Threshold voltage is that gate voltage at which the surface band bending is twice ,Where
We know that the depth of depletion region for is between 0 and and is given by,
Charge in depletion region at is given by where
Beyond threshold, the total charge QD in the seminconductor has to balance the charge on gate electrode, Qs i.e. where we define the charge in the inversion layer as a quantity which needs to be determined.
This leads to following expression for gate voltage-
,Where
is between 
and is given by, 
is given by 
where
where we define the charge in the inversion layer as a quantity which needs to be determined.
