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Modeling of mechanical stress in silicon isolation technology and its influence on device characteristics

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Modeling of mechanical stress in silicon isolation technology and its influence on device characteristics
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Rueda, Hernan A., 1970-
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x, 140 leaves : ill. ; 29 cm.

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Cantilevers ( jstor )
Compressive stress ( jstor )
Mechanical stress ( jstor )
Oxidation ( jstor )
Oxides ( jstor )
Sexually transmitted diseases ( jstor )
Silicon ( jstor )
Simulations ( jstor )
Structural strain ( jstor )
Thermal stress ( jstor )
Dissertations, Academic -- Electrical and Computer Engineering -- UF ( lcsh )
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Thesis:
Thesis (Ph.D.)--University of Florida, 1999.
Bibliography:
Includes bibliographical references (leaves 133-139).
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Typescript.
General Note:
Vita.
Statement of Responsibility:
by Hernan A. Rueda.

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MODELING OF MECHANICAL STRESS IN SILICON ISOLATION
TECHNOLOGY AND ITS INFLUENCE ON DEVICE CHARACTERISTICS












By

HERNAN A. RUEDA


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


1999





























Copyright 1999

By

Hernan Rueda































To my family














ACKNOWLEDGMENTS


I would like to express my sincere gratitude to Dr. Mark Law,

chairman of my advisory committee, for his patience and guidance. He has

introduced me to the research area of process modeling, within which this

dissertation falls, and has provided valuable advice and direction throughout

my master's and doctorate studies.

Thanks also go to Drs. Gijs Bosman, Toshikazu Nishida, Kenneth K. O

and Kevin Jones for their interest and participation in serving on my

committee and their suggestions and comments.

I would also like to thank the University of Florida Graduate School

for its support during my master's program and the Semiconductor Research

Corporation for its support of my doctorate studies.

I have been very lucky to work with many industry mentors who have

provided invaluable assistance for my graduate studies. I acknowledge Drs.

Jim Slinkman and Dureseti Chidambarrao of IBM for their suggestions and

direction of the STI SKPM experiment. I also thank Dr. Len Borucki of

Motorola for assistance in design of the diode bending experiment. Thanks








also go to Drs. Paul Packan and Steve Cea of Intel for many valuable

discussions and suggestions on modern device concerns and strain modeling.

I wish to thank my friends who have made my time, over ten years at

the University of Florida, a very enjoyable experience. I've been very lucky to

meet so many good friends such as all the old TCAD research assistants, the

SWAMP group, my many roommates through all the years, and my 'old

school' friends back in the 'hood in Miami.

Last but not least, I express my love to my parents, Hernan Sr. and

Gloria, and my brother, Camilo, for their never-ending support, love and

encouragement throughout my whole life.














TABLE OF CONTENTS


page

ACKNOW LEDGM ENTS................................................................................... iv

A B STR A C T ......................................................................................................viii

CHAPTERS

1 INTRODU CTION ......................................................................................... 1

1.1 M motivation ............................................................................................ 1
1.2 Stress-Induced Effects in Silicon Fabrication................................... 3
1.2.1 Oxidation Influences ................................................................ 4
1.2.2 Diffusion Influences ............................................................... 6
1.3 Stress-Induced Effects in Silicon Device Operation ......................... 7
1.3.1 Carrier Mobility Influences.................. .......... ............ 7
1.3.2 Energy Band Influences.............................................. ........... 11
1.4 G oals ........................................................... ................................ 12
1.5 Organization................................................................................. ....... 14

2 PROCESS-INDUCED MECHANICAL STRAIN MODELS..................... 16

2.1 Continuum Mechanics ............................................................. 16
2.1.1 The Stress Tensor......................................................... 17
2.1.2 The Strain Tensor............................................................ 20
2.1.3 Stress-Strain Relationships ........................................ ........... 23
2.2 Strain Sources ....................................................... ................... 30
2.2.1 Film Stress................................................. ................... 31
2.2.2 Dopant Induced Stress ............................................ .......... .. 35
2.2.3 Oxidation Volume Expansion ....................................... .......... 39
2.3 Strain Computation Methods ............................................... .......... .. 40
2.3.1 Boundary Loading Method....................................... .......... .. 45
2.3.2 Fully-Integrated Method......................................... .......... ... 47
2.4 Sum m ary ...................................................................................... 48








3 APPLICATION EXAMPLES AND COMPARISONS............................... 50

3.1 FEM Com parisons......................................................................... 50
3.1.1 LO COS ................................................................................. 55
3.1.2 Post-STI Process Re-Oxidation............................................... 59
3.2 Raman Spectroscopy Measurements and Comparisons................. 65
3.2.1 Raman Simulation Method.................................... ........... 66
3.2.2 Nitride Film Edge-Induced Stress.......................................... 67
3.3 Boron-Doped Cantilever Bending Comparisons ............................... 70
3.3.1 Silicon Bulk Micro-Machining ............................................. 70
3.3.2 Cantilever Bending Simulations .......................................... 73
3.4 3D Boundary Loading Method Example (LOCOS)......................... 80
3.5 Sum m ary ....................................................................................... 82

4 KELVIN PROBE FORCE STI EXPERIMENT......................................... 85

4.1 Scanning Kelvin Probe Force Microscopy....................................... 86
4.2 Work Function Influence ............................................. ........... 88
4.3 STI Experiment.......................................................... ........... .. 93
4.4 SKPM Measurements ............................................................. 94
4.5 STI Strain Simulations............................................... .......... ..... 99
4.6 Results and Discussion .................................................................. 101
4.7 Sum m ary ....................................................................................... 105

5 STRESS INFLUENCES IN DEVICE OPERATION............................. 107

5.1 Uniaxial Stress Influence Experiment.......................................... 107
5.1.1 Stress-Inducing Apparatus Design..................................... 109
5.1.2 Stress-Wafer Deflection Relationship................................. 112
5.2 Experimental Procedure ................................................................ 114
5.3 Experimental Results .................................................................... 118
5.4 Sum m ary .......................................................................................... 124

6 SUMMARY AND FUTURE WORK ...................................................... 126

6.1 Sum m ary ....................................................................................... 126
6.2 Future W ork .................................................................................. 129
6.2.1 SKPM Strain Measurement Calibration Studies .......... 129
6.2.2 Effect of Stress on Dislocations............................................. 130
6.2.3 Silicidation-Induced Stress ................................................. 131
6.2.4 Three-Dimensional Modeling of the STI Process.............. 131

REFEREN CES........................................ ................................................ 133

BIOGRAPHICAL SKETCH........................................................................... 140














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

MODELING OF MECHANICAL STRESS IN SILICON ISOLATION
TECHNOLOGY AND ITS INFLUENCE ON DEVICE CHARACTERISTICS

By

Hernan Rueda

May 1999

Chairman: Dr. Mark E. Law
Major Department: Electrical and Computer Engineering

One of the challenges the semiconductor industry faces as it attempts

and continues the scaling of silicon integrated circuits is understanding and

control of mechanical strain resulting from silicon fabrication technology.

High magnitudes of strain can be induced under standard fabrication

conditions and may produce deleterious effects in device behavior, such as

increased current leakage. Current leakage has been identified as a critical

device characteristic for future sub-micron dynamic random access memory

(DRAM) and complementary metal oxide semiconductor (CMOS)

technologies, as it is a limiting factor for increasing switching speeds and

decreasing power consumption. The following are various known sources of

stress in silicon technology: thermal expansion mismatch, intrinsic stress,








and oxidation volume dilation. This work results from an examination, by

modeling, experiment, and simulation, of the contribution of stress due to

these sources using the Florida Object-Oriented Process Simulator

(FLOOPS).

The contributions of each source can be simulated using different

models that represent or approximate the physics involved. After the models

are described and presented, example applications are provided to

distinguish the advantages and limitations for each model.

Coupling experiment along with process simulation then validates the

results and allows for a better understanding of the problem. One such

problem examined in this work is the strain induced by the shallow trench

isolation (STI) process. STI has become an essential isolation scheme for

present and future sub-micron processes. It consists of several sequential

steps that exert stress in the silicon active area by each of the previously

described sources. Scanning Kelvin probe force microscopy (SKPM) is then

applied as a new technique to characterize the strain exerted from STI

processes through measurements of strain-induced work function variations

in silicon. Qualitative agreement is demonstrated between the SKPM

measurements and the work function influence due to finite element based

STI induced mechanical strain computations.

Finally, a wafer bending experiment is performed that quantifies the

influence of tensile and compressive uniaxial stress on forward current of pn-








junction devices. This effect is then modeled primarily through the strain

influences in the silicon bandgap.














CHAPTER 1
INTRODUCTION

1.1 Motivation


During the quest for increasing device density in integrated circuits,

many problems are encountered and need to be solved. As some problems are

alleviated, new issues emerge. One of the problems gaining importance in

silicon fabrication is process-induced mechanical stress. Many of the

processes used in silicon IC fabrication individually and cooperatively

contribute to the development of stress in the silicon active areas.

Of prime interest is the mechanical stress generated in the isolation

process flow. Isolation technology is continually challenged as design rules

are scaled. It is well known that high stress magnitudes in certain silicon

dioxide structures cause a decrease in oxidation rate [1, 2].

In LOCOS (LOCal Oxidation of Silicon) isolation processes, silicon

nitride is deposited over a thin pad oxide and patterned to mask the

oxidation of silicon. In order to cope with the necessary electrical isolation at

the ever-decreasing linewidths, increasing the stress during oxidation is

exploited [3]. This has been achieved by increasing the nitride thickness and

decreasing the pad oxide thickness. These techniques provide the sharp








transition from isolation field oxide to active area that is necessary in

decreasing pitch lengths for 0.35 micron technologies.

The main problem associated with this trend is that the silicon

substrate yields under the increased magnitudes of strain produced [4, 5].

Dislocations that are generated will degrade device performance [5, 6].

Dislocations serve as gettering sites that attract metal atoms introduced

during subsequent processes. Junctions are continually becoming shallower,

and therefore these nucleated dislocations have an even greater probability of

lying across the device junctions lined with metal atoms. Unacceptably

increased magnitudes of leakage current then result. Off-state currents are

critical design characteristics in logic and memory circuits, limiting the

switching speeds, causing increasing power consumption and limiting

reliability.

Shallow Trench Isolation (STI) is steadily becoming the predominant

isolation technology as minimum feature sizes decrease below the minimum

attainable pitch lengths of LOCOS-based technologies. The general STI

process flow includes a nitride-pattered reactive ion etch, sacrificial sidewall

oxidation, oxide deposition and finally a chemical mechanical polish.

However, stress-induced dislocation generation is not exclusive to LOCOS

based isolation technologies and is also a factor in designing STI processes

since the steps in the STI process may cooperatively strain the enclosed

silicon active areas [7].







Aside from isolation processes, other fabrication processes also induce

strain in the silicon crystal. Processes such as thin film deposition and

dopant introducing processes induce strain that can generate dislocations. As

feature sizes decrease, all these different strain-generating sources become

closer together and their influences overlap each other. Under these extreme

circumstances, it is important to understand strain fields generated by

multiple sources neighboring the silicon active areas.

Strain fields may also play a role in affecting process and device

behavior other than generating dislocations. For example, in silicon micro-

machining, high residual stresses of boron induce bending sensor structures

composed of diaphragms and cantilevers [8]. Strained regions have also been

shown to affect diffusion of dopants [9]. Also, it is well known that crystal

strain affects the energy band structure [10] and carrier mobilities [11]. Both

are major parameters influencing the electrical properties of a device. These

are just a few of the many concerns related to the strain state of the crystal.


1.2 Stress-Induced Effects in Silicon Fabrication


Stress concerns in process design first became significant in LOCOS

isolation technologies. As thermal oxidation temperatures are reduced,

dislocation densities in silicon increased due to this isolation technology [4].

These effects were attributed to the increased stress magnitudes induced at

lower oxidation temperatures due to the increased viscosities of the silicon







dioxide. Since then, the correlation between increased dislocation densities

and high stress magnitudes have caused isolation technologies to be a major

stress related concern.

Since then the diffusion process has also been reported to be influenced

by film-induced stress [12]. This affect has been explained by stress

influences on point defect concentrations as well as extended defect size and

concentrations such as dislocation loops. As junction depths continue to

decrease in the evolution of sub-micron technologies, the stress influences

generated by surface films have a greater effect.

1.2.1 Oxidation Influences

It is well known that the growth rate of thermal oxidation of silicon

has a stress dependence [13]. This stress dependence occurs in nonplanar

regions. In thermal oxidation, a volume of silicon reacts into a volume of

silicon dioxide that is 2.2 times in volume. For planar oxidation this does not

induce a large stress. This is due to the newly grown oxide lifting the old

oxide perpendicular to the surface. Since the surface of the oxide is not

constrained the oxide is free to move in this direction so negligible stress is

induced normal to the Si-SiO2 interface.

In nonplanar regions such as convex and concave corners of silicon, a

strain is exerted in the silicon dioxide. For the convex corners the strain is

laterally tensile since the old volume of oxide is stretched around a longer

periphery. In concave corners the strain is laterally compressive since the old







volume of oxide is compressed into a smaller periphery. Convex corners occur

at the top corners of a trench oxidation and concave corners result at the

bottom corners of a trench oxidation and at the Bird's Beak in a LOCOS

oxidation.

It was reported through cylinder structure oxidation experiments that

the stress induced in these structures reduced the amount of oxide grown as

compared to planar oxidation [1, 2, 13]. This behavior has been observed in

highly stressed LOCOS processes and in corners of trench oxidation. It has

been documented that the stress induced in the oxide alters the oxidation

reaction rate, oxidant diffusion of reactant, and the oxide viscosity [2]. All

these influences affect the oxidation growth rate.

The stresses generated in the silicon dioxide are relaxed by exerting

force on the neighboring films and the silicon substrate. This leads to

strained regions in the silicon due to the oxide growth. Silicon is an elastic

material for a wide range of strain. However, as the induced stress exceeds

the critical yield stress, dislocations in the silicon crystal are generated to

relax the strain [14]. The generated dislocations then present problems in

device behavior. Therefore, the strain induced in the silicon by oxidation

becomes a concern.

Trenches are popular isolation technologies that also exhibit

dislocation generation problems [7, 15]. The stress induced by trench

structures is due to other sources also. After the reactive ion etch, an







oxidation is performed to provide a low interface state density and a low

oxide trap density. This oxidation is the first source of stress in the silicon

substrate in the trench fabrication process. Next the trench is filled with a

deposited film. This film influences a stress due to its built in intrinsic stress

and thermal expansion mismatch. The three sources all act simultaneously

influencing strain in the surrounding silicon substrate. Again, the primary

concerns are dislocations generated to alleviate the strain in the substrate

induced by the trench process.

1.2.2 Diffusion Influences

It has been observed that both phosphorus and boron diffusion

behavior under silicon nitride films is different than in inert conditions [9]. It

has been established that boron and phosphorus diffusion is governed by

interactions with point defects, namely silicon interstitials. Ahn attributed

the measured diffusion reduction to a vacancy supersaturation and self-

interstitial undersaturation that may be due to the compressive stress under

the silicon nitride films [12]. Osada later confirmed this by performing

similar experiments [16].

These experiments were performed with junction depths of

approximately 0.8 microns. Scaling for sub-micron devices has lead to

decreasing junction depths. The magnitudes of strain in silicon due to film

stress are at a maximum at the film interface. Therefore, it is believed that








an even greater effect is encountered for the shallower junctions that are

getting more prevalent in modern technologies.


1.3 Stress-Induced Effects in Silicon Device Operation


Interest in mechanical properties of silicon was sparked with the

discovery of its piezoresistive effect [17, 18]. This led to the increased interest

of the use silicon as a pressure sensor. Sensor research also extended

investigation of the mechanical influences on other electrical parameters

such as the energy bands of semiconductor crystals [10, 19-22]. As a result,

there is satisfactory understanding of how applied mechanical forces affect

electrical device behavior. These principles are used mainly in the design of

semiconductor mechanical sensors. In the present age mechanical stress is

becoming a limitation in silicon device fabrication, these principles also need

to be incorporated into microelectronic device behavior.

1.3.1 Carrier Mobility Influences

The influence of mechanical stress on carrier mobility is starting to

gain importance in modern microelectronics. It has been shown that in

CMOS transistors the transconductance will vary dependent on the

magnitude of applied stress and the behavior is dependent on crystal

direction of the current flow as well as the type of carrier [23]. This effect is

was also noticed and believed to be due to LOCOS-induced stress in SOI








CMOS devices [24]. The variation in transconductance is attributed to the

piezoresistance effect on the carrier mobilities.

The piezoresistance effect of semiconductors describes how the

resistivity is influenced by mechanical stress. The electric field vector is

proportional to the current vector by a symmetrical resistivity tensor of rank

two with nine components:


El pA P4 P6 1
E2 P4 2 5 2 *
2 = p, p2 p (1-1)
E3 P6 P5 P3_ Ji35

If the system axis is aligned along the <100> crystal directions, the

normal resistivity components p1, p2, and p3 relate the e field vector

component to the current vector i component in the same direction. The cross-

resistivity components p4, P5, and p6 relate the E field vector component to the

current vector i component in a perpendicular direction. If in the unstressed

state, the normal components have the same magnitude p and the cross

components are equal to zero, this reduces to the following isotropic

relationship:

E = pi. (1-2)

When the crystal is under mechanical stress, the resistivity

components change as the following:








P1 p App
P2 P Ap2

P3 P AP3
=3 + A3 (1-3)
P4 0 AP4
P5 0 AP5
P6 0 Ap6

The piezoresistive coefficients relate the stress-induced changes in the

resistivity components to the stress tensor influencing the change. This

matrix relating the six resistivity components to six stress components

consists of 36 piezoresistive coefficients nij. Due to the cubic crystal

symmetries in silicon, the piezoresistance coefficient matrix reduces to three

independent components, Rn1, 7C12, and x44:


Ap, ~n t12 X2 0 0 0 a
Ap2 2 711 r 12 0 0 0 02
1 Ap3 7_ 2 12 1 0 0 0 03 (
*- (1-4)
p Ap4 0 0 0 4 0 0 a4
Ap5 0 0 0 0 7C,4 0 5
_Ap6 0 0 0 0 0 7r44 0C6

The stress components are also referenced with the system axis oriented in

the <100> directions. Smith initiated the investigations of these

piezoresistive coefficients and found the following values for silicon at room

temperature displayed in Table 1-1 [17, 18].

The piezoresistance coefficients are also dependent on dopant

concentration as well as temperature. Later it was found that they would







decrease as the temperature increases and/or the dopant concentration

increases.

Table 1-1: Piezoresistive coefficients for silicon [17, 18].

Material p 711 7X12 744

(4-cm) (10-12 cm2/dyne)

p-Si 7.8 +6.6 -1.1 +138.1

n-Si 11.7 -102.2 +53.4 -13.6


The accepted explanation for this phenomenon is the many-valley

model [17, 18]. Anisotropic conditions exist when the mobility in one crystal

direction is different than the mobility in the other crystal lattice directions.

This results when the semiconductor is in a stressed state. The stress tensor

distorts the conduction energy bands of the unstressed semiconductor in

different magnitudes depending on direction. The energy levels and

curvatures of the band energies corresponding to the perpendicular directions

are influenced differently by the applied strain. The effective masses of the

carriers are proportional to the energy bands' curvatures in reciprocal k

space. Since the carrier mobilities are functions of the carrier effective

masses, the strain influence on the energy band level curvatures results in

directionally dependent influences on the carrier mobilities and therefore the

resistivities of the semiconductor. The energy band shifts are also influenced

on dopant concentration and temperature. Therefore the energy band's

sensitivity to stress will also be dependent on these influences.








1.3.2 Energy Band Influences

The mechanical stress state's influence on the energy bands also

affects the electrical behavior of p-n junction devices such as diodes and

bipolar transistors. In these devices the operation is governed by the flow of

minority carriers. Using a diode as an example, the forward bias current is

described by the following relation:

V V
IF = I exp() + IRO exp( ) (1-5)
VT 2VT

where the saturation current is


I,= qA po coth + Dn coth (1-6)
L L" L.


and the recombination current is


qAniW
IRo -qAn (1-7)
270

The saturation current term is linearly related to the minority carrier

densities npo and pno. The minority carrier densities are directly dependent to

the square of the intrinsic carrier density

2
no =- (1-8)
PpO

The intrinsic carrier concentration is exponentially dependent to the stress

dependent band gap Eg








-E
ni =KT 3/2 exp( -g). (1-9)
2kT

The stress induced shifts in the conduction and valence energy bands will

alter the band gap and therefore ultimately influence the saturation current.

Wortman initiated the quantification of the effects of uniaxial and

hydrostatic compressive applied external stresses to forward and reverse

biased diodes [10, 22].

Mechanical stress also influences the generation/recombination

current component of p-n junction devices. Rindner attributed the effect of

uniaxial compressive stresses to increased dislocation densities that

decreased the carrier lifetimes and therefore increased the

generation/recombination current component [25]. This effect becomes the

greater influence under higher magnitudes of stress due to the dislocation

generation to relax the applied stress.


1.4 Goals


The goals of this work are primarily to develop a system where strain

can be computed from multiple sources simultaneously. Silicon IC fabrication

involves a sequential flow of many processes that introduce and alter the

strain in the crystal. These process-induced strain fields influence the

behavior of processes later in the fabrication flow as well as device operation







once the process flow is completed. An accurate strain solution is necessary

for further investigation of its effects.

Once the strain in the system is understood, strain dependent models

can be developed to help understand unexplained behavior that has been

observed that may be due to strain. Such areas may include point defect and

extended crystal defect interactions, diffusion kinetics, and band-gap and

mobility influences.

Strain simulations also could aid in the development and analysis of

isolation process technologies. Often in fabrication process development, the

stress levels generated and dislocation densities produced may decide the

isolation process required. A strain field simulator could reduce the amount

of experimental work necessary for solving these problems.

Another goal of this work is to validate the process induced strain

models with experimental measurements. Currently, this is a major hurdle

due to the few characterization techniques available for localized sub-micron

strain measurement. Micro-Raman spectroscopy has been the most suitable

method for investigating localized strains [26]. But as technologies continue

to scale towards the 0.1pm generation, this may even surpass micro-Raman's

spatial resolution limits. Therefore, in this work Scanning Kelvin probe force

microscopy (SKPM) has been investigated as a new method for analyzing

localized strains through detecting influences in the silicon work function.







One last goal is to quantify the influence of tensile and compressive

stress on pn-junction device current. This would then provide some insight

into the mechanical strain influence on leakage currents.


1.5 Organization


Chapter two provides descriptions of the various process induced strain

sources and discusses how they are modeled in this work. Afterward, the

finite element methods that were developed for strain computation are then

described.

Results and comparisons between the methods for various processes

are included in chapter three. Example applications are provided to

distinguish the advantages and limitations for each model. Next, the film

edge-induced stress solutions are validated with published micro-Raman

measurements. Finally, three-dimensional applications are demonstrated.

Scanning Kelvin probe force microscopy (SKPM) is then explored as a

technique for characterizing STI induced strain in chapter four. An STI

experiment is performed and strain influence is measured by SKPM. These

measurements are then compared with simulations of band gap influences

using the models described in chapter two.

An experiment relating mechanical stress to pn-junction device

operation is then described in chapter five. This wafer bending experiment

addresses uniaxial stress influences on the forward current. This allows for





15

quantifying the influence on the reverse leakage current through observation

in the forward bias.

Finally chapter six concludes with a summary of the research work

accomplished and addresses topics for future work.













CHAPTER 2
PROCESS-INDUCED MECHANICAL STRAIN MODELS

The solution of strain present in a particular device technology is

computed using a finite element method (FEM) formulated to solve for the

strain induced by various sources in silicon technology [27, 28]. In a

fabricated device, the strain field in the silicon is generated due to various

processes at different steps along the fabrication flow. The most critical

sources for inducing stress are deposited and grown films. Sources in the

silicon crystal such as dopants and extended defects are becoming more

important as technologies advance.

In this chapter, a brief review of continuum mechanics is first

provided. Next the individual strain sources are then described. The

algorithms used to model the strain generated from each source are also

discussed. Afterward the finite element methods used to integrate the

various strain sources are described.


2.1 Continuum Mechanics


It is the intention that this review refreshes the reader with the theory

and notation of continuum mechanics. References are provided for a more








complete description. The stress tensor is first introduced in this section.

Next the strain tensor is described. Finally this section concludes with

different stress-strain relationships descriptions and how they may relate in

silicon processing.

2.1.1 The Stress Tensor

Stress is the distribution of internal body forces of varying intensity

due to externally applied forces and/or heat [29]. The intensity is represented

as the force per unit area of surface on which the force acts. To illustrate this

concept, consider an arbitrary continuous and homogeneous body (Figure 2-1)

under the applied external forces, F1, F2, F3, and F4. If the body is sliced into

two smaller volumes V1 and V2, then V2 exerts force on V1 at their surface





F,
F2


+z +Y V

+x S


Figure 2-1 Continuous body with external forces applied.






18

interface S to remain in equilibrium. The resulting force may be of varying

intensities along the surface.

At any point P on the surface between V1 and V2, the forces can be

reduced to a force and a moment, which can be described by a stress vector

acting on that surface. Three stress vectors acting on three mutually

orthogonal planes intersecting at that point can then determine the stress

state at any point P (Figure 2-2). The stress tensor is composed of the three

stress vectors and, according to Cauchy's equations of motion, is sufficient to

define the stress state in any element in a body [30].

To illustrate the tensor nature of stress present at point P in the

continuous body, consider a cubic element (Figure 2-2) of infinitesimal






Txx





+z Y" t 0

I /





+X

Figure 2-2 An infinitesimal cubic element located within a continuous body
with stress tensor components shown.






19

dimensions located at point P in Figure 2-1. For simplicity of notation, let the

cube be aligned perpendicular with the system axis. The stress vector Tx

acting on the plane normal to the x-direction is the following:

T, = a, + f-+- (2-1)


Let the surface AS be the plane of the cube normal to the x-direction. The

stress vector Tx is defined as the ratio of force acting on that surface area AS:

AF dFP
T = lim (2-2)
s -4 ASO dSX


The stress tensor on that volume is defined by nine stress components acting

on the three surfaces of the cubic element, which make up the three stress

vectors Ti:



9,= x T xyy Ty Z (2-3)



In the definition above, ii are the normal stress components acting on the

faces perpendicular to i-direction and zj are the shear stress components

oriented in the j-direction on the face with normal in the i-direction. At

mechanical equilibrium, it can be shown that the stress tensor is symmetrical

[29],


Tij = Tj .


(2-4)






20
A column vector of six independent components can then describe the state of

stress at a point:

UT =[UXX ayy U z, Tu Ty ]. (2-5)

2.1.2 The Strain Tensor

The application of stress to a body in equilibrium causes it to undergo

deformation and/or motion. A measure of deformation is strain. Two common

measures for strain are the Lagrangian and the Eulerian definitions. Both

are functions of the initial and final dimensions. When the displacement

between the final and initial measurement is referenced to the original

position dimensions then it is known as the Lagrangian definition. The

Eulerian definition describes the deformation displacement referenced with

respect to the deformed position.

Figure 2-3 shows a one-dimensional example of the deformation in a



(a) (b)
Unstretched Stretched





Ax t Ax + Au v


Figure 2-3 One-dimensional deformation of a spring: (a) original length
(Ax), (b) deformed length (Ax+Au).






