Abstract: X-ray diffraction is used to show that the structural distortion of magnetite below 125 kelvin is to a first approximation caused by charge ordering of its constituent iron ions, but that the localized electrons are distributed over three iron sites to form ‘trimeron’ quasiparticles. In a letter to Nature in 1939, Evert Johannes Willem Verwey described the first example of a low-temperature charge-ordering transition in a solid — the Verwey transition — in the mineral magnetite, Fe3O4 (see go.nature.com/3h2pp1 ). This phenomenon has since been observed in other transition metal oxides, yet despite decades of study, the precise structure of the charge-ordered state in magnetite has remained elusive. The complex structural distortions that characterize the Verwey state have now been determined, revealing an anomalous shortening of some inter-atomic distances. These are suggestive of an unusual charge configuration in which the localized electrons are each distributed over three neighbouring Fe sites. The mineral magnetite (Fe3O4) undergoes a complex structural distortion and becomes electrically insulating at temperatures less than 125 kelvin. Verwey proposed in 1939 that this transition is driven by a charge ordering of Fe2+ and Fe3+ ions1, but the ground state of the low-temperature phase has remained contentious2,3 because twinning of crystal domains hampers diffraction studies of the structure4. Recent powder diffraction refinements5,6,7 and resonant X-ray studies8,9,10,11,12 have led to proposals of a variety of charge-ordered and bond-dimerized ground-state models13,14,15,16,17,18,19. Here we report the full low-temperature superstructure of magnetite, determined by high-energy X-ray diffraction from an almost single-domain, 40-micrometre grain, and identify the emergent order. The acentric structure is described by a superposition of 168 atomic displacement waves (frozen phonon modes), all with amplitudes of less than 0.24 angstroms. Distortions of the FeO6 octahedra show that Verwey’s hypothesis is correct to a first approximation and that the charge and Fe2+ orbital order are consistent with a recent prediction17. However, anomalous shortening of some Fe–Fe distances suggests that the localized electrons are distributed over linear three-Fe-site units, which we call ‘trimerons’. The charge order and three-site distortions induce substantial off-centre atomic displacements and couple the resulting large electrical polarization to the magnetization. Trimerons may be important quasiparticles in magnetite above the Verwey transition and in other transition metal oxides.

Abstract: In this contribution to the special issue on multiferroics we focus on multiferroicity driven by different forms of charge ordering. We will present the generic mechanisms by which charge ordering can induce ferroelectricity in magnetic systems. There is a number of specific classes of materials for which this is relevant. We will discuss in some detail (i) perovskite manganites of the type (PrCa)MnO3, (ii) the complex and interesting situation in magnetite Fe3O4, (iii) strongly ferroelectric frustrated LuFe2O4 and (iv) an example of a quasi-one-dimensional organic system. All these are ‘type-I’ multiferroics, in which ferroelectricity and magnetism have different origins and occur at different temperatures. In the second part of this article we discuss ‘type-II’ multiferroics, in which ferroelectricity is completely due to magnetism, but with charge ordering playing an important role, such as (v) the newly discovered multiferroic Ca3CoMnO6, (vi) possible ferroelectricity in rare earth perovskite nickelates of the type RNiO3, (vii) multiferroic properties of manganites of the type RMn2O5, (viii) perovskite manganites with magnetic E-type ordering and (ix) bilayer manganites. (Some figures in this article are in colour only in the electronic version)

Abstract: Multiferroics were considered to be rare because magnetism and ferroelectricity require entirely different criteria for the materials. Several multiferroic oxides have, however, been discovered in the past few years by virtue of novel operating mechanisms, the most effective one being ferroelectricity driven by magnetism itself. Many such oxides where the magnetic and electric order parameters interact also exhibit magnetoelectric or magnetodielectric properties. In this Perspective, properties of manganites, ferrites, and other monophasic multiferroic oxides with spin-induced electric polarization are described. Multiferroic properties arising from charge ordering are examined. The present status of BiMnO3, which is an unusual example of a ferromagnetic–ferroelectric, is presented. Recent findings suggest that it is likely that many more multiferroic and magnetoelectric oxide materials exhibiting magnetically induced ferroelectricity will be found in the future.

