Homework 2 with solutions added Jan 30.

I am hoping you can start this pretty soon, but I realize that is not always possible because you may have other plans or commitments. I like the idea of homework as an opportunity to learn something rather than as a task to be completed by a certain date. So, work on it when you can, and you might not finish it in one sitting, but hopefully it will be a learning experience. I welcome your feedback via email and/or comments here!

Added solution note, Feb 2. What I was looking for in problem 5 was the relationship:
\(n = KT  D e^{-(E_c -E_F)/KT}\)
where D is the (constant) Density of States (near the bottom of the conduction band) and K is K Boltzmann.

Consider a semiconductor with the following density of states:
\(D(E) = 10^{22}/\sqrt{4-(E-3)^2}\)  for 1 eV < E < 5 eV ...  (conduction band)
where E is in eV, the 4 has units of eV squared and the 3 has units of eV.
Additionally in the range -4 eV to 0 eV,
\(D(E) = 10^{22}/\sqrt{4-(E+2)^2}\)  for -4 eV < E < 0 eV ...  (valence band).

1. a) Sketch D(E) as a function of E. Show that there is a gap, with zero density of states (DOS) between 0 and 1 eV.  The top of the valence band is at E=0 eV and the bottom of the conduction band is at 1 eV. (The choice of putting the zero of energy at the top of the valence band is pretty common. The zero of energy is arbitrary.)




2. Suppose the Fermi energy is \(E_F=0.5 eV\) and that KT=.025 eV.
a) Integrate* the product \(D(E) e^{-(E-E_F)/.025 eV}\) as a function of E from 1 to 5 eV. (Email me what you get soon after you do this so I can get a sense how you are doing.)
b) What is the total number of states in the conduction band? What is the total number of electrons in the conduction band for kT=0.025 eV and \(E_F=0.5 eV\)?
c) What is the total number of electrons in the lower half of the conduction band? (from 1 to 3 eV)
d) What is the total number of electrons in the lowest quarter of the conduction band? (from 1 to 2 eV)
e) I think those numbers will be all very similar. What does that tell you about which states are most likely to be occupied by electrons? (email me)
Added note: Note that density of states has units of states/eV, so when you integrate just DOS over a band you get the total number of states in the band. That is different from the number of electrons in the band because the number of electrons is the number of occupied states. Note that F(E) has units of electrons per state, so when you multiply F(E) and D(E) together you get units of: electrons per eV. And when you integrate that as a function of energy (over the band) you get the number of electrons. Thus, integrating f(E)*D(E) gives you the total number of filled states in the band, that is, the number of electrons in the band. In problems 2 and 3 we are using an approximate form, a simple exponential, instead of the true Fermi function. Why is that okay in these cases? (email me you thoughts on that if you like.)

3. What is the total number of electrons in the conduction band for kT=0.025 eV and \(E_F=0.7 eV\)? Is it more or less than what you got in 2? (email me)

*You may want to do these integrals numerically. See example below.

4. Change the DOS in the conduction band to be just a constant value from 1 to 5 eV (zero elsewhere) and with same total area, that is, the same total number of states as above.
a) recalculate each of the things above. It should be a bit easier. How do the results compare?

5. Major extra credit: Show that by ignoring the upper limit of integration you can get a simple analytic relationship between E_F and n that is pretty accurate for the constant DOS case.


Solutions: