Citation
Adaptive simulation of dopant implantation and diffusion for advanced semiconductor process simulation

Material Information

Title:
Adaptive simulation of dopant implantation and diffusion for advanced semiconductor process simulation
Creator:
Lin, Chih-Chuan ( Dissertant )
Law, Mark E. ( Thesis advisor )
Burk, Dorothea E. ( Reviewer )
Fossum, Jerry G. ( Reviewer )
Fox, Robert M. ( Reviewer )
Davis, Tim A. ( Reviewer )
Lowther, Rex E. ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1995
Language:
English
Physical Description:
viii, 237 leaves : ill. ; 29 cm.

Subjects

Subjects / Keywords:
Dosage ( jstor )
Estimators ( jstor )
Grid generation ( jstor )
Interpolation ( jstor )
Modeling ( jstor )
Oxidation ( jstor )
Oxides ( jstor )
Silicon ( jstor )
Simulations ( jstor )
Triangles ( jstor )
Dissertations, Academic -- Electrical Engineering -- UF
Electrical Engineering thesis, Ph. D
Semiconductors -- Mathematical models ( lcsh )
Semiconductors -- Simulation methods ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
Modern numerical process simulators are becoming increasingly complicated in both physical models and domain shape. Grid generation is difficult for these simulators because of the inherent transient nature of the problems being solved. This work addresses adaptive grid refinement for use in solving diffusion problems. It additionally investigates higher order approximations to the discretized diffusion flux. Several methods of both adaptive grid refinement and discretization are investigated and compared in terms of CPU time and final discretization error. A new nodal error estimator is implemented and benchmarked. This allows fully functional mesh generation as well as mesh adaptation capability in two-dimensions. The global smoothing and solution interpolation scheme are used to restore the grid quality after performing local refinement. All the methods are directly applied to the Florida object Oriented Process Simulator (FLOOPS). The entire two-dimensional adaptive algorithm for simulation dopant in ion implantation and dopant simulation is applied to the simulation of a CMOS twin-tub process. This adaptive approach allows the spatial discretization under control so that the solution accuracy can be ensured. The automatic mesh generation provides the user friendly interface. The optimal mesh in adaptive simulation results in the efficient computation.
Thesis:
Thesis (Ph. D.)--University of Florida, 1995.
Bibliography:
Includes bibliographical references (leaves 223-236).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Chih-Chuan Lin.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
021602854 ( ALEPH )
33393615 ( OCLC )
AKN3378 ( NOTIS )

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ADAPTIVE SIMULATION OF DOPANT IMPLANTATION AND DIFFUSION
FOR
ADVANCED SEMICONDUCTOR PROCESS SIMULATION















BY

CHIH-CHUAN LIN


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1995
















ACKNOWLEDGEMENTS


I wish to express my deepest gratitude to Dr. Mark E.

Law, the chairman of my supervisory committee, for his

support during my research. I also want to thank Dr. D. E.

Burk, Dr. Jerry G. Fossum, Dr. Robert M. Fox, and Dr. Tim

Davis, members of my supervisory committee, for their time

and advice.

Special thanks should go to Dr. Rex Lowther for his

insightful discussion on discretization method and error

estimation. Also, I like to thank Tim Crandle for the experi-

ence of integrating my work into ATHENA/SSPREM in Silvaco

International.

I would like to thank my parents and the people from

Happy Hour.

This work is supported by SRC, the Semiconductor

Research Corporation.
















TABLE OF CONTENTS

page

ACKNOWLEDGEMENTS ........................................ ii

ABSTRACT ................................................ vii

CHAPTERS ..............................................

1 INTRODUCTION .................................... 1

1.1 Motivation and Challenge.................... 1

1.2 Predictive Design for VLSI Fabrication..... 5

1.3 Overview of Adaptive Simulation............ 7

1.4 Organization ............................ 14

2 PHYSICS AND MODELING FOR PROCESS INTEGRATION ... 19

2.1 Introduction ............................. 19

2.2 Etch and Deposition ...................... 21
2.2.1 The Etch Process in Modern VLSI
Technology ..................... 22
2.2.2 The Deposition Process in Modern
VLSI Technology ................. 24
2.2.3 The String Algorithm for Modeling
Etch and Deposition............... 25

2.3 Ion Implantation ......................... 26
2.3.1 Modeling the Ion Implantation ..... 27
2.3.2 Deviation form Gaussian
Distribution .................... 28
2.3.3 The Ion Implant Modeling in FLOOPS. 30

2.4 Dopant Diffusion ......................... 31
2.4.1 Modeling for the Dopant Diffusivity 32
2.4.2 Fair's Model for Dopant-defect Pairs
Diffusion......................... 33
2.4.3 Modeling of Oxide Enhanced
Diffusion......................... 34
2.4.4 Modeling of Transient Enhanced
Diffusion ...................... 36

iii










2.4.5 Advanced Diffusion Modeling in
FLOOPS ..........................

2.5 Oxidation ...............................
2.5.1 Deal-Grove Model for Oxidant
Diffusion and Reaction ..........
2.5.2 The Oxide Flow Characteristics ....
2.5.3 Factors Influencing Oxidant
Diffusivity ......................
2.5.4 Oxidation and Silicidation in FLOOPS

2.6 Conclusion ...............................

3 THE SOFTWARE SYSTEM ............................

3.1 Introduction .............................

3.2 The Implementation of Field Server ........

3.3 The System Structure of FLOOPS ...........
3.3.1 The Building Blocks ...............
3.3.2 The Implementation of Adaptive
Spatial Discretization ..........
3.3.3 The Implementation of Flux
Discretization Methods ..........

3.4 Conclusion ...............................

4 SPATIAL DISCRETIZATION AND FLUX DISCRETIZATION
FOR DOPANT DIFFUSION IN ONE-DIMENSION ...........

4.1 Introduction .............................

4.2 Grid Refinement Schemes ...................
4.2.1 Contour Refinement Method .........
4.2.2 Heuristic Method Based on


4.2.3
4.2.4


4.3 Higher
t- ir-n


Concentration Ratio of Adjacent
Nodes ..........................
Refinement Based on Error Estimator
Comparison of Different Autogrid
Schemes .........................

Order Correction of Flux Discretiza-


4.3.1 Different Flux Discretization
Schemes .........................
4.3.2 Comparison of Different Flux
Discretization Schemes ..........


4.4 Conclusion ...............................










5 LOCAL REFINEMENT IN TWO-DIMENSIONS ............. 88

5.1 Introduction ............................. 88
5.2 The Error Source ........................... 89
5.2.1 The Evaluation of Line Integral ... 91
5.2.2 The Evaluation of Area Integral ... 93

5.3 Two-dimensional Nodal Error Estimator...... 96

5.4 Numerical Performance of Nodal Error Esti-
mator ........................ ............ 103

5.5 Conclusion ........... ................... 109

6 MESH GENERATION AND ADAPTION WITH MOVING BOUNDARY
IN TWO-DIMENSIONS ........................... 110
6.1 Introduction ............................ 110

6.2 Higher Order Flux discretization in two-di-
mension .................................... 111
6.3 Local Refinement Strategies .............. 113
6.3.1 The Numerical Error due to Obtuse
Triangle ....................... 113
6.3.2 Element-based Local Refinement
Approach ........................ 117
6.3.3 Node-based Local Refinement
Approach ........................ 119
6.4 Automatic Mesh Generation during Implant .. 123
6.5 Mesh Adaption during Diffusion ........... 129
6.6 Mesh Adaption with Moving Boundary ....... 134
6.7 Mesh Smoothing and Solutions Interpolation 137
6.8 Conclusion ............................... 141

7 GRID QUALITY ISSUES ON LOCAL REFINEMENT......... 144
7.1 Introduction ............................. 144

7.2 Grid Quality Issues for Local Refinement... 146

7.3 Example ................................ 151
7.4 Conclusion ............................... 162

8 EXAMPLE ........................................ 165
8.1 Introduction ............................. 165

8.2 Simulation of the Twin-tub CMOS Process ... 166










8.3 The Formation of Twin-tub and Isolation
Structure ................................. 169

8.4 The NMOS and PMOS Structures ............ 193
8.4.1 NMOS Device ...................... 193
8.4.2 PMOS Device ....................... 204

8.5 Conclusion ............................... 204

9 SUMMARY AND FUTURE WORK.......................... 207

9.1 Summary .................................... 207

9.2 Recommendation for Future Work ............ 212

APPENDIX A THE COMPLETE CMOS STRUCTURE BY A TWIN-WELL 216
PROCESS ................................
221
APPENDIX B THE MACHINE DEFINITIONS FOR THE TWIN-TUB
CMOS PROCESS ............................ 223

REFERENCES .................... ....................... 237

BIOGRAPHICAL SKETCH ..................................
















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


ADAPTIVE SIMULATION OF DOPANT IMPLANTATION AND DIFFUSION
FOR ADVANCED SEMICONDUCTOR PROCESS SIMULATION

By Chih-Chuan Lin

May 1995

Chairman: Dr. Mark E. Law
Major Department: Electric Engineering

Modern numerical process simulators are becoming

increasingly complicated in both physical models and domain

shape. Grid generation is difficult for these simulators

because of the inherent transient nature of the problems

being solved. This work addresses adaptive grid refinement

for use in solving diffusion problems. It additionally

investigates higher order approximations to the discretized

diffusion flux. Several methods of both adaptive grid

refinement and discretization are investigated and compared

in terms of CPU time and final discretization error. A new

nodal error estimator is implemented and benchmarked. This

allows fully functional mesh generation as well as mesh

adaption capability in two-dimensions. The global smoothing

and solution interpolation scheme are used to restore the

grid quality after performing local refinement. All the

methods are directly applied to the FLorida Object Oriented

Process Simulator (FLOOPS).









The entire two-dimensional adaptive algorithm for sim-

ulation dopant in ion implantation and dopant simulation is

applied to the simulation of a CMOS twin-tub process. This

adaptive approach allows the spatial discretization under

control so that the solution accuracy can be ensured. The

automatic mesh generation provides the user-friendly inter-

face. The optimal mesh in adaptive simulation results in

efficient computation.















CHAPTER 1
INTRODUCTION


1.1 Motivation and Challenges


Modern numerical process simulators have been used

extensively for minimizing the dopant diffusion to produce

the shallow junctions required for scaled VLSI technology or

the optimization of new devices [1]. Since the device size

has been reduced below one micron, the two-dimensional

effect of dopant distribution, e.g. the lateral diffusion of

source/ drain, becomes significant. A two-dimensional pro-

cess simulator is capable of producing the two-dimensional

device structure (impurity profile and device geometry)

according to the process recipes. This will allow the device

engineer to investigate and test the electrical characteris-

tics of a newly designed device via a device simulator based

on real process input, e.g. dopant implantation, thermal

anneal temperature, oxidation, etc. With this capability of

producing realistic device structures, the device engineer

can look at interesting device problems, e.g. CMOS device

latchup [2], which is critical for VLSI circuit performance.

As a result of scaling of the device, the three-dimen-

sional effects are also becoming important. The narrow width

effects in determining the latchup immunity was reported

[3]. This is mainly due to the three-dimensional current









flow at the corner of the source/ drain of CMOS devices. For

the purpose of simulating and characterizing this effect in

device simulation, a three-dimensional device simulation is

required as well as a three-dimensional process simulator

for producing the three-dimensional device structure. There

are many three-dimensional device simulators [4]-[9], but no

three-dimensional general purpose process simulator. Some

work regarding coupling two-dimensional device simulator to

two-dimensional or one-dimensional process simulator have

been reported [10]-[11]. For a three-dimensional device sim-

ulator, a two-dimensional process simulator can be linked in

a similar fashion to support the three-dimensional device

structure. This approach allows simulation of the quasi-

three-dimensional effects. Whether or not this approach is

sufficient for simulating the latchup problem in three-

dimensions remains to be seen. The robustness of the modern

two-dimensional process simulator should be reexamined and

the physics and numerical performance should be improved in

order to cope with the requirement of simulating three-

dimensional device structures. The experience on the effort

for improving the two-dimensional process simulator will

also help the development of robust three-dimensional pro-

cess simulators.

For the motivation of this work, an inverter fabricated

in a twin-tub CMOS process is shown in Fig. 1.1. The device

structure includes the twin-wells, source/drain implant for

NMOS and PMOS, channel stop implant and oxide for isolation,

substrate contacts, poly gate, and double metal. The sche-

matic circuit of latchup is also shown in the twin-wells and















Vin


Vout



im 2 .m 4 im
--- 1_. ..-1--


Figure 1.1


The inverter in circuit schematic diagram
and its structure representation in VLSI
semiconductor process.


Vcc


41J


GND


-V









substrate. In order to understand the latchup immunity of

this circuit, the conventional approach is to build the

device in the laboratory and perform electrical measure-

ments. Then device parameter extraction [12] can be used to

model this device for circuit simulation. An alternative

approach is to prepare the structure by process simulation

and use the device simulator to characterize the device for

the circuit simulation. The later approach is inexpensive

and efficient. The challenges for this approach in the area

of process simulation are as follows:

1) The accurate representation of the nonplanar device

geometry and the accurate representation of multiple

dopant profiles after ion implantation and thermal

anneal.

2) The advanced physics for modeling the phenomenon of

interest and to link this process information to the

device characteristic, e.g. leakage current, latchup

immunity, etc.

3) The user environment for generation of adequate grid

for the representation of this complicate structure.

Adequate computational grid for complicate struc-

tures is not accessible manually (need automatic

refinement based on some error criteria.)

4) For the dopant representation through all the ther-

mal process steps, the adaptive capability is

required for the boundary representation and the

dopant redistribution for each time step in the sim-

ulation.









5) An efficient approach to simulate this large struc-

ture (10 x 6 pm). With 100 A fix grid spacing which

is insufficient, this structure would require total

grid points of (1000x600).



1.2 Predictive Design for VLSI Fabrication


Due to the dramatic scaling of devices, the design of a

new technology becomes increasingly difficult in that the

coupling between the process conditions, device characteris-

tics, and the circuit performance complicate the advance of

VLSI technology. For instance, the requirement of high pack-

ing density for Dynamic RAM (DRAM) circuits results in

advanced technology of the trench capacitor. However, the

leakage current [13] via the dielectric layer in this tech-

nology becomes the dominant factor to determine the circuit

performance. The design for optimal performance of this

device then becomes nontrivial. The traditional approach of

design by fabrication via trial-and error is expensive in

term of laboratory cost and the design turn-around time.

With the emergence of Technology for Computer Aided

Design (TCAD,) a different design methodology become effi-

cient in technology development for IC fabrication. The

basic building blocks for the TCAD design methodology is

shown in Fig. 1.2. The process simulator (e.g. FLOOPS [14])

allows the optimization of the process conditions to produce

the desired device structure. The device simulator (e.g.

FLOODS [14] can be used to characterize the newly designed

device to produce a compact model which is to be implemented















process steps _._ Process Simulator
(FLOOPS)


device structure
(geometry, dopant,...)



Device Simulator
I, V,
(FLOODS)



I-V characteristics
parameter extraction


Circuit Simulator
I, V,
(SPICE)



Circuit performance
(frequency response...)




Figure 1.2 The basic building blocks for the IC pre-
dictive design methodology.


in the circuit simulator. Thereafter, the circuit simulation

(e.g. SPICES [15]) will be performed with these well-char-

acterized device models. The system performance will be the

output of this predictive design methodology. This utiliza-

tion of TCAD to develop new technology has reduced the cost

and turn-around time. In this TCAD methodology, process sim-

ulation becomes increasingly important due to the device









scaling. The usage of an analytical solution for dopant

specification as the input for device simulation will be

inadequate to characterize some 2D and 3D effects. The pre-

dictive design as shown in Fig. 1.2 becomes critical for

accurate and efficient design methodology.

TCAD for predictive design is not just helpful for

design of advanced technology, but it's necessary for many

specific applications. For the circuit in Fig. 1.3, one DRAM

cell with an application circuit and the device structure

are shown. The stress effect upon the capacitor on the edge

of LOCal Oxidation of Silicon (LOCOS,) is important to iden-

tify the characteristic of leakage current [16]. Since the

leakage current or the stress effect are three-dimensional,

a three-dimensional process simulator will be critical in

the near future. Moreover, with complicated structures like

this, the design through fabrication will need lots of

steps, which is expensive and has long turn around.



1.3 Overview of Adaptive Simulation


Over the past 25 years, adaptive simulation has been an

active research area [17] in the finite element method for

computing problems such as fluid mechanics [18], material

property analysis [19], electromagnetism [20], etc. The

quantitative error estimator [21]-[23], automatic error con-

trol (adaptivity) [24]-[25], and automatic mesh generation

[26]-[27] are the most important subjects under investiga-

tion for the application of this approach in different
















word line


access transistor

F-n-

storage
capacitor


Vcc


Vout -.:- GND


bit line


4 pm 2 tm
I- L- -I-


-I I -I


4 pm


access transistor


n+ substrate


storage node


storage plate


Figure 1.3 The inverter and one DRAM cell in circuit
schematic diagram and it's structure repre-
sentation in VLSI semiconductor process.










fields of science and engineering. The basic algorithm of

adaptive approach for solving a physical problem is illus-

trated in Fig. 1.4. The initial mesh generation will provide






Mesh Generation








error estimation
and
grid Adaption
next
iteration





Solver










Figure 1.4 The basic algorithm of adaptive approach
for solving a physical problem.




the optimal grid for representation the initial solution.

The grid refinement is used to adapt the grid according to

the newly computed solution so that the new grid will repre-









sent the current solution best. The solver is to solve the

Partial Differential Equations system which incorporates the

physics into the computational grid.

There are several different approaches for grid refine-

ment: h refinement [28], p-refinement [29] and h-p refine-

ment [30]. The h-refinement improves the solution accuracy

by refining the element (e.g. subdivide a triangular element

into four triangular sub-elements.) while the p-refinement

increases the accuracy of solution by increasing the order

of the approximation shape function. The h-p refinement is a

hybrid approach including both the h-refinement and p-

refinement in the refinement scheme. The purpose of imple-

menting an adaptive approach is to produce the optimal mesh

for representing the solution of interest automatically at

all time through the numerical computation. This will allow

the usage of minimum grid size to achieve the required accu-

racy in the minimum time. Shephard [31] discussed the sub-

ject of this adaptive finite element analysis and CAD. The

idea is suitable for application in all disciplines of engi-

neering.

The numerical simulation of VLSI technology has been an

area of intensive research. This is mainly due to the scaling

of device dimensions. An accurate simulator for semiconduc-

tor devices and processes becomes indispensable in designing

new technology. Together with improvement in physical model-

ing, the accuracy of computational results is strongly

related to the underlying grid and discretization scheme for









the partial differential equations to be solved. In order to

meet the requirement for robustness, the adaptive finite

element method has been introduced to the semiconductor sim-

ulation area. With the grid adaption ability, the numerical

simulation becomes less memory and time intensive.

The adaptive grid method has been applied to device sim-

ulation [32] to simulate CMOS latchup and better discretiza-

tion schemes for drift-diffusion equations have also been

studied [33], [34]. In the area of DC device simulation, sev-

eral good algorithms for mesh adaption have been proposed

[35]-[37]. Bank and Weiser [38] have developed a means of

estimating the local spatial truncation error which has been

used successfully to develop a refinement strategy for

device simulators [32]. This work can be used as a base to

develop refinement algorithms for process simulation.

For process simulation, the mesh adaption is complicated

by the moving boundaries of oxidation/ silicidation process

steps and the time dependent nature of the dopant profile.

Some work on simulation of dopant diffusion using an adap-

tive mesh scheme has been published by Pichler et al. [39]

and Ismail and Amaratunga [40], however without addressing

the ability to simulate complex microstructure such as

LOCOS. Other researchers have presented mesh generation

ability [41]-[43] to resolve the geometry representation for

multilayer structure but lack the emphasis on the solving of

dopant diffusion simultaneously via discretization error









control. For the purpose of implementing a robust process

simulator, both features need to be considered.

Mesh generation for process simulation is a difficult

task, complicated by both the moving boundaries and the time

dependence of the solutions. Especially for multilayer

process simulation [41], [42], different requirements of

grid spacing are needed for different areas of the

structure. Many existing grid generation methods for devices

simulation [44] are not necessarily applicable to grid

generation in process simulation due to the different

requirements for initial grid. The grid generation in device

simulation generally is to fit device geometries and

describe junctions while the process simulation requires

grid generation to represent the dopants. For both

computational and ease of use reasons, automatic mesh

generation and discretization error control is desirable.

A modern process simulator must cope with the gradually

increasing complexity of physical models [45]-[47].

Recently, research has been dedicated to the improvement of

the physical models for dopant diffusion. The contribution

of these advanced physical models is better modeling of both

defect and dopant diffusion. However, these improved models

also significantly increase the amount of CPU time required.

As device dimensions shrink, it is also necessary to model in

two and three dimensions. The improvement of computational

efficiency, therefore, is an important task for all simula-

tion tools. The application of adaptive grid refinement in









solving dopant diffusion will allow optimal grid throughout

the transient simulation.

In general, the application of an adaptive approach in

the semiconductor area is critical due the complexity of the

device structure and the upcoming requirement of three-

dimensional process simulation capability. In the device

simulation, most mesh generation is for the boundary

description of the device. However, the generated mesh

should provide dopant information as well as describe the

device geometry. To meet this requirement, the mesh genera-

tion can utilize the self-adaptive approach to refine

according to the dopant [48]. For device simulation [32],

the mesh can be regenerated for each bias point according to

solutions, i.e current, potential. However, the generated

structure is not from the real process flow. It's then neces-

sary to incorporate process simulator for supporting the

real device structure from process conditions.

To cope with the same problem, mesh generation and

adaption requirement for process simulation is not like that

of device simulation. Its time dependency requires the

update of grid in each time-step. The moving boundary and

surface evolvement nature complicate the geometry represen-

tation of a device. The unstructured grid [49], [50] will

allow better representation for nonplanar structure over the

structured grid [43], [51], [52]. For the grid adaption dur-

ing the transient of dopant diffusion, the degree of refine-

ment should be limited so that the overhead of computing the

grid adaption will be minimal compared to the solution time.

Yang [53] proposes the level control concept to be imple-









mented in a quad-tree based refinement algorithm for process

simulation.

Both Yerry [54] and Hitschfeld [44] discussed the auto-

matic three-dimensional mesh generation based on modified-

octree technique for device simulation. Many other three-

dimensional automatic mesh generation techniques in other

discipline of engineering are reported [55], [56]. The grid

for two-dimensional adaptive approach is generally easier

than three-dimensional counterpart. However, the capability

of adaptive simulation is more critical for three-dimen-

sional in term of computation efficiency due to adaptive

simulation capability. The three-dimensional automatic mesh

generation for process simulation is even harder than device

simulation. A robust adaptive approach developed in two-

dimensional process simulation should be able to extend to

three-dimensions naturally.



1.4 Organization


The goal of this work is to implement an adaptive

approach for simulating dopant in the bulk of semiconductor

during ion implantation and thermal process step. The adap-

tation during ion implantation enables the automatic mesh

generation according to the implantation condition. With

this capability of grid adaptation for each thermal step of

process integration, the mesh will be optimal and solution

will be accurate.

In Chapter 2, the basic physics and the modeling for

these major process step will be discussed to lay the ground









work for the development of suitable adaptive algorithm.

Since this adaptive algorithm is to be implemented in

FLOOPS, the discussion will focus on the feature of FLOOPS.

To simulate these major steps, the process simulator

requires adaptive feature in the structure representation.

The relation between computational grid and the advanced

physic modeling is addressed in the discussion of modeling

for each individual process module.

Chapter 3 describes the software system with regard to

this work of adaptive simulation for dopant implantation and

dopant diffusion in semiconductor process simulation. The

entire algorithm and methodology of adaptive simulation is

implemented in FLOOPS. The hierarhical representation of the

mesh, the class and functions associated to the adaption of

grid and the programming environment for solving the partial

differential equation system for process simulation are

briefly discussed.

