View source for SRH PN Diode (1D) - full deck

Step 1: Build a one dimensional silicon diode with a constant p-type doping of 1e16 and an n-type Gaussian w/ peak concentration of 1.0e20 and a junction depth of 0.1um. Think carefully about the grid spacing needed to resolve the problem! Use constant mobility of 1000 for electrons and 200 for holes.  Ignore recombination (for now). Implement a simple version of the three device equations and plot current v. voltage.  Cut the grid spacing in half, and rerun the current / voltage plot.  Does the current change appreciably?  What does that say about the choice of grid?


Step 2: Add a simple SRH recombination with carrier lifetimes of 1 nanosecond.  Use 1.0e10 for the intrinsic carrier concentration.  How does the reverse bias current change?  


Solution: 

Step 1: 


[[File:Transient_0.5e-10_without_SRH.png]] [[File: Gaussian_Implant_Profile.png]]


CODE: 

    #establish diffusion parameters and solution variable "diode"
    DevicePackage
    math diffuse dim=1 umf none col !scale
    solution add name=diode solve !negative
    solution name=Potential nosolve
    solution add name=DevPsi solve negative damp
    solution add name=Elec solve !negative
    solution add name=Hole solve !negative

    pdbSetDouble Silicon diode Abs.Error 1.0e-8
    #absolute error range more important for quasi fermi levels
    pdbSetDouble Silicon diode Rel.Error 1.0e-2

    #establish the grid and assign a material
    line x loc=0.0 spac=0.0002 tag=Top
    line x loc=1.0 spac=0.001 tag=Bottom
    region silicon xlo=Top xhi=Bottom
    init

    #Gaussian implant profile
    #fill in the file name for as-implanted profile here
    sel z=1.0e20*exp(-(x)*(x)/(1.086e-3)) name=diode
    #plot the the as-implanted profile
    sel z=log10(abs(diode+1))
    plot.1d label=implant

    #setting the mobility and other variables
    set eps [expr 11.8 * 8.854e-14 / 1.619e-19]
    solution name=qfn add silicon const val= "DevPsi - 0.025*log(Elec/1.0e10)"
    solution name=qfp add silicon const val= "0.025*log(Hole/1.0e10) + DevPsi"
    set Emob "1000.0"; #/ (sqrt(1 + (200*diff(DevPsi)/1.0e7)^2))"
    set Hmob "200.0"; #/ (sqrt(1 + (100*diff(DevPsi)/1.0e7)^2))"
    set T 300.0
    set k 1.38066e-23
    set q 1.619e-19
    set ni "1e10"
    set Na "1.0e16"
    set Nd "10"
    set Vt "[expr {$k*$T/$q}]"
    set ni "1.1e10"

    #continuity equations
    set eqnP "$eps*grad(DevPsi)+Doping-Elec+Hole"
    set eqnE "ddt(Elec)-$Emob*0.025*sgrad(Elec, DevPsi/0.025)"
    set eqnH "ddt(Hole)-$Hmob*0.025*sgrad(Hole, -DevPsi/0.025)"

    pdbSetDouble Si DevPsi DampValue 0.025
    pdbSetDouble Si DevPsi Abs.Error 1.0e-9
    pdbSetString Si DevPsi Equation $eqnP
    pdbSetString Si Elec Equation $eqnE
    pdbSetDouble Si Elec Abs.Error 1.0e-5
    pdbSetString Si Hole Equation $eqnH
    pdbSetDouble Si Hole Abs.Error 1.0e-5

    pdbSetDouble top Hole Abs.Error 1.0e-8
    pdbSetDouble top Elec Abs.Error 1.0e-8
    pdbSetDouble top DevPsi Abs.Error 1.0e-6
    pdbSetDouble top Hole Rel.Error 1.0e-2
    pdbSetDouble top Elec Rel.Error 1.0e-2
    pdbSetDouble top DevPsi Rel.Error 1.0e-2

    pdbSetDouble ReflectBottom Hole Abs.Error 1.0e-8
    pdbSetDouble ReflectBottom Elec Abs.Error 1.0e-8
    pdbSetDouble ReflectBottom DevPsi Abs.Error 1.0e-6
    pdbSetDouble ReflectBottom Hole Rel.Error 1.0e-2
    pdbSetDouble ReflectBottom Elec Rel.Error 1.0e-2
    pdbSetDouble ReflectBottom DevPsi Rel.Error 1.0e-2

