Relative importance of crystal field versus bandwidth to the high pressure spin transition in transition metal monoxides

The crystal field splitting and d bandwidth of the 3d transition metal monoxides MnO, FeO, CoO and NiO are analyzed as a function of pressure within density functional theory. In all four cases the 3d bandwidth is significantly larger than the crystal field splitting over a wide range of compressions. The bandwidth actually increases more as pressure is increased than the crystal field splitting. Therefore the role of increasing bandwidth must be considered in any explanation of a possible spin collapse that these materials may exhibit under pressure. At low pressures the 3d transition metal oxides (TMOs) are magnetic and high spin. The existence and nature of a high spin to low spin transition for these materials at high pressure has been discussed widely in the literature. The earliest theoretical studies of the system with density functional theory (DFT) predicted that magnetic moments on the metal ions switch to a much smaller value at experimentally accessible pressures[1]. Subsequent experiments claimed to confirm this prediction[2] only to be contradicted by later experiments and more refined electronic structure calculations[3; 4]. The question of the mechanism for the spin collapse in 3d transition metal oxides has recently been revisited, with differing conclusions from dynamical mean field theory[5] and a model fit to experimental results[6]. These papers explain the spin transition in terms of the crystal field splitting and the 3d bandwidth with Kunes et al. arguing for the crystal field splitting driving the transition and Mattila et al. suggesting that there is a competition between the crystal field splitting and the bandwidth. We address this situation by calculating the crystal field splitting and 3d bandwidth for four TMOs as a function of pressure within DFT[7]. To drive a spin collapse via the crystal field splitting, the decreased exchange energy available to electrons in a high spin configuration must be offset by the change in the single particle energy levels due to the crystal field splitting. The effect of pressure on the exchange energy is small whereas the crystal field splitting does significantly change the energy of the d orbitals as the material is compressed. Typically the crystal field splitting is treated as being dominated by the electrostatic interaction of the neighboring oxygen atoms with the metal sites. However, a much greater effect is the increasing overlap and subsequent hybridization of the oxygen 2p states with the metal 3d states[3]. This change in single particle energy levels favors the pairing of the d electrons in the lower energy d orbitals, resulting in a reduced magnetic moment. The other explanation for the spin collapse is that as the pressure is increased, the band energy sacrificed to break symmetry and form a magnetic state eventually becomes larger than the energy to be gained the magnetic interaction. This competition can be analyzed in light of the extended Stoner theory of magnetism[8; 9] which allows these competing effects to be quantified. The extended Stoner theory works equally well for ordered and disordered magnetic states and can predict the optimal magnetic moment in contrast to the simple Stoner model that only determines if an instability to magnetism exists. The essential ingredients to this analysis are the change in the interaction energy with respect to magnetic moment and the change in magnetism with respect to the exchange splitting. Both of these quantities are driven by a change in the bandwidth near the Fermi level, which in this case is the 3d bandwidth. In this case an increase in the bandwidth will raise the energy necessary to form a high spin state, eventually causing a transition to a low spin state. We compare these mechanisms by studying an idealized system with a purely cubic structure at zero temperature in order to isolate the electronic contributions to the phenomenon (there is significant magnetoelastic coupling in the real monoxides[10]). Despite the well known failings of DFT withing the local density approximation to capture properties such as band gap of the TMOs, it is nonetheless useful for gaining a qualitative intuition into the physics of these materials. Additionally, nonmagnetic LDA is the typical starting point for both the LDA+U [11; 12] and LDA+DMFT[13] approaches, giving a special importance to the details of this solution. We will also comment on how the strong correlations present affect these quantities by examining LDA+U results. We use a full potential linear muffin tin orbital (LMTO) method, that allows the results of the DFT calculations to be transformed directly into a tight binding Hamiltonian with no approximations[14]. In this tight binding approach, we consider a set of atomic-like wavefunctions centered on the ions where the each state has energy ǫα = 〈Ψα| Ĥ |Ψα〉 (1) and states are coupled by hopping matrix elements tα,β = 〈Ψα| Ĥ |Ψβ〉 (2) where Ĥ is an effective Hamiltonian determined from the results of the DFT calculation. As a first application of this technique, we consider the on site energies of the oxygen 2p orbitals and metal 3d state in Fig.1. The plot shows that as the number of electrons in the problem is increased, the energy gap between the oxygen and metal 3d states decreases and the hopping between those states also becomes less favorable. Also the hopping between d states on nearest neighbor metal ions is included showing it to be non-negligible compared to the p − d hopping. These calculations also provide estimates of the crystal field splitting and d bandwidth in the TMOs as a function of pressure, determining the relative size of these terms and how they behave under pressure. In this case the bands near the Fermi energy that are important to the low energy physics of the system are formed by the hybridization of the metal 3d orbitals with the surrounding oxygen 2p orbitals. For the cubic NaCl structure, there are two different symmetries of d-orbital to be considered. The first are the two eg orbitals that point towards the nearest neighbor oxygen atoms. These orbitals form bands that are raised in energy versus the isolated d orbital because of the electrostatic interaction of the electrons on the metal ion with the negatively charged O ion as well as the hybridization of the p and d orbitals. The second class of orbital which is of interest are the three t2g orbitals that do not point towards the nearest neighbor oxygens. These orbitals will form bands which are relatively lower in energy than the eg orbitals. The difference in energy between these two sets of orbitals is conventionally termed crystal field splitting.