Homework 7 with solution.

You can discuss these problems here. Especially the methods that you might use. Discussion of 4, how to approach it, might help everyone. 
(These problems should not be too hard. It is okay if we have them due on Friday?)
errors corrected Thursday. question about thermal speed deleted.

1. Consider a conduction band like we derived the first week with a dispersion relation \(E = Eo - 2t cos(a k_x)\).
a) What is k_f for a half filled band (in terms of a and B, the bandwidth)?
b) What is the Fermi velocity for a half-filled band?
c) If a is 0.08 nm and B is 5 eV, what is the Fermi velocity (in cm/sec)?
d) If a is 0.08 nm and B is 5 eV, what is the (unit-less) effective mass?

2. For a semiconductor with a scattering rate (transport) of 10^-12 sec and an effective mass of 0.5.
a) what is the average speed of an electron for an applied electric field of 1 Volt/cm?
b) how does the speed from a compare with that of an electron at the Fermi surface in problem 1. 
c) what is the mobility for this semi-conductor? (cm^2/Volt-sec). Is that very good?

3. Back to the case of problem 1 with a = 0.08 nm and B = 5 eV:
a) calculate v_f for the cases of a 1/4-filled, half-filled, and 3/4-filled band. How do they compare?
b) What is E_f for each case?

4. For a 2-dimensional cubic crystal where \(E(k) = E_o - 2t(cos(a k_x) + cos(a k_y))\) sketch the Fermi surface for the following cases:
a) 1/8 filled band.
b) 1/4 filled band.
c) 1/2 filled band.
d) 3/4 filled band.
e) which is most like a circle in k_x, k_y space.
f) What is E_f in each case? (extra credit)
 Note on 4: There is no easy, magic way to do this that I know of. You might have to get your hands dirty and explore. The 3/4 case is really unusual and requires your interpretation and understanding. The shape of the 1/2 filled case surprised me.
Here is an example of the sort of thing you could try to explore with:

I think in that plot the B Z would extend from -pi to pi and be squared shaped. In k-space states are evenly distributed, so 3/4 filled means the area inside the Fermi surface is 3/4 of the total area of the B Z. (Of course, this represents a k-space plot. I used x and y cause Wolfram probably prefers that.)

Here is a page of solution notes. This are pretty short and may not be so clear, so please feel free to ask questions here (and to answer other peoples questions). Your comments will be much appreciated. This will be probably on the final.