computational studies of stress and structure development
TRANSCRIPT
Computational studies of stress andstructure development resulting from
the coalescence of metallic islandsby
Andrew Rikio Takahashi
B.S. Materials Science and Engineering
Massachusetts Institute of Technology, 1999
SUBMITTED TO THE DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING IN PARTIALFULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE IN MATERIALS SCIENCE AND ENGINEERING AT THE MASSACHUSETTSINSTITUTE OF TECHNOLOGY
SEPTEMBER 2007
COPYRIGHT 2007 MASSACHUSETTS INSTITUTE OF TECHNOLOGY. ALL RIGHTS RESERVED
Signature of Author .......................... :...-...;.... .. ...................Department of Materials Science and Engineering
August 3rd, 2007
Certified by ............................ ....... ...- . . ............Carl V. Thompson
Stavros Salapatas Professor of Materials Science and EngineeringThesis Supervisor
/7
Accepted by.............. S.................. ........ ........... .............Samuel M. Allen
POSCO Professor of Physical MetallurgyChair, Departmental Committee on Graduate Students
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Abstract
Thin film component properties are critical design elements in almost all industries. These filmsare particularly important in the performance of micro- and nano-electromechanical systems(MEMS and NEMS). Residual stress in thin film components is often treated as an unavoidableside effect of processing steps and the degree of residual stress can drastically affect theperformance and properties of the final product. While high levels of residual stress are oftendetrimental to performance, control of the stress and stress gradients can also be used toenhance performance and even generate new capabilities.
The work presented in this thesis examines the role of island coalescence in the development ofstructure and stress in thin films. The primary methods of investigation are molecular dynamics(MD) and finite element analysis (FEA). The semi-empirical MD calculations show thatcoalescence is a very rapid process for unconstrained spheres and for hemispheres allowed toslide on a frictionless substrate. Particle rotations are commonly observed during thecoalescence calculations. The extent of neck formation between 2 particles is consistent withcontinuum models even down to length scales which would normally be outside the range inwhich the models might be expected to be applicable. The MD calculations also show thatinternal island defects may be induced by the island coalescence process, but only for aparticular range of island sizes. We present an energetic model for the existence of such a sizerange and have located experimental evidence in the literature for such defects. Our FEA workextends an earlier study on the effects of contact angle on island coalescence. Our FEA study ofislands with greater than 90 degree contact angle coalescence shows that neck formation occursvery similarly to the free sphere coalescence case.
We conclude that MD and FEA calculations are useful tools in analyzing the island coalescenceprocess and can provide mechanistic insight beyond what is available from the more generalcontinuum models.
Table of ContentsA bstract .......................................................................................... ..................................... 2Table of Figures ..................................................................................................................... . . ...... .. 51 Acknow ledgm ents ........................................................ ............................................... 62 M otivatio n ............................................................................... ........................................... 8
2.1 Applications of metallic thin films................................................. ......................... 82.2 Stress engineering .................................................................................... ................... 102.3 Reliability and stress................................................. ................................................ 112.4 Fundamentals of film formation .................................................................................... 13
3 Backgro und ......................................................................................... ............................. 153.1 Stresses in thin film s....................................................... .......................................... 15
3.1.1 Therm al stress .................................................................... .............................. 153.1.2 G row th stress ......................................................................................................... 17
3.2 Particle sintering and coalescence .................................................... 193.2.1 Continuum-based modeling ............................................................................. 203.2.2 Com putational studies .................................................................... ................. 26
3.3 Mechanical deformation of islands .......................................................................... 273.4 Stress relief mechanisms of supported atomic clusters ....................................... 27
4 -M etho ds ........................................................................................... ................................ 324.1 Molecular dynamics .................................................. 32
4.1.1 Embedded atom method ........................................................... ................. 324.1.2 Predictor-corrector iterators............................................................................... 344.1.3 Lennard-Jones potential ....................................................................................... 344.1.4 Thermodynamic averages .......................................................................... 354.1.5 Metrics for local atomic environments ........................................ ......... 36
4.2 Finite elem ent m ethod..................................................... ........................................ 375 Calculations of time required for coalescence.......................... ................ 39
5.1 Initial structure setup ...................................................... ......................................... 395.2 Structure equilibration ................................................................................................... 395.3 Structure placem ent ........................................................................... ..................... 405.4 Evolution over tim e ............................................................................ ...................... 40
6 Calculations of length scales during coalescence ............................................................ 426.1 Analysis of neck form ation .......................................................................................... 42
6.1.1 Binning algorithm for neck size determination ................................................... 426.1.2 Computed neck size via MD ............................................................................. 43
7 Defects induced by boundary formation ..................................... ........ ....... 447.1 Internal structure of isolated clusters............................................ ................. 447.2 Internal structure of islands on substrates ......................................... ...... 447.3 Structural analysis of coalesced structures................ ...... ....... 45
7.3.1 Types of stacking faults .................................................................... ................ 457.3.2 Size window for stacking fault formation ........................................ .... 467.3.3 Model for the size dependence of defect formation............................. ... 47
8 FEA of coalescence with a greater than 90 degree contact angle ..................................... 539 Conclusions ................................................................................................... . ... ............. 5910 Future w ork ......................................................................................................... .... ............ 61
11 Bibliography ...................................................................................... ................................ 63
Table of Figures
Figure 2.1: The structures shown in this micrograph are self-assembled interconnect arrayprobes generated using the StressedMetal process from Xerox-PARC. Image taken from theXerox-PA RC w ebsite........................................................................................... ................................... 11Figure 3.1: Typical behavior of the stress thickness product during the deposition of a metal onan amorphous substrate. This particular case is Au on silicon nitride at room temperature and adeposition rate of 1 A/s ................................................................................................................................... 17Figure 3.2: Schematic drawing of coalescence for various contact angles .............................................. 19Figure 3.3: Neck growth behavior predicted by Cabrera for a 1 nm sphere of material withdifferent values of B ................................................................................................................. . . .... 21Figure 3.4: Bulk diffusion enhancement from reduction in particle size for Au ......................................... 30Figure 5.1: Evolution of center of mass distances over time. The coalescence process iscom plete in a m anner of nanoseconds....................................................... .......................................... 40Figure 6.1: Comparison of the atomistic calculation data with continuum models. The diamondsdenote the atomic calculation data and the lines denote existing continuum models. The 2-DFreund and Chason model and the Seel FEM model produce very similar results for the zippingheight ....................................................................................................................................................................... 43Figure 7.1: Representation of a coalescing cluster structure. The lighter atoms denote locallyhcp atom s w hich com prise a stacking fault. ........................................................................ .................. 45Figure 7.2: Histogram showing the number of calculations which included stacking faults for thedifferent sizes. These calculations were tabulated for the free space coalescence case............... 46Figure 7.3: Illustration of the unfaulted and faulted sphere configurations .............................................. 47Figure 7.4: Projection of the fault plane in the sphere showing the relevant distances used in themodel. ........................................................................................................................................................................ 48Figure 7.5: Critical stress for generation of an induced stacking fault for Ag. The predictedcoalescence stresses for the most applicable continuum models are also shown. The insetshows the small thickness strengthening effect ..................................................................................... . 51Figure 8.1: FEA mesh used to analyze the strains required for island zipping. ..................................... 54Figure 8.2: Strain energy mapping using up and down zipping distances as parameters ....................... 55Figure 8.3: Contour plot showing the strain energy required to zip in each direction for a contactangle of 110 degrees. ................................................................................................................................... 57Figure 8.4: Plot of the energy minimizing zipping distances for a 5 nm island with a contact angleof 110 degrees................................................................ .......................................... ............. ..... 58Figure 8.5: Zipping stress as a function of contact angle ................................................................. 59
1. Acknowledgments
First of all, I am deeply indebted to the Singapore-MIT Alliance and to the NSF for financial
support of this project over the years.
I would also like to express my appreciation to the entire Materials Science and Engineering
Department at MIT. Nearly everything I know about science and engineering was drilled into me
during my undergraduate years. Everything else, I learned during my year at Stanford and in the
subsequent years in graduate education at MIT.
