ECE 695 Numerical Simulations Lecture 13: Drift-Diffusion ...pbermel/ece695/Lectures/ECE695-Lectu… · Lecture 13: Drift-Diffusion Simulations Prof. Peter Bermel February 8, 2017

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


( ) ( )

( ) ( )

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)


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.


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:


φi (x,y,z,t)

p (x,y,z,t)

n (x,y,z,t)

8

FEM Electronic Transport Model


• 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


• 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


12

Poisson Solver• Perturbing potential by d yields:


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


13

Poisson Solver

• Renormalized form


2

2

2

2

new old

new old

dp n C p n

dx

dp n p n C p n

dx

d

d


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


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:


- Forcing function



Test maximum


Maximum voltage exceeded?

Calculate current

STOP yes

no

> tolerance

< tolerance

Non-Equilibrium solver

> tolerance

< tolerance

V is a fraction of the thermal voltage VT


21

Current Discretization


• 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

)()()()(


22

Naïve Linearization Scheme


• 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


23

Scharfetter-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

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


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



Update:


- Forcing function

Test maximum


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:


- Forcing function



Test maximum


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

Documents

ECE 695 Numerical Simulations Lecture 13: Drift-Diffusion ...pbermel/ece695/Lectures/ECE695-Lectu… · Lecture 13: Drift-Diffusion Simulations Prof. Peter Bermel February 8, 2017