Homework 2 (some solution notes added)

Please point out any errors, and ask questions and share thoughts here freely. Problem 7 is now finished (updated) following a discussion of that in the comments section (and it includes a special extra-credit part!).
     The assignment had two distinct parts. Problems 1-4 relate to finding the bands (of electron states) for our microscopic 1D model. If you get stuck on 4, which is pretty difficult, I would recommend skipping it and going to 5. Also, you can ask questions (about 4) here, and that is encouraged.
Problem 5 is a transitional problem. It introduces the Fermi function. It is important at this stage to be able to visualize the Fermi function as a function of energy! The energy dependence is the important part.
Problems 6-8 use a somewhat simples DOS and involve calculations of what states are occupied, using the Fermi function. While this DOS is simpler that the one you would get in problem 4, it is still not trivial and understanding the juxtaposition of bands (regions of non-zero DOS) and gaps (regions of zero DOS) is very important. In fact, that, along with visualizing the Fermi function vs energy, is probably the most important thing.
PS. I think 8 is a pretty difficult problem, really 2 hard problems molded into one. Leave yourself enough time for that. Get actual numbers; accurate numbers. The part where you calculate the number of empty states in the valence band involves an approximation for the Fermi function that is somewhat more difficult than the approximation we use for the Fermi function in the conduction band.

1. Our key result from Tuesday's class was:
 \( E(k) = E_1 - 2t \, cos(a k) \)
where
\( t = - \int^\infty_{-\infty} \phi_1(x) v(x-a) \phi_1(x-a) dx   \)
 a) What does \(E(k) \) represent?
b) What does \(\phi_1(x) \) represent?
c) What does \(v(x) \) represent?
d) What does \(E_1\) represent? Thinking about its origins, would you think that \(E_1\) would be positive or negative? Why?
e)  What does \(\psi(k) \) represent?

2. Assume that t is positive and that the magnitude of \(E_1\) is about 10x larger that the magnitude of 2t.
a) With just your understanding of the cos function and the relative magnitude of \(E_1\) and t, sketch a thoughtful plot of \(E(k)\). Graph only from \(k= - \pi/a\) to \(+\pi/a\). Take your time on this. Do it thoughtfully and look at it.
b) Where is \(E(k)\) largest?  Where is it smallest?
c) What is the difference, in energy, between those highest and lowest points?
d) Looking at this graph, what might you call the bandwidth? (That is, how would you define bandwidth?)
e) extra credit. How would you define the effective mass associated with this band? To what part of the band does effective mass refer? How are effective mass and bandwidth related?

3. Extra credit: Suppose that \(\phi_1(x) = \frac{1}{\sqrt{b}} e^{-|x|/b} \) (This is a reasonable atomic wave-function in the sense that it decays exponentially away from a cusped center (like an H-atom ground state). Additionally, suppose that  \(v(x) = -\alpha \delta (x) \) (a delta function).
a) Do you know how to do an integral involving a delta function? If no, ask someone about it or pst a question here. (A lot of people don't seem to know how to do this.) Computationally, it is really easy to do the integrals once you know what to do. Conceptually, think of the delta function potential as a very, very narrow square well. So narrow that \(\phi_1(x)\) is pretty much the same everywhere inside the well, which is all you need to know to integrate since \(v(x)\) is zero outside the well.
b)  Calculate t. (The overlap integral)
c) Use t to obtain an expression for \( E(k)\) that is based on the approximation of keeping just the n=j=1 and n=j=-1 terms from "the sum".
d) Plot \(E(k)\) from \(k= - \pi/a\) to \(+\pi/a\). What is the bandwidth?  Assuming \(E_1\) is -16 eV, what is the lowest energy state in the band and what is the highest?
e) What are the largest terms form the sum involving \(v(x\) and also from the other sum (over just n). Explore whether they are actually smaller than the terms we kept (e.g., n=j=1) and how including them might alter the band, \(E(k)\).

4. Starting with the 1D band-structure:
\( E(k) = E_1 - 2t \, cos(a k) \)
where  \(E_1\) is the atom state energy of the atom state from which the band is derived,
t is not time but is an overlap integral, as we discussed in class on Tuesday (except that we changed the sign of t so that it would be positive for a band made from a symmetric state. a is the lattice parameter, and k is the key variable.
a) What are the units of density of states as a function of energy for our 1-dimensional crystal?
b) What is the density of states as a function of energy? To calculate this you can start by assuming that states are uniformly distributed along the k axis. Also, you can assume that the total number of the states in the band, per cm, is equal to the number of atoms per cm.
[thoughts on problem 4: One can find the energy dependence by differentiating followed by substitution to eliminate k (and get things just in terms of E). I think you will get a density of states (as a function of energy) with "integrable singularities" (at the band edges). The integral of D(E) over the band should be related to the number of atoms in the crystal, since there are two states per atom if you include spin (one state per atom if not). Please post questions, thoughts or comments here.] There are some solutions here and in the link.

https://drive.google.com/file/d/0B_GIlXrjJVn4eXg0NlNzV21JMk0/edit?usp=sharing
There are more problems below this break.



