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Advances in Condensed Matter Physics
Volume 2011, Article ID 298928, 107 pages
Review Article

Electronic Structure of Strongly Correlated Systems

1Andersen Department, Max-Planck-Institut für Festkörperforschung, Heisenberg Straße 1, 70569 Stuttgart, Germany
2Computational Physics Department, Institute of Metal Physics, 36 Vernadskii street, 03142 Kiev, Ukraine

Received 23 March 2011; Accepted 12 July 2011

Academic Editor: P. Guptasarma

Copyright © 2011 V. N. Antonov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The article reviews the rich phenomena of metal-insulator transitions, anomalous metalicity, taking as examples iron and titanium oxides. The diverse phenomena include strong spin and orbital fluctuations, incoherence of charge dynamics, and phase transitions under control of key parameters such as band filling, bandwidth, and dimensionality. Another important phenomena presented in the article is a valence fluctuation which occur often in rare-earth compounds. We consider some Ce, Sm, Eu, Tm, and Yb compounds such as Ce, Sm and Tm monochalcogenides, Sm and Yb borides, mixed-valent and charge-ordered Sm, Eu and Yb pnictides and chalcogenides R4X3 and R3X4 (R = Sm, Eu, Yb; X = As, Sb, Bi), intermediate-valence YbInCu4 and heavy-fermion compounds YbMCu4 (M = Cu, Ag, Au, Pd). Issues addressed include the nature of the electronic ground states, the metal-insulator transition, the electronic and magnetic structures. The discussion includes key experiments, such as optical and magneto-optical spectroscopic measurements, x-ray photoemission and x-ray absorption, bremsstrahlung isochromat spectroscopy measurements as well as x-ray magnetic circular dichroism.

1. Introduction

Determination of the energy band structure of solids is a many-body problem. Band theory, a mean-field theory to treat this problem, in the framework of the local spin density approximation (LSDA), has been successful for many kinds of materials and has become the de facto tool of first-principles calculations in solid-state physics. It has contributed significantly to the understanding of material properties at the microscopic level. However, there are some systematic errors which have been observed when using the LSDA. In particular, the LSDA fails to describe the electronic structure and properties of the electron systems in which the interaction among the electrons is strong. These are now called strongly correlated electron systems, and many new concepts to address these phenomena have been constructed. However, the understanding of these systems is not complete. A wide variety of physical properties arise from the correlations among 3𝑑 and 4𝑓 electrons: metal-insulator transitions, valence fluctuations in the Kondo effect, heavy-fermion behavior, superconductivity, and so on.

The investigation of this class of systems goes back to the early 1960s. The main motivations at the time came from experiments on transition metal oxides, from the Mott metal-insulator transition, and from the problem of itinerant ferromagnetism. Theoretical progress in the field has been impeded however by the extreme difficulty of dealing with even the simplest model Hamiltonians appropriate for these systems, such as the Hubbard model and the Kondo lattice model [1]. Only in the one-dimensional case do we have a variety of theoretical tools at our disposal to study these models in a systematic manner. For two- and three-dimensional models, one is often unable to assess confidently whether a given physical phenomenon is indeed captured by the idealized Hamiltonian under consideration or whether a theoretical prediction reflects a true feature of this Hamiltonian, rather than an artifact of the approximation used in its solution. These difficulties originate in the nonperturbative nature of the problem and reflect the presence of several competing physical mechanisms for even the simplest models. The interplay of localization and lattice coherence, that of quantum and spatial fluctuations and that of various competing types of long-range order are important examples.

With the aim of undertaking a systematic investigation of the trends in some mixed-valent and charge- and orbital-ordering compounds, we present the theoretically calculated electronic structure, optical, magneto-optical, and photo-emission spectra for some 3𝑑 and 4𝑓 compounds. The main idea of the paper is to show how the modern band structure methods which take into account strong electron-electron interaction can properly describe the electronic structure and physical properties of strongly correlated electron systems.

In the first part of the article we concentrate on the description of the methods and the results for the 3𝑑 transition metal oxides which possess different fascinating physical properties including charge and orbital ordering as well as metal-insulator transition (MIT). Metal-insulator transitions are accompanied by huge resistivity changes, even over tens of orders of magnitude, and are widely observed in condensed-matter systems [2]. Especially important are the transitions driven by correlation effects associated with the electron-electron interaction. The insulating phase caused by the correlation effects is categorized as the Mott Insulator. Near the transition point the metallic state shows fluctuations and orderings in the spin, charge, and orbital degrees of freedom. The properties of these metals are frequently quite different from those of ordinary metals, as measured by transport, optical, and magnetic probes. We consider the Fe3O4, Fe2OBO3, Ti4O7, and 𝛼-NaV2O5.

The second part of the paper is devoted to the strongly correlated 4𝑓 systems such as CeFe2, Ce, Sm, and Tm monochalcogenides RX (R = Ce, Sm, Tm; X = S, Se, Te), Sm and Yb borides SmB6, YbB12, mixed-valent and charge-ordered Sm, Eu and Yb pnictides and chalcogenides R4X3 and R3X4 (R = Sm, Eu, Yb; X = As, Sb, Bi), intermediate-valence YbInCu4 and heavy-fermion compounds YbMCu4 (M = Cu, Ag, Au, Pd).

In rare-earth compounds, where 4𝑓 levels are relatively close to the Fermi energy, various anomalous phenomena frequently appear. Most of them can be attributed to the hybridization between the 4𝑓 states and conduction bands. A mixed-valence (MV) state is one of these phenomena. The MV phenomenon has attracted a great deal of interest during the last several decades in connection with valence fluctuations [37]. In the gas phase most rare earths are divalent, but in the solid state most are trivalent, due to the large cohesive energy gained by promoting a 4𝑓 electron into an extended bonding state. The rare-earth compounds based on Sm, Eu, Tm, and Yb ions frequently exhibit a mixed-valence state consisting of divalent and trivalent valences. In the mixed-valence compounds, therefore, one must also consider the charge degrees of freedom of the 4𝑓 ions in addition to the spin and orbital degrees of freedom.

It is necessary to distinguish between homogeneously mixed-valence compounds and inhomogeneously mixed-valence compounds. In the former, all the rare-earth ions occupy crystallographically equivalent sites, and, therefore, this is essentially a single ion property where the magnetic ion hybridizes with the sea of the conduction electrons, causing an exchange of the inner 4𝑓 electron with the conduction band at the Fermi level. Such effects are expected to arise in systems where two electron configurations corresponding to 4𝑓 occupation numbers 𝑛 and 𝑛1 have nearly degenerate energies. So the ground state of a homogeneously mixed-valence compound is a quantum mechanical mixture of both the 4𝑓𝑛 and the 4𝑓𝑛1𝑑 configuration on each rare-earth ion. Typical compounds exhibiting homogeneously mixed-valence phenomena are rare-earth materials TmSe, SmS (high-pressure golden phase), SmB6, YbB12, and YbInCu4.

In the case of inhomogeneously or static mixed-valence compounds, rare-earth ions with different valency occupy clearly different sites. However, at high temperatures they become homogeneously mixed-valence semimetals or valence-fluctuating insulators. Their 4𝑓 electrons are strongly correlated and close to localization, that is, having a low effective kinetic energy. The 4𝑓 electrons can hop between the magnetic ions with different valences due to thermal activation (a thermal valence-fluctuating state). If the intersite Coulomb repulsion is large enough, it may dominate the kinetic energy and, once the charge-disorder entropy due to hopping is low enough, lead to a charge-ordered transition at a critical temperature 𝑇co below which the valence fluctuation suppressed. The resulting inhomogeneously mixed-valence state consists of two species of ions with the 4𝑓𝑛 and the 4𝑓𝑛1𝑑 configurations. This transition may be compared to a Wigner crystallization on a lattice [8], and its earliest example is the Verwey transition in magnetite Fe3O4 [9], although this picture turned out to be too simplified for this compound [10, 11]. There are several charge-fluctuating inhomogeneous mixed-valence compounds containing rare-earth ions. They are the rare-earth pnictides Yb4As3, Sm4Bi3, and Eu4As3 with the cubic anti-Th3P4 structure and rare earth chalcogenides Sm3X4 (X = S, Se or Te) and Eu3S4 with the Th3P4 structure.

2. Theoretical Framework

It is well known that the LSDA fails to describe the electronic structure and properties of the systems in which the interaction among the electrons is strong. In recent years more advanced methods of electronic structure determination such as LSDA plus self-interaction corrections (SIC-LSDA) [12], the LSDA+𝑈 [13] method, the GW approximation [14], and dynamical mean-field theory (DMFT) [1, 15, 16] have sought to remedy this problem and have met with considerable success. The LSDA+𝑈 method is the simplest among them and most frequently used in the literature therefore; we describe the method in detail.

2.1. LSDA+𝑈 Method

A rigorous formulation for the quasiparticle properties of solids is the Green function approach. The self-energy Σ=𝐺01𝐺1 of the single-particle Green function 𝐺 is energy dependent and yields the correlation corrections to the single-particle (mean-field) approximation to the quasiparticle excitation spectrum described by 𝐺0. With a number of plausible assumptions, the LSDA+𝑈 approach has been related to the so-called GW approximation to Σ in [17]. Already the simplest random phase approximation applied to Σ for the Hubbard model yields a jump of Σ(𝜖) at the Fermi level 𝜖𝐹 by the Hubbard 𝑈. The more elaborate analysis of [17] results in a correlation correction to the mean-field approximation of the self-energy, which is 𝑈e/2 downward below the Fermi level and 𝑈e/2 upward above the Fermi level. As mean-field theory in a crystal describes always a delocalized situation and the LSDA Kohn-Sham potential is a well-proved approximation to the self-energy of weakly correlated situations [18], the suggestion isΣ𝑟,𝑟;𝜖𝛿𝑟𝑟𝑣LSDA𝑟+𝑃𝑚𝑈e2𝜃𝜖𝜖𝐹𝜖𝜃𝐹𝑃𝜖𝑚,(1) where 𝑃𝑚 is the projector onto a strongly correlated 𝑚 state.

The LSDA+𝑈 approach simply uses (1) to replace the LSDA Kohn-Sham potential in the self-consistency loop. This can be considered as a rough approximation to Σ. Since the potential shift is taken to be constant in space, it does not deform the Kohn-Sham orbital 𝜙𝑚. However, it shifts the levels of strongly correlated motion away from the Fermi level and thus removes incorrect hybridization with conduction states, which would spoil the calculated ground-state spin density. On the other hand, being also understood as an approximation to Σ, it hopefully yields for the Kohn-Sham band structure the same quality of a working approximation to the quasiparticle excitation spectrum as it does in the case of weakly correlated metals.

The main idea of the LSDA+𝑈 is the same as in the Anderson impurity model [19]: the separate treatment of localized 𝑓-electrons for which the Coulomb 𝑓𝑓 interaction is taken into account by a Hubbard-type term in the Hamiltonian (1/2)𝑈𝑖𝑗𝑛𝑖𝑛𝑗 (𝑛𝑖 are 𝑓 orbital occupancies) and delocalized 𝑠,𝑝,𝑑 electrons for which the local density approximation for the Coulomb interaction is regarded as sufficient.

The spectrum of excitations for the shell of an 𝑓 electron system is a set of many-body levels describing processes of removing and adding electrons. In the simplified case, when every 𝑓 electron has roughly the same kinetic energy 𝜀𝑓 and Coulomb repulsion energy 𝑈, the total energy of the shell with 𝑛 electrons is given by 𝐸𝑛=𝜀𝑓𝑛+𝑈𝑛(𝑛1)/2 and the excitation spectrum is given by 𝜀𝑛=𝐸𝑛+1𝐸𝑛=𝜀𝑓+𝑈𝑛.

Let us consider 𝑓 ion as an open system with a fluctuating number of 𝑓 electrons. The correct formula for the Coulomb energy of 𝑓𝑓 interactions as a function of the number of 𝑓 electrons 𝑁 given by the LDA should be 𝐸=𝑈𝑁(𝑁1)/2 [20]. If we subtract this expression from the LDA total energy functional and add a Hubbard-like term (neglecting for a while exchange and nonsphericity), we will have the following functional:𝐸=𝐸LDA𝑈𝑁(𝑁1)2+12𝑈𝑖𝑗𝑛𝑖𝑛𝑗.(2) The orbital energies 𝜀𝑖 are derivatives of (2):𝜀𝑖=𝜕𝐸𝜕𝑛𝑖=𝜀LDA1+𝑈2𝑛𝑖.(3) This simple formula gives the shift of the LDA orbital energy 𝑈/2 for occupied orbitals (𝑛𝑖=1) and +𝑈/2 for unoccupied orbitals (𝑛𝑖=0). A similar formula is found for the orbital-dependent potential 𝑉𝑖(𝑟)=𝛿𝐸/𝛿𝑛𝑖(𝑟) where variation is taken not on the total charge density 𝜌(𝑟) but on the charge density of a particular 𝑖th orbital 𝑛𝑖(𝑟):𝑉𝑖𝑟=𝑉LDA1𝑟+𝑈2𝑛𝑖.(4) Expression (4) restores the discontinuous behavior of the one-electron potential of the exact density-functional theory.

The functional (2) neglects exchange and nonsphericity of the Coulomb interaction. In the most general rotationally invariant form the LDA+𝑈 functional is defined as [21, 22]𝐸LDA+𝑈𝜌𝑟,̂𝑛=𝐸L(S)DA𝜌𝑟+𝐸𝑈(̂𝑛)𝐸dc(̂𝑛),(5) where 𝐸L(S)DA[𝜌(𝑟)] is the LSDA (or LDA as in [20]) functional of the total electron spin densities, 𝐸𝑈(̂𝑛) is the electron-electron interaction energy of the localized electrons, and 𝐸dc(̂𝑛) is the so-called “double counting” term which cancels approximately the part of an electron-electron energy which is already included in 𝐸LDA. The last two terms are functions of the occupation matrix ̂𝑛 defined using the local orbitals {𝜙𝑙𝑚𝜎}.

The matrix ̂𝑛=𝑛𝜎𝑚,𝜎𝑚 generally consists of both spin-diagonal and spin-nondiagonal terms. The latter can appear due to the spin-orbit interaction or a noncollinear magnetic order. Then, the second term in (5) can be written as [2123]𝐸𝑈=12𝜎,𝜎,{𝑚}𝑛𝜎𝑚1,𝜎𝑚2𝑈𝑚1𝑚2𝑚3𝑚4𝑛𝜎𝑚3,𝜎𝑚4𝑛𝜎𝑚1,𝜎𝑚2𝑈𝑚1𝑚4𝑚3𝑚2𝑛𝜎𝑚3,𝜎𝑚4,(6) where 𝑈𝑚1𝑚2𝑚3𝑚4 are the matrix elements of the on-site Coulomb interaction which are given by𝑈𝑚1𝑚2𝑚3𝑚4=2𝑙𝑘=0𝑎𝑘𝑚1𝑚2𝑚3𝑚4𝐹𝑘,(7) with 𝐹𝑘 being screened Slater integrals for a given 𝑙 and𝑎𝑘𝑚1𝑚2𝑚3𝑚4=4𝜋2𝑘+1𝑘𝑞=𝑘𝑙𝑚1||𝑌𝑘𝑞||𝑙𝑚2𝑙𝑚3||𝑌𝑘𝑞||𝑙𝑚4.(8) The 𝑙𝑚1|𝑌𝑘𝑞|𝑙𝑚2 angular integrals of a product of three spherical harmonics 𝑌𝑙𝑚 can be expressed in terms of Clebsch-Gordan coefficients, and (8) becomes𝑎𝑘𝑚1𝑚2𝑚3𝑚4=𝛿𝑚1𝑚2+𝑚3,𝑚4𝐶𝑙0𝑘0,𝑙02×𝐶𝑙𝑚1𝑘𝑚1𝑚2,𝑙𝑚2𝐶𝑙𝑚4𝑘𝑚1𝑚2,𝑙𝑚3.(9)

The matrix elements 𝑈𝑚𝑚𝑚𝑚 and 𝑈𝑚𝑚𝑚𝑚 which enter those terms in the sum in (6) which contain a product of the diagonal elements of the occupation matrix can be identified as pair Coulomb and exchange integrals𝑈𝑚𝑚𝑚𝑚=𝑈𝑚𝑚,𝑈𝑚𝑚𝑚𝑚=𝐽𝑚𝑚.(10)

The averaging of the matrices 𝑈𝑚𝑚 and 𝑈𝑚𝑚𝐽𝑚𝑚 over all possible pairs of 𝑚, 𝑚 defines the averaged Coulomb 𝑈 and exchange 𝐽 integrals which enter the expression for 𝐸dc. Using the properties of Clebsch-Gordan coefficients, one can show that1𝑈=(2𝑙+1)2𝑚𝑚𝑈𝑚𝑚=𝐹0,1𝑈𝐽=2𝑙(2𝑙+1)𝑚𝑚𝑈𝑚𝑚𝐽𝑚𝑚=𝐹012𝑙2𝑙𝑘=2𝐶𝑙0𝑛0,𝑙02𝐹𝑘,(11) where the primed sum is over 𝑚𝑚. Equation (11) allows us to establish the following relation between the average exchange integral 𝐽 and Slater integrals:1𝐽=2𝑙2𝑙𝑘=2𝐶𝑙0𝑛0,𝑙02𝐹𝑘,(12) or explicitly1𝐽=𝐹142+𝐹41for𝑙=2,𝐽=6435286𝐹2+195𝐹4+250𝐹6for𝑙=3.(13)

The meaning of 𝑈 has been carefully discussed by Herring [24]. In, for example, am 𝑓 electron system with 𝑛𝑓 electrons per atom, 𝑈 is defined as the energy cost for the reaction2(𝑓𝑛)𝑓𝑛+1+𝑓𝑛1,(14) that is, the energy cost for moving an 𝑓 electron between two atoms where both initially had 𝑛𝑓 electrons. It should be emphasized that 𝑈 is a renormalized quantity which contains the effects of screening by fast 𝑠 and 𝑝 electrons. The number of these delocalized electrons on an atom with 𝑛+1𝑓 electrons decreases, whereas their number on an atom with 𝑛1𝑓 electrons increases. The screening reduces the energy cost for the reaction given by (14). It is worth noting that because of the screening the value of 𝑈 in L(S)DA+𝑈 calculations is significantly smaller than the bare 𝑈 used in the Hubbard model [25, 26].

Some aspects of currently used LDA+𝑈 formulation and, in particular, of the determination of the parameters entering the model have been so far tied to the LMTO approach. The reformulation of the method for different basis sets has been made recently [2729]. Pickett et al. [27] present a reformulation of the LDA+𝑈 method based on a local-orbital expansion (linear combination of atomic orbitals). The implementation of the LDA+𝑈 method by Bengone et al. [28] is based on the projector augmented wave (PAW) method [30], an efficient all-electron method without shape approximations on the potential or electron density. Based on a Car-Parrinello-like formalism [31], the PAW method allows complex relaxations and dynamical properties in strongly correlated systems. Cococcioni and De Gironcoli [29] reexamine the LDA+𝑈 method in the framework of a plane-wave pseudopotential approach (PWPP). A simplified rotational-invariant formulation was adopted. They demonstrate the accuracy of the method, computing structural, and electronic properties of a few systems including transition and rare-earth correlated metals, transition metal mono-oxides, and iron silicate [29], transition metal centers Fe2,Fe2,FeO [32], and iron heme complexes [33].

In principle, the screened Coulomb 𝑈 and exchange 𝐽 integrals can be determined from supercell LSDA calculations using Slater’s transition-state technique [34] or from constrained LSDA calculations [27, 35, 36].

To obtain Hubbard 𝑈 and exchange parameter 𝐽, Anisimov and coworkers [34, 36] propose to perform LMTO calculations in supercells in which the occupation of the localized orbitals of one atom is constrained. The localized orbitals of all atoms in the supercell are decoupled from the remainder of the basis set. This makes the treatment of the local orbitals an atomic-like problem making it easy to fix their occupation numbers and allows them to use Janak’s theorem [37] to identify the shift in the corresponding eigenvalue with the second-order derivative of the LDA total energy with respect to orbital occupation. It has, however, the effect of leaving a rather artificial system to perform the screening, in particular when it is not completely intra-atomic. In elemental metallic iron, for instance, only half of the screening charge is contained in the Wigner-Seitz cell [34]. This fact, in addition to a sizable error due to the atomic sphere approximation used, could be at the origin of the severe overestimation of the computed on-site Coulomb interaction. For example, Anisimov and Gunnarsson computed the effective on-site Coulomb interaction in metallic Fe and Ce. For Ce the calculated Coulomb interaction was found to be about 6 eV in good agreement with empirical and experimental estimates ranging from 5 to 7 eV [38], while the result for Fe (also about 6 eV) was surprisingly high since 𝑈 was expected to be in the range of 1-2 eV for elemental transition metals, with the exception of Ni [39, 40].

Cococcioni and De Gironcoli [29] provide an internally consistent, basis-set-independent, method based on a linear response approach for the calculation of the effective interaction parameters in the LDA+𝑈 method. They estimate the Hubbard 𝑈 in elemental iron at the experimental lattice parameter to be equal to 2.2±0.2 eV in good agreement with the experimental data [39, 40].

Alternatively, the value of 𝑈 estimated from the photo-emission spectroscopy (PES) and X-ray Bremsstrahlung isochromat spectroscopy (BIS) experiments can be used. Because of the difficulties with unambiguous determination of 𝑈, it can be considered as a parameter of the model. Then its value can be adjusted to achieve the best agreement of the results of LDA+𝑈 calculations with PES or optical spectra [28]. While the use of an adjustable parameter is generally considered an anathema among first principles practitioners, the LDA+𝑈 approach does offer a plausible and practical method to approximately treat strongly correlated orbitals in solids.

2.2. Magneto-Optical Effects

Magneto-optical (MO) effects refer to various changes in the polarization state of light upon interaction with materials possessing a net magnetic moment, including rotation of the plane of linearly polarized light (Faraday, Kerr rotation) and the complementary differential absorption of left and right circularly polarized light (circular dichroism). In the near visible spectral range these effects result from excitation of electrons in the conduction band. Near X-ray absorption edges, or resonances, magneto-optical effects can be enhanced by transitions from well-defined atomic core levels to transition symmetry selected valence states.

For a crystal of cubic symmetry, where the magnetization 𝐌 is parallel to the 𝑧-axis, the dielectric tensor is composed of the diagonal 𝜀𝑥𝑥 and 𝜀𝑧𝑧 and the off-diagonal 𝜀𝑥𝑦 components in the form𝜀𝜀=𝑥𝑥𝜀𝑥𝑦0𝜀𝑥𝑦𝜀𝑥𝑥000𝜀𝑧𝑧.(15) A complete description of the MO effects is given by the four nonzero elements of the dielectric tensor or, equivalently, by the complex refractive index 𝑛(𝜔)𝑛(𝜔)𝜀(𝜔)=1𝛿(𝜔)+𝑖𝛽(𝜔)(16) for several normal modes corresponding to the propagation of pure polarization states along specific directions in the sample, which can be obtained by solving Maxwell’s equations [41]. Two of these modes are for circular components of opposite (±) helicity with the wave vector 𝐡𝐌 and have indices𝑛±=1𝛿±+𝑖𝛽±=𝜀𝑥𝑥±𝑖𝜀𝑥𝑦.(17)

The other two modes are for linear polarizations with 𝐡𝐌 [42]. One has the electric vector 𝐄𝐌 and index 𝑛=1𝛿+𝑖𝛽=𝜀𝑧𝑧. The other has 𝐄𝐌 and 𝑛=1𝛿+𝑖𝛽=(𝜀2𝑥𝑥+𝜀2𝑥𝑦)/𝜀𝑥𝑥.

At normal light incidence the relation between the polar complex Kerr angle and the dielectric tensor components is given by the expression [43]𝜃+𝑖𝜂𝜀𝑥𝑦𝜀𝑥𝑥1𝜀𝑥𝑥.(18)

X-ray magnetic circular dichroism is given by 𝛽+𝛽 and is first-order in 𝐌. Magnetic linear dichroism 𝑛𝑛 (also known as the Voigt effect) is quadratic in 𝐌. The Voigt effect is present in both ferromagnets and antiferromagnets, while the first order MO effects in the forward scattering beam are absent with the net magnetization in antiferromagnets.

Within the one-particle approximation, the absorption coefficient 𝜇𝜆𝑗(𝜔) for incident X-ray of polarization 𝜆 and photon energy 𝜔 can be determined as the probability of electronic transitions from initial core states with the total angular momentum 𝑗 to final unoccupied Bloch states𝜇𝑗𝜆(𝜔)=𝑚𝑗𝑛𝐤|||Ψ𝑛𝐤||Π𝜆||Ψ𝑗𝑚𝑗|||2𝛿𝐸𝑛𝐤𝐸𝑗𝑚𝑗𝐸𝜔×𝜃𝑛𝐤𝐸𝐹,(19) where Ψ𝑗𝑚𝑗 and 𝐸𝑗𝑚𝑗 are the wave function and the energy of a core state with the projection of the total angular momentum 𝑚𝑗; Ψ𝑛𝐤 and 𝐸𝑛𝐤 are the wave function and the energy of a valence state in the 𝑛th band with the wave vector 𝐤; 𝐸𝐹 is the Fermi energy.

Π𝜆 is the electron-photon interaction operator in the dipole approximationΠ𝜆=𝑒𝜶𝐚𝝀,(20) where 𝜶 are the Dirac matrices, 𝐚𝜆 is the 𝜆 polarization unit vector of the photon vector potential, with 𝑎±=1/2(1,±𝑖,0), 𝑎=(0,0,1). Here, + and denote, respectively, left and right circular photon polarizations with respect to the magnetization direction in the solid. Then, X-ray magnetic circular and linear dichroisms are given by 𝜇+𝜇 and 𝜇(𝜇++𝜇)/2, respectively.

The Kerr effect has now been known for more than a century, but it was only in recent times that it became the subject of intensive investigations. The reason for this recent development is twofold: first, the Kerr effect gained considerable interest due to modern data storage technology, because it can be used to “read” suitably stored magnetic information in an optical manner [44], and, second, the Kerr effect has rapidly developed into an appealing spectroscopic tool in materials research. The technological research on the Kerr effect was initially motivated by the search for good magneto-optical materials that could be used as information storage medium. In the sequence of this research, the Kerr spectra of many ferromagnetic materials were investigated. An overview of the experimental data collected on the Kerr effect can be found in the review articles by Buschow [45], Reim and Schoenes [46], and Schoenes [43].

The quantum mechanical understanding of the Kerr effect began as early as 1932 when Hulme [47] proposed that the Kerr effect could be attributed to spin-orbit (SO) coupling (see also Kittel [48]). The symmetry between left- and right-hand circularly polarized light is broken due to the SO coupling in a magnetic solid. This leads to different refractive indices for the two kinds of circularly polarized light, so that incident linearly polarized light is reflected with elliptical polarization, and the major elliptical axis is rotated by the so-called Kerr angle from the original axis of linear polarization. The first systematic study of the frequency-dependent Kerr and Faraday effects was developed by Argyres [49] and later Cooper presented a more general theory using some simplifying assumptions [50]. The very powerful linear response techniques of Kubo [51] gave general formulas for the conductivity tensor which are being widely used now. A general theory of frequency dependent conductivity of ferromagnetic (FM) metals over a wide range of frequencies and temperatures was developed in 1968 by Kondorsky and Vediaev [52].

The main problem afterward was the evaluation of the complicated formulas involving MO matrix elements using electronic states of the real FM system. With the tremendous increases in computational power and the concomitant progress in electronic structure methods, the calculation of such matrix elements became possible, if not routine. Subsequently much earlier, simplified calculations have been shown to be inadequate, and only calculations from “first principles” have provided, on the whole, a satisfactory description of the experimental results [53]. The existing difficulties stem either from problems using the local spin density approximation (LDA) to describe the electronic structure of FM materials containing highly correlated electrons or simply from the difficulty of dealing with very complex crystal structures. For 15 years after the work of Wang and Callaway [53] there was a lull in MO calculations until MO effects were found to be important for magnetic recording and the computational resources had advanced. Different reliable numerical schemes for the calculation of optical matrix elements and the integration over the Brillouin zone have been implemented, giving essentially identical results [54]. Prototype studies have been performed using modern methods of band theory for Fe, Co, and Ni [5559]. Following the calculations for the elemental 3𝑑 ferromagnets, a number of groups have evaluated the MO and XMCD spectra for more interesting compounds [10, 6088] and multilayers [89102].

3. Iron Oxides

3.1. Cubic Fe3O4

The problem of a theoretical description of metal-insulator transitions has a challenging history of almost 70 years. It was first addressed by Verwey, de Boer, and Peierls in the late 1930s; they pointed out the extremely important role of electron-electron correlations in a partially filled 𝑑 electron band in transition metal oxides [103, 104], for example, nickel oxide (NiO) and magnetite (Fe3O4). In both systems the metal-insulator transition occurs, violating the Bloch-Wilson band-insulator concept, the only one known at that time [105108]. These earlier observations launched the long and continuing history of the field of strongly correlated electrons. In the past 80 years, much progress has been achieved from both theoretical and experimental sides in understanding strongly correlated electrons and metal-insulator transitions [2]. However, the charge ordering proposed by Verwey behind the metal-insulator transition [9, 109, 110] in Fe3O4 remains at the focus of active debate [111, 112].

Historically, magnetite, discovered before 1500 B.C., is the first known magnet and is extensively used for industrial applications, notably in magnetic recording. Most of the properties of magnetite have been thoroughly studied and are well documented [113, 114]. However, the electronic structure of Fe3O4 as well as that of many other MIT compounds is still a subject of debate [115119].

