Supplemental Information
Experimental
:
ALD Reaction Conditions
D
epositing a thin conformal hard coating on high
-
aspect ratio structures is non
-
trivial, with ALD being particularly suitable for these types of coatings as it offers
atomic
-
level control of
the depositing species on
e monolayer at a
time (Figure 1c). Figure
1c shows a schematic of this process for an initially uncoated copper pillar
whose surface
is terminated with
oxygen atoms . In the first step (1), a precursor gas is added into an
ALD
syst
em
where the precursor bonds with the oxygen
to form a monolayer of the
corresponding oxide
. The following purge step (2) removes the remaining extra precursor
as well as any additional reaction products from the chamber. This surface layer is then
functio
nalized through reaction in a plasma (3) to produce a
reactive oxygenated
surface .
The final step (4) is another purge step to remove the remaining reaction products. This
process is repeated until the desired thickness is achieved. All reactions in our p
rocess
were performed in an Oxford OpAL ALD
system
(Oxfordshire, UK); whereby 3nm of
alumina
wa
s deposited with a precursor of trimethyl aluminum
(
SAFC Hitech
,
Allentown,
PA)
and the remaining thickness is titania
formed
from a titanium tetra
-
iso
-
propoxide
precursor
(
SAFC Hitech
)
.
The initial
3nm
-
thick
Al
2
O
3
layer
was
deposited
first (1) with a reactant dose of
precursor of
trimethyl aluminum for 30ms at 120C. This
wa
s followed by (2) a 2 second
purge followed by (3) a total of 6 seconds in a 300W plasma,
2 seconds for gas
stabilization and 4 seconds for plasma
power on
. Finally, (4) the last purge step also
lasted
two seconds
[
1
]
. This process
wa
s repeated until the 3nm layer
wa
s complete. The
following TiO
2
layer was added with a titanium tetra
-
iso
-
propoxide precursor at 200C.
The remaining thickness
es
of 2nm at pillar diameters
,
D, 75nm
-
150nm
;
7nm at D~200nm,
and 22nm at D ~ 500nm
-
1000nm
,
respectively
,
w
ere
deposited
via a
process
similar
to
the alumina de
position
[
2
]
. In both of these procedures, a
remote oxygen
plasma
functionalized
the surface with oxygen atoms such that the surface
wa
s identical to the
initial conditions.
It is expected that oxygen atoms and ozone are the most likely reactive
species as there was a showerhead separating the plasma from the substrate.
Results
Metho
ds of Calcu
lating
Axial
Stress in the Copper Pillar:
In the case where the coating shares load with the copper pillar,
we use
a simple
iso
-
strain model:
!
!
!"!#$
=
퐴
!
+
1
−
퐴
!
!
!
!
!""
!
!"##$%
휎
!"##$%
[
3
]
. Here,
퐹
is the applied force,
퐴
!"!#$
is the cross
-
sectional area
of the coated pillar
.
퐴
!
is th
e
cross
-
sectional
area
fraction of the copper pillar or
퐴
!
=
퐴
!"##$%
퐴
!"!#$
,
퐸
!
!
!""
and
퐸
!"##$%
are the elastic
moduli for the coating and the copper pillar, respectively. This shared load model is most
relevant for understanding th
e stress
-
state in the copper pillar at small strains
, i.e.
prior to
catastrophic cracking of the coating.
On the opposite end of the spectrum,
we assume that
the coating supports no load, and therefore the stress in the copper pillar is measured by
휎
!"##$%
=
퐹
퐴
!"##$%
where
퐹
is still the applied force and
퐴
!"##$%
is the area of the copper
pillar only. This model best estimates the stress
-
state in the copper pillar after the coating
has cracked in multiple locations and is effectivel
y “going along for the ride” as it is not
capable of supporting any appreciable elastic stresses.
Discussion
:
Strengthening from Dislocation Storage
Building upon analytical models for single arm sources
[
4
-
6
]
, the general
equation for overall shear stress in a small
-
scale sample is comprised of the lattice
friction stress (first term on RHS in Eq. 1), the elastic interactions stress (2
nd
term on the
RHS in Eq. 1), and the line tension stress (last term in Eq. 1):
,
(1)
where
휏
!
is the resolved shear stress for the activation of a single arm dislocation source,
is the lattice friction stress,
is the isotropic shear
modulus,
is the magnitude of
the Burgers vector, and
is the total dislocation density: material and microstructural
parameters with identical applicability in the framework of classical bulk dislocation
theory
[
7
]
. On the other hand,
is a parameter relevant to single arm sources in pillars,
as it represents the shortest distance be
tween the source’s pinning point and the free
surface within the same elliptical slip plane.
τ
i
=
τ
0
+
0
.
5
μ
b
ρ
t
o
t
+
μ
b
4
π
λ
i
l
n
λ
i
b
!
"
#
$
%
&
0
τ
μ
b
tot
ρ
i
λ
In order to calculate the activation stress for a given nanopillar diameter, we first
estimate the number of available single arm sources by following Parthasara
thy’s model,
and by assuming that each dislocation segment represents a single
-
arm source, the
average length of a dislocation segment is equal to the pillar diameter. The estimated
number of single arm sources,
n
, can then be represented as:
푛
=
Integer
휌
!"!
×
!"
!
!
,
(2)
where
and
h
are the pillar diameter and height, and
is the total dislocation density.
The pinning points are then randomly distributed within a pillar, and the shortest length
from each pinning point to the pillar surface,
휆
!
, is
calculated.
The stress required to
operate the weakest (i.e. longest) glissile dislocation source is taken as the stress required
to produce
the first strain burst. Statistics were calculated over 100 simulations. For each
diameter, 100 simulations were performed, and the average and standard deviation of
those results are shown in Figure 6
, main text,
along with the experimental results from
Figure 2, main text.
To calculate the resolved shear stress,
휏
!
, we adopt the model
introduced by Parthasarathy
et al
.
[
6
]
modified appropria
tely to suit the <111>
orientation of our pillars, i.e. a multiple
-
slip condition. In such orientations, any one of
the 12 equivalent slip systems
–
as opposed to a single one
-
can be randomly assigned to
a dislocation segment as proposed by Ng and Ngan
[
5
]
.
It is reasonable to assume that the lattice friction stress is negligible in fcc metals
and the shear modulus and the magnitude of the Burge
rs vector are taken as 44 GPa and
2.55 Å, respectively. Using these quantities, we calculate the stresses at the first strain
burst for simulated samples with diameters
~200, 250, 500, and 1000 nm and 3:1 (
h:D
)
aspect ratio, and compare them with the experimental data. We intentionally did not
include the samples with diameters smaller than
D
~200 nm because the plasticity
mechanism at such small sizes is governed by dislocation nucleation at the free surfaces
a
t the strain rates used in our experiments
[
8
]
.
Hoop Stress
In these experiments, the source of the pressure normal to the coating and thus the
hoop stress is primarily due to (1) the Poisson expansion of the copper pillar during the
nearly elastic compressive loading and (2) the dislocati
on motion and resulting localized
D
tot
ρ
D
plastic flow of the copper pillar. As has been ubiquitously shown, the stress for
dislocation motion and thus plastic flow in pillar compressions is size
-
dependent,
suggesting that the strength for coated pillars may also
size
-
dependent. The size
-
dependence in coated pillars appears to be reflected in the experimental results seen in
Figure 2d and is also seen in other experimental and simulation
work
[
9
-
11
]
.
Threading Dislocations
Interestingly, in our investigations of initial pillar microstructure, TEM analysis of
coated 500nm pillars reveals an extensive netw
ork of equally
-
spaced, ~40nm, threading
dislocations originating from the pillar
-
coating interfaces, as shown in Figure 7. This
observation may not be surprising since threading dislocations emanate from misfit
dislocations which are typically found at thi
n film interfaces alleviating the strain across
that interface
[
12
]
. The observed equilibrium threading dislocation spacing is a result of
the balance between the energy required to produce threading dislocat
ions and the
resulting relieved elastic energy from the interfacial strain. As a result, the relative
thickness of the coating with respect to the pillar diameter determines whether or not the
threading dislocations will form. If the coating is too thin, f
ew or no threading
dislocations will be created because the elastic strain energy is insufficient to overcome
the energy required to produce a misfit dislocation. Analogously, for sufficiently small
pillars, the amount of stored elastic energy may be insuf
ficient to drive the formation of
dislocations, as consistent with our observations that misfit dislocations are not seen in
pillars with
퐷
<
500
푛푚
. However, the lack of threading dislocations may result in an
additional internal strain that may influence
the pillar deformation resulting in an
interesting, yet likely complex problem in smaller pillars. One consequence of threading
dislocations in 500nm pillars may be that these threading dislocations weaken the coating,
resulting in a smaller increase in h
ardening relative to that seen in 200nm pillars.
Bauschinger Analytical Model Detail
We take Cu as the representative materials and use it for all material properties.
Beginning with the single
-
source in a dislocation
-
free plane, we incrementally apply a
shear stress at a rate
Δ
휏
=
±
0
.
01
휏
!"#$%&
. When the force concentrated at the sou
rce
reaches
퐹
=
휏
!"#$%&
푏
, a loop is emitted and two oppositely oriented point segments are
introduced into the slip plane at a distance
훿푥
=
0
.
2
퐿
on either side of the source, where
L is the length of the slip plane. The equilibrium positions of th
ese dislocations are then
determined through balance of the Peach
-
Kohler forces from long
-
range dislocation
interactions and image forces caused by the hard coating. As an approximation, we
truncate the image forces to the first image field. The Peach
-
Kohl
er force on the
i
th
dislocation in a system of
N
dislocations is given by:
퐹
!
=
휏
!""
푏
−
휇
푏
!
2
휋
푠푖푔푛
(
푛
,
푖
)
푥
!
−
푥
!
!
!
!
!
!
!
!
!
Where we have treated the dislocations as screw
-
type, b is the Burgers vector and
푠푖푔푛
(
푛
,
푖
)
is read as the direct
ion of the force on the
i
th
dislocation by the
n
th
dislocation.
The summation is performed over both the real and image dislocations for
2N
total
dislocations. Following the calculation of the equilibrium position of the dislocations, the
force on the two
dislocations closest to the coating is calculated to check if either exceeds
their respective coating strength. This condition is:
휏
!""
−
휇푏
2
휋
1
푥
!
−
푥
!
!
!
!
!
≥
휏
!"#$%&'
If this condition is met, this dislocation escapes through the coating and
the equilibrium
positions of the remaining dislocations are recalculated. If the condition is not met, the
applied stress is incremented until the sum of the applied stress and the back
-
stresses are
larger than the source strength:
휏
!""
−
휇푏
2
휋
1
푥
!
!
!
!
!
≥
휏
!"#$%&
This process is iterated through loading/unloading and at each stress, the total strain is
calculated by the elastic strain given by Hooke’s law and the plastic strain that is
proportional to the distance swept out by the dislocations:
푑
휀
!"#$%&'
=
1
2
푏
푑퐴
푉
For
y
indicating the direction normal to the plane and
z
directed out of Figure #, we can
take a representative volume,
푉
=
퐿
!
퐿
!
퐿
!
=
3
퐿
!
퐿
!
,
퐿
!
being arbitrary for the 1
-
dimensional problem. With
푑퐴
=
퐿
!
푑푥
, the expre
ssion for plastic strain simplifies to
푑
휀
!"#$%&'
=
1
6
푏
퐿
!
푑푥
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