Letter pubs.acs.org/NanoLett
Compression of Nanowires Using a Flat Indenter: Diametrical Elasticity Measurement Zhao Wang,*,†,‡ William M. Mook,‡,§ Christoph Niederberger,‡ Rudy Ghisleni,‡ Laetitia Philippe,‡ and Johann Michler‡ †
Frontier Institute of Science and Technology, Xi’an Jiaotong University, 710054, Xi’an, China EMPA, Swiss Federal Laboratories for Materials Testing and Research, Laboratory for Mechanics of Materials and Nanostructures, Feuerwerkerstrasse 39, CH-3602 Thun, Switzerland § Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, United States ‡
S Supporting Information *
ABSTRACT: A new experimental approach for the characterization of the diametrical elastic modulus of individual nanowires is proposed by implementing a micro/nanoscale diametrical compression test geometry, using a flat punch indenter. A 250 nm diameter single crystal silicon nanowire is compressed inside of a scanning electron microscope. Since silicon is highly anisotropic, the wire crystal orientation in the compression axis is determined by electron backscatter diffraction. In order to analyze the loaddisplacement compression data, a two-dimensional analytical closed-form solution based on a classical contact model is proposed. The results of the analytical model are compared with those of finite element simulations and to the experimental diametrical compression results and show good agreement. KEYWORDS: Nanoindentation, indentation, nanowires, mechanical test, EBSD, finite element normal to the flat punch is attenuated, since the experiments are conducted inside of a scanning electron microscope (SEM), while the misalignment due to the nonintersection of the nanowire axis and the flat punch axis is not as critical as in the case of vertical wire indentation, since the indenter used is a flat punch with a diameter that is much larger than the diameter of the nanowire. Classical contact theory is well suited to diametrical compression data analysis. This allows one to solve this problem in a closed form in the two-dimensional (2D) simplified case, determining the lateral elastic modulus of the wire from the load−displacement data obtained experimentally. This test geometry is inspired by the macroscale Brazilian test19 that was introduced back in 1950s to study ceramic fracture behavior. Below we outline a 2D closed-form analytical solution to determine the diametrical elastic modulus of the nanowire compressed with a flat indenter, by analyzing the load− displacement (P−δ) relationship. We define the diametrical elastic modulus as the wire cross-section stiffness. Considering the cross section of the system in two physical dimensions, the total displacement δ measured from the compression experiment consists of three terms:
N
anowires have attracted great interest in the past decade due to their interesting mechanical, electrical, and optical properties.1,2 This has enabled nanowires to gain advantages in a wide range of applications, such as nanoelectromechanical systems,3 sensors,4 transistors,5 light sensors,6 photovoltaic cells,7 and atomic force microscopy (AFM) and scanning tunneling microscopy (STM) tips.8 The main experimental techniques employed to determine the mechanical properties of nanowires are mechanical resonant methods,9 AFM-based bending methods,10,11 tensile testing,12 and nanoindentation methods.13,14 As pointed out by many research groups, nanowire indentation has the great advantage of little specimen preparation; in fact the nanowire does not need to be held in place as with all the other techniques. Not only does the gripping require intensive preparation but it can also influence the measured mechanical response, since the mechanical and adhesive properties of the gripping element are generally neglected. However nanowire indentation has a couple of drawbacks, such as the alignment of the nanoindenter tip with respect to the nanowire15 and the nonapplicability of the Oliver and Pharr method,16,17 since the nanowire cannot be considered as a half-space.18 To overcome these issues, it is introduced in this Letter a diametrical compression test performed at the micro/nanoscale. In this approach, the misalignment defined as the nonperfect perpendicularity between the nanowire axis and the © 2012 American Chemical Society
Received: January 10, 2012 Revised: February 25, 2012 Published: March 20, 2012 2289
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Figure 1. Left: Schematic of the contact model. Right: 2D diagram of an indentation test on a nanowire with cylindrical cross-section. It illustrates qualitatively the strain energy distribution in a typical experimental configuration.
δ = δi + δw + δs
The penetration of the nanowire into the indenter and the substrate leads to a typical problem of calculating the compression of a half-space under a Hertzian pressure, assuming that the dimension of the substrate and the indenter is much larger than that of the contact interface. A classical solution20 to this problem involves integration along the depth of the substrate (hs) or indenter (hi). It yields
(1)
where the subscripts i, w, and s indicate the indenter, the nanowire, and the substrate, respectively. To calculate the deformation, we seek the two components of stress (σx and σy) distributed at an arbitrary point A(x,y) at x = 0 in the cylinder (wire cross section of radius R) under a pressure P (Figure 1). By assuming a Hertzian traction distribution (= 2P(1 − x2/a2)1/2/aπ in case that R ≫ a) at the interface,20 the stress can be written as ⎧ ⎡ ⎤ 2(a 2 + 2y 2 ) 4y ⎥ P⎢1 ⎪ + 2 ⎪ σx = π ⎢ R − 2 2 a ⎥⎦ a a + y2 ⎣ ⎪ ⎨ ⎡ ⎤ ⎪ 2 2 ⎥ ⎪σ = P ⎢ 1 − + 2 2 ⎥ ⎪ y π ⎢⎣ R 2R − y + a y ⎦ ⎩
⎧ ⎛ ⎪ δi = P ⎜2ln πE i* ⎝ ⎪ ⎨ ⎪ P ⎛ ⎜2ln ⎪ δs = πEs* ⎝ ⎩
a=
4PR /πE
(2)
−1 ⎞ 2hs vs ⎞⎤ 1 ⎛ − 1⎟ + − ⎜2ln ⎟⎥ as 1 − vs ⎠⎥⎦ Es* ⎝ ⎠
(3)
where E stands for the interface elastic modulus, which is defined as
(8)
This relationship between the load P and the displacement δ is not totally linear since a depends on P. It is worth noting that the 2D analysis assumes a straight wire with no in-plane curvature in the contact region, this assumption could lead to significant inaccuracy in the case that the indenter size is much smaller than the one used in this study. The compression of nanowires is simulated using the finiteelement (FE) classic contact model, in which frictionless contact and isotropic elastic material properties analogous to the analytical model were assumed. The geometry of the simulated system is shown in Figure 1 right. The bottom of the substrate is fixed in the simulations, and an imposed vertical force (downward) is applied to the top of the punch indenter. The object sizes and the material properties of the simulation system are set with respect to our experimental setup in nanoscale. For the mesh, a plane−strain Lagrangian model is used (approximately 40 000 triangle meshes). The interface is set to be in general contact without friction. The 250 nm diameter Si nanowire used in our experiments was prepared by the vapor−liquid−solid (VLS) method using chemical vapor deposition (CVD) with silane. The substrate was heated to about 580 °C for 10 min. The temperature was then reduced to 510 °C, and a mixture of Ar (10 sccm) and
(4)
where the subscripts iw and sw indicate the indenter−wire and the substrate−wire interfaces, respectively. E* = E/(1 − v2), where E is the elastic modulus and v is the Poisson’s ratio. We note that, in case of anisotropic materials, the diametrical elastic modulus Ew defined in this model is not the component of the elastic modulus in the direction of the loading axis but is strongly correlated to this. The vertical deformation εy at A(x = 0,y) under biaxial stress can be written as εy =
1 ⎛⎜ v ⎞⎟ σy − σx ⎝ E* 1 − v⎠
(5)
The compression of the wire δw is given by integrating εy in each half cylinder ∫ 0Rεydy. The result of the integration (using eqs 1−5) gives δw =
⎞ 2P ⎛ 4R 4R + ln − 1⎟ ⎜ln as πEw* ⎝ a i ⎠
(7)
⎡ 1 ⎛ 2h vi ⎞ 2 ⎛ 4R 4R P = δπ⎢ ⎜2ln i − + ln ⎜ln ⎟+ ⎢⎣ E i* ⎝ ai 1 − vi ⎠ as Ew* ⎝ a i
int
int ⎧ * * ⎪1/ E iw = 1/ E i + 1/ E w ⎨ ⎪ int ⎩1/Esw = 1/Es* + 1/Ew*
vs ⎞ 2hs − ⎟ as 1 − vs ⎠
where h is thickness of the indenter or the substrate. In summary, the load−displacement relationship can be expressed as
where a is the transverse semicontact length (Figure 1), which can be calculated according to a Hertzian pressure distribution at the interface as int
2h i vi ⎞ − ⎟ ai 1 − vi ⎠
(6) 2290
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SiH4 (5 sccm) was introduced for the nanowire growth for 15 min. The nanowire was compressed with a Hysitron PicoIndenter that was built for SEM in situ experiments.21 The experiment was conducted in a Zeiss DSM 962 SEM operated at an accelerating voltage of 10.0 kV. All compressions were conducted such that the substrate tilt was between 8° to 10° with respect to the electron beam. The tip is a boron-doped diamond that was milled by a focus ion beam (FIB) for shaping into a flat-punch geometry. The nominal indenter tip displacement rate was 2 nm/s using a closed feedback loop which adequately represents displacement control during the elastic segment of the compressions. An SEM image of the experimental configuration can be seen in Figure 2a.
Figure 3. Comparison of the load−displacement curves from a nanoindentation experiment (load: elastoplastic and unload: elastic), the analytical model (eq 8), and a FE simulation.
longitudinal contact length is very important since it must be used to normalize the load, P, in eq 8. It should be noted that the contact length within the first 10 nm of displacement in Figure 3 is not constant but increases to the final recorded length of 1.34 μm. This is due to small but unavoidable misalignments between the flat punch, the nanowire, and the substrate. Excluding these first few nanometers, the remainder of the elastic portion of the first compression is then used in order to fit the effective elastic modulus of the nanowire using eq 8. The EBSD mapping confirms that our Si nanowire is a single crystal. The crystal orientation of the nanowire and the underlying substrate are visualized in Figure 2c. The ⟨100⟩ crystal directions of the nanowire (red) and the Si wafer substrate (blue) are displayed in a stereographic projection in Figure 2b. The configuration of the nanowire is drawn schematically into the pole figure. The crystal orientation of the nanowire is such that the compression direction y expressed in the crystal coordinate system is (0.17, 0.92, 0.34). The lateral direction x is (0.87, −0.39, 0.29). According to the measured crystal orientations, the Young’s modulus and the Poisson’s ratio of the nanowire are estimated to be Ew = 147 GPa and vw= 0.233 based on the data of Wortman and Evans.22 In the substrate, the principal direction of the stress varies and strongly depends on the position within the substrate. Therefore, based on indentation results, an average value of 153 GPa has been assumed for the modulus of the substrate.24 Figure 3 shows a comparison between the compression experiment, the theoretical prediction, and the FE simulation. The offset between −10 to −12 nm is associated with irregularities along the wire surface and the fact that the wire is not in perfect contact with the indenter and the substrate surface at the beginning of compression. It can be seen that the experimental elastic unloading curve roughly follows the track predicted by the simulation using the effective Young’s modulus and Poisson’s ratio estimated with EBSD. At the small deformation range, the load−displacement ratio predicted by eq 8 is close to that of the experiment and the simulation. The difference becomes more important when the deformation increases. We note that using the unloading curve is a better operational procedure than choosing the loading one, in order to avoid the influence of inelastic deformation and contact problems at the beginning of compression.
Figure 2. (a) SEM image showing a nanowire lying on the Si wafer substrate and the diamond flat-punch indenter tip. A real-time video recording is available in Supporting Information. (b) Stereographic projection of the ⟨100⟩ directions illustrating the crystal orientation of the Si nanowire (blue dots) and the substrate (red dots). (c) EBSD mapping of the Si nanowire and the substrate. The color code gives the crystallographic orientation of the surface normal.
The analytical model (eq 8) for the deformation of the nanowire requires accurate values of the Young’s modulus E and the Poisson coefficient v for all involved materials. However, the elastic behavior of single crystal silicon is anisotropic and strongly depends on the orientation of the crystal.22 In order to determine the effective values of E and v in our experiments, the crystallographic orientation of the compression axis of the specific nanowire has been determined by means of electron backscatter diffraction (EBSD), which can be used to identify the local crystal orientation with a spatial resolution as good as 10 nm and an angular orientation resolution better than 0.5°.23 The EBSD analysis was conducted on a Hitachi S-4800 cold field emission highresolution SEM equipped with a TSL/EDAX EBSD system at an acceleration voltage of 20.0 kV. The crystal orientation mappings of the Si nanowire and the underlying Si wafer can be seen in Figure 2. In our experiment, two consecutive compressions were conducted on the wire. The first compression was elastic such that there was minimal difference between the loading and unloading slopes of the load−displacement graph seen in Figure 3. A second compression was then conducted at which point plasticity occurred. The plasticity enabled the length of the contact to be estimated as 1.34 μm by subsequent SEM plan-view imaging of the deformed region of the nanowire. This 2291
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measurement and the analytical solution showed good agreement, confirming the feasibility of such experiments to determine the diametrical elastic modulus, which is strongly correlated to the wire elastic modulus component in the compression direction. The stability and reliability of the analytical solution was further proven by comparing with numerical simulation results.
The stability of this simplified Hertzian model has been checked by using FE simulations. Our experimental configuration (flat punch + wire + substrate) was rebuilt in a plane− strain Lagrangian model applied to four different sets of material parameters (see inset in Figure 4). A vertical load was
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ASSOCIATED CONTENT
S Supporting Information *
Real-time video recordings of the compression experiment and the FE simulation. These materials are available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected] Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank S. Christiansen at Max-Planck-Institute for the Science of Light at Erlangen in Germany for providing the nanowires. The authors acknowledge the SFOE (Swiss Federal Office for Energy) project, European Commission within the frame of the FP7 project ROD-SOL (project reference 227497), and the National Basic Research Program of China (grant no. 2012CB619402).
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Figure 4. (a) Left: Geometry meshes used in our FE indentation simulations. Color scale represents the distribution of strain energy (N·m/m). A real-time video recording of FE simulation is available in Supporting Information. (b) Comparison between the load−displacement curves from eq 8 (lines) and those from simulations (symbols) with four different sets of parameters as that shown in the inset: E (GPa) and v (hi = hs = 100R = 10 μm).
applied progressively to the top of the indenter (see Figure 1 right), and the deformation was then measured to compare with that predicted by eq 8. The comparison of the P−δ curves is shown in Figure 4. We can see that the stiffness increases with increasing elastic modulus of the materials. The stiffness obtained by simulations is slightly lower than that predicted analytically when the deformation is small. However, the difference becomes more significant in the case of large deformation. This is due to the fact that the R ≫ a Hertzian assumption is no longer valid in the large deformation case. In conclusion, in situ SEM diametrical compression at the nanoscale has been introduced to measure the elastic modulus of individual nanowires with cylindrical cross sections. The experimental compression load−displacement response of a 250 nm diameter single crystal Si nanowire was used to validate a closed-form analytical solution derived from classical contact mechanical theory, where the experiment geometry is simplified as the compression of an infinite long cylinder between two half-spaces. Well-characterized materials (diamond, silicon wafer, and nanowire) were chosen for the three contact elements in order to have known material property values for comparison. The comparison between the experimental 2292
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