Stoner criterion explained

The Stoner criterion is a condition to be fulfilled for the ferromagnetic order to arise in a simplified model of a solid. It is named after Edmund Clifton Stoner.

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 (N _{\uparrow-N} _{\downarrow})/(N)} , E _{\downarrow(k) = \epsilon(k) + I (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 density^[1] 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 forID(E _{\rm} ) > 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} ) .

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 obtainH = 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} (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} )U > 1 .

Notes

1. Having 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)} .

References

• Stephen Blundell, Magnetism in Condensed Matter (Oxford Master Series in Physics).
• Teodorescu. C. M.. Lungu, G. A. . Band ferromagnetism in systems of variable dimensionality. Journal of Optoelectronics and Advanced Materials. November 2008. 10. 11. 3058–3068. 24 May 2014.
• Stoner. Edmund Clifton. Collective electron ferromagnetism. Proc. R. Soc. Lond. A. April 1938. 165. 922. 372–414. 10.1098/rspa.1938.0066. 1938RSPSA.165..372S.