Fe3O4 is a strongly correlated 3𝑑 compound which is ferrimagnetically ordered below a high transition temperature (~850 K). The valence of various atoms is described by the formal chemical formula, Fe𝐴3+[Fe2+Fe3+]𝐵(O2)4. The tetrahedral lattice sites (𝐴 sites) in the inverse spinel structure are occupied by Fe3+ ions, whereas the octahedral lattice sites (𝐵 sites) are occupied alternately by equal numbers of Fe2+ and Fe3+. At 𝑇𝑉=120 K Fe3O4 undergoes a first-order phase transition (Verwey transition) [9]. This is a particular MIT that has been studied for quite some time [120]. The Verwey transition is characterized by an abrupt increase in the electrical conductivity by two orders of magnitude on heating through 𝑇𝑉 [121123]. Verwey and co-workers [9, 110] were the first to point out that this transition is associated with an electron localization-delocalization transition. The Fe2+ ion can be regarded as an “extra” electron plus an Fe3+ ion. When all 𝐵 sites are equivalent, the “extra” electron is moving between Fe3+B ions and the system is a mixed-valent metal, with average FeB-valence, 𝑍=2.5. The Verwey phase transition below 𝑇𝑉 is accompanied by long-range charge ordering (LRCO) of Fe3+ and Fe2+ ions on 𝐵1 and 𝐵2 sites of the 𝐵 sublattice. Indeed, studies by electron and neutron diffraction and nuclear magnetic resonance [124126] show that below 𝑇𝑉 the 𝐵1 and 𝐵2 sites are structurally distinguishable with the crystal structure slightly distorted because of the charge ordering. Just how these charges arrange themselves has been the subject of debate [120] since Verwey first proposed that, below 𝑇𝑉, all Fe3+B and Fe2+B sit on different chains [9].

The electronic structure of Fe3O4 has been investigated experimentally by means of soft X-ray spectroscopy [127129], Seebeck-effect measurements [130, 131], photoelectron spectroscopy [132139], optical [140, 141] and MO spectroscopies [142149], and magnetic dichroism [150, 151]. The optical data [141] indicate a gap of 0.14 eV between occupied and empty electronic states and also show a strong temperature dependence of the optical conductivity in the energy region of 0 to 1 eV.

The interpretation of the optical and MO spectra of Fe3O4 is very difficult due to the existence of three kinds of iron atoms, that is, Fe𝐵2+, Fe𝐵3+, and Fe𝐴3+. The substitution for one of the types of iron ions by nonmagnetic ions provides a possibility for distinguishing transitions from various sites. There are several such experimental studies in the literature. Simsa and coworkers reported the polar Kerr rotation and ellipticity of Fe3O4 as well as the influence of a systematic substitution of Fe2+ by Mn2+ in the 0.5–3.0 eV energy range [144, 145]. Zhang and coworkers reported the polar Kerr spectra and the off-diagonal element of the dielectric tensor of Fe3O4 between 0.5 and 4.3 eV. They also reported spectra of Li0.5Fe2.5O4 and MgFe2O4 between 0.5 and 5.0 eV [142]. Finally, recent investigation of the optical and MO spectra of Fe3O4 and Al3+ and Mg2+ substitution has been carried out in [143].

Energy band structure calculations for Fe3O4 in the high-temperature cubic phase have been presented in [152] using the self-consistent spin-polarized augmented plane wave method with the local spin-density approximation. The calculations show that Fe3O4 is a half-metallic ferrimagnet. The Fermi level crosses only the minority-spin energy bands consisting of 𝑡2𝑔 orbitals on the Fe(𝐵) sublattice. There is an energy gap for the majority-spin bands at the Fermi level. A similar energy band structure of Fe3O4 was obtained in [153] using the LMTO method. The energy band structure for charge ordering in the low-temperature phase of Fe3O4 has been calculated in [10, 11, 154, 155] using the LMTO method in the LSDA+𝑈 approximation.

3.1.1. Crystal Structure and Charge Ordering

Fe3O4 crystallizes in the face-centered cubic (FCC) inverse spinel structure (Figure 1) with two formula units (14 atoms) per primitive cell. The space group is 𝐹𝑑3𝑚 (no. 227). The oxygen atoms form a close-packed face-centered cubic structure with the iron atoms occupying the interstitial positions. There are two types of interstitial sites both occupied by the iron atoms. One site is called the 𝐴 or 8𝑎 site, tetrahedrally coordinated by four O2 ions composing a diamond lattice. The interstices of these coordination tetrahedra are too small for larger Fe2+ ions, and this site is occupied only by Fe3+ ions. Another cation site is called the 𝐵 or 16𝑑 site, and is coordinated by six O2 ions forming slightly distorted octahedra, which line up along the 110 axes of the cubic lattice sharing edges. The point symmetry of the 𝐵 site is 𝐷3𝑑. This site forms exactly one-half of a face-centered cubic lattice. The lattice of the 𝐵 site can be considered as a diamond lattice of cation tetrahedra, sharing corners with each other. All the tetrahedra on the same (e.g., 𝑥𝑦) plane are isolated. In the following, we refer to 𝑏1 axes or 𝑏1 chains and 𝑏2 axes or 𝑏2 chains. The 𝑏1 direction is [110], 𝑏2 is [110] and the 𝑐-axis is [001]. All the Fe octahedral or 𝐵 sites lie on either 𝑏1 or 𝑏2 chains. It should be mentioned also that the distances FeA-O1 and FeB-O2 are different and equal to 1.876 and 2.066 Å, respectively.

Figure 1: Crystal structure of Fe3O4.

Details concerning the mechanism of the Verwey transition and the type of LRCO are still unclear. Many elaborate theories for the Verwey transition have been proposed. Anderson [156] pointed out the essential role of short-range charge ordering (SRCO) in the thermodynamics of the transition. The observed entropy change in the transition [(~0.3 to 0.35)𝑅/𝐵-site mole] is decisively smaller than the 𝑅ln2=0.69𝑅 expected in a complete order-disorder transition. Anderson interpreted the Verwey transition as a loss of the LRCO of the “extra” electrons on the 𝐵 sublattice at temperatures above 𝑇𝑉, while the short-range charge order is maintained across the transition. Using a Hartree-Fock analysis, Alben and Cullen and Callen [157159] showed that an ordering transition could occur in Fe3O4 as a function of the ratio 𝑉/𝐵, where 𝐵 is the bandwidth of the “extra” electrons in the absence of disorder. The transition in this case is of second order, while experimentally a first-order transition is observed. The conduction mechanisms in Fe3O4 have been reviewed by Mott [160162] (see also [120, 163]). A basic problem concerning magnetite is that both the localization of the valence states and the mixing of the oxygen 𝑝 states and iron 𝑑 states are considerable.

In the disordered high-temperature phase the 𝐵 sites are occupied by equal numbers of Fe2+ and Fe3+ ions randomly distributed between 𝐵1 and 𝐵2 sites. Below 𝑇𝑉 the system undergoes a first-order transition accompanied by long-range charge ordering of Fe3+ and Fe2+ ions on the 𝐵 sites. Verwey from the very beginning proposed a rather simple charge separation: 𝑏1 chains occupied only by Fe2+ ions and 𝑏2 chains by Fe3+ ions (or vice versa) [9]. Since that time the type of charge ordering has been the subject of debate [120]. As an example, in Mizoguchi’s model [164, 165] ions run in pairs of Fe2+ followed by Fe3+ along each 𝑏 chain. Another charge ordering considers three Fe2+ alternating with one Fe3+ on one half of the chains, and a sequence of three Fe3+ alternating with one Fe2+ on the other half and so on [120]. Despite the wealth of effort devoted to investigating the low-temperature phase of magnetite, there is still no completely satisfactory description of the ordering of the Fe atoms on the octahedral or 𝐵 sites in this spinel structure. In addition some experimental measurements disproved the orthorhombic Verwey CO model. These experiments have clearly established the rhombohedral distortions of the cubic unit cell first detected by Rooksby et al. from X-ray powder diffraction [166, 167]. Furthermore, observations of superstructure reflections revealed half-integer satellite reflections, indexed as (,𝑘,𝑙+1/2) on the cubic unit cell, which points to a doubling of the unit cell along the 𝑐-axis and shows the symmetry to be monoclinic [168, 169]. The observation of monoclinic lattice symmetry was also confirmed by a single-crystal X-ray study [170], whereas the observation of a magnetoelectric effect indicated even lower 𝑃1 symmetry in the low-temperature phase [171]. Although clear evidence of the monoclinic lattice symmetry below 𝑇𝑉 was obtained, small atomic displacements have not been fully resolved so far. The absence of a definitive, experimentally determined structure gives rise to many theoretical models proposed for the low-temperature (LT) phase of magnetite [172]. In particular, purely electronic [158, 159, 164, 173] and electron-phonon [174176] models for CO, as well as a bond-dimerized ground state without charge separation [177], have been proposed.

Recent bond-valence-sum analysis [178] of high-resolution neutron and X-ray powder diffraction data results in a small charge disproportion of only 0.2 𝑒 between Fe𝐵 cations with the 2+ and 3+ formal valency [111, 179]. This interpretation has been the subject of much controversy [112, 180]. However, the smallness of the charge-order parameter was reproduced in an electronic structure study of the refined low-temperature crystal structure using the local-spin-density approximation (LSDA)+𝑈 method [181, 182]. In particular, a more complicated charge-ordering pattern inconsistent with the Verwey CO model was obtained. In addition to that, the 𝑡2𝑔 occupancy self-consistently obtained in the LSDA+𝑈 calculations is strongly modulated between the Fe𝐵2+ and Fe𝐵3+ cations, yielding a distinct orbital order with an order parameter that reaches 70% of the ideal value [181]. Since no direct experimental confirmation of this charge- and orbital-order pattern is so far available, the interpretation of these results is still open to debate. However, this behavior seems to be universal and has recently been found in several other charge-ordered mixed-valent systems [182186].

In order to check the pertinence of the CO model obtained self-consistently in [181], authors of [187] carried out a detailed theoretical study of exchange coupling constants, optical conductivity, magneto-optical (MO) Kerr effect, and X-ray absorption at the O 𝐾 edge of low-temperature Fe3O4 and compared the results of the calculations to the available experimental data.

3.1.2. LSDA Band Structure

Figure 2 shows the partial density of states of Fe3O4 obtained from the LSDA calculation. The occupied part of the valence band can be subdivided into several regions separated by energy gaps. The oxygen 2𝑠 bands, which are not shown in the figure, appear between −20.0 and −19.7 eV for both spins with the exchange splitting of about 0.2 eV. The next group of bands in the energy region −7.4 to −3.4 eV is formed mostly by oxygen 2𝑝 states. The Fe 𝑑 energy bands are located above and below 𝐸𝐹 at about −4.0 to 3.0 eV. As indicated from Figure 2, the exchange splitting between the spin-up and spin-down 𝑑 electrons on the Fe atom is about 3.5 eV. In addition to the exchange splitting, the five 𝑑 levels of the Fe atom are split due to the crystal field. At the 𝐴 site (𝑇𝑑 point symmetry) in the spinel structure the crystal field causes the 𝑑 orbitals to split into a doublet 𝑒 (3𝑧21 and 𝑥2𝑦2) and a triplet 𝑡2 (𝑥𝑦, 𝑦𝑧, and 𝑥𝑧). The octahedral component of the crystal field at the 𝐵 site is strong enough that the 𝑡2𝑔 (𝑥𝑦, 𝑦𝑧, and 𝑥𝑧) and 𝑒𝑔 (3𝑧21 and 𝑥2𝑦2) orbitals form two separate nonoverlapping bands. At the 𝐵 site the crystal field is trigonal (𝐷3𝑑), as a result the 𝑡2𝑔 orbitals split into a singlet 𝑎1𝑔 and a doublet 𝑒𝑔. However, the 𝑎1𝑔𝑒𝑔 splitting of the 𝑡2𝑔 band is negligible in comparison with its width in LSDA calculations; therefore, in the following we will denote the states formed by 𝑎1𝑔 and 𝑒𝑔 orbitals as 𝑡2𝑔 states. Accordingly, we present in Figure 2 the DOS of “𝑡2𝑔” orbitals as a sum of the 𝑎1𝑔 and 𝑒𝑔 ones. The crystal-field splitting ΔCF is approximately 2 eV for the Fe𝐵 atom and 1 eV for the Fe𝐴 atom. This difference may be attributed to the large covalent mixing of the Fe𝐵 orbitals with its six nearest neighbors of the same kind. The spin-polarized calculations show that Fe3O4 in the high-temperature phase is a half-metallic ferrimagnet. The Fermi level crosses only the majority spin energy bands, consisting of spin-up 𝑡2𝑔 orbitals on the Fe𝐵 sublattice (Figure 2). There is an energy gap for the minority spin bands at the Fermi level. Spin-orbit splitting of the 𝑑 energy band at Γ is about 0.02 eV and much smaller than the crystal-field splitting.

Figure 2: LSDA partial DOS of Fe3O4 [10].

In Fe3O4 the magnetic moments within the 𝐴 and the 𝐵 sublattices are ferromagnetically aligned while the two sublattices are antiferromagnetic with respect to each other. This magnetic structure was first proposed by Neel [188] to explain the magnetization data and was later confirmed by neutron scattering measurements [189]. Measurements indicated that the magnetic moment of an iron atom on the 𝐴 site is much smaller than the 5.0 𝜇𝐵 of a pure Fe3+ ion [190]. This is an indication of strong hybridization between the 3𝑑 orbitals of Fe𝐴. The orbital magnetic moment is rather small for all the atoms due to small spin-orbit coupling.

3.1.3. LSDA+𝑈 Band Structure

The application of LSDA calculations to Fe3O4 is problematic, because of the correlated nature of 𝑑 electrons in this compound. The intersite Coulomb correlation is well described by the LSDA. However, the on-site Coulomb interaction, which is a driving force for Mott-Hubbard localization, is not well treated within LSDA. As a result, LSDA gives only a metallic solution without charge ordering. The LSDA+𝑈 calculations [10] started from a 𝑑5(𝑡32𝑒2) configuration for Fe𝐴3+ ions on the tetrahedral site of the sublattice 𝐴 and 𝑑6(𝑡32𝑔𝑒2𝑔𝑎11𝑔) and 𝑑5(𝑡32𝑔𝑒2𝑔) for Fe𝐵2+ and Fe𝐵3+ ions on octahedral site of the sublattices 𝐵1 and 𝐵2, respectively. 𝑈e was applied to all the 𝑑 states, and the occupation numbers were obtained as a result of the self-consistent relaxation.

