# MODELLING OF SEMICONDUCTOR DEVICES FOR ICS AND VLSI

**ABSTRACT** This thesis is about Simulation of semiconductor devices. Simulation of

semiconductor devices aims at solving the semiconductor partial differential equations

(PDE) for different device geometry and parameters. Device simulators help the device

engineer to create, test and verify novel structures. This increases our physical knowledge

without even going to the laboratory. In the era of the ever-decreasing device structures

simulation methods must be validated and the device engineer must rely on it, since the

cost of manufacturing a device to test is high.

This thesis discusses the requirements needed to implement such a simulator. Several

computational challenges and how they can be solved are presented. It was the aim of this

thesis to implement a device simulator using an object-oriented programming (OOP)

methodology. OOP is a modern programming approach which makes it easy to increase

the program capabilities while keeping the program neat and tight.

The thesis was divided into two main components: geometry and computation. The

geometry component deals with the geometrical aspects ( geometry intersection, point

inclusion, mesh generation and neighborhood adjacency relationships ). We used

Semiconductor Wafer Representation (SWR)model, the well known standard by

Stanford, for implementing the geometry component. SWR does not provide details about

the implementation, it mainly concerned about giving the interface. We tried to keep

close to the standard as we could. Since our simulator is designed to be used for different

device structures, It is important for the mesh generation procedures to deal with

irregularities in the device. Triangles are chosen as the mesh elements. Since we use

OOP, a triangle is inherited from a more abstract polygon class so that different polygonal

elements can be used as well. The simulator is implemented to deal with polygonal

devices but, curved boundaries can be added by inheriting a curve from the basic Edge

class as well.

The geometry algorithms: point inclusion, region intersections, storage of device, etc..

are chosen to be efficient and general and that is the real challenge since, efficiency

doesn't always go well with generality. Therefore, we are biased to generality when

generality is a must, otherwise efficiency has the higher priority. Since the variables

(electric field, electric current, potentials,..etc..) may change with different orders of

magnitude in small regions, adaptive mesh is required. A quad-tree algorithm is used for

generating the mesh. The algorithm goes by spatially decomposing the device domain

into squares when the level of a nodal point is higher than the level of the square. This

will generate an adaptive spatial decomposition. Triangulation in the interior of the device is

done using certain patterns. The boundary zone (known as delta zone) is isolated from the

device region and triangulated using Constrained Delaunay Triangulation ( CDT )

algorithm. CDT is a complex and expensive (from the computational point of view )

algorithm. Fortunately, it is used only in the delta-zone.

The computational component is a real challenge as well. For different device

structures as well as in different modes of operation certain parameters and functions may

dominate while others may be negligible. The simulator has to deal with this. Different

mobility, recombination and generation models can be incorporated to the simulator. The

simulator user has to be free to use the model he wants. Not only that, but the user can

even implement his own models and test them. Again, OOP plays the important role here.

Models are inherited from a general, abstract function class. The user can do what ever he

wants inside the function, but since all functions have the same interface, the simulator

will not distinguish between them.

There are different device models, Hydrodynamic, Energy Transport, Drift-

Diffusion,..etc.. Even the Drift-Diffusion model, which is the simplest model, is complex

in its own. The Drift-Diffusion model consists of three coupled nonlinear partial

differential equations. There are different discretization schemes for discretizing the

system of equations. We used a hybrid mixed finite element scheme. This scheme gives

us some sort of continuity for the electric flux ( required when using different materials )

and current density. It is common for a practical device consisting of different sub-regions

to be discretized using thousands of elements. This will lead to thousands of variables and

millions of matrices elements. Fortunately, the finite element scheme is sparse in nature.

Using different sparse techniques depending on the nature of the matrix is another use of

OOP. Nonlinear Poisson's equation is linearized. The linearized system was solved using

a global Newton method algorithm. Global Newton method converges to the exact

solution whatever the initial guess is.

Different results, a PN junction and a MOS device are shown. Different channel

lengths are entered to the simulator and the I-V characteristics are drawn. The results

shown are highly compatible with the analytical formulae.

