The following tour physical components of that affect threshold voltage of a MOS structure.

1. Work junction difference between the gate and channel.
2. The gate voltage components to change the surface potential
3. The gate voltage component to offset the depletion region charge.
4. the voltage component to offset the fixed charge in the gate oxide and in silicon-oxide interface.

Depending on the gate material the work function difference is

$\phi_{GC}=\phi_F\textrm{(substrate)}-\phi_M \textrm{for metal gate}\\ \phi_{GC}=\phi_F \textrm{(substrate)}-\phi_F(gate)\textrm{for polysilicon gate}$

Another component of the applied gate voltage is necessary to offset the depletion region charge, which is due to fixed acceptor ions located in the depletion region near surface. We can calculate the depletion.

Charge density at surface inversions

$Q_{BO}=-\sqrt{2q.N_A.E_{si}|-2\phi _F|}$

If substrate is biased at different voltage level than source then it can be expressed as

$Q_{B}=-\sqrt{2q.N_A.E_{si}|-2\phi _F+V_{SB}|}$

The component that offsets the depletion region charge is then equal to -QB/Cox. Where Cox is the gate oxide capacitance per unit area

$C_{ox}=\dfrac{E_{ox}}{t_{ox}}$

The gate voltage component that is necessary to offset this positive charge on the interface is -Qox/Cox

It can also be written as

$V_T=\phi_{GC}-2\phi_f-\dfrac{Q_B}{C_{ox}}-\dfrac{Q_{ox}}{C_{ox}}$

It can also be written as,

$V_T=\phi_{GC}-2\phi_F-\dfrac{Q_{BO}}{C_{ox}}-\dfrac{Q_{ox}}{C_{ox}}-\dfrac{Q_B-Q_{BO}}{C_{ox}}$

$=V_{T0}-\dfrac{Q_b-Q_{BO}}{C_{ox}}$

The substrate term is a simple function of the material constant and of source to substrate voltage VSB.

$\dfrac{Q_B-Q_{Bo}}{C_{ox}}=\dfrac{-\sqrt{2q.N_A.E_{si}}}{C_{ox}}.(\sqrt{1-2\phi_f+V_SB})-\sqrt{|2\phi_p|}$

The most generalized express for threshold voltage VT is:

$V_T = V_{T0}+r.(\sqrt{|-2\phi_f+V_{SB}|}-\sqrt{|2\phi_F|})$

where $r=\dfrac{\sqrt{2q.N_B.E_{si}}}{C_{ox}}$