# Buckling of Single-Crystal Silicon Nanolines under Indentation

Authors: Min K. Kang; Bin Li; Paul S. Ho; Rui Huang
Journal: Journal of Nanomaterials
 (2008)

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## Abstract

Atomic force microscope-(AFM-) based indentation tests were performed to examine mechanical properties of parallel single-crystal silicon nanolines (SiNLs) of sub-100-nm line width, fabricated by a process combining electron-beam lithography and anisotropic wet etching. The SiNLs have straight and nearly atomically flat sidewalls, and the cross section is almost perfectly rectangular with uniform width and height along the longitudinal direction. The measured load-displacement curves from the indentation tests show an instability with large displacement bursts at a critical load ranging from
480μN to 700 μN. This phenomenon is attributed to a transition of the buckling mode of the SiNLs under indentation. Using a set of finite element models with postbuckling analyses, we analyze the indentation-induced buckling modes and investigate the effects of tip location, contact friction, and substrate deformation on the critical load of mode transition. The results demonstrate a unique approach for the study of nanomaterials and patterned nanostructures via a combination of experiments and modeling.


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## Body

## 1. Introduction

Silicon (Si)-based nanomaterials [1–9] have
drawn much attention recently due to continual miniaturization of semiconductor
devices in microelectronics and development of nanoelectromechanical systems (NEMSs). One of the
persisting challenges for practical applications of nanomaterials has been the
measurement of their mechanical properties, which as well known can be
different from their counterparts of bulk materials. A variety of metrology
methodologies have been developed, such as nanoscale tensile tests [10–12],
resonance methods [13, 14], bending techniques [1, 2, 4–6], and nanoindentation tests [7, 15]. In particular, the nanoindentation tests have been well established for
measuring mechanical properties such as elastic modulus, hardness, and fracture
toughness, for both bulk and thin-film materials [16, 17], with very high force/displacement
resolution and excellent controllability. Recently, the nanoindentation
technique has been adopted for mechanical characterization of various nanomaterials
[15], including nanotubes [18, 19], nanowires [20, 21], nanobelts [22], and
nanoparticles [7]. However, interpretation of the nanoindentation results for
nanomaterials is nontrivial and often requires sophisticated modeling effort [7, 21, 23, 24].In this paper, we carry out nanoindentation
experiments to characterize the mechanical properties of single-crystal Si nanolines
(SiNLs). These SiNLs are fabricated by a process combining electron beam
lithography (EBL) and anisotropic wet etching [25, 26]. They have atomically
flat sidewalls and almost perfectly rectangular cross sections. The measured
indentation load-displacement curves show a peculiar character with a large
displacement burst at a critical load. A finite element model is developed to
simulate the indentation test, which reveals a buckling mechanism of the SiNLs
under indentation. A systematic modeling effort is presented here to elucidate the
effects of the indenter tip location, contact friction, and substrate
deformation on the critical load. The results demonstrate a potential
methodology to study buckling, friction, and fracture of nanomaterials through a
combination of experiments and modeling.


