# Mechanical Researches on Young's Modulus of  SCS Nanostructures

Authors: Qinhua Jin; Tie Li; Ping Zhou; Yuelin Wang
Journal: Journal of Nanomaterials
 (2009)

---

## Abstract

Nanostructures of SingleCrystalSilicon (SCS) with superior electrical, mechanical, thermal, and optical properties are emerging in the development of novel nanodevices. Mechanical properties especially Young's modulus are essential in developing and utilizing such nanodevices. In this paper, experimental researches including bending tests, resonance tests, and tensile tests on Young' s modulus of nanoscaled SCS are reviewed, and their results are compared. It was found that the values ofE  measured by different testing methods cannot match to each other. As the differences cannot be explained as experimental errors, it should be understood by taking surface effect into account. With a simplified model, we qualitatively explained the difference in E value measured by tensile test and by resonance test for Si nanobeams.


---

## Body

## 1. Introduction

Being the most important semiconductor used in microelectronics and
microelectromechanical systems, Single Crystalline Silicon (SCS) is still a
research hot spot in nanoscience and nanotechnology, as current trend pushes
sciences and technologies to nanometer scale. Researchers have found thatSi
structures would exhibit unique electrical [1], mechanical [2], thermal [3],
and optical [4] properties at nanoscale. This makes Si nanostructures
(nanobeams and nanowires) attractive in developing Field-effect transistors (FETs),
field-emission devices, chemical sensors, nanoresonators, and photonics, and so forth.
However, in order to make these devices reliable, mechanical properties,
especially Young’s modulus (E) of Si nanostructures are essential and critical
to study.Being defined as the ratio of tensile stress to tensile
strain,E is always considered as an intrinsic material property and
fundamentally related to internal atomic bonding. The stronger the atomic
bonding is, the larger the Young’s modulus is. Besides the direct testing
method, tensile test, there are many other techniques for obtaining E, for
example, bending/curvature, resonance, indentation, and so forth. But in nanoscale, it is
in question that whether these testing methods are still effective, and where
surface effect cannot be ignored.In this review, we present the recent experimental researches
on Young’s modulus of SCS nanostructures and compare theirE results. It was
found that different testing methods showed different values of E. With a
simplified model taking surface effect into account, we qualitatively explained
the difference that for Si nanobeams Et measured by tensile test
are not the same as Er measured by resonance test, it may realize
that the measurement of E in nanoscale is more complex than in macroscale.


## 2. Current Experimental Tests on SCS Nanostructures

Mechanical property measurements of nanostructures, such as
nanobeams and nanowires, are very challenging because of the difficulties in
(1) sample preparation and nanomanipulating and (2) measuring force and
displacement (strain and stress) with nanoscale resolution. But in these years
researchers have demonstrated some mechanical tests on SCS nanobeams and
nanowires, which cover bending tests, resonance tests, and tensile tests.


### 2.1. Bending Tests

Atomic force microscopy (AFM) is usually employed to give 3D
image of the topography of the sample surface, control and apply a specified
amount of force on the sample. It is suitable to use AFM in mechanical tests on
a single- (double-) clamped nanobeam (nanowire) by applying force to the
specimen and measure deformation simultaneously. By using beam bending
equations, the mechanical properties can be deduced. For double-clamped beam
being applied as a force in its middle point,  Young’s modulus can be expressed
as(1)E=l3192Im, where l is the beam/wire length, I is the moment of inertia for the beam cross-section, and m is the gradient of the force-displacement curve during bending
[5].Namazu et al. patterned Si nanobeams by means of
field-enhanced anodization on SOI wafer using AFM and obtained the nanobeams
by anisotropic wet etching, which are <110>-oriented with thickness of
255 nm and widths from 200 to 800 nm. By applying bending tests on these double
clamped nanobeams, the measured values of Young’s modulus are determined to be
169 GPa on average. These results indicate that the specimen size has no
influence onE yet [5].  Sundararajan et al. also followed the experimental
procedure and presented Young’s modulus values of 182±11 GPa [6, 7].Virwani et al. patterned some 200–400 nm-wide193/255 nm-thick Si nanobeam on SOI wafer using e-beam lithography, and their
bending tests by AFM gave an Si Young’s modulus of 174 GPa along <100> direction
[8].Paulo et al. also used AFM to characterize the
mechanical elasticity of their Si nanowires synthesized by Vapor-Liquid-Solid method.
The nanowires are horizontally grown between the two facing Si sidewalls of
microtrenches. The values ofE were estimated to be 186 GPa and 207 GPa, respectively, for single- and double-clamped Si nanowires [9].


