UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Interaction of electron beams with carbon nanotubes Alam, Md. Kawsar 2011

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
24-ubc_2011_fall_alam_md.pdf [ 3.14MB ]
Metadata
JSON: 24-1.0072019.json
JSON-LD: 24-1.0072019-ld.json
RDF/XML (Pretty): 24-1.0072019-rdf.xml
RDF/JSON: 24-1.0072019-rdf.json
Turtle: 24-1.0072019-turtle.txt
N-Triples: 24-1.0072019-rdf-ntriples.txt
Original Record: 24-1.0072019-source.json
Full Text
24-1.0072019-fulltext.txt
Citation
24-1.0072019.ris

Full Text

  Interaction of Electron Beams with Carbon Nanotubes by Md. Kawsar Alam  B.Sc., Bangladesh University of Engineering and Technology, 2005 M.Sc., Bangladesh University of Engineering and Technology, 2007   A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Electrical and Computer Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) August 2011  © Md. Kawsar Alam, 2011  ii  Abstract Carbon nanotubes have great potential for nanoscale devices. Previous studies have shown the prospects of carbon nanotubes as stable, low-voltage electron emitters for vacuum electronic applications. Yet, their electron emission mechanisms are far from being fully understood. For example, it is not completely clear how nanotubes interact with an external electron beam and generate secondary electrons. In addition to its fundamental scientific importance, understanding these mechanisms and properties will facilitate the engineering of nanotube-based devices for applications such as vacuum transistors, electron multipliers, X- ray devices for medical imaging, etc. This thesis presents an experimental and theoretical investigation into the interaction of electron beams with carbon nanotubes. First-principles simulations are carried out to qualitatively analyze the possible direct interaction mechanisms of electron beams with nanotubes. An experimental study of electron yield (total, backscattered and secondary electron yields) from individual nanotubes and collections of nanotubes is reported. The experiments reveal low secondary electron yield from individual nanotubes. A different backscattered electron emission behaviour compared to that in bulk materials is observed in nanotube forests due to unusually high electron penetration range in them. A semi-empirical Monte Carlo model for the interaction of electron beams with collections of nanotubes is presented. Physically-based empirical parameters are derived from the experimental data. The secondary electron yield from individual nanotubes is first investigated in the light of the commonly used energy loss model for solids. Finally, the problems of using the traditional models for individual nanotubes are identified and an approach to modeling secondary and backscattered electron emission from such nanostructures is presented. The experiments and analysis presented in this thesis provide a platform for investigating backscattered and secondary electron emission also from nanostructures other than nanotubes. iii  Preface The following is a list of the papers published based on the contributions of this thesis: Journal Papers 1. M. K. Alam, S. P. Eslami, and A. Nojeh, “Secondary Electron Emission from Single- Walled Carbon Nanotubes,” Physica E, vol. 42, pp. 124-131, 2009. (Part of Chapter 3) 2. M. K. Alam, P. Yaghoobi, and A. Nojeh, “Unusual Secondary Electron Emission Behavior in Carbon Nanotube Forests,” Scanning, vol. 31, pp. 221-228, 2009. (Part of Chapter 4) 3. M. K. Alam, P. Yaghoobi, and A. Nojeh, “Monte Carlo Modeling of Electron Backscattering from Carbon Nanotube Forests,” Journal of Vacuum Science and Technology B, vol. 28, pp. C6J13-C6J18, 2010. (Part of Chapter 5) 4. M. K. Alam, P. Yaghoobi, M. Chang, and A. Nojeh., “Secondary Electron Yield of Multiwalled Carbon Nanotubes,” Applied Physics Letters, vol. 97, pp. 261902-1– 261902-3, 2010. (Part of Chapter 6) 5. M. K. Alam, R. F. W. Pease, and A. Nojeh, “Comment on “Ultrahigh Secondary Electron Emission of Carbon Nanotubes” [Appl. Phys. Lett. 96, 213113 (2010)],” Applied Physics Letters. vol. 98, pp. 066101-1–066101-1, 2011. (Part of Chapter 6) 6. M. K. Alam and A. Nojeh, “Monte Carlo Simulation of Electron Scattering and Secondary Electron Emission in Individual Multiwalled Carbon Nanotubes: A Discrete- Energy-Loss Approach,” Journal of Vacuum Science and Technology B, vol. 29, pp. 041803-1–041803-7, 2011. (Part of Chapter 7)  Conference Papers 1. M. K. Alam and A. Nojeh, “Monte Carlo Modeling of Electron Backscattering from Carbon Nanotube Forests,” 54th International Conference on Electron, Ion and Photon  Preface iv  Beam Technology and Nanofabrication, Anchorage, Alaska, USA, June 1-4, 2010. (Part of Chapter 5)  In addition, in writing Chapters 1 and 2, usage (with occasional exact quotation) has been made of my above publications. The necessary permissions have been obtained from the publishers. Here, I clarify my contributions to these publications. I was the principal researcher and author for all of the publications. In all cases, I performed the literature review, identified research problems, developed research ideas and wrote all the codes. I performed all the first-principles simulations (except for the data given in Tables 3.1 and 3.2 and Figure 3.13) and Monte Carlo simulations presented in this thesis. I also prepared all the experimental setups for electron yield measurements, conducted all the experiments, and analyzed the data. All the manuscripts were co-authored by my supervisor, Prof. Alireza Nojeh, who closely supervised the entire project. He provided the general research problem and ideas and continuous input/feedback on each aspect of the work, and assisted in the writing of the manuscripts. S. P. Eslami was involved in one paper (related to Chapter 3). He provided the data given in Tables 3.1 and 3.2 and Figure 3.13. P. Yaghoobi was involved in three journal papers (related to Chapters 4, 5 and 6). He fabricated the carbon nanotube forests for those works (the steps related to Section 4.1). M. Chang performed the chemical vapour deposition for a nanotube forest in the work related to Chapter 6. Prof. R. F. W. Pease participated in writing the comment reported to Applied Physics Letters (related to Chapter 6).  v  Table of Contents Abstract .................................................................................................................................. ii Preface ................................................................................................................................... iii Table of Contents ................................................................................................................... v List of Tables ....................................................................................................................... viii List of Figures ....................................................................................................................... ix Acknowledgements ............................................................................................................. xvi Dedication ........................................................................................................................... xvii Chapter 1  Introduction ........................................................................................................ 1 1.1 Carbon Nanotubes ......................................................................................................... 2 1.2 Literature Review .......................................................................................................... 4 1.3 Research Objectives ....................................................................................................... 6 1.4 Methodology .................................................................................................................. 7 1.5 Thesis Overview ............................................................................................................ 9 Chapter 2  Theoretical Overview ....................................................................................... 10 2.1 Monte Carlo Simulation .............................................................................................. 10 2.2 Elastic Scattering ......................................................................................................... 11 2.3 Inelastic Scattering and Energy Loss ........................................................................... 13 2.4 First-Principles Simulations......................................................................................... 15 2.4.1 The Hartree-Fock Method .................................................................................... 16 Chapter 3  Secondary Electron Emission from Carbon Nanotubes ............................... 20 3.1 The Model .................................................................................................................... 21 3.2 Estimation of the Ionization Energy and Interaction Depth ........................................ 22 3.3 Results and Discussion ................................................................................................ 25 3.3.1 The Non-Relaxed (5,5) SWNT ............................................................................. 26 3.3.2 The Relaxed (5,5) SWNT ..................................................................................... 36  Table of Contents vi  3.3.3 The (8,0) SWNT ................................................................................................... 37 3.4 Summary ...................................................................................................................... 37 Chapter 4  Electron Yield of Carbon Nanotube Forests .................................................. 38 4.1 Carbon Nanotube Forest Fabrication ........................................................................... 38 4.2 Experimental Procedure ............................................................................................... 39 4.3 Results and Discussion ................................................................................................ 44 4.4 Summary ...................................................................................................................... 52 Chapter 5  Modeling of Electron Yield from Carbon Nanotube Forests ....................... 53 5.1 The Model .................................................................................................................... 54 5.2 Results and Discussion ................................................................................................ 59 5.2.1 Secondary Electron Yield ..................................................................................... 65 5.3 Summary ...................................................................................................................... 69 Chapter 6  Electron Yield of Suspended Nanotubes ........................................................ 70 6.1 Fabrication of Suspended Nanotubes .......................................................................... 70 6.2 Experimental Setup for Electron Yield Measurement ................................................. 71 6.3 Results and Discussion ................................................................................................ 73 6.4 Comment on “Ultrahigh Secondary Electron Emission of Carbon Nanotubes” [Appl. Phys. Lett. 96, 213113 (2010)] .......................................................................................... 77 6.5 Summary ...................................................................................................................... 80 Chapter 7  Discrete Monte Carlo Simulation.................................................................... 81 7.1 The Model .................................................................................................................... 82 7.1.1 Scattering Cross-Sections and Mean Free Paths .................................................. 83 7.1.2 Energy Loss .......................................................................................................... 85 7.1.3 Scattering Angle Distributions ............................................................................. 87 7.1.4 Monte Carlo Procedure ......................................................................................... 88  Table of Contents vii  7.2 Secondary Electron Model for MWNTs ...................................................................... 90 7.3 Results and Discussion ................................................................................................ 92 7.4 Summary ...................................................................................................................... 98 Chapter 8  Conclusion ......................................................................................................... 99 8.1 Contributions ............................................................................................................... 99 8.2 Future Work ............................................................................................................... 101 Bibliography ....................................................................................................................... 102 Appendix A  Scripts for First-Principles Simulations .................................................... 110 Appendix B  Empirical Fit Routine Used in Chapter 5 ................................................. 122  viii  List of Tables Table 1.1 Comparison of the Monte Carlo simulators   ......................................................... 6 Table 3.1 Distance (in Å) between different atoms. The indices correspond to the atom numbers in the coordinate file of the simulation.   ............................................... 25 Table 3.2 Energy (in Hartrees) of HOMO and LUMO.   ..................................................... 25 Table 5.1 List of parameters for the Monte Carlo simulations.   .......................................... 63   ix  List of Figures Figure 1.1 Electron beam-specimen interaction showing the primary electron trajectory, backscattered and secondary electrons.   ................................................................ 1 Figure 1.2 Illustration of the relation between a graphene sheet and a nanotube (a) A graphene sheet and (b) 5 unit cells of a (4,4) nanotube.   ....................................... 3 Figure 1.3 An electron passing through a nanotube.   ............................................................. 4 Figure 2.1 100 electron trajectories in bulk graphite at 5 keV simulated using Casino [38].   ............................................................................................................................ 11 Figure 2.2 Schematic representation of the concept of differential elastic scattering cross- section.   ................................................................................................................ 12 Figure 2.3 A typical molecular system (i, j are electrons and A, B are atoms).   .................. 15 Figure 3.1 (a) (5,5) SWNT with various positions of the extra electron. In each simulation, the electron is placed in one of these positions. They are located at the centre, 1 Å away from the centre, 2 Å away from the centre, 5 Å away from the centre and 10 Å away from the centre. Once the electron is located 5Å or more away from the centre, it is outside of the nanotube. (b) (8,0) SWNT with the same electron positions as in (a).   ................................................................................. 23 Figure 3.2 (a) Mulliken charge distribution (scale given in atomic units) and (b) HOMO of (5,5) SWNT in the absence of external electron.   ............................................... 26 Figure 3.3 Directions along which potential profiles are plotted: 1x - in the positive horizontal direction through the center of the nanotube; 2x - in the positive horizontal direction two rings below the centre; 1y - into the page through the center of the nanotube; 2y - into the page two rings below the center. The position of the extra electron was moved along the 1x  direction.   ...................... 27 Figure 3.4 Energy levels and potential profiles of the (5,5) SWNT along the chosen directions without any external electron. Horizontal axis indicates distance from the nanotube axis.   ............................................................................................... 28  List of Figures x  Figure 3.5 Energy levels and potential profiles of the (5,5) SWNT along the chosen directions with external electron at the center of the nanotube. Horizontal axis indicates distance from the nanotube axis.   ......................................................... 28 Figure 3.6 Energy levels and potential profiles of the (5,5) SWNT in the 1x  direction for various positions of the external electron. Horizontal axis indicates distance from the nanotube axis.   ...................................................................................... 29 Figure 3.7 Energy levels and potential profiles of the (5,5) SWNT in the 2x  direction for various positions of the external electron. Horizontal axis indicates distance from the nanotube axis.   ...................................................................................... 30 Figure 3.8 Energy levels and potential profiles of the (5,5) SWNT in the 1y  direction for various positions of the external electron. Horizontal axis indicates distance from the nanotube axis.   ...................................................................................... 30 Figure 3.9 Energy levels and potential profiles of the (5,5) SWNT in the 2y  direction for various positions of the external electron. Horizontal axis indicates distance from the nanotube axis.   ...................................................................................... 31 Figure 3.10 Electron density profiles (in atomic units) in the 1x  direction for various positions of the external electron. Horizontal axis indicates distance from the nanotube axis.   ..................................................................................................... 33 Figure 3.11 Estimated average energy loss for different primary beam energies. Inset shows the primary beam energy limits (dotted lines) for ~5 eV and ~3 eV energy loss.   ............................................................................................................................ 33 Figure 3.12 Potential profile in the 1x  direction showing the possible tunneling event for a nanotube electron when the external electron is at 1 Å from the center. Horizontal axis indicates distance from the nanotube axis.  ................................ 35 Figure 3.13 Energy levels and potential profile of the (5,5) SWNT in the 1x  direction showing the effect of relaxation with an external electron at 2 Å from the center. Horizontal axis indicates distance from the nanotube axis.  ................................ 36 Figure 4.1 (a and b) Scanning electron micrographs of patterned nanotube forests. In (a) forests with a diameter of 500 µm and lengths of approximately 1 mm are  List of Figures xi  shown. (b) shows a zoomed-in view of a 50-µm-diameter forest (field of view = 51.3 µm). (c) Transmission electron micrograph of one individual nanotube extracted from the forest, revealing more detail about the nature of the grown nanotubes (mutiwalled with diameters of the order of 10 nm). ......................... 39 Figure 4.2 Schematic of the experimental setup in the SEM (arrows show the assumed direction of current for which formulas are given in the calculations). The specimen (forest) is connected to the Keithley 6517A electrometer (internal voltage source and current meter are in series) using vacuum feed-throughs. The body of the SEM column and the chamber wall are grounded.   ......................... 40 Figure 4.3 TEY from a carbon nanotube forest with a diameter of 500 µm. Legends show the corresponding applied voltages to the specimen. The dotted curve (diamond marker) represents the backscattering coefficient. The vertical line indicates that we had to exchange the anode in-between the low-keV and high-keV experiments. The 5 keV data points were measured in both experiments. They were very close (difference of less than 3%) and here the values obtained with the low-keV anode are plotted for the 5-keV point.   ........................................... 44 Figure 4.4 SE coefficient of the nanotube forest with a diameter of 500 µm (obtained from Figure 4.3 by calculating the difference between the curves of TEY at –50 V and +50 V).   ................................................................................................................ 45 Figure 4.5 Monte Carlo simulations showing the electron trajectories for comparison of electron range between a solid material and nanotube forest at 5 keV. Among the 10000 simulated trajectories, the first 50 are shown. (a) Top view of the trajectories in the solid graphite cylinder of 1.4 µm, (b) The same as (a) but for a cluster of nano graphite cylinders with a diameter of 20 nm each, having an overall cluster diameter of 1.4 µm, representing the nanotube forest, (c) Side view of the trajectories in solid graphite, and (d) Side view of the trajectories in the nanotube forest. (Red lines in (a) and  (b) and blue lines in (c) and (d) indicate that electrons are moving in empty spaces and the straight lines at the end of the trajectories show the flight of the backscattered electrons.)   .............. 47  List of Figures xii  Figure 4.6 TEY from a carbon nanotube forest with a diameter of 50 µm. Legends show the corresponding applied voltages to the specimen. The dotted curve (diamond marker) represents the backscattering coefficient. The 5-keV data points were measured in both low-keV and high-keV experiments. They were very close (within 0.2%) and here the values obtained with the low-keV anode are plotted for the 5-keV point.   ............................................................................................ 49 Figure 4.7 SE coefficient of the nanotube forest with a diameter of 50 µm (obtained from Figure 4.6 - the difference between the curves of TEY at –50 V and +50 V).   .. 50 Figure 4.8 A comparison of SE coefficient between nanotube forests and other forms of carbon and several other materials. All the data have been taken from Joy’s database [40,72] and references therein except for the DLC film [76] and the nanotube forest (this study).   ............................................................................... 51 Figure 5.1 Electron trajectories in a typical Monte Carlo simulation, showing the randomness of beam-specimen interaction and (b) one small step in a single trajectory (s = step length, θ = scattering angle and ϕ = azimuthal angle).   ........ 55 Figure 5.2 Micrograph and schematic of a circularly patterned nanotube forest and (b) top view of the nanotube grid used in Monte Carlo simulations. (Nanotube diameter and inter-nanotube spacing have been exaggerated in the schematics for clarity)   ............................................................................................................................ 56 Figure 5.3 Typical distribution of the scattering angle (for 10000 scattering events with 100 energy bins) and change in the distribution with the change of screening parameter.   ........................................................................................................... 58 Figure 5.4 Schematic representation of the nanotube grid (R-1 = hollow region inside the shell; R-2 = in the shell; R-3 = outside the shell) and proposed change in scattering angle distribution for the screened part of the fitted differential cross- section in a particular step of the simulation, where the scattering site falls within a nanotube shell (R-2). ei, ef and  ec are the directions of the incident electron, center of the scattering angle distribution for the bulk model and the shifted center of the scattering angle distribution for the forest, respectively.   ... 58  List of Figures xiii  Figure 5.5 Comparison of experimental and simulation results for the electron backscattering coefficient from 500-μm-diameter nanotube forest using the adjusted bulk model.   ........................................................................................... 60 Figure 5.6 100 electron trajectories (primary energy: 20 keV) through a 500-μm-diameter nanotube forest, obtained using the described Monte Carlo program.   ............... 61 Figure 5.7 Comparison of experimental and simulation results for the electron backscattering coefficient from the 500-μm-diameter nanotube forest.   ............. 62 Figure 5.8 Comparison of experimental and simulation results for the electron backscattering coefficient from the 200-μm-diameter nanotube forest.   ............. 62 Figure 5.9 Comparison of experimental and simulation results for the electron backscattering coefficient from 200-μm-diameter nanotube forest using the adjusted bulk model.   ........................................................................................... 64 Figure 5.10 Comparison of the proposed and adjusted bulk model for the predicted electron backscattering coefficient from the 500-μm-diameter nanotube forest using the extracted empirical parameters. Angles with the axes for oblique incident are: X=90̊, Y= 159.5̊ and –Z=69.5̊.   ............................................................................ 65 Figure 5.11 The semi-empirically modeled SE emission data and the experimental data for the 500-μm-diameter nanotube forest ( SEε =32 eV, ρ=0.019 g/cm 3 , λ1=300 nm and λ2=300 nm).   .................................................................................................. 67 Figure 5.12 The semi-empirically modeled SE emission data and the experimental data for the 200-μm-diameter nanotube forest ( SEε =32 eV, ρ=0.035 g/cm 3 , λ1=200 nm and λ2=75 nm).   .................................................................................................... 68 Figure 6.1 (a) Schematic diagram of the experimental setup for extracting nanotubes from the forests, (b) micrograph of the nanotube forests and (c) extracted nanotubes on tungsten tips.   .................................................................................................. 71 Figure 6.2 Schematic of the experimental setup for electron yield measurement in the SEM. IP, IT, ITE and VT are the same as those defined in Figure 4.2.   ............................ 72  List of Figures xiv  Figure 6.3 Specimen current at 1 keV as the beam was moved across the nanotubes [along the arrow indicated on Figure 6.1(c)]. The horizontal axis indicates the recorded time as the beam was moved along the arrow.   ................................................... 73 Figure 6.4 Electron yield measured from the suspended nanotubes. Fluctuations in the beam current or specimen current are included in error bars.   ............................ 74 Figure 6.5 Comparison of the experimental data with the theoretically estimated SE yield. Inset shows the behaviour of electron energy loss.   ............................................ 76 Figure 7.1 MFPs for the different scattering processes and the effective MFP as a function of electron energy.   .............................................................................................. 85 Figure 7.2 A comparison of the equivalent energy loss rate calculated from discrete processes with the CSDA energy loss. Inset shows the energy loss rate on a log- log scale.   ............................................................................................................. 87 Figure 7.3 Probability of different scattering processes at different primary beam energies.   ............................................................................................................................ 90 Figure 7.4 Electron trajectories in the MWNT for a 0.5-keV beam perpendicular to the nanotubes (No. of walls = 6). Only 70 trajectories are plotted for clarity.   ......... 92 Figure 7.5 Electron trajectories in the MWNT for a 5-keV beam (No. of walls = 6). Only 70 trajectories are plotted for clarity. (a) Beam perpendicular to the nanotube and (b) beam oblique to the nanotube (angles with the axes: X=78°, Y=90° and – Z=168°).   .............................................................................................................. 93 Figure 7.6 Simulated SE coefficients for different MWNTs, together with the experimental data presented in Chapter 6.   ............................................................................... 94 Figure 7.7 Simulated SE coefficients showing the effect of incorporating core-shell excitations.   .......................................................................................................... 95 Figure 7.8 Energy distribution of the transmitted electrons: (a) primary beam energy = 0.5 keV and (b) primary beam energy = 5 keV.   ....................................................... 95 Figure 7.9 Energy loss distribution of the electrons (primary beam energy = 5 keV).   ....... 96 Figure 7.10 Simulated SE coefficients for a MWNT, with and without corrections for the lower energy plasmon and elastic scatterings, and comparison with the experimental data presented in Chapter 6.   ......................................................... 