ECE 695 Numerical Simulations Lecture 13: Drift-Diffusion...
Transcript of ECE 695 Numerical Simulations Lecture 13: Drift-Diffusion...
ECE 695Numerical Simulations
Lecture 13: Drift-Diffusion Simulations
Prof. Peter Bermel
February 8, 2017
Outline
• Drift-Diffusion Model
– Electronic transport theory
– Solution scheme
– Poisson solver
– Scharfetter-Gummel scheme
– Newton’s method
2/8/2017 ECE 695, Prof. Bermel 2
Physics-Based Device Modeling
2/8/2017
D. Vasileska and S.M. Goodnick, Computational Electronics, published by Morgan & Claypool , 2006.
ECE 695, Prof. Bermel 3
Electronic Transport Theory• Will assume electronic bandstructures known,
and take a semiclassical approach
• Electrostatics modeled via Poissson’s equation:
• Charge conservation is required:
2/8/2017 ECE 695, Prof. Bermel
D AV p n N N
1
1
J
J
n n
p p
nU
t q
pU
t q
S. Selberherr: "Analysis and Simulation of Semiconductor Devices“, Springer, 1984.
4
Electronic Transport Theory
• Both p-type and n-type currents given by a sum of two terms:
– Drift term, derived from Ohm’s law
– Diffusion term, derived from Second Law of Thermodynamics
2/8/2017 ECE 695, Prof. Bermel
( ) ( )
( ) ( )
n n n
p p p
dnJ qn x E x qD
dx
dnJ qp x E x qD
dx
S. Selberherr: "Analysis and Simulation of Semiconductor Devices“, Springer, 1984.
5
Poisson/Laplace Equation SolutionPoisson/Laplace Equation
No knowledge of PDE solvers
Method of images
With knowledge ofPDE solvers
Theoretical Approaches
Numerical Methods:
finite difference
finite elementsPoisson
Green’s function method
Laplace
Method of separation of variables
(Fourier analysis)
2/8/2017 ECE 695, Prof. Bermel 6
Numerical Solution Details
Governing
Equations
ICS/BCS
Discretization
System of
Algebraic
Equations
Equation
(Matrix)
Solver
Approximate
Solution
Continuous
Solutions
Finite-Difference
Finite-Volume
Finite-Element
Spectral
Boundary Element
Hybrid
Discrete
Nodal
Values
Tridiagonal
SOR
Gauss-Seidel
Krylov
Multigrid
φi (x,y,z,t)
p (x,y,z,t)
n (x,y,z,t)
D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes,
Arizona State University, Tempe, AZ.
2/8/2017 ECE 695, Prof. Bermel 7
FEM Electronic Transport Model
• Much like in earlier work, will employ the following strategy:– Specify problem parameters, including bulk and
boundary conditions
– Construct finite-element mesh over spatial domain
– Generate a linear algebra problem
– Solve for key field variables:
2/8/2017 ECE 695, Prof. Bermel
φi (x,y,z,t)
p (x,y,z,t)
n (x,y,z,t)
8
FEM Electronic Transport Model
2/8/2017 ECE 695, Prof. Bermel
• Regarding the grid set-up, there are several points that need to be
made:
In critical device regions, where the charge density varies
very rapidly, the mesh spacing has to be smaller than the
extrinsic Debye length determined from the maximum doping
concentration in that location of the device
Cartesian grid is preferred for particle-based simulations
It is always necessary to minimize the number of node points
to achieve faster convergence
A regular grid (with small mesh aspect ratios) is needed for
faster convergence
2
maxeN
TkL B
D
D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes, Arizona State
9
Example for Meshing
The function below is used to generate non-uniform mesh with constant mesh aspect ratio r. Input parameters are initial mesh size (X0), total number of mesh points (N) and the size of the domain over which we want these mesh points distributed (XT).
This is determined by the
maximum doping in the
device in a particular region.
