University of Groningen
Plastic deformation of freestanding thin films
Nicola, L.; Xiang, Y.; Vlassak, J. J.; van der Giessen, Erik; Needleman, A.
Published in:
Journal of the Mechanics and Physics of Solids
DOI:
10.1016/j.jmps.2006.04.005
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Nicola, L., Xiang, Y., Vlassak, J. J., Van der Giessen, E., & Needleman, A. (2006). Plastic deformation of
freestanding thin films: Experiments and modeling. Journal of the Mechanics and Physics of Solids, 54(10),
2089-2110. DOI: 10.1016/j.jmps.2006.04.005
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ARTICLE IN PRESS
Journal of the Mechanics and Physics of Solids
54 (2006) 2089–2110
www.elsevier.com/locate/jmps
Plastic deformation of freestanding thin films:
Experiments and modeling
L. Nicolaa, Y. Xiangb, J.J. Vlassakb, E. Van der Giessenc,
A. Needlemana,
a
Division of Engineering, Brown University, Providence, RI 02912, USA
Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA
c
Department of Applied Physics, University of Groningen, Nyenborgh 4, 9747 AG Groningen, The Netherlands
b
Received 19 October 2005; received in revised form 7 March 2006; accepted 8 April 2006
Abstract
Experimental measurements and computational results for the evolution of plastic deformation in
freestanding thin films are compared. In the experiments, the stress–strain response of two sets of Cu
films is determined in the plane-strain bulge test. One set of samples consists of electroplated Cu
films, while the other set is sputter-deposited. Unpassivated films, films passivated on one side and
films passivated on both sides are considered. The calculations are carried out within a twodimensional plane strain framework with the dislocations modeled as line singularities in an isotropic
elastic solid. The film is modeled by a unit cell consisting of eight grains, each of which has three slip
systems. The film is initially free of dislocations which then nucleate from a specified distribution of
Frank–Read sources. The grain boundaries and any film-passivation layer interfaces are taken to be
impenetrable to dislocations. Both the experiments and the computations show: (i) a flow strength
for the passivated films that is greater than for the unpassivated films and (ii) hysteresis and a
Bauschinger effect that increases with increasing pre-strain for passivated films, while for
unpassivated films hysteresis and a Bauschinger effect are small or absent. Furthermore, the
experimental measurements and computational results for the 0.2% offset yield strength stress, and
the evolution of hysteresis and of the Bauschinger effect are in good quantitative agreement.
r 2006 Elsevier Ltd. All rights reserved.
Keywords: Dislocations; Thin films; Bauschinger effect; Size effects; Computer simulation
Corresponding author.
E-mail address: [email protected] (A. Needleman).
0022-5096/$ - see front matter r 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jmps.2006.04.005
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1. Introduction
The decreasing dimensions of micro-electronic and micro-mechanical devices have
motivated research on the micro- and nano-scale mechanical behavior of materials. It has
become clear that, at such scales, the plastic response of crystalline solids can exhibit a
strong size dependence, e.g. Fleck et al. (1994), Ma and Clark (1995), Stölken and Evans
(1998). Various mechanisms can lead to this size dependence. One well-appreciated source
of size dependence is associated with plastic strain gradients and geometrically necessary
dislocations, e.g. Fleck et al. (1994). A size effect can also arise even when the
macroscopically applied deformation is uniform if dislocation motion is constrained due to
an internal interface such as a grain boundary; leading, for example, to a grain sizedependent flow strength, i.e. the Hall–Petch effect. Recently, a strong size effect has been
seen in single crystal compression tests (Uchic et al., 2004; Dimiduk et al., 2005; Greer et
al., 2005), where dislocation glide is relatively unconstrained.
Motivated by these experimental observations, a variety of theoretical frameworks have
been developed to incorporate one or more material length scales into a theory of plastic
flow, e.g. Gao et al. (1999), Acharya and Bassani (2000), Fleck and Hutchinson (2001),
Gurtin (2002). These phenomenological theories describe size dependence arising from the
presence of geometrically necessary dislocations and, for some theories, from boundary
constraints on plastic flow. However, neither size dependence arising from ‘‘dislocation
starvation’’ (Uchic et al., 2004; Dimiduk et al., 2005; Greer et al., 2005) nor from sourcelimited plasticity is modeled by current phenomenological plasticity theories. On the other
hand, a framework for solving boundary value problems has been developed, where plastic
flow occurs through the collective motion of discrete dislocations (Van der Giessen and
Needleman, 1995). In this discrete dislocation plasticity framework, size dependence
emerges as a natural outcome of the boundary value problem solution. Size effects occur
due to dislocation starvation and due to limited dislocation sources, in addition to arising
from geometrically necessary dislocations and boundary constraints (Cleveringa et al.,
1999; Shu et al., 2001; Nicola et al., 2003, 2005a; Deshpande et al., 2005).
Micro-scale devices typically contain thin films, either freestanding or bonded to a
substrate. Accordingly, much attention has been directed to experimentally characterizing
the mechanical properties of thin films by a variety of techniques, e.g. (Vlassak and Nix,
1992; Shen et al., 1998; Hommel and Kraft, 2001; Espinosa et al., 2003, 2004; Dehm et al.,
2003; Lou et al., 2003; Phillips et al., 2004; Florando and Nix, 2005; Xiang et al., 2005),
and to modeling their size-dependent behavior, e.g (Hutchinson, 2001; Haque and Saif,
2003; Pant et al., 2003; Groh et al., 2003; Von Blanckenhagen et al., 2004; Ghoniem and
Han, 2005; Hartmaier et al., 2005; Nicola et al., 2003, 2005a–c). Additional references can
be found in the works cited. The experiments, carried out under either thermal or
mechanical loading, have shown that the inelastic response depends strongly on film
thickness and on whether or not the films are passivated. Calculations of the sizedependent response carried out using phenomenological continuum plasticity constitutive
relations and discrete dislocation descriptions of plastic flow have qualitatively reproduced
aspects of the plastic response seen in the experiments. What has been lacking is a direct
quantitative comparison between experiment and theoretical predictions that covers a
broad range of film thicknesses and both passivated and unpassivated films. Here, we
present such a direct quantitative comparison between a series of experiments on
freestanding Cu films, Xiang et al. (2002, 2004, 2005), Xiang and Vlassak (2005a, b),
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involving cyclic as well as monotonic loading, and the predictions of discrete dislocation
plasticity.
