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A study of silicon/silicon dioxide interface roughness evolution during microwave electron cyclotron resonance plasma and thermal oxidation processes

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R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
A STUDY OF Si/Si02 INTERFACE ROUGHNESS EVOLUTION
DURING MICROWAVE ELECTRON CYCLOTRON RESONANCE
PLASMA AND THERMAL OXIDATION PROCESSES
by
Changyi Zhao
A dissertation submitted to the faculty of the University of North Carolina at Chapel
Hill, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in
the Department of Chemistry.
Chapel Hill
1997
Approved
Advisor: Profess
Eugene A. Irene
Reader: Professor John J. Boland
_______
Read'
dpSrofiss
essor Frank Tsui
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
UMI Number: 9818447
Copyright 1997 by
Zhao, Changyi
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Changyi Zhao
ALL RIGHTS RESERVED
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ABSTRACT
Changyi Zhao: A Study of Si/Si02 Interface Roughness Evolution During Microwave
Electron Cyclotron Resonance Plasma and Thermal Oxidation Processes
(Under the direction of Professor Eugene A. Irene)
The properties o f the Si/Si02 interface are crucial to the proper functioning of
microelectronics devices. Microroughness at the interface is known to deteriorate both
device performance and reliability. The influence of this interface roughness becomes
increasingly critical as the device dimension shrinks. Understanding this interface region
is thus of both scientific importance and technological urgency. Despite decades of
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research, many questions about the interface remain unanswered. The purpose of this
research is to employ several roughness characterization techniques to independently
investigate the Si/Si02 interface roughness during electron cyclotron resonance (ECR)
plasma and thermal oxidation processes. One new technique is spectroscopic immersion
ellipsometry (SIE), which is an interface-enhanced optical technique capable of obtaining
information about the buried interface without physically removing the oxide overlayer.
Atomic force microscopy (AFM) and the fractal analysis were used to directly image the
morphology of the oxidized Si surfaces after the oxide overlayer films were physically
removed by a brief HF dip, and extract such roughness parameters as the rms values and
the fractal dimensions of the revealed Si surfaces. The results are concordant and show
that during both ECR plasma and thermal oxidation processes, there are two competing
mechanisms that are responsible for the Si/Si02 interface roughness evolution, i.e., both
interface smoothing and roughening occur and which is dominant depends on whether the
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original Si surfaces were rough or smooth. The smoothing is believed mainly due to the
diffusion and/or reaction of Si atoms driven by the reduction of free energy, and also by
the interfacial stress. In the thermal oxidation, the smoothing is enhanced by the high
temperature chemical reaction and viscous flow at the interface, while in the ECR plasma
oxidation it is enhanced by the locally intensified external electric field. The roughening
is speculated due to a chemical reaction mediated roughening transition. The results
demonstrate that SIE has emerged as a powerful tool for non-destructive investigation of
film-covered interfaces. Also, when using AFM to study surface roughness, a more
complete quantitative representation of roughness can be achieved by combining the uses
o f rms values and fractal dimensions.
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Acknowledgements
First of all, I would like to thank my adviser, Professor Eugene Irene, for
introducing me into this wonderful world of microelectronics materials, and patiently
guiding me through my graduate study like the way he maneuvers his beloved yacht. It
has been such a pleasure and a memorable experience to team from him and work for
him. Without him, I would probably still be pondering in the dark where my future career
lies.
I want to thank Dr. J. J. Boland, Dr. Frank Tsui, Dr. R. W. Linton, and Dr. L. G.
Pedersen for reading this dissertation and the helpful suggestions. Also, I appreciate Dr.
L. E. McNeil for her guidance and advice during my graduate study.
My gratitude also goes to Dr. Yao-Zhi Hu, who broadened my viewpoint and
diversified my interests beyond my own research project. In addition, I was fortunate
enough to have many opportunities to enjoy his excellent cooking skills.
I want to thank the past and present members of the Irene group, especially Dr.
Yiqiong Wang, from whom I first learned about this group, and Dr. Lycourgas Spanos.
Dr. Jennifer Wall, Dr. Qian Liu, Dr. Kelly Hebert, Dr. Brian Augustine, Debbie Diehl,
Telmo Labayen, Pierre Lefebvre, Catherine Basa, Lilean Lai, Imran Aftab, and Dr. Ying
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Gao. They have been so kind in helping me to settle into this new environment when I
first joined the group, and have been generously helping me in all the aspects ever since.
This is a group full of friendship, and I feel lucky to be a member of it.
The people that I think about the most at this moment are my parents and my
brother, who have been waiting for this day for so long. Their patience may have started
to wear out as my seemingly forever-going school life extended again and again, but their
faith on me, their love and support have never been shaken. To them I want to dedicate
this dissertation.
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There is one special person who has been supporting me with all her strength.
Without her love and support, I might have never gotten this far. To her, my lovely wife,
Xiangmei Wang, I am forever indebted. This dissertation is also dedicated to her, and my
little naughty Peter, who has provided me a lot of pleasure even though he might be more
of a source of fatigue than the research work.
Lastly, I would like to thank the National Science Foundation (NSF) for the
financial support of this research work.
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R e p r o d u c e d w ith p e r m is s io n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
To my parents, Jian Zhao and Shuqing Cai,
my brother, Changhong,
my wife, Xiangmei Wang,
and my little Peter.
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TABLE OF CONTENTS
LIST OF TABLES.................................................................................................
. xi
LIST OF FIGURES ...............................................................................................
.xii
CHAPTER I
.
INTRODUCTION.....................................................................
1
1.1 A Historic Perspective.......................................................................
.1
1.2 Motivation..........................................................................................
. 4
1.3 Strategy..............................................................................................
1.4 Dissertation Overview.......................................................................
8
. 9
1.5 References..........................................................................................
.12
.
ROUGHNESS AND ROUGHNESS CHARACTERIZATION
.13
2.1 Introduction........................................................................................
.13
2.2 Roughness Characterization Techniques...........................................
.18
CHAPTER II
2.2.1
Electron Microscopy..........................................................
.18
2.2.2
Stylus Profiler....................................................................
.23
2.2.3
Scanning Probe Microscopies............................................
.23
2.2.4
Optical Techniques............................................................
.27
2.3 Roughness Characterization Parameters............................................
.30
2.4 References..........................................................................................
.40
CHAPTER III
ATOMIC FORCE MICROSCOPY AND FRACTAL
ANALYSIS..............................................................................
.43
3.1 Basic Principles for Atomic Force Microscopy..................................
.43
3.1.1
Precision Control and Scanning Tunneling Microscope ....
.43
3.1.2
A Different Interaction - Force ..........................................
.46
3.1.3
Force Detection Mechanisms..............................................
.48
3.1.4
Operation Modes of AFM ...................................................
.55
3.1.5
AFM Artifacts....................................................................
.58
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3.2
A New Concept - Fractal Geometry......................................................... 61
3.3
Fractal Dimensions................................................................................... 64
3.3.1
Box Counting Method ...............................................................68
3.3.2
Power Spectrum Method............................................................ 70
3.3.3 Variation M ethod........................................................................ 71
3.4
References..................................................................................................76
CHAPTER IV
ELLIPSOMETRY AND IMMERSION ELLIPSOMETRY............78
4.1
Introduction................................................................................................78
4.2
Polarized Light.......................................................................................... 79
4.3
Basic Ellipsometry Systems....................................................................... 84
4.4
Measurements in Ellipsometry ................................................................. 86
4.4.1
Propagation o f Light Through Optical Elements....................... 86
4.4.2
Null Ellipsometry.......................................................................90
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4.4.3 Rotating Analyzer Ellipsomtry...................................................96
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4.4.4 Immersion Ellipsometry.............................................................. 100
4.4.5 Alignment and Calibration.......................................................... 102
4.5
Analysis of Ellipsometric D ata..................................................................111
4.5.1 Reflection and Refraction at the Interface of Two Isotropic
M edia..........................................................................................I l l
4.5.2 Light Reflection from an Ambient-film-substrate System
114
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4.5.3
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Isotropic Stratified Planar Structure........................................... 116
4.6
Heterogeneous Systems and Effective Medium Approximation...............123
4.7
Model Optimization...................................................................................126
4.8
References..................................................................................................128
CHAPTER V
OXIDATION OF SILICON .............................................................129
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5.1 Introduction..................................................................................................129
1
5.2 Thermal Oxidation of Si ............................................................................. 129
5.3
5.2.1
Thermal Oxidation Mechanisms and Kinetics........................... 129
5.2.2
Thermal Oxidation System and Procedure................................ 134
Microwave ECR Plasma Oxidation...........................................................137
5.3.1
Introduction to Plasma............................................................... 137
5.3.2
Generation of ECR Plasm a........................................................ 139
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5.3.3
ECR Plasma Oxidation Kinetics.................................................142
5.3.4 ECR Plasma System and Oxidation Procedure .......................... 146
5.4 References...................................................................................................149
CHAPTER VI
SI/SI02 INTERFACE ROUGHNESS EVOLUTION DURING
OXIDATION PROCESSES............................................................. 151
6.1
Introduction................................................................................................. 151
6.2
Experimental Procedures............................................................................152
6.3
Roughness Measurements and Data A nalysis........................................... 154
6.4
Experimental Results and Discussion.........................................................160
6.4.1
Si/SiO, Interface Roughness for Initially Rough Si
Surfaces........................................................................................166
6.4.2
Si/Si02 Interface Roughness for Initially Smooth Si
Surfaces........................................................................................176
6.4.3
6.5
Summary......................................................................................184
References...................................................................................................187
CHAPTER VII
CONCLUSIONS AND FUTURE DIRECTIONS........................... 191
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LIST OF TABLES
Table 6.1
Initial Si surface roughness and oxidation conditions for different
sets of samples studied...............................................................................161
Table 6.2
(a) Oxide thickness versus oxidation time for samples thermally
oxidized at 1000°C, 1 atm. (b) Oxide thickness versus oxidation time
for several sets of ECR plasma oxidized samples ..................................... 162
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LIST OF FIGURES
Figure 1.1 Si lattice and the dangling bonds on Si (100) surface................................. 3
Figure 1.2 (a) Schematic drawing of an n-channel MOSFET device (in the
"on" state), (b) The I*V characteristic curve o f a MOSFET
device....................................................................................................... 6
Figure 2.1 Interface trapped charge density (Djt) increases as the interface
roughness increases....................................................................................15
Figure 2.2 The shape of a surface is one aspect o f roughness, (a) High
aspect ratio spikes on an otherwise relatively smooth surface, (b)
A surface with features of large spatial wavelengths................................. 17
Figure 2.3 Schematic drawing of a scanning electron microscope...............................19
a
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Figure 2.4 Schematic diagram of the major components of a transmission
electron microscope....................................................................................20
Figure 2.5 Schematic illustration of a stylus profiler................................................... 24
Figure 2.6 A scanning probe microscope employs a sharp tip to probe the
surface, and a piezoelectric stage for high precision positioning
of the surface (or tip)................................................................................. 26
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Figure 2.7 Ellipsometry measures the relative changes o f the two
eigenpolarization states of the light reflected from a sample
surface.......................................................................................................29
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Figure 2.8 Artifacts of rms roughness illustrated by computer simulated
profiles: (a) its dependence on measuring scales (from ref. 48),
and (b) its inability to distinguish different features...................................32
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Figure 2.9 FFT spectrum for an AFM im age.............................................................. 34
Figure 3.1 Schematic drawing of a scanning tunneling microscope............................45
Figure 3.2 Forces in A FM ........................................................................................... 47
Figure 3.3 Fabrication of Si3N4 AFM cantilever and tip with Si technology.............. 50
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Figure 3.4
Illustration of an AFM that employs the tunneling current to
detect the cantilever deflection.................................................................51
Figure 3.5
Capacitance probe is used in this AFM as the force detection
mechanism................................................................................................53
Figure 3.6
Illustration of using optical lever method as the force detection
mechanism for A F M ................................................................................. 54
Figure 3.7
The three operating modes of AFM: (a) contact mode, (b) noncontact mode, and (c) tapping m ode......................................................... 56
Figure 3.8
AFM artifacts caused by the tip geometry................................................59
Figure 3.9
Tip approaching angle may also create artifacts in AFM images............ 60
Figure 3.10 Some sample induced AFM artifacts: (a) A streaky image caused
by soft surface, (b) Steps or jumps caused by tip picking up loose
particles on the surface.............................................................................. 62
Figure 3.11 Fractal dimension vs. Euclidean dimension (from ref. 2 8 ).......................65
Figure 3.12 Computer simulated profiles showing the correlation between the
fractal dimension and roughness (from ref. 2 8 ) ........................................ 66
Figure 3.13 The fractal dimension o f a surface is a measure of its complexity,
as demonstrated by these computer simulated surfaces (from ref.
2 8 )..............................................................................................................67
Figure 3.14 Illustration of box counting method for calculating the fractal
dimension of a rough profile..................................................................... 69
Figure 3.15 Illustration of the 2-D variation method (after ref. 2 8 ) .............................72
Figure 3.16 The resemblance of the upper (or lower) blanket to the actual
surface depends on the tile size (from ref. 28)........................................... 74
Figure 3.17 The slope of the linear region in the log-log (Richardson) plot (b)
is the fractal dimension of rough surface (a )............................................. 75
Figure 4.1
A monochromatic uniform TE plane wave traveling in the
positive-z direction.....................................................................................81
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Figure 4.2
Jones matrices o f some polarization states: (a) - (c) linearly
polarized states, (d) left-handed circular polarization state, (e)
left-handed elliptical polarization state (after ref. 1 )...........................
83
Figure 4.3
Basic ellipsometry system arrangement..............................................
85
Figure 4.4
The effect of (a) one optical element, or (b) several such elements
in series, on the polarization states o f light can be represented by
its (their) characteristic Jones matrix (matrices).................................
87
Figure 4.5
Effects of (a) an isotropic medium, (b) a linear polarizer, and (c)
a compensator, on the polarization state of a linearly polarized
89
(a) A PCSA null ellipsometer system, (b) The rotation of
reference system for different optical elem ents..................................
91
(a) A rotating analyzer ellipsometer system, (b) Rotation of
analyzer azimuth angle A ....................................................................
97
Schematic diagram of rotating analyzer ellipsometer automation
(after ref. 6 ) .........................................................................................
99
Schematic comparison of ellipsometry measurements carried out
in air ambient and in liquid am bient...................................................
101
Figure 4.10 Illustration of our spectroscopic immersion ellipsometer system......
103
Figure 4.11 Illustration of ellipsometer system alignment process: (a)
straight-through alignment, and (b) sample alignment.......................
104
Figure 4.12 Illustration of several cases of sample misalignment..........................
106
Figure 4.13 A typical plot of residual calibration of ellipsometry offsets using
a gold standard.....................................................................................
no
Figure 4.14 Reflection and refraction of light at the interface of two isotropic
media follow Snell's la w ......................................................................
112
Figure 4.15 Light reflection and transmission on an ambient-film-substrate
sample system ......................................................................................
115
Figure 4.16 Reflection and transmission of light on a stratified planar films
sample system ......................................................................................
117
Figure 4.6
Figure 4.7
Figure 4.8
Figure 4.9
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Figure 4.17 Reflection and transmission of light at the interface of two
arbitrary media a and b: (a) when the light is propagating from
medium a to medium b, (b) when the light is propagating from
medium b to medium a. (c) Reflection and transmission of light
in an isotropic film of thickness d .............................................................120
Figure 5. 1 Illustration of oxygen fluxes in thermal oxidation process........................131
Figure 3.2 A resistance heated, double-wall fused silica tube furnace for
thermal oxidation (after ref. 2 ) ..................................................................136
Figure 5.3 Illustration of electron movement in ECR plasma, in which an
electron gyrates along the magnetic field line with cyclotron
frequency / ............................................................................................... 140
Figure 5.4 Schematic illustration o f our microwave ECR plasma system ................. 147
Figure 6.1 Flow chart of experimental procedures......................................................153
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Figure 6.2 The optical model used in the present study for SIE data analysis........... 156
Figure 6.3 Comparison of the refractive indices of CC14 solution and Si02
films. The refractive index of air/vacuum is also plotted as a
reference................................................................................................... 158
Figure 6.4 Oxide growth data for a thermally oxidized Si sample............................. 163
Figure 6.5 Typical oxide growth data for ECR plasma oxidized Si samples............. 164
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Figure 6.6 Experimental (scatters) and calculated (lines) ellipsometric
parameter A for the ECR plasma oxidized initially smooth Si
surfaces .................................................................................................... 165
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Figure 6.7 Si/Si02 interface region thickness, derived from SIE
measurements, for the initially rough Si surfaces at different
stages of ECR plasma oxidation under different oxidation
temperatures and bias voltages ................................................................. 167
Figure 6.8
Rms roughness of the exposed Si surfaces after the initially
rough surfaces were ECR plasma oxidized under different
conditions................................................................................................. 168
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Figure 6.9
The fractal dimensions o f the exposed Si surfaces after the
initially rough surfaces were ECR plasma oxidized under
different conditions...................................................................................169
Figure 6.10 Some typical AFM images of the exposed Si surfaces at different
stages of ECR plasma oxidation process..................................................170
Figure 6.11 Rms roughness for the initially rough Si surfaces thermally
oxidized in a conventional furnace at 1000°C with ultra-pure dry
0 2 ............................................................................................................. 172
Figure 6.12 The fractal dimensions for the initially rough Si surfaces
thermally oxidized at 1000°C ...................................................................173
Figure 6.13 Normalized rms roughness for the purposely roughened samples ..........177
Figure 6.14 Interface region thickness, from SIE analysis, for the initially
smooth Si surfaces oxidized under different conditions...........................178
Figure 6.1S The rms values and the fractal dimensions for the initially
smooth Si that were ECR plasma oxidized at 200°C, +60V
applied bias, show interface roughening...................................................179
Figure 6.16 Rms roughness for smooth Si surfaces oxidized at 200°C and at
400°C, +60V b ias..................................................................................... 180
Figure 6.17 Some typical AFM images of the initially smooth Si surfaces at
different stages of ECR plasma oxidation process....................................181
Figure 6.18 SIE analysis results of the interface region thickness for both
initially rough and initially smooth Si surfaces during ECR
plasma oxidation...................................................................................... 185
Figure 7.1
Schematic illustration of interface roughness evolution during
oxidation process: the roughness for different Si surfaces
eventually converges and reaches a steady roughness..............................193
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C H A PTER I
INTRODUCTION
1.1
A Historical Perspective
At the turn of this century the electronics industry started to form with the
invention of the first electronics devices, the spark-gap transmitters and crystal detectors.
The inventions of the diode vacuum tube by Fleming in 1905 and three-electrode tube by
deForest in 1906 really jump-started the industry with the widespread commercialization
o f such applications as radio and telegraphy. Since then, device miniaturization has
emerged as one of the most critical issues, and a main target for the development of the
industry, with the purpose of making smaller, faster, cheaper, less power-consuming and
more reliable electronic devices.
In 1947 the industry saw a revolutionary advance, the transformation from the
dominance o f thermionic devices to the era of solid-state devices with the invention of a
silicon transistor by Bardeen, Brattain1 and Shockley2 in Bell Labs. In 1952, driven by
the desire o f further miniaturizing devices, the concept of device integration was raised
by Dummer who stated. "At this stage, I would like to take a peep into the future. With
the advent of the transistor and the work in semiconductors generally, it seems now
possible to envisage electronic equipment in a solid block with no connecting wires. The
block may consist of layers o f insulating, conducting, rectifying and amplifying
materials, the electrical functions being connected directly by cutting out areas of the
various layers".3 His vision was realized in 1959 when the first silicon-integrated circuit
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— "a novel miniaturized electronic circuit fabricated from a body of semiconductor
material containing a diffused p-n junction wherein all components of the electronic
circuit are completely integrated into the body of semiconductor material" was invented
by Kilby of Texas Instruments.4 In the same year Hoemi, and Fairchild Semiconductor's
Noyce and Moore, developed planar process to fabricate transistors, which was quickly
applied to the design of integrated circuits.
During the same period, the entire industry gradually shifted its focus from Ge to
Si as the material of choice for semiconductor devices. This transition was partly due to
the more favorable working conditions and easier availability o f silicon, but more
importantly, it was due to the superior properties and the excellent compatibility of
silicon and its oxide, and to its compliance with the planar process for large scale
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other semiconductor materials. Due to the abrupt termination of the lattice structure at the
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surface, there exists a large number (~ 1015/cm2) of unsatisfied chemical bonds, also
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called dangling bonds, from the surface silicon atoms (see Figure 1.1). These dangling
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bonds introduce surface states into the otherwise forbidden band-gap which adversely
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affect the functioning and performance of the devices made on such surfaces. However,
integration. The ability of Si surface passivation by Si02 is unparalleled by any of the
when a S i02 film is formed on the Si surface the number o f dangling bonds can be
reduced by up to 5 orders of magnitude by bonding with Si02, and thus the electrical
properties of Si surface can be drastically improved.5 Additionally, the Si02 film is an
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excellent dielectric, and very stable under a wide range of working conditions.
Integrated circuits, the planar process, and Si techniques together set the stage for
i.
the microelectronics industry which rapidly grew at an unprecedented pace into one of the
largest industries in the world.
Since being invented to replace individual components, integrated circuits (ICs or
"chips" as they are conventionally called) have progressed from the early 1960's "Small-
9
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Surface
Figure 1.1
Dangling bonds
Si lattice and the dangling bonds on Si(100) surface.
3
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Scale-Integration" (SSI) of less than 20 devices per chip, to late 1960's "Medium-ScaleIntegration" (MSI) of several hundred devices per chip, to the "Large-Scale-Integration"
(LSI) of 1970's that has several thousand devices on one chip, to the "Very-Large-ScaleIntegration" (VLSI) of 1980's of tens of thousands of devices per chip, and nowadays'
"Ultra-Large-Scale-Integration" (ULSI) having up to one billion devices on a single chip.
The industry is driven by the desire to minimize the size and increase the density of ICs
so as to achieve smaller, faster and cheaper devices. From the early ICs of 1960 to today's
ULSI, the density of devices per chip has increased by more than 7 orders of magnitude
in less than 40 years! The smallest dimension of each individual device on today's latest
ULSI chip is 0.25 pm, while 0.18 pm technology has already been developed. At the time
of this writing, the world's first one gigabit dynamic random access memory (1 Gb
DRAM) chip (which roughly contains 109 devices on a single chip) was just announced
by Samsung, and the first 4 Gb DRAM fabricated with 0.15 pm technology is scheduled
to be unveiled by NEC in just a matter of days. The trend is likely to continue to at least
0.10 pm or even less, depending on the advancement of science and technology.
This everlasting miniaturization has brought down the cost, improved device
performance and speed. This improvement is best demonstrated by the evolution of
j
computers, from the size of a small building for the early computers which only few big
companies and government agencies could afford, to computers that fit in one's palm and
I
can be owned by millions o f households across the world.
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1.2
j
Motivation
In the process of downsizing, many scientific and technological problems have
been encountered and mostly solved. However, as the size of a device approaches the
quantum limits, another problem arises, namely, surface/interface roughness. Because the
functioning of microelectronics devices depends on the surface of the semiconductor
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
materials they are built on, it is obvious that surface roughness will have its effect on the
performance of such devices. This can be best illustrated using one important type of
device, the Metal-Oxide-Semiconductor-Field-EfFect-Transistor or MOSFET6, as an
example.
The MOSFET is the most important device in vast majority of today's ULSI
circuits. The structure of a N-channel MOSFET is shown in Figure 1.2a. The NMOSFET is built on a boron doped (p-type) single crystalline Si substrate. Two n+
regions are formed by heavily doping the p-type Si substrate with phosphorous to work as
a source and drain, respectively. When a positive voltage is applied to the metal gate,
which is separated from the substrate by a thin dielectric film (typically Si02), the
majority carriers o f the p-type substrate, holes, are pushed away from the Si/SiO,
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interface to create a depletion layer in between the source and drain. When the applied
|
gate voltage exceeds certain threshold value, the holes in the depletion region are further
|
depleted and the minority carriers, electrons, are attracted into the region. The region
*
!
close to the Si/Si02 interface in between the source and drain is therefore "inverted" from
p-type to n-type and connects the source and drain like a channel. With a negative source
voltage applied, electrons can flow from source to drain through the channel, and the
'
device is "turned on" like a switch (Figure 1.2b). The "on" and "off' states provide the
basic operations of all binary-based digital systems. The flow o f electrons or holes from
I
source to drain is a major factor that determines the speed of the device. Therefore it is
p
obvious that by shortening the channel length faster device can be achieved,
f
When the size of a MOSFET is reduced, it is required that the reduction follows
:
certain "scaling rules"7,8 so that the characteristics o f the device are preserved with the
reduced geometry. Depending on the purpose of the device, different scaling rules may
apply. Usually keeping the internal electric field or the capacitance constant, the
dimension of channel length, channel width, gate oxide thickness, gate voltage, etc. all
5
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Gate dielectric
Source
fei-SX i
(a)
(b)
Figure 1.1
(a) Skemadc drawing of an n-channel MOSFET device (in the "on" state),
(b) The I-V characteristic curve o f a MOSFET device.
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scale accordingly. The smallest dimension in the MOSFET is the thickness of gate
dielectric, usually an oxide film. For the 0.25 pm device, the gate oxide film will be only
4 - 4.5 nm thick, and only 3.5 - 4 nm for the next 0.18 pm generation ICs. The small
thicknesses required for the ultra-thin gate oxide film have given rise to several problems
such as thickness measurement and control. Moreover, the Si/Si02 interface roughness
also became a very serious problem for devices with such ultra-thin gate oxide, because
the interface roughness now comprises a considerable part of the ultra-thin film and the
working region of the devices. Interface roughness would cause the variation of the
electric field sustained by the gate oxide due to the electric field enhancement effect,
which in turn can cause premature breakdown of the gate oxide and thus detract from the
reliability of the device.9 Interface roughness also increases the number o f interface
trapped charges, which will alter the tum-on threshold voltage and cause soft turn-on.10
Interface roughness also affects the mobility of hot carriers and thus the current flow in
the channel due to carrier scattering, which will slow down the speed of the device.10 A
great deal of effort has been expended on interface roughness studies, in order to improve
the device performance and reliability for the future generations of ICs, However, due to
the extremely small scale involved and the lack of capable exploration techniques until
recently, there remain many unanswered questions, including many fundamental science
questions, about the nature of a rough interface and the ability to control the interface
roughness. These questions motivate this research on the Si/Si02 interface roughness
evolution during the microwave electron cyclotron resonance (ECR) plasma oxidation
process and thermal oxidation process, under different oxidation conditions, and the
mechanisms behind the changes.
7
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1.3
Strategy
The recent invention of scanning probe microscopes (SPM)11,12 has opened the
door for scientists to readily explore the world of the nanometer scale. The new
techniques, especially atomic force microscopy or AFM, can be applied to the studies of
surface topography of many materials, including microelectronics materials, to provide
actual images of the surface topography and quantitative measurement of surface
morphology.