2/8/2017 ECE 695, Prof. Bermel 10
Poisson Equation
• The Poisson equation is of the following general form:
It accounts for Coulomb carrier-carrier interactions in the
Hartree approximation
It is always coupled with some form of transport simulator
except when equilibrium conditions apply
It has to be frequently solved during the simulation procedure
to properly account for the fields driving the carriers in the
transport part
There are numerous ways to numerically solve this equation
that can be categorized into direct and iterative methods
)()(2 rfr
2/8/2017 ECE 695, Prof. Bermel 11
Poisson Solver
2/8/2017 ECE 695, Prof. Bermel
• The 1D Poisson equation is of the form:
2
2
exp exp( / )
exp exp( / )
D A
F ii i T
B
i Fi i T
B
d ep n N N
dx
E En n n V
k T
E Ep n n V
k T
D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes, Arizona State
12
Poisson Solver• Perturbing potential by d yields:
2/8/2017 ECE 695, Prof. Bermel
2/ /
2
/ /
2/ / / /
2
/ /
/
/
T T
T T
T T T T
T T
V Vii
V Vi
V V V Vnewi ii
V V oldi
new old
ende e C n
dx
ene e
en ende e e e C n
dx
ene e
d
d
D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes, Arizona State
13
Poisson Solver
• Renormalized form
2/8/2017 ECE 695, Prof. Bermel
2
2
2
2
new old
new old
dp n C p n
dx
dp n p n C p n
dx
d
d
D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes, Arizona State
14
Finite Difference Representation
1 1 1
1 12 2 2
1 2 1
( ) ( )
n n n
i i i i i
n
i i i i i i
n p
p n C p n
Equilibrium:
exp( ), exp( )n n
i i i in p
Non-Equilibrium:
n calculated using PM coupling and p still calculated as in equilibrium case (quasi-equilibrium approximation)
2/8/2017 ECE 695, Prof. Bermel 15
Criterion for Convergence
There are several criteria for the convergence of the iterative procedure when solving the Poisson equation, but the simplest one is that nowhere on the mesh is the absolute change in potential greater than 10-5 V.
This criterion has shown to be sufficient for all device simulations that have been performed within the Computational Electronics community.
2/8/2017 ECE 695, Prof. Bermel 16
Boundary Conditions
• There are three types of boundary conditions that are specified during the discretization process of the Poisson equation:
Dirichlet (this is a boundary condition on the potential)
Neumann (this is a boundary condition on the derivative of the potential, i.e. the electric field)
Mixed boundary condition (combination of Dirichlet and Neumann boundary conditions)
• Note that when applying the boundary conditions for a particular structure of interest, at least one point MUST have Dirichlet boundary conditions specified on it to get the connection to the real world.
2/8/2017 ECE 695, Prof. Bermel 17
1D Discretization
• The resultant finite difference equations can be represented in a
matrix form Au= f, where:
x0 x1 x2 x3 x4
)(
2;
2;
)(
2
where
10
0
0
2
,),,,,,(,),,,,,(
444
333
222
111
00
543210543210
ei
wi
ei
iwi
ei
iei
wi
wi
idxdxdx
cdxdx
bdxdxdx
a
cba
cba
cba
cba
cb
fffff
A
fu
Neumann Dirichletx5
2/8/2017 ECE 695, Prof. Bermel 18
2D Discretization
• In 2D, the finite-difference discretization of the Poisson equation
leads to a five point stencil:
N=5,M=4
Dirichlet: 0,4,5,9,10,14,15,19
Neuman: 1,2,3,16,17,18
xi-1 xi xi+1
xi-N
xi+N
widx
eidx
sidx
nidx
xi-1 xi xi+1
xi-N
xi+N
widx
eidx
sidx
nidx
0 1 2 3 4
5 6 7 8 9
10 11 12 13 14
15 16 17 18 19
Va
0 1 2 3 4
5 6 7 8 95 6 7 8 9
10 11 12 13 1410 11 12 13 14
15 16 17 18 1915 16 17 18 19
Va
2/8/2017 ECE 695, Prof. Bermel 19
2D Discretization (cont’d)
Dirichlet: 0,4,5,9,10,14,15,19
Neuman: 1,2,3,16,17,18
0
0
0
0
1
2
2
2
1
1
1
1
1
1
2
2
2
1
18
17
16
13
12
11
8
7
6
3
2
1
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
18181818
17171717
16161616
1313131313
1212121212
1111111111
88888
77777
66666
3333
2222
1111
f
f
f
V
f
f
f
V
f
f
f
V
f
f
f
V
dcba
dcba
dcba
edcba
edcba
edcba
edcba
edcba
edcba
edcb
edcb
edcb
a
a
a
a
2/8/2017 ECE 695, Prof. Bermel 20
Poisson Solver
2/8/2017 ECE 695, Prof. Bermel
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
Test maximum
absolute error update
Equilibrium solver
VA = VA+V
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
Test maximum
absolute error update
Maximum voltage exceeded?
