Title of Publication Edited by
TMS (The Minerals, Metals & Materials Society), Year
DISLOCATION FORMATION DURING VAPOR DEPOSITION OF
EPITAXIAL MULTILAYERS
X. W. Zhou, H. N. G. Wadley
Department of Materials Science and Engineering;
School of Engineering and Applied Science;
University of Virginia; Charlottesville, VA 22903, USA
Keywords: Dislocation, Vapor deposition, Molecular Dynamics, Multilayers
Abstract
A molecular dynamics method has been used to study the formation of dislocations during the
vapor deposition of NiFe/Au/NiFe/Au and CoFe/NiFe/CoFe/Cu multilayers. We observed the
direct nucleation of misfit dislocations (with zero critical thickness) at (111) interfaces in the
NiFe/Au/NiFe/Au system. Both the dislocation configuration and density observed in the
simulations were similar to those reported in HRTEM experiments. A misfit energy increasing
dislocation structure was found in simulated CoFe/NiFe/CoFe/Cu multilayers. These
dislocations were formed during deposition on the (111) surface of the f.c.c. lattice. Deposited
adatoms were able to either occupy f.c.c. or h.c.p. sites if the energy difference between these
sites were minor (e.g., ~ 2meV). This resulted in islands with h.c.p. stacking and the creation of
partial dislocations at the f.c.c and h.c.p. domain boundaries. Since these boundaries were filled
the last and tended to have missing atoms, the surface layer tended to have less planes
compared to the underlayer even the surface atoms were slightly smaller. This phenomenon is
negligible for large lattice mismatch interfaces such as NiFe/Au where the lattice mismatch
dominates the dislocation nucleation, but is relatively significant for small lattice mismatch
interfaces such as CoFe/Cu.
Introduction
Multilayers composed of thin (~30-50 Å) ferromagnetic metal (e.g., Ni1-xFex or Co1-xFex) layers
separated by thin (~20 Å) conductive metal layers (e.g., Cu or Au) often exhibit giant
magnetoresistance (GMR) manifested by a large change in electrical resistance in the absence
and presence of an external magnetic field1,2,3. These multilayers are used as magnetic sensors
in hard disk drives4,5. They are also being explored for nonvolatile magnetic random access
memories6. Magnetotransport in these multilayers is sensitive to the composition of multilayer
system, the thickness of the individual layers, and the method (processing conditions) used for
their synthesis. The dependence upon the process is thought to result from a sensitivity of
magnetic properties and electron transport to the crystal defect populations trapped during their
vapor phase processing7,8. Because the volume density of defects generally rises when devices
shrink in size, the existence of defects can become a limiting factor in device miniaturization.
The dislocation formation in vapor deposited nanoscale multilayers is particularly interesting
because the growth of such multilayers often introduces significant dislocation densities due to
the lattice mismatch between the different layers9.
The formation of misfit dislocations during the epitaxial growth of films has been extensively
investigated using a continuum mechanics approach9,10. These studies indicated that under
thermodynamic equilibrium conditions, there was a critical film thickness below which misfit
dislocations did not form at the interface. After the film had grown to a critical thickness, other
dislocations (such as threading dislocations) glided to the interface to form the misfit
dislocations9. The critical film thickness, hc, for misfit dislocation formation increased as lattice
mismatch δ decreased9. For small δ, hc was large, and reasonably thick dislocation-free films
could be grown. However, as the lattice mismatch increased, the critical thickness fell and it
became energetically favorable to form dislocations. The continuum theories do not address the
mechanism of dislocation nucleation. They usually also assume the pre-existence of threading
dislocations that propagate into the film from substrate.
The continuum mechanics approach does not treat the atomic assembly events11,12,13,14 and it
also does not address kinetic phenomena controlling the complex atomic assembly processes
that can result in dislocation formation. Here we use a molecular dynamics (MD) atomistic
simulation approach to study dislocation formation during GMR multilayer deposition. Both
large (δ ~15%) lattice mismatch (Ni0.8Fe0.2/Au) and small (δ <3%) lattice mismatch
(Co0.9Fe0.1/Cu) systems are studied.