Figures 3 and 4 show the energy band structure along the symmetry lines and the total and partial density of states obtained from the LSDA+𝑈 calculation. In contrast to LSDA, where the stable solution is a metal with a uniform distribution of the 𝑡2𝑔 electrons on the 𝐵 octahedral sites, the LSDA+𝑈 gives a charge-ordered insulator with a direct energy gap value of 0.19 eV at the Γ point. The experimental optical measurements [141] gave a gap of 0.14 eV at 𝑇=10 K. The energy gap occurs between the Fe𝐵2+𝑎1𝑔 (the top of valence band) and Fe𝐵3+𝑡2𝑔 (bottom of empty conduction band) states (Figure 4). Actually, the LSDA+𝑈 band structure calculations support the key assumption that Cullen and Callen have made earlier [157159] in proposing the one-band model Hamiltonian, where it was assumed that the “extra” electron moves in the 𝑎1𝑔 band split off below the rest of the 𝑑 bands of other symmetries. Two electrons at the Fe𝐵2+𝑎1𝑔 orbitals situated in the close vicinity of the Fermi level are mostly localized. Other electrons at the 𝐵1 site are well hybridized with oxygen 𝑝-electrons (Figure 4). The screening of the Coulomb interaction in Fe3O4 is very effective and the system is close to the metallic state. Even a small change in the ratio (Fe2+/Fe3+)oct. (which can be modeled by changing of the occupation numbers of the 𝑎1𝑔 orbital at the 𝐵1 and 𝐵2 sites in the framework of the “virtual crystal approximation”) leads to the closing of the energy gap and a suppression of the metal-insulating transition [10].

Figure 3: LSDA+𝑈 energy band structure and total DOS (in states/(unit cell eV)) of Fe3O4 [10].
Figure 4: LSDA+𝑈 partial DOS of Fe3O4 [10].
3.1.4. Optical and MO Properties

After the consideration of the above band structure properties, we turn to the optical and MO spectra. In Figure 5 experimental optical reflectivity and the diagonal part of the dielectric function 𝜀𝑥𝑥 of Fe3O4 are compared to the theoretical ones calculated within the LSDA and LSDA+𝑈 approaches. Better agreement between the theory and the experiment was found when the LSDA+𝑈 approximation was used. As was mentioned above, the LSDA theory produces the metallic solution and, therefore, gives a wrong asymptotic behavior for the optical reflectivity and the dispersive part of the dielectric function 𝜀1𝑥𝑥 as 𝜔0. In Figure 6 we show the calculated and experimental absorptive part of the diagonal optical conductivity spectra 𝜎1𝑥𝑥 in a wide energy range. The characteristic features of the LSDA calculation of 𝜎1𝑥𝑥 is an erroneous peak at 1.9 eV which is absent in the experimental measurement. The absence of this peak in the experiment indicates that the LSDA calculations produce incorrect energy band positions. Accounting for the Coulomb repulsion 𝑈e strongly influences not only the electronic structure but also the calculated optical spectra of Fe3O4. The LSDA+𝑈 calculations make better job in describing the optical spectra of Fe3O4 than the LSDA approach (see Figures 5 and 6). The calculated optical conductivity spectrum (Figure 6) can be sorted into the following groups of interband transitions: (1) the interband transitions between the Fe 3𝑑 bands below 2.5 eV, (2) the transitions from O 2𝑝 to Fe 3𝑑 bands in the region of 2.5 to 9 eV, and (3) Fe 3𝑑4𝑝 and O 2𝑝Fe4𝑠 interband transitions above 9 eV. To avoid misunderstanding, we should mention that here and in the following when talking about 𝑑𝑑 transitions we mean that the energy bands involved in the transitions have predominantly 𝑑 character; however, the contribution of 𝑝 or 𝑓 states to these bands is sufficient to provide a significant transition probability through optical dipole matrix elements.

Figure 5: Optical reflectivity and diagonal parts of the dielectric function 𝜀𝑥𝑥 of Fe3O4 calculated in the LSDA and LSDA+𝑈 approximation [10] compared with experimental data [140] (open circles).
Figure 6: The absorptive part of the diagonal optical conductivity of Fe3O4 calculated in LSDA (dashed line) and LSDA+𝑈 (solid line) approximations [10] compared with experimental data [141] (solid squares) and [140] (open circles).

Let us consider now the magneto-optical properties of Fe3O4. In Figure 7 we show the experimentally measured [146] Kerr rotation 𝜃𝐾(𝜔) and Kerr ellipticity 𝜖𝐾(𝜔) MO spectra of Fe3O4, as well as the off-diagonal parts of the dielectric function calculated with the LSDA and LSDA+𝑈 approximations [10]. This picture clearly demonstrates that the better description is achieved with the LSDA+𝑈 approach.

Figure 7: Calculated Kerr rotation and Kerr ellipticity spectra of Fe3O4 [10] compared with experimental data (circles) [146].

One should mention that, although the O 2𝑝 → Fe 3𝑑 interband transitions, which start already from about 2.5 eV, play an important role in the formation of the optical spectra of Fe3O4, the Kerr spectra are mostly determined by transitions between energy bands which have predominantly Fe 3𝑑 character. The reason for this is that the spin-orbit and exchange splitting of O 2𝑝 states is much smaller in comparison with the Fe 3𝑑 ones. The minimum in the Kerr rotation spectrum at 0.9 eV is due to the Fe𝐵2+(𝑎1𝑔)Fe𝐵3+ (𝑡2𝑔) interband transitions. The second maximum at about 2 eV is associated with the Fe𝐵2+(𝑎1𝑔)Fe𝐵3+(𝑒𝑔) interband transitions. The minimum in the Kerr rotation spectrum between 3 and 4 eV can be associated with the Fe𝐵3+(e𝑔)Fe𝐴3+(𝑒,𝑡2) transitions.

All the experimental measurements of the Kerr spectra of Fe3O4 [142147] have been performed at room temperature. The LSDA+𝑈 calculations, in comparison with the LSDA ones, describe better the electronic structure, optical and MO properties not only in the low-temperature semiconducting phase but also in the high-temperature metallic phase of Fe3O4. This leads to a conclusion that Fe 𝑑 electrons remain “correlated” above 𝑇𝑣. The main effect of heating through 𝑇𝑣 is a disappearance of the long-range charge order on the 𝐵 sublattice. This leads to the rearrangement of the electronic states in a small vicinity of the Fermi level and to the closing of the energy gap. However, high-energy Hubbard bands, whose energy position is mainly determined by on-site exchange and correlation interactions, remain almost unaffected (𝑈e4 to 5 eV is much larger than 𝑇𝑣0.01 eV). This picture is supported by recent optical measurements [141], which show a strong temperature dependence of the optical properties of Fe3O4 only in the range from 0 to 1 eV. The absolute value of the measured prominent peak in the optical absorption for photon energies around 0.6 eV determined by the Fe𝐵2+(𝑎1𝑔)Fe𝐵3+(𝑡2𝑔) interband transitions gradually decreases by about 30% when changing the temperature from 10 K to 490 K. However, the other parts of the spectrum change very little.

Finally, we would like to point out that, while the LSDA+𝑈 approach does a better job than the LSDA in the treatment of correlation effects, it is still unclear how well it performs in evaluating the subtle energies and interactions affecting the charge-ordered ground state and the higher-temperature short-range ordered states.

3.1.5. X-Ray Magnetic Circular Dichroism

The XMCD technique developed in recent years has evolved into a powerful magnetometry tool to separate orbital and spin contributions to element-specific magnetic moments. XMCD experiments measure the absorption of X-rays with opposite (left and right) states of circular polarization. The XMCD spectra in core level absorption are element specific and site selective, thus providing valuable information on the energy position of empty states in a wide energy interval.

The interpretation of the experimental XMCD spectra of Fe3O4 [150, 151] is very difficult due to the existence of three kinds of iron atoms, that is, Fe𝐵2+, Fe𝐵3+, and Fe𝐴3+. The substitution for one of the types of iron ions by another transition metal ion provides a possibility for distinguishing transitions from various sites. There are several such experimental studies in the literature. Koide and coworkers reported the XMCD spectra at the FeM2,3 and Co M2,3 core-absorption edges in Fe3O4 and CoFe2O4 [151]. For Fe3O4 the M2,3 prethreshold MCD spectra were measured above and below the Verwey transition temperature. Van Der Laan et al. [192] reported the XMCD spectra at the Ni 𝐿2,3 edges of NiFe2O4 (trevorite). The Ni 𝐿2,3 edge magnetic circular dichroism measurements of ferrimagnetic Zn𝑥Ni1𝑥Fe2O4 (𝑥=0.0, 0.26, 0.50, and 0.75) were reported by Pong et al. [193]. Magnetic circular dichroism is reported for Mn2/3Zn1/3Fe2O4 ferrite in [194] with the measurements performed on the 2𝑝 and 3𝑝 core levels of Mn and Fe. The electronic structure, spin and orbital magnetic moments, and XMCD spectra of the series Fe3O4, CoFe2O4, NiFe2O4, and MnFe2O4 are presented in [191]. The XAS and XMCD spectra at 𝐾, 𝐿2,3 and 𝑀2,3 edges for transition metals sites were calculated.

In Figure 8 the experimentally measured Fe 𝐿2,3 XMCD spectra [150] in Fe3O4 are compared to the theoretical ones calculated within the LSDA+𝑈 approach [191]. The dichroism at the 𝐿2 and 𝐿3 edges is influenced by the spin-orbit coupling of the initial 2𝑝 core states. This gives rise to a very pronounced dichroism in comparison with the dichroism at the 𝐾 edge. Two prominent negative minima of Fe 𝐿3 XMCD spectrum are derived from iron ions at octahedral 𝐵 sites. The major positive maximum is from Fe𝐴3+ ions. In the LSDA+𝑈 calculations of the charge-ordered Fe3O4, the 𝐿23 XMCD spectra have slightly different shape for the Fe𝐵2+ and Fe𝐵3+ ions. The LSDA+𝑈 calculations are not able to produce the small positive shoulder at the high-energy side of the main peaks of the Fe 𝐿3 XMCD spectrum.

Figure 8: Fe 𝐿2,3 XMCD spectra in Fe3O4 calculated with the LSDA+𝑈 approximation [191] in comparison with the experimental data (circles) [150].

The XMCD spectra at the 𝐿2,3 edges are mostly determined by the strength of the SO coupling of the initial 2𝑝 core states and spin-polarization of the final empty 3𝑑3/2,5/2 states, while the exchange splitting of the 2𝑝 core states as well as the SO coupling of the 3𝑑 valence states are of minor importance for the XMCD at the 𝐿2,3 edge of 3𝑑 transition metals [195].

To investigate the influence of the initial state on the resulting XMCD spectra, the XAS and XMCD spectra of Fe3O4 at the 𝑀2,3 edge were also calculated. The spin-orbit splitting of the 3𝑝 core level is about one order of magnitude smaller than for the 2𝑝 level in Fe3O4. As a result the magnetic dichroism at the 𝑀2,3 edge is smaller than at the 𝐿2,3 edge. In addition the 𝑀2 and 𝑀3 spectra are strongly overlapped, and the 𝑀3 spectrum contributes to some extent to the structure of the total 𝑀2,3 spectrum in the region of the 𝑀2 edge. To decompose a corresponding experimental 𝑀2,3 spectrum into its 𝑀2 and 𝑀3 parts will therefore be quite difficult in general.

In Figure 9 the experimentally measured Fe 𝑀2,3 XMCD spectrum [151] in Fe3O4 is compared to the theoretical one calculated within the LSDA+𝑈 approach [191]. In Fe3O4 the magnetic moments within the 𝐴 and the 𝐵 sublattices are ferromagnetically aligned while the two sublattices are antiferromagnetic with respect to each other. The XMCD spectra are positive at the 𝑀3 and negative at the 𝑀2 edge at the tetrahedral 𝐴 sites and vice versa for the octahedral 𝐵 ones. The interpretation of the experimental Fe 𝑀2,3 XMCD spectrum is very difficult without a knowledge of the band structure and corresponding transition matrix elements because this spectrum is a superposition of six 𝑀2,3 spectra (from Fe𝐴3+, Fe𝐵2+, and Fe𝐵3+ sites) appearing simultaneously in a rather small energy range.

Figure 9: Fe 𝑀2,3 XMCD spectra in Fe3O4 calculated with the LSDA+𝑈 approximation [191] in comparison with the experimental data (circles) [151].
3.2. Low-Temperature Monoclinic Fe3O4
3.2.1. Crystal Structure

The low-temperature structure was shown to have a 2𝑎×2𝑎×2𝑎 supercell with space group 𝐶𝑐 from X-ray and neutron diffraction [111, 179]. However, recent structural refinement (at 90 K) was only possible in the centric monoclinic space group 𝑃2/𝑐 with 𝑎/2×𝑎/2×2𝑎 of the cubic spinel subcell and eight formula units in the primitive unit cell [111, 179]. Since the refinement for the 𝑃2/𝑐 space group was found to be unstable, additional 𝑃𝑚𝑐𝑎 orthorhombic symmetry constraints were also applied. This is equivalent to averaging the true superstructure over the additional symmetry operators; that is, each 𝐵 site in the 𝑃2/𝑐 unit cell is averaged over four nonequivalent subsites in the large 2𝑎×2𝑎×2𝑎𝐶𝑐 supercell. Note, however, that such an approximation is robust in the sense of smallness of any distortions from the 𝑃2/𝑐 subcell to the 𝐶𝑐 monoclinic cell (according to [179] these are of ~0.01 Å). Previous structure refinement below 𝑇𝑉 obtained by Iizumi et al. resulted in a 𝑎/2×𝑎/2×2𝑎 subcell of the 𝐶𝑐 unit cell and imposed orthorhombic symmetry constraints on the atomic positions [196]. In particular, a refinement based on an approximation of the true crystal structure by a centric space group 𝑃𝑚𝑐𝑎 or polar 𝑃𝑚𝑐21 was proposed. But a charge-ordered arrangement has not been identified in this refinement, although large atomic displacements of Fe and O atoms were found. This is in strong qualitative contrast to the recent structure refinement proposed by Wright et al. where clear evidence of CO below the transition has been found [111, 179].

According to the refinement the octahedral Fe𝐵 sites are split into two groups with different values of the averaged Fe–O bond distances, with 𝐵2 and 𝐵3 sites being significantly smaller than 𝐵1 and 𝐵4 (𝐵1𝐵4 are crystallographically independent Fe𝐵 sites according to the notation in [111, 179]). A different averaged Fe–O bond distance is a sensitive experimental indicator of the cation charge state. Quantitative analysis of the valence state of both Fe𝐵 groups using the bond-valence-sum (BVS) method shows that the octahedral Fe𝐵 sites fall into two clear groups with respect to the estimated value of valence. The result is a charge disproportion of 0.2 𝑒 between large (𝐵1 and 𝐵4) and small (𝐵2 and 𝐵3) sites (which has been referred as the class I CO model). Another possible class of CO arises from the symmetry-averaging orthorhombic constraint. There are 32 charge-ordered models which are referred to as class II CO because large (𝐵1 and 𝐵4) and small (𝐵2 and 𝐵3) sites could be averaged over (3Fe2++Fe3+) and (Fe2++3Fe2+) subsites, respectively. The symmetry averaging results in decrease of the more pronounced charge separation of 0.4𝑒 in the full 𝐶𝑐 superstructure (class II CO) down to 0.2𝑒 in the 𝑃2/𝑐 subcell. The Anderson criterion is not satisfied by any of the class I or class II CO models. This is remarkable because the Anderson criterion has been widely used in many CO models [164, 173]. However, class II, as was shown from electrostatic repulsion energy estimations, appears to be more plausible than the class I arrangement.

