HW 7.

Some people asked for extra time and I think it would be okay to just turn this in Wednesday in class.

For any of these problems, assume a scattering rate of \(\tau = 10^{-13} s\) unless otherwise specified.

1. a) Show that you can combine two 1D Bloch states to obtain a standing wave.
b) 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 \(+\pi/a\) ?
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. Some people 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) Argue for or against that as a good approximation. What other factor do you need?
b) What is the fermi velocity for a half-filled band (n=N)?
c) What is the fermi velocity for a quarter-filled band (n=N/2)?
d) How does that compare with: the thermal velocity of electron in a semiconductor with an effective mass of 1? the average speed of an electron in a semiconductor conduction band with an electric field of 10 V/cm applied?

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?
* If doing this for a specified filling fraction (1/8, 1/4...) is too difficult, then maybe we could use a sequence of fermi energy values instead. Would that be more manageable? What do you think? What might be a reasonable selection of values of Ef that meet our goal of exploring?

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??

6.2 (problem added on Wednesday)  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

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). What structure would you guess that this could be the and structure for? (educated guess --based on symmetry considerations perhaps)

Here are some possible suggestions:
b) Figure out and illustrate the Fermi boundary for the case where the fermi energy is \(E_o -b/2\).
b) Figure out and illustrate the Fermi boundary for the case where the fermi energy is \(E_o)
b) Figure out and illustrate the Fermi boundary for the case where the fermi energy is \(E_o + b/2\)
b) Figure out and illustrate the Fermi boundary for the case where the fermi energy is \(E_o + 0.9 b\)
f) Life changing extra credit opportunity: To what 2D lattice structure do you think this E vs k relationship corresponds? Explain your thinking.

10. Special project opportunity: Suppose you have a solar cell like that in the previous homework, problem 7. Suppose you want to design a circuit to be able to efficiently generate heat in the resistor (which is a stand-in for energy use).  The system should be able to work in a range of brightness of the incident light.  Let's say that the number of photons absorbed per second in the junction might range from 10^16 to 2*10^17 photons/second. Can you find a single value of R that would provide pretty good energy conversion over that entire range?