Figure 63: De Haas-van Alphen effect for gold as a good approximation for a free electron gas. The
magnetic moment is plotted as function of B.
So, when one calculates the magnetization by taking the derivative as shown above, the magnetization
will oscillate. This is the De Haas-van Alphen effect. Fig. 63 shows this effect, using gold as a good
approximation for a free electron gas. As one can see, the magnetic moments oscillate in dependence
of the magnetic field applied.
One can think about that, using the picture of the cylinders of the Landau levels. The diameter of
those cylinder increases with the magnetic field and then the cylinders move through the fermi surface.
This causes a big change in the density of states at that point, which leads to large consequences for
all thermodynamic quantities. So the effect can not only be seen in the magnetization, but in all
thermodynamic quantities, as they are all functions of the free energy.
Another effect is the Quantum Hall Effect. In this case, one varies the magnetic field and measures
the resistivity (green line in fig. 64). There are oscillations in conductivity, called Shubnikov-De Haas
oscillations. The red line depicted in the figure shows the Hall voltage (measured perpendicular to
the applied voltage and the applied field), for a 2-dimensional electron gas (electrons on a surface).
Here we can see steps in this voltage and a constant voltage for regions where the resistivity drops
to zero. Normally electrons travel for a while and then they get scattered off in different directions,
which causes the resistance. When there is a high magnetic field of the right amplitude applied, the
electrons move at the edge of the sample, they don’t get scattered any more. Electrons in the middle
of the sample might also move in circles, but then they don’t participate in the conduction process.
9.5 Ferromagnetism
A ferromagnetic material shows spontaneous magnetization below a certain critical temperatureTc.
Below that temperature the magnetic moments in the solid (i.e. spins, magnetic moments due to
electron movement,... ) align in a preferential direction by themselves. From a classical point of view
this is in deed very puzzling because two neighboring magnetic moments should align antiparallel to
minimize their energy in their mutual magnetic fields. Ferromagnetism is a quantum effect which is