Calculate current
STOP yes
no
> tolerance
< tolerance
Non-Equilibrium solver
> tolerance
< tolerance
V is a fraction of the thermal voltage VT
D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes, Arizona State
21
Current Discretization
2/8/2017 ECE 695, Prof. Bermel
• The discretization of the continuity equation in conservative form requires the knowledge of the current densities
on the mid-points of the mesh lines connecting neighboring grid nodes. Since solutions are available only on the grid nodes, interpolation schemes are needed to determine the solutions.
• There are two schemes that one can use:
(a)Linearized scheme: V, n, p, and D vary linearly between neighboring mesh points
(b) Scharfetter-Gummel scheme: electron and hole densities follow exponential variation between mesh points
peDExepxJneDExenxJ
ppp
nnn
)()()()(
D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes, Arizona State
22
Naïve Linearization Scheme
2/8/2017 ECE 695, Prof. Bermel
• Within the linearized scheme, one has that
• This scheme can lead to substantial errors in regions of high electric fields and highly doped devices.
2/12/11
2/12/12/1
iii
iiiii neD
a
VVenJ
i
i
i
iiii
i
i
i
iiiii
a
eD
a
VVen
a
eD
a
VVenJ
2/112/1
2/112/112/1
2
2
D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes, Arizona State
23
Scharfetter-Gummel Scheme
2/8/2017 ECE 695, Prof. Bermel
• One solves the electron current density equation
for n(V), subject to the boundary conditions
• The solution of this first-order differential equation leads to
x
V
V
neD
a
VVen
x
neD
a
VVenJ
ii
iii
ii
iiii
2/11
2/1
2/11
2/12/1
11
)(and)(
iiiinVnnVn
Vt
VVBn
Vt
VVBn
a
eDJ
e
eVgVgnVgnVn
iii
iii
i
ii
VtVV
VtVV
iiii
i
111
2/12/1
/)(
/)(
11
1)(),()(1)(
1
1)(
xe
xxB is the Bernouli function
D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes, Arizona State
24
Gummel’s Method (cont’d)
initial guess
of the solution
solve
Poisson’s eq.
Solve electron eq.
Solve hole eq.
nconverged?
converged?n
y
y
initial guess
of the solution
solve
Poisson’s eq.
Solve electron eq.
Solve hole eq.
nconverged?
converged?n
y
y
initial guess
of the solution
Solve Poisson’s eq.
Electron eq.
Hole eq.
Update
generation rate
nconverged?
converged?n
y
y
initial guess
of the solution
Solve Poisson’s eq.
Electron eq.
Hole eq.
Update
generation rate
nconverged?
converged?n
y
y
Original Gummel’s scheme Modified Gummel’s scheme
2/8/2017 ECE 695, Prof. Bermel 25
• The three equations that constitute the DD model, written in residual form are:
• Starting from an initial guess, the corrections are calculated by solving:
0),,( 0),,( 0),,( pnvFpnvFpnvF pnv
p
n
v
ppp
nnn
vvv
F
F
F
p
n
v
p
F
n
F
v
Fp
F
n
F
v
Fp
F
n
F
v
F
kkk
kkk
kkk
ppp
nnn
VVV
1
1
1
Newton’s Method
2/8/2017 ECE 695, Prof. Bermel 26
Newton’s Method (cont’d)
p
n
V
n
Fp
F
n
F
F
F
F
p
n
V
p
F
n
F
V
Fn
F
V
FV
F
n
vv
p
n
v
ppp
nn
v
000
00
0
0
00
• The method can be simplified by the following iterative scheme:
111
11
1
kpkpp
kp
kknn
kv
kvkvv
kv
nn
FV
V
FFp
p
F
pp
FnV
V
FFn
V
F
pp
Fn
n
FFV
V
F
k+1 k
2/8/2017 ECE 695, Prof. Bermel 27
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
Test maximum
absolute error update
Equilibrium solver
VA = VA+V
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
Test maximum
absolute error update
Maximum voltage exceeded?
Calculate current
STOP yes
no
> tolerance
< tolerance
Non-Equilibrium solver
> tolerance
< tolerance
V is a fraction of the thermal voltage VT
2/8/2017 28ECE 695, Prof. Bermel
Next Class
• We will continue with electronic band structure modeling
2/8/2017 ECE 695, Prof. Bermel 29