Stoner criterion

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

A schematic band structure for the Stoner model of ferromagnetism. An exchange interaction has split the energy of states with different spins, and states near the Fermi energy E_F are spin-polarized.

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 density^{[note 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 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^{[note 1]} 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^{[note 1]}

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_{i,\sigma }^{\dagger }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

^ ^a ^b ^c Having a lattice model in mind, ${\textstyle N}$ 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. (November 2008). "Band ferromagnetism in systems of variable dimensionality". Journal of Optoelectronics and Advanced Materials. 10 (11): 3058–3068. Retrieved 24 May 2014.
Stoner, Edmund Clifton (April 1938). "Collective electron ferromagnetism". Proc. R. Soc. Lond. A. 165 (922): 372–414. Bibcode:1938RSPSA.165..372S. doi:10.1098/rspa.1938.0066.

[:0-1] Having a lattice model in mind, ${\textstyle N}$ 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 $D(E)=\sum _{i}\delta (E-\epsilon _{i})$ .

[note 1]