The different grid refinement schemes and flux discret-

ization methods are discussed and the results are compared

in chapter 4. Three different schemes for automatic grid

refinement have been implemented in a one-dimensional ver-

sion of SUPREM-IV for investigation. The three methods are a

contour based technique, a heuristic method and the error

estimator based method [37], which is referred to as the

Bank-Weiser (BW) method. For all methods, grid refinement

occurred prior to the diffusion solution. Better accuracy

can also be achieved with higher order flux discretization.

Three different discretization methods were investigated.

The first is the simple linear approximation to the flux









terms. The second involves assuming the concentration gradi-

ents vary exponentially and that the logarithms of the con-

centration are linear across the element.

Using the one-dimensional work in SUPREM-IV as a base,

chapter 5 describes the implementation of a nodal error

estimator and its application in the two-dimensional process

simulation for dopant implantation and dopant diffusion. The

physical insight of using this nodal error estimator is dis-

cussed. The benchmark of the CPU time consumption and accu-

racy is presented and discussed. The performance evaluation

of this error estimator is done in FLOOPS.

Chapter 6 proposes an advanced scheme [57] for mesh

generation and adaption for dopant diffusion based on local

discretization error estimates derived from chapter 5. The

new nodal error estimate in 2-D simplifies computation and

applies to both implantation and annealing in two-dimen-

sions. A self-adaptive mechanism for initial mesh generation

during dopant implantation has been developed based on the

new nodal error estimate. This initial mesh generation

method will automatically generate an accurate mesh repre-

senting an accurate initial dopant profile for nonplanar

structures. The mesh adaption scheme also applies to simula-

tion of dopant diffusion based on local discretization error

estimates. This scheme for adaptive simulation of dopant

diffusion enables accurate and faster simulations.The grid

smoothing procedure is suggested to restore the mesh quality

degraded due to the local refinement. An appropriate solu-

tion interpolation algorithm is required to interpolate new

solutions onto new grid nodes after mesh smoothing.









Chapter 7 discusses the grid quality issue and solution

interpolation scheme in detail. The grid smoothing procedure

is used to restore mesh quality degraded by the local

refinement. The appropriate solution interpolation

algorithms are investigated to interpolate new solutions

onto new grid positions after mesh smoothing. The

interpolation techniques should compute smooth solution

while the grid quality is significantly improved. The

ability for solving the moving boundary problem

simultaneously is demonstrated.

The example in Chapter 8 shows the application of this

adaptive approach in the area of simulating dopants for pro-

cesses integration of IC fabrication. The front-end to fab-

ricate a CMOS inverter by a twin-well process [58] is

simulated to demonstrate the robustness of the proposed grid

adaptation and generation algorithm. The process simulation

of the wells formation and the isolation structure illus-

trate the capability of multiple dopants and multilayer

thermal process simulation. It also shows the compatibility

with the solving procedure for simulating the moving bound-

ary along the oxide/ silicon interface. The feature of auto-

matic mesh generation in the dopant implantation has the

advantage of ease of use and the computational efficiency of

using optimal mesh to resolve accurate dopant profile.

Finally, the summary of the contributions of this work

toward the TCAD community in the area of process integration

are in chapter 9. The direction for future work and the rec-

ommendation for three-dimensional process simulation are







18

also discussed. Few challenges remain in the implemented

two-dimensional adaptive approach. Several issues need to be

addressed prior to this algorithm ready for three-dimen-

sional platform.















CHAPTER 2
PHYSICS AND MODELING FOR PROCESS INTEGRATION



2.1 Introduction


The purpose of a process simulator is to simulate a mod-

ern IC fabrication process flow. The FLorida Object Oriented

Process Simulator (FLOOPS) has been developed to fulfill

this requirement. It will allow quick development of a new

technology for IC design and fabrication. A robust and pre-

dictive process simulator therefore has become an indispens-

able tool for the semiconductor industry. The major modules

for VLSI process integration are shown in Fig. 2.1. Ion

implantation, dopant diffusion and oxidation are the major

modules used to introduce and redistribute dopant. Moreover,

oxidation is required for the device isolation structure,

e.g. LOCal Oxidation of Silicon (LOCOS.) Photolithography is

the step used to define the pattern of circuit layout from

the mask, while etch and deposition are the steps used to

transfer the pattern to the wafer. To simulate these major

steps, a process simulator requires adaption to represent

the structure as shown in Fig.2.2. In a process simulator,

the deposition and etch modules need a capability of simu-

lating the surface evolvement. The oxidation module requires

the adaptive algorithm to resolve the moving boundary

between the silicon/ oxide interface as well as the surface

evolvement. The ion implantation and dopant diffusion mod-











Process modules

SIm lant
ingot ( Q T /a Oxidation

wafer Diffusion
"7$ (Deposit



I photolithography


wafer probe Etch
and dice
<--- 25> I

IC packaging
and testing




Figure 2.1 Process modules required for simulation of
VLSI process integration. The modules
inside the solid box are the major process
steps of interest in FLOOPS.


ules also require the adaptive capability for representation

of accurate dopant solution after implantation and through-
out the anneal.

In this chapter, the basic physics and the modeling for

these major process steps will be discussed to lay the ground

work for the development of a suitable adaption algorithm.

Since this algorithm is to be implemented in FLOOPS, the dis-
cussion will focus on the feature of FLOOPS. Section 2.1

describes the etch and deposition. Section 2.2 and 2.3

describe the dopant implantation and diffusion. The oxida-












mask

Etch


Photolithograph

photoresist
oxide


Surface grid activity


silicon


Bulk grid activity


Figure 2.2 The process simulation modules that require
the adaptive capability to simulate the
VLSI process integration.


tion capability is discussed in section 2.4. The photoli-

thography is very simple in FLOOPS, therefore it won't be

discussed. More advanced algorithms for surface evolvement

[59], [60] will provide better insight of physics and model-

ing for the photolithography.









2.2 Etch and Deposition


Etch and deposition are important in the pattern trans-

fer of VLSI circuits from mask onto the wafer [61]. The fine

linewidth of interconnect and device structure, e.g. the

formation of oxide spacer to construct the Lightly Doped

Drain (LDD) structure, are required. The accuracy in the

pattern transfer will have significant impact on the charac-

terization of the performance of resulting device and cir-

cuit via numerical simulation.



2.2.1 The Etch Process in Modern VLSI Technology


For the etch process, the degree of anisotropy(A) and

selectivity(S) are defined as Equation 2.1 and 2.2

RH
A= 1 (2.1)
Rv

where RH is the horizontal etch rate and Rv is the vertical

etch rate. The anisotropy is 1 when RH is much smaller than

Rv and approaches zero for the case: Rv = RH.

RA
SAB -= (2.2)
AB R

where RA is the etch rate for the material A and Rg is the

etch rate for material B. The anisotropy of etch determines

the final shape of etch. The purely isotropic etch results in

the shape in Fig. 2.3 (a) while the purely anisotropic etch

results in the shape in Fig. 2.3 (b). In the real etch condi-

tions, both components are important for the final shape.

The selectivity is required to leave the mask intact, and



















(a) isotropic etch (b) anisotropic etch


Figure 2.3 The process steps that require the adaptive
capability to simulate the VLSI process
integration.


therefore, it should be high enough to produce the required

minimum linewidth.

The categories of etch in IC process are generally wet

and dry. The wet etch process is done by immersing the wafer

into liquid etchant. It's typically purely isotropic with

high selectivity as in Fig. 2.3 (a). The dry etch is impor-

tant for producing fine line width due to its anisotropic

nature. Many advanced approaches are reported [62]-[64].

There are different kinds of dry etch processes: physical

sputtering, plasma etching and reactive ion etching(RIE).

The physical sputtering process uses ions to bombard the

wafer thus it's anisotropic etch and introduces damage eas-

ily. The plasma etch is similar to wet etch except its

etchant is gas instead of a liquid. It's anisotropic with

high selectivity The most popular dry etch is the RIE system

due to the combination of plasma etch and ion bombard. It has

high degree of anisotropy with good selectivity. The combi-

nation of the isotropic and anisotropic component in the









etching system will determine the shape of the final struc-

ture as described in Fig. 2.3.



2.2.2 The Deposition Process in Modern VLSI Technology


The deposition process is used to introduce a layer of

material to the wafer. The deposition coverage can be

described as isotropic and anisotropic as in the etch pro-

cess. The isotropic deposition has same deposition rate at

each point on the surface as shown in Fig. 2.4 (a). For the



Imfp smfp







(a) isotropic (b) anisotropic (c) anisotropic
deposition deposition deposition


Figure 2.4 The process steps that require the adaptive
capability to simulate the VLSI process
integration.



isotropic deposition, it is assumed that the surface migra-

tion rate of particle is large and therefore results in a

uniform deposited layer. When the surface migration rate is

small, a local point on the surface will show the anisotropic

deposition due to the local visibility(o) is different point

by point. The local visibility is defined as the angle(O) of

particle collection as shown in Fig. 2.4 (b). The Figs 2.4









(b) and (c) show the result of anisotropic deposition on a

trench. The long-mean-free-path(Imfp) anisotropic deposition

in Fig. 2.4 (b) results in the thinner deposition layer on

the bottom of trench while the short-mean-free-path results

in a bump on the edge corner due to the larger local visibil-

ity as shown in Fig. 2.4. (c).

For the deposition process the often used systems are

Chemical Vapor Deposition(CVD) and Physical Vapor Deposition

(PVD). There are three main types of CVD systems: Low pres-

sure CVD(LPCVD), Atmospheric pressure CVD(APCVD) and Plasma

enhanced CVD(PECVD). Different deposition systems have dif-

ferent characteristics in the mean-free-path of gas reac-

tants and the surface migration. These conditions will

result in different final step coverage as described in Fig.

2.4. Therefore, the main interest in these systems is for

modeling of the surface evolvement for the fabrication of

device structure.



2.2.3 The String Algorithm for Modeling Etch and Deposition


In FLOOPS the modeling of the evolvement of the surface

node during the simulation of material etch and deposit is a

string based algorithm [65], [66]. In Fig. 2.5, this algo-

rithm is illustrated via the simulation of isotropic etch,

e.g. plasma etch, and anisotropic etch, e.g. ion milling. It

calculates the isotropic etch rate and anisotropic etch rate

on each surface node according to etch rate projection of the

surface normal and the ion incident direction. The more

sophisticated implementation of surface evolvement and the
























(a) isotropic etch (b) anisotropic etch


Figure 2.5 The surface nodes evolvement of a) isotro-
pic etch and b) anisotropic etch in the
string algorithm.



characterization of system are required to simulate an accu-

rate final structure. In FLOOPS, a more advanced algorithm

is implemented and reported [67].



2.3 Ion Implantation


Ion implantation is an important process step used to

introduce dopant into the wafer. Compared to the two-step

diffusion [68] for dopant introduction, ion implantation has

the advantage of using a lower process temperature with less

lateral profile spreading. Moreover, the ion implantation

has more precise control of dopant introduction. This is

especially important for small devices. As the device scale

reduced, the more advanced technique is required to cope

with the challenge in formation of a shallow junction. Also,








the technique for precisely profiling the dopant [69] and
accurate computation for the dopant profile after implant in
process simulation are required. Moreover, the capability of
providing the damage information [70] is becoming increas-
ingly demanded in that the generated point-defects and loops
after implant has strong influence to the dopant redistribu-
tion during following thermal process [71], [72].



2.3.1 Modeling the Ion Implantation


In the most simple model for implantation, the
implanted impurity profile can be described as a Gaussian
distribution function as in Equation 2.3:


[-(x-R2]
N(x) = Npexp (2.3)
2 AR (2.3)

where Rp is the projected range and the ARp is the straggle.
The Np is the peak concentration N(R) The area under the
impurity distribution curve is the total dose Q as in Equa-
tion 2.4

Q = NiNpARp (2.4)

In most cases, the implantation onto a masked area as in
Fig. 2.6 is required for IC process. The two-dimensional
distribution can then be modeled as the Equation 2.5. (2.5)
N(x,y) = N(x) F(y)

F(y) = 0.5 erfe{ -a- }-erfe{ Y+a}] (2.5)
"- F2AR F2AR 5








The N(x) is as in Equation 2.3 and the AR, is the tran:
straggle.


l a







F(y) =0.5 erfe{ a} -erfe{f -}
2ARJ_ FARj_


-\
0.5

AR y
IHI- I


Figure 2.6 The two-dimensional distribution can then be
modeled by including the lateral straggle.







