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Advances in Condensed Matter Physics
Volume 2011 (2011), Article ID 298928, 107 pages
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.
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 and 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 . 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 and 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 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 . 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 , , , and .
The second part of the paper is devoted to the strongly correlated 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 and (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 levels are relatively close to the Fermi energy, various anomalous phenomena frequently appear. Most of them can be attributed to the hybridization between the 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 [3–7]. 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 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 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 electron with the conduction band at the Fermi level. Such effects are expected to arise in systems where two electron configurations corresponding to occupation numbers and have nearly degenerate energies. So the ground state of a homogeneously mixed-valence compound is a quantum mechanical mixture of both the and the 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 electrons are strongly correlated and close to localization, that is, having a low effective kinetic energy. The 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 below which the valence fluctuation suppressed. The resulting inhomogeneously mixed-valence state consists of two species of ions with the and the configurations. This transition may be compared to a Wigner crystallization on a lattice , and its earliest example is the Verwey transition in magnetite , 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 , , and with the cubic anti- structure and rare earth chalcogenides (X = S, Se or Te) and with the 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) , the LSDA+  method, the GW approximation , and dynamical mean-field theory (DMFT) [1, 15, 16] have sought to remedy this problem and have met with considerable success. The LSDA+ method is the simplest among them and most frequently used in the literature therefore; we describe the method in detail.
2.1. LSDA+ Method
A rigorous formulation for the quasiparticle properties of solids is the Green function approach. The self-energy of the single-particle Green function is energy dependent and yields the correlation corrections to the single-particle (mean-field) approximation to the quasiparticle excitation spectrum described by . With a number of plausible assumptions, the LSDA+ approach has been related to the so-called GW approximation to in . Already the simplest random phase approximation applied to for the Hubbard model yields a jump of at the Fermi level by the Hubbard . The more elaborate analysis of  results in a correlation correction to the mean-field approximation of the self-energy, which is downward below the Fermi level and 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 , the suggestion is where is the projector onto a strongly correlated state.
The LSDA+ 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 . Since the potential shift is taken to be constant in space, it does not deform the Kohn-Sham orbital . 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 , 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+ is the same as in the Anderson impurity model : the separate treatment of localized -electrons for which the Coulomb – interaction is taken into account by a Hubbard-type term in the Hamiltonian ( are orbital occupancies) and delocalized electrons for which the local density approximation for the Coulomb interaction is regarded as sufficient.
The spectrum of excitations for the shell of an electron system is a set of many-body levels describing processes of removing and adding electrons. In the simplified case, when every electron has roughly the same kinetic energy and Coulomb repulsion energy , the total energy of the shell with electrons is given by and the excitation spectrum is given by .
Let us consider ion as an open system with a fluctuating number of electrons. The correct formula for the Coulomb energy of – interactions as a function of the number of electrons given by the LDA should be . 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: The orbital energies are derivatives of (2): This simple formula gives the shift of the LDA orbital energy for occupied orbitals () and for unoccupied orbitals (). A similar formula is found for the orbital-dependent potential where variation is taken not on the total charge density but on the charge density of a particular th orbital : 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+ functional is defined as [21, 22] where is the LSDA (or LDA as in ) functional of the total electron spin densities, is the electron-electron interaction energy of the localized electrons, and is the so-called “double counting” term which cancels approximately the part of an electron-electron energy which is already included in . The last two terms are functions of the occupation matrix defined using the local orbitals .
The matrix 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 [21–23] where are the matrix elements of the on-site Coulomb interaction which are given by with being screened Slater integrals for a given and The angular integrals of a product of three spherical harmonics can be expressed in terms of Clebsch-Gordan coefficients, and (8) becomes
The matrix elements and 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 integrals
The averaging of the matrices and over all possible pairs of , defines the averaged Coulomb and exchange integrals which enter the expression for . Using the properties of Clebsch-Gordan coefficients, one can show that where the primed sum is over . Equation (11) allows us to establish the following relation between the average exchange integral and Slater integrals: or explicitly
The meaning of has been carefully discussed by Herring . In, for example, am electron system with electrons per atom, is defined as the energy cost for the reaction that is, the energy cost for moving an electron between two atoms where both initially had electrons. It should be emphasized that is a renormalized quantity which contains the effects of screening by fast and electrons. The number of these delocalized electrons on an atom with electrons decreases, whereas their number on an atom with 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 in L(S)DA+ calculations is significantly smaller than the bare used in the Hubbard model [25, 26].
Some aspects of currently used LDA+ 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 [27–29]. Pickett et al.  present a reformulation of the LDA+ method based on a local-orbital expansion (linear combination of atomic orbitals). The implementation of the LDA+ method by Bengone et al.  is based on the projector augmented wave (PAW) method , an efficient all-electron method without shape approximations on the potential or electron density. Based on a Car-Parrinello-like formalism , the PAW method allows complex relaxations and dynamical properties in strongly correlated systems. Cococcioni and De Gironcoli  reexamine the LDA+ 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 , transition metal centers , and iron heme complexes .
In principle, the screened Coulomb and exchange integrals can be determined from supercell LSDA calculations using Slater’s transition-state technique  or from constrained LSDA calculations [27, 35, 36].
To obtain Hubbard and exchange parameter , 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  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 . 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 , while the result for Fe (also about 6 eV) was surprisingly high since 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  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+ method. They estimate the Hubbard in elemental iron at the experimental lattice parameter to be equal to eV in good agreement with the experimental data [39, 40].
Alternatively, the value of 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 , 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+ calculations with PES or optical spectra . While the use of an adjustable parameter is generally considered an anathema among first principles practitioners, the LDA+ 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 is parallel to the -axis, the dielectric tensor is composed of the diagonal and and the off-diagonal components in the form 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 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 . Two of these modes are for circular components of opposite (±) helicity with the wave vector and have indices
The other two modes are for linear polarizations with . One has the electric vector and index . The other has and .
