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Drift-Diffusion Modeling
Prepared by
Dragica VasileskaProfessor
Arizona State University
Outline of the Lecture
Classification of PDEs Why Numerical Analysis? Numerical Solution Sequence Flow-Chart of Equilibrium Poisson Equation
Solver Discretization of the Continuity Equation Numerical Solution Techniques for Sparse
Matrices Flow-Chart of 1D Drift-Diffusion Simulator
Classification of PDEs
Classification of PDEs
Different mathematical and physicalbehaviors: Elliptic Type Parabolic Type Hyperbolic Type
System of coupled equations for several variables: Time : first-derivative (second-derivative for
wave equation) Space: first- and second-derivatives
Classification of PDEs (cont.)
General form of second-order PDEs ( 2 variables)
PDE Model Problems
Hyperbolic (Propagation)
• Advection equation (First-order linear)
• Wave equation (Second-order linear
)
PDE Model Problems (cont.)
Parabolic (Time- or space-marching)
• Burger’s equation (Second-order nonlinear)
• Fourier equation (Second-order
linear )
(Diffusion / dispersion)
PDE Model Problems (cont.)
Elliptic (Diffusion, equilibrium problems)
• Laplace/Poisson (second-order linear)
• Helmholtz equation
Well-Posed Problem
Numerically well-posed Discretization equations Auxiliary conditions (discretized
approximated)
• the computational solution exists (existence)• the computational solution is unique (uniqueness)• the computational solution depends continuously
on the approximate auxiliary data• the algorithm should be well-posed (stable) also
Boundary and InitialConditions
R
s
n
R
Initial conditions: starting point for propagation problems
Boundary conditions: specified on domain boundaries to provide the interior solution in computational domain
Numerical Methods
Complex geometry Complex equations (nonlinear,
coupled) Complex initial / boundary
conditions
No analytic solutions Numerical methods needed !!
Why Numerical Analysis?
Coupling of Transport Equations to Poisson and Band-Structure Solvers
D. Vasileska and S.M. Goodnick, Computational Electronics, published by Morgan & Claypool , 2006.
Drift-Diffusion Approach
Constitutive Equations
Poisson
Continuity Equations
Current Density Equations
1
1
J
J
n n
p p
nU
t q
pU
t q
D AV p n N N
( ) ( )
( ) ( )
n n n
p p p
dnJ qn x E x qD
dxdn
J qp x E x qDdx
S. Selberherr: "Analysis and Simulation of Semiconductor Devices“, Springer, 1984.
Poisson/Laplace Equation Solution
Poisson/Laplace Equation
No knowledge of solving of PDEs
Method of images
With knowledge for solving of PDEs
Theoretical Approaches
Numerical Methods:finite differencefinite elements
Poisson
Green’s function method
Laplace
Method of separation of variables(Fourier analysis)
Numerical Solution Sequence
Numerical Solution Details
Governing Equations ICS/BCS
DiscretizationSystem of Algebraic Equations
Equation (Matrix) Solver
ApproximateSolution
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.
What is next?
MESHFinite Difference DiscretizationBoundary Conditions
MESH TYPE
The course of action taken in three steps is dictated by the nature of the problem being solved, the solution region, and the boundary conditions. The most commonly used grid patterns for two-dimensional problems are
Common grid patterns: (a) rectangular grid, (b) skew grid, (c) triangular grid, (d) circular grid.
Mesh Size
• 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
maxeNTk
L BD
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 themaximum doping in thedevice in a particular region.
Finite Difference Schemes
Before finding the finite difference solutions to specific PDEs, we will look at howone constructs finite difference approximations from a given differential equation. This essentially involves estimating derivatives numerically. Let’s assume f(x) shown below:
Estimates for the derivative of f (x) at P using forward, backward, and central differences.
Finite Difference Schemes
We can approximate derivative of f(x), slope or the tangent at P by the slope of the arc PB, giving the forward-difference formula,
x
xfxxfxf
)()()( 00
0
or the slope of the arc AP, yielding the backward-difference formula,
x
xxfxfxf
)()()( 00
0
or the slope of the arc AB, resulting in the central-difference formula,
x
xxfxxfxf
2
)()()( 00
0
Finite Difference Schemes
We can also estimate the second derivative of f (x) at P as
x
xxfxf
x
xfxxf
x
x
xxfxxfxf
)()()()(1
)2/()2/()(
0000
000
or
2000
0
)()(2)()(
x
xxfxfxxfxf
Any approximation of a derivative in terms of values at a discrete set of points iscalled finite difference approximation.