97  List of Figures xv  Figure B.1 Flow diagram of simulation procedure.   ............................................................ 123 Figure B.2 (a) Effect of changing the screening parameter, (b) Effect of changing the effective nanotube thickness and (c) Effect of changing the density.   .............. 124 xvi  Acknowledgements I am very grateful to my supervisor, Prof. Alireza Nojeh, for giving me the opportunity to be a part of his wonderful research group. I would like to express my sincere gratitude for his unquestionable support during the past few years. He always directed me toward the right course of my research. His enthusiasm, dedication and friendly attitude provided the perfect environment for me to achieve my research goals. I have learnt a lot from his unique research management and teaching methods and hope to learn much more in the future. I am indebted to Parham Yaghoobi for his continuous support throughout my Ph.D. program. I would also like to thank Seyed Payam Eslami for the first tour of the program Gaussian and Mike Chang for his part in my research. My deep appreciation goes to my colleagues in the Nojeh Nanostructure Group who always listened to me and gave useful feedback on my research. I am also thankful to Mary Fletcher for her support while working in the Materials Science and Engineering electron microscope laboratory. Sincere thanks go to the members of my doctoral qualifying and final examination committees: Prof. Andre Ivanov, Prof. Edmond Cretu, Prof. Vijay K. Bhargava, Prof. Shahriar Mirabbasi, Prof. Konrad Walus, Prof. Kenichi Takahata, Prof. Joshua A. Folk, Prof. Chad W. Sinclair, and Prof. Jeff F. Young. I would also like to acknowledge the financial support I received through a UBC University Graduate Fellowship (Four-Year Doctoral Fellowship) and the research assistantship which made this study possible. All the first-principles simulations and some of the Monte Carlo simulations were performed on the Compute Canada/WestGrid high speed computing facility. Special thanks to Waqas Khalid, Imtiaz Ahmed, Shankhanaad Mallick and Toufiqul Islam who made my days more exciting in Vancouver. Last but not least, I thank my family members for their love and support from thousands of miles away.  xvii  Dedication  To my family…        1  Chapter 1 Introduction The interaction of an electron beam with a material is complex and involves multiple phenomena and mechanisms. A high energy electron (known as primary electron) entering a specimen is scattered by both nuclei and other electrons inside the specimen. Interaction with the nuclei is elastic in nature for non-relativistic energies, in which the energy transfer is negligible due to the large difference of mass between the electron and the nucleus; it only changes the direction of movement of the primary electron. The collision of a primary electron with other electrons in the specimen involves inelastic scattering, in which the primary electron loses energy and causes phenomena such as plasmon excitation, excitation of valence electrons or inner-shell ionization. Elastic and inelastic scattering processes result in zig-zag-like trajectories of the primary electron in the specimen until it comes to rest by gradual deceleration or leaves the sample. The deflection angle at each step can be large and the primary electron may escape from the specimen after one or more elastic scattering events. The primary electron may also escape from the other side of the specimen if it has high energy or the specimen is thin. In addition, some excited electrons generated through the inelastic scattering events may carry enough energy to overcome the surface potential barrier and escape the specimen (Figure 1.1).   Backscattered Electron (E > 50 eV) Primary Electron Secondary Electron (E ≤ 50 eV)   Figure 1.1 Electron beam-specimen interaction showing the primary electron trajectory, backscattered and secondary electrons.  Chapter 1. Introduction 2  Traditionally [ISO 18115:2001/5.49], electrons exiting with a kinetic energy larger than 50 eV are called backscattered electrons (BSE), while those with an energy less than or equal to 50 eV are called secondary electrons (SE). The study and measurement of electron emission from solids have long been of interest for various reasons such as understanding and modeling different aspects of electron beam- specimen interaction [1-7], as well as building new devices and equipment like vacuum tubes, electron multipliers, micro-channel plates, electron microscopes [8, 9] and crossed- field devices (e.g. magnetrons and crossed-field amplifiers used in radar systems) [10]. Electron emission is intensively studied in electron microscopy and electron probe analysis, electron-beam lithography and Auger spectroscopy [11]. The development and improvement of new diagnostic tools for nanoelectronics with feature sizes in the nanometre scale also require knowledge of the spatial distribution of electrons as a result of interaction with a given material, their energy loss and the electron emission properties. For instance, scanning electron microscopy continues to play a vital role in many disciplines, both as an integral part in research and development and as a metrology tool in fabrication lines, due to its several advantages: ease of sample preparation, simplicity of operation, intuitive image interpretation and high spatial resolution approaching the atomic scale [12]. Study of BSE and SE emission from nanostructures is very significant in all of the above contexts. 1.1 Carbon Nanotubes Carbon nanotubes are hollow cylinders made of carbon with a diameter in the nanometre scale. They represent one of the stable structures that carbon can adopt in the solid state. Every lattice point of a graphene1 sheet can be represented by multiples of two unit vectors 1a  and 2a   (Figure 1.2). Using similar multiples, the chiral vector of a nanotube can be defined:  1 Individual layers of graphite are known as graphene.  Chapter 1. Introduction 3   1 2hC na ma= +     (1.1) Consequently, every nanotube can be characterized by two indices (n,m). Within a good approximation, a single-walled carbon nanotube (SWNT) is metallic if the value n–m is divisible by 3; otherwise, the nanotube is semiconducting. Carbon nanotubes have been shown to enable robust, stable, low-voltage and high-brightness electron emitters [13-17], as well as high electron gain (ratio of the number of electrons emitted to the number of primary electrons) [18-21]. SE emission plays an important role in the imaging of nanotubes [13, 14, 22-29]. Understanding the interaction between a primary electron beam and nanotubes is crucial in designing electron emission devices, imaging/characterizing nanotube-based devices or calculating their X-ray spectra for biomedical applications such as diagnostic radiology, mammography, etc [30]. Given the nanoscale and hollow structure of carbon nanotubes, their interaction with electron beams and imaging mechanisms in electron microscopy [22, 23, 26, 31] are expected to be quite different than those of bulk materials. An electron beam passing through a nanotube (Figure 1.3) is likely to experience a very  (a) (b) hC   (4,3) armchair (n=m) zigzag (n≠m) 2a   1a    Figure 1.2 Illustration of the relation between a graphene sheet and a nanotube (a) A graphene sheet and (b) 5 unit cells of a (4,4) nanotube.   Chapter 1. Introduction 4  small interaction area to excite SEs. Still, nanotubes are readily visible in the scanning electron microscope (SEM).  1.2 Literature Review To the best of our knowledge, there are few reports in the literature related to BSE and SE emission from carbon nanotubes. Mostly, experimental works on the imaging of nanotubes in scanning electron microscopy have been reported. Brintlinger et al. demonstrated electron microscopy of nanotubes lying on insulating substrates [22]. They assumed that the direct interaction between the nanotube and primary electrons was negligible and that nanotubes could acquire charge only through the insulating substrate. They believed imaging was possible because of the dynamic voltage contrast between the nanotubes and the substrate. Nojeh et al. reported direct interaction of primary electrons with the tip of a nanotube [25, 26]. Finnie et al. reported voltage-contrast imaging of suspended nanotubes, assuming that the nanotubes can be positively or negatively charged through direct SE emission from them [31]. Homma et al. observed another mechanism for SE imaging of nanotubes on insulators. They found that the electron-beam-induced current on the SiO2 surface around nanotubes creates a discharge path and modifies SE emission from the SiO2, thus providing a contrast  e-  Figure 1.3 An electron passing through a nanotube.   Chapter 1. Introduction 5  mechanism for imaging [23]. Wong et al. showed that the contrast and brightness of SEM images of SWNTs depend strongly on many factors including primary beam landing energy, history of imaging, substrate electrical conductivity and electron beam-induced contamination [28]. Although the above works have attempted to explain their observed contrast mechanisms in terms of SE emission from nanotubes or the substrate, they have not reported any experimental data or model for direct SE emission from nanotubes. Rivacoba and Garcia de Abajo calculated the electron energy loss spectra in carbon nanostructures by considering the polarizability of each carbon atom in the structure [32]. Kyriakou et al. proposed a semi- empirical model for the dielectric response function of nanotubes, which was used to calculate inelastic mean free paths [33] and Emfietzoglou et al. employed the Bethe approximation to calculate inelastic cross-sections and average energy transfer in single inelastic interactions [34]. Electron yield from multiwalled carbon nanotube (MWNT) forests (arrays of vertically aligned MWNTs with macroscopic dimensions) coated with different materials such as MgO, CsI and ZnO was reported [18, 20, 35, 36]. These studies demonstrated the potential of coated carbon nanotubes as electron multipliers and sensors. For example, maximum electron yield from MgO-coated MWNTs was reported to be in the range of 21~22000 from different MgO/MWNT samples [18, 20, 36]. Recently, Luo et al. reported ultrahigh SE yield from the sidewall of SWNTs [37], which is in contradiction with the previous reports predicting low yield from nanotubes or the assumption of negligible direct interaction between electron beams and nanotubes [18, 21, 23-25, 29, 31]. They claimed that even a single nanotube can have an SE yield of more than 100 without any external stimulus. However, our analysis and experimental data show that the claim is unfounded (a detailed discussion is presented in Chapter 6). Overall, there is much to be learnt from bare nanotubes and no direct explanation, measurement or model has yet been reported for the BSE or SE yield of nanotubes.  Chapter 1. Introduction 6  1.3 Research Objectives The main objective of this research was to study the interaction of an electron beam with nanotubes and develop a semi-empirical model of electron emission from them. The focus was on the experimental measurement of the BSE and SE yields from individual nanotubes and collections of nanotubes, and on building a versatile Monte Carlo program that can simulate BSE and SE emission from individual nanotubes or a collection of nanotubes, equipped with all the capabilities available in a standard Monte Carlo simulator for bulk solids. Table 1.1 shows a list of the available simulators and their capabilities for the simulation of electron trajectories in solids, including the simulator we developed during this project. While few previous simulators can handle complex geometries (such as cylinders, spheres, etc) and/or SE emission (from selected solids), none of them is capable of simulating nanostructures like nanotubes and nanowires. Table 1.1 Comparison of the Monte Carlo simulators Name Bulk Layers Complex geometries SE model Nanostructures Casino [38] Yes Yes No No No Win-Xray [39] Yes Yes No No No David Joy’s [40] Yes Yes No Yes No MC-SET [41] Yes Yes No No No Electron flight simulator [42] Yes Yes No No No Penelope [43] Yes Yes Yes Yes No NISTMONTE [44] Yes Yes Yes No No Metrologia [45] Yes Yes No Yes No Our simulator Yes Yes Yes Yes Yes   Chapter 1. Introduction 7  1.4 Methodology The research described in this thesis can be divided into two main streams: a) Performing first-principles simulations and experiments in order to gain insight into electron-nanotube interaction, obtain data and estimate the parameters to be used in the Monte Carlo model and b) Monte Carlo modeling of electron trajectories, SE and BSE coefficients based on available theoretical models and empirical parameters obtained from the experiments. The following approach was followed to carry out the above work: 1. Given the hollow and nanoscale nature of nanotubes, there seems to be little chance for an incoming electron to significantly interact with them (in particular for SWNTs or MWNTs with only a few walls). Modeling the dynamics of this interaction based on accurate first-principles approaches would be an extremely difficult, as well as computationally expensive, task and was beyond the scope of this project. Rather, the objective in this step was to use static first-principles calculations to gain insight into the electron-nanotube interaction by (a) evaluating the effect of an external electron on the energy levels and ionization energies of nanotubes and determining whether the electrostatic interaction could be significant or not, and (b) estimating the value of possible electron energy loss in a single nanotube and unravelling probable mechanisms of SE emission from nanotubes. 2. The experimental measurement of electron yield is indispensable because all analytical and Monte Carlo models depend on experimental data for the empirical parameters [4, 46]. While individual nanotubes can provide crucial information on electron-nanotube interaction, it can be difficult to measure electron yield from a single nanotube. A collection of nanotubes, such as a carbon nanotube forest (an array of vertically-aligned nanotubes – Figure 4.1), is an ideal structure as a starting point. Moreover, a nanotube forest is a richer system than individual nanotubes and could have additional interesting properties due to its multiscale structure. Hence, we measured electron yield (BSE, SE) from nanotube forests in this step. Electron yield  Chapter 1. Introduction 8  measurements were performed on various sizes of carbon nanotube forests in an SEM using a typical experimental configuration [18, 20, 36]. 3. The next step was the modeling of electron backscattering yield from nanotube forests using Monte Carlo simulations. As pointed out in Section 1.3, the existing simulators are not capable of simulating electron trajectories in nanostrcutures and also not open- source. We needed a new Monte Carlo simulation tool, which had the ability to take into account the internal structure of nanotube forests and could be modified according to our requirements, in addition to having the standard capabilities of present bulk simulators. We started by writing a code to implement a widely accepted (especially for carbon) three dimensional (3D) bulk model [47, 48] using MATLAB [49]. This step was necessary in order to ensure that our basic Monte Carlo simulator worked correctly. The simulation results were compared with those reported using existing simulators and experimental data for bulk materials in order to test the accuracy of the new tool. Then, the bulk simulator was modified to account for the anisotropy and porous nature of nanotube forests, introducing physically-meaningful empirical parameters. The results were compared with experimental data obtained in the previous stage to find the values of the defined empirical parameters and to test the validity of the model. 4. The Monte Carlo model was extended to calculate the SE yield. Two approaches were followed: (a) Semi-empirical and (b) so-called fast secondary electron (FSE) modeling. The applicability of each model to the SE emission of nanotube forests was analyzed. 5. Having investigated the interaction of electron beams with nanotube forests and found the SE and BSE yields, the next step was to find the intrinsic emission properties of individual nanotubes. For this, experiments on suspended nanotubes were performed. Suspended nanotubes were made by extraction of a few nanotubes from nanotube forests using a precise microprobe. Measurements of electron yield from individual suspended nanotubes were carried out by modifying our previous measurement setup. In addition, electron energy loss in a single nanotube was calculated using the model  Chapter 1. Introduction 9  proposed in the first step and SE yield was estimated based on the known empirical parameters. 6. As the widely used simulation methods (which work based on the so-called continuous slowing down approximation) for bulk materials may not be directly applicable to nanostructures like a nanotube, a discrete-energy-loss Monte Carlo simulation2 formalism was used to analyze the interaction of electron beams with individual nanotubes. The results were compared with the experimental data obtained in the previous step. 1.5 Thesis Overview This thesis is divided into eight chapters. Thesis objectives, related works and methodology are described in this chapter (Chapter 1). Chapter 2 gives a brief overview of the terminologies and theoretical methods used heavily in this thesis. Chapter 3 discusses the first-principles calculations and the possible mechanisms of SE emission due to electron- nanotube interaction. Chapter 4 explains the experimental method of measuring the electron yield and the results obtained from nanotube forests. Chapter 5 proposes an electron backscattering model for nanotube forests based on the experimental data presented in Chapter 4 and discusses the possible SE modeling techniques for nanotube forests. Chapter 6 provides a description of a simple method for measuring the electron yield from individual suspended nanotubes and the measurement results obtained using this method. Chapter 7 proposes a “discrete” Monte Carlo simulation framework for nanotubes. The last chapter (Chapter 8) summarizes the findings and contributions made in the context of this thesis and provides directions for future work.   2 A discrete-energy-loss Monte Carlo simulation method incorporates all the energy loss processes separately in contrast to the average energy loss formula used in the traditional models for solids. 10  Chapter 2 Theoretical Overview This chapter provides a brief description of the terminologies and methods used commonly throughout this thesis. A detailed explanation has been given in Chapters 3 to 7 along with the corresponding results and discussion of the findings. 2.1 Monte Carlo Simulation The study of electron-specimen interaction (and the resulting SE and BSE yields, electron range, energy and momentum distribution, etc) involves statistical uncertainties. At the very atomistic level (one electron and one atom interacting) the progression of scattering events can be determined provided that sufficient information (position, energy, cross-section, energy loss, etc) about the electron and the atom is available [4]. However, from a macroscopic point of view, the trajectory of any given electron appears to be entirely random to an outside observer as it travels through the solid or collection of atoms. Each electron goes through a unique set of scattering events and every trajectory is different [4] due to the variation of inter-scattering distance, scattering angle and energy loss along the trajectory (Figure 2.1). However, it is possible to find probabilities for specific events, such as the chance of an electron being scattered by a certain angle or being transmitted through the specimen. The Monte Carlo method uses the concept of sampling by using random numbers and empirical parameters to compute one possible set of scattering events for an electron as it travels through the solid in a random-walk fashion [4]. By repeating this process for many electrons and using statistical averages, various physical quantities can be obtained. This method has been used extensively for studying the interaction of electrons with bulk solids and shown to produce results in good agreement with experimental data, for physical quantities of interest such as SE and BSE yields [4].  Chapter 2. Theoretical Overview 11   2.2 Elastic Scattering The interaction between the nucleus of an atom and the primary electron can be considered elastic for non-relativistic energies. The elastic scattering of electrons by the nuclei of the specimen affects electron diffusion and SE and BSE yields. In such an event, the electron energy is conserved and the momentum only changes directions. The elastic scattering cross-section of a scattering center (atom) is needed in finding the elastic scattering angle and step length for the primary electron at each small step of a Monte Carlo simulation. The interaction of two particles is generally described in terms of the differential scattering cross-section d d σ   Ω  . This quantity gives a measure of the probability of a scattering event with a given angle to occur and can be calculated if the underlying physics of the interaction  Figure 2.1 100 electron trajectories in bulk graphite at 5 keV simulated using Casino [38].   Chapter 2. Theoretical Overview 12  between the colliding particles are known. It is defined as the average number of the particles scattered into the solid angle Ωd  per unit flux through area σd  and scattered by angle θ  [5, 50] (Figure 2.2). In general, the value of the cross-section varies with the energy of the primary electron, the type of the scattering center (i.e. atom) and the angle at which the particle (electron) is scattered. The differential cross-section determines the scattering angle distribution in a Monte Carlo simulation. In classical mechanics, the angle of scattering is determined by the impact parameter b (perpendicular distance between the nucleus and the original path of the incident electron). An increase in the impact parameter leads to reduced effect of the atom’s nucleus on the incoming electron resulting in a decrease in the scattering angle. The quantity which is used to determine the random step length in a Monte Carlo simulation is the total elastic scattering cross-section which can be calculated by integrating the differential cross- section over all solid angles. It gives a hypothetical area which reflects the probability of an electron being scattered by the nucleus of an atom. The classical model of elastic scattering between an electron and an atom is governed by the Rutherford scattering cross-section [5]. A more accurate calculation takes into account the wave nature of electrons and the screening of the nuclear charge by the atom’s electrons. Obviously, an exact description of scattering can only be done by a quantum mechanical model in which both the spin of electrons and spin-orbit coupling are considered. The effects  2 sind dπ θ θΩ = 2d bdbσ π= Atom θ dθ  Impact parameterb =  Figure 2.2 Schematic representation of the concept of differential elastic scattering cross- section.   Chapter 2. Theoretical Overview 13  of neighbouring atoms also need to be considered (such as what is accomplished in the muffin-tin model [5]). A more accurate cross-section (the Mott cross-section) for elastic scattering is calculated by substituting a screened Coulomb potential in the Dirac equation and solving it using the Born approximation or partial-wave analysis [5, 51]. While the Mott cross-section is the most accurate one, no simple analytical form is available for the Mott differential and total cross-sections. Often, empirical forms of the cross-sections derived from tabulated Mott cross-sections are used in the Monte Carlo analysis of electron yield [48]. The total elastic scattering cross-section (empirical fit to the Mott total cross-section) used to define the mean free path (MFP)3 between elastic scattering events is [48]:  18 1.7 1.7 0.5 2 0.5 3 10 , 0.005 0.0007 /T Z E Z E Z E σ −× = + +  (2.1) where Tσ is in 2cm , E is the incident electron energy in keV and Z is the atomic number. 2.3 Inelastic Scattering and Energy Loss The interaction between the atomic electrons and the primary electron is inelastic in nature. The electron kinetic energy and momentum in an inelastic collision are not conserved and the kinetic energy is converted to the excitation of the atom’s electrons. The energy loss of the primary electron to the nucleus (phonon excitation) can be neglected for the electron energies used in scanning electron microscopy [5]. For example, when 30-keV electrons are scattered from a Cu nucleus through the maximum angle (180˚), the transferred energy is only ~1 eV and the probability of such scattering is negligible compared to other excitation processes (less than 1 out of 10,000 [4]). The following excitation mechanisms are the most important for SE generation [5]: 1. Excitations of the outer-shell atomic electrons 2. Excitation of collective oscillations (plasmons) 3. Ionization of core-shell electrons  3 Distance travelled by the primary electron between two scattering events.  Chapter 2. Theoretical Overview 14  The probability of inelastic scattering is determined by the differential inelastic cross- section. The differential inelastic scattering cross-section is generally expressed in terms of the energy loss of the primary electron. Assuming a two-body collision, the cross-section d dW σ       determines the energy loss4 of the primary electron and the corresponding inelastic scattering angle distribution in a Monte Carlo simulation due to an inelastic event. In a single scattering-type Monte Carlo model, only the elastic scattering angle is considered, which is much larger than the inelastic scattering angle. However, in a “discrete-energy- loss” Monte Carlo model each process is considered separately (Chapter 7). The inelastic MFP is obtained in a similar manner to how the elastic MFP is determined. The differential cross-section for inelastic electron-electron scattering has been calculated using quantum mechanics. A relativistic theory that considers the indistinguishability of electrons and spin interactions leads to the Møller scattering cross-section (widely used for outer-shell electrons) [5]. Several other cross-sections such as Gryzinski, Vriens, and Evans have also been used [4, 5, 52]. Gryzinski is mainly used for core-shell electron ionization. Theoretical expressions for calculating the MFP of plasmon creation have also been reported [53-55]. The energy loss along the trajectory of an electron is calculated assuming the energy transfer in each single scattering event to be small. The progression of collisions can then be described by Bethe’s continuous-slowing-down approximation (CSDA). For non-relativistic energies the energy loss rate has been calculated by Bethe [56]. Subsequent models have been built upon this to address the issue of non-physical behaviour of the Bethe’s logarithmic term at low energies based on experimental observations. An example is the one proposed by Joy and Luo [57], which is widely used and has been implemented in several simulators [38, 39, 44]:  4 W is the energy loss of the primary electron due to the inelastic collision.  Chapter 2. Theoretical Overview 15   1.166( )785 ln / Å,dE Z E tJ eV ds AE J ρ + =      (2.2) where s is the interaction length in the travelling direction of the primary electron, ρ is the density of the target (g/cm3), Z is the atomic number, A is the atomic weight, E is the incident electron energy (eV), J is the mean excitation potential (eV) [50] and t is an empirical factor, which is usually 0.77~0.85 [5]. 2.4 First-Principles Simulations First-principles methods are approaches to solving the quantum many-particle problem (where the particles interact with one another) without resorting to the use of empirical parameters (Figure 2.3) [58]. The aim is usually to solve the Schrödinger equation5 for the system of many electrons and nuclei. This forms the basis of the molecular orbital theory.   5 An analytical solution of the Schrödinger equation can be found only for two-particle systems (e.g. the hydrogen atom). B j i A ri rj RA RB rij RAB RAi Y X Z  Figure 2.3 A typical molecular system (i, j are electrons and A, B are atoms).  Chapter 2. Theoretical Overview 16  The Hartree-Fock formalism and the Density Functional Theory are some of the most commonly used first-principles methods. In the present work, the Hatree-Fock method was used for simulating the electrostatic effect of primary electrons on a carbon nanotube (Chapter 3). Therefore, a brief introduction to the Hartree-Fock method is given in the next section. 2.4.1 The Hartree-Fock Method The goal is to find an approximate solution to the non-relativistic many-body Schrödinger equation [58]:  ,H EΨ = Ψ  (2.3) where H, E and Ψ  are the many-body Hamiltonian operator, energy of the system and its wavefunction, respectively. The many-body Hamiltonian operator for M nuclei and N electrons in atomic units is expressed as follows:  n e nn ee ne ,H T T V V V= + + + +  (2.4) where 2n 1 1(Kinetic energy of nuclei) 2 M A A A T M= = − ∇∑ ; 2 e 1 1(Kinetic energy of electrons) 2 N i i T = = − ∇∑ ; nn 1 (Coulombic energy between nuclei) M M A B A B A AB Z ZV R= > =∑∑ ; ee 1 1(Coulombic energy between electrons) N N i j i ij V r= > =∑∑  and ne 1 1 (Coulombic energy between nuclei and electrons) M N A A i Ai ZV r= = = −∑∑ . In Eq. (2.4), RA,B…M and ri,j…N are the position vectors of the nuclei and electrons, respectively, MA is the ratio of the mass of nucleus A to the mass of an electron and ZA is the  Chapter 2. Theoretical Overview 17  atomic number of nucleus A. Equation (2.3) is solved under the Born-Oppenheimer approximation6, which allows for treating the nuclear and electronic wavefunctions separately. By applying the variational principle, the ground state electronic wavefunction is calculated self-consistently in the form of a single Slater determinant [58]:  1 1 1 2 2 2 ½ 1 2 N N N (x ) (x )  ...  (x ) (x ) (x )  ... (x )     .            .                . (x , x ,  , x )  (1/ N!)     .            .                .     .            .                . (x ) (x ) ... i j k i j k i j k χ χ χ χ χ χ χ χ χ Ψ … = N , (x )  (2.5) where , ,...i j kχ  are the single-electron spin orbitals [ ]1 1 1 1 1(x ) (r ) ( ) or (r ) ( )i i iχ ψ α ω ψ β ω= ( 1(r )iψ  is the spatial orbital, α and β  are spin functions and x defines a four coordinate system including three spatial coordinates and one spin coordinate collectively). Spin orbitals can be expressed as the linear combination of basis functions. The Slater determinant satisfies the antisymmetry principle:  1 2 i j N 1 2 j i N(x , x ,  .x , x ..., x ) (x , x ,  x , x ..., x )Ψ … = −Ψ …  (2.6) The wavefunction vanishes if two electrons occupy the same spin orbital (i = j). The antisymmetry of the Slater determinant with respect to the exchange of the coordinates (spin and position) of any two electrons satisfies the Pauli Exclusion Principle. Applying the variational principle in order to obtain, from the Schrodinger equation, the best possible solution to the ground state in the form of the expression (2.5) leads to the Hartree-Fock equation for the spin orbitals [58]:  ,i i if χ ε χ=  (2.7)  6 Since nuclei are much heavier than electrons, one can assume that in a molecule electrons move in a fixed field created by the nuclei.  Chapter 2. Theoretical Overview 18  where 1(x )f  is the Fock operator 1 1 1(x ) (x ) (x ) N j j j h    + −     ∑ J K and iε is the orbital energy associated with spin orbital iχ . Here, 1(x )h , 1(x )jJ  and 1(x )jK  are the core- Hamiltonian, Coulomb and exchange operators, respectively. They are defined as [58]:  21 1 1 1(x ) 2 A A Zh r = − ∇ −  (2.8)  11 1 2 2 12 2 1(x ) (x ) x (x ) (x ) (x )j i j j id rχ χ χ χ ∗ − =  ∫J  (2.9)  11 1 2 2 12 2 1(x ) (x ) x (x ) (x ) (x )j i j i jd rχ χ χ χ ∗ − =  ∫K  (2.10) For a closed-shell system with an even number of electrons, every two of which are paired in a single spatial orbital, the restricted Hartree-Fock equation will take the form [58]:  1 1(r ) (r ),i i ifψ εψ=  (2.11) where 1(r )f  is the closed-shell Fock operator [ ] /2 1 1 1(r ) 2 (r ) (r ) N a a a h J K + −    ∑ . The closed- shell Coulomb [ ]1(r )aJ  and exchange [ ]1(r )aK  operators have similar forms to Eqs. (2.9) and (2.10). Eq. (2.11) can be solved by introducing a set of known spatial basis functions. For example, a set of K known basis functions { }1,2,.....Kµφ µ =  can be used to expand the spatial orbitals as:  1 , K i iCµ µ µ ψ φ = =∑  (2.12) where iCµ  are the expansion coefficients. Using Eq. (2.12) in Eq. (2.11), the matrix representation of the Hartree-Fock equation is obtained (Roothaan equations) as [58]:  FC=SC ,ε  (2.13)  Chapter 2. Theoretical Overview 19  where F is the Fock matrix ( )1 1 1 1F r (r ) (r ) (r )d fµν µ νφ φ∗= ∫ , S is the overlap matrix ( )1 1 1S r (r ) (r )dµν µ νφ φ∗= ∫ , ε  is a diagonal matrix of the orbital energies and C is a K K× square matrix of the expansion coefficients. Eq. (2.13) can be solved self-consistently to obtain the expansion coefficients and hence the spatial orbitals. The formalism leads to an accurate treatment of exchange interaction, while it neglects electron-electron correlation. The Hartree-Fock method can effectively be looked at as treating each electron in the average field resulting from other electrons.  20  Chapter 3 Secondary Electron Emission from Carbon Nanotubes7 Several experimental studies have been carried out on the SEM imaging mechanisms of SWNTs [23, 24, 26, 28, 31]. Yet it is quite surprising that no direct explanation has yet been given for the secondary emission mechanism itself in nanotubes. Homma et al. completely ignored direct interaction of the electron beam and nanotubes in explaining their electron- beam-induced-current contrast mechanism [23], whereas Finnie et al. explained their results assuming the nanotubes are charged, presumably due to some direct interaction with the primary beam [31]. Due to the extremely small interaction area of the nanotube with the electron beam, which does not provide much possibility for scattering the high-energy electrons, SEM imaging of nanotubes may not be explained by traditional multi-scattering beam-bulk models. Electron beam stimulated field-emission from the tip of a nanotube under external bias has been investigated before. In normal SEM imaging, however, secondary electron emission from the sidewall of a nanotube, without a notable external field, must be studied, which is the subject of the present chapter. The primary objective of this chapter is to investigate whether an incoming electron can affect a nanotube in a way that may lead to some significant direct interaction between the two. We describe first-principles simulations performed in order to gain some preliminary insight into this issue. We obtain the electron-nanotube interaction length and the energy loss of primary electrons as they pass through the electronic cloud of the nanotube orbitals. We show that the direct interaction of high energy electrons with individual nanotubes could be significant and the common electron energy loss mechanisms present in solids could play a role in direct SE emission from nanotubes. This provides only a qualitative insight into the  7 A version of this chapter has been published in a peer-reviewed journal (Reused with permission from “M. K. Alam, S. P. Eslami, and A. Nojeh, ‘Secondary Electron Emission from Single-Walled Carbon Nanotubes,’ Physica E, vol. 42, pp. 124-131, 2009”, Copyright 2009, Elsevier).  Chapter 3. Secondary Electron Emission from Carbon Nanotubes 21  interaction of electron beams with individual nanotubes, and mainly serves as a motivation for the rest of the project, rather than being directly used in subsequent chapters (except for limited use in Chapter 6). Sections 3.1 and 3.2 present the model used for the calculation of energy loss. In Section 3.3, it is shown that there could be enough energy transfer via inelastic scattering to allow the emission of secondaries, and a qualitative description of the direct interaction mechanisms is presented. In addition, it is also revealed that the primary electron could cause a major upward shift in the occupied energy levels of the nanotube, effectively reducing the ionization energy and increasing the secondary emission probability. It is also predicted that if the nanotube has enough time to relax in response to the electric field of the primary electron, its ionization energy is lowered even more, further enhancing secondary emission. 3.1 The Model Existing Monte Carlo models for SE emission are based on the multiple scatterings of the primary electrons with many atoms in a solid. A SWNT may not be adequately modeled with these methods as it is a single layer of atoms with a very small interaction area. An alternative approach is to directly study the electronic structure of the nanotube and find the charge density profile from which the interaction length and energy loss can be estimated. The ionization energy of the nanotube can also be estimated from the electronic structure to investigate how it may lead to secondary emission. A primary electron travelling with a typical energy of 0.3–30 keV might have both elastic (due to interaction with nuclear potentials) and inelastic (due to interaction with other electrons) scattering. Radiation loss (Bremsstrahlung) does not lead to SE emission (as the name implies this loss is associated with photon emission). In the case of secondary emission, we are interested in inelastic scattering that leads to energy transfer to the nanotube electrons. This is directly related to the effective interaction volume between the primary electron and the nanotube. As mentioned previously, this volume is extremely small due to the nanoscale diameter and hollow structure of the nanotube. However, if the primary  Chapter 3. Secondary Electron Emission from Carbon Nanotubes 22  electron has a chance to experience multiple changes of direction due to elastic scatterings and bounce around inside the nanotube for some time, the effective interaction volume will be increased. Therefore, we first look at the likelihood of multiple elastic scattering events. The possibility of elastic scattering due to atomic potentials is determined by the elastic scattering cross-section i.e. Mott/fitted Mott cross-section, which can be calculated from the empirical equation given by Browning et al. [Eq. (2.1)]. For a carbon atom, in the energy range of interest for typical primary beams in the SEM, this equation gives a value in the range of 10-20~10-22 m2, which means a small area around an atom available for elastic scattering in a SWNT (~16% in the best case for a C-C distance of ~1.4 Å). Also, depending on the density of the nanotube, this yields an elastic MFP of several nanometres, which is larger than the typical diameter of SWNTs. Moreover, the peak of the elastic angle distribution and most of the scattering occurs at very small angle [4]. As a result, it is unlikely that the primary electron could become “trapped” inside the nanotube through multiple elastic reflections from the inside of the nanotube wall. Therefore, we looked at the energy loss of the primary electron as it makes one pass through the nanotube, perpendicular to it. The energy loss by the primary electron is often calculated under Bethe’s CSDA. In the case of SWNTs, we estimated the average energy loss using the interaction length obtained from the charge density distribution and Eq. (2.2). 3.2 Estimation of the Ionization Energy and Interaction Depth Semi-empirical or continuum modeling of SWNTs are computationally efficient, but may not capture all the nanoscale effects accurately. Therefore, first-principles quantum mechanical modeling was used to explore the interaction of an electron beam with SWNTs. Small sections of nanotubes were modeled due to the high computational cost of first- principles calculations. Here, 7½ unit cells of a (5,5) SWNT (metallic) and 4 unit cells of a (8,0) SWNT (semiconducting) were chosen as the structures to be simulated. A similar size structure has been used in several previous studies on nanotubes and has been observed to reproduce the electronic structure quite accurately [25, 59, 60]. The SWNTs were terminated with hydrogen atoms to avoid dangling bonds and simulated in the presence of a single  Chapter 3. Secondary Electron Emission from Carbon Nanotubes 23  electron fixed at various positions (Figure 3.1). It is reasonable to consider the primary-beam electrons from the SEM only one at a time because one can perform SEM imaging of nanotubes with primary-beam current and energy conditions where electrons in the beam are on average several metres apart from each other. For instance, at 5 keV and 0.5 pA, there is only one electron in every 13 m of beam length [25]. The electron was placed at five different locations from the centre to the outside of the nanotube. The Hartree-Fock method in the software package Gaussian 03 [61] was primarily used for the simulations8. This method, when used with an appropriate basis set such as 6-31G(d), has been observed to reproduce the occupied energy levels of such nanotube systems quite accurately, and is therefore suitable for problems involving ionization energies and electron emission [25]. This method has also been compared with Density Functional Theory (DFT) calculations and found to be in good agreement for such problems [25]. Hence, we used the Hartree-Fock method in this work for better computational efficiency. The extra electron  8 A sample Gaussian script and the procedure for job submission on the supercomputer WestGrid are given in Appendix A.  (a) (b)  Figure 3.1 (a) (5,5) SWNT with various positions of the extra electron. In each simulation, the electron is placed in one of these positions. They are located at the centre, 1 Å away from the centre, 2 Å away from the centre, 5 Å away from the centre and 10 Å away from the centre. Once the electron is located 5Å or more away from the centre, it is outside of the nanotube. (b) (8,0) SWNT with the same electron positions as in (a).   Chapter 3. Secondary Electron Emission from Carbon Nanotubes 24  was treated like a fixed nucleus in the electronic Hamiltonian. It modifies the term Vne in Eq. (2.4) to 1 1 1 M N A A i Ai Z r + = = −∑∑ , where ZA= −1 for A=M+1. However, the size of the Slater determinant remains unchanged. For each nanotube, the initial atomic coordinates were generated using the software Nanotube Modeler [62]. Before calculating the electronic structure of the nanotube in each case, one needs to find the relaxed atomic structure of the system. This ‘geometry optimization’ step is computationally very expensive, and it is desirable to perform it with a minimal basis set such as STO-3G, which enables much faster relaxation simulations than the more accurate 6-31G(d). We performed comparison simulations with both basis sets on a small sub-system [one unit cell of the (5,5) nanotube]. The distances between various pairs of atoms and the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) energy levels were compared to check whether the results provided by STO-3G would be reasonably close to those provided by 6-31G(d) (Tables 3.1 and 3.2). In all compared cases, the difference in bond lengths was less than 0.8% and the difference in orbital energies was less than 0.5%. Moreover, our interest in this problem is in electronic structure, and not in bond lengths themselves. Obviously a 0.8% inaccuracy in atomic coordinates would have a minimal effect in the electronic structure (of course this would not be appropriate for a problem where small changes in the bond lengths and atomic coordinates are the main topic of study, such as the electromechanical actuation of nanotubes [63]). Therefore, for the rest of the simulations, we used STO-3G for geometry optimization in order to have better computational efficiency. However, the single point energy calculations after each geometry optimization step were all carried out using the 6- 31G(d) basis set to preserve a higher level of accuracy. The ionization energy (from potential profiles) and electron-electron interaction depth (from the charge density profile) were then estimated from the ab-initio calculation results. Average energy transfer from the primary beam of the SEM to the SWNT was estimated for a range of primary beam energies in order to gain insight into the mechanism of secondary emission from SWNTs.  Chapter 3. Secondary Electron Emission from Carbon Nanotubes 25  Table 3.1 Distance (in Å) between different atoms. The indices correspond to the atom numbers in the coordinate file of the simulation. Atoms (arbitrarily chosen) 6-31G(d) STO-3G Difference (%) C10 to C26 6.95452 6.95969 –0.0789 C10 to C13 2.3541 2.36724 –0.5581 C10 to C24 7.34215 7.35126 –0.1241 C10 to C28 7.12276 7.13985 –0.2399 C8 to C11 1.50502 1.5163 –0.7494  Table 3.2 Energy (in Hartrees) of HOMO and LUMO. Molecular Orbitals 6-31G(d) optimized STO-3G optimized Difference (%) HOMO –0.224 –0.225 –0.4464 LUMO 0.046 0.046 0.0000  3.3 Results and Discussion The structure of the nanotube might be deformed due to the electric field of the incoming electron. The timescale of nuclear dynamics is in femtoseconds [64]. Therefore, the likelihood that the nanotube’s nuclear structure will have enough time to fully respond and change to a new relaxed configuration as the incoming electron passes through it is low. For example, the velocity of a 1-keV electron is 1.87×107 m/sec, which means it takes about ~ 0.053 fs to cross a nanotube with a diameter of about 1 nm. (On the other hand, electronic transition timescales are in attoseconds [64]. Thus, change in the electronic structure of the nanotube is expected [25].) Nonetheless, we have carried out simulations on both the relaxed and non-relaxed structures in the presence of the additional electron in order to cover both ends of the spectrum of possibilities. Also, the (5,5) and (8,0) SWNTs gave generally similar results. So in this section, results for (5,5) SWNT are mainly presented.  Chapter 3. Secondary Electron Emission from Carbon Nanotubes 26  3.3.1 The Non-Relaxed (5,5) SWNT The results discussed below are for the case where a single extra electron was fixed at various positions and single point energy calculations were performed without carrying out a relaxation (geometry optimization). First, the energy levels were calculated without any primary beam (i.e. no extra point charge in the simulations). The occupied levels were all below the vacuum level, as expected. The HOMO was at –5.1 eV. This value agrees closely with the ionization energy of the nanotube, which is known to be around 5 eV, and this further justifies our model and use of the Hartree-Fock level of theory [25]. Figure 3.2 shows the Mulliken charge distribution and HOMO for the (5,5) nanotube with no background charge. The molecular orbital is evenly distributed as expected. Also, the Mulliken charge diagram shows that the effect of the hydrogen termination is limited to the first ring of carbon atoms on either side (the local dipole is due to the difference in the electro-negativity of hydrogen and carbon), and that the majority of the length of the nanotube is unaffected by the edges. This further confirms that the length of the cluster is appropriate for these simulations.   (a) (b) –0.21 a.u. +0.21 a.u.  Figure 3.2 (a) Mulliken charge distribution (scale given in atomic units) and (b) HOMO of (5,5) SWNT in the absence of external electron.  Chapter 3. Secondary Electron Emission from Carbon Nanotubes 27  Figure 3.3 shows the directions along which electrostatic potential energy profiles were investigated. The purpose of analyzing the potential profiles in different directions is to provide an insight into where electrons are more likely to tunnel out or overcome the vacuum barrier of the nanotube. The directions are 1x , 2x , 1y  and 2y  (Figure 3.3 ). Figures 3.4 and 3.5 illustrate the potential profiles with no background charge and charge at the center of the nanotube, respectively, in the four chosen directions. The potential wells (where the orbital energies are plotted as horizontal lines) in these potential profiles correspond to the right edge of the SWNT of Figure 3.3 for all the directions. It can be seen that the profiles are rather similar in all these directions. The potential profiles were shifted on each figure to keep the HOMO constant in all cases for ease of comparison. The insets depict the potential profile over a longer distance to show the vacuum level outside the nanotube and the effective barrier heights. As can be seen on Figure 3.5, due to the presence of the extra (external) electron, the gap between HOMO and the vacuum level is lowered by more than 2 eV (Figure 3.5 inset) compared to Figure 3.4. Although there still exists a local potential barrier of almost 5 eV in Figure 3.5, this barrier is less than 2.5 nm thick and the nanotube electrons can potentially tunnel through it. Therefore, the ionization energy has effectively been reduced from ~5 to ~3 eV due to the presence of the extra electron.  1y  1x  2x  2y  Figure 3.3 Directions along which potential profiles are plotted: 1x - in the positive horizontal direction through the center of the nanotube; 2x - in the positive horizontal direction two rings below the centre; 1y - into the page through the center of the nanotube; 2y - into the page two rings below the center. The position of the extra electron was moved along the 1x  direction.   Chapter 3. Secondary Electron Emission from Carbon Nanotubes 28     4 5 6 7 8 -7 -6 -5 -4 -3 -2 -1 0 1 2 Horizontal position [Å] En er gy , P ot en tia l [ eV ]   x 1  direction x 2  direction y 1  direction y 2  direction 5 10 15 20 25 -6 -4 -2 0 2 ~5eV HOMO  Figure 3.4 Energy levels and potential profiles of the (5,5) SWNT along the chosen directions without any external electron. Horizontal axis indicates distance from the nanotube axis. 4 5 6 7 8 -7 -6 -5 -4 -3 -2 -1 0 1 2 Horizontal position [Å] En er gy , P ot en tia l [ eV ]   x 1  direction x 2  direction y 1  direction y 2  direction 5 10 15 20 25 -6 -4 -2 0 ~3eV  Figure 3.5 Energy levels and potential profiles of the (5,5) SWNT along the chosen directions with external electron at the center of the nanotube. Horizontal axis indicates distance from the nanotube axis.  Chapter 3. Secondary Electron Emission from Carbon Nanotubes 29  Figures 3.6 and 3.7 show the potential profiles and energy levels along 1x  and 2x  directions, respectively for different positions of the primary electron. Obviously the potential rises to infinity at the location of the extra electron (Figure 3.6 inset). As can be seen from the barrier heights and shapes in various cases, the ionization energy is smallest when the extra charge is inside the tube. In particular when the extra charge is inside the nanotube, 2 Å from the center (relatively close to the nanotube wall), even the local barrier has been considerably lowered. Similar profiles were found for cases where the extra electron was placed at 1.5~3.0 Å from the center (not shown here). As expected, there was no significant decrease in the potential barrier in 1y  and 2y  directions (Figures 3.8 and 3.9).  3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 -10 -8 -6 -4 -2 0 2 4 Horizontal position [Å] En er gy , P ot en tia l [ eV ]   No charge Charge at center Charge at 1Å from center Charge at 2 Å from center Charge at 5 Å from center Charge at 10 Å from center 0 5 10 0 50 100  Figure 3.6 Energy levels and potential profiles of the (5,5) SWNT in the 1x  direction for various positions of the external electron. Horizontal axis indicates distance from the nanotube axis.   Chapter 3. Secondary Electron Emission from Carbon Nanotubes 30    4 5 6 7 8 -10 -8 -6 -4 -2 0 2 4 Horizontal position [Å] En er gy , P ot en tia l [ eV ]   No charge Charge at center Charge at 1 Å from center Charge at 2 Å from center Charge at 5 Å from center Charge at 10 Å from center  Figure 3.7 Energy levels and potential profiles of the (5,5) SWNT in the 2x  direction for various positions of the external electron. Horizontal axis indicates distance from the nanotube axis. 4 5 6 7 8 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 Horizontal position [Å] En er gy , P ot en tia l [ eV ]   No charge Charge at center Charge at 1 Å from center Charge at 2 Å from center Charge at 5 Å from center Charge at 10 Å from center  Figure 3.8 Energy levels and potential profiles of the (5,5) SWNT in the 1y  direction for various positions of the external electron. Horizontal axis indicates distance from the nanotube axis.  Chapter 3. Secondary Electron Emission from Carbon Nanotubes 31  The observed effective decrease of ~2 eV in ionization energy due to the presence of the extra electron (representing the primary beam) in certain locations suggests a significant increase in the emission of electrons from the nanotube, and might provide a partial explanation for SE emission. In order to quantify the effect of this decrease in ionization potential on the secondary emission current, one possibility is to look at thermionic emission (for instance, at a very low workfunction, there could be significant thermionic emission even at room temperature). The thermionic emission current density can be calculated using the Richardson-Dushman equation,  2 ,T R kTJ A T e           Φ − =  (3.1) where AR is the Richardson constant, k is the Boltzmann constant, T is absolute temperature and Φ  is the work function. Eq. (3.1) suggests that the emission current rises dramatically 4 5 6 7 8 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 Horizontal position [Å] En er gy , P ot en tia l [ eV ]   No charge Charge at center Charge at 1 Å from center Charge at 2 Å from center Charge at 5 Å from center Charge at 10 Å from center  Figure 3.9 Energy levels and potential profiles of the (5,5) SWNT in the 2y  direction for various positions of the external electron. Horizontal axis indicates distance from the nanotube axis.   Chapter 3. Secondary Electron Emission from Carbon Nanotubes 32  (on the order of 1025- 1035) due to the workfunction lowering (to be more precise the lowering of the ionization energy in our simulation) because of the presence of an extra electron. However, the actual value of the current might still be very small (for instance, at room temperature, this leads to an increase from ~10-70 A/m2 to ~10-35 A/m2 in the best case). Therefore, even under the assumption that there is enough time for thermionic emission during the primary electron’s pass, it is doubtful that thermionic emission can provide an explanation for the observed SE emission in nanotubes. Below, other possibilities are investigated. The main energy loss mechanism for the primary electron is from electron-electron inelastic scattering, which could lead to SE emission, as in the case of bulk solids. The difference is that multiple scattering events are not expected in the nanotube as discussed previously. The question, then, is whether one pass through the nanotube (perpendicular to it) will lead to enough energy transfer from the primary to the nanotube electrons via inelastic scattering. Whether a continuous-slowing-down approximation can be applied to the case of a primary electron passing through a nanotube is not obvious, since the electron effectively “sees” only two separate layers of atoms. Nonetheless, we used Eq. (2.2) to gain a first-order insight into the problem. For this, the effective thickness of the nanotube, determined by the extension of its electronic cloud, was needed. We used the ab-initio simulation results discussed previously to plot the electronic density distribution along the primary beam path (Figure 3.10). An important issue to consider is that the primary electron can affect this distribution as it moves. To gain further insight, we show this distribution for various locations of the primary beam on Figure 3.10. It can be seen that although the primary electron affects the electron density distribution, the total effective thickness of the nanotube obtained from this distribution is not significantly changed (less than ~5% change in the width and, therefore, in the energy loss in the worst case).The energy loss was then calculated from Eq. (2.2) as a function of the primary beam energy (Figure 3.11). It can be seen that a low-energy primary electron (a few hundred eV) can lose up to an average of ~40 eV during one full pass through the nanotube. This energy loss quickly decreases as the initial energy of the primary electron increases.  Chapter 3. Secondary Electron Emission from Carbon Nanotubes 33    -8 -6 -4 -2 0 2 4 6 8 0 0.05 0.1 0.15 0.2 0.25 Horizontal position [Å] De ns ity  [a .u .]   No charge Charge at center Charge at 1 Å from center Charge at 2 Å from center Charge at 5 Å from center  Figure 3.10 Electron density profiles (in atomic units) in the 1x  direction for various positions of the external electron. Horizontal axis indicates distance from the nanotube axis. 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 40 Primary electron energy [keV] A ve ra ge  e ne rg y lo ss  [e V ]   Primary electron at the center Primary electron out of the nanotube 10 20 30 0 2 4  Figure 3.11 Estimated average energy loss for different primary beam energies. Inset shows the primary beam energy limits (dotted lines) for ~5 eV and ~3 eV energy loss.  Chapter 3. Secondary Electron Emission from Carbon Nanotubes 34  As we saw from the ab-initio simulations discussed earlier, the ionization energy of the nanotube is ~5 eV, and it reduces to ~3 eV when the primary electron reaches inside the nanotube. Thus, according to the energy loss results of Figure 3.11, we expect SEs to be generated through one of the following scenarios: - When the primary beam has reached around the center of the nanotube, the ionization energy of the nanotube has been reduced to around 3 eV, although a local barrier of 5 eV still remains for secondary emission (see Figure 3.5). According to Figure 3.11, if the primary electron energy has been less than ~3.5 keV, the energy loss is greater than ~5 eV. If enough of this energy is transferred to a given nanotube electron, that electron can be emitted as a secondary by directly overcoming the vacuum barrier. - If the primary energy is between 3.5 and 7.5 keV, the energy loss by the time the primary reaches the center of the nanotube is less than 5 eV, but more than 3 eV. This means, according to Figure 3.5, that if this entire energy is transferred to the HOMO electron of the nanotube, that electron could be excited to a high-enough level to be able to subsequently tunnel out of the nanotube (Figure 3.12). Note that the lifetime of the excited state carriers in single nanotubes could be larger than the typical attempt frequency of tunnelling and tunnelling time through such a narrow barrier (less than 2.5 nm in width) [65-67]. The lifetime has been reported to be on the order of a few tens of femtoseconds (for higher excited states) to picoseconds (for the lowest excited states) [65], whereas the typical attempt frequency in such a narrow (~ 2-4 Å) quantum well is ~234-117 THz (corresponding to a few femtoseconds between subsequent attempts) [66] and the quantum tunnelling time is on the order of a few femtoseconds [67]. Therefore, an electron could potentially tunnel through the barrier, although the probability of such an event may be low. - After the primary electron has completed a full pass and exited from the other side of the nanotube, the ionization energy has gone back to 5 eV. But according to Figure 3.11, for primary beam energies of up to approximately 10 keV, there is enough energy loss to allow for the emission of secondaries.  