Philosophical Magazine, 11 January 2004
Vol. 84, No. 2, 193–212
Misfit dislocations in gold/Permalloy multilayers
X. W. Zhouy and H. N. G. Wadley
Department of Materials Science and Engineering, School of Engineering and
Applied Science, University of Virginia, Charlottesville, Virginia 22903, USA
[Received 16 May 2003 and accepted in revised form 8 September 2003]
Abstract
Several groups have reported the misfit dislocation structures in Au/Ni0.8Fe0.2
multilayers where the lattice parameter misfit is very large. To explore the factors
controlling such structures, molecular dynamics simulations have been used to
simulate the vapour-phase growth of (111)-oriented Au/Ni0.8Fe0.2 multilayers.
The simulations revealed the formation of misfit dislocations at both the goldon-Ni0.8Fe0.2 and the Ni0.8Fe0.2-on-gold interfaces. The dislocation configuration
and density were found to be in good agreement with previously reported highresolution transmission electron microscopy observations. Additional atomicscale simulations of a model nickel–gold system indicated that dislocations are
nucleated as the first nickel layer is deposited on gold. These dislocations have
an (a/6)h112i Burgers vector, typical of a Shockley partial dislocation. Each
dislocation creates an extra {220} plane in the smaller lattice parameter nickel
layer. These misfit-type dislocations effectively relieve misfit strain. The results
also indicated that the dislocation structure is insensitive to the energy of
the depositing atoms. Manipulation of the deposition processes is therefore
unlikely to reduce this component of the defect population.
} 1. Introduction
Multilayers composed of thin (about 50 Å) ferromagnetic layers (such as
Ni1xFex or cobalt) separated by thin (about 20 Å) conductive metal layers (such as
copper or gold) often exhibit large changes in their electrical resistance upon the
application of an external magnetic field. This giant magnetoresistance (GMR)
(Parkin et al. 1991, Egelhoff and Kiel 1992, Honda et al. 1993) was first discovered
in 1988 (Baibich et al. 1988, Binasch et al. 1989). GMR multilayers are widely used
as magnetic sensors in hard disk drives (Tsang et al. 1994, Prinz 1998). They are also
being explored for non-volatile magnetic random-access memories (Wang et al.
2000). The magnetotransport properties of these multilayers are sensitive to the
composition of the multilayer system, the thickness of the individual layer, and
the method (processing conditions) used for their synthesis. The dependence upon
the process is thought to result from a sensitivity of the magnetic and electron
transport properties to the crystal defect population trapped in the layers during
their vapour-phase processing (Ranjan et al. 1987, Nozières et al. 1993, Nicholson
et al. 1994, Butler et al. 1995, Wellock et al. 1995).
y Author for correspondence. Email: [email protected].
Philosophical Magazine ISSN 1478–6435 print/ISSN 1478–6443 online # 2004 Taylor & Francis Ltd
http://www.tandf.co.uk/journals
DOI: 10.1080/14786430310001623579
194
X. W. Zhou and H. N. G. Wadley
Ross et al. (1996) have used high-resolution transmission electron microscopy
(HRTEM) to examine the defect structure of a Ni0.8Fe0.2/Au/Ni0.8Fe0.2 multilayer
deposited by molecular-beam epitaxy. In this process, the thermalized vapour atoms
are deposited at a low energy (about 0.1 eV). The atomic positions of a portion of
the HRTEM image of the multilayer reported by Ross et al. are shown in figure 1.
Several defect populations were observed, including twins and dislocations (Ross
et al. 1996). Here, we concentrate upon the population of misfit dislocations that
create extra half-planes in the Ni0.8Fe0.2 layers (shown by white arrow heads in
figure 1). There are four such dislocations at both the Ni0.8Fe0.2-on-gold and the
gold-on-Ni0.8Fe0.2 interfaces. This is a very high dislocation density and could therefore affect the magnetotransport properties by impeding domain wall migration
and contributing to electron scattering at the interfaces (Ranjan et al. 1987,
Hayashi and Echigoya 2001).
The multilayer shown in figure 1 has a fcc structure with a [111] growth direction.
Since the bcc iron has an atomic spacing (2.482 Å) very close to that of the fcc nickel
(2.489 Å), the lattice mismatch between gold and the Ni0.8Fe0.2 alloy layers can be
well estimated from the lattice mismatch between gold and pure nickel. The lattice
parameters of gold and nickel are 4.080 Å and 3.520 Å respectively. The significant
dislocation density observed in figure 1 is therefore a result of a large (15.91%) lattice
mismatch.
Numerous groups have sought to understand the formation of misfit dislocations in epitaxial films of various misfit strains (van der Merwe 1963, Hartley 1969,
Matthews et al. 1970, Steihardt and Schafer 1971, Matthews and Blakeslee 1974,
1975, 1976, Matthews 1975, Steihardt and Hassen 1978, Dodson and Tsao 1987,
Nix 1989, Gosling et al. 1992, Jain et al. 1992, Stiffler et al. 1992a, b, Jesser and Kui
1993, Hirth 2000). These studies utilize a continuum mechanics analysis to identify
y <111>
Ni0.8Fe0.2
Au
Ni0.8Fe0.2
10 Å
Figure 1.
x
<112>
HRTEM image of a {110} cross-section of the Au/Ni0.8Fe0.2 multilayer.
