWs are particularly intriguing because of the mutual coupling to the structural antiphase DWs and antiferromagnetic DWs. Moreover, the cloverleaf-like six-fold multiferroic vortices and anti-vortices have also stimulated intensive research interests as a novel topological defect.

5.1 Interlocked ferroelectric and structural antiphase domain wall

As mentioned in Section 2.1, the order parameter of the trimerization mode K3 consists of two components Q and Φ. In the ferroelectric P63cm phase, Φ equals image file: d0cp02195d-t11.tif, which correspond to three trimerization centers with two alternative tilting directions. Therefore, three types of structural antiphase domains (α±, β±, γ±) emerge as shown in Fig. 9, where “±” indicates the direction of ferroelectric polarization along the c-axis due to the associated displacive mode. Choi and co-workers first reported the interlocked ferroelectric and structural antiphase DWs in h-YMnO3 using transmission electron microscopy (TEM) and conductive atomic force microscopy (CAFM).138 Both the CAFM and TEM results show a domain pattern in which six domains in a sequence of [α+, β, γ+, α, β+, γ] or the reversed order, grow from a single point to form a vortex or anti-vortex structure [Fig. 10(a)]. There is no such adjacent domain as [α+, α] or [α+, β+], indicating the absence of pure ferroelectric DWs or antiphase DWs. Even during the poling process, antiphase DWs always couple with ferroelectric DWs, therefore, the mutual locking of ferroelectric and antiphase DWs is robust. Using aberration-corrected annular-bright-field and HAADF STEM, Zhang et al. investigated the domain vortex in h-YMnO3 on the atomic scale, showing the high-resolution images of DWs in which the atomic displacements are clearly distinguished, as demonstrated in Fig. 10(b).139 This six-fold cloverleaf-like domain pattern not only appears in the in-plane direction, but also is isotropic, as evidenced by the observation of transverse and longitudinal DWs. Spherical aberration-corrected STEM was also used to visualize the antiparallel ferroelectric DW and tail-to-tail ferroelectric DW in h-YMnO3, which exhibit a double-arc shape of reversed curvatures.140 The ferroelectric/antiphase DWs in h-TmMnO3 and h-LuMnO3 ceramics were also imaged at the atomic scale.141 Besides, Du et al. have successfully grown highly-cleavable Sc-stabilized h-Lu0.6Sc0.4FeO3 single crystals and visualized their vortex domain patterns.142 These experimental observations show that the cloverleaf-like domain patterns are universal in h-RMnO3 and h-RFeO3.
image file: d0cp02195d-f9.tif
Fig. 9 (a) Six structural domains in ferroelectric h-YMnO3 viewed along the c-axis. Here image file: d0cp02195d-t7.tif and image file: d0cp02195d-t8.tif correspond to image file: d0cp02195d-t9.tif and image file: d0cp02195d-t10.tif, respectively. Arrows indicate the displacement of apical oxygen ions (small yellow spheres), which represent the tilting direction of the MnO5 bipyramids. Reprinted with permission from ref. 120. Copyright (2014) Macmillan Publishers Ltd: Nature Communications. (b) Side views of the six structural domains. Y and Mn ions are sketched with green and grey spheres, respectively, and oxygen ions are not shown.

image file: d0cp02195d-f10.tif
Fig. 10 (a) TEM dark-field image of the vortex domain structure in ferroelectric h-YMnO3 with a sketch of the six domains. Reprinted with permission from ref. 138. Copyright (2010) Macmillan Publishers Ltd: Nature Materials. (b) High-angle annular-dark-field STEM image of the anti-vortex domain structure. Yellow and blue arrows indicate the direction of local polarization. Red dotted lines are the boundaries of different domains, and the circle is the core of the anti-vortex. Type I DWs and type II DWs are marked, where type I DWs span 1/3 unit cell and type II DWs have a thickness of 2/3 unit cell. The three structural domains [α, β, γ] are presented at the bottom schematically. Reprinted with permission from ref. 139. Copyright (2013) Macmillan Publishers Ltd: Scientific Reports. (c) Optical microscopy image of stripe domain patterns in h-ErMnO3, which are also observed in h-HoMnO3, h-TmMnO3, h-YbMnO3, and h-LuMnO3 crystals. Inset shows the atomic force microscopy (AFM) image. (d) Optical microscopy image of vortex domain patterns in h-YMnO3, the inset shows the AFM image. (c and d) Reprinted with permission from ref. 143. Copyright (2012) the American Physical Society.

Besides the vortex and anti-vortex domain patterns in h-RMnO3, Chae et al. discovered another domain pattern which has stripe domain walls.143 In their experiments, the stripe domain patterns span over the whole sample surface with no observed vortices in any h-RMnO3 samples except h-YMnO3. It is suggested that when the crystal is grown below the structural transition temperature TC (not the nonferroelectric-to-ferroelectric transition temperature), the stripe patterns arise in accordance with a long-range ordered phase. When the growth temperature is above TC or the stripe domains are exposed to the temperature above TC, the vortex and anti-vortex domain structures begin to shape as the so-called Kosterlitz–Thouless (KT) phase. Chae et al. proposed that the long-range ordered stripe domains are the true ground state of h-RMnO3 instead of the KT phase. It thus also provides a way to determine the structural transition temperature TC for h-RMnO3 by checking whether the stripe domain patterns emerge.142,144

Artyukhin et al. built the Landau free energy model for topological defects in h-YMnO3 with the coefficients extracted from first-principles calculations.65 Later the model was further developed for h-DyMnO3, h-ErMnO3 and h-TmMbO3.145 It was found that the discrete sequence of vortex or anti-vortex meeting at a point is topologically protected. It was also predicted that the stripe domain walls can be induced by strain, which was realized in h-ErMnO3 by employing shear strain with a large gradient.146 Kumagai and Spaldin investigated the ferroelectric and antiphase DWs in hexagonal manganites at the atomic scale using first-principles calculations and showed that the {210} ferroelectric/antiphase domain walls have the lowest energy.147 Through analyses on atomic structures with different types of DWs, they revealed that the pure antiphase DW is actually a composition of two ferroelectric/antiphase DWs and the pure ferroelectric DW is a composition of three ferroelectric/antiphase DWs, therefore, pure antiphase DWs and pure ferroelectric DWs would likely decay into ferroelectric/antiphase DWs due to the entropy at high temperatures, which accounts for the only observation of ferroelectric/antiphase DWs in experiments. Their findings also explained the observation of the stripe domain walls in h-RMnO3 grown below TC.143