2.3.2 Deviation form Gaussian Distribution

Based on the LSS theory [73],the implanted ion profile
can be models in the sense of first order accuracy by Gauss-









ian distribution for amorphous materials. However, the real

dopant profile has deviation from the Gaussian distribution

as described in Fig. 2.6. The profile is not symmetrical as




z/
a Pearson-IV






Rp
I I I x
G auCsian
--,

ARp ARp
Depth


Figure 2.7 The profile is not symmetrical as predicted
by Gaussian profile. The more advanced
model, Pearson IV distribution, uses four
"moments" to allow good agreement between
predicted and experimental profiles




predicted by Gaussian profile. The more advanced model,

Pearson IV distribution, uses four "moments" to allow good

agreement between predicted and experimental profiles for

all the amorphous materials.

Still, the simple LSS theory can't be applied to the

implantation on the nonplanar or multilayer structure. This

requires the Boltzmann Transport Equation(BTE) [74] theory

or the Monte Carlo method(MC) [75] to simulate the scatter-

ing event of implanted ion. The BTE theory will allow to pre-

dict the profile in amorphous silicon in the case when thin









film exists. The MC method is based on the simulation of

individual particles through their successive collision with

the target atoms. This will allow to predict profile in both

amorphous and crystalline materials. Moreover, the damage

profile [76] generated after these more advanced theory will

be useful for modeling complicated dopant diffusion phenome-

non associated with defect.



2.3.3 The Ion Implant Modeling in FLOOPS


As discussed in the previous section, different levels

of modeling for ion implantation are required for predicting

the profile to achieve better agreement with the experimen-

tal profile. In FLOOPS, the available models for ion implan-

tation are Gaussian distribution, Pearson IV distribution

and Tasch model [77] which is currently only effective for

the BF2 and boron impurity.

In order to utilize the advantage of these advanced

models, the impurity distribution after implantation need to

be computed on an adequate initial grid for numerical simu-

lation. The Fig. 2.8 illustrates the important of the ini-

tial grid for computing the accurate initial dopant

distribution. It shows the arsenic implant profiles

(dose=5.0el5, energy=50 keV) on a grid spacing of 500 A and

100 A using the Pearson IV model in FLOOPS. The dash line

represents the implanted profile computed on a 500 A grid

spacing. The inadequate initial grid introduces significant

dose error and the prediction of dopant profile is incor-

rect. This has lead to the requirement of the self-adaptive











500 A
0 FLOOPS 93.1

1 0 2 1 ....... .... ...........................................................................................................
1021
1020
1019
108 ............
107



.. ...... ......................
1016
1015
10'4
10'3
1012
Depth (gLm)
0.0 0.5

Figure 2.8 The dash line represents the implant com-
puted on a 500 A grid spacing. The inade-
quate initial grid introduce significant
dose error and the prediction of dopant
profile is incorrect.



approach [41] to predict an accurate profile after dopant
implantation. For modern IC processes, the representation of
dopant in the complicated structure won't be easily solved
without this capability. Only with an adequate initial grid,
can advanced models for predicting dopant implant profile be
valid. To implement this capability for ion implantation
simulation in FLOOPS and justify the predictivity of model-

ing dopant implantation will be one of the major goals of
this work.


2.4 Dopant Diffusion









The diffusion of dopant occurs in the thermal processes

of IC fabrication. The methods for diffusion are 1)diffusion

from a dopant source on the wafer surface [78] and 2)the dif-

fusion and annealing from the ion-implanted layer [79]. The

diffusion process is critical to produce desired junction

depth for the source and drain in CMOS device or the base,

emitter, and collector in BJT device. Therefore, it's

required to understand the complicated physics involved in

order to model the diffusion for numerical simulation.

The basic diffusion modeling for impurities in silicon

can be described as in Equation 2.6. The first term is the

concentration gradient driven flux and the second term is

the electric field driven flux.

C a DC + (DZNDEx) (2.6)
a t: xx kTx D x

where C is the dopant concentration D is the effective dopant

diffusivity, ND is the donor with charge state Z assuming all

donor are ionized, and Ex is the electric field.



2.4.1 Modeling for the Dopant Diffusivity


By neglecting the electric driven term and assuming a

constant diffusivity, Equation 2.6 has a simpler form
2
ac C
t ax2 (2.7)


This is Fick's Law for diffusion. However, the constant dif-

fusivity assumption is not adequate to model most of the phe-

nomenon occurring in dopant diffusion. At high

concentration, the impurity profile can be represented by a









concentration-dependent diffusivity which can be approxi-

mated [80] as

(C r
D = 2D (2.8)


where Di is the constant diffusivity at low concentration, ni

is the intrinsic carrier concentration and r is a constant.

The temperature dependency of the diffusivity can be deter-

mined experimentally over a range of temperature for diffu-

sion and can be expressed as

D = D exp k-T (2.9)


where Do is the frequency factor, E is the activation energy.

Both DO and E can be derived from atomic diffusion theory

[81]. As device shrinks, more other physics will be required

to model the diffusivity for predicting the accurate dopant

profile during diffusion.



2.4.2 Fair's Model for Dopant-defect Pairs Diffusion


It is well known that the dopant in silicon diffuses by

interaction with point defects. Fair's models [82] for dif-

fusion are based on the analysis of the diffusion rate of

impurity-vacancy pair. The diffusion mechanism can be illus-

trated in Fig. 2.9 and the diffusion flux is described in

Equation 2.10

ac ac ac ac ac
A AV AV A? AV
D- =D +D +D +D (2.10)
ax AV Tx AV dx AVOX AV+ Ox

where C is the concentration of impurity A-charge state x
vacancy pairs, and D is the diffusivity of impurity A-
vacancy pairs, and D is the diffusivity of impurity A-
AV
























Figure 2.9


0 0 0 0 Silicon

O ** O Vacancy

0 0 0 Impurity

0000


The diffusion via the dopant-vacancy pair
mechanism to assist dopant diffusion.


charge state x vacancy pairs. Equation 2.10 is simplified to
an effective diffusivity by using the relative population
statistics given by Shockley and Last [83]. The effective
diffusivity can be expressed as


D=D +D +D- +D +D
AV n.i AV n. AV AV n


(2.11)


This will allow a model for vacancy assisted diffusion via
the effective diffusivity accounting for dopant-vacancy
pairs


2.4.3 Modeling of Oxide Enhanced Diffusion

Oxidation Enhanced Diffusion (OED) is the well known
effect of diffusion enhancement by defect injection. Fair's



















Dopant profile"" ---






Figure 2.10 The interstitial is injected under the
oxidized portion of the interface. The
dopant diffusion is enhanced there.



model can't describe this effect since the enhancement is

from an injection of silicon interstitial during oxidation.

As shown in Fig. 2.10, the interstitial is injected under the

oxidized portion of the surface, and therefore the dopant,

diffusion is enhanced by an interstitial assisted diffusion

mechanism as illustrated in Fig. 2.11

To model OED, the empirical effective diffusivity can

be expressed as

D = D* fIC+ (1-f) (2.12)
IC C (2.12)


where the C. and C, are the interstitial and vacancy

concentration respectively and the CI* and C* are the

equilibrium concentration of defects. The f. is the por-

tion of interstitial and D* is the diffusivity under an

inert boundary.












0 0 0 0




0 000


O Silicon

SSilicon
self-interstitial

0 Impurity


Figure 2.11 The Interstitial Kick-Out mechanism. This
will assist the dopant diffusion.




2.4.4 Modeling of Transient Enhanced Diffusion

Transient enhanced diffusion (TED) is complicated due
to the nonequalibrium effects through the concentration
dependent diffusivity D(C) which is believed to be mediated
most by the points defects (interstitial and vacancy).
Therefore, it's necessary to calculate the defect concentra-
tion so that it can be coupled into the solution of dopant
diffusion. Equation 2.13 and 2.14 characterize the defect
concentration as continuity equations.
SCI _D DCI |( \
tac a- r-K( C c; C C (2.13)
at Ox Ox Kr rIv I v(


ac D SD^C ( N
at aox -Kr CICV- C)*C (2.14)
'i-: jx _









where Kr is the recombination constant. The second term in

Equation 2.13 and 2.14 is the bulk recombination for both

type of point defects.

There are various methods to solve the system equations

for dopants and defects according to the degree of coupling

between the equations. There are Fermi model, Two.dim model

and Full.cpl model [84]. The Fermi model uses the analytical

expression for dopant dependent diffusion as in Equation

2.11 to describe effective diffusivity accounting defect

assisted effect. The Two.dim model allows the loosely cou-

pling between point defect and the dopant stream while the

Full.cpl model allows the fully coupling between the system

equations for defect and dopant diffusion. The analytical

solution is required for its computational efficiency while

the fully coupled approach gives the more accurate solution

after computation.



2.4.5 Advanced Diffusion Modeling in FLOOPS


In FLOOSP, some more advanced modeling of the defect is

implemented, e.g. the pressure dependency [85] due to stress

from dislocation loop. Damage enhanced diffusion has also

been investigated [86]. Currently, the stress dependency of

diffusivity is under investigation. As the device scaled

down, the material characteristic such as stress will play

an important role to mediate the dopant diffusion as well as

defects.

All the efforts for improving the physics associated

with the dopant diffusion are resulting in more complicated









expressions of effective diffusivity to account for the

physical factors of interest. For numerical simulation of

dopant diffusion, the discretization of the equations has to

be improved. Since these advance models are second order

effects, the discretization has to be sufficiently accurate

to examine the effects. The conventional linear or logrith-

mic discretization for diffusion flux is inadequate. More-

over, the computational grid for solving the system equation

should be updated through the transient of diffusion. The

adaptive approach therefore become attractive and necessary

in term of computational efficiency and solution accuracy.

The adaptive approach [87] of this work is mainly to investi-

gate the suitable approach to reduce the error from the equa-

tion discretization and the spatial discretization of grid

so that the advance modeling for dopant diffusivity will be

accessible from the computed solution during dopant simula-

tion.



2.5 Oxidation


The oxidation process step of IC fabrication is impor-

tant in that it has many applications for VLSI technology:

the gate thin oxide for MOS device, the dielectric layer of

capacitor for Dynamic RAM (DRAM), the field oxide in the

LOCOS, and the passivation of the wafer, etc. Generally, a

thermal grown oxide is used due to its good material property

along Si/ Si02 interface and well-controlled thickness rang-

ing from 60 A to 10,000A. For the formation of thin oxides,

dry oxidation [88] (02 ambient) is often used while for thick









oxide, e.g. LOCOS, wet oxide [89] (H20 ambient) is preferred

to reduce the thermal cycle so that the additional spreading

of the existing dopant profile will be reduced.



2.5.1 Deal-Grove Model for Oxidant Diffusion and Reaction


The Deal-Grove model [90] provides a general model for the

diffusion and reaction of oxidant during oxidation. The one-

dimensional model for oxidation is shown in Fig. 2.12.



N

Co
i F



UCi


O SiO2 Si

X
Distance from surface



Figure 2.12 Del-Grove model for one-dimensional oxida-
tion.



According to the Fick's Law, the diffusion of oxidant can be

expressed as


F = -D C(x, t) (2.15)
ax









where D is the oxidant diffusivity and the C is the oxidant

concentration. By assuming the flux is constant inside

oxide, Equation 2.15 becomes

(Ci-Co)
F = -D -) (2.16)
xo

where Co and Ci are the oxidant concentration at the

surface and Si/ SiO2 interface respectively. xo is the

oxide thickness at given time. By assuming that the oxida-

tion rate at Si/ Si2 interface is proportional to the oxidant

concentration, the flux at the interface becomes




Fi = -ksC (2.17)


where ks is the rate constant for the reaction at inter-

face. By canceling Ci the Equation 2.16 and 2.18, the flux

can be expressed as

CO
F = D (2.18)

x0 +ks


The change rate of oxide thickness then can be derived as

following


dx0 F D CO (2.19)
dt M M( D~
x0+


where M is the number of molecules of oxidant that dif-

fuses into the volume of the grown oxide. Solving the Equa-

tion 2.19 by the boundary condition of xo(O) = xi the

solution of thickness of oxide is








2
Xo Xo
X0 X0
t +-- t (2.20)
B B/A 0
2
x x
where A = 2D/k B = 2DCO/M and tO + -0
S 0 B B/A
x0(0) = xi is the initial thickness of wafer. The solution
for the Equation 2.20 then becomes




4B 2
x0(t) = 0.5A[ {l+ (t+to)} (2.21)
A



To approximate the solution in Equation 2.21, for short time

with (t+ to) A2/(4B) the Equation 2.21 becomes

B
xo(t) = A(t+ t) (2.22)


this is the linear rate region. The growth rate is limited by

the reaction at the interface.

