Three dimensions and the de Haas-van Alphen effect

How does all this go over into three dimensions? One now needs to add $k_z$, so the semi-classical orbits, which were circles in two dimensions, are now cylinders. One such cylinder is shown in Fig. 6. As $H$ increases the radius increases. However, one does not expect a dramatic change in the magnetization until the radius increases to $k_F$, (shown dashed in Fig. 6), beyond which the cylinder does not intersect the Fermi sphere at all. When the cylinders "pop out" of the Fermi sphere one expects oscillatory behavior in the magnetization just as in two dimensions. This can be confirmed by a mathematical analysis, see e.g. Peierls, pp. 144-149. Clearly then, the periodicity of the magnetization with $1/H$ involves the maximum area formed by the intersection of the Fermi sphere with a plane perpendicular to $H$. If one has a general Fermi surface from some complicated band structure, then it is also possible that, on varying $k_z$, the intersection of the Fermi surface with the plane of constant $k_z$ might have a minimum. These orbits will also contribute to the oscillatory behavior because the part of the cylinder inside the Fermi sphere will decrease rapidly when the radius of the cylinder passes this extremal radius.

Figure 6: The dashed circle represents the Fermi sphere in three dimensions. The cylinder represents the semi-classical orbit corresponding to a single Landau level. As the strength of the magnetic field increases, the radius of the cylinder increases and will eventually exceed the Fermi wave vector, $k_F$. This ``popping out'' of the Landau levels from the Fermi surface leads to oscillatory behavior of the energy and magnetization at sufficiently low temperature.

Thus we conclude that the magnetization will show oscillations with periods given by

$$\Delta \left({1\over H}\right) = {2 \pi e \over \hbar c} {1\over A_{\rm ext}} ,$$ (48)

where $A_{\rm ext}$ is the area of any extremal orbit in the plane perpendicular to the field. If there is more than one extremal area then several periods will be superimposed.

A determination of oscillations in $M$ as a function of $1/H$ (the de Haas-van Alphen effect) for different orientations orientations of the field has been the most successful method for mapping out the shape of the Fermi surface of metals. It is discussed in AM, Ch. 14