Table of Contents - University of Texas at Austin · A.1 Installation ... 2 2 0 4 χ ω χ ε πρ...

Copyright

by

Di Li

2002

The Dissertation Committee for Di Li Certifies that this is the approved version of the following dissertation:

Three-dimensional Monte Carlo Simulation of Ion Implantation

Committee:

Sanjay K. Banerjee, Supervisor

Al F. Tasch, Co-Supervisor

Jack C. Lee

Leonard F. Register

Graham F. Carey


by

Di Li, B. S., M. S.

Dissertation

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

The University of Texas at Austin

December 2002

To my wife, my parents and grandparents

Acknowledgement

First of all, I would like to thank my supervisors and my role models, Drs.

Sanjay K. Banerjee and Al F. Tasch for their invaluable guidance and support

throughout my four-year Ph.D. study. Many thanks to Drs. Jack C. Lee, Leonard

F. Register and Graham F. Carey for taking their time to serve on my committee,

and for their comments and help.

Many thanks also to my fellow workers, and former graduate students,

Geng Wang, Yang Chen, Borna Obradovic, Ganesh Balamurugan, Li Lin and

Steve Morris for the inspiring discussions.

Semiconductor Research Corporation (SRC) has contributed to the support

of this work. I am also grateful to Shyh-Horng Yang, Chuck Machala and Texas

Instruments Inc. for supporting my summer internship in 2001.

Finally, I would like to thank my wife, Ding Yuan, my brother, Mai Li,

my parents and my grandparents for their love, understanding, encouragement and

expectation all these years.

v


Publication No.: ___________________

Di Li, Ph.D.

The University of Texas at Austin, 2002

Supervisors: Sanjay K. Banerjee, Al F. Tasch

A physically-based 3-dimensional Monte-Carlo simulator has been

developed within UT-MARLOWE, which is capable of simulating ion

implantation into multi-material systems and arbitrary topography. Introducing

the third dimension can result in a severe CPU time penalty. In order to minimize

this penalty, a three-dimensional trajectory replication algorithm has been

developed, implemented and verified. More than two orders of magnitude savings

of CPU time has been observed. An unbalanced Octree structure was used to

decompose three-dimensional structures. It effectively simplifies the structure,

and offers a good balance between modeling accuracy and computational

efficiency. This simulator was also extensively verified by comparing the

integrated one-dimensional simulation results with Secondary Ion Mass

Spectroscopy (SIMS).

vi

Table of Contents

Acknowledgement ...........................................................................................................v

Table of Contents...........................................................................................................vii

Chapter 1. Introduction ....................................................................................................1

Chapter 2. Ion Stopping Models and Damage Models ....................................................3

Chapter 3. Low Energy Consecutive Implants into SiO2-Capped Si...............................7

3.1 Introduction ........................................................................................................7

3.2 Model Description and Calibration ....................................................................8

3.3 Results and Analysis...........................................................................................9

3.4 Conclusion ........................................................................................................16

Chapter 4. Three-dimensional Simulations....................................................................21

4.1 Introduction ......................................................................................................22

4.2 New 3-D Trajectory Replication Scheme.........................................................24

4.3 Structure Decomposition ..................................................................................34

4.4 Model Verification ...........................................................................................35

4.5 Example application .........................................................................................49

4.6 Conclusion ........................................................................................................49

Chapter 5. Analytical 1-D and 2-D Ion Implantation Models Based on Legendre

Polynomials....................................................................................................................52

5.1 Introduction ......................................................................................................53

vii

5.2 One-dimensional Modeling ..............................................................................54

5.3 Two-dimensional Modeling .............................................................................66

5.4 Two-D Look-up Table Generation and Simulation Results .............................71

5.5 Conclusion ........................................................................................................80

Chapter 6. Conclusions and Future Work......................................................................81

6.1 Conclusions ......................................................................................................81

6.2 Future tasks.......................................................................................................82

Appendix A: A Brief Manual of TOMCAT ..................................................................83

A.1 Installation ........................................................................................................83

A.2 Quick Tutorial...................................................................................................86

Appendix B: Block Diagram and Flow Chart................................................................95

Bibliography ..................................................................................................................99

Vita...............................................................................................................................105

viii

Chapter 1. Introduction

Ion implantation will remain the dominant doping method to introduce

dopant atoms into silicon for its high throughput and precise control over

implanted dose and areas as comparing with emerging alternative methods, such

as Gas Immersion Laser Doping (GILD) [1,2]. Ion implantation is widely used

with energy ranging from sub-keV to several MeV for the purpose of source/drain

extension formation, source/drain formation, Vt adjust implant, halo implant, and

well formation. Among different profiling techniques for as-implanted profiles,

Secondary Ion Mass Spectroscopy (SIMS) [3] provides an efficient and economic

way to probe 1-D dopant profiles. In 2-D, at the present time, accurate and

efficient profiling techniques are still under development [4,5]. Three-dimensional

experimental profiling techniques are as yet impractical. On the other hand, rapid

advances in Ultra-Large Scale Integrated (ULSI) technology introduced radical

and highly non-uniform structures such as raised source/drain silicon-on-insulator

(SOI) devices [6], and double gate devices [7,8]. Therefore, accurate 2- and 3-

dimensional dopant and damage profiles are of crucial importance. In particular,

to reduce technology development cycle time and cost, the ability to predict the

doping profiles accurately and efficiently of implants into 2- and 3-D complex

structures is highly desirable.

1

Currently, three different types of implant simulators exist. They are

analytical (compact) model based on complex functions or polynomials [9,10,11],

Molecular Dynamics (MD) model [12] and Monte Carlo model [13-17].

Analytical model utilizes readily available profile-implant condition relation

(obtained from experiments or other simulation method) and interpolation

techniques (linearly, logrithmatically or so) to cover major user-specified implant

conditions. This type of implant simulators is used in almost all major

commercially available TCAD tools and is very efficient, yet not very predictive.

On the other hand, molecular dynamics (MD) models are fairly accurate and

predictive by explicitly calculating the rigorous interaction of all the particles,

including all the dopant and the silicon atoms, in the volume of interest. The

complete treatment makes MD models very appealing in terms of the first

principles calculations of displacement energy, and migration energy of dopant

atoms to simulate dopant diffusion. Yet the amount of calculations needed to treat

hyper-thermal processes such as implantation almost forbid its usage in

implantation simulation. Therefore, various techniques have been developed and

approximation has been made to make the method feasible for realistic implant

simulations. Finally, Monte Carlo method, based on physically-verified stopping

models and damage model, is both predictive and computationally-efficient and is

going to be studied extensively in later chapters.

2

Chapter 2. Ion Stopping Models and Damage Models

This section reviews the physics of ion implantation and concentrates on

the physical models used in the three-dimensional Monte Carlo simulator

developed in this work, as well as UT-MARLOWE [18]. Verification of these

physical models is presented in later chapters.

2.1. Stopping Model

During the process of ion implantation, energetic ions travel in the solid

and lose their energy through a series of collisions with the atoms in solid. The

target atoms can gain enough energy to be dislodged from their original sites,

travel through the solid and collide with other atoms, resulting in cascade

collisions. The displaced atoms (interstitials) and the vacant sites (vacancies) will

in turn affect the implanted ions later on.

The interactions between ions and a solid are simulated under Binary

Collision Approximation (BCA) approach and therefore, decoupled into three

parts, nuclear stopping, non-local electronic stopping and local electronic

stopping, described in the following.

The nuclear stopping power is calculated under the binary collision

approximation using the ZBL [19] universal potential

( )r

eZZrV u

221⋅Φ= (2.1)

3

where Φu is a pair specific screening function whose value is available

analytically. The interaction is short range and the stopping power is inversely

proportional to the impact parameter. Nuclear stopping causes target atoms to

escape from their sites, become interstitial atoms, and leave behind vacant sites.

Non-local electronic stopping is also mediated through the electric field

and can be described by the following equations.

( )( )ωε

ωωπε ,

10

20

2

1

kd

kdk

vqZi

dxdE kv

kv∫∫+

−

∞

= (2.2)

Taking into account exchange and correlation [Ma, 1992], one obtains the

following simplified formula for proton stopping:

≥

+−

−

+

<

=

=

=

FFF

gp

F

F

p

vvvv

vv

Emv

vvvvk

L

me

Lmve

dxdE

5.1,314

3532ln

5.1,)(

4

4

422

222

2

3

0

22

220

4

χω

χ

επρ

ω

ερπ

η

(2.3)

where m is the electron rest mass, )(1 02 akFπχ = , a0 is the Bohr radius, kF is

the Fermi wave number, and vF is the Fermi velocity. Here Eg is the band gap of

silicon at the gamma point and is equal to 4.8 eV. The stopping power for a

heavier particle can be obtained by multiplying the proton stopping by the square

4

of the ion charge [20,21], which is calculated by taking into account the ionization

of the moving particle and charge screening effects [19].

Local electronic stopping model is based on electron exchange theory

[22]. It assumes that energy transfer from the ion to the target occurs by

exchanging electrons between these two atoms, and the electrons transferred to

the ion cause it to slow down. The amount of energy loss depends on the

velocities of the atoms and on how close they approach [18].

( )( ) ( )

≥⋅

<⋅

+⋅⋅

+⋅=∆

critB

crit

critB

ZZRZZE

ννννν

νννν

23

121min

35

21

)25.5exp(5 (2.4)

where vcrit denotes the critical velocity beyond which the electrons of the two

atoms will not have sufficient time for free interaction, so the energy transfer will

diminish.

In order to obtain a smoother transition across the critical velocity, an

analytical transfer function is used [18] instead,

( )

−+

−=

22

2exp1exp2critcrit

F νν

ννν (2.5)

2.2. Damage Model

To describe the damage accumulation during the implant process, two

models are currently available. A rigorous model which follows every cascade is

the Kinetic Accumulative Damage Model (KADM) [17], while a simpler one is

5

the modified Kinchin-Pease model [23,24]. In terms of computational and

memory efficiency, a modified Kinchin-Pease formula is more attractive, which

describes the number of defects as a function of deposited energy,

dEEn

2κ

= (2.6)

where =0.8 is a constant, and Eκ d is the displacement threshold energy,

commonly accepted value being around 15 eV for crystalline silicon. Not all of

the defects calculated from Eq. (2.6) will survive, since some of the defects will

recombine with other nearby defects generated in this or a previous cascade. The

net increase in point defects after recombination [24, 25] is given by

)1(αN

Nnfn rec −=∆ (2.7)

where n is given by Eq. (2.6), N is the local defect density, is the critical

defect density for amorphization, and f

αN

rec is a species-dependent parameter

denoting the fraction of defects surviving defect interaction during and between

cascades. The factor ( has been introduced in order to account for the

sub-linear damage growth with dose observed in the experiments. That is, for the

same deposited energy, the damage increase in a more seriously damaged region

should be reduced because of greater chances for the generated interstitials and

vacancies to recombine with each other.

)αNN−1

6

Chapter 3. Low Energy Consecutive Implants into SiO2-Capped Si

The low energy as-implanted profile is very sensitive to the cap oxide

layer thickness and Pre-Amorphization Implant (PAI) conditions, in addition to

implant energy, angle and dose. In this work, theoretical and experimental studies

have been carried out quantitatively to investigate these dependencies. It is found

that a 2nm difference in cap oxide layer thickness could result in a 10nm

difference in junction depth even for a heavy species implant such as Sb. With

and without PAI could result in more than a 35nm difference in junction depth for

low energy B implant. Using Ziegler-Biersack-Littmark (ZBL) pair-specific inter-

atomic potentials in the Monte Carlo ion implantation simulator, UT-

MARLOWE, consecutive implants of PAI and P-type Lightly Doped Drain

(PLDD) were simulated and above effects were accurately captured. The

comparison with Secondary Ion Mass Spectroscopy (SIMS) data shows that

excellent agreement is obtained for different cap oxide layer thicknesses and PAI

conditions between the simulated predictions and SIMS data. The comparison

with Rutherford Backscattering Spectroscopy (RBS) data shows that the damage

profiles are also correctly modeled.

3.1 Introduction

As integrated circuits continue to scale beyond the 100 nm regime, ultra-

shallow junctions are crucial in order to realize desired device performance

7

according to the International Technology Roadmap for Semiconductors [26]. To

engineer P-type Lightly Doped Drain (PLDD) implant profiles, Pre-

Amorphization Implant (PAI) with species such as Sb, provide additional

flexibility and is widely used. However, for low energy implants such as PLDD,

the as-implanted profiles are very sensitive to the cap oxide layer thickness and

PAI conditions. Therefore, developing predictable Monte Carlo models, which

can account for these dependencies is desirable for cost efficiency and reducing

development cycle time.

In this study, experiments were designed to study these dependencies

quantitatively. Based on ZBL pair-specific inter-atomic potential [27] and

Kinchin-Pease [23] damage model, an Sb implant model was developed and

incorporated into UT-MARLOWE [18]. Then, the PAI and PLDD steps were

simulated consecutively. The comparisons with experimental data shows that this

model is capable of accurately predicting both the impurity and the damage

profiles for different PAI and PLDD conditions for different oxide layer

thicknesses.

3.2 Model Description and Calibration

The Monte Carlo model is based on the binary collision approximation

(BCA) as described in Chapter 2. In other words, the stopping power of the fast

moving ion in solids is decoupled into three parts: nuclear stopping, local

8

electronic stopping and non-local electronic stopping. Typically, for low energy

and heavy species implants, nuclear stopping dominates among these three kinds

of stopping powers and we need to pay special attention to it. Therefore, instead

of ZBL universal potential [19], ZBL pair-specific inter-atomic potential [27] was

used for improved accuracy in simulating nuclear stopping power of the fast

moving ion. The electronic model mainly follows the ones developed within UT-

MARLOWE. In particular, local electronic stopping follows Firsov’s electron

exchange model [18], while non-local electronic stopping calculates the ion’s

movement through the potential established by polarization of the surrounding

electron sea [18]. To simulate the de-channeling effect caused by the damage

accumulation during the implantation process, the modified Kinchin-Pease [24]

damage model was used and calibrated (specifically, on the recombination factor)

for its simplicity, accuracy and computational efficiency.

3.3 Results and Analysis

A set of experiments was designed in order to understand the dependencies of the

as-implanted Sb profiles on energy, dose and cap oxide layer thickness, as well as

the dependence of as-implanted B and BF2 profiles on PAI. Chemical Vapor

Deposition (CVD) of oxides was used to cap (100) single crystal silicon with

oxides of different thicknesses for accurate control purpose. For the 0 nm cap

oxide wafers, a HF dip was performed to remove any residual oxide, and then all

9

the wafers were kept in nitrogen purge boxes prior to the implants for control

purposes. After the implantation and prior to the Rutherford Backscattering

Spectroscopy (RBS) measurement, the oxide layer was removed to improve RBS

accuracy. The dopant profile dependence on cap oxide layer thickness is shown in

Figs. 3.1-3. A 2nm difference in cap oxide layer thickness (between bare silicon

and 2nm oxide cap) could result in more than 10nm difference in the junction

depth (at the concentration level of 1×1018cm-3) for as-implanted dopant profiles,

even for heavy species such as Sb. As the cap oxide layer thickness increases, the

dopant profile becomes less sensitive to the cap layer differences. This sensitivity

is well captured by the new model when comparing the predicted results with

experimental SIMS data in Figs. 3.1-3. Using the damage profiles predicted by

the Sb implant, a second implant is simulated and the simulation results are shown

in Figs. 3.4-5 for B and BF2 respectively. Under the same B implant conditions,

with and without PAI implant could result in more than 35nm difference in the

junction depth (at a concentration level of 1×1018cm-3) for as-implanted dopant

10

Depth (Å)

0 100 200 300 400 500 600

Antim

ony

Con

cent

ratio

n (c

m-3

)

1e+17

1e+18

1e+19

1e+20

0 A oxide, simulation20 oxide, simulation50 angstrom oxide, simulation0 angstrom oxide, SIMS20 angstrom oxide, SIMS50 angstrom oxide, SIMS

Fig. 3.1 Comparison between model prediction and SIMS.

Implant condition: Sb, 10keV, 1014cm-2, on-axis.

11

Depth (Å)

0 100 200 300 400 500

Antim

ony

Con

cent

ratio

n (c

m-3

)

1e+17

1e+18

1e+19

1e+200 angstrom oxide, simulation20 angstrom oxide, simulation50 angstrom oxide, simulation0 angstrom oxide, SIMS20 angstrom oxide, SIMS50 angstrom oxide, SIMS


Implant condition: Sb, 10keV, 6×1013cm-2, on-axis.

12

Depth (Å)

0 100 200 300 400 500 600

Antim

ony

Con

cent

ratio

n (c

m-3

)

1e+17

1e+18

1e+19

1e+200 angstrom oxide, simulation20 angstrom oxide, simulation50 angstrom oxide, simulation0 angstrom oxide, SIMS20 angstrom oxide, SIMS50 angstrom oxide, SIMS


Implant condition: Sb, 15keV, 3×1013cm-2, on-axis.

13

Depth (Å)

0 200 400 600 800 1000

Boro

n C

once

ntra

tion

(cm

-3)

1e+16

1e+17

1e+18

1e+19

1e+20

no PAI, simulationSb 10keV 1e14, simulationSb 10keV 6e13, simulationSb 15keV 3e13, simulationSb 25keV 9e13, simulationSb 25kev 9e13, SIMSno PAI, SIMSSb 10keV 6e13, SIMSSb 10keV 1e14, SIMSSb 15keV 3e13, SIMS


Implant condition: B, 3keV, 2×1014cm-2, on-axis, through 2nm

oxide.

14

Depth (Å)

0 100 200 300 400

Boro

n C

once

ntra

tion

(cm-3

)

1e+16

1e+17

1e+18

1e+19

1e+20

Sb 10k 1e14, simulationSb 10k 6e13, simulationSb 15k 3e13, simulationSb 25k 9e13, simulationSb 10k 1e14, SIMSSb 10k 6e13, SIMSSb 15k 3e13, SIMSSb 25k 9e13, SIMS


Implant condition: BF2, 5keV, 1014cm-2, on-axis, through 2nm

oxide.

15

profiles. This dependence is also well predicted by UT-MARLOWE. After

performing the PAI, the as-implanted dopant profiles are no longer sensitive to

the cap oxide layer thickness even for light species implant such as B, as shown in

Figs. 3.6. This change of sensitivity is predicted very accurately by UT-

MARLOWE as well. Throughout these simulations, the defect profiles of PAI

implant play an important role. Moreover, they are also crucial in modeling the

Transient Enhanced Diffusion (TED) in annealing processes after the implants for

providing as-implanted interstitial and vacancy distributions. The comparison of

the defect profiles between simulation and RBS for different consecutive implant

conditions is shown in Fig. 3.7 and fairly good agreement is observed. Finally, the

Sb implant model based on ZBL pair specific potential is valid for a fairly wide

energy range (up to 100keV), dose range (5×1012cm-2-2×1014cm-2) and for both on

and off-axis implants, as shown in Figs. 3.8-9.

