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Radiation Effects in a PIN Diode
This tutorial performs steady-state and transient analysis of the response of a PIN diode to constant and pulsed radiation, respectively. The effect of radiation is modeled as spatially uniform generation of electron-hole pairs within the device. At high dose rates the separation of the generated charges causes the reduction of the interior electric field and prolonged storage of excess carriers. A quantitative prediction of this phenomenon is only possible with numerical simulation, since analytical solution is unattainable. Several techniques for achieving convergence in the cases of high reverse bias, field-dependent mobility, and time-dependent studies are demonstrated. The computed carrier concentrations and electric field distribution agree well with the reference paper.
Introduction
Radiation effects in semiconductor devices are of great scientific and engineering interests. This tutorial focuses on the steady-state and transient responses of a reverse biased PIN diode to ionizing radiation, following the treatment of Ref. 1, which models the radiation as a spatially uniform generation rate for electron-hole pairs within the device. At high dose rates, the interplay of electrostatics with contribution from the space charge of separated carriers, and charge transport with field-dependent mobilities, results in the intricate situation of much reduced interior electric field and extended storage of excess carriers. The inclusion of the field-dependent mobilities makes the equation system highly nonlinear and difficult to solve. Several techniques for achieving convergence are employed in the study settings with detailed discussions in the Modeling Instructions section.
Model Definition
Following Ref. 1, the PIN diode is constructed from a 300 um thick silicon wafer with heavy doping diffused into its two faces. The doping profile is shown in Figure 1, with parameters guessed from Fig. 3 of Ref. 1. As mentioned in the paper, recombination does not play a major role but is still included for completeness with a carrier lifetime of 100 us. Maxwell-Boltzmann statistics is used as suggested by Eq. (4) and (5) in Ref. 1.
The mobility models given by Eq. (15) and (16) in Ref. 1 differ by many orders of magnitude from the curves of carrier velocity vs. electric field shown in Fig. 2B in the same paper. Here we use similar expressions with modified coefficients to better match the curves in Fig. 2B.
In addition to the Semiconductor interface, an Events interface is added for the transient study to mark the end of the radiation pulse.
See the Modeling Instructions section for all the parameter, function and variable definitions, and detailed discussions on the techniques used to achieve convergence.
Figure 1: Doping profile.
Results and Discussion
Figure 2 and Figure 3 show the hole distribution and electric field distribution at various revers bias voltages, which compare well with Fig. 4A and 4B in the reference paper, respectively.
Figure 2: Hole distribution at various reverse bias voltages.
Figure 3: Electric field distribution at various reverse bias voltages.
Figure 4 and Figure 5 show the steady-state electric field and hole distribution for several ionization rates, which compare well with Fig. 5A and 5B in Ref. 1, respectively.
Figure 4: Steady-state electric field distribution for several ionization rates.
Figure 5: Steady-state hole distribution for several ionization rates.
Figure 6 through Figure 9 show the hole, electron, and electric field distributions at several times, and the photocurrent response at two ionization levels, which compare well with Fig. 7A–7C and Fig. 8 in the reference paper, respectively.
Figure 6: Hole distributions at several times.
Figure 7: Electron distributions at several times.
Figure 8: Electric field distributions at several times.
Figure 9: Photocurrent response at two ionization levels.
Reference
1. C.W. Gwyn, D.L. Scharfetter and J.L. Wirth, “The Analysis of Radiation Effects in Semiconductor Junction Devices,” presented at IEEE Conference on Nuclear and Space Radiation Effects, Columbus, Ohio, July 10–14, 1967.
Application Library path: Semiconductor_Module/Photonic_Devices_and_Sensors/pin_radiation_effects
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  1D.
2
In the Select Physics tree, select Semiconductor>Semiconductor (semi).
3
Click Add.
Add an Events interface to capture the abrupt change in the slope at the end of the applied radiation pulse.
4
In the Select Physics tree, select Mathematics>ODE and DAE Interfaces>Events (ev).
5
Click Add.
6
Click  Study.
The Semiconductor Equilibrium study can be used to obtain a good initial condition for subsequent stationary or transient studies.
7
In the Select Study tree, select Preset Studies for Some Physics Interfaces>Semiconductor Equilibrium.
8
Click  Done.
Geometry 1
The Model Wizard exits and starts the COMSOL Desktop at the Geometry node. We can set the length scale here right away. Then enter some model parameters. In particular, the time parameter t lets stationary solvers recognize the built-in time variable of the same name and makes it convenient to set up the model using the same time-dependent expression for the generation rate due to the radiation dosage.
1
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
2
In the Settings window for Geometry, locate the Units section.
