Resistor example (1D)

This example shows how to create and simulate a simple 1D n-type resistor. After creating a 1D structure, we plot the output current as an external bias is applied. Once you run and understand this example, you should study the PN Diode example (1D).

New Concepts

• Creating a structure (i.e. grid) with 1 material
• Defining the 3 basic device solution variables
• Defining tcl variables (set)
• Math in tcl (expr)
• Using a constant ionized dopant profile
• Storing "Equation" strings in the parameter database (pdb)
• The importance of initial conditions, guesses, and solves
• "Ramping," i.e. consecutive DC solves

1D Resistor Deck

Create 1D struture explanation

line x loc=0.0 spac=0.01 tag=Top
line x loc=1.0 spac=0.01 tag=Bottom
mater add name=Silicon
region Silicon xlo=Top xhi=Bottom
init

Create contacts explanation

contact name=VSS Silicon xlo=-0.1 xhi=0.1 add
contact name=GND Silicon xlo=0.9  xhi=1.1 add

Define solution variables explanation

DevicePackage
solution add name=DevPsi pde solve negative damp
solution add name=Elec   pde solve !negative
solution add name=Hole   pde solve !negative

Define constants explanation

set T 300.0
set k 1.38066e-23
set q 1.619e-19
set Vt [expr {$k*$T/$q}]
set ni 1.1e10
set esi [expr 11.8 * 8.85418e-14]
set eps [expr $esi / $q]
set Emob 350.0
set Hmob 150.0
set small 1.0e-10

Ionized dopant profile explanation

set Nd 1.0e19
set Na 1.0e15
set Doping [expr {$Nd-$Na}]

#sel z=1.0e19 name=ND
#sel z=1.0e15 name=NA
#sel z=ND-NA name=Doping

Bulk Equations (Poisson, Electron/Hole Continuity Equations) explanation

set eqnP "$eps * grad(DevPsi) + $Doping - Elec + Hole"
set eqnE "ddt(Elec) - ($Emob) * $Vt * sgrad(Elec, DevPsi/$Vt)"
set eqnH "ddt(Hole) - ($Hmob) * $Vt * sgrad(Hole, -DevPsi/$Vt)"

#set eqnP "$eps * grad(DevPsi) + Doping - Elec + Hole"
#set eqnE "ddt(Elec) - ($Emob) * $Vt * sgrad(Elec, DevPsi/$Vt)"
#set eqnH "ddt(Hole) - ($Hmob) * $Vt * sgrad(Hole, -DevPsi/$Vt)"

pdbSetDouble Silicon DevPsi DampValue $Vt
pdbSetString Silicon DevPsi Equation $eqnP
pdbSetString Silicon Elec   Equation $eqnE
pdbSetString Silicon Hole   Equation $eqnH 

Ohmic contact equations using a procedure explanation

proc OhmicContact {Contact} {
    #set vt 0.02558357
    #set ni 1.1e10
    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 $Contact DevPsi Equation "$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

Initial conditions explanation

#Initial Guess of DevPsi, Elec and Hole
proc InitialGuess {Doping} {
	sel z= {(Doping>0.0)  
		?  ( 0.025*log( (Doping+1.0e10) / 1.0e10))
		:  (-0.025*log(-(Doping+1.0e10) / 1.0e10))} name = DevPsi
	sel z=1.0e10*exp(DevPsi/0.025) name=Elec
	sel z=1.0e10*exp(-DevPsi/0.025) name=Hole
} 
#InitialGuess Doping

sel z=(-$Vt*log(-($Doping+$small)/$ni)) name=DevPsi
sel z=$ni*exp(DevPsi/0.025) name=Elec
sel z=$ni*exp(-DevPsi/0.025) name=Hole

#Bias Voltage on the Contacts
contact name=VSS voltage supply=0.0
contact name=GND voltage supply=0.0

DC solve / plot Id-Vg as output explanation

set Win [CreateGraphWindow]
set bias 0.0
for {set bias 0.0} {$bias < 1.01} {set bias [expr $bias+0.1]} {
	contact name=VSS supply = $bias
	device
	set cur [expr abs([contact name=VSS sol=Elec flux] - [contact name=VSS sol=Hole flux])]
	AddtoLine $Win I $bias $cur
}

Notes

This deck was successfully run by Nicole on 10/26/10 using TEC ~flooxs/linux64