Simulation Methodology
A crystal is composed of discrete atoms located at lattice sites held together by interatomic
forces. If an adatom is deposited on a surface, it interacts with other atoms. This interaction
defines an array of surface sites that result in either global or local energy minimum for the
system. While some of these sites are lattice sites, some of them may not coincide with the
perfect lattice sites. They are hence the defect sites. Upon deposition, adatoms occupy initially
either lattice or defective sites. Although defective sites usually become energetically more
unfavorable as additional atoms are deposited around them, atomic reconstruction may not be
kinetically feasible. Reconstruction must occur by thermal vibration in which atoms jump from
one site to another. The jump frequency depends on temperature and the energy barrier along
the jump path. Typically, the energy barriers along the jump paths on the surface are smaller
than those in the bulk. As a result, surface atoms are much more mobile than bulk atoms. The
atomic structure obtained during deposition is then a result of a combination of the sites the
adatoms initially occupy and the extent to which surface atoms jump before they are buried into
the bulk by later deposited atoms. The deposition rate contributes to the structure evolution by
controlling the time available for atoms to jump while they remain on the surface. Because
many surface jump paths with different activation barriers are encountered, and the population
of such paths depends upon the local atomic configuration, atomic assembly during deposition
is a very complicated process to understand13.
Defects can easily form during deposition. For instance, a surface asperity can create incident
atom flux shadow on the surface, promoting the formation of surface roughness and voids when
other surface atoms are not mobile enough to migrate to the shadowed regions15. If atoms that
are initially deposited into the defective sites are buried within the bulk before they can hop out
of these sites, various defects can form, including twins, dislocations and grain boundaries. The
energy carried by the incident adatoms can assist structure modification of the deposited films.
For instance, the latent heat release during an atom’s condensation together with dissipation of
the remote kinetic energy of the adatom can both contribute to “extra” local surface
reconstruction11, resulting in surface flattening and mixing of atoms across interfaces in
multilayers13. High-energy adatoms impacting a smooth surface at an oblique angle can have a
significant biased diffusion on the surface11. Depending on the incident angle, hyperthermal
adatoms can also be reflected or cause resputtering12. Molecular dynamics (MD) simulations
are now beginning to realistically capture all these atomic assembly processes and to provide
new insights into the mechanisms of dislocation formation.
In a MD simulation of vapor deposition, a computational substrate crystal is created by
assigning coordinates to an assembly of atoms based on the equilibrium bulk lattice of the
substrate. Periodic boundary conditions are typically used in the horizontal coordinate
directions. A free boundary condition is used for the third coordinate direction. Vapor
deposition is simulated by continuously injecting atoms towards the top free surface of the
crystal at a frequency determined by the deposition rate. By calculating the forces between
atoms using an interatomic potential, Newton’s equations of motion are then used to solve for
the positions of all the substrate and deposited atoms as a function of time. The results therefore
provide precise time history of atomic assembly of a film during its vapor deposition.
Because Newton’s equation of motion preserves the total energy, the initial kinetic energy of
hot adatoms together with their latent heat release during condensation causes a continuous
increase of temperature in the simulated crystal. Isothermal deposition like that encountered in
most experiments can be achieved by applying a “heat sink” algorithm to a region below the
surface13. Since the time step used in MD simulations must be less than the shortest lattice
vibration period (typically around 10-15 sec., the real time of deposition analyzed by MD is
usually limited to about 1-100 nanoseconds. Consequently, an accelerated rate of deposition is
normally used in order to deposit enough atoms to reveal microstructural features. To prevent
the simulated crystal from shifting due to the momentum transfer during adatom impact, several
monolayers of atoms at the bottom (free) surface are fixed. Experience indicates that while the
accelerated deposition reduces the time available for kinetic processes, the rapid energy
(temperature) accumulation at the free surface due to the high adatom flux accelerates them.
The two effects to some degree cancel, and the simulations give results reasonably approximate
to those seen in experiments16.
MD simulations require an interatomic potential. For study of dislocation formation in
mismatched lattices, the potential must accurately predict lattice constants, elastic constants,
cohesive energies, and vacancy formation energies. The embedded atom method (EAM)
potentials originally developed by Daw and Baskes have successfully reproduced these
properties for many metal elements17. In addition to pairwise energies, the EAM potential
incorporates the many-body effect of the potential by including an embedding energy term.
This embedding energy can be thought of as accounting for the extra binding energy to embed
each atom into a local electron density background present at its position.
