PII:
Acta mater. Vol. 47, No. 3, pp. 1063±1078, 1999
# 1999 Published by Elsevier Science Ltd
On behalf of Acta Metallurgica Inc. All rights reserved
Printed in Great Britain
S1359-6454(98)00403-0
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TWIN FORMATION DURING THE ATOMIC DEPOSITION
OF COPPER
X. W. ZHOU and H. N. G. WADLEY{
Department of Materials Science, School of Engineering and Applied Science, University of Virginia,
Charlottesville, VA 22903, U.S.A.
(Received 27 March 1998; accepted 13 July 1998)
AbstractÐVapor deposited copper ®lms with h111i growth texture usually contain twin plates stacked normal to the growth direction. However, twins are rarely seen when growth occurs in other principle crystallographic directions. Atomistic modeling indicated that during h111i growth, adatoms occupied either
parent or twin surface lattice sites with almost equal probability, resulting in a high nucleation density of
twin domains. During growth on either {110} or {100} surfaces, adatoms were only able to occupy parent
lattice sites, and no twin nucleation occurred. A phase ®eld method was used to model the twin domain
evolution during h111i growth as a function of deposition rate and temperature. Simulation results indicated that twin domains evolved by rapid lateral expansion with very little vertical thickening. The lateral
expansion was found to be fast compared with the deposition rate, and as a result twin domains usually
grew to occupy the entire width of a growth column, in good agreement with experimental observations.
The model indicated that the twin structures formed during the h111i growth of copper are not directly
controllable by either the processing temperature or the deposition rate. # 1999 Acta Metallurgica Inc.
Published by Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
Vapor deposited copper has a low electrical resistivity, a high electromigration resistance, and as a
result is beginning to replace aluminum alloys for
the metal interconnects in microelectronic
devices [1, 2]. It is also used as the conducting
(spacer) layer in giant magnetoresistive (GMR)
multilayers for building magnetic ®eld sensors, magnetic disc drive read heads and nonvolatile electronic data storage devices [3±10]. These
applications all require copper ®lms containing low
concentrations of electron scattering sites such as
point defects and twin boundaries [11]. A wide variety of vapor deposition methods are being investigated for synthesizing the low resistivity copper
needed for these applications. They include sputtering, thermal evaporation, molecular beam epitaxy
and emerging directed/jet vapor deposition
processes [12±14]. These deposition routes provide
access to a very wide range of deposition rates, incident atom energies/angles, and substrate temperatures. These conditions in turn control the atomic
scale processes responsible for the trapping of
defects in the deposited ®lms. Usually, extensive experiments were used to ®nd combinations of processing conditions that result in high quality ®lms.
However, because of the high cost and long development time associated with the experimental process optimization approaches, interest is growing in
{To whom all correspondence should be addressed.
the use of modeling methods to establish the relations between process conditions and the defect
content of vapor deposited ®lms [15±37].
Transmission
electron
microscopy
(TEM)
studies of the microstructure of copper ®lms [13]
have indicated that ®lms deposited at 2508C usually
have a h111i texture and contain extensive twin
plates normal to the h111i growth direction.
However, growth at higher temperatures (e.g.
05008C) often results in a h110i growth texture
without signi®cant twin formation. It is unclear
why this change in twin density occurs and if the
possibility exists to reduce the incidence of twins
and their electron scattering interfaces by modifying
the h111i growth conditions. Since the precise mechanisms of twin nucleation and growth, and their
dependence on the deposition process conditions
are dicult to experimentally elucidate, we have
pursued an atomistic modeling approach to analyze
the atomic scale processes active during copper ®lm
growth.
Molecular dynamics (MD) can be used to study
the atomistic assembly processes involved in vapor
deposition. Molecular dynamics determines the positions of atoms during their thermal vibration by
applying Newton's equations of motion. When a
precise description of interatomic forces is used, the
method is well able to predict atomic scale structure
and stress. However, because the time scales of the
lattice's dynamics are short, a large numerical calculation must be repeated every femtosecond or so,
and this technique is extremely computationally ex-
1063
1064
ZHOU and WADLEY: TWIN FORMATION
pensive. Molecular dynamics calculations of vapor
deposition can usually only be applied to small systems (up to a few thousand atoms) for short
periods (up to a few nanoseconds) of real deposition time. Under these conditions, the e€ective deposition rates of the calculation are accelerated by
about nine orders of magnitude compared to those
used in practice [22].
Setting aside the drawback of accelerated deposition rates, three-dimensional MD simulations
ought to enable the mechanisms of growth twin
nucleation during vapor deposition to be identi®ed
correctly. The kinetics of the subsequent evolution
of these nucleated twin domains might also be
observable within the real time treatable by the calculation (a few nanoseconds) if they are fast.
However, to explore the wide ranges of practical
processing parameter space, the coupling of an atomistic and a continuum analysis is required. A
method has recently been proposed for linking a
MD simulation of vacancy formation during vapor
deposition with a continuum (partial di€erential
equation) model of vacancy di€usion [14]. This
approach extended the real time achievable by the
analysis and enabled the determination of the
dependence of the vacancy content upon deposition
rate, substrate temperature, and incident ¯ux angle/
energy [15]. We noted that a di€use interface
approximation [38, 39] exploiting the notion of an
excess interfacial free energy has been used to formulate a time-dependent Ginzburg±Landau (Allen±
Cahn) equation [40, 41] for analyzing the evolution
of interfaces. This di€use interface (or phase ®eld)
method has been successfully used to analyze the
kinetics of grain growth [38, 39], displacive phase
transformations in the zirconia±yttria system [42],
and the development of coherent tetragonal
precipitates [43, 44]. The method appears well suited
to the analysis of the twin growth.
Here, atomistic (i.e. molecular dynamics and statics) calculations have been used to simulate copper
deposition, analyze twin nucleation, establish twin
growth mechanism, and determine the energetic and
kinetic parameters controlling these processes. The
results of these atomistic simulations were then
coupled with a Ginzburg±Landau phase ®eld theory
to simulate the structural evolution that occurred
during low rate deposition. From this we predict
the twin microstructures that form over length and
time scales representative of realistic deposition processes, and determine the e€ect of processing conditions upon the twin structures of vapor deposited
copper ®lms.