Abstract: We report the first experimental observation of the spin Seebeck effect in magnetite thin films. The signal observed at temperatures above the Verwey transition is a contribution from both the anomalous Nernst (ANE) and spin Seebeck effects (SSE). The contribution from the ANE of the Fe3O4 layer to the SSE is found to be negligible due to the resistivity difference between Fe3O4 and Pt layers. Below the Verwey transition the SSE is free from the ANE of the ferromagnetic layer and it is also found to dominate over the ANE due to magnetic proximity effect on the Pt layer.

Abstract: $\ensuremath{\alpha}$-(BEDT-TTF)${}_{2}$I${}_{3}$ exhibits a metal to insulator transition around 135 K that has been ascribed to charge ordering in the donor layers containing three different donors (A-A dimers, B and C). First-principles density-functional theory (DFT) calculations provide a description of the electronic structure of this system in agreement with the presently available experimental information above and below the transition. A new mechanism of the charge ordering transition in which the anions play a major role is proposed. It is shown that lowering the temperature induces a differentiation of the two I${}_{3}^{\ensuremath{-}}$ zigzag chains which leads to an increase/decrease of the donor-anion hydrogen bonding with the two initially identical A donors. This induces a redistribution of holes mostly affecting the A donors (thus becoming inequivalent A, A${}^{\ensuremath{'}}$), giving rise to the charge ordering. The interrelation between the strength of the hydrogen bonding with the anions and the concentration of holes in the HOMOs involves a subtle polarization mechanism among the $\ensuremath{\sigma}$ and $\ensuremath{\pi}$ electron densities of the donors. The redistribution of holes leads to the opening of a band gap via the modulation of the site potential of donors A/A${}^{\ensuremath{'}}$ and the increased band mixing because of the symmetry lowering.

Abstract: The present paper reports detailed structural and magnetic characterization of the low-bandwidth manganite Pr1−xCaxMnO3 (with x = 0.0–0.5) (PCMO) polycrystalline samples. With increasing Ca content, reduction of the unit cell volume and improvement in perovskite structure symmetry was observed at room temperature. Magnetic characterization shows the signature of coexisting AFM–FM ordering and spin-glass phase at the low doping range (x = 0.0–0.2) while increased hole doping (x = 0.3–0.5) leads to charge ordering, training effect and an irreversible metamagnetic phenomenon. The large irreversible metamagnetism in the CO phase of PCMO and the corresponding spin memory effect is a direct consequence of hysteretic first-order phase transition arising from the weakening of the CO state under the external magnetic field and trapping of the spins due to a strong pinning potential in the material.

Abstract: We theoretically investigate the interplay between charge ordering and magnetic states in quasi-one-dimensional molecular conductors TMTTF(2)X, motivated by the observation of a complex variation of competing and/or coexisting phases. We show that the ferroelectric-type charge order increases two-dimensional antiferromagnetic spin correlation, whereas in the one-dimensional regime two different spin-Peierls states are stabilized. By using first-principles band calculations for the estimation for the transfer integrals and comparing our results with the experiments, we identify the controlling parameters in the experimental phase diagram to be not only the interchain transfer integrals but also the amplitude of the charge order.

Abstract: Magnetic reversal of an artificial square ice pattern subject to a sequence of magnetic fields applied slightly off the diagonal axis is investigated via magnetic force microscopy of the remanent states that result. Sublattice independent reversal is observed via correlated incrementally pinned flip cascades along parallel dipolar chains, as evident from analysis of vertex populations and dipolar correlation functions. Weak dipolar interactions between adjacent chains favour antialignment and give rise to weak charge ordering of 'monopole' vertices during the reversal process.

Abstract: Fermi-level pinning of C${}_{60}$ (sub)-monolayers on a sexithiophene (6T) bilayer grown on Ag(111) is shown to induce electron transfer from the metal to a fraction of the C${}_{60}$ molecules. The electrostatic potential resulting from the charge transfer process is responsible for a potential drop within the 6T interlayer and, more remarkably, for dipole-dipole repulsion, leading to a disproportionation into coexisting neutral and charged C${}_{60}$ molecules. We suggest that charge ordering phenomena may occur for such systems.

Abstract: The spin reorientation temperature TSR of stoichiometric Fe3O4, as well as of magnetite with a small number of vacancies and magnetite containing a low concentration of Ti, Zn, Al and Ga was measured on single-crystal samples using the ac susceptibility. In the same experiment the temperature TV? of the Verwey transition was also found. The results show that a correlation between TSR and TV? exists. The electronic structure of the compounds studied was determined using the density-functional-based GGA?+?U method. For stoichiometric magnetite the first and second cubic anisotropy constants were calculated, while for magnetite with defects the distribution of electron density using the ?atoms in molecules? approach was determined. Based on a combination of experimental results with the electronic structure calculations an explanation of the temperature dependence of the magnetocrystalline anisotropy of magnetite is suggested.

