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I'm trying to set up a system for solving these 5 coupled PDEs in FyPi to study the dynamics of electrons and holes in semiconductors

The system of coupled PDEs

I'm struggling with defining the terms highligted in blue as they're products of one variable with gradient of another. For example, I'm able to define the third equation like this without error messages:

eq3 = ImplicitSourceTerm(coeff=1, var=J_n) == ImplicitSourceTerm(coeff=e*mu_n*PowerLawConvectionTerm(var=phi), var=n) + PowerLawConvectionTerm(coeff=mu_n*k*T, var=n)

But I'm not sure if this is a good way. Is there a better way how to define this non-linear term, please?

Also, if I wanted to define a term that would be product of two variables (say p and n), would it be just:

ImplicitSourceTerm(p, var=n)

Or is there a different way?

josefT
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1 Answers1

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I am amazed that you don't get an error from passing a PowerLawConvectionTerm as a coefficient of an ImplicitSourceTerm. It's certainly not intended to work. I suspect you would get an error if you attempted to solve().

You should substitute your flux equations into your continuity equations so that you end up with three second-order PDEs for electron drift-diffusion, hole drift-diffusion, and Poisson's equation. It will hopefully then be a bit clearer how to use FiPy Terms to represent the different elements of those equations.

That said, these equations are challenging. Please see this issue and this notebook for some pointers on how to set up and solve these equations, but realize that we provide no examples in our documentation because we haven't been able to come up with anything robust enough. Solving for pseudo-Fermi levels has worked a bit better for me than solving for electron and hole concentrations.

ImplicitSourceTerm(p, var=n) is a reasonable way to represent the n*p recombination term.

jeguyer
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  • Thank you for the link for the notebook. Although, I'll need to apply the boundary conditions for the current densities. Does it mean, that I would define J_n as a new CellVariable, e.g. `eq = ImplicitSourceTerm(var=J_n) == -TransientTerm(coeff=e, var=n)` and then specify the boundary conditions? – josefT Jun 30 '20 at 08:08
  • No, you should not define an equation in terms of `J_n`. A current density boundary condition can be achieved by [constraining the boundary flux](https://www.ctcms.nist.gov/fipy/documentation/USAGE.html#applying-fixed-flux-boundary-conditions). – jeguyer Jun 30 '20 at 15:51