Upon my return to MIT, I was very fortunate to find immediate challenges that enabled me to
become engaged with the work. For computational assistance for the early portion of the work,
I am particularly grateful to Prof. Carter and Prof. Yip. Of course, I owe a huge debt of gratitude
to my advisor, Prof. Carl Thompson, who gave me the freedom to pursue the challenges posed
by the project in my own way.
For both technical and moral support, I would not have progressed very far without the
assistance of the post-docs and fellow graduate students in the research group. From the old
guard, Mauro Kobrinsky, Steve Seel, Rob Bernstein, and Cody Friesen provided a tremendous
resource of information regarding thin film stress. From the 'middle' guard, I drew a great deal
of motivation from the discussions I had with Zung-sun Choi, Frank Wei, Amanda Giermann, and
Reiner Moenig. When I started to hit the doldrums in the latter part of the project, the new
students arriving in the group reenergized my interest. I feel particularly fortunate to have
gotten to know Steve Boles and Jeff Leib, both through scientific discussion and Wii parties.
My proudest non-academic achievement at the Institute was the foundation of the MIT
Micropoker Club. Even during my hardest weeks at the Institute, the club always provided a
stress relief valve for me and I expect the friendships formed during the club 'meetings' will last
me the rest of my lifetime.
For the rare occasion when the Micropoker Club counselors were not sufficient in reducing my
stress level, I found the trained professionals at MIT Mental Health to be quite capable. I am
particularly grateful to Dr. Hsi for getting me through a rough patch right before my wedding.
Although I did not really appreciate it much until the very end, my family's support is the main
reason I have made it this far and is also a huge influence in my life moving forward. My core
values are based largely on my parents' teachings and much of my development as a person
comes from chasing after my brother's accomplishments and exploits.
Lastly and most importantly, I want to thank my wife, Melody, who has been a dominant
influence in my life for nearly 10 years now. While my return to MIT from Stanford was driven
mostly by the desire for new academic challenges, my desire to share geography with my closest
friend was probably just as important in my decision to return to Cambridge. Her love and
support have kept me going even during my darkest moments. Now that we no longer need to
be separated, I am very much looking forward to joining her and starting our new life together
in North Carolina.
2 Motivation
The primary motivation for this project is the universal need for metallic thin film components
and the ability to tailor the performance of those components through processing conditions.
While metallic thin film components are typically employed as passive devices, these thin films
are essential to the operation of critical active devices such as the transistors in
microprocessors. Even in microprocessors, the limiting factor in circuit performance is now the
RC delay in the thin film interconnects. In mechanical applications, the performance
characteristics of thin film components are heavily dependent on the processing conditions of
the thin film, while in electrical applications the reliability characteristics are more affected.
Typically, the important thin film properties altered by processing are the microstructure and
internal stress. The ability to engineer the internal stress through control of processing
parameters has lead to the development of novel devices and has the potential to introduce
technological breakthroughs while using standard thin film deposition equipment. This project
is aimed at understanding the fundamentals of thin film formation through computation in
order to enable thin film engineers to more effectively control stress and structure in the pursuit
of new technology.
2.1 Applications of metallic thin films
Thin metallic films are an essential component for a wide variety of products. From applications
such as reflective coatings in food packaging up to performance critical interconnects in
microprocessors, ultrathin metal films are ubiquitous. In many cases, thin films have properties
which are fundamentally different from their bulk counterparts. In other cases, thin films are
used simply because the geometry of the application precludes the use of bulk material. Thin
films are also used in many applications where the material required is prohibitively expensive
so economics drives product designers to use the minimum amount of material required.
One of the major driving industries for the commercialization of thin films is the computer
microprocessor market. Ever since the first integrated circuit was developed to replace vacuum
tube processors, the industry has been driven to high density, low cost devices. With over 300
million transistors and miles of connecting wire on a typical chip, planar thin film processing is
nearly synonymous with semiconductor microfabrication (1).
In a similar vein, magnetic storage companies are constantly trying to improve the bit density on
hard drive platters or find an alternative technology which provides improved storage capacity
(2). Careful engineering of the grain size distribution has allowed for the creation of magnetic
media with densities near the super-paramagnetic limit.
The media used for optical data recording is also comprised of thin films. While the phase
change materials are chacogenide glasses, new schemes for reading and writing information to
the media involve nanoheaters manufactured from purely metallic Pt thin films (3).
Another interesting application for metallic thin films is their use as thermal barrier coatings. As
an example, silver-based multilayers are often used to coat large windows which are in direct
sunlight. The optical properties of the thin silver films allow optical frequencies of light to pass
through the window while reflecting away infrared frequencies which would lead to undesirable
heating of the building interior (4). Earlier work highlights the different requirements for
energy-saving windows in both warm and cold climates and outlines the theory for generating
selectively reflective windows (5).
Typically, hard coatings for tools are fashioned out of ceramics but these brittle thin films are
often in danger of cracking. In order to improve the crack resistance, the hard ceramic coatings
are often layered with ductile metals. Inelastic yielding in the metal thin films increases the
fracture toughness of the hard coating without having noticeable deleterious effects on the
hardness (6).
On the more exotic side of applications, much research has been devoted to the use of metal
particles as plasmon waveguides. One route for generating these metal particles would be from
the dewetting of a thin film. The potential for generating ordered arrays of particles could have
implications for energy storage and transfer (7)(8).
2.2 Stress engineering
The properties of thin films depend significantly on the manner in which those films are
processed. One of the most critical characteristics of a thin film is the amount of internal stress.
Different processing conditions can lead to different levels of stress within the film. A deeper
understanding of how this stress develops allows process engineers to tailor thin film structures
by manipulating the internal stress.
Researchers at Xerox have managed to form robust interconnect systems and on chip inductors
by engineering stress gradients (9). Adjustments of the sputtering power can lead to
development of large stress gradients through the thickness of the film. The binding of the film
to the substrate restricts the amount of curvature that the film can experience. However, if the
film is partially released from the substrate via a chemical etch, the resulting curvatures can be
quite large. Figure 2.1 shows an example of a structure generated using the above mentioned
chemical release process.
The intrinsic stress level of a metal thin film can also be controlled via thermal processing.
Different annealing temperatures can lead to precipitate formation which will alter the level of
stress within a thin film. The thermal mismatch between film and substrate can also lead to
inelastic deformation yielding of the film and repeated thermal cycles can be used to set the
appropriate level of stress within a film.
The reliability of thin film components depends strongly on the level of internal stress within the
film. Component failure can be the result of standard fatigue processes in the case of dynamic
loading. In the case of static loading, electromigration or plastic yielding can also lead to
component failure. On the more optimistic side, tailoring of the internal stress can also result in
beneficial effects on the component reliability.
For dynamically loaded thin films, accumulated fatigue damage occurs as the result of surface
cracks, voids and extrusions (10). This behavior differs from the bulk behavior since the thin film
geometry inhibits the formation of dislocations and persistent slip bands. For thin film fatigue,
the formation of free surface and grain boundary defects is more energetically favorable than
the generation of dislocation cells. The reason for the change in mechanism is that the thin film
geometry alters the localized stress state.
For microprocessor interconnects subjected to DC electrical loads, component failure occurs due
to electromigration. The impact of electromigration voiding in interconnects can be greatly
amplified due to the thermal stress voiding which occurs during interconnect processing (11).
The existence of stress gradients within a line is an important factor in determining whether the
critical stress required for void nucleation will be reached or if the steady state will result in an
immortal line(12)(13).
One of the most dramatic demonstrations of reliability enhancement due to stress engineering
is the Prince Rupert's drop experiment. The Prince Rupert drop is essentially a thermally
engineered glass bead. The thermal processing of a Rupert's drop results in a highly
compressive stress near the surface of the drop-head. Internally, the drop is under extreme
tension. The compressive nature of the drop head makes the glass highly resistant to impact
damage, but releasing the tensile stress by cracking the tail of the drop results in explosive
decomposition of the glass(14). While the behavior will likely be quite different in ductile
materials, a metallic analog of the Prince Rupert's drop could be quite useful in enhancing the
reliability of thin film coatings.
