NMOS IV (2D) - Scharfetter Gummel Method

2. 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

2. Equations #
3. 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

1. 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)

1. 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)"

1. set eqnP "$epo * grad(DevPsi)"

pdbSetDouble Oxide DevPsi DampValue $Vt pdbSetDouble Oxide DevPsi Abs.Error 1.0e-9 pdbSetString Oxide DevPsi Equation $eqnP

1. 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))"

1. 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"

2. Define Grid #
3. 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

1. 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

2. 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

2. Solve IV Curve #

set xval 0.0001 set yval 0.2

1. 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