Stoner criterion

Stoner model of ferromagnetism

Ferromagnetism ultimately stems from Pauli exclusion. The simplified model of a solid which is nowadays usually called the Stoner model, can be formulated in terms of dispersion relations for spin up and spin down electrons,

: E_\uparrow(k)=\epsilon(k)-I\frac{N_\uparrow-N_\downarrow}{N},\qquad E_\downarrow(k)=\epsilon(k)+I\frac{N_\uparrow-N_\downarrow}{N},

where the second term accounts for the exchange energy, I is the Stoner parameter, N_\uparrow/N (N_\downarrow/N) is the dimensionless densityHaving a lattice model in mind, N is the number of lattice sites and N_\uparrow is the number of spin-up electrons in the whole system. The density of states has the units of inverse energy. On a finite lattice, \epsilon(k) is replaced by discrete levels \epsilon_i and then D(E)=\sum_i \delta(E-\epsilon_i). of spin up (down) electrons and \epsilon(k) is the dispersion relation of spinless electrons where the electron-electron interaction is disregarded. If N_\uparrow +N_\downarrow is fixed, E_\uparrow(k), E_\downarrow(k) can be used to calculate the total energy of the system as a function of its polarization P=(N_\uparrow-N_\downarrow)/N. If the lowest total energy is found for P=0, the system prefers to remain paramagnetic but for larger values of I, polarized ground states occur. It can be shown that for

: ID(E_{\rm F}) 1

the P=0 state will spontaneously pass into a polarized one. This is the Stoner criterion, expressed in terms of the P=0 density of states at the Fermi energy D(E_{\rm F}).

A non-zero P state may be favoured over P=0 even before the Stoner criterion is fulfilled.

Relationship to the Hubbard model

The Stoner model can be obtained from the Hubbard model by applying the mean-field approximation. The particle density operators are written as their mean value \langle n_i\rangle plus fluctuation n_i-\langle n_i\rangle and the product of spin-up and spin-down fluctuations is neglected. We obtain

: H = U \sum_i [n_{i,\uparrow} \langle n_{i,\downarrow}\rangle +n_{i,\downarrow} \langle n_{i,\uparrow}\rangle

• \langle n_{i,\uparrow}\rangle \langle n_{i,\downarrow}\rangle] - t \sum_{\langle i,j\rangle,\sigma} (c^{\dagger}{i,\sigma}c{j,\sigma}+h.c).

With the third term included, which was omitted in the definition above, we arrive at the better-known form of the Stoner criterion

: D(E_{\rm F})U 1.

Notes

References

• Stephen Blundell, Magnetism in Condensed Matter (Oxford Master Series in Physics).