21

spring. The Lagrangian strain and Eulerian strain then, respectively, become

the following over the length of the spring:

increase in length Au
E- (2-6)
S original length Ax

increase in length Au
Ei (2-7)
deformed length Ax +Au (

For the case of infinitesimal deformation, the Eulerian and Lagrangian

descriptions become equivalc. ,. An infinitesimal deformation approximation

requires that the maximum deformations involved be much smaller than the

smallest dimension of the deformed body. A second requirement is that the

deformation gradient is much less than one. When these assumptions hold,

the infinitesimal strain can be defined by the following relationship for any

point in the spring [31]:

A. Au Qu
E=lim -(2-8)
Ax-o Ax ax

By expanding this definition in three dimensions, the strain is related to the

displacements by the following strain components [30]:

Sdu (du v
E =- E = E +--

E dv v = E = (2-9)
e", C =i +-) (2-9)

dw 1(du dw
cE E E i-+-
U U x 2 dx





22
where u, v, w are the displacements in the x, y, and z directions, respectively.

These components make up the strain tensor (s i) that is analogous to the

stress tensor:


Ek = yx
EZ


xy 'xzl
Eyy Cyz
SZ zz


(2-10)


A three-dimensional example of a body undergoing deformation due to

an externally applied force is illustrated in Figure 2-4. In the example a

compressive force is applied to the infinitesimal cube in the x-direction. A

negative displacement compressivee strain) results in the x-direction and

positive displacements (tensile strains) result in the y- and z- directions. The


dv



/Zf 7\


/*

+z +
+x


Before
deformation


Compressive Force in
x direction


S----- du +
du/dx
i du/dx < 0
dw/dz > 0
dw +
------dw/dz dv/dv > 0
<- dv + dv/dy -

After
deformation


Figure 2-4 Strain reference example displaying a compressive stress in the
x-direction that generates normal strains in the x-, y-, and z-directions.






23

relationship of how the strain in each dimension results from the applied

force will be discussed next.


2.1.3 Stress-Strain Relationships

Bodies of different materials but of same dimensions may deform

differently under the same stress application. The relationship between the

stress tensor and the deformation is known as a constitutive relation. It may

vary for a given material depending on conditions such as temperature and

pressure.

Elasticity. All structural materials possess, to some extent, the

property of elasticity. Elastic bodies possess memory during deformation. For

example, when a force is applied on an elastic solid, it will deform until it

reaches its elastic yield limit or until the load is released. Microscopically the

bonds between atoms that compose the solid 'stretch' during elastic

deformation. When the force is removed the body will return to its original

shape if it is an ideal elastic body and it had not reached its yield stress,

similarly to an ideal spring. When the load is removed the bonds return to

their unstressed lengths corresponding to the original environmental

conditions.

For a Hookean elastic solid, the stress tensor is linearly proportional to

the strain tensor over a specific range of deformation:


(2-11)


T = CqklE k






24

by the tensor of elastic constants cykl [30]. In order to relate each of the nine

elements of the second rank strain tensor to each of the nine elements of the

second rank stress tensor, ijki consists of a fourth rank tensor of 81 elements.

However due to the symmetries involved for the stress and strain tensors

under equilibrium, ciki is reduced to a tensor of 36 elements.

Crystal silicon has diamond cubic crystal geometry resulting from its

strong directional covalent bonds. For such crystals, cijki has the following

form due to their cubic symmetry [32]:

Ct, c12 c12 0 0 0
C12 CtL C12 0 0 0
c,2 cL2 C1, 0 0 0
c = (2-12)
C*j =0 0 0 c4 0 0 ( )
0 0 0 0 c, 0
0 0 0 0 0 c4

Thus, for silicon the tensor of elastic stiffness constants reduces to the three

independent components: c1, c12, and c44. Due to the crystal's lattice

temperature dependence, the elastic constants are also thermally dependent.

At room temperature (25C) the elastic constants have been measured

as the following for silicon [33, 34]:

Ct, = 1.657 x 102dyn/cm2
C12 = 0.639 x 1012 dyn/cm2
c. = 0.7956 x1012 dyn/cmr2.
The elastic constants' thermal dependence has been documented as the

following linear relationship for silicon [35]:








Tc1 -L =-75X106 /0K
C11
Tc12 -At2 =-24.5x106 /Ko
Cl2
Tc A4 -55.5 x 10 /K.
C44
From the above linear dependence, it can be seen that the elastic constants

do not change significantly for large temperature changes. Also the degree of

anisotropy does not change significantly for the range of 100-9000K [36].

These studies support that silicon acts as an anisotropic elastic material over

a wide temperature range frequently encountered in silicon IC fabrication.

Although silicon is an anisotropic crystal, it is sometimes desirable to

approximate it with isotropic elastic properties for simplification. When the

components of elastic constants for a material are equal for any rotation of

the reference axis, the material is said to be isotropic. This means that the

elastic properties of the material are the same in all directions. The tensor of

elastic constants for an isotropic material reduces to the following:

E(1- v)
cL = (2-13)
(1+ v)( 2v)
Ev
cL2 = (2-14)
(1 + v)(1 2v)
E
c ( = v (2-15)
(1+ v)
where E represents the Young's modulus and v represents the Poisson's ratio.

As was demonstrated in Figure 2-4, contractions in one dimension may be

accompanied by dilations in other dimensions. The Young's modulus is a

measure relating stress and strain for an elastic material when stress is






26

applied in one direction and the other directions are free to deform as in

Figure 2-4. The Poisson's ratio is a material property describing the ratio of a

strain perpendicular to the applied stress to the strain oriented in the

direction of the applied stress.

Due to its anisotropy, E and v vary depending in the direction of the

applied stress and plane acted upon for silicon. At room temperature E may

vary from 1.3e12 dyn/cm2 (for <100> directions) to 1.875e12 dyn/cm2 (for

<111> directions). Poisson's ratio also may vary from 0.06 to 0.34 for the

same conditions. The following are measured values for E and v in silicon at

room temperature [32, 33, 35]:

E[ooI =1.31 x 102 dyn/cm2
E11o1 = 1.69 x 10L dyn/cm2
EI L1 = 1.875 x 102 dyn/cm2
voo = 0.279.
Viscosity. Although silicon deforms elastically over a wide temperature

and load range, other materials used in silicon fabrication behave differently.

Some deform elastically at temperatures near room temperature and flow at

higher temperatures with fluid behavior. Silicon dioxide (Si02) and silicon

nitride (Si3N4) are examples of this these type of materials. Fluids resist

deformation with viscous behavior.

As was previously mentioned, Hookean elastic solids will return to

their original shape after an applied stress is removed. Materials with

viscous constitutive properties may not. Viscous bodies relax or minimize the






27

strain associated with an applied stress. As the strain is relaxed the stress

also reduces. Due to the reduction in strain, when the applied stress is

removed, the body will retain its currently deformed shape.

Microscopically, in bodies with viscous mechanical properties, the

bonds between atoms that compose the solid break during deformation. New

bonds are then formed and re-broken as the body deforms under an applied

force. When the applied force is removed the bonds remain in their current

configurations.

A common constitutive relationship for a viscous body is the

Newtonian fluid. In a Newtonian relationship, the shear stress on the surface

is linearly proportional to the rate of deformation [30]:


Crj oc pkijki (2-16)

where /jki is the tensor of viscosity coefficients. The components of ijkl for

fluids are not as well known as cijki for elastic solids. However, most fluids

appear to behave isotropically. An isotropic approximation with the

restriction of incompressibility (constant density) allows for the following

constitutive relationship known as a Stokes fluid [30]:

aj = -pS, +pij (2-17)

where p now is a scalar that represents the viscosity of the fluid. The normal

stress components are dependent on the static pressure p of the fluid where

y& represents the Kronecker delta function.






28

Viscoelasticitv. It has been recognized that some materials deform

with a combination of elastic and viscous properties. There are various

models that have been formulated to describe the mechanical behavior of

viscoelastic bodies. In the Maxwell model of viscoelasticity, the total strain is

simply the sum of the strain due to elastic deformation and the strain due to

viscous deformation:

E = EE +C. (2-18)

This relationship can then be expanded to formulate the well-known

Maxwellian viscoelastic relationship between stress and strain:


e=--+-- (2-19)
E p

where E is the elastic modulus and g. is the viscosity of the material. Figure

2-5 illustrates the differences in deformation responses among a Hookean

elastic, Newtonian viscous and a Maxwellian viscoelastic material given the

same applied pulsed stress. Notice that the viscous and viscoelastic response

are time-dependent. Instantaneously after the stress is applied, the

viscoelastic body deforms elastically. Later when the applied stress is

constant, the viscoelastic body begins to display a linear viscous deformation.

As the applied stress is removed, the viscoelastic body then elasticallyy'

deforms towards its original strain state. However due to the viscous

relaxation component, it does not return to its original strain state.






29

Therefore, for the Maxwellian viscoelastic relationship, the short-term

deformation is elastic and the long-term deformation is viscous.


Applied Stress


t2 t


Hookecn Elcstic Solid


't1 't2

Newtonian Viscous Fluid


ET Mcv wellicn Viscoelastic Fluid


Figure 2-5 Comparison in loading response among Hookean elastic,
Newtonian viscous and Maxwellian viscoelastic relationships.


e, = co(t,-t,)/-






30

In silicon IC fabrication, oxide (SiO2) is considered to behave

nonlinearly viscoelastic at midrange (800-11000C) anneal temperatures [2,

14]. The nonlinearity is due to the fact that the degree of its viscosity is also

stress dependent. Nitride has been recognized to behave as a viscous body in

this same temperature range [37].

Not very many materials behave exactly as a Hookean elastic solid, a

Newtonian viscous fluid, or a Maxwellian viscoelastic fluid. However in

limited ranges of stress, strain, and temperature, these constitutive relations

can approximate their deformation behavior quite well. As an example silicon

behaves elastically for a wide temperature and stress range. However under

high stresses, silicon will yield and nucleate dislocations and defects in the

crystal to relax the stress present. Therefore it important to learn under what

conditions this will result.


2.2 Strain Sources


The individual strain sources included for strain computation are

discussed next. Three different strain sources are modeled and integrated:

film-induced, dopant-induced, and oxidation volume expansion induced

strain. Each strain source is discussed and explanations of how they induce

strain in the bulk are included. Following each is a discussion on how the

strain induced is modeled using the finite element method.







2.2.1 Film Stress

Physics. As silicon technology progresses, layers upon layers of

different materials are grown and deposited on top of and adjacent to each

other. When materials that have different structural, mechanical and

thermal properties are attached to each other, strains in each of the

materials can result. Adjacent material films relax or expand differently

based on their material properties. This causes one film to stretch or contract

the other film in a manner that will cause a local strain in each film. Large

localized stresses can result due to discontinuous films in regions such as at

the film edges and in non-planar sections.

According to Hu [14], stresses result in thin films due to two different

mechanisms. The first is referred to as an 'extrinsic' stress and is primarily

due to thermal expansion mismatch of neighboring materials. The process

used to deposit the film is done at an elevated temperature ranging from 150-

12000C. After the process is over and the thin film is deposited, subsequent

thermal cycles will cause the film to expand and contract. If the film was

attached to or restrained by a rigid material that does not thermally deform,

the amount of strain induced is linearly proportional to the temperature

difference:

E, = a,, AT (2-20)

where acth is the linear thermal expansion coefficient of the film material. The

body experiences an increase in volume in each normal direction. No shear






32

components result from the thermal difference. For many materials the

thermal expansion coefficient is not necessarily constant or linear over a wide

temperature range. For silicon, ath varies 2.5-4.5x10-6 /K over the 300-900K

temperature range. Oxygen content and dopant concentration are factors

leading to ath variations.

If the material that the film is attached to also expands due to a

thermal increase then the local strain at the interface that is produced is due

to the difference in thermal expansion coefficients:

,t = (a,i -a,,).AT. (2-21)

Thermal mismatch stress is often incorrectly shortened and referred to

as thermal stress. However, thermal stress is due to thermal gradients

within a material. This often occurs in the wafers during before and after

temperature cycles. As the wafers cool down, the maximum stress is due to

surface tension. However to maintain force equilibrium the interior of the

wafer must be in compression. As the wafer heats up, the surface proceeds to

expand due to thermal expansion. However, the cooler interior of the wafer

restricts this expansion causing a compressive stress at the surface and a

tensile stress in the interior. Thermal stress primarily occurs in the substrate

as its thickness allows for a greater thermal gradient between the surface

and the interior. The thermal stress in the substrate becomes a problem

during temperature ramp rates encountered in Rapid Thermal Annealing

(RTA). At higher temperature ramp rates, the thermal stress built up from






33

the high temperature gradient may exceed the yield stress and cause the

wafer to shatter. Normally the thicknesses of grown and deposited films in IC

fabrication are so thin that a negligible thermal gradient exists across them.

The other source of stress encountered in thin films is the 'intrinsic'

stress. Several researchers attribute this stress as due to growth mechanism

of the material during the process. For grown oxides, the intrinsic stress

results from the planar volume expansion resulting from the oxidation

reaction. This stress should not be confused with the stress induced at

isolation edges, which is non-planar and is a multidimensional problem

discussed later. Other material films such as polysilicon, silicon nitride, and

silicides exhibit intrinsic stress also.

After a film is deposited or grown, the wafer will warp according to the

total stress in the film. A highly tensile film will bend the wafer's edges

towards the film and the reverse for a compressive film. A popular technique

for measuring film stress is to measure the amount of wafer curvature

optically. The total film stress is then proportional to the radius of curvature

by the following relation [38]:


E t 2
= -.' (2-22)
f 6(1-v,)R t,

where Es and Vs are the elastic properties of the substrate and t. and tf are

the thicknesses of the substrate and film respectively. The film stress

measured is the total stress due to thermal mismatch and its intrinsic





34

components. The intrinsic stress can then be derived from this measurement

and the known thermal mismatch stress from the previous relationship.

Even though high stresses can result from the sum of thermal

mismatch and intrinsic stress in a film, if the film is uniformly planar then

the stress resulting in the substrate due to the film will be orders of

magnitude smaller. This is due to the large difference in thicknesses between

the film (tf) and the substrate (ts):

t
a, = -4Q f -. (2-23)
ts

Higher local stresses result in the substrate due to discontinuities in the film.

Such discontinuities include etched film edges from masking and

nonplanarities as in trench fill depositions.

Film-induced Strain Model. The stress due to deposited films is

modeled as an initial condition before deformation. The planar

measurements of intrinsic stress are used as the initial condition for the

finite element solution. The stresses are input at the nodes as the film is

deposited. They are directed in a biaxial tangential orientation along the

growing interface as is shown in Figure 2-6. To handle the stress components

in nonplanar interfaces, the stresses are translated from the planar system

axis to the axis perpendicular to the normal of the growing film. After the

film is deposited or grown, the stresses at the nodes are then averaged to

their neighboring triangular (2D) or tetrahedral (3D) elements.






35

The thermal mismatch stress is modeled as a hydrostatic stress. Each

normal component of strain exerted is equal in magnitude. No shear

components result from thermal mismatch. For each element in the film, the

three normal components of strain are added by superposition to the previous

state of stress due to other sources. This presents a problem in plane strain

FEM formulations since the strain in one direction is set to zero. This

problem will be addressed in section 2.3.

2.2.2 Dopant Induced Stress

As dopants are introduced to the silicon substrate, the mechanical

state of the substrate will change. The dopants may substitute for the silicon

positions in the lattice. Silicon atoms are displaced forming extended defects

that are lodged in the crystal lattice. Different dopants have varying atomic

sizes and therefore have different mechanical behavior in the crystal.

Precipitates and other atoms present such as oxygen and carbon also alter

the mechanical properties of the crystal.

Boron. Boron is well known as a substitutional dopant. Its atomic size

is smaller than that of silicon. When it locates into a substitutional site,

lattice contraction results due to its smaller size. This presents an atomically

localized strain in the crystal due to each boron atom. Figure 2-7 exemplifies

this in three- and two-dimensional illustrations. For high concentrations of

boron in silicon, these atomic strains add up significantly and result in a

larger localized region of strain in the boron doped silicon lattice. The non-






36

boron doped silicon region will resist the diffused boron layer from

contracting and thus result in a tensile strain field. This effect has been

demonstrated in silicon micro-machining applications [8, 39, 40], where

boron-doped cantilevers have exhibited bending due to strain induced by the

boron.





2D planar film

These nodes' stresses are (7
oriented parallel to the -
deposition interface
-^-7- ^--Y / D --7, -


+v
+x
2D nonplanar
film nx'yC,

n, *,+ n,, *---- x ,\ ^ y

These nodes'
stresses are
translated
around covers
by the no* n,* T,-
component of
the unit surface n,*oax *
normal Y



Figure 2-6 Intrinsic film stresses are oriented parallel to the interface on
which the film is grown or deposited.

























2D approximation
0 0 000 00 00 0
0 0 0 0 0 0 0000

o o o o O o oOo o
00000 00000
0 0 000 0 0000

Figure 2-7 Two- and three-dimensional illustrations of lattice deformation
due to boron substitution.



Densitometric studies have been done with boron doped silicon

crystals. Horn measured the silicon lattice constant variation as a function of

boron concentration [41]. From his measurements, the induced strain (Aa/a)

was extracted and given as function of boron concentration (Figure 2-8).

Boron-induced Strain Model. The empirical relationship introduced by

Horn is used as the contribution of strain in silicon due to substitutional

boron dopant. According to his measurements, 0.0141 A of lattice contraction






38

results per percentage of B in Si at room temperature. Using this figure, the

following relationship is derived:


Aa 0.0141 C8
E, = E= = Eu -100
asi as, Ns,


(2-24)


where asi is the silicon lattice constant (5.4295A at 25C) and Nsi is the

density of Si atoms (5.02e22 cm-3). The average concentration of Boron (CB) is

computed for each element from its node quantities. Dopant-induced strain is

also hydrostatic (as in thermal mismatch strain) and again presents a

problem in plane strain FEM formulations since the strain in one direction is

set to zero. This problem will be addressed in section 2.3.


0.0009
0.0008
0.0007
S0.0006
E 0.0005
S0.0004
g 0.0003
9 0.0002
0.0001
0.0001


0 5E+19 1E+20 1.5E+20
Boron Concentration (cm-3)


2E+20


Figure 2-8 Hydrostatic strain as a linear function of boron concentration
[41]






39

Other dopants. The strain contributions of other common dopants such

as phosphorus and arsenic are not as great per atom as that due to boron.

Because arsenic has a much larger atomic mass than silicon, one would tend

to believe that arsenic would induce a large compressive strain in the silicon

lattice. However, it has been reported that heavily doped arsenic (5x1021 cm-3)

only induces a lattice compression (Aa/asi) of approximately 0.0019 [42]. The

atomic size of phosphorus is more closely matched to that of silicon and

therefore the P-Si bond lengths are of the same magnitude as the Si-Si bond

lengths.

2.2.3 Oxidation Volume Expansion

The oxidation process also induces strain in the substrate due to the

net volume expansion of the oxidation reaction. It is well known that silicon

oxidizes to a volume of oxide that is 2.2 times larger. For planar oxidation,

this presents no problem since the newly acquired volume pushes the old

oxide upward towards the unconstrained surface perpendicular to the

interface. However, in nonplanar regions such as in trench corners and in

constrained regions such as in LOCOS edges, this presents a problem. For

these regions, the boundaries are constrained and therefore the newly

acquired oxide volume compresses against the earlier grown oxide. Since the

oxide has no place to move, large compressive strains build up in these

regions. The strains are somewhat relaxed by applying pressure to the silicon






40

interface. The forces applied to the silicon are high enough to surpass the

yield stress of silicon.


2.3 Strain Computation Methods


Two different finite element based methods are developed to solve and

compute the previously described models of the various strain sources. These

algorithms are developed in the process simulator FLOOPS [43] and are

integrated with and derived from methods developed to model stress-

dependent oxidation and silicidation [44].

Newton's second law of motion governing deformation is stated as the

following [30]

do..
pa, =d + b (2-25)


where p is the density of the body, ai is the acceleration, oiy is the local stress

tensor, and bi is the body force.

The equivalent nodal force for each element (qe) may be represented as the

following:

q' = BTad(vol) J Nbd(vol) (2-26)
V' V,

where b represents local body forces (e.g. gravitational or electromagnetic

forces), the B matrix relates the strain rate to the displacement rate (velocity)

of element and N represents the shape functions of the element. This






41

statement is valid quite generally for any stress-strain relationships. The

assumption of negligible body forces and negligible acceleration for each

element allow the equivalent nodal force equation to reduce to the following:

q' = fB'd(vol). (2-27)
V'

For mechanical equilibrium where the body is not under rigid body motion,

net force is equal to zero.

An Hookean elastic element with the following constitutive relation

a = D(e- e,)+ o (2-28)

would be modeled by substituting the constitutive relation into the

equivalent nodal forces equation:


q = [B'DEd(vol) BDoDCd(vol) + BTd(vol). (2-29)
V' V V'

The strain e is related to the unknown displacement Aa through the B matrix

e = BAae (2-30)

and may also be substituted into the elastic equivalent nodal force equation:


q' = LBTDBd(vol) Aae JBTDeod(vol) + B'Taod(vol). (2-31)
v I V V,

Under mechanical equilibrium, the elastic equivalent nodal force equation

reduces to the following discretized form:


[BTDBA]Aae = BTADE0A BTaoA


(2-32)





42

where A is the area (volume in 3D) of the element. The left-hand side

represents the stiffness matrix of the element. The right hand side represents

the initial stress and strain state of the element. From this relation, the

displacement Aa is solved for globally and the current stress and strain state

can be derived from the displacement Aa.

A viscoelastic body is handled in the same manner. The Maxwell

viscoelastic constitutive relation

-+-= E (2-33)
G r

has the following solution for the stress oa


a= 7 1-exp( -t + Coexp(F (2-34)


where r is the relaxation time constant and is the ratio of the viscosity r7 to

the elastic modulus G

=--. (2-35)
G

The Maxwellian viscoelastic constitutive relation can then be substituted into

the equivalent nodal force equation:


q' = [fB T 1-exp A Dd(vol) + f BT exp (vol). (2-36)


The strain rate e is related to the unknown change in velocity Av through the

B matrix





43
e = BAv' (2-37)

and may also be substituted into the viscoelastic equivalent nodal force

equation:


q' =[ B T 1-expt DBd(vol) Av' + fBT o exp (vol). (2-38)
Lv,. I v ).lJ _T v] I T

Under mechanical equilibrium, the viscoelastic equivalent nodal force

equation reduces to the following discretized form:


B {l 1-exp(1 DBA Ave = -BTa exp( -At (2-39)

where A is the area (volume in 3D) of the element. The left-hand side

represents the stiffness matrix of the element. The right-hand side represents

the initial stress and strain state of the element. From this relation, the

unknown change in velocity Av is solved for globally and the current stress

and strain rate can then be derived from the new velocity change Au.

The viscoelastic formulation reduces to the elastic formulation for

large relaxation time constant z. If z >>At, then

exp =l- A~ (2-40)

The viscoelastic formulation then becomes

SBT tDBA Ave = -B'o(1- At (2-41)
f T I I T )


and reduces to the elastic formulation by allowing AtAve=Aae.






44

To model two-dimensional problems, the plane strain formulation is

used. This formulation can be used for problems where the strain component

in the z-direction is zero or negligible [28]:

EZ = E = E=c =0. (2-42)

This can be approximated solving problems with infinitely long dimensions in

the z-direction. Therefore, the strain in the z-direction will approach zero. As

was mentioned before in the earlier sections, a problem arises using the

plane strain approximation while computing thermal mismatch and dopant

induced stress. These sources include a hydrostatic strain field Eo described as

the following:



D, = EC = E. (2-43)



Then plane strain presumption implies that stresses in the z-direction will

still occur even if there is no z-component of strain. These stresses occur due

to dopant and thermal expansion and are affected by the elastic constants. To

account for this using a plane strain approximation, the following expression

is used instead for an elastic relationship [28]:



o = E] =(l+v) e (2-44)
_C 0.


where v is the Poisson's ratio.






45

The FEM formulation uses three noded triangular elements called

'faces' for two-dimensional applications and four noded tetrahedral elements

called 'volumes' for three-dimensional applications. Linear shape functions

are used for interpolation of the strain solution within each element.

Reflecting boundary conditions are used to handle the strain solution

at the boundary of the simulation field. The normal component of the velocity

field or displacement rate is set to zero across this interface. Physically this

corresponds to a 'mirror reflected' symmetrical structure across the boundary.

2.3.1 Boundary Loading Method

The boundary loading method (BL) uses the strain solution calculated

during the oxidation to drive the elastic solution in the silicon. This technique

has been used in the SUPREM IV process simulator to calculate the elastic

strain in silicon due to oxidation [2]. Chidambarrao also used this technique

to investigate strains resulting from isolation trenches [45]. It is fully

decoupled and is performed in two sequential steps. This is demonstrated in

Figure 2-9. In the first step, the nonlinear viscoelastic oxide flow is computed

with the surface films modeled as viscoelastic materials and with silicon

acting as a rigid body [44]. This assumption allows for a more efficient

technique for solving the oxide growth, since only the surface films are

iterated over in the nonlinear stress dependent oxidation solution.

The stress tensor in the silicon dioxide elements along the silicon -

silicon dioxide interface are averaged to interface nodes. The averaged node








Step 1-Calculate the stress in the upper surface films to compute the stress-
dependent oxidation growth
Surface Film Forces
Oxide Growth__ -- I
Forces )k
Step 2-Calculate the stress induced in the substrate due to the forces
generated during oxidation and strain
generated from dopants. _




Strain -- dopants, defects





Figure 2-9 The boundary loading method is composed of two separate
steps for computing the strain in silicon.

stresses are then converted into boundary forces along the interface. The

boundary forces are simply the product of the stress tensor with the unit

surface normal of each node:

F, a,, aT zXr n.
F: = 1 (Y a n* (2-45)
z o' yz azJ _z

These forces are then input as boundary loading forces that drive the

calculation of the elastic substrate strain solution. Silicon is modeled as an






47

isotropic elastic material in the second step. Strain may be included that is

exerted from dopants and defects within the silicon for this solution.

2.3.2 Fully-Integrated Method

The other technique developed to investigate the strain in silicon is

similar to the finite element method implemented by Senez [46]. This fully-

integrated method couples the strain solution in the silicon along with the

surface films to compute the oxidation growth. This method allows for the

oxide stresses to relax by exerting forces on the silicon substrate. In the BL

method, oxide stresses can't relax due to the rigid body boundary condition

imposed on the silicon. Therefore, lower magnitudes of stresses are expected

in using this method. This method is extended to include strains from other

sources simultaneously (Figure 2-10).

This technique becomes very computationally intensive because the

silicon elements are also assembled into the nonlinear oxidation equation.

This usually results in an order of magnitude more elements included to

adequately define the substrate and reduce the effect of the reflecting

boundaries on the solution. Therefore, the FI method's Newton iteration

involves a much larger matrix than the BL method and results in much

slower performance overall.

As in the previously described method, the surface films are modeled

as viscoelastic bodies and the silicon substrate is modeled as an isotropic

elastic material. The boundary condition at the oxidant reaction interface is








6_- -_Surface Film Forces

Oxide Growth, -~
Forces \_





Strain -- dopants, defects




Figure 2-10 The fully integrated method couples the solutions of strain in
the upper films along with the substrate simultaneously.

similar to a polysilicon boundary condition: the silicon flows and is consumed.
Additionally, strain from dopants and intrinsic film stress may be exerted in
the silicon substrate and also is included as a stress source in this method.