MATLAB was originally used for displaying the geometry, mesh as well as plotting

the I-V characteristics. Another version is implemented which uses the windows

application interfaces for entering the device, setting the different parameters as well as

generating the mesh.

SEMILAB is the result of this thesis. It is a mini simulator, which shows how

sophisticated a device simulator is.

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**ABSTRACT:**A model hierarchy for semiconductor devices is presented, in particular kinetic models (Boltzmann and quantum Boltzmann equations) and fluid dynamical models (hydrodynamic, energy-transport and drift-diffusion equations). Fluid dynamical models including quantum correction terms are also discussed. The links between the various models are given. - [Show abstract] [Hide abstract]

**ABSTRACT:**We present a modification to the divide-and-conquer algorithm of Guibas & Stolfi [GS] for computing the Delaunay triangulation of n sites in the plane. The change reduces its &THgr;(n log n) expected running time to &Ogr;(n log n) for a large class of distributions which includes the uniform distribution in the unit square. The modified algorithm is significantly easier to implement than the optimal linear-expected-time algorithm of Bentley, Weide & Yao [BWY]. Unlike the incremental methods of Ohya, Iri & Murota [OIM] and Maus [M] it has optimal &Ogr;(n log log n) worst-case performance. The improvement extends to the composition of the Delaunay triangulation in the Lp metric for 1 < p ≤ ∞. Experimental evidence presented demonstrates that in the Euclidean case the modified algorithm performs very well for n ≤ 216, the range of the experiments. We conjecture that its average running time is no more than twice optimal for n less than seven trillion.01/1986; - SourceAvailable from: So-Hsiang Chou[Show abstract] [Hide abstract]

**ABSTRACT:**Given the anisotropic Poisson equation ¡r ¢ Krp = f, one can convert it into a system of two first order PDEs: the Darcy law for the flux u = ¡Krp and conservation of mass r ¢ u = f. A very natural mixed finite volume method for this system is to seek the pressure in the nonconforming P1 space and the Darcy velocity in the lowest order Raviart-Thomas space. The equations for these variables are obtained by integrating the two first order systems over the triangular volumes. In this paper we show that such a method is really a standard finite element method with local recovery of the flux in disguise. As a consequence, we compare two approaches in analyzing finite volume methods (FVM) and shed light on the proper way of analyzing non co-volume type of FVM. Numerical results for Dirichlet and Neumann problems are included.01/2001;

Page 1

MODELLING OF SEMICONDUCTOR DEVICES

FOR ICS AND VLSI

A Thesis

Presented to the Graduate School

Faculty of Engineering, Alexandria University

In Partial fulfillment of the

Requirements for the Degree

Of

Master of Science

In

Electrical Engineering

By

Sameh Ahmed Yousry

2006

Page 2

MODELLING OF SEMICONDUCTOR DEVICES

FOR ICS AND VLSI

Presented by

For the Degree of

Master of Science

In

Electrical Engineering

By

Sameh Ahmed Yousry

Examiners’ Committee Approved

Prof. Ahmed K. Abouelsoud ……………..

Prof. Hany Fekry Ragaie ………..…...

Prof. Mohamed I. Elbanna ……………..

Page 3

Advisor’s Committee

Prof. Mohamed Ismail Elbanna

Dr. Tawfik Nammour

Page 4

ACKNOWLEDGMENTS

I would like to give my deep thanks to Prof. Mohamed Elbanna who initiated this work

and whose persistent guidance was behind its completion.

The members of the evaluation committee are deeply thanked for giving their

invaluable time to read and evaluate this thesis.

I would like to express my gratitude to Prof. S. Micheletti and Prof. F. Brezzi,

Universita di Pavia, who helped me in the numerical solution part of the DD model.

Their guidance was of an important help in the completion of this work.

I would like to thank Prof. S. Holst, Universitat Trier, for his help when I faced real

problems with the solution of Poisson's equation. He gave me the clue for obtaining

the suitable Jacobian of the system.