3
From the Length unit list, choose µm.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
L
300[um]
3E-4 m
Device length
V0
1000[V]
1000 V
Applied voltage
Nds
1e20[cm^-3]
1E26 1/m³
Surface concentration of N-plus and P-plus doping
dj
25[um]
2.5E-5 m
Junction depth of N-plus and P-plus doping
Npi
2e12[cm^-3]
2E18 1/m³
Intrinsic doping concentration
tau
100[us]
1E-4 s
Carrier lifetime
mup0
480[cm^2/V/s]
0.048 m²/(V·s)
Low field, low doping, hole mobility
mun0
1350[cm^2/V/s]
0.135 m²/(V·s)
Low field, low doping, electron mobility
cp
1
1
Continuation parameter for mobility model
RadSi
0
0
Radiation dose rate in Rad(Si)/s
t
0[s]
0 s
Time parameter
tp
0.5[ns]
5E-10 s
Radiation pulse duration
gR
4.03e13*RadSi[1/cm^3/s]*pw1(t/tp)
 
Generation rate due to radiation dose
index
1
1
Parameter for solution indexing
area
1[mm^2]
1E-6 m²
Cross-section area of the 1D model
The expression for the generation rate gR is red colored because the pulse function pw1 is not yet defined, which we will do next. Create a piecewise function without smoothing for the triangular pulse. Make the argument and the function unitless to avoid confusion.
Piecewise 1 (pw1)
1
In the Home toolbar, click  Functions and choose Global>Piecewise.
2
In the Settings window for Piecewise, locate the Definition section.
3
Find the Intervals subsection. In the table, enter the following settings:
Start
End
Function
0
1
x
1
2
0
4
Locate the Units section. In the Arguments text field, type 1.
5
In the Function text field, type 1.
Create a 300 um long interval to represent the 300 um thick wafer.
Geometry 1
Interval 1 (i1)
1
In the Model Builder window, under Component 1 (comp1) right-click Geometry 1 and choose Interval.
2
In the Settings window for Interval, locate the Interval section.
3
In the table, enter the following settings:
Coordinates (µm)
0
L
Create a local variable node for the impact ionization and mobility models. The mobility models given by Eq. (15) and (16) in the reference paper differ by many orders of magnitude from the curves of carrier velocity vs. electric field shown in Fig. 2B in the same paper. Here we use similar expressions with modified coefficients to better match the curves in Fig. 2B. Note that the continuation parameter cp is used to control the amount of field dependence for aiding the solution process.
Definitions
Variables 1
1
In the Model Builder window, under Component 1 (comp1) right-click Definitions and choose Variables.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
E
semi.normE/1[V/cm]
 
Electric field intensity in V/cm
alphap
1.8e7[cm^-1]*exp(-3.2e6/E)
1/m
Ionization coefficient, holes
alphan
2.4e7[cm^-1]*exp(-1.6e6/E)
1/m
Ionization coefficient, electrons
NDt
(semi.Ndplus+semi.Naminus)/1[cm^-3]
 
Total ionized dopant concentration in 1/cm^3
mun
mun0/(1+81*NDt/(NDt+3.24e18))^0.5/(1+cp*(E/8e3)^4.9*(E+1.64e5)/(E+1))^(1/4.9)
m²/(V·s)
Electron mobility
mup
mup0/(1+350*NDt/(NDt+1.05e19))^0.5/(1+cp*(E/8.72e4)^1.15*(E+8.51e5)/(E+8.12e4))^(1/1.15)
m²/(V·s)
Hole mobility
Add the silicon material properties from the list of built-in materials. Some properties will be changed in the physics settings according to the reference paper.
Add Material
1
In the Home toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Semiconductors>Si - Silicon.
4
Click Add to Component in the window toolbar.
5
In the Home toolbar, click  Add Material to close the Add Material window.
Configure the physics settings. Enter the cross-section area for the 1D model. Use the default Maxwell-Boltzmann statistics as suggested by Eq. (4) and (5) in the reference paper. Use the local variables mun and mup defined above for the electron and hole mobilities.
Semiconductor (semi)
1
In the Model Builder window, under Component 1 (comp1) click Semiconductor (semi).
2
In the Settings window for Semiconductor, locate the Cross-Section Area section.
3
In the A text field, type area.
Semiconductor Material Model 1
1
In the Model Builder window, under Component 1 (comp1)>Semiconductor (semi) click Semiconductor Material Model 1.
2
In the Settings window for Semiconductor Material Model, locate the Mobility Model section.
3
From the μn list, choose User defined. In the associated text field, type mun.
4
From the μp list, choose User defined. In the associated text field, type mup.
Add intrinsic and surface doping using the parameters defined earlier. For the surface doping remember to make a selection for the surface (boundary) from which dopants are diffused into the bulk.
Analytic Doping Model 1: Intrinsic
1
In the Physics toolbar, click  Domains and choose Analytic Doping Model.
2
In the Settings window for Analytic Doping Model, type Analytic Doping Model 1: Intrinsic in the Label text field.
3
Locate the Domain Selection section. From the Selection list, choose All domains.
4
Locate the Impurity section. In the NA0 text field, type Npi.
Geometric Doping Model 1: P-plus
1
In the Physics toolbar, click  Domains and choose Geometric Doping Model.
2
In the Settings window for Geometric Doping Model, type Geometric Doping Model 1: P-plus in the Label text field.
3
Locate the Domain Selection section. From the Selection list, choose All domains.
4
Locate the Distribution section. From the Profile away from the boundary list, choose Error function (erf).
5
Locate the Impurity section. In the NA0 text field, type Nds.
6
Locate the Profile section. In the dj text field, type dj.
7
From the Nb list, choose Acceptor concentration (semi/adm1).
Geometric Doping Model 2: N-plus
1
Right-click Geometric Doping Model 1: P-plus and choose Duplicate.
2
In the Settings window for Geometric Doping Model, type Geometric Doping Model 2: N-plus in the Label text field.
3
Locate the Impurity section. From the Impurity type list, choose Donor doping (n-type).
4
In the ND0 text field, type Nds.
Boundary Selection for Doping Profile 1
1
In the Model Builder window, expand the Component 1 (comp1)>Semiconductor (semi)>Geometric Doping Model 1: P-plus node, then click Boundary Selection for Doping Profile 1.
2
Select Boundary 2 only.
Boundary Selection for Doping Profile 1
1
In the Model Builder window, expand the Component 1 (comp1)>Semiconductor (semi)>Geometric Doping Model 2: N-plus node, then click Boundary Selection for Doping Profile 1.
2
Select Boundary 1 only.
Add generation and recombination features using the parameters and variables defined above.
Trap-Assisted Recombination 1: SRH
1
In the Physics toolbar, click  Domains and choose Trap-Assisted Recombination.
2
In the Settings window for Trap-Assisted Recombination, type Trap-Assisted Recombination 1: SRH in the Label text field.
3
Locate the Domain Selection section. From the Selection list, choose All domains.
4
Locate the Shockley-Read-Hall Recombination section. From the τn list, choose User defined. In the associated text field, type tau.
5
From the τp list, choose User defined. In the associated text field, type tau.
Impact Ionization Generation 1
1
In the Physics toolbar, click  Domains and choose Impact Ionization Generation.
2
In the Settings window for Impact Ionization Generation, locate the Domain Selection section.
3
From the Selection list, choose All domains.
4
Locate the Impact Ionization Generation section. From the Impact ionization model list, choose User-defined.
5
Find the User-defined parameters subsection. In the αn text field, type alphan.
6
In the αp text field, type alphap.
User-Defined Generation 1: Radiation effect
1
In the Physics toolbar, click  Domains and choose User-Defined Generation.
2
In the Settings window for User-Defined Generation, type User-Defined Generation 1: Radiation effect in the Label text field.
3
Locate the Domain Selection section. From the Selection list, choose All domains.
4
Locate the User-Defined Generation section. In the Gn,0 text field, type gR.
5
In the Gp,0 text field, type gR.
Finally create ohmic contacts, one grounded and the other driven with the voltage parameter V0.
Metal Contact 1: Ground
1
In the Physics toolbar, click  Boundaries and choose Metal Contact.
2
In the Settings window for Metal Contact, type Metal Contact 1: Ground in the Label text field.
3
Select Boundary 2 only.
Metal Contact 2: V0
1
In the Physics toolbar, click  Boundaries and choose Metal Contact.
2
In the Settings window for Metal Contact, type Metal Contact 2: V0 in the Label text field.
3
Select Boundary 1 only.
4
Locate the Terminal section. In the V0 text field, type V0.
Finish the physics setup by adding an explicit event at the end of the triangular pulse marked by the pulse duration tp.
Events (ev)
In the Model Builder window, under Component 1 (comp1) click Events (ev).
Explicit Event 1
1
In the Physics toolbar, click  Global and choose Explicit Event.
2
In the Settings window for Explicit Event, locate the Event Timings section.
3
In the ti text field, type tp.
Use physics-controlled mesh. It is always recommended to double check the mesh resolution and to perform mesh refinement studies.
Mesh 1
1
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Physics-Controlled Mesh section.
3
From the Element size list, choose Extra fine.
The field-dependent mobility model makes the equation system very nonlinear and difficult to solve. So we first assume field-independent mobility by setting the continuation parameter cp to zero. This helps an easier ramp up of the applied voltage V0 from equilibrium to the operating voltage of 1000 V. Disable the Events interface which is only needed for the transient study of the pulsed radiation. Show default solver to change the continuation predictor from the default Constant to Linear. The Linear option helps accelerate the voltage sweep by using a linear extrapolation scheme to estimate the initial guess for the next swept parameter. The default Constant option uses the current solution as the initial guess for the next swept parameter, which is a more conservative approach and in most cases is appropriate for the highly nonlinear semiconductor equation system - for this model it is too conservative.
Study 1a: Ramp V0 with field-independent mobility
1
In the Model Builder window, click Study 1.
2
In the Settings window for Study, type Study 1a: Ramp V0 with field-independent mobility in the Label text field.
Step 1: Semiconductor Equilibrium
1
In the Model Builder window, under Study 1a: Ramp V0 with field-independent mobility click Step 1: Semiconductor Equilibrium.
2
In the Settings window for Semiconductor Equilibrium, locate the Physics and Variables Selection section.
3
In the table, clear the Solve for check box for Events (ev).
Stationary
1
In the Study toolbar, click  Study Steps and choose Stationary>Stationary.
2
In the Settings window for Stationary, locate the Physics and Variables Selection section.
3
In the table, clear the Solve for check box for Events (ev).
4
Click to expand the Study Extensions section. Select the Auxiliary sweep check box.
5
From the Sweep type list, choose All combinations.
6
Click  Add.
7
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
cp (Continuation parameter for mobility model)
0
 
8
Click  Add.
9
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
V0 (Applied voltage)
0 0.5 10 50 100 250 1000
V
10
In the Model Builder window, click Study 1a: Ramp V0 with field-independent mobility.
11
In the Settings window for Study, locate the Study Settings section.
12
Clear the Generate default plots check box.
Solution 1 (sol1)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 1 (sol1) node.
3
In the Model Builder window, expand the Study 1a: Ramp V0 with field-independent mobility>Solver Configurations>Solution 1 (sol1)>Stationary Solver 2 node, then click Parametric 1.
4
In the Settings window for Parametric, click to expand the Continuation section.
5
From the Predictor list, choose Linear.
6
In the Study toolbar, click  Compute.
Now that the voltage sweep is completed with field-independent mobility, we can use the set of solutions as the initial guess for the case of the full mobility model with field dependence. This is done by using a Parametric Sweep node to pair the swept voltage V0 with a swept parameter index. Then use the parameter index to select the correct solution corresponding to each V0 value, using the Manual option in the Step 1: Stationary node.
Add Study
1
In the Study toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Physics interfaces in study subsection. In the table, clear the Solve check box for Events (ev).
4
Find the Studies subsection. In the Select Study tree, select General Studies>Stationary.
5
Click Add Study in the window toolbar.
6
In the Study toolbar, click  Add Study to close the Add Study window.
Study 2
Step 1: Stationary
1
In the Settings window for Stationary, click to expand the Values of Dependent Variables section.
2
Find the Initial values of variables solved for subsection. From the Settings list, choose User controlled.
3
From the Method list, choose Solution.
4
From the Study list, choose Study 1a: Ramp V0 with field-independent mobility, Stationary.
5
From the Parameter value (V0 (V),cp) list, choose Manual.
6
In the Index text field, type index.
Parametric Sweep
1
In the Study toolbar, click  Parametric Sweep.
2
In the Settings window for Parametric Sweep, locate the Study Settings section.
3
Click  Add.
4
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
V0 (Applied voltage)
0 0.5 10 50 100 250 1000
V
5
Click  Add.
6
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
index (Parameter for solution indexing)
range(1,7)
 
7
In the Model Builder window, click Study 2.
8
In the Settings window for Study, type Study 1b: Ramp V0 with full mobility model in the Label text field.
9
In the Study toolbar, click  Compute.
Plot the doping profile, hole distribution, and electric field distribution, to be compared with Fig. 3, 4A, and 4B in the reference paper, respectively.
Results
Fig. 3 Doping Profile
1
In the Model Builder window, under Results click Electric Potential (semi).
2
In the Settings window for 1D Plot Group, type Fig. 3 Doping Profile in the Label text field.
3
Locate the Data section. From the Parameter selection (V0, index) list, choose First.
Line Graph 1
1
In the Model Builder window, expand the Fig. 3  Doping Profile node, then click Line Graph 1.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type abs(semi.Ndoping).
4
In the Unit field, type 1/cm^3.
5
Select the Description check box. In the associated text field, type Doping density.
6
Locate the x-Axis Data section. From the Parameter list, choose Expression.
7
In the Expression text field, type x.
Fig. 3 Doping Profile
1
In the Model Builder window, click Fig. 3  Doping Profile.
2
In the Settings window for 1D Plot Group, locate the Plot Settings section.
3
Clear the y-axis label check box.
4
Locate the Axis section. Select the y-axis log scale check box.
5
Select the Manual axis limits check box.
6
In the x minimum text field, type 0.
7
In the x maximum text field, type 300.
8
In the y minimum text field, type 1e10.
9
In the y maximum text field, type 1e20.
10
In the Fig. 3 Doping Profile toolbar, click  Plot.
Fig. 4A Hole Distribution
1
Right-click Fig. 3  Doping Profile and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Fig. 4A Hole Distribution in the Label text field.
3
Locate the Data section. From the Parameter selection (V0, index) list, choose All.
4
Locate the Axis section. In the x maximum text field, type 1.
5
In the y maximum text field, type 1e19.
6
Locate the Legend section. From the Position list, choose Upper left.
Line Graph 1
1
In the Model Builder window, expand the Fig. 4A  Hole Distribution node, then click Line Graph 1.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type semi.P.
4
Clear the Description check box.
5
Locate the x-Axis Data section. In the Expression text field, type x/L.
6
Select the Description check box. In the associated text field, type Fractional distance.
7
Click to expand the Legends section. Select the Show legends check box.
8
In the Fig. 4A Hole Distribution toolbar, click  Plot.
Fig. 4B Electric Field Distribution
1
In the Model Builder window, right-click Fig. 4A  Hole Distribution and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Fig. 4B Electric Field Distribution in the Label text field.
3
Locate the Axis section. In the y minimum text field, type 1e2.
4
In the y maximum text field, type 1e5.
5
Locate the Legend section. From the Position list, choose Upper middle.
Line Graph 1
1
In the Model Builder window, expand the Fig. 4B  Electric Field Distribution node, then click Line Graph 1.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type semi.normE.
4
In the Unit field, type V/cm.
5
In the Fig. 4B Electric Field Distribution toolbar, click  Plot.
With the diode fully reverse biased at 1000 V, we are ready to study the response of the device to steady-state radiation. In previous studies the time parameter t was 0[s] as defined earlier, thus the pulse function pw1(t/tp) was zero and the generation rate due to radiation gR was also zero (Refer to the Parameters 1 node for the definition). Now to specify nonzero radiation, just set the time parameter t to equal to the pulse duration tp, so that the pulse function pw1(t/tp) is unity. Then the dosage rate can be specified by the parameter RadSi directly in units of Rad(Si)/s.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Physics interfaces in study subsection. In the table, clear the Solve check box for Events (ev).
4
Find the Studies subsection. In the Select Study tree, select General Studies>Stationary.
5
Click Add Study in the window toolbar.
6
In the Home toolbar, click  Add Study to close the Add Study window.
Study 2: Steady-state radiation effect
1
In the Model Builder window, click Study 3.
2
In the Settings window for Study, type Study 2: Steady-state radiation effect in the Label text field.
3
Locate the Study Settings section. Clear the Generate default plots check box.
Step 1: Stationary
1
In the Model Builder window, under Study 2: Steady-state radiation effect click Step 1: Stationary.
2
In the Settings window for Stationary, locate the Values of Dependent Variables section.
3
Find the Initial values of variables solved for subsection. From the Settings list, choose User controlled.
4
From the Method list, choose Solution.
5
From the Study list, choose Study 1b: Ramp V0 with full mobility model, Stationary.
6
Locate the Study Extensions section. Select the Auxiliary sweep check box.
7
From the Sweep type list, choose All combinations.
8
Click  Add.
9
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
t (Time parameter)
tp
s
10
Click  Add.
11
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
RadSi (Radiation dose rate in Rad(Si)/s)
{0.25,2.5,5.0,5.7}*1e8
 
As we will see, the field-dependent mobility gives rise to an interesting result for high dose rates, however it also makes the equation system very nonlinear and difficult to solve. In the previous pair of studies for voltage ramping, the difficulty is overcome by separating the solution process into two stages, with field-independent mobility in the first stage and full mobility in the second. In this current study of dose rate ramping, we show an alternative method to overcome the difficulty using a single study with full mobility.
The nonlinearity causes the Automatic Newton solver to converge slower than the ideal quadratic convergence behavior, which in turn causes the continuation solver to take too small steps for the swept parameter RadSi. So use the Constant Newton option instead, with an appropriate damping factor. In addition, provide a better initial step size for the continuation solver under the Parametric 1 node to prevent it from taking too large a first step and then waste time in back tracking. Finally, use Anderson acceleration to further improve performance (by taking advantage of the information from earlier history of the nonlinear iteration) and use a smaller max number of iterations to reduce the time wasted in back tracking.
Solution 12 (sol12)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 12 (sol12) node.
3
In the Model Builder window, expand the Study 2: Steady-state radiation effect>Solver Configurations>Solution 12 (sol12)>Stationary Solver 1 node, then click Fully Coupled 1.
4
In the Settings window for Fully Coupled, click to expand the Method and Termination section.
5
From the Nonlinear method list, choose Constant (Newton).
6
In the Damping factor text field, type 0.6.
7
In the Model Builder window, under Study 2: Steady-state radiation effect>Solver Configurations>Solution 12 (sol12)>Stationary Solver 1 click Parametric 1.
8
In the Settings window for Parametric, locate the Continuation section.
9
Select the Tuning of step size check box.
10
In the Initial step size text field, type 5e7.
11
In the Maximum step size text field, type 1e8.
12
From the Predictor list, choose Linear.
13
In the Model Builder window, under Study 2: Steady-state radiation effect>Solver Configurations>Solution 12 (sol12)>Stationary Solver 1 click Fully Coupled 1.
14
In the Settings window for Fully Coupled, locate the Method and Termination section.
15
In the Maximum number of iterations text field, type 20.
16
From the Stabilization and acceleration list, choose Anderson acceleration.
17
In the Dimension of iteration space text field, type 5.
18
In the Study toolbar, click  Compute.
Plot the steady-state electric field distribution and hole distribution for several ionization rates, to be compared with Fig. 5A and 5B in the reference paper, respectively.
Results
Fig. 5A Steady-State Electric Field Distribution vs. Dose Rate
1
In the Model Builder window, right-click Fig. 4B  Electric Field Distribution and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Fig. 5A Steady-State Electric Field Distribution vs. Dose Rate in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 2: Steady-state radiation effect/Solution 12 (sol12).
4
Locate the Axis section. In the y maximum text field, type 1e6.
5
Locate the Legend section. From the Position list, choose Lower middle.
6
In the Fig. 5A Steady-State Electric Field Distribution vs. Dose Rate toolbar, click  Plot.
Fig. 5B Steady-State Hole Distribution vs. Dose Rate
1
In the Model Builder window, right-click Fig. 4A  Hole Distribution and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Fig. 5B Steady-State Hole Distribution vs. Dose Rate in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 2: Steady-state radiation effect/Solution 12 (sol12).
4
Locate the Axis section. In the y minimum text field, type 1e11.
5
In the y maximum text field, type 1e15.
6
Locate the Legend section. From the Position list, choose Lower right.
