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...

Lasers and LEDs

Comments, questions and discussion are encouraged.

Why n+ to p to p+ ?
Where is the p-n junction?
Where is the light emitted?

What role(s) does the (GaAl)As play?
How does it help?

Circuit elements

Here are some pictures of "circuit elements" that use FET's. Please comment, question, etc.
Beaucoup extra credit for any ones who explain what these are and how they function.

Last HW assignment

For this week's HW perhaps you could write a two page report outlining what you have learned in this class, (including what you expect to learn about lasers and photo-detectors next week). You can also post outlines of this here or comments, questions etc. related to this matter.

Also, suggested problems for the final are encouraged.

Key moments in the life of a MOSFET

Key moments for a MOSFET tend to occur as a function of z, distance from the oxide (in the semicondoctor), or V (applied gate voltage). They generally involve the relationship of the bands to $\mu$. Focusing on the phenomenology of the space-charge:
1) one obvious "key moment" is when the valence band has moved a few kT away from $\mu$ indicating the onset of depletion,
2) a second, less obvious "key moment", which generally occurs at a smaller value of z, is when the the electron concentration in the conduction band, n(z), becomes greater than $N_a$. At that point one is leaving the depletion regime, in which the space charge can be approximated as a constant ($-e N_a$), and entering a realm in which the space charge will be strongly (exponentially) dependent on z.

(Actually, n(x) always depends exponentially in $E_c (z)$, but that is "hidden" until the rubicon of "2)" is crossed as I understand it. )

(Comments welcome.)

PS. a third key moment might be when $E_c(z)$ crosses $\mu$, but we should probably focus on understand the first 2 before delving into that.

MOSFET picture clarified

Maybe this will help clarify some of the issues that came up in today's class. This focuses on the nature of a metal-oxide-semiconductor(p-doped) sandwich, where-in the chemical potentials of the metal and doped semiconductor are perfectly aligned. In the left-hand picture the voltage is zero. If you apply a positive voltage, that leads to depletion in the p-doped semiconductor (center picture). All the equations pertain to the center illustration and to the use a a depletion ansatz for that case. It is interested to see how the applied voltage, V, influences the depletion width, $z_0$, and the band bending, $\Delta \Phi$.

Looking at these relationships, which seem pretty understandable, i am thinking we should have started with this simpler zero chemical potential difference case. For extra-credit, who can figure out/ calculate how a higher chemical ptential in the metal would effect the equations for the depletion width and $\Delta \Phi$.