# A Cylindrical Poisson equation for semiconductors

1. Sep 8, 2017

### chimay

In a cylindrical symmetry domain $\Phi(r,z,\alpha)=\Phi(r,z)$. Does anyone can point me what can be found in literature to solve, even with an approximate approach, this equation?
$$\nabla^2 \Phi(r,z)=-\frac{q}{\epsilon} \exp(-\frac{\Phi(r,z)-V}{V_t})$$
Where $q, \epsilon, V$ and $V_t$ are some (very well known) constants.

2. Sep 8, 2017

### phyzguy

First off, don't you want to include a background lattice charge density on the right hand side in addition to the mobile charge? To solve this, you'll probably need to use numerical methods. I've attached an old paper from Dutton and Pinto outlining what I think is the accepted technique. I have some expertise in this area, so if you have more questions, don't hesitate to ask.

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3. Sep 8, 2017

### chimay

Thank you for your help, I will take a look at the paper for sure. Anyway, what I really need is an analitical solution; for example I have found in literature some papers about
$$\nabla^2 \Phi(r)=-\frac{q}{\epsilon} \exp(- \frac{\Phi(r)-V}{V_t})$$

Ps: At the moment I do not need to inclued any fixed charge.

Thank you again!

4. Sep 8, 2017

### phyzguy

I don't know of an analytic solution. If you find one, let me know!

5. Sep 8, 2017

### chimay

Sure!
Thank you for the paper.