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*x*).
In the bulk Si, far from the surface, charge neutrality condition for uniformly
doped p-type silicon must exist. (Here, we assume the p-type bulk Si is
uniformly doped for simplified solution)

*Mathematical Analysis*

The potential y, as well as the electric
field
*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 Poisson’s equation can be written as

Eq. 1.

where e_{si} 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.

In addition, the potential y(*x*)
= y_{i} (*x*) - y_{i}
(*x* = ¥) is defined as the amount
of band bending at position *x*. Where *x*= 0 for the potential
at silicon surface, and *x* = ¥
for the intrinsic potential in the Si bulk. And the boundary condition
can be given by y = 0 in the bulk Si, and y
= y(0) = y_{s}
at the Si surface.

To solve the equation, we need to find more information for r(.
Eq. 2

And for p(*x*) and n(*x*) we can express them in terms of
potential y using Boltzmann’s relations.

Eq. 3

Therefore, using Eq.3 the n(*x*) and p(*x*) can be rewritten
as

Eq. 4

and . Eq. 5

And then, substituting Eq. 2, 4, 5 into Eq. 1 we have

.
Eq. 6

From the Eq. 6, we may derive the relation between the electric field
*E*
and the potential y as

.
Eq. 7

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.

**Reference**

S.M. Sze, '*Physics of Semiconductor Devices'* 2^{nd}
Ed. , John Wiley & Sons , 1981, pp 367-368