Fractal theory13 has emerged from an abstract mathematical and geometrical
concept to one that has found usefulness in many practical applications. Fractal theory
has been successfully applied to semiconductor surfaces to provide quantitative
information about the complexities of surfaces.
Ellipsometry,14 on the other hand, is an old optical technique that has benefited a
lot from the rapid advancement of microelectronics industry. Many problems that
ellipsometry should have been able to resolve in principle but had actually been
nonpractical due to the complex nature of analysis and the lack of computing power can
now be solved almost instantaneously with the help o f the more and more powerful
computers. Spectroscopic ellipsometry (SE) has been employed to study the physical
properties of thin films, surface roughness, and to some degree, interfaces of
semiconductor materials. But its sensitivity to film/substrate interface is, at best,
questionable and ambiguous when the measurements are carried out in air ambient. The
newly developed spectroscopic immersion ellipsometry15 (SIE), however, improves the
sensitivity of ellipsometry to such interface by about a factor of 10, simply by carrying
out measurements in a liquid whose refractive index matches that of the transparent
overlayer film.
In this study we employ these three novel techniques, AFM, fractal analysis, and
SIE, to investigate Si/Si02 interface roughness resulting from ECR plasma oxidation and
8
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thermal oxidatiou processes, two processes that have been used to produce high quality
gate oxides. By studying the change of Si surface roughness (or Si/Si02 interface
roughness in the case of SIE) during these processes, we can not only determine at what
processing conditions better interface can be obtained, but also acquire a great deal of
knowledge on the physical nature of what is happening at the interface. In this study we
will put our emphasis on the effects o f an external electric field on the oxidation at the
interface and on its effects on the interface roughness, which is an important reason we
chose ECR plasma oxidation process, because an external electric field can be established
with DC substrate biasing during the oxidation process. The present study on interface
roughness in thermal oxidation is an extension o f previous work16,17 done in our group,
and also as a reference to the ECR plasma oxidation. The independent measurements of
j
SIE and AFM (together with fractal analysis) serve as mutual confirmation,
f
r
1.4
Dissertation Overview
The objective of this dissertation is to study the evolution of surface/interface
j
roughness of the Si-Si02 system in ECR plasma and thermal oxidation processes, and the
relationship of roughness to some physical parameters. AFM, fractal analysis, and SIE
|
are the primary techniques used in this study.
,
I
}
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r
In Chapter 2, different ways of characterizing surface/interface roughness are
discussed, including descriptions of different parameters and techniques that have been
f
In Chapter 3, basic principles of AFM are presented, such as its physical
j.
used and the ones that we chose to use in our study.
»
t
foundations, structure design, and instrumental set up. The commonly encountered
artifacts of AFM are also discussed. The second part of this chapter is an overview of the
fractal theory. In this part, we will show what fractal dimension is and why it is needed.
We will also describe the algorithms to calculate fractal dimension from profiles and
9
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i
surfaces, emphasizing on the variation method that we choose to use in this study to
extract fractal dimension values from AFM measurements o f Si surfaces.
Chapter 4 is devoted to the optical study of roughness by ellipsometry. Using
Jones vector and matrix representation, we start from polarized light and show how the
polarization state of light changes when passing through a series of optical elements, from
which it is demonstrated what ellipsometry measures and how it makes the
measurements. We then explain briefly how physical information about the system under
measurement is extracted from the ellipsometric measurements with the construction of
an optical model, analyzing detailed interaction of polarized light with stratified planar
structure and utilizing the effective medium approximation theory. In this chapter we also
illustrate the instrumentation setup o f several ellipsometry systems, including the null
ellipsometer, rotating analyzer ellipsometer, and spectroscopic immersion ellipsometer.
Chapter 5 is a simple review on the oxidation of Si. The first part covers the
thermal oxidation process with the introduction to the oxidation mechanism and the DealGrove model for oxidation kinetics for oxides thicker than 30 nm. A brief introduction to
the early stages of oxidation for ultra-thin films is also included in the first part, followed
by the experimental considerations and procedure for the thermal oxidation process. The
second part of the chapter deals with microwave ECR plasma oxidation of Si. We first
describe the generation of the ECR plasma, then the mechanisms of plasma oxidation and
the kinetic model. Finally, we will show the configuration o f our ECR plasma system and
experimental procedure.
In Chapter 6 we will apply the three novel techniques, AFM, fractal analysis, and
SIE, to study the surface/interface roughness of the Si-SiO, system during the oxidation
processes. Several series of Si surfaces, some of them are purposely roughened by wet
etching beforehand, are thermally and ECR plasma oxidized to grow oxide of different
thickness and under different experimental conditions. Before and after oxidation, the
10
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roughness of Si surface (or Si/Si02 interface) is characterized by the three techniques.
Relationships between the roughness and the oxidation extent, oxidation temperature, DC
bias voltage, as well as initial Si surface roughness (whether the Si surface is pre­
roughened or not before oxidation) are investigated. Results from thermal oxidation and
ECR plasma oxidation are compared to find out the effect of external electric field.
Chapter 7 summarizes the results of this work and provides some directions for
future research on this subject.
11
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1.5
References
1.
J. Bardeen and W.H. Brattain, Phys. Rev., 74,230 (1948).
2.
W. Shockley, Bell Syst. Tech. J., 2 8 ,435 (1949).
3.
G.W.A. Dummer, Proc. IRE Symposium on Progress in Quality Electronic
Components, (Washington DC, May 1952), p. 19.
4.
J.S. Kilby, U.S. Patent No. 3,138,743, filed February 6th, 1959.
5.
E.A. Irene, CRC Crit. Rev. Solid State Mater. Sci., 1 4 ,175, (1988).
6.
W.N. Carr and J.P. Mize, MOS/LSI Design and Application, edited
by R.E.
Sawyer and J.R. Miller, (McGraw-Hill, 1972), p. 2.
7.
R.H. Dennard, F.H. Gaensslen, H. Yu, V.L. Rideout, E. Bassons, and A.R.
LeBlanc, IEEE J. Solid State Circuits, SC-9,256 (1974).
8.
Y. El-Mansy, IEEE Trans. Electron Devices, ED-29,567 (1982).
9.
D.J. DiMaria and D.R. Kerr, Appl. Phys. Lett., 27,505 (1975).
10.
P.O. Hahn and M. Henzler, J. Vac. Sci. Technol. A, 2 ,574 (1984).
11.
G. Binnig, H. Rohrer, Ch. Gerber, and E. Weibel, Phys. Rev. Lett., 49, 57 (1982).
12.
G. Binnig, C.F. Quate, and Ch. Gerber, Phys. Rev. Lett., 56,930 (1986).
13.
B.B. Mandelbrot, The Fractal Geometry o f Nature, (W.H. Freeman, New York.
1982).
14.
R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, (NorthHolland - Elsevier Science, Amsterdam, 1987).
15.
V.A. Yakovlev and E.A. Irene, J. Electrochem. Soc., 139,1450 (1992).
16.
Q. Liu, J.F. Wall, and E.A. Irene, J. Vac. Sci. Technol. A, 12, 2625 (1994).
17.
Q. Liu, L. Spanos, C. Zhao, and E.A. Irene, J. Vac. Sci. Technol. A, 13, 1977
(1995).
12
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CHAPTER II
ROUGHNESS AND ROUGHNESS CHARACTERIZATION
2.1
Introduction
Roughness is a term that is used to describe the topographic state of material
termination. No natural objects are perfectly smooth at all observation and measurement
scales. Roughness of materials can come from the manufacturing process and the various
t
treatments (mechanical, physical, chemical, etc.) afterwards. Even when a material is so
carefully prepared to achieve a near perfect surface, roughness persists in microscopic
scale. An example is the single crystalline Si which can be manufactured nearly defectfree, yet there will still be at least atomic scale roughness on its surface.
Roughness is one of the most important physical aspects in material applications
|
It is through its surface that any material interacts with its environment, and therefore, the
i
properties of the material are closely related to its surface states. Roughness is one of
those surface states that can determine many properties of a material. In today's
microelectronics applications roughness has become a great concern for the continuing
1
advancement of technologies. Take the MOSFET device for example, the operation of
|
MOSFET devices1 depends greatly on the surface states of the substrate, and also on the
I
I
electric field sustained in the gate oxide. Therefore, excessive roughness at the interface
will adversely affect the performance and reliability of the device. First of all, roughness
at the metal-dielectric or dielectric-substrate interface in the form of high aspect ratio
(vertical : lateral) spikes will greatly change the distribution of electric field in the gate
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dielectric film, which is the main reason for the leakage current through the dielectric
film, premature breakdown of the gate dielectric film and eventually the destruction of
the device.2'8 Roughness is also a main contributor to the interface trapped charges (Dit),
the existence of which can alter the strength of the electric field in the dielectric film and
thus change the tum-on voltage of the device, resulting in the undesired soft tum-on. Our
results (displayed in Figure 2.1) from capacitance-voltage (C-V) and AFM measurements
on MOS capacitors fabricated on purposely roughened Si substrate with thermally grown
Si02 as the gate dielectric film agree with others9,10 in that the rougher the interface, the
higher the interface trapped charge density Djt. It has also been found that the mobility of
the hot carriers is affected by the interface roughness having short spatial wavelengths
relative to the mean free path for the carriers.10*13 The carriers are scattered and thus
slowed down by the rough surface features during transport through the rough interface.
The lower the carrier mobility, the larger would be the intrinsic switch time for a device.
As the dimensions of devices shrink, the scale of roughness of interest also
decreases to the atomic range. What used to be considered smooth may become
unacceptably rough in today's applications. The most visible o f such transformation can
be also found in the roughness for the interface between the gate dielectric film and Si
substrate in MOSFET devices. The thickness of the gate dielectric Film in today’s most
advanced MOSFET devices is well below 10 nm, rendering the interface roughness
region between the oxide and the substrate a relatively large fraction of the oxide and the
working area o f the device. Nonuniform distribution of electric field in the oxide due to
interface roughness becomes more important. In addition, because the device dimensions
are smaller, the inversion layer also gets thinner, and thus the harmful effect of interface
roughness is much more pronounced.
In order to study the effects of roughness on materials application, we must first
know how to measure and characterize roughness. Roughness, however, is a concept that
14
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1.4
1.2
3*
N
“
o
^
Q
''1
1.0
1 *W
0.9
0.8
0.7
0.6
0
2
4
6
8
10
12
14
16
18
Rms Roughness (nm)
j
[
Figure 2.1
Interface trapped charge density (Dit) increases as the interface roughness
increases.
15
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seems intuitive at first, but when a quantitative description is attempted it becomes
ambiguous and relative. One obvious ambiguity o f roughness is its dependence on the
scale of observation and measurement. One surface that is smooth to the naked eye and
human touch may be very rough under higher magnification. Another example is the
texture or shape of a surface (or profile), that is whether the surface consists o f high
aspect ratio spikes on an otherwise relatively smooth surface as shown in Figure 2.2a, or
of the more uniformly distributed features of large spatial wavelengths as shown in
Figure 2.2b. Whether an object is considered rough or smooth also depends on the
application purpose of the object. Surfaces with exactly the same roughness can work
perfectly well in one part of an integrated circuit, but be totally unacceptable in another
part.
Even though no single definition has been found to clearly and completely
describe every aspect of roughness, many attempts have been made to measure and define
roughness by a variety of means and parameters, and the results, even though not
k
I
completely satisfactory, have served specific purposes. The methods that have been used
to measure surface/interface roughness can generally be grouped into two categories:
t
I
direct and indirect methods. Direct methods usually employ some type of probe to image
the surface topography. For example, scanning electron microscopy (SEM)14 and
transmission electron microscopy (TEM)15 use electron beams to obtain images of sample
surfaces, as do the more recently developed family o f scanning probe microscopies
s
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(SPM)16,17 which form the topographic images of surfaces by raster scanning across the
surfaces with a sharp tip placed in close proximity to the sample and monitoring such
interactions as the tunneling current or forces between the surface and the probing tip.
Indirect methods such as optical techniques measure the optical or other physical
responses from sample surfaces and extract information about surface or interface
roughness by using a model to best-fit the experimental data.
16
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Figure 2.2
The shape of a surface is one aspect o f roughness, (a) High aspect
ratio spikes on an otherwise relatively smooth surface, (b) A surface
with features of large spatial wavelengths.
17
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Like there are many measurement techniques to obtain information about
surface/interface roughness, there are also many parameters and methods that have been
used to qualitatively and quantitatively describe the roughness o f measured
surfaces/interfaces, yet no one of them can provide a complete representation of the
roughness.
Several commonly used experimental techniques in acquiring roughness
information of surfaces and interfaces, and the parameters and methods to provide
quantitative and qualitative description o f roughness will be briefly overviewed in the
following sections.
2.2
■
Roughness Characterization Techniques
2.2.1 Electron Microscopy
Electron microscopy, including scanning and transmission electron microscopies
(SEM and TEM) make use of a finely-focused beam o f electrons as the probe to obtain
microscopic structural information about the specimen under measurement. In an electron
microscope, electrons emitted from a filament or a field emission source are accelerated
to 20 - 200 KeV or even higher (MeV for high voltage TEM) and focused by a set of
;
electromagnetic lenses into a fine beam before impinging onto the specimen being
studied. Several processes can happen due the interaction between the impinging
1
energetic electrons and the electrons and atoms in the specimen, and different particles
can be emitted from the specimen in addition to the transmission and back-scattering of
the incident electrons. Depending on the type of detected particles, electron microscopy
|
I
can be sorted into either SEM where images are formed mainly by secondary electrons
emitted back from sample surface (see Figure 2.3), or TEM in which transmitted and
diffracted electrons form a diffraction pattern as well as a micrograph at the other side of
the sample as shown in Figure 2.4.
18
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Scanning
Generator
Electron Gun
CRT
Condenser Lens
Scan Coils
Objective Lens
Specimen
Detector
To
Vacuum
Pump
>
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Figure 2.3
Schematic drawing of a scanning electron microscope.
19
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LYJ
Electron Gun
Condenser Lens 1
Aperture
I3=gl
Condenser Lens 2
Aperture
IS] ISI
Specimen
Objective Lens
Aperture
Selected Area Aperture
Intermediate Lens
Projector Lens
Screen
Figure 2.4
Schematic diagram o f the major components of a transmission electron
microscope.
20
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SEM14 has long been used to study the morphology of materials surfaces. In
SEM, the focused electron beam scans the sample in raster-style, and a detector collects
the secondary electrons back-emitted from sample surface. The detected signal is
displayed on a cathode ray tube (CRT) synchronously scanned with the probing beam.
Because the yield of secondary electrons depends on the sample curvature (in addition to
its dependence on material) and the orientation of surface features with respect to the
detector, the flux of these secondary electrons at different point of the scanned area is
different, and thus forms a contrasted image of surface morphology. Due to the one-toone correspondence in scanning but different scan size on the CRT and on the specimen
surface, the image displayed on CRT is a magnified replication of the surface area
scanned on the specimen. Due to the large depth of field (~ pm) and high magnification
(up to 300,000*) of SEM, it enables direct observation of the surface morphology of
!
materials. Resolution of SEM can be up to about 2 nm though it depends on the materials
|
among other factors. Even though SEM can provide excellent measurement in the lateral
9
feature sizes on surface, its vertical measurements are still not satisfactory. Since the
images are not acquired in digital format for SEM (as opposed to SPM), statistical
analysis cannot be easily performed. Therefore, quantitative roughness analysis is in
general not the strong suit for SEM. Also, SEM measurements need to be done in vacuum
to reduce electron scattering in free space. Electron charging is another concern in SEM.
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It can seriously limit the SEM measurement of non-conducting surfaces unless some sort
of conducting film is coated over the surface and connected to ground, which, however,
may not be appropriate for every application.
TEM15 can provide better resolution (up to 0.1 nm) and higher magnification than
SEM, which is crucial for the study of the ultra-fine structure in today's ULSI devices.
When the finely focused and highly energetic electron beam passes through very thin
samples, the diffraction and transmission of the electrons are modulated by the
21
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t
differences in sample thickness, phase composition, crystal structures, and crystal
orientation, form both a diffraction pattern and a contrasted image containing the
modulation information of the specimen at planes of different distances from the
specimen. The image or the diffraction pattern is magnified by a set of lenses to achieve
high magnification and high resolution. One special type of TEM technique called crosssection TEM (or XTEM) has been increasingly used in analyzing the interface structures,
including roughness. If the cross-sectioned specimen contains a crystalline phase of
known lattice constant, the thickness of the interface layer can be measured from the high
resolution micrographs of the specimen using the distance between lattice points as the
measuring scale. The major advantage of the technique is that crystal lographic
information can be obtained in addition to interface roughness, such as crystal orientation,
defects, grain size, etc. However, since XTEM is performed only on a section o f the
specimen, what is obtained is only a localized picture and may not be representative of
the whole specimen. Moreover, because the diffraction patterns and micrographs are
formed by electrons penetrating through the specimen, and the electrons in the energy
range of TEM have limited travel range, the specimen has to be thinned by mechanical,
physical, and/or electrochemical means to a thickness of less than ~ 1 pm. This indicates
that the technique is destructive by nature, which is its main disadvantage. The
preparation of TEM specimen is also tedious and time-consuming, and in some cases is
suspected of altering the interface state or other structural information. Like SEM. TEM
also operates in high vacuum, and the images are not acquired in digital format (though
post-acquisition digitization can be performed on the images), and thus statistical analysis
cannot be easily done.
22
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
2.2.2 Stylus P rofiler
Before the invention of scanning tunneling microscope, stylus profilers18'20 (or
profilometers) had been the most powerful instruments for directly measuring surface
topography. It is in principle the forefather for the now widely used atomic force
microscope, but working in a different force range (10‘3 - 1C6 N as opposed to 10'8 - 1011
N for AFM). Stylus profilers use a diamond stylus, resting on the sample by the
gravitational force of a load, to draw a fine line across the sample surface at constant
pace, as illustrated in Figure 2.5. The change of vertical position of the stylus due to the
variation of surface topography will be picked up by monitoring the electrical signal from
the output of a transducer. The signal is then amplified and converted into a digitized
profile of surface height which enables statistical analysis of the roughness. Stylus
j
profilers can achieve a vertical sensitivity of 0.1 nm, and ahorizontalresolution of ~ 1
I
pm which depends on the slopes of thesurface irregularities, thestylus radius andits
included angle. The acquired profiles are convolutions of the actual surface features and
the shape of the stylus. The smaller the stylus radius and the included angle, the better the
acquired profiles represent the actual surfaces. Usually the stylus has a radius of several
'
micrometers but can be as small as 0.1 pm. The load on the stylus also can change the
profile, and heavier load usually causes more severe damage to the surface. Due to the
fast development of today's microelectronics industries, traditional stylus profilers are
phasing out and gradually replaced by SPMs which have a much better lateral resolution
|
than stylus profilers.
I
[
\
2.2.3 Scanning P robe Microscopies
Rapid development in microelectronics technologies pushed the need for
experimental instruments capable of the atomic scale observation of surface topography,
electronic structure and atomic structure. Such experimental means simply did not exist
23
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Amplifier
A/D
Converter
Computer
Diamond
Stylus "
Sample
{
(
I
Figure 2.5
Schematic illustration o f a stylus profiler.
i
24
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
until 1982 when Binnig and co-workers invented the first scanning tunneling microscope
(STM).21 The ability of STM is best demonstrated by its revealing o f the (7x7)
reconstructed Si (111) surface in real space.22 Since then it has rapidly expanded into a
family of scanning probe microscopies and these SPMs have become one o f the most
powerful tools in surface science.
The common characteristics o f the SPM family are that all the microscopes
employ a sharp tip as the local probe that is placed in close proximity to the object being
investigated, while the positioning and scanning of the probe across the surface are
performed through a piezoelectric ceramic stage system to which either the probing tip or
the sample is attached. The strongly distance-dependent interaction between the local
probe and the object is monitored, and the recorded signal is then converted to desired
I
information about the surface such as its topography, electronic structure, etc.
The probe-sample interactions being monitored during the measurements can be
tunneling current, forces or others physical properties. Different interactions provide a
variety of information about the properties of the surface. STM, the first member of SPM.
is based on the quantum mechanical tunneling through the vacuum barrier between the tip
of the metal probe and surface atoms in close proximity under an applied electric field.
!
Because this current reflects the interaction between the overlapping wavefunctions of
electrons on the tip and surface, the electronic structure of the surface can be directly
[
1
•
mapped with atomic resolution. Because of the distance dependence of the tunneling.
|
microscopy (AFM) which was invented in 1986, also by Binnig and co-workers23, on the
]
basis of STM, uses the interatomic force between the probing tip and the sample surface
topographic information can also be obtained for homogeneous surfaces. Atomic force
rather than tunneling current as the control interaction, and therefore can be used to study
the surface of insulating materials which STM is unable to measure. In AFM, the probe is
attached to a rectangular or triangular spring cantilever, as illustrated in Figure 2.6. When
25
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Cantilever
Piezo
ceramic
stage
Figure 2.6
A scanning probe microscope employs a sharp tip to probe the surface,
and a piezoelectric stage for high precision positioning o f the surface (or
tip).
26
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
the tip is close enough to the object being studied, the cantilever will be deflected by the
interatomic force which is a reflection o f the topography and atomic structure of the
surface. This deflection can be detected by several methods such as tunneling,
capacitance, interferometry, interference, and the optical lever method. Via a feed-back
loop mechanism, a constant force (i.e. constant spacing) between the tip and the surface
can be maintained by changing the voltage that drives the piezoelectric ceramic stage to
move the sample along the z direction during the scanning. The signal (driving voltages
for the piezoelectric ceramics) can then be translated into the coordinates of the surface at
each sampling point so that a topographic mapping of the surface can be obtained. Other
modes of acquiring surface topographic information are also available. More details about
the operation principles of AFM will be presented in Chapter 3.
Other SPM include the friction force microscope (FFM) which can be used to
study the atomic nature of friction and wear of materials, the magnetic force microscope
(MFM) which has been successfully applied to the study of ferromagnetic and
superconducting materials with high resolution, and many others using different
interactions as the control signal to obtain different information in the nanometer scale.
The advantages of SPM include the unsurpassed high resolution, the accessibility
to a wide variety of properties because of the different interactions, and the broad range
of operating environments from liquid to ultra-high vacuum. The images obtained with
SPM are all in digital format and therefore statistical analysis can be readily performed on
the acquired images to extract useful quantitative information.
2.2.4 O ptical Techniques
Many optical techniques24 have been used to obtain information about surface
and/or interface roughness, such as light scattering,25 x-ray diffraction,26,27 etc. One
optical technique that has been increasingly applied to roughness analysis is
27
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
ellipsometry,28*30 which is an old technique rejuvenated by the rapid development of
microelectronics industry, and the fact that the complex calculations can be performed
with the much less expensive computers. Ellipsometry31 employs a beam of polarized
light impinged on the sample being studied, and it measures the relative change o f the
two eigenpolarization states o f the light reflected from (or transmitted through) the
sample to obtain two ellipsometric measurables, ¥ and A (see Figure 2.7). Other
parameters can also be derived from A, which is the relative phase change between the
electric field vector components perpendicular (s) and parallel (p) to the plane of
incidence, and T , whose tangent is the relative change o f amplitudes of the two
components upon reflection. One of the most useful parameters is the dielectric function c
(or pseudo-dielectric function <e>), defined as:
e
1 -P
or (e) = sin2<|) l + tan2<j>
( 2 . 1)
where <{> is the angle of incidence for the light beam with respect to the surface normal
and p = tan^exp(/A) is the complex reflectance coefficient (for reflection ellipsometry).
The (pseudo-)dielectric function is important because it is directly related to the physical
properties of the sample.
Like most other optical techniques, ellipsometry measurements do not directly
provide the desired physical, chemical, and structural information about the sample. In
order to obtain such information from the ellipsometry measurements of the change of
polarization state upon reflection (or transmission), it is necessary to construct an optical
model with such desired information incorporated as unknown parameters. These
unknown parameters can then be determined by trial-and-error comparison of the
experimental data with the theoretical values calculated from the model. For the study of
surface or interface roughness, spectroscopic ellipsometry (SE) measurements usually
28
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Sample
Surface
Figure 2.7
i
l
Ellipsometry measures the relative changes o f the two eigenpolarization
states o f the light reflected from a sample surface.
29
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
I
have to be done, i.e., at multiple wavelengths so that more physical parameters can be
added into the model. In the model calculation, the surface roughness can be represented
as a mixed layer of substrate material and voids (or overlayer film for interface
roughness), and an effective medium approximation (EMA)32 theory can then be
employed. A more detailed discussion of ellipsometric analysis can be found in Chapter
4.
The advantage of ellipsometry is that it is non-destructive and thus it can be used
study interface roughness without destroying the sample or altering the state of the
interface, which is a huge advantage over the other techniques that we have mentioned so
far. Also, ellipsometry measurements can be performed in situ, even in real time if
properly arranged. But ellipsometry depends on an optical model to obtain useful
information. Therefore the interpretation of ellipsometric data can be subjective, and also
requires some knowledge of the sample system beforehand, usually obtained from other
i
techniques, in order to build the optical model.
\-
2.3
Roughness Characterization Parameters
As mentioned in the beginning of this chapter, many parameters have been used
for a roughness description, yet none of these parameters can cover every aspect of
roughness. For some techniques like ellipsometry, roughness is represented as a
heterogeneous film of mixed materials. The thickness of this film and the volume fraction
of each constituent provide a quantitative picture of surface or interface roughness, albeit
i
j
an average value over a relative large area (the area of the probing light beam is about 1
mm2). Further details of structure are difficult to obtain with these indirect methods. For
|
direct surface mapping techniques like AFM, information can be extracted from the
digitized images of surface topography through statistical analysis of the images. These
statistical analysis methods have been described in great details in several references.33'35
30
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
In this section we will briefly present several parameters and methods that are commonly
encountered in roughness analysis.
The Root-Mean-Square (rms) Roughness
The rms roughness for a digitized image is the statistical evaluation of the surface
height distribution with respect to a reference plane. Assuming that a scanned image
consists of N*N points (i.e., there are N scan lines for the image and on each scan line the
surface is sampled N times), the rms roughness is defined as:
rms = ^ Z ( 2' j - 2‘»')2 '
<2-2>
where ztj is the height or vertical position of the Ith point on the / h scan line [i.e., at the
coordinates of (r, y;)], and : w is the arithmetic average of ztj over the whole area. From
the definition it is clear that rms value is in fact the standard deviation from the average
of the z-value (height) for the surface at each data point. Therefore, even though it has
'
been the parameter most frequently given as a quantitative indication of roughness, the
[
rms value does not provide information in the lateral directions. Sometimes it is even a
!
misleading indicator of vertical surface features size in that on one hand its value depends
on the measuring scale, i.e., for the same object the rms value may be very different for
|
different measuring scale (Figure 2.8a), and on the other hand, different surface images or
t
profiles having drastically different characteristics can have exactly the same rms value,
I
as illustrated in Figure 2.8b. Therefore judging only by the rms values one can not
i
differentiate the two completely different surfaces without help from some other
parameters.
Some other often quoted roughness parameters are the mean roughness which is
the arithmetic average of the absolute values of surface height relative to the center plane:
31
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
1 •
Scale
rms
L/2
L
12.065
20.733
1 •
1
■ '
-
1 ■ 1
0
■ 1 ■ 1 ■ 11
L
Profile Length (a.u.)
10
rms = 0.66
s
C&
w
•§>
U
X
ju
E
o
5
0
Profile Length (a.u.)
Figure 2.8
Artifacts of rms roughness illustrated by computer simulated profiles: (a)
its dependence on measuring scales (from ref. 48), and (b) its inability to
distinguish different features.
32
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
1
N -lN -l
<2 - 3 >
jsO 7*0
where N, zfj have the same meanings as defined in Eqn. 2.2; the max height which is the
difference of the highest peak and the lowest valley in the scanned area; and ten-pointmean roughness which is defined as the average difference in height between the five
highest peaks and five lowest valleys. Like the rms roughness, these roughness
parameters do not depict the shape or texture o f surface features, and only provide very
limited information on the roughness in the vertical sense. Also, they all suffer from the
same problem of being non-intrinsic properties of the surfaces and therefore they all
depend on the measuring scale. This means that if the scan areas are not properly chosen,
the results from these parameters can be very misleading. Because o f these problems for
the above mentioned vertical roughness parameters, people have been trying to find other
|
ways to address the problem of depicting roughness. The following parameters and
|
methods all provide at least some information related to the shape/lateral/spatial
(
dimensions of roughness. Many involve Fourier transformation of the original
topographical images or profiles, especially when information related to the surface
[
|
features in the lateral directions is desired. Since the Fourier transformation by nature is a
transformation from one space (spatial frame) to its reciprocal space (frequency frame), it
contains the same rich amount of information as the original image, and additional
ii
calculation is usually needed to obtain quantitative information.
i
\j
i
|
Fast Fourier Transformation (FFT) Spectrum
The FFT spectrum of a surface profile displays the spatial distribution of features
on the profile on a 2-0 diagram as shown in Figure 2.9, so that the lateral dimensions
(spatial frequencies) of dominant surface features can be studied. Averaging the spectra
over selected area gives the average distribution over the whole area. The practice is
33
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
SpectruM
DC
!
i
i
f
I
Figure 2.9
FFT spectrum for an AFM image.
34
R e p r o d u c e d w ith p e r m is s io n o f th e co p y r ig h t o w n e r . F u rth er r ep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
especially useful for comparison of surfaces undergoing sequential processing,36 and
when periodic features exist on the surface. One disadvantage of analyzing the FFT
spectrum is that the spectrum depends on the orientation of the selected profile, i.e.,
whether the section is drawn parallel, perpendicular, or at an angle to the actual scan line,
due to the artifacts in the scanned images. Also, except for the sinusoidal shaped features,
the frequency distribution in the spectrum can often be misleading, because a real surface
has a FFT spectrum that is a composite o f almost infinite number of different frequencies,
which makes the distribution too complicated to interpret.
Autocovariance
Autocovariance function (ACF) analysis can yield information about the inherent,
periodic features that exist on the surface which may not be obvious otherwise, and yield
the average size and separation of surface features. The ACF of an image or profile is
defined as the inverse Fourier transformation of the product of the FFT and its complex
conjugate, or:
ACF = F F T 1{F F T ■FFT} = F F T 1{FFT2}
(2.4)
There is an alternate method of calculating ACF.37 For a surface profile z(.r), a
relative height function H(x) is defined as:
I
H{x) = z{x )-zine
I
where z
(2.5)
is the arithmetic average height over the whole profile. The autocovariance
function ACF is then defined as:
G(l) = lim^-JJ H{x')H{l+ x')dx'
35
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(2.6)
where a is the length of the profile and I is called the "lag length". For a discrete profile of
finite length a and N data points, the ACF can be calculated as:
(2.7)
in which the lag length / = n8, n = 0, 1, 2, ..., N-l, and 8 is the minimum step distance
between the two neighboring points. From the above definitions it is seen that ACF is the
product of the original image with the same image but laterally shifted by a distance, the
lag length /, from its original. When G(t) is plotted against the lag length /, the initial part
of the ACF generally agrees reasonably well with a Gaussian function, and thus can be
expressed in the form of:
(2 .8)
The a in Eqn. 2.8 is the rms value o f the profile and the t is called the autocorrelation
length which is indicative of the lateral feature size. Therefore, from ACF analysis one
can get statistical information about the vertical magnitude as well as the lateral size of
surface features.
In addition to the ACF, there are some other similar and related functions such as
the autocorrelation function and structure function from which both lateral and vertical
information can be extracted about the surface roughness.
Power Spectrum Density
Power spectrum density (PSD) analysis is a widely used method to characterize
surface roughness by analyzing the spatial frequencies or wavelengths o f surface features,
and the relative contribution (power) of each frequency to the whole surface. 3839 It is
defined as the product of FFT of the original image and its complex conjugate, or:
36
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
PSD = FFT* ■FFT = \FFT\2
(2.9)
From Eqn. 2.9 it is seen that PSD can be calculated directly from the Fourier
transformations of the original image or profile. There is an alternate way, as given in
Eqn. 2.4, in which one can simply perform Fourier transform to the autocovariance
function once ACF is calculated.
For a surface image of scan size (Z, XL) with N*N total data points, the twodimensional (2-D) FFT of the surface height function z(x^ yn) is:
( 2 . 10 )
in which 5 is the increment step between two neighboring data points, m and n are
integers, and kx and k are the wave vectors in the reciprocal space. The power spectrum
density yik^ ky) can then be calculated from the 2-D FFT as follows:
(2 . 11)
In actual applications usually only the angular average of the PSD is displayed as a
function of spatial frequencies, and the averaging is done by defining k 1 = k] + k ] . The
available wave vector k is limited to a range from 2n/L to Nn/L by the scan size (L) and
the number of sampling points (N) on each scan line. Like the autocovariance function
analysis, from the PSD analysis not only the spatial or lateral roughness information can
be obtained, but also the vertical rms roughness, which is related to the PSD by:
(2 . 12)
37
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
where v = kiln is the spatial frequency. Existence of a linear region often found in the
log-log PSD curve is an indication o f the fractal nature for the surface, and the fractal
dimension can be derived from the slope of the linear region3*-45 (see next section and
also Chapter 4 for more details on fractal dimension). However, the rms roughness can be
more easily obtained from the original image without complicated transformation, and
the fractal dimension extracted this way is less precise compared to some other
algorithms due to the fact that data points are more scattered in the PSD curve.
Fractal Dimension
The concept of fractal dimension, Dj, was first introduced by Mandelbrot46 to
describe certain types of irregular objects that the traditional Euclidean geometry (of
i
dimensions 0, 1, 2, and 3 for points, lines, planes and cubes, respectively) are no longer
;
appropriate to describe the irregularity of the objects. The fractal dimension is a measure
i
of the irregularities of rough or "fractal" objects by examining the space filling ability of
the objects. The magnitude of fractal dimension is thus a reflection of the shape
;
complexity of the object. Being an intrinsic property of an object, the fractal dimension is
scaling-invariant. Therefore, using fractal dimension enables us to compare the
!
complexity (one important aspect of roughness) of different rough objects with a simple,
single-value parameter.
I
The variation method47,48 is used in this study among many other algorithms to
i.
extract the fractal dimension of rough Si surfaces from AFM micrographs. A more
r
r
t
detailed overview on the fractal concept of nature and calculation algorithms will be
5
it
presented in Chapter 4.
In this study we decided to choose the combination of rms roughness and the
fractal dimension (calculated by the 2-D variation method from the AFM images) to
38
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
represent the state of surface roughness. The reason to choose this combination is that it
provides information on both the vertical scales or magnitudes of the features (from rms
values) and the shapes or complexities of these features (from the fractal dimension) on
surface being studied. Also, both rms roughness and the fractal dimension have been
shown linked to other physical properties. Therefore, representing roughness with these
two parameters can have the potential of incorporating roughness into physical models
for studies of other surface roughness related physical phenomena.
i
t
!
39
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
2.4
1.
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G. Binnig, C.F. Quate, and Ch. Gerber, Phys. Rev. Lett., 56,930 (1986).
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J.M. Bennett, Opt. News, 1 1 ,17 (1985).
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C. Amra, C. Grezes-Besset, P. Roche, and E. Pelletier, Appl. Opt., 28, 2723
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|
26.
W. Weber and B. Lengeler, Phys. Rev. B, 4 6 ,7953 (1992).
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M.T. Tang, K.W. Evans-Lutterodt, G.S. Higashi, and T. Boone, Appl. Phys. Lett..
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D.E. Aspnes, J. Vac. Sci. Technol., 18,289 (1981).
5
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3 1.
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Holland - Elsevier Science, Amsterdam, 1987).
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R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
34.
J.C. Russ, Computer-Assisted Microscopy, The Measurement and Analysis o f
Images, (Plenum, New York, 1990).
35.
J.M. Bennett and L. Mattsson, Introduction to Surface Roughness and Scattering,
(Optical Society of America, 1989).
36.
Q. Liu, L. Spanos, C. Zhao, and E.A. Irene, J. Vac. Sci. Technol. A, 13, 1977
(1995).
37.
M. Rasigni, G. Rasigni, J.P. Palmari, and A. Llebaria, J. Opt. Soc. Am. 71, 1124
(1981).
38.
M.W. Mitchell and D.A. Bonnell, J. Mater. Res., 5 , 2244 (1990).
39.
Ph. Dumas, B. Bouffakhreddine, C. Amra, O. Vatel, E. Andre, R. Galindo, and F.
Salvin, Europhys. Lett., 22,717 (1993).
i
40.
B. Mandelbrot, D. Passoja, and A. Paullay, Nature, 308,721(1984).
<
41.
R. Voss, The Science o f Fractal Images, edited by O.Peitgen
|
t
I
f
i
|
and D. Sampe.
(Springer-Verlag, Berlin, 1991), Chapter 1.
42.
C. Tricot, J.Chim. Phys., 85,379 (1988).
43.
A. Majumdar and C.L. Tien, Wear, 136,313(1990).
44.
J.P. Carrejo, T. Thundat, L.A. Nagahara, S.M. Lindsay, and A. Majumdar, J. Vac.
Sci. Technol. B, 9,955 (1991).
45.
J. Krim, I. Heyvaert, C. Van Haesendonck, and Y. Bruynseraede, Phys. Rev. Lett..
70,57(1993).
1
46.
j
;
B.B. Mandelbrot, The Fractal Geometry o f Mature, (W.H. Freeman, New York,
1982).
47.
B. Dubuc, S.W. Zucker, C. Tricot, J.F. Quiniou, and D. Wehbi, Proc. R. Soc.
Lond. A, 425,113(1989).
48.
L. Spanos and E.A. Irene, J. Vac. Sci. Technol. A, 12,2646 (1994).
42
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
CHAPTER III
ATOM IC FORCE MICROSCOPY AND FRACTAL ANALYSIS
3.1
Basic Principles for Atomic Force Microscopy
The first atomic force microscope (AFM) was built, literally, on basis of the
newly invented scanning tunneling microscope (STM) and by the same group of
researchers. The basic elements are the same for the two (and also for many of the other
SPMs developed alter STM and AFM): both have a local probe that is placed in close
proximity to the sample surface, both employ a strong distance-dependent interaction to
obtain surface information, and the precision positioning and scanning systems are the
same for both. In order to better understand the basic principles o f AFM, it is helpful to
start with a brief introduction to STM, the first SPM.
3.1.1 Precision Control and Scanning Tunneling Microscope
The invention of STM (as well as other SPMs) was driven by the need to study
the surface electronic structure and topography with atomic resolution. Some of the main
obstacles for atomic scale imaging had been precision position control and vibration
isolation. In 1972 Russel D. Young1 of National Bureau o f Standards succeeded in
manipulating an object in three dimensions with a precision o f about a nanometer using a
position control system built with piezoelectric ceramic materials. Piezoelectric ceramics
change dimension very slightly when a potential is applied and the change is close to a
linear function of the applied potential. Sample stages built with piezoelectric ceramics
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
I
are capable of position control down to 0.001 nm or even better, with a dynamic range of
several micrometers. In 1982, with a piezoelectric ceramic scanner, Binnig and Rohrer2’3
finally realized atomic scale imaging of sample surfaces with their new instrument, STM.
The fundamental principle of STM is based on the quantum tunneling of electrons
from the atoms at the apex of a metal probe to the atoms at the nearby sample surface
when a small bias voltage is applied to the probe to help the electrons overcome the
vacuum gap potential barrier. Quantum tunneling reflects the fact that when the atoms are
brought close to each other their wavefimctions overlap, and the interaction between the
overlapping wavefimctions depends strongly on the separation of the atoms. Therefore,
scanning at constant interaction (tunneling) across the surface with sub-angstrom
precision will give us a constant-interaction contour that reflects the surface topography
|
of sample (provided that the sample surface is homogeneous).
The most critical part in STM is the position control of the probe with a sub­
angstrom precision, because in order to obtain detectable tunneling current, the gap
between the probe and surface has to be small enough (less than several angstroms). The
?
position control in the first STM was accomplished with a tripod piezoelectric ceramic
scanner. A tube scanner4 was later developed to improve the vibration resistance and
I
linearity of position control. A feedback mechanism can be used to keep the gap spacing
between the probe and sample surface constant, as illustrated in Figure 3.1. First the
probe is brought close enough by coarse adjustment screws with ultrafine threads to be
j
within the dynamic range of the piezoelectric ceramic stage, which then takes over to
j
further reduce the gap between the probe and sample until the probe tip is "engaged" to
!
the surface, at which point detectable quantum tunneling occurs. While raster scanning
the surface in the x-y (lateral) plane, the fluctuation of tunneling current due to the
varying surface height at different lateral positions is used to modulate the voltage
applied to the z (vertical) post of the piezoelectric scanner. The corresponding movement
44
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
. k*Y.;' ^
Display
Piezoelectric
C ontrol
Bias
Voltage
^tun
Feedback
G enerator
Figure 3.1
lref
Schematic drawing of a scanning tunneling microscope.
45
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
of the probe (which is attached to the piezoelectric ceramic scanner) compensates for the
change of gap spacing, so that constant gap (or current) can be maintained. The voltages
that drive the piezoelectric scanner in x-y-z directions can then be converted directly to
images of surface topography. Another STM operation mode is to fix the height o f the
probe, and the fluctuation of tunneling current is directly used to reconstruct the image of
surface structure. No feedback is needed for this so-called constant height mode.
3.1.2 A D ifferent Interaction - Force
Because of the nature of its operating principle, STM cannot be used to probe
nonconducting surfaces because electrons cannot readily tunnel through bulk insulating
materials. Some other localized interactions have to be used in place of tunneling in order
to image these insulating materials with atomic scale resolution. In 1985 Binnig and his
colleagues came up with the idea of replacing tunneling current with the interatomic force
between the atoms at the apex of a sharp tip and the atoms on the sample surface as the
control interaction, and with that they built the first AFM.5 This interatomic force (see
Figure 3.2a) between two atoms can either be the ionic repulsive force or Van der Waals
force, and it can be characterized by the Lennard-Jones potential:6
(3.1)
where r is the distance between two atoms, and C;. C2 are constants. For long range the
f
Van der Waals interaction (the r 6 term) prevails. This is an attractive interaction which
originates mainly from the induced dipole-dipole interaction. If the two atoms are brought
even closer, the wavefunctions of the electrons in the two atoms will overlap and create a
repulsive interaction due to the Pauli exclusion principle, and the potential is
characterized by the r '12 term in Eqn. 3.1. From this discussion it is easily seen that this
46
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(a)
Cantilever
Fluid Layer
(b)
/
+'
/
/ /
/ / / /
Figure 3.2
/
4
4 4/ // a/ 4 ' 4 V
4.
/ / / / /
Sample
/
/
s
s
/
/
/ / / /
/
Electrostatic
Charges
Forces in AFM.
47
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
interatomic force is determined by the localized atomic structure close to the probe tip,
and it depends strongly on the distance that separates the tip and the surface. Therefore it
qualifies as the interaction that a local probe method needs for atomic scale imaging.
More importantly, the interatomic force does not depend on the conductivity of the
materials, and therefore can be used to probe both conducting and nonconducting
surfaces.
It has to be noted, however, that the interatomic force characterized by LennardJones potential is an idealized representation. In reality it is much more complicated. One
of the complications is that other forces may be involved during the scanning. Assuming
that the surface is nonmagnetic, when the operation is carried out in ambient air, there is
usually an absorbed layer of gases on the surface, mostly condensed water vapor and
other contaminants, which is about several nanometers thick. When the tip reaches this
layer during the engaging process, the surface tension caused by the capillary effect will
pull the tip towards the surface as shown in Figure 3.2b. Sometimes, there are also
trapped charges on the tip and sample surface, and the electrostatic forces will further
increase the total forces exerted on the surface. Also during the scanning, especially in the
traditional contact mode (see Section 3.1.4) there are lateral shear forces when the tip is
moved along the surface. These forces can cause image distortion, and in the worst case,
cause serious damage to the surface and the probe.
i
3.1.3 Force Detection M echanisms
The detection of this interatomic force which is in the range of 10'® - 10'n N or
even smaller is the most critical part in AFM, and it is achieved by attaching the probe to
a spring cantilever and monitoring the deflection of this cantilever upon the application of
the force. From Hooke's law (F = - k x wherek is the spring constant) we see that in order
to have the largest deflection with forces of such magnitude, the cantilever needs to be as
48
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soft as possible, i.e., a small k is necessary. However, in order to minimize its
susceptibility to vibration, the spring cantilever also needs to be stiff with a high
resonance frequency. The resonance frequency for a spring cantilever is known to be:
where m is the effective mass of the cantilever. So in order to have a softer spring
(smaller k) but with higher yj, we need to decrease the effective mass m to keep the ratio
of k/m large. Cantilevers used in the earlier days were usually made of metal foils or
wires and the tips attached to the end o f the cantilever were mostly small pieces of
diamond. Now the cantilevers and tips are usually mass fabricated of Si3N4 as an integral
1
i
part with advanced Si processing technologies (illustrated in Figure 3.3). Highly
j
reproducible results can be yielded with these new cantilever-tips due to the standardized
i
fabrication procedure.
When the tip is brought close enough (i.e. "engaged") to the sample surface it will
experience the interatomic force which will be sensed by the cantilever and cause the
|
cantilever deflection. The problem of force detection is now transformed into detection of
cantilever deflection which can be easily implemented. Several techniques have been
employed to detect the small deflection of the cantilever. Binnig et al sandwiched the Au-
I
foil cantilever between the sample and the tunneling tip of an STM as illustrated in Figure
I
I
f
i
\t
3.4. When the STM tip is brought within the tunneling range of the conducting cantilever,
any movement of the cantilever will result in fluctuation of the tunneling current which
can then be used to control the AFM feedback circuit to maintain a constant force
(spacing) between the AFM tip and sample surface by adjusting the AFM piezoelectric
sample stage accordingly.
49
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(b)
glass
saw cut
/ / /J..S /
Cr
(c)
saw cut
(d)
Au coating
\
r sr
glass and Cr
removed
cantilever
(e)
Figure 3.3
Fabrication of Si3N4AFM cantilever and tip with Si technology.
50
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Control
Piezoelectric
Cantilever
STM Tip
Sample
Piezoelectric
Sample Stage
Figure 3.4
Illustration of an AFM that employs the tunneling current to detect the
cantilever deflection.
51
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Another method of detecting cantilever deflection is to measure the capacitance
between the back of the cantilever and a capacitance probe (Figure 3.5). Change of
capacitance due to sub-angstrom cantilever displacement can be measured with a
capacitance bridge.7 Through a feedback mechanism the variation of capacitance controls
the tip-to-sample spacing and keeps it constant.
The most widely used deflection detection technique is the optical lever method8
which employs a laser diode and a split-cell photodiode detector. The light emitted from
the laser diode is impinged onto the reflective back of the cantilever and reflected back
onto the split-cell photodiode detector, as illustrated in Figure 3.6. The output of the
photodetector, (A-B)/(A+B), where A and B are signals detected by the two respective
cells, depends on the relative position of the laser beam spot on the detector. A software
|
defined set-point reference voltage determines the equilibrium position of the cantilever
after the tip is engaged, i.e., brought within the force range of the sample. It is obvious
that this equilibrium position is decided by the force between the surface and the probe.
Therefore, choosing a proper set-point value is important in minimizing the force damage
!
to the surface as well as to the probe. During the scanning, the cantilever deviates from its
equilibrium position when the variation of tip-to-surface distance at different surface
positions causes change in the interatomic force, which in turn moves the spot of the
reflected laser beam in the detector, and thus alters the differential signal (A-B)/(A+B).
|
r
.l
The differential signal can then be used to control the piezoelectric stage via feedback
circuit to adjust the sample until the force is returned to the set-point value and the
equilibrium position of the reflected laser beam spot is restored. Most, if not all, of the
commercially available AFMs use this optical lever technique as the mechanism for
force/deflection detection.
In addition to the above mentioned force/deflection detection techniques, there are
some others that have been used by various researchers, such as the interferometry910 and
52
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Control
Piezoelectric
Cantilever
Capacitor
d
Sample
Piezoelectric
Sample Stage
Figure 3.5
Capacitance probe is used in this AFM as the force detection mechanism.
53
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
position sensitive
detector
.A
3
O.
■M
3
O
CD CD
o(0 - t <+
c
d)
0)
to feedback
sharp
tip
flexible cantilever
piezoelectric
sample stage
Figure 3.6
Illustration of using optical lever method as the force detection mechanism
for AFM.
54
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
laser interference11 techniques. However, these techniques are not as widely used as the
ones described above.
3.1.4 O peration Modes o f AFM
As have been described above, the operation of AFM centers on the deflection of
a cantilever upon experiencing the interatomic forces when the tip is brought close to the
surface. If detected, the deflection can be used with or without a feedback circuit to
reconstruct surface topography. Depending on the way the force deflection is detected.
AFM can operate in several modes.
In the "constant force" mode, the tip-to-surface spacing (and thus force) is kept
constant by the control signal (i.e., deflection of cantilever) via the feedback circuit.
i
|
Scanning over the surface with constant force/spacing allows the tip to follow surface
topography. The recorded voltages that drive the x-y-z piezoelectric sample stage can then
directly be converted into the x-y-z coordinates of a surface at each sampling point.
In the "constant h eig h t" mode, the position of the tip and the z-piezoelectric
sample stage are not changed during the scanning, so the spacing between the tip and the
F
surface varies from point to point depending on the surface topography. The
f
corresponding fluctuation of the detected deflection can therefore be used, without the
feedback circuit, to directly reconstruct the surface topography based on the fact that the
1
force/cantilever deflection/detected signal is a single-valued function o f the separation
IS
between the tip and the surface. This mode, however, is not suitable for large area
scanning of rough objects, because the tip may either fall out o f the force range (i.e..
’
withdrawn or disengaged from surface) or crash into some high surface features.
In both the constant force mode and constant height mode, AFM works basically
in the repulsive force range, and the tip is in contact with sample surface as shown in
Figure 3.7a. The dragging of a tip across the surface, combined with adhesive forces
55
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
'
(a)
/
/
/
/
f
/
/
/
/
/
/
/
/
/
/
/
Sample
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
Contact
/
/
Moisture
Non-contact
Sample
(b)
Moisture
/
(C)
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
Sample
/
/
/
✓
/
Tapping
/
/ /
/ / / / / / / / / / / / / / / /
Figure 3.7
The three operating modes of AFM: (a) contact mode, (b) non-contact
mode, and (c) tapping mode.
56
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
between the tip and the surface can cause substantial damage to both sample and probe
and create artifacts in the acquired image, especially for soft samples. In order to lower
the interaction between the probe and the surface, some other AFM operation modes have
been developed. One such mode is the "non-contact" mode (see Figure 3.7b) in which
the cantilever, instead of statically sitting on the surface, is modulated near its resonance
frequency when the tip is brought close to the surface. Any deviation from the
equilibrium tip-to-surface distance during the scanning will change the oscillation
frequency and amplitudes accordingly, which can be detected and converted into surface
topography. This non-contact mode employs mainly the long range Van der Waals force
rather than the repulsive contact forces so the total force exerted onto the surface is much
smaller and the damage to the surface and the probe is minimized. However, the weaker
I
I
Van der Waals force gives rise to instability in sustaining the required tip-to-surface
engagement, and resolution of AFM working in this mode is lower than that in the
contact mode.
A newly developed "tapping" mode12 (Figure 3.7c) which combines the merits of
the contact mode and the non-contact mode is getting more and more popular. In this
mode, a piezoelectric crystal drives the tip vibrating in and out o f the surface region with
'
large amplitude (typically greater than 20 nm) at or near the resonance frequency (50
KHz to 500KHz) of the cantilever. The tip is alternately placed in contact with the surface
f
<
("tapped" into the repulsive force range) to provide high resolution imaging and then
I
pulled away from the surface (out of the force range) when the sample is being moved to
[
the next position to avoid dragging the tip across the surface. As the vibrating tip starts to
[
contact the surface, the amplitude of oscillation is reduced due to the tip-surface
interaction, and the reduction of the amplitude is used to reconstruct the surface
topography. Because the tip vibrating in and out of the surface region, the tapping mode
allows high resolution of the traditional contact mode but with much reduced damage to
57
R e p r o d u c e d w ith p e r m is s io n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
the surface and the tip because adhesive and shear forces effects are minimized by the tip
pulling away from surface during the lateral movement of sample. Even when the tip is
tapped onto the surface, the contact forces are greatly reduced as compared with the
traditional contact mode, because the interaction is brief due to the high oscillation
frequency. The damage to the surface is so much smaller that it allows AFM to be used to
study such extremely soft samples as polymers and biological specimens. Compared with
the non-contact modulation mode which can also be applied to soft samples, the tapping
mode is much more stable and has a much better resolution.
3.1.5
AFM Artifacts
AFM images are reconstructed from the detected signal generated by cantilever
;
deflection, but how close an acquired image represents the real surface depends on many
I
factors. Instrumental noise from the electric circuits and environment vibrations can show
i
up in the images. In addition, the probing tip, piezoelectric sample stage, even the sample
surface itself can all give rise to various artifacts in the AFM images. Another important
source of artifacts is the improper post-acquisition manipulation of AFM data.13
The most significant source of AFM artifacts is the probe-cantilever. Because of
the way AFM operates, an AFM image is the convolution of the tip shape and the real
surface topography. Therefore the geometry of the tip shows up in the AFM images, as
1
t
illustrated in Figure 3.8. It is obvious that in order to get a more accurate representation of
|
surface, the tips of smaller radius of curvature and higher aspect ratio are preferred for
!
AFM measurements. However, a long sharp probe is more prone to probe-bending which
j.
will also cause image distortions. Also, geometry irregularities of probes will show up on
surface features. For example, images acquired using a probe with double tips will show
double features for each real one on the surface. The angle at which the probe approaches
the surface may also create image artifacts, and a probe perpendicular to the surface can
58
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
N|
\
W
Figure 3.8
Some AFM artifacts caused by the tip geometry.
59
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n
Figure 3.9
Tip approaching angle may also create artifacts in AFM images.
60
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
reproduce the real surface more accurately, as shown in Figure 3.9. Artifacts caused by
the probe can usually be minimized with proper design. Nevertheless, they cannot be
totally eliminated unless exact geometry of each tip is known beforehand.
The piezoelectric ceramic scanner is another source of artifacts for AFM images.
The movement of piezoelectric ceramics, upon application of a voltage, is not linear,
which may cause image distortion, creep, drift, and bowing. These artifacts also depend
on the design of the piezoelectric scanner in addition to the mentioned non-ideal behavior
o f the material. The piezoelectric scanner related artifacts can be eliminated or minimized
through proper design, careful calibration, and post-acquisition data manipulation
(filtering). However, if not performed correctly, post-acquisition data manipulation can
introduce other artifacts, sometimes very serious ones. Images may be altered to be
'
\
different from the actual surfaces. It is therefore extremely important to understand the
nature of the manipulations and apply them with great care.
Surfaces being scanned can introduce some artifacts. If not properly placed, the
sample surface may have tilt, warp and bowing that will show up in the image. Even
worse, some regions of the surface may be out of the dynamic range of the piezoelectric
scanner. If the surface under scanning is soft, the images acquired with contact mode may
show streaks (Figure 3.10a) and images may not be reproducible since surface conditions
may easily be altered by the probe. Loose particles may be picked up by the tip and cause
;
large "jumps" in the image (Figure 3.10b). These and some other sample related artifacts
*
can be corrected by choosing proper operation mode (e.g. the tapping mode), careful
[
adjustment of samples and careful application of filtering.
i
f
3.2
A New Concept - Fractal Geometry
The geometry of various objects has long been described with Euclidean
dimension of 0, 1, 2, 3 for points, lines, planes, and bodies, respectively. These integer
61
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(a)
(b)
Figure 3.10
Some sample induced AFM artifacts: (a) A streaky image caused
by soft surface, (b) Steps or jumps caused by tip picking up loose
particles on the surface.
62
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
II
dimensions, in a strict sense, are mathematical concepts for perfectly ordered or regular
systems, and real objects in nature are rarely as ordered. Roughness is a kind of disorder
since a rough profile (or surface) is a deviation from a straight line (or plane) which is a
perfectly ordered system. When the deviation is not large, the disordered system can still
inherit substantial properties of the ordered system, and its geometry can still be
represented by the geometry that describes the ordered system, i.e., the Euclidean
geometry. However, when the degree of disorder becomes large and the irregular aspect
of these objects is the center of their applications, the properties of such disordered
systems can no longer be derived from that o f the ordered systems. For example, the
shape of a mountain, or a cloud, or a coastline cannot be described accurately with
traditional Euclidean geometry. Some other geometrical quantity similar to the Euclidean
dimension is needed to quantitatively describe this disordering/irregularity/complexity of
these strongly disordered objects. The fractal approach is exactly what is needed for these
systems. The embryo o f the fractal concepts started around the turn of this century (1875
- 1925) by some mathematicians who tried to understand the nonvanishing roughness of
some constructed geometric shapes, which they labeled as "monsters" because these
geometric shapes could not be explained by the traditional Euclidean geometric concepts.
In 1975 B.B. Mandelbrot14 first introduced a new geometric language, the fractal
geometry, as a new tool aimed towards the study of diverse aspects of diverse objects,
either mathematical as those geometric monsters or natural, that are not smooth but rough
and fragmented to the same degree at all scales. By 1982 the fractal theory15 has been
systematized and found its usefulness in a broad spectrum o f applications, including
artistic, scientific, and engineering.
In fractal theory, disorder is considered as an intrinsic property rather than a
perturbation to the perfectly ordered system. A system is termed fractal when it shows
scaling-invariant behavior at all length scales, i.e., if the system looks the same and has
63
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
i
the same degree of disorder at ever smaller scales or ever larger scales. In reality, though,
a system only exhibits fractal behavior over a finite range o f scales. Depending on the
scaling behavior there are two types of fractals: self-similar and self-affine. For self­
similar fractal systems (sometimes also called genuine fractals) the systems are invariant
when scaling in all directions. Each part of the system is a linear geometric reduction of
the whole, with the same reduction ratios in all directions. A natural fractal system is self­
similar if the system is statistically similar at varying scales. One mostly quoted example
of self-similar fractal system is a coastline. Self-affine fractal systems, on the other hand,
are anisotropic. Each part of the system is still a linear reduction of the whole system but
the scaling ratios are different in different directions. Many natural surfaces are self-affine
surfaces. The fractal concepts have been successfully applied to the study of the
complexities of natural objects, including the roughness of semiconductor materials.
3.3
F r a c ta l D im ensions
Fractal dimension, Dj, is the quantity that describe the intrinsic scaling invariant
property of strongly disordered (fractal) systems. It is a measure of the space-filling
ability/complexity/irregularity/roughness for such fractal systems. It has a fractional value
in between the integer Euclidean dimensions, ranging from 1 to 2 for an irregular profile
or 2 to 3 for a fractal surface (see Figure 3.11). Fractal dimension for a disordered system
is defined such that the higher the value of the fractal dimension, the more spacefilling/complex/irregular/rough is the system, as illustrated in Figure 3.12 by computersimulated profiles, and in Figure 3.13 by computer-generated surfaces.
There are many definitions of fractal dimension, depending on the algorithm that
one uses to calculate it. If the systems are continuous and strictly fractal, the fractal
dimension calculated from different algorithms should be identical. However, for discrete
systems that have a finite number of data points, such as a digitized image or profile.
64
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
D „= l
Df = 1.x
De = 2
1
Df =2.x
2 < D f <3
Dc = 3
Figure 3.11
Fractal dimension versus Euclidean dimension (from ref. 28).
65
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
0.155
0.150
i.e
0.145
I
0.130
18
0.61
082 083 0.54 0.65 088 0.67 0.68 089 0.7 0
161
0.62 0.65 0.64 0.65 0.66 a67 0.68 0.69
0.61
0.62 0.63 0.64 085
0.70
1.9
18
I
2.0
1.7
1.6
081
0.62 083 0.64 a65
0.66 0.67 0.68 0.69 170
2.8
0.66 0.67 0.68 0.69 0.70
38
15
17
14
2.6
13
28
11
12
f
10
14
28
28
17
28
28
0.61
081
0.62 0.63 0.64 165
166
0.70
0.67 088 089 170
161
0.61
082 163
083 084 0.65 086 0.67 0.68
0.70
C.69
0.70
1.8
D r 1.7
081
0.69
0.82 0.63 0.64 165
0.66 0.67 088 089
0.70
0.61
0.62
0.63
084
085
0.66 087
088
x
Figure 3.12
Computer simulated profiles showing the correlation between the fractal
dimension and roughness (from ref. 28).
66
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
i
Figure 3.13
The fractal dimension of a surface is a measure of its complexity, as
demonstrated by these computer simulated surfaces (from ref. 28).
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
different algorithms can yield results of different precision. Some of the commonly used
algorithms for calculating the fractal dimensions from rough profiles and surfaces are the
box counting method, the power spectrum method, the slit island method.
3.3.1 Box Counting Method
The box counting method, illustrated in Figure 3.14, is very visual, straight­
forward and simple to understand, even though it is more subject to errors than many
other algorithms. For a rough profile, a mesh of square cells of size e is laid over the
profile. It is obvious that the number of cells that the profile intercepts, M is a function of
the cell size, and the relationship between N and e is found to be16:
N x e ' 0'
(3.3)
where Dj is the fractal dimension. The definition holds for a rough surface, except that the
mesh of square cells is replaced by a stack of cubic boxes and e is the size of the box.
From Eqn. 3.3 it is seen that from a log-log plot (Richardson plot) of N as a function of
1/e, the slope of the curve will be the fractal dimension {Dj) of the rough object. For any
object, Dtop < D j< d , where d is the dimension of the embedding Euclidean space {d
equals to 2 for a profile and 3 for a surface), and Dtop is the topological dimension (which
is 1 for profiles and 2 for surfaces).
For example, if the profile is a straight line, the number o f cells it intercepts
increases linearly with the cell size, i.e.,
W oce' 1
(3.4)
Therefore, the profile has a dimension of 1, which equals to its topological dimension,
D.top . The fact that D,=
D,top indicates that the system
is not a fractal one, but Euclidean. A
]
J
system can only be fractal if Dj is different from Dlop, and the difference between the
68
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
££
I
ji
I
j
log(l/e)
--------------------------------------------------------------------
Figure 3.14
Illustration of box counting method for calculating the fractal dimension
of a rough profile.
69
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
fractal dimension and the topological dimension, or
- Dlop, is a measure of system
disorder.
3.3.2 Pow er Spectrum M ethod
Power spectrum analysis has long been used to study the roughness of surfaces
and profiles. For a profile f(x) its power spectrum is defined as follows:
(3.5)
where to is the spatial frequency. When P(g>) is plotted against to in the log-log plot, it is
often observed the existence of a linear region, which is an indication of fractal nature for
the objects.38-41 The decay rate of power spectrum when to increases, has been found
directly related with the fractal dimension of the object. Voss17 suggested the following
relationship between the PSD and Dj for self-affine rough objects:
P(tiJ) ocTU-^I-p,, where P = 7 - 2Df
(3.6)
Similar relationships are also reported by Mandelbrot et a/,18 Tricot,19 Majumdar and
Tien.20
Calculating the fractal dimension by power spectrum algorithm, however, has
several problems which will introduce errors. Firstly, in estimating the power spectrum at
a particular discrete frequency there is always a tendency for the leakage of power from
neighboring frequencies, which may cause errors in the value of the power at a discrete
frequency.21 Also the data points in the power spectrum plots are much scatterous and
rarely fall onto a straight line, which makes the evaluation of the fractal dimension
difficult. Secondly, only the fractal dimension of self-affine objects can be calculated
70
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
using power spectrum method.19 The fact that there are only a finite number of data
points makes it difficult to obtain a good fit.
3.3.3 V ariation M ethod
The algorithm we choose to use in this study is the variation method proposed by
Dubuc et al.2Ui It has been applied to many self-affine surfaces and profiles,23'27 and has
been proven to be one of the simplest yet most accurate algorithms for the fractal
dimension calculation.23
Like box-counting method, the variation method starts with the concept that the
fractal dimension is a measure of the space-filling ability of rough objects, and the
extracting of the fractal dimension from a digitized image (or profile) is accomplished by
calculating the volume the rough object fills as a function of "observation" scale.23,25
;
|
A digitized surface image consisted of N*N data points can be defined by
z,t = /(/,_/) where / and j are the lateral coordinates with i,j e [J, 2,..., N], and ztj is the
measured vertical height at point (/, j). If we use two flat, horizontal square "tiles" of size
2e x 2e and centered at point (/, j) to approach the surface from both above and below, the
smallest volume enclosed by the two tiles is the one when the top tile rests on the local
maximum and the bottom tile reaches the local minimum of the surface within the
covered region, i.e., the part of the surface./(/, m) where /-£ < / < /+e, j - e < m < y+e, as
^
illustrated in Figure 3.15. The upper and lower tiles are denoted as uc(i, j) and bz(i, j),
|
respectively. This can be expressed mathematically as follows:
|
ME(/,y) = max{/(/,m); / e [ / - s , / + e ] , m e [ / - e , y + e]}
1
(3.7)
6E(/,y) = min{/(/,m); / e [ t - e , / + e ] , m e [ j - z , y'+e]}
71
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Local Maximum
„ Local Minimum
i
i
ck
f
Figure 3.15
Illustration of the 2-D variation method (after ref. 28).
72
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
The difference, ut ( i,j) - b t (i,j) is called the e-oscillation, and the sum of e-oscillation
over the whole surface is called the e-variation:
^(e ) *
(3.8)
/»i
i
If all the upper tiles are connected they form a surface which is call the "upper blanket"
whose shape resembles the actual surface / / / ) . A similar "lower blanket" is formed by
connecting all the lower tiles. The E-variation is in fact the volume enclosed by the two
blankets in space, which is a reflection of the space-filling ability or complexity of the
surface / / / ) , and thus is related to its fractal dimension. It is obvious that the volume
enclosed by the blankets or the E-variation or the goodness that the blankets represent the
actual surface depends on the size of the tiles, or e . Changing
e
is therefore equivalent to
changing the observation scale. The smaller is the tile size, the better the upper and lower
blankets represent the actual surface//,/), and the smaller the volume or the E-variation,
as shown in Figure 3.16. When we plot
F (e )/e 3
(equivalent to the number of boxes
intercepted by the surface in the box-counting method) as a function of
1 /e
in the log-log
plot, if the curve shows linear behavior, then the slope of that linear region will be the
fractal dimension of the surface (see Figure 3.17).
The variation method is a robust and efficient algorithm for the calculation of
fractal dimension for both self-affine and self-similar objects. It is also proven to be more
accurate.
73
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
i
Figure 3.16
The resemblance of the upper (and lower) blankets to the actual surface
depends on the tile size (from ref. 28).
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
20
19
18
17
r*i
U
16
15
14
00
o
13
12
11
10
2
3
4
6
5
log[l/e]
(b)
Figure 3.17
The slope o f the linear region in the log-log (Richardson) plot (b) is the
fractal dimension o f the rough surface (a).
75
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
3.4
R eferen ces
1.
R.D. Young, J. Ward, and H.F. Scire, Rev. Sci. Instrum., 43,999 (1972).
2.
G. Binnig, H. Rohrer, Ch. Gerber, and E. Weibel, Appl. Phys. Lett., 40, 178
(1982).
3.
G. Binnig, H. Rohrer, Ch. Gerber, and E. Weibel, Phys. Rev. Lett., 49,57 (1982).
4.
G. Binnig and D.P.E. Smith, Rev. Sci. Instrum., 57,1688 (1986).
5.
G. Binnig, C.F. Quate, and Ch. Gerber, Phys. Rev. Lett., 56,930 (1986).
6.
C. Kittel, Introduction to Solid State Physics, 4th Edition, (John Wiley & Sons,
1971), Chapter 3.
7.
T. Goddenhenrich, H. Lemke, U. Hartmann, and C. Heiden, J. Vac. Sci. Technol.
A, 8, 383 (1990).
8.
Y. Martin, C.C. Williams, and H.K. Wickramasinghe, J. Appl. Phys., 61, 4723
(1987).
8.
G. Meyer and N.M. Amer, Appl. Phys. Lett., 53,1045 (1988).
9.
Y. Martin, C.C. Williams, and H.K. Wickramasinghe, J. Appl. Phys., 61, 4723
(1987).
10.
P.C.D. Hobbs, D.W. Abraham, and H.K. Wickramasinghe, Appl. Phys. Lett., 55.
2357 (1989).
11.
D. Sarid, D.A. lams, J.T. Ingle, V. Weissenberger, and J. Ploetz, J. Vac. Sci.
Technol. A, 8,378 (1990).
|
12.
C.B. Prater, P.G. Maivald, K.J. Kjoller, and M.G. Heaton, Tapping Mode
Imaging, Applications and Technology, Digital Instruments, Santa Barbara.
California.
13.
Artifacts in SPM, TopoMetrix Corporation, Santa Barbara, California (1993).
76
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
14.
B.B. Mandelbrot, Les Objets Fractals: Forme,
Hasard et Dimension,
(Flammarion, Paris, 1975).
15.
B.B. Mandelbrot, The Fractal Geometry o f Nature, (W.H. Freeman, New York,
1982).
16.
P. Pfeifer and M. Obert, The Fractal Approach to Heterogeneous Chemistry.
edited by D. Avnir, (John Wiley & Sons, 1989) p. 11.
17.
R. Voss, The Science o f Fractal Images, edited by O. Peitgen and D. Sampe,
(Springer-Verlag, Berlin, 1991), Chapter 1.
18.
B. Mandelbrot, D. Passoja, and A. Paullay, Nature, 308,721 (1984).
19.
C. Tricot, J. Chim. Phys., 8 5 ,379 (1988).
20.
A. Majumdar and C.L. Tien, Wear, 136,313 (1990).
21.
J.P. Carrejo, T. Thundat, L.A. Nagahara, S.M. Lindsay, and A. Majumdar, J. Vac.
Sci. Technol. B, 9,955(1991).
22.
B. Dubuc, J.F. Quiniou, C. Roques-Carmes, C. Tricot, and S.W. Zucker, Phys.
Rev. A, 39,1500(1989).
23.
B. Dubuc, S.W. Zucker, C. Tricot, J.F. Quiniou, and D. Wehbi, Proc. R. Soc.
Lond. A, 42 5 ,113(1989).
I
i
24.
S. Miller and R. Reifenberger, J. Vac. Sci. Technol. B, 1 0 ,1203 (1992).
25.
L. Spanos and E.A. Irene, J. Vac. Sci. Technol. A, 12,2646 (1994).
26.
L. Spanos, Q. Liu, E.A. Irene, T. Zettler, B. Homung, and J.J. Wortman, J. Vac.
j
j
Sci. Technol. A, 12,2653 (1994).
27.
!
Q. Liu, L. Spanos, C. Zhao, and E.A. Irene, J. Vac. Sci. Technol. A, 13, 1977
(1995).
28.
L. Spanos, Ph.D. Dissertation, University of North Carolina at Chapel Hill, 1994.
77
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
CHAPTER IV
ELLIPSOMETRY AND IMMERSION ELLIPSOMETRY
4.1
Introduction
Ellipsometry is a non-destructive, non-invasive, yet very sensitive optical
technique that has been widely used in surfaces and thin films studies for over 100 years
since Drude introduced the two ellipsometric measurables (¥ , A).1 Ellipsometry analysis
!
has two major components, i.e., experimental measurements and theoretical analysis.
Ellipsometry experiments measure the two ellipsometric parameters T and A, which are
the relative changes in amplitudes and phases of the two eigenpolarization states of the
monochromatic light reflected from (or transmitted through) a sample. These two
ellipsometric parameters are functions of the optical and structural properties of the
sample. The objective of the theoretical analysis is, therefore, to extract the desired
optical and/or structural information about the sample from the two ellipsometric
measurables. This is generally done by constructing an optical model that incorporates
i
both known and desired physical properties based on the previously available knowledge
i
about the sample, varying the unknown parameters in the model until the calculated T
|
and A from the theoretical model match their measured values, and finally making sure
I
that the best-fit model is physically acceptable.
i
The extreme sensitivity of ellipsometry measurements, up to submonolayer
changes on the surface or thin films, comes from the fact that it measures the relative
changes of polarization state (both phase and amplitude) of the reflected (or transmitted)
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
I
light. In microelectronics applications, ellipsometry (especially single wavelength
ellipsometry, or SWE) is particularly useful for measuring the thickness and/or optical
index of thin transparent films on reflective substrates, e.g. Si02 on Si, and for
performing real-time in situ process-monitoring and quality control.2,3 In many
applications, however, the samples being measured have much more complicated
structures that multilayer films must be used for the proper modeling of such structures,
in which case spectroscopic ellipsometry (SE) can be used to obtain much more
information about the measured sample than SWE, because ellipsometry measurement is
done at each wavelength over a wavelength range, and thus more experimental data are
available for the modeling.
In this chapter, we briefly describe the basic principles of ellipsometry, i.e., what
j
ellipsometry measures and how ellipsometric data is measured and analyzed. Unless
explicitly mentioned, all subsequent discussions consider only reflection ellipsometry
.
which is the more popular configuration for semiconductor applications.
4.2
i
f
Polarized Light
As mentioned above, ellipsometry measures the modification of polarization state
of a polarized monochromatic light by the optical system (i.e., sample) under study.
Polarization refers to the behavior with time of one o f the field vectors of a vector wave
(e.g. the electric field vector E or magnetic field vector B for the electromagnetic wave)
|
I
observed from a fixed point in space. Monochromatic light is, in general, a uniform
transverse-electric (TE) electromagnetic plane wave. It is either unpolarized, which
'
means that the end trace of its field vector is random and does not follow any pattern
when observed from a point in space, or elliptically polarized, which means that the end
of its field vector traces an ellipse when observed from a point in space. Circularly and
linearly polarized lights are two special cases of the elliptically polarized light.
79
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Since light is an electromagnetic wave, it must satisfy the Maxwell's equations:
v -e
=£
V-« = 0
'
Vx J? = p 0
(4.1)
dE'
^17
V x£ = - —
dt
where e0 and (iQare the permittivity constant and permeability constant of vacuum, and p
and J are the charge density and displacement current density, respectively. For a light
wave traveling in free space, p and / are all zero. From the Maxwell's equations, the
electric vector £ of a light wave of frequency ®, traveling in the positive-z direction in
free space (see Figure 4.1) can be solved and represented mathematically as follows:
Ex cos^w t ~
2n
z+8
I , 2lt z+8sA y
x + Ey co sj (o t
\
X
(4.2)
j
where x and y are the orthogonal unit vectors in the wave-front plane whose normal is
£, k is the wavelength, Ex, Ey, 8 and 8 ' are the amplitude and phase components o f E
along x and y directions, respectively. The magnetic field vector B, which is orthogonal
to the electric field vector, can also be obtained from Maxwell's equations. From
Maxwell's equations it is seen that E and B are not independent of each other, and thus,
only one of the two is needed to represent the electromagnetic wave. Conventionally the
electric field E is chosen as the vector to represent the polarization state of a
monochromatic, uniform TE plane wave.
If the wave is traveling in a medium of index o f refraction N instead of free space,
then the electric field vector becomes:
80
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
B (z , t)
Figure 4.1
A monochromatic uniform TE plane wave traveling in the positive-z
direction.
81
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
„
( 2 xN
1 .
=
(
2x N
.A
£.coslo)/— — z + o j jc+ Z^coslw/— — z + 8 ' l
(4.3)
where N = n - i k , n and k being the refractive index and the extinction coefficient
respectively. For vacuum, N= 1, and Eqn. 4.3 is identical to Eqn. 4.2.
Equation 4.2 can be rewritten in the form of:
/ , 2xN:\
/Id}/----E(z,t) = Re E(0) ,1 X )
where
riyM
E(0) = c lS,
\E
y e /
(4.4)
(4.5)
The vector E(0) as expressed in Eqn. 4.5 can therefore be used to represent the
polarization state of the plane wave without loosing any integrity since the phase factor in
the exponential term in Eqn. 4.4 is the same for both x and y components, which can be
added in when needed to reconstruct the full mathematical representation. The field
vector E can be further simplified as:
E=
(4.6)
where
(4.7)
E y = \ E y\e
Thus we see that a plane wave can be represented simply by a vector in the form of Eqn.
4.5 or Eqn. 4.7, which is called Jones vector, specifying only the amplitudes and phases
of the two orthogonal components in the wave plane. Since the Jones vector contains
complete information about the phases and amplitudes of the field vector E, it can be
used, therefore, as a representation of the polarization state of the wave. Figure 4.2 shows
82
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(a)
'
(t>)
E
mW
E
(C)
i
1
>
({)
(d)
1
J .
(?)
cosa *1
sina J
[
t
-©
M
Figure 4.2
f a + ib]
I c - id J
)
Jones matrices of some polarization states: (a) - (c) linearly polarized
states, (d) left-handed circular polarization state, (e) left-handed elliptical
polarization state (after ref. 1).
83
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
the Jones vectors of some typical polarization states, of which the elliptical polarization is
the most generalized polarization state. For circular and elliptical polarization, there are
two different types: left-handed and right-handed, depending on whether the electric field
vector E proceeds clockwise (right-handed) or counterclockwise (left-handed) when
looking into the beam. In Figure 4.2 only the left-handed circular and left-handed elliptic
polarization states are shown.
4.3
Basic Ellipsometry Systems
A basic ellipsometry system consists o f a light source, polarizer, compensator,
sample, analyzer, and photodetector, as illustrated in Figure 4.3. For some systems the
compensator is placed in between the sample and the analyzer, and sometimes can even
be omitted, as for the case of rotating analyzer ellipsometry systems. The optical elements
(polarizer, compensator, and analyzer) can be rotated either manually or by a computer
controlled automation system during the measurement in order to achieve desired
polarization states.
The light source can either be monochromatic or wide-band. A monochromatic
light source can be obtained by either using a laser or with a monochromator for other
wavelengths not available from laser (light of some specific wavelength is sometimes
preferred in some process monitoring applications4). A wide-band light source is usually
used for spectroscopic measurements. The polarization state of the light emitted from the
j
i
i
I
light source can either be unpolarized, partially polarized, or circularly/elliptically
polarized. After it passes through the linear polarizer, the light becomes linearly
polarized. A compensator can change the polarization state from linear into generally an
elliptical one. The combination of a linear polarizer and a compensator can provide all the
possible states of polarization. For a null ellipsometry system, for example, any
polarization state can be obtained by rotating the polarizer and compensator such that the
84
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Light
Source
Detector
Polarizer
Analyzer
Optional '
Compensator
'
Sample
Figure 4.3
Basic ellipsometry system arrangement.
85
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light becomes linearly polarized after reflection from the sample surface. The polarization
state of the light reflected from the sample surface can then be determined by finding the
null emanating from the analyzer, which is a linear polarizer. The photodetector, usually a
photomultiplying tube, is used to determine the amplitude (intensity) o f the reflected
light. In the following section, the details of ellipsometric measurement o f polarization
state changes will be presented.
4.4
M e a s u re m e n ts in E llip so m etry
4.4.1
Propagation of Light Through Optical Elements
Whenever light interacts with an optical element its polarization state changes. If
we disregard the details of interaction between the wave and the element and just
consider the polarization state of the wave before and after it interacts with the element,
as illustrated in Figure 4.4a, the effect of an element on the light wave can be represented
by a (2x2) transformation matrix which is characteristic to that element as follows:
E,= TE,
(4.8)
where Et and E0 are the field vectors of the impinging and out-going waves, respectively,
and the matrix
T=
%
U
V ^2I
^22
(4.9)
J
is called Jones matrix of the element. When the light passes a series of such elements as
seen in Figure 4.4b, the effect is that the vectors of the initial impinging wave and the
final exiting wave are related by a combined Jones matrix, Tcomb, as follows:
E0 - Ts Tn_, •••TltTlEi - TcambEi
86
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(4.10)
(a)
Tn
?
t
(b)
Figure 4.4
The effect of (a) one optical element, or (b) several such elements in series,
on the polarization states o f light can be represented by its (their)
characteristic Jones matrix (matrices).
87
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The Jones matrices of some most frequently encountered optical elements in ellipsometry
systems will be presented next.
a. Isotropic Medium
When a light of wavelength X propagates through an isotropic medium of
thickness d and refractive index n (see Figure 4.5a), the field is then transformed by a
Jones matrix as follows:
f e -i2 itn d /k
q
(4.H)
J
If the medium is isotropically absorbing, the effect can be represented simply by
replacing the refractive index n in the above matrix with the complex refractive index N
of the medium.
b. Linear Polarizer
A linear polarizer has two orthogonal axes, i.e., a transmission axis and an
extinction axis. When an unpolarized or elliptically polarized light passes through a linear
polarizer, the light is transformed into linearly polarized light with a field vector that is
parallel to the transmission axis o f the polarizer, as shown in Figure 4.5b. The effect of a
linear polarizer (of thickness d and index n) can be represented by the following Jones
matrix:
'tlitniifX,
88
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(4.12)
(a)
H
db -
e,
(b)
- H db -
E,
f
I
(C)
I
H
1►
dh -
t
Figure 4.5
Effects of (a) an isotropic medium, (b) a linear polarizer, and (c) a
compensator, on the polarization state o f a linearly polarized light.
89
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I
c. Compensator
A compensator is an anisotropic phase plate. It has a fast and a slow axis that are
orthogonal to each other. Waves travel at different speeds due to different refractive
indices along these two axes. Therefore, when light passes through a linear compensator
the phase of the component of the electric vector that is parallel to the slow axis is
retarded by 5Cand the amplitude attenuated by Tc with respect to the component parallel
to the fast axis, as illustrated in Figure 4.5c. The Jones matrix for a compensator can thus
be written as:
T = KC n
n .
0 PJ
Pc = Tce*‘
(4-13)
where Kc is a constant that accounts for the common attenuation and phase shift along
both the fast and slow axes of the compensator.
Knowing these optical elements, we now can examine how basic ellipsometry
systems measure the change of polarization states.
4.4.2
Null Ellipsometry
In a null ellipsometry system, the optical elements are arranged in sequence of
linear polarizer, compensator, sample, analyzer, in between the monochromatic light
source and the photodetector. This configuration is called the PCSA configuration and
illustrated in Figure 4.6a. The compensator can also be put in between the sample and
analyzer to form the so-called PSCA system which can be treated in the similar way to
the PCSA system to be discussed here.
From the previous section it is known that light exiting from a linear polarizer is
linearly polarized and can be represented by a Jones vector:
90
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Light
Source
Polarizer
Detector
Analyzer
Compensator
Sample
e
►x
(b)
Figure 4.6
(a) A PCSA null ellipsometer system, (b) The rotation o f reference system
for different optical elements.
91
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(4.14)
where Ac is the amplitude attenuation constant that determines the light intensity. The
subscript index PO refers to the polarizer (P) output (O), and the superscript te refers to
the transmission-extinction principal reference system of the polarizer. This output wave
is the input for the compensator. However, the fast-slow (fs) reference system o f the
compensator in general is not aligned with that of the polarizer. Hence a rotation of
coordinate system from te of polarizer to fs of the compensator is necessary, as shown in
Figure 4.6b. This can be accomplished by multiplying a coordinate rotation matrix R(a )
(where a is the angle of rotation from the old system to the new one) as:
cosa
sm a
sina
cosa
(4.15)
The input wave for the compensator after rotation is then:
(4.16)
where P and C are the azimuth angles of the polarizer and compensator, respectively,
defined by the angle of transmission axis (for the polarizer) or fast axis (for the
compensator) rotating counterclockwise from the positive-x axis of the laboratory
reference system of the sample. Then we can obtain the output of compensator by
applying the Jones matrix, T fs, for the compensator (Eqn. 4.13) to the input wave to
yield:
(4.17)
92
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It is proven from Eqn. 4.17 that light exiting the compensator is in general elliptically
polarized as opposed to the linear input before the compensator, and by changing the
azimuths angles of the polarizer and the compensator, all possible polarization states can
be obtained, as have been mentioned previously.
The output of the compensator then becomes the input to the sample surface,
which has an x-y reference system, so another rotation from the fs system of compensator
to the xy system of sample is needed:
E * = E £ = f?(-C )£*
(4.18)
The light will then interact with the sample under measurement, and at least part
of the light will be reflected back from the sample surface. When light is obliquely
reflected from the interface between two isotropic and non-gyrotropic media, the linear
polarizations parallel (p) and perpendicular (5) to the plane o f incidence (which is defined
by the beam of incidence and the surface normal of the optical system) are reflected with
their polarization unchanged.3 Therefore, if a reference system is chosen so that for both
input and output waves the x-axis is always in the plane o f incidence and y-axis is always
perpendicular to the plane of incidence, which actually align with the p and s directions
(as seen in Figure 4.6a), the effect o f the reflection from sample surface can be expressed
by a Jones matrix, R$ , as follows:5
(4.19)
I
Rp and Rs are the reflection coefficients for the p and s components of the light wave,
which are the ratios of overall reflected wave amplitudes to the incident wave amplitudes
in the p and j directions, respectively, and can be derived from the Fresnel equations with
93
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known angle of incidence and refractive indices of the media involved (see section 4.5 for
details).
The final optical element the light interacts with before reaching the photodetector
is the analyzer, which is also a linear polarizer and thus has a t-e reference frame. The
light reflected from the sample surface therefore needs to be rotated to this t-e reference
frame (from the x-y system of the sample) before considering the effects o f the element
on the light:
E " ^ E i,= K ( A ) E?’SO
‘*y
(4.20)
where A is the azimuth angle of the analyzer similarly defined in the same way as P for
the polarizer. The output wave from the analyzer can thus be obtained by applying the
Jones matrix that is characteristic o f an analyzer (which has the same form as that for a
polarizer):
(4.21)
in which KA is the attenuation constant.
To sum it up, the wave vector that reaches the photodetector can be expressed as
follows:
ETm = TZR(A)T?R(-C )T'’R (P -C )A c
.
\j /
=K
'
I v
(4.22)
,
where
K = K aKcK p
E, = Rp cos /fjcosC cos^ - C ) - p csinC sin(P-C )]
and
+/?, sin /l[sin A cos( P - C) + pc cosCsin(P - C)]
Finally, the intensity of the light collected by the photodetector is:
94
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(4.23)
(4.2:
(4.25)
which is a function of P, C, A, pe Rp and Rs. For a null ellipsometry system, usually a
quarter wave plate is chosen as the compensator, so pc is known (because Tc and 5C are
known from calibration). During the measurements, P, C, and A are arranged so that the
light intensity detected by the photodetector becomes zero (null), which means that E, =
0. With this condition, we finally obtain from Eqn. 4.24 the following relationship:
tan C + p ctan(/>- C )
Rg
p = — = -ta n /f
l - p etanCtan(/>-C )
R.
(4.26)
where p is called the complex reflection coefficient of the sample. The two complexamplitude reflection coefficients (Rp and Rs) can be written as:
rb =
k k 5'
p 1 p|
(4.27)
With Eqn. 4.27, the complex reflection coefficient can be written in the following form:
p = tan4VA
(4.28)
|R ,y
(4.29)
in which
A = 5 /, - 8 t
Therefore, it is seen from Eqns. 4.26-4.29 that null ellipsometry measures V
F and A which
are the relative changes in amplitudes and phases o f the two eigenpolarization states
when light is reflected from sample surface.
95
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4.4.3
Rotating Analyzer Ellipsomtry
In rotating analyzer ellipsometry (RAE), the configuration of optical elements
differs from that for null ellipsometry in that the analyzer is rotated synchronously, as
illustrated in Figure 4.7. In null ellipsometry the azimuth angles P, C, A are varied in
order to reduce the light intensity exiting the analyzer to zero, and the physical and
structural information about the sample lies in the values of P, C, A, the relative
retardation of compensator (5C), and the angle o f incidence <j>. In rotating analyzer
ellipsometry, however, the azimuth angles o f the polarizer and compensator are not
changed during the measurements. The intensity of the light exiting from the analyzer
rotating with frequency o is a function o f the instantaneous orientation o f the analyzer,
besides the fixed values of P, C, 8Cand <(). Because the null condition is not required in
rotating analyzer ellipsometry, the compensator is optional in the setup and often omitted.
For a PSA arrangement of a rotating analyzer ellipsomtry system, the electric field vector
of the light entering the photodetecting device can be expressed similarly to that for null
ellipsometry discussed in the previous section:
f\
EpuT ~ E*
0^ f cos A
,0 0, K-sinA
sin A ' ( Rp
<0 f cos P
sin F Y
cos A t l o
R ,l ^-sin F
cos P
(4.30)
where EQ is a constant. The azimuth angle of the analyzer is varying with time with a
frequency of co so
A =o) t+ 8
(4.31)
where 5 is a phase constant. Equation 4.30 can be simplified as:
Ep.MT
- E0(Rp cos AcosP + R, sin A sin P)
c o s/f
sin/I
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(4.32)
Light
Source
Polarizer
Detector
Analyzer
Compensator
Sample
y
CD t + 8
i
I
!
I
Figure 4.7
(a) A rotating analyzer ellipsometer system, (b) Rotation o f analyzer
azimuth angle A.
97
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Since the light intensity detected by PMT is / = \Enrr\2, by applying Eqns. 4.27 and 4.29,
this intensity can be expressed as:
( tan24*-ta n 2 P}
cos2 A +
2 tan 7*cos A tan ¥
or
7 = 70(l + a cos2 A + P sin2 A)
(4.34)
The constants l0, a and p can be resolved from a Fourier analysis of the detected signal,
and thus the two ellipsometric parameters 4* and A, where
B
- --
cosA =
(4.35)
can be obtained directly from the detected intensity.
c
Usually, the rotating analyzer ellipsometry system is automated so that data
acquisition and analysis can be done simultaneously with the help of a computer. This
automation is especially important for in situ real-time monitoring and for spectroscopic
measurements. Typical automation is illustrated in Figure 4.8, in which an optical angular
encoder synchronized with the rotating analyzer is added to digitize the signal and also
provide the start pulse at the beginning of each analyzer revolution. For each analyzer
revolution, the optical encoder generates one start pulse and 360 equally spaced data
pulses so that the analog-to-digital (A/D) converter digitizes the analog output signal
from the PMT into 360 data points at the trigger of the start pulse. The digitized signal is
then collected and stored by a computer which then performs the Fourier analysis to
calculate a , P, (and thus ¥ and A), and extracts physical and structural information about
the sample by model calculation. The computer also has a digital-to-analog (D/A)
98
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Mono­
chromater
Angular
encoder
PMT
To monochromater, PMT,
and electronic shutter
controllers
W
Encoder
outputs
Data pulses
(360 )
A/D
convener
converter
Start pulse (1)
Digital
data input
Figure 4.8
Schematic diagram o f rotating analyzer ellipsometer automation (after ref.
6).
99
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converter for issuing commands to control PMT voltage, the electronic shutter for
measurement of background noise, and also for controlling the monochromator if the
RAE is used in the spectroscopic mode.
4.4.4
Immersion Ellipsometry
Immersion ellipsometry is essentially a special form of spectroscopic ellipsometry
in which the SE measurements are carried out in a liquid instead of in air or vacuum
ambient. It was introduced with a purpose to increase the sensitivity of ellipsometry to the
interface of transparent-film-on-substrate systems by reducing the ambiguity involved in
the interpretation of ellipsometric data. As illustrated in Figure 4.9, when such a
|
i
r
transparent-film-on-substrate sample (e.g., Si0 2 on Si) is measured in air, light will be
reflected from both the air/fllm and film/substrate interfaces. If the point o f our interest is
the very narrow but non-zero film/substrate interfacial region covered by a thin
transparent overlayer film, isolating the optical response of that interface from the total
reflected light will be at least ambiguous, and the uncertainty in the determined
parameters will be high. This is caused by the progressive insensitivity of ellipsometry to
the film covered interface as half a period is approached.2 This uncertainty, however, can
i
'
be greatly reduced by carrying out the measurements in a liquid whose refractive index
j
closely matches that of the overlayer film in the energy range of the measurements, so
t
that the ambient/film interface is optically eliminated or its effects minimized. For
i
samples with an interface region that is optically distinct from the bulk film, as is the case
for Si0 2 film on Si substrate, measurements carried out in liquid ambient (such as CC14
for Si0 2 on Si substrate) has been proven to increase the sensitivity of ellipsometry to the
film/substrate interface by as much as an order of magnitude.7
100
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V
Liquid (Nq' w N j )
Film (N.)
SIE
Figure 4.9
Schematic comparison o f ellipsometry measurements carried out in air
ambient and in liquid ambient
101
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The immersion ellipsometry system used in this study is converted from a
conventional rotating analyzer spectroscopic ellipsometry system by adding a specially
designed cell to house the immersion liquid and sample under measurement, as illustrated
in Figure 4.10. While the two ports of the cell should have certain flexibility to ensure
change and precision adjustment of the angle o f incidence, in order to avoid light
dispersion care must be taken to make sure that the windows from which light enters and
exits the cell must be perpendicular to the light beam, and the liquid must be filled
properly. More discussion on this technique can be found in Chapter 6.
4.4.5 Alignment and Calibration
4
Because ellipsometry measures the relative changes in amplitude and phase of the
I
light reflected from a sample surface, it can have extremely high precision and sensitivity.
To attain such high precision, however, an ellipsometer system needs to be well aligned
and calibrated.
The purpose of aligning an ellipsometer system is to ensure that all the optical
components (such as polarizer, analyzer, filters, irises, etc.) are coaxial with the light
beam, and that the main optical axis of the system falls onto the plane of incidence (or
POI, defined by the incident beam and surface normal of sample) with correct angle of
incidence (AOI). Considering the rotating analyzer ellipsometry system for example,
typical alignment process8 consists of two major steps, i.e., the straight-through
1
j
alignment and the sample alignment as illustrated in Figure 4.11.
|
Firstly, the two arms of the ellipsometer are adjusted to a straight-through
'
position, i.e., at 90° AOI, with a He-Ne laser as the light source on one arm and the
photodetector on the other arm. All other optical components are removed. Using either
an iris or a special alignment target the laser source is adjusted so that the light beam is
exactly parallel to the optical bench rails, and falls on the center of the photodetector.
102
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Immersion cell
Figure 4.10
Illustration of our spectroscopic immersion ellipsometer system.
103
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Mono­
chromator
He-Ne
f t
# P
# A
I
i
I
i
-2 -
Photode“ ? “ e
system
Z jr —
t
fc»~
(a)
Photo­
detecting
system
Xe arc
lamp
(b)
Figure 4 . 11
Illustration o f ellipsometer system alignment process: (a) straight-through
alignment, and (b) sample alignment
104
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Fine adjustment can be made to maximize the output signal from the photodetector. The
light source and the photodetector then define the main optical axis o f the ellipsometer.
The remaining optical components are then placed onto the optical bench arms
sequentially. Adjustments are made accordingly to make sure that light does not deviate
from the defined main optical axis with the placement of each component.
The He-Ne laser source can then be replaced with the actual light source, which is
adjusted similarly to align with the system.
After all optical elements are placed onto the system, the two arms are then raised
to the desired angle (i.e., AOI or 4»), with a sample placed on the sample stage. The main
optical axis is now divided into two sections, one along the source-polarizer arm and the
other along the analyzer-detector arm. The two sections intercept at the principal axes of
x
j
the ellipsometry system and form an angle of 2(j>. Only when the principal axes of the
sample and the ellipsometry system coincide, the reflected light beam can follow the
optical axis of the analyzer-detector arm and fall onto the photodetector, as illustrated in
Figure 4.12. The orientation of the sample can be adjusted through the sample stage that
has three degrees of freedom as shown in Figure 4.11b. Sample alignment is done with
the assistance of an autocoilimating alignment telescope, from the eyepiece of which a
bright spot is centered when the sample is correctly aligned with the principal axes of the
i
ellipsometry system. To fine tune the sample alignment, the reflected signal (which is
collected by a data acquisition program) is analyzed to make sure that all unwanted
J
Fourier components are minimized, because if the sample is aligned perfectly, the light
[
flux detected by the photodetector should be a symmetric sinusoidal and show no
difference between the values of the successive maxima and minima.
After these steps, the ellipsometry system is considered well aligned. Usually the
straight-through alignment needs to be done only once unless one of the optical
105
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\ '
/\
Light
Source
Detector
Polarizer
Analyzer
Iris
Sample
rm
S;
/ \
Detector
Light
Source
Analyzer
Polarizer
Iris
Sample
Detector
Light
Source
Analyzer
Polarizer
Iris
Sample
i
Figure 4.12
Illustrations o f several cases o f sample misalignment.
106
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components is moved. However, sample alignment has to be performed every time a
sample is placed on the stage.
In order to accurately calculate the two ellipsometric measurables, Y and A,
accurate values of the azimuth angles of the optical components are needed. These angles
are defined by the primary axes of the optical components with respect to the POI, and
can be read from the scale dials of the components. However, these readings are not
sufficiently accurate and thus need to be corrected. In addition, the signal processing
circuit
introducesdelay and attenuation when the actual light flux is converted into
electric signal.Therefore, an optical calibration process is necessaryto determine these
offsets and the attenuation factor. In this study the residual calibration method introduced
by Aspnes and Studna9 is used.
From section 4.3.3 we learned that the light flux reaching the photodetector of a
RAE system can be expressed as follows:
/ = / 0(l+acos2v4 + Psin2^)
(4.36)
where I0 is the average intensity, A is the instantaneous azimuth of the rotating analyzer
measured with respect to a zero reference, a and P are the normalized Fourier coefficients
describing the phase and relative amplitude of the ac component of the flux incident on
the photodetector. As mentioned above, the conversion from the light flux to the electric
output introduces delay and attenuation. Let l/r| be the relative attenuation, and AF be the
offset of A which combines the delay and system error in analyzer reading, the measured
signal then can be written as:
/ = /„[l +r|a 2cos2(^ + AF)+r\b2sin2(^ + AF)]
(4.37)
where a2 and b2 are the second order Fourier coefficients o f the measured signal. The two
intensities should be equal, which leads to:
107
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
a =r|(a 2cos 2Ah- +b2sin2^A)
(4.38)
P =il(-fl 2sin 2 4? +b2cos2 AF)
If a residual function r(P) is defined as follows:
r =\-a \-b \
(4.39)
Using Eqn. 4.38 we will have the following:
/•(/») = l - r f 2( a 2+ P 2)
(4.40)
= ( l - t f 2)+TV 2R( P)
where
/?(/») = 1- a 2 - p
(4.41)
It has been shown10 that R(P) depends quadratically upon (P-Ps) for small values of (PPs), where P3 is the polarizer offset (the azimuth of polarizer with respect to the POI), and
R(P) reaches its minimum value R = 0 at P = Ps. Therefore, r(P) should also have a
minimum:
^i„ = l-T l 2
(4.42)
at the same P. Therefore, if a series of measurements are made with different azimuth
angles P around Ps, by fitting the data points {r, fV} (j = 1,2, ..., w) and finding the
minimum of the curve, we can obtain the polarizer offset Ps, as well as the attenuation q
from Eqn. 4.42. Also, if we define:
(4.43)
or
108
R e p r o d u c e d w ith p e r m is s io n o f th e co p y r ig h t o w n er . F u rth er r ep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Eqn. 4.36 can be rewritten as:
rC os(2^-0)
1+
(4.44)
•y/a2 + P
and similarly, Eqn. 4.37 can be rewritten as:
/ = /„ i +7 J L _ c « ( 2 ( ^ + ^ ) - e ’]
\° 2
(4.45)
+ b2
where
0 ' = tan' V
(4.46)
Ka 2 )
Since Eqns. 4.44 and 4.45 are equal, we have:
2 /1 -0 = 2(A + Af )-Q '
(4.47)
or —0 ' = A.. — 0
From the definition of a and P (Eqn. 4.33.), we have the following linear relationship
between the polarizer azimuth and 0’ in the vicinity of P :
1 .
= k ( P - Ps)+ A h
-ta n
2
\°2J
V
(4.48)
1
where it is a constant. Therefore, if we plot - ta n ~l{b2/a 2) (calculated from calibration
measurements) as a function of the polarizer azimuth P, and fit the data with a straight
line, the value on the line at the azimuth angle P = Ps will be the analyzer offset Ar as
illustrated in Figure 4.13.
109
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Ellipsometric Offsets Calibration
0.4
Au / 405nm
-28
-29.79 ♦ 0.1978P
0.3
0.04204 + 0.002473P + 0.003723P
a
►
-30
8
oe
0.1
-31
0.0
-8
-6
-4
-2
-0
2
4
6
-32
8
Polarizer Azimuth Angle, P (degree)
ia
i
t
lr
!
Figure 4.13
A typical plot of residual calibration o f ellipsometry offsets using a gold
standard.
110
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(1/2)tan \bJaJj
•29
4.5
Analysis of Ellipsometric Data
In the previous sections, such questions as what ellipsometry measures and how
ellipsometry measures are answered. Next we will see how the measured ellipsometric
data are analyzed and how physical information is extracted from the measured data to
solve practical problems. Therefore, instead of examining the initial and final polarization
states o f light before and after it interacts with an optical element or a series of elements,
our concentration in this section will be on the details o f light interaction with the optical
system under measurement, viz. the sample.
4.5.1
Reflection and Refraction at the Interface of Two Isotropic Media
When light propagates to the interface o f two isotropic media, part o f the light is
reflected back to the medium it comes from and part is refracted into the other medium as
seen in Figure 4.14. The reflection and refraction o f the light are governed by Maxwell's
equations, since light waves are electromagnetic waves. The wave vectors at both side of
the interface must satisfy the boundary conditions for Maxwell's equations. Assuming
that the media are isotropic absorbing and their optical indices are Nn and iVr
\
respectively, the incident, reflected, and refracted beams should all be in the plane of
incidence (POI), and the angular relationships should satisfy Snell's law as follows:
N0sm §0 = Ar, sin<f>,
(4.49)
where <t>0is the angle of incidence, <(>rthe angle of reflection, and <(>, the angle of refraction.
Satisfaction of the boundary conditions for Maxwell's equations also requires the
tangential components of E and H across the interface to be equal. If the reference system
is chosen as seen in Figure 4.14, i.e., p (parallel to the POI), s (perpendicular to the POI)
and the wave propagation direction form a right-handed Cartesian coordinate system, the
R e p r o d u c e d w ith p e r m is s io n o f th e co p y r ig h t o w n e r. F u rth er r ep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
1
Figure 4.14
Reflection and refraction o f light at the interface o f two isotropic media
follow Snell's law.
112
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following relations between the p and s components of electric vectors o f incidence (£,),
reflected (£,) and transmitted (£f) beams can be derived:
r -
E f p=
r _
c o s < j>o ~ W q C o M * !
iVt cos<t>0+ iV0cos<(»,
P E,p
E„^ iVoCOS<t>0- A/~, cos<{>,
£„ A^0cos(ji0+ N! cosij>|
(4.50)
, _
_
2Af0cos({i0
' Eip Af, cos^o + A^oCos^,
, - £ » -r
2A/0cos({>0
£„ Af0cos<{»o + Af, cos(j>,
Equations 4.S0 are called the Fresnel equations, and
fp, /s called Fresnel reflection
and transmission coefficients for the p and s polarizations, respectively. Generally these
Fresnel coefficients are complex and thus can also be written as:
t p =!/I p \e'&
\
rp= |I rp |\ A
(4.51)
r,= h \e'K
', = k , k 5“
Defining
f
p s -£ -s ta n 4 V A
r.
(4.52)
we get:
tanT = ?4
(4.53)
A = 8 rp- 8
rt
113
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Equation 4.53 demonstrates that tan¥ represents the relative amplitude change and A the
relative phase change of the two components of the reflected light.
4.5.2
Light Reflection from an Ambient-film-substrate System
So far ellipsometry has been mostly applied to the so-called ambient-film-
substrate structure, to measure the thickness and optical index of the overlayer film. What
differentiates this system with the one in the previous section is the multiple reflection
between the two interfaces, as illustrated in Figure 4.15. Each time light hits any of the
two interfaces, reflection and refraction occur. The final wave reflected from the system
is thus the sum of the initial zero-order reflection and all the subsequent partially reflected
waves that are refracted back into the ambient. For both p and s polarizations, the total
reflected amplitude R can be expressed as follows:
(4.54)
where rtj and r are the Fresnel reflection and transmission coefficients for the interface
between medium i and j (i,j = 0, 1,2), and
(4.55)
represents the common phase retardation due to light traveling in the isotropic film during
the multiple reflection process. The total reflected amplitudes Ro and Rs are generally
complex, and thus can be written as follows:
114
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Ambient (N0)
Substrate (N2)
Figure 4.15
\
B
4ft
Light reflection and transmission on an ambient-film-substrate sample
system.
iI
115
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We can then define the complex reflection coefficient p as:
Ra
p s — s ta n T e
R.
(4.57)
which leads to
tanT =
(4.58)
Once again we see that the two ellipsometric parameters OF and A) measure the relative
changes of polarizations in p and s components of light.
4.5.3
Isotropic Stratified Planar Structure
A more general sample system is a series of m isotropic planar films on a
reflective substrate, as shown in Figure 4.16. The film thickness for the ilh layer (/ = 1....
, m ) is dp and its refractive index is Nr The ambient and substrate are semi-infinite, and
their refractive indices are N0 and Nm+I, respectively.
When a monochromatic light is shone on the system from the ambient, part of the
light will be transmitted into the successive layers of films, and at each interface between
two media, part of the light will be reflected. While in the ambient and inside the i'h layer
there is a forward-traveling plane wave which is denoted by a "+" sign, and a backwardtraveling plane wave, denoted by a
sign, in the substrate there is only one forward-
traveling plane wave. At an arbitrary plane z, the total electric field can therefore be
represented by a Jones vector as follows:
116
w ith p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
0 (ambient)
1
2
m
m+1 (substrate)
Figure 4.16
Reflection and transmission o f light on a stratified planar films sample
system.
117
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I
(4.59)
with the two elements representing the forward- and backward-traveling plane wave,
respectively, and at two different z planes the field vectors can be transformed by a (2x2)
Jones matrix as has been described previously. By choosing the two planes appropriately
we can study the behavior of light at the interface between two media, inside a layer, and
at both ends of the film stack.
If we designate
as the Jones matrix characteristic of the interface between the
(i-l),h and i'h layer, and L t as the Jones matrix characteristic of the i,h layer, then the total
field in the ambient before the light enters the first layer, Ea, and the field in the substrate
immediately after it exits the last film, Es, can be related as follows by a combined matrix
S which represents the overall reflection and transmission properties of the system:
(4.60)
where
(4.61)
V^l
22/
From Eqns. 4.60 and 4.61 the reflection coefficient R can be written as:
(4.62)
from which we can get the complex coefficient of reflection p as follows:
p = RP = S2IP S||.t
118
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(4.63)
>
It is therefore clear that if we can find out the Jones matrix S, we can calculate the
complex coefficient and compare that with the measured value.
Equation 4.61 shows that in order to determine S, it is necessary to determine the
characteristic Jones matrix for each interface and layer.
Light reflection and refraction at the interface between media a and b can be
represented as follows:
I \
'11 '12
( I
'
e
; '
(4.64)
U 'J
<E a j
From the definitions in the Fresnel relationships (see Eqn. 4.50),
E: =rahE l
i f E* incident (Figure 4.17a)
e;
e; «
o
(4.65)
E l = rbuEb
E„ = thaE/,
e := o
i f Eb incident (Figure 4.176)
and also the fact that rha = rab and t^ = ( l -r^,)ftab, we can obtain the Jones matrix
representing the interface:
r l V i
J VC*
ah
1
(4.66)
The effects of an isotropic layer of thickness d and refractive index N on light can
be simply described as:
119
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(a)
(b)
i
z
=
0
z = d
(c)
Figure 4.17
Reflection and transmission of light at the interface o f two arbitrary media
a and b: (a) when the light is propagating from medium a to medium b, (b)
when the light is propagating from medium b to medium a. (c) Reflection
and transmission of light in an isotropic film of thickness d.
120
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
where E0 and Ed are the field vectors at two ends of the layer and
p = Z l ^ C0S(jl
(4.68)
with <(>representing the angle between the direction of propagation in the layer and the r
direction, as shown in Figure 4.17c.
With / and L known for each interface and layer, it is straightforward to construct
the overall effect Jones matrix S using Eqn. 4.61 for specific systems.
a.
Ambient - substrate structure
The simplest structure is a bare substrate. For such a system, the Jones matrix S =
From Eqns. 4.62 and 4.66 we have 5,, = 1/tQl, Sv =r01//0, , and R = r01. Therefore,
the complex reflection coefficient for a bare substrate is simply:
P = 'b i,/roi*-
I
|
b.
Ambient - film - substrate structure
For the commonly encountered ambient-film-substrate system, the Jones matrix S
|
1
(4-69)
=
2, where subscripts 0, 1, 2 represent ambient, film, and substrate, respectively.
By substituting Eqns. 4.66 and 4.67 we have:
121
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
and therefore,
-<2P
£ — roi ^ rne
11 ^+ ror
r r12*
(4.71)
and
_
R po _
p
a j
r 01
0 ! pO + r ii 2VOe " 2 P
l + r0, / | 2je
-<2P
(4.72)
1+ rr01pri
r 2pg”1'^ 'ro i l +/*
g”1’^
M 2.i
1 ^ '0 1 p M 2 p e
where
2rc JV,*/,
(4.73)
P = ------------— -C O S(j>|
with Mv d v <j>, being the index o f refraction, thickness and angle o f refraction of the film,
respectively.
c.
ambient - film 1 - film 2 - substrate system
For a two-film system, the overall Jones matrix
S - h\L\I\lLnJ.23
1 rn.
V ^01^12^23 . V r oi
0
r i
-<p ,
Vri2
° v i
(4.74)
'2 3
Jvu
and,
n
-$21 _ ( r° l + r »2g',2B' ) + ( ^ l2 -t- g ' ,2Pl>23g~,2Bl
SU
( l+ '- 0 .^ ''2P,) + (ri2 + ^ ' ' 2P,)r23^'2Pl
122
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(4.75)
I
and from which we can get p straightforward, as a function of wavelength o f the incident
beam (>.), angle of incidence (<|>0), thickness and indices of refraction for each film (</,, N v
d2, N2, a n d refractive index of the substrate (N$), i.e.,
p = p (X ^ 0,dl,N l,ti2,N 2, - , K )
(4.76)
More complex structures can be constructed similarly by multiplying in sequence
the characteristic Jones matrices.
4.6
Heterogeneous Systems and Effective Medium Approximation
In the previous section it is shown that for homogeneous and isotropic multilayer
systems, the theoretical values of 4* and A can be calculated directly based on the optical
models constructed. However, if a system contains heterogeneous materials as most
applications do have at least some form o f heterogeneity in the microscopic scale, i.e., the
films or substrates are composed o f mixtures of materials with different dielectric
responses, direct model calculation as discussed above cannot be applied because the
refractive indices are not known for such heterogeneous materials. Several effective
medium approximation (EMA) theories11 have been developed to deal with this problem
by linking the compositional and structural information of composite layers with the
dielectric response of each constituent so that the heterogeneous layers can be treated as
homogenous with an effective dielectric function (pseudo-dielectric function). The same
procedure discussed in the previous section can then be applied to extract physical and
structural information about the samples containing heterogeneous structures.
When a light wave interacts with a dielectric medium, this external
electromagnetic field will cause redistribution of charges within the medium, and thus
induce dipoles and cause electric polarization in the medium which will contribute to the
total internal electric field. The dipole moments will oscillate in the same frequency as
123
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the external electromagnetic wave, and thus cause the dielectric dispersion of the
medium. Therefore, the macroscopic dielectric function e is closely related to the
microstructures of the medium. By using a simple model of cubic lattice with polarizable
atoms sitting at each lattice points (representing the case of polarizable points placed
uniformly in vacuum), Clausius-Mossotti derived the relationship as given by equation:
6 “ 1 471
— - = — na.
e +2
3
(4.77)
where a is the atomic polarizability and n the number of polarizable points, i.e., atoms,
per unit volume.12 If the medium is not homogenous, but a uniform mixture of two
atomic species, a and b, with different atomic poiarizabilities, a a and a b, then atoms from
both species will contribute to the electric polarization and thus Clausius-Mossotti
equation can be extended as follows:
6 - 1
4 * /
—
r = -z-{nao.a
+nho.i \)
8+ 2
3
, A - 7 C>\
(4.78)
to accommodate such mixtures. By applying the Clausius-Mossotti equation to each of
the constituents of known dielectric functions, Eqn. 4.78 becomes the Lorentz-Lorenz
effective medium expression12,13:
I
|i
8 +2
e a +2
eb +2
(4.79)
I
i
\
f
f
where f a = nj(na+nb) and f b = (1 - f a) are the volume fractions of species a and b.
respectively.
Usually, the two species are not mixed atomically but form regions large enough
to have their own dielectric identities. In this case the polarizable species are not points
124
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
embedded in vacuum anymore, but some large enough regions in a host medium of
dielectric function of eh, for which the Lorentz-Lorenz expression can be modified
accordingly as:
^
e
=
+ 2
e
(4-80)
e a +2eh
*
e A + 2 e A
There are two cases that are most common in practical applications. The first one is that
one species is dilutely mixed within the other, with the latter naturally becomes the host
medium. For such cases of dilute solutions, Eqn. 4.80 becomes:
- - - - = f h— —
b + 2 e „
e* + 2e a
when
e
=
e h
or
(4.81)
e-g* =
_ fa
f gq-g*
g + 2
e a
e a + 2
w h en zb =z h
e a
which is the Maxwell-Gamett effective medium approximation. The second case is that
when the two species are comparably mixed so neither one can act as the host medium. In
this case, the best choice is to have the effective medium itself as the host, and thus Eqn.
4.80 becomes:
I
0 = / q - ^ + / * - £ if ^
e
0 + 2
e
e a
+ 2
(4.82)
e
I
i
[
which is called the Bruggeman effective medium approximation or BEMA.14
|
i
i
which is to effectively represent microroughness of semiconductor surfaces or interfaces.
i
BEMA has been successfully used in many microelectronics applications, one
o f
where the scale of microstructures is small compared to the wavelength of probing
light.15 BEMA is also the basis of this study when applying spectroscopic immersion
ellipsometry to study the Si/Si0 2 interface roughness during oxidation processes.
125
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
4.7
Model Optimization
Thus far we have demonstrated what and how ellipsometry measures, and also the
way to construct optical models and perform theoretical calculations, if we have previous
information about the sample from other sources. Through the comparison of model
calculations with highly accurate ellipsometric measurements, optical or other physical
properties about the system under study can be obtained, either through p as defined by
Eqn. 4.28 or through the dielectric functions e (or pseudodieiectric functions <e>) as
defined by:
(1-
e
V
or < e > = sin~ <J>0+sin 2<j)0 tan' (j>0 [ — U +p;
(4.83)
or directly through the two ellipsometric parameters, 4* and A, which is what we will do
in this work. This procedure, called model optimization, is usually performed using some
standard algorithm to minimize a merit function. In our case, the Levenberg-Marquardt
method16will be used to minimize the following merit function:
(4.84)
/=i
where £/ is the photon energy, P is a vector of unknown parameters, superscripts cal and
exp correspond to the calculated and experimental (values o f 4* and A), respectively.
In summary, ellipsometry analysis can be performed to obtain physical and
structural information about the sample under measurement with a general procedure as
follows:
•
experimentally measure the two ellipsometric parameters 4/ and A at either a
single wavelength (SWE) or over a spectral range (SE).
126
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• construct an optical model based on the previously available information
about the sample, with the desired physical and structural information
incorporated in the model as unknown variables.
•
give an initial trial value to each unknown parameter and calculate the
theoretical values of T and A (or p or
<e> )
from the optical model, compare
the calculated ¥ and A (or p or < e > ) with the experimental data by evaluating
a merit function (such as Eqn. 4.84).
• use one o f the optimization algorithms to vary the unknown variables, and
find the model that best fits the experimental data set, i.e., when the merit
function is minimized.
ti
i
•
verify that the values of the unknown variables o f this best-fit model are
physically acceptable.
Following this general procedure, samples with complex multilayer film structures can be
analyzed for desired physical and structural information.
127
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
4.8
R eferen ces
1.
R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, (NorthHolland - Elsevier Science, Amsterdam, 1987), p. 1.
2.
E.A. Irene, Thin Solid Films, 233,96 (1993).
3.
E.A. Irene and J.A. Woollam, Mater. Res. Soc. Bulletin, 20,24 (1995).
4.
J. Joseph, Y.Z. Hu, and E.A. Irene, J. Vac. Sci. Technol. B, 10,611 (1992).
5.
R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, (NorthHolland - Elsevier Science, Amsterdam, 1987), p. 78.
1
6.
P.S. Hauge and F.H. Dill, IBM J. Res. Devel., 17,472 (1973).
7.
V.A. Yakovlev and E.A. Irene, J. Electrochem. Soc., 139,1450 (1992).
8.
Rudolph Research, Alignment Instructions, Rudolph P/N A8901, Rev. 2.1.
9.
D.E. Aspnes and A.A. Studna, Appl. Opt., 14,220 (1975).
10.
D.E. Aspnes, J. Opt. Sco. Am., 6 4 ,812 (1974).
11.
D.E. Aspnes, Thin Solid Films, 8 9 ,249 (1982).
12.
L. Lorenz, Ann. Phys. Chem. (Leipzig), 11,70 (1880).
13.
H.A. Lorentz, Theory o f Electrons, 2nd edtion, (Teubner, Leipzig, 1916).
14.
D.A.G. Bruggeman, Ann. Phys. (Liepzig), 2 4 ,636 (1935).
15.
D.E. Aspnes, J.B. Theeten and F. Hottier, Phys. Rev. B, 2 0,3292 (1979).
16.
W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical
Recipe in C, The Art o f Scientific Computing, 2nd edition, (Cambridge University
Press, 1992), p. 683.
128
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CHAPTER V
OXIDATION OF SILICON
5.1
Introduction
Being able to form high quality oxide thin films on Si for the purpose of surface
passivation and as a gate dielectric has relegated Si to be the most important
semiconductor material. Formation o f an oxide layer can reduce the surface states on
single crystalline Si from 1015/cm3 down to 1010/cm3,1,2 which is essential for the use of
Si in microelectronics devices. As an insulating material, Si0 2 can sustain a high electric
field which is necessary for MOSFETs to function properly. Si0 2 can also be used in
areas such as masking for dry etching and implantation, device isolation, tunneling oxide,
and planarization. The highly compatible and stable Si02-Si system can be formed in a
variety of ways. In this chapter, we are going to describe two important processes that
have been used to grow high quality oxide on Si surfaces, i.e., thermal oxidation and
microwave ECR plasma oxidation processes.
5.2
Thermal Oxidation of Si
5.2.1 T herm al O xidation M echanisms a n d Kinetics
When freshly prepared Si surface is exposed to an oxidizing ambient such as 0 2
or H20, a film of Si0 2 is formed by the following chemical reactions:
Si (s) + 0 2(g ) -> Si02(s)
(5.1)
Si{s)+ 2H ,0{g)-> S/O, (s)+2H, (g)
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
This process is called oxidation process, as opposed to other ways of producing oxide
such as by vapor deposition and anodization. Oxidation of Si can happen at room
temperature in atmosphere, but to grow high quality oxide o f controllable thickness that
is needed for microelectronics applications, the process is usually carried out in a furnace
at elevated temperature of 800-1200°C. Thermal oxidation process, as such a process
being called, can produce the highest quality oxide that can be used in the most stringent
applications such as the gate dielectric film in the MOSFET devices.
It has been well demonstrated3*9 that during thermal oxidation the oxidant (e.g. 0 2
or H20 ) diffuses through the growing oxide and then reacts with Si at the Si/Si0 2
interface. The overall growth rate of Si0 2 is controlled by the slowest process. Numerous
experimental data of thermal oxidation show that the thickness versus oxidation time for
oxide film growth follows a linear-parabolic law for oxide thickness greater than several
tens of nanometers, i.e., at the start of the oxidation, the oxide grows linearly, but as the
oxide grows thicker, the thickness increases parabolically with time. Deal and Grove10
proposed a model that employs a steady state analysis of the physical and chemical
processes involved during the oxidation to explain the kinetics of thermal oxidation
process. The Deal-Grove model shows excellent agreement with the experimental data
j
over a broad range of temperatures (700~1200°C) and pressures (0.1-1 atm) in both dry
and wet ambients.11'18 Figure 5.1 illustrates these two processes (diffusion and reaction)
^
t
|
in the Deal-Grove model in terms of two fluxes in the oxidation process.
|
is driven by the concentration gradient of 0 2 in the growing oxide. The diffusion follows
The first flux, F v describes the diffusion o f 0 2 through the growing oxide, which
Fick’s first law:
Fx= - D ^ ax
130
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(5.2)
o 2
Si
Substrate
c ,
Gas
F,
Figure 5. 1
Illustration of oxygen fluxes in thermal oxidation process.
131
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where D is the diffusion coefficient and C is the concentration of oxidant. In a steady
state, the concentration gradient
— =
dx
(5.3)
x
where C, is the solubility of 0 2 in Si02, C2 is its concentration at the Si/Si0 2 interface,
and x is the thickness of the growing oxide.
The second flux, F2, depicts the reaction of oxidants and Si atoms at the Si/Si0 2
interface. Because of the abundance of available Si atoms at the interface, the reaction is
limited by the number of arriving oxidant species, and thus flux F2 is proportional to the
concentration of 0 2at the interface, i.e.:
F2 =k,C2
(5.4)
where ks is the kinetic rate constant for the reaction. Under steady state conditions the two
fluxes are equal at all times:
fJ = F2 = F = f i ^ dt
(5.5)
where Q is a constant that converts the growth rate of oxide to oxidant flux. Solve the
Eqns. 5.2-5.5 with the boundary condition that at time t0, the thickness of oxide is .t0, we
get:
x 2-x„
'" ° =i
r
x -x 0
+i
r
(
1
where kp (= 2DC,/£2) and k, (= A^C/Q) are the parabolic and linear rate constants,
respectively. The above derivation has a small deviation from the original Deal-Grove
132
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I
model in that the flux from gaseous oxygen through the 0 2/S i0 2 interface is omitted
because it is very rapid compared to the other two oxygen fluxes and thus has no
significant impact on the oxidation kinetics.2
Using the Deal-Grove model, the linear-parabolic behavior of thermal oxidation
can be explained in the following way. At the early stage of oxidation when there is no or
little oxide on the Si surface, there is sufficient oxidant available at the interface for
reaction, and therefore the oxide growth at this stage is only limited by the rate of
chemical reaction which is assumed to be linear. After the oxide grows beyond a certain
thickness, the oxidant species have to penetrate the oxide barrier and reach the interface
where the available oxidant is no longer sufficient to achieve the maximum reaction rate.
The oxidation becomes diffusion limited and shows predominantly the parabolic growth
:
i
behavior.
Even though the Deal-Grove model has been successful for the basic aspects of
thermal oxidation of Si, it does not explain the initial regimes o f oxidation as well as
other factors that have been found to affect the oxidation kinetics such as film stress and
substrate orientation, which leads to the development of many new models and revisions
1
to the original Deal-Grove model.
The boundary condition, i.e., x = x0 at t = t0, sets the lower limit upon which the
!
Deal-Grove model stands. However, current technology requires gate oxides below the
,
range that the Deal-Grove model covers, namely less than 10 nm.
|
Experimental studies10'12,18,20"22 show that this early stage o f oxidation (for oxide
[
i
!
I
thinner than 25 nm) can be divided into two regimes: (a) the very initial stage in which
the oxide grows rapidly from 0 to ~1 nm; and (b) from 1 nm to several tens o f nanometers
in which the oxide growth follows a linear-parabolic law similar to that for the thicker
oxide but with a rate higher than predicted from the Deal-Grove model. Based on their
extensive experimental data, Massoud et al developed a model21,22 in which the overall
133
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i
oxide growth rate is a combination of the Deal-Grove growth rate and two enhancement
factors that decay exponentially with oxide thickness. The enhancement is believed
related to surface impurities. Many other models2,19have also been proposed in attempt to
explain the enhanced oxidation rate in the thin film regime, such as models based on
structural defects (e.g. micro-pores or micro-channels) in oxide, on stress, on the
formation of an interface layer, on electrochemical reactions, etc., yet none of these
models successfully explains all aspects of the experimental dependence on process
conditions. The new findings and models help us gain more understanding of the initial
regime of the thermal oxidation process, nevertheless, many mechanistic details remain
undiscovered.
5.2.2 T herm al Oxidation System and Procedure
To grow high quality oxide for microelectronics devices, the Si surface has to be
extremely clean, because contaminants such as organic residue and metal impurities can
greatly affect the surface states, oxidation behavior, and thus the electric properties of the
device. Therefore, silicon wafers out of the box need to be cleaned to remove these
organic and inorganic contaminations before oxidation, usually by a slightly modified
RCA procedure23 which essentially consists o f a 5 min basic bath step and a 5 min acid
cleaning step, both performed at ~70°C with ultrasonic agitation. The basic cleaning step
is in NfyOH-HjOj-l^O (1:1:5) which removes the organic contamination on wafer
*
(
surface by oxidizing and dissolving the contaminants. The acid cleaning is in HC1-H,0-,-
;
H20 (1:1:5) which removes the metallic contamination on the wafer surface. The RCA
]
cleaning is usually followed by a brief HF dip to remove any native oxide and oxide
1
r.
grown from the previous RCA steps. Before each step the wafers are rinsed at room
temperature in deionized (DI) water of resistance 16 MQ or higher. After the HF dip and
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the final DI water rinse, the wafers are dried using dry N2, loaded vertically onto a fused
silica boat, and inserted into a furnace for oxidation.
As we have mentioned above, thermal oxidation is usually carried out at elevated
temperatures of 800~1200°C in a resistance heated furnace. The thermal oxidation
furnace, like the wafers, has to be extremely clean and free of particles in order to grow
high quality oxide. Therefore, the furnace is usually made of fused silica and has a
double-wall structure as illustrated in Figure 5.2. The use of fused silica for the tube and
all other handling tools during the oxidation process ensures high temperature durability
and low contamination. The double-wall tube structure with gas flushing in between
prevents the diffusion of harmful impurities such as Na and
into the oxidation
environment. Before oxidation, the furnace needs to be steam ("bubble") cleaned. The
cleaning is done by first flowing dry N2 (or other gases such as 0 2 and Ar) through the
furnace via a bubbler containing filtered DI water for 24 hours and then shutting down
the bubbler and flowing only dry N2 through the furnace for another 48 hours. During the
cleaning process the furnace temperature is set at 1000°C. In addition to the cleaning of
furnace, all the wafer handling tools, including the beakers (for RCA cleaning), the
tweezers, the wafer boat, the push rod, and the end-cap of the furnace, have to be cleaned
using the so-called "white etch" solution, which is a mixture of HN0 3 (80%) and HF
(20%), followed by a thorough rinse in filtered DI water.
Generally ultra-pure dry 0 2 is preferred as ambient over H20 steam for better
control and higher oxide quality. It is established24 that even trace amount of H ,0 will
alter the oxidation rate. Therefore, before being inserted into the furnace which is at set
oxidation temperature, the wafers and the boat are usually left in the end cap for -10 min
with dry N2 as the purging gas flowing through the furnace. This can also prevent wafer
warpage that is due to the drastic temperature change. After oxidation for cooling down,
135
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Flush
Gas
Outlet
Resistance Heating
End Cap
Wafers
Wafer Boat
Figure 5.2
A resistance heated, double-wall fused silica tube furnace for thermal
oxidation (after ref. 2).
i
i
136
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the boat and wafers are also left in the end cap for -10 min with N2 purging before being
removed from the furnace.
5.3
M ic ro w a v e E C R P la sm a O x id a tio n
5.3.1 Introduction to Plasma
Thermal oxidation produces the highest quality of Si0 2 films. However, because
thermal oxidation is usually carried out at high temperatures, it can cause problems that
may prevent its use in some applications. One such problem is that high temperatures can
produce defects such as stacking faults in the silicon wafers. It may also change the
doping profile and thus alter the designed properties of devices. Wafer warping at high
temperature is also a concern. To reduce such problems, low temperature oxidation
techniques were developed, one of which is plasma oxidation. Besides low oxidation
temperature, plasma oxidation is usually carried out in vacuum and thus enable system
integration and reduce wafer contamination from the environment.
Plasma, sometimes called glow discharge, is a term that is used to describe a
partially ionized gas which contains equal amount of positive and negative charges,
together with neutral particles. To generate a plasma, one of the many power coupling
schemes needs to be used to energize existing free electrons which might be ionized
initially from gas molecules, for example, by high energy cosmic rays. The energized
electrons then collide with gas molecules. When the electron-molecule collision is
inelastic, energy is transferred from the electron to the molecule. If the transferred energy
is less then the ionization energy of the valence electron of the molecule, then the
molecule is simply excited to a higher energy level from which it will later relax back to
the ground state by emitting visible light, i.e., the so-called "glow discharge". When
enough energy is transferred from the electron during the collision, however, valence
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electron will be ionized and the newly freed electron will also participate in the ionization
process. The cascading of the ionization process will sustain the glow discharge.
The generation of electron-ion pairs in the plasma is not unrestricted. As their
densities increase, so will the probability of electron-ion recombinations. In addition,
electrons and ions in the plasma may also be lost by drift and diffusion to the electrodes
and chamber walls. Eventually the generation and loss of electrons and ions will reach a
dynamic equilibrium.
In addition to the generation and loss o f electrons and ions, there are several other
processes occurring in plasma. One such process is the dissociation of gas molecules,
mainly by electron impact. This is an important process for the purpose of technological
applications, because often the atoms and radicals from the dissociation of molecules are
more chemically reactive than the original gas molecules.
Because of the abundance of ionic and chemically active neutral species in the
plasma from molecules ionization and dissociation, plasma can be used in many different
processing steps such as etching, deposition and oxidation, simply by introducing
different gases into the plasma chamber and setting up the corresponding working
conditions such as pressure, temperature, and bias. Plasma applications are designed to
make use o f either the ionic or the neutral atomic and radical species, or the combination
of both for the processings. In other words, plasma can be used as a source of ions, or
reactive neutral species such as atoms and free radicals, or both. For example, sputter
etching and ion milling are based mainly on the physical sputtering using the ionic
species in the plasma, and plasma etching and plasma oxidation depend mostly on the
neutral atomic species and radicals.
Many types of plasma sources have been developed. Based on the plasma density
(i.e., the density of electrons or positive ions) these sources can be generally divided into
two groups, i.e., low density sources such as DC and radio-frequency (RF) sources, and
138
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high density sources that include ECR plasma, helicon plasma, and inductively coupled
plasma or ICP (sometimes called transformer coupled plasma or TCP) sources.
Different applications have different requirements for the plasma conditions, but
usually high density plasma sources are preferred because high density plasma system
can be used to achieve high yield and high aspect ratio (height:width) at a low pressure
for devices of smaller dimensions.
5.3.2 G eneration o f EC R Plasm a
For a typical ECR system, a 2.45 GHz microwave generated by a magnetron
source is guided through the microwave circuit to a coupler. The choice of microwave
frequency (2.45 GHz) is because of the inexpensive and readily available magnetron
sources that are widely used for domestic microwave applications. The microwave
propagates through the quartz window to the vacuum plasma source chamber which is
filled with the selected gas. The electromagnets around the plasma source chamber
generate a nonuniform static magnetic field inside the chamber. Electrons moving in the
magnetic field inside the plasma source chamber will gyrate (see Figure 5.3) under the
influence of Lorentz force (F = -ev*B) with a frequency:
where me is the mass of electron. In a magnetic field of B = 875 Gauss, the cyclotron
frequency of electrons, / = — = 2.45 GHz, resonates with the input microwave
2n
frequency, and thereby enables very efficient energy transfer. Therefore, by carefully
placing and adjusting the electromagnets a very thin ECR region (of field strength 875
Gauss) can be created in the plasma chamber. Driven by a higher (than the 875 Gauss)
but gradually decreasing magnetic field, the microwave propagates into this ECR region
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Figure S.3
Illustration of electron movement in ECR plasma, in which an electron
gyrates along the magnetic field line with cyclotron frequency f
140
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where microwave power is strongly absorbed by the gyrating electrons in the plasma gas.
Typical electron energy in an ECR system is about 5 eV, and the gyration radius of
electron is about 0.01 cm which is much smaller than the dimensions of the vacuum
system. The energized electrons will gyrate through the plasma, and in the course of their
movement they will collide with the gas molecules and transfer their energy to excite,
ionize and dissociate these gas molecules. The extended stay and the elongated path of
electrons in the plasma region due to gyration and the mirror symmetry o f the magnetic
field greatly increase the efficiency of the plasma source (up to - 20%) and raise the ion
density to about 10n- l 0 12 cm'3 at a pressure of -1 mTorr. Aided by the diverging
magnetic field at the other end of the plasma source chamber the electrons diffuse down
stream along the magnetic field lines to the process chamber where wafer processing is
performed, bringing with them the positive ions by the pulling o f electrostatic potential
created by the fast movement o f electrons. Neutral reactive species including both
molecules and atoms (from the dissociation of molecules) will be transported to the
process chamber mainly by diffusion.
Technologically, ECR plasma processing has found wide applications in the
microelectronics industry, particularly in etching and chemical vapor deposition, and also
in oxidation. Many processing steps in device fabrication can be done using ECR plasma,
at low pressure, low temperature, and with high efficiency. High plasma density ensures
high etching or deposition rates. Low pressure minimizes reactants loading effects which
i
I
|
t
i
can cause non-uniform etching or deposition. In addition, low pressure means fewer
collisions in the plasma sheath region. Therefore, the active ions have very good
directionality, which is desired for anistropic etching o f smaller and high aspect ratio
v
features in today's ULSI devices. In terms of oxidation, it has been shown that ECR
plasma grown oxide films show comparable properties to that of the thermally grown
oxide. Damage due to ion bombardment can be minimized because of the low plasma
141
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potential, and the low oxidation temperature ensures that results from previous processing
steps (e.g. doping profiles) will not be altered by the oxidation process. Another
advantage is that the plasma generation region is separated from the wafer processing
region in ECR systems, which reduces ion energies and minimizes ion bombardment
damage to the wafer.
Even though ECR plasma (as well as other types of plasma) processing has been
widely used in technological applications, scientifically speaking, many fundamental
questions about processes employing plasma are still unanswered. Plasma processing has
been more of an art than a science. Considering the ECR plasma oxidation for example,
some aspects of the oxidation mechanisms are not understood. Fundamental
understanding about the interface during oxidation process is catching up but still far
behind technological application. One reason for us to choose ECR plasma to grow oxide
I
i
is that it enables us to study the effects of electric field on interface roughness in the
s
oxidation.
<
§
{
i
5.3.3 ECR Plasma Oxidation Kinetics
When oxygen gas is let into the source chamber and ECR plasma is generated, the
vacuum chamber will be filled with a mixture of rather high density electrons, positive
oxygen ions and neutral oxygen atoms, together with the oxygen molecules that are not
dissociated. The reactive species in the oxygen plasma are:
0 2 +e-+ 0 2 +2e
0 2 + e ->2 O+e
0 2+e—* 0 +O
O +e-* O'
0 ;+ e ^2 0
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(5.8)
These reactive species can readily react with the Si substrate under favorable conditions
to form an oxide layer without the necessity for high thermal temperatures.
As in the thermal oxidation process, the plasma oxidation process is also limited
by the transport of oxidants after the very initial stage of the oxidation which is
dominated by the surface reaction of Si atoms with oxygen. Therefore, the oxide growth
also shows a linear-parabolic characteristic with respect to oxidation time as for the
thermal oxidation process. However, the ECR plasma oxidation differs from the thermal
oxidation in that not only neutral oxidants but also charged species participate in the
oxidation process. Studies25'28 show that in ECR plasma oxidation a neutral or positively
charged oxygen atom diffuses from the plasma source to the sample surface where it
acquires an electron to become a negatively charged oxygen atom. The negatively
i
charged oxygen atom then drifts through the growing oxide to react with Si at the Si/SiO,
interface. When a positive DC bias is applied to the Si substrate, the electric field
generated in the growing oxide layer assists the drift of these negatively charged oxygen
ions, and thus enhances the oxide growth. The actual mechanism for the diffusion of
;
these negatively charged oxygen atoms, though, is still under debate. On the other hand,
!
if negative bias is applied to the Si substrate, the oxide growth is retarded. Moreover, the
potential difference between the plasma and the substrate surface increases with the
applied negative bias voltage,29 which accelerates the positive ions towards the Si
substrate and induces surface damage on the substrate by ion sputtering rather than
oxidation.
The kinetics of the oxide growth can be similarly treated as for the thermal
oxidation, with the addition of the field assisted drifts of the oxidants. Figure 5.1 can be
i
!
used again to illustrate the kinetics. Considering the field assisted drifts of oxygen ions
(CTs), the flux that transports the CTs through the oxide becomes:
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(
dC \
Fx= - \iCE + D - p
dxj
(5.9)
where C is the concentration of 0", E is the external electric field generated by the
applied DC bias to the Si substrate, p and D are the mobility and diffusivity of the oxygen
ions, respectively. The first term explains the field assisted drifts and the second is due to
the Fickian diffusion. Assuming that the electric field is constant across the oxide layer,
the solution of Eqn. 5.9 is:
C(x) = Cle‘,l£l/0 - j ^ ( l - e - M&/D)
(5.10)
where C, is the concentration of O" at the gas/Si0 2 interface (r = 0). Now let x be the
thickness of the oxide layer, hence C(x) = C2, i.e., the concentration of 0~ at the Si/Si0 2
!
I
interface. Equation 5.10 can be rewritten as:
{
I
C p ~^V‘’Id —C
!
<5I1)
in which the electric field in the oxide is replaced by E = VMj x where Vgx is the voltage
drop across the oxide.
The second flux, which is the flux at the Si/Si0 2 interface due to oxidation
reaction, should have the same form as:
E2 = kC2
(5.12)
where k is the reaction constant. It should be equal to the first flux at equilibrium:
Fx = F2 = F = n ^ dt
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(5.13)
where Q is a constant that converts the flux to the growth rate of the oxide. Substituting
Eqn. 5.13 into Eqn. 5.11 yields:
e ^ yjD\ 1
d* = VVax C , - C' X, _________
-VVJD
dt
Q i, 1- e
(5.14)
The solution for this equation is:
(5.15)
x = Bt + A
where A is a constant and B is the parabolic coefficient,
B=
2\iVm C2 - C ,e » VJD
Q \ l ~ e -IiVJD
(5.16)
Applying the Einstein relationship, qD = \ikBT, in Eqn. 5.16, and taking into account that
the diffusion coefficient,
JL
D = D0e *k *. T
(5.17)
where E is the activation energy, the parabolic coefficient can be rewritten as:
E.
Q -C je
~ * .T
(5.18)
1- e k,T
After the very initial stage of oxidation, the exponential term in Eqn. 5.18.
exp -
« 1, therefore, the parabolic coefficient B can be approximated as:
B=
Q kJ
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(5.19)
This kinetic model of ECR plasma oxidation is very similar to a model proposed
by Cabrera and Mott30 which is widely used to explain the oxidation o f metals. It has
been proven to be successful in explaining the experimental data except for the very
initial stage of the oxidation. From Eqn. S. 19 we see that during the ECR plasma
oxidation process the oxide growth rate is not only a function of oxidation temperature,
but more importantly, it is a function of the voltage drop across the growing oxide layer
(or equivalently, the electric field in the oxide layer). If we can increase this voltage drop
the growth o f oxide can be enhanced greatly. Since this voltage drop, Fox, increases
monotonically with the positive DC bias applied to the Si substrate,28 it can greatly
enhance the oxidation rate during the ECR plasma oxidation process.
5.3.4 ECR Plasma System and Oxidation Procedure
The custom made microwave ECR plasma system that is used in this research is
illustrated in Figure 5.4. The whole system consists of a microwave system and a vacuum
system. The microwave system, which includes the magnetron source, power supply and
microwave guide, can provide 2.45 GHz microwaves of up to 500 Watts power.
Microwaves generated by the magnetron source are guided through the microwave circuit
to a coupler. A four-stub tuner is placed just before the coupler for impedance matching
to minimize the power reflection. In case of reflected power surge, a ferrite circulator
directs any reflected microwaves to a dummy load in order to protect the magnetron
source. The stainless steel vacuum system can also be divided into two parts: the plasma
source chamber and the process chamber. The plasma source chamber is lined with a
fused silica tube and surrounded by two external electromagnets to provide the ECR field
required for plasma excitation, with one end connected to the microwave coupler by a
fused silica window and the other end opened to the process chamber. During the plasma
146
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
ferrite
circulatoL
l^ l S 3
magnetron
source &
power supply
directional
coupler
dummy
load
reflected
power monitor
O,
electromagnets
load
lock
r*
DC bias
high vacuum valve
It
ion gauge
Figure 5.4
Schematic illustration of our microwave ECR plasma system.
147
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oxidation, ultra-pure oxygen is let into the plasma chamber through a gas line. A flow
regulator in the gas line regulates the flow rate of the oxygen (or any other desired gases
in other applications) and maintains certain working pressure in the ECR system. Excited
oxygen plasma then diffuses into the process chamber through the opening that connects
the two chambers. Wafers to be processed are placed on a stainless steel holder plate
which is then loaded into the process chamber via a load lock mechanism and inserted
into a slotted sample stage that is fixed onto an XYZ-sample manipulator. The wafers,
kept about 20 cm away from the plasma source, can be heated by a halogen lamp at the
back o f the sample holder to the desired processing temperature which is monitored by a
thermocouple attached on the sample stage. Temperature calibration is needed to account
for the temperature difference between the sample stage and sample substrate surface.
{
The sample stage is also electrically isolated from the chamber and wired to an electric
feed-through so that a DC bias can be applied to the substrate by contact with the holder.
The process chamber is connected via a high vacuum gate valve to a turbomolecular
pump that is capable of maintaining a background pressure o f 10'8 Torr and also capable
o f pumping large gas flow at processing pressures of up to several millitorrs. An
ionization gauge and a baratron gauge are used to measure pressure at high vacuum and
low vacuum ranges, respectively. A residual gas analyzer (RGA) can be used to analyze
the gas species in the vacuum chamber using a quardrupole mass spectrometer.
Like the case of thermal oxidation, ultra-cleanliness of Si wafer surfaces and
|
oxidation environment is of utmost importance for the quality o f oxidation. Therefore.
[.
before being transferred into the ECR system the wafers are cleaned using the modified
*
|
RCA procedure as described earlier in the thermal oxidation section, followed by wet
etching in concentrated HF solution to remove oxide. By carefully handling the wafers
and managing oxidation conditions, high quality oxide films comparable to those
thermally grown oxide films can be and have been obtained with ECR plasma process.
148
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
5.4
References
1.
M.M. Atalla, E. Tannenbaum, and E.J. Scheiber, Bell Syst. Tech. J., 38, 749
(1959).
2.
E.A. Irene, CRC Crit. Rev. Solid State Mater. Sci., 1 4 ,175 (1988).
3.
J.R. Ligenza and W.G. Spitzer, J. Phys. Chem. Solids, 1 4 ,131 (1960).
4.
W.A. Pliskin and R.P. Gnall, J. Electrochem. Soc., 129,867 (1982).
5.
P.J. Jorgenson, J. Chem. Phys., 37,874 (1962).
6.
K.N. Karube, K. Yamamoto, and M. Kamiyama, Jpn. J. Appl. Phys., 2,11 (1963).
7.
E. Rosencher, A. Straboni, S. Rigo, and G. Amsel, Appl. Phys. Lett., 34, 254
(1979).
8.
F. Rochet, B. Agius, and S. Rigo, J. Electrochem. Sco., 131,914 (1984).
9.
J.C. Mikkelsen, Jr., Appl. Phys. Lett., 4 5 ,1187 (1984).
10.
B.E. Deal and A.S. Grove, J. Appl. Phys., 3 6 ,3770 (1965).
11.
E.A. Irene and Y.J. van der Meulen, J. Electrochem. Soc., 123,1380 (1976).
12.
M.A. Hopper, R.A. Clarke, and L. Young, J. Electrochem. Soc., 122, 1216
(1975).
13.
W.A. Pliskin, IBM J. Res. Dev., 1 0 ,198 (1966).
14.
A.G. Revesz, K.H. Zaininger, and R. J. Evans, Appl. Phys. Lett., 8,57 (1966).
15.
T. Nakayama and F.C. Collins, J. Electrochem. Soc., 113,706 (1966).
16.
T. Smith and A.J. Carlan, J. Appl. Phys., 43 ,2455 (1972).
17.
E.A. Irene and D.W. Dong, J. Electrochem. Soc., 125,1146 (1978).
18.
H.Z. Massoud, J.D. Plummer, and E.A. Irene, J. Electrochem. Soc., 132, 1745
(1985).
19.
E.A. Lewis and E.A. Irene, J. Vac. Sci. Technol. A, 4 ,916 (1986).
20.
E.A. Lewis and E.A. Irene, J. Electrochem. Soc., 134,2332 (1987).
149
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I
21.
H.Z. Massoud, J.D. Plummer, and E.A. Irene, J. Electrochem. Soc., 132, 2685
(1985).
22.
H.Z. Massoud, J.D. Plummer, and E.A. Irene, J. Electrochem. Soc., 132, 2693
(1985).
23.
W. Kern and D.A. Poutinen, RCA Rev., 3 1 ,187 (1970).
24.
E.A. Irene, J. Electrochem. Soc., 121,1613 (1974).
25.
D.A. Carl, D.W. Hess, M.A. Lieberman, T.D. Nguyen, and R. Gronsky, J. Appl.
Phys., 70,3301 (1991).
26.
K. Eljabaly and A. Reisman, J. Electrochem. Soc., 138,1071 (1991).
27.
Y.Z. Hu, J. Joseph, and E.A. Irene, Appl. Phys. Lett., 5 9 ,1353 (1991).
28.
Y.Z. Hu, Y.Q. Wang, M. Li, J. Joseph, and E.A. Irene, J. Vac. Sci. Technol. A,
11,900(1993).
29.
J.L. Vossen, J. Electrochem. Soc., 126,319 (1979).
30.
N. Cabrera and N.F. Mott, Rep. Prog. Phys., 1 2 ,163 (1948).
i
150
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
CHAPTER VI
SI/SI02 INTERFACE ROUGHNESS EVOLUTION DURING
OXIDATION PROCESSES
6.1
Introduction
At the beginning of this dissertation the importance of Si/Si0 2 interface properties
to the microelectronics device applications was discussed. An abrupt, smooth interface is
obviously desirable for high device performance and reliability. It has been found.
;
/
however, that instead of an atomically abrupt interface between the Si0 2 film and the Si
substrate there exists a transitional region in which the structure and properties of the
interface change gradually from that of the crystalline Si substrate to that of the
amorphous Si0 2 film. This transitional region could be considered as the microscopic
roughness (or microroughness) at the Si/Si0 2 interface. The formation o f this transitional
k
region (or interface roughness) depends on the process type and process conditions to
form the S i0 2 film on the Si substrate. Under normal process conditions for the smooth Si
wafers, this interface region is of scale of less than several nanometers. Therefore, when
\
the Si0 2 film is very thick, compared to the interface region, the influence of the interface
f
roughness can be negligible, as for the earlier generations of ICs. However, as electronic
1
[
i
device dimensions scale down, the thickness of gate dielectric film is in the range of
several nanometers, which makes the interface region a relatively large fraction of the
dielectric film and the active region of the Si substrate. Thus the interface region may
greatly affect the proper functioning of devices. Understanding this interface roughness
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
region is therefore not only of scientific importance but also technologically necessary.
Much work has been done to study this interface region. Yet it remains ill understood.
In this chapter we apply several roughness characterization techniques, such as the
nondestructive optical technique of ellipsometry, and the direct but destructive AFM
technique with fractal analysis to study the evolution of Si/Si0 2 interface roughness
during microwave ECR plasma and thermal oxidation processes under different oxidation
conditions. From the analysis results we attempt to uncover the driving mechanisms for
interface roughness evolution.
6.2
Experimental Procedures
The experimental procedures and flow can be illustrated in a chart as shown in
Figure 6.1. The Si wafers used in the study were commercially available p-type (boron
doped) single crystalline Si of (100) orientation with a resistivity of ~ 2 Q-cm.
In order to study the effects of the initial surface feature size on the interface
roughness change, and also to get a more distinguishable measurement of the roughness
change, we purposely roughened some of the wafers. The roughening was performed,
after a brief deionized (DI) water rinse, in the chemical solution of HN03-HFCH3COOH (30:20:40) at room temperature with ultrasonic agitation, which etched the Si
surface isotropically and yielded rougher Si surfaces than the original, out-of-the-box Si
surfaces.
All wafers, including the wet-etch roughened Si wafers and the smooth out-of-thebox wafers, were subjected to the slightly modified RCA cleaning process (as described
in Chapter 5) before being oxidized. Each wafer was then cut into several pieces. One of
the pieces was used as a reference from which the initial Si surface roughness was
obtained using the three roughness characterization techniques (AFM, fractal analysis,
and ellipsometry), while the others were oxidized for different lengths of time.
152
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p-type S87
,
]
Etch in
HNO,-HF-CHjCOOH
RCA Cleaning
I
Initial Si
Surface
Roughness
HF Dip
Thermal Oxidation
ECR Plasma Oxidation
Roughness Characterization
Figure 6.1
Flow chart of experimental procedures.
153
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1
The ECR plasma oxidation was performed in the custom-made ECR plasma
system which was described in Chapter 5. Ultra-pure oxygen (99.997%) was let into the
plasma source chamber at a controlled flow rate o f 20 seem while maintaining a working
pressure o f 1 mTorr in the vacuum chamber. The microwave input power was 300 W.
The Si substrate to be oxidized was about 20 cm down stream from the plasma source
chamber opening. During the oxidation process the Si substrate was heated to several
different temperatures, from the floating temperature (heated only by the plasma itself to
<100°C) up to 400°C (heated by the halogen lamp behind sample holder), and with
different positive DC bias voltages (+30 - +90 V) applied to the sample holder. No
negative bias studies were performed in this study because it has been found previously
that while positive DC bias increases the oxidation rate, sputtering of the substrate
dominates with negative bias.1,2
All thermal oxidations were carried out in a conventional resistance-heated
horizontal furnace following the procedure as described in Chapter S, at 800°C and
1000°C with ultra-pure dry 0 2 of atmospheric pressure.
6.3
Roughness Measurements and Data Analysis
After the oxidation, the oxidized sample was first characterized non-destructively:
using a conventional spectroscopic ellipsometry (SE) to obtain the thickness of the grown
oxide layer, and then using the interface-sensitive spectroscopic immersion ellipsometry
(S1E) to obtain information about the Si/Si02 interface layer thickness.
The SIE measurements were performed in the system adapted from a
conventional SE system, as illustrated in Figure 4.10. Before measurements, the system
was carefully aligned in air. The offsets of the polarizer and rotating analyzer as well as
the attenuation factor of the circuit system were calibrated using a gold standard sample
at an incidence angle of 70° and at the wavelength of 405 nm following the procedure
154
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described in Chapter 4. Since the SIE measurements were done with the add-on fused
silica cell which has two optical windows, the system was checked by comparing the
measurement results for a freshly HF dipped Si sample with and without the empty cell to
ensure that there was no obvious stress effect from the windows. The system was
calibrated again with the immersion liquid in the cell at the incidence angle of 72° (at
which angle all the SIE measurements were performed) to make sure that the system
offsets agreed with the in-air values. During the measurements, a long pass filter (A-cutofr=
400 nm) was added at 425 nm to eliminate the second harmonic effects of the high energy
photons.
The acquired SIE spectrum contains the wavelength (or photon energy) dependent
¥ and A values. From the discussion in Chapter 4 it is known that the data reflect
structural and composition configuration of the sample, which can be extracted following
the general procedure outlined in Chapter 4, that is, by constructing an optical model that
incorporates the unknown parameters of the interface, fitting the values of T and A
calculated from the theoretical model with their corresponding experimentally acquired
values to minimize the merit function (Eqn. 4.84) and verifying that the final fitted
parameters are physically acceptable.
The optical model we used in this study for the SIE data analysis is illustrated in
Figure 6.2. The modelconsists of the liquid ambient,the Si0 2overlayer,
region and
the Sisubstrate.For the Si0 2 covered
theinterface
Si substrate, wechose
carbon
tetrachloride (CC14) as the immersion liquid. Over the range from near UV to visible,
CC14 is basically non-absorbing (i.e., its extinction coefficient k - 0) and its the refractive
index can be calculated using the following Cauchy dispersion formula:
°
n = na +—
r + f a AT
X dT
a
t
155
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
«
n
(6.1)
■ox
■inf
Figure 6.2
The optical model used in the present study for SIE data analysis.
156
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I
where X is the wavelength of light, nn ~ 1.4427, a = 5.15 x 103 nm2 at T = 24.8°C, and
dnldT = 0.00055 at T= 20°C for pure CC14.3 The thickness of the Si0 2 overlayer film was
obtained from a one-film model calculation of the conventional SE measurement in air
ambient. In our SIE analysis, the refractive index of the Si0 2 thin film was calculated
using a single term Sellmeier approximation:
n =A+ -- k2- k
(6.2)
where A = 1.0, fl = 1.15, and X0 = 92.3 nm .4 It is known that the refractive index o f SiO,
depends on the thickness of the film. The set of parameters we chose is for thermally
grown oxide film of 20 nm thick. We believe that to the first order for our SIE data
analysis, it is an suitable approximation for the thickness range o f our oxide films.
As can be seen in Figure 6.3, the refractive index of CC14 matches that of the Si0 2
film very well. From Snell's law (Eqn. 4.49) and the Fresnel relationships (Eqn. 4.50) it is
known that if the refractive indices of the two media N0 = N V the incident and refractive
angles will be equal, i.e., <|>0 = <(>,, which leads to the reflection coefficient r 01 = 0 (for both
p and s polarizations). There will be no reflection at the interface o f these two media.
However, when the refractive indices are not exactly equal but approximate, i.e., when (1
- NJNX) «
1, then (1 - <(>(/<)>,) «
1, and thus instead of r 01 = 0, we have r0I » 0. The
reflection at the interface of the two media is not totally eliminated but minimized. Since
the refractive index of CC14 matches that of the SiO, films very well, the optical response
from the liquid/film interface is essentially removed, and therefore, the detected signal
greatly signifies the effect of the film/substrate interface. In the SIE data analysis using a
C program we wrote, the actual refractive indices for CC14 solution and Si0 2 thin film
(based on Eqns. 6.1 and 6.2) were used for the theoretical calculation o f the reflection
157
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1.7
------ 1—
1—---- 1 '
~1-------- 1------ - 1- ■■—1-------- 1 " ~ 1-------- 1-----....... CCI4
• •• • Si0 2 (20 nm)
------ Si0 2 (bulk)
1.6
-
c 1.5
©
-
1 14 ■
.1
|
1.3 -
-
1.2
-
-
1.1
-
I
Air / Vacuum
1.0
1
2.4
-.1- ___ 1_____ 1_____ 1____ _ l ---- 1-------J-------- 1--------!— .
2.6
2.8
3.0
3.2
3.4
3.6
3.8
4.0
4.2
Energy (eV)
Figure 6.3
Comparison of the refractive indices o f CCI4 solution and Si0 2 films. The
refractive index of air/vacuum is also plotted as a reference.
158
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coefficients, and thus the minimization of the CCl4/Si0 2 interface effects was
automatically taken care of in the analysis.
The interface region was modeled as an isotropic film consisted of uniformly
mixed Si0 2 and amorphous Si (a-Si). Because of the heterogeneity of the region, its
refractive index is not readily available and thus we can not employ the Fresnel equations
to calculate the theoretical ellipsometric parameters directly. Therefore, the Bruggeman
effective medium approximation (BEMA) theory5 was applied to calculate the
pseudodielectric function of the layer, with the volume fractions of Si0 2 and a-Si as the
variable parameters (in addition to the thickness of this effective medium layer) in the
fitting algorithm. The theory assumes that the size of both components are large enough
that they have their respective dielectric functions, yet small enough compared to the
wavelength of the probing light beam.
After the non-destructive optical measurements, the sample was then dipped in
concentrated HF (49%) for less than 10 seconds to remove the oxide film and reveal the
buried Si/Si0 2 interface. The etch rate of Si0 2 film in HF is very high (3000 - 5000
nm/min) while that for the Si it is very slow (-0.03 nm/min at 25°C).6 Studies have
demonstrated7,8 that immersing Si in HF for a very short period of time (less than 1 min)
does not make detectable changes to the Si surface roughness. The morphology o f the
exposed Si surface was then directly imaged using a commercial AFM (NANOSCOPE II
from Digital Instruments) operating in contact mode. During the AFM scanning the
interatomic force between the tip and the surface was carefully administered to a
minimum value while maintaining tip engagement to the surface in order to minimize
damage to sample surface, and dry N2 gas was flowed through the AFM apparatus
enclosure in order to reduce the influence of moisture (see Chapter 3 for more details on
the effect of moisture). For each sample at least three different areas were scanned to
ensure surface uniformity and reduce statistical error.
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From Chapters 2 and 3, we know that statistical analysis on the acquired AFM
images of the exposed Si surfaces yields the rms values among several other roughness
describing parameters, while fractal analysis of the images extracts the fractal dimensions
of the surfaces. In this study we will combine the use o f rms values which show the
vertical magnitude of the surface features, and the fractal dimensions which give us the
shape or complexity of the surfaces, to obtain a more complete representation o f the
surface roughness. Details about calculation of the rms values and the 2-D variation
method that we used to extract fractal dimensions from an AFM image have been
discussed in Chapter 3.
6.4
E x p e rim e n ta l R esu lts a n d D iscussion
Several sets of Si samples with initial rms values ranging from 0.1 to 8.2 nm were
|
oxidized in thermal and ECR plasma oxidation processes. Each set of samples came from
I
the same wafer. During the oxidation process the experimental conditions were kept the
same for each sample set except for the time of oxidation which was varied to grow Si0 2
films of different thicknesses. The initial Si roughness and oxidation conditions for
different sets are listed in Table 6.1.
For the thickness range of our films, the oxide growth clearly shows the mainly
i
parabolic growth characteristic for both thermally and ECR plasma oxidized samples.
The thicknesses of the oxide films, calculated based on a single film model o f the SE data
i
j>
(in air ambient), for several selected sets of oxidized samples are listed in Table 6.2 and
|
results are also plotted in Figures 6.4 and 6.5. The parabolic growth indicates that the
i
oxidations of these samples were dominated by the diffusion of oxidant species according
f
to the kinetic models for the two processes.
The results of SIE measurements can be best illustrated in Figure 6.6 in which the
ellipsometric measurable A is plotted as a function of photon energy. Changes in A curve
160
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Table 6.1
Initial Si surface roughness and oxidation conditions for different sets of
samples studied.
Samples
Init. Si Surf,
rms (nm)
Oxidation
Process Type
Oxidation
Temperature
Bias (V)
Set 1
3.93
ECR
400°C
+90
Set 2
8.24
ECR
400°C
+60
Set 3
3.93
ECR
400°C
+30
Set 4
5.91
ECR
200°C
+60
Set 5
0.10
ECR
200°C
+60
Set 6
0.10
ECR
floating
+30
Set 7
4.46
Thermal
1000°C
N/A
Set 8
2.92
Thermal
1000°C
N/A
Set 9
5.22
Thermal
1000°C
N/A
Set 10
6.18
Thermal
1000°C
N/A
Set 11
6.40
Thermal
1000°C
N/A
Set 12
0.10
Thermal
800°C
N/A
161
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Table 6.2a
Oxide thickness versus oxidation time for samples thermally oxidized at
1000°C, 1 atm.
Oxidation
Time (min)
Oxide
Thickness (nm)
0
60
180
300
0
98
190
237.5
Table 6.2b
Oxide thickness versus oxidation time for several sets of ECR plasma
____________ oxidized samples.______________________________________________
Oxidation Conditions
Oxidation Time (min)
Oxide Thickness (nm)
400°C, +60V
0
1.50*
400°C, +60V
10
14.69 ± 1.43
400°C, +60V
30
26.54 ± 0.48
400°C, +60V
40
31.58 ±0.39
400°C, +60V
60
37.57 ±0.31
200°C, +60V
0
1.50*
200°C, +60V
10
8.87 ±0.14
200°C, +60V
20
14.05 ±0.18
200°C, +60V
30
16.66 ±0.19
200°C, +60V
40
19.40 ±0.20
200°C, +60V
60
25.86 ±0.17
400°C, +30V
0
1.50*
400°C, +30V
10
10.75 ±0.46
400°C, +30V
20
15.36 ±0.19
400°C, +30V
30
18.21 ±0.45
400°C, +30V
40
21.59 ±0.23
A 1.5 nm thick native oxide layer is assumed on the Si substrate before oxidation.
162
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240
1000°C
220
200
g
180
3
160
M
140
0)
120
y
io o
(0
O
an
0
50
100
150
200
250
300
250
300
Time (min)
60000
50000
E
40000
(0 30000
20000
10000
-10000
0
50
100
150
200
Time (min)
Figure 6.4
Oxide growth data for a thermally oxidized Si sample.
163
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40
30
(A
©
4 0 0 °C ,3 0 V ^ 200°C, 60V
in
c
20
o
10
0
0
10
20
40
30
50
60
Time (min)
1400
c
900
4 0 0 °C ,3 0 V ^ 200°C, 60V
JC
400
-100
0
10
20
30
40
SO
60
70
Time (min)
Figure 6.S
Typical oxide growth data for ECR plasma oxidized Si samples.
164
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170
0 min (exp) .
30 min (exp)
60 min (exp) *
Smooth Si, ECR: 200*C, +60V
160
150
140
0
s
130
01 120
tt
2,
<
no
100
2.5
2.7
2.9
3.1
3.3
3.5
3.7
3.9
Energy (eV)
fii
Figure 6.6
Experimental (scatters) and calculated (lines) ellipsometric parameter A
for the ECR plasma oxidized initially smooth Si surfaces.
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are obvious with oxidation. Since in SIE the effect of the Si0 2 overlayer film is optically
removed, these changes in the A spectrum mainly reflect the changes in the Si/Si0 2
interface as the oxidation proceeds.
From the analysis of the SIE measurements o f the Si/Si0 2 interface, the AFM
images of exposed Si surfaces, and the fractal analysis of these AFM images, it is found
that based on the changing trend of the interface roughness, the results can be divided into
two categories: those for the initially rough Si surfaces and those for the initially smooth
Si surfaces. Next we will first present and discuss our results in each category, and then
summarize both results to get the overall picture.
6.4.1 S i/S i02 Interface Roughness fo r Initially Rough Si Surfaces
5
;
The change in the thickness o f Si/Si0 2 interface layer during ECR plasma
oxidation process for the purposely roughened Si surfaces was analyzed from the SIE
measurements and the results are plotted in Figure 6.7. It is clearly seen that regardless of
oxidation conditions (temperature, bias), the interface region became thinner as the
oxidation proceeds. The thinning of the interface region during ECR plasma oxidation for
the initially rough Si surfaces is also evidenced from the decreasing trend in the rms
1
;
values (Figure 6.8) o f the exposed Si surfaces measured by AFM. From the fractal
analysis of the AFM images of the Si surfaces, it is also found that the shape (or
i
1
complexity) of the Si surfaces became simpler in ECR plasma oxidation, as quantitatively
ii
:
indicated by the decreasing trend in the fractal dimension of the surfaces as seen in Figure
6.9. Viewing the AFM images provides direct, but only qualitative information, as
?
exemplified in Figure 6.10.
Smoothing of Si/Si0 2 interface in terms of the vertical scale (or the interface
region thickness) and the complexity of the interface was also observed for the initially
rough Si surfaces in thermal oxidation in a previous study9,10 done in our group and also
166
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Interface Region Thickness (nm)
8
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Si02 Overlayer Thickness (nm)
i"I
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Figure 6.7
Si/Si0 2 interface region thickness, derived from SIE measurements, for
the initially rough Si surfaces at different stages of ECR plasma oxidation
under different oxidation temperatures and bias voltages.
167
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Figure 6.8
Rms roughness of the exposed Si surfaces after the initially rough surfaces
were ECR plasma oxidized under different conditions.
168
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The fractal dimensions o f the exposed Si surfaces after the initially rough
surfaces were ECR plasma oxidized under different conditions.
169
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Before ECR oxidation
After 40 min oxidation
After 90 min oxidation
rms = 5.91 nm
Df = 2.68
rms = 3.98 nm
Df = 2.58
rms = 3.43 nm
Dr = 2.52
Oxidation condition: 200°C, +60V
Figure 6.10
Some typical AFM images of the exposed Si surfaces at different stages
of ECR plasma oxidation process.
170
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
in this extended study. Figures 6.11 and 6.12 show the change in rms values and the
fractal dimensions, respectively, for several sets of initially rough Si surfaces during
thermal oxidation at 1000°C.
From the previous investigations9,11 on the behavior of rough Si surfaces during
thermal processes, it was concluded that the smoothing o f interface roughness is due to
the interfacial diffusion of Si atoms, predominantly driven by the reduction of free energy
dictated by Kelvin equation. Assuming that a surface feature has a spherical shape with a
radius of curvature R, its molar free energy can be expressed as follows:
8G = 2 yV /R
(6.3)
where V is the molar volume and y is the surface free energy of the substance. It is
therefore obvious that the feature has a tendency to increase its radius of curvature so as
to reduce its free energy. Assuming that the radius of curvature for such a feature changes
from Rx to R2 where R2 > Rv the free energy o f the system will be reduced by the amount:
AG = 2Vy
[i~k)
Therefore, sharper features (i.e., features with smaller R) have higher free energy. In our
case of Si in thermal oxidation, it means that sharper features have higher chemical
activity and higher tendency to flatten compared to their more rounded counterparts. In
addition to the reduction of free energy, high interface stress at the Si protrusions also
provides an additional driving force for the diffusion of Si atoms.
It was also found9'11that due to the high temperature nature of thermal oxidation
process, the interface smoothing is enhanced by several high temperature effects. One
such effect is the viscous flow of Si0 2 at the interface which becomes more significant at
171
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
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Figure 6.11
Rms roughness for the initially rough Si surfaces thermally oxidized in a
conventional furnace at 1000°C with ultra-pure dry 0 2.
172
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
1000°C
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Figure 6.12
The fractal dimensions for the initially rough Si surfaces thermally
oxidized at 1000°C.
173
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
temperatures above 950°C. Another possibility is the interfacial chemical reaction
through the decomposition and formation of Si0 2 near the interface. These mechanisms
explain the change of interface roughness during thermal oxidation of initially rough Si
surfaces in the previous and the current studies very well.
For the smoothing of initially rough Si surfaces during microwave ECR plasma
oxidation process, however, there must be some other enhancement mechanisms because
of the low temperature nature o f the process. Controllable growth of oxide at low
temperature, usually < 400°C, is one major reason for using the ECR plasma process for
oxidation applications. Even though the smoothing of interface can still be considered
from the point of view interfacial diffusion of Si atoms driven by the reduction o f free
energy and also by the stress, the high temperature effects found in thermal oxidation
process that enhance the interfacial diffusion are less likely to be operational for the
change o f Si/Si0 2 interface roughness during ECR plasma oxidation. Based on our
results, we believe that electric field at the interface plays an important role in interface
smoothing during ECR plasma oxidation.
During ECR plasma oxidation, the oxidant species have been found12'15 to be
predominantly the negatively charged oxygen ions formed at the gas/oxide interface,
which move through the growing oxide layer to the Si/Si0 2 interface to react with Si
atoms. As in thermal oxidation, oxide growth in the ECR plasma oxidation process is
limited by the transport of oxidant species after the very initial stage of oxidation. If a
bias is applied to the Si substrate, the transport of the oxidant species will be assisted (or
retarded, depending on the polarity of the applied bias) by the external electric field
established in the growing oxide layer, in addition to the Fickian diffusion driven by the
concentration gradient. The oxide thickness shows parabolic relationship with respect to
oxidation time after the initial stage of oxidation (more details can be found in Chapter
5):
174
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
1
x 2 = Bt + A
(6.5)
where A is a constant, and B is the parabolic coefficient. It has been shown in Chapter 5
that the parabolic coefficient B is proportional to the voltage drop (or equivalently the
electric field E) across the growing oxide film:
V
kBT
kaT
( 6 .6 )
where kB is the Boltzmann constant, T the oxidation temperature, and Ea is the activation
energy related to the diffusion of O’s under the influence of the external electric field.
From Eqns. 6.5 and 6.6, it is apparent that the higher the external electric field, the higher
the overall oxidation rate, which has been proven by the kinetic studies of the ECR
plasma oxidation data.I4 IS
If we shift our concentration to the interface, which is not flat but consists of Si
microprotrusions, we know that the electric field will not be uniformly distributed across
the interface, but rather will be distorted and its strength intensified at the asperities. It
has been shown1617 that the intensification depends on the shape and distribution of the
protrusions in addition to the distance from the protrusions. Sharper and more isolated
protrusions see a higher intensification factor. Associating this locally enhanced electric
field with the oxidation kineticmodel, it is logical to expect that the oxidation rate around
Si protrusions will be higher,and that sharper and more isolated features will beoxidized
faster than the duller features, which will on one hand lead to the reduction in the overall
vertical size of the features (i.e., smaller rms values and thinner interface layer) and on
the other hand reduce the complexity of the Si surfaces (i.e., decreasing fractal
dimensions). Our experimental results agree well albeit qualitatively with these
expectations. Yet any attempt at quantitative analysis is difficult to accomplish because
175
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
!
the complexity of the real interface yields very complicated field enhancement effects
near the interface.
Because the initial Si surface roughnesses were different for different sets of
samples, in order to see how oxidation conditions (temperature and applied DC bias)
affect the change in interface roughness, we normalized the rms roughness o f each
sample from the same wafer set by the initial reference Si surface roughness for that set,
and plot the new results in Figure 6.13. It is now clearly seen that the change of interface
roughness is more bias dependent than temperature dependent. This is understandable
from the mechanism discussed above. It was shown15 that in ECR plasma process, the
voltage drop across the oxide is approximately proportional to the applied positive DC
bias. Therefore, higher positive DC bias induces larger field disparity among Si features
of different sizes, which in turn results in a more drastic reduction of interface roughness.
6.4.2 S i/S i0 2 Interface Roughness for Initially Sm ooth Si Surfaces
For the initially smooth Si surfaces, our results show an opposite trend to that for
the initially rough Si, i.e., the interface roughness increases as the oxide grows. This is
evidenced by the change in the interface region thickness from SIE analysis which is
displayed in Figure 6.14, and by the increasing rms values and the increasing fractal
dimensions of the exposed Si surfaces as shown in Figure 6.15. The Si/Si0 2 interface
roughening is observed for both ECR plasma and thermally oxidized smooth Si surfaces,
and it seems that the increase of roughness is enhanced at higher oxidation temperatures,
as shown in Figure 6.16. Visually inspecting the AFM images reveals the roughening
effect directly, and also that the roughness is uniform across the scanned surfaces (Figure
6.17).
Previously, both interface roughening and smoothing during thermal oxidation
process have been observed for initially smooth Si surfaces. Using LEED spot profile
176
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
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Si02 Overlayer Thickness (nm)
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Figure 6.13
Normalized rms roughness for the purposely roughened samples.
177
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Figure 6.14
Interface region thickness, from SIE analysis, for the initially smooth Si
surfaces oxidized under different conditions.
178
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Figure 6.15
The rms values and the fractal dimensions for the initially smooth Si that
were ECR plasma oxidized at 200°C, +60V applied bias, show interface
roughening.
179
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Rms roughness for smooth Si surfaces oxidized at 200°C and at 400°C,
+60V bias.
180
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Before EC R oxidation
After 20 m in oxidation
A fter 60 m in oxidation
rms = 0.10 nm
Df = 2.61
rms = 0.19 nm
Df = 2.66
rms = 0.27 nm
Df = 2.71
i
Oxidation condition: 200°C, +60V
Figure 6.17
Some typical AFM images o f the initially smooth Si surfaces at different
stages of ECR plasma oxidation process.
181
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
analysis, Hahn and Henzler18 studied the step atom density (which corresponds to
roughness) o f the oxidized Si surface alter the oxide layer was removed, and found that
the interface roughness strongly depended on the oxidation conditions. High oxidation
rate (for oxide thicknesses < 20 nm or for wet oxidation) would increase the Si/SiO,
interface roughness and low oxidation rate (for oxide thicknesses > 20 nm or in non­
oxidizing ambient) would lower the roughness, while increasing oxidation temperature
lead to a decrease in roughness. They further explained the roughening as the result of
random process o f diffusion and reaction of oxygen at the interface, i.e., as a statistical
effect. As to what really drives the roughening, it is still unclear. The smoothing was
explained from the point of view that a smooth, step-free surface is energetically favored
in thermal equilibrium. In a previous study9 done in our group using eilipsometry with a
interface model that contains a chemical interface (a suboxide layer) and a physical
i
interface (Si protrusions), it was found that the thickness of the chemical interface
increased yet the physical interface decreased with oxidation. Another eilipsometry study
by Jellison4 showed interface roughening. Similar studies were carried out on interface of
thermally oxidized Si using a variety of techniques, such as Rutherford scattering,19 AES
>
depth profiling,20,21 SIMS depth profiling,22 XPS,23-25 UPS,26 Low-energy ion
scattering,27 TEM,28,29 X-ray diffraction,30 etc., however, no consensus conclusion on the
roughness changes or mechanisms existed,
i
For the ECR plasma oxidized surfaces, some previous studies31,32 also reported
^
roughening and suggested that it was possiblv due to the plasma ion bombardment
I
damage. This ion bombardment mechanism does not seem operative in our study. Firstly.
!
the potential difference between the plasma and the substrate is about 20 V in ECR
plasma oxidation with positive bias applied, and thus the ions hitting the Si substrate have
rather low kinetic energy. Even if low energy ion damage is possible, most damage
should occur at the very initial stage of oxidation, i.e., when the bare Si substrate is
182
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
exposed to the plasma. Alter that, as the oxidation proceeds the growing oxide film
should act as an ion mask and protect the underlying Si substrate from further ion
bombardment. Therefore, the Si/Si0 2 interface roughness should not increase with
oxidation time/extent, if ion bombardment is the only mechanism. This is contradictory to
our result, i.e., a continued increase in roughness with oxidation time. Ion bombardment,
therefore, is excluded as a dominant source for the interface roughening.
It is known that some localized factors such as local surface defects and metal
contaminants may cause the interface roughening by altering the oxidation kinetics and
enhancing oxidation locally. These localized factors can be excluded as the sources for
interface roughening in the current study based on the fact that the AFM images show no
localized roughening, but rather a uniform increase in roughness (Figure 6.17).
Electric field or bias effect is not believed to be a factor, either. The intensification
of field for such uniformly distributed small features is minimized except for the extreme
proximity to the surface.16 Furthermore, if the field had any effect it would have been
smoothing rather than roughening, for the same reason discussed in the case for the
initially rough Si surfaces.
The actual mechanism for the interface roughening during oxidation process
remains not fully understood. In this study, we speculate that the roughening may be the
result of defects generation at the Si/Si0 2 interface during oxidation, as we know that a
rough surface can be considered consisting of large number of surface defects (steps,
edges, etc.). Based on a roughening transition model proposed by Burton and Cabrera.33
the number of surface defects is proportional to a Boltzmann factor as follows:
N oce*fc'-/*»r
(6.7)
where Ea is the activation energy for the defects generation, kB is the Boltzmann constant,
and T is the temperature. For bare Si surfaces such a roughening transition is not expected
183
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
to be observable at the temperatures used in the present study (< 400°C for ECR plasma
oxidation) since the roughening transition usually has a high activation energy for bare
surfaces. However, we speculate that the ongoing oxidation reaction at the interface
couples with this defect-generating roughening transition, and the coupling may lower the
activation energy so that the roughening transition can be observed at reduced
temperatures. The generated defects are uniformly distributed, and they will change the
oxidation kinetics around the defected sites which can lead to the uniform increase of
interface roughness. To our knowledge, this chemical reaction mediated roughening
transition mechanism has not been previously reported. Therefore, even though this
mechanism can explain the roughening trend that we observe and also the temperature
dependence of the roughening, further studies are necessary to substantiate this
mechanism.
6.4.3 Sum m ary
We have seen that the Si/Si0 2 interface roughness evolution during thermal and
ECR plasma oxidation processes depends on whether the initial Si surfaces are rough or
smooth. The interface roughness decreases for the initially rough Si surfaces while
increases for the initially smooth Si surfaces during oxidation processes. It is thus natural
to expect, and indeed is seen from Figure 6.18, that the two trends will converge and
reach a dynamic equilibrium as the oxide grows thicker, which yields a steady final
interface roughness. This leads to our conclusion that during oxidation process the two
mechanisms, smoothing and roughening, affect the interface roughness simultaneously.
Which one of the two competing mechanisms is dominant is determined by the initial Si
surface roughness. The smoothing of interface dominates for the initially rough Si
surfaces, and is believed mainly due to the diffusion and/or reaction of Si atoms, driven
by the reduction of free energy and also by the stress, enhanced by the high temperature
184
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
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♦ ■
--□ —
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o
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20
30
40
SiO^ Overlayer Thickness (nm)
Figure 6.18
SIE analysis results of the interface region thickness for both initially
rough and initially smooth Si surfaces during ECR plasma oxidation.
185
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
effects such as viscous flow and chemical reaction at the interface for the thermal
oxidation, and by the local electric field effect for the ECR plasma oxidation. Interface
roughening, which dominates the roughness change for the initially smooth Si surfaces, is
o f a much smaller scale compared to the smoothing, and it is believed due to a chemical
reaction mediated roughening transition that generates surface defects during oxidation
and alter the oxidation kinetics.
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186
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6.5
R eferen c es
1.
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2.
Y.Z. Hu, Y.Q. Wang, M. Li, J. Joseph, and E.A. Irene, J. Vac. Sci. Technol. A,
11,900(1993).
3.
Techniques o f Chemistry, edited by A. Weissberger, Organic Solvents Vol. II,
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G.E. Jellison, Jr., J. Appl. Phys., 69,7627 (1991).
5.
D.E. Aspnes, Thin Solid Films, 89,249 (1982).
6.
Thin Film Processes, edited by J.L. Vossen and W. Kern, (Academic Press, New
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7.
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9.
Q. Liu, J.F. Wall, and E.A. Irene, J. Vac. Sci. Technol. A, 12,2625 (1994).
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11.
V.A. Yakovlev, Q. Liu, and E.A. Irene, J. Vac. Sci. Technol. A, 10,427 (1992).
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D.A. Carl, D.W. Hess, M.A. Lieberman, T.D. Nguyen, and R. Gronsky, J. Appl.
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13.
K. Eljabaly and A. Reisman, J. Electrochem. Soc., 138,1071 (1991).
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Y.Z. Hu, Y.Q. Wang, M. Li, J. Joseph, and E.A. Irene, J. Vac. Sci. Technol. A.
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16.
T.J. Lewis, J. Appl. Phys., 26,1405 (1955).
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17.
M.C.V. Lopes, S.G. dos Santos F., and C.M. Hasenack, J. Electrochem. Soc., 143.
1021 (1996).
18.
P.O. Hahn and M. Henzler, J. Vac. Sci. Technol. A, 2, 574 (1984).
19.
L.C. Feldmann, P.D. Silvermann, I. Stensgard, and N.W. Cheung, Appl. Phys.
Lett., 35,859 (1979).
20.
J.S. Johannessen, W.E. Spicer, and J.E. Strausser, J. Appl. Phys., 4 7 ,3028 (1976).
21.
C.R. Helms, N.M. Johnson, S.A. Schwarz, and W.E. Spicer, J. Appl. Phys., 50,
7007 (1979); C.R. Helms, N.M. Johnson, S.A. Schwarz, and W.E. Spicer,
Proceedings o f the International Topical Conference on the Physics o f S i0 1 and
Its Interfaces, edited by S.T. Pantelides (Pergamon, New York, 1983), p. 366.
22.
23.
I. Barsony and J. Giber, Appl. Surf. Sci., 4 , 1 (1980).
F.J. Grunthaner and J. Maseijan, Proceedings o f the Conference on the Physics o f
5 /0 , and Its Interface, (Yorktown Heights, 1978).
I
24.
25.
R. Flitsch and S.I. Raider, J. Vac. Sci. Technol, 12,305 (1975).
R.S. Bauer, J.C. Menamin, H. Petersen, and A. Bianconi, Proceedings o f the
\
|
International Topical Conference on the Physics o fS i0 2 and Its Interfaces, edited
j
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Lett., 27,644 (1975).
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International Topical Conference on the Physics o f SiO, and Its Interfaces,
(Pergamon, New York, 1978), p. 356.
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M.T. Tang, K.W. Evans-Lutterodt, G.S. Higashi, and T. Boone, Appl Phys. Lett.
62,3144(1993).
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A.K. Ray and A. Reisman, J. Electrochem. Soc., 128,2466 (1981).
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Froment, and B. Agius, J. Vac. Sci. Technol. B, 13,227 (1995).
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W.K. Burton, N. Cabrera, and F.C. Frank, Philos. Trans. R. Soc. London Sect. A,
243,299(1951).
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189
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CHAPTER VH
CONCLUSIONS AND FUTURE DIRECTIONS
The Si-Si0 2 system is the foundation of the modem microelectronics industry,
and as was pointed out earlier, the functioning o f microelectronics devices depends on the
properties of the Si/Si0 2 interface. The scaling down of semiconductor device
dimensions in pursuit of smaller, faster, cheaper, less power-consuming, and more
reliable devices put increasingly stringent requirements on the tolerable roughness of
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surfaces and interfaces. The roughness effects are especially critical for the interface
I
between the Si0 2 gate dielectric film and Si substrate, considering that the thickness of
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gate dielectric film is now in the range of 5 nm and is being further reduced. Eventually,
we may run into such a situation that only the interface is left. It is therefore o f utmost
importance to understand the interface roughness behavior during the oxidation process
and the mechanisms involved.
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In the present study, we applied a novel non-destructive optical technique,
spectroscopic immersion eilipsometry (SIE), to study the Si/Si0 2 interface region during
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thermal and microwave electron cyclotron resonance (ECR) plasma oxidation processes
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for Si surfaces of different initial roughnesses. By immersing the film covered sample in
t
CC14 solution to optically "remove" the effect of the Si0 2 overlayer, the SIE technique
greatly enhances the sensitivity of eilipsometry to the thin Si0 2 film covered interface.
From the analysis of the SIE data we obtained the thickness of the Si/Si0 2 interface.
Results show both smoothing and roughening effects for the interface region during both
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
thermal and ECR plasma oxidation processes, and whether smoothing or roughening is
the dominant mechanism depends on whether the original Si surface was purposely
roughed or smooth. For rough Si surfaces, oxidation reduces the interface thickness,
while for smooth Si surfaces the interface roughens during oxidation. This indicates that
there are two competing mechanisms occurring simultaneously at the interface.
Regardless of the initial surface roughness, the two mechanisms eventually reach a
dynamic equilibrium that leads to a steady final interface roughness, as schematically
illustrated in Figure 7.1.
The morphology of oxidized Si surfaces was measured directly using atomic force
microscopy (AFM) after the oxide overlayer was chemically removed to reveal the buried
interface. Statistical analysis of these AFM images yields Si surface roughness in terms
of the rms values, which corresponds to the magnitude of the vertical feature size on the
Si surface, and the results were found to agree with the interface region thickness results
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from SIE measurements. Additional information on the shape of the oxidized Si surfaces
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was obtained by fractal analyzing the AFM images. The fractal analysis yields the fractal
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dimension, which is an intrinsic property of a rough (fractal) surface that describes the
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complexity or space-filling ability of the surface. It was found that the fractal dimensions
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of oxidized surfaces are also dependent on the initial Si surface roughness. Oxidation
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r
;
simplifies the interface for the initially rough Si surfaces, but increases the surface
complexity for the initially smooth Si surfaces.
Based on the results of SIE. AFM and fractal analysis, it is concluded that the
dominating interface smoothing effect for the initially rough Si surfaces is due to the
1i
diffusion of Si atoms along the interface, driven by the reduction of free energy and also
by the interface stress. In thermal oxidation, the smoothing is enhanced by the high
temperature effects such as viscous relaxation and chemical reaction at the interface.
191
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
O)
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Oxidation Extent
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Figure 7.1
Schematic illustration of interface roughness evolution during oxidation
processes: the roughness for different Si surfaces eventually converges and
reaches a steady roughness.
192
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
while in ECR plasma oxidation it is enhanced by the local intensification o f external
electric field around Si protrusions at the interface.
On the other hand, interface roughness evolution for the smooth Si surfaces is
dominated by the roughening effect. The cause o f roughening is not totally understood,
but based on our results it is speculated to be related to the generation of defects, and the
results can be explained using a chemical reaction mediated roughening transition model.
The present study together with previous studies demonstrate that SIE is a very
powerful technique for the non-destructive study of film-covered interfaces. It is also
shown that rms values and the fractal dimension provide information on the different
aspects of roughness. The rms value calculated from direct AFM measurement provides
information on the vertical magnitude of roughness, while fractal dimension describes the
shape or complexity of roughness. When rms roughness is combined with fractal
dimensions, a more complete quantitative representation of surface roughness is
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achieved.
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The present study leads to several future research directions:
I
(1).
Further application of fractal dimension analysis
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Through the results of the present study and also of previous research work, the
fractal dimension as a roughness parameter has shown its usefulness in providing the
!
shape or complexity of rough surfaces. However, one of the main reasons we choose
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fractal dimension over other parameters to represent roughness is that it is a single
parameter that has the potential of being incorporated into some other roughness studies,
for example, used as a parameter in optical models for other techniques. This potential,
however, has not yet been realized.
(2).
Further study of the roughening effect for smooth Si surfaces
193
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
I
As mentioned the mechanism for roughening remains not fully understood. The
chemical reaction mediated roughening transition needs to be confirmed by results from
other experimental techniques. More extensive work needs to be done to substantiate the
proposed model.
(3).
Interface roughness evolution for ultra-thin oxide films on smooth Si surfaces
This represents a more realistic situation, and thus future work should be
concentrated more on the subject since it will be more technologically beneficial.
(4).
Correlation of electrical properties and the structural properties of the interface
The present work concentrated on the structural properties of the interface. In
microelectronics applications, it is the electrical properties that really matter. It is
therefore necessary to relate the structural study results with the electrical properties and
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find out under what conditions the best interface can be obtained. Some preliminary work
i
has been done on thermally oxidized Si surfaces, but more extensive study is needed for
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better understanding.
(5).
Different interface models
In the present study we choose to use a simple interface optical model for the SIE
analysis. It is also possible to use other more complicated models. However, more
complicated models often mean that more unknown parameters need to be fitted. It is
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known for model optimization that having more variable parameters in the fitting often
yields more serious correlation among the variable parameters and thus often reduces the
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reliability of the fitted result. However, it is always desirable to find a more realistic
!
physical model (not necessarily more fitting parameters) to achieve a better
understanding of the physics.
194
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
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