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,

$E_{\uparrow }(k)=ϵ(k)-I\frac{N_{\uparrow }-N_{\downarrow }}{N},E_{\downarrow }(k)=ϵ(k)+I\frac{N_{\uparrow }-N_{\downarrow }}{N},$

where the second term accounts for the exchange energy, $I$ is the Stoner parameter, $N_{\uparrow }/N$ ($N_{\downarrow }/N$) is the dimensionless density of spin up (down) electrons and $ϵ(k)$ 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

$ID(E_{\mathrm{F}})>1$

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 at the Fermi energy $D(E_{\mathrm{F}})$.

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 $⟨n_i⟩$ plus fluctuation $n_i-⟨n_i⟩$ and the product of spin-up and spin-down fluctuations is neglected. We obtain

$H=U\sum _i[n_{i,\uparrow }⟨n_{i,\downarrow }⟩+n_{i,\downarrow }⟨n_{i,\uparrow }⟩-⟨n_{i,\uparrow }⟩⟨n_{i,\downarrow }⟩]-t\sum _{⟨i,j⟩,\sigma }(c_{i,\sigma }^{†}{c}_{j,\sigma }+h.c).$

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

$D(E_{\mathrm{F}})U>1.$

References

• Stephen Blundell, Magnetism in Condensed Matter (Oxford Master Series in Physics).
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