5. Have you heard of the Fermi function? I think it is \( f(E) = \frac {1} {e^{(E-E_F )/KT} +1}\) or, better,  \( f(E) = \frac {1} {e^{(E-\mu )/KT} +1}\), where \(\mu\) is called the chemical potential.
a) Plot the fermi function as a function of E.
b) Consider and describe how the function is influenced by \( \mu\).
c) Consider and describe how the function is influenced by kT.

---- part II
6. The density of states that you would get from problem 4 is relevant to some physical phenomena, and emerges naturally from our simple microscopic 1D-crystal model. However, it is not the simplest thing to use in conjunction with the Fermi function.

Using the Fermi function along with a density of states is essential to a lot of physics, including p-n junctions which we will cover soon. So to facilitate understanding the role and use of the Fermi Function, let us now use a density of states (DOS) that is mathematically simple.  Let's use a DOS that is constant within each band (and zero between bands). That is, for the ground state derived band let us suppose that \(D(E)=D_1\) for \((E_1-2t_1)\leq E \leq (E_1+2t_1)\).
Additionally, we'll suppose that the density of states contribution for the band that comes from the atomic 1st-exited state is:  \(D(E)=D_2\) for \((E_2-2t)\leq E \leq (E_2+2t_2)\),
and \(D(E)=D_3\) for \((E_2-2t_3)\leq E \leq (E_3+2t_3)\) is the DOS contribution from the 2nd-excited state derived band. These 3 non-overlapping contributions constitute the total DOS of a crystal. Does that make sense? Can you picture that?
a) Sketch a plot of the total D(E). (Assume the bands don't overlap.)
b) Suppose each constituent atom, when isolated, has only one electron. Where would \(\mu\) tend to be? (at low T)
c) Suppose each atom had two electrons. Where would \(\mu\) tend to be? (at low T)
d)  Suppose each constituent atom had 4 electrons. Where would \(\mu\) tend to be? (at low T) What bands would be full?

7. Continuing with the case from d), which is an important case for semi-conductor physics, suppose each constituent atom has exactly 4 electrons to start and these atoms come together to form a crystal. Use the density of states form from the previous problem. (Constant within in band; zero outside/between  bands.)
a) Suppose kT = 25 meV.  What else would you need to know in order to calculate the density of electrons in the 3rd band? This is a pretty open ended, vague question. Let's discuss this here so that we can collectively create a precise question.
b) (precise version: added April 7) Suppose kT = 25 meV and, suppose further that \(E_3-2t_3\), the bottom of the 3rd band, is 0.5 eV higher energy than \(E_2+2t_2\), the top of the 2nd band. Can you picture that? We can call that the gap between the 2nd and 3rd bands. Dos that make sense? (see, for reference, your plot from 6a). Further suppose that \(D_2 = D_3 = 4 \times 10^{21} \frac {states}{eV-cm^3}\). I think with all that given, it should be possible to calculate the number of electrons in the 3rd band. Let me know if you need something else, if I have left anything out.  [Hint: Where is \(\mu\) on the energy axis? That is a good place to start. Feel free to post thoughts, guesses, results, etc regarding where \(\mu\) is.
c) Extra credit challenging question. If you get this, and I think you can, I will be impressed. Based in what you are given in part b), what can your infer about \(t_2\) and \(t_3\)?  (email me at zacksc@gmail.com if you get this.)

8. (This is a hard problem.) Doping in 3 dimensional semiconductor. There are ways to influence the position of \(\mu\), to some degree, that play a very big role in semiconductor physics (mostly doping by chemical substitution). Suppose that \(\mu\) is 0.7 eV above the top of the 2nd band and 0.3 eV below the bottom of the 3rd band. Further suppose that with each band the DOS is \(10^{22}\) states per \(cm^3\) and that both bands have a band-width of 4 eV.
a) At room temperature how many electrons would there be in the 3rd band?
b) At room temperature how many unoccupied states would there be in the 2nd band?
c) How would those numbers be different (and what would they be) if \(\mu\) were instead 0.2 eV above the top of the 2nd band and 0.8 eV below the bottom of the 3rd band?
d) how would you describe the two cases above? What names could you give them? 
e) What is the band gap for this semiconductor?