(Reconstructed from the work by Ross et al. (1996).)
195
Misfit dislocations in gold/Permalloy multilayers
equilibrium dislocation configuration. Essentially, it is assumed that a material with a
lattice constant ad is deposited on a substrate with a lattice constant as (figure 2 (a)).
If the substrate is much thicker than the deposited film, the substrate strain is
negligible during deposition. The lattice mismatch ¼ (as ad)/ad between the
deposited material and the substrate is then accommodated by an elastic strain "
in the thin film: " ¼ (figure 2 (b)).
However, if an array of edge-type misfit dislocations (with Burgers vector
magnitude b and spacing S) are formed along the substrate–deposit interface,
(figure 2 (c)), the elastic strain can be reduced to " ¼ (b/S ). The residual elastic
strain energy (per unit film area) of the film can then be written as (Nix 1989)
Mh"2 Mh
b 2
EE ¼
¼
:
2
S
2
ð1Þ
Here, M is the biaxial elastic modulus of the film and h is the film thickness.
[010]
(a)
Film lattice
ad
[100]
a s – ad (misfit)
as
a s – ad
(b)
(c)
Substrate lattice
b = ad
S
Edge dislocations
Figure 2.
Illustration of a misfit dislocation. (a) before epitaxial growth; (b) elastic accommodation; (c) accommodation with misfit dislocations.
196
X. W. Zhou and H. N. G. Wadley
While the misfit elastic strain energy is reduced by misfit dislocation formation,
the line energy of the dislocations contributes to the total energy of the system. Since
the total dislocation length per unit film area is 1/S, the dislocation energy ( per unit
film area) can be approximated by
b2
h 1
ED ¼
,
ð2Þ
ln
b S
4pð1 Þ
where and are the average shear modulus and Poisson’s ratio near the substrate–
deposit interface respectively. The total energy of a thin film containing misfit dislocations is then
Mh
b 2
b2
h 1
ET ¼ EE þ ED ¼
:
ð3Þ
ln
þ
2
S
b S
4pð1 Þ
This simple model reveals that, because the misfit strain energy increases with
increasing film thickness, a critical film thickness hc must be reached before it is
energetically favourable for the introduction of misfit dislocations. At any film
thickness h, the equilibrium dislocation density can be determined by energy minimization with respect to the total dislocation length 1/S per unit film area (by setting
the derivative of the energy with respect to 1/S equal to zero). This leads to the
expression
b
b
h
ln
Mh ¼
:
ð4Þ
S
4pð1 Þ
b
At the onset of misfit dislocation formation, S is very large. The critical film thickness hc can then be found by solving equation (4) for the special case when S ! 1:
hc
b
:
¼
ln ðhc =bÞ 4pð1 ÞM
ð5Þ
It can be seen that, for a small misfit , the right-hand side of equation (5) is large.
Equation (5) then predicts a very large positive hc. This means that, under thermodynamic equilibrium conditions, misfit dislocations do not form upon deposition
but are rather introduced after the film has grown to a considerable thickness
above the critical thickness hc. This critical thickness generally decreases as increases. However, there is a critical misfit c above which no hc can be defined
by equation (5). The smallest hc ( 2.72 b) is obtained at c. This is a result of the
singularity of a continuum model which fails to account for the dislocation core
structure. When > c, the virtual hc is very small and misfit dislocations are
presumed to form immediately upon deposition. When c, hc is large and misfit
dislocations are not directly nucleated at the interface. Instead, dislocations are
taken to have nucleated elsewhere. These dislocations are then converted to misfit
dislocations after being driven to the interface at the critical film thickness or
above. A mechanism is therefore required to account for the migration of dislocations from the sites of their nucleation to the interface. A threading dislocation
model has been widely accepted for this mechanism (Nix 1989).
Little is known about the dislocation nucleation when epitaxial films with a
large are grown on to a dislocation-free substrate. The continuum mechanics
approach described above does not treat the atomistic nature of crystals that is
critical for dislocation nucleation and it also is unable to provide insight about the
Misfit dislocations in gold/Permalloy multilayers
197
kinetic phenomena controlling the atomic assembly processes resulting in dislocation formation during the early stages of a new layer growth. Here we use a molecular dynamics (MD) simulation of the growth of an Au/Ni0.8Fe0.2 multilayer to
explore the defective structures of a multilayer system similar to that experimentally
studied by Ross et al. (1996). We analyse these simulations to identify the mechanism
of dislocation nucleation at the interfaces.