The experiments were carried out on polycrystalline films made either by an
electroplating or sputtering process, with thicknesses from 4:2 to 0:34 mm. Films passivated
on one or both sides as well as unpassivated films were used in the experiments. Strong
effects of size and passivation were seen on the flow strength and on the Bauschinger effect
on unloading.
The discrete dislocation formulation follows that of Nicola et al. (2005a) for films on
stiff substrates, except that here the loading is mechanical rather than thermal. A plane
strain model of the polycrystalline film is used. In this formulation, plastic flow arises from
the collective motion of discrete edge dislocations, represented as line singularities in an
elastic solid, with the long-range interactions between dislocations directly accounted for.
Drag during dislocation motion, dislocation nucleation and dislocation annihilation are
incorporated through a set of constitutive rules. The boundary conditions are enforced by
solving for an image field. This two-dimensional framework has predicted (Nicola et al.,
2003), at least qualitatively, a range of features observed experimentally including, for
example, the bilinear hardening seen in sufficiently thin films (Florando and Nix, 2005).
In this study, three parameters entering the dislocation constitutive rules (the density of
dislocation sources, their average nucleation strength and the standard deviation of the
nucleation strengths) are set to fit the measured stress–strain response of an electroplated
film passivated on both sides. Then, with all constitutive parameters fixed, the response, i.e.
the evolution of the Bauschinger effect as well as of the flow strength, is predicted for the
remaining cases, including unpassivated films as well as films passivated on one side. The
calculations give insight into the mechanisms leading to the observed size dependence and
the predictions are compared quantitatively with the experimental observations.
2. Experiments
2.1. Methodology
The mechanical behavior of two sets of Cu films was investigated using the plane-strain
bulge test (Xiang et al., 2005). In this test, silicon micromachining techniques are used to
fabricate long rectangular membranes out of the film of interest. These membranes are
deformed in plane strain by applying a uniform pressure to one side of the membrane as
illustrated in Fig. 1. The average stress s and strain in the membrane are determined from
the applied pressure, p, and the corresponding membrane deflection, d, using the following
expressions (Xiang et al., 2005):
s¼
pða2 þ d2 Þ
;
2dh
¼ 0 þ
ða2 þ d2 Þ
2ad
arcsin
1,
2ad
ða2 þ d2 Þ
(1)
where h is the film thickness, 2a the width of the membrane and e0 the residual strain in the
film. Using this technique, it is possible to measure the plane strain stress–strain curve of a
thin film, as well as its residual stress. With some care, films as thin as 100 nm can be tested.
The first set of samples consisted of electroplated Cu films with thicknesses varying from
1:0 to 4:2 mm, but with constant microstructure. These films were fabricated by
electroplating 5:4mm-thick Cu films on Si substrates using a commercial plating process
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Fig. 1. Perspective views of a typical bulge test sample: (a) as prepared, the Si-framed freestanding membrane is
flat and under residual tension; (b) when a uniform pressure is applied to one side of the membrane, the film is
deformed in plane-strain tension with zero strain along the longitudinal direction. Only a section of the bulge test
sample is shown.
(Xiang et al., 2004). Prior to the plating process, the Si substrates were coated with a 75 nm
Si3 N4 film using low-pressure chemical vapor deposition, a 20 nm TaN adhesion layer
using reactive sputtering, and a Cu seed layer using magnetron sputtering. The
electroplated films were annealed at 400 C for 15 min in vacuum to obtain a uniform
and stable microstructure. The films were subsequently thinned to the desired thickness by
means of chemical–mechanical planarization. Further details of the materials preparation
process are given in Xiang et al. (2004).
Freestanding Cu membranes were fabricated by opening long rectangular windows in
the substrate using Si micromachining techniques (Xiang et al., 2005) and by etching the
exposed Si3 N4 and TaN coatings. Some of the membranes were passivated by sputter
coating 20 nm Ti onto either one or both surfaces of the freestanding Cu membranes.
The second set of samples consisted of sputter-deposited Cu films with thicknesses
ranging from 0:34 to 0:89 mm. For each thickness, Cu was sputtered directly onto
freestanding bilayer membranes that consisted of 75 nm of Si3 N4 and 20 nm of TaN.
Immediately prior to the Cu deposition, the TaN surface of the membranes was sputtercleaned in situ using an Ar plasma. After deposition, all membranes were annealed in
vacuum to increase the grain size of the films. Annealing temperatures were selected to
ensure that the membranes would not buckle during the annealing step as a consequence of
the differential thermal expansion between the membrane and the Si frame. The resulting
grain sizes were measured using transmission electron microscopy (TEM) and are listed in
Table 1. For some samples, the Si3 N4 /TaN layer was etched away using reactive ion
etching (RIE) in order to create freestanding Cu membranes. Other samples were tested as
annealed without removing the Si3 N4 /TaN layer.
