Download Straight-forward derivation of Kittel Eq. 9.37 It is possible to derive

Straight-forward derivation of Kittel Eq. 9.37
It is possible to derive this without reference to the vector potential and Bohr-Sommerfeld
quantization, if we assume that the cross-section of the Fermi surface is circular.
First, assume a two-dimensional electron gas in the x-y plane with a magnetic field in the zdirection. Then, as discussed in class, the electron energy is
1
1 ℏ𝑒𝐵
𝐸𝑛 = (𝑛 + ) ℏ𝜔𝑐 = (𝑛 + )
.
2
2 𝑚
1
The allowed states in k-space now fall on a series of circles with radii 𝑘 = ℏ √2𝑚𝐸 (see Fig.
24). As the magnetic field increases, these circles increase in radius. As a given circle “pops”
through the Fermi energy, the states contained on that circle go from full to empty. We are
interested in those magnetic fields 𝐵𝑛 for which 𝐸𝑛 is equal to the Fermi energy:
1 ℏ𝑒𝐵𝑛 ℏ2 𝑘𝐹2
ℏ2 𝑆
(𝑛 + )
=
=
,
2 𝑚
2𝑚
2𝑚𝜋
where 𝑆 = 𝜋𝑘𝐹2 is the area of the Fermi circle. In 3D, this would be the cross-sectional area of
the Fermi surface.
Rearrange this expression as follows:
ℏ𝑆
1
− .
2𝜋𝑒𝐵𝑛 2
Evaluating for 𝑛 and 𝑛 + 1 and taking the difference:
𝑛=
ℏ𝑆
1
1
(
− ),
2𝜋𝑒 𝐵𝑛+1 𝐵𝑛
1
so that any quantity that depends on field will be periodic in 𝐵 with period
1
2𝜋𝑒
Δ( ) =
.
𝐵
ℏ𝑆
1=
Note that this proof is not limited to 2D. If we go to 3D, we can take the z-axis to be the
magnetic field direction and then
1
ℏ2 𝑘𝑧2
𝐸𝑛 = (𝑛 + ) ℏ𝜔𝑐 +
.
2
2𝑚
In other words, motion along the magnetic field direction is described by planes waves (or Bloch
waves), which are not modified by the magnetic field. The rest of the argument proceeds as
above, where 𝑆 is the cross-section in the x-y plane.
I admit that it is less obvious what to do in the general case where the Fermi surface no longer
has a circular cross-section. The Bohr-Sommerfeld rule (Eq. 26) allows one to generalize the
above results.
The key point is that if the magnetic field is large enough and the sample clean enough, then just
about any electronic property that depends on magnetic field will oscillate with a period
determined by the extremal cross-sectional area of the Fermi surface. If you are willing to wade
through the quantum mechanics and are looking for a rigorous proof, I recommend Ziman,
Theory of Solids, pp. 313 – 324. He discusses quite clearly why only the extremal orbits need to
be considered. I addressed this using pictures in class, but of course it can be proven.