Full text of DOI:10.1557/mrs2002.17

www.mrs.org/publications/bulletin
film, and ꢁp is the equi-biaxial plastic strain,
then rates of change are related by
˙
ꢀ˙/M ꢂ ꢁ˙p ꢂ (ꢃfilm ꢄ ꢃsub)Tꢀ 0,
Dislocation Plasticity
ꢀ(0) ꢀ ꢀ0,
(1)
where ꢃ is a coefficient of thermal expan-
sion. M ꢀ E/(1 ꢄ ꢅ) is the biaxial elastic
film modulus, where E is Young’s modu-
lus, and ꢅ is the Poisson ratio. The tempera-
ture history T(t) is presumed to be imposed,
and ꢀ0 is the stress prior to temperature
change.
in Thin Metal Films
O. Kraft, L.B. Freund, R. Phillips, and E. Arzt
Figure 1 illustrates stress evolution for
two cases of Cu films deposited onto Si
substrates with silicon nitride under layers
as barriers to the diffusion of Cu into the
substrate; the film thickness is 0.5 ꢆm in
both cases. In one case, the Cu film has a
free surface, whereas in the other case it has
a thin aluminum oxide layer on its surface;
this layer was formed by self-passivation
of a Cu-1at.%Al film.19 After sputter-
deposition of the films on their substrates
and annealing at 600ꢇC, the films are under
tensile stress in excess of 400 MPa at room
temperature; this defines the initial stress
ꢀ0 in Equation 1. As the temperature is in-
creased from room temperature, the film
material tends to expand. It is constrained
from doing so by the substrate, and in
order to enforce this constraint, the stress
in the film decreases linearly with increas-
ing temperature. The slope of the curve is
given by (ꢃfilm ꢄ ꢃsub)M. At roughly 150ꢇC,
the film stress is zero, and upon further
heating, the film stress becomes compres-
sive. In this range, the behaviors of the free
Abstract
This article describes the current level of understanding of dislocation plasticity in thin
films and small structures in which the film or structure dimension plays an important
role. Experimental observations of the deformation behavior of thin films, including
mechanical testing as well as electron microscopy studies, will be discussed in light of
theoretical models and dislocation simulations. In particular, the potential of applying
strain-gradient plasticity theory to thin-film deformation is discussed. Although the
results of all studies presented follow a “smaller is stronger” trend, a clear functional
dependence has not yet been established.
Keywords: mechanical properties, metals, thin films.
Introduction
The behavior of thin metal films has
deformation of a given material depends
on many factors, including temperature,
material microstructure, and material vol-
ume. It is the aim of this article to describe
the current level of understanding of the
mechanisms for plastic deformation in
thin films and small structures, for which
the connection between mechanical prop-
erties and material volume is a special
characteristic. These issues are particu-
larly important for assessing the reliability
of metal structures integrated into small-
scale systems during subsequent process-
ing and long-term service.
been the subject of intense study over the
past decade or more, driven largely by the
importance of small-scale metal features
in the fabrication and performance of
microelectronic and optoelectronic devices,
multilayer configurations for magnetic re-
cording, and microelectromechanical sys-
tems (MEMS). It is primarily the electrical
conductivity and chemical bonding char-
acteristics of metals that are exploited in
these applications. The low resistivity of
aluminum and copper, along with tech-
niques that have been developed for the
deposition of patterned structures of these
metals (lithographic processing technol-
ogy), have put these particular materials
in the research spotlight.
Measuring Plastic Deformation
Because of the difficulties inherent in im-
posing stress on a thin film in a controlled
way and in measuring film deformation,
specialized techniques have been devel-
oped to study the mechanical behavior of
films.1–3 The configuration that has received
the most attention is a metal film of nomi-
nally uniform thickness in the micrometer
range deposited on and bonded to the flat
surface of a substrate; usually, the substrate
is relatively thick, compared with the film,
and it has a relatively small coefficient of
thermal expansion. In this case, stress can
be imposed on the film by changing the
temperature of the system.4–18
Although metals in the applications men-
tioned do not serve a load-bearing struc-
tural role, they are invariably subjected to
high levels of mechanical stress as a result
of the constraint on deformation imposed
by other materials to which they are joined.
Stress arises most commonly due to tem-
perature change and is a consequence of
the relatively high coefficients of thermal
expansion of metals as compared with those
of the semiconductors, glasses, and ce-
ramics to which they are typically bonded.
In general, metals under stress exhibit
a propensity for inelastic deformation. At
relatively low homologous temperatures,
the physical mechanism giving rise to this
inelastic deformation is predominantly
dislocation nucleation and glide. Plastic
Figure 1. Stress-evolution as a function
of temperature for Cu films with a
thickness of 0.5 ꢆm. The film stress
was measured by the wafer-curvature
technique (data from Reference 19).
Solid circles (ꢀ) refer to a film with a
cap layer, open circles (ꢁ) refer to a film
with a free surface.
The constraint of the substrate implies
that the generated thermal strain is essen-
tially offset by the generation of some com-
bination of elastic and plastic strain in the
film. If ꢀ is the equi-biaxial stress in the
30
MRS BULLETIN/JANUARY 2002

Dislocation Plasticity in Thin Metal Films
and passivated films are nearly identical.
