of 12
Heterogeneous dislocation nucleation from surfaces and interfaces
as governing plasticity mechanism in nanoscale metals
Andrew T. Jennings and Julia R. Greer
a)
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125
(Received 10 May 2011; accepted 29 September 2011)
We report the results of constant strain rate experiments on electroplated, single crystalline copper
pillars with diameters between 75 and 525 nm. At slow strain rates, 10

3
s

1
, pillar diameters
with 150 nm and above show a size-dependent strength similar to previous reports. Below 150 nm,
we
fi
nd that the size effect vanishes as the strength transitions to a relatively size-independent
regime. Strain rate sensitivity and activation volume are determined from uniaxial compression tests
at different strain rates and corroborate a deformation mechanism change. These results are
discussed in the framework of recent in situ transmission electron microscopy experiments
observing two distinct deformation mechanisms in pillars and thin
fi
lms on
fl
exible substrates:
partial dislocation nucleation from stress concentrations in smaller structures and single arm source
operation in larger samples. Models attempting to explain these different size-dependent regimes
are discussed in relation to these experiments and existing literature revealing further insights into
the likely small-scale deformation mechanisms.
I. INTRODUCTION
Investigations into strengthening mechanisms in met-
als have demonstrated many different pathways in which
the same elemental material can increase its strength. In
large systems, standard examples include work harden-
ing, whereby the strength increases due to the evolv-
ing dislocation density through the Taylor relation
r
}
l
b
ffiffiffi
q
p
,
1
and decreasing grain size to
;
100 nm,
whereby strengthening occurs via dislocation pile-ups
against grain boundaries, known as the Hall
Petch mecha-
nism.
2,3
In addition, in the last
;
5 years it has been
demonstrated that strengthening in metals can be achieved
by reducing sample dimensions to the micro- and nano-
scale.
4
7
Speci
fi
cally, single crystalline metals containing
initial dislocations have been shown to attain much greater
strengths compared with bulk as a result of one or more of
their external dimensions decreasing to the micron and
below dimensions.
8
At these small length scales, the
strength of cylindrical metallic pillars under uniaxial
compression and tension scales with their critical dimen-
sions as
r
}
D
n
, where
r
is the pillar
fl
ow stress at some
characteristic strain,
D
is the pillar diameter, and
n
is
a value found to lie between

0.5 and

1.0 for face-
centered cubic (FCC) metal.
4
7,9
12
In addition to these
1-dimensional
samples, single crystalline metallic thin
fi
lms on
fl
exible substrates exhibit similar size-dependent
mechanical properties, whereby their
fl
ow strength scales
with the
fi
lm thickness in a power-law fashion, with the
exponent of approximately

0.5.
13,14
At both the sub-micron and micron scales, several
groups have proposed that collective dislocation behav-
ior is responsible for the size effect, with the principal
mechanism arising from the operation of truncated
dislocation sources, also known as single arm sources,
whose strength scales as
ln
L
L
, where
L
is their pinned
segment length.
5
7,9,15
18
The average pinned segment
length has been shown by several authors to decrease with
decreasing pillar diameter or
fi
lm thickness leading to
higher strengths in smaller-sized structures.
15,19
This
mechanism has been demonstrated through analytical
models,
9,15,18
dislocation dynamics simulations in two
and three dimensions,
6,20
23
and in situ transmission
electron microscopy (TEM) testing of Au thin
fi
lms
14
and Al wires.
24
Few works have been published on assessing metallic
strengths with critical sample dimensions on the order
of
;
100 nm and below, and those that exist reveal
a relatively size-independent response of
fl
ow strength in
both pillar
25,26
and thin
fi
lm geometries,
13,14
sometimes
deforming at nearly theoretical strengths.
26
In situ TEM
investigations have postulated a plasticity mechanism
transition from single arm source operation to partial
dislocation nucleation from free surfaces and interfaces near
;
100 nm in size.
14,27
Also, several groups have noticed
a transition from size-dependent strength above
;
100 nm to
relatively size-independent strength below
;
100 nm,
13,25
a transition predicted by Zhu et al.
28
as a result of the emer-
gence of partial dislocation nucleation as the dominant
a)
Address all correspondence to this author.
e-mail: jrgreer@caltech.edu
This paper has been selected as an Invited Feature Paper.
DOI: 10.1557/jmr.2011.338
J. Mater. Res., Vol. 26, No. 22, Nov 28, 2011
Ó
Materials Research Society 2011
2803
plasticity carrier. Thes
e experiments are corroborated
with the results of molecular dynamics (MD) simula-
tions, which also suggest that at these very small length
scales, dislocation nucleation from surfaces and inter-
faces controls deformation.
28,29
Although numerous
computational studies a
ddressing nanowire (NW) de-
formation exist, the focus of this article is to present our
experimental
fi
ndings on the deformation of sub-600-nm
single crystalline Cu nanopillars and then relate these
results in the context of reported experimental inves-
tigations of NW and thin
fi
lm deformation. The state of
the art overview concerning plasticity in small-scale
metallic systems can be found in four recent reviews on
small-scale plasticity.
4
7
Here, we examine heterogeneous dislocation nucle-
ation, i.e., nucleation of dislocations from imperfections,
in plastically deforming single crystalline pillars with
diameters below 600 nm and compare our results with
those reported for similar-thickness thin
fi
lms on com-
pliant substrates. We discuss this deformation behavior in
the framework of previously reported in situ TEM
investigations, which clearly illustrate that in these small
structures, partial dislocations preferentially nucleate
from local stress concentrations. We highlight the variety
of nanomechanical testing sample fabrication routes
reported to date and their resulting initial microstructures,
which have been found to have a signi
fi
cant impact on
deformation behavior.
4
7,25
27,30
37
We compare our
experimental results to the existing models that attempt
to explain heterogeneous dislocation nucleation in pillar
and thin
fi
lm geometries. Further, we measure the
experimental activation volumes as a means to test the
applicability of these models to the deformation of our Cu
nanopillars.
II. EXPERIMENTAL
To investigate size effects in pillar geometries,
several fabrication routes have been utilized: focused
ion beam (FIB) fabrication,
4
7
vapor
liquid
solid
growth,
26
directional solidi
fi
cation,
33
35
nanoimprint-
ing,
36,37
cold welding,
27
and electroplating
25,30
32
.FIB-
fabricated pillars have been studied most extensively,
with average pillar diameters ranging from
;
100 nm up
to several microns. This fabrication technique has lim-
itations in that the smallest pillar sizes
;
100 nm are
extremely cumbersome to produce and often suffer from
FIB damage and substantial vertical taper.
38,39
Some of
the alternative fabrication routes do not suffer from
these limitations. For example, two different sets of
single crystalline copper pillars have been produced
without the use of FIB,
26,30
32
creating nearly taper-free
samples with diameters as small as 75 nm.
In our studies, single crystalline copper pillars
were prepared through a combination of electron beam
lithography and electroplating.
30,31
In this procedure,
cylindrical holes, with the aspect ratio of the desired pillar
geometry, are
fi
rst patterned on a polymethylmethacrylate
(PMMA) substrate via electron beam lithography. This
template is then used as the cathode in the electroplating
of copper into the prepatterned pores.
25,30
32
The resulting
cylindrical
;
[111] oriented copper pillars are single crys-
talline and have a non-zero initial dislocation density
and a non-negligible surface roughness.
31
TEM investiga-
tions of the microstructure indicate that the dislocation
densities in these pillars are on the order of 10
14
m

2
in
100-nm-diameter samples.
31
These dislocation densities
would be considered as large in bulk samples; however,
they are reasonable considering the very small pillar vol-
umes. For example, in a typical
;
100-nm-diameter pillar,
a single 2-nm-long dislocation loop results in a dislocation
density of
;
10
12
m

2
, whereas two 100-nm-long dislo-
cation loops give the dislocation density of 10
14
m

2
.
We perform uniaxial compression tests at a con-
stant nominal strain rate in two different instruments:
(i) Agilent G200 nanoindenter (Agilent, Santa Clara, CA)
with a 7-
l
m
fl
at punch in the Dynamic Contact Module
and (ii) SEMentor, a custom-built in situ nanomechanical
deformation instrument
fi
tted with a
;
10-
l
m diamond
fl
at
punch.
40
Although both machines are inherently load
controlled, in our experiments a custom-written software
method utilizing a feedback loop controls the applied load
to maintain a constant displacement rate, and therefore
a constant nominal strain rate. We conducted the compres-
sion tests at a series of strain rates, ranging between
;
10

4
and
;
10
0
s

1
, which correspond to a displacement rate
range of
;
0.1 to
;
1000 nm/s. Although the lower bound of
attainable displacement rates of 0.1 nm/s seems unphysical,
the load and displacement versus time response is quite
robust. Examples of the load and displacement versus time
response can be found in Fig. 2 of Ref. 25, which shows
a constant average slope at the measured displacement
marked by rapid displacement bursts. Displacement rate of
0.1 nm/s was the smallest displacement rate attempted, and
as a result limits the strain rates accessible in the smallest
diameter samples.
III. RESULTS
Representative stress
strain curves for the compres-
sions of pillars with diameters of 125 and 250 nm can be
seen in Fig. 1. There are two stress
strain curves for each
pillar diameter, corresponding to compression tests at two
different strain rates: 10

3
and 10

1
s

1
. These four curves
have a typical stochastic signature with intermittent strain
bursts and are examples of two global trends found in
Figs. 2 and 3. The
fi
rst is that at a constant strain rate,
smaller pillars reach higher strengths, and the second is that
for a constant diameter, faster strain rates result in higher
strengths. Further examples of these curves can be found
A.T. Jennings et al.: Heterogeneous dislocation nucleation from surfaces and interfaces as governing plasticity mechanism in nanoscale metals
J. Mater. Res., Vol. 26, No. 22, Nov 28, 2011
2804
in Ref. 31. The stress
strain behavior in these pillars is
similar to that reported in other literature on pillar com-
pressions in similar instruments.
5
7
Figure 2 shows a log
log plot of the strength at 10%
strain versus pillar diameter for
fi
ve different samples with
diameters between 75 and 525 nm, deformed at different
constant strain rates spanning four orders of magnitude.
Examining the strength as a function of pillar diameter at
the slowest strain rate accessible for all pillar diameters,
10

3
s

1
,we
fi
nd that the largest pillar diameters
D
$
150 nm
obey a power law with the slope of

0.54 similar to
ubiquitous reports in the literature on the size dependence
of FIB-fabricated pillars.
4
7
Furthermore, at small sizes
D
,
150 nm, the strength versus size appears to
fl
atten out
relative to diameter, power-law slope

0.5, suggesting
that there is a transition to a different deformation
mechanism. The transition diameter is de
fi
ned here as
the smallest pillar diameter whose strength can be accu-
rately de
fi
ned through the power-law scaling seen in larger
pillars. At the strain rate of 10

3
s

1
, this transition
diameter is
;
150 nm. We
fi
nd that this transition diameter
is a strong function of strain rate, with faster strain rates
resulting in smaller transition diameters.
25
For example,
for strain rates between 10

3
and 10

1
s

1
, the transition
diameter shifts from
D
;
150 to
D
;
75 nm or smaller as
seen in Fig. 2 and Ref. 25.
Data analysis: Measurement of activation volumes
To explore the strain rate dependence, the current
authors measured the strain rate sensitivity,
m
, from the
phenomenological dependence of stress on strain rate:
r
¼
r
0
_
e
m
. The results at each pillar diameter can be seen
in Fig. 3(a), a log
log plot of
fl
ow stress at 10% strain versus
applied strain rate, where the lines correspond to the
fi
ts of
m.
There is considerable scatter in the measured strength of
small volume pillars.
4
7
As a result, we assume here that each
type of stress versus diameter signature seen in Fig. 2, either
size-dependent behavior in larger pillars or relatively size-
independent behavior in smaller pillars, corresponds to
a single strain rate sensitivity regime, taking into account
this inherent scatter in these types of measurements. The
three largest pillar diameters consistently have strengths well
described by power-law behavior and as a result, a single
value of
m
is measured across the range of strain rates tested.
This value of
m
increases from
;
0.027 at 500 nm to
;
0.04
at 150 nm. At the two smallest diameters, two
fi
ts for
the strain rate sensitivity are plotted, with the transition
diameter inferred from Fig. 2. At slow strain rates, the
strain rate sensitivity drama
tically increases from 0.04 in
150-nm-diameter pillars to
;
0.11 in both 125- and 75-nm
pillars. At high strain rates, the increase in strain rate
sensitivity is more subdued, corresponding to again
;
0.04
at 150 nm and
;
0.057 at 125 and 75 nm.
This strength versus strain rate data can be used to
estimate experimental activation volumes associated with
nanoscale plasticity in these structures. The activation
volume measured by assuming the shear strain rate is
controlled through a dislocation nucleation process,
consistent with previous reports in the literature,
28,41
43
and can therefore be described by an Arrhenius form:
_
c
¼
_
c
o
exp
Q

sXs
;
T
ðÞ
k
B
T

;
ð
1
Þ
where
_
c
o
is a constant prefactor,
Q
is the energy barrier,
s
is the resolved shear stress,
k
B
is Boltzman
s constant, and
T
is the temperature. The activation volume,
X
(
s
,
T
), is
de
fi
ned as a change in the activation energy with applied
FIG. 2. True stress at 10% strain versus pillar diameter at three
different strain rates: 10

3
,10

2
, and 10

1
s

1
. Bottom left inset is an
atomistic image for surface source nucleation from a free surface in
a square pillar (from Ref. 28). Top right inset is an atomistic image of
two single arm sources in a circular pillar (from C. Weinberger and
W. Cai, personal communication). (Reproduced from Ref. 25 with
permission from Elsevier.)
FIG. 1. Representative stress
strain curves for two different electroplated
Cu pillars with diameters 125 and 250 nm. Two different strain rates: 10

1
and 10

3
s

1
are shown for each pillar diameter.
25
(Reproduced from
Ref. 25 with permission from Elsevier.)
A.T. Jennings et al.: Heterogeneous dislocation nucleation from surfaces and interfaces as governing plasticity mechanism in nanoscale metals
J. Mater. Res., Vol. 26, No. 22, Nov 28, 2011
2805
stress. We assume here that the resolved shear stress
dominates the deformation mechanism such that the
activation volume can be described as
Xs
;
T
ðÞ
5
dQ
d
s


T
.
Rearranging this equation for activation volume results in:
Xs
;
T
ðÞ
¼
k
B
T
d
ln
_
c
d
s
;
ð
2
Þ
which can be measured from the stress dependence on
strain rate at a constant temperature.
The data in Fig. 3(a) represent
;
150 successful
compression tests with each point having error bars con-
taining on average approximately
fi
ve data points. The
activation volumes as a function of diameter can be found
in Fig. 3(b), where they are plotted for two different strain
rates, 10

