Appendix E
BEYOND THE DEPLETION
APPROXIMATION
In the depletion approximation the contribution of mobile charges to the electrostatics of the
depletion region was neglected. This allowed one to accurately define depletion region edges be-
yond which the material was neutral. A schematic of this structure is shown below in figure E.1.
However, this picture is not physical because the mobile charges cannot abruptly go to zero
but will decrease in a manner predicted by the law of the junction where
n=nn 0 e−kqΨ
BT (E.1)
whereΨis the band bending measured from the bulk. This is shown schematically in fig-
ure E.2.
As the mobile charge concentration decreases exponentially with band bending the net charge
in the regions close to the depletion region edge is no longer given by the depletion charge, but
as is always the case in general, the sum of all mobile and fixed charges. Studying thep-side of
the junction
ρ=−eNA−+epp(Ψ)
and∂∂xEderivates from the linear relationship when the charge is constant. This leads to “skirts”
in theEvs.xrelationship. It also raises the question, “what is the depletion region edge?”
The depletion region edge is defined by extrapolating the linear region of the curve (where the
mobile charges are negligible) to zero. We recognize that the area under theEvs.xis the built-in
voltage of the junction,Vbi. This is obviously larger than the area of the triangle, specifically by
the area of the “skirts” shown shaded in figure E.2. We will show shortly that each of the areas
is of the orderkBeT, the thermal voltage. Hence the area under the triangularEvs.xcurved
bounded by−Wpand+Wnis
V′
bi=Vbi−kBT
e−
kBT
e=Vbi−2 kBT
e