Magnetic Exchange Interactions

exponentially in space. Therefore the overlapping of wave functions of magnetic ions is very small and hence it leads to very weak direct exchange interaction.

1.4.2 Superexchange Interaction

This is an indirect exchange interaction where the exchange interaction between two non-neighboring magnetic ions is mediated by means of a non-magnetic ion located between them. Superexchange interaction is a long range interaction. According to Kramers’ model [49], interaction between cations having more than half filled d-shells gives rise to the AFM interaction, while such interaction among cations having less than half filled d-shells gives rise to FM interaction. On the other hand, Slater [50], Goodenough and Loeb [51, 52]

proposed that the AFM interaction is also allowed among cations having less than half filled d-shell.

To understand the mechanism of superexchange interaction we have taken the example of interaction between two Mn - 3d orbitals. When core spins of magnetic cations are antiparallel to each other across an intermediate non-magnetic ion like oxygen, AFM interaction is facilitated due to the strong Hund’s coupling. Following Hund’s rule, each of the five 3d electrons of Mn²⁺ occupy a different 3d orbital in order to align parallel to each other. If the left-most Mn²⁺ ion has up-spin as shown in Fig. 1.6(a), the neighboring oxygen ion donates its down-spin 2p electron and form a partial covalent bond. This covalent bond is possible only when the oxygen ion donates its down spin electron; because all the Mn²⁺

orbitals (left side) contain up-spin electrons already. Similarly the up spin electron left in the oxygen 2p orbital is donated to the next Mn²⁺ ion in the chain. By the same argument, bonding can only occur if the electrons on the next Mn²⁺ ion are down-spin. This oxygen- mediated interaction leads to an overall AFM alignment between the two Mn²⁺ ions [53].

Figure 1.6: Schematic diagram of (a), (b) AFM and (c) FM superexchange interactions [53].

If the 3d orbital of Mn is less than half filled, the oxygen-mediated coupling between neighboring Mn ions can be either FM or AFM, depending upon whether the empty or the filled sub level of Mn - 3d orbitals points towards the electrons in oxygen 2p orbital. If Mn³⁺ ions in both side of oxygen point their respective empty d orbital towards oxygen 2p orbital, then this leads to AFM interaction as shown in Fig. 1.6(b). On the other hand, if one of the Mn³⁺ ions points the partially filled (i.e. with an electron) 3d - sub level towards oxygen 2p orbital while the other Mn³⁺ ion pointing its empty 3d - sub level towards oxygen 2p orbital, then this arrangement will lead to FM coupling as shown in Fig. 1.6(c).

Here in both cases, the coupling are permitted such that Hund’s rule and Pauli’s exclusion principle are not violated [53].

Superexchange can also be described by a Heisenberg Hamiltonian, in which the sign of J_ij is determined by the metal – oxygen – metal bond angle and the d electron configuration on the transition metal. According to semi-empirical Goodenough – Kanamori – Anderson rules 180º metal – oxygen – metal angles between identical metal ions with both

orbitals either filled or empty lead to AFM interactions whereas 90º angle results in FM [53].

In spinel compounds, the cations in tetrahedral (A) and octahedral (B) sites are bonded via O²⁻ ions. In spinels with magnetic ions in both A and B sites, superexchange interaction through A – O²⁻ − B networks is the strongest one and gives rise to the FIM ordering. Interactions through other networks such as B – O²⁻ − B and A – O²⁻ − A are also possible. In ferrites, A – O²⁻ − B angle is around 120° and it gives rise to AFM superexchange interaction. On the other hand, B – O²⁻ − B angles are close to 90° and hence they interact ferromagnetically. In some spinel chromites AFM interaction also occurs through B – O²⁻ − B networks due to spin canting. For example in NiCr₂O₄, AFM interaction exists within the B sites due to canting of the B site moments; but this interaction is weaker compared to that of between A and B site ions [54].

1.4.3 Double Exchange Interaction

Double exchange interaction occurs in mixed valent materials, i.e. in materials where the magnetic ions can exist in more than one oxidation states. It is also an indirect exchange interaction but with a transfer of electron from one magnetic ion to another magnetic ion through a non-magnetic intermediate ion and is popularly known as Zener double exchange interaction [55, 56]. Typical example of double exchange interactions between Mn³⁺ and Mn⁴⁺ ions through O^2– ion is shown in Fig. 1.7(a). Here, O^2– gives up its spin-up electron to Mn⁺⁴ and this is followed by the transfer of a spin up electron from Mn³⁺ to O^2–. At the end of the process Mn⁴⁺ reduces to Mn³⁺ and Mn³⁺ oxidized to Mn⁴⁺. Due to the two step process of exchange of electrons, it is called double exchange interaction. Zener model predicts that the carrier electrons can jump between two Mn ions only if their core electron spins are parallel to each other. If the Mn spins are not parallel, the electron transfer becomes difficult due to strong onsite Hund’s coupling as shown in Fig. 1.7(b). The electron transfer is also more difficult if the Mn – O – Mn bond is considerably bent away from 180°. Such deviation of bond angles affects the overlapping of Mn - d and oxygen - 2p orbitals. Thus double exchange interaction always gives rise to ferromagnetism. The movement of electron can be shown as,Mn O₁³_↑⁺ _{2 ,3↑ ↓}Mn⁴⁺ →Mn O⁴⁺ _{1 ,3↑ ↓}Mn³_2↑⁺, where the electron spins are labeled as