The EAM potentials developed for the individual elements cannot be simply used to model
multilayers or alloys18. A recently developed “alloy EAM” potential model and database of
sufficient generality has enabled alloy potentials to be created for any combinations of a list of
16 elemental metals (Cu, Ag, Au, Ni, Pd, Pt, Al, Pb, Fe, Mo, Ta, W, Mg, Co, Ti, and Zr)16,19.
The potentials predict heats of solution and stacking fault energies that are in good agreement
with experimental values. They have also been successfully applied to simulate the deposition
of a number of metal multilayer systems13,16,20. This interatomic potential database has been
utilized in this work.
Dislocation Formation During Thin Film Growth
The MD simulation approach described above was used to “grow” a (20Å)Ni0.8Fe0.2/
(20Å)Au/(20Å)Ni0.8Fe0.2 multilayer on a (111) gold substrate at a substrate temperature of 300
K, an adatom incident angle normal to the surface, a deposition rate of 5 nm/ns, and various
adatom energies. The initial gold substrate had 120 (22 4 ) planes in the x direction, 3 (111)
planes in the y (growth) direction, and 16 (2 2 0) planes in the z direction. To illustrate the
results, the x-y {110} atomic planes of the multilayer deposited at the adatom energy of 0.5 eV
are projected in Fig. 1(a). It clearly reveals four edge type misfit dislocations at both the
Ni0.8Fe0.2-on-Au and the Au-on-Ni0.8Fe0.2 interfaces. Each dislocation is associated with one
extra Ni0.8Fe0.2 plane. This is a significant dislocation density and could impede domain wall
migration and contribute to electron scattering at the interfaces21,22. Multilayers deposited at
higher adatom energies were also found to contain similar dislocation configurations. These
results indicated that the dislocation structures are unlikely to be changed by using
hyperthermal energy deposition methods such as magnetron sputtering or ion beam deposition.
The lattice mismatch between gold and Ni0.8Fe0.2 is at least 15%. The significant dislocation
density observed in Fig. 1(a) is therefore a result of a large lattice mismatch.
Fig. 1 {110} atomic planes of the Ni0.8Fe0.2/Au/Ni0.8Fe0.2 multilayer obtained by (a) MD
simulation and (b) HRTEM experiment.
Ross et al.23 have used high-resolution transmission electron microscopy (HRTEM) to directly
examine dislocations in a Ni0.8Fe0.2/Au/Ni0.8Fe0.2 multilayer deposited by molecular beam
epitaxy where only low adatom energies (~0.1 eV) could be used. The {110} atomic planes of
the multilayer reported by Ross et al. are shown in Fig. 1(b). A remarkable similarity exists
between the simulated image, Fig. 1(a), and the experimental one, Fig. 1(b). In particular, both
Fig. 1(a) and Fig. 1(b) show four dislocations at each of the Ni0.8Fe0.2-on-Au and the Au-onNi0.8Fe0.2 interfaces. The average dislocation spacing is about 20 Å in both figures. The
dislocation configuration of Fig. 1(a) is also virtually indistinguishable from that shown in Fig.
1(b). Clearly, the misfit dislocations shown in Fig. 1 effectively increase the lateral dimension
of the Ni0.8Fe0.2 layer and therefore help relieve the elastic misfit strain.
The MD simulation approach has been used to then grow (10Å)Co0.9Fe0.1/(20Å)Ni0.82Fe0.18/
(20Å)Co0.9Fe0.1 multilayer on a (111) Cu substrate at substrate temperature of 300 K, a normal
adatom incident angle, a deposition rate of 10 nm/ns, and various adatom energies. This
multilayer system is similar to the ones widely utilized in GMR sensors24. The initial Cu
substrate had 68(2 2 0) planes in the x direction, 3(111) planes in the y (growth) direction, and
24( 2 2 4) planes in the z direction. To simplify the observation of edge type misfit dislocations,
the x direction was chosen to be in line with the Burgers vector of a typical (a/2<110>) unit
dislocation in an f.c.c. structure. The x-y {112} atomic planes of a selected region of a
Co0.9Fe0.1-on-Cu interface deposited at a 0.2 eV adatom energy are projected in Fig. 2(a), where
the bottom three monolayers belong to the Cu layer. To analyze the structure from an additional
dimension, three consecutive x-z {111} atomic planes A, B, and C, marked by white, gray and
black colors respectively, are viewed from top in Fig. 2(b). Fig. 2(a) shows two edge
dislocations near the Co0.9Fe0.1-on-Cu interface.
