And Nano- Particles in Fluids: Tunable Macroscopic Material Properties and Characterization of Individual Nanoparticles

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FIELD-BASED MANIPULATION OF MICRO- AND NANO- PARTICLES IN FLUIDS: TUNABLE MACROSCOPIC MATERIAL PROPERTIES AND CHARACTERIZATION OF INDIVIDUAL NANOPARTICLES

by WUHAN YUAN

A dissertation submitted to the School of Graduate Studies Rutgers, The State University of New Jersey In partial fulﬁllment of the requirements For the degree of Doctor of Philosophy Graduate Program in Mechanical and Aerospace Engineering

Written under the direction of Jerry W. Shan and Liping Liu And approved by

New Brunswick, New Jersey May, 2019 ABSTRACT OF THE DISSERTATION

Field-Based Manipulation of Micro- and Nano- Particles in Fluids: Tunable Macroscopic Material Properties and Characterization of Individual Nanoparticles

By WUHAN YUAN

Dissertation Directors: Jerry W. Shan and Liping Liu

Field-based particle manipulation has provided us unique opportunities to remotely manipulate fluid-suspended micro- and nanoparticles in a highly efficient and controllable manner. Specifically, an external electric/magnetic field can be used to control (1) the macroscopic physical properties of the materials and (2) the microscopic movement of micro- or nanoparticles in a suspending fluid, both of which have given rise to a wide variety of novel applications.

In terms of controlling the macroscopic material properties, we have investigated a novel type of smart material with anisotropic and tunable acoustic properties. To elaborate, the microstructure (e.g., particle orientation and chaining) of suspensions of non-spherical ferromagnetic particles can be controlled by an external magnetic field, making it possible to change the acoustic properties of the suspension. Here, we experimentally demonstrate that dilute suspensions of subwavelength-sized oblate-spheroidal nickel particles exhibit up to a 35% change in attenuation coefficient at MHz frequencies upon changing the direction of an external magnetic field, for particle volume fractions

ii of only 0.5%. Comparison is made to suspensions of spherical particles, in which the attenuation is smaller and nearly isotropic. Optical transmission measurements and analysis of the characteristic timescales of particle alignment and chaining are also per- formed to investigate the reasons for this acoustic anisotropy. The alignment of the oblate-spheroidal particles is found to be the dominant mechanism for the anisotropic and tunable acoustic attenuation of these suspensions. A mathematical model has also been developed to describe the acoustic properties of the above mentioned microstructured suspensions using eﬀective medium theory as well as Eshelby’s analytical solutions for ellipsoidal inclusions.

For individual micro- or nanoparticles, we take advantage of the strong dependence between AC-ﬁeld induced particle rotation and particle properties to eﬀectively characterize individual particles. In detail, we utilize the contactless, high-throughput electro-orientation spectroscopy (EOS) technique to measure the electrical conductivity of Si nanowires with atomic-layer-deposition- (ALD-) induced Al2O3 layers in a statistically meaningful manner. Initial surface treatments with HF and chemical-induced oxide (SC1) before ALD passivation were both investigated. We show that nanowire properties and variability are dominated by the surfaces, given that the ALD-induced

Al2O3 layer changes the electrical conductivity by 5-orders-of-magnitude while reducing the variation in conductivity by half. These changes in the nanowire conductivity are believed to be due to a combination of chemical passivation and the ﬁeld-eﬀect passivation introduced by the ALD treatment.

Finally, we extend the EOS technique to enable characterization of nanowires with more complex structures. Specifically, we show that the EOS technique can determine the conductivity and dimensions of two distinct segments in individual Si nanowires with axially encoded dopant profiles. Our analysis combines experimental measurements and computational simulations to determine the electrical conductivity of the nominally undoped segment of two-segment Si nanowires, as well as the ratio of the segment lengths. We demonstrate the efficacy of our approach by comparing results generated by EOS with conventional four-point-probe measurements. Our work provides

iii new insights into the control and variability of semiconductor nanowires for electronic applications and is a critical ﬁrst step toward the high-throughput interrogation of complete nanowire-based devices.

iv Preface

Much of the content in Chapters 2, 4 and 5 is taken verbatim from published or soon- to-be-submitted papers with collaborators, and other chapters contain wording similar or identical to that found in these papers. Permissions have been obtained from the collaborators to include the content in this thesis.

References

[1] W. Yuan, L. Liu, and J. W. Shan. Tunable acoustic attenuation in dilute suspensions of subwavelength, non-spherical magnetic particles. Journal of Applied Physics, 121(4):045110, 2017.

[2] W. Yuan, G. Tutuncuoglu, A. T. Mohabir, L. Liu, L. C. Feldman, M. A. Filler, and J. W. Shan. Contactless electrical and structural characterization of semiconductor nanowires with axially modulated doping proﬁles. Small, 15(15):1805140, 2019.

[3] W. Yuan, G. Tutuncuoglu, A. T. Mohabir, L. Liu, L. C. Feldman, M. A. Filler, and J. W. Shan. Contactless, high-throughput analysis on electrical conductivity of si nanowires with surface passivation. in preparation, 2019.

v Acknowledgements

I would like to start by expressing my sincere gratitude to my advisors, Prof. Jerry W. Shan and Prof. Liping Liu, for their valuable guidance and never-ending support. I thank Prof. Shan for oﬀering me the opportunity to work in a multidisciplinary environment and allowing me to grow as an experimental researcher. I could not have accomplished my research goals without him sharing his experience and wisdom with me. I would also like to thank Prof. Liu, as he helped me greatly with his intelligence and knowledge in shaping the theoretical end of my research. His patience and encour- agement throughout my PhD studies are invaluable. It is truly a privilege for me to have both Prof. Shan and Prof. Liu as my advisors, as they will be my role models through my life.

I want to extend my gratitude to the rest of my dissertation committee members, Prof. Leonard C. Feldman and Prof. Stephen D. Tse, for their insightful comments and brilliant suggestions on my research. I am also very grateful for the assistance from Prof. Michael Filler and Dr. Gozde Tutuncuoglu at GA Tech on our collaborative project.

I would like to recognize some of my previous and current colleagues, Cevat Akin, Richard J. Castellano, Semih Cetindag, Yasir Demiryurek, Gabriel Giraldo, Lixin Hu, Xin Liu, Hanxiong Wang and Kaiyan Yu, for their kind assistance and counseling in my research work. I also want to thank John Petrowski for helping me build my experimental setups.

Finally, I want to express my deepest gratitude to my mother, Jun Zhou and my grandparents, Huanzhi Bao and Wenbin Zhou. I could not have done it without their unconditional love and support. I also want to thank my girlfriend, Xin Ai, for always being there for me and believing in me.

vi Special thanks to my father, Naichao Yuan, who passed away during the second year of my PhD. Thank you, dad, for everything.

vii Table of Contents

Abstract ...... ii

Preface ...... v

Acknowledgements ...... vi

List of Tables ...... xi

List of Figures ...... xii

1. Introduction ...... 1

1.1. Particle manipulation by electric/magnetic ﬁelds ...... 1

1.1.1. Electric/magnetic ﬁeld responsive smart materials ...... 2

1.1.2. Microscopic particle manipulation and characterization using electric/magnetic ﬁeld ...... 4

1.2. Electrical characterization of semiconductor nanowires ...... 5

1.3. Dissertation outline ...... 6

2. Tunable Acoustic Attenuation in Dilute Suspensions of Non-spherical Magnetic Particles: Experiments ...... 10

2.1. Introduction ...... 10

2.2. Experiment ...... 12

2.3. Results and discussion ...... 15

2.4. Conclusions ...... 25

3. Acoustic Properties of Microstructured Suspensions: Theory .... 27

3.1. Introduction ...... 27

3.2. Acoustic properties of a homogeneous viscoelastic medium ...... 29

viii 3.3. Homogenization limit of mixtures of viscoelastic materials ...... 32

3.4. Eﬀective viscoelastic properties of two-phase suspensions ...... 36

3.4.1. Suspensions in the dilute limit ...... 37

3.4.2. Non-dilute suspensions ...... 39

3.4.3. Suspensions with random particle directions ...... 40

3.5. Eﬀective acoustic properties of a two-phase medium with spheroidal inclusions ...... 41

3.5.1. Inﬂuence of particle aspect ratio ...... 41

3.5.2. Inﬂuence of particle volume fraction ...... 43

3.5.3. Comparison with the experimental results ...... 43

3.6. Conclusions ...... 47

4. Contactless, High-throughput Analysis on Electrical Conductivity of Si Nanowires with Surface Passivation ...... 48

4.1. Introduction ...... 48

4.2. Method ...... 50

4.2.1. Electro-orientation ...... 50

4.2.2. Electro-orientation Spectroscopy ...... 53

4.2.3. EOS measurements ...... 54

4.2.4. Nanowire processing ...... 54

4.3. Results and discussion ...... 56

4.4. Conclusions ...... 62

5. Contactless Electrical and Structural Characterization of Semiconduc- tor Nanowires with Axially Modulated Doping Proﬁles ...... 65

5.1. Introduction ...... 65

5.2. Methods ...... 66

5.3. Results and discussion ...... 69

5.4. Conclusions ...... 74

ix 6. Conclusions and Future Works ...... 76

6.1. Conclusions ...... 76

6.2. Future works ...... 78

6.2.1. Expanding the EOS measurement range ...... 78

6.2.2. Radiation damage on conductivity of semiconducting nanowires . 79

Appendix A. Attenuation measurement of suspensions with no magnetic ﬁeld applied ...... 80

Appendix B. Volume fraction dependance of the measured attenuation coeﬃcient ...... 82

Appendix C. Scale analysis of the thermal attenuation eﬀect ...... 84

Appendix D. Scale analysis on the eﬀect of acoustic radiation force .. 85

Appendix E. Elliptic Integrals ...... 86

Appendix F. Fabrication of two-segment Si nanowires ...... 89

Appendix G. Doping dependent selective etching ...... 90

Appendix H. EOS measurements on two-segment Si nanowires ..... 91

Appendix I. Numerical simulations of two-segment Si nanowires ... 92

Appendix J. Potential EOS applications in more complex nanowires . 96

x List of Tables

2.1. Attenuation coeﬃcient ratio of parallel and perpendicular alignment(αparl/αperp) for suspensions with oblate and spherical particles...... 17

2.2. Timescales estimation ...... 21

I.1. Physical properties used in the numerical calculation ...... 92

xi List of Figures

1.1. Schematic of MR and ER ﬂuid ...... 3

1.2. Schematic of nanowire/nanotube based ﬁeld eﬀect transistors(FETs) . . 5

2.1. Suspension microstructure controlled by an external magnetic ﬁeld: (a) Field direction perpendicular to the wave-propagation direction, and (b) ﬁeld direction parallel to the wave-propagation direction...... 12

2.2. Schematic diagram of experimental set up...... 13

2.3. Validation of the experimental method. Attenuation coeﬃcients were measured in two ﬂuids at the indicated frequencies and compared to values in the literature...... 14

2.4. The measured attenuation coefficient of suspensions of nickel flakes and nickel spheres. Results for samples at 0.1%, 0.2% and 0.5% volume fraction with both parallel and perpendicular particle alignment are all included in the figure...... 16

2.5. Dynamic measurement of both acoustic and optical transmission rates through suspensions when exposed to a magnetic ﬁeld of 885 gauss for 5 secs...... 19

2.6. Comparison between experimental data and Ahuja and Hendee’s model. Only the parallel alignment cases are considered...... 22

2.7. Linear regression of the measured attenuation data of the suspensions with oblate particles, good linearity is observed...... 23

2.8. Extracted y-intercept data α0 is plotted against the measured ﬂuid at-

tenuation data αf , the error is less than 5%...... 24

xii 2.9. Comparison between experimental data and Ahuja and Hendee’s model. Here the diﬀerence in attenuation coeﬃcient for parallel and perpendicular alignment are compared with that calculated from theory...... 24

3.1. A specimen of dilute suspension ...... 36

3.2. Eﬀects of particle aspect ratio on eﬀective viscosity. Particle volume fraction is taken to be 5%...... 42

3.3. Eﬀects of particle aspect ratio on attenuation. Particle volume fraction is taken to be 5%...... 43

3.4. Eﬀects of particle aspect ratio on sound speed of suspensions (parallel alignment). Particle volume fraction is taken to be 5%...... 44

3.5. Eﬀects of particle aspect ratio on sound speed of suspensions (perpendicular alignment). Particle volume fraction is taken to be 5%...... 44

3.6. Eﬀects of volume fraction on attenuation of suspensions. Particle aspect ratio is taken to be 100 and acoustic wave frequency is ﬁxed at 2 MHz. . 45

3.7. Eﬀects of volume fraction on sound speed of suspensions. Particle aspect ratio is taken to be 100 and acoustic wave frequency is ﬁxed at 2 MHz. . 45

3.8. The comparison between this theory and measured attenuation: parallel alignment...... 46

3.9. The comparison between this theory and measured attenuation: diﬀer- ence between the two parallel and perpendicular alignment...... 46

4.1. Schematic of an ellipsoidal shaped nanowire suspended in the ﬂuid under an external ﬁeld ...... 50

4.2. Schematic of the experimental setup, uniform AC signal is generated by the function generator and ampliﬁed before reaching the electrodes, the ﬁeld direction is switched back and force to keep nanowires aligning and the alignment rate is extracted from the videos taken by the high speed camera connected to the microscope. Electrical conductivity is calculate from the collected alignment rate data...... 55

4.3. Measured conductivity of Si nanowires in mineral oil and DPG . . . . . 55

xiii 4.4. The measured conductivity of as-grown intrinsic nanowires are plotted in both linear and log scale...... 57

4.5. The measured conductivity of nanowires with ALD deposited Al2O3 are shown. Here we use either HF or SC1 for initial nanowire surface treatment. 58

4.6. The measured conductivity of Si nanowires are plotted against (a) the

number of ALD cycles and (b)the Al2O3 layer thickness...... 60

4.7. The thickness of the Al2O3 layer are characterized by X-ray photoelectron spectroscopy (XPS) on planar Si samples with the same treatment as Si nanowires ...... 61

4.8. The COV of the measured conductivity are plotted against the Al2O3 thickness for both HF and SC1 treated Si nanowires...... 62

4.9. The measured conductivity and the COV are plotted against the nanowire diameters for as-grown Si nanowires...... 63

5.1. Electro-orientation spectroscopy (EOS) is shown to enable highly efficien- t electrical characterization of the properties and structure of individual semiconductor nanowires with homogeneous or axially programmed dopant profiles. The contactless, solution-based method enhances fundamental statistical understanding of highly variable 1D nanomaterials, and serves as a first step toward the high-throughput interrogation of complete nanowire-based devices...... 66

xiv 5.2. Simulations of the EO spectra of nanowires with axially homogeneous

and two-segment dopant proﬁles with diﬀerent Lundoped/Ltotal ratios. The nominally undoped and doped segments are assumed to have the same permittivity and only vary in electrical conductivity. (a) EO spec-

tra for varying Lundoped/Ltotal ratio, showing the change in plateau height with increasing doped segment length. Schematic ﬁgures of nanowires

with diﬀerent Lundoped/Ltotal ratios are also shown. (b) EO spectra for undoped segments of increasing conductivity and all other parameters unchanged, showing the increasing frequency of the ﬁrst (lower- frequency) crossover. (c) EO spectra for doped segments of increasing conductivity and all other parameters unchanged, showing the increasing frequency of the second (higher-frequency) crossover...... 68

5.3. Measured EO spectra for two-segment Si nanowires with fabricated ra-

tios of Lundoped/Ltotal of 1/3, 1/4 and 1/6, compared to that for an

axially homogeneous, nominally undoped (i.e., Lundoped/Ltotal = 1) Si nanowire. The error bars indicate the standard deviation for 3 repeated measurements on the same nanowire...... 70

5.4. Measured length ratios of two-segment Si nanowires. (a) Comparison

of Lundoped/Ltotal as determined via EOS and electron microscopy. (b)

Scanning electron microscope image of a Si nanowire with Lundoped/Ltotal = 1/3 in which the undoped region is selectively etched with KOH. . . . 70

xv 5.5. Measured conductivity distributions for nominally undoped nanowires. (a) Comparison of one- and two- segment Si nanowires. The conductivity distribution of the undoped segments of two-segment nanowires (blue) overlaps with that undoped, one-segment Si nanowires synthesized under the same conditions. The arrows indicate four-point measurements of the conductivity of 5 undoped one-segment nanowires. The direct- contact measurements show similar variability and, with one exception, fall within the range of the EOS-measured conductivities. (b) Repre- sentative I − V curve from a four-point measurement of an individual Si nanowire. Inset: Representative scanning electron microscope image of four lithographically fabricated electrodes contacting an individual Si nanowire...... 71

6.1. Conductivity of Si nanowires measured by EOS method in DI water . . 79

A.1. Comparison of random and oblate alignment case for attenuation coeﬃ- cient of nickel-ﬂake suspensions...... 81

B.1. Measured attenuation coeﬃcient of suspensions with nickel micro-ﬂakes against volume fraction...... 83

B.2. Measured attenuation coeﬃcient of suspensions with nickel micro-spheres against volume fraction...... 83

I.1. Simulation set up in Comsol Multiphysics. Nanowire diameter and length are set to be respectively 50 nm and 10 µm to match our experiments. . 93

I.2. Validation of the numerical method. The nanowires are assumed to be prolate spheroidal particles with semi-major and semi-minor axis being 5 µm and 25 nm. The calculated EO spectra of the homogeneous particles, with conductivities of 1.56−3 S/m or 10 S/m, were compared with the corresponding analytical solutions...... 94

I.3. An example of the best-ﬁtting simulated EO spectrum and the corre-

sponding experimental data for a two-segment nanowire with LI /(LI +

LD) = 0.31 and σI = 0.0038 S/m...... 95 J.1. The simulated EO spectrum of a three-segment nanowire...... 97

xvi J.2. The simulated EO spectra of two-segment nanowires with conductivity gradients at the interface...... 97

xvii 1

Chapter 1

Introduction

1.1 Particle manipulation by electric/magnetic ﬁelds

The ability to effectively manipulate micro- or nanoparticles in a fluid medium using an applied electric/magnetic field opens up new possibilities in a wide variety of applications. From a macroscopic viewpoint, electric/magnetic fields can actively control the microstructure and hence the physical properties of fluid-particle media, making it possible to develop a series of stimuli-responsive or “smart“ materials. Some of the most widely-used examples include magnetorheological (MR) fluids, electrorheological (ER) fluids and ferrofluids, all of which have received significant attention for their remotely and spatio-temporally controllable properties. From a microscopic viewpoint, on the other hand, an external field can also be used to control the movement of individual particles in a predictable and programmable manner. The field-induced mechanisms such as electrophoresis (EP), dielectrophoresis (DEP) and magnetophoresis enable effective trapping, transport, property-based sorting and separation of micro- and nanoparticles, for a wide variety of electronic, environmental and biomedical applications. Furthermore, these particle-field interactions can be in turn used to efficiently characterize individual micro- and nanoparticles, providing us with unique access to particle properties and their variations that are otherwise difficult to obtain. However, despite the extensive attention it has received, there are still many potential areas where field- based particle manipulation remains unexplored and can be useful. The goal of this dissertation is to further explore the applications of electric/magnetic field based micro- and nanoparticle manipulation, specifically in areas of field-responsive smart materials and the non-contact, high-throughput characterization of individual nano-particles. 2

1.1.1 Electric/magnetic ﬁeld responsive smart materials

Electric/magnetic-field-responsive or “smart” materials have found more and more applications recently [1]. One example is the magnetorheological (MR) fluid where micron- sized ferromagnetic particles are dispersed in aqueous solution or organic liquids. The microstructure of MR fluid can be dramatically, yet reversibly, changed by the dipole moment induced by the external magnetic field, thus allowing active control of its properties. MR fluids have been studied extensively ever since their inception in the 1940s [2]. The apparent viscosity of MR fluid exhibits several orders-of-magnitude of change upon applying a magnetic field of 1T strength [3]. This amazing property makes it ideal for applications such as automotive clutches, brakes, polishing fluids, seat dampers, artificial knee damper, actuator systems, shock absorbers and earthquake dampers [4]. In addition to its outstanding rheological properties, the large controllability of MR fluid’s microstructure also reflects on other physical properties, making it promising in areas like tunable thermal-energy transfer [4, 5, 6, 7], control of sound propagation [8, 9, 10, 11] and high-precision finishing [12, 13, 14].

