1 Direct, indirect exchange and superexchange

the electrons spend most of their time in between neighboring atoms when the

interatomic distance is small. This gives rise to antiparallel alignment and therefore

negative exchange. (antiferromagnetic), Fig. 12.



Fig. 12. Antiparallel alignment for small interatomic distances.

If the atoms are far apart the electrons spend their time away from each other

in order to minimize the electron-electron repulsion. This gives rise to parallel

alignment or positive exchange (ferromagnetism), Fig. 13.



Fig. 13. Parallel alignment for large interatomic distances.

For direct inter-atomic exchange j can be positive or negative depending on

the balance between the Coulomb and kinetic energies. The Bethe-Slater curve

represents the magnitude of direct exchange as a function of interatomic distance.

Cobalt is situated near the peak of this curve, while chromium and manganese are

on the side of negative exchange. Iron, with its sign depending on the crystal

structures probably around the zero-crossing point of the curve, Fig. 14.



•



Fig. 14. The Bethe-Slater curve.

Indirect exchange



Indirect exchange couples moments over relatively large distances. It is the

dominant exchange interaction in metals, where there is little or no direct overlap

37



between neighboring electrons. It therefore acts through an intermediary, which in

metals are the conduction electrons (itinerant electrons). This type of exchange is

better known as the RKKY interaction named after Ruderman, Kittel, Kasuya and

Yoshida. The RKKY exchange coefficient j oscillates from positive to negative as

the separation of the ion changes and has the damped oscillatory nature shown in

Fig. 15. Therefore depending on the separation between a pair of ions their

magnetic coupling can be ferromagnetic or antiferromagnetic. A magnetic ion

induces a spin polarization in the conduction electrons in its neighborhood. This

spin polarization in the itinerant electrons is felt by the moments of other magnetic

ions within the range leading to an indirect coupling.

In rare-earth metals, whose magnetic electrons in the 4f shell are shielded by

the 5s and 5p electrons, direct exchange is rather and indirect exchange via the

conduction electrons gives rise to magnetic order in these materials.



Fig. 15: The coefficient of indirect (RKKY) exchange versus the interatomic spacing

a.

MnO

• Superexchange in antiferromagnetic



38



Figure 16: Antiferromagnetic MnO [Reprinted from Blundell, Stephen;

1st

Magnetism in Condensed Matter,

Edition; Oxford Univ. Press]

• electrons interact with neighbouring electrons

⇒ direct exchange, no intermediary

•



BUT: direct exchange often not possible



⇒ insufficient direct overlap between orbitals

⇒ for example, 4f electrons in rare earths strongly localized, close to the nucleus,

so direct exchange interaction uneffective.

The exchange interaction described in the previous section stated a direct

exchange, so electrons would interact only with their next neighbours. But if the

electrons are strongly localised there is only a small probability for them to interact

with electrons an neighbouring atoms. This is for example the case for the 4f

electrons in the rare earths but MnO still exhibits the characteristics of an

antiferromagnet. The reason is that in the MnO-lattice (see Fig. 16) the Oxygen



O2



takes up the rôle of an intermediary transmitting the exchange forces. This is then

called superexchange.

4.2 Anisotropic



exchange: Crystalline anisotropy



By looking at the Heisenberg Hamiltonian:



39



ur uu

r

µ = −2

H

J

S

.

S

∑ i> j ij i j ,



It is obvious that the asscociated exchange energy and therefore the

magnetization should be totally isotropic, as



µ

H



is invariant with respect to choice



of coordinate systems. So the experimental fact, that the magnetization curve of

iron depends on the positioning of the lattice in respect to the external magnetic

field, is very surprising. How can the Heisenberg Hamiltonian, which is only spin

dependent, be affected by the structure of the lattice? The answer is simple: Even

though the spin part



χ



Φ



and the spatial part



of the wave function do not depend



on the same variables they are connected by the spin-orbit coupling! So

transmitts the structure of the environment (symmetry of the lattice) onto



χ



Φ

via



LS-coupling (→ [Bl])!

exchangeuu

depends on scalar

renergy

uu

r

S1.S 2

product

⇒ invariant with respect to choice of

coord. sys.

⇒ magnetization of ferromagnets

considered isotropic

 BUT: experimentally was found:

⇒ magnetization lies along certain axes

 Anisotropy is caused by spin-orbit

coupling





Figure 17: Magnetization curve of

iron [Reprinted from Heiko Lueken,

Magnetochemie, 1. Auflage;

Teubner Verlag]



40



•



spatial wave functions reflect symmetry of the lattice because of the

interactions between the lattice atoms that arise from the crystalline electric



•



fields and the overlap of the wave functions

via spin-orbit coupling, spins are made aware of this anisotropy



41



CONCLUSIONS

We have seen that magnetic properties of matter originate essentially from

the magnetic moment of electrons in incomplete shells in the atoms and from

unpaired electrons. The incomplete shell may be, for example, the 3d shell in the

case of the elements of the iron group, or the 4f shell in the rare earths (Sections 1

and 2). The cause for some solids, called (anti)ferromagnets, to exhibit

macroscopic magnetic characteristics was given by the exchange interaction, which

is quantum mechanical in nature. Dipolar interaction, which has a tendency to align

microscopic magnetic moments as well, does not present a cause for

(anti)ferromagnetism due to energetical reasons (Section 3). This text was then

concluded by qualitatively explaining why the exchange interaction, though short

range in nature, is able to cause ferromagnetic characteristics in compounds like

MnO where there is no direct overlap of the Mn-orbitals and how anisotropy can

occure in crystals even though the Heisenberg hamiltonian, describing

(anti)ferromagnetism, is totally isotropic (Section 4).



42



REFERENCES

[1]

[2]



Lueken, Heiko; Magnetochemie, 1. Auflage; Teubner Verlag

Blundell, Stephen; Magnetism in Condensed Matter,



1st



Edition; Oxford



Univ. Press

[3]



H. Haken, H.C. Wolf; Atom- und Quantenphysik, 8. Auflage; Springer



Verlag

[4] Demtröder, Wolfgang; Experimentalphysik Band 3, 3. Auflage; Springer

Verlag

1st



[5]



J. Stöhr, H.C. Siegmann; Magnetism,



Edition; Springer Verlag



[6]



Bringer, Andreas; Heisenberg Model; Institut für Festkörperforschung



and Institut for Advanced Simulation, Forschungszentrum Jülich, D52425 Jülich,

Germany

[7]



Aharoni, Amikam; Introduction to the Theory of Ferromagnetism,



2nd



Edition; Oxford Univ. Press

[8]



Guimarães, A. P.; Magnetism and Magnetic Resonance in Solids,



1st



John Wiley & Sons, Inc.

[9]



Simonds, J. L. (1995), “Magnetoelectronics Today and Tomorrow”,



Physics Today 48, 26-32.



43



Edition;