Science of Nanotechnology

THE SCIENCE OF NANOTECHNOLOGY: AN INTRODUCTORY TEXT

THE SCIENCE OF NANOTECHNOLOGY: AN INTRODUCTORY TEXT

LUANNE TILSTRA S. ALLEN BROUGHTON ROBIN S. TANKE DANIEL JELSKI VALENTINA FRENCH GUOPING ZHANG ALEXANDER K. POPOV ARTHUR B. WESTERN AND

THOMAS F. GEORGE

Nova Science Publishers, Inc. New York

Copyright © 2008 by Nova Science Publishers, Inc.

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA George, Thomas F., 1947Science of nanotechnology : an introductory text / Thomas F. George. p. cm. Includes index. ISBN-13: 978-1-60692-870-7 1. Nanotechnology--Textbooks. I. Title. T174.7.G46 2006 620'.5--dc22 2006030796

Published by Nova Science Publishers, Inc.

New York

CONTENTS Preface

vii

Chapter 1

The Nanoscale World

1

Chapter 2

Investigating the Nanoscale

19

Chapter 3

Making the Nanoworld

49

Chapter 4

Describing the Geometry of the Carbon NanoFamily

67

Chapter 5

Mechanical and Magnetic Properties of Nanoparticles

85

Chapter 6

Optical Properties

103

Appendix:

Activities for Nanotechnology

143

Glossary

155

References

163

Index

169

PREFACE This book arose from the desire of a group of chemistry, physics and mathematics faculty to bring nanotechnology into the undergraduate classroom. We were fortunate to secure funding that made the project possible. Our initial effort was to teach one-credit classes at the three campuses where we work: Rose-Hulman Institute of Technology, Indiana State University and University of Wisconsin−Stevens Point. To teach these classes, we worked together to create course materials appropriate for sophomore science majors. Those lecture notes have evolved into this book. Any book with nine co-authors is a complicated project. We decided early on that we did not simply want to produce an edited volume, but instead to create a textbook. The goal is teaching rather than monographs, and the intent is that there be a progression from one chapter to the next, so that the book can usefully be read from cover to cover. At the same time, the expertise represented by our group varied widely: experimentalists and theoreticians; mathematicians and laboratory chemists; and much in between. There is no point in trying to hide this fact, and hence, while we’ve tried to create a coherent text, we have made no effort to produce an homogeneous one. The casual reader will note variations in style and content that reflect the personalities and interests of the principal authors of that chapter. It is assumed that all students reading this book will have completed general chemistry. Hence, terms like stoichiometry and Ångstrom are used with abandon. It is supposed that students are completely familiar with metric and SI units. Concepts such as molecular orbital theory are described in more detail, but it is still assumed that the student has seen some of this before. For some chapters, familiarity with calculus will prove useful. Chapter One serves as an introduction and discusses scales, sizes and some history of nanoparticles. Chapter Two concentrates on optical methods for studying nanoparticles, especially absorption, light scattering and diffraction methods. Chapter Three discusses various ways to make nanoparticles, including laser ablation, chemical vapor deposition and self-assembly techniques. These three chapters can be considered as introductory material and should be covered by any student using this text. Anyone with a general chemistry background will readily understand the material.

viii

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Chapter Four concerns the structure of carbon nanotubes. The primary author is a mathematician, and thus mathematically-inclined students will find it most rewarding. However, the prerequisite knowledge for this chapter is minimal—a good knowledge of trigonometry and vector algebra will be useful. Chapter Five is about the mechanical and magnetic properties of nanoparticles, and will be of greatest interest to those who may intend to be engineers. A year of physics will also be helpful before reading this chapter, as is some familiarity with calculus. Chapter Six, covering optical properties, is the most difficult chapter in the book, and definitely requires a good background in general physics. Arguably, it is more appropriate for senior students than sophomores, but the well-prepared or hard-working sophomore will find it rewarding. The book contains exercises and hands-on activities. The latter, some of which may be considered laboratory experiments, are collected in an appendix. As with most textbooks, a comprehensive literature review is not provided, but the principal references and articles that may be of further interest to students are cited at the end of the book. There is a comprehensive glossary. Terms in the text are italicized if they appear in the glossary. The authors acknowledge the National Science Foundation for funding under Grant No. DMR-0304487. We thank Rose-Hulman sophomore Ross Poland for assistance in developing the hands-on projects. And finally, we acknowledge our copy editor, Dr. Bob Rich (www.bobswriting.com), for technical assistance, wise counsel and good friendship. For a book with nine authors, his efforts were crucial in bringing this text to fruition. There are a great many diagrams within this book. Some were created by the authors. Other figures were copied or adapted from various sources. Each such source is cited within the relevant figure caption. The copyright holders of all these drawings have kindly given permission to have their material reproduced.

SOME USEFUL CONSTANTS Speed of light Boltzmann constant Gas constant Planck’s constant Avogadro’s Number Bohr magneton

c = 2.9979 × 108 m/s kB = 1.3807 × 10-23 J/K R = 8.3145 J/mol K h = 6.6261 × 10-34 J s NA = 6.0221 × 1023 1/mol μB = 9.27 x 10-24 A m2 109 nm = 106 μm = 103 mm = 1 m

Chapter 1

THE NANOSCALE WORLD 1.1. A MATTER OF SCALE It’s all a Matter of Scale Beginning around 1600, people began to design instruments. The pendulum clock was invented by Huygens Christiaan in 1656. Pendulum clocks were rather large and not especially portable (think of your grandmother’s grandfather clock). They also had to stay upright in order to function properly, which rendered them useless for shipboard navigation. The invention of spring-driven clocks changed this by making clocks both smaller and more portable. The most important immediate application was the development of the marine chronometer. This accurate clock permitted the determination of longitude by comparing local time with Greenwich Mean Time, read from the chronometer. During the 19th century, people manufactured exceedingly fine chronometers, easily accurate to the nearest second. Today this technology exists only as a luxury item, in the form of elegant Swiss watches. The development of clocks, and the desire to make them as small and as portable as possible, required the ability to manipulate very small objects. Fingers were too big and so forceps were created. Fine machining was developed—think of all the little screws, springs and hinges that are necessary for a mechanical wristwatch. But the key limitation was human eyesight. In those days it was impossible to manipulate things too small to see—and this indeed defined the scale of the technology—millimeter scale. Just for fun, let’s call it “millitechnology”. While we no longer use mechanical clocks, we still use a lot of millitechnology— engines, toasters and pianos, for example, along with the moving parts of all sorts of items you would never associate with the 19th century: CD players, computer keyboards and photocopiers, to name just a few. Millitechnology continues to be important to this very day. Not all millitechnology has served useful purposes; some of it is done simply for fun. Think of the ship models built in bottles, or model train sets. These entertaining though impractical endeavors demonstrate just how proficient 19th century millitechnologists were. One of the most important products of the millimeter era was the invention of the microscope. A Dutch clockmaker (and millitechnologist) named Anton Van Leeuwenhoek invented one of the earliest microscopes near the end of the 17th century. We normally associate this invention with developments in medicine, and certainly those are important, but for our purposes here the microscope takes on a larger (or smaller?) meaning, namely

2

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

removing the limitation on human eyesight. In principle, it permitted technology 1000 times smaller than a millimeter: on the order of a micrometer, μm, frequently referred to as a micron. Microtechnology was born... but it took a long time to develop. Of course the famous Swiss watchmakers used magnifying lenses in their work from early on. Elegant binoculars, opera glasses, and cameras were manufactured; the famous German optics firm, Zeiss, was founded in 1846. But none of this is really microtechnology; it is simply the application of simple optics to millimeter-scale problems. So what was the hang-up? To actually do microtechnology requires not only being able to see at the micron scale, but also to be able to manipulate objects at that scale. A pair of forceps no longer works. Oddly enough, the solution to this problem evolved from another crucial invention of the millimeter age—lithography. While type-setting required manipulating individual letters (using forceps) into position to make lines of words, lithography involved using available materials and a clever application of simple chemistry. Lithography involves five steps. (1) the image is laid down on a print block (originally limestone) with an oil-based medium. (2) The block is etched with an acid and gum arabic (a highly branched carbohydrate whose constitutive monomers are primarily D-galactose and salts of D-glucuronic acid). No etching occurs where the image is laid down. (3) The oilbased medium is washed off with turpentine. A raised salt matrix remains that is the ‘negative’ of the image. (4) An oil-based ink is rolled over the moist print block, filling the recessed areas where the image was laid down. (5) Paper is pressed on the block to produce multiple copies of the same image. The ink in step 4 can be any available color, and so this process allows production of multiple copies of multi-colored prints. In 1948 William Shockley, John Bardeen and Walter Brattain invented the solid state transistor; this led the way to integrated circuits that make up today’s microtechnology. A modern computer chip contains roughly 100,000 transistors, connected by “wires” that are roughly 0.5 to 1.0 μm in width. The process by which these transistors are connected uses photolithography. Fundamentally similar to lithography, this process allows micrometer-level precision when positioning transistors and their connectors. The components of a microchip can be readily discerned using an optical microscope—this really is microtechnology. Another early invention of microtechnology (1939) was the electron microscope. Optical microscopes are limited because of the wave nature of light—visible light has a wavelength on the order of half a micron. To see objects much smaller than that, one needs to “look” at them with something having a much shorter wavelength. It turns out that electrons have a wavelength on the order of one one-millionth of a micron. This permits objects of that size to become “visible”. The invention of the electron microscope, though not realized at the time, was as important in its own way as the invention of the optical microscope. Whereas the optical microscope increased resolution over the naked eye by a factor of 1000 or so, the electron microscope increased resolution over the optical microscope by an additional factor of 100,000. Thus a new distance scale is in order: a nanometer, abbreviated nm, is 10-9 m in size. The prefix “nano” denotes objects on this scale, and nano-objects are a hundred million times smaller than what can be seen with the naked eye. The electron microscope allows one to look at nanoscale objects—nanotechnology becomes possible. It’s taking a long time. As with the development of microtechnology, actually doing nanotechnology requires the ability to manipulate nanoscale objects. Development of these skills is very much in the forefront of current research and engineering, and constitutes the

The Nanoscale World

3

major topic of this book. However, it is now necessary to introduce another strand of the tale, specifically, chemistry.

1.1.1. Chemistry and Nanotechnology Parallel to the history just described, similar progress was being made in chemistry. The reader is probably familiar with Dalton’s Laws, formulated in 1808, that postulated the existence of “atoms”, and stated the empirically demonstrated laws of definite and multiple proportions. In 1828, Friedrich Wöhler published a paper in which he reported the synthesis of urea from inorganic molecules. He reported the stoichiometry of that substance as follows: Nitrogen Carbon Hydrogen Oxygen

46.78 20.19 6.59 26.24 99.80

4 atoms 2 atoms 8 atoms 2 atoms

Thus even in Wöhler’s time the term “atom” was widely used. So how big is an atom? Chemists usually think in terms of Ångstroms, where an Ångstrom is 1 × 10-10 m = 0.1 nm (nanometers). Thus it seems that chemists have been doing better than nanotechnology for nearly two centuries now—at least since 1808! They’ve been doing Ångstrom-technology, and furthermore, they are very good at it. So what’s the big deal? Why do we treat nanotechnology as if it is something new and important? There are two answers to this question. The first, and less significant one, is that the meaning of the word “atom” has changed. For Dalton and Wöhler, “atom” simply meant “basic chemical unit”, rather like we use “aliquot” or “portion” today. They did not necessarily believe in the actual physical existence of particles called “atoms”, and they certainly had no conception of atomic structure as we understand it today. For Dalton, an “atom” was simply what permitted the laws of definite and multiple proportions. This is also the way Wöhler thought of it; indeed, he even mentions the water atom. The very existence of atoms was controversial even into more modern times. Ludwig Boltzmann, who near the end of the 19th century developed statistical mechanics on the assumption that atoms really existed, committed suicide in 1906 in despair that atoms didn’t really exist and his life’s work was for naught. Only a few years later (1909), Jean Perrin was the first to measure Avogadro’s number and thus decisively prove the existence of atoms. The moral is this: it is hard to give early chemists credit for Ångstrom-technology if they didn’t believe in the existence of atoms. The second reason nanotechnology is really important is that nanotechnology is not chemistry, at least not as understood by Dalton or Wöhler. Nanotechnology derives not from chemistry, but rather from the spirit of the clock makers of yore. Imagine, for example, if Christiaan Huygens had attempted to synthesize a clock using the tools of chemistry: Mix 3 moles of screws, 2 moles of hinges and 0.5 moles of springs in a container with suitable solvent. Heat to 350°C stirring continuously for 48 hours. Extract the product. Air dry, and add an excess of clock-faces… This procedure results in a 5% yield of clocks, albeit with two enantiomers—half of them run counter-clockwise.

4

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Of course not! Clocks are manufactured one at a time with nearly 100% yield. Even if a chemical synthesis could be devised, it would be so inefficient as to be not worth doing. Put another way, the entropy cost of manufacturing clocks occurs in the human brain and on the factory floor, not in the chemical reaction chamber. And nanotechnologists live in the spirit of clockmakers and microchip makers—they want nothing less than to produce their molecules one at a time, with 100% yield, and with whatever custom-made appurtenances the customer may desire. This is as different from traditional chemistry as it possibly could be, and this is why nanotechnology is such a big deal. Nanotechnology really is technology, and is in some ways more akin to mechanical engineering than traditional chemistry. Indeed, the goal of nanotechnology is to make oldfashioned, synthetic chemistry obsolete. Big words, that. Unfortunately it hasn’t come about—yet. In the current state of the art, nanotechnology still has to use a lot of chemistry, but it isn’t by choice. An example illustrates: In the following pages you will read much about carbon nanotubes, strips of graphite rolled up into a tube. These tubes have many characteristics, the details of which you will learn, but for the moment we simply mention that important variables include tube length, radius, chirality (whether it has a right- or left-hand twist) and purity. Figure 1-1 shows a big jumble of nanotubes that were synthesized using chemistry. Recently a research group from Rice University took such a jumble of nanotubes and separated them from each other by sonication (shaking up with sound waves). A detergent was added to the tube suspension so that a single nanotube was in each bubble. The scientists were then able to do spectroscopy on individual nanotubes by looking at individual soap bubbles. This is nanotechnology, because instead of looking at a mole of particles, or even particles in parts per million concentration, they examined and were able to characterize individual nanotubes. Additional results of manipulation of individual nanoparticles are shown in figures 1-2 and 13. Figure 1-4 is not really at the nanometer scale, but illustrates the desired precision in fabrication.

Figure 1-1. A Scanning Electron Microscope photo of nanotubes, from www.iljinnanotech.co.kr/en/material/r-4-1.htm.

The Nanoscale World

5

Figure 1-2. A Scanning Electron Microscope photo of iron atoms on a copper surface. These are the Japanese (Kanji) letters spelling “atom.” The colors are created by the artist, but the resolution and placement of the individual atoms is real. (From http://www.almaden.ibm.com/vis/stm/atomo.html).

Figure 1-3. (From http://www.ipt.arc.nasa.gov/nanoflag_lowres.html) Caption included in figure.

6

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Figure 1-4. A nanotube farm, though note the large scale.

To be considered nanotechnology, the substance must be produced with 100% yield and there must be an absolute control of the structure. Nanostructures are built one atom at a time, as described in Feynman’s speech (Exercise 1.1). The current state of the art is not very far developed, and the actual implementation of the technology is still mostly in the future. Work is proceeding on two fronts. In the first case, we are trying to improve our ability to do the nanoengineering, and in the second case we are looking for possible applications of nanoparticles. A very clear definition comes from M. Meyyappan of Ames Labs, “Nanotechnology is the creation of USEFUL/FUNCTIONAL materials, devices and systems through control of matter on the nanometer length scale and exploitation of novel phenomena and properties (physical, chemical, biological) at that length scale,” (Meyyappan, http://ipt.arc.nasa.gov/nanotechnology.html). Having provided a sense of what nanotechnology is, it is appropriate to introduce you to some specific nanoparticles. Nanoparticles can be roughly divided into carbon-based, and not carbon-based. The following sections will introduce you to these materials, a brief history of their development, and some applications—anticipated and realized.

1.2. CARBON-BASED NANOPARTICLES To understand some of the features of carbon-based nanoparticles, it is useful to review some features of the different forms, or allotropes, of pure carbon. The most stable allotrope of carbon is graphite, shown in figure 1-5a. Graphite, such as found in pencil lead, has a large number of such sheets stacked on top of each other. It is, however, fairly straightforward to isolate a single sheet of graphite, known as graphene. In graphite, the carbon atoms have trigonal planar geometry. You may recall from your general chemistry course that this means the orbitals are sp2 hybridized to form the planar structure. Another well-known allotrope of carbon is diamond, shown in figure 1-5b. In diamond, each atom has four nearest neighbors located at tetrahedral positions; the orbitals are sp3 hybridized and the resulting structure is not planar. Although difficult to see, the carbon atoms of diamond exhibit a regular repeating pattern in three dimensions. Both graphite and diamond are pure carbon, yet they have very

The Nanoscale World

7

different properties and appearance. Carbon-based nanoparticles can be considered to be additional allotropes of carbon; not surprisingly, they have unique and sometimes amazing physical and chemical properties.

Figure 1-5. Two allotropes of carbon. Each dot is a carbon atom. (a) graphite; the dotted lines identify a single unit cell (see Chapter 2). This sketch shows portions of three parallel graphene sheets. (b) diamond: one unit cell is shown. From www.chem.wisc.edu/~newtrad/CurrRef/BDGTopic/BDGtext/BDGGraph.html

1.2.1. Bucky Ball Bucky ball, short for buckminsterfullerene, which in turn is the name for the C60 molecule shown in figure 1-6, plays an important historical role in the development of nanotechnology. C60, as the chemical formula implies, contains 60 carbon atoms arranged in the form of a soccer ball. All atoms are chemically identical, i.e., they will show up in the same place on an NMR (nuclear magnetic resonance) spectrum. C60 is historically important for a couple of reasons: arguably, it was the first molecule to be explicitly denoted as “nano”. Also, the discovery of C60 led within a couple of years to the discovery of nanotubes, which are now an integral part of nanotechnology. At the time of its discovery, C60 was described as another allotrope of carbon, distinct from graphite or diamond. Today this picture is muddier—there is a whole class of molecules known as fullerenes, and yet another class of molecules known as carbon nanotubes or onions. The story of buckminsterfullerene began when a British astrochemist, Harry Kroto, wanted to test the hypothesis that small carbon particles existed in interstellar dust clouds. Our data from such dust clouds is necessarily spectroscopic, and the species are identified by comparing such spectra with those generated by molecules on earth. Kroto learned that a group at Rice University in Houston headed by Richard Smalley was able to experimentally reproduce the environment of an interstellar dust cloud. Smalley’s group had designed their experiment for other purposes, but eventually Kroto was able to prevail on them to attempt to generate carbon clusters.

8

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Figure 1-6. A schematic diagram of buckminsterfullerene. Each vertex on the surface of the ball is the location of a carbon atom. The lines represent bonds between carbon atoms.

The result was published in a now famous paper in 1985 (Kroto et. al., 1985). Figures 1-7 and 1-8 are reproduced from that paper. As pointed out in the caption to figure 1-8, the C60 structure appears to be relatively stable. This means that as carbon clusters are placed in an environment where they are more able to exchange energy with their surroundings and to collide with other carbon clusters, then C60 will be the predominant product. On the other hand, at low helium pressures as shown in figure 1-8c, where there is less interaction with the surroundings, a variety of carbon fragments are formed.

Figure 1-7. Schematic diagram of the pulsed supersonic nozzle used to generate carbon cluster beams. The integrating cup can be removed at the indicated line. The vaporization laser beam (30-40 mJ at 532 nm in a 5 ns pulse) is focused through the nozzle striking a graphite disk which is rotated slowly to produce a smooth vaporization surface. The pulsed nozzle passes high-density helium over this vaporization zone. This helium carrier gas provides the thermalizing collisions necessary to cool, react and cluster the species in the vaporized graphite plasma, and the wind necessary to carry the cluster products through the remainder of the nozzle. Free expansion of this cluster-laden gas at the end of the nozzle forms a supersonic beam which is probed 1.3 m downstream with a time-of-flight mass spectrometer. Figure and caption from Kroto et. al, (1985).

The Nanoscale World

9

Figure 1-8. Time-of-flight mass spectra of carbon clusters prepared by laser vaporization of graphite and cooled in a supersonic beam. Ionization was effected by direct one-photon excitation with an Ar-F excimer laser (6.4 eV, 1 mJ cm-2). The three spectra shown differ in the extent of helium collisions occurring in the supersonic nozzle. In c, the effective helium density over the graphite target was less than 10 torr - the observed cluster distribution here is believed to be due simply to pieces of the graphite sheet ejected in the vaporization process. The spectrum in b was obtained when roughly 760 torr helium was present over the graphite target at the time of laser vaporization. The enhancement of C60 and C70 is believed to be due to gasphase reactions at these higher clustering conditions. The spectrum in a was obtained by maximizing these cluster thermalization and cluster - cluster reactions in the “integration cup” shown in figure 2 (figure 1-7). The concentration of cluster species in the especially C60 form is the prime experimental observation of this study. Figure and caption from Kroto, et. al, (1985).

The other noticeable fact about figure 1-8 is that carbon clusters with an even number of atoms are found. This was the crucial structural clue that led to the hypothesis that the clusters formed closed polyhedral cages. It is easy to show that such cages must have an even number

10

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

of atoms, and further, there is no reason why graphite or diamond fragments should necessarily have an even number of atoms. So Kroto et al’s hypothesis, subsequently verified, was that these small carbon clusters formed polyhedral structures with five- and six-membered rings. For carbon, the sixmembered, hexagonal ring is most stable, whereas the five-membered, pentagonal ring is less stable. It turns out that 60 atoms is the smallest such polyhedron that can be made where pentagonal rings are not adjacent to each other. This, the authors suggested, is what leads to the unique stability of C60. Further, for species larger than 60 atoms, the next smallest structure with no adjacent pentagonal rings is C70, which shows up as the second largest peak in all spectra in figure 1-8. These hypotheses imply that a large class of fullerenes will exist, with stable ones having even numbers of carbon atoms with 60, 70 or more atoms. A fullerene is any closed polyhedron made up of carbon atoms. This has turned out to be true, and fullerenes with 84, 92 and 96 atoms are now known to exist. For several years, the only evidence for fullerenes was the squiggly lines generated by a mass spectroscope, as shown in figure 1-8. Needless to say, the very existence of C60 was controversial. This changed in 1990 when a group in Switzerland synthesized C60 in bulk quantities (Krätschmer et. al., 1990). They did this in a remarkably simple way: they took graphite and vaporized it in an inert atmosphere. C60 was deposited on the walls of the container and could be isolated by dissolving it in toluene. The result was a flowering of research on fullerenes and fullerene-like molecules. Science magazine chose C60 as the “Molecule of the Year” in 1991. When they were first discovered, much importance was ascribed to fullerenes. They were hailed as 3-dimensional molecules that potentially opened up a brand new functional group for organic chemistry. Much of this hype has proved disappointing—there are to date few practical applications for C60. However, the discovery of C60 has directly led to the development of nanotechnology, and so the historical importance of the molecule is immense.

1.2.2. Nanotubes Shortly after the synthesis of fullerenes, Japanese scientists used a scanning tunneling electron microscope and found nanotubes (Iijima, 1991). To envision a nanotube, take a bucky ball and cut it in half. Now roll up a graphene sheet and insert it between the two halves; the result is a nanotube. The actual geometry of nanotubes is considerably more complicated and will be considered in detail in Chapter 4. Some pictures are shown in figure 1-9, and here it is seen that one can put nanotubes inside of other nanotubes—sort of like stacks of graphite rolled up. Thus one can distinguish between multiwalled nanotubes (MWNT) and singlewalled nanotubes (SWNT).

The Nanoscale World

11

Figure 1-9. Electron micrographs of microtubules of graphitic carbon. Parallel dark lines correspond to the (002) lattice images of graphite. A cross-section of each tubule is illustrated. a. Tube consisting of five graphitic sheets, diameter 6.7 nm. b. Two-sheet tube, diameter 5.5 nm. c. Seven-sheet tube, diameter 6.5 nm, which has the smallest hollow diameter (2.2 nm). Figure and caption from Iijima (1991).

Depending on the specific geometry, carbon nanotubes can conduct electricity, in which case they are known as metallic nanotubes, or they can be semiconducting. A quantity known as chiral angle, discussed in Chapter 4, which measures the twisting of a nanotube, partially determines the conductivity of carbon nanotubes. “Chirality” refers to the fact that the structure can occur with a right or left hand twist. Semiconducting nanotubes are potentially important for electronic applications. Unfortunately, as of this writing, it is only possible to make chiral mixtures of nanotubes, though semiconducting and metallic tubes are fairly easily separated. Most nanotubes are MWNTs—these tend to be rigid. They can easily be grown with reproducible length, diameter and location, though the metallic or semiconducting properties are not yet controllable. Figure 1-10 shows the state of the art as of 1998—it is more advanced now. SWNTs are harder to manufacture. Because they are not as rigid, they tend to stick to surfaces and form spaghetti. Nevertheless, it is now possible to control more carefully the structure of SWNTs. It has recently been reported that an SWNT with a 1 nm diameter has been grown to a length of 1.5 cm!

12

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Figure 1-10. Images of MWNT’s as produced at a laboratory at Boston College. Since these photos were taken, the laboratory has been able to make more uniform samples with greater control. From Ren et al (1998).

Carbon nanotubes have already found some important applications. They serve as efficient catalysts in Ni/Cd batteries, extending the duty cycle by several multiples. They play a similar role in the venerable lead/acid battery used in your car—new car batteries should last longer. They form the tips in AFM instruments. They have the greatest tensile strength of any substance known. Unlike C60, there are already commercial applications for nanotubes, and private companies are manufacturing them to user-defined specifications. Although the ability to control the diameter, length and number of walls in nanotubes exists, we are not yet able to control the chirality. As alluded to earlier, a chiral molecule is one which has a distinct mirror image, in the same way that your left and right hands are mirror images of each other. Conversely, a ball is not chiral because its mirror image is identical to the original ball, i.e., it is not distinct. A molecule that has a distinct mirror image is said to be chiral, or to have the property of chirality. Manufacturing nanotubes in precisely reproducible ways is the hallmark of nanotechnology. Ultimately, we want to be able to mass-produce nanotubes just like we mass-produce clocks or cars—with near 100% yield and to precise specification. This is the world of nanoengineering.

1.3. NON-CARBON BASED NANOPARTICLES Inorganic (non-carbon) substances have also been used to make nanotubes. In particular WS2 (that’s right—tungsten disulfide) has been formed into nanotube structures. In principle, any substance that forms a two-dimensional lattice is a candidate to make a nanotube. Most

The Nanoscale World

13

non-carbon based nanoparticles are not nanotubes, but rather nanosize particles of metals or semiconductors. Like buckyballs, these non-carbon based nanoparticles were synthesized and studied by chemists before there was a real awareness of their potential applications. In 1856, Michael Faraday at the Royal Institution of Great Britain prepared gold “fluids” that were ruby, blue or purple in color by the reduction of gold salts with phosphorous. Faraday found that these “fluids” were like solutions in that they did not settle readily over time, but unlike solutions, the “fluids” dispersed light as suspensions as shown in figure 1-11. Faraday is credited as the discoverer of metallic colloids, particles from 0.5-500 nm that do not readily settle out from solution. Unfortunately, the microscopic tools available to Faraday did not allow him to “see” the gold particles and consequently he could not answer questions about their behavior. Nevertheless, Faraday described a method for preparing metallic nanoparticles and suggested that changes in particle size or shape may influence the properties of particles. These methods (described in Chapter 3) allow preparation of nanoparticles that are within rigid size parameters.

Figure 1-11. A beam of light passes through a solution dispersed and a colloidal mixture. Photo taken by Ryan D. Tweney, Bowling Green State University. Permission to take this image was kindly given by the Royal Institution of Great Britain.

In addition to the nanoparticles of pure metals, nanocrystals of semiconducting materials such as CdS and GaAs have been prepared with sizes ranging from 1000 nm to 10 nm. These nanosize fragments of semiconductors are called quantum dots. Conductivity and color all

14

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

vary as the size of these particles is reduced from the macroscopic crystal to the molecular regime. The electrical and optical properties of quantum dots depend strongly on the size of the fragments because of a phenomenon known as quantum confinement, which will be discussed in Chapter 2. The special topic of the optical properties of nanoparticles is discussed in Chapter 6. For now, it is sufficient to mention that the electrical properties of quantum dots allows the development of tunable LEDs (light emitting diodes), optical switches, and maybe even quantum lasers (Alivisatos, 1996). Because nanoparticles have a very low heat capacity, they are being studied for a possible role in cancer treatments. Gold nanoparticles are bound to antibodies that are selective for proteins found only on the membranes of cancerous cells. The energy of a brief laser pulse, tuned to the gold nanoparticles, is absorbed. The low heat capacity results in very localized extreme heating that in turn kills cells with gold nanoparticles attached to their membrane. Many physical properties are dramatically altered when inorganic materials are clustered in nanosize particles. Figure 1-12 presents results of how the melting temperature (and the heat capacity) of gold varies with particle size. There is clearly a dramatic change as one enters the nanoscale regime. In addition, the mechanical properties of nanoscale materials are modified. Tungsten carbide, tantalum carbide, and titanium carbide are much harder, more wear-resistant, and more erosion resistant than their large-grain counterparts. It is not surprising, then, that they are currently used as cutting tools and may become part of new spark plug designs for automobiles. The special mechanical and magnetic properties of nanomaterials are discussed in Chapter 5.

Figure 1-12. The melting point of gold depends on the radius of gold particle size most noticeably when articles have a radius smaller than 4 nm. From Koper and Winecki, (2001).

The Nanoscale World

15

1.4. SPECIAL FEATURES OF NANOPARTICLES 1.4.1. Dimensionality In our earlier discussion of the allotropes of carbon, we mentioned that diamond has a three-dimensional regular repeating pattern, but graphene—a single sheet of graphite—is flat. Being the thickness of only one atom of carbon, a sheet of graphene can be said to be nanoscale in one dimension. Alternatively, we can say that in the macroscopic world, graphene is a two-dimensional structure, i.e., its dimensionality is 2. In a similar fashion, nanotubes—whether they are made of carbon or any other material—are nanoscale in two dimensions. They are one-dimensional in the macroscopic world. It is not a large stretch to imagine a material that is nanoscale in three dimensions, e.g., a bucky ball. Particles that are nanoscale in three dimension are said to be 0-D structures in the macroscopic sense. This does not mean the particles occupy no volume, but rather that they are very, very small scale structures. The macroscopic properties of materials depend on their nanoscale structure, and dimensionality is one way of describing that structure. Diamond has a dimensionality of three, graphite has a dimensionality of two. Diamond is hard; graphite is soft. Diamond is an electrical insulator, graphite is a conductor. While defining dimensionality is not equivalent to defining the macroscopic properties, defining the dimensionality does introduce possible sets of macroscopic behavior to study.

1.4.2. Surface Effects Surfaces are unstable. This is because matter is almost always attractive, and hence atoms and molecules are lower in energy when surrounded by neighbors. You may recall the old chemistry expression “like likes like.” This means that similar substances will dissolve in one another, whereas dissimilar substances will segregate. The ultimate separation is a welldefined surface, which further suggests that surfaces are unstable. Nanosize particles have an inordinately large surface area to volume ratio, so surface effects are very important for nanoscale materials. We now spend some time discussing their properties. Surfaces are usually stabilized in some way. To illustrate, consider a diamond crystal, shown in figure 1-5b. Each atom has four nearest neighbors (we say it has a coordination number of 4), located at tetrahedral positions. For a diamond crystal of the size you might put on your ring, the typical atom is very far away from the surface. However, at the surface, the atoms no longer have nearest neighbors and are therefore unstable. There are two ways to stabilize them: passivation—that is, to provide material for surface atoms to bond to, and surface rearrangement (also known as surface reconstruction). Saturation is a term taken from organic chemistry, and generally means filling all dangling bonds with hydrogens. An example of saturation is given by figure 1-13a, and shows that saturated surfaces tend to be more stable. An extreme example of saturation is Teflon. That surface is so stable that almost nothing sticks to it at all. Typically, the surfaces on a diamond ring are saturated, though perhaps not only with hydrogen atoms. A surface may be passified by organic compounds, hydroxyl or amino groups, or whatever else is scavenged from the environment.

16

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

A clean surface refers to one where the saturation molecules are removed. A clean surface can usually only exist in a vacuum or in an inert atmosphere. Since a clean surface cannot be stabilized via saturation, then surface rearrangement occurs. In this case, surface atoms will bond with each other to tie up the dangling bonds, or in the case of diamond, they may form multiply-bonded structures, i.e., double and triple bonds. Thus, the geometry around each atom at the surface will no longer be tetrahedral or sp3. Some elements, such as clean silicon, form consistent and well-characterized rearrangements. A rearranged surface is usually unstable when exposed to the atmosphere—bonds will tend to saturate, reducing the strain energy implicit in rearrangement. An example of a possible rearrangement of a diamond surface is shown in figure 1-13b, which shows that the surface assumes a graphitelike structure. Surfaces are very important and will affect optical and chemical properties. Nanostructures have a large surface area per unit mass, and thus for nanoparticles the properties of surfaces is crucial. It doesn’t matter if it is a diamond-air surface or a diamondaluminum surface—there still will be surface effects.

Figure 1-13a. The effect of surface saturation. The purpose of this study was to determine the effect of surface saturation on a diamond-aluminum interface. The gray balls represent diamond, while the darker ones are aluminum. For an unsaturated surface the circumstance is as depicted in (b). When the diamond surface is saturated with (smaller) hydrogens, the situation is as depicted in (c). Note that saturation stabilizes the diamond surface, which destabilizes diamond-aluminum interactions, forcing the aluminum further away. From Qi and Hector (2003).

The Nanoscale World

17

Figure 1-13b. This shows the reconstruction that takes place at a diamond surface. The dark balls represent surface atoms that are four-fold coordinated, as is bulk diamond. The lighter balls on the surface are only three-fold coordinated, and look more like graphite. The graphitic-like structure stabilizes the surface. From Petukhov et. al., (2000).

A graphite sheet has two kinds of surfaces. The surface of the sheet is relatively stable— there are no dangling bonds. Graphite, however, is reactive in the sense that free radical species can relatively easily attack the aromatic rings that make up the sheet. For this reason, graphite readily burns at elevated temperatures; it is a primary constituent of charcoal. Nevertheless, the planar surface of graphite may be considered stable under atmospheric conditions at room temperature. The other kind of surface is the edge of the sheet. Here there are dangling bonds, and these must be saturated or rearranged as described with diamond. Nevertheless, clean graphite is less reactive than clean diamond, simply because only the edge atoms require saturation or rearrangement.

1.4.3. Optical and Electrical Properties A complete understanding of the effect of existence in the nanoscale regime requires a review of quantum mechanics. You may recall from General Chemistry that quantum mechanics only becomes important when one is dealing with material within the size/mass regime of electrons. This is the nanoscale regime. The purpose of this section is to introduce you to what the special properties are without providing a justification. First, nanoparticles have unique electrical conductivity properties. This is particularly noticeable for nanocrystals of semi-conductors. As will be demonstrated in Chapter 2, the

18

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

ability of a semiconductor to elevate electrons to higher energy levels decreases as the particles get smaller. This means that regulating particle size may result in very small tunable laser systems. On another front—and as has already been mentioned—the conductivity of carbon nanotubes depends on their chirality. What this means, and the nature of this relationship, will be spelled out in Chapter 4. Secondly, the reflectivity and/or color of particles depends on their size in the nanoscale regime. This is apparent from the changing color of gold colloids already mentioned. Although metals tend to be good conductors all the way down to the nanoscale, the confinement of the electrons to the nanoparticle results in a lower heat capacity than the metal would have in the bulk. Any energy put into the nanoparticles can not be readily dissipated and so the temperature of the particles rises dramatically with a relatively small influx of energy. This is the source of the utility of gold nanoparticles for cancer cell death described earlier in this chapter. The purpose of this chapter has been to introduce you to the possibilities of a world filled with nanotechnology. To really understand how and why nanoparticles have the properties they have, you’ll need to dig into the following chapters. Some of it will seem straightforward and some of it will be confusing. It’s going to take some time, but considering the developments we’ve experienced in going from millitechnology to microtechnology and the beginning of great benefits we see from nanotechnology, it’ll be worth the effort.