2. ATOMISTIC SIMULATIONS
2.1. Interatomic potential
An embedded atom method (EAM) potential
developed by Daw and Baskes [45, 46] was used for
the three-dimensional MD atomistic scale calculations. A signi®cant advantage of the EAM over
traditional pair potentials [47] is that the ``local environment'' dependence of the EAM potential
allows a realistic calculation of the energies of
defective lattice regions (such as free surfaces, dislocations, twin/stacking faults, grain boundaries, and
point defects). The standard EAM functions of copper determined by Foiles et al. [48] were used for all
the calculations. These functions were (empirically)
calibrated by ®tting to measured bulk crystal properties, including the equilibrium lattice constant [49],
the sublimation energy [50], the bulk modulus [51],
the elastic constants [51], and the vacancy formation energy [52]. A cut-o€ distance of 4.95 AÊ was
used for the potential. This is larger than the third
neighbor spacing (of 4.43 AÊ based on a lattice constant of copper, 3.615 AÊ), yielding a realistic representation of the stacking fault and various twin
boundary energies.
2.2. MD simulation of vapor deposition
The three-dimensional methodology used here for
simulating vapor deposition is similar to a twodimensional model described in detail elsewhere [22].
In the two-dimensional model, adatoms were introduced as a group at random locations above the
growing ®lm. To prevent atomic interactions in the
vapor phase, the introduced adatoms were widely
separated and were given the same incident energy
and direction. We have modi®ed this model to
accelerate the calculation and also to remove constraints on the incident energy and direction. In
particular, the adatoms were added one by one with
a time interval corresponding to the overall deposition rate. These adatoms were ®rst injected into
the system at random positions high above the surface. They were then ballistically moved to the substrate until they started to interact with at least one
of the surface atoms (i.e. once they reached the cuto€ distance of the EAM potential away from the
nearest surface atom). The subsequent motion of
the atoms was then determined by integration of
the equations of motion. To minimize anomalous
free surface e€ects of small crystal systems, periodic
boundary conditions are normally used. Two versions of MD are available. In one version [53], the
periodic lengths in the coordinate directions are
®xed, and the positions and velocities of atoms are
determined solely by the interatomic potentials and
the classical Newton's equations. In other words,
the equations of motion are constructed from a
Lagrangian that is de®ned only by the interatomic
potential energy and the atomic kinetic energy [54].
Because the crystal volume is ®xed in this approach,
it does not allow crystal strain imposed by external
stresses or thermal expansion. In the second version
of MD, the strain energy and a boundary kinetic
energy for the computational cell are added to the
Lagrangian so that both the atomic positions and
ZHOU and WADLEY: TWIN FORMATION
1065
Fig. 1. Computational crystal used for deposition along the [111] direction.
the instantaneous size of the crystal cell (i.e. its
strain) can be simultaneously solved [55]. Both versions were used in the present work and almost
identical results were obtained with each method.
Molecular dynamics simulations of vapor
deposition were conducted for growth in the
[111], [110], and [100] directions. An example of the
crystal geometry is shown in Fig. 1 for the [111]
growth.
In
general,
the
size
of
the
computational crystal is notated as lx ly lz
where la (a ˆ x, y, and z) refers to the
length in the a-direction. The initial substrates
had dimensions of 132d…224† 3d…111† 4d…220† ,
54d…002† 4d…220† 4d…220† , or 76d…022† 3d…200† 4d…022† for growth in the [111], [110], or [100] directions, respectively. Here d…hkl † is the spacing between
adjacent (hkl) planes. During deposition, the positions of the bottom layer of atoms were ®xed to
prevent the crystal from shifting upon atom impact.
The incident kinetic energy and the latent heat
release during the condensation of each adatom led
to a continuous increase in thin ®lm temperature
during the simulation of vapor deposition. To
mimic the isothermal deposition condition normally
achieved in practice, a temperature control method
was applied to a region below the surface to keep
Fig. 2. A simulated atomic …220† plane formed during deposition along the [111] direction.
1066
ZHOU and WADLEY: TWIN FORMATION
the deposition at a stable temperature [22], while
the atoms above the temperature controlled region
were left free so that their motion was completely
determined by the force law. In this way, phenomena associated with the atom impact process could
be accurately revealed. Naturally, this algorithm
also created a region with a thermal gradient near
the surface which caused the thermal energy to be
conducted to the temperature controlled region and
then removed. To minimize the change in the width
of this thermal gradient, the temperature controlled
region was expanded upwards during deposition.
Since surface roughness often developed during deposition, the expansion rate of the thermostatically
controlled volume was taken to be 75% of the deposition rate to prevent the temperature controlled
region from passing beyond any part of the deposited surface boundary.
3. ATOMISTIC ORIGINS OF TWINNING
Figure 2 shows an example of a single atomic
…220† plane obtained from a copper ®lm grown in
the [111] direction at an incident atom kinetic
energy, Ei=1.0 eV, a substrate temperature,
T = 100 K, a deposition rate, R = 10 nm/ns, an
incident atom angle, y = 08, and a deposition time,
t = 0.4 ns. In Fig. 2, dark spheres represent the
(original) substrate atoms, while light spheres correspond to the deposited atoms. The shading of Fig. 2
is used to highlight the extensive numbers of
nucleated twins. It is interesting that no twin
nucleations were observed in our simulations for
the ®lms grown in either the [110] or the [100] direction. A detailed analysis indicated that the reason
for this distinction lay in the di€erent atomic stacking sequences. The stacking sequence is
ABCABC . . . for the (111) planes, but is ABAB . . .
for both the (200) and the (220) planes. When an
atom is deposited onto an ``A'' plane during [111]
growth, it can either occupy the parent lattice ``B''
sites or the twin lattice (stacking faulted) ``C'' sites.
Subsequent atoms that bond to this atom then
form a nucleus for either a regular lattice or a twin
faulted one. However, during either [110] or [100]
growth, the adatom can only occupy the parent
``B'' sites when dropped onto an ``A'' plane. As a
result, twin nucleation is an intrinsic phenomenon
accompanying deposition in the [111] direction but
does not occur during growth in other principal
crystal directions.