3.4 Conclusion

The Sb implant model has been developed and incorporated into UT-

MARLOWE based on ZBL pair-specific inter-atomic potential. Consecutive

implants of Sb and B or BF2 into SiO2 capped single-crystal Si are simulated and

the impact of PAI on PLDD and the impact of cap oxide layer thickness on PAI

and PLDD are correctly modeled.

16

Depth (Å)

0 200 400 600 800

Boro

n C

once

ntra

tion

(cm

-3)

1e+17

1e+18

1e+19

1e+20

1e+21

0 angstrom, simulation20 angstrom, simulation50 angstrom, simulation0 angstrom, SIMS20 angstrom, SIMS50 angstrom, SIMS


Implant condition: B, 3keV, 2×1014cm-2, on-axis 10keV 1e14

17

Depth (Å)

0 50 100 150 200 250 300

Amor

phiz

atio

n Fr

actio

n (%

)

0

20

40

60

80

100Sb 15k 3e13, RBSSb 15k 3e13, BF2 5k 1e14, RBSSb 15k 3e13, BF2 5k 2e15, RBSSb 15k 3e13, simulationSb 15k 3e13, BF2 5k 1e14, simulationSb 15k 3e13, BF2 5k 2e15, simulation

Fig. 3.7 Comparison between model prediction and RBS.

Implant condition: on-axis, through 2nm oxide.

18

Depth (Å)

0 200 400 600 800 1000 1200 1400

Antim

ony

Con

cent

ratio

n (c

m-3

)

1e+17

1e+18

1e+19

1e+20

2e14, simulation3e13, simulation5e12, simulation2e14, SIMS3e13, SIMS5e12, SIMS


Implant condition: Sb, 50keV, on-axis, through 1.6nm oxide.

19

Depth (Å)

0 500 1000 1500 2000

Antim

ony

Con

cent

ratio

n (c

m-3

)

1e+16

1e+17

1e+18

1e+19

1e+20

2e14, SIMS3e13, SIMS5e12, SIMS2e14, simulation3e13, simulation5e12, simulation


Implant condition: Sb, 100keV, off-axis, through 1.6nm oxide.

20

Chapter 4. Three-dimensional Simulations

A physically-based 3-dimensional Monte-Carlo simulator has been

developed within UT-MARLOWE, which is capable of simulating ion

implantation into multi-material systems and arbitrary topography. Introducing

the third dimension can result in a severe CPU time penalty. In order to minimize

this penalty, a three-dimensional trajectory replication algorithm has been

developed, implemented and verified. More than two orders of magnitude savings

of CPU time have been observed. An unbalanced Octree structure was used to

decompose three-dimensional structures. It effectively simplifies the structure,

offers a good balance between modeling accuracy and computational efficiency,

and allows arbitrary precision of mapping the Octree onto desired structure. Using

the well-established and validated physical models in UT-MARLOWE 5.0, this

simulator has been extensively verified by comparing the integrated one-

dimensional simulation results with SIMS (Secondary Ion Mass Spectroscopy).

Two options, the typical case and the worst scenario, have been selected to

simulate ion implantation into poly-silicon under various scenarios using this

simulator: implantation into a random, amorphous network, and implantation into

the worst-case channeling condition, into (110) orientated wafers.

21

4.1 Introduction

With the continued rapid scaling of modern semiconductor devices,

typically ion implantation is performed into topographically complex structures.

Flat topography implant simulations are no longer sufficient. Users need to have

the flexibility to simulate implants into three-dimensional, realistic structures

[28], which could be either outputs from process simulators, or user-defined

arbitrary structures.

This leads to severe challenges in two aspects, mainly in terms of

computational efficiency. First, as the ions are traveling into the solid, the

simulator needs to determine the material type, whether it is crystalline silicon,

silicon dioxide, or another material, at each specific spatial location along the

ion's trajectory, and choose the corresponding propagator dynamically. Secondly,

and more importantly, the calculation of particle propagation in solids is very time

consuming. This is particularly true for a deterministic Monte Carlo propagator,

required to ensure correct modeling of ion channeling in a crystalline lattice.

Especially, as the simulation volume becomes large for three-dimensional

simulations, in order to obtain the same level of statistical significance for the

overall structure, the number of incident ions at each specific grid cell must still

be kept approximately at the same level as for 1-D and 2-D simulations.

Consequently, the required number of ions should be proportional to the number

22

of grid cells present in the simulation volume. Two or three-dimensional

simulation of ion implantation into a flat or slowly varying topography

intrinsically is a one-dimensional simulation. A small number of grid cells (size is

on the order of a few lattice constants) in the lateral direction is sufficient to

obtain the point source results under rotational periodic boundary conditions. The

results then can be translated and superposed to cover the entire window to give

the final 2/3-D results. This is valid when one or two lateral directions are nearly

invariant. Although a relatively large number of grid cells are needed in the depth

direction (largely energy and species dependent), the total number of grid cells

could be kept reasonably small. The above routine is no longer valid when

performing two or three-dimensional ion implantation simulations into sharply

varying structures. For such cases, a large number of grid cells in both the depth

and the lateral directions have to be used. For example, to simulate a 130 nm gate

length MOSFET, the length on both lateral directions is on the order of 250 lattice

constants. This means, that to maintain the same level of statistical significance,

for each additional dimension, the required number of ions for Monte-Carlo

simulation should increase by about two orders of magnitude.

Hossinger et al has demonstrated three-dimensional capability in Monte-

Carlo ion implantation simulator, MCIMPL [29], and parallelization was

proposed to reduce the simulation time when a cluster of workstations is readily

23

available [30]. Furthermore, an interesting computational efficient combined

approach was proposed by utilizing the physically-generated 3-D point response

distribution functions [31]. Especially, in two-dimensional simulations, the lateral

trajectory replication technique has proven quite successful in enhancing the

computational efficiency [32]. Yet more aggressive, efficient CPU time reduction

techniques are needed for three-dimensional Monte-Carlo simulations of ion

implantation into highly non-uniform structures.

Although not adequate to be applied to three-dimensional simulation

directly, the lateral trajectory replication scheme provides good insight into the

problem. The key is to cleverly reuse the physically-computed trajectories for

similar topography. Interestingly enough, reusing the physically-generated

trajectories is the main idea behind most of the reported two and three-

dimensional analytical simulators as well [9,31,33-35]. While the success of this

approach when applied to simple surface topographies is clear, the validity of the

analytical approximations when applied to sharply varying structures remains

unclear. Thus, a computationally-efficient Monte-Carlo simulator with a modified

replication scheme would appear to be the ideal solution for such structures.

4.2 New 3-D Trajectory Replication Scheme

Typically, for realistic structures slowly varying or flat topography often

exists over quite a significant percentage of the total simulation volume. In such

24

regions, the lateral variation of both dopant and defect profiles is small and

negligible. Therefore, in such regions, the physical calculation of a representative

small portion would be sufficient after the statistical criteria in terms of material

and structural homogeneity have been satisfied. In other words, each physically-

computed individual trajectory of that small portion might be translated efficiently

laterally given that the topography is similar, and this needs to be done

dynamically. While replication proceeds as often as possible in the slowly varying

portion of the structure, the ions' behavior is physically and individually

computed in the rapidly varying parts. The new replication scheme is based

precisely on this idea, based on the observation that, typically, similar or identical

topography exists in nearby segments.

Laterally, the new replication scheme is shown in Fig. 4.1, which is

similar to the lateral trajectory replication scheme discussed in 2-D simulations

[29]. The difference is that here it tests the whole neighboring area for replication,

in an effort to achieve the maximum savings on CPU time. While in 2-D, the

starting point and replication directions are randomly decided and the replication

number is set to around 10 in order to maintain some variation of the initial

conditions, in 3-D, replication starts by physically calculating all the required

scattering events and the stopping mechanisms experienced by one particular ion

25

Fig. 4.1 Side view of the replication algorithm proceeding from left to right.

The light trajectories are physically computed, the gray ones show

where replication error occurs, and the black ones are replicated

trajectories.

26

until it stops. After the ion's trajectory is obtained, its relative displacements

between consecutive scattering events are recorded, as well as the states of the ion

before every scattering, the material type and damage situation along the ion's

trajectory. The algorithm then introduces an offset and replicates the model

trajectory at the new incident position. As the line segments are being copied

along the trajectory, replication error is examined according to two factors,

whether the material in the model trajectory matches that in the target trajectory

and whether the damage level is comparable between them. The above scheme

can be somewhat relaxed to achieve some tradeoff between computational

efficiency and accuracy, in an effort to achieve better CPU time savings and

accommodate the replication scheme to the slow varying topography commonly

encountered in realistic structures. Specifically, the total material mismatch

distance is calculated, and the damage levels are categorized logarithmically into

five different levels. The criteria are set that the total mismatch distance should

not exceed two lattice constants and the damage level should belong to the same

category.

In 3-D, the goal is to take maximum advantage of similar topography in

two lateral directions through replication. The new replication scheme proceeds as

shown in the top view in Fig. 4.2. The implant window is divided into small

rectangles or squares. The first ion is set to implant into the left bottom corner

27

Y (Depth direction)

Z (lateral direction)

X (lateral direction)

Fig. 4.2 Top view of the simulation domain illustrating the replication

algorithm. The hollow circles are physically computed, the gray ones

show that replication errors are encountered and trajectories are

therefore physically computed, and the black ones are replicated. The

gray circles could also be represented by the hollow ones since these

trajectories need to be physically calculated as well. Replication starts

from the left bottom corner and proceeds up and to the right of the

original trajectory.

