Avalanche Breakdown

Thanks to Juan’s efforts on developing the DEVSIM. Especially after version 1.5, a lot of models can be implemented with the support of edge/element node volume based integration.
Here I wanna share an example of Avalanche Breakdown by implementing impact ionization model.

The model comes from the work in [1]. By using the constant temperature assumption, the ionization coefficient fomula can be simplified to:
Imp_coeff = a*F_ava*exp(-(b/F_ava)^2)
where a and b are empirical parameters and F_ava is the driving force.

The driving force can be defined as the absolute value of the gradient of quasi-fermi level:
F_ava = |grad(EFN or EFP)|

Since the quasi-fermi levels for electrons and holes (EFN and EFP) are already defined in devsim_bjt_example (devsim_bjt_example/simdir/physics/new_physics), we can just append their gradient and derivative definations at the end of function CreateQuasiFermiLevels:

edge_average_model(device=device,region=region,node_model="EFN",edge_model="dEFN",average_type='gradient')
edge_average_model(device=device,region=region,node_model="EFP",edge_model="dEFP",average_type='gradient')
for v in variables:
   edge_average_model(device=device,region=region,node_model="EFN",edge_model="dEFN",derivative=v,average_type='gradient')
   edge_average_model(device=device,region=region,node_model="EFP",edge_model="dEFP",derivative=v,average_type='gradient')

EFN:Holes and EFH:Electrons can be set to zero.

The driving forces and ionization coefficients for electrons and holes should be:

Fava_n = "abs(dEFN)"
Fava_p = "abs(dEFP)"
Ion_coeff_n = "a_n*Fava_n*exp(-((b_n/(Fava_n+1e-30))^2))"
Ion_coeff_p = "a_p*Fava_p*exp(-((b_p/(Fava_p+1e-30))^2))"

Note that both driving force and current are along the edge. We assume they were constant in the area corresponding to the “EdgeNodeVolume” defined in [2]. Thus the impact ionization rate in this area can be calculated as:

Imp_rate = "(Ion_coeff_n*(abs(Jn))+Ion_coeff_p*(abs(Jp)))/q"

which is also an edge model. The above physical quantities and their derivatives to the variables should be defined before current continuity equation:

CreateEdgeModel(device,region,"Fava_n",Fava_n)
CreateEdgeModel(device,region,"Fava_p",Fava_p)
CreateEdgeModel(device,region,"Ion_coeff_n",Ion_coeff_n)
CreateEdgeModel(device,region,"Ion_coeff_p",Ion_coeff_p)
CreateEdgeModel(device,region,"qGn_imp","q * %s" % Imp_rate)
CreateEdgeModel(device,region,"qGp_imp","-q * %s" % Imp_rate)
for v in variables:
  CreateEdgeModelDerivatives(device,region,"Fava_n",Fava_n,v)
  CreateEdgeModelDerivatives(device,region,"Fava_p",Fava_p,v)
  CreateEdgeModelDerivatives(device,region,"Ion_coeff_n",Ion_coeff_n,v)
  CreateEdgeModelDerivatives(device,region,"Ion_coeff_p",Ion_coeff_p,v)
  CreateEdgeModelDerivatives(device,region,"qGn_imp","q * %s" % Imp_rate,v)
  edge_model(device=device,region=region,name="qGp_imp:%s" % v,equation="-qGn_imp:%s" % v)

Next, put the “qGn_imp” and “qGp_imp” to field edge_volume_model in ECE and HCE, respectively.

Then you can insert a big resistor in series with your device and sweep the votlage to simulate the avalanche breakdown.

Ref:
[1] Okuto, Y., and C. R. Crowell. “Threshold energy effect on avalanche breakdown voltage in semiconductor junctions.” Solid-State Electronics 18.2 (1975): 161-168.
[2] Sanchez, Juan, and Qiusong Chen. “Element Edge Based Discretization for TCAD Device Simulation.” (2021).

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Thanks for sharing your approach, and being the first user post on the new forum. I hope to have more to share concerning [2] soon.

I have followed the DEVSIM project for years. The symbolic equation system and python-based interface really impressed me.
Hope more people can join the community and share their experience on using this tool.

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