To explore the evolution of the twins nucleated
during deposition (under the same conditions as
above), a simulation of deposition in the [111] direction was conducted using a 120d…224† 3d…111† 70d…220† substrate with a larger dimension in the x±z
plane [see Fig. 3(a)]. The observed twin evolution is
displayed in Figs 3(b)±(d) which show ``snapshots''
of atom positions on three adjacent (111) planes
(including one substrate plane). In Fig. 3, the yellow
spheres represent the substrate atoms, the red
spheres represent the ®rst plane of deposited atoms,
and the blue spheres represent the next plane of
deposited atoms. Figure 3 indicates that after 30 ps
of deposition, adatoms had aggregated in numerous
islands on the substrate. Some islands had ABC
stacking (all spheres are observable), indicating that
they were local f.c.c. domains, while others had
ABA stacking (only red and blue spheres are observable), indicating that they were local h.c.p. (stacking faulted or twin faulted) domains. The formation
of these local f.c.c. and h.c.p. domains resulted
from the random selection of the B or C sites onto
which the adatoms fell. The areas between the f.c.c.
and h.c.p. domains were incompletely ®lled after
30 ps of deposition.
During continued deposition, the un®lled regions
were progressively ®lled, and boundaries developed
between the f.c.c. and h.c.p. domains. The domains
also changed their sizes. In general we observed
that the largest domains expanded while smaller
domains shrank. Little or no thickening of twin
domains occurred in the growth direction. This indicates that the domain growth on the (111) plane
is controlled by the energy of the boundaries dividing the domains, and the energy of the (111) stacking fault or the (111) twin boundary is too small to
result in a signi®cant driving force for twin plate
thickening. We also noticed that in the (111) plane,
the nucleated domains developed with triangular
shapes whose boundaries were parallel to {110}
planes, suggesting that these boundaries have relatively lower energies than any others in this plane.
Twin domain evolution occurred by competitive
growth. The locally larger domains were observed
to consume their neighboring smaller domains
rapidly, while the size of the surviving domains
increased quickly with time. The planes shown in
Fig. 3 happen to have f.c.c. domains surrounded by
h.c.p. domains after the initial domain growth.
Hence, the h.c.p. domains expanded and consumed
the f.c.c. domains [even though, as shown below,
the (111) stacking fault energy ``slightly'' favors the
f.c.c. domains]. As the domains grew, the blue
atoms eventually fully occupied the A sites on the
(111) plane, and a single h.c.p. domain eventually
®lled the entire simulated area (at t = 100 ps). This
led to the formation of a completely twinned plane.
Simulations repeated with the same conditions but
with di€erent random number seeds indicated that
the f.c.c. domains could also readily expand to occupy the entire plane area. The likelihood of forming a fully twinned (111) plane [i.e. a (111) stacking
fault] then depended on the initial distribution of
the nucleated twin domains.
4. MS CALCULATION OF TWIN ENERGETICS
To develop a more detailed understanding of
twin formation during the [111] growth of copper
ZHOU and WADLEY: TWIN FORMATION
1067
Fig. 3. Time evolution of three (111) planes during deposition along the [111] direction at 08, 1.0 eV,
10 nm/ns, and 100 K: (a) three-dimensional crystal; (b) top view of three atomic planes at 30 ps; (c) top
view of three atomic planes at 50 ps; (d) top view of three atomic planes at 60 ps; (e) top view of three
atomic planes at 100 ps.
requires a quantitative knowledge of the energetics
controlling the atomic assembly process. This
includes knowledge of the surface (g), stacking fault
(gsf), twin boundary (gt), and adatom binding energies (Eb) as well as the activation energy of bulk
di€usion (Qb). The stacking fault energy, gsf, is distinguished from the (111) twin boundary energy,
gt …111†, because a stacking fault is a special case of
a twin just one atomic plane thick. A molecular
statics method [15, 56] was used with the EAM
functions of copper to deduce the relevant energetic
parameters.
To calculate the (111) surface energy, g…111†, and
the stacking fault energy, gsf, a 60d…224† 18d…111† 4d…220† crystal was constructed. The bulk crystal was
approximated by applying periodic boundary conditions in all three coordinate directions. A crystal
with two (111) surfaces was simulated by using periodic boundary conditions in the x- and z-directions
and a free surface boundary condition in the y-
1068
ZHOU and WADLEY: TWIN FORMATION
Table 1. Surface energies, stacking fault energies, twin boundary energies, binding energies of a copper atom on (111) surface, and activation energy of bulk di€usion
g(110) (eV/AÊ2)
0.088
gsf (eV/AÊ2)
0.0011
g(100) (eV/AÊ2)
g(111) (eV/AÊ2)
Eb(f.c.c.) (eV)
Eb(h.c.p.) (eV)
Qb (eV)
0.080
0.074
2.567
2.565
0.7
gt(111) (eV/AÊ2)
0.0006
gt …224† (eV/AÊ2)
gt …220† (eV/AÊ2)
no boundary vacancies
boundary vacancies
no boundary vacancies
boundary vacancies
0.122
0.096
0.100
0.072
direction. A crystal containing stacking faulted
planes was created by shifting the upper half of the
crystal by the stacking fault vector and employing
periodic boundary conditions in all three coordinate
directions (the number of stacking faulted planes
thus introduced is two due to the boundary condition). The total energies of these crystals were
determined by molecular statics using a conjugate
gradient energy minimization procedure [56]. The
(111) surface energy and the stacking fault energy
were then calculated from the energy di€erence of
the faulted and perfect bulk crystal and the total
area of the fault plane. A similar procedure was
used to determine the (110) and (100) surface energies, g…110† and g…100†, using the 22d…002† 20d…220† 32d…220† and 32d…220† 14d…002† 32d…220†
crystals, respectively. The results are summarized in
Table 1.
Table 1 reveals that the (111) surface has the lowest surface energy, consistent with the experimental
observation that this surface is the most likely to
develop during copper deposition at ambient temperatures. Table 1 also shows a very small value for
the stacking fault energy, suggesting that stacking
faults are very likely to develop in copper. It should
be pointed out that for a realistic simulation of
twin formation, it is important for the interatomic
potential to yield a realistic value of stacking fault
energy. The calculated stacking fault energy of
0.0011 eV/AÊ2 found in Table 1 is well within the
error bar of the experimental value of 0.0025 eV/
AÊ2 [57].
The (111) twin boundary energy, gt …111†, was calculated using a 54d…224† 12d…111† 30d…220† crystal.