EXERCISES Exercise 1-1: After you have read the transcript of the speech printed at http://www.zyvex.com/nanotech/feynman.html, choose one of Feynman’s examples of nanotechnology and write a paragraph that elaborates on a possible application. Exercise 1-2: Suppose atoms are spherical with a radius of 1 Å. Suppose a collection of such atoms is arranged in a spherical ball that has a radius of 1 μm. Estimate the fraction of atoms located at the surface of the ball. Repeat this calculation for a ball with a 1 nm radius.

Chapter 2

INVESTIGATING THE NANOSCALE 2.1. INTRODUCTION One take-home message from the first chapter is that nanoparticles are really small. So, how can scientists ‘see’ what they’ve made? The electron microscope mentioned in Chapter 1 allows scientists to observe materials on the nanometer scale, however there are several additional methods that have been developed to look at individual molecules. These techniques are described in this chapter with enough detail that, when you read about results, you will both understand them and accept their validity. Most of the principles described in this chapter require at least a basic understanding of quantum mechanics. Quantum theory is also necessary for understanding the properties of nanoparticles; and so quantum mechanics is also introduced in this chapter. The purpose of this chapter is to describe methods that can be used to investigate nanosize materials. All of the methods involve, in some way, the interaction of energy and matter. It is useful, then, to first present some fundamental ideas associated with the nature of energy and of matter. Let us first look at energy.

2.1.1. The Nature of Energy Energy is transmitted as a wave. Consider the nature of light and the extended electromagnetic spectrum shown in figure 2-1. The term ‘electromagnetic’ arises because the waves described in figure 2-1 are comprised of an electric field component and a magnetic field component. Electromagnetic radiation is the propagation of energy in the form of electromagnetic waves. Included in the types of electromagnetic radiation are x-rays, ultraviolet rays, visible light, infrared light, microwaves and radio waves. As can be seen by looking at figure 2-1, the wavelength of these various types of electromagnetic radiation is inversely related to their frequency. Expressing wavelength in meters (m) and frequency in Hz (s-1), the speed of light (3 x 108 m/s) is the proportionality constant ν = c/λ

(2.1)

20

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Figure 2-1. The electromagnetic spectrum. From Rubinson and Rubinson, (1998), page 317.

If all the waves emanating from a source hit their peak at the same point in time, then the waves are said to be ‘in phase’. In other words, the waves are coherent. When waves are not coherent, then it is possible for destructive interference to occur. Two waves that are perfectly out of phase, that is one reaches its maximum while the other reaches its minimum, will interfere with each other, resulting in a ‘wave’ with zero magnitude at all points in space. If two waves are coherent, their constructive interference will yield a wave with maxima that has a magnitude equal to the sum of the two original waves. Understanding of the nature of electromagnetic energy is made more complicated with the realization, first proposed by the German physicist Max Planck, that light has a particulate nature. In 1900, Planck proposed that energy must be quantized; the small, discrete units of energy were called quanta (singular: quantum). Later, Albert Einstein proposed the name photon to describe a quantum of light. The energy of photons is proportional to their frequency E = hν = hc/λ.

(2.2)

Investigating the Nanoscale

21

where h, known as Planck’s constant, has the value 6.626 x 10-34 Joules. It is instructive to look at the electromagnetic spectrum from the point of view of the magnitude of energy. From equation 2.2, it is apparent that small wavelength is associated with large energy. In units of J per event (J), chemical reactions produce energy on the order of 10-19 J (which translates to about 10,000 J/mole of substance reacted), whereas nuclear reactions are roughly 10-12 J or higher in energy. The energy of X-rays is in between these two values: 1-100 x 10-16 J. Electromagnetic energy isn’t the only kind of energy that has both wave and particulate nature. The thermal energy of an atom in a solid is also quantized. Manifested as elastic vibrations of a particle around its equilibrium position in a solid, these vibrational waves are excited spontaneously in a crystal by the thermal agitation of atoms. Their amplitude increases as temperature rises. Phonon is the name given to a quantum of vibrational energy hν for a frequency ν. Another commonly used notation for this quantum of energy is ħω for a frequency ω; where ħ = h/2π and ω = 2πν. The maximum frequency of a phonon is around 1012 Hz. While this corresponds to the frequency of infrared light, the velocity of phonons is considerably less than the velocity of light. The velocity of propagation for these elastic waves is of the order of 5000 m/s.

2.1.2. The Nature of Matter Most of our understanding of matter comes from our own observations: matter has mass, occupies space, when in motion it tends to stay in motion. These observations have been clearly defined in the laws of Classical Mechanics. There’s just one hitch: as the mass of particles gets smaller and smaller, classical mechanics begins to fail. ‘Particles’ begin to behave like waves. In 1924, the French physicist Louis de Broglie suggested that the wavelength of a particle, λ, must be inversely related to its mass, m, and velocity, v. λ = h/(mv)

(2.3)

This equation suggests that for massive particles like baseballs, the wavelength is very small and the wave nature is difficult to discern. However, for particles with very small mass (e.g., electrons), the wave nature is a significant part of their behavior. The behavior of a photon of green light is well described by equations of wave mechanics. Until 1900, energy and light were thought to be entirely wave-like in nature. Indeed, their behavior is mostly wave-like; the theories and equations of wave mechanics are sufficient to describe and explain most observations about light. Electrons, having a mass of 9.1 x 10-31 kg (rest mass), exhibit both wave and particle nature. The only way to explain the behavior and properties of electrons is to use a theory that includes both particle and wave properties. The theories and equations of quantum mechanics must be used to describe and explain observations about electrons.

2.1.2.1. Quantum Mechanics During the 1920s, Erwin Schrödinger found a way to combine the two fields of classical mechanics and wave mechanics in an equation named after him.

22

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al. ĤΨ = EΨ

(2.4)

In this equation, Ĥ is called the Hamiltonian operator and E is the energy of the wavefunction Ψ. The solutions to the Schrödinger equation are the wave functions Ψ and the corresponding energies E that describe the state of a particle. From these solutions came an understanding of how electrons exist around the nuclei of atoms. As you may recall from your General Chemistry course, orbitals (s, p, d, f, g, etc.) describe the three-dimensional regions of space around the nucleus where it is most likely that the electrons exist. Each orbital is correlated with one solution to equation 2.4. Orbitals have a shape defined by squaring the wave function Ψ and the energy of an orbital is the energy E associated with the wave function Ψ of that orbital.

2.1.2.2. Molecular Orbital Theory and Band Theory Something interesting happens when two atoms, each with their own sets of orbitals, approach each other. As the wave functions overlap in space, they interfere with each other, resulting in the formation of new wave functions. These new wave functions have different shapes and different energies than the wave functions of the isolated atoms. They are called molecular orbitals (MO). According to Molecular Orbital Theory, when two atomic orbitals overlap, two molecular orbitals will form. One of the molecular orbitals will have lower energy than the atomic orbitals and the other will have higher energy. In the ground (lowest energy) state, electrons occupy the lowest energy orbitals possible. If occupied, the lower energy molecular orbital would lower the energy of a system relative to the energy of the isolated atoms; it is called a bonding MO. The higher energy molecular orbital is called an antibonding MO. These are represented in figure 2-2a for a diatomic of copper atoms. Because it is reasonable that only the orbitals that are furthest away from the nucleus will have significant overlap with orbitals from another atom, figure 2-2 presents only the valence atomic orbital of copper (4s) which—in the copper atom’s ground state—has one electron.

Figure 2-2. As copper atoms approach each other, their valence orbitals overlap, resulting in the formation of molecular orbitals. Bonding molecular orbitals are represented with a solid line, and antibonding molecular orbitals are represented with a broken line. (a) If two atoms have overlapping atomic orbitals, the resulting molecular orbitals each have a single defined energy. However, as more and more atoms approach each other (b) the molecular orbitals have a distribution of energies manifested as a band. The width of the band widens as the number of atoms increases.

Investigating the Nanoscale

23

Based on figure 2-2a, it is apparent that diatomic copper has lower energy than a copper atom, yet diatomic copper does not exist. We know, from our own experience, that copper exists as a metal, i.e., many copper atoms held together. In order to explain why, we continue our trip across figure 2-2 and introduce Band Theory. One way to understand why the MOs have different energies from the atomic orbitals is to recognize that the bonding MO feels the nuclear charge from both of the atoms and the antibonding MO feels less nuclear charge than the original atomic orbitals. (To convince yourself of this, check out the three-dimensional shape of MOs in a general chemistry textbook.) When many atoms are near each other, the MOs that form experience forces from many more sources than just two nuclei. The variation in the magnitude of these forces results in the formation of MOs with a distribution of energy levels rather than the two, defined energies that result when only two atoms bond. The distribution of energy levels results in what are known as ‘bands’ of energy. It turns out that the stronger the interactions between neighboring atoms, the wider the distribution. Additionally, the greater the number of nearest neighbors, the stronger their interactions will be. The combination of these effects results in what is shown in the rest of figure 2-2; namely, as the number of particles increases, the width of the bands increases. In the case of metals, the wider bands overlap. The result is one continuous band, the lower half of which is completely occupied in the ground state of the metal. The occupied level with the maximum energy is known as the Fermi level. For metals, the Fermi level is in the middle of a continuous state, even at relatively small cluster sizes (tens or hundreds of atoms) (Alivisatos, 1996). One practical consequence of this is that the electrons can readily move about. Because of the mobility of their electrons, metals are good conductors. The picture is somewhat altered for non-metals. Diamond crystals, for example, contain many carbon atoms, therefore the MOs must have a distribution of energies. But diamond does not conduct a current. This is because the bands do not overlap. The distribution of energy for the bonding MOs, also known as the valence band, is energetically separated from the distribution of energy for the antibonding MOs, also know as the conducting band. There is a forbidden region between the two bands. Electrons with energy between the valence band and conducting band cannot exist in diamond. The Fermi level is at the top of the valence band. The valence band is full, so electrons cannot readily move. Nor can they readily attain the energy of the conducting band. Diamond is an insulator. In the case of diamond, the energy difference between the two bands is large, > 1 eV. There also exist materials for which the energy difference between the two bands is not very big. These substances are called semiconductors. A semiconductor has an energy gap of less than 1 eV between the valence and conducting band. Adding energy to a semiconductor, whether by increasing the temperature or some other means, will promote an electron from the valence band to the conducting band. Electrons in the conducting band can move freely and so the material can conduct, albeit not as well as a metal; there are not as many mobile electrons in a semiconductor. Silicon and germanium are two examples of semiconductors.

24

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Figure 2-3. Comparison of the energy gaps between valence band and conduction band in a conductor, semiconductor, and insulator. In a conductor the energy gap is virtually non-existent, which means that when an electrical potential is applied across the material, electrons can move easily. In a semi-conductor the energy gap is small, so that some electrons may be promoted to the conduction band to become mobile upon application of an electrical potential. However, in an insulator the energy gap is very large; electrons are not promoted to the conduction band and remain immobile.

One method that has been used to enhance the conductivity of semiconductors is a process called doping. Consider the semiconductor silicon. Silicon forms crystals that are very similar to diamond. Each silicon atom is bonded to four other silicon atoms so that the four are at the corners of a tetrahedron. Although complex, this is a regular repeating pattern in the atomic-level structure of silicon. You may recall that the repeat pattern of a crystalline solid is called a unit cell. It turns out that if a few silicon atoms in this regular repeating structure are replaced with a similarly sized atom, such as arsenic, there is no effect on the structure. There is also no effect on the energies of the valence band or of the conducting band. However, arsenic atoms have one more valence electron than silicon does. Because the valence band is full, that electron must go into the conducting band (see figure 2-4). The few electrons in the conducting band (as many as arsenic atoms have been incorporated) are very mobile and so the conductivity is enhanced. Doping a semiconductor with atoms that have a greater number of valence electrons results in what is called an n-type semiconductor. Alternatively, silicon atoms can be replaced with an atom with one fewer electron such as aluminum. This results in an empty spot known as a ‘hole’ in the valence band. Figure 2-4 represents a system in which three of the silicon atoms have been replaced by atoms with three valence electrons. The presence of a hole (is that an oxymoron?) means the electrons in the valence band are more mobile. Doping a semiconductor with atoms that have fewer valence electrons results in what is called a p-type semiconductor. If too many silicon atoms are replaced by aluminum atoms, the result will be aluminum metal with a few silicon atoms in it. This material is not a semiconductor.

Investigating the Nanoscale

25

Figure 2-4. Semiconductors can be doped by replacing atoms in the matrix with other atoms that have more valence electrons (n-type) or fewer valence electrons (p-type). Because most of the atoms that form the matrix are identical, the energy of the bands is not affected when the semiconductor is doped. The additional electrons in the system for the n-type semiconductor must therefore occupy the conducting band. The p-type semiconductor is said to have ‘holes’ in the valence band.

2.1.2.3. Quantum Dots As the size of semiconductors is reduced, the band gap gets wider and wider (see figure 2-2). In semiconductors, the Fermi level lies between the valence band and the conducting band. Consequently, the electrical transport properties of semiconductor nanoparticles depends strongly on size as do their optical properties. Quantum dots are fragments of semiconductors consisting of hundreds to thousands of atoms having the same unit cell they have in macroscale materials. Among the interesting properties of quantum dots is the fact that their electrons are constrained to stay within the nanoparticles. If an electron in the valence band is somehow excited to the conducting band, it leaves an empty spot called a hole in the valence band. This excited electron-hole pair is called an exciton. When the electron returns to the valence band, it returns to the hole from which it came. This means that when there is a collection of nanoparticles of the same (or very similar) size, the energy emitted after an excitation event will be exactly (or very nearly so) the same for all of the particles. At the very worst the energy will have an extremely narrow distribution. This has marvelous applications for optics. 2.1.2.4. Types of Molecular Energy Within the bands discussed above, the energy levels of electrons are so closely spaced that they form a continuum. This does not remove the fact that the energy of electrons is quantized. The existence of a forbidden region between the valence band and conducting band in insulators and semiconductors attests to the requirement that electrons exist as wavefunctions with defined energies. When we talk about the energy of matter, it is convenient to define some specific types of energies in terms of the molecules. The discrete energy levels observed for electrons associated with particular atoms and molecules are presented schematically in figure 2-5. The bold lines represent electronic

26

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

energy levels; these energies are associated with the atomic and molecular orbitals reviewed in the previous section. The narrower lines above the bold lines represent vibrational energy levels.

Figure 2-5. Energy can only be absorbed by a substance if the energy matches the spacing between the energy levels of the particle. Once energy has been absorbed (a), the substance can return to its ground state through (b) collisional deactivation, (c) intersystem crossing, or (d) emission. Energy can be emitted as (d1) fluorescence or (d2) phosphorescence.

Molecular vibrations, i.e., fluctuations in interatomic distances between atoms of a molecule, occur at defined frequencies that are a function of bond length, bond strength, and atomic mass. Gas-phase atoms do not have vibrational energy; gas phase molecules, e.g., O2, do have vibrational energy. The energy difference between vibrational energy levels (ΔEvib) is proportional to the difference between the frequencies of the two vibrational modes (Δν = νvib1 - νvib0). The proportionality constant is Planck’s constant, h (see equation 2-2). One can also consider the rotational energy levels of particles if the particles are not spherically symmetrical, however analysis of rotational quantum states is beyond the scope (and need) of this book. The energy associated with the movement of a particle through space is called translational energy. For most particles in containers that are larger than nanoscale in at least one dimension, the translational energy levels are so close to each other, the apparent translational energy is a continuum. An electron confined to the space of an orbital has large spacing between energy levels because of the small volume of the orbital. The mathematical justification for this is presented in section 2.1.5.

2.1.3. When Matter and Energy Interact Any attempt to examine the nature of particles on the nanoscale requires a consideration of the nature of the interaction of energy and the particles. When energy impinges on a material it can be transmitted, reflected, absorbed, or diffracted (scattered). A stop light looks red through a car’s windshield because the glass transmits red light. Grass looks green

Investigating the Nanoscale

27

because chlorophyll pigments in the grass absorb many wavelengths, but reflect green light. Finally, the rainbow after a thunderstorm appears because the water droplets remaining in the sky scatter or diffract light of different wavelengths at different angles. Which of these phenomena occur, and to what degree, depends both on the wavelength of the incident energy and the nature of the material. Consequently, monitoring the effect of interactions of energy with particles reveals a great deal about the structure and/or appearance of the material.

2.1.3.1. Absorption When the electrons of a particle are all in the lowest possible energy levels, the particle is said to be in its ground state. When energy strikes the particle, it may interact in such a way as to increase the energy of the electrons. However, the only energy that will be absorbed by a substance (path a in figure 2-5) is energy that corresponds to the spacing between two energy levels. The absorption of a photon to increase the vibrational energy of atoms in a solid is an example of photon-phonon interaction; consistent with the frequency of phonons, the vibrational modes of solids are best determined using infrared light. Measuring the frequency of the energy absorbed reveals information about the spacing between the levels depicted in figure 2-5. Energy that is not absorbed is transmitted, scattered, or reflected. Electrons that are in an excited state may return to their ground state by collisional deactivation, emission of a photon, or intersystem crossing. A return to the ground state by way of collisions with other particles usually does not result in the emission of photons and so the only observable phenomenon is absorption of incident energy. If the photon is emitted from the lowest vibrational level of the excited electronic state, the phenomenon is called fluorescence (path d1). In this case, the energy of the emitted photon is somewhat less than the energy of excitation. Intersystem crossing (path c), which means there is a change in the spin state of the particle, is not—according to the ‘rules’ of quantum mechanics—allowed. However, in quantum mechanics, this really means it is improbable and/or it takes a long time to occur. If a photon is emitted after intersystem crossing occurs, the phenomenon is called phosphorescence (path d2). As with fluorescence, the energy of the emitted photon is less than the energy of excitation. But while fluorescence occurs within nanoseconds of the excitation event, phosphorescence may occur hours after excitation. Fluorescence and phosphorescence spectra reveal details about the vibrational energy levels in the ground and excited electronic states. 2.1.3.2. Scattering Frequently, the energy that impinges on a substance is not of a frequency that exactly matches any of the transitions described in the previous paragraph. Whether absorbed or not, energy interacts with matter. The electrons of atoms and molecules are held in the vicinity of their particle by the attractive forces of the nucleus (or nuclei). A molecule has a dipole moment if the electron density is larger at one end of the molecule than at the other. Electrons feel an effective nuclear charge. The smaller this is, the less rigidly the electrons are held in place; the particle is said to have a large polarizability. The electrons will respond to an external electric field. The oscillating electric field of electromagnetic radiation that impinges on a particle may interact with the electrons of a particle, resulting in an oscillating dipole moment. The magnitude of this oscillating dipole moment will be proportional to the polarizability of the particle and the strength of the oscillating field. One might say that the particle is in a virtual electronic state, where the energy of the particle with an oscillating

28

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

dipole does not correspond to any of the energy levels presented in figure 2-5. The oscillating dipole radiates light at all angles, a phenomenon called scattering. If the frequency of the scattered light is the same as the frequency of the incident light, the scattering is said to be elastic. The intensity of the scattered light is a quantifiable function of the angle of observation (relative to the incident light) and the mass of the particle.

2.1.3.3. Rayleigh Scattering When light is scattered by particles that are much smaller than the wavelength of the incident radiation, the phenomenon is called Rayleigh scattering. The intensity of scattered light increases as wavelength decreases, consequently short wavelength radiation is scattered more intensely than long wavelength radiation. The sky appears blue because particles in the atmosphere scatter the shorter wavelength blue light more intensely than the longer wavelength red light. Another property of Rayleigh scattering is that the intensity of the scattered light is proportional to the size of the particle. Additionally, intensity varies with the angle at which the scattered light is observed. When the size of the particle is comparable to the wavelength of the incident light, scattering may occur from different sites of the same molecule. Frequently, scattering measurements are done on solutions of particles, and the fact that not all solutions behave ideally results in a modification of the observed scattering. Nevertheless, light scattering experiments can reveal useful information about the size and shape of particles that are close to the size of the wavelength of the exciting light. The size of particles detected, therefore, depends on the excitation source wavelength. Clearly, if the scattering of x-rays is being monitored, information will be learned about particles that are considerably smaller than if visible light scattering is monitored. 2.1.3.4. Raman Scattering As molecules vibrate, their interatomic distances change. In some cases, this results in a change of the polarizability of the molecule. We have said that the oscillating electric field of incident radiation induces an oscillating dipole moment in the particle. The size of this depends on the polarizability of the particle. If the polarizability changes while the dipole is oscillating, then the magnitude of the oscillating dipole moment will also change. The light radiated by the oscillating dipole will be at a different frequency than the incident light. This phenomenon is called Raman scattering. Because the frequency of the radiated light is different that the frequency of the incident light, Raman scattering is said to be inelastic. Because these are rare incidents, Raman scattering typically has very low intensity; if there is any fluorescence, Raman scattering is masked. Raman scattering reveals details about the vibrational modes of particles. 2.1.3.5. X-Ray Diffraction When the wavelength of incident light is approximately the same as the spacing between atoms in a solid, the light reflected off the surface experiences constructive and destructive interference. X-ray diffraction takes advantage of the fact that the X-ray wavelength is approximately the same as the spacing between atoms in a solid. Thus, when X-rays are reflected off a solid, an array of dots is formed that is directly related to the orientation of the particles relative to each other. To illustrate the method, consider a two-dimensional system in which the particles (which are represented as dots) are arranged in a regular repeating

Investigating the Nanoscale

29

pattern. It is possible to envision sets of parallel lines to represent planes of particles; for the following discussion, consider the horizontal planes.

Figure 2-6. Diffraction of an x-ray by two parallel planes of particles. The text defines θ as the angle between the source and the horizontal plane. Note that, because DB is perpendicular to the planes defined by ‘a’ and ‘b’, both
Consider the x-ray that is diffracted or scattered by two parallel planes of particles. If the incident light is coherent and is from a point source that is very far away, then the angle between the source and the parallel planes (θ) will be the same for all the planes. The difference between the distance traveled by a light beam that is scattered from plane a and one that is scattered from plane b is AB + BC . When this distance is an integral product of the wavelength of the incident light, i.e.,

AB + BC = nλ

(2.5)

then there will be constructive interference; in other words a point of high intensity. It can be shown that the distance AB + BC is also related to the angle between the source and the detector (180 - 2θ) and the distance between adjacent planes. If we define d as the length of vector DB , then the relationship is

AB + BC = 2d sin θ

(2.6)

If equations 2.5 and 2.6 are combined, the result is called the Bragg equation. nλ = 2d sin θ

(2.7)

The final result is that the scattered light forms a pattern of dots that is related to both the relative orientation of the particles that scattered the light and the orientation of the detector relative to the source. Expanding this development to three dimensions, it is possible to map out the exact relative positions of atoms in three dimensions.

30

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

2.1.4. Sources of Energy 2.1.4.1. Photons Photons are frequently used to study small particles; there exist a variety of reliable sources. Some materials glow at defined frequencies when heated to high temperatures, others glow when a current is passed through them (tungsten, argon, deuterium). Lenses and/or monochromators are used to select the desired wavelength. Lasers produce light of defined wavelengths and significantly higher intensity than lamps. Lasers and lamps are practical solutions if the desired wavelength is in the visible or infrared region of the spectrum. However, if one wishes light of shorter wavelengths, x-ray tubes or synchrotron radiation are required. Special methods must be employed to generate higher energy photons in the x-ray region. In large atoms such as uranium, the 1s electrons are known as K-shell electrons. If somehow one of the K-shell electrons has gone missing (for example through the nuclear decay process known as electron capture), then one of the valence electrons will fall down into the K-shell, or 1s orbital. Using uranium as an example, the valence electrons are in a 5f orbital, so the transition, from 5f to 1s state is a large energy gap—this transition will produce an X-ray. Xrays are thus produced by bombarding metals with high-energy electrons, ionizing K-shell electrons, and then collecting and focusing the resulting X-ray beam. This can result in Xradiation of high intensity. However, the light is difficult to collect, partly because air absorbs light in this portion of the spectrum. A method is required that produces light of the desired wavelength in a vacuum to avoid absorption by air. Charged particles radiate when they are accelerated. Electrons constrained to move in a circle are accelerating inward. Using a synchronously increasing magnetic field, a synchrotron constrains electrons to move in a circular (not spiral) path at a velocity near the speed of light. The result is a stable and continuous source of light in the region 1014 to 1017 Hz, i.e., from ultraviolet to x-radiation. Accessing this source of light requires obtaining permission to use a station at one of the sites. There exist about 40 synchrotron radiation sources around the world. 2.1.4.2. Electrons At times, the best way to study the nanostructure of a substance is to examine the interaction of high energy matter with low energy matter. For example, when electrons— which have a significant wave nature, see equation 2-3—are accelerated to high velocity, they will have an energy that is directly proportional to that velocity squared. When certain materials, e.g., tungsten, are heated, they emit electrons. Those electrons can be accelerated through a potential gradient. The larger the gradient, the faster the electrons move. It is not unreasonable to expect an electron source to provide electrons with a frequency of 1.2 x 1020 Hz and wavelengths of 0.002 nm. It is worth noting that accelerated electrons can reveal a great deal of information about the material they interact with. Because they have wave properties, electrons can be diffracted, refracted, and reflected. These ‘optic’ effects become profoundly important to nanoparticles, because resolution is limited by the wavelength of illumination. It is possible to resolve two objects only if the wavelength of illumination is smaller than the distance between the objects. The wavelength of accelerated electrons is 250,000 times smaller than that of visible light and so the resolution of the

Investigating the Nanoscale

31

electron microscope is much finer than that of a light microscopy. In addition to their wavelength, electrons have charge; this also affects the way they interact with other matter.

2.1.5. Additional Concepts In order to understand some of the techniques used to analyze nanoparticles, it is necessary to have at least a rudimentary understanding of a few of the consequences of the dual nature of matter. The purpose of this section is to give the reader a minimal introduction to two of these concepts.

2.1.5.1. Particle-in-a-Box Electrons are quantum particles whose behavior is described by the Schrödinger equation (eq. 2.4, page 22). The allowed states and energies for an electron in a one-dimensional box for which the walls are infinitely high can be obtained by solving this equation. In reality the “boxes” are 3-dimensional, but nothing is lost if we only consider one dimension at a time. For one dimension the solution to equation 2.4 yields

En =

h 2n 2 8ma 2

(2.8)

where h is Planck’s constant, a is the size of the one-dimensional box, and m is the particle mass. The quantum number, n, is a positive integer for the state with energy En. If n = 0, then there is no particle in the box. Any real particle will have n = 1, 2, 3… Thus the energy levels increase stepwise, though the steps are not equally sized.

2.1.5.2. Tunneling According to classical mechanics, the location of a particle can be defined. According to quantum mechanics, there exists a function that describes the probability of the particle being in a particular region of space. If one restricts the particle with walls or boundaries of finite height, then—according to classical mechanics—the particle must be within the boundaries. According to quantum mechanics, however, there is a non-zero probability that the particle may exist outside those boundaries. Because electrons have a very low mass, they follow the laws of quantum mechanics. They are not truly restricted by the boundaries described by the box! Consider a valence electron on each of two particles. The region of space that is allowed for these electrons has boundaries that are defined by the attraction of each particle’s nuclei and the repulsion of other electrons. If the two particles are brought close to each other, it is possible for their valence electrons to exchange or interact from a greater distance than classical mechanics would predict because—according to quantum mechanics—the electrons have a non-zero probability of existing outside their defined boundaries. The phenomenon of electron interaction that results from electrons existing beyond their boundaries is called tunneling. Tunneling can only be observed and measured if the two materials are good conductors, and if the two materials are very close to each other. The magnitude of tunneling is very dependent on the distance between the sample and the probe.

32

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

2.2. MICROSCOPY Microscopy involves using a lens or a combination of lenses to form a virtual image of a small object so that a normal eye can focus sharply on that object. There are two approaches for creating the virtual image: bright field and dark field microscopy. In bright field microscopy, light is shone through the sample. Materials within the light path scatter or absorb light to a degree that depends on the optical properties of the material. The support medium for the sample tends to transmit light while the material being studied absorbs or scatters the light. What one sees is mostly light. While this approach makes good intuitive sense, it is not effective when the optical properties of the material are similar to the optical properties of the supporting medium. In this case, dark field microscopy is more effective. For dark field microscopy, the incident beam is blocked from the objective. The only light that can pass into the objective is light that has been scattered. If nothing is in the field, no light reaches the eyepiece. However, light is scattered into the objective if the material being studied is under it. Hence, what one sees is light where there is sample, and dark where there is only support medium.

Figure 2-7 (Continued on next page).

Investigating the Nanoscale

33

Figure 2-7. Schematic of a microscope identifying the lenses and light path. (a) The dashed lines represent light that does not enter the objective when the light stop is in place. (b) When the light stop is in place and there are no particles in the light path, no light gets to the objective lens. (c) However, when there are particles in the light path, light is scattered in the objective. Particles appear light while the absence of particles is dark..

With any microscope the limiting factor is resolution, which is the minimum distance two objects can be separated and still be observed as distinct from each other. Resolution is directly proportional to the wavelength of light. With visible light microscopes, particles can be resolved down to 100 nm. To resolve particles of approximately 1 nm, the wavelength of the light must be on the order of one nanometer, i.e., x-rays. Alternatively, one can use accelerated electrons; then one can take advantage not only of the wave-nature of the electrons, but also of the fact that their charge causes them to be refracted in ways that—after detailed interpretation of the results—result in a better understanding of the structure of the material. Electron microscopes function in a manner similar to light microscopes in that the incident energy must be well-characterized before it passes through the sample, and in that— once it has passed through the sample—it must be passed through a series of lenses in order to produce an image that the human eye can discern. The biggest difference is in the nature of the lens; electrons are focused by either magnetic or electric fields rather than by pieces of carefully shaped glass.

2.2.1. Transmission Electron Microscopy (TEM) Figure 2-8 presents a schematic of an electron microscope. The particles to be studied are placed on the specimen tray located at the top of the figure. Although the diagram does not

34

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

include any part of the instrument that precedes the specimen tray, it is valuable to the reader to know that the purpose of the excluded part of the instrument is to force the electrons to follow a parallel path, to control the size of the beam, and to ensure that the beam is coherent, i.e., its wavefront is a plane wave. The diagram does show the objective, intermediate, and projector lenses. These are magnetic fields whose strengths can be controlled by varying the current passing through a wire wrapped around an iron core; their purpose is to magnify the image. The objective aperture is used to control the amount of contrast due to diffraction, and the area aperture is used to select a specific area on a sample from which an electron diffraction pattern is obtained. When the electron beam passes through the specimen, the waves are transmitted or diffracted, depending on whether a particular electron is unaffected by the sample (transmission) or is redirected relative to the optical axis (diffraction). Because the waves of the electron beam are coherent, their destructive and constructive interference reveals important information about the relative location of atoms. Figure 2-8 identifies three locations along the optical axis at which images are formed. The images, located on planes that are perpendicular to the optical axis, get progressively larger. These bright-field images are formed as a result of electrons that either are or are not transmitted by the sample. Also identified in figure 2-8 are two locations where diffraction patterns are formed. Although the images on these planes appear to be nothing more than a series of dots of various intensities, the pattern of these dots is very revealing when it comes to determining the relative locations of atoms in a nanostructure. This is very closely related to the earlier discussion of x-ray diffraction.

Figure 2-8. An overview of the optical path of an electron microscope.

Investigating the Nanoscale

35

Not only does TEM provide the satisfaction of ‘seeing’ what nanoparticles look like, it supplies valuable information to the field of nanotechnology. For example, the twodimensional image can be used to determine the size and shape of carbon nanotubes (figure 29a) and to determine if the nanotubes are single walled (figure 2-9b). In addition, the crystal structure can be determined from the elastically scattered electrons (see section 2.1.3). Moreover, analysis of the characteristic x-rays given off by the electrons as they move to a lower energy state reveals the identity of the elements by either energy dispersion x-ray analysis (see section 2.1.4) or electron energy loss spectroscopy (EELS, discussed in section 2.3.6). These capabilities are summarized in figure 2-10. A downside of TEM as an analytical method is that only a small portion of the entire sample can be seen at a time. One must always question whether or not the sample is truly representative. In addition, multiple inelastic collisions occur, resulting in either destruction or modification of the sample under study.

a.

b. Figure 2-9. Transmission electron microscopy images of (a) carbon MWNTs (scale bar = 500 nm), and (b) SWNTs (scale bar = 10 nm). From Sun et. al., 2002).

36

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Figure 2-10. A generalized view of Transmission Electron Microscopy. The idea for this figure comes from Williams and Carter (1996).

High resolution transmission electron microscopy (HRTEM) is very similar to TEM, except that it uses electrons with high spatial frequencies. Each point in the HRTEM image has contributions from many points in the specimen because both direct and diffracted beams are combined to form the image. This results in atomic level resolution.

2.2.2. Scanning Probe Microscopy (SPM) There are two types of scanning probe microscopy. In both methods, a probe is moved systematically across the surface to be studied. The probe’s movement over the surface tracks a grid that is stored electronically. The changing height of the probe provides detail about the third-dimensional shape of the surface. The two methods differ with respect to how the probe reveals information about the surface.

2.2.2.1. Atomic Force Microscopy In AFM, a sharpened stylus attached to a beam is scanned across the surface. The force exerted by the surface pushes or pulls on the stylus and deflects the beam. The instrument is comprised of three main parts: a cantilever, a tip, and a detection scheme (see figure 2-11). The cantilever has a very small spring constant. When the tip is brought close to atoms on the surface, the static interaction between the tip and atoms exerts a force on the tip, causing it to deflect. Because the deflection of the tip is very small, it is necessary to amplify the signal. This is accomplished using an optical method. The back of the cantilever is highly reflective. A laser focuses on a point just above the probe’s tip; the beam is reflected to a diode array photodetector. As the cantilever moves up and down in response to the changing topography of the surface, the reflected beam reaches different diodes on the photodetector. The change in location of the beam reflected onto the photodetector, ΔB, is related to the deflection of the cantilever, Δz, according to the equation Δz = ΔB*L/(2*x)

(2.9)

where L is the length of the cantilever and x is the distance from the cantilever to the detector, in other words the length of the optical path of the reflected laser beam. Small changes in topography (Δz) can best be seen if the cantilever is short (L is small) and the light path (x) is small.

Investigating the Nanoscale

37

In Atomic Force Microscopy (AFM), the probe actually touches the surface and the degree to which the probe is deflected from some original position is measured with an interferometer. While the surface does not need to be modified for this method, it can be damaged in the process. On the other hand—because of the direct contact with surfaces— AFM can also be used to manipulate nanoparticles.

Figure 2-11. The atomic force microscope, in which the probe tip moves up and down on the surface of the material being studied. The deflection of the probe tip is measured as the location of the laser’s reflected beam on a photodiode detector.