Finite Difference Schemes
)()(!3
1)()(
!2
1)()()( 0
30
2000 xfxxfxxfxxfxxf
The approach used above in obtaining finite difference approximations is ratherintuitive. A more general approach is using Taylor’s series. According to the wellknownexpansion,
)()(!3
1)()(
!2
1)()()( 0
30
2000 xfxxfxxfxxfxxf
and
Upon adding these expansions,4
02
000 )()()()(2)()( xxfxxfxxfxxf
where O(x)4 is the error introduced by truncating the series. We say that this error is of the order (x)4 or simply O(x)4. Therefore, O(x)4 represents terms that are not greater than ( x)4. Assuming that these terms are negligible,
2000
0
)()(2)()(
x
xxfxfxxfxf
Finite Difference Schemes
)()(!3
1)()(
!2
1)()()( 0
30
2000 xfxxfxxfxxfxxf
)()(!3
1)()(
!2
1)()()( 0
30
2000 xfxxfxxfxxfxxf
3000 )()()(2)()( xxfxxxfxxf
Subtracting
from
We obtain
and neglecting terms of the order (x)3 yields
x
xxfxxfxf
2
)()()( 00
0
This shows that the leading errors of the order (x)2. Similarly, the forward and backward difference formula have truncation errors of O(x).
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
Poisson Equation Linearization
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
Φ => Φ +
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
dxen
e e
en ende e e e C n
dxen
e e
Renormalized Form LD=sqrt(qni/εVT)
2
2
2
2new old
new old
dp n C p n
dx
dp n p n C p n
dx
Finite Difference Representation
1 1 11 12 2 2
1 2 1
( ) ( )
n n ni i i i i
ni 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)
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 the absolute value of the potential update is larger than 1E-5 V.
This criterion has shown to be sufficient for all device simulations that have been performed within the Computational Electronics community.
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.
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
100
02
,),,,,,(,),,,,,(
444
333
222
111
00
543210543210
ei
wi
ei
iwi
ei
iei
wi
wi
idxdxdx
cdxdx
bdxdxdx
a
cbacba
cbacba
cb
fffff
A
fu
Neumann Dirichletx5
2D Discretization
• In 2D, the finite-difference discretization of the Poisson equation leads to a five point stencil:
N=5,M=4
Dirichlet: 0,4,5,9,10,14,15,19Neuman: 1,2,3,16,17,18
xi-1 xi xi+1
xi-N
xi+N
widx e
idx
sidx
nidx
xi-1 xi xi+1
xi-N
xi+N
widx e
idx
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
2D Discretization (cont’d)
Dirichlet: 0,4,5,9,10,14,15,19Neuman: 1,2,3,16,17,18
0
0
0
0
12
22
11
11
11
22
21
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
fffV
fffV
fff
V
fff
V
dcbadcba
dcba
edcbaedcba
edcba
edcbaedcba
edcba
edcbedcb
edcb
a
a
a
a
Flow-Chart of Equilibrium Poisson Equation 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
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 decompositionSolve 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
Discretization of the Continuity Equation
Sharfetter-Gummel Discretization Scheme• 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
pppnnn
)()()()(
• 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
aVV
enJ
21 ii nn
i
iia
nn 1
i
i
i
iiii
i
i
i
iiiii
aeD
aVVe
n
aeD
aVVe
nJ
2/112/1
2/112/112/1
2
2
(a) Linearized Scheme
(b) Sharfetter-Gummel Scheme• 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
xV
Vn
eDa
VVen
xn
eDa
VVenJ
ii
iii
ii
iiii
2/11
2/1
2/11
2/12/1
11
)(and)(
iiii
nVnnVn
VtVV
BnVt
VVBn
aeD
J
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
Bernouli Function Implementation Others as Well …
Numerical Solution Techniques for Sparse Matrices
Numerical Methods
Objective: Speed, Accuracy at minimum cost Numerical Accuracy (error analysis) Numerical Stability (stability analysis) Numerical Efficiency (minimize cost) Validation (model/prototype data, field data,
analytic solution, theory, asymptotic solution) Reliability and Flexibility (reduce preparation
and debugging time) Flow Visualization (graphics and animations)
Solution Methods
Direct methods
• Gaussian elimination• LU decomposition method
Iterative methods
• Mesh relaxation methods- Jacobi- Gaus-Seidel- Successive over-relaxation method (SOR)- Alternating directions implicit (ADI) method
• Matrix methods- Thomas tridiagonal form- Sparse matrix methods: Stone’s Strongly Implicit Procedure
(SIP), Incomplete Lower-Upper (ILU) decomposition method- Conjugate Gradient (CG) methods: Incomplete Choleski
Conjugate Gradient (ICCG), Bi-CGSTAB- Multi–Grid (MG) method
• The variety of methods for solving Poisson equation include:
LU Decomposition (cont’d)
• If LU decomposition exists, then for a tri-diagonal matrix A, resulting from the finite-difference discretization of the 1D Poisson equation, one can write
where
Then, the solution is found by forward and back substitution:
n
n
nnn
nnn c
cc
abcab
cabca
1
22
11
3
2
111
222
11
...
......
1......
...1
1
.........
nkcab
a kkkkk
kk ,...,3,2,,, 1
111
1,2,...,1,,
,,...,3,2,,
1
111
nixcg
xg
x
nigfgfg
i
iiii
n
nn
iiii
Iterative Methods
Iterative (or relaxation) methods start with a first approximation which is successively improved by the repeated application (i.e. the “iteration”) of the same algorithm, until a sufficient accuracy is obtained.
In this way, the original approximation is “relaxed” toward the exact solution which is numerically more stable.
Iterative methods are used most often for large sparse system of equations, and always when a good approximation of the solution is known.
Error analysis and convergence rate are two crucial aspects of the theory of iterative methods.
Error Equation
• The simple iterative methods for the solution of Ax = f proceed in the following manner:
ri = f - Avi
ei = x - vi.
Aei = ri
A sequence of approximations v0,v1,...,vn,..., of x is construc-ted that converges to x. Let vi be an approximation to x after the i-th iteration. One may define the residual
as a computable measure of the deviation of vi from x. Next, the algebraic error ei of the approximation vi is defined by
From the previous equations one can see that ei obeys the so-called residual equation:
Jacobi, Gauss-Seidel MethodsThe expansion of Ax=f gives the relation
In Jacobi’s method the sequence v0,v1,...,vn,... is computed by
Note that one does not use the improved values until after a complete iteration. The closely related Gauss-Seidel’s method solves this problem as follows:
0;,...,2,1,,1
kk
kk
n
kjjkjki
k anka
fxa
x
0;,...,2,1,,11
kk
kk
n
kjjk
ijkj
ik ank
a
fva
v
0;,...,2,1,
1
1 1
1
1
kk
kk
k
jk
n
kj
ijkj
ijkj
ik ank
a
fvava
v
SOR Method
By a simple modification of Gauss-Seidel’s method it is often possible to improve the rate of convergence. Following the definition of residual ri = f - Avi, the Gauss-Seidel formula can be written i
kik
ik rvv 1
0;,...,2,1,
1
1
1
kkkk
k
jk
n
kj
ijkj
ijkj
ik ank
a
fvava
r
, where
The iterative methodik
ik
ik rvv 1
is the so-called successive over-relaxation (SOR) method. Here, , the relaxation parameter, should be chosen so that the rate of convergence is maximized. The rate of convergence of the SOR is often surprisingly higher than the one of Gauss-Seidel’s method. The value of depends on the grid spacing, the geometrical shape of the domain, and the type of boundary conditions imposed on it.
Convergence
Any stationary iterative method can be written in the general
form
xk+1 = Bxk + c
A relation between the errors in successive approximation can be derived by subtracting the equation x=Bx+c :
xk+1 - x = B(xk - x)=…=Bk+1(x0-x)
Now, let B have eigenvalues 1,2,...,n, and assume that the corresponding eigenvectors u1,u2,...,un are linearly independent. Then we can expand the initial error as
x0 - x = u1 + u2 +…+un
• Convergence of the iterative methods:
Convergence (cont’d)
and thus
This means that the process converges from an arbitraryapproximation if and only if |i|<1, i=1,2,...,n.
