How to quantify fringing fields around a finite width plate electrode?

[deusexlumina]

Homework Statement

This isn't homework or coursework as such, but i thought it may be the best place to ask this question. The last time i posted in the other section it was deleted!

Im considering the case of an electrode of finite width L in the x direction. The y direction is perpendicular to the electrode. The electrode is in contact with an N doped region, donor concentration is N_D. The region is depleted so that these are the fixed charges. As i approach the end of the electrode in the x direction, the potential drops off rapidly. This is particularly important for the work I am doing. The problem is i cannot find any analytical expressions for the field in this region. The potential should be a "flat bottomed well", with the width of the "flat bottom" determined by the width of the electrode. I tried solving Poissons equation, but it gave me a normal symmetric quadratic potential well. Any ideas?

Homework Equations

Poissons equation

∇²∅ = (-qN_d)/ε_si

The Attempt at a Solution

solving it, with the boundary condition that the field is 0 at the edge of the electrode gives:

∅(y) = (-qN_Dy²)/e_si +C₁y + C₂

The Cs are constants of integration.

 

[Nate810]

The problem is, this gives a normal symmetric quadratic potential well, not the "flat bottomed" one i was expecting. Any ideas?