Notably, Huang et al. discovered a partially undistorted antipolar (PUA) phase in the ferroelectric DWs in h-InMnO3.63 In transverse DWs, the polarization have a smooth transition across adjacent domains, leaving the polarization of the middle layer at zero. In other words, in the DW area, the displacements of different In ions are cancelled, inducing a centrosymmetric structure. In h-TmMnO3 and h-LuMnO3, these DWs were observed at the surfaces of samples using HAADF imaging.141 While first-principles calculations found that these DWs have higher energy than the sharp DWs. The PUA phase observed in DWs can be stabilized in Ga doped h-InMnO3.148 Actually, this PUA phase at ferroelectric DW is similar to the proposed antipolar phase in Fig. 3(d).43

Explorations on ferroelectric DWs in hexagonal ferrites are relatively rare because of the instability of the bulk hexagonal phase. In thin films of h-LuFeO3 and h-ErFeO3 grown by metal–organic chemical vapor deposition, the longitudinal and transverse ferroelectric/antiphase DWs observed in h-YMnO3 have also been observed using aberration-corrected HAADF STEM.149 However, the domains and polarization switching are random to some extent, which may be due to the defects preventing the long-range ferroelectric order. In In-stabilized h-LuFeO3, cloverleaf-like domain patterns are distinguishable and the atomic-scale structures of type I and type II DWs are imaged by the aberration-corrected HAADF STEM.150 Besides, using both piezoresponse force microscopy (PFM) and electrostatic force microscopy (EFM), Yang et al. imaged the ferroelectric/charge order DWs of LuFe2O4, but the domains have irregular patterns with a size of 100 nm.151 More work on the coupled ferroelectric/charge order DWs in LuFe2O4 are needed to understand the coupling at DWs, which might help to comprehend the controversial conclusions on the charge-order induced ferroelectricity in LuFe2O4.

5.2 Conductance at ferroelectric domain walls

With the first observation of interlocked ferroelectric/antiphase DWs in h-YMnO3, Choi et al. also found the cloverleaf-like pattern visible in the CAFM images.138 Domains with downward polarization and upward polarization have sharp contrast in the brightness, clearly indicating different conductance, which might be caused by the distinct Schottky barriers formed at the metallic-tip-semiconductor interface. Besides, ferroelectric DWs (FDWs) naturally trap defects such as oxygen vacancies, which influence the conductance as well as the mobility of FDWs.152 Moreover, unlike BiFeO3 and Pb(Zr,Ti)O3 films in which the FDWs have good electric conduction for the reduced band gap,153,154 the FDWs in h-YMnO3 have an insulating nature. Nevertheless, the decreased conductivity at FDWs is not always the case in h-RMnO3. In h-TbMnO3 film, using CAFM, Kim et al. found neutral DWs which show enhanced conductivity.155 This phenomenon is explained by the defects at the DW where back-to-back Schottky diode structures form. In h-ErMnO3, the FDW conductance was found to be highly anisotropic, varying in a range of more than one order of magnitude.135 The conductance at FDWs can be either higher or lower than the bulk conductance.

In conventional ferroelectrics, tail-to-tail and head-to-head FDWs are usually energetically unfavorable owing to the electrostatic interaction, therefore FDWs are mostly charge-neutral. In hexagonal h-RMnO3, due to the interlocking of structural antiphase DWs and FDWs, the situation is different. Recently, Schoenherr et al. demonstrate the robustness of charged domain walls in h-Er0.99Ca0.01MnO3 using EFM at low temperatures of 4.5 K.156 In the vortex and anti-vortex structures, head-to-head and tail-to-tail FDWs are topologically protected and bound charge accumulates due to the discontinuity of the normal component of polarization. The FDWs in h-RMnO3 thus possess diverse conductance and novel functionalities. In p-type semiconductor h-HoMnO3, using in situ CAFM, PFM, and Kelvin-probe force microscopy, Wu et al. found that the conductance of tail-to-tail FDWs can be largely (slightly) enhanced by the forward (reverse) bias, while the conductance of head-to-head FDW is suppressed by the high reverse bias or unchanged via high forward bias.157 The reason for the different behaviors is related to the interaction between carriers and bound charges at different FDWs. The tail-to-tail FDWs are negatively charged and attract holes in the p-type h-HoMnO3, hence the carrier density is larger than that of the bulk. At forward bias, the positive carriers easily transmit from the surface to the metal tip. On the contrary, the positively charged head-to-head FDWs attract electrons which eliminate the original hole like carriers, and are thus less conductive than the bulk. By the same token, p-type semiconductor h-ErMnO3 also has high conductance at tail-to-tail FDWs and poor conductance at head-to-head FDWs as a consequence of different carrier accumulation and band-structure shifting at the FDWs.135 Theoretical tools including phenomenological electrostatic modelling and ab initio density functional theory were also used to illuminate the mechanism in the above work.

The crystal structure, energetics, and electronic structures of head-to-head and tail-to-tail FDWs in h-YMnO3, h-InMnO3, and h-YGaO3 were theoretically studied using first-principles calculations combined with phenomenological modeling.158 A Zener-like electrostatic breakdown model was proposed to determine the charge compensation at the FDWs, resulting in a crucial distance defined as image file: d0cp02195d-t13.tif, where Eg is the band gap of the bulk and ε is the dielectric constant. According to the model, charge transfer to the charged FDWs occurs when the distance between the FDWs exceeds the crucial distance. Under this circumstance, local electronic structure will bend to allow the Fermi level to cross the valence bands at tail-to-tail FDWs, or to reach the conduction bands at head-to-head FDWs. Therefore, the guidance on engineering n-type and p-type DW conductivity can be established based on the domain size, polarization, and electronic band gap (Fig. 11).

image file: d0cp02195d-f11.tif
Fig. 11 Calculated atomically resolved density of states (DOS) for [α|β+] domain wall in h-YMnO3, h-InMnO3 and h-YGaO3via density functional theory. In this calculation, the distances between neighboring FDWs for the three systems are set to be equal. The top panels are the DOS for head-to-head FDWs, the middle panels are for the DOS of β+ domains, and the lower panels are the DOS for tail-to-tail FDWs. In h-YMnO3, the distance d between FDWs, the band gap Eg and spontaneous polarization P satisfy the relation image file: d0cp02195d-t12.tif, and charge transfer thus occurs with the Fermi level located above the conduction band edge (n-type) at the head-to-head FDWs, and the Fermi level located below the valence band at the tail-to-tail FDWs (p-type). While the charge transfer at charged FDWs does not happen in h-InMnO3 or h-YGaO3, owing to the wide band gap in h-YGaO3, and small polarization in h-InMnO3. Reprinted with permission from ref. 158. Copyright (2018) the American Physical Society.

6 Non-stoichiometric hexagonal manganites and ferrites

This review mainly concerns the multiferroic properties of stoichiometric hexagonal rare-earth manganites and ferrites. In this section we give a glance into recent work on doped or defective compounds.