    pdbSetBoolean top Elec Fixed 1
    pdbSetBoolean top Hole Fixed 1
    pdbSetBoolean top DevPsi Fixed 1
    pdbSetBoolean top Elec Flux 1
    pdbSetBoolean top Hole Flux 1
    pdbSetBoolean top DevPsi Flux 1
    pdbSetString top Elec Equation {Doping - Elec + Hole}
    pdbSetString top Hole Equation {DevPsi + 0.025*log((Hole+1.0e-10)/1.0e10) - top}
    pdbSetString top DevPsi Equation {DevPsi - 0.025*log((Elec+1.0e-10)/1.0e10) - top}
    pdbSetDouble top Elec Flux.Scale 1.619e-19
    pdbSetDouble top Hole Flux.Scale 1.619e-19

    pdbSetBoolean ReflectBottom Elec Fixed 1
    pdbSetBoolean ReflectBottom Hole Fixed 1
    pdbSetBoolean ReflectBottom DevPsi Fixed 1
    pdbSetBoolean ReflectBottom Elec Flux 1
    pdbSetBoolean ReflectBottom Hole Flux 1
    pdbSetBoolean ReflectBottom DevPsi Flux 1
    pdbSetString ReflectBottom Hole Equation "Doping-Elec+Hole"
    pdbSetString ReflectBottom Elec Equation "DevPsi-0.025*log((Elec+1.0e-10)/1.0e10)"
    pdbSetString ReflectBottom DevPsi Equation "DevPsi+0.025*log((Hole+1.0e-10)/1.0e10)"
    pdbSetDouble ReflectBottom Elec Flux.Scale 1.619e-19
    pdbSetDouble ReflectBottom Hole Flux.Scale 1.619e-19

    line x loc=0.0 spac=.01 tag=Top
    line x loc=0.4 spac=0.001
    line x loc=0.7 spac=0.001
    line x loc=2.0 spac=0.01 tag=Bottom
    region silicon xlo=Top xhi=Bottom
    init

    contact name=top silicon xlo=-0.1 xhi=0.1 add
    contact name=top voltage supply=0.0
    #contact name=GND Silicon xlo=0.9  xhi=1.1 add
    sel z=1.0e21*(x<0.5)-1.0e17 name=Doping
    sel z=0.5*(Doping+sqrt(Doping*Doping+4.0e20))/1.0e10 name=arg
    sel z=0.025*log(arg) name=DevPsi
    sel z=1.0e10*exp(DevPsi/0.025) name=Elec
    sel z=1.0e10*exp(-DevPsi/0.025) name=Hole

    device init; #saves old physics unless you have init
    set plt [CreateGraphWindow]
    contact name=top voltage supply = 1.0
    device time=0.5e-10 movie = {
        set time [simGetDouble Device time]
        set cur [expr ([contact name=top sol=Elec flux] - [contact name=top sol=Hole flux])]
        puts "$time $cur"
        AddtoLine $plt 1.0IV $time $cur
    }
    puts $cur


Step 2:
Implement a simple version of the three device equations and plot current v. voltage.  Cut the grid spacing in half, and rerun the current / voltage plot.  Does the current change appreciably?  What does that say about the choice of grid?
Add a simple SRH recombination with carrier lifetimes of 1 nanosecond.  Use 1.0e10 for the intrinsic carrier concentration.  How does the reverse bias current change?  



The Implant Profile does not change. However the transient plots are different. In the figures below, the blue lines are the reverse bias voltages. The lighter blue line is a smaller applied reverse bias with magnitude of 0.05 V whereas the darker blue line has a reverse bias magnitude of 1.0 V. In reverse bias, the model with shorter carrier lifetimes will have more recombination current and therefore a larger negative overall current, because more carriers can recombine rather then travel to the other end of the material. The difference between the two figures is difficult to see and the magnitudes of current is extremely small, but Figure 3 has a smaller negative overall current than Figure 2 because of the large difference in carrier lifetimes. After the current has reached its peak, the graph stabilizes to close to zero.


This project also experimented with different hole and electron carrier lifetimes (the lifetimes of the holes and electrons were not equal) and the results of the current versus time plots were similar to the larger lifetime behavior; smaller recombination current. This is believed to be true because its takes both particles to recombine and if some of the particles cannot recombine due to a longer lifetime, then the current of the recombination rate will be smaller, as if both carriers had longer lifetimes.