Eventually, the dependence of stress on
temperature deviates from linear behavior,
indicating the onset of plastic deforma-
tion. On cooling from the maximum tem-
perature change of 500ꢇC, the film tends to
contract more than the substrate, and the
initial slope is again the elastic slope. Con-
sequently, the film stress changes from
compression to tension and increases to
more than 450 MPa when the temperature
has been reduced to room temperature.
However, the slope of the curves in the
tensile-stress range is smaller than the
thermoelastic slope, indicating that the film
is being plastically deformed in the re-
verse direction in this range.
The stress in the passivated film is ob-
served to be higher at comparable tem-
peratures than in the unpassivated film
throughout the regions of plastic straining.
At temperatures higher than 300–400ꢇC,
this difference has been attributed to con-
strained diffusional creep via grain bound-
aries,19 which is described in detail in the
article by Josell et al. in this issue. Often,
the stress–temperature evolution is regarded
as a measure of film strength as a function
of temperature; see, for example, Refer-
ence 7. However, this picture is incomplete,
as the film stress relaxes quite substan-
tially when temperature is held at a constant
value; this tendency is more pronounced
at the higher end of the temperature range
being considered here.8,18 Therefore, the
rate of temperature change has an influ-
ence on the stress–temperature depend-
ence observed, and the time-dependence
of the plastic deformation needs also to be
taken into account for a complete descrip-
tion. Furthermore, it has been discussed
that the stress–temperature evolution is
also influenced by strain-hardening or a
Bauschinger effect14,20,21 as the film is plas-
tically strained, despite the fact that the
imposed plastic strains of about 0.5% are
rather small.
which plastic deformation is dominated
opposite direction. It appears that on re-
verse loading, the yield strength is some-
what smaller compared with the initial
loading, indicative of a Bauschinger effect.
by thermally activated glide of dislocations
past lattice obstacles; the trend is reversed
when the temperature is high enough for
grain-boundary diffusion to contribute
significantly to inelastic strain.28
Observation of Dislocation Motion
The behavior of an isolated dislocation
on its glide plane in a stressed single-crystal
thin film bonded to a substrate is illus-
trated schematically in Figure 3a for the
case of a film with a free surface through
which dislocations can leave the material.
The driving force on the dislocation arises
through the component of resolved shear
stress on the glide plane acting in the di-
rection of the Burgers vector of the dis-
location. As the threading segment of the
dislocation glides along through the film
under the action of this driving force, it
leaves behind an ever-increasing length
of misfit dislocation on the film–substrate
interface. The amount of plastic strain ac-
cumulated in this process is proportional
to the total length of misfit dislocation.
The same plastic strain is detected macro-
scopically for the case of many short seg-
ments as is detected for the case of a few
long segments, provided only that the
There are some fundamental differences
between experiments on freestanding films
and on films bonded to thick substrates.
An obvious difference is that freestanding
films can be stressed only in tension but at
fixed temperature, whereas films bonded
to substrates with relatively low coefficients
of thermal expansion can be stressed in
both tension and compression but with
varying temperature. A second, more
subtle difference is that plastic deformation
in the film is constrained kinematically by
the substrate in the case of bonded films,
but is not for freestanding films. Some
possible implications of this difference
will be addressed.
It has recently been demonstrated that
the restriction to tensile stress in cases of
direct mechanical loading can be overcome
by depositing the thin metal film on a sub-
strate that is elastically soft in compari-
son.29 The stress–strain response of a Cu
film that is 0.7 ꢆm thick deposited on a
polyimide substrate is shown in Figure 2.
The stress was measured by in situ x-ray
diffraction in this experiment. On loading,
the film deforms first elastically until
yielding at about 250 MPa is observed.
Then, on straining to 0.5%, the flow stress
increases to 400 MPa, indicating the pres-
ence of strain-hardening effects. On un-
loading of the specimen, the contracting
elastic substrate compresses the film, which
then undergoes plastic deformation in the
Figure 3. (a) Schematic representation
of the motion of a single dislocation in a
single-crystal film according to the model
of Freund and Nix for a film with a free
surface. It indicates that a dislocation
segment with a length dx needs to be
created when the dislocation moves by
dx. (b) Cross-sectional transmission
electron micrograph showing a
In the configuration just discussed,
stress is induced in the film as a conse-
quence of the constraint of the substrate
when a thermoelastic mismatch strain is
introduced in the film by a change in its
temperature. Observations of the plastic
response of thin films under direct tensile
loading have also been reported for Ni,22
Al,23,24 Cu,22,25 and multilayers.26,27 Collec-
tively, the data obtained support the view
that the flow stress at a given level of
plastic strain is higher in thin films than it
is in chemically identical bulk samples
and that flow stress usually increases with
decreasing film thickness (or layer thick-
ness in multilayers) and with decreasing
grain size. The dependence on grain size
is restricted to the temperature range for
dislocation in a 350-nm-thick Al film,
which was grown epitaxially on a
(0001)-oriented Al2O3 substrate. Glide
of a dislocation on the Al plane, which is
inclined ꢀ70ꢇ to the Al(111)ꢁAl2O3(0001)
interface, created a dislocation segment
nearly parallel to the interface. Contrast
near the film–substrate interface indicates
the presence of other, possibly misfit,
dislocations. The diffraction vector is
indicated by g111 (from Reference 31).
Figure 2. Stress-evolution as a function
of applied strain for a Cu film with a
thickness of 0.7 ꢆm on a polyimide
substrate. The film stress was
measured by in situ x-ray diffraction
(data from Reference 29).
MRS BULLETIN/JANUARY 2002
31

Dislocation Plasticity in Thin Metal Films
total lengths are comparable. This mecha-
nism has been observed in polycrystalline
Al films with very large in-plane grain di-
mensions deposited on amorphous sub-
strates7 and in epitaxial Al films on Si30
or Al2O331 substrates. A dislocation for
which the threading segment has traveled
a distance of several micrometers in the
Al/Al2O3 system is shown in Figure 3b.
The misfit segment is seen in the micro-
graph to stand off from the film–substrate
interface by as much as 100 nm. The con-
trast variations close to the interface seen
the threading dislocation. The yield strength
ꢈy is defined by the so-called Matthews–
Blakeslee condition when the driving
force and retarding force just balance each
other. As a result, the yield strength varies
with film thickness h roughly as hꢄ1.
The foregoing model is clearly not ap-
plicable in the case of polycrystalline films
for which the underlying assumption of
the steady advance of an isolated threading
dislocation is precluded by the presence of
grain boundaries. For polycrystalline films,
Chaudhari35 and Thompson36 pointed out
that if dislocations are confined to individ-
ual grains, then dislocation segments must
be created at the grain boundaries as well
as at the film–substrate interface. This
leads to a flow-stress estimate that is pro-
portional to hꢄ1 for a fixed grain size and
to Dꢄ1 for a fixed film thickness. It is note-
worthy that this estimate, too, is based on
the behavior of a single isolated disloca-
tion. An experimental validation of these
models based on the behavior of isolated
dislocations has been given by Venkatraman
and Bravman7 for coarse-grained Al films
and by Dehm et al.31 for epitaxial thin Al
films and Al2O3 substrates.
Figure 4. Dislocations in Cu grains at
in Figure 3b suggest that the stand-off
may be due to the stress fields of interface
misfit dislocations that arrived at the inter-
face earlier in time.
(a) 600ꢇC and (b) 130ꢇC. No interfacial
dislocations deposited by advancing
threading dislocations are discernible in
the images. At elevated temperatures,
(a) dislocations appear to be longer and
more mobile than at lower temperatures
(b), where dislocation motion became
jerky and dislocation tangles formed.
Plan-view transmission electron
microscopy images are from
This picture of plastic-strain accumula-
tion by the motion of threading dislocations
on their glide planes can be generalized to
account for other dislocations on parallel
or intersecting glide planes, but all within
the context of elastic homogeneity and
planar glide. The uniformity of mechanical
conditions along the glide plane necessary
for this mode of relaxation is clearly not
present in the case of polycrystalline thin
films with a grain size of the order of the
film thickness. In such a case, grain bound-
aries can serve as sources of dislocations
as well as barriers to their motion. A goal
for the field at the present time is to estab-
lish which physical mechanisms of plastic
deformation are controlling the plastic
response of the material for polycrystal-
line films. Kobrinsky and Thompson18 have
described dislocation motion in Ag films
with thicknesses of about 200 nm and
strongly textured microstructures at 150ꢇC.
Their in situ transmission electron micros-
copy (TEM) observations are consistent
with the textbook picture of thermally ac-
tivated dislocation glide, which leads to a
jerky dislocation motion at low tempera-
tures. The typical spacing between glide
barriers and the typical glide distance
from barrier to barrier were both reported
to be between 50 nm and 100 nm. These
distances are somewhat smaller than the
film thickness or grain size, but by less
than an order of magnitude. Similar TEM
observations on polycrystalline Cu films
on Si substrates with an intervening layer
of amorphous SiNx lead to the same gen-
eral picture of plastic relaxation.31,32 The
typical dislocation distribution observed
after heating to 600ꢇC followed by cooling
to 130ꢇC is shown in Figure 4. For tempera-
tures of less than 220ꢇC, short segments of
tangled dislocations and a jerky disloca-
tion motion were observed, confirming
the behavior reported by Kobrinsky and
Thompson.18 Collectively, these observa-
tions lend strong support to the hypothe-
sis of thermally activated dislocation glide
Reference 32.
as the dominant plastic-deformation mecha-
nism in these materials.
It was also pointed out by Dehm et al.31
that no interfacial misfit dislocations were
observed, which is at odds with the gen-
eral notion of relaxation suggested in Fig-
ure 3 in which the film–substrate interface
presents an impenetrable barrier to glide.
If this is so, then dislocation lines should
accumulate at such barriers in the course
of plastic relaxation. In situ cross-sectional
TEM observations also revealed that dis-
locations in the Cu/SiNx/Si system are at-
tracted by the Cu/SiNx interface, but again,
no evidence of the interface misfit disloca-
tions was found.32 In contrast, Weihnacht
and Brückner33 and Shen et al.21 observed
interfacial dislocations in plan-view TEM
and dislocation pile-ups in cross-sectional
TEM for the Cu/SiOx/Si and Cu/SiOx
materials systems, respectively. This be-
havior appears to be connected to the
quality of the interface bond; a model of a
weak interface will be discussed.
