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[edit]

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,


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, N_\uparrow/N (N_\downarrow/N) is the dimensionless density[1] 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_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[1] at the Fermi level D(E_F).

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[edit]

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[1]


 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}.

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


  D(E_F)U > 1.

References[edit]

Footnotes[edit]

  • 1. Having 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).