There are no new hw problems that i have to assign today, but here are some thoughts on what you might do "for fun" to follow up on our exploration of doping and mu today.
Try doing our problem today, i.e. calculating chemical potential, mu, for a few different values of doping (i.e., phosphorous substitution concentration). Like maybe: 10^17 and 10^18. See where our approximations break down... (like the apprroximation of f(E). was that okay today? when would it not be okay?
(hint: if one finds that mu is venturing into, or really close to the CB, that would not be good, right?) What if we started out saying that n_i was 10^11 (cm-3) instead of 10^10? How would they effect the breakdon pt and why?...
What about lower dopings like 10^14 or so? Does anything get weird there?...
So anyway, one could continue to explore. Try making Nc and Nv different, see what effect that has. Try specifying Nc and Nv explicitly, instead of implicitly through n_i or ...
PS. What would you think of a surprise pop quiz on this on Tuesday in class?
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Hi - it looks like I can get values for "style 1" where we are given ni and Nc...
ReplyDeleteUsing a bunch of values for Nc:
Nc = {{10^14, 10^15, 10^16, 10^17, 10^18, 10^19, 10^20}}
and ni = 10^16
I can find several, successive mu's:
mu = {{-4.60517 k T, -2.30259 k T, 0., 2.30259 k T, 4.60517 k T, 6.90776 k T, 9.21034 k T}}
I'm still trying to figure out how to get a sensible mu using "style 2" where we start with a value for Nc and a relationship between Nc and Nv...
The mu's all seem to be within the 1eV bandgap, but low values of Nc retrieve negative values for mu. Hmm... Saying that one could lower the chemical potential by adding donor impurities would be a lot like saying that one could make a solution more basic by adding an acid. In an aqueous solution, this is like trying to change the inherent concentration of 10^-7 H+ (hydrogen ions) with 10^-8 H+. In our crystal, I think we would be neglecting that the initial ni would swamp the low values of Nc.
For the high values of Nc, the chemical potential didn't raise above 0.4eV - so mu never reached the conduction band.
Without making judgements about resistivity and the likelihood of electrons falling into trap-states, I can't really come up with any ideas about what would happen if the chemical potential had gotten really close to the CB.
I think I remember someone saying that a large enough Nc could move the chemical potential above the CB resulting in a metallic crystal - where the HOMO overlapped and had access to the closely spaced levels in the conduction band.
For ni := 10^14, all the mu's move upward - qualitatively, it is easier to move the chemical potential (the chemical potential doesn't have to move as far to accommodate as many carriers).
mu = 0., 2.30259 k T, 4.60517 k T, 6.90776 k T, 9.21034 k T, 11.5129 k T, 13.8155 k T