What did we learn today? (and more)

Well, several things, depending on how you look at it:

1) How to start problems in which the proximity of mu to the band edges is "given" (specified), instead of Na and Nd.

2) How Ec(x) behaves and looks for an asymmetric p-n junction (i.e., a junction in which Nd =/ Na). (And also rho(x) and E(x) (electric field).)

3) That things can become a little grungy, and definitely more difficult computationally, when we have less symmetry.

4) Anything else?? (your comments are welcome)

(Oh, i thought of one other thing. With regard to our process and the failure of our original method. What we (I) originally thought was that we could divide the "step" change in Ec(x) into two equal parts. That would have made the problem easier, but it was wrong and so we had to retool when we realized that.
In physics we often try to use intuition and symmetry to simplify; in this case we went too far and had to reconsider our simplifying assumptions. I think we learned the meaning of the phrase: "things should be made as simple as possible, but not simpler".)

Perhaps even more important than these specific things, we got a chance to spend more time in the world of band edges (Ec(x) is a "band edge"). This experience will serve us well when we take on the surprisingly daunting task of understanding exponential current flow in a biased p-n junction (that is, the exponential dependence of I on V).

At this point Ec(x), Ev(x) and mu should be your 3 closest companions (your new bff's, if you will). Otherwise the I-V analysis of a p-n junction will seem hopelessly complicated, when really it is merely hopelessly subtle and mysterious.