Recently this interpretation of the refined crystal structure has been found to be controversial. The lack of atomic long-range CO and, as a result, an intermediate valence regime below the Verwey transition were proposed [112, 180]. It is argued that the difference of the average Fe–O distances between compressed and expanded FeO6 octahedra, which could be considered as a maximum limit of charge disproportionation, has the same order as the total sensitivity (including experimental errors) of the bond-valence-sum method. This remarkable controversy shows that the understanding of the system is far from satisfactory.

3.2.2. Band Structure

In order to account for the strong electronic correlations in the Fe 3𝑑 shell, at least on the static Hartree-Fock level, the authors of [155] calculated the electronic structure of the LT phase of Fe3O4 using the LSDA+𝑈 method. The value of the 𝑈 parameter for Fe cations estimated using different experimental and theoretical technics lies in the range of 4.5–6 eV [154, 173, 197]. A reasonably good agreement of the calculated gap value of 0.18 eV with the experimental value [141] of 0.14 eV at 10 K was obtained using the 𝑈 value of 5 eV. Note, however, that the charge and orbital order derived from the LSDA+𝑈 calculations does not depend on the exact 𝑈 value when it is varied within the above-mentioned limits. The value of the Hund’s coupling 𝐽=1 eV was estimated from constrained LSDA calculations [27]. In the following all results presented in the paper were obtained using a 𝑈 value of 5 eV.

Figure 10 shows the LSDA+𝑈 band structure and the total DOS [155] calculated self-consistently for the low-temperature structure of Fe3O4 using the Coulomb interaction parameter 𝑈=5 eV and exchange coupling 𝐽=1 eV. The corresponding partial Fe𝐵3𝑑 DOSs are shown in Figure 11. The LSDA+𝑈 calculations give results qualitatively distinct from those of the LSDA. An indirect energy gap of 0.18 eV opens in the minority spin channel between M and Γ symmetry points. One of the minority spin 𝑡2𝑔 states of Fe𝐵1 and Fe𝐵4 ions becomes occupied while the Fe𝐵2 and Fe𝐵3𝑡2𝑔 states are pushed above the chemical potential. Although, as will be discussed below, the calculated disproportion of Fe𝐵3𝑑 charges is significantly less than 1, in the following we use the notations Fe2+ and Fe3+ for Fe𝐵1,𝐵4 and Fe𝐵2,𝐵4 cations, respectively, having in mind the difference of their 𝑡2𝑔 occupations. The top of the valence band is formed by the occupied 𝑡2𝑔 states of 𝐵1 and 𝐵4Fe2+ cations. The bottom of the conduction band is formed predominantly by the empty 𝑡2𝑔 states of 𝐵2 and 𝐵3Fe3+ cations. The remaining unoccupied 𝑡2𝑔 states of 𝐵1 and 𝐵4Fe2+ cations are pushed by the strong Coulomb repulsion to energies above 2.5 eV. Majority-spin 3𝑑Fe𝐵 states are shifted below O 2𝑝 states, which form a wide band in the energy interval between −7 and −2 eV. This is in strong contrast with the uniform half-metallic solution obtained by the LSDA.

Figure 10: Total DOS and band structure of the 𝑃2/𝑐 phase of Fe3O4 self-consistently obtained by the LSDA+𝑈 with 𝑈=5 eV and 𝐽=1 eV [155]. The Fermi level is denoted by the horizontal line and is taken as the zero of energy. A bandgap of 0.18 eV opens between M and Γ symmetry points. The energy bands predominantly originating from the Fe𝐵3+𝑡2𝑔 and 𝑒𝑔 states are shown in dark gray (red) and light gray (green) colors, respectively, whereas the gray (blue) color corresponds to Fe𝐵2+𝑡2𝑔 bands. For the majority spin an energy gap of ~2 eV opens between Fe𝐵2+𝑒𝑔 and Fe𝐴 bands shown in dark gray (red) and light gray (green) colors, respectively. The corresponding contributions to the total DOS are shown (a).
Figure 11: Partial DOS obtained from the LSDA+𝑈 calculations with 𝑈=5 eV and 𝐽=1 eV for the 𝑃2/𝑐 phase of Fe3O4 [155]. The Fermi level shown by dotted lines. A charge-ordered insulating solution is obtained. Fe 3𝑑 minority states corresponding to 𝐵1 and 𝐵4 sites are occupied (Fe2+) and located just below the Fermi level, whereas 𝐵2 and 𝐵3 are empty (Fe3+). The charge difference between 2+ and 3+ Fe𝐵 cations is found to be 0.23𝑒.

Bands corresponding to the Fe𝐴3+ cations are fully occupied (empty) for minority- (majority-) spin states, respectively, and already in the LSDA do not participate in the formation of bands near the Fermi level. The LSDA+𝑈 method does not strongly affect these bands, which lie in the energy interval of −6 eV below and 1-2 eV above the Fermi level.

3.2.3. Charge Ordering

The obtained solution for CO of Fe2+ and Fe3+ cations on the 𝐵 sublattice is described by a dominant [001]𝑐 charge (and spin) density wave, which originates from alternating chains of Fe2+ ions on octahedral 𝐵1 sites and Fe3+ ions on 𝐵2 sites (see Table 1 and Figure 2 in [181]). A secondary [001/2]𝑐 modulation in the phase of CO, which is formed by the chain of alternately “occupied” Fe2+ ions on the 𝐵4 sites and “empty” Fe3+ ions on 𝐵3 sites, was found. This is consistent with a [001] nesting vector instability at the Fermi surface in the Fe𝐵 minority electron states which has been recently revealed by the LSDA calculations for the cubic phase [198]. The calculated CO scheme coincides with the class I CO model proposed by Wright et al. [111, 179]. All the tetrahedra formed by Fe𝐵 cations have either a 3 : 1 or 1 : 3 ratio of Fe2+ and Fe3+ ions. Thus, the LSDA+𝑈 calculations confirm that the Anderson criterion is not satisfied in the LT phase. However, it should be pointed out that the Anderson criterion was introduced under the assumption of equal interatomic distances within each tetrahedron, whereas in the distorted LT structure the iron-iron distances vary from 2.86 to 3.05 Å. The same CO pattern has been recently confirmed by other LSDA+𝑈 calculations [182].

An analysis of the 3𝑑 minority occupation matrices of Fe𝐵 cations confirms very effective charge disproportion within the Fe𝐵𝑡2𝑔 minority-spin subshell. In particular, one of the 𝑡2𝑔 states of Fe2+𝐵1 and Fe2+𝐵4 cations is almost completely filled with the occupation 𝑛0.8. On the other hand, the other two 𝑡2𝑔 orbitals of the Fe𝐵2+ cations have significantly smaller population of about 0.04. The occupation numbers of 𝑡2𝑔 orbitals for Fe3+𝐵2 and Fe3+𝐵3 cations do not exceed 0.1–0.17, which gives a value of about 0.7 for the largest difference of the populations of Fe𝐵2+ and Fe𝐵3+𝑡2𝑔 states. The occupation numbers of the minority-spin Fe𝐵3𝑑 orbitals and the net occupations of the 𝑡2𝑔 and 𝑒𝑔 states are given in the last two columns of Table 1.

Table 1: 3𝑑 orbital contribution to the formation of Fe𝐵 minority spin states with occupancy 𝑛 evaluated by diagonalization of the occupation matrix [155]. Although one of the 𝑡2𝑔 states of 𝐵1 and 𝐵4 sites is almost occupied with 𝑛0.7𝑒 the 𝑡2𝑔 minority spin occupancies of 𝐵2 and 𝐵3Fe3+ cations are less than 0.1𝑒. The occupied 𝑡2𝑔 states of 𝐵1 and 𝐵4Fe2+ cations are predominantly of 𝑑𝑥𝑧±𝑑𝑦𝑧 and 𝑑𝑥2𝑦2 characters, respectively. The sum of 𝑡2𝑔 (𝑒𝑔) occupations is given in the last column.

The change of the 𝑡2𝑔 occupations caused by the charge ordering is very effectively screened by the rearrangement of the other Fe electrons. A significant contribution to the screening charge is provided by Fe𝐵𝑒𝑔 states. Although the bands originating from these states are located well above the energy gap, the minority spin 𝑒𝑔 orbitals form relatively strong 𝜎 bonds with 2𝑝 states of the oxygen octahedron and, as a result, give an appreciable contribution to the occupied part of the valence band. The energy of Fe𝐵3+𝑒𝑔 states is lower than the energy of corresponding Fe𝐵2+ states, and the former give a significantly larger contribution to the part of the valence band formed mainly by O 2𝑝 states. Because of the stronger covalency of the Fe𝐵3+𝑒𝑔–O 𝑝 bonds, the net occupation of Fe𝐵3+𝑒𝑔 states becomes 0.25𝑒 larger (see the last column of Table 1). The resulting 3𝑑 charge difference (0.23) and disproportionation of the total electron charges inside the atomic spheres of Fe𝐵2+ and Fe𝐵3+ ions (0.24) are in reasonably good agreement with the value of 0.2 estimated from a BVS analysis of the 𝑃2/𝑐 structure [111, 179]. The above-mentioned screening of the changes in the Fe𝐵𝑡2𝑔 minority occupations reduces the energy loss due to the development of charge order incompatible with the Anderson criterion in the LT phase of Fe3O4.

Hence, due to the strong screening effects, the order parameter defined as the difference of the net 3𝑑 charges of Fe𝐵 cations does not provide conclusive evidence for CO. This explains why the BVS analysis does not give a convincing proof of CO existence. Apparently, a well-defined order parameter is the difference of the occupations of the 𝑡2𝑔 minority-spin states for Fe𝐵3+ and Fe𝐵2+ cations which amounts to 70% of the ideal ionic CO model and clearly indicates the existence of a charge-ordered ground state below the Verwey transition.

The LSDA+𝑈 calculations were also performed for the assumption of Verwey charge order in the 𝑃2/𝑐 structure [155]. However, instead of the assumed Verwey CO, the same self-consistent solution as the one described above was found. Therefore, the Verwey CO model is unstable in the distorted 𝑃2/𝑐 structure. It is well known that with increasing 𝑈 value localization is effectively increased. Remarkably, even for a 𝑈 value increased up to 7-8 eV no Verwey-like CO pattern was found self-consistently in the distorted 𝑃2/𝑐 structure. On the contrary, the LSDA+𝑈 calculations performed for an undistorted 𝑃2/𝑐 supercell of the 𝐹𝑑3𝑚 structure result in an insulating CO solution which is compatible with the Verwey CO model. Altogether this implies that the Verwey CO model is unstable under a structure distortion from the high-symmetry cubic into the low-symmetry 𝑃2/𝑐 phase.

Also authors performed LSDA+𝑈 calculations [155] with the same 𝑈 and 𝐽 parameters (5 and 1 eV, resp.) for the assumption of one of the 32 class II CO models within 𝐶𝑐 supercell of 𝑃2/𝑐, which is shown in Figure 2 in [179]. But it was found that this kind of CO is unstable and the self-consistent solution coincides with the one found previously for the 𝑃2/𝑐 structure.

Comparing the LSDA+𝑈 results for the undistorted and distorted 𝑃2/𝑐 unit cells, we can conclude that the charge-ordering pattern of Fe𝐵2+ and Fe𝐵3+ cations in the LT phase of Fe3O4, derived from the BVS analysis in [179] and confirmed by our study, is mainly forced by the local distortions of the crystal structure. The results of [155] consistently indicate the importance of the small amplitude of atomic displacements (almost of 0.07 Å) recently resolved by X-ray and neutron powder diffraction [111, 179]. The additional displacements leading to the 𝐶𝑐 supercell were estimated to be of ~0.01 Å but have not been fully resolved so far. They also may be important for full understanding of the CO in Fe3O4. In particular, in the 𝑃2/𝑐 subcell the true atomic positions are averaged over the corresponding number of subsites in the 𝐶𝑐 cell. Therefore, the actual arrangement of the locally Fe𝐵O6 octahedra in the true 𝐶𝑐 structure can be more complex, probably resulting in a more complicated charge and/or orbital order for the LT structure. The calculations in [155] indicate that the competition of the “elastic” and electrostatic energy contributions in the total energy appears to be responsible for the CO, which is realized in the LT structure of Fe3O4 [181]. Because of this, the Verwey CO model, which has the lowest electrostatic but significant “elastic” energy contribution in the total energy, becomes less favorable than other arrangements.

3.2.4. Orbital Ordering

The self-consistent solution obtained by the LSDA+𝑈 is not only charge but also orbitally ordered. Table 1 clearly presents which presents the contribution of 3𝑑 cubic harmonics to the formation of Fe𝐵 minority-spin states with an occupancy 𝑛 (next to last column in Table 1) evaluated by diagonalization of the corresponding occupation matrix self-consistently obtained by the LSDA+𝑈 [155].

As shown in the table the most occupied Fe2+3𝑑 minority orbitals are centered on the 𝐵1𝑎, 𝐵1𝑏, and 𝐵4 iron sites and have 𝑑𝑥𝑧𝑑𝑦𝑧, 𝑑𝑥𝑧+𝑑𝑦𝑧, and 𝑑𝑥2𝑦2 characters, respectively. Remarkably, the occupied 𝑡2𝑔 orbitals of Fe𝐵 cations are almost orthogonal to each other; that is, their relative orientation corresponds to an anti-ferro-orbital order. Since all Fe𝐵 cations are ferromagnetically coupled the obtained orbital order conforms with the anti-ferro-orbital ferromagnetic state, which is the ground state of the degenerate Hubbard model according to the Kugel-Khomskii theory [199, 200]. This orbital order is consistent with the corresponding distortions of FeO6 octahedra. In particular, using simple considerations which take into account only the change of the Fe–O bond lengths and neglect the bending of the bonds, it was previously concluded that the calculated orbital order is mainly determined by the distortions of oxygen octahedra surrounding Fe𝐵 sites [183].

Also this simple analysis shows a remarkable difference between Fe2+𝐵1 and Fe2+𝐵4 cations; namely, the average Fe𝐵1𝑎–O distance in the plane of occupied 𝑑𝑥𝑧𝑑𝑦𝑧 orbital is 2.087 Å, whereas in the planes of two other 𝑡2𝑔 orbitals they are only 2.063 and 2.067 Å. This difference between the average cation-anion distance in the planes of occupied and unoccupied orbitals is remarkably larger for Fe𝐵1𝑎,𝑏 (more than 0.02 Å), although for Fe𝐵4 they are 2.074 and 2.067 Å for occupied 𝑑𝑥2𝑦2 and unoccupied 𝑑𝑥𝑧±𝑑𝑦𝑧 orbitals, respectively, which gives a difference of only 0.007 Å. This small difference can be changed by applying a uniaxial stress to the 𝑃2/𝑐 unit cell resulting in modification of the electronic properties [201]. In particular, a few percent of magnitude elongation of the 𝑃2/𝑐 unit cell along the 𝑐-axis with simultaneous (in order to preserve the same unit cell volume) compression in the 𝑎𝑏 plane gives rise to orbital-order crossover on the Fe𝐵4 site from a 𝑑𝑥2𝑦2 to a 𝑑𝑥𝑧 occupied orbital. At the same time the charge order and occupied orbitals on the Fe2+𝐵1𝑎,𝑏 sites remain the same. The pressure-induced spatial reorientation of the occupied Fe𝐵4𝑡2𝑔 orbital was proved by the LSDA+𝑈 calculations for the strained 𝑃2/𝑐 unit cell. Note, however, that these rough estimations do not take into account the elastic anisotropy in Fe3O4. Moreover, the analysis was performed for the “averaged” 𝑃2/𝑐 structure. However, they provide insight into the orbital-ordering phenomena behind the Verwey transition in magnetite as well as the problem of an external parameter-controlled electron state (e.g., orbital ordering) in solids [202].

