164.67.234.104 at 05:05, 10 December 2011

2011-12-10T05:05:02Z

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A condition to be fulfilled for the ferromagnetic order to arise in a simplified model of a solid.
Named after [[Edmund Clifton Stoner]].

==Stoner model of ferromagnetism==

Ferromagnetism ultimately stems from electron-electron interactions. 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,

:<math>
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},
</math>

where the second term accounts for the exchange energy, <math>N_\uparrow/N</math> (<math>N_\downarrow/N</math>)
is the dimensionless density[[#Footnotes|[1]]] of spin up (down) electrons and <math>\epsilon(k)</math> is the dispersion
relation of spinless electrons where the electron-electron interaction is disregarded. If <math>N_\uparrow
+N_\downarrow</math> is fixed, <math>E_\uparrow(k), E_\downarrow(k)</math> can be used to calculate the total energy
of the system as a function of its polarization <math>P=(N_\uparrow-N_\downarrow)/N</math>. 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

:<math>
2ID(E_F) > 1
</math>

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[[#Footnotes|[1]]] at the Fermi level <math>D(E_F)</math>.

Note that 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 <math>\langle n_i\rangle</math> plus fluctuation <math>n_i-\langle n_i\rangle</math> and the product of spin-up and spin-down fluctuations is neglected. We obtain[[#Footnotes|[1]]]

:<math>
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 +
\sum_{i,\sigma} \epsilon_i n_{i,\sigma}.
</math>

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

:<math>
D(E_F)U > 1.
</math>

==References==

* Stephen Blundell, Magnetism in Condensed Matter (Oxford Master Series in Physics).
*[http://igorbarsukov.com/stoner.html Demonstrative derivation of the Stoner Criterion]

==Footnotes==

*1. Having a lattice model in mind, N is the number of lattice sites and <math>N_\uparrow</math> 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, <math>\epsilon(k)</math> is replaced by discrete levels <math>\epsilon_i</math> and then <math>D(E)=\sum_i \delta(E-\epsilon_i)</math>.

[[Category:Magnetism]]

Semicomputable function - Revision history

164.67.234.104 at 05:05, 10 December 2011