HW2. Due Wednesday.

This may be a pretty difficult and long homework. Going from Bloch states to density of states to occupation, and then finally calculating numbers of thermal electrons in the conduction band will probably take some time to sort out. I would suggest starting as soon as you can and spending a couple of sessions working on this and thinking about it over the next few days. Comments and questions are very welcome here. If you are stuck ask questions here, don't wait until class. Also, please reply to other students questions, e.g., David S, below.

PS. All email parts are rescinded. You can just include that in your written work.

part 1. Bloch states.

1.1 Using \(\psi_{atom} (x) = \frac{1}{\pi^{1/4}\sqrt{b}}e^{-x^2/2b^2}\) as your atom state with b= .05 nm:

a) Plot  \(\psi_{atom} (x) \) as a function of x.
For each of the next plots, let the range of your plot cover 5 cells, that is, n=-2 to n=+2, and let the crystal lattice parameter, that is, the center to center distance between atoms, be a=.1 nm (= 2b). You can use a computer to learn about these, but your plots must be hand drawn.

b) Plot the Bloch state made from this state for \(k=\pi/a\).

c) Plot the Bloch state made from this state for \(k=\pi/2a\).
d) Plot the Bloch state made from this state for \(k=2\pi/a\).
e) Plot the Bloch state made from this state for \(k= - \pi/2a\).
f) Plot the Bloch state made from this state for \(k=-\pi/a\).
g) Plot the "probability density" for each of the Bloch states above.
h) which of these plots actually require 2 plots and which can be done with only one plot?  What is the difference between c) and e)? What is the difference between b) and f)?

1.2 Can you figure out a general relationship between the Bloch state \(\psi_{n,k}\) and the Bloch state \(\psi_{n,k+2\pi}\)?

1.3 Post a comment on the post "3X..." that addresses some of the issues raised by that problem or ask a question.

Part 2. Density of states
2.1 For a crystal, a total of N energy eigenstates form from each atom energy eigenstate. These states can be kept track of using the crystal quantum number k. Allowed values of k are pretty much \(j 2 \pi/L\)  where j is an integer ranging from -N/2 to +N/2 where N is the number of atoms in our 1D crystal and L is the overall length of the crystal (L=Na).
a) Sketch a k axis and draw a small circle each allowed value of k for N=10.
Suppose the energy eigenvalues for each of the states at these discrete values of k are given by the equation \(E_{n,k} = E_n -(B/2) cos(ak)\)
b) Divide the k-axis into 5 equal sections: How many allowed k states are in each of these 5 sections for N=20? (this part is easy once you understand it.)
c) Divide the energy axis into 3 equal sections, one from the bottom of the band to B/3 above the bottom;  etc. How many states are in each of these 3 sections for N=20?
d) Calculate the density of states, D(E), for the of large N.
e) Do a graph of D(E) with the vertical scale set for 100. Use units of eV and assume that B=2 eV.

2.2 Sketch the density of states for \(E_2 = -10 eV\) and \(B_2 = 4 eV\),  \(E_3 = -4 eV\) and \(B_3 = 6 eV\). Suppose that the E2 band is filled and the E3 band is empty. What is the band gap? How much energy does it take to excite an electron from the top of the filled band to the bottom of the empty band? (email me this number)

2.3 For the same bands, \(E_2 = -10 eV\) and \(B_2 = 4 eV\):  \(E_3 = -4 eV\) and \(B_3 = 6 eV\).
a) How many states are occupied if T=0 and \(E_f = -4 eV\).
b) Plot the density of states vs E and show which parts are occupied and which are not.

2.4 For the same bands, \(E_2 = -10 eV\) and \(B_2 = 4 eV\):  \(E_3 = -4 eV\) and \(B_3 = 6 eV\). 

a) How many states are occupied if T=0 and \(E_f = -7.5 eV\).

b) Plot the density of states vs E and show which parts are occupied and which are not.

c) What is notable about -7.5 eV ?

(email me your a very short description of the situation of 2.3 and 2.4, respectively (ideally just a few words) and your thoughts on the essence of the difference between them.)

2.5 For the same bands, \(E_2 = -10 eV\) and \(B_2 = 4 eV\):  \(E_3 = -4 eV\) and \(B_3 = 6 eV\). 

a) How many conduction band states are occupied if kT = 25 meV (which corresponds to T= about 295 K or so) and \(E_f = -7.5 eV\).

Actually, that is not a very good way to ask the question. let's ask instead how many thermal electrons there are, on average, in conduction band states.

2.6 Is there an easier way to calculate how many thermal electrons there are, on average, in the conduction band? An approximation you can use in problem 2.5? If so, what is it? How accurate is it in this case?