3.2.5. Magnetic Moments

The strong variation of the occupancies of the minority-spin Fe𝐵𝑡2𝑔 states leads to a pronounced modulation of the spin magnetic moments on the 𝐵 sublattice. While the total moment per formula unit remains at 4 𝜇𝐵, the magnetic moments of the Fe2+𝐵1 (3.50 𝜇𝐵) and 𝐵4 (3.48 𝜇𝐵) cations become appreciably smaller than Fe𝐵2 (3.94 𝜇𝐵) and Fe𝐵3 (3.81 𝜇𝐵) moments. The [001]𝑐 charge and spin modulation on the 𝐵 sublattice is accompanied by formation of a weak spin modulation on the oxygen ions caused by different strengths of the hybridization of O 2𝑝 states with the minority-spin 3𝑑 states of Fe𝐵2+ and Fe𝐵3+ ions. In particular, the oxygen magnetic moment reaches its maximal value of ~0.1 𝜇𝐵 for O3 and O4 ions, which lie in the plane of Fe3+𝐵2 cations. It substantially decreases for other oxygen ions and approaches minimum for O1 and O2 lying in the plane of Fe2+𝐵1 cations (~0.04 𝜇𝐵).

Recently, an anomalously large value of the Fe𝐵 orbital magnetic moment reaching 0.33 𝜇𝐵 has been deduced by applying sum rules to experimental 𝐿2,3 X-ray magnetic circular dichroism spectra of Fe3O4 [203]. In addition, the unquenched Fe𝐵 orbital moment was also reported to be confirmed by the LSDA+𝑈 calculations. Later, however, this experimental finding was questioned by Goering et al. [204]. The average orbital moments between 0.001𝜇𝐵 and 0.06𝜇𝐵 were found from X-ray magnetic circular dichroism sum rules depending on the integration range. From the spin-polarized relativistic LSDA+𝑈 calculations [155] for the LT structure it was estimated orbital moments of 0.19𝜇𝐵 and 0.014𝜇B for Fe𝐵1 and Fe𝐵2 ions, respectively. Somewhat larger values of 0.039𝜇𝐵 and 0.22𝜇B were calculated for Fe𝐵3 and Fe𝐵4 cations, respectively. Taking into account the negative Fe𝐴 orbital moment of 0.021𝜇𝐵, this gives the value of 0.07𝜇𝐵 for the average orbital moment. Thus, in agreement with the previous theoretical results of [10] and XMCD sum rule data of [204], the calculations give the value of Fe orbital moment of 0.07𝜇𝐵 which is much smaller than reported in [203].

3.2.6. Exchange Coupling Constants

The calculations of the exchange interaction parameters 𝐽𝑖𝑗 were also performed [155] via the variation of the ground-state energy with respect to the magnetic moment rotation angle [205]. The exchange coupling parameter 𝐽𝑖𝑗 represents the effective pair exchange interaction between the 𝑖th and 𝑗th Fe atoms with effective Heisenberg Hamiltonian 𝐻=𝑖>𝑗𝐽𝑖𝑗𝐒𝐢𝐒𝑗. Here, 𝐒𝑖 and 𝐒𝑗 are the spins at sites 𝑖 and 𝑗 (5/2 and 2 for Fe3+ and Fe2+ cations, resp.). Positive (negative) values of 𝐽𝑖𝑗 correspond to the ferromagnetic (antiferromagnetic) coupling between sites. As shown in Table 2 the exchange couplings between 𝐴 and 𝐵 iron sublattices are rather large, of about −70 K, and antiferromagnetic. The Fe𝐴Fe𝐴 interactions are weakly antiferromagnetic with the maximal absolute value of 9.3 K (not shown in Table 2). The exchange couplings between the Fe𝐵 sites (|𝐽𝐵𝐵|27.5 K) are substantially smaller than Fe𝐴Fe𝐵 ones and almost all of them are ferromagnetic (see Figure 12). Weak antiferromagnetic couplings with |𝐽𝐵𝐵|11.6 K are also obtained (mainly between the sites with the same 2+ or 3+ valence state, shown by the thin (blue) lines in Figure 12). The spatial representation of these exchange couplings is presented in Figure 12. Other couplings that are not shown in Table 2 are weaker than 10 K.

Table 2: Exchange couplings 𝐽𝑖𝑗 (all with |𝐽𝑖𝑗|>10 K) are presented [155]. The values are given in Kelvin. The spatial representation of the Fe𝐵Fe𝐵 exchange couplings is schematically shown in Figure 12. 𝐽𝑖𝑗 were calculated between the sublattices formed by the translations of the following Fe sites: Fe𝐴1 (1/4,0.0034,0.06366), Fe𝐴2 (1/4,−0.4938,0.18867), Fe𝐵1𝑎 (0,1/2,0), Fe𝐵1𝑏 (1/2,1/2,0), Fe𝐵2𝑎 (0,0.0096,1/4), Fe𝐵2𝑏 (1/2,0.0096,1/4), Fe𝐵3 (−1/4,0.2659,0.1198), Fe𝐵3 (1/4,−0.2659,−0.1198), Fe𝐵3 (1/4,0.2659,0.3801), Fe𝐵4 (1/4,0.2479,−0.1234), Fe𝐵4 (−1/4,−0.2479,0.1234), and Fe𝐵4 (1/4,−0.2479,0.3765).
Figure 12: Sketch of the spatial arrangement of exchange interaction parameters between the octahedral Fe𝐵 sites [155]. Orbitals approximate the occupied 3𝑑 minority orbitals of Fe𝐵2+ cations. Fe𝐵3+ cations are shown by large (blue) spheres. Oxygen atoms are shown by small (green) spheres. Ferromagnetic couplings between Fe𝐵 cations are shown by the thick (red) lines, whereas antiferromagnetic exchanges depicted by the thin (blue) lines.

Experimental estimation of the exchange couplings in Fe3O4 was first performed by Néel on the basis of the two-sublattice collinear model [188]. From analysis of the temperature behaviour of the saturation magnetization and paramagnetic susceptibility, he obtained 𝐽𝐴𝐴=17.7, 𝐽𝐴𝐵=23.4, and 𝐽𝐵𝐵=0.5 K, where 𝐴 and 𝐵 refer to the tetrahedral and octahedral Fe sites, respectively. These values are qualitatively in accordance with the results presented in Table 2; namely, as in Neel’s model, the calculations result in strong antiferromagnetic coupling between the 𝐴 and 𝐵 sublattices; 𝐽𝐴𝐴 couplings (not shown in Table 2) are considerably smaller than 𝐽𝐴𝐵; the exchange couplings in the 𝐵 sublattice are weak and almost all of them are ferromagnetic. On the other hand, the small antiferromagnetic Fe3+𝐵2𝑎Fe3+𝐵2𝑏 exchange interaction (see Table 2) is in exact agreement with recent estimations using the two-sublattice model [206]. Three-sublattice model calculations give an overall similar result, except, however, the Fe𝐵2+Fe𝐵2+ exchange coupling, which seems to be overestimated [207].

3.2.7. Summary

The LSDA+𝑈 study of the 𝑃2/𝑐 model of the LT phase of Fe3O4 [155] shows a charge- and orbitally ordered insulator with an energy gap of 0.18 eV. The obtained charge-ordered ground state is described by a dominant [001]𝑐 charge density wave with a minor [001/2]𝑐 modulation on the Fe𝐵 sublattice. A weak [001]𝑐 spin/charge modulation on the oxygen ions was also obtained. The CO coincides with the earlier proposed class I CO [111, 179] and confirms violation of the Anderson criterion [156]. While the screening of the charge disproportion is so effective that the total 3𝑑 charge disproportion is rather small (0.23), the charge order is well pronounced with an order parameter defined as the difference of 𝑡2𝑔 occupancies of 2+ and 3+ Fe𝐵 cations (0.7). This agrees well with the result of BVS analysis for a monoclinic structure (0.2). The orbital order is in agreement with the Kugel-Khomskii theory [199] and corresponds to the local distortions of oxygen octahedra surrounding Fe𝐵 sites.

Calculations of the effective exchange coupling constants between Fe spin magnetic moments show that the dominating interaction is an antiferromagnetic coupling between Fe𝐴 and Fe𝐵 moments. The coupling between Fe𝐵2+ and Fe𝐵3+ moments is found to be weaker and ferromagnetic.

3.3. Fe2OBO3

Iron borate (Fe2OBO3) is a semivalent oxide. It belongs to the homometallic warwickite family with formal chemical formula MMOBO3, where M and M are, respectively, a divalent and trivalent metal ions. Surprisingly, the homometallic (M=M) warwickites are known only for Fe [208] and Mn [209]. In both compounds the metal has octahedral coordination. These octahedra share edges to form ribbons of four infinite along crystallographic 𝑎 direction chains of octahedra linked by corner sharing and the trigonal BO3 groups (see Figure 13).

Figure 13: The Fe2OBO3 structure projected on (100) plane (𝑏 vertical, 𝑐 horizontal). Two structurally distinct Fe(1)O6 and Fe(2)O6 octahedra shown by light and dark sharing, respectively. Plus and minus signs indicate the relative orientation of the moments within each Fe(1) and Fe(2) chains in the magnetically ordered phase.

There are two crystallographically inequivalent sites of the metal ions Fe(1) and Fe(2). Fe2OBO3 is 𝐿-type ferrimagnetic with drastically smaller in comparison with Fe3O4 critical temperature of 𝑇𝑐155 K, the Fe(1) magnetic moments being aligned antiparallel to the Fe(2) moments. It is almost antiferromagnetic, but a small ferrimagnetic moment of 0.03𝜇𝐵 per Fe atom in a 0.05 T field was found [208]. At room temperature Fe2OBO3 is a semiconductor with a thermoactivated conductivity below 𝑒𝐸𝑎/𝑘𝑇 with 𝐸𝑎0.35 eV [208]. Upon farther heating a broad semiconductor-to-semiconductor transition occurs at 𝑇co317 K, where resistivity drops down by a factor of 3, and, as a result, a small decrease of the activated energy up to 𝐸𝑎0.31 eV above 350 K is observed [208]. The 317 K transition is assigned to charge ordering of 2+ and 3+ Fe cations on Fe(1) and Fe(2) sites, and accompanied by a structural transition from monoclinic 𝑃21/𝑐 to orthorhombic 𝑃𝑚𝑐𝑛 symmetry with increasing temperature. This structural transition is attributed by modification of the 𝛽 angle from 𝛽=90.220(1) at 3 K to 𝛽=90 at 337 K [208]. The change in conductivity and structure are small. But the 57Fe Mössbauer spectra at around 317 K clearly result in the charge localization at the transition with an equal distribution of Fe2+ and Fe3+ cations over the two structurally distinct Fe(1) and Fe(2) sites with formal chemical formula Fe(1)2+0.5Fe(1)3+0.5Fe(2)2+0.5Fe(2)3+0.5OBO3 [208, 210]. Although, there are two types of distorted FeO6 octahedra with Fe–O bond length varying between 1.92 and 2.23 Å for 3 K, the average Fe(1)–O and Fe(2)–O distances are 2.085 and 2.082 Å, respectively, that is, equal within experimental errors [208]. Such a small difference results in the extremely small value of deviation (≤0.01) from the average 2.5+ value of valence of Fe cations estimated by the bond valence sum method. While an electronic transition between charge-ordered and charge-disordered states occurs at around 317 K, as evidenced by the Mössbauer spectroscopy and resistivity measurements, no long-range Fe2+/Fe3+ ordering is directly observed by X-ray, neutron, or electron diffraction. Thus, a long-range charge ordering such as the simple alternating scheme proposed in [208] destroys the mirror symmetry, which leads to a tilting of the Fe ribbons, consistent with the observed enlargement of the 𝛽 angle below the transition. However, there is no observation of the increasing of 𝑎-axis periodicity (it should increase by a factor of two or another integer factor below 𝑇co). Thus, below the transition, a charge ordering is not implicit in the atom coordinates, although it is indirectly evidenced by other experiments. This ambiguity is resolved in our electronic structure study, which reveals an arrangement of Fe2+ and Fe3+ cations alternately ordered within the chains along the 𝑎 direction.

Theoretical investigation of the electronic structure and magnetic properties of Fe2OBO3 in the low-temperature 𝑃21/𝑐 structure was reported in [211]. An order parameter, defined as the difference between 𝑡2𝑔 minority-spin occupancies of Fe(1)2+ and Fe(1)3+ as well as the difference between 𝑡2𝑔 majority-spin occupancies of Fe(2)2+ and Fe(2)3+ cations was propose. This order parameter was found to be quite large, although the total 3𝑑 charge difference between 2+ and 3+ cations is small.

The band structure calculations have been carried out for the low-temperature monoclinic structure of Fe2OBO3. The corresponding 𝑃21/𝑐 unit cell contains four Fe2OBO3 formula units. The LSDA calculations give only a metallic ferrimagnetic solution without charge separation where partially filled bands at the Fermi level originate from the 𝑡2𝑔 orbitals of Fe cations (see Figure 14).

Figure 14: Total density of states (DOS) obtained from the LSDA calculations for the low-temperature 𝑃21/𝑐 phase of Fe2OBO3 [211]. The top of the valence band is shown by dotted line.

The lower part of the valence band (below −3.5 eV) is mainly formed by O 2𝑝 states with a bonding hybridization with Fe 3𝑑 states. Fe 3𝑑 states give predominant contribution to the bands at −3.5 eV below and up to 2.5 eV above the Fermi level. The exchange splitting between the spin-up and spin-down Fe 3𝑑 states is roughly 3 eV which results in a net magnetic moment of 0.31 𝜇𝐵 per formula unit. Additionally, the fivefold 3𝑑 levels are split by the crystal field into 𝑡2𝑔 and 𝑒𝑔 subbands. The oxygen octahedra in Fe2OBO3 are strongly distorted and the local symmetry of Fe sites is, of course, lower than cubic. Nevertheless, the cubic component of the ligand field, which is determined by the relative strength of Fe𝑑O𝑝 hybridization of 𝜋- and 𝜎-type, remains dominant, whereas the splitting within “𝑡2𝑔” and “𝑒𝑔” subbands is smaller than the corresponding band-width. This allows one to label the corresponding states as 𝑡2𝑔 and 𝑒𝑔. The crystal-field splitting is roughly 2 eV, which is less than the exchange splitting. This is consistent with the high-spin state of the Fe cations. The symmetry inequivalence of Fe(1) and Fe(2) sites leads to an inexact cancellation of magnetic moments and results in a small ferrimagnetic moment of 0.31𝜇𝐵 per formula unit.