Two (111) twin boundaries were introduced by
shifting the upper half of the crystal into the twin
orientation. This was achieved by exchanging the xand y-coordinates of the atoms on two adjacent
planes. Notice that the stacking sequence of an
f.c.c. crystal along the [111] (y) direction is
ABCABC . . . , so an exchange of any two planes,
say, A and B, leads to a shift in which A ÿ4B and
B ÿ4A. The stacking sequence then becomes
BACBACBAC which is in a twin orientation. After
relaxing the crystals using a conjugate gradient
energy minimization method under periodic boundary conditions, gt …111† was calculated as the energy
di€erence between crystals with and without the
twin boundaries divided by the total area of the
boundaries. The calculated result is included in
Table 1. It can be seen that the (111) twin boundary
energy of 0.0006 eV/AÊ2 is even smaller than the
stacking fault energy. Twin domain growth is therefore more likely to be driven by a reduction of the
boundaries separating twin domains in a given
plane rather than by a reduction of the area of the
(111) twin boundary. Because of the very small
value of gt …111†, a structure consisting of twin plates
stacked along the [111] direction is likely to be a
relatively stable structure.
The coordinate exchange method was also used
to create …224† and …220† twin boundaries separating
twin domains. Since the exchange led to abnormal
spacings between some atoms near the boundaries,
these boundaries are unstable. This is in contrast to
the (111) twin boundary where the nearest spacing
between atoms does not change across the boundary. To calculate realistic energies for the …224† and
…220† twin boundaries, gt …224† and gt …220†, use of
crystals with a large dimension in the x-direction
normal to the twin boundary were necessary. Such
crystals can be divided into three regions as shown
in Fig. 4: the parent region I, the twin region II,
and the boundary region III. The twins were initially created between the two center planes of the
two boundary regions. To relax the boundary
regions, MS calculations were conducted using periodic boundary conditions in all three coordinate
directions and a constraining condition that allowed
only atoms in region III to move while all the other
atoms remained in position. The twin boundary
energies were calculated as the energy di€erence
(per unit of twin boundary area) between relaxed
crystals with and without the twins. A 360d…224† 12d…111† 6d…220† crystal shown in Fig. 4(a) was used
for the calculation of gt …224†. The unrelaxed (original) twin region contained 180 …224† planes and the
boundary region (d) contained 20 …224† planes. A
200d…220† 12d…111† 12d…224† crystal shown in
Fig. 4(b) was used for the calculation of gt …220†
where the unrelaxed twin region contained 100
…220† planes and the boundary region contained 12
…220† planes. Further increasing the boundary
region caused only a minor change in the calculated
energy. The calculated results are listed in Table 1.
The …224† and …220† twin boundary energies are sig-
ZHOU and WADLEY: TWIN FORMATION
1069
Fig. 4. Geometry of the crystal used to calculate the vertical twin boundary energies: (a) …224† twin
boundary; (b) …220† twin boundary.
ni®cantly bigger than the (111) twin boundary
energy. These high energies provide a driving force
for lateral twin domain growth on a (111) plane.
Twin domain boundaries formed during deposition sometimes contain vacancies. To explore the
e€ects of vacancies on the …224† and …220† twin
boundary energies, vacancies were created by
removing one [111] atomic row of atoms on a plane
in the boundary region. The corresponding twin
boundary energies gt …224† and gt …220† were then calculated according to the procedure described earlier
and are given in Table 1. Vacancies were found to
reduce the twin boundary energies and tended to
therefore stabilize the boundaries.
The MD simulation of vapor deposition has
shown that twin nucleation occurs because adatoms
have roughly equal probabilities of occupying either
f.c.c. or h.c.p. sites on a growing (111) copper surface. This suggests a small binding energy di€erence
for adatoms falling on the two sites. A 54d…224† 6d…111† 30d…220† crystal was used to calculate these
energies. A copper atom was added to either an
f.c.c. or an h.c.p. site on the (111) copper surface.
The equilibrium crystals were obtained by energy
minimization using periodic boundary conditions in
the x- and z-directions, a ®xed boundary condition
(at the equilibrium bulk atomic positions) for the
bottom two atomic planes, and a free boundary
condition for the top (y) surface. The binding energies Eb …f:c:c:† and Eb …h:c:p:† were calculated as the
energy changes associated with placing an isolated
atom at the f.c.c. and the h.c.p. sites on the top
(111) crystal surface, and the results are shown in
Table 1. Table 1 indicates nearly equal binding
energies for an atom at the f.c.c. site and at the
h.c.p. site. The very high density of twin nuclei on
the (111) surface is therefore consistent with the
notion of equal probabilities of the adatom's initial
occupancy of either f.c.c. or h.c.p. site as a result of
very small binding energy di€erence between the
two sites. The absence of signi®cant twin thickening
in the growth direction is also consistent with the
very small stacking fault energy, while the rapid lateral expansion is apparently a consequence of the
large energy of the boundaries between twin
domains in the (111) plane.
1070
ZHOU and WADLEY: TWIN FORMATION
A 18d…224† 6d…111† 10d…220† periodic crystal was
used to determine the activation energy of bulk diffusion, Qb. A vacancy was created in the bulk and
by constraining the bottom (111) plane to prevent
the crystal from moving, the standard procedure [15]
was used in the calculation. The calculated result is
again shown in Table 1.
5. TWIN GROWTH KINETICS
The molecular dynamics simulations presented
earlier suggest that the morphology of growth twins
in ®lms grown in h111i directions is controlled by
the kinetics of twin domain expansion or shrinkage
on {111} planes. A molecular dynamics simulation
can be used to deduce the e€ective values of the
parameters governing the evolution kinetics.
The shrinkage of an isolated domain was studied
at various domain thicknesses using 120d…224† hd…111† 70d…220† crystals, where the crystal height,
h, was set to contain either six or eight (111) planes.
Domains were created in the center of the top (y)
free (111) surfaces of the crystals. These domains
had volumes of 80d…224† o d…111† 46d…220† , where
the domain thickness, o, was set to be either one or
six for the crystal with h equal to six, and three for
the crystal with h equal to eight. Use of a domain
thickness, o, of one, three, or six then allowed an
examination of domains one, three or an in®nite
(by using the periodic boundary condition along the
y-direction) number of layers thick. To separately
study the e€ects of gt …111† (or gsf) on the growth of
domains of ®nite thickness, two types of single
domains were considered: (i) a domain with the
twin orientation surrounded by a lattice with the
parent orientation; (ii) a domain with the parent
orientation surrounded by a lattice with the twin
orientation. The MS energy minimization procedure
described earlier was used to relax the boundary
regions, and a MD simulation of the domain's time
dependent shrinkage was then performed. For crystals with ®nite twin thicknesses, a free boundary
condition was used for the top (y) surface while the
two lowest atomic layers were ®xed at the equilibrium bulk positions. The latter condition minimized e€ects of the pseudo surface at the bottom.