Abstract: We show that a Fermi surface in underdoped YBa2Cu3O6+x yielding the distribution of quantum oscillation frequencies observed over a broad range of magnetic field can be reconciled with the wavevectors of charge modulations found in nuclear magnetic resonance and x-ray scattering experiments within a model of biaxial charge ordering occurring in a bilayer CuO2 planar system. Bilayer coupling introduces the possibility of different period modulations and quantum oscillation frequencies corresponding to each of the bonding and antibonding bands, which can be reconciled with recent experimental observations.

Abstract: We review 35 years of structural studies of quasi-1D organic conductors during which the concepts of 2kF and 4kF BOW and CDW have been elaborated. In strongly correlated quarter filled band systems these instabilities give rise to SP, DM and CO ground states. We relate these structural features to the instabilities of the 1D electron gas. To stabilize the different ground states the nature of the electron-phonon coupling has to be considered together with the coupling of the organic stacks with the anion sublattice. New results concerning the classification of the SP phase in connection with the adiabatic or antiadiabatic phonon field and its competition with the CO are also introduced.

Abstract: We theoretically investigate the interplay between charge ordering and magnetic states in quasi-one-dimensional molecular conductors TMTTF(2)X, motivated by the observation of a complex variation of competing and/or coexisting phases. We show that the ferroelectric-type charge order increases two-dimensional antiferromagnetic spin correlation, whereas in the one-dimensional regime two different spin-Peierls states are stabilized. By using first-principles band calculations for the estimation for the transfer integrals and comparing our results with the experiments, we identify the controlling parameters in the experimental phase diagram to be not only the interchain transfer integrals but also the amplitude of the charge order.

Abstract: The occurrence of antiferromagnetism (AFM) in κ-(ET)2X can be understood within an effective 1/2-filled band with dimers of ET molecules containing one hole each. We argue that while this effective model can describe the presence of AFM, a complete description for these materials requires the correct carrier density of one-half per molecule. For dimerized and strongly frustrated 1/4-filled lattices we show that a singlet-paired statecoexisting with charge ordering occurs that we have termed the paired electron crystal (PEC). Here we investigate the 1/4-filled model on a dimerized lattice, showing regions where AFM, PEC, and the Wigner-crystal occur. We point out the need to go beyond quantum spin liquid concepts for highly frustrated materials such as κ-(ET)2Cu2(CN)3 and β′-EtMe3Sb[Pd(dmit)2]2 which we believe are PECs at low temperatures.

Abstract: A review is given for recent theoretical studies on phase transitions in quasi-one-dimensional molecular conductors with a quarter-filled band. By lowering temperature, charge transfer salts exhibit a variety of transitions accompanying symmetry breaking, such as charge ordering, lattice dimerization, antiferromagnetic transition, spin-Peierls distortion, and so on. Analyses on microscopic quasi-one-dimensional models provide their systematic understandings, by the complementary use of different analytical and numerical techniques; they can reproduce finite-temperature phase transitions, whose results can be directly compared with experiments and give feedbacks to material design.

Abstract: Charge disproportion at octahedral Fe sites in magnetite was observed at low temperature using two inversion-symmetry related three-wave resonant x-ray diffraction, 022-311 and $002\mathrm{\text{\ensuremath{-}}}\overline{3}\overline{1}1$, near the iron $K$ edge. Both of the three-wave cases involve the (002) forbidden-weak reflection. The self-normalized three-wave to two-wave (002) diffraction intensity ratio automatically cancels the self-absorption effect and leads to direct determination of charge disproportion for magnetite below 120 K. This approach provides a more direct and effective way for extracting charge-ordering information.

Abstract: In this review we present results of our theoretical study of charge and spin ordering in strongly correlated electron systems obtained within various generalizations of the Falicov-Kimball model. The primary goal of this study was to identify crucial interactions that lead to the stabilization of various types of charge ordering in these systems, like the axial striped ordering, diagonal striped ordering, phase-separated ordering, phase-segregated ordering, etc. Among the major interactions that come into account, we have examined the effect of local Coulomb interaction between localized and itinerant electrons, long-range and correlated hopping of itinerant electrons, long-range Coulomb interaction between localized and itinerant electrons, local Coulomb interaction between itinerant electrons, local Coulomb interaction between localized electrons, spin-dependent interaction between localized and itinerant electrons, both for zero and nonzero temperatures, as well as for doped and undoped systems. Finally, the relevance of resultant solutions for a description of rare-earth and transition-metal compounds is discussed.