Many thanks to my friend Abd Elhamid Salah Eldin for his help in working with the

Visual C++ tool.

I owe many thanks to all my family, especially my parents for their continuing

sacrifice, support, and encouragement during the course of my research.

Finally, I am grateful to my wife, for her unwavering support and her patience while I

was spending many hours working on my PC.

i

Page 5

ABSTRACT

This thesis is concerned about Simulation of semiconductor devices. Simulation of

semiconductor devices aims at solving the semiconductor partial differential equations

(PDE) for different device geometry and parameters. Device simulators help the device

engineer to create, test and verify novel structures. This increases our physical knowledge

without even going to the laboratory. In the era of the ever-decreasing device structures

simulation methods must be validated and the device engineer must rely on it, since the

cost of manufacturing a device to test is high.

This thesis discusses the requirements needed to implement such a simulator. Several

computational challenges and how they can be solved are presented. It was the aim of this

thesis to implement a device simulator using an object-oriented programming (OOP)

methodology. OOP is a modern programming approach which makes it easy to increase

the program capabilities while keeping the program neat and tight.

The thesis was divided into two main components: geometry and computation. The

geometry component deals with the geometrical aspects ( geometry intersection, point

inclusion, mesh generation and neighborhood adjacency relationships ). We used

Semiconductor Wafer Representation (SWR)model, the well known standard by

Stanford, for implementing the geometry component. SWR does not provide details about

the implementation, it mainly concerned about giving the interface. We tried to keep

close to the standard as we could. Since our simulator is designed to be used for different

device structures, It is important for the mesh generation procedures to deal with

irregularities in the device. Triangles are chosen as the mesh elements. Since we use

OOP, a triangle is inherited from a more abstract polygon class so that different polygonal

elements can be used as well. The simulator is implemented to deal with polygonal

devices but, curved boundaries can be added by inheriting a curve from the basic Edge

class as well.

The geometry algorithms: point inclusion, region intersections, storage of device, etc..

are chosen to be efficient and general and that is the real challenge since, efficiency

doesn't always go well with generality. Therefore, we are biased to generality when

generality is a must, otherwise efficiency has the higher priority. Since the variables

(electric field, electric current, potentials,..etc..) may change with different orders of

magnitude in small regions, adaptive mesh is required. A quad-tree algorithm is used for

generating the mesh. The algorithm goes by spatially decomposing the device domain

into squares when the level of a nodal point is higher than the level of the square. This

will

ii

Page 6

generate an adaptive spatial decomposition. Triangulation in the interior of the device is

done using certain patterns. The boundary zone (known as delta zone) is isolated from the

device region and triangulated using Constrained Delaunay Triangulation ( CDT )

algorithm. CDT is a complex and expensive (from the computational point of view )

algorithm. Fortunately, it is used only in the delta-zone.

The computational component is a real challenge as well. For different device

structures as well as in different modes of operation certain parameters and functions may

dominate while others may be negligible. The simulator has to deal with this. Different

mobility, recombination and generation models can be incorporated to the simulator. The

simulator user has to be free to use the model he wants. Not only that, but the user can

even implement his own models and test them. Again, OOP plays the important role here.

Models are inherited from a general, abstract function class. The user can do what ever he

wants inside the function, but since all functions have the same interface, the simulator

will not distinguish between them.

There are different device models, Hydrodynamic, Energy Transport, Drift-

Diffusion,..etc.. Even the Drift-Diffusion model, which is the simplest model, is complex

in its own. The Drift-Diffusion model consists of three coupled nonlinear partial

differential equations. There are different discretization schemes for discretizing the

system of equations. We used a hybrid mixed finite element scheme. This scheme gives

us some sort of continuity for the electric flux ( required when using different materials )

and current density. It is common for a practical device consisting of different sub-regions

to be discretized using thousands of elements. This will lead to thousands of variables and

millions of matrices elements. Fortunately, the finite element scheme is sparse in nature.

