# Boundary conditions electrostatic potential

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I'm modelling a system with a nanosized semiconductor in 1d, inside which I want to find the electrostatic potential. Having found this I am unsure what boundary conditions to put on this, when it is connected to a metal on one side and to vacuum on the other. So far I have put that it is continuous at the metal interface (inside which it is a constant). But what about at the boundary to vacuum. I want to say that it should be simply continiuous which then gives that V(x) has a finite value, non constant, outside the semiconductor. But on the other hand this seems unphysical, since it would imply that there is a finite electric field outside the semiconductor. Should I simply put that V(x)=0 at the boundary to vacuum?

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Simon Bridge
Homework Helper
The potential continues into the vacuum the same as any potential does ... i.e. if you were modelling a sheet of charge with a metal on one side and a vacuum on the other how would you do it?

A perfect metal just stamps it's own potential on everything where it is. You can get a step there... a lump of metal in a vacuum is often modelled as a finite square well.

What level are you doing this at?

I am modelling the band structure of a semiconductor-metal hetero junction, by solving the Schrödinger equation in the conduction band (in the effective mass approximation), calculating the electron density and then calculating the electrostatic potential in the semiconductor using Poissons equation. This is then plugged back into the Schrödinger equation and the procedure is reiterated until a self-consistent solution is found.
When I calculate the electrostatic potential in the heterostructure I get a decay towards the vacuum edge of the semiconductor (on the right). I don't know if this is physical or if it comes from my numerical method failing. Physically I expect that if the semiconductor is large that the potential would approach a constant at the edge to vacuum. What do you think?

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Simon Bridge