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 ↑ ( k ) = ϵ ( k ) − I N ↑ − N ↓ N , E ↓ ( k ) = ϵ ( k ) + I N ↑ − N ↓ N , {\displaystyle 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 {\displaystyle I} is the Stoner parameter, N ↑ / N {\displaystyle N_{\uparrow }/N} ( N ↓ / N {\displaystyle N_{\downarrow }/N} ) is the dimensionless density of spin up (down) electrons and ϵ ( k ) {\displaystyle \epsilon (k)} is the dispersion relation of spinless electrons where the electron-electron interaction is disregarded. If N ↑ + N ↓ {\displaystyle N_{\uparrow }+N_{\downarrow }} is fixed, E ↑ ( k ) , E ↓ ( k ) {\displaystyle 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 ↑ − N ↓ ) / N {\displaystyle P=(N_{\uparrow }-N_{\downarrow })/N} . If the lowest total energy is found for P = 0 {\displaystyle P=0} , the system prefers to remain paramagnetic but for larger values of I {\displaystyle I} , polarized ground states occur. It can be shown that for
I D ( E F ) > 1 {\displaystyle ID(E_{\rm {F}})>1}
the P = 0 {\displaystyle P=0} state will spontaneously pass into a polarized one. This is the Stoner criterion, expressed in terms of the P = 0 {\displaystyle P=0} density of states at the Fermi energy D ( E F ) {\displaystyle D(E_{\rm {F}})} . A non-zero P {\displaystyle P} state may be favoured over P = 0 {\displaystyle 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 ⟩ {\displaystyle \langle n_{i}\rangle } plus fluctuation n i − ⟨ n i ⟩ {\displaystyle n_{i}-\langle n_{i}\rangle } and the product of spin-up and spin-down fluctuations is neglected. We obtain
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