Magnetic properties

Magnetism in transition metal compounds is considerably more complex than that of isolated atoms because of the interaction (coupling) between atomic moments. Coupling between moments is respon- sible for cooperative magnetism. A pair of electrons of like spin, each localized on a different atom, is lower in energy than a pair with opposite spin by an amount called the inter-atomic exchange energy.

This phenomenon results from exchange interactions, and is expressed by the exchange parameterJ defined by the equation

Hex=−

∑

i j

JSiSj (4.6)

whereHexis the exchange energy of atomsiandjwhich have total spinsSiandSj, respectively. The exchange parameterJhas a positive value for a ferromagnetic compound (collinear order of moments, JFM>0), whereas it attains a negative value for antiferromagnetic ordering (moments of neighboring atoms are exactly opposed,J_AFM<0).

One can distinguish two classes of exchange.Direct exchangeoccurs between moments on atoms that are close enough to have significant overlap of their wave functions; exchange coupling is strong but decreases rapidly with increasing interatomic distance. Indirect exchangeon the other hand cou- ples moments over relatively large distances. It can act through an intermediary nonmagnetic ion (superexchange), or through itinerant electrons (RKKY). Superexchange generally occurs in insula- tors, whereas the RKKY (Ruderman, Kittel, Kasuya, and Yoshida) coupling is important in metallic systems [1].

Superexchange, or interaction between localized moments of ions in insulators that are too far apart to interact by direct exchange, operates through the intermediacy of a nonmagnetic ion (e.g. metal- oxygen-metal). A typical example is the 180^◦cation-anion-cation interaction in oxides of rock salt structure, where the antiparallel orientation of spins on the neighboring cations is favored by covalent mixing of the anionporbital with the cationdorbitals on each side [Fig.4.8(a)]. Different types of superexchange interaction are possible, depending on the structure of the compound and the electronic configuration of the cations.

Two important types of superexchange are thecorrelation superexchangeand thedelocalization superexchange.

Delocalization superexchange involves transfer from one cation to another as a result of cation-cation or cation-anion-cation interaction. Correlation superexchange is restricted to cation-anion-cation in- teraction. Goodenough, Kanamori, and Anderson (GKA) have described rules governing the sign and magnitude of superexchange [79]. Very briefly, the GKA rules state that a 180^◦cation-anion-cation interaction between half-filleddorbitals is antiferromagnetic [Fig.4.8(a)], whereas for an overlap be- tween an occupied and an emptydorbital the resulting 180^◦-exchange is ferromagnetic [Fig.4.8(b)].

On the other hand, a 90^◦cation-anion-cation interaction between half-filleddorbitals is ferromagnetic, provided that thedorbitals are bonded to orthogonal anion orbitals [Fig.4.8(c)].

In order to understand visually how the GKA rules work, let us start with Fig.4.8(a). The electron of the metal ion (M) has spin up. Although the ligand (L)porbital formally contains two electrons, it can nonetheless be involved in bonding with the metal orbital. Such bonding means a pairing between the metal unpaired electron and that of opposite spin in theporbital. This pairing leaves an electron in theporbital, which is of the same spin as that at the left-hand side of the metaldorbital. Pairing between this electron and the metaldelectron in the orbital at the right-hand side of the diagram requires that this latterdelectron has its spin in the opposite direction to that of thedelectron in the orbital at the left-hand side of the diagram. That is, the two metal ions are antiferromagnetically linked by the ligand. In this example, the two metal ions were taken to be identical.

Suppose now that the left-hand side metal ion has its unpaired electron in anegorbital (we assume octahedral symmetry), whereas the right-hand side metal ion has an unpaired electron in at2g level [Fig.4.8(b)]. Starting with the left-hand sidedorbital, the argument follows that one given above until we reach the right-hand sidedorbital, which is orthogonal (small overlap) to the ligandporbital.

Unpaired electrons in orthogonal orbitals tend to align themselves with parallel spins due to Hund’s rule exchange coupling. Therefore, the electron at the right-hand sidedorbital is of the same spin as that at the left-hand side. The two metal ions are ferromagnetically linked. Key to the development of ferromagnetic coupling in the argument above was a step involving the orthogonality of two orbitals.

The orthogonality need not be between metal and ligand orbitals. Fig.4.8(c) shows an interaction pathway which includes twoporbitals on the same atom; theseporbitals are mutually orthogonal.