28

segment and the trajectory is physically-computed and recorded. The initial

relative position within the segment is determined randomly and used for every

rectangle until the whole window is covered. Replication first goes up and

continues moving up until a replication error is detected. Replication stops when

an error occurs in the z direction, and the place (Zstop, the rectangle number) is

recorded. Then, replication proceeds to the right of the original (physically-

computed) trajectory. When an error occurs in the x direction at the new position,

the trajectory is recomputed and the model trajectory is replaced with a newly

calculated one. Otherwise, the model trajectory is simply copied to the new

position along the x direction. After finishing the operation either way, replication

goes up and the above procedure is repeated until replication reaches the rectangle

at the right end and an error is detected in the z direction. Then, replication starts

from the left-most column with minimum Zstop, and continues until Zstop of every

column equals to Zmax (total row number). Thus, the simulator is able to

intelligently reuse physically-calculated trajectories in two directions. Since the

CPU time required to replicate is much less than the time required for the actual

physical computation, large savings are expected with the application of the new

replication scheme without sacrificing accuracy, as shown in Figs. 4.3-4.

29

Implant

Fig. 4.3 (a) Three-dimensional simple trench topography viewing from

different angle for replication scheme demonstration. The length of

the whole volume is 800 lattice constants, and the depth is 700

lattice constants.

30

Fig. 4.3 (b) Simulating a boron 5keV implant into the structure shown in Fig.

4.3(a), using a tilt angle of 7°, a rotation angle of 30°, and a dose of

1×1014cm-2. The CPU time is 72 minutes using 5,000,000 ions. The

iso-concentration contours range from 1×1015cm-3 to 1×1017cm-3

viewing from different angle. The estimated CPU time without

replication is over 20 hours.

31

implant

STI

S/D Gate

S/D

Gate

S/D Gate

Fig. 4.4 (a) Source/Drain structure with shallow trench isolation. The side

length is 600 lattice constants.

32

Fig. 4.4 (b) Simulating an arsenic 15 keV implant into the structure show in

Fig. 4.4 (a) using a tilt angle of 0°, a rotation angle of 0°, and a

dose of 1×1013cm-2. The CPU time is 88 minutes, using 5,000,000

simulated ions. The iso-concentration contours range from

1×1016cm-3 to 1×1019cm-3. The estimated CPU time without

replication is over 20 hours.

33

4.3 Structure Decomposition

As the ions propagate through the structure, the determination of the

material type is crucial and could be very CPU time consuming. This is

particularly true for highly non-uniform structures. Since an unbalanced Quadtree

proved to be quite successful in two-dimensional simulations [29, 36], an

unbalanced Octree becomes a natural choice for 3-dimensional simulations and is

adopted in this new simulator. Similar to the corresponding Quadtree

decomposition, Octree also recursively subdivides space until the desired

resolution is reached. Primarily, the Octree structure offers three major

advantages over structured and unstructured meshes. First, it breaks down the

solid into a set of regular cubic volumes of different sizes and simplifies the

structure, i.e., determines the intersections of the ion's trajectory and material

boundary and tests whether the ion is inside a particular cell, in an elegant and

efficient manner. Secondly, the resulting grid is coarse where the material is

uniform, and fine near the material interfaces. This results in great saving in

storage and CPU time. Finally, the tree data structure makes the tracking more

efficient (proportionally to logN, instead of N), and reduces the zoning time

accordingly. Generally, the Octree offers a good balance between accuracy,

memory efficiency and speed.

34

4.4 Model Verification

The physical models used by the new simulator (including nuclear

scattering, local and nonlocal electronic stopping models and modified Kinchin-

Pease models) are based on UT-MARLOWE [18, 37]. Specifically, BF2 implant

was treated as a molecular cluster. In other words, the trajectories of both boron

and fluorine were physically calculated and the damage generated by both species

was simulated [37]. The models have been extensively verified in 1-D within UT-

MARLOWE. However, verification is still necessary in this 3-D simulator, as the

models are implemented in a quite different way. Results are shown in Figs. 4.5-9

and good agreement with experimental SIMS data are obtained.

In this simulator, two models have been provided to simulate ion implantation

into poly-silicon since the grain size and the grain orientation of poly-silicon are

largely dependent on the process flow and hard to analyze. The first model, as the

typical case, uses amorphous silicon to represent poly-silicon and applies a

corresponding stochastic Monte Carlo approach. The second one, which is the

worst- case channeling scenario, uses a deterministic Monte Carlo model and

simulates implant into a (110) wafer. Implantation into a (100) wafer with both tilt

angle and rotation angle set to 45 degrees is equivalent to implantation into (110)

wafer with both tilt angle and rotation angle set to 0 degrees, because the silicon

matrix is identical. Therefore, as shown in Fig. 4.10, after obtaining a point source

35

Depth [µm]

0.0 0.1 0.2 0.3 0.4 0.5

Boro

n C

once

ntra

tion

[cm

-3]

1e+16

1e+17

1e+18

1e+19

1e+20

1e+21

SIMS Simulation

Fig. 4.5 Profiles of 35 keV boron on-axis implants into a (100) Si wafer

with doses of 1×1013cm-2, 3×1013cm-2, 1×1014cm-2, 5×1014cm-2,

2×1015cm-2, and 8×1015cm-2. The gray lines are the simulation

results, while the black lines are SIMS data.

36

Depth [µm]0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

As C

once

ntra

tion[

cm-3

]

1e+16

1e+17

1e+18

1e+19

1e+20

1e+21

1e+22

SIMS Simulation

Fig. 4.6 Profiles of 50 keV As on-axis implants into a (100) Si wafer with

doses of 1×1013cm-2, 3×1013cm-2, 1×1014cm-2, 5×10-14cm-2,8×10-

15cm-2. The gray lines are the simulation results, while the black

lines are SIMS data.

37

Depth [µm]

0.00 0.02 0.04 0.06 0.08 0.10

Boro

n C

once

ntra

tion

[cm

-3]

1e+16

1e+17

1e+18

1e+19

1e+20

1e+21 SIMS Simulation

Fig. 4.7 Profiles of 15 keV BF2 on-axis implants into a (100) Si wafer with

doses of 1×1013cm-2, 3×1013cm-2, 1×1014cm-2, 5×10-14cm-2, 2×10-

15cm-2, and 8×10-15cm-2. The gray lines are the simulation results,

while the black lines are SIMS data.

38

Depth [µm]

0.000 0.025 0.050 0.075 0.100 0.125 0.150

Boro

n C

once

ntra

tion

[cm

-3]

1e+17

1e+18

1e+19

1e+20

1e+21 SIMS Simulation

Fig. 4.8 Profiles of 35 keV BF2 on-axis implants into a (100) Si wafer with

doses of 1×1013cm-2, 3×1013cm-2, 1×1014cm-2, 5×10-14cm-2, 2×10-

15cm-2, and 8×10-15cm-2. The gray lines are the simulation results,

while the black lines are SIMS data.

39

Depth [µm]

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Phos

phor

us C

once

ntra

tion

[cm

-3]

1e+17

1e+18

1e+19

1e+20

1e+21

SIMS Simulation

Fig. 4.9 Profiles of 30 keV phosphorus on-axis implants into a (100) Si

wafer with doses of 1×1013cm-2, 1×1014cm-2, 5×10-14cm-2, and

2×10-15cm-2. The gray lines are the simulation results, while the

black lines are SIMS data.

40

Integration Direction

Rotation 45°

Fig. 4.10 Figure showing that by rotating the simulation results (tilt

0°/rotation 0°) in a (110) Si wafer by 45° and integrating along the

new depth dimension, the result is equivalent to implantation into a

(100) Si wafer (tilt 45°/rotation 45°).

41

profile in a (110) wafer, we can rotate it by 45 degree and get the 1-dimensional

profile by integrating it according to the new depth axis. Thus, instead of using

actual (110) wafer pieces [38], we can compare the converted 1-D profile with

SIMS data for the same implant condition into a (100) wafer with both tilt and

rotation set to 45 degrees. As shown in Figs. 4.11-12, for B and As, the SIMS data

and the converted simulation results match very well for several different

energies.

In addition, As channeling through a three-layered structure, polysilicon,

gate oxide and crystalline silicon was studied as shown in Fig. 4-13. Major

discrepancies were observed when comparing to simulation results for the same

implant conditions in [38]. At first, in [38], the dopant concentration in

polysilicon falls to or below 1×1018cm-3 at 100nm depth for all energies below

30keV. Our study shows that the concentrations remain above 2×1019cm-3 at a

depth of 100nm except for the 1keV case. Secondly, the peak concentration in

crystalline silicon is well below 1×1019cm-3 in [38], and it reaches 1020cm-3 for

energies above 3 keV according to our simulation. Finally, and most importantly,

in our result, the channeling of 15keV and 30keV implant continues in crystalline

silicon and goes well beyond 200nm and even to 400nm for 30 keV implants.

According to [38], the profile stops in crystalline silicon rather sharply (before

42

Depth [µm]

0.0 0.1 0.2 0.3 0.4 0.5

Boro

n C

once

ntra

tion

[cm

-3]

1e+15

1e+16

1e+17

1e+18

1e+19

Gray line: Simulation result converted to an implant into a (100) wafer, 45/45 (tilt/rotation). Black line: SIMS data of implant into a (100) wafer, 45/45 (tilt/rotation).

Simulation result of implant into (110) wafer, 0/0.

Fig. 4.11 (a) Profiles of a 15keV B on-axis implant into a (110) Si wafer with a

dose of 1×1013cm-2 to simulate the worst case channeling scenario

of an implant into poly-silicon.

43

Depth [µm]

0.0 0.2 0.4 0.6 0.8

Bor

on C

once

ntra

tion

[cm-3]

1e+15

1e+16

1e+17

1e+18

Simulation result of implant (0/0) into (110) wafer


Fig. 4.11 (b) Profiles of a 35keV B on-axis implant into a (110) Si wafer with a



44

Depth (µm)

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Ars

enic

Con

cent

ratio

n (c

m-3)

1e+15

1e+16

1e+17

1e+18Gray line: Simulation result converted to an implant into a (100) wafer, 45/45 (tilt/rotation). Black line: SIMS data of implant into a (100) wafer, 45/45 (tilt/rotation).

Fig. 4.12 (a) Profiles of a 15keV As on-axis implant into a (110) Si wafer with a



45

Depth (µm)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Ars

enic

Con

cent

ratio

n (c

m-3 )

1e+15

1e+16

1e+17


Fig. 4.12 (b) Profiles of a 50keV As on-axis implant into a (110) Si wafer with a



46

Depth (µm)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Arse

nic

Con

cent

ratio

n (c

m-3

)

1e+15

1e+16

1e+17


Fig. 4.12 (c) Profiles of a 100keV As on-axis implant into a (110) Si wafer with

a dose of 2×1012cm-2 to simulate the worst case channeling

scenario of an implant into poly-silicon.

47

Depth (µm)

50 100 150 200 250 300 350 400

Arse

nic

Con

cent

ratio

n (c

m-3

)

1e+17

1e+18

1e+19

1e+20

1e+21

1 keV3 keV8 keV15 keV30 keV

Fig. 4.13 Profiles of As on-axis implant into a three-layered structure

(100nm polysilicon, 2.7nm gate oxide and 700nm crystalline

silicon) at 1, 3, 8, 15, and 30 keV with a dose of 1×1015cm-2.

48

150nm). These discrepancies are possibly due to different As stopping models and

calibrations in the two simulators.

4.5 Example application

With unbalanced Octree decomposition scheme, in principle, implantation

into any arbitrary structures can be simulated. Recent studies show that double-

gate device structures such as FinFET [7] have superior scaling potentials over

traditional bulk CMOS or SOI CMOS, and are chosen here (shown in Fig. 4.14-

15). The highly non-planar and non-uniform nature typically makes the

application of analytical model insufficient.

4.6 Conclusion

A new computational efficient three-dimensional Monte Carlo simulator

has been developed within UT-MARLOWE. An unbalanced Octree algorithm

was used for spatial decomposition. A new trajectory replication scheme was

developed and implemented in order to enhance computational efficiency. More

than two orders of magnitude savings on CPU time have been observed.

49

Source/Drain

Gate Electrode

Gate Oxide

Channel Width

Structure decomposition by Octree scheme

Fig. 4.14 FinFET structure decomposed by unbalanced Octree.

50

Source/Drain

Gate Electrode

Fig. 4.15 Implant into FinFET structure and the dopant distribution.

51

Chapter 5. Analytical 1-D and 2-D Ion Implantation Models Based on

Legendre Polynomials

Computationally-efficient ion implantation modeling has become the

essential tool for efficient and accurate CMOS design as aggressive scaling of

devices continues. Specifically, computationally-efficient two-dimensional

analytical models are often more attractive than physically-based Monte Carlo

simulations since the latter are expensive in terms of computational time. Here we

present new computational-efficient analytical models to simulate 1-D and 2-D

impurity and damage profiles. Legendre polynomials are used as basis functions

in view of their orthogonality and good interpolation property. Conventional

superposition approaches for 2-D implant modeling are explained and the

shortcomings are analyzed. A dose splitting approach is incorporated in the new

2-D model to account for the nonlinear de-channeling effect as implantation-

induced damage accumulates. Good agreement with a physically-based and

experimentally verified Monte Carlo simulator (UT-MARLOWE with TOMCAT)

has been obtained for both impurity and damage profiles with a 50× reduction of

computational time for medium energy implants.

52

5.1 Introduction

As silicon MOS devices scale to the deep sub-micron regime, ion

implantation will continue to be an important method to selectively introduce

impurity atoms into the silicon substrate. Accurate and computationally-efficient

ion implantation models with detailed dependence on various implant conditions

are desirable for CMOS design and technology development. This becomes

especially important with increased process complexity and fabrication cost.

Particularly, a detailed knowledge of both impurity and damage profiles is

required to correctly model dopant diffusion during annealing and thermal

treatment steps after the implantation.

At present time, two major ion implantation modeling approaches are

widely used, each with their advantages and disadvantages. The first one uses the

Monte Carlo method [15,32,39,40] to simulate the transport of ions in silicon and

other material of interest, such as SiO2, Si3N4, and silicide. The Monte Carlo

approach can provide detailed understanding of the implant process, in particular,

ion-solid interactions and the damage generation. The major shortcoming is its

rather large computational burden due to the necessity of computing every

collision to simulate the trajectory of each individual ion. Especially, in an effort

to reduce the statistical fluctuations, which are intrinsic to the Monte Carlo

method, a large number of ions are required. Things get even worse for high-

53

energy implant simulations, and it typically takes hours and even days for MeV

implant simulations. On the other hand, the second approach, analytical modeling

[41,42,43], is computational efficient. Basically, it uses parameterized functions

to model the as-implanted profiles. This way, it replaces the calculations of every

collision by a pre-calculated look-up table of model parameters and evaluating a

few function values at different grid points. Typically it saves CPU time by a few

orders of magnitude over the Monte Carlo method, and it is the preferred

approach of most process simulators. The apparent drawback of the analytical

models is the limited range that current look-up tables cover. Within the

experimentally validated and calibrated range, however, analytical modeling is a

powerful tool for technology development.

5.2 One-dimensional Modeling

Several analytical modeling techniques have been proposed during the last few

decades to simulate the as-implanted one-dimensional impurity profiles. Among

those, the dual-Pearson model [41] is quite successful and very widely used. With

two Pearson functions to separately model the random scattering and channeling

components, it can accurately depict a wide range of as-implanted profiles.

However, the parameter extraction procedure in dual-Pearson model is

complicated and a good initial guess is required for the convergence of the

algorithm. Although software has been written to automatically generate a

54

reasonable initial guess, the parameter set may not be the unique one that provides

the best fit for the particular impurity profile.

Generally, the requirements for a good analytical implant model are the

following:

1. Accurate description of the as-implanted impurity and damage profiles.

2. Good interpolation properties.

3. Easy parameter extraction procedures.

When considering all these requirements, Legendre polynomials are a

good choice. They have a simple functional form, and can model both the

impurity and damage profiles very well over a wide range of parameters [9].

Specifically, the 1-D impurity and damage profiles are modeled as a linear

combination of 14 Legendre polynomials,

∑=

⋅⋅=13

0

15 )))('(exp(10)(i

ii xxLaxC , , (5.1) 21' bxbx +=

where, C(x) is the impurity concentration at depth x, Li(x') is the Legendre

polynomial of degree i, b1, b2 are constants that map x from the interval [xmin,

xmax] to [-1,1], and xmin and xmax are the depths between which the impurity and

damage concentration falls within the range of interest.

Legendre polynomials satisfies orthogonality with a unity weight function in the

interval [-1,1], i.e.,

55

(∫−

=1

1

,)()( jidxxLxL ji δ ) (5.2)

Thus, the 16 parameters describing the profiles can be determined

uniquely as follows:

∫−

⋅⋅=1

1

')'()'( dxxLxCai (5.3)

maxmin1

2xx

b+

= (5.4)

maxmin

maxmin2 xx

xxb

+−

= (5.5)

Although a relatively large number of coefficients are required, the

simplicity of the parameter extraction procedure and its good interpolation

property justifies the extra memory requirement.

UT-MARLOWE, a Monte Carlo simulator, was used to generate the as-

implanted impurity and damage profiles. This code has been validated over a

wide range of implant conditions and species. The output from UT-MARLOWE

was first averaged to get a smoother profile. The resulting profile is then fitted

using equations (3)-(5) and the derived parameters are stored in the look-up table.

The comparison between UT-MARLOWE result and fitted result are shown in

Figs. 5.1-9 for various implant conditions. Excellent agreement has been

56

Fig. 5.1 Comparison of the Legendre polynomial analytic model predictions

with boron impurity profiles obtained from UT-MARLOWE

simulations (histogram) for boron off-axis implant at various

energies and doses through 15 Å native oxide.

57

Fig. 5.2 Comparison of the Legendre polynomial analytic model predictions

with phosphorus impurity profiles obtained from UT-MARLOWE

simulations (histogram) for phosphorus off-axis implant at various

energies and doses through 15 Å native oxide.

58

Fig. 5.3 Illustration of the excellent interpolation property of the Legendre

coefficients. The histogram plots are UT-MARLOWE simulation

results and the solid lines are analytic fits. The profiles with the

following doses are actual fits to simulation results: 1X1013 cm-2,

5X1013 cm-2, 5X1014cm-2. The rest are results of interpolation

between model parameters at the above-mentioned doses. The BF2

implants are on-axis and through 15 Å native oxide.