In all the calculations, periodic boundary conditions
were used in both the x- and z-directions.
The shrinkage of a single domain can be most
clearly observed by viewing the top three (111)
planes of the crystal from above as a function of
annealing time. An example of the shrinkage of one
layer thick domains at a temperature of 100 K is
shown in Fig. 5. Figure 5(a) corresponds to the case
of a twin domain, while Fig. 5(b) corresponds to
the case of a parent domain. The ®gure shows that
the lateral shrinkage kinetics of domains is very
fast: both domains disappeared in under 3 ps. The
calculations also show that the kinetics of the two
domains are roughly the same, suggesting that the
twin domain evolution is not sensitively a€ected by
the presence of the stacking fault or the (111) twin
boundary. This is also consistent with the likelihood
of formation of an entire stacking faulted/twinned
plane during the competitive domain growth
observed in Fig. 3.
To further quantify the kinetics, a relationship
between the area, S, of a surrounded domain and
the annealing time, t, was obtained. To simplify
analyses of results, an atomic con®guration without
a domain (i.e. after a domain had completely disappeared) was ®rst set up as a template. The di€erence between atomic positions of any con®guration
and those of the template was then calculated. If
the magnitude of this di€erence was larger than
half the magnitude of the Burgers vector of a
Shockley partial dislocation, the corresponding
atom was counted as belonging to the domain. The
total area of the domain was calculated as the total
number of domain atoms multiplied by the area associated with a single domain atom. The latter can
be simply obtained from the con®guration at t ˆ 0
where both the domain area and the number of
domain atoms are known. The dependence of the
area, S, upon time at a temperature of 100 K is
plotted in Fig. 6 for both the twin and parent
domains each with a thickness of one atomic plane.
A linear kinetic relationship between S and time
was observed in both cases, analogous to the kinetics of grain growth [38]. The rate of area change,
_ was about 1860 and 1660 AÊ2/ps for the parent
jSj,
and twin domains, respectively, consistent with a
slight retarding e€ect upon the shrinkage of the
parent domain due to the stacking fault.
A growing surface usually has some degree of
surface roughness, and the domains formed on a
rough surface can be thicker than one atomic plane.
S_ was found to decrease by about an order of magnitude as the domain thickness was increased from
one to three atomic planes, and then remained approximately independent of the depth thereafter
(see the table in Fig. 6). This occurred because
when a domain was one atomic plane thick (such as
stacking fault), the motions of the atoms jumping
from the twin lattice sites to the parent lattice sites
(or vice versa) were at the surface and were unconstrained (i.e. no other atoms were on top of them
to interfere with their jumps). The kinetics of a
three atomic plane thick domain were slowed
because the jumping atoms were severely constrained by atoms above. The e€ect of a further
increase in the domain thickness is negligible due to
the periodic symmetry (ABCABC . . .) of the stacking sequence in the thickness direction.
To investigate the e€ect of temperature on
domain growth, the kinetics of three atomic layer
thick domains were analyzed at temperatures
between 10 and 100 K. These low temperatures
were chosen so that the initial assigned velocity distribution of atoms can be easily equilibrated (by
ZHOU and WADLEY: TWIN FORMATION
1071
Fig. 5. Molecular dynamics results of kinetics at 100 K for one atomic layer thick twin domain. Atoms
on the substrate surface, the ®rst deposited monolayer and next deposited monolayer are marked by
yellow, red, and blue colors, respectively: (a) twin domain; (b) parent domain.
MD calculation) without causing the domain to
shrink. By averaging the values of the parent
domain and twin domain, the temperature dependence of the rate of domain growth or shrinkage
_ ˆ 370 exp…ÿ0:006=kT † AÊ2/ps,
was identi®ed as jSj
where k is the Boltzmann constant. The low value
of the e€ective activation energy is related to the
high energy of the twin domain boundaries. It can be
seen that at the temperature of 100 K,
_
jSj1180
AÊ2 =ps (i.e. 12.0 106 mm2/s). If deposition
is conducted at a typical rate of 1 mm/min, about
0.01 s is required to deposit each monolayer, and a
domain can grow to cover an area of about
2.0 104 mm2 in that time. Growth columns in thick
copper ®lms are about 1±10 mm in diameter and
because domain boundary mobility is high, twins are
able to spread laterally to ®ll the entire cross section
of growth columns.
1072
ZHOU and WADLEY: TWIN FORMATION
Fig. 6. Domain size as a function of time obtained from a molecular dynamics calculation for a one
atomic layer thick domain.
6. PHASE-FIELD ANALYSIS OF TWIN GROWTH
The accelerated rate MD simulations of copper
vapor deposition have shown that twin nuclei are
randomly formed on {111} surfaces during deposition because of the very small di€erence in binding energies of atoms on f.c.c. and h.c.p. lattice
sites. The subsequent evolution of the twin structure
has been shown to be governed by the lateral
expansion or shrinkage of these nucleated twin
domains. However, the high computational cost of
MD calculations precludes the simulation of
domain growth (and the prediction of the twin
microstructure) on the length and time scales typical
of real deposition processes. Like grain growth,
twin domain growth is driven by a combination of
the curvature of the boundaries dividing domains
and the associated excess boundary energy. A
phase-®eld method has been used by Chen and
Yang [38, 39] for simulating grain growth driven by
the (grain) boundary curvature and the (grain)
boundary excess energy. This method has been
adapted to the twining problem and combined with
atomistic (MD and MS) results to quantitatively
simulate the evolution of twin structures for realistic
vapor deposition processing conditions.
6.1. Principles
To apply a phase ®eld method to twin domain
growth, the crystallographic orientation is represented by an ordering parameter, Z, that lies on
the interval [ÿ1, 1] [38, 39]. Here Z ˆ 1 refers to the
parent orientation, while Z ˆ ÿ1 indicates the twin
orientation. The value of Z continuously changes
from ÿ1 to 1 across a twin boundary. In this notation, any twin microstructure can be described by Z
as a function of space. If the free energy of a system
is not at a minimum with respect to a local variation in Z, the Z distribution will change to minimize the free energy. The resulting evolution of the
ordering parameter is governed by a time dependent
Ginzburgh±Landau (Allen±Cahn) equation [38, 39]:
dZ
df …Z†
ˆ ÿL
ÿ kr2 Z
…1†
dt
dZ
where L is a coecient characterizing the kinetics
of evolution, f(Z) is the local free energy density,
and k is a coecient accounting for the e€ect of the
energy gradient on the local change of Z. Since isolated crystals in the parent and twin orientations
are structurally indistinguishable and are both equilibrium phases, the local free energy density f(Z)
should achieve the same minimum value when
Z ˆ 21. f(Z) can then be approximated by [38, 39]
a
1
f …Z† ˆ ÿ Z2 ÿ Z4
…2†
2
2
where a is a coecient related to the energy and
width of the boundary between domains.