2,A Fundamentals of film formation
The formation of thin films on substrates depends strongly on the surface energy of the material
being deposited and of the substrate. In cases where full wetting occurs, growth occurs by a
layer by layer process and is known as the Frank-van der Merwe growth mode. For the case of
partially wetting systems, three dimensional islands are formed on the substrate and full
continuity is achieved after island impingement occurs. This type of growth mode is called
Volmer-Weber. A third growth mode occurs for systems which are initially wetting but
transition to island growth. This type of growth mode is known as Stranski-Krastanow
(15)(16)(17).
The deposition of metals on amorphous substrates typically occurs via the Volmer-Weber
growth mode. Well documented cases of this growth mode are seen for low magnitude
cohesive energy metals such as Au, Ag, Cu, Al, Ni, Pt, Fe, Pd, Co, and Pb on higher magnitude
cohesive energy solids such as ionic crystals and stable oxides and nitrides (15).
Deposition via electron beam evaporation is well described by the Langmuir equation. The flux
of material out of the source is given by
N PO
AT 27rmkBT
Here N is the number of arriving atoms, A is an area, i is a time interval. PO is the equilibrium
vapor pressure at the source temperature, m is the atomic mass, kB is Boltzmann's constant and
T is the source temperature. The arrival flux on the substrate depends on both the source to
substrate distance and the source temperature. In practice, the arrival flux is most commonly
altered through changes in the source temperature. In most e-beam systems, the input power
of the electron beam is easier to adjust than the sample to source distance.
The theory behind the nucleation behavior of Volmer-Weber thin films has been developed in
some detail with a rate equation approach (18). Recent modifications of the rate equation
approach have allowed for the treatment of surfaces which have defect sites or mobile clusters
(19). Still, film formation via Volmer-Weber growth is a complex process and the island
structures are often quite complex. In-situ TEM experiments of metals on ionic crystals have
been performed by Pashley and co-workers (20)(21)(22). The observed films started as isotropic
nuclei with equiaxed lateral dimensions, but continued growth quickly led to anisotropic island
shapes due to island coalescence and the formation of channels between the larger islands.
These channels can exist for much longer than would be expected from isotropic growth
conditions. Eventually, the channels fill and full continuity is achieved.
3 Background
This computational effort deals mainly with the stress and structural changes induced during
island coalescence. This chapter will start with a brief review of the general stress sources and
characteristics of high atomic mobility metallic thin films on amorphous substrates. This
overview will be followed by a more detailed description of the stress and structural models
which are particular to the island coalescence process. We will also review computational
efforts similar to ours in order to highlight the extensions and new developments that the
current work provides. Towards the end of the chapter, we discuss the current understanding
of mechanical deformation of nanoscale islands and some of the stress relief mechanisms which
are also expected to be active during island coalescence.
3.1 Stresses in thin films
3.1. Thermal strless
One of the major sources of stress in thin films is the stress generated by differences in the
linear coefficient of thermal expansion. A dissimilarity between the thin film and the thick
substrate leads to an additional stress if the system is allowed to change temperature.
Conventionally, we assume that the thin film accommodates the thermal strain in the substrate
layer.
athermal = AaAT
Here, A a is the difference between the expansion coefficients and A T is the temperature
change from the initially strain free deposition condition. Typically the thermal expansion
coefficient for the metal is 3-4 times the value for the amorphous insulator. If the substrate is
composed of multiple layers such as nitrided or oxidized silicon, the thermal expansion will be
largely dictated by the thicker component. Some representative values of thermal expansion
coefficients are given in Table 1.
Table 1.: Thermal coefficient of expansion for selected materials. Values for metals are obtained fromWebElements, Values for compounds and glasses are obtained from the ASM Engineered Materials Handbook,
Material a (10 - 6 K- 1 )
Ag 18.9
Au 14.2
Cu 16.5
Si 2.6
AI203 8.0
SiO2 8.6
Si3N4 2.9
Borosilicate glass 3.5
Almost any type of manufacturing process will involve large magnitude thermal cycles. This
statement is particularly true for the microelectronics industry where high temperature anneals
to control dopant concentration are often required. These thermal cycles must be accounted
for in design processes and the effects on the internal structure and performance of the film can
be dramatic.
Consideration must also be given to the operating conditions of the thin film. For applications
such as laser-based actuators and fast switching MEMS, the local temperature of the film during
operation conditions can cause large thermal stresses. If these stresses are not accounted for
appropriately, positioning and deflection errors can occur.
3.,2 Growth stress
The development of intrinsic stress within a growing film is a complex process. The general
behavior is shown below in Figure 3.1.
·z
CO
F-~ZCOCo~
C)
CoL~
-1'0 100 200 300 400
Thickness (Angstrom)
Figure 3.1: Typical behavior of the stress thickness product during the deposition of a metal on an amorphoussubstrate, This particular case is Au on silicon nitride at room temperature and a deposition rate of I A/s.
500
3.1.2.1 P recoalescence stress
In the earliest stages of film formation, many researchers have noted a precoalescence
compressive stress for fcc metals on amorphous substrates. Most notably, Au and Al on SiO2
show a clear compressive dip at the beginning of deposition. This dip is absent in observations
of Ag on SiO2. The mechanism most often referred to for the precoalescence stress phenomena
is a capillary pressure argument developed by Cammarata et al (23). Other explanations have
included the effects of step density and the force dipole induced by adatoms on the substrate
(24). Our studies of Au on nitride and Ag on nitride have not shown this behavior to be
reproducible and as indicated in the figure above, we often find the initial transient to be tensile
rather than compressive.
3.:.2.2 Stress arising during coalescence
Previous studies have shown that the tensile rise during film deposition is due to island
coalescence (25). The existing models for stress generation due to island coalescence will be
discussed in detail in the next section.
To some extent tensile behavior during the period of island impingement and coalescence is
also expected due to grain growth and densification. Any process which leads to densification of
the thin film will also result in tensile strain.
3.1.2.3 Sre.ss behavior durinq continued deposition
After a high mobility metal thin film reaches the point of continuity, the growth stress becomes
more compressive. The source of this later stage compressive stress is still under debate.
Possible explanations for the asymptotic compressive stress have included the incorporation of
excess adatoms at surface ledges (26), the compressive nature of adatom force-dipoles(27), and
an excess atomic concentration within grain boundaries (28). However, recent atomistic
calculations show that the adatom concentrations necessary for the force-dipole model are
unlikely to be achieved under normal deposition conditions (29).
3.2 Particle sintering and coalescence
Island coalescence has a close analog in the ceramics sintering literature. In its own right, the
behavior of islands coalescing on a substrate has been considered within a purely mechanical
framework as well.
Figure 3.2: Schematic drawing of coalescence for various contact angles.
Figure 3.2 is a schematic representation of how island coalescence is typically modeled. The
first case shows coalescence of islands with 90 degree contact angles. This case is the most
common case used in models since the equations describing a semi-circular or hemispherical
island have convenient mathematical forms. However, the strain field arising from the point of
Zf--
impingement may be heavily dependent on the local geometry near the contact. The second
case in the illustration shows coalescence between islands with less than 90 degree contact
angles. This type of coalescence is expected for materials which have a high degree of
wettability on the substrate. The third case in the illustration shows the coalescence for islands
with a greater than 90 degree contact angle. The greater than 90 degree contact angle case is
expected when the deposited material wets the substrate poorly. Most high mobility metals on
amorphous substrates will fall into the third case.
3.2.1 Continuuim-based modeling
The behavior of spherical particles in contact at a single point has been analyzed by many
researchers in an attempt to understand sintering. Kuczynski found that the neck radius
between spherical particles could be fit to a power law in the early regime where the neck size is
much smaller than the particle radius. Kuczynski and later Cabrera studied the phenomena with
fairly large particles and Cabrera arrived at a slightly different power law than Kuczynski (30).