[[File:Transient_with_SRH_lifetime_e-11.png]] [[File:Transient_with_SRH_lifetimes_of_e-6.png]]

CODE: 

    #establish diffusion parameters and solution variable "diode"
     DevicePackage
     math diffuse dim=1 umf none col !scale
     solution add name=diode solve !negative
     solution name=Potential nosolve
     solution add name=DevPsi solve negative damp
     solution add name=Elec solve !negative
     solution add name=Hole solve !negative

     pdbSetDouble Silicon diode Abs.Error 1.0e-8
    #absolute error range more important for quasi fermi levels
     pdbSetDouble Silicon diode Rel.Error 1.0e-2

    #establish the grid and assign a material
     line x loc=0.0 spac=0.0002 tag=Top
     line x loc=1.0 spac=0.001 tag=Bottom
     region silicon xlo=Top xhi=Bottom
     init

    #Gaussian implant profile
    #fill in the file name for as-implanted profile here
     sel z=1.0e20*exp(-(x)*(x)/(1.086e-3)) name=diode

    #plot the the as-implanted profile
     sel z=log10(abs(diode+1))
     plot.1d label=implant

    #setting the mobility and other variables
     set eps [expr 11.8 * 8.854e-14 / 1.619e-19]
     solution name=qfn add silicon const val= "DevPsi - 0.025*log(Elec/1.0e10)"
     solution name=qfp add silicon const val= "0.025*log(Hole/1.0e10) + DevPsi"
     set Emob "1000.0"; #/ (sqrt(1 + (200*diff(DevPsi)/1.0e7)^2))"
     set Hmob "200.0"; #/ (sqrt(1 + (100*diff(DevPsi)/1.0e7)^2))"
     set T 300.0
     set k 1.38066e-23
     set q 1.619e-19
     set ni "1e10"
     set Na "1.0e16"
     set Nd "10"
     set Vt "[expr {$k*$T/$q}]"
     set ni "1.1e10"

    #SRH Recombination (Simplified) -
     set tau_max_e 1.0e-5
     set tau_max_h 3.0e-6
     set Nref 1.0e16
    
    #Doping Dependence
     solution name=tau_n add silicon const val= "$tau_max_e/(1+((($Na+$Nd)/($Nref))))"
     solution name=tau_p add silicon const val= "$tau_max_h/(1+((($Na+$Nd)/($Nref))))"
     solution name=SRHrate add silicon const val= "((Elec*Hole)-(1.0e20))/((tau_p*(Elec+$ni))+(tau_n*(Hole+$ni)))"

    #continuity equations
     set eqnP "$eps*grad(DevPsi)+Doping-Elec+Hole"
     set eqnE "ddt(Elec)-$Emob*0.025*sgrad(Elec, DevPsi/0.025)"
     set eqnH "ddt(Hole)-$Hmob*0.025*sgrad(Hole, -DevPsi/0.025)"

     pdbSetDouble Si DevPsi DampValue 0.025
     pdbSetDouble Si DevPsi Abs.Error 1.0e-9
     pdbSetString Si DevPsi Equation $eqnP
     pdbSetString Si Elec Equation $eqnE
     pdbSetDouble Si Elec Abs.Error 1.0e-5
     pdbSetString Si Hole Equation $eqnH
     pdbSetDouble Si Hole Abs.Error 1.0e-5

     pdbSetDouble top Hole Abs.Error 1.0e-8
     pdbSetDouble top Elec Abs.Error 1.0e-8
     pdbSetDouble top DevPsi Abs.Error 1.0e-6
     pdbSetDouble top Hole Rel.Error 1.0e-2
     pdbSetDouble top Elec Rel.Error 1.0e-2
     pdbSetDouble top DevPsi Rel.Error 1.0e-2

     pdbSetDouble ReflectBottom Hole Abs.Error 1.0e-8
     pdbSetDouble ReflectBottom Elec Abs.Error 1.0e-8
     pdbSetDouble ReflectBottom DevPsi Abs.Error 1.0e-6
     pdbSetDouble ReflectBottom Hole Rel.Error 1.0e-2
     pdbSetDouble ReflectBottom Elec Rel.Error 1.0e-2
     pdbSetDouble ReflectBottom DevPsi Rel.Error 1.0e-2