The reason for leaving the thin Si3 N4 /TaN layer in place is twofold: (i) The Si3 N4 /TaN
layer serves to passivate one of the surfaces of the Cu film by blocking dislocations from
exiting the film. Comparison of the stress–strain curves of these films with those of
unpassivated Cu films provides information on the effect of a free surface or interface on
the mechanical behavior of thin films. (ii) The Si3 N4 /TaN layer makes it possible to
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Table 1
Deposition method, thickness, h, and average grain size, d, of the films tested
Deposition method
h (mm)
d (mm)
h=d
Electroplated
4:20 0:05
1:90 0:05
1:00 0:05
1:50 0:05
1:51 0:04
1:50 0:05
2:80 0:13
1:26 0:07
0:67 0:04
Sputtered
0:89 0:01
0:67 0:01
0:61 0:01
0:44 0:01
0:35 0:01
0:46 0:02
0:46 0:01
0:54 0:02
0:39 0:01
0:33 0:01
1:93 0:11
1:43 0:05
1:14 0:06
1:13 0:05
1:06 0:06
For the electroplated films, twins are counted as separate grains; the actual mean grain size, not counting twins as
separate grains, is about 2:7 mm.
alternate the direction of plastic flow in the Cu film during a bulge test experiment as
described in Xiang and Vlassak (2005a). This is realized by first loading the composite
membrane in the bulge test until the Cu flows plastically. The Si3 N4 /TaN layer only
deforms elastically and a large tensile stress is built up in this layer. Upon unloading, the
tensile stress in the Si3 N4 /TaN drives the Cu film into compression, while the overall stress
in the composite membrane is kept tensile to prevent buckling of the membrane. The
stress–strain curve of the Cu film is then obtained by subtracting the elastic contribution of
the TaN/Si3 N4 coating from the stress–strain curve of the composite film. The contribution
of the TaN/Si3 N4 coating is readily determined by bulge testing the membrane after the Cu
film has been dissolved in dilute nitric acid. It should be noted that this technique
circumvents the problem of buckling when testing freestanding thin films in compression
because the average stress in the membrane is kept tensile. As such, it is the only technique
currently available to test the same freestanding film in both tension and compression.
Table 1 shows the film thickness h and average grain size d for all films tested. Crosssection TEM micrographs of the electroplated films, Fig. 2(a), reveal that the thinnest films
have a grain size on the order of the film thickness with grain boundaries traversing the
entire film; thick films typically have several grains through the thickness. This is not
surprising given that the average grain size of the electroplated films is approximately onethird the thickness of the thickest film. These films contain many twins, which give rise to
internal boundaries (Xiang et al., 2004, 2006). The actual mean grain size of these films is
approximately 2.7 mm, but counting the twins as separate grains gives an average grain size
of 1:5 mm as shown in Table 1, independent of film thickness. X-ray diffraction (XRD)
measurements by Xiang et al. (2004) show that the crystallographic texture of the
electroplated films is independent of thickness and that it has ð1 1 1Þ, ð1 0 0Þ and ð1 1 0Þ
components, with the plane-strain tension Taylor factor ranging between 2.97 and 3.00.
The sputtered films have a mixed texture that consists mainly of ð1 1 1Þ and ð1 0 0Þ fiber
components, with the ð1 0 0Þ component slightly weaker for the thinner films. The planestrain tension Taylor factor of the thinnest sputtered film is 3.17 and that of the thickest
sputtered film 3.11. The grain size of the sputtered films is a weak function of film
thickness. TEM micrographs show that the sputtered films have a columnar grain structure
with equiaxed grains in the plane of the film. Fig. 2(b) presents a TEM micrograph of an
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Fig. 2. Cross-section TEM micrographs of freestanding electroplated Cu films after deformation showing:
(a) grain size on the order of the film thickness in a 1:0mm-thick unpassivated film; (b) a region of high dislocation
density at the Cu–Ti interface in a 1:9 mm passivated film.
electroplated film with a Ti passivation layer that was deformed in the plane of the film. A
50 nm region of high dislocation density can be observed near the Cu/Ti interface,
indicating that this interface prevents dislocations from exiting the film, which has
important consequences for the plastic response of the film.
2.2. Experimental results
Typical stress–strain curves for the electroplated Cu films are presented in Fig. 3.
Fig. 3(a) shows the effect of the Ti passivation layer on the stress–strain curve of a 1:0 mm
Cu film, while Fig. 3(b) illustrates the effect of film thickness for films with both surfaces
passivated by 20 nm Ti. For a given film thickness, the passivation layer clearly increases
the work hardening rate of the film.
Fig. 4 shows typical stress–strain curves for a 0:6mm-thick sputter-deposited film. Films
both with and without TaN/Si3 N4 passivation are shown. The film with passivation shows
significantly more work hardening than the unpassivated film. Moreover, the film with
passivation is in compression at the end of the last two unloading cycles.
All unloading/reloading cycles for this film show significant hysteresis with reverse
plastic flow occurring even when the stress in the film is still tensile. By contrast, the film
without passivation shows little or no reverse plastic deformation when fully unloaded; i.e.,
passivated films show hysteresis and a strong Bauschinger effect, while unpassivated films
do not.
3. Modeling
3.1. Problem formulation
The formulation and solution procedure follow that used by Nicola et al. (2005a) for
films on stiff substrates, except that here the loading is mechanical rather than thermal. A
more complete description of the analysis procedure and further references are given in
Nicola et al. (2005a). As sketched in Fig. 5, the polycrystalline film is modeled as an infinite
array of planar rectangular grains of height h and width d. Plane strain conditions are
assumed and the elastic anisotropy of the grains is neglected. Plasticity in the film
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400
400
two surfaces passivated
one surface passivated
unpassivated
350
300
<σ11>[MPa]
250
200
150
h=1µm
h=1.9µm
h=4.2µm
350
300
<σ11>[MPa]
2095
250
200
150
100
100
50
50
0
0
0
0.002
0.004
ε
(a)
0.006
0
0.002
0.004
0.006
0.008
0.01
ε
(b)
Fig. 3. Typical stress–strain curves for the freestanding electroplated Cu films with constant microstructure:
(a) the effect of passivation by 20 nm Ti on 1:0 mm films and (b) the effect of film thickness on films passivated on
both surfaces with 20 nm Ti. All curves are offset by the equi-biaxial residual strains in the films, as represented by
the dashed lines starting from the origins. Experimental data from Xiang et al. (2004).
