HW 6 wi solutions.

This week's classes will focus on FETs, so you are on your own, so to speak, with problems 1-3, which involve things we have already covered (but might be pretty challenging). (i.e., don't wait thinking that we will cover them in class.)

These problems are tricky to design and to get the parameters in a range that works. I am interested to see your results for 1c and 2a, which are the easier parts. Please email me those as soon as you get something if you like. (Then we can see if the problem is going to work out okay.  The voltage range should hopefully stay below the band bending.)
1. Consider a semiconductor with an energy gap of 1.5 eV, \( D_2 = 0.8 \times 10^{22} \, states/(eV-cm^3)\) and \( D_3 = 0.8 \times 10^{22} \, states/(eV-cm^3)\), \(D_n = D_p = 100 \, cm^2/sec\) and \(\tau_r = 10^{-10} \, sec\). Suppose you make an n-p junction with an area of 1 cm^2 where \(\mu\) is 0.2 eV away from the majority carrier band on each side. [1a) and b) are warm-up problems and should be pretty easy and quick to do. c) is where it gets more difficult.]
a) What is the total band bending in this case (the difference between E_c on the left and on the right)?
b) What are the majority and minority carrier concentrations on each side? (away from the interface region) [Don't be discouraged if the minority carrier density seems too small; that might be a natural consequence of the larger gap (1.5 eV).]
c) If you connect a battery to the junction, how much bias voltage would you need to get a current of 1 amp?
d) How much voltage would you need to get a current of 0.5 amps?
e) Which way does the current go (most easily)? Sketch a picture.

2. Suppose you get rid of the battery and instead have a resistor, R, in its place. Then you illuminate the junction with \(10^{19}\) photons per second per cm^2. Each photon has an energy of 1.5 eV.
a) If half the photons are absorbed, in the junction region, then how much current would you get for R=0 Ohms?
b) Which way does the current go through the resistor? Sketch a picture.
c) How much current would you get for R = 1 Ohm?
d) Is this difficult? Why? (discuss here) Would it be easier if you were asked to get an estimate of I to just one or two sig figs?

https://drive.google.com/file/d/0B_GIlXrjJVn4djlsd2ExZElKNHc/edit?usp=sharing

3. We can think of the resistor as a stand-in for something that we would like to provide power to.


a) For an illumination of \(10^{19}\) photons per second per cm^2, what value of R provides the most power? (assume 50% absorption in an electron-hole pair generating process in the junction region) What value of V, the voltage across the resistor, does that correspond to?
b) For an illumination of \(.25 \times 10^{19}\) photons per second per cm^2, what value of R provides the most power? (assume 50% absorption, as before) What value of V, the voltage across the resistor, does that correspond to?
[Hint: You may not be able to do something elegant here. Focus on getting an actual value of R (and V). Imagine you are working for a company or group and they just want to know which resistor to put in the circuit (what impedance to put in the circuit). They want a number in Ohms accurate to 5 or 10%. Don't let them down.]

These problems may not all be do-able in the usual straightforward manner that you may be used to. Please discuss where you get stuck here.

You might want to do 5 before 4, though you would have to read 4 first.
4.  Consider a semiconductor with an energy gap of 1.0 eV, \( D_2 = 0.5 \times 10^{22} \, states/(eV-cm^3)\) and \( D_3 = 0.2 \times 10^{22} \, states/(eV-cm^3)\). Suppose this semiconductor is doped so that \(\mu\) is 0.35 eV above the valance band edge. Suppose you grow, evaporate or in some way create an insulator 20 nm thick (a material with a large band gap, say 5 eV) on a surface of this semiconductor, and that on top of the insulator you evaporate a metal layer (roughly 100 nm thick). Then you attach a wire to the metal and to the back of the semiconductor.
Additionally, suppose that, by design (i.e., choice of metal, doping in the semiconductor...) the Fermi energy of the metal is the same as the chemical potential (mu) of the semiconductor. Additionally, suppose that the minimum energy to excite an electron from the conduction band edge of the semiconductor to the vacuum is 4 eV.
a) What is the density of holes in the semiconductor?
b) What is the minimum energy to excite an electron from the valence band of the semiconductor to the vacuum?
c) What is the work function of the metal? Can that be determined from what you are given? If so, how?
d) As a warm-up for the next part, sketch the position of the Fermi level of the metal and of the semiconductor band edges, E_c and E_v, as a function of x, where x is the distance from the metal-insulator interface.
e) Insert a battery in the wire and do another band sketch for the case where the gate voltage is 0.5 eV higher than the voltage at the back end (far from the interface) of the semiconductor. You can assume that voltage drop (and variation of mu) occurs primarily in the insulator.
f) Do you see anything possibly occurring in the vicinity of the interface that might be interesting?

5. Suppose we regard the metal-insulator-semiconductor structure of the previous problem as a capacitor, where one plate is the metal and the other is the semiconductor. Applying a voltage across them should lead to a charging in which there is equal and opposite charge on the two sides. One could calculate C, but for this problem let's take C as given. Recall that Q=CV.
a) What sign of V would lead to a negative charge in the semiconductor?
b) Where might that charge reside in the semiconductor?
c) Assuming the same doping level, etc as in the previous problem, how thick a depletion layer would you need for a given voltage V?
d) Do a sketch (graph) as a function of x (distance from the interface) showing the space charge density for a voltage that depletes the valance band near the interface and also,
e) mu and the band edges as a function of x.  
(possibly one more problem to follow. )