2.4 Summary

One of the goals of this work are to develop a system where strain can
be computed from multiple sources independently and simultaneously. In
this chapter, a brief review of continuum mechanics was first provided to
refresh the reader with the necessary terminology. Next the individual strain
sources and the algorithms used to model the mechanical strain generated






49

from each source were described. Finally the boundary loading (BL) and

fully-integrated (FI) finite element were introduced and discussed for

modeling the strain due to oxidation volume dilation. The optimized BL

method is enforces a rigid body assumption for silicon. This allows it to be

useful for analyzing oxidation-induced strains in three-dimensions, as will be

discussed in the chapter three. However, this assumption may produce

influences in the oxide growth and stress solutions. Therefore, in chapter

three, the two methods will be analyzed for several applications to find if and

when there are influences. Also, in chapter three, applications using the

film-induced and boron-induced strain models are demonstrated and they

are compared and validated with experimental studies.














CHAPTER 3
APPLICATION EXAMPLES AND COMPARISONS



Due to the fully integrated method's computational intensiveness, the

boundary loading method is more commonly used. This brings into question

how well the solution from each method agrees. Comparisons are performed

between each method for computing the oxidation-induced stresses for

different processes in silicon fabrication. Next, solutions of nitride film-

induced stresses are compared with micro-Raman spectroscopy

investigations. Afterwards, boron-induced cantilever bending simulations are

performed and compared with previous experiments. Finally, a three-

dimensional example is demonstrated for simulating oxidation-induced stress

using the boundary loading method.


3.1 FEM Comparisons


To compare the two methods for computing oxidation-induced stress,

two different oxidation processes are analyzed. The first is a standard

LOCOS process. The second is a re-oxidation process over an oxide filled

shallow trench. These examples detail when the boundary loading method








solutions fail to match the solutions computed by the fully-integrated

method.

Each of the following simulations is two-dimensional and use the plane

strain approximation with reflecting boundary conditions. This means that

the dimensions in the z-direction (which extends outward perpendicular to

the plane of the page) are infinitely long. The reflecting boundary conditions

indicate that the structure is periodic along the right and left boundaries of

the field.

Hydrostatic pressure contours are plotted for each process to compare

solutions computed by each method. This allows for a scalar interpretation of

the stress tensor's variation in the two-dimensional field. The hydrostatic

pressure definition used is simply the negative average of the normal

components or negative one-third of the trace of the stress tensor:


CXX 0 0
P = 0 Y Y (3-1)
0 0 CTZ_

For each wet ambient simulation, material properties calibrated at

1000C by Cea and Senez [44, 46] are utilized. These parameters are listed in

Table 3-1.

For the STI simulations where dry oxidations are also performed, the

oxide material properties were calibrated to Kao's dry oxidation cylinder








Table 3-1 Material coefficients at 1000C [44, 46]

Elastic Parameters Viscosity
Parameters
material Elastic Poisson's Viscosity Vo
Modulus Ratio (Dyn/cm2)
(Dyn/cm2)
oxide 6.6ell 0.17 2.0e14 450e-24
nitride 3.89e12 0.3 7.0e14 100e-24
silicon 1.7e12 0.3 -___


experiments [13], similarly to how the wet oxidation parameters were

calibrated [44].

In Kao's experiment, cylinders and holes of varying radii were oxidized

to grow approximately 3500A at various temperatures in a dry ambient. In

both the cylinders and holes the oxide growth was retarded when compared

to plane wafer oxide growth. It was concluded that this decrease in oxide

growth was due to the stress dependence on the diffusivity of oxidant

reactant, reaction rate, and viscosity of oxide. As the radii of the cylinders

and holes decrease, the amount of oxide growth also decreases. A standard

calibration procedure adopted by Rafferty [2] and Cea [44] is used to obtain

the oxide parameters that fit the curves for each temperature and radius

geometry. The results of the cylinder and hole simulations are plotted in

Figure 3-1. The dry oxide parameters obtained from the calibration

simulations are listed in Table 3-2 along with the wet oxide parameters

obtained by Cea [44].









Dry Oxidation Calibrations


Figure 3-1


0 1 2 3 4 5
1/R (Si) (um)
Results of dry oxidation calibrations for holes and cylinders
calibrated to Kao's measurements [131


The calibrated low stress viscosity of dry oxide is plotted over

temperature in Figure 3-2 along with the wet oxide viscosities calibrated by

Cea [44]. It can be seen that the dry oxide low stress viscosities are about an

order of magnitude less than the corresponding viscosity at each

temperature. This is expected since dry oxidations produce a denser oxide.

Also, the viscosities follow along the same slope in the Arrhenous plot with

similar activation energies.









Calibrated dry and wet [44] oxide parameters.


Oxide Viscosity Diffusivity and
Parameters Reaction Rate
Parameters
T(C) Tlo Vo VD VK

(dyn/cm2) (A3) (A3) (A3)
1100 W 3.3e13 1100 75 10
1000 W 2.0e14 450
900 W 1.5e15 300
1100 D 3.5e14 250
1000 D 2.2e15 150
900 D 2.5e16 6.25 "



Oxide Viscosity versus Reciprocal Temperature


0.74 0.76 0.78 0.8
1000o r (1/K)


0.82 0.84 0.86


Figure 3-2 Calibrated dry oxide viscosity compared with the wet oxide
viscosity calibrated by Cea [44].


1017

106 ....




S105




1014




10'13
0.72


Table 3-2







3.1.1 LOCOS

The LOCOS process is simulated to compare between two-dimensional

stress solutions computed by each method. Figure 3-3 exemplifies

simulations of a 10000C eighty-minute wet oxidation process at atmospheric

pressure. A 180nm thick nitride film with a two-micron linewidth is

patterned over a 20nm thick pad oxide. The grid spacing used for the

simulations is 0.05am at nitride edge, 0.15 m at the left and right side

boundaries, and 0.25pm at the bottom boundary.

Figure 3-3 displays the stress solution for each method. By analyzing

the average pressure contours displayed, it is observed that there is

qualitative agreement between the two methods for this particular LOCOS

process and geometry. Corresponding regions are in tension (negative

pressure) and in compression (positive pressure). However the relative areas

occupied by the corresponding pressure contours indicate that their

magnitudes disagree. This is expected since the fully integrated method

allows for some stress relaxation through a 'flexible' silicon base. The

boundary loading method does not allow oxide stress relaxation through the

silicon because the silicon is modeled as a rigid body. The presence of both

compressive and tensile regions in the silicon is due to its elastic behavior.

When a compressive force is applied to an elastic body (such as at the bird's

beak in silicon), a neighboring region in the material tends to deform in

tension due to the Poisson's ratio effect.







The solutions in the oxide agree more closely in magnitude than in the

silicon. It is for this reason that the bird's beak heights (BBH) are almost

identical as can be seen in Figure 3-4. This indicates that the rigid body

boundary condition used for computing the oxide growth, as modeled in [44],

does not affect the oxide stress solution enough to alter the oxide growth

significantly at longer nitride linewidths. However at shorter linewidths, it

becomes evident that the BB curvature underneath the nitride edge is

influenced by the strain computing method. In Figure 3-4, an overlay of

simulations using both strain computing methods is shown. The FI method

solution shows a little more oxidation under the nitride edge. This indicates

that the BL method computed a higher stress in this region that retarded the

oxide growth. A matrix of LOCOS simulations is performed to analyze the

differences of the stress and oxide growth solutions for shorter nitride

linewidth simulations using each FEM method.

Table 3-3 represents a matrix of simulations with varying nitride

linewidths. The first set with linewidth of 2gm is demonstrated in Figure 3-3.

Depending on the amount of elements in the field, the boundary loading

method is about two to three times faster than the fully integrated method.

Table 3-3 demonstrates that the oxide thickness and BBH is consistent

between the two methods and the variation in oxide shape is small until the

nitride linewidth decreases to 0.5pm. Below this linewidth, the oxide punch-

through effect begins to occur and the BBH varies more significantly








depending on the strain computing method used. The maximum tensile and

compressive stress magnitudes in the silicon are also noted for each

simulation and tend agree.


Table 3-3 Matrix of LOCOS simulations depicting how the two FEM
solutions correlate with decreasing nitride linewidth.

linewidth BBH Max Oxide Max Max CPU
(urn) (um) Growth (um) Tension Compression time (s)

2.0 FI 0.276 0.499 -1.5e9 4.9e9 1077.4

BL 0.268 0.499 -3e9 5e9 633.9

1.0 FI 0.233 0.453 -3.7e9 7.5e9 1638.5

BL 0.235 0.46 -3.4e9 5.1e9 552.5

0.7 FI 0.215 0.415 -6.5e9 6.9e9 895.6

BL 0.215 0.42 -8e9 5e9 343.0

0.5 FI 0.179 0.367 -5e8 8e9 821.6

BL 0.190 0.368 -5e8 8e9 300.2











Y Axis BL Method


0.00 +,09 0.00
+108 1 9 108 +109





-108
1.o00 -108 1.00 +108 1






.................... 10%
2.00 1 8j Compression 2.00

-_1.0x10o Tension ___ 10io8 t Compression

_i O Tnin8 8 Tension
---l -1.0x10

3.00 __ 3.00


-1.00 0.00 X Axis 1.00 -1.00 0.00 X Axis 1.00

Figure 3-3 Comparisons for a LOCOS-induced stress computed by the boundary loading method (left) and the
fully-integrated method (right).


FI Method


Y Axis






59
Oxide Shape Comparisons


x in microns

-0.50











0.00










0.50


-0.60 0.00 y in microns 0.60

Figure 3-4 Overlaying oxide shape solutions comparisons between the BL
method and FI method for an example LOCOS simulation with 2gm nitride
linewidth.

3.1.2 Post-STI Process Re-Oxidation

Strain Solution Comparisons. An oxidation process over a previously

processed oxide filled shallow trench isolation (STI) is simulated to compare

between two-dimensional stress solutions computed by each method. Figure

3-5 exemplifies a simulation of a 10000C three-minute wet oxidation at

atmospheric pressure. The trench with 0.5pm depth and 0.5gm width is filled






60
with deposited oxide. The nitride mask in this simulation has been previously

stripped as in a CMP process. The grid spacing used for the simulation is

0.02pm at the trench edges, 0.20am at the left and right side boundaries, and

0.10pm at the bottom boundary. The initial field stress solution is set to zero

to compare the stress induced by the oxidation alone.

Figure 3-5 is an example of a 'wedge driving effect' that is an oxidation

performed over a fabricated trench that is not masked. A vertical bird's beak

forms at the upper corners of the trench due to reactant diffusion into the

trench. This produces a very high compressive stress in the silicon substrate

along the side of the trench as oxide is grown on the trench sidewall.

Therefore this is analogous to a wedge being driven in at the trench sidewall.

By observing the average pressure contours displayed, it is observed

that there is qualitative agreement between the two methods for this process.

The tensile and compressive regions again correspond between the two

method's solutions. The relative areas occupied by the corresponding pressure

contours indicate that their intensity variations disagree. The BL method

computes that the entire field is in compression with the exception of the

silicon at the bottom trench corner. Therefore, the two methods agree

qualitatively in the strain solution computed for this process. However, for

STI re-oxidation processes the strain solution is only part of the concern.












Y Axis BL Method


Y Axis FI Method


0.00 '00 1 -. 0.00


+109 +109 +109 +109


0.50 0.50
0. -10 109

0108
+108
y +108 -o108 +108


1.0 1.0


S109
----A
Compression 10
10 7
-1.OxlO7
1.5 109 1.5 Tension -1.0xlO
1-o8 Compression -1.oxo
107
-1.0xl0
S -1 .x i o Tension
_1 Ox10-'

2.0 21 2.0


0.00 0.50 XAxis 1.00 0.00 0.50 X Axis 1.00

Figure 3-5 Comparisons for post STI process-induced stress computed by the boundary loading method (left) and
the fully-integrated method (right).








Corner Rounding Comparisons. One of the most critical design aspects

of the STI is the top corner shape. It is desirable for the top corners to be

rounded for several reasons. The most important being that sharp corners

lead to high electric fields and result in undesirable electrical characteristics

[47, 48]. Another reason for avoiding sharp corners is they lead to higher

mechanical stress magnitudes that generate dislocations also deleterious to

device operation [7].

Re-oxidation after fabrication of the STI has been introduced as a

method to round the upper trench corners [48]. The increased radius of

curvature at the upper corners alleviates these problems. Selecting the most

appropriate oxidation process for this technique can be simplified through

oxidation process simulation [49].

Viscoelastic flow models are used for the grown and deposited films

during the re-oxidation process [44]. Stress-dependent models for oxide

viscosity, reaction rate, and diffusivity are utilized for simulating the re-

oxidation anneals. Two-dimensional plane strain simulations are performed

for a re-oxidation process to study the evolution of the radius of curvature of

the upper STI corners. The re-oxidation occurs after the planarization step of

the STI process. Therefore the oxide fills the trench up to the surface of the

nitride film (Figure 3-6). The STI dimensions have a depth of 300nm and a

width of 400nm. A series of re-oxidation simulations is performed varying the

nitride thickness from 70nm to 180nm over a 30nm pad oxide. The re-









x in microns FLOOPS


Si3N4
Nitnde Lift-ng__ ~,' -._
0.00 i Bird' s Bak Le gt
NI Raius of Curmtwe

0.10

Si02
Silicon
020


0.30



-0.20 0.00 y in microns 0.40

Figure 3-6 10000C 300 minute dry re-oxidation simulation.


oxidation processes simulated and analyzed are a 300 minute 10000C dry

anneal, 100 minute 11000C dry anneal, and a 30 minute 9500C wet anneal.

The oxidation processes result in about 150nm of oxide grown on a blank

wafer. Each process is examined for each of the nitride film thicknesses.

The radius of curvature relationship to nitride thickness is consistent

with the experimental results of Chang [48] (Figure 3-7). The amount of

nitride lifting also varies linearly with the nitride thickness. The bird's beak

lengths are fairly constant over this range so there is little trade-off against

the thinner nitride process.










70o (a) -22 -----.......--.---.------- (b) --

6020
55 18
50 4 16
45 114
40
35
30 50 70 90 110 130 150 170 190
50 70 90 110 130 150 170 190
50 70 iD NiNde Thickne. (m) 0 1 1 Nitride Thickess (nm)


Figure 3-7 (a) The radius of curvature of the upper STI corner and (b)
nitride lifting at the bird's beak increases with decreasing nitride mask
thickness for the 1100C dry oxidation.


Figure 3-8 depicts how the various process simulations examined

resulted for a given nitride thickness using each oxidation strain computation

method. It can be seen that for the higher temperature 11000C dry oxidation

resulted in the greatest amount of corner rounding and stress relaxation.



x n microns FLOOPS 98.1


-0.15

-0.10


0.00


0.10 950C wet
(FI)
950C Wet
(BL)
0.20 1100C Dry
(BL)


0.30


0.00 in microns
Figure 3-8 Overlay of the re-oxidation profiles for each anneal process
with 120nm nitride film thickness.








Also, due to less stress relaxation in the lower temperature 9500C wet

oxidation, less corner rounding is resulted by the BL method and an

overhang results by the FI method. The overhang also agrees with re-

oxidation experiments [48]. However its shape is different since the

simulated shape may be locally grid dependent. The higher stresses reduce

the oxidation rate at the upper silicon corner just below the nitride edge and

result in excessive compressive stress magnitudes in the neighboring silicon

region.

From this application, it is found that for the low temperature wet re-

oxidation simulations, the FI method results in an overhang at the trench

corner due to the severe oxide growth retardation. However for the same

simulation, the BL method returns a rounded corner. Since in experiment

[48], this process does result in a slight overhang, it is believed that including

the silicon deformation while calculating the oxide growth plays an important

role in modeling this effect at the trench corner.


3.2 Raman Spectroscopy Measurements and Comparisons


In the previous section qualitative comparisons were performed for

different oxidation processes between the two FEM methods implemented. In

this section, efforts are made to compare the nitride film edge-induced stress

with measured strain data. Currently, the best method for measuring local







process-induced mechanical strains in the substrate at sub-micron

dimensions is micro-Raman spectroscopy [26].

Raman spectroscopy is most commonly utilized for crystallographic

and chemical composition analysis. Mechanical strain or stress affects the

frequencies of the Raman modes of the crystal. The Raman spectrum of the

silicon crystal is sensitive to strain in the lattice and therefore has been used

to measure the stress state induced by microelectronics fabrication processes

[50]. Micro-Raman stress measurements involve focusing laser light onto a

sample. The scattered light of the sample is then collected and analyzed. The

light scattered from a (100) unstressed silicon sample has a Raman peak

frequency of about 520 Rcm-1. Stressed samples will influence a shift in the

Raman peak. The amount and direction of the shift corresponds to the

magnitude and sign compressivee or tensile) of the stress in the laser spot

region of the sample. A negative shift corresponds to a tensile stress and a

positive shift indicates a compressive stress.

3.2.1 Raman Simulation Method

A method for determining the expected frequency shift in Raman

modes due to process-induced strain was introduced by Jones [51] and De

Wolf [52]. This method is implemented here to validate the stress solution

from the nitride film edge-induced stress model with micro-Raman

measurement data. The method involves first evaluating the shift in Raman







modes for a given strain tensor solution. For an elastic material, the strain

tensor is related to the stress tensor through the compliance tensor sijkl:


E- = Sik kk. (3-2)

The compliance tensor sijkl is simply the inverse of the stiffness tensor cijkl

introduced in chapter two. Once the strain solution is known, the three

Raman mode shifts (AoI, Ao)2, Ao3) are computed by solving the following

eigenvalue problem [52]:

PE +q' E22 +q33 (P-q)e12 2rE,3
(P- q)eL2 E22+ q33+ q' El- 2rE23=0 (3-3)
2rE,3 2rE23 p33 +q(eL +E22)-

where Ao = 22oo.

Next, the Raman mode shifts are convoluted to account for the

penetration of the laser light into the silicon crystal and the width of the

laser spot. An exponential decay of light intensity and gaussian intensity

distribution over the spotwidth is assumed. These corrections average the

Raman mode shifts over a volume of the crystal corresponding to each spot

reading.

3.2.2 Nitride Film Edge-Induced Stress

A simulation of the stress induced by a nitride-poly stack is used to

compare with a Raman measurement experiment. A 0.24gim thick nitride

film with intrinsic stress of 1.2e10 dyn/cm2 is deposited over a 0.05pm poly

film with intrinsic stress of-3e9 dyn/cm2. The stack rests over a 0.01 m pad









oxide and is then patterned for a linewidth of five microns. The stress

induced by the film edges is then solved for and illustrated in Figure 3-9 by

stress contours of hydrostatic pressure. This represents the stress state of a

Poly-Buffered LOCOS structure before oxidation. Finally using the method

described in the previous section the three Raman mode shifts are solved for

and are displayed in Figure 3-10.


This simulation is then compared to Raman measurement data

introduced by De Wolf [52] for the same process. Since the magnitudes of the


x in microns


.oo /



1.00 8 I x -1.0x108
-1.Ox10 i


2.00



3.00 8




4.00
4.00



5.00

0.0 y in microns 10.0

Figure 3-9 Simulation of the stress induced by the intrinsic stress of a
nitride poly stack for a linewidth of 5gm. Units are in Dyn/cm2.












f0.3


0.1


-0.3


-0.3


Position (cm)
Figure 3-10 Frequency shift simulation for the three Raman modes due to
nitride-poly stack edge induced strain compared with Raman measurement
[52].

Raman mode shifts are small, only one shift can be recovered in experiment,

which is an averaged function of the three Raman mode shifts. This

simulated Raman shift is plotted along with the measured Raman shift [52]

in Figure 3-10. It is noticeable that frequency shift predicted correlates well

with the De Wolfs measurements. The resulting compressive strain induced

shift is consistently lower than the measured Raman shifts. This indicates

that the compressive strain magnitudes under the nitride film may be under-

predicted. Other reasons for the discrepancy may be film-related effects on

the incident signal reflection, or convolution averaging errors due to light

intensity decay and spot width averaging.







3.3 Boron-Doped Cantilever Bending Comparisons


Next, an application using the boron-induced strain model is presented

in order to validate its solutions. The strain-induced bending of boron doped

cantilevers are simulated and compared with previous experimental

investigations.

3.3.1 Silicon Bulk Micro-Machining

Silicon bulk micro-machining is important for fabricating silicon-based

sensors and transducers. Silicon sensors are often composed of thin

membranes, bridges, cantilevers, and beams. These structures can be

fabricated by various bulk micro-machining methods. Anisotropic wet

chemical etching is often used to develop sensor structures due to its

simplicity and convenience as well as providing very accurate dimensional

control [18].

Boron etch stops are often used as a method for controlling etch depth

in silicon substrates. Thin silicon film structures can be fabricated by

thermally diffusing or implanting boron on one surface of the silicon wafer

and then by etching through a mask window on the other side of the wafer.

For wet chemical etchants such as KOH, the etch rate decreases significantly

as the etch front approaches boron concentrations greater than 7x1019 cm-3. It

is believed that the strong B-Si bond tends to bind the crystal more

stringently, therefore requiring more energy to release the silicon atom [53].







It is then possible to design thin silicon film structures with the

desired thickness by controlling the diffusion of the boron dopant profile so

that the etch stop will occur at depth where the boron concentration

approaches ~7x1019 cm-3. However due to these levels of boron concentration,

high levels of residual stress are generated. Since micro-machined thin

membranes are critical components of silicon sensors and transducers,

residual stresses in these structures may lead to mechanical failure of the

device and/or deteriorate its performance. Figure 3-11 illustrates the bulk

micro-machining process performed by Yang [39].



# Si02

Oxidation
Si

Si02

Thermally Diffused Boron




Backside Etch



Stain-induced
Bending

Figure 3-11 Bulk micro-machining process flow for fabrication of the
cantilever.







First boron is introduced by thermal diffusion and is masked by an

oxide layer on the backside of the wafer. Next the front side is masked and

the backside is chemically etched. The etch slows as the surface approaches

the high concentration of boron (~6x1019 cm-3). Next the front side mask is

stripped off and the cantilever is developed. In Yang's experiment, a series of

front side etches then performed to study the bending behavior of cantilevers

of different thicknesses doped with the same boron profile. The relative depth

of the peak of the boron profile then shifts for each cantilever etch.

The high concentrations of boron necessary to produce the etch stop

behavior results in residual tensile stress with magnitudes approaching and

exceeding levels of 1x109 dyn/cm2. To relieve these high levels of stress the

silicon crystal yields and may generate dislocations that may be deleterious

to device and sensor performance [8]. This is one of the main reasons for

studying residual stress and its origins.

The residual stress resulting is dependent on the gradient and

maximum magnitude of the boron dopant profile as well as the thickness of

the cantilever resulting after the backside etch. Since the boron dopant

profile is not uniform the stress distribution varies with depth causing the

cantilever to bend in order to relieve the resulting residual stress. This is

evident in previous studies analyzing positive and negative bending of boron-

doped cantilevers under varying diffusion conditions [39, 40].









3.3.2 Cantilever Bending Simulations


Simulations are performed using the finite element models previously

described for the cantilever process described in Figure 3-11 [54]. First boron

is introduced by thermal diffusion. The resulting profile has a peak

concentration of 8x1019 cm-3 (Figure 3-12). A backside wet chemical etch is

then performed. A boron concentration of 6x109cm-3 is chosen as the stop for

this etch. The resulting cantilever thickness is about 1.4gm.


Next a series of 2d plane strain and 3d elastic deformation simulations

is performed for the resulting boron-doped cantilever structures. The 2d

cantilever has a length of 50pm. The grid spacing in the x-direction is 0.05gm

and in the y-direction is 0.5gm. Since the x-direction spacing is limited by the

necessary resolution to represent the boron profile, it becomes a challenge to

preserve element quality as the cantilever length is increased while


Boro

8x10
6x10

4x10


2x10


nCoi.c(-o) FLOOPS____
20? ____



)19



*,1-
19 -_________________



19? _____________ ____


-200 Dep1 (micrni) -1 00

Figure 3-12 Resulting cantilever boron profile with a thickness of 1.4pm.








maintaining the number of elements constant. The element aspect ratio

problem is magnified further in three dimensions. The 3d cantilever has

dimensions of 1am width and 10m length. The 3d cantilever grid spacing is

0.3pm in the z-direction (width), 0.5am in the y-direction (length), and O.1pm

in the x-direction (thickness).

The following mechanical material properties were utilized for all

simulations performed: Young's modulus (Ep+si = 1.22x1012 dyn/cm2) [55],

Poisson's ratio (v = 0.3).

A series of varying front side etch simulations were performed to

analyze how the shift in boron profile and decrease in thickness of the

cantilever affected the deflection solution. These series of etch simulations

model the experiment performed by Yang [39].

The results of the 2d plane strain simulations are displayed in Figure

3-13a. An example of the deflection simulation for the 0.62pm thickness beam

is shown in Figure 3-14. The structure is regridded after the nodal

displacements are solved for. Figure 3-13 displays the nodal displacements

along the top surface of each cantilever structure simulated. The differences

between each structure thickness relates to the amount etched off the top

surface. Notice that all the cantilever beams except the thickest deflected in

the negative x- direction (upward in Figure 3-14). Generally as the

cantilevers were etched thinner, the amount of deflection increased.










A 3d simulation for the 1.37pm thickness cantilever is demonstrated in


Figure 3-15. It is more difficult to examine the deflection visually in the 3d


simulations due to the shorter length of the cantilevers. The deflection


solution plot for the 3d simulations is displayed in Figure 3-13b. The same


general trend also results.


(a) 20 Simulation
50 um Length Deflections
80E-06 T
60E-06



..: ..


.- ,b- .. -"
4 -rF -0



., 3 2-

1-t .... .......... .............. ............. ... ...... ....
0 000i 00o0 0003 0004
Y-Position along the Cantilever (cm)


00a0


(b)

40E-07
3OE-07
20E-07
I OE-07
E
e
- 000.00


-2 0E-07
0 -30E-07
-40E-07
-50E-07
-6.OE-07


3D Simulation
10 um Length Deflections



s^


1~


--" -a'-
*-092
-- 1I7 T ,
--1 22
137

0 0.0002 00004 00006 0.0000 0001
Y-Position along Cantilever (cm)


Figure 3-13 (a) Two- and (b) Three-dimensional simulation results for
deflection curves for cantilevers of varying thickness.






The simulation lengths of the cantilever beams are much shorter than


those fabricated in various experiments [39, 40]. Typically cantilever beams


are fabricated with lengths up to 1mm in order to have an accurate


measurement of the deflection. The element quality problem limits the length


of the cantilevers simulated before a significant error is resulted in the elastic


solution.








x in microns FLOOPS
-1.40 -
12'' ... '-" 'i2 i'K '" .. :;'7










Figure 3-14 Two-dimensional plane strain simulation of the boron strain
induced bending of a cantilever with thickness 0.62pm.


To compare the simulations performed with experiments in the

literature, a parabolic curve fit was used to extrapolate the expected

deflections for longer cantilever beams. This is possible because beams

processed the same with varying lengths would all have the same bending



FLOOPS
.. ._- ... .. _
0. 000393











-1







0. 000000000


Figure 3-15 Three-dimensional simulation of the bending of 1.37um
thickness cantilever. The maximum deflection is at the bottom corner of the
cantilever beam.
cantilever beam.






77

moment [40]. Therefore, the cantilever deflection simulation results are then

extrapolated to amounts corresponding to 1250km to compare with

experiments performed by Yang [39]. These results are presented in the

histogram shown in Figure 3-16.


Several points can be deduced from the results obtained. First is the 2d

and 3d simulations resulted in roughly the same amount of deflection. This

confirms that the plane strain approximation does not affect the result of the

simulation and that cantilever width is not a factor in the stress solution.