500
400
<σ11> [MPa]
300
200
100
0
-100
unpassivated
one side passivated
-200
0
0.002
0.004
0.006
0.008
ε
Fig. 4. Typical stress–strain curves for a 0:6mm-thick sputter-deposited Cu film: the effect of the TaN/Si3 N4
passivation. Both curves are offset by the equi-biaxial residual strains in the films, as represented by the dashed
line starting from the origin. Experimental data from Xiang and Vlassak (2005a).
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x2
w
p
+
ε
h
ε
−
φ
d
x1
Fig. 5. Two-dimensional model of a freestanding film passivated on both sides under tensile loading. The
calculations are carried out for a cell consisting of eight grains (w ¼ 8d in the calculations) and for films passivated
on one or both sides as well as for unpassivated films. The left most grain shows the sign convention for
dislocations on the slip system oriented at f and positive dislocations on each of the three slip systems are shown
in the center grain.
originates from the motion of edge dislocations in the x1 –x2 plane which are treated as line
singularities in an elastic solid. The calculations are carried out for a unit cell of width w
consisting of eight grains (w ¼ 8d). Each grain has three sets of slip planes on which edge
dislocations having Burgers vector b, can nucleate and glide. The slip systems are oriented
at f, f þ 60 and f þ 120 from the x1 -axis. The orientation of each grain, f, is chosen
randomly from the range ½0; pÞ. Films passivated on one or both sides as well as
unpassivated films are considered. Any passivation layers remain elastic, are of thickness p
and are taken to have elastic constants identical to those of the film. The grain boundaries
as well as any film-passivation layer interfaces are taken to be flat and impenetrable to
dislocations.
Tension is imposed by a prescribed displacement difference between unit cells. In
particular, for the unit cell for which 0px1 pw,
ui ð0; x2 Þ ¼ ui ðw; x2 Þ UðtÞdi1 ,
(2)
where UðtÞ is a prescribed function of time and dij is the Kronecker delta. The imposed
strain is e ¼ U=w. The top and bottom surfaces are traction-free so that
s12 ðx1 ; 0Þ ¼ s22 ðx1 ; 0Þ ¼ 0;
s12 ðx1 ; h þ kpÞ ¼ s22 ðx1 ; h þ kpÞ ¼ 0; k 2 0; 1; 2
(3)
where k is the number of passivation layers (counting from the bottom x2 ¼ 0). In
addition, to prevent rigid body motions
u1 ð0; 0Þ ¼ u2 ð0; 0Þ ¼ 0.
(4)
Although the boundary conditions, Eqs. (2)–(4), completely specify a boundary value
problem, the highly nonuniform distribution of plastic deformation leads to extensive
bending of the film. The actual films do not exhibit this behavior due to the constraint
imposed by grains located in front of and behind the plane of deformation considered in
the two-dimensional analysis. We mimic this constraint by imposing
h
(5)
u2 x1 ; þ kp ¼ 0; k 2 0; 1
2
along the mid-plane of the film.
The solution of the boundary value problem is based on using superposition (Van der
Giessen and Needleman, 1995) of the singular elastic fields of individual dislocations
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together with the solution of a boundary value problem for nonsingular image fields
(Nicola et al., 2005a). The finite element method is used to obtain the image field solutions.
In this approach, the long-range elastic interactions between dislocations are accounted for
directly, and short-range interactions are added through a set of constitutive rules of the
type suggested by Kubin et al. (1992) for dislocation glide, dislocation annihilation and
dislocation nucleation.
The simulations start from a dislocation free state. Dislocation sources are randomly
distributed on the slip planes with each source characterized by a nucleation strength tnuc ,
chosen out of a Gaussian distribution of strengths. A source becomes active when the
Peach–Koehler force on dislocation I, given by
!
X ðJÞ ðIÞ
ðIÞ
ðIÞ
f ¼ ni s^ ij þ
sij bj ,
(6)
JaI
exceeds btnuc for a time span tnuc . Here, nðIÞ
i is the normal to the slip plane of dislocation I
ðJÞ
~
and bðIÞ
its
Burgers
vector.
Also,
s
is
the
singular stress field of dislocation J and s^ ij is the
j
ij
image stress field. When the nucleation criterion is met, a dislocation dipole is nucleated:
two dislocations of opposite sign are introduced on the slip plane at a distance Lnuc which
is such that the attractive stress field that the dislocations exert on each other is
equilibrated by tnuc , so that
Lnuc ¼
E
b
,
2
4pð1 n Þ tnuc
(7)
where E is Young’s modulus and n is Poisson’s ratio.
The glide velocity V ðIÞ
gln along the slip plane of dislocation I is taken to be linearly related
to the Peach–Koehler force through the drag relation
V ðIÞ
gln ¼
1 ðIÞ
f ,
B
(8)
where B is the drag coefficient.
When two opposite signed dislocations come closer to each other than the critical
distance Lann they annihilate and are removed from the calculation. Dislocations that exit
the film through a free surface leave a step at the surface.
The deformation history is calculated incrementally. At time t, the stress and
displacement fields are known at each material point along with the positions of all
dislocations. An increment of the applied displacement U is prescribed and calculation of
the state at time t þ dt involves: (i) determining the Peach–Koehler forces on the
dislocations; (ii) determining the rate of change of the dislocation structure caused by the
motion of dislocations, the generation of new dislocations and their mutual annihilation
and (iii) determining the stress and strain state for the updated dislocation arrangement.
3.2. Choice of parameters
The Burgers vector is taken to be b ¼ 0:25 nm, Young’s modulus is E ¼ 110 GPa and
Poisson’s ratio is n ¼ 0:34, values that are representative of copper. The elastic constants of
any passivation layers are taken to be identical to those of the film. The annihilation
distance, Lann , is fixed at 6b and the drag coefficient at B ¼ 104 Pa s. Dislocation sources
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are evenly distributed among the eight grains in the unit cell and are randomly positioned
on the slip planes. Parallel slip planes are spaced at 200b.
