Set of supplementary notes.

9.2. MAGNETISM 139
2
∆E ~ − t /U
Mn O Mn
Figure 9.9: An illustration of the superexchange mechanism in antiferromagnetic insulators.
Two magnetic moments are separated by a non-magnetic ion. The large separation rules out
a direct exchange interaction between the magnetic moments. The excited states (top left
and top right), which involve double occupancy of an atomic orbital on the magnetic ions, are
only possible, if the original spin state of the two magnetic ions was anti-aligned. Only for
anti-aligned magnetic moments can the system therefore benefit from the lower energy, which
admixing an excited state brings in second order perturbation theory. This results in a ground
state energy, which depends on the mutual spin orientation of the two magnetic moments.
the ground state, it can be admixed to the initial ground state and will – in second order perturbation
theory – always cause the new, perturbed, ground state energy to be lowered. This
admixture is not possible, if the two magnetic moments were aligned. We arrive, therefore at
a total energy for the system, which depends on the mutual orientation of the two magnetic
moments.
Second order perturbation theory suggests that this effective superexchange interaction is of
order J ∼ −t 2 /U < 0, where t is the matrix element governing hopping between the magnetic
moment and the non-magnetic ion, and U is the Coulomb repulsion energy on the magnetic
moment. When extended to a lattice, it favours an antiferromagnetic ground state, in which
alternate sites have antiparallel spins. On complicated lattices, very complex arrangements of
spins can result.
The magnitude of this interaction is often quite small, in the range of a few to a few
hundred degrees Kelvin. Consequently, these systems will often exhibit phase transitions from
a magnetically ordered to a disordered paramagnetic state at room temperature or below.
Band magnetism in metals
Let us start with Pauli paramagnetism – the response of a metal to an applied magnetic field.
We consider a Fermi gas with energy dispersion ɛ k in a magnetic field H. In a magnetic field,
the spin-up and spin-down bands will be Zeeman-split (see Fig. 9.10):
ɛ k↑ = ɛ k − µ B B a ,
ɛ k↓ = ɛ k + µ B B a . (9.24)