Second, both the simulations agree with the published experiment in the

direction of the deflection for each beam thickness. However the measured


Cantilever Deflection Comparisons
200 ........................ ............................... ..... .. ..................................................................... .............
M Measured [6]
e 100-- Extapolated 2D Simulation
3 OExtapolated 3D Simulation
c -

100


1 -200

S-300


N 0 0 I
~400 ------------------


Various Thicknesses


Figure 3-16 Results of both sets of simulations compared with Yang
experiment [391.







deflection results are consistently about two to three times the magnitude of

the simulations. Also the measured quantities have a relative maximum

negative deflection for the 0.92pm thick beams, while the simulations showed

relatively constant deflections for cantilevers of less thickness. It is believed

that these differences are due to errors in the boron profile and a variation in

the lattice strain coefficient.

Cantilever self-loading (due to its weight) has also been suggested as a

possible source of extra strain. However the maximum bending moment due

to a constant distributed load on a cantilever supported on one end is

described as the following relationship [56]:

WL2
max = (3-4)
max 2

where L is the length of the cantilever and W is the uniformly distributed

load per unit length. For a body force such as gravity, W is represented as the

following:

W = Apg (3-5)

where A is the cross-sectional area of the cantilever, p is the density of silicon

(2.33 gm/cm3 for 5el9cm-3 boron-doped silicon [41]) and g is gravitational

acceleration constant. For a cantilever with rectangular cross section and

thickness t, the longitudinal stress a is proportional to M by the following

relationship:





79

6M
a -A. (3-6)
At

The following gravity-induced stress relationship then results by substituting

(3-4) and (3-5) into (3-6):

3pgL2
a -3 (3-7)
t

It can then be shown that for a cantilever of 1000m length and lgm

thickness, the maximum stress produced is about 7e5 dyn/cm2, which is at

least three orders of magnitude less than the stress magnitudes induced by a

concentration of boron at 5el9cm-3.

The most probable cause for the difference between the simulations

and experiment is differences in the boron profile and lattice strain

parameter. Since the strain is computed directly from the local concentration

of boron and the lattice strain parameter, a shift in these results in a linear

shift in the strain and therefore the stress. Therefore scaling the lattice

parameter by a factor of three results in much closer agreement with the

deflection magnitudes in Yang's experiment. This also results if the boron

concentration is scaled by the same factor. It is therefore believed that the

differences in both factors result in the observed disagreement in deflection

magnitudes. This is highly probable since the accuracy in the concentration

of boron reduces as the depth increases and the lattice parameter may be

different for the thermally diffused case.








Another method to estimate the lattice contraction parameter, is to

subtract the covalent radius of boron (0.88A) from that of silicon (1.17A) and

then normalizing the difference to the silicon atomic radius. This results in

an estimate of the normal radius contraction when a boron atom substitutes

for a silicon atom:

R R 1.17-0.88
= R 1.17 -0.88 = 0.248. (3-8)
Rs, 1.17

This lattice contraction parameter results very close to that extracted from

Horn's measurements [41] (0.014A/5.43A)*(100).


3.4 3D Boundary Loading Method Example (LOCOS)


The computational advantage of the boundary loading technique

allows for less CPU intensive three-dimensional strain simulations. For two

dimensional oxidation simulations, the BL method averaged 2-5 times faster

depending on the number of elements in the silicon. This time savings is even

much greater for three-dimensional simulations since the number of

elements for a typical 3D simulation is an order of magnitude greater than in

2D.

Figure 3-17 illustrates a three-dimensional simulation of the stress

induced in a LOCOS process computed by the boundary loading method. The

process underwent a 10000C wet oxidation for 15 minutes at atmospheric

pressure. The pad oxide thickness is 10nm and the dimensions of the nitride


































Figure 3-17 Three-dimensional simulation of LOCOS-induced stress in the
substrate.


film are 1.0pm x 1.5pm x 0.15pm. The iso-stress contour lines demonstrate

how the stress levels are dependent on the geometry of the nitride film. In

the 1.5pim length dimension the stress level is greater than in the 1.0pm

dimension. This is evident by comparing the area occupied by corresponding

iso-contour lines.

Figure 3-18 and Figure 3-19 are top views of the same simulation. The

stress tensor component in the x-direction (directed normal to the surface) is



























Figure 3-18 YZ plane view of the tensile oxx component of stress induced
during LOCOS oxidation.

plotted as contour lines. Figure 3-18 demonstrates that the maximum tensile

stress is at the corner of the nitride film. Figure 3-19 demonstrates that the

maximum compressive stress is located at the edges of the nitride film and is

greatest along the longer 1.5pm side of the nitride film.


3.5 Summary


The purpose of this chapter was to demonstrate through application

examples the capabilities of the strain models described in chapter two.

Another intention was to validate the results of various simulations with

experiments that have been reported. In the first section of this chapter,

solutions of oxidation-induced stress computed by the BL method and FI



























Figure 3-19 YZ plane view of the compressive on component of stress
induced during LOCOS oxidation.


method were analyzed The LOCOS simulation example applications showed

that at higher nitride linewidths, the two method's solutions agreed.

However, at short linewidths, the two method's solutions began to disagree.

This lead to believe that the stress relaxation in the silicon becomes a critical

factor in solving for the strain as dimensions are scaled. Also, by analyzing

the re-oxidation simulations it was found that the simulations performed

with the FI method computed an overhang in the low temperature wet

process at the trench corner. The BL method simulation did not. Since in

experiment it was found that an overhang is resulted, it is believed that the

stain solution computed using the FI method was more accurate in retarding

the growth for that process.







Next, an application solving for the film edge-induced stress of a

nitride poly stack was demonstrated. The results of these simulations were

then used to predict the Raman signal measured over the nitride-poly stack

structure using a method outlined by De Wolf [52]. The expected frequency

shift in Raman modes then agreed both qualitatively and quantitatively with

Raman measurement experiments for the same structure.

Next, the boron-induced strain model was used to simulate the

bending of boron-doped cantilevers due to their boron-induced residual stress.

The simulations were then compared with studies of this effect and produced

qualitatively favorable results. The simulations agreed with experiment in

comparative bending between the different thickness cantilevers. The

difference in deflection magnitudes can be attributed to the lattice strain

parameter used in the model. By simply scaling this parameter, simulation

deflection magnitudes would coincide better with the measurements. One

possible explanation may be that the lattice strain parameter used may not

apply since it was measured in bulk silicon and may have a cantilever

thickness dependence.

Finally, three-dimensional simulations of oxidation induced stress

were demonstrated for a LOCOS process using the BL method. The BL

method's efficiency allows for three dimensional strain simulations to be

performed rapidly.













CHAPTER 4
KELVIN PROBE FORCE STI EXPERIMENT



In the previous chapter, several applications were simulated using the

strain models described in chapter two. Those applications analyzed

individual sources of mechanical stress and how the modeled strain fields

compared with various experimental studies. In this chapter, the strain field

due to the shallow trench isolation (STI) process sequence is investigated.

STI has become an essential isolation scheme as CMOS technologies

are scaled down below the 0.25pm generation. However the basic STI process

sequence involves several sources of mechanically straining the enclosed

silicon region. Such sources include the thermal mismatch strain between the

isolation dielectric and the silicon substrate, intrinsic stress of the nitride

mask, and volume expansion-induced stress of the sidewall oxidation (Figure

4-1). It is therefore important to study each source and their combined strain

influences. As pitch lengths continue to scale, the distance between transistor

active areas and the STI sidewall also narrows. Therefore they continue to

approach the more highly stressed regions at the sidewall interface. This in

turn influences the device characteristics such as leakage through higher

dislocation densities and band gap deformation. Therefore for next



















Figure 4-1 Major sources of stress in STI fabrication include the following:
(a) Nitride film edge-induced, (b) Insulator thermal expansion mismatch-
induced, and (c) Sidewall oxidation volume expansion-induced stress. Arrows
indicate force vectors.

generation process design, methods are needed to measure the stress

quantitatively. Up until now, micro-Raman spectroscopy has been the most

suitable method for investigating the stress [26]. But as technologies

continue to scale towards the 0.lpm generation, this may even surpass micro-

Raman's spatial resolution limits. In this chapter, scanning Kelvin probe

force microscopy (SKPM) is investigated as a new method for measuring

stress-induced influences in the silicon work function AO and for inferring the

stress magnitudes through coupling with mechanical strain simulation.


4.1 Scanning Kelvin Probe Force Microscopy


Scanning Kelvin probe force microscopy (SKPM) has previously been

researched as method for profiling 2D dopant concentration [57-62]. This has

been achieved by detecting the electrostatic potential difference (EPD)








AFM Cantilever Probe
I I
+z \V

v V Rastered laterally

FE(Z)



Figure 4-2 Cross-sectional sketch of SKPM measurement.

between the SKP tip and sample (Figure 4-2) through their electrostatic and

van der Waals force interaction.

The method involves rastering a cantilever probe over a cross-sectional

surface of the sample. At each step, both a surface topography measurement

and a surface work function measurement is performed. For measuring

surface topography the technique simply works as an atomic force microscope

(AFM). Under the AFM mode of operation, the mean distance between

cantilever probe and sample is kept constant over each step position. The

cantilever probe then is oscillated mechanically at a particular frequency

[58]. As the cantilever-sample mean distance changes, the van der Waals

force also changes causing a change in the oscillation amplitude. The

cantilever position is then corrected in the z-direction to return the cantilever

to its original oscillation amplitude. The corrected cantilever position is then

recorded as the cantilever is rastered across the sample and therefore maps

the surface topography as demonstrated in Figure 4-2.








Laboraory STI Array Sample
Laboratory Cross Section
Frame of Reference


Z [iTO]

Y [110]

x [00T]


Figure 4-3 Sketch of Kelvin probe scanning laterally (y-direction) and in
depth (x-direction). Variation in work function (EPD) is measured by
monitoring the electrostatic force contribution to the cantilever deflection.

Surface work function measurement is performed under the SKPM

mode of operation. First, an AC bias is also applied to the cantilever along

with the mechanical oscillation. An electrostatic force then results between

the cantilever tip and sample. A DC flat-band voltage bias is then applied to

the cantilever probe to null the electrostatic force. This DC bias is

proportional to EPD of the cantilever-sample system. The DC flat-band

voltage is then recorded at each raster step and provides a two-dimensional

map of the work function variation of the sample (Figure 4-3).


4.2 Work Function Influence


The EPD scanned by the SKPM method is a measure of the difference

in work functions of the cantilever tip and sample. This can best be described

with the energy band diagram of the system (Figure 4-4). It is similar to that

of a metal-insulator-semiconductor (MIS) device band diagram, with the







airspace between the cantilever and sample representing the insulator. In

this application, the cantilever tip is constituted of silicon with a layer of

titanium silicide grown on the surface for improved signal detection. When

the tip and sample are electrically connected, their Fermi levels become

aligned. The existing electrostatic force between the tip and the sample

causes the silicon bands to bend at the surface. Additional band bending is

also due to the presence of surface states and charges on the semiconductor

surface.

In SKPM mode, a DC flat-band bias is applied to null the electrostatic

force (Figure 4-5). This applied bias represents the EPD at each rastered



Ti-Silicide Tip vacuum Silicon


Figure 4-4 Energy band diagram of the cantilever-sample system at zero
applied DC bias.







point along the cross-sectional surface of the sample. The measured VEPD can

then be expressed as the sum of the flat-band correcting voltage VFB and

surface potential terms [61]:


VEPD = Vf +V, +- sE s,].
Coins


(4-1)


The flat-band voltage is the difference between the work functions of the tip

and sample:


(4-2)


The work function of the sample Osi is the energy difference between the

vacuum level and the Fermi level:


Ti-Silicide Tip


vacuum Silicon


qXsi

I EC
A-X ---


Figure 4-5 Energy band diagram after VEPD is applied and the electrostatic
force is nulled.


V = Ti-Si Os,.




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MODELING OF MECHANICAL STRESS IN SILICON ISOLATION
TECHNOLOGY AND ITS INFLUENCE ON DEVICE CHARACTERISTICS
By
HERNAN A. RUEDA
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
1999

Copyright 1999
By
Hernán Rueda

To my family

ACKNOWLEDGMENTS
I would like to express my sincere gratitude to Dr. Mark Law,
chairman of my advisory committee, for his patience and guidance. He has
introduced me to the research area of process modeling, within which this
dissertation falls, and has provided valuable advice and direction throughout
my master's and doctorate studies.
Thanks also go to Drs. Gijs Bosman, Toshikazu Nishida, Kenneth K. O
and Kevin Jones for their interest and participation in serving on my
committee and their suggestions and comments.
I would also like to thank the University of Florida Graduate School
for its support during my master’s program and the Semiconductor Research
Corporation for its support of my doctorate studies.
I have been very lucky to work with many industry mentors who have
provided invaluable assistance for my graduate studies. I acknowledge Drs.
Jim Slinkman and Dureseti Chidambarrao of IBM for their suggestions and
direction of the STI SKPM experiment. I also thank Dr. Len Borucki of
Motorola for assistance in design of the diode bending experiment. Thanks
IV

also go to Drs. Paul Packan and Steve Cea of Intel for many valuable
discussions and suggestions on modern device concerns and strain modeling.
I wish to thank my friends who have made my time, over ten years at
the University of Florida, a very enjoyable experience. I’ve been very lucky to
meet so many good friends such as all the old TCAD research assistants, the
SWAMP group, my many roommates through all the years, and my ‘old
school’ friends back in the ‘hood in Miami.
Last but not least, I express my love to my parents, Hernán Sr. and
Gloria, and my brother, Camilo, for their never-ending support, love and
encouragement throughout my whole life.
v

TABLE OF CONTENTS
page
ACKNOWLEDGMENTS iv
ABSTRACT viii
CHAPTERS
1 INTRODUCTION 1
1.1 Motivation 1
1.2 Stress-Induced Effects in Silicon Fabrication 3
1.2.1 Oxidation Influences 4
1.2.2 Diffusion Influences 6
1.3 Stress-Induced Effects in Silicon Device Operation 7
1.3.1 Carrier Mobility Influences 7
1.3.2 Energy Band Influences 11
1.4 Goals 12
1.5 Organization 14
2 PROCESS-INDUCED MECHANICAL STRAIN MODELS 16
2.1 Continuum Mechanics 16
2.1.1 The Stress Tensor 17
2.1.2 The Strain Tensor 20
2.1.3 Stress-Strain Relationships 23
2.2 Strain Sources 30
2.2.1 Film Stress 31
2.2.2 Dopant Induced Stress 35
2.2.3 Oxidation Volume Expansion 39
2.3 Strain Computation Methods 40
2.3.1 Boundary Loading Method 45
2.3.2 Fully-Integrated Method 47
2.4 Summary 48
vi

3 APPLICATION EXAMPLES AND COMPARISONS 50
3.1 FEM Comparisons 50
3.1.1 LOCOS 55
3.1.2 Post-STI Process Re-Oxidation 59
3.2 Raman Spectroscopy Measurements and Comparisons 65
3.2.1 Raman Simulation Method 66
3.2.2 Nitride Film Edge-Induced Stress 67
3.3 Boron-Doped Cantilever Bending Comparisons 70
3.3.1 Silicon Bulk Micro-Machining 70
3.3.2 Cantilever Bending Simulations 73
3.4 3D Boundary Loading Method Example (LOCOS) 80
3.5 Summary 82
4 KELVIN PROBE FORCE STI EXPERIMENT 85
4.1 Scanning Kelvin Probe Force Microscopy 86
4.2 Work Function Influence 88
4.3 STI Experiment 93
4.4 SKPM Measurements 94
4.5 STI Strain Simulations 99
4.6 Results and Discussion 101
4.7 Summary 105
5 STRESS INFLUENCES IN DEVICE OPERATION 107
5.1 Uniaxial Stress Influence Experiment 107
5.1.1 Stress-Inducing Apparatus Design 109
5.1.2 Stress-Wafer Deflection Relationship 112
5.2 Experimental Procedure 114
5.3 Experimental Results 118
5.4 Summary 124
6 SUMMARY AND FUTURE WORK 126
6.1 Summary 126
6.2 Future Work 129
6.2.1 SKPM Strain Measurement Calibration Studies 129
6.2.2 Effect of Stress on Dislocations 130
6.2.3 Silicidation-Induced Stress 131
6.2.4 Three-Dimensional Modeling of the STI Process 131
REFERENCES 133
BIOGRAPHICAL SKETCH 140
vii

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
MODELING OF MECHANICAL STRESS IN SILICON ISOLATION
TECHNOLOGY AND ITS INFLUENCE ON DEVICE CHARACTERISTICS
By
Heman Rueda
May 1999
Chairman: Dr. Mark E. Law
Major Department: Electrical and Computer Engineering
One of the challenges the semiconductor industry faces as it attempts
and continues the scaling of silicon integrated circuits is understanding and
control of mechanical strain resulting from silicon fabrication technology.
High magnitudes of strain can be induced under standard fabrication
conditions and may produce deleterious effects in device behavior, such as
increased current leakage. Current leakage has been identified as a critical
device characteristic for future sub-micron dynamic random access memory
(DRAM) and complementary metal oxide semiconductor (CMOS)
technologies, as it is a limiting factor for increasing switching speeds and
decreasing power consumption. The following are various known sources of
stress in silicon technology: thermal expansion mismatch, intrinsic stress,
Vlll

and oxidation volume dilation. This work results from an examination, by
modeling, experiment, and simulation, of the contribution of stress due to
these sources using the Florida Object-Oriented Process Simulator
(FLOOPS).
The contributions of each source can be simulated using different
models that represent or approximate the physics involved. After the models
are described and presented, example applications are provided to
distinguish the advantages and limitations for each model.
Coupling experiment along with process simulation then validates the
results and allows for a better understanding of the problem. One such
problem examined in this work is the strain induced by the shallow trench
isolation (STI) process. STI has become an essential isolation scheme for
present and future sub-micron processes. It consists of several sequential
steps that exert stress in the silicon active area by each of the previously
described sources. Scanning Kelvin probe force microscopy (SKPM) is then
applied as a new technique to characterize the strain exerted from STI
processes through measurements of strain-induced work function variations
in silicon. Qualitative agreement is demonstrated between the SKPM
measurements and the work function influence due to finite element based
STI induced mechanical strain computations.
Finally, a wafer bending experiment is performed that quantifies the
influence of tensile and compressive uniaxial stress on forward current of pn-
IX

junction devices. This effect is then modeled primarily through the strain
influences in the silicon bandgap.
x

CHAPTER 1
INTRODUCTION
1.1 Motivation
During the quest for increasing device density in integrated circuits,
many problems are encountered and need to be solved. As some problems are
alleviated, new issues emerge. One of the problems gaining importance in
silicon fabrication is process-induced mechanical stress. Many of the
processes used in silicon IC fabrication individually and cooperatively
contribute to the development of stress in the silicon active areas.
Of prime interest is the mechanical stress generated in the isolation
process flow. Isolation technology is continually challenged as design rules
are scaled. It is well known that high stress magnitudes in certain silicon
dioxide structures cause a decrease in oxidation rate [1, 2].
In LOCOS (LOCal Oxidation of Silicon) isolation processes, silicon
nitride is deposited over a thin pad oxide and patterned to mask the
oxidation of silicon. In order to cope with the necessary electrical isolation at
the ever-decreasing linewidths, increasing the stress during oxidation is
exploited [3]. This has been achieved by increasing the nitride thickness and
decreasing the pad oxide thickness. These techniques provide the sharp
1

2
transition from isolation field oxide to active area that is necessary in
decreasing pitch lengths for 0.35 micron technologies.
The main problem associated with this trend is that the silicon
substrate yields under the increased magnitudes of strain produced [4, 5].
Dislocations that are generated will degrade device performance [5, 6].
Dislocations serve as gettering sites that attract metal atoms introduced
during subsequent processes. Junctions are continually becoming shallower,
and therefore these nucleated dislocations have an even greater probability of
lying across the device junctions lined with metal atoms. Unacceptably
increased magnitudes of leakage current then result. Off-state currents are
critical design characteristics in logic and memory circuits, limiting the
switching speeds, causing increasing power consumption and limiting
reliability.
Shallow Trench Isolation (STI) is steadily becoming the predominant
isolation technology as minimum feature sizes decrease below the minimum
attainable pitch lengths of LOCOS-based technologies. The general STI
process flow includes a nitride-patterned reactive ion etch, sacrificial sidewall
oxidation, oxide deposition and finally a chemical mechanical polish.
However, stress-induced dislocation generation is not exclusive to LOCOS
based isolation technologies and is also a factor in designing STI processes
since the steps in the STI process may cooperatively strain the enclosed
silicon active areas [7].

3
Aside from isolation processes, other fabrication processes also induce
strain in the silicon crystal. Processes such as thin film deposition and
dopant introducing processes induce strain that can generate dislocations. As
feature sizes decrease, all these different strain-generating sources become
closer together and their influences overlap each other. Under these extreme
circumstances, it is important to understand strain fields generated by
multiple sources neighboring the silicon active areas.
Strain fields may also play a role in affecting process and device
behavior other than generating dislocations. For example, in silicon micro¬
machining, high residual stresses of boron induce bending sensor structures
composed of diaphragms and cantilevers [8]. Strained regions have also been
shown to affect diffusion of dopants [9]. Also, it is well known that crystal
strain affects the energy band structure [10] and carrier mobilities [11]. Both
are major parameters influencing the electrical properties of a device. These
are just a few of the many concerns related to the strain state of the crystal.
1.2 Stress-Induced Effects in Silicon Fabrication
Stress concerns in process design first became significant in LOCOS
isolation technologies. As thermal oxidation temperatures are reduced,
dislocation densities in silicon increased due to this isolation technology [4],
These effects were attributed to the increased stress magnitudes induced at
lower oxidation temperatures due to the increased viscosities of the silicon

4
dioxide. Since then, the correlation between increased dislocation densities
and high stress magnitudes have caused isolation technologies to be a major
stress related concern.
Since then the diffusion process has also been reported to be influenced
by film-induced stress [12]. This affect has been explained by stress
influences on point defect concentrations as well as extended defect size and
concentrations such as dislocation loops. As junction depths continue to
decrease in the evolution of sub-micron technologies, the stress influences
generated by surface films have a greater effect.
1.2.1 Oxidation Influences
It is well known that the growth rate of thermal oxidation of silicon
has a stress dependence [13]. This stress dependence occurs in nonplanar
regions. In thermal oxidation, a volume of silicon reacts into a volume of
silicon dioxide that is 2.2 times in volume. For planar oxidation this does not
induce a large stress. This is due to the newly grown oxide lifting the old
oxide perpendicular to the surface. Since the surface of the oxide is not
constrained the oxide is free to move in this direction so negligible stress is
induced normal to the Si-SiCh interface.
In nonplanar regions such as convex and concave comers of silicon, a
strain is exerted in the silicon dioxide. For the convex comers the strain is
laterally tensile since the old volume of oxide is stretched around a longer
periphery. In concave comers the strain is laterally compressive since the old

5
volume of oxide is compressed into a smaller periphery. Convex corners occur
at the top corners of a trench oxidation and concave corners result at the
bottom corners of a trench oxidation and at the Bird’s Beak in a LOCOS
oxidation.
It was reported through cylinder structure oxidation experiments that
the stress induced in these structures reduced the amount of oxide grown as
compared to planar oxidation [1, 2, 13]. This behavior has been observed in
highly stressed LOCOS processes and in corners of trench oxidation. It has
been documented that the stress induced in the oxide alters the oxidation
reaction rate, oxidant diffusion of reactant, and the oxide viscosity [2], All
these influences affect the oxidation growth rate.
The stresses generated in the silicon dioxide are relaxed by exerting
force on the neighboring films and the silicon substrate. This leads to
strained regions in the silicon due to the oxide growth. Silicon is an elastic
material for a wide range of strain. However, as the induced stress exceeds
the critical yield stress, dislocations in the silicon crystal are generated to
relax the strain [14]. The generated dislocations then present problems in
device behavior. Therefore, the strain induced in the silicon by oxidation
becomes a concern.
Trenches are popular isolation technologies that also exhibit
dislocation generation problems [7, 15]. The stress induced by trench
structures is due to other sources also. After the reactive ion etch, an

6
oxidation is performed to provide a low interface state density and a low
oxide trap density. This oxidation is the first source of stress in the silicon
substrate in the trench fabrication process. Next the trench is filled with a
deposited film. This film influences a stress due to its built in intrinsic stress
and thermal expansion mismatch. The three sources all act simultaneously
influencing strain in the surrounding silicon substrate. Again, the primary
concerns are dislocations generated to alleviate the strain in the substrate
induced by the trench process.
1.2.2 Diffusion Influences
It has been observed that both phosphorus and boron diffusion
behavior under silicon nitride films is different than in inert conditions [9]. It
has been established that boron and phosphorus diffusion is governed by
interactions with point defects, namely silicon interstitials. Ahn attributed
the measured diffusion reduction to a vacancy supersaturation and self¬
interstitial undersaturation that may be due to the compressive stress under
the silicon nitride films [12]. Osada later confirmed this by performing
similar experiments [16].
These experiments were performed with junction depths of
approximately 0.8 microns. Scaling for sub-micron devices has lead to
decreasing junction depths. The magnitudes of strain in silicon due to film
stress are at a maximum at the film interface. Therefore, it is believed that

7
an even greater effect is encountered for the shallower junctions that are
getting more prevalent in modem technologies.
1.3 Stress-Induced Effects in Silicon Device Operation
Interest in mechanical properties of silicon was sparked with the
discovery of its piezoresistive effect [17, 18]. This led to the increased interest
of the use silicon as a pressure sensor. Sensor research also extended
investigation of the mechanical influences on other electrical parameters
such as the energy bands of semiconductor crystals [10, 19-22], As a result,
there is satisfactory understanding of how applied mechanical forces affect
electrical device behavior. These principles are used mainly in the design of
semiconductor mechanical sensors. In the present age mechanical stress is
becoming a limitation in silicon device fabrication, these principles also need
to be incorporated into microelectronic device behavior.
1.3.1 Carrier Mobility Influences
The influence of mechanical stress on carrier mobility is starting to
gain importance in modern microelectronics. It has been shown that in
CMOS transistors the transconductance will vary dependent on the
magnitude of applied stress and the behavior is dependent on crystal
direction of the current flow as well as the type of carrier [23]. This effect is
was also noticed and believed to be due to LOCOS-induced stress in SOI

8
CMOS devices [24]. The variation in transconductance is attributed to the
piezoresistance effect on the carrier mobilities.
The piezoresistance effect of semiconductors describes how the
resistivity is influenced by mechanical stress. The electric field vector is
proportional to the current vector by a symmetrical resistivity tensor of rank
two with nine components:
V
Pi Pa Pó
*l"
¿2
=
Pa Pi P5
•
h
1
m
CO
1
_P6 Pi Pl_
h.
If the system axis is aligned along the <100> crystal directions, the
normal resistivity components pi, p2, and p3 relate the £ field vector
component to the current vector i component in the same direction. The cross¬
resistivity components p4, p5, and p6 relate the £ field vector component to the
current vector i component in a perpendicular direction. If in the unstressed
state, the normal components have the same magnitude p and the cross
components are equal to zero, this reduces to the following isotropic
relationship:
£ = pi. (1-2)
When the crystal is under mechanical stress, the resistivity
components change as the following:

9
Pi
P
Ap,"
P2
P
A p2
P3
P
+
A P3
P4
0
Ap4
P5
0
A P5
_Pfi_
0
Ap6.
(1-3)
The piezoresistive coefficients relate the stress-induced changes in the
resistivity components to the stress tensor influencing the change. This
matrix relating the six resistivity components to six stress components
consists of 36 piezoresistive coefficients 7tij. Due to the cubic crystal
symmetries in silicon, the piezoresistance coefficient matrix reduces to three
independent components, Tin, 7112, and JI44:
A P,
ft11
ft 12
ft 12
0
0
0 ‘
A p2
ft\2
ft\i
ft 12
0
0
0
0-2
1
A P3
ft Vi
ft 12
ft\i
0
0
0
—
~
•
P
Ap4
0
0
0
ft 44
0
0
A p5
0
0
0
0
^44
0
.Ap6.
_ 0
0
0
0
0
ft 44 _
.^6.
(1-4)
The stress components are also referenced with the system axis oriented in
the <100> directions. Smith initiated the investigations of these
piezoresistive coefficients and found the following values for silicon at room
temperature displayed in Table 1-1 [17, 18].
The piezoresistance coefficients are also dependent on dopant
concentration as well as temperature. Later it was found that they would

10
decrease as the temperature increases and/or the dopant concentration
increases.
Table 1-1: Piezoresistive coefficients for silicon [17, 18].
Material
P
(Q-cm)
Jill 7112
(10 12 cm2/dyne)
7144
p-Si
7.8
+6.6
-1.1
+138.1
n-Si
11.7
-102.2
+53.4
-13.6
The accepted explanation for this phenomenon is the many-valley
model [17, 18]. Anisotropic conditions exist when the mobility in one crystal
direction is different than the mobility in the other crystal lattice directions.
This results when the semiconductor is in a stressed state. The stress tensor
distorts the conduction energy bands of the unstressed semiconductor in
different magnitudes depending on direction. The energy levels and
curvatures of the band energies corresponding to the perpendicular directions
are influenced differently by the applied strain. The effective masses of the
carriers are proportional to the energy bands’ curvatures in reciprocal k
space. Since the carrier mobilities are functions of the carrier effective
masses, the strain influence on the energy band level curvatures results in
directionally dependent influences on the carrier mobilities and therefore the
resistivities of the semiconductor. The energy band shifts are also influenced
on dopant concentration and temperature. Therefore the energy band’s
sensitivity to stress will also be dependent on these influences.

