In the previous blog post, 1D diode junction I showed some simulation results, here I share some of the actual script for the physics behind this example.

For the poisson equation, the implementation is:

pne = "-ElectronCharge*kahan3(Holes, -Electrons, NetDoping)"
CreateNodeModel(device, region, "PotentialNodeCharge", pne)
CreateNodeModelDerivative(device, region, "PotentialNodeCharge", pne, "Electrons")
CreateNodeModelDerivative(device, region, "PotentialNodeCharge", pne, "Holes")


equation(device=device, region=region, name="PotentialEquation", variable_name="Potential",
node_model="PotentialNodeCharge", edge_model="PotentialEdgeFlux",
time_node_model="", variable_update="log_damp")

where “PotentialEdgeFlux” was defined elsewhere in the script as “Permittivity * ElectricField”. The “kahan3” function is provided to add 3 numbers with additional precision.

For the electron and hole current densities, the Scharfetter-Gummel method is used to calculated the current along each edge connecting the nodes in the mesh. The Bernoulli function, B(x) and its derivative, dBdx(x) are provided in the DEVSIM interpreter

vdiffstr="(Potential@n0 - Potential@n1)/V_t"
CreateEdgeModel(device, region, "vdiff", vdiffstr)
CreateEdgeModel(device, region, "vdiff:Potential@n0", "V_t^(-1)")
CreateEdgeModel(device, region, "vdiff:Potential@n1", "-vdiff:Potential@n0")
CreateEdgeModel(device, region, "Bern01", "B(vdiff)")
CreateEdgeModel(device, region, "Bern01:Potential@n0", "dBdx(vdiff) * vdiff:Potential@n0")
CreateEdgeModel(device, region, "Bern01:Potential@n1", "-Bern01:Potential@n0")
Jn = "ElectronCharge*mu_n*EdgeInverseLength*V_t*kahan3(Electrons@n1*Bern01, Electrons@n1*vdiff, -Electrons@n0*Bern01)"
CreateEdgeModel(device, region, "ElectronCurrent", Jn)
for i in ("Electrons", "Potential", "Holes"):
  CreateEdgeModelDerivatives(device, region, "ElectronCurrent", Jn, i)

In this example, the @n0, and @n1 refers to the variables on each node of the edge. For an edgemodel, “foo”, a derivative with respect to variable on the first node, “bar”, would be “foo:bar@n0”.

For the Shockley Read Hall recombination model, the implementation is:

USRH="(Electrons*Holes - n_i^2)/(taup*(Electrons + n1) + taun*(Holes + p1))"
Gn = "-ElectronCharge * USRH"
Gp = "+ElectronCharge * USRH"
CreateNodeModel(device, region, "USRH", USRH)
CreateNodeModel(device, region, "ElectronGeneration", Gn)
CreateNodeModel(device, region, "HoleGeneration", Gp)
for i in ("Electrons", "Holes"):
  CreateNodeModelDerivative(device, region, "USRH", USRH, i)
  CreateNodeModelDerivative(device, region, "ElectronGeneration", Gn, i)
  CreateNodeModelDerivative(device, region, "HoleGeneration", Gp, i)

The purpose of this post is to describe some of the physics used in the previous example, and show how they are implemented within the DEVSIM software using a scripting interface. Once a set of physical equations has been implemented, it can be placed in modules that can be reused for other simulations.