General tips to solve divergence problem in metal-insulator-metal (MIM) structures

MIM structures are one of the most challenging devices to simulate and obtaining stable and converged results require patience. While there can be different reasons why a simulation diverges, most often, PML layers cause some artefacts in the areas where they interfere with metal objects. This can be observed by adding a movie monitor to capture electric field behaviour as a function of time. The artificial gain in these sections cause electric field enhancements, which eventually drives simulation to diverge. Here are a few tips that a user can try to solve the divergence problem:

Increase the FDTD simulation region: Most often, the divergence problem will be solved if the electric field is weak enough when it reaches the PML. This can be done by increasing the FDTD span so that lossy waveguide/plasmonic modes decay as they propagate before the PML.

PML type and number of layers: Start with 8 layers of standard PML and then increase the number of layers to its maximum of 64 layers. If you still had divergence problem, use stablized PML and increase the number of layers. Bare in mind that stablized PML has lower absorption compared to standard PML and might cause some unexpected reflection (which should be improved as you increase the number of layers). If you still had problem, you can use custom PML and set the parameters to be identical to stablized PML. Then you can start decreasing the value of alpha from 0.9 to 0 by small increments, hoping PML absorption to be improved:

Here are a few additional note:

Please note that every simulation requites convergence testing. Of particular importance for MIM structures is to use stair case mesh with fine mesh sizes rather than conformal meshing. A hand waving explanation is that conformal mesh uses some index averaging that might not be very suitable for MIM cases where there is a large contrast between metal and insulator indices. A fine mesh is strongly recommended if cavity structures are present in the simulations where can trap light.

To summarize, divergence problem in MIM structure in most cases is due to field enhancement in the region where metal and PML interfere. A user needs to try a few different things or combination of them to solve the problem. Here are the list of few cases that our team have worked on: