Semiconductor Process/Device
Simulation (EEE533)
Assignment 5
1D Drift Diffusion simulator for modeling
pn-junction Diode
Due date – October 13 2008
Dr. Dragica Vasileska
Student
Pinakpani Nayak (995030989)
Assignment 5
1D Drift Diffusion pn Diode simulator
TABLE OF CONTENT
1
2
3
4
5
Equilibrium Solver ..................................................................................................................... 2
1.1
NA = 1016 and ND = 1016 .................................................................................................. 2
1.2
NA = 1016 and ND = 1018 .................................................................................................. 4
1.3
NA = 1018 and ND = 1018 .................................................................................................. 6
Non–Equilibrium Solver ............................................................................................................ 8
2.1
Objective .............................................................................................................................. 8
2.2
Specifications ....................................................................................................................... 8
2.3
Methodology ........................................................................................................................ 8
2.4
Flow chart for 1D Drift Diffusion Solver ............................................................................... 10
2.5
Finite Difference Representation of Linearized Poisson’s Eqn ............................................ 11
2.6
Scharfetter – Gummel Discretization Scheme .................................................................... 11
Simulation Results................................................................................................................... 13
3.1
Conduction Band vs Device Length.................................................................................... 13
3.2
Electron and Hole Densities vs Device Length ................................................................... 14
3.3
IV Characteristics ............................................................................................................... 15
3.4
Electron and Hole Quasi Fermi Levels ............................................................................... 16
3.5
Electric Field ...................................................................................................................... 17
3.6
Equilibrium Condition ......................................................................................................... 17
APPENDIX A ............................................................................................................................ 19
4.1
Results from Larger Bias steps = VT. .................................................................................. 19
APPENDIX B ............................................................................................................................ 20
5.1
Matlab Code for 1D drift Diffusion simulator for modeling pn-junction (diodes) ................... 20
Pinakpani Nayak (995030989)
1
Assignment 5
1D Drift Diffusion pn Diode simulator
1 Equilibrium Solver
1.1
NA = 1016 and ND = 1016
POTENTIAL
MATLAB
NanoHub
Potential vs Position - at Equilibrium
15
10
Potential [eV]
5
0
-5
-10
-15
0
1
2
3
4
5
6
7
x [um]
ELECTRIC FIELD
MATLAB
4
0.5
NanoHub
Field Profile vs Position - at Equilibrium
x 10
0
X: 3.215
Y: -34.66
X: 3.985
Y: -80.89
Electric Field [V/cm]
-0.5
-1
-1.5
Width of Deplition
Region = 0.766um
-2
Peak Electric Field
= -3.117e4 V/cm
-2.5
X: 3.617
Y: -3.117e+004
-3
-3.5
0
1
2
3
4
5
6
7
x [um]
Width of Depletion Region = 0.766um and Peak electric field = - 3.117E4 V/cm
Pinakpani Nayak (995030989)
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Assignment 5
1D Drift Diffusion pn Diode simulator
ELECTRON AND HOLE DENSITIES
MATLAB
NanoHub
Electron & Hole Densities vs Position - at Equilibrium
16
10
Electron & Hole Densities [1/cm3]
14
10
n
p
12
10
10
10
8
10
6
10
4
10
0
1
2
3
4
5
6
7
x [um]
TOTAL CHARGES
MATLAB
-3
2
NanoHub
Total Charge Density vs Position - at Equilibrium
x 10
Total Charge Density [C/cm3]
1.5
1
0.5
0
-0.5
-1
-1.5
-2
0
1
2
3
4
5
6
7
x [um]
Thus Nanohub and Matlab results match.
Pinakpani Nayak (995030989)
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Assignment 5
1D Drift Diffusion pn Diode simulator
NA = 1016 and ND = 1018
1.2
POTENTIAL
MATLAB
NanoHub
Potential vs Position - at Equilibrium
20
X: 5.024
Y: 18.05
15
Potential [eV]
10
5
0
-5
-10
-15
X: 0.7983
Y: -13.44
0
1
2
3
4
5
6
7
x [um]
ELECTRIC FIELD
MATLAB
4
1
NanoHub
Field Profile vs Position - at Equilibrium
x 10
0
X: 2.813
Y: -31.7
X: 3.366
Y: -6.017
Electric Field [V/cm]
-1
-2
-3
-4
-5
X: 3.326
Y: -6.641e+004
-6
-7
0
1
2
3
4
5
6
7
x [um]
Depletion region = 0.553 um
Peak Electric Field = -6.641x104 V/cm
Pinakpani Nayak (995030989)
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Assignment 5
1D Drift Diffusion pn Diode simulator
ELECTRON AND HOLE DENSITIES
MATLAB
NanoHub
Electron & Hole Densities vs Position - at Equilibrium
Electron & Hole Densities [1/cm3]
X: 1.25
Y: 1e+016
X: 4.46
Y: 1e+018
15
10
n
p
10
10
X: 1.443
Y: 2.103e+004
5
10
X: 4.911
Y: 210.2
0
1
2
3
4
5
6
7
x [um]
TOTAL CHARGES
MATLAB
NanoHub
Total Charge Density vs Position - at Equilibrium
0.12
0.1
Total Charge Density [C/cm3]
0.08
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
0
1
2
3
4
5
6
7
x [um]
Pinakpani Nayak (995030989)
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Assignment 5
1.3
1D Drift Diffusion pn Diode simulator
NA = 1018 and ND = 1018
POTENTIAL
MATLAB
NanoHub
Potential vs Position - at Equilibrium
20
X: 0.5981
Y: 18.05
15
Potential [eV]
10
5
0
-5
-10
X: 0.2185
Y: -18.05
-15
-20
0
0.1
0.2
0.3
0.4
x [um]
0.5
0.6
0.7
0.8
ELECTRIC FIELD
MATLAB
5
0.5
NanoHub
Field Profile vs Position - at Equilibrium
x 10
0
X: 0.3674
Y: -491.8
Electric Field [V/cm]
-0.5
X: 0.4495
Y: -894.1
-1
-1.5
-2
-2.5
-3
X: 0.4098
Y: -3.651e+005
-3.5
-4
0
0.1
0.2
0.3
0.4
x [um]
0.5
0.6
0.7
0.8
Width of Depletion Region = 0.0821 um
Peak Electric Field = -3.651E5 V/cm
Pinakpani Nayak (995030989)
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Assignment 5
1D Drift Diffusion pn Diode simulator
ELECTRON AND HOLE DENSITIES
MATLAB
NanoHub
Electron & Hole Densities vs Position - at Equilibrium
20
10
Electron & Hole Densities [1/cm3]
X: 0.1594
Y: 1e+018
X: 0.5963
Y: 1e+018
15
10
n
p
10
10
5
10
X: 0.1705
Y: 210.3
X: 0.5835
Y: 210.2
0
10
0
0.1
0.2
0.3
0.4
x [um]
0.5
0.6
0.7
0.8
TOTAL CHARGES
MATLAB
NanoHub
Total Charge Density vs Position - at Equilibrium
0.2
Total Charge Density [C/cm3]
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
0
0.1
0.2
0.3
0.4
x [um]
0.5
0.6
0.7
0.8
As Doping density of a particular section increases the depletion region in that section
decreases.
Pinakpani Nayak (995030989)
7
Assignment 5
1D Drift Diffusion pn Diode simulator
2 Non–Equilibrium Solver
2.1
Objective
Goal of this project is to develop a one-dimensional (1D) drift-diffusion simulator for modeling pnjunctions (diodes) under forward bias conditions. Both types of carriers have to be included in the
model (electrons and holes). Finite-difference expressions should be used for the electron and hole
current continuity equations using Sharfetter-Gummel discretization scheme.
2.2
Specifications
εsc = 1.05x10-10
ni = 1.5x1010 cm-3 , T = 300K.
Electron mobility = 1500 cm2/V-s
Hole mobility
= 1000 cm2/V-s
Doping : NA = 1016 cm-3 for p-region and ND = 1016 cm-3 for n-region.
Numerical Methods used – LU Decomposition method for the solution of 1D Poisson and 1D
electron and Hole continuity equation.
2.3
Methodology
The flow chart explains the methodology followed. Following steps were followed to simulate the nonequilibrium condition of a 1D pn-diode.
1. Calculate the equilibrium condition of the diode by solving the linearized Poisson’s equation.
2. Define an initial Mesh, which should be fine enough to give correct result. If the Mesh size is
too large then the results obtained will not be correct. The mesh size is less than the Debye
Length.
3. Initialize all discretization co-efficients, doping density and potential based on charge density.
4. Solve for the updated potential using LU decomposition method.
5. Update coefficient of the linearized Poisson’s equation.
6. Check for convergence against a predefined error value(10-5).
7. Equilibrium solution gives electric field, electric potential and net doping values (n and p
concentration) to be used as the initial values for the non-equilibrium solver.
Pinakpani Nayak (995030989)
8
Assignment 5
1D Drift Diffusion pn Diode simulator
8. Based on the equilibrium values we solve the continuity equation for electron and holes to get
the current.
9.
Calculate the low field mobility. mun0 = 1500 cm2/V-s and mup0 = 1000 cm2/V-s. Store the
mobility values in an array for each node points.
10. Calculate the field dependant mobility based on the local filed at each node. mun(E) for
electron and mup(E) for holes. (FDM)
11. Calculate Bernauli’s function that depends on potential. (BF)
12. Determine the discretization coefficients (product of FDM and BF) of continuity eqns for
electron and holes.
13. Update the generation and recombination rate i.e. the forcing function of the continuity
equation.
p.n  ni
G
 p (n  ni )   n ( p  pi )
2
14. Solve electron continuity eqn using LU decomposition and we get “n”.
15. Solve hole continuity eqn using LU decomposition and we get “p”.
16. Update central coefficient and forcing function in Poisson’s equation (now use n and p instead
of eφ and e-φ).
17. Solve Poisson’s equation using LU decomposition.
18. Check tolerance, if not valid then go to step 9 and repeat (as we have updated the potential
the field dependant mobility has changed).
Pinakpani Nayak (995030989)
9
Assignment 5
2.4
1D Drift Diffusion pn Diode simulator
Flow chart for 1D Drift Diffusion Solver
Initialize parameters:
-Mesh size
-Discretization coefficients
-Doping density
-Potential based on charge neutrality
Solve for the updated potential
given the forcing function using LU decomposition
Update:
- Central coefficient of the linearized Poisson Equation
- Forcing function
> tolerance
Equilibrium solver
Test maximum
absolute error update
< tolerance
Non-Equilibrium solver
VA = VA+V
V is a fraction of the
thermal voltage VT
Calculate coefficients for:
- Electron continuity equation
- Hole continuity equation
- Update generation recombination rate
Solve electron continuity equation using LU decomposition
Solve hole continuity equation using LU decomposition
Update:
- Central coefficient of the linearized Poisson Equation
- Forcing function
Solve for the updated potential
given the forcing function using LU decomposition
> tolerance
Test maximum
absolute error update
< tolerance
Calculate current
no
STOP
Pinakpani Nayak (995030989)
Maximum voltage exceeded?
yes
10
Assignment 5
2.5
1D Drift Diffusion pn Diode simulator
Finite Difference Representation of Linearized Poisson’s Eqn
1 n1  2
1

 i 1   2  pi  ni  in1  2  in11   pi  ni  Ci    in  pi  ni 
2




ni = eφ
At Equilibrium :
2.6
and
pi = e-φ
Scharfetter – Gummel Discretization Scheme
Electron Current is given by
J
n
i 1 / 2
  i 1   i
eDin1 / 2 

ni 1 B
 
 VT

    i 1 
  ni B i

V
T



J
n
i 1 / 2
   i
eDin1 / 2    i   i 1 
  ni 1 B i 1

ni B
   VT 
 VT



Hole Current is Given by
J
p
i 1 / 2
J
p
i 1 / 2
  i   i 1 
   i
eDip1 / 2 
  p i B i 1

 p i 1 B
 
 VT 
 VT

eDip1 / 2




   i 1   i 
    i 1 
  p i 1 B i

 p i B
V
V
T
T



 
The electron Continuity equation is given by
  i 1   i
1 n

D
B
i

1
/
2
2
 VT
1

    i 1  1 n
    i 1 
   i
1
ni 1   2 Din1 / 2 B i
  2 Di 1 / 2 B i
 ni  2 Din1 / 2 B i 1


 VT  
 VT 
 VT


ni 1  Gi

The hole Continuity equation is given by
1 p
  i   i 1 
  i 1   i  1 p
  i 1   i 
    i 1 
1 p
1





 p i  2 Dip1 / 2 B i
 p i 1  Gi
D
B
p

D
B

D
B

i

1
/
2
i

1
i

1
/
2
i

1
/
2
2
2
2




V
V
V
V




T
T
T
T









2

D
K BT
p.n  nint
, ' 
,

 VT  D  VT
VT
 p (n  ni )   n ( p  pi )

q
x
B( x)  x
, x = normalized, nint = intrinsic carrier concentration, Δ is normalized by LDi
e 1
Gi 
Pinakpani Nayak (995030989)
11
Assignment 5
1D Drift Diffusion pn Diode simulator
For electron
  in1 / 2VT

 in1 / 2VT






B

'


'
n
n

B

'


'

B

'


'

i 1
i
i 1 int
i
i 1
i
i 1  n i n i int
2
2 2
2
LDi 2
LDi 2
 LDi 

 in1 / 2VT

 in1 / 2VT
LDi 2
2
B i 1 ' i 'ni 1 ni int
pi .ni nint  nint

 p (ni  1)   n ( pi  1) nint
2
2


 in1 / 2 B i 1 ' i 'ni 1   in1 / 2 B i ' i 1 '   in1 / 2 B i ' i 1 'ni   in1 / 2 B i 1 ' i 'ni 1
p i .ni  1
L Di 2
 p (ni  1)   n ( p i  1) VT
2



So the coefficients of electron Continuity eqn are
ci   in1 / 2 B i 1 ' i '


bi   in1 / 2 B i ' i 1 '   in1 / 2 B i ' i 1 '
ai  in1 / 2 B i 1 ' i '
pi .ni  1
LDi 2
 p (ni  1)   n ( pi  1) VT
2
fi 


nint = Intrinsic carrier concentration
LDi = Intrinsic Debye Length
Δ = normalized mesh
VT = Thermal Voltage
ni pi = electron and hole
concentration at node i
Similarly for Holes the coefficients can be determined as
ci   ip1 / 2 B i ' i 1 '


bi   ip1 / 2 B i 1 ' i '   ip1 / 2 B i 1 'i '
ai  
p
i 1 / 2
B i ' i 1 '
pi .ni  1
LDi 2
 p (ni  1)   n ( pi  1) VT
2
fi 


The simulation was performed by varying Anode voltage from 0 to 0.625V in 0.33*V T
steps. All the results were plotted as mentioned below.
NOTE :
It was observed that by varying the DC voltage from 0 to 0.625 with smaller bias steps
like 0.33VT gives better results compared to larger bias steps like VT . The results
obtained from smaller bias steps exactly matche with NanoHub plots.
Appendix A contains the graphs for Larger Bias steps and the difference can be
observed.
Pinakpani Nayak (995030989)
12
Assignment 5
1D Drift Diffusion pn Diode simulator
3 Simulation Results
Anode Voltage varied from 0 to 0.625V in steps of 0.33*Vt
3.1
Conduction Band vs Device Length
MatLab (Va = 0.625V)
Nano Hub (Va = 0.625V)
Conduction Band vs Position - at Applied Bias (0.625)
1
0.8
Conduction Band Energy (eV)
0.6
X: 3.088
Y: 0.303
0.4
0.2
X: 3.413
Y: 0.1457
0
-0.2
-0.4
-0.6
-0.8
-1
0
1
2
3
4
5
6
7
x [um]
MatLab (Equilibrium)
Nano Hub (Equilibrium)
Conduction Band vs Position - at Equilibrium
1
0.9
Conduction Band Energy (eV)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
x [um]
Pinakpani Nayak (995030989)
13
Assignment 5
3.2
1D Drift Diffusion pn Diode simulator
Electron and Hole Densities vs Device Length
MatLab (Equilibrium)
Nano Hub (Equilibrium)
Electron & Hole Densities vs Position - at Equilibrium
18
10
n
p
16
Electron & Hole Densities [1/cm3]
10
14
10
12
10
10
10
8
10
6
10
4
10
2
10
0
1
2
3
4
5
6
7
x [um]
MatLab (Va = 0.625V)
Nano Hub (Va = 0.625V)
Electron & Hole Densities vs Position - at Applied Bias(0.625V)
18
10
16
Electron & Hole Densities [1/cm3]
10
X: 3.365
Y: 2.602e+013
14
10
X: 3.126
Y: 2.237e+014
12
10
10
10
8
10
6
10
X: 0.1936
Y: 2.25e+004
n
p
4
10
X: 6.475
Y: 2250
2
10
0
1
2
3
4
5
6
7
x [um]
The electron and hole densities calculated at 0.625V in Matlab matches with Nanohub values.
Pinakpani Nayak (995030989)
14
Assignment 5
3.3
1D Drift Diffusion pn Diode simulator
IV Characteristics
Total Current - MatLab (Va = 0.625)
I vs V Plot
Total Current - Nano Hub (Va = 0.625V)
X: 0.6226
Y: 4.177
4.5
4
Total Current Density [Amp/cm2]
3.5
3
2.5
2
1.5
1
0.5
0
-0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
VA [V]
The final current at Va=0.6226 is 4.177 A/cm2 . The Matlab and Nanohub values match.
IV Plot for Electron, Hole and Total Current
Current Continuity
I vs V Plot
Total Current Density vs Position - at Applied Bias(0.625V)
5
6
Jtotal
Jhole
Jelec
5
Total Current Density [A/cm2]
Total Current Density [Amp/cm2]
4
3
2
1
0
-1
4
3
2
1
0
0.1
0.2
0.3
0.4
VA [V]
0.5
0.6
0.7
0
0
1
2
3
4
5
6
7
x [um]
Total Current (Electron + Hole) going
from Cathode to Anode is Constant.
Total Current (sum of electron and Hole currents) for 0.625 V is found to be 4.177 A/cm2
which matches with NanoHub Total current for the same bias voltage. The total current is
constant through the entire device from cathode to anode as shown in the plot. This proves
current continuity.
Pinakpani Nayak (995030989)
15
Assignment 5
3.4
1D Drift Diffusion pn Diode simulator
Electron and Hole Quasi Fermi Levels
MatLab (Va = 0.625V)
Nano Hub (Va = 0.625V)
Quasi Fermi Levels (Efn & Efp) vs Position - at Applied Bias(0.625V)
1
Ec
Ev
Ei
Efn
Efp
0.8
0.6
Energy [eV]
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
1
2
3
4
5
6
7
x [um]
The quasi Fermi levels are calculated as given below
Net Charge Density
MatLab (Va = 0.625V)
-3
12
x 10
Nano Hub (Va = 0.625V)
Total Charge Density vs Position - at Applied Bias(0.625V)
10
Total Charge Density [C/cm3]
8
6
4
X: 3.354
Y: 0.004766
2
0
-2
-4
-6
X: 3.322
Y: -0.003202
0
1
2
3
4
5
6
7
x [um]
Pinakpani Nayak (995030989)
16
Assignment 5
3.5
1D Drift Diffusion pn Diode simulator
Electric Field
MatLab (Va = 0.625V)
4
0.5
Nano Hub (Va = 0.625V)
Field Profile vs Position - at Applied Bias(0.625V)
x 10
0
X: 3.054
Y: -51.58
-0.5
X: 3.432
Y: -18.2
Electric Field [V/cm]
-1
-1.5
X: 3.322
Y: -1.719e+004
-2
-2.5
-3
-3.5
-4
-4.5
0
1
2
3
4
5
6
7
x [um]
3.6
Equilibrium Condition
4
Potential vs Position - at Equilibrium
0.5
0.5
0.4
0
-0.5
0.3
-1
Electric Field [V/cm]
Potential [eV]
0.2
0.1
0
-0.1
-2
-2.5
-3.5
-0.3
-4
0
1
2
3
4
5
6
7
x [um]
-3
16
-1.5
-3
-0.2
-0.4
Field Profile vs Position - at Equilibrium
x 10
-4.5
0
1
2
3
4
5
6
7
x [um]
Total Charge Density vs Position - at Equilibrium
x 10
14
Total Charge Density [C/cm3]
12
10
8
6
4
2
0
-2
-4
0
1
2
3
4
5
6
7
x [um]
Pinakpani Nayak (995030989)
17
Assignment 5
1D Drift Diffusion pn Diode simulator
From this project I conclude that I learnt more by designing the simulator myself compared to
generating the plots on Nanohub. Nanohub is a wonderful tool to explore device properties
and verify results. But designing one’s own simulator gives greater insight into the actual
working of the commercial simulator and Nanohub. I can better understand how the
equations are solved, how it is better to take smaller biases and finer mesh sizes to solve an
equilibrium condition.
Pinakpani Nayak (995030989)
18
Assignment 5
1D Drift Diffusion pn Diode simulator
4 APPENDIX A
4.1
Results from Larger Bias steps = VT.
The output plots do not match with the actual NanoHub results when larger Bias steps are
used for Anode voltage increment. Hence finer bias steps give better results.
-3
Electron & Hole Densities vs Position - at Applied Bias(0.625V)
18
10
20
n
p
16
10
15
14
Total Current Density [Amp/cm2]
Electron & Hole Densities [1/cm3]
I vs V Plot
x 10
10
12
10
10
10
8
10
6
10
10
5
0
4
10
2
10
0
1
2
3
4
5
6
-5
7
0
0.1
0.2
0.3
x [um]
The charge density values are lower than actual
0.5
0.6
0.7
The total current plot is distorted and lesser.
Quasi Fermi Levels (Efn & Efp) vs Position - at Applied Bias(0.625V)
Potential vs Position - at Applied Bias(0.625V)
1
0.5
Ec
Ev
Ei
Efn
Efp
0.8
0.6
0.4
0.4
0.3
0.2
Potential [eV]
Energy [eV]
0.4
VA [V]
0
-0.2
-0.4
0.2
0.1
0
-0.6
-0.1
-0.8
-1
0
1
2
3
4
5
6
7
x [um]
Quasi Fermi levels are lesser than actual
Pinakpani Nayak (995030989)
-0.2
0
1
2
3
4
5
6
7
x [um]
Potential values for 0 to 3um is less than actual
19
Assignment 5
1D Drift Diffusion pn Diode simulator
5 APPENDIX B
5.1
Matlab Code for 1D drift Diffusion simulator for modeling pn-junction (diodes)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%%
%%
1D Drift Diffusion Model for pn Diodes
%%
%%
Equilibrium and Non Equilibrium Solver
%%
%%
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all;
close all;
% Defining the Fundamental and Material Constants %
q
kb
eps
T
ni
Vt
RNc
TAUN0
TAUP0
mun0
mup0
=
=
=
=
=
=
=
=
=
1.602E-19;
1.38E-23;
1.05E-12;
300;
1.5E10;
kb*T/q;
2.8E19;
0.1E-6;
0.1E-6;
= 1500;
= 1000;
% C or [J/eV]
% [J/K]
% This includes the eps = 11.7 for Si [F/cm]
% [K]
% Intrinsic carrier concentration [1/cm^3]
% [eV]
% This is 2.8e20 in the FORTRAN file
% Electron SRH life time
% Hole SRH life time
% Electron Mobility in cm2/V-s
% Hole Mobility in cm2/V-s
dEc = Vt*log(RNc/ni);
% Define Doping Values %
Na = 1E16;
Nd = 1E17;
% [1/cm^3]
% [1/cm^3]
% Calculate relevant parameters for the simulation %
Vbi = Vt*log(Na*Nd/(ni*ni));
W
= sqrt(2*eps*(Na+Nd)*Vbi/(q*Na*Nd))
Wn = W*sqrt(Na/(Na+Nd))
Wp = W*sqrt(Nd/(Na+Nd))
Wone = sqrt(2*eps*Vbi/(q*Na))
E_p = q*Nd*Wn/eps
Ldn = sqrt(eps*Vt/(q*Nd));
Ldp = sqrt(eps*Vt/(q*Na));
Ldi = sqrt(eps*Vt/(q*ni))
%
%
%
%
%
[cm]
[cm]
[cm]
[cm]
[V/cm]
% Calculate relevant parameters in an input file %
% Write to a file
save input_params.txt Na Nd Vbi W Wn Wp E_p Ldn Ldp
%Material_Constants
%Define some material constants
% Setting the size of the simulation domain based
% on the analytical results for the width of the depletion regions
Pinakpani Nayak (995030989)
20
Assignment 5
1D Drift Diffusion pn Diode simulator
% for a simple pn-diode %
x_max = 0;
if(x_max < Wn)
x_max = Wn;
end
if(x_max < Wp)
x_max = Wp;
end
x_max = 20*x_max
% Setting the grid size based on the extrinsic Debye lengths %
dx = Ldn;
if(dx > Ldp)
dx=Ldp;
end
dx = dx/20;
% Calculate the required number of grid points and renormalize dx %
n_max = x_max/dx
n_max = round(n_max);
dx = dx/Ldi;
% Renormalize lengths with Ldi
% Set up the doping C(x)=Nd(x)-Na(x) that is normalized with ni %
for i = 1:n_max
if(i <= n_max/2)
dop(i) = - Na/ni;
elseif(i > n_max/2)
dop(i) = Nd/ni;
end
end
% Initialize the potential based on the requirement of charge
% neutrality throughout the whole structure
for i = 1: n_max
zz = 0.5*dop(i);
if(zz > 0)
xx = zz*(1 + sqrt(1+1/(zz*zz)));
elseif(zz < 0)
xx = zz*(1 - sqrt(1+1/(zz*zz)));
end
fi(i) = log(xx);
n(i) = xx;
p(i) = 1/xx;
end
delta_acc = 1E-5;
% Preset the Tolerance
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%%
%%
EQUILIBRIUM SOLUTION PART BEGINS
%%
%%
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%(A) Define the elements of the coefficient matrix for the internal nodes and
%
initialize the forcing function
Pinakpani Nayak (995030989)
21
Assignment 5
1D Drift Diffusion pn Diode simulator
dx2 = dx*dx;
for i = 1: n_max
a(i) = 1/dx2;
c(i) = 1/dx2;
b(i) = -(2/dx2+exp(fi(i))+exp(-fi(i)));
f(i) = exp(fi(i)) - exp(-fi(i)) - dop(i) - fi(i)*(exp(fi(i))+exp(-fi(i)));
end
%(B) Define the elements of the coefficient matrix and initialize the forcing
%
function at the ohmic contacts
a(1) = 0;
c(1) = 0;
b(1) = 1;
f(1) = fi(1);
a(n_max) = 0;
c(n_max) = 0;
b(n_max) = 1;
f(n_max) = fi(n_max);
%(C)
%
Start the iterative procedure for the solution of the linearized Poisson
equation using LU decomposition method:
flag_conv = 0;
k_iter= 0;
while(~flag_conv)
k_iter = k_iter + 1;
% convergence of the Poisson loop
alpha(1) = b(1);
for i=2:n_max
beta(i)=a(i)/alpha(i-1);
alpha(i)=b(i)-beta(i)*c(i-1);
end
% Solution of Lv = f %
v(1) = f(1);
for i = 2:n_max
v(i) = f(i) - beta(i)*v(i-1);
end
% Solution of U*fi = v %
temp = v(n_max)/alpha(n_max);
delta(n_max) = temp - fi(n_max);
fi(n_max)=temp;
for i = (n_max-1):-1:1
%delta%
temp = (v(i)-c(i)*fi(i+1))/alpha(i);
delta(i) = temp - fi(i);
fi(i) = temp;
end
delta_max = 0;
for i = 1: n_max
xx = abs(delta(i));
if(xx > delta_max)
delta_max=xx;
end
%sprintf('delta_max = %d',delta_max)
%'k_iter = %d',k_iter,'
end
Pinakpani Nayak (995030989)
22
Assignment 5
1D Drift Diffusion pn Diode simulator
%delta_max=max(abs(delta));
% Test convergence and recalculate forcing function and
% central coefficient b if necessary
if(delta_max < delta_acc)
flag_conv = 1;
else
for i = 2: n_max-1
b(i) = -(2/dx2 + exp(fi(i)) + exp(-fi(i)));
f(i) = exp(fi(i)) - exp(-fi(i)) - dop(i) - fi(i)*(exp(fi(i)) + exp(-fi(i)));
end
end
end
xx1(1) = dx*1e4;
for i = 2:n_max-1
Ec(i) = dEc - Vt*fi(i);
%Values from the second Node%
ro(i) = -ni*(exp(fi(i)) - exp(-fi(i)) - dop(i));
el_field1(i) = -(fi(i+1) - fi(i))*Vt/(dx*Ldi);
el_field2(i) = -(fi(i+1) - fi(i-1))*Vt/(2*dx*Ldi);
n(i) = exp(fi(i));
p(i) = exp(-fi(i));
xx1(i) = xx1(i-1) + dx*Ldi*1e4;
end
Ec(1) = Ec(2);
Ec(n_max) = Ec(n_max-1);
xx1(n_max) = xx1(n_max-1) + dx*Ldi*1e4;
el_field1(1) = el_field1(2);
el_field2(1) = el_field2(2);
el_field1(n_max) = el_field1(n_max-1);
el_field2(n_max) = el_field2(n_max-1);
nf = n*ni;
pf = p*ni;
ro(1) = ro(2);
ro(n_max) = ro(n_max-1);
figure(1)
plot(xx1, Vt*fi,'r','LineWidth',2)
xlabel('x [um]');
ylabel('Potential [eV]');
title('Potential vs Position - at Equilibrium');
figure(2)
plot(xx1, el_field1,'r','LineWidth',2)
hold on;
plot(xx1, el_field2,'r','LineWidth',2)
xlabel('x [um]');
ylabel('Electric Field [V/cm]');
title('Field Profile vs Position - at Equilibrium');
figure(3)
%plot(xx1, nf,'g','LineWidth',2)
semilogy(xx1, nf,'g','LineWidth',2)
hold on;
%plot(xx1, pf,'r','LineWidth',2)
semilogy(xx1, pf,'r','LineWidth',2)
xlabel('x [um]');
ylabel('Electron & Hole Densities [1/cm^3]');
Pinakpani Nayak (995030989)
23
Assignment 5
1D Drift Diffusion pn Diode simulator
title('Electron & Hole Densities vs Position - at Equilibrium');
legend('n','p');
%axis([0 6.75 0 10.2e17])
figure(4)
%plot(xx1, ro,'r','LineWidth',2)
plot(xx1, q*ro,'r','LineWidth',2)
xlabel('x [um]');
%ylabel('Total Charge Density [1/cm^3]');
ylabel('Total Charge Density [C/cm^3]');
title('Total Charge Density vs Position - at Equilibrium');
%axis([0.5 5 -3e17 8e17])
figure(5)
plot(xx1, Ec,'r','LineWidth',2)
xlabel('x [um]');
%ylabel('Total Charge Density [1/cm^3]');
ylabel('Conduction Band Energy (eV)');
title('Conduction Band vs Position - at Equilibrium');
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
END OF EQUILIBRIUM SOLUTION PART
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%%
%%
NON-EQUILIBRIUM SOLUTION PART BEGINS
%%
%%
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
1. Calculate Low filed mobility
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
Prameters for Low field mobility calculation %%
TL = 300;
N = Na + Nd;
MU1N.CAUG
MU2N.CAUG
ALPHAN.CAUG
BETAN.CAUG
GAMMAN.CAUG
DELTAN.CAUG
NCRITN.CAUG
%
%
%
%
%
%
%
%
% Temp in Kelvin
% Local (total) impurity concentration
=
=
=
=
=
=
=
55.24;
1429.23;
0.0;
-2.3;
-3.8;
0.73;
1.072*10^17;
%
%
%
%
%
%
%
cm2/(V.s)
cm2/(V.s)
unitless
unitless
unitless
unitless
cm-3
MU1P.CAUG
=
MU2P.CAUG
=
ALPHAP.CAUG =
BETAP.CAUG =
GAMMAP.CAUG =
DELTAP.CAUG =
NCRITP.CAUG =
BETAN = 2.0;
BETAP = 1.0;
49.7;
479.37;
0.0;
-2.2;
13.7;
0.70;
1.606*10^17;
%
%
%
%
%
%
%
cm2/(V.s)
cm2/(V.s)
unitless
unitless
unitless
unitless
cm-3
mun0 = ( MU1N.CAUG*((TL/300)^ALPHAN.CAUG) ) ...
+ (( (MU2N.CAUG*((TL/300)^BETAN.CAUG)) - (MU1N.CAUG*((TL/300)^ALPHAN.CAUG)) ) ...
/ ( 1 + ((TL/300)^GAMMAN.CAUG) * ((N/NCRITN.CAUG)^DELTAN.CAUG) ))
Pinakpani Nayak (995030989)
24
Assignment 5
%
%
%
%
%
%
%
%
1D Drift Diffusion pn Diode simulator
mup0 = ( MU1P.CAUG*((TL/300)^ALPHAP.CAUG) ) ...
+ (( (MU2P.CAUG*((TL/300)^BETAP.CAUG)) - (MU1P.CAUG*((TL/300)^ALPHAP.CAUG)) ) ...
/ ( 1 + ((TL/300)^GAMMAP.CAUG) * ((N/NCRITP.CAUG)^DELTAP.CAUG) ))
VSATN = (2.4*10^7) / (1 + 0.8*exp(TL/600));
VSATP = VSATN;
% Saturation Velocity of Electrons
% Saturation Velocity of Holes
%%%%%%%%%%%%%%%%%%% END of Low Field Mobility Calculation %%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
2. Start the main Loop to increment the Anode voltage by Vt=KbT/q %%
%%
till it reaches 0.625V.
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
vindex=0;
for VA = 0:0.33*Vt:0.625
VA
% Start VA increment loop
Each_Step
= 0.33*Vt
Total_Steps = 0.625/(0.33*Vt)
vindex = vindex +1
Vplot(vindex) = VA;
fi(1) = fi(1) + VA;
%fi(1)
flag_conv2 = 0;
k_itern= 0;
% Apply potential to Anode (1st node)
% Convergence of the Poisson loop
%% Initialize the First and Last Node for Poisson's eqn
a(1) = 0;
c(1) = 0;
b(1) = 1;
f(1) = fi(1);
a(n_max) = 0;
c(n_max) = 0;
b(n_max) = 1;
f(n_max) = fi(n_max);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 3. Start the Poisson equation solver loop to calculate the
%%
%%
potential for each Anode voltage increase
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
while(~flag_conv2)
% Start Poisson's eqn
k_itern = k_itern + 1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 3.1 . Calculate Field Dependant Mobility for each value of 'fi'
%%
%%
at each node point of the PN diode.
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% To test with Constant Mobility without field dependancy.
% %
for i = 1:n_max
% Start Loop for Field Dep Mobility
% %
mup(i) = mup0;
% %
mun(i) = mun0;
% %
end
Pinakpani Nayak (995030989)
25
Assignment 5
1D Drift Diffusion pn Diode simulator
%% Calculate the Electric Field at each Node
for i = 2:n_max-1
Ef(i) = abs(fi(i) - fi(i+1))*Vt/(dx*Ldi);
end
Ef(1)
= Ef(2);
Ef(n_max) = Ef(n_max-1);
%% Calculate the Field Dependant Mobility at each Node
for i = 1:n_max
pdeno = (mup0 * Ef(i) / VSATP) ^ BETAP;
mup(i)
= mup0 * ( (1/(1 + pdeno)) ^(1/BETAP));
ndeno = (mun0 * Ef(i) / VSATN) ^ BETAN;
mun(i) = mun0 * ( (1/(1 + ndeno)) ^(1/BETAN));
end
mup(1)
= mup(2);
mup(n_max) = mup(n_max-1);
mun(1)
= mun(2);
mun(n_max) = mun(n_max-1);
%%%%%%%%%%% END of FIELD Dependant Mobility Calculation %%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 3.2 Solve Continuity Equation for Electron and Holes using LU Decomposition %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%(A) Define the elements of the coefficient matrix and initialize the forcing
%
function at the ohmic contacts for ELECTRON and HOLE Continuity Eqns
an(1) =
bn(1) =
cn(1) =
ap(1) =
bp(1) =
cp(1) =
%fnp(1)
0;
%Co-ef
1;
%Co-ef
0;
%Co-ef
0;
%Co-ef
1;
%Co-ef
0;
%Co-ef
= (Ldi*Ldi*dx2/Vt) * (
for electron at
for electron at
for electron at
for hole
at
for hole
at
for hole
at
p(1)*n(1) - 1 )
Anode
Anode
Anode
Anode
Anode
Anode
/ ( TAUP0*(n(1) + 1 ) + TAUN0*(p(1)
+ 1 ) );
fn(1) = n(1);
fp(1) = p(1);
an(n_max) =
bn(n_max) =
cn(n_max) =
ap(n_max) =
bp(n_max) =
cp(n_max) =
%fnp(n_max)
+ TAUN0*(p(n_max) +
fn(n_max) =
fp(n_max) =
0;
%Co-ef
1;
%Co-ef
0;
%Co-ef
0;
%Co-ef
1;
%Co-ef
0;
%Co-ef
= (Ldi*Ldi*dx2/Vt)
1) );
n(n_max);
p(n_max);
for
for
for
for
for
for
* (
electron at Cathode
electron at Cathode
electron at Cathode
hole
at Cathode
hole
at Cathode
hole
at Cathode
p(n_max)*n(n_max) - 1 ) / ( TAUP0*(n(n_max) + 1)
%(B) Define the elements of the coefficient matrix for the internal nodes and
%
initialize the forcing function
Pinakpani Nayak (995030989)
26
Assignment 5
1D Drift Diffusion pn Diode simulator
for i = 2: n_max-1
munim1by2 = (mun(i-1)+mun(i))/2;
munip1by2 = (mun(i)+mun(i+1))/2;
mupim1by2 = (mup(i-1)+mup(i))/2;
mupip1by2 = (mup(i)+mup(i+1))/2;
%% Co-efficients for HOLE Continuity eqn
cp(i) = mupip1by2 * BER(fi(i) - fi(i+1));
ap(i) = mupim1by2 * BER(fi(i) - fi(i-1));
bp(i) = -( mupim1by2 * BER(fi(i-1) - fi(i)) + mupip1by2 * BER(fi(i+1) - fi(i)));
%% Co-efficients for ELECTRON Continuity eqn
cn(i) = munip1by2 * BER(fi(i+1) - fi(i));
an(i) = munim1by2 * BER(fi(i-1) - fi(i));
n(i) = -( munim1by2 * BER(fi(i) - fi(i-1)) + munip1by2 * BER(fi(i) - fi(i+1)));
%% Forcing Function for ELECTRON and HOLE Continuity eqns
fn(i) = (Ldi*Ldi*dx2/Vt) * ( p(i)*n(i) - 1 ) / ( TAUP0*(n(i) + 1) + TAUN0*(p(i)+1));
fp(i) = (Ldi*Ldi*dx2/Vt) * ( p(i)*n(i) - 1 ) / ( TAUP0*(n(i) + 1) + TAUN0*(p(i)+1));
end
%(C)
%
Start the iterative procedure for the solution of the linearized Continuity
equation for "ELECTRONS" using LU decomposition method:
alphan(1) = bn(1);
for i=2:n_max
betan(i)=an(i)/alphan(i-1);
alphan(i)=bn(i)-betan(i)*cn(i-1);
end
% Solution of Lv = f %
vn(1) = fn(1);
for i = 2:n_max
vn(i) = fn(i) - betan(i)*vn(i-1);
end
% Solution of U*fi = v %
tempn = vn(n_max)/alphan(n_max);
%deltan(n_max) = tempn - n(n_max);
n(n_max)=tempn;
for i = (n_max-1):-1:1
%delta%
tempn = (vn(i)-cn(i)*n(i+1))/alphan(i);
% deltan(i) = tempn - n(i);
n(i) = tempn;
end
%%%%%%%%%%%%%%%%%%%%%%% END of ELECTRON Continuty Solver %%%%%%%%%%%
%(D)
%
Start the iterative procedure for the solution of the linearized Continuity
equation for "HOLES" using LU decomposition method:
alphap(1) = bp(1);
for i=2:n_max
betap(i)=ap(i)/alphap(i-1);
alphap(i)=bp(i)-betap(i)*cp(i-1);
end
% Solution of Lv = f %
Pinakpani Nayak (995030989)
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Assignment 5
1D Drift Diffusion pn Diode simulator
vp(1) = fp(1);
for i = 2:n_max
vp(i) = fp(i) - betap(i)*vp(i-1);
end
% Solution of U*fi = v %
tempp = vp(n_max)/alphap(n_max);
%deltap(n_max) = tempp - p(n_max);
p(n_max)=tempp;
for i = (n_max-1):-1:1
%delta%
tempp = (vp(i)-cp(i)*p(i+1))/alphap(i);
%
deltap(i) = tempp - p(i);
p(i) = tempp;
end
%%%%%%%%%%%%%%%%%%%%%%% END of HOLE Continuty Solver %%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 3.3 Calculate potential fi again with new values of "n" and "p"%%
%%
and check for convergence
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Recalculate forcing function and central coefficient b for fi
for i = 2: n_max-1
b(i) = -(2/dx2 + n(i) + p(i));
f(i) = n(i) - p(i) - dop(i) - (fi(i)*(n(i) + p(i)));
%% here values of n(i) and p(i) are used in place of exp(fi(i))
end
% Solve for Updated potential given the new value of Forcing
% Function using LU decomposition
alpha(1) = b(1);
for i=2:n_max
beta(i)=a(i)/alpha(i-1);
alpha(i)=b(i)-beta(i)*c(i-1);
end
% Solution of Lv = f %
v(1) = f(1);
for i = 2:n_max
v(i) = f(i) - beta(i)*v(i-1);
end
% Solution of U*fi = v %
temp = v(n_max)/alpha(n_max);
delta(n_max) = temp - fi(n_max);
fi(n_max)=temp;
for i = (n_max-1):-1:1
%delta%
temp = (v(i)-c(i)*fi(i+1))/alpha(i);
delta(i) = temp - fi(i);
fi(i) = temp;
end
delta_max = 0;
Pinakpani Nayak (995030989)
28
Assignment 5
for i = 1: n_max
xx = abs(delta(i));
if(xx > delta_max)
delta_max=xx;
end
end
1D Drift Diffusion pn Diode simulator
%% Calculate the max error
%delta_max=max(abs(delta));
% Test convergence and start the loop if necessary else increase
% the applied potential
%% delta_max
if(delta_max < delta_acc)
flag_conv2 = 1;
end
end
% End of WHILE Loop for Poisson's eqn solver
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
CALCULATE CURRENT
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Electron Current
for i=2:n_max-1
Jnim1by2(vindex,i) = (q*mun(i)*Vt/(dx*Ldi)) * ni*( n(i)*BER(fi(i)-fi(i-1)) \
- n(i-1)*BER(fi(i-1)-fi(i)) );
Jnip1by2(vindex,i) = (q*mun(i)*Vt/(dx*Ldi)) * ni*( n(i+1)*BER(fi(i+1)-fi(i)) \
- n(i)*BER(fi(i)-fi(i+1)) );
Jelec(vindex,i) = (Jnip1by2(vindex,i) + Jnim1by2(vindex,i))/2;
% %
% %
% Electron Current with only one node
Jnim1by2 = (q*mun(2)*Vt/dx*Ldi) * ( (n(2)*ni)*BER((fi(2)-fi(2-1)))
- (n(2-1)*ni)*BER((fi(2-1)-fi(2))) );
Jnip1by2 = (q*mun(2)*Vt/dx*Ldi) * ( (n(2+1)*ni)*BER((fi(2+1)-fi(2)))
- (n(2)*ni)*BER((fi(2)-fi(2+1))) );
Jelec(vindex) = (Jnip1by2 + Jnim1by2)/2;
% %
% %
% Hole Current
Jpim1by2 = (q*mup(2)*Vt/dx*Ldi) * ( (p(2)*ni)*BER((fi(2-1)-fi(2)))
- (p(2-1)*ni)*BER((fi(2)-fi(2-1))) );
Jpip1by2 = (q*mup(2)*Vt/dx*Ldi) * ( (p(2+1)*ni)*BER((fi(2)-fi(2+1)))
- (p(2)*ni)*BER((fi(2+1)-fi(2))) );
Jhole(vindex) = (Jpip1by2 + Jpim1by2)/2;
% %
% %
% %
Jpim1by2(vindex,i) = (q*mup(i)*Vt/(dx*Ldi)) * ni*( p(i)*BER((fi(i-1)-fi(i))) \
- p(i-1)*BER((fi(i)-fi(i-1))) );
Jpip1by2(vindex,i) = (q*mup(i)*Vt/(dx*Ldi)) * ni*( p(i+1)*BER((fi(i)-fi(i+1))) \
- p(i)*BER((fi(i+1)-fi(i))) );
Jhole(vindex,i) = (Jpip1by2(vindex,i) + Jpim1by2(vindex,i))/2;
end
%%
end
Jtotal = Jelec + Jhole;
Jtotal(vindex) = Jelec;
% End of main FOR loop for VA increment.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
END OF NON-EQUILIBRIUM SOLUTION PART
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Pinakpani Nayak (995030989)
29
Assignment 5
1D Drift Diffusion pn Diode simulator
% Write the results of the simulation in files %
xx1(1) = dx*1e4;
for i = 2:n_max-1
Ec(i) = dEc - Vt*fi(i);
%Values from the second Node%
ro(i) = -ni*(n(i) - p(i) - dop(i));
el_field1(i) = -(fi(i+1) - fi(i))*Vt/(dx*Ldi);
el_field2(i) = -(fi(i+1) - fi(i-1))*Vt/(2*dx*Ldi);
xx1(i) = xx1(i-1) + dx*Ldi*1e4;
end
Jtotal(:,1) = Jtotal(:,2);
Jelec(:,1) = Jelec(:,2);
Jhole(:,1) = Jhole(:,2);
Jtotal(:,n_max) = Jtotal(:,(n_max-1));
Jelec(:,n_max) = Jelec(:,(n_max-1));
Jhole(:,n_max) = Jhole(:,(n_max-1));
Ec(1) = Ec(2);
Ec(n_max) = Ec(n_max-1);
xx1(n_max) = xx1(n_max-1) + dx*Ldi*1e4;
el_field1(1) = el_field1(2);
el_field2(1) = el_field2(2);
el_field1(n_max) = el_field1(n_max-1);
el_field2(n_max) = el_field2(n_max-1);
nf = n*ni;
pf = p*ni;
ro(1) = ro(2);
ro(n_max) = ro(n_max-1);
%% Calculate Quasi Fermi Level - Efn Efp
for i = 1:n_max
Ei(i)
= Ec(i) - 0.56;
Efn(i) = Ei(i) + Vt*log(nf(i)/ni);
Efp(i) = Ei(i) - Vt*log(pf(i)/ni);
end
Ev = Ec - 1.12;
figure(14)
plot(xx1, Ec,'black','LineWidth',2.5);
hold on;
plot(xx1, Ev,'black','LineWidth',2.5);
hold on;
plot(xx1, Ei,'--black','LineWidth',2.5);
hold on;
plot(xx1, Efn,'r','LineWidth',2.5);
hold on;
plot(xx1, Efp,'b','LineWidth',2.5);
xlabel('x [um]');
ylabel('Energy [eV]');
title('Quasi Fermi Levels (Efn & Efp) vs Position - at Applied Bias(0.625V)');
legend('Ec','Ev','Ei','Efn','Efp');
axis([0 7 -1 1]);
figure(6)
plot(xx1, Ec,'b','LineWidth',2)
xlabel('x [um]');
ylabel('Conduction Band Energy (eV)');
title('Conduction Band vs Position - at Applied Bias (0.625)');
Pinakpani Nayak (995030989)
30
Assignment 5
1D Drift Diffusion pn Diode simulator
figure(7)
plot(xx1, Vt*fi,'b','LineWidth',2)
xlabel('x [um]');
ylabel('Potential [eV]');
title('Potential vs Position - at Applied Bias(0.625V)');
figure(8)
plot(xx1, el_field1,'b','LineWidth',2)
hold on;
plot(xx1, el_field2,'b','LineWidth',2)
xlabel('x [um]');
ylabel('Electric Field [V/cm]');
title('Field Profile vs Position - at Applied Bias(0.625V)');
figure(9)
%plot(xx1, nf,'g','LineWidth',2)
semilogy(xx1, nf,'g','LineWidth',2)
hold on;
%plot(xx1, pf,'b','LineWidth',2)
semilogy(xx1, pf,'b','LineWidth',2)
xlabel('x [um]');
ylabel('Electron & Hole Densities [1/cm^3]');
title('Electron & Hole Densities vs Position - at Applied Bias(0.625V)');
legend('n','p');
%axis([0 6.75 0 10.2e17])
figure(10)
%plot(xx1, ro,'b','LineWidth',2)
plot(xx1, q*ro,'b','LineWidth',2)
xlabel('x [um]');
%ylabel('Total Charge Density [1/cm^3]');
ylabel('Total Charge Density [C/cm^3]');
title('Total Charge Density vs Position - at Applied Bias(0.625V)');
%axis([0.5 5 -3e17 8e17])
figure(11)
plot(Vplot, Jtotal(:,2),'r','LineWidth',2)
hold on
plot(Vplot, Jhole(:,2),'g','LineWidth',2)
hold on
plot(Vplot, Jelec(:,2),'b','LineWidth',2)
xlabel('VA [V]');
ylabel('Total Current Density [Amp/cm^2]');
title('I vs V Plot');
legend('Jtotal','Jhole','Jelec','2');
figure(12)
plot(Vplot, Jtotal(:,2),'r','LineWidth',2)
xlabel('VA [V]');
ylabel('Total Current Density [Amp/cm^2]');
title('I vs V Plot');
%legend('Jtotal','Jhole','Jelec','2');
figure(13)
plot(xx1,Jtotal((round((Total_Steps)-1)),:),'b','LineWidth',2)
xlabel('x [um]');
ylabel('Total Current Density [A/cm^2]');
title('Total Current Density vs Position - at Applied Bias(0.625V)');
axis([0 7 0 6]);
Pinakpani Nayak (995030989)
31
Assignment 5
1D Drift Diffusion pn Diode simulator
%figure(5)
%plot(xx1, n)
%hold all
%plot(xx1, p)
save
save
save
save
save
cond_band xx1 Ec;
tot_charge xx1 ro;
el_field xx1 el_field1 el_field2;
np_data xx1 nf pf;
pot_1 fi;
Pinakpani Nayak (995030989)
32