## 2. Experimental

Figure1 shows an
array of parallel SiNLs fabricated by a process combining EBL and anisotropic
wet etching. The fabrication process began with chemical vapor deposition of an
oxide layer on a Si (110) wafer, followed by depositing a thin chromium layer. A
positive photoresist was then spun on the wafer and patterned using an electron
beam exposure system. Pattern transfer from the resist was performed by plasma
etching of the chromium and oxide down to the silicon surface. Subsequently,
the residual resist was removed and tetra-methyl-ammonium hydroxide (TMAH) was
used for anisotropic etching of the Si, with the remaining chromium and oxide
layers as the hard mask. When the EBL patterned lines were aligned with one of
the 〈112〉 directions on the Si surface, parallel SiNLs were produced by
the anisotropic etching along the {111} crystalline planes, with vertical and atomically
flat sidewalls. Finally, the residual chromium and oxide were removed by
chromium etchant and buffered oxide etchant (BOE), respectively. Figure 1(a)
shows highly uniform line width along the longitudinal directions of the SiNLs,
and Figure 1(b) shows a perspective view near one end of the SiNLs where a
trench pattern was specially designed for the purpose of SEM imaging. The line
width of the SiNLs shown in Figure 1 is 74 nm, and the height is 510 nm, corresponding
to an aspect ratio of 6.9 for the rectangular cross section. The pitch distance
is 180 nm. Following the same process, we have successfully fabricated
SiNLs with line widths ranging from 40 nm to 500 nm. The height of the SiNLs
can be controlled by TMAH etching time within the range of 100–2000 nm, and the
length is defined by the exposed area for electron-beam patterning, which was
50 μm for the present study. The high crystal quality and well-defined line geometry,
along with the atomically flat sidewalls and highly uniform line width, make these
SiNLs well suited for accurate experimental measurements and modeling.SEM images of parallel silicon nanolines, with 74 nm line
width and 510 nm height; the pitch distance is 180 nm. (a) A plan view, (b) a
perspective view with60° tilt angle. A trench pattern is specially designed at one end of the line for
the SEM image in (b), showing the cross sections of the nanolines with sharp
edges due to anisotropic etching.
(a)(b)An atomic force
microscope (AFM)-based nanoindentation system (Triboscope by Hysitron,
Inc.) was used to characterize the mechanical properties of the SiNLs. A conically shaped diamond indenter
with the tip radius around 5μm was used to probe the SiNLs, as schematically
illustrated in Figure 2(a). The tip radius was more precisely determined
by scanning electron microscope (SEM) to be 4.6 μm. During each indentation test, the indenter
was placed directly above an array of parallel SiNLs, patterned in a 50 μm by 50 μm area of the wafer surface. The
indenter was brought into contact with the SiNLs, monitored by a build-in transducer.
Subsequently, loading and unloading of the indenter were performed with the force
and displacement measured simultaneously. Figure 2(b) shows three load-displacement
curves obtained from the indentation tests on the 74 nm SiNLs. In the first
test (thick yellow curve in the plot), a small indentation load of 500 μN was applied and subsequently unloaded, which
shows an elastic response with coinciding curves for loading and unloading.
With the indentation load increased beyond 550 μN for the second and third tests, a large
displacement burst was observed. This indicates the occurrence of instability under
indentation. Similar displacement bursts were observed in nanoindentation
experiments of metal films, which were attributed to a dislocation mechanism as
an intrinsic material instability [24]. Here, however, the magnitude of the
displacement burst is much larger (over 60 nm in comparison with a few
nanometers for the metal films). As noted in a previous study [2],
plastic deformation of nanoscale single-crystal Si beams was observed only at
elevated temperatures (> 373 K), thus the dislocation mechanism is not expected to operate at the
room temperature. Furthermore, it was found that, after the displacement burst,
unloading of the indenter fully recovers the displacement, with no observable residual
deformation. This suggests a likely instability mechanism due to buckling of
the SiNLs, to be confirmed by modeling simulations as discussed in the later
sections.(a) Schematic of the
nanoindentation test on parallel SiNLs. (b) A set of three load-displacement
curves from the nanoindentation tests on the 74 nm SiNLs, with no residual
deformation after unloading of the indenter.
(a)(b)Additional
indentation tests with the SiNLs were carried out with various indentation loads,
as shown in Figure3(a). Here, however, irrecoverable residual deformation was observed after
unloading of the indenter. It is noted that the maximum indentation
displacement for each of these tests is greater than 150 nm, while in Figure
2(a) the maximum displacement was less than 100 nm. It is thus conjectured that
the larger indentation displacements in this set caused more significant
bending of the SiNLs after the buckling instability, which in turn induced higher
tensile stresses that eventually fractured the SiNLs. Indeed, debris of
fractured Si were observed, as shown in Figure 3(b) for one
of these tests. Remarkably, the debris of the fractured SiNLs are of isosceles
triangular shape, with two sides well aligned in the 〈110〉 directions and
the base in the 〈112〉 direction parallel to the SiNLs. This indicates a
primary cleavage mechanism of the close packed {111} planes along the
〈110〉 directions [17]. This opens a possibility to study fracture of Si at the nanoscale.(a) A set of five load-displacement curves from the nanoindentation tests on
the 74 nm SiNLs, with irrecoverable displacements due to fracture of the SiNLs.
(b) A SEM image of the fractured SiNLs, with debris of isosceles triangular
shape.
(a)(b)