Fe(1) and Fe(2) 𝑡2𝑔 and 𝑒𝑔 states with the opposite spin projections share nearly the same energy intervals. Thus, Fe 3𝑑 states between −3.5 and −2.0 eV originate predominantly from majority-spin Fe(1) and minority-spin Fe(2) 𝑡2𝑔 states whereas the states between −2.0 and −0.5 eV are mainly of 𝑒𝑔 character. Partially occupied bands crossing the Fermi level are formed by minority spin Fe(1) and majority-spin Fe(2) 𝑡2𝑔 states. The nominal occupation of these bands is 1/6. In the majority-spin channel, however, the Fe(2) 𝑡2𝑔 state, which is oriented in the plane perpendicular to the shortest Fe(2)–O bond, forms quasi-one-dimensional bands with a strong dispersion along the 𝑎 direction. The one-dimensional character of the dispersion is determined by the existence of only two nearest neighbours of the same kind around each Fe(2) ion. The other two Fe(2) 𝑡2𝑔 states are shifted to higher energy and the corresponding bands are completely unoccupied. As a result, the majority-spin bands crossing the Fermi level turn out to be half filled.

An Fe(1) ion, in contrast to Fe(2) one, has four Fe(1) neighbours at close distances. As a result of the hybridisation between Fe(1) 𝑡2𝑔 states the situation in the minority-spin channel is more complicated. Twelve 𝑡2𝑔 bands are split into three groups of 4 bands each. The Fermi level is crossed by lowest bands which show a rather strong dispersion along 𝑎 but with a two times smaller period.

It should be noted that in contrast to experimental data [212] LSDA predicts Fe2OBO3 to be metallic with substantial magnetic moment per unit cell. Apparently, the electron-electron correlations, mainly in the 3𝑑 shell of Fe cations, play a significant role.

The LSDA+𝑈 calculations have been performed for the 𝑃21/𝑐 unit cell as well as for double (2𝑎×𝑏×𝑐) and triple (3𝑎×𝑏×𝑐) 𝑃21/𝑐 supercells of Fe2OBO3 (without putting in any local displacements of oxygen atoms around Fe2+/Fe3+ sites). Thus, for the 2𝑎×𝑏×𝑐 CO pattern proposed in [208] using the classical value of Coulomb and exchange interaction parameters for Fe 5 eV and 1 eV, respectively, a charge-ordered insulator with an energy gap of 0.13 eV was found. This is in a strong contrast with metallic solution without CO obtained by the LSDA. This is a notable result because a CO is not implicit in the atom coordinates, and it shows that LSDA+𝑈 calculations can assist experiments in revealing CO arrangements. To obtain a reasonably good agreement of the calculated gap of 0.39 eV with experimental value of 0.35 eV, the 𝑈 value has to be increased up to 5.5 eV (see Figures 15 and 16). It does not exceed 10% of the 𝑈 value, which is in an accuracy of the 𝑈 calculation. Note, however, that the CO obtained by LSDA+𝑈 within 2𝑎×𝑏×𝑐 supercell does not depend on the 𝑈 value of 5–5.5 eV. Here and in the following all results are presented for double along 𝑎-direction 𝑃21/𝑐 supercell of Fe2OBO3.

Figure 15: The total DOS obtained from LSDA+𝑈 calculations with 𝑈=5.5 eV and 𝐽=1 eV for the low-temperature 𝑃21/𝑐 phase of Fe2OBO3 [211]. The top of the valence band is shown by dotted lines.
Figure 16: The partial DOSs for different Fe cations are shown. The gap is opened between Fe(2)2+ and Fe(2)3+ for majority spin and Fe(1)2+ and Fe(1)3+ cations for minority spin states. The gap value of 0.39 eV was obtained by LSDA+𝑈 with 𝑈=5.5 eV and 𝐽=1 eV [211]. The Fermi level is shown by dotted line.

After self-consistency each of two groups of Fe(1) and Fe(2) atoms is split out in two subgroups of 2+ and 3+ Fe cations with equal number of 2+ and 3+ cations. Thus, one of 𝑡2𝑔 majority-/minority-spin states of Fe(2)/Fe(1) atom becomes completely occupied, whereas all the rest of 𝑡2𝑔 states are pushed by strong Coulomb interaction at the energies above 3 eV. The gap is opened between occupied and unoccupied 𝑡2𝑔 states of Fe(1)2+ and Fe(1)3+ for spindown and Fe(2)2+ and Fe(2)3+ for spinup. Majority-spin 3𝑑 states of Fe(1)3+ and minority-spin states of Fe(2)3+ cations are shifted below the O2𝑝 states, which form the band in the energy range of −8 and −2 eV. In contrast to Fe3+ states, the majority-spin Fe(1)2+ and minority-spin Fe(2)2+3𝑑 states form the broad bands between −8 and −1 eV.

The obtained magnetic structure is almost antiferromagnetic (without spin moment per unit cell) with nearly the same spin moment per Fe(1)2+ and Fe(2)2+ as well as per Fe(1)3+ and Fe(2)3+ cations. Using the moment populations in Table 3, the calculated net moment is 0.03𝜇𝐵 per Fe atom, in exact agreement with the experimental value [208].

Table 3: Total and 𝑙-projected charges, magnetic moments, and occupation of the most populated 𝑡2𝑔 orbitals calculated for inequivalent Fe atoms in the low-temperature 𝑃21/𝑐 phase of Fe2OBO3 [211].

The charge order obtained by LSDA+𝑈 in 2𝑎×𝑏×𝑐𝑃21/𝑐 supercell is consistent with observed enlargement of the 𝛽 angle below the transition and coincides with charge-ordering scheme proposed earlier by Attfield et al. [208] It is described by the sloping 2+ and 3+ Fe cation lines alternately stacked along 𝑎 direction and could be considered as a quasi-one-dimensional analog of the Verwey CO model in pyrochlore lattice of Fe3O4. An additional self-consistent LSDA+𝑈 calculations for 𝑃21/𝑐 unit cell as well as for double and triple along 𝑎 direction 𝑃21/𝑐 supercells was performed using the same 𝑈 and 𝐽 values. But only self-consistent solutions with larger value of the total energy or with substantial magnetic moment per unit cell, which contradicts the experimental data, were found. Also it was found that other charge arrangements in 2𝑎×𝑏×𝑐𝑃21/𝑐 supercell are unstable, and the stable one coincides with the CO found previously. Thus, the CO obtained for certain value of 𝑈 and 𝐽 does not depend on the initial charge arrangement. It is not possible to check all possible CO arrangements including more complex CO scenarios, but our results consistently indicate that the obtained CO solution is more favourable than other simple alternatives and is the ground state of Fe2OBO3 in the low-temperature phase.

Although the corresponding total 3𝑑 charges difference (0.34𝑒) and disproportion of the total electron charges inside the atomic spheres of Fe2+ and Fe3+ cations (0.24𝑒) are small, an analysis of occupation matrices of 3𝑑 Fe(1)/Fe(2) minority-/majority-spin states confirms substantial charge disproportionation. Thus, as shown in Table 3, one of the 𝑡2𝑔 states of Fe(1)2+ and Fe(2)2+ cations is almost completely filled with the occupation numbers 𝑛0.9, whereas the remaining two 𝑡2𝑔 orbitals of the Fe2+ cations have significantly smaller population of about 0.1. According to [181] an order parameter was defined as the largest difference between Fe2+ and Fe3+𝑡2𝑔 populations. While, due to strong static “screening” effects, the order parameter introduced as the total 3𝑑 charge difference between 2+ and 3+ Fe cations is ill defined, the well-defined order parameter is the difference of 𝑡2𝑔 occupancies for Fe3+ and Fe2+ cations, which amounts to 80% of ideal ionic CO model and clearly pronounces the existence of CO below the transition. The occupation matrices analysis shows that the change of the 𝑡2𝑔 occupations caused by the charge ordering is very effectively screened by the rearrangement of the other Fe electrons. Thus, significant contribution to the charge screening is provided by Fe 𝑒𝑔 states due to relatively strong 𝜎 bonds with 2𝑝 O states and, as a result, appreciable contribution to the occupied part of the valence band.

The occupied 𝑡2𝑔 states of Fe2+ cations are predominantly of 𝑑𝑥𝑦 character in the local cubic frame (according to that we later mark the orbital as 𝑑𝑥𝑦 orbital). This is illustrated in Figure 17, which shows the angular distribution of the majority and minority spin 3𝑑 electron density of the Fe(2) and Fe(1) cations, respectively. Thus, occupied Fe2+ and unoccupied Fe3+ cations are ordered alternately within the chain which is infinite along 𝑎 direction. The angular distribution of charge density of the Fe(1) and Fe(2) cations, which correspondingly belongs to different Fe ribbons being formed a cross in the Fe2OBO3 structure projected on (100) plane (see Figure 13), is shown in Figure 18.

Figure 17: The angular distribution of the majority and minority spin 3𝑑 electron density of the Fe(2) and Fe(1) cations, respectively, within Fe ribbon [211]. The size of orbital corresponds to its occupancy. Oxygen atoms are shown by small spheres. 𝑥-𝑦-𝑧 coordinate system corresponds to the local cubic frame.
Figure 18: The angular distribution of the majority and minority spin 3𝑑 electron density of the Fe(2) and Fe(1) cations, respectively, from different Fe ribbons [211]. The size of orbital corresponds to its occupancy. The frame of four Fe(1) atoms from the Fe ribbon presented in Figure 17 is shown by dashed lines.

Using the LSDA+𝑈 method the exchange interaction parameters have been calculated via the variation of ground-state energy with respect to the magnetic-moment rotation angle [211]. In Table 4 we have shown the total set of different intraribbon exchange parameters as well as a contribution of different subbands into exchange interactions. The spatial representation of all these exchanges is schematically presented in Figure 19. Surprisingly, only the exchange interaction parameter between Fe(2)2+ and Fe(2)3+ cations is ferromagnetic with relatively small value of 𝐽6=25 K. In contrast, the nearest sites in quasi-one-dimensional Fe(1) chain are coupled antiferromagnetically with noticeably larger exchange absolute value of |𝐽5|=275 K. Furthermore, the exchange parameters between the nearest sites of two Fe(1) chains are relatively strong and antiferromagnetic (see 𝐽2, 𝐽4, and 𝐽8 in Table 4). Therefore, the Fe(1) sublattice is highly frustrated, while the relatively weak frustrations in the Fe(2) sublattice considerably reduce ferromagnetic interaction within Fe(2) chain. Also it is interesting to note that relatively strong ferromagnetic intrachain interaction between 𝑡2𝑔 subbands of Fe(2)2+ and Fe(2)3+ cations (see 𝐽6 in Table 4) is strongly suppressed by the substantial antiferromagnetic 𝑡2𝑔𝑒𝑔 and 𝑒𝑔𝑒𝑔 exchange.

Table 4: Total and partial intraribbon exchange interaction parameters are shown [211]. The values are given in kelvin. The spatial representation of all these exchanges is schematically presented in Figure 19.
Figure 19: The sketch of the arrangement of exchange interaction parameters within the ribbon of iron atoms. Open circles correspond to Fe3+, while Fe2+ cations are noted by the closed circles. The spin moment direction on each Fe site is shown by an arrow.

On the other hand, the interribbon exchange interaction parameters between Fe(1) and Fe(2) atoms are considerably larger. The values of these interactions are shown in Table 5, whereas the spatial representation is schematically presented in Figure 20. Thus, the exchange parameters between Fe(1)3+ and Fe(2)2+ cations are antiferromagnetic with values of 𝐽10=917 K and 𝐽14=827 K (see Table 5). Such an appreciable difference between 𝐽10 and 𝐽14 arise from geometry. Thus, the former corresponds to the exchange interaction between Fe(1) and Fe(2) atoms, which belongs to an edge, whereas the latter corresponds to the diagonal interaction. It seems that such geometrical reason is also responsible for decrease of absolute value of the exchange interactions between Fe(1)2+ and Fe(2)3+ cations from|𝐽12|=837 K to |𝐽15|=586 K (see Table 5). Also it is interesting to note that the exchange interaction between Fe(1)3+ and Fe(2)3+ cations are considerably larger than between Fe(1)2+ and Fe(2)2+ (𝐽11 and 𝐽13, correspondingly). We find that the interribbon exchange interactions play predominant role and determine the whole 𝐿-type ferrimagnetic spin structure below 𝑇𝑐 in contrast with the ferromagnetic intrachain order due to 𝑑5𝑑6 superexchange [208].

Table 5: Total and partial interribbon exchange interaction parameters are shown [211]. The values are given in kelvin. The spatial representation of all these exchanges is schematically presented in Figure 20.
Figure 20: The sketch of the arrangement of interribbon exchange interaction parameters [211]. Open circles correspond to Fe3+, while Fe2+ cations are noted by the closed circles. The spin moment direction on each Fe site is shown by an arrow. Note that 𝐽𝑖 exchange parameters presented here have the same total values as 𝐽𝑖, while subband contributions are different.
3.3.1. Summary

In the LSDA+𝑈 study of the low-temperature 𝑃21/𝑐 phase of Fe2OBO3 [211] was found a charge-ordered insulator with an energy gap of 0.39 eV. While the screening of the charge disproportion is so effective that the total 3𝑑 charge disproportion is rather small (0.34), the charge order is well pronounced with an order parameter defined as a difference of 𝑡2𝑔 occupancies of 2+ and 3+ Fe cations (0.8). The occupied Fe2+ andFe3+ cations are ordered alternately within infinite along 𝑎-axis chains of Fe atoms. This result is remarkable in view of the absence of directly observed CO atomic displacements in the experimental coordinates and demonstrates the utility of the LSDA+𝑈 method as an aide to experimental studies of CO structures. However, the charge order obtained by LSDA+𝑈 is consistent with observed enlargement of the 𝛽 angle and coincides with charge-ordering scheme proposed earlier by Attfield et al. [208]. It seems certain that Fe2OBO3 is charge ordered below 𝑇co, and the absence of the long-range charge ordering from X-ray, neutron, or electron diffraction arises from formation of charge order within small domains, which have been termed “Wigner nanocrystals” [213]. Thus, the superstructure peaks are too weak and broad to be observed against background in diffraction patterns, whereas the observed long-range monoclinic lattice distortion can arise despite a large concentration of defects as these preserve the direction of the monoclinic distortion but do not propagate the coherent doubling of the lattice periodicity. An analysis of the exchange interaction parameters obtained by LSDA+𝑈 method inevitably results in predominance of the interribbon exchange interactions which determine the whole 𝐿-type ferrimagnetic spin structure below 𝑇𝑐, in contrast with the ferromagnetic intrachain order due to 𝑑5𝑑6 superexchange proposed earlier in [208].

