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Review Article

# Mechanics of Crystalline Nanowires: An Experimental PerspectivePUBLIC ACCESS

[+] Author and Article Information
Yong Zhu

Department of Mechanical
and Aerospace Engineering,
North Carolina State University,
Raleigh, NC 27502
e-mail: yong_zhu@ncsu.edu

Manuscript received June 24, 2016; final manuscript received December 11, 2016; published online January 12, 2017. Assoc. Editor: Xiaodong Li.

Appl. Mech. Rev 69(1), 010802 (Jan 12, 2017) (24 pages) Paper No: AMR-16-1054; doi: 10.1115/1.4035511 History: Received June 24, 2016; Revised December 11, 2016

## Abstract

A wide variety of crystalline nanowires (NWs) with outstanding mechanical properties have recently emerged. Measuring their mechanical properties and understanding their deformation mechanisms are of important relevance to many of their device applications. On the other hand, such crystalline NWs can provide an unprecedented platform for probing mechanics at the nanoscale. While challenging, the field of experimental mechanics of crystalline nanowires has emerged and seen exciting progress in the past decade. This review summarizes recent advances in this field, focusing on major experimental methods using atomic force microscope (AFM) and electron microscopes and key results on mechanics of crystalline nanowires learned from such experimental studies. Advances in several selected topics are discussed including elasticity, fracture, plasticity, and anelasticity. Finally, this review surveys some applications of crystalline nanowires such as flexible and stretchable electronics, nanocomposites, nanoelectromechanical systems (NEMS), energy harvesting and storage, and strain engineering, where mechanics plays a key role.

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## Introduction

With the advance of nanotechnology, a plethora of nanostructures such as zero-dimensional (0D) structures like nanoparticles, one-dimensional (1D) structures like nanotubes, nanowires (NWs), and nanofibers, and two-dimensional (2D) materials like graphene and transition metal dichalcogenide monolayers have emerged. Numerous studies have reported that these nanostructures typically possess ultrahigh mechanical strengths, close to their ideal strengths [1,2]. Among them, 1D nanostructures have been used as building blocks for a wide range of applications including nanoelectronics, nanosensors, nanocomposites, energy harvesting/storage, and nanoelectromechanical systems (NEMS). For example, carbon nanotubes (CNTs) have been used in conductive and high-strength composites [3,4]. ZnO NWs have been demonstrated to harvest mechanical energy through piezoelectric effect [5] and used in highly efficient dye-sensitized solar cells [6]. Si NWs have been used to harvest waste heat using thermoelectric effect [7] and can potentially serve as an excellent class of materials as anodes in lithium-ion batteries [8]. SiC NWs have been used in ultrahigh-frequency nanoresonators with high-quality factor [9]. Ag NWs are outstanding conductors and have been successfully used in flexible, transparent electrodes [10,11] and stretchable electrodes [1214].

It is known that material properties can change with elastic strain including electronic band gap, carrier mobility, phononic band gap, thermal transport, ferroic transition, catalytic activity, etc. Hence, the ultrahigh strength offers an unprecedented opportunity to tune the properties of crystalline NWs through the elastic strain engineering [1]. For example, for Si NWs the electron–hole recombination rate can increase sixfold under a strain of 5% [15]; for Ag NWs, axial or bending strain was found to significantly affect the surface plasmon resonance [16,17]. For device applications including elastic strain engineering, it is of critical relevance to measure, understand, and eventually design mechanical properties of 1D nanostructures.

Mechanical properties of materials at the nanoscale significantly deviate from their bulk counterparts. This is true not only for nanostructured materials (e.g., nanocrystalline or nanotwinned materials) but also for nanostructures (surface-dominated structures like NWs). Nanoindentation on bulk crystals or thin films is a well-characterized technique for probing mechanics at the nanometer length scale [18,19], manifesting so-called indentation size effects, i.e., an increase in hardness with decreasing depth of penetration [20]. However, nanoindentation introduces complicated stress state and is ineffective in evaluating the effect of free surfaces on mechanical properties. In the past decade, mechanical properties under uniaxial loading (e.g., using micro/nanopillars with diameters well above 100 nm) have received significant interests, manifesting another type of size effects—“smaller is stronger” (an increase in strength with decreasing diameter) [2125]. A plethora of theories were proposed to explain the observed size effects; a now commonly accepted theory hypothesizes that in such pillars the dislocations multiply and form complex networks through the operation of single-arm sources as supported by dislocation dynamics simulations [22,2629] and in situ transmission electron microscopy (TEM) observations [30]. With further shrinkage of the size into the regime where surface effects become dominant (e.g., ∼100 nm or below), NWs have received much interest. The NWs often exhibit near ideal strength. Metal NWs tend to deform via dislocation nucleation at the surface, glide, and subsequent annihilation at the free surfaces as supported by molecular dynamics (MD) simulations [3133] and in situ TEM observations [34].

In this review, we provide a summary of recent advances in mechanics of crystalline NWs focusing on the experimental aspects. Mechanics of carbon nanotubes has been well reviewed [35], while modeling on mechanics of crystalline NWs has been an extremely active field in the last decade or so and deserves its own review articles [1,36,37]. This review is organized as follows: In Sec. 2, we summarize major experimental methods to measure mechanical behaviors of NWs employing tools such as atomic force microscope (AFM) and electron microscopes. While this review is limited to crystalline NWs, these experimental methods can be readily applied to other 1D nanostructures such as amorphous NWs, nanotubes, and nanofibers. In Sec. 3, we discuss a number of mechanics topics of crystalline NWs including elasticity, fracture, plasticity, and anelasticity. In Sec. 4, we survey some mechanical applications of crystalline NWs ranging from stretchable electronics to energy applications to strain engineering. Before closing, challenges and future prospects of experimental mechanics of crystalline NWs are discussed.