At normal light incidence the relation between the polar complex Kerr angle and the dielectric tensor components is given by the expression 
X-ray magnetic circular dichroism is given by and is first-order in . Magnetic linear dichroism (also known as the Voigt effect) is quadratic in . 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 for incident X-ray of polarization and photon energy can be determined as the probability of electronic transitions from initial core states with the total angular momentum to final unoccupied Bloch states where and are the wave function and the energy of a core state with the projection of the total angular momentum ; and are the wave function and the energy of a valence state in the th band with the wave vector ; is the Fermi energy.
is the electron-photon interaction operator in the dipole approximation where are the Dirac matrices, is the polarization unit vector of the photon vector potential, with , . 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 and , 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 , 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 , Reim and Schoenes , and Schoenes .
The quantum mechanical understanding of the Kerr effect began as early as 1932 when Hulme  proposed that the Kerr effect could be attributed to spin-orbit (SO) coupling (see also Kittel ). 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  and later Cooper presented a more general theory using some simplifying assumptions . The very powerful linear response techniques of Kubo  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 .
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 . 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  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 . Prototype studies have been performed using modern methods of band theory for Fe, Co, and Ni [55–59]. Following the calculations for the elemental ferromagnets, a number of groups have evaluated the MO and XMCD spectra for more interesting compounds [10, 60–88] and multilayers [89–102].
3. Iron Oxides
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 electron band in transition metal oxides [103, 104], for example, nickel oxide (NiO) and magnetite (). In both systems the metal-insulator transition occurs, violating the Bloch-Wilson band-insulator concept, the only one known at that time [105–108]. 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 . However, the charge ordering proposed by Verwey behind the metal-insulator transition [9, 109, 110] in 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 as well as that of many other MIT compounds is still a subject of debate [115–119].
is a strongly correlated compound which is ferrimagnetically ordered below a high transition temperature (~850 K). The valence of various atoms is described by the formal chemical formula, . The tetrahedral lattice sites ( sites) in the inverse spinel structure are occupied by ions, whereas the octahedral lattice sites ( sites) are occupied alternately by equal numbers of and . At K undergoes a first-order phase transition (Verwey transition) . This is a particular MIT that has been studied for quite some time . The Verwey transition is characterized by an abrupt increase in the electrical conductivity by two orders of magnitude on heating through [121–123]. Verwey and co-workers [9, 110] were the first to point out that this transition is associated with an electron localization-delocalization transition. The ion can be regarded as an “extra” electron plus an ion. When all sites are equivalent, the “extra” electron is moving between ions and the system is a mixed-valent metal, with average FeB-valence, . The Verwey phase transition below is accompanied by long-range charge ordering (LRCO) of and ions on and sites of the sublattice. Indeed, studies by electron and neutron diffraction and nuclear magnetic resonance [124–126] show that below the and 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  since Verwey first proposed that, below , all and sit on different chains .
The electronic structure of has been investigated experimentally by means of soft X-ray spectroscopy [127–129], Seebeck-effect measurements [130, 131], photoelectron spectroscopy [132–139], optical [140, 141] and MO spectroscopies [142–149], and magnetic dichroism [150, 151]. The optical data  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 is very difficult due to the existence of three kinds of iron atoms, that is, , , and . 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 as well as the influence of a systematic substitution of by 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 between 0.5 and 4.3 eV. They also reported spectra of and between 0.5 and 5.0 eV . Finally, recent investigation of the optical and MO spectra of and and substitution has been carried out in .
Energy band structure calculations for in the high-temperature cubic phase have been presented in  using the self-consistent spin-polarized augmented plane wave method with the local spin-density approximation. The calculations show that is a half-metallic ferrimagnet. The Fermi level crosses only the minority-spin energy bands consisting of orbitals on the Fe() sublattice. There is an energy gap for the majority-spin bands at the Fermi level. A similar energy band structure of was obtained in  using the LMTO method. The energy band structure for charge ordering in the low-temperature phase of has been calculated in [10, 11, 154, 155] using the LMTO method in the LSDA+ approximation.
3.1.1. Crystal Structure and Charge Ordering
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 (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 or site, tetrahedrally coordinated by four ions composing a diamond lattice. The interstices of these coordination tetrahedra are too small for larger ions, and this site is occupied only by ions. Another cation site is called the or site, and is coordinated by six ions forming slightly distorted octahedra, which line up along the axes of the cubic lattice sharing edges. The point symmetry of the site is . This site forms exactly one-half of a face-centered cubic lattice. The lattice of the site can be considered as a diamond lattice of cation tetrahedra, sharing corners with each other. All the tetrahedra on the same (e.g., ) plane are isolated. In the following, we refer to axes or chains and axes or chains. The direction is , is and the -axis is . All the Fe octahedral or sites lie on either or 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.
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  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)-site mole] is decisively smaller than the expected in a complete order-disorder transition. Anderson interpreted the Verwey transition as a loss of the LRCO of the “extra” electrons on the sublattice at temperatures above , while the short-range charge order is maintained across the transition. Using a Hartree-Fock analysis, Alben and Cullen and Callen [157–159] showed that an ordering transition could occur in as a function of the ratio , where 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 have been reviewed by Mott [160–162] (see also [120, 163]). A basic problem concerning magnetite is that both the localization of the valence states and the mixing of the oxygen states and iron states are considerable.