11
1.3.2 Energy Band Influences
The mechanical stress state’s influence on the energy bands also
affects the electrical behavior of p-n junction devices such as diodes and
bipolar transistors. In these devices the operation is governed by the flow of
minority carriers. Using a diode as an example, the forward bias current is
described by the following relation:
7f = /Jexp(^) + /ROexp(^-) (1-5)
where the saturation current is
Is=qA
Dp
— Pn oCOth
V
+~np0co\h
(1-6)
and the recombination current is
_ qAnW
2to
(1-7)
The saturation current term is linearly related to the minority carrier
densities ripo and pno. The minority carrier densities are directly dependent to
the square of the intrinsic carrier density
n
P o
(1-8)
The intrinsic carrier concentration is exponentially dependent to the stress
dependent band gap Eg

12
n, = KlTy2cx p(-^). (1-9)
2 kT
The stress induced shifts in the conduction and valence energy bands will
alter the band gap and therefore ultimately influence the saturation current.
Wortman initiated the quantification of the effects of uniaxial and
hydrostatic compressive applied external stresses to forward and reverse
biased diodes [10, 22],
Mechanical stress also influences the generation/recombination
current component of p-n junction devices. Rindner attributed the effect of
uniaxial compressive stresses to increased dislocation densities that
decreased the carrier lifetimes and therefore increased the
generation/recombination current component [25]. This effect becomes the
greater influence under higher magnitudes of stress due to the dislocation
generation to relax the applied stress.
1.4 Goals
The goals of this work are primarily to develop a system where strain
can be computed from multiple sources simultaneously. Silicon IC fabrication
involves a sequential flow of many processes that introduce and alter the
strain in the crystal. These process-induced strain fields influence the
behavior of processes later in the fabrication flow as well as device operation

13
once the process flow is completed. An accurate strain solution is necessary
for further investigation of its effects.
Once the strain in the system is understood, strain dependent models
can be developed to help understand unexplained behavior that has been
observed that may be due to strain. Such areas may include point defect and
extended crystal defect interactions, diffusion kinetics, and band-gap and
mobility influences.
Strain simulations also could aid in the development and analysis of
isolation process technologies. Often in fabrication process development, the
stress levels generated and dislocation densities produced may decide the
isolation process required. A strain field simulator could reduce the amount
of experimental work necessary for solving these problems.
Another goal of this work is to validate the process induced strain
models with experimental measurements. Currently, this is a major hurdle
due to the few characterization techniques available for localized sub-micron
strain measurement. Micro-Raman spectroscopy has been the most suitable
method for investigating localized strains [26]. But as technologies continue
to scale towards the 0.1pm generation, this may even surpass micro-Raman’s
spatial resolution limits. Therefore, in this work Scanning Kelvin probe force
microscopy (SKPM) has been investigated as a new method for analyzing
localized strains through detecting influences in the silicon work function.

14
One last goal is to quantify the influence of tensile and compressive
stress on pn-junction device current. This would then provide some insight
into the mechanical strain influence on leakage currents.
1.5 Organization
Chapter two provides descriptions of the various process induced strain
sources and discusses how they are modeled in this work. Afterward, the
finite element methods that were developed for strain computation are then
described.
Results and comparisons between the methods for various processes
are included in chapter three. Example applications are provided to
distinguish the advantages and limitations for each model. Next, the film
edge-induced stress solutions are validated with published micro-Raman
measurements. Finally, three-dimensional applications are demonstrated.
Scanning Kelvin probe force microscopy (SKPM) is then explored as a
technique for characterizing STI induced strain in chapter four. An STI
experiment is performed and strain influence is measured by SKPM. These
measurements are then compared with simulations of band gap influences
using the models described in chapter two.
An experiment relating mechanical stress to pn-junction device
operation is then described in chapter five. This wafer bending experiment
addresses uniaxial stress influences on the forward current. This allows for

15
quantifying the influence on the reverse leakage current through observation
in the forward bias.
Finally chapter six concludes with a summary of the research work
accomplished and addresses topics for future work.

CHAPTER 2
PROCESS-INDUCED MECHANICAL STRAIN MODELS
The solution of strain present in a particular device technology is
computed using a finite element method (FEM) formulated to solve for the
strain induced by various sources in silicon technology [27, 28]. In a
fabricated device, the strain field in the silicon is generated due to various
processes at different steps along the fabrication flow. The most critical
sources for inducing stress are deposited and grown films. Sources in the
silicon crystal such as dopants and extended defects are becoming more
important as technologies advance.
In this chapter, a brief review of continuum mechanics is first
provided. Next the individual strain sources are then described. The
algorithms used to model the strain generated from each source are also
discussed. Afterward the finite element methods used to integrate the
various strain sources are described.
2.1 Continuum Mechanics
It is the intention that this review refreshes the reader with the theory
and notation of continuum mechanics. References are provided for a more
16

17
complete description. The stress tensor is first introduced in this section.
Next the strain tensor is described. Finally this section concludes with
different stress-strain relationships descriptions and how they may relate in
silicon processing.
2.1.1 The Stress Tensor
Stress is the distribution of internal body forces of varying intensity
due to externally applied forces and/or heat [29]. The intensity is represented
as the force per unit area of surface on which the force acts. To illustrate this
concept, consider an arbitrary continuous and homogeneous body (Figure 2-1)
under the applied external forces, Fi, F2, F3, and F4. If the body is sliced into
two smaller volumes Vi and V2, then V2 exerts force on Vi at their surface
Figure 2-1 Continuous body with external forces applied.

18
interface S to remain in equilibrium. The resulting force may be of varying
intensities along the surface.
At any point P on the surface between Vi and V2, the forces can be
reduced to a force and a moment, which can be described by a stress vector
acting on that surface. Three stress vectors acting on three mutually
orthogonal planes intersecting at that point can then determine the stress
state at any point P (Figure 2-2). The stress tensor is composed of the three
stress vectors and, according to Cauchy’s equations of motion, is sufficient to
define the stress state in any element in a body [30].
To illustrate the tensor nature of stress present at point P in the
continuous body, consider a cubic element (Figure 2-2) of infinitesimal
+ +y
+x
Figure 2-2 An infinitesimal cubic element located within a continuous body
with stress tensor components shown.

19
dimensions located at point P in Figure 2-1. For simplicity of notation, let the
cube be aligned perpendicular with the system axis. The stress vector Tx
acting on the plane normal to the x-direction is the following:
Tx=°xxX+oxy-y+oxiz. (2-1)
Let the surface AS be the plane of the cube normal to the x-direction. The
stress vector Tx is defined as the ratio of force acting on that surface area AS:
A^o AS dS,
(2-2)
The stress tensor on that volume is defined by nine stress components acting
on the three surfaces of the cubic element, which make up the three stress
vectors Tí:
(2-3)
In the definition above, on are the normal stress components acting on the
faces perpendicular to ¿-direction and v.j are the shear stress components
oriented in the ./-direction on the face with normal in the ¿-direction. At
mechanical equilibrium, it can be shown that the stress tensor is symmetrical
[29],
T.. = Z...
ij ji
(2-4)

20
A column vector of six independent components can then describe the state of
stress at a point:
0„ (Ta Tyz tJ. (2-5)
2.1.2 The Strain Tensor
The application of stress to a body in equilibrium causes it to undergo
deformation and/or motion. A measure of deformation is strain. Two common
measures for strain are the Lagrangian and the Eulerian definitions. Both
are functions of the initial and final dimensions. When the displacement
between the final and initial measurement is referenced to the original
position dimensions then it is known as the Lagrangian definition. The
Eulerian definition describes the deformation displacement referenced with
respect to the deformed position.
Figure 2-3 shows a one-dimensional example of the deformation in a
(a)
Unstretched
(b)
Stretched
Figure 2-3 One-dimensional deformation of a spring: (a) original length
(Ax), (b) deformed length (Ax+Au).

21
spring. The Lagrangian strain and Eulerian strain then, respectively, become
the following over the length of the spring:
increase in length _ Au
original length Ax
(2-6)
_ increase in length _ Au
deformed, length Ax + Au
(2-7)
For the case of infinitesimal deformation, the Eulerian and Lagrangian
descriptions become equivalcut. An infinitesimal deformation approximation
requires that the maximum deformations involved be much smaller than the
smallest dimension of the deformed body. A second requirement is that the
deformation gradient is much less than one. When these assumptions hold,
the infinitesimal strain can be defined by the following relationship for any
point in the spring [31]:
Am du
£ = lim — = —.
A)t^° Ax dx
(2-8)
By expanding this definition in three dimensions, the strain is related to the
displacements by the following strain components [30]:
du
i
p — p —
(du
dv)
dx
ry y* 2
r
dx,
dv
dy
1
£ =£ = —
K * 2
' dv + dw'
ydz dy j
dw
!k
1
£ =£ = —
ix xz 2
' du dw'
ydz dx J
(2-9)

22
where u, v, w are the displacements in the x, y, and z directions, respectively.
These components make up the strain tensor (&/) that is analogous to the
stress tensor:
£
kl —
xy
yy
zy
(2-10)
A three-dimensional example of a body undergoing deformation due to
an externally applied force is illustrated in Figure 2-4. In the example a
compressive force is applied to the infinitesimal cube in the x-direction. A
negative displacement (compressive strain) results in the x-direction and
positive displacements (tensile strains) result in the y- and z- directions. The
Figure 2-4 Strain reference example displaying a compressive stress in the
x-direction that generates normal strains in the x-, y-, and z-directions.

23
relationship of how the strain in each dimension results from the applied
force will be discussed next.
2,1.3 Stress-Strain Relationships
Bodies of different materials but of same dimensions may deform
differently under the same stress application. The relationship between the
stress tensor and the deformation is known as a constitutive relation. It may
vary for a given material depending on conditions such as temperature and
pressure.
Elasticity. All structural materials possess, to some extent, the
property of elasticity. Elastic bodies possess memory during deformation. For
example, when a force is applied on an elastic solid, it will deform until it
reaches its elastic yield limit or until the load is released. Microscopically the
bonds between atoms that compose the solid ‘stretch’ during elastic
deformation. When the force is removed the body will return to its original
shape if it is an ideal elastic body and it had not reached its yield stress,
similarly to an ideal spring. When the load is removed the bonds return to
their unstressed lengths corresponding to the original environmental
conditions.
For a Hookean elastic solid, the stress tensor is linearly proportional to
the strain tensor over a specific range of deformation:
(2-11)

24
by the tensor of elastic constants ajki [30]. In order to relate each of the nine
elements of the second rank strain tensor to each of the nine elements of the
second rank stress tensor, ajki consists of a fourth rank tensor of 81 elements.
However due to the symmetries involved for the stress and strain tensors
under equilibrium, cijki is reduced to a tensor of 36 elements.
Crystal silicon has diamond cubic crystal geometry resulting from its
strong directional covalent bonds. For such crystals, ajki has the following
form due to their cubic symmetry [32]:
cll
c12
C12
0
0
0
cl2
Cll
cl2
0
0
0
C12
cl2
cu
0
0
0
0
0
0
C44
0
0
0
0
0
0
C44
0
0
0
0
0
0
C44
(2-12)
Thus, for silicon the tensor of elastic stiffness constants reduces to the three
independent components: cu, C12, and cm. Due to the crystal’s lattice
temperature dependence, the elastic constants are also thermally dependent.
At room temperature (25°C) the elastic constants have been measured
as the following for silicon [33, 34]:
cn= 1.657 x\0l2dyn/cm2
cl2= 0.639 x\0l2dyn/cm2
c„ =0.7956x\0n dyn/cm2.
The elastic constants’ thermal dependence has been documented as the
following linear relationship for silicon [35]:

25
III
>
â–  =-75X10-6
/K°
cu
hi
&
Ac12
= -24.5 x 10-6
/K°
C12
III
3
ft
AC.4
= -55.5 x 10-6
/K°
From the above linear dependence, it can be seen that the elastic constants
do not change significantly for large temperature changes. Also the degree of
anisotropy does not change significantly for the range of 100-900°K [36].
These studies support that silicon acts as an anisotropic elastic material over
a wide temperature range frequently encountered in silicon IC fabrication.
Although silicon is an anisotropic crystal, it is sometimes desirable to
approximate it with isotropic elastic properties for simplification. When the
components of elastic constants for a material are equal for any rotation of
the reference axis, the material is said to be isotropic. This means that the
elastic properties of the material are the same in all directions. The tensor of
elastic constants for an isotropic material reduces to the following:
E( 1-V)
c12 —
(l + v)(l-2v)
Ev
(l + v)(l-2v)
E
C 44 ~
(1 + v)
(2-13)
(2-14)
(2-15)
where E represents the Young’s modulus and v represents the Poisson’s ratio.
As was demonstrated in Figure 2-4, contractions in one dimension may be
accompanied by dilations in other dimensions. The Young’s modulus is a
measure relating stress and strain for an elastic material when stress is

26
applied in one direction and the other directions are free to deform as in
Figure 2-4. The Poisson’s ratio is a material property describing the ratio of a
strain perpendicular to the applied stress to the strain oriented in the
direction of the applied stress.
Due to its anisotropy, E and v vary depending in the direction of the
applied stress and plane acted upon for silicon. At room temperature E may
vary from 1.3el2 dyn/cm2 (for <100> directions) to 1.875el2 dyn/cm2 (for
<111> directions). Poisson’s ratio also may vary from 0.06 to 0.34 for the
same conditions. The following are measured values for E and v in silicon at
room temperature [32, 33, 35]:
£■[[00] =1.31x10n dyn/cm2
£[110] = 1.69 xlO12 dyn/cm2
£;m] = 1.875 xlO12 dyn/cm*
'/[ioo] = 0.279.
Viscosity. Although silicon deforms elastically over a wide temperature
and load range, other materials used in silicon fabrication behave differently.
Some deform elastically at temperatures near room temperature and flow at
higher temperatures with fluid behavior. Silicon dioxide (SÍO2) and silicon
nitride (SÍ3N4) are examples of this these type of materials. Fluids resist
deformation with viscous behavior.
As was previously mentioned, Hookean elastic solids will return to
their original shape after an applied stress is removed. Materials with
viscous constitutive properties may not. Viscous bodies relax or minimize the

27
strain associated with an applied stress. As the strain is relaxed the stress
also reduces. Due to the reduction in strain, when the applied stress is
removed, the body will retain its currently deformed shape.
Microscopically, in bodies with viscous mechanical properties, the
bonds between atoms that compose the solid break during deformation. New
bonds are then formed and re-broken as the body deforms under an applied
force. When the applied force is removed the bonds remain in their current
configurations.
A common constitutive relationship for a viscous body is the
Newtonian fluid. In a Newtonian relationship, the shear stress on the surface
is linearly proportional to the rate of deformation [30]:
a a « (2-16)
where ¡Mjki is the tensor of viscosity coefficients. The components of /Ujki for
fluids are not as well known as djki for elastic solids. However, most fluids
appear to behave isotropically. An isotropic approximation with the
restriction of incompressibility (constant density) allows for the following
constitutive relationship known as a Stokes fluid [30]:
o>j = -pSij+^ij (2-17)
where p now is a scalar that represents the viscosity of the fluid. The normal
stress components are dependent on the static pressure p of the fluid where
5ij represents the Kronecker delta function.

28
Viscoelasticity. It has been recognized that some materials deform
with a combination of elastic and viscous properties. There are various
models that have been formulated to describe the mechanical behavior of
viscoelastic bodies. In the Maxwell model of viscoelasticity, the total strain is
simply the sum of the strain due to elastic deformation and the strain due to
viscous deformation:
£ = £E+£V. (2-18)
This relationship can then be expanded to formulate the well-known
Maxwellian viscoelastic relationship between stress and strain:
ó o
£ = -+- (2-19)
E ju
where E is the elastic modulus and g is the viscosity of the material. Figure
2-5 illustrates the differences in deformation responses among a Hookean
elastic, Newtonian viscous and a Maxwellian viscoelastic material given the
same applied pulsed stress. Notice that the viscous and viscoelastic response
are time-dependent. Instantaneously after the stress is applied, the
viscoelastic body deforms elastically. Later when the applied stress is
constant, the viscoelastic body begins to display a linear viscous deformation.
As the applied stress is removed, the viscoelastic body then ‘elastically’
deforms towards its original strain state. However due to the viscous
relaxation component, it does not return to its original strain state.

29
Therefore, for the Maxwellian viscoelastic relationship, the short-term
deformation is elastic and the long-term deformation is viscous.
Figure 2-5 Comparison in loading response among Hookean elastic,
Newtonian viscous and Maxwellian viscoelastic relationships.

30
In silicon IC fabrication, oxide (SÍO2) is considered to behave
nonlinearly viscoelastic at midrange (800-1100°C) anneal temperatures [2,
14]. The nonlinearity is due to the fact that the degree of its viscosity is also
stress dependent. Nitride has been recognized to behave as a viscous body in
this same temperature range [37].
Not very many materials behave exactly as a Hookean elastic solid, a
Newtonian viscous fluid, or a Maxwellian viscoelastic fluid. However in
limited ranges of stress, strain, and temperature, these constitutive relations
can approximate their deformation behavior quite well. As an example silicon
behaves elastically for a wide temperature and stress range. However under
high stresses, silicon will yield and nucleate dislocations and defects in the
crystal to relax the stress present. Therefore it important to learn under what
conditions this will result.
2.2 Strain Sources
The individual strain sources included for strain computation are
discussed next. Three different strain sources are modeled and integrated:
film-induced, dopant-induced, and oxidation volume expansion induced
strain. Each strain source is discussed and explanations of how they induce
strain in the bulk are included. Following each is a discussion on how the
strain induced is modeled using the finite element method.

31
2.2.1 Film Stress
Physics. As silicon technology progresses, layers upon layers of
different materials are grown and deposited on top of and adjacent to each
other. When materials that have different structural, mechanical and
thermal properties are attached to each other, strains in each of the
materials can result. Adjacent material films relax or expand differently
based on their material properties. This causes one film to stretch or contract
the other film in a manner that will cause a local strain in each film. Large
localized stresses can result due to discontinuous films in regions such as at
the film edges and in non-planar sections.
According to Hu [14], stresses result in thin films due to two different
mechanisms. The first is referred to as an ‘extrinsic’ stress and is primarily
due to thermal expansion mismatch of neighboring materials. The process
used to deposit the film is done at an elevated temperature ranging from 150-
1200°C. After the process is over and the thin film is deposited, subsequent
thermal cycles will cause the film to expand and contract. If the film was
attached to or restrained by a rigid material that does not thermally deform,
the amount of strain induced is linearly proportional to the temperature
difference:
£lh=aih-AT (2-20)
where ccth is the linear thermal expansion coefficient of the film material. The
body experiences an increase in volume in each normal direction. No shear

32
components result from the thermal difference. For many materials the
thermal expansion coefficient is not necessarily constant or linear over a wide
temperature range. For silicon, ath varies 2.5-4.5xl0 6 /°K over the 300-900K
temperature range. Oxygen content and dopant concentration are factors
leading to octh variations.
If the material that the film is attached to also expands due to a
thermal increase then the local strain at the interface that is produced is due
to the difference in thermal expansion coefficients:
^th = (®thi ~^1*2)'AT1. (2-21)
Thermal mismatch stress is often incorrectly shortened and referred to
as thermal stress. However, thermal stress is due to thermal gradients
within a material. This often occurs in the wafers during before and after
temperature cycles. As the wafers cool down, the maximum stress is due to
surface tension. However to maintain force equilibrium the interior of the
wafer must be in compression. As the wafer heats up, the surface proceeds to
expand due to thermal expansion. However, the cooler interior of the wafer
restricts this expansion causing a compressive stress at the surface and a
tensile stress in the interior. Thermal stress primarily occurs in the substrate
as its thickness allows for a greater thermal gradient between the surface
and the interior. The thermal stress in the substrate becomes a problem
during temperature ramp rates encountered in Rapid Thermal Annealing
(RTA). At higher temperature ramp rates, the thermal stress built up from

33
the high temperature gradient may exceed the yield stress and cause the
wafer to shatter. Normally the thicknesses of grown and deposited films in IC
fabrication are so thin that a negligible thermal gradient exists across them.
The other source of stress encountered in thin films is the ‘intrinsic’
stress. Several researchers attribute this stress as due to growth mechanism
of the material during the process. For grown oxides, the intrinsic stress
results from the planar volume expansion resulting from the oxidation
reaction. This stress should not be confused with the stress induced at
isolation edges, which is non-planar and is a multidimensional problem
discussed later. Other material films such as polysilicon, silicon nitride, and
silicides exhibit intrinsic stress also.
After a film is deposited or grown, the wafer will warp according to the
total stress in the film. A highly tensile film will bend the wafer’s edges
towards the film and the reverse for a compressive film. A popular technique
for measuring film stress is to measure the amount of wafer curvature
optically. The total film stress is then proportional to the radius of curvature
by the following relation [38]:
<*/ =
6(1-v )/? t,
(2-22)
where Es and vs are the elastic properties of the substrate and ts and tf are
the thicknesses of the substrate and film respectively. The film stress
measured is the total stress due to thermal mismatch and its intrinsic

34
components. The intrinsic stress can then be derived from this measurement
and the known thermal mismatch stress from the previous relationship.
Even though high stresses can result from the sum of thermal
mismatch and intrinsic stress in a film, if the film is uniformly planar then
the stress resulting in the substrate due to the film will be orders of
magnitude smaller. This is due to the large difference in thicknesses between
the film (tf) and the substrate (ts):
tf
*s
Higher local stresses result in the substrate due to discontinuities in the film.
Such discontinuities include etched film edges from masking and
nonplanarities as in trench fill depositions.
Film-induced Strain Model. The stress due to deposited films is
modeled as an initial condition before deformation. The planar
measurements of intrinsic stress are used as the initial condition for the
finite element solution. The stresses are input at the nodes as the film is
deposited. They are directed in a biaxial tangential orientation along the
growing interface as is shown in Figure 2-6. To handle the stress components
in nonplanar interfaces, the stresses are translated from the planar system
axis to the axis perpendicular to the normal of the growing film. After the
film is deposited or grown, the stresses at the nodes are then averaged to
their neighboring triangular (2D) or tetrahedral (3D) elements.

35
The thermal mismatch stress is modeled as a hydrostatic stress. Each
normal component of strain exerted is equal in magnitude. No shear
components result from thermal mismatch. For each element in the film, the
three normal components of strain are added by superposition to the previous
state of stress due to other sources. This presents a problem in plane strain
FEM formulations since the strain in one direction is set to zero. This
problem will be addressed in section 2.3.
2,2,2 Dopant Induced Stress
As dopants are introduced to the silicon substrate, the mechanical
state of the substrate will change. The dopants may substitute for the silicon
positions in the lattice. Silicon atoms are displaced forming extended defects
that are lodged in the crystal lattice. Different dopants have varying atomic
sizes and therefore have different mechanical behavior in the crystal.
Precipitates and other atoms present such as oxygen and carbon also alter
the mechanical properties of the crystal.
Boron. Boron is well known as a substitutional dopant. Its atomic size
is smaller than that of silicon. When it locates into a substitutional site,
lattice contraction results due to its smaller size. This presents an atomically
localized strain in the crystal due to each boron atom. Figure 2-7 exemplifies
this in three- and two-dimensional illustrations. For high concentrations of
boron in silicon, these atomic strains add up significantly and result in a
larger localized region of strain in the boron doped silicon lattice. The non-

36
boron doped silicon region will resist the diffused boron layer from
contracting and thus result in a tensile strain field. This effect has been
demonstrated in silicon micro-machining applications [8, 39, 40], where
boron-doped cantilevers have exhibited bending due to strain induced by the
boron.
2D planar film
2D nonplanar
film
These nodes’
stresses are
translated
around comers
by the
component of
the unit surface
normal
+x
Figure 2-6 Intrinsic film stresses are oriented parallel to the interface on
which the film is grown or deposited.

