Summary/preparation post.

In addition to understanding how to do problems, I would suggest seeking an understanding of the underlying reasoning and concepts. This may be obvious, but I though I would mention it anyway. Also, I recommend that you save time by making appropriate approximations wherever you can. The ability to recognize and use approximations shows understanding and confidence (or as someone once said "if you see a fork in the road, take it").

You'll be asked to calculate n for a given mu or visa versa.  You will wish to have a good understanding of the fermi function and its approximations -- including how to use approximations to get n to very few significant figures quickly. If you don't have a calculator you may want to bring a  table of e^{-n} where n is an integer. Also, be able to sketch, understand, visualize and estimate the area of integrands. Understand how the Fermi function pertains to metals as well as semiconductors.

Understand Fermi surfaces and how to construct and visualize them in novel 2D k-space.

Understand band bending and how it pertains to junctions and FETs. Understand "everything" about a forward biased or illuminated p-n junction. Understand circuits, the relationships between current and voltage.

What else? Your input, comments and questions are welcome and encouraged!

Draft practice problems:
1. Consider a crystal for which the density of states is given by a half cycle of a cos function centered at \(E=E_o\). That is,
 \( D(E) =D_o cos ((E-E_o)/b) \, states/(cm^3-eV) \).
Outside that half cycle, D(E) is defined to be zero.
a) plot the density of states. what is the band width?
b) Suppose that the total number of states in the band is 10^22 cm^{-3} and KT=0.025 eV. Estimate, calculate or determine the carrier density to just one sig fig for \(\mu\) :
0.5 eV below the band edge
0.2 eV below the band edge
0.2 eV above the band edge
1.0 eV above the band edge
\(\mu = E_o\)
Make each calculation as simple as you can by making appropriate approximations.

2. Sketch a Fermi function with \(\mu = 0.5 \, eV\) from 0 to 1 eV for: a) KT=0.05 eV,  and, b) for KT = 0 (or .00001 eV if you prefer).
c) Sketch the integrand for a calculation of n.

3. Sketch the bands and chemical potential for an unbiased and a biased n-p junction.

4. Describe the journey of: a) an electron that is created in an illuminated n-p junction, and b) one that travels through a biased junction in "LED mode".

5. Explain the concepts associated with an illuminated n-p junction in series with a simple resistor. What does the resistor represent? What does it do? Why does its R value matter to how much power is generated? How does the resistor effect the n-p junction? How does the junction effect the resistor? What are the concepts involved in optimizing power generation?

6. Sketch the bands of a MOSFET, with an inversion layer, that is "on" due to:
a) an applied bias voltage
b) due to a work-function "mismatch" issue (normally on).
c) what is an inversion layer?

7. Illustrate and explain the origins of a ferromagnetic instability of a half-filled band. What is the possible benefit, to a metal, of becoming ferromagnetic?

8. Illustrate and explain the origins of a mott-hubbard instability of a half-filled band. What benefit can such a Mott insulator gain from having its spins arranged antiferromagnetically?

9. Sketch the structure of graphene. Why are there 2 pz derived bands and not just one? How many total filled bands are there in graphene? Explain, and put them in groups according to nature or function.

10. Here is a problem related to ferromagnetism. I am not sure how computationally difficult and time consuming it is, and it may not be worth your time, but on the other hand, it might illustrate the nature of ferromagnetism in an interesting way. Using the D(E) from problem 1 above, and assuming the band is half-filled in the normal state, with a density of electrons, n. We can divide the density of states into two identical parts, one for spin-up and the other for spin-down electrons.
a) Calculate the band energy by integration. How does the band energy depend on \(n_{\uparrow}-n_{\downarrow}\)?
b) For what range of bandwidth would a ferromagnetic state \(n_{\uparrow} \neq n_{\downarrow}\) have a lower overall energy than the normal state \(n_{\uparrow} = n_{\downarrow}\) , given that there is an iteracttion energy in addition to the band energy that can be approximated as (4 eV) n - \((U/n) (n_{\uparrow}-n_{\downarrow})^2\) where U = 2 eV.
c) Explain where such an interaction energy could come from and how that origin involves Fermi statistics.
d) extra credit: Approximately what value of bandwidth would lead to \(n_{\uparrow} = 6 n/10\). [Hint: expand the band-energy term around \(x= n_{\uparrow}-n_{\downarrow}=0\) and keep only the terms you need (probably x^2 and x^4).]