# Modelling ion Drift-Diffusion under an external electric potential, convergency in PDE?

Ion drift-diffusion under the effect of an external electric field is a phenomenon with a huge relevancy in light emitting devices, neuromorphic architectures and in general molecular electronics

Up to now, the most advanced approaches considered two steady-state operative models: Electrodynamical model (ED) and Electrochemical Doping model (ECD). But the key point to solve the resulting set of Partial differential equations is related to their non-linearity and time-dependence. These set include Drift and Diffusion equations for each type of carriers (ions, electrons and holes) and Poisson equation:

\begin{align}{\delta{n_e} \over \delta{t}} &= \mu_{e} \left[{kT \over e} \nabla^2n_e - n_e \nabla^2 \phi- (\nabla n_e)(\nabla \phi)\right] -k_{eh}n_en_h\tag{1}\\{\delta{n_h} \over \delta{t}} &= \mu_{h} \left[{kT \over e}\nabla^2n_h - n_h \nabla^2 \phi- (\nabla n_h)(\nabla \phi)\right] -k_{eh}n_en_h\tag{2}\\{\delta{n_a} \over \delta{t}} &= \mu_{a} \left[{kT \over e} \nabla^2n_a - n_a \nabla^2 \phi- (\nabla n_a)(\nabla \phi)\right]\tag{3}\\{\delta{n_c} \over \delta{t}} &= \mu_{c} \left[{kT \over e} \nabla^2n_c - n_c \nabla^2 \phi- (\nabla n_c)(\nabla \phi)\right]\tag{4}\\{\nabla^2 \phi} &= {-e \over \epsilon} (n_h + n_c -n_e -n_a)\tag{5}\end{align}

My question is related to programmable solving methods for highly non-linear PDEs. Even in the case that we have the algebraic expression of the PDEs, which methodologies are powerful enough to deal with such a complicated problem?

• In our experience, achieving convergence in this PDE is extremely complex when your boundary conditions assume that the flux of electrons ($j_e$) and holes ($j_h$) at both sides of the junction is also a function of the electric field (E) at such point ( i.e. Fowler-Nordheim tunneling process). In fact, many contributions have stated their inability to process and converge such a complicated system even for a steady state solution. (deMello, J. C., Phys. Rev. B, 2002, 66, 235210). May 4, 2020 at 13:24