Homework 5
(due Friday, October, 31 - spooky)
1. Here we examine the de Haas–van Alphen effect for free electrons. For free electrons
with a magnetic field in the z-direction the energy is E = (h̄2 kz2 /(2m)) + h̄ωc (ν + 1/2).
We showed in class and it is shown in the book that the degeneracy of the Landau
levels per unit area is 2eB/h
(a) Show that the density of states at energy E is given by
g(E) =
νX
max
ν=0
h̄ωc
m3/2
q
√
,
3
2π 2 h̄ E − h̄ωc (ν + 1/2)
(1)
where νmax is the largest integer for which h̄ωc (ν + 1/2) < E.
(b) In the limit of the magnetic field going to zero show that the above sum reduces
to an integral, which is equal to the free electron three dimensional density of
states.
(c) Make an argument that for h̄ωc E the density of states, g(E), is periodic in
E/(h̄ωc ). What is the period?
(d) Use the previous result to derive that the period in 1/B, where B is the magnetic
field, is indeed 2πe/(h̄A). A is the area of the semiclassical extremal orbit: πk 2
for free electrons.
(e) Show that for this case ∆A = 2πeB/h̄, which is equation 14.12 in Ashcroft and
Mermin.
(f) Estimate the degeneracy of Landau levels using 2∆A/(2π)2 . The prefactor of 2
is for spin, and the density in k-space is d2 k/(2π)2 . How does your result for the
degeneracy compare with the one you used in part (a)?
(g) Bonus: Plot the density of states as a function of E/(h̄ωc ) for fixed energy much
larger than h̄ωc . Show that it is indeed periodic in E/(h̄ωc ) ∝ 1/B.
2. Chapter 16, Problem 1.