Because of the deviation from Z ˆ 21, the local
free energy exhibits a maximum at the boundary,
which gives rise to an excess boundary energy. The
incorporation of the twin domain boundary energy
in the formulation is the key for equation (1) to
describe the domain evolution where the boundary
area is reduced.
Molecular statics calculations of the …224† and
…220† twin boundaries have shown their excess
boundary energies to be similar (see Table 1). To
simplify the evolution analysis, the di€erence in
twin boundary energies can be ignored (note this
ZHOU and WADLEY: TWIN FORMATION
a€ects the shape of the domains but has little e€ect
on the overall kinetics), and a is therefore treated as
constant in the present work. With this simpli®cation, substitution of equation (2) into equation (1)
gives
dZ
ˆ ÿL‰ÿa…Z ÿ Z3 † ÿ kr2 ZŠ:
dt
…3†
Integration of equation (3) yields the time evolution
of twin domains in an isolated crystal layer.
During the vapor deposition of copper ®lms, the
growth of a single domain will also be a€ected by
the presence (or absence) of the (111) twin boundary (or stacking fault) above or below the domain
since the (111) twin boundary energy (or stacking
fault energy) also provides the driving force for the
domain growth. A general driving force, f(111), in
the presence or absence of a (111) twin boundary,
can then be expressed as
gt…111†
f…111† ˆ
‰…Z ÿ Z1 †2 ‡ …Z ÿ Z2 †2 Š
…4†
4
where Z1 and Z2 refer, respectively, to the ordering
parameters of the neighboring crystal positions on
the layers above and below the crystal position
where Z is evaluated. It can be seen from
equation (4) that when there is a sharp (111) twin
boundary (say, jZ ÿ Z1 j ˆ 2 and jZ ÿ Z2 j ˆ 0),
f…111† ˆ gt…111† , whereas when no (111) twin boundary
exists …jZ ÿ Z1 j ˆ jZ ÿ Z2 j ˆ 0†, f…111† ˆ 0.
The migration of the (vertical) domain boundary
walls due to the (111) twin boundary energy starts
in the regions near the (111) twin boundary and
then propagates along the walls. Because this migration must come to completion for the entire
thickness Dy, the overall migration kinetics for the
domain walls is inversely proportional to Dy. To incorporate the e€ect of f(111) on the average kinetics
of the entire domain wall migration, equation (3)
was extended as
gt…111†
dZ
ˆ ÿL ÿ a…Z ÿ Z3 † ‡
‰…Z ÿ Z1 † ‡ …Z
dt
2Dy
ÿ Z2 †Šb ÿ kr2 Z
…5†
where the unitless parameter b has been introduced
to account for the di€erent e€ects of the (111) twin
boundary energy and the (vertical) twin boundary
energy on the kinetics of the domain wall migration.
By dividing the crystal into a three-dimensional
grid (grid size Dx Dy Dz) and using a periodic
boundary condition in both the x- and z-directions
(see Fig. 1 for the coordinate system), equation (5)
can be readily solved numerically.
6.2. Phase ®eld simulation of twin formation
Four model parameters, a, k, b, and L are introduced in the phase ®eld simulation theory. The
1073
details of a procedure to determine these parameters, as well as the choice of the numerical parameters Dx, Dy, and Dz, are described in the
Appendix A. With all the parameters known, the
formation of a twin microstructure can be simulated by the phase ®eld method for arbitrary deposition temperatures and rates. To illustrate, we
divided horizontal 3000 5000 AÊ2 copper layers
into 120 200 grids. A substrate was simulated by
assigning Z ˆ 1 for a left half (60 200) grid and
Z ˆ ÿ1 for a right half grid. This represents an initial substrate containing equal areas of a parent
and a twin phase. To start the phase ®eld analysis,
a second layer of copper was placed on top of the
®rst. Random twin nucleation was simulated by
assigning random values of Z from the interval
[ÿ0.5, ÿ0.5] to each grid element of the second
layer. A deposition rate, R, was speci®ed and the
twin microstructure was allowed to evolve using
equation (5) for a period Dy=R. A new layer was
then added, and the same procedure repeated until
the ®lm was grown to a desired thickness.
An example of twin domain nucleation and
growth for the ®rst layer deposited on a copper
substrate at a temperature of 520 K and a deposition rate of 2.5 mm/min is shown in Fig. 7, where
the value of Z is displayed using a scale in which
the light region represents the twin while the dark
region designates the parent phase. The many
domains nucleated can be seen at t = 60 ps. As
time elapsed, the locally smaller domains were consumed, and the sizes of the surviving domains grew.
These larger domains continued this same competitive process of minimizing the boundary area
between domains. This eventually resulted in a
single embedded elliptical domain with entirely convex boundaries at t = 9000 ps. Thereafter, this elliptical domain continuously shrank until it
completely disappeared at t = 13 200 ps, forming a
uniformly oriented plane.
Considering the boundary energy isotropy
assumption used in the simulation, a remarkable
similarity between Figs 3 and 7 was found. Since
the original substrate contained half parent phase
and half twin phase, the eventual formation of an
entire twin plane shown in Fig. 7 indicates that the
twin evolution is dominated by the lateral expansion of the nucleated domains and the e€ect of
gt…111† is, as anticipated, relatively minor. Figure 7
again is consistent with very fast twin domain
mobility. Unless there are obstacles available to pin
the domain boundaries, the time for one domain to
occupy the entire 3000 5000 AÊ2 (0.3 0.5 mm2)
computational area is on the order of 10ÿ8 s. Since
the growth of copper ®lms at 520 K results in polycrystals with grain diameters of a few micrometers,
domain evolution on a ¯at plane is highly likely to
result in the domain fully occupying the grain
width.
1074
ZHOU and WADLEY: TWIN FORMATION
Fig. 7. Continuum calculation results of time evolution of domains during deposition at a temperature
of 520 K: (a) 60 ps; (b) 3000 ps; (c) 9000 ps; (d) 13 200 ps.
