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