} 2. Simulation methodology
During vapour deposition, a variety of atomic assembly processes can occur. For
instance, an atom that lands on a surface can hop from one lattice site to another at
a rate determined by the temperature, the locally available transition paths and the
energy barriers along these paths. Many such jump paths are encountered depending
upon the local atomic configuration (Yang et al. 1997). Defects are readily entrained
in the growth process. For example, a surface asperity can shadow incident vapour
atoms, promoting the formation of surface roughness which then leads to vacancies
and voids if the mobilities of surface atoms are insufficient to compensate for the
‘lost’ flux in the region of a shadow (Zhou and Wadley 2000). There is a variety of
other phenomena that can be used to counteract such processes. For instance, the
latent heat release during an atom condensation event together with the adatom’s
kinetic energy can result in a local thermal spike on the surface during impact
(Zhou and Wadley 1999a). This local thermal spike can induce extra local surface
reconstruction above that expected purely by thermal activation and contribute to
surface flattening. It can also lead to undesired mixing at the interfaces of multilayers
(Zhou and Wadley 1998). High-energy impacting atoms can also skip on a smooth
surface for a long distance before coming to rest (Zhou and Wadley 1999a), or they
can reflect or cause resputtering (Zhou and Wadley 1999b). The growth of a thin film
is therefore a complex process. However, each of the events described above is
captured in a MD simulation. MD simulations therefore provide a means for gaining
significant physical insight about these complex phenomena and their consequence
for the nucleation of misfit dislocations. New accelerated MD approaches have also
been developed (Voter 1997, Sprague et al. 2002). These new methods allow simulations on realistic time scales by ignoring details of the dynamics.
In a MD simulation of vapour deposition, a computational substrate crystal is
created by assigning the 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. To simulate a free surface, a free boundary
condition is used for the third coordinate direction (the growth direction). Vapour
deposition is simulated by continuously injecting atoms towards the top (free) surface of the crystal at a frequency determined by the deposition rate. By defining 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 atoms as well as those of
the deposited atoms as a function of time. The time step used in the MD simulations
must be less than the shortest lattice vibration period (typically around 1015 s).
As a result, the real time of deposition that can be analysed is usually limited to
about 1–100 ns. Consequently, an accelerated rate of deposition is normally used
in order to deposit enough atoms in the available computational time to reveal
structural features. It is noted that, while the accelerated deposition reduces the
time available for kinetic reconstruction, the high adatom flux elevates the energy
(temperature) accumulation at the free surface which significantly accelerates
198
X. W. Zhou and H. N. G. Wadley
these processes. The two effects to some degree cancel, and the simulations give
results reasonably approximate to those seen in experiments (Zhou et al. 2001).
To prevent the crystal from shifting (owing to the absorbed momentum of
adatom impact), several atomic layers of atoms at the bottom free surface are
fixed during simulations (when the fixed region is sufficiently thick to span the
interaction range between atoms, the simulated crystal can be viewed as extended
in the thickness direction into a bulk substrate). Either fixed or relaxed periodic
lengths can be used in the simulations (Anderson 1980). The former assumes that
the crystal is not deformable while the latter assumes that the crystal is fully
deformed to reduce the strain.
The transfer of the adatom’s kinetic energy and its latent heat release during
condensation cause a continuous increase in a simulated crystal’s temperature.
Isothermal deposition conditions are simulated by applying damping forces to
atoms in a selected ‘isothermal region’ below the surface (Zhou and Wadley 1998).
This leaves a free surface region whose atomic motion is purely determined by
Newton’s equations of motion while creating a thermal conduction path between
the surface and the isothermal region. To avoid overheating due to the accelerated
rate deposition, the change in the thickness of the thermal conduction region is
minimized by allowing the isothermal region to expand during simulation to keep
pace with the growth surface front. For the vapour deposition simulation described
in this work, the upper bound of the thermostated region was initially set between
the surface monolayer and the second monolayer from the surface at the start of
the simulation. To prevent the thermostated region from surpassing any points of
a rough surface that can possibly form during deposition, the upper boundary
of the thermostated region propagated at about 97% of the deposition rate. Test
runs indicated that this approach guaranteed a very small temperature fluctuation
(less than 3%) during the simulated depositions.
MD simulations resolve atomic processes based upon Newton’s equation of
motion. The results are therefore only reliable as interatomic potentials used to
define the interaction forces between atoms. The embedded-atom method (EAM)
potential originally developed by Daw and Baskes (1984) is now widely used for
MD simulations of crystal defects in close-packed metal systems. In addition to a
pairwise energy term, the EAM potential incorporates the many-body dependence of
the potential by introducing an embedding energy term. The embedding energy can
be thought of as the extra binding energy to embed each atom into its lattice site with
an associated local electron density. With this improvement, EAM potentials
describe well many metal properties such as the lattice constants, cohesive energy,
elastic constants and vacancy formation energy (Daw and Baskes 1984).
EAM potentials developed for the monatomic elements cannot be simply used
to model multilayers or alloys (Johnson 1989). By fitting the parameters of EAM
potentials to alloy properties (such as the heat of solution), several ‘alloy’ EAM
potentials have been devised (Foiles 1985, Foiles et al. 1986, Asta and Foiles
1996). However, these alloy potentials are of restricted utility because they are
not intercompatible and cannot be simply combined with other EAM potentials
to study other alloys. A recently developed alloy EAM potential and database of
sufficient generality has enabled alloy potentials to be created directly from as many
as 16 elemental metals (copper, silver, gold, nickel, palladium, platinum, aluminium,
lead, iron, molybdenum, tantalum, tungsten, magnesium, cobalt, titanium and zinc)
without any further fitting (Wadley et al. 2001, Zhou et al. 2001). The potentials
Misfit dislocations in gold/Permalloy multilayers
199
have been used to calculate the heats associated with mixing a single nickel atom in a
large gold crystal and a single gold atom in a large nickel crystal. The results indicated that the heats of mixing are around 0.08 eV for nickel in gold and around
0.7 eV for gold in nickel. These values are consistent with experiments and other
EAM calculations (Foiles et al. 1986). Based on the calculation of energy difference
between (relaxed) perfect crystals and crystals with stacking faults, the potentials
have also been used to deduce the stacking-fault energy. It predicted that gold has
a stacking-fault energy of 0.0029 eV Å2, and that nickel has a stacking-fault energy
of 0.0074 eV Å2. These are in very good agreement with the experimental data of
0.002 eV Å2 for gold (Balk and Hemker 2001) and 0.008 eV Å2 for nickel (Murr
1975, Mishin et al. 1999). The interatomic potentials utilized in this work were taken
from this EAM database reported by Zhou et al. (2001).
} 3. Dislocation formation during film growth
The MD simulation approach described above was used to investigate the
growth of a multilayer stack similar to that characterized by Ross et al. (1996).
It consists of a Ni0.8Fe0.2 (20 Å)/Au (20 Å)/Ni0.8Fe0.2 (20 Å) multilayer deposited
on a gold (111) substrate (figure 3). The initial gold substrate has 120 ð224 Þ planes
in the x direction, three (111) planes in the y (growth) direction and 16 ð22 0Þ planes
in the z direction. A fixed-periodic-length MD algorithm was used for the simulations. It approximated well the case of a very thick (with respect to the film) substrate
where the crystal’s lattice parameter remained fixed (at the equilibrium value of the
bulk gold) and was not affected by the stresses that developed in the deposited layer.
Simulations were performed at a substrate temperature of 300 K. The vapour
atoms were injected normal to the surface. The frequency of vapour atom addition
corresponded to a film growth rate of 5 nm ns1. Owing to the computational
expense of MD simulations, the real-time simulated was about 1.2 ns. Various
incident atom energies were investigated.
Figure 3 shows the atomic structures of two simulations. One used a low incident
atom energy of 0.5 eV (figure 3 (a)). The second used a relatively high incident atom
energy of 5.0 eV (figure 3 (b)). Here, the black, white and golden spheres correspond
to iron, nickel and gold atoms respectively. The gold concentration profiles obtained
from the two simulations are shown in figure 3 (c). It can be seen that the Ni0.8Fe0.2
free surface is smoother in figure 3 (b) than in figure 3 (a). This resulted from highenergy impact-induced flattening (Zhou and Wadley 1998, Zhou et al. 2001). In
addition, some of the deposited gold atoms have been mixed into the subsequently
deposited Ni0.8Fe0.2 layers. This mixing is slightly more significant in figure 3 (b)
than in figure 3 (a). These phenomena are reflected in figure 3 (c), which indicates
that the interfaces obtained at the higher adatom energy are slightly more diffuse
than those obtained at the lower adatom energy. Also, the Ni0.8Fe0.2-on-gold interfaces are more diffuse than the gold-on-Ni0.8Fe0.2 interfaces. Similar effects have
been observed with Ni/Cu multilayer deposition (Zhou and Wadley 1998). They
are caused by a high-energy impact-induced atom exchange mechanism (Zhou and
Wadley 1998, Zhou et al. 2001). The mixing of gold atoms in the subsequently
deposited layer suggests a continuous gold segregation on the surface during deposition (Zhou and Wadley 1998, Zhou et al. 2001). The primary cause for this lies in
atomic size difference. Because the size of gold atoms is much larger than that of
either nickel or iron atoms, gold preferentially segregates to the surface and relieves
surface tension and misfit strain. On the other hand, the smaller nickel and iron
200
X. W. Zhou and H. N. G. Wadley
(a)
y
T = 300K
normal incidence
[111]
Fe
Au
Ni
Au substrate
z
x
[110]
[112]
o
10A
(b)
(c)
Au concentration
1.0
Ei = 0.5eV
0.8
0.6
Ei = 5.0eV
0.4
0.2
0.0
0
10
20
30
40
50
60
Thickness in growth direction (Å)
Figure 3. MD simulations of the Au/Ni0.8Fe0.2 multilayer deposited at a temperature of
300 K, an adatom incident direction normal to the surface, and two different deposition energies of 0.5 and 5.0 eV: (a) atomic structure at 0.5 eV; (b) atomic structure
at 5.0 eV; (c) gold concentration profile.
201
Misfit dislocations in gold/Permalloy multilayers
y <111>
Ni0.8Fe0.2
Au
Ni0.8Fe0.2
Au
10 Å
Figure 4.
x <112>
Reconstructed image of a {110} cross-section of the Au/Ni0.8Fe0.2 multilayer
simulated using a 0.5 eV adatom energy.
atoms depositing on a surface composed of larger gold atoms can easily penetrate
the lattice through lattice interstices, resulting in their position exchange with gold
atoms. The continuous ejection of gold atoms on to the top of the growth surface
then results in alloying of gold atoms in the subsequently deposited Ni0.8Fe0.2 layers.
Many misfit dislocations were found at the gold-on-Ni0.8Fe0.2 and the Ni0.8Fe0.2on-gold interfaces. They are present in multilayers deposited using both low
(figure 3 (a)) and high (figure 3 (b)) energies. Figure 3 (a) is used as a representative
case to analyse further the dislocation structures. The x–y {110} atomic planes in
figure 3 (a) were projected and the result is shown in figure 4. It clearly indicates the
formation of numerous edge-type misfit dislocations aligning parallel to the interface. Four dislocations are found at both Ni0.8Fe0.2-on-gold and gold-on-Ni0.8Fe0.2
interfaces. Each dislocation is associated with one extra Ni0.8Fe0.2 plane. The image
shown in figure 4 utilized an adatom energy of 0.5 eV. Similar dislocation configurations were obtained using 5.0 eV adatom energy. These results indicate that the
dislocation structures are unlikely to be changed by using hyperthermal energy
deposition methods such as magnetron sputtering or ion-beam deposition.
A remarkable similarity exists between the simulated image (figure 4) and that
obtained experimentally (figure 1). In particular, both figure 4 and figure 1 show four
dislocations at each of the Ni0.8Fe0.2-on-gold and the gold-on-Ni0.8Fe0.2 interfaces.
The average dislocation spacing is about 20 Å in both figure 1 and figure 4. The
dislocation configuration of figure 4 is also virtually indistinguishable from that
shown in figure 1. Clearly, the misfit dislocations shown in figures 4 and 1 effectively
increase the lateral dimension of the Ni0.8Fe0.2 layer and therefore help to relieve
the elastic misfit strain.
} 4. Dislocation formation mechanisms
Deducing the mechanisms of dislocation nucleation is not simple even in a
MD simulation where arrest and detailed examination of the assembly process
202
X. W. Zhou and H. N. G. Wadley
are feasible. Firstly, tangled dislocation configurations often occur during 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 analysing a more ideal atomistic model.
Consider a gold surface that is partially covered by an epitaxial layer of
nickel atoms. 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. To illustrate, a gold crystal containing 68 (22 0)
planes in the x direction, five (111) planes in the y direction, and 24 (2 2 4) planes in
the z direction was constructed. A fraction of the gold surface was then covered by a
single epitaxial atomic layer of nickel. This three-dimensional crystal is shown in
figure 5 (a), where the light-coloured balls represent nickel atoms and the darkcoloured balls are gold atoms. Here the x direction was chosen to be parallel to
the Burgers vector of an (a/2)h110i unit dislocation in the fcc lattice. This facilitates
the analysis of the dislocation configuration.
Details of plane stacking for a part of the crystal are displayed in figure 5 (b).
The top part of the figure is a front (112) view of the crystal while the bottom part is
a top (111) view of the top three crystal planes (the nickel atomic layer and the two
subsurface gold atomic layers). To distinguish the top three planes, the nickel atoms
are now represented by small white circles, and the next two layers of gold atoms
are shown by larger white and grey circles respectively. Figure 5 (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. . . (fcc) 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 figure 5 (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 figure 5 (c). Clearly, the front view now shows two extra
(22 0) planes (i.e. dislocations) in the nickel atomic layer. In the top view, the middle
region between the dislocations still has fcc stacking. However, both regions to
the left and the right have hcp (i.e. ABAB. . .) stacking (the top nickel atoms are
aligned with the second ‘grey’ plane of gold atoms). The transition from the fcc
region to the hcp region corresponds to an atomic shift of (a/6)½12 1 on the left
and (a/6)½2 11 on the right, where a refers to the local lattice constant. It can also
be seen that the boundaries between the fcc and the hcp regions correspond well to
the locations of the extra planes shown in the front view. Hence, the observed
dislocations are Shockley partial dislocations. These dislocations have a Burgers
vector of (a/6)h112i and are separated by either fcc or hcp (stacking-fault) regions.
Figure 5 (c) indicates a dislocation spacing of about 11.5 Å. The appearance of these
dislocations in figure 5 indicates that the critical thickness for misfit dislocation
formation is one atomic layer or less in the nickel–gold system.
Formation of fcc and hcp domains bounding partial dislocations are also likely
to occur in systems with small lattice mismatch, especially when these systems
have a small energy difference between fcc and hcp adsorption sites for adatoms.
In the case studied here, the energy difference between fcc and hcp adsorption sites
was not the dominant mechanism for the domain formation. However, it may affect
the initial nucleation of fcc and hcp domains. It should be noted that there is no
203
Misfit dislocations in gold/Permalloy multilayers
y [111]
(a)
Ni
Au
x [110]
z [112]
(c)
[111]
T
(b)
T
Ni
Au
[110]
[112]
[110]
~11.5 Å
1
6
[211]a
1
6
[121]a
Figure 5. Formation of edge misfit dislocations on the nickel-on-gold surface: (a) threedimensional epitaxial crystal; (b) front and top views of the epitaxial crystal; (c) front
and top views of the relaxed crystal.
explicit cut-off distance in our potential. However, all potential functions are
virtually cut off around fifth-nearest-neighbour distance as their values at this distance are two orders of magnitude smaller than those at fourth-nearest-neighbour
distance. The inclusion of at least fourth-nearest neighbours captures the geometry
difference between fcc and hcp sites. This allowed us to calculate realistically the
energies for adatoms to adsorb on different sites on the surface. We found that, on a
gold surface, nickel at a hcp site is about 0.02 eV more stable than at a fcc site and,
on a nickel surface, gold at a fcc site is about 0.01 eV more stable than at a hcp site.
204
X. W. Zhou and H. N. G. Wadley
These values are very small, consistent with other EAM simulations (Liu et al.
1991). It indicates that the nucleation rates for fcc and hcp domains are about the
same in nickel-on-gold or gold-on-nickel systems.
To examine dislocation initiation on a more realistic composition surface, the
early stages of deposition of a Ni0.8Fe0.2 layer on a gold substrate were simulated.
The growth conditions were the same as those used in figure 3 (a). The gold substrate
has 68 (22 0) planes in the x direction, three (111) planes in the y direction, and
24 (2 2 4) planes in the z direction. The orientation is the same as that in figure 5.
The x direction was normal to the extra {220} planes of the (a/6)h112i dislocations
again to facilitate dislocation visualization.
The x–y projections of the atomic crystals after each of the first four Ni0.8Fe0.2
monolayers was deposited are shown in figures 6 (a)–(d ), where the black, white and
(a)
y <111>
Fe
Ni
Au
x <110>
(b)
(c)
(d)
10 Å
Figure 6. MD simulated Ni0.8Fe0.2-on-gold structures as a function of deposition time t
(deposition temperature, 300 K; adatom incident direction normal to the surface;
adatom energy, 0.5 eV): (a) t ¼ 50 ps; (b) t ¼ 100 ps; (c) t ¼ 150 ps; (d ) t ¼ 200 ps.
Misfit dislocations in gold/Permalloy multilayers
205
orange circles represent iron, nickel and gold atoms respectively. It can be seen
that, after one Ni0.8Fe0.2 monolayer (about 2.5 Å) was deposited (figure 6 (a)),
many gold atoms exchanged with the depositing iron or nickel atoms, resulting in
extensive mixing at the Ni0.8Fe0.2-on-gold interface. This is largely caused by the
larger gold size as has been discussed above. 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 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 dislocated atoms form dislocation
nuclei even when the first monolayer of Ni0.8Fe0.2 was deposited. It should be
pointed out that the mixing at the Ni0.8Fe0.2-on-gold interface can change the misfit
strain at least during the deposition of first a few monolayers of Ni0.8Fe0.2.
Molecular statics simulations of relaxed (Ni0.8Fe0.2)1xAux bulk crystals indicated
that their lattice parameter is very close to a linear function of x and can be expressed
as a ¼ 3.534 þ 0.546x. The lattice mismatch strain for a (Ni0.8Fe0.2)1xAux-on-gold
interface can decrease from 15.45% at x ¼ 0.0 to 13.68% at x ¼ 0.1.
As more Ni0.8Fe0.2 monolayers were deposited, the {220} planes in the Ni0.8Fe0.2
became gradually clearer (figures 6 (b)–(d )). 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 the dislocations at the Ni0.8Fe0.2-on-gold interface. Analysis of the crystal structure after the entire Ni0.8Fe0.2 (20 Å)/Au (20 Å)/
Ni0.8Fe0.2 (20 Å) multilayer was deposited indicated eight to nine extra {220}
planes over a length scale of about 98 Å. Each extra {220} plane corresponds to
an (a/6)h112i partial dislocation. The average dislocation spacing is around 11–12 Å,
in agreement with the results shown in figure 5 (c).
} 5. Discussion
HRTEM experiments have indicated extensive misfit dislocations at the interfaces of vapour deposited Ni0.8Fe0.2/Au multilayers. Atomistic simulations revealed
a similar misfit dislocation configuration. Both experiments and simulations have
an average distance between dislocations of about 20 Å in the h112i direction.
Atomistic simulations further indicated that, owing to the large lattice mismatch
between gold and Ni0.8Fe0.2 layers, the epitaxial Ni0.8Fe0.2 on gold is not stable
even when only one Ni0.8Fe0.2 monolayer is deposited, and misfit dislocations
directly nucleate on the gold surface when Ni0.8Fe0.2 is deposited. Further atomistic
simulations using crystals with coordinate direction parallel to the (a/2)h110i Burgers
vector indicated that the observed dislocations are (a/6)h112i Shockley partial
dislocations dissociated from the (a/2)h110i unit dislocations. The average spacing
of the (a/6)h112i dislocations is about 11–12 Å in the h110i direction.
For the Ni0.8Fe0.2/Au system, the result that the misfit dislocation directly nucleates on substrate surface means that the critical film thickness for misfit dislocation
formation is one monolayer or below. It is interesting to compare the results of
atomistic simulations and continuum calculations for both the critical film thickness
of misfit dislocation formation and the equilibrium misfit dislocation spacing.
The deposition of nickel on gold is explored as an example.
Appendix A calculates the line energy of an (a/6)h112i dislocation in elemental
nickel and gold crystals. To calculate the critical film thickness, the biaxial modulus
M is needed. M in principle can be derived from other elastic moduli. However,
for the extremely thin films studied here, the M dependence on the film thickness
206
X. W. Zhou and H. N. G. Wadley
needs to be considered. One way to derive M as a function of film thickness is
to calculate the energy change of a film with various thicknesses during biaxial strain.
If the energy of the relaxed film is E0, and the energy of the same film subject to a
uniform biaxial strain " is E", then the biaxial modulus can be written as
M¼
2ðE" E0 Þ
,
V"2
ð6Þ
where V is the volume of the film. The (111) film energies E0 and E" were calculated
using molecular statics. The biaxial modulus of nickel film during a tensile strain was
then calculated using equation (6). The calculated biaxial modulus as a function
of film thickness is shown in figure 7. It can be seen that the biaxial modulus
quickly approaches a saturated (bulk) biaxial modulus value as the film thickness
is increased. In fact, a nickel film thickness of one atomic layer is the only case where
the biaxial modulus is significantly larger than that of other film thicknesses. This
reflects the effect of surface stress on the apparent biaxial modulus. The inclusion
of the surface stress effect in biaxial modulus naturally incorporated the surface
stress effect in the analysis of dislocation for very thin films.
For an (a/6)h112i misfit dislocation formation during deposition of nickel on
gold, ¼ 0.1593 and b ¼ 1.437 Å. The shear modulus and Poisson’s ratio of bulk
nickel are ¼ 0.5917 eV Å3 and ¼ 0.2529 respectively. No hc can be solved from
equation (5) using these values, indicating hc to be one monolayer or less. For one
atomic layer of nickel on gold, h ¼ 2.032 Å and M ¼ 6.384 eV Å3. Using an approximate nickel dislocation line energy of 0.15 eV Å1 (see appendix A) to substitute
the ½b2 =4pð1 Þ lnðh=bÞ term in equation (4), equation (4) gives an equilibrium
dislocation spacing of about 9.5 Å. This is slightly shorter than the value predicted
by atomistic simulations described above. We believe that the difference mainly
comes from the neglect of the discrete lattice nature in the continuum calculations.
8.0
Biaxial modulus M (eV/Å3)
7.0
6.0
5.0
4.0
3.0
2.0
Bulk value
1.0
0.0
0
1
2
3
4
5
6
7
8
9
10
11
12
Number of Ni monolayers
Figure 7.
Biaxial elastic modulus of nickel as a function of the number of nickel layers.
Misfit dislocations in gold/Permalloy multilayers
207
} 6. Conclusions
MD simulations have been used to study the atomistic mechanisms responsible
for misfit dislocation formation during Ni0.8Fe0.2 (20 Å)/Au (20 Å)/Ni0.8Fe0.2 (20 Å)
multilayer deposition. The following conclusions have been obtained.
(i) Extensive misfit dislocations that create extra half-planes in the Ni0.8Fe0.2
layer formed during the deposition. Both dislocation configuration and
dislocation spacing are in good agreement with the HRTEM experiment
on the same multilayer system.
(ii) (a/6)h112i Shockley misfit partial dislocations have been identified to nucleate directly during the multilayer deposition, indicating a zero critical film
thickness for their formation. The average spacing of these dislocations
in the h110i direction is about 11–12 Å. This was 10–20% higher than that
predicted by continuum models.
(iii) The critical film thickness for misfit dislocation formation is consistent
with that predicted by continuum calculations.
(iv) The dislocation configurations are unaffected by high energy deposition
conditions.
Acknowledgement
We are grateful to the Defense Advanced Research Projects Agency (D. Healy
and S. Wolf, Program Managers) for support of this work.
A P PE N DI X A
} A1. Dislocations energetics
Molecular statics simulations indicated that the ‘direct’ nucleation of the
(a/6)h112i misfit dislocations at an interface in the nickel–gold system is energetically preferred. To better understand dislocations for this simple system and to
relate the atomistic simulations to continuum models, we deduce the line energy of
the (a/6)h112i dislocations. Because the misfit energy and dislocation energy cannot
be separated in the nickel–gold system, we can only deduce the (a/6)h112i dislocation
energies in elemental nickel and gold crystals using a molecular statics method.
The (a/6)h112i partial dislocations are dissociated from unit (a/2)h110i dislocations. Edge dislocations with an (a/2)h110i Burgers vector can be created in a crystal
as long as one of its coordinate axes aligns with the h110i direction. In a traditional
dislocation model, an (a/2)h110i edge dislocation was created by inserting two
adjacent {220} planes in the upper part of the crystal (von Boehm and Nieminen
1996). The insertion of the two {220} planes maintains the ABAB. . . stacking along
the h110i direction and can naturally lead to two (a/6)h112i partial dislocations
bounding a stacking-faulted region after molecular statics simulation of energy
minimization (von Boehm and Nieminen 1996). However, this model has a serious
problem. Imagine that the upper half of the crystal has extra planes with respect
to the lower half of the crystal. The upper half of the crystal therefore is subject
to a compressive stress while the lower half experiences a tensile stress. This stress
is essentially constant when the thicknesses of the upper and lower halves of the
crystal are simultaneously increased. However, since dislocations are linear defects,
208
X. W. Zhou and H. N. G. Wadley
the stress field should diminish as the distance from the dislocation core increase.
It should be noted that the created dislocations in fact form an array of dislocations owing to the periodic boundary conditions. The problem arises because the
superimposition of the strain field of this array of edge dislocations adds up to a
constant in the far-field. A model where the far-field stress of the dislocation array
diminished was developed to calculate the dislocation energy accurately.