For long time with (t+ to) A2/4B and (t > to) the

approximation becomes


x0(t) = JB (2.23)


This is in the parabolic rate region, the oxide growth rate

is diffusion-limited.



2.5.2 The Oxide Flow Characteristics


Based on the previous derivation for one-dimension oxi-

dation, the two-dimensional oxidation also can be computed

for the system shown in Fig. 2.13. There are various

approaches for solving two-dimensional oxidation [43]. The











DxC(x, t) =0 C =


d d a
aa



DaC(x, t) = -kC(x, t)






Figure 2.13 The equation system for solving oxidation
in two-dimension



approach of incorporating the oxide flow model is preferred

in that it account for the oxide flow during oxidation. The

oxide flow can be characterized with different flow models

as shown in Fig. 2.14. The viscous flow model is modeled by a

dash pot while the elastic flow is modeled by a spring. The

viscous flow model is suitable for temperature above 9600

while for temperature below 9600, a elastic model will be

more appropriate. The simulation with linear flow modeling

in the proper temperature shows good agreement with experi-

mental result.

For the calculation of the stress-dependent oxidation

growth the linear elastic or the linear viscous model will be

inadequate, therefore nonlinear flow modeling is required

for simulating stress-dependent oxidation [91].


2.5.3 Factors Influencing Oxidant Diffusivity





















(a)


Figure 2.14 Modeling the oxide flow for simulating
oxidation in two-dimensions: a) elastic
flow, b) viscous flow c) viscous and elas-
tic flow in parallel and d) viscous and
elastic flow in series.


The oxidation growth by Deal-Grove model presents the

Arrhenius relation between the parabolic rate constant and

the temperature. It can be expressed as


D = Doexp( -'


(2.24)


where DO is a constant and EA

depending on the oxidation ambient.


is the activation energy


The growth of nonplanar oxide gives a velocity gradient

in the oxide layer and produces stress. The stress not only









has influence on electrical property of the device struc-

ture, e.g. the rounded corner of trench capacitor on a DRAM

cell, it will also influence the diffusivity [92], viscos-

ity, solubility, reaction rate of the oxidant. The resulting

shape of oxidation will be dramatically different. The

stress from the pressure can be modeled simply by including

the pressure volume work PAV The Equation 2.24 will be

modified to include the stress effects on the oxidant diffu-

sivity as following



(- EA + PA V
D = D0exp K-
KT I



This combined with more advance oxide flow modeling will

allow to incorporate other physics during oxidation so that

the simulation for oxidation will result in better agreement

with the experimental results, e.g the length of bird's beak

of a LOCOS structure.



2.5.4 Oxidation and Silicidation in FLOOPS


In FLOOPS, the oxide grows in the linear flow model as

well as nonlinear flow modeling. The oxidation by Del-Grove

model also apply to solve the silicidation problem [93]. The

stress-dependency in oxidation simulation not only predicts

more accurate final shape of LOCOS structure, it also indi-

cates the high stress location on a isolation structure

[93]. This is valuable for advanced device design,e.g. DRAM

cell.









The algorithm of numerical simulation for oxidation is
basically compute the moving boundary along the Si/ SiO2
interface as shown in Fig. 2.15. Since during the growth of
oxide, the volume of oxide expand, the upper part of the
oxide boundary is essentially a surface evolvement problem.
By including the physics into the oxidant diffusivity and
accounting for the oxide flow model, the velocity of node
movement is calculated. With this solution available, the
string algorithm discussed earlier for etch/ deposition will
be applied to evolve the surface and the interface of oxide.







0.54Tox


)1I t 0.46Tox





adaptive dopant diffusion




Figure 2.15 By including the physics into the oxidant
diffusivity and accounting for the oxide
flow model, the velocity of node movement
is calculated. The string algorithm will
be applied to expand the volume of oxide.









The algorithm ready for simulating surface evolvement for

material etch/ deposition will also be applied to resolve

the better shape representation of bird's beak. With this

capability to resolve the fine geometry of nonplanar struc-

ture, the stress effect and influence of flow model will be

the dominant driving force for the formation of final shape

and therefore be more predictive. Moreover, the adaptivity

of simulating dopant diffusion in the Si bulk grid should be

compatible with the simulation of moving boundary during

oxidation. Besides, the quality of the grid inside the oxide

volume will be important when utilizing the nonlinear flow

for simulating oxidation. Therefore, it's necessary to

ensure the bulk grid quality inside the volume of oxide is in

good condition. This will allow the faster convergence, i.e.

computational efficiency.



2.6 Conclusion


The physics and modeling for the major components of

the VLSI process integration are discussed. The more

advanced modeling for physics required in each module of

process integration will improve the accuracy of simulated

results to allow the predictive design by a process simula-

tion. Although the advanced models are available in the pro-

cess simulation, the accuracy of the computational results

still strongly depends on the underlying computational grid.

As the physic has been improved, the approach for domain dis-

cretization and equation discretization have to be advanced.

Therefore, the adaptive approach for process simulation of







47

dopant becomes indispensable in term of solution accuracy

and computational efficiency. This work will mainly focus on

the adaptive simulation of dopant in the bulk of device as

described in Fig. 2.1. The development of the adaptive algo-

rithm will discussed through the rest of chapters.















CHAPTER 3
THE SOFTWARE SYSTEM


3.1 Introduction


Object-oriented programming has become increasingly

popular due to its nature of data hiding and class inherit-

ance. The abstract data type allows the security and modu-

larity in data representation while the class derivation

capability allows the reusability of an elementary class

[94]. With the inheritance features, a variety of closely

related objects can be created from one generic object. In

order to handle the ever increasing structural complexity

and to speed the prototyping of the physical model in semi-

conductor process/device simulation, object-oriented pro-

gramming becomes an invaluable approach [95].

This chapter describes the software system with regard

to this work of adaptive mesh generation for dopant implan-

tation and diffusion in semiconductor process simulation.

The entire algorithm and methodology of adaptive simulation

is implemented in the FLorida Object Oriented Process Simu-

lator (FLOOPS). The hierarchical representation of the mesh,

the class and functions associated with the adaption of

grid, and the programming environment for solving the par-

tial differential equation system for process simulation

will all be discussed in the following sections.









3.2 The Implementation of Field Server


A field in FLOOPS is a hierarchical representation of

geometry, grid and data information. It uses points, nodes,

edges, faces and volumes to represent the mesh instead of

using the node-element table approach in SUPREM-IV. This

field server in FLOOPS is implemented as in Fig. 3.1. The

field server consists of coordinate objects, data objects

and mesh objects. Many other objects contained by the Mesh

object are also shown in Fig. 3.1.

The coordinate is derived from a Location object. It

describes a physical point and a list of nodes associated to

this point. The relationship of node and coordinate can be

multiple to one. For example, there are three nodes sitting

on the coordinate of the boundary of Silicon/Oxide, i.e. a

Silicon node, Oxide node and Silicon/Oxide interface node.

The data objects have a variety of forms in derivation

such as RealData, IntData and LocationData. They are all

derived from a BaseData object. These data objects are man-

aged by a DataManger object inside the FieldServer. The data

storage in the field is created by a DataStorage object in

any Element object. This will allow the data to be stored in

any Element object depending on the nature of the data. In

Fig.3.2, for instance, the coupling coefficient is stored on

the edge el while the area of the Box Discretization Method

associated to a specific grid point no is stored on the node.




















I Location


/ Node


SVolume


OEdge
\


Element)


Figure 3.1 The FieldServer consists of Coordinate
object, Mesh object and Data object.



















Area



n
el
Flux









Figure 3.2 The DataStorage object allows the Area of
Box method on no and the coupling coeffi-
cient on e1 to be stored into node type
data and edge type data respectively.








From the Data objects, The Solution object is con-

structed. The various type of solution objects for the dif-

fusion solver are derived from this base Solution to

simplify the management of data structure. As shown in Fig.

3.3, the Dopant solution object, Loop solution object and

Trap solution object are derived from base Solution object.

From the Dopant solution, the Defect and Active solution

objects are further derived.























































Figure 3.3


The Solution object is constructed from
Data objects. The various solution type
objects regarding diffusion solver are
derived from this base Solution to simplify
the management of data structure.









The Mesh object consists of Node, Edge, Face and Volume

objects which are all derived from an Element object. These

elements describe the spatial discretization of a domain of

interest. The hierarchical representation is most obviously

seen through the interrelationship between these derived

elements. Fig. 3.4 shows this derivation of objects inside a

Mesh object. The Material object of a mesh defines the corre-

sponding parameter object which specifies the physical mod-

els. The OEdge (order edge object) is an object containing a

list of Edge objects with a directed end. This makes sequen-

tial access to Edge data possible. The capability of object

derivation and inheritance significantly reduces the time

and effort in building this FieldServer as shown in Fig. 3.3

and Fig. 3.4.

Fig. 3.5 shows the structure of the field in FLOOPS.

Each mesh is composed of many Nodes, Edges, Faces and Vol-

umes.The relationship between the objects is indicated along

the direction of arrow in the diagram. For instance, the

relation between a Mesh object and Node objects is l:n, i.e.

n nodes per mesh. The OEdge allows sequential access to data

information stored on Edge objects associated to different

type of element. In Fig. 3.6, the OEdge regarding a Node

object is the edges connected to this node in counter-clock-
















( Field Server)

SData


Mesh

^Material^


( OEdge )


Figure 3.4


The derivation relationship between the
objects inside field of FLOOPS. The Mesh
consists of Node, Edge, Face and Volume
objects which are all derived from an Ele-
ment object.
























































Figure 3.5 The structure of the Field in FLOOPS. Each
mesh is composed of many nodes, edges,
faces and volumes.The relationship between
the objects is indicated along the direc-
tion of arrow.









wise (CCW) sequence. For a Face object, the OEdge object

allows to walk edges of face sequentially.





CCW
N


CCW



Figure 3.6 The OEdge on Node and Face. The OEdge
regarding a Node object is the edges con-
nected to this node in CCW sequence. For a
Face object, the OEdge object allows to
walk edges of face sequentially


The object inheritance capability allows the quick der-

ivation of many similar objects. This characteristic greatly

improves the efficiency of programming in constructing a

complex objects. The data encapsulation provides the secu-

rity of those data which are not to be modified upon deriva-









tion. It also provides the convenience of later modification

on a base object without much further update on the derived

objects. This concept can be exemplified via the derivation

relationship of solution objects in Fig. 3.3. The base Solu-

tion object can be modified and the derived objects such as

Active solution object and Defect solution object need not

to be changed.

The combined advantage from objected-oriented program-

ming and hierarchical mesh concept can be illustrated in the

work of extending the 2D field to 3D. As shown in Fig. 3.6,

the 3D related object is the Volume. The extension from the

2D field is straightforward by creating a Volume object from

the base Element object. The base Element object needs to be

enhanced and the interrelationship between these 3D related

objects and the existing objects needs to be constructed.

The new features of the FieldServer in FLOOPS allows effi-

cient development of this 3D extension.



3.3 The System Structure of FLOOPS


This section provides a brief description of the system

structure of FLOOPS so that the work regarding the advanced

flux discretization methods and the adaptive spatial dis-

cretization can be introduced. The tool command language

(TCL) is used to implement this process simulator. The

advantage of using TCL in writing an application program is

that tcl provides many utility commands and the general pro-

gramming interface [14]. This will reduce the effort in

building complex command procedures such as those in FLOOPS.