In the disordered high-temperature phase the sites are occupied by equal numbers of and ions randomly distributed between and sites. Below the system undergoes a first-order transition accompanied by long-range charge ordering of and ions on the sites. Verwey from the very beginning proposed a rather simple charge separation: chains occupied only by ions and chains by ions (or vice versa) . Since that time the type of charge ordering has been the subject of debate . As an example, in Mizoguchi’s model [164, 165] ions run in pairs of followed by along each chain. Another charge ordering considers three alternating with one on one half of the chains, and a sequence of three alternating with one on the other half and so on . 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 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 on the cubic unit cell, which points to a doubling of the unit cell along the -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 , whereas the observation of a magnetoelectric effect indicated even lower symmetry in the low-temperature phase . Although clear evidence of the monoclinic lattice symmetry below 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 . In particular, purely electronic [158, 159, 164, 173] and electron-phonon [174–176] models for CO, as well as a bond-dimerized ground state without charge separation , have been proposed.
Recent bond-valence-sum analysis  of high-resolution neutron and X-ray powder diffraction data results in a small charge disproportion of only 0.2 between 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)+ method [181, 182]. In particular, a more complicated charge-ordering pattern inconsistent with the Verwey CO model was obtained. In addition to that, the occupancy self-consistently obtained in the LSDA+ calculations is strongly modulated between the and cations, yielding a distinct orbital order with an order parameter that reaches 70% of the ideal value . 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 [182–186].
In order to check the pertinence of the CO model obtained self-consistently in , authors of  carried out a detailed theoretical study of exchange coupling constants, optical conductivity, magneto-optical (MO) Kerr effect, and X-ray absorption at the O edge of low-temperature 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 obtained from the LSDA calculation. The occupied part of the valence band can be subdivided into several regions separated by energy gaps. The oxygen 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 states. The Fe energy bands are located above and below at about −4.0 to 3.0 eV. As indicated from Figure 2, the exchange splitting between the spin-up and spin-down electrons on the Fe atom is about 3.5 eV. In addition to the exchange splitting, the five levels of the Fe atom are split due to the crystal field. At the site ( point symmetry) in the spinel structure the crystal field causes the orbitals to split into a doublet ( and ) and a triplet (, , and ). The octahedral component of the crystal field at the site is strong enough that the (, , and ) and ( and ) orbitals form two separate nonoverlapping bands. At the site the crystal field is trigonal (), as a result the orbitals split into a singlet and a doublet . However, the splitting of the band is negligible in comparison with its width in LSDA calculations; therefore, in the following we will denote the states formed by and orbitals as states. Accordingly, we present in Figure 2 the DOS of “” orbitals as a sum of the and ones. The crystal-field splitting is approximately 2 eV for the atom and 1 eV for the atom. This difference may be attributed to the large covalent mixing of the orbitals with its six nearest neighbors of the same kind. The spin-polarized calculations show that in the high-temperature phase is a half-metallic ferrimagnet. The Fermi level crosses only the majority spin energy bands, consisting of spin-up orbitals on the sublattice (Figure 2). There is an energy gap for the minority spin bands at the Fermi level. Spin-orbit splitting of the energy band at Γ is about 0.02 eV and much smaller than the crystal-field splitting.
In the magnetic moments within the and the sublattices are ferromagnetically aligned while the two sublattices are antiferromagnetic with respect to each other. This magnetic structure was first proposed by Neel  to explain the magnetization data and was later confirmed by neutron scattering measurements . Measurements indicated that the magnetic moment of an iron atom on the site is much smaller than the 5.0 of a pure ion . This is an indication of strong hybridization between the orbitals of . The orbital magnetic moment is rather small for all the atoms due to small spin-orbit coupling.
3.1.3. LSDA+ Band Structure
The application of LSDA calculations to is problematic, because of the correlated nature of 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+ calculations  started from a configuration for ions on the tetrahedral site of the sublattice and and for and ions on octahedral site of the sublattices and , respectively. was applied to all the 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+ calculation. In contrast to LSDA, where the stable solution is a metal with a uniform distribution of the electrons on the octahedral sites, the LSDA+ gives a charge-ordered insulator with a direct energy gap value of 0.19 eV at the Γ point. The experimental optical measurements  gave a gap of 0.14 eV at K. The energy gap occurs between the (the top of valence band) and (bottom of empty conduction band) states (Figure 4). Actually, the LSDA+ band structure calculations support the key assumption that Cullen and Callen have made earlier [157–159] in proposing the one-band model Hamiltonian, where it was assumed that the “extra” electron moves in the band split off below the rest of the bands of other symmetries. Two electrons at the orbitals situated in the close vicinity of the Fermi level are mostly localized. Other electrons at the site are well hybridized with oxygen -electrons (Figure 4). The screening of the Coulomb interaction in is very effective and the system is close to the metallic state. Even a small change in the ratio (which can be modeled by changing of the occupation numbers of the orbital at the and 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 .
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 of are compared to the theoretical ones calculated within the LSDA and LSDA+ approaches. Better agreement between the theory and the experiment was found when the LSDA+ 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 as . In Figure 6 we show the calculated and experimental absorptive part of the diagonal optical conductivity spectra in a wide energy range. The characteristic features of the LSDA calculation of 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 strongly influences not only the electronic structure but also the calculated optical spectra of . The LSDA+ calculations make better job in describing the optical spectra of 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 bands below 2.5 eV, (2) the transitions from O to Fe bands in the region of 2.5 to 9 eV, and (3) Fe and O interband transitions above 9 eV. To avoid misunderstanding, we should mention that here and in the following when talking about transitions we mean that the energy bands involved in the transitions have predominantly character; however, the contribution of or states to these bands is sufficient to provide a significant transition probability through optical dipole matrix elements.
