NMOS IV (2D) - Scharfetter Gummel Method
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###################################################################
# Constants #
###################################################################
set T 300
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 eox [expr 3.9 * 8.85418e-14]
set eps [expr $esi / $q]
set epo [expr $eox / $q]
set Emob 350.0
set Hmob 150.0
set small 1.0e-10
set Nc 3.2e+19
set Nv 1.8e+19
###################################################################
# Equations #
###################################################################
# math
math device dim=1 row bcgs ilu tol=1e-30
math diffuse dim=1 umf none col !scale
pdbSetDouble Math iterLimit 100 ;# set itteration limit
#add solutions
DevicePackage ;# Call for Device Simulations
solution add name=DevPsi pde solve negative damp continuous
solution add Silicon name=Elec pde solve !negative
solution add Silicon name=Hole pde solve !negative
solution add Oxide name=Elec const solve val=($small)
solution add Oxide name=Hole const solve val=($small)
#set solution equations
set eqnP "$eps * grad(DevPsi) + Doping - Elec + Hole"
set eqnE "ddt(Elec) - 400.0 * 0.025 * sgrad(Elec, DevPsi/0.025)"
set eqnH "ddt(Hole) - 200.0 * 0.025 * sgrad(Hole, -DevPsi/0.025)"
pdbSetDouble Silicon DevPsi DampValue 0.025
pdbSetDouble Silicon DevPsi Abs.Error 1.0e-10
pdbSetString Silicon DevPsi Equation $eqnP
pdbSetDouble Silicon Elec Abs.Error 1.0e5
pdbSetString Silicon Elec Equation $eqnE
pdbSetDouble Silicon Hole Abs.Error 1.0e5
pdbSetString Silicon Hole Equation $eqnH
set eqnP "$epo * grad(DevPsi) - (1.0e-10)"
#set eqnP "$epo * grad(DevPsi)"
pdbSetDouble Oxide DevPsi DampValue $Vt
pdbSetDouble Oxide DevPsi Abs.Error 1.0e-9
pdbSetString Oxide DevPsi Equation $eqnP
# Solve For Energy Band Structure
pdbSetDouble Silicon Eg 1.1
pdbSetDouble Silicon Affinity 4.05
solution add name=Econd solve Silicon const val = "((-[pdbGetDouble Silicon Affinity])-(DevPsi))"
solution add name=Eval solve Silicon const val = "((-[pdbGetDouble Silicon Affinity])-([pdbGetDouble Silicon Eg])-(DevPsi))"
set e " Econd - ($Vt) * log( $Nc / Elec ) "
solution add name=Qfn solve Silicon const val = "($e)"
set e " Eval + ($Vt) * log( $Nv / Hole ) "
solution add name=Qfp solve Silicon const val = "($e)"
pdbSetDouble Oxide Eg 9
pdbSetDouble Oxide Affinity 0.95
solution add name=Econd solve Oxide const val = "((-[pdbGetDouble Oxide Affinity])-(DevPsi))"
solution add name=Eval solve Oxide const val = "((-[pdbGetDouble Oxide Affinity])-([pdbGetDouble Oxide Eg])-(DevPsi))"
# Contact Equations
pdbSetBoolean sub Elec Fixed 1
pdbSetBoolean sub Hole Fixed 1
pdbSetBoolean sub DevPsi Fixed 1
pdbSetString sub Hole Equation {Doping - Elec + Hole}
pdbSetString sub DevPsi Equation {DevPsi + 0.025*log((Hole+1.0e-10)/1.0e10) - sub}
pdbSetString sub Elec Equation {DevPsi - 0.025*log((Elec+1.0e-10)/1.0e10) - sub}
pdbSetDouble sub Elec Flux.Scale 1.619e-19
pdbSetDouble sub Hole Flux.Scale 1.619e-19
pdbSetBoolean drain Elec Fixed 1
pdbSetBoolean drain Hole Fixed 1
pdbSetBoolean drain DevPsi Fixed 1
pdbSetString drain Elec Equation {Doping - Elec + Hole}
pdbSetString drain Hole Equation {DevPsi + 0.025*log((Hole+1.0e-10)/1.0e10) - drain}
pdbSetString drain DevPsi Equation {DevPsi - 0.025*log((Elec+1.0e-10)/1.0e10) - drain}
