# Field-dependent mobility model convergence

#1

Hi all,

Background
I’m trying to model current in a reverse-biased lateral p-i-n diode around an etched silicon waveguide (as would be made by ion-implantation around an etched waveguide on an SOI (Silocon-on-Insulator) chip. The diode is meant to be strongly reverse biased near, but not quite at avalanche breakdown, which for this structure (~800-900nm wide intrinsic region) should occur at ~-20V. I would like to model the transient electron and hole distributions transverse to the optical axis of the waveguide ( so n(x,y,t) and p(x,y,t)) when a pulse of light generates electron-hole pairs by two-photon absorption (TPA). To do this I am solving for the mode profile I(x,y) using Mode-Solutions, calculating the electron-hole pair density generation rate due to TPA g(x,y) [cm^-3 s^-1] ~I(x,y)^2, importing that generation profile into Device, and modeling the transport in two steps: 1) without any optical generation I run a 2D steady-state charge solver that ramps up the reverse bias from 0V to -15V. 2) I run a transient simulation based on the steady-state solution using a ‘pulse-on’ optical shutter to ramp the optical intensity up to a specified value and then back to zero. Ideally this should give me the steady state electron and hole distributions in the structure as well as the transients n(x,y,t) and p(x,y,t). Basically this is just a 2D simulation of a reversed biased silicon p-i-n diode, 60nm wide (transverse to the junction) and 6um long with a 850nm-long intrinsic region where the width increases to 215nm for 450nm of length (the waveguide ridge).

The Problem
This all seems to work quite well if I do not turn on the field-dependent mobility model in Silicon. Unfortunately the results are not physical, as high DC E-fields cause unreasonably high electron and hole velocities in the waveguide. If I turn on the field-dependent mobility models for Silicon (E dot J, monotonic, default parameter values), the steady-state model (no optical generation), stops converging. I know that field-dependent mobility can make the numerical Boltzmann transport problem unstable, and as discussed in previous posts (ex: GaAs diode example: Error with field dependent mobility simulation) I have enabled fast gradient mixing and increased the global iteration limit to 250. So far, refining the mesh near the p-plus/p, p/i, i/n and n/n-plus interfaces to <5nm and decreasing the boundary condition step size to 1mV (so 15001 steps 0V to -15V) still has not helped the simulation converge for reverse biases larger than ~-1V. This makes me think that I am doing something completely wrong here, because I am fairly confident it should be possible to simulate this system/effect. So far I’m wondering whether my doping profiles are too abrupt, causing very strong fields at the interfaces between differently doped regions. I haven’t found any obvious indication of the problem in the partially finished simulation data, but I’m probably just not looking in the right places.

Again, this is all before any optical generation is turned on. I explained that part of it both for context and because I am hoping the solution that allows the steady-state simulation to converge will be robust enough to also handle optical generation.

I’m attaching the .lsf script file that generates the structure and controls the simulation. The .ldev file it creates with the data generated before the simulation fails is too big to attach but here’s a Google Drive link. Hopefully that works. Please let me know if you can’t access it and I can find another way to share it.

If anybody could suggest how I should change the structure geometry, mesh or simulation settings I would be very grateful.

Thanks!
Dodd

carrier_sweep_out_dev_ss.lsf (9.9 KB)

Screenshot of geometry
purple is Silicon, most of the grey is Glass, however Aluminum contacts (hard to see here) are present above the thin Silicon layer on the left and right edges of the simulated region.

#2

Hi @dodd, First of all thanks a lot for posting such a nice and detailed question. The problem you are facing is a common one since achieving convergence with field dependent mobility turned on is always hard. That said between the two options of calculating the gradient of electric field (grad_phi versus EdotJ) the E_dot_J option is much harder to handle. The grad_phi option on the other hand behaves much better and is able to give you accurate results in a single system simulation (no heterojunction). Since this is case in your device I would recommend using the grad_phi option in the field dependent mobility model rather than E_dot_J. I have tried your script file and I was able to run the simulation up to -15 V with field dependent mobility turned on for Silicon using the grad_phi option. Let me know if this information helps.