Magnetic Properties from the Perspectives of Electronic Hamiltonian

Abstract

In this chapter, we review the quantitative and qualitative aspects of describing the properties of magnetic solids on the basis of electronic Hamiltonian that describes the energy states of a magnetic system using both orbital and spin degrees of freedom. To quantitatively discuss a magnetic property of a given magnetic system, one needs to generate the spectrum of its energy states and subsequently average the properties of these states with each state weighted by its Boltzmann distribution factor. Currently, this is an impossible task to achieve on the basis of an electronic Hamiltonian, so it is necessary to resort to a simple model Hamiltonian, that is, a spin Hamiltonian that describes the energy states of a magnetic system using only the spin degree of freedom. We show that a spin Hamiltonian approach becomes consistent with an electronic Hamiltonian approach if the spin lattice and its associated spin exchange parameters, to be used for the spin Hamiltonian, are determined by the energy‐mapping analysis based onDFTcalculations. The preferred spin orientation (i.e., the magnetic anisotropy) of a magnetic ion is not predicted by a spin Hamiltonian because it does not include the orbital degree of freedom explicitly. In contrast, the magnetic anisotropy is readily predicted by electronic structure theories employing both orbital and spin degrees of freedom, if one takes into consideration the spin–orbit coupling (SOC), , of a magnetic ion where and are, respectively, the spin and orbital operators and λ theSOCconstant. It was shown that the preferred spin orientation of a magnetic ion can be predicted and understood in terms of theHOMO–LUMOinteractions of the magnetic ion by takingSOC, , as perturbation. A spin Hamiltonian gives rise to the spin‐half misconception, namely, the blind belief that spin‐half magnetic ions do not possess magnetic anisotropy that arise fromSOC. This misconception contradicts not only experimental observations on spin‐half ions but also theoretical results based onDFTcalculations and perturbation theory analyses based on an electronic Hamiltonian. This misconception is a direct consequence from the limitedness of a spin Hamiltonian that it lacks the orbital degree of freedom. We show that the magnetic properties of 5dion oxides are better explained by theLS‐coupling than by thejj‐coupling scheme ofSOC, and the spin‐orbital entanglement of 5dions is not as strong as has been assumed.

Orbital Magnetic Moments of the High-Spin Co2+ Ions at Axially-Elongated Octahedral Sites: Unquenched as Reported from Experiment or Quenched as Predicted by Theory?

“…When the axial Co–X bond is very long, the 3 z 2 – r 2 ↓ state can be lowered below the xz ↓/ yz ↓ states so that the HOMO and LUMO occur within the degenerate ( xz , yz ) set (Figure d). In the latter case, |Δ L z | = 0, which predicts the || z spin direction. ,− So far, the || z spin direction has not been observed experimentally for the Co 2+ ions of CoO 4 X 2 octahedra.…”

“…The d-states of the Co 2+ ion in an CoO 4 X 2 octahedron, determined in the absence of SOC, can interact among themselves under the action of SOC, and hence introduce a small nonzero orbital moment on the Co 2+ ion. In terms of the perturbation theory in which the SOC is treated as the perturbation, the most important interaction is the one between the HOMO and LUMO states of CoO 4 X 2 . ,− In the spin-polarized description of electronic structure for a magnetic ion, the up-spin states lie, by convention, below the down-spin states. This makes the HOMO and LUMO for the Co 2+ ion of CoO 4 X 2 occur in the down-spin states.…”

“…Thus, in the configuration Ψ 1 , the HOMO is given by the xz ↓/ yz ↓ states and the LUMO by the xy ↓ state (Figure b). The SOC-induced interactions between states of identical spins are described by the Hamiltonian, ,− where the spin is oriented along the z ′-axis defined by the polar/azimuthal angles (θ, ϕ) with respect to the Cartesian coordinate, in which the 4-fold rotational axis of CoO 4 X 2 is chosen as the z -axis. (Hereafter, the directions parallel and perpendicular to the z -axis will be referred to as the || z and ⊥ z directions, respectively.)…”

“…(Hereafter, the directions parallel and perpendicular to the z -axis will be referred to as the || z and ⊥ z directions, respectively.) Then, the minimum difference, |Δ L z |, in the magnetic quantum numbers L z of the HOMO and LUMO is 1, so the selection rule ,− for the preferred spin orientation predicts that the spin orientation of the Co 2+ ion is the ⊥ z direction. In agreement with this prediction, the neutron diffraction studies − show that the Co 2+ spins of all compounds composed of CoO 4 X 2 octahedra have the ⊥ z direction.…”

“…As discussed above, a small nonzero orbital moment can be introduced into the Co 2+ ion by the interaction of the xy ↓ state with the yz ↓ state. The strength of this interaction between two states of different energies is inversely proportional to the energy gap Δ e 1 , ,− but this gap is nearly independent of R . Consequently, the orbital moment μ L , and hence the total moment μ tot , of the Co 2+ ion is predicted to be independent of the bond length ratio R .…”