HW 7 modified for 2016

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)

Final discussion post.

Not sure is many people are still looking at this blog, but, just in case, here is a place to discuss the final if you like. Some people had trouble with #5. (Illuminated n-p junction and resistor..) Does anyone want to explain how to handle that? #6 as well.

Overall, I was impressed by how much people learned. Nice work!

Physics 156 Final. Wednesday, 12:00 noon

The final should provide an opportunity for you to show your understanding of semiconductors and metals. E vs k relationships play a big role. Where do they come from? Can you utilize them to calculate fermi velocity, fermi boundaries,…? Do you understand the significance of the Brillouin zone? Can you distinguish occupied and unoccupied states?

For semiconductors, the fermi energy is in an energy gap. Do you understand what an energy gap is and what the density of states as a function of energy typically looks like for a semiconductor? Special approximations can be used to estimate carrier density for cases where Ef is in a gap. Other approximations can often be used when Ef is not in a gap. The ability to recognize and distinguish those cases is important. Understanding the nature of those approximations, and what they rely on, is also important.

What is the essential nature and phenomenology of an n-p junction?

What role does shifting of the fermi boundary play in the conductivity of a metal?

Do you understand the origin of ferromagnetism?

I’ll post more here later. I just wanted to get the ball rolling and provide a place for questions and discussion here.
Added  notes: Perhaps it would be good to have a problem on illuminated n-p junctions. How do you feel about that? What about a ferromagnetism problem.
    It might be a good idea to test yourself with Fermi boundaries. You may wish to test your ability to identify where a Fermi boundary crosses the kx axis, realizing that the ky term is not zero when ky=0. (Same thing for the ky axis.)

Equations are show here: (What else do we need? I can't think of very many basic equations from the second part of the class.)

Conductivity of a Metal

Here is a video on conductivity of a metal that talks about the shift of the Fermi boundary and the electron speed. Please feel free to discuss and ask questions here.

Homework 7 solutions.

Here is a link. When you show a Fermi boundary, it is important to also show the extent of the (1st) Brillouin zone. The (1st) Brillouin zone is a domain in k-space in which each state can be found exactly once. Going beyond that would lead to encompassing some states more than once (over counting). The point of the Fermi boundary is to be able to specify which states are occupied (and which states are not occupied).

https://drive.google.com/file/d/0B_GIlXrjJVn4R09ISFh6Nl92NU0/view?usp=sharing

Superconductivity

A number of people have indicated they would like to learn about superconductivity. Perhaps on Monday we can discuss that.  In superconductor there electrons near the fermi surface pair with each other. This is pretty surprising.  (Electrons typically repel each other.  Why would electrons pair?)    

Anyway, these pairs of electrons are bosons and a superfluid bose condensate can form that is able to transport  current (charge) with no resistance. Superconductivity is an unusual and completely unexpected phenomenon. The idea that "more is different" in physics are partly motivated by the emergence of the superconductivity and the difficulty of explaining it in a reductionist manner.

Ferromagnetism and anti-ferromagnetism.

    The relationship between ferromagnetism and what is called "anti-ferromagmetism" is not simple.   Ferromagnetism occurs in metals and is driven by the tendency of electrons to, in some circumstances, use spin alignment in order to avoid each other. In this way the overall coulomb energy associated with electron-electron repulsion can be diminished.
     Antiferromagnetism, on the other hand, usually involves electrons that are confined (that is, electrons in non-metallic systems). Confinement has increased the kinetic energy of these electrons. The electrons seek a way to lower their confinement-related kinetic energy.   Communicating through their spins, the electrons find a way to lower their KE through an inherently collective, cooperative behavior involving spin alignment. In an antiferromagnet, each spin tends to point in the opposite way of its nearest neighbors. Antiferromagnetism is not just the alternating of classical spins.  It is a sophisticated many-electron quantum collective state.
    The names, FM and AFM,  tend to be rather misleading. The magnetism part makes sense, however, ferro is just from an ancient greek name for a material in which magnetism was observed long ago. It tells us nothing about the underlying origin. One might instead call it magnetism from spontaneous spin alignment, or just "spontaneous magnetism".
     More importantly, anti-ferromagnetism is not really the opposite of ferromagnetism. It is its own collective phenomenon with its own origins. These are very different and probably even more intriguing and subtle than those of ferromagnetism. Anti-ferromagetism involves a collective state with broken symmetry which arises to enable electrons to lower their quantum kinetic energy.