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A small signal solution can be obtained by linearizing poisson’s equation. The resulting solution is of interest when the potential across the semiconductor is small.
We start from the charge density r in a semiconductor for the general case where electrons, holes, ionized acceptors and ionized donors are present
where f is the potential in the semiconductor. The potential is chosen to equal zero deep into the semiconductor. For an n-type semiconductor without acceptors or free holes this can be further reduced to:
assuming the semiconductor to be non-degenerate and fully ionized. Poisson's law can then be rewritten as:
For small values of the potential |f| < Vt this equation can be linearized yielding:
where LD is the Debye length given by:
The resulting potential, electric field and charge density are:
where fs is the potential at the surface.
© Bart Van Zeghbroeck 1997