Resistor example 1D - Ohmic contact procedure explanation

This page explains the following code snipped from the 1D Resistor example:

Ohmic contact equations using a procedure

proc OhmicContact {Contact} {
    global Vt ni Nd Na
    pdbSetBoolean $Contact Elec Flux 1
    pdbSetBoolean $Contact Hole Flux 1
    pdbSetBoolean $Contact DevPsi Flux 1
    pdbSetBoolean $Contact Elec Fixed 1
    pdbSetBoolean $Contact Hole Fixed 1
    pdbSetBoolean $Contact DevPsi Fixed 1
    pdbSetDouble $Contact Elec Flux.Scale 1.619e-19
    pdbSetDouble $Contact Hole Flux.Scale 1.619e-19
    pdbSetString $Contact DevPsi Equation "$Nd - $Na - Elec + Hole"
    pdbSetString $Contact Elec Equation "DevPsi - $Vt*log((Elec)/$ni) -$Contact"
    pdbSetString $Contact Hole Equation "DevPsi + $Vt*log((Hole)/$ni) -$Contact"
}
OhmicContact VSS
OhmicContact GND

proc - the tcl procedure

The "function" or "routine" in tcl is called the "procedure," or "proc." Use the procedure to accomplish repetitive tasks. For example, this 1D resistor needs 2 ohmic contacts, one with the name "VSS," (which we'll presumably bias later), and one with the name "GND," (which we'll presumably hold at 0V). Rather than type the 11 lines necessary to set the Ohmic contact equations twice, changing only the name of the contact, we make a procedure with the 11 lines inside it, then call it twice, with the name of the contact as the argument:

OhmicContact VSS
OhmicContact GND

Inside the OhmicContact procedure

Inside a proc, all variables are local. This means that to use previously-defined tcl variables we need use the "global" command to tell tcl to retrieve said variables from the outside:

global Vt ni Nd Na

Fixed and Flux are required booleans (yes/no expressed as 1/0) that must be set in the parameter database (pdb) for every contact. You can read the section, What does "Fixed" and Flux" mean?, if you'd like to know what these mean, and what physics they represent. Fixed and Flux are both "1" (yes) for an ohmic contact for all 3 pde solution variables:

pdbSetBoolean $Contact Elec Flux 1
pdbSetBoolean $Contact Hole Flux 1
pdbSetBoolean $Contact DevPsi Flux 1
pdbSetBoolean $Contact Elec Fixed 1
pdbSetBoolean $Contact Hole Fixed 1
pdbSetBoolean $Contact DevPsi Fixed 1

The pdb also contains an optional double under all contacts called "Flux.Scale". Here we have chosen to scale the electron (/cm^3) and hole (cm^3) flux through the contact nodes by q (C), elementary charge, to make the units of I (current) work out.

pdbSetDouble $Contact Elec Flux.Scale 1.619e-19
pdbSetDouble $Contact Hole Flux.Scale 1.619e-19

Finally, these lines store the 3 equation strings for the 3 pde variables in the pdb, setting the boundary conditions on these contact or boundary nodes:

pdbSetString $Contact DevPsi Equation "$Nd - $Na - Elec + Hole"
pdbSetString $Contact Elec Equation "DevPsi - $Vt*log((Elec)/$ni) -$Contact"
pdbSetString $Contact Hole Equation "DevPsi + $Vt*log((Hole)/$ni) -$Contact"

The first "$Contact" after "pdbSetString"

pdbSetString $Contact

is interpreted by flooxs as the *contact name*, whereas the "$Contact" in the equation strings

"DevPsi - $Vt*log((Elec)/$ni) -$Contact"
"DevPsi + $Vt*log((Hole)/$ni) -$Contact"

is interpreted as the *voltage value* on that contact (in Volts) - a number.

Ohmic Contact Physics

Contact equations are boundary conditions. These are necessary to find a specific solution to pde's. In our 1D resistor case, we need 3 boundary conditions - 1 for each pde variable that exists in the Silicon.

When a contact is ohmic, several conditions are true at that edge:

• 1) charge neutrality - there is no net charge. In other words:

<math>Q = q(n - p + N_d^{-} - N_a^{+}) = 0.</math>

This is expressed in floods via the line

pdbSetString $Contact DevPsi Equation "$Nd - $Na - Elec + Hole"

Remember that "Equation" strings are set to zero by the flooxs parser.

• 2 and 3) the quasi-fermi levels are equal, and equal to the voltage on the contact

<math>E_{fn} = E_{fp} = V_{applied}.</math>
How do we get an expression for the fermi levels? First we assume Maxwell-Boltzmann statistics. Then, electron concentration can be expressed as:
<math>n=ni*exp(\frac{E_{fn}-E_i}{kT})</math>
We can rearrange to get an expression for <math>E_{Fn}</math>:
<math>E_{Fn}=kT*log(\frac{n}{ni})</math>
Now we want to substitute in for n (electron concentration) with something we know. We can use the fact that at an ohmic contact the minority carriers are negligible, which gives us:
<math>Q = q(n - p + N_d^{-} - N_a^{+}) = 0</math>

<math>q(n - p + N_d^{-} - N_a^{+}) = 0</math>

<math>n - p + N_d^{-} - N_a^{+} = 0</math>

<math>n = -(+ N_d^{-} - N_a^{+}) = -Doping</math> (for n-type Si)

<math>p = +(+ N_d^{-} - N_a^{+}) = +Doping</math> (for p-type Si)

Efp=$Contact Efn=$Contact

Advanced

• BEWARE: Do not use "solution const val=" variables in your contact Equation strings. This is not relevant in the simple 1D resistor example, but keep it in mind for later.