37
2D approximation
o
o
o
o
o
o
o
o
o
o
o
0
0
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
Figure 2-7 Two- and three-dimensional illustrations of lattice deformation
due to boron substitution.
Densitometric studies have been done with boron doped silicon
crystals. Horn measured the silicon lattice constant variation as a function of
boron concentration [41]. From his measurements, the induced strain (Aa/a)
was extracted and given as function of boron concentration (Figure 2-8).
Boron-induced Strain Model. The empirical relationship introduced by
Horn is used as the contribution of strain in silicon due to substitutional
boron dopant. According to his measurements, 0.0141 A of lattice contraction

38
results per percentage of B in Si at room temperature. Using this figure, the
following relationship is derived:
A a 0.0141 CB
ac
as¡
Nr
•100
(2-24)
where asi is the silicon lattice constant (5.4295A at 25°C) and Nsi is the
density of Si atoms (5.02e22 cm 3). The average concentration of Boron (Cb) is
computed for each element from its node quantities. Dopant-induced strain is
also hydrostatic (as in thermal mismatch strain) and again presents a
problem in plane strain FEM formulations since the strain in one direction is
set to zero. This problem will be addressed in section 2.3.
Hydrostatic strain as a linear function of boron concentration
[41]
Figure 2-8

39
Other dopants. The strain contributions of other common dopants such
as phosphorus and arsenic are not as great per atom as that due to boron.
Because arsenic has a much larger atomic mass than silicon, one would tend
to believe that arsenic would induce a large compressive strain in the silicon
lattice. However, it has been reported that heavily doped arsenic (5xl021 cm 3)
only induces a lattice compression (Aa/asO of approximately 0.0019 [42], The
atomic size of phosphorus is more closely matched to that of silicon and
therefore the P-Si bond lengths are of the same magnitude as the Si-Si bond
lengths.
2.2.3 Oxidation Volume Expansion
The oxidation process also induces strain in the substrate due to the
net volume expansion of the oxidation reaction. It is well known that silicon
oxidizes to a volume of oxide that is 2.2 times larger. For planar oxidation,
this presents no problem since the newly acquired volume pushes the old
oxide upward towards the unconstrained surface perpendicular to the
interface. However, in nonplanar regions such as in trench comers and in
constrained regions such as in LOCOS edges, this presents a problem. For
these regions, the boundaries are constrained and therefore the newly
acquired oxide volume compresses against the earlier grown oxide. Since the
oxide has no place to move, large compressive strains build up in these
regions. The strains are somewhat relaxed by applying pressure to the silicon

40
interface. The forces applied to the silicon are high enough to surpass the
yield stress of silicon.
2.3 Strain Computation Methods
Two different finite element based methods are developed to solve and
compute the previously described models of the various strain sources. These
algorithms are developed in the process simulator FLOOPS [43] and are
integrated with and derived from methods developed to model stress-
dependent oxidation and silicidation [44],
Newton’s second law of motion governing deformation is stated as the
following [30]
(2-25)
where p is the density of the body, ai is the acceleration, Oy is the local stress
tensor, and bi is the body force.
The equivalent nodal force for each element (qe) may be represented as the
following:
q' = J BTod{vol)~ J NTbd(vol)
(2-26)
where b represents local body forces (e.g. gravitational or electromagnetic
forces), the B matrix relates the strain rate to the displacement rate (velocity)
of element and N represents the shape functions of the element. This

41
statement is valid quite generally for any stress-strain relationships. The
assumption of negligible body forces and negligible acceleration for each
element allow the equivalent nodal force equation to reduce to the following:
q' = J BTod(vol). (2-27)
For mechanical equilibrium where the body is not under rigid body motion,
net force is equal to zero.
An Hookean elastic element with the following constitutive relation
o = D{£-£0) + ct0 (2-28)
would be modeled by substituting the constitutive relation into the
equivalent nodal forces equation:
q =
J BTD£d{vol) - j BTD£0d(vol)+\ B'o0d(vol). (2-29)
V' V* ve
The strain e is related to the unknown displacement Aa through the B matrix
£ = BAae
(2-30)
and may also be substituted into the elastic equivalent nodal force equation:
Q =
J BTDBd(vol) Aae - J BTD£0d(vol)+ J BTo0d{vol). (2-31)
V' V’ V'
Under mechanical equilibrium, the elastic equivalent nodal force equation
reduces to the following discretized form:
[BTDBA]Aae = BrD£0A-BTo0A
(2-32)

42
where A is the area (volume in 3D) of the element. The left-hand side
represents the stiffness matrix of the element. The right hand side represents
the initial stress and strain state of the element. From this relation, the
displacement Aa is solved for globally and the current stress and strain state
can be derived from the displacement Aa.
A viscoelastic body is handled in the same manner. The Maxwell
viscoelastic constitutive relation
a o
1 = £
G ij
has the following solution for the stress (2-33)
a = r¡
1-exp
-At
T J
£ + (70 exp
'-At
l r
(2-34)
where r is the relaxation time constant and is the ratio of the viscosity r¡ to
the elastic modulus G
T = —.
G
(2-35)
The Maxwellian viscoelastic constitutive relation can then be substituted into
the equivalent nodal force equation:
Q =
\t A
(-At V
)
+ Í B'g0 exp
(-At]
f BT\r¡
1 - exp
>Ded{vol)
V' 1
l T )\
J
V'
l T J
d(vol). (2-36)
The strain rate ¿ is related to the unknown change in velocity Av through the
B matrix

43
¿ = BAve
(2-37)
and may also be substituted into the viscoelastic equivalent nodal force
equation:
q =
KÍ’
V l
1-exp
'-At
V T
jDBdiyol)
AU + JzTcFoexp^——^jd(vol). (2-38)
Under mechanical equilibrium, the viscoelastic equivalent nodal force
equation reduces to the following discretized form:
r T[
'-At V
1
/
b \ q
l-exp
\DBA
AU = -Bt {
-
l T )\
)
V
A
(2-39)
where A is the area (volume in 3D) of the element. The left-hand side
represents the stiffness matrix of the element. The right-hand side represents
the initial stress and strain state of the element. From this relation, the
unknown change in velocity Au is solved for globally and the current stress
and strain rate can then be derived from the new velocity change Au.
The viscoelastic formulation reduces to the elastic formulation for
large relaxation time constant t. If r »At, then
exp
(-At
\ r J r
The viscoelastic formulation then becomes
r ri At)
<
B \ri—\DBA
AU = -Bto0
1 —
. 1 T J
l T )
A
(2-40)
(2-41)
and reduces to the elastic formulation by allowing AtAve=Aae.

44
To model two-dimensional problems, the plane strain formulation is
used. This formulation can be used for problems where the strain component
in the z-direction is zero or negligible [28]:
£u=ez*=£n =°- (2-42)
This can be approximated solving problems with infinitely long dimensions in
the z-direction. Therefore, the strain in the z-direction will approach zero. As
was mentioned before in the earlier sections, a problem arises using the
plane strain approximation while computing thermal mismatch and dopant
induced stress. These sources include a hydrostatic strain field Eo described as
the following:
(2-43)
Then plane strain presumption implies that stresses in the 2-direction will
still occur even if there is no 2-component of strain. These stresses occur due
to dopant and thermal expansion and are affected by the elastic constants. To
account for this using a plane strain approximation, the following expression
is used instead for an elastic relationship [28]:
e'
£
XX
£yy
= (l + v)
£
0
(2-44)
where v is the Poisson’s ratio.

45
The FEM formulation uses three noded triangular elements called
‘faces’ for two-dimensional applications and four noded tetrahedral elements
called ‘volumes’ for three-dimensional applications. Linear shape functions
are used for interpolation of the strain solution within each element.
Reflecting boundary conditions are used to handle the strain solution
at the boundary of the simulation field. The normal component of the velocity
field or displacement rate is set to zero across this interface. Physically this
corresponds to a ‘mirror reflected’ symmetrical structure across the boundary.
2.3.1 Boundary Loading Method
The boundary loading method (BL) uses the strain solution calculated
during the oxidation to drive the elastic solution in the silicon. This technique
has been used in the SUPREM IV process simulator to calculate the elastic
strain in silicon due to oxidation [2]. Chidambarrao also used this technique
to investigate strains resulting from isolation trenches [45]. It is fully
decoupled and is performed in two sequential steps. This is demonstrated in
Figure 2-9. In the first step, the nonlinear viscoelastic oxide flow is computed
with the surface films modeled as viscoelastic materials and with silicon
acting as a rigid body [44]. This assumption allows for a more efficient
technique for solving the oxide growth, since only the surface films are
iterated over in the nonlinear stress dependent oxidation solution.
The stress tensor in the silicon dioxide elements along the silicon -
silicon dioxide interface are averaged to interface nodes. The averaged node

46
Step 1-Calculate the stress in the upper surface films to compute the stress-
dependent oxidation growth
Surface Film Forces
Oxide Growth
Forces V
Step 2-Calculate the stress induced in the substrate due to the forces
generated during oxidation and
generated from dopants.
\ + /
Strain — dopants, defects
S i \
Figure 2-9 The boundary loading method is composed of two separate
steps for computing the strain in silicon.
stresses are then converted into boundary forces along the interface. The
boundary forces are simply the product of the stress tensor with the unit
surface normal of each node:
(2-45)
These forces are then input as boundary loading forces that drive the
calculation of the elastic substrate strain solution. Silicon is modeled as an

47
isotropic elastic material in the second step. Strain may be included that is
exerted from dopants and defects within the silicon for this solution.
2,3.2 Fullv-Integrated Method
The other technique developed to investigate the strain in silicon is
similar to the finite element method implemented by Senez [46]. This fully-
integrated method couples the strain solution in the silicon along with the
surface films to compute the oxidation growth. This method allows for the
oxide stresses to relax by exerting forces on the silicon substrate. In the BL
method, oxide stresses can’t relax due to the rigid body boundary condition
imposed on the silicon. Therefore, lower magnitudes of stresses are expected
in using this method. This method is extended to include strains from other
sources simultaneously (Figure 2-10).
This technique becomes very computationally intensive because the
silicon elements are also assembled into the nonlinear oxidation equation.
This usually results in an order of magnitude more elements included to
adequately define the substrate and reduce the effect of the reflecting
boundaries on the solution. Therefore, the FI method's Newton iteration
involves a much larger matrix than the BL method and results in much
slower performance overall.
As in the previously described method, the surface films are modeled
as viscoelastic bodies and the silicon substrate is modeled as an isotropic
elastic material. The boundary condition at the oxidant reaction interface is

48
Oxide Growth
Forces A V'' _
Surface Film Forces
â– -L X
\\ -
\ +
Strain — dopants, defects
/ +
Figure 2-10 The fully integrated method couples the solutions of strain in
the upper films along with the substrate simultaneously.
similar to a polysilicon boundary condition: the silicon flows and is consumed.
Additionally, strain from dopants and intrinsic film stress may be exerted in
the silicon substrate and also is included as a stress source in this method.
2.4 Summary
One of the goals of this work are to develop a system where strain can
be computed from multiple sources independently and simultaneously. In
this chapter, a brief review of continuum mechanics was first provided to
refresh the reader with the necessary terminology. Next the individual strain
sources and the algorithms used to model the mechanical strain generated

49
from each source were described. Finally the boundary loading (BL) and
fully-integrated (FI) finite element were introduced and discussed for
modeling the strain due to oxidation volume dilation. The optimized BL
method is enforces a rigid body assumption for silicon. This allows it to be
useful for analyzing oxidation-induced strains in three-dimensions, as will be
discussed in the chapter three. However, this assumption may produce
influences in the oxide growth and stress solutions. Therefore, in chapter
three, the two methods will be analyzed for several applications to find if and
when there are influences. Also, in chapter three, applications using the
film-induced and boron-induced strain models are demonstrated and they
are compared and validated with experimental studies.

CHAPTER 3
APPLICATION EXAMPLES AND COMPARISONS
Due to the fully integrated method’s computational intensiveness, the
boundary loading method is more commonly used. This brings into question
how well the solution from each method agrees. Comparisons are performed
between each method for computing the oxidation-induced stresses for
different processes in silicon fabrication. Next, solutions of nitride film-
induced stresses are compared with micro-Raman spectroscopy
investigations. Afterwards, boron-induced cantilever bending simulations are
performed and compared with previous experiments. Finally, a three-
dimensional example is demonstrated for simulating oxidation-induced stress
using the boundary loading method.
3.1 FEM Comparisons
To compare the two methods for computing oxidation-induced stress,
two different oxidation processes are analyzed. The first is a standard
LOCOS process. The second is a re-oxidation process over an oxide filled
shallow trench. These examples detail when the boundary loading method
50

51
solutions fail to match the solutions computed by the fully-integrated
method.
Each of the following simulations is two-dimensional and use the plane
strain approximation with reflecting boundary conditions. This means that
the dimensions in the z-direction (which extends outward perpendicular to
the plane of the page) are infinitely long. The reflecting boundary conditions
indicate that the structure is periodic along the right and left boundaries of
the field.
Hydrostatic pressure contours are plotted for each process to compare
solutions computed by each method. This allows for a scalar interpretation of
the stress tensor’s variation in the two-dimensional field. The hydrostatic
pressure definition used is simply the negative average of the normal
components or negative one-third of the trace of the stress tensor:
P = -
l
3
0
0
0
0
0
0
(3-1)
For each wet ambient simulation, material properties calibrated at
1000°C by Cea and Senez [44, 46] are utilized. These parameters are listed in
Table 3-1.
For the STI simulations where dry oxidations are also performed, the
oxide material properties were calibrated to Kao’s dry oxidation cylinder

52
Table 3-1 Material coefficients at 1000°C [44, 46]
Elastic Parameters
Viscosity
Parameters
material
Elastic
Modulus
(Dyn/cm2)
Poisson’s
Ratio
Viscosity
(Dyn/cm2)
Vo
oxide
6.6ell
0.17
2.0el4
450e-24
nitride
3.89el2
0.3
7.0el4
100e-24
silicon
1.7el2
0.3
-
-
experiments [13], similarly to how the wet oxidation parameters were
calibrated [44].
In Kao’s experiment, cylinders and holes of varying radii were oxidized
to grow approximately 3500A at various temperatures in a dry ambient. In
both the cylinders and holes the oxide growth was retarded when compared
to plane wafer oxide growth. It was concluded that this decrease in oxide
growth was due to the stress dependence on the diffusivity of oxidant
reactant, reaction rate, and viscosity of oxide. As the radii of the cylinders
and holes decrease, the amount of oxide growth also decreases. A standard
calibration procedure adopted by Rafferty [2] and Cea [44] is used to obtain
the oxide parameters that fit the curves for each temperature and radius
geometry. The results of the cylinder and hole simulations are plotted in
Figure 3-1. The dry oxide parameters obtained from the calibration
simulations are listed in Table 3-2 along with the wet oxide parameters
obtained by Cea [44].

53
Dry Oxidation Calibrations
Figure 3-1 Results of dry oxidation calibrations for holes and cylinders
calibrated to Kao’s measurements [13]
The calibrated low stress viscosity of dry oxide is plotted over
temperature in Figure 3-2 along with the wet oxide viscosities calibrated by
Cea [44]. It can be seen that the dry oxide low stress viscosities are about an
order of magnitude less than the corresponding viscosity at each
temperature. This is expected since dry oxidations produce a denser oxide.
Also, the viscosities follow along the same slope in the Arrhenous plot with
similar activation energies.

Viscosity (dyn/cm2)
54
Table 3-2 Calibrated dry and wet [44] oxide parameters.
Oxide Viscosity
Parameters
Diffusivity and
Reaction Rate
Parameters
T(C)
Tl0
Vo
VD
Vk
(dyn/cm2)
(A3)
(A3)
(A3)
1100 W
3.3el3
1100
75
10
1000 W
2.0el4
450
u
u
900 W
1.5el5
300
u
u
HOOD
3.5el4
250
a
u
1000 D
2.2el5
150
u
u
900 D
2.5el6
6.25
u
u
Oxide Viscosity versus Reciprocal Temperature
looorr (1/°K)
Figure 3-2 Calibrated dry oxide viscosity compared with the wet oxide
viscosity calibrated by Cea [44].

55
3.1.1 LOCOS
The LOCOS process is simulated to compare between two-dimensional
stress solutions computed by each method. Figure 3-3 exemplifies
simulations of a 1000°C eighty-minute wet oxidation process at atmospheric
pressure. A 180nm thick nitride film with a two-micron linewidth is
patterned over a 20nm thick pad oxide. The grid spacing used for the
simulations is 0.05pm at nitride edge, 0.15pm at the left and right side
boundaries, and 0.25pm at the bottom boundary.
Figure 3-3 displays the stress solution for each method. By analyzing
the average pressure contours displayed, it is observed that there is
qualitative agreement between the two methods for this particular LOCOS
process and geometry. Corresponding regions are in tension (negative
pressure) and in compression (positive pressure). However the relative areas
occupied by the corresponding pressure contours indicate that their
magnitudes disagree. This is expected since the fully integrated method
allows for some stress relaxation through a ‘flexible’ silicon base. The
boundary loading method does not allow oxide stress relaxation through the
silicon because the silicon is modeled as a rigid body. The presence of both
compressive and tensile regions in the silicon is due to its elastic behavior.
When a compressive force is applied to an elastic body (such as at the bird’s
beak in silicon), a neighboring region in the material tends to deform in
tension due to the Poisson’s ratio effect.

56
The solutions in the oxide agree more closely in magnitude than in the
silicon. It is for this reason that the bird’s beak heights (BBH) are almost
identical as can be seen in Figure 3-4. This indicates that the rigid body
boundary condition used for computing the oxide growth, as modeled in [44],
does not affect the oxide stress solution enough to alter the oxide growth
significantly at longer nitride linewidths. However at shorter linewidths, it
becomes evident that the BB curvature underneath the nitride edge is
influenced by the strain computing method. In Figure 3-4, an overlay of
simulations using both strain computing methods is shown. The FI method
solution shows a little more oxidation under the nitride edge. This indicates
that the BL method computed a higher stress in this region that retarded the
oxide growth. A matrix of LOCOS simulations is performed to analyze the
differences of the stress and oxide growth solutions for shorter nitride
linewidth simulations using each FEM method.
Table 3-3 represents a matrix of simulations with varying nitride
linewidths. The first set with linewidth of 2pm is demonstrated in Figure 3-3.
Depending on the amount of elements in the field, the boundary loading
method is about two to three times faster than the fully integrated method.
Table 3-3 demonstrates that the oxide thickness and BBH is consistent
between the two methods and the variation in oxide shape is small until the
nitride linewidth decreases to 0.5pm. Below this linewidth, the oxide punch-
through effect begins to occur and the BBH varies more significantly

57
depending on the strain computing method used. The maximum tensile and
compressive stress magnitudes in the silicon are also noted for each
simulation and tend agree.
Table 3-3 Matrix of LOCOS simulations depicting how the two FEM
solutions correlate with decreasing nitride linewidth.
linewidth
BBH
Max Oxide
Max
Max
CPU
(um)
(um)
Growth (um)
Tension
Compression
time (s)
2.0
FI
0.276
0.499
-1.5e9
4.9e9
1077.4
BL
0.268
0.499
-3e9
5e9
633.9
1.0
FI
0.233
0.453
-3.7e9
7.5e9
1638.5
BL
0.235
0.46
-3.4e9
5.1e9
552.5
0.7
FI
0.215
0.415
-6.5e9
6.9e9
895.6
BL
0.215
0.42
-8e9
5e9
343.0
0.5
FI
0.179
0.367
-5e8
8e9
821.6
BL
0.190
0.368
-5e8
8e9
300.2

Y Axis
BL Method
Y Axis
FI Method
cn
Oo
-1.00
0.00
X Axis
1.00
Figure 3-3 Comparisons for a LOCOS-induced stress computed by the boundary loading method (left) and the
fully-integrated method (right).
58

59
x in microns Oxide Shape Comparisons
Figure 3-4 Overlaying oxide shape solutions comparisons between the BL
method and FI method for an example LOCOS simulation with 2pm nitride
linewidth.
3.1.2 Post-STI Process Be-Oxidation
Strain Solution Comparisons. An oxidation process over a previously
processed oxide filled shallow trench isolation (STI) is simulated to compare
between two-dimensional stress solutions computed by each method. Figure
3-5 exemplifies a simulation of a 1000°C three-minute wet oxidation at
atmospheric pressure. The trench with 0.5pm depth and 0.5pm width is filled

60
with deposited oxide. The nitride mask in this simulation has been previously
stripped as in a CMP process. The grid spacing used for the simulation is
0.02pm at the trench edges, 0.20pm at the left and right side boundaries, and
0.10pm at the bottom boundary. The initial field stress solution is set to zero
to compare the stress induced by the oxidation alone.
Figure 3-5 is an example of a ‘wedge driving effect’ that is an oxidation
performed over a fabricated trench that is not masked. A vertical bird’s beak
forms at the upper corners of the trench due to reactant diffusion into the
trench. This produces a very high compressive stress in the silicon substrate
along the side of the trench as oxide is grown on the trench sidewall.
Therefore this is analogous to a wedge being driven in at the trench sidewall.
By observing the average pressure contours displayed, it is observed
that there is qualitative agreement between the two methods for this process.
The tensile and compressive regions again correspond between the two
method’s solutions. The relative areas occupied by the corresponding pressure
contours indicate that their intensity variations disagree. The BL method
computes that the entire field is in compression with the exception of the
silicon at the bottom trench corner. Therefore, the two methods agree
qualitatively in the strain solution computed for this process. However, for
STI re-oxidation processes the strain solution is only part of the concern.

Y Axis
BL Method
Y Axis
FI Method
0.00
0.50 X Axis 1.00 0.00
0-50 x Axis 1.00
Figure 3-5 Comparisons for post STI process-induced stress computed by the boundary loading method (left) and
the fully-integrated method (right).

62
Córner Rounding Comparisons. One of the most critical design aspects
of the STI is the top corner shape. It is desirable for the top comers to be
rounded for several reasons. The most important being that sharp corners
lead to high electric fields and result in undesirable electrical characteristics
[47, 48]. Another reason for avoiding sharp corners is they lead to higher
mechanical stress magnitudes that generate dislocations also deleterious to
device operation [7].
Re-oxidation after fabrication of the STI has been introduced as a
method to round the upper trench comers [48]. The increased radius of
curvature at the upper corners alleviates these problems. Selecting the most
appropriate oxidation process for this technique can be simplified through
oxidation process simulation [49].
Viscoelastic flow models are used for the grown and deposited films
during the re-oxidation process [44]. Stress-dependent models for oxide
viscosity, reaction rate, and diffusivity are utilized for simulating the re¬
oxidation anneals. Two-dimensional plane strain simulations are performed
for a re-oxidation process to study the evolution of the radius of curvature of
the upper STI comers. The re-oxidation occurs after the planarization step of
the STI process. Therefore the oxide fills the trench up to the surface of the
nitride film (Figure 3-6). The STI dimensions have a depth of 300nm and a
width of 400nm. A series of re-oxidation simulations is performed varying the
nitride thickness from 70nm to 180nm over a 30nm pad oxide. The re-

63
x in microns FLOOPS
Figure 3-6 1000°C 300 minute dry re-oxidation simulation.
oxidation processes simulated and analyzed are a 300 minute 1000°C dry
anneal, 100 minute 1100°C dry anneal, and a 30 minute 950°C wet anneal.
The oxidation processes result in about 150nm of oxide grown on a blank
wafer. Each process is examined for each of the nitride film thicknesses.
The radius of curvature relationship to nitride thickness is consistent
with the experimental results of Chang [48] (Figure 3-7). The amount of
nitride lifting also varies linearly with the nitride thickness. The bird’s beak
lengths are fairly constant over this range so there is little trade-off against
the thinner nitride process.

64
Figure 3-7 (a) The radius of curvature of the upper STI corner and (b)
nitride lifting at the bird’s beak increases with decreasing nitride mask
thickness for the 1100°C dry oxidation.
Figure 3-8 depicts how the various process simulations examined
resulted for a given nitride thickness using each oxidation strain computation
method. It can be seen that for the higher temperature 1100°C dry oxidation
resulted in the greatest amount of corner rounding and stress relaxation.
x in microns FLOOPS 98.1
Figure 3-8 Overlay of the re-oxidation profiles for each anneal process
with 120nm nitride film thickness.

65
Also, due to less stress relaxation in the lower temperature 950°C wet
oxidation, less corner rounding is resulted by the BL method and an
overhang results by the FI method. The overhang also agrees with re¬
oxidation experiments [48]. However its shape is different since the
simulated shape may be locally grid dependent. The higher stresses reduce
the oxidation rate at the upper silicon corner just below the nitride edge and
result in excessive compressive stress magnitudes in the neighboring silicon
region.
From this application, it is found that for the low temperature wet re¬
oxidation simulations, the FI method results in an overhang at the trench
corner due to the severe oxide growth retardation. However for the same
simulation, the BL method returns a rounded corner. Since in experiment
[48], this process does result in a slight overhang, it is believed that including
the silicon deformation while calculating the oxide growth plays an important
role in modeling this effect at the trench corner.
3.2 Raman Spectroscopy Measurements and Comparisons
In the previous section qualitative comparisons were performed for
different oxidation processes between the two FEM methods implemented. In
this section, efforts are made to compare the nitride film edge-induced stress
with measured strain data. Currently, the best method for measuring local

66
process-induced mechanical strains in the substrate at sub-micron
dimensions is micro-Raman spectroscopy [26].
Raman spectroscopy is most commonly utilized for crystallographic
and chemical composition analysis. Mechanical strain or stress affects the
frequencies of the Raman modes of the crystal. The Raman spectrum of the
silicon crystal is sensitive to strain in the lattice and therefore has been used
to measure the stress state induced by microelectronics fabrication processes
[50]. Micro-Raman stress measurements involve focusing laser light onto a
sample. The scattered light of the sample is then collected and analyzed. The
light scattered from a (100) unstressed silicon sample has a Raman peak
frequency of about 520 Rem'1. Stressed samples will influence a shift in the
Raman peak. The amount and direction of the shift corresponds to the
magnitude and sign (compressive or tensile) of the stress in the laser spot
region of the sample. A negative shift corresponds to a tensile stress and a
positive shift indicates a compressive stress.
3.2.1 Raman Simulation Method
A method for determining the expected frequency shift in Raman
modes due to process-induced strain was introduced by Jones [51] and De
Wolf [52]. This method is implemented here to validate the stress solution
from the nitride film edge-induced stress model with micro-Raman
measurement data. The method involves first evaluating the shift in Raman

67
modes for a given strain tensor solution. For an elastic material, the strain
tensor is related to the stress tensor through the compliance tensor Sijki:
Ey = siju^kr (3-2)
The compliance tensor sijki is simply the inverse of the stiffness tensor ajki
introduced in chapter two. Once the strain solution is known, the three
Raman mode shifts (Acoi, Ao)2, ACO3) are computed by solving the following
eigenvalue problem [52]:
p£l{ + C¡ £22 + CjE^ A (P
(P~y)£ 12 P ^22 *7^33 Q — ^
2re13 2 r£23
Ire
Ire
13
23
p£33 + + £22) ^
= 0
(3-3)
where Aco = A/2coo.
Next, the Raman mode shifts are convoluted to account for the
penetration of the laser light into the silicon crystal and the width of the
laser spot. An exponential decay of light intensity and gaussian intensity
distribution over the spotwidth is assumed. These corrections average the
Raman mode shifts over a volume of the crystal corresponding to each spot
reading.
3.2,2 Nitride Film Edge-Induced Stress
A simulation of the stress induced by a nitride-poly stack is used to
compare with a Raman measurement experiment. A 0.24pm thick nitride
film with intrinsic stress of 1.2el0 dyn/cm2 is deposited over a 0.05pm poly
film with intrinsic stress of -3e9 dyn/cm2. The stack rests over a 0.01pm pad

68
oxide and is then patterned for a linewidth of five microns. The stress
induced by the film edges is then solved for and illustrated in Figure 3-9 by
stress contours of hydrostatic pressure. This represents the stress state of a
Poly-Buffered LOCOS structure before oxidation. Finally using the method
described in the previous section the three Raman mode shifts are solved for
and are displayed in Figure 3-10.
This simulation is then compared to Raman measurement data
introduced by De Wolf [52] for the same process. Since the magnitudes of the
x in microns
0.00 y in microns 10.00
Figure 3-9 Simulation of the stress induced by the intrinsic stress of a
nitride poly stack for a linewidth of 5pm. Units are in Dyn/cm2.

69
Figure 3-10 Frequency shift simulation for the three Raman modes due to
nitride-poly stack edge induced strain compared with Raman measurement
[52].
Raman mode shifts are small, only one shift can be recovered in experiment,
which is an averaged function of the three Raman mode shifts. This
simulated Raman shift is plotted along with the measured Raman shift [52]
in Figure 3-10. It is noticeable that frequency shift predicted correlates well
with the De Wolfs measurements. The resulting compressive strain induced
shift is consistently lower than the measured Raman shifts. This indicates
that the compressive strain magnitudes under the nitride film may be under¬
predicted. Other reasons for the discrepancy may be film-related effects on
the incident signal reflection, or convolution averaging errors due to light
intensity decay and spot width averaging.

70
3.3 Boron-Doped Cantilever Bending Comparisons
Next, an application using the boron-induced strain model is presented
in order to validate its solutions. The strain-induced bending of boron doped
cantilevers are simulated and compared with previous experimental
investigations.
3.3.1 Silicon Bulk Micro-Machining
Silicon bulk micro-machining is important for fabricating silicon-based
sensors and transducers. Silicon sensors are often composed of thin
membranes, bridges, cantilevers, and beams. These structures can be
fabricated by various bulk micro-machining methods. Anisotropic wet
chemical etching is often used to develop sensor structures due to its
simplicity and convenience as well as providing very accurate dimensional
control [18].
Boron etch stops are often used as a method for controlling etch depth
in silicon substrates. Thin silicon film structures can be fabricated by
thermally diffusing or implanting boron on one surface of the silicon wafer
and then by etching through a mask window on the other side of the wafer.
For wet chemical etchants such as KOH, the etch rate decreases significantly
as the etch front approaches boron concentrations greater than 7xl019 cm 3. It
is believed that the strong B-Si bond tends to bind the crystal more
stringently, therefore requiring more energy to release the silicon atom [53].

71
It is then possible to design thin silicon film structures with the
desired thickness by controlling the diffusion of the boron dopant profile so
that the etch stop will occur at depth where the boron concentration
approaches ~7xl019 cm 3. However due to these levels of boron concentration,
high levels of residual stress are generated. Since micro-machined thin
membranes are critical components of silicon sensors and transducers,
residual stresses in these structures may lead to mechanical failure of the
device and/or deteriorate its performance. Figure 3-11 illustrates the bulk
micro-machining process performed by Yang [39].
* SiQ2
Oxida tiorL
Si
*
Si02
Thermally Diffused Boron
nns?
Backside Etch
Strain-induced
Bending
Figure 3-11 Bulk micro-machining process flow for fabrication of the
cantilever.

72
First boron is introduced by thermal diffusion and is masked by an
oxide layer on the backside of the wafer. Next the front side is masked and
the backside is chemically etched. The etch slows as the surface approaches
the high concentration of boron (~6xl019 cm 3). Next the front side mask is
stripped off and the cantilever is developed. In Yang’s experiment, a series of
front side etches then performed to study the bending behavior of cantilevers
of different thicknesses doped with the same boron profile. The relative depth
of the peak of the boron profile then shifts for each cantilever etch.
The high concentrations of boron necessary to produce the etch stop
behavior results in residual tensile stress with magnitudes approaching and
exceeding levels of lxlO9 dyn/cm2. To relieve these high levels of stress the
silicon crystal yields and may generate dislocations that may be deleterious
to device and sensor performance [8]. This is one of the main reasons for
studying residual stress and its origins.
The residual stress resulting is dependent on the gradient and
maximum magnitude of the boron dopant profile as well as the thickness of
the cantilever resulting after the backside etch. Since the boron dopant
profile is not uniform the stress distribution varies with depth causing the
cantilever to bend in order to relieve the resulting residual stress. This is
evident in previous studies analyzing positive and negative bending of boron-
doped cantilevers under varying diffusion conditions [39, 40].

73
3.3.2 Cantilever Bending Simulations
Simulations are performed using the finite element models previously
described for the cantilever process described in Figure 3-11 [54], First boron
is introduced by thermal diffusion. The resulting profile has a peak
concentration of 8xl019 cm 3 (Figure 3-12). A backside wet chemical etch is
then performed. A boron concentration of 6xl019cnr3 is chosen as the stop for
this etch. The resulting cantilever thickness is about 1.4pm.
Next a series of 2d plane strain and 3d elastic deformation simulations
is performed for the resulting boron-doped cantilever structures. The 2d
cantilever has a length of 50pm. The grid spacing in the x-direction is 0.05pm
and in the y-direction is 0.5pm. Since the x-direction spacing is limited by the
necessary resolution to represent the boron profile, it becomes a challenge to
preserve element quality as the cantilever length is increased while
Figure 3-12 Resulting cantilever boron profile with a thickness of 1.4pm.

74
maintaining the number of elements constant. The element aspect ratio
problem is magnified further in three dimensions. The 3d cantilever has
dimensions of 1pm width and 10pm length. The 3d cantilever grid spacing is
0.3pm in the z-direction (width), 0.5pm in the y-direction (length), and 0.1pm
in the x-direction (thickness).
The following mechanical material properties were utilized for all
simulations performed: Young’s modulus (EP+si = 1.22xl012 dyn/cm2) [55],
Poisson’s ratio (v= 0.3).
A series of varying front side etch simulations were performed to
analyze how the shift in boron profile and decrease in thickness of the
cantilever affected the deflection solution. These series of etch simulations
model the experiment performed by Yang [39].
The results of the 2d plane strain simulations are displayed in Figure
3-13a. An example of the deflection simulation for the 0.62pm thickness beam
is shown in Figure 3-14. The structure is regridded after the nodal
displacements are solved for. Figure 3-13 displays the nodal displacements
along the top surface of each cantilever structure simulated. The differences
between each structure thickness relates to the amount etched off the top
surface. Notice that all the cantilever beams except the thickest deflected in
the negative x- direction (upward in Figure 3-14). Generally as the
cantilevers were etched thinner, the amount of deflection increased.

75
A 3d simulation for the 1.37pm thickness cantilever is demonstrated in
Figure 3-15. It is more difficult to examine the deflection visually in the 3d
simulations due to the shorter length of the cantilevers. The deflection
solution plot for the 3d simulations is displayed in Figure 3-13b. The same
general trend also results.
40E-O6
2D Simulation
50 um Length Deflections
0 001 0.002 0 003 0.004 0 005
Y-Position along the Cantilever (cm)
(b) 3D Simulation
Figure 3-13 (a) Two- and (b) Three-dimensional simulation results for
deflection curves for cantilevers of varying thickness.
The simulation lengths of the cantilever beams are much shorter than
those fabricated in various experiments [39, 40]. Typically cantilever beams
are fabricated with lengths up to 1mm in order to have an accurate
measurement of the deflection. The element quality problem limits the length
of the cantilevers simulated before a significant error is resulted in the elastic
solution.

76
x in
-1 40
1 20
-1 00
-0 80
-0 60
-0 40
-0 20
0.00
Figure 3-14 Two-dimensional plane strain simulation of the boron strain
induced bending of a cantilever with thickness 0.62pm.
To compare the simulations performed with experiments in the
literature, a parabolic curve fit was used to extrapolate the expected
deflections for longer cantilever beams. This is possible because beams
processed the same with varying lengths would all have the same bending
microns F LOO PS
0.00 y ¡n microns 50.00
0.000000000
FLOOPS
1.0 4.0 5,0
8.(5
Maximum displacement
Displacement X (cm)
Figure 3-15 Three-dimensional simulation of the bending of 1.37um
thickness cantilever. The maximum deflection is at the bottom corner of the
cantilever beam.

77
moment [40]. Therefore, the cantilever deflection simulation results are then
extrapolated to amounts corresponding to 1250pm to compare with
experiments performed by Yang [39]. These results are presented in the
histogram shown in Figure 3-16.
Several points can be deduced from the results obtained. First is the 2d
and 3d simulations resulted in roughly the same amount of deflection. This
confirms that the plane strain approximation does not affect the result of the
simulation and that cantilever width is not a factor in the stress solution.
Second, both the simulations agree with the published experiment in the
direction of the deflection for each beam thickness. However the measured
Cantilever Deflection Comparisons
200 -r
100 --
c 0
o
*-»
0 -100
v-
0)
a -200
0)
C7>
S -500
-400 â– L-
r-
ro
0 Measured [6]
m Extapolated 2D Simulation
â–¡ Extapolated 3D Simulation
L ■ M 1 1 i tasa—r 1 ! „■ i 1 i 1 i i i 1 «sa
CM
CM
CM
O
CM
Oh
o
CM
CD
d
Various Thicknesses
IT
CM
Figure 3-16 Results of both sets of simulations compared with Yang
experiment [39].

78
deflection results are consistently about two to three times the magnitude of
the simulations. Also the measured quantities have a relative maximum
negative deflection for the 0.92pm thick beams, while the simulations showed
relatively constant deflections for cantilevers of less thickness. It is believed
that these differences are due to errors in the boron profile and a variation in
the lattice strain coefficient.
Cantilever self-loading (due to its weight) has also been suggested as a
possible source of extra strain. However the maximum bending moment due
to a constant distributed load on a cantilever supported on one end is
described as the following relationship [56]:
WL2
A*ma*=— (3-4)
where L is the length of the cantilever and W is the uniformly distributed
load per unit length. For a body force such as gravity, W is represented as the
following:
W = Apg (3-5)
where A is the cross-sectional area of the cantilever, p is the density of silicon
(2.33 gm/cm3 for 5el9cnv3 boron-doped silicon [41]) and g is gravitational
acceleration constant. For a cantilever with rectangular cross section and
thickness t, the longitudinal stress o is proportional to M by the following
relationship:

79
The following gravity-induced stress relationship then results by substituting
(3-4) and (3-5) into (3-6):
<7 =
3 pgL2
t
(3-7)
It can then be shown that for a cantilever of 1000pm length and 1pm
thickness, the maximum stress produced is about 7e5 dyn/cm2, which is at
least three orders of magnitude less than the stress magnitudes induced by a
concentration of boron at 5el9cnr3.
The most probable cause for the difference between the simulations
and experiment is differences in the boron profile and lattice strain
parameter. Since the strain is computed directly from the local concentration
of boron and the lattice strain parameter, a shift in these results in a linear
shift in the strain and therefore the stress. Therefore scaling the lattice
parameter by a factor of three results in much closer agreement with the
deflection magnitudes in Yang’s experiment. This also results if the boron
concentration is scaled by the same factor. It is therefore believed that the
differences in both factors result in the observed disagreement in deflection
magnitudes. This is highly probable since the accuracy in the concentration
of boron reduces as the depth increases and the lattice parameter may be
different for the thermally diffused case.

80
Another method to estimate the lattice contraction parameter, is to
subtract the covalent radius of boron (0.88A) from that of silicon (1.17A) and
then normalizing the difference to the silicon atomic radius. This results in
an estimate of the normal radius contraction when a boron atom substitutes
for a silicon atom:
S
Si-B
RSi-RB _ 1.17-0.88
” 1-17
0.248.
(3-8)
This lattice contraction parameter results very close to that extracted from
Horn’s measurements [41] (0.014A/5.43A)*(100).
3.4 3D Boundary Loading Method Example (LOCOS)
The computational advantage of the boundary loading technique
allows for less CPU intensive three-dimensional strain simulations. For two
dimensional oxidation simulations, the BL method averaged 2-5 times faster
depending on the number of elements in the silicon. This time savings is even
much greater for three-dimensional simulations since the number of
elements for a typical 3D simulation is an order of magnitude greater than in
2D.
Figure 3-17 illustrates a three-dimensional simulation of the stress
induced in a LOCOS process computed by the boundary loading method. The
process underwent a 1000°C wet oxidation for 15 minutes at atmospheric
pressure. The pad oxide thickness is lOnm and the dimensions of the nitride

81
Figure 3-17 Three-dimensional simulation of LOCOS-induced stress in the
substrate.
film are 1.0pm x 1.5pm x 0.15pm. The iso-stress contour lines demonstrate
how the stress levels are dependent on the geometry of the nitride film. In
the 1.5pm length dimension the stress level is greater than in the 1.0pm
dimension. This is evident by comparing the area occupied by corresponding
iso-contour lines.
Figure 3-18 and Figure 3-19 are top views of the same simulation. The
stress tensor component in the x-direction (directed normal to the surface) is

82
Figure 3-18 YZ plane view of the tensile Oxx component of stress induced
during LOCOS oxidation.
plotted as contour lines. Figure 3-18 demonstrates that the maximum tensile
stress is at the corner of the nitride film. Figure 3-19 demonstrates that the
maximum compressive stress is located at the edges of the nitride film and is
greatest along the longer 1.5pm side of the nitride film.
3.5 Summary
The purpose of this chapter was to demonstrate through application
examples the capabilities of the strain models described in chapter two.
Another intention was to validate the results of various simulations with
experiments that have been reported. In the first section of this chapter,
solutions of oxidation-induced stress computed by the BL method and FI

83
Figure 3-19 YZ plane view of the compressive Oxx component of stress
induced during LOCOS oxidation.
method were analyzed The LOCOS simulation example applications showed
that at higher nitride linewidths, the two method’s solutions agreed.
However, at short linewidths, the two method’s solutions began to disagree.
This lead to believe that the stress relaxation in the silicon becomes a critical
factor in solving for the strain as dimensions are scaled. Also, by analyzing
the re-oxidation simulations it was found that the simulations performed
with the FI method computed an overhang in the low temperature wet
process at the trench corner. The BL method simulation did not. Since in
experiment it was found that an overhang is resulted, it is believed that the
stain solution computed using the FI method was more accurate in retarding
the growth for that process.

84
Next, an application solving for the film edge-induced stress of a
nitride poly stack was demonstrated. The results of these simulations were
then used to predict the Raman signal measured over the nitride-poly stack
structure using a method outlined by De Wolf [52], The expected frequency
shift in Raman modes then agreed both qualitatively and quantitatively with
Raman measurement experiments for the same structure.
Next, the boron-induced strain model was used to simulate the
bending of boron-doped cantilevers due to their boron-induced residual stress.
The simulations were then compared with studies of this effect and produced
qualitatively favorable results. The simulations agreed with experiment in
comparative bending between the different thickness cantilevers. The
difference in deflection magnitudes can be attributed to the lattice strain
parameter used in the model. By simply scaling this parameter, simulation
deflection magnitudes would coincide better with the measurements. One
possible explanation may be that the lattice strain parameter used may not
apply since it was measured in bulk silicon and may have a cantilever
thickness dependence.
Finally, three-dimensional simulations of oxidation induced stress
were demonstrated for a LOCOS process using the BL method. The BL
method’s efficiency allows for three dimensional strain simulations to be
performed rapidly.

CHAPTER 4
KELVIN PROBE FORCE STI EXPERIMENT
In the previous chapter, several applications were simulated using the
strain models described in chapter two. Those applications analyzed
individual sources of mechanical stress and how the modeled strain fields
compared with various experimental studies. In this chapter, the strain field
due to the shallow trench isolation (STI) process sequence is investigated.
STI has become an essential isolation scheme as CMOS technologies
are scaled down below the 0.25pm generation. However the basic STI process
sequence involves several sources of mechanically straining the enclosed
silicon region. Such sources include the thermal mismatch strain between the
isolation dielectric and the silicon substrate, intrinsic stress of the nitride
mask, and volume expansion-induced stress of the sidewall oxidation (Figure
4-1). It is therefore important to study each source and their combined strain
influences. As pitch lengths continue to scale, the distance between transistor
active areas and the STI sidewall also narrows. Therefore they continue to
approach the more highly stressed regions at the sidewall interface. This in
turn influences the device characteristics such as leakage through higher
dislocation densities and band gap deformation. Therefore for next
85

86
Figure 4-1 Major sources of stress in STI fabrication include the following:
(a) Nitride film edge-induced, (b) Insulator thermal expansion mismatch-
induced, and (c) Sidewall oxidation volume expansion-induced stress. Arrows
indicate force vectors.
generation process design, methods are needed to measure the stress
quantitatively. Up until now, micro-Raman spectroscopy has been the most
suitable method for investigating the stress [26]. But as technologies
continue to scale towards the 0.1pm generation, this may even surpass micro-
Raman’s spatial resolution limits. In this chapter, scanning Kelvin probe
force microscopy (SKPM) is investigated as a new method for measuring
stress-induced influences in the silicon work function A(p and for inferring the
stress magnitudes through coupling with mechanical strain simulation.
4.1 Scanning Kelvin Probe Force Microscopy
Scanning Kelvin probe force microscopy (SKPM) has previously been
researched as method for profiling 2D dopant concentration [57-62], This has
been achieved by detecting the electrostatic potential difference (EPD)

87
AFM Cantilever Probe
Figure 4-2 Cross-sectional sketch of SKPM measurement.
between the SKP tip and sample (Figure 4-2) through their electrostatic and
van der Waals force interaction.
The method involves rastering a cantilever probe over a cross-sectional
surface of the sample. At each step, both a surface topography measurement
and a surface work function measurement is performed. For measuring
surface topography the technique simply works as an atomic force microscope
(AFM). Under the AFM mode of operation, the mean distance between
cantilever probe and sample is kept constant over each step position. The
cantilever probe then is oscillated mechanically at a particular frequency
[58]. As the cantilever-sample mean distance changes, the van der Waals
force also changes causing a change in the oscillation amplitude. The
cantilever position is then corrected in the z-direction to return the cantilever
to its original oscillation amplitude. The corrected cantilever position is then
recorded as the cantilever is rastered across the sample and therefore maps
the surface topography as demonstrated in Figure 4-2.

88
Figure 4-3 Sketch of Kelvin probe scanning laterally (y-direction) and in
depth (x-direction). Variation in work function (EPD) is measured by
monitoring the electrostatic force contribution to the cantilever deflection.
Surface work function measurement is performed under the SKPM
mode of operation. First, an AC bias is also applied to the cantilever along
with the mechanical oscillation. An electrostatic force then results between
the cantilever tip and sample. A DC flat-band voltage bias is then applied to
the cantilever probe to null the electrostatic force. This DC bias is
proportional to EPD of the cantilever-sample system. The DC flat-band
voltage is then recorded at each raster step and provides a two-dimensional
map of the work function variation of the sample (Figure 4-3).
4,2 Work Function Influence
The EPD scanned by the SKPM method is a measure of the difference
in work functions of the cantilever tip and sample. This can best be described
with the energy band diagram of the system (Figure 4-4). It is similar to that
of a metal-insulator-semiconductor (MIS) device band diagram, with the

89
airspace between the cantilever and sample representing the insulator. In
this application, the cantilever tip is constituted of silicon with a layer of
titanium silicide grown on the surface for improved signal detection. When
the tip and sample are electrically connected, their Fermi levels become
aligned. The existing electrostatic force between the tip and the sample
causes the silicon bands to bend at the surface. Additional band bending is
also due to the presence of surface states and charges on the semiconductor
surface.
In SKPM mode, a DC flat-band bias is applied to null the electrostatic
force (Figure 4-5). This applied bias represents the EPD at each rastered
Ti-Silicide Tip vacuum Silicon
Figure 4-4 Energy band diagram of the cantilever-sample system at zero
applied DC bias.

90
point along the cross-sectional surface of the sample. The measured Vepd can
then be expressed as the sum of the flat-band correcting voltage Vfb and
surface potential terms [61]:
Vepd = Vjb + Vs + j-£slEI[\lfI]. (4-1)
The flat-band voltage is the difference between the work functions of the tip
and sample:
4/b = tpTí-si ~ The work function of the sample (¡>sí is the energy difference between the
vacuum level and the Fermi level:
Ti-Silicide Tip vacuum Silicon
Figure 4-5 Energy band diagram after Vepd is applied and the electrostatic
force is nulled.

91
Qs¡ ~ XSi />• (4-3)
The chemical potential (Fp represents the energy difference between the
Fermi level and the valence level of a p-doped semiconductor and is a
function of the crystal properties such as dopant concentration and the
energy band structure:
Â¥P
= -kT\ n
(4-4)
Variations in the dopant concentration Na or the energy band structure Eg
influence a variation in the silicon work function AQsi. In previous SKPM
investigations [57-62], the focus was to correlate influences in AQsi to
variations in dopant concentration Na and Nd. For characterizing strain in
the silicon, the dopant concentration Na is kept constant and variations in
energy band structure Eg are detected.
In chapter one, it was introduced that mechanical strain affects device
characteristics through influences in the energy band of the semiconductor.
Through quantum mechanical calculations [19, 21] and experiment [10, 63],
it has been recognized that both hydrostatic and anisotropic stresses induce
shifts and splits in the energy bands of silicon. The outcome of these
investigations were deformation potential constants E that characterize the
magnitude of the shifts with respect to the strain applied:
AE » Se
LXL^C,V w°ij•
(4-5)

92
Wortman [10] tabulated the deformation potentials that Goroff and
Kleinman calculated [19, 21]. These constants are utilized to model the
influence of a general strain tensor a, on the conduction and valence band
levels of silicon and are summarized in Table 4-1.
The strain tensor in Table 4-1 is described in the crystal lattice frame
of reference and the normal x-, y-, and z-components are oriented along the
<100>, <010>, and <001> crystal directions respectively. This is important to
note when comparing with finite element simulations, since the modeled
strain tensor is described in the laboratory frame of reference (Figure 4-3)
and is oriented in the <001>, <110>, and <110> crystal directions. Therefore,
Table 4-1 Deformation potential constants for the conduction and valence
energy band levels in silicon [10] [19, 21].
Energy Band
influence
Energy shift (eV)
AEv
Ar25(j=l/2)
[en + £22 + £33](-2.09)
Ar25(j=3/2)
[Eli + £22 + £33](-2.09) + [£ll - £33](±2.49) +
[£22 - £33](±2.49) + [£23](±3.28) + [£i3](±3.28)
+ [£i2](±3.28)
AEc
AXi([100])
[£ll + £22 + £33](-1.8) + [£ll - £33](6.38) + [£22 -
E33K-3.19) + [£23](±15.7) + [£i3](0) + [£12](0)
AXi([010])
[£ll + £22 + £33](-1.8) + [£ll - E33K-3.19) + [£22
- £33](6.38) + [£23](0) + [£is](±15.7) + [£i2](0)
AXi([001])
[Ell + £22 + £33](-1.8) + [£ll - E33K-3.19) + [£22
- £33](-3.19) + [£23](0) + [£13](0) + [£i2](±15.7)

93
a transformation of axis [31] is necessary for the strain tensor components
when calculating the energy band shifts from the modeled strain tensor.
The energy band influences have deformation potential constants with
a choice in sign, indicating that the level is split under the corresponding
strain component. When this occurs, the minimum of the conduction levels
AXi and the maximum of the valence energy levels AT25 is chosen. Since the
conduction level influence is direction dependent, the average of the
minimums of the three conduction level influences AXi is computed for the
AEc influence.
4.3 STI Experiment
The STI process outlined in Figure 4-6 is performed. The densification
anneal is done in both dry oxygen and nitrogen ambients for 5 minutes at
RIE
Sidewall oxidation
TEOS Gap Fill
900°C 5 min
N2 ambient
Densification
900°C 5 min
02 ambient
Densification
Sample A1
Sample A2
1050°C 30 min
1050°C 30 min
N2 ambient
02 ambient
Densification
Densification
Sample B1 Sample B2
Figure 4-6 Four densification anneals are performed to investigate
differences in the stress state due to each.

94
900°C and 30 minutes at 1050°C. The contribution of stress due to the
oxidation volume expansion is then investigated and compared to a process
that underwent the same thermal densification cycle. Also, the stress
differences between a short densification and a long densification anneal can
be studied.
Dopants have not been introduced by either implantation or thermal
diffusion before the STI process. Therefore it is assumed that the boron
concentrations are at the constant background levels found in the
prefabricated wafer. Under these conditions it is postulated that influences in
the silicon work function due to dopant concentration are constant over the
scanned area. The substrate is lightly boron doped (~lel6 cm 3). The SKP
used is a Digital Instruments Nanoscope III AFM outfitted for Kelvin
measurement and the silicon probe tips have been coated with titanium
silicide for improved signal detection. Samples are prepared by polishing the
cross-sections of a DRAM STI array [64],
4.4 SKPM Measurements
Topographical AFM images and SKPM images are scanned for each
sample. An example of each is shown in Figure 4-7 for sample Bl. The AFM
scan is first performed to locate the desired region to be Kelvin scanned.
Figure 4-7a is an example of a topographical scan. During sample
preparation, the different materials in the sample are polished to different

95
depths. The variation for a good sample are within a few nanometers. The
AFM scan is then able to detect the different material’s locations from the
topography without undergoing a selective etch. In Figure 4-7a regions with
shallower topography show up with darker intensity such as the silicon.
Regions such as the oxide between the silicon and the nitride above the
silicon image with lighter intensity. A selective etch would improve the
contrast and decorate the interfaces but would add another source for
altering the EPD signal by removing more material that would change the
stress state.
Once the desired region to be studied is located, an SKPM scan is
performed over the same region as in Figure 4-7b. This image represents the
detected EPD signal over the scanned region. In the SKPM image, only the
EPD signal measured in the silicon region is of interest since we are
concerned with a strain-induced variation in the work function. The work
functions of the oxide and nitride are different in magnitude than that of
silicon and therefore their EPD measurements result with a darker intensity.
The most interesting observation from Figure 4-7b is that the EPD
signal in the silicon region between the STIs (Si active area) results with a
higher intensity than that below the STI region. This indicates that the work
function in this region varies from that in the silicon below. This behavior is
more apparent when viewing the SKPM image as a perspective EPD plot

96
Figure 4-7 (a) AFM topographical image of STI array cross-section of
sample Bl, and (b) corresponding SKPM scan
The same signal pattern was observed among the four samples as that
measured for sample A1 in Figure 4-8. This indicates that for each

Figure 4-8 SKPM perspective EPD plot for Sample A1
densification anneal process, the silicon active area was the most highly
strained region. Next the work function variations are compared for each
sample. This is done by comparing cross-sectional line profiles of the EPD
signal for each sample at the middle of the silicon active area (Figure 4-9).
For samples A1 and A2, the difference in maximum and minimum EPD in
silicon is less than 0.03V. For samples B1 and B2, the difference in maximum
and minimum EPD in silicon is on the order of 0.2V. The peak influence in
EPD signals is at the top surface of the silicon just below the nitride mask.

98
Figure 4-9 Cross-sectional profiles of EPD measurements by SKPM in the
silicon active area for each sample. Arrows indicate the region of silicon
enclosed by STIs.

99
At
4.5 STI Strain Simulations
A quantitative relationship between the EPD measurements and
theoretical strain calculations is then determined. Silicon strain calculations
are determined through 2D plane strain finite element analysis implemented
in the process simulator FLOOPS with the models discussed in chapter two.
Materials are modeled isotropic elastic for the low temperature nitride edge-
induced strain and thermal expansion mismatch-induced strain conditions.
The surface films are modeled Maxwell viscoelastic for the higher
temperature oxidation-induced strains. The FI method is utilized for
computing the strain due to thermal oxidation.
Simulations of the stress induced by each of the STI stress sources are
demonstrated in Figure 4-10. The stress due to the nitride film edge, thermal
FLOOPS 98
Figure 4-10 (a) Calculated lateral stress due to edges of nitride film w/
intrinsic stress 1.4el0 dyn/cm2. (b) Calculated lateral stress due to thermal
mismatch of TEOS for 600°C temperature difference, (c) Calculated lateral
stress due to oxidation-volume expansion by the 1050°C dry O2 30 minute
anneal.

100
FLOOPS 98 FLOOPS 98
Figure 4-11 Resulting lateral stress after superposition of the solutions due
to (a) nitride film edge and thermal expansion mismatch and due to (b)
nitride film edge, thermal expansion mismatch, and thermal oxidation.
expansion mismatch of the insulator, and oxidation-volume expansion each
provides a compressive lateral stress in the isolated silicon active area.
The final strain state is then computed by the superposition of each
applicable individual solution previously mentioned in the STI process
(Figure 4-11). The computed strain tensor is oriented in the laboratory frame
Z [110] Z’|001]
Y 11101
Y’ [0101
< - <
X[001] ^X’ [100]
[e] _â–º [ef
Figure 4-12 Strain referenced in laboratory frame is transformed into the
crystal lattice frame.