Some work concentrates on R-site doping with other cations. For example, the ferroelectric and ferromagnetic domain patterns in Er1−xZrxMnO3 and Er1−xCaxMnO3 are investigated via different techniques.156,159,160 The magnetic phase diagram in Ho1−xDyxMnO3 is also studied, showing that the magnetic structures A1 and B2 dominate the phase transition.161 In h-LuMnO3, both R-site doping and Mn-site doping with trivalent ions are explored theoretically.162 The Sc-doping on the Lu site in h-LuFeO3 is found to result in a stable bulk hexagonal phase which makes it possible to compare the multiferroicity between bulk and thin-film forms of h-LuFeO3.163 And comprehensive studies on the magnetic and ferroelectric properties as well as the domain patterns in Lu1−xScxFeO3 can be found in ref. 142, 164 and 165. Besides, the In-doping at the Lu site in h-LuFeO3 can also stabilize the hexagonal phase in the bulk state. It is reported that the TN of Lu1−xInxFeO3 (x = 0.4–0.6) is around 350 K and the polarization is 1.73 μC cm−2, which makes the system a room-temperature multiferroic with antiferromagnetic order.150

The transition metal Fe/Mn can also be substituted with different metal elements. There is quite a few research articles on RMn0.5Fe0.5O3, e.g. ErMn0.5Fe0.5O3 and LuMn0.5Fe0.5O3.165–168 In h-YMnO3, the substitution of Mn with Cr increases the magnetic transition temperature.169 The inner steric effects were investigated by doping Al and Ga respectively, into Mn sites in h-YMnO3. As a consequence, a dramatic doping effect with Al-doping arises while no obvious change is observed with Ga-doping up to 50%, which can be explained by the sharp contrast between the ionic size of Al and Mn, and the close ionic size of Ga and Mn.170 In h-LuMnxO3 with x ∼ 0.98, the Mn vacancy breaks the geometric frustrated state and introduces a weak ferromagnetic moment below 90 K, which also prompts an additional electric polarization contribution and an enhancement of the magnetoelectric coupling.171 The ferroelectric transition temperature is also sensitive to the cation ratio x in h-LuMnxO3.144

The oxygen content in hexagonal manganites and ferrites is another degree of freedom to modulate the magnetic and ferroelectric properties since the oxygen ion plays a vital role in the superexchange interaction. In h-YMnO3, oxygen vacancies can trigger the off-center displacements of Mn which lead to the changing of the magnetic ground state from B1 to the weak ferromagnetic A2 phase,172,173 and transition the vortex domain walls to the stripe domain walls.152 The interstitial oxygen defects enhance the conductivity in h-RMnO3, leading to p-type electronic conductance without reducing the spontaneous ferroelectric polarization.174 Interestingly, the conductance manipulation by the interstitial oxygen migration can be easily achieved in the ab-plane, but is prohibited along the c-axis.175 This phenomenon is explained using density functional calculations as follows: the migration barrier along the in-plane direction for interstitial oxygen is lower than that along the out-of-plane direction. The redox properties of h-HoMnO3+δ and h-DyMnO3+δ have also been investigated.176 As with other metal oxides, the control of oxygen content is an effective method to tune the magnetism, conductivity and chemical activity of h-RMnO3 and h-RFeO3.

7 Conclusion

We have reviewed the research on the ferroelectric and magnetic properties of hexagonal rare-earth manganites and ferrites which are well-known type-I multiferroics. Hexagonal RMnO3 with R = Y, Ho–Lu are room-temperature geometric ferroelectrics. The nature of the two-step ferroelectric phase transition is clear: the trimerization of MnO5 bipyramids is seen first, followed by ionic displacements leading to the polarization. Hexagonal RFeO3, only stabilized in the thin-film form or chemically doped bulk form, and are similar to h-RMnO3 in structure and ferroelectric mechanism. On the other hand, the compounds of RFe2O4 are charge-ordered systems and ferroelectricity could arise from the charge ordering, however, this is still under discussion. At about 60–150 K, h-RMnO3 and h-RFeO3 undergo a magnetic phase transition from the paramagnetic state to the antiferromagnetic state with an in-plane triangular magnetic order. h-RFeO3 usually has higher transition temperature than h-RMnO3 and yields a weak out-of-plane ferromagnetism. An overview of magnetoelectric effects in hexagonal manganites and ferrites are presented with a focus on the two major mechanisms: magnetoelectric coupling from spin–orbit effects and from spin–lattice effects. In addition, the unique cloverleaf-like domain patterns observed in h-RMnO3 and h-RFeO3 with six domains arranged in a vortex or anti-vortex are discussed, which are composed of interlocked ferroelectric and structural antiphase domain walls. Magnetoelectric coupling is also expected at these domain walls, and the related work is summarized. At last, recent work on the non-stoichiometric effect in h-RMnO3 and h-RFeO3 are briefly introduced, showing that the cation substitution can bring on improved magnetism while the oxygen defects can tune the stable magnetic phase as well as the conductivity.

Conflicts of interest

There are no conflicts to declare.


We thank Zhirong Liu and Meng Ye for their critical reading of the manuscript and helpful discussions. This work was supported by the National Natural Science Foundation of China (under grants no. 11704038 and 51788104), and the Beijing Advanced Innovation Center for Materials Genome Engineering.