Abstract: The transition metal ions in spinel LiNi2O4 are commonly assumed to be in a single charge state and uniformly distributed throughout the crystal lattice. However, multivalent Ni can exist in a number of different charge states simultaneously within the crystal, whilst still maintaining overall charge neutrality. By carrying out a systematic study of possible charge states of Ni and their distribution in LiNi2O4 using first-principles calculations within the framework of density functional theory, we show that two charge-ordered phases are more stable than the charge-disordered spinel phase: (a) an orthorhombic phase containing Ni3+ and Ni4+ ions in a 1 : 1 ratio resulting in Imma symmetry, and (b) a cubic phase with Ni2+ and Ni4+ in a 1 : 3 ratio with P4332 symmetry. Our results indicate that charge ordering is an important process in stabilizing the spinel phase of LiNi2O4, and should not be ignored in any quantitative study. Our findings are of particular importance for elucidating the relative stabilities of the spinel phase and layered phase of Li1−xNi2O4, a delithiated form of lithium secondary battery cathode material LiNiO2.

Abstract: We have investigated the spontaneous polarization in Fe$_3$O$_4$ thin films by using dynamic and static pyroelectric measurements. The magnetic and dielectric behavior of Fe$_3$O$_4$ thin films grown on Nb:SrTiO$_3$(001) substrates was consistent with bulk crystals. The well-known metal-insulator (Verwey) transition was observed at 120 K. The appearance of a pyroelectric response in the Fe$_3$O$_4$ thin films just below the Verwey temperature shows that spontaneous polarization appeared in Fe$_3$O$_4$ at the charge ordering transition temperature. The polar state characteristics are consistent with bond- and site- centered charge ordering of Fe$^{2+}$ and Fe$^{3+}$ ions sharing the octahedral $B$ sites. The pyroelectric response in Fe$_3$O$_4$ thin films was dependent on the dielectric constant. Quasi-static pyroelectric measurement of Pd/Fe$_3$O$_4$/Nb:SrTiO$_3$ junctions showed that magnetite has a very large pyroelectric coefficient of 735 nC/cm$^{2}$K at 60 K.

Abstract: A series of oxygen deficient LuFe2O4-delta materials have been prepared under a controlled oxygen partial-pressure atmosphere. Measurements of magnetization reveal that the increase of oxygen deficiencies could evidently depress the ferrimagnetic phase transition temperature (T-N). In additional to the well-known charge ordering within the (1 (1) over bar0) crystal plane, a visible structural modulation with q = (0, 1/4.2, 7/8) commonly appears on the (100) plane in the oxygen deficient samples. An aberration-corrected transmission electron microscopy study on the oxygen deficient samples demonstrates the presence of oxygen vacancies and local structural distortion. The atomic structural features in correlation with the structural modulation, distortion of the FeO5 polyhedron and the (001) twinning domains have been also examined.

Abstract: The presence of complex charge ordering phenomena in La1−xSrxFeO3−δ (x = 0.55, 0.60, 0.67, and 0.72) has been studied by kilohertz mechanical spectroscopy. With the decrease in temperature, a step increase of the modulus and decrease of internal friction were observed around the charge ordering transition temperature in La0.33Sr0.67FeO3−δ. This characteristic behavior preserves in La1−xSrxFeO3−δ (x = 0.55, 0.60, and 0.72), which suggests the existence of charge ordering. An internal friction peak around 160 K in La1−xSrxFeO3−δ (x = 0.55 and 0.60) is identified as an elastic manifestation of magnetic freezing. The description of the electronic phase diagram is obtained and it supports an ionic picture.