The values of the source density, the average nucleation strength tnuc and its standard
deviation were set by fitting the measured stress–strain response of the electroplated film
with h ¼ 4:2 mm that is passivated on both sides. The flow strength is mainly sensitive to
the source strength while the hardening rate is mainly sensitive to the source density. To
account for the constraint on slip due to the twins, the grain size in calculations for the
electroplated films is taken to be d ¼ 1:5 mm (see Table 1).
Curves of average stress in the film, hs11 i, versus imposed strain, e, are shown in Fig. 6,
with
Z
1
s11 ðw; x2 Þ dx2 ,
(9)
hs11 i ¼
h h
where the integral over the film thickness excludes any passivation layer. Note that from
equilibrium hs11 i is independent of x1 . To facilitate comparison between the computed and
experimental stress–strain curves, the experimental curves from Fig. 3(b) are also plotted.
The calculations in Fig. 6, which shows the response for three values of film thickness, were
carried out using a source density of 15 mm2 and a mean nucleation strength tnuc ¼
350
model
exp
h=1µm
h=1.9µm
300
h=4.2µm
<σ11> [MPa]
250
200
150
100
50
0
0
0.002
0.004
0.006
ε
Fig. 6. Curves of average stress, hs11 i, given by Eq. (9) versus applied strain e for films with both surfaces
passivated. The 0.2% yield strength, sy , is obtained from the intersection of the dash-dotted line (elastic slope)
with the computed stress–strain curve. The grain size is taken as d ¼ 1:5 mm to account for the constraint on slip
imposed by the twins and, for comparison purposes, the experimental stress–strain data from Fig. 3(b) are also
plotted. The values of the dislocation nucleation parameters (specified in the text) were chosen to match the
experimental stress–strain response of the h ¼ 4:2 mm film.
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100 MPa with a standard deviation of 20 MPa. With tnuc ¼ 100 MPa, from Eq. (7)
Lnuc ¼ 0:0247 mm. The films are passivated on both sides by elastic coatings of thickness
p ¼ 20 nm and subjected to a displacement rate of U_ ¼ 3 104 mm=s (this very high strain
rate, e_ ¼ 2500 s1 , is used to limit the computation time).
The 0.2% offset yield strength, denoted by sy , is obtained from the intercept of the dashdotted line in Fig. 6, which has slope E=ð1 n2 Þ, with the computed stress–strain curves.
The values of sy are 175, 219 and 245 MPa for the 4.2, 1.9 and 1 mm films, respectively. The
corresponding experimental values are 179, 204 and 237 MPa. The initial portions of all
three computed and experimental stress–strain curves agree quite well and a good fit is
obtained for the values of sy . However, the hardening becomes linear for the calculations
with h ¼ 1:9 and 1 mm while the experimental curves continue to exhibit parabolic
hardening. The lack of an accurate hardening description as deformation proceeds most
likely arises from idealizations of the analysis. In particular: (i) the dislocation constitutive
rules used in the calculations do not account for effects such as the creation of dislocation
sources and obstacles during the deformation history (the extended dislocation constitutive
rules developed by Benzerga et al. (2004) provide a more accurate hardening description)
and (ii) the grain boundaries and interfaces are idealized as being impenetrable to
dislocations and the possibility of grain boundaries and interfaces acting as sources and/or
sinks for dislocations is not accounted for.
The same dislocation constitutive parameters are employed in all calculations reported
on here; for unpassivated as well as passivated films, and both for the sputtered and the
electroplated films.
3.3. Numerical results
Both monotonic and cyclic loading calculations were carried out for a range of grain
sizes corresponding to those in the experiments. For the sputtered films, calculations were
carried out with both h ¼ d (square grains) and with d fixed at 0:5 mm to give an indication
of the effect of grain aspect ratio on the response. In this section, results are presented to
illustrate the qualitative features that emerge from the calculations; in particular, the
effects of passivation, grain size and grain aspect ratio. We emphasize that all
results presented here and subsequently use the fixed set of material parameters given in
Section 3.2.
Fig. 7 illustrates the effect of passivation on the film stress–strain response. Computed
curves of hs11 i versus e are shown for films with no passivation, with one side passivated
and with both sides passivated. For all three films h ¼ 1 mm. Consistent with the
experimental data, shown for comparison purposes in Fig. 7, the flow strength increases
with an increasing number of passivation layers. However, for the passivated films the
computations give linear hardening for strains greater than 0:003 while the experiments
continue to show parabolic hardening. Also, the computational results somewhat
overestimate the effect of the passivation. For example, the experiments give 0.2% offset
yield strength values of 196, 210 and 233 MPa, while the corresponding computed values
are 182, 214 and 245 MPa (the computation with two surfaces passivated in Fig. 7 has a
different realization of the statistically distributed dislocation sources than the curve with
h ¼ 1 mm shown in Fig. 6 and the corresponding stress–strain curves in the two figures
differ slightly). Although the stress level is lower, the calculation for the unpassivated film
in Fig. 7 exhibits strain hardening similar to that seen in the experiments. With the
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300
<σ11> [MPa]
250
200
150
100
model
exp
two surfaces passivated
50
one surface passivated
unpassivated
0
0
0.002
0.004
0.006
ε
Fig. 7. Curves of average stress, hs11 i, versus applied strain e showing the effect of the presence of passivation
layer(s) on the film response. The films have h ¼ 1 mm and d ¼ 1:5 mm. For comparison purposes, the
experimental stress–strain data from Fig. 3(a) are also plotted.
dislocation constitutive rules used, hardening arises from dislocations remaining inside the
film due to being pinned either at the interface between the film and any passivation layers
or at grain boundaries. For the unpassivated film in Fig. 7, where h ¼ 1 mm and
h=d ¼ 0:67, many of the dislocations exit through the film surfaces, so that a high density
of dislocations has not developed over the range of strain considered. The calculations
underestimate the 0.2% yield strength of the unpassivated film and overestimate the 0.2%
yield strength of the film with two surfaces passivated. Thus, the increase in hardening due
to passivation is overestimated by the modeling. This suggests that the discrepancy
between the computed and observed stress–strain behavior mainly arises from the
idealization of grain boundaries and interfaces as being completely impenetrable to
dislocations.