1
and 10

3
s

1
represented by
X
s and
O
s,
respectively. At faster strain rates, 10

1
s

1
, the activation
volume scales linearly with diameter, as recently reported
by Jennings et al.
25
and as might be expected for single
arm sources.
25
It should be noted that while the observed
linear trend is in reasonable agreement with this theory, the
predicted and observed magnitudes are substantially differ-
ent implying the need for further improvements in theoret-
ical investigations. Slower strain rates demonstrate a drop in
the activation volumes for the two smallest pillar diameters
to below 10b
3
.
25
These two diameters, 75 and 125 nm, also
experience a deviation from the commonly observed power
law, transitioning to a size-independent strengthening re-
gime, suggesting a deformation mechanism transition.
IV. DISCUSSION
A. Experiments on NW deformation
Electroplating copper pillars is not the only fabrica-
tion route developed to study size effects in FIB-less
geometries. For example, Buzzi et al.
36
and Dietiker
et al.
37
both used an embossing method on Ag and Au,
respectively, whereby a patterned Si template was pushed
into a square platelet of the desired material at elevated
temperatures. The metal is then formed into the mold,
producing pillars with diameters ranging from
;
150 nm
up to several microns.
36,37
The resulting pillars had
different crystallographic orientations, with the smallest
pillars
D
,
200 nm in Au and
D
,
500 nm in Ag being
single crystalline. Larger pillars were a mix of single
crystals and polycrystals. The results of these experi-
ments show a clear size effect similar to FIB-fabricated
pillars.
36,37
In fact, Dietiker et al.
37
shows no substantial
difference between Au pillars produced by the FIB and
those produced through the embossing method.
Richter et al.
26
grew pristine [110]-oriented copper
NWs through vapor
liquid
solid method with diameters
between 75 and 400 nm. In contrast to pillars produced
through electroplating, these NWs result in an equilibrium
Wulff shape, with atomically smooth side surfaces and
virtually nonexistent initial dislocations, resulting in
a similar microstructure as the micron-sized wires origi-
nally investigated by Brenner.
44
Similar to Brenner
s now-
classical results on Cu whiskers, these NWs were pulled in
tension and exhibited very large strengths: 2
6 GPa and
failed predominantly via brittle fracture.
26
The size
dependence of these wires has been described through
Weibull statistics, calculating the probability of
fi
nding
a defect on the surface or in the wire volume, which would
lead to brittle failure.
6
Stress
strain behavior demon-
strating the large strengths and subsequent fracture in
Richter et al.
s NW experiments can be found in Fig. 4.
Uniaxial compression experiments were also per-
formed on Mo
alloy pillars, also produced without the
FIB through eutectic solidi
fi
cation.
33
35
As grown, these
FIG. 3. (a) Log
log plot of stress at 10% strain versus strain rate for
fi
ve different pillar diameters:
;
500, 250, 150, 125, and 75 nm. Lines are
fi
ts to
the strain rate sensitivity,
m
. Data replotted from Ref. 25 with permission from Elsevier (b) Log
log plot of activation volume versus diameter at two
different strain rates denoting the change in activation volume for the smallest diameters.
A.T. Jennings et al.: Heterogeneous dislocation nucleation from surfaces and interfaces as governing plasticity mechanism in nanoscale metals
J. Mater. Res., Vol. 26, No. 22, Nov 28, 2011
2806
square pillars contain zero initial dislocation densities and
scanning electron microscopy micrographs appear to have
pristine surfaces. As a result, these pillars have a size-
independent strength under compression as they all attain
nearly theoretical strengths at all sizes.
33
35
What emerges is that in the pillar and NW tests, there
is a clear distinction in the mechanical behavior and
deformation between the initially pristine samples and
those that contain defects: dislocations and nonatomically
smooth surfaces. The truly and nearly pristine samples
attain strengths that are very high and nearly size
independent, whereas those with defects have lower
strengths, with considerable scatter in their measured
values. In the former, the failure is catastrophic at near
theoretical strengths. In contrast, pillars with initial
defects show a mechanism transition: larger samples
deform at strengths according to the widely observed
power law, and samples with smaller diameters deform at
a relatively size-independent strength that is signi
fi
cant-
ly lower than the ideal strength.
Several groups have performed in situ TEM inves-
tigations to understand the origins of the deformation
mechanisms unique to one-dimensional nanopillar and
NW geometries. These investigations revealed two
different mechanisms: the operation of single arm sour-
ces
24
in larger pillars and wires and partial dislocation
nucleation in smaller pillars.
27
In the former, 460-nm-wide
single crystalline Al wires with rectangular cross-sections
were cut by the FIB from nonpristine single crystalline
Al
fi
lms on polyimide substrates. An unobstructed view
of the Al wire was obtained by selectively fracturing
the polyimide around the viewi
ng area. Tensile straining
of these wires unambiguously demonstrated the sequen-
tial nucleation of concentric
dislocation half-loops em-
anating from an operating single arm source. Tests at
higher strain rates revealed a build up of dislocation
density as the dislocation generation rate exceeded their
annihilation rate.
24
Investigations into thinner wires
were limited due to the stability of the samples after
processing.
On the opposite end of the size spectrum, in very small
Au NWs with diameters of 10 nm and below, high
resolution transmission electron microscopy (HRTEM)
investigations revealed neck formation and partial dislo-
cation emission under tensile loading.
27
These NWs were
produced through the reduction of AuCl (oleylamine)
complex on the TEM holder. The Au NWs were loaded
into the TEM and attached to an Au substrate through
compression cold welding,
27
a process whereby two
crystals of the same material bond together without the
introduction of external heat such that no interface exists at
the
weld
site.
45
These Au NWs have a very complex
surface composed of low energy {111} facets.
27
During
tensile tests on [001] oriented wires, partial dislocations
were noticed to nucleate at local stress concentrators like
slip offsets along the wire surface. A series of HRTEM
images showing the time progression of these partial
dislocation emissions can be found in Fig. 5. Figure 5(a)
shows that the initial atomic structure contains surface
steps and a twin boundary (TB) intersecting the surface.
Figure 5(b) shows the stacking fault left behind after the
leading partial dislocation has nucleated, and Fig. 5(c)
displays the disappearance of the stacking fault after the
trailing partial dislocation is nucleated. NWs with
[110] loading orientations were also tested and favored
twinning as their deformation mechanism. The authors
ascribe this preference for twinning to the large
Schmid factor difference in the leading versus trailing
partials in [110] tension. The result may be the preference
for the repeated nucleation of leading partials on
adjacent slip planes as opposed to the nucleation of
trailing partials.
Analyzing the HRTEM images, the authors were able
to capture local stress and strain information at the
speci
fi
c site the partial dislocation nucleates. Signi
fi
-
cantly, the authors
fi
nd that the local stress at the
nucleation site is generally higher than the rest of the
pillar demonstrating the important role that stress
concentrations have in the nucleation of partial disloca-
tions. For example, the leading partial dislocation shown
in Fig. 5(b) nucleated at the intersection of the surface and
the TB. Furthermore, in contrast to bulk materials, the
dislocations nucleated in these small wires immediately
pass through the wire diameter and escape at the opposite
surface. This type of dislocation starvation has also been
shown in other in situ TEM investigations
38,46
and also in
MD simulations of circular gold NWs.
47
FIG. 4. Stress
strain curves from tensile tests of pristine single
crystalline copper nanowires (NWs) performed by Richter et al.
26
Stress
strain curves show very high strengths, on the order of 2
6GPa.
(Reproduced from Ref. 26 with permission from American Chemical
Society.)
A.T. Jennings et al.: Heterogeneous dislocation nucleation from surfaces and interfaces as governing plasticity mechanism in nanoscale metals
J. Mater. Res., Vol. 26, No. 22, Nov 28, 2011
2807
B. Experiments on thin film deformation on
flexible substrates
As described above, the unique deformation mecha-
nisms found in one-dimensional pillar and NW geome-
tries arise due to the reduced sample sizes. It is useful to
discuss the deformation mechanisms in two-dimensional
structures, i.e., thin
fi
lms, whose thicknesses are reduced
to nanoscale dimensions. Most mechanical tests on single
crystalline, as opposed to polycrystalline thin
fi
lms, have
been conducted for samples on stiff substrates, which
have been shown to greatly in
fl
uence the observed
mechanical response.
48,49
To discuss the deformation
mechanisms inherent to small-scale single crystals with
minimal added constraining effects of the substrates,
here we focus on tensile tests of single crystal Au thin
fi
lms on
fl
exible polyimide substrates.
13,14
These thin
gold
fi
lms were prepared by
fi
rst growing an epitaxial
Au
fi
lm on a single crystal of NaCl followed by the
deposition of polyimide on top of the Au layer. Sub-
sequently, the seed NaCl layer was dissolved in water,
resulting in the
upside-down
Au
fi
lm on a
fl
exible
substrate. The resulting test samples were [001] oriented,
5-mm wide, 8-mm long, and ranged from 30 nm to nearly
1-
l
m thick. These samples were not pristine, but con-
tained several initial dislocations, as well as a small
number of growth twins and pores in samples with
thicknesses below 50 nm as a result of the growth process.
Further details of the sample preparation procedure can be
found in Refs. 13 and 14.
Films with thicknesses ranging from
;
30 to
;
868 nm
were tested in tension along the [001] direction. Tensile
tests were performed by applying a displacement in steps
from 30 to 90
l
m up to a total length of 1000
l
m, and the
total strain was measured via a laser extensometer. In
between each extension, Laue patterns were obtained with
an exposure time between 15 and 120 s to determine the
complete stress state in the
fi
lms. As a result of the
different testing methodology, constant strain rate com-
parisons are dif
fi
cult. In contrast to pillar compressions,
the resulting stress
strain curves of the
fi
lms are quite
smooth and exhibit no noticeable bursts, likely due to the
fi
lm
s much larger length and width as compared to
pillars.
13
The resulting
fl
ow stresses at 0.5% strain are
plotted as a function of
fi
lm thickness in Fig. 6, which
shows a log
log plot of stress as a function of thickness.
The
fl
ow stress reported here is the resolved shear stress on
the perfect dislocation slip direction (111)[1-10]. These
authors
fi
nd that for larger
fi
lm thicknesses,
t
.
60 nm, the
strength follows a power law with the exponent
n
;

0.53,
similar to that seen in FIB-fabricated pillars and quite close
to that seen in larger electroplated copper pillars. Similarly
to the electroplated copper pillars, when the
fi
lm thickness
decreased below a critical value, in this instance
t
;
60 nm,
the
fi
lms
strengths remained constant or even decreased
with decreasing
fi
lm thickness.
13
To understand the origins of the deformation mecha-
nisms in thin
fi
lms, Oh et al.
14
performed in situ TEM
tensile tests on similar single crystalline gold samples
supported by polyimide substrates. In those experiments,
fi
ve different
fi
lm thicknesses between 160 and 40 nm
were tested and the resulting size-dependent behavior is
also measured outside the TEM via x-ray diffraction and
the results are similar to that seen in Fig. 6.
14
In these
experiments, thicker
fi
lms,
t
$
;
80 nm obey a power law
with the exponent
;

0.5, whereas thinner
fi
lms deform at
a relatively size-independent
fl
ow stress. In contrast to the
ex situ experiments discussed above, these
fi
lms are sub-
ject to additional thermal effects due to the polyimide layer
heating up during electron beam exposure.
14
These
researchers found that samples with the larger
fi
lm thick-
nesses,
t
.
80 nm, deformed via multiplication and glide
FIG. 5. High resolution transmission electron microscopy image of a [001] gold NW in tension. (a) Before and (b) after leading partial dislocation
nucleation. Inset in (a) and (b) show Fourier-
fi
ltered images of the stacking sequence highlighting the stacking fault. (c) After trailing dislocation
nucleation. Scale bar is 3 nm. (Reproduced from Ref. 27 with permission from Nature Publishing Group.)
A.T. Jennings et al.: Heterogeneous dislocation nucleation from surfaces and interfaces as governing plasticity mechanism in nanoscale metals
J. Mater. Res., Vol. 26, No. 22, Nov 28, 2011
2808
of perfect dislocations. Frequently though not always, these
perfect dislocations were observed to deposit interfacial
segments at the Au/polyimide interface, a deformation
mechanism characteristic of thin
fi
lms on stiff substrates as
demonstrated in the Matthews, Blakeslee, Freund, and Nix
model.
48,49
At large strains, this deposition of threading
segments leads to the so-called
cube glide
or glide of [001]
dislocations. Also, the authors see evidence of ample
dislocation emission from single arm sources pinned at
grown-in defects, in this instance growth twins or epitaxial
gold nanoparticles. Single arm sources are also seen through
in situ tests of pillar geometries
24
and have been shown to
manifest in a size-dependent yield stress.
5,9,15,18,21,22
In thinner
fi
lms,
t
#
;
80 nm, the glide of threading
dislocations still contributes to deformation; however,
these dislocations no longer deposit interfacial segments.
Furthermore, the authors note that partial rather than
perfect dislocation nucleation becomes the dominant de-
formation mechanism, whereby these partial dislocations
predominantly nucleate at stress concentrations at the
internal interfaces: square pores, twins, and surfaces
defects, produced during
fi
lm growth. An example of
partial dislocation emission at a TB is shown in Fig. 7.
Figure 7(a) shows the crystal prior to dislocation emission,
whereas Fig. 7(b) shows the stacking fault after the leading
partial has nucleated and passed through. The time differ-
ence between Figs. 7(a) and 7(b) is 1 frame or 0.4 s, an
amount of time suf
fi
cient to obscure the mechanistic details.
Figure 7(c) shows the trailing partial emitted from the
same location 8 s later and the subsequent annihilation of
the stacking fault.
14
C. Dislocation starvation
These and several other in situ TEM investigations in
pillars and thin
fi
lms have provided insight into how the
behavior of individual dislocations changes as a function
of size.
38,46
Separate in situ investigations have shown
that not only does the individual dislocation behavior
change but also the collective dislocation behavior
changes with critical thickness. As the critical length
scale decreases, dislocations no longer tend to form
FIG. 6. Resolved shear stress for
fl
ow stress at 0.5% strain for single
crystalline gold thin
fi
lms on polyimide substrates. (Reproduced from
Ref. 13 with permission from Elsevier.)
FIG. 7. (a)
(c) Successive transmission electron microscopy images
of partial dislocation nucleation. (a) Initial state. (b) 0.4 s after initial
state showing a stacking fault extending between twins a-a
and b-b
.
(c) 8.5 s later the stacking fault closes. (d) Schematic of nucleation process.
(Reproduced from Ref. 14 with permission from Elsevier.)
A.T. Jennings et al.: Heterogeneous dislocation nucleation from surfaces and interfaces as governing plasticity mechanism in nanoscale metals
J. Mater. Res., Vol. 26, No. 22, Nov 28, 2011
2809
substructures; instead they readily escape at free surfaces,
which result in the crystal becoming starved of disloca-
tions.
38,46
This concept of dislocation starvation was
fi
rst
introduced by Greer and Nix
50
to explain the deformation
behavior of nanoscale Au pillars, and post-deformation
TEM behavior on nanoscale samples has suggested its
operation.
31,50
Most convincingly, this phenomenon was
demonstrated during in situ TEM compressions of Ni
pillars with diameters at or below
;
200 nm.
46
In those
experiments, the dislocations likely produced by the FIB
damage and escape the sample resulting in a signi
fi
cant
reduction of the dislocation density in a process coined as
mechanical annealing.
46
A similar set of in situ TEM
compressions performed by a subset of the same team of
researchers also observed this phenomenon in smaller
[001] oriented copper pillars.
38
In these experiments, stre-
ngthening after the yield point was associated with a de-
creasing dislocation density. The authors postulate that as
a result of fewer dislocations being present in the pillar,
higher strength dislocation sources are required to sustain
deformation. The lack of dislocation multiplication via
double cross-slip processes and substructure formation, as
would be the case in bulk crystal deformation, is unique to
this nanometer length scale and increases the likelihood
that dislocation nucleation from alternative sources con-
tributes substantially to deformation. In pillars and thin
fi
lms, a prime candidate as an alternative source of dis-
location becomes heterogeneous dislocation nucleation as
the surface to volume ratio increases with decreasing pillar
diameter or
fi
lm thickness.
Thin
fi
lm geometries have only one dimension reduced
to the nanometer scale, their thicknesses, and therefore
might be expected to also experience dislocation starva-
tion; however, the constraints imposed by the supporting
substrates lock the mobile dislocations into the
fi
lm,
thereby hindering starvation effects.
13,14
In addition,
extensive cross-slip has been observed during in situ
experiments on thin
fi
lms, which may further effect
starvation.
Dislocation starvation model
A model based on the balance between nucleation and
annihilation rates of dislocations at surfaces in circular
pillars was recently developed by Nix and Lee.
42
This
relatively simple phenomenological model describes the
physics of pillar compressions through a kinetic law:
_
c
¼
1
l
d
s
dt
þ
q
m
b

v
;
ð
3
Þ
where
_
c
is the shear strain rate,
l
corresponds to the shear
modulus,
d
s
dt
is the shear loading rate,
b
is the Burgers
vector,

v
is the average dislocation velocity, and
q
m
is the
mobile dislocation density. Equation (3) describes the
shear strain rate as composed of two terms corresponding
to the elastic loading term and the subsequent plastic
deformation term. Through the comparison of the rates of
dislocation nucleation and annihilation at a free surface,
these authors derive an equation for the stress required to
maintain a constant shear strain rate as:
s
ss
¼
s
th
_
c
b
px
o
D

m
;
ð
4
Þ
where
s
ss
corresponds to the steady state applied shear
stress,
s
th
is the theoretical shear stress,
D
is the pillar
diameter, and
x
o
is the nucleation frequency at the
theoretical shear stress.
42
Notably, the stress here depends
on both the pillar diameter through
D

m
and on the strain
rate as
_
c
m
suggesting that as pillar diameter becomes
smaller or the strain rate increases, higher stresses will
be required to maintain steady state deformation. Further,
the size-dependent strength and strain rate dependence are
linked through the exponent
m
. In larger pillars, where the
size-dependent strength obeys a power law with exponent
between

0.5 and

1.0 the resulting value of the strain
rate sensitivity is of 0.5
1, unreasonably large values for
FCC metals.
Applying this analytical framework to our experiments on
Cu nanopillars, we see good agreement in regard to the
expected strain rate sensit
ivity and diameter-dependent
strength. Substituting our experi
mentally determined strain
rate sensitivity of
;
0.1 for the 125- and 75-nm pillar
diameters at the slowest strain rates of 10

3
s

1
into
Eq.(4),we
fi
nd the corresponding strength dependence on
diameter of
;

0.1 a reasonably weak strength dependence
on diameter, which compares favorably with our smallest
samples, as well as with the results of investigations presented
here and other theoretical works.
13,14,25,28
Although this phe-
nomenological model accurately captures the correlation
between strain rate dependenc
e, size, and strength, it cannot
explore the details of disloca
tion nucleation. Therefore, we
next discuss atomistic models
that attempt to capture the
physics and the stresses required for partial dislocation
nucleation.
D. Partial dislocation nucleation models
Two different models for heterogeneous nucleation of
dislocations in FCC metallic thin
fi
lms and pillars have
been reported.
13,14,28
Despite the differences in sample
geometries between these models and our samples, it is
reasonable to expect that the governing mechanisms in
both small-scale structures are similar.
1. Classical dislocation source model
The model recently published by Chen et al.
51
provides
an estimate for the transition diameter between perfect
and partial dislocation nucleation at grain boundaries in
A.T. Jennings et al.: Heterogeneous dislocation nucleation from surfaces and interfaces as governing plasticity mechanism in nanoscale metals
J. Mater. Res., Vol. 26, No. 22, Nov 28, 2011
2810
nanocrystalline Al. The shear stress to expand a partial
dislocation with the Burgers vector
b
p
and the stacking
fault energy
c
SF
is correspondingly written as:
s
p
¼
2
al
b
p
D
þ
c
SF
b
p
:
ð
5
Þ
The stress to expand a perfect dislocation loop is:
s
N
¼
2
al
b
N
D
:
ð
6
Þ
Here,
s
p
is the resolved shear stress to operate a partial
dislocation source,
a
is a coef
fi
cient, between 0.5 and 1.5,
re
fl
ecting to the orientation dependence of the line energy,
l
is the shear modulus,
b
N
is the perfect Burgers vector, and
D
/2
is the critical radius for thi
s dislocation source where
D
is the
critical length scale, originally grain size
51
but recently
extended to
fi
lm thickness.
13,14
This simple model compares
the strength to operate a Frank
Read source emitting a perfect
dislocation loop versus a partial dislocation and an accom-
panying stacking fault as functio
n of the critical length scale.
By setting Eqs. (5) and (6) equal to each other, these authors
fi
nd the critical length scale for the transition.
D
¼
2
al
b
N

b
p

b
p
c
SF
:
ð
7
Þ
This model has been extend
ed more recently to single
crystalline Au
fi
lms
6,13,14
to explain the observed transition
in strength as a function of
fi
lm thickness, as seen in Fig. 6.
This partial dislocation nucl
eation model captures the trend
in observed strength reasonably well.
6,13,14,51
However, in the
case of Au thin
fi
lms on polyimide this model overpredicts
the stresses by a factor of
.
2 which the authors hypothesize
may be due to defects enabling l
ower strengths for partial
dislocation nucleation.
13
Frank
Read and single arm source
operation requires the existe
nce of a pinned dislocation
segment; however, in single crystals, partial dislocation
nucleation appears to operate in the absence of such sessile
dislocation segments. In fa
ct, as shown previously, the
existing in situ TEM studies sugge
st that partial dislocations
nucleate at external surface
s or internal interfaces.
14,27
In the
absence of a pinned dislocation segment, the critical length
scale,
D
, is not necessarily associated with the
fi
lm thickness.
Rather, atomistic and analy
tical models studying homoge-
neous nucleation
52
and heterogeneous dislocation nucle-
ation
28,53,54
showed much smaller critical dislocation loops,
with the critical radius on the order of a few nanometers.
Such a small critical radius further suggests an important
contribution from thermal effects. Including thermal
effects would result in lowering the strength necessary
for dislocation nucleation, an effect possibly manifested as
the larger transition diameter found in in situ thin
fi
lm
studies versus ex situ tests.
2. Heterogeneous dislocation nucleation
Zhu et al.
28
recently investigated the probabilistic
nature of thermally activated surface dislocation nucle-
ation through the development of a general analytical
model. This model requires the knowledge of the thermal
activation parameters: activation energy and activation
volume, to make predictions regarding speci
fi
c nucleation
processes. These activation parameters were then deter-
mined through
fi
xed end nudged elastic band (FENEB)
method, in square Cu nanopillars.
28
In their analytical
model, the nucleation frequency,
v
, of a given site is taken
to have an Arrhenius form as in Eq. (1) where
v
is
substituted for shear strain rate,
_
c
. Then, to describe the
probabilistic nature of thermal activation, they de
fi
ne
a survival probability,
f
(
t
), as the fraction of a set of pillars
that have not nucleated a dislocation by a time
t
. The
change in
f
(
t
) is related by the nucleation frequency,
v
,
through:
df t
ðÞ
dt
5

vf t
ðÞ
. The most probable time that a NW
will nucleate a dislocation can then be found by
fi
nding the
maximum of
df t
ðÞ
dt
. The resulting equation describes the
most likely time at which a pillar will nucleate a disloca-
tion; however, in experiments, a more relevant measure is
applied stress. The survival probability analysis can be re-
written in terms of the applied stress by relating time and
stress through the elastic modulus and applied strain rate:
r
5
E
_
e
t
. Rearranging the above and linearizing the result
for clarity, we
fi
nd that the activation stress can be
represented as:
r
¼
r
athermal

k
B
T
X
ln
k
B
TNv
o
E
_
e
e
X
:
ð
8
Þ
Here, the
fi
rst term,
r
athermal
, corresponds the athermal
stress or the stress required to nucleate a dislocation at zero
temperature. The second term comprises the thermal
contribution to nucleation stress where
k
B
is Boltzman
s
constant,
T
is temperature,
N
is the number of equivalent
nucleation sites,
m
o
is the atomic vibration frequency,
E
is
the Young
s modulus, and
_
e
e
is the applied strain rate. This
term is dominated by the constant prefactor outside the
logarithm and will be large at high temperatures and small
activation volumes.
Zhu et al.
28
determined the activation parameters for
surface source nucleation in an initially perfect square
[001] Cu wire through an atomistic FENEB calculations.
The predicted size-dependent stresses are shown in Fig. 8.
The expected size effect due to partial dislocation nucle-
ation is quite weak with the exponent of
;
0.1
0.2 related
to the number of equivalent sites
N
inside the logarithm.
This weak size effect re
fl
ects well the observed trend in
experimental literature on thin
fi
lms
6,13,14
and, more
recently, in pillar compressions.
25
Furthermore, they esti-
mate the transition diameter to be between 10
100 nm and
to depend on strain rate, both of which are in reasonable
A.T. Jennings et al.: Heterogeneous dislocation nucleation from surfaces and interfaces as governing plasticity mechanism in nanoscale metals
J. Mater. Res., Vol. 26, No. 22, Nov 28, 2011
2811
agreement with our experiments on pillars and those
discussed on thin
fi
lms. These authors
calculations
fi
nd
the activation volumes for surface source nucleation to be
within the range of 1
10b
3
, which extends a range from
nucleation at a sharp corner (
;
1b
3
) to the
fl
at side of the
NW (
;
10b
3
).
28
These small activation volumes correlate
well with the ones we
fi
nd in our pillar experiments as seen
in Fig. 5 in Ref. 25 and result in a large thermal
contribution to strength as seen in Eq. (8).
The two models above, the classical source model and
heterogeneous dislocation nucleation model, differ in
their assumption of the state of the crystal prior to partial
dislocation nucleation. The classical source model
assumes the pre-existence of a pinned dislocation seg-
ment, whereas the heterogeneous nucleation model does
not. As a result of this assumption, the expected
activation volumes of the two processes are drastically
different. The classical source model is Frank
Read type
model and would therefore likely result in an activation
volume on the order of 100
1000b
3
,
55
58
whereas the
heterogeneous nucleation model results in activation
volumes between 1 and 10b
3
is in good agreement with
our pillar compressions.
25
This discrepancy between the
activation volumes of the classical source model and the
measured activation volumes suggests that the heteroge-
neous dislocation model may be a more accurate depiction
of the nucleation physics during pillar compressions.
E. Effects of imperfections on dislocation
nucleation
TEM studies on partial dislocation nucleation have
highlighted the role that various defects play in disloca-
tion nucleation. This is most obvious in the case of thin
fi
lms where dislocations preferentially nucleate at square
voids as they serve as sites of stress concentrations during
tensile experiments.
13,14
Furthermore, partial dislocation
nucleation at TBs has also been observed in both thin
fi
lms
and NWs.
14,27
This preference for nucleation at stress
concentrations has also been observed in recent MD
simulations of bulk nanotwinned copper
59
and twinned
NWs.
29
In large-scale MD simulations of a bulk nano-
twinned sample, dislocation emission is strongly preferred
at the intersection of TBs and grain boundaries, which are
characterized by local stress concentrations.
59
Similar
results were found in MD simulations of twinned nano-
pillars, where partial dislocations preferentially nucleate at
the intersection of the TB and the NW surface.
29
This last
type of stress concentration has been observed experimen-
tally in gold NWs, as shown in Fig. 5. Further, in these
in situ experiments, the pillar surface is characterized by
the intersection of distinct {111} planes resulting in
a faceted surface along the NW length promoting nucle-
ation through stress concentrations at the kinks in NW
surface and surface steps.
27
All this evidence for the
preference of heterogeneous dislocation nucleation at
inhomogeneities also corroborates the work of Richter
et al.
26
In the absence of surface defects or internal
dislocations, failure via brittle fracture is more energeti-
cally favorable than dislocation nucleation from a pristine
surface.
In our pillar tests, the surface roughness also likely
affects the local surface stress state. In tension, surface
roughness may be expected to create stress concentra-
tions similar to a crack tip with the initial crack length
controlled by the local roughness; however, the experi-
ments presented here were performed in compression,
likely posing a more subtle in
fl
uence of surface roughness
on the local stress state. Tensile tests on similar electro-
plated copper pillars showed deformation through immedi-
ate necking.
32
Comparisons of these same pillars under both
tension and compression did not demonstrate a tension
compression asymmetry suggesting that in these pillars
the surface roughness did not act as a crack tip, further
emphasizes the subtle in
fl
uence the surface roughness
may play.
32
It should be noted that the strengths of these pillars
always showed size-dependent strength; however, the
effects of a constant displacement rate, as opposed to
a constant strain rate, and the relatively few samples
tested precludes a de
fi
nitive conclusion. Further, as the
critical dislocation nucleation radius is on the order of
a few nanometers,
28,53,54
variations over these small
distances such as individual surface steps may play
a key role in determining a material
s resistance to
heterogeneous dislocation nucleation.
F. Partial versus perfect dislocation nucleation
Our discussion up to this point has been dominated by
heterogeneous partial dislocation nucleation at internal
FIG. 8. Log
log plot of strength versus diameter showing the predicted
transition from collective dislocation dynamics to surface source
nucleation. (Reproduced from Ref. 28 with permission from American
Physical Society.)
A.T. Jennings et al.: Heterogeneous dislocation nucleation from surfaces and interfaces as governing plasticity mechanism in nanoscale metals
J. Mater. Res., Vol. 26, No. 22, Nov 28, 2011
2812
surfaces or interfaces. The preference for partial dis-
location nucleation as opposed to perfect dislocation
nucleation is primarily due to the low stacking fault
energy of the materials tested so far: Au and Cu. In single
crystals of low stacking fault materials, perfect disloca-
tions readily split into ribbons bounded by the leading
and trailing partials as this con
fi
guration has a lower
energy. In metals with high stacking fault energies,
perfect dislocations dominate deformation as the energy
does not decrease through the separation of partials. The
details of this preference for perfect versus partial
dislocation nucleation have been discussed in the MD
simulations of nanocrystalline materials, where it has
been shown that the choice between perfect and partial
nucleation depends critically on the complete generalized
stacking fault curve.
60
As Au and Cu both have low
stacking fault energies, it is not surprising that partial
rather than perfect dislocation nucleation has been exper-
imentally observed. Further experimental and theoretical
investigations into FCC metals with high stacking fault
energies, like Al or Ni, will help elucidate the speci
fi
c role
stacking fault energy plays in nanoscale plasticity.
V. SUMMARY
We discuss the role of heterogeneous nucleation of
partial dislocations at local stress concentrators found on
the surfaces and interfaces of nanosized pillars and
fi
lms
on plastic deformation. Our experiments on single crys-
talline Cu nanopillars, as well as other experimental
studies on thin
fi
lms demonstrate a transition in size-
dependent strength when the critical length scale dips
below
;
100 nm. At this size, the strength becomes
independent of size and deviates from the commonly
observed power law. In situ TEM tests on thin
fi
lms and
NWs reveal the likelihood that this observed transition
results from a change in preference from perfect disloca-
tion multiplication through internal source operation to
partial dislocation nucleation from local inhomogeneities
on the surfaces and interfaces like voids, TBs, and surface
roughness. These
fi
ndings help explain the lower strengths
for pillars and thin
fi
lms containing initial defects in
contrast with the much higher strengths exhibited by
pristine and nearly pristine small-scale geometries. We
fi
nally discuss the combined strain rate and size-dependent
experimental data in the context of several models con-
cerning partial dislocation nucleation and combined
through comparisons of activation volumes.
ACKNOWLEDGMENT
The authors gratefully acknowledges the
fi
nancial
support of the National Science Foundation through
ATJ
s NSF Graduate Research fellowship and JRG
s
CAREER grant (DMR-0748267).
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