3.3.1 The Building Blocks


There are three major blocks in the FLOOPS, i.e. Mesh,

Assembly and Solver as shown in Fig. 3.7. The mesh block

describes the spatial discretization of the domain of inter-

est for the process simulation. The specification of initial

mesh can be done through line and region command or via init

command to source an existing structure and mesh. As dis-

cussed in section 3.2, this mesh is managed by the Field-

Server. The grid command has been implemented to specify

parameters for controlling the quality of the spatial dis-

cretization of this mesh. For instance, the minDist, minAdd,

minAngle, shrinkRemove, shrinkSmall, shrinkLarge are the

parameters for grid removal control during simulation of the

moving boundary. These parameters are used by the grid

removal functions to prevent the generation of obtuse trian-

gles or bad connectivity along the moving boundary.

With a well discretized structure, i.e. mesh, the

Assembly block describes the procedure to assemble the

stiffness matrix on each solution of each element for solv-

ing the partial differential equations (PDE) in the semicon-

ductor. The PDEinMaterial object contains the methods for

assembling the Jacobian on each element. To describe assem-

bly methods for different semiconductor PDEs, different

assembly objects are derived from base PDEinMaterial. For

solving the dopant diffusion equations, the ConstantDiffu-

sivity object, DefectinSemiconductor object and AnalyticAc-

tive object are derived from the PDEinMaterial object to

assemble the stiffness matrix for dopant, defect and acti-














Grid Command,
Parameters.


Physical
" Modeling.
(diffuse,
oxidation
)


print.ld
plot.2d
contour ...


Figure 3.7 The building blocks of FLOOPS. There are
three major blocks in FLOOPS, i.e. Mesh,
Assembly and Solver. The post processing
allows printing or plotting the solution
data and visualizing the structure to be
solved.









vated dopant. Various physical modeling for process simula-

tion are accessed via PDEinMaterial during the assembly

(e.g., the LoopinSemiconductor is the assembly object to

access the parameters in LoopParams object for modeling dis-

location loop and to produce the stiffness matrix.)

The assembly object produces a stiffness matrix and

right-hand side (RHS) vector required by the sparse matrix

solver. The Stiff and Vector objects provide communication

between the matrix solver object and the assembly object.

Due to concerns of computational efficiency, most sparse

solvers are written in programming languages other than C++.

The Matrix object provides the program interface between

FLOOPS and the mathematical package. This also allows quick

upgrades for the mathematical package since the solver is

not hard coded into FLOOPS.



3.3.2 The Implementation of Adaptive Spatial Discretization


The purpose of implementing the adaptive spatial dis-

cretization is to control the discretization error. With the

spatial discretization error under control, the solution

accuracy improvement due to the advanced physical models

will be more obvious. Figure 3.8 illustrates the concept of

this adaptive spatial discretization in FLOOPS. The old mesh

in Fig. 3.8 is a core mesh from the initial mesh specifica-

tion. The spatial discretization error evaluation is used to

indicate the location where the mesh need to be adjusted. The

mesh adaption function carries out the operation of this

adjustment according the estimated discretization error in
























































Figure 3.8


The adaptive spatial discretization algo-
rithm. It allows to control the spatial
discretization error and update the mesh
according to the specified error criteria.









each mesh. The new mesh which satisfies the error criteria is

then obtained after these two operations. The additional

work regarding mesh regularization is required to polish the

new mesh. This is mainly because the adaption of the mesh is

carried out based on local refinement which tends to deteri-

orate the quality of a mesh (e.g., in a two-dimensional mesh,

obtuse triangles might be generated after local refinement).

With a mesh regularization function, a well discretized mesh

can be obtained. This algorithm for adaptive spatial dis-

cretization proves to be efficient and also has been imple-

mented in another process simulator, ATHENA/SSUPREM [96].

In FLOOPS, the adaptive spatial discretization is

applied to both the dopant implantation and dopant diffusion

simulation. The adaptive meshing is necessary for solving

dopant diffusion due to the time dependent nature of dopant

diffusion. The correct spatial discretization should be

guaranteed for each time step so that the error won't be

accumulated through transient simulation. For adaptive

dopant implantation, this algorithm also provides a solution

for automatic mesh generation. Not only has the mesh been

well discretized to represent the dopant distribution, also

the users no longer need to manually generate mesh.

To implement this adaptive spatial discretization algo-

rithm, the grid command is written to control the adaption of

mesh, evaluation of discretization and the access to the

grid parameters. It allows to specify the error criteria for

refinement, such as errorAdd, errorDel, absdos, minConc,

which are parameters defined in the GridParams object. The

functions to operate the adaption of mesh during dopant









implantation are ErrNode, ImpAdaptNode and ImpSmooth. The

ErrNode function returns the computed discretization of the

old mesh based on nodal error estimation. The ImpAdaptNode

and ImpSmooth will adapt the old mesh according to the eval-

uated spatial discretization and regularize the new mesh

after adaption. For the dopant diffusion simulation, the

discretization is also evaluated by ErrNode,and the mesh is

adapted by DiffAdaptNode. The mesh regularization for dopant

diffusion is different from algorithm in ImpSmooth. During

simulation of dopant diffusion, the smoothing of a mesh is

accomplished by solving new solution and new grid location

at same time via the upwinding technique in each time step. A

SmoothWind is created to fulfill this requirement. All of

these functions are created and collected by the fieldutil-

ity which is a tool box for field management in FLOOPS.



3.3.3 The Implementation of Flux Discretization Methods


The flux discretization methods in FLOOPS are specified

via the diffuse command. This allows users to choose between

simple discretization methods and more complicated methods.

The linear and log flux discretizations assume the dopant

concentration gradients vary linearly and exponentially sep-

arately. The advanced flux discretization method in FLOOPS

is the summation flux method [97] which assumes (N-l) nodes

between the two actual nodes. This high order flux method is

represented by S2 (summation method with N=2.) and S4. The

use of this advanced discretization method will improve the

accuracy of the discretization for diffusion equation as









well as the efficiency of the adaptive algorithm for dopant

diffusion simulation.

Fig. 3.9 shows the implementation of these flux dis-

cretization methods in FLOOPS. The mesh after adaptive spa-

tial discretization provides the data and elements

associated to this mesh. During the assembly of the stiff-

ness matrix, the flux discretization is evaluated via the

EdgeFlux object. This assembles the diffusion flux on each

edge according to the specified method and writes the evalu-

ated edge flux into the associated stiffness matrix. During

assembly, the physics of process modeling and the discreti-

zation of dopant diffusion flux on edge will be incorporated

into the stiffness matrix on each element. The small stiff-

ness matrices will be added up to form the sparse matrix by

AddElement in the Matrix object. This sparse matrix will be

solved by mathematical package of sparse solver. The update

of the solution vector and the right-hand side vector are

managed by the Vector object. The factorization of a sparse

matrix is typically the most computationally intensive work.

In FLOOPS, UMFPACK [98] and DMF [99] are some more efficient

packages for factorization of a sparse matrix. The DMF pack-

age allows the parallel computation capability to take the

full advantage of the computation resources.



3.4 Conclusion


This chapter describes the hierarchical field represen-

tation in FLOOPS and the relationship of the derived objects

inside this field representation. The FieldServer is dis-






























































Figure 3.9 The diffusion flux discretization algo-
rithm and the solving procedure of PDE sys-
tem.









cussed to assist the appreciation of the management for this

hierarchical field. This lays the ground work for the adap-

tion of a field to allow the spatial discretization control.

The hierarchical field also provides the flexibility in the

management of the field so that the increasingly complex

geometry of the microstructure can be represented, e.g. the

bird's beak of a LOCal Oxidation of Silicon (LOCOS.)

The adaptive spatial discretization algorithm is dis-

cussed and applied in the area of dopant implantation and

dopant diffusion. The advanced flux discretization methods

are implemented to allow the more accurate computational

result for dopant profile after anneal. The implementation

of these concepts are described and the related solution

procedures for PDE's are briefly discussed.

Object-oriented programming is becoming a popular

approach for large software system. The modern IC process

simulation can become more robust in system management and

code development by utilizing this object-oriented program-

ming concept. The nature of data hiding and class inherit-

ance assist the implementation of the hierarchical mesh

concept in the field of FLOOPS. The class derivation feature

also simplifies the work to prototype the new physical model

regrading IC process.















CHAPTER 4
SPATIAL DISCRETIZATION AND FLUX DISCRETIZATION FOR DOPANT
DIFFUSION IN ONE-DIMENSION


4.1 Introduction


A modern process simulator must cope with the gradually

increasing complexity of physical models [45]-[47].

Recently, research has been dedicated to the improvement of

the physical models for dopant diffusion. The contribution

of these advanced physical models is better modeling of both

defect and dopant diffusion. However, these improved models

also significantly increase the amount of CPU time required.

As device dimensions shrink, it is also necessary to model in

two and three dimensions. The improvement of computational

efficiency, therefore, is an important task for all

simulation tools. The application of adaptive grid

refinement in solving dopant diffusion will allow optimal

grid throughout the transient simulation.

The flux evaluation can be made more accurate by using a

higher order flux discretization. The standard

discretization method assumes a linear variation of

concentration between two adjacent nodes, which is the

piecewise linear approximation of the dopant profile. This

assumption, however, will not be valid when grid spacing is

too coarse. This is due to the nonlinear nature of dopant

concentration. In a simulation with fixed grid spacing, the









grid spacing is limited by the nonlinearity of the dopant

profile. A higher order flux discretization on a fixed grid

shows a significant increase in the allowed grid spacing

[100]. It is then interesting to investigate the interaction

between adaptive grid refinement and higher order flux

correction.

In order to compare the resulting profile of different

methods of grid refinement and different flux

discretization, each of the methods were benchmarked using a

1014 atoms/cm2 dose of boron and a 1 hour 10000 C anneal. The

initial dopant profile after implantation is a Gaussian

distribution with a range of 0.3 pm and a standard deviation

of 100 A. Three errors have been evaluated to indicate the

performance of different methods. First, the dose

integration error. Throughout the diffusion, the total dose

should be conserved. The dose can be lost during grid

addition and subtraction, and interpolation errors of grid

adaption must be kept small. The second error index is the

peak concentration of the final profile. The third is the

junction depth error at the dopant concentration of 1014 cm

. The error was computed by comparing the resulting profiles

with a computation on a fixed grid spacing of 10 A. It is

important to reduce these errors in order to extract

accurate electrical parameters.



4.2 Grid Refinement Schemes


Three different schemes for automatic grid refinement

have been implemented in a one-dimensional version of









SUPREM-IV. The three methods are a contour based technique,

a heuristic method and the error estimator based method

[38], which is referred to as the Bank-Weiser (BW) method.

For all methods, grid refinement occurred prior to the

diffusion solution as shown in Fig. 4.1.



4.2.1 Contour Refinement Method


This method attempts to maintain a fixed concentration

ratio between adjacent grid nodes. At the end of each

diffusion step, the maximum concentration is calculated and

grid points are placed at a fixed concentration ratio away

from this point. This method is straightforward and easy to

implement. However, due to the rearrangement of all the grid

points after each time step, this method is subject to a

large amount of interpolation error. Due to the Gaussian

nature of the profile, it's convenient to process grid

refinement in a logarithmic concentration. The criterion,

therefore, is expressed as:


C.
= (4.1)
ci-1



where the F is the specified constant dependent on the

required concentration ratio of adjacent nodes. A F = 1.1 is

used in the comparison section to evaluate this method. (For

convenience in this discussion, let i be the node at the

higher concentration such that Ci > Ci-_.)










Algorithm of Grid Adaption


Figure 4.1


The flow chart of automatic grid
refinement scheme. The three different
autogrid methods have been implemented
in the function block for Grid
Refinement.









4.2.2 Heuristic Method Based on Concentration Ratio of

Adjacent Nodes


This heuristic method is a modification of the contour

method in that it relies on the concentration ratio of

adjacent grid points. This ratio is used to decide whether

refinement or coarsening of the local mesh is required. Some

related work had been discussed to present the efficiency of

this class of application [40], [41]. This is said to be of

an a priori nature due to the computation of ratio criterion

before solving the PDE's. In this work, the algorithm is

simply implemented as:


C.
> 1 (4.2)
i-



C < (4.3)
C-i 2



where IF is an empirical criterion for adding grid point

between two nodes while F2 is for removal of nodes and

assumes Ci > Ci-1. It is suggested [40] that the F's be a

function of time-step size and diffusivity of impurity. The

relation is expressed as:



Fo 1 (4.4)
D At
maxa



where Dmax is the maximum diffusivity at the pair of nodes.

In this case F is spatially varying. By assuming that the









adaptive time-step in SUPREM-IV is efficient enough to

resolve the time discretization, the F's can be picked

empirically dependent on diffusivity. In the case of boron

diffusion, values of F1 = 1.99, F2 = 1.1 are chosen as the

criteria for grid refinement.

In this heuristic method, three different regions of

the profile were identified: the peak, tail and body--each

of which used different regridding criteria. With a peak

dopant concentration Cpeak the peak region is then defined

as the area where the dopant concentration is between Cpeak

and 0.1 Cpeak. In the peak region, the contour method is

applied. In this case, a smaller criterion, F = 1.02, in the

contour method is used to give reasonably fine grid around

the concentration peak. This will guarantee better flux

continuity around this nonlinear area, therefore better dose

conservation is observed.

The tail region is identified by the slope of the dopant

profile in a semi-logarithmic scale. As the profile falls

off, the slope decreases. A lower limit of this slope can be

chosen empirically to define the boundary of the body and the

tail region. A value of 105 (cm-~) is used in this work. By

this criterion, the dopant concentration in the tail region

is roughly below 1.5x105 cm3. It should be pointed out that

the background dopant concentration in SUPREM-IV is 105 cm-3.

In the tail of the dopant profile, the grid refinement is

implemented by allocating grid points at gradually

increasing spacing with increasing distance from the

boundary of the body region and the tail region. This









implementation will efficiently place allowable coarse grid

in the tail region. The criterion can be expressed as:


(xi+1-xi) = d(xi-xi 1) (4.5)



where d is a specified constant, and x is the distance from

surface in that xi-1 < xi < xi+1. A value of d = 2 is used for

discussion. This value is safe in this work because the first

grid spacing, which is about 10 A, is taken from the body

region at the boundary. As the grid spacing increases, the

dopant concentration has reduced to the background

concentration. At this flat region of the profile, a very

coarse grid spacing is acceptable. However, in an

application where disjoint peak regions exist, e.g., for a

double implant with different energies, the dopant

concentration in the identified tail region is not

necessarily close to the background dopant concentration,

e.g., the area between two peaks. In this case, a smaller d

can be used to insure the grid spacing does not vary too

rapidly.



4.2.3 Refinement Based on Error Estimator


The Bank-Weiser (BW) error estimator is based on

solving a local Neumann problem in each mesh element.

Consequently, it can be applied to 2-dimensional and 3-

dimensional problems in a straightforward way. This

technique has been successfully used for refinement of

device problems [32]. The error should be evaluated before a









solution is obtained. This local-error-based autogrid scheme

is based on the finite element method of lines (FEMOL)

discussed by Biterman and Babuska [101], [102]. However, in

applying this FEMOL technique, the local error estimator is

built by the approach proposed by Bank and Weiser [38].

The BW error estimator approach is a posteriori by its

very nature. In order to implement this method in the a

priori sense as shown in Fig. 4.1, an extra single time-step

Trapezoidal Rule (TR) integration is carried out prior to

the TR-BDF [103] multiple time-step integration in each

time-step. The BW refinement is then applied and based on the

error estimate at the end of this time-step (to+At). The one-

dimensional BW scheme is implemented by utilizing a one-

dimensional finite line-element approach for time dependent

grid refinement. For each element, the endpoint

concentrations are held fixed in time, (to) to (to+At), and a

solution for the midpoint concentration is obtained. This is

compared to the interpolated midpoint value at time (t0+At)

to estimate the error. The error estimated in an element is

used as an indication for refinement. If the error estimate

is larger than the criterion, the refinement is carried out

by splitting the element. Thereafter, by applying TR-BDF

time-step integration, the solution at time (to+At) can be

recomputed with a new grid optimized for accuracy in the

profile at the end of the time-step. Though more CPU time

will be spent on this additional TR integration, the more

accurate concentrations can be used as initial guesses for

the following TR-BDF to accelerate the rate of convergence.

Therefore, the increase of CPU time will be alleviated by









accurate initial guesses for concentration. The

specification of the criteria for refinement and coarsening

can be summarized as the following:

Cmid- Cacc p (4.6)
C 1
acc



mid acc 2 (4.7)
C 2
acc



where Cmid is the estimated midpoint concentration by

interpolation, and Cacc is the accurate concentration at the

midpoint by solving the local Neumann problem in an element.

In this work the criteria for refinement and coarsening are

F1 = 0.06 and F2 = 0.001 respectively.



4.2.4 Comparison of Different Autoarid Schemes


The results in Fig. 4.2(a) and Table I show the

concentration errors in which the contour method gives an

order of magnitude larger error in peak concentration as

well as total dose than the heuristic method. Both methods

give junction errors less than 1% at a dopant concentration

of 1014 cm-3 as shown in Fig. 4.2(b). In general, this shows

that the ratio-based criterion gives good control of

junction error. The inefficiency in resolving the region at

the peak and tail area can be improved by adding the

techniques used in the heuristic method.

Comparing the BW method to the heuristic method in Fig.

4.2(a). and Fig. 4.2(b), the BW method shows a factor of 5











Concentration Error


C-3
cm-3
1019
1018
1017
1016

1014
1013
1012
1011
1010
109
108
107
106
105


Error(%) %

50. 1.00
50.
0.80
40. 0.60

30. 0.40
30.
0.20
20. -0.00
-0.20
-0.40
0. -0.60
-0.80
-10.
-10 -1.00

.0 p.m


1012


1015


1018 cm-3


#grid

700
650
600
550
500
450
400
350
300
250
200
150
100
50


Figure 4.2


Grid Points Number


0 2000 3600sec
The comparison of resulting profiles by different
methods of grid refinement, each of the resulting
profiles were benchmarked using a 1014 atoms/cm2
dose of boron and a 1 hour 10000 C anneal. (a). The
dopant concentration error of different autogrid
methods. (b). The junction depth error of different
autogrid methods. (c). The progression of grid
points number of different autogrid methods during
diffusion.


Junction Depth Error
.... ........................................ ....................................... ..............




.. ... ..... .................. .. ........




Contour
-- --- --------- --------- -----


I ;








Heuristic
................................. O.
................ .....




.. ...B .. .............. ... ... .. .. ...I .. ... .. .. ... .. ..
... .............................. ................. ............................. I............ ...









less error in the peak concentration. A better junction

accuracy of 0.2% error at the dopant concentration of 1014

cm-3 is also reached. As shown in Table 4.1, the results also



Table 4.1- Errors of Different Autogrid Methods


demonstrate the excellent control of dose which is improved

by a factor of 3 compared to the heuristic method.

The number of grid points in a 1 hour diffusion are

shown in Fig. 4.2(c). The BW method uses an average 200 grid

points while the heuristic method uses about 300. In

addition, the BW method is much more accurate than the

heuristic one. The average number of grid points of the

contour method is around 400. The CPU time consumed is also

shown in Table 4.1. The heuristic method is the most

efficient due to its simplicity and speed of computation; it

benefits from its simple evaluation compared to the more

complicated BW method. However, in this one-dimensional


Contour Heuristic BW


Peak Conc. -8.40 % -1.41% -0.31%



Junction 0.95 % 0.43 % 0.21 %


Total Dose -10.29% -1.51% -0.51%


CPU Time 184 sec. 83 sec. 88 sec.









case, the CPU times of the heuristic and BW methods are about

the same order. As shown in Fig. 4.2(c), the average grid

size of the heuristic method is about 2 times larger than the

BW method. In a two-dimensional case, this means 4 times more

nodes are required by the heuristic method.



4.3 Higher Order Correction of Flux Discretization


4.3.1 Different Flux Discretization Schemes


Better accuracy can also be achieved with higher order

flux discretization. Three different discretization methods

were investigated. The first is the simple linear

approximation to the flux terms. The second involves

assuming the concentration gradients vary exponentially and

that the logarithms of the concentration are linear across

the element. The simple Fick's law flux from nodes b to a,

Fa,b, is expressed as:



F = D (nCb nC) (4.8)
Fab h (InCb a




where D is the diffusivity and h is the distance between

nodes a and b.

The third method, proposed previously by Lowther [100],

also assumes that the flux varies exponentially in space.

This method is based on two ideas: The first is that the

concentration between two adjacent nodes is best

approximated by linear interpolation of the logarithms of









the concentrations. The second is that the flux between two

adjacent nodes should be treated as if there were an infinite

number of virtual nodes between the two actual nodes; and

that the total flux should be calculated by combining the

fluxes between each pair of adjacent virtual nodes as would

be done with resistors in series. This results in the

following equation for the flux Fa,b from nodes b to a:



h DCaCb (lnCb nCa)2 (4.9)
a, b f( f( 1 ) h (Cb- Ca)




where f(x) is the flux at x based on the assumed

interpolations. This evaluation of Fa,b, combined with an

analogous formula for drift, allows an approximate doubling

or tripling (per dimension) of the grid spacing needed.

Equation (4.9) is indeed the formula used previously.

However, for larger two-dimensional or three-dimensional

problems where a fine grid spacing is not guaranteed, ratios

of Ca/Cb > 104 may occur. This greatly increases the non-

linearity of the problem in regions far down on the tail of

the impurity distribution, It was necessary in (4.9),

therefore, to mix back in a small percentage (0.01%) of the

standard linear method to minimize nonlinearities associated

with these large ratios. An alternate method of compromising

between (4.9) and the standard linear method is now

proposed. Simply change the integral in (4.9) to a summation

over a finite, instead of an infinite number of temporary

nodes. The analogous calculation is then












F N (4.10)
a,b N

n=y1



where, based on the logarithmic interpolation of the

concentration, fn is the flux between virtual nodes n and n-

1. Now if the form for fn is chosen to be the simple linear

form, (4.10) becomes:


DN2/h DCaN2 (a- 1) (4.11)
a,b N h N

h 1-n
=1 n-1 n=1



where Cn = CaCn, and a = (Cb/ Ca) /N. Note that for N = 1,
(4.11) reduces to the standard linear discretization and to

(4.9) for N approaching infinity. The choice of N will

therefore be based on a trade-off between linearity and

accuracy. Since we are only looking for an allowed doubling
or tripling of the grid spacing (per dimension), it doesn't

make sense to allow N much larger. The value of N = 4 is

therefore ideal since it does not limit the accuracy and

because it is small enough to prevent problems with

nonlinearity. In fact, profiles obtained with N = 4 are

nearly identical to the final results obtained in the

previously published result [100]. Another advantage of this

form is in the limit where Cb approaches Ca. In (4.9), Fa,b

does approach zero but computationally this must be handled

as a separate case because the denominator also approaches









zero (not as fast as the numerator). With (4.11) this

complexity is eliminated. This method will be referred to as

the Summation method or the SN method where the subscript

N is the number of terms in the summation in (4.10) and

(4.11). Likewise the method proposed in [100], (4.9), will

be referred to as the S. method.



4.3.2 Comparison of Different Flux Discretization Schemes


In order to illustrate the results of different flux

discretization methods, simulations based on a fixed grid

are carried out with different grid spacings. The

simulations were done using the same example used

previously. In Table 4.2, the third method uses N = 4 for the


Table 4.2 Junction Depth of Different Discretizations
y--------------- ------- ---------- ------- ----------
acing 100 A 50 A 20 A


Linear 0.722 um 0.719 gm 0.718 gm


Logarithm 0.719 gm 0.718 gm 0.718 gm


S4 method 0.716 pm 0.718 gm 0.718 pm


summation in (4.10). The junction depths at concentration of

1014 cm-3 are compared as shown in Table 4.2. These results

should be compared to a junction depth of 0.718 pLm obtained

by simulation using a 10 A fixed core grid. It is obvious









that the S4 method defines a shallower junction due to its

smaller flux description. The benefit of this method is the

allowing of a coarser fixed grid spacing for a specified

error. As shown in Table 4.2, a junction depth of 0.718 pm

requires a fixed grid spacing of 20 A by standard flux

discretization while in the S4 method, a fixed grid spacing

of 50 A is used to achieve the same accuracy in junction

depth. This shows that the allowed increase of spacing is

between doubled and tripled. With the fine grids used here,

the S4 and Smethods are essentially the same.