The effect of passivation on the dislocation structure that develops and on the stress
distribution is shown in Fig. 8. Superposed on the contours of s11 is the distribution of
dislocations and sources. For the unpassivated film, Fig. 8(c), relatively few dislocations
pile up at the grain boundaries. With passivation, dislocations pile up at the filmpassivation layer interface, but also the number of dislocations piled up at grain
boundaries increases. The significantly higher stress level in the passivated films, as well as
the large grain-to-grain variation, can be seen in Fig. 8. Large grain-to-grain variations (in
the number of slip lines per grain) are also seen in the experiments, Fig. 2, although it is
possible that different diffraction conditions for different grains may be responsible for at
least some of the grain-to-grain variation. It is also worth noting that an absence of
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Fig. 8. Distributions of s11 and dislocations at e ¼ 0:005 for films with h ¼ 1 mm that are (a) passivated on both
sides; (b) passivated on one side; or (c) unpassivated. The solid circles mark the locations of the dislocation
sources, the þ and symbols represent dislocations (with the sign convention of Fig. 5). All distances are in mm.
dislocations, as in the grain in Fig. 8(c) lying between x1 ¼ 6 and 7:5 mm, does not imply a
lack of dislocation activity: dislocations nucleated in this grain have exited through the free
surfaces. The dislocation structure seen in Figs. 8(a) and (b) that develops due to the
restriction on dislocation glide by the film-passivation layer (or layers) and by the grain
boundaries is what gives rise to having a back stress that inhibits subsequent nucleation at
the dislocation sources, which in turn leads to increased stress levels.
The predicted stress–strain response is sensitive to the grain aspect ratio, as illustrated
for unpassivated films in Fig. 9. Curves of average stress, hs11 i, versus applied strain, e, are
shown for two grain aspect ratios with the film thickness fixed at h ¼ 1 mm. In one
calculation, the grain size is at the reference value d ¼ 1:5 mm, so that h=d ¼ 0:67; in the
other calculation d ¼ 0:3 mm so that h=d ¼ 3:33 (note that even with d ¼ 0:3 mm, the grain
size is still more than an order of magnitude greater than Lnuc ). With the larger value of the
grain aspect ratio, h=d, the number of active slip planes terminating at grain boundaries
increases, leading to an increase in the number of dislocations piled up at the grain
boundaries and, as a consequence, to hardening.
The effects of grain aspect ratio and of passivation on hysteresis and on the Bauschinger
effect are shown in Fig. 10. Fig. 10(a) shows results for the reference values h ¼ 1 mm and
d ¼ 1:5 mm so that h=d ¼ 0:67, while in Fig. 10(b), h ¼ 0:6 mm, d ¼ 0:5 mm giving
h=d ¼ 1:2. In Fig. 10(a), the unpassivated film exhibits nearly no hysteresis and no
Bauschinger effect, whereas the film passivated on one side shows a Bauschinger effect and
hysteresis loop that increases with increasing pre-strain. Reverse plastic flow starts when
the film is still in tension. The high back stress from the dislocation pile-ups provides the
driving force for dislocation activity during the early stages of unloading. First, dislocation
pile-ups collectively stretch out, with dislocations gliding back on their slip planes and
subsequently new dislocations (with an opposite sign from those generated during loading)
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500
d=0.3µm
d=1.5µm
<σ11> [MPa]
400
300
200
100
0
0
0.002
0.004
0.006
0.008
0.01
ε
Fig. 9. Average stress, hs11 i, versus applied strain e for two unpassivated films with h ¼ 1 mm; h=d ¼ 0:67 for one
and h=d ¼ 3:33 for the other.
are nucleated. On reloading, there is a significant knee in the stress–strain curve due to the
dislocations present in the film.
The evolution of the hysteresis and the Bauschinger effect in Fig. 10(a) reproduces the
trends seen in Fig. 4: (i) for the film passivated on one side the width of the hysteresis loop
and the Bauschinger effect increase with increasing pre-strain while (ii) no hysteresis or
Bauschinger effect is evident for the unpassivated film. The agreement between theory and
experiment is remarkable; the evolution of the size and shape of hysteresis loops is difficult
to model within a phenomenological plasticity framework. Discrete dislocation plasticity is
apparently able to capture the generation of the internal stress that is responsible for the
hysteresis. In Fig. 10(b), with a higher grain aspect ratio, h=d ¼ 1:2, some hysteresis is seen
even for an unpassivated film. This is due to increased piling-up of dislocations against the
grain boundaries of higher aspect ratio grains. It is worth noting that in the experiments,
the films are pre-strained during deposition so that there are initial stresses due to the
dislocation structure whereas in the calculations the initial state is taken to be stress free.
Nevertheless, the same features are seen in the experimental and computational results.
In Fig. 10(b), for the passivated film with h=d ¼ 1:2, the hardening is higher and a larger
hysteresis loop develops than for the unpassivated film. This suggests that hardening in
thin films with small grain size occurs mainly because the back stress leads to the
suppression of dislocation nucleation. When the film thickness and grain size of the films is
reduced, there are insufficient sources available on favorable slip planes to nucleate
dislocations (this effect has been identified in Nicola et al., 2003, 2005c to be responsible
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1000
one side passivated
unpassivated
one side passivated
unpassivated
400
800
h=0.6µm, d=0.5µm
<σ11> [MPa]
<σ11> [MPa]
h=1µm, d=1.5µm
300
200
100
600
400
200
0
0
0
(a)
2103
0.005
0.01
ε
0
0.015
(b)
0.005
0.01
0.015
ε
Fig. 10. Average stress, hs11 i versus applied strain e for two films with: (a) h ¼ 1 mm and d ¼ 1:5 mm (h=d ¼ 0:67);
(b) h ¼ 0:6 mm and d ¼ 0:5 mm (h=d ¼ 1:2). In each plot, one of the films is passivated on one side.
for the enhanced hardening rate in very thin single crystal films on stiff substrates).