## 3. Finite Element Model

In this section, we calibrate a finite element model
to simulate the nanoindentation test on SiNLs. The model system consists of a
spherical indenter and parallel SiNLs standing on a substrate, as illustrated
in Figure4. Since the tip radius (R = 4.6 μm) is much greater than the line width (w = 74 nm) and the pitch (S = 180 nm), the conical indenter used in the experiments is effectively modeled
by the spherical indenter of the same tip radius. Furthermore, the indenter is
modeled as a rigid body. This approximation is justified by noting that Young’s
modulus of the diamond indenter (1140⁢GPa) is much greater than that of Si
(e.g., 169⁢GPa in the [110] direction). It was confirmed that a similar model
with an elastic diamond indenter predicts a load-displacement curve nearly
identical to the rigid indenter model, as shown in Figure 5. On the other hand,
the elastic indenter model requires significantly higher computational effort,
with a larger number of elements, especially for the contact between the
indenter and the SiNLs. Next, the Si substrate is modeled as a rigid foundation
to the SiNLs. This approximation is made by considering that the patterned
SiNLs, with air gaps in between, are more compliant than the solid substrate.
The effect of elastic deformation in the substrate will be further discussed in
the next section. Finally, the material of SiNLs is assumed to be linearly
elastic and isotropic. As suggested by Vlassak et al. [27], modeling
indentation on elastically anisotropic materials can be simplified by using an
equivalent isotropic model. To determine the equivalent isotropic elastic
modulus for the SiNLs, we compare the load-displacement curve obtained from the
model to the experimental curves shown in Figure 2(b). With the Poisson’s ratio
fixed as 0.27, we found that a Young’s modulus of 140⁢GPa gives a
load-displacement curve in good agreement with the experimental curve, as shown
in Figure 5. This modulus is lower than the Young’s modulus of bulk Si in the
[110] direction, but higher than that in the [001] direction.Figure 4
Illustration
of the parallel SiNLs under spherical indentation (not to the scale), showing
the cross sections of two lines and the contact forces. The tip alignment is
indicated by an offsetδ from the
trench center.Figure 5
Calibration of
the finite element model using the experimental data. The elastic indenter
model and the rigid indenter model give nearly identical load-displacement
curves, both in good agreement with the experimental data.Three-dimensional finite element (FE) models are
constructed for the SiNLs using ABAQUS [28]. Figure 6 shows an example model,
with the tip of the indenter aligned with a trench center (i.e., δ = 0 in Figure 4). By symmetry, three half-lines
on one side of the tip are modeled for indentation on six parallel SiNLs. The
number of lines and the indenter tip location are varied to investigate their
effects, as discussed in the next section. The length of the SiNLs in the
experiments was 50 μm, much greater than the line width (w = 74 nm) and height (h = 510 nm). In the FE model, the lines are 6 μm long. The length is chosen such that the boundary
condition at the ends has negligible effect on the simulation results,
mimicking infinitely long lines. Only half of each line is modeled, with a
symmetry boundary condition for the cross section at z = 0.Figure 6
A finite
element model of parallel SiNLs under indentation, showing three half-lines
with symmetric boundary conditions at the planesz = 0 and x = 0. The tip
of the indenter moves along the y-axis
during the simulation for a symmetric trench-center loading.To model the indentation load, contact surfaces are
defined between the indenter and the lines. The contact property, either
frictionless or frictional, is specified. As discussed in the next section, the
friction between the indenter tip and the SiNLs is very important in
determining the critical buckling load. On the other hand, contact and friction among
SiNLs are ignored because of the relatively large spacing between the lines (S/w>2). For columnar structures with narrow spacing
such as those in thermal barrier coatings [28, 29], the inter-columnar contact
and friction are critical. To simulate buckling and post-buckling behavior of
the SiNLs, the modified Riks method [30, 31] is adopted for the FE simulations.
The essence of the Riks method is that the solution is viewed as a
single equilibrium path in a space defined by both the nodal displacements and
the loading parameters. This approach solves simultaneously for loads and
displacements, thus providing solutions regardless of whether the response is
stable or unstable. The contact between the indenter and the lines and possible loss of the
contact in the present model often causes divergence in the numerical simulations.
A spring-dashpot element [30]
has been used as a numerical damper to stabilize the numerical simulations,
which effectively adjusts the step size to relieve the sudden force change upon
contact and buckle initiation.


## 4. Results and Discussions

### 4.1. Symmetric Loading with Frictionless Contact

We begin with a simplest model, assuming frictionless
contact between the indenter and the lines. The tip of the indenter is aligned
with a trench center between two lines (i.e.,δ = 0 in Figure 4). The number of lines in the model is varied from
2 to 8 by including the neighboring lines, all placed symmetrically with
respect to the tip location. The condition of symmetry is used to reduce the
number of elements and thus the computational time. Figure 7 plots the indentation
load-displacement curves from the FE simulations. By comparing the curves for
different numbers of lines, we can determine the initiation of multiple
contacts at different indentation depths, as indicated by the vertical arrows
in Figure 7. The first contact occurs between the indenter and the two center
lines, with a very small displacement of the tip from its initial position at
the same height as the lines. The indentation force is zero before the first
contact and increases with the tip displacement afterwards. This part is the
same for all the curves, as the first contact occurs in all the four models.
The curve for the two-line model deviates from the other curves after the tip
displacement reaches 5.9 nm, at which point the indenter makes contact with the
next pair of lines adjacent to the center lines. This second contact is absent
in the two-line model, which gives a more compliant force-displacement curve thereafter.
Similarly, the curve of the four-line model deviates from the other two after
the third contact at the tip displacement of 18.6 nm, and the six-line curve
deviates from the eight-line curve after the fourth contact at the tip displacements
of 38.4 nm. The tip displacements for the initiation of each contact can be
confirmed from a pure geometric consideration using the given radius of the
rigid spherical indenter along with the line width and the pitch distance.Figure 7
Indentation
load-displacement curves from finite element models with different numbers of
lines. The indenter tip is aligned with the trench center between two center
lines. The initiation of contact with the first four pairs of the lines is
indicated by the arrows with numbers 1–4.All the force-displacement curves in Figure7 exhibit
an unstable behavior, with the force decreasing after a critical load. While
the critical load for the two-line model is significantly lower, the other
three models give the same critical load, as the instability occurs after the
second contact but before the third contact. In other words, at the critical
load, only the four lines nearest to the tip of the indenter are in contact and
thus subject to the contact forces. The detail of the instability is elucidated
by the six-line model in Figure 8. Figure 8(a) shows the deformation of the
SiNLs and the stress distribution immediately before the instability, indicated
as point A on the load-displacement curve (see Figure 7). At this point, the
center two lines are bent symmetrically, with the cross section taking a
half-wave shape and nearly perpendicular to the surface of the indenter at the
contact. This is similar to the buckling mode of a column with one end fixed
and the other end constrained of rotation (see the inset). Apparently, the
frictionless contact allows lateral sliding of the lines along the spherical surface, but the
compressing rigid surface constrained the rotation of the top surface of the
lines. Such constraint however is lost in Figure 8(b), corresponding to point B
on the load-displacement curve (see Figure 7). Now, the cross section of the
center two lines takes a quarter-wave shape, similar to the buckling mode of a
column with one end fixed and the other end free of any constraint (see the
inset). The transition of the buckling mode is thus responsible for the
instability shown in Figure 7. The quarter-wave mode is more compliant
(less constrained) than the half-wave mode, leading to the force drop from A to
B in the load-displacement curve. Meanwhile, the next two lines are still bent
in a half-wave mode at point B. A similar transition of the buckling mode for
these two lines gives the second peak in the load-displacement curve. After
that, all four lines nearest to the indenter tip are bent in the quarter-wave
mode, as shown in Figure 8(c). It can be seen that the transition of the
buckling mode is accompanied by significant sliding and partial loss of the
contact between the indenter and the SiNLs.Buckling of
SiNLs and mode transition simulated by the finite element model. The contours show
the distributions of normal stress in the 2-direction. (a)–(c) correspond to
the points A, B, C on the load-displacement curve in Figure7, respectively.
The insets in (a) and (b) show analogous buckling modes in columns.
(a)(b)(c)While the Riks method used in the numerical
simulation gives a single equilibrium path for each model, experimental curves
depend on the control of the load or displacement. If the experiments were under
a displacement control, similar curves as shown in Figure7 would be expected.
Under a load control, however, the equilibrium path in Figure 7 would predict a
displacement burst at the critical load, that is, from point A to point C for
the six-line model. This qualitatively agrees with the experimental curves
shown in Figure 2(b). However, it is noted that, for models with six or more
lines, the displacement burst starts at point A with only four lines in contact
with the indenter but stops at point C with six lines in contact. This implies
a possible dynamic contact process during the displacement burst, not captured
in the present static analysis. This may account for the quantitative discrepancy in
terms of the magnitude of the displacement jump between the experimental curves
in Figure 2(b) and the equilibrium path in Figure 7. The former shows
displacement jumps of over 60 nm, but the latter predicts only about 10 nm for the
simulations with six or eight lines. Interestingly, a much larger displacement
burst is predicted by the four-line model, about 55 nm from A to D. Apparently,
the postbuckling behavior under the load-control experiments is complicated by
the dynamic contact and friction processes, which is beyond the present static
model and will be left for further investigations. In the present study, we
focus on the critical load for the onset of the displacement burst, for which
the static analysis of the present model suffices. Figure 7 shows that a
minimum of four lines is sufficient for the modeling of the indentation test up
to the critical load, because the contact with additional lines occurs only afterwards.
The modeling however predicts a relatively low critical load at about 400 μN, while the critical load from the
experimental curves (see Figures 2(b) and 3(a)) ranges from 480 μN to 700 μN. In the following, we discuss possible causes
of this discrepancy by analyzing the effects of relative tip location, contact
friction, and elastic deformation of the substrate.