Notes and references

  1. J. Valasek, Phys. Rev., 1921, 17, 475 CrossRef CAS.
  2. O. Auciello, J. F. Scott and R. Ramesh, et al. , Phys. Today, 1998, 51, 22 CrossRef CAS.
  3. M. Dawber, K. Rabe and J. Scott, Rev. Mod. Phys., 2005, 77, 1083 CrossRef CAS.
  4. N. Setter, D. Damjanovic, L. Eng, G. Fox, S. Gevorgian, S. Hong, A. Kingon, H. Kohlstedt, N. Park and G. Stephenson, et al. , J. Appl. Phys., 2006, 100, 051606 CrossRef.
  5. H. Schmid, Ferroelectrics, 1994, 162, 317 CrossRef.
  6. G. Smolenskii, V. Isupov, N. Krainik and A. Agranovskaya, Izv. Akad. Nauk SSSR, Ser. Fiz., 1961, 25, 1333 CAS.
  7. G. Smolenskii and V. Bokov, J. Appl. Phys., 1964, 35, 915 CrossRef CAS.
  8. E. Ascher, H. Rieder, H. Schmid and H. Stössel, J. Appl. Phys., 1966, 37, 1404 CrossRef CAS.
  9. G. Smolenskii and I. Chupis, Sov. Phys. – Usp., 1982, 25, 475 CrossRef.
  10. M. Fiebig, J. Phys. D: Appl. Phys., 2005, 38, R123 CrossRef CAS.
  11. N. A. Hill, J. Phys. Chem. B, 2000, 104, 6694 CrossRef CAS.
  12. J. Wang, J. Neaton, H. Zheng, V. Nagarajan, S. Ogale, B. Liu, D. Viehland, V. Vaithyanathan, D. Schlom, U. Waghmare, N. Spaldin, K. Rabe, M. Wutting and R. Ramesh, Science, 2003, 299, 1719 CrossRef CAS PubMed.
  13. T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T.-h. Arima and Y. Tokura, Nature, 2003, 426, 55 CrossRef CAS PubMed.
  14. B. B. Van Aken, T. T. Palstra, A. Filippetti and N. A. Spaldin, Nat. Mater., 2004, 3, 164 CrossRef CAS PubMed.
  15. S.-W. Cheong and M. Mostovoy, Nat. Mater., 2007, 6, 13 CrossRef CAS PubMed.
  16. Y. Miyamoto and M. Shindo, J. Phys. Soc. Jpn., 1993, 62, 1423 CrossRef CAS.
  17. N. Ikeda, K. Kohn, N. Myouga, E. Takahashi, H. Kitôh and S. Takekawa, J. Phys. Soc. Jpn., 2000, 69, 1526 CrossRef CAS.
  18. N. Ikeda, H. Ohsumi, K. Ohwada, K. Ishii, T. Inami, K. Kakurai, Y. Murakami, K. Yoshii, S. Mori, Y. Horibe and H. Kitô, Nature, 2005, 436, 1136 CrossRef CAS PubMed.
  19. M. Angst, Phys. Status Solidi RRL, 2013, 7, 383 CrossRef CAS.
  20. D. Khomskii, Physics, 2009, 2, 20 CrossRef.
  21. G. Catalan and J. F. Scott, Adv. Mater., 2009, 21, 2463 CrossRef CAS.
  22. J. Silva, A. Reyes, H. Esparza, H. Camacho and L. Fuentes, Integr. Ferroelectr., 2011, 126, 47 CrossRef CAS.
  23. W. Prellier, M. Singh and P. Murugavel, J. Phys.: Condens. Matter, 2005, 17, R803 CrossRef CAS.
  24. J. Fontcuberta, C. R. Phys., 2015, 16, 204 CrossRef CAS.
  25. S. Hosokawa, Front. Chem., 2019, 7, 8 CrossRef CAS PubMed.
  26. V. M. Goldschmidt, Naturwissenschaften, 1926, 14, 477 CrossRef CAS.
  27. C. Li, K. C. K. Soh and P. Wu, J. Alloys Compd., 2004, 372, 40 CrossRef CAS.
  28. X. Xu and W. Wang, Mod. Phys. Lett. B, 2014, 28, 1430008 CrossRef CAS.
  29. S. Hosokawa, H.-J. Jeon, S. Iwamoto and M. Inoue, J. Am. Ceram. Soc., 2009, 92, 2847 CrossRef CAS.
  30. T. Nishimura, S. Hosokawa, Y. Masuda, K. Wada and M. Inoue, J. Solid State Chem., 2013, 197, 402 CrossRef CAS.
  31. A. A. Bossak, I. E. Graboy, O. Y. Gorbenko, A. R. Kaul, M. S. Kartavtseva, V. L. Svetchnikov and H. W. Zandbergen, Chem. Mater., 2004, 16, 1751 CrossRef CAS.
  32. E. Magome, C. Moriyoshi, Y. Kuroiwa, A. Masuno and H. Inoue, Jpn. J. Appl. Phys., 2010, 49, 09ME06 CrossRef.
  33. T. Matsumoto, N. Mori, J. Iida, M. Tanaka, K. Siratori, F. Izumi and H. Asano, Phys. B, 1992, 180, 603 CrossRef.
  34. M. Isobe, N. Kimizuka, J. Iida and S. Takekawa, Acta Crystallogr., Sect. C: Cryst. Struct. Commun., 1990, 46, 1917 CrossRef.
  35. X. Xu, J. De Groot, Q.-C. Sun, B. C. Sales, D. Mandrus, M. Angst, A. Litvinchuk and J. Musfeldt, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 82, 014304 CrossRef.
  36. C. J. Fennie and K. M. Rabe, Phys. Rev. B: Condens. Matter Mater. Phys., 2005, 72, 100103 CrossRef.
  37. H. Yakel, W. Koehler, E. Bertaut and E. Forrat, Acta Crystallogr., 1963, 16, 957 CrossRef CAS.
  38. I. Ismailzade and S. Kizhaev, Sov. Phys. Solid State, 1965, 7, 236 Search PubMed.
  39. K. uukaszewicz and J. Karut-Kalicińska, Ferroelectrics, 1974, 7, 81 CrossRef.
  40. T. Katsufuji, M. Masaki, A. Machida, M. Moritomo, K. Kato, E. Nishibori, M. Takata, M. Sakata, K. Ohoyama and K. Kitazawa, et al. , Phys. Rev. B: Condens. Matter Mater. Phys., 2002, 66, 134434 CrossRef.
  41. H. T. Stokes and D. M. Hatch, 2002, http://www.physics.byu.edu/;stokesh/isotropy.html.
  42. T. Lonkai, D. Tomuta, U. Amann, J. Ihringer, R. Hendrikx, D. Többens and J. Mydosh, Phys. Rev. B: Condens. Matter Mater. Phys., 2004, 69, 134108 CrossRef.