1, 2 and 3. Anderson and Hasegawa

by visualizing a second order process in which the electron transfer takes as follows

3 4 3 3 4 3

1 2 ,3 1 3 2 1 ,3 2

Mn O_↑⁺ _{↑ ↓}Mn ⁺ →Mn O Mn_↑⁺ _{↓ ↑}⁺→Mn O⁺ _{↑ ↓}Mn_↑⁺

Figure 1.7: (a) Sketch of double exchange O ion. (b) The mobility of

parallel to each other.

1.4.4 RKKY Interaction

The RKKY interaction occurs in metals with localized magnetic moments and the exchange interaction between the magnetic ions is mediated via conduction electrons. A localized magnetic moment of an ion, spin polarizes the conduction electrons and this polarization couples to the neighboring localized magnetic moment at a distance

Since direct coupling between the magnetic moments do not takes place, this interaction is an indirect one. The coupling takes the form of an

given by [46]

1, 2 and 3. Anderson and Hasegawa [57] presented the double exchange mechanism in detail by visualizing a second order process in which the electron transfer takes as follows

3 4 3 3 4 3

1 2 ,3 1 3 2 1 ,3 2

Mn O_↑⁺ _{↑ ↓}Mn ⁺ →Mn O Mn_↑⁺ _{↓ ↑}⁺→Mn O⁺ _{↑ ↓}Mn⁺_↑.

(a) Sketch of double exchange interaction which involves two Mn ions and one O ion. (b) The mobility of eg electrons improves if the localized spins are polarized

RKKY Interaction

The RKKY interaction occurs in metals with localized magnetic moments and the exchange interaction between the magnetic ions is mediated via conduction electrons. A localized magnetic moment of an ion, spin polarizes the conduction electrons and this

ization couples to the neighboring localized magnetic moment at a distance

Since direct coupling between the magnetic moments do not takes place, this interaction is an takes the form of an r-dependent exchange interactio

mechanism in detail by visualizing a second order process in which the electron transfer takes as follows

which involves two Mn ions and one electrons improves if the localized spins are polarized and

The RKKY interaction occurs in metals with localized magnetic moments and the exchange interaction between the magnetic ions is mediated via conduction electrons. A localized magnetic moment of an ion, spin polarizes the conduction electrons and this ization couples to the neighboring localized magnetic moment at a distance r away.

Since direct coupling between the magnetic moments do not takes place, this interaction is an dependent exchange interaction J_RKKY (r)

( ) ( )

3

cos 2

_F

RKKY

J r k r

α r

(1.2) where, k_F is the radius of the spherical Fermi surface. This interaction has oscillatory dependence of the distance between the magnetic ions and it is in long range. Such interaction gives rise to either FM or AFM coupling depending on the distance between the magnetic ions.

1.4.5 Anisotropic Exchange Interaction

Anisotropic exchange interaction was proposed by Dzyaloshinsky-Moriya and hence it is also known as Dzyaloshinsky–Moriya (DM) interaction [58, 59]. This exchange mechanism leads to the tilting of the magnetic moments of antiparallel spins towards each other and hence gives rise to a net magnetic moment. However, such a canted spin arrangement is possible only if the magnetic crystal symmetry remains same as that of the original one having antiparallel spins. The exchange energy between the magnetic moments at this canted configuration can be written as follow,

(

^{1 2}

)

DM

.

H = D S × S

(1.3)

Where S₁

and S₂

are the spins of two interacting magnetic ions and D

is DM vector. Vector D

vanishes when the crystal field has inversion symmetry with respect to the middle point connecting the two magnetic ions. Usually D

lies either parallel or perpendicular to the line connecting the two spins, depending on the symmetry. The DM interaction tries to align S₁

and S₂

at right angle to each other in a plane perpendicular toD

. Its effect is therefore very often to cant (i.e. slightly rotate) the spins by a small angle. It commonly occurs in antiferromagnets and then results in a small FM component of the moments which is produced perpendicular to the spin-axis of the antiferromagnet. The effect is known as weak ferromagnetism. Some of the AFM crystals like α-Fe2O3, Cr2O3, MnCO3, CoCO3, etc. exhibit such weak FM behavior [46, 60].