7
In the Fig. 5B Steady-State Hole Distribution vs. Dose Rate toolbar, click  Plot.
In addition, plot the mobility versus the ionized impurity concentration and the drift velocity versus the electric field intensity, to be compared with the curves in Fig. 2A and 2B in the reference paper, respectively. Add datasets to plot the impurity-dependent and field-dependent curves on top of the simulated data points.
Fig. 2A Mobility vs. Doping
1
Right-click Fig. 5B  Steady-State Hole Distribution vs. Dose Rate and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Fig. 2A Mobility vs. Doping in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 1b: Ramp V0 with full mobility model/Parametric Solutions 1 (sol4).
4
Click to expand the Title section. From the Title type list, choose None.
5
Locate the Plot Settings section.
6
Select the x-axis label check box. In the associated text field, type Doping (cm<sup>-3</sup>).
7
Select the y-axis label check box. In the associated text field, type Mobility (cm<sup>2</sup>/V-s).
8
Locate the Axis section. Select the x-axis log scale check box.
9
Clear the y-axis log scale check box.
10
In the x minimum text field, type 1e13.
11
In the x maximum text field, type 1e20.
12
In the y minimum text field, type 0.
13
In the y maximum text field, type 1400.
14
Locate the Legend section. From the Position list, choose Upper right.
Line Graph 1
1
In the Model Builder window, expand the Fig. 2A  Mobility vs. Doping node, then click Line Graph 1.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type semi.mun.
4
From the Unit list, choose cm^2/(V*s).
5
Locate the x-Axis Data section. In the Expression text field, type NDt.
6
In the Description text field, type Ionized impurity concentration.
7
Click to expand the Coloring and Style section. Find the Line style subsection. From the Line list, choose None.
8
Find the Line markers subsection. From the Marker list, choose Cycle.
9
From the Positioning list, choose In data points.
10
Locate the Legends section. Clear the Show legends check box.
Line Graph 2
1
Right-click Results>Fig. 2A  Mobility vs. Doping>Line Graph 1 and choose Duplicate.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type semi.mup.
4
Locate the Coloring and Style section. Find the Line markers subsection. From the Marker list, choose Cycle (reset).
Grid 1D 1: logNDt
1
In the Results toolbar, click  More Datasets and choose Grid>Grid 1D.
2
In the Settings window for Grid 1D, type Grid 1D 1: logNDt in the Label text field.
3
Locate the Data section. From the Source list, choose Function.
4
From the Function list, choose Piecewise 1 (pw1).
5
Locate the Parameter Bounds section. In the Name text field, type logNDt.
6
In the Minimum text field, type 13.
7
In the Maximum text field, type 20.
Line Graph 3
1
In the Model Builder window, right-click Fig. 2A  Mobility vs. Doping and choose Line Graph.
2
In the Settings window for Line Graph, locate the Data section.
3
From the Dataset list, choose Grid 1D 1: logNDt.
4
Locate the y-Axis Data section. In the Expression text field, type mun0[V*s/cm^2]/(1+81*10^logNDt/(10^logNDt+3.24e18))^0.5.
5
Select the Description check box. In the associated text field, type Electron mobility.
6
Locate the x-Axis Data section. From the Parameter list, choose Expression.
7
In the Expression text field, type 10^logNDt.
8
Locate the Coloring and Style section. From the Color list, choose Cyan.
9
From the Width list, choose 2.
10
Locate the Legends section. Select the Show legends check box.
11
Find the Include subsection. Clear the Solution check box.
12
Select the Description check box.
Line Graph 4
1
Right-click Line Graph 3 and choose Duplicate.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type mup0[V*s/cm^2]/(1+350*10^logNDt/(10^logNDt+1.05e19))^0.5.
4
In the Description text field, type Hole mobility.
5
Locate the Coloring and Style section. From the Color list, choose Green.
6
In the Fig. 2A Mobility vs. Doping toolbar, click  Plot.
Fig. 2B Velocity vs. Electric Field
1
In the Model Builder window, right-click Fig. 2A  Mobility vs. Doping and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Fig. 2B Velocity vs. Electric Field in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 2: Steady-state radiation effect/Solution 12 (sol12).
4
Locate the Plot Settings section. In the x-axis label text field, type Electric field (V/cm).
5
In the y-axis label text field, type Drift velocity (cm/s).
6
Locate the Axis section. Clear the x-axis log scale check box.
7
In the x minimum text field, type 0.
8
In the x maximum text field, type 1e5.
9
In the y maximum text field, type 9e6.
10
Locate the Legend section. From the Position list, choose Lower right.
Line Graph 1
1
In the Model Builder window, expand the Fig. 2B  Velocity vs. Electric Field node, then click Line Graph 1.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type semi.mun*semi.normE.
4
From the Unit list, choose cm/s.
5
Locate the x-Axis Data section. In the Expression text field, type semi.normE[cm/V].
6
Clear the Description check box.
Line Graph 2
1
In the Model Builder window, click Line Graph 2.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type semi.mup*semi.normE.
4
From the Unit list, choose cm/s.
5
Locate the x-Axis Data section. In the Expression text field, type semi.normE[cm/V].
6
Clear the Description check box.
Grid 1D 1: E1
1
In the Model Builder window, right-click Grid 1D 1: logNDt and choose Duplicate.