  43. S. H. Skjærvø, Q. N. Meier, M. Feygenson, N. A. Spaldin, S. J. Billinge, E. S. Bozin and S. M. Selbach, Phys. Rev. X, 2019, 9, 031001 Search PubMed.
  44. L. Wang, S. Feng, J. Zhu, R. Yu, C. Jin, W. Yu, X. Wang and L. Li, Appl. Phys. Lett., 2007, 91, 172502 CrossRef.
  45. A. Ghosh, J. R. Sahu, S. V. Bhat and C. Rao, Solid State Sci., 2009, 11, 1639 CrossRef CAS.
  46. S. Abrahams, Acta Crystallogr., Sect. B: Struct. Sci., 2001, 57, 485 CrossRef CAS PubMed.
  47. H. Chen, J. Bai, Y. Wang, T. Tyson and S.-W. Cheong, APS Meeting Abstracts, 2010.
  48. N. Fujimura, T. Ishida, T. Yoshimura and T. Ito, Appl.Phys. Lett., 1996, 69, 1011 CrossRef CAS.
  49. C.-Y. Ren, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 79, 125113 CrossRef.
  50. A. Bosak, C. Dubourdieu, J.-P. Sénateur, O. Y. Gorbenko and A. Kaul, J. Mater. Chem., 2002, 12, 800 RSC.
  51. A. Akbashev, A. Semisalova, N. Perov and A. Kaul, Appl. Phys. Lett., 2011, 99, 122502 CrossRef.
  52. Y. K. Jeong, J.-H. Lee, S.-J. Ahn, S.-W. Song, H. M. Jang, H. Choi and J. F. Scott, J. Am. Chem. Soc., 2012, 134, 1450 CrossRef CAS PubMed.
  53. W. Wang, J. Zhao, W. Wang, Z. Gai, N. Balke, M. Chi, H. N. Lee, W. Tian, L. Zhu and X. Cheng, et al. , Phys. Rev. Lett., 2013, 110, 237601 CrossRef PubMed.
  54. V. V. Roddatis, A. R. Akbashev, S. Lopatin and A. R. Kaul, Appl. Phys. Lett., 2013, 103, 112907 CrossRef.
  55. Y. K. Jeong, J.-H. Lee, S.-J. Ahn and H. M. Jang, Chem. Mater., 2012, 24, 2426 CrossRef CAS.
  56. S.-J. Ahn, J.-H. Lee, H. M. Jang and Y. K. Jeong, J. Mater. Chem. C, 2014, 2, 4521 RSC.
  57. H. Yokota, T. Nozue, S. Nakamura, H. Hojo, M. Fukunaga, P.-E. Janolin, J.-M. Kiat and A. Fuwa, Phys. Rev. B: Condens. Matter Mater. Phys., 2015, 92, 054101 CrossRef.
  58. C. Xu, Y. Yang, S. Wang, W. Duan, B. Gu and L. Bellaiche, Phys. Rev. B: Condens. Matter Mater. Phys., 2014, 89, 205122 CrossRef.
  59. T. Yu, P. Gao, T. Wu, T. Tyson and R. Lalancette, Appl. Phys. Lett., 2013, 102, 172901 CrossRef.
  60. M. F. Bekheet, I. Svoboda, N. Liu, L. Bayarjargal, E. Irran, C. Dietz, R. W. Stark, R. Riedel and A. Gurlo, J. Solid State Chem., 2016, 241, 54 CrossRef CAS.
  61. A. A. Belik, S. Kamba, M. Savinov, D. Nuzhnyy, M. Tachibana, E. Takayama-Muromachi and V. Goian, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 79, 054411 CrossRef.
  62. Y. Kumagai, A. A. Belik, M. Lilienblum, N. Leo, M. Fiebig and N. A. Spaldin, Phys. Rev. B: Condens. Matter Mater. Phys., 2012, 85, 174422 CrossRef.
  63. F.-T. Huang, X. Wang, Y. S. Oh, K. Kurushima, S. Mori, Y. Horibe and S.-W. Cheong, Phys. Rev. B: Condens. Matter Mater. Phys., 2013, 87, 184109 CrossRef.
  64. A. Cano, Phys. Rev. B: Condens. Matter Mater. Phys., 2014, 89, 214107 CrossRef.
  65. S. Artyukhin, K. T. Delaney, N. A. Spaldin and M. Mostovoy, Nat. Mater., 2014, 13, 42 CrossRef CAS PubMed.
  66. H. Xiang and M.-H. Whangbo, Phys. Rev. Lett., 2007, 98, 246403 CrossRef CAS PubMed.
  67. S. Cheng, C. Xu, S. Deng, M.-G. Han, S. Bao, J. Ma, C. Nan, W. Duan, L. Bellaiche and Y. Zhu, et al. , Sci. Adv., 2018, 4, eaar4298 CrossRef PubMed.
  68. M. Angst, R. P. Hermann, A. D. Christianson, M. D. Lumsden, C. Lee, M.-H. Whangbo, J.-W. Kim, P. Ryan, S. E. Nagler and W. Tian, et al. , Phys. Rev. Lett., 2008, 101, 227601 CrossRef CAS PubMed.
  69. J. Scott, J. Phys.: Condens. Matter, 2007, 20, 021001 CrossRef.
  70. D. Viana, R. Gotardo, L. Cótica, I. Santos, M. Olzon-Dionysio, S. Souza, D. Garcia, J. A. Eiras and A. Coelho, J. Appl. Phys., 2011, 110, 034108 CrossRef.
  71. J. Y. Park, J. H. Park, Y. K. Jeong and H. M. Jang, Appl. Phys. Lett., 2007, 91, 152903 CrossRef.
  72. J. d. de Groot, T. Mueller, R. Rosenberg, D. Keavney, Z. Islam, J.-W. Kim and M. Angst, Phys. Rev. Lett., 2012, 108, 187601 CrossRef CAS PubMed.
  73. J. A. Mundy, C. M. Brooks, M. E. Holtz, J. A. Moyer, H. Das, A. F. Rébola, J. T. Heron, J. D. Clarkson, S. M. Disseler and Z. Liu, et al. , Nature, 2016, 537, 523 CrossRef CAS PubMed.
  74. H.-J. Noh, H. Sung, J. Jeong, J. Jeong, S. B. Kim, J.-Y. Kim, J. Kim and B. Cho, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 82, 024423 CrossRef.
  75. T. Nagata, P.-E. Janolin, M. Fukunaga, B. Roman, K. Fujiwara, H. Kimura, J.-M. Kiat and N. Ikeda, Appl. Phys. Lett., 2017, 110, 052901 CrossRef.
  76. T. Nagata and N. Ikeda, AIP Adv., 2018, 8, 075312 CrossRef.
