Potential Discontinuities and Dipole Layers

The Possion equation in its simplest form is
e0erd2V/dx2 = -r(x).
Differentiating the potential including the discontinuities thus will gives us the charge distribution. We can do that very easily in a qualitative way a shown on the left hand side below.

Differentiating potentials

However, an infinitely sharp discontinuity will not be noticed in the dV/dx curves, The curves we get are identical to the old curves without a discontinuity.
Infinitely sharp discontinuities, or singularities in general, often do not make sense in physics. All we have to do therefore, is to redraw the potential with the discontinuity spread over a small distance (obviously at the minimum in the order of the atom size)
Differentiating graphically in a qualitative way now is easy, this is shown on the right hand side.
We now get a sharp "wiggle" in the charge distribution, corresponding to a dipole layer right at the interface.
Much can be learned from this. Here are a few suggestions for investigations of your own:
Look at the other type of discontinuity
Look at the case of extremely heavily doped semiconductors
Now look at the junction between two different metals. Can you understand why such a junction is not "felt" electronically?
Can you guess on much charge is transferred from one material to the other one? On the field strength that we encounter in these dipole layers?

zu_zu.gif 5.3.1 Heterojunctions