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Stoner criterion

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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

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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 EF 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 ↑ ( 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}},} {\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} {\displaystyle I} is the Stoner parameter, N ↑ / N {\displaystyle N_{\uparrow }/N} {\displaystyle N_{\uparrow }/N} ( N ↓ / N {\displaystyle N_{\downarrow }/N} {\displaystyle N_{\downarrow }/N}) is the dimensionless density[note 1] of spin up (down) electrons and ϵ ( k ) {\displaystyle \epsilon (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 }} {\displaystyle N_{\uparrow }+N_{\downarrow }} is fixed, E ↑ ( k ) , E ↓ ( k ) {\displaystyle E_{\uparrow }(k),E_{\downarrow }(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} {\displaystyle P=(N_{\uparrow }-N_{\downarrow })/N}. If the lowest total energy is found for P = 0 {\displaystyle P=0} {\displaystyle P=0}, the system prefers to remain paramagnetic but for larger values of I {\displaystyle I} {\displaystyle I}, polarized ground states occur. It can be shown that for

I D ( E F ) > 1 {\displaystyle ID(E_{\rm {F}})>1} {\displaystyle ID(E_{\rm {F}})>1}

the P = 0 {\displaystyle 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} {\displaystyle P=0} density of states [note 1] at the Fermi energy D ( E F ) {\displaystyle D(E_{\rm {F}})} {\displaystyle D(E_{\rm {F}})}.

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

Relationship to the Hubbard model

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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 } {\displaystyle \langle n_{i}\rangle } plus fluctuation n i − ⟨ n i ⟩ {\displaystyle n_{i}-\langle n_{i}\rangle } {\displaystyle n_{i}-\langle n_{i}\rangle } and the product of spin-up and spin-down fluctuations is neglected. We obtain[note 1]

H = U ∑ i [ n i , ↑ ⟨ n i , ↓ ⟩ + n i , ↓ ⟨ n i , ↑ ⟩ − ⟨ n i , ↑ ⟩ ⟨ n i , ↓ ⟩ ] − t ∑ ⟨ i , j ⟩ , σ ( c i , σ † c j , σ + h . c ) . {\displaystyle 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).} {\displaystyle 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 F ) U > 1. {\displaystyle D(E_{\rm {F}})U>1.} {\displaystyle D(E_{\rm {F}})U>1.}

Notes

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  1. ^ a b c Having a lattice model in mind, N {\textstyle N} {\textstyle N} is the number of lattice sites and N ↑ {\displaystyle N_{\uparrow }} {\displaystyle 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, ϵ ( k ) {\displaystyle \epsilon (k)} {\displaystyle \epsilon (k)} is replaced by discrete levels ϵ i {\displaystyle \epsilon _{i}} {\displaystyle \epsilon _{i}} and then D ( E ) = ∑ i δ ( E − ϵ i ) {\displaystyle D(E)=\sum _{i}\delta (E-\epsilon _{i})} {\displaystyle D(E)=\sum _{i}\delta (E-\epsilon _{i})}.

References

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