Theorem: A necessary and sufficient condition for a stationary iterative method xk+1 = Bxk + c to converge for an arbitrary initial approximation x0 is that
where (B) is called the spectral radius of B. For uniform mesh and Dirichlet boundary conditions, one has for the SOR method
....222111 nknn
kkk uuuxx
,1|)(|max)(1
BB ini
211
2
Multi-Grid Method
1. pre-smoothing on n
2. restriction of r to n-1
3. solution of Ae=r on n-1
4. prolongation of e to n
5. post-smoothing on n
two-grid iteration to solve Av=f on grid n5.
2.
3.
4.
1.n
n-1
2.
3.
4.
1.n
n-1
Coarsening Techniques
• Coarsening techniques: importance of boundaries
18 points 17 points
When the number of points in one dimension is 2N+2 (N being a natural number), a geometric mismatch is generated in the coarser grids, which show pronounced in-homogeneity. The convergence of the method is severely slowed down in these cases.
initial grid
good coarseningbad coarsening
Coarsening Techniques (cont’d)
initial grid
post-smoothing often required on these points
propagation of a contact (Neumann BC) through grids
no post-smoothing is required
• Coarsening techniques: importance of boundaries
Restriction
Restriction is used to transfer the value of the residual from a grid n to a grid n-1; relaxation is then a fine-to-coarse process.In a two-dimensional scheme is convenient to use two different restriction operators, namely the full-weighting and the half-weighting. In the simple case of a homogeneous, square grid, one has:
16
1
8
1
16
18
1
4
1
8
116
1
8
1
16
1
08
10
8
1
2
1
8
1
08
10
full-weighting half-weighting
The criteria to choose the most effective scheme in a given situation depend on the relaxation method, as will be discussed in the section devoted to relaxation.
Prolongation
a
ab
The prolongation is used to transfer the computed error from a grid n-1 to a grid n. It is a coarse-to-fine process and, in two dimensions, can be described as follows (see figure below):
(1) Values on points on the fine grid, which correspond to points on the coarse one (framed points) are just copied.
(2) Values on points of type a are linearly interpolated from the two closer values on the coarse grid.
(3) Values on points of type b are bilinearly interpolated from the four closer values on the coarse grid.
The Coarsest Grid Solver
As shown by the algorithm description, the final coarsest grid has just a few grid-points. A typical grid has 3 up to 5 points per axis.
On this grid, usually called W0, an exact solution of the basic equation Ae = r is required.
The number of grid points is so small on W0 that any solver can be used without changing the convergence rate in a noticeable way.
Typical choices are a direct solver (LU), a SOR, or even a few iterations of the error smoothing algorithm.
Relaxation Scheme
The relaxation scheme forms the kernel of the multigrid method.Its task is to reduce the short wavelength Fourier components of the
error on a given grid. The efficiency of the relaxation scheme depends sensitively on
details such as the grid topology and boundary conditions. Therefore, there is no single standard relaxation scheme that can be applied.
Two Gauss-Seidel schemes, namely point-wise relaxation and line relaxation, can be considered. The correct application of one or more relaxation methods can dramatically imp-rove the convergence. The point numbering scheme plays also a crucial role.
Relaxation Scheme (cont’d)
(1) Relaxation scheme: point Gauss-Seidel
The approximation is relaxed (i.e. the error is smoothed) on each single point, using the values on the point itself and the ones of the neighbors.
This is equivalent to solve a single line of the system Au=f.
Relaxation Scheme (cont’d)
The approximation is relaxed on a complete row (column) of points, using the values on the point itself and the ones of the other points on that row (column).
Successive rows (columns) are visited using a “zebra” numbering scheme. Different combinations of row/column relaxation can be used.
This technique, which processes more than a point a the same time is called block relaxation. This is equivalent to solve an one-dimensional problem with a direct method, that is to solve a tridiagonal problem.
This method is very efficient when points are inhomogeneous in one direction.