Fig. 2 Atomic planes of a Co0.9Fe0.1-on-Cu interface region obtained from MD simulation.
(a) front view and (b) top view.
The edge type dislocations seen in Fig. 2(a) at first sight appear similar to the misfit
dislocations that accommodate lattice mismatch. However, there are fewer (2 2 0) planes in the
Co0.9Fe0.1 layer than in the copper substrate. This is surprising because copper atoms are bigger
than the cobalt, nickel, or iron atoms that are in the layer deposited on copper. As a result, extra
(2 2 0) planes would have been expected from lattice misfit considerations. Furthermore, we
found that these dislocations are not equilibrium structures. For instance, the dislocation density
decreased as the adatom energy was increased, and no dislocations were observed in
simulations with adatom energies above 2.0 eV. We did discover the formation of conventional
“misfit” dislocations when we deposited Cu atoms (with a lattice constant of 3.615 Å) on the
(111) surface of Co atoms (lattice constant 3.549 Å) under similar kinetic conditions.
Dislocation Formation Mechanism
Even though a MD simulation records atom coordinates as atoms assemble, deducing the
mechanism of dislocation nucleation is still not simple. First, dislocation loops rather than
straight lines often nucleate in the simulations. Secondly, dislocations are not easily seen in a
three-dimensional crystal without first identifying plane alignment. Finally, the transition state
that initiates a dislocation is usually unclear in a continuously distorted lattice. The problems
are simplified by analyzing an idealized Ni-on-Au surface.
Consider a gold crystal containing 68 (2 2 0) planes in the x direction, 5 (111) planes in the y
direction, and 24 ( 2 2 4) planes in the z direction. A fraction of the (111) gold surface was then
covered by a single epitaxial atomic layer of nickel. This three-dimensional crystal is shown in
Fig. 3(a), where the light and dark colored balls represent nickel and gold atoms respectively.
By minimizing the total energy of the system (using molecular statics), it is possible to identify
the transition of a dislocation free nickel atomic layer to a dislocation-containing atomic layer.
Fig. 3 Relaxation of one monolayer Ni on Au. (a) crystal geometry; (b) front and top views
of epitaxial Ni-on-Au; and (c) front and top views of relaxed Ni-on-Au.
Details of plane stacking for a part of the unrelaxed crystal are shown in Fig. 3(b), where the
top frame is a front (112) view of the crystal while the bottom frame is a top (111) view of the
top three monolayers (the nickel atomic layer and the two subsurface gold atomic layers). To
distinguish different monolayers, the nickel atoms are shown by small white circles, and the
next two layers of gold atoms are displayed by larger white and gray circles respectively. Fig.
3(b) shows that for the epitaxially grown nickel atomic layer, the front view contains no extra
planes and the top view shows three distinctive sites for the nickel and the two gold atomic
layers indicative of ABCABC... (f.c.c.) stacking. However, this (unrelaxed) configuration is not
the lowest energy state of the system.
A molecular statics method was used to minimize the energy of the crystal shown in Fig. 3(b)
under the constraint that the periodic boundary lengths are fixed (at the size of an equilibrium
gold bulk). The corresponding front and top views of the crystal are shown in Fig. 3(c). Clearly,
the front view now shows two extra (2 2 0) planes (i.e., dislocations) in the nickel atomic layer.
In the top view, the middle region between the dislocations still has f.c.c. stacking. However,
both regions to the left and the right have h.c.p. (i.e., ABAB...) stacking (the top nickel atoms
are aligned with the second “gray” plane of gold atoms). The transition from the f.c.c. region to
the h.c.p. region corresponds to an atomic shift of a/6[1 2 1] at the left and a/6[ 2 11] at the right,
where “a” refers to the local lattice constant. It can also be seen that the boundaries between the
f.c.c. and the h.c.p. regions correspond well to the locations of the extra planes shown in the
front view. Hence, the observed dislocations are Schockley partial dislocations. These
dislocations have a Burgers vector of a/6<112> and are separated by either f.c.c. or h.c.p.
(stacking fault) regions. Fig. 3(c) indicates a dislocation spacing of ~11.5 Å. The appearance of
these dislocations in Fig. 3 indicates that the critical thickness for misfit dislocation formation is
one atomic layer or less in the nickel/gold system.
Stop-action analysis was used to reveal dislocation initiation during the growth of a Ni0.8Fe0.2
layer on a gold substrate. The growth conditions were the same as those used in Fig. 1(a). The
crystal orientation was chosen to be the same as that in Fig. 3 to facilitate the analysis of
dislocations with an a/2<110> Burgers vector. The x-y projections of the atomic crystals as a
function of time are shown in Figs. 4(a)-(c), where the black, white and gray circles represent
iron, nickel and gold atoms respectively. It can be seen that after one Ni0.8Fe0.2 monolayer (~2.5
Å) was deposited, Fig. 4(a), many gold atoms exchanged with the depositing iron or nickel
atoms, resulting in extensive mixing at the Ni0.8Fe0.2-on-Au interface. Interestingly, the {220}
planes in the deposited layer are not well defined and many deposited atoms are clearly not at
the positions of the {220} planes when epitaxially extended from the substrate. This indicates
that for this multilayer system (with a large lattice mismatch), many adatoms are directly
deposited at the wrong sites. These misplaced atoms form dislocation nuclei when the first
monolayer of Ni0.8Fe0.2 was deposited. It should be pointed out that mixing at the Ni0.8Fe0.2-onAu interface can change the misfit strain at least during the deposition of the first few
monolayers of Ni0.8Fe0.2. If the lattice parameter is a linear function of composition, the lattice
mismatch strain for a (Ni0.8Fe0.2)1-xAux-on-Au interface can decrease from 15.45% at x = 0.0 to
13.68% at x = 0.1.
Fig. 4 Dislocation formation during deposition Ni0.8Fe0.2 on gold. (a) deposition time 50 ps;
(b) deposition time 150 ps; and (c) deposition time 200 ps.
As more Ni0.8Fe0.2 monolayers were deposited, the {220} planes in the Ni0.8Fe0.2 became
gradually clearer, Fig. 4(b)-(c). When about four Ni0.8Fe0.2 monolayers were deposited, the
lattice of the deposited layer was clear enough to reveal the extra {220} planes characteristic of
dislocations at the Ni0.8Fe0.2-on-Au interface, Fig. 4(c). The final (20Å)Ni0.8Fe0.2/(20Å)Au/
(20Å)Ni0.8Fe0.2 multilayer contained 8-9 extra {220} planes (a/6<112> partial dislocation) over
a length scale of ~98 Å. The average dislocation spacing was therefore around 11-12 Å, in
agreement with the results shown in Fig. 3(c).
Based on the top view, Fig. 2(b), the two mismatch-enhancing dislocations shown in Fig. 2(a)
were identified to be the a/6[1 2 1] and a/6[211] partial dislocations. To understand how these
dislocations are nucleated, stop-action pictures were used to examine the early stage of the
deposition of the (10Å)Co0.9Fe0.1/(20Å)Ni0.82Fe0.18/(20Å)Co0.9Fe0.1/Cu multilayer. An adatom
energy of 0.2 eV was used and the complete multilayer structure is shown Fig. 5(a), where
different atom species are distinguished by different gray scales. The front (x-y) and the top (xz) projections of the atomic planes are shown in Fig. 5(b)-5(d) as a function of time, where
different gray scales are used to distinguish different atomic planes.
Fig. 5 Dislocation formation during deposition of Co0.9Fe0.1 on Cu. (a) a complete
multilayer; (b) deposition time 86 ps; (c) deposition time 111 ps; and (d) deposition
time 186 ps.
At a deposition time t = 46 ps, Fig. 5(b), it can be seen from the top view image that the (111)
planes had evolved to consist of three distinct regions. In the first region, all three differently
shaded atoms can be observed, consistent with local f.c.c (ABCABC...) stacking sequence. In
the second region, only white and gray atoms can be observed, indicative of local h.c.p.
(ABAB...) stacking. In the third region, no white or gray atoms can be observed, simply
revealing that these domains had not yet been completely filled with atoms. At t = 111 ps, Fig.
5(c), all the three planes had been almost filled. This left the (111) planes either occupied by
f.c.c. or h.c.p. domains. Also the boundaries between the f.c.c. and the h.c.p. domains appeared
to nucleate a missing (2 2 0) plane. Finally at t = 186 ps in Fig. 5(d), the missing plane
representing a dislocation became clear in the front view, and its horizontal location coincided
exactly with the boundary between the f.c.c. and the h.c.p. domains. Comparing Figs. 5(c) and
5(d) indicates that such dislocations were highly mobile on the (111) slip plane as a significant
lateral shift of dislocation location occurred within a very short period of time (75 ps).
During deposition on some surfaces, individual adatoms may suffer only a small energy penalty
when they occupy the wrong (e.g., h.c.p. instead of f.c.c.) sites. For example, the binding
energies of a single copper atom on the f.c.c. and h.c.p. sites of a (111) copper surface differ by
only 2 meV25. As a result, there is a high probability to form different stacking fault domains
during growth on the (111) surface of such f.c.c. lattices. When the lateral boundaries (normal
to the surface) of these fault domains meet, dislocations can be nucleated on the surface. While
these dislocations may have high energies and are likely to be unstable, it requires a significant
thermal activation for them to be annealed out. As a result, kinetically constrained atomic
assembly can result in the direct nucleation and retention of dislocations on the growth surface.
Unlike continuum models9, these dislocation may form without pre-existing threading
dislocations.
Since the fault/unfaulted domain boundaries are filled last during each layer growth, they are
likely not to be completely filled, resulting in missing planes, at least at the initial contact of
domain boundaries. If the growth rate is low so that these boundaries are at the surface, the
filling of the missing planes would require squeezing of extra rows of atoms into these locations
in the top surface monolayer. This is energetically unfavorable, at least locally, because the
local compressive strain energy caused by the discrete lattice will then be significantly larger
than the more global misfit strain energy that is very small for a thin film thickness. If the
growth rate is high enough so that the junctions of these domain boundaries are buried below
the surface, then the filling of these places would require atoms to diffuse into these regions.
These events are associated with significant activation energy barriers and less frequently occur
under kinetically constrained growth conditions. As a result, edge dislocations with missing
planes in the later deposited layers often form during epitaxial deposition of the (111) surface of
the f.c.c. structures, even when the later deposited atoms have slightly smaller lattice constants.
Under conditions that promote equilibrium such as high adatom energy, high substrate
temperature, or low deposition rate, the impact induced and thermally activated diffusion events
can proceed sufficiently to anneal out defects. As a result, no dislocations were observed in
simulations with adatom energies above 2.0 eV.
Conclusions
Molecular dynamics simulations have been used to study the atomic assembly mechanisms
responsible for dislocation formation during the vapor deposition of large and low lattice
mismatch (20Å)Ni0.8Fe0.2/(20Å)Au/(20Å)Ni0.8Fe0.2/Au and (10Å)Co0.9Fe0.1/(20Å)Ni0.82Fe0.18/
(20Å)Co0.9Fe0.1/Cu multilayers. The following conclusions can be made:
a) Extensive misfit dislocations that create extra half planes in the Ni0.8Fe0.2 layer are formed
during the deposition of the Ni0.8Fe0.2/Au/Ni0.8Fe0.2/Au multilayers. Both dislocation
configuration and dislocation spacing are in good agreement with the HRTEM experiments.
b) The dislocation configurations in the Ni0.8Fe0.2/Au/Ni0.8Fe0.2 multilayer are unaffected by
deposition conditions.
c) The edge dislocations formed at the interfaces of a Co0.9Fe0.1/Ni0.82Fe0.18/Co0.9Fe0.1/Cu
multilayer deposited under kinetically constrained conditions tend to have missing planes in
the later deposited layers. They can either relieve or increase the misfit strain energy.
d) During the growth on the (111) f.c.c. surface, adatoms may either occupy f.c.c. or h.c.p.
sites, resulting in the formation of f.c.c and h.c.p. domains on surface. The misfit energy
increasing dislocations directly nucleate at the domain boundaries. The incomplete filling of
the domain boundaries under kinetically limited growth conditions is responsible for the
missing planes in the later deposited layer.
e) Conditions promoting atom diffusion, such as increasing adatom energy or ion assistance,
greatly reduce the probability of forming the mismatch increasing dislocations.
Acknowledgment
We are grateful to the Defense Advanced Research Projects Agency and Office of Naval
Research (C. Schwartz and J. Christodoulou, Program Managers) for the support of this work.
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