Set of supplementary notes.

92 CHAPTER 6. EXPERIMENTAL PROBES OF THE BAND STRUCTURE
e.g. Kittel ch. 9):
∮
pdr =
(
n + 1 )
h (6.1)
2
Here, p is the canonical momentum, conjugate to the position r. The canonical momentum
can be written as the sum of the kinetic (or mv-) momentum, mv, and the field momentum,
qA.
Particles with charge q moving in a strong magnetic field B are forced into an orbit by the
Lorentz force: m ˙v = qṙ × B. This relation connects the components of velocity and position
of the particle in the plane perpendicular to B and can be integrated: mv ⊥ = qr × B, where r
is measured from the centre of the orbit.
This allows us to write p = mv + qA = q(r × B + A), and using ∮ Adr = Φ (the magnetic
flux), we obtain:
∮
∮
pdr = q
r × Bdr + qΦ = −qΦ , (6.2)
because ∮ r × Bdr = −B ∮ r × dr = −2BA r , where A r is the real space area enclosed by
the orbit’s projection onto the plane perpendicular to B.
We arrive at the conclusion that the flux threading the real space orbit is quantised:
Φ n = A (n)
r B n = (n + 1 2 )h e
(6.3)
Can we relate the motion of the electron in real space to the accompanying motion in k-
space From our earlier result for the relation between momentum and position, mv ⊥ = h¯k ⊥ =
qr × B, we find that the k-space orbit has the same shape as the real space orbit, but is turned
by 90 degrees and stretched by Bq . This means that the area enclosed by the k-space orbit A h¯ k
is
( e
) 2
A k = B 2 A r , (6.4)
h¯
where the electron charge e has been inserted for the more general q. Combining this result
with Eqn. 6.3, we find
6.3.2 Density of states oscillations
A k = 2πe
h¯ B(n + 1 2 ) (6.5)
In a magnetic field, the allowed k-states no longer form a regular lattice in reciprocal space, as
k is no longer a good quantum number. All the k-states in the vicinity of a k-orbit superimpose
to form the orbital motion of the electrons. The electrons now ’live’ on a set of cylinders, the
Landau tubes, with quantised cross-sectional areas.
These cylinders, whose cross-sectional area expands with increasing field B, cut through
the zero-field Fermi surface of the metal. What effect will this have on the B-dependence of