New homework post. I would really like you to do this by Wednesday. That way we can keep moving. Does that seem reasonable? Can you do it?? Plus, problem 1 provides the underpinning for this class and your future understanding of solid state physics, so it is worth wasting some time on I hope.
This requires some discussion and interaction to get us going. Please post your comments, thoughts, questions and contributions here! Hopefully someone can do the square well problem soon to help everyone get started.
I think the parameters I proposed for our square well "atom" were not good at all. I am thinking that to learn relevant things about the bands that evolve from it, it would probably be preferable to make the wells much narrower and, consequently, get into higher energy scales. My new thought is for each of us to work with a well about 0.6 Angstroms wide, i.e., L=.06 nm. And then make the well deep enough to accommodate 4 bound states. Can someone work on that preliminary part here and post results for energies and wave-functions that everyone can use. (It is a boundary condition problem. "Guessing" wave-function forms and then matching at one of the boundaries should lead to approximate energies and states. You only need to match on one side; the two side are redundant by symmetry. The slope and value can be matched by numerical "fiddling". You need a well depth to start with. Somewhat less than 16 times the kinetic energy of the ground state of an infinite well will have at least 4 bound states, right? (Ask if you don't see why?) I think each state can be specified with two normalization coefficients and two length scales (one for inside, one for outside).
This is basically the atom that we use to make our crystal. Having this in common for everyone will help us focus on the nature of the 4 independent bands that arise from these 4 atom states.
Problem 1: 4-Bands.
a) sketch the "atom" bound state wave functions as a
function of x. Pay attention to nodes and the length scale associated
with the evanescent part for each state.
b) do a graph (E vs q) of the
bands that you would expect to get from these bound states for an
infinite 1-dimensional crystal made of these wells. This is a graph of E
vs q (the convention is to make E the vertical axis and q the
horizontal axis and E negative with the zero for E outside the well.
i.e. so the ground state is at about -1400 eV, not +100 eV
c) Do
sketches of the overlap integrals that would play a role in an actual
calculation of these bands. Discuss the sign of each one and discuss
their relative magnitudes. Are this things relevant?
d) redo your graph of E vs q. Think about it. Redo it again.
e) What do you learn from this. Discuss. How close together would the wells have to be in the crystal in order to get a bandwidth of 5 eV for one of your bands? Which band?
f) How close together would the wells have to be in the crystal in order to
get a bandwidth of 10 eV for one of your bands? Which band?
g) what are some better things to ask about this?
Problem 2: Phonon (lattice) specific heat.
Remember how we calculated E vs q for the 1D lattice. I think it was a linear relationship at low q? Use that starting point to calculate the low temperature specific heat associated with the lattice motion degree of freedom. I think at low enough T you can approximate E vs q as linear since only the lowest energy modes are important. Do that. Just get the low temperature form. If you run into an integral, make it unit-less so that we can get the T dependence before we do the integral. If you want to do this problem in a post here that is fine. (That will give other people more time to work on problem 1.) What is the starting integral for E vs T? The thing that we would take the derivative of to get the specific heat? What is the form of the density of states? What is the temperature dependent occupation-related factor?
Please post comments and questions. I haven't really thought this out much, so let's do that here.
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ReplyDeleteSo, here is what I got as result for the preliminary part:
ReplyDeleteThe ground state energy, n=1, of a finite well with width a = 0.6 Å:
\(E_n = \frac{\hbar ^2 \pi^2}{2ma^2} n^2 \)
\(E_1 = 104 eV \)
To fit four bound states in our well it does not have to be deeper than \( 4^2 * 104 = 1664 eV \). In the calculations below I use \( V_0 = 1660 eV \).
We solve the Schrödinger equation and look at the even and odd solutions respectively, with the center of the well set at x = 0 and to make it less messy I set \( \frac{a}{2} = b \).
Even states are described by:
\( \phi (x) = \left\{\begin{matrix}
D e^{\kappa x} & x < -b\\
Bcos(kx) & |x| < b\\
D e^{-\kappa x} & x > b
\end{matrix}\right. \)
\( \kappa = \sqrt{\frac{2m(-E)}{\hbar^2}} \)
\( k = \sqrt{\frac{2m(E+V_0)}{\hbar^2}} \)
Fitting those by making the function and its derivative continuous at the well boundary gives us the following equation:
\( \sqrt{\frac{-E}{E+V_0}} = tan(\sqrt{0.02eV(E+V_0)}) \)
Where \( 0.02eV = \frac{2mb^2}{\hbar^2} \)
From here Wolfram Alpha did the job and the solution for the 3rd state is: \( E_3 = -878 eV \)
The odd states are described by:
\( \phi (x) = \left\{\begin{matrix}
-D e^{\kappa x} & x < -b\\
A sin(kx) & |x| < b\\
D e^{-\kappa x} & x > b
\end{matrix}\right. \)
Which gives the following equation and energies:
\( -\sqrt{\frac{E+V_0}{-E}} = tan(\sqrt{0.02eV(E+V_0)}) \)
\( E_2 = -324 eV, E_4 = -1306 eV \)
So, if you want the derivations I can take pictures of my notes and send by email, just let me know. Also let me know if there are mistakes, questions, anything!
Wow, this looks awesome. Nice work!
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DeleteLooking at these results, I would like to suggest a fine tune. That is, that we use a well depth of 1500 eV instead of 1660 eV.
DeleteThen we need to think about the question of how close we should put these wells to each other in the crystal. What should the distance between the wells be? I think that picking a well-chosen distance and then calculating E vs q, with our approximate method with actual numbers for the overlap integrals, will be instructive. Any ideas?
use Nellie's square well. (see her separate post above) Pick specific numbers for everything including well separation, and do integrals numerically. Does that make sense?
ReplyDeleteI'm not sure on how big we want the well separation to be, as wide as possible but still small enough so that the wave function for the ground states (and then automatically the wave functions of the rest states as well) overlap?
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