Please comment on HW 2 and start working these problems.

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. Please post your results, progress and questions here.

Here is a video discussing how to approach some problems:


--Understanding Bloch states.
1. Using \(\psi_{atom} (x) = \frac{1}{\pi^{1/4}\sqrt{b}}e^{-x^2/2b^2}\) or \(\psi_{atom} (x) = \frac{1}{\sqrt{b}}e^{-|x|/b}\)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)?

--Density of states
2. 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=12.
Suppose the energy eigenvalues for each of the states at these discrete values of k are given by the equation \(E_{n,k} = E_n -(B/2) cos(ak)\)
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=12?

 3. a) For a 1D band,  \(E_{n,k} = E_n -(B/2) cos(ak)\), calculate the density of states, D(E), for the limit of large N. You can do that using the idea that D(E) is proportional to the inverse of the derivative of E(k) with respect to k, and the constraint that the integral of D(E) over the band is equal to 2N. You will need to go from k to E after you take the derivative. If that, or anything, is not clear, please ask a question here!
b) Do a graph of D(E) with the vertical scale set for N=100. Use units of eV and assume that B=2 eV.

--working with Density of states
4.1 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)

4.2 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.

4.3 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 \(E_f = -7.5 eV\).
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.)

4.4 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 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.

4.5 Is there an easier way to calculate how many thermal electrons there are, on average, in the conduction band? Approximations you can use in problem 4.4? If so, what are they? How accurate are they in this case?