The Pennsylvania State University

The Graduate School

College of Earth and Mineral Science

FLEXURE STRENGTH AND FAILURE PROBABILITY OF

SILICON NANOWIRES

A Dissertation in

Materials Science and Engineering

by

Rebecca Kirkpatrick

© 2010 Rebecca Kirkpatrick

Submitted in Partial Fulfillment of Requirements for the Degree of

Doctor of Philosophy

August 2010

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The dissertation of Rebecca Kirkpatrick was reviewed and approved* by the following: Christopher L. Muhlstein Associate Professor of Materials Science and Engineering Dissertation Advisor Chair of Committee Joan M. Redwing Professor of Materials Science and Engineering Associate Head for Graduate Studies James H. Adair Professor of Materials Science and Engineering Srinivas Tadigadapa Associate Professor of Electrical Engineering *Signatures are on file in the Graduate School.

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Abstract Silicon nanowires are used in a variety of small scale applications where mechanical

reliability predictions are based on bulk, instead of length-scale dependent materials

properties. This research presents a study of the mechanical behavior of silicon

nanowires using an atomic force microscope to fracture samples in centrally loaded,

fixed-fixed beam bending. Silicon nanowires 43-83 nm in diameter were grown using

the vapor-liquid-solid technique, and the crystallographic orientation of each nanowire

([100], [110], [111], and [112] growth directions) was characterized using electron

backscatter diffraction patterns (EBSD). Nanowires in flexure exhibited large deflection,

nonlinear elastic behavior followed by brittle fracture. Flexure strengths ranged from

5.10 to 20.01 GPa, with an average value of 13.74 GPa, and displayed no clear

dependence on diameter or single crystal orientation. Numerical analyses were also used

to evaluate the effect of the boundary conditions and the implications of weakest link

statistical theories on the measurement of mechanical properties. For the flexible beams

loaded in fixed-fixed bending it is not possible to achieve a high localization of stresses,

therefore there is a lower probability of approaching the theoretical strength of materials.

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Table of Contents List of Figures.................................................................................................................. vii List of Tables ................................................................................................................... xii Acknowledgements ........................................................................................................ xiii Chapter 1 Introduction and Background ......................................................................................... 1 

1.1  Mechanical Characterization of Nanowires ......................................................... 2 1.2  Silicon................................................................................................................... 8 

1.2.1  Silicon Nanoscale Modeling ......................................................................... 9 1.2.2  Silicon Nanoscale Mechanical Properties..................................................... 9 

1.3  Analysis of Brittle Fracture ................................................................................ 10 1.4  Structure of Thesis ............................................................................................. 13 References..................................................................................................................... 14

Chapter 2 Materials and Methods................................................................................................... 20 

2.1  Silicon Nanowires .............................................................................................. 20 2.1.1  Nanowire Growth........................................................................................ 20 2.1.2  Nanowire Release ....................................................................................... 22 2.1.3  Nanowire Manipulation .............................................................................. 25 

2.1.3.1  Microtweezers ..................................................................................... 26 2.1.3.2  Microprobe .......................................................................................... 28 2.1.3.3  Field-Assisted Alignment .................................................................... 31 2.1.3.4  Solution Deposition ............................................................................. 33

2.2  Mechanical Test Fixture..................................................................................... 33 2.2.1  Fixture Design............................................................................................. 34 

2.3  Nanowire Characterization................................................................................. 37 2.3.1  TEM Characterization................................................................................. 38 2.3.2  EBSD Characterization............................................................................... 40 

2.4  Nanowire Preparation......................................................................................... 43 2.5  Mechanical Testing Method............................................................................... 47 

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2.5.1  AFM Cantilever Calibration Methods ........................................................ 48 2.5.1.1  Thermal Noise Calibration Method..................................................... 49 

2.5.2  Asylum AFM Cantilever Calibration Procedure ........................................ 50 2.5.3  Minimizing Instrument Drift....................................................................... 52 2.5.4  Centrally Loaded, Fixed-Fixed Beam Bending Procedure ......................... 54 2.5.5  Centrally Loaded, Fixed-Fixed Beam Bending Analysis ........................... 58 

References..................................................................................................................... 66 Chapter 3 Analytical Modeling........................................................................................................ 69 

3.1  Linear Elastic Analytical Models....................................................................... 70 3.1.1  Off-Center Loading..................................................................................... 70 3.1.2  Ledges ......................................................................................................... 72 

3.1.2.1  Variable Inner Span Error Analysis..................................................... 73 3.1.2.2  Variable Outer Span Error Analysis.................................................... 75 

3.2  Non-linear Analytical Model for Large Deflection............................................ 79 3.2.1  Large Deflection Center Loading ............................................................... 80 3.2.2  Large Deflection Off-Center Loading ........................................................ 82 

References..................................................................................................................... 86 Chapter 4 Results and Discussion.................................................................................................... 87 

4.1  Influence of Adhesive Behavior......................................................................... 87 4.2  Silicon Nanowire Flexure Strength from Force Measurements......................... 93 

4.2.1  Experimental Curve Fits ............................................................................. 93 4.2.2  Deformation and Failure Behavior of Silicon Nanowires ........................ 102 4.2.3  Statistical Analysis.................................................................................... 104 

4.2.3.1  Standard Weibull Analysis ................................................................ 105 4.2.3.2  Adaptation of Flexure Strength to Uniaxial Tensile Testing............. 106 4.2.3.3  Effective Surface Area under Large Deflection ................................ 109 

References................................................................................................................... 119 Chapter 5 Conclusions and Future Work..................................................................................... 121 

5.1  Conclusions ...................................................................................................... 121 5.2  Future Work ..................................................................................................... 122

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Appendix A Fixture Processing......................................................................................................... 124 

A.1  Fixture Masks................................................................................................... 124 A.1.1  Photoresist................................................................................................. 126 A.1.2  Oxide......................................................................................................... 128 

A.2  Fixture Etch Techniques................................................................................... 128 A.2.1  Xenon Diflouride Etching......................................................................... 128 A.2.2  Reactive Ion Etching................................................................................. 133 

A.3  Focused Ion Beam Milled Fixture.................................................................... 139 A.4  TEM Grid with Holey Silicon Nitride Membrane ........................................... 143 References................................................................................................................... 147

Appendix B Linear Elastic Analytical Model: Ledges.................................................................... 148 Appendix C Nonlinear Elastic Analytical Model for Large Deflection: Axial Tension............... 150 Appendix D Nonlinear Elastic Analytical Model for Large Deflection: Elastic Curve ............... 153 Appendix E Maximum Liklihood Estimation for Weibull Parameters ........................................ 155 Appendix F Effective Surface Area under Large Deflection ......................................................... 158 

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List of Figures Figure 2.1 Scanning electron micrographs of silicon nanowires grown from (a) an alumina membrane and (b) an oxidized silicon surface. Images courtesy of Sarah Eichfeld. ............................................................................................................................ 21 Figure 2.2 Scanning electron micrograph of a silicon nanowire released from an alumina membrane using a sodium hydroxide etch. The nanowire is etched and contains remnant membrane.......................................................................................................................... 23 Figure 2.3 Transmission electron micrographs of a silicon nanowire released from an oxidized silicon surface using ultrasonic agitation demonstrating (a) the complete nanowire supported on lacey carbon and (b) smooth sample surfaces at high magnification. ................................................................................................................... 25 Figure 2.4 Optical micrograph showing silicon nanowires extending from an oxidized silicon growth substrate with microtweezers approaching for individual manipulation. The image was captured at an unknown magnification.................................................... 27 Figure 2.5 Scanning electron micrographs of the process involved in individually manipulating a silicon nanowire sample across a fixture test span using a microprobe in the dual beam instrument, including the sample (a) atop of lacy carbon membrane, (b) attached to the tungsten microprobe using a platinum-based deposit, (c) approaching the fixture span, and (d) resting across the device fixture gap. .............................................. 30 Figure 2.6 Optical micrograph of silicon nanowires aligned between two tungsten microprobes using the field-assisted alignment technique. The image was taken at an unknown magnification. ................................................................................................... 32 Figure 2.7 Schematic illustration of a cross-section of the proposed fixture design, shown in on angle, from above, and from the front, where the labels denote (a) the knife edge supports, (b) the TEM transparent window, and (c) the lithographically defined electrodes for field-assisted alignment of the nanowire samples...................................... 34 Figure 2.8 Scanning electron micrographs of the final fixture layout (a) as seen from above and (b) in cross-section........................................................................................... 37 Figure 2.9 Transmission electron micrograph images of a [112] silicon nanowire (a) diameter for accurate dimensional measurements and (b) diffraction pattern used to characterize the sample growth direction. ........................................................................ 39 Figure 2.10 Series of <100> single crystal silicon diffraction pattern images collected at 20 kV accelerating voltage and 18 mm working distance showing (a) a low magnification mode diffraction pattern at 2000 × (b) a distorted high magnification mode diffraction

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pattern at 80,000 × and (c) the high magnification pattern with an array of lines used to create the undistorted pattern shown in (d)....................................................................... 42 Figure 2.11 Schematic representation of the layout used for the determination of nanowire crystal orientation inside the FESEM chamber................................................. 43 Figure 2.12 Scanning electron micrograph of a silicon nanowire fixed across a test span............................................................................................................................................ 45 Figure 2.13 Scanning electron micrograph of silicon nanowires fixed across the testing gap using the platinum-based adhesive, where (a) the adhesive contaminated a significant area around the intended deposit area and (b) the adhesive was deposited away from the edge of the gap, creating a ledge in the test span.............................................................. 47 Figure 2.14 Pre-test AFM tapping mode scan of fixed nanowire................................... 55 Figure 2.15 AFM force curves collected during a fixed three-point bend test which resulted in (a) nanowire fracture and (b) no nanowire fracture. The information is originally collected using deflection volts as a function of the linear variable differential transformer (LVDT) sensor. The data is converted into applied force as a function of nanowire deflection using instrument calibration information. The blue line represents the cantilever approach and extension onto the sample, while the red line shows the cantilever retraction. ......................................................................................................... 57 Figure 2.16 Scanning electron micrograph of a fractured silicon nanowire. .................. 59 Figure 2.17 Scanning electron micrographs showing examples of the three types of nanowire fracture which occurred during experimental testing; (a) center fracture, (b) edge fracture, (c) section fracture. .................................................................................... 60 Figure 2.18 Applied force as a function of nanowire deflection data collected for an entire flexure test. The dashed red lines indicate the area of interest for the nanowire deflection and fracture. ..................................................................................................... 61 Figure 2.19 Applied force as a function of nanowire deflection for a fixed nanowire tested in three-point bending, with the extraneous data eliminated and the axes re-set for the beginning of the nanowire deflection.......................................................................... 62 Figure 2.20 Applied force as a function of nanowire deflection showing the experimental data from Figure 2.19 (solid black line), the linear elastic curve fit (dashed red line), and the non-linear elastic curve fit (dotted blue line) utilizing established analytical theories for the fixed beam mechanical behavior............................................. 64 Figure 3.1 Linear elastic analytical model for the effect of off-center loading on a nanowire in a fixed-fixed bending configuration, including (a) a schematic representation and (b) the resulting elastic curves for increasingly inaccurate load placement............... 72

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Figure 3.2 Linear elastic analytical model for the effect of ledges within the testing span on a nanowire in a centrally loaded, fixed-fixed beam bending configuration, including (a) a schematic representation, (b) the resulting elastic curves as the inner testing span is reduced, and (c) a closer view of the effect of the ledges at the fixed edge. .................... 75 Figure 3.3 Linear elastic analytical model for the effect of ledges on a nanowire in a centrally loaded, fixed-fixed beam bending configuration, including (a) a schematic representation, (b) the resulting elastic curves as the location of nanowire fixation is changed, and (c) a closer view of the effect of the ledges at the fixed edge..................... 78 Figure 3.4 Non-linear elastic analytical model for the effect of large deflection on a nanowire in a centrally loaded, fixed-fixed beam bending configuration with increasing applied load....................................................................................................................... 82 Figure 3.5 Non-linear elastic analytical model for the effect of off-center loading on a nanowire in a fixed-fixed bending configuration with large deflection. Elastic curve results for (a) 5 nN applied load and (b) 5 µN applied load with increasingly inaccurate load placement. ................................................................................................................. 84 Figure 4.1 Applied force as a function of nanowire deflection for all silicon nanowires successfully tested to failure using centrally loaded, fixed-fixed beam bending. ............ 88 Figure 4.2 Applied force as a function of nanowire deflection for nanowires tested to failure shown over (a) the complete test and (b) the area where slip occurred. The dashed blue line represents a test completed exhibiting the predicted force-deflection behavior for a fixed-fixed beam in bending. The solid red line shows a test where the platinum-based adhesive has slipped and subsequently yielded, resulting in inaccurate deflection data.................................................................................................................................... 90 Figure 4.3 Scanning electron micrographs illustrating (a) overlap of two fractured ends of a nanowire test sample after fracture and (b) secondary fracture of the tested nanowire sample. Both results of testing were caused by slip or yielding of the platinum-based adhesive used to fix the nanowire sample across the testing gap. .................................... 92 Figure 4.4 Scanning electron micrograph of NW1 after fracture. .................................. 94 Figure 4.5 Applied force as a function of nanowire deflection for NW1 illustrating (a) raw data collected during centrally loaded, fixed-fixed bend testing and (b) the elastic curve fits based on measured nanowire deflection. .......................................................... 95 Figure 4.6 Scanning electron micrograph of NW2 after fracture. .................................. 97 Figure 4.7 Applied force as a function of nanowire deflection for NW2 illustrating (a) raw data collected during centrally loaded, fixed-fixed bend testing and (b) the elastic curve fits based on measured nanowire deflection ........................................................... 98

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Figure 4.8 Applied force as a function of nanowire deflection for (a) NW1 and (b) NW2. The solid black lines are the experimentally measured force and deflection. The dashed blue lines are the nonlinear elastic curve fit, using calculated values of nanowire deflection......................................................................................................................... 101 Figure 4.9 Plot of silicon fracture strength in bending as a function of (a) nanowire radius and (b) nanowire growth direction, as determined through EBSD...................... 104 Figure 4.10 Weibull plot of the experimental results for silicon nanowires tested in centrally loaded, fixed-fixed bending configuration. The series of fracture strength values determined using the experimentally measured force and calculated deflection are shown using black squares. The series of fracture strength values determined using both the experimentally measured force and deflection are shown using blue circles. .......... 106 Figure 4.11 Weibull plot showing comparison of experimentally evaluated flexure strength (σflexure, black squares) to equivalent tensile strength (σtensile,SS, blue circles) derived using the effective surface area calculation for a simply supported beam bending configuration. .................................................................................................................. 109 Figure 4.12 Effective surface area for model silicon nanowire. The solid line represents the SE,total calculated with the new model, accounting for both the bending and axial tension components. The dotted line represents the SE,bending, which only accounts for the bending tension in the nanowire. .................................................................................... 112 Figure 4.13 Information from NW1 used to interpret the effective surface area for a centrally loaded, fixed-fixed nanowire in bending, including (a) the experimental applied force as a function of deflection and (b) the elastic curve. ............................................. 114 Figure 4.14 Weibull plot showing comparison of experimentally evaluated flexure strength (σflexure, black squares) to equivalent tensile strength (σtensile,FF, blue circles) derived using the effective surface area calculation for a centrally loaded, fixed-fixed beam bending configuration. .......................................................................................... 116 Figure 4.15 Effective surface area as a function of applied force showing the dependence of SE on Weibull modulus for a model nanowire with increasing applied load.................................................................................................................................. 117 Figure A.1 Optical micrograph of the fan mask patterned in photoresist on a silicon wafer with an expansion view to illustrate the thin lines that are used for nanowire mechanical evaluation. The images were captured at unknown magnification............. 125 Figure A.2 Optical micrograph of one device on the bridge mask patterned in photoresist on a silicon wafer. The functional region of the device is located between the two thin lines in the center of the image. ........................................................................ 126

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Figure A.3 The (a) 15 µm2 3D rendering and (b) line profile for an AFM tapping mode image of the fixture support columns created using a 1 minute total time XeF2 etch. The scans were collected using a DI 3000 Nanoscope. ......................................................... 131 Figure A.4 Scanning electron micrograph images of the XeF2 etch sequence, taken after (a) 30 seconds, (b) 1 minute, (c) 2 minutes, and (d) 3 minutes of etch were completed.132 Figure A.5 Series of schematics illustrating the formation of the peaked support columns using a combination of anisotropic and isotropic etch techniques with a photoresist mask.......................................................................................................................................... 134 Figure A.6 AFM tapping mode line profile for a bridge mask sample. The trenches were creating using the combination of anisotropic and isotropic etching..................... 135 Figure A.7 Scanning electron micrographs illustrating the formation of the support columns with (a) 2 minute (b) 3 minute and (c) 4 minute anisotropic etch times. ......... 137 Figure A.8 Scanning electron micrograph images illustrating the progression of the isotropic etch. Image (a) is the starting point, where only the anisotropic etch has been completed. (b) and (c) occur as the polymer layer builds on the upper walls of the pillars, confining the majority of the etch to the lower half of the fixture until the columns pinch off at the base (d). ........................................................................................................... 139 Figure A.9 Optical micrograph of a thermo-mechanical fatigue sample. The interim FIB fixture is developed in between two of the large gold pads............................................ 140 Figure A.10 (a) Design schematic and (b) scanning electron micrograph of the interim FIB fixture. In (b) the fixture design is milled into the wafer on the top of the image and tungsten lines are deposited using ion beam deposition to connect the fixture to the gold lines from the existing structure (Figure A.9)................................................................. 141 Figure A.11 Scanning electron micrograph of a TEM grid coated with holey silicon nitride membrane showing (a) the entire gird area and (b) a closer view of the individual holes. ............................................................................................................................... 144 Figure A.12 Scanning electron micrograph of a silicon nanowire sample fixed across a hole in the silicon nitride membrane............................................................................... 145 

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List of Tables Table 1-1 Elastic modulus values of single crystal silicon according to crystallographic growth direction [77]. ......................................................................................................... 8 

Table 1-2 Review of experimental testing results for elastic modulus of silicon nanoscale samples............................................................................................................. 10 

Table 1-3 Review of experimental testing results for strength of silicon nanoscale samples.............................................................................................................................. 10 

Table 3.1 Overview of the different linear elastic analytical model configurations and the total resulting deflection associated with each............................................................ 78 

Table 3.2 Overview of the different linear elastic analytical model configurations and the accumulated error in the final deflection measurement. ............................................. 79 

Table 3.3 Results for the non-linear analytical model of silicon nanowire in a centrally loaded, fixed-fixed beam bending configuration, accounting for large deflection and increasing applied loads.................................................................................................... 81 

Table 3.4 Overview of the measured deflection error associated with increasing applied off-center loading using the non-linear elastic analytical model. ..................................... 83 

Table 4-1 Summary of flexure strengths from silicon nanowires experimentally tested in centrally loaded, fixed-fixed beam bending.................................................................... 103 

Table 4-2 Summary of effective surface area and strength for several experimentally tested silicon nanowires examples. ................................................................................. 115 

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Acknowledgements I would like to thank my thesis advisor, Dr. Christopher Muhlstein, for his guidance and

encouragement. I would also like to thank my dissertation committee, Dr. Redwing, Dr.

Adair, and Dr. Tadigadapa, for their time and consideration. For the significant technical

assistance provided over the course of this research, thanks in particular to Trevor Clark,

Nik Duarte, and David Sarge at Penn State and to Ed Fuller, Steve Stranick, and Koo-

Hyun Chung at NIST. Financial support was provided by the National Science

Foundation.

To my friends and ever-expanding family, thanks for helping me enjoy life during my

time at school. Thank you to my parents for your endless patience, encouragement, and

support. Especially for your patience. I cannot express how grateful I am for all that you

have given me. Finally, I would like to thank Ryan for challenging and supporting me,

both in work and in life. Your belief in me (and persistence) is the reason I am finally

finished.

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1 Introduction and Background One dimensional structures such as nanowires and nanotubes can function as building

blocks for nanoscale electronic and mechanical devices and allow for higher device

packing densities than many current conventional fabrication methods. An extensive

collection of small scale device applications have been introduced and a wide variety of

devices are now commercially available, including accelerometers, optical switches,

pressure sensors, ink-jet systems, and micro-pumps for biomedical devices [1, 2].

Naturally, reliability and the ability to predict behavior while in service are critical for the

performance of commercial devices. In many cases the small size is supposed to provide

for unique mechanical behavior, but often there is little supporting empirical evidence for

such claims.

As the size of materials decreases from the bulk to the nanoscale, intrinsic material

properties can change. Several of these properties are enhanced, others degrade, and

some do not appear to be affected. To complicate matters, the effects are different

between materials systems. Due to the existence of size effects, it may not be possible to

accurately predict the performance of micro- and nanoscale devices using conventional

theories. Therefore, there is a need for testing at the nanoscale in order to determine the

trend in properties with the reduction in sample size. Computational and theoretical

simulations are routinely applied to predict material behavior, however these models are

only as good as the constitutive material models which they draw upon.

Experimental measurement of the mechanical properties of nanowires poses multiple

challenges, including the manufacture of a test fixture, the fabrication of similar test

specimens, the manipulation of samples into correct locations and alignment, and the

need for high resolution force and displacement sensing. The inability to visualize the

sample and unknown boundary conditions in many configurations adds to the difficulty.

There have been significant developments in the instrumentation used for nanoscale

research, however there are still considerable challenges to adapting the techniques for

dependable and repeatable mechanical characterization of nanowires.

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The two primary objectives of this study are to 1) establish a methodology to reliably

evaluate the flexure strength of nanowires and 2) determine if the experimentally

measured flexure strength of a nanowire can approach the theoretical limits established

for the material. The research presented in this dissertation will use silicon as a baseline

material system. Mechanical evaluation will be completed for the centrally loaded, fixed-

fixed beam bending configuration using an atomic force microscope to interpret the

applied force and resulting nanowire deflection. Each silicon nanowire sample will be

loaded to fracture and experiments will focus on establishing fracture strengths.

The following chapter presents a brief introduction to the theories, materials, and

methods involved in the research for the remainder of the study. It begins by introducing

the theories and experimental techniques of nanoscale mechanical evaluation, followed

by an overview of the behavior, properties, and current results relating specifically to

silicon. The remainder of the chapter reviews current ceramic statistical analysis

methodologies and the implications for nanomechanical research.

1.1 Mechanical Characterization of Nanowires Some mechanical properties of materials are intrinsic to each individual system, while

others are sensitive to the testing method and sample size. To complicate matters, it has

become clear that the relationships between size scale, geometry, surface effects,

microstructure, and mechanical properties at the nanoscale do not consistently follow the

theory developed for bulk materials. Computer simulations and experimental testing of

nanoscale materials have been investigated to characterize the possible changes in

material behavior to derive an understanding of reliability and limits of use for

experimental applications.

There have been no standards established to measure mechanical properties at the

nanoscale, with the exception of instrumented indentation (nanoindentation). The

existing literature on the evaluation of individual, freestanding specimens covers a large

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range of materials, instrumented methods, sample sizes, and testing configurations.

Many groups use a scaled version of the standards set for bulk material testing and high

resolution force and displacement measurement systems. Unfortunately, in nanoscale

mechanical tests it is difficult, or in some cases not possible, to directly measure strain in

the specimens. Additionally, accurate dimensional measurements are also increasingly

difficult to make with decreasing size scale. In particular, the cross sectional dimensions

of the specimens may approach the tolerances of the most sensitive of instruments, and

variations of sample dimensions are common. Finally, the boundary conditions of

experiments cannot always be confirmed. As a result, large variability and

inconsistencies exist in measured properties for supposedly identical materials.

The most commonly investigated property of nanoscale materials is the elastic modulus,

E. Elastic modulus is ultimately a measure of the stiffness of the interatomic bonds and

should be invariant for pure materials at a given temperature, with the exception of the

effect of crystal orientation. However, nanowire literature has observed E values that are

both consistent with and different from the bulk. The range in E may be due to

insufficient interpretation of experimental data and boundary conditions, but has also

shown a particular dependence on sample size.

As the size of a sample material is reduced to the nanoscale, the increasing surface-to-

volume ratio that results is the prevailing theory for changes in mechanical properties.

Nanowires have a significantly higher percentage of surface atoms than bulk

counterparts. Surface atoms have fewer bonding neighbors than bulk atoms therefore

charge density is redistributed [3]. This changes the nature of chemical bonding and the

interatomic distances in comparison to the bulk [4], creating differences in stresses and

energies at the surface. As the percentage of surface atoms can be much larger than the

bulk at the nanoscale, the influence of surface properties may significantly contribute to

the overall behavior of the material.

Various methods have been employed to explain the surface effects on mechanical

properties, including surface stresses [5-11], energies [6, 12, 13], and tension [4, 14].

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Each of the studies predicted that surface effects will dominate mechanical properties

after the surface-to-volume ratio reaches a critical size. Some experimental research on

nanowires reflected the simulation results, with elastic modulus changing as the size of

the nanowire decreases. However, this trend depended on the material and was not

consistent within material systems. For example, in metal nanowires the E of silver

increased with decreasing nanowire diameter [14, 15]. In contrast, gold and chromium

showed E softening with decreasing sample size [16, 17]. Ceramic materials displayed

the same type of scatter. Zinc oxide, tungsten oxide, and copper oxide showed the elastic

modulus increasing with smaller diameter nanowire samples. The E of silicon, gallium

nitride nanowires decreased with decreasing sample size [18-20].

The dimensions in which the size effect of the mechanical properties was evident in

experimental research was much larger than predicted in the aforementioned theories.

Additionally, some materials exhibited contradictory experimental results. Nanoscale

silicon and silver, for example, have also been reported with bulk values of E [21-26] and

E different from the bulk but not dependent upon size [27, 28]. Though there may be

other unique phenomena to account for small scale elastic modulus behavior that is not

influenced by surface effects, variation in property measurements within material and

between systems may also be attributed to the range of experimental methods,

interpretation of data, and different processes used for nanowire development.

The E of nanowires has been most commonly measured using resonance, bending, and

tensile methods, though other techniques have been applied [24, 29-34]. In resonance the

sample is subjected to an alternating electric field at varying frequencies until the correct

frequency can be found to induce mechanical resonance, which is monitored in-situ.

Depending on the particular configuration, the fundamental resonant frequency of the

wire or the amplitude response to the applied electric field can be measured. Tests are

often conducted in an electron microscope, therefore accurate measurements of the

nanowire diameter and length can be established and used along with the nanowire

response to determine the elastic modulus of the sample. This method has been

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demonstrated for silicon [19, 27], boron [35], germanium [36], tungsten [37], silica and

SiO2/SiC composites [38-40], SiC [41], and zinc oxide nanowires [21, 42, 43].

Tensile and bending techniques have been applied to determine elastic modulus of

nanoscale materials, but also have the advantage of including the possibility of high

resolution force and deflection measurements, from which strain and fracture strength

may be determined. Nanoscale tensile testing has been used to determine E, fracture

strength, and strain using MEMS [44-49] and individual probe techniques [18, 35, 50-

52]. The method is typically designed to be run in-situ in scanning or transmission

electron microscopes, where it is possible to observe deformation and failure of

nanoscale samples. The loading and stress state of a uniaxial tensile test is analytically

straight forward, however it is experimentally challenging to adapt to the nanoscale.

Nanowire manipulation, alignment, and gripping technique may all impact error and

uncertainty in the measured mechanical properties. Challenges also exist with the

individual technique measurement of stress and strain.

Flexure of beams and plates is a well established method of mechanical testing. In bulk

samples, it is used in particular for ceramics (ASTM C1684-08), where it is difficult to

machine, grip, and align tensile specimens with precision. Knowledge of the specimen

size and testing configuration, as well as accurate monitoring of the applied force and

subsequent displacement during testing, make it possible to determine the elastic modulus

of a material and the flexural strength if a sample is tested to failure.

In experiments conducted on nanoscale specimens, elastic modulus and fracture strength

can be explored through various bending configurations. The atomic force microscope

(AFM) has become a preferred method for flexure testing of nanomaterials, due to its

high force and displacement resolution, and much of the reported flexure literature was

performed using this instrument. The basic operation of the AFM involves measuring

forces between a sample surface and a sharp tip, which is attached to a cantilever spring.

The tip is positioned at the end of the cantilever and scans over the sample surface. Any

detected vertical motion of the cantilever as a result of the tip-surface interaction is

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measured by the reflection of a laser beam aimed at the end of the cantilever. The

reflected laser beam is collected by a position-sensitive photodetector, which consists of

split photo-diodes. Depending on the angular placement of the cantilever, one photo-

diode will collect more light than another, creating a signal which is proportional to the

deflection of the cantilever. There are a variety of operating modes for the AFM which

depend upon the characteristics and information desired from the sample [53].

Cantilever [24, 26, 37, 54-57], simply supported [58-63], and fixed-fixed bending

configurations [22-25, 37, 64-71] have been used for mechanical evaluation of nanowires

over a wide range of materials. Similar issues to the nanoscale adaption of tensile testing

exist with flexure measurements as well, including nanowire manipulation and fixation

techniques. And while the AFM has superior force and deflection sensing resolution, it

commonly lacks the direct visualization capability during testing because the force is

applied via the cantilever that is traditionally used to image the specimen. Observing

experiments makes it possible to ensure that a test is performed correctly, the boundary

conditions are known, and the collected data is accurate. In-situ testing capabilities are a

great benefit to any nanoscale characterization, but are not commonly available in the

AFM, creating uncertainty in the collected flexure data. While commonly used, there are

still a wide variety of fundamental questions about the accuracy of data derived from

AFM-based experiments. In this research we will use a silicon nanowire model system to

explore these issues.

Mechanical testing of nanowires in this research was completed using the fixed-fixed

beam bending configuration. There are benefits and drawbacks to each of the various

methods which have been utilized to determine the fracture strength of nanoscale

materials. For instance, it is difficult to generate the necessarily high forces required to

cause individual nanowire failure via the resonance testing technique. The applied force

is also not measured directly, but modeled, which enhances uncertainty of the value.

Uniaxial tensile testing is a straight forward method used to determine the mechanical

strength of a material, however high accuracy in sample manipulation, alignment, and

strain measurements can prove to be difficult. This test configuration is being explored

7

simultaneously within the research group. In flexure, cantilevered nanowire experiments

have poorly defined fixed-end boundary conditions and fracture commonly occurs at the

fixed location. Maintaining a stationary position of applied load, predicting the location

of fracture, and interpreting the resulting data can be complicated. Similar issues with

data interpretation arise for non-cantilevered flexure test geometries, in addition to

difficulties with nanowire manipulation and fixation. The current nanowire mechanical

testing methodologies all result in data that requires interpretation and can lead to

significant uncertainties in the final analysis. The choice of using a fixed-fixed bending

configuration for the mechanical evaluation of silicon nanowires in this research was

largely a matter of convenience. The nanowires, which were individually grown, could

be tested in tension and compared to results obtained for the same nanowire sets in

bending. Additionally, testing in flexure with fixed boundary conditions allows for

access to smaller geometries and the possibility of observing size effects.

There are benefits and drawbacks to each of the mechanical testing methods which have

been utilized to determine the fracture strength of nanoscale materials. For instance, it is

difficult to generate the necessarily high forces required to cause individual nanowire

failure via the resonance testing technique. The applied force is also not measured

directly, but modeled, which enhances its uncertainty. Uniaxial tensile testing is a

straight forward method used to determine the mechanical strength of a material.

However, high accuracy in sample manipulation, alignment, and strain measurements can

be difficult to achieve. In spite of these challenges, this test configuration is being

explored simultaneously within our research group [72]. In flexure, cantilevered

nanowire experiments have poorly defined fixed-end boundary conditions, and fracture

commonly occurs at or near the fixed end. Additionally, maintaining a stationary

position of applied load (due to slipping at large deflections), identifying the location of

fracture, and interpreting the resulting data can be complicated. Similar issues with data

interpretation arise for non-cantilevered flexure test geometries, in addition to difficulties

with nanowire manipulation and fixation. Each of the nanowire mechanical testing

methodologies can provide insights into the mechanical behavior. The choice of using a

fixed-fixed bending configuration to evaluate the silicon nanowires in this research was

8

largely a matter of experimental practicality. Additionally, testing in flexure with fixed

boundary conditions allows the characterization of smaller geometries and for the

possibility of observing size effects.

1.2 Silicon Silicon nanowires have been used in a variety of micro- and nano- electronic and

mechanical devices, including resonators [73], sensors [74], probes for microscopy [75],

and field effect transistors [76]. For the numerous applications, operation and reliability

depend upon the mechanical properties of the nanoscale silicon. Even when the primary

function of the nanowire is non-structural, the mechanical strength is still involved in

maintaining the structural integrity of the system or device. However, while silicon is a

widely utilized material in many different engineering applications, the strength,

properties, and fracture mechanics at the nanoscale remain unclear.

Silicon is a well characterized bulk system and is therefore an ideal material for

fundamental research of nanoscale mechanics. It has a diamond cubic crystal structure,

which causes anisotropic behavior in the structurally dependent properties, including the

elastic modulus. For common single crystal growth directions, estimates of E have been

previously derived (Table 1-1 [77]) and are widely accepted as standard in the bulk. Bulk

silicon behaves as a brittle, ceramic material which follows linear elastic fracture

mechanics and does not experience fatigue [78]. As a brittle ceramic, bulk silicon is

subject to strength limitations according to the largest flaw present in the sample [79] and

as a single crystal, it demonstrates anisotropy in fracture events, favoring the {111} and

{110} crystallographic planes [78].

Table 1-1 Elastic modulus values of single crystal silicon according to crystallographic growth direction [77].

[100] [111] [110] E (GPa) 130.2 187.5 168.9

9

1.2.1 Silicon Nanoscale Modeling

Predicted nanoscale mechanical behavior of silicon is significantly different than

behavior of the bulk material [80-83]. Strength is predicted to move toward theoretical

values and various computer simulations for the elastic modulus of silicon result in a

softening effect with the reduction in nanowire diameter [9, 82-85]. However, these

models are not internally consistent within the material and depend on the type of

modeling procedure used, the method of surface reconstruction, and the direction of the

single crystal. There is also debate involved in what causes the change in mechanical

properties, with the dominant theory focused on surface effects [86-88]. Additionally,

while computer simulations demonstrated a reduction in E as nanowire diameters

dropped below 30 nm [82] or even 4 nm [9, 84], experimental counterparts exhibited

differing trends. While some aspects of models obviously deviate from reality, including

defect-free crystals and complete control of instrument and experimental conditions,

there exist large discrepancies between theory and testing results.

1.2.2 Silicon Nanoscale Mechanical Properties

Nanoscale mechanical testing of single crystal silicon has been performed with each of

the experimental techniques mentioned in the previous section. However, the results

varied widely between research groups and did not necessarily follow the trends

predicted by computer simulations or traditional beam theory. For example, the trends

for elastic modulus variations with sample size have been reported as essentially invariant

[22, 24, 26, 89] or even decreasing with decreasing sample size [18, 19, 90]. The

strength of silicon nanowires tested to failure, using a variety of techniques, encompassed

a large range of values from 30 MPa [91] to over 18 GPa [65]. And surprisingly, the

fracture of nanowire samples at room temperature commonly followed linear elastic,

brittle behavior [23, 65, 91], but a significant amount plastic deformation was found

during in-situ TEM tensile testing by one research group [50]. An overview of elastic

modulus and strength results for silicon nanoscale testing using a range of sample sizes

and testing techniques are outlined in Table 1-2 and Table 1-3, respectively.

10

Table 1-2 Review of experimental testing results for elastic modulus of silicon nanoscale samples.

Mechanical Test Method

Silicon Direction

Nanowire Diameter/Thickness

(nm)

Measured E

(GPa)

Reported Bulk E (GPa)

Reference

100 300-500 80-110 - [27]

Resonance 110

300 38.5 12

167 68 53

170 [19]

100 193-233 171.8 179 [26]

111 120-190 186 169 [24]

111 140 93 185 [25] Cantilever Bending

111 100-700 100-180 - [91]

Uniaxial Tension 110/112 111

15-30 30-60

90-170 170-190

169 187

[90]

110 255 169 169 [23]

110 255 171-195 169 [65]

111 150-200 150-250 185 [25]

111 120-190 207 169 [24]

Fixed-Fixed Bending

111 50-100 158 169 [22]

Table 1-3 Review of experimental testing results for strength of silicon nanoscale samples.

Mechanical Test Method

Silicon Direction

Nanowire Diameter/Thickness

(nm)

Measured Strengths

(GPa) Reference

111 140 0.82 [25] Cantilever Bending 111 100-425 0.03-4 [91]

110/112/111 15-60 5.1-12.2 [90] Uniaxial Tension 110/112/111 150-420 6.67-12.5 [72]

110 255 11.56-17.53 [23]

110 255 15-21 [65] Fixed-Fixed Bending 111 200 0.3-0.56 [25]

1.3 Analysis of Brittle Fracture Fracture strength values of ceramics are considered as distributions rather than fixed

numbers [92], so standards have been established using statistics to analyze the strength

behavior of ceramics (ASTM C1683-08 and ASTM C1239-07). The standards begin

with the assumption that ceramic samples inherently contain flaws and that the largest

flaws (for a given loading condition and crack orientation) will cause failure of the

sample. This assumption is a form of extreme value statistics, where the weakest link

11

initiates specimen failure. The analysis of multiple samples with one type of flaw will

form a distribution of strength. Two parameters are commonly needed to describe the

width and magnitude of the strength distribution, and because the exact distribution is not

known before hand, the strength of ceramic materials are fit to a Weibull distribution

[93]. The Weibull distribution allows for the prediction of fracture strength at a specified

applied stress.

The probability of failure F in a Weibull distribution is determined using Equation 1-1

[94].

Equation 1-1 ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−=

m

o

fractureFσ

σexp1

where m is the Weibull modulus, σo is the characteristic strength, and σfracture is the

fracture strength. Experimental strength values can be ranked and assigned a probability

of failure [93], while the Weibull fit parameters are most accurately obtained using a

maximum likelihood estimation (ASTM C 1239-07). The characteristic strength

provides a value for strength below which the probability of failure occurring is 63%.

The Weibull modulus is a measure of the variability in the distribution, with higher

values indicating narrow distributions of strength. With one exception [23], ceramic

nanowire fracture data has not been presented using the Weibull statistical analysis. For

similar testing methodologies and sample behavior, the only other known set of literature

which addresses Weibull statistics approaching a similar size scale and deformation

behavior, involve the mechanical evaluation of glass fibers [95-100].

As stated previously, reliability is a concern for nanoscale components. Reported

fracture strengths for silicon nanowires cover a wide range of values, reaching beyond

theoretical strengths (Table 1-3). However each set of experimental tests and

corresponding fracture strength values must be considered as its own defect distribution.

The probability of failure for that individual distribution can only be compared with

samples of the same size which were tested using the same techniques. This means, for

12

example, that the fracture strength of a nanowire measured in uniaxial tension cannot be

directly compared to one tested in resonant bending. In principle, the Weibull statistical

analysis strategy can be adapted to account for the variability that exists in nanowire

sample size and the method can be used to determine the effect of sample size and testing

configuration on the apparent strength (probability of failure).

Because a distribution of flaws exist in a specimen, as the size of the sample is reduced

there is a lower probability of finding a flaw to cause failure. Consequently, the

predicted fracture strength of the sample increases. For example, silicon fracture

strengths measured using a centrally loaded, fixed-fixed beam bending configuration

increased from 530 MPa for millimeter-scale samples, to 4-8 GPa for micrometer-scale

samples, to 12-18 GPa using nanometer-scale samples [23]. This trend is particularly

true for single crystal nanowire samples, which are anticipated to contain a very small

number of flaws. In the case of a defect-free sample, the only method of fracture would

be interatomic bond failure, resulting in strengths that approach the theoretical fracture

strength σTH of the material [79].

Like the defect population, the loading conditions have an important effect on the

measured fracture strength. Uniaxial tensile testing evenly distributes stress over the

entire sample. On the other hand, simply supported three-point bending produces a stress

field which varies linearly from zero at the edge supports to a maximum at the center

location of the load [93]. As a consequence, the fracture strength predicted from a

uniaxial tensile test will be lower than for the bending configuration for the same

specimen due to the higher probability of finding a flaw in the uniaxial case. To directly

compare the probability of failure of silicon nanowires tested in tension to the same

samples tested in bending, a correction factor is applied to account for the different

amount of the sample affected by the stress (ASTM C1684-08) [101].

As previously mentioned, nanowires have large length to diameter ratios which can

generate structures that are very flexible. A result of the flexibility is the possibility for

large deflection in samples during mechanical testing, which was clearly seen in

13

resonator and bend test results. While the additional flexibility does not necessarily

affect uniaxial tensile, resonator, or nanoindentation methods of testing, all bending

methods must be carefully evaluated. Large deflection numerical models were originally

derived for bending in a variety of simply supported and cantilever boundary

configurations [102-107] and later developed for flexible bars that were fixed at both

ends [108]. For fixed bars in bending, axial tensile forces can be induced along the bar as

the sample stretches in large deflections. As the amount of deflection relative to the

sample size increases, the response becomes increasingly non-linear due to geometric

effects. Though nanowires can be extremely flexible, only one research group which

tested silicon samples in the fixed-fixed bending configuration accounted for the

nonlinear behavior at large deflections [22].

1.4 Structure of Thesis There is a large variation in reported values and trends for mechanical properties as

sample sizes are reduced from the bulk to the nanoscale. While many larger scale

methods can be miniaturized and applied to small samples such as nanowires, the details

of the testing and analysis methods have not been standardized. It is therefore not

surprising that measurements of basic materials properties, such as the elastic modulus

and ultimate strength, have been so inconsistent. The research presented in the remainder

of this document is focused on reliably establishing the flexure strength of silicon

nanowires and assessing the viability of measuring theoretical material strength for a

nanowire in the fixed-fixed beam bending configuration. Chapter 2 will review the

materials and methods applied to accomplish the research, including the techniques of

structural and mechanical characterization for the silicon nanowire samples. Chapter 3

presents numerical models used to evaluate the impact of various data analysis techniques

and the uncertainties that emerge from changes in experimental boundary conditions.

Chapter 4 will provide an interpretation of the experimental data collected and present a

theoretical model to advance current probability statistics of ceramic nanoscale behavior.

Finally, a summary of the conclusions and implications derived from this research will be

presented in Chapter 5, followed by a brief discussion of possible future work which

could be used to further explore the key issues presented.

14

References 1. Bryzek, J., K. Peterson, and W. McCulley, Micromachines on the march.

Spectrum, IEEE, 1994. 31(5): p. 20-31. 2. Muller, R.S., MEMS: Quo vadis in century XXI? Microelectronic Engineering,

2000. 53(1-4): p. 47-54. 3. Sun, W., E.L. Chaikof, and M.E. Levenston, Numerical Approximation of

Tangent Moduli for Finite Element Implementations of Nonlinear Hyperelastic Material Models. Journal of Biomechanical Engineering, 2008. 130(6): p. 061003-7.

4. Haiss, W., Surface stress of clean and adsorbate-covered solids. Reports on Progress in Physics, 2001. 64(5).

5. Agrawal, R., et al., Elasticity Size Effects in ZnO Nanowires - A Combined Experimental-Computational Approach. Nano Letters, 2008. 8(11): p. 3668-3674.

6. Cammarata, R.C. and K. Sieradzki, Surface and Interface Stresses. Annual Review of Materials Science, 1994. 24: p. 215-234.

7. Dingreville, R., J. Qu, and C. Mohammed, Surface free energy and its effect on the elastic behavior of nano-sized particles, wires and films. Journal of the Mechanics and Physics of Solids, 2005. 53(8): p. 1827-1854.

8. Gurtin, M. and A.I. Murdoch, A continuum theory of elastic material surfaces. Archive for Rational Mechanics and Analysis 1975. 57: p. 291-323.

9. Kang, K. and W. Cai, Brittle and ductile fracture of semiconductor nanowires - molecular dynamics simulations. Philosophical Magazine, 2007. 87(14): p. 2169 - 2189.

10. Miller, R.E. and V.B. Shenoy, Size-dependent elastic properties of nanosized structural elements. Nanotechnology, 2000. 11: p. 139-147.

11. Sharma, P., S. Ganti, and N. Bhate, Effects of Surfaces on the Size-Dependent Elastic State of Nano-Inhomogeneities. Applied Physics Letters, 2003. 82: p. 535-537.

12. Park, H., S. , P. Klein, A. , and G. Wagner, J. , A surface Cauchy-Born model for nanoscale materials. International Journal for Numerical Methods in Engineering, 2006. 68(10): p. 1072-1095.

13. Sun, C.Q., et al., Bond-order-bond-length-bond-strength (bond-OLS) correlation mechanism for the shape-and-size dependence of a nanosolid. Journal of Physics Condensed Matter, 2002. 14(34): p. 7781-7795.

14. Cuenot, S., et al., Surface tension effect on the mechanical properties of nanomaterials measured by atomic force microscopy. Physical Review B, 2004. 69(16): p. 165410.

15. Jing, G.Y., et al., Surface effects on elastic properties of silver nanowires: Contact atomic-force microscopy. Physical Review B, 2006. 73(23): p. 235409.

16. Nilsson, S.G., X. Borrise, and L. Montelius, Size effect on Young's modulus of thin chromium cantilevers. Applied Physics Letters, 2004. 85.

17. Petrova, H., et al., Crystal structure dependence of the elastic constants of gold. Journal of Materials Chemistry, 2006. 16: p. 3957-3963.

15

18. Kizuka, T., et al., Measurements of the atomistic mechanics of single crystalline silicon wires of nanometer width. Physical Review B, 2005. 72(3): p. 035333.

19. Li, X., et al., Ultrathin single-crystalline-silicon cantilever resonators: Fabrication technology and significant specimen size effect on Young's modulus. Applied Physics Letters, 2003. 83(15): p. 3081-3083.

20. Nam, C.-Y., et al., Diameter-Dependent Electromechanical Properties of GaN Nanowires. Nano Letters, 2006. 6(2): p. 153-158.

21. Chen, C.Q., et al., Size Dependence of Young's Modulus in ZnO Nanowires. Physical Review Letters, 2006. 96(7): p. 075505-4.

22. Heidelberg, A., et al., A Generalized Description of the Elastic Properties of Nanowires. Nano Lett., 2006. 6(6): p. 1101-1106.

23. Namazu, T., Y. Isono, and T. Tanaka, Evaluation of size effect on mechanical properties of single crystal silicon by nanoscale bending test using AFM. Journal of Microelectromechanical Systems, 2000. 9(4): p. 450-459.

24. San Paulo, A., et al., Mechanical elasticity of single and double clamped silicon nanobeams fabricated by the vapor-liquid-solid method. Applied Physics letters, 2005. 87.

25. Tabib-Azar, M., et al., Mechanical properties of self-welded silicon nanobridges. Applied Physics Letters, 2005. 87.

26. Virwani, K.R., et al., Young's modulus measurements of silicon nanostructures using a scanning probe system: a non-destructive evaluation approach. Smart Materials and Structures, 2003. 12(6): p. 1028-1032.

27. Gupta, A., et al., Novel fabrication method for surface micromachined thin single-crystal silicon cantilever beams. Microelectromechanical Systems, Journal of, 2003. 12(2): p. 185-192.

28. Wu, B., et al., Microstructure-Hardened Silver Nanowires. Nano Lett., 2006. 6(3): p. 468-472.

29. Fang, T.-H. and W.-J. Chang, Nanolithography and nanoindentation of tantalum-oxide nanowires and nanodots using scanning probe microscopy. Physica B: Condensed Matter, 2004. 352(1-4): p. 190-199.

30. Feng, G., et al., A study of the mechanical properties of nanowires using nanoindentation. Journal of Applied Physics, 2006. 99.

31. Hsin, C.-L., et al., Elastic Properties and Buckling of Silicon Nanowires. Advanced Materials, 2008. 20(20): p. 3919-3923.

32. Li, X., et al., Direct nanomechanical machining of gold nanowires using a nanoindenter and an atomic force microscope. Journal of Micromechanics and Microengineering, 2005. 15(3): p. 551-556.

33. Tao, X., X. Wang, and X. Li, Nanomechanical Characterization of One-Step Combustion-Synthesized Al4B2O9 and Al18B4O33 Nanowires. Nano Letters, 2007. 7(10): p. 3172-3176.

34. Li, X., et al., Nanoindentation of Silver Nanowires. Nano Letters, 2003. 3(11): p. 1495-1498.

35. Ding, W., et al., Mechanics of crystalline boron nanowires. Composites Science and Technology, 2006. 66(9): p. 1112-1124.

16

36. Smith, D.A., et al., Young's Modulus and Size-Dependent Mechanical Quality Factor of Nanoelectromechanical Germanium Nanowire Resonators. Journal of Physical Chemistry C, 2008. 112(29): p. 10725-10729.

37. Cimalla, V., et al., Nanomechanics of Single Crystalline Tungsten Nanowires. Journal of Nanomaterials 2008. 2008: p. 9.

38. Wang, Z.L., et al., Measuring the Young's modulus of solid nanowires by in situ TEM. Journal of Electron Microscopy, 2002. 51(suppl_1): p. S79-85.

39. Wang, Z.L., et al., Nano-Scale Mechanics of Nanotubes, Nanowires, and Nanobelts. Advanced Engineering Materials, 2001. 3(9): p. 657-661.

40. Wang, Z.L., et al., Mechanical and electrostatic properties of carbon nanotubes and nanowires. Materials Science and Engineering: C, 2001. 16(1-2): p. 3-10.

41. Perisanu, S., et al., Mechanical properties of SiC nanowires determined by scanning electron and field emission microscopies. Physical Review B (Condensed Matter and Materials Physics), 2008. 77(16): p. 165434-12.

42. Huang, Y., X. Bai, and Y. Zhang, In-situ mechanical properties of individual ZnO nanowires and the mass measurement of nanoparticles. Journal of Physics: Condensed Matter, 2006. 18(15): p. L179-L184.

43. Huang, Y., et al., Size Independence and Doping Dependence of Bending Modulus in ZnO Nanowires. Crystal Growth & Design, 2009. 9(4): p. 1640-1642.

44. Brown, J.J., A.I. Baca, and V.M. Bright. Tensile Measurement of a Single Crystal Gallium Nitride Nanowire. in Micro Electro Mechanical Systems, 2009. MEMS 2009. IEEE 22nd International Conference on. 2009.

45. Demczyk, B.G., et al., Direct mechanical measurement of the tensile strength and elastic modulus of multiwalled carbon nanotubes. Materials Science and Engineering A, 2002. 334(1-2): p. 173-178.

46. Desai, A.V. and M.A. Haque, Mechanical properties of ZnO nanowires. Sensors and Actuators A: Physical, 2007. 134(1): p. 169-176.

47. Espinosa, H.D., Z. Yong, and N. Moldovan, Design and Operation of a MEMS-Based Material Testing System for Nanomechanical Characterization. Microelectromechanical Systems, Journal of, 2007. 16(5): p. 1219-1231.

48. Kiuchi, M., et al., Mechanical Characteristics of FIB Deposited Carbon Nanowires Using an Electrostatic Actuated Nano Tensile Testing Device. Microelectromechanical Systems, Journal of, 2007. 16(2): p. 191-201.

49. Zhu, Y., N. Moldovan, and H.D. Espinosa, A microelectromechanical load sensor for in situ electron and x-ray microscopy tensile testing of nanostructures. Applied Physics Letters, 2005. 86(1): p. 013506-3.

50. Han, X.D., et al., Low-Temperature In Situ Large-Strain Plasticity of Silicon Nanowires. 2007. p. 2112-2118.

51. Yu, M.-F., et al., Tensile Loading of Ropes of Single Wall Carbon Nanotubes and their Mechanical Properties. Physical Review Letters, 2000. 84(24): p. 5552.

52. Yu, M.-F., et al., Strength and Breaking Mechanism of Multiwalled Carbon Nanotubes under Tensile Load. Science, 2000. 287(5453): p. 637-640.

53. MFP-3D Intallation and Operation Manual, in Asylum Reseach: Atomic Force Microscopes. 2004.

54. Chen, C.Q. and J. Zhu, Bending strength and flexibility of ZnO nanowires. Applied Physics Letters, 2007. 90(4): p. 043105-3.

17

55. Hoffmann, S., et al., Fracture strength and Young's modulus of ZnO nanowires. Nanotechnology, 2007. 18(20): p. 205503.

56. Hoffmann, S., et al., Measurement of the Bending Strength of Vapor-Liquid-Solid Grown Silicon Nanowires. Nano Lett., 2006. 6(4): p. 622-625.

57. Song, J., et al., Elastic Property of Vertically Aligned Nanowires. Nano Lett., 2005. 5(10): p. 1954-1958.

58. Lee, S.-H., C. Tekmen, and W.M. Sigmund, Three-point bending of electospun TiO2 nanofibers. Materials Science and Engineering A, 2005. 398: p. 77-81.

59. Chen, Y., et al., Mechanical elasticity of vapour-liquid-solid grown GaN nanowires. Nanotechnology, 2007. 18(13): p. 135708.

60. Ni, H. and X. Li, Young's modulus of ZnO nanobelts measured using atomic force microscopy and nanoindentation techniques. Nanotechnology, 2006. 17(14): p. 3591-3597.

61. Ni, H., X. Li, and H. Gao, Elastic modulus of amorphous SiO2 nanowires. Applied Physics Letters, 2006. 88(4): p. 043108-3.

62. Ni, H., et al., Elastic modulus of single-crystal GaN nanowires. Journal of Materials Research, 2006. 21(11): p. 2882-2887.

63. Chen, Y., et al., On the importance of boundary conditions on nanomechanical bending behavior and elastic modulus determination of silver nanowires. Journal of Applied Physics, 2006. 100(10): p. 104301-7.

64. Ngo, L.T., et al., Ultimate-Strength Germanium Nanowires. Nano Letters, 2006. 6(12): p. 2964-2968.

65. Sundararajan, S., et al., Mechanical property measurements of nanoscale structures using an atomic force microscope. Ultramicroscopy, 2002. 91(1-4): p. 111-118.

66. Tan, E.P.S., et al., Crystallinity and surface effects on Young's modulus of CuO nanowires. Applied Physics Letters, 2007. 90(16): p. 163112-3.

67. Wen, B., J.E. Sader, and J.J. Boland, Mechanical Properties of ZnO Nanowires. Physical Review Letters, 2008. 101(17): p. 175502-4.

68. Wu, B., A. Heidelberg, and J.J. Boland, Mechanical properties of ultra-high strength gold nanowires. Nature Materials: Letters, 2005. 4: p. 525-529.

69. Wu, X., et al., Synthesis and Electrical and Mechanical Properties of Silicon and Germanium Nanowires. Chemistry of Materials, 2008. 20(19): p. 5954-5967.

70. Xiong, Q., et al., Force-Deflection Spectroscopy: A New Method to Determine the Young's Modulus of Nanofilaments. Nano Lett., 2006. 6(9): p. 1904-1909.

71. Zhu, Y., et al., Annealing effects on the elastic modulus of tungsten oxide nanowires. Journal of Materials Research, 2008. 23(8).

72. Steighner, M.S., Tensile Strength of Silicon Nanowires, in Department of Materials Science and Engineering. 2009, The Pennsylvania State University: University Park.

73. Cleland, A.N. and M.L. Roukes, Fabrication of high frequency nanometer scale mechanical resonators from bulk Si crystals. Applied Physics Letters, 1996. 69.

74. Fritz, J., et al., Translating Biomolecular Recognition into Nanomechanics. Science, 2000. 288(5464): p. 316-318.

75. Stowe, T.D., et al., Attonewton force detection using ultrathin silicon cantilevers. Applied Physics Letters, 1997. 71.

18

76. Cui, Y. and C.M. Lieber, Functional Nanoscale Electronic Devices Assembled Using Silicon Nanowire Building Blocks. Science, 2001. 291(5505): p. 851-853.

77. Brantley, W.A., Calculated elastic constants for stress problems associated with semiconductor devices. Journal of Applied Physics, 1973. 44(1): p. 534-535.

78. Hull, R., ed. Properties of Crystalline Silicon. Electronic Materials Information Service, ed. B.L. Weiss. Vol. 20. 1999, INSPEC: London.

79. Green, D.J., An Introduction to the Mechanical Properties of Ceramics. Cambridge Solid State Science Series, ed. D.R. Clarcke, S. Suresh, and I.M. Ward. 1998, Cambridge: Cambridge University Press.

80. Cahn, J.W., Surface stress and the chemical equilibrium of small crystals--I. the case of the isotropic surface. Acta Metallurgica, 1980. 28(10): p. 1333-1338.

81. Law, M., J. Goldberger, and P. Yang, SEMICONDUCTOR NANOWIRES AND NANOTUBES. Annual Review of Materials Research, 2004. 34(1): p. 83-122.

82. Park, H.S., Surface stress effects on the resonant properties of silicon nanowires. Journal of Applied Physics, 2008. 103(12): p. 123504-10.

83. Zhang, W.W., Q.A. Huang, and L.B. Lu, Size-Dependent Elasticity of Silicon Nanowires. Advanced Materials Research 2009. 60-61: p. 315-319.

84. Lee, B. and R.E. Rudd, First-principles calculation of mechanical properties of Si<001> nanowires and comparison to nanomechanical theory. Physical Review B (Condensed Matter and Materials Physics), 2007. 75(19): p. 195328-13.

85. Wang, G. and X. Li, Predicting Young's modulus of nanowires from first-principles calculations on their surface and bulk materials. Journal of Applied Physics, 2008. 104(11): p. 113517-8.

86. Chuang, T.-j., On the Tensile Strength of a Solid Nanowire, in Nanomechanics of Materials and Structures. 2006, Springer: Netherlands. p. 67-78.

87. Park, H.S., Quantifyinf the size-dependent effect of the residual surface stress on the resonant frequencies of silicon nanowires if finite deformation kinematics are considered. Nanotechnology, 2009. 20.

88. Park, H.S., et al., Mechanics of Crystalline Nanowires. MRS Bulletin, 2009. 34: p. 178-183.

89. Stan, G. and W. Price, Quantitative measurements of indentation moduli by atomic force acoustic microscopy using a dual reference method. Review of Scientific Instruments, 2006. 77.

90. Zhu, Y., et al., Mechanical Properties of Vapor-Liquid-Solid Synthesized Silicon Nanowires. Nano Letters, 2009. 9(11): p. 3934-3939.

91. Gordon, M.J., et al., Size Effects in Mechanical Deformation and Fracture of Cantilevered Silicon Nanowires. Nano Letters, 2009. 9(2): p. 525-529.

92. Lawn, B., Fracture of Brittle Solids. Second Edition ed. Cambridge Solid State Science Series, ed. A.E. Davis and I.M. Ward. 1993, Cambridge: Cambridge University Press.

93. Wachtman, J.B., W.R. Cannon, and M.J. Matthewson, Mechanical Properties of Ceramics. Second Edition ed. 2009: John Wiley & Sons, Inc.

94. Munz, D. and T. Fett, Ceramics: Mechanical Properties, Failure Behavior, Materials Selection. Springer Series in Materials Science, ed. R. Hull, et al. 1999, Berlin: Springer.

19

95. Annovazzi-Lodi, V., et al., Statistical Analysis of Fiber Failures Under Bending-Stress Fatigue. Journal of Lightwave Technology, 1997. 15(2): p. 288-92.

96. Gupta, P., K. and C.R. Kurkjian, Intrinsic failure and non-linear elastic behavior of glasses. Journal of Non-Crystalline Solids, 2005. 351: p. 2324-2328.

97. Karabulut, M., et al., Mechanical and Structural Properties of Phosphate Glasses. Journal of Non-Crystalline Solids, 2001. 288: p. 8-17.

98. Matthewson, J.M., C.R. Kurkjian, and S. Gulati, Strength Measurement of Optical Fibers by Bending. Journal of the American Ceramics Society, 1986. 69(11): p. 815-21.

99. Medrano, R.E. and P.P. Gillis, Weibull Statistics: Tensile and Bending Tests. Journal of the Amerian Ceramic Society, 1987. 70(10): p. C230-C232.

100. Muraoka, M., The Maximum Stress in Optical Glass Fibers Under Two-Point Bending. Journal of Electronic Packaging, 2001. 123(1): p. 70-73.

101. Quinn, G.D., Weibull Effective Volume and Surfaces for Cylindrical Rods Loaded in Flexure. Journal of the American Ceramics Society, 2003. 86(3): p. 475-79.

102. Mitelman, M., Large Deflection of Cantilever Beams in Bending. Israel Journal of Technology, 1968. 6(3): p. 227-230.

103. Mitelman, M., Large Deflection of simply-supported beams in bending under symmetrical pair of forces. Israel Journal of Technology, 1969. 7(4): p. 367-70.

104. Paolinelis, S.G., S.A. Paipetis, and P.S. Theocaris, Three-point bending at large deflections of beams with different moduli of elasticity in tension and compression. Journal of Testing and Evaluation, 1979. 7(3): p. 177-82.

105. Schile, R.D. and R.L. Sierakowski, Large Deflections of a Beam Loaded and Supported at Two Points. International Journal of Nonlinear Mechanics, 1967. 2: p. 61-68.

106. Theocaris, P.S., S.A. Paipetis, and S. Paolinelis, Three-point bending at large deflections. Journal of Testing and Evaluation, 1977. 5(6): p. 427-36.

107. Wang, T.M., S.L. Lee, and O.C. Zienkiewicz, A numerical analysis of large deflections of beams. International Journal of Mechanical Sciences, 1961. 3(3): p. 219-228.

108. Landau, L.D. and E.M. Lifshitz, Theory of Elasticity. 3 ed. Course of Theoretical Physics. Vol. 7. 1986, Oxford: Pergamon Press.

20

2 Materials and Methods

2.1 Silicon Nanowires Silicon remains the dominant structural material in many small scale systems, however

the understanding of mechanical behavior for components at the nanoscale is limited.

Moreover, the data that are reported are often inconsistent with each other and prevailing

mechanical behavior theories. Silicon nanowires were used for all evaluations completed

in this research. Samples were specifically grown, released, and manipulated to

ultimately prepare for mechanical characterization.

2.1.1 Nanowire Growth

The vapor-liquid-solid (VLS) synthesis technique for silicon nanowire growth, developed

by Wagner and Ellis [1], utilizes gold (Au) as the catalyst for the decomposition of a

silicon-containing gas source, such as silane (SiH4). Si and Au form a liquid phase alloy

above the eutectic temperature of ~363°C. When the alloy becomes supersaturated, a

silicon nanowire is precipitated [2-4]. Among other methods, silicon nanowires can be

grown directly from a Au coated oxidized silicon substrate or with the use of a porous

alumina membrane to control the wire diameter and density. Nanowires grown using

either technique consist of a crystalline silicon core with a thin native oxide coating (< 5

nm) [3-5].

Silicon nanowires were provided for this work by Dr. Joan Redwing’s research group

(Pennsylvania State University, Department of Materials Science and Engineering) and

were grown using both production techniques, Figure 2.1. The wires were initially grown

in anodic alumina membranes, 60 nm thick with 300 nm diameter pores containing 200

nm Au plugs. Nanowires were also grown from oxidized silicon substrates using a 3 nm

Au catalyst layer resulting in average diameters of 80 – 100 µm (± 50 µm). The length

and diameter of the nanowires are a function of time, temperature, and partial pressure

used during growth. Experimental conditions for nanowires grown from the Au coated

21

substrate in a low pressure, hot walled chemical vapor deposition reactor were 500°C and

13 Torr using a 10% mixture of SiH4 in H2 as the silicon precursor.

Figure 2.1 Scanning electron micrographs of silicon nanowires grown from (a) an alumina membrane and (b) an oxidized silicon surface. Images courtesy of Sarah Eichfeld.

22

2.1.2 Nanowire Release

In order to individually manipulate and test the silicon nanowires of Figure 2.1, it was

necessary to first release them from the growth substrate. Releasing the VLS nanowires

required specialized procedures for each growth technique, both of which had specific

disadvantages. It was crucial to ensure that the nanowires remained undamaged, however

the release process introduced defects that were detrimental to the strength of the wires.

Silicon nanowires grown in an alumina membrane were released using a wet etch

technique in a basic environment. Sodium hydroxide has an etch selectivity that is higher

toward alumina than silicon and is used to dissolve the membrane from the silicon

nanowires [6, 7]. A piece of alumina membrane containing nanowires, approximately 2

mm2, was placed into a centrifuge tip filled with 1 M sodium hydroxide (J.T. Baker).

The membrane was submerged in solution for 2 hours and subjected to ultrasonic

agitation causing fragmentation of the membrane surrounding the nanowires. The

solution was then soaked in the etchant for an additional 30 minutes to ensure the

remaining membrane was released from the nanowire surface. The newly released

nanowires were then centrifuged to the bottom of the container allowing the etchant

solution to be removed with a pipette. The etchant was replaced with anhydrous alcohol

(J.T. Baker) and agitated to fully dilute the etchant with the newly added alcohol. This

process was repeated twice more to ensure that no significant amount of sodium

hydroxide was left in solution.

Following the release process, the nanowire containing solution was pipetted onto a

copper TEM grid covered with a lacey carbon membrane (Electron Microscopy

Sciences), which allowed for the alcohol in the solution to either seep through the grid or

evaporate, leaving individual silicon nanowires suspended on the carbon film. The

nanowires were then inspected using a field emission SEM (FEI Phillips XL-20), Figure

2.2. There was a fine margin between fully releasing the nanowires from the alumina

membrane and etching into the silicon. There was often both remnant membrane and

partially etched nanowires together in solution. Variables were adjusted in an effort to

optimize the release process, including the length of time that the membrane was left in

23

the etchant and the concentration of the etchant solution used. However, the yield of

acceptable nanowires produced and released was determined to be too low to feasibly

enable a full mechanical testing suite. Silicon nanowires grown and released from the

porous alumina membranes were therefore dismissed from further experimentation in this

study.

Figure 2.2 Scanning electron micrograph of a silicon nanowire released from an alumina membrane using a sodium hydroxide etch. The nanowire is etched and contains remnant membrane.

The second type of silicon nanowires provided, grown from Au coated oxidized silicon

substrate, were released using ultrasonic agitation in alcohol. Without the Al2O3

membrane, the basic environment etch step, which is potentially harmful to the silicon,

was circumvented. In the simplified release process, an approximately 2 mm2 piece of

the substrate was submerged in a centrifuge tip filled with an anhydrous alcohol (J.T.

Baker) and placed into a ultrasonic bath for 1 second. The substrate was removed from

solution and placed into a separate centrifuge tip filled with fresh alcohol and agitated for

1-2 seconds. The alcohol in the second tip served as the final solution of nanowires used

24

for characterization and testing. The solution was ultrasonically agitated and the

nanowires were distributed onto a lacey carbon TEM grid, as described previously.

Released nanowires were inspected under the field emission SEM (Ziess Ultra-60) to

determine viability for mechanical testing. Using only a single ultrasonic agitation step

resulted in a large amount of debris remaining in the nanowire solution. The addition of

an initial exposure to the ultrasonic bath with fresh alcohol, while sacrificing some

nanowires, removed excess debris that would have ultimately contaminated the final

solution. The solution of alcohol and nanowires after the final immersion contained a

sufficient concentration of nanowire samples, and a reduced concentration of debris from

the substrate surface. While all debris is undesirable, managing a small amount in a final

solution containing mostly pristine nanowires is preferable to generating etched

nanowires or having large pieces of residual membrane present, as was the case with the

alumina membrane growth substrate. Additionally, the release method for nanowires

grown on the oxidized silicon substrate was consistent regardless of nanowire size and

was used to release nanowire samples over a large range of diameters and lengths.

Nanowires grown on the Au coated oxidized silicon were easily and reliably released,

with the condition of the resulting silicon nanowires being acceptable for mechanical

testing, Figure 2.3.

25

Figure 2.3 Transmission electron micrographs of a silicon nanowire released from an oxidized silicon surface using ultrasonic agitation demonstrating (a) the complete nanowire supported on lacey carbon and (b) smooth sample surfaces at high magnification.

2.1.3 Nanowire Manipulation

The precise placement of individual nanowires is a considerable challenge in all testing

schemes. In this work, it was necessary for individual nanowires to be positioned across

26

fixture gaps to perform mechanical tests. Several methods were evaluated in this effort.

The goal of the multiple placement techniques reviewed was to ultimately generate as

little damage to each sample as possible while reliably positioning the wires over the test

fixtures.

2.1.3.1 Microtweezers

Manual manipulation of individual nanowires provides the potential for precise sample

placement across fixture testing spans. Silicon microtweezers, which can be mounted in

vacuum-secured micromanipulators and viewed under a high magnification optical

microscope (Figure 2.4) was one method employed for sample manipulation. Nanowire

samples grown on an oxidized silicon surface extended roughly perpendicular to the

growth substrate. This growth configuration made it possible to utilize microtweezers to

pluck individual nanowires off of the substrate, avoiding the issue of nanowire release

altogether. The microtweezers could be closed or opened by alternatively applying and

removing a current to a electrothermal actuator. In order to test this method, the fixture

was first placed on the optical microscope stage along with an oxidized silicon substrate

coated with silicon nanowire samples. The microtweezers were then moved into position

at the edge of the growth substrate, opened, moved again to surround a single nanowire,

and then closed. While in the closed position, the microtweezers were slowly retracted,

plucking a nanowire from the growth substrate. The nanowire was then moved to a test

fixture, with the gap aligned perpendicularly to the wire. The tweezers were then opened,

in an attempt to allow the nanowire to fall across the fixture test span.

27

Figure 2.4 Optical micrograph showing silicon nanowires extending from an oxidized silicon growth substrate with microtweezers approaching for individual manipulation. The image was captured at an unknown magnification.

Ideally, plucking a nanowire and placing it in the desired location would have been

straight forward. However, the practical application created several issues due to

adhesion and electrostatic forces. As the tweezers were brought into the vicinity of the

substrate containing an array of nanowires, the wires would often bend away from or

snap onto the tweezer arms. Due to the configuration of the particular set of

microtweezers being employed for this work, it was not possible to directly ground the

tool to eliminate any potential charge on the surface of the arms.

In some cases, the nanowires were positioned to enable the microtweezers to successfully

close around and pluck individual samples from the growth substrate. The microtweezers

were then manipulated over a mechanical testing gap and opened to release the sample.

However, similar problems to the plucking process were encountered with the release and

placement of the nanowire. Rather than dropping onto the test fixture surface, the

nanowires consistently remained adhered to the tweezer arms. Several attempts to

28

resolve the stiction issue were made. First, a Nucleo-Spot™ ionizing radiation source

(commonly known as a “static buster”) was placed next to the apparatus in order to

eliminate static charge from the tips of the tool as well as the nanowire sample, but this

did not prove to be effective. The tips of the microtweezers were also carefully dragged

across the substrate surface to dislodge the attached nanowire, however when the sample

did finally come off of the tool, it was no longer in the correct position and there was a

significant possibility of mechanical damage due to the physical contact with the

substrate. Overall, this promising method proved to be ineffectual in reliably

manipulating individual nanowires.

2.1.3.2 Microprobe

The second technique evaluated for the manipulation of individual nanowires utilized a

microprobe mounted within a dual beam SEM/FIB instrument (FEI DualBeam Quanta

200 3D FIB). Silicon nanowire samples released from a growth substrate into an alcohol

solution were deposited onto a 200 mesh copper TEM grid coated with a lacey carbon

film (Electron Microscopy Sciences). This method of distribution enabled inspection and

structural characterization of the individual nanowire samples in the TEM, however the

nanowires must then be transferred to a separate fixture for mechanical testing. The

devised manipulation process was to use deposited metal-based adherent to weld the

nanowire to a microprobe tip, move it to the test fixture gap, align the sample in the

proper position, and then mill the nanowire from the microprobe.

In practice, the TEM grid with deposited nanowires and a substrate containing multiple

test fixtures were loaded into the dual beam instrument vacuum chamber and situated at

the microscope eucentric height of 15 mm. The eucentric height is the location where the

tilt of the stage will not result in translation of the sample. The electron beam was

operated using an accelerating voltage of 30 keV with a 0.67 nA current. The gallium ion

beam was focused with a 10 pA current. After locating a characterized nanowire of

interest on the TEM grid, a tungsten microprobe was introduced into the chamber and

brought into contact with one end of the nanowire using a micromanipulator system

(Omniprobe AutoProbe 200), where the combination of the electron and ion beam

29

imaging allowed for rapid and precise positioning of the probe in the chamber. A

precursor gas injection system (GIS) nozzle was then brought 100 µm above the chamber

eucentric height, close to both the microprobe and the nanowire of interest. Ion beam

induced deposition (IBID) of a platinum-based precursor gas was used to attach the

silicon nanowire to the tip of the microprobe, maintaining a power of approximately 5

pA/µm2 for optimal deposition. The microprobe with the attached nanowire was then

slowly retracted, detaching the nanowire from the lacey carbon film. The lacey carbon

film exhibits a tendency to adhere to the nanowire, preventing it from being lifted off of

the film surface without additional action. It was often necessary to use the ion beam to

mill the lacey carbon and sever a large portion of the film surrounding the nanowire.

However, film debris remained attached the sample and there was an increased

possibility of causing further damage to the nanowire with exposure to the ion beam.

When a nanowire was successfully removed from the TEM grid, the microprobe was

fully retracted, the test fixture span was brought into the center position of the chamber.

The fixture gap was aligned perpendicular to the nanowire angle at the eucentric height of

the instrument and the microprobe with the attached nanowire was then reintroduced into

the chamber. The microprobe and attached nanowire were carefully positioned above

and brought into contact with the test fixture. Each end of the nanowire spanning the test

gap was then affixed to the fixture surface with 0.5 µm2 rectangular pads of IBID

deposited platinum, which were approximately 1 µm thick. To maintain the optimal

deposition power, and to avoid milling into the fixture surface, extra pads of platinum

were simultaneously deposited away from the sample. When the nanowire was secured

to the surface, the ion beam was used to mill the nanowire from the attached microprobe.

The manipulation process is illustrated in Figure 2.5 a through d.

30

Figure 2.5 Scanning electron micrographs of the process involved in individually manipulating a silicon nanowire sample across a fixture test span using a microprobe in the dual beam instrument, including the sample (a) atop of lacy carbon membrane, (b) attached to the tungsten microprobe using a platinum-based deposit, (c) approaching the fixture span, and (d) resting across the device fixture gap.

The use of microprobes to manipulate individual nanowires was time consuming and

only moderately successful. While individual physical manipulation made it possible for

the nanowires to be characterized under the TEM prior to testing to obtain growth

direction and accurate diameters for each wire and allowed for the selection of specific

samples to be chosen from the grid for testing, it exposed the nanowires to the gallium

ion beam in the dual beam instrument. The penetration depth of the ion beam into the

silicon sample has the ability to cause significant structural damage to the thin nanowires

and possibly alter the intrinsic mechanical properties prior to testing. A 30 keV Ga+ ion

beam can penetrate a silicon surface and cause ion implantation as well as structural

damage to approximately 27 nm deep and 10 nm laterally, which is a significant portion

31

of the entire nanowire sample [8]. It was therefore preferable to not require any ion beam

use in the manipulation of the nanowire samples.

2.1.3.3 Field-Assisted Alignment

Manipulation of nanowires that required direct physical contact between a tool and the

samples were unreliable, unrepeatable, or caused an unacceptable amount of damage to

the nanowires. Two methods were devised to eliminate direct physical contact with the

samples in manipulating the nanowires onto test fixtures.

The first methodology used dielectrophoretic alignment, a field-assisted alignment

technique that exploits a non-uniform electric field to manipulate a material suspended in

a liquid. This method was originally pioneered by Pohl [9, 10] to align and separate

various particles and has been more recently investigated for nanowire manipulation [4,

11-19]. The force of the dielectrophoretic effect is dependent upon the oscillation

frequency, the applied electric field, the size of the samples, the dielectric constant of the

suspension, as well as the conductivity of the suspension and the material [18].

Ultimately, the parameters are adjusted until the nanowires align in response to the

generated electric field, following the field lines between the two electrodes.

In practice, two tungsten microprobes mounted in vacuum-secured micromanipulators

were placed in close proximity under an optical microscope. The probes were then

positioned just above a glass substrate located on the microscope stage. An electric field

was applied across the probes, varying at a specific frequency, before a drop of the

solution containing the silicon nanowires was flooded onto the substrate surface

surrounding the microprobes. The applied voltage, frequency, distance between the

probe tips, nanowire solution concentration, and the nanowire solution solvent were

varied to optimize nanowire alignment between the probes.

The most successful alignment of the silicon nanowires was achieved using a Vrms of 83

V and 1 kHz oscillating frequency with an isopropanol nanowire solution (Figure 2.6).

Nanowires also aligned to various degrees with voltages higher than 27 V and using 1 –

32

10 kHz frequencies. While there was a certain amount of success using this technique,

several drawbacks were also observed. In many of the alignment tests the samples

agglomerated. For mechanical testing, it is necessary isolate nanowires with a significant

amount of separation. This issue was mitigated, but not eliminated by using a solution

with a lower concentration of nanowires. Another complication arose as the alcohol

solution evaporated. The aligned nanowires suspended in the liquid were violently

agitated just prior to final evaporation and did not consistently remain in the aligned

position after evaporation.

Figure 2.6 Optical micrograph of silicon nanowires aligned between two tungsten microprobes using the field-assisted alignment technique. The image was taken at an unknown magnification.

While manipulation was demonstrable at optimized frequencies and voltages and weak

manipulation was possible even without optimization, factors such as Brownian motion,

thermal effects, and short-range surface forces can all have a strong effect on the motion

and final alignment of the nanowires [18]. Alignment of a single, isolated nanowire was

also an unresolved issue. A solution which included incorporating electrodes on the

fixture using various lithography techniques was expected to be marginally successful,

33

but not promising enough to devote design resources for fixture production. Similar

conclusions have been drawn based on the difficulty of method and statistical yield of

individual nanowire placement in previous work [11, 12, 16, 20].

2.1.3.4 Solution Deposition

A final, and extremely simple method, was devised as a non-contact technique.

Nanowires dispersed in alcohol solution, as described in Section 2.1, were ultrasonically

agitated and deposited via pipette drop onto a substrate containing 48 fixtures over 60

mm2. Upon evaporation of the solute alcohol, a certain number of nanowires remained in

the properly align position based on probability [21-24]. By increasing the number of

fixtures, the possible sites for nanowire placement was increased to improve the yield.

This method of nanowire positioning resulted in the most efficient placement of

individual nanowires across the mechanical testing spans, inducing little to no damage to

the samples. Though simple, the solution deposition method was ultimately the preferred

method for nanowire placement and was employed for this research.

2.2 Mechanical Test Fixture In the absence of a standardized mechanical testing fixture geometry developed to

evaluate the flexure strength of silicon nanowires, it was necessary to design, develop,

and process a platform for testing. Several experimental regiments were considered to

adequately characterize the structural and mechanical properties. The most constrictive

elements of the design are requirements imposed so that nanowires are placed and located

in configurations amenable to the particular instruments used for testing. Samples must

be suspended over a void in order to perform mechanical evaluation of flexural strength

in a three-point bending configuration using the AFM. In addition, structural information

including the nanowire growth direction, oxide thickness, and any defects present prior to

mechanical testing can be identified using a TEM for a complete understanding of the

nanowire behavior. With these two criteria in mind, an initial fixture was designed to

incorporate the size and other geometric and electron optics restrictions that are

34

associated with the TEM along with inspection under the mechanical integrity that was

needed for flexure testing in the AFM.

2.2.1 Fixture Design

The initial design for the silicon nanowire centrally loaded fixed-fixed bending test

fixture incorporated three key features; two closely spaced supports upon which the

nanowire sample would be suspended for mechanical testing, a TEM transparent window

in between the supports for visual inspection and structural characterization, and field-

assisted alignment capabilities for nanowire manipulation and placement, Figure 2.7.

Figure 2.7 Schematic illustration of a cross-section of the proposed fixture design, shown in on angle, from above, and from the front, where the labels denote (a) the knife edge supports, (b) the TEM transparent window, and (c) the lithographically defined electrodes for field-assisted alignment of the nanowire samples.

35

Each design feature was first processed and assessed separately before combining the

methodologies and parameters into one fixture, ultimately being produced using a silicon-

on-insulator (SOI) wafer. Various methods, including photolithography using several

types of photoresist and oxide masks, as well as numerous etch techniques were

employed in the manufacturing efforts.

The first and most important design feature was specified by the technique used to

determine the ultimate bending strength of the silicon nanowire samples. For this

research, an AFM cantilever was the instrument utilized to apply force in a three-point

bending configuration. The support span for the test must obviously be incorporated

within the fixture. The support columns were originally designed around a simply-

supported bend test, meaning that the nanowire samples rest on top of the supports with

no additional methods used to restrict motion. The calculation for flexure strength in this

testing configuration depends on the length of the support span (Equation 2-1) [25],

where σ is the ultimate flexure strength of the sample, F is the applied force, L the testing

span, and r is the radius of the nanowire.

Equation 2-1 3rFLπ

σ =

The distance between the two supports was constrained by the length of the nanowire

samples and the acceptable amount of deflection for the test. The amount of deflection at

failure in a standard, simply-supported three-point bend test depends upon the sample

diameter and testing span, as is evident in Equation 2-2 [25], where δ is the sample

deflection and E is the modulus of elasticity. The total applied force in the test nanowire

is limited by the AFM and the cantilevers used. Equation 2-2 therefore set a minimum

support span length and an acceptable range of spacing between the support columns was

determined. The top of each column must come to a point with a tight radius to assure

that support span did not change as the test progressed. If the radius of the peak was too

large, the bend test span would become smaller as the nanowire exhibited large

deflection, which would in turn generate uncertainty in the final interpretation of the data.

36

Equation 2-2 4

3

12 rEFLπ

δ =

The second design feature, integration of the fixture into the TEM chamber and a

transparent window to visualize nanowire samples, was set by the dimensions of the

nanowire samples, automatically limiting the instruments which could be used for

characterization prior to mechanical testing. The TEM is a well established and accurate

method for investigating small features. A TEM transparent window or gap was

therefore designed in between the two support columns to visualize the nanowire and

characterize several aspects of the sample, such as diameter, growth direction, defect

structure, and oxide content. TEM compatible samples are restricted to a maximum

feature length of 3 mm and height of 200 µm in order to insert the fixture into the

instrument chamber (Phillips EM420T). Fixtures processed for TEM compatibility

would also be compatible with AFM instrumentation.

The last design feature was the inclusion of lithographically defined electrodes on either

side of the TEM window to incorporate the ability to manipulate nanowires into

amenable testing positions using field-assisted alignment. As described previously,

preliminary work demonstrated that this manipulation method was not efficient, and the

design feature was eliminated to reflect those findings.

The final fixture design was drastically different than the initial design after the

incorporation of processing considerations, described in detail in Appendix A. The final,

operable design is essentially a testing span of 1.4 µm with parallel supports that were

each 4.3 µm wide and 300 µm long. The fixture, as manufactured and utilized in this

work, is shown in Figure 2.8.

37

Figure 2.8 Scanning electron micrographs of the final fixture layout (a) as seen from above and (b) in cross-section.

2.3 Nanowire Characterization Silicon is an anisotropic material with crystallographically-dependent mechanical

properties. It is therefore necessary to determine the crystallographic growth direction of

each individual nanowire tested to examine the effect of orientation on the ultimate

38

flexure strength of silicon at the nanoscale. Two methods were utilized to characterize

the growth direction of the samples in this research, transmission electron microcopy

selected area diffraction (TEM SAD) and electron backscatter diffraction (EBSD).

2.3.1 TEM Characterization

For several of the previously described release and manipulation techniques, nanowire

samples were deposited from an alcohol solution onto a lacey carbon coated TEM grid.

The silicon nanowires were left suspended on the film when the alcohol evaporated, and

were then imaged in a SEM (FEI Philips XL-20). Acceptable nanowire samples were

located, identified for mechanical testing at intermediate magnification (1000 ×), and

mapped on the grid in order to locate, following transfer to the TEM sample chamber.

The physical specifications desired for nanowire flexure testing include sufficient length

to span a testing gap, maximum diameter of approximately 150 nm, and the absence of

any kinks or other obvious defects to the structure surface. While it was not possible to

identify all defects at this magnification, many of the nanowires could be eliminated from

consideration with this basic screening process.

Nanowires of interest were imaged at high magnification (30-35,000 ×) in order to obtain

an accurate measure of the diameter (Figure 2.9 (a)) after the inspected grid was placed

into the TEM (Phillips EM420T). Then, using selected area diffraction, the nanowire

was aligned along the zone axis and overfocused in order to produce a diffraction pattern

indicative of the particular nanowire growth direction (Figure 2.9 (b)). The distance

between spots on the pattern were measured and compared with known distances

between crystallographic planes in silicon. The growth directions of silicon nanowires

provided for this research were identified as [100], [110], [111], and [112]. There was

also evidence of [112] nanowires that were bicrystals with a [111] twin boundary running

along the length.

39

Figure 2.9 Transmission electron micrograph images of a [112] silicon nanowire (a) diameter for accurate dimensional measurements and (b) diffraction pattern used to characterize the sample growth direction.

While this method of characterization was an accurate way to determine the growth

direction and diameter of each nanowire sample, the transfer of individual nanowires

onto a test fixture required physical manipulation using microprobes in the FIB, as

discussed previously. As the final fixture was not compatible with TEM observation, the

nanowires that were mechanically evaluated were not characterized in this manner. TEM

results did, however, provide the likely possible growth directions and flaws present

40

within the silicon nanowires. These observations are used to confirm an alternative

method of crystallographic characterization.

2.3.2 EBSD Characterization

To characterize the growth direction of the silicon nanowire samples used for mechanical

evaluation, and deposited directly onto the solid silicon fixture surface, electron

backscatter diffraction (EBSD) was employed. While EBSD is a well known technique

for grain orientation determination, phase identification, and strain mapping [26], it is not

a common technique for the characterization of nanoscale features. Challenges to this

application arose in differentiating the silicon nanowire from the silicon support fixture

with the electron beam, as well as collecting diffraction patterns at very high

magnifications.

To collect electron diffraction patterns from the silicon nanowire samples prior to

mechanical testing, the entire test fixture was mounted onto an aluminum stub and loaded

into a field emission SEM chamber (Hitachi S-4700). The fixture surface was then tilted

to form a 70° angle from the normal plane of the microscope chamber so the incident

electron beam made a small angle with the sample surface to align the sample in a

manner that produced electron backscatter diffraction patterns on a phosphor screen in

the FESEM chamber. The phosphor screen was introduced to the chamber to between 18

– 22 mm from the sample to capture the diffracted electrons and transmit the electron

backscatter diffraction patterns to a CCD camera. The diffraction pattern is a collection

of Kikuchi lines (Figure 2.10 a) where each band, or pair of nominally parallel lines,

corresponds to a distinct crystallographic plane. The width between the lines is inversely

proportional to the lattice spacing and the intersection of the bands are projections of the

zone axes of the sample crystal structure [27]. In that manner, the Kikuchi patterns

reflect the crystal symmetry of the sample. Completion of pattern indexing is now

conventional by modern computer algorithms (HKL) and further image analysis was

performed to solve specific issues, including high-magnification pattern distortion

(Lispix). Geometric deduction of nanowire growth direction from indexing information

41

was performed via Excel calculation devised by Mark Vaudin at the National Institute of

Standards and Technology (NIST).

The Kikuchi patterns were collected using a FESEM equipped with a Nordlys detector

(Oxford Instruments, HKL). In many applications of EBSD it is not necessary to work at

high magnification (< 2 kx) and magnetic lenses are not used to focus the electron beam.

However the nanowire samples, typically less than 100 nm in diameter, could not be

accurately isolated by the electron beam using the lower magnification mode, which was

free of a magnetic lens. The magnetic field produced by the lenses at high magnification

interfered with the projection of the Kikuchi patterns on the phosphor screen, which lead

to a distortion of the diffraction pattern, Figure 2.10 b. The pattern image could be

undistorted using Lispix software, an image analysis and manipulation software

developed and maintained at NIST (David Bright). By collecting diffraction patterns of a

known, single crystal sample at both low and high magnification and at a single, specified

working distance, it was possible to characterize the amount of distortion present in the

high magnification pattern. The two patterns were then compared and a map of the

distortion was created. The software can then apply that map to the distorted diffraction

pattern to produce an undistorted equivalent pattern from the high magnification state.

This new image was reconstructed to represent the pattern collected at the lower

magnification, Figure 2.10 d. Partial loss of the diffraction pattern in the final undistorted

image was a consequence of this routine, however it did not hinder indexing for further

analysis.

42

(a) (b)

(c) (d)

Figure 2.10 Series of <100> single crystal silicon diffraction pattern images collected at 20 kV accelerating voltage and 18 mm working distance showing (a) a low magnification mode diffraction pattern at 2000 × (b) a distorted high magnification mode diffraction pattern at 80,000 × and (c) the high magnification pattern with an array of lines used to create the undistorted pattern shown in (d).

HKL Channel 5 software was used to quantify the spatial relationships of the Kikuchi

bands by applying a Hough transform. The physical location of the phosphor screen, the

sample-to-screen distance, and the phases present in the sample, which in this case

included only silicon, were all used by the program to apply band detection algorithms to

determine the correct Kikuchi patterns for the orientation of the nanowire sample. The fit

resolved by the software described the orientation of the crystal in space by the Euler

angles, representing the angular rotation of the sample to a base coordinate system. This

analysis method is well documented [26, 27], as EBSD is regularly used for grain

orientation mapping.

43

The indices of the nanowire growth direction are determined using the aforementioned

program developed by Mark Vaudin (NIST). The Euler angles from each individual

diffraction pattern, the angle of the nanowire on the sample when the information was

collected, the orientation of the sample in the chamber, and information on the crystal

system of silicon were all provided as input. After applying specific constraints on

allowable error, the program was able to calculate the growth direction of each nanowire

sample. The process is outlined schematically in Figure 2.11.

Figure 2.11 Schematic representation of the layout used for the determination of nanowire crystal orientation inside the FESEM chamber.

2.4 Nanowire Preparation A nano-scale simply supported three-point bend test follows established descriptive

mathematics and is analogous to macro-scale testing. Preliminary tests on nanowires

with < 100 nm average diameters demonstrated that the excessive flexibility of thin

silicon samples exceeded the established fixture constraints. To complete a mechanical

flexure test to failure, it was necessary to alter the experimental conditions to a fixed

three-point bend test by binding the ends of each nanowire to the support beams of the

44

test fixture. The alteration of experimental boundary conditions provided some

advantages, including further simplification of the fixture design. Such changes are

described in detail in Appendix A.

Silicon nanowire samples, deposited onto the fixture surface using the solution deposition

method and aligned across testing spans, were fixed to the support columns with a

“metal” deposition process to provide stability during testing (Figure 2.12). This was

done using electron beam induced deposition (EBID) of platinum, performed in a dual

beam FIB (FEI Nova™ Nanolab 600), where the design of the instrument allowed for

easy introduction of the precursor gas to the chamber. The sample was cleaned in oxygen

plasma (Fischione Instruments Model 1020 Plasma Cleaner) for 2 minutes, mounted into

the FIB vacuum chamber, and focused at 5 mm working distance using an electron

accelerating voltage of 5 keV with either 98 or 25 pA current. For EBID in this system, a

precursor gas injection nozzle was brought approximately 150 µm above the surface of

the sample where the deposition was to take place. The electron beam was then scanned

over the desired area in a specific pattern where secondary electrons caused the precursor

gas, methylcyclopentadienyl platinum (CH3)3Pt(CpCH3), to decompose and locally

deposit metal on the substrate surface. The remaining precursor gas was removed from

the chamber through the vacuum system [28]. EBID is similar to IBID and performed in

the same type of instrument, however the process is completed without the use of an ion

beam and therefore does not cause structural damage to the silicon nanowire. While the

silicon nanowire samples were successfully fixed to the device surface using this process,

two experimental conditions arose from utilizing electron beam deposited platinum which

resulted in subsequent complications for data analysis.

45

Figure 2.12 Scanning electron micrograph of a silicon nanowire fixed across a test span.

The platinum-based deposit composition included a significant amount of carbon as a

result of using organometallic precursor gases [29]. The low purity level of the deposit is

speculated to be due to incomplete decomposition of the precursor, chamber

contamination, or a combination thereof [30]. Depending on the precursor and the

deposition conditions used, experiments have reported between 60-75 atomic % carbon in

the deposit [31]. The microstructure of which consist of fcc platinum nanocrystals either

surrounded by [31] or embedded within [32-34] amorphous carbon. Given the high

proportion of carbon within the metal-based deposit, it was reasonable to assume that the

adhesive had a lower elastic modulus and yield strength than pure platinum metal, and

the performance as an adhesive would therefore be significantly inferior. While it is

possible to reduce the amount of carbon in the deposit through post-treatments [29], it

would require exposing the sample to conditions that could alter the mechanical

properties of the nanowire while improving the purity of the platinum-based deposit.

46

Another issue encountered in binding the nanowires using EBID was the unintentional

spread of the platinum-based adhesive, Figure 2.13 (a). Adhesive deposited outside the

area exposed to the electron beam is speculated to come from two sources. The first is

due to the secondary electron assisted decomposition of the precursor gas. Monte Carlo

simulations predict that the minimum feature size of the deposit must be significantly

larger than the beam diameter. An additional contribution to the broadening of the

adhesive film is a result of thermally assisted diffusion, where the substrate undergoes

local heating due to the impinging electron beam [33]. The result was platinum-carbon

contamination on the surface of the silicon nanowires over a significant portion of each

sample. The platinum-based deposit appeared to wick from the desired deposition

location up a portion of the nanowire testing span, creating cone-like cross section

profiles on the ends of each nanowire, which could possibly change the interpretation of

nanowire strength. To minimize the extent of the platinum-silicon composite in the

effective testing span the adhesive was deposited at a slight offset from the edge of either

side of the gap. While this solution did result in a cleaner testing span, it also created

ledges between where the nanowire was fixed to the surface and the gap for the testing

span began, as seen in Figure 2.13 (b). Each ledge averaged 93 nm in length, which

added over 13% to the measured testing span and had a pronounced, unanticipated effect

on the mechanical test results.

47

Figure 2.13 Scanning electron micrograph of silicon nanowires fixed across the testing gap using the platinum-based adhesive, where (a) the adhesive contaminated a significant area around the intended deposit area and (b) the adhesive was deposited away from the edge of the gap, creating a ledge in the test span.

2.5 Mechanical Testing Method The atomic force microscope (AFM) was used to investigate the mechanical properties of

the silicon nanowire samples. An advantage of using the AFM was the ability to track

48

both force and displacement with high resolution during testing. All testing was

conducted using a fixed-fixed three-point bending configuration, where the nanowire was

attached to the top of the fixture support columns. The Asylum MFP-3D-BIO™ AFM

was used with an inverted optical microscope. It was mounted on top of a Herzan TS-

140 isolation table to actively damp frequencies between 0.7 and 1000 Hz and passively

damp frequencies beyond 1000 Hz. The entire system was then placed in an acoustic

enclosure (Herzan BCH-45), which offered further noise reduction and environmental

stability during testing. The laboratory that housed the system was also maintained at

23.5 ± 0.15° C and 40.0 ± 0.9 % humidity, which was monitored for a 24 hour interval

several times throughout the testing period.

2.5.1 AFM Cantilever Calibration Methods

Accurate calibration of the AFM cantilever tip used for force measurements is crucial, as

uncertainties in the spring constant values can be a major source of error in quantitative

experimental data [35, 36]. Spring constant specifications reported by the manufacturer

are typically given in a range over a factor of 10, which introduces a high level of

uncertainty.

There are numerous accepted approaches to cantilever calibration, each having different

advantages and limitations. Published methods for spring constant measurements have

uncertainties that range from 10 – 40% [35, 36]. Two of the more common techniques

are the Cleveland method [37] and the Sader method [38, 39]. The Cleveland method is

an added mass resonance technique, where a spring constant is determined by first

measuring the natural resonant frequency of the cantilever, spheres of known mass are

attached to the free end of the cantilever, and the change in resonant frequency as a result

is determined. This calibration technique claims an accuracy of within approximately

10% and is applicable to cantilevers of any shape, but is unfortunately destructive and not

particularly easy to perform [37].

The Sader method also uses the resonant frequency of the cantilever to determine the

spring constant, however it requires dimensional measurements of the cantilever and the

49

calculation of a quality factor of the fundamental flexural mode in air, which is not

available with every instrument. The Sader method of calculating the spring constant for

rectangular cantilevers is presented in Equation 2-3 [38], where k is the spring constant, ρf

is the fluid density, b and L are the cantilever width and length, Qi is the quality factor in

air, Γi is the imaginary component of the hydrodynamic function, and ωf is the cantilever

fundamental mode resonant frequency. It should be noted that Γi can be calculated with

knowledge of the fluid viscosity, and both Qi and ωf can be measured in the Asylum

AFM. The fluid density and viscosity for air, 1.18 kg/m2 and 1.86E-5 kg/m·s

respectively, are applied for all calibration calculations using this method [38].

The Sader method is a more user-friendly calibration technique, which results in usable

cantilevers. However, the accuracy of the method ranges from 10-20% and depends

greatly on the correct determination of dimensions, the effective mass for the cantilever,

and is only valid for rectangular cantilevers [35, 39]. The Sader method of calibration

was used in this research to double check the in-situ spring constant calculations that

were ultimately performed on each cantilever.

Equation 2-3 ( ) ffiff LQbk ωωρ Γ= 21906.0

2.5.1.1 Thermal Noise Calibration Method

To collect force data from AFM experiments, the method of calibration used involved

measuring the thermal fluctuation characteristics of each cantilever. It is a nondestructive

method and can be applied in-situ. This ‘thermal’ method, which was first proposed by

Hutter and Benchhoefer [40] and later refined by Butt and Jaschke [41], uses the

experimental thermal power spectrum of each individual cantilever to determine stiffness

and is based on the equipartition theorem of statistical mechanics. The basis of the

equipartition theorem is that in thermal equilibrium, energy is shared equally among all

of its various forms. More specifically, each quadratic degree of freedom contributes ½

kBT to the total energy of a system, where kB is the Boltzmann constant and T is the

absolute temperature. The potential energy of a simple harmonic oscillator is ½ kx2,

50

therefore for the AFM cantilever ½ kBT = ½ k <x2>, where k is the cantilever spring

constant and <x2> is the mean square deflection of the cantilever caused by thermal

vibrations.

There are several correction factors that must be taken into account when using the

thermal method for calibrating a cantilever [41]. First, an actual cantilever does not

behave like an ideal spring; therefore the potential energy is not simply ½ kx2. Second is

the measurement method of the cantilever deflection. Cantilevers are mounted at an

angle to the specimen and the deflection of the cantilever is monitored using an optical

lever technique, where a laser is reflected off of the free end of the back side of the

cantilever and the angle of reflection is measured by a photodiode. Due to the geometry

of cantilever placement, a correction must be made for measuring the angle of the

cantilever, rather than the pure deflection normal to the long axis of the beam.

Additionally, each time a new cantilever is mounted or the laser spot repositioned, the

photodiode of the AFM must be calibrated by performing a force curve on a rigid

substrate to find the ratio between the vertical displacement of the scanner and the

voltage of the photodiode. When the thermal fluctuations of each cantilever are

measured, it is important to account for both thermal noise and deflection sensitivity [41,

42].

Monitoring of these variables leads to a correction factor which can be used with the

equipartition theorem to accurately measure the spring constant when cantilever

deflection is measured with the optical lever technique in any vibrational mode. The

thermal method of calibration has a reported 5-10% accuracy [40, 42] and approximately

5% precision [40, 42, 43].

2.5.2 Asylum AFM Cantilever Calibration Procedure

The accuracy of the cantilever calibration is essential in this research, as both force and

deflection data collected during mechanical testing are directly dependent on the

measured spring constant. The thermal noise calibration method was utilized to

determine the spring constant of each cantilever. The Sader method calculation then

51

confirmed each calibration. The following section provides a summary of the detailed

cantilever calibration procedure that was used prior to each mechanical test.

A fixture containing nanowire samples was positioned on a glass microscope slide and

secured in place on the AFM base using magnets on either end of the slide. The silicon

tapping mode cantilever (PPP-NCH Nanosensors™) was mounted in a supplied holder at

11°, attached to the AFM head, and positioned over the fixture, ensuring sufficient

clearance from the cantilever tip. The cantilever was then brought into focus and the

super luminescent diode (SLD) spot was moved to the free end of the cantilever,

maintaining a high sum voltage, which indicated the intensity of the reflected light on the

position sensitive segmented photodiode. This deflected signal corresponded to the angle

of the cantilever and any movement of the spot was tracked by the photodiode and

translated into a deflection signal in volts. The amount of signal generated with the spot

movement is called the optical lever sensitivity (OLS).

When the SLD spot was in the correct position the AFM head was manually lowered

until the cantilever was 1-2 mm above the sample surface. A bubble level was positioned

on top of the AFM head to ensure that the instrument was kept level throughout this

process. After closing the hood, the tapping mode cantilever was tuned to a target

amplitude (typically 1 V) and target percentage (-5%). The tapping piezo achieved 1.05

V of amplitude at resonance and drove the tip at a frequency less than resonance,

resulting in the 1 V target. The target percentage was set to lower the drive amplitude in

order to engage the tip on the surface in repulsive mode, which protected both the tip and

the surface from damage. This tuning process also determined an approximate resonant

frequency for the cantilever. The resonant frequency was recorded and later used to

determine the spring constant of the cantilever for the Sader method. The thermal power

spectral density (PSD) was then collected to once again measure the resonant frequency

of the cantilever and also to confirm that the cantilever and light source were properly

aligned.

52

The inverse optical lever sensitivity (InvOLS) of the system was evaluated by performing

an indent on a hard, noncompliant surface and tracking the cantilever deflection. This

provided a baseline for all future deflection using the specific cantilever and PSD

location. The InvOLS was necessary to convert the data collected in a force plot from

volts to distance.

To perform an indent the tip was brought into contact with the sample surface. In tapping

mode the engagement setpoint was set to 950 mV, or 95% of the drive amplitude. The

hood was then opened and the head was manually lowered toward the surface, tracking

the progression of the cantilever amplitude until it reached the previously chosen

setpoint. The software control was then used to withdraw the tip from the surface and the

hood was closed. After re-engaging the tip, the setpoint was lowered incrementally with

the software to increase the force on the surface until the cantilever achieved acceptable

tracking. This entire process was completed to ensure that only the smallest possible

vertical forces were applied to both the tip and the sample while finding the surface,

minimizing any possible damage. The tip was then retracted from the surface, the

software was used to switch the operation to contact mode, and the tip was re-engaged to

complete an indent and establish the deflection sensitivity of the cantilever. The

calibration included all system variations, so changes to any component required

repetition of the method to calculate a new sensitivity. Upon withdrawing the tip from

the surface, the hood was opened and the tip was manually moved away from the surface.

A second measure of the thermal PSD was used to confirm the resonant frequency of the

cantilever and ensure no damage incurred during the collection of the force curve. This

resonant frequency, along with the InvOLS value, could be used to calculate the

cantilever spring constant that was used in the initial interpretation of all force-deflection

data.

2.5.3 Minimizing Instrument Drift

Upon preliminary scans with the calibrated Asylum instrument, a noticeable drift existed,

which would cause tip placement and stability to be extremely difficult for nanowire

mechanical testing. Initial drift in the instrument was expected. After start-up, the

53

system experienced a slight increase in temperature, which created drift. Additionally, it

was proven helpful to warm up the piezo used to for the scanning movement by

performing air scans for at least one hour prior to use. However, while a certain amount

of drift in the instrument was expected and could be avoided with these simple

procedures, the drift over a 12 hour period after the initial instrument start up and

temperature stabilization remained over 1 µm in both planes of the scan. Much of the

vertical drift was eliminated with a steady temperature. After 4 hours, the temperature of

the instrument and surrounding enclosed environment varied less than 0.1° C over a 24

hour period.

Following extensive trouble-shooting, several key changes were made to both the

mechanical system and procedure in order to minimize the extent of drift present in the

instrument. First, the cantilever holder was replaced. The stock cantilever holder when

the instrument was purchased was made from Kel-F®, or polychlorotrifluoroethylene.

The replacement holder was PEEK, which is a fiber reinforced polyetherether-ketone. At

the temperatures and environments used in this research, this change of material should

not have a significant impact, however it was recommended as an initial step in the

attempt to reduce drift. Next, the lower bushings on the instrument head legs were

tightened, both the legs and companion contact points were cleaned, and the instrument

head was leveled, removing the remainder of the drift that may be associated with the

mechanical parts of the AFM. Finally, the computer controlled long-range stage

movement system was switched off. Though this last step was an inconvenience for

testing, the electrical instabilities of the system caused significant stage motion while at

rest.

The drift in the system was re-calibrated following each change noted above. After

allowing four hours for temperature and instrument stabilization, the final drift

measurements were under 200 nm in all directions over a 12 hour period. The

temperature within the acoustic and environmental isolation enclosure was stable to

within 0.2° C over the same time period. While it would be ideal for the system to

54

contain no drift at all, the minimal extent of drift remaining following the outlined

changes was sufficient for the purposes of this research.

2.5.4 Centrally Loaded, Fixed-Fixed Beam Bending Procedure

After the mounted cantilever was calibrated, an individual nanowire sample was

positioned under the tip using an optical microscope mounted above the AFM head. This

system did not have the capability to provide high magnification, however the specific

testing spans and nanowire sample sites were previously identified by SEM during the

process of fixing the nanowire to the support column. Therefore visually approximating

the position of the cantilever tip over the test span under low magnification was sufficient

for initial contact to the surface. The fixtures were aligned with the support columns,

parallel to the cantilever. Once the sample was in place and the hood was closed around

the instrument, the system was left for 2 hours to stabilize the temperature. The

cantilever tip was then brought into light contact with the surface while in tapping mode,

as described previously, and scanned over the surface using an 8 µm2 grid. The initial

scan size was chosen to encompass both support columns and the test gap in between.

While scanning, the setpoint voltage was slowly decreased to improve the surface-tip

interaction. The scan rate and gain were also varied to reduce noise and provide clear

images. The cantilever was adjusted in the plane of the nanowire and device fixture to

locate the nanowire sample using successive 8 µm2 scans. When the sample position was

established, the scan size and rate were reduced to collect an image of the nanowire and

testing span prior to mechanical testing (Figure 2.14).

55

Figure 2.14 Pre-test AFM tapping mode scan of fixed nanowire.

Following the collection of a pre-testing image, the cantilever was disengaged from the

surface, the drive software was changed from tapping to contact mode, and the cantilever

tip position was moved over the nanowire to the center of the testing gap using the

instrument computer software. To break the nanowire in flexure, the vertical motion

piezo in the instrument head (Z piezo) was then lowered a predetermined distance or until

a set maximum voltage associated with cantilever deflection was reached, whichever

came first. The instrument recorded the cantilever deflection and the distance traveled by

the Z piezo, which could be converted into deflection and applied force through the

previous instrument calibration. The bend test was set up and performed immediately

following the tapping mode scan in order to minimize the possibility of accumulated drift

in the system, improving accuracy of tip placement at the center of the testing gap and in

the middle of the nanowire. The accuracy was time dependent due to systematic changes

in vibration and temperature. In case of interruption during the testing process, the tip

must be relocated to the center of the nanowire sample. Ultimate failure of the nanowire

was observed via the force curve (Figure 2.15 (a)). A sudden drop indicated either

successful failure of the nanowire in bending or the disengagement of the cantilever tip

from the sample. To discern between the test outcomes, the AFM was set to tapping

56

mode and the nanowire was rescanned. Fracture was easily identified. If the tip was

dislodged prior to failure, it was relocated and the test was repeated. Force curves

performed which resulted in no dramatic effect to the nanowire (Figure 2.15 (b)) were

repeated after changing the parameters to either increase the distance traveled by the Z

piezo or the maximum allowable cantilever deflection, thereby increasing the applied

force to the nanowire. There were some nanowire and cantilever combinations that did

not result in fracture. This might have been due to nanowire thickness, span length,

cantilever stiffness, inadequate nanowire constraints, or possibly poor tip placement

during testing. If a nanowire did not fail in flexure after three attempts, the sample was

abandoned to avoid the possibility of accumulated damage affecting the test results.

57

Figure 2.15 AFM force curves collected during a fixed three-point bend test which resulted in (a) nanowire fracture and (b) no nanowire fracture. The information is originally collected using deflection volts as a function of the linear variable differential transformer (LVDT) sensor. The data is converted into applied force as a function of nanowire deflection using instrument calibration information. The blue line represents the cantilever approach and extension onto the sample, while the red line shows the cantilever retraction.

A single cantilever may have been used to collect data from several nanowire samples

until it was apparent that the tip had been damaged, which was readily evident during

58

pre- and post- testing tapping mode scans. Before each cantilever was removed from the

instrument, a second series of calibration experiments were performed to verify the initial

calibration. Post calibration testing was initiated by a thermal PSD to confirm the

resonant frequency. A significant change from the initial measured value was indicative

of severe damage or contamination of the tip. The tip was then positioned over a clean

silicon surface and the deflection sensitivity of the cantilever was measured using large

deflection and a minimum of ten force curves, in contrast with the two pre-testing

calibration force curves performed, where small deflection was used to protect the

cantilever tip. The initial calibration provided high confidence values for applied force

and deflection of each nanowire sample during mechanical testing. However, post-

testing force curve calibration was used in the data analysis, as it was composed of an

average sensitivity value over a larger range of cantilever deflection.

2.5.5 Centrally Loaded, Fixed-Fixed Beam Bending Analysis Mechanical evaluation and analysis were performed to obtain a realistic value of the

silicon nanowire flexure strength. An understanding of nanowire strength would allow

for the calcuation of a failure probability, and therefore reliability, for silicon components

at the nanoscale. Elastic modulus was another parameter of interest, as there is debate

over the trends of the property at the nanoscale. However, although it was principally

possible to extract a numerical value for E from the mechanical test data, it was not a

reliable measure in this research, for reasons that will be discussed later.

Prior to any data analysis, fractured nanowire samples were inspected in a FESEM (Zeiss

Ultra-60), Figure 2.16. The resulting images were used to measure the test span,

nanowire diameter, and location of fracture. The images also provided insight into the

amount of metal that was deposited onto the nanowire when fixing the samples to the

surface, as well as the size of the ledge created by binding the samples a small distance

back from the edge of the gap.

59

Figure 2.16 Scanning electron micrograph of a fractured silicon nanowire.

The location of nanowire fracture was relevant to the final interpretation of data.

Mechanical tests were run in a three-point bending configuration, where the applied force

was located in the center of the testing span. Samples evaluated in this manner typically

fracture at the point of applied force or in the location of a flaw, which could cause

reduced mechanical strength. The tested nanowires fractured in three ways; a clean break

at the approximate center of the testing span, a clean break at the edge of the testing span,

or a break which resulted in a section of the sample completely missing (Figure 2.17 (a),

(b), and (c)). Data from samples which broke at the edge of the testing span were not

included in the final analysis, as the location of fracture is not valid for a three-point bend

test.

60

Figure 2.17 Scanning electron micrographs showing examples of the three types of nanowire fracture which occurred during experimental testing; (a) center fracture, (b) edge fracture, (c) section fracture.

Raw data resulting from mechanical tests in the AFM included the force applied to the

sample through the vertical motion of the instrument head and attached cantilever, as well

as a combination of the deflection from the nanowire and cantilever tip. This data was

first converted into the applied force onto and deflection of only the nanowire sample

using prior knowledge of the calibrated cantilever behavior. Because information was

collected before, during, and after the actual mechanical evaluation of each nanowire,

extraneous data was eliminated from the files using graphical interpretation. A rough

estimate of the location of cantilever contact and nanowire fracture were the bounds

established on either end of each data set (Figure 2.18).

61

Figure 2.18 Applied force as a function of nanowire deflection data collected for an entire flexure test. The dashed red lines indicate the area of interest for the nanowire deflection and fracture.

To obtain accurate final measures of deflection and applied force from the bend test, it

was necessary to establish a zero point in the test data, where the nanowire actually began

to deflect. For small deflections, on the order of the sample radius, the centrally loaded

fixed-fixed bend test should comply with linear elastic beam theory [44, 45]. Therefore

the slope at the beginning of the applied force versus deflection curve will approximate

the applied forces and deflections calculated using Equation 2-4 [46] for a center loaded

beam, where I is the nanowire area moment of inertia. All data collected before this was

discarded as approach, noise, or initial ‘snap-on’ of the cantilever tip to the nanowire

surface. Establishing a zero point for both the force and deflection enabled the accurate

interpretation of an ultimate force at fracture and maximum nanowire deflection, Figure

2.19.

Equation 2-4 EI

FL192

3−=δ

62

Figure 2.19 Applied force as a function of nanowire deflection for a fixed nanowire tested in three-point bending, with the extraneous data eliminated and the axes re-set for the beginning of the nanowire deflection.

The mechanical testing results initially displayed the expected linear elastic behavior

between the applied force and nanowire deflection. With larger deflection, the response

of the applied force became increasingly non-linear, ultimately resulting in brittle

fracture. Silicon does not plastically deform at room temperature, therefore the majority

of the force-deflection data collected occurred within the non-linear elastic beam

deflection regime. Because the nanowires were attached to the fixture surface, the

applied bending force on the nanowire caused an extension of the sample. Following

linear elastic deflection, this fixed nanowire extension produced an axial force along the

length of the nanowire, in addition to the applied transverse bending force. At greater

nanowire deflections, components of both the bending and axial tension contributed to an

enhanced apparent stiffness and the overall fracture strength of the sample. Using the

theory of large deflections it was possible to describe the entire elastic curve, through

both the linear and non-linear behavior [47].

To accurately interpret the behavior of the fixed nanowire in bending necessitated the use

of a transcendental equation. Approximated solutions have been developed in previous

research [21, 22]. The applicable equations provided generalized solutions for the

63

nanowire flexure strength or the elastic modulus, both as a function of the experimentally

measured force and deflection. The ultimate strength (σ) of the nanowires were

determined using Equation 2-5 [22]. These series of equations exploited the

experimentally measured applied force and deflection only, without requiring the

material elastic modulus as input.

Equation 2-5 ( )απ

σ gr

FL32

=

where

( )( ) ( )

( )

21

2 4cosh

2sinh62cosh2

4tanh4

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛−+

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

ααααα

αα

αg

( )εεεα

33501406++

=

22⎟⎠⎞

⎜⎝⎛=

rδε

Similarly, Equation 2-6 [21] predicted the nanowire elastic modulus based on the bend

test data. As these models were approximate numerical solutions to a transcendental

function, there was a reported maximum 2.1% error associated with the resulting curve

fit.

Equation 2-6 ( )δαfL

EIF 3

192=

where

64

( ) ( )α

ααα

4tanh19248 −=f

The interpretation of all the mechanical data, as well as the Sader method of cantilever

calibration, relied on the measurement of sample, fixture, and cantilever dimensions.

These measurements were performed with image analysis software (ImageJ) using

FESEM images. Error associated with image analysis measurement was on the order of

2%, which included nanowire diameter and testing span for all of the calculations of

force, strength, or deflection. The force-deflection curves fit the theoretical solutions

when the range of error associated with dimensional and force measurements was

included, Figure 2.20. Dimensional measurements provided a main contribution to the

error associated with the final data.

Figure 2.20 Applied force as a function of nanowire deflection showing the experimental data from Figure 2.19 (solid black line), the linear elastic curve fit (dashed red line), and the non-linear elastic curve fit (dotted blue line) utilizing established analytical theories for the fixed beam mechanical behavior.

While the test data from the force-deflection curve of the fixed nanowire in bending

could be adequately interpreted using the existing theories and methods outlined above,

65

there were several sources of error which did not originate from physical dimension

measurements or experimental testing error that may have had significant impact on the

interpretation of the results. Analytical models were employed to explore the effect of

several parameters in the nanowire mechanical bending tests, including the possibility of

off-center loading, the development of ledges during sample preparation, and the impact

of axial tension on the allowable ultimate strength.

66

References

1. Wagner, R.S. and W.C. Ellis, Vapor-Liquid-Solid Mechanism of Single Crystal Growth. Applied Physics Letters, 1964. 4(5): p. 89-90.

2. Lew, K.-K. and J.M. Redwing, Growth characteristics of silicon nanowires synthesized by vapor-liquid-solid growth in nanoporous alumina templates. Journal of Crystal Growth, 2003. 254(1-2): p. 14-22.

3. Lew, K.-K., et al., Template-directed vapor--liquid--solid growth of silicon nanowires. Journal of Vacuum Science & Technology B: Microelectronics and Nanometer Structures, 2002. 20(1): p. 389-392.

4. Redwing, J.M., et al. Synthesis and properties of Si and SiGe/Si nanowires. in Quantum Dots, Nanoparticles, and Nanoclusters. 2004. San Jose, CA, USA: SPIE.

5. Bogart, T.E., S. Dey, K.-K. Lew, S. E. Mohney, J. M. Redwing,, Diameter-Controlled Synthesis of Silicon Nanowires Using Nanoporous Alumina Membranes. Advanced Materials, 2005. 17(1): p. 114-117.

6. Hull, R., ed. Properties of Crystalline Silicon. Electronic Materials Information Service, ed. B.L. Weiss. Vol. 20. 1999, INSPEC: London.

7. Williams, K.R., K. Gupta, and M. Wasilik, Etch Rates for Micromachining Processing - Part II. Journal of Microelectromechanical Systems, 2003. 12(6): p. 761-78.

8. Ziegler, J.F., M.D. Ziegler, and J.P. Biersack, SRIM: The Stopping Range of Ions in Matter. 1984.

9. Pohl, H.A., The Motion and Precipitation of Suspensoids in Divergent Electric Fields. Journal of Applied Physics, 1951. 22(7): p. 869-871.

10. Pohl, H.A., Dielectrophoresis : the behavior of neutral matter in nonuniform electric fields. 1978, Cambridge: Cambridge University Press.

11. Englander, O., Electric-field assisted growth and self-assembly of intrinsic silicon nanowires. Nano Letters, 2005. 5(4): p. 705-708.

12. Fan, D.L., et al., Manipulation of nanowires in suspension by ac electric fields. Applied Physics Letters, 2004. 85(18): p. 4175-4177.

13. Fan, D.L., et al., Controllable High-Speed Rotation of Nanowires. Physical Review Letters, 2005. 94(24): p. 247208-4.

14. Mohney, S.E., et al., Measuring the specific contact resistance of contacts to semiconductor nanowires. Solid-State Electronics, 2005. 49(2): p. 227-232.

15. Raychaudhuri, S., et al., Precise Semiconductor Nanowire Placement Through Dielectrophoresis. Nano Letters, 2009. 9(6): p. 2260-2266.

16. Smith, P.A., et al., Electric-field assisted assembly and alignment of metallic nanowires. Applied Physics Letters, 2000. 77(9): p. 1399-1401.

17. Suehiro, J., et al., Dielectrophoretic fabrication and characterization of a ZnO nanowire-based UV photosensor. Nanotechnology, 2006. 17(10): p. 2567-2573.

18. Uran, C., et al., On-Chip-Integrated Nanowire Device Platform With Controllable Nanogap for Manipulation, Capturing, and Electrical Characterization of Nanoparticles. Selected Topics in Quantum Electronics, IEEE Journal of, 2009. 15(5): p. 1413-1419.

67

19. Motayed, A., et al., Realization of reliable GaN nanowire transistors utilizing dielectrophoretic alignment technique. Journal of Applied Physics, 2006. 100(11): p. 114310/1-114310/9.

20. Ramos, C.A., E. Vasallo Brigneti, and M. Vázquez, Self-organized nanowires: evidence of dipolar interactions from ferromagnetic resonance measurements. Physica B: Condensed Matter, 2004. 354(1-4): p. 195-197.

21. Heidelberg, A., et al., A Generalized Description of the Elastic Properties of Nanowires. Nano Lett., 2006. 6(6): p. 1101-1106.

22. Ngo, L.T., et al., Ultimate-Strength Germanium Nanowires. Nano Letters, 2006. 6(12): p. 2964-2968.

23. Wu, B., A. Heidelberg, and J.J. Boland, Mechanical properties of ultra-high strength gold nanowires. Nature Materials: Letters, 2005. 4: p. 525-529.

24. Wu, B., et al., Microstructure-Hardened Silver Nanowires. Nano Lett., 2006. 6(3): p. 468-472.

25. Green, D.J., An Introduction to the Mechanical Properties of Ceramics. Cambridge Solid State Science Series, ed. D.R. Clarcke, S. Suresh, and I.M. Ward. 1998, Cambridge: Cambridge University Press.

26. Schwartz, A.J., et al., eds. Electron Backscatter Diffraction in Materials Science. Second Edition ed. 2009, Springer: New York.

27. Randle, V., Microtexture Determination and its Applications. Second Edition ed. 2003, London: Maney Publishing.

28. Bieber, J.A., J.F. Pulecio, and W.A. Moreno. Applications of Electron Beam Induced Deposition in nanofabrication. in Devices, Circuits and Systems, 2008. ICCDCS 2008. 7th International Caribbean Conference on. 2008.

29. Botman, A., M. Hesselberth, and J.J.L. Mulders, Improving the conductivity of platinum-containing nano-structures created by electron-beam-induced deposition. Microelectronic Engineering, 2008. 85(5-6): p. 1139-1142.

30. Botman, A., J.J.L. Mulders, and C.W. Hagen, Creating pure nanostructures from electron-beam-induced deposition using purification techniques: a technology perspective. Nanotechnology, 2009. 20(37): p. 372001.

31. Langford, R.M., T.X. Wang, and D. Ozkaya, Reducing the resistivity of electron and ion beam assisted deposited Pt. Microelectronic Engineering, 2007. 84(5-8): p. 784-788.

32. Tham, D., C.-Y. Nam, and J.E. Fischer, Microstructure and Composition of Focused-Ion-Beam-Deposited Pt Contacts to GaN Nanowires. Advanced Materials, 2006. 18(3): p. 290-294.

33. Gopal, V., et al., Metal delocalization and surface decoration in direct-write nanolithography by electron beam induced deposition. Applied Physics Letters, 2004. 85(1): p. 49-51.

34. Xie, G., et al., Characterization of nanometer-sized Pt-dendrite structures fabricated on insulator Al2O3 substrate by electron-beam-induced deposition. Journal of Materials Science, 2006. 41(9): p. 2567-2571.

35. Gibson, C.T., G.S. Watson, and S. Myhra, Determination of the spring constants of probes for force microscopy/spectroscopy. Nanotechnology, 1996. 7(3): p. 259-262.

68

36. Holbery, J.D., et al., Experimental determination of scanning probe microscope cantilever spring constants utilizing a nanoindentation apparatus. Review of Scientific Instruments, 2000. 71(10): p. 3769-3776.

37. Cleveland, J.P., et al., A nondestructive method for determining the spring constant of cantilevers for scanning force microscopy. Review of Scientific Instruments, 1993. 64(2): p. 403-405.

38. Sader, J.E., J.W.M. Chon, and P. Mulvaney, Calibration of rectangular atomic force microscope cantilevers. Review of Scientific Instruments, 1999. 70(10): p. 3967-3969.

39. Sader, J.E., et al., Method for the calibration of atomic force microscope cantilevers. Review of Scientific Instruments, 1995. 66(7): p. 3789-3798.

40. Hutter, J.L. and J. Bechhoefer, Calibration of atomic-force microscope tips. Review of Scientific Instruments, 1993. 64(7): p. 1868-1873.

41. Butt, H.J. and M. Jaschke, Calculation of thermal noise in atomic force microscopy. Nanotechnology, 1995. 6(1): p. 1-7.

42. Burnham, N.A., et al., Comparison of calibration methods for atomic-force microscopy cantilevers. Nanotechnology, 2003. 14(1): p. 1-6.

43. Matei, G.A., et al., Precision and accuracy of thermal calibration of atomic force microscopy cantilevers. Review of Scientific Instruments, 2006. 77(8): p. 083703-6.

44. Beer, F.P. and J. Johnston, E. Russell, Mechanics of Materials. Second ed. 1992, New York: McGraw-Hill, Inc.

45. Frisch-Fay, R., Flexible Bars. 1962, Washington D.C.: Butterworth Inc. 46. Young, W.C. and R.G. Budynas, Roark's Formulas for Stress and Strain. 2002:

McGraw Hill, Inc. 47. Landau, L.D. and E.M. Lifshitz, Theory of Elasticity. 3 ed. Course of Theoretical

Physics. Vol. 7. 1986, Oxford: Pergamon Press.

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3 Analytical Modeling The development of theoretical mathematical models to mimic mechanical testing

enables the interpretation of behavior that may not be readily apparent from the

experimental data. Analytical models are a useful way to control and observe the effect

of multiple variables, which includes not only re-evaluating experimental data, but also

analyzing anomalies that have not been accounted for in the previously established

systems. Analytical models are also particularly useful for analysis at the nano- length

scale, as it is difficult to visualize experimental mechanical tests or the resulting fracture

surfaces, which are commonly used to aid in data interpretation of macro- and micro-

scale ceramic flexure tests. In this research, models were used to compare applied theory

with mechanical test data and to explore additional parameters that were not initially

included in the analyses.

Analytical models were used to assess the error associated with individual aspects of non-

ideal preparation and mechanical testing that did not precisely align with convention,

including the possibility of off-center loading, the addition of ledges to the testing span,

and the effect of axial tension due to large deflection of the nanowire samples. The

models were developed based on established theory for a centrally loaded, fixed-fixed

beam bending configuration with loading in both the linear elastic and large deflection

regimes [1, 2]. Each analysis was used to evaluate the possible magnitude of individual

errors and the impact the errors have on the determination of an accurate flexure strength

from experimental measurements, where ideal boundary conditions were assumed. The

models were constructed utilizing a 60 nm diameter silicon nanowire grown in the [110]

direction, with an elastic modulus of 168.9 GPa [3]. In all linear elastic models, the

maximum allowable deflection did not exceed 75% of the nanowire radius, remaining

well within the limits of small-deflection theory [1, 4]. The large deflection analytical

model was not subject to the same constraints. All analysis was performed using

scientific computing software [Mathmatica 7.0, Mathcad 12].

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3.1 Linear Elastic Analytical Models The simple model of a brittle material with linear elastic behavior is applied to the design

and use of many ceramics. For most ceramic samples linear elasticity adequately

describes the behavior [5]. They also are a logical starting point for understanding

nanoscale specimens. The established principal mechanics equations are applied here to

generate a basic appreciation for the accuracy of results based on ideal testing.

3.1.1 Off-Center Loading

There are several variables used in the interpretation of flexural strength where relatively

minor errors may lead to significant misinterpretation of the available data. The test

conditions of three-point bending require that the load be applied to the specimen at the

center of the testing gap. Experimentally, using an AFM to accurately place a 7 nm

radius cantilever tip in the exact center of a < 2 µm gap atop of a curved sample was

problematic. In addition, accurate cantilever placement could not be confirmed without

in-situ high magnification imaging, which was not available in the testing system used for

this research. It was safe to assume that over a 1-2 µm span, the cantilever was placed

with some accuracy in the test span center. However, it is important to determine the

magnitude of the effect of violating the assumption and imparting off-center loading on

the deflection of the nanowire during testing.

Using an established beam mechanics equation (Eq 2-4 [6]), a linear elastic analytical

model of a 60 nm diameter fixed silicon nanowire was established in three-point bending

with a 2 µm testing span and 50 nN applied load. The load location was systematically

changed to reflect increasingly inaccurate cantilever tip placement (Figure 3.1 (a)). With

an accurately placed 50 nN center load, the calculated nanowire deflection was 19.39 nm.

The point of contact was shifted until an arbitrary threshold of a 1% reduction (0.20 nm)

in the calculated nanowire deflection was reached. This threshold was achieved a

distance of 67 nm away from the center of the span along the nanowire axis. The

location of the maximum deflection is also shifted toward the applied load, but did not

track directly beneath the tip, as shown in Figure 3.1 (b). Over the limited test span size,

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67 nm in either direction away from center is a significant portion of the total length,

therefore it was reasonable to assume that the cantilever tip can be repeatedly placed

within this range of error from the exact center of the span. Additional sources of error

were possible, including the calibration of the instrument optics and computer controls.

As the potential range of cantilever tip misalignment was widened to 100 nm, the

maximum deflection of the nanowire was only reduced by a total of 0.43 nm. In other

words, a 5% off-center load would produce 2.22% error in the resulting deflection

measurement. In addition, this was a linear elastic model, which will only mimic the

experimental behavior for a small portion of the overall deflection. Therefore the error in

the deflection measurement associated with off-center loading will be at a maximum.

72

(a)

fixed nanowire ends

center of test gap

applied load

Figure 3.1 Linear elastic analytical model for the effect of off-center loading on a nanowire in a fixed-fixed bending configuration, including (a) a schematic representation and (b) the resulting elastic curves for increasingly inaccurate load placement.

3.1.2 Ledges

Non-ideal boundary conditions on a sample necessarily generate error in calculations and

assumptions associated with analysis of that sample. While adhering the silicon

73

nanowires across the fixture test spans the samples were bound slightly back from the

edge of the gap in order to reduce the amount of platinum-based composite deposited on

the nanowire surface within the test span. The alteration of experimental conditions

mitigated the surface contamination, but it generated a new set of boundary conditions for

the fixed-fixed beam bending test. When a nanowire lies across the testing gap, but is

fixed at a point beyond the edge, the deflection of the nanowire does not conform to the

conventional behavior of a three-point bend test. The additional “ledges” affect the

defection of the flexible silicon nanowires. Therefore accuracy in the measurement of the

experimental testing span and the effect of the ledges on the nanowire deflection must be

well understood to assess the error associated with and properly interpret the

experimental data under this altered boundary condition.

Working within the small deflection linear elastic regime, an analytical model was

developed for this second source of experimental error. To account for the ledges within

a fixed-fixed beam bending scheme, the model required adapting the basic linear elastic

mechanics equation. A statically indeterminate solution was required to interpret

nanowire deflection at the multiple support and load locations. This model is detailed in

Appendix B.

3.1.2.1 Variable Inner Span Error Analysis

There are two possible methods to account for the overall testing span of a nanowire

fixed with ledges. In this first configuration, the length of the nanowire and the location

of the platinum tape remained the same, setting the outer testing span to a constant 2 µm.

However the total length of the ledges on either end of the test gap was increased from

zero to 100 nm. Therefore, while the total outer testing span was fixed at 2 µm, the inner

test span, where the nanowire was able to deflect under loading, was reduced from 2 µm

to 1.8 µm with the addition of two 100 nm ledges (Figure 3.2 (a)). The nanowire

dimensions remained the same as in the previous model, with the total nanowire diameter

fixed at 60 nm and the 50 nN load applied only in the center, following an ideal fixed-

fixed bending configuration.

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With no ledges, the nanowire deflected the same distance as the previous model under a

50 nN center load, at 19.39 nm. As the ledges were extended into the testing span gap,

the maximum calculated nanowire deflection was reduced. With 50 nm ledges (1.9 µm

inner test span) the nanowire deflected 17.27 nm. This was 10.92% less than the

deflection that occurred using the ideal 2 µm span, with no ledges. Increasing the ledge

length to 100 nm, essentially reducing the gap by 10% from 2 µm to 1.8 µm, the

nanowire deflection was reduced over 21% to 15.28 nm (Figure 3.2 (b)). If, in addition

to the reduction in the inner testing span, the nanowire was also loaded off-center as in

the previous model, the overall deflection was further reduced. A 10% reduction in the

testing span, to 1.8 µm, and a load applied 100 nm to the left of center, or 5.56% off-

center, produced a 23.31% error in the measured deflection. In this analytical model, it

was clearly shown that the development of ledges had a large effect on the measured

nanowire deflection. It was therefore important to accurately assess how the total testing

span was accounted for in the calculation of bending strength.

(a)

outer test span

fixed nanowire end ledge supports

no ledges

small ledges

large ledges

75

Figure 3.2 Linear elastic analytical model for the effect of ledges within the testing span on a nanowire in a centrally loaded, fixed-fixed beam bending configuration, including (a) a schematic representation, (b) the resulting elastic curves as the inner testing span is reduced, and (c) a closer view of the effect of the ledges at the fixed edge.

3.1.2.2 Variable Outer Span Error Analysis

The following analysis utilized the same numerical model for ledges, however with this

configuration, rather than containing the total outer testing span to 2 µm and developing

the ledges within that total length as before, the inner testing span (ledge to ledge) was set

76

to 2 µm and the ledges were added as extended length to the outer testing span (Figure

3.3 (a)). As the ledges increased in length and the effective testing span increased, it

allowed for larger nanowire deflection. This analytical model mimicked the actual

experimental testing scheme more closely than the previous configuration, as the

experimental testing span was measured as only the gap in the device fixture. During

actual experiments, the total gap designed for testing on a fixture was set and could not

change, but the location where the nanowires were pinned down varied with each sample,

thereby changing the individual ledge length in each sample produced.

For the variable outer span model, a less dramatic response in the nanowire deflection

was calculated than with the previous variable inner span configuration. For 50 nm

ledges the nanowire deflected 20.11 nm is 0.72 nm (or 3.7%) more than a nanowire in the

ideal fixed configuration with no ledges. Ledges 100 nm in length added 10% to the

outer testing span. The resulting deflection was 20.81 nm, or a 7.32% increase over the

ideal configuration. In comparison, when the ledge length was compensated by the same

amount in the variable inner span model, the nanowire deflection was reduced by over

21%. Additionally, with 100 nm ledges in the variable outer span model, a 5% off-center

load increased the deflection by 5%, which was a significantly smaller error than the

previous configuration. The effect of changing the ledge length on the measured

nanowire deflection is shown in Figure 3.3 (b). The changes in testing span, load

location, and measured deflection for each linear elastic analytical model is summarized

in Table 3.1. The errors associated with the calculations are presented in Table 3.2.

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(a)

fixed nanowire ends

ledge supports

inner test span

no ledges

small ledges

large ledges

78

Figure 3.3 Linear elastic analytical model for the effect of ledges on a nanowire in a centrally loaded, fixed-fixed beam bending configuration, including (a) a schematic representation, (b) the resulting elastic curves as the location of nanowire fixation is changed, and (c) a closer view of the effect of the ledges at the fixed edge.

Table 3.1 Overview of the different linear elastic analytical model configurations and the total resulting deflection associated with each.

Model Configuration Inner – Outer Span (µm) Load Offset (nm)

Deflection, δ (nm)

2.0 – 0 0 19.39

2.0 – 0 100 18.96

1.8 – 2.0 0 15.28

1.8 – 2.0 100 14.87

2.0 – 2.2 0 20.81

2.0 – 2.2 100 20.36

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Table 3.2 Overview of the different linear elastic analytical model configurations and the accumulated error in the final deflection measurement.

Model Configuration Error from Original Span

Error in Load Location

Error in Measured Deflection

0 % 0 -

0 % 5 % 2.22 %

- 10 % 0 21.20 %

-10 % 5.56 % 23.31 %

10 % 0 7.32 %

10 % 5 % 5.00 %

3.2 Non-linear Analytical Model for Large Deflection Euler-Bernoulli beam theory is used for determination of the deflection characteristics of

linear elastic beams over a well defined range, as demonstrated in the previous analytical

models. For a flexible bar that is fixed at both ends and loaded vertically at or near the

center (i.e. in three-point bending), solutions for bending moment and deflection have

been determined and are widely used for small deflections of the beam in comparison to

the beam thickness, where there exists a linear relationship with the load. However,

experimental observations of the nanowires in bending confirm that fracture occurred at

much higher loads and larger deflections than are allowable in the linear elastic models.

As the applied load of the fixed-fixed beam in bending is increased and the beam

undergoes larger deflection, axial forces develop along the beam due to the constrained

horizontal movement of the fixed configuration. The addition of axial forces enhances

the rigidity of the beam and the strength of the beam consists of both bending and axial

tension [2]. A derivation that included the axial term in a fixed-fixed bending model with

a center load was completed in previous research to determine the ultimate strength of a

nanowire sample independent of elastic modulus, or vice versa [7, 8]. The previous

derivation was employed in this research to analyze force-deflection experimental data

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gathered from brittle silicon nanowires, which included non-linear behavior as the

deflection of the nanowires reached well beyond the beam radius.

The final model examined the effect of large deflection, rather than containing the

analysis to the linear elastic bending regime. The addition of the axial tension in the

nanowire reduced the extent of nanowire deflection as the applied load was increased,

while creating large internal stresses. This analytical model was again applied to a 60 nm

diameter, [110] silicon nanowire fixed over a 2 µm testing span, with the addition of

variable applied loads and load locations.

3.2.1 Large Deflection Center Loading

The governing beam equation (Equation 3.1 [2]) for the series of large deflection

calculations was based in linear elastic beam theory, with the addition of the effects of

axial forces. In Equation 3.1, F is the applied force, E is the elastic modulus, I is the

moment of inertia, u is the transverse deflection of the nanowire, z is the spatial

coordinate along the length of the nanowire, and T is the axial force along the nanowire.

The model, presented in full in Appendix C and Appendix D, calculated the axial tension

and overall nanowire deflection based on an externally applied load and the material

elastic modulus. The extent of deflection and developed tension within the nanowire had

a strong effect on the overall calculated bending strength.

Equation 3.1 2

2

4

4

dzudT

dzudEIF +=

This model subset was constrained to only the center loaded, large deflection behavior of

the fixed nanowire. It was expected that the application of 50 nN would result in the

same amount of deflection as the linear elastic model, due to the fact that the maximum

deflection of the nanowire was significantly less than the nanowire radius and the

deflection should be within the purely linear elastic bending regime, with the stress

contribution emerging due to the tension from bending alone. However, even with this

small amount of deflection, the analytical model predicted the development of a

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significant amount of axial tension and, consequently, the maximum deflection of the

nanowire was 17.86 nm, or 1.52 nm less than the linear elastic analytical model described

previously. The applied load must be decreased to 5 nN before the effects of axial

tension were not readily distinguishable in the deflection measurement using this model,

Table 3.3. This result suggested that the assumptions of the linear elastic, small

deflection model were violated much earlier than was previously anticipated and that the

axial tension developed in the fixed configuration was important from almost

immediately after a load was applied to the nanowire. High applied loads under fixed

loading conditions resulted in a significant reduction in the nanowire deflection when

compared to the linear elastic model, as expected.

Table 3.3 Results for the non-linear analytical model of silicon nanowire in a centrally loaded, fixed-fixed beam bending configuration, accounting for large deflection and increasing applied loads.

Load (nN)

Axial Tension (nN)

Deflection (nm)

Difference from LE Deflection

1 0.04 0.39 0.00 %

5 1.07 1.94 0.10 %

10 4.27 3.86 0.40%

20 16.70 7.64 1.53 %

30 36.26 11.25 3.26 %

40 61.59 14.67 5.42 %

50 91.33 17.86 7.87 %

100 272.79 30.93 20.23 %

1000 2988.43 104.48 73.06 %

5000 10101.82 196.40 89.87 %

82

Figure 3.4 Non-linear elastic analytical model for the effect of large deflection on a nanowire in a centrally loaded, fixed-fixed beam bending configuration with increasing applied load.

3.2.2 Large Deflection Off-Center Loading

The second series in the non-linear elastic beam bending model examined the combined

effect of off-center loading and large nanowire deflection. Using the same nanowire and

analytical model, the accuracy of the cantilever placement location of the applied force

on the nanowire sample was varied to observe the magnitude of error associated with this

possible experimental scenario.

Under the large deflection model, a 50 nN center load resulted in a nanowire deflection of

17.87 nm. An offset of 50 nm reduced the nanowire deflection by 0.09 nm, or 0.49%. A

100 nm offset at this load changed the deflection by 0.34 nm, or 1.93%, compared to the

slightly higher 2.21% change in the linear elastic model under the same set of altered

experimental conditions. As the load was increased, the magnitude of error associated

with the off-center loading decreased, Table 3.4. For example, at 5 µN of applied load, a

100 nm offset load created only 0.8% error in the deflection measurement. Figure 3.5 (a)

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and (b) represent the elastic curves for 50 nN and 5 µN of applied load at several offsets,

respectively.

Table 3.4 Overview of the measured deflection error associated with increasing applied off-center loading using the non-linear elastic analytical model.

Applied Load (nN)

Deflection with 5% Off-Center Load

(nm)

Error in Deflection Measurement

50 17.53 1.93 %

100 30.45 1.60 %

1000 103.55 0.91 %

5000 194.88 0.79 %

84

Figure 3.5 Non-linear elastic analytical model for the effect of off-center loading on a nanowire in a fixed-fixed bending configuration with large deflection. Elastic curve results for (a) 5 nN applied load and (b) 5 µN applied load with increasingly inaccurate load placement.

The various analytical models examined here emphasize the importance of assessing all

of the assumptions made for the final calculation of nanowire bending strength. Small

errors made in the development of the samples or during the bending experiments may

lead to dramatic changes in the proper interpretation of collected data. Discrepancies in

85

the dimensional measurements compound these errors. Without explicit knowledge of

each aspect of testing, it is difficult to minimize or eliminate the uncertainties in the

resulting calculated ultimate nanowire strength. This series of problems extends to all

testing at the nanoscale, where in-situ observation is restricted and dimensional

measurements become increasingly important.

Uncertainties will exist in the final flexure strength values of silicon nanowires analyzed

in this research and the model data presented here represented extreme cases. That said,

ledges were present to some extent in all of the samples tested and cantilever tip

placement and dimensional measurements introduce instrument and operator error.

These models and the constraints of the experiments have established that the individual

error for ledges and off-center loading should not exceed 10% of the ultimate strength

value and the total predicted error will remain below 15%. Additionally, due to the

development of axial tension along the nanowire and the subsequent restrictions on

deflection, the extent of error associated with each of the aforementioned sources is

reduced with increasing applied force in the case of large deflections and are therefore

conservative.

86

References 1. Beer, F.P. and J. Johnston, E. Russell, Mechanics of Materials. Second ed. 1992,

New York: McGraw-Hill, Inc. 2. Landau, L.D. and E.M. Lifshitz, Theory of Elasticity. 3 ed. Course of Theoretical

Physics. Vol. 7. 1986, Oxford: Pergamon Press. 3. Brantley, W.A., Calculated elastic constants for stress problems associated with

semiconductor devices. Journal of Applied Physics, 1973. 44(1): p. 534-535. 4. Frisch-Fay, R., Flexible Bars. 1962, Washington D.C.: Butterworth Inc. 5. Wachtman, J.B., W.R. Cannon, and M.J. Matthewson, Mechanical Properties of

Ceramics. Second Edition ed. 2009: John Wiley & Sons, Inc. 6. Roark, R.J. and W.C. Young, Roark's formulas for stress and strain. 6th ed. 1989,

New York: McGraw Hill. 7. Heidelberg, A., et al., A Generalized Description of the Elastic Properties of

Nanowires. Nano Lett., 2006. 6(6): p. 1101-1106. 8. Ngo, L.T., et al., Ultimate-Strength Germanium Nanowires. Nano Letters, 2006.

6(12): p. 2964-2968.

87

4 Results and Discussion For experimental bending tests a fixture which was amenable to inspection,

characterization, and mechanical testing of silicon nanowire samples was developed. A

silicon wafer was patterned with photolithographic masks and etched using various

reactive ion etch techniques to define the device features. Nanowires were flooded onto

the fixture from an alcohol dispersion and wires located across testing spans were

identified for mechanical testing using an optical microscope. The growth directions of

mechanically tested nanowires were characterized using EBSD Kikuchi patterns

collected in a FESEM. The samples were then fixed in place using EBID with a platinum

precursor gas. Mechanical testing was then performed on an AFM in a centrally loaded,

fixed-fixed beam bending configuration, where nanowires were loaded to fracture,

monitoring the associated force and displacement.

4.1 Influence of Adhesive Behavior Nineteen silicon nanowires were successfully tested to failure in bending using

displacement controlled AFM. Applied force and resulting nanowire displacement data

were collected throughout the entire test. The zero-point of each test, where the

cantilever tip applied a measureable force to the nanowire sample and established the

initial point of contact, was established using linear elastic beam theory (Chapter 2). The

cantilevers used to apply the loads were calibrated prior to and after testing, ensuring

accurate and precise measurements. The experimentally applied force measurements

ranged from approximately 1 – 6.5 µN, which covered a large range, but was not

unexpected as ceramic materials typically exhibit a significant scatter in measured

strength. However, as the measured nanowire deflection of the brittle ceramic material

extended beyond 400 nm, it became apparent that there was a significant source of error

in the experiments.

The silicon nanowire ultimate flexure strength calculated using Equation 2-4 and the

force and deflection measured from the original test data ranged from 3.70-16.98 GPa,

88

with an average of 8.36 GPa. These values were reasonable in considering the theoretical

strength of silicon. However when coupled with the calculated elastic moduli, which

ranged between 4.13 and 169.69 GPa, it was clear that there was a non-systematic error

in the measurement and/or calculation. A further review of the test results (Figure 4.1)

revealed anomalies in the nanowire deflection curves.

Figure 4.1 Applied force as a function of nanowire deflection for all silicon nanowires successfully tested to failure using centrally loaded, fixed-fixed beam bending.

Instead of tracking only the deflection of the nanowire sample, it was a reasonable

possibility that during a bend test, the platinum-based adhesive used to affix the nanowire

to the test fixture may have yielded, allowed the sample to slip, or both. This would

result in exaggerated values of measured nanowire deflection, and consequently low

calculated values of elastic modulus. Due to size and instrumentation restrictions, it was

not possible to visualize the bend tests in-situ, which made it difficult to quantify any

systematic experimental error possibly caused by the adhesive. However there were

several indicators that suggested the hypothesis of platinum-based adhesive deformation

was valid.

89

A number of the force-deflection curves collected during bend tests displayed what

appeared to be classic slip behavior. Figure 4.2 illustrates the difference between a bend

test where the platinum-based adhesive appeared to have allowed for nanowire slip, and

one that was not affected. As the cantilever ran through the scheduled deflection, the

applied force abruptly dropped before continuing along the same curve with additional

deflection. Because the force-deflection curve continued along the same path after the

interruption, it was clear that this small drop in applied force did not represent sample

failure. In addition, the degree of load drop in these type of tests was typically only a

fraction of what was observed for a complete nanowire fracture event (0.3 μN versus 6

μN, respectively in Figure 4.2).

90

Figure 4.2 Applied force as a function of nanowire deflection for nanowires tested to failure shown over (a) the complete test and (b) the area where slip occurred. The dashed blue line represents a test completed exhibiting the predicted force-deflection behavior for a fixed-fixed beam in bending. The solid red line shows a test where the platinum-based adhesive has slipped and subsequently yielded, resulting in inaccurate deflection data.

Anomalously large deflection measurements were not exclusively a consequence of the

slip behavior. A second indicator of yielding or slip was observed in post-test imaging.

91

After the bend test was completed, each fractured nanowire was examined in the FESEM

in order to confirm fracture occurred and to identify the fracture location, testing span,

and any possible experimental issues at the sample site that would impact results and

analysis. During this inspection, two separate observations suggested the nanowire

sample did not remain absolutely fixed during testing. In the absence of nanowire yield

or slip, the fracture surfaces were expected to align after the applied force was removed,

however that did not occur, as is evident in Figure 4.3 (a). The overlap of the fractured

nanowire ends, which was observed for all in-tact samples, was consistently larger than

expected and observed roughness of a fracture surface and, as there was no evidence of

plastic deformation in the silicon, the overlap must have resulted from the extension of

sample into the gap during testing. Figure 4.3 (b) exhibits another consequence, the

section or complete failure of the nanowire, which was indicative of a secondary fracture

resulting from the impact of two fracture surfaces after the initial sample failure.

92

Figure 4.3 Scanning electron micrographs illustrating (a) overlap of two fractured ends of a nanowire test sample after fracture and (b) secondary fracture of the tested nanowire sample. Both results of testing were caused by slip or yielding of the platinum-based adhesive used to fix the nanowire sample across the testing gap.

The combination of experimental results and post-testing observation created a

convincing argument for yielding and slip effects in the platinum-based adhesive. The

93

extended deflection measurement also explained the poor fit of established nonlinear

elastic models to much of the experimental data [1-3].

4.2 Silicon Nanowire Flexure Strength from Force Measurements Three-point bend testing evaluates the mechanical strength of a material through the

generation of tensile forces at bottom surface of the specimen. For the fixed testing

scheme used in the mechanical evaluation of the silicon nanowire samples, deflection

beyond the distance of the nanowire radius produced an additional tensile force in the

axial direction [2]. The bottom surface of a bend test is nominally the area of maximum

tensile stress, however due to the fixed boundary conditions, significant tensile stresses

can be generated axially and at the fixed nanowire ends. The failure of the platinum-

based adhesive during testing inhibited an accurate approximation of the flexure and

extension for individual nanowires. The large experimental deflection measurements,

which resulted from adhesive yield or slip, created an overestimate in the calculation of

generated axial tensile forces of the fixed nanowire in bending.

4.2.1 Experimental Curve Fits

The following examples demonstrate the significant difference between nanowire bend

tests in which the platinum-based adhesive did and did not have an effect on the

experimentally measured deflection, and therefore the resulting silicon flexure strength.

The first nanowire, labeled NW1 and shown after fracture testing in Figure 4.4, had a

measured radius of 41.33 nm and testing span of 1763.18 nm. The measured testing span

neglected the ledges extending from the gap to the beginning of the platinum-based

adhesive. The sample was grown in the [110] direction, which corresponds to a elastic

modulus of approximately 168.9 GPa [4]. The raw force-displacement data collected

during the bend test is shown in Figure 4.5 (a), illustrating increasingly nonlinear

behavior throughout the test and ending with a sharp drop in load, indicative of brittle

failure. In Figure 4.5 (b) the experimental data is shown with the elastic curve fits, which

were calculated for a center loaded nanowire using Equation 2-4. The initial portion of

the experimental curve was fit to the slope of the linear elastic curve fit, as the nanowire

94

behavior should fit along the linear elastic curve for small nanowire deflection. In this

research, small deflections were taken as the radius of each nanowire sample, or slightly

less. Due to the fixed boundary conditions, larger deflections resulted in the stretching of

the sample. As stated previously, this produced axial tensile forces within the nanowire

in addition to the tensile forces associated with flexure, therefore as the deflection of the

sample moved beyond the dimensions of the radius, the linear elastic curve fit predicted

in Equation 2-4 and the experimental data diverged. The nonlinear elastic curve fit

shown in Figure 4.5 (b) was calculated with Equation 2-6, using the experimental

deflection and known elastic modulus to predict resulting force. With this method the

predicted failure load was approximately 0.9 µN higher than the experimentally

measured load at failure, which represented a 16.3% increase in force.

Figure 4.4 Scanning electron micrograph of NW1 after fracture.

95

Figure 4.5 Applied force as a function of nanowire deflection for NW1 illustrating (a) raw data collected during centrally loaded, fixed-fixed bend testing and (b) the elastic curve fits based on measured nanowire deflection.

In contrast to NW1, sample NW2 illustrated a bend test that was significantly affected by

the deformation of the platinum-based adhesive. NW2 had a radius of 27.22 nm, a

testing span of 1429.34 nm, and was grown in the [112] direction, corresponding to an E

of 174.0 GPa [4] (Figure 4.6). Figure 4.7 (a) shows the raw data collected during testing.

96

Unlike NW1, this bend test did not result in the predicted force-deflection curve typical

of a centrally loaded, fixed-fixed bend test. Instead, there was clear evidence of nanowire

slip and possible additional yielding of the platinum-based adhesive throughout the

experiment, which led to very large values for deflection with comparatively moderate

applied loads. The beginning of the test, where the AFM cantilever overcame initial

contact with the nanowire and began to apply an appreciable force, was estimated using

the linear elastic curve fit, as before. However, unlike NW1, the nonlinear elastic curve

fit calculated using the experimental nanowire deflection (in red) predicted a failure load

for this nanowire that was far beyond any realistic value for silicon. Where the nanowire

fractured under 4.17 µN of applied force, the model predicted that it would withstand

almost 60 µN of applied force prior to failure. The predicted applied force was not

remotely achievable with the AFM testing system utilized in this research. Additionally,

using 60 µN of force and 359 nm of nanowire deflection, the calculated flexure strength

of NW2 was 94.7 GPa, which far exceeds the theoretical strength of silicon. In contrast,

with the experimentally applied force and deflection, the flexure strength of NW2 was

6.58 GPa. This strength was more consistent with the theoretical limit and

characterization of the small silicon specimen.

97

Figure 4.6 Scanning electron micrograph of NW2 after fracture.

98

Figure 4.7 Applied force as a function of nanowire deflection for NW2 illustrating (a) raw data collected during centrally loaded, fixed-fixed bend testing and (b) the elastic curve fits based on measured nanowire deflection

In recent studies and models, many research groups report changes in elastic modulus

with sample size [5-7]. It was therefore of interest to determine whether or not the silicon

nanowires demonstrated a change in behavior for the limited range of nanowire diameters

used in this work. The model proposed by Heidelberg [1] was used to measure E as it

99

describes the elastic modulus as a function of force and deflection over the entire elastic

range, through linear and nonlinear deflection. However, the values of elastic modulus

derived from experimentally measured force and deflection for most nanowires tested

were not consistent within the same nanowire sizes in the study, nor were they realistic

according to predictive computer models [8-10].

For the two previous example tests (NW1 and NW2) the nonlinear curve fit used the

experimental deflections and expected elastic modulus (determined by the orientation of

the wire established through EBSD) to predict the applied force at fracture, according to

Equation 2-6. As demonstrated by Figure 4.7 (b), the deflection data collected during the

AFM bending experiments was not accurate for every sample, creating unrealistic values

of predicted failure loads according to the established theories. It was therefore necessary

to invert the calculation and use the experimentally applied force to determine the

corresponding nanowire deflection. Both methods utilized a value of elastic modulus in

order to determine the resulting behavior. Unfortunately, using the force to calculate the

nanowire deflection response eliminated the ability to independently determine the

nanowire elastic modulus for this research. However, the minimum sample diameter

tested was 43 nm, which is significantly larger than the 4 nm diameter predicted for

elastic modulus softening to begin [8, 9]. As a result, elastic modulus values established

from nanowire growth direction were assumed to be constant over the entire range of

nanowire diameters tested. The accurate in-situ cantilever stiffness calibration allowed

for well quantified motion of the cantilever during testing, and therefore the applied force

was regarded as reliable experimental data. Hence, experimentally determined E and

force were used to calculate the corresponding nanowire deflection, as the measured

deflection was not consistently accurate.

For the linear elastic curve fit (Equation 2-4), using force as input rather than deflection

was a simple change. And because the tested samples followed linear elastic behavior at

low values of applied load and deflection, it did not result in a change to the shape or

location of the curve, only the total length along the abscissa. However, to use the

applied load as the input for the nonlinear curve fit was a much more involved process.

100

Rather than directly applying previously derived equations which provided numerical

approximations for a transcendental equation (Equation 2-6 [1, 3]), it was necessary to

return to the original theory of a fixed beam in bending [2] in order to solve for the

nanowire deflection that results from various applied loads while accounting for the

development of both the bending and axial tension within the nanowire (see previous

descriptions in Chapter 3 and Appendices C and D).

Using the applied force rather than the measured nanowire deflection produced a model

which followed the same trend as the previous nonlinear curve fit shown in Figure 4.5 (b)

and Figure 4.7 (b), beginning with a linear fit and eventually resulting in a cubic

dependence between force and deflection. For NW1, the new nonlinear curve fit, based

on experimentally measured applied force and assumed elastic modulus, was similar to

the original curve fit (Figure 4.8 (a)). The evaluation of NW1 did not show any

significant yielding or slip of the platinum-based adhesive, therefore there was a small

(5.6%) error between the measured deflection (135.7 nm) and calculated deflection

(128.1 nm). When the platinum-based adhesive deformed during testing, as was the case

for NW2, there was a dramatic difference in the nonlinear curve fit (Figure 4.8 (b)). The

deflection was reduced over 62%, from experimentally measured 359.0 nm to the

predicted 134.8 nm. With this test, it was readily evident that the measured deflection

was a byproduct of more than just strain in the nanowire.

101

Figure 4.8 Applied force as a function of nanowire deflection for (a) NW1 and (b) NW2. The solid black lines are the experimentally measured force and deflection. The dashed blue lines are the nonlinear elastic curve fit, using calculated values of nanowire deflection.

While force in the experiments was reliable, there were only four samples tested that did

not appear to be significantly affected by slip or deformation of the platinum-based

adhesive. In each case, the displacement and load at failure tracked reasonably well with

the predicted behavior, resulting in high failure strength and relatively sensible values of

102

calculated elastic modulus. This not only indicated that the model of large deflection

used for the strength calculations was correct, but also that the deviations from the

predicted elastic response observed in many of the tested samples was a consequence of

sample preparation and testing configuration and not an unexplained sample/material

behavior.

4.2.2 Deformation and Failure Behavior of Silicon Nanowires

The experimentally measured applied force was used to calculate the large deflection

flexure strength of the 19 nanowires successfully tested to failure. The fracture strength

of the silicon nanowires, calculated using experimental force measurements and expected

elastic modulus values (determined by the orientation of the nanowires) ranged from 5.10

GPa to 20.01 GPa, with an average flexure strength of 13.74 GPa. The results of each

nanowire sample are listed in Table 4-1. In contrast, the values of flexure strength

calculated using experimentally measured deflection, which was determined to be

inaccurate due to yielding or slip of the platinum-based adhesive, ranged from 3.70 to

16.98 GPa, with an average strength of 8.36 GPa. The change in individual values of

flexure strength varied according to the degree of deformation in the adhesive. For NW1,

the calculated deflection was similar to the experimentally measured deflection and the

strength was increased by only 1.23%. However, the average increase in strength over

the whole series of nanowires tested was 37.39% with largest being 72.06%, increasing

from 3.70 to 13.23 GPa. There was no evidence of the bend strength changing as a

function of nanowire growth direction (Figure 4.9(a)). Nor was there evidence of a size

effect on the bending strength, within the limited range of nanowire radii tested (Figure

4.9 (b)).

The flexure strength results showed a significant amount of scatter, which is not

uncommon in ceramic materials and was interpreted to be the result of randomly

distributed flaws over the samples. Scatter in the strength creates uncertainty in

engineering design. It is therefore of interest to be able to describe and compare the

mechanical information quantitatively through statistics.

103

Table 4-1 Summary of flexure strengths from silicon nanowires experimentally tested in centrally loaded, fixed-fixed beam bending.

Growth

Direction Measured Force

(µN) Calculated Deflection

(nm) Flexure Strength

(GPa)

[110] 0.96 83.48 5.10 [100] 1.64 105.61 9.72 [100] 2.42 113.35 10.32 [100] 2.37 141.76 10.95 [110] 5.51 128.10 11.20 [112] 2.23 117.52 11.33 [100] 3.68 118.36 11.61 [112] 3.53 96.95 13.23 [111] 4.05 101.33 13.87 [112] 3.05 122.49 14.19 [110] 4.90 121.06 14.58 [100] 4.48 166.01 14.76 [100] 3.86 166.98 14.85 [112] 4.17 134.84 16.20 [112] 6.07 168.10 16.40 [110] 6.45 133.79 16.93 [110] 6.15 139.25 17.22 [112] 3.21 142.69 18.69 [110] 6.44 164.65 20.01

104

Figure 4.9 Plot of silicon fracture strength in bending as a function of (a) nanowire radius and (b) nanowire growth direction, as determined through EBSD.

4.2.3 Statistical Analysis

The fundamental assumption of the Weibull statistical distribution is a weakest link

hypothesis; the failure of weakest single flaw will cause failure of the entire specimen.

Therefore the strength of brittle materials depends on the size of the largest flaw, which

will vary between specimens and is based on specimen size. Because silicon is a brittle

105

material, it is useful to describe the strength using a two parameter Weibull distribution

function. Weibull statistics also aide in the interpretation of samples evaluated using

different testing methods, samples of various sizes, and results which exhibit a large

amount of variability. Silicon nanowire testing falls under all of these categories, as is

evident from the previous literature summarized in Chapter 1. Silicon fracture strength

has been reported between 30 MPa – 21 GPa [11, 12], evaluated using tensile methods

and various types of flexure methods [7, 11-14], and each nanowire experiment was

performed on a sample of unique dimensions. Statistical analysis can be a powerful tool

for the interpretation and comparison of nanowires.

4.2.3.1 Standard Weibull Analysis

The Weibull modulus m and characteristic strength σ0 are parameters in the Weibull

distribution function. The parameters may be determined by two methods, a least-

squares fit to the linearized form of the data or the maximum likelihood method, which is

recommended by ASTM standards [ASTM C 1239-07]. The least-squares linear

regression method employs a relatively simple graphical interpretation of the data, which

is sufficient as a first approximation for the Weibull parameters. However the method

does not apply to some data sets and results in somewhat wide confidence intervals for

small numbers of samples. In contrast, using the maximum likelihood estimators (MLE)

of m and σ0 provides a more precise assessment of the probability distribution across a

wide range of data sets [15]. Weibull parameters for this research were calculated using a

maximum likelihood procedure with confidence intervals calculated via Monte Carlo

simulation using a nonparametric bootstrap procedure, detailed in Appendix E [16]. The

bootstrap method provided an estimate of experimental uncertainty to the small data set

completed. The resulting Weibull modulus was 4.59 and the characteristic strength was

15.04 GPa with 90% confidence levels.

The MLE Weibull parameters m and σ0 from the series of flexures strengths calculated

using inaccurate deflection measurements were 2.80 and 9.38 GPa, respectively. By

changing the analysis of the data and calculating the strength using experimental force

and E values, the flexure strength was improved by 5.38 GPa, or 64%. The Weibull

106

modulus also increased significantly, from 2.80 to 4.59, which indicated a tighter

distribution for the probability of failure. The Weibull plot of the two separate

interpretations of the flexure strength data (Figure 4.10) provided a visual illustration of

the improvement in strength and reliability attained by excluding the experimental

deflection results.

Figure 4.10 Weibull plot of the experimental results for silicon nanowires tested in centrally loaded, fixed-fixed bending configuration. The series of fracture strength values determined using the experimentally measured force and calculated deflection are shown using black squares. The series of fracture strength values determined using both the experimentally measured force and deflection are shown using blue circles.

4.2.3.2 Adaptation of Flexure Strength to Uniaxial Tensile Testing

Weibull statistics developed for flexure testing are based on simply supported boundary

conditions with a center load, which is the primary configuration for macro-scale

samples. Using this assumption, the stress field varies linearly from zero at the edge

supports to a maximum at the center location of the load, leaving only a small amount of

material along the sample that is exposed to the maximum stress during testing. While

the stress increases with increasing load, the shape of the stress field is invariant [15].

For silicon nanowires grown using the VLS technique it is uncommon to find a volume

107

defect, with the exception of gold inclusions. Therefore all silicon nanowire fracture

events in this research were assumed to have initiated from a surface flaw. By neglecting

the possibility of volume defects, the amount of material under maximum load during a

simply supported flexure experiment was reduced to only the lower surface of the sample

at the location of the load. In contrast, the stress on a uniaxial tensile specimen is

distributed evenly over the entire surface. The experimental strength analysis represented

in Figure 4.10 was performed for data collected using a centrally loaded, fixed-fixed

bending scheme. It was previously established that the fixed boundary conditions added

an axial component to the stresses that must be considered. Therefore it would seem that

the flexure strength of the nanowires tested in this configuration would fall in between

the uniaxial tensile strength and the simply supported bending strength, as the stress is

distributed over the sample in a combination of the two methods. There are no known

examples of silicon nanowires tested in a simply supported, three-point configuration, but

reported tensile strengths of silicon nanowires range from 5.10-12.5 GPa [7, 17] and the

flexure strengths in this research were within a similar range, from 5-20 GPa. The

fracture strength of the silicon nanowires during uniaxial tensile loading can be estimated

from the flexure strength of nanowires tested in this research. The Weibull estimates

allow for a direct comparison of the probability of failure between testing methods. The

analysis involves deriving an effective surface area SE for each nanowire tested.

The stress applied to each nanowire during a test is distributed according to the testing

method. The stress on a uniaxial tensile specimen is distributed evenly over the entire

surface, therefore the effective surface is equal to the surface area of the entire nanowire.

For a flexure test sample, the SE is significantly reduced. To directly compare values of

strength between tensile and flexure samples, the same amount of surface area must

experience the applied stress. In that way, there exists the same probability to failure, as

the likelihood of finding a flaw within the surface area is the same. Therefore, for a

flexure test, the SE is the surface area of a hypothetical uniaxial tensile specimen which

has the same probability of failure.

108

The majority of past experimental work on ceramics has been completed on much larger

scales, where there was no need to perform bend tests using fixed boundary conditions

due to the size and rigidity of the samples (with the exception of glass fibers in flexure

[18, 19]. As a result, the existing statistical theory does not extend to include the testing

configuration used in this research. The effective surface area for a cylindrical rod in

simply supported three-point bending has been previously derived (Equation 4-1 [20]),

where S is the sample surface area and G is a combined gamma function.

Equation 4-1 GmmSS ptE ⎟

⎠⎞

⎜⎝⎛

++

=12

23, π

where

⎟⎠⎞

⎜⎝⎛ +

Γ

⎟⎠⎞

⎜⎝⎛Γ⎟

⎠⎞

⎜⎝⎛ +

Γ=

24

23

21

m

m

G

The measured three-point bending strength σ3pt can then be used to predict the strength of

the nanowires in uniaxial tension σT with the ratio stated in Equation 4-2 [20].

Equation 4-2 mptE

T

pt

SS

13,3

⎟⎟⎠

⎞⎜⎜⎝

⎛=

σ

σ

The flexure strengths of silicon nanowires from this research were converted to predicted

uniaxial strengths using this analysis. The conversion of strength was completed for each

fractured sample, applying individual nanowire radii and testing span. Ideally, the test

specimens within a Weibull statistical series will have the same dimension and span.

However that was not the case for these experiments and each sample effective surface

area was individually calculated. The results (Figure 4.11) demonstrated that changing

the testing scheme shifted the average ultimate strength of the material, from 13.74 GPa

in flexure to 5.97 GPa for the equivalent strength in uniaxial tension, but the shape of the

Weibull curve remained the same. It is important to note that the analysis in Equation

109

4-1 was derived for a simply supported bending configuration, which has a different

stress field than the centrally loaded, fixed-fixed bending used to establish flexure

strengths for this research. The conversion of strength only accounted for the change in

the stressed surface area between a simply supported bend test and a uniaxial tensile test.

Therefore, there was no consideration toward the evolution of stresses along the sample

that would occur under fixed boundary conditions.

Figure 4.11 Weibull plot showing comparison of experimentally evaluated flexure strength (σflexure, black squares) to equivalent tensile strength (σtensile,SS, blue circles) derived using the effective surface area calculation for a simply supported beam bending configuration.

4.2.3.3 Effective Surface Area under Large Deflection

While it was possible to complete the previous analytical comparison for each nanowire,

it was not conceptually accurate. Initial assumptions made by the comparative testing

configuration analysis included that the ceramic sample was fractured in simply

supported three-point bending where the bending moment and deflection follow a linear

relationship with the applied load. For a nanowire fixed at both ends, the elastic curve of

the deflected beam was longer than a straight line between the supports. The elongation

created axial tensile forces within the nanowire in addition to the existing bending

moments, therefore in this case, the effective surface area was not constant as a function

110

of applied load. As the load was increased and the nanowire extended, larger axial

stresses were developed along the length of the sample. The effective surface area

changed to one where a larger portion of the nanowire experienced a tensile stress.

Additionally, since the length of the nanowire did not remain the same during testing, it

was necessary to account for the extension of the sample with increasing applied load.

To more accurately compare uniaxial tensile testing with the fixed-fixed bend testing

configuration used in this research, it was necessary to derive a new effective surface

area. Utilizing the derivation scheme laid out by Quinn [20], the theory was extended to

include large deflection of fixed beams in bending by first determining the shape of the

nanowire elastic curve at fracture. This was achieved in several steps. Using an

analytical model described in the previous chapter and detailed in Appendix C, the axial

tension developed along the nanowire during a test was determined for each nanowire,

accounting for the testing span, radius, growth direction, and varying applied load. With

the tension, it was possible to calculate the deflection as a function of position along the

nanowire (Appendix D). The elastic curve was then fit with a 9th order polynomial

function and the constants which defined the curve fit were then used to evaluate the first

and second derivatives of the elastic curve, the local slope and curvature, respectively.

Assuming that the neutral axis of the nanowire did not shift under large deflection, the

calculated curvature of the beam was related to the strain with Equation 4-3,

Equation 4-3 cbendε

κρ

−==

1

where ρ is the radius of curvature, κ is the curvature, εbend is the bending strain, and c is

the distance from the neutral axis. Equation 4-4 could then be used to explicitly calculate

the bending strain.

Equation 4-4

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

23

2

2

2

1dxdy

dxydcbendε

111

The maximum bending strain εbend,max was assessed at the center of the testing span and

when the neutral axis was equal to the radius of the nanowire. The axial strain εaxial was

determined using the change in nanowire length as a result of stretching during loading.

The final length of the nanowire was evaluated with Equation 4-5, where l0 and l are the

original and deflected length of the nanowire sample, respectively.

Equation 4-5 ∫ ⎟⎠⎞⎜

⎝⎛+=

0

0

2

1l

dxdxdyl

Finally, as it was assumed that silicon is a linear elastic material, stress and strain are

proportional and can simply be added to combine the axial and bending contributions to

strength. Therefore the effective surface area for a centrally loaded, fixed-fixed bend test

sample was given by Equation 4-6,

Equation 4-6 dcdx

rc

SmL r total

ptE

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎠⎞

⎜⎝⎛−

⎟⎟⎠

⎞⎜⎜⎝

⎛= ∫ ∫

21

20 0 max3,

1

1εε

where L is the test span, r is the nanowire radius, εtotal is the sum of εbend and εaxial, and

εmax is the sum of εbend,max and εaxial.

There is no significant change in surface area of the nanowire due to elastic extension and

Poisson’s contraction during large deflection flexure conditions. Using the previously

described model nanowire dimensions and properties for the case of uniaxial tension, the

increase in total surface area is on the order of 3-5%. In contrast to the uniaxial loading

case, the stress applied to a nanowire tested in the fixed-fixed bending configuration is

distributed over a minute fraction of the total sample surface area. Consequently,

deformation-induced changes in sample surface area have been neglected.

112

The derivations involved in the determination of a new effective surface area are detailed

in Appendix F. To demonstrate the effect of fixed boundary conditions on the SE, a series

of calculations were completed using the same model nanowire used in the previous

chapter, with a 30 nm radius, 2 µm testing span, and 168.9 GPa elastic modulus. By

systematically increasing the applied load, it was possible to observe the influence of the

developing axial tension on the nanowire SE (Figure 4.12). The effective surface area of

the fixed nanowire increased with increasing applied load until a plateau was reached,

after which the SE remained the same. Conceptually, this showed that for a uniaxial

tensile nanowire with the same failure probability as the model nanowire, the size of the

sample increased until a certain applied load, after which the maximum probability of

failure was reached. Or in other words, the axial stretching contribution to the flexure

strength developed with increasing applied load until the centrally loaded, fixed-fixed

bend test essentially mimicked a small scale uniaxial tensile test.

Figure 4.12 Effective surface area for model silicon nanowire. The solid line represents the SE,total calculated with the new model, accounting for both the bending and axial tension components. The dotted line represents the SE,bending, which only accounts for the bending tension in the nanowire.

The behavior of the effective surface area with increasing applied load for the centrally

loaded, fixed-fixed beam bending configuration was significantly different from the

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existing model for simply supported, three-point bending. The existing model resulted in

a constant value for each Weibull modulus, reflecting the fact that the stress field is

constant under that testing configuration. The behavior of the newly derived SE with

increasing load was also highly dependent upon the addition of axial tension in the

nanowire. If the axial tension contribution was not considered, the resulting effective

surface area only accounted for bending stresses and continued to increase with

increasing applied load (Figure 4.12).

While some fixed-fixed nanowire bend testing can be completed with low total

deflection, the tensile contribution evolved prior to the evolution of nonlinear behavior,

as was shown in the large deflection center loading model of the Chapter 3. Therefore

the standard methods of predicting probability of failure cannot be extended to large

deflection loading conditions and nanowire tests conducted in centrally loaded, fixed-

fixed bending, even with small total deflection, may result in incorrect values of

measured stain. The calculation of a new SE was completed for several of the

experimentally tested nanowires to ensure the applicability of the technique beyond the

analytical model. NW1, described in a previous section, fractured at the center of the

testing span under 5.51 µN of applied force and 128 nm of deflection. The equivalent

flexure strength for a nanowire under those conditions was 11.20 GPa. The experimental

force-deflection data and elastic curve representing the nanowire deflection as a function

of position are shown in Figure 4.13 (a) and (b), respectively. Using the shape of the

elastic curve at maximum deflection, it was determined that the nanowire stretched

approximately 22 nm prior to fracture. This extension produced a 3.75% total bending

strain and 1.25 % strain in the axial direction of the nanowire.

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Figure 4.13 Information from NW1 used to interpret the effective surface area for a centrally loaded, fixed-fixed nanowire in bending, including (a) the experimental applied force as a function of deflection and (b) the elastic curve.

The total surface area for NW1 within the undeflected testing span was 0.458 µm2. The

effective surface area calculated for a simply supported bending test was 0.0144 µm2,

using Equation 4-1 and the previously determined Weibull modulus of 4.59. However,

the effective surface area for the centrally loaded, fixed-fixed bend test determined with

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Equation 4-6 was 0.0963 µm2, or just 21% of the total surface area within the testing

span. Utilizing the established ratio (Equation 4-2) to compare the ultimate strength of

various testing configurations, the flexure strength of NW1 was reduced from 11.20 GPa

to an equivalent uniaxial tensile strength of 7.97 GPa. This is in comparison to the 5.27

GPa determined using the simply supported bending effective surface area. The new

analytical model for the effective surface area in fixed-fixed beam bending was applied to

several additional experimentally fractured nanowires, with a sample of results

summarized in Table 4-2 and a Weibull plot of the shift in strength displayed in Figure

4.14.

Table 4-2 Summary of effective surface area and strength for several experimentally tested silicon nanowires examples.

Sample NW1 (110)

NW3 (110)

NW4 (112)

NW5 (100)

Radius, r nm 41.33 28.60 32.54 23.32

Span, L nm 1763.2 1514.8 1760.2 1268.58

Elastic modulus, E GPa 168.9 168.9 174.0 130.2

Applied Force, F µN 5.51 6.44 6.07 1.64

Deflection, δ nm 128.10 164.65 168.10 105.61

Extension nm 22.11 38.43 34.74 19.18

Bending strain, εbend % 3.76 4.78 4.07 3.48

Axial strain, εaxial % 1.25 2.54 1.97 1.51

Effective surface area, SE,3pt µm2 0.0963 0.1180 0.1483 0.0685

Fraction of the total surface area % 21.04 43.34 41.19 36.84

Experimental flexure strength, σflexure GPa 11.20 20.01 16.40 9.72 Equivalent tensile strength, σT GPa 7.97 16.68 13.52 7.82

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Figure 4.14 Weibull plot showing comparison of experimentally evaluated flexure strength (σflexure, black squares) to equivalent tensile strength (σtensile,FF, blue circles) derived using the effective surface area calculation for a centrally loaded, fixed-fixed beam bending configuration.

The new derivation of effective surface area for a centrally loaded, fixed-fixed beam in

bending is an important contribution for the description and comparison of nanoscale

ceramic data in a quantitative manner. Weibull statistics are used to adapt the probability

of failure over different stress fields and sample sizes. The strength of silicon nanowires

has been derived using a variety of methods, covering a large range of values, however

there has been little effort made to directly compare and utilize the knowledge gained

from previous researchers. The equivalent uniaxial tensile strength derived from silicon

nanowires tested in fixed-fixed bending for this research ranged between 3.50 to 16.67

GPa. Previously reported studies of nanowires tested in tension reported fracture

strengths from 5.1 to 12.5 GPa [7, 17]. While there is significant scatter in each data set,

the series of flexure strength values from this research corresponded well with past

reported tensile strengths.

It is important to note that the evaluation of the effective surface area calculation is

highly dependent on the Weibull modulus derived for the individual sample set. The

Weibull modulus describes the variability of the data, which means that errors in sample

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preparation, testing, and analysis may significantly contribute to the parameter, and

therefore the probability of failure. The 90% confidence limits calculated for m using

MLE provided a range from 3.52 to 6.74. The considerable change in effective surface

area with those bounds is shown in Figure 4.15, using the model nanowire to demonstrate

the effect with increasing applied force. In addition, the derivation of SE in this research

used the assumption that the location of the neutral axis did not shift as a result of the

large deflection.

Figure 4.15 Effective surface area as a function of applied force showing the dependence of SE on Weibull modulus for a model nanowire with increasing applied load.

Ceramic macroscale samples are often tested in flexure, as the method involves simple

sample shapes and stress evaluations for brittle materials with no issues of alignment or

gripping. For simply-supported three-point bending, the very small amount of sample

under maximum stress allows for the approach of theoretical flexure strengths.

Nanoscale testing has removed many of the perceived conveniences of the bend test.

Due to the high flexibility of silicon nanowire samples, it was essential to grip both ends

to achieve fracture in the configuration utilized. Large deflection and fixed ends resulted

in more complex stress fields within the samples, as it was not possible to constrict the

load to the center point. As a consequence of the evolution of axial tension, there is a

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lower probability of achieving the theoretical strength of a sample using the centrally

loaded, fixed-fixed beam bending configuration than for the simply supported three-point

bending configuration.

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References 1. Heidelberg, A., et al., A Generalized Description of the Elastic Properties of

Nanowires. Nano Lett., 2006. 6(6): p. 1101-1106. 2. Landau, L.D. and E.M. Lifshitz, Theory of Elasticity. 3 ed. Course of Theoretical

Physics. Vol. 7. 1986, Oxford: Pergamon Press. 3. Ngo, L.T., et al., Ultimate-Strength Germanium Nanowires. Nano Letters, 2006.

6(12): p. 2964-2968. 4. Brantley, W.A., Calculated elastic constants for stress problems associated with

semiconductor devices. Journal of Applied Physics, 1973. 44(1): p. 534-535. 5. Kizuka, T., et al., Measurements of the atomistic mechanics of single crystalline

silicon wires of nanometer width. Physical Review B, 2005. 72(3): p. 035333. 6. Li, X., et al., Ultrathin single-crystalline-silicon cantilever resonators:

Fabrication technology and significant specimen size effect on Young's modulus. Applied Physics Letters, 2003. 83(15): p. 3081-3083.

7. Zhu, Y., et al., Mechanical Properties of Vapor-Liquid-Solid Synthesized Silicon Nanowires. Nano Letters, 2009. 9(11): p. 3934-3939.

8. Kang, K. and W. Cai, Brittle and ductile fracture of semiconductor nanowires - molecular dynamics simulations. Philosophical Magazine, 2007. 87(14): p. 2169 - 2189.

9. Lee, B. and R.E. Rudd, First-principles calculation of mechanical properties of Si<001> nanowires and comparison to nanomechanical theory. Physical Review B (Condensed Matter and Materials Physics), 2007. 75(19): p. 195328-13.

10. Park, H.S., Surface stress effects on the resonant properties of silicon nanowires. Journal of Applied Physics, 2008. 103(12): p. 123504-10.

11. Gordon, M.J., et al., Size Effects in Mechanical Deformation and Fracture of Cantilevered Silicon Nanowires. Nano Letters, 2009. 9(2): p. 525-529.

12. Sundararajan, S., et al., Mechanical property measurements of nanoscale structures using an atomic force microscope. Ultramicroscopy, 2002. 91(1-4): p. 111-118.

13. Namazu, T., Y. Isono, and T. Tanaka, Evaluation of size effect on mechanical properties of single crystal silicon by nanoscale bending test using AFM. Journal of Microelectromechanical Systems, 2000. 9(4): p. 450-459.

14. Tabib-Azar, M., et al., Mechanical properties of self-welded silicon nanobridges. Applied Physics Letters, 2005. 87.

15. Wachtman, J.B., W.R. Cannon, and M.J. Matthewson, Mechanical Properties of Ceramics. Second Edition ed. 2009: John Wiley & Sons, Inc.

16. Fuller, E.R., R. Kirkpatrick, Editor. 2010. 17. Steighner, M.S., Tensile Strength of Silicon Nanowires, in Department of

Materials Science and Engineering. 2009, The Pennsylvania State University: University Park.

18. Annovazzi-Lodi, V., et al., Statistical Analysis of Fiber Failures Under Bending-Stress Fatigue. Journal of Lightwave Technology, 1997. 15(2): p. 288-92.

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19. Matthewson, J.M., C.R. Kurkjian, and S. Gulati, Strength Measurement of Optical Fibers by Bending. Journal of the American Ceramics Society, 1986. 69(11): p. 815-21.

20. Quinn, G.D., Weibull Effective Volume and Surfaces for Cylindrical Rods Loaded in Flexure. Journal of the American Ceramics Society, 2003. 86(3): p. 475-79.

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5 Conclusions and Future Work

5.1 Conclusions The primary objectives of this study were to establish a reliable methodology to evaluate

the flexure strength of nanowires and to assess the possibility of approaching the

theoretical strength of a material tested in flexure. Using silicon as a baseline material

system, the mechanical behavior of nanowire samples was evaluated with a centrally

loaded, fixed-fixed bending configuration. Analytical models were developed to

illustrate the impact of fundamental errors in sample preparation and testing. Numerical

analyses were also used to evaluate the effect of the boundary conditions and the

implications of weakest link statistical theories on the measurement of basic mechanical

properties such as elastic modulus and strength. The key results of this work are

summarized below based on experimental and numerical analysis.

• The single crystal silicon nanowires had flexure strengths that ranged from 5.10 to

20.01 GPa, with an average value of 13.74 GPa. The strength of the wires did not

show a significant dependence on diameter (43-83 nm) or crystallographic

orientation ([111], [110], and [112] were evaluated).

• The platinum-based adhesive deposited using the electron beam did not provide

the desired “fixed” or “built in” boundary condition at the stresses required to fail

the nanowires.

• Changes in experimental boundary conditions resulting from error in the locations

of nanowire fixation and applied bending force can lead to uncertainty of over

15% in measured fracture strength for a typical nanowire sample. The impact of

these effects becomes less pronounced with larger nanowire deflection and higher

effective stiffness.

• From the earliest stages of deformation, axial tension develops in the nanowire as

a result of the fixed boundary conditions and has a measureable impact on

deflection. The influence can be seen for nanowire deflection equivalent to just 5-

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10% of the sample diameter, values well within the traditional boundaries of

linear elastic beam theory.

• A new effective surface area correction factor was derived for Weibull statistical

analysis for the case of fixed beams and large deflection. The failure to consider

the evolution of the stress field with increasing nanowire deflection leads to

predicted tensile strength values which are 31-44% lower than the distribution of

nanowire flexure strengths analyzed in this research

• There is a lower probability of approaching the theoretical strength of a nanowire

analyzed using the fixed-fixed bending configuration in comparison to the simply

supported bending configuration, due to the presence of axial tension in the beam.

The equivalent surface area for compliant nanowires under fixed boundary

conditions can be over an order of magnitude higher than what is predicted based

on small deflection theory. Where simply supported samples in three-point

bending can focus the applied force to one location, the fixed-fixed beam bending

configuration produces distributed stress state behavior that, in the limit, evolves

in a way that is analogous to a uniaxial tensile test.

5.2 Future Work

During the course of this research several issues arose which would be interesting to

address in future work. The first involves more closely examining the platinum-based

adhesive. A thorough investigation into the mechanical behavior of the deposited layer

and methods to increase its yield and interfacial shear strengths would benefit the

nanomechanical properties research community. Because this fixation method is

commonly employed and has few alternatives, it represents a critical issue that must be

addressed. In addition to developing better materials for fixation of nanowires, the

fracture behavior observed in this work raised a number of questions that could be

answered with additional experimental studies. These include 1) perform the flexure tests

within a more controlled environment, such as vacuum or liquid immersion, to examine

possible environmental effects, 2) test silicon nanowires with smaller diameters in an

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attempt to observe a size effects in the mechanical behavior, and 3) explore the behavior

of other material systems.

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A Fixture Processing

A.1 Fixture Masks Fixture processing of the three-point bend test supports began by experimenting with

existing masks in order to get a sense of what sizes and spacing would be needed for the

final design. Two masks with features that include closely spaced lines were borrowed

from research assistants (The Pennsylvania State University, Department of Electrical

Engineering). The lines written on each mask could be exploited as supports for a

nanowire in three-point bending. The first mask, originally designed as a 4-probe test

structure, contained thin, closely spaced lines in the center of the feature which then

radially expanded to larger, thicker lines around the edge. At the center of the design the

lines are approximately 1 µm thick with 2 µm spacing. The second mask was designed

to create a parallel heater bridge device. It contains individual sets of 5 µm thick lines,

each spaced 1 µm apart. These masks designs are referred to as “fan” and “bridge”, and

are pictured in Figure A.1 and (b), respectively. Both mask designs were subjected to a

series of coating and etch trials to determine the applicability of each individual design

feature and processing method.

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Figure A.1 Optical micrograph of the fan mask patterned in photoresist on a silicon wafer with an expansion view to illustrate the thin lines that are used for nanowire mechanical evaluation. The images were captured at unknown magnification.

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Figure A.2 Optical micrograph of one device on the bridge mask patterned in photoresist on a silicon wafer. The functional region of the device is located between the two thin lines in the center of the image.

A.1.1 Photoresist

A significant number of trials were needed in manufacturing the fixture to determine

appropriate feature sizes and etch times. To test several configurations efficiently,

processing began using photoresist as a mask on silicon test wafers. Photoresist has been

reported to create a substantial barrier to etching in many different processing

environments [1-3]. Individual wafers were coated with photoresist using a spinner

(Headway Research, Inc. Bowl Model) to produce a thin, uniform layer on the wafer

surface and then soft baked at a prescribed temperature to remove all solvents from the

photoresist and enable the UV exposure sensitivity of the coating. A mask was then used

to selectively expose the photoresist to high intensity UV light on an aligner (Electronic

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Visions EV620) for a specified time. The exposed photoresist was then removed with a

developer solution, leaving the desired pattern masked by exposed photoresist on the

silicon wafer. The remaining photoresist was exposed to an additional heat treatment at a

specified time and temperature, depending on which photoresist was used, in order to

harden the photoresist and increase adhesion between the coating and the wafer. Type

and thickness of photoresist, time of exposure in the aligner, and time of development

were all variables that needed to be optimized in the procedure.

Five separate photoresists were used in an attempt to achieve a balance between coating

thickness and developed edge precision for this particular application. The fan and

bridge design masks were used to perform UV exposure and development time trials on

photoresist coated silicon wafers. Trials included AZ® 9260 photoresist (Clariant),

Megaposit™ SPR™ 220 positive photoresist (Rohm and Haas), Microposit™ SC™ 1827

positive photoresist (Shipley), Megaposit™ SPR™ 3012 positive photoresist (Shipley),

and Microposit™ S1805™ positive photoresist (Rohm and Haas) with corresponding

developer solutions. By varying spinner speed, exposure time, and developer time, it was

possible to narrow the photoresist choices down to one that was acceptable for this

application. Microposit™ SC™ 1827 positive photoresist with Microposit™ MF™ 351

developer (Shipley) produced approximately 2.9 µm thick coatings and consistently

developed into precise mask features using the bridge design. The final procedure began

by cleaning the silicon wafers using an HF dip (brand 49%) then dehydrating the surface

for five minutes at 200° F. The wafer was then placed into a spinner and coated with

photoresist before being spun for 5 seconds at 500 rpm followed by 40 seconds at 4000

rpm. The freshly coated wafer was heated to 110° F for 1 minute, exposed to the mask

design for 12 seconds, submerged into the developer solution for approximately 1 minute,

rinsed with DI water, and finally hard baked for 5 minutes at 110° F. Microposit™

S1805™ positive photoresist created a thinner final mask and produces precise features

with the more finely spaced lines in the fan mask design, however the final photoresist

layer thickness was subsequently found insufficient in providing a significant barrier

during etch processing. The bridge design also had well spaced, identifiable fixtures

which made it easier to locate and track individual NW samples during characterization

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and testing. For these reasons, the bridge design was used for the majority of future

testing.

A.1.2 Oxide

Silicon dioxide was also utilized as a mask material. The oxide provided a more robust

mask than photoresist. To prepare the mask, a 250 nm thick oxide was grown on a set of

silicon test wafers in a dry oxygen environment at 1100°C using a Thermco® 4-stack

thermal oxidation horizontal atmospheric furnace. The silicon wafers with a SiO2

thermal layer were then spin coated with photoresist, which was exposed and developed

using the bridge mask in the same manner described in the previous section. With the

desired pattern developed in hard baked photoresist atop of the SiO2 layer, the wafer was

exposed to a plasma etch (PlasmaTherm SLR Series) for 310 seconds to selectively

remove the oxide layer in the mask design. The wafer was then cleaned with acetone and

2-propanol rinses to remove the photoresist layer. This was followed by a 45 minute

Piranha etch to remove any remaining contaminants. The Piranha etch had a 50:1 ratio

between H2SO4 and H2O2. As with photoresist, the oxide mask using the bridges design

was developed with sufficient precision for the desired feature sizes.

A.2 Fixture Etch Techniques Two etch techniques were evaluated to create the pointed three-point bending support

column features in the silicon wafer; xenon diflouride (XeF2) and reactive ion etching

(RIE). For each etch trial a silicon wafer patterned with a mask, either photoresist or

oxide, was diced to approximately 5 × 12 mm in size, with each sample containing 48 test

fixtures from the bridges mask to examine the results of the etch process.

A.2.1 Xenon Diflouride Etching

For the XeF2 etch trials, each diced silicon sample was placed individually into the

instrument (Xactix) vacuum chamber. At room temperature and vapor pressures between

1 – 4 Torr, solid XeF2 sublimates and is adsorbed to the silicon surface. Fluorine then

dissociates and reacts with the silicon, forming SF4. Both SF4 product and the Xe desorb

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from the substrate surface and are removed from the chamber by the vacuum system.

The primary reaction for this isotropic dry etch is 2XeF2 + Si 2Xe +SiF4, which has a

reported etch selectivity to silicon over silicon dioxide and photoresist of greater than

1000:1 [1-3]. Using the same etch recipe and sample size, the total etch time is adjusted

to determine optimal conditions to obtain the support columns desired on the fixture.

Prior to each etch trial the samples were subjected to oxygen plasma at 100 Torr and 80

Watts for 2 minutes to remove any remaining photoresist from the exposed silicon. The

samples were then dehydrated for 2 minutes at 110° F. This was completed to ensure that

all samples entered the chamber under the same conditions, and to avoid the possibility of

creating HF vapor, which is a product of the XeF2 process in the presence of water. The

XeF2 etch was run at 2 Torr in pulsed etch mode, where the sample was alternatively

exposed to the XeF2 vapor and vacuum conditions. The number of cycles and the time

per cycle were specified in the instrument software and are varied in order to investigate

the applicability of the XeF2 etch in producing the peaked support columns. The etch rate

of this technique was very load dependant and displayed variations in etch depth

depending on the location of the sample in the chamber with respect to the XeF2 inlet [3].

All samples were therefore diced to roughly the same size and placed at the center of the

chamber for each etch trial. To remove the photoresist mask after the etch was complete,

the samples are cleaned with acetone and rinsed with methanol.

The lines for the support columns on the bridge mask design are 5 µm thick with 1 µm

spacing. After the isotropic XeF2 etch, the support span was between 5 – 6 µm, peak to

peak. Figure A.3 illustrates the results of an AFM scan on a test fixture exposed to 1

minute of XeF2 vapor. While the supports clearly developed into rounded peaks, the

depth of the valley in between the peaks was only approximately 600 nm, which was the

deepest gap produced for all XeF2 etch time trials. The silicon nanowire samples are very

flexible, therefore to complete a centrally loaded, simply supported bend test a larger

vertical clearance than the 600 nm was needed. The final surface of the silicon fixture

after the XeF2 etch was also extremely rough. Figure A.4 a – d show the development of

the support columns, imaged at the same magnification, after a total etch time of between

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30 seconds and 3 minutes. The support columns partially developed into peaks after 30

seconds, but were completely etched away with a severely pitted surface after 3 minutes.

It was possible to use N2 gas to dilute the potency of the XeF2 etch, however instead of

reducing the amount of pitting that occurred, the diluted etch simply slowed the etch

process and eventually resulted in the same damage. In addition, when the support peaks

were formed, the surface of each peak was rough and a nanowire sample did not rest in

plane on top of the columns.

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Figure A.3 The (a) 15 µm2 3D rendering and (b) line profile for an AFM tapping mode image of the fixture support columns created using a 1 minute total time XeF2 etch. The scans were collected using a DI 3000 Nanoscope.

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Figure A.4 Scanning electron micrograph images of the XeF2 etch sequence, taken after (a) 30 seconds, (b) 1 minute, (c) 2 minutes, and (d) 3 minutes of etch were completed.

The etch selectivity of XeF2 to photoresist was much lower in experimental results than

was reported in the literature [3]. The XeF2 etch rate depends on the chamber pressure

and volume as well as the total exposed surface of the sample. Under the conditions used

in these etch trials, the etch rate for the Si was approximately 2-3 µm/min. Therefore

with almost 3 µm of photoresist, there should have been no appreciable etch of the mask

layer. Instead, the mask was completely removed by the XeF2 etch within a minute. This

photoresist removal contributed to the shallow valleys in between the support peaks. The

XeF2 etch technique was not appropriate for processing the support columns in this

application. Even if the more robust SiO2 mask had been used, creating peaks and

valleys of acceptable depths for testing, the surface quality resulting from this method

was not sufficient for the small diameter nanowire samples used for mechanical testing.

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A.2.2 Reactive Ion Etching

Reactive ion etching (RIE) was used as a second etch technique to create the fixture

support columns, in an effort to increase the height of the columns and reduce the final

surface roughness. This technique can be used to create both isotropic and anisotropic

etch profiles. RIE uses fluorine-based plasma (SF6) to isotropically etch silicon and

several types of fluorine-based inhibitors, such as O2 and C4F8 gases, can be used to

create directional, or anisotropic, etching. Using a pulsed mode etch technique

alternatively exposes the silicon to the isotropic plasma etch and the deposition of a

chemically inert passivation layer, which builds up along the side walls of an etched

feature, making it possible to create deep sided features with high aspect ratios [4, 5].

For this research, inductively coupled plasma reactive ion etch (ICP RIE) was completed

using an Alcatel ADIXEN AMD, 100 I-Speeder. ICP is a plasma created with a radio

frequency (RF) power magnetic field and can create a very high plasma density, and

therefore potentially high etch rates.

The instrument used for RIE requires a 4 inch wafer sample size, therefore individual

diced 5 x 12 µm samples were affixed to a silicon carrier wafer using a small amount of

photoresist, which was then heated at 80° F for 5 minutes to act as an adhesive. The

sample chamber was prepared before each use by completing 20 minute cleaning and

conditioning processes in order to maintain the same chamber conditions at the start of

every etch. To create the three-point bending test span a combination of isotropic and

anisotropic etches were employed. First, the sample was exposed to an anisotropic etch

to create a trench with a sufficient depth for the nanowire deflection during mechanical

testing. Subsequently, an isotropic etch was performed to produce the desired peak

feature for the fixture supports (Figure A.5). After each sample etch was completed, the

chamber was subjected to a 10 minute conditioning treatment to ensure the same base

chamber conditions exist for prior to etching the next sample. Time, pressure, and gas

flow rates were varied to optimize the process.

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Figure A.5 Series of schematics illustrating the formation of the peaked support columns using a combination of anisotropic and isotropic etch techniques with a photoresist mask.

When each etch sequence was completed, the carrier wafer was removed from the

instrument chamber and individual samples were released by dissolving the photoresist

adhesive in acetone. Samples with a photoresist mask were then cleaned with acetone to

strip the majority of the photoresist, rinsed in methanol, and any remaining photoresist

was removed under oxygen plasma exposure at 80 W for 2 minutes using a O2 flow rate

of 200 sccm. Samples that had a SiO2 mask were exposed to a 10 minute HF etch to

remove the oxide layer followed by rinses in water, acetone, and methanol.

A series of trials were completed to analyze the etch method and resulting support peaks

on the device fixtures. First, samples were exposed to an anisotropic etch for different

lengths of time in order to determine an estimated etch rate of the recipe in the machine,

which began at approximately 1100 µm/min, slowing slightly as the trench depth

increased. The calculations were made based on tapping mode AFM scans of the final

fixture surfaces. In the initial etch sequence the anisotropic etch was performed first in

order to produce the trench depth, followed by an isotropic etch to form the peak features,

as demonstrated schematically in Figure A.5. This etch combination created a relatively

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smooth final surface with the peaked support columns separated by 5 µm and a gap depth

of approximately 591 nm. The first etch series, shown in Figure A.6, was more suited for

use as a test fixture than any of the XeF2 etched samples, however the depth of the valley

in between the support columns was surprisingly shallow and was still insufficient for

centrally loaded, simply supported beam bend tests of the silicon nanowires.

Figure A.6 AFM tapping mode line profile for a bridge mask sample. The trenches were creating using the combination of anisotropic and isotropic etching.

In the initial series of etches, photoresist was being used as the design mask. When the

fixture was removed from the instrument after the etch sequence was completed, there

was little to no photoresist mask remaining on the silicon surface. It was concluded that

during the anisotropic etch, the photoresist mask was slowly removed, leaving a very thin

mask layer remaining for the isotropic etch. As the isotropic etch sequence was run, the

mask layer was completely removed. Therefore, while forming the peaks, the etch

process also reduced the height of the columns established in the anisotropic etch.

To increase the selectivity between the silicon and the photoresist mask, a high aspect

ratio, low frequency (HARLF) anisotropic etch recipe was substituted for the original

anisotropic etch process. While the first anisotropic recipe utilized O2 to form a

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passivation layer on the silicon, HARLF used C4F8 which created a Teflon-like polymer

layer. The etch rate of the HARLF recipe was estimated at 1.1 µm/min based on SEM

images taken of sample cross-sections (Figure A.7). Using the new anisotropic recipe,

the same etch sequence of an anisotropic etch followed by an isotropic etch was

completed on a series of sample fixture, as before.

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Figure A.7 Scanning electron micrographs illustrating the formation of the support columns with (a) 2 minute (b) 3 minute and (c) 4 minute anisotropic etch times.

Initial results of the HARLF and isotropic etch combination did not produce a significant

isotropic etch profile. The support column profile that developed after the anisotropic

etch remained basically the same shape after the completion of the isotropic etch. It was

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also visually apparent that the photoresist mask was still being removed too quickly to

achieve the desired depth profiles for the columns. The polymer layers deposited on the

sidewalls during the anisotropic etch built up to the point where the isotropic etch was

only effective at the bottom of each trench. To restrict the build-up of the polymer layer

during the anisotropic etch, the HARLF recipe was altered to include a longer O2 step

during each sequence. In addition, a short O2 clean wais added in between the

anisotropic and isotropic etch steps to try and strip any remaining polymer from the

sidewalls before the isotropic etch began. The photoresist mask was also replaced by a

more robust, thermal oxide mask for the subsequent etch series.

The results of the changes to the etch recipe, sequence, and mask are shown in Figure A.8

a-d. The replacement of the photoresist mask significantly improved the consistency of

the trench depth, however the passivation layer created by the inclusion of the C4F8 gas

proved to be too robust to fully remove within the instrument chamber prior to the

isotropic etch. As the anisotropic etch proceeded, creating deeper trenches, the polymer

layer built up along the sidewalls. This progression therefore left a thin protective layer

at the base of the trench, while multiple layers build up the polymer thickness toward the

top of the column. During the isotropic etch, the base of each trench was actively etched

and the top was protected by the polymer layer. This effect was stronger outside surface

of the support columns. Eventually the isotropic plasma etched through both sides at the

base of the columns and pinched off the entire fixture (Figure A.8 (d)).

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Figure A.8 Scanning electron micrograph images illustrating the progression of the isotropic etch. Image (a) is the starting point, where only the anisotropic etch has been completed. (b) and (c) occur as the polymer layer builds on the upper walls of the pillars, confining the majority of the etch to the lower half of the fixture until the columns pinch off at the base (d).

A.3 Focused Ion Beam Milled Fixture While the etch trials to develop the support columns for use on the final fixture was being

conducted, a series of fixtures were also created in the FIB (FEI DualBeam Quanta 200

3D). These interim fixtures were designed to investigate the AFM as a testing method for

three-point flexure and to attempt field-assisted alignment for nanowire placement across

a test span. The fixture design was based on a thermo-mechanical fatigue test device

(Figure A.9) consisting of gold lines approximately 4.5 µm wide and 250 nm thick

deposited onto an oxidized silicon substrate. The lines of the existing device were

connected to 125 µm2 gold electrode pads developed as contact points for external

stimulation.

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Figure A.9 Optical micrograph of a thermo-mechanical fatigue sample. The interim FIB fixture is developed in between two of the large gold pads.

Using a gallium ion beam, a portion the gold lines on the existing fixture were removed,

leaving separate circuits on either side of the gap. A three-point bending fixture design

(Figure A.10 (a)) was then milled into the wafer surface just above the gap in the

terminated gold lines. The design was milled approximately 1 µm into the SiO2 layer of

the wafer, leaving posts 0.5 µm thick and 5 µm long at the same level as the substrate

surface. Using the ion beam thin tungsten lines were then deposited along the posts and

connected to the pre-existing gold lines (Figure A.10 (b)). The tungsten deposition was

uneven, but continuous and therefore sufficient for the nanowire alignment trials.

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7 µm

8 µm

2 µm

5 µm

0.5 µm

2 µm

(a)

Figure A.10 (a) Design schematic and (b) scanning electron micrograph of the interim FIB fixture. In (b) the fixture design is milled into the wafer on the top of the image and tungsten lines are deposited using ion beam deposition to connect the fixture to the gold lines from the existing structure (Figure A.9).

The placement trials of the silicon nanowire samples onto the fixture made within the FIB

were performed using a similar technique to the field-assisted nanowire manipulation

142

described in Chapter 2. The substrate containing the fixtures was placed under an optical

microscope where two tungsten microprobes mounted in micro-manipulators were

brought into contact with the gold electrode pads at either end of one fixture. An electric

field was then applied across the probes, varying at a set frequency. The electric field

parameters of the first two trials were set according to the most successful alignment

achieved in previous testing, 83 Vrms and 1 kHz oscillating frequency [6]. This electrical

stimulation alone caused complete, rapid failure of the ion beam deposited tungsten lines

before any nanowires were deposited onto the surface. The voltage was then drastically

reduced to approximately 1.5 Vrms and slowly increased to 4 Vrms to observe the effect on

the deposited tungsten. The tungsten deposits visually darkened, but remained in place

on the fixture. Beginning at 4 Vrms and 1 kHz oscillating frequency the solution

containing the silicon nanowires was flooded onto the substrate surface over the fixture.

The voltage was slowly increased with no visual alignment of the nanowire samples,

however at approximately 13 Vrms the tungsten lines failed, completely separating from

the surface.

Due to the method of metal deposition in the FIB, it was likely that a significant amount

of tungsten was deposited in an overspray onto the surface surrounding the lines. A

series of new fixtures were created in the same manner as before, however after the

deposition is complete and the tungsten gas had dissipated from the instrument vacuum

chamber, the ion beam was used to mill a shallow depth into the substrate SiO2 layer,

ensuring that all of the metal on the surface from the two sides of the circuit was

electrically isolated. The series of nanowire alignment tests was repeated. The tungsten

lines remained intact past 13 Vrms, however by 20 Vrms the tungsten peeled off the surface

of the substrate with no visible nanowire alignment observed prior to failure. After the

second series of complete device failure, in addition to the results of the previous field-

assisted manipulation trials, it was concluded that using the FIB to continue to develop

the interim set of fixtures should be abandoned.

143

A.4 TEM Grid with Holey Silicon Nitride Membrane Rather than processing an entire fixture, pre-fabricated silicon chucks with a holey silicon

nitride (SiN) membrane are evaluated as possible devices that may achieve a combination

of TEM compatibility for nanowire characterization and structural integrity for

mechanical testing. The purchased grids (Ted Pella, Inc.) are made from 3 mm diameter

silicon with a 0.5 µm square hole in the center. The silicon base structure is then coated

with a 200 nm thick silicon nitride membrane on one side of the sample, which is

patterned with approximately 2.9 µm diameter holes in a 100 X 100 grid (Figure A.11).

The holes in the membrane acted as the mechanical testing span, while also allowing for

TEM characterization of each individual nanowire. Nanowire growth direction,

diameter, and span could be accurately determined prior to testing, while the fractured

nanowire and incurred damage could be evaluated afterwards.

144

Figure A.11 Scanning electron micrograph of a TEM grid coated with holey silicon nitride membrane showing (a) the entire gird area and (b) a closer view of the individual holes.

145

The silicon nitride membrane was sputter coated with a 2 nm gold-palladium film

(Denton Vacuum Desk IV) to eliminate charging under the electron microscope. The

method of silicon nanowire sample preparation was similar to that of the silicon wafer

bridge samples reported in Chapter 2. Nanowires dispersed in an alcohol solution were

flooded onto the membrane grid and visual inspection with an optical microscope was

sufficient to determine whether multiple nanowires were spanning the membrane holes.

Electron beam targeted deposition of platinum-containing adhesive in a dual beam FIB

(FEI Nova™ Nanolab 600) was used to affix the ends of each spanned nanowire to the

membrane surface, Figure A.12.

Figure A.12 Scanning electron micrograph of a silicon nanowire sample fixed across a hole in the silicon nitride membrane.

This fixture appeared to simplify and improve the accuracy of nanowire characterization

during for mechanical testing by allowing for TEM inspection prior to and after sample

fracture. However, in practice the SiN membrane was not a robust support for nanowire

146

flexure testing. As the cantilever tip was used to perform a three-point bend test, both the

nanowire and membrane deflected at high applied loads. This combination created

uncertainty in establishing pure nanowire deflection. The additional membrane

movement also caused difficulty in fracturing the nanowire samples prior to the limits of

cantilever deflection. Therefore, the fixture containing a holey SiN membrane was

eliminated as a possible testing platform for the silicon nanowires.

147

References 1. Hoffman, E., et al. 3D structures with piezoresistive sensors in standard CMOS.

in Micro Electro Mechanical Systems, 1995, MEMS '95, Proceedings. IEEE. 1995.

2. Madou, M.J., Pattern Transfer with Dry Etching Techniques, in Fundamentals of Microfabrication: The Science of Miniaturization. 2002, CRC Press: Boca Raton.

3. Chang, F.I., et al., Gas-phase silicon micromachining with xenon difluoride Proceedings of SPIE-The International Society for Optical Engineering, 1995. 2641: p. 117-28.

4. Jansen, H.V., et al., Black silicon method X: a review on high speed and selective plasma etching of silicon with profile control: an in-depth comparison between Bosch and cryostat DRIE processes as a roadmap to next generation equipment Journal of Micromechanics and Microengineering, 2009. 19(3).

5. Zou, H., Anisotropic Si deep beam etching with profile control using SF6/O2 Plasma. Microsystem Technologies, 1994. 10: p. 603-607.

6. Motayed, A., et al., Realization of reliable GaN nanowire transistors utilizing dielectrophoretic alignment technique. Journal of Applied Physics, 2006. 100(11): p. 114310/1-114310/9.

148

B Linear Elastic Analytical Model: Ledges

Thin beam in centrally loaded, fixed-fixed bending configuration with adjustable ledge

length and load location.

Variables

disp = displacement (nm)

sl = location of left hand support (nm)

sf = location of applied load (nm)

sr = location of right hand support (nm)

soa = location of right hand side fixed end (nm)

F = applied force (nN)

modulus = elastic modulus (GPa)

moment = moment of inertia for the nanowire

��������� � ������ � � ��� ����� � ������ � ����������� � ��� � ��� � �������� � ������� � ���� !� � ���"���� � ���#�$������� � ������ � %����& '

�����()�*+ � � ������ � � ��� � �,��� � ��� #�$������� � ������ � %���& ����� ������� �� -$������� �� � ���������$������� � .������ � %������� �� ���������� � %������ � %����� �� ��#$������������%��� � �� � ���� ������� � %�� �� ��������%������� � ������� � %�� �&� �� !�� ���� � %����� ���%������� � %��� � ������� � ,����� � ,�� �� ���#�����%��� � �� � %������� � ��&"/0

149

����(1)�*+ � � ������ � � ��� � �,��� � ��� #�$������� � ������ � %���& ����� ������ � ��#�� �$������� � %���� ���� � �� � ���� � $���������� ������� � ,����� � %����%������� � �� � ������ � ��� �&� ����#%��������������� � ����� � ������ ����������� � ���������� � ����� � %����� �&� �� !�%�����#���� � ������� � ��&� ��#%����� � ��������� � ������ � � �&"0

����12)�*+� � ������ � � ��� 3���� � ������ � ��������� � ���� � ���������� � �� ��� � ������ � ������� � ���#�$������� � ������ � %����& 4

�� 5 �� 5� �� 5 ��� 5� � 5 ������ 5 � ��� 5 67�8 � 9 � �: ����� 5 ;������������< =�< 6< ��> ����� 5 ;���������(��< =�< ��< ��> ����% 5 ;��������(1��< =�< ��< ��> ����$ 5 ;��������12��< =�< ��< ���> ?@�A������< �����< ����%< ����$< ;���B��C� D ��� �E7 F���)����G��� � ������(��< =�< ��< ��>+ �� 5 H �����(��� ����;��� 5 I����J�K���L�< 2M������)��������+N< =�< 6< ��>, J�K���L�< 2M������)�����(��+N< =�< ��< ��>, J�K���L�< 2M������)����(1��+N< =�< ��< ��>, J�K���L�< 2M������)����12��+N< =�< ��< ���> 2������OPQRSTUVS7WRXO< ����;���< OYZ[O

Creates a two column table from graph, x (x-axis) and disp (y-axis).

Exports columns into excel file named “filename.xls”.

150

C Nonlinear Elastic Analytical Model for Large Deflection:

Axial Tension

Determination of Young’s modulus (E) from experimental measurements of load-point

deflection (ua) and applied load (Fa) and an approximate value of the load application

point (d).

Edwin R. Fuller, Jr. and Rebecca Kirkpatrick

January 13, 2010

Load applied at:

\��� 5 �� ] �� � �� �� � \� 5 �� ] �� � ��

Define the functions:

1�^� � �^ ] XQT_�^� � � ] �`aX_�^� � �� 1b�\< ^� � � � `aX_�^ ] �� � \� � `aX_�^� � `aX_�^ ] \� � ^ ] �� � \� ] XQT_�^� 1 �\< ^� � XQT_�^ ] �� � \� � XQT_�^ ] \� � XQT_�^� � ^ ] \ � ^ ] �� � \� ] `aX_�^�� ^ ] `aX_�^ ] �� � \�

�cd 5 Fe ] �f�.� ] 2 ] g

Accordingly, the transverse deflection of a beam in tension and clamped at both ends

normalized by um0 (i.e. Un = u/um0) is:

hi�j< \< ^� � �.�^e ] 1�^�] �1 �\< ^� ] �`aX_�^ ] k� � �� � 1b�\< ^� ] �^ ] k � XQT_�^ ] k�� � �.�] ^e ] �XQT_�^ ] �k � \� � �^ ] �k � \� ] l�k � \�

At the position of the applied transverse force, k 5 \, the transverse deflection of the

beam is:

him 5 hi�\< \< ^� 5 �f�cd

him�\< ^� � �.�^e ] 1�^�] �1 �\< ^� ] �`aX_�^ ] \� � �� � 1b�\< ^� ] �^ ] \ � XQT_�^ ] \��

151

The normalized tangent to the beam is: 1hi�j< n< o� � �.�o ] p�o� ] �p �n< o� ] XQT_�o ] j� � pb�n< o� ] �� � `aX_�o ] j�� � �.�] o ] �`aX_�o ] �j � n� � � ] �l�k � \�

Assume E, know Fa and d.

Calculate um0 from :

�cd 5 Fqe ] �f�.� ] 2 ] g

given

Lb, Rb, g 5 b: ] 9 ] Bq:, �cd 5 rstuv

� � \��� � �� ] �� � ��

Enter the following data: \ � Bq � Fq � �f � 2 � g � 67�8 ] 9 ] Bq:

�cd � Fqe ] �f�.� ] 2 ] g �cd 5 w � Bq�cd w 5

Determine ^ from the root of:

Kq�^< \< w� � � ] x 1hi�j< n< o� �jbd � �w ] ^�

@ � �����Kq�@< \< w�< @� 5 �f�^� � him�\< ^� ] �cd

152

�f#�����Kq�@< \< w�< @�& 5

ua is the calculated deflection (in nm), given E and Fa

^ � �����Kq�@< \< w�< @� ^ 5

T is the axial tension in the beam as a result of large deflection.

J � ^ ] 2 ] gFq

J 5

153

D Nonlinear Analytical Model for Large Defection:

Elastic Curve

Thin beam under large deflections for fixed-fixed bending configuration

Variables:

L = nanowire length (nm)

radius = nanowire radius (nm)

modulus = elastic modulus (GPa)

F = applied force (nN)

a = location of applied force (nm)

ρ = normalized parameter for the evaluation of axial tension

T = axial tension (nN)

u = displacement (nm)

F 5� ������ 5� � ��� 5 9 ] ������:$ � 5� ������ 5 � 5 F� ^ 5�

J 5 ^ ] ������ ] � ���F

�cd 5 Fe��.� ] ������ ] � ��� \ 5 �F

k�y � yF

1d 5 ^?��@�^ � ��(��@�^ � �� 1b 5 � � (��@�^�� � \� � (��@�^ � (��@�^ ] \ � ^�� � \��?��@�^ 1 5 ?��@�^�� � \� � ?��@�^ ] \ � ?��@�^ � ^ ] \� �^�� � \��(��@�^ � ^�(��@�^�� � \��

154

��k< z � �.���cd^e1d !1 �(��@�^ ] k � �� � 1b�^ ] k � ?��@�^ ] k�� 1d !?��@�^��k � \� � #^��k � \�&"{��M�����J@����k � \"�E7 JD z

;������k�y< J< =y< 6< F> ?@�A�=H>< ;���B��C� D ��� y��E7 F��������G��� � ���k�y< J< =y< 6< F> y� 5 2M������)��k�y�< J+ ;�����J ����;��� 5 J�K��)Ly< 2M������)��k�y< J+N< =y< 6< F< F $66| >+ 2������OPQRSTUVS7WRXO< ����;���< OYZ[O

Creates two column table from graph, z (x-axis) and u(y-axis), from z=0 to L in

increments of L/400.

Exports columns into excel file named “filename.xls”.

155

E Maximum Liklihood Estimation for Weibull Parameters

Bootstrap Strength Data

Obtain the Maximum Likelihood Estimators of the Weibull parameters, and Bootstrap

Estimators of the Confidence Intervals for the Weibull parameters of a strength

distribution (sf).

Read in the Strength Data from data file, “Strength_IN.prn”:

?����C�@g} � B2�1;B}�O?����C�@*g}7 ���O�

Failure Stresses (GPa) ~� � ?����C�@g}�d� ] �;�

G � ���C�@#~�& Number of strength specimens G 5

Display input Failure Stresses: M � ����� -G� / M 5 � � 6<�7 7 �M � �� ?�� � ��K �����#~� < � ] �< � ] � � ,<6<6& ?�� � ��K �����#~� < � ] M<G � �<6<6& ?�d�= ?�b�= ?� � 5

~�� 5

Maximum Likelihood Estimators of the Weibull distribution parameters from the failure

stress distribution.

Weibull Parameters

Define:

?cd�~< � � �G ] ��~��c�b��d

156

GF2�~< � � G G � � ���~�� � � ��~��c ] ���~���b��d ?cd�~< ��b��d

Finds the value of m that makes the function MLE(s,m) equal to zero. [TOL=1e-007]

d � m0: initial guess for Weibull modulus

b � ���� !GF2 ! ����f < d" < d" m1: 1st approximation b 5 GF2 ! ����f < b" 5

� ���� !GF2 ! ����f < b" < b" m2: 2nd

approximation 5 GF2 ! ����f < " 5

� � ���� !GF2 ! ����f < " < " me: Weibull modulus estimator � 5 GF2 ! ����f < �" 5

This equation calculates the Weibull scale parameter once me has been calculated

?�d � ?cd ! ~��;� < �"b c�� ��;� ?�d 5

Maximum Likelihood Estimators of the Weibull distribution parameters

Weibull modulus: � 5

Weibull Scale Parameter: ?�d 5

Bootstrap the calculation to determine 90% confidence intervals

i is a counter � � 67 7 �G � �� Number of bootstrap samples Gq �10000

k is the bootstrap counter � � 67 7 �Gq � �� Number of strength samples G 5

Mb bootstrap samples of the failure stresses �~�<� � ~�������!�u����"

157

Order-statistics estimate of the failure probability F for the ranked data Sfr ��� � � � 678G ~�� � ����#~�&

Define: ���� � �� ! bb�"

� � Use me as the initial guess for the Weibull modulus

K� � ���� 3GF2 3�~����;� < 4 < 4

?K� � ?cd 3�~����;� < K�4b cq��

Weibull modulus me and 90% confidence interval calculated from MLE

� 5 B����� K � ����� K� Gq 5 �6666 ������6768 ] Gq� 5 866 68H � B����� K������d7d�]��� 68H 5 .8H � B����� K������d7 �]��� .8H 5

Weibull scale parameter Sd0 and 90% confidence interval calculated from MLE

?�d 5 B������K � ������K� Gq 5 �6666 ������67.8 ] Gq� 5 .866 ��668H � B������K������d7d�]��� ��668H 5 ��6.8H � B������K������d7 �]��� ��6.8H

158

F Effective Surface Area under Large Deflection

Model uses the Adaptive Monte Carlo method (a numerical integration technique) to

determine the effective surfaces of the sample.

The fit to the elastic curve of a circular cross section nanowire is given by a 9th

degree

polynomial defined by constants C0 to C9, x is the position along the horizontal axis and

y is the deflection (downward in the negative direction).

Variables and Definitions:

C0 to C9 = 9th

order polynomial defined constants

m = Weibull modulus

radius = radius of the nanowire in the undeformed state (m)

length0 = length of the nanowire in the undeformed state (m)

surfacearea0 = surface area of the undeformed nanowire (m2)

dydx = first derivative of the elastic curve (i.e., the local slope)

d2ydx2 = second derivative of the elastic curve (i.e., the curvature)

strainbend = bending strain from the shape of the elastic curve, including geometric

nonlinearities (i.e., using the real definition of curvature)

strainbendmax = maximum bending strain of the fixed nanowire

lengthdef = length of the elastic curve (for the deformed nanowire)

strainax = axial strain

effsurfbend = effective surface area considering both bending and axial tension

surfareafrac = a comparison of the effective to uniaxial tension surface area

(6 5 (� 5 (� 5 (% 5 ($ 5 (8 5 (, 5 (� 5 (¡ 5 (. 5 5 ������ 5 ���C�@6 5 ��������6 5 � � 9 � ������ � ���C�@6 ¢ 5 (6 � (��� � (��� � (%��e � ($��: � (8��� � (,��£ � (���¤ � (¡��¥ � (.�� 

159

�¢�� 5 1�¢< =�< �> ��¢��� 5 1�¢< =�< �> ������K��� 5 � � - ��¢������ � ��¢��� �b7��/

(����B�������¢���< =�< ���C�@6> ����¦���� 5 H ��������� 5 ����g�����¦����¢��� 55 6< �< B����< ����¦���� ?����H ���� 5 � 7��H| �����EEJ�K����� ������K��� �� 5 � � - ��¢������ � ��¢��� �b7��/ E7 §� D ���C�@6� < � D ������¨ �������K����

5 }g���C���� ©3 �ª�� � �� ������| � �4� - �������K���������K��� ��/

c < =�< 6< ������>< L�< 6< ����)��+N<G��@��D O������M�G����(����O«

�������K����5 }g���C���� ©3 �ª�� � �� ������| � �4� - �������K���������K��� ��/

c < =�< 6< ������>< L�< ����)��+< �������N<G��@��D O������M�G����(����O«

�������K���%5 }g���C���� ©3 �ª�� � �� ������| � �4� - �������K���������K��� ��/

c < =�< 6< ������>< L�< ����)��+< ������%N<G��@��D O������M�G����(����O«

�������K��� 5 �������K���� � �������K���� � �������K���% ���C�@��� 5 }g���C���� ¬ª�� � ��¢��� �< =�< 6< ���C�@6>­

160

�������� 5 3����C�@��� � ���C�@6����C�@6 4

�������K������5 }g���C���� ©3 �ª�� � �� ������| � �4� 3 ��������K��� � ����������������K��� �� � ���������4

c < =�< 6< ������>< L�< 6< ����)��+N<G��@��D O������M�G����(����O« �������K������5 }g���C���� ©3 �ª�� � �� ������| � �4� 3 ��������K��� � ����������������K��� �� � ���������4

c < =�< 6< ������>< L�< ����)��+< �������N<G��@��D O������M�G����(����O« �������K�����%5 }g���C���� ©3 �ª�� � �� ������| � �4� 3 ��������K��� � ����������������K��� �� � ���������4

c < =�< 6< ������>< L�< ����)��+< ������%N<G��@��D O������M�G����(����O« �������K����� 5 �������K������ � �������K������ � �������K�����% �����������¦ 5 �������K�������������6

Vita

Rebecca Kirkpatrick

Education

Ph.D., Materials Science and Engineering

The Pennsylvania State University, University Park, PA

Dissertation Title: Flexure Strength and Failure Probability of Silicon Nanowires

M.S., Materials Science and Engineering

The Pennsylvania State University, University Park, PA

Thesis Title: Durability of Octadecyltrichlorosilane Coated Borosilicate Glasses

B.S., Materials Science and Engineering

Lehigh University, Bethlehem, PA

B.A., Asian Studies

Lehigh University, Bethlehem, PA

Select Publication

Kirkpatrick, R. and C.L. Muhlstein, Performance and durability of octadecyltrichlorosilane

coated borosilicate glass. Journal of Non-Crystalline Solids, 2007. 353(27): p. 2624-2637.

Antolino, N.E., G. Hayes, R. Kirkpatrick, C.L. Muhlstein, M.I. Frecker, E.M. Mockensturm, and

J.H. Adair, Lost Mold Rapid Infiltration Forming of Mesoscale Ceramics: Part 1, Fabrication.

Journal of the American Ceramic Society, 2009. 92(s1): p. S63-S69.

Antolino, N.E., G. Hayes, R. Kirkpatrick, C.L. Muhlstein, M.I. Frecker, E.M. Mockensturm, and

J.H. Adair, Lost Mold-Rapid Infiltration Forming of Mesoscale Ceramics: Part 2, Geometry and

Strength Improvements. Journal of the American Ceramic Society, 2009. 92(s1): p. S70-S78.

Academic Honors and Activities

• Alpha Sigma Mu Materials Science and Engineering Honor Society

• Keramos National Ceramics Honor Society

• Wilson Research Award, The Pennsylvania State University

• Materials Research Institute Fellowship, The Pennsylvania State University

• ASM International, Chair, The Pennsylvania State University

• Service/Leadership Award, Department of Materials Science and Engineering,

The Pennsylvania State University

• Student Advisory Council, Department of Materials Science and Engineering,

The Pennsylvania State University

• NSF Community Outreach

• Women in Science and Engineering (WISE) Summer Camp counselor,

The Pennsylvania State University