For sphere-sphere coalescence events, the Cabrera model shows
-- 25Bt.a
2
Here, x is the neck radius while a is the particle radius. t is time and B is the constant defined as
Dsyvf 2
B=kBT
where Ds is the surface diffusivity, y is the surface tension, v is the number of atoms per area,
and 0 is the atomic volume.
The extracted value for B obtained from sintering experiments on Cu is approximately 10-29
cm4 /s (31). Mullins used known chemical data in his earlier paper to arrive at an estimate for B
of 10-18 cm 4/s (32).
0.4
0.3
Ct/-ItC
0.2
0.1
n0 le-09 2e-09
time (s)3e-09 4e-09
Figure 3.3: Neck growth behavior predicted by Cabrera for a I nm sphere of material with different values of B.
While considering more general surface diffusion and surface morphology issues, Mullins
developed a comprehensive mathematical theory of morphological evolution based on the
different mechanisms of surface diffusion, bulk diffusion, grain boundary diffusion, and
evaporation/condensation.
B=le-29 cm4/s
B=le-27 cm 4/s
B=le-24 cm 4/s
- -
1
a
ir. .
*.
. .. . . . . .
m
. -.
dlmo~w l m lW i •~ l
)
Nichols went on to implement the derived theory within a computational framework and using
finite difference methods to solve the morphological diffusion equation. Nichols's numerical
results produced yet another power law for the neck radius between equal sized particles. The
form used by Nichols was
an B B aK- = y -as)at y as asHere n is the surface normal, c s is the differential surface segment and K is the curvature (33).
In the sintering literature, the energetics of particle arrays has been reviewed by Kellett and
Lange (34). In their consideration of the energetics, particles were assumed to retain their
energy minimizing surface curvature during interpenetration. The energetic model does not
accurately capture the transient behavior during neck formation but does provide a reasonable
estimate of the energy minimizing particle configurations.
Stress in the Kellett and Lange model is based on the dihedral angle and size of spherical
particles in a compact. The sintering stress defined in the ceramics literature is not a true stress
but is really an energetic parameter which captures the capillarity driving force for sintering.
The sintered compact is assumed to be stress free since the mechanisms for sintering should all
lead to a reduction in stress. For a linear array of spherical particles, the sintering stress is
Similarly, the limiting shrinkage strain of a linear array of spherical particles is given in terms of
the equilibrium dihedral angle as
1(2cos2() 3
Within the context of thin films, Hoffman was the first to propose that islands coalescence
would lead to tensile film stresses (35). Hoffman's original argument was presented as a one
dimensional model where the total system energy could be reduced by closing the gap between
two islands. In order to close this gap, each island would have to generate a spontaneous
tensile strain and form a grain boundary. For block-like islands, the resulting tensile stress was
given by
EAa'Hoffman -
Here E is the Young's modulus of the island, L is the island size and A a is the size of the gap
between islands. The gap size is conventionally set to the atomic dimension of the element
comprising the island.
Nix and Clemens chose to refine Hoffman's original argument by considering more realistic
island geometries (36). The Nix and Clemens analysis modeled two-dimensional elliptical islands
at the point of impingement. By altering the starting geometry of Hoffman's original model, Nix
and Clemens were able to eliminate the use of the atomic gap parameter. By performing an
energy balance between strain energy and surface energy, the resulting expression for the
zipping height is
zo [36(1 - v)(1 + v) 3 (2ysv - Ygb
b Ea
where z o is the zipping height, b is the y-axis length of the ellipse, v is the Poisson ratio, Ysv is
the island-vapor surface energy, Ygb is the grain boundary energy and a is the x-axis length of
the ellipse. The average stress within the island can be expressed as
[(1 + v) E(2ysv - Ygb)21=--v) a
Or alternatively in terms of the zipping height,
< U > E 1 1 ZO 2 b arcsin Y_1V2 2 b (2zo)
While the Nix and Clemens model(36) captures the essentials of the balance between surface
and strain energy, the energetic criteria used by those authors gives an upper bound on the
zipping height and average stress in the film. Experimental evidence showed that typical metal
films on amorphous substrates have an order of magnitude less stress than predicted by the Nix
and Clemens model.
In order to improve the predictions of the model, Seel et al (37) used FEA and a different energy
minimization criteria to find the equilibrium island configuration in terms of zipping height and
stress. The Seel model excludes the effects of surface and bulk diffusion since on an elastic time
scale under the assumption that these processes should be slow.
In a further extension to the numerical model, Seel and Thompson (38) also considered the
coalescence of islands with a less than 90 degree contact angle. In this case, the elastic strains
for zipping are increased greatly over the 90 degree contact angle case. The critical parameter is
recast as the maximum in-plane strain and the results for any contact angle can be mapped onto
a single curve using this parameter.
In an analytical approach starting from Hertzian contact theory, Freund and Chason examined
the impingement of periodic arrays of islands (39). The consideration of cohesion during the
impingement of islands requires invocation of the Johnson-Kendall-Roberts(JKR) theory (40).
For the impingement of 3 dimensional islands, Freund and Chason found that the stress of
impingement is unrelated to the mechanical properties of the island and only depends on the
island geometry. However, in a realistic coalescence scenario, mixing of 2-D and 3-D
coalescence events will occur and the elastic properties of the solid will again become relevant.
The general expression derived for the stress at impingement is
< > N YSV 2Y gb
Note that y is defined differently here so I should either alter the equation or make a note of the
notation change. R is the particle radius. N refers to the dimensionality of the coalescence
events. c1=1/2, c2=2/3 and c3=1. A1=0.82, A2=0.44, and A3=4. For the Freund and Chason
model, the dimensionless neck size at zero load is constant at 4.46 for 3 dimensional
coalescence and constant at 2.2 for 2 dimensional coalescence.
Rajamani et al developed a model which includes the effects of continuous tensile stress
development during film growth(41). This model may be applicable to metals with low mobility
but does not accurately capture the phenomenology for the high mobility metals such as Ag, Au,
and Cu. The FE model presented is most applicable for faceted semiconductor growth. The
model captures the local stresses at the grain boundary triple junction and in the absence of
stress relaxation and diffusion, Rajamani et al predict that the majority of tensile stress
generation occurs continuously during boundary formation rather than instantaneously as
predicted by the other continuum models. While the instantaneous effect at island
impingement is still present, Rajamani et al claim that the more significant contribution to the
tensile stress development is the continuous advancement of the island faces during growth.
3,2.2 Computational studies
The sintering of copper nanoparticles was previously studied by Zhu and Averback (42)(43).
Their research showed that the particles exhibited extremely high shear forces during
coalescence and that these forces were often large enough to induce defects. They did not
however examine the dependence of defect formation on the size of the particle.
A more comprehensive study of Au particles was undertaken by Lewis et al (44). These
researchers examined the effects of melting, freezing and island coalescence for Au particles.
The study incorporated clusters of different sizes but fewer calculations were performed on the
large clusters. The coalescence portion of the study focused on the differences between liquid-
liquid, liquid-solid, and solid-solid coalescence events. Surprisingly, the solid-solid coalescence
events occurred on timescales which were just as rapid as the liquid-liquid and liquid-solid
coalescence events.
Mazzone's study of island coalescence used the most detailed potential model and many
rigorous metrics for cluster structure (45). However, Mazzone restricted his study to cluster
sizes'in the 50-300 atom range of Ag, Cu, and Pb. For the size ranges studied, Mazzone
concluded that the continuum theories for sintering were inadequate in determining the extent
of neck formation and the timescales for sintering. The simulated clusters formed necks at rates
orders of magnitude faster than would be predicted using a Cabrera model and standard values
for surface diffusivity. Mazzone noted that the failure of continuum sintering theories involving
diffusion suggests that nanoparticle sintering occurs by a different mechanism. Athermal
sintering was not explicitly defined in Mazzone's work, but an athermal sintering mechanism is a
possible alternative to the diffusive mechanisms which were identified as inadequate.
3.3 Mechanical deformation of islands
For bulk materials, the dependence of the yield strength on grain size is well-known. The Hall-
Petch relation is used to justify the dependence on yield stress. The basic argument for the
Hall-Petch effect in the bulk is that the grain boundaries serve as barriers to dislocation motion.
This impedes plastic deformation and results in increased yield strength. As we approach thin
film dimensions, the situation changes. Due to the reduced dimensionality present in a thin film
1 1the scaling law follows a dependence rather than a 1 dependence. Thompson used an
energetic model to explain why the dependence for thin films would differ from that of a bulk
material (46). If we view nanocrystals as an even further reduced dimensionality, we notice that
the dependence of yield strength on grain size changes drastically. In the case of nanocrystalline
materials, the yield strength starts to again decrease as grain size is decreased. This inverse
Hall-Petch effect has been attributed to grain boundary sliding or other deformation processes
not involving the standard dislocation cell approach(47)(48).
3.4 Stress relief mechanisms of supported atomic clusters
In our discussion of the coalescence stress, we shall be referring to the instantaneous stress
which elastic models such as the NC and FC models predict will result from boundary formation.
This elastic stress will be an upper bound on the actual measured stress since several stress
relief mechanisms will occur simultaneously with the boundary formation stress. These
mechanisms may include any combination of surface diffusion, bulk diffusion, grain boundary
creep, island-substrate interfacial creep, viscous and plastic flow, dislocation creep, or other
complex processes.
Surface diffusion is likely to be the dominant relaxation path for high curvature islands. The
governing differential equation for surface diffusion is 4th order. Data on surface diffusion is
scarce and different surface diffusion mechanisms can lead to estimates of surface diffusion
which vary over many orders of magnitude. However, redistribution of the material at the
surface of the clusters does not greatly reduce the bulk stress in the film.
In order for surface diffusion to be an effective stress relief pathway, grain boundary diffusion
must also occur and will possibly be rate limiting. Typically grain boundary diffusion is studied
under the A-,B-, and C-kinetic regimes based on the Fisher model for a grain boundary (49). The
C-kinetic regime refers to the situation where the enhanced diffusion region is completely
limited to the grain boundary width. The C-regime is most commonly observed when the grain
size is much larger than the grain boundary width. The B-kinetic regime occurs when the
enhanced diffusion region extends deeper into the grain but the profile is still identifiable. In
the A-kinetic regime, the grain size is small enough such that the enhanced diffusion regions of
each grain boundary interact and the grain boundary diffusion becomes inseparable from the
bulk and surface diffusive behavior. To our knowledge, all recorded values of grain boundary
diffusivity are in the B- or C-kinetic regimes while we expect the small island sizes at coalescence
to be dominated by the A-kinetic regime.
Normally, atomic diffusion through the bulk of the island would be a relatively slow process.
However, the coupling of bulk diffusion to surface and grain boundary diffusion along with the
high levels of stress in the coalesced islands may lead to atomic creep in the bulk. Bulk diffusion
coefficients are also known to scale with the melting temperature of the solid in question and
the decreasing melting temperature with decreasing particle size could lead to homologous
temperatures high enough to make bulk atomic transport a significant stress relaxation
mechanism. Brown and Ashby (50) have shown that a good estimate for the bulk diffusion
constant in fcc metals is
D - 5.4 x 10- s exp (-18.4 ) in units of m2/s.
Buffat and Borel (51) have shown that the melting temperature as a function of particle radius is
given by
2
Tm = Ti 1 - sysv - 1ý
where T° is the bulk melting point, L is the latent heat, and Ps and pi are the specific gravities of
the solid and liquid phase, respectively. Using material parameters for Au,
Tm = 1336 x (1- 4.24710-10)
where r is input in units of m and melting point is returned in K. Combining the above equations
gives
D(r, T) = 5.4 x 10-5 exp (2452.4 1 - 4.247 ))
The magnitude of the bulk diffusion enhancement for Au is plotted in Figure 3.4.
X-1
2 4 0 I)
radius (nm)
Figure 3.4: Bulk diffusion enhancement from reduction in particle size for Au.
Diffusion enhancement at room temperature begins to have a significant effect for sizes below
4nm. While the average particle size at coalescence is usually larger than 4nm, collisions of
larger islands with smaller islands which may be present due to secondary nucleation may be
affected by enhanced diffusion in the smaller particle. Also, the diffusion enhancement for very
small sizes should be considered for early stage growth and nucleation models as the
assumption of a constant diffusion coefficient will not apply to small nuclei.
Stress relaxation is not limited to the above mentioned diffusive mechanisms. The generation of
1-D and 2-D defects within the coalescing islands can also lead to a reduction of stress.
Localization of the strain field by the formation of dislocations can result in a net reduction of
energy and a reduction of the bulk stress. Movement or dissociation of these dislocations can
further reduce the stress in the islands. If the dislocation moves through the bulk of the island,
the process is similar to dislocation creep in bulk materials. If the dislocation moves along the
island-substrate interface, interfacial slip will occur, which could result in rigid body translations
or rotations of the islands.
4 Methods
4.1 Molecular dynamics
Molecular dynamics refers to the computational method of solving the real-space trajectory of
molecular units. In classical MD, the force law that governs the trajectory of the molecules is
simply a matter of applying Newton's second law, F = md. For any reasonably sized system,
the number of molecules involved in the calculation is quite large and computer automated
methods are required to solve the equations of motion. Typically the equations of motion are
solved using iterative methods and standard algorithms from the area of applied mathematics.
The most accurate method of calculating the forces acting on the molecules is to use density
functional theory (DFT). However, DFT calculations are prohibitively time consuming for large
systems.
4.1.1 Embedded atomr method
The embedded atom method (EAM) was developed by Daw, Foiles and Baskes at Sandia(52).
The core principle of the EAM is that the binding energy of an atom is a simple function of the
electron density provided by the surrounding atoms in the matrix. While the physical derivation
of the EAM is unique, the final derived form for the energy functional is not unique. Other
researchers have used alternate physical reasoning to arrive at a potential function which is
identical in practice to the EAM. Among these approaches are the 'glue' potentials of Ercolessi
and the second moment approximation to tight binding potentials provide by Rosato, Guillope,
and Legrand (53).
In the original formulation of EAM
Ec Gi (I pa (rij)) + 2 U1 (rqj)i j*i i,j(j*i)
Here Ec is the cohesive energy, Gi is the embedding energy, p7a is the spherically averaged
electron density around atom j. rij is the distance between atom i and j. Uij is the electrostatic
interaction between atom i and atom j.
The key to effective use of the EAM lies in the selection of the appropriate interaction term Uij
and the exact shape of the embedding functional. Typically, the interaction term requires at
least two adjustable parameters and the embedding functional can use any number of fitting
parameters. It is not uncommon to see EAM calculations which use upwards of 13 parameters
in order to capture the real behavior of a material.
4-..111 RGL potential
This study uses the potential form devised by Rosato, Guillope, and Legrand.
1
Ec=[[- - 2 exp [-2q (r'j- 1)] + A exp [-p ( 1)]]i j
k is the effective hopping integral, q is an effective distance parameter for the attractive portion
of the potential, ro is the atomic radius, A is a parameter describing the strength of the Born-
Mayer repulsion term and p is a parameter related to the distance interaction of the repulsive
term.
In practice this potential has 4 parameters and for the particular case of silver, the parameters
have been fit to the elastic constants and cohesive energy of the bulk solid. The calculated
surface energy using these parameters results in values which are close to the experimentally
observed values.
In order to compare with the form of Equation 3.1, the term UUj has been replaced with the
Born-Mayer repulsive potential. The first term on the RHS of Equation 3.2 represents the
embedding functional.
The key elements in an MD calculation are the equations of motion, the interatomic potential
and the time integrator which will be used to solve the equations of motion(54). Some of the
simplest integrators use numerical derivatives and fixed time steps to advance the equations of
motion. The direct Verlet and Verlet leapfrog algorithms fall into this category. More complex
algorithms which make use of higher-order derivatives have been shown to produce faster
convergence and greater accuracy for certain classes of problems. The Gear predictor-corrector
is a common integrator in MD calculations which takes advantage of higher-order derivative
information(55).