Let us consider now the magneto-optical properties of . In Figure 7 we show the experimentally measured  Kerr rotation and Kerr ellipticity MO spectra of , as well as the off-diagonal parts of the dielectric function calculated with the LSDA and LSDA+ approximations . This picture clearly demonstrates that the better description is achieved with the LSDA+ approach.
One should mention that, although the O → Fe interband transitions, which start already from about 2.5 eV, play an important role in the formation of the optical spectra of , the Kerr spectra are mostly determined by transitions between energy bands which have predominantly Fe character. The reason for this is that the spin-orbit and exchange splitting of O states is much smaller in comparison with the Fe ones. The minimum in the Kerr rotation spectrum at 0.9 eV is due to the () interband transitions. The second maximum at about 2 eV is associated with the interband transitions. The minimum in the Kerr rotation spectrum between 3 and 4 eV can be associated with the transitions.
All the experimental measurements of the Kerr spectra of [142–147] have been performed at room temperature. The LSDA+ 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 . This leads to a conclusion that Fe electrons remain “correlated” above . The main effect of heating through is a disappearance of the long-range charge order on the 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 ( to 5 eV is much larger than eV). This picture is supported by recent optical measurements , which show a strong temperature dependence of the optical properties of 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 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+ 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 [150, 151] is very difficult due to the existence of three kinds of iron atoms, that is, , , and . 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 and Co M2,3 core-absorption edges in and . For the M2,3 prethreshold MCD spectra were measured above and below the Verwey transition temperature. Van Der Laan et al.  reported the XMCD spectra at the Ni edges of (trevorite). The Ni edge magnetic circular dichroism measurements of ferrimagnetic (, 0.26, 0.50, and 0.75) were reported by Pong et al. . Magnetic circular dichroism is reported for ferrite in  with the measurements performed on the and core levels of Mn and Fe. The electronic structure, spin and orbital magnetic moments, and XMCD spectra of the series , , , and are presented in . The XAS and XMCD spectra at , and edges for transition metals sites were calculated.
In Figure 8 the experimentally measured Fe XMCD spectra  in are compared to the theoretical ones calculated within the LSDA+ approach . The dichroism at the and edges is influenced by the spin-orbit coupling of the initial core states. This gives rise to a very pronounced dichroism in comparison with the dichroism at the edge. Two prominent negative minima of Fe XMCD spectrum are derived from iron ions at octahedral sites. The major positive maximum is from ions. In the LSDA+ calculations of the charge-ordered , the XMCD spectra have slightly different shape for the and ions. The LSDA+ calculations are not able to produce the small positive shoulder at the high-energy side of the main peaks of the Fe XMCD spectrum.
The XMCD spectra at the edges are mostly determined by the strength of the SO coupling of the initial core states and spin-polarization of the final empty states, while the exchange splitting of the core states as well as the SO coupling of the valence states are of minor importance for the XMCD at the edge of transition metals .
To investigate the influence of the initial state on the resulting XMCD spectra, the XAS and XMCD spectra of at the edge were also calculated. The spin-orbit splitting of the core level is about one order of magnitude smaller than for the level in . As a result the magnetic dichroism at the edge is smaller than at the edge. In addition the and spectra are strongly overlapped, and the spectrum contributes to some extent to the structure of the total spectrum in the region of the edge. To decompose a corresponding experimental spectrum into its and parts will therefore be quite difficult in general.
In Figure 9 the experimentally measured Fe XMCD spectrum  in is compared to the theoretical one calculated within the LSDA+ approach . In the magnetic moments within the and the sublattices are ferromagnetically aligned while the two sublattices are antiferromagnetic with respect to each other. The XMCD spectra are positive at the and negative at the edge at the tetrahedral sites and vice versa for the octahedral ones. The interpretation of the experimental Fe 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 spectra (from , , and sites) appearing simultaneously in a rather small energy range.
3.2. Low-Temperature Monoclinic
3.2.1. Crystal Structure
The low-temperature structure was shown to have a supercell with space group from X-ray and neutron diffraction [111, 179]. However, recent structural refinement (at 90 K) was only possible in the centric monoclinic space group with of the cubic spinel subcell and eight formula units in the primitive unit cell [111, 179]. Since the refinement for the space group was found to be unstable, additional orthorhombic symmetry constraints were also applied. This is equivalent to averaging the true superstructure over the additional symmetry operators; that is, each site in the unit cell is averaged over four nonequivalent subsites in the large supercell. Note, however, that such an approximation is robust in the sense of smallness of any distortions from the subcell to the monoclinic cell (according to  these are of ~0.01 Å). Previous structure refinement below obtained by Iizumi et al. resulted in a subcell of the unit cell and imposed orthorhombic symmetry constraints on the atomic positions . In particular, a refinement based on an approximation of the true crystal structure by a centric space group or polar 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 sites are split into two groups with different values of the averaged Fe–O bond distances, with and sites being significantly smaller than and (– are crystallographically independent 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 groups using the bond-valence-sum (BVS) method shows that the octahedral sites fall into two clear groups with respect to the estimated value of valence. The result is a charge disproportion of 0.2 between large ( and ) and small ( and ) 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 ( and ) and small ( and ) sites could be averaged over () and () subsites, respectively. The symmetry averaging results in decrease of the more pronounced charge separation of in the full superstructure (class II CO) down to in the 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 shell, at least on the static Hartree-Fock level, the authors of  calculated the electronic structure of the LT phase of using the LSDA+ method. The value of the 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  of 0.14 eV at 10 K was obtained using the value of 5 eV. Note, however, that the charge and orbital order derived from the LSDA+ calculations does not depend on the exact value when it is varied within the above-mentioned limits. The value of the Hund’s coupling eV was estimated from constrained LSDA calculations . In the following all results presented in the paper were obtained using a value of 5 eV.