101
of reference. This strain tensor is then transformed into the lattice frame
(Figure 4-12) so that its influence on the valence and conduction levels can be
evaluated (Figure 4-13). Calculated deformation potentials E relate the
lattice strain tensor to the crystal energy band shifts. A 2D distribution of the
predicted EPD signal is resulted from the simulated general strain solution.
The calculated work function influence can then be compared with the SKPM
measured influence to verify the strain simulations and learn what
magnitudes of stress may be present for each STI process.
4.6 Results and Discussion
The results of the SKPM measurements of EPD are encouraging. The
resulting EPD signal correlates well qualitatively with strain-influenced EPD
simulations. Each results with the highest magnitude work function
influence at the silicon active area between the STIs. The SKPM
measurements of samples B1 and B2 resulted with higher magnitude EPD
influences than those of A1 and A2. This may be due to the greater amount of
densification of the TEOS resulting in these samples. However, the
magnitude measured is too great to be correlated solely due to influences in
the local band gap. It is believed that certain artifacts such as surface states
and differences in AC signal biases between each sample measurement may
cause difficulties in extracting consistent EPD measurements. However, even
though these difficulties may present problems in sample to sample

102
(a)
FLOOPS 98
(c)
-0.001 V
FLOOPS 98
-0.005 V
Si Active Area
Si Below STI
-0 009 V
Depth (microns)
(b) FLOOPS 98
(d)
FLOOPS 98
Figure 4-13 Resulting band gap variations due to (a) nitride film edge and
thermal expansion mismatch and due to (b) nitride film edge, thermal
expansion mismatch, and thermal oxidation, (c) and (d) represent one¬
dimensional cuts through the silicon active areas and below the STIs for (a)
and (b) respectively.
quantitative comparisons, this technique allows for a qualitative analysis of
strain over different regions of the same sample.
Figure 4-14 demonstrates a comparison for sample A1 between the
SKPM measured EPD and the simulated EPD influence due to the strain
induced by the STI process. The strain is first computed due to the sum of

103
thermal expansion mismatch of TEOS and silicon for a 900°C temperature
difference and the intrinsic stress of the nitride film. The EPD is then
calculated from the band gap influence that is due to the resulting strain
state. Figure 4-15 illustrates a corresponding comparison for sample A2. This
simulation also includes the strain due to the 5 minute 900°C dry oxidation.
The EPD measurements for A1 have been scaled so that the peak
influence corresponds with the simulated maximum. This allows for a
qualitative comparison. The peak influence occurs at the top silicon interface
Sample A1
Figure 4-14 Scaled profile of EPD through the active area of Sample A1
overlayed with simulated strain-induced EPD profile.

104
just below the nitride and pad oxide. It then decays as the depth increases.
This trend is also observed in the simulations. The magnitude of the EPD
measurements does not agree for simulations of sample A1 but does agree for
A2. It is believed that for Al, measurement problems discussed earlier were a
greater influence than in A2. Another theory may be that even for the
nitrogen densification anneal of Al, there is still oxidation at the interface
due to oxygen supplied by the TEOS. This would cause the greater influence
that is observed with sample A2.
Sample A2
Figure 4-15 Scaled profile of EPD through the active area of Sample A2
overlayed with simulated strain-induced EPD profile.

105
Calibration studies are still necessary for a better quantitative
interpretation of the results. Such an experiment could consist of limiting the
strain source to just one source as in a nitride film edge or thermal expansion
mismatch source. Then the stress state could be controlled more easily and
work function variations can be extracted that are due only to one source.
4.7 Summary
One goal of this work is to validate the process induced strain models
described in chapter two with experimental measurements. Currently this is
a major hurdle as technologies continue to scale towards the 0.1pm
generation, due to the few characterization techniques available for localized
sub-micron strain measurement Therefore, in this chapter SKPM has been
investigated as a new method for analyzing localized strains through
detecting influences in the silicon work function.
The result of SKPM measurements is an associated change in the work
function of silicon crystal. To relate this influence in work function, the
measurements are coupled with strain simulations. Through this technique
once the work function influences match, it is hoped that the mechanical
strain that influenced the shift in work function is understood.
The strain simulations provided a predicted work function influence
that was qualitatively in agreement with the SKPM measurements. The
maximum influence was resulted at the top interface of the silicon active area

106
and decayed through depth, similarly to the SKPM measurements. However
the magnitudes were in the neighborhood but no attempt has been made to
match exactly the results of SKPM magnitudes due to the difficulties still
present. Also, the simulations involve a number of material parameters that
were estimated because they are not very well known currently, such as the
mechanical and oxidant diffusion properties of TEOS. Varying these
parameters can change the strain state simulations resulted considerably
and therefore change the anticipated effect on the band gap. Considerable
progress has been achieved in utilizing SKPM as a characterization
technique of stress. However, calibration studies are still necessary for a
better quantitative interpretation of the results.

CHAPTER 5
STRESS INFLUENCES IN DEVICE OPERATION
The influence of externally applied stress on electrical characteristics
has been investigated for many years. Most of the research applies to sensor
development rather than digital or analog IC influences. It has long been
recognized that the externally applied stress has a direct influence on the
energy band structure of a semiconductor [10]. This influence on the bandgap
is mostly noticeable in minority carrier devices such as diodes and bipolar
transistors since the minority carrier densities are strong functions of the
band gap. Quantifying this effect allows a better understanding of strain
influence on reverse saturation current levels and therefore supports leakage
issues in CMOS and DRAM circuits, which are majority carrier devices.
5.1 Uniaxial Stress Influence Experiment
Experiments have been performed relating stress effects on pn-
j unction operation to band structure influences based on quantum
mechanical calculations [10, 19, 21]. In these studies the changes in the
various semiconductor energy levels were quantified for hydrostatic and
uniaxial compressive applied stress. It was shown that for silicon, band-gap
107

108
variation due to a uniaxial compressive stress is greatest in the <111>
direction. Also, it was recognized that a significant band-gap variation began
as stress levels increased above le9 dyn/cm2 (Figure 5-1).
These derivations were then correlated to experiments involving the
application of stress directly on the surface of the device with an indentation
[Oil] Uniaxial Compressive Stress (Dyn/cm2)
- [Ill]
[100]
" Hydrostatic
Figure 5-1 Ratio of stressed minority carrier density to unstressed minority
carrier density as a function of uniaxial stress as forecasted by Wortman’s
model [10].

109
mechanism [25]. This method allows for application of a compressive stress in
the region of the pn-junction but may lead to some discrepancies in the
electrical measurements due to the creation of recombination centers because
of the extremely high stress levels encountered. Also this mechanism does not
allow for a method of exerting tensile strain in the pn-junction region.
An experiment was performed to measure the influences of an
externally applied compressive and tensile stress on various types of pn-
junction devices. This allows for comparison with a previous experiment [10]
and for determining if the same results held for modern pn-junction devices
with shallower junctions. Also it allows for measurement of tensile influences
which were not performed in the earlier experiments.
5.1.1 Stress-Inducing Apparatus Design
The uniaxial stress experiment involved the development of an
apparatus that mechanically exerts a known stress on fabricated devices. As
the stress is applied, the device operation could be monitored to observe the
stress-induced variations. The apparatus was designed so that it could fit on
the chuck for the probe station and not interfere with the probe pins and
microscope of the probe station. This tool was also designed so that it bends a
wafer strip up and down and therefore exerts a uniaxial compressive and
tensile stress on the top surface of the wafer.

110
(a) (b)
Figure 5-2 A CAD sketch of the stress-inducing tool. The left figure (a) is
the side view and the right figure (b) is a top view. The wafer strip is place
between the notches of the micrometer heads as is shown in (a).
Other constraints involved in the tool design were that it accommodate
wafer strips of varying widths, lengths and thicknesses. Precision in the

Ill
amount of bending displacement at the ends of the wafer strip needed to be
on the order of less than 0.01mm. The wafer-bending tool was constructed
with a spacing of 1.6 cm at the center between reaction points. The wafer
bending apparatus is sketched in Figure 5-2.
Configuration for exerting compressive stress on the top of the wafer
y
Reaction points on p
the top of the wafer
strip I
l
| <— a —jj s (e— a
F,
4
side view ofthewafer
strip
Forces exerted by the
Micrometer heads
moving up
Configuration for exerting tensile stress on the top of the wafer
Forces exerted by the
Micrometer heads
moving down
side view ofthewafer ——
strip
L Á
F,
a —j| s |«— a —> |
t Reaction points on
I the bottom of the
Ri wafer strip
Figure 5-3 Free-body diagrams displaying the applied and reacting forces
producing tensile and compressive stress along the top of the wafer strip.

112
5,1,2 Stress-Wafer Deflection Relationship
Standard elastic beam bending analysis is used to relate the amount of
wafer bending at the edge to the stress resulting in the center. The free body
diagram in Figure 5-3 shows the configuration of the forces acting on the
wafer strip.
The geometry represented in Figure 5-3 is explained as the following:
The length of the wafer strip is L, s is the distance between reaction forces
and a is the distance from the edge to the fulcrum reaction and is equal to
(L-s)/2. For Fi = F2, by the condition of static equilibrium Ri = R2 = Fi = F2.
This holds true for equal edge deflections on both sides of the beam. Using
the concentrated force approximation at the beam edges as singularity
functions, the following relation can be derived [56]:
(5-1)
where yn¿s is the vertical displacement at the wafer edge referenced to the
W afer strip under compression at top surface
1 8 R- 1
M
â– > x
Figure 5-4 The moment function exerted along the wafer strip is at a
maximum and constant between the reaction forces.

113
reaction positions which are fixed. The Elastic modulus E is oriented in the
direction along the beam, and I represents the moment of inertia of the wafer
strip. The moment M is at a maximum within the reaction forces as shown in
Figure 5-4.
From beam bending analysis the moment is related to the axial stress by the
following relation:
o
xx
My
I
(5-2)
where y is the vertical distance from the center of the beam. As an example,
if the wafer thickness t is 600pm then at the top surface y = +300pm. From
equations (5-1) and (5-2) the axial stress can be expressed as a function of
vertical beam edge displacement and vertical position
= <5-3)
For the force configuration shown above, the uniaxial stress can be
determined at the top and bottom surface of the center of the wafer strip from
the following expressions respectively:
o- (v f = (5-4)
"0W [4a2 -3Z,2]
“0W [4a2-3L2]
(5-5)

114
These equations show equally tensile and compressive magnitudes of stress
result on both surfaces of the wafer strip.
5.2 Experimental Procedure
DC measurements were performed to record the stress-induced
forward bias current influence. In forward bias, the diode forward current If
is proportional to the saturation current L and the recombination current Iro
by the following relations:
IF = Is exp(^) + Im exp(^-)
(5-6)
h = qA
PnO C°th
kl,j
+—nnn coth
j PV
V y
(5-7)
_ qAn,W
R0 = 2r '
0
(5-8)
The variation in current in forward bias can then be attributed through the
stress influence on the current components L and Iro. The minority carrier
densities pno and nPo and the intrinsic carrier concentration m are influenced
by the externally applied stress through its effect on the bandgap Eg.
n¿Ei=eJi^r\ (5.9)
n,0 V kT
The carrier lifetimes to are also influenced by stress through the creation of
dislocations or recombination centers to relieve the strain induced. The

115
influences of stress on the minority carrier densities are proportional to its
effects on the total current.
The following diodes were electrically characterized under varying
applied stresses:
Lateral P+ Area Diode (250pm x 750pm A=187500pm2 P=2000pm)
Lateral P+ Periphery Diode (250pm x 750pm A=187500pm2 P=2000pm)
N+/P-Sub Area Diode (250pm x 750pm A= 187500pm2 P=2000pm)
P+/N-Well Area Diode (250pm x 750pm A=187500pm2 P=2000pm)
N+/P-Sub Periphery Diode (A=92434pm2 P=45414pm)
P+/N-Well Periphery Diode (A=92434pm2 P=45414pm).
The wafers were cleaved into strips of varying lengths and widths in
the <110> directions perpendicular and parallel to the wafer flat. The
average dimensions of each strip were about five centimeters in length and
two centimeters in width.
To measure the influence of stress on each diode, the DC voltage is
stepped up from -1.0V to 0.5V on the P-sub terminal of the N+ diodes and
1.0V to -0.5V on the N-Well terminal of the P+ diodes. The bias voltage is
stepped in 30mV increments. Current is measured on both terminals of the
diode. Forward biased measurements were taken at various edge deflection
readings (Figure 5-5). These readings were then converted to the appropriate
stress levels generated by the previously introduced compressive (5-4) and
tensile relationships (5-5).

116
1.6E-07
(a)
1.4E-07
1.2E-07
1.0E-07
8.0E-08
6.0E-08
4.0E-08
2.0E-08
O.OE+OO *
Lateral P+ Area Diode
♦— -170E+09
-1.24E+09
-8.90 E+08
Unstressed
*— 7.33E+08
+— 1.42E+09
•— 1.76E+09
-2.0E-08
0.85
°'4 DC Bias (V) °'45
0.5
(b)
Lateral P+ Area Diode
i t r r 11.0E-06
-1 -0.8 -0.6 -0.4 -0.2 0
0.2
0.4
DC Voltage (V)
Figure 5-5 (a) Forward and (b) semi-log IV Characteristic for the lateral P+
large area diode under applied uniaxial stress (dyn/cm2).

117
The forward biased operation is observed to compare the influence
between different stress states. The stress-induced variation AI(o) is
measured as the following:
Ai(o)=m-,(uns,ressed\im)
¡{unstressed.)
(5-10)
This variation is calculated for each measured bias point. It is then
averaged over the interval from 0.1V to 0.5V forward bias points. In this
interval, the current magnitudes were high enough to show consistent
variations due to the applied stress (Figure 5-6).
1.0E-08
Lateral P+ Area Diode
c
0
D
Ü
o
Q
0.35
1.0E-09
0.37
--1.70E+09
-->"88—
-1.24E+09
-8.90E+08
-M~
Unstressed
—*-
- 7.33E+08
-1.42E+09
1
-1.76E+09
0.45
DC Voltage (V)
Figure 5-6 Magnification of the semi-log IV Characteristic for the lateral
P+ large area diode under applied uniaxial stress (dyn/cm2) resulting in
shifted current measurements with same slope.

118
5.3 Experimental Results
Stress levels ranging from -2e9 to +2e9 dyn/cm2 were applied to each
diode. This corresponds to maximum levels of one-third the yield strength of
dislocation free silicon at room temperature [65]. Exceeding this stress range
caused the wafer strip samples to shatter consistently due to micro-fractures
along the wafer strip edges resulting from the preparation. Figures 5-7 to 5-9
display how the applied external stress influenced the forward biased DC
behavior for the different diodes tested. The reverse bias operation was also
analyzed under applied stress. However, the reverse bias current magnitudes
were too low to detect influences due to the applied stress.
It is found that the applied stress affected the DC operation
consistently for each of the diodes. Compressive stress increases the current
for an applied DC voltage bias. Tensile stress decreases the current as
compared to the unstressed case. For compressive stresses of about 2e9
dyn/cm2, N+ diodes current increase was about 10-15 percent. Over the same
stress levels, P+ diodes current increase was about 7-10 percent. Tensile
stresses just below 2e9 dyn/cm2 caused both N+ and P+ diodes’ currents to
decrease about 6-8 percent. These results remained consistent when the
stress is applied along the perpendicular <011> directions.

CD
CD
05
C=
CD
O
q;
Q.
<
-2.5E+09
N+ Diode Stress-Induced Current Influence
20
Í
Â¥
15
10
-1.5E+09 -5.0E+08 | 5.0E+08
-5
â– 10-
Stress (Dyn/cm2)
â–² N+ Areal
X N+ Area2
u N+ Periphery
1.5E+09
2.5E+09
Figure 5-7 Applied stress-induced influence on N+ diodes.
119

P+ Diode Stress-Induced Current Influence
Figure 5-8 Applied stress-induced influence on P+ diodes.
120

Figure 5-9 Applied stress-induced influence on Lateral P+ diodes.
121

122
Some samples were measured several times at the same applied stress levels
to determine the repeatability of the influence. An example of this is shown
in Figure 5-9. It is observed that the measured Al(a) influences were within
one and a half to two percent variation. The micrometer head’s reading
accuracy for the edge deflection measurement is estimated to be about
±0.02mm. For a sample of 5cm length this translates to about ±4.6e7
dyn/cm2.
The following trend-lines were linearly extracted from the N+ diodes
influence measurements for compressive and tensile stresses respectively:
A/(cj) = -6.74e -lie (5-11)
A/(c) = -3.63e -11 • Similarly for P+ diodes, the following linear relationships were extracted:
A/( A/( If it is assumed that the change in current is due to the stress influence on
the Is current component alone, the linear relationships (5-11) - (5-14) can be
entered into expression (5-9) to solve for the band gap influence ÁEg:
AEg =-mn[A/( (5-15)

123
(a)
— p+ Extracted Compressive Stress (dyn/cm2)
"â–  N+Extracted
~3~ Wortrnan's Model [10]
P+ Extracted Tensile Stress (dyn/cm2)
â– " N+Extracted
Figure 5-10 Extracted Band gap influences due to (a) compressive and (b)
tensile <011> uniaxial stress.

124
The results for the band gap influences extracted from the measurements are
illustrated in Figure 5-10 along with Wortman’s model [10] predictions for
compressive uniaxial <011> stress.
Applying Wortman’s model, a <110> uniaxial compressive of about 2e9
dyn/cm2 would result in about 30-35 percent increase in L (Figure 5-1). The
corresponding influence extracted from the preceding measurements resulted
in about 8-15 percent increase. This explains why in Figure 5-10 the model
also predicts a larger AEg than was extracted from the experiment. The
assumption that L and Iro are similarly affected by mechanical stress may
lead to some of the error in correlating the measurements with the
experiment This assumption neglects influences in carrier mobilities and
lifetimes. Also, the relationship between wafer edge deflection and applied
stress, equations (5-4) and (5-5), may induce some error in determining the
stress state. However, the measurements provide at the very least a
minimum bound on the influence of both tensile and compressive stress on
the band gap Eg .
5.4 Summary
According to the results plotted, the largest degree of measurement
error encountered was at the lower magnitude stress levels. However at both
high compressive and tensile stress levels a more consistent stress influenced
measurement was observed. For the range of stresses produced, the

125
relationship looks linear. However, the highest stress magnitudes achieved
using this technique were in the ±2e9 dyn/cm2 range. When passing this
range the samples usually shattered due to exceeding the yield strength.
It was also observed that compressive stresses increased the current
flow while tensile stresses decreased the current flow. Also a trend that was
consistently encountered was that for the same magnitude of stress applied,
the compressive state had a more predominant effect as compared to the
tensile state. The N+ diodes under compressive stress resulted in a
consistently greater influence (almost double) than the corresponding P+
diodes.
Unfortunately, the stress influence on the reverse bias current was not
analyzed due to current magnitudes being too low to measure at room
temperature. However, a lower bound can be extracted from the forward bias
influence studied. At the lower to mid-range stress levels studied, a
maximum of 15 percent current influence was measured. The reverse bias
current influence should be greater due to the fact that the recombination
current component is more influenced by stress than the saturation current
component.

CHAPTER 1
SUMMARY AND FUTURE WORK
6.1 Summary
This thesis explores three different realms of mechanical stresses in
silicon fabrication. The first is examining two modeling techniques for
computing oxidation volume dilation-induced stress and integrating it along
with other sources of stress. The main problem in this analysis is validating
the models with experimental measurements. Since very few methods are
available for this purpose, SKPM is investigated as a new method for
characterizing localized stresses induced in STI processes. This involves
understanding how the mechanical strain affects the crystal properties that
can be measured, such as work function and band gap. To help understand
these influences, another experiment is performed using pn-junction device
electrical measurements to extract the strain influences on silicon crystal
properties. The results of each task benefits the others as each has the
common long term goal for improving the understanding how mechanical
stress affects silicon technology and developing methods for alleviating it.
Isolation processes are given special consideration as most mechanical stress
problems in silicon technology are encountered there.
126

127
First, two multidimensional finite element methods are implemented
to analyze mechanical stress generated by the oxidation growth process. The
strain solutions from either method can be added to contributions from other
sources, such as boron dopant, film, and thermal expansion mismatch. The
two methods’ solutions have been shown to agree qualitatively when solving
for the strain induced due to the LOCOS process. At smaller nitride
line widths, the oxidation growth solutions begin to show differences due to
the method utilized. Also for lower temperature re-oxidations, the STI corner
rounding simulations disagreed. Under these conditions it then becomes
necessary to solve for the strain in the silicon simultaneously, since it affects
the strain solution in the oxide. Both techniques allow for solution from
multiple strain sources simultaneously. This is necessary for analyzing stress
generated from modem isolation processes such as STI and in DRAM
technologies. The BL method is computationally optimized and therefore
allows it to be useful for analyzing oxidation-induced strains in three-
dimensions.
The solutions of stress induced by deposited nitride films are compared
to Raman investigations via techniques introduced by De Wolf [52] and Jones
[51]. The simulated Raman frequency shift for the nitride stack agrees both
qualitatively and quantitatively with the measured data [52],
Simulations are also performed of the bending of boron-doped
cantilevers due to their residual stress to examine the boron-induced strain

128
model. The simulations were then compared with studies of this effect and
produced qualitatively favorable results. The simulations agreed with
experiment in comparative bending between the different thickness
cantilevers. The difference in deflection magnitudes can be attributed to the
lattice strain parameter used in the model. By simply scaling this parameter,
simulation deflection magnitudes would coincide better with the
measurements. One possible explanation may be that the magnitude of the
lattice strain parameter used may not apply since it was measured in
samples with boron dopant introduced during silicon crystal growth instead
of by thermal diffusion as in the cantilever experiment. Also there may be
strain relaxation with a cantilever thickness dependence.
Next, SKPM is applied as a new technique to characterize the strain
exerted from STI processes through measurements of work function
variations in silicon. The STI experiment performed demonstrates a change
in work function of silicon that is likely due to the strain induced by the STI
process. The change in work function is also found to vary in magnitude
depending on the STI densification anneal performed. Strain simulations are
then performed to correlate the differences in work function influences to the
strain induced by the densification anneals. It is found that the simulations
of work function influence agree qualitatively with the measured influences.
However, the measured influences’ magnitudes are greater than the expected
changes. This is believed to be due to the surface and biasing effects in the

129
SKPM measurements. Scaling the measurements to the simulated levels
show the qualitative agreement over the scanned STI regions.
Finally, an experiment for quantifying the amount of uniaxial stress-
induced current variation for various diode types has also been accomplished.
Stress influenced measurements are performed for influences in the forward
biased operation of pn-junctions. The results demonstrate that the measured
influence of the applied mechanical stress was consistent among the different
types of diodes. N+ diodes resulted in a greater influence due to the stress
than the P+ diodes. It was also observed that compressive stresses increased
the current flow while tensile stresses decreased the current flow. These
measurements demonstrate how much low to mid stress levels influence pn-
junction currents. The measured data provides a lower bound for how critical
the uniaxial stress affects reverse current or leakage levels.
6.2 Future Work
The following are proposed ideas for future work related to modeling of
mechanical strain induced by silicon fabrication processes:
6.2.1 SKPM Strain Measurement Calibration Studies
It would be useful to follow up the SKPM study with another
experiment using a simpler more controllable stress source. An example of
such could be depositing nitride with a high intrinsic stress directly on the
silicon wafer and then pattern the nitride into stripes with varying

130
thicknesses and widths. Performing an SKPM scan below the nitride regions
would allow for analyzing the silicon work function influences due to different
levels of stress controlled by the thickness of the nitride films. Then the
amount of work function shift can be correlated to strain by finite element
simulations and micro-Raman investigations. Also, it could be possible to put
multiple structures on the same sample to reduce work function influences
due to sample preparation and measurement dependencies such as surface
potentials and biasing influences.
6.2.2 Effect of Stress on Dislocations
In this study, the work was focused on solving for the stress
magnitudes due to various processes in silicon fabrication of integrated
circuits. One consequence of highly stressed regions is the yielding of
dislocations to relieve the strain in the silicon crystal. The dislocations
formed then act as generation/recombination centers which worsen the device
performance. It would be beneficial to begin correlating the amount of stress
necessary to form dislocation densities which ultimately result in
unacceptably high leakage levels. Such a study could perhaps be performed
by intentionally stressing large pn-junction devices locally through either the
isolation process or with films of high intrinsic stress. Then measuring
influences in the leakage currents due to the stress applied as was done in
chapter 5.

131
6.2.3 Silicidation-Induced Stress
Modern integrated circuit processes involve the growth of metal
silicide materials for improved conductivities for interconnects between
devices and contacts. During the titanium silicide growth process, a net
volume contraction results due to the newly formed silicide occupying less
volume than the reactant volumes of silicon and titanium. This is opposite
behavior to that of silicon dioxide growth where a net volume dilation results.
This growth process may produce significant local mechanical stresses
in regions such as at the gate and source/drain regions of CMOS devices
formed by the salicide process. It would then be necessary to include titanium
silicide growth as another strain inducing source when analyzing the strain
in these regions which are critical for CMOS electrical characteristics. It is
important to have an accurate strain field calculation in this active area
region for examining the impact of mechanical strain on the evolution of the
extended defects in the critical source/drain regions.
6.2.4 Three-Dimensional Modeling of the STI Process
STI technologies are rapidly becoming the standard isolation process
in modem integrated circuits as minimum transistor lengths decrease below
0.25-micron. This scaling then leads to an increase in importance of three-
dimensional isolation effects on device characteristics such as electric fields
and leakage currents.

132
The main obstacle encountered in three-dimensional modeling of the
STI process is a moving boundary grid handling problem. STI processes
include multiple etch and depositions. These processes as well as the
material growth processes encounter occasional problems in three-
dimensions for handling the grid as the boundary front advances.
Developments for correcting these problems are continuously resulting and
has accounted for an increasingly robust grid handling in two-dimensions
[66, 67]. Improving the three-dimensional grid handling routines [68] will
allow for strain analysis of three-dimensional problems such as STI and
DRAM structures.

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BIOGRAPHICAL SKETCH
Hernán Adolfo Rueda was born in Chicago, Illinois, on November 9,
1970, to Hernán and Gloria Rueda. He has one brother, Camilo. Hernán grew
up in Miami, Florida. He attended North Miami Senior High School and
graduated in June 1988.
He then enrolled at the University of Florida in fall 1988 and earned
his Bachelor of Science degree in electrical engineering in August 1992.
During this time he developed interest in the fields of VLSI circuit design and
fabrication and in computer programming.
In fall 1992, he began his graduate studies at the University of
Florida, and soon thereafter he began his master's research in IC process
modeling focusing on moving boundary problems. His doctorate research has
focused on modeling and characterizing mechanical strain in silicon
integrated circuit fabrication processes.
140

I certify that I have read this study and that in my opinion it conforms
to acceptable standards of scholarly presentation and is fully adequate, in
scope and quality, as a dissertation for the degree of Doctor of Philosophy.
lark E. Law, Chairman
Professor of Electrical and
Computer Engineering
I certify that I have read this study and that in my opinion it conforms
to acceptable standards of scholarly presentation and is fully adequate, in
scope and quality, as a dissertation for the degree of^Doctor of^Plplosophy.
Kevin S. Jones/
Professor of Materials Science and
Engineering
I certify that I have read this study and that in my opinion it conforms
to acceptable standards of scholarly presentation and is fully adequate, in
scope and quality, as a dissertation for the degraeof Doctor of Philosophy.
Gijs Bosman
Professor of Electrical and
Computer Engineering
I certify that I have read this study and that in my opinion it conforms
to acceptable standards of scholarly presentation and is fully adequate, in
scope and quality, as a dissertation for the degree of Doctop of Philosophy^
A
Kenneth
Associate Professor of Electrical and
Computer Engineering
I certify that I have read this study and that in my opinion it conforms
to acceptable standards of scholarly presentation and is fully adequate, in
scope and quality, as a dissertation for the degree of Doctor of Philosophy.
Toshikazu Nishida
Associate Professor of Electrical and
Computer Engineering

This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate School and was accepted as partial
fulfillment of the requirements for the degree of Doctor of Philosophy.
May 1999
Winfred M. Phillips
Dean. College of Engineering
M. J. Ohanian
Dean, Graduate School

U>
1599
UNIVERSITY OF FLORIDA
3 1262 08554 4681