### 2.2. Resonance Tests

According to Euler-Bernoulli theory, dynamic studies on the
resonant frequency of nanobeams and nanowires can provide  Young’s modulus when the geometry is determined accurately.In the work of Li et al. [10], the SCS untratine cantilevers
were patterned on SOI wafer and dry etched along <110> crystalline
direction with thicknesses ranging from 12–300 nm. Resonant properties of the
cantilevers were characterized in high vacuum chamber by laser vibrometer. A
pulsed light beam from a laser diode excites the cantilever to resonate and
another fiber-coupled laser to pick up the vibration signal quality factor Q.
The fundamental resonant frequency  f0 was obtained and  Young’s
modulus was deduced using following equation: (2)f0≈0.162(Eρ)1/2tl2,where ρ is the material density, t
is the thickness of the cantilever, and l is the length. The E was fitted to be
68 GPa for 38.5-nm-thick cantilever, and 53 GPa for 12-nm-thick cantilever, as
shown in Figure 1, it demonstrated that  Young’s modulus decreased
monotonously as the cantilevers become thinner [10].(a) Resonant frequency data for a set of
38.5 nm-thick cantilevers (with varied length) are fitted into theoretical curve
based on resonant frequency expression. Young’s modulus of 68 GPa is obtained
from the fitted result. (b) The monotonous decrease in Young’s modulus value of
the ultrathin SCS cantilevers is obtained. The thickness of the cantilevers is
in the range of 300–12 nm. (Reprinted
with permission from [10]. 2003 American Institute of Physics).
(a)(b)


### 2.3. Tensile Tests

Among all the methods ofE in nanoscale, tensile test is more challenging since the specimens must be free-standing, clamped at both ends, stretched uniaxially, and measured its elongation with nanometer resolution. To facilitate tensile testing on nanosturctures, various
nanomanipulators, based on multiaxes actuation, were designed to work inside scanning
electron microscope (SEM) or transmission electron microscopy (TEM). With nanostructure
being stretched and tensile force being measured by nanomanipulators, sample
elongation being observed by SEM or TEM, insitu tensile tests are carried out
inside SEM and TEM.By using a high-resolution TEM equipped inside with
nanomanipulators for sub-nano-Newton force measurements (an AFM) and electronic
conductance measurements (an STM) [13],  Kizuka et al. synthesized <111>-oriented
Si wires with nanometer widths inside the TEM by nanometer-tip contact and successive retraction. Then insitu tensile tests were carried out on the nanowire between two tips of AFM and STM. Figure 2 gives time-sequent series of high-resolution images of a tensile deformation process of an SCS nanowire with
diameter of 4±2 nm.  Young’s modulus was found to be 18±2 GPa
[11].Time-sequent series of high-resolution images
of a tensile deformation process of a single crystalline Si nanowire. The tip of
the cantilever (the left-hand side) is retracted toward the left as indicated
by the bold arrows. (Reprinted with permission from [11]. 2005
American physical Society).
(a)(b)(c)(d)Micro-electro-mechanical systems (MEMSs) can be advantageously
employed in the testing of nanoscale samples. It usually consists of three
parts: actuator for nanomanipulating sample, sensor for measuring force on the
sample, and a cofabricated (or later-attached) sample. By integrating MEMS chip in SEM or TEM, researchers have successfully performed several tensile
tests on nanostructures [14–16].We have designed and fabricated an MEMS tensile-testing chip, a
TEM holder with electric terminals, and carried out insitu TEM tensile testing
on SCS nanobeam [12, 17]. The holder and chip fit well in the 0.9 mm pole piece gap of JOEL 2010. The
chip (5 mm × 9 mm × 0.5 mm) is fabricated by means of Si bulk
micromachining and wafer bonding. Figure 3 gives the SEM view of the MEMS chip, with a sample (SCS
nanobeam), a comb drive actuator, a force sensor beam, and a TEM electron beam
window being integrated into it, and those parts are fabricated in one process.
As the electrostatic force actuated comb drive actuator causes no current, it
will cause no interference to TEM measurements.Figure 3
SEM views of the MEMS tensile-testing chip. The
structures of beams and combs are shown. Inset is the enlarged view of the sample, seen through the electron beam window. The <110>-oriented SCS
thin film is connected on its both ends to the supporting structures. (Reprinted
with permission from [12]. 2008 IEEE).When actuating voltage was applied, with the on-chip comb drive
actuator stretching the SCS nanobeam and insitu TEM observation, tensile tests are performed on the 33μm-long 8.2 μm-broad 90 nm-thick <110>-oriented SCS nanobeam. For
actuating voltages from 50 V to 100 V, incremented in 10 V steps, we snapshot the
movements of the two ends of the nanobeam, A and B (indicated by the arrows),
as shown in Figure 4. By taking the 4 corner marks, which were the images of 4
fixing film clamps in TEM, as reference positions, we measured the
displacements l1 (of A) and l2 (of B). The elongation of
the nanobeam Δl was l2-l1 and the deflection of the force sensor beam can be considered as l1.
The tensile force on the SCS nanobeam was calculated as 210 N/m×l1,
with the elastic constant of the force sensor beam being 210 N/m. By fitting the
strain-stress relationship under different actuating voltages, we obtained a
Young’s modulus of 167 Gpa.Figure 4
TEM bright field (BF) images of the structures
and the SAD pattern of the SCS nanobeam (inset) TEM images of the movements of
the two ends of the nanobeam,A and B (indicated by arrows). (Reprinted with permission from [12]. 2008 IEEE.)