Figure 10 shows the LSDA+ band structure and the total DOS  calculated self-consistently for the low-temperature structure of using the Coulomb interaction parameter eV and exchange coupling eV. The corresponding partial DOSs are shown in Figure 11. The LSDA+ 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 states of and ions becomes occupied while the and states are pushed above the chemical potential. Although, as will be discussed below, the calculated disproportion of charges is significantly less than 1, in the following we use the notations and for and cations, respectively, having in mind the difference of their occupations. The top of the valence band is formed by the occupied states of and cations. The bottom of the conduction band is formed predominantly by the empty states of and cations. The remaining unoccupied states of and cations are pushed by the strong Coulomb repulsion to energies above 2.5 eV. Majority-spin states are shifted below O 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.
Bands corresponding to the 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+ 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 and cations on the sublattice is described by a dominant charge (and spin) density wave, which originates from alternating chains of ions on octahedral sites and ions on sites (see Table 1 and Figure 2 in ). A secondary modulation in the phase of CO, which is formed by the chain of alternately “occupied” ions on the sites and “empty” ions on sites, was found. This is consistent with a  nesting vector instability at the Fermi surface in the minority electron states which has been recently revealed by the LSDA calculations for the cubic phase . The calculated CO scheme coincides with the class I CO model proposed by Wright et al. [111, 179]. All the tetrahedra formed by cations have either a 3 : 1 or 1 : 3 ratio of and ions. Thus, the LSDA+ 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+ calculations .
An analysis of the minority occupation matrices of cations confirms very effective charge disproportion within the minority-spin subshell. In particular, one of the states of and cations is almost completely filled with the occupation . On the other hand, the other two orbitals of the cations have significantly smaller population of about 0.04. The occupation numbers of orbitals for and cations do not exceed 0.1–0.17, which gives a value of about 0.7 for the largest difference of the populations of and states. The occupation numbers of the minority-spin orbitals and the net occupations of the and states are given in the last two columns of Table 1.
The change of the 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 states. Although the bands originating from these states are located well above the energy gap, the minority spin orbitals form relatively strong bonds with states of the oxygen octahedron and, as a result, give an appreciable contribution to the occupied part of the valence band. The energy of states is lower than the energy of corresponding states, and the former give a significantly larger contribution to the part of the valence band formed mainly by O states. Because of the stronger covalency of the –O bonds, the net occupation of states becomes larger (see the last column of Table 1). The resulting charge difference (0.23) and disproportionation of the total electron charges inside the atomic spheres of and ions (0.24) are in reasonably good agreement with the value of 0.2 estimated from a BVS analysis of the structure [111, 179]. The above-mentioned screening of the changes in the minority occupations reduces the energy loss due to the development of charge order incompatible with the Anderson criterion in the LT phase of .
Hence, due to the strong screening effects, the order parameter defined as the difference of the net charges of 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 minority-spin states for and 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+ calculations were also performed for the assumption of Verwey charge order in the structure . 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 structure. It is well known that with increasing value localization is effectively increased. Remarkably, even for a value increased up to 7-8 eV no Verwey-like CO pattern was found self-consistently in the distorted structure. On the contrary, the LSDA+ calculations performed for an undistorted supercell of the 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 phase.
Also authors performed LSDA+ calculations  with the same and parameters (5 and 1 eV, resp.) for the assumption of one of the 32 class II CO models within supercell of , which is shown in Figure 2 in . But it was found that this kind of CO is unstable and the self-consistent solution coincides with the one found previously for the structure.
Comparing the LSDA+ results for the undistorted and distorted unit cells, we can conclude that the charge-ordering pattern of and cations in the LT phase of , derived from the BVS analysis in  and confirmed by our study, is mainly forced by the local distortions of the crystal structure. The results of  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 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 . In particular, in the subcell the true atomic positions are averaged over the corresponding number of subsites in the cell. Therefore, the actual arrangement of the locally octahedra in the true structure can be more complex, probably resulting in a more complicated charge and/or orbital order for the LT structure. The calculations in  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 . 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+ is not only charge but also orbitally ordered. Table 1 clearly presents which presents the contribution of cubic harmonics to the formation of minority-spin states with an occupancy (next to last column in Table 1) evaluated by diagonalization of the corresponding occupation matrix self-consistently obtained by the LSDA+ .
As shown in the table the most occupied minority orbitals are centered on the , , and iron sites and have , , and characters, respectively. Remarkably, the occupied orbitals of cations are almost orthogonal to each other; that is, their relative orientation corresponds to an anti-ferro-orbital order. Since all 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 sites .
Also this simple analysis shows a remarkable difference between and cations; namely, the average –O distance in the plane of occupied orbital is 2.087 Å, whereas in the planes of two other 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 (more than 0.02 Å), although for they are 2.074 and 2.067 Å for occupied and unoccupied orbitals, respectively, which gives a difference of only 0.007 Å. This small difference can be changed by applying a uniaxial stress to the unit cell resulting in modification of the electronic properties . In particular, a few percent of magnitude elongation of the unit cell along the -axis with simultaneous (in order to preserve the same unit cell volume) compression in the plane gives rise to orbital-order crossover on the site from a to a occupied orbital. At the same time the charge order and occupied orbitals on the sites remain the same. The pressure-induced spatial reorientation of the occupied orbital was proved by the LSDA+ calculations for the strained unit cell. Note, however, that these rough estimations do not take into account the elastic anisotropy in . Moreover, the analysis was performed for the “averaged” 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 .
3.2.5. Magnetic Moments
The strong variation of the occupancies of the minority-spin states leads to a pronounced modulation of the spin magnetic moments on the sublattice. While the total moment per formula unit remains at 4 , the magnetic moments of the (3.50 ) and (3.48 ) cations become appreciably smaller than (3.94 ) and (3.81 ) moments. The charge and spin modulation on the sublattice is accompanied by formation of a weak spin modulation on the oxygen ions caused by different strengths of the hybridization of O states with the minority-spin states of and ions. In particular, the oxygen magnetic moment reaches its maximal value of ~0.1 for O3 and O4 ions, which lie in the plane of cations. It substantially decreases for other oxygen ions and approaches minimum for O1 and O2 lying in the plane of cations (~0.04 ).
Recently, an anomalously large value of the orbital magnetic moment reaching 0.33 has been deduced by applying sum rules to experimental X-ray magnetic circular dichroism spectra of . In addition, the unquenched orbital moment was also reported to be confirmed by the LSDA+ calculations. Later, however, this experimental finding was questioned by Goering et al. . The average orbital moments between and were found from X-ray magnetic circular dichroism sum rules depending on the integration range. From the spin-polarized relativistic LSDA+ calculations  for the LT structure it was estimated orbital moments of and for and ions, respectively. Somewhat larger values of and were calculated for and cations, respectively. Taking into account the negative orbital moment of , this gives the value of for the average orbital moment. Thus, in agreement with the previous theoretical results of  and XMCD sum rule data of , the calculations give the value of Fe orbital moment of which is much smaller than reported in .
3.2.6. Exchange Coupling Constants
The calculations of the exchange interaction parameters were also performed  via the variation of the ground-state energy with respect to the magnetic moment rotation angle . The exchange coupling parameter represents the effective pair exchange interaction between the th and th Fe atoms with effective Heisenberg Hamiltonian . Here, and are the spins at sites and (5/2 and 2 for and cations, resp.). Positive (negative) values of correspond to the ferromagnetic (antiferromagnetic) coupling between sites. As shown in Table 2 the exchange couplings between and iron sublattices are rather large, of about −70 K, and antiferromagnetic. The interactions are weakly antiferromagnetic with the maximal absolute value of 9.3 K (not shown in Table 2). The exchange couplings between the sites ( K) are substantially smaller than ones and almost all of them are ferromagnetic (see Figure 12). Weak antiferromagnetic couplings with 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.
Experimental estimation of the exchange couplings in was first performed by Néel on the basis of the two-sublattice collinear model . From analysis of the temperature behaviour of the saturation magnetization and paramagnetic susceptibility, he obtained , , and K, where and 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 and sublattices; couplings (not shown in Table 2) are considerably smaller than ; the exchange couplings in the sublattice are weak and almost all of them are ferromagnetic. On the other hand, the small antiferromagnetic exchange interaction (see Table 2) is in exact agreement with recent estimations using the two-sublattice model . Three-sublattice model calculations give an overall similar result, except, however, the exchange coupling, which seems to be overestimated .
The LSDA+ study of the model of the LT phase of  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 charge density wave with a minor modulation on the sublattice. A weak 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 . While the screening of the charge disproportion is so effective that the total charge disproportion is rather small (0.23), the charge order is well pronounced with an order parameter defined as the difference of occupancies of 2+ and 3+ 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  and corresponds to the local distortions of oxygen octahedra surrounding sites.
Calculations of the effective exchange coupling constants between Fe spin magnetic moments show that the dominating interaction is an antiferromagnetic coupling between and moments. The coupling between and moments is found to be weaker and ferromagnetic.
Iron borate () is a semivalent oxide. It belongs to the homometallic warwickite family with formal chemical formula , where and are, respectively, a divalent and trivalent metal ions. Surprisingly, the homometallic () warwickites are known only for Fe  and Mn . In both compounds the metal has octahedral coordination. These octahedra share edges to form ribbons of four infinite along crystallographic direction chains of octahedra linked by corner sharing and the trigonal BO3 groups (see Figure 13).
There are two crystallographically inequivalent sites of the metal ions Fe(1) and Fe(2). is -type ferrimagnetic with drastically smaller in comparison with critical temperature of K, the Fe(1) magnetic moments being aligned antiparallel to the Fe(2) moments. It is almost antiferromagnetic, but a small ferrimagnetic moment of per Fe atom in a 0.05 T field was found . At room temperature is a semiconductor with a thermoactivated conductivity below with eV . Upon farther heating a broad semiconductor-to-semiconductor transition occurs at K, where resistivity drops down by a factor of , and, as a result, a small decrease of the activated energy up to eV above 350 K is observed . 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 to orthorhombic symmetry with increasing temperature. This structural transition is attributed by modification of the angle from at 3 K to at 337 K . The change in conductivity and structure are small. But the Mössbauer spectra at around 317 K clearly result in the charge localization at the transition with an equal distribution of and cations over the two structurally distinct Fe(1) and Fe(2) sites with formal chemical formula [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 . 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 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  destroys the mirror symmetry, which leads to a tilting of the Fe ribbons, consistent with the observed enlargement of the angle below the transition. However, there is no observation of the increasing of -axis periodicity (it should increase by a factor of two or another integer factor below ). 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 and cations alternately ordered within the chains along the direction.
Theoretical investigation of the electronic structure and magnetic properties of in the low-temperature structure was reported in . An order parameter, defined as the difference between minority-spin occupancies of and as well as the difference between majority-spin occupancies of and cations was propose. This order parameter was found to be quite large, although the total charge difference between 2+ and 3+ cations is small.
The band structure calculations have been carried out for the low-temperature monoclinic structure of . The corresponding unit cell contains four 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 orbitals of Fe cations (see Figure 14).