Plastic Rate Equations
For fine-grained polycrystalline films,
on the other hand, these models tend to
underestimate the yield strength, as was
demonstrated by References 9, 14, 37, and
38, for example. This is not surprising, as
many interacting dislocations are neces-
sary to give rise to detectable levels of
plastic deformation in fine-grained films.
Under such conditions, the physical mecha-
nism of plastic deformation is expected to
be the thermally activated motion of dis-
locations past lattice obstacles, as it is for
bulk samples of these materials, but with
values of flow stress reflecting the special
conditions that prevail in small-scale struc-
tures. This expectation was reflected in
the early work of Flinn et al.4 and Volkert
et al.8 on Al and Cu films, and it has been
confirmed through direct observations by
Kobrinsky and Thompson,18 as noted earlier.
The theory of thermally activated glide
of dislocations in fcc metals has its origins
in statistical mechanics, but its implemen-
tation in continuum plasticity theory is
largely empirical. A constitutive equation
for dislocation motion on a particular glide
system is customarily expressed in terms
of an activation energy ꢉF and a reference
resolved shear stress ꢈˆ that by itself would
induce the dislocation to bypass the ob-
stacle if applied at zero absolute tempera-
ture T. The activation energy is a repulsive
energy of interaction between a represen-
tative dislocation and an obstacle to glide
in its path. The macroscopic plastic-strain
Dislocations in Modeling
Discrete Dislocation Models
Freund34 and Nix1 have discussed the
mechanics of glide of a single isolated
threading dislocation in a single-crystal
thin film, as depicted in Figure 3. As the
threading segment of the dislocation
glides through the film, it draws energy
from the background elastic field arising
from the applied or mismatch stress. The
advancing dislocation leaves behind one
interface misfit dislocation segment of in-
creasing length, and the work expended
in doing so imposes a retarding force on
32
MRS BULLETIN/JANUARY 2002

Dislocation Plasticity in Thin Metal Films
rate due to stress-assisted glide on that glide
system is then expressed as
The particular form of back-stress-
which are comparable to the experimental
results, were obtained by adjusting ꢈˆ to
about 250 MPa. Using Orowan’s classical
result, that the critical shear stress depends
on the pinning-point distance L according
to ꢈˆ ꢀ Gb/L, where G is the shear modu-
lus and b is the displacement distance, the
corresponding pinning-point distance is
50 nm. This value is indeed much smaller
than the film thickness and of the same
order of magnitude as those observed by
in situ TEM on Ag films by Kobrinsky and
Thompson.18 However, the approach pre-
sented here does not take into account that
the pinning-point distance changes dur-
ing heating and cooling (see Figure 4) as a
result of strain-hardening and possibly re-
covery processes.
dependence on plastic strain introduced
by Shen et al. was linear, essentially
ꢈb ꢀ ꢈˆꢊp/ꢊ*, where a value of the parame-
ter ꢊ* was deduced from the data.21 The
stress and strain parameters in Equation 3
can be interpreted as equivalent strain and
effective stress parameters for isotropic
response, or as resultant shear stress
and shear strain on a particular slip sys-
tem. In the latter case, ꢈb would include
self-hardening and latent hardening in
different ways.
ꢉF
kT
ꢄꢈꢄ
ꢈˆ
ꢊ˙p ꢀ ꢊ˙0 exp ꢄ
1 ꢄ
,
(2)
ꢂ ꢃ ꢅꢆ
where ꢊ˙0 is a phenomenological constant,
ꢈ is the applied shear stress on the glide
planes, k is the Boltzmann constant, and T
is absolute temperature.
Once Equation 2 has been incorporated
into Equation 1 by relating the shear on
the glide systems to overall film strain and
resolved shear stress to the biaxial film
stress, it becomes an ordinary differential
equation for stress as a function of time.
Up to this point, the description of plastic
deformation of a thin film includes no
influence of deformation history on the in-
stantaneous response to stress and tempera-
ture. This feature is often adopted on the
basis of the behavior of films under cyclic
stressing for which the response becomes
cyclic after a small number of stress cycles.
It has been pointed out by Shen et al.,21
however, that there is persuasive evidence
for deformation-history-dependence of
response within each stress cycle. This
history effect can arise principally in two
ways. If a result of prior plastic deforma-
tion is to increase the number of obstacles
to subsequent glide of dislocations in any
shearing direction, the effect is viewed as
isotropic strain-hardening, and it is in-
cluded by considering ꢈˆ in Equation 2 to
depend on plastic-strain history in some
way. On the other hand, if a result of prior
plastic deformation is to build up num-
bers of blocked dislocations in a certain di-
rection of shearing, which can be released
if the shearing direction is reversed, the ef-
fect is viewed as kinematic hardening, that
is, hardening in one direction of stressing
accompanied by softening in the opposite
direction. It is included in constitutive mod-
eling by referring stress to a nonzero “back
stress” in some way. For a plastic rate
equation of the form of Equation 2, the in-
fluence of kinematic hardening or back
stress can be incorporated by referring the
applied stress ꢈ to a back stress ꢈb that also
depends on accumulated plastic strain.
Shen et al.21 analyzed their data on
temperature cycling of Cu films on silica
substrates by appealing to the classical
plasticity theory of kinematic hardening
with a well-defined elastic range. Their
general idea is also readily incorporated
into the plastic rate equation framework,
which preserves the connection to disloca-
tion dynamics, by rewriting Equation 2 as
To illustrate the influence of the back-
stress contribution, consider Equations 2
or 3 along with a simplified form of
˙
Equation 1 as ꢈ˙/G ꢂ ꢊ˙p ꢂ ꢃT ꢀ 0 with
ꢈ(500ꢇC) ꢀ 0. Note that these differential
equations now form a nonlinear coupled
system of equations for ꢈ(t) and ꢊ(t). Fig-
ure 5 shows that the behavior computed
with this approach is, overall, in reason-
able agreement with experimental data
such as are shown in Figure 1. The influ-
ence of back stress is also illustrated in Fig-
ure 5, where the behavior leading to the
two stress–temperature curves is identical
except for the effect of the back stress
(solid curve). The effect of the back stress
leads to the commonly observed asymme-
try in the magnitude of the stresses on
heating and cooling. The high stresses,
Strain-Gradient Plasticity
The discussion of continuum plasticity
models for the deformation of thin metal
films has been restricted up to this point to
spatially homogeneous states of stress and
strain. This is a natural consequence of the
translationally invariant geometry of thin
films and the absence of a length scale in
the formulation. Motivated primarily by
an accumulation of compelling observa-
tional evidence that such formulations are
deficient, a search for a modified frame-
work for describing plastic deformation in
small volumes is being pursued actively.
Experiments done with sample configura-
tions as diverse as indentation (indenta-
tion sizes of 1–100 ꢆm) and torsion of
round wires (diameters of 10–200 ꢆm) re-
veal a strong dependence of inferred flow
stress on indentation size or wire diameter,
mainly following the “smaller is stronger”
trend.
The phenomenological strain-gradient
plasticity theories are generalizations of
classical continuum theories. The feature
added is that the nonlinear relationship
between stress and plastic strain at a point
also involves the spatial gradient of plastic
strain at that point.39 The appearance of
the strain gradient in the constitutive rela-
tion implies the existence of a character-
istic length in the constitutive equations,
thereby introducing a type of nonlocality.
Furthermore, the appearance of the strain
gradient leads naturally to couple stresses
as force variables work-conjugate to the
strain gradients. The predictions of this
theory have been compared with avail-
able experimental data, and the character-
istic length implied seems to fall in the
range of 0.1–5 ꢆm, well within the domain
of thin-film plasticity.
Figure 5. Stress versus temperature
change for a film on a substrate,
illustrating the difference between
response according to the standard
plastic rate equation (Equation 2,
dashed curve) and a rate equation
modified by the inclusion of a back
stress depending on plastic strain
(Equation 3, solid curve). Start with
ꢈ ꢀ 0 was at 500ꢇC; parameters used
were for bulk Cu:58 ꢊ˙0 ꢀ 1 ꢋ 106 sꢄ1
,
Consider the deformation of a single
crystal, large compared with atomic di-
mensions, due to dislocation motion on
a single slip system. What is meant by
ꢉF ꢀ 3.5 ꢋ 10ꢄ19 J. The Schmid factor
s ꢀ 0.27 is for (111)-oriented grains;
ꢈˆ was chosen to be 265 MPa and
ꢊ* to be 0.005.
ꢉF
kT
ꢄꢈ ꢄ ꢈbꢄ
ꢊ˙p ꢀ ꢊ˙0 exp ꢄ
1 ꢄ
.
(3)
ꢂ ꢃ ꢅꢆ
ꢈˆ
MRS BULLETIN/JANUARY 2002
33

Dislocation Plasticity in Thin Metal Films
“plastic strain” of this material sample?
Plastic deformation from an arbitrary ini-
tial configuration is a measure of the net
number of dislocation lines that have passed
completely through the sample, thereby
altering the configuration. This number,
along with the length of the Burgers vec-
tor, the geometry of the slip system, and
the size of the sample, make it possible to
identify a homogeneous, simple shear
strain as a starting point for a continuum
description of the process. In arriving at
this description, there is no reference to
dislocations remaining in the sample, but
only to those passing completely through
it. If the sample includes a random distri-
bution of dislocations at the outset that
does not change statistically during plastic
straining, then the argument is unaffected
and a homogeneous deformation can still
be identified. On the other hand, if the
Burgers circuit with dimensions on the
scale of the sample size leads to a net off-
set that changes in the course of plastic
deformation, then the interpretation as a
homogeneous deformation becomes am-
biguous. Furthermore, the situation becomes
increasingly ambiguous as the sample size
is made smaller. Finally, it is noted that
some cases of plastic deformation require
that a certain density of dislocations must
be present for reasons of geometric com-
patibility; these are the so-called geometri-
cally necessary dislocations introduced by
Cottrell40 and discussed by Ashby.41
The foregoing ideas underlie the devel-
opment of strain-gradient plasticity theo-
ries. The reasoning is as follows: If some
net density of dislocations is geometrically
necessary in a particular plastically de-
forming crystal, then the deformation of
a small sample of that material cannot be
represented as being locally homogeneous.