The approach used here is illustrated in figure A 1. Figure A 1 (a) shows a
dislocation-free nickel or gold crystal containing 68 (22 0) planes in the x direction,
15 (111) planes in the y direction, and 24 (2 2 4) planes in the z direction. To eliminate
lateral surfaces, periodic boundary conditions were used in the x and z directions and
a free surface boundary condition was used in the y direction. To create dislocations,
the shaded block of atoms was shifted by a vector, (a/6)[12 1]. This created two
unrelaxed partial dislocations at the locations indicated by the white circles. The
y [111]
(a)
1
6
[121]a
x [110]
z [112]
(b)
y
T
(c)
[
T
x
-z
x
Figure A1. Ideal atomistic model of edge dislocations: (a) three-dimensional perfect crystal;
(b) front view of the relaxed crystal containing two (a/6)h112i dislocations; (c) top view
of the relaxed crystal containing two (a/6)h112i dislocations.
Misfit dislocations in gold/Permalloy multilayers
209
Burgers vectors of the two dislocations were (a/6)[12 1] and (a/6)[12 1] respectively.
Because the two dislocations were of the opposite sign, the stress field of the
alternating dislocation array diminished in the far-field. Relaxed dislocation configurations were obtained by molecular statics simulation of energy minimization.
To prevent the two dipolar dislocations from combining under their attractive
interaction, atoms in three small regions near the dislocation slip plane, marked by
the small outlined squares and rectangle in figure A 1 (a), were fixed during energy
minimization. It is known that dislocations may be trapped to the closest Peierls–
Nabarro minimum during molecular statics simulation. Test runs were hence carried
out to verify that the model can fully relax dislocations. A nickel crystal containing
dislocations five monolayers below the surface was used for this study. All the atoms
that were not fixed during the molecular statics simulations were randomly displaced
before simulations. The molecular statics simulations of the displaced crystals
indicated that they were relaxed to the same configuration with almost the same
energy regardless of the way that the atoms were displaced.
Energy minimization was carried out with periodic lengths fixed at the
equilibrium bulk crystal size. This best reflected the dislocation strain fields because
these dislocations were assumed to lie in a thin film that was deposited on a much
thicker equilibrium bulk crystal. As an example, the relaxed configuration of
the crystal shown in figure A 1 (a) are examined in figure A 1 (b) and (c). Figure
A 1 (b) is a front view, and figure A 1 (c) is a top view of three adjacent planes
indicated by white, grey and black colours in figure A 1 (b). Figure A 1 (b) indicates
that the two partial dislocations have an extra (22 0) plane above and below the
slip plane. Figure A 1 (c) shows that the two dislocations bound a region with a
stacking fault.
Assume that the energies of the relaxed crystals with and without dislocations are
Ed and E0 respectively. The line energy of a dislocation can be expressed as
G¼
Ed E0 ld dsf
,
2ld
ðA 1Þ
where ld is the dislocation length (i.e. the crystal dimension along the z direction), d
is the separation distance between the two dislocations (i.e. the stacking-fault band
width) and sf is the stacking-fault energy. The dislocation energy can be directly
calculated using equation (A 1). Since the thin-film thickness lies in the dislocation
core regime, the dependence of the dislocation energy on the film thickness needs to
be addressed. This was achieved by studying crystals with different deposited layer
thicknesses. The dislocation line energies calculated using equation (A 1) at various
film thicknesses are shown in figures A 2 (a) and (b) for nickel and gold respectively
using the full circles.
According to continuum theory, the dislocation energy in a crystal can be
expressed as (Nix 1989)
"
# b2e
b2s
h
E¼
þ
,
ðA 2Þ
ln
b
4pð1 Þ 4p
where b, be and bs are the total length, the edge component and the screw component respectively of the dislocation Burgers vector, h is the film thickness, is the
shear modulus and is Poisson’s ratio. For our potentials, ¼ 0.5917 eV Å3
and ¼ 0.2529 for nickel, and ¼ 0.1938 eV Å3 and ¼ 0.4125 for gold. The line
X. W. Zhou and H. N. G. Wadley
(a)
0.5
Dislocation line energy (eV/Å)
210
0.4
MD simulation
0.3
0.2
Eq. (A2)
0.1
0.0
0
1
2
3
4
5
6
7
8
9
10
11
12
11
12
0.5
Dislocation line energy (eV/Å)
Number of Ni overlayers
(b)
0.4
MD simulation
0.3
0.2
0.1
Eq. (A2)
0.0
0
1
2
3
4
5
6
7
8
9
10
Number of Au overlayers
Figure A 2.
Dislocation line energy as a function of the number of deposited layers: (a) nickel
crystal; (b) gold crystal.
energy of the (a/6)[12 1] dislocations shown in figure A 1 was then calculated
using equation (A 2) for both nickel and gold crystals, and the results are plotted
in figure A 2 (a) and (b) respectively. It can be seen that for both nickel and
gold crystals, the trend of dislocation energy as a function of the film thickness
calculated from atomistic simulations is very similar to that calculated using
continuum theory. Over all the film thickness range, the dislocation energy predicted
by the continuum calculations is slightly lower than that predicted by the MD
simulations. This difference can account for the dislocation core energy that is
ignored in the continuum theory.
Misfit dislocations in gold/Permalloy multilayers
211
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