  77. M. Fiebig, D. Fröhlich, K. Kohn, T. Lottermoser, V. Pavlov and R. Pisarev, et al. , Phys. Rev. Lett., 2000, 84, 5620 CrossRef CAS PubMed.
  78. M. Fiebig, T. Lottermoser and R. V. Pisarev, J. Appl. Phys., 2003, 93, 8194 CrossRef CAS.
  79. I. Munawar and S. Curnoe, J. Phys.: Condens. Matter, 2006, 18, 9575 CrossRef CAS.
  80. F. Yen, C. Dela Cruz, B. Lorenz, E. Galstyan, Y. Sun, M. Gospodinov and C. Chu, J. Mater. Res., 2007, 22, 2163 CrossRef CAS.
  81. X.-B. Chen, N. T. M. Hien, K. Han, J.-Y. Nam, N. T. Huyen, S.-I. Shin, X. Wang, S.-W. Cheong, D. Lee and T. Noh, et al. , Sci. Rep., 2015, 5, 1 Search PubMed.
  82. J. R. Sahu, A. Ghosh, A. Sundaresan and C. Rao, Mater. Res. Bull., 2009, 44, 2123 CrossRef CAS.
  83. Y. Liu, J. Wang, X. Sun, J.-S. Zhou, Z. Xia, Z. Ouyang, M. Yang, C. Liu, R. Chen and J.-G. Cheng, et al. , Phys. Rev. B, 2018, 97, 214419 CrossRef CAS.
  84. D. Meier, H. Ryll, K. Kiefer, B. Klemke, J.-U. Hoffmann, R. Ramesh and M. Fiebig, Phys. Rev. B: Condens. Matter Mater. Phys., 2012, 86, 184415 CrossRef.
  85. A. Munoz, J. Alonso, M. Martnez-Lope, M. Casáis, J. Martnez and M. Fernández-Daz, Chem. Mater., 2001, 13, 1497 CrossRef CAS.
  86. O. Vajk, M. Kenzelmann, J. Lynn, S. Kim and S.-W. Cheong, Phys. Rev. Lett., 2005, 94, 087601 CrossRef CAS PubMed.
  87. J. Magesh, P. Murugavel, J. Krishnamurthy, V. Adyam and W. Prellier, J. Appl. Phys., 2015, 117, 074104 CrossRef.
  88. X. Fabreges, I. Mirebeau, P. Bonville, S. Petit, G. Lebras-Jasmin, A. Forget, G. André and S. Pailhes, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 78, 214422 CrossRef.
  89. J. Fontcuberta, M. Gospodinov and V. Skumryev, J. Appl. Phys., 2008, 103, 07B722 CrossRef.
  90. R. Eremina, T. Gavrilova, I. Yatsyk, R. Zaripov, A. Sukhanov, V. Shustov, N. Lyadov, V. Chichkov and N. Andreev, Appl. Magn. Reson., 2016, 47, 869 CrossRef CAS.
  91. S. Chattopadhyay, V. Simonet, V. Skumryev, A. Mukhin, V. Y. Ivanov, M. Aroyo, D. Dimitrov, M. Gospodinov and E. Ressouche, Phys. Rev. B, 2018, 98, 134413 CrossRef CAS.
  92. E. Standard, T. Stanislavchuk, A. Sirenko, N. Lee and S.-W. Cheong, Phys. Rev. B: Condens. Matter Mater. Phys., 2012, 85, 144422 CrossRef.
  93. A. Midya, S. Das, P. Mandal, S. Pandya and V. Ganesan, Phys. Rev. B: Condens. Matter Mater. Phys., 2011, 84, 235127 CrossRef.
  94. H. Tan, C. Xu, M. Li, S. Wang, B.-L. Gu and W. Duan, J. Phys.: Condens. Matter, 2016, 28, 126002 CrossRef PubMed.
  95. I. Solovyev, M. Valentyuk and V. Mazurenko, Phys. Rev. B: Condens. Matter Mater. Phys., 2012, 86, 054407 CrossRef.
  96. S. M. Disseler, J. A. Borchers, C. M. Brooks, J. A. Mundy, J. A. Moyer, D. A. Hillsberry, E. L. Thies, D. A. Tenne, J. Heron and M. E. Holtz, et al. , Phys. Rev. Lett., 2015, 114, 217602 CrossRef PubMed.
  97. W. Wang, J. A. Mundy, C. M. Brooks, J. A. Moyer, M. E. Holtz, D. A. Muller, D. G. Schlom and W. Wu, Phys. Rev. B, 2017, 95, 134443 CrossRef.
  98. P. Suresh, K. V. Laxmi, A. Bera, S. Yusuf, B. L. Chittari, J. Jung and P. A. Kumar, Phys. Rev. B, 2018, 97, 184419 CrossRef CAS.
  99. H. Iida, T. Koizumi, Y. Uesu, K. Kohn, N. Ikeda, S. Mori, R. Haumont, P.-E. Janolin, J.-M. Kiat and M. Fukunaga, et al. , J. Phys. Soc. Jpn., 2012, 81, 024719 CrossRef.
  100. S. Cao, K. Sinha, X. Zhang, X. Zhang, X. Wang, Y. Yin, J. Wang, D. J. Keavney, T. R. Paudel and Y. Liu, et al. , Phys. Rev. B, 2017, 95, 224428 CrossRef.
  101. X. Wang, Z. Xiao, X. Xu, W. Wang, D. Keavney, Y. Liu and X. Cheng, APS March Meeting Abstracts, 2014.
  102. W. Wu, V. Kiryukhin, H.-J. Noh, K.-T. Ko, J.-H. Park, I. W. Ratcliff, P. Sharma, N. Harrison, Y. J. Choi and Y. Horibe, et al. , Phys. Rev. Lett., 2008, 101, 137203 CrossRef PubMed.
  103. S. Dong, J.-M. Liu, S.-W. Cheong and Z. Ren, Adv. Phys., 2015, 64, 519 CrossRef CAS.
  104. S. Dong, H. Xiang and E. Dagotto, Natl. Sci. Rev., 2019, 6, 629 CrossRef CAS.
  105. M. Fiebig, T. Lottermoser, M. Kneip and M. Bayer, J. Appl. Phys., 2006, 99, 08E302 CrossRef.
  106. Z. Huang, Y. Cao, Y. Sun, Y. Xue and C. Chu, Phys. Rev. B: Condens. Matter Mater. Phys., 1997, 56, 2623 CrossRef CAS.
  107. N. Iwata and K. Kohn, J. Phys. Soc. Jpn., 1998, 67, 3318 CrossRef CAS.
  108. H. Sugie, N. Iwata and K. Kohn, J. Phys. Soc. Jpn., 2002, 71, 1558 CrossRef CAS.
  109. J. R. Sahu, A. Ghosh, A. Sundaresan and C. Rao, Mater. Res. Bull., 2009, 44, 2123 CrossRef CAS.
  110. N. Hur, I. Jeong, M. Hundley, S. Kim and S.-W. Cheong, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 79, 134120 CrossRef.
