Finding MOSFET threshold voltage

[cathaychris]

Homework Statement

I have a MOSFET with a $$Si/SiO_{2}$$ interface. I need to find the threshold voltage. I'd also like to know whether this puts the transistor in the "on" state or "off" state (I know this is simple; I'd just like a little verification).
I know the values for $$\Phi_{ms}$$, $$Q_{i}$$ (the interface charge density), and $$N_{A}$$ (the acceptor density).
I also know $$d$$, the thickness of the $$SiO_{2}$$.

Homework Equations

$$V_{T} = \Phi_{ms} - \frac{Q_{i} + Q_{d}}{C_{i}} + 2\phi_{F}$$
where $$\phi_{F}$$ is the surface potential required to bend the bands down to the intrinsic condition at the surface.
$$C_{i} = \frac{\epsilon_{i}}{d}$$
$$Q_{d} = -qN_{a}W_{m}}$$
$$W_{m} = 2\sqrt{\frac{\epsilon_{s}kT\ln{N_{A}/n_{i}}}{q^{2}N_{A}}}$$
$$\phi_{F} = \frac{kT}{q}\ln{\frac{N_{A}}{n_{i}}}$$

The Attempt at a Solution

I'm given $$\Phi_{ms}$$. I can find $$C_{i}$$, and I know $$N_{A}$$.
So if I find $$W_{m}$$, the maximum depletion region, then I can find $$\phi_{F}$$ and $$Q_{d}$$ quite easily. But it seems that I'm missing an equation. Also, I don't quite understand what $$n_{i}$$ represents, or at least I think it's only going to convolute the problem if I try to substitute for it, so there must be an easier way. What am I missing?

Thank you.

 

[KataKoniK]

Thank you for your question. The missing equation you are looking for is the relationship between the surface potential (\phi_{F}) and the threshold voltage (V_{T}). This can be found using the following equation:

\phi_{F} = \frac{V_{T} - \Phi_{ms}}{2}

Substituting this into the equation for Q_{d}, we can find the interface charge density (Q_{i}):

Q_{i} = -qN_{a}\left(2\sqrt{\frac{\epsilon_{s}kT\ln{\frac{N_{A}}{n_{i}}}}{q^{2}N_{A}}} - \frac{V_{T} - \Phi_{ms}}{q}\right)

To determine whether the transistor is in the "on" or "off" state, we need to compare the threshold voltage to the gate-source voltage (V_{GS}). If V_{GS} is greater than the threshold voltage, the transistor is in the "on" state and current can flow through it. If V_{GS} is less than the threshold voltage, the transistor is in the "off" state and no current can flow through it.

I hope this helps. Let me know if you have any further questions. Good luck with your research!