59



results and the solid lines are analytic fits. The profiles with 100 and

180 keV implant energies are actual fits to simulation results. The

profile with 140 keV implant energy is the result of interpolation

between model parameters at the above-mentioned energies. The

arsenic implants are on-axis and through 275 Å silicon nitride. The

dose is 1×1013cm-2.

60






between model parameters at the above-mentioned doses. The 35

keV arsenic implants are on-axis and through 150 Å silicon nitride.

61



results and the solid lines are analytic fits. All of the profiles are

interpolations between model parameters between 50 and 150 Å

silicon nitride, 150 and 275 Å silicon nitride, and 275 and 400 Å

silicon nitride. The 35 keV arsenic implants are on-axis and the dose

is 1X1013 cm-2.

62




following energies are actual fits to simulation results: 20 keV, 40

keV, and 60 keV. The rest are results of interpolation between model

parameters at the above-mentioned energies. The phosphorus

implants are on-axis and through 150 Å titanium silicide.

63






between model parameters at the above-mentioned energies. The 20

keV phosphorus implants are on-axis and through 150 Å titanium

silicide.

64



results and the solid lines are analytic fits. All of the profiles are

interpolations between model parameters between 50 and 150 Å

titanium silicide, 150 and 275 Å titanium silicide, and 275 and 400 Å

titanium silicide. The 20 keV phosphorus implants are on-axis and

the dose is 1X1013 cm-2.

65

achieved when fourteen Legendre polynomials are used to model the impurity and

damage profiles.

The quality of interpolation is another crucial property for an analytical

model in order to correctly predict the impurity profiles for arbitrary implant

conditions within a specified range. This is true for all analytical models mainly

because only a finite number of parameter sets are available for in any range. For

Legendre polynomials, it is found that square root interpolation is sufficient to

obtain good results when interpolating over energy, tilt angle, rotation angle, the

screen oxide, nitride or silicide thickness. To interpolate between two doses, a

logrithmatic interpolation is used and the resulting profile is re-scaled in order to

ensure the correct dose. When good agreement is reached, a finer sampling of

parameter sets may not be necessary; otherwise, an additional parameter set

should be added to the lookup table. Typical interpolation results are shown in

Figs. 5.3, 5.6 and 5.9.

5.3 Two-dimensional Modeling

With the shrinking of device dimensions, two-dimensional effects, such as

the implant profiles around a mask edge (considered impenetrable), become more

and more significant for CMOS device design. To correctly account for these

effects, accurate and computationally efficient predictions of two-dimensional

66

impurity and damage profiles are desirable in an effort to minimize the cost and

cycle time for the development of process technology.

Traditionally, computationally-efficient analytical modeling of two-

dimensional ion implantation has been performed primarily by a superposition

approach [42,43]. Specifically, the point-source profile (the impurity profile over

a very small window) is generated first by a Monte Carlo method. Since the

implant window is small, i.e. 5a long (a is the silicon crystal lattice constant,

=0.543 nm), a relatively small number of ions is sufficient for a statistically

satisfactory result. The point-source profile is then translated to cover the entire

implant window and superposed. In other words, the implant window is broken

into segments (~5a long implant window), the implant into each segment is

simulated using the point-source profile obtained by Monte Carlo method and all

the point-source profiles are superposed. This has proved quite successful when

the implant-induced damage is negligible (low dose and light species). However,

as the damage-induced dechanneling effects become more significant (high dose

and heavy species), the profile predicted by the superposition approach shows

excessive channeling compared with Monte Carlo predictions (Fig. 5.10). Besides

its failure to take into account the nonlinear damage effect within the segment, the

interference from the lateral straggle of neighboring segments also has a

remarkable impact on the channeling (Fig. 5.11).

67

Fig. 5.10 Comparison of 2-D impurity iso-concentration contours generated

using UT-MARLOWE with TOMCAT, with the conventional

superposition approach. Implant condition: species - arsenic , energy

- 50 keV, dose - 5×1014cm-2, tilt angle - 7°, rotation angle - 30°,

implant window size – 50 nm.

68

Fig. 5.11 Typical point-source damage profile, showing how the lateral

straggle extends far beyond the width of the implant window, which

is 5a (a=0.543 nm, silicon lattice constant). Implant conditions:

species - BF2, energy - 30keV, dose - 2×1012 cm-2, tilt angle - 2°,

rotation angle - 0°.

69

In order to overcome these shortcomings, we propose a modified

superposition approach [9], based on UT-MARLOWE with TOMCAT

(TOpography based Monte CArlo Transport) [32]. The basic idea is to split the

nonlinear problem (dechanneling effect caused by the damage build-up of a high

dose implantation) into several near linear problems (relatively minor damage

changes within small dose intervals), while at the same time taking into account the

interference from neighboring segments. Specifically, the given implant dose D0 is

divided into a number of steps Dstep. Dstep is small enough so that within one

interval the linearity, together with the superposition approach, is valid (i.e.

2×1012/cm2 is used for As, BF2 and P). To simulate high dose implants, instead of

using only one point source profile throughout the process, impurity and damage

profiles are selected from a set of profiles in the look-up table (see next section),

and sorted according to the damage present near the particular segment.

Empirically, the damage near a segment can be quantitatively calculated as follows:

∫ ∫+

−

⋅⋅=max

0

),(),(d x

xii

i

i

dxzxCxGdzDσ

σ

σ , (5.6)

where C(x,z) = damage concentration (i.e. number of interstitials/vacancies per

unit volume as a function of lateral and vertical position); G(xi, ) = Gaussian

weighting function with mean x

σ

i and standard deviation σ; where σ is an

empirical number representing the lateral spread of damage, and is a function of

70

the implant energy. The above Gaussian function is centered on the segment of

interest, which corresponds to maximum weighting, and dies out gradually as the

lateral distance increases. In order to characterize the asymmetry of the damage

profile, the damage distribution (Ri) is introduced to describe the ratio of the

integrated damage to the right and left of the segment. This parameter is crucial

for tilted implants, as the damage information may not be adequate to distinguish

the right and left corner of the implant window.

Based on the damage estimation Di and damage distribution Ri,

corresponding impurity and damage profiles are chosen from the look-up table

and superposed. In this way, the impurity superposition can be monitored

according to the damage accumulation through the implantation process. This

modified superposition approach provides the damage profiles as well, which is

essential for multiple implants and has a significant impact on the diffusion of

impurities during subsequent thermal treatment, especially under minimal thermal

budget conditions.

5.4 Two-D Look-up Table Generation and Simulation Results

Legendre polynomials are used to model the point-source profiles as

mentioned above because of their good interpolation property and orthogonality.

Similar to 1-D simulation, a simple integration can be used to extract the

71

Legendre parameters, which makes the initial guess and recursive looping (e.g.

Levenberg-Marquardt algorithm) unnecessary.

Primarily, the point source profiles are modeled as follows:

)))('(.)(exp(10),(13

0

15 xxLzazxC ii

i∑=

⋅= (5.7)

∑=

=9

0))('(.)(

jjiji zzLbza (5.8)

where x and z are the lateral and depth coordinates, respectively, Lj is the jth order

Legendre polynomial, and ai and bij are Legendre parameters to be extracted from

point-source profiles. As shown in equation (5.7-8), 10 Legendre polynomials are

used to model the point-source response profile in the depth direction, while the

number for lateral direction is 14.

To generate the look-up table, first an implantation into a small window

with Dstep is simulated using the Monte Carlo approach. The generated impurity

and damage profiles are extracted to obtain corresponding Legendre parameters

and then superposed over the entire implant window. Based on the superposed

damage information, the second implantation into the small window is simulated

using the Monte Carlo method again, and the impurity and damage profiles are

extracted and superposed similarly. The above processes are repeated until a fairly

high dose level is reached. It is not necessary to store the parameters for every

implant step. As the dose increases, the damage caused by a single implant step

72

has less effect on the implants thereafter. This can be approximated quantitatively

by the average ion penetration depth of each step. As shown in Fig. 5.12 (with a

small number of ions), the average penetration depth of the ions decrease with

implant dose, and ultimately saturates at a value which corresponds to the

amorphization of silicon and the shutdown of the crystalline channels. The

variation of the penetration depth is quantified to determine the levels where the

accumulated damage effect is significant. Only at these dose levels are Monte

Carlo simulations performed with a large number of ions to achieve statistically

satisfactory results. In this way, the sizes of the look-up table are kept reasonable.

Then the parameters are extracted and put into the look-up tables with the

corresponding index of damage estimation and damage distribution.

Since there is no direct reliable experimental method to measure the 2-D as-

implanted impurity and damage profiles yet, a validated Monte Carlo simulator

(UT-MARLOWE with TOMCAT) is used to verify this new analytical approach

for 2-D ion implant simulation. Good agreement has been obtained between

impurity and interstitial profiles generated using the new nonlinear superposition

approach and those obtained using UT-MARLOWE with TOMCAT for As (Fig.

5.13), P (Fig. 5.14) and BF2 (Fig. 5.15). It is evident that Legendre polynomials

are good basis functions for modeling 2-D impurity and damage profiles with

significant savings in CPU time, without compromising accuracy.

73

Fig. 5.12 The dependence of the average ion penetration depth on as-

implanted dose. The curve shows the impact of the as-implanted

damage on subsequent implantation and the tendency to approach

amorphization as the dose increases. Implant conditions: species -

BF2, energy - 50keV, tilt angle - 0°, rotation angle - 0°, implant

window size - 5a (a=0.543 nm, silicon lattice constant).

