Magnetic and orbital ordering in cuprates and manganites

We address the role played by orbital degeneracy in strongly correlated transition metal compounds. The mechanisms of magnetic and orbital interactions due to double exchange (DE) and superexchange (SE) are presented. Specifically, we study the effective spin-orbital models derived for the d ions as in KCuF3, and for the d ions as in LaMnO3, for spins S = 1/2 and S = 2, respectively. The magnetic and orbital ordering in the undoped compounds is determined by the SE interactions that are inherently frustrated, carrying both antiferromagnetic (AF) and ferromagnetic (FM) channels due to low-spin and high-spin excited states, respectively. As a result, the classical phase diagrams consist of several magnetic phases which all have different orbital ordering: either the same orbitals (x − y or 3z − r) are occupied, or two different linear combinations of eg orbitals stagger, leading either to G-AF or to AAF order. These phases become unstable near orbital degeneracy, leading to a new mechanism of spin liquid. The model for d Mn ions in collosal magnetoresistance compounds provides an explanation of the observed A-AF phase, with the orbital order stabilized additionally by the Jahn-Teller effect. Possible extensions of the model to the doped compounds are discussed both for the insulating polaronic regime and for the metallic phase. It is shown that the spin waves are well described by SE in the insulating regime, while they are explained by DE for degenerate eg orbitals in the metallic FM regime. Orbital excitations contribute to the hole dynamics in FM planes of LaMnO3, characterized by new quasiparticles reminiscent of the t-J model, and a large redistribution of spectral weight with respect to mean-field treatments. Finally, we point out some open problems in the present understanding of doped manganites. I CORRELATED TRANSITION-METAL OXIDES WITH ORBITAL DEGENERACY Theory of strongly correlated electrons is one of the most challenging and fascinating fields of modern condensed matter. The correlated electrons are responsible for such phenomena as magnetic ordering in transition metals, heavy-fermion behavior, mixed valence, and metal-insulator transitions [1–3]. They play also a prominent role in transition metal oxides, where they trigger such phenomena as superconductivity with high transition temperatures in cuprates and collosal magnetoresistance (CMR) in manganites. At present, most of the current studies of strongly correlated electrons deal with models of nondegenerate orbitals, such as the Hubbard model, Kondo lattice model, Anderson model, and the like. Strong electron correlations lead in such situations to new effective models which act only in a part of the Hilbert space and describe the low-energy excitations. A classical example is the t-J model which follows from the Hubbard model [4,5], and describes a competition between the magnetic superexchange and kinetic energy of holes doped into an antiferromagnetic (AF) Mott insulator. The realistic models of correlated electrons are, however, more complex than the Hubbard or Kondo lattice model. Transition metal oxides crystallize in a threedimensional (3D) perovskite structure, where the oxygen ions occupy bridge positions between transition metal ions, as in LaMnO3, or in similar structures with two-dimesional (2D) planes built by transition metal and oxygen ions, as in CuO2 planes of high temperature superconductors. The oxygen ligand 2p orbitals play thereby a fundamental role in these systems, and determine both the electronic structure and actual interactions between the electrons (holes) which occupy correlated 3d orbitals of transition metal ions. The bands in transition metal oxides are built either by pσ or by pπ oxygen orbitals which hybridize with the respective 3d orbitals of either eg or t2g symmetry. Taking an example shown in Fig. 1, it is clear that the overlap between the pσ orbitals and dx2−y2 orbitals is larger than that between the pπ orbitals and the corresponding d orbitals of t2g symmetry. Therefore, the t2g and pπ states are filled in the cuprates, and the relevant model Hamiltonians known as charge transfer models include frequently only the eg orbitals of transition metal ions and the pσ oxygen orbitals between them. There are two crucial parameters which decide about the physical properties of a transition metal oxide, provided the d− p hybridization elements are much smaller than the value of the on-site Coulomb interaction U . The latter parameter has to be compared with the splitting between the 3d and 2p orbitals, given by the so-called charge-transfer energy, ∆ = |εp − εd|, where εd and εp are the energies of an electron (hole) in these states, respectively. These systems are called MottHubbard insulators (MHI) when U < ∆, and it is in this limit that the Hubbard model would apply directly for the description of a metal-insulator transition. In FIGURE 1. Examples of configurations for transition-metal 3d orbitals which are bridged by ligand 2p orbitals in transition metal oxides (after Ref. [1]). the opposite case, one deals instead with charge-transfer insulators, as introduced by Zaanen, Sawatzky and Allen fifteen years ago [6,7]. Both classes of correlated (in contrast to band) insulators have quite different spectral properties, but in the strongly correlated regime the charge-transfer insulators resemble MHI, with a charge-transfer energy ∆ playing a role of the effective U [8]. In reality, however, many oxides are found close to the above qualitative boarder line between Mott-Hubbard and charge-transfer systems (Fig. 2), and one might expect that the only relevant description has to be based on the charge-transfer models which include explicitly both d and p orbitals. Nevertheless, a reduction of such models to the effective simpler Hamiltonians dealing only with correlated d-like orbitals is possible, and examples of such mapping procedure have been discussed in the literature [9–12]. Unfortunately, there is no general method which works in every case, but the principle of the mapping procedure is clear, at least in perturbation theory. We will follow this idea in the present paper and concentrate ourselves on such simpler models which describe interactions between 3d electrons, determined by the effective hopping between transition metal ions which follows from intermediate processes involving charge-transfer excitations at the 2p oxygen orbitals [13]. It will be clear from what follows that while this simplification is allowed, there is in general no way to reduce these models any further to those of nondegenerate d orbitals, at least not for the oxides with a single electron or hole occupying (almost) degenerate eg orbitals. We concentrate ourselves on a class of insulating strongly correlated transition metal compounds, where the crystal field leaves the 3d orbitals of eg symmetry explicitly degenerate and thus the type of occupied orbitals is not known a priori , while the effective magnetic interactions between the spins of neighboring transition metal ions are determined by orbitals which are occupied in the ground state [14–17]. The most interesting situation occurs when eg orbitals are partly occupied, which results in rather strong magnetic interactions, accompanied by strong JahnTeller (JT) effect. Typical examples of such ions are: Cu (d configuration, one hole in eg-orbitals) [18], low-spin Ni 3+ (d configuration, one electron in eg-orbitals) [19–21], as well as Mn [22] and Cr ions (high-spin d configuration with one eg electron). The situation encountered for d 9 (or d) transition metal ions is simpler, as the t2g orbitals are filled. The effective interactions may then be derived by considering only eg orbital degrees of freedom and spins s = 1/2 at every site, and were first considered by Kugel and Khomskii more than two decades ago [18]. In the case of d configuration one needs instead to consider larger spins S = 2 which interact with each other, due to virtual excitation processes which involve either eg or t2g electrons [23]. Finally, the early transition-metal compounds with d or d ions give also some interesting examples of degenerate t2g orbitals [24–29]. In general, the magnetic superexchange and the coupling to the lattice are weaker in such cases due to a weaker hybridization between 3d and 2p orbitals (Fig. 1). Moreover, this problem is somewhat different due to the symmetry of the orbitals involved, and we will not discuss it here. The collective behavior of eg electrons follows from their interactions. The models of interacting electrons in degenerate 3d states are usually limited to the leading on-site part of electron-electron interaction given by the Coulomb and exchange elements, U and JH , respectively. The model Hamiltonian which includes these interactions is of the form,