Using different sparse techniques depending on the nature of the matrix is another use of

OOP. Nonlinear Poisson's equation is linearized. The linearized system was solved using

a global Newton method algorithm. Global Newton method converges to the exact

solution whatever the initial guess is.

Different results, a PN junction and a MOS device are shown. Different channel

lengths are entered to the simulator and the I-V characteristics are drawn. The results

shown are highly compatible with the analytical formulae.

MATLAB was originally used for displaying the geometry, mesh as well as plotting

the I-V characteristics. Another version is implemented which uses the windows

application interfaces for entering the device, setting the different parameters as well as

generating the mesh.

SEMILAB is the result of this thesis. It is a mini simulator, which shows how

sophisticated a device simulator is.

iii

Page 7

TABLE OF CONTENTS

ACKNOWLEDGEMENT........................................................................................i

ABSTRACT............................................................................................................ii

TABLE OF CONTENTS.........................................................................................iv

LIST OF FIGURES.................................................................................................vii

LIST OF SYMBOLS...............................................................................................x

LIST OF ABBREVIATIONS..................................................................................xii

CHAPTER 1 INTRODUCTION ............................................................................1

1-1 Device Models...................................................................................1

1-2 Semiconductor Wafer Representation.................................................2

1-3 General Features................................................................................3

1-4 Geometry Server................................................................................3

1-5 Grid Server........................................................................................3

1-6 The Solver.........................................................................................4

CHAPTER 2 THEORETICAL ASPECTS OF

GEOMETRY AND GIRD SERVERS.................................................5

2-1 Introduction......................................................................................5

2-2 SWR information model....................................................................5

2-3 Geometry Server...............................................................................6

2-3-1 Boundary Orientation................................................................9

2-3-2 Point Inclusion Algorithm.........................................................10

2-3-3 Intersection detection Algorithm...............................................11

2-4 Grid Server.......................................................................................12

2-4-1 Quad-tree generation...............................................................12

2-4-1-1 Quad-tree algorithms.........................................................14

2-4-2 Delta-zone detection................................................................18

2-4-3 Interior triangulation................................................................20

2-4-4 Delta-zone triangulation...........................................................20

2-4-5 Divide and Conquer Delaunay triangulation..............................21

iv

Page 8

CHAPTER 3 THE DRIFT-DIFUSSION MODEL....................................................33

3-1 Introduction.......................................................................................31

3-2 The DD system of equations..............................................................32

3-3 Change of variables............................................................................33

3-4 Mobility and recombination-generation rates......................................33

3-5 Scaling...............................................................................................34

3-6 Boundary conditions...........................................................................35

3-7 Historical Perspective.........................................................................37

CHAPTER 4 NUMERICAL SOLUTION

OF THE DD-MODEL..........................................................................39

4-1 Introduction.......................................................................................39

4-2 The DD-model...................................................................................39

4-3 Nonlinear iterations............................................................................40

i. Gummel Iteration method.................................................................40

ii. Full Newton method.........................................................................42

4-4 Simulator challenges...........................................................................42

4-5 Discretization techniques....................................................................43

4-5-1 Finite difference method............................................................43

4-5-2 Finite element discretization......................................................44

4-5-3 Galerkin finite element method..................................................44

4-5-4 Notation...................................................................................45

4-5-5 Mixed formulation.....................................................................46

4-5-6 Discretization of the continuty equations...................................48

4-5-7 Examples of mixed finite elements.............................................49

4-5-8 Hybridization of the mixed finite formulation.............................52

4-5-9 Algebraic treatment of problem (4-5-47 )..................................53

4-5-10 Application of the continuity equation.....................................57

4-6 Laplace and Poisson equations.........................................................62

4-7 Global approximate Newton method...............................................65

CHAPTER 5 SOFTWARE IMPLEMENTATION

OF THE DD MODEL.........................................................................66

5-1 Introduction....................................................................................66

5-2 Model architecture..........................................................................67

i. Functions.......................................................................................67

ii. Matrix............................................................................................69

iii.Gauss Quadrature..........................................................................72

iv.Linear Solver.................................................................................72

v. Gummel Class................................................................................72