pdbSetDouble drain Elec Flux.Scale 1.619e-19
pdbSetDouble drain Hole Flux.Scale 1.619e-19
pdbSetBoolean src Elec Fixed 1
pdbSetBoolean src Hole Fixed 1
pdbSetBoolean src DevPsi Fixed 1
pdbSetString src Elec Equation {Doping - Elec + Hole}
pdbSetString src Hole Equation {DevPsi + 0.025*log((Hole+1.0e-10)/1.0e10) - src}
pdbSetString src DevPsi Equation {DevPsi - 0.025*log((Elec+1.0e-10)/1.0e10) - src}
pdbSetDouble src Elec Flux.Scale 1.619e-19
pdbSetDouble src Hole Flux.Scale 1.619e-19
set WFN -0.0 ;#FB=0.40295, Al=-0.5
pdbSetBoolean front DevPsi Flux 1
pdbSetBoolean front DevPsi Fixed 1
pdbSetString front DevPsi Equation "DevPsi+$WFN-front"
###################################################################
# Define Grid #
###################################################################
#line x loc=-0.02 spac=0.01 tag=M
line x loc=-0.01 spac=0.001 tag=TopOx
line x loc=0.0 spac=0.001 tag=TopSi
line x loc=0.025 spac=0.005
line x loc=0.04 spac=0.01 tag=Jn
line x loc=0.6 spac=0.1 tag=Bottom
line y loc=0.0 spac=0.1 tag=Left
line y loc=0.1 spac=0.01 tag=GL
line y loc=0.2 spac=0.01
line y loc=0.3 spac=0.01 tag=GR
line y loc=0.4 spac=0.1 tag=Right
#add material and region
mater add name=Oxide
mater add name=Silicon
region Oxide xlo=TopOx xhi=TopSi ylo=GL yhi=GR
region Silicon xlo=TopSi xhi=Bottom ylo=Left yhi=Right
init
contact name=drain silicon xlo=-0.005 xhi=0.005 ylo=0.32 yhi=0.51 add supply=0.0
contact name=front oxide xlo=-0.015 xhi=-0.005 ylo=0.09 yhi=4.41 add supply=0.0
contact name=sub silicon xlo=0.595 xhi=0.605 ylo=-0.05 yhi=0.55 add supply=0.0
contact name=src silicon xlo=-0.005 xhi=0.005 ylo=-0.01 yhi=0.08 add supply=0.0
###################################################################
# Doping Profile #
###################################################################
set buff 1.0e10
sel z=(1.0e21*((y>0.32)?1.0:(exp(-(y-0.32)*(y-0.32)/(0.0002)))*(exp(-(x*x)/(0.0002))))) name=DrainDoping
sel z=(1.0e21*((y<0.08)?1.0:(exp(-(y-0.08)*(y-0.08)/(0.0002)))*(exp(-(x*x)/(0.0002))))) name=SourceDoping
sel z=(5e16+$buff) name=NA
sel z=DrainDoping+SourceDoping-NA name=Doping
###################################################################
# Solve IV Curve #
###################################################################
set xval 0.0001
set yval 0.2
#setup window to prepare to plot
window xwin col=3
for {set Vgs 0} {$Vgs<=8} {set Vgs [expr $Vgs+1]} {
# set initial conditions at each Vgs
sel z=0.5*(Doping+sqrt(Doping*Doping+4.0e20))/1.0e10 name=arg
sel z=0.025*log(abs(arg)+1) name=DevPsi
sel z=1.0e10*exp(DevPsi/0.025) name=Elec
sel z=1.0e10*exp(-DevPsi/0.025) name=Hole
# Solve for proper electrostatics at each Vgs
contact name=front supply=$Vgs
device ;# Call device 3 times to ensure convergence
device
device
for {set Vds 0.0} {$Vds<10.0} {set Vds [expr $Vds+0.2]} {
# Sweep the Drain Voltage
contact name=drain supply=$Vds
device
device
sel z=Econd
plot2d graph=shades clear
plot2d bound levels=100 graph=shades xmin=0.001 xmax=0.25 ;#line=20
plot3d graph=persp azimuth=140 altitude=50 xmin=0.001 xmax=0.15
set cur [expr abs([contact name=drain sol=Elec flux] - [contact name=drain sol=Hole flux])]
chart graph=IV curve=Vgs$Vgs xval=$Vds yval=$cur
}
}
puts "Press Enter to Continue"
gets stdin