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.
For number 6, I got that the minima were at \( \pm n \sqrt{28.8} / 32 \) , if I recall correctly... That's either a 32 or a 64, something like that.
ReplyDeleteFor 5 b) I (think) got a maximum power at R=1/15
ReplyDeleteOnce you realize that the \(e^{qV/kT}\) term can be ignored below about 1.3 eV, you can pretty easily solve for the maximum power in closed form, since \(I = -I_{Ill} +\gamma V^2\) and so P (in the resistor) can be written as \(P=IV= -I_{Ill}V +\gamma V^3\). Then you can just take the derivative and set it equal to zero, right?
DeleteThe key to the problem was to think to question whether you really needed to consider both terms and realize that one would have almost no effect on this answer.
I think that would work. I used the V I graph and tried to estimate a box that maximize the area within the I(V) curve.
ReplyDeleteAre you going to have office hours this week?