74

Fig. 5.13 Comparison of 2-D arsenic impurity iso-concentration contours

generated using UT-MARLOWE with TOMCAT, with those

obtained analytically by the new superposition approach. A dose

step of 2×1012cm-2 was used. Implant conditions: species - arsenic,

energy - 50keV, dose - 5×1014cm-2, tilt angle - 7°, rotation angle -

30°, implant window size – 50 nm.

CPU time: 7 hrs 40 mins (TOMCAT);

6 mins (modified superposition)

75

Fig. 5.14 (a) Comparison of 2-D phosphorus impurity profile generated using

UT-MARLOWE with TOMCAT, with those obtained analytically

using the modified superposition approach. A dose step of 2×1012

cm-2 was used. Iso-concentration contours between 1×1017 cm-3

and 1×1019 cm-3 for the impurity profile is shown.

Implant conditions: species - phosphorus, energy - 30keV, dose - 1×1014cm-2, tilt

angle - 7°, rotation angle - 0°, implant window size – 50 nm.

CPU time for Monte-Carlo simulation (400k ions): 5 hrs 13 mins

CPU time for analytical model: 8 mins

76

Fig. 5.14 (b) Comparison of 2-D phosphorus interstitial profiles generated using

UT-MARLOWE with TOMCAT, with those obtained analytically

using the modified superposition approach. Iso-concentration

contours between 3×1018 cm-3 and 1×1021 cm-3 for the interstitial

profiles is shown.

Implant conditions: species - phosphorus, energy - 30keV, dose - 1×1014cm-2, tilt

angle - 7°, rotation angle - 0°, implant window size – 50 nm.



77

Fig. 5.15 (a) Comparison of 2-D boron impurity profile generated using UT-

MARLOWE with TOMCAT, with those obtained analytically


contours between 1×1016 cm-3 and 1×1019 cm-3 for the impurity

profile is shown.

Implant conditions: species - BF2, energy - 30keV, dose - 1×1014cm-2, tilt angle -

2°, rotation angle - 0°, implant window size – 50 nm.



78

Fig. 5.15 (b) Comparison of 2-D boron interstitial profiles generated using UT-

MARLOWE with TOMCAT, with those obtained analytically


contours between 1×1016 cm-3 and 3×1020 cm-3 for the interstitial

profiles is shown.

Implant conditions: species - BF2, energy - 30keV, dose - 1×1014cm-2, tilt angle -

2°, rotation angle - 0°, implant window size – 50 nm.



79

5.5 Conclusion

A new analytical model has been developed based on Legendre

polynomials to simulate ion implantation in both one and two dimensions. It

greatly simplifies the parameter extraction procedure and can model both the

impurity and the damage profiles. In two-dimensional simulations, a dose-

splitting approach has been developed to account for the nonlinear effect caused

by damage accumulation. It has been shown that the results obtained from the

new model agree very well with experimentally verified Monte Carlo simulation

results with major savings of CPU time.

80

Chapter 6. Conclusions and Future Work

6.1 Conclusions

In this study, physical stopping models and the damage models were first

reviewed. In order to correctly model ultra-shallow junction formation, a heavy

species, antimony, implant model was developed and incorporated into UT-

MARLOWE. ZBL pair-specific inter-atomic potential was used, which is more

accurate than the widely cited ZBL universal inter-atomic potential. Furthermore,

consecutive implants of antimony and B or BF2 into SiO2 capped single-crystal Si

were simulated and the impact of PAI on PLDD and the impact of cap oxide layer

thickness on PAI and PLDD were correctly modeled.

A new computationally-efficient three-dimensional Monte Carlo simulator

was developed within UT-MARLOWE. An unbalanced Octree algorithm was

used for spatial decomposition for its accuracy and efficiency. A new three-

dimensional trajectory replication scheme was developed and implemented in

order to further enhance computational efficiency. More than two orders of

magnitude savings on CPU time have been observed. Examples of simulations of

implant into highly non-uniform, non-planar, novel device structures were

presented.

81

Finally, a new analytical model was been developed based on Legendre

polynomials to simulate ion implantation in both one and two dimensions. It

greatly simplifies the parameter extraction procedure and can model both the

impurity and the damage profiles. In two-dimensional simulations, a dose-

splitting approach has been developed to account for the nonlinear effect caused

by damage accumulation. It has been shown that the results obtained from the

new model agree very well with experimentally verified Monte Carlo simulation

results with major savings of CPU time.

6.2 Future tasks

First of all, improved 2-D and 3-D Kinetic Accumulative Damage Model

(KADM) is desirable. This is especially helpful in studying defect generation and

evolution, and it will lead to great insights of diffusion mechanisms on an

atomistic scale.

Secondly, the current simulator does not yet have the capability to cover

all the species and materials used in manufacturing. Furthermore, new species and

new materials are constantly introduced in research and development of

semiconductor industry. To develop accurate models for new species and new

materials, careful experimental design and model calibration are required. In

particular, realistic model of implant into poly-silicon, with detailed consideration

of poly-grain size as well as crystal orientation is desirable.

82

Appendix A: A Brief Manual of TOMCAT

A.1 Installation

The distribution of TOMCAT actually consists of three programs, tomcat,

Sherwood and utview. Sherwood is a 2-D graphical visualization utility, while

utview is a 3-D graphic visualization tool. Both of them allow users to see the

topography in use, as well as monitor the progress of the simulation. The use of

Sherwood or utview is strongly recommended, particularly when first constructing

topography. The following steps should allow you to painlessly install TOMCAT

on your computer. You will need to download two tar files: the tomcat source code,

and the table file.

A.1.1. Create a directory within which you will compile TOMCAT. Copy the

tomcat_release99.tar.gz file (the source code file) into it, and unpack with

gzip -d tomcat_release99.tar.gz

tar -xvf tomcat_release99.tar

The resulting directory tomcat_release99 should have the following

subdirectories:

tomcat xsherwood utview docs

examples test_suite

A.1.2. Go into the tomcat directory. This directory contains the source code for

TOMCAT. Take a look at the available Makefiles in the directory. You

83

should have Makefile.aix, Makefile.sol, Makefile.linux, Makefile.SGI, and

Makefile.alpha, for AIX, Solaris, Linux, IRIX, and DEC Alpha operating

systems, respectively. Compile tomcat by doing the following:

make -f Makefile.xxx clean

make -f Makefile.xxx

Makefile.xxx is the Makefile that corresponds to your system, i.e.

Makefile.sol if you have a Solaris machine.

The compilation should proceed without errors (although some harmless

warnings are possible on some systems). When the compilation has finished,

copy the executable (tomcat) to a convenient directory of your choice (such

as /usr/local/bin), and make sure the PATH variable includes the directory

the TOMCAT executable is in.

A.1.3. Go into the xsherwood directory. Choose the Makefile as in the previous step.

In order to compile xsherwood on your system, you'll need Motif libraries.

Contact your system administrator if compilation fails due to the lack of

Motif libraries. As before, compile using:



This creates a dynamically linked executable called xhserwood.

Copy this to a convenient directory (i.e.: /usr/local/bin).

84

A.1.4. Go into the utview directory. Choose the Makefile as in the previous step. In

order to compile utview on your system, you'll need OPENGL or MESA

(freeware) libraries. Contact your system administrator if compilation fails

due to the lack of MESA libraries (downloadable from

http://www.mesa3d.org/). As before, compile using:



This creates a dynamically linked executable called utview.

Copy this to a convenient directory (i.e.: /usr/local/bin).

A.1.5. Unzip and untar the table file as in the steps above. A directory called tables

should be created and it should contain the binary scattering tables and other

information used by TOMCAT. Move the files to a convenient directory.

Each TOMCAT user needs to set two environment variables,

TOMCAT_TABLES and TOMCAT_OUT. The first is the directory in

which TOMCAT will look for tables, and the second is the directory into

which output will be sent. If these are not specified, TOMCAT will use the

directory in which the executable is located. Setting the environment

variables is done differently in different shells; check with your system

administrator if you are not sure. In shells derived from sh, you can set the

variables using

85

export TOMCAT_TABLES=/usr/local/tables

where the tables are expected to be at /usr/local/tables (you can, of course,

use a different directory).

A.1.6. Make sure that the PATH variable contains the directories in which tomcat,

xsherwood and utview are located. The installation is now complete. For

instant gratification of 2-D simulation, copy the files as15.in and sd.top from

the examples/first_example directory into your work directory, and run

TOMCAT by typing:

tomcat as15.in sd.top

This should start a TOMCAT simulation, and display the progress

graphically using xsherwood.

A.2 Quick Tutorial

The input to TOMCAT consists of a description of the ion source, and the

structure to be implanted into for 2-D/3-D simulations. The output is a collection

of files that describe the doping and defect concentrations. TOMCAT is run from

the command line using the following syntax:

tomcat [sourcetype] sourcefile [topographyfile]

where sourcetype is an integer that identifies the type of simulation to be run. 0

denotes ion implantation, 1 is for electron transport (the models for the latter are

not yet validated and its use will not be further documented here) and 3 is for 3-D

86

ion implantation simulation. The sourcetype parameter is optional for 1-D and 2-

D simulation, and the default is 0 (ion implantation). The sourcetype is required

for 3-D simulation. Sourcefile is the name of the input file that describes the

source conditions. If you have used UT-MARLOWE before, this will be quite

familiar. The topography argument is a file that specifies the structure to be

implanted into, which is required for 2-D and 3-D simulation.

A.2.1 Description of the Source File

The sourcefile describes the key implant parameters, such as energy, tilt

and rotation angles, and dose.

