# 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.
|  | 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?

5.3.1 Heterojunctions