     pdbSetBoolean top Elec Fixed 1
     pdbSetBoolean top Hole Fixed 1
     pdbSetBoolean top DevPsi Fixed 1
     pdbSetBoolean top Elec Flux 1
     pdbSetBoolean top Hole Flux 1
     pdbSetBoolean top DevPsi Flux 1
     pdbSetString top Elec Equation {Doping - Elec + Hole}
     pdbSetString top Hole Equation {DevPsi + 0.025*log((Hole+1.0e-10)/1.0e10) - top}
     pdbSetString top DevPsi Equation {DevPsi - 0.025*log((Elec+1.0e-10)/1.0e10) - top}
     pdbSetDouble top Elec Flux.Scale 1.619e-19
     pdbSetDouble top Hole Flux.Scale 1.619e-19

     pdbSetBoolean ReflectBottom Elec Fixed 1
     pdbSetBoolean ReflectBottom Hole Fixed 1
     pdbSetBoolean ReflectBottom DevPsi Fixed 1
     pdbSetBoolean ReflectBottom Elec Flux 1
     pdbSetBoolean ReflectBottom Hole Flux 1
     pdbSetBoolean ReflectBottom DevPsi Flux 1
     pdbSetString ReflectBottom Hole Equation "Doping-Elec+Hole"
     pdbSetString ReflectBottom Elec Equation "DevPsi-0.025*log((Elec+1.0e-10)/1.0e10)"
     pdbSetString ReflectBottom DevPsi Equation "DevPsi+0.025*log((Hole+1.0e-10)/1.0e10)"
     pdbSetDouble ReflectBottom Elec Flux.Scale 1.619e-19
     pdbSetDouble ReflectBottom Hole Flux.Scale 1.619e-19

     line x loc=0.0 spac=.01 tag=Top
     line x loc=0.4 spac=0.001
     line x loc=0.7 spac=0.001
     line x loc=2.0 spac=0.01 tag=Bottom
     region silicon xlo=Top xhi=Bottom
     init

     contact name=top silicon xlo=-0.1 xhi=0.1 add
     contact name=top voltage supply=0.0
    #contact name=GND Silicon xlo=0.9  xhi=1.1 add
     sel z=1.0e21*(x<0.5)-1.0e17 name=Doping
     sel z=0.5*(Doping+sqrt(Doping*Doping+4.0e20))/1.0e10 name=arg
     sel z=0.025*log(arg) name=DevPsi
     sel z=1.0e10*exp(DevPsi/0.025) name=Elec
     sel z=1.0e10*exp(-DevPsi/0.025) name=Hole

     device init; #saves old physics unless you have init
     set plt [CreateGraphWindow]
     contact name=top voltage supply = 1.0
     device time=0.5e-10 movie = {
         set time [simGetDouble Device time]
         set cur [expr ([contact name=top sol=Elec flux] - [contact name=top sol=Hole flux])]
        puts "$time $cur"
        AddtoLine $plt 1.0IV $time $cur
    }
     puts $cur
    #look into log plots

     device init
     contact name=top voltage supply = -1.0
     device time=0.5e-10 movie = {
        set time [simGetDouble Device time]
         set cur [expr ([contact name=top sol=Elec flux] - [contact name=top sol=Hole flux])]
         puts "$time $cur"
         AddtoLine $plt n1.0IV $time $cur
    }

     device init
     contact name=top voltage supply = 0.5
     device time=0.5e-10 movie = {
        set time [simGetDouble Device time]
         set cur [expr ([contact name=top sol=Elec flux] - [contact name=top sol=Hole flux])]
         puts "$time $cur"
         AddtoLine $plt 0.5IV $time $cur
    }

     contact name=top voltage supply = 0.05
     device time=0.5e-10 movie = {
        set time [simGetDouble Device time]
         set cur [expr ([contact name=top sol=Elec flux] - [contact name=top sol=Hole flux])]
         puts "$time $cur"
         AddtoLine $plt 0.05IV $time $cur
    }

     contact name=top voltage supply = -0.05
     device time=0.5e-10 movie = {
         set time [simGetDouble Device time]
         set cur [expr ([contact name=top sol=Elec flux] - [contact name=top sol=Hole flux])]
         puts "$time $cur"
         AddtoLine $plt n0.05IV $time $cur
    }