2
In the Settings window for Grid 1D, type Grid 1D 1: E1 in the Label text field.
3
Locate the Parameter Bounds section. In the Name text field, type E1.
4
In the Minimum text field, type 0.
5
In the Maximum text field, type 1e5.
Line Graph 3
1
In the Model Builder window, under Results>Fig. 2B  Velocity vs. Electric Field click Line Graph 3.
2
In the Settings window for Line Graph, locate the Data section.
3
From the Dataset list, choose Grid 1D 1: E1.
4
Locate the y-Axis Data section. In the Expression text field, type E1*mun0[V*s/cm^2]/(1+(E1/8e3)^4.9*(E1+1.64e5)/(E1+1))^(1/4.9).
5
In the Description text field, type Electron velocity.
6
Locate the x-Axis Data section. In the Expression text field, type E1.
7
From the Unit list, choose m.
8
Locate the Coloring and Style section. From the Color list, choose Red.
Line Graph 4
1
In the Model Builder window, click Line Graph 4.
2
In the Settings window for Line Graph, locate the Data section.
3
From the Dataset list, choose Grid 1D 1: E1.
4
Locate the y-Axis Data section. In the Expression text field, type E1*mup0[V*s/cm^2]/(1+(E1/8.72e4)^1.15*(E1+8.51e5)/(E1+8.12e4))^(1/1.15).
5
In the Description text field, type Hole velocity.
6
Locate the x-Axis Data section. In the Expression text field, type E1.
7
From the Unit list, choose m.
8
Locate the Coloring and Style section. From the Color list, choose Magenta.
9
In the Fig. 2B Velocity vs. Electric Field toolbar, click  Plot.
After completing the steady-state study, we are now ready to investigate the effect of pulsed radiation with the waveform given by the inset of Fig. 8 in the reference paper. This waveform is provided by the pulse function pw1(t/tp) in the model with a normalized height of unity. The peak dose rate is specified by the parameter RadSi directly in units of Rad(Si)/s. Tighten the tolerance to 1e-8. Since the solution of the steady-state study is used as the initial condition, use the option of Initial value based for the scaling of the dependent variables, to get a proper scaling for the error estimate.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies>Time Dependent.
4
Click Add Study in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 4
Step 1: Time Dependent
1
In the Settings window for Time Dependent, locate the Study Settings section.
2
From the Time unit list, choose ns.
3
In the Output times text field, type 0 0.1 0.2 0.3 0.5 0.6 0.8 1 1.3 1.7 2 2.4 3 3.5 4 4.3 4.7 5 6 7 8 10.
4
From the Tolerance list, choose User controlled.
5
In the Relative tolerance text field, type 1.0E-8.
6
Click to expand the Values of Dependent Variables section. Find the Initial values of variables solved for subsection. From the Settings list, choose User controlled.
7
From the Method list, choose Solution.
8
From the Study list, choose Study 1b: Ramp V0 with full mobility model, Stationary.
9
Click to expand the Study Extensions section. Select the Auxiliary sweep check box.
10
Click  Add.
11
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
RadSi (Radiation dose rate in Rad(Si)/s)
2.5e8 2.5e10
 
12
In the Model Builder window, click Study 4.
13
In the Settings window for Study, type Study 3: Pulsed radiation effect in the Label text field.
14
Locate the Study Settings section. Clear the Generate default plots check box.
Solution 13 (sol13)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 13 (sol13) node, then click Dependent Variables 1.
3
In the Settings window for Dependent Variables, locate the Scaling section.
4
From the Method list, choose Initial value based.
5
In the Study toolbar, click  Compute.
Plot the hole, electron, and electric field distributions at several times, and the photocurrent response at two ionization levels, to be compared with Fig. 7A ~ 7C and Fig. 8 in the reference paper, respectively.
Results
Fig. 7A Hole Distribution at Several Times
1
In the Model Builder window, right-click Fig. 5B  Steady-State Hole Distribution vs. Dose Rate and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Fig. 7A Hole Distribution at Several Times in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 3: Pulsed radiation effect/Solution 13 (sol13).
4
From the Parameter selection (RadSi) list, choose Last.
5
Locate the Axis section. In the y maximum text field, type 1e16.
6
Locate the Data section. From the Time selection list, choose From list.
7
In the Times (ns) list, choose 0.6, 1, 2, and 5.
8
In the Fig. 7A Hole Distribution at Several Times toolbar, click  Plot.
Fig. 7B Electron Distribution at Several Times
1
Right-click Fig. 7A  Hole Distribution at Several Times and choose Duplicate.
2
In the Model Builder window, click Fig. 7A  Hole Distribution at Several Times 1.
3
In the Settings window for 1D Plot Group, type Fig. 7B Electron Distribution at Several Times in the Label text field.
4
Locate the Legend section. From the Position list, choose Lower left.
Line Graph 1
1
In the Model Builder window, click Line Graph 1.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type semi.N.
4
In the Fig. 7B Electron Distribution at Several Times toolbar, click  Plot.
Fig. 7C Electric Field Distribution at Several Times
1
In the Model Builder window, right-click Fig. 7B  Electron Distribution at Several Times and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Fig. 7C Electric Field Distribution at Several Times in the Label text field.
