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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 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.

In addition, the potential y(x) = yi (x) - yi (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) = ys at the Si surface.
To solve the equation, we need to find more information for r(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)

.                               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' 2nd Ed. , John Wiley & Sons , 1981, pp 367-368