Instead, the deformation of a small sample
also depends on the density of geomet-
rically necessary dislocations, which is
reflected macroscopically through a plastic-
strain gradient. These ideas, which are
discussed only loosely here, have been
given a precise mathematical structure by
Fleck and Hutchinson.39 The structure of
the theory, which continues to evolve, will
not be discussed further here. Instead,
some dislocation concepts on which it is
based will be reviewed in the context of
thin-film deformation.
free surface and the other side gliding
sion for energy per unit area incorporating
such a penalty contribution is
toward the film–substrate interface, thereby
resulting in elastic-strain relief. The ends
of the loops form threading dislocations,
which tend to move apart. The dislocation
lines traveling toward the interface cannot
penetrate the interface; the substrate is as-
sumed to be very hard, compared with the
film, and the interface boundary to be per-
fectly bonded for the time being. The first
arriving dislocations are therefore blocked
at the interface. Dislocations formed sub-
sequently are repelled in their progress
toward the interface by those arriving ear-
lier, and dislocation pile-ups are formed,
producing a back stress. Dislocation spac-
ing in the pile-ups is smaller near the
interface than it is further away. This de-
formation can be interpreted in terms of
an effective extensional plastic strain ꢁp(y)
in the direction parallel to the interface.
This plastic strain is zero at the interface
and increases with distance from the inter-
face, with its gradient diminishing with
distance from the surface. Loosely speak-
ing, the gradient of plastic strain in the
y direction is proportional to the local den-
sity of dislocations ꢌ(y). Finally, for any fi-
nite level of stress, no dislocations can
exist at points close to the free surface so
that the density is zero there, which
implies that the gradient of plastic strain
vanishes at the surface, or ꢁpꢍ(h) ꢀ 0. An
obvious implication of this simple line of
reasoning is that the combination of an
interface that is impenetrable to disloca-
tions and the development of a dislocation
pile-up due to the induced back stress
leads to a plastic-strain distribution that
necessarily varies through the thickness of
the film.
h
1
2
W ꢀ
Mꢇꢊꢁm ꢄ ꢁpꢇyꢈꢋ2
ꢉ
0
ꢂ
ꢎ
2ꢁpꢍꢇyꢈ2ꢈ dy,
(4)
where ꢎ is a material parameter with the
dimensions of length that represents the
sensitivity of the total energy to plastic-
strain gradients. The measure W is now
minimized if
y
ꢎ
ꢁpꢇyꢈ ꢀ ꢁm 1 ꢄ cosh
ꢃ ꢅ
ꢃ
y
ꢎ
h
ꢎ
ꢂ sinh
tanh
,
ꢃ ꢅ ꢃ ꢅ
ꢅ
(5)
which is the solution of a simple, ordinary
differential equation obtained as a necessary
condition for the minimizing strain distri-
bution. This result satisfies the boundary
conditions imposed on the basis of dis-
location arguments, and it also reduces to
the expected uniform strain result when
ꢎ/h l 0. The foregoing discussion is in-
tended to illustrate that plastic deformation
can be spatially nonuniform in a material
system for which it is commonly assumed
to be uniform from the outset, that is, in a
material system for which strain gradients
may be an essential feature. This is done
by appealing to only the most rudimentary
aspects of the physics of plastic deformation.
Strain-gradient plasticity models have
only recently been introduced into the
discussion of plastic response of small
material structures. This conceptual frame-
work has the potential for providing quan-
titative predictions of the dependence of
plastic-flow characteristics of small metal
structures that arise from dislocation be-
havior, but without the need to confront
the enormous complexity of the behavior
of individual dislocations. While analysis
of experimental data supporting the idea
that plastic-strain gradients play an im-
portant role in small structures, film plas-
ticity remains to be examined systematically
from this point of view.
How is this nonuniform strain distribu-
tion to be estimated, even in this simple
case of symmetric double slip? From the
perspective of dislocation mechanics, an ad-
ditional strain-relieving dislocation can be
inserted in a strained thin film whenever
the interaction energy (of the dislocation
with the equilibrium stress field existing
prior to its formation) that is recovered
balances the self-energy of formation. This
is essentially an energy-minimization argu-
ment, so the same general approach might
be adopted to determine ꢁp(y). In addition
to the overall elastic energy for a given
plastic strain with density 12 M[ꢁm ꢄ ꢁp(y)]2,
there is also a configurational energy
stored in dislocation pile-ups. The closer
together the dislocations are in a pile-up,
the higher the value of this energy is. In
effect, this pile-up results in an energetic
penalty on the system for having closely
spaced dislocations in its configuration
that is not taken into account by macro-
scopic strain energy alone. Thus, an expres-
Weak Interface Effect
As has been noted, plastic deformation
of a metal film consists largely of driving
dislocations through the film toward the
film–substrate interface. TEM observations
of the interfaces of plastically deformed
films show that interface dislocations are
present in some cases, but not in others. It
was also reported that interface disloca-
tions disappear during TEM imaging.42 In
virtually all cases, the substrate material is
Consider a single-crystal thin film sub-
jected to an in-plane tensile stress. This
stress is to be relaxed by symmetric double
slip, that is, by glide of dislocations on
symmetrically disposed glide planes. Strain-
relieving dislocation loops are presumed
to form at nucleation sites on the glide
planes. These loops then expand, with one
side passing out of the crystal through the
34
MRS BULLETIN/JANUARY 2002

Dislocation Plasticity in Thin Metal Films
not deformed plastically, so the difference
between these observations must be a con-
sequence of variations in interface quality.
