## What is Stoner instability?

In my earlier post on multiferroic materials, I mentioned Stoner instability. Here is the explanation of Stoner instability (and, this explanation is based on Chapter 7 (p.63) of this text: Physics of magnetism and magnetic materials (E-book)
K H J Buschow and F R de Boer

Consider a 3d transition metal, in which the 3d electrons give rise to magnetism; since the electrons are itinerant (and delocalised) in the metal, the magnetism stems from 3d electron bands. For simplicity’s sake, let us further assume that the 3d bands are rectangular (which means that we are assuming that the density of electron states is a constant over the entire range spanned by the width of the band). The band itself consists of two sub-bands — one for up-spin electrons and another for down-spin electrons. If there are less than ten 3d-electrons in the system, the 3d-band will be partially filled. Further, if the system fills these bands without discrimination, then both the sub-bands will be equally filled. However, if suppose we can define an interaction energy which indicates a reduction in energy if the electrons from one of the sub-bands, say those corresponding to down-spin can be transferred to the up-spin band, then, under certain circumstances it can be shown that this will lead to an instability as discussed below. However, what prevents such an emptying of one of the sub-bands in favour of another is the resultant increase in the kinetic energy of the electrons. In fact, the total variation in energy in such sub-band transfer of electrons can be shown to be equal to $\Delta E = \frac{n^{2} p^{2}}{N(E_F)} [1 - U_{eff} N(E_F)]$, where, $n$ is the total number of 3d electrons per atom, $p$ is the fraction of atoms that move from down-spin sub-band to up-spin sub-band, $U_{eff}$ is the effective interaction energy, and $N(E_{F})$ is the density of energy states at the Fermi level. Thus, if the quantity in square brackets is positive, the state of lowest energy corresponds to $p = 0$ — or, in other words, the metal is non-magnetic. However, if the quantities in the square bracket is negative, the band is “exchange split” — $p > 0$, and hence the metal is ferromagnetic. This is known as the Stoner instability, or sometimes ferromagnetic instability. From the equation, it is clear that such band splitting is favoured for large exchange interaction energy as well as for large density of states. Since the density of states for s– and p-bands are considerably smaller, which, in turn explains why such band magnetism is restricted to elements with partially filled d-band.

Here are the schematics explaining band magnetism in partially filled d-electron systems (based on Fig. 7.1.1 of the reference above):

Paramagnetic:

Weak ferromagnetism:

Strong ferromagnetism with systems in which the d-electrons per atom are less than five, and great than five respectively.

### 4 Responses to “What is Stoner instability?”

1. hopefulp Says:

Thanks for very good explanation.

2. Mierk Says:

Thanks for this explanation!

3. ejaz Says:

superb explaination. thanks.

4. riyaz Says:

What is the cause of this exchange interaction?