Supporting Information
Failure Mechanisms in Vertically Aligned Dense
Nanowire Arrays
Rebecca A. Gallivan*, Julia R. Greer
Division of Engineering and Applied Science, California Institute of Technology, Pasadena,
California 91125, United States
1.
Hydrothermal Growth Substrate and Apparatus
In this creating the substrates for ZnO nanowire bundle growth, a 10 nm platinum adhesion layer
and 100 nm gold layer is deposited via e
-
beam evaporation (Lesker Labline) onto a
silicon
substrate. Next, a 1.5 μm thick poly(methyl methacrylate) resist (950 PMMA A9) is spin
-
coated
onto the substrate and patterned into an array of holes via e
-
beam lithography (Raith EBPG
5000+). This holey PMMA coating then acts as a template whic
h allows nanowire growth only in
the exposed cylindrical holes, resulting in the bundles. The substrate is then loaded into the
cathode side of a custom electrolysis holder (
Figure
SI1). The holder provides electrical
connection to the gold film through
a platinum wire. A platinum mesh acts as the counter
electrode.
Figure S
I
1
: Hydrothermal growth apparatus and corresponding schematic with custom chip holder.
Demonstrates method by which gold coated substrate is submerged int
o growth solution and
connected to electrical source.
2.
TEM Sample Preparation
The top surface of the bundle is protected with a 100
nm
-
thick layer of platinum (Pt) deposited via
a Gas Injection System (GIS) using an electron beam in an FEI Versa DualBeam
SEM followed
by a 400
nm
-
thick layer of Ga+ FIB
-
deposited Pt in the same chamber. Next, a Ga+ ion beam is
used to carve out 5 μm x 3 μm trenches into the substrate forming a U
-
cut to free the silicon base
from the rest of the substrate. Using a tungsten
needle (EZlift program), the detached silicon base
with the nanowire bundle on the top surface is transferred and glued via FIB
-
deposited Pt to a
copper halfmoon grid with the long axis of the bundle parallel aligned with the grid post using the
same GIS
Pt deposition. After detaching the tungsten needle, a series of cuts are made with a
decreasing Ga+ voltage/current (30kV, 100pA; 30kV, 10pA; 16kV 23pA) are used to
progressively thin the cross
-
section of the bundle stru
cture to <100
nm for TEM imaging.
3.
Tilt Angles of Nanowires
Figure S
I3
:
Histogram showing the relatively uniform distribution of tilt angles measured from as
fabricated bundle samples with a mean of 2.8
°
+/
-
1.2
°
.
4.
Burst Initiation Stress
Figure SI4: Histogram compares the
distribution of stress associated with initial burst event for 2
μm (left) and 4
μm (right) bundles. Distributions show similar range (values from 0.3
-
1.3 GPa)
and mean values (0.64 +/
-
0.31 and 0.58 +/
-
0.27 GPa respectively).
5.
Euler Buckling of Nanowire
s
Assuming a general column form
with a hexagonal cross
-
section
for the individual nan
owires, the
standard Euler Buckling relationship applies:
휎
푏푢푐푘푙푒
≥
휋
2
퐸
2
푘
2
(
푙
푟
⁄
)
2
With
푙
being the height,
푟
being the radius,
푘
being the end condition factor
(i.e. friction condition)
,
and
퐸
the elastic modulus in the direction of loading.
The aspect ratio of the nanowires in this
experiment lies between 26 and 27.
We assume a fixed based condition due to surface bonding
of
th
e nanowires to the substrate. The interaction between the top surf
ace of the nanowires and the
in
denter is best described by a free rotation condition (k = 0.7) based on SEM images during in
-
situ compression.
Therefore, the critical Euler Buckling stress
for these nanowires is 2 GPa.
To
highlight the range of values that could possibly be seen in our system, we include calculations for
both fixed (k = 0.5) and free translation (k = 1.0)
friction conditions at the indenter
-
nanowire
interface as demonstrat
ed in Figure SI5. The figure highlights the region of our observed
nanowires as well as the aspect ratios required for buckling to occur in the range of failure initiation
stresses.
Figure SI
5
:
Critical buckling stress for an unsupported hexagonal ZnO
nanowire of particular
aspect ratio given a fixed base and fixed (k = 0.5), pinned (k
–
0.7), or free (k = 1.0) boundary
condition
.
Grey area reflects experimentally average +/
-
a standard deviation in observed
dimensions of the nanowires.
Table SI5
further identifies the specific aspect ratios require for critical buckling failure to occur
at the stress values associated with our onset of nanowire interfacial splitting. Values bolded
indicate conditions in which buckling would likely be a dominant m
echanism in our system.
Critical
Stress
(GPa)
Fixed
(k
=
0.5)
Free
Rotation
(k
=
0.7)
Free
Translation
(k
=
1)
0.3
96.0
68.6
48.0
1.0
52.6
37.4
26.5
1.3
46.1
33.0
23.0
Table
SI5:
Specific
aspect
ratios
required
for
critical
buckling
In
order
for
buckling
to
explain
the
range
of
values
seen
during
localized
failure
with
the
free
translation
condition,
some
nanowires
would
need
to
be
either
10.4
μm
taller
than
average
(a
260%
increase)
or
134
nm
thinner
than
average
(a
44%
decrease).
F
or
the
free
rotation
condition,
nanowires
with
dimensions
either
16.6
μm
taller
than
average
(a
415%
increase)
or
183.4
nm
thinner
than
average
(a
61%
decrease)
would
need
to
be
present.
All
these
values
are
far
outside
the
observed
range
of
301
+/
-
31nm
and
4.00
+/
-
0.15
μm.
6.
Calculation of tilt angle and flaw size impact on failure stress
In evaluating the impact of angle on failure stress we look at the relationship between
flaw size at
first failure, a, the stress at failure initiation, and the angle,
휃
, described in equation 4 from the
Results and Discussion section through the following relationship:
푎
=
퐾
퐼퐶
2
푏
2
ℎ
2
12
(
휎
퐴
푠푖푛
휃
)
2
−
ℎ
2
3
where b is the nanowire thickness for 300 nm, h is the interfacial thickness of 5Å,
퐾
퐼퐶
is the
interfacial toughness of
0.58
MPa
√
m
,
and A is the area of the top surface of a 2
μ
m diameter
bundle. If we assume the same flaw distribution in order to isolate the effect of simply the angle
on failure stress, a will take on the same value in both bundles as they will both have the same
largest critical flaw size for initiation of failure. Looking at a variation in
휃
of 1
°
in two bundles
we show the maximum impact of the variation using a fl
aw size of 300
nm and tilt angles of 2
°
and 3
°
, Solving for stress we see that the stress of failure initiation is 1.3 GPa for the bundle with
2
°
tilt and 0.87 GPa for the bundle with 3
°
tilt. This represents an upper bound of a 0.43 GPa
difference in in
itial failure stress.
For comparing the impact of flaw size on failure stress observed, we use the equation above but
with a fixed failure stress and identify the crack length associated with a particular tilt angle. We
determine that both a 217 nm defect
with a tilt of 3
°
and a 163nm with a tilt of 4
°
will initiate
failure at 0.7
GPa
.
7.
Derivation of
probability function for failure
Probability that an element of the bundle can survive is
푓
̌
. The initiation rate of a particular
failure can be describes as the fractional reduction rate of intact elements within the next
time interval.
푛
̇
푖
=
−
1
푓
̌
푖
푑
푓
̌
푖
푑푡
looking along a common strain path,
ln
(
푓
̌
푖
)
=
−
∫
푛
̇
푖
휀
̇
푑퐿
which is equivalent to
ln
(
푓
̌
)
=
−
∫
∑
푛
̇
휀
̇
푑휀
The initiation rate can also be defined by the probability of a particular element reaching the
failure strain of the material or failure mechanism
푛
̇
푖
=
휈
푑
푁
푖
exp
[
−
(
휀
푐
휀
)
푚
]
with
휀
푐
being equal to the critical failure strain as a funct
ion of
휀
being the current strain.
푚
is the Weibull exponent and relates to the failure prediction of the material or mechanism.
휈
is the attempt frequency. Combining this definition with the above equation for
푓
̌
gives
ln
(
푓
̌
)
=
−
휈
∫∫
1
휀
̇
exp
[
−
(
휀
푐
휀
)
푚
]
푑푁
푑휀
The probability of failure can be defined as
푓
=
1
−
푓
̌
and thus
푓
=
1
−
exp
[
−
휈
∫∫
1
휀
̇
exp
[
−
(
휀
푐
휀
)
푚
]
푑푁
푑휀
]
Further manipulation of this equation assuming linear elasticity gives rise to a solution
assuming uniform strain in compression of the structure and
assuming a constant strain rate.
푓
(
휎
)
=
1
−
푒푥푝
[
휈푁
휎
̇
∫
푒푥푝
[
−
(
휎
푐
휎
′
)
푚
]
푑
휎
′
휎
0
]
with
휎
̇
as the stress loading rate,
휈
as the attempt frequency for failure, N as the total number
of nanowires, σc as the critical failure stress, and m as the Weibull exponent.
The
휈
휎
̇
⁄
ratio
ultimately determines the "stress step size" over which failure attempts occur.
F