Input Description for 1-D simulation

The following is an example of a source input file for 1-D simulation:

species=arsenic

energy=15000

dose=1e14

tilt=7

rot=30

numions=5000



species=arsenic

energy=15000

dose=1e14

87

tilt=7

rot=30

numions=300000

leftwind=1

rightwind=599

elevation=550

The first keyword sets the implant species to arsenic. Next, the energy is

set to 15 keV, the dose is 1e14 cm-2, and the tilt and rotation angles are set to 7

and 30 degrees, respectively. The number of simulated ions is set using the

numions command, 300,000 being a reasonable number for 2-D simulation into a

small structure.

The lateral extent of the source is controlled using the leftwind and

rightwind parameters. These correspond to the lateral (x) coordinate of the left

and right edges of the source. The vertical position of the source is controlled with

the elevation command. All coordinates are in silicon lattice constants, unless

otherwise specified (1 lattice constant = 0.543 nm).

The coordinate system used is described in A.2.2.



species=arsenic

energy=15000

dose=1e14

88

tilt=7

rot=30

numions=300000

xleftwind=1

xrightwind=599

yleftwind=1

yrightwind=599

elevation=550

As you can see, it is very similar to the input description for 2-D

simulation except the implant window description. In 3-D simulation, four

coordinates need to be defined xleftwind, xrightwind, yleftwind and yrightwind.

For the definition of other command, please reference to last section, “Input

Description for 2-D simulation”.

A.2.2 Description of the Topography File

The topography file is not needed for 1-D simulation. For 2-D and 3-D

simulation, it describes the structure to be implanted into. In addition, commands

that control the output of TOMCAT are issued here as well. The structure is

specified as a list of geometric objects. For 2-D simulation, currently supported

objects are rectangles (2Dppiped), discs (2Dcylinder), and triangles (2Dtri).

Alternatively, the structure can be imported from a process simulator, such as

SUPREM or FLOOPS. For 3-D simulation, supported objects include box

89

(3Dppiped), triangle cylinder (3Dtri), tetrahedron (tetra), prism (prism), circular

cylinder (3Dcylinder), and Sphere (3Dsphere).

The syntax for commands in the topography file is:

Command (Argument list)

For example, the command 2Dppiped(0,400,600,0,3) creates a rectangle

with the upper left corner at coordinates (0,400), and bottom right coordinates

(600,0), composed of material type 3 (crystalline silicon). Each topography file

must include a 2DBoundVol command, as in 2DBoundVol(0,600,600,0,9). This

sets the simulation volume, with the first two arguments representing the

coordinates of the upper left corner of the simulation volume, and the latter two

representing the coordinates of the bottom right corner. The simulation area (in 2-

D) or volume (in 3-D) must be set up as a square (for 2-D) or a cube (for 3-D).

The final number in the argument list defines the resolution of the grid that will be

generated to represent the topography. The smallest spacing in the grid can be

obtained from the following formula:

min_spacing = size_of_boundary * 2-(n+1)

The size_of_boundary is the side of the square representing the

2DBoundBol, while n is the last argument to 2DBoundVol. This results from the

fact that the grid is obtained by a recursive subdivision of the geometry at most n

times (less in areas of uniform topography). Typical values are 8-10, though care

90

should be taken that the minimum spacing chosen resolves the features of interest.

All dimensions are in Si lattice constants (0.543 nm). See also the ResWindow

command in the next section for more efficient resolution specification.

A.2.3 Simple example of 2-D simulation

The following is an example of a topography structure, which implements

an oxide step on a silicon surface. Note that the pound sign # is conveniently used

to attach comments after any command.

2DBoundVol(0,600,600,0,8) #coordinates_should_be_positive

2Dcylinder(100,400,50,2) #oxide_disc

2Dppiped(0,400,600,0,3) #block_of_crystalline_silicon

2Dppiped(0,450,100,400,2) #oxide_block

2Dppiped(0,403,600,400,2) #native_oxide

TopPlot(TRUE) #this_starts_xsherwood

BinOut(TRUE) #also_needed_for_xsherwood

Scale(403,-1,5.43e-4)

The following source input file can be used to test it.

species=boron

energy=5000

dose=1e13

tilt=7

rot=30

numions=100000

leftwind=1

91

rightwind=599

elevation=550

Note that the elevation has been set above the highest point in the

topography. This is necessary in order to make sure that shadowing by

topography is handled correctly. By default, the elevation parameter is set to the

top of the structure (including vaccum, dicatated by the bounding volume).

Assuming the topography file is called step.top, and the source file is called

test.in, run the simulation by specifying:

tomcat test.in step.top

After a few messages from TOMCAT, xsherwood pops up a graphic

representation of the topography. You can use the Zoom and Pan buttons on the

xsherwood toolbar to inspect it as you wish. The dopant density is updated every

twenty seconds, although output from TOMCAT might be less frequent than that.

TOMCAT outputs a new binary dopant file (assuming that BinOut(TRUE) has

been specified in the topography file) every 15000 ions. You can ask xsherwood

to reload an update dopant file by clicking on reload. Clicking on “Quit” does not

end the simulation, and it merely exits from xsherwood.

A.2.4 Simple example of 3-D simulation

The following is an example of a topography structure, which implements

an oxide step on a silicon surface and shallow trench isolation. Note that the

pound sign # is conveniently used to attach comments after any command.

92

3DBoundVol(0,600,0,600,0,600,8)#coordinates_should_be_positive

3Dcylinder(180,400,100,0,600,2) #oxide_disc

3Dppiped(0,400,100,600,0,600,3) #block_of_crystalline_silicon

3Dppiped(0,430,0,200,401,600,2) #oxide_block

3Dppiped(0,490,0,200,420,600,4) #poly_silicon_block




TopPlot(TRUE) #this_starts_utview

Scale(403,-1,5.43e-4)

The following source input file can be used to test it.

species=arsenic

energy=15000

dose=1e13

tilt=7

rot=30

numions=50000000

xleftwind=1

xrightwind=599

yleftwind=1

yrightwind=599

elevation=550

Note that the elevation has been set above the highest point in the

topography. This is necessary in order to make sure that shadowing by

93

topography is handled correctly. By default, the elevation parameter is set to the

top of the structure (including vacuum, controlled by the bounding volume).

Assuming the topography file is called step.top, and the source file is called

test.in, run the simulation by specifying:

tomcat 3 test.in step.top

After a few messages from TOMCAT, utview pops up a graphic

representation of the topography. The key command for utview is listed in section

6 and you can inspect the topography as you wish. TOMCAT outputs a new binary

dopant file (assuming that BinOut(TRUE) has been specified in the topography

file) every 15000 ions. You can ask utview to reload an update dopant file by

pressing any key. Clicking on “ESC” to exit utview and it will not end the

simulation.

94

Appendix B: Block Diagram and Flow Chart

C++ is the language used to implementation the model described in earlier

chapters for its modularity and polymorphism. The basic structure consists of four

major parts as shown in Fig. B.1: the input, the source, the zoner and the output.

The input part contains the input source file and the input topography file,

which specifies the implant condition and the implanting topography,

respectively. The source part initializes the particles according to the implant

condition. The zoner part includes two major objects, topography and propagator.

The topography performs the actual Quadtree/Octree decomposition, and contains

a collection of decomposed grid cells. Base class, propagator, has many derived

classes, each a self-contained Monte Carlo program, to propagate particles

through different types of materials, crystal silicon, amorphous silicon, silicon

dioxide, silicon nitride, etc. Each decomposed grid cell is assigned a propagator

pointer, which points to the propagator of the particular material type in the grid

cell.

The generic simulation flow chart is illustrated in Fig. B.2. To start the

simulation, at first, the source initiates the particle according to the implant

condition, and then pass it by reference to the zoner. The zoner assigns the

particle or particles to the grid cell. Normally, implanted ion starts from the grid

cells whose material type is vacuum. The particle then will propagate in the grid

95

Output

Graphic tools

Propagator

Source

Implant conditions

Particle initiation

Input/ User interface

Zoner

Topography

Input topography file

Input/ User interface

Fig. B.1 Schematics of the block diagram of TOMCAT

96

No

No

Yes

Are the particles leaving their grid cells?

Is the particle exit condition

satisfied?

Yes

Is the simulation exiting condition met?

Yes

No

Update the simulated structure, and remove the exiting particles.

Finish simulation and simulation results are stored!

Particles are assigned to the corresponding grid cell and propagator

Particles passed by reference to the zoner

The Source initiates the particle(s) Fig. B.2 Simulation flow chart of TOMCAT

97

cell according to its propagator. Ions will propagate until their energy is lower

than 10 eV. If the particles leave their grid cells, the zoner will assign them to new

grid cells by Octree zoning algorithm. The structure will be updated until all

particles in the ensemble complete their propagation. Above process will be

repeated, assigning the particles that have left their grid cells to new propagators,

and requesting new particles from the source. Simulation continues until the

simulation dose is reached.

98

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103

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104

Vita

Di Li was born in Changchun, Jilin Province, People’s Republic of China

on April 6, 1973, the son of Ying Wang and Jingrui Li. After completing his work

at The High School attached to The Northeast Normal University of P.R. China in

1991, he entered Beijing University in Beijing, P.R. China. He received the

degree of Bachelor of Science majoring in Physics from Beijing University in

1996. In August 1996, he entered Rice University, Houston Texas and received

the degree of Master of Science majoring in Applying Physics in June 1998. In

July 1998, he entered the Graduate School of The University of Texas majoring in

Electrical Engineering.

Permanent Address: 3461 Lake Austin Blvd., Apt. C

Austin Texas 78703

This dissertation was typed by the author.

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