In the MD literature, the algorithm referred to as the Gear predictor-corrector is actually a
special case of a more general predictor-corrector algorithm. The general method of solution is
known as the Adams-Bashforth predictor-corrector but Gear was able to optimize the algorithm
coefficients for solving Newtonian equations of motion. Unfortunately, the optimization of the
algorithm depends on the particular type of motion being analyzed and two sets of Gear
coefficients are commonly used. For atomistic calculations, the value of the second order
coefficient is set to 3/20 instead of 3/16.
As a model for the substrate, the Lennard-Jones potential provides the necessary characteristics.
Initially, the Lennard-Jones potential was found to be an adequate model for the interaction of
gases via Van der Waals forces (56). The form of the potential is physically justified for the
treatment of noble gases and the convenient mathematical form makes it popular with the
study of other systems as well. The general form of the potential is given by
ULJ=- 4 c1 6 U) -
where co is a characteristic distance at which the potential takes on a zero value. The actual
energy minimum occurs at rij = 21/6a o.The U potential is a pairwise potential which is quite
successful in predicting the properties of a non-interacting gas like Argon. For fcc solids, the
potential reproduces the packing correctly but the surface relaxations and mechanical
properties derived from the potential often run contrary to experiment. Nevertheless, this
potential is quite useful due to its mathematical simplicity and is effective as a boundary
condition for atomistic calculations with a rigid wall.
4.L4 Thermodynamic averages
In simple terms, the surface energy is the excess free energy due to the fact that the crystal is
not infinite and terminates at a free boundary. For particular crystal facets, this quantity can be
evaluated computationally. One of the key considerations when performing this calculation is
that the initial slab thickness should be chosen such that the two free surfaces do not interact
with each other. Typically the length of convergence is determined by running the calculation
for various thicknesses until the energy per area becomes constant with some tolerance for
error.
These types of calculations allow for the location of the free surface in the sense of a Gibbs
dividing surface. The Gibbs interface does not necessarily coincide with the physical termination
of the crystal.
The situation becomes far more complex when considering a general surface of the crystal or in
considering a crystal which is not semi-infinite. The analogous quantity for a liquid is the surface
tension, which is measurable both experimentally and computationally. Definition of the
surface energy requires that the total free energy can be partitioned into a surface component
and a bulk component. This type of partitioning is equivalent to saying that the free energy of a
planar defect is localized to a region surrounding the defect and that the characteristic length
scale of the localization is much smaller than the thickness of the bulk region.
While the true energy that drives any physical process is the free energy, we can gain some
insight into the energetic changes that occur in the system by studying the total internal energy.
Strain is an inherently continuum concept but analogous measures have been developed for the
analysis of atomistic data. Typically, a Langrangian strain is defined in terms of a reference
crystal structure. The properties of the Langrangian strain are not entirely compatible with the
continuum mechanics definition of strain.
In order to interpret the wealth of data available in a fully atomistic calculation, we must define
appropriate metrics which reduce the amount of information to a manageable level. Often we
will be primarily interested in the characteristics of the local atomic environment around a
specific location in the system. For instance, the location of a free surface seems like an easy
thing to visualize or to quantify in a macroscopic system but given a list of unsorted atomic
coordinates, the problem of locating that surface becomes a much more difficult problem.
Similarly, a broad view of a 2-D crystal will often draw our attention to areas which are dissimilar
or defective in some way. When our system is fully 3-D, we can identify similar locations by
taking slices through the crystal at each atomic plane but this process becomes cumbersome.
Also, the definition of the defective area using only atomic coordinate data is problematic.
With this and similar problems in mind, practitioners of computational materials have
developed a toolbox of metrics for examining the structure in solids. One of the primary metrics
is to examine the local coordination number around any atom. This metric is useful in
identifying certain types of defective areas such as free surfaces or grain boundaries. However,
this metric will fail if there are non-equivalent local environments which have the same
coordination number. This problem arises often in the case of fcc crystals since the local atomic
coordination number for an fcc atom is degenerate with the local coordination number for a hcp
crystal.
The next higher order consideration for local atomic environment takes into account the bond
angles present around a particular atom. Our study takes a simple binning approach to the
analysis of bond angle coordination since we are primarily using the metric to distinguish
between two very different local atomic structures.
4.2 Finite element method
For our FE calculations, we use a commercial code developed by Prof. K.J. Bathe and his
research group at MIT. The ADINA program implements the methods described in Prof. Bathe's
book(57). In essence, the FE method requires partitioning the system into pieces which can
each. be solved as a coupled linear elasticity problem. The solution of the coupled problem is
achieved through a variational energy principle. The nodes in each element are displaced
according to an energy criterion and the calculation is considered complete when the global
energy is minimized to within a set tolerance.
5 Calculations of time required for coalescence
Within the MD framework, we were able to calculate the time required for island coalescence.
In all of the cases considered, coalescence was completed in a matter of nanoseconds. The time
required is only weakly dependent on the size of the islands. The size range of islands studied
varied from 1.5nm in diameter to 6.5 nm in diameter. Both coalescence in free space and on
flat frictionless surfaces was considered.
5.1. Initial structure setup
The initial cluster structures were generated by extracting a spherical section of atoms from a
perfect fcc crystal. The fcc crystal was generated using a simple utility written in the C
programming language (may need to include the code in an appendix). The algorithm for
extracting the spherical cluster utilized a simple radial distance cutoff to delete any atoms which
were too far away from the intended center of mass.
5.2 Structure equilibration
Prior to use in an MD calculation, each structure had to be equilibrated to the desired
temperature. Practically, the equilibration is achieved by assigning random velocities consistent
with the desired temperature and then evolving the structure via the equations of motion.
Since the initial structure is not in equilibrium, evolving the system microcanonically would
result in a large conversion of thermal energy to stored potential energy and an undesirable
decrease in kinetic energy. To avoid this problem, the velocities of the atoms are periodically
rescaled via a Gaussian thermostat to maintain the desired temperature. This Gaussian
thermostat is only applied during the equilibration process since the actual coalescence process
that we are modeling will occur microcanonically.
For free space structure, the structures were placed with 1 to 2 atomic separation distances.
This spacing assured that the interaction forces of the potential would be active and that the
particles would be attracted to each other. Variation of the initial separation did not drastically
impact the coalescence behavior in either time or length scale.
5.4 Evolution over time
09
E0
__JZ5-j
0 500 1000 1500 2000Time (picoseconds)
2500
Figure 5.1: Evolution of center of mass distances over time. The coalescence process is complete in a manner ofnanoseconds.
5.3 Structure placement
Figure 5.1 shows the typical evolution of the center of mass distance between two particles
during a coalescence process. The approach velocities vary from 1 m/s to 3 m/s which is a quite
modest variation considering that the limiting elastic velocity would be on the order of 1000
m/s. The microcanonical nature of the calculation shows up in the recoil effects seen after the
initial island collision. The oscillation in the center of mass distance is expected since the latent
energy of boundary formation is converted directly into kinetic energy. Over time, the
oscillations are damped as the kinetic energy is scattered more randomly over the coalesced
structure.
6 Calculations of length scales during coalescence
6.1 Analysis of neck formation
6.1. Binuing algorithm for neck size determination
For a discrete calculation, the radius and location of the coalescence neck is not completely
defined. For our purposes, we will select a definition and remain consistent with our
terminology for the treatment of neck analysis. We begin by noting that the neck should form
at the perpendicular bisector of the segment connecting the two particle centers of mass. We
also note that the neck is defined as the minimum radial distance measured along this segment.
Since the radial distances we consider in an atomic calculation are discrete instead of
continuous, we divide the segment into several bins and consider the radius to be the maximum
radial distance of all atoms within that bin. We find that this method accurately predicts the
location and thickness of the neck.