  111. J. Vermette, S. Jandl, M. Orlita and M. Gospodinov, Phys. Rev. B: Condens. Matter Mater. Phys., 2012, 85, 134445 CrossRef.
  112. T. Lottermoser, T. Lonkai, U. Amann, D. Hohlwein, J. Ihringer and M. Fiebig, Nature, 2004, 430, 541 CrossRef CAS PubMed.
  113. H. Wang, I. V. Solovyev, W. Wang, X. Wang, P. J. Ryan, D. J. Keavney, J.-W. Kim, T. Z. Ward, L. Zhu and J. Shen, et al. , Phys. Rev. B: Condens. Matter Mater. Phys., 2014, 90, 014436 CrossRef.
  114. M. Ye and D. Vanderbilt, Phys. Rev. B: Condens. Matter Mater. Phys., 2015, 92, 035107 CrossRef.
  115. J. Varignon, S. Petit, A. Gellé and M. Lepetit, J. Phys.: Condens. Matter, 2013, 25, 496004 CrossRef CAS.
  116. S. Holm, A. Kreisel, T. Schäffer, A. Bakke, M. Bertelsen, U. Hansen, M. Retuerto, J. Larsen, D. Prabhakaran and P. Deen, et al. , Phys. Rev. B, 2018, 97, 134304 CrossRef.
  117. S. Lee, A. Pirogov, M. Kang, K.-H. Jang, M. Yonemura, T. Kamiyama, S.-W. Cheong, F. Gozzo, N. Shin and H. Kimura, et al. , Nature, 2008, 451, 805 CrossRef CAS PubMed.
  118. X. Fabrèges, S. Petit, I. Mirebeau, S. Pailhès, L. Pinsard, A. Forget, M. Fernandez-Diaz and F. Porcher, Phys. Rev. Lett., 2009, 103, 067204 CrossRef PubMed.
  119. Y. Geng, H. Das, A. L. Wysocki, X. Wang, S.-W. Cheong, M. Mostovoy, C. J. Fennie and W. Wu, Nat. Mater., 2014, 13, 163 CrossRef CAS PubMed.
  120. H. Das, A. L. Wysocki, Y. Geng, W. Wu and C. J. Fennie, Nat. Commun., 2014, 5, 1 Search PubMed.
  121. I. Dzyaloshinsky, J. Phys. Chem. Solids, 1958, 4, 241 CrossRef CAS.
  122. T. Moriya, Phys. Rev., 1960, 120, 91 CrossRef CAS.
  123. T. Moriya, Phys. Rev. Lett., 1960, 4, 228 CrossRef CAS.
  124. Y. Geng, N. Lee, Y. J. Choi, S.-W. Cheong and W. Wu, Nano Lett., 2012, 12, 6055 CrossRef CAS PubMed.
  125. K. Singh, M.-B. Lepetit, C. Simon, N. Bellido, S. Pailhès, J. Varignon and A. De Muer, J. Phys.: Condens. Matter, 2013, 25, 416002 CrossRef.
  126. C. J. Howard, B. J. Campbell, H. T. Stokes, M. A. Carpenter and R. I. Thomson, Acta Crystallogr., Sect. B: Struct. Sci., Cryst. Eng. Mater., 2013, 69, 534 CrossRef CAS.
  127. S. Lee, A. Pirogov, M. Kang, K.-H. Jang, M. Yonemura, T. Kamiyama, S.-W. Cheong, F. Gozzo, N. Shin and H. Kimura, et al. , Nature, 2008, 451, 805 CrossRef CAS.
  128. I. Solovyev and S. Nikolaev, Phys. Rev. B: Condens. Matter Mater. Phys., 2014, 90, 184425 CrossRef.
  129. D. A. Allwood, G. Xiong, C. Faulkner, D. Atkinson, D. Petit and R. Cowburn, Science, 2005, 309, 1688 CrossRef CAS PubMed.
  130. M. Li, Y. Gu, Y. Wang, L.-Q. Chen and W. Duan, Phys. Rev. B: Condens. Matter Mater. Phys., 2014, 90, 054106 CrossRef.
  131. Y. Gu, M. Li, A. N. Morozovska, Y. Wang, E. A. Eliseev, V. Gopalan and L.-Q. Chen, Phys. Rev. B: Condens. Matter Mater. Phys., 2014, 89, 174111 CrossRef.
  132. Z. Wu, N. Huang, Z. Liu, J. Wu, W. Duan and B.-L. Gu, J. Appl. Phys., 2007, 101, 014112 CrossRef.
  133. W. Wang, W. Sun, G. Zhang, Z. Cheng and Y. Wang, J. Mater. Chem. C, 2019, 7, 10059 RSC.
  134. Y. Geng, N. Lee, Y. J. Choi, S.-W. Cheong and W. Wu, Nano Lett., 2012, 12, 6055 CrossRef CAS PubMed.
  135. D. Meier, J. Seidel, A. Cano, K. Delaney, Y. Kumagai, M. Mostovoy, N. A. Spaldin, R. Ramesh and M. Fiebig, Nat. Mater., 2012, 11, 284 CrossRef CAS PubMed.
  136. D. R. Småbråten, Q. N. Meier, S. H. Skjærvø, K. Inzani, D. Meier and S. M. Selbach, Phys. Rev. Mater., 2018, 2, 114405 CrossRef.
  137. T. Mitsui and J. Furuichi, Phys. Rev., 1953, 90, 193 CrossRef CAS.
  138. T. Choi, Y. Horibe, H. Yi, Y. J. Choi, W. Wu and S.-W. Cheong, Nat. Mater., 2010, 9, 253 CrossRef CAS PubMed.
  139. Q. Zhang, G. Tan, L. Gu, Y. Yao, C. Jin, Y. Wang, X. Duan and R. Yu, Sci. Rep., 2013, 3, 2741 CrossRef.
  140. T. Matsumoto, R. Ishikawa, T. Tohei, H. Kimura, Q. Yao, H. Zhao, X. Wang, D. Chen, Z. Cheng and N. Shibata, et al. , Nano Lett., 2013, 13, 4594 CrossRef CAS PubMed.
  141. Q. Zhang, L. Wang, X. Wei, R. Yu, L. Gu, A. Hirata, M. Chen, C. Jin, Y. Yao and Y. Wang, et al. , Phys. Rev. B: Condens. Matter Mater. Phys., 2012, 85, 020102 CrossRef.
  142. K. Du, B. Gao, Y. Wang, X. Xu, J. Kim, R. Hu, F.-T. Huang and S.-W. Cheong, npj Quant. Mater., 2018, 3, 1 CrossRef CAS.
  143. S. Chae, N. Lee, Y. Horibe, M. Tanimura, S. Mori, B. Gao, S. Carr and S.-W. Cheong, Phys. Rev. Lett., 2012, 108, 167603 CrossRef CAS PubMed.
  144. A. Baghizadeh, P. Mirzadeh Vaghefi, D. Alikin, J. Amaral, V. Amaral and J. Vieira, J. Phys. Chem. C, 2019, 123, 6158 CrossRef CAS.