Because of the high back stress, plastic flow occurs more readily when the sign of the
loading is reversed.
4. Comparison of experimental and computational results
The computational results and experimental observations for the effect of film thickness
on the yield strength are shown in Figs. 11 and 12. In these figures the 0.2% offset yield
strength, sy , is plotted versus 1=h. We recall that all computations were carried out using
the same values of the three dislocation constitutive parameters—the density of dislocation
sources, their average nucleation strength and the standard deviation of the nucleation
strengths—which were chosen to fit the experimental stress–strain response for an
electroplated film with h ¼ 4:2 mm and passivated on both sides. The experimental results
for both the passivated (one-sided passivation, Fig. 11) and unpassivated (Fig. 12) films
show a 1=h scaling. A similar scaling is seen for the computational results for the large
grained electroplated films, but there is more scatter in the computational results for the
small grain size sputtered films than in the experimental results and, particularly for the
unpassivated films, the agreement between the experimental and computational results is
not as good as for the films with larger grains. One contribution to this is that all
calculations are carried out for eight grains in the unit cell analyzed. Since the dislocation
source density is the same for all films, thinner films and films with a smaller grain size,
which have a smaller volume of material in the computational cell, have fewer dislocation
sources and are therefore more affected by statistical variations. For some grain sizes,
calculations were carried out for several realizations of the source distribution and,
particularly for small grain sizes where there may be only one dislocation source per grain,
the variation in yield strength is large. This is seen particularly in Fig. 12 for h=d ¼ 1 and
h ¼ 0:3 mm where the value of sy varies between 287 and 560 MPa. Nevertheless, in
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electroplated (d=1.5µm)
sputtered
simulations (d=0.5µm)
simulations (h/d=1)
800
simulations (d=1.5µm)
σy [MPa]
600
400
200
0
0
0.5
1
1.5
2
2.5
3
3.5
1/h [µm-1]
Fig. 11. The 0.2% offset yield strength, sy , plotted versus the reciprocal of the film thickness 1=h for films
passivated on one side. The filled symbols show experimental values while computational results are denoted with
open symbols.
addition to a clear agreement between computation and experiment for the trend of the
variation, considering the idealizations in the modeling, the quantitative agreement in
Figs. 11 and 12 is remarkably good. This agreement suggests that the main contribution to
the size dependence in the experiments comes from the constraint on slip by grain
boundaries (including the twin boundaries in the electroplated films) and by any
passivation layers.
In both the experiments and the calculations four distinct slopes in the sy versus 1=h
curves are seen (Figs. 11 and 12): (i) for the passivated electroplated films; (ii) for the
passivated sputtered films; (iii) for the unpassivated electroplated films and (iv) for the
unpassivated sputtered films. The difference in crystallographic texture between the
electroplated and sputtered films leads to only a small difference (about 5%) in the planestrain tension Taylor factors so that the change in slope in the experiments cannot be
attributed to a change in texture. Since the same material properties are used in the
calculations for the electroplated films and for the sputtered films, the agreement between
the experimental and computed values indicates that the different slopes seen in the
experiments are not due to a difference in material properties between the electroplated
and sputtered films. In the calculations, the difference between the electroplated films
(hX1 mm) and the sputtered films (ho1 mm) is the difference in grain size; the electroplated
films are taken to have the fixed grain size 1:5 mm, whereas the grain size for the sputtered
films is taken to be either 0:5 mm or the film thickness h. Hence, the agreement between the
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2105
electroplated (d=1.5µm)
sputtered
simulations (d=0.5µm)
simulations (h/d=1)
800
simulations (d=1.5µm)
σy [MPa]
600
400
200
0
0
0.5
1
1.5
2
2.5
3
3.5
1/h [µm-1]
Fig. 12. The 0.2% offset yield strength, sy , versus the reciprocal thickness 1=h for unpassivated films. The filled
symbols show experimental values while the open symbols show computational results.
calculations and the experiments indicates that the rather abrupt change in slope at 1 mm
in Figs. 11 and 12 is mainly associated with a grain size difference between the two
sets of films.
The plots of dislocation density at a stage of deformation where hs11 i is approximately
sy versus film thickness in Fig. 13 give insight into the means by which grain size affects the
yield strength. In the computations for the electroplated films (hX1 mm), the dislocation
density at yield is comparatively low and increases slightly with decreasing film thickness.
On the other hand, for the sputtered films (ho1 mm) the dislocation density at yield is
much higher but tends to decrease with decreasing film thickness. There is a lower
dislocation density in the computations modeling the larger grain size electroplated films
because many of the nucleated dislocations are able to exit the film through an
unpassivated surface. For the smaller grain size sputtered films, the grain boundaries
prohibit many of the dislocations from gliding to the free surface, so the dislocation density
is higher. In these films, increasing sy corresponds to decreasing dislocation density due to
a combination of two effects: (i) the hindering of slip by the grain boundaries and
interfaces leads to dislocation pile-ups that induce a back stress which inhibits further
dislocation nucleation (Nicola et al., 2005a) and (ii) as the grain size is decreased, the
number of dislocation sources per grain decreases. Passivation affects the magnitude of the
dislocation density but the variation with film thickness in Fig. 13 exhibits similar trends
for the passivated and unpassivated films.