## Experimental Methods

Mechanical testing of 1D nanostructures has seen significant progress but remains challenging due to the fact that their diameters are typically less than 100 nm and lengths ∼10 μm. Apart from a few specific methods, most testing methods involve tension or bending loading mode. The available testing methods can be mainly grouped into two categories based on the instruments involved: AFM/nanoindentation testing and electron microscopy testing. Figure 1 lists the key developments in the testing methods for 1D nanostructures in these two categories in the past two decades (most of these methods are first of their kinds).

A fundamental difference exists between AFM/nanoindenter and electron microscopes—AFM or nanoindenter is an instrumented testing tool in addition to being an imaging tool, while an electron microscope is an imaging tool only. For AFM/nanoindentation testing of 1D nanostructures, the imaging capability is used to locate the position for testing followed with the testing capability. AFM/nanoindenter is limited as the “imaging” is realized by touching the samples, in analogy to the blind. AFM/nanoindentation testing employs commercially available instruments to apply load and measure deformation. During the mechanical testing, AFM or nanoindenter cannot perform the in situ imaging simultaneously but can be easily switched to the imaging mode before or after the mechanical testing.

For electron microscopy testing, a separate mechanical testing tool must be employed inside the electron microscope, which is often custom made. The electron microscopes can provide real-time imaging of defect nucleation and propagation. Hence, in situ electron microscopy mechanical testing is probably the most powerful in elucidating deformation mechanisms. But electron microscopes, either scanning electron microscope (SEM) or TEM, are stilled somewhat limited for imaging as only one electron beam (“eye”) is used.

###### Atomic Force Microscopy and Nanoindenter.

AFM can be operated in mainly four modes for mechanical characterization of 1D nanostructures—(normal) contact mode, lateral force mode, nanoindentation mode, and contact resonance mode. Figure 2 shows the schematics of the four modes. Mechanical properties are extracted from the AFM data based on (1) the continuum beam bending theory (contact and lateral force modes) and (2) the elastic contact theory (AFM nanoindentation mode and contact resonance mode). For an AFM, the cantilever deflection can be measured with a resolution of 0.02 nm, thus the force resolution can reach 0.2 nN (for a typical cantilever stiffness of 10 N/m). The load resolution of a nanoindenter is circa 72 nN, and the position of the tip can be determined with a resolution of ∼0.1 nm [57,58].

###### Contact Mode.

In the contact mode, samples are dispersed randomly over perforation patterns on a substrate; commonly used substrates include anodic aluminum oxide membrane [38], rectangular silicon gratings for AFM scanner calibration (e.g., TGZ-04 from MikroMasch) [59], or patterns fabricated by FIB or microfabrication [60]. In this mode, AFM is used to deflect vertically a suspended sample to obtain the load–deflection signature. Upon contact with the sample, the AFM cantilever deflects up; the cantilever deflection is monitored using a four-quadrant photodiode, which gives rise to the applied load if the cantilever stiffness is known. The sample deflection is equal to the difference between the movement of the piezo-actuator (or substrate for some AFMs) and the cantilever deflection. With the load–deflection curve measured, elasticity theory (e.g., Euler–Bernoulli beam theory) can be used to extract the Young's modulus and the yield or fracture strength of the NW.

More specifically, the contact mode can be implemented in two ways: (1) the deflection at a particular position as a function of the applied force and (2) the deflection profile of the entire NW by scanning the AFM tip along its length at a constant force. For a cantilevered beam and a double-clamped beam, the deflection at the loading point is given by Display Formula

(1a)$d(F,x)=Fx33EI$
Display Formula
(1b)$d(F,x)=Fx3(L−x)33EIL3$

respectively, where x is the distance from the fixed end, F is the applied force, E is the Young's modulus, I is the moment of inertia, and L is the NW length. As an example, Fig. 3(a) shows the deflection measurements on a cantilevered NW following the first way. The cantilever deflection as a function of the vertical piezoposition, at several different positions along the NW length, is plotted in Fig. 3(a) (top). Curve A is obtained when applying the AFM tip directly on the substrate as a reference with the known slope of one, i.e., the cantilever deflection is equal to the movement of the piezo-actuator. With the known cantilever stiffness, the applied force versus the NW deflection is obtained, as shown in Fig. 3(a) (bottom). The slope of the linear fit yields the stiffness of the NW, $k(x)=3EI/x3$. Using this approach, one can check the consistency of the Young's moduli measured at different locations and verify the single-clamped or double-clamped boundary condition.

To obtain the NW deflection profile d(x) following the second way, an image of the NW is acquired under a given nonzero set-point force. The NW deflection profile is then extracted from a cross-sectional cut of the image along the NW length. Figure 3(b) shows the deflection profiles of a cantilever NW and a double-clamped NW. In case that the NW might be initially slacked, a second image is acquired under a zero set-point force, which is subtracted from the first image, leading to the desired deflection profile.

Note that in the example shown in Fig. 3, the NWs were horizontally grown between two facing Si (111) sidewalls of prefabricated microtrenches, resulting in the ideal single-clamped or double-clamped boundary condition for the NWs [43]. However, typically the NWs are dispersed on top of perforation patterns, relying on only van der Waals interaction between the NWs and the substrate, which might cause ambiguous boundary conditions. Even clamping of the NWs by electron beam induced deposition (EBID) of platinum or carbon materials cannot ensure ideal boundary conditions [61,62]. In Sec. 2.4.3, discussions on how to treat such boundary conditions will be given. Another issue in the contact mode is the slippage of the AFM tip off the tested specimen. To mitigate this effect, FIB milling has been used to produce a toothlike groove that can secure the AFM tip on the NW during the scanning [63]. It is also possible that the AFM tip might slip along the NW especially at large bending of the NW. The contact mode has been widely used for testing mechanical properties of CNTs [38], and later for a variety of NWs such as Si, Ag, ZnO, LaB6, and amorphous SiO2 [43,60,6365]. In addition to bottom-up synthesized NWs, AFM has been used to test mechanical properties of top-down fabricated NWs [43].

###### Lateral Force Mode.

In the lateral force mode, two ways of sample preparation have been used. In the first way, samples are randomly dispersed on a substrate and some of them are pinned by microfabricated islands at one end [66]. Then, AFM is used to bend the free end of a cantilevered nanostructure laterally. In this method, it is difficult to eliminate the friction force from the substrate. In the second way, samples are suspended over an FIB-fabricated trench [42]. After a desirable sample across the trench is found, electron beam induced deposition (EBID) of hydrocarbon or platinum can be used to clamp it. It is also possible to prepare suspended samples by etching a trench under a selected sample following the random dispersion [39].

In this mode, the load–deflection signature and hence the Young's modulus and strength of the sample are obtained similar to the contact mode, with the lateral force instead of the vertical contact force. Figure 4 shows the force–displacement curves during a sequence of repeated loading and unloading cycles of a Au NW. Before the mechanical testing, a tapping-mode image of the suspended NW sample was recorded. The first two curves show that the NW was elastically loaded as seen from the linear slope and reversible loading and unloading. Increased loading leads to plastic deformation in the NW. The third and fourth curves showed the same initial linear region followed by a clear turning point corresponding to the yielding. Both the elastic and plastic deformations were confirmed by subsequent AFM images of the NW sample [42].

Typically, the contact mode has better displacement (and force) resolution than the lateral force mode, but slippage of the AFM tip off the NW sample is less likely a problem for the lateral force mode. In the lateral force mode, the AFM tip moves along the centerline orthogonal to the NW sample to prevent possible slippage along the NW. The lateral force mode has been used to test mechanical properties of CNTs and SiC NWs [66], Au NWs [42], Si NWs [67], ZnO NWs [68], and Ge NWs [69].

###### AFM Nanoindentation Mode.

According to the Hertz contact theory, the applied force F is related to the indentation displacement d by Display Formula

(2)$F=43E*R1/2d3/2$

where $E*$ is the reduced Young's modulus $E*=[(1−ν1)/E1+(1−ν2)/E2]−1$ with $Ei$ and $νi$ as the Young's modulus and Poisson's ratio for the AFM tip (i = 1) or the NW (i = 2), respectively, and R is the AFM tip radius. F and d in the AFM nanoindentation mode can be measured similar to those in the contact mode. Hence, the reduced Young's modulus can be measured, giving rise to the Young's modulus of the NW (assuming the Young's modulus of the AFM tip is known). The contact stiffness is given by Display Formula

(3)$k=∂F∂d=(6E*2RF)1/3$

which is often times used to analyze the F–d curve and extract the Young's modulus of the NW. Note that in Ref. [44], NW stiffness $kNW$ was used in Eq. (3), which should be contact stiffness as it involves both the sample and the AFM tip. This method is easy to implement but the analysis is based on several assumptions: (1) The Hertz contact theory is assumed, which is applicable to elastic half space and might not be valid for NWs especially thin NWs; (2) The substrate is assumed to be much more rigid than the NWs, which might not be true for stiff NWs; (3) The radius of the AFM tip is assumed to be spherical; and (4) The NWs are assumed to be isotropic [44]. In the AFM nanoindentation, the force–indentation displacement curve can be converted to an indentation stress–strain curve, from which the plastic behavior can be evaluated [70].

###### Contact Resonance Mode.

In the contact resonance mode, the elastic modulus of a sample can be extracted from the change in the resonant frequency of an AFM cantilever as a result of the sample-cantilever interaction [7174]. Free vibration of an AFM cantilever (Euler–Bernoulli beam) can be described by the dynamic beam equation (i.e., Euler–Lagrange equation) Display Formula

(4)$EI∂4y∂x4+ρA∂2y∂t2=0$

with the boundary conditions

In the above equations, E, I, ρ, and A are the Young's modulus, moment of inertia, mass density, and cross-sectional area of the AFM cantilever, respectively. Solving Eq. (4) with the boundary conditions yields a series of resonance frequencies. When the AFM cantilever is in contact with a sample with the contact stiffness k, the second boundary condition at $x=L$ changes to $∂3y/∂x3=ky/EI$ [71]. Now solving Eq. (4) gives rise to a modified series of resonance frequencies, which depend on the contact stiffness k.

Contact stiffness following the Hertz contact model is also used in this mode, i.e., Eq. (3), as in the AFM nanoindentation mode. Hence, the Young's modulus of the sample can be obtained from the shift of the resonance frequencies. Note that sometimes Eq. (4) includes a damping term, solving which can provide the damping or dissipative property of the contact based on measurement of the vibration amplitude [74].

###### Other AFM-Based Methods.

AFM can be used to directly pull 1D nanostructures or biomolecules, in the so-called force spectroscopy mode. In this mode, the specimen is attached between the AFM tip and a substrate. This technique was used to measure the quantized plastic deformation of gold NWs [75]. Both the AFM tip and the substrate were coated with a thin layer of gold. An atomic chain of gold (or gold NW) was formed when the AFM tip was pressed against the substrate and then pulled out. A piezopositioner was used to move the substrate and pull the NW in tension. The force and NW elongation were measured, similar to the force and NW deflection measured in the contact mode. This method has been used in measuring the mechanical properties of single biomolecules and proteins [76,77].

AFM probe has also been used to directly bend or break NWs lying on a substrate under an optical microscope [7880]. A bent NW can be used to study the friction and shear strength at the NW–substrate interface and electromechanical coupling in the NW. Here, the AFM is used as an actuator (or manipulator) only. Indeed, a micro- or nanomanipulator can be used instead of an AFM to bend NWs [81].

###### Nanoindentation.

Nanoindentation is a widely used method to characterize mechanical properties of bulk materials and thin films [18] and has enabled some early mechanical characterization of NWs, nanobelts, and other nanostructures [40,8284]. A nanoindentation test of NWs using a nanoindenter is similar to the AFM nanoindentation test but different in the force range and related analyses. Note that in this paper by default nanoindentation is meant for nanoindentation using a nanoindenter, to differentiate from AFM nanoindentation. The applied force in the AFM nanoindentation is much smaller such that the NW sample undergoes only the elastic deformation. As a result, the Hertz contact theory is used in the AFM nanoindentation to extract the Young's modulus of an NW sample. But in the nanoindentation test, the NW sample likely undergoes plasticity. Then, the commonly used Oliver–Pharr model is used to extract the Young's modulus and hardness of an NW sample. However, since the indentation of an NW differs significantly from that of a half space in terms of geometry, the Oliver–Pharr model may not be readily applicable [85]. Using finite-element simulation, Shu et al. found that the Young's modulus and hardness from nanoindentation of an NW using the Oliver–Pharr model without corrections may be significantly underestimated [86]. Also because the applied force is relatively large and the NWs are typically thin, the substrate effect might need to be considered [87].

###### Summary.

Table 1 summarizes the main AFM-based testing methods. For both the contact and lateral force modes, the samples are typically dispersed over a hole or trench and sometimes clamped by EBID especially for the lateral force mode. In both cases, the simple beam theory is used to extract the mechanical properties including Young's modulus, yield strength, and fracture strength. For the double-clamped samples, the simple beam theory should be corrected to take into account the axial tension along the samples especially under large strain [88]. In the contact mode, eccentric force might be applied on the NW sample, leading to slippage of the AFM tip off the NW, while in the lateral force mode, the lateral force calibration is more challenging than in the contact mode [8991]. In both modes, the AFM tip might slip along the NW sample especially under large deformation.

For the AFM nanoindentation and contact resonance modes, the samples are simply dispersed on a rigid substrate. In both cases, only Young's modulus can be measured and the data reduction is based on a number of assumptions. For example, the Hertz contact theory is used, which might not apply to the tip-NW contact configuration; in addition, the substrate effect is typically neglected. Also, the AFM tip might slip off the NW sample. Of note is that Si NWs were tested using both the AFM contact mode and the AFM nanoindentation mode. It was found that the contact mode provided more accurate measurement of the mechanical properties [88].

For the force spectroscopy mode, the sample is loaded in tension, and thus, the data reduction is simple. However, this method is not widely used, likely due to the challenge in creating stable contacts between the bottom-up synthesized NWs and the AFM tip or the substrate.

For the nanoindentation by a nanoindenter, the sample preparation is the same as the AFM nanoindentation, but the Oliver–Pharr model is used instead of the Hertz contact model. Both Young's modulus and hardness can be measured, which however can be both underestimated using the Oliver–Pharr model [86]. Also, the substrate could influence the measured mechanical properties.

###### Electron Microscopy.

The major advantage of performing nanomechanical testing inside an electron microscope is the in situ imaging capability. SEM imaging can record the specimen deformation (strain), fracture morphology, and some evidence of plasticity (e.g., shear band), while TEM imaging is more powerful in observing the defect dynamics. A number of testing methods have been developed for mechanical characterization of crystalline NWs inside electron microscopes including bending [9294] and buckling [9597], tension [47,51,52,95,98,99], and vibration/resonance [45,46,50,61]. Most of these methods can trace back to some methods based on AFM or nanoindenter, but with external actuator and load sensor.

###### Bending and Buckling.

Few quantitative bending tests of 1D nanostructures have been reported inside SEM or TEM, which might be because AFM-based bending tests are quite popular. Desai and Haque used a tipless AFM cantilever to bend ZnO NWs [100]. Deflections of the NW and the AFM cantilever were measured simultaneously using the SEM. Adhesion and friction forces were measured in this study. The Young's moduli of the ZnO NWs can be calculated too. Cheng et al. used an MEMS device to bend individual ZnO and Si NWs [101]. They did not investigate the elastic and fracture properties of the ZnO NWs. Instead, they investigated the anelasticity (i.e., time-dependent recovery) during the unloading. Figure 5(a) shows bending test of a ZnO NW in SEM.

Due to the large ratio of length over diameter, an NW typically buckles under compression. The buckling method has been used to measure Young's moduli of several NWs such as ZnO [95], Si [97], and Boron [96]. Buckling is essentially is bending, but is simpler to implement for 1D nanostructures. Similar to the tension test, the setup for the buckling test includes an actuator and a load sensor, with the load sensor much more compliant than that used in the tensile test. Figure 5(b) shows buckling test of a ZnO NW in SEM, and Fig. 6(a) shows the force–displacement curve.

###### Tension.

Tensile testing is the most straightforward among all the mechanical testing methods [103]. In situ TEM tensile testing of top-down fabricated microscale samples (e.g., using focused ion beam) have been recently reported [104106]. However, top-down fabrication of feature sizes of 100 nm or below is challenging, hence for tensile testing most tested nanostructures are bottom-up synthesized. A major challenge in testing such nanostructures is nanomanipulation and high-resolution measurement of force and displacement.

Similar to the method to create thin NWs in the AFM force spectroscopy, thin metal NWs were formed with stable contacts between a metal rod and a scanning tunneling microscope (STM) probe inside TEM, especially under applied voltage [107,108]. Stable contacts can also form between thin metal NWs inside TEM by cold welding [109].

Ruoff and coworkers reported the first tensile testing of individual CNTs inside SEM [47]. Two AFM cantilevers were used in the test with a stiff and a compliant cantilever used as the actuator and the load sensor, respectively. EBID of hydrocarbon was used to create stable contacts between an individual CNT and two AFM tips. This work could be viewed as extending the AFM force spectroscopy into SEM [75].

The stiff AFM cantilever (actuator) used in Ruoff's work can be replaced with an even stiffer and sharp tungsten probe, which, when attached to a nanomanipulator, can be directly used to manipulate 1D nanostructures. Zhu et al. used such a tungsten probe for manipulating and conducting tensile testing of Si NWs in SEM, with a compliant AFM cantilever as the load sensor [51]. Assuming SEM resolution of 1 nm, strain resolution of 0.03% can be obtained for an NW of 3 μm in length. For an AFM cantilever (load sensor) with stiffness of 1 N/m, the load resolution of 1 nN is obtained, leading to stress resolution on the order of 1 MPa for a typical NW with diameter <100 nm. Subpixel resolution can be obtained using digital image correlation (DIC), resulting in better strain and stress resolutions [110]. Figure 5(c) shows tension test of a Si NW in SEM, and Fig. 6(b) shows the stress–strain curve. However, the compliant AFM cantilever (load sensor) could rotate especially under high force, introducing a bending component into the tensile loading condition.

A double-clamped, microfabricated flexure beam has also been used as the load sensor, which can eliminate the end rotation problem of the AFM cantilever. Using this load sensor and a nanomanipulator probe, tensile tests of Cu [52] and Ag NWs were performed.

###### Vibration/Resonance.

Resonance is a simple yet widely used method to measure Young's modulus of 1D nanostructures According to a simple beam theory, the nth mode resonance frequency of a cantilevered beam is given by Display Formula

(5)$ωn=βn2L2EIρA$

where E is the Young's modulus, I is the moment of inertia, L is the beam length, A is the cross-sectional area, and ρ is the beam density. The βn term is the eigenvalue from a characteristic equation in Ref. [111]. When the resonance frequency is measured, the Young's modulus can be calculated according to Eq. (5).

The resonance can be excited by thermal [45], electrostatic [46,50], or mechanical [48,61] means. Figure 5(d) shows resonance test of a ZnO NW in SEM excited by mechanical vibration, and Fig. 6(c) shows the resonance peak from which the resonance frequency can be determined. The NWs can be directly grown on a substrate or clamped onto a probe using EBID. NWs attached to a substrate by van der Waals force only have been tested using the resonance method, which however might lead to underestimation of the Young's modulus [61]. It is important to distinguish so-called forced resonance and parametric resonance. Assuming the fundamental natural frequency ω0, forced resonance occurs at driving frequency ω = ω0, while parametric instability occurs at ω = 2ω0/n (n integer not smaller than 1). More details on the parametric resonance can be found elsewhere [50,112].

###### MEMS-Based Testing Methods.

It is worth discussing MEMS-based mechanical testing methods in a separate section. MEMS consist of micrometer-scale components but offer nanometer displacement and nano-Newton force resolutions. MEMS have been widely used in mechanical testing of thin films [113116]. They can also be used for nanomechanical characterization by controlled actuation with high precision and high speed, high-resolution force/displacement measurements, and integrated multifunctions. In addition, their tiny size makes them ideal for in situ SEM/TEM testing. There has been extensive interest in the past decade in developing MEMS-based stage for in situ SEM/TEM testing of nanostructures [49]. More details on MEMS-based nanomechanical testing have been reviewed elsewhere [117,118]. In this review, only a brief summary of the major advances in this area will be given.

Haque and Saif reported an MEMS stage to characterize nanoscale thin films that are cofabricated with the stage inside SEM and TEM [129,136,167]. The stage was actuated by an external piezo-actuator with a mechanism that can mitigate misalignment. Zhu and Espinosa have developed the first MEMS stage that includes an on-chip actuator and an electronic load sensor with a gap in between [49,121123]. Two types of MEMS actuators were used: thermal actuator for displacement control and comb-drive actuator for force control; the electronic load sensor was based on differential capacitive sensing. The main novelty of their work was the introduction of an electronic load sensor; displacement of the load sensor was measured electronically instead of commonly used microscope imaging. Figure 7(a) shows the MEMS stage with the thermal actuator and the capacitive load sensor. A number of similar MEMS stages have been developed thereafter for testing a wide variety of 1D nanostructures [125132].

MEMS testing stages consisting of two separated capacitive sensors to record both the specimen displacement and load have been developed [123,124,133]. In this case, specimen elongation was also measured electronically (instead of microscope imaging) in addition to load sensor displacement. As a result, such stages are uniquely suited for fatigue tests in situ or ex situ; Fig. 7(b) shows one such stage. Fatigue test of Au ultrathin films (nanobeams) has been conducted using this stage [124].

It is of relevance to characterize the thermomechanical behavior of 1D nanostructures for their device applications. Chang and Zhu recently developed an MEMS thermomechanical stage with an on-chip heater that can heat from room temperature up to 600 K [54]. The MEMS stage consists of a comb-drive actuator, a capacitive load sensor, and a heater based on Joule heating next to the specimen area, as shown in Fig. 7(c). The entire stage is symmetric to ensure that the temperatures on both sides of the specimen are equal so as to prevent temperature gradient and heat flow through the specimen. Note that the capacitive sensor is also in the form of comb drive, identical in geometry to the comb-drive actuator. Chen et al. achieved temperature control for mechanical testing of NWs by placing their MEMS stage inside a vacuum cryostat including a heater, a cooling channel with liquid nitrogen circulation, and a temperature controller [134]. Kang and Saif fabricated a novel MEMS stage for in situ uniaxial testing at high temperatures [135]. The stage was fabricated out of SiC, which can sustain temperature up to 700 °C, much higher than those made of Si.

It is of relevance to investigate the multiphysical coupling of nanostructures, especially how mechanical strain can tune other physical properties—so-called elastic strain engineering. Bernal et al. developed an MEMS stage to characterize electromechanical coupling of NWs, combining four-point electric measurement and tensile loading [136]. Zhang et al. [132] fabricated an electromechanical MEMS stage with a SiO2 layer beneath the structural layer for insulation. Murphy et al. reported thermal conductivity of Si NWs as a function of tensile strain [137]. While the MEMS stage was used to apply tensile strain to the specimen, Raman spectroscopy was used to measure its thermal conductivity. Of note is that most MEMS-based in situ testing has been performed inside SEM or TEM. Integration of MEMS stages with spectroscopy such as Raman and photoluminescence (PL) could offer exciting new opportunities for mechanical testing of nanostructures especially multiphysical testing.

###### Experimental Issues.

Several commonly encountered experimental issues are discussed here. Specific ones such as potential sample heating in MEMS testing stages using thermal actuators [138] are not included.

###### Sample Preparation.

A key but challenging step in nanomechanical testing is to manipulate and position specimens with nanometer resolution and high throughput. For tensile testing, this step becomes even more challenging as the specimens must be freestanding, aligned with the loading direction and clamped at both ends. Methods for manipulation and positioning of 1D nanostructures include “pick-and-place” by a nanomanipulator [49] and dielectrophoresis [131,139] in addition to cofabrication and direct synthesis. EBID of amorphous carbon or platinum is commonly used to clamp the nanostructures. However, recent studies showed that such clamps have certain compliance and could affect the measured mechanical properties [61,62]; see Sec. 2.4.3 for more details. To alleviate this issue, displacement markers can be deposited on the NW for local displacement measurement [55,62,140]. Adhesives (e.g., epoxy) have also been used to clamp nanostructures like polymer nanofibers [141], CNTs [142], and Ni nanobeams [124]. However, in this case individual samples can only be manipulated under an optical microscope, which might limit this method to relatively large sample size. Also, compliance of the adhesives is of potential concern. More details on the sample preparation can be found elsewhere [57,118,143].

###### Vertically Aligned NWs.

For all the experimental methods discussed so far, individual NW samples are either mounted in a freestanding manner or dispersed on a substrate, which require some sample preparation. However, it is quite common that as-synthesized crystalline NWs especially semiconductor and metal NWs are vertically aligned on a substrate. Hence directly measuring the mechanical properties of such as-synthesized NWs would be of great interest to eliminate sample preparation and avoid ambiguous boundary condition. The AFM lateral force mode was used to deflect the vertically aligned ZnO NWs and measure the Young's modulus of individual ZnO NWs [144]. In another case, the substrate with vertically aligned Si NWs was cleaved to expose an array of cantilevered NWs at the cleaved edge. Then, AFM was used to deflect multiple locations along the NW length in the contact mode [145]. As-synthesized ZnO NWs were also excited to resonance by an electrostatic field [50].

Some vertically aligned NWs are grown on a substrate by epitaxy. These NWs strongly bond to the substrate. Bending tests have been used to measure their fracture strengths [93]. In the bending test, an AFM cantilever attached to a nanomanipulator system was used to deflect an NW in the lateral direction. This method was used to measure the bending strength of [111]-oriented Si NWs synthesized by the popular vapor–liquid–solid (VLS) method. Using a rigid nanomanipulator tip instead of the AFM cantilever, Chen and Zhu measured the fracture strain of ZnO NWs under bending [92].

###### Effect of Boundary Condition.

One challenge for AFM contact mode or lateral force mode is the uncertainly of the boundary conditions, which can range from simply supported to doubly clamped. For NWs without EBID-based clamps, the exact boundary condition depends on adhesion between the sample and the substrate and relative stiffness of the sample and the substrate. To overcome this challenge, one method is to probe at multiple locations along the sample length. Therefore, the deflection profile can be obtained instead of isolated midpoint bending measurement. For NWs with small diameters, the deflection profiles were fitted best with doubly clamped boundary condition, while for NWs with large diameters, the deflection profiles were fitted better with simply supported boundary condition [59] (Fig. 8(a)).

Even in the case of EBID-based clamping, the clamps are still relatively compliant. It has been found necessary to investigate the effect of the clamping on the measured mechanical properties. Using the resonance test, Xu and coworkers studied the effect of clamping on the measured Young's modulus of ZnO NWs [61]. The NWs were clamped on a tungsten probe in the configuration of a cantilever by EBID of hydrocarbon for in situ SEM resonance tests. EBID was repeated several times to deposit more hydrocarbons at the same location. The resonance frequency was found to increase with the increasing clamp size until approaching that under the “fixed” boundary condition (Fig. 8(b)). The critical clamp size was fitted into a function of NW diameter and NW Young's modulus, which can provide valuable guidance for testing other NWs using EBID-based clamping.

Gianola and coworkers reported that the stiffness of the EBID clamps commonly used for nanomechanical testing is comparable to that of inorganic NWs [62]. As a result, significant errors can be introduced in measurements of displacement, and hence strain and Young's modulus; the extent of the errors would depend on the stiffness of the sample and the geometry of the clamp. In addition, the clamps can be deformed permanently, which might be misinterpreted as plastic deformation in the NWs.

The elasticity size effect can be generally attributed to two mechanisms [37]: surface elasticity [146150] and bulk nonlinear elasticity (as a result of the surface stress) [151]. Under different loading modes, the elasticity size effect manifests differently for different mechanisms [53,146, 152]. For instance, in the case of surface elasticity, the elasticity size effect would be stronger under bending than under tension as the surface plays a more important role under bending. More specifically, the NW Young's modulus $E=Ec+8(S/D)$ and $E=Ec+4(S/D)$, respectively, under bending and tension, where $Ec$ is the Young's modulus of the core and D is the NW diameter [53]. Therefore, for probing the underlying mechanism of the elasticity size effect, it is of relevance to measure the Young's moduli under different loading modes. Zhu and coworkers measured the elasticity size effect of ZnO NWs under both tension and buckling [53] (Fig. 9); the buckling test should give rise to the Young's modulus under bending. However, in the bulking test the measured Young's modulus sensitively depends on the NW diameter (fourth power in contrast to square in the case of tension), thus the buckling method could lead to larger error in measuring the Young's modulus [112]. Resonance test can be a more accurate method to measure the Young's modulus under bending. Recently, the same group measured the Young's modulus of the same Ag NW using the resonance test followed by the tensile test. After the resonance test, the same NW was transferred to an MEMS stage for the tensile test [102].

## Mechanical Properties of Crystalline Nanowires

Crystalline NWs exhibit considerably different mechanical behaviors from their bulk counterparts as a result of large surface-to-volume ratio. Free surfaces can lead to many interesting properties such as size-dependent elasticity [37,149,151], size-dependent yield strength [153], and surface dislocation nucleation [3234], to name a few.

###### Elasticity.

It would be illustrative to think that the atoms on the free surfaces are created by cleaving a bulk material. As a result, the surface atoms have fewer bonding neighbors, or a lower coordination number, than do atoms in the bulk. Such a coordination number deficiency leads to surface stress and an effective Young's modulus of surface atoms, which is different from that of bulk atoms [37].

The surface atoms can be stiffer or softer than the bulk atoms [149]. The softening effect is primarily due to the bond loss (i.e., loss of neighboring atoms on the surface). In contrast, the stiffening effect can be attributed to the electron redistribution (often called bond saturation) [149,154]. For NWs made of covalent or ionic bonds, the surfaces tend to reconstruct, forming periodic patterns. For these NWs, the surface stresses are typically compressive, which leads to an increase in NW length at equilibrium. At the same time, the surface atoms significantly contract toward the bulk, leading to overall smaller interatomic spacing and volume contraction near the surface [103,154]. For NWs of covalent bonding, the smaller interatomic spacing indicates a higher electron density (assuming bonding electrons are localized in covalent systems), which can contribute to surface stiffening [154]. For NWs of ionic bonding, the smaller interatomic spacing additionally causes stronger (long-range) Coulomb force (electrostatic force), which can also contribute to surface stiffening [98].

The surface stress is typically tensile for FCC metals, which leads to contraction of the NW length at equilibrium. The contraction allows surface atoms to increase their coordination number and their electron density, contributing to stiffening effect. But whether a particular surface is softer or stiffer depends on the competition between bond loss and electron redistribution on the surface [37]. In the case of Cu NWs, Zhou and Huang found that the Young's modulus along $〈110〉$ direction on {100} surface is larger than its bulk counterpart, but smaller along $〈100〉$ direction on {100} surface [149].

In addition to the aforementioned surface elasticity, there is another mechanism that contributes to the overall stiffening or softening of NWs—so-called bulk nonlinear elasticity. Liang et al. found that the axial compressive strain caused by the tensile surface stresses in metallic NWs is large enough to induce a nonlinear increase in the Young's modulus of bulk atoms. For Cu NWs along $〈001〉$, $〈110〉$, and $〈111〉$ crystallographic directions, while the surface is always softer than an equivalently bulk, and the overall NW softening or stiffening is determined by orientation-dependent core elasticity, more specifically, increase in the $〈110〉$ direction but decrease in the $〈001〉$ and $〈111〉$ directions [151].

As a result of the surface stress and surface elasticity, the crystalline NWs can exhibit marked elasticity size effect, i.e., Young's modulus as a function of NW diameter. The surface stress and surface elasticity can influence the mechanical behaviors of NWs under tension, bending, buckling, and other deformation modes [155160].

###### Covalent Bonding.

Several types of semiconductor NWs with covalent bonding have been investigated. Here, Si NWs will be discussed as an example. Zhu et al. reported the quantitative stress–strain measurements of Si NWs using tensile testing for the first time [51]. The Si NWs were synthesized by the VLS process, and the NWs tested ranged from 15 to 60 nm in diameter. Their results showed clear size effects—the Young's modulus decreases with the decreasing diameters when the diameter is below about 30 nm [51]. However, Heidelberg et al. found essentially no size effects [67], while Gordon et al. found a stiffening trend [145]. Figure 10(a) shows the available Young's modulus data of Si NWs from both experiments and simulations.

Several mechanisms have been put forth to account for size effects in elasticity of NWs including: (1) surface effects (surface stress and surface elasticity), (2) nonlinear elastic response of the NW core, and (3) increasing importance of the oxide layer. But for the Si NWs in the work of Zhu et al., high-resolution TEM images showed that there was little or no visible amorphous oxide on the surface [51]. So, the oxide layer likely played a negligible role in the measured Young's modulus of Si NWs. Using density function theory (DFT) calculations of $〈100〉$ Si NWs, Lee and Rudd found that the nonlinear bulk elasticity had a negligible effect on the Young's modulus of Si NWs [161]. Thus, a possible mechanism for the elasticity size effect of Si NWs was the surface elasticity, more specifically, the softening size effect due to the bond loss [149,154]. Of note is that multiscale resonance calculations also predicted similar elastic softening in $〈100〉$ Si NWs in the similar diameter range [172].

###### Ionic Bonding.

The extent of ionic character in a bond is related to the electronegativity difference of the elements in the bond. As in most group II–VI materials, the bonding in ZnO is largely ionic with certain extent of being covalent.

Chen et al. performed resonance tests to measure the Young's modulus of [0001]-oriented ZnO NWs with diameters ranging from 17 to 550 nm [50]. When the diameters of the ZnO NWs were smaller than ∼120 nm, the Young's modulus increased with the decreasing diameter (i.e., elasticity size effect). When the diameters were larger than 120 nm, the Young's modulus approached to that of bulk ZnO. Such a size effect was attributed to a surface stiffening effect as a result of bond length contraction near free surfaces, which extends several layers deep into the bulk and fades off slowly [173].

Agrawal et al. performed in situ TEM tensile tests on [0001]-oriented ZnO NWs using an MEMS testing platform [98]. A similar size effect in Young's modulus to Chen et al. was reported; more specifically, the modulus increased as the diameters decreased below ∼80 nm, while NWs with larger diameters showed a Young's modulus close to the bulk value (140 GPa). MD simulations found that the surface atoms significantly contract toward the bulk during surface reconstruction, leading to overall smaller interatomic spacing near the surface and stronger Coulomb force, which is the main cause of the stiffening size effect. Using DFT-based ab initio calculations, Zhang and Huang found that the stiffening size effect is primarily due to surface bond saturation (electron redistribution) instead of bulk nonlinear elasticity [174]; bulk nonlinear elasticity was proposed by several atomistic simulation studies [175,176]. Surface electron redistribution is known to be a common mechanism for surface elasticity in covalent materials [37,149]. Hence, Zhang and Huang's result is not surprising in view that ZnO bonding is partially covalent.

Xu et al. performed in situ SEM tests on [0001]-oriented ZnO NWs to measure their Young's modulus under both uniaxial tension and buckling [53]. The experimental setup involved a nanomanipulator probe (actuator) and an AFM cantilever (load sensor). Using the same setup, both uniaxial tension and compression (buckling) were conducted. Note when buckled, an NW is effective under bending. It has been reported that different loading modes can manifest the size dependence of the Young's modulus differently, due to the greater influence of surface elasticity under the bending mode [146]. Xu et al. found that for ZnO NWs, the Young's modulus under bending was larger than that under tension, which revealed that the surface has a higher elastic modulus [53]. This result is consistent with the previously results mentioned above that generally favor the surface elasticity effect instead of the nonlinear bulk elasticity effect. Figure 10(b) shows the available experimental data on Young's modulus of ZnO NWs.

Bernal et al. found a similar stiffening size effect in GaN NWs, though much smaller in magnitude than ZnO NWs [177]. The difference was attributed to the lower Young's modulus of ZnO; a material of lower Young's modulus can deform more by the (relaxation-induced) same level of surface stress.