Physics of Magnetism

SECTION 7.3. STRONG AND WEAK FERROMAGNETISM 67

moments have a certain size. It will be shown below that the presence of a region in the 3d

band with a low density of states can lead to two different situations.

Let us assume that the 3d band has a general shape of the type as indicated above, as

is schematically shown in Fig. 7.3.1. In the case of simple ferromagnetism and relatively

where

strong moments, it is possible to compute directly from the expression:

In analogy with the more simple case of rectangular bands, one can identify the first

two terms in Eq. (7.3.1) as representing the loss in kinetic energy and the third term as

representing the gain in exchange energy.

In order to have a ferromagnetic phase that is more stable than the paramagnetic phase,

one has to meet the following condition:

When applied to Eq. (7.3.1), this means that a stable ferromagnetic state is found if

Generally, the maximum moment that can be obtained for a given number n of 3d electrons
equals for more than half-filled bands (see Fig. 7.1.1.c) and for less than half-
filled bands (see Fig. 7.1.1d). However, when taking account of the kinetic-energy increase,
the energy minimum of in Fig. 7.3.1 may be reached for values considerably
smaller than the maximum values of just mentioned (in analogy to the situation shown
in Fig. 7.1.1b).
The situation shown for rectangular bands in Figs. 7.1.1b and d is shown for the more
general band shapes in Figs. 7.3.2a and b, respectively. It is unphysical to have the Fermi
energy at a higher level in the majority electron band than in the minority electron band.
For this reason, the two subbands have been shifted relative to each other after electron
transfer so as to have the same Fermi energy. This can also be interpreted by stating that the
subband containing the larger number of electrons with parallel spins has been stabilized by
the exchange energy with respect to the subband containing the lower number of electrons
with parallel spins.
The situation in Fig. 7.3.2a corresponds to the equality sign in Eq. (7.3.4): For a given
magnitude of th e optimum band shift and the concomitant optimum electron transfer
between the two subbands has been reached, the low density of states in the minority spin
band preventing further electron transfer because of the too high kinetic energy expendi­
ture. Note that both spin subbands remain partially depleted even though there are enough
3d electrons available for complete filling of the majority spin subband. The situation
shown in Fig. 7.3.2a is referred to as weak ferromagnetism. The situation represented in