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

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,

:$E_\uparrow(k)=\epsilon(k)-I\frac,\qquad E_\downarrow(k)=\epsilon(k)+I\frac,$

where the second term accounts for the exchange energy, $I$ is the Stoner parameter, $N_\uparrow/N$ ($N_\downarrow/N$) is the dimensionless densityHaving 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 $D(E)=\sum_i \delta(E-\epsilon_i)$. 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

:$ID(E_) > 1$

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 $D(E_)$.

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

:
H = U \sum_i _ \langle n_\rangle
+n_ \langle n_\rangle
- \langle n_\rangle \langle n_\rangle- t
\sum_ (c^_c_+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

:$D(E_)U > 1.$

Notes

References

* Stephen Blundell, Magnetism in Condensed Matter (Oxford Master Series in Physics).
*
* {{cite journal, last=Stoner, first=Edmund Clifton, title=Collective electron ferronmagnetism, journal=Proc. R. Soc. Lond. A, date=April 1938, volume=165, issue=922, pages=372–414, doi=10.1098/rspa.1938.0066, bibcode=1938RSPSA.165..372S, doi-access=free
Ferromagnetism