6.1.2 Computed neck size via MD
nrt
E 20
D 18
16cD 14
cz 1210
c 8
Fl 64
220 25 30 35 40 45 50
Diameter (Angstrom)
I Simulated 0 NC v FC3D , Seel et al. - FC2D
Figure 6.1: Comparison of the atomistic calculation data with continuum models. The diamonds denote the atomiccalculation data and the lines denote existing continuum models. The 2-D Freund and Chason model and the SeelFEM model produce very similar results for the zipping height.
Figure 6.1 shows the atomistic calculation data along with the predictions from the continuum
models. The atomistic data matches well with best with the 2-dimensional FC model and the
Seel model. Geometrically, the setup for the MD calculations is more compatible with the
assumptions of the 3-dimensional FC model. The FC models and Seel model all predict a zipping
height which scales with particle radius to the 0.67 power. The extracted power from fitting the
atomistic data is 0.57.
7 Defects induced by boundary formation
Stacking faults are formed during the boundary formation process. The energetics of these
stacking faults is analyzed using a simple energetic model.
Experimentally, the coalescence process is known to generate dislocations and stacking faults
(20).
7,1 Internal structure of isolated clusters
Clusters in free space do not show much deviation from the original fcc structure that they were
assigned. Surface atoms experience short distance oscillations as well as longer scale
migrations, but the internal structure of the islands is unaffected. At the temperatures used in
the study, point defects such as vacancies and interstitials were limited to the surface layers of
the clusters.
7.2 Internal structure of islands on substrates
Islands generated on Lennard-Jones substrates will adjust their structure in the region nearest
to the substrate in response to the substrate potential. Typically, this results in 2-3 atomic
layers becoming disordered while in contact with the substrate. Some wetting behavior is
observed. The atomic layers closest to the substrate will often rearrange into a close packed
structure if they were not already in that arrangement prior to interacting with the substrate.
This rearrangement process can drive defects through the internal structure of the island. The
adjustment of the shape of the island increases the number of surface/interface atoms relative
to the bulk atoms so at times the distinction between bulk atomic structure and interface
atomic structure becomes blurred.
7.3 Structural analysis of coalesced structures
7.3.1 Types of stacking faults
Figure 7A.: Representation of a coalescing cluster structure, The lighter atoms denote locally hcp atoms whichcomprise a stacking fault.
The stacking fault illustrated in Figure 7.1 was induced by the shear stress generated during the
coalescence process. The particles in Figure 7.1 are not constrained to a substrate but the
behavior for particles on frictionless substrates was seen to be similar. The initial
misorientations of the coalescing particles did not have a discernible effect on the defect
formation behavior. Other types of defects can form just due to the misalignment of the
clusters prior to coalescence. A stacking fault generated due to misfit can act as a source of
stress but does not require any additional material transport upon its formation (i.e. motion of a
partial dislocation is not required to generate a misfit fault).
7.3.2 Size vvindow for stacking fault formation
# with stackingfault
U # without stack-ing fault
3.4 3.9diameter (nm)
5.9
Figure 7.2: Histogram showing the number of calculations which included stacking faults for the different sizes.These calculations were tabulated for the free space coalescence case.
Our examination of the post coalescence island structures showed that the occurrence of
induced stacking faults is restricted to a particular size window. These observations are
summarized in Figure 7.2. Our initial expectation was that the larger initial clusters would
experience less faulting since the stress generated during coalescence is much less than for
smaller clusters. The absence of defects in the smallest coalescing islands is a bit more
complicated and we will elaborate on this in the next section.
8
7-
6-
C 5-0
> 4-L.
: 3-0
2-
1-
0A I 5.9
7,3.3 Model for the size dependence of defect formation
Figure 7.3: Illustration of the unfaulted and faulted sphere configurations.
In order to understand the apparent size window for defect formation we have developed an
energetic model for the creation of defects within an initially defect-free sphere. The absence of
defects for the larger island sizes is easily explainable by the fact that all of the continuum stress
models predict a decreasing coalescence stress with increasing size. The larger sized islands lack
the necessary stress to nucleate a planar defect. The strengthening of the islands at smaller
sizes is more complex and our model addresses this issue specifically.
Figure 7.3 shows the unfaulted sphere and the expected configuration if a planar defect is
allowed to form within the sphere. The location of the defect plane is illustrated as crossing
through the equator but in reality the defect plane will be projected along the most accessible
slip plane near the newly formed boundary. This complication alters the model only slightly
since the circular geometry of the defect is still preserved.
If we assume an initial hydrostatic stress, a, in a reference sphere, the strain energy is
Estrain,unfaulted = - (7.))
where B is the bulk modulus. The sum of the strain and surface energy is then
22 R3 1 2y.Etot,unfaulted 3B + 4 2y. (7.2)
Figure 7.4: Projection of the fault plane in the sphere showing the relevant distances used in the model.
Figure 7.4 shows the projected area of the defect in the sphere. After the defect has been
allowed to pass through the sphere, an atomic shift will occur between the upper and lower
halves of the sphere. We represent the atomic shift by the Greek symbol A. The excess surface
energy generated by the fault process is given by
Esurf,faulted = YsfAoverlap + YAledg e . (7.3)
Here Aoverlap is the overlapping area of the two intersecting circles, Aledge is the area of the
non-intersecting ledges. Simple geometry leads us to
Aoverlap = 2 Rarccos2R) 2 - 4
(7.1)
(7.4)
and
Aledge = 2 (R2 - 2 R2 arccos 2 -f) (7.5)
We expect Ysf to be roughly one order of magnitude less than y. Generally, the values of
surface energies are close to 1 J/m 2 in metals and the stacking fault energies are close to 0.1
J/m2. Some representative values of these energies are given in Table 2. Additionally, we can
calculate the amount of strain energy relieved by the formation of the stacking fault by
analyzing the energy required to move a partial dislocation through the fault. The strain energy
relief is given as
Estrain relief = TbAoverlap (7.6)
T is the shear stress in the cluster prior to the passage of the partial dislocation and b is the
Burger's vector of the dislocation. The magnitude of the Tb term should be on the order of 0.1
J/m 2. The strain energy of the faulted sphere is
2 bA (7.7) 3Estrain,faulted = 3 overlap (7.7)
and the total energy of the faulted structure is
2 U2tf le3 2 y.tot,faulted 3B - bAoverlap Ysf Aoverlap + YAledge + 47R2 (7.8)
The total energy change from the unperturbed structure to the perturbed structure is achieved
by taking the difference of Equation 7.8 and Equation 7.2.
AEsf = (Ysf - T-b)Aoverlap + YAledge (7.9)
Table 2: Values of stacking fault and surface energy from Hirth and Lothe, Theory of Distocations.
Metal Ysf (J/m2 ) y (J/m2)
Ag .016 1.14
Al .166 0.98
Au .032 1.49
Cu .045 1.73
We can use an energy equivalence criteria between the faulted and unfaulted structure to
determine the critical stress for a given particle size. We set the energy change described in
Equation 7.9 to zero and solve for -.
Tcrit = YsfAoverlap+yAledge = Ysf ( 1 + YAledgebAoverlap b YsfAoverlap/
(7.10)
Using the stacking fault and surface energies for Ag from Table 2 and a Burger's vector length of
0.5nm, the behavior of rcrit is shown in Figure 7.5. As particle size decreases, the critical stress
for which the faulted structure is energetically favorable increases dramatically.
Vt)
Co
"O 100 200 300 400 500 600Radius (nm)
Figure 7.5: Critical stress for generation of an induced stacking fault for Ag. The predicted coalescence stresses forthe most applicable continuum models are also shown. The inset shows the small thickness strengthening effect.
The values of coalescence stress predicted by the continuum models are less than the critical
stress for the smallest island sizes. For larger islands, coalescence stress is greater than the
critical value and the occurrence of induced stacking faults is expected. The limiting value of the
critical stress is , which would be 32 MPa for Ag. The Seel model predicts the upper limit for
induced defect formation at 100nm while the FC2D model predicts that the transition will occur
at 450nm. Both of these predicted upper limits are well above the limit we observe from the
• •J',
MD calculations of around 5nm but the existence of the defect size window is qualitatively
justified.
