1. Introduction

Determination of the energy band structure of solids is a many-body problem. Band theory, a mean-field theory to treat this problem, in the framework of the local spin density approximation (LSDA), has been successful for many kinds of materials and has become the de facto tool of first-principles calculations in solid-state physics. It has contributed significantly to the understanding of material properties at the microscopic level. However, there are some systematic errors which have been observed when using the LSDA. In particular, the LSDA fails to describe the electronic structure and properties of the electron systems in which the interaction among the electrons is strong. These are now called strongly correlated electron systems, and many new concepts to address these phenomena have been constructed. However, the understanding of these systems is not complete. A wide variety of physical properties arise from the correlations among 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 [1]. Only in the one-dimensional case do we have a variety of theoretical tools at our disposal to study these models in a systematic manner. For two- and three-dimensional models, one is often unable to assess confidently whether a given physical phenomenon is indeed captured by the idealized Hamiltonian under consideration or whether a theoretical prediction reflects a true feature of this Hamiltonian, rather than an artifact of the approximation used in its solution. These difficulties originate in the nonperturbative nature of the problem and reflect the presence of several competing physical mechanisms for even the simplest models. The interplay of localization and lattice coherence, that of quantum and spatial fluctuations and that of various competing types of long-range order are important examples.

With the aim of undertaking a systematic investigation of the trends in some mixed-valent and charge- and orbital-ordering compounds, we present the theoretically calculated electronic structure, optical, magneto-optical, and photo-emission spectra for some 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 [2]. Especially important are the transitions driven by correlation effects associated with the electron-electron interaction. The insulating phase caused by the correlation effects is categorized as the Mott Insulator. Near the transition point the metallic state shows fluctuations and orderings in the spin, charge, and orbital degrees of freedom. The properties of these metals are frequently quite different from those of ordinary metals, as measured by transport, optical, and magnetic probes. We consider the , , , 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 [8], and its earliest example is the Verwey transition in magnetite [9], although this picture turned out to be too simplified for this compound [10, 11]. There are several charge-fluctuating inhomogeneous mixed-valence compounds containing rare-earth ions. They are the rare-earth pnictides , , 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) [12], the LSDA+ [13] method, the GW approximation [14], and dynamical mean-field theory (DMFT) [1, 15, 16] have sought to remedy this problem and have met with considerable success. The LSDA+ 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 [17]. 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 [17] 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 [18], 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 [19]: 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 [20]. If we subtract this expression from the LDA total energy functional and add a Hubbard-like term (neglecting for a while exchange and nonsphericity), we will have the following functional: 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 [20]) 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 [24]. 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. [27] 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. [28] is based on the projector augmented wave (PAW) method [30], an efficient all-electron method without shape approximations on the potential or electron density. Based on a Car-Parrinello-like formalism [31], the PAW method allows complex relaxations and dynamical properties in strongly correlated systems. Cococcioni and De Gironcoli [29] reexamine the LDA+ method in the framework of a plane-wave pseudopotential approach (PWPP). A simplified rotational-invariant formulation was adopted. They demonstrate the accuracy of the method, computing structural, and electronic properties of a few systems including transition and rare-earth correlated metals, transition metal mono-oxides, and iron silicate [29], transition metal centers [32], and iron heme complexes [33].

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

To obtain Hubbard 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 [37] to identify the shift in the corresponding eigenvalue with the second-order derivative of the LDA total energy with respect to orbital occupation. It has, however, the effect of leaving a rather artificial system to perform the screening, in particular when it is not completely intra-atomic. In elemental metallic iron, for instance, only half of the screening charge is contained in the Wigner-Seitz cell [34]. This fact, in addition to a sizable error due to the atomic sphere approximation used, could be at the origin of the severe overestimation of the computed on-site Coulomb interaction. For example, Anisimov and Gunnarsson computed the effective on-site Coulomb interaction in metallic Fe and Ce. For Ce the calculated Coulomb interaction was found to be about 6 eV in good agreement with empirical and experimental estimates ranging from 5 to 7 eV [38], while the result for Fe (also about 6 eV) was surprisingly high since was expected to be in the range of 1-2 eV for elemental transition metals, with the exception of Ni [39, 40].

Cococcioni and De Gironcoli [29] provide an internally consistent, basis-set-independent, method based on a linear response approach for the calculation of the effective interaction parameters in the LDA+ 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 [28]. 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 [41]. 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 [42]. 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 [43]

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 [44], and, second, the Kerr effect has rapidly developed into an appealing spectroscopic tool in materials research. The technological research on the Kerr effect was initially motivated by the search for good magneto-optical materials that could be used as information storage medium. In the sequence of this research, the Kerr spectra of many ferromagnetic materials were investigated. An overview of the experimental data collected on the Kerr effect can be found in the review articles by Buschow [45], Reim and Schoenes [46], and Schoenes [43].

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

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

3.1. Cubic

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 [2]. 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) [9]. This is a particular MIT that has been studied for quite some time [120]. The Verwey transition is characterized by an abrupt increase in the electrical conductivity by two orders of magnitude on heating through [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 Fe_B-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 [120] since Verwey first proposed that, below , all and sit on different chains [9].

