Electric Field

The electric field is calculated from the electric potential for each grid point $(i,j,k)$ when calling calculate_electric_field!(sim). The field vector components on each grid point are the means of the electric field in each direction calculated as finite differences.

On a Cartesian grid, this results in:

\[\vec{E}^{i,j,k} = \left( \mathcal{E}_x^{i,j,k}, \mathcal{E}_y^{i,j,k}, \mathcal{E}_z^{i,j,k} \right)^{\mathsf{T}}~\hspace{10pt},\]

where

\[\begin{aligned} \mathcal{E}_x^{i,j,k} &= \dfrac{1}{2}\left( \dfrac{\Phi_{i+1,j,k}-\Phi_{i+1,j,k}}{x_{i+1} - x_{i}} + \dfrac{\Phi_{i,j,k}-\Phi_{i-1,j,k}}{x_{i} - x_{i-1}} \right)\hspace{10pt},\\ \mathcal{E}_y^{i,j,k} &= \dfrac{1}{2}\left( \dfrac{\Phi_{i,j+1,k}-\Phi_{i,j,k}}{y_{j+1} - y_{j}} + \dfrac{\Phi_{i,j,k}-\Phi_{i,j-1,k}}{y_{j} - y_{j-1}} \right)\hspace{10pt},\\ \mathcal{E}_z^{i,j,k} &= \dfrac{1}{2}\left( \dfrac{\Phi_{i,j,k+1}-\Phi_{i,j,k}}{z_{k+1} - z_{k}} + \dfrac{\Phi_{i,j,k}-\Phi_{i,j,k-1}}{z_{k} - z_{k-1}} \right)\hspace{10pt}. \end{aligned}\]

On a cylindrical grid, the calculation is:

\[\vec{E}^{i,j,k} = \left( \mathcal{E}_r^{i,j,k}, \mathcal{E}_{\varphi}^{i,j,k}, \mathcal{E}_z^{i,j,k} \right)^{\mathsf{T}}~\hspace{10pt},\]

where

\[\begin{aligned} \mathcal{E}_r^{i,j,k} &= \dfrac{1}{2}\left(\dfrac{\Phi_{i+1,j,k}-\Phi_{i,j,k}}{r_{i+1} - r_{i}} + \dfrac{\Phi_{i,j,k}-\Phi_{i-1,j,k}}{r_{i} - r_{i-1}}\right)\hspace{10pt},\\ \mathcal{E}_{\varphi}^{i,j,k} &= \dfrac{1}{2 r_{i}}\left(\dfrac{\Phi_{i,j+1,k}-\Phi_{i,j,k}}{\varphi_{j+1} - \varphi_{j}} + \dfrac{\Phi_{i,j,k}-\Phi_{i,j-1,k}}{\varphi_{j} - \varphi_{j-1}}\right)\hspace{10pt},\\ \mathcal{E}_z^{i,j,k} &= \dfrac{1}{2}\left( \dfrac{\Phi_{i,j,k+1}-\Phi_{i,j,k}}{z_{k+1} - z_{k}} + \dfrac{\Phi_{i,j,k}-\Phi_{i,j,k-1}}{z_{k} - z_{k-1}} \right)\hspace{10pt}. \end{aligned}\]

This discrete electric field is interpolated (via Interpolations.jl) during the drift in order to get the electric field at the current position of the charge carrier.

The calculated electric field is stored as a field in the Simulation object, i.e. sim.electric_field, consisting of the actual field vectors for each grid point stored in sim.electric_field.data and the corresponding grid in sim.electric_field.grid.