Sorry I need to make a correction to my previous comment: The thermal mode needs to be set to coupled for the temperature monitor to record any heating. This is because non-isothermal mode, heating is not simulated; this mode is helpful for importing spatial temperature map. More information on the thermal mode is provided here.
Obtaining convergence in coupled mode, is indeed more difficult since the solution becomes more complex. We don’t have an online example for coupled Avalanche Detector simulation. To troubleshoot the error, I recommend testing the behavior for smaller bias voltages and fewer steps (e.g. 0, -0.5, and -1V) to see if convergence can be obtained at all. Adjusting some of the advanced settings might be helpful. More information for addressing convergence issues is mentioned here.
As you suggested, the convergence error at breakdown voltage is normal.
I will update my previous comment to include the correct information about CHARGE thermal modes.
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