An interface that is fully or partially co-
herent will likely preserve the structure of
nearby dislocations. In particular, the lat-
tice distortion near the dislocation line
that results in the contrast seen in TEM
images will be preserved. If the interface is
incoherent, on the other hand, its intrinsic
resistance against slip can be much smaller
than the flow strength of the film. Macro-
scopically, no traction is transmitted across
the interface for a uniform film, and there-
fore the background film stress does not
test the strength of the interface. In the
case of a weak interface, the stress field of
a dislocation near the interface, or perhaps
of a group of similar dislocations, will re-
sult in a traction on the interface with the
potential for inducing slip. The slip, in
turn, softens the response detected by the
dislocations to their environment and
leads to an attractive force on the disloca-
tions. Consequently, the configuration is
unstable, and a dislocation is spontaneously
drawn toward a weak interface. The range
of the effect from the interface is roughly
Gb/2ꢏꢈ0, where ꢈ0 is the shear stress re-
quired to induce slipping on the interface.
For G ꢀ 100 GPa, b ꢀ 0.3 nm, and ꢈ0 ꢀ
100 MPa, this distance is about 50 nm.
The interaction between a dislocation and
a weak interface is inherently nonlinear
because the slipping zone at first expands
as a dislocation is attracted to the inter-
face. As the dislocation moves closer, the
slipping zone necessarily contracts, leaving
behind a slipped but no longer slipping
portion. This nonlinear interaction has been
described exactly within the framework of
elastic dislocation theory by Hurtado and
Freund.43 The main consequence of the
interaction is that the dislocations near
weak interfaces are essentially drawn into
the interface as slipping progresses. As a
result, the large lattice distortions associated
with dislocation cores give way to small
lattice distortions over a relatively large
part of the interface; the strain contrast of
the core region is no longer seen in TEM
observations, giving the impression that
the dislocations have vanished. This is a
possible explanation for the fact that inter-
face misfit dislocations are not seen in some
systems. Complete absorption of a disloca-
tion by an interface requires minor atomic
rearrangement at the interface, but this is
quite possible, especially for an amorphous
substrate material at elevated temperature.
As pointed out in Reference 44, this be-
havior is reminiscent of the dislocation
behavior in oxide-dispersion-strengthened
materials, where the attraction of disloca-
tions to the weak oxide/metal interface
leads to the high-temperature strength of
these materials.
Dislocation Simulation Studies
Analytical models for fundamental thin-
film issues of dislocation interaction on
parallel and intersecting glide planes have
been discussed by Willis et al.45 and Freund46
and more recently in References 33 and 47.
Even for such apparently simple configu-
rations, the three-dimensional analyses are
not immediately transparent. Consequently,
substantial effort is being devoted to the
development of computer-simulation meth-
ods, or so-called dislocation-dynamics
methods, to understand multiple disloca-
tion interactions both in thin films and in
bulk materials. Schwarz48,49 implemented
the Peach–Koehler theory to simulate the
motion and interaction of multiple dis-
locations in general three-dimensional
strained heteroepitaxial film–substrate
configurations. At an even more detailed
level of modeling, interesting recent
dislocation-dynamics simulations have
also been undertaken, with the aim of
examining plastic deformation induced in
a metal film subject to indentation.50
Figure 6. Illustration of the dislocation
loops nucleated beneath an indenter, as
computed using molecular dynamics
(from Reference 52).
dislocations, a critical unresolved challenge
remains the determination of systematic
criteria for the nucleation of dislocations
in thin films, as well as in the bulk.
Pant et al.54 adopted the approach of
Schwarz to study specific dislocation inter-
actions in fcc metal films on amorphous
substrates. In contrast to epitaxial films,
the film–substrate interface in this case is
assumed to be impenetrable to dislocation
motion. Many configurations of dislocations
on parallel and intersecting glide planes
with different combinations of Burgers
vectors have been considered. It was found
that the stress needed to drive a threading
dislocation past an interface misfit disloca-
tion in its path may be 30% larger than the
stress needed to advance the isolated
threading dislocation. For threading dis-
locations bypassing each other on parallel
glide planes, the increase in flow stress is
even stronger.