Chapter 3. Secondary Electron Emission from Carbon Nanotubes 35  - If the energy transfer is not sufficient for any of the above to take place, but greater than the bandgap of the nanotube, that is enough for HOMO-LUMO or any occupied- unoccupied state transition, a multiple-step process might take place (several nanotube electrons being excited to higher energy levels due to multiple primary-beam electrons and all of them relaxing almost simultaneously and the sum of their energies helping another nanotube electron to overcome the vacuum barrier). Obviously, these events would occur with a very low probability, although with increased primary beam current (more primary electrons present at once) this probability would also increase. It should also be noted that a smaller average energy loss at higher energies arising from larger inelastic MFPs indicates a lower probability of electronic excitations and, therefore, lower SE emission at higher primary energies.  5 10 15 20 25 -10 -8 -6 -4 -2 0 2 4 Horizontal position [Å] En er gy , P ot en tia l [ eV ]    Figure 3.12 Potential profile in the 1x  direction showing the possible tunneling event for a nanotube electron when the external electron is at 1 Å from the center. Horizontal axis indicates distance from the nanotube axis.   Chapter 3. Secondary Electron Emission from Carbon Nanotubes 36  3.3.2 The Relaxed (5,5) SWNT In this section, the results are discussed for the case where the extra electron was fixed at various positions and a further geometry optimization was carried out before the single point energy calculation. The results were compared to that of the non-relaxed structure and Figure 3.13 shows the most important case (when the charge was just inside the nanotube). It can be seen that when the nanotube nuclear structure itself is assumed to “respond” to the field of the incoming electron, the ionization energy is lowered by an additional ~1 eV. Correspondingly, the probability of secondary emission is even higher according to the mechanisms discussed previously. In reality, given that the nanotube phonon modes are in the THz range, corresponding to response times of tens of femtoseconds, and the fact that the incoming primary electron spends only a fraction of a femtosecond in the vicinity of the nanotube, it is likely that the nanotube will not have enough time to significantly relax in the field of the incoming electron.  4 5 6 7 8 -8 -7 -6 -5 -4 -3 -2 -1 0 1 Horizontal position [Å] En er gy , P ot en tia l [ eV ]   Charge at 2 Å from center (Non-relaxed) Charge at 2 Å from center (Relaxed) 5 10 15 20 25 -6 -4 -2 0HOMO ~3.6eV ~2.9eV  Figure 3.13 Energy levels and potential profile of the (5,5) SWNT in the 1x  direction showing the effect of relaxation with an external electron at 2 Å from the center. Horizontal axis indicates distance from the nanotube axis.  Chapter 3. Secondary Electron Emission from Carbon Nanotubes 37  3.3.3 The (8,0) SWNT The above simulations were also performed on an (8,0) SWNT. The results are not discussed in detail here since the general behaviours are very similar to those observed for the (5,5) SWNT. We expect less secondary emission from semiconducting nanotubes at higher scanning energies (> ~7.5 keV) as more energy is needed for HOMO-LUMO or other occupied-unoccupied state transitions than for metallic nanotubes. 3.4 Summary In summary, the effect of primary electrons on the electronic structure of carbon nanotubes was investigated using ab-initio simulations in order to gain insight into the possible scenarios of SE emission from nanotubes. The addition of an extra electron as a background charge particle to the system, mimicking the primary beam of an SEM, causes an upward shift in the HOMO and could make it easier for nanotube electrons to overcome the vacuum barrier or tunnel out. This effect is present regardless of where the extra electron is positioned, unless it is too far away (~10 Å or more from the center of the nanotube). It was observed that primary electrons with lower energies (a few hundred eV to ~10 keV) can transfer enough energy to the nanotube electrons to allow secondary emission. This is consistent with the fact that in experiments it is usually easier to observe nanotubes at low primary beam energies (for example at 500 eV). For primary electron energies of more than ~10 keV, the probability of SE emission from single electronic excitation is low. In such cases, either a multiple-step process or other mechanisms such as energy absorption by surface plasmons and subsequent transfer to other electrons to be emitted might be the root cause.  38  Chapter 4 Electron Yield of Carbon Nanotube Forests9 In this chapter, experimental electron yield data obtained from nanotube forests are presented. Electron yield was measured for a wide range of primary beam energies (400– 20,000 eV) in an SEM. Sections 4.1 and 4.2 describe the nanotube growth process and measurement setup in detail and the next section explains the SE and BSE coefficients obtained through the experiments. It was observed that SE and BSE emission behaviours in these forests are quite different than in bulk materials. This seems to be primarily because of the increased penetration range of electrons due to the porous nature of the forests and dependent on their structural parameters, namely nanotube length, diameter and inter- nanotube spacing. 4.1 Carbon Nanotube Forest Fabrication The nanotube forests were fabricated using the following process: Photoresist was coated on a highly doped p-type silicon substrate and circular patterns with diameters varying from 50 to 500 µm were defined using photolithography. About 10 nm of aluminum and 2 nm of iron were then deposited using electron-beam evaporation. Circularly patterned catalyst islands of aluminum/iron were thus obtained after lifting-off the photoresist.  Nanotube growth was done using chemical vapour deposition (CVD). A typical growth entailed heating the sample up to 750 o C while flowing argon at 1000 sccm until the temperature was stabilized. At 750 o C, the sample was annealed for 3 minutes while reducing the flow rate of argon to 200 sccm and adding hydrogen at 500 sccm. Ethylene (at a rate of 20 sccm) was then introduced in the reaction chamber and the flow of argon and hydrogen were reduced to 120 sccm and 80 sccm, respectively for 30 minutes for nanotube growth. This recipe grew  9 A version of this chapter has been published in a peer-reviewed journal (Reused with permission from “M. K. Alam, P. Yaghoobi, and A. Nojeh, ‘Unusual Secondary Electron Emission Behavior in Carbon Nanotube Forests,’ Scanning, vol. 31, pp. 221-228, 2009”, Copyright 2009, John Wiley and Sons).  Chapter 4. Electron Yield of Carbon Nanotube Forests 39  forests as tall as 1 mm long consisting of MWNTs as confirmed by electron microscopy (Figure 4.1).  4.2 Experimental Procedure The forests were then placed inside a Philips 525M SEM chamber for the measurement of the SE and BSE yield. The schematic of the experimental apparatus is shown in Figure 4.2.  Figure 4.1 (a and b) Scanning electron micrographs of patterned nanotube forests. In (a) forests with a diameter of 500 µm and lengths of approximately 1 mm are shown. (b) shows a zoomed-in view of a 50-µm-diameter forest (field of view = 51.3 µm). (c) Transmission electron micrograph of one individual nanotube extracted from the forest, revealing more detail about the nature of the grown nanotubes (mutiwalled with diameters of the order of 10 nm).   Chapter 4. Electron Yield of Carbon Nanotube Forests 40  We followed a procedure similar to that used by others for the measurement of total electron yield from various materials [18, 20, 36], as well as for SE and BSE coefficients [35, 68]. The primary beam current (IP) was measured using a Faraday cup and the specimen current (IT) was measured with a high precision programmable Keithley 6517A electrometer (interfaced with a PC) at various primary beam energies (0.4–20 keV) for different bias values (+50 V, 0 V, –50 V, –100 V, –150 V, –200 V) applied by the electrometer’s internal voltage source. The SEM detector was turned off throughout the experiments (except for initially bringing a forest under the beam and focusing on it). In all experiments, the primary beam was scanned over an area of ~20×15 µm2 at the speed of 31.25 ms per frame (62.5 lines per frame and 0.5 ms per line) and positioned at the center of the top surface of the nanotube forest. Therefore, given that the smallest-diameter forest used in the experiments  Keithley 6517A IT El ec tro n G un  A IP ITE Specimen VT IP  = Primary beam current ITE  = Total emitted electron current IT  = Specimen current VT  = Applied voltage to the specimen  Figure 4.2 Schematic of the experimental setup in the SEM (arrows show the assumed direction of current for which formulas are given in the calculations). The specimen (forest) is connected to the Keithley 6517A electrometer (internal voltage source and current meter are in series) using vacuum feed-throughs. The body of the SEM column and the chamber wall are grounded.   Chapter 4. Electron Yield of Carbon Nanotube Forests 41  had a diameter of 50 µm (see the results and discussion – Section 4.3), the beam was always far away from the vertical side walls of the forest for all the specimens. The movement between the Faraday cup and the specimen was controlled by the automatic stage control system of the SEM. Each experiment was performed on a single forest, which was kept the same for the entire energy range for consistency. At each value of specimen bias, the total electron yield (TEY), δT, (note that historically, in the literature the total electron yield is sometimes referred to as secondary electron yield [18, 20, 36]. Here, to avoid confusion with SE coefficient, we chose the term total electron yield) was calculated using Eq. (4.1).  TE TT P P 1I I I I δ = = +  (4.1) IT and IP were defined above (Figure 4.2) and ITE is the total emitted electron current. This is deduced from the fact that ITE=IP+IT. The BSE coefficient, that is the ratio of the number of electrons exiting the specimen having kinetic energy greater than 50 eV to the number of primary electrons hitting the specimen, was calculated by measuring the primary and BSE current. The BSE current was measured by applying a bias, VT, of +50 V to the specimen, which retains any electron having a kinetic energy of less than 50 eV. The SE coefficient is defined as the ratio of the number of electrons exiting the specimen having a kinetic energy less than 50 eV to the number of primary electrons hitting the specimen. To measure this, the difference between the values of total yield with bias voltages of –50 V (ensuring that all secondaries along with BSEs were emitted from the specimen) and +50 V (ensuring that no secondaries escaped the specimen) was used [5]. (Note that the value of 50 eV used for separating SEs from BSEs is somewhat arbitrary and historical, but is widely accepted in the literature and we used it for consistency). SE and BSE coefficients were then calculated from the Eqs. (4.2) and (4.3).  T T P ( 50 )BSE coefficient : 1 I V V I η = + = +  (4.2)  Chapter 4. Electron Yield of Carbon Nanotube Forests 42   T T T T P ( 50 ) ( 50 )SE coefficient : I V V I V V I δ = − − = + =  (4.3) It is important to note that there are three main sources of secondary electrons: SE1- SE generated directly by primary electron in the specimen, SE2- SE generated by BSE in the specimen and SE3- SE generated by BSE hitting the SEM column or chamber wall. Sometimes an outer collector electrode and a grid biased at –50 eV relative to the collector are used in order to prevent the SE3 contribution [5, 69]. The accuracy of this method mainly depends on the backscattering coefficient of the collector and the transparency of the grid [5, 7, 70, 71]. Another method to minimize the SE3 contribution is to have a very small exposed area of the specimen [5, 68, 70]. As the specimen surface (which is in our case ~ < 8×8 mm2) is much smaller than the collector surface (the entire SEM chamber wall in our setup), an SE3 generated at the SEM column or chamber wall by a BSE is very unlikely to hit the specimen again, but likely to be collected by the opposite side of the chamber wall and thus not contribute to the specimen current [5]. Moreover, at negative specimen bias (– 50 V or more negative), SE3 cannot make it back to the specimen also and the only concern is for the positive bias value of +50 V [68]. However, as reported by Sim and White [68], the contribution of such electrons toward TEY is very low (for gold - atomic number = 79 - it is less than 3% and for silicon - atomic number = 14 - about 1%). Given that BSE yield decreases with the decrease of the atomic number [5], we believe that SE3 is not significant for nanotubes (less than 1%) and Eqs. (4.2) and (4.3) are valid in our experimental setup where exposure to SE3 was minimized. In addition, in the literature there is a very large variation in reported values of TEY, SE and BSE coefficients. For example, for carbon, at 5 keV, TEY values in the range of 0.211~0.501- a variation of up to ~60% - have been reported [40, 72]. Also, even with the grid-collector method there could be an error due to the scattering from grid and collector (e.g. 0.3% error was estimated for a sample with no more than 3 mm in diameter [70, 71]. Given the above, we believe that the use of this experimental configuration (without grid and outer collector electrode) is justified. Furthermore, a similar experimental configuration was used for measuring SE and BSE coefficients by other authors [35, 68].  Chapter 4. Electron Yield of Carbon Nanotube Forests 43  To characterize the experimental errors arising from the current measurements, we performed a series of assessments on the measurement setup. The nominal noise margin of the electrometer (Keithley 6517A) is 0.75 fAp-p (peak-to-peak). However, we observed a fluctuation of up to ~9 fAp-p over a period of 30 minutes (which is longer than the duration of any of our measurements) in the steady state when the electrometer was connected to the experimental setup with the primary beam of the SEM being off. The same measurement was also done with the primary beam on and no additional effect was seen on the noise level. The built-in filters (averaging and median filters) of the electrometer were enabled throughout the experiments to minimize the low current measurement errors. The averaging filter computes moving averages over 10 data points taken at 16.67-ms intervals. The median filter takes the median of each set of 3 data points obtained from the averaging filter. To further increase the accuracy of the current measurements, each data point (for both primary and specimen current) in our experiment was calculated from the average of ~100 of those filtered measurements, each taken at one-second intervals. The primary beam and specimen currents were in the pico-ampere range (0.2–3 pA), which is much higher than the inherent noise level (~< 9 fA) of the setup. The primary current was measured both before and after the experiments at each primary energy to observe the effect of beam current fluctuations. For most of the primary energy range (1.5–20 keV), the beam currents were in the range of 1–3 pA and a variation of less than 4% was measured over the course of each experiment. For primary energies less than 1.5 keV, the beam currents were in the range of 0.2–1 pA and a variation of less than 10% was measured. The above errors and fluctuations were used to obtain the corresponding errors (standard error of the mean) in our calculated TEY, BSE and SE coefficients, which appear as error bars on Figures 4.3, 4.4, 4.6 and 4.7. (For most data points the error bars are smaller than the curve markers and are masked by them). A note is also in order on the issue of the repeatability of the behaviours observed: The experiments were performed on three different sizes of the forests (50, 200 and 500 µm in diameter). Some of the data points were also repeated several times for each specimen and a variation of less than 5% was seen among the different experiments. No significant charging from the primary beam was observed as the specimen was connected to a charge reservoir.  Chapter 4. Electron Yield of Carbon Nanotube Forests 44  4.3 Results and Discussion Figures 4.3 and 4.4 show the TEY and SE coefficient, respectively, from a nanotube forest having a diameter of 500 µm and length of ~1 mm for different biases and primary beam energies. The dotted vertical line indicates that the experiments in the 5–20 keV range were done in a separate sitting compared to the 0.4–5 keV experiments. The reason is that we had to change the SEM’s anode after low-keV experiments to enable high-keV experiments. Therefore we had to vent the chamber in-between the two experiments. We kept all the conditions (sample location, working distance, etc) exactly the same in both experiments to prevent potential variations. To ensure that the conditions were the same in both cases, we measured the data points for 5 keV during both experiments and observed a difference of less than 3%. Nonetheless, we added this dotted vertical line in the interest of providing the full experimental details.  0 5 10 15 20 0 0.5 1 1.5 2 2.5 3 Primary beam energy [keV] TE Y   VT= +50V (Backscattered) VT= 0V VT= –50V VT= –100V VT= –200V5–20 keV 0.4–5 keV  Figure 4.3 TEY from a carbon nanotube forest with a diameter of 500 µm. Legends show the corresponding applied voltages to the specimen. The dotted curve (diamond marker) represents the backscattering coefficient. The vertical line indicates that we had to exchange the anode in-between the low-keV and high-keV experiments. The 5 keV data points were measured in both experiments. They were very close (difference of less than 3%) and here the values obtained with the low-keV anode are plotted for the 5-keV point.  Chapter 4. Electron Yield of Carbon Nanotube Forests 45  The dotted curve (diamond marker) in Figure 4.3 (obtained with a bias of + 50 V) represents the backscattering coefficient for the 500-µm-diameter forest. On the other hand, negative voltage applied to the sample favours the escape of SEs. As can be seen on the figure, as the negative bias is increased, more SEs acquire enough energy to overcome the vacuum barrier and a higher electron yield is obtained at each primary beam energy. Figure 4.4 was obtained from Eq. (4.3), namely by calculating the difference between the two curves with square (TEY at –50 V) and diamond (TEY at +50 V) markers in Figure 4.3. It is known that energy loss inside a bulk material decreases with increasing the primary beam energy [5, 73]. Hence, the total yield and SE coefficient should decrease with increasing beam energy after the maximum yield point, which is usually in the range of 0.1~1 keV for solid materials [5, 7, 73]. A 500-µm-diameter solid cylinder should behave like a bulk for the energy range under consideration since the penetration range (in depth but also laterally) of electrons at such energies is only a few micrometres in solids (e.g. ~5 µm 0 5 10 15 20 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Primary beam energy [keV] SE  c oe ff ic ie nt  Figure 4.4 SE coefficient of the nanotube forest with a diameter of 500 µm (obtained from Figure 4.3 by calculating the difference between the curves of TEY at –50 V and +50 V).   Chapter 4. Electron Yield of Carbon Nanotube Forests 46  for graphite at 20 keV [68, 73, 74]). Indeed, even for the nanotube forest the SE coefficient seems to eventually decreases with primary energy after 18 keV (although we have only one data point beyond 18 keV for this specimen, the decrease in SE coefficient is quite obvious from the experimental results for the 50-µm-diameter forest, given later in this section). However, the nanotube forest exhibits a peculiar behaviour in the intermediate range of primary energies: as can be seen from Figures 4.3 and 4.4, both the BSE and SE coefficients increase after 10 keV, in contrast with solid materials (the BSE coefficient in solids also decreases with the increase of primary beam energy for materials with low atomic number such as carbon [4, 5, 40, 72]). We believe this happens because both high energy electrons (BSEs) and low energy electrons (SEs) can escape from the side walls in addition to the surface of the nanotube forest. This unusual behaviour was not observed in the SE coefficient of a nanotube mat (a wide, unpatterned nanotube forest), where the lateral dimensions are so large that electrons in this primary energy range cannot escape from the sidewalls, as measured by Huang et al. [35]. However, given the 500 µm diameter of the forest used in our experiments, for this hypothesis to be true still a very high penetration range for primary electrons in the nanotube forest is necessary. We believe this to be the case because of the porosity of the structure made up of many individual MWNTs. To estimate the inter-nanotube spacing, we performed an experiment of liquid-induced shrinkage of MWNT forests and the forests were found to shrink approximately 4.5 times after the liquid was introduced. Assuming that all the nanotubes in the forest have a diameter on the order of 10 nm (see Figure 4.1(c) and that the nanotubes were fully packed (hexagonal packing) after shrinking, we estimate an average distance of ~36 nm between two neighbouring nanotubes in the original sample. A similar liquid-induced shrinkage was demonstrated by Futaba et al. for SWNT forests [75]. To investigate the hypothesis of unusually high electron range in the nanotube forest, we performed Monte Carlo simulations using the program NISTMONTE [44]. A 1.4-µm- diameter solid graphite cylinder was chosen to be simulated to compare with a cluster of 20- nm-diameter cylinders (representing nanotubes) having a gap of 40 nm between neighbouring cylinders and an overall diameter of 1.4 µm (Figure 4.5). The diameter of the  Chapter 4. Electron Yield of Carbon Nanotube Forests 47  “nanotubes” and the overall diameter were chosen according to computational limitations and do not correspond exactly to the experimental values. However, as the simulations were performed at a primary beam energy of 5 keV at which the electron range is approximately 0.5 µm for graphite, they could still provide a qualitative insight into the variation of electron range between the two simulated structures.    (b) (a) (c) 1 μm 0.5 μm (d) 1 μm 0.5 μm  Figure 4.5 Monte Carlo simulations showing the electron trajectories for comparison of electron range between a solid material and nanotube forest at 5 keV. Among the 10000 simulated trajectories, the first 50 are shown. (a) Top view of the trajectories in the solid graphite cylinder of 1.4 µm, (b) The same as (a) but for a cluster of nano graphite cylinders with a diameter of 20 nm each, having an overall cluster diameter of 1.4 µm, representing the nanotube forest, (c) Side view of the trajectories in solid graphite, and (d) Side view of the trajectories in the nanotube forest. (Red lines in (a) and  (b) and blue lines in (c) and (d) indicate that electrons are moving in empty spaces and the straight lines at the end of the trajectories show the flight of the backscattered electrons.)  Chapter 4. Electron Yield of Carbon Nanotube Forests 48  10000 electron trajectories were simulated for each structure at a 5 keV primary beam energy and it was found that the total BSE coefficient (including surface backscattering and sidewall escape) increased from ~0.08 in solid graphite (no sidewall scattering as the range is smaller than the diameter of the cylinder) to ~0.4 in the cluster mimicking the nanotube forest. In the case of the nanotube forest, both surface backscattering (~45% of the total backscattered electrons) and sidewall escape (~55% of the total backscattered electrons) happened, in agreement with our explanation for the increase in TEY, BSE and SE coefficients. In addition, it was also seen that surface backscattering was increased by ~10% because of the porous nature of the structure. For further investigation, the simulations were repeated at 6 keV primary beam energy (note that the range of primary electrons is still less than 1.4 µm in solid graphite at this energy). It was found that the total BSE coefficient was decreased to 0.077 in solid graphite, whereas it was increased to 0.65 in the cluster (surface backscattering decreased to ~38.5%, but sidewall escape increased to ~61.5%), again consistent with our explanation of the observed trend  in BSE and SE coefficients based on the porosity of the forest. The experiment was repeated with a nanotube forest of 50 µm in diameter for further validation of this hypothesis. The results are presented in Figure 4.6 and 4.7. It is seen that the minima of the TEY are shifted down to approximately 1.5 keV (for the 500-µm-diameter forest they were at about 10 keV). This shift seems to be primarily due to a similar shift in the SE coefficient (Figure 4.7). This behaviour is consistent with the previously discussed conjecture that the increase in the SE coefficient beyond a certain energy is due to the escape of secondaries from the sidewalls of the porous forest structure: for a forest with smaller diameter, it is expected that primary electrons with less energy will be able to reach the sidewalls of the forest to generate secondaries (for solid graphite the range is less than 0.1 µm at 1.5 keV). Also, eventually the SE coefficient starts decreasing with primary energy, like in bulk materials and in the case of the wider forest.  However, this decreasing trend begins at a smaller primary energy (5 keV) compared to the wider forest (where the decreasing trend started at 18 keV).  Chapter 4. Electron Yield of Carbon Nanotube Forests 49  0 2.5 5 7.5 10 12.5 15 0 0.5 1 1.5 2 2.5 3 Primary beam energy [keV] TE Y   VT= +50V (Backscattered) VT= 0V VT= –50V VT= –100V VT= –150V 5–15 keV 0.4–5 keV  Figure 4.6 TEY from a carbon nanotube forest with a diameter of 50 µm. Legends show the corresponding applied voltages to the specimen. The dotted curve (diamond marker) represents the backscattering coefficient. The 5-keV data points were measured in both low- keV and high-keV experiments. They were very close (within 0.2%) and here the values obtained with the low-keV anode are plotted for the 5-keV point.  Chapter 4. Electron Yield of Carbon Nanotube Forests 50  Interestingly, and contrary to the case of the wider forest, the BSE coefficient does not increase with primary energy and stays almost constant slightly after the minimum point. We believe the reason behind this is that the length of the narrower forest is also much less than that of the wider one (about 100 microns as opposed to 1 mm: The shorter length in this case is due to the fact that the small-diameter forest cannot stand straight if made too long). Therefore, we expect that a large number of electrons easily reach the substrate and are captured by it, as opposed to escaping from the sidewalls of the forest and contributing to the emitted current. This also helps further explain the decrease in the SE coefficient after 5 keV: it appears that the BSE coefficient stays almost constant with increase in primary energy, implying that a constant number of primary or BSEs reach the sidewalls. However, as the energy of these electrons is increased, their energy loss to the sample is reduced, leading to a decrease in SE emission, like in bulk materials. As a final note, obviously with an almost constant BSE coefficient and decreasing SE coefficient with primary energy beyond 5 keV, the total electron yield also decreases, as seen on Figure 4.6. 0 2.5 5 7.5 10 12.5 15 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Primary beam energy [keV] SE  c oe ff ic ie nt  Figure 4.7 SE coefficient of the nanotube forest with a diameter of 50 µm (obtained from Figure 4.6 - the difference between the curves of TEY at –50 V and +50 V).   Chapter 4. Electron Yield of Carbon Nanotube Forests 51  Nanotube forests with a diameter of 200 µm and a length of ~ 1 mm also showed electron emission behaviour similar to the case of the 500-µm-diameter forest. The data for the 200- µm-diameter forests are presented in the next chapter along with the Monte Carlo simulation results. It is worth mentioning that in addition to the qualitative deviation from secondary emission behaviour in bulk materials discussed above, in general, electron yield from these forests was found to be higher than carbon, graphite and diamond-like-carbon (DLC) films [7, 40, 72, 76]. Figure 4.8 shows a comparison between the SE yield from the nanotube forests and those other forms of carbon, as well as a few other materials. As can be seen from Figure 4.8, nanotube forests do not have the highest pure SE emission yield (SE coefficient) of all the materials. However, they may have other advantages due to their structural strength and chemical stability. Moreover, as the total yield can be increased 1 2 3 4 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Primary beam energy [keV] SE  co ef fic ie nt   500-µm nanotube forest 50-µm nanotube forest Carbon Graphite DLC SiO2 Au MgO GaAs  Figure 4.8 A comparison of SE coefficient between nanotube forests and other forms of carbon and several other materials. All the data have been taken from Joy’s database [40,72] and references therein except for the DLC film [76] and the nanotube forest (this study).   Chapter 4. Electron Yield of Carbon Nanotube Forests 52  significantly by the application of bias (see Figures 4.3 and 4.6), such nanotube structures can be considered as candidates for electron multipliers and vacuum transistors. Previously, highly enhanced secondary emission (electron yield greater than 100) was observed from the tip of an individual nanotube lying on a dielectric surface and biased near the threshold of field-emission [19]. One of the long-term goals is to create structures using collections of nanotubes to obtain even higher electron gains as a potential candidate for vacuum nano- electronics. This work was focused on the characterization of the TEY, SE and BSE coefficients as a preliminary step in that direction. If the observed behaviour in our experiments is indeed because of the porosity of the structure as we suggested, then by controlling the nanotube density in the forest, one may be able to control the electron yield at different energies, which could have applications in energy-selective electron detectors and multipliers. In other words, the internal structure of the forest may provide an additional degree of freedom compared to bulk materials in designing electron emission devices. In addition, electron microscopy of nanoscale materials in itself can be an interesting and challenging problem, since charging and contamination could play a major, even dominant, role [22, 23, 28, 31]. A study of electron yield from nanostructures such as presented here can provide additional insight into imaging mechanisms and optimal imaging conditions (for example, the choice of the primary beam energy to reduce or prevent sample charging). 4.4 Summary A systematic experimental study of electron yield from nanotube forests was presented. Total yield, backscattered and secondary coefficients were measured from patterned nanotube forests and a different electron emission behaviour compared to bulk materials was observed. This is believed to be because of the unusually high range of electrons in nanotube forests, which could be ascribed to the porous nature of the nanotube forests. These results may present a useful step in the study of the interaction between electrons and nanotube collections for the development of various vacuum nano-electronic devices such as electron multipliers. 53  Chapter 5 Modeling of Electron Yield from Carbon Nanotube Forests10 In this chapter, a new Monte Carlo tool, capable of simulating electron trajectories in nanotube forests, taking into account the underlying nanoscale nature of the material is presented. The Monte Carlo method has been used extensively for the simulation of electron trajectories and yield from bulk materials [4, 77]. The material is typically implemented as a homogeneous, isotropic structure for the purpose of these Monte Carlo simulations. However, a nanotube forest is a semi-regular array of a large number of sub-structures (individual nanotubes) with empty spaces in-between, making it a non-homogeneous and anisotropic structure. One would expect that such a structure cannot be treated in the same manner as a regular bulk material in a physically meaningful way. In the previous chapter, we presented BSE and SE emission data from nanotube forests. The electron emission behaviour from the forests was found to be different than that from bulk materials. This was attributed to the porous nature of the structure that leads to an unusually high electron penetration range (distance travelled by the primary electron inside a specimen before it loses all of its kinetic energy). Here, a semi-empirical model for electron backscattering from nanotube forests is presented using physically-meaningful empirical parameters. These parameters bring additional degrees of freedom into the simulation that can be directly correlated with the internal structure of the forests. It is also of fundamental interest to investigate the possibility of building a Monte Carlo tool for macroscopic-size structures, incorporating their internal micro/nanostructure explicitly into the simulation. The results of Monte Carlo simulations based on this model are compared with the experimental data. The  10 A version of this chapter has been published in a peer-reviewed journal (Reused with permission from “M. K. Alam, P. Yaghoobi, and A. Nojeh, ‘Monte Carlo Modeling of Electron Backscattering from Carbon Nanotube Forests,’ Journal of Vacuum Science and Technology B, vol. 28, pp. C6J13-C6J18, 2010”, Copyright 2010, American Vacuum Society). Parts of this chapter were also presented in the 54th International Conference on Electron, Ion and Photon Beam Technology and Nanofabrication (EIPBN), Anchorage, Alaska, USA, June 1-4, 2010.  Chapter 5. Modeling of Electron Yield from Carbon Nanotube Forests 54  applicability of the semi-empirical SE model for solids to the nanotube forest is also investigated. 5.1 The Model Our model is based on the widely accepted Monte Carlo model by Browning et al.[47]. In this model, the step length of an electron is obtained from [4, 77]:  1ln ,s Rλ= −  (5.1) where s is the step length (which follows a Poisson distribution), λ is the elastic MFP and R1 is a uniform random number between 0 and 1. The elastic MFP can be estimated from the density of the target and its scattering cross-section, ( , A T A N λ ρ σ = where A is the atomic weight, NA is the Avogadro number, ρ is the density of the material and Tσ , given by Eq. (2.1),  is the total elastic cross-section) [4]. The scattering angle (θ) is estimated from a fitted Mott differential cross-section. The fitted cross-section is composed of two parts: the screened Rutherford and isotropic distributions [47, 48], which lead to the following forms for scattering angle distribution:  2 2 2cos 1 (Rutherford scattering), 1 R R α θ α = − + −  (5.2)  3cos 1 2  (Isotropic scattering),Rθ = −  (5.3) where α is the screening parameter and R2 and R3 are uniform random numbers between 0 and 1. Only the screened part of the cross-section depends on the screening parameter. For a pure bulk material, α is calculated from:  37 10 , E α −× =  (5.4)  Chapter 5. Modeling of Electron Yield from Carbon Nanotube Forests 55  where E is the incident electron energy in keV. The azimuthal angle (ϕ) is calculated from a uniform distribution:  42 ,Rφ π=  (5.5) where R4 is a uniform random number between 0 and 1. Figure 5.1 shows typical electron trajectories and a single step of a Monte Carlo simulation for a bulk material. At each small step of the simulation, the primary electron is scattered away by a distance s. The scattering angles are calculated from Eqs. (5.2)–(5.5). As the scattering cross-section and the screening parameter depend on the incident energy of the electron, the energy loss at each small step is also needed. The incident energy at each step is calculated based on the energy loss rate obtained from Eq. (2.2). If an electron exits from the material with an energy greater than 50 eV, it is counted as a BSE. This method has been extensively used for bulk materials [38, 44]. On the other hand, as mentioned earlier, a nanotube forest is an array of many individual nanotubes (Figure 5.2), and may behave differently from a bulk solid. For the purposes of this work, we used a standard procedure with the following modifications:  (a) (b)  e- s θ  ø  Figure 5.1 Electron trajectories in a typical Monte Carlo simulation, showing the randomness of beam-specimen interaction and (b) one small step in a single trajectory (s = step length, θ = scattering angle and ϕ = azimuthal angle).   Chapter 5. Modeling of Electron Yield from Carbon Nanotube Forests 56  We defined the forest as a rectangular array of nanotubes [Figure 5.2 (b)], although this does not diminish the generality of the model. Each grid point indicates the center of a nanotube in the forest. We estimated the average diameter of the nanotubes (10 nm) and the spacing between nanotube walls (36 nm) from electron microscopy observations and experiments on liquid-induced shrinkage of the forests, as discussed in the previous chapter (Chapter 4). Another important parameter is the density of the forests. It is known that backscattering from the surface of a material is not highly sensitive to density [4]. However, scattering observed from the side of these forests at high primary beam energies seems to be due to the low density and porous nature of the forests. For a given chirality of nanotubes and number of walls per MWNT, the density can be obtained from:  TA AL lDensity( )= ,N N mρ × ×  (5.6) where NTA is the number of nanotubes per unit area, NAL is the number of atoms along the nanotube axis per unit length of the nanotube and ml is the mass of a carbon atom. Due to the low level of control in existing fabrication processes, not only do these parameters vary from forest to forest, even within one forest the nanotubes will have different chiralities and numbers of walls. Therefore, in practice the density of MWNT forests can be estimated to be  Cross-section of the forest (a) (b)  Figure 5.2 Micrograph and schematic of a circularly patterned nanotube forest and (b) top view of the nanotube grid used in Monte Carlo simulations. (Nanotube diameter and inter- nanotube spacing have been exaggerated in the schematics for clarity)   Chapter 5. Modeling of Electron Yield from Carbon Nanotube Forests 57  on the order of 210−  g/cm3 for an inter-nanotube spacing of ~36 nm (distance between the outer walls of adjacent nanotubes). For example, for a 4-wall MWNT including chiralities of (95,0), (104,0), (113,0) and (122,0), the density is 0.0296 g/cm3. Futaba et al. estimated the density of their SWNT forests to be on the order of 0.029 g/cm3 (calculated for an inter- nanotube spacing of 16.4 nm and nanotube diameter of 2.8 nm) [75]. Given these variations, we kept the density as an empirical parameter to be obtained by fitting to the experimental data. Next we turn to the treatment of the scattering angle, which constitutes the main difference between the proposed model for nanotube forests and the bulk model. Figure 5.3 shows a typical distribution of scattering angles (obtained from 10000 scattering events) and its dependence on the screening parameter (α). A low value of α indicates a higher probability of low angle scattering and vice versa. Given a fixed value of α for the simulation, the scattering angle probability distribution depends only on the incident energy. For nanotube forests, which consist of individual nanotubes with empty spaces in-between, the idea is to adaptively shift the scattering angle distribution at each step of the simulation based on the position of the primary electron. The rationale is that, for a given step of the Monte Carlo simulation, larger scattering angles are to be favoured if an electron scatters from the wall (Figure 5.4, R-2) of a nanotube compared to when it scatters in the empty spaces between nanotubes (Figure 5.4, R-3) or inside a nanotube (Figure 5.4, R-1). One way to achieve the required modification may be to adaptively vary the width of the scattering angle distribution by adopting a higher value for the screening parameter for steps where the primary electron ends up on the nanotube wall (Figure 5.4, R-2) and a lower value when the scattering site is far from the nanotube wall (R-1 and R-3 in Figure 5.4). However, this approach still would not take the anisotropic nature of the forest into account. Therefore, in the proposed model, rather than changing the width of the scattering angle distribution, we shifted the center of the distribution according to the location of the scattering site in each step. To implement this, the effective nanotube shell thickness (Figure 5.4) was introduced as an empirical parameter.  Chapter 5. Modeling of Electron Yield from Carbon Nanotube Forests 58    5 10 15 20 25 30 0 500 1000 1500 2000 2500 Scattering angle (degree) Pr ob ab ili ty  d is tr ib ut io n   9×10-3/E 7×10-3/E 6×10-3/E  Figure 5.3 Typical distribution of the scattering angle (for 10000 scattering events with 100 energy bins) and change in the distribution with the change of screening parameter.   4.75 nm 5 nm  ec ei ef R-1 R-2 (in the shell) R-3 Effective nanotube thickness  Figure 5.4 Schematic representation of the nanotube grid (R-1 = hollow region inside the shell; R-2 = in the shell; R-3 = outside the shell) and proposed change in scattering angle distribution for the screened part of the fitted differential cross-section in a particular step of the simulation, where the scattering site falls within a nanotube shell (R-2). ei, ef and  ec are the directions of the incident electron, center of the scattering angle distribution for the bulk model and the shifted center of the scattering angle distribution for the forest, respectively.  Chapter 5. Modeling of Electron Yield from Carbon Nanotube Forests 59  The position of the primary electron relative to the surrounding nanotubes was checked at each step of the simulation. Note that we did not need to save the positions of all the nanotubes in the memory, which would have been prohibitive due to the extremely high number of nanotubes forming the forest. Instead, having them on a regular grid allowed us to easily determine the relative position of the electron and the nanotubes in its close proximity at every step. This makes the simulator memory-efficient and allows the simulation of macroscopically-sized forests in a reasonable time on a regular computer. If the primary electron fell within the shell of a nanotube (Figure 5.4, R-2), the center of the scattering angle distribution (the direction of forward movement) was changed according to the classical elastic collision law to favour large angle scattering. This is illustrated in Figure 5.4 where the direction of forward movement is shifted from the ef direction to the ec direction. In this way, the inhomogeneous and anisotropic nature of nanotube forests was incorporated in the simulation and the empirical parameters were found by fitting to experimental data. Note that in reality the nanotubes in the forest, although overall highly aligned, are not perfectly straight and have wiggle and physical entanglement with each other. Since the exact geometry of all the nanotubes in the forest cannot be known, we include this effect by assuming that the (straight) nanotubes are in an effective background space (R-1 and R-3 in Figure 5.4), where electrons do scatter, although without shifting the center of the scattering angle distribution. In addition, a portion of the side-wall scattered electrons hit the substrate. Hence, the substrate was also included in the simulation and treated with the standard bulk model [47]. A detailed flow diagram of the Monte Carlo procedure is given in Appendix B (Figure B.1). 5.2 Results and Discussion At first, we tried to match the experimental data using the traditional bulk model. The result is shown in Figure 5.5 for the 500-μm-diameter forest. In order to obtain a good fit with experimental data, the screening parameter needed to be increased to 9×10-3/E with a density of 0.019 g/cm3. The higher value of the screening parameter indicates a higher probability of large-angle scattering in the forest compared to bulk solids. This seems to be nonphysical  Chapter 5. Modeling of Electron Yield from Carbon Nanotube Forests 60  because the nanotube forests are full of empty spaces and on an average smaller-angle scatterings are expected to be favoured. Large angle scatterings are expected to be favoured only if an electron hits a nanotube. Next, we used the proposed model to simulate the BSE yield of nanotube forests and estimated the empirical parameters. Figure 5.6 shows 100 electron trajectories simulated by our Monte Carlo simulator at a primary electron energy of 20 keV. The simulator was implemented in MATLAB. The straight lines at the end of the trajectories indicate that electrons are escaping from the surface or side of the forest. As can be seen, sidewall escape is the major source of BSE at high energies as hypothesized in the previous chapter. 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 Primary beam energy [keV] Ba ck sc at te rin g co ef fic ie nt   Experimental data Simulation  Figure 5.5 Comparison of experimental and simulation results for the electron backscattering coefficient from 500-μm-diameter nanotube forest using the adjusted bulk model.   Chapter 5. Modeling of Electron Yield from Carbon Nanotube Forests 61  Figures 5.7 and 5.8  show experimental data for nanotube forests with diameters of 500 μm and 200 μm, respectively, and the corresponding simulation results using our proposed model. Most of the error bars in the experimental data are masked by the curve markers. Trajectories of 10,000 electrons were simulated at each primary electron energy in all the simulations. 10,000 electron trajectories were enough to make the statistical variations negligible compared to the actual yield for the quantities of interest. The experimental dataset shown in Figure 5.7 was initially used for finding the empirical parameters. The same values of screening parameter and shell thickness were then used in order to simulate the data of Figure 5.8; however, a new value of density (Table 5.1) was found to improve the fit, in particular in the high-energy region. The detailed procedure for finding the empirical parameters is discussed in Appendix B.  Figure 5.6 100 electron trajectories (primary energy: 20 keV) through a 500-μm-diameter nanotube forest, obtained using the described Monte Carlo program.   Chapter 5. Modeling of Electron Yield from Carbon Nanotube Forests 62   0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 Primary beam energy [keV] B ac ks ca tte ri ng  c oe ff ic ie nt   Experimental data Simulation  Figure 5.7 Comparison of experimental and simulation results for the electron backscattering coefficient from the 500-μm-diameter nanotube forest. 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 Primary beam energy [keV] B ac ks ca tte ri ng  c oe ff ic ie nt   Experimental data Simulation  Figure 5.8 Comparison of experimental and simulation results for the electron backscattering coefficient from the 200-μm-diameter nanotube forest.  Chapter 5. Modeling of Electron Yield from Carbon Nanotube Forests 63  The structural and empirical parameters used for fitting the simulations to experiments are summarized in Table 5.1. Table 5.1 List of parameters for the Monte Carlo simulations. Parameters Value Diameter of the nanotube (nm) [Chapter 4] 10 Nanotube spacing (nm) [Chapter 4] 36 Effective nanotube thickness 5 % of the nanotube radius (0.25 nm) Screening parameter ][ 106 3 keVE −× Total trajectories 10000 Density of the forest (g/cm3) 0.019 for 500-μm-diameter forest  and 0.0356 for 200-μm-diameter forest Beam direction Perpendicular to the top surface  As can be seen, the simulated results are in good agreement with our measured data. The r.m.s. error was found to be less than 0.05 (0.046 for the 500-μm-diameter forest and 0.0139 for the 200-μm-diameter forest). The large angle scatterings from nanotube walls in our model were taken care of by the shift of the distribution (Figure 5.4). Therefore, the lower value of the screening parameter found in our simulation does not contradict with the increased probability of large angle scattering from MWNT walls (Table 5.1). In fact, a lower value of the screening parameter compared to the bulk materials is more physical for regions R-1 and R-3 than the one found from the adjusted bulk model. The new value of the density for 200-μm-diameter forest also gives a reasonable fit with the bulk model (Figure 5.9), but the value needed for the screening parameter is still higher, 9×10-3/E. To understand the reason why the bulk model works with a larger screening parameter, one has to consider the fact that when the forest diameter is significantly larger than the MFP of the electrons, which is the case in the experimental data used here, each  Chapter 5. Modeling of Electron Yield from Carbon Nanotube Forests 64  electron undergoes many scatterings. Due to the anisotropic nature of the structure, some of these scatterings will be to one side of a particular nanotube and some to the other. Although most of the scatterings in the forest are likely to be small-angle scattering because of the high percentage of empty spaces in nanotube forests (large-angle scattering happens where the number of scattering centers, i.e. nanotube sidewalls, are higher), the effect is averaged out over many scattering events and results in an effective increase in the probability of large-angle scattering in all directions for all regions. This is why the bulk model works for this system, albeit with a larger screening parameter for all regions than usual. In addition, our experimental data so far pertain to a case with cylindrical symmetry around the forest axis as the primary electrons impinge on the top surface of the forest perpendicular to it. The situation may be different when this symmetry is broken, such as when the primary beam is incident on the surface of the forest at an oblique angle. It was observed that the proposed model would deviate noticeably from the adjusted bulk model for an 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 Primary beam energy [keV] B ac ks ca tte ri ng  c oe ff ic ie nt   Experimental data Simulation  Figure 5.9 Comparison of experimental and simulation results for the electron backscattering coefficient from 200-μm-diameter nanotube forest using the adjusted bulk model.   Chapter 5. Modeling of Electron Yield from Carbon Nanotube Forests 65  obliquely incident beam even with the new screening parameter (keeping all the parameters the same as before for both models). Figure 5.10 shows a comparison of the predictions of our proposed model with those of the adjusted bulk model for an oblique beam11. It is seen that the adjusted bulk model predicts higher backscattering yield than the proposed model for primary energies up to approximately 15 keV for an obliquely incident beam, whereas it predicts otherwise for perpendicular incidence above 5 keV.  5.2.1 Secondary Electron Yield The SE generation process is very complex in nature. The so-called fast secondary electron (FSE) model was proposed for solids, incorporating only one mechanism of SE generation –  11 The direction of the oblique incident beam was chosen arbitrarily, only as an example to show the effect on the BSE yield. 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 Primary beam energy [keV] Ba ck sc at te rin g co ef fic ie nt   Proposed model (Oblique beam) Adjusted bulk model (Oblique beam) Proposed model (Perpendicular beam) Adjusted bulk model (Perpendicular beam)  Figure 5.10 Comparison of the proposed and adjusted bulk model for the predicted electron backscattering coefficient from the 500-μm-diameter nanotube forest using the extracted empirical parameters. Angles with the axes for oblique incident are: X=90̊, Y= 159.5̊ and – Z=69.5̊.   Chapter 5. Modeling of Electron Yield from Carbon Nanotube Forests 66  outer-shell excitation [4]. As pointed out by Joy, the FSE model is only good for predicting the trends in SE emission behaviour [4]. Quantitatively, FSE gives much lower values than the actual yield for all solid materials. We also verified this issue and found similar results for nanotubes forests. It was found that the FSE model is also insensitive to the density. An improvement has been seen with so-called “first-principles” Monte Carlo models (a form of discrete Monte Carlo simulation) or by adding plasmon excitations into the simulation process [4, 78]. Historically, the spread in experimental data has been very large making the assessment of accuracy of SE emission models a difficult task. Discrete Monte Carlo simulation requires the availability of all the elastic and inelastic cross-sections, knowledge of SE generation process and significant computational resources. One such simulation scheme for individual nanotubes is presented in Chapter 7. In general, researchers have been mainly relying on empirical models of SE generation and diffusion process for solids. For nanotube forests, since we were able to estimate the backscattering data also using a semi- empirical model (and an adjusted bulk model with new screening parameters), an empirical bulk SE model might also be applicable. In this model, the SE yield is calculated using an empirical equation [4]:  1 ,SE z SE dEA e dz λδ ε − = ∫  (5.7) where dE/dz is the electron energy loss rate [calculated from Eq. (2.2)], A is a parameter related to the percentage of SEs going in the direction of the surface of the material, SEε is the average energy needed for the generation of one SE or initiating the cascade process of multiple SE generation and λSE is the escape depth determining the escape probability of SEs. For a nanotube forest, sidewall scattering was also observed in addition to the surface scattering. Although the exact details of the diffusion processes along the nanotube axis direction and the radial direction may not be the same, a first-order simplification could be made, assuming a similar form for the SE diffusion processes in the two directions. Therefore, the above equation takes the following form:  Chapter 5. Modeling of Electron Yield from Carbon Nanotube Forests 67   1 21 2 1 1 , z r forest SE SE dE dEA e A e dz dz λ λδ ε ε − − = +∫ ∫  (5.8) where r is the radial distance from the sidewall of the forest. This equation was applied to the forest along with the BSE model presented in this chapter, assuming A1=0.5 and A2=1 (50% of the SEs are directed to the surface direction whereas electrons can escape through any side of the forests in the radial direction). The results for the two different forests are shown in Figures 5.11 and 5.12. The values of the parameters that were used to match the model results to the experimental data are given in the figure captions. 5 10 15 20 25 30 0 0.5 1 1.5 2 2.5 3 Primary beam energy [keV] SE  y ie ld   SE yield [simulation] SE yield [experiment]  Figure 5.11 The semi-empirically modeled SE emission data and the experimental data for the 500-μm-diameter nanotube forest ( SEε =32 eV, ρ=0.019 g/cm 3 , λ1=300 nm and λ2=300 nm).   Chapter 5. Modeling of Electron Yield from Carbon Nanotube Forests 68  As can be seen from the figures, a reasonable agreement is achievable for SEs escaping through the top surface rather than the sidewall (that is up to the value of primary electron energy where trends yield starts to increase). Although this simple model does show the increase of SE yield at high energies, the exact trend of this increase does not follow that of the experimental data. One reason could be the highly anisotropic nature of the forest, leading to a different behaviour in the regime where sidewall escape becomes dominant. Among the empirical parameters used, it is interesting to note the value of SEε  (32 eV). The smaller value of SEε  (around 3 times lower than that of bulk graphite) suggests that on an average less energy is needed for SE generation from nanotube forests than from bulk carbon. The physical reason for this is not known to us, but it might be related to the role of nanotube tips (where easier SE emission is expected [19]) or the different nature of plasmon excitations in nanotubes compared to bulk carbon (in fact, even in different nanotubes the 0 5 10 15 20 25 30 0 0.5 1 1.5 2 2.5 3 SE  y ie ld Primary beam energy [keV]   SE yield [simulation] SE yield [experiment]  Figure 5.12 The semi-empirically modeled SE emission data and the experimental data for the 200-μm-diameter nanotube forest ( SEε =32 eV, ρ=0.035 g/cm 3 , λ1=200 nm and λ2=75 nm).   Chapter 5. Modeling of Electron Yield from Carbon Nanotube Forests 69  plasmon modes are different, for instance two plasmon modes have been observed in MWNTs compared to one for SWNTs [79]). Also, following the difference in nanotube density for these two forests, λ1 and λ2 were found to be different: The value was smaller for the smaller forest (200-μm-diameter). This directly corresponds to the density of the forest – indicating a smaller escape depth for higher-density material. It can also be noted that side wall escape parameters would vary with the shape of the specimen cross section (e.g. circular or rectangular or any arbitrary shape) and might not have a simple functional form such as the one assumed here. However, if the lateral dimensions of the specimen are greater than the electron penetration range, the second term can be neglected. For example, a 0.019 g/cm3 forest has a penetration range of ~850 μm at 30 keV. Therefore, SE emission could be modeled with a one-term semi-empirical model [Eq. (5.7)] if the lateral size of the specimen is greater than 1.7 mm × 1.7 mm for the typical SEM energy range of up to 30 keV. 5.3 Summary A model for electron backscattering from nanotube forests, taking into account the internal nanoscale nature of the forest structure was proposed and parameterized using experimental data. It was also seen that the traditional bulk model could generate similar results, although with an unusual value for the screening parameter. The proposed model yields a more physical value of the screening parameter. In addition, if the microscopic aspects of the electron movement through the forest and phenomena similar to channelling are of interest, the bulk model cannot provide an adequate representation and more elaborate models, taking the porous forest structure into account, are needed. The model proposed here is a first step in this direction. These results may also be used for modeling SE emission or X-ray microanalysis from nanotube-based structures.    70  Chapter 6 Electron Yield of Suspended Nanotubes12 In this chapter, an effective and simple experimental procedure is presented to directly measure the intrinsic electron yield of individual nanotubes. By using a simple experimental setup under an optical microscope, we made suspended nanotubes, which were free from interaction with the substrate during electron yield measurements. It was found that the secondary electron yield from isolated suspended nanotubes is less than unity and decreases as a function of energy. Recently, Luo et al. reported ultrahigh SE yield from the sidewall of SWNTs lying on a dielectric surface [37]. However, the apparent high yield seems to be an artefact of the measurement setup and analysis used: secondary electrons originating from a large area of the substrate around the nanotube seem to be counted as being from the nanotube13. Here, a detailed discussion is also presented on the short-comings of the work by Luo et al. 6.1 Fabrication of Suspended Nanotubes We used nanotube forests grown using CVD in all the experiments to make suspended nanotubes. The fabrication procedure of nanotube forests has been described in Section 4.1. To measure the intrinsic SE yield from nanotubes, one needs to isolate the nanotubes from the forests and the substrate. Here, we describe the procedure to make suspended nanotubes (Figure 6.1).  12 A version of this chapter has been published in a peer-reviewed journal (Reused with permission from “M. K. Alam, P. Yaghoobi, M. Chang, and A. Nojeh., ‘Secondary Electron Yield of Multiwalled Carbon Nanotubes,’ Applied Physics Letters, vol. 97, pp. 261902-1–261902-3, 2010”, Copyright 2010, American Institute of Physics).  13 An article has also been published as a comment in ‘Applied Physics Letters’ regarding this issue (M. K. Alam, R. F. W. Pease, and A. Nojeh, “Comment on “Ultrahigh Secondary Electron Emission of Carbon Nanotubes” [Appl. Phys, Lett. 96, 213113 (2010)],” Applied Physics Letters, vol. 98, pp. 066106-1–066101-1, 2011).  Chapter 6. Electron Yield of Suspended Nanotubes 71  We adopted a method of attaching nanotubes to sharp conductive tips under optical microscope that has been mainly used for generating nanotube-tips for atomic force microscopy [Figure 6.1(a)] [80]. For the purpose of our experiments, we performed this procedure with tungsten tips (tip diameter: 0.2–20 µm). Two micro-manipulators were used to bring a tip and a forest close to each other and an electric arc discharge was induced by using an external DC source to attach one or a few suspended nanotubes to the tip [Figure 6.1(c)]. 6.2 Experimental Setup for Electron Yield Measurement The tips were then placed in a Hitachi S–570 SEM for SE yield measurement and connected to a Keithley 6517A electrometer capable of biasing the specimen and measuring currents with femto-ampere accuracy (Figure 6.2). The measurement procedure has been described in Section 4.2. We used the same experimental configuration with an extra Faraday cup (grounded) under the suspended nanotubes to block backscattered or secondary electrons  Forest (b) (c) (a) +  - + Optical microscope stage Micro-manipulators Source Tip (ii) 2.7 μm  (i) 3.0 μm  30 μm  Figure 6.1 (a) Schematic diagram of the experimental setup for extracting nanotubes from the forests, (b) micrograph of the nanotube forests and (c) extracted nanotubes on tungsten tips.   Chapter 6. Electron Yield of Suspended Nanotubes 72  from the surface underneath the device (due to the small and hollow nanotube structure, most of the “backscattered” electrons are, in fact, scattered with a small scattering angle and collected in the Faraday cup, making SE generation from the chamber walls negligible). The small size of the specimen further reduces the probability of being hit by SEs generated from the surrounding walls. Another Faraday cup was used for primary current measurements. As the individual nanotubes are hollow and made of a single/a few layer(s) of carbon (with inter-wall distance of ~0.34 nm for MWNTs [81]), electron energy loss in them is expected to be extremely small and the primary electrons are expected to leave the nanotube with an energy far greater than 50 eV, thus effectively being counted as “backscattered” electrons. In other words, no high-energy electron is captured by the nanotube and the backscattered yield for suspended nanotubes is unity. We measured the total yield [Eq. (4.1)] from the nanotubes while applying –50 V to the specimen (to ensure that all electrons, including secondaries, escape the sample [5]). We then subtracted the backscattered yield (unity) to calculate the true SE yield:  SE yield  Total yield –  Backscattered yield=  (6.1)    El ec tro n G un  A Nanotube Faraday cup Tungsten tip VT IP ITE IT  Figure 6.2 Schematic of the experimental setup for electron yield measurement in the SEM. IP, IT, ITE and VT are the same as those defined in Figure 4.2.  Chapter 6. Electron Yield of Suspended Nanotubes 73  We performed the experiments on samples with a few individual nanotubes. The beam was slowly moved along a line [arrow in Figure 6.1(c)] across the nanotubes in the spot mode and the specimen current was recorded (Figure 6.3). The electrometer interfaced with a PC was used for real-time data acquisition. A running average was used for current measurements to reduce noise and fluctuations (analog-to-digital converter integration time: 16.67 msec, moving averaging: 10 points and median filtering: 3 points). A peak in Figure 6.3 indicates that the beam is crossing a nanotube (The typical inner diameters of our nanotubes were around 10 nm based on transmission electron microscopy, whereas the typical value of the beam spot size is ~5 nm).  6.3 Results and Discussion Figure 6.4 shows the total and the SE yield calculated from Eqs. (4.1) and (6.1). As can be seen, the SE yield from individual nanotubes is less than unity and decreases as a function of 0 5 10 15 0 2 4 6 8 10 12 x 10 -12 Time [Sec] Sp ec im en  c ur re nt  [A ] Current from nanotubes  Figure 6.3 Specimen current at 1 keV as the beam was moved across the nanotubes [along the arrow indicated on Figure 6.1(c)]. The horizontal axis indicates the recorded time as the beam was moved along the arrow.   Chapter 6. Electron Yield of Suspended Nanotubes 74  energy for the measured energy range. The experiment was repeated on another specimen [Figure 6.1(c)-ii] at several primary energies and similar values of electron yield were obtained. The low yield from individual nanotubes can be attributed to the very weak stopping power (corresponding to small energy loss of the primary electron) of the nanotube. A high energy electron cannot lose a significant amount of energy through inelastic collisions because of the hollowness, small dimension and thin atomic wall of the nanotube. Also, it is very unlikely for the primary electron to go through multiple scatterings process given that the mean free path is expected to be much higher than the thickness of the nanotube sidewalls [4, 33]. It has been predicted using first-principles calculations that the molecular orbitals of the nanotube can be raised in energy due to the presence of an external electron beam, thus effectively reducing the workfunction. This can help in emitting SEs and making the nanotubes visible in an SEM [82]. Nonetheless, a small SE yield is obtained (unless a strong external field is also applied and electron-stimulated field-emission occurs [21]). The decreasing trend of the electron yield is because of the decreasing energy loss of the primary electron with the increase of the beam energy [82].  0 5 10 15 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Primary beam energy [keV] El ec tro n yi el d   Total yield SE yield  Figure 6.4 Electron yield measured from the suspended nanotubes. Fluctuations in the beam current or specimen current are included in error bars.  Chapter 6. Electron Yield of Suspended Nanotubes 75  We used the equation provided by Joy and Luo [Eq. (2.2)] to calculate the energy loss in a single nanotube (the Bethe energy loss formula has previously been used to calculate the average energy transfer through a single inelastic scattering in nanotube bundles [34]). We then used the following parametric equation to calculate the SE yield [4]:  1 ( ) b SE a dEA p z dz dz δ ε = ×∫  (6.2) Here SEε  is an empirical parameter corresponding to the average energy needed for SE generation, dE/dz is the energy loss rate and p(z) is the escape probability (for solids, an exponential function of the depth [Eq. (5.7)]). A is an empirical parameter, representing the percentage of the electrons emitted toward the outside of the material. Since SEs can escape from any side of a suspended nanotube, the empirical constant A can be taken as 1 for our case. Also, given the hollow nanotube structure with a sidewall thickness smaller than the typical escape depth in solids, the escape probability can be assumed to be approximately 100%. The value of SEε  for carbon has been reported to be 125 eV [4]. Using this value of SEε , an interaction length [integration range, b–a, in Eq. (6.2)] of 2 nm gives a reasonable fit to the experimental data (Figure 6.5). The predicted value of a few nanometres for the interaction length correlates well with the sidewall thickness of our nanotubes as observed using transmission electron microscopy. For instance, a 4-wall MWNT has a total sidewall thickness of ~2 nm.  Chapter 6. Electron Yield of Suspended Nanotubes 76  Note that, the interaction length would vary with the chiralities and number of walls in MWNTs and may not be exactly equal to the thickness of the nanotube wall14. Nevertheless, it is expected to be in the order of the wall thickness and the trend of the SE yield as a function of primary energy will not be affected by small changes in the interaction length; only an upward or downward shift of the curve is to be seen with the change of this or other empirical parameters. Therefore, our measured data follow a trend consistent with the predictions of established models for bulk solids. It should also be noted that a nanotube tends to be positively charged under electron irradiation as it loses electrons and hardly captures any primary electron. However, if the nanotube is connected to an external source (the present case), there should not be significant charging as the lost electrons are supplied from the biasing source. If the  14 A more rigorous treatment is presented in the next chapter incorporating discrete energy loss processes. 0 5 10 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Primary beam energy [keV] SE  c oe ff ic ie nt   0 5 10 15 0 20 40 60 Primary beam energy [keV] En er gy  lo ss  [e V ]   Estimated SE yield Experimental data  Figure 6.5 Comparison of the experimental data with the theoretically estimated SE yield. Inset shows the behaviour of electron energy loss.   Chapter 6. Electron Yield of Suspended Nanotubes 77  nanotubes are electrically isolated, they would soon become positively charged to the point of preventing further secondary emission. In general, charging effects play an important role in the imaging of nanotubes. For example, if there is a substrate under a charged suspended nanotube, the charge distribution on the surface of the substrate below the nanotube could change and therefore, the SE yield of the substrate at that location could also change, or the charged nanotube could deflect the secondaries emitted by the substrate underneath [31]. If the nanotube lies on a surface, depending on its potential relative to the substrate, it could increase or decrease SE emission from the substrate in its immediate surroundings [22]. 6.4 Comment on “Ultrahigh Secondary Electron Emission of Carbon Nanotubes” [Appl. Phys. Lett. 96, 213113 (2010)]15 Luo et al. studied the SE yield of SWNTs lying on SiO2 [37]. The value of the average specimen current during the scan of an area including the nanotube was used to calculate the intrinsic SE yield of nanotubes (Eq. (1) of [37]). The conclusion of the paper is that carbon nanotubes connected to a reservoir have an ultrahigh SE yield of up to 123 without any stimulus other than the primary electron. The high yield was justified on the basis of the raising of the HOMO above the vacuum barrier in the presence of the primary electron. In this section, we argue that the claim of ultrahigh yield is not supported by the data and analysis presented in [37]. We also provide a possible alternative explanation for the experimental data. Previous theoretical and experimental works on the interaction of electron beams with nanotubes imply that the SE yield of pristine nanotubes is very low at the primary beam energies used in scanning electron microscopy [18, 21, 23-25, 29, 31]. A high-energy beam hitting a nanotube perpendicular to it is likely to pass through it without encountering any significant scattering (change in the direction and energy): on average the primary electron  15 A version of this section has been published as a comment in Applied Physics Letters.  Chapter 6. Electron Yield of Suspended Nanotubes 78  loses only a few tens of eV as it goes through the nanotube [34, 82]. Reference [37] seems to be in contradiction with all of the above reports. We also note that the argument made by the authors on the rise of the HOMO above the surface barrier is only applicable to the nanotube tip in the presence of a strong external electric field as shown by Nojeh et al.[25]. In the absence of a strong external field or other stimulation, which is the case in [37], the HOMO rise is only 1–2 eV. Although this could provide a partial explanation for SE emission from nanotubes, it does not support the claim of a high SE yield [25, 82], and a strong external stimulation is required for a high yield [21]. The reported high SE yield of up to more than a 100 by Luo et al. [37] is also in contradiction with the electron stopping power of nanotubes as estimated based on existing theories. Using the CSDA, one obtains only ~35 eV of energy loss for a 1 keV primary electron passing through a nanotube, assuming the interaction length to be equal to the diameter of the nanotube (to obtain an optimistically high value for energy loss). It is not reasonable that ~35 eV of energy loss from the primary electron could lead to the emission of more than a hundred secondary electrons, especially given the workfunction of nanotubes in the 4–5 eV range and the fact that SEs can have a kinetic energy of up to 50 eV. Since for carbon it has been reported that an average energy loss of 80–125 eV by the primary beam is needed in order to generate one SE [4, 46], the above 35 eV loss clearly cannot explain a SE yield of more than 100. In fact, even under the unrealistically optimistic assumption that the emission of each SE only requires an energy equal to the nanotube workfunction (~ 4–5 eV), and assuming a quantum efficiency of 1, still 35 eV loss could not explain such a high yield. We suggest an alternative explanation for the experimental data presented by Luo et al. Note that the high yield deduced from the data results from the fact that the average specimen current during the scan of an area containing nanotubes was higher than that measured on a similar area but without nanotubes. The small (less than 5% of primary current) difference in current was attributed to SE emission from the nanotubes. We argue that, instead, this could  Chapter 6. Electron Yield of Suspended Nanotubes 79  be due to nanotubes providing a conductive path and enhancing SE emission from the surrounding oxide. Previously, it has been suggested that a nanotube lying on SiO2 provides a conductive path around it and supplies extra electrons to the insulator surface within the electron-beam- induced current (EBIC) range, which is much larger than the nanotube diameter even at 1 keV [23, 24]. This provides an explanation for the visibility of nanotubes on insulating substrates in the SEM and for the fact that they appear much thicker than they actually are. We believe a similar effect could explain the high average specimen current in the presence of nanotubes as observed in by Luo et al. In other words, the current is the aggregate effect of SE emission from many points on the oxide surface near the nanotube. Without such a conductive path to ground, the SE current emitted from the oxide surface is simply not measured. Reference [37] does not present any control experiment with materials other than nanotubes to establish that the nanotubes are playing any role other than simply providing a path for the current. The authors did argue that the experimental conditions were such that the oxide surface was not charged, even in the absence of nanotubes. We agree that, if that is the case (implying unity yield from the oxide surface), the oxide surface will not contribute to the specimen current. However, the data presented in the paper do not support the argument that there is no oxide charging. The experiment was performed at 1 keV. It was argued that surface charge on the SiO2 was zero because the specimen current was zero. However, given that the leakage resistance through 312 nm of oxide is expected to be very high, in steady state (no displacement current present) one expects to measure zero specimen current even though the surface is charged to a static potential. In fact, the surface will charge to the point of making the total electron yield effectively equal to unity in steady state. Indeed, it has been reported that the SE yield of SiO2 is greater than unity at 1 keV and positive charging of the specimen occurs [31, 40, 72]. In Reference [37], no direct measurement of surface charging or electron yield was performed. The authors themselves mentioned that the SE yield of SiO2 is 1.18 at 1 keV, which directly supports our charging argument.  Chapter 6. Electron Yield of Suspended Nanotubes 80  As a final note, we point out that even if nanotubes did have an unusually high electron yield, in order to observe it in such an experiment, one would need to choose the measurement parameters (scan rate; current sampling, filtering and averaging) very carefully and synchronize the current sampling with the scan so as to ensure that the electrometer does take enough samples when the beam hits the nanotube. No such detail was given by the authors. These issues become all the more critical given that the claimed high yield is a product of the difference of very small leakage currents (only a few pA) and a very large scan length compared to the nanotube diameter (Eq. (1) in [37]). 6.5 Summary In summary, SE yield from isolated suspended MWNTs was measured systematically and a low yield was obtained. The method can also be used for single-walled nanotubes. These results have important implications on our understanding of the interaction of electron beams with nanotubes. The data may also be used in Monte Carlo simulations of structures made of collections of nanotubes (with or without other materials) to predict their electron yield for potential device applications. The results can also assist in finding the optimum imaging conditions of nanotube-based devices in electron microscopy to control artefacts such as charging. In addition, we believe that the high SE yield reported by Luo et al.[37] is an artefact of their analysis of the experimental data. Although the data appear to be correct (notwithstanding the fact that in the analysis the effect of SE emission from the amorphous carbon layer appears to have been assumed to be the same with and without the nanotubes, which is not fully justified), the conducted experiment was not capable of revealing the SE yield of nanotubes and the results presented do not support the claim of high SE yield - more careful experiments would be required. An example of such experiments was presented here in this chapter.  81  Chapter 7 Discrete Monte Carlo Simulation16 Monte Carlo simulation has been the most common approach to modeling the interaction of electron beams with solids. In this method, each electron trajectory is created from a series of random scattering events [4, 77]. This technique has been used to investigate various phenomena such as electron backscattering and energy dissipation in solids with considerable success [4, 38, 39, 44, 47, 48, 77, 83]. However, secondary electron (SE) emission has proven more difficult to deal with. Conventional Monte Carlo simulators typically use Bethe’s energy loss model (or a modified version of it) based on the CSDA. Hence, they cannot be directly applied when the structure is smaller than the random step length and an incoming electron occasionally loses a large fraction of its energy in a single collision or a few collisions. Examples of such structures are carbon nanotubes and nanowires. Nanotubes are hollow cylindrical structures made of carbon with diameters of a few nanometres to a few tens of nanometres only. Therefore, a Monte Carlo model incorporating discrete energy loss events seems to be a more appropriate approach for individual nanotubes. Hybrid models have also been proposed in order to include more details of the various phenomena in the simulation [52]. An early effort was made by Shimizu et al. who introduce discrete energy loss events in a Monte Carlo simulation of bulk materials [83, 84]. To the best of our knowledge, no model has yet been reported for simulating electron trajectories in individual nanotubes. We presented experimental SE data for MWNTs in Chapter 6. Here, we present the results of a discrete Monte Carlo simulation for such MWNTs, calculating energy loss as a result of discrete scattering events, and compare the  16 A version of this chapter has been published in a peer-reviewed journal (Reused with permission from “M. K. Alam and A. Nojeh, ‘Monte Carlo Simulation of Electron Scattering and Secondary Electron Emission in Individual Multiwalled Carbon Nanotubes: A Discrete-Energy-Loss Approach,’ Journal of Vacuum Science and Technology B, vol. 29, pp. 041803-1–041803-7, 2011”, Copyright 2011, American Vacuum Society)  Chapter 7. Discrete Monte Carlo Simulation 82  results with experimental data. We also discuss the energy distribution of the transmitted electrons and the effect of the number of nanotube walls on SE emission. 7.1 The Model We defined a MWNT using its inner diameter and number of walls. The outer diameter was calculated based on the inner diameter and number of walls using an inter-wall distance of 0.34 nm [81]. Electron trajectories in individual nanotubes (presented in the results and discussion section) were calculated using a discrete-energy-loss approach, discussed below. We used MWNTs with an inner diameter of 10 nm in this work, which is typical of the nanotubes used in our experiments (Chapter 4). Our approach is based on the one proposed by Shimizu et al.[83, 84]. We used a different cross-section to define the elastic scattering events [48]. For inelastic scattering, we considered the following three most significant mechanisms: Outer-shell electron ionization, core-shell electron ionization and plasmon excitation. In addition, we added the SE emission simulation capability. Several experimental and theoretical works have been reported on the electron energy loss spectra of nanotubes, revealing the fundamental excitation processes in them [85, 86]. In addition to the most common and prominent loss mechanisms (considered here), some very- low-energy loss peaks close to the elastic peak have been observed. For example, peaks at 90 meV and 170 meV for SWNTs have been recorded by reflection energy loss spectroscopy [86]. These side peaks have been attributed to the excitation of various phonons [86, 87]. Thus, a primary electron can lose only a few tens of meV in such processes, which is negligible compared to its kinetic energy (~ 0.5 keV to 30 keV for scanning electron microscopy). Therefore, the energy loss due to phonons can be neglected and electron-nucleus interaction was considered as entirely elastic here, similar to the case of bulk solids in typical Monte Carlo simulations of electron trajectories. A brief description of the different cross-sections, the energy loss and secondary emission model, and the Monte Carlo procedure is given in the next sub-sections.  Chapter 7. Discrete Monte Carlo Simulation 83  7.1.1 Scattering Cross-Sections and Mean Free Paths The MFPs for different types of scattering determine the random step length and the probability of each scattering process in the Monte Carlo simulation. The empirical total cross-section proposed by Browning et al. was used for calculating the elastic MFP [4, 48]:  ,el A el A N λ ρ σ =  (7.1) where A is the atomic weight, NA is the Avogadro number,  ρ is the density of the material and elσ , given by Eq. (2.1),  is the total elastic cross-section. The Møller scattering formula in the non-relativistic form was used for calculating the differential inelastic cross-section for outer-shell ionization [5, 52] as  4 2 2 2 0 1 1 1 , (4 ) ( ) ( ) d e dW E W E W W E W σ π πε   = + − − −   (7.2) where E is the initial energy of the incident electron, W is the energy loss suffered by it, e is the charge of the electron and ε0 is the permittivity of  vacuum. The Gryzinski differential cross-section was used for the corer-shell ionization process. It incorporates the binding energy of the shell and has proven successful for several solids [52, 84, 88]. It is calculated as [5]: 3 1 4 2 2 2 3 0 41 1 ln 2.7 , (4 ) 3 b b E E W b b b b b E Ed e E W W E W dW EW E E E E E E σ π πε +       −     = − − + +         +             (7.3) where all the variables are the same as in Eq. (7.2) and the additional variable Eb is the binding energy of the shell.  Chapter 7. Discrete Monte Carlo Simulation 84  The total scattering cross-sections for the Møller and Gryzinski scattering were found by integrating Eqs. (7.2) and (7.3) with respect to W. The binding energy of the inner-shell was taken to be 284.2 eV [89]. The MFP for these scattering events can be found from  M/G M/G , A A N Zγ λ ρ σ =  (7.4) where M/Gσ is the total inelastic cross-section (M for Møller and G for Gryzinski), Zγ is the number of electrons in the corresponding shell (for carbon, 4 in the outer shell and 2 in the inner shell). The MFP for plasmon creation has been derived theoretically by Quinn [54, 55] as  11 2 1 2 2 (1 ) 1 2 ln , ( ) p p B p p yEa x x y λ ω −   + − =    − −    (7.5) where , F Ex E = ,pp F y E ω =  Ba is the Bohr  radius, pω is the plasmon energy and FE is the Fermi energy. The plasmon energy was taken to be 20 eV for nanotubes. Experimentally, it has been reported that the plasmon peak occurs between 15 eV and 26.9 eV [85]. The effective MFP for each scattering event was calculated from:  1 1 1 1 1 T el M G pλ λ λ λ λ = + + +  (7.6) Figure 7.1 shows the different MFPs calculated from the above expressions and the effective MFP as a function of electron energy.  Chapter 7. Discrete Monte Carlo Simulation 85   7.1.2 Energy Loss Unlike in the CSDA energy loss calculation, in our model the energy loss was calculated for discrete scattering events according to the following rules: a. No energy is lost during an electron-nuclei (elastic) scattering process. b. A fixed amount of energy, pω , is lost during a plasmon excitation process. c. For core-shell and outer-shell ionization, the random nature of the energy loss is implemented using the probability integral transform theorem [84, 88]:  max , c c WW w W W d ddW R dW dW dW σ σ′ ′= ′ ′∫ ∫  (7.7) where Wc is the cut-off energy (taken as 10 eV for Møller scattering and equal to the binding energy for Gryzinski scattering), Wmax (the maximum amount of energy that can be lost for a 10 3 10 4 10 0 10 1 10 2 10 3 Electron energy [eV] M FP  [n m ]   Elastic scattering Outer-shell ionization Core-shell ionization Plasmon excitation Effective MFP  Figure 7.1 MFPs for the different scattering processes and the effective MFP as a function of electron energy.   Chapter 7. Discrete Monte Carlo Simulation 86  particular process) is equal to E/2 for Møller and to E for Gryzinski scattering, and Rw is a uniform random number between 0 and 1. Before proceeding to use the described scattering cross-sections and MFPs for nanotubes, it is instructive to estimate how they perform in a bulk material (for example, graphite) compared to Bethe’s modified CSDA formula [Eq. (2.2)]. We can calculate an equivalent average energy loss arising from the discrete events using the following [5]:  max minM/G , W A W dE Z dN WdW ds A dW σρ= ∫  (7.8)  P ,p p dE ds ω λ =   (7.9) where the plasmon energy was taken to be 26 eV for graphite [90]. As can be seen on Figure 7.2, for the chosen parameters the sum M G P dE dE dE ds ds ds + + closely follows dE/ds (CSDA), which indicates that the discrete energy loss scheme with these cross-sections may also be used for bulk materials.  Chapter 7. Discrete Monte Carlo Simulation 87   7.1.3 Scattering Angle Distributions The scattering angle of the incident electron for each individual scattering process was calculated as follows. The elastic scattering angle was estimated from a fitted Mott differential cross-section [Eqs. (5.2) and (5.3)]. The Møller scattering angle of the primary electron was calculated from [4, 88]:  2 2sin , 2 n M n E E θ µ µ = + −  (7.10) where μ is the kinetic energy of the electron in the units of its rest mass (511 keV) and En equals W/E. The Gryzinski  and plasmon scattering angles were calculated from [53, 54, 88]:  ( )2 /G P Wsin E θ =  (7.11) 0 5 10 15 0 2 4 6 8 10 Electron energy [keV] En er gy  lo ss  [e V /Å ]   CSDA Outer-shell ionization Core-shell ionization Plasmon excitation Total (outer-shell+core- shell+plasmon) 10 2 10 4 10 -2 10 0 Electron energy [eV] E ne rg y lo ss  [ eV /Å ]  Figure 7.2 A comparison of the equivalent energy loss rate calculated from discrete processes with the CSDA energy loss. Inset shows the energy loss rate on a log-log scale.   Chapter 7. Discrete Monte Carlo Simulation 88  Note that the elastic cross-section for very-low-energy scatterings does not have an analytical form and the cross-section proposed by Browning et al. is reported for >100 eV. Nonetheless, in one approach we used Eqs. (5.2) and (5.3) due to their simple analytical form. Therefore, a correction was needed for the small portion of the elastic scattering processes happening below 100 eV. An alternative solution may be to use the tabulated data for the Mott cross-section [91]. Thus, we also used the interpolated values (from Mott numerical data given down to 20 eV [92]) of elastic scattering angles and MFPs for higher accuracy in the lower energy range. Although the Browning scattering cross-section is reported to be valid over 100 eV, we used the Mott cross-section up to 1 keV for better accuracy. Here, we present the results for both cases (with or without these corrections). We also note that there are some ambiguities present in the plasmon MFP below 80 eV where the logarithmic term becomes negative [see Eq. (7.5)]. The effect of this ambiguity reflects on the probability of plasmon scattering at lower energies and thus on the energy loss of the slow electrons. One simple solution might be to use extrapolated values down to the plasmon energy. It is also worth mentioning that Eq. (7.11) was used for calculating the scattering angles based on a classical two-body collision. For low energies, this could significantly overestimate the scattering angles for plasmon excitations. In fact, for plasmon excitations, typically the deflection of the primary electron is very small. Nonetheless, we used Eqs. (7.5) and (7.11) for plasmons in the absence of any better model for nanotubes. All of these limitations just remind one of the fact that the scattering at low energies is not yet completely understood and further fundamental study is needed. It may be possible to correct the cross-sections for such anisotropic nanostructures using empirical parameters. However, more fundamental experimental work is also needed in order to incorporate such corrections in a physically meaningful manner. 7.1.4 Monte Carlo Procedure The step length was calculated from:  ln ,Ts Rλ= −  (7.12)  Chapter 7. Discrete Monte Carlo Simulation 89  At each step of the simulation, the primary electron was scattered a distance s away. The energy loss was calculated according to the description provided in section 7.1.2 and the entire loss was assumed to happen at the point of interaction. The scattering angles were estimated from Eqs. (5.2), (5.3), (7.10) and (7.11). To find the energy loss and the scattering angle we also need to determine the type of scattering at each step. It is known that the scattering probability is proportional to the inverse MFP. As the MFPs are known from Eqs. (7.4) – (7.6), we define the probability of each scattering process based on a uniform random number Rs, as follows: 1/elastic scattering : 0 1/ 1/ 1/1/plasmon excitation : 1/ 1/ 1/ 1/ 1/ 1/ 1/ outer-shell ionization : 1/ 1/ 1/ 1/ 1/ 1/ 1/ 1/ core-shell ionization : 1/ el s T el pel s T T el p el p M s T T el p M el p M s T R R R R λ λ λ λλ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ < ≤ + < ≤ + + + < ≤ + + + + + < ≤ 1/ 1 1/ G T λ λ                      =      (7.13) Based on the above inequalities, the type of scattering was determined and the corresponding set of equations was used for calculating the energy loss and the scattering angle. Figure 7.3 shows the probability limits of different scatterings for different primary beam energies [according to Eq. (7.13)]: A significant number of the primary electrons are elastically scattered without losing any energy. Very few of them excite the electrons in the core shell. Elastic scattering and plasmon excitation have comparable probabilities.  Chapter 7. Discrete Monte Carlo Simulation 90   7.2 Secondary Electron Model for MWNTs For nanotubes, the role of each scattering process in generating SEs is still not clear. The most common mechanism of generating SEs seems to be valence electron ionization. In this work, we considered both outer-shell and core-shell ionization processes (the contribution from the core shell is very small as can be seen from Figure 7.3). Also, there is no report in the literature on SE generation from plasmon decay in individual nanotubes. Furthermore, theoretical work on one-dimensional systems (quantum wires) suggests an infinite lifetime for plasmons and, therefore, no decay into single-particle excitations [93]). Therefore, we neglected the SE generation probability from plasmon decay, that is we assumed that the primary electron energy loss causes the plasmon oscillation only, without exciting any single electron to a high enough energy to overcome the vacuum energy barrier. 0 5 10 15 0 0.2 0.4 0.6 0.8 1 Electron energy [keV] Pr ob ab ili ty  li m its   elastic scattering plasmon excitation outer-shell ionization core-shell ionization  Figure 7.3 Probability of different scattering processes at different primary beam energies.   Chapter 7. Discrete Monte Carlo Simulation 91  It is also important to consider whether phonons can play a role in SE emission. Although phonons have low energy as discussed before, in principle it is possible for multiple phonons to transfer energy to one nanotube electron, assisting it in the emission process. Indeed, phonon-assisted electron emission has been reported for SWNTs biased between two electrodes [94]. However, such a phenomenon has not been reported for SE emission from nanotubes. Typically, it is also neglected in bulk solids for the considered energy range. Moreover, since the MFP for electron-phonon collision (on the order of micrometers for acoustic phonons and tens of nanometers for optical phonons [95]) is much larger than the nanotube diameter, the chance of multiple phonon generation as the primary electron crosses the nanotube is very low, suggesting negligible probability of non-equilibrium phonon build-up and subsequent contribution to SE emission. For example, if a given nanotube electron gains 90 meV of kinetic energy from phonon absorption, at least 50 phonon absorption events are needed to accumulate a kinetic energy equal to the minimum ionization energy of nanotubes (~ 4.5 eV) to overcome the vacuum barrier. Therefore, electron-phonon interaction was not considered in the present work. Nevertheless, this phenomenon, if proven important in future, can be incorporated into the proposed simulation framework easily if the excitation cross-sections for phonon generation and absorption are theoretically known. For each SE, the initial energy was assumed to be equal to the energy lost by the primary electron minus the corresponding binding energy (for core-shell or outer-shell). The initial scattering angle ( )SEθ  for the electrons generated by the outer-shell ionization can be estimated from [4, 88]:  2 , 2(1 )sin , 2 n SE M n E E θ µ − = +  (7.14) where μ is the kinetic energy of the electron in the units of its rest mass (511 keV) and En is equal to W/E. For core-shell ionization:  Chapter 7. Discrete Monte Carlo Simulation 92   ( )2 ,cos SE G WEθ =  (7.15) Once a new electron was generated as a result of a scattering process, it was tracked down using the same Monte Carlo procedure. If an electron exited the nanotube with an energy of less than 50 eV, it was counted as an SE. Otherwise, it was counted as a backscattered electron. 7.3 Results and Discussion Electron trajectories for primary energies of 0.5 keV and 5 keV are shown in Figures 7.4 and 7.5, respectively, for an incident beam impinging perpendicular to the tube axis or at an angle (for the 5-keV case). As can be seen from the figures, the perpendicular beam goes almost straight through for higher energies. However, the number of scattering events inside the nanotube increases if the beam hits the nanotube at an angle relative to the surface normal [Figure 7.5(b)].   Figure 7.4 Electron trajectories in the MWNT for a 0.5-keV beam perpendicular to the nanotubes (No. of walls = 6). Only 70 trajectories are plotted for clarity.  Chapter 7. Discrete Monte Carlo Simulation 93  The SE yield calculated (for a beam perpendicular to the nanotube) from the simulation along with the available experimental data (Chapter 6) is presented in Figure 7.6. 10,000 trajectories were simulated at each primary beam energy. The effect of number of walls is negligible in the range of 5–9 walls because the MFP (i.e. the random step length) is larger than the sidewall thickness for most of the energy range and, therefore, the number of scatterings does not depend on the number of walls. It was also seen that with the increase of the number of walls above 9 (not shown here), or its decrease below 5 walls, the SE coefficient decreases in the lower energy range. With a higher number of walls the probability of SEs escaping from the specimen decreases. On the other hand, the energy loss decreases for fewer walls because the number of scattering events decreases.  (b) (a)  Figure 7.5 Electron trajectories in the MWNT for a 5-keV beam (No. of walls = 6). Only 70 trajectories are plotted for clarity. (a) Beam perpendicular to the nanotube and (b) beam oblique to the nanotube (angles with the axes: X=78°, Y=90° and –Z=168°).   Chapter 7. Discrete Monte Carlo Simulation 94  The effect of core-shell electron ionization on SE emission is shown in Figure 7.7. The effect is negligible because the probability of core-shell ionization is very low (<1%) [Figure 7.3]. The energy distribution of the transmitted electron through the nanotube (for 6 walls) is shown in Figure 7.8. As expected, most of the primary electron escapes with a very minimal energy loss for higher beam energies. However, the energy has a wide distribution for lower incident beam energies [Figure 7.8(a)]. 10 3 10 4 0 0.2 0.4 0.6 0.8 1 Primary beam energy [eV] SE  c oe ff ic ie nt   Experimental data No. of walls = 2 No. of walls = 5 No. of walls = 7 No. of walls = 8 No. of walls = 9  Figure 7.6 Simulated SE coefficients for different MWNTs, together with the experimental data presented in Chapter 6.   Chapter 7. Discrete Monte Carlo Simulation 95     0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 250 Transmitted electron energy normalized  to primary beam energy N um be r o f e le ct ro ns Primary beam energy = 0.5 keV 0.2 0.4 0.6 0.8 1 0 1000 2000 3000 4000 5000 N um be r o f e le ct ro ns Transmitted electron energy normalized  to primary beam energy Primary beam energy = 5 keV (a) (b)  Figure 7.8 Energy distribution of the transmitted electrons: (a) primary beam energy = 0.5 keV and (b) primary beam energy = 5 keV.  10 3 10 4 0 0.2 0.4 0.6 0.8 1 Primary beam energy [eV] SE  c oe ff ic ie nt   10 3 0.16 0.18 0.2 0.22   Core-shell ionization excluded Core-shell ionization included No. of walls =7  Figure 7.7 Simulated SE coefficients showing the effect of incorporating core-shell excitations.  Chapter 7. Discrete Monte Carlo Simulation 96  Multiple peaks can be noticed on the energy loss distribution of a 6-wall nanotube (Figure 7.9): the zero-loss peak due to elastic scattering, the plasmon loss peak and a weak peak around the Møller cut-off energy. Similar peaks were theoretically calculated for and observed in electron-energy-loss-spectroscopy of nanotubes [32, 85]. It can also be noted that there are several strong peaks at integer multiples of the plasmon energy. These peaks arise because of the multiple scatterings of the primary electron (number of scatterings up to 5 in this case) in the nanotube. As expected the relative strength of the peaks decreases with energy due to the lower probability of multiple scatterings. The energy loss due to outer- shell ionization extends to a wide energy range starting at the Møller cut-off energy. Figure 7.10 shows that a better agreement with the experimental data in the higher energy range is achieved when the low energy corrections (discussed in section 7.1.3) are incorporated. However, the difference with the experimental data is still maximal at 500 eV [underestimated by the simulation (Figure 7.6)] and further increased by ~35%. Plasmon excitation might play a very important role in this region and contribute to SE emission. 0 0.02 0.04 0.06 0.08 0.1 0 500 1000 1500 2000 Energy loss [keV] N um be r o f e le ct ro ns Primary beam energy = 5 keV  Figure 7.9 Energy loss distribution of the electrons (primary beam energy = 5 keV).   Chapter 7. Discrete Monte Carlo Simulation 97  Incorporating the contribution of plasmon excitations (if there is any) could potentially improve the low-energy fit. However, as mentioned earlier the contribution of plasmon excitation to the SE emission of nanotubes is not well known. Also, inelastic scattering cross-sections for SE excitation from various materials are still not known very accurately. Nonetheless, the agreement is fairly good in the range of 1–15 keV, with a root-mean-square difference of 0.0406 between simulated and experimental results. Similar differences are also seen between simulated and experimental data even for bulk materials [7], and even between values reported in different experiments on the same material [72]. We thus believe that the present work is a useful preliminary step toward the modeling of SE emission from nanotubes. The framework presented here can also be used for nanotubes lying on a substrate or larger structures made of several nanotubes or other nanostructures such as nanowires (although for larger structures, such as nanotube bundles, CSDA-based models could also be used 10 3 10 4 0 0.2 0.4 0.6 0.8 1 Primary beam energy [eV] SE  c oe ff ic ie nt   Experimental data No. of walls = 7 [no correction] No. of walls =7 [elastic scattering and plasmon corrected]  Figure 7.10 Simulated SE coefficients for a MWNT, with and without corrections for the lower energy plasmon and elastic scatterings, and comparison with the experimental data presented in Chapter 6.   Chapter 7. Discrete Monte Carlo Simulation 98  [Chapter 5]). We note that our simulation results suggest that SE yield is significantly lower at low primary beam energies for nanotubes with less than 5 walls (Figure 7.6). Therefore, one would expect less SE emission from SWNTs due to the outer-shell electron ionization. In addition, two plasmon modes have been reported for MWNTs compared to one for SWNTs [79]. This suggests that the contribution of plasmon excitation and, therefore, the total SE yield of SWNTs should be less than those of MWNTs. This also agrees with the experimental observation that MWNTs are typically more readily visible in scanning electron microscopy than SWNTs. 7.4 Summary We used a discrete-energy-loss approach for the Monte Carlo simulation of SE emission from individual nanotubes in order to overcome the limitation of the CSDA-based models. Such a model, although previously used for bulk materials, may not have been in widespread use so far because CSDA-based  models could explain the interaction of electron beams with bulk solids reasonably well (at least in terms of the overall trends) with a fraction of the computational complexity. However, our approach seems to be suitable for studying the interaction between electron beams with nanostructures. A set of known cross- sections were used to calculate the discrete energy losses and the method was validated by comparing the results with the available experimental data. Further experimental investigation is needed in order to gain a better understanding of the role of each scattering process toward SE emission.  99  Chapter 8 Conclusion In this thesis, an experimental and theoretical study of the interaction of electron beams with carbon nanotubes was presented. A summary of the contributions made and some proposed ideas for future research are given here. 8.1 Contributions The following contributions were made: 1. In Chapter 3, we showed that an external electron beam can raise the HOMO of a nanotube being imaged in an SEM. This enhances the tunnelling of electrons and facilitates the generation of SEs when a primary electron passes around the nanotube. We also discussed some possible mechanisms of SE emission. We observed that a primary electron can lose enough energy in a nanotube, leading to SE emission making the nanotube visible in an SEM. Although the analysis presented in this chapter was qualitative, it served as a strong motivation for further experimental and theoretical study into the direct interaction of electron beams with carbon nanotubes. 2. In Chapter 4, a systematic study and measurement of electron yield from vertically aligned nanotube forests was reported. The study revealed that electron penetration range and MFP in these structures are unusually high. At the primary electron energies involved in SEM, electrons can escape through the sidewall of nanotube forests of several hundreds of micrometres in diameter. 3. In Chapter 5, a new Monte Carlo tool was described to simulate the electron trajectories and backscattered yield of nanotube forests. The tool provided the possibility of taking the internal, nanostructured nature of the forest into account through a model that considered the interaction of the electrons with each individual nanotube in the forest explicitly.  Chapter 8. Conclusion 100  4. In Chapter 6, a method to measure the electron yield of individual nanotubes was presented. The experiments confirmed the expectation of low yield from nanotubes. We also discussed the results of a previously reported measurement method and the possible reasons that lead to the incorrect conclusion of ultrahigh yield from individual nanotubes. 5. In Chapter 7, we presented a discrete Monte Carlo model applicable to nanostructures. The method overcomes the shortcomings of the CSDA models and seems to be suitable for nanotubes. Several properties such as SE yield and energy distribution were investigated using this model. SE yield was also compared with the experimental data. Aside from the specific contributions described above, perhaps the simplest way to summarize the work presented in this thesis is as follows: when this work began, the interaction of electron beams with carbon nanotubes was a rather puzzling topic. It was quite surprising that, given the small interaction area with the electron beam provided by individual nanotubes, nanotubes were so readily visible in the SEM. The SE and BSE yield of individual nanotubes or collections of nanotubes such as nanotube forests were not known and there was no model or simulation tool capable of estimating these quantities for nanotubes. Most of the previous studies had neglected direct interaction between the electron beams and nanotubes and implied negligible/low yield. Further, a claim of ultrahigh yield from nanotubes made the topic more perplexing. Through both experimental and theoretical work accomplished in the context of this thesis, we are now in a position where we have a good level of understanding of these phenomena: we have measured the quantities of interest and created models and simulation tools for predicting them, and compared the obtained results with experimental data. Our results show that nanotubes have low intrinsic SE yield. However, the yield is enough to make them visible in the SEM. It might be fair to say that BSE and SE emission from nanotubes are not puzzling anymore.  Chapter 8. Conclusion 101  8.2 Future Work Theoretically, an interesting direction would be to investigate the energy loss mechanisms with time dependant first-principles simulations. Another obvious extension of this work is to study different shapes of nanotube forests experimentally and compare the results with the predictions of the model developed here. In addition, electron yield for oblique incidence could be explored both theoretically and experimentally. Modeling and studying the effect of high applied bias on SE emission could enable the use of nanotube forests as electron multipliers. A more advanced task would be to incorporate SE yield estimation into the simulation of different shapes and densities of nanotube forests. Preliminary simulations suggest that nanotube forests need a lower average energy to produce SEs, which is very interesting. Another promising study would be to simulate nanotube forests using a discrete Monte Carlo method such as the one presented for single nanotubes in Chapter 7. The SE yield of single-walled nanotubes can also be studied. The role of direct interaction mechanisms on the so-called electron-stimulated field-emission [19] can be better studied using suspended nanotubes in the presented experimental configuration. Although it poses a great challenge, incorporating the role of plasmons into SE generation would be very useful. A starting point for this task could be to find a way to study the phenomenon experimentally using the methods developed for solids. Also, more precise and general values for the empirical parameters used in the presented models might be deduced if the experiments could be done on many different nanotubes or nanotube forests with precisely known structural parameters, but such a work remains impractical at this stage, given the significant limitations and uncertainties involved in the existing fabrication processes for carbon nanotubes and structures based on them.  102  Bibliography [1] K. G. McKay, Secondary Electron Emission, New York: Academic Press, 1948. [2] E. Baroody, “A Theory of Secondary Electron Emission from Metals,” Physical Review, vol. 78, pp. 780-787, 1950. [3] A. Dekker, “Variation of Secondary Electron Emission of Single Crystals with Angle of Incidence,” Physical Review B, vol. 4, pp. 55-57, 1960. [4] D. C. Joy, Monte Carlo Modeling for Electron Microscopy and Microanalysis, New York: Oxford University Press, 1995. [5] L. Reimer, Scanning Electron Microscopy: Physics of Image Formation and Microanalysis, Berlin: Springer, 1998. [6] J. Schou, “Secondary Electron Emission from Solids by Electron and Proton Bombardment,” Scanning Microscopy, vol. 2, pp. 607-632, 1988. [7] C. G. H. Walker, M. M. El-Gomati, A. M. D. Assa'd, and M. Zadraz ̌ il, “The Secondary Electron Emission Yield for 24 Solid Elements Excited by Primary Electrons in the Range 250-5000 eV: A Theory/Experiment Comparison,” Scanning, vol. 30, pp. 365- 380, 2008. [8] S. Pendyala, “Detection of Low Energy Positrons Using Spiraled Electron Multipliers,” Review of Scientific Instruments, vol. 45, pp. 1347-1348, 1974. [9] D. Shapira, T. Lewis, L. Hulett, and Z. Ciao, “Factors Affecting the Performance of Detectors that Use Secondary Electron Emission from a Thin Foil to Determine Ion Impact Position,” Nuclear Instruments and Methods A, vol. 449, pp. 396-407, 2000. [10] A. Shih, “Secondary Electron Emission Studies,” Applied Surface Science, vol. 111, pp. 251-258, 1997. [11] D. C. Joy, “Scanning Electron Microscopy: Second Best No More,” Nature Materials, vol. 8, pp. 776-777, 2009. [12] Y. Zhu, H. Inada, K. Nakamura, and J. Wall, “Imaging Single Atoms Using Secondary Electrons with an Aberration-Corrected Electron Microscope,” Nature Materials, vol. 8, pp. 808-812, 2009. [13] N. de Jonge, M. Allioux, J. Oostveen, K. Teo, and W. Milne, “Optical Performance of Carbon-Nanotube Electron Sources,” Physical Review Letters, vol. 94, pp. 186807-1– 186807-4, 2005. [14] N. de Jonge and J. M. Bonard, “Carbon Nanotube Electron Sources and Applications,” Philosophical Transactions of the Royal Society A, vol. 362, pp. 2239-2266, 2004.  Bibliography 103  [15] B. Hu, P. Li, J. Cao, H. Dai, and S. Fan, “Field Emission Properties of a Potassium- Doped Multiwalled Carbon Nanotube Tip,” Japanese Journal of Applied Physics, vol. 40, pp. 5121-5122, 2001. [16] P. Yaghoobi, M. K. Alam, K. Walus, and A. Nojeh, “High Subthreshold Field- Emission Current Due to Hydrogen Adsorption in Single-Walled Carbon Nanotubes: A First-Principles Study,” Applied Physics Letters, vol. 95, pp. 262102-1–262102-3, 2009. [17] P. Yaghoobi and A. Nojeh, “Electron Emission from Carbon Nanotubes,” Modern Physics Letters B, vol. 21, pp. 1807-1830, 2007. [18] W. S. Kim, W. Yi, S. G. Yu, J. Heo, T. Jeong, J. Lee, C. S. Lee, J. M. Kim, H. J. Jeong, Y. M. Shin, and Y. H. Lee, “Secondary Electron Emission From Magnesium Oxide on Multiwalled Carbon Nanotubes,” Applied Physics Letters, vol. 81, pp. 1098- 1100, 2002. [19] A. Nojeh, W.-K. Wong, E. Yieh, R. F. Pease, and H. Dai, “Electron Beam Stimulated Field-Emission from Single-Walled Carbon Nanotubes,” Journal of Vacuum Science and Technology B, vol. 22, pp. 3124-3127, 2004. [20] W. Yi, S. Yu, W. Lee, I. T. Han, T. Jeong, Y. Woo, J. Lee, S. Jin, W. Choi, J. Heo, D. Jeon, and J. M. Kim, “Secondary Electron Emission Yields from MgO Deposited on Carbon Nanotubes,” Journal of Applied Physics, vol. 89, pp. 4091-4095, 2001. [21] M. Michan, P. Yaghoobi, B. Wong, and A. Nojeh, “High Electron Gain from Single- Walled Carbon Nanotubes Stimulated by Interaction with an Electron Beam,” Physical Review B, vol. 81, pp. 195438-1–195438-8, 2010. [22] T. Brintlinger, Y.-F. Chen, T. Dü rkop, E. Cobas, M. S. Fuhrer, J. D. Barry, and J. Melngailis, “Rapid Imaging of Nanotubes on Insulating Substrates,” Applied Physics Letters, vol. 81, pp. 2454-2456, 2002. [23] Y. Homma, S. Suzuki, Y. Kobayashi, M. Nagase, and D. Takagi, “Mechanism of Bright Selective Imaging of Single-Walled Carbon Nanotubes on Insulators by Scanning Electron Microscopy,” Applied Physics Letters, vol. 84, pp. 1750-1752, 2004. [24] Y. Homma, D. Takagi, S. Suzuki, K. Kanzaki, and Y. Kobayashi, “Electron- Microscopic Imaging of Single-Walled Carbon Nanotubes Grown on Silicon and Silicon Oxide Substrates,” Journal of Electron Microscopy, vol. 54, pp. i3-i7, 2005. [25] A. Nojeh, B. Shan, K. Cho, and R. Pease, “Ab Initio Modeling of the Interaction of Electron Beams and Single-Walled Carbon Nanotubes,” Physical Review Letters, vol. 96, pp. 056802-1–056802-4, 2006.  Bibliography 104  [26] A. Nojeh, W.-K. Wong, A. W. Baum, R. F. Pease, and H. Dai, “Scanning Electron Microscopy of Field-Emitting Individual Single-Walled Carbon Nanotubes,” Applied Physics Letters, vol. 85, pp. 112-114, 2004. [27] L. Qin, X. Zhao, K. Hirahara, Y. Ando, and S. Iijima, “Electron Microscopic Imaging and Contrast of Smallest Carbon Nanotubes,” Chemical Physics Letters, vol. 349, pp. 389-393, 2001. [28] W. K. Wong, A. Nojeh, and R. F. W. Pease, “Parameters and Mechanisms Governing Image Contrast in Scanning Electron Microscopy of Single-Walled Carbon Nanotubes,” Scanning, vol. 28, pp. 219-227, 2006. [29] Y. A. Kasumov, I. I. Khodos, M. Kociak, and A. Y. Kasumov, “Scanning and Transmission Electron Microscope Images of a Suspended Single-Walled Carbon Nanotube,” Applied Physics Letters, vol. 89, pp. 013120-1–013120-3, 2006. [30] M. R. Ay, M. Shahriari, S. Sarkar, M. Adib, and H. Zaidi, “Monte Carlo Simulation of X-ray Spectra in Diagnostic Radiology and Mammography Using MCNP4C,” Physics in Medicine and Biology, vol. 49, pp. 4897-4917, 2004. [31] P. Finnie, K. Kaminska, Y. Homma, D. G. Austing, and J. Lefebvre, “Charge Contrast Imaging of Suspended Nanotubes by Scanning Electron Microscopy,” Nanotechnology, vol. 19, pp. 335202-1–335202-6, 2008. [32] A. Rivacoba and F. J. García de Abajo, “Electron Energy Loss in Carbon Nanostructures,” Physical Review B, vol. 67, pp. 085414-1–085414-8, 2003. [33] I. Kyriakou, D. Emfietzoglou, R. Garcia-Molina, I. Abril, and K. Kostarelos, “Electron Inelastic Mean Free Paths for Carbon Nanotubes from Optical Data,” Applied Physics Letters, vol. 94, pp. 263113-1–263113-3, 2009. [34] D. Emfietzoglou, I. Kyriakou, R. Garcia-Molina, I. Abril, and K. Kostarelos, “Analytic Expressions for The Inelastic Scattering and Energy Loss of Electron and Proton Beams in Carbon Nanotubes,” Journal of Applied Physics, vol. 108, pp. 054312-1– 054312-5, 2010. [35] L. Huang, S. P. Lau, H. Y. Yang, and S. F. Yu, “Local Measurement of Secondary Electron Emission from ZnO-Coated Carbon Nanotubes,” Nanotechnology, vol. 17, pp. 1564-1567, 2006. [36] J. Lee, J. Park, K. Sim, and W. Yi, “Double Layer-Coated Carbon Nanotubes: Field Emission and Secondary-Electron Emission Properties Under Presence of Intense Electric Field,” Journal of Vacuum Science and Technology B, vol. 27, pp. 626-630, 2009.  Bibliography 105  [37] J. Luo, J. H. Warner, C. Feng, Y. Yao, Z. Jin, H. Wang, C. Pan, S. Wang, L. Yang, Y. Li, J. Zhang, A. A. R. Watt, L.-M. Peng, J. Zhu, and G. A. D. Briggs, “Ultrahigh Secondary Electron Emission of Carbon Nanotubes,” Applied Physics Letters, vol. 96, pp. 213113-1–213113-3, 2010. [38] D. Drouin, A. R. Couture, D. Joly, X. Tastet, V. Aimez, and R. Gauvin, “CASINO V2.42—A Fast and Easy-to-use Modeling Tool for Scanning Electron Microscopy and Microanalysis Users,” Scanning, vol. 29, pp. 92-101, 2007. [39] R. Gauvin, E. Lifshin, H. Demers, P. Horny, and H. Campbell, “Win X-ray: A New Monte Carlo Program that Computes X-ray Spectra Obtained with a Scanning Electron Microscope,” Microscopy and Microanalysis, vol. 12, pp. 49-64, 2006. [40] D. C. Joy, “ http://web.utk.edu/~srcutk/htm/simulati.htm,” Joy’s Database and Monte Carlo Program, 1995. [41] E. Napchan, “MC-SET,” http://www.mc-set.com, 2010. [42] Small World LLC, “Electron Flight Simulator,” http://www.small-world.net/index.htm, 2004. [43] F. Salvat, J. M. Fernandez-Varea, and J. Sempau, “PENELOPE,” OECD Nuclear Energy Agency,http://www.nea.fr/abs/html/nea-1525.html, 2009. [44] N. W. M. Ritchie, “A New Monte Carlo application for Complex Sample Geometries,” Surface and Interface Analysis, vol. 37, pp. 1006-1011, 2005. [45] Spectel Research Corporation, “METROLOGIA,” http://www.spectelresearch.com/metrolog.html, 2004. [46] Y. Lin and D. C. Joy, “A New Examination of Secondary Electron Yield Data,” Surface and Interface Analysis, vol. 37, pp. 895-900, 2005. [47] R. Browning, T. Z. Li, B. Chui, J. Ye, R. F. W. Pease, Z. Czyzewski, and D. C. Joy, “Low-Energy Electron/Atom Elastic Scattering Cross Sections from 0.1-30 keV,” Scanning, vol. 17, pp. 250-253, 1995. [48] R. Browning, T. Z. Li, B. Chui, J. Ye, R. F. W. Pease, Z. Czyz ̇ ewski, and D. C. Joy, “Empirical Forms for the Electron/Atom Elastic Scattering Cross Sections from 0.1 to 30 keV,” Journal of Applied Physics, vol. 76, pp. 2016-2022, 1994. [49] The MathWorks Inc., “MATLAB 2009b,” USA, 2009. [50] W. R. Leo, Techniques for Nuclear and Particle Physics Experiments: A How to Approach, 2nd Revised ed., pp. 24-25, Berlin: Springer, 2005.  Bibliography 106  [51] N. F. Mott and H. S. W. Massey, The Theory of Atomic Collisions, 3rd ed., London: Oxford Unversity Press, 1965. [52] K. Murata, M. Yasuda, and H. Kawata, “Effects of The Introduction of the Discrete Energy Loss Process into Monte Carlo Simulation of Electron Scattering,” Scanning, vol. 17, pp. 228-234, 1995. [53] R. Ferrell, “Angular Dependence of the Characteristic Energy Loss of Electrons Passing Through Metal Foils,” Physical Review, vol. 101, pp. 554-563, 1956. [54] J. Quinn, “Range of Excited Electrons in Metals,” Physical Review, vol. 126, pp. 1453-1457, 1962. [55] N. V. Smith and W. E. Spicer, “Photoemission Studies of the Alkali Metals. I. Sodium and Potassium,” Physical Review, vol. 188, pp. 593-605, 1969. [56] H. Bethe, “Zur Theorie des Durchgangs Schneller Korpuskularstrahlung durch Materie,” Ann. de Phys., vol. 5, pp. 325-400, 1930. [57] D. C. Joy and S. Luo, “An Empirical Stopping Power Relationship for Low-Energy Electrons,” Scanning, vol. 11, pp. 176-180, 1989. [58] A. Szabo and N. S. Ostlund, Modern Quantum Chemistry : Introduction to Advanced Electonic Structure Theory, New York: Dover Publications, 1996. [59] C. Kim, B. Kim, S. Lee, C. Jo, and Y. Lee, “Electronic Structures of Capped Carbon Nanotubes Under Electric Fields,” Physical Review B, vol. 65, pp. 165418-1–165418- 6, 2002. [60] S. Han, M. Lee, and J. Ihm, “Dynamical Simulation of Field Emission in Nanostructures,” Physical Review B, vol. 65, pp. 085405-1–085405-7, 2002. [61] M. J. Frisch, G. Trucks, and H. B. Schlegel et al., “Gaussian 03, Revision C.02,” Gaussian Inc. (http://www.gaussian.com), 2004. [62] JCrystalSoft, “Nanotube Modeler,” http://jcrystal.com/products/wincnt/, 2010. [63] T. Mirfakhrai, R. Krishna-Prasad, A. Nojeh, and J. D. W. Madden, “Electromechanical Actuation of Single-Walled Carbon Nanotubes: An ab-initio Study,” Nanotechnology, vol. 19, pp. 315706-1–315706-8, 2008. [64] P. Baum and A. H. Zewail, “Attosecond Electron Pulses for 4D Diffraction and Microscopy,” Proceedings of the National Academy of Sciences, vol. 104, pp. 18409- 18414, 2007.  Bibliography 107  [65] S. Reich, M. Dworzak, A. Hoffmann, C. Thomsen, and M. Strano, “Excited-State Carrier Lifetime in Single-Walled Carbon Nanotubes,” Physical Review B, vol. 71, pp. 033402-1–033402-4, 2005. [66] P. J. Price, “Attempt Frequency in Tunneling,” American Journal of Physics, vol. 66, pp. 1119-1122, 1998. [67] P. C. W. Davies, “Quantum Tunneling Time,” American Journal of Physics, vol. 73, pp. 23-27, 2005. [68] K. S. Sim and J. D. White, “New Technique for In-situ Measurement of Backscattered and Secondary Electron Yields for the Calculation of Signal-to-Noise Ratio in a SEM,” Journal of Microscopy, vol. 217, pp. 235-240, 2005. [69] H. Drescher, L. Reimer, and H. Seidel, “Backscattering and Secondary Electron Emission of 10–100 keV Electrons and Correlation to Scanning Electron Microscopy,” Z. Angew. Physik, vol. 29, pp. 331-336, 1970. [70] L. Reimer and C. Tolkamp, “Measuring the Backscattering Coefficient and Secondary Electron Yield Inside a Scanning Electron Microscope,” Scanning, vol. 3, pp. 35-39, 1980. [71] A. M. D. Assa'd and M. M. El-Gomati, “Backscattering Coefficients for Low Energy Electrons,” Scanning Microscopy, vol. 12, pp. 185-192 1998. [72] D. C. Joy, “A Database on Electron-Solid Interactions,” Scanning, vol. 17, pp. 270- 275, 1995. [73] J. L. Goldstein, D. E. Newburry, P. Echlin, J. D. C., and J. A. D. Roming, et al, Scanning Electron Microscopy and X-Ray Microanalysis: A Text for Biologists, Material Scientists and Geologists, New York: Plenum Press, 1992. [74] K. Kanaya and S. Okayama, “Penetration and Energy-Loss Theory of Electrons in Solid Targets,” Journal of Physics D: Applied Physics, vol. 5, pp. 43-58, 1972. [75] D. N. Futaba, K. Hata, T. Yamada, T. Hiraoka, Y. Hayamizu, Y. Kakudate, O. Tanaike, H. Hatori, M. Yumura, and S. Iijima, “Shape-Engineerable and Highly Densely Packed Single-Walled Carbon Nanotubes and their Application as Super- Capacitor Electrodes,” Nature Materials, vol. 5, pp. 987-994, 2006. [76] K. Yamamoto, T. Shibata, N. Ogiwara, and M. Kinsho, “Secondary Electron Emission Yields from the J-PARC RCS Vacuum Components,” Vacuum, vol. 81, pp. 788-792, 2007. [77] M. Dapor, Electron-Beam Interactions with Solids: Application of Monte Carlo Method to Electron Scattering Problems, New York: Springer, 2003.  Bibliography 108  [78] J. R. Lowney, “Monte Carlo Simulation of Scanning Electron Microscope Signals for Lithographic Metrology,” Scanning, vol. 18, pp. 301-306, 1996. [79] O. Stéphan, D. Taverna, M. Kociak, K. Suenaga, L. Henrard, and C. Colliex, “Dielectric Response of Isolated Carbon Nanotubes Investigated by Spatially Resolved Electron Energy-Loss Spectroscopy:From Multiwalled to Single-Walled Nanotubes,” Physical Review B, vol. 66, pp. 155422-1–155422-6, 2002. [80] R. M. D. Stevens, N. A. Frederick, B. L. Smith, D. E. Morse, G. D. Stucky, and P. K. Hansma, “Carbon Nanotubes as Probes for Atomic Force Microscopy,” Nanotechnology, vol. 11, pp. 1-5, 2000. [81] X. Liang, Z. Fu, and S. Y. Chou, “Graphene Transistors Fabricated via Transfer- Printing In Device Active-Areas on Large Wafer,” Nano Letters, vol. 7, pp. 3840- 3844, 2007. [82] M. K. Alam, S. P. Eslami, and A. Nojeh, “Secondary Electron Emission from Single- Walled Carbon Nanotubes,” Physica E, vol. 42, pp. 124-131, 2009. [83] R. Shimizu, Y. Kataoka, T. Matsukawa, T. Ikuta, K. Murata, and H. Hashimoto, “Energy Distribution Measurements of Transmitted Electrons and Monte Carlo Simulation for Kilovolt Electron,” Journal of Physics D: Applied Physics, vol. 8, pp. 820-828, 1975. [84] R. Shimizu, Y. Kataoka, T. Ikuta, T. Koshikawa, and H. Hashimoto, “A Monte Carlo Approach to the Direct Simulation of Electron Penetration in Solids,” Journal of Physics D: Applied Physics, vol. 9, pp. 101-113, 1976. [85] M. M. Brzhezinskaya and E. M. Baitinger, Trends in Nanotube Research, pp. 223-228, New York: Nova Science Publishers, 2006. [86] G. Chiarello, E. Maccallini, R. G. Agostino, V. Formoso, A. Cupolillo, D. Pacile, E. Colavita, L. Papagno, L. Petaccia, R. Larciprete, S. Lizzit, and A. Goldoni, “Electronic and Vibrational Excitations in Carbon Nanotubes,” Carbon, vol. 41, pp. 985-992, 2003. [87] B. J. LeRoy, S. G. Lemay, J. Kong, and C. Dekker, “Electrical generation and absorption of phonons in carbon nanotubes,” Nature, vol. 432, pp. 371-374, 2004. [88] L. N. Pandey and M. L. Rustgi, “A Comparative Study of Electron Transport Phenomenon in the keV Range,” Journal of Applied Physics, vol. 66, pp. 6059-6064, 1989. [89] F. Arezzo, N. Zacchetti, and W. Zhu, “X-ray Photoelectron Spectroscopy Study of Substrate Surface Pretreatments for Diamond Nucleation,” Journal of Applied Physics, vol. 75, pp. 5375-5381, 1994.  Bibliography 109  [90] M. H. Gass, U. Bangert, A. L. Bleloch, P. Wang, R. R. Nair, and A. K. Geim, “Free- Standing Graphene at Atomic Resolution,” Nature Nanotechnology, vol. 3, pp. 676- 681, 2008. [91] D. Drouin, P. Hovington, and R. Gauvin, “CASINO: A New Monte Carlo Code in C Language for Electron Beam Interactions-Part II: Tabulated Values of the Mott Cross Section,” Scanning, vol. 19, pp. 20-28, 1997. [92] Z. Czyzewski, D. O. N. MacCallum, A. Romig, and D. C. Joy, “Mott Numerical Cross-section Data,” http://web.utk.edu/~srcutk/Mott/Calculations/06.dat, 1990. [93] S. D. Sarma and E. Hwang, “Dynamical Response of A One-Dimensional Quantum- Wire Electron System,” Physical Review B, vol. 54, pp. 1936-1946, 1996. [94] X. Wei, D. Golberg, Q. Chen, Y. Bando, and L. Peng, “Phonon-Assisted Electron Emission from Individual Carbon Nanotubes,” Nano Letters, vol. 11, pp. 734-739, 2011. [95] J.-Y. Park, S. Rosenblatt, Y. Yaish, V. Sazonova, H. Üstünel, S. Braig, T. A. Arias, P. W. Brouwer, and P. L. McEuen, “Electron-Phonon Scattering in Metallic Single- Walled Carbon Nanotubes,” Nano Letters, vol. 4, pp. 517-520, 2004.  110  Appendix A Scripts for First-Principles Simulations An example set of scripts for submitting simulations to WestGrid is given below for a (5,5) nanotube where an extra electron is placed at 2 Å from the center of the nanotube (bold lines are not a part of the scripts). Script for Submitting Jobs to WestGrid:  #The following lines should be in a text file and the file extension should be changed to *.pbs #For example job.pbs # Sample Gaussian job script cd $PBS_O_WORKDIR echo "Current working directory is `pwd`" echo "Running on `hostname`" # Set up the Gaussian environment using the module command module load gaussian # Run g09 g09 < job.com  [Gaussian script file name]  Code for Gaussian Simulations on WestGrid:  The following code should be saved in a *.com file (e.g. job.com for the above submission script)  %chk=CNT55_15AUCEC5A.chk %mem=120MW %nproc=2 # rhf/6-31g(d) scf=(qc,maxcycle=1000) geom=connectivity charge sp test formcheck nosymm  [Defines the method, basis set, output parameters, etc.]  Appendix A. Scripts for First-Principles Simulations 111  Title Card Required  0 1    [Charge and spin multiplicity]  C                 -4.94933206   -2.93338882    1.79632427  C                 -6.14737516   -2.53736436    2.38519242  C                 -7.36088273   -3.03428622    1.77965420  C                  0.00000667   -2.92557761    1.76647914  C                 -1.24690593   -2.51176858    2.36483419  C                 -2.44843098   -2.94146811    1.79840446  C                 -3.69356270   -1.32306760    3.13487361  C                 -3.69355020   -2.46477128    2.34593734  C                 -4.94936524   -0.64331041    3.37889864  C                 -6.14737521   -1.33418343    3.21666706  C                 -7.36093528    0.75492244    3.43562455  C                 -7.36092295   -0.59207922    3.46740051  C                  4.94934264   -2.93336768    1.79632277  C                  3.69355224   -2.46473999    2.34591885  C                  2.44843926   -2.94143558    1.79840126  C                  1.24691759   -1.32407027    3.18553692  C                  1.24689748   -2.51176082    2.36484304  C                 -0.00001450   -0.61804938    3.36106976  C                 -1.24690735   -1.32408644    3.18555053  C                 -2.44846623    0.80144706    3.35310094  C                 -2.44845735   -0.64232887    3.38718187  C                 -3.69359516    1.46950806    3.06897781  C                 -4.94938093    0.80199003    3.34480734  C                 -6.14743633    2.64702963    2.26285027  C                 -6.14740728    1.48441054    3.15016537  C                 -7.36094608    3.11477934    1.63453001  C                  7.36089665   -3.03430385    1.77963340  Appendix A. Scripts for First-Principles Simulations 112   C                  6.14739395   -1.33411025    3.21660260  C                  6.14737781   -2.53735033    2.38517304  C                  4.94936241   -0.64324351    3.37883552  C                  3.69355908   -1.32303713    3.13486366  C                  2.44845841    0.80148862    3.35309837  C                  2.44844796   -0.64232065    3.38717850  C                  1.24688950    1.47296083    3.11946176  C                 -0.00002364    0.77597814    3.32811567  C                 -1.24694289    2.62056935    2.24365930  C                 -1.24692786    1.47294214    3.11951641  C                 -2.44849354    3.02303876    1.65759423  C                 -3.69360821    2.57269415    2.22702648  C                 -4.94940473    3.42904302    0.27087664  C                 -4.94940375    3.01483555    1.65593171  C                 -6.14742453    3.45482312   -0.43832389  C                 -7.36093988    3.50082448    0.34361678  C                  7.36089268    0.75503220    3.43551560  C                  7.36090406   -0.59198765    3.46730485  C                  6.14736696    1.48449806    3.15013752  C                  4.94936538    0.80203221    3.34472412  C                  3.69348246    2.57273198    2.22696639  C                  3.69351882    1.46954334    3.06891813  C                  2.44840928    3.02309393    1.65759673  C                  1.24687066    2.62053266    2.24363492  C                 -0.00005867    3.40526422    0.29049255  C                 -0.00006264    3.00572296    1.62644384  C                 -1.24693243    3.42221833   -0.43684420  C                 -2.44849535    3.43683011    0.27397020  C                 -3.69357420    2.91315833   -1.75855349  C                 -3.69359609    3.37303794   -0.44919667  Appendix A. Scripts for First-Principles Simulations 113   C                 -4.94936812    2.50659795   -2.35555568  C                 -6.14740966    2.97016719   -1.81823894  C                 -7.36086120    1.40861677   -3.22327066  C                 -7.36090623    2.51710801   -2.45728381  C                  7.36080637    3.11487689    1.63446153  C                  6.14735854    2.64715781    2.26277365  C                  4.94927174    3.42915929    0.27083567  C                  4.94929265    3.01491586    1.65583335  C                  3.69346955    3.37310917   -0.44925157  C                  2.44839518    3.43688633    0.27392311  C                  1.24685882    2.94378948   -1.79890743  C                  1.24686129    3.42223115   -0.43685339  C                 -0.00002332    2.47572216   -2.35590016  C                 -1.24691092    2.94377790   -1.79891112  C                 -2.44842118    1.32251382   -3.18361027  C                 -2.44846686    2.51075747   -2.36280822  C                 -3.69350502    0.61507510   -3.34651855  C                 -4.94932416    1.31717978   -3.17727024  C                 -6.14733484   -0.81144163   -3.38655442  C                 -6.14732997    0.65071526   -3.42106928  C                 -7.36082818   -1.55915831   -3.15319011  C                  7.36079727    3.50107490    0.34355407  C                  6.14733918    2.97051549   -1.81832143  C                  6.14730774    3.45520260   -0.43836809  C                  4.94932767    2.50687941   -2.35567980  C                  3.69351014    2.91329713   -1.75862051  C                  2.44848303    1.32259516   -3.18368850  C                  2.44842447    2.51077987   -2.36282737  C                  1.24695344    0.64204119   -3.38938657  C                  0.00003570    1.32847682   -3.14854866  Appendix A. Scripts for First-Principles Simulations 114   C                 -1.24687613   -0.80124296   -3.35542424  C                 -1.24687246    0.64200955   -3.38943685  C                 -2.44840599   -1.47137982   -3.11793632  C                 -3.69349752   -0.77233344   -3.31386500  C                 -4.94929995   -2.61493656   -2.23466122  C                 -4.94930094   -1.46574890   -3.11174914  C                 -6.14734261   -3.05269581   -1.67609342  C                 -7.36082595   -2.63028647   -2.33574334  C                  7.36100058    1.40859968   -3.22282754  C                  7.36091231    2.51735507   -2.45717061  C                  6.14752566    0.65076496   -3.42080502  C                  4.94947412    1.31724599   -3.17714704  C                  3.69359638   -0.77224598   -3.31360843  C                  3.69362582    0.61510727   -3.34644984  C                  2.44849032   -1.47131859   -3.11779333  C                  1.24694638   -0.80124011   -3.35544175  C                  0.00004128   -2.58408032   -2.23645587  C                  0.00004652   -1.47562560   -3.08248321  C                 -1.24686746   -3.02532546   -1.65802098  C                 -2.44840439   -2.61939813   -2.24171153  C                 -3.69351412   -3.39046889   -0.28956902  C                 -3.69350084   -2.99289558   -1.61918021  C                 -4.94932493   -3.41248911    0.43234517  C                 -6.14734166   -3.47171595   -0.27483989  C                 -7.36086506   -3.48082527    0.50841454  C                  7.36096899   -1.55940839   -3.15301612  C                  6.14750925   -0.81146754   -3.38603522  C                  4.94935103   -2.61515211   -2.23464467  C                  4.94944750   -1.46567764   -3.11142891  C                  3.69356619   -2.99298404   -1.61913854  Appendix A. Scripts for First-Principles Simulations 115   C                  2.44847194   -2.61941540   -2.24169831  C                  1.24690328   -3.43894523   -0.27485884  C                  1.24692389   -3.02535724   -1.65803884  C                  0.00001381   -3.38768474    0.45084095  C                 -1.24688272   -3.43890355   -0.27489061  C                 -2.44842906   -3.42006188    0.43581568  C                  7.36087194   -2.63063562   -2.33577925  C                  6.14733738   -3.47188328   -0.27483685  C                  6.14735563   -3.05294650   -1.67605323  C                  4.94931926   -3.41258903    0.43237602  C                  3.69354685   -3.39052251   -0.28952643  C                  2.44842114   -3.42008738    0.43586103  C                  7.36085924   -3.48099077    0.50843574  H                 -8.29852158   -2.93359509    2.30951764  H                 -8.29854852    1.28999196    3.50371381  H                 -8.29852486   -1.12336924    3.56063185  H                 -8.29857826    3.03921589    2.16856071  H                  8.29852486   -2.93363053    2.30951348  H                 -8.29856688    3.73082797   -0.14424602  H                  8.29851497    1.29010270    3.50351106  H                  8.29852846   -1.12323640    3.56048400  H                 -8.29846676    1.01571525   -3.59283912  H                 -8.29854164    3.00167858   -2.22045882  H                  8.29849216    3.03911604    2.16837155  H                 -8.29847480   -1.18410918   -3.54076756  H                  8.29847623    3.73103223   -0.14424098  H                 -8.29847015   -3.10312588   -2.07631008  H                  8.29863726    1.01559329   -3.59217169  H                  8.29854185    3.00192479   -2.22033931  H                 -8.29848327   -3.73365838    0.03196355  Appendix A. Scripts for First-Principles Simulations 116   H                  8.29863557   -1.18429789   -3.54038452  H                  8.29843469   -3.10372288   -2.07649120  H                  8.29843919   -3.73401448    0.03201854   1 2 1.5 8 1.5 110 1.5  2 10 1.5 3 1.5  3 112 1.5 131 1.0  4 121 1.5 5 1.5 17 1.5  5 6 1.5 19 1.5  6 8 1.5 123 1.5  7 8 1.5 21 1.5 9 1.5  8  9 10 1.5 23 1.5  10 12 1.5  11 12 1.5 25 1.5 132 1.0  12 133 1.0  13 14 1.5 29 1.5 127 1.5  14 15 1.5 31 1.5  15 17 1.5 129 1.5  16 17 1.5 18 1.5 33 1.5  17  18 19 1.5 35 1.5  19 21 1.5  20 21 1.5 22 1.5 37 1.5  21  22 23 1.5 39 1.5  23 25 1.5  24 41 1.5 26 1.5 25 1.5  25  26 43 1.5 134 1.0  Appendix A. Scripts for First-Principles Simulations 117   27 130 1.5 29 1.5 135 1.0  28 29 1.5 30 1.5 45 1.5  29  30 47 1.5 31 1.5  31 33 1.5  32 34 1.5 49 1.5 33 1.5  33  34 35 1.5 51 1.5  35 37 1.5  36 37 1.5 38 1.5 53 1.5  37  38 39 1.5 55 1.5  39 41 1.5  40 41 1.5 42 1.5 57 1.5  41  42 43 1.5 59 1.5  43 136 1.0  44 45 1.5 46 1.5 137 1.0  45 138 1.0  46 47 1.5 63 1.5  47 49 1.5  48 49 1.5 50 1.5 65 1.5  49  50 51 1.5 67 1.5  51 53 1.5  52 54 1.5 53 1.5 69 1.5  53  54 55 1.5 71 1.5  55 57 1.5  56 57 1.5 58 1.5 73 1.5  Appendix A. Scripts for First-Principles Simulations 118   57  58 59 1.5 75 1.5  59 61 1.5  60 61 1.5 77 1.5 139 1.0  61 140 1.0  62 79 1.5 63 1.5 141 1.0  63 65 1.5  64 65 1.5 66 1.5 81 1.5  65  66 67 1.5 83 1.5  67 69 1.5  68 69 1.5 70 1.5 85 1.5  69  70 71 1.5 87 1.5  71 73 1.5  72 73 1.5 89 1.5 74 1.5  73  74 75 1.5 91 1.5  75 77 1.5  76 93 1.5 77 1.5 78 1.5  77  78 95 1.5 142 1.0  79 81 1.5 143 1.0  80 81 1.5 82 1.5 97 1.5  81  82 83 1.5 99 1.5  83 85 1.5  84 85 1.5 86 1.5 101 1.5  85  86 87 1.5 103 1.5  Appendix A. Scripts for First-Principles Simulations 119   87 89 1.5  88 89 1.5 105 1.5 90 1.5  89  90 91 1.5 107 1.5  91 93 1.5  92 93 1.5 94 1.5 109 1.5  93  94 95 1.5 111 1.5  95 144 1.0  96 97 1.5 98 1.5 145 1.0  97 146 1.0  98 99 1.5 114 1.5  99 101 1.5  100 101 1.5 116 1.5 102 1.5  101  102 118 1.5 103 1.5  103 105 1.5  104 105 1.5 106 1.5 120 1.5  105  106 122 1.5 107 1.5  107 109 1.5  108 109 1.5 123 1.5 110 1.5  109  110 111 1.5  111 112 1.5  112 147 1.0  113 124 1.5 114 1.5 148 1.0  114 116 1.5  115 116 1.5 117 1.5 126 1.5  116  Appendix A. Scripts for First-Principles Simulations 120   117 118 1.5 128 1.5  118 120 1.5  119 121 1.5 129 1.5 120 1.5  120  121 122 1.5  122 123 1.5  123  124 126 1.5 149 1.0  125 126 1.5 127 1.5 130 1.5  126  127 128 1.5  128 129 1.5  129  130 150 1.0  131  132  133  134  135  136  137  138  139  140  141  142  143  144  145  146  Appendix A. Scripts for First-Principles Simulations 121   147  148  149  150  0.0 2.0 0.0 -1.0 [Position of the extra electron]  Both files (job.pbs and job.com) should be copied in a single directory and the scripts can be submitted for simulation by using the Terminal/Command Window of an SSH Client with the following command: qsub -l walltime=300:00:00 job.pbs Gaussian is currently available on the server “checkers.westgrid.ca” on WestGrid. Matlab simulations can be submitted to “glacier.westgrid.ca” using a similar method.   122  Appendix B Empirical Fit Routine Used in Chapter 5 The empirical fit procedure is shown in the flow diagram (Figure B.1). To find an optimal method, we first simulated the distinct effects of each empirical parameter on the backscattering coefficient of a 500-µm-diameter forest. As can be seen from Figure B.2(a), the screening parameter affects the entire energy range. The backscattering yield increases with the increase of the screening parameter and vice versa. The reason is that large angle scattering increases with the increase of the screening parameter (as can be seen from Figure 5.3 and its associated discussion in Chapter 5). Figure B.2(b) shows that a smaller effective nanotube thickness (5% of the nanotube radius) produces less fluctuation, which is more realistic (note from Figure 5.7 that the experimental values do not show much fluctuations) compared to a higher value of the nanotube thickness. It is also known that surface backscattering is not highly sensitive to the density of a material [4] (confirmed by our simulation [Figure B.2(c)]). Figure B.2(c) shows that sidewall scattering (which affects the yield at high primary energies, as discussed in Chapter 5) increases with the decrease of the density while the surface backscattering remains almost the same. Therefore, it is reasonable and advantageous to initiate the empirical fit routine to match the surface backscattering data (Figure 5.7 before 12 keV) by sweeping the screening parameter and shell thickness, while keeping the density fixed at a reasonable value such as the theoretically estimated value (0.0296 g/cm3). The r.m.s. error was minimized for the surface backscattering data (Figure 5.7 before 12 keV) using the Monte Carlo routine shown in the flow diagram (Figure B.1) (in the case of the adjusted bulk model, the steps from checking the electron’s position to changing the direction cosines using the elastic collision law do not apply. Also, there is no shell thickness parameter and only the screening parameter is swept at this stage). Once the error was minimized with respect to the screening parameter and effective shell thickness and these two parameters were known, the density was adjusted in order to improve the fit  Appendix B. Empirical Fit Routine Used in Chapter 5 123  (minimize the r.m.s error), in particular for the high energy backscattering data that show a rising trend due to sidewall scattering (Figure 5.7 beyond 12 keV).   Calculate electron’s new energy Save the position, reset direction cosines and position variables  Calculate scattering angles Calculate new position of the electron  Change the direction cosines using the elastic collision law  F Electron in the shell T T F Electron backscattered Return Count BSE Calculate step length Check the electron’s position Start Monte Carlo routine  Calculate initial direction cosines Nanotube diameter, dCNT ; density, ρ ; Grid parameters; Primary energy, Screening parameter, α; Shell thickness  T F E >Ecutoff Sweep the screening parameter and shell thickness (Fixing the density to a reasonable value, e.g. theoretically estimated 0.0296 g/cm3) to minimize the r.m.s. error for the surface backscattering data  Monte Carlo routine End Total r.m.s. error minimized T F Monte Carlo routine r.m.s. error minimized T F Start main routine (parameter extraction)  Sweep the density to minimize the r.m.s. error for the entire dataset   Figure B.1 Flow diagram of simulation procedure.  Appendix B. Empirical Fit Routine Used in Chapter 5 124  At this step, the entire dataset was taken into consideration to minimize the r.m.s. error. Thus, the empirical parameters were estimated from the dataset for the 500-µm-diameter forest (Figure 5.7). The same set of parameters (screening parameter, shell thickness, density) was then used to predict the backscattered yield of the 200-µm-diameter forest and the results were compared with the experimental data. It was found that the experimental surface backscattering data (Figure 5.8 before 10 keV) fit the predicted results very well. This indicates that the extracted screening parameter and the effective nanotube thickness, derived from the 500-µm-thickness forest, are also applicable to other structures. However,  (b) 0 5 10 15 20 0.05 0.1 0.15 0.2 Primary energy [keV] BS E co ef fic ie nt   Nanotube thickness = 50% of the radius Nanotube thickness = 15% of the radius Nanotube thickness = 5% of the radius α = 6×10-3/E; ρ = 0.0296 g/cm3 (a) 0 5 10 15 20 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 Primary energy [keV] BS E co ef fic ie nt   α=6×10-3/E α=9×10-3/E α=4×10-3/E Nanotube thickness= 5% of the radius; ρ= 0.0296 g/cm3 0 5 10 15 20 0.05 0.1 0.15 0.2 0.25 0.3 Primary energy [keV] BS E co ef fic ie nt   ρ = 0.019 g/cm3 ρ = 0.0296 g/cm3 ρ = 0.035 g/cm3 ρ = 0.01 g/cm3 Nanotube thickness= 5% of the radius; α = 6×10-3/E (c) Figure B.2 (a) Effect of changing the screening parameter, (b) Effect of changing the effective nanotube thickness and (c) Effect of changing the density.   Appendix B. Empirical Fit Routine Used in Chapter 5 125  we needed to change the density to 0.0356 g/cm3 (corresponding to a difference of 26% in the average distance between the nanotubes assuming everything else remains the same) to improve the fit for the sidewall scattering part of the data (Figure 5.8 after 10 keV). Note that our two forests with the different diameters were grown during different CVD runs. Due to the current limitations in nanotube fabrication, the forest density varies from growth to growth depending upon the uniformity and size of the catalyst particles created during the annealing of the catalyst films (deposited using e-beam evaporation). It also depends on the ambient pressure, gas flow rate and growth dynamics. Therefore, a 26% change in the nanotube spacing or a change in the chirality (which also contributes to the density) is very reasonable in the CVD growth of nanotube forests.        

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.24.1-0072019/manifest

Comment

Related Items