3
Locate the Data section. In the Times (ns) list, choose 0, 0.6, 1, 2, and 5.
4
Locate the Axis section. In the y minimum text field, type 1e2.
5
In the y maximum text field, type 1e6.
6
Locate the Legend section. From the Position list, choose Upper middle.
Line Graph 1
1
In the Model Builder window, expand the Fig. 7C  Electric Field Distribution at Several Times node, then click Line Graph 1.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type semi.normE.
4
In the Unit field, type V/cm.
5
In the Fig. 7C Electric Field Distribution at Several Times toolbar, click  Plot.
Fig. 8 Transient Photocurrent Response
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Fig. 8 Transient Photocurrent Response in the Label text field.
3
Locate the Data section. From the Dataset list, choose None.
4
Locate the Plot Settings section. Select the Two y-axes check box.
Global 1
1
Right-click Fig. 8  Transient Photocurrent Response and choose Global.
2
In the Settings window for Global, locate the Data section.
3
From the Dataset list, choose Study 3: Pulsed radiation effect/Solution 13 (sol13).
4
From the Parameter selection (RadSi) list, choose First.
5
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
semi.I0_2/area
A/cm^2
Current density
6
Locate the x-Axis Data section. From the Axis source data list, choose Time.
7
Click to expand the Legends section.
Global 2
1
Right-click Global 1 and choose Duplicate.
2
In the Settings window for Global, locate the Data section.
3
From the Parameter selection (RadSi) list, choose Last.
4
Locate the y-Axis section. Select the Plot on secondary y-axis check box.
5
In the Fig. 8 Transient Photocurrent Response toolbar, click  Plot.
Finally present the time evolution of the hole density profile and the drift velocity as the main mechanism of such evolution in one single condensed graph using the Parametric Extrusion 1D dataset.
Parametric Extrusion 1D 1
1
In the Results toolbar, click  More Datasets and choose Parametric Extrusion 1D.
2
In the Settings window for Parametric Extrusion 1D, locate the Data section.
3
From the Dataset list, choose Study 3: Pulsed radiation effect/Solution 13 (sol13).
4
From the Parameter selection (RadSi) list, choose Last.
Evolution of Hole Density Profile and Drift Velocity
1
In the Results toolbar, click  2D Plot Group.
2
In the Settings window for 2D Plot Group, type Evolution of Hole Density Profile and Drift Velocity in the Label text field.
3
Click to expand the Title section. From the Title type list, choose Manual.
4
In the Title text area, type x:Distance y:Time z:log<sub>10</sub>(Hole Density(cm<sup>-3</sup>)) color:Drift Velocity(cm/s).
Surface 1
1
Right-click Evolution of Hole Density Profile and Drift Velocity and choose Surface.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type semi.normE*semi.mup.
4
From the Unit list, choose cm/s.
Height Expression 1
1
Right-click Surface 1 and choose Height Expression.
2
In the Settings window for Height Expression, locate the Expression section.
3
From the Height data list, choose Expression.
4
In the Expression text field, type semi.log10P.
Filter 1
1
In the Model Builder window, right-click Surface 1 and choose Filter.
2
In the Settings window for Filter, locate the Element Selection section.
3
In the Logical expression for inclusion text field, type semi.log10P>11 && semi.log10P<16.
Contour 1
1
In the Model Builder window, right-click Evolution of Hole Density Profile and Drift Velocity and choose Contour.
2
In the Settings window for Contour, locate the Expression section.
3
In the Expression text field, type x.
4
Locate the Levels section. In the Total levels text field, type 10.
5
Locate the Coloring and Style section. From the Coloring list, choose Uniform.
6
From the Color list, choose Gray.
7
Clear the Color legend check box.
Height Expression 1
1
In the Model Builder window, under Results>Evolution of Hole Density Profile and Drift Velocity>Surface 1 click Height Expression 1.
2
In the Settings window for Height Expression, locate the Axis section.
3
Select the Scale factor check box.
4
Right-click Results>Evolution of Hole Density Profile and Drift Velocity>Surface 1>Height Expression 1 and choose Copy.
Height Expression 1
In the Model Builder window, right-click Contour 1 and choose Paste Height Expression.
Filter 1
In the Model Builder window, under Results>Evolution of Hole Density Profile and Drift Velocity>Surface 1 right-click Filter 1 and choose Copy.
Filter 1
In the Model Builder window, right-click Contour 1 and choose Paste Filter.
Contour 2
1
Right-click Contour 1 and choose Duplicate.
2
In the Settings window for Contour, locate the Expression section.
3
In the Expression text field, type t.
4
From the Unit list, choose ns.
5
Locate the Levels section. From the Entry method list, choose Levels.
6
In the Levels text field, type 0.1 0.2 0.5 1 2 3 4 5 8.
7
Locate the Coloring and Style section. From the Coloring list, choose Gradient.
8
From the Top color list, choose Cyan.
9
From the Bottom color list, choose Magenta.
10
Select the Color legend check box.
11
In the Evolution of Hole Density Profile and Drift Velocity toolbar, click  Plot.
12
Click the  Zoom Extents button in the Graphics toolbar.