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Localized ferromagnetism refers to materials where the magnetic moments are primarily associated with localized atomic orbitals.

Ferromagnets, such as those made of iron or nickel, are called itinerant because the electrons whose spins aligned to create the magnetic state are extended and are the same as the ones responsible for conduction.

I don't understand how can localized ferromagnetism exist at all. From my understanding, a large gap insulator will always be non-ferromagnetic because the full valence band always have electrons paired up and leave no net electronic magnetic moment (while nuclear magnetic moment is negligible for ferromagnetism). The energy favorability of spin alignment is much smaller than overcoming the large band gap, so the spins are always paired.

This means ferromagnetism can only occur in metal or small gap insulator where the energy favorability of having aligned spins is larger than the energy favorability of strictly filling the spin-unpolarized states below the fermi-level (before considering the interaction of electron spins).

Is my understanding of the necessary condition for ferromagnetism correct? (ie, energy favorability of aligned electron being larger than band gap)? If this is correct, how can localized ferromagnetism exist at all?

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You are thinking that one model of magnetism can capture them all. This is false.

energy favorability of aligned electron being larger than band gap

This is correct in the itinerant ferromagnetism model.

But this is not the only model possible.

Consider that you have deep, core, d or f electrons shells that are not fully filled. They are trapped in at their ionic positions, maybe ringed by oxygen, before the next nearest neighbours are the same ions.

Then these are the kinds of electrons that need a treatment by tight-binding approximation. They are quite localised, but there could be a simultaneous jump between the electrons from these ions, and the oxygen ions. By doing this, their spins can become aligned, generating an observable magnetic field.

The stereotypical situation where this is most easy to see, is when the unpaired core electrons are 3/4 filled. That is, for every two ions, there is one filled full and one half filled. The alignment is of $$\uparrow\_\quad\downarrow\uparrow\quad\downarrow\uparrow\qquad\Leftrightarrow\qquad \uparrow\downarrow\quad\uparrow\downarrow\quad\_\uparrow$$ where the middle filled pair is the oxide electrons, just participating in the collective jumping, that ends up causing the alignment.

Again, there are plenty of different possibilities, this is but one of them.

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  • $\begingroup$ "Consider that you have deep, core, d or f electrons shells that are not fully filled. They are trapped in at their ionic positions, maybe ringed by oxygen, before the next nearest neighbours are the same ions" Why would there be deep core states that are not fully filled? Shouldn't it be highly thermodynamically unfavorable and irrelevant? Electrons are very quantum mechanical object. Shouldn't they be able to tunnel and equilibrate so all states deep below the fermi level are filled? $\endgroup$
    – Bohan Xu
    Jun 8 at 5:14
  • $\begingroup$ It is commonly the case that d and f electrons become such that part of the shell is deeply under the Fermi level, and another part of the same shell is above the Fermi level, making separate bands and having a band gap. They are quite complicated. For example, if an element is faced between filling up the valence p subshell or the incompletely filled d or f subshell, it is often the case that the atom would steal electrons from d or f subshell to quickly achieve the closing of the p subshell instead. The ordering of the energy levels is just not so simple. $\endgroup$
    – naturallyInconsistent
    Jun 8 at 6:07
  • $\begingroup$ Those mostly d f states that are deeply under the Fermi level, are still practically never dynamical, no? All the spin dynamics involving partially filled states, can only be around Fermi level...because if they are too deep, they simply happen too infrequently thermodynamically to be relevant. In other words, those spin dynamics can only in metal where the electrons are not localized, or undoped semiconductor where electrons in the conduction band are still not localized....or I guess you can have p-doped semiconductor whose states near Fermi level is somewhat localized? $\endgroup$
    – Bohan Xu
    Jun 8 at 6:41
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    $\begingroup$ The case I gave is for an insulator doped a lot. The binding is just enough to be called tight, yet able to jump from the oxides to the metal and vice versa $\endgroup$
    – naturallyInconsistent
    Jun 8 at 7:22

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