Introduced in 1960s [15], ferrofluids are another typical example of magnetic-field- responsive materials. With magnetic nanoparticles suspended in liquids by Brownian motion, ferrofluids enjoy much better suspension stability than MR fluids thanks to the greatly inhibited particle settling. Particle agglomeration and clustering are also very limited in ferrofluid due to thermal motion and the surfactant coating of nanoparticles. While MR fluids behave almost like solids in the presence of a magnetic field, ferrofluids always remain fluid even under high field strengths, making them ideal candidates for a number of applications in heat-transfer intensification [16, 17, 18], mass-transfer enhancement [19, 20], energy storage [21, 22], friction reduction [23, 24], magnetic seal- ing [25, 26], biomedical applications [27, 28, 29] and optical-transmission manipulation [30, 31].

Electrorheological (ER) fluids are electric-field-responsive suspensions with polar- izable particles dispersed in fluid media. They were first proposed by Winslow in the late 1940s [32]. Similar to their magnetic counterparts, in ER fluids, the field-induced 3

Figure 1.1: Schematic of MR and ER fluid dipole moments cause particles to aggregate and form chain-like structures along the field direction, thus increasing the apparent viscosity of the ER fluid [33]. Given their remarkable properties, ER fluids have been found to be particularly useful in applications like actuators, dampers, torque transducers and artificial muscle[34, 35]. However, limitations with respect to temperature stability, liquid leakage, energy consumption and most importantly insufficient yield strength hampered further development of ER fluids, until the discovery of the giant electrorheological (GER) effect by Wen et al. [36, 37]. They found that using coated nanoparticles in ER fluids can result in a yield strength that is higher than the theoretical upper bound for static yield stress in conventional ER fluids. The strong ER response of GER fluids makes them strong candidates for various microfluidic applications [38, 39]. A schematic illustrating the particle-chaining mechanism for MR and ER fluids is shown in Figure 1.1.

The above-mentioned examples involve manipulation of spherical or near-spherical particles. When it comes to highly non-spherical particles, additional shape-related alignment effects will be introduced, making the particle-field interactions significantly different from those of spherical particles. To broaden the applicability of field- responsive materials, it is of intrinsic interest to study how different particle shapes affect the macroscopic properties of the material in the presence of an external field. In recent years, researchers have turned their attention to field manipulation of non- spherical particles. For instance, attempts have been made to manipulate prolate- spheroidal carbon nanotubes (CNT) in an effort to transfer their exceptional properties to a macroscopic matrix [40]. It has been reported that polymer matrices with 4

aligned CNTs exhibit enhanced properties (mechanical, thermal, electrical, transport, etc.) [41, 42, 43]. Thus, considerable effort has been put into gaining better control and understanding of the alignment process before polymer curing [44, 45, 46]. For example, Castellano et al. [47] used a combination of AC and DC fields to fabricate composite thin films containing vertically aligned carbon nanotubes (VACNTs). On the other hand, manipulation of oblate-spheroidal particles has also been explored by Seitel et al. [48], for its effect on the acoustic properties of a suspension. Seitel et al. experimentally studied the sound propagation speed in suspensions with micron- sized nickel flakes, nickel nanowires and nickel-coated multi-wall nanotubes (MWNTs) under external magnetic fields. They found that the change in sound speed is of the same order of magnitude as the volume fraction. Overall, field-responsive materials with highly non-spherical particles are much less investigated compared to those with spherical particles, even though the former often have the potential to have greater controllability and enhanced properties for certain applications. With the ever-growing demand for novel and enhanced material properties, more studies on field-responsive materials with highly non-spherical particles are very much desired.

1.1.2 Microscopic particle manipulation and characterization using electric/magnetic ﬁeld

In addition to controlling macroscopic materials properties, electric/magnetic fields can also be used to guide individual micro- and nanoparticles at a microscopic scale. One example is the manipulation of cells in microfluidic systems using dielectrophoresis (DEP) [49, 50]. DEP has been widely used in transporting [51, 52], trapping [53, 54], sorting [55, 56] and separation [57, 58] of cells and other particles or macromolecules for biological applications. Other field-based mechanisms such as electrophoresis [59] and magnetophoresis [60] have also been used for cell manipulation. On the other hand, the strong dependence of cell behavior on the external field can in turn be used to characterize the cell properties [61, 62].

Field-based manipulation is also ﬁnding applications for manipulating and characterizing nanowires and nanotubes. Akin et al. developed a microﬂuidic device that can 5

Figure 1.2: Schematic of nanowire/nanotube based field effect transistors(FETs) sort semiconducting nanowires based on their conductivity [63] . Krupke et al. have used AC DEP to separate metallic from semiconducting CNTs from suspension [64]. Fan et al. [65] have proposed a systematic method to manipulate metallic nanowires, including assembly, dispersion, chaining, acceleration and concentration onto designat- ed places. They have also demonstrated a precise control scheme where nanowires can be moved to any locations along any prescribed path and with any specified orientation [66]. A number of attempts have been made to fabricate high-performance field-effect- transistors (FETs) by depositing the nanowires or nanotubes on an insulating substrate surface, with their ends in contacts with source and drain electrodes. [67, 68, 69, 70, 71]. A schematic of nanowire/nanotube based FETs are shown in Figure 1.2.

1.2 Electrical characterization of semiconductor nanowires

Semiconductor nanowires have been studied extensively over the past two decades [72], due to their potential in areas like nanowire electronics [73], photonics [74], thermo- electrics [75] and photovoltaics [76]. Despite how important it is in these applications, the electrical conductivity of a nanowire is usually poorly known, due to the huge time and labor cost of the existing direct-contact characterization methods. On the other hand, nanowires often exhibit massive variations even within the same sample as a result of diﬀerent surface states, growth conditions and intrinsic variations due to discrete numbers of charge carriers [77, 78, 79].

To address this issue, Akin et al. [80, 63] introduced the new electro-orientation 6

spectroscopy (EOS) technique for the rapid, contactless, and quantitative determination of electrical conductivity of individual nanowires. They extracted the electrical conductivity of individual nanowires by monitoring the alignment rate of the targeted nanowires in a liquid under an external AC electric ﬁeld. The non-contact EOS measurements were validated by comparing the results for Si nanowires with direct-contact I − V measurements. Signiﬁcant variations in electrical conductivity were observed even for Si nanowires fabricated from the same wafer, which emphasizes the necessity of quantifying, and controlling the causes of, these variations.

Thanks to the EOS technique, the statistics of the semiconductor nanowires has become available to us for the first time. A systematic study that connects the statistics of semiconductor nanowires to their growth conditions and surface passivation is very much needed in order to fabricate reliable nanowire devices. On the other hand, nanowires with inhomogeneous doping profiles (and hence a complicated conductivity distribution) have found more and more applications recently [81]. A characterization technique that is capable of measuring complex conductivity profiles is highly desired. In this thesis, we will try to accomplish these two goals, to understand semiconductor nanowire properties and variation as a function of surface passivation, and to measure the properties and structure of two-segment (doped-undoped) nanowires.

1.3 Dissertation outline

The speciﬁc research hypotheses, approach, and signiﬁcance, of each part of this dissertation are outlined below:

We first study a magnetic-field-controllable acoustic metamaterial enabled by manipulating nickel flakes in dilute suspensions in Chapters 2 and 3. Hypothesis: The acoustic properties, in particular the absorption coefficient, of suspensions with nickel microflakes are anisotropic and tunable by an external magnetic field. Approach: We designed, constructed and validated an acoustic measurement system based on piezoelectric transducers. Using this apparatus, the attenuation coefficient of 7

various suspensions was measured at different acoustic frequencies and particle volume fractions, with magnetic fields in two directions (parallel and perpendicular to the wave propagation direction). The experimental results were compared with the predictions of an analytical model. Additional experiments were designed and conducted with the aid of UV-Vis spectroscopy to clarify the main mechanism(s) for the anisotropic, controllable attenuation. We also developed an analytical model for the acoustic properties of this material by solving for its effective viscosity tensor. The influence of frequency, aspect ratio and volume fraction on the sound speed and attenuation coefficient are presented. Outcomes: A 30% difference in attenuation coefficient was observed for the proposed material with magnetic-fields parallel and perpendicular to the wave propagation direction, at particle volume fractions of 0.1%, 0.2% and 0.5%. Particle alignment, rather than particle chaining, was identified as the main mechanism for this large controllability. The measured attenuation coefficient is in good qualitative but poor quantitative agreement with existing theoretical models. On the other hand, our newly developed analytical model underpredicts the contribution of volume fraction, while grasping the general trends of the attenuation coefficient. Significance This work highlights the significance of particle shape on the acoustic properties of magnetic-field-responsive materials, and demonstrates the possibility of a controllable smart media able to attenuate and shape acoustic energy in a programmable manner.

In Chapter 4, we experimentally study the eﬀect of surface passivation on the electrical conductivity and variability of Si nanowires using the new EOS technique. Hypothesis The electrical properties of Si nanowires are dominated by surface traps, and, by passivating, we expect to be able to eﬀectively control the properties and variability of the nanowires. Approach: Taking advantage of the recently developed EOS technique, we measured and statistically analyzed the electrical conductivity of Si nanowires with 0, 3, 6, 12, 24 cycles of Al2O3 passivation induced by atomic layer deposition (ALD). The nanowires 8

were treated with both HF dipping and chemically induced oxide (SC1 treatment) to study the effect of initial condition, before ALD deposition. We also characterizaed Si nanowires of different diameters (50nm, 100nm, 150nm) to further investigate the role of nanowire surfaces on their electrical conductivity. Outcomes: The measured nanowire conductivities were highly skewed, following a log-normal distribution. As more cycles of the Al2O3 layers were deposited, the electrical conductivity of Si nanowires increased by orders of magnitude, while the variation of the conductivity decreased. The measured conductivities were higher for otherwise similar nanowires with larger diameters, again indicating the importance of the surface traps. The larger-diameter wires also showed smaller variation in the conductivity, also consistent with their smaller surface to volume ratio. Significance: Our measurements reveal the connection between nanowire processing and properties, laying the foundation for the understanding and control of the properties and variability of nanowires that is needed for fabricating reliable nanowire devices.

Then in Chapter 5, we extend the EOS technique to enable characterization of two- segment Si nanowires with a distinct dopant density in each segment. Hypothesis: The non-contact EOS technique can be used to achieve both electrical and structural characterization of two-segment Si nanowires. Approach: We developed an extension of the EOS technique and experimentally characterized two-segment Si nanowires with nominally undoped and doped segments. With the help of numerical simulations, we extracted the electrical conductivity of the undoped segment as well as the length ratios of the two segments of the nanowires. The obtained conductivities were found to be in good agreement with those measured by EOS for homogeneous Si nanowires with the same growth conditions, and as well with those obtained by traditional direct-contact I −V measurements. The measured length ratios, on the other hand, were validated by the growth time of each sections as well as a selective etching study that distinguished diﬀerent dopant densities. Outcomse: We demonstrated the capability of the extended EOS technique to provide highly eﬃcient characterization for nanowires with two-segment Si nanowires. The 9

measured conductivities of the undoped segments of the Si nanowires follow a non- Gaussian distribution and displays a 2-3 orders-of-magnitude variation, consistent with our findings in Chapter 4 for one-part nanowires. In contrast to the large variability in conductivity, the variation in measured length ratio of the two segments was only around 10%. Significance To the author’s best knowledge, this work is the first to efficiently characterize the electrical properties of individual semiconductor nanowires with functional heterogeneity with a non-contact, solution-based method. It also serves as a first step towards the characterization and separation of full nanowire devices with complex dopant profiles or heterostructures.

The dissertation concludes in Chapter 6 with a summary of the most signiﬁcant results, and a discussion of potential future work. 10

Chapter 2

Tunable Acoustic Attenuation in Dilute Suspensions of Non-spherical Magnetic Particles: Experiments

2.1 Introduction

The availability of engineered “smart” media having actively controllable acoustic properties would open new possibilities for guiding and attenuating acoustic energy. An a- coustic medium having actively controllable and anisotropic sound speed, for instance, could be used to design novel, reconﬁgurable acoustic lenses to dynamically shape a- coustic pulses. Such tunable acoustic lenses may have applications in diverse areas like medical diagnostics [82], underwater sonar, non-invasive treatment of urinary s- tones (lithotripsy) [83] and cancer [84], or even in acoustic canons for non-lethal dispersal of animals from airports [85]. An acoustic media having actively controllable, frequency-dependent attenuation constant could similarly enable the development of new acoustic devices like tunable ﬁlters to reduce acoustic noise, or to eliminate alias- ing of bandwidth-limited hydrophones [86], for example.

Recent work has shown that dilute suspensions of highly non-spherical, ferromagnetic micro-particles, can function as such a tunable acoustic medium, with controllable, anisotropic sound speeds under external magnetic ﬁelds. The change in sound speed, although larger than predicted by classical theory, was on the order of the particle volume fraction, i.e., 0.45% for 0.5% volume fraction. Here, we extend the previous work to show that dilute suspensions of non-spherical magnetic particles have widely tunable acoustic attenuation for ultrasonic waves. The tunability of the attenuation coeﬃcient is much larger than that of the sound speed, being two-orders of magnitude larger than the volume fraction of the suspensions (20% for 0.2% particle concentration). The change in attenuation is both much greater than that of spherical-particle 11

suspension of the same volume fraction, and also greater than the prediction of Ahuja and Hendee’s theory [87].

The basic concept of this tunable medium is shown in Fig. 2.1, with disk-shaped ferromagnetic particles aligned and chained in different directions by an external magnetic field. In general, when exposed to a magnetic field, the induced magnetization of the suspended particles causes changes in the microstructure of the suspension, thus leading to possible changes in the thermophysical properties of the suspension; the properties may also become anisotropic. Ferrofluids are one common example, in which spherical nanoscale particles are kept suspended in a fluid by Brownian motion. In addition to applications in areas like heat-transfer enhancement [88, 89, 90], optical-transmission manipulation [30, 31, 91] and biomedicine [27, 28, 29], ferrofluids have been studied through experiments and modeling for their potential in shaping acoustic fields [92, 93, 94, 95]. Magnetorheological (MR) fluids, having micron-sized particles, are another example of ferromagnetic suspensions. Most previous research on MR fluids has focused on their applications in mechanical control, e.g., [3, 4, 96, 97, 98, 99, 100, 101, 102, 103], with only a few authors having chosen to investigate their acoustic performance. Rodriguez- Lopes et al. [8, 9] and Bramantya et al. [10] have experimentally studied the acoustic properties of spherical-particle MR fluids, and shown that a magnetic field is capable of changing (as well as inducing an anisotropy in) the sound speed of the samples. However, the change in sound speed is only about 5% which is too small for applications such as adaptive matching layers for acoustic cloaking [104]. The effect of the magnetic-field intensity on the change of acoustic properties was also investigated and hysteretic behavior was observed. For typical MR fluids and ferrofluids, the particle volume fractions are on the order of 10% and the particles are spherical in shape.

In the following, we describe the dilute suspensions of highly non-spherical ferromagnetic particles that we prepare, and our measurements of acoustic attenuation in this medium. Comparison is made between suspensions with and without magnetic-ﬁeld- induced microstructure, as well as with spherical-particle suspensions. The measured attenuations are also compared with the theoretical values predicted by Ahuja and Hendee’s model [87], which has been used as a reference under several circumstances 12

Figure 2.1: Suspension microstructure controlled by an external magnetic ﬁeld: (a) Field direction perpendicular to the wave-propagation direction, and (b) ﬁeld direction parallel to the wave-propagation direction.

[9, 48, 105, 106, 107]. We discuss two possible mechanisms, particle alignment and particle chaining, for the change of acoustic attenuation in these non-spherical suspensions, and analyze their relative importance with the aid of optical measurements and scaling analysis of the characteristic timescales of the particle dynamics.

2.2 Experiment

We prepared suspensions of nickel flakes (Alfa Aesar -325 mesh) in SAE 30 oil at particle volume fractions of 0.1%, 0.2% and 0.5%. These particles can be approximated as oblate-spheroidal particles, with a thickness of 0.37 µm and measured mean diameter of 21 µm. We also prepared suspensions of spherical nickel particles (Alfa Aesar APS 5-15 µm diameter) in the same oil at a volume fraction of 0.2% for comparison to the micro- flake suspensions. The primary particle size of both nickel flakes and spherical particles was much smaller than the shortest acoustic wavelengths considered (approximately 0.4 mm at 5 MHz in oil). Vigorous shaking followed by 10 minutes of tip sonication (Misonix Sonicator 3000) were applied on all samples to mix and homogenize them 13

Figure 2.2: Schematic diagram of experimental set up. immediately before measurements.

We adopted a through-transmission method to measure the acoustic attenuation, in which a 10-wave pulse train travels through the target medium and is received on the other side. A specialized measurement chamber was designed and constructed in which the distance between transmitter and receiver could be altered, as shown in Fig. 2.2. The uniform magnetic field needed in the experiments was provided by a custom-fabricated Helmholtz coil. By changing the current output of the power supply (EMS 60-80), the field strength was adjustable up to approximately 1000 gauss. The orientation of the magnetic field relative to the acoustic-propagation direction could be adjusted by changing the placement of the measurement chamber within the Helmholtz coil. Piezo-ceramic discs (Steiner & Martins, Inc.) with diameters of 20mm and 5mm were chosen as transmitter and receiver, respectively. The transducer generating the acoustic signal was driven by a function generator (Tektronix AFG3200C) connected to a high-frequency amplifier (TREK 2100HF). Ultrasonic signals received by the other peizo-transducer were recorded with a high-speed digital oscilloscope (Picoscope 5203) 14

Figure 2.3: Validation of the experimental method. Attenuation coefficients were measured in two fluids at the indicated frequencies and compared to values in the literature. and then stored in a computer. The time course of the experiments was controlled by a National Instruments DAQ system running LabVIEW to trigger the magnetic field, initiate the acoustic signal, and begin data acquisition. A recirculating bath with constant-temperature water (Endocal RTE-Series Refrigerated Bath/Circulator) was used to keep the suspensions at 20 degree Celsius. A schematic diagram of the experimental set-up is shown in Fig. 2.2.

The experiments were carried out at three different frequencies (3 MHz, 4 MHz and 5 MHz), with the sound-propagation direction and the magnetic-field direction either parallel or perpendicular to each other. A trigger signal was generated by LabVIEW and sent from the DAQ board to the power supply to enable the magnetic field for a duration of 5 seconds. After the end of the magnetic field, another trigger signal was sent to the function generator to produce the 10-wave pulse packet, which was amplified and sent to the piezo transmitter. Observations with an optical microscope suggest that these acoustic pulses are not inducing any visible movement to the suspended particles, and a scale estimation also shows the forces resulted from acoustic wave is negligible. The time-separation between the magnetic field and the acoustic signal was intended to eliminate any possible electromagnetic interference in the acoustic measurement. Both transmitted and received acoustic signals were recorded by a Picoscope digital-storage 15

oscilloscope, and a custom MATLAB program was used to post-process the data and extract the amplitude of the signals. The attenuation coeﬃcient of the samples was then calculated as, 1 A α = − ln 1 , (2.1) x A0 where x is the distance between transmitter and receiver, and A0 and A1 are the measured amplitudes of transmitted and received signals, respectively.

Calibration tests were carried out on pure fluids with known attenuation coefficients at specific frequencies in order to verify the accuracy of the experimental method. The error bars are calculated from at least 5 sets of experimental results assuming Student’s t-distribution with 95% confidence. Measurements were conducted on castor oil and Dow Corning 710 fluid, and the results were compared with published values [108, 109] as shown in the Fig. 2.3. The differences at three frequencies for the fluids were within 5%, which validates the accuracy of the apparatus and procedure for measuring sound-attenuation coefficient.

2.3 Results and discussion

We have measured the attenuation coefficient of the nickel micro-sphere and micro- sphere suspensions with an 885 magnetic gauss field either parallel or perpendicular to the wave-propagation direction. As seen in Figure 2.4, the measured acoustic attenuation increases with frequency and volume fraction of particles for both kind of suspensions, as expected. When the magnetic field was applied, a 20% to 35% d- ifference in attenuation coefficient was observed for nickel micro-flake suspensions at volume fractions of 0.1%, 0.2% and 0.5%, as seen in Table 2.1. This difference is only very weakly dependent on volume fraction and frequency. For suspensions with spherical particles, on the other hand, the difference between the two alignments is not only much smaller but also of opposite signs compared with that of nickel micro-flake suspensions. It is clear that the acoustic attenuation of the nickel micro-flake suspensions is anisotropic and controllable through an external magnetic field, and that the oblate- spheroidal shape of the nickel micro-flakes is an important factor that results in the 16

Figure 2.4: The measured attenuation coefficient of suspensions of nickel flakes and nickel spheres. Results for samples at 0.1%, 0.2% and 0.5% volume fraction with both parallel and perpendicular particle alignment are all included in the figure. 17

large tunability of these suspensions. We notice that the tunability of the attenuation of the nickel micro-flakes is two-orders of magnitude larger than the volume fraction of the suspensions (20% change in attenuation for a 0.2% particle concentration). This stands in contrast to what has been observed for sound speed in the same suspensions, in which the change in speed was the same order of magnitude as the particle volume fraction (c.f., the ∼0.2% change in sound speed observed by Seitel et al. for 0.2% volume-fraction nickel-microflake suspensions [48]). It should also be noted that suspensions with nickel flakes enjoy a up to 50% higher attenuation coefficient than those with spherical particles.

Table 2.1: Attenuation coeﬃcient ratio of parallel and perpendicular alignment(αparl/αperp) for suspensions with oblate and spherical particles.

Oblate particles 3 MHz 4 MHz 5 MHz 0.1% 1.35 ± 0.09 1.29 ± 0.10 1.22 ± 0.08 0.2% 1.20 ± 0.10 1.24 ± 0.06 1.19 ± 0.06 0.5% 1.27 ± 0.07 1.27 ± 0.07 1.32 ± 0.07 Spherical particles 3 MHz 4 MHz 5 MHz 0.1% 0.94 ± 0.09 0.92 ± 0.16 0.95 ± 0.11 0.2% 0.99 ± 0.11 0.93 ± 0.07 0.96 ± 0.04 0.5% 0.90 ± 0.05 0.94 ± 0.04 1.04 ± 0.06

To better understand the physical mechanism(s) behind this large anisotropy in attenuation, we designed another experiment to study the formation of field-induced microstructure in the suspensions. The optical transmission through a 1-mm-path- length sample under the magnetic field was dynamically recorded by a UV-Vis spec- trophotometer (Ocean Optics) and compared with the acoustic-transmission percentage through the same suspension. In these measurements we used nickel-flake suspensions of 0.1% particle volume fraction and the frequency of acoustic transmission was fixed at 2 MHz. A 885 gauss magnetic field was turned on at t = 0 and removed at t = 5 sec, with the field direction parallel to the wave direction for the acoustic transmission, and perpendicular for the light transmission. As seen in Figure 2.5, the optical response has a rapid (within 0.5 sec) initial drop, followed by a gradual increase over much longer time scales. The initial decrease in optical transmission is believed to be due to the 18

alignment of the micro-flakes, which presents a larger scattering cross-section when oriented (by the magnetic field) broadside to the transmitted light. The gradual recovery in transmitted optical intensity is due to longer-timescale chaining (and possibly sedi- mentation) of the particles under the magnetic field. As seen in Figure 2.5, the acoustic transmission had a similar immediate drop (within the time-resolution of the acoustical measurements) upon the application of the magnetic field, and then remained almost constant thereafter. Thus, based on comparison of timescales, it would appear that the primary mechanism for the large acoustic-attenuation anisotropy in these suspensions is the rapid alignment of the highly non-spherical particles, rather than their longer-time chaining.

To further clarify the roles of these two mechanisms in changing the sound attenuation, a simple analysis was done to estimate the time scales of the particle aligning and chaining processes. The magnetic interaction energy between two neighboring particles is given by [110, 111], µ U = 0 V 2[M~ · M~ − 3(M~ · rˆ)(M~ · rˆ)], (2.2) 4πr3 1 2 1 2 where V is the particle volume, µ0 is the permeability of free space, r is distance between the particles,r ˆ is the unit vector pointing from one particle to another and M~ is the magnetization, which for a linearly magnetized particle is given by, 1 χ M~ = ( )B,~ (2.3) µ0 1 + nχ where χ is the susceptibility and n is the demagnetization factor, which depends on particle shape and orientation relative to the ﬁeld and is deﬁned by elliptical integrals. With the radius of the oblate-spheroidal particle along the x, y and z axes being re-

p 2 2 2 spectively a, a and b, and Rs ≡ (s + a ) (s + b ), the demagnetization factor is then given by [110], a2b Z ∞ ds n = n = , x y 2 (s + a2)R 0 s (2.4) a2b Z ∞ ds nz = 2 = 1 − 2nx. 2 0 (s + b )Rs The speed u at which two particles will move together under dipolar interactions can be estimated with the balance between the interaction force Fmagn and the Stokes-ﬂow 19

Figure 2.5: Dynamic measurement of both acoustic and optical transmission rates through suspensions when exposed to a magnetic ﬁeld of 885 gauss for 5 secs.

drag force FD. The drag force for an oblate spheroid in edgewise translation is given by [112],

FD = 32µau/3, (2.5) where µ is the dynamic viscosity of the ﬂuid, while the magnetic force can be conve- niently calculated as the gradient of the potential,

dU F = . (2.6) magn dr

Note that under our previous assumption the disk-shaped particles should already be aligned when the chaining process begins, thus in calculating U we shall assume that the magnetic-ﬁeld direction is parallel to one of the semi-major axis of the spheroid. Then the characteristic time for a nickel particle to reach the a neighboring particle under the magnetic ﬁeld can be estimated as,

τinteraction ∼ r/u,¯ (2.7)

1 wherer ¯ = (V/φ) 3 is the characteristic spacing between particles, φ being the particle volume fraction. These estimates show that in our experimental setting the time scale for two particles to interact via dipolar attraction is on the order of 1 sec. For macroscopic chains to form containing many particles, it is reasonable to assume that the time scale will be at least an order of magnitude larger, i.e. tens of seconds. Such a long 20

timescale for chaining is consistent with the slow recovery in optical attenuation seen in Figure 2.5, but not the rapid response of the acoustic transmission to the applied ﬁeld.

On the other hand, the aligning torque for a magnetic disk-shaped particle aligning in a magnetic ﬁeld is given by [110],

2 χ χ Tx = V µ0H~ ( − )cycz, (2.8) 1 + χny 1 + χnz

where cx and cy are the direction cosines of the ﬁeld and H~ is the magnetic-intensity vector. Similar to the chaining timescale, the alignment timescale for a particle to rotate 90◦ around its semi-major axis can be estimated from the balance between hy- drodynamic friction and the magnetic torque as,

π ζR τalignment ∼ , (2.9) 2 Tx where ζR is the rotational friction coeﬃcient for an oblate spheroid given by [112, 113],

16µV ζ = . (2.10) R 3πb/a

For the parameters of our experiment, the alignment timescale of the particles is calculated to be on the order of milliseconds. This is consistent with the rapid initial response of both the acoustic and optical transmission rates upon application of the magnetic ﬁeld, as seen earlier in Figure 2.5.

The comparison between optical and acoustic responses, together with the estimates of various timescales, strongly suggests that particle alignment, rather than chaining, is the primary mechanism responsible for the large change in acoustic attenuation observed in these dilute suspensions of nickel micro-flakes. This also explains the weak acoustic anisotropy observed in suspensions of spherical nickel particles, in which particle chaining, but no particle alignment, can take place when the field is applied. We further note that the acoustical-transmission percentage remains the same even after the field is turned off (Figure 2.5), indicating that the field-induced microstructure remains, and that the post-field suspensions have the same acoustic-attenuation characteristics as the suspensions under an active magnetic field. This hysteresis is consistent 21

with what has been previously observed for sound speed in similar suspensions under a magnetic field [48]. In these suspensions, the residual magnetization and low rotary diffusivity of the particles appears to be sufficient to keep them aligned for substantial times (∼ 10 sec) after the field is turned off. To further verify rotary-stability timescale of the system, we calculate the rotational diffusion coefficient DR given by

kBT DR = , (2.11) ζR

where kB is the Boltzmann constant and T is the absolute temperature. The rotary

Brownian timescale of the system can be estimated by τstability = 1/DR. Under our

6 current experimental environment, τstability is calculated to be in the order of 10 seconds, and presumably would be even longer if the effect of residual magnetization is considered. This long timescale is consistent with our observations that the acoustic attenuation of the microstructured suspensions remained unchanged for long times after the field is turned off (Figure 2.5).

Table 2.2: Timescales estimation

Particle aligning timescale 10 seconds Particle chaining timescale 10−3 seconds Rotary Brownian timescale 106 seconds

Next, we compare the measured attenuation coeﬃcients with the predictions of Ahuja and Hendee’s classical model for the acoustic properties of suspensions of oriented spheroidal particles. This model was derived in the long-wavelength limit by considering the conservation of mass and momentum as well as an equation of motion for the particles [87]. The attenuation coeﬃcient is predicted to be,

1 ω (ρ0/ρ − 1)2s α = φ , (2.12) 2 c (ρ0/ρ + τ)2 + s2 where ω is the angular frequency of sound wave, c is the sound speed in the matrix, ρ0 and ρ are densities of the particle and the matrix respectively, and

9 2 τ = Linert + (δ/b)k , 4 (2.13) 9 s = k2(1 + (1/k)(δ/a)). 4 22

Figure 2.6: Comparison between experimental data and Ahuja and Hendee’s model. Only the parallel alignment cases are considered.

1 The parameter δ is the depth of penetration given by (2µ/(ρω)) 2 , k is the shape factor and Linert is the inertial coefficient giving the added mass that comes about because each oscillating particle must also accelerate some of the surrounding fluid. We note that the experimental situation in which the magnetic field is parallel to the wave- propagation direction can be directly compared to this model, by taking the case with the symmetric axis (the short axis of the micro-flakes) of the particle to be perpendicular to the wave propagation direction. However, the model cannot be directly applied to the experimental arrangement with the field perpendicular to the wave-propagation direction, because the magnetic field in this case does not ensure that every particle is broadside to the wave, as assumed by the theory.

For the situation in which the model applies (field parallel to acoustic-propagation direction), Ahuja and Hendee’s theory qualitatively predicts the observed increase in acoustic attenuation with particle volume fraction, as well as acoustic frequency, as seen in Figure 2.6. However, quantitative agreement between the experimental data and model is poor, with experiments showing approximately 2-3 times of the attenuation predicted by theory. This is consistent with previous work, which has reported that this model generally differs from the obtained experimental results for sound speed [48] and attenuation coefficient [107]. 23

Figure 2.7: Linear regression of the measured attenuation data of the suspensions with oblate particles, good linearity is observed.

To further explain this difference, we applied linear curve fitting on the measured attenuation data of the suspensions with oblate particles against the volume fraction with 95% confidence bounds and compared it with the theory, only the parallel alignment case was considered here for the same reason mentioned above. As seen in Figure 2.7, we can see that the fitted data line, while exhibiting a satisfactory linearity with R-square greater than 0.999, did not go through the origin as the theory predicts. It is reasonable that the attenuation coefficient of the suspensions is that of the fluid at zero particle volume fraction. We verified this by comparing the extracted y-intercept

α0 with the measured fluid attenuation coefficient αf , where we found good agreement with errors less than 5%, as seen in Figure 2.8. On the other hand, the experimental data line and the theoretical data line in Figure 2.7 are almost parallel with each other, with the former having a slightly greater gradient. The absence of a linear constant term in the theoretical formula contributes to the relatively large difference between the theory and experiments. Considering that Ahuja and Hendee’s theory only considered viscous loss due to the relative motion between particles and fluid, we argue that the absence of fluid viscous contribution in their formulation of the attenuation coefficient is the major contributor to this discrepancy [114]. 24

Figure 2.8: Extracted y-intercept data α0 is plotted against the measured ﬂuid attenuation data αf , the error is less than 5%.

Figure 2.9: Comparison between experimental data and Ahuja and Hendee’s model. Here the diﬀerence in attenuation coeﬃcient for parallel and perpendicular alignment are compared with that calculated from theory. 25

From the above analysis, it can be argued that a more reasonable comparison between experiments and theory can be made by considering the difference in attenuation coefficients of two different alignments, such that the attenuation of the fluid, along with other possible mechanisms not considered in the model may be be excluded from the comparison [107]. In order to do this, the theoretical value of sound attenuation of the “perpendicular” alignment case (in which the particle’s long axis is perpendicular, but not necessarily broadside, to the wave-propagation direction) can be found from the relations below [87, 106],

αparl = α0◦ , (2.14) αperp = (α90◦ + α0◦ )/2, where α0◦ and α90◦ are the model predictions of attenuation coefficients when the particle’s axis of symmetry is respectively parallel and perpendicular to the wave propagation direction. The comparison is shown in Figure 2.9. The results show that the difference we observed between the two alignments in the experiments is qualitatively consistent with the theory. Furthermore, while still deviating from the theory, the measured values are much closer to the model predictions than those of the absolute values shown in Figure 2.6. Here we conclude that the viscous attenuation of the fluid is the main cause of the gap between the experiments and the theory, while other factors including possible long-range magnetic interactions between particles, particle aggregation and inter-particle contact, and variability in the particle shapes and sizes are also likely to have contributed to the observed difference.

2.4 Conclusions

Acoustic-attenuation coefficients in the MHz range have been measured in dilute (0.1%, 0.2% and 0.5% volume fraction) suspensions of sub-wavelength oblate-spheroidal particles aligned by an external magnetic field. Differences of up to 30% were observed in attenuation coefficients measured with wave-propagation directions parallel and perpendicular to the magnetic-field direction, with the attenuation greater in the former case. This anisotropy is two orders-of-magnitude greater than found for attenuation in 26

either spherical-particle suspensions, or for sound speed in identical micro-flake suspensions [48]. The field-induced microstructure, and the acoustic-attenuation properties, persist for long times after the magnetic field is turned off in these suspensions. By dynamically monitoring and comparing the acoustic and optical transmission rates, as well as the expected timescales for particle alignment and chaining, the primary mechanism for the large controllability of the acoustic attenuation is shown to be alignment of the highly anisotropic particles, rather than particle chaining. This stands in contrast to the rheological properties of dilute suspensions of anisotropic particles under external electromagnetic fields, which have been previously shown to be largely dependent on particle chaining [115]. The measured attenuation coefficients in these suspensions of non-spherical ferromagnetic particles are in reasonable qualitative, but poor quantitative, agreement with Ahuja and Hendee’s model [87], which we believe mainly comes from the absence of fluid viscous contribution in the formulation. These results highlight the importance of particle shape and orientation in long-wavelength acoustic attenuation, and show the need for a more comprehensive model of acoustic transmission in suspensions of highly anisotropic particles with field-induced microstructure. 27

Chapter 3

Acoustic Properties of Microstructured Suspensions: Theory

3.1 Introduction

Sound propagation in suspensions has long been a major concern in acoustic society given its complex nature and numerous applications. Numerous theoretical models have been proposed to describe the phenomenon under the assumption of spherical suspending particles. Urick [116, 117] obtained a formula for sound speed by regarding the suspension as a single phase with volume-averaged effective properties of the two components. He also derived the attenuation coefficient by considering the velocity potentials of waves scattered from the particle. The model is simple but inaccurate. Urick and Ament [118] developed a complex propagation equation using scattering theory. The model’s superiority over the previous one arises from the fact that it incorporated particle radius, sound frequency and fluid viscosity in the equation. But as they only used zero- and first- order scattering coefficients, applications of the model are thus limited. By considering mass, momentum and energy conservation equations, Allegra and Hawley [119] also obtained a complex equation for sound propagation. Their derivation follows that of Epstein and Carhart [120] closely and the novelty lies in the consideration of thermal transport effects. Ahuja [121] formulated the wave equation for suspension by applying mass and momentum balances on a sufficiently small volume element which is assumed homogeneous and compressive. The added mass effect was taken into account and shown to have modified the wave equation. The deduced sound speed agrees with the result from scattering theory. With the similar approach, Ahuja and Hendee [87] extended the derivation to spheroidal suspending particles. Prolate and oblate particles with their axes of symmetry parallel and perpendicular to the sound 28

propagation direction are considered and the eﬀects of particle shape, orientation, and sound wave frequency on the speed of sound are explored. This is, to the author’s knowledge, the only theoretical study on sound propagation in suspensions with non- spherical particles.

As a useful tool of characterizing suspensions, modeling of the effective viscosity has remained a challenge for researchers since Einstein’s pioneering attempt [122] to derive the effective viscosity of suspensions of rigid spheres. Einstein’s formula, assuming volume fraction of particles Φ, viscosity of ambient fluid µ0 and effective viscosity µe, is given by:

µe = µ0(1 + 2.5Φ). (3.1)

The above formula is valid for dilute suspensions where the distance between particles are much greater than the particle size, implying no particle interactions take place. Einstein’s result can be interpreted as the first-order approximation of an infinite asymptotic expansion and thus can be extended by adding higher order volume-fraction terms. Plenty of research have been carried out to obtain more accurate and rigorous formulas that can be applied to different types of suspensions, as attested by the following survey.

By assuming that the particles are subject to no forces other than those from fluid pressure, Jeffery [123] extended Einstein’s theory to spheroidal particles. He claims that the increase of viscosity may still yield a formula of Einstein’s type but with a modified numerical factor, though definite values for which are not possible to give, due to a certain degree of indeterminacy in characterizing the motion of particles. However, it is possible to specify an upper and a lower limit for the factor, and the difference between which will vanish as the particles approach spherical. Based on Jeffery’s work, Leal and Hinch [124, 125] took rotary Brownian motion into consideration and gave a complete evaluation of bulk stress of suspensions containing non-spherical particles. Brenner [126] pointed out that fundamental tensors can be deduced to characterize intrinsic geometric properties of the particle surfaces. With this idea, he presented a general dynamic theory of rheological properties of dilute suspensions of arbitrary-shaped rigid particles in a Newtonian fluid, with the effects of rotary Brownian motion and external couples included. 29

Recently, taking advantage of Eshelby’s ellipsoidal inclusion model [127, 128], Hsueh and Becher [129] established a model for the eﬀective viscosity of dilute suspensions of spherical particles. Based on this work, Hsueh and Wei [130] took one step further by modelling the eﬀective viscosity of semi-dilute suspensions of rigid ellipsoids using similar methods.

In this paper, we proposed an novel approach to modeling ultrasonic propagation in suspensions with ellipsoidal particles. First we developed a wave equation for homogeneous anisotropic ﬂuid. We then established a model to obtain eﬀective properties of the suspensions using Eshelby’s ellipsoidal inclusion model and substitute the results into the wave equation we previously derived, then we are able to calculate the acoustic properties. Numerical results are obtained and discussed.

Notation: | | denotes the volume, etc

3.2 Acoustic properties of a homogeneous viscoelastic medium

We consider acoustic waves propagating through a viscoelastic medium. Without loss of generality, assume the displacement is given by u(x, t) = uˆ(x)eiωt, where ω is the frequency, and x represents the Lagrange coordinates of material points instead of the Euler coordinates in the usual setting of ﬂuid dynamics. This is convenient for investigating small-amplitude acoustic properties of ﬂuids, solids and mixtures of them.

The viscoelastic properties of a linear medium can in general be described by a stress-strain (or strain rate) relation:

σˆ = Cve∇uˆ,

n×n n×n where σˆ = σˆ(x) is the local stress and Cve : C → C is the viscoelastic tensor.

For a linear elastic medium, the tensor Cve = Celst is the forth-order, real, symmetric, and positive stiﬀness tensor; for a compressible Newtonian ﬂuid, the viscoelastic tensor is given by

Cve = ıωCvisc + κI ⊗ I, (3.2) where Cvisc is the real, symmetric, and positive viscosity tensor, and κ is the bulk 30

modulus of the fluid. If the elastic medium and fluid are assumed to be isotropic, the stiffness tensor and viscosity tensor are given by

(Celst)piqj = G(δpqδij + δpjδiq) + λδpiδqj, (3.3)

2 (C ) = µ(δ δ + δ δ ) + (µ − µ)δ δ , (3.4) visc piqj pq ij pj iq v 3 pi qj respectively. Here, (G, λ) are the Lam´econstants; µ (µv) is the shear (bulk) viscosity. In general we can write the complex viscoelastic tensor of a linear viscoelastic medium as

Cve = Celst + ıωCvisc,

where the forth-order viscosity tensor Cvisc and elastic tensor Celst may depend on the frequency ω as well.

For small amplitude acoustic motions, we can safely neglect the nonlinear convection eﬀect and write the equation of motion as

2 div(Cve∇uˆ) = −ω ρ0uˆ, (3.5) where ρ = ρ(x) is the mass density in the reference conﬁguration. From the above equation, we can immediately ﬁnd the sound speed and attenuation of a bulk acoustic waves that propagate through a homogeneous viscoelastic medium. To see this, we

ık·x consider a trial solution uˆ(x) = uˆ0e for a complex wave vector k = ke and constant n n vector uˆ0 ∈ C , where the unit vector e ∈ IR represents the propagating direction and the complex wave number k = kR + ıkI characterizes the sound speed and attenuation. Inserting this trial solution into (3.5) we obtain

2 2 (−k Ae + ρ0ω I)uˆ0 = 0, (3.6) where the n × n acoustic tensor with respect to the propagating direction e is deﬁned as

(Ae)pq = (Cve)piqj(e)i(e)j. (3.7) 31

The system of linear equations (3.6) admits a nonzero solution if and only if

2 2 det(−k Ae + ρ0ω I) = 0. (3.8)

Upon specifying the viscoelastic tensor Cve of the medium and the propagating direction e, we can solve the above algebraic equation and determine a relation between the complex wave vector and frequency as k = kR(ω) + ikI (ω). Then the sound speed and attenuation constant are given by ω V = ,A = kI (ω), (3.9) kR(ω) respectively.

For an isotropic Newtonian ﬂuid with viscoelastic tensor given by (3.2) and (3.4) and density ρ0, the acoustic tensor deﬁned by (3.7) is given by µ A = ıωµI + [ıω(µ + ) + κ]e ⊗ e. (3.10) e v 3 Then equation (3.8) implies

2 2 2 2 2 2 2 (ρ0ω − ıωk µ) (ω ρ0 − ıωk (µv + 4/3µ) + ρ0c k ) = 0. (3.11)

Let k = kR + ıkI with kR, kI ∈ IR. Then the above equation implies that   2 2 2 2 c (k − k ) − 2(µv + 4/3µ)kI kRω/ρ0 = ω , 2 2  R I ρ0ω − ıω(kR + ıkI ) µ = 0 or (3.12)  2 2 2 2c kI kR + (µv + 4/3µ)(kR − kI )ω/ρ0 = 0.

Solving the latter of the above equation for kR, kI , we ﬁnd that

p 2 2 p 2 2 p 2 2 1 + 1 + η ρ0ω 2 ( 1 + η − 1) ρ0ω 1 + η − 1 2 kR = , kI = = kR, (3.13) 2(1 + η2) κ 2(1 + η2) κ p1 + η2 + 1 where

4 ω( µ + µv) η = 3 κ is a dimensionless number. For acoustic waves with frequency at the order of MHz or below, η is of the order of 10−6 1 for ﬂuids like water. Then to the leading order, equation (3.13) can be written as rρ η ηω rρ ω2 4 rρ k ≈ 0 ω, k ≈ k ≈ 0 = ( µ + µ ) 0 . R κ I 2 R 2 κ 2κ 3 v κ 32

We remark that the above result is consistent with that in the textbook of Thompson

[131]. Also, the ﬁrst of (3.12) implies solutions of k with kR = −kI , meaning that the waves are non-propagating and hence neglected.

Further, if the medium is transverse isotropic on 12-plane, then

(C ) − (C ) (C ) = (C ) , (C ) = (C ) , (C ) = elst 1111 elst 1122 . elst 1111 elst 2222 elst 1313 elst 2323 elst 1212 2

Similar relations hold for the viscosity tensor Cvisc. Then for in-plane propagating direction e1 and out-of-plane propagating direction e3, by (3.7) the acoustic tensors are given by

Ae1 = diag[(Cve)1111, (Cve)1212, (Cve)1313],

Ae3 = diag[(Cve)1313, (Cve)2323, (Cve)3333]. respectively. Solving (3.8) for a complex wave number k, we ﬁnd that for in-plane propagating direction e1, the wave number of longitudinal waves with displacement along e1, shear waves with displacement along e2 and e3 direction is given by

r ρ r ρ r ρ k = ±ω , k = ±ω , k = ±ω . (3.14) (Cve)1111 (Cve)1212 (Cve)1313

Similarly, for propagating direction e3, the wave number of longitudinal waves with

displacement along e3, shear waves with displacement along e1 and e2 direction is given by

r ρ r ρ r ρ k = ±ω , k = ±ω , k = ±ω . (3.15) (Cve)3333 (Cve)1313 (Cve)2323

We remark that the above procedure of computing the sound speed and attenuation in a viscoelastic medium has ignored the coupling between the dissipation of mechanical energy and heat transport. The local thermodynamic process is assumed to be adiabatic.

3.3 Homogenization limit of mixtures of viscoelastic materials

We now consider mixtures of viscoelastic materials, e.g., suspension of solid particles in a Newtonian ﬂuid. The acoustic property of such mixtures can be quite complicated 33

because of the interactions of particles mediated by the ambient ﬂuid. In regard of the simplicity of a homogeneous viscoelastic medium, it is desirable to develop a homogenization theory for viscoelastic mixtures that enables the calculation of the eﬀective viscoelastic properties and density of the mixture that subsequently determine the effective sound speed and attenuation constant by the procedure discussed in the previous section. However, this “coarse-grained” viewpoint is not always valid, particularly for waves with wavelength comparable to the lengthscale of the inhomogeneities. To be precise, we shall introduce a lengthscale ε such that a Representative Volume Element (RVE) of size ε essentially captures the relevant microscopic information of the mixture and the macroscopic properties of the medium is invariant for translations larger than ε. For periodic mixtures, the length scale ε can be chosen as the size of the (irreducible) unit cell.

By introducing the concept of RVE, we can now deﬁne the microstructure of the mixture. Without loss of generality and upon rescaling, assume the representative

n volume element of the mixture is the unit cell Y = (0, 1) with Ωα ⊂ Y occupied by the

α αth phase with viscoelastic tensor Cve. In other words, the materials in the mixture are distributed periodically with unit cell εY , ε << 1:

x x Cε (x) = Cper( ), ρε(x) = ρper( ), ve ve ε ε

per per where (Cve , ρ ) are periodic functions that describe the material distribution in the rescaled RVE Y :

per α α α Cve (y) = Cve = Celst + iωCvisc, (3.16)

per α ρ (y) = ρ0 if y ∈ Ωα, α = 0, 1, ··· ,N, (3.17)

α α and (Cve, ρ0 ) are the viscoelastic tensor and mass density of the αth phase in the mixture. For this mixture, we seek for a solution uˆε,ω to the equation of motion:

ε (ε) 2 ε (ε) div(Cve∇uˆ ) + ω ρ uˆ = 0, (3.18)

where the superscript “ε” reﬂects that the solution depends on the lengthscale ε.

It is clear that the solution uˆ(ε) to the equation of motion (3.18) will be oscillating

ε ε at the scale of ε since the viscoelastic tensor Cve and density ρ oscillates at the scale 34

of ε. Macroscopically, the effect of these fine scale oscillations can be captured by the effective properties of the mixture. and that our goal is to find the asymptotic viscoelastic and acoustic properties of the mixture as ε → 0

To this end, we employ the formal method of multiscale expansion, introduce an additional variable, the fast variable “y = x/ε”, to characterize the ﬁne scale motions, and assume the original solution can be written as

uˆ(ε)(x) = uˆ(0)(x, y) + εuˆ(1)(x, y) + ε2uˆ(2)(x, y) + ··· , (3.19)

where uˆ(m)(x, y) is a periodic function of y variable with period Y for m = 0, 1, ··· , and Z uˆ(m)(x, y)dy = 0 if m = 1, 2, ··· . Y Inserting the above expansion (3.19) into (3.18) and arranging terms according to the order of ε, we obtain that for any (x, y) ∈ Ω × Y , 1 div [Cper(y)∇ uˆ(0)(x, y)]+ ε2 y ve y 1n per (1) per (0) per (0) o divy[Cve (y)∇yuˆ (x, y) + Cve (y)∇xuˆ (x, y)] + divx[Cve (y)∇yuˆ (x, y)] + ε (3.20) per (0) (1) per (1) (2) divx[Cve (y)(∇xuˆ (x, y) + ∇yuˆ (x, y)] + divy[Cve (y)(∇xuˆ (x, y) + ∇yuˆ (x, y)]

ω2ρper(y)uˆ(0)(x, y) + εω2ρper(y)uˆ(1)(x, y) + ··· = 0. As ε → 0, equation (3.20) implies that the terms associated with each diﬀerent power of ε shall vanish. In particular, the leading order terms associated with ε−2 implies that

per (0) divy[Cve (y)∇yuˆ (x, y)] = 0 ∀ (x, y) ∈ Ω × Y.

Since the solution to the above equation has to be constant on Y , we conclude that uˆ(0) = uˆ(0)(x) is in fact independent of the fast variable y. Then, the next order term associated with ε−1 implies

per (1) per (0) divy[Cve (y)∇yuˆ (x, y) + Cve (y)∇xuˆ (x)] = 0 ∀ (x, y) ∈ Ω × Y. (3.21)

For a given F ∈ C n×n, we introduce the unit cell problem for w(· ; F): Y → C n:   per divy[Cve (y)(∇yw(y; F) + F)] = 0 in Y, (3.22)  periodic boundary conditions on ∂Y. 35

Comparing (3.21) with (3.22), we can represent a solution to (3.21) as

(1) (0) uˆ (x, y) = w(y, ∇xuˆ (x)). (3.23)

Further, it is clear that w(x; F) depends linearly on F and hence determines a linear mapping:

Z per e F 7→ − Cve (y)(∇yw(y; F) + F) =: CveF. (3.24) Y

Finally, the term associated with ε0 is given by

per (0) (1) per 2 (0) divx[Cve (y)(∇xuˆ (x) + ∇yuˆ (x, y)] + ρ (y)ω uˆ (x)

per (1) (2) +divy[Cve (y)(∇xuˆ (x, y) + ∇yuˆ (x, y)] = 0 ∀ (x, y) ∈ Ω × Y.

Integrating the above equation over the unit cell Y , by (3.23) and (3.24) we obtain

e (0) e 2 (0) divx[Cve∇uˆ (x)] + ρ ω uˆ (x) = 0 ∀ x ∈ Ω. (3.25)

where

Z ρe = − ρper(y). (3.26) Y

We remark that the above formal calculation is by now standard and can be made rigorous.

Since viscoelastic tensor of the mixture is given by (3.16), we can show the the eﬀective viscoelastic tensor deﬁned by (3.24) can also be written as

e e e Cve(ω) = Celst(ω) + iωCvisc(ω),

e where the frequency ω-dependence of the effective viscoelastic tensor Cve is to be de- e termined. Further, it can be shown that the effective viscoelastic tensor Cve depends analytically on the viscoelastic tensors of the constituent phases [132]. If we find the

e e α α formua Cve = Cve(Cve) with Cve being real elastic stiﬀness tensors, then the formula is α also valid for general complex viscoelastic tensor Cve. This is the so-called the principle of correspondence [132, 133, 134]. Therefore, to calculate the real and imaginary parts

e of the eﬀective viscosity tensor Cve, it is suﬃcient to solve the unit cell problem (3.22) 36

Fig. 3.1: A specimen of dilute suspension

α α for real viscoelastic tensors Cve = Celst (α = 0, 1, ··· ,N). This is a classic problem in the context of elastic composites. For self-containedness, in the next section we present the detailed calculation of the eﬀective viscoelastic tensors of two-phase mixtures and compute the eﬀective acoustic properties of suspensions of aligned and random particles.

3.4 Eﬀective viscoelastic properties of two-phase suspensions

We consider the unit cell problem (3.22) for two-phase mixtures as shown in Figure 3.1:   0 C in Ω0 = Y \ Ω1, per  ve Cve (y) = (3.27)  1 Cve in Ω1, where Ω1 (Ω0) will be referred to as the inclusion (matrix). The unit cell problem (3.22) can be alternatively written as   per div[Cve (y)(∇w + F)] = 0 in Y,   per [[Cve (y)(∇w + F)]]n = 0 on ∂Ω1, (3.28)    periodic boundary conditions on ∂Y,

where [[ · ]] = ( · )| + − ( · )| − denotes the jump across the interface ∂Ω1. Let θ1 = ∂Ω1 ∂Ω1

vol(Ω1)/vol(Y ) be the volume fraction of the inclusion, θ0 = 1 − θ1 be the volume 37

fraction of the matrix, and F1, F0 Z Z F1 := − (∇w + F), F0 := − (∇w + F). (3.29) Ω1 Ω0 be the average strain in the inclusion and ambient medium, respectively. It is clear that

θ0F0 + θ1F1 = F. (3.30)

From the definition (3.24), the effective viscoelastic tensor satisfies Z e 0 1 CveF = − (Cve + CveχΩ1 )(∇w + F) Y (3.31) 0 1 = CveF + θ1MCveF1 = CveF − θ0MCveF0.

1 0 where MCve = Cve − Cve, and χΩ1 , equal to one on Ω1 and zero otherwise, is the char-

acteristic function of Ω1. From the above equation we see that the eﬀective viscoelastic

e tensor Cve is determined if the average strain in the inclusion phase or matrix phase is known for any average strain F.

Further, the solution to the unit cell problem (3.28) for any F ∈ C n×n determines a linear relation between the average strain in the inclusion phase and matrix phase that can be formally represented as

F1 = TF0, (3.32) where the fourth-order tensor T may be called the “concentration tensor” [135, 136].

−1 In terms of the concentration tensor, by (3.30) and (3.32) we ﬁnd that F1 = (θ0T +

−1 −1 θ1II) F, F0 = (θ1T + θ0II) F, and

e 0 −1 −1 1 −1 Cve = Cve + θ1MCve(θ0T + θ1II) = Cve − θ0MCve(θ1T + θ0II) . (3.33)

3.4.1 Suspensions in the dilute limit

We need to solve the unit cell problem (3.28) to ﬁnd the eﬀective viscosity tensor. A closed-form solution can be obtained by the celebrated Eshelby equivalent inclusion method for ellipsoidal inclusions[127][137]. In the dilute limit, we can replace the unit

n Pn 2 2 cell Y by the entire space IR and let Ω1 be an ellipsoid given by i=1 xi /ai = 1. 38

By the Eshelby equivalent inclusion method, we shall ﬁrst consider the homogeneous problem for an eigenstress P∗ ∈ C n×n:   0 n div[Cve∇v + PχΩ] = 0 in IR , (3.34)  |∇v(x)| → 0 as |x| → ∞.

By Fourier transformation, we ﬁnd that a solution to (3.34) satisﬁes the Eshelby uniformity property, i.e., the strain in the ellipsoidal inclusion Ω1 is given by

∗ ∇v(x) = −RP in Ω1, (3.35) where the components of the Eshelby tensor R : C n×n → C n×n is given by

Z det(Λ) (R)piqj = − n (N)pq(e)i(e)jde, (3.36) Sn−1 |Λe|

n−1 n n−1 n S is the unit sphere {e ∈ R : |e| = 1}, e ∈ S denote a unit vector in IR , the 0 n × n matrix (N)pq is the inverse of the acoustic tensor (Ae)pq = [Cve]piqj(e)p(e)q, and

Λ = diag[a1, ··· , an]. In particular, if the matrix phase is an isotropic Newtonian ﬂuid

0 with Cve of form (3.2) with Cvisc given by (3.4), then by (3.10) we ﬁnd that

µ 1 κ + iω(µv + ) N = I + 3 e ⊗ e. (3.37) e 2 4 2 iωµ ω (µµv + 3 µ ) − iωµκ From (3.36), (3.37) and Appendix E, the components of the above Eshelby tensor R can be explicitly written as µ 1 κ + iω(µv + ) (R) = I + 3 J , 1111 1 2 4 2 11 iωµ ω (µµv + 3 µ ) − iωµκ µ 1 κ + iω(µv + ) (R) = I + 3 J , 1212 2 2 4 2 12 (3.38) iωµ ω (µµv + 3 µ ) − iωµκ µ κ + iω(µv + ) (R) = 3 J . 1122 2 4 2 12 ω (µµv + 3 µ ) − iωµκ

By comparing the interfacial conditions on ∂Ω1, we ﬁnd that a solution to (3.34) is also a solution to the unit cell problem (3.28) if

∗ ∗ MCveF = MCveRP + P . (3.39)

In the setting of linear viscoelasticity, the viscoelastic tensors shall satisfy the minor

α α symmetries (Cve)piqj = (Cve)pijq. Therefore, it will be suﬃcient to consider symmetric 39

∗ n×n average strain and eigenstress. For F, P ∈ C sym , the above algebraic equation (3.39) is equivalent to

ˆ ∗ ∗ MCveF = MCveRP + P , (3.40) where Rˆ is the symmetrized tensor of R with components given by

1 (Rˆ ) = [(R) + (R) + (R) + (R) ]. piqj 4 piqj ipqj pijq ipjq

By (3.35), (3.39) and the deﬁnition (3.29), we ﬁnd that

ˆ ∗ ˆ ˆ −1 ˆ −1 F1 = F − RP = [II − R(MCveR + II) MCve]F = (II + RMCve) F, (3.41)

where the last equality follows from the observation that

ˆ ˆ −1 ˆ ˆ ˆ −1 ˆ ˆ R(MCveR + II) MCve(II + RMCve) = R(MCveR + II) (II + MCveR)MCve = RMCve.

Therefore, by (3.31) we ﬁnd that the eﬀective viscoelastic tensor

e 0 ˆ −1 0 −1 ˆ −1 Cve = Cve + θ1MCve(II + RMCve) = Cve + θ1(MCve + R) . (3.42)

Further, it is clear the average strain in the matrix phase deﬁned by the latter

of (3.29) is the same as the overall average strain in the dilute limit, i.e., F0 = F. Therefore, the concentration tensor deﬁned by (3.32) is given by

ˆ ˆ −1 ˆ −1 T = II − R(MCveR + II) MCve = (II + RMCve) , (3.43)

3.4.2 Non-dilute suspensions

For non-dilute two-phase mixtures, the solution to the unit cell problem (3.28) is complicated because of interactions between particles. Nevertheless, the effective viscosity tensor may be extracted by the “mean-field” approach proposed by Mori and Tanaka [135]. In this approach, though the actual strain in the inclusion and ambient medium are unknown, we assume the average strain in each inclusion is only influenced by the mean field in the ambient matrix, and hence the concentration tensor remains independent of the volume fraction. For ellipsoidal inclusions, the concentration tensor T is 40

given by (3.43). By (3.33) the eﬀective viscoelastic tensors of the non-dilute mixtures in the Mori-Tanaka approximation shall be given by

e 0 −1 −1 0 ˆ −1 −1 Cve = Cve + θ1MCve(θ0T + θ1II) = Cve + θ1(θ0R + MCve ) . (3.44)

We remark that the above formula agrees with (3.42) to the linear order of θ1 for

θ1 << 1. We now specialize our results to suspensions of elastic particles in a Newtonian fluid. Assume the elastic particles are of ellipsoidal shape with viscoelastic (i.e., elastic) tensor being real and given by (3.3) whereas the fluid is isotropic with viscoelastic tensor given by (3.2) and (3.4). Then by (3.36) and (3.42) we can calculate the effective viscoelastic tensor explicitly. The result is too lengthy to be presented here for general ellipsoidal particles. Nevertheless, for spherical particles, i.e., Λ = I, by (3.38) we find that Rˆ is

an isotropic tensor given by (I1 = 1/3,J12 = 1/15)

ˆ (R)piqj = rµ(δpqδij + δpjδiq) + rλδpiδqj, µ µ (3.45) I [κ + iω(µv + )]J12 [κ + iω(µv + )]J12 r = 1 + 3 , r = 3 . µ 2 4 2 λ 2 4 2 i2ωµ ω (µµv + 3 µ ) − iωµκ ω (µµv + 3 µ ) − iωµκ By (3.44) and tedious algebraic calculation, we ﬁnd the eﬀective viscoelastic tensor are

e e e e given by Cve = Celst +iωCvisc, where Celst is an isotropic elasticity tensor with eﬀective Lam´econstants given by

Ge = Re[a], λe = Re[b], θ (G − ıωµ) a = ıωµ + 1 , 4θ0rµ(G − ıωµ) + 1 λ − ıω(µv − 2µ/3) − κ − 2θ0rλ(G − ıωµ)(2G + 3λ − 3ıωµv − 3κ) b = ıω(µv − 2µ/3) + κ + θ1 , [4θ0rµ(G − ıωµ) + 1][(2rµ + 3rλ)θ0(2G + 3λ − 3ıωµv − 3κ) + 1] e and Cvisc is an isotropic viscosity tensor with dynamic viscosity and bulk viscosity given by

Im[a] Im[b] 2 µe = , µe = + µe. ω v ω 3

3.4.3 Suspensions with random particle directions

In this section, we consider suspensions in which particles are randomly aligned. Let the angle between the symmetric axis of ellipsoid and e3 direction and e1 direction be 41

α and ϕ, respectively. Then the angle-averaged concentration factor can be written as

(T(l))piqj = ΛikΛjl(T)pkql,   cos α − sin α cos ϕ sin α sin ϕ     where Λ is the rotation matrix about e1 axis, given by sin α cos α cos ϕ − cos α sin ϕ.     0 sin ϕ cos ϕ Then we can calculate the angle-averaged strain rate on the inclusion,

Z Z h − ∇ui = hTih − ∇ui, Ω1 Ω0 where

1 Z hTi := T(l)dl. 4π S2

Based on the Eshelby’s solution, the eﬀective viscoelastic tensors of a dilute suspension shall be given by

e 0 Cve = Cve + θ1MCvehTi, if the volume fraction of the particulate phase is non-dilute, based on the mean-ﬁeld type theory of Mori-Tanaka, the eﬀective viscoelastic tensors shall be given by

e 0 −1 −1 Cve = Cve + θ1MCve(θ0hTi + θ1II) .

3.5 Eﬀective acoustic properties of a two-phase medium with spheroidal inclusions

3.5.1 Inﬂuence of particle aspect ratio

We start by showing the normalized effective viscosity plotted against particle aspect ratio in Figure 3.2, with axis of symmetry of the particle parallel and perpendicular to the sound propagation direction (referred to as parallel and perpendicular alignment in the remaining text). For perpendicular alignment, the effective viscosity first slightly decreases before increasing dramatically after the aspect ratio reaches 1, while for parallel alignment the effective viscosity monotonically decreases with the increasing 42

Figure 3.2: Effects of particle aspect ratio on effective viscosity. Particle volume fraction is taken to be 5%. aspect ratio. Note that effective viscosities measured from two different directions meet at an aspect ratio of 1, where the value coincides with that computed from Einstein’s formula [122]. This serves as a validation of our formulation of effective viscosity tensor.

The effective acoustic attenuation coefficient of a two phase medium, normalized by frequency squared, is shown in Figure 3.3 at a volume fraction of 5% with particle aspect ratio ranging from 0.1 to 10. For parallel alignment, the attenuation coefficient increases with increasing aspect ratio, while for perpendicular alignment the opposite trend has been observed. We can also see that suspensions of prolate particles display a more pronouncing anisotropy in terms of acoustic attenuation.

In Figure 3.4 and Figure 3.5, the normalized sound speed of the two phase medium is plotted against the particle aspect ratio at frequencies of 1 MHz, 3 MHz and 5 MHz. For parallel alignment, the sound speed almost remains the unchanged as we increase the aspect ratio before it starts to increase at an aspect ratio of 10. For perpendicular alignment, on the other hand, the sound speed decreases dramatically as aspect ratio increases from 0.01 to 0.1 and then remains almost constant as the aspect ratio further goes up to 100. When compared with Figure 3.3, it is obvious that the acoustic attenuation of the suspension is more sensitive to aspect ratio than the sound 43

Figure 3.3: Eﬀects of particle aspect ratio on attenuation. Particle volume fraction is taken to be 5%. speed.

3.5.2 Inﬂuence of particle volume fraction

The effects of particle volume fraction on sound speed and attenuation in a two phase medium are shown in Figure 3.6 and Figure 3.7 for prolate particles with aspect ration of 100 at a sound frequency of 2 MHz. Figure 3.6 suggests that the increasing volume fraction in general enhances the acoustic attenuation of the medium, as one might expect, with the parallel alignment having a much greater attenuation coefficient. On the other hand, the volume fraction has a more complicated influence on the sound speed. Specifically, the sound speed decreases as the volume fraction increases form 0 to 6.5% for both perpendicular and parallel alignment. As volume fraction further goes up to 20%, however, the sound speed of the former keeps decreasing monotonically while that of the latter is experiencing a slow recovery.

3.5.3 Comparison with the experimental results

The comparison between this theory and the experimentally measured sound attenuation results from Chapter 2 is presented Figure 3.8 and Figure 3.9. The prediction 44

Figure 3.4: Eﬀects of particle aspect ratio on sound speed of suspensions (parallel alignment). Particle volume fraction is taken to be 5%.

Figure 3.5: Eﬀects of particle aspect ratio on sound speed of suspensions (perpendicular alignment). Particle volume fraction is taken to be 5%. 45

Figure 3.6: Eﬀects of volume fraction on attenuation of suspensions. Particle aspect ratio is taken to be 100 and acoustic wave frequency is ﬁxed at 2 MHz.

Figure 3.7: Eﬀects of volume fraction on sound speed of suspensions. Particle aspect ratio is taken to be 100 and acoustic wave frequency is ﬁxed at 2 MHz. 46

Figure 3.8: The comparison between this theory and measured attenuation: parallel alignment.

Figure 3.9: The comparison between this theory and measured attenuation: diﬀerence between the two parallel and perpendicular alignment. 47

from Ahuja and Hendee’s model [87] is also plotted in the figures. While grasping the general trends of the frequency dependence of sound attenuation, our model greatly underpredicts the influence of volume fraction in Figure 3.8. The calculated attenuation difference between the two alignments shown in Figure 3.9 is also smaller than the experimental data. These differences between our model prediction and the experimental data is possibly due to the absence of an inertial term in the modeling.

3.6 Conclusions

In this work we have developed a analytical model for a two-phase medium with aligned spheroidal inclusions based on effective media theory, assuming a dilute two-phase media in a long wavelength limit. We have presented the calculated sound speed and attenuation coefficient as a function of inclusion aspect ratio, acoustic wave frequency and medium volume fraction. The comparison of the model and the experiments is less than satisfactory, with the model in general underpredicting the effect of volume fraction. We believe this is possibly due to the absence of an inertial term in the modeling. Improvements are needed in the model to better predict the acoustic properties of a two phase medium with spheroidal inclusions. 48

Chapter 4

Contactless, High-throughput Analysis on Electrical Conductivity of Si Nanowires with Surface Passivation

4.1 Introduction

Al2O3 nanolayers induced by atomic layer deposition (ALD) technique have been shown to provide excellent passivation for Si surfaces with the electron-hole recombination at the interface effectively suppressed by two different mechanisms. Chemical passivation refers to satisfying the open bonds at the interface and thus reducing the number of recombination centers, while field-effect passivation is where the high density negative

12 13 −2 charges (∼ 10 − 10 cm ) induced by the Al2O3 layers repel the electrons from the interface and thus separate them from the recombination centers.

ALD deposited Al2O3 nanolayers have been extensively studied in the past decades, particularly for their striking performance in passivating Si solar cells where extending the bulk carrier lifetime is critical [138, 139, 140]. It has been shown by Hoex et al.

[141] that Al2O3 films deposited by plasma-assisted ALD has much higher fixed charge density than those deposited by thermal ALD. Werner et al. [142] have found that the negative charges are located within 1 nm of the Al2O3 films. Terlinden et al. [143] have

studied the effect of Al2O3 layer thickness on the passivation performance of crystalline silicon (c-Si) surfaces. They have concluded that the field-effect passivation is unaffected

by the Al2O3 layer beyond a thickness of 2nm. It has also been demonstrated that the Si surface condition before the ALD treatments can also greatly impact the passivation performance. Bordihn et al. [144] have measured the surface recombination rate for Si surface treated by HF, HNO3,H2SO4/H2O2 and HCl/H2O2, with post annealing and

ﬁring. Excellent Seff value of ∼ 10 cm/s has been obtained. Simon et al. [145] have

shown that the ﬁxed charge density can be readily controlled by introducing a HfO2 49

layer, which suppresses the negative charge induced by Al2O3. The HfO2/Al2O3 stacks actually provide excellent surface passivation for Si even without ﬁeld eﬀect passivation.

However, most of the existing work on ALD-induced Al2O3 passivation consider planar surfaces. Thus, its effect on Si nanowires have not been properly studied due to the additional difficulty in characterizing nanowires. With Si nanowires finding more and more applications in a wide variety of areas [72], it is critical to be able to better control their properties. Furthermore, large variations, possibly caused by nanowire surface states, have often been observed in the electrical conductivity of Si nanowires [80, 63]. Considering the extremely high surface to volume ratio, one would expect the surface passivation to be much more effective (and needed) for the Si nanowires in controlling their electrical transport properties. At this point, however, the studies of ALD passivations on nanowires are far less than those on planar Si. Gaboriau et al. [146] studied the ALD coating of highly doped Si nanowires used as electrodes for

capacitors. Kato et al. have demonstrated that the ALD-Al2O3 can increase the carrier lifetime as well as the effective diffusion length of minority carriers in Si nanowire arrays, which is of great importance for applications like Si solar cells [147]. To the author’s best knowledge, no systematic results have been reported on the statistics of Si nanowire properties in the presence of ALD passivation. With the recently developed EOS method, we are able to perform the first statistical study on the electrical

conductivity of Si nanowires with ALD-induced Al2O3 layers. In the following, we introduce the technical background and the detailed procedures of the EOS method, with which we measure the electrical conductivity of Si nanowires

grown by a vapor liquid solid (VLS) technique with ALD-enabled Al2O3 passivation.

The effect of pre-ALD surface conditions, ALD cycle numbers and Al2O3 layer thickness are investigated. We then discuss the importance of the nanowire surface on the nanowire conductivity and variability and further explore the subject by electrically characterizing as-grown Si nanowires with different diameters (hence different surface- to-volume ratios). 50

Figure 4.1: Schematic of an ellipsoidal shaped nanowire suspended in the ﬂuid under an external ﬁeld

4.2 Method

4.2.1 Electro-orientation

Consider an ellipsoidal shaped nanowire suspended in a ﬂuid medium with semi-axes lengths given respectively by a, b and c, as shown in Figure 4.1. The conductivity and permittivity of the ﬂuid and the nanowire are given by (σf , εf ) and (σp, εp) respectively.

A spatially uniform AC field with strength E0 and angular frequency ω is applied to the nanowire so that its long axis tends to rotate into alignment in parallel with the field direction. To determine the alignment rate, we first calculate the torque on the nanowire imposed by the external field. According to Jones [110], the effective moment is given by,

4πabc (~p ) = ε K E , eff i 3 f i 0,i where i represents the three axis x1, x2 and x3 and K is the complex Clausius-Mossotti factor,

εp − εf Ki = , εf + (εp − εf )Li 51

σf the complex permittivity for ﬂuid and nanowire are given by εf = εf − ω  and εp =

σp εp − ω . The demagnetization factor L is deﬁned as, abc Z ∞ ds Lx = , 1 2 p 2 2 2 2 0 (s + a ) (s + a )(s + b )(s + c ) abc Z ∞ ds Lx = , 2 2 p 2 2 2 2 0 (s + b ) (s + a )(s + b )(s + c ) abc Z ∞ ds Lx = , 3 2 p 2 2 2 2 0 (s + c ) (s + a )(s + b )(s + c )

Lx1 + Lx2 + Lx2 = 1.

Assuming a b = c, the time-averaged torque along the longest axis on the nanowire is given by Jones [110]

1 < T~ e >= Re(~p × E~ + ~p × E~ ), 2 eff,k 0,⊥ eff,⊥ 0,k where k and ⊥ represent the parallel and perpendicular components (of the electric field and the effective moments). Following the previous definition, the effective moments can be written as,

2 4πab εp − εf ~peff,k = εf ( E0,k), 3 εf + (εp − εf )Lk 2 4πab εp − εf ~peff,⊥ = εf ( E0,⊥). 3 εf + (εp − εf )L⊥ With the above assumption, the demagnetization factor L can be further approximated as,

1 L = (ln(2β) − 1),L = (1 − L )/2, k β2 ⊥ k where β = a/b is the aspect ratio of the nanowire. Then the time averaged torque can be written as,

2 ε − ε ε − ε ~ e πab 2 p f p f |Tx3 | = εf E0 sin 2θRe( − ), 3 εf + (εp − εf )Lk εf + (εp − εf )L⊥ πab2 = ε E2 sin 2θRe(K), 3 f 0 where

ε − ε ε − ε K = p f − p f , εf + (εp − εf )Lk εf + (εp − εf )L⊥ 52

and θ is the angle between the longest axis and the ﬁeld direction. The Maxwell-Wagner relaxation time scale is deﬁned as follows,

(1 − Lk)εf + εpLk τmw,k = , (1 − Lk)σf + σpLk (1 − L⊥)εf + εpL⊥ τmw,⊥ = . (1 − L⊥)σf + σpL⊥ ~ e Then |Tx3 | can be rearranged into, " 2 2 πab2 (εp − εf )ω τ |T~ e | = ε E2 sin 2θ mw,k x3 f 0 2 2 3 (1 + ω τmw,k)(εf + (εp − εf )Lk) σp − σf + 2 2 (1 + ω τmw,k)(σf + (σp − σf )Lk) 2 2 (εp − εf )ω τmw,⊥ − 2 2 (1 + ω τmw,⊥)(εf + (εp − εf )L⊥) # σp − σf − 2 2 . (1 + ω τmw,⊥)(σf + (σp − σf )L⊥)

In the low frequency limit (ωτmw,k 1, ωτmw,⊥ 1) and high frequency limit(ωτmw,k ~ e 1, ωτmw,⊥ 1), |Tx3 | becomes respectively, 2 ~ e πab 2 σp − σf σp − σf |Tx3 | = εf E0 sin 2θRe − , 3 σf + (σp − σf )Lk σf + (σp − σf )L⊥ 2 ~ e πab 2 εp − εf εp − εf |Tx3 | = εf E0 sin 2θRe − . 3 εf + (εp − εf )Lk εf + (εp − εf )L⊥ It is clear that in the low frequency limit, the conductivity is dominating the electro- orientation torque, while in the high frequency limit, the permittivity dominates. The transition frequency, which is called crossover frequency, is the inverse of Maxwell- Wagner relaxation time scale deﬁned above and is a strong function of the particle conductivity.

As the nanowire is rotating into alignment by the induced torque described above, it is also experiencing a friction force from the ﬂuid. The balance of the friction torque and the ﬁeld induced torque determines the rotation rate Ω,

~ e ζRΩ − |Tx3 | = 0, where ζR is the rotational friction coeﬃcient. Thus the rotation rate is given by, |T~ e | Ω = x3 . ζR 53

For an prolate ellipsoidal particle ζR is given by [148, 149], 1 + β2 ζR = 4V µ 2 , 1 + Lk(2β − 1) where V = 4πab2/3 is the volume of the nanowire and µ is the viscosity of the ﬂuid.

Upon assuming high aspect ratios (β 1), ζR can be simpliﬁed into, 16π ζ = a3µ(2 ln(2β) − 1)−1. R 3

Then the alignment rate can be written as,

ε E2 sin 2θ(2 ln(2β) − 1)Re(K) Ω = f 0 . 16β2µ

4.2.2 Electro-orientation Spectroscopy

Based the previous derivation, we know that the crossover frequency is strongly dependent on the conductivity of the nanowires. The crossover frequency can be obtained by measuring the alignment rate of a single nanowire under a series of uniform electric ﬁeld with the same magnitude and diﬀerent frequencies. Thus, the conductivity of the nanowire can be obtained by experimentally monitoring the electro-orientation of nanowires. From the last section we have the crossover frequency given by,

1 (1 − Lk)σf + σpLk ωcrossover = = . τmw,k (1 − Lk)εf + εpLk For high-aspect-ratio-particles (β ∼ 50), the demagnetization factor becomes very small

−3 (Lk ∼ 10 ). Then if the conductivity of the ﬂuid is very small such that (1−Lk)σf

σpLk, the crossover frequency can be written as,

σp ωcrossover = , (1/Lk − 1)εf + εp

since εf and εp are of the same order of magnitude, then (1/Lk − 1)εf εp and

ωcrossover can be further simpliﬁed into,

σp ωcrossover = . (1/Lk − 1)εf Note that the conductivity of the nanowire is only a function of aspect ratio and ﬂuid permittivity. Experimentally, the crossover frequency can be obtained from ﬁtting the

2 alignment rate data using Ω = c/(1 + (ω/ωcrossover) ), where c is a ﬁtting constant. 54

4.2.3 EOS measurements

The experimental method of the EOS technique can be described as follows. We dis- perse the nanowires in mineral oil and dipropylene glycol (DPG) (dielectric constant 2.1 and 21 respectively) and measure the alignment rate of individual nanowires under an external AC electric field at a series of frequencies, from which the conductivity of nanowires can be extracted. A specialized electrode is designed and built by aligning two pairs of wire electrodes into a square to provide two electric fields that are perpendicular to each other, so that we can align nanowires continuously by switching the field direction back and forth and obtain the averaged alignment rate at a single frequency. The uniform AC signal is generated by a function generator (Tektronix AFG3200C) and amplified by a high-frequency amplifier (TREK 2100HF) to the required amplitude. Nanowire rotation is recorded by a high-speed monochrome CCD camera (pco.edge sC- MOS, PCO AG) connected to an inverted optical microscope (Olympus IX71, Olympus Corp.) with a 40x objective lens (Olympus LUCPLFN 40x, N.A. 0.6, Olympus Corp). The alignment rate of individual nanowires under external AC electric field with different frequencies are extracted from the video by a MATLAB program, from which the conductivity of nanowires can be calculated. A schematic of the experimental set up is shown in Figure 4.2.

To validate the independence of the EOS technique on the suspending ﬂuid, we have measured the conductivity of Si nanowires fabricated from metal-assisted chemical etching (MACE) in both mineral oil and DPG, the results are shown in Figure 4.3. The measured conductivity of Si nanowires are almost the same for these two suspending ﬂuid in terms of both average values and distribution.

4.2.4 Nanowire processing

The Si nanowires studied in this work are synthesised by collaborators at Georgia Institute of Technology (Prof. Michael Filler and Dr. Gozde Tutuncuoglu) using a vapor-liquid-solid (VLS) technique [150, 151, 152], with a diameter of 50 nm and a 55

Figure 4.2: Schematic of the experimental setup, uniform AC signal is generated by the function generator and ampliﬁed before reaching the electrodes, the ﬁeld direction is switched back and force to keep nanowires aligning and the alignment rate is extracted from the videos taken by the high speed camera connected to the microscope. Electrical conductivity is calculate from the collected alignment rate data.

Figure 4.3: Measured conductivity of Si nanowires in mineral oil and DPG 56

total length of 20 microns. The as-grown Si nanowires are first dipped in buffered hy- drofluoric acid (Buffered oxide etch (6:1), CMOS) for 5 seconds to remove the somewhat unpredictable native oxide, in order to gain total control of the nanowire surface. Then

ALD technique is used to deposit Al2O3 nanolayers on the nanowires with cycle numbers of 0, 3, 6, 12 and 24. Speciﬁcally for 0 cycle, the nanowires are immediately put in oil and measured within a few hours in order to avoid the regrowth of native oxide. For comparison, we also grow chemically induced oxide on the HF treated nanowires before the ALD deposition. It is achieved by a SC1 step using hydrogen peroxide and ammonia solution. Nanowires with 0, 3, 6 and 12 cycles of ALD deposition are studied for this group of samples. A separate group of Si nanowires with diameters of 50 nm, 100 nm and 150 nm are also fabricated with no intentional surface treatment, to study the eﬀect of surface-to-volume ratio on nanowire properties and variability.

4.3 Results and discussion

We start by showing the electrical conductivity of as-grown intrinsic Si nanowires measured by the EOS method. A variation of 2-3 orders-of-magnitude is observed in nanowire conductivity as shown in Figure 4.4. This variation has been reported previously by Akin [80, 63] and can be attributed to the variation of the nanowire surface states, as we will discuss later. We also notice that the measured nanowire conductivity follows a lognormal distribution, with a long tail on the linear scale. This lognormal distribution is still valid for conductivity of Si nanowires after the ALD treatment, as shown in Figure 4.5. Therefore, in order to avoid the nanowire conductivity statistics being overly aﬀected by a few large data points, we will base the rest of our analysis on lognormal distribution and focus mainly on the statistics of the log value of conductivity. On the other hand, despite the large variations in Figure 4.5, we can still see a gradual increase of nanowire conductivity with increasing ALD cycles in a overall range of 5-orders-of-magnitude. Hence, we argue the electrical properties of Si nanowires are dominated by their surface states.

In order to further study how Al2O3 deposition aﬀects nanowire conductivity, we 57

Figure 4.4: The measured conductivity of as-grown intrinsic nanowires are plotted in both linear and log scale. 58

Figure 4.5: The measured conductivity of nanowires with ALD deposited Al2O3 are shown. Here we use either HF or SC1 for initial nanowire surface treatment. 59

plot the nanowire conductivity against both ALD cycle numbers and Al2O3 layer thickness in Figure 4.6. As mentioned before, the plotted average here is calculated based on the log of the measured conductivity. It is clear that HF- and SC1- treated nanowires exhibit diﬀerent ALD-dependent electrical behaviors, with the latter increasing more rapidly with the ALD cycles within our measurement range according to Figure 4.6a.

The Al2O3 thickness is estimated by measuring the planar Si samples with the same treatments as the nanowires using X-ray photoelectron spectroscopy (XPS), as shown in Figure 4.7. One thing to notice is that the HF-treated nanowires have thinner Al2O3 layers for the same number of ALD cycles. This is possibly because the hydrogen- terminated nanowire surfaces are less reactive to the ALD precursors. Taking advantage of this XPS measurement and assuming the nanowire surfaces behave similarly to

planar surfaces, we can then show the eﬀect of Al2O3 layer thickness on the nanowire conductivity in Figure 4.6b. The nanowire conductivity increases with the increasing

Al2O3 thickness and tends to saturate toward a thickness of 3.5nm. This is consistent with previous observations on planar Si samples where the surface recombination velocity only decreases with the first few nanometers of Al2O3 layer growth before reaching saturation [142]. This ALD-induced increase in nanowire conductivity likely results from a combination of chemical passivation and field-effect passivation.

In Figure 4.8 we show the eﬀect of Al2O3 layers on the variation of nanowire conductivity, where the lognormal-based coeﬃcient of variation (COV) is calculated by the formula below [153], q σ2 COV = e log − 1,

here σlog is the standard deviation of the natural log of the measured conductivity. Assuming the sample number is n, the 100(1 − α)% two-sided conﬁdence interval for the COV is given by [153], v v u 2 u 2 uσlogn/(n − 1) uσlogn/(n − 1) [L, U] = [t 2 , t 2 ]. χ(n−1),(1−α/2) χ(n−1),(α/2)

The COV of the nanowire conductivity in general decreases with increasing thickness of the ALD-induced layers for both SC1 and HF treated nanowires. This can be explained 60

Figure 4.6: The measured conductivity of Si nanowires are plotted against (a) the number of ALD cycles and (b)the Al2O3 layer thickness. 61

Figure 4.7: The thickness of the Al2O3 layer are characterized by X-ray photoelectron spectroscopy (XPS) on planar Si samples with the same treatment as Si nanowires

by the mechanisms of ALD-Al2O3 passivation on Si surfaces. The induced Al2O3 layers decrease the number of trapped carriers by satisfying the surface traps (chemical passivation) and by driving the carriers away from the surface traps (ﬁeld-eﬀect passivation). This decrease in the number of trapped carriers leaves less room for variation, and thus decreases the COV of the number of active carriers which determines the nanowire conductivity.

To further demonstrate the importance of surface traps in controlling the conductivity of Si nanowires, we electrically characterize as-grown Si nanowires with diameters of 50 nm, 100 nm and 150 nm. As the surface-to-volume ratios get smaller, a higher percentage of carriers are in the bulk of the nanowires and remain free from the traps on the surfaces, resulting in higher conductivity for thicker nanowires, as shown in Figure 4.9a. This is consistent with the formulation by Schmidt et al. [154, 155], where they show that nanowires with smaller diameters are more likely to be fully depleted than those with greater diameters at a same doping concentration. As we believe the variations in nanowire conductivity mainly come from the variations in the surface traps, the minimized role of the nanowire surfaces can also explain the decrease in COV for Si nanowires with bigger diameters. These results further support our argument that 62

Figure 4.8: The COV of the measured conductivity are plotted against the Al2O3 thickness for both HF and SC1 treated Si nanowires.

the conductivity of nominally undoped and lightly doped Si nanowires are dominated by the surfaces.

4.4 Conclusions

In this work we have measured the electrical conductivity of nominally undoped Si

nanowires with HF- and SC1- treated surfaces plus ALD deposited Al2O3 layers in a statistically meaningful manner. The measured electrical conductivity in general follows a lognormal distribution with or without any surface treatment, and is heavily

affected by the ALD passivation. Specifically, the ALD deposited Al2O3 layers seem to increase the nanowire conductivity while decrease its variation. On the other hand, the initial surface condition can also significantly change the ALD deposition and thus

the nanowire conductivity, with HF-treated nanowires growing Al2O3 thickness and increasing conductivity more slowly than those treated with SC1. We have also measured the electrical conductivity of Si nanowires with diﬀerent diameters, again ﬁnding that nanowire conductivity and variability are dominated by surface traps in this regime. This work serves as an important step towards fully understanding and controlling the 63

Figure 4.9: The measured conductivity and the COV are plotted against the nanowire diameters for as-grown Si nanowires. 64

electrical properties of Si nanowire and eventually nanowire devices. 65

Chapter 5

Contactless Electrical and Structural Characterization of Semiconductor Nanowires with Axially Modulated Doping Proﬁles

5.1 Introduction

Semiconductor nanowires are promising building blocks for future electronics [68, 156, 157], photonics [158, 159, 160], sensors [161, 162, 163], and energy-conversion technologies [164, 165]. In the vapor-liquid-solid (VLS) mechanism, a liquid seed droplet collects atoms from precursors in the gas-phase and, upon reaching supersaturation, precipi- tates a single-crystalline nanowire. Central to the promise of the VLS mechanism is the ability to change gas-phase composition during growth, thus allowing segments with distinct structures or compositions to be axially (or radially) encoded. Direct control of crystal structure [166], dopant proﬁle [167, 168], alloy composition [169, 170], and morphology [171, 172] along the nanowire length have all been reported.

Accurate and eﬃcient methods are needed to statistically characterize the structure and properties of every segment encoded along a nanowire’s length. A suite of such capabilities are likely to be necessary to fully understand VLS growth, engineer nanowire properties, and translate nanowires into future technologies. In particular, knowledge of segment electrical conductivity is crucial for p-n or p-i-n diodes – the simplest device structures. While direct-contact electrical characterization methods [173, 174] yield accurate conductivities, they are labor intensive. These methods are further constrained when assessing multi-segment nanowires, as each segment requires two or more contacts. Other techniques, including electrostatic force microscopy [175], laser-assisted atom probe tomography [176] and Kelvin probe force microscopy [177], can map carrier- density distribution. However, these methods require specialized facilities and largely 66

Figure 5.1: Electro-orientation spectroscopy (EOS) is shown to enable highly efficient electrical characterization of the properties and structure of individual semiconductor nanowires with homogeneous or axially programmed dopant profiles. The contactless, solution-based method enhances fundamental statistical understanding of highly variable 1D nanomaterials, and serves as a first step toward the high-throughput interrogation of complete nanowire-based devices. remain even more time-consuming than direct-contact methods.

Here, we advance electro-orientation spectroscopy (EOS) to enable the eﬃcien- t, non-contact characterization of individual two-segment Si nanowires. As depicted in Figure 5.1, each nanowire contains one nominally undoped and one doped segment. We combine experiment and simulation to assess not only the electrical conductivity of the undoped segment, but also the length ratio Lundoped/Ltotal, where

Ltotal = Lundoped + Ldoped with Lundoped and Ldoped being respectively the length of the undoped and undoped segments of Si nanowires. Measured undoped segment conductivities are compared to those obtained from direct-contact measurements, as well as the EOS-measured conductivity of homogeneous Si nanowires grown under the same

conditions. The ratio Lundoped/Ltotal is also measured with EOS and compared with the expected length ratios based on growth times.

5.2 Methods

As previously described in Chapter 4, EOS relies on the transient alignment rate of individual, fluid-suspended nanowires under external AC electric fields. Upon application of an external field, the nanowire develops an induced dipole and rotates into alignment with the electric field in a manner that depends on the properties of the 67

nanowire and suspending fluid, and the frequency of the applied field [110]. Nanowire conductivity can be quantitatively extracted by observing the field-induced rotational speed of individual nanowires at different frequencies in a liquid of known properties [80, 63]. For nanowires with axially homogeneous carrier densities, the conductivity is directly related to the so-called “crossover frequency” by the relation

(1 − L||)σf + σpL|| ωcrossover = , (5.1) (1 − L||)εf + εpL|| where σ is the conductivity, ε is the (real) permittivity, L|| is the geometric depolariza- tion factor, and the subscripts f and p refer to the ﬂuid and particle (i.e., nanowire), respectively [110]. The crossover frequency occurs at the transition between conductivity- dominated, low-frequency alignment and permittivity-dominated, high-frequency alignment. For suﬃciently long (≥ 1µm) homogeneous nanowires, the transient alignment process (and hence crossover frequency) can be observed with a standard optical microscope and used to determine conductivity over a seven orders-of-magnitude range (10−5-102 S/m) [80, 63]. The EOS method is also compatible with other solution- based processing, sorting, and positioning operations used in nanodevice fabrication [178, 66, 179, 71, 180].

Analytic relationships analogous to Equation 5.1 do not exist for nanowires with axially modulated conductivities. The closest existing models are for the dielectric response of biological cells with an ellipsoidal core-shell structure (e.g., erythrocytes with a cell membrane, cytoplasm, and nucleus) [181, 182, 183]. Because of this, we numer- ically evaluate nanowire polarizability as a function of frequency via 2D axisymmetric

ﬁnite-element simulations (Appendix I). The length ratio Lundoped/Ltotal and the conductivity of the undoped segment of the nanowires are extracted by least-square ﬁtting the measured EO spectrum with a look-up table of simulated spectra (Appendix I).

To experimentally test the ability of EOS to quantitatively measure two-segment semiconductor nanowires, Si nanowires are grown using the gold-seeded VLS mechanism in a previously described low pressure chemical vapor deposition (CVD) system

[171]. The Lundoped/Ltotal ratio is synthetically controlled to be 1/3, 1/4 and 1/6 with

Ltotal = 10µm. Fully undoped Si nanowires (Lundoped/Ltotal = 1) are also synthesized 68

Figure 5.2: Simulations of the EO spectra of nanowires with axially homogeneous and two-segment dopant profiles with different Lundoped/Ltotal ratios. The nominally undoped and doped segments are assumed to have the same permittivity and only vary in electrical conductivity. (a) EO spectra for varying Lundoped/Ltotal ratio, showing the change in plateau height with increasing doped segment length. Schematic figures of nanowires with different Lundoped/Ltotal ratios are also shown. (b) EO spectra for undoped segments of increasing conductivity and all other parameters unchanged, showing the increasing frequency of the first (lower-frequency) crossover. (c) EO spectra for doped segments of increasing conductivity and all other parameters unchanged, showing the increasing frequency of the second (higher-frequency) crossover. 69

using the same growth conditions as the undoped segment of two-segment nanowires for comparison. Hydrogen passivation of the sidewalls of the nanowire during growth prevents sidewall deposition and unwanted radial dopant incorporation (Appendix F) [152, 151].

For EOS measurements, these nanowires are dispersed in mineral oil (dielectric constant of 2.1) and placed between electrodes on a glass substrate, as shown in Figure 5.1. Spatially uniform AC electric ﬁelds are applied at a series of frequencies for repeated measurements to determine the EO spectra. As a nanowire rotates into alignment with the ﬁeld, the frequency-dependent EO alignment rates are collected and analyzed for nanowire characterization (Appendix H).

5.3 Results and discussion

Simulated electro-orientation (EO) spectra for two-segment nanowires exhibit a number of interesting features, as shown in Figure 5.2. All EO spectra for two-segment nanowires feature two crossover frequencies and an intermediate “plateau” in the alignment rate. As seen in Figure 5.2a, the height of the plateau in the alignment rate increases as the length of the undoped segment decreases relative to the total length of the nanowire. Figures 5.2b and 5.2c show that each crossover frequency is determined by the conductivity of the segment responsible for it. The lower and higher crossover frequencies increase with the conductivity of the doped and doped segments, respectively. While the alignment-rate plateau is largely independent of each segment’s conductivity, it is very sensitive to the segment lengths. By determining the plateau and both crossover frequencies in the alignment rate, the conductivity and length of each nanowire segment can be determined.

Figure 5.3 shows experimentally measured EO spectra for one- and two- segment

Si nanowires with diﬀerent values of Lundoped/Ltotal. For a one-segment nanowire (i.e.,

Lundoped/Ltotal = 1), a single crossover frequency is observed and the high-frequency alignment rate drops to zero. This behavior is consistent with the analytical solution of Equation 5.1 as well as the simulations shown in Figure 5.2. For two-segment Si 70

Figure 5.3: Measured EO spectra for two-segment Si nanowires with fabricated ratios of Lundoped/Ltotal of 1/3, 1/4 and 1/6, compared to that for an axially homogeneous, nominally undoped (i.e., Lundoped/Ltotal = 1) Si nanowire. The error bars indicate the standard deviation for 3 repeated measurements on the same nanowire.

Figure 5.4: Measured length ratios of two-segment Si nanowires. (a) Comparison of Lundoped/Ltotal as determined via EOS and electron microscopy. (b) Scanning electron microscope image of a Si nanowire with Lundoped/Ltotal = 1/3 in which the undoped region is selectively etched with KOH. 71

Figure 5.5: Measured conductivity distributions for nominally undoped nanowires. (a) Comparison of one- and two- segment Si nanowires. The conductivity distribution of the undoped segments of two-segment nanowires (blue) overlaps with that undoped, one-segment Si nanowires synthesized under the same conditions. The arrows indicate four-point measurements of the conductivity of 5 undoped one-segment nanowires. The direct-contact measurements show similar variability and, with one exception, fall within the range of the EOS-measured conductivities. (b) Representative I − V curve from a four-point measurement of an individual Si nanowire. Inset: Representative scanning electron microscope image of four lithographically fabricated electrodes contacting an individual Si nanowire. nanowires, we observe a prominent low-frequency crossover and a nonzero plateau in EO alignment rate as predicted by the simulation. However, the second, higher-frequency crossover lies outside the measurable frequency range for the present experiments. The height of the plateau in the alignment rate increases as the length of the doped segment increases, again consistent with the simulations shown in Figure 5.2. The excellent repeatability of the EOS measurement is seen in the small standard deviations shown by the error bars in Figure 5.3 for multiple measurements of the same nanowire [80, 63].

Figure 5.4a compares the length ratios determined by EOS against the grown length ratios. As seen in the ﬁgure, the EOS-measured and expected values of Lundoped/Ltotal agree to within 0.02. Figure 5.4b shows a scanning electron microscope image of one selectively etched nanowire with Lundoped/Ltotal = 1/3 in which the undoped region is 72

preferentially etched with KOH [184] (Appendix G). The error bars in Figure 5.4a represent the standard deviation of Lundoped/Ltotal from 5 individually measured nanowires for each value of Lundoped/Ltotal. The observed variability may be due to several reasons, including slight differences in the length of each segment, uncertainties in the EO spectrum fitting process, and variations in surface traps. In particular, we only measure nanowires optically determined to be unbroken (i.e., 10 µm long). However, variation- s in total length . 0.5µm (e.g., due to differences in the point at which nanowires break during sonication) are optically undetectable and can contribute to the observed variability.

Figure 5.5a shows that the actual conductivity of the undoped segment of the nanowires varies over 2-3 orders-of-magnitude, as described previously in Chapter 4. At the lower end, the nanowires are seen to be pseduo-intrinsic, with effective carrier densities in 1010 cm−3 range. The low effective carrier concentration of the silicon nanowires is due largely to surface traps, and previous work has shown that surface passivation can increases the effective conductivity by orders of magnitude [80]. The distribution of conductivities is non-Gaussian (note the semi-log axes) and skewed toward higher conductivities. The coefficient of variation (standard deviation divided by the mean) for the conductivity of the nanowire ensemble exceeds 2. Direct-contact four-point measurements of Si nanowires corroborate the conductivity variation shown by the EOS measurements. Figure 5.5b shows a representative four-point I − V measurement of a nominally undoped, axially homogeneous Si nanowire grown under the same conditions as the undoped segment of two-segment Si nanowires. The conductivities of five Si nanowires measured in this manner are indicated by arrows in Figure 5.5a. Two-orders-of-magnitude variations in conductivity are seen in the four-point measurements, which is consistent with the EOS results. We also observe somewhat higher conductivities with the direct-contact measurements than the EOS technique. We believe the slight conductivity difference is the result of the procedure used to form ohmic contacts to the Si nanowires. In particular, we annealed the 4-point nanowire devices at 500 ◦C for 20 min under forming gas, which has previously been reported to passivate dangling bonds at the Si-SiO2 interface [185, 186]. Such passivation would 73

increase nanowire conductivity, as consistent with the measurements shown in Figure 5.5a.

Our EOS measurements indicate that segment length is better controlled than the actual nanowire conductivity. In particular, the variability in the value of Lundoped/Ltotal is approximately 10%, in contrast to the two-orders-of-magnitude variability for the conductivity of the undoped segment (Figures 5.4a and 5.5). The large conductivity variability, which occurs despite the fact that the nanowires were grown in the same reactor run, is consistent with our prior studies of undoped, axially homogeneous Si nanowires [80, 63]. The within-batch variability may be attributed to nanowire-to- nanowire differences in surface trap density. Recent pump-probe experiments on similar VLS-grown Si nanowires show that surface trap densities can vary by as much as two orders-of-magnitude between individual nanowires from the same batch, but only by a factor of 2-3 within a given nanowire [187]. Such nanowire-to-nanowire differences in surface-trap densities could lead to large variations in effective conductivity when the number of traps is significant compared to the number of carriers. The reduced varia-

tion in Lundoped/Ltotal, which is a result of the capability for rapid precursor switching in modern chemical vapor deposition systems, is also consistent with the experiments of Cating et al. [187]. As long as the intrawire variations in surface traps do not significantly change the spatial distribution of eﬀective carrier density along the nanowire, the nominal segment lengths will not change. We anticipate that the eﬀect of variable surface trap density on the conductivity of the nanowire may be reduced when the number of carriers greatly exceeds the number of surface traps, as would be the case for highly doped nanowires, or when nanowires are well passivated.

The EOS technique can, in principle, be extended to measure more complex nanowires as well as segments with higher conductivity. The ability to determine the structure and conductivity of nanowires with multiple axial or radial heterostructures will ultimately depend on the frequency resolution required to observe increasingly ﬁne features in an EO spectrum. For instance, an N-segment axially modulated nanowire would require the accurate determination of N crossover frequencies and N − 1 plateaus in the EO spectrum, which would become challenging for highly complex nanowires (Appendix 74

J). A conductivity gradient at the interface between two differently doped sections could also in principle be detected and estimated, if sufficient experimental resolution is available in the measurements of the EO spectrum (Appendix J). However, we do not believe that the present measurements are affected by possible carrier-density gradients at the interface between the two segments. Since the axial length of any gradient would be no more than the diameter of the nanowire (50 nm) [167], any changes to the EO spectrum would be imperceptible with our current setup (Appendix J). Nanowires of higher conductivities can be measured by utilizing a suspending fluid of higher dielectric constant and/or higher-frequency electronics. For instance, dipropylene glycol (DPG), with a dielectric constant 10× greater than that of mineral oil, can increase the maximum EOS-measurable conductivity by shifting the spectra to lower, more easily measurable frequencies. Unfortunately, the higher fluid conductivities associated with fluids like DPG can also induce ion-double-layer shielding that affects orientation at low frequencies; this can adversely affect the lower bound of the measurement range [63, 188]. At present, with a combination of DPG and mineral oil to cover different frequency regimes, the overall measurement range of EOS is approximately 7 orders of magnitude, 10−5 to 102 S/m, corresponding to effective bulk Si carrier densities of 1010 − 1017 cm−3.

5.4 Conclusions

In summary, we have developed and validated an eﬃcient, non-contact technique for quantitatively determining the electrical conductivity and dimensions of two-segment Si nanowires grown with the VLS mechanism. More speciﬁcally, EOS can provide, for individual nanowires, both the conductivity of the undoped segment and the ratio of the lengths of the undoped and doped segments, as well as their statistical distributions. EO spectra for two-segment nanowires are distinct from their axially homogeneous counterparts, exhibiting two crossover frequencies and a plateau in the alignment rate that depends on the conductivity and lengths of the two segments, respectively. Nanowire segment conductivity and dimensions measured via EOS agree with those determined from conventional four-point I − V measurements and growth times, respectively. A 75

non-Gaussian, two-orders-of-magnitude variation in conductivity of the undoped segments is observed whereas the variation in segment-length ratio is only about 10%. EOS is an increasingly powerful tool for the eﬃcient statistical characterization of semiconductor nanowires with structural and functional heterogeneity, and represents a ﬁrst step toward the measurement and separation of complete nanowire devices. 76

Chapter 6

Conclusions and Future Works

6.1 Conclusions

We have measured the acoustic attenuation coefficient of dilute suspensions of nickel microflakes in the presence of an external magnetic field at frequencies in the MHz range. We observe differences up to 30% in attenuation coefficient for field direction parallel and perpendicular to the acoustic wave prorogation direction, with the former case being higher. Additional experiments were designed to explore the origin of this large anisotropy and controllability, in which we measure the suspensions of nickel microspheres and observe much less anisotropy then those of nickel flakes. This indicates that particle chaining is not significant in changing the attenuation of the suspensions compared to particle chaining. This is further validated by the optical measurement using UV-Vis spectroscopy as well as a time-scale analysis on particle alignment and particle chaining. When comparing our measured data to a slightly modified theoretical model developed by Ahuja and Hendee [87], the results are in good qualitative but poor quantitative agreement. A more comprehensive model is very much needed.

We then develop a mathematical model to describe the acoustic properties of microstructured suspensions using effective media theory. We start by modelling the sound speed and attenuation in a homogeneous viscoelastic media, followed by the derivation of effective viscoelastic tensor of a two-phase medium with spheroidal inclusions. We then present the dependence of medium acoustic properties on particle aspect ratio, ultrasonic frequency and particle volume fraction. The theoretical predictions are also compared with the attenuation coefficient measured experimentally in Chapter 2. The difference between the theory and the experiments is believed to be due to the absence of an inertial term in the modeling. 77

Taking advantage of the recently developed high-throughput EOS method, we are able to perform the first statistically significant analysis on the electrical conductivity of Si nanowire with different surface-treatment strategies (HF+ALD, SC1+ALD). A lognormal distribution was identified for the measured conductivity. We also demonstrate that the nanowire conductivities are dominated by the surface states; when surface-trap density is modified with passivation, a 5-orders-of-magnitude increase in conductivity is observed. Additional Al2O3 layer thickness in general increases the nanowire conductivity and decreases the conductivity variation, due to both chemical and field-effect passivation. The HF- and SC1- treated nanowires also have different responses to the varying thickness of Al2O3 layers, with the conductivity of the former more susceptible to ALD passivation. The importance of the nanowire surface to their electrical-transport properties is further illustrated by a series of EOS measurements on Si nanowires with diameters of 50nm, 100nm and 150 nm. Conductivity increases, and variation decreases, as the surface to volume ratio decreases.

We have also developed an extension of the EOS approach to quantitatively determine the electrical conductivity and dimensions of two-segment Si nanowires grown with the VLS mechanism. More specifically, EOS can provide, for individual nanowires, both the conductivity of the undoped segment and the ratio of the lengths of the undoped and doped segments, as well as their statistical distributions. EO spectra for two-segment nanowires are distinct from their axially homogeneous counterparts, exhibiting two crossover frequencies and a plateau in the alignment rate that depends on the conductivity and lengths of the two segments, respectively. Nanowire segmen- t conductivity and dimensions measured via EOS agree with those determined from conventional four-point I − V measurements and electron microscopy, respectively. A two-orders-of-magnitude variation in conductivity of the undoped segments is observed whereas the variation in segment-length ratio is only about 10%. EOS is an increasingly powerful tool for the efficient statistical characterization of semiconductor nanowires with structural and functional heterogeneity, and represents a first step toward the measurement and separation of complete nanowire devices. 78

6.2 Future works

6.2.1 Expanding the EOS measurement range

The current measurement range of EOS method (10−5 S/m to 10 S/m) only covers nominally undoped or lightly doped semiconductor nanowires. It is often the case that semiconductor nanowires with higher conductivity are preferable in many applications. Therefore, it is crucial to increase the current range of EOS method in order to make it a more useful characterization tool. The crossover frequency is given by the following formula,

∼ σp ωcrossover = . (1/Lk − 1)εf

At this point the limitation of the EOS measurement range lies in the available bandwidth of the electronics. It is clear that increasing the dielectric constant of the fluid would shift the crossover frequency of the nanowires that are originally out of range back into our measurable frequency range, enabling us to measure nanowires with higher conductivities and increasing the EOS measuring range. To that end, we have used deionized (DI) water as suspending fluid in our recent EOS measurement, as shown in Figure 6.1. Dielectric constant of DI water is 80, 4 times as high as that of DPG and 40 times as high as that of mineral oil. On the other hand, nanowires can be aligned in water without amplifier due to its low viscosity, such that frequency range can be pushed up to 20 MHz. Combined we can increase the measurement range of EOS by an order of magnitude. However, the electrical double layer (EDL) charging effect, where the EDL is formed on the nanowire surface and effectively shield the nanowire from the external field, dramatically narrows the EOS range in suspending fluid with high conductivity, putting significant limitations on DI water based EOS measurements. Moreover, it has been suspected that DI water can change the conductivity of the suspended Si nanowire by surface passivation. Therefore, better strategies are required to extend the EOS measurement range in order to better serve the characterization need of semiconductor nanowires. 79

Figure 6.1: Conductivity of Si nanowires measured by EOS method in DI water

6.2.2 Radiation damage on conductivity of semiconducting nanowires

We investigate the radiation effects on nanowire conductivity for two reasons. First of all, if we can establish a reliable dependence between ion beam dosage and the change in nanowire conductivity, we can quantitatively control nanowire conductivity on a post-processing base. On the other hand, the radiation effects on nanowires have not been systematic studied to the authors’ best knowledge, it would be intrinsically interesting to understand how radiation affect nano-scaled materials differently than bulk materials. 80

Appendix A

Attenuation measurement of suspensions with no magnetic ﬁeld applied

It is observed in Figure A.1 the measured attenuation of suspensions with random aligned nickel ﬂakes are fairly close to that of the suspensions with perpendicularly aligned particles. It is consistent with our discussions in the paper, where we argue that the theoretical value of sound attenuation of the “perpendicular” alignment case (in which the particle’s long axis is perpendicular, but not necessarily broadside, to the wave-propagation direction) can be found from the relations below [87, 106],

αparl = α0◦ , (A.1) αperp = (α90◦ + α0◦ )/2, where α0◦ and α90◦ are the model predictions of attenuation coeﬃcients when particle’s axis of symmetry is respectively parallel and perpendicular to the wave propagation direction. 81

Figure A.1: Comparison of random and oblate alignment case for attenuation coeﬃcient of nickel-ﬂake suspensions. 82

Appendix B

Volume fraction dependance of the measured attenuation coeﬃcient

The measured attenuation coefficient of suspensions with oblate and spherical nickel micro-particles have been presented here respectively in Figure B.1 and Figure B.2, along with that of the fluid. We can see that the attenuation coefficient increases with the increasing volume fraction for both parallel and perpendicular alignments in both kind of suspensions as we expected. We also noticed that the increment can be considered linear within a reasonable range of error. This is consistent with our argument that the suspensions are in dilute limit where the linear single-particle model is applicable. 83

Figure B.1: Measured attenuation coeﬃcient of suspensions with nickel micro-ﬂakes against volume fraction.

Figure B.2: Measured attenuation coeﬃcient of suspensions with nickel micro-spheres against volume fraction. 84

Appendix C

Scale analysis of the thermal attenuation eﬀect

The thermal attenuation can be estimated by[114],

3ωφ AtρCp 2 αt = [(γ − 1) 1 − 0 0 0 (δt/R)(1 + δt/R)], (C.1) 4c Atρ Cp where ω is the angular frequency of the sound wave, φ is the particle volume fraction of the matrix, c is the sound speed, γ is the adiabatic constant of the matrix, A and Cp are p respectively the coeﬃcient of thermal expansion and the speciﬁc heat, δt = 2σ/ρCpω is the thermal boundary layer thickness with σ being the thermal conductivity of the matrix, and R is the nominal radius of the particles. The thermal contribution is estimated to be in the order of 10−5cm−1, which is 4 order of magnitude smaller than the measured attenuation. So here we conclude that the it’s reasonable to neglect the thermal attenuation under our experimental settings. 85

Appendix D

Scale analysis on the eﬀect of acoustic radiation force

We estimated the acoustic radiation force exserted on the particles by a progressing wave, which can be calculated by [189],

3 2 2 8π Vp E0 c 2 2 Fr = − 4 2 3 − (2 + λρ)λρ 2 + 2(1 − λρ) , (D.1) λ (2 + λρ) c0 where V is the volume of the particle, λ is the wavelength, E0 is the acoustic wave energy

0 density, c and rho are the respectively the sound speed and density, and λρ = ρ/ρ . (The primed quantities denote the matrix properties and the unprimed ones denote the particle properties.)Under our current experimental settings, Fr caused by the acoustic wave pulse is estimated to be at the order of 10−17N, while the magnetic force is calculated to be in the order of 10−9N. Hence we conclude that the forces resulted from acoustic wave is negligible. 86

Appendix E

Elliptic Integrals

For an n-tuple nonnegative integer index (α1, ··· , αn) and a vector x = (x1, ··· , xn), we denote by

|α| α α1 αn α ∂ |α| = α1 + ··· + αn, x = x1 ··· xn ,Dx = α1 αn . ∂x1 ··· ∂xn

Consider elliptic integrals of the form:

Z det(A)kˆα Iα = − dω(kˆ), (E.1) Sn−1 |Akˆ|n where A = diag[a1, a2, ··· , an] is a diagonal matrix. From symmetry, we infer that If |α| = 0, I0 = 1; if any entry in the multi-index α is odd, Iα = 0, and in particular, if

α Pn α+2ςj α |α| is odd, I = 0. If |α| is even , j=1 I = I , where ςj is the multi-index with

|ςj| = 1 and the only nonzero occurs at the jth entry. Consider change of variables:

Akˆ 1 kˆ → kˆ0 = (⇒ |A−1kˆ0| = ). |Akˆ| |Akˆ|

Since the mapping x 7→ Ax maps a cone in a unit sphere to a cone of height |Akˆ|, we have

det Adω(kˆ) = |Akˆ|ndω(k0).

Therefore,

−1 Akˆ α ˆ |α| Z det(A)(A ˆ ) |Ak| Iα = − |Ak| dω(kˆ) Sn−1 |Akˆ|n Z (A−1kˆ0)α = − dω(kˆ0). (E.2) Sn−1 |A−1kˆ0||α| 87

If n = 3 and |α| = 2, Iα has three independent components which are deﬁned as

Z ˆ2 Z ˆ2 det(A)ki ki αi 0 Ii = − dω(kˆ) = − dω(kˆ ) 2 ˆ 3 n−1 ˆ2 ˆ2 ˆ2 S |Ak| S α1k1 + α2k2 + α3k3 Z +∞ a1a2a3 du = 2 , 2 0 (ai + u)∆

2 where αi = 1/ai , the last equation follows from Eshelby [127] and Routhe(1897), and

2 2 2 1/2 ∆ = [(a1 + u)(a2 + u)(a3 + u)] .

If n = 3 and |α| = 4, Iα has six independent components which are deﬁned as

Z ˆ2ˆ2 Z ˆ2ˆ2 det(A)ki kj αiαjki kj Jij = − dω(kˆ) = − dω(kˆ) 2 ˆ 3 n−1 ˆ2 ˆ2 ˆ2 2 S |Ak| S (α1k1 + α2k2 + α3k3)   d −αj dα Ii if i 6= j, = j −α2 d Ii if i = j.  i dαi αi

Therefore,

Z +∞ a1a2a3 udu J12 = 2 2 ; 4 0 (a1 + u)(a2 + u)∆ Z +∞ 3a1a2a3 udu J11 = 2 2 . 4 0 (a1 + u) ∆

By (E.1) we have

J11 + J12 + J13 = I1.

By (E.2) we have

Z ˆ0 2 2 2 2 (k1/a1) ˆ0 a1J11 + a2J12 + a3J13 = − dω(k ) Sn−1 |A−1kˆ0|4

From Mura [137], if a1 > a2 = a3 (i.e., e = a1/a3 > 1),we have

e h i I = I = e(e2 − 1)1/2 − cosh−1 e ,I = 1 − 2I , 2 3 2(e2 − 1)3/2 1 2 e2I − I J = J = 1 2 ,J = I − 2J , (E.3) 12 13 2(e2 − 1) 11 1 12

J22 = 3J23, 4J23 = I2 − J12, 88

if a1 < a2 = a3 (i.e., e = a1/a3 < 1),we have

e h i I = I = cos−1 e − e(1 − e2)1/2 ,I = 1 − 2I , 2 3 2(1 − e2)3/2 1 2 I − e2I J = J = 3 1 ,J = I − 2J , (E.4) 12 13 2(1 − e2) 11 3 12

J22 = J33 = 3J23, 4J23 = I3 − J13,

if a1 = a2 = a3 (i.e., e = a1/a3 = 1),we have

1 I = I = I = , 1 2 3 3 1 J = J = J = , (E.5) 11 22 33 5 1 J = J = J = . 12 23 13 15

On the other hand, calculations of eﬀective viscoelastic tensors frequently involve multiplication and inversion of fourth-order isotropic tensor of form

(C)piqj = µ(δpqδij + δpjδiq) + λδpiδqj, where µ, λ ∈ C are the Lam´econstant in linear elasticity. For any symmetric matrix

n×n E ∈ C sym , we have CE = 2µE + λTr(E)I. Further, it is straightforward to verify that

1 λ (C−1) = δ δ + µαδ δ − δ δ . piqj 4µ pq ij 2 pj iq 2µ(2µ + nλ) pi qj

1 λ In other words, the inverted tensor is isotropic with Lam´econstants given by ( 4µ , − 2µ(2µ+nλ) ) 89

Appendix F

Fabrication of two-segment Si nanowires

Si nanowires with an nominally undoped segment and a doped segment are grown via the gold-assisted vapor-liquid-solid (VLS) mechanism in a low-pressure chemical vapor deposition (CVD) system (First Nano, EasyTube 3000). 3-triethoxysilylpropylamine (APTES) is used to attach 50 nm gold colloid (Ted Pella) to a Si(100) substrate coated with 200 nm of thermal oxide. These substrates are then rinsed in deionized (DI) water, dried with nitrogen, and treated with UV-ozone prior to loading into the CVD reactor. The CVD system consists of a graphite susceptor inside a rectangular quartz chamber, surrounded by infrared lamps that enable heating and cooling rates up to ± 10 ◦C/s. Three K-type thermocouples embedded in diﬀerent sections of the susceptor provide edge-to-edge temperature uniformity better than ± 3 ◦C. Nanowires are grown

◦ at 500 C and 10 Torr, using 50 sccm silane (SiH4) and 600 sccm hydrogen (H2). Phos-

phine (PH3) diluted to 5% in nitrogen (N2) is used for n-type doping. A PH3:SiH4

ratio of 1:200 is used for the doped segments. The Lundoped/Ltotal ratio is synthetically controlled to be 1/3, 1/4 and 1/6 by adjusting growth times with and without phos- phine for n-type doping . A total growth time of 8 min yields nanowires of 10 µm in

length. Homogeneous, nominally undoped Si nanowires (Lundoped/Ltotal = 1 ) are also synthesized using the same growth conditions as the undoped segment for comparison. Hydrogen passivation of the sidewalls of the nanowire during growth prevents sidewall deposition and unwanted radial dopant incorporation.[152, 151] 90

Appendix G

Doping dependent selective etching

Undoped-selective nanowire etching is accomplished with an aqueous solution of 10% potassium hydroxide (KOH) and 23% isopropanol at room temperature for 60 seconds. A 10 second dip in 1:10 buﬀered oxide etchant (J.T. Baker) is used to remove the native oxide prior to KOH treatment. Scanning electron microscopy (SEM) images are taken with a Zeiss Ultra60 microscope. 91

Appendix H

EOS measurements on two-segment Si nanowires

The nanowires are dispersed in mineral oil (Drakeol 7 LT Mineral Oil, Calumet Specialty Products and Partners, L.P.) using a bath sonicator (Fisher Scientiﬁc - FS60 Ultrasonic

Cleaner). To ensure that the nanowires do not break (and thus have variable LI /Ltotal ratio) as a result of the sonication process, only nanowires which remained 10 µm in length after dispersal are studied. A spatially uniform AC field is generated on a microscope slide with wire electrodes driven by a function generator (Tektronix AFG3200C) and a high-frequency amplifier (TREK 2100HF), as shown in Figure 4.2 from Chapter 4. The spacings between the electrodes are 750 µm and the applied AC voltage is set be be 75 V rms. Field-induced nanowire rotation is captured by a high-speed monochrome CCD camera (pco.edge sCMOS, PCO AG) connected to an inverted optical microscope (Olympus IX71, Olympus Corp.) with a 40× objective lens (Olympus LUCPLFN 40x, N.A. 0.6, Olympus Corp). To obtain EO spectra, the alignment rate of individual nanowires is measured under an external AC electric field at a series of frequencies between 600 ≤ ω ≤ 2 × 107 rad/s. The maximum alignment rates (45◦ relative to the field direction) of individual nanowires at different AC frequencies are extracted from the recorded video using an image-analysis routine written in MATLAB[80, 63]. 92

Appendix I

Numerical simulations of two-segment Si nanowires

The normalized electro-orientation (EO) alignment rate spectra of semiconductor nanowires can be closely approximated by calculating the effective dipole moment[110], Z − ~peff = (εp − εf )E~ , (I.1) V where ε is the complex permittivity of the particle and E~ − represents the electric field inside the particle. p and f denote the properties of the particle and the fluid, respectively. The above equation was calculated in Comsol Multiphysics (version 4.4) via a 2D axisymmetric finite-element simulation, as shown in Figure I.1. The properties of the particle (Si nanowire) and the suspending fluid (mineral oil) used in simulation are shown in Table I.1. The simulation was first tested by comparing the results for homogeneous particles with the analytical result given by Jones[110]. Good agreement is observed, as seen in Figure I.2.

Table I.1: Physical properties used in the numerical calculation

Particle relative permeability 11.7 Fluid conductivity 10−10 S/m Fluid relative permeability 2.1

In order to obtain material properties from the experimental data, we simulate the EO spectra of two-segment nanowires with the same overall dimensions as the fabricated ones (10 µm length and 50 nm diameter) at the frequencies measured in the

−3 experiments. The conductivity of the undoped part(σundoped) is swept from 10 S/m

−1 0.02 to 10 S/m with a log scaled 10 S/m step, while that of the doped part(σdoped) is set to be 100 S/m since it does not aﬀect the EO spectra of the low frequency region. The

Lundoped/Ltotal ratio is varied from 0.05 to 0.4 with a 0.01 increment. The measured EO spectra of each nanowires are least-square-ﬁtted to all the simulation data to extract the 93

Figure I.1: Simulation set up in Comsol Multiphysics. Nanowire diameter and length are set to be respectively 50 nm and 10 µm to match our experiments.

conductivity of the undoped part(σundoped) and the Lundoped/Ltotal ratios. An example is shown in Figure I.3. 94

Figure I.2: Validation of the numerical method. The nanowires are assumed to be prolate spheroidal particles with semi-major and semi-minor axis being 5 µm and 25 nm. The calculated EO spectra of the homogeneous particles, with conductivities of 1.56−3 S/m or 10 S/m, were compared with the corresponding analytical solutions. 95

Figure I.3: An example of the best-ﬁtting simulated EO spectrum and the corresponding experimental data for a two-segment nanowire with LI /(LI + LD) = 0.31 and σI = 0.0038 S/m. 96

Appendix J

Potential EOS applications in more complex nanowires

The EO spectrum of a three-segment nanowire can be obtained from similar numerical simulations described above, as shown in Figure J.1. Two plateaus and three crossover frequencies are observed in the EO spectrum.

We have also simulated the EO spectra of nanowires with a conductivity gradient at the interface between two differently doped segments, as shown in Figure J.2. The EO spectra are different between nanowires having a sharp interface, and nanowires having 1 µm- and 0.2 µm- wide interfaces. Thus, in principle, it should be possible to detect and estimate the conductivity gradient at an interface between two differently doped sections. In practice, however, detection of conductivity gradients at interfaces is likely to be experimentally challenging, and would require substantial improvements in the frequency range and resolution of the EOS measurements. For the two-segment nanowires studied in this paper, we do not believe that the measurements are affected by possible interfaces for the following two reasons. Firstly, the length of these interfaces should be of the same order as the diameter of the nanowire [167], thus the changes in the EO spectra due to the gradients are very limited, as seen in Figure J.2. Furthermore, the resulting changes in the EO spectrum are mainly in the vicinity of the second crossover frequency, which is out of our present experimental frequency range, and not used to extract the length ratio of the doped and undoped segments, nor the conductivity of the undoped segment. 97

Figure J.1: The simulated EO spectrum of a three-segment nanowire.

Figure J.2: The simulated EO spectra of two-segment nanowires with conductivity gradients at the interface. 98

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