Homework 3, more solutions

Notes:

• 1 and 2 soln's were posted earlier.
• Regarding 3, in response to this sort of question you are expected to express n(x) as $\color{white} N_c e^{-(E_c(x) - \mu)/(k_B T)}$ . It is also helpful to note that: $\color{white} N_c e^{-(E_c(\infty) - \mu)/(k_B T)} = N_d$. Then what remains is to evaluate a function defined in sections (piecewise) using a band-edge energy you have calculated from an assumed charge density in the depletion region $\color{white} E_c(x) = E_c(\infty) -e(\Phi(x)-\Phi(\infty))}$, where $\color{white} \Phi(x)}$ is related to the charge density. You do an analagous calculation to get p(x). n(x) and p(x) are important because they help us understand J(x).
• In #4 you may notice that i tend to calculate values in a quirky way that avoids mass in kg by using c^2 factors. There are many other ways to get numbers. Also, I am wondering if my result for part b) is off by 100 or so. Extra credit if you find this error (or any error in HW solutions.)
• For #5, I was thinking that, unlike a metal where there is "fixed" Fermi velocity (usually about 5x10^7 cm/sec), in semiconductors the carrier speed tends to depend on electric field. Then for a given scattering time*, the mean-free-path would be field-strength dependent (v=u E = e tau E /M*) and, since it depends of E, not that useful.
• In 6 the sign of the terms is important. I think there was perhaps a sign error in the problem originally? Is that true? One has to think carefully about gradients and flow and the sign of charge carried to get these signs correct.