1. a) Show that you can combine two 1D Bloch states to obtain a standing wave. [How are their k values related?]
2. Suppose the E vs k relation for the partially filled band of a metal is:
\(E(k) = E_o - (B/2) cos(ak)\) where k ranges from \(-\pi/a\) to \(+\pi/a\) and a = 0.157 nm.
a) What is the relationship between the density of atoms, N, and the integral of g(k) dk from \(-\pi/a\) to \(+\pi/a\) ? (this is a warm-up question. ask right away if you are not sure.)
b) If the density of electrons is N, what is \(k_f\)?
c) If the density of electrons is N/2, what is \(k_f\)?
3. (Same system) When an electric field is applied, the Fermi boundaries (in k space) shift.
a)
For an electric field of 10 Volts/cm applied, by what amount is the
Fermi boundary shifted if the density of electron is N? Illustrate.
b) For an electric field of 10 Volts/cm applied, by what amount is the Fermi boundary shifted if the density of electron is N/2?
4.
When the Fermi boundary is shifted, electrons can be viewed as
belonging to two classes: those that have an occupied partner state with
which they can form a standing wave state and those that do not. What
percentage electrons fall into each class for:
a) n=N
b) n=N/2
5. One can say that the current density of metal is proportional to:
the density of electrons in states that have no partner, multiplied by
the fermi velocity. (Let's use a bandwidth of 6 eV for this problem.)
a) What is the fermi velocity for a half-filled band (n=N)?
b) What is the fermi velocity for a quarter-filled band (n=N/2)?
Fermi Boundaries:
A Fermi boundary
in k-space is the set of points that separate a regions of occupied
states from a region of unoccupied states. In 1D it is just two points
-- one at kf, the other at -kf. In order to explore the concept of a
Fermi boundary we therefore need to go up to at least 2D.
6.
Consider a two-dimensional (2D) metal for which the E vs k relationship
for the conduction band is: \(E(k) = E_o - b cos(ak_x) - b
cos(ak_y)\) where kx and ky each range from \(-\pi/a\) to \(+\pi/a\) and
a = 0.157 nm. This range of k includes all the states of the band (2N
states), however, they are not all filled.
a) What is the bandwidth of this band?
b-e)
Figure out and illustrate the Fermi boundary for the cases where the
band is roughly: 1/8, 1/4, 1/2 and 3/4 filled, respectively.*
f)
Which one of these is simplest? Which one of this is not like the
others? Describe and discuss your results and illustrations.
g) To what 2D lattice structure do you think this E vs k relationship corresponds? How come?
* You may to use a numerical integration for some of these. Can the first one to do that post the value of Ef corresponding to each of these fillings? Also, please post here any that you can guess or do quickly.
6.2
I think it might be interesting to also examine the fermi boundaries
for an "upside down" E vs k relationship. That is, suppose:
\(E(k) = E_o + b cos(ak_x) + b cos(ak_y)\) where kx and ky each range from \(-\pi/a\) to \(+\pi/a\) and a = 0.157 nm.
a-d) Figure out and illustrate the Fermi boundary for the cases where
the band is roughly: 1/8, 1/4, 1/2 and 3/4 filled, respectively.
e) Discuss
7. One can make this anisotropic in the following way:
\(E(k) = E_o - b cos(a_xk_x) - \frac{b}{4} cos(a_yk_y)\)
a) Figure out and illustrate the Fermi boundary for the case where the fermi energy is \(E_o -b/8\).
b) Figure out and illustrate the Fermi boundary for the case where the fermi energy is \(E_o -b/4\).
c) Figure out and illustrate the Fermi boundary for the case where the fermi energy is ... your choice.
d) What would you guess is a lattice structure to which that this might correspond??
8.
The previous problems are for a simple structure, but it is not a very
realistic or stable structure (because the atoms are not very
efficiently (closely) packed). A more realistic 2D metal could have an E
vs k relationship for the conduction band of:
\(E(k) = E_o - b
cos(ak_x) - b cos(\frac{1}{2}ak_x + \frac{\sqrt{3}}{2}a k_y) - b
cos(\frac{-1}{2}ak_x + \frac{\sqrt{3}}{2}a k_y) \).
a) Explore the
nature of the Fermi boundaries for this band for a some particular
examples of filling (corresponding to different values of Ef). For what
structure would you guess that this could be the and E vs k relationship?
(think about symmetry considerations perhaps)
