CuO2 bands.

This post deals with the nature of the CuO2 bands (within the compound \(La_2CuO_4 \) ). The 3 bands that one can construct from the \(O-p_x, O-p_y, Cu-d_{x^2-y^2}\) orbitals are somewhat interesting in their own right, and they also provide a format in which to understand how bands acquire "mixed character" and what it means to say a band is 70% Cu / 30% oxygen or something like that.

This first video derives the LCAO (linear combination of atomic orbitals) nearest-neighbor matrix. Maybe later I can do another one that looks at and interprets the eigenvectors. Feel free to do that on your own also. What eigenvectors do you get at k=0 and how would you interpret them in terms of, e.g., percent of Cu content? This is just exploratory, so feel free to make assumptions about the magnitude of t (maybe around 1 to 6 eV?) and about the diagonal terms, Ed and Ep.



Hmmm... I see now that my k=o suggestion has some problems and limitations. Why is that? What happens at k=0? Something kind of interesting, but a bit special compared to everywhere else in the BZ.

Here is another idea. There might be some specific zone-boundary spots that would be interesting to explore. Places where something is pi/a; or both things are pi/a. That might be more interesting and better. I think for this compound it is the zone-boundary regions that are most important and interesting.

Notes on the final.

A couple of notes:
On problem 1 b) an estimate just based on the location of \(E_F\) within the band is pretty accurate. There is no advantage to integration. (By the way, for 1a, if you did not have a calculator, but you got the form correct that is going to get you most of the points. I think 1a was the only problem where you needed a calculator.)  What do you get for problem 1?

In problem 5a the voltage across the resistor or junction is essentially the open circuit voltage which is 1.5 V.  The \(e^{-60} I_{ill}\) tells you that the highest voltage possible is 1.5 volts since (1.5 eV/KT) = 60. I would say this is a "large resistance" in this context. What do you think? What did/do you get for I?

In problem 5b you can ignore the exponential part of the I-V relation. The \(V^2\) term is dominant everywhere below 1.3 volts or so, I think. With just the constant negative term, \(I_{Ill}\), and the quadratic term, I think you can solve for the maximum power in closed form. (differentiate and set to zero). That is the approximation that would really help for that one. I am not sure how many people got that. What R value do you get for maximum power?

Problem 6 I believe the total energy has two off-center minima representing ferromagnetic states for alpha greater than one eV.  The x=0 point is the stable equilibrium point for alpha less than 1 eV, but becomes unstable above that value of alpha. This illustrates the nature of the instability leading to the ferromagnetic state when the bandwidth is narrow and the interaction is strong. Does anyone have an idea where those off-center minima are for alpha = 1.4?

Any comment or thoughts you have, feel free to share them here.

Final thing: 2 page email pdf.

As we discussed, please send me a roughly one or two page document in pdf or plain text format, equations are fine, but not required at all. Please focus on one or two things that you found particularly interesting --what they are, why you found it interesting including a clear and cogent explanation of your chosen topic and anything else related to that. (The length suggestion is just a rough guideline. Whatever length seems appropriate to you is fine.)
zacksc@gmail.com    (pdf format)

Thanks very much. I really enjoyed this quarter.
Best,
Zack

Last minute advice and thoughts.

There will be no problem on FETs.

There are a number of short problems that put emphasis on conceptual understanding and descriptions, as well as some longer problems where appropriate approximations are important. Sleep is good.

1. What essential aspect allows a semi-conductor to function as a semi-conductor?
--a gap and a malleable (easily move-able) chemical potential. you can dope it.

2. What is one essential key thing you need to understand to be able to understand and model inhomogeneous semi-conductor systems like junctions and FETs?
--band bending

3. What essential characteristics do metals manifest?
--a Fermi boundary (inside which essentially all states are filled). Very quantum. Pauli exclusion plays a big role.

4. Why are some metals weird?
--electron-electron interactions can make metals do strange things. Unusual collective behavior, for example ferromagnetism, anti-ferromagnetism in Mott-Hubbard insulators and superconductivity, results from electron interactions.
(Normal metal are sometimes unstable. What is the origin of instability.)
--e-e interaction 

Summary/preparation post II.

I was trying to think of something profound to say about the test. How it will test your understanding and will be the kind of test where getting a lot of sleep the night before will be really important. But I haven't been able to conjure up the right way to say that yet. Anyway, regarding Wednesday, please come prepared with your questions, and also feel free to ask more questions here.

And as you study your more detailed list of topics remember these essential things:
1. What essential aspect allows a semi-conductor to function as a semi-conductor?

2. What is one essential key thing you need to understand to be able to understand and model inhomogeneous semi-conductor systems like junctions and FETs?

3. What essential characteristics do metals manifest?

4. Why are some metals weird?
(Normal metal are sometimes unstable. What is the origin of instability.)

Polls are broken.

There is a system-wide blogspot polling problem. If you would peruse and remember the choices on the problem preferences poll and give me your feedback tomorrow in class, that would be very helpful and appreciated. Also, please comment here on your problem preferences, positive and negative.

Tuesday Class.

I am thinking that this Tuesday's class should be at least partially a question-driven review of key topics. Please come prepared with your questions! Also, you can post your questions here.