Abstract: The structural, magnetic, and electrical properties of Ln 0.5 Sr 0.5 Mn 0.9 Cu 0.1 O 3 (Ln = La, Pr, Nd, or Ho) perovskite manganites have been investigated to explore the influence of A-site cation radius (〈 r A 〉) and the A-site cation size-disorder ( σ 2 ) on the various interdependent phenomena such as ferromagnetism (FM), phase separation (PS), and charge ordering (CO). The temperature dependence magnetization ( M – T ) curve of La-based sample shows four distinct points at ∼269 K, 255 K, 200 K, and 148 K corresponding to strong FM, cluster glass (CG), weak FM, and charged ordered antiferromagnetic (COAFM) transitions, respectively. Our investigation shows that Neel temperatures ( T N ) increases, whereas Curie ( T C ) and irreversibility temperatures ( T irr ) decrease with decreasing 〈 r A 〉, i.e., with increasing σ 2 . Furthermore, the value of the magnetization decreases and resistivity increases with decreasing 〈 r A 〉. All samples exhibit insulating behavior in the temperature range 77–300 K and above 110 K the electronic conduction mechanism has been described within the framework of the variable range hopping (VRH) model.

Abstract: The candidate magnetoelectric Pb3Mn7O15 has a structure consisting of one-third filled kagome layers linked by ribbons of edge sharing octahedra in the stacking direction. Previous reports have indicated a complex hexagonal–orthorhombic structural transition upon cooling through ~335 K, although its origins are uncertain. Here both structures are revisited using a combination of neutron and synchrotron x-ray diffraction data. Large shifts of oxygen positions are detected, which show that the interlayer sites and those which occupy voids in the kagome lattice are trivially charge ordered in both phases. The symmetry breaking is found to occur due to Mn3+ orbital ordering on the ribbon sites and charge ordering of the subset of layer sites which make up a kagome network.

Abstract: The influence of Na doping upon orbital and charge ordering in Nd0.5Sr0.5MnO3 manganites has been investigated combining magnetic measurements with X-ray powder diffraction at room temperature. It is shown that Na doping induces or reinforces the ferromagnetic metallic state, and destroys the charge ordering one and its associated antiferromagnetism whatever is its nature. The maximum magnetic entropy change , in a magnetic field change of 4T is found to be 0.49J/kgK and 3.1J/kgK for Nd0.5Sr0.5MnO3 and Nd0.5Sr0.3Na0.2MnO3 respectively.

Abstract: We have studied experimentally the responses of high-quality single crystals of stoichiometric synthetic magnetite to applied weak dc and ac magnetic fields in the range of 6--60 K, far below the Verwey transition. The results can be compared to so-called magnetic after effects (MAE) measurements, which are the most extensive magnetic measurements of magnetite at these temperatures. We present a novel point of view on the relaxation phenomena encountered at these temperatures---the low-temperature anomaly, addressing the striking difference between the results of conventional ac susceptibility measurements and those accompanying MAE measurements, i.e., periodic excitations with strong magnetic pulses. We also draw a connection between this anomaly and the so-called glasslike transition, and discuss possible mechanisms responsible for these effects.

Abstract: An investigation on the rearrangement of charge density distribution in response to the magnetic behavior is done on the nanosized Sn 1− x Mn x O 2 systems prepared via sol–gel technique. The samples were analyzed using X-ray diffraction, SEM and VSM measurements. The charge ordering is found as a signature to its magnetic response from the reconstruction of the charge density in the unit cell derived from the MEM/Rietveld analysis of the X-ray structure factors. Quantitative investigation is done by analyzing the 3-dimensional, 2-dimensional and 1-dimensional charge density profiles drawn along the bonding directions. The observed para- to dia-magnetic transition with respect to doping concentration of Mn is explained using defect chemistry and the charge ordering. The MEM/Rietveld analysis is used for the first time to analyse the charge and magnetic ordering in the chosen system.

Abstract: We report the results of magnetization and electron paramagnetic resonance (EPR) studies on nanoparticles (average diameter ∼ 30 nm) of Bi0.25Ca0.75MnO3 (BCMO) and compare them with the results on bulk BCMO. The nanoparticles were prepared using the nonaqueous sol-gel technique and characterized by XRD and TEM analysis. Magnetization measurements were carried out with a commercial physical property measurement system (PPMS). While the bulk BCMO exhibits a charge ordering transition at ∼230 K and an antiferromagnetic (AFM) transition at ∼130 K, in the nanoparticles, the CO phase is seen to have disappeared and a transition to a ferromagnetic (FM) state is observed at Tc ∼ 120 K. However, interestingly, the exchange bias effect observed in other nanomanganite ferromagnets is absent in BCMO nanoparticles. EPR measurements were carried out in the X-band between 8 and 300 K. Lineshape fitting to a Lorentzian with two terms (accounting for both the clockwise and anticlockwise rotations of the microwave field) w...