HW 5 with Solutions.

1. Using the equation we obtained for n(x) last Thursday (April 24) for an n-p junction with an applied voltage that pushes electrons to the right:
a) Calculate the diffusion current in the region \(x \ge 2d\).
b) How does the size of this current depend on the doping level of the p-type material? How does it depend on the applied voltage?
c) How does it depend on the recombination time?
d) Does our result from problem 1 for the overall magnitude (prefactor) of the current seem correct? I haven't really checked any books regarding this. Is there anything wrong with this result for the current prefactor? Does our result agree with what other people get?

There are more solutions, as well as the original problems, below the break. I think for 5 it should say "electric field". Please post any comments below.

[between 2 and 3, which problem do you think should come first?]
2. Current bottleneck problem:  (I probably should have thought of asking this problem earlier (last week). I think it plays a key? role in explaining why the current through an entire n-p junction can be estimated by calculating the current at just one spot.  Also, feel free to skip this and go on to the next problem if it seems confusing. I wasn't really sure what problem order would be best.)
Suppose you start with a homogeneous semiconductor doped to \(10^{17} cm^{-3}\). Suppose its dimensions are 4.5 cm x 5 mm x 5 mm, except that in the middle there is a region 5 mm long etched to 0.1 mm x 0.1 mm. Suppose you applied a voltage of 1 Volt across the long direction and just for convenience lets put the right hand side at ground.
note added:  \(\tau_{tr} = 10^{-14} \, sec, \: m^* = .2\)  (what does "tr" stand for?)
a) estimate the current through this semiconductor.
b) Use your estimate to calculate the voltage at each end of the narrow region. What do you get for the voltage drop across the fat sections? How does that compare with that across the narrow middle region?
c) (extra) Sketch your best guess of the current density throughout. (There is not necessarily one right answer,  but this might be fun to try.)
d) Discuss transport scattering time and recombination time. To what sorts of events is each one related?

3. For somewhat complex reasons, one can approximate the total current through a biased n-p device from the value of the diffusion current at x= 2d.  (We assume that at x=2d the built-in electric field associated with the junction region is zero and therefore that drift current is zero; we also assume that for \(x \ge 2d\) the decrease in this minority-carrier diffusion current is compensated by an increase in majority-carrier drift current, With these two assumptions, one can argue that that diffusion current at x=2d provides a pretty accurate estimate of the total current through the biased device.)  With that in mind explain why:
a) a short recombination time might enhance this current.
b) Also explain the intriguing dependence of this current on doping \(N_a\) (on the p side) and,
c) the stunning dependence of this current on the applied voltage.

4. Consider an n-p device of dimensions 1mm x 1mm x 1cm. Each side is doped at \(10^{17} cm^{-3}\) and the energy gap is 1.5 eV. Let's assume that every recombination event leads to a photon.
a) What else would you need to know in order to be able to calculate the radiated power as a function of bias voltage? (Let's work this out here in the comments.)
b) Calculate the radiated power at 0.5, 1.0, 1.3 and 1.4 Volts (bias). (Make a table.)

note added:  \(D_n = 100 \, cm^2/sec\),  \(\tau_r = 10^{-14}\), and use the D_2 and D_3 (density of states) same as in last week's HW.

5. Consider an unbiased n-p device just like the one in the previous problem (but unbiased).
a) Suppose a photon excites an electron from the valence band to the conduction band. What happens to the electron? What happens to the empty state it leaves behind?
b) Suppose you connect the n and p ends with a wire of negligible resistance. How much current flow could you get through the wire if there were \(10^{18}\) photons per second exciting electrons from the VB to the CB?
c) what kind of current is that? Which way does it go?
d) extra credit: what constitutes negligible resistance in this context?

6. Suppose that the resistance is R, and maybe it is not completely negligible. How would that effect things? (Things like voltage and current.)
a) Can you think of the current as having two parts? One associated with the excitations from incident photons and another due to a voltage across the resistor (which creates a voltage across the junction that was not there before). Do those add or do they oppose each other? What you say about the nature of each current (drift or diffusion)? [I think that in E&M there is a superposition principle, associated with linearity, that is helpful here.]
b) Is this really the best way to look at (parse) the current? I am really not sure.

7. [This problem is exploratory. I am not really sure how it will work out or what it will show.] It may be of interest to explore how long it takes for a photo-excited electron to get out of the junction region.
a) For example, suppose it starts at t=0 right at the interface. Which side would it go to and how long would it take to get there? (What other input parameter(s) might one need to calculate this?)
b) Based on that, roughly how much does the illumination (as in problem 5) effect the carrier density in the interface region? How about in the other regions? Let's use \(\tau_{r} = 10^{-13} \, sec\) (this is the recombination time) for this problem, or perhaps you have a different suggestion?).

8. (added April 29) Draw a picture of a field effect transistor (FET). Any FET. Your choice. Pay some attention to the nature and details of the source and drain contact regions.