(2) Relaxation scheme: line Gauss-Seidel
Comparison of Relaxation Schemes
s
t
s
0. 1
0.1
0.1
0. 1
0.2
0.2
0.2
0.3
0. 3
0.3
0.4
0.5
0.6
0.7
0.8
0.2
0.3
0.1
0.2
0.6
0.4
0.5
0.9
0.9
Point Gauß-Seidel Line Gauß-Seidel
Smoothing factor 0.5 Smoothing factor 0.14
(3) Relaxation schemes: Comparison
Complexity of linear solvers
2D 3D
Sparse Cholesky: O(n1.5 ) O(n2 )
CG, exact arithmetic:
O(n2 ) O(n2 )
CG, no precond: O(n1.5 ) O(n1.33 )
CG, modified IC: O(n1.25 ) O(n1.17 )
CG, support trees: O(n1.20 ) -> O(n1+ ) O(n1.75 ) -> O(n1.31 )
Multigrid: O(n) O(n)
n1/2 n1/3
Time to solve model problem (Poisson’s equation) on regular mesh
Validity of the Drift-Diffusion Model(a) Approximations made in its derivation
• Temporal variations occur in a time-scale much longer than the momentum relaxation time.
• The drift component of the kinetic energy was neglected, thus removing all thermal effects.
• Thermoelectric effects associated with the temperature gradients in the device are neglected, i.e.
• The spatial variation of the external forces is neglected, which implies slowly varying fields.
• Parabolic energy band model was assumed, i.e. degenerate materials can not be treated properly.
peDxepx
neDxenx
ppp
nnn
EJ
EJ
)()(
)()(
(b) Extension of the capabilities of the DD model
• Introduce field-dependent mobility (E) and diffusion coefficient D(E) to empirically extend the range of validity of the DD Model.
• An extension to the model, to take into account the overshoot effect, has been accomplished in 1D by adding an extra term that depends on the spatial derivative of the electric field
1. K.K. Thornber, IEEE Electron Device Lett., Vol. 3 p. 69, 1982.
2. E.C. Kan, U. Ravaioli, and T. Kerkhoven, Solid-State Electron., Vol. 34, 995 (1991).
xn
EeDxE
ELEExenxJ nnnn
)()()()()()(
Validity of the Drift-Diffusion Model (cont’d)
Time-Dependent Simulations
The time-dependent form of the drift-diffusion equations can be used both for
steady-state, and transient calculations.
Steady-state analysis is accomplished by starting from an initial guess, and letting the numerical system evolve until a stationary solution is reached, within set tolerance limits.
Time-Dependent Simulations (cont’d)
This approach is seldom used in practice, since now robust steady-state simulators are widely available.
It is nonetheless an appealing technique for beginners since a relatively small effort is necessary for simple applications and elementary discretization approaches.
If an explicit scheme is selected, no matrix solutions are necessary, but it is normally the case that stability is possible only for extremely small time-steps.
Time-Dependent Simulations (cont’d)
The simulation of transients requires the knowledge of a physically meaningful initial condition.
The same time-dependent numerical approaches used for steady-state simulation are suitable, but there must be more care for the boundary conditions, because of the presence of displacement current during transients.
Displacement Current
In a true transient regime, the presence of displacement currents manifests itself as a potential variation at the contacts, superimposed to the bias, which depends on the external circuit in communication with the contacts.
Displacement Current (cont’d)
Neglect of the displacement current in a transient is equivalent to the application of bias voltages using ideal voltage generators, with zero internal impedance.
In this arrangement, one observes the shortest possible switching time attainable with the structure considered.
Hence, a simulation neglecting displacement current effects may be useful to assess the ultimate speed limits of a device structure.
When a realistic situation is considered, it is necessary to include a displacement term in the current equations. It is particularly simple to deal with a 1-D situation. Consider a 1-D device with length W and a cross-sectional area A. The total current flowing in the device is
(1)
The displacement term makes the total current constant at each position x. This property can be exploited to perform an integration along the device
(2)
The 1-D device, therefore, can be studied as the parallel of a current generator and of the cold capacitance which is in parallel with the (linear) load circuit.
At every time step, Eq. (1) has to be updated, since it depends on the charge stored by the capacitors.
( , )( ) ( , )D n
E x tI t I x t cA
t
*
0
1( ) ( , )
W
D n
cA VI t I x t dx
W W t
Displacement Current (cont’d)
Example: Gunn Diode
• Consider a simple Gunn diode in parallel with an RLC resonant load containing the bias source.
