X W Γ

Phys. 7440
Problem set 6
Due Oct. 15
Problem 1.
a) 25 pts The picture below shows the first Brillouin zone in k-space.
kz
W
Γ
X
ky
kx
2
Plot the free electron energies, ε
=
(k − G) 2 along the three lines, Γ-X,
2m
X-W and W-Γ. Plot contributions from all G such that the energy is less than 5ε X0
 2π 2
where ε X0 =
is the free electron energy at the X-point. Label the energy
2ma 2 €
curves with the particular associated values of G.
€
0
k −G
At each point where the energy curves touch a zone boundary, show the number
€ of degenerate free electron levels and the associated G that need to be treated
perturbatively to get nearly free electron energy bands.
b) 25 pts Assume that we have a weak periodic potential and use the results from
class to plot the nearly free electron energy bands. Note the amount by which the
various free electron degeneracies are lifted. Comment particularly on the
differences between what happens at the X-point verses the W-point. You may
use Mathematica, Matlab or smth. similar to evaluate the determinants.
Problem 2
A two-dimensional metal has one atom of valency one in a simple rectangular
primitive cell a=2Å; b=4Å.
a) 5 pts. Draw the first Brillouin zone. Give its dimensions in Å-‐1. b) 5 pts. Calculate the radius of the free electron Fermi surface. c) 5 pts. Draw this surface to scale on a drawing of the first Brillouin zone. Make another sketch to show the first few periods of the free electron band in the periodic zone scheme, for both the first and second energy bands. Assume there is a small gap at the zone boundary.

Download Report