## 2.1. Bending Tests

Atomic force microscopy (AFM) is usually employed to give 3D
image of the topography of the sample surface, control and apply a specified
amount of force on the sample. It is suitable to use AFM in mechanical tests on
a single- (double-) clamped nanobeam (nanowire) by applying force to the
specimen and measure deformation simultaneously. By using beam bending
equations, the mechanical properties can be deduced. For double-clamped beam
being applied as a force in its middle point,  Young’s modulus can be expressed
as(1)E=l3192Im, where l is the beam/wire length, I is the moment of inertia for the beam cross-section, and m is the gradient of the force-displacement curve during bending
[5].Namazu et al. patterned Si nanobeams by means of
field-enhanced anodization on SOI wafer using AFM and obtained the nanobeams
by anisotropic wet etching, which are <110>-oriented with thickness of
255 nm and widths from 200 to 800 nm. By applying bending tests on these double
clamped nanobeams, the measured values of Young’s modulus are determined to be
169 GPa on average. These results indicate that the specimen size has no
influence onE yet [5].  Sundararajan et al. also followed the experimental
procedure and presented Young’s modulus values of 182±11 GPa [6, 7].Virwani et al. patterned some 200–400 nm-wide193/255 nm-thick Si nanobeam on SOI wafer using e-beam lithography, and their
bending tests by AFM gave an Si Young’s modulus of 174 GPa along <100> direction
[8].Paulo et al. also used AFM to characterize the
mechanical elasticity of their Si nanowires synthesized by Vapor-Liquid-Solid method.
The nanowires are horizontally grown between the two facing Si sidewalls of
microtrenches. The values ofE were estimated to be 186 GPa and 207 GPa, respectively, for single- and double-clamped Si nanowires [9].


## 2.2. Resonance Tests

According to Euler-Bernoulli theory, dynamic studies on the
resonant frequency of nanobeams and nanowires can provide  Young’s modulus when the geometry is determined accurately.In the work of Li et al. [10], the SCS untratine cantilevers
were patterned on SOI wafer and dry etched along <110> crystalline
direction with thicknesses ranging from 12–300 nm. Resonant properties of the
cantilevers were characterized in high vacuum chamber by laser vibrometer. A
pulsed light beam from a laser diode excites the cantilever to resonate and
another fiber-coupled laser to pick up the vibration signal quality factor Q.
The fundamental resonant frequency  f0 was obtained and  Young’s
modulus was deduced using following equation: (2)f0≈0.162(Eρ)1/2tl2,where ρ is the material density, t
is the thickness of the cantilever, and l is the length. The E was fitted to be
68 GPa for 38.5-nm-thick cantilever, and 53 GPa for 12-nm-thick cantilever, as
shown in Figure 1, it demonstrated that  Young’s modulus decreased
monotonously as the cantilevers become thinner [10].(a) Resonant frequency data for a set of
38.5 nm-thick cantilevers (with varied length) are fitted into theoretical curve
based on resonant frequency expression. Young’s modulus of 68 GPa is obtained
from the fitted result. (b) The monotonous decrease in Young’s modulus value of
the ultrathin SCS cantilevers is obtained. The thickness of the cantilevers is
in the range of 300–12 nm. (Reprinted
with permission from [10]. 2003 American Institute of Physics).
(a)(b)


## 2.3. Tensile Tests

Among all the methods ofE in nanoscale, tensile test is more challenging since the specimens must be free-standing, clamped at both ends, stretched uniaxially, and measured its elongation with nanometer resolution. To facilitate tensile testing on nanosturctures, various
nanomanipulators, based on multiaxes actuation, were designed to work inside scanning
electron microscope (SEM) or transmission electron microscopy (TEM). With nanostructure
being stretched and tensile force being measured by nanomanipulators, sample
elongation being observed by SEM or TEM, insitu tensile tests are carried out
inside SEM and TEM.By using a high-resolution TEM equipped inside with
nanomanipulators for sub-nano-Newton force measurements (an AFM) and electronic
conductance measurements (an STM) [13],  Kizuka et al. synthesized <111>-oriented
Si wires with nanometer widths inside the TEM by nanometer-tip contact and successive retraction. Then insitu tensile tests were carried out on the nanowire between two tips of AFM and STM. Figure 2 gives time-sequent series of high-resolution images of a tensile deformation process of an SCS nanowire with
diameter of 4±2 nm.  Young’s modulus was found to be 18±2 GPa
[11].Time-sequent series of high-resolution images
of a tensile deformation process of a single crystalline Si nanowire. The tip of
the cantilever (the left-hand side) is retracted toward the left as indicated
by the bold arrows. (Reprinted with permission from [11]. 2005
American physical Society).
(a)(b)(c)(d)Micro-electro-mechanical systems (MEMSs) can be advantageously
employed in the testing of nanoscale samples. It usually consists of three
parts: actuator for nanomanipulating sample, sensor for measuring force on the
sample, and a cofabricated (or later-attached) sample. By integrating MEMS chip in SEM or TEM, researchers have successfully performed several tensile
tests on nanostructures [14–16].We have designed and fabricated an MEMS tensile-testing chip, a
TEM holder with electric terminals, and carried out insitu TEM tensile testing
on SCS nanobeam [12, 17]. The holder and chip fit well in the 0.9 mm pole piece gap of JOEL 2010. The
chip (5 mm × 9 mm × 0.5 mm) is fabricated by means of Si bulk
micromachining and wafer bonding. Figure 3 gives the SEM view of the MEMS chip, with a sample (SCS
nanobeam), a comb drive actuator, a force sensor beam, and a TEM electron beam
window being integrated into it, and those parts are fabricated in one process.
As the electrostatic force actuated comb drive actuator causes no current, it
will cause no interference to TEM measurements.Figure 3
SEM views of the MEMS tensile-testing chip. The
structures of beams and combs are shown. Inset is the enlarged view of the sample, seen through the electron beam window. The <110>-oriented SCS
thin film is connected on its both ends to the supporting structures. (Reprinted
with permission from [12]. 2008 IEEE).When actuating voltage was applied, with the on-chip comb drive
actuator stretching the SCS nanobeam and insitu TEM observation, tensile tests are performed on the 33μm-long 8.2 μm-broad 90 nm-thick <110>-oriented SCS nanobeam. For
actuating voltages from 50 V to 100 V, incremented in 10 V steps, we snapshot the
movements of the two ends of the nanobeam, A and B (indicated by the arrows),
as shown in Figure 4. By taking the 4 corner marks, which were the images of 4
fixing film clamps in TEM, as reference positions, we measured the
displacements l1 (of A) and l2 (of B). The elongation of
the nanobeam Δl was l2-l1 and the deflection of the force sensor beam can be considered as l1.
The tensile force on the SCS nanobeam was calculated as 210 N/m×l1,
with the elastic constant of the force sensor beam being 210 N/m. By fitting the
strain-stress relationship under different actuating voltages, we obtained a
Young’s modulus of 167 Gpa.Figure 4
TEM bright field (BF) images of the structures
and the SAD pattern of the SCS nanobeam (inset) TEM images of the movements of
the two ends of the nanobeam,A and B (indicated by arrows). (Reprinted with permission from [12]. 2008 IEEE.)


## 3. Discussion

The experimental results on Young’s modulus of SCS nanobeams
and nanowires are summarized in Table1 on the catalog of their crystal
directions. As the structures dimension decreasing, the resonance test shows a
decreasing tendency of Young’s modulus on the dimension, which is also shown clearly
in Figure 5. The result of 18 GPa obtained by tensile test on an <111>-directed
SCS nanowire supports this tendency too. All these suggest that for crystal
silicon the size effect of E is decreasing as the dimension decreasing.Table 1
Experimental results ofE for SCS nanobeams (nanowires).
DirectionDimension (nm)Young’s modulus (GPa)Testing methodReferenceNanobeamNanowireWidthThicknessdiameter<110>200255—169Bending[5]200255—182Bending[6, 7]N.A.300—170Resonance[10]N.A.170—135Resonance[10]N.A.38.5—68Resonance[10]N.A.12—53Resonance[10]3000090—167Tensile[12, 17]<100>200193—174Bending[8]<111>——418Tensile[11]N.A.——120186/207Bending[9]Figure 5
Values of Young’s modulus versus
nanostructures critical dimension, withE measured by bending, resonance, and tensile tests.It is also interesting to note that the measured values of Young’s modulus by different
testing methods have a little difference. Size effect did not appear at 200 nm-thick nanobeam in bending test, while in resonance test the tendency curve
indicates that it should appear at around 250 nm. As for tensile test,  Young’s modulus still keeps the bulk value at 90 nm-thick nanobeam.Compared to the
tensile test result,  Young’s modulus by resonance test shows its size effect more early. The different behaviors ofE in tensile and resonance test cannot be treated as experimental error, and we think it must be surface effect
working.Taking nanobeam,
for instance, a model to explain the difference is constructed, in which the nanobeam
is treated as a composite beam with a silicon middle layer and two surface layers. The difference of Young’s modulus caused by the two different methods can
be described as(3)Et−Er=2(1−EsE0)(2hsh−6hs2h2+hs3h3)E0,  where Et is  Young’s
modulus of the composite beam by tensile test, and Er is  Young’s
modulus by resonance test [18].Since many theoretical [19–24] and experimental results
[10, 11] show that the silicon Young’s modulus decreases monotonously as the SCS
nanostructures dimensions decreasing, it can be deduced that ES<E0. The difference between Et and Er is
plotted schematically in Figure 6. When h is much larger than hs,
every testing method gives a same value E0 for Young’s modulus. That
is the case at macro- or microscale. As h decreases, E decreases, and the
value of E by different testing methods will be different from each other. Er shows size effect more early as h decreases, compared with Et.Figure 6
Illustration of the different
values ofEt (in tensile test) and Er (in resonance test)
deduced from a simplified model considering nanobeam as a composite beam with
two surfaces of thickness hs and surface E0.


## 4. Conclusion

In this paper, the existing
experimental tests of Young’s modulus, including bending tests, resonance
tests, and tensile tests, on SCS nanostructures are reviewed. The results suggest
that at nanoscale, the silicon Young’s modulus exhibits obvious size effect of a
decreasing tendency on the nanostructures decreasing dimension. Different
testing methods may give different Young’s modulus values. The size effect of
Young’s modulus appears earlier in resonance test than in the other methods.
By taking surface effect into account, this difference may be qualitatively
explained for SCS nanobeam. It indicates that current experiments are still
not enough to understand the behavior of SCS Young’s modulus at nanoscale thoroughly,
more research work should be done for it.



---

*Source: 319842-2009-05-11.xml*