As noted, the physics of both the nuclea-
tion and interaction of dislocations cannot
be handled naturally in an elastic theory,
and ad hoc rules were adopted for this
purpose. To overcome some of the limita-
tions of phenomenological plasticity mod-
els and of elastic-dislocation simulations,
Phillips et al.51 have studied atomistic
models of the nucleation and interaction
of dislocations that require no assumptions
beyond those embodied in the interatomic
potential. They emphasized the critical
role of boundary conditions in simulating
such events, and raised the important
question of how more phenomenological
simulation models can become better in-
formed through atomistic modeling. From
the standpoint of modeling dislocation-
mediated plasticity in thin films, the key
unknowns concern how to supplement
the already convincing elastic analyses of
the energetics of isolated dislocations with
insights into the nucleation and inter-
action of such dislocations. An indication
of the types of results that have been pro-
duced on the question of dislocation nu-
cleation is illustrated in Figure 6, which
shows dislocation loops nucleated in a
molecular-dynamics calculation as a re-
sult of indentation.52 Also, it is noted that
Miller et al.53 observed abundant disloca-
tion nucleation at grain boundaries in an
atomistic simulation of the advance of a
crack in a grain of a polycrystal toward a
grain boundary. Though such results indi-
cate the basic mechanistic underpinnings
for the deformation-induced nucleation of
Interactions among large numbers of
dislocations in a thin film under two-
dimensional conditions have been simu-
lated by Nicola et al.55 for the case of a film
with a free surface in tension and by Shu
et al.56 for the case of a film with passi-
vated faces in simple shear. In the latter
case, the problem was also modeled using
strain-gradient plasticity. In both cases,
strong gradients in plastic strain were
found near boundary surfaces that are
impenetrable to dislocation motion, simi-
lar to the behavior suggested by Equa-
tion 5. An example of a final equilibrium
dislocation array in a film with a free sur-
face obtained by Nicola et al. is shown in
Figure 7.
A discrete dislocation simulation model
for the study of film plasticity at the level
of a single grain has been developed by
von Blanckenhagen et al.,57 also within the
framework of isotropic linear elasticity. All
boundaries of the columnar grain, includ-
MRS BULLETIN/JANUARY 2002
35

Dislocation Plasticity in Thin Metal Films
Concluding Remarks
All studies presented and discussed
here indicate clearly that the motion of
dislocations is constrained in thin metal
films, due to the reduced material volume,
the presence of the interfaces to a sub-
strate, and/or passivation. As a result, the
mechanical behavior of thin metal films
can be qualitatively summarized by say-
ing that smaller is stronger. Up to now,
however, a clear functional dependence
had not been established, either empiri-
cally or theoretically. This ambiguity is
further accentuated by the fact that the
grain size of polycrystalline films as well
as subtle changes in the chemistry of the
interfaces present play an important role
for dislocation plasticity in thin films. The
general area would benefit from greater
emphasis on understanding how individual
dislocations interact with or are generated
from physically realistic grain boundaries
and incoherent interfaces. This will require
Figure 7. Dislocation distribution as calculated by two-dimensional dislocation dynamics in a
0.5-ꢆm-thick Al film after cooling by 300 K (from Reference 55).
ing the lateral grain boundaries and the
interface with the substrate and the passi-
vated surface, are assumed to be impene-
trable to dislocations. The generation of
dislocations by a Frank–Read source
within the grain and the formation of the
pile-ups on the glide planes were simu-
lated. The optimal size of the source was
found to be between one-fourth and one-
third of the smallest grain dimension. Be-
cause the stress required to activate the
source scales with source size, this implies
a strong connection between plastic-flow
stress and microstructural dimensions.
The effect of source size also decreases as
the number of dislocations produced in-
creases. The result that the dislocation
source size scales with the smaller grain
dimension, either the film thickness or the
in-plane grain size, implies that a critical
microstructural size exists below which
the flow stress depends on the inverse
grain size instead of on the square root of
this dimension.
ever, because the simulation does not
include any processes like cross-slip, dis-
location core-spreading, or other relaxa-
tion mechanisms.
This simulation has been extended re-
cently to take into account a distribution
of potential Frank–Read sources. A strain
is imposed on the grain, and the evolution
of stress, plastic strain, and dislocations
are calculated. As an example of such a
simulation, Figure 8 shows the dislocation
configuration in an fcc grain of width
512 nm after it was plastically strained up
to 0.12%. A parallel dislocation array
(PDA) and a pile-up (PU) of dislocations
at the interface with the substrate can be
readily identified. Furthermore, tangled
dislocations (T) and dislocation sources
(DS) that bow out without producing a
full dislocation loop are also evident.
These dislocation configurations lead to a
strong increase in flow stress as a function
of plastic strain, and the predicted stresses
are even higher than those measured ex-
perimentally for a comparable situation
(see, e.g., Figure 2). This is plausible, how-
Figure 8. Dislocation distribution as calculated by three-dimensional dislocation dynamics
in a 512-nm cubic grain.59 The simulation was conducted as described in Reference 57; the
interfaces and grain boundaries were assumed to be impenetrable for dislocations, and the
parameters used were for Cu. DS is a dislocation source, T is a tangled dislocation, PDA is
a parallel dislocation array, and PU is a pile-up of dislocations.
36
MRS BULLETIN/JANUARY 2002

Dislocation Plasticity in Thin Metal Films
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rely on insight and judgment in problem
definition. For the latter, a vivid exchange
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ꢀ
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