The potential y, as well as the electric
E, as a function of distance
x can be obtained by solving
the Poisson equation. If we consider an n-MOS capacitor that has the p-type
Si substrate, the Poissonís equation can be written as
where esi is the semiconductor (Si) permittivity and r(x) is the total space-charge density given by . are the densities of the ionized donors and acceptors, respectively.
And for p(x) and n(x) we can express them in terms of
potential y using Boltzmannís relations.
Therefore, using Eq.3 the n(x) and p(x) can be rewritten
and . Eq. 5
And then, substituting Eq. 2, 4, 5 into Eq. 1 we have
From the Eq. 6, we may derive the relation between the electric field
and the potential y as
And also, one can find the total charge per unit area, Qs, induced in
the silicon using the above equation 7 . We leave it to a Homework
assignment for students.
S.M. Sze, 'Physics of Semiconductor Devices' 2nd Ed. , John Wiley & Sons , 1981, pp 367-368