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 [141] indicate a gap of 0.14 eV between occupied and empty electronic states and also show a strong temperature dependence of the optical conductivity in the energy region of 0 to 1 eV.

The interpretation of the optical and MO spectra of 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 [142]. Finally, recent investigation of the optical and MO spectra of and and substitution has been carried out in [143].

Energy band structure calculations for in the high-temperature cubic phase have been presented in [152] using the self-consistent spin-polarized augmented plane wave method with the local spin-density approximation. The calculations show that 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 [153] 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 Fe_A-O₁ and Fe_B-O₂ are different and equal to 1.876 and 2.066 Å, respectively.

Figure 1: Crystal structure of .

Details concerning the mechanism of the Verwey transition and the type of LRCO are still unclear. Many elaborate theories for the Verwey transition have been proposed. Anderson [156] pointed out the essential role of short-range charge ordering (SRCO) in the thermodynamics of the transition. The observed entropy change in the transition [(~0.3 to 0.35)-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) [9]. Since that time the type of charge ordering has been the subject of debate [120]. As an example, in Mizoguchi’s model [164, 165] ions run in pairs of 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 [120]. Despite the wealth of effort devoted to investigating the low-temperature phase of magnetite, there is still no completely satisfactory description of the ordering of the Fe atoms on the octahedral or 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 [170], whereas the observation of a magnetoelectric effect indicated even lower symmetry in the low-temperature phase [171]. 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 [172]. 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 [177], have been proposed.

Recent bond-valence-sum analysis [178] of high-resolution neutron and X-ray powder diffraction data results in a small charge disproportion of only 0.2  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 [181]. Since no direct experimental confirmation of this charge- and orbital-order pattern is so far available, the interpretation of these results is still open to debate. However, this behavior seems to be universal and has recently been found in several other charge-ordered mixed-valent systems [182–186].

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

Figure 2: LSDA partial DOS of [10].

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 [188] to explain the magnetization data and was later confirmed by neutron scattering measurements [189]. Measurements indicated that the magnetic moment of an iron atom on the site is much smaller than the 5.0  of a pure ion [190]. 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 [10] 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 [141] 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 [10].

Figure 3: LSDA+ energy band structure and total DOS (in states/(unit cell eV)) of [10].

Figure 4: LSDA+ partial DOS of [10].

3.1.4. Optical and MO Properties

After the consideration of the above band structure properties, we turn to the optical and MO spectra. In Figure 5 experimental optical reflectivity and the diagonal part of the dielectric function 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.

Figure 5: Optical reflectivity and diagonal parts of the dielectric function of calculated in the LSDA and LSDA+ approximation [10] compared with experimental data [140] (open circles).

Figure 6: The absorptive part of the diagonal optical conductivity of calculated in LSDA (dashed line) and LSDA+ (solid line) approximations [10] compared with experimental data [141] (solid squares) and [140] (open circles).

Let us consider now the magneto-optical properties of . In Figure 7 we show the experimentally measured [146] 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 [10]. This picture clearly demonstrates that the better description is achieved with the LSDA+ approach.

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

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 [141], 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 M_2,3 core-absorption edges in and [151]. For the M_2,3 prethreshold MCD spectra were measured above and below the Verwey transition temperature. Van Der Laan et al. [192] reported the XMCD spectra at the Ni 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. [193]. Magnetic circular dichroism is reported for ferrite in [194] 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 [191]. The XAS and XMCD spectra at , and edges for transition metals sites were calculated.

In Figure 8 the experimentally measured Fe XMCD spectra [150] in are compared to the theoretical ones calculated within the LSDA+ approach [191]. 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.

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

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 [195].

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 [151] in is compared to the theoretical one calculated within the LSDA+ approach [191]. 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.

Figure 9: Fe XMCD spectra in calculated with the LSDA+ approximation [191] in comparison with the experimental data (circles) [151].

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 [179] 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 [196]. 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 [155] 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 [141] 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 [27]. 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 [155] 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.

Figure 10: Total DOS and band structure of the phase of self-consistently obtained by the LSDA+ with  eV and  eV [155]. The Fermi level is denoted by the horizontal line and is taken as the zero of energy. A bandgap of 0.18 eV opens between M and Γ symmetry points. The energy bands predominantly originating from the and states are shown in dark gray (red) and light gray (green) colors, respectively, whereas the gray (blue) color corresponds to bands. For the majority spin an energy gap of ~2 eV opens between and bands shown in dark gray (red) and light gray (green) colors, respectively. The corresponding contributions to the total DOS are shown (a).

Figure 11: Partial DOS obtained from the LSDA+ calculations with  eV and  eV for the phase of [155]. The Fermi level shown by dotted lines. A charge-ordered insulating solution is obtained. Fe minority states corresponding to and sites are occupied () and located just below the Fermi level, whereas and are empty (). The charge difference between 2+ and 3+ cations is found to be .

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 [181]). 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 [001] 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 [198]. 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 [182].

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.

Table 1: orbital contribution to the formation of minority spin states with occupancy evaluated by diagonalization of the occupation matrix [155]. Although one of the states of and sites is almost occupied with   the minority spin occupancies of and cations are less than . The occupied states of and cations are predominantly of and characters, respectively. The sum of  () occupations is given in the last column.