### 4.2. Effect of Tip Location

The indentation tests of the present study were not
equipped with sufficient lateral resolution for accurate positioning of the
indenter with respect to individual SiNLs. The possible location of the indenter tip
therefore could vary from a trench center between two lines to the center of
one line (another case of symmetric loading). As illustrated in Figure4, the eccentricity
from the trench center, δ, varies
from 0 to S/2, with S being the pitch distance between two
adjacent lines. Figure 9 plots the critical load as a function of δ from the FE model with six lines. For
the cases of asymmetric loading, six half-lines are meshed with one symmetric
plane at z = 0 (refer to the
coordinates in Figure 6). The contact between the indenter and the lines
remains frictionless for the time being. It is found that shifting the
relative tip location has negligible influence on the initial elastic response
and the critical load. The no-effect on the elastic response before the
critical load may be understood by noting the fact that the radius of the
indenter is significantly larger than the pitch distance, and thus the
variation of the contacts due to a shift of the tip location within the pitch
distance is hardly noticeable by the large indenter. This is consistent with
the experimental curves that follow a nearly identical path up to the critical
load despite the uncertainty of the tip locations. On the other hand, the
contact forces acting on each individual line vary asymmetrically with the tip
location. For the center two lines that contribute the most to the total
indentation force, the contact force is higher on the line closer to the tip
and lower on the other one. Meanwhile, the critical load for each of these
lines also varies with the tip location. As the tip moves closer (or farther)
to the line, the contact angle θ decreases (or increases), and the critical load for the buckling mode
transition of the individual line increases (or decreases). This trend has been
confirmed by FE simulations with only one line under the indenter. Consequently, as the tip
location shifts from the trench center towards the right side, the critical
load increases for the right center lines but decreases for the left center
line, as shown in Figure 9. Together with the redistribution of the contact
forces on each line, the buckling instability occurs almost simultaneously for
the two center lines, and the sum of the two individual critical loads is
nearly independent of the tip location. The contact forces acting on the next
two lines farther away from the tip account for the difference between the sum
of the single-line critical loads and the total critical loads, also nearly independent
of the tip location as a result of similar force redistribution.Figure 9
The critical load of buckling transition
versus the indenter tip location, obtained from the finite element models. The
critical loads for the left and right center lines are obtained from a
single-line model.As a special case, when the tip of the indenter is aligned
exactly with the line center (i.e.,δ = S/2), the line directly underneath
the tip is subjected to pure compression and does not buckle at all. In this
case, the critical load is 500 μN, considerably higher than those for tip
locations off the line center, as noted in Figure 9. A large portion of this
critical load is due to the pure compression of the center line, while the
buckling instability occurs for the two lines immediately adjacent to the
center line. From a single-line model with the indenter tip located 180 nm off
the line center (or, δ = 90 nm for
the left center line), the critical load for each of these two lines is
obtained as 116 μN, twice of which contributes less than half of the total
critical load from the six-line model.