The lower part of the valence band (below −3.5 eV) is mainly formed by O states with a bonding hybridization with Fe states. Fe 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 states is roughly 3 eV which results in a net magnetic moment of 0.31 per formula unit. Additionally, the fivefold levels are split by the crystal field into and subbands. The oxygen octahedra in 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 hybridization of - and -type, remains dominant, whereas the splitting within “” and “” subbands is smaller than the corresponding band-width. This allows one to label the corresponding states as and . 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 per formula unit.
Fe(1) and Fe(2) and states with the opposite spin projections share nearly the same energy intervals. Thus, Fe states between −3.5 and −2.0 eV originate predominantly from majority-spin Fe(1) and minority-spin Fe(2) states whereas the states between −2.0 and −0.5 eV are mainly of character. Partially occupied bands crossing the Fermi level are formed by minority spin Fe(1) and majority-spin Fe(2) states. The nominal occupation of these bands is 1/6. In the majority-spin channel, however, the Fe(2) 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 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) 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) states the situation in the minority-spin channel is more complicated. Twelve 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 but with a two times smaller period.
It should be noted that in contrast to experimental data  LSDA predicts to be metallic with substantial magnetic moment per unit cell. Apparently, the electron-electron correlations, mainly in the shell of Fe cations, play a significant role.
The LSDA+ calculations have been performed for the unit cell as well as for double () and triple () supercells of (without putting in any local displacements of oxygen atoms around / sites). Thus, for the CO pattern proposed in  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+ 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 value has to be increased up to 5.5 eV (see Figures 15 and 16). It does not exceed 10% of the value, which is in an accuracy of the calculation. Note, however, that the CO obtained by LSDA+ within supercell does not depend on the value of 5–5.5 eV. Here and in the following all results are presented for double along -direction supercell of .
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 majority-/minority-spin states of Fe(2)/Fe(1) atom becomes completely occupied, whereas all the rest of states are pushed by strong Coulomb interaction at the energies above 3 eV. The gap is opened between occupied and unoccupied states of and for spindown and and for spinup. Majority-spin states of and minority-spin states of cations are shifted below the states, which form the band in the energy range of −8 and −2 eV. In contrast to states, the majority-spin and minority-spin 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 and as well as per and cations. Using the moment populations in Table 3, the calculated net moment is per Fe atom, in exact agreement with the experimental value .
The charge order obtained by LSDA+ in supercell is consistent with observed enlargement of the angle below the transition and coincides with charge-ordering scheme proposed earlier by Attfield et al.  It is described by the sloping 2+ and 3+ Fe cation lines alternately stacked along direction and could be considered as a quasi-one-dimensional analog of the Verwey CO model in pyrochlore lattice of . An additional self-consistent LSDA+ calculations for unit cell as well as for double and triple along direction supercells was performed using the same and 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 supercell are unstable, and the stable one coincides with the CO found previously. Thus, the CO obtained for certain value of and 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 in the low-temperature phase.
Although the corresponding total charges difference and disproportion of the total electron charges inside the atomic spheres of and cations are small, an analysis of occupation matrices of Fe(1)/Fe(2) minority-/majority-spin states confirms substantial charge disproportionation. Thus, as shown in Table 3, one of the states of and cations is almost completely filled with the occupation numbers , whereas the remaining two orbitals of the cations have significantly smaller population of about 0.1. According to  an order parameter was defined as the largest difference between and populations. While, due to strong static “screening” effects, the order parameter introduced as the total charge difference between 2+ and 3+ Fe cations is ill defined, the well-defined order parameter is the difference of occupancies for and 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 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 states due to relatively strong bonds with O states and, as a result, appreciable contribution to the occupied part of the valence band.
The occupied states of cations are predominantly of character in the local cubic frame (according to that we later mark the orbital as orbital). This is illustrated in Figure 17, which shows the angular distribution of the majority and minority spin electron density of the Fe(2) and Fe(1) cations, respectively. Thus, occupied and unoccupied cations are ordered alternately within the chain which is infinite along 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 structure projected on (100) plane (see Figure 13), is shown in Figure 18.
Using the LSDA+ method the exchange interaction parameters have been calculated via the variation of ground-state energy with respect to the magnetic-moment rotation angle . 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 and cations is ferromagnetic with relatively small value of K. In contrast, the nearest sites in quasi-one-dimensional Fe(1) chain are coupled antiferromagnetically with noticeably larger exchange absolute value of K. Furthermore, the exchange parameters between the nearest sites of two Fe(1) chains are relatively strong and antiferromagnetic (see , , and 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 subbands of and cations (see in Table 4) is strongly suppressed by the substantial antiferromagnetic and exchange.
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 and cations are antiferromagnetic with values of K and K (see Table 5). Such an appreciable difference between and 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 and cations from K to K (see Table 5). Also it is interesting to note that the exchange interaction between and cations are considerably larger than between and ( and , correspondingly). We find that the interribbon exchange interactions play predominant role and determine the whole -type ferrimagnetic spin structure below in contrast with the ferromagnetic intrachain order due to superexchange .
In the LSDA+ study of the low-temperature phase of  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 charge disproportion is rather small (0.34), the charge order is well pronounced with an order parameter defined as a difference of occupancies of 2+ and 3+ Fe cations (0.8). The occupied and cations are ordered alternately within infinite along -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+ method as an aide to experimental studies of CO structures. However, the charge order obtained by LSDA+ is consistent with observed enlargement of the angle and coincides with charge-ordering scheme proposed earlier by Attfield et al. . It seems certain that is charge ordered below , 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” . 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+ method inevitably results in predominance of the interribbon exchange interactions which determine the whole -type ferrimagnetic spin structure below , in contrast with the ferromagnetic intrachain order due to superexchange proposed earlier in .