4. Titanium and Vanadium Oxides

4.1. Ti4O7

The aforementioned phenomena of sharp metal-insulator transitions associated with pronounced charge and/or orbital ordering are characteristic for the Magnéli phases M𝑛O2𝑛1 (M=Ti,V). These compounds form a homologous series and have been studied recently to understand the differences in crystal structures and electronic properties between the end members MO2 (𝑛) and M2O3 (𝑛=2) [214]. In particular, the metal-insulator transition of VO2 discovered some fifty years ago still is the subject of ongoing controversy and is another “hot topic” in solid-state physics. LSDA calculations have revealed strong influence of the structural degrees of freedom on the electronic properties of VO2 and neighbouring rutile-type dioxides [215217]. In this scenario the characteristic dimerization and antiferroelectric displacement of the metal atoms translate into orbital ordering within the 𝑡2𝑔 states and a Peierls-like singlet formation between neighbouring sites. Recently, this was confirmed by LDA+DMFT calculations, which suggested to regard the transition of VO2 as a correlation-assisted Peierls transition [218].

Ti4O7 titanium oxide is another remarkable member of the Magnéli phases with 𝑛=4 which shows metal-insulator transitions associated with the spatial charge ordering. It is a mixed-valent compound which has an even mixture of 3+ and 4+ Ti cations (Ti23+Ti24+O7), corresponding to an average 3𝑑 occupation of 1/2 electron per Ti site. Electrical resistivity, specific heat, magnetic susceptibility, and X-ray diffraction data reveal two first-order transitions in the temperature range of 130–150 K [219]. At 150 K a metal-semiconductor transition occurs without measureable hysteresis in resistivity and specific heat. It is followed by a semiconductor-semiconductor transition at 130–140 K, which again is characterized by an almost two orders of magnitude abrupt increase in electrical resistivity and has a hysteresis of several degrees [219]. The magnetic susceptibility shows a sharp enhancement when heating through 150 K. However, it is small and temperature independent below this temperature and does not show any anomaly at 140 K.

The crystal structure of Ti4O7 (see Figure 21) can be viewed as rutile-type slabs of infinite extension and four Ti sites thickness, separated by shear planes with a corundum-like atomic arrangement. Below 130 K it crystallizes in a triclinic crystal structure with two formula units per primitive unit cell [220]. Four crystallographically inequivalent Ti sites are found at the centers of distorted oxygen octahedra. They form two types of chains, namely, (a) 1-3-3-1 and (b) 2-4-4-2, which run parallel to the pseudo-rutile 𝑐-axis and are separated by the crystallographic shear planes. Although interatomic distances in the (b) chain are almost uniform (3.01 and 3.07 Å between 4-4 and 2-4 Ti sites, resp.) they are remarkably different for the (a) chain (3.11 and 2.79 Å between 3-3 and 1-3 Ti sites).

Figure 21: The low-temperature crystal structure of Ti4O7. Chains of four Ti sites run parallel to the pseudorutile 𝑐-axis. Red and blue (light and dark on the black and white image) chains of four Ti atoms correspond to the (a) and (b) chains of Ti atoms, respectively. Further gradation of red and blue on light and dark subsets indicates inequivalent Ti sites in (a) and (b) chains.

Accurate determination of the crystal structure allowed to elucidate the nature of the three phases distinguished by the two first-order transitions [220]. In particular, in the metallic phase the average Ti–O bond lengths for crystallographically inequivalent TiO6 octahedra are very similar which results in the average valence state of 3.5+ per each Ti cation. Below 130 K charge has been transferred from the (b) to the (a) chains. In addition, Ti3+ cations in alternate (a) chains are paired to form nonmagnetic metal-metal bonds, whereas in the intermediate phase pairing also persists but its long-range order calls for a fivefold supercell [219]. Thus, the 130–140 K transition is associated with a transition to the phase with a long-range order of Ti3+-Ti3+ pairs, whereas above 150 K 3+ and 4+ Ti cations are disordered. The presence of the Ti3+-Ti3+ pairs strongly differentiates Ti4O7 from Fe3O4 and results in two steep first-order transitions found in the electrical resistivity.

Recent LSDA band structure calculations of both high- and low-temperature phases of Ti4O7 results in significant 𝑡2𝑔 charge separation between crystallographically independent 3+ and 4+ Ti sites in the low-temperature phase, whereas a rather isotropic occupation of the 𝑡2𝑔 states has been found at room temperature [221]. While, in addition, an orbital order at the Ti 𝑑1 chains originating from metal-metal dimerization was found, the LSDA gave only metallic solution with semimetallic-like band overlap instead of the semiconducting gap. This problem is overcome in our work taking into account strong electronic correlations in Ti 3𝑑 shell using the LSDA+𝑈 method.

In [187] the authors investigate the electronic structure of the low-temperature phase using the LSDA+𝑈 approach. The LSDA+𝑈 calculations result in a charge- and orbitally ordered insulator with an energy gap of 0.29 eV, which is in a good agreement with an experimental gap value of 0.25 eV. From the results of [187], an orbital-order parameter was proposed as the difference between 𝑡2𝑔 majority-/minority-spin occupancies of Ti(1)3+/Ti(3)3+ and Ti(2)4+/Ti(4)4+ cations respectively. This order parameter is found to be quite large, although the total 3𝑑 charge difference between 3+ and 4+ cations, remains small. Also it is interesting to note that the total charge separation between 3+ and 4+ Ti cations is completely lost due to efficient screening by the rearrangement of the other Ti electrons. In addition, we find a strong antiferromagnetic coupling of 𝐽1700 K of the local moments within the dimerized Ti3+-Ti3+ pairs, whereas an interpair coupling is only of 40 K. This is in a good agreement with small and temperature-independent magnetic susceptibility in the low-temperature phase of Ti4O7.

4.1.1. LSDA Band Structure

The LSDA band structure calculations [187] for the low-temperature 𝑃1 structure gives a nonmagnetic metallic solution with substantial charge separation between crystallographically independent Ti(1)/Ti(3) and Ti(2)/Ti(4) cations. The lower part of the valence band below −3 eV is predominantly formed by O 2𝑝 states with a bonding hybridization with Ti 3𝑑 states. Crystal -field splitting of the latter is roughly of 2.5 eV. Ti 𝑡2𝑔 states form the group of bands at and up to 2 eV above the Fermi energy, whereas Ti 𝑒𝑔 states give a predominant contribution to the bands between 2.5 and 4.5 eV. Within the 𝑡2𝑔 group of bands, the symmetry inequivalence of Ti(1)/Ti(3) and Ti(2)/Ti(4) sites leads to substantial 𝑡2𝑔 charge separation between these two groups of Ti atoms. In addition, an analysis of the partial density of states reveals significant bonding-antibonding splitting of 𝑑𝑥𝑦 (in local cubic frame) states of about 1.5 eV for Ti(1)/Ti(3) cations, whereas Ti(2)/Ti(4) cations show a relatively weak substructure. This substantial bonding-antibonding splitting of Ti(1)/Ti(3) 𝑡2𝑔 states agrees well with the concept of formation of Ti3+-Ti3+ spin-singlet pairs proposed earlier by Lakkis et al. [219]. However, the LSDA calculations fail to reproduce an insulating spin-singlet ground state of the low-temperature phase of Ti4O7. Apparently, the electron-electron correlations, mainly in the 3𝑑 shell of Ti cations, play a significant role.

4.1.2. LSDA+𝑈 Results and Charge Ordering

In order to take into account strong electronic correlations in Ti 3𝑑 shell, authors of [187] perform LSDA+𝑈 calculations for Ti4O7 in the low-temperature 𝑃1 structure. The LSDA+𝑈 calculations result in a charge and orbitally ordered insulator with an energy gap of 0.29 eV (see Figure 22). This is in a strong contrast with the metallic solution with a substantial charge disproportionation between crystallographically inequivalent Ti(1)/Ti(3) and Ti(2)/Ti(4) cations obtained by LSDA and in a reasonably good agreement with an experimental gap value of 0.25 eV. Note, however, that the charge- and orbital-order pattern remains exactly the same for 𝑈 in the range 2.5–4.5 eV, whereas the energy gap increases considerably up to 1.12 eV for 𝑈=4.5 eV. This remarkable increase of the gap value is accompanied by the enhancement of the spin magnetic moment from 0.56 up to 0.8 𝜇𝐵 per 3+ Ti(1)/Ti(3) cation as 𝑈 is increased from 2.5 to 4.5 eV.

Figure 22: The total DOS obtained from LSDA+𝑈 calculations with 𝑈=3.0 eV and 𝐽=0.8 eV for the low-temperature 𝑃1 phase of Ti4O7 [187]. The top of the valence band is shown by dotted lines.

In addition, authors perform LSDA+𝑈 calculations for high-temperature metallic phase of Ti4O7 [187]. In particular for 𝑈 of 2.5 eV a metallic self-consistent solution with substantial density of states (76 states/Ry) at the Fermi level has been found, whereas for 𝑈 of 3 eV the LSDA+𝑈 solution becomes unstable but remains metallic. With further increase of the 𝑈 value the metallic solution collapses into insulating one.

After self-consistency four crystallographically independent Ti atoms are split out in two subgroups with respect to the spin magnetic moment per Ti site: Ti(1)/Ti(3) with a moment of 0.66/−0.67 𝜇𝐵, respectively, and Ti(2)/Ti(4) with 0.04/−0.02 𝜇𝐵. Thus, one of 𝑡2𝑔 majority-/minority-spin states of Ti(1)/Ti(3) becomes occupied (𝑑1), whereas all other 𝑡2𝑔 states are pushed by strong Coulomb interaction above the Fermi level. In contrast, all 𝑡2𝑔 states of Ti(2) and Ti(4) are almost depopulated (𝑑0) and form bands up to 2.5 eV above the Fermi level. The occupied Ti(1)/Ti(3) states are strongly localized and form a prominent structure with a bandwidth of 0.25 eV just below the Fermi level (see Figure 23). The strong Coulomb interaction does not affect much the empty Ti 𝑒𝑔 states, which give predominant contribution between 2.5 and 4.5 eV. The obtained magnetic structure is almost antiferromagnetic with the spin magnetic moments within Ti(1)3+-Ti(3)3+ as well as Ti(2)4+-Ti(4)4+ pairs being of the same magnitude with opposite sign.

Figure 23: The partial DOSs for Ti(1)3+ and Ti(2)4+ cations are shown [187]. The gap value of 0.29 eV was obtained by LSDA+𝑈 with 𝑈=3.0 eV and 𝐽=0.8 eV. The Fermi level is shown by dotted line.

An analysis of occupation matrices of Ti(1)3+/Ti(3)3+ majority/minority 3𝑑 spin states confirms substantial charge disproportionation within the Ti 3𝑑 shell. As shown in Table 6, one of the 𝑡2𝑔 states of Ti3+ cations (𝑑1) is occupied with the occupation number of 0.74, whereas the remaining two 𝑡2𝑔 orbitals have a significantly smaller population of about 0.08. Thus, according to [181] we define an orbital-order parameter as the largest difference between 3+ and 4+ Ti 𝑡2𝑔 populations which amounts to 66% of ideal ionic charge-ordering model. The orbital-order parameter clearly shows the existence of substantial charge disproportionation in the Ti 3𝑑 shell of Ti4O7 which is remarkable because of the complete lack of the total charge separation (see column 𝑞 in Table 6) between 3+ and 4+ Ti cations. The occupation matrices analysis shows that the change of the 𝑡2𝑔 occupations is very efficiently screened by the rearrangement of the other Ti electrons. A significant portion of the screening charge is provided by Ti 𝑒𝑔 states due to formation of relatively strong 𝜎 bonds with O 2𝑝 states, which results in appreciable contribution of the former to the occupied part of the valence band. Ti 4𝑠 and 4𝑝 states give additional contributions to the screening of the difference in 𝑡2𝑔 occupations which leads to complete loss of the disproportionation between the charges at 3+ and 4+ Ti sites.

Table 6: Total (𝑞) and 𝑙-projected (𝑞𝑠,𝑝,𝑑) charges, magnetic moments (𝑀), and occupation of the most populated 𝑡2𝑔 orbitals (𝑛) calculated for inequivalent Ti atoms in the low-temperature 𝑃1 phase of Ti4O7 [187].

The occupied 𝑡2𝑔Ti3+ states are predominantly of 𝑑𝑥𝑦 character in the local cubic frame (according to that we later mark the orbital as 𝑑𝑥𝑦 orbital). This is illustrated in Figure 24, which shows the angular distribution of the majority- and minority-spin 3𝑑 electron density of Ti cations, marked by red and cyan color (or light and dark on the black and white image), respectively. Since Ti(1)3+ and Ti(3)3+ cations are antiferromagnetically coupled, the obtained ferro-orbital order is consistent with the formation of a bonding spin-singlet state from the 𝑑𝑥𝑦 orbitals of two neighboring Ti(1) and Ti(3) sites. The orientation of occupied Ti3+𝑡2𝑔 orbitals is consistent with the largest average Ti-O distance in the plane of 𝑡2𝑔 orbitals. As shown in Table 7 the average Ti(1)–O distance (2.061 Å) in the plane of 𝑑𝑥𝑦 orbital is considerably larger than average distances in the other two 𝑦𝑧 and 𝑧𝑥 planes (2.032 and 2.045 Å, resp.). The same is also true for the Ti(3) cation but in this case the variation of the average Ti(3)–O distances is much smaller (2.047 versus 2.041 and 2.042 Å), and, as a consequence, the out-of-plane rotation of the occupied 𝑡2𝑔 minority spin orbital is stronger.

Table 7: The averaged Ti–O distances in the plane of 𝑡2𝑔 orbitals (𝑑orb.) and in the oxygen octahedra (𝑑av.) for 𝑃1 structure of Ti4O7. 𝑑𝑥𝑦 approximates to the occupied orbital of the 3𝑑1 Ti(1) and Ti(3) 3+ states [187].
Figure 24: Structure of Ti4O7 showing the angular distribution of the majority and minority spin 3𝑑 electron density of Ti cations [187]. Red and cyan (light and dark, resp., on the black and white image) orbitals correspond to the majority and minority 3𝑑 spin states, respectively. Oxygen atoms are shown by small spheres. The size of orbital corresponds to its occupancy. 𝑥-𝑦-𝑧 coordinate system corresponds to the local cubic frame.

In addition, hopping matrix elements were evaluated via Fourier transformation from reciprocal to real space of the Ti 𝑡2𝑔 LSDA Wannier Hamiltonian [222]. Remarkably, for the low-temperature phase the Ti(1)-Ti(3) intra-pair 𝑑𝑥𝑦-𝑑𝑥𝑦 hopping matrix element is found to be of 0.61 eV, whereas all other hoppings are 3-4 times smaller. This strong inhomogeneity of the hopping matrix elements disappears in the high-temperature phase. Thus, according to our calculations hopping elements in the high-temperature phase are 0.23, 0.21, 0.39, and 0.33 eV between 1-3, 2-4, 3-3, and 4-4 Ti sites, respectively.

Estimation of exchange interaction parameters via the variation of the ground-state energy with respect to the magnetic moment rotation angle [17] results in a strong antiferromagnetic coupling of −1696 K between Ti(1)3+ and Ti(3)