### 4.3. Effect of Friction

As illustrated
in Figure4, the normal contact force acting on the line tends to push the top
of the line sliding against the indenter surface. This has been confirmed by
the FE simulations assuming frictionless contact between the indenter and the
SiNLs (see Figure 8). With friction, however, the sliding would be hindered,
and thus the response of the SiNLs under indentation would be altered. By
using a simple Coulomb friction model [28], we define a frictional contact
surface between the indenter and the SiNLs in the FE models to analyze the effect of the
friction coefficient μ. First, it is
noted that the frictional contact property has negligible influence on
the load-displacement
curve before the critical load, for the coefficients of friction between 0 and
0.02 used in the present study. On the other hand, the critical load for the buckling
transition of the SiNLs increases with the friction coefficient, as shown in Figure
10.
With the indenter tip aligned at the trench center, the critical load increases
from 400 μN for frictionless contact to about 510 μN with a friction coefficient μ=0.012. Clearly, the friction delays the
transition of the buckling model that requires significant sliding of the
contact surfaces. Further increasing the friction coefficient to 0.015 sees a
sudden jump of the critical load to over 800 μN. This happens because of a change in the
scenario. As the sliding of the center two lines is highly constrained by the
frictional force, the transition of buckling mode occurs with the next two
lines farther away from the tip while the center two lines are locked
underneath the indenter without buckling. The total critical load in this case
thus includes a large portion from the compression of the center two lines,
plus the critical load for the next two lines. It is noted that the initial
contact angle between the indenter and the center lines is about 0.0115, close
to the coefficient of friction that locks down the sliding of the center line.
For the next two lines, the initial contact angle is around 0.05, which requires
a much larger coefficient of friction to lock down their sliding.Figure 10
The critical load of buckling
transition as a function of the friction coefficient at the contact, obtained
from the finite element models with tip locations at the trench center, line center, and
3 nm off the line center.When the indenter
tip is aligned with the line center, higher critical loads are obtained except
for the case with the large friction coefficientμ=0.015, as shown in Figure 10. As discussed for the frictionless contact, with
the perfect alignment, the center line is under pure compression and does not
buckle. The friction then delays the transition of the buckling mode for the
two lines closest to the center line, giving higher critical loads for larger
coefficients of friction. For the case of the large friction coefficient (μ=0.015), the critical load for the line
center indentation becomes lower than that for the trench center indentation,
because the former has only one line locked underneath the indenter while the
latter has two. Interestingly, when the location of the indenter tip is
slightly off the line center, the center line can still be locked by the
frictional contact, while it always slides and buckles with a frictionless
contact. This is illustrated in Figure 11, which shows buckling of the SiNLs
from the FE models with the indenter tip aligned 3 nm off the line center, one
with frictionless contact and the other with a coefficient of friction μ=0.002. As shown in Figure 10, the
former gives a critical load close to that under the symmetric trench-center
loading, and the latter gives a critical load close to that under the exact
line-center loading for the same coefficient of friction. Therefore, the
friction at the contact leads to a change in the critical load and its
dependence on the tip location.Simulated
buckling modes of SiNLs under indentation with the indenter tip aligned 3 nm
off the line center. (a) With frictionless contact between the indenter and the
lines, (b) with coefficient of friction,μ=0.002, at the contact.
(a)(b)The critical
loads from the experiments, ranging from 480μN to 700 μN, fall within the range shown in Figure 10 from the FE
modeling. The scattering in the experimental data thus may be attributed to the
combined effect of contact friction and tip location. The contact friction may
vary with local surface conditions for the indenter and the SiNLs, the tip
location varies from test to test due to the lack of lateral resolution for
exact positioning. Therefore, refined experiments with sufficient lateral
resolution would reduce the data scattering and give a better measure of the
friction properties. It is noted that the coefficients of friction used in the
present study are about one order of magnitude lower than those obtained
from a tribological test using a spherical diamond tip (tip radius 20 μm) on
single-crystal Si (100) wafers [32]. The indentation tests on the SiNLs may
offer a new approach to studying friction at the nanoscales.