5-3 Simulation results............................................................................73

i. PN results......................................................................................73

ii. MOSFET results...........................................................................76

v

Page 9

CHAPTER 6 CONCLUSTION AND FUTURE WORK............................................81

APPENDEX A. OBJECT ORIENTED PROGRAMMING........................................84

APPENDEX B. SEMILAB........................................................................................86

REFERENCES..........................................................................................................98

vi

Page 10

LIST OF FIGURES

1. Figure 1-1 Device Models hierarchy..............................................................1

2. Figure 1-2 SEMILAB Information Model......................................................2

3. Figure 2-1 SWR Information Model.............................................................6

4. Figure 2-2 Example of CSG..........................................................................6

5. Figure 2-3 B-rep geometry model..................................................................7

6. Figure 2-4 B-rep MOSFET model.................................................................7

7. Figure 2-5 Two dimensional tree...................................................................8

8. Figure 2-6 Point Inclusion.............................................................................10

9. Figure 2-7 Point inclusion degenerate case....................................................10

10. Figure 2-8 A Quad and its children...............................................................12

11. Figure 2-9 A Typical Quad-tree....................................................................12

12. Figure 2-10 Boundary parameters...................................................................14

13. Figure 2-11 Example of a Quad-tree...............................................................15

14. Figure 2-12 Element container........................................................................16

15. Figure 2-13 Hypothetical device geometry......................................................17

16. Figure 2-14 Quad-tree of the hypothetical device............................................17

17. Figure 2-15 Delta-zone of leaf quads..............................................................18

18. Figure 2-16 Slim elements near the boundary..................................................18

19. Figure 2-17 Improvement of delta-zone method..............................................19

20. Figure 2-18 Delta-zone of the hypothetical device...........................................19

21. Figure 2-19 Interior triangulation pattern.........................................................20

vii

Page 11

22. Figure 2-20 Triangles adjacency.......................................................................21

23. Figure 2-21 Edge pointer.................................................................................21

24. Figure 2-22 Vertices in the plane......................................................................21

25. Figure 2-23 Decomposition of the vertices.......................................................22

26. Figure 2-24 Merging between halves...............................................................22

27. Figure 2-25 Potential and Next potential candidate..........................................23

28. Figure 2-26 In-Circle test................................................................................23

29. Figure 2-27 In-Circle test conditions...............................................................24

30. Figure 2-28 In-Circle test for the two halves..................................................24

31. Figure 2-29 Merging of halves........................................................................25

32. Figure 2-30 In-Circle algorithm.......................................................................25

33. Figure 2-31 Adding constrains........................................................................26

34. Figure 2-32 Delaunay triangulation of the hypothetical device.........................27

35. Figure 2-33 Delta-zone triangulation of the hypothetical device......................27

36. Figure 2-34 Triangulation of the hypothetical device.......................................28

37. Figure 2-35 MOSFET device..........................................................................28

38. Figure 2-36 MOSFET Quad-tree....................................................................29

39. Figure 2-37 Delta quads of MOSFET.............................................................29

40. Figure 2-38 Delta zone regions.......................................................................30

41. Figure 2-39 Triangulation of MOSFET...........................................................30

42. Figure 3-1 Boundary conditions....................................................................36

43. Figure 4-1 Finite difference...........................................................................43

44. Figure 4-2 Barycenter coordinates................................................................45

45. Figure 4-3 Triangle parameters.....................................................................50

46. Figure 5-1 SWR-Solver interaction...............................................................67

47. Figure 5-2 Function Class hierarchy..............................................................68

