Croaker
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I'm not really sure what you are trying to ask in the first part.
The energy band diagrams are for electrons. Conventional potential is for positive charges. The electron has the highest energy where the potential for a positive charge (this is the convention for voltage) would be lowest. When going from electron energy to voltage, a negative sign must be applied.
E.g. if there is a band energy difference of 10 eV, an electron has to experience a 10 V drop in potential to move to the higher band
In other words, the electron energy band diagrams are inverted relative to voltage diagrams; moving up on the electron energy diagram is the same as moving down on the voltage diagram. *IF* voltage meant the potential of a negative charge, they would not have an inverted relationship.
-qV = E
For an NMOS device we have a p-substrate. We apply a positive voltage to the gate to bend the bands. Ef cannot change because the device is still in equilibrium (no current flows), so Ei bends.
For an inverted NMOS device, at the surface, Ei is below Ef by |Ei-Ef|; the total energy difference is 2*|Eib-Efb|, where Eib and Efb are the energy levels in the bulk.
With the way you've defined Vb:
Ei - Ef = -q*(Vi - Vf) (Ei - Ef)/q = Vf - Vi = Vb
For what you presented, both 1) and 4) are equivalent.
The key is just understanding the inverted relationship between electron energy and voltage.
Cheers, Marc
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