The computed value of sy for the unpassivated electroplated films (hX1 mm) is nearly
independent of film thickness and is consistent with the resolved value of sy being
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250
sputtered (d=0.5µm)
electroplated (d=1.5µm)
200
200
150
150
ρ [µm-2]
ρ [µm-2]
250
100
50
sputtered (d=0.5µm)
electroplated (d=1.5µm)
100
50
0
0
0
0.5
1
1.5
2
1/h [µm-1]
(a)
2.5
3
3.5
0
(b)
0.5
1
1.5
2
2.5
3
3.5
1/h [µm-1]
Fig. 13. Calculated dislocation density at approximately sy versus the reciprocal of the film thickness, 1=h. For
the calculations modeling the sputtered films (ho1 mm), only films with a fixed grain size, d, of 0:5 mm are shown.
(a) Films passivated on one side. (b) Unpassivated films.
approximately equal to tnuc (a mean nucleation strength of 100 MPa corresponds to
sy ¼ 231 MPa; the values for the electroplated films in Fig. 12 are somewhat below this but
with a standard deviation of 20 MPa, sources weaker than the mean are available). Values
of sy in Figs. 11 and 12 for both the electroplated and sputtered films can be much above
the stress required for dislocation nucleation. The strengthening mechanism seen here
differs from that for ‘‘dislocation starvation’’ which also gives rise to increasing strength
with decreasing dislocation density, but where the increase in flow strength with decreasing
size stems from dislocations more readily exiting from small single crystal specimens and
where the nucleation stress acts as an upper bound to the flow strength, as seen in the
calculations by Deshpande et al. (2005).
As sketched in the inset to Fig. 14, the Bauschinger effect for unpassivated films and
films passivated on one side is quantified in terms of the difference between the actual
residual strain ep and the residual strain, eq , that would have been present if the unloading
were purely elastic. The Bauschinger effect measure is then
eq ep
,
ey
where ey ¼ sy ð1 n2 Þ=E.
Due to experimental difficulties with unloading films at small pre-strains, in the
experiments the unloading from some of the earlier cycles is not complete and the value of
ðeq ep Þ is taken at the lowest load in the cycle, which underestimates the value of the
Bauschinger strain at lower values of e0 . In Fig. 14, this measure of the Bauschinger effect
is plotted versus pre-strain e0 . In both the experiments and computations, the unpassivated
films show nearly no Bauschinger effect while for the passivated films there is a significant
Bauschinger effect that increases with increasing pre-strain, e0 . For passivated films, the
predicted Bauschinger effect tends to be on the high side of what is seen in the experiments.
This is consistent with the calculations somewhat overestimating the back stress because of
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0.6
h=0.34; h=0.61; h=0.89 µm (unpassivated)
h=0.60 µm (unpassivated)
h=0.34 µm
h=0.44 µm
h=0.61 µm
h=0.67 µm
h=0.89 µm
h=0.30 µm
h=0.60 µm
h=0.80 µm
ε0
0.2
σ
(εq-εp)/εy
0.4
2107
0
0
0.4
0.8
1.2
ε0/εy
1.6
2
2.4
εp
ε
εq
Fig. 14. Bauschinger effect versus applied pre-strain e0 in unpassivated films and films passivated on one side. The
Bauschinger effect is quantified in terms of ðeq ep Þ=ey , where ep is the residual strain, eq is the residual strain
corresponding to a hypothetical purely elastic unloading, ey is the plane strain yield strain and e0 is the pre-strain,
as sketched in the inset. The filled symbols show experimental values while computational results are denoted with
open symbols. The grain size of the simulated films is d ¼ 0:5 mm, the grain size of the films in the experiments is
listed in Table 1.
the idealization that grain boundaries and interfaces are completely impenetrable to
dislocations.
5. Concluding remarks
A quantitative comparison has been carried out between experimental measurements
and discrete dislocation plasticity predictions for the thickness dependence of the tensile
stress–strain response of freestanding passivated (on one or both sides) and unpassivated
Cu films. Films with thicknesses ranging from 0:3 to 4:2 mm and having a columnar grain
structure were considered. The values of the dislocation source density, average source
strength and standard deviation were chosen to fit the experimentally measured
stress–strain response of a film passivated on both sides and with a thickness of 4:2 mm.
With no further adjustable parameters, the discrete dislocation plasticity calculations: (i)
reproduce the trends seen in the experiments and (ii) are in remarkably good quantitative
agreement with the experimental measurements for the 0.2% offset yield strength, and for
the evolution of hysteresis and of the Bauschinger effect. For example,
The yield strength of unpassivated films with sufficiently large grains is nearly
independent of film thickness.
For sufficiently thin unpassivated films and for films passivated on one side, the yield
strength increases with decreasing film thickness as h1 . The slope of the variation with
h1 is much greater for the passivated films than for the unpassivated films.
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In the passivated films, there is hysteresis and a Bauschinger effect, and the size of the
hysteresis loop increases with increasing pre-strain. The evolution of such hysteresis
loops is difficult to predict using a phenomenological plasticity framework. In the
unpassivated films, the hysteresis is small or absent and there is little or no Bauschinger
effect.
In the calculations for passivated films, linear hardening eventually occurs whereas
parabolic hardening is seen to continue in the experiments. This difference is most likely
due to the interfaces and grain boundaries taken to be completely impenetrable to
dislocations. It is also possible that the use of simple dislocation constitutive rules that do
not account for effects such as cross-slip and the dynamic creation of dislocation sources
and obstacles plays a role.
The calculations indicate that the increase in yield strength with decreasing film
thickness mainly arises from the restriction on slip due to any passivation layers and grain
boundaries leading to a dislocation structure with a back stress that inhibits dislocation
nucleation.
Acknowledgments
LN and AN are grateful for support from the Materials Research Science and
Engineering Center at Brown University (NSF Grant DMR-00520651). YX and JV would
like to acknowledge financial support from the National Science Foundation (NSF Grant
DMR-0133559) and from the Materials Research Science and Engineering Center of the
National Science Foundation under NSF Grant DMR-0213805. The work by LN and
EVdG is partly funded by the Netherlands Institute for Metals Research.
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