48. Figure 5-3 Matrix Class hierarchy.................................................................69

49. Figure 5-4 Sparse matrix data structure........................................................70

50. Figure 5-5 Matrix object Data-hiding............................................................71

51. Figure 5-6 PN device....................................................................................73

viii

Page 12

52. Figure 5-7 PN triangulation..........................................................................73

53. Figure 5-8 Thermal equilibruim....................................................................74

54. Figure 5-9 Electrons concentation in the thermal equilibruim case................75

55. Figure 5-10 IV characteristics for PN junction...............................................75

56. Figure 5-11 MOSFET device.........................................................................76

57. Figure 5-12 MOSFET triangulation................................................................76

58. Figure 5-13 VGS=0.4V and VDS=0 .......................................................77

59. Figure 5-14 Different biasing voltages............................................................77

60. Figure 5-15 IV characteristics of the MOSFET...............................................78

61. Figure B-1 A shot of SEMILAB classes.........................................................86

62. Figure B-2 MATLAB front-end.....................................................................87

63. Figure B-3 SEMILAB IDE.............................................................................87

64. Figure B-4 Start Drawing...............................................................................88

65. Figure B-5 Entering the number of regions.....................................................89

66. Figure B-6 Type and sub-regions selection.....................................................90

67. Figure B-7 Device Structure...........................................................................91

68. Figure B-8 Drawing the device..........................................................................92

69. Figure B-9 Doping Level...................................................................................93

70. Figure B-10 Boundary Conditions.....................................................................94

71. Figure B-11 Mesh parameters...........................................................................95

72. Figure B-12 The mesh generated.......................................................................96

73. Figure B-13 Global Newton Method.................................................................97

ix

Page 13

LIST OF SYMBOLS

SymbolDescription

δ

E

Deltazone value

Electric Field

Electrical Dielectric constant

ε

n

p

Electrons concentration density

Holes concentration density

Jn

Jp

Electrons Current density

Holes Current density

Total charge density

ρ

µn

Electrons mobility

µp

Dn

Dp

Holes mobility

Electrons diffusion coefficient

Holes diffusion coefficient

k,Coh

Doping Level

ψ

Potential Energy

φn

Electrons imref

φp

Holes imref

ρn

Electrons Slotboom variable

ρp

RSHR

RAU

RII

R

Vth

Holes Slotboom variable

Shockley-Hall-Reed recombination

Auger recombination

Impact ionization recombination - generation

Total recombination- generation

Thermal voltage

x

Page 14

Symbol

ni

LD

Description

Intrinsic concentration level

Intrinisic Debye length

σ

Flux vector

τ

Vector finite element

¯ nj

¯ tj

Normal unit vector to the edge j

Tangent unit vector to the edge j

L2(Ω)

M

Sobolev space

Stiffness Matrix

Barycenter coordinate for the vertex i

λi

Th

Triangulation of the device region

xi

Page 15

CHAPTER 1

INTRODUCTION

1-1 Device Models

The device simulator aims at solving the semiconductor partial differential equations

(PDE) for different device geometry and parameters. Device simulators help the device

engineer to create, test and verify novel structures. This increases our physical knowledge

without even going to the laboratory. Device simulators are not only useful for advanced

device engineers, but undergraduate students can benefit from using the simulator. Indeed,

the student can understand more about the theoretical device aspects by logging to a

simulator and visualizing how carriers move, electric field, breakdown voltages,..etc.

Semiconductor devices can be modeled using the well known Boltzmann Transport

Equation (BTE). BTE is a nonlinear partial differential equation (PDE) in a probability

distribution function (PDF) which characterizes the device region. Taking the moments of

the PDF provides us with the carriers concentrations as well as other data [1]. However the

solution of BTE is not easy. Monte Carlo simulation was used to solve such a problem as

shown in [1]. By taking the moments of BTE a much more suitable model is derived,

namely the Hydrodynamic model. The Energy transport (ET) is derived from the

Hydrodynamic model. ET can be simplified further to the Drift-Diffusion model (DD). The

following figure (Figure 1-1) shows to us the hierarchy of the device models[1]. It worth to

mention that there is a parallel model hierarchy , the quantum models , quantum correction

factors together with Schrodinger equation are coupled to the classical models shown in the

figure (Figure 1-1).

Figure 1-1 Device Model hierarchy

1

Boltzmann Transport Equation

Energy Transport model

Hydrodynamic model

Drift-Diffusion model