A transmission electron microscope picture of the
cross view of a copper ®lm grown in the [111] direction at a temperature of 520 K and a deposition
rate of 2.5 mm/min [13] is shown in Fig. 8(a). It can
be seen that the growth twins observed in the experiment usually extend fully across a layer (rather
than existing as small domains within a layer). To
directly compare with this experiment, the twin
structure was simulated under the same conditions,
and the cross view is shown in Fig. 8(b). It can be
seen from Figs 8(a) and (b) that the alternative twin
plates along the growth (y) direction revealed in the
simulation are in good agreement with the twin
structure within the growth columns observed in
the experiment.
Because the twin domain evolution on a ¯at
plane is dominated by the lateral migration of
domain boundaries, and because this migration is
always fast, the e€ects of processing conditions
such as deposition rate and substrate temperature
upon the twin morphology are likely to be relatively
minor. Simulated twin structures for a lower deposition temperature of 300 K and a higher deposition
rate of 250 mm/min are shown in Figs 8(c) and (d).
Clearly, varying these temperature and deposition
rates does not cause a signi®cant variation in the
twin structures when growth occurs by a layer-bylayer mode. This is consistent with experimental observations of twin structures in samples grown
under di€erent conditions [58].
We occasionally observed that a domain sometimes propagated across the entire length of the cal-
culated area. Since periodic boundary conditions
were used for the x±z plane, the corresponding
domain boundary was then in®nitely long, and
eventually became straight. This type of boundary
cannot be eliminated since a straight boundary is
meta-stable and can no longer migrate. This
resulted in the twin steps in the simulation (shown
by the arrows) in Figs 8(b)±(d). Interestingly, such
steps are also found in experiments as indicated by
the arrow in Fig. 8(a).
7. DISCUSSION
Molecular dynamics simulations have shown that
the vapor deposition of copper on {111} planes
during h111i growth results in extensive twin
domain nucleation because the {111} plane is
stacked with an ABCABC . . . sequence, and adatoms can deposit onto either parent lattice (B
plane) or twin lattice (C plane) sites on an A plane.
Twin nucleation was not observed in experiments
or during molecular dynamics depositions on {110}
and {100} planes because their ABAB . . . stacking
sequence prevents the adatoms from occupying
twined sites during deposition. The experimentally
observed twin structure for h111i growth results
from the lateral expansion (or shrinkage) of densely
nucleated domains. Because the {111} twin boundary energy (or stacking fault energy) is quite low
for copper, the boundary motion in the growth
direction has a low driving force and the evolution
of the nucleated twin domains is dominated by the
ZHOU and WADLEY: TWIN FORMATION
1075
Fig. 8. Cross view of twin structure along the growth direction [111]: (a) TEM image of a copper ®lm
deposited at a temperature of 520 K and a deposition rate of 2.5 mm/min; (b) simulated image at the
same conditions as those of the experiment; (c) simulated image at the same deposition rate as that of
the experiment, but at a lower deposition temperature of 300 K; (d) simulated image at the same deposition temperature as that of the experiment, but at a higher deposition rate of 250 mm/min.
high inplane boundary energy between adjacent
domains. As a result of the rapid kinetics of the
domain expansion (or shrinkage), the domains on
¯at {111} planes usually evolve into a single parent
or twin layer occupying an entire growth column
(grain) width. Since the time required for this is
small compared with the time for a realistic deposition process, growth via a layer by layer mode is
almost always likely to lead to a completed layer by
layer twin structure, and this type of twin structure
is not sensitively a€ected by processing conditions
such as deposition rate or temperature.
If growth by an island mechanism occurs, the
resulting surface roughness could have important
e€ects on twin evolution and thus twin microstructure. Several mechanisms potentially exist. For
example, the continuous ¯at {111} planes for which
the twin domain evolution has been modeled are
not present on a rough surface. The lateral convergence of columns could result in the delayed formation of highly defective twin/parent interfaces.
The mobility of these interfaces could be di€erent
from that analyzed. Also the atomic scale structure
of ®lms with rough surfaces usually has associated
defects such as vacancies and voids that can impede
or arrest the migration of the domain boundaries.
In addition, di€erent parts on a rough surface such
as surface peaks, side walls, and valleys, etc. are
1076
ZHOU and WADLEY: TWIN FORMATION
Fig. 9. Rs as a function of b.
usually associated with di€erent planes. Some of
these planes may provide additional adatom attachment sites (other than the twin sites or the parent
sites). For instance, {112} planes are stacked in the
ABCDEFABCDEF . . . sequence, and our MD
simulations indicated that adatoms deposited on the
{112} plane can occupy the lattice sites of di€erent
grain orientations to form low and high angle
boundaries. The frequent formation of low/high
angle grain boundaries can in turn ``freeze'' a twin
region in the deposited ®lm. Hence, processing conditions such as temperature, deposition rate, incident energy and incident angle might indirectly
modify twin structures through their e€ects on the
surface morphology and defect populations.
Finally, it has been observed in Fig. 3 that
vacancies can be created during domain growth. In
general whenever an atom switches from an f.c.c.
(h.c.p.) to an h.c.p. (f.c.c.) site, a vacancy can be
created. When this occurs at the periphery of a
domain, a row of vacancies can form. Unlike grain
growth where vacancies can be eliminated at the
grain boundaries, twin domain evolution results in
an elimination of domain boundaries, and the
vacancies therefore remain in the bulk crystal.
When domains annihilate below the growing surface, the relatively high activation energy for bulk
vacancy di€usion (0.7 eV) could result in signi®cant
vacancy trapping. When these vacancies are created
at the growing surface, they can be easily removed
(®lled) due to the high surface mobility of copper
atoms. Nonetheless, the mechanism may still be important because the density of twin nuclei is very
high, creating a potent vacancy formation source.
8. CONCLUSIONS
Atomistic and phase ®eld methods have been
combined to study the formation of twins during
the deposition of copper under growth conditions
typical of practical deposition processes. The analysis has revealed that:
1. The low binding energy di€erence for atoms that
occupy either f.c.c. or stacking faulted lattice
sites is small and this results in the extensive
nucleation of twin domains during copper
growth in h111i directions. No faulted lattice
sites are available during growth in either h110i
or h100i directions, and hence no twin nucleation
was observed.
2. The subsequent evolution of domains nucleated
on the {111} surfaces during the h111i growth
occurs by the lateral expansion (or shrinkage) of
the domains. Little thickening of the twins
occurs in the growth direction.