The discrepancy between our atomistic calculation data and the upper range predicted by the
continuum models is not fully understood. As an energetic model, our explanation for the
existence of the defect size window does not include kinetic effects. Our atomistic calculations
may include additional activation barriers to the defect formation which are not accounted for
in the continuum model.
8 FEA of coalescence with a greater than 90 degree contact angle
By expanding on the earlier work of Seel and Thompson with islands of less than 90 degree
contact angle (38), we find that when the point of calescence occurs above the substrate, the
mechanical process is similar to the coalescence of clusters in free space. In these cases, the
contact and deformation behavior is similar to the 90 degree contact angle calculations except
for the extreme cases when the fixed motion boundary point interferes with the deformation.
Seel and Thompson (38) used the maximum in plane zipping distance,y o, rather than zo in order
to quantify the effects of zipping for contact angles less than 90 degrees. Irrespective of contact
angle, the normalized strain energy of the zipped configuration is given by
Er 2 4
Here, r is the radius of the contact area between the particle and substrate. The average stress
within an island was found to be
) E LYo,eq1-v2 Y w
where YO,eq was obtained numerically.
!:i!:uv U ,I- t'E~i mesh used to •ima/ze the stte.ins r eq.,niwed for island %ipping,
Figure 8.1 shows the general setup for the FE problem formulation. By symmetry, we only need
to consider one half of the impinging island geometry. The initial structure is a plane strain
cylinder which is set to have perfect traction at the cylinder-substrate interface. By applying
nodal displacements to form different contact surfaces, we obtain the system strain energy as a
function of two spatial variables. The spatial variables used in this study are the distance zipped
up from the point of initial contact and the distance zipped down from the initial contact. The
initial contact angles we have examined are 110, 120, 130, 150, and 180 degrees.
Strain Energy
6e-10
5e-10
4e-10
3e-10
2e-10
le-10
0
1.2
Figure 8.2: Strain energy mapping using up and down zipping distances as parameters.
Figure 8.2 shows a strain energy map generated from a FE island coalescence calculation. The
energy surface is well represented by
Estrain . E A(R, 6)(coz% + cZdozwn),
where zup and Zdown are the up and down zipping distances normalized to island radius,
respectively, and
A(R, 0) (-' - sin(6)(1 + cos(S)) - (1+cos(O)))R22 3 8sin(8)
The constants, co and c1, are extracted from the FE calculation and characterize the strain
energy penalty for upward and downward zipping, respectively. The extracted values for co and
cl are given in Table 3.
Table 3: Extracted fitting parameters for the strain energy surface
Contact angle (degrees) co cl
110 0.047 0.049
120 0.044 0.044
130 0.040 0.040
150 0.034 0.034
180 0.027 0.027
By minimizing the sum of the island strain energy and the newly formed boundary energy, we
can obtain both the normalized up and down zipping distances.
1 1
z = (z ( Iucp = 4EA(R, O)co ; down 4EA(R, 0)cl
2
Since the area function scales with R2 , the total zipping distance will scale with R3. This scaling
matches well with the earlier derived scaling for coalescence of islands with 90 degree contact
angles.
0.6
0.4
0,2
ۥ
0 0.2 0.4 0.6 0.8Z downi nm)
Figure 8.3: Contour plot showing the strain energy required to zip in each direction for a contact angle of 110degrees.
Figure 8.3 shows a contour plot of the strain energy surface for the case of a 110 degree contact
angle and allows for a more visual interpretation of zipping distances. The contours are nearly
symmetric in both the up and down distances. This symmetry is only broken when the position
of the cylinder-substrate contact restricts the range of the down zipping parameter.
th8
0,25
0.2
0 15
cl
A1
0 0.5 1 1.5 2
Surface Energy (JI/m)
Figure 8.4: Plot of the energy minimizing zipping distances for a 5 nm island with a contact angle of 110 degrees.
Figure 8.4 shows the energy minimizing zipping distances in both directions for a variety of
different surface energies. For large surface energy driving forces, the actual energy minimizing
configuration shows some asymmetry but this deviation is quite small. Under the boundary
conditions used in our calculations, the islands will have only a slight preference to zip away
from the constraint of the substrate.
250
Figure 8.5: Zipping stress as a function of contact angle.
Figure 8.5 shows our results for the average island stress along with the results from Seel and
Thompson(38). The Seel and Thompson results show that the zipping stress increases with
increasing contact angle up to the case of 90 degree contact angles. Our results show that the
90 degree contact angle case represents the maximum stress that can be generated in the island
through zipping.
The lack of strong asymmetry in the zipping behavior of greater than 90 degree contact angle
islands suggests that the contact process occurs in a manner similar to the coalescence of free
space spheres.
9 Conclusions
In conclusion, computational techniques such as MD and FEA are valuable tools in studying the
stress and structural evolution of films formed via island coalescence. The timescales for island
coalescence are tractable by the MD method and the resulting structures are in agreement with
200
150
o 100
50
0
--0-- Seel 10nm
S--Seel 20nm
-- &--Takahashi 10nm
***X*** Takahashi 20nm
0 50 100 150 200
0 (degrees)
1_^______X_____~II__________1_~111111111
the continuum models. The fact that the fundamental energetic continuum models can be used
to describe the behavior of clusters which consist of discrete collections of atoms is a bit
surprising but provides some relief to device designers who can continue using the existing
models.
The presence of defects induced by island coalescence is another interesting result for MEMS
and NEMS designers. The elevated defect density in vacuum deposited thin films is well known,
and the population of coalescence-induced defects may be an explanation. From calculations
and our energetic model, the quantity of these induced defects will be lessened for larger
islands at coalescence and will be absent for the smallest islands. Control over the island
structure prior to coalescence should allow thin film engineers to tailor the defect density for
their particular application.
Coalescence of islands with greater than 90 degree contact angle can be studied using FEA. For
all practical purposes, the fact that initial impingement occurs above the plane of the substrate
results in coalescence behavior which is quite similar to the coalescence of free spheres. Zipping
from the impingement point is symmetric in the up and down directions unless the contact
angle is very near 90 degrees. The importance of the boundary condition definition becomes
increasingly important as the contact angle approaches 90 degrees.
10 Future work
From a computational perspective, the most interesting extension of this work would be to
consider the effects of partial and complete traction on the coalescence behavior. The limiting
case of complete traction could be investigated by freezing the positions of the first layer of
atoms in each island. The dynamic layers in each island would still be allowed to coalesce but
the atomic configuration of the layers in contact with the substrate would be fixed. Special care
would have to be taken with atoms which are initially dynamic but random walk into an atomic
position in the first layer which should be frozen. The optimal number of layers to freeze in
order to represent the effect of traction is also not known a priori.
The effects of partial traction can be treated for both the isotropic case and the case for
preferred orientations. The most appropriate treatment for the isotropic case would be to
introduce a damping term on the motion of atoms in the first layer of the island. The form of
this damping term is not known explicitly but the strength of the term should correlate with the
dynamic coefficient of friction for the island material on the substrate material.
Another possible extension to the work presented here is to repeat the analysis for a wider
range of material models and potentials. This study has focused on Ag, but extension to other
metals and more advanced interatomic potentials should be straightforward. Ideally, a similar
approach could also be taken with a DFT model of the solid clusters.
Also, now that the presence of induced defects has been confirmed in both computational and
experimental studies, the incorporation of these defects into a continuum stress model is
desirable. The defect generation model that we presented can be merged with the continuum
island zipping models to provide improved estimates of the actual stress generated in
experiments. Before the effects of induced defects can be quantitatively added into the
continuum models, further experimental characterization of the defects is necessary. In-situ
TEM characterization of the actual size window in which defects occur should allow for the
extraction of the critical stress required for defect formation and the minimum critical island
radius.
Ultimately, it may be possible to quantitatively simulate the entire stress-thickness evolution
during thin film growth. Seel et al have already shown that qualitative agreement can be
achieved with a kinetic island growth model utilizing FE estimates of stress(37). Modification of
the kinetic model to include defect formation and other stress relief mechanisms would be a
significant step in taking the model from qualitative to quantitative.
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