# MOSFET - Seemingly equation contradiction (or my misunderstanding)

#### bobpease4ever

Joined Jul 28, 2019
23
Hello all,

If you remember, most textbooks say that the depletion charge in an N-MOSFET's channel in strong inversion is given by this equation:

Qb = -Sqrt( 2 * q * N_a * Epsilon_Si * 2 * Phi_F), where N_a is the substrate doping, Phi_F is the Fermi potential at the bulk.

The textbooks say that this charge is in the channel exactly when the gate voltage is equal to the threshold voltage (which itself has that charge term in it divided by C_ox). It also says this is the strong inversion charge ( so n-type carriers).

On the other hand, later on when developing the drain characteristics, it says that Q = Cox(Vgs - V_t) (with drain voltage at zero for this example). But if you look at this equation, the charge is exactly ZERO when Vgs = V_t.

How do we solve this apparent contradiction? On the one hand it says that when Vgs is equal to Vt, the charge is given as my first equation. On the other hand the charge is also ZERO when Vgs = Vt for the drain characteristics.

Could it be that even though the channel is inverted at Vgs = Vt, there is still too little charge and so they consider it as ZERO charge? But clearly there isn't zero charge when Vgs = Vt. There should be some charge there, no?

What on earth is going on here?

Thank you so much

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#### Dodgydave

Joined Jun 22, 2012
10,591

#### bobpease4ever

Joined Jul 28, 2019
23

#### bobpease4ever

Joined Jul 28, 2019
23

#### pcolarusso

Joined Jul 10, 2019
1
I think Qb and Q are different things. Qb is the bulk charge so it is the charge that is there from the doping of the semiconductor. This is the charge that needs to be depleted to create the channel. Q is zero at the threshold voltage because Qb has been depleted so there is no charge there

#### bobpease4ever

Joined Jul 28, 2019
23
I think Qb and Q are different things. Qb is the bulk charge so it is the charge that is there from the doping of the semiconductor. This is the charge that needs to be depleted to create the channel. Q is zero at the threshold voltage because Qb has been depleted so there is no charge there
I have thought about that, but it still does not make sense because the threshold voltage is composed of BOTH the depletion charge voltage, AND the 2Phi_F potential, which IS the potential to invert the channel !!

I think this equation is an extreme simplification since it says that for Vgs < Vt the charge is zero.

But that is so wrong.... I am doing a calculation now to find the correct amount of charge at the channel... Let's see if it is zero or not!

Any any more input to this serious issue?

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#### bobpease4ever

Joined Jul 28, 2019
23
I think Qb and Q are different things. Qb is the bulk charge so it is the charge that is there from the doping of the semiconductor. This is the charge that needs to be depleted to create the channel. Q is zero at the threshold voltage because Qb has been depleted so there is no charge there

I just did the calculation, and it says that the concentration of negative channel in the channel (p-channel) is exactly Nb = concentration at the bulk, when the channel reaches strong inversion. Hence there should be Nb of charge in the channel and not zero...

There is also ni^2 / Nb holes in the channel......

Makes sense because ni^2 / Nb * Nb = ni^2.

So there is charge in the channel. It's not zero charge.

#### bobpease4ever

Joined Jul 28, 2019
23
I just did the calculation, and it says that the concentration of negative channel in the channel (p-channel) is exactly Nb = concentration at the bulk, when the channel reaches strong inversion. Hence there should be Nb of charge in the channel and not zero...

There is also ni^2 / Nb holes in the channel......

Makes sense because ni^2 / Nb * Nb = ni^2.

So there is charge in the channel. It's not zero charge.

The formula is that that Qi = -Cox(Vgs - 2Phi_F - Gamma*Sqrt(2Phi_F)),

because we need Cox*Gamma*Sqrt(2PHI_F) in order to create the depletion charge, and the charge across the Cox cap is [Vgs -GammaSqrt(2PHI_F)] - 2Phi_F.

I know that PHI_S is the surface potential with respect to the bulk. But there is no potential unless we apply some at the gate(ignoring parasitics). If Vgs = 0, then the surface potential is zero. If Vgs = -Qi/Cox + Vt is calculated with Vgs = Vt, then Qi=0.

So I must conclude that this Vgs is a gate potential for any charges ABOVE the inversion charges, and NOT for the charges below threshold.

So really this equation must mean that Qi is any charge inverted AFTER strong inversion and not before. I think ........... Eureka!

Anyone could confirm this ?