3. The inplane boundary energy between twin
domains was about 100 times larger than the
{111} twin boundary or the stacking fault energies, and the observed high mobility of the
inplane domain boundaries was a result of this
high interdomain boundary energy. Little driving
force existed for coarsening in the h111i growth
direction. As a result, the layer by layer growth
of {111} copper led to a ¯at plate-like twin structure with the twin plates stacked normal to the
{111} planes.
4. The lateral evolution of twin domains was found
to be very fast, and twin domain evolution on
¯at {111} surfaces was therefore not signi®cantly
a€ected by processing conditions such as temperature or deposition rate.
5. The annihilation of domain boundaries is a
potent source of vacancies. When this occurs on
the surface, the high surface mobility of copper
may be sucient to ensure ®lling of empty lattice
sites. When domains annihilate below the growing surface, the relatively high activation energy
ZHOU and WADLEY: TWIN FORMATION
for bulk vacancy di€usion (0.7 eV) could result
in signi®cant vacancy trapping.
AcknowledgementsÐWe are grateful to the Defence
Advanced Research Projects Agency (A. Tsao, Program
Manager) and the National Aeronautics and Space
Administration for support of this work through NASA
grants NAGW1692 and NAG-1-1964.
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APPENDIX A
Four parameters, a, k, b, and L have been introduced
in the phase ®eld theory. All these model parameters were
determined by ®tting the results of phase ®eld theory to
those of the atomistic simulations.
Since the anisotropy of the domain boundary energy is
not considered, an average value of gt…224† and gt…220† of
gt=0.111 eV/AÊ2, was used as the domain boundary
energy. In the present work, the twin domain boundary
1078
ZHOU and WADLEY: TWIN FORMATION
thickness w was chosen to be 50 AÊ, and the grid size was
chosen to be Dx ˆ Dy ˆ Dz ˆ 25 AÊ which is smaller than
the twin domain boundary thickness.
A ¯at twin boundary was analyzed to determine a and
k. In this case, Z is a function of x only, where x is the
coordinate measured from the boundary. When equilibrium is achieved, the left-hand side of equation (3) is
equal to zero. Z as a function of x can then be solved
from
k
d2 Z
‡ a…Z ÿ Z3 † ˆ 0:
dx 2
…A1†
By noticing that dZ=dx ˆ 0 at Z ˆ 21, dZ=dxr0 (corresponding to Z<0 at x<0 and Z > 0 at x > 0), and Z ˆ 0 at
x ˆ 0, equation (A1) can be solved to give a relation
between Z and a/k
r
1‡Z
2a
ˆ
x:
…A2†
ln
1ÿZ
k
In the present work, the boundary thickness w was de®ned
as the distance between Z ˆ ÿ0:5 and 0.5. From
equation (A2), it is easy to show that
r
2k
w ˆ ln 3
:
…A3†
a
The twin boundary energy, gt, is de®ned as the excess
energy across the boundary [39, 41]
…1 k dZ 2
gt ˆ
f…Z† ÿ fmin ‡
dx
…A4†
2 dx
ÿ1
where fmin represents the minimum of f(Z). Using the Z
de®ned by equation (A2), equation (A4) can be solved to
yield a relation between the known domain boundary
energy, gt, and the two unknown model parameters
r
4 ak
:
…A5†
gt ˆ
3 2
Using the w and gt values, equations (A3) and (A5) can be
directly solved to give a = 0.0037 eV/AÊ3, and k = 3.79 eV/
AÊ.
To determine the mobility parameters L, a kinetics calculation comparable to the one studied using the MD
method (Fig. 5) was carried out using the continuum
equation. The results of the phase ®eld calculation were
then ®tted to their MD counterparts to yield an appropriate L. The ®tting is easy because according to equation (5),
the value of S_ is linearly proportional to L. Two adjacent
layers 25 AÊ thick (Dy) and 1250 1250 AÊ2 big on the x±z
plane were divided into grids. A parent orientation was
assigned to the lower layer by letting Z ˆ 1 for all of its
grids. An embedded domain 400 400 AÊ2 in size was created in the middle of the upper layer so that a parent
domain corresponds to Z ˆ 1 for grids within the domain
and Z ˆ ÿ1 for the rest of grids, and a twin domain corresponds to Z ˆ ÿ1 for grids within the domain and Z ˆ 1
for the rest of grids. With Z ®xed for the lower layer, the
time evolution of Z for the upper layer was solved using
equation (5). Since MD simulations have indicated that
the kinetics of a twin domain and a parent domain are
quite close, calculations were ®rst carried out to determine
L at b ˆ 0 at which the twin domain and the parent
domain are equivalent in the continuum model. The
domain area was found to linearly decrease with evolution
time as has been seen in the MD simulation. By ®tting the
S_ values to those obtained from MD simulations, L as a
function of temperature was derived:
0:006
L ˆ 23:0 exp ÿ
A3 =eV ps:
…A6†
kT
It can be seen that the apparent activation of twin domain
growth is small (00.006 eV), suggesting that twin domain
growth is an almost athermal (fast) process.
The value of the mobility parameter b re¯ects the di€erent in¯uences of the (horizontal) (111) twin boundary and
the (vertical) boundary between domains on twin domain
growth, and di€erent values of b change the di€erence in
growth kinetics between a parent domain and a twin
domain. If this di€erence is quanti®ed by a parameter Rs
(de®ned as the ratio of the S_ of a twin domain to that of
a parent domain), Rs is only dependent on b. With all the
other parameters known, the continuum simulations were
used to calculate the values of Rs at di€erent b between 0
and 10, and the results are shown in Fig. 9. The data were
well ®tted by a parabolic expression under the constraint
that Rs …b ˆ 0† ˆ 1:
Rs ˆ 1 ‡ 3:68125 10ÿ2 b ‡ 6:46221 10ÿ3 b2 :
…A7†
Since the average e€ect of the twin boundary energy
gt…111† diminishes for an in®nitely thick domain (i.e. Rs ˆ 1
at Dy ÿ41), Rs must be dependent on the domain thickness, and was approximated by
Rs ˆ 1 ‡
C
Dy
…A8†
where C is a constant. The value of Rs was accurately
determined to be 1.04 at Dy ˆ 3d…111† from analysis of MD
results. C can then be determined as C ˆ 0:25 A. Hence,
Rs at Dy ˆ 25 A was evaluated as 1.01. Substituting this
value into equation (A7) yields b ˆ 0:255.