  145. Q. N. Meier, M. Lilienblum, S. M. Griffin, K. Conder, E. Pomjakushina, Z. Yan, E. Bourret, D. Meier, F. Lichtenberg and E. K. Salje, et al. , Phys. Rev. X, 2017, 7, 041014 Search PubMed.
  146. X. Wang, M. Mostovoy, M.-G. Han, Y. Horibe, T. Aoki, Y. Zhu and S.-W. Cheong, Phys. Rev. Lett., 2014, 112, 247601 CrossRef CAS PubMed.
  147. Y. Kumagai and N. A. Spaldin, Nat. Commun., 2013, 4, 1 Search PubMed.
  148. F.-T. Huang, X. Wang, S. M. Griffin, Y. Kumagai, O. Gindele, M.-W. Chu, Y. Horibe, N. A. Spaldin and S.-W. Cheong, Phys. Rev. Lett., 2014, 113, 267602 CrossRef.
  149. V. V. Roddatis, A. R. Akbashev, S. Lopatin and A. R. Kaul, Appl. Phys. Lett., 2013, 103, 112907 CrossRef.
  150. J. Liu, T. L. Sun, X. Q. Liu, H. Tian, T. T. Gao and X. M. Chen, Adv. Funct. Mater., 2018, 28, 1706062 CrossRef.
  151. I. Yang, J. Kim, S. Lee, S.-W. Cheong and Y. Jeong, Appl. Phys. Lett., 2015, 106, 152902 CrossRef.
  152. Y. Du, X. Wang, D. Chen, Y. Yu, W. Hao, Z. Cheng and S. X. Dou, Phys. Chem. Chem. Phys., 2013, 15, 20010 RSC.
  153. J. Seidel, L. W. Martin, Q. He, Q. Zhan, Y.-H. Chu, A. Rother, M. Hawkridge, P. Maksymovych, P. Yu and M. Gajek, et al. , Nat. Mater., 2009, 8, 229 CrossRef CAS PubMed.
  154. J. Guyonnet, I. Gaponenko, S. Gariglio and P. Paruch, Adv. Mater., 2011, 23, 5377 CrossRef CAS PubMed.
  155. D. J. Kim, J. Connell, S. Seo and A. Gruverman, Nanotechnology, 2016, 27, 155705 CrossRef CAS PubMed.
  156. P. Schoenherr, K. Shapovalov, J. Schaab, Z. Yan, E. D. Bourret, M. Hentschel, M. Stengel, M. Fiebig, A. Cano and D. Meier, Nano Lett., 2019, 19, 1659 CrossRef CAS PubMed.
  157. W. Wu, Y. Horibe, N. Lee, S.-W. Cheong and J. Guest, Phys. Rev. Lett., 2012, 108, 077203 CrossRef PubMed.
  158. D. R. Småbråten, Q. N. Meier, S. H. Skjærvø, K. Inzani, D. Meier and S. M. Selbach, Phys. Rev. Mater., 2018, 2, 114405 CrossRef.
  159. M. E. Holtz, K. Shapovalov, J. A. Mundy, C. S. Chang, Z. Yan, E. Bourret, D. A. Muller, D. Meier and A. Cano, Nano Lett., 2017, 17, 5883 CrossRef CAS PubMed.
  160. E. Hassanpour, V. Wegmayr, J. Schaab, Z. Yan, E. Bourret, T. Lottermoser, M. Fiebig and D. Meier, New J. Phys., 2016, 18, 043015 CrossRef.
  161. J. Magesh, P. Murugavel, J. Krishnamurthy, V. Adyam and W. Prellier, J. Appl. Phys., 2015, 117, 074104 CrossRef.
  162. A. Sousa, A. Lima and M. d. S. Rezende, J. Alloys Compd., 2016, 689, 977 CrossRef CAS.
  163. L. Lin, H. Zhang, M. Liu, S. Shen, S. Zhou, D. Li, X. Wang, Z. Yan, Z. Zhang and J. Zhao, et al. , Phys. Rev. B, 2016, 93, 075146 CrossRef.
  164. J. C. Leiner, T. Kim, K. Park, J. Oh, T. Perring, H. Walker, X. Xu, Y. Wang, S.-W. Cheong and J.-G. Park, Phys. Rev. B, 2018, 98, 134412 CrossRef CAS.
  165. S. M. Disseler, X. Luo, B. Gao, Y. S. Oh, R. Hu, Y. Wang, D. Quintana, A. Zhang, Q. Huang and J. Lau, et al. , Phys. Rev. B: Condens. Matter Mater. Phys., 2015, 92, 054435 CrossRef.
  166. T. Sarkar, K. Manna, S. Elizabeth and P. Anil Kumar, J. Appl. Phys., 2017, 121, 084102 CrossRef.
  167. A. Ahlawat, S. Satapathy, D. Phase, K. Varma and P. Gupta, Appl. Phys. Lett., 2015, 106, 252903 CrossRef.
  168. H. S. Nair, Z. Fu, C. Kumar, V. Pomjakushin, Y. Xiao, T. Chatterji and A. M. Strydom, Europhys. Lett., 2015, 110, 37007 CrossRef.
  169. J. Cui, A. Zhang, J. Shi, H. Cao, X. Wu and Y. Zhang, J. Appl. Phys., 2019, 126, 114103 CrossRef.
  170. H. Sim, H. Kim, K. Park, M. Lilienblum, M. Fiebig and J.-G. Park, Phys. Rev. B, 2018, 98, 085132 CrossRef CAS.
  171. F. Figueiras, D. Karpinsky, P. Tavares, J. Gonçalves, S. Yañez-Vilar, A. M. Dos Santos, A. Franz, M. Tovar, J. A. Moreira and V. Amaral, Phys. Chem. Chem. Phys., 2017, 19, 1335 RSC.
  172. S. Cheng, M. Li, S. Deng, S. Bao, P. Tang, W. Duan, J. Ma, C. Nan and J. Zhu, Adv. Funct. Mater., 2016, 26, 3589 CrossRef CAS.
  173. S. Cheng, M. Li, Q. Meng, W. Duan, Y. Zhao, X. Sun, Y. Zhu and J. Zhu, Phys. Rev. B, 2016, 93, 054409 CrossRef.
  174. S. H. Skjærvø, E. T. Wefring, S. K. Nesdal, N. H. Gaukås, G. H. Olsen, J. Glaum, T. Tybell and S. M. Selbach, Nat. Commun., 2016, 7, 1 Search PubMed.
  175. X. Wang, D. Yang, H.-M. Zhang, C. Song, J. Wang, G. Tan, R. Zheng, S. Dong, S.-W. Cheong and J. Zhang, Phys. Rev. B, 2019, 99, 054106 CrossRef CAS.
  176. K. Świerczek, A. Klimkowicz, K. Nishihara, S. Kobayashi, A. Takasaki, M. Alanizy, S. Kolesnik, B. Dabrowski, S. Seong and J. Kang, Phys. Chem. Chem. Phys., 2017, 19, 19243 RSC.

This journal is © the Owner Societies 2020