That extra MOSFET problem

Thinking more about the advanced problem we spoke about in class, which had to do with charge as a function of voltage in a MOSFET, I realize it could be pretty difficult. I think we're imagining starting with a p-doped semi-conductor starting out in a flat band situation in which there is no space charge and no band bending. The initial application of positive voltage leads to the development of a depletion layer, associated with the valence band bending away from the chemical potential. The width of this depletion region will tend to grow linearly with voltage, I think, and in that way Q=CV is satisfied (with a constant C, which I would expect depends simply on the width of the insulating oxide and its dielectric coefficient). Does that make sense? I hope so. Because, really, that's the easy part.

Then, as more voltage is applied, the conduction band will begin to get too close to the chemical potential to be ignored. And at that point, a large electron concentration, associated with what we might call deep inversion, will develop close to the interface between the semiconductor and the oxide.

I guess I was thinking that if one could assume some sort of exponential dependence of the charge density on z and then one could integrate that twice to get a potential that depends also exponentially on the z, and that somehow that might be self consistent. But then I realized, the charge density, which in that region is associate with the conduction electron density (n(z)), depends exponentially on the potential so we have an exponential of an exponential which doesn't seem very self consistent or easy to work with at all. HMMM…

So anyway, strike one, I guess. If anybody has any ideas how to model the inversion region self-consistently, or how to make a convincing argument that the width of the depletion region stops increasing when strong inversion occurs, that would be really great. Then at least we would have a pretty clear idea where the charge goes as a function of voltage. I'm still wondering how one might characterize the functional form and width of the strong depletion region. Maybe its not that easy.

Final, please comment, Due Weds noon...