Again, ferromagnetic coupling is realized.

I have already mentioned that a system with a partially filled electronic band is susceptible to structural distortions so as to remove its orbital degeneracy (cooperative Jahn-Teller effect). Since the kind of orbitals which are occupied determine the magnitude and the sign of the exchange interaction J, such orbital ordering is closely related with the magnetic one. Given the type of orbital ordering, and by using the aforementioned GKA rules, one can predict the type of magnetic structure realized.

4.2 Exchange interactions in transition metal compounds

M

M

M

M L

L

L

M

M

(a)

(b)

(c)

180 antiferromagnetic exchange

o

180 ferromagnetic exchange

o

90 ferromagnetic exchange

o

Figure 4.8: Schematic illustration of the Goodenough-Kanamori-Anderson rules, regarding the cation- anion-cation superexchange interactions: (a) Ligand (L)σ-orbital mediated antiferromagnetic coupling between two metal (M) ions; (b) and (c) show two ways in which orbital orthogonality can lead to ferromagnetic coupling. The small arrows in the orbitals represent electrons. After Ref. [77].

Starting from simple considerations, and within the Hubbard model, it can be shown that the same superexchange interaction which is responsible for the magnetic ordering, leads at the same time, in the case of an orbitally degenerate system, into an orbital ordering closely related to the type of magnetic ordering.

Let us consider a case with two-fold orbital degeneracy (Fig.4.9). We have four possible configura- tions: same orbital-same spin, different orbital-same spin etc. For situation Fig.4.9(a), Pauli’s exclusion principle forbids a transfer of the electron, hence no change in energy will take place (∆E=0). In cases (b) and (d), the gain in energy is∝2t²/U(see the last paragraph in Section 2.2 of this Chapter). If in the case of Fig.4.9(c) the electron hops from left to right, it has to overcome the Coulomb energyU, but gains the Hund’s coupling energyJH. Hence, configuration (c) with antiferro-orbital ordering and ferromagnetic spin is the energetically most favorable state. Thus, whereas in the non-degenerate case the superexchange gives antiferromagnetic ordering, in the case of orbital degeneracy it leads simulta- neously to both spin and orbital ordering, and the magnetic ordering may turn out to be ferromagnetic.

Besides the types of ordering of moments discussed hitherto, various magnetic excitations can oc- cur in the different magnetic states. One of the elementary excitations is a quantized spin wave in a classical ferromagnet [94,95]. In the ground state of a ferromagnet, all the spins are oriented paral- lel giving rise to some spontaneous magnetization. We can form an excited state by reversing one of the spins. Alternatively, we can let all of the spins to share the reversal, in which case the spins take on a wavelike arrangement. The elementary excitations of a spin system having a wavelike form are calledmagnons. Magnons are quasi-particle excitations which obey the Bose-Einstein statistics, like phonons. The dispersion relations of magnons in a solid can be determined by inelastic neutron scat- tering [96]. At the center of the Brillouin zone (|kkk|=0), the magnon frequencyωmagnon(kkk)has a finite

ΔE = 0 (a)

ΔE = -2t /U² (b)

ΔE = -2t /U² (d) ΔE = -2t /(U-J )² _H

(c)

Figure 4.9: The various configurations of nearest neighbors for doubly-degenerate orbitals. JHis the Hund’s rule exchange constant. After Ref. [79].

value [ωmagnon(0)>0] due to static magnetization [94,95]. This practically means that one-magnon scattering can be probed by means of Raman (or Brillouin) spectroscopy. Furthermore, magnons can be observed also for antiferromagnetic compounds. In particular, assuming the simplest antiferromagnetic case where the spins form two magnetic sublattices of oppositely directed spins, and without the pres- ence of an applied magnetic field, the magnon frequency spectrum splits into two degenerate branches ω^±(kkk)[94,95]; application of a magnetic field can lift this magnon-frequency degeneracy. The ap- pearance of two-magnon features in the Raman spectra is also possible, as long as the two magnons participating in the scattering process have equal and opposite wavevectorskkk. Much more theoretical and experimental aspects on the magnon scattering processes in the various types of magnetic solids (ferromagnets, antiferromagnets, ferrimagnets etc.) are covered in Ref. [95].