### 4.4. Effect of Elastic Substrate

In the FE simulations
discussed above, the SiNLs have
been assumed to be supported by a rigid foundation. Consideration of an elastic
substrate requires a significant increase in the number of FE elements and the computational
time. To illustrate the effect of the elastic substrate, a FE model is
constructed with six lines under a symmetric loading at the trench center. The dimensions
of the substrate are 490 nm (height) by 900 nm (width) by 3000 nm (length). The
material properties for the substrate are the same as those for the SiNLs, with
Young’s modulus140⁢GPa and Poisson’s ratio 0.27. The contact surface is
defined as frictionless for this case. Figure 12 compares the load-displacement
curves from the models with the elastic and rigid substrates. Clearly, the
model with the elastic substrate is more compliant, and the critical load for
the onset of instability is lower. A higher Young’s modulus could be used in
the FE model with an elastic substrate to match the load-displacement curve
with the experimental data, which would also increase the critical load. The
final result for the critical load would be close to that of the rigid
substrate model.Figure 12
A comparison of
indentation load-displacement curves from the finite element models of parallel
SiNLs on a rigid substrate and on an elastic substrate, respectively.The stress distribution after the buckling transition,
obtained from the FE model with the elastic substrate, is shown in Figure13.
As expected, stress concentration near the roots of the SiNLs is observed. In
addition, relatively tensile high stresses in the direction parallel to the
SiNLs exist at the top of center line, due to the localized buckling
deformation. The locations of high tensile stresses, along with the specific
crystal orientation and cleavage planes, would be useful for understanding
fracture of the SiNLs observed in the experiments (see Figure 3).Stress
distributions from a finite element model with SiNLs on an elastic substrate
under a symmetric trench center loading. (a) and (b) show contours of the
normal stresses in the 2- and 3-directions, respectively.
(a)(b)


## 4.1. Symmetric Loading with Frictionless Contact

We begin with a simplest model, assuming frictionless
contact between the indenter and the lines. The tip of the indenter is aligned
with a trench center between two lines (i.e.,δ = 0 in Figure 4). The number of lines in the model is varied from
2 to 8 by including the neighboring lines, all placed symmetrically with
respect to the tip location. The condition of symmetry is used to reduce the
number of elements and thus the computational time. Figure 7 plots the indentation
load-displacement curves from the FE simulations. By comparing the curves for
different numbers of lines, we can determine the initiation of multiple
contacts at different indentation depths, as indicated by the vertical arrows
in Figure 7. The first contact occurs between the indenter and the two center
lines, with a very small displacement of the tip from its initial position at
the same height as the lines. The indentation force is zero before the first
contact and increases with the tip displacement afterwards. This part is the
same for all the curves, as the first contact occurs in all the four models.
The curve for the two-line model deviates from the other curves after the tip
displacement reaches 5.9 nm, at which point the indenter makes contact with the
next pair of lines adjacent to the center lines. This second contact is absent
in the two-line model, which gives a more compliant force-displacement curve thereafter.
Similarly, the curve of the four-line model deviates from the other two after
the third contact at the tip displacement of 18.6 nm, and the six-line curve
deviates from the eight-line curve after the fourth contact at the tip displacements
of 38.4 nm. The tip displacements for the initiation of each contact can be
confirmed from a pure geometric consideration using the given radius of the
rigid spherical indenter along with the line width and the pitch distance.Figure 7
Indentation
load-displacement curves from finite element models with different numbers of
lines. The indenter tip is aligned with the trench center between two center
lines. The initiation of contact with the first four pairs of the lines is
indicated by the arrows with numbers 1–4.All the force-displacement curves in Figure7 exhibit
an unstable behavior, with the force decreasing after a critical load. While
the critical load for the two-line model is significantly lower, the other
three models give the same critical load, as the instability occurs after the
second contact but before the third contact. In other words, at the critical
load, only the four lines nearest to the tip of the indenter are in contact and
thus subject to the contact forces. The detail of the instability is elucidated
by the six-line model in Figure 8. Figure 8(a) shows the deformation of the
SiNLs and the stress distribution immediately before the instability, indicated
as point A on the load-displacement curve (see Figure 7). At this point, the
center two lines are bent symmetrically, with the cross section taking a
half-wave shape and nearly perpendicular to the surface of the indenter at the
contact. This is similar to the buckling mode of a column with one end fixed
and the other end constrained of rotation (see the inset). Apparently, the
frictionless contact allows lateral sliding of the lines along the spherical surface, but the
compressing rigid surface constrained the rotation of the top surface of the
lines. Such constraint however is lost in Figure 8(b), corresponding to point B
on the load-displacement curve (see Figure 7). Now, the cross section of the
center two lines takes a quarter-wave shape, similar to the buckling mode of a
column with one end fixed and the other end free of any constraint (see the
inset). The transition of the buckling mode is thus responsible for the
instability shown in Figure 7. The quarter-wave mode is more compliant
(less constrained) than the half-wave mode, leading to the force drop from A to
B in the load-displacement curve. Meanwhile, the next two lines are still bent
in a half-wave mode at point B. A similar transition of the buckling mode for
these two lines gives the second peak in the load-displacement curve. After
that, all four lines nearest to the indenter tip are bent in the quarter-wave
mode, as shown in Figure 8(c). It can be seen that the transition of the
buckling mode is accompanied by significant sliding and partial loss of the
contact between the indenter and the SiNLs.Buckling of
SiNLs and mode transition simulated by the finite element model. The contours show
the distributions of normal stress in the 2-direction. (a)–(c) correspond to
the points A, B, C on the load-displacement curve in Figure7, respectively.
The insets in (a) and (b) show analogous buckling modes in columns.
(a)(b)(c)While the Riks method used in the numerical
simulation gives a single equilibrium path for each model, experimental curves
depend on the control of the load or displacement. If the experiments were under
a displacement control, similar curves as shown in Figure7 would be expected.
Under a load control, however, the equilibrium path in Figure 7 would predict a
displacement burst at the critical load, that is, from point A to point C for
the six-line model. This qualitatively agrees with the experimental curves
shown in Figure 2(b). However, it is noted that, for models with six or more
lines, the displacement burst starts at point A with only four lines in contact
with the indenter but stops at point C with six lines in contact. This implies
a possible dynamic contact process during the displacement burst, not captured
in the present static analysis. This may account for the quantitative discrepancy in
terms of the magnitude of the displacement jump between the experimental curves
in Figure 2(b) and the equilibrium path in Figure 7. The former shows
displacement jumps of over 60 nm, but the latter predicts only about 10 nm for the
simulations with six or eight lines. Interestingly, a much larger displacement
burst is predicted by the four-line model, about 55 nm from A to D. Apparently,
the postbuckling behavior under the load-control experiments is complicated by
the dynamic contact and friction processes, which is beyond the present static
model and will be left for further investigations. In the present study, we
focus on the critical load for the onset of the displacement burst, for which
the static analysis of the present model suffices. Figure 7 shows that a
minimum of four lines is sufficient for the modeling of the indentation test up
to the critical load, because the contact with additional lines occurs only afterwards.
The modeling however predicts a relatively low critical load at about 400 μN, while the critical load from the
experimental curves (see Figures 2(b) and 3(a)) ranges from 480 μN to 700 μN. In the following, we discuss possible causes
of this discrepancy by analyzing the effects of relative tip location, contact
friction, and elastic deformation of the substrate.


## 4.2. Effect of Tip Location

The indentation tests of the present study were not
equipped with sufficient lateral resolution for accurate positioning of the
indenter with respect to individual SiNLs. The possible location of the indenter tip
therefore could vary from a trench center between two lines to the center of
one line (another case of symmetric loading). As illustrated in Figure4, the eccentricity
from the trench center, δ, varies
from 0 to S/2, with S being the pitch distance between two
adjacent lines. Figure 9 plots the critical load as a function of δ from the FE model with six lines. For
the cases of asymmetric loading, six half-lines are meshed with one symmetric
plane at z = 0 (refer to the
coordinates in Figure 6). The contact between the indenter and the lines
remains frictionless for the time being. It is found that shifting the
relative tip location has negligible influence on the initial elastic response
and the critical load. The no-effect on the elastic response before the
critical load may be understood by noting the fact that the radius of the
indenter is significantly larger than the pitch distance, and thus the
variation of the contacts due to a shift of the tip location within the pitch
distance is hardly noticeable by the large indenter. This is consistent with
the experimental curves that follow a nearly identical path up to the critical
load despite the uncertainty of the tip locations. On the other hand, the
contact forces acting on each individual line vary asymmetrically with the tip
location. For the center two lines that contribute the most to the total
indentation force, the contact force is higher on the line closer to the tip
and lower on the other one. Meanwhile, the critical load for each of these
lines also varies with the tip location. As the tip moves closer (or farther)
to the line, the contact angle θ decreases (or increases), and the critical load for the buckling mode
transition of the individual line increases (or decreases). This trend has been
confirmed by FE simulations with only one line under the indenter. Consequently, as the tip
location shifts from the trench center towards the right side, the critical
load increases for the right center lines but decreases for the left center
line, as shown in Figure 9. Together with the redistribution of the contact
forces on each line, the buckling instability occurs almost simultaneously for
the two center lines, and the sum of the two individual critical loads is
nearly independent of the tip location. The contact forces acting on the next
two lines farther away from the tip account for the difference between the sum
of the single-line critical loads and the total critical loads, also nearly independent
of the tip location as a result of similar force redistribution.Figure 9
The critical load of buckling transition
versus the indenter tip location, obtained from the finite element models. The
critical loads for the left and right center lines are obtained from a
single-line model.As a special case, when the tip of the indenter is aligned
exactly with the line center (i.e.,δ = S/2), the line directly underneath
the tip is subjected to pure compression and does not buckle at all. In this
case, the critical load is 500 μN, considerably higher than those for tip
locations off the line center, as noted in Figure 9. A large portion of this
critical load is due to the pure compression of the center line, while the
buckling instability occurs for the two lines immediately adjacent to the
center line. From a single-line model with the indenter tip located 180 nm off
the line center (or, δ = 90 nm for
the left center line), the critical load for each of these two lines is
obtained as 116 μN, twice of which contributes less than half of the total
critical load from the six-line model.


## 4.3. Effect of Friction

As illustrated
in Figure4, the normal contact force acting on the line tends to push the top
of the line sliding against the indenter surface. This has been confirmed by
the FE simulations assuming frictionless contact between the indenter and the
SiNLs (see Figure 8). With friction, however, the sliding would be hindered,
and thus the response of the SiNLs under indentation would be altered. By
using a simple Coulomb friction model [28], we define a frictional contact
surface between the indenter and the SiNLs in the FE models to analyze the effect of the
friction coefficient μ. First, it is
noted that the frictional contact property has negligible influence on
the load-displacement
curve before the critical load, for the coefficients of friction between 0 and
0.02 used in the present study. On the other hand, the critical load for the buckling
transition of the SiNLs increases with the friction coefficient, as shown in Figure
10.
With the indenter tip aligned at the trench center, the critical load increases
from 400 μN for frictionless contact to about 510 μN with a friction coefficient μ=0.012. Clearly, the friction delays the
transition of the buckling model that requires significant sliding of the
contact surfaces. Further increasing the friction coefficient to 0.015 sees a
sudden jump of the critical load to over 800 μN. This happens because of a change in the
scenario. As the sliding of the center two lines is highly constrained by the
frictional force, the transition of buckling mode occurs with the next two
lines farther away from the tip while the center two lines are locked
underneath the indenter without buckling. The total critical load in this case
thus includes a large portion from the compression of the center two lines,
plus the critical load for the next two lines. It is noted that the initial
contact angle between the indenter and the center lines is about 0.0115, close
to the coefficient of friction that locks down the sliding of the center line.
For the next two lines, the initial contact angle is around 0.05, which requires
a much larger coefficient of friction to lock down their sliding.Figure 10
The critical load of buckling
transition as a function of the friction coefficient at the contact, obtained
from the finite element models with tip locations at the trench center, line center, and
3 nm off the line center.When the indenter
tip is aligned with the line center, higher critical loads are obtained except
for the case with the large friction coefficientμ=0.015, as shown in Figure 10. As discussed for the frictionless contact, with
the perfect alignment, the center line is under pure compression and does not
buckle. The friction then delays the transition of the buckling mode for the
two lines closest to the center line, giving higher critical loads for larger
coefficients of friction. For the case of the large friction coefficient (μ=0.015), the critical load for the line
center indentation becomes lower than that for the trench center indentation,
because the former has only one line locked underneath the indenter while the
latter has two. Interestingly, when the location of the indenter tip is
slightly off the line center, the center line can still be locked by the
frictional contact, while it always slides and buckles with a frictionless
contact. This is illustrated in Figure 11, which shows buckling of the SiNLs
from the FE models with the indenter tip aligned 3 nm off the line center, one
with frictionless contact and the other with a coefficient of friction μ=0.002. As shown in Figure 10, the
former gives a critical load close to that under the symmetric trench-center
loading, and the latter gives a critical load close to that under the exact
line-center loading for the same coefficient of friction. Therefore, the
friction at the contact leads to a change in the critical load and its
dependence on the tip location.Simulated
buckling modes of SiNLs under indentation with the indenter tip aligned 3 nm
off the line center. (a) With frictionless contact between the indenter and the
lines, (b) with coefficient of friction,μ=0.002, at the contact.
(a)(b)The critical
loads from the experiments, ranging from 480μN to 700 μN, fall within the range shown in Figure 10 from the FE
modeling. The scattering in the experimental data thus may be attributed to the
combined effect of contact friction and tip location. The contact friction may
vary with local surface conditions for the indenter and the SiNLs, the tip
location varies from test to test due to the lack of lateral resolution for
exact positioning. Therefore, refined experiments with sufficient lateral
resolution would reduce the data scattering and give a better measure of the
friction properties. It is noted that the coefficients of friction used in the
present study are about one order of magnitude lower than those obtained
from a tribological test using a spherical diamond tip (tip radius 20 μm) on
single-crystal Si (100) wafers [32]. The indentation tests on the SiNLs may
offer a new approach to studying friction at the nanoscales.


## 4.4. Effect of Elastic Substrate

In the FE simulations
discussed above, the SiNLs have
been assumed to be supported by a rigid foundation. Consideration of an elastic
substrate requires a significant increase in the number of FE elements and the computational
time. To illustrate the effect of the elastic substrate, a FE model is
constructed with six lines under a symmetric loading at the trench center. The dimensions
of the substrate are 490 nm (height) by 900 nm (width) by 3000 nm (length). The
material properties for the substrate are the same as those for the SiNLs, with
Young’s modulus140⁢GPa and Poisson’s ratio 0.27. The contact surface is
defined as frictionless for this case. Figure 12 compares the load-displacement
curves from the models with the elastic and rigid substrates. Clearly, the
model with the elastic substrate is more compliant, and the critical load for
the onset of instability is lower. A higher Young’s modulus could be used in
the FE model with an elastic substrate to match the load-displacement curve
with the experimental data, which would also increase the critical load. The
final result for the critical load would be close to that of the rigid
substrate model.Figure 12
A comparison of
indentation load-displacement curves from the finite element models of parallel
SiNLs on a rigid substrate and on an elastic substrate, respectively.The stress distribution after the buckling transition,
obtained from the FE model with the elastic substrate, is shown in Figure13.
As expected, stress concentration near the roots of the SiNLs is observed. In
addition, relatively tensile high stresses in the direction parallel to the
SiNLs exist at the top of center line, due to the localized buckling
deformation. The locations of high tensile stresses, along with the specific
crystal orientation and cleavage planes, would be useful for understanding
fracture of the SiNLs observed in the experiments (see Figure 3).Stress
distributions from a finite element model with SiNLs on an elastic substrate
under a symmetric trench center loading. (a) and (b) show contours of the
normal stresses in the 2- and 3-directions, respectively.
(a)(b)


## 5. Summary

A buckling instability is observed in the
nanoindentation tests of the parallel silicon nanolines. A systematic analysis
of the buckling modes is presented based on finite element modeling. In particular,
the effects of indenter tip location and contact friction on the critical load
are discussed as possible causes for the scattering of the experimental data.
The result from the present study demonstrates a potential methodology to study
buckling, friction, and fracture at the nanoscale via a combination of
experiments and modeling.



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*Source: 132728-2008-04-14.xml*