Stoner criterion

Stoner model of ferromagnetism

Ferromagnetism ultimately stems from electron-electron repulsion. 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,

where the second term accounts for the exchange energy (I is the Stoner parameter) , N ↑ / N ( N ↓ / N ) is the dimensionless density[1] of spin up (down) electrons and ϵ ( k ) is the dispersion relation of spinless electrons where the electron-electron interaction is disregarded. If N ↑ + N ↓ is fixed, E ↑ ( k ) , E ↓ ( k ) can be used to calculate the total energy of the system as a function of its polarization P = ( N ↑ − N ↓ ) / 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

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[1] at the Fermi level D ( E F ) .

Note that 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 ⟨ n i ⟩ plus fluctuation n i − ⟨ n i ⟩ and the product of spin-up and spin-down fluctuations is neglected. We obtain[1]

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