By applying the BW grid refinement scheme, the results
of three flux discretization methods have been compared as

shown in Fig. 4.3. The concentration errors in Fig. 4.3(a)

show that the error near the tail portion has been largely

reduced by using a higher order flux discretization method.

This is due to the more accurate description of the diffusion

flux which is smaller than a standard linear flux. The final

dopant profile from higher order flux discretization is more

accurate as shown in Fig. 4.3(b). As before, the results of

S4 and S, are basically the same.

Fig. 4.3(c) shows the progression of the number of grid

points during the transient simulation. In general, the

higher order methods require more grid points. This is due to

several factors. First, the higher order methods give a more

accurate flux which, in general, results in a steeper

profile. A steeper profile requires more grid points to

represent it to the same accuracy. A second reason is due to

the linear approximations used in the error estimator. This











Concentration Error


cm-3
1019
1018
1016

1015
1014

1013
1012
1011
1010

109
108
107
106
105


Error(%) %

S60. 1.00
0.80
50. 0.60

40. 0.40
40.
0.20
30. -0.00

-0.20
20.
-0.40
10. -0.60
-0.80
0. -1.00


1018 cm-3


Grid Points Number


Figure 4.3


The comparison of resulting profiles by different
methods of flux discretization, each of the
resulting profiles were benchmarked using a 1014
atoms/cm2 dose of boron and a 1 hour 1000 C anneal.
(a). The dopant concentration error of different
methods. (b). The junction depth error of different
methods. (c). The progression of grid points number
of different methods during diffusion.


Junction Depth Error

.. ........................................ .........................





Log Linear



... ............................................................. .. ..... .1.......... ...
.. ........................................ ....................................... ..............

....................................... .... ................................. ..............

..r


1012


1015


#grid

360
340
320
300
280
260
240
220
200
180
160
140
120
100


-- -------- ........ .......... 11 ------~~
...................................................... .........................................
... .................................................. .......................................... ..
.. .. .. .. .. .. .. .. ..I .. .. .. .. .. .. .. ............, ...........................



.. ........................ .................~.... I ............... ......~. .........~....
...............................................*...... ........................................ .
... ................................................... ......................................... .




-------- ---------'" ^ ; ........... -- "

^^^- ^-^- *'4 S
... .. ...... .....-.............^. ......^^ ...............................
Liear ^-^ ^^^ -


2000


3600sec









tends to increase the error estimate when nonlinear fluxes

are involved.

It should be pointed out that the application of the

Summation method in the adaptive grid scheme will largely

improve the accuracy of the dopant profile for the same

number of grid points. By using the adaptive gridding

scheme, the junction depth error was reduced by a factor of

10 and the peak concentration error was improved by a factor

of 5 in comparison to a fixed 100 A grid spacing which

requires a grid size of 200. The results in Table 4.3 show



Table 4.3 Errors of Different Discretizations


that the higher order flux discretizations have roughly the

same accuracy and CPU time. Compared with the standard flux

discretization, the S, method improves the accuracy by an

order of magnitude while the CPU time increases by a factor

of 2.5. It is interesting to note that the previous result in

the fixed gridding case [100] shows a smaller ratio of 1.1 in


Linear Log S4 method

Peak Cone. -0.11% -0.04 % -0.01%


Junction 0.31 % 0.05 % -0.03 %


Total Dose -0.27 % -0.28 % -0.29 %


CPU Time 54 sec. 138 sec. 132 sec.









speed between S, and linear method. The larger increase in

computation time found here is due to several factors.

First, more grid points are used for the higher order

methods. Second, in these one-dimensional simulations, the

CPU time is dominated by the assembly of the Jacobian. The

assembly of the S4 method is slower due to evaluations of

powers and logarithms of the concentrations that are

required.

In considering the trade-off between the accuracy and

efficiency, the junction depth error (Errorj in Table 4.4) as

an accuracy indicator and the CPU time as an efficiency

indicator are shown in Table 4.4. The adaptive grid scheme


Table 4.4 Trade-off between Accuracy and Efficiency


Grid Size 200 400 1000 Adaptive

Methods (100 A) (50 A) (20 A) Grid
m -------------- mm-

Errorj 0.51% 0.15 % 0.03 % 0.32%
Linear

CPU time 22 sec. 56 sec. 176 sec. 54 sec.

Errorj 0.06 % 0.02 % 0.004 % 0.05 %
Logarithm

CPU time 57 sec. 140 sec. 409 sec. 138 sec.


Errorj -0.33 % -0.07 % -0.009 % -0.03 %
S4 method
CPU tim 63 sec. 184 sec. 399 sec. 132 sec.









uses the BW method and the final grid size is around 100 grid

points. For the linear discretization, the CPU time

consumption in the adaptive grid case is about the same as a

simulation by fixed grid spacing of 50 A which requires grid

size of 400 grid points. The great benefit of an adaptive

scheme is the resulting grid size at the end of diffusion.

The fixed grid case requires 4 times more grid points. The

optimal grid obtained in the adaptive grid simulation of

dopant diffusion will largely increase the computational

efficiency in the multiple-step process of simulation.



4.4 Conclusion


Three different automatic refinement strategies are

implemented in a one-dimensional diffusion solver and

compared. The Bank-Weiser method appears the most promising

in one-dimension and has a simple extension to multiple

dimensions. It has better performance in computation error

with only slightly more CPU time than the heuristic method.

The results of the heuristic method is favorable in

consideration of the computational speed due to the simpler

evaluation of the criterion for grid refinement. However,

the strong dependence of the tuned parameters will hinder

the application in various classes of process simulation.

The linear flux discretization tends to overestimate

the amount of diffusion. The logarithmic flux will be

effective enough to correct the flux discretization and

improve the accuracy. The S4 and S, methods give more

accuracy than linear and logarithmic flux under the









condition of coarse fixed grid spacing; and, for a given

accuracy, they also allow a coarser grid spacing. The new

method presented here, S4 achieves the same accuracy as the

S, method, but it is much simpler to code and gives better

computational efficiency.

On a fixed grid, substantial accuracy improvement is

available by utilizing a higher order discretization such as

the Summation method or logarithmic discretization. The

Summation method offers the best accuracy of the higher

order flux methods without substantial additional CPU time.

There is a CPU penalty in comparison to the linear

discretization due to more Newton iterations, but given the

difficulties in the fixed grid generation, it is still

desirable to use the S4 method.

However, in the case of an adaptive grid, the linear

discretization appears preferable. Because the grid itself

adapts to solve the profile accurately, little additional

accuracy is obtained using the higher order flux

discretization. CPU times are increased because of the

additional Newton iterations as well as the additional grid

points required to resolve the more steeply varying fluxes.

Overall, an adaptive grid with a linear discretization

appears to be the best approach.















CHAPTER 5
LOCAL REFINEMENT IN TWO-DIMENSIONS


5.1 Introduction


This chapter describes the implementation of a nodal

error estimator and its application in the two-dimensional

process simulation for dopant implantation and dopant diffu-

sion. The benchmark of the CPU time consumption and accuracy

is presented and discussed. In order to access the accurate

solution, the testing structure and the design of a two-

dimensional stripe mesh with fixed grid spacing of 10 A is

used to compute the solutions for comparison with the solu-

tions obtained by adaptive algorithm.

This work continues the development of grid adaption

and flux discretization in one-dimension. The previous one-

dimensional results are used as a base to develop refinement

algorithms for two-dimensional process simulation. A robust

method utilizing automatic mesh refinement and higher order

flux discretizations [104] for diffusion modeling has been

developed for dopant simulation after the investigation in

one dimension in chapter 4. The new nodal error estimator

discussed here will enhance the adaptive algorithm and allow

it to extend to two dimensions. Moreover, a novel initial

mesh generation for dopant implantation based on this new









nodal error estimate is first introduced here and will be

elaborated in next chapter.



5.2 The Error Source



In chapter 4, the concept of discretization error esti-

mation is realized in the one-dimensional investigation. It

illustrates the requirement for grid adaption and the higher

order discretization to improve the accuracy of numerical

solution [30], [105]. To extend the idea into two-dimen-

sional work, the derivation of some appropriate two-dimen-

sional error estimator requires more rigorous mathematical

examination on the problem to be solved. For the process sim-

ulation, the most important partial differential equations

system (PDEs) is the dopant diffusion equations. In the sim-

plified form, it is the transient diffusion equation



ac
-VF
t (5.1)


with F = DVC




,where C is dopant concentration, D is the diffusivity of

dopant and F is the diffusion flux. By integrating and apply-

ing Gauss's theorem, Equation (5.1) becomes the following



A = F. (5.2)
A 1









For the numerical simulation of semiconductor process

in two-dimension, the numerical error sources are mainly due

to: 1) the discretization error of the Equation (5.2), e.g.

the evaluation of diffusion flux F. 2) the spatial discreti-

zation error of the domain of interest for the PDEs to be

solved, i.e. the mesh. In Fig. 5.1, a mesh illustrates the

simple two-dimensional discretization. The domain of inter-

est is divided into triangles. The area associated to each

node is bounded by the perpendicular bisectors of the edges

connected to this node at the shaded area as in the Fig. 5.1.


xi+l


h
--


Figure 5.1 A simple two-dimensional discretization of
space.The shaded area is the nodal area
associated with node b.









The following two subsections will discuss the error contri-

bution due to these two factors via the method for evaluation

the area integral (the left-hand-side of Equation (5.2)) and

the line integral (the right-hand-side of Equation (5.2)).

For the simplicity in the derivation, only the one-dimen-

sional discretization along the path a-b-c is considered.



5.2.1 The Evaluation of Line Integral


In Fig. 5.1, considering the edge flux Fba, between node

b and node a, the line integral in Equation (5.2) can be

evaluated as




F.i- = Fbal (5.3)
1


,where 1 is the length of the perpendicular bisector of the

edge between node b and node a. The diffusion edge flux Fba
in Equation (5.3) then can be evaluated as



ac
Fa = DVC =D (5.4)
ba -ax




The problem about the accurate evaluation of this edge flux

Fba has been discussed and solved in chapter 4. The better

accuracy in diffusion flux evaluation improves the accuracy

of the line integral approximation.









In Equation (5.4), the more rigorous analysis of the

error associated with the approximation to the edge flux can

be carried out. Simply by approximating the partial differ-

ential operator with Taylor series expansions about the mid-

point between node b and node a, the Fba in Equation (5.4)

can be expressed as





Sa Cb -Ca (h/2)- (h/2)dc (h/2)2 2+ (h/2)2d c
Dx h 2 dx2 6 dx3

3
SCb-Ca (h/2)2d c +O(h4
-h 6+ 6 d3


C -C
= Db a+O(h2)
h


SFba (5.5)





,where Cb and Ca are dopant concentration at node b and node

a respectively, h is the grid spacing between node b and node

a. From Equation (5.5), the popular linear flux discretiza-

tion has the can be expressed as



Cba D a
F hDb a (5.6)
ba h




Full Text
xml record header identifier oai:www.uflib.ufl.edu.ufdc:UF0008236100001datestamp 2009-01-28setSpec [UFDC_OAI_SET]metadata oai_dc:dc xmlns:oai_dc http:www.openarchives.orgOAI2.0oai_dc xmlns:dc http:purl.orgdcelements1.1 xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.openarchives.orgOAI2.0oai_dc.xsd dc:title Adaptive simulation of dopant implantation and diffusion for advanced semiconductor process simulation dc:creator Lin, Chih-Chuandc:publisher Chih-Chuan Lindc:date 1995dc:type Bookdc:identifier http://www.uflib.ufl.edu/ufdc/?b=UF00082361&v=0000133393615 (oclc)002045454 (alephbibnum)dc:source University of Florida



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