(3)
where I0(t) is the particle current given by the first term on the right hand side of Eq. (1), calculated at the given time step with drift-diffusion. It is also
(4)
• Upon time differencing this last equation, with the use of finite differences, we obtain
(5)
*( )( ) ( )o o
V tI t C I t
t
** ( )( )( ) bV t VV t
I t dtR L
* *( ) ( ) ( ) ( )oo
tV t t V t I t I t
C
* **( ) ( )
( ) ( ) ( ) )b
V t t V t tI t t I t V t V
R L
A robust approach for transient simulation should be based on the same numerical apparatus established for purely steady-state models.
It is usually preferred to use fully implicit schemes, which require a matrix solution at each iteration, because the choice of the time-step is more likely to be limited by the physical time constants of the problem rather than by stability of the numerical scheme.
Typical calculated values for the time step are not too far from typical values of the dielectric relaxation time in practical semiconductor structures.
Some General Comments
Solution of the Coupled DD Equations
There are two schemes that are used in solving the coupled set of equations which comprises the Drift-Diffusion model: Gummel’s method Newton’s method
Gummel’s relaxation method, which solves the equations with the decoupled procedure, is used in the case of weak coupling:
• Low current densities (leakage currents, subthreshold regime), where the concentration dependent diffusion term in the current continuity equation is dominant
• The electric field strength is lower than the avalanche threshold, so that the generation term is independent of V
• The mobility is nearly independent of E
The computational cost of the Gummel’s iteration is one matrix solution for each carrier type plus one iterative solution for the linearization of the Poisson Equation
Gummel’s Method
Gummel’s Method (cont’d)The solution strategy when using Gummel’s relaxation
scheme is the following one:
• Find the equilibrium solution of the linearized Poisson equation
• After the solution in equilibrium is obtained, the applied voltage is increased in steps VVT
• Now the scaled Poisson equation becomes:
i
DAi
i
nNN
VVNn
xd
Vd
VVVNn
xd
Vd
expexp
expexp
2
2
2
2
i
DApn
in
NNVV
Nn
xd
Vdexp)exp(exp)exp(
2
2
Gummel’s Method (cont’d)The 1D discretized electron current continuity equation (as
long as Einstein’s relations are valid) is:
For holes, one can obtain analogous equations by substituting:
021
1 111121
11121
iiiiiiiiii
i
iiiiiii
i
RGaaVVBnVVBna
D
VVBnVVBna
D
/
/
pnVV ,The decoupled iteration scheme goes as follows:
(1) Solve the Poisson equation with a guess for the quasi-Fermi levels (use the applied voltage as initial guess)
(2) The potential is used to update the Bernouli functions
(3) The above equations are solved to provide an update for the quasi-Fermi levels, that enter into the Poisson equation
Gummel’s Method (cont’d)The criterion for convergence is:
In the case of strong coupling, one can use the extended Gummel’s scheme
kTT
kk
kTT
kk
kk
p
pVV
VV
pp
n
nVV
VV
nn
VVV
lnexp
lnexp
1
1
1
1
1
11
1k
k
Tk
k
Tkk
p
pV
n
nVVV lnmax,lnmax,max
Gummel’s Method (cont’d)
initial guessof the solution
solvePoisson’s eq.
Solve electron eq.Solve hole eq.
nconverged?
converged?n
y
y
initial guessof the solution
solvePoisson’s eq.
Solve electron eq.Solve hole eq.
nconverged?
converged?n
y
y
initial guessof the solution
Solve Poisson’s eq.Electron eq.
Hole eq.
Updategeneration rate
nconverged?
converged?n
y
y
initial guessof the solution
Solve Poisson’s eq.Electron eq.
Hole eq.
Updategeneration rate
nconverged?
converged?n
y
y
Original Gummel’s scheme Modified Gummel’s scheme
• 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
Fn
FvF
pF
nF
vF
kkk
kkk
kkk
ppp
nnn
VVV
1
1
1
Newton’s Method
Newton’s Method (cont’d)
pnV
nFpF
nF
FFF
pnV
p
F
n
F
V
FnF
VFVF
n
vv
pnv
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
VF
FnVF
ppF
nnF
FVVF
k+1 k
Flow-Chart of 1D Drift-Diffusion Simulator
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 decompositionSolve 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