4. Titanium and Vanadium Oxides
The aforementioned phenomena of sharp metal-insulator transitions associated with pronounced charge and/or orbital ordering are characteristic for the Magnéli phases (). 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 () and () . In particular, the metal-insulator transition of 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 and neighbouring rutile-type dioxides [215–217]. In this scenario the characteristic dimerization and antiferroelectric displacement of the metal atoms translate into orbital ordering within the 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 as a correlation-assisted Peierls transition .
titanium oxide is another remarkable member of the Magnéli phases with 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 (), corresponding to an average 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 . 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 . 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 (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 . 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 -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).
Accurate determination of the crystal structure allowed to elucidate the nature of the three phases distinguished by the two first-order transitions . 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, 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 . Thus, the 130–140 K transition is associated with a transition to the phase with a long-range order of - pairs, whereas above 150 K 3+ and 4+ Ti cations are disordered. The presence of the - pairs strongly differentiates from 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 results in significant charge separation between crystallographically independent 3+ and 4+ Ti sites in the low-temperature phase, whereas a rather isotropic occupation of the states has been found at room temperature . While, in addition, an orbital order at the Ti 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 shell using the LSDA+ method.
In  the authors investigate the electronic structure of the low-temperature phase using the LSDA+ approach. The LSDA+ 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 , an orbital-order parameter was proposed as the difference between majority-/minority-spin occupancies of and cations respectively. This order parameter is found to be quite large, although the total 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 K of the local moments within the dimerized - 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 .
4.1.1. LSDA Band Structure
The LSDA band structure calculations  for the low-temperature 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 states with a bonding hybridization with Ti states. Crystal -field splitting of the latter is roughly of 2.5 eV. Ti states form the group of bands at and up to 2 eV above the Fermi energy, whereas Ti states give a predominant contribution to the bands between 2.5 and 4.5 eV. Within the group of bands, the symmetry inequivalence of Ti(1)/Ti(3) and Ti(2)/Ti(4) sites leads to substantial 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 (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) states agrees well with the concept of formation of - spin-singlet pairs proposed earlier by Lakkis et al. . However, the LSDA calculations fail to reproduce an insulating spin-singlet ground state of the low-temperature phase of . Apparently, the electron-electron correlations, mainly in the shell of Ti cations, play a significant role.
4.1.2. LSDA+ Results and Charge Ordering
In order to take into account strong electronic correlations in Ti shell, authors of  perform LSDA+ calculations for in the low-temperature structure. The LSDA+ 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 in the range 2.5–4.5 eV, whereas the energy gap increases considerably up to 1.12 eV for 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 per 3+ Ti(1)/Ti(3) cation as is increased from 2.5 to 4.5 eV.
In addition, authors perform LSDA+ calculations for high-temperature metallic phase of . In particular for 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 of 3 eV the LSDA+ solution becomes unstable but remains metallic. With further increase of the 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 , respectively, and Ti(2)/Ti(4) with 0.04/−0.02 . Thus, one of majority-/minority-spin states of Ti(1)/Ti(3) becomes occupied , whereas all other states are pushed by strong Coulomb interaction above the Fermi level. In contrast, all states of Ti(2) and Ti(4) are almost depopulated 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 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 - as well as - pairs being of the same magnitude with opposite sign.
An analysis of occupation matrices of majority/minority spin states confirms substantial charge disproportionation within the Ti shell. As shown in Table 6, one of the states of cations is occupied with the occupation number of 0.74, whereas the remaining two orbitals have a significantly smaller population of about 0.08. Thus, according to  we define an orbital-order parameter as the largest difference between 3+ and 4+ Ti 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 shell of which is remarkable because of the complete lack of the total charge separation (see column in Table 6) between 3+ and 4+ Ti cations. The occupation matrices analysis shows that the change of the occupations is very efficiently screened by the rearrangement of the other Ti electrons. A significant portion of the screening charge is provided by Ti states due to formation of relatively strong bonds with O states, which results in appreciable contribution of the former to the occupied part of the valence band. Ti and states give additional contributions to the screening of the difference in occupations which leads to complete loss of the disproportionation between the charges at 3+ and 4+ Ti sites.
The occupied states are predominantly of character in the local cubic frame (according to that we later mark the orbital as orbital). This is illustrated in Figure 24, which shows the angular distribution of the majority- and minority-spin electron density of Ti cations, marked by red and cyan color (or light and dark on the black and white image), respectively. Since and cations are antiferromagnetically coupled, the obtained ferro-orbital order is consistent with the formation of a bonding spin-singlet state from the orbitals of two neighboring Ti(1) and Ti(3) sites. The orientation of occupied orbitals is consistent with the largest average Ti-O distance in the plane of orbitals. As shown in Table 7 the average Ti(1)–O distance (2.061 Å) in the plane of orbital is considerably larger than average distances in the other two and 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 minority spin orbital is stronger.
In addition, hopping matrix elements were evaluated via Fourier transformation from reciprocal to real space of the Ti LSDA Wannier Hamiltonian . Remarkably, for the low-temperature phase the Ti(1)-Ti(3) intra-pair - 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  results in a strong antiferromagnetic coupling of −1696 K between and cations. All other couplings are two orders of magnitudes smaller. This indicates a possible formation of the spin-singlet pairs via direct antiferromagnetic exchange between neighboring Ti(1) and Ti(3) sites. The contribution of the superexchange via O orbitals to the Ti(1)-Ti(3) exchange coupling is found to be negligible. This was verified by calculating the exchange coupling constants with the subblocks of the LMTO Hamiltonian responsible for the Ti-O hybridization being set to zero. This calculation gave qualitatively same results for the exchange constants although the possibility for the superexchange via O orbitals was eliminated.
In the study  of the low-temperature phase of