Figure 2-12. (A) TEM picture of silver nanodisks prepared by photoreduction of silver ions irradiated for 90 minutes at 0.8 W/cm2 at λ=457 nm. (B) AFM picture of product particles. (C) height profile along the line in Figure B. (From Maillard et. al., 2003).

38

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

2.2.2.2. Scanning Tunneling Microscopy Scanning Tunneling Microscopy (STM) uses the phenomenon of electron tunneling to determine the surface shape of material. The probe does not actually touch the surface. However, it is necessary that the surface be conductive, so that when a voltage is applied between the probe and the surface, electrons tunnel between the two, producing a current (figure 2-13). If the material is not intrinsically conducting, the surface must be modified to make it so. In 1986 two pioneering physicists, Gerd Binnig and Heinrich Rohrer, shared the Nobel Prize in Physics for inventing the STM. The central component of STM is a platinumrhodium or tungsten needle, which is scanned across the surface of a conducting solid. When the tip is very close to the surface, electrons tunnel across the intervening space. If the current is held constant, then the stylus moves up and down, following the topography of the surface. In the constant-z mode, the vertical position of the stylus is held constant and the current is monitored. Because the tunneling probability is very sensitive to the size of the gap, the microscope can detect tiny, atom-scale variations in the height of the surface. STM has been called the ultimate tool for imaging single wall nanotubes (SWNTs), because it is capable of achieving atomic-level resolution. An example of this is shown in figure 2-14, which shows an STM image of DNA-gold nanoparticles complex film drop-coated on a conducting silicon substrate. The plot shows surface height variation along the length of one line of the image. The nanoscale resolution is apparent. Like SEM, then, STM provides the nanotechnologist with a valuable tool for determining exactly what has been assembled.

Figure 2-13. Scanning probe microscopy is a method in which a probe scans over the surface of a material and measures the topography. Scanning tunneling microscopy takes advantage of electrons’ propensity to exist beyond the boundaries of the substance to which they ‘belong’. A voltage is applied to the scanning

Investigating the Nanoscale

39

needle tip and the magnitude of current is monitored. Either the probe is held at a fixed height and current is monitored, or the current is held fixed and the required movement of the probe is monitored.

Figure 2-14. An STM of DNA-gold nanoparticles complex, drop-coated onto a conducting silicon surface to produce a film. From http://physics.unipune.ernet.in/~spm/dna.html.

2.3. SPECTROSCOPY 2.3.1. Absorbance Spectroscopy Absorption spectrometers require a light source that produces light over a broad spectrum, and a monochromator to select light of a very narrow range. A detector monitors the intensity of light not transmitted by a sample as a function of wavelength. As presented in figure 2-15, these double-beam instruments measure the percentage of light transmitted. Transmittance, T, is defined by T = I/Io where I is the intensity of light transmitted by the sample and Io is the intensity of light transmitted by a cuvette containing no sample. Absorbance, A, is equal to the negative logarithm of transmittance (A = - log10T). The amount of light absorbed is a function of how many absorbing species are encountered by the light; which—in turn—is a function of both the distance the light traverses through the sample, b, and the concentration, c, of the absorbing species in the sample. The proportionality constant is called the absorptivity, absorption coefficient, or extinction coefficient and is usually represented as ε. The resulting equation is known as the LambertBeer law, A = εbc

(2.10)

A measure of the strength of a transition of a substance is its oscillator strength, which is proportional to the integrated absorption coefficient.

40

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Figure 2-15. Generalized diagram of an absorption spectrometer. Note that the light source may be replaced with an ‘electron’ source to increase the energy. The transducer is any device that translates the intensity of light that reaches it to an electronic signal.

The electrons of nanocrystals are confined to the small volume of their nanocrystal. Consequently, the energy of transition, observed as the frequency of light absorbed by the particle, is confined to a very narrow range of values. This phenomenon is referred to as quantum confinement. Of particular interest to the field of nanotechnology is the observation that decreasing the radius of nanocrystals results in a shift of absorption to higher energies (shorter wavelengths) (figure 2-16). Moreover, as particles get smaller, the oscillator strength is compressed into a smaller number of transitions. The increase in band gap and the compression of oscillator strength to fewer frequencies with decreasing size are both consequences of quantum confinement. If the wave function of the confined electron overlaps with its hole, theory predicts that the oscillator strength should increase as 1/r3. This nonlinear behavior has great potential for applications in the design of optical switches.

Figure 2-16. UV-Vis spectra of isolated clusters. Cluster sizes are (a) 6.4 Ǻ, (b) 7.2 Ǻ, (c) 8.0 Ǻ, (d) 9.3 Ǻ, (e) 11.6 Ǻ, (f) 19.4 Ǻ, (g) 28 Ǻ, (h) 48 Ǻ. Note that with decreasing cluster size, the excitonic transition is shifted toward higher energies and the molar absorption coefficient, which refers to the concentration of Cd, increases. From Vossmeyer et al. (1994).

Investigating the Nanoscale

41

Absorption spectroscopy has also been useful for distinguishing between metallic and semiconducting single-wall nanotubes (SWNTs). The transition from the highest energy conducting band to the lowest energy valence band for a semiconducting tube is 0.6 eV, while the same transition for a metallic SWNT is 1.8 eV. Finally, interband transition energies depend on the diameter and chirality of SWNTs. (Chen, 1998 and Jost 1999).

2.3.2. Fluorescence Spectroscopy Fluorescence spectrometers (figure 2-17) are similar to absorbance spectrometers in that the sample is irradiated with light of a known wavelength. The intensity of emitted light is monitored as a function of wavelength. Although fluorescence will leave the sample in all directions, only light emitted perpendicular to the light source is detected; this is to avoid detection of the incident beam. For emission scans, the excitation monochromator is fixed at a wavelength where the sample is known to absorb and the emission is scanned. Alternatively, the emission monochromator can be fixed at a wavelength where emission is known to occur, and the excitation can be scanned.

Figure 2-17. Generalized schematic of a fluorescence spectrometer.

X-rays provide higher energy for excitation. In fact, the energy is sufficient to remove electrons from inner shells of atoms. When samples are scanned with x-rays of continuously varying wavelength, the spectra are monitored for discontinuities. These discontinuities occur when the x-rays have sufficient energy to eject an inner electron. The spectra are simple because there are only a few electrons in the inner levels, giving rise to very few permitted transitions.

42

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

2.3.3. Raman Spectroscopy When monochromatic radiation is scattered by molecules, a small fraction of the scattered radiation is observed to have a different frequency from that of the incident radiation; this inelastic scattering is known as the Raman effect. The Raman effect arises when the sample contains molecules that undergo a change in molecular polarizability as they vibrate. Recall that as light (an oscillating electric field) interacts with the electrons of molecules, it induces an oscillating dipole moment. Raman spectroscopy requires intense monochromatic radiation, i.e., a laser is required for the light source. The instrument design is similar to that of a fluorescence spectrometer, (figure 2-17), however to eliminate substrate scattering background from the weak signal, the experimental set-up shown in figure 2-18 can also be used. This is called backscattering confocal Raman spectroscopy.

Figure 2-18. Experimental setup for backscattering confocal Raman spectroscopy. From Yu and Brus, 2001.

Investigating the Nanoscale

43

Figure 2-19. Two photon modes of a nanotube.

Raman scattering is a valuable tool for investigation of the vibrational properties of SWNTs. Nanotubes have two phonon modes. The radial breathing mode (RBM), which occurs near 200 cm-1, involves the in-phase motion of all carbon atoms in the radial direction. It is predicted that the RBM frequency depends sensitively on the diameter of tubes, but not on chirality. Raman spectra also show a strong band at around 1580 cm -1: the tangential mode. Although the frequency shift of this mode remains relatively constant with the diameter of the nanotube, the shape of the band depends on whether the tube is metallic or semi-conducting. Figure 2-20 shows a representative Raman spectrum of a single SWNT. Notice that the x-axis identifies the shift in frequency relative to the incident frequency; for this spectrum, the incident frequency was at 632 nm (Yu, 2001).

Figure 2-20. Raman spectrum of a single SWNT bundle obtained by using backscattering confocal Raman spectroscopy with 20 kW/cm2 632 nm excitation. From Yu and Brus, 2001.

44

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

2.3.4. X-Ray Diffraction Advances in both instrumentation and data analysis allow evaluation of physical structures on the nanoscale when those structures are in powder form. This method is called x-ray powder diffraction. Moreover, when analysis is done at values of θ (see section 2.1.3.5) that approach zero, it is possible to learn details about structure for larger values of d. This method is called small angle x-ray scattering or SAXS. Complex structures may appear to be amorphous if one is required to use a small measuring device. When θ is small, it is possible to detect patterns with very large spacing. For example, Vossmeyer et al. (1994) describe a process in which they used SAXS to determine the distance between CdS nanoclusters. Figure 2-21 shows a decrease in 2-theta going down the plot. According to the Bragg equation (equation 2-7), this corresponds to a decrease in the distance between scattering planes. In this work, the peak angle maxima were converted to nearest neighbor distances of clusters in a powdered sample. The closer the clusters, the larger they must be. The values are in reasonable agreement with measurements made using TEM.

Figure 2-21. Small angle powder X-ray diffractograms of CdS nanoclusters. The calculated radii are (a) 6.4 Å, (b) 7.2 Å, (c) 9.3 Å, (d) 11.6 Å, (e) 19.4 Å, (f) could not be determined by this method. From Vossmeyer et al., 1994.

When θ is large, only repeat patterns with small spacing (small d) can be detected. This has proved useful for the larger CdS nanoparticles. The method is sometimes called wideangle x-ray diffraction or WAXD. Figure 2-22 shows results from WAXD on the same samples shown in figure 2-21. The peak angle maxima can be related to specific unit cells for the larger particles.

Investigating the Nanoscale

45

Figure 2-22. Wide angle powder X-ray diffractograms of CdS nanoclusters. The smaller nanoclusters are too small to produce an identifiable diffraction pattern. However, particles larger than about 20 Å (as determined by TEM) have unit cells that depend on the preparation method and the identity of the compound used for passivation. Sample (f) (R = 19-22 Å) has a hexagonal unit cell and (h) (r = 48 Å) has a cubic unit cell. The identity of samples (a-e) is the same as that identified in figure 2-20. From Vossmeyer, et al., 1994.

2.3.5. Electron Energy Loss Spectroscopy When electrons—accelerated to the x-ray energy levels—are fired into nanotubes, the electrons undergo inelastic collisions with the electrons in the nanotubes, and lose energy. The energy lost by the electrons can be monitored. The technique is called electron energy loss spectroscopy (EELS). When the energy loss corresponds to the promotion of an internal electron to one of the lowest unoccupied molecular orbitals (LUMO), the resulting spectrum reveals useful information about the nanotube’s structure. Because the energy of a molecule’s LUMOs is very sensitive to the nature of the molecule, EELS is a very sensitive method.

Figure 2-23 (Continued on next page).

46

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Figure 2-23. The energy of x-radiation is sufficient to eject electrons from the inner core of atoms. When this happens, outer shell electrons fall to fill the empty inner orbitals. The resulting release of energy may be emitted from the particle (a) in a process known as X ray fluorescence spectroscopy (XFS) or it can cause the ejection of electrons from the outer orbitals of the particle (b) in a process called Auger emission spectroscopy (AES).

EELS can distinguish between a nitrogen atom that is part of the nanotube’s skeleton from a nitrogen atom adsorbed onto the inside or outside of the skeleton. Also, since the incoming electron beam can be fairly precisely aimed, we can tell where the nitrogen atom is located in the nanoparticle. Thus EELS allows the detection of individual atoms along with information about their chemical environment. It is thus a very powerful tool for probing the structure of nanotubes. The disadvantage of EELS is that it is destructive; shooting highenergy electrons at a target tends to degrade it.

2.4. CONCLUSION A lot of theory has been presented in this chapter. If you understood it all, congratulate yourself. Be aware, however, that this was a very cursory overview. The goal was to present just enough detail to help you understand both the techniques and a little more about what makes the field of nanotechnology so full of potential. Chapter 6 has a lot more detail about most of the theories presented here. Many analytical techniques have been presented, but—once again—the presentation is not complete. While the techniques most often used in nanotechnology research have been described, there are others not mentioned here. Many of these techniques are used in combination with each other to allow insight into a large number of properties of the particles being studied. Moreover, the type of analytical instrumentation available is always changing and growing. As questions arise that cannot be answered by existing techniques, some very creative scientists find and build new ways to answer their questions. Finally, a lot of results have been presented in this chapter. For each technique, some tidbit of knowledge about nanoparticles has been presented. By now, you should have a

Investigating the Nanoscale

47

clearer picture of what nanoparticles are and some ideas about what they can do. But you’ve yet to learn just how they are made. That is the topic of the next chapter.

EXERCISES Exercise 2-1: What is the wavelength of a photon with a frequency of 7.5 × 1014 s-1? What is the energy of that photon in Joules? Exercise 2-2: How fast must a proton be moving in order to have a wavelength comparable to the photon in exercise 2-1? Exercise 2-3: How fast must an electron be moving in order to have the wavelength comparable to the photon in exercise 2-1? Exercise 2-4a: The gap between the valence and conduction bands in silicon is 1.12 eV. Calculate the maximum wavelength of a photon that can excite an electron from the top of the valence to the bottom of the conduction band. In what region of the electromagnetic spectrum is this wavelength? Exercise 2-4b: Explain the differences between a good electrical conductor, an insulator, and a semiconductor. Explain why semiconductor materials like germanium and silicon become insulators at very low temperatures and good conductors at very high temperatures. Exercise 2-5: Phonons that can be propagated through a crystal have a minimum wavelength on the order of twice the linear dimension of the crystal unit cell and a maximum frequency of around 1012 Hz. If the velocity of propagation for these vibrational waves is 5000 m/s, what is the corresponding minimum wavelength? What is linear dimension of the crystal unit cell? Exercise 2-6: Describe a circumstance under which an atom releases x-rays. Exercise 2-7: List the kinds of information that can be obtained by using a transmission electron microscope to analyze a sample. Exercise 2-8: Discuss the limitations of microscopy methods. Exercise 2-9: Although named a scanning tunneling “microscope”, the instrument is not really a microscope. Explain why it is not a microscope and how the STM image is actually obtained. Exercise 2-10: Describe the tunnel effect. Exercise 2-11: List and describe three ways the energy absorbed by matter may be released. Exercise 2-12: How can absorption spectroscopy be used to determine the size of nanoparticles? Exercise 2-13: Describe Raman Spectroscopy and discuss how it can be used to characterize SWNTs. Exercise 2-14: What is TEM and how does it work? Exercise 2-15: How is x-ray crystallography done and what is required for it to be performed? Exercise 2-16: What forces are involved in AFM? Exercise 2-17: What are fluorescence and phosphorescence and how do they occur? Exercise 2-18: How are x-rays produced? Exercise 2-19: How does a dipole moment affect the path of light? Exercise 2-20: What is electron energy loss spectroscopy?

48

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Exercise 2-21: Why is AFM not useful for imaging single-walled nanotubes? Exercise 2-22: How do electron microscopes work in comparison to normal light microscopes? Exercise 2-23: How is it possible to move a single nanoparticle in a controlled way?

FURTHER READING Bube, R. R., Electrons in Solids: An Introductory Survey, New York: Academic Press, (1981). Christman, J. R., Fundamentals of Solid State Physics, New York: John Wiley and Sons, Inc, (1988). Dresselhaus, M. S., Dresselhaus, G. and Eklund, P.C., Science of Fullerenes and Carbon Nanotubes, New York: Academic Press, (1996). Elliott, S., The Physics and Chemistry of Solids, New York: John Wiley and Sons, Inc., (1998). Kittel, C. Introduction to Solid State Physics, New York: John Wiley and Sons, Inc., (1996).

Chapter 3

MAKING THE NANOWORLD Many techniques have been employed to prepare nanostructured materials—both simple chemical methods developed in the 1850s, and methods developed by the electronics industry to produce semiconductors. This chapter explores a variety of fabrication techniques, discusses the need for surface saturation (termination groups), and introduces properties of self-assembled systems.

3.1. PREPARATION OF CARBON NANOSTRUCTURES In Chapter 1, the discovery of C60, bucky ball, was introduced. C60 was initially synthesized in very small amounts by laser ablation, i.e., vaporizing carbon atoms with a laser. The carbon atoms were allowed to combine and they resulted in a structure that favored the molar mass of C60 to all others; C70 was also produced in disproportionate amounts. When Smalley and coworkers initially employed this technique, only very small amounts of C60 were produced, detectable only on a mass spectrometer.

3.1.1. Carbon Arc Vaporization In 1990, macroscopic quantities of C60 were produced by carbon arc vaporization. This consists of a carbon (graphite) anode and cathode inside a helium-filled chamber. A current runs from the anode to the cathode and carbon atoms vaporize in a helium atmosphere. By varying the current and pressure in the chamber, scientists can produce C60, C70 or MWNTs (multi-walled nanotubes). SWNT (single-walled nanotubes) may be produced by this method if the anode contains a catalytic amount of a metal such as nickel or cobalt.

3.1.2. Chemical Vapor Deposition Another method to produce SWNTs that allows one to control nanotubes’ diameter is chemical vapor deposition (CVD). This is a synthetic technique common in the preparation of microelectronic devices and protective coatings. It requires a gas phase reactant to come in

50

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

contact with a heated surface. A chemical reaction occurs on the surface, resulting in the formation of a thin film, a powder or a nanostructure. The gas phase reactant may react thermally, or react at lower temperatures on the surface of a catalyst. This technique has been used to prepare a variety of nanostructures, especially carbon nanotubes.

Figure 3-1. A simple CVD apparatus. From Fahlman, (2002).

A simple CVD apparatus for the preparation of SWNTs is shown in figure 3-1. The reactant methane, CH4, is mixed with argon, an inert gas, and passed through a quartz tube containing catalyst powders. A tube furnace heats the quartz tube to 1000°C. Argon, unreacted methane and hydrogen leave the quartz tube through mineral oil bubblers. The nitrogen stream is applied to cool the exiting gases. The SWNTs form on the catalyst surface as shown in figure 3-2.

Figure 3-2. (a-f) TEM images of SWNTs grown on nanoparticle catalysts showing the “nanoparticlenanotube” relationship. The arrows show the ends of the nanotube and the dark circle is the nanoparticle catalyst. The scale bar is 10 nm. (g) A cartoon representing SWNT growth. (h) AFM image of a 50 μm nanotube. From Dai (2002).

Making the Nanoworld

51

The images in figure 3-2 (a-f), obtained using a transmission electron microscope (TEM) described in Chapter 2, show a number of important features of carbon nanotube expitaxy (the growth of the carbon nanotubes on crystalline iron oxide, Fe2O3). First, the images support the claim that the nanotube width is similar to the Fe2O3 nanoparticle diameter. Second, the images provide evidence that the iron oxide nanoparticle remains on the surface and the carbon nanotube grows out from the iron oxide surface as illustrated in the cartoon in figure 3-2(g). Consequently, the catalyst dimensions and length of time of reaction can control the width and the length of the SWNTs. SWNTs can be isolated from the catalyst by sonication and centrifugation in a micelle containing aqueous solution. These are just a few examples of how carbon nanostructures may be prepared. Researchers have great interest in learning to control nanotube diameter and chirality, a topic discussed at length in Chapter 4.

3.2. PREPARATION OF NON-CARBON NANOSTRUCTURES BY CHEMICAL METHODS Several synthetic techniques have been employed to prepare nanoparticles and nanotubes in the laboratory. In this section, a few techniques that employ chemical reactions for nanostructure formation will be introduced.

3.2.1. Preparation of Colloids In 1856, Michael Faraday at the Royal Institution of Great Britain prepared gold “fluids” that were ruby, blue or purple in color by the reduction of gold salts with phosphorous. He is credited as the discoverer of metallic colloids, particles from 0.5-500 nm that do not readily settle out from solution. Faraday observed that upon heating, the ruby colloidal mixture became violet and with prolonged heating, gold particles precipitated. He wondered how the nature of the gold particles changed when heated. Did the gold particles change in size or shape or was there a change in the distance between gold particles? Unfortunately the microscopic tools available to Faraday did not allow him to “see” the gold particles and consequently he could not answer his questions. Moreover, the theories that explain his observations were not well developed in the 1800s. Nevertheless, Faraday described a method for preparing metallic nanoparticles and suggested that changes in particle size or shape may influence the properties of particles. Today, we understand that the colors Faraday observed arise from the interaction of visible light with the gold nanoparticles. The frequency of light that is most strongly absorbed is called the resonant frequency, which is discussed thoroughly in Chapter 6. The colored colloidal mixture results from the absorption of resonant frequency which, has been shown to depend on the nanoparticle’s size, shape, surrounding medium and distance from other nanoparticles. The different colors Faraday observed resulted from a change in the resonant frequency due to one or more of these factors. A variety of nanoparticles can be prepared by reduction of metal ions. Generally, metal salts react with a reducing agent, a source of electrons. The reduction half reaction for gold (III) is shown below in Equation 3.1.

52

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al. Αu3+(aq) + 3e− Æ Αu (s)

(3.1)

The reducing agent may be hydrogen, sodium borohydride, or sodium citrate. The reducing agent is oxidized; the oxidation half reaction for hydrogen is illustrated in Equation 3.2. Η2 (g) Æ 2 Η+ (aq) + 2e−

(3.2)

The overall redox reaction is shown in Equation 3.3. 2 Αu3+(aq) + 3 Η2 (g) Æ 2 Αu (s) + 6 Η+ (aq)

(3.3)

As the gold atoms form, being reduced from gold(III) ions, they cluster together to form nucleation sites. The clusters grow as more and more gold(III)) ions are reduced to form gold atoms. The challenge with this approach to nanoparticle synthesis is controlling the nucleation and growth processes such that all the particles are the same size and contain (approximately) the same number of atoms. In figure 3-3, every dark spot represents a single cluster. Notice that the dark spots in each frame are very close to the same size. This indicates that the authors succeeded in maintaining very good control over the size of gold nanoparticles.

Figure 3-3. TEM image of Au nanoparticles produced with good size control. From T. Shimizu et. al. (2003).

After creating particles of the same size, the next challenge is to prevent undesired aggregation or gathering of particles to produce a larger cluster. The thermodynamically favored state of a system is the state with the lowest free energy. Figure 3-4 illustrates that thermodynamics favor aggregation and formation of bulk material; consequently, scientists have had to discover ways to kinetically stabilize the nanoparticles against aggregation.

Making the Nanoworld

53

Figure 3-4. Schematic energy diagram illustrating that nanoparticles are thermodynamically less favorable than the bulk phase.

3.2.2. Stabilization of Nanoparticles with Termination Groups One of the ways kinetic stabilization has been achieved is by passivation (see section 1.3), i.e., using a termination group to make the surface less reactive. Termination groups are often organic monolayers, a single layer of molecules interacting either through covalent bonds or intermolecular forces with atoms on the nanoparticle surface. Figure 3-5 shows the surface atoms of a gold nanoparticle bonding to organic molecules that contain a sulfur atom; however, nanoparticle surfaces may interact with a variety of elements including oxygen, nitrogen, phosphorous. Due to the interaction with the sulfur atom, the gold nanoparticle is prevented from forming larger gold nanoparticles. The termination group is not only important for the stabilization of the nanoparticle, but also for many of the applications of nanoparticles. For example, the drawings of the gold nanoparticles in figure 3-6 have oligonucleotides, which are DNA fragments, as part of the termination group. Some of the oligonucleotides have one DNA base pairing sequence; the other oligonucleotides have a complementary DNA base pairing sequence. When the complementary base pairs aggregate, the gold nanoparticles are close together and this causes changes in the absorption spectra as seen in figure 3-7.

54

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Figure 3-5. Cartoon of a gold nanoparticle stabilized by a thiol.

Figure 3-6. Aggregation of gold nanoparticles with oligonucleotide termination groups. From Elghanian et. Al, (1997).

Making the Nanoworld

55

Figure 3-7. (A) comparison of absorption spectra for Au nanoparticles in 1 mL total of aqueous solution and gold nanoparticles functionalized with oligonucleotides. (B) Comparison of Au nanoparticles functionalized with 5'- oligonucleotides before and after treatment with a complementary base oligonucleotide. From Storhoff et. Al. (1998).

3.2.3. Formation and Stabilization of Nanoparticles within Cavities Another way to control and stabilize nanoparticle size is to form the nanoparticle in a cavity or cage. Although there are a number of materials that exist in nature with nanosize cavities, purveyors of nanotechnology have had some success synthesizing substances with cavities of a particular size and shape. One type of material that contains cavities of uniform structure is a zeolite—a framework silicate consisting of interlocking tetrahedrons of SiO4 and AlO4—and similar mesoporous (larger pore) materials. Of special interest to nanotechnology is the observation that the size and shape of the cavities can be controlled by varying the method of zeolite synthesis. Figure 3-8 shows a cartoon of a molecule inside the cavity of a zeolite. In practice, however, mesoporous materials are used more often than zeolites. Solutions of metal ions can penetrate the channels; then a reducing agent is added to yield nanoparticles with a controlled size and shape.

56

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Figure 3-8. A molecule inside the cavity of a zeolite.

Another type of nanocavity is a micelle, an aggregation of molecules that form a submicroscopic cluster. Molecules capable of forming micelles are rather large with a hydrophilic (water-loving) group and a hydrophobic (water-hating) side chain. When placed in water, the hydrophobic portions aggregate so that they are surrounded by the hydrophilic portions of the molecule, as shown in figure 3-9. A reverse micelle forms when the surfactant is dissolved in an organic solvent; then the center of the micelle is the hydrophilic region. The cavity formed by the micelle is a nanocavity and can be used to prepare nanoparticles of a specific size. For example a solution of an organic soluble metal complex forms the center of the micelle. Then the metal complex is reacted with a reducing agent to form the nanoparticles whose size is controlled by the micelle cavity. Micelles have also been used to isolate individual single walled nanotubes. This work by Rice University scientists was significant because it allowed them to study one nanotube by itself, instead of generating nanotubes in bulk.

Figure 3-9. Soap molecules in water forming a micelle and a nanocavity.

Making the Nanoworld

57

A final example of nanocavity used to control nanoparticle size is a protein. Ferritin is a protein that stores iron and is important for the proper regulation of iron concentrations in humans and other animals. Figure 3-10 illustrates the nanocavities of ferritin. Ferritin has an 8 nm cavity and can hold a maximum of 4500 iron atoms. Importantly, the number of iron atoms can be controlled and ferritin has been used to prepare iron oxide nanoparticles containing about 200 iron atoms that are about 1.5 nm in diameter. Iron oxide nanoparticles prepared by this method were used for the CVD production of single-walled carbon nanotubes with controlled diameters.

a.

b. Figure 3-10. Nanocavities in Ferritin. From Donlin et. al., (1998).

58

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

3.2.4. Electrochemical Deposition of Nanowires Electrochemical deposition, or the use of an electric current to deposit a material, has been used to prepare nanowires. In one example, Zach and coworkers used a graphite surface to prepare molybdenum nanowires as shown in figure 3-11.

Figure 3-11. Depiction of the method of preparation for molybdenum nanowires. From Zach et. al., (2002).

The scientists began preparing the nanowires by purchasing a graphite electrode. The graphite surface was not smooth; imperfections and breaks in individual graphene sheets resulted in steps and terraces as shown in figure 3-11. The graphite electrode was placed in a solution of sodium molybdate (Na2MoO4) along with two other electrodes supplying a source of electrons. The molybdate was reduced according to the half reaction in equation 3.4. ΜοΟ42− + 2 Η2Ο + 2 e− Æ ΜοΟ2 + 4 ΟΗ−

(3.4)

As the insoluble molybdenum oxide (MoO2) formed, it preferentially nucleated on the steps rather than the terraces and grew to form nanowires—as long as the reduction voltage was carefully controlled. The molybdenum oxide nanowires were reduced with hydrogen to form nanowires of molybdenum metal. The nanowires were removed from the graphite

Making the Nanoworld

59

electrode by casting a solution containing polystyrene (from a Styrofoam cup), allowing the solvent to evaporate, and peeling the polystyrene with embedded nanowires from the electrode surface. Molydenum nanowires with a variety of widths were produced as shown in the scanning electron micrograph (SEM) images in figure 3-12. Electrochemical deposition has been used to produce a number of different nanomaterials. In addition, the nanomaterials produced often serve as a precursor or template for other nanomaterials.

Figure 3-12. SEM images of MoO2 nanowires of various diameters. Reprinted with permission from M. P. Zach et. al., (2002).

3.3. PREPARATION OF NANOSTRUCTURES BY PHYSICAL TECHNIQUES The approaches to nanostructure preparation in this section are referred to as physical techniques because they use light and mechanical means to prepare these nanostructured materials.

3.3.1. X-Ray Lithography X-ray lithography is a technique for making nanostructures using a mask or template to transfer a specific pattern onto a resist, a coating that reacts with X-rays on the surface of a

60

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

material. X-ray lithography is a projection lithography technique similar to optical lithography, which is used to make integrated circuits. Both lithography techniques use a setup similar to the one shown in figure 3-13. The mask needs to be made of a material that absorbs X-rays in some areas and transmits X-rays in others. The resist is usually a carbonbased polymer and is adhered to a silicon wafer. The resist becomes more or less soluble in a suitable solvent upon exposure to light. If the polymer becomes more soluble on exposure to light, it is removed only from the areas exposed to light, thereby transferring the pattern from the mask to the resist-coated silicon wafer. An advantage of projection lithography is that once a mask is made, the pattern can be produced again and again on multiple resists. This is like making multiple photocopies of an original drawing.

Figure 3-13. General layout for projection lithography. Light or X-rays are projected on a mask to form a pattern unto a resist. A mechanism to align the mask with the resist is also required but not shown. Permission from Franco Cerrina, director, CN Tech (2003).

Because the wavelength of an X-ray is 0.1-1.0 nm, compared to the wavelength of visible light between 450-800 nm, X-ray lithography can pattern a resist with a higher resolution than optical lithography or UV lithography. Consequently, more information can be stored in a smaller amount of space using the X-ray technique. Several methods have been developed to prepare the masks, including chemical vapor deposition (CVD), nanoparticle self-assembly and mechanical pushing. The disadvantage of X-ray lithography is that it is expensive to acquire the X-ray source needed to produce high-resolution resists. Figure 3-14 shows a resist formed from X-ray lithography.

Making the Nanoworld

61

Figure 3-14. A resist prepared to 100 nm resolution using X-ray lithography. Permission from Franco Cerrina, director, CN Tech (2003).

3.3.2. Laser Ablation When a material is irradiated with a high density of photons as provided by a laser, small pieces of matter, referred to as plasma, are ejected from the material. This process is known as laser ablation, and has a multitude of applications from cutting paper to correcting vision. The size of the particles that make up the plasma is on the nanoscale; therefore, laser ablation provides another technique for the preparation of nanostructures. Mafuné and coworkers used laser ablation to produce aqueous solutions of platinum nanoparticles. They immersed a platinum plate in an aqueous solution containing surfactant and in pure water and irradiated the metal plate with a laser. During the irradiation process, the solutions turned brown. Figure 3-15 shows the TEM images of the platinum nanoparticles produced with surfactant present and in pure water, respectively. Interestingly, platinum nanoparticles of 2-5 nm in diameter were observed with the surfactant, whereas larger nanoparticles of 4-7 nm in diameter were produced in pure water. These results are a clear example of the value of termination groups for the control of nanoparticle size. Although this particular example illustrates the formation of colloidal mixtures using laser ablation, laser ablation can also be used to place nanoparticles on a solid support. Laser ablation is also used to create magnetic thin films and magnetic nanostructures described in Chapter 5.

62

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Figure 3-15: Electron micrographs of platinum nanoparticles produced by laser ablation of a platinum plate immersed in a surfactant solution (panel a) and in pure water (panel b). The size distributions of the nanoparticles shown in panels a and b are plotted in panels c and d, respectively. From Mafune et. al., (2003).

3.3.3. Mechanical Pushing Similar to how a macroscopic structure is designed, nanotubes can be “pushed” into place with the tip of the atomic force microscope (AFM). Recall from Chapter 2 that the atomic force microscope works by moving a tip over the surface of interest. The tip is at the end of a cantilever and is attracted to or repelled by the surface. As the tip interacts with the surface, the cantilever also responds. The cantilever is monitored by a laser and attached to a feedback loop that adjusts the cantilever position relative to the surface. The tip of an AFM has been used to “push” atoms, nanoparticles or nanorods around the surface. In the example shown in figure 3-16, gold nanorods were pushed on a surface coated with molecules containing sulfur. The “pushing” occurs because as the tip comes in contact with the nanorod, the tip begins to move upward. As the tip continues contact with the molecule, the cantilever will stop moving and begin to flex. At this time the feedback loop that normally adjusts the cantilever is turned off. Consequently, the bent cantilever exerts a force on the nanorod. The nanorod

Making the Nanoworld

63

will be “pushed” as long as cantilever feedback loop remains turned off. This “pushing” of the nanorod is like a person using a stick to move a roll of carpeting. As the techniques for mechanical pushing mature, an efficient method for the preparation of masks for X-ray lithography may result.

Figure 3-16. A sequence of images displaying the manipulation of four gold nanorods. The arrows in each image show the manipulation direction that results in the rod configuration in the next image. (a) Initial arrangement of the rods; (b) result of translational manipulations of "1" along its longitudinal axis and "2" across this axis; (c) results of rotational manipulation operations of all 4 rods by ±45 relative to their original orientation. The height scale from black to white is 10 nm for (a-c). From Hsieh et. al., (2002).

64

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

3.4. PREPARATION OF NANOSTRUCTURES USING SELF-ASSEMBLY TECHNIQUES In the literature, authors write about self-assembly, meaning that a supramolecular species or complex structure has been obtained spontaneously by combining simpler substructures. The goal of a researcher designing a self-assembled system is to make the desired supramolecular species in perfect yield. Nature is quite proficient at self-assembled systems. Can humans design systems that operate like enzymes or (nano)machines? On a fundamental level, intermolecular (or interparticle) forces (IMFs), the forces between molecules (or particles), have a significant role in the design of a self-assembled system, largely because these forces must overcome the entropy barriers associated with any organized system. IMFs are noncovalent forces and include ion-dipole, dipole-dipole, hydrogen bonding, and London forces. Supramolecular species may also assemble due to metal-ligand bonds, charge transfer complexes or steric (geometric) effects. IMFs are responsible for many of the physical properties of molecules and solutions and the 2-D or 3-D structure of many important biological molecules. The double helix structure of DNA depends on hydrogen bonding between complementary base pairs. DNA is not the only molecule where IMFs are important both to structure and function. Subunits of proteins self-assemble to form the active species. Then, this supramolecular structure depends on IMFs to recognize the proper substrate and perform the appropriate signaling operations. Consequently, if scientists want nanostructures to “self-assemble”, they use chemical bonds and IMFs to build the structure they desire. Figure 3-17 shows a cartoon of the self-assembly of nanoscale containers designed by J. Rebek.

Figure 3-17. Schematic representations of self-folding cavitands (top) and self-folding nanoscale containers (bottom), i.e., unimolecular capsules. From Lucking et. al., (2000).

Making the Nanoworld

65

The 1987 Nobel Prize in Chemistry was awarded to Cram, Lehn, and Peterson for their contributions to host-guest chemistry for preparing some of the first man made self-assembled systems. Peterson’s work demonstrated that metal ions and organic molecules could self organize. The molecules which fold into the nanocavities and containers are shown in figure 3-18. The dashed lines represent hydrogen bonds; the rings control the size of the cavity and thereby the size of the metal ion the cavity can contain. The metal ions are not shown, but form ion-dipole interactions with the oxygen atoms in the container. This is one of many examples of self assembled systems that demonstrate great control of assembly, but are a long way from the naturally occurring self-assembled systems we observe in nature.

