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286 Magnetic materials

3rd Landau level
impurity level
Energy
Fig. 11.23
Discrete energy levels in high 2nd Landau level
magnetic fields.

of the Hall resistance at these plateaus turn out to be dependent only on the
fundamental constants h and e and on the number of Landau levels filled.

11.10 Magnetoresistance
The subject has become very popular in the last decade or so but it is not new.
It has been around for a century and a half. The effect was first observed by
Lord Kelvin (William Thomson at the time) in 1857, when he found a few per
cent change in electrical resistance depending on the direction of the applied
magnetic field, whether it was in the same direction or transverse to the flow of
the electrons. The effect has become known as anisotropic magnetoresistance.
A qualitative description can be based on the Lorenz force. In the presence of
a magnetic field the electron flow is deflected, hence there can be a change in
scattering which leads to a changed resistance. In good metals like iron and
cobalt the magnetoresistance is indeed a small effect. The change in resistivity
is of the order of 0.5 to 3%.
∗
∗ Nobel Prize in Physics in 2007. It was found in 1988 by Fert and Grunberg (working independently)
that magnetoresistance increases significantly in multilayer ferromagnetic thin
films. They called it ‘giant magnetoresistance’, abbreviated as GMR. It is not
obvious that ‘giant’ is the right adjective to use because the resistance changed
† †
In Greek mythology, you may remem- by no more than a factor of two. On the other hand, if we think of giants
ber, giants challenged the gods but it was we imagine them just about twice the size of homo sapiens. So after all we
not a wise move. The gods, reinforced by
Heracles, killed them all off. can use the term with clear conscience. We should also mention here Colossal
Magnetoresistance, which can cause changes in current by a factor of several
thousand. It has though an entirely different mechanism, related to a magnetic-
ally induced metal-to-insulator phase transition. It seems less significant at the
moment because it has not been harnessed for practical applications.
Let us now think of an explanation based on quantum mechanics. We may
first ask the question of how the band structure of magnetic materials is related
to resistance. As we know (see Table 4.1 and Fig. 11.16), magnetic elements
from chromium to nickel have partially filled 3d bands whereas in copper the
3d band is filled. If we apply a voltage to a specimen of these materials we find
that the magnetic materials have high resistivity in contrast with copper which
is close to having the lowest resistivity of all materials. The number of electrons
available in the conduction band is not much different, so what is the reason? It
must be low mobility or in other words low relaxation time. In copper when an
electron bumps into the lattice or scatters for any other reason, it has nowhere
to go; well nowhere relative to an electron in, say, nickel. The scattered electron
has then a temporary resting place in the partially filled 3d band so it does not
participate in conduction. Its mobility is reduced. Interestingly this effect also
applies inside a ferromagnetic material. Since the filling of the 3d band is spin