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.
- Citations (22)
- Cited In (0)
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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