Figure 3-18. Self-folding cavitand 1 (side and top views) and self-folding containers 2 and 3. From Lucking et. al., (2000).

A large number of techniques have been developed to prepare nanomaterials. A few of them have been presented in this chapter. Synthesized nanoparticles have been used as tools to build microscopes with higher resolution, as imaging agents to help detect disease states (in animals), and as detectors of viruses to help combat bioterrorism.

66

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

EXERCISES Exercise 3-1: Discuss which synthetic techniques for preparing nanostructures seem most applicable to the electronics industry. Structure your answer by first discussing requirements of the electronics industry. Exercise 3-2: Some colloidal mixtures do not show the Faraday-Tyndall effect with using visible light. Discuss under what circumstances this will occur. Exercise 3-3: Platinum colloids may be produced by laser ablation of Pt metal or by reduction of platinum (II) salts. Discuss the differences between these two approaches and how you would attempt to control particle size using either approach. Exercise 3-4: Optical lithography uses resists made of a material that reacts with visible light. Consequently, the mask must be made of materials that absorb visible light (opaque matter) and transmit visible light (transparent matter). In X-ray lithography, what materials are appropriate for mask construction? (What materials absorb and transmit Xrays?) Exercise 3-5: Reduction of metal salts is a common method for making nanoparticles. Pt(II)Cl2 can be reduced with hydrogen to give Pt colloids. Write the oxidation and reduction half reactions and the overall balanced equation. Exercise 3-6: Calculate the % yield when 2.3 moles of methane produces 5.23 grams of carbon nanotubes. Exercise 3-7: Discuss what a laser is and why it is an effective surgical tool. Exercise 3-8: Define molecular recognition and discuss how it relates to the concept of self assembly. Exercise 3-9: Give an example of a host-guest interaction and discuss why the interaction is favorable (ΔG is negative.) Exercise 3-10: Discuss how an understanding of molecular recognition and intermolecular forces can be used in drug development.

Chapter 4

DESCRIBING THE GEOMETRY OF THE CARBON NANOFAMILY Nanoparticles can be made from semiconductors, metals and carbon. This chapter will focus on nanoparticles made from carbon in part because their structure is well-defined, and the effect of structural variations on specific properties has been well-documented. Moreover, carbon-based nanoparticles are already in use for a wide variety of applications. We are going to restrict ourselves even more by focusing on single wall nanotubes (SWNTs), because their geometrical structure is the easiest to understand even while they have great structural variations that lead to exploitable new properties. Though the carbon nanodots (fullerenes) have many interesting properties arising from nanoscale phenomena, they do not have many practical applications. Therefore, following the theme of learning about applicable nanotechnology, we won’t discuss them here. On the other hand, although diamond and graphite are immensely valuable and useful, we will not study them either because their value is already well understood and is not based on nano properties.

4.1. DIMENSIONALITY AND THE CARBON NANOFAMILY Four types of stable, purely carbon materials were described in Chapter 1. They were differentiated by the number of bulk dimensions. Diamond has three bulk dimensions, i.e., three independent directions in which there is a repetition of carbon atoms for a distance of thousands or millions of atoms. A sheet of graphene has two bulk dimensions and one nanoscale dimension. The bulk dimensions correspond to two independent directions in the plane of the graphene sheet and one nanoscale dimension perpendicular to the sheet. Graphite, which is several parallel graphene sheets held together, also has two bulk dimensions and one nanoscale dimension, because each sheet has a nanoscale dimension. A nanotube or nanowire has only one bulk dimension along the length or axis of the tube or wire and two nanoscale dimensions in directions perpendicular to the axis. Even if we have several nanotubes held together concentrically as in the multiple wall nanotube (MWNT), the material is only a few atoms thick in the nanoscale directions. This is illustrated in figure 1-9

68

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

(page 11). Finally, the buckyball, C60, has three nanoscale dimensions and no bulk dimensions. The same is true of the larger fullerenes. Note that we have, essentially by definition: #bulk dimensions + #nanoscale dimensions = 3. Because of space limitations, in this chapter we are going to concentrate on the case of nanotubes and nanowires, where the number of bulk dimensions is one. The notion of nanodimension and the possibility of interesting nanoscale properties and applications can occur in any material capable of forming a 2-dimensional structure. For instance, as mentioned in Chapter 1, a sheet of tungsten disulphide (WS2) can form a nanotube. Consequently, much of what is learned in this initial study of carbon SWNTs can be directly applied to other materials that have one bulk dimension and two nanoscale dimensions.

4.2. PROPERTIES OF SINGLE-WALL NANOTUBES Single wall nanotubes (SWNTs) have macroscopic properties that one would not predict based on the properties of diamond or graphite. The purpose of this section is to define and describe some macroscopic properties of SWNTs, thereby establishing a basis for the theory that will be presented in the rest of this chapter.

4.2.1. Density and Strength Density is a straightforward way of differentiating between forms of a substance; moreover, it’s tabulated in many handbooks. For example, Lange’s Handbook of Chemistry identifies the density of graphite as 2.267 g/cm3, and the density of diamond as 3.5153 g/cm3. However, if one tries to determine the density of nanotubes, values range from 1.3 to 1.4 g/cm3. Two comments can be made about this observation. First, the carbon atoms in nanotubes must be packed considerable less densely than for either graphite or diamond. This might lead one to suppose that carbon nanotubes are not very strong. Such a supposition would be dramatically mistaken. Young’s modulus is a measure of the strength of a material. Because it is reported as force per unit area, the units are units of pressure. The larger the value of Young’s modulus, the more resistant a material is to a deforming stress. The Young’s modulus for wood (Douglas Fir) is 13 GPa (13 x 109 Pa) compared to copper which has a large value for Young’s modulus, around 128 GPa. The Young’s modulus for SWNTs is closer to 1000 GPa. SWNTs are the strongest and most flexible molecular material known. Their strength to weight ratio is 500 times greater than that of aluminum. How can something with such a low density have so much strength? The second comment that can be made about the density of SWNTs is the variability in its value. It turns out that the density of the nanotubes depends on exactly how the graphene

Describing the Geometry of the Carbon Nano-Family

69

sheet rolls. This is related to the chirality that was mentioned in earlier chapters. One important purpose of this chapter is to establish a way of precisely defining the chiral angle of nanotubes. If variations in strength are correlated with chiral angle, then perhaps the definitions developed in this chapter can set the groundwork for a theory to explain the remarkable strength exhibited by SWNTs.

4.2.2. Conductivity (Electrical and Thermal) Another interesting property of SWNTs is their conductivity. In Chapter 2, the electrical conductivity of material is described as being related to the spacing between the valence band and the conducting band. In graphite, the MOs that comprise the valence band and conducting band are formed from the overlap of the many parallel, unhybridized p orbitals of the sp2 hybridized carbon atoms. Graphite is a conductor because the valence band and conducting band overlap. The electrons are very mobile. As described in Chapter 1, SWNTs can be thought of as graphene sheets that have been rolled up. When a graphene sheet is rolled, its parallel 2p orbitals become... well, not parallel. And yet SWNTs are good conductors, although not as good as graphite or copper. Resistivity is the inverse of conductivity. The smaller the resistivity, the better a conductor the material is. The resistivity of graphite is very small (1 x 10-8 Ω cm), the resistivity of copper is a little larger (1.7 x 10-6 Ω cm) and the resistivity of SWNTs, although larger than that of copper, is still very small (1 x 10-4 Ω cm). Another interesting property of SWNTs is their ability to transfer energy as heat. A measure of a material’s ability to transfer energy as heat is its thermal conductivity. The larger the value of the thermal conductivity, the greater the material’s ability to conduct energy as heat. Thermal conductivity is a property of the material. Like electrical conductivity, thermal conductivity is also greater for nanotubes than for copper, although not as dramatically enhanced. These properties are summarized in the table 4.1. Many of them can be explained by establishing an understanding of geometry of nanotubes. Table 4.1. Properties of SWNTs compared to other well-known materials. From http://www.pa.msu.edu/cmp/csc/ntproperties/ Property Density (g/cc)

Resistivity (Ω cm) Young’s modulus (G/Pa) Thermal conductivity (W/m/K)

SWNT 1.33 (10, 10) 1.34 (17, 0) 1.40 (12, 6) 1 x 10-4 1000 2000

Au 19.3

C (graphite) 2.26

Cu 8.92

2.2 x 10-6 78 320

1 x 10-8 1000* 140

1.7 x 10-6 130 400

* This is the modulus if the stress is applied parallel to the graphene sheets, that is to say such that the stress would cause the sheets to buckle.

70

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

4.3. CONSTRUCTING CARBON NANOTUBES GEOMETRICALLY So what is a carbon-nanotube? Just as we can roll up paper to form a circular tube, in the nanoworld we can “roll up a sheet of graphene” to make a nanotube. A schematic drawing of a portion of such a nanotube is given in figure 4-1. For a given nanotube, the length of the tube will be much greater than the girth (circumference). To better understand the content of this section, it is useful to acquire a sheet of paper— or better yet a transparency—which has been imprinted with the hexagon lattice pattern of graphene. (An enlarged copy of figure 4-2 would work very well.) Roll the sheet into a tube to model a carbon nanotube. As you work with this rolled paper model, be aware that you cannot roll up the tube in just any fashion. As you go around the girth of the tube, the hexagon patterns must match. Now think of creating a molecule where each vertex of the rolled up pattern is replaced by a carbon atom and the edges between them represent nearest neighbor bonds. If the curvature of the tube is reasonably low (large diameter) then the molecule will locally look like the graphene sheet and inherit the stability of graphene. The ends of the tube are sometimes finished off somewhat like knitting the toe-end of a sock. We could also think of cutting some enormous bucky ball of the right diameter into two pieces and compatibly gluing them onto the ends. Obviously the patterns of the end caps are going to depend very much on the girth and the twisting of the nanotube along its length. A typical nanotube is about one micrometer long; that is, it is about 1000 times longer than it is wide. Consequently, the ends have little to no effect on the properties of the nanotube. We will, therefore, ignore the ends in this development.

Figure 4-1. A schematic portion of a single walled carbon nanotube with (n,m) = (9,1). The significance of the values of n and m is explained in the text.

Describing the Geometry of the Carbon Nano-Family

71

4.3.1. Defining the Geometry of Graphene and the NanoTube Even though the compatible rolling up of the graphene sheet has restrictions, there are an infinite number of ways to roll the sheet. The differences depend on the direction of the rolling and how tightly the sheet is rolled. To get to a precise model for a nanotube, we are going to need to establish some coordinates on a graphene sheet. To this end, consider the hexagon pattern in figure 4-2. All the hexagons are regular and the side length, which we denote by a, represents the carbon-carbon distance in graphene. The value of a is 1.421 Å.

Figure 4-2. A portion of a graphene sheet with x,y coordinate axes.

It is traditional in crystallography to set the center of coordinates at an atom instead of the center of a hexagon. However, for this system, using the center of the hexagon simplifies the exposition. The derived formulas are the same; the only difference is that defined vectors and unit cells will be slightly offset. The graphene sheets shown in figures 4-2 and 4-3 are aligned so that the hexagons have vertical sides. Because these are regular repeating patterns, any of the hexagons can be

72

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

selected as the base hexagon, H0. The origin of our coordinate system will be at the center of H0. Observe that the horizontal axis of the coordinate system perpendicularly bisects the vertical sides of the hexagon, and the vertical axis of the coordinate system passes through the highest and lowest carbon. For illustrative purposes, the unit scale of the axes equals the distance between two hexagon centers. We are going to define two vectors U1 and U2 , shown in figure 4-3. The vector U1 starts at the origin and ends at the center of the adjacent hexagon, crossing over a vertical side of a hexagon on the horizontal axis. The vector U2 is rotated 60º counterclockwise from U1 and it, too, is the displacement vector between the center of the base hexagon and the adjacent hexagon. The length of the basic vectors will be denoted by ||U1|| and ||U2||. Our next task is to develop expressions for the vectors U1 and U2 in terms of the Cartesian (x,y) coordinate system. This requires application of some basic trigonometry. First, recognize that the perfect hexagons of our graphene sheet can each be divided into six equilateral triangles with sides of length a and angles of 60º. Now divide one of these triangles into two right triangles such that the hypotenuse is from the center of the hexagon to a corner, and the longest of the two remaining sides follows U1. The length of the hypotenuse is a, and the lengths of the two sides are a/2 and ||U1||/2. Using the Pythagorean theorem, we find a2 = (a/2)2 + (||U1||/2)2 ||U1|| = (3a2)1/2 =

3a .

Similarly, ||U2|| =

3a

This is the length of the two vectors relative to the carbon-carbon bond distance, a. Because U1 lies on the horizontal x axis, U1=

3a (1,0) = a( 3 ,0).

(4.1)

The angle between U2 and horizontal x axis is 60º, therefore the length of its projection on the x axis is ||U2||cos 60º, and its projection on the y axis is ||U2||sin 60º, so

U2 =

3a (cos 60°, sin 60°) =a(

3 3 , ). 2 2

(4.2)

The location of the center of any of the hexagons can be defined by the expression nU1+mU2 where n and m are integers. For instance the centers of the six hexagons adjacent to the basic hexagon, in counter clockwise order are U1, U2, U2-U1, -U1, -U2, U1-U2. When reading articles that discuss the geometry of SWNTs, be aware that the basic vectors are often chosen to emanate from a base atom rather than the center of a hexagon. Moreover, the notation for base vectors and other vectors soon to be defined also vary from

Describing the Geometry of the Carbon Nano-Family

73

paper to paper. It is useful to locate a diagram in the paper similar to figures 4-3 or 4-4 to clarify the notation before becoming immersed in the article.

Figure 4-3. A portion of a graphene sheet with basis vectors.

Now let us use this knowledge to construct a nanotube. Again using the rolled paper model, roll the paper until the base hexagon overlaps some chosen hexagon H1. Mark the centers of the two hexagons and draw a vector C from the center of H0 to the center of H1. The vector C is called the chiral vector or roll up vector and is pictured in figure 4-4. When the paper is rolled up, the line segment along C will go around the circumference of the tube exactly once. Thus the direction of rolling is the direction of the chiral vector C, and the circumference or girth of the tube is the length of the chiral vector. The vector C determines the geometry of the tube. Any line drawn on the graphene sheet parallel to C will form a circle around the tube. In analogy to a rolled up Mercator map, we call these circles latitudes. Any line perpendicular to C will form a straight line running along the length of the tube. Again in an analogy to the Mercator map we call these lines meridians. Any other line will form a line that spirals around the tube. We will call these skew lines. A skew line will meet all latitudes at the same angle and all meridians at the same angle.

74

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Figure 4-4. The chiral vector C and the unit cell vector T.

Now consider the skew lines that are parallel to the horizontal axis. They may also be characterized as skew lines that are perpendicular to the vertical bonds. The chiral angle θ is the acute angle between one of these skew lines and any latitude. On the graphene sheet, θ is simply the angle between the chiral vector and the horizontal axis.

4.3.2. The (n,m) Notation The chiral vector may be expressed in terms of the basic vectors, U1 and U2: C= nU1+mU2

(4.3)

Applying equations 4.1 and 4.2, we get

C = ( 3an +

3 3 3 3a am, 0an + am) = ( (2n + m), am) 2 2 2 2

(4.4)

where n and m are positive integers. In figure 4-4, n = 2 and m = 3. A formula for calculating n and m can be derived (Activity 7, Part D). It turns out that no matter what the shape of the nanotube, n and m can be chosen to be positive. You may choose values of n and m that are

Describing the Geometry of the Carbon Nano-Family

75

not positive, however there will always exist positive values of n and m that define a geometrically equivalent nanotube. Now, assuming that we have a chiral vector C given as above, let us calculate a few of its geometrical invariants in terms of n and m . First let us compute the nanotube’s girth L. The girth is simply the length of the chiral vector so that, using dot products and noting U1•U1 = ||U1||2 = 3a2 , U2•U2 = ||U2||2 = 3a2 U1•U2 = ||U1|| ||U2|| cos 60° =

3 2 a 2

we obtain L2 = ||C||2 = (nU1+mU2)•( nU1+mU2) = n2 U1•U1 +2mn U1•U2 + m2 U2•U2 = 3a2(n2 +mn+ m2) or 2

2

L = a 3(n + nm + m ) .

(4.5)

Obviously the diameter of the nano tube will be given by D = L/π = a

3(n2 + nm + m 2 ) /π.

Next let us compute a formula for the chiral angle, θ, namely the angle between the vectors C and U1. Form a right angle triangle with C as hypotenuse by dropping a vertical line from the tip of C to the line on which U1 lies, i.e., the horizontal axis. The length of this triangle’s side that is opposite θ will be the length of the vertical rise in the vector C (see equation 4.4). The length of this triangle’s side that is adjacent to θ will be the length of the horizontal displacement of the vector C. Using equation 4.4 and the definition of tan θ from trigonometry,

opposite = tan θ = adjacent

3 am 2 3an +

3 am 2

=

3m 2n + m

or θ = arctan( 3m /(2n+m))

(4.6)

In Exercise 4-6 you are invited to compute values of θ in terms of m, n and arccosine.

76

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

4.4. THE UNIT CELL Solids are materials in which individual atoms are not free to move about, but are confined to small oscillations about fixed equilibrium positions. Nanoparticles are solids on a very small scale. There are two major groups of solid materials: crystalline and amorphous. In crystalline materials, the atomic equilibrium positions form a pattern that is repeated throughout; they are said to have a periodic structure. In an amorphous solid, the atomic equilibrium positions do not form a repeating pattern. The pattern of atomic positions in a crystal is called a unit cell, which has two key requirements. First, it must be as small as possible, and second it must have translational periodicity. For a substance with a regular repeating pattern in three bulk dimensions, the unit cell will be some parallelepiped that, when replicates are stacked together, completely fills up space. In addition, the location of atoms within each parallelepiped must be exactly the same; in other words, when the unit cell is translated one unit in any of the three directions, the atomic positions must perfectly overlap. Many metals and salts have cubic unit cells; however, other parallelepiped shapes are possible. Now consider the two-dimensional graphene sheet. Here it is necessary to satisfy the translation requirement in two directions. A single hexagon is a unit cell because if you visually slide one hexagon such that its left vertical side moves to where its right vertical side was, you will perfectly overlap the existing pattern of circles and lines. Another possible unit cell for graphene is presented in figure 4-5 wherein the solid lines mark the edges of unit cells. If you translate any of the nine parallelograms one unit in the direction of either of the vectors, the dots and lines of the unit cell overlap perfectly. Both of these two different unit cells (hexagon and parallelogram) have the same area Ah, a fact left to the student as an exercise (Exercise 4-2).

Figure 4-5. A schematic representation of one possible unit cell for a graphene sheet. This diagram shows nine unit cells. Notice that translating one unit in the direction of either arrow results in a new parallelogram with bonds (lines) and atoms (circles) in exactly the same pattern as the original parallelogram.

Describing the Geometry of the Carbon Nano-Family

77

Nanotubes are one-dimensional structures. Therefore, the translation requirement of the unit cell needs to be satisfied in only one dimension, along the length of the nanotube. As we move along the surface of a nanotube, we see that a pattern eventually repeats itself. Thus, the nanotube could be constructed by piecing together many of these unit cells just as we can piece together fixed lengths of pipe to form a pipeline. Notice that the structures at the ends of the pipes have to match each other. This is the key to determining the unit cell for a nanotube. As with all nanotube concepts, the unit cell is best conceptualized when we unroll the tube back onto the graphene sheet. If we cut out the unit cell by snipping a nanotube at a circumference and at its first repeat, we will obtain something that looks like a piece of pipe or a tin can. If we cut a tin can along a meridian and flatten it out, we get a rectangle. Therefore we seek a unit cell that looks like a rectangle on the graphene sheet. An example is given in figure 4-6 with the corresponding unit cell on the nanotube given in figure 4-7.

Figure 4-6. Two adjacent unit cells in a graphene sheet.

78

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Figure 4-7. Two adjacent unit cells on a (2,4) nanotube.

To emphasize the concept of translation, we have actually shown two unit cells meeting along the chiral vector or girth of the nanotube. If you were to cut out one of the rectangles in figure 4-6 and lay it on top of the other, you would see that they are exact copies of each other. The asterisk * (very faint) in the nanotube picture corresponds to the origin, which is the center of the base hexagon H0 on the graphene sheet. From our construction of the unit cell on the graphene sheet, we may take one of the sides of the rectangle to be a chiral vector C. This would correspond to a circumference on the nanotube as in figure 4-4. Let T be the vector meeting C in a corner of the rectangle, as pictured in figure 4-4. On the nanotube, the vector T corresponds to a cut along a meridian. As with all vectors on the graphene sheet, T can be defined in terms of the unit vectors. We write T = t1U1+t2U2, where t1 and t2 are integers. In figure 4-4, t1 = -8 and t2 = 7.

(4.7)

Describing the Geometry of the Carbon Nano-Family

79

We identify T as follows. Consider the meridian line along T issuing from the base hexagon H0. Find the closest point (there are two equidistant from the center of the H0) on this line that passes through the center of another hexagon H2. Then just as we formed C, we form T as the vector from the center of H0 to the center of H2. Shifting the graphene sheet along T brings the hexagon pattern back to itself and we have our unit cell. Because shifting along T brings the hexagon pattern back onto itself, t1 and t2 must be integers. We will call such a vector T a unit cell vector. Notice that there are two possible unit cell vectors. They are the same length and point in opposite directions. By convention, the unit cell vector T is the one that is 90º counterclockwise from the chiral vector C. In the forgoing it was not clear how to calculate t1 and t2; an algebraic approach will work here. Because C and T are—by definition—perpendicular, their dot product must be zero. It follows that 0 =C•T= (nU1+mU2)•(t1U1+t2U2) = nt1 U1•U1 +(m t1 +n t2)U1•U2 + m t2 U2•U2 Using the equations preceding equation 4.5, the equation simplifies to

0= =

3 2 a (2nt1 + m t1 + nt2+2mt2) 2

3 2 a ((2n+m)t1 + (n+2m)t2). 2

Because (2 n+m)t1 + (n+2m) t2 = 0 the solutions on the meridian line will have the form t1 = -(n+2m)c , t2 = (2n+m)c, where c is some constant. Obviously, if we pick c = 1 we will get integer points. However if n+2m and 2n+m have a common divisor e then picking c = 1/e also yields integer solutions. The shortest vector we can get is when e is the greatest common divisor of n+2m and 2n+m. In other words, e = d =gcd(n+2m,2n+m) Thus we have t1 = -(n+2m)/d, t2 = (2n+m)/d

80

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

From these formulas we can compute the length, ||T||, of T, the area, A=||C||•||T||, of a unit cell and the number of atoms, N = 2A/Ah, in a unit cell, where Ah is the area of a hexagon. The formulas are: ||T|| =

3 L/d

(4.8a)

3 3 (n + mn + m ) d 2 2 4(n + mn + m ) N =2A/Ah = d 2

A = L||T|| =

2

3 L2/d =

(4.8b) (4.9)

In the last equation, A/Ah is the number of parallelograms (or hexagons) covering the rectangular unit cell and there are two unique atoms inside each parallelogram bonds. See Exercises 4-1, 4-2, and 4-7.

4.5. TYPES OF NANOTUBES The properties of nanotubes depend on the values of n and m. Before proceeding, let us note that every nanotube is geometrically equivalent to a nanotube where the (n,m) parameters satisfy 0 ≤ m ≤ n. Thus the set of possible nanotubes may be visualized by a sector of hexagon centers in the graphene sheet. This is pictured in figure 4-8. In that figure, the parameters n, m are placed in the hexagon whose center corresponds to the endpoint of the chiral vector C = nU1+mU2. Note that the sector extends out to infinity away from the (0,0) point. The (0,0) point and other points close to the apex of the sector are not feasible.

Figure 4-8. Metallic and nonmetallic nanotubes (grey = infeasible, black = metallic, red = semi-conducting).

Describing the Geometry of the Carbon Nano-Family

81

There are four interesting classes of nanotubes classified according to geometric or electronic properties. As shown in table 4.2, these properties usually can be characterized by a condition on n and m. Particular interest has been focused on armchair and zigzag nanotubes. The armchair nanotube has n = m and chiral angle 30° degrees and the zigzag nanotube has m = 0 and chiral angle 0°. Pictures of armchair and zigzag nanotubes are given in figures 4-9 and 4-10. Two adjacent unit cells are pictured in both figures. In figure 4-8, the armchair and zigzag nanotubes correspond to the upper and lower boundary, respectively, emanating from the (0,0) apex of the sector. Table 4.2. Four categories of nanotubes based on geometry and/or electronic properties Type

Characterization

armchair

Geometric: max chiral angle of 30 degrees Geometric: chiral angle of 0 degrees metallic-like properties along the length of the nanotube Semi-conductor properties along the length of the nanotube

zigzag metallic

semiconducting

(n,m) condition n=m

Chiral Angle

Circumference

30 degrees

3an

m=0

0 degrees

3 divides 2n+m

arctan(

3m /(2n+m))

A

3(n 2 + mn + m 2 )

3 does not divide 2n+m

arctan(

3m /(2n+m))

A

3(n 2 + mn + m 2 )

3 an

Figure 4-9. Two unit cells of the armchair nanotube with (n,m) = (7,7).

82

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Figure 4-10. Two unit cells of the zigzag nanotube with (n,m) = (7,0).

Another important classification of nanotubes is by their conduction properties along the nanotube, either as a metallic conductor or as a semiconductor. Theory predicts that the conducting properties of nanotubes are dependent on their geometry, i.e, the values of (n,m), the chiral angle, and the diameter of the nanotube. A complete discussion of this theory is very much beyond the scope of this chapter. Interesting results, however, are not. The theory allows for classification of a nanotube as metallic or semi-conducting. Namely an (n,m) nanotube is metallic if 2n+m is divisible by 3. Thus an armchair nanotube, for which 2n+m=3n, will always be metallic. On the other hand, a zigzag nanotube for which 2n+m=2n is divisible by 3 only if n is divisible by 3. Thus only 1/3 of zigzag nanotubes are metallic. The condition on (n,m) is only a theoretical prediction. The actual conductivity depends on the diameter, dislocations from perfect geometry, and any twisting, bending or other geometric deformation of the nanotubes. Quantitatively determining the actual conductivity is a topic of recent and current research. In figure 4-8, the predicted metallic and semi-conducting properties are differentiated by the color of the (n,m) parameter.

4.6. CONCLUSION This chapter has given you an opportunity to get a feel for the structure of carbon nanotubes at the atomic level. Should you choose to read an article that talks about the specific geometries of nanotubes, you are now able to understand what all the numbers mean.

Describing the Geometry of the Carbon Nano-Family

83

At the very least, you should be aware that there are different geometries and that the geometry affects the conductivity. Experiment has shown that different nanotubes have different electrical properties. Theory predicts that nanotubes with different geometries should have different electrical properties. A reasonable hypothesis would be that the experimentally observed differences in electrical properties arise from differences in the geometries of nanotubes. A goal of nanotechnologists is to demonstrate the validity of this hypothesis and—ultimately— synthesize the specific nanotubes with exactly the desired properties. We’re not there yet, but we’re getting closer. It’s going to take some time.

NOTATION Symbol C D d L n, m N T U1 , U2 θ

Description chiral or rollup vector nanotube diameter gcd(2n+m,n+2m) nanotube girth or circumference parameters describing a nanotube rollup vector in terms of the basis vectors number of atoms in a unit cell unit cell vector basis vectors chiral angle

Where defined or described Eq 4.4 immediately after Eq 4.5 paragraph before Eq 4.8a Eq 4.5 Eqs 4.3, 4.4 Eq 4.9 Eq 4.7 Eqs 4.1, 4.2 Eq 4.6

EXERCISES Note: Students will find working Activity 7 (in the Appendix) extremely useful for understanding the content of this chapter. You are strongly encouraged to work on Activity 7 before attempting these exercises. Exercise 4-1: On a copy of figure 4-5, subdivide a hexagon and a parallelogram into congruent 30°-60°-90° triangles. Then, by counting triangles, show that the parallelogram and the hexagon have the same area Ah. Exercise 4-2: Suppose that a large convex region of total area A is drawn on a graphene sheet. Estimate the number of carbon atoms contained within the region. Suggestion: First cover the region by non-overlapping translates of the unit cell parallelogram, then estimate the number of parallelograms in terms of A and Ah, and then count up the number of atoms in each parallelogram. Exercise 4-3: Suppose that a nanotube has length l and diameter t. Estimate the mass of the nanotube, in terms of l and t. Assume that the nanotube is so long that you may ignore the ends. Exercise 4-4: Show that equations (4.1) and (4.2) hold. Suggestion: Subdivide two adjacent hexagons into 6 equilateral triangles each. This will help determine the length of U1, and U2.

84

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Exercise 4-5: For practice with dot products verify the following formulas U1•U1 = ||U1||2 = 3a2 , U2•U2 = ||U2||2 = 3a2 U1•U2 =||U1||·||U2|| cos 60° =

3 2 a 2

Exercise 4-6: The calculus formula for the angle between two vectors gives cosθ = C•U1/(||C|| ||U2||) Use this to compute the chiral angle in terms of a cosine and n and m. Exercise 4-7: Verify equations 4.8a, 4.8b and 4.9. Namely, compute the length of T and the area of a unit cell in terms of n and m. Then compute the number of carbon atoms in a unit cell. Exercises 1 and 2 will be helpful. Exercise 4-8: Demonstrate formula 4.10 by simple computation. Exercise 4-9: Consider the various directions defined below: V1 = U1, V2 = U2, V3 =U2-U1, V4 = -U1, V5 = -U2, V6 = U1-U2. Demonstrate that the atoms in the hexagon whose center is C = nU1+mU2 are located at C + P1, … , C + P6 where P1 = (V1+V2 )/4, P2 = (V2+V4)/4, etc. Part of the exercise is to define the remaining vectors Pi. Exercise 4-10: Two vectors in the plane V1 ,V2 form a positively oriented base if the clockwise angle from V1 to V2 is acute. Alternatively, the triple of (V1,V2, k) is a right handed system. Now (V1,V2, k) is a right handed system if and only the triple scalar product (V1xV2)•k > 0. In turn, (V1xV2)•k =det([V1,V2]) where [V1,V2] is the 2x2 matrix whose columns are the vectors V1,V2. Using this discussion show that C and T as defined form a right handed system.

FURTHER READING AND INTERNET LINKS Dresselhaus, Mildred, Dresselhaus, Gene, Saito, Riichiro, Carbon nanotubes, http://physicsweb.org/articles/world/11/1/9 Harris, P. F. J., Carbon Nanotubes and Related Structures: New materials for the Twenty First Century, Cambridge University Press, (1999). The VRML gallery of chiral NanoTubes: http://jcrystal.com/steffenweber/gallery/ NanoTubes/NanoTubes.html The Nanotube website: http://www.pa.msu.edu/cmp/csc/nanotube.html A site showing folding nanotubes: http://www.photon.t.u-tokyo.ac.jp/~maruyama/kataura/ chirality.html The SCAN website http://www.rose-hulman.edu/SCAN/

Chapter 5

MECHANICAL AND MAGNETIC PROPERTIES OF NANOPARTICLES The primary motivation for studying nanoparticles is the remarkable set of properties that these materials possess. The reduced dimensionality of nanoparticles results in properties that present new technological opportunities and applications. Enhanced mechanical strength and unprecedented magnetic properties are just two examples. This chapter will provide background information about how scientists and engineers characterize mechanical properties and then explore some of the phenomenal mechanical properties exhibited by nanoparticles. The second section of the chapter will review some fundamental ideas associated with the magnetic properties of materials, and then explore a few of the amazing magnetic properties of nanomaterials. The chapter will review the state of the art with respect to high density magnetic data storage systems and propose ways in which special magnetic properties of nanomaterials may enhance the density of data storage.

5.1. MECHANICAL PROPERTIES 5.1.1. Stress-Strain Relationships When a force is applied to a material, the material experiences stress; any deformation that results from the stress is called strain. For macroscopic materials, the phenomena of stress and strain can be precisely defined for a bar-shaped portion of the material being studied. First, envision the force being applied outwardly from the ends of a bar (figure 5-1 (a)), where the force is centered on the axis of the bar. If A is the cross-sectional area of the bar perpendicular (normal) to the axis of the applied force, Fn, then stress, σ, is defined as σ = Fn/A

(5.1)

Note that the value of A may change if the bar is deformed by the force. The units of stress must be force per unit area, so that in SI units stress is Newton per squared meter (N/m2) which is the definition of a Pascal (Pa). If the force, Fn , is a pulling force, σ is a tensile stress. If the force is a pushing force—that is, a negative value for the vectors in figure

86

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

5-1 (a)—then σ is a compressive stress. The deformation that results from the applied force is quantified in the concept of strain, ε, which is defined for tensile stress as the change in the length of the bar (∆L = Lfinal – Lo) relative to the initial length, Lo. ε = ∆L/Lo

(5.2)

a.

b. Figure 5-1. A force, F, applied to a bar-shaped sample of the material being studied results in stress experienced by the material. Stress, σ, is recorded as force per cross-sectional area of the sample. In the figure, the relevant cross-sectional area is shaded grey. (a) Tensile strain, ε, is recorded as the change in length of the sample relative to its initial length (∆L/Lo). Compression tests apply force on the sample in the opposite direction of the arrows in the diagram. This may be considered a negative force. (b) Shear strain, γ, is displacement in the direction of applied force relative to the initial height of the bar.

Now envision a force applied parallel to the cross-section area, Fs (see figure 5-1(b)). The force is applied is such a manner that it appears to be trying to slide one plane of the surface across the plane of molecules adjacent to it. This effect is called shear. The stress that results from a force applied in this manner is called shear stress, τ. τ = Fs/A

(5.3)

Shear strain is the ratio of displacement in the direction of the applied force relative to the initial height of the bar (see figure 5-1(b)). Any force acting on a material must be the vector sum of the force normal to the surface, Fn, and a force parallel to the surface, Fs. Let the angle between Fn and the actual applied force be θ, then Fn = F cos θ and Fs = F sin θ (figure 5-2).

Mechanical and Magnetic Properties of Nanoparticles

87

Figure 5-2. A force, F, applied to a material that is not normal to the cross-sectional area is the vector sum of the portion that is normal to the cross-sectional area, Fn, and a force that is parallel to the cross-sectional area, Fs. The normal force results in tensile stress, σ, while the parallel force results in shear stress, τ.

The different mechanical tests performed on materials involve applying some combination of these two stresses (tensile and shear) and observing the response of the material. For example, to determine a material’s flexural rigidity, one end of a beam is clamped in place while force is applied to the opposite end to bend the beam, perpendicular to the long axis of the beam. This causes a tensile stress on one side, a compressive stress on the opposite side, and shear stress as the layers of material slide past each other. Some material tests result in no permanent deformation, while others apply stress until the material reaches a failure point, for example, stretching a bar until it yields and then breaks, or compressing a sample until it buckles and then folds. One frequently measured material property is Young’s modulus, E. Young’s modulus is the constant of proportionality between tensile stress and strain and therefore has units of stress (strain is unitless). σ = Eε

(5.4)

Although a plot of stress versus strain is linear near the origin, the slope changes at higher values. Tensile tests are performed by clamping the ends of the sample and pulling with a measured force. The relationship between the applied force and degree of deformation can either be converted to stress and strain or plotted directly. For a sufficiently small force, the plot of force, F, versus deformation ∆L is linear; the material follows Hooke’s Law K = F/∆L

(5.5)

where K is called the spring constant or stiffness. The point at which behavior deviates from linearity is called the proportional limit of the material. Applying the given definitions to equation 5.4,

88

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al. F/A = E∆L/Lo = EF/LoK

(5.6)

Finally, solving equation 5.6 for the spring constant, K = AE/Lo

(5.7)

For materials below their proportional limit, the spring constant is a measure of the value of Young’s modulus and is therefore sometimes reported in place of the Young’s modulus. Note, too, that the spring constant is related to the cross-sectional area of the material being tested. Consequently, if we know the value of the spring constant for a material with a known cross-sectional area, we can scale its value according to the area for a smaller sample. Consider the specific example of a single-walled nanotube. First, assume that the spring constant for a cylinder of solid graphite is the spring constant for a force applied parallel to the graphite sheets, i.e., the area of the cylinder is the area of a circle normal to the plane of the graphite sheets. Next recognize that the cross-sectional area of a nanotube is the cross sectional area of a cylinder less the area of the hole in the center of the nanotube (see figure 53). Then,

( Acylinder − Ahole ) (rcyl − rhole ) A = swnt ∗ K cylinder = ∗ K cylinder = ∗ K cylinder Acylinder Acylinder rcyl2 2

K swnt

2

(5.8)

Figure 5-3. A cross-sectional view of a nanotube. The black circles represent carbon atoms. The value of rcyl is the radius to the outer edge of the nanotube, and rhole is the internal radius.

If one uses the thickness of a single graphite sheet as the value for the difference between the radius of the cylinder and the radius of the hole—specifically rcyl – rhole = 0.34 nm (Puech, 2006)—one can approximate a relative value for the spring constant of a nanotube. A force applied parallel to graphite sheets is also perpendicular to the cross-sectional area of a cylinder made by rolling one of the sheets into a nanotube. The Young’s modulus for this system is E = 1060 * 109 Pa = 1060 GPa. Using this value and the relationship K1/K2 = E1/E2, the modulus for a SWNT (single walled nanotube) can be predicted. For example, a SWNT

Mechanical and Magnetic Properties of Nanoparticles

89

with an exterior radius of 1.0 nm will have a modulus of 0.56*Egraphte = 594 GPa. Note that this approach represents a macroscopic treatment of a nanosize system.

5.1.2. Stress-Strain Relationships on a Nanoscale When materials are subjected to stress, the molecules in one part of the structure transfer the stress to another part in some potentially definable way. The resulting strain in the overall structure is modified because of that transfer from one part of the material to another. In many cases, engineers take advantage of the stress transfer by incorporating strong, brittle components into the matrix of a more flexible, less brittle material. Materials comprised of two or more components are called composites. A very early example of a composite is the combining of straw and mud to make the building bricks used for pyramid construction. Composites often have properties that are greater than the sum of their parts. For many years, titanium metal was used for golf clubs and bicycles because of its strength and light weight. However, a composite made from carbon fiber and epoxy results in a material that has a higher tensile strength and one-third the weight of titanium. Yet a golf club made of pure carbon fiber would snap in two the first time it was used. All of the methods described in the previous section are based on macroscopic behaviors. In that sense, they represent what happens on an average scale. Such averaging is not an option if one wishes to characterize the mechanical strength of individual nanoparticles. One must consider the stress applied to—for example—a single carbon nanotube and the resulting deformation as defined by the displacement or deformation of the hexagonal rings that comprise the walls of the nanotube. For example, a more precise means of expressing strain can be established by defining the displacement of a given atom from the pre-stressed position as u, then the infinitesimal strain is reported as

ε=

∂u ∂X

(5.9)

where X is the position of the atom. The definition of stress is modified as well. In a limiting definition, stress can be thought of as force over an infinitesimal area as the area tends to zero. The smallest appropriate volume is a volume over which the force is uniform. For nanoparticles, this volume will be smaller than for macroscopic materials. There currently exist three approaches to analyzing stress on carbon nanotubes. The Virial approach is used if the stress is homogeneous in the entire volume of the portion of the molecule being simulated. A Lutsko stress is computed if partial volumes must be considered. Finally, a BDT (Basinski, Deusberry, and Taylor) stress is computed if the desire is to analyze with respect to a single atom. These theoretical methods reveal that both MWNTs (multiwalled nanotubes) and SWNTs should possess very high axial stiffness, with a Young modulus in the range of 1000 - 2000 GPa; that’s 1 x 1012 -2 x 1012 Pa, which is comparable to the reported elasticity of graphite, 1060 GPa. BDT theory also predicts that the stress of a nanotube at zero strain is not zero. The magnitude of this intrinsic residual stress, which results from the distortion of the graphene sheet as it is rolled, depends on the curvature of the nanotube, approaching zero for an infinite

90

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

radius (i.e., planar graphene sheet). The smaller the radius, the greater the residual stress. It is interesting to correlate this observation with the macroscopic model described in the previous section, which predicts that the Young’s modulus for nanotubes is relative to the modulus of graphite (Eo) and to the radius of the nanotube according to the equation

E=

(

2 E 0 rcyl2 − rhole

)

2 cyl

r

(5.10)

where rcyl is the external radius of the nanotube, and rhole is the size of the hole through the center of the nanotube (see figure 5-3). Assuming a constant wall thickness (rcyl – rhole), one can see that the smaller the radius, the stiffer the nanotube. (See Exercise 5-1 for practice. ) It is not unreasonable to conclude that the greater stiffness of small radius nanotubes is somehow correlated with the greater residual stress. It has also been observed that the force required to bend MWNTs exceeds the force required to bend SWNTs. In fact, when MWNTs bend, they do so by buckling and kinking. High resolution TEM has shown that SWNTs are more pliable, exhibiting high curvature bends without buckling. On the other hand, there is also experimental evidence that MWNTs withstand compression better than single-walled nanotubes. Experimentally determined results on isolated nanotubes approach the theoretical with reported results from 900 - 2000 GPa. However, when measurements are made on clusters of 105 to 106 nanotubes (called fibers), the measured mechanical properties are considerably less. Fibers of SWNTs have reported moduli from 50 to 120 GPa, a full order of magnitude less than that determined for individual nanotubes! These trends are also apparent in systems where nanotubes are incorporated into other materials. Although the modulus and the strength of the materials increase, the level of improvement is well below that predicted by theory. It has been noticed that the more the nanotubes cluster within the matrix, the less notable the improvement in mechanical properties. One proposed explanation is that when nanotubes are clustered, individual tubes slide past each other when stress is applied. While this may improve flexibility, it does little to minimize strain. Most of the information presented so far has focused on theories to predict and explain the mechanical properties of carbon nanotubes. That’s because this theory is an area of active research among chemists, mathematicians, mechanical engineers and physicists. Non-carbon nanoparticles also have amazing mechanical properties. These are described in the next section.

5.1.3. Nanoscale Mechanical Properties and Applications Materials with nanoscale grain size have enhanced mechanical properties. For example, cutting tools made of nanomaterials such as tungsten carbide, tantalum carbide and titanium carbide are much harder, more wear-resistant, and more erosion resistant than their conventional large-grain counterparts. Combine these properties with the small size and you have the micro drill bits needed for construction of microelectronic circuits. This enhanced hardness, which is closely related to a greater wear-resistance, makes nanomaterials excellent candidates for use in automotive parts such as spark plugs.

Mechanical and Magnetic Properties of Nanoparticles

91

Material scientists have long been aware that fatigue strength increases with a reduction in the grain size of the material. Nanomaterials provide such a significant reduction in the grain size over conventional materials that the fatigue life is increased by an average of 200 300 %. This is an area of great interest to those designing components for spacecraft, where fatigue strength at high temperatures is a crucial property. Ceramics are hard, brittle solids used in many applications because of their hardness. They are, however, very difficult to machine, even at very high temperatures. On the other hand, when the grain size of ceramics reaches the nanoscale, they can be machined at significantly lower temperatures. The final product retains the excellent physical, chemical, and mechanical properties of its large-grain counterpart. Carbon nanotubes are among the nanomaterials incorporated into materials known as composites. Their strength, combined with their ability to rebound (return to their original shape) after compression, means they play a very useful role in the dissipation of energy for a composite material. They are also used as tips in atomic force microscopy. In the realm of up and coming applications for nanoparticles, consider the possibility of a memory chip designed by a company called Nantero. In this chip, arrays of nanotubes connect two wires. When a field is applied to the nanotubes in one direction, the nanotubes flex to touch a metal electrode (considered the ‘1’ state). Once flexed, they remain in contact with the electrode (even when the power is turned off) allowing for storage of the logic of the ‘1’ state. Not until a field is applied in the opposite direction do the nanotubes straighten, losing contact with the metal electrode. This is called the ‘0’ state. The nanotubes act as electromechanical switches on the nanoscale. Now let’s enter the realm of dreams. Due to the cylindrical structure of nanotubes, they would make ideal axles for use in nanogears and nanorobots. Imagine the surface of two adjacent nanotubes covered with evenly spaced, covalently attached molecules that protrude outward. When one nanotube rotates, it will transmit its rotation to another nanotube. These nanogears would be the most important element of a nanorobot. One can imagine nanorobots carrying a particular drug to a precise location. Another possibility might be to leave a nanorobot in one location in the body to watch for abnormal symptoms, providing the kind of early detection that leads to more effective healing. A piece of news that makes this dream worth pursuing is the recent evidence that functionalized nanotubes pose little to no risk to living organisms. Pharmacokinetic studies on mice reveal that functionalized carbon nanotubes administered intravenously are cleared from blood and excreted in urine with no change to their structure. (Chemical and Engineering News, 20 February 2006, p. 8.) The mechanical properties of nanomaterials have already led to many useful applications. As our understanding of these materials increases and our ability to manipulate them improves, more and more elegant applications are being developed. It is only reasonable, then, that we strive to continue the dreams that were initiated by Feynman’s speech in 1959. We’re not there yet, but we’re getting closer.

5.2. MAGNETIC PROPERTIES This section explores the magnetic properties of nanomaterials. The opening section provides a brief background to magnetic properties of materials. The intent is to provide just

92

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

enough background to allow the reader to understand the special magnetic properties of nanomaterials. The reader is referred to any physics textbook or books on solid state materials for more complete discussions.

5.2.1. Theoretical Background To understand the source of magnetic properties of materials, it is best to start with the role of an individual electron. Whether free or bound to the nucleus of a single atom, an electron has an intrinsic angular momentum which gives it a magnetic moment. The magnitude of the magnetic moment of one electron is called a Bohr magneton, μB, and its value is μB = eh/4πme = 9.27 x 10-24 A m2

(5.11)

In this equation, e is the charge of one electron, h is Planck’s constant, and me is the mass of one electron. The magnetic moment that results from the intrinsic angular momentum of an electron is called the spin magnetic moment. An additional contribution to the magnetic moment of atoms arises because electrons that are bound to a nucleus are in motion about the nucleus. An electron in motion about the nucleus creates a magnetic field just as a small current loop does. The resulting magnetic moment is called the orbital magnetic moment. The total contribution of an electron to the magnetic moment of an atom is the sum of the spin and orbital magnetic moments. Moreover, the contributions from different electrons of an atom add together. Because the orbital magnetic moment is quenched in bulk materials, the spin magnetic moment plays the predominant role; it will be the focus of the remaining discussion. To understand how electron spin affects the magnetic moment of an atom, it is appropriate to review some ideas about electron configuration. Electrons occupy orbitals, filling those orbitals from lowest energy up (Aufbau Principle). The number of electrons per orbital is limited to two, because no two electrons in an atom can have all the same quantum numbers (Pauli exclusion Principle). The two electrons in a single orbital have the same energy and the same values for quantum numbers n, l, and ml, but different values for ms which is know as the ‘spin quantum number’. The spin quantum number can be either +½ or -½. When constructing electron configuration diagrams, the two electrons of a single orbital are often distinguished by drawing arrows: one pointing up (spin-up) and one pointing down (spin-down) to distinguish the two values of ms. The process of determining the electron configuration of an atom is made somewhat more complicated by the fact that there may be more than one orbital at a given energy level. For example, an atom has three p orbitals with the same energy and five d orbitals with the same energy; that means there can be a total of six electrons on p-type levels and ten electrons on d-type levels. Multiple orbitals with the same energy are called degenerate orbitals. Recall from General Chemistry that as the energy of orbitals increases, the types of orbitals (s, p, d, f, etc.) that exist increases. Consequently, the number of degenerate orbitals increases with increasing energy. You might think of this as all the people in a classroom standing on rungs of a very strong ladder. First, two people stand on the bottom rung, then the next two people stand on

Mechanical and Magnetic Properties of Nanoparticles

93

the next rung, and so on until everyone is standing on the ladder. The existence of degenerate orbitals is like have multiple rungs at a given height, so there can be a total of six people on ptype levels and ten people on d-type levels. The ladder becomes more like an upside down pyramid; hmmm...our ladder analogy is getting a bit precarious. Another aspect about the filling of degenerate orbitals is very important to the discussion of magnetic properties of atoms. If there is more than one orbital with the same energy, electrons will singly occupy degenerate orbitals until double occupancy is absolutely necessary. Only when all the degenerate orbitals are singly occupied will there begin to be two electrons per orbital; this is known as Hund’s Principle. Moreover, there is an energetic advantage for electrons of singly-occupied degenerate orbitals to have the same spin. In other words, when multiple electrons singly occupy orbitals of the same energy, the electrons will all have the same spin.

5.2.1.1. From Electron Configuration of Atoms to Magnetic Properties of Materials Let’s say we’ve assigned the electrons of an atom to orbitals while following all the rules and principles listed in the above paragraphs. If, when we are done, the number of electrons with spin-up is equal to the number of electrons with spin-down, then we say there is no net spin. Atoms for which all the electrons are paired not only have zero net spin, they also have a zero net magnetic orbital moment. When these atoms are exposed to an external magnetic field, there is essentially no effect on the orbitals of the atoms. They may acquire a very small moment, the direction of which is opposite that of the applied field. This phenomenon is called diamagnetism. Materials comprised of atoms with zero magnetic moment are called diamagnetic materials. If the number of electrons with spin-up is not equal to the number of electrons with spindown, then the atoms will have a non-zero magnetic moment. At 0K, when an external magnetic field is applied to these materials, the energy of electrons with spin aligned with the magnetic field will decrease while the energy of electrons with spin opposite to the magnetic field will increase. If the system temperature differs from 0K, a thermal fluctuation (coming from nuclear motion or lattice vibration) will affect the distribution of spin. This thermal field will compete with the magnetic field to determine the system magnetic property. How can we determine which field (thermal or magnetic) will have the predominant effect on the system? There is an equation called the Bose-Einstein distribution that allows us to calculate the relative population of two states of a system as a function of the difference in the energy of the two states and the temperature of the system. Essentially, the relative population of two energy levels is proportional to exp (-∆E/kBT) where ∆E is the energy difference, kB is the Boltzmann constant (1.38 x 10-23 J/K), and T is the temperature of the system. For most materials whose atoms have a non-zero magnetic moment, the energy difference is so small that the difference in population of the two levels is only about 0.01%. In other words, the tendency to align magnetic moments is swamped by the thermal fluctuation. These materials are called paramagnetic materials. When an external magnetic field is applied to a bulk solid, the solid may or may not become magnetic. The degree of magnetization of the solid, M, will be proportional to the strength of the external magnetic field, H. The proportionality constant is called the magnetic susceptibility. M = χmH.

(5.12)

94

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

The value of the magnetic susceptibility increases as temperature decreases; the proportionality constant is C and the resulting relationship is called Curie’s law. The following equation combines these concepts.

χm =

M C = H T

(5.13)

It should be noted that when the external field is very strong or the temperature is very low, Curie’s law is not valid. Under these conditions, the material reaches saturation and a maximum value for magnetization is notated as Ms. Values for the magnetic susceptibility have been determined under conditions in which Curie’s law holds. It has been found that for diamagnetic materials, χm is on the order of -1 x 10-6 while for paramagnetic materials χm is about +1 x 10-6 (magnetic susceptibility is a unitless quantity). There are some materials, however, that have magnetic susceptibilities that are much larger, on the order of 102. These are the ferromagnetic materials. They not only magnetize to a high degree under an applied external field, they also maintain a portion of the magnetization after the field is removed. What is so unique about these materials? For some transition metals—in particular iron, cobalt, and nickel—the global order of aligned spins is more favored than for other materials. For these materials, the effective magnetic field is enhanced by the magnetization of the material, such that the relationships in equation 5.13 become

χm =

M C = H + λM T − Tc

(5.14)

Notice that as Tc approaches zero, the equation becomes Curie’s law. Equation 5.14 is the Curie-Weiss law, and the Curie temperature, Tc, is a constant for the material. The value of Tc is well above room temperature for ferromagnetic materials: for nickel, Tc = 631 K, for iron, Tc = 1043 K, and for cobalt Tc = 1404 K. When T = Tc, the value of χm is infinite. This means that the magnetization need not go to zero even when the external field (H) goes to zero. When T is less than Tc, the material is ferromagnetic and can spontaneously magnetize. Figure 5-4 shows the theoretical plot of spontaneous magnetization versus the relative temperature of the system. Note that as temperature decreases, the magnitude of spontaneous magnetization increases. When T exceeds Tc, the material will become paramagnetic.

Mechanical and Magnetic Properties of Nanoparticles

95

Figure 5-4. The magnitude of spontaneous magnetization (Msp) that forms in the absence of an external magnetic field depends on the temperature, T. Above the critical temperature, Tc, the material is unmagnetized in the absence of an external field and behaves like a paramagnetic.

A very important question remains unanswered: Why does spontaneous magnetization occur in ferromagnetic materials below their Curie temperature? The short answer is, because of magnetic interactions between electron spin and orbital moments. In ferromagnetic materials, the energy of electrons with parallel magnetic moments is lower than if the moments are anti-parallel. This is because of something known as the exchange interaction, which is of purely quantum mechanical nature. The existence of an exchange interaction means that one kind of spin is favored over the other. The Coulomb interaction is a repulsion or attraction between two charges. The magnitude of the Coulomb interaction can be calculated by integrating the overlap of four orbitals at two neighboring atoms. Exchange interaction is calculated by exchanging two orbitals at two atoms. The value of this integral and the strength of the exchange interaction depends on the interatomic distance, and overlap between electron orbitals. In order for a material to be ferromagnetic, three conditions must be satisfied. 1. The atoms must have a magnetic moment. 2. The exchange interaction must support parallel moment configurations (determined by the sign of the above-mentioned integral). 3. The magnitude of the exchange interaction must be large compared to kBT; that is, large compared to the energy of thermal agitation so that the ferromagnetic alignment can persist. To summarize, in ferromagnetic materials, magnetic moments are parallel to each other when the material is below its Curie temperature because the thermal energy can not compete with the magnetic energy. One reasonable explanation for this is that the coupling interaction is related to the energy difference between the states (∆E), and that for ferromagnetic materials below their Curie temperature, the tendency to align magnetic moments is more important than the disordering effect of thermal agitation. There are also materials in which the exchange interactions favor alternating spins in neighboring atoms. These are known as

96

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

antiferromagnetic materials and include NiO and Fe2O3. External magnetic fields have no effect on antiferromagnetic materials.

5.2.1.2. Magnetic Properties in Bulk Materials On a microscopic level, a ferromagnetic material consists of small regions of parallel or nearly parallel magnetic moments called magnetic domains. Within the same domain, the magnetic moment’s direction is nearly the same, but the spin directions of neighboring domains are not necessarily aligned with each other. When one sums over the magnetic moments of all the magnetic domains, the result is the total magnetic moment of the system. For some materials, despite the existence of domains with magnetic moments, the total magnetic moment of the system is zero. For other materials, the total magnetic moment is not zero. It is the interplay between exchange energy (which favors parallel spin) and the energy required to maintain the external magnetic field (which favors anti-parallel spin) that determines the most stable (lowest energy) domain size. This domain size will be in the range of 1 – 100 μm. For bulk metals, the magnetic moment is not an integral multiple of the Bohr magneton. To understand this, recall that in metals, not all the valence electrons are fully localized near a particular nucleus. For one single iron atom, there are eight valence electrons. The maximum number of unpaired electrons in a ground-state iron atom is four; the two 4s electrons are paired, and when six electrons are distributed among five 3d orbitals, only four of them can remain unpaired for a predicted moment of 4μB. In the bulk iron, due to the exchange interaction, electrons with spin up have different energy than electrons with spin down. The net effect is that there will be more electrons with one spin than the other. If, on average, there are 7.78 electrons in the 3d band of iron, then— on average—five will have spin up and 2.78 will have spin down. The resulting magnetism is the experimentally observed net spin, 5 – 2.78 = 2.22 μB.

5.2.2. The Effect of Cluster Size on Magnetic Properties Nanoparticles are on the order of 100 nm. This is smaller than the average domain size for ferromagnetic materials, which suggests that a single nanoparticle made from a ferromagnetic material will be a single magnetic domain. The energy barrier to magnetization reversal, ∆E, is proportional to 1/kBTB where kB is Boltzmann’s constant and TB is called the blocking temperature. Note that materials with a large TB require very little energy to reverse magnetization. For a ferromagnetic material,

χm =

C T − TB

(5.15)

If T is much, much larger than the blocking temperature, then χm = C/T, which is the behavior observed for paramagnetic materials. If T is less than TB, then the particle is a single domain ferromagnetic material. It turns out that the smaller the particle, the lower the value of TB. So as ferromagnetic materials get smaller and smaller, their behavior goes from being ferromagnetic to what is

Mechanical and Magnetic Properties of Nanoparticles

97

called superparamagetic. A practical definition of superparamagnetism is the state of a ferromagnetic particle that is so small that the energy barrier to magnetization reversal is smaller than the thermal energy that is available to the particle. Imagine applying an external magnetic field to an array of very small ferromagnetic particles. When the field is removed, the magnetization of these very small particles may randomly reverse at room temperature, resulting in a loss of the bulk magnetic property of the material. The ferromagnetic material exhibits paramagnetic behavior. As described in a previous section, the strength of the exchange interaction is the single most important factor in determining the magnetic properties of a material. If there is some way to alter the nature of the exchange interaction, then the magnetic properties of the material could be altered. One way to accomplish this is by doping semiconductors. For example, doping silicon with manganese not only results in an unbalanced spin, but also introduces different exchange interactions that affect the magnetic properties of the semiconductor. Reduced dimensionality does not always favor an increase in the exchange interaction; sometimes it is detrimental to magnetic properties. One famous example is surface magnetism. Experiments suggest that the magnetic moment is zero on the surface of ferromagnetic materials. This is inconsistent with theory that indicates a non-zero magnetic moment on the surface of ferromagnetic materials. Very careful experimental analysis has revealed that the magnetic moment is—in fact—enhanced on the surface. The magnetic moments can be switched from being in the plane of the surface to perpendicular to the surface. This shows great promise for magnetic storage systems.

5.2.3. Magnetoresistance For some materials, electrical resistance increases or decreases when a magnetic field is applied. This phenomenon, known as magnetoresistance (MR), was discovered in 1857 by Lord Kelvin (aka William Thomson). More recently, materials processing has resulted in materials with greatly enhanced magnetoresistance or giant magnetorsistance (GMR), a phenomenon that has been lucrative for reading head, spin valve and sensor technologies. In 1988, the GMR effect was discovered independently by two research groups, one led by Albert Fert of the University of Paris-Sud and the other led by Peter Grünberg of the Jülich Research Center. The materials are Fe/Cr sandwiched or layered structures. Their discoveries opened a completely new field in physics and enginerring, and directly started spintronics investigation. Figure 5-5 schematically shows the trilayer structure. The left and right layers are both ferromagnetic, but their magnetization directions are either in parallel or antiparallel. The middle layer, also called spacer, is nonmagnetic, and must be kept thin—about a few nanometers—because the electron will lose its spin orientation if it travels too far. An external bias or the battery supplies the current to the circuit. GMR works as follows. If the magnetizations of the left and right layers are in parallel or point in the same direction, the electron can move freely through this trilayer structure, and the resistance is very low. On the other hand, if their magnetizations are antiparallel, the electron will experience a huge resistance when it passes through from the left to the right. Therefore, depending on whether the magnetization directions of these two ferromagnetic layers, the resistance can be changed dramatically, about 10% for GMR compared to less than 1% for MR.

98

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

-e

Figure 5-5. Giant magnetoresistance. This device has three layers. The first and third layers are magnetic. The second layer is nonmagnetic, and is normally very thin. The electron flows from the left and first enters the magnetic layer. After it goes through the middle layer or spacer, it enters the last layer. It is this last layer that determines how large the resistance can be.

To understand the underlying physics principles, look at figure 5-6 which shows what happens to an electron in these two situations, namely parallel and anitparallel configurations. Electrons transport in a nonmagnetic material differently from in a magnetic material. In nonmagnetic materials, the electron’s spin orients randomly, and it can point any direction in space. It is a principle of quantum mechanical that in nonmagnetic materials the spin does not have a fixed axis or quantum axis to align. This is why nonmagnetic materials have no net magnetic moment, simply because individual spins of the electron cancel out; this cancellation is at the atomic level. The electron has the same energy regardless of spin orientation. When the electrons enter the ferromagnetic material (first magnetic layer), they feel a difference in energy for different spin orientations. If the electron has a spin oriented along the magnetization direction of the material, its energy is lowered. On the other hand, if they orient themseleve with a finite angle with respect to the magnetization, the energy increases. As a result, more electrons will align their spin along the magnetization direction. In physics, we normally say electrons become spin-polarized. When these spin-polarized electrons enter the nonmagnetic spacer, their spin polarization gradually decreases, but since the spacer is very thin, a majority of the electrons still have a well-defined spin orientation when they enter the last magnetic layer. How easily these electrons can pass through this second magnetic layer depends on whether the magnetization of the first layer is in the same direction as or opposite direction to the spin polarization of the second layer. If in the same direction, the electrons can easily pass through the last layer since the electronic states are available to them. However, if their spin is in the opposite direction, the electrons cannot find a suitable state to occupy, and they experience a strong scattering or resistance. In order to pass, the electrons must change their spin orientation to the magnetization of the material. This results a huge resistance to those electrons. This is the physical reason for the giant magnetoresistance. The difference between these two resistances is a good indicator of the GMR materal’s performance.

Mechanical and Magnetic Properties of Nanoparticles

99

(a)

(b)

Figure 5-6. The physical principles of giant magneto-resistance in trilayer structures. (a) Both ferromagnetic layers have the same magnetization direction. The polarized electron experiences a smaller resistance. (b) The first and last ferromagnetic layers have different magnetization directions. The polarized electron experiences a much stronger resistance.

5.2.4. Applications for Magnetic Recording Media The enhanced magnetoresistance created by carefully assembled nanostructures allows consideration of a myriad of potential applications. Industrial applications include process monitors, magnetic-field testing for machinery and engines, speed sensing for gears, and position sensors for ferromagnetic parts. Automotive applications might include antilock brakes, shock absorbers, ignition timing and control systems. One can imagine sensors used for currency sorting and counting, as well as applications in medical devices. It appears that the most technologically challenging application is computer disk-drive read heads. To explain why this is challenging, it is useful to review/explain how disk drives work. Disk drives have three key components: the magnetic disk medium, the write-head element, and the read-head element. The magnetic disk medium stores the information. A single ‘bit’ of information on a magnetic disk consists of a small magnetized region that produces a minute magnetic field just above that spot on the disk. In binary code, there are exactly two acceptable values for a bit, usually identified as “1” or “0”. One way of thinking about this bit is to imagine that if the magnetic field is pointing up out of the disk, the value is “1” and if the magnetic field is point down into the disk, the value is “0”. Every square inch of a modern disk drive has about 20 billion of these bits. That gives a density of 20 gigabits per square inch (20 Gb/in2). The existence of the phenomenon of superparamagnetism introduces a limit to the standard approach of data storage. When the ‘bits’ become too small, the blocking temperature falls below room temperature. This results in randomization of the direction of the magnetic moment in the bit and consequent loss of information. The theoretical limit imposed by the phenomenon of superparamagnetism is 100 Gb/in2.

100

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

The write-head element of a magnetic disk writes information onto the disk, while the read-head element reads the information. As the size of the bit is reduced to increase data storage density, not only must the read-head shrink, but its sensitivity must also increase, because the smaller bit will have a weaker magnetic field. Moreover, the head must also respond to the field faster, because a smaller bit on a rotating disk spends less time under the read head. There is an additional consideration about shrinking bit size with respect to how the readheads sense information. When evaluating the effectiveness of a read-head, one considers not just the magnitude of the signal, but the magnitude of the signal relative to the noise detected. Typically, MR sensors that use magnetized materials have a significant amount of magnetic noise; that is to say, because the magnetism is generated by magnetic atoms, the atomic magnetic fields will be roughly but not perfectly aligned. If the volume of the material is large, the fluctuations average out to be negligible. However, if the volume decreases, the proportion of noise increases, introducing inaccuracies. On the other hand, if the read heads take advantage of the EMR effect described above, they can be made with nonmagnetic nanoparticles, thus eliminating the magnetic noise limitations.

5.2.5. Applications for Biotechnology The magnetic properties of nanoparticles are used to facilitate the separation of bioparticles. Nanoparticles of Fe3O4, which is ferromagnetic in the bulk, are attracted to a magnetic field, but retain no residual magnetism after the field is removed; they are superparamagnetic. The nanoparticles are tagged with an antibody that is specific to an antigen on the surface of the cell to be separated from the other cells in the system. The nanoparticles bind to the surface of the desired cell, and these cells can then be collected in a magnetic field. Superparamagnetic Fe3O4 nanoparticles are also useful as magnetic resonance imaging (MRI) contrast agents. MRI, which is proton NMR done on tissue, provides images of the tissue that correspond to proton density; high proton density results in a bright image. Contrast agents work by changing the strength of the MRI signal at a desired location; regions containing the superparamagnetic contrast agent appear darker in an MRI than regions without the agent. When superparamagnetic nanoparticles are delivered to the liver, healthy liver cells can uptake the particles while diseased cells cannot. Consequently, the healthy regions are darker while the diseased regions remain bright. This chapter has introduced and described some of the basic language of mechanical and magnetic properties and enlarged on those that are most significant to nanotechnology. A limited number of applications to nanosystems have been described. Because this field is growing so rapidly, those described here may be old news. Yet understanding how these applications work may facilitate understanding of even the most recent developments. And that, after all, is the goal of this book.

Mechanical and Magnetic Properties of Nanoparticles

101

EXERCISES Exercise 5-1: For calculations on double-walled carbon nanotubes, Puech et al. used a wall thickness of 0.34 nm. Using this value, determine the stiffness of a nanotube with an exterior diameter of a) 4 nm and b) 10 nm. Exercise 5-2: For some applications, nanotubes must be stiff. For other applications, flexibility is important. Using calculations like those in exercise 5-1, explain why it is important to be able to control the size of nanotubes. Exercise 5-3: The text describes a memory chip design that uses carbon nanotubes. Referring to Chapter 4, identify at least two important features of the structure of nanotubes used for this application. Exercise 5-4: A recording material has an areal density of 3 Gb/in2. Assume the magnetic bits are uniformly distributed and estimate the size of a single magnetic bit. Report your answer in nm2.

Chapter 6

OPTICAL PROPERTIES Optics at the nanoscale is attracting a growing interest because of its exciting applications in information technology, photochemistry, biology, medicine, and other areas. The general press has reported promising applications for metal nanoparticles in biological systems that take advantage of their unique optical properties. In addition, the recent creation of nanocomposites, which provide magnetization at optical frequencies and a negative refractive index, promises revolutionary changes in optics and its applications. This chapter summarizes the basic principles of optical physics as applied to optical processes at the nanoscale. Phenomena related to surface effects are presented along with their elevated importance for very small particles. The specific features of both near-field optics and size-effects are outlined. Special emphasis is placed on the effect of electron confinement, demonstrating—once again—the value and importance of being able to control particle size at the nanometer level. In addition, the analysis of nanoscale clusters as fractal aggregates is presented. All in all, the focus is on basic principles of the breakthroughs that nanoscience makes possible, for example the idea of ‘left-handed’ nanomaterials, currently a subject of competitive research, is introduced. Some of the concepts presented in this chapter are better understood in the context of a more detailed mathematical development. Unfortunately, it is easy to lose track of the big picture when focusing on these details. For this reason, topics that benefit from the mathematical details are first presented as a bigger picture. The mathematical details follow in sections entitled “Mathematical development of...” In all cases, the mathematical development can be omitted without loss of general understanding.

6.1. PROPAGATION OF ELECTROMAGNETIC WAVES: BASIC RELATIONSHIPS WITH MATERIAL PROPERTIES Light is electromagnetic radiation, i.e., it consists of electromagnetic waves. Oscillating electric, E, and magnetic, H (in free space) or B (in substances), fields compose an electromagnetic wave. In nonmagnetic materials, H and B are equal. The simplest example is an electromagnetic light wave in free space, shown in figure 6-1. The electric, E, and magnetic, H, components of an electromagnetic wave are orthogonal to each other and periodic in space and in time. Their magnitude and direction oscillate in time with frequency,

104

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

ν. In free space and natural materials, electromagnetic waves propagate in the same direction as would an ordinary (right) screw rotated from E toward B.

Figure 6-1. An electromagnetic wave showing the perpendicular electric (E) and magnetic (B) fields.

Figure 6-1(a) depicts electric and magnetic fields on a plane perpendicular to the direction of propagation at two different moments in time, but the same location. Figure 61(b) depicts a light wave frozen in time. It shows how these fields vary along the propagation direction at a given instant. The phase of a wave is a particular stage in the periodic phenomenon. It determines the value of the oscillating quantity at a specified point in time or space. Phase is represented by the fraction of a complete cycle elapsed as measured from a specified reference point and often expressed as an angle. The angle can be conveniently expressed as a multiple of π. At one moment in time, the quantity oscillates in space; at one point in space, the quantity oscillates with time. A spatial period of oscillations along the propagation direction, λ, is referred to as wavelength. A temporal period, T, and the oscillation frequency, ν, are related as ν = 1/T. The electric and magnetic components of the wave complete each oscillation cycle in time and space through the corresponding periods T and λ. The reader may recognize the notation ‘T’ as having been used to denote temperature in Chapter 2 and the chiral vector in Chapter 4. One of the challenges of learning a field as interdisciplinary as nanotechnology is learning to speak the languages of the separate disciplines that comprise the new field. In the language of optics, ‘T’ means temporal period. Every attempt is made to make the meaning of ‘T’ obvious from the context. For light, the oscillation frequency is on the order of ν ≈ 1014 … 1015 s-1 (oscillations per second). Consequently, the oscillation period is extremely short and falls into the femtosecond range (1 fs = 10-15 s). This opens avenues for many applications, including ultrafast information technologies. Angular frequency, ω = 2πν, is also often used in calculations. The periodicity in space of an electromagnetic field at any given instant of time is caused by its finite propagation velocity (phase velocity) and by the related phase difference along the propagation direction. Optical radiation (a.k.a., light) corresponds to the interval of wavelengths that stretches approximately between 100,000 nm and 100 nm. Recall from Chapter 2 that radio waves have wavelengths of about 1 meter, while visible light has wavelengths in the hundreds of nanometers: the wavelength is about 700 nm for dark red, 400 nm for violet, and 500 nm for green light. In the long wavelength, low (radio) frequency range, where ν ≈ 109 Hz, a direct

Optical Properties

105

measurement of the strength of electric and magnetic fields and of their frequencies can be performed. In the visual optical range, where ν ≈ 1015 Hz, the oscillations are too fast. Therefore, the energy flow and the radiation wavelength become the primary measured values. Orientation of the electric component of an electromagnetic wave is referred to as the polarization of light. It is linearly polarized if the tip of the vector E moves along a line. If the tip of the vector moves along a circle or ellipse (clockwise or counter clockwise ), the polarization is circular or elliptic. Linearly polarized light can be converted into circular or elliptically polarized radiation and vice versa. A surface, each point of which corresponds to the fields at the same phase of the oscillation, is referred to as wave front. The vector orthogonal to the wave front at a given point in space represents the propagation direction (i.e., light rays in the corresponding space areas). Figure 6-1 depicts linearly polarized light with a plane wave front that corresponds to parallel rays along a z-axis. Optical instruments, such as lenses and curve mirrors, convert a plane wave front into a spherical one that corresponds to converging (focusing) or divergent (defocusing) light rays. Coherence and interference (defined in section 2.1.1.) are among the most fundamental concepts of physics. Interference may lead to counterintuitive processes. Two oscillations of the same frequency and same phase may enhance each other, which is an example of constructive interference. Alternatively, oscillations with opposite phases may totally suppress each other, which is an example of destructive interference. Two oscillations are called coherent if they maintain the same phase difference over the entire observation time. A stable interference indicates coherent oscillations. A standing wave created by the interference of two counter-propagating waves, y1 and y2, with equal frequencies, amplitudes and phases is the simplest example of interference (figure 6-2). The maxima correspond to constructive and the minima to destructive interference. Hence, instead of twofold illumination, completely dark spots form at the expense of a fourfold increase of the illumination (twofold increase of the amplitudes of oscillations) in other spots. This counter-intuitive result is due to the fact that unlike incoherent processes, the amplitudes of oscillations combine in coherent processes.

Figure 6-2. Representation of a standing wave made by the combination of two waves defined by the functions y1 and y2 where y1 = a sin (kz - ωt), y2 = a sin (kz+ωt), y = y1+y2 = (2a sin kz) cos (ωt), and k = 2π/λ.

106

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

The classical way to produce coherent waves is to split a light beam into two parts by the slits in a screen (see figure 6-3). Constructive or destructive interference occurs at different points on the second screen, depending on the difference of the paths traveled by the waves emitted by two different slits. As a result, multiple maxima and minima appear at the second screen. They are modulated by the envelope (dashed line) that is determined by the diffraction. Indeed, diffraction is a source of divergent beams behind the slits, which gives rise to the interference patterns.

Figure 6-3. Interference of two light waves.

6.1.1. The Effect of Matter on Properties of Light Waves Oscillating electric fields create electric and magnetic fields in the neighboring points of space. Likewise, oscillating magnetic fields create electric fields. These phenomena cause v propagation of the fields with the velocity v in the direction along the movement of a righthanded screw, which is rotated from E to H (figure 6-1). The speed v indicates the magnitude of the velocity without consideration for the direction of propagation. The term phase velocity refers to how fast a defined phase of the electric and magnetic fields propagates in space. In free space, v = c = 2.99x108 m/sec, which is a fundamental constant known as the speed of light. In materials, the propagation speed is usually less than the speed of light by a factor of n, v = c/n

(6.1)

The factor n is known as the refractive index, which is frequency dependent, i.e., n = n(ν). In dense transparent materials, it may have values of about n = 2, whereas in low density gases it is close to the vacuum value of n = 1. Recall that wavelength is the distance that light travels by the end of the oscillation period. It is related to the propagation velocity by the temporal period, T, according to the equation λ = vT = cT/n. For a given frequency of light, the wavelength varies in different materials and is different from that in vacuum. The refractive index, then, is a measure of the effect of a given material on the speed of electromagnetic radiation.

Optical Properties

107

The squared strength of the electric or magnetic fields, E2 or B2, averaged over a period much longer than the temporal period, T, is proportional to the density of the light energy flow, I, which is usually along the direction of propagation in isotropic media. Light propagating in a substance may experience depletion due to interaction with and consequent transfer of its energy to the substance. The dependence of such depletion on the material length z is usually exponential (figure 6-4),

I ( z ) = I 0 exp[−α (ω ) z ]

(6.2)

where I0 is the density of energy flux (often referred to as the light intensity) at the medium entrance. The factor α(ω) is called the absorption index, which is frequency-dependent. At z = za = 1/α, referred to as absorption length, the light intensity decreases by the factor I0 /I = e = 2.72. The greater the absorption index, the steeper the depletion of light as a function of distance z will be. The absorption index is a measure of the effect of material on the intensity of the electromagnetic radiation.

Figure 6-4. Exponential depletion of the light power flux density in absorbing media.

The magnitude and frequency dependence (spectral properties) of the refractive and absorption indices are stipulated by the interaction of light and matter. The greater the change of the refractive index at the interface between two media, the higher is the reflectivity of such an interface and the greater the refraction (bending) of light rays that advance into the second medium at an angle different from 90°. We will return to this concept in Section 6.3.

108

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

6.1.2. Mathematical Development of the Effect of Matter on Waves The equations below show how the described concepts can be treated mathematically. This material can be omitted without loss of general understanding. The simplest harmonic electromagnetic wave traveling in a vacuum is given by the equation

E ( z , t ) = E 0 cos[ Φ ( z , t )] = E 0 cos( ω t − k 0 z ) = E 0 cos[ ω ( t −

z )] c

(6.3)

Here, E(z,t) represents the electric field of the light wave, ω is the angular frequency, k0 = 2π/λ0 is the propagation constant or wave vector, and λ0 is the wavelength in a vacuum. The phase Φ does not depend on x and y, which indicates a plane wave with the wave front orthogonal to the z-axis. However, the oscillation phase varies along the z – axis, along which the wave travels. At some point in space and time, the phase will be zero, i.e., there exist values of z and t for which Φ(z,t) = 0. The instantaneous field value corresponding to this phase travels in free space with the speed c ≡ z/t = ω/k0

(6.4)

This is why such a speed is called the phase velocity. The field does not change when z changes by ± j λ0 , and the time changes by ± j 2π/ω (where j is any integer number). Hence, λ = 2 π / k 0 = 2 π c / ω = cT .

(6.5)

In a given medium, an electromagnetic wave may experience depletion and travel with a different phase velocity: E = E 0 exp[ − α ( ω ) z / 2 ] cos( ω t − kz ).

(6.6)

Here, α(ω) is the absorption index, which represents an exponential depletion of the radiation power flux density I ~ E2 along the medium due to its conversion to heat. A change of the phase velocity in a medium is accounted for by the following set of equations k = 2π/λ λ = vT = cT/n(ω) k = n(ω )ω / c

ω t − kz = ω ( t −

z ) c /n

(6.7)

where n(ω) is the refractive index, which determines the phase velocity, v = c/n, and wavelength, λ = λ0/n, of the electromagnetic wave in a medium.

Optical Properties

109

6.2. A CLASSICAL MODEL OF THE INTERACTION OF LIGHT AND MATTER Materials are made of bound positive and negative charges. Most optical properties of materials originate from the interaction of the electric field component of the light with the outermost or valence electrons in the material. While many basic properties of this interaction can be understood using the simple classical model of forced electron oscillations, the explanation of some important optical features requires a quantum approach. The following subsections summarize classical models of the interaction of light with matter and use them to describe specific features of optics at the nanoscale. The quantum approach will be presented in section 6.4.

6.2.1. Resonance Frequency The electric field that surrounds a charge follows it when the charge moves with a constant velocity. Acceleration or deceleration of a charge causes emission of electromagnetic (EM) radiation; a charge oscillating about an equilibrium position periodically accelerates and decelerates. A confined particle is a particle held in place by mutual forces of interaction, which oppose any disturbance from an equilibrium position. An electron confined in an atom, molecule or in some other nanostructure begins oscillating about a fixed equilibrium position if it is slightly displaced by a disturbance. The frequency of such free oscillations, ω0, called the natural or resonance frequency, is determined by the electric potential that confines the electron within its area of localization. The resonance frequency is determined by the dependence of the restoring force on the displacement. The steeper such dependence, the larger is the oscillation frequency. For the sake of simplicity, it is assumed that the confinement (restoring) force, F(x), is proportional to the displacement, x, from the equilibrium position, as shown in figure 6-5.

F ( x ) = − Kx

(6.8)

Here, K is the force constant, similar to the elastic constant of a spring. In the given simplified model, the force constant represents the confinement of the electron and determines the resonance frequency,

ω0 ≈ K m

(6.9)

where m is the electron mass. Thus, the stronger the confinement force at a given displacement, the larger is the natural oscillation frequency, ω0. The resonance (natural) frequency of a nanostructure depends on its size in a similar way.

110

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Figure 6-5. Mechanical oscillator.

When light advances into a material, its alternating electric field forces the confined electrons to oscillate with the frequency of the applied electric field ω. However, the phase of the forced oscillation is not necessarily equal to the phase of the applied field. Both phase delay and advance may occur. The amplitude and phase of the forced oscillation depend on the difference between the frequency of the applied field and the natural (resonance) frequency. The amplitude of the forced oscillations reaches a maximum when ω = ω0 , which depicts resonance enhancement of optical processes. The greater the amplitude of the forced oscillations, the greater is the dissipation of the oscillation energy due to the interaction of the oscillator with the environment and conversion of that energy to heat. If you’ve ever managed to light a fire by rapidly rubbing a stick within a groove of wood, you have experienced this phenomenon first hand. The energy dissipated by an oscillating electron is taken from the light that induces the oscillations. This explains the resonance behavior of the absorption index. Amplitude decreases with the growth of resonance detuning, ω − ω 0 , symmetrically in lower- and higher-frequency wings of the resonance peak [figure 6-7(a)]. On the other hand, phase delay or advance of the forced oscillation depends on whether the applied frequency is less or greater than the resonance frequency. Radiation at a frequency less than the resonance value induces phase-delayed oscillation [figure 6-7(b)]. Such delay causes reduced phase velocity for light propagating through a dense medium. If the applied frequency is greater than the resonance frequency, the result is a phase-advanced oscillation.

6.2.2. Damping and Dispersion When the applied force is turned off, the amplitude of forced oscillation dampens as exp(-γt) (see fig. 6-6). An oscillating electron emits radiation at the oscillation frequency. Consequently, the oscillator energy depletes and the oscillation dampens. As pointed out above, the interaction with surrounding particles leads to conversion of the oscillation energy

Optical Properties

111

to heat and may substantially contribute to damping. The damping rate γ, measures how fast the oscillator loses its energy due to transfer to the environment. It also determines how sharply the refractive and absorption indices depend on the frequency in the vicinity of the resonance. Since the dissipation of oscillation energy leads to a decrease in the oscillation amplitude, the faster the damping, the less the resonance enhancement of the optical response of the material. A typical dependence of the absorption and refractive indices on frequency in the vicinity of a resonance is given in figure 6-7.

Figure 6-6. Damping oscillation.

Figure 6-7 (Continued on next page).

112

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Figure 6-7. Schematic representation of (a) absorption index and (b) refractive index. The dashed curves correspond to a twofold decrease and the dashed-dotted curves to a twofold increase in the relaxation (damping) rate, γ.

ω

As shown in figure 6-7(a), the absorption index decreases by a factor of two at = ω0 ± γ / 2 , compared to its resonance value at ω0. Consequently, γ is referred to as the

resonance full-width at half maximum (FWHM) and γ/2 as half-width at half-maximum (HWHM). Figure 6-7(b) depicts the frequency dependence of the refractive index. It shows that the phase speed of light is less than c at the low-frequency wing of the resonance and exceeds c at the higher-frequency wing of the resonance. Light experiences only absorption and no change of the phase speed at resonance. Notice that the refractive index grows with the frequency increase except for a narrow interval in the neighborhood of the resonance frequency. This dependence is called normal dispersion. The decrease of the refractive index with frequency in the vicinity of resonance is called anomalous dispersion. The visible frequency range is below the resonance frequency in most transparent materials. Consequently, the dispersion is normal and phase velocity is slower than in vacuum, and the refractive index is larger than unity. The resonance properties of light-matter interaction have a wide range of applications of great importance. For example, absorption spectroscopy, described in Chapter 2, is often used to identify and characterize substances. Since the resonance frequency is determined by the electron confinement and the resonance width by the environment, information about the structure and size of the emitting or absorbing objects (e.g., nanostructures) and about properties of the environment can be derived from a measurement of the frequency (spectral) dependence of the absorption and refractive indices. The corresponding investigative method is called spectroscopy. Alternatively, unusual optical properties of the materials can be realized and manipulated through the design of artificial nanostructures and the variation of their sizes and shapes. From this presentation, it can be seen that even a simplified model of forced oscillations of the confined electron is sufficient to explain the basic principles underlying the optical properties of nanostructured materials.

Optical Properties

113

6.2.3. Mathematical Development of Concepts: Damping and Dispersion As follows from the equations of classical mechanics, the displacement of an electron of mass m that oscillates about its equilibrium position is given by the equation

x(t ) = x0′ exp( −γt ) sin(ω ′t + ϕ ′) where the oscillation amplitude, x0′ , and phase,

(6.10)

ϕ ′ , are determined by the initial

displacement, and γ is the damping rate. The oscillation frequency, ω', is defined as

ω ′ = ω 0 2 − γ 2 , ω0 = K m

(6.11)

where ω0 is the natural frequency. When the damping rate γ is much less than the natural frequency (i.e. γ, << ω0), ω' is fully determined by the confinement force:

ω ′ ≈ ω0

(6.12)

When light advances into a material, its alternating electric field

E (t ) = E0 cos ωt

(6.13)

forces the confined electrons to oscillate with the frequency of the electric field. If the electron charge is e, the electric field force on an electron is FE = eE(t). It is assumed that E(t) is aligned along the x-axis. Then the electron displacement is expressed in the form

x ( t ) = x 0′ exp( − γ t ) sin( ω ′t + ϕ ′) + x 0 cos( ω t − ϕ )

(6.14)

where the first term represents the damping free oscillation, and the second term gives the forced oscillation. The free oscillation disappears very quickly. For optical oscillations, the corresponding lifetime, τ0 = 1/γ , is on the order of 10-8 s. The quantities x0 and φ are found from the equations of classical mechanics and are frequency dependent as described above. For dilute materials like gases, where the interactions between oscillators can be neglected, and in the vicinity of a resonance (i.e., at | ω − ω0 |<< ω + ω0 ≈ 2ω0), the frequency dependence of the absorption index can be expressed as α (ω ) = α

γ /2 (ω0 − ω ) 2 + γ 2 / 4

(6.15)

where α = Ne 2 / 2mε 0c is a constant dependent on the number density of oscillators N, and ε0 is a fundamental electrodynamics constant called the permittivity of free space. As shown in figure 6.7(a), the absorption index decreases by a factor of two at ω = ω0 ± γ / 2 as compared

114

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

to its resonance value at ω0. This also determines important optical properties of materials. Due to the frequency-dependent phase shift of the induced oscillations, the secondary reemitted electromagnetic waves are also phase shifted with respect to the primary waves. As a result, the propagation speed of light is different in a material as compared to free space. The refractive index also changes most significantly in the vicinity of the absorption resonance, where its frequency dependence for dilute materials is given by the equation

n(ω ) = 1 + n

ω0 − ω (ω0 − ω ) 2 + γ 2 / 4

(6.16)

where n = cα / 2ω0 = Ne2 / 4ω0ε 0m is a constant dependent on the material. Figure 6.7(b) depicts the frequency dependence of the refractive index. Thus, optical processes experience enhancement in the vicinity of the resonance frequency. The smaller the damping rate γ, the larger and sharper the resonance enhancement will be.

6.2.4. Nonlinear Spectroscopy As outlined in the previous sections, the position of a resonance, its width and shape provide information about the structure and physical processes in a substance. Additional information can be gleaned from the polarization of the emitted light. An enhanced, forced oscillation can be excited not only by one resonant field, but also by a combination of several fields whose frequency-combination coincides with one of many resonances inherent to the confined electrons. The amplitude of such oscillations appears proportional to a product of amplitudes of the contributing fields or to higher powers of amplitude of one of the fields. The corresponding forced oscillations are referred to as nonlinear oscillations and the associated optical processes as nonlinear-optical processes. Nonlinear optics is a branch of optics that studies nonlinear optical processes. The investigation method based on resonance nonlinear-optical processes is referred to as nonlinear spectroscopy. Nonlinear spectroscopy is an extremely powerful extension of linear spectroscopy.

6.2.5. Optics of Dense Dielectrics For dilute materials—like a rarefied gas—where the interaction between the nearest particles is negligible, equations (6.15) and (6.16) provide a good approximation for the absorption and refractive indices in the vicinity of a resonance. However, in dense dielectric materials, the assumptions made in deriving these equations do not hold. Because the electric field of oscillating dipoles contributes to the electric field that forces oscillations of neighboring dipoles, the resonance frequency of the dense medium, ωod, is red-shifted proportionally to the number density of oscillators N

Optical Properties

ω 02d = ω 02 −

e2 N 3mε o

115

(6.17)

where e is the charge on an electron, m is its mass and εo is the permittivity of free space. For the frequency ranges |ω – ω0d| >> γ/2 the refractive index can be expressed by the dispersions equation

n 2 (ω ) = 1 +

Ne 2

mε o ω −ω Ne ≈ 1+ 2 2 2 2 2 2 mε 0 (ω 0 d − ω ) + γ ω ω0d − ω 2 2

2 0d

2

(6.18)

According to equation (6.18), for dense materials n2 may exceed unity at ω < ω0d and become negative at ω > ω0d. In fact, the squared value of the refractive index for solids may become negative within a certain frequency interval. A negative magnitude of n2 (imaginary magnitude of n) means that light cannot penetrate a material at a depth substantially exceeding its wavelength. Consequently, the medium becomes totally reflective for light in the corresponding frequency interval for light striking the material interface at any angle of incidence. Figure 6-8 illustrates this dependence for the frequency range where absorption can still be neglected. A medium becomes totally reflective within the frequency interval ω r > ω >> ω0 d + γ / 2 . Here, γ/2 is the half-width of the resonance at half-maximum (HWHM).

Figure 6-8. Schematic dependence of the squared value of the refractive index n2 for dense dielectric materials in the frequency range of small absorption.

6.2.6. Optics of Metals The characteristic feature of a bulk-like metal is the presence of unbound electrons. According to Eqs. (6.8) and (6.9), coupling with unbound electrons corresponds to a

116

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

vanishing resonance frequency. If we set ω0 = 0 and neglect damping by taking γ = 0, equation (6.18) reads as

⎛ω ⎞ n = 1 − ⎜⎜ p ⎟⎟ , ⎝ω ⎠ 2

2

(6.19)

where we have defined the plasma frequency as

ωp =

Ne 2 mε 0

(6.20)

Plasma frequency is a parameter that plays an important role in many areas of physics, not only in optics of metals (see, e.g., Eqs. (6.15)-(6.18)). In metals, the plasma frequency usually falls in the ultraviolet wavelength range. Equation (6.19) shows that when ω > ωp the refractive index is less than unity, which means that the phase velocity is greater than that in free space. Alternatively, when the optical frequency ω is below the critical value ωp, the squared magnitude of the refractive index becomes negative. This indicates strong reflection of light by the metal, which leads to a very steep attenuation of light as it advances in the metal even without any dissipation caused by its conversion to heat:

E = E 0 exp( −

z ) cos(ωt ) 2zs

(6.21)

where

z s = c /( 2 − n 2 ω ) = λ0 /( 4π − n 2 )

(6.22)

As light progresses into a conductor, its flux density (which is proportional to |E|2) drops by the factor 1/e ≈ 1/2.7 after it has propagated a distance zs, which is called the skin layer or penetration depth. For a material to be transparent, its thickness must be small in comparison to the penetration depth.

6.2.7. The Effect of Particle Size on the Optics of Metals The penetration depth for a typical metal is usually at the nanometer scale. For example, for copper at ultraviolet wavelengths (λ0 ≈ 100 nm), zs is only 0.6 nm, while in the infrared (λ0 ≈ 10,000 nm) the value of zs is about 6 nm. These numbers are within the size scale of nanoparticles. Hence, unlike bulk metal samples, nanostructures can be nearly transparent for optical electromagnetic radiation. Therefore, strong nonlinear-optical processes can develop in the skin layer near a metal surface. Even more unusual optical processes may develop at a

Optical Properties

117

surface covered by a mixture of metal and dielectric islands of a nanoscale size. Such artificial materials (metamaterials) are referred to as metal-dielectric nanocomposites. When electrons become confined in small metal particles or in thin metal layers, the confinement force, F, must be accounted for by Eqs. (6.8), (6.9), and (6.17). Therefore, one can readily expect the appearance of a geometrical resonance, called the plasmonic resonance, which is specific for very small metal particles. The plasmonic resonance depends on the shape, size and material of the particles; it does not exist in bulk metal samples. The characteristic mean-square velocity of electrons in the solid state (Fermi velocity) is about vF = 108 cm/s. Hence, the oscillation period for an electron confined within a metal ball of d = 10 nm diameter is about d/ vF = 10-14 s, which roughly corresponds to the optical frequency domain. This represents another example of a typical size effect in the nanoworld. Corresponding rapidly-developing fields of research and technology—plasmonic nanomaterials, plasmonic nanooptics and plasmonic nanophotonics currently attract much attention.

6.3. BREAKING DIFFRACTION LIMITS Light tends to bend behind any impediment, a phenomenon known as diffraction. The smaller the light beam cross-section, e.g., due to focusing, the stronger is the opposing process of diffraction. Diffraction sets the limit (diffraction limit) for the minimum achievable size of a spot of light to have a diameter at about half the wavelength of the light. Thus, the fundamental laws of physics seem to limit the best spatial resolution achievable with coherent light to values of approximately several hundred nanometers. In this section, we discuss how nanoscience enables breaking such seemingly fundamental limits, thus opening new avenues for numerous exciting applications.

6.3.1. Electric Dipole Radiation: Near-Field and Far-Field Radiation Zones A pair of equal and opposite electric charges, whose centers do not coincide, form an electric dipole. An atom in which the center of the negative cloud of electrons is shifted slightly away from the nucleus by an external electric field also forms an electric dipole. The dipole moment is a vector directed from the positive to the negative charge and is given by d = ex (t ) , where e is the charge on an electron and x is the separation between the centers of two charges. Notice that the equation allows x to vary with time. The simplest source of an electromagnetic wave is an oscillating electric dipole, where the electron vibrates with respect to the positive charge, back and forth along a straight line. One can consider oscillations of a confined electron as oscillations of an electric dipole. Focusing on this source of electromagnetic waves sets the stage for an amazing breakthrough in spatial resolution. A positive charge repels while a negative one attracts another positive charge along the Coulomb’s force lines depicted in figure 6-9(a, b). Drawing the force line is a way to picture the effects that electric charges have on one another. Instead of talking about the force a positive (+) charge exerts on an electron, we can say the charge creates a force "field" in the empty space around it. An electron put down at any place in this force field is pulled toward

118

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

the + charge; a positive charge set down at the same place is pushed away. Coulomb’s force lines depict the paths followed by an electric charge free to move in an electric field. When positive and negative charges are separated by a small distance, electric force lines stretch as shown in figure 6-9(c). The number of the force lines that run through a unit surface is a measure of the strength of the electric field. The strength of the electric field is greatest in the vicinity of a charge. As the distance between the charges and their relative position varies, the spatial distribution of the electric field varies too.

Figure 6-9. Coulomb’s force lines for a positive and a negative charge and for an electrical dipole.

Figure 6-10. Electric field (force lines) around an oscillating electric dipole.

Optical Properties

119

Figure 6-10 shows the distribution of the electric field for an oscillating dipole at several sequential instants. The charges move toward each other in figure 6-10(a), converge in figure 6-10(b), change their positions, and begin to move in opposite directions in figure 6-10(c). Very near the dipole, in the near-field zone, the electric field lines retain a localized form characteristic of a static dipole; farther out, they form closed loops. These lines describe the field that travels out; however, there is no specific wavelength, λ that characterizes the periodicity in space of this field. At further distances (figure 6-10(c)), in the far-field zone or radiation zone, a fixed λ is established, and the radiation wave front of the alternating electric field approaches a plane shape with polarization along the orientation of the emitting dipole (figure 6-10(d)). Figure 6-11 schematically depicts the distribution of an oscillating electric field in the region of an electric dipole. The lines correspond to equal electric field strengths. Greater density corresponds to the strength maxima with the alternate direction of the field. The magnified portion of the figure depicts the force lines in greater detail.

Figure 6-11. Near and far-field zones.

From these figures one can see that in the near-field zone (i.e., at distances r << λ), the radiation field possesses unusual properties. It is confined around the dipole in a very small size volume that is much less than the radiation wavelength. The smaller the dipole, the smaller the volume with unusual properties. In contrast, conventional optics, i.e., the optics of microscopic and macroscopic objects, usually deals with radiation in a far-field zone; this radiation is composed of traveling electromagnetic waves. Indeed, the diffraction limits are applied to the far-field zone. Thus, figure 6-11 shows that in the near-field zone around nanoparticles, the optical field can be localized in a space with nanoscale dimensions, much smaller than the radiation wavelength. This gives rise to a breakthrough in physics and technology because it allows extremely high spatial resolution and concentration of optical radiation below the wavelength

120

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

(diffraction) limit. A clear understanding of one powerful application of this great resolution requires knowing what an evanescent field is. That is the topic of the next section.

6.3.2. Evanescent Waves A clear understanding of possible options to achieve a resolution better than the diffraction limit requires knowing what an evanescent field is. Even when light is totally reflected, the electric field penetrates the material to some extent. The part of the field that penetrates is called an evanescent wave. The field strength of an evanescent wave decreases sharply (exponentially) with respect to depth in the material. In such cases, light is predominantly concentrated within a layer; the thickness of this layer may become less than the wavelength of the light that exists within it. Light is totally reflected by a medium when n2 is less than one. Light may also experience a total reflectivity from the media with positive values of n2 if the refractive index of the second medium is less than that of the first one. The value of the refractive index in a particular medium depends on the radiation wavelengths as schematically shown in figure 68. Very close to resonance, at ω < ω 0 d + γ / 2 , absorption becomes too strong, and figure 68 ceases to be valid. In most transparent crystals, visible light falls in the frequency range ω < ω0 d , and the refractive index values lie in the interval between 1.5 (glass) and 2, reaching 2.4 in diamond. The following section discusses the details of how the evanescent waves can be created with such materials.

6.3.3. Mathematical Development of Evanescent Waves Light is totally reflected by a medium when n2 is less than one. Light may also experience a total reflectivity from the media with positive values of n2 if the refractive index of the second medium is less than that of the first one. When a light beam strikes the interface between two dielectrics with refractive indices n1 and n2, it splits into a reflected beam in the n1 medium and a beam refracted in the n2 medium. The angle of incidence, θ1, and the angle of refraction, θ2, with respect to the normal to the interface are related according the Snell’s law:

n1 sin θ1 = n2 sin θ 2

(6.23)

If n1 < n2, the angle of refraction is less than the angle of incidence ((figure 6-12(a)). Alternatively, if n1 > n2 (or for the ray propagating in the opposite direction), the angle of refraction is greater than the angle of incidence. In this case, at

θ1 = θ c = sin −1 ( n2 / n1 )

(6.24)

Optical Properties

121

the refracted beam becomes directed along the interface (θ2 ≈ 90°). At θ1 > θc, sin θ2 would become greater than unity, which is not possible. This represents the phenomenon known as total internal reflection. For n1 ≈ 1.5 (glass) and n2 ≈ 1 (air), θc ≈ 48.3° (or 0.73 radians).

a.

b.

Figure 6-12. Reflection and refraction of light at the interface of two media with different refractive indices.

Assume that the interface aligns along the x-axis and the normal to the interface along the z-axis. Then it can be shown that for an angle of incidence greater than the critical angle θc, (i.e., n1sinθ1 > n2) the electric field component in the second medium is given by the equation

E 2 = E 20 cos(ωt −

ω c

n1 sin θ1 x ) exp[ −

ω c

z n1 sin 2 θ1 − n2 ] 2

2

(6.25)

which represents a wave propagating in the +x-direction with its amplitude decaying exponentially along the z-direction. A wave front corresponding to the phase

Φ = ωt −

ω

n1 sin θ1 x = 0 propagates along the x-direction as x = ct/ n1sinθ1. Such a sharply c decaying wave that exists only in a thin layer near the medium interface is known as an evanescent wave (see figure 6-13). Its amplitude decreases by a factor e and the density of power flow by e2 at a distance from the interface equal to

z e = c /(ω n1 sin 2 θ1 − n2 ) = λ0 /( 2π n12 sin 2 θ1 − n22 ) 2

2

(6.26)

122

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

This distance may become less than half the radiation wavelength in a vacuum, λ0/2, which falls into the tens to hundred nanometer scale.

Figure 6-13. Evanescent wave. For θ1 > θc, the field in the second medium is an evanescent wave (shown by the arrows), which propagates along the x-direction and decays exponentially along the z-direction.

6.3.4. Near-Field Optical Nanoscopy and Nanolithography: Overcoming Diffraction Limits Microscopy, in its various forms, plays a most important role in the analysis of small objects. The popularity of the method stems from several advantages: it is relatively inexpensive, easy to use, reliable, and requires little or no sample preparation. The disadvantage imposed by the fundamental phenomenon of light diffraction is a lack of resolution, which is restricted to half the wavelength of the light that is used. The same limitation is applied to the inverse process of light focusing, which is used in lithography. A breakthrough to overcome the diffraction limits of optical microscopy and lithography has become possible through the application of the principles of near-field optics and the invention of near-field scanning optical microscopy (NSOM). NSOM is a technique that allows one to achieve resolution to an order of magnitude below normal optical microscopy by overcoming the diffraction limit. It retains the advantages of optical microscopy while simultaneously eliminating its major disadvantage. The resolution of this method is determined by the size of the aperture used to collect the light. The nanometer-scale light source is often obtained by the propagation of conventional laser light through an aluminumcoated fiber tip sharpened at the end (see figure 6-14). The tip is also used for detection of near-field light, while the sample is illuminated by another source of evanescent light.

Optical Properties

123

Figure 6-14. A fiber tip with nanoscale aperture. (from http://ceas.eth.ch/zenobi/projects/SNOM/tips/fibertips.html)

Figures 6-15 and 6-16 depict a typical NSOM setup, which rely on the local detection of the optical near-field intensity of a sample by a sharpened optical fiber tip. The illumination of the sample is accomplished by a laser-induced evanescent field. The evanescent field appears above and close to the prism surface as the result of the total internal reflection of the laser beam from the prism interface with air. The prism and laser create a wide spot with a very thin layer of evanescent field. The sample to be examined is placed on this surface and is therefore immersed in the very thin layer of evanescent light. This determines the resolution along the z-axis.

Figure 6-15. Scheme of near-field optical microscopy. (from http://nanooptics.unigraz.at/ol/work/pstm/pstom.html)

124

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

There are materials that change their shape in a well-defined manner that is proportional to an applied voltage; this phenomenon is called the piezo effect. The prism is moved in very small increments by taking advantage of the piezo effect. The piezo tube on which the prism rests therefore brings different illuminated zones under the tip, which picks up light from very small subwavelength areas. This results in amazing resolution in the xy plane. The light signal picked up by the fiber tip is detected with a photomultiplier tube (PMT). The feedback controlling the fiber tip position relative to the sample surface (via the exponentially decaying evanescent light field) is activated only to decrease the distance from the fiber tip to the sample surface between two consecutive constant height scans.

Figure 6-16. Typical view of the near-field optical microscope setup.

Implementation of nanometer-scale apertures that are smaller than the wavelength of the illuminating light provides for sub-wavelength spatial resolution that supplements frequencyselective characterization and modification methods. One application that takes advantage of sub-wavelength spatial resolution is nanolithography, which is widely used in modern electronics. The price for such high resolution is that the character of the light changes dramatically when it propagates through the aperture. The localization of the light waves is due to the formation of evanescent waves, which is discussed in sections 6.3.2 and 6.3.3. The intensity of an evanescent wave decays rapidly as the distance from the aperture increases. Here, it is sufficient to note that the intensity of an evanescent wave decays rapidly as the distance from the aperture increases. Therefore, the aperture has to be close to the object, often only a fraction of a wavelength away, which is the regime of near-field optics. The occurrence of resolution below diffraction limits allows one to refer to the above-described near-field technique as nanoscopy rather than microscopy.

Optical Properties

125

6.3.5. Nanoantennas The NSOM techniques have many applications in solid state physics, where substantial efforts are made to design electronic devices with features on the nanometer scale. One possible avenue is the creation of optical nanoantennas. These can be small metal particles, or sharp metal structures at surfaces (metal-dielectric nanocomposite), or sharpened optical fibers (fiber tips) with a diameter of a few nanometers. Under appropriate conditions, the irradiance, I, is proportional to the time-averaged squared acceleration (deceleration) of an electron. In this case, ‘appropriate conditions’ means that emission occurs radially outward from the dipole under the angle θ. The total power W is radiated over a sphere of radius r in the far-field zone. For the case under consideration, these relationships can be expressed in the form: 2

2 d 02ω 4 sin 2 θ 2e 2 d 0 ω 4 sin 2 θ 2 I (θ ) = a(t ) = 2 = ,W = 2 (6.27) 32π 2c 3ε 0 r 2 3c ε 0 3c ε 0 16π 2 c 4ε 0 r 2

e 2 a(t )

2

Here, a( t ) is squared acceleration averaged over time, d0 is the amplitude of the dipole oscillations, c is the speed of light in a vacuum, and ε0 is the permittivity of free space. Light-induced oscillations re-radiate or (i.e., scatter) electromagnetic waves of the same frequency as that of the incident wave. Along with the amplitude of induced oscillations, d0, this process experiences great enhancement as the light frequency approaches the resonance frequency. The above equation can be considered in terms of antenna theory, which usually implies a short dipole case, x<<λ, where x is the dipole length. For optical radiation, where λ is several hundred nanometers, this dipole represents a nanoantenna. If one neglects ohmic losses and assumes that the transmitter supplies the antenna with enough power to compensate for the losses to the radiation field so that there is no damping, then the term eω in Eq. (6.27) represents a current, i0, and the equation can be written as 2 i R 4π 2 2 ⎛ x0 ⎞ W= i0 ⎜ ⎟ = 0 eff , 3cε 0 ⎝ λ ⎠ 2 2

(6.28)

where

8π 2 ⎛ x0 ⎞ 2⎛ x ⎞ = ⎜ ⎟ = 80π ⎜ 0 ⎟ [ohms ] . 3cε 0 ⎝ λ ⎠ ⎝λ⎠ 2

Reff

2

(6.29)

Here, Reff represents an effective “radiation resistance” for an antenna of length x0, i.e., the resistance, which with a current i0 presents the same load as the surrounding electromagnetic field. The numeric value in the right side of Eq. (6.29) is given in ohms. It is

126

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

seen that the emitted power decreases as the square of the ratio of the dipole size to the radiation wavelength. The amplitude of the electron oscillations, x0, is usually much smaller than the size of a nanostructure. That means that for nanoparticles, the resistance to radiation is very small. This shows the importance of the resonance enhancement of induced oscillations (x0) and, consequently, optical processes at the nanoscale.

6.4. QUANTUM PHYSICS AND THE OPTICS OF NANOSTRUCTURES Many optical phenomena can be understood with the aid of classical models, as described above. However, some important features of physical processes in the nanoworld require a quantum approach. In quantum physics, the state of an electron is described by a wave function that varies in time and space. The squared modulus of such a function represents the probability of finding an electron in a specified energy state at a given instant in a specific space area. As outlined above, acceleration or deceleration of electrons causes emission and absorption of light. Hence, the forces that return a charge to the equilibrium position in the material, and therefore cause a change of their velocity, have a key impact on the optical properties of materials. The simplest example is a confining potential in the form of a box, which implies that the returning force becomes infinitely large very sharply at a certain distance from the equilibrium position. This means that the electron can not reach the wall, and, consequently, the wave function must be a standing wave having zero values at the box walls. Therefore, only a discrete set of states, referred to as quantum states, is allowed for a confined electron, so that the spatial period of the corresponding wave function must be an integer part of the box length. Different states usually correspond to different energies. Hence, the allowed energies of a confined charge (its energy spectrum) are discrete as well. The energy gap between two adjacent states is

ΔE j = E j +1 − E j =

(2 j + 1)h 2 . 8ma 2

(6.30)

Here m is mass of the electron, a is the size of the confinement zone, and h is a fundamental constant called Plank’s constant (see section 2.1.4). This shows that the most important physical features, such as allowed and forbidden energies, depend on the size of the object that is related to a. In the real word, spatial dependence of the confining potential is more complex than rectangular. However, even this simplified model enables an understanding of important basic features regarding to nanotechnology. Light is emitted and absorbed by electrons confined in atoms. Since their energy may change only in a discrete way, light also consists of discrete portions called quanta. According to quantum physics, the classical oscillations considered in the preceding subsections are stipulated by the change in the electron energy, i.e., by the quantum transitions. The corresponding oscillation frequencies, ωij, are defined by the equation

Optical Properties

hω jl = ( E j − E l )

127 (6.31)

where ћωjl is the energy of an emitted (absorbed) quantum. For small displacements of an electron in atoms and molecules, the potential is of a parabolic shape (i.e., it is proportional to the square of the displacement). Consequently, the restoring force is proportional to the displacement, which is in agreement with the classical approach to optical properties of materials based on the model of forced oscillations. For two adjacent states, j = 1 and l = 0, the resonance wavelength, λ, associated with a quantum transition (with oscillations of the confined electron) can be estimated as

λ = 8mca 2 / h ≈ 400nm −1 a 2

(6.32)

The units of λ are nm when a is also given in nm. The characteristic size of atoms and molecules is about 1 nm, which determines the resonance frequency for such electrons to fall into the optical range. Taking into account the simplicity of both models, this is also in reasonable agreement with the estimate for the plasmonic resonance frequency given in section 6.2.6. Various nanostructures, such as metal-dielectric nanocomposites and semiconductor nanostructures made in the shapes of a sphere, a wire, a ring, etc., (respectfully referred to as ‘quantum dots’, ‘quantum wires’, ‘quantum rings’, etc.) attract much attention from the photonics community. With the size on the order of just a few nanometers, they confine electrons, and are characterized by a quantum wave function, which resembles that for an electron confined in an atom or molecule. In many respects, such nanostructures behave as artificial atoms and molecules.

6.5. NANOSTRUCTURED FRACTAL METAL AGGREGATES The optical properties of an aggregate composed of more than several hundred nanoparticles do not, for the most part, change with further growth of the aggregate. These aggregates usually possesses fractal properties. Fractals are objects that are characterized by two important properties. They possess a scale invariant symmetry, i.e., each of their sections has the same structure as the whole object; often these are the rarefied and branching structures. Their second important property is that a number of elementary chunks inside a circle or sphere grow as a non-integer power of the radius. This non-integer number is smaller than the usual integer number attributed to the corresponding one-, two-, or three-dimensional object. For example, the number of particles inside a circle of radius R grows as N ~ R2 for a common two-dimensional object. For a two-dimensional fractal, it grows as N ~ RD, where D is called the fractal dimension. It is a non-integer number less than 2. The fractal dimension is an important parameter that determines the distribution of distances between the chunks inside the fractal. Fractal structures are often found in nature, with examples in figure 6-17.

128

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

a.

b.

c. Figure 6-17. Fractals in nature:(a) Romanesco broccoli (from http://www.fourmilab.ch/images/Romanesco/), (b) Mountains (from http://classes.yale.edu/fractals/Panorama/Nature/MountainsReal/Mountains5.gif), (c) High voltage dielectric breakdown within a block of plexiglas (from http://en.wikipedia.org/wiki/Image: Square1.jpg).

Optical Properties

129

A three-dimensional fractal aggregate of metal nanoparticles is statistically scale invariant. Instead of R3, a number of nanoparticles inside the sphere of radius R varies approximately as RD, independent of the radius. Here, D is a non-integer number, usually between 1 and 2 (often it is equal to 1.78). A typical image of such an aggregate obtained with electron microscope is depicted in figure 6-18. Fractals are referred sometimes to as objects with null density. For example, the density of particles N ~ RD inside the volume V ~ R3 is N/V ~ RD-3. It tends to zero with an increase of the radius, because D < 3. Hence, fractal nanostructures possess strikingly different properties from those of both gases and solids.

Figure 6-18. Fractal aggregate of silver nanoparticles. The diameter of each particle is between several nanometers and several tens of nanometers. (From A. K. Popov, et al., Nanotechnology 17 1901-1905 (2006))

Consider their optical properties. Near-field radiation produced by the nanoparticles may interfere constructively in certain areas of the aggregates. The amplitude of electromagnetic radiation emitted by an antenna in the direction of constructive interference grows as the number of emitting dipoles N, which compose the antenna. Consequently, the intensity of the radiation emitted in the direction of the interference maximum grows as N 2. An aggregate of metal nanoparticles works as a nanoantenna. A multi-order enhancement of the intensity of the local electromagnetic field of about 105 times may occur in small “hot” zones sized about ten nanometers (figure 6-19). Such hot zones occur for a wide variety of different wavelengths in different sections of a fractal. A corresponding high concentration of energy of electromagnetic radiation becomes possible due to breaking the diffraction limit by means of near-field optics. There are examples given below of very important applications made possible by these findings.

130

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Figure 6-19. Zones of giant local enhancement of fluctuating optical fields by an aggregate of metal nanoparticles. (from A.K. Sarychev and V. M. Shalaev, Physics Reports, 335, 275 (2000).)

As outlined in Section 6.2.4, an electromagnetic field may excite nonlinear forced oscillation of an electron confined in a molecule. The amplitude of an oscillation generated in this manner is proportional to more than one order of amplitude of the applied field(s). Corresponding optical processes, referred to as nonlinear-optical processes, are commonly used for material characterization, optical sensing and laser control in physics, chemistry and biology. Nonlinear-optical processes in molecules attached to the metal nanostructures experience even greater gain in the locally enhanced electromagnetic fields compared to the linear optical processes. For example, for nonlinear optical process referred to as four-wave mixing (FWM), an enhancement of the generated field may exceed 1019 (figure 6-20, parameter gFWM). Such immense enhancement offers opportunities for detecting single molecules, and promises numerous breakthroughs in optics, photochemistry, photobiology, and in various optics-based technologies. An example of an application in photonics is presented in figures 6-21 and 6-22.

Figure 6-20. Giant enhancement of nonlinear optical processes at the nanoscale. (from A.K. Sarychev and V. M. Shalaev, Physics Reports, 335, 275 (2000).)

Optical Properties

Figure 6-21. Enhancement of photoluminescence by R6G molecules after metal aggregates are added in water solution. From Kim et. al. (1999).

Figure 6-22 (Continued)

131

132

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Figure 6-22. Enhancement of photoluminescence of R6G molecules inside a glass microcavity (left) and microlasing (right). From Kim et. al. (1999).

The word ‘laser’ is an acronym for light amplification by the stimulated emission of radiation. The word lasing is often used to express the occurrence of stimulated emission. Molecules of dyes are often used as the lasing medium excited by an external light source. This enables conversion of readily available light at fixed wavelength into frequency-tunable laser radiation. Figure 6.21 shows enhancement of photoluminescence of dye molecules, Rhodamine 6G (R6G), in a water solution after silver fractal aggregates are added. The enhancement is due to molecules adsorbed by such aggregates. This enables the creation of a microlaser. The enhancement is so strong that lasing becomes possible even in microscopic volumes inside the microcavity. At that, the excitation power occurs as low as 5 x 10 -4 W, where an ordinary low-power He-Ne laser is used for excitation (figure 6.22). A quartz capillary with inner diameter of several parts of a mm was used as the laser cavity (resonator). In conclusion, fractal aggregates composed of hundreds of metal nanoparticles enable a giant concentration of optical radiation in near-field zones with a size of about ten nanometers, which is below the diffraction limit. This occurs due to constructive interference of near fields generated by the nanoparticles inside the aggregate. The corresponding multiorder enhancement of various optical processes and laser effects on atoms and molecules offers numerous breakthroughs in photonics, spectroscopy, sensing of single molecules, photochemistry, and photobilogy. The sizes and properties of such nanostructures can be manipulated through photophysical and chemical processes.

Optical Properties

133

6.6. OPTICS AT THE NANOSCALE: APPLICATIONS AND CHALLENGES This section supplements the summary presentation of specific applications that arise because of the special effects of nanosize on the optical properties of materials. While this list is not exhaustive, it is a good start.

6.6.1. Dots, Wires, Rings, and Artificial Solid-State Atoms and Molecules In marked contrast to bulk semiconductor systems, the electrons in nanostructures are confined in all three dimensions. Therefore, they may exist only in certain quantized states that can be manipulated by changing the size, shape, composition and position of such structures, as well as the number of active confined electrons. Consider a multi-electron quantum dot that is electrically neutral due to the positively-charged nuclei of the crystalline lattice. When one of the electrons is optically excited to the upper energy state, the positivelycharged “hole” appears in the ground state of the dot. Such a coupled electron-hole pair is equivalent to a quasiparticle known as an exciton, which spreads through a dot. Recombination of an electron and hole leads to the release of the stored excitation energy in the form of a photon (i.e., a quantum of light). This process can be externally controlled. It is even possible to build up quantum-dot analogues of the periodic table of atoms. Besides electrical charge, which reveals itself quantum-mechanically through confinement, electrons possess magnetic properties that are characterized by spin. An electron’s spin behaves like a small magnet. However, being quantized, it can exist only in two discrete orientation states—either ‘up’ or ‘down’. This gives rise to a pure quantummechanical interaction of electrons known as the exchange interaction. Hence, quantum states and optical processes associated with quantum dots depend also on the relative spins of electrons. Electronic excitation in quantum dots may exist for a relatively long time when they interact in a restricted way with their environment. As a result, electrons may exist in a simultaneous coherent superposition of several states (wave functions), which are called entangled states. An entangled state is presented by the superposition of the corresponding wave functions. Constructive and destructive interference, associated with the entangled states and with several simultaneous quantum transitions, exhibits itself through unusual counterintuitive optical effects. This leads to advanced technologies based on quantum coherence and interference. Among the prospective revolutionary applications of entangled states is quantum computing, which promises a fundamental breakthrough in computational productivity. As outlined in the section 6.3.3., external space- and state-selective control of quantum wave functions can be performed in a semiconductor nanostructure with coherent laser near-field radiation. This also applies to related quantum interference processes. Quantum semiconductor nanostructures, such as dots, wires, rings, etc, can be combined into one-, two- and three-dimensional complexes that are similar to molecules built from atoms. The interaction between such coupled artificial solid-state atoms results in diverse novel properties. Thus, a new avenue is open for designing and constructing a great variety of artificial solid-state quantum-mechanically engineered nanomaterials with novel electrical and optical properties. This enables new avenues for creating various optoelectronic

134

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

semiconductor devices at the nanoscale, such as lasers and nonlinear optical elements, optical cavities and integrated optical elements, electronic and magnetic switches and gates, interconnected by quantum wires.

6.6.2. “Left-Handed” Metamaterials: A Revolution in Optics In free space and in known ordinary materials, the dielectric permittivity, ε, magnetic permeability, μ, and consequently refractive index, n, are positive quantities. As a result, the electric field E vector, magnetic field B vector, and k wave vector comprise a right-handed system as shown in figure 6-23(a). This means that electromagnetic waves propagate in the same direction of k as would an ordinary (right) screw rotated from E toward B. Energy flow and phase velocity appear co-directed with k. All the basic laws of electrodynamics of continuous media and, hence, optics are derived on the assumption that ε and μ may not become negative together, and n is positive. Magnetization at optical frequencies was supposed impossible and, hence, μ was conventionally substituted by 1 in all corresponding equations.

Figure 6-23. Orientation of the electric and magnetic components of an EM wave and its propagation direction in right-handed (a) and left-handed (b) materials.

In 1967, the Russian physicist Victor Veselago published several papers where he considered the properties of electromagnetic waves in what was at that time a hypothetic medium where both ε and μ become negative. The refractive index in such a medium becomes negative too. It turned out that this medium would possess extremely unusual properties that assumed revolutionary changes of the most basic concepts of optics and, therefore, would offer exciting applications. First, since the wave vector k is proportional to the refractive index, the orientation of the B, E, and k vectors in a medium with negative refraction index should correspond to a left screw seen in figure 6-23(b). Therefore, such materials should be referred to as left-handed materials (LHM). The phase velocity and flow of energy in such media become oppositely directed. Instead of pushing (light pressure), light in such a medium would pull a surface. The Doppler effect and Vavilov-Cherenkov effect would become the inverse of the usual. Practically all laws of geometrical, physical and nonlinear optics require revision for LHM. The papers by V. Veselago did not attract much attention until early 2000, when the rapid development of nanotechnology made possible the creation of artificial materials with a

Optical Properties

135

negative refraction index. One of the possible realizations of such a metamaterial for the optical range theoretically proposed in 2000-2003 is a set of pairs of silver nanowires with diameter of about ten nm and length of about half the designated optical wavelength. Such pairs of nanowires should be immersed in a dielectric, so that the electric component of an electromagnetic wave is directed along the wires (figure 6-24). This type of nanocomposite enables magnetization at optical frequencies, which was thought to be impossible and was always neglected in all equations of physical optics.

Figure 6-24. “Left-handed” metamaterial with ε <0, μ < 0 and n < 0.

Theoretical and experimental studies on left-handed optics are underway. The first experimental realization of a negative refractive index in the optical wavelength range occurs in nanocomposites in the near infrared wavelength range, which is optimum for optical telecommunications (see the articles by Shalaev et. al. (2005), Zhang et. al. (2005), and Shalaev (2007)). One simple but striking example of the impact of nanotechnology through the creation of LHM concerns the fundamental Snell’s law that relates the angle of incidence, θ1, and angle of refraction, θ2, with the change in the refraction index, n, at the interface of two media: n1 sin θ1 = n2 sin θ2 (figure 6-25). If the index n2 changes its sign, the quantity sin θ2 must have changed its sign too. Hence, a light beam must become bent on the opposite side compared to that in an ordinary RHM (figure 6-25(b)). Then a plate made of LHM should work as an ideal focusing lens opposing diffraction (figure 6-26).

136

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Figure 6-25. Refraction of a light beam in (a) ordinary (right handed) and (b) left-handed media.

Figure 6-26. Propagation of a divergent light beam through a plate made of RHM (black arrows) and LHM (gray arrows).

Further information about the experimental realization of a superlens can be found in the article by Fang et. al. (2005).

Optical Properties

137

6.6.3. Giant Enhancements of Optical Processes at the Nanoscale with the Aid of Aggregates of Metal Nanoparticles Unlike in bulk metals, the properties of electrons in small metal particles are strongly affected by the surface that imposes their confinement in a small volume. The electric field of the light wave that would produce an electron current in a bulk sample shifts electrons of a nanoparticle from their equilibrium position (figure 6-27). A spherical surface causes a confinement potential, and the charge separation leads to the appearance of a restoring force. Hence, the behavior of electrons in the ball resembles a classical oscillating dipole. Its resonance frequency depends on the shape and size of the particle.

Figure 6-27. Dipole-type distribution of charges in a metal nanosphere.

For a silver ball with a diameter of about ten nanometers, the natural resonance wavelength corresponding to such a geometrical resonance (plasmonic resonance) is about 400 nm. The width of the resonance is several tens of nanometers. Two closely-spaced balls present a new structure, which possesses new resonance properties due to the interaction between two dipoles. The new resonance properties depend on the distance between the particles (figure 6-28) and on the orientation of such a pair relative the polarization of the probe light.

138

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Figure 6-28. Plasmonic resonance for isolated metal nanosphere (0), absorption spectrum for two closelyspaced particle (1), and modification of the absorption index with an increase of the interparticle distance (from 1 to 10). From Karpov et. al. (2002).

Various distances and orientations can be realized in aggregates composed of several hundreds of silver nanoparticles. This leads to modification of the shape and broadening of the absorption resonance for such aggregates compared with the resonance of the isolated particles.

a. Figure 6-29 (Continued on next page).

Optical Properties

139

b. Figure 6-29. Aggregation stages (a) and corresponding modification of the absorption spectrum (b). From Karpov et. al. (2002).

Figure 6-29(b) depicts such a broadened resonance. Unlike a sample composed of isolated nanoparticles, it stretches over almost the entire visible range. Such broadening is referred to as inhomogeneous broadening, because pairs with different interparticle distances are responsible for the absorption at different frequencies. The long-wavelength tail of this resonance corresponds to the most closely-spaced and strongly-interacting nanoparticles. It is obvious that the shape of the absorption resonance, and consequently optical properties of such aggregates, depend on the specific distribution of nanoparticles over the distances between them. A water solution of silver nanoparticles called a hydrosol or colloid can be easily produced, e.g., through a chemical reduction of the silver-containing salts. Owing to Brownian motion, such particles can approach each other and slowly build up the aggregates. This is possible due to the very short-ranged attractive force (Van-der-Waals’s force) existing between the neutral particles. However, in real situations, e.g., in colloids, such nanoparticles usually become coated with other ions or polar molecules, which results in an electrical double layer and a repulsive forces between the particles. This may lead to stabilization, and cessation of the aggregation. Light at the appropriate wavelength may produce the photoeffect by taking out electrons from the nanospheres at the expense of photon energy. This breaks a counterbalance of the attractive and repulsive forces and thus may trigger up to 108 times faster photoinduced aggregation compared with other methods of synthesis. An increased aggregation rate determines different properties of the aggregates. Consequently, manipulating optical properties of such aggregates of metal nanoparticles (nanoengineering) can be controlled chemically, through the electrolytic properties of hydrosol, with polymers wrapping the nanoparticles and with light at different wavelengths

140

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

and intensities. Modification of the absorption spectrum, caused by the aggregation, leads to a change of the colloid color (figure 6-30). This gives us the long-awaited explanation for the color change in gold colloidal systems first noted by Michael Faraday in 1856.

Figure 6-30. Characteristic color changes after irradiation of silver hydrosol by light. (A) Original sample. (B) Sample after exposure to UV light. (C) Sample after exposure to an Argon-Ion laser (experiments on photoinduced aggregation of silver nanoparticles carried out by UWSP undergraduate students).

It’s taken a long time; the development began in 1856 and continues today. Many theories, concepts, and ideas have been introduced in this book. Many areas of science, mathematics and engineering had to develop to the right point and come together in the right way to permit the establishment of nanotechnology as a field in its own right. There is still a long way to go, but we’ll get there. Considering the amazing possibilities and realities you’ve already learned, we can be confident that it’s worth the time and effort to continue the trip.

EXERCISES Exercise 6-1: Describe an electromagnetic (EM) wave and explain why it propagates in free space. Define wave, wavefront, rays, refraction and absorption indices, and a plain EM wave. Exercise 6-2: Give the definition of absorption and refractive indices. Explain why absorption and refractive indices for materials made of atoms, molecules, and nanostructures possess resonance properties. Exercise 6-3: Explain how confinement determines the oscillatory motion of a bound charge. Assuming that the return force tends to infinity while the charge displaced from the equilibrium position approaches the confinement boundaries, discuss how the resonance frequency depends on the dimensions of the confinement zone and on the mass of the confined charge.

Optical Properties

141

Exercise 6-4: The potential energy of electron confined in atom, U, and restoring force, F, are approximated as U= Kr2/2; F= -Kr, where r is displacement from the equilibrium position and K is a spring constant. At the displacement r = L =5 Å, which is about the size of the atom (ten Bohr’s radii), U = 5 eV. Estimate the electron oscillation resonance frequency ν0 =ω0/2π and corresponding wavelength of emitted radiation, based on the model of a harmonic oscillator (ω02 = K/me). To what part of the visible spectrum does this correspond? The conversion factors are me = 9.1x10-31 kg, 1 eV = 1.6x10-19 J, 1nm = 10 Å. Exercise 6-5: What is required in order to force a charge to emit electromagnetic radiation? Explain why an oscillating charge emits electromagnetic radiation. Exercise 6-6: What is the resonance width? How does it depend on the rate of loss of the oscillation energy? Indicate several most typical processes that cause depletion of the oscillation energy. Exercise 6-7: What are normal and anomalous dispersions? Exercise 6-8: With the aid of the model of harmonic oscillation, explain why atoms, molecules and nanostructures absorb light and cause a difference between the light velocity in free space and in substances. Exercise 6-9: Estimate the critical number density of free electrons that corresponds to total reflection of light at λ = 500 nm by a plasma, which is a model for bulk metal. How many electrons per cube of (1 nm)3 does this makes up? Use the atomic constants e= 1.6x10-19 C, me = 9.1x10-31 kg, ε0 = 8.85x10-12 F/m. Exercise 6-10: Based on the model of forced oscillations, explain why the optical properties of bulk metals and metal nanostructures are so different. How are the skin-effect, plasmonic resonance, and near-field optics relate to optical processes at the nanoscale, and how they can be applied to nanotechnology? Exercise 6-11: What are electric force lines? What is electric dipole radiation? What are nearfield and far-field radiation zones? What are the main differences in radiation properties in these zones? Exercise 6-12: What is the diffraction limit in optics? How does contemporary nanotechnology enable the breaking of this limit? Discuss the importance and several applications of such a technique. Exercise 6-13: Explain the basic principles of breaking the diffraction limit and of giant local enhancement of optical processes by metal nanoaggregates with the aid of the concepts of interference and near-field dipole radiation. Exercise 6-14: What are evanescent waves? Describe their properties. How can an evanescent wave be created in a transparent medium? Exercise 6-15: Describe how total reflection can be realized. How can the phenomenon of total inner reflection be applied in nanotechnology? Exercise 6-16: Estimate the minimum penetration depth for an evanescent wave formed with the aid of thallium bromoiodide crystal at λ = 600 nm (refractive index n ≈ 2.6) and with the aid of KBr at λ = 300 nm (n ≈ 1.8). Assume the penetration depth as a depth where light power flow density decreases in e2 ≈ 7 times. Exercise 6-17: Describe the principles of quantum nanoengineering associated with semiconductor crystalline nanostructures in terms of the wave properties of the confined electrons, which are represented by the wave function.

142

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Exercise 6-18: Draw a picture showing a light ray in the second medium if it strikes the medium interface under the angle of 45 degrees, where the refractive index of the first medium is negative, and that of the second medium is (a) positive, (b) negative with less absolute value, and (c) negative with greater absolute value. Exercise 6-19: In real, highly-conductive metals, the refractive index is determined by their conductivity and depends on the specific metal and radiation wavelength. Estimate the penetration depth (thickness of the skin layer) for green light at λ ≈ 500 nm striking a surface of Al, Cu, and Ag. The corresponding values for (-n2)1/2 are 4.8, 2.37, and 2.72. Exercise 6-20: With the aid of the potential box and quantum approach, explain why the electron confinement determines optical properties of materials and how the resonance frequency depends on the size of the confinement zone.

FURTHER READING Useful explanations of various physical concepts can be found at the Web site http://science.howstuffworks.com/ (go to "physical science" and then click "more") and http://en.wikipedia.org/wiki/Portal:Science (type any key word at the Search window at the left side of the webpage). For information on fractals, see http://en.wikipedia.org/wiki/Fractal and http://local.wasp.uwa.edu.au/~pbourke/fractals/selfsimilar/ For infomation on silver nanoparticals and their aggregates see http://arxiv.org/abs/physics/0301081; http://arxiv.org/abs/physics/0511147; http://arxiv.org/abs/physics/0510252. For information on left-handed metamaterials see http://arxiv.org/abs/physics/0504091; http://arxiv.org/abs/physics/0504208; http://arxiv.org/abs/physics/0601055; Nature Photonics, 1: 41-47 (2007). For information on hyperlens see http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-188247

APPENDIX: ACTIVITIES FOR NANOTECHNOLOGY If a picture is worth a thousand words, then an experience is worth a million. With that thought in mind, we present to you the following activities. The primary motivation in all cases is to enhance your understanding of the material in the book. Most of the background for each activity is in the body of the book. The chapters enhanced by each activity are identified in the contents list presented below. Name of Activity

Related chapter(s)

page

1. Getting a grip on matters of scale

1

143

2. From meters to nanometers

1

193

3. Atomic force microscopy

2

146

4. Spontaneous assembly of soda straws

3

147

5. Ferrofluid synthesis and analysis

3,5

148

6. Silver nanodisk growth

3,6

149

7. Building a mental picture of the carbon nanotube

4

150

8. Electrochromic Prussian blue thin films

3

152

ACTIVITY 1. GETTING A GRIP ON MATTERS OF SCALE Materials Marked 100 yard football field Ten (10) markers. Orange cones, footballs, basically anything that is clearly visible from 100 yards and won’t blow away in the wind.

144

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Procedure Part A. Letting each marker represent one planet, try to arrange the markers so that their location relative to the end zone is proportional to their respective distances from the Sun. One end zone represents the sun, the other represents Pluto. Use the data in the table below to help you. Notice that the distance of Pluto to the sun is approximately two orders of magnitude larger than all the planets inside the asteroid belt. Notice, too, that there are two orders of magnitude difference between the distance from the sun to Mercury and the distance from the sun to Pluto.

Concepts Explained For this part of this activity, the 100 yard football field represents 39.5 AU (about 3.7 billion miles). Jupiter is the largest planet. The diameter of Jupiter is 88,836 miles. If 100 yards represents 4 billion miles, then—on this scale—the diameter of Jupiter is between 1/16th and 1/8th of an inch. (0.081 inches). Go ahead, convince yourself; do the math. Part B. Letting each marker represent one planet, try to arrange the markers so that their location relative to the end zone represents the diameters of the planets. Jupiter—as the largest planet—will have its cone on the opposite end zone. Use the values in the table below to help you. Planet

Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto

Distance from the sun Actual distance (AU)1,2 0.39 0.72 1 1.5 5.2 9.5 19.2 30.1 39.5

Relative distance (yards) 0.99 1.82 2.53 3.80 13.16 24.05 48.61 76.20 100

Diameter Actual diameter (miles) 3032 7519 7926 4194 88736 74978 32193 30775 1423

Relative diameter (yards) 3.42 8.47 8.93 4.73 100 84.5 36.3 34.7 1.6

1

One AU (astronomical unit) (93 million miles) is equal to the average distance of the Earth from the Sun. 2 The distances recorded here are average distances; at any given time the planets may be slightly closer or farther away from the sun because the planets follow elliptical orbits.

Concepts Explained Notice that the radius of Earth is about one order of magnitude smaller than that of Jupiter. Part C. Now let’s start getting to a smaller scale.

Appendix: Activities for Nanotechnology

145

1. Using the value for the diameter of the earth as the full length of the football field, where would you put a marker to represent the width of North America? [The distance from Sacramento, CA to Washington, D.C. is about 3000 miles.] 2. Now, let the full length of the football field be representative of the width of North America; where would you put a marker to represent 100 miles? 3. If the full length of the football field is representative of one (1) mile, then one meter would extend two inches from the end zone. Now let the length of the football field represent the width of a child’s finger (1 cm). Set up markers to help you visualize the items in the table below. Notice that in decreasing the size from the width of a child’s finger to a red blood cell, we have decreased by a little more than three orders of magnitude. Object

Width μm

mm blood vessel human hair red blood cell

1 0.1 0.006

1000 100 6

Part D. If the full length of a football field is representative of one millimeter, which item on the table above is representative of one nanometer? In other words, which of the items in the table above is six orders of magnitude smaller than a football field?

ACTIVITY 2. FROM METERS TO MILLIMETERS Materials 1 Square meter of paper (or another material of choice) Scissors Meter Stick (ruler)

Procedure Start by getting one of the square meters of paper. Alternatively, draw a square meter on the board. Using the meter stick, cut a square of paper that is 10 cm on a side. Then cut another square that is 1 cm on a side. Finally, cut a square that is 1 mm on a side.

Concepts Explained Each successively smaller square is smaller by one order of magnitude. Odds are that no one can cut the square into anything smaller than a square millimeter. On the order of millimeters, the square has been reduced by three orders of magnitude. Three more are

146

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

required to achieve the square nanometer. This should help the students get a better grip on how small a nanometer actually is.

ACTIVITY 3. ATOMIC FORCE MICROSCOPY Materials Required Laser pointer A small mirror or piece of reflective metal, about the size of a pen Lego® wheel and axle combination plus one or two small Lego® bricks A small piece of cardboard or paper with a grid marked on its surface. (Grid lines should not be closer than 1 cm) Small objects to attach to the grid.

Procedure Using glue or double-stick tape, attach several small objects to the surface marked with a grid. Attach the Lego® wheel to one end of the piece of metal, creating a sort of wheelbarrow (hereafter referred to as the ‘lever’). The area directly above the wheel should be reflective. Attach the end of the lever furthest from the wheel to a support rod (ring stands work well) so that the lever is angled as shown in the picture below. This attachment should allow the lever to flex up and down. Position the laser pointer and a dark piece of paper (detector) so that the laser light reflects from the reflective surface of the lever onto the detector.

Move the grid under the wheel and note how the reflected light moves up and down on the detector. If done correctly, all changes in elevation of the surface will cause the laser to

Appendix: Activities for Nanotechnology

147

either rise or fall. With a partner, come up with a way to “map” the surface under the lever. When you’ve finished, compare the drawing to the actual surface. If you have several teams, you may want to have a contest to see which team’s design is best for mapping the various surfaces.

Concepts Explained Because nanoparticles are so very small, observation can be extremely hard. As discussed in Chapter 2, atomic force microscopy is one method of observing nanoparticles.

The Challenge What could you do to your design to turn it into something that could push items across the surface? This would be equivalent to the nanomanipulation described in the text.

Reference There is a similar activity presented at http://www.cns.cornell.edu/cipt/labs/lab-index.html

the

following

web

site.

ACTIVITY 4. SPONTANEOUS ASSEMBLY O SODA STRAWS Materials 500 mL glass beaker (or any convenient container of similar volume) 1 straw Scissors Water Liquid soap

Procedure Cut the straw into segments of approximately 2 cm each. Make sure that the cuts are perpendicular to the length of the straw. Fill the container about halfway with water. Drop all the straw segments into the water, and make sure to submerge them all. They will float back to the top. Take note of the arrangement of the straws after gentle agitation. Add some soap to the water, mix it up, and allow the straws to assemble themselves again. Take note of the how the straws arrange with the soap in the water.

148

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Concepts Explained The straws are driven to arrange themselves lengthwise against each other and end to end because the hydrophobic surfaces of the straws make them more attractive to each other than to the water in which they are floating. These hydrophobic interactions cause the straws to undergo spontaneous assembly. The spontaneous assembly maximizes the surface area of the straws that is interacting with other straws and minimizes the surface are of the straws that is interacting with the water. The addition of soap provides a coating for the straws that makes their surface less hydrophobic. Hence the self-assembly is affected.

Reference D. J. Campbell, E. R. Freidinger, J. M. Hastings, and M. K. Querns, "Spontaneous Assembly of Soda Straws" J. Chem. Ed. 79, 201-202 (2002).

ACTIVITY 5. FERROFLUID SYNTHESIS AND ANALYSIS Materials Required Cow Magnet Strong Magnet (neodymium-iron-boron) Centrifuge 2.0 M FeCl2 in 2 M HCl 1.0 M FeCl3 in 2 M HCl 0.7 M NH3 25% aqueous (CH3)4NOH solution

Procedure Combine 1.0mL of the FeCl2 solution and 4.0mL of the FeCl3 solution. Add a magnetic stirring bar and begin vigorous stirring. Add 50mL of 0.7 M NH3. A slow rate of addition is critical to the reaction occurring properly. Magnetite will form as a product of the reaction. It should be stirred throughout the addition of ammonia. After the addition is complete, allow the mixture to settle for 5-10 minutes. Decant the liquid, then stir the remaining solution and centrifuge (NOTE: approximately 15-20 mL should be left to obtain an adequate ferrofluid sample). Divide 8mL of a 25% aqueous (CH3)4NOH solution between however many test tubes were used in the centrifugation. Stir these tubes until all the solid is suspended in liquid, and pour them all into a vacuum filtration flask. Add a magnetic stirring bar, then vacuum for 30 minutes to remove any excess ammonia. Once completed, the magnet should be coated in black sludge. Put the magnet and its sludge in a plastic weighing boat, then hold the stronger magnet beneath it. Rotate the stirring bar on its axis until all the ferrofluid has become

Appendix: Activities for Nanotechnology

149

attached to the stronger magnet (NOTE: BE SURE TO WEAR GLOVES). Remove the weaker magnet from the boat while making sure that all the ferrofluid has been removed.

Concepts A ferrofluid is a colloidal suspension of magnetic particles. Basically, it is a magnetic liquid. It also exhibits superparamagnetism. A superparamagnetic material is one that is made up of discrete particles which are individually smaller than the size of a magnetic domain for that material. This causes the blocking temperature to be extremely low, and thus the energy to magnetization reversal is smaller than the energy available to the particle. Assemblies of some nanoparticles exhibit superparamagnetic properties.

Reference P. Berger, N. B. Adelman, K. J. Beckman, D. J. Campbell, A. B. Ellis, and G. C. Lisensky, "Preparation and Properties of an Aqueous Ferrofluid" J. Chem. Ed. 76, 943-948 (1999).

ACTIVITY 6. SILVER NANODISK GROWTH Materials Required 25 mM sodium borohydrate 10mM citrate 5mM silver nitrate Ultrapure water Refrigerator Freezer or ice bath Argon laser Volumetric equipment (flasks, micropipettes)

Procedure Getting Ready A lot of equipment is necessary to perform this experiment. Make sure to obtain the necessary volumetric equipment, glassware, and reagents prior to starting. Seed Colloid Synthesis Chill the sodium borohydrate called for in the materials section to 0oC. Add 20 mL citrate to a container. Add a stirring rod to said container and engage the stirring apparatus. To this solution, add 0.5 mL of the chilled sodium borohydrate. Stir the solution for 5 minutes, then cover it and let it sit for 24 hours. This is the seed colloid.

150

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Growth Solution Synthesis Mix 100 μL of seed colloid solution with 50 μL citrate, 50 μL silver nitrate and 800 μL of ultrapure water. Refrigerate this solution in the dark. The solution is then irradiated at 23oC in a 4.3 mm optical cell with a linearly polarized argon laser beam. Irradiate the solution in aliquots of 200 μL for various times. Characterization Characterization of these nanoparticles can be performed via TEM, AFM, absorption spectroscopy, and measuring silver nitrate concentration using a silver electrode. It’s probable that most labs will be using absorption as a means of characterization. Absorption—Take a spectrum of the seed colloid, then compare this spectrum to the spectrum of the various growth solutions. Notice the difference in appearance of peaks.

Concepts Explained This activity demonstrates one of the many methods of nanoparticle preparation, which is a form of self assembly. Although a laser is used, this is NOT a form of laser ablation. The laser in this experiment provides the nanoparticles with the energy necessary to attract one another and begin to grow. As noted in characterization, the longer the irradiation, the higher the concentration of the particles and the lower the concentration of starting materials. Moreover, the color of the solution should change and the absorption peak should shift.

References A. K. Popov, R. S. Tanke, J. Brummer, G. Taft, M. Loth, R. Langlois, A. Wruck, and R. Schmitz, “Laser-stimulated synthesis of large fractal silver nanoaggregates” Nanotechnology 17, 1901-1905 (2006). To prepare gold colloids with a monolayers and understand some facets of self-assembly, see C. D. Keating, M. D. Musick, M. H. Keefe, and M. J. Natan, "Kinetics and Thermodynamics of Au Colloid Monolayer Self-Assembly" J. Chem. Ed. 76, 949-955 (1999) for a series of experiments.

ACTIVITY 7. BUILDING A MENTAL PICTURE OF THE CARBON NANOTUBE Part A Print out several copies of graphene sheets from the website or photocopy and enlarge Figure 4-2. You may find it useful to copy the page onto a transparency. Roll up the paper into a tube with the hexagon pattern on the outside. Make sure the hexagon patterns match. Now mark the centers of two pairs of aligned hexagons. Unroll the paper and connect the

Appendix: Activities for Nanotechnology

151

pairs of centers with a straight line. Draw several lines parallel to the horizontal axis that bisect the vertical sides of the hexagon. For each pair of centers: 1. 2. 3. 4. 5.

Determine (n,m) Determine the chiral angle (with a protractor) Compute the chiral angle from the formula and compare it to the answer in 2 What is the circumference of the nanotube (measure line segment)? Use the circumference formula (derived from equation 4.5) to calculate a value for the diameter and compare the value with your answer to question 4 6. What is the relation between the line segment in question 4 and the line segment in question 5? How is this relevant to the answers to questions 2, 3, 4, 5? 7. What do you observe about the lines that you drew parallel to the horizontal axis?

Part B Repeat Activity 1 with a different direction of rolling and a different diameter.

Part C Construct a nanotube as in Part A. Unroll and draw out the chiral vector C. Draw in the perpendicular line and find the point on the line that is the center of a hexagon closest to the center of the base hexagon. Determine t1 and t2 and verify the formulas. Do this twice, once when n , m have no common divisor and again when n and m do have a common divisor.

Part D Suppose that you make any trip whatsoever from the center of the base hexagon to the center of a second hexagon, stopping at the center of each hexagon along the trip. Now in moving from the center of one hexagon to the center of an adjacent hexagon, you may travel in one of the following six directions: V1 = U1, V2 = U2, V3 =U2-U1, V4 = -U1, V5 = -U2, V6 = U1-U2. In this trip, suppose that you travel in the direction V1, s1 times… V6, s6 times. Then the chiral vector from the center of the base hexagon to the center of the second hexagon is given by: C = (s1- s3 - s4 + s6)U1+(s2 + s3 – s5 - s6)U2 Thus m and n may be easily determined for any chiral vector. Pick a hexagon and make two separate trips to the second hexagon. Verify that you get the same m and n for both trips.

152

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Part E (Easy) Look at armchair and zigzag nanotubes (figures 4-9 and 4-10) and discover why they have those names.

ACTIVITY 8. ELECTROCHROMIC PRUSSIAN BLUE THIN FILMS Materials Required 5 mL of 0.05 M HCl (Dilute 2 mL conc HCl to 500 mL with water.) 10 mL of 0.05 M K3[Fe(CN)6] (Dissolve 1.65 g in 100 mL water.) 10 mL of 0.05 M FeCl3.6H2O (Dissolve 1.35 g in 100 mL water with a drop of 0.05 M HCl) 25 mL of 1.0 M KCl (Dissolve 18.6 g in 250 mL water with two drops of 0.05 M HCl) Conductive glass can either be purchased from Hartford Glass Co. (1" x 1" x 2.3 mm TEC 15 glass) or, if you have the equipment, can be prepared by sputtering a gold thin film on a standard microscope slide (Note: the glass can be reused by removing Prussian blue coatings with concentrated ammonia. Solubility increases with pH.) Ohmmeter Pt wire or graphite electrode 1.5 Volt battery, holder, and 50 K ohm variable resistor for controlling deposition current, or an automatic DC power supply to complete the circuit to produce a constant current Electrochemical apparatus for obtaining cyclic voltammograms (optional)

Procedure Add 5 mL 0.05 M HCl, then 10 mL 0.05 M K3[Fe(CN)6], and then 10 mL 0.05 M FeCl3.6H2O to a 50 mL. You should observe a distinct color change upon addition of FeCl3.6H2O. This solution should be prepared just before use. Obtain a piece of conductive glass with known dimensions, and, with an ohmmeter, determine which side of the glass is the conductive side. The gold-plated (or tin oxide) side of the glass should have a lower resistance than the untreated side of the glass slide. Face the conducting side of the glass towards a platinum wire coil or graphite electrode. Connect the negative lead of a voltage source to the glass and the positive lead to a platinum wire coil or graphite electrode. Dip the electrodes (but not the alligator clips) into the solution prepared above and quickly adjust the voltage to produce a current density of 40µA/cm2 for 60 seconds. Rinse the electrode with deionized water. The approximately 50 nm coating (see calculation) after 60 seconds of deposition can be seen by eye on the conducting surface.

Appendix: Activities for Nanotechnology 40 x10 −6 A coulomb mole electrons mole Fe 7 (CN ) 18 6.022 x10 23 molecules (1.02 nm ) 3 x x x x x A sec mole molecule cm 2 96485 coulomb 3 mole electrons

153 2

⎛ 10 −7 cm ⎞ 0.88nm ⎟⎟ = x⎜⎜ nm sec ⎠ ⎝

(The sixth term in the equation above comes from the cubic unit cell dimensions. Inorg. Chem., 16(11), 2704-2710.) Longer electrolysis times give thicker coatings. The rinsing steps are repeated after every 30 seconds to show the changing color but would not be needed otherwise unless you are recording visible absorbance (690 nm) as a function of layer thickness. Place the rinsed electrodes in 25 mL 1.0 M KCl, keeping the clip above the solution. Changing the applied voltage changes the color of the thin film. Either obtain the cyclic voltammogram (next paragraph) or remove the battery from the circuit and alternately connect the electrodes as follows: 1. Glass electrode connected to battery (–) and platinum or graphite electrode connected to battery (+); negative voltage applied. 2. Glass electrode directly connected to platinum or graphite electrode; zero volts applied. 3. Glass electrode connected to battery (+) and platinum or graphite electrode connected to battery (–); positive voltage applied. 4. Glass electrode directly connected to platinum or graphite electrode; zero volts applied. 5. Repeat. Record the cyclic voltammogram of the coated glass at 20 mV/sec from +550 mV to -250 mV to +1200 mV to +550 mV versus a Ag/AgCl reference electrode. When the voltage matches a redox reaction the current increases.

Concepts Explained In this experiment K3[Fe(CN)6] is electrochemically reduced at a glass electrode to produce K4[Fe(CN)6]. The K4[Fe(CN)6] at the electrode reacts with FeCl3 in solution and deposits insoluble Prussian Blue, Fe4[Fe(CN)6]3, on the electrode (conductive glass). The approximately 100 nm thick layer of Prussian Blue exhibits visible electrochromism. Chromism is the changing of color based on a stimulus, in this case and electrical stimulus. When the oxidation state of certain metals in a complex compound or ion changes, the color of that complex also changes due to the difference in energies of the newly populated orbitals for the complex.

The Challenge What area electrode (conductive glass) did you use? How much current for how long? How thick is the film you produced? What color is the reduction product for the unbalanced redox process below? Balance the reaction in acidic solution. Fe(III)4[Fe(II)(CN6]3 Fe(II)4[Fe(II)(CN)6]3 What color is the oxidation product for the unbalanced redox process below? Balance the

154

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al. reaction in acidic solution. Fe(III)4[Fe(II)(CN6]3 K4Fe(III)4[Fe(III)(CN)6]3

Reference This experiment was adapted from the following citation by Rebecca DeVasher, Assistant Professor of Chemistry, Rose-Hulman Institute of Technology, Terre Haute, IN: Garcia-Jareño, J. J., Benito, D., Navarro-Laboulais, J. and Vicente, F., "Electrochemical Behavior of Electrodeposited Prussian Blue Films on ITO Electrodes," J. Chem. Educ., 75, 881-884 (1998) Applications of Prussian Blue thin films and other electrochromic materials can be found at the following web site. http://newsroom.spie.org/x2318.xml

GLOSSARY absorption—when energy that interacts with matter is of a frequency that corresponds to the spacing between two energy levels, then the energy may be absorbed by the matter. absorption index (coefficient)—Value which represents an exponential depletion of the radiation power as light propagates through a medium. Light drops its power by a factor of 1/e≈1/2.7 after the wave has propagated a distance which is the reciprocal absorption index. absorption length—Distance which is the reciprocal absorption index. absorptivity (see also extinction coefficient)—the proportionality constant that relates the amount of energy absorbed to the amount of matter the energy encountered. aggregation—gathering of particles to produce a larger cluster. allotrope—a form in which an element is found in nature. Carbon, for example, comes in three common allotropes: graphite, diamond and fullerene. amorphous—matter in which the particles (atoms or molecules) are arranged in a totally random way; i.e., there is not a pattern to the relative orientation of particles. Angular frequency—Oscillation frequency multiplied by 2π. Anomalous dispersion—Gradually-decreasing dependence of the refractive index on the light frequency. aperture—a small opening, sized in such a way as to regulate the nature of the energy that passes through it. armchair nanotube—nanotube with n = m and chiral angle 30 degrees. buckminsterfullerene—a molecule with the formula C60. It is shaped like a soccer ball. Known as bucky ball for short. bucky ball—see buckminsterfullerene. carbon nanotubes—long tubes made up entirely of carbon. These are sheets of graphite rolled up. They come in various diameters and chiral angles. chemical vapor deposition (CVD)—a chemical method that involves the reaction of gases with a heat surfaces to form a solid. chiral angle—The angle between the chiral vector and the horizontal axis when the nanotube is unrolled. chiral vector—The vector formed from the base hexagon when a nanotube is unrolled to form a graphene sheet. Any chiral vector rolls up to form a circumference of the of the nano tube.

156

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

chirality— A chiral molecule is one which has a distinct mirror image, in the same way that your left and right hands are mirror images of each other. Conversely, a ball is not chiral because its mirror image is identical to the original ball, i.e., it is not distinct. A molecule that has a distinct mirror image is said to be chiral, or have the property of chirality. Any carbon nanotube that has a mirror image will be chiral. clean surface—a surface where there is no saturation. Coherence—Stability of the oscillation phase difference over the entire observation time. coherent—a term from wave mechanics, this refers to the situation in which all the light that passes through a defined wave is at the same point in its oscillation. colloid—a particle from 0.5-500 nm in diameter that does not readily settle out from solution. compressive stress—stress applied to cause uniaxial compression of a material. constructive interference—when two waves interact in such a way that the amplitude of their sum is greater than the amplitude of the separate waves. Coulomb’s force line—Path followed by an electric charge free to move in an electric field. Curie’s law—a mathematical expression of the empirical observation that magnetic susceptibility is inversely related to the temperature of the material. Dalton's Laws—John Dalton (1766-1844) developed a set of chemical laws that are similar to those used in chemical stoichiometry. This includes the notion of "atoms" as a fundamental chemical unit, and the law of definite proportions that follows from that idea. damping rate (damping constant)—The value which represents an exponential decrease of the oscillation amplitude in time. The amplitude of such oscillations drops by a factor of 1/e ≈ 1/2.7 after the period of time which is the reciprocal of the damping constant. degenerate orbitals—orbitals with the same energy. destructive interference—when two waves interact in such a way that the amplitude of their sum is less than the amplitude of the separate waves. diamagnetic—incapable of enhancing or otherwise responding to an applied magnetic field. The atoms of diamagnetic materials have no spin magnetic moment. diffraction—another term for the phenomenon of scattering, this term refers to what occurs when energy interacts with matter only to be re-emitted by the matter. This re-emission occurs in all directions, but the magnitude of the energy is a function of separation from the direction of the incident radiation. diffraction limit—Consequence of diffraction that does not allow the focusing of electromagnetic radiation on a spot with diameter less the half its wavelength. diffraction pattern—when energy is scattered from crystals, the waves will experience constructive and destructive interference such that when monitoring the intensity of the energy at defined locations, one can back-calculate the structure of the crystal. The pattern of energy intensities is called diffraction pattern. dipole moment—when the electron density around a particle is not uniform, i.e., when there is a greater density at one end of the particle than at the other, the particle is said to have a dipole moment. domain—a subsegment of a ferromagnetic material in which the atomic magnetic moments are all aligned. elastic collision—a collision that results in no loss of kinetic energy, although kinetic energy may be transferred from one particle to another. electric dipole—Pair of equal and opposite charges separated by some distance.

Appendix: Activities for Nanotechnology

157

electron energy loss spectroscopy (EELS)—a method that monitors the magnitude of energy lost by high energy electrons when they interact with a material. electronic energy levels—the atomic and molecular orbitals presented in early chemistry classes have defined energy. These energies are the electronic energy levels. i.e., the energies that electrons surrounding a particular set of nuclei are “allowed” to have. Entangled state—State that is represented by a wave function which is a superposition of several wave functions attributed to the allowed states of a quantum system. epitaxy—the growth of one material (such as carbon) on the crystal face of another material (such as iron oxide). evanescent wave—Wave propagating along the interface of two media with its amplitude decaying exponentially as it penetrates one of the media. Such a wave can be generated, e.g., by total internal reflection of light from the less dense medium. Rapidly (exponentially) decaying space waves (within a distance less than the characteristic optical wavelength), such as near fields, are often generally referred to as evanescent waves. exchange interaction—the mechanism by which the atomic magnetic moments of neighboring atoms are correlated to be either parallel (as in ferromagnetic materials) or not parallel (as occurs in paramagnetic materials). Although calculable using quantum mechanics, the exact nature of the interaction is not well-defined. extinction coefficient (see also absorptivity)—the proportionality constant that relates the amount of energy absorbed to the amount of matter the energy encountered. far-field zone or radiation zone—Region at large distances from the oscillating dipole, where the field is transverse and definite wavelength has formed. ferromagnetic —the atoms of ferromagnetic materials have spin magnetic moment. Their response to an applied magnetic field is to align the atomic magnetic moment of every atom parallel with the applied magnetic field. Much of the resulting magnetic character is retained after the magnetic field is removed. fluorescence—after an electron has been excited to an elevated energy level (as a result of the interaction of energy and matter), it will return to its ground state. This is the process that occurs when the return to the ground state occurs via a transition from an excited electronic energy level to the lowest energy electronic energy level. fractal—Object that possesses scale invariance, non-integer dimensionality, and a density that tends to zero with the increase of the fractal size. Scale invariance means that each section of the fractal possesses the same symmetry or statistical properties as the whole object. Non-integer dimensionality means that a number of particles inside the sphere of a radius R inside the fractal varies as RD, where D is a non-integer number less than the dimension of space. frequency—ν, how often a given amplitude of the wave passes a fixed point; usually expressed in hertz (Hz; 1 Hz = 1 s-1) fullerenes—a polyhedral molecule consisting only of carbon usually arranged in hexagonal or pentagonal rings, such as in buckminsterfullerene. grapheme—a single sheet of graphite. inelastic collision—a collision that results in a loss of kinetic energy. Inhomogeneous broadening (of a resonance)—Broadening caused by the distribution of the resonance shifts for the species comprising the sample.

158

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

interference—Net effect of the combination of two or more wave trains moving on intersecting or coincident paths. Interference between waves traveling in opposite directions produces standing waves. interferometry—a method of filtering or measuring the properties of light by taking advantage of constructive and destructive interference. intermolecular force (IMF)—force between molecules (or particles), noncovalent forces including hydrogen-bonding, London forces, and electrostatic interactions. laser ablation—the application of a laser to a surface producing a plasma of particles. lifetime—Reciprocal oscillation damping constant. lithography—a process for printing multiple copies. The process involves five steps. (1) the image is laid down on a print block (originally limestone) with an oil-based medium. (2) The block is etched with an acid and gum arabic (a highly branched carbohydrate whose constitutive monomers are primarily D-galactose and salts of D-glucuronic acid). No etching occurs where the image was laid down. (3) The oil-based medium is washed off with turpentine. A raised salt matrix remains that is the ‘negative’ of the image. (4) An oil-based ink is rolled over the moist print block, filling the recessed areas where the image was laid down. (5) Paper is pressed on the block to produce multiple copies of the same image. magnetic permeability—Degree of magnetization of a material that responds linearly to an applied magnetic field. Negative permeability indicates that the sign of the induced magnetic moment becomes opposite to the applied magnetic field which is caused by the large phase delay equal to π. magnetic susceptibility—the proportionality constant that relates the magnitude of magnetism (M) to the strength of the applied external magnetic field (H). Notation: χm magneton—the smallest unit of magnetic moment, μB = 9.274 x 10-24 (J/T). magnetoresistance—the application of a magnetic field induces a transverse electrical field, which increases the resistance of the material. mask—a template used in optical lithography to project an image on a photoresist material. metallic nanotubes—carbon nanotubes that conduct electricity. All nanotubes of sufficiently large radius are metallic. metamaterials—artificial materials that do not exist naturally. micelle—an aggregation of molecules to form a submicroscopic cluster. micron—short for micrometer, abbreviated μ, and equivalent to 10-6 m. microscopy—the process of making virtual images of very small objects such that details of the small objects’ appearances become apparent. modulus—the ratio of stress to strain. multiwalled nanotubes—a series of carbon nanotubes with one rolled inside another. natural or resonance frequency—Oscillatory frequency of a momentarily disturbed undriven system (e.g., bound electron). near-field (or static) zone—Region about a dipole that is commensurate with its size and much smaller than the wavelength of the emitted dipole radiation. near-field scanning optical microscopy (NSOM)—Technique that allows one to achieve resolution to an order of magnitude below normal optical microscopy by overcoming the diffraction limits.

Appendix: Activities for Nanotechnology

159

nonlinear optics—branch of optics that describes the behavior of light in nonlinear media, that is, media in which forced electric dipole oscillations respond nonlinearly to the electric field E of the light. Nonlinear optics gives rise to a host of optical phenomena. nonlinear spectroscopy—Branch of spectroscopy that employs resonant nonlinear-optical processes. normal dispersion—Gradually-increasing dependence of the refractive index on the light frequency. orbital magnetic moment—the magnetic moment that results from the motion of an electron around a nucleus. oscillation frequency—Number of cycles or vibrations occurring during one unit of time. paramagnetic—the atoms of paramagnetic materials have spin magnetic moment. Their response to an applied magnetic field is to align the atomic magnetic moment of every atom parallel with the applied magnetic field. The resulting magnetic character is lost after the external magnetic field is removed. passivation—the process of bonding the surface atoms to another material to remove the structural reconstruction that occurs as surface atoms seek stability. phase (of a wave)—particular stage in a periodic phenomenon. It determines the value of the oscillating quantity at a specified point in time or space. Phase is represented by the fraction of a complete cycle elapsed as measured from a specified reference point and often expressed as an angle. The angle can be conveniently expressed as a multiple of π. phase velocity—Speed at which a wavefront moves. phosphorescence—in the process of being excited to an elevated energy level, the spin-state of the electron is unaltered. After being excited, the electron’s spin may flip. In order to return to its ground state, the electron’s spin must flip again. Because both ‘spin flips’ are disallowed, this pair of processes takes a rather long time. The return of an electron to the ground state under these conditions is called phosphorescence. photolithography—using the same principles as in lithography, except that image results from photochemistry. Photoresist, a material that hardens when exposed to light, is applied to a surface. A mask laid over the photoresist protects defined regions from exposure to light. Exposed regions harden; soft regions are washed away to result in a well-defined pattern of hardened, often conductive, material on a surface. This method is used in the preparation of silicon chips. photon—a quantum of light. photon scanning tunneling microscope (PSTM)—Microscope where the tip is used only for the detection of emitted light photons, while the sample is illuminated by another source of evanescent light generated by total internal reflection inside a glass prism. Piezo effect—change in the shape or size of a specimen that is proportional to an applied voltage.

ωp = plasma frequency—parameter defined as and mass, N is electron number density,

Ne 2 mε 0

, where e and m are electron charge

ε 0 is permittivity of free space. Plasma totally ωp

reflects electromagnetic waves at all frequencies less than

.

160

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

plasmonic resonance—resonance behavior attributed to a metal nanoparticle, the resonance frequency of which depends on its shape and size. polarization (of light)—orientation of the electric component of an electromagnetic wave. polarization density (or electric polarization, or simply polarization)—vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. The polarization vector P is defined as the dipole moment per unit volume. polarizability—when the electrons of a particle are not densely packed, it is possible for an external electric field to induce movement of the electrons from one end of the particle to another. How readily this can be done is called the polarizability. potential gradient—an electric field of continually increasing magnitude. propagation constant or wavevector—wavenumber multiplied by 2π. proportional limit—identifies the maximum stress and strain on a system before the material loses its linear response. quantum dot—a nanoscale particle of a semiconductor. The dot has electrical or optical properties that are significantly different from bulk semiconductor. quantum state—state of submicroscopic systems described by quantum mechanics. quantum transition—process of changing quantum states. quantum wires—A wire that is macroscopic length but nanoscale in diameter. Rayleigh ratio—the ratio of scattered light to incident light. rays—orthogonal trajectories of the wavefront. refractive index—also called index of refraction:—measure of the bending of a light ray when passing from one medium into another. This index is equal to the velocity of light in empty space divided by the velocity of light of a given wavelength in a substance. resolution—in microscopy, minimum distance two objects can be separated and still be observed as distinct from each other. The shorter the wavelength used to observe the objects, the smaller the distance can be; resolution is improved with the use of shorter wavelength. resist—short for photoresist, a material that reacts upon irradiation. resonance enhancement—tendency of a system to increase its response when the frequency of the applied force approaches the system's natural frequency (its resonant frequency). resonance full-width at half-maximum (FWHM)—frequency detuning from its resonance value, where the frequency-dependent quantity (such as absorption index) decreases by a factor of two as compared with its resonant magnitude. resonance half-width at half-maximum (HWHM) is ½ of (FWHM). resonance Raman scattering—spectroscopic method that monitors the small fraction of scattered monochromatic radiation that has a different frequency from that of the incident radiation. roll up vector—same as the chiral vector. saturation—a molecule or solid crystal in which all possible chemical sites are occupied is said to be saturated. self-assembly—a process by which a supramolecular system forms spontaneously from its components. semi-conducting nanotube—nanotube with semi-conducting conduction properties along the length of the nanotube. semiconducting nanotubes—carbon nanotubes that behave electrically like a semiconductor. Such nanotubes must be of small radius and of a particular chiral angle.

Appendix: Activities for Nanotechnology

161

shear —if a sample of solid material is conceptually divided into parallel planes, this is the process of sliding the planes past each other so that two adjacent planes are moving at different velocities. singlewalled nanotubes—SWNT, a carbon nanotube with walls comprised of a single layer of graphene, rolled into a tube. skin or penetration depth—Reciprocal of the attenuation coefficient. For a material to be transparent, the penetration depth must be larger than its thickness. spatial period (of oscillations), also named wavelength—space interval that one oscillation cycle completes along the propagation direction. Spatial period, λ, is referred to as wavelength. spectroscopy—technique that derives the information about the structure and size of emitting or absorbing agents (e.g., nanostructures) and about properties of their environment from a measurement of the frequency (spectral) dependence of emission, and absorption and refraction of electromagnetic radiation by such an agent. speed of light—phase velocity of electromagnetic waves in empty space. spin magnetic moment—the magnetic moment that arises from an electron’s intrinsic angular momentum. strain—the relative deformation of the sample under a given stress. stress—the force applied to the sample per unit cross-sectional area. structure factor, Pθ—the manner in which a particle’s shape and size alters the observed Rayleigh ratio. superparamagnet—a small cluster of a ferromagnetic material for which the blocking temperature is below operational temperature so that the observed behavior of a collection of the particles is that of a paramagnetic material. supramolecular —molecular assemblies that include many components. surface rearrangement—the surface of a crystal that is unsaturated will rearrange in order to minimize surface energy. Atoms at the surface will form bonds with each other, and thus probably move from what their equilibrium position would be if they were in the bulk. temporal period (of oscillations)—interval of time the oscillation completes one cycle at one point in space. A temporal period, T, and the oscillation frequency, ν, are related as ν = 1/T. tensile strength—the force required to break a sample by uniaxial extension per cross sectional area of the sample. tensile stress—a stress applied to induce uniaxial extension of a material. termination group—a group on the surface of a nanoparticle design to stabilize the particle against undesired aggregation. tetrahedron— A polyhedron with four sides faces and four corners. In diamond, the four carbons adjacent to a given carbon form a regular or symmetric tetrahedron in which the corners are placed at angles as far from each other as possible. The tetrahedral angle is 109°. temporal period (of oscillations)—interval of time the oscillation complete one oscillation cycle at one point in space. A temporal period, T, and the oscillation frequency, ν, are related as ν = 1/T. total internal reflection—total reflection that occurs when the incident light is traveling within the medium with higher refractive index toward the medium with less refractive index.

162

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

transmission—when light passes through matter without any apparent alteration of the light’s direction, intensity, or frequency. tunneling probability—how likely it is that an electron can transfer from one type of matter to another. unit cell vector—A vector in the graphene sheet perpendicular to the chiral vector and whose length equal the length of the unit cell. unit cell—the basic unit from which the nanotube can be constructed by translation. vibrational energy levels—the energies that correspond to the natural vibrational frequencies of the various bonds of molecules according to the equation E = hν, where ν is the natural vibrational frequency. wave motion—Propagation of disturbances from place to place in a regular and organized way. An electromagnetic wave represents the propagation of oscillating disturbances of the electromagnetic field. wavefront—Imaginary surface representing corresponding points of a wave that vibrate in unison, i.e., with the same phase at any instant. A surface drawn through all the points of the same phase constitutes a wavefront. wavefunction—Function whose squared modulus gives the probability of finding a submicroscopic system in a certain quantum state. wavelength—spatial period, i.e., the distance between repeating units of a wave pattern. It is commonly designated by the Greek letter lambda (λ). The wavelength is equal to the phase velocity of the wave multiplied by the temporal period (or divided by the frequency) of the wave. wavenumber—Number of waves in a unit distance, which is equal to the true frequency divided by the speed of light. X-ray diffraction—a phenomenon in which x-rays are scattered by particles in a crystal. Used to determine the structure of solids. X-ray lithography—a technique for making nanostructures using X-rays through a template to transfer a specific pattern onto a surface. zeolite—glassy minerals consisting of hydrated aluminum silicates of alkali and alkali earth elements that frequently contain cavities. zigzag nanotube—nanotube with m= 0 and chiral angle 0 degrees.

REFERENCES Alivisatos, A.P., “Semiconductor Clusters, Nanocrystals, and Quantum Dots”, Science, 271: 933-937, (1996). Banyai. L. and Koch, S. W., Semiconductor Quantum Dots, Singapore: World Scientific, (1993). Berman, G. P., Doolen, G. D., Mainieri, R. and Tsifrinovich, V. I., Introduction to Quantum Computers, Singapore: World Scientific, (1998). Binnig, G. and Rohrer, H., “The Scanning Tunnel Microscope”, Scientific American, 253: 50, (1985). Bonnell, D. A., Ed., Scanning Probe Microscopy and Spectroscopy: Theory, Techniques, and Applications, New York: Wiley, (2000). Bozhevolnyi, S. I. and Shalaev, V. M., “Nanophotonics with Surface Plasmons”, Part I, Photonics Spectra, January 2006, p. 58. Bube, R. R., Electrons in Solids: An Introductory Survey, New York: Academic Press, (1981). Chen, J., Hamon, M. A., Hu, H., Chen, Y., Rao, A. M., Eklund, P. C. and Haddon, R. C., “Solution Properties of Single-Walled Carbon Nanotubes”, Science, 282: 95, (1998). Christman, J. R., Fundamentals of Solid State Physics, New York: John Wiley and Sons, Inc, (1988). Dai, H. J., "Carbon Nanotubes: Synthesis, Integration, and Properties", Accounts of Chemical Research, 35: 1035-1044, (2002). Donlin, M. J., Frey, R. F., Putnam, C., Proctor, J. and Bashkin, J. K., “Analysis of Iron in Ferritin, the Iron-Storage Protein: A General Chemistry Experiment”, Journal of Chemical Education, 75 (4): 437-441, (1998). Dresselhaus, M. S., Dresselhaus, G., Saito, R. “Carbon nanotubes”, Physics World 33-38, (January 1998). Dresselhaus, M. S., Dresselhaus, G. and Eklund, P.C. Science of Fullerenes and Carbon Nanotubes, New York: Academic Press, (1996). Elghanian, R., Storhoff, J. J., Mucic, R. C., Letsinger, R. L. and Mirkin, C. A., "Selective Colorimetric Detection of Polynucleotides Based on the Distance-Dependent Optical Properties of Gold Nanoparticles", Science, 277: 1078-1081, (1997). Elliott, S., The Physics and Chemistry of Solids, New York: John Wiley and Sons, Inc, (1998).

164

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Fahlman, B. D., “Chemical Vapor Deposition of Carbon Nanotubes: An Experiment in Materials Chemistry”, Journal of Chemical Education, 79 (2): 202-206, (2002). Fang, N., Lee, H., Sun, C. and Zhang, X. “Sub–diffraction-limited optical imaging with a silver superlens,” Science, 308: 534-537, (2005). Gammon, D., “Semiconductor physics: Electrons in artificial atoms,” Nature, 419, 278-281, (2002). Garcia-Jareño, J. J., Benito, D., Navarro-Laboulais, J. and Vicente, F., "Electrochemical Behavior of Electrodeposited Prussian Blue Films on ITO Electrodes," J. Chem. Educ., 75, 881-884 (1998) Ghatak, A. K. and Thayagarjan, K., Contemporary Optics, New York and London: Plenum Press (1978). Guinier, A. and Jullien, R., The Solid State From Superconductors to Superalloys, Oxford Science Publications, International Union of Crystallography, (1989). Hamilton, A. D., “Molecular Recognition”, J. Chem. Ed., 67: 812-828, (1990). Harris, P. F. J., Carbon Nanotubes and Related Structures: New materials for the Twenty First Century, Cambridge University Press, (1999). Hecht, E., Optics, Second Edition, Reading, Mass.: Addison-Wesley, (1987). Hsieh, S., Meltzer, S., Wang, C. R. C., Requicha, A. A. G., Thompson, M. E. and Koel, B. E., "Imaging and Manipulation of Gold Nanorods with an Atomic Force Microscope", J. Phys. Chem. B, 106: 231-234, (2002). Hummel, R. E., Electronic Properties of Materials, Third Edition, Springer, New York, (2001). Iijima, S., “Helical microtubules of graphitic carbon”, Nature, 354: 56, (1991). Jacob, J, Alekseyev, L. V., and Narimanov, E, "Optical Hyperlens: Far-field imaging beyond the diffraction limit," Opt. Express 14: 8247-8256 (2006) http://www.opticsinfobase. org/abstract.cfm?URI=oe-14-18-8247 Karpov, S. V., Bas’ko, A. L., Popov, A. K., Slabko, V. V. and George, T. F., “Optics of nanostructured fractal silver colloids,” pp. 427-463 in Recent Research Developments in Optics Vol. 2, Trivandrum, Kerala, India: Research Signpost, (2002). http://arxiv.org/abs/physics/0301081 Kim, W., Safonov, V. P., Shalaev, V. M. and Armstrong, R. L., “Fractals in microcavities: Giant coupled, multiplicative enhancement of optical responses”, Physical Review Letters, 82: 4811-4814, (1999). Kittel, C. Introduction to Solid State Physics, New York: John Wiley and Sons, Inc, (1996). Koch, S. W. and Knorr, A., “Optics in the nano-world,” Science, 293, 2217-2218, (2001). Koper, O. and Winecki, S. “Specific Heats and Melting Points of Nanocrystalline Materials”, Chapter 8 in K. J. Klabunde, ed., Nanaoscale Materials in Chemistry, N.Y.: Wiley Interscience, (2001). Krätschmer, W., Lamb, L. D., Fostiropoulos K. and Huffman, D. R., “Fullerenes: C60's smallest cousin”, Nature, 347: 354, (1990). Kroto, H. W., Heath, J. R., O’Brien, S. C., Curl R. F. and Smalley, R. E., “C60: Buckminsterfullerene”, Nature, 318: 162, (1985). Lindoy, L. F. and Atkinson, I. M., Self-Assembly in Supramolecular Systems, London: RSC, (2000). Lucking, U., Tucci, F. C., Rudkevich, D. M. and Rebek, J. Jr., “Self-Folding Cavitands of Nanoscale Dimensions”, J. Am. Chem. Soc, 122 (37): 8880-8889, (2000).

Appendix: Activities for Nanotechnology

165

Mafune, F., Kohno, J. Y., Takeda, Y. and Kondow, T., "Formation of Stable Platinum Nanoparticles by Laser Ablation in Water", Journal of Physical Chemistry B, 107: 42184223, (2003). Maillard, M., Huang, P. and Brus, L., “Silver Nanodisk Growth by Surface Plasmon Enhanced Photoreduction of Adsorbed [Ag+]”, NanoLetters, 3: 1611-1615, (2003). Garbuny, M., Optical Physics, New York and London: Academic Press, (1970). Murphy, C. J. and Coffer, J. L., "Quantum Dots: A Primer", Applied Spectroscopy, 56: 16a27a, (2002). Nalwa, H. S., Handbook of Nanostructured Materials and Nanotechnology, San Diego: Academic Press, (2000). O’Connell, M. J., Bachilo, S. M., Huffman, C. B., Moore, V. C., Strano, M. S., Haroz, E. H., Rialon, K. L., Boul, P., Noon, W., Kitrell, C., Ma, J. P., Hauge, R. H., Weisman, R. B. and Smalley, R. E., "Band Gap Fluorescence from Individual Single-Walled Nanotubes", Science, 297: 593-596, (2002). Paesler, M. A. and Moyer, P., Near-Field Optics: Theory, Instrumentation, and Applications, New York: Wiley, (1996). Petukhov, A. V., Passerone, D., Ercolessi, F. Tosatti, E. and Fasolino, A., “(Meta)stable reconstructions of the diamond (111) surface: Interplay between diamond and graphitelike bonding”, Phys. Rev. B, 61: R10590, (2000). Popov, A. K., and Shalaev, V. M., “Negative-index Metamaterials: Second Harmonic Generation, Manley-Rowe Relations and Parametric Amplification”, Applied Physics, B84: 131-137 (2006); http://arxiv.org/abs/physics/0601055. Puech, P., Hubel, H., Dunstan, D. J., Bacsa, R. R., Laurent, C. and Bacsa, W. S., “Discontinuous Tangential Stress in Double Wall Carbon Nanotubes,” Physical Review Letters, 93 (9): 095506-1 - 095506-4, (2004). Qi, Y. and Hector, L. G. Jr., “Hydrogen effect on adhesion and adhesive transfer at aluminum/diamond interfaces”, Phys. Rev. B., 68: 201403(R), (2003). Rao, C. N. R., Satishkumar, B. C., Govindaraj, A. and Nath, M., "Nanotubes", Chemphyschem, 2: 78-105, (2001). Ren, Z. F., Huang, Z. P., Xu, J. W., Wang, J. H., Bush, P., Siegal, M. P. and Provencio, P. N., " Synthesis of Large Arrays of Well-Aligned Carbon Nanotubes on Glass", Science, 282: 1105-1107, (1998). Rubinson, J. F. and Rubinson, K. A., Contemporary Chemical Analysis, Prentice-Hall, Upper Saddle River, NJ, (1998). Saito, R., Fujita, M., Dresselhaus, G. and Dresselhaus, M. S., “Electronic Structure of Chiral Graphene Tubules”, Appl. Phys. Lett., 60 (18): 2204-2206, (1992). Shalaev, V. M., “Optical Negative-index Metamaterials”, Nature Photonics, 1: 41-48, (2007). Shalaev, V M, Cai, W, Chettiar, U, Yuan, H-K, Sarychev, A K, Drachev, V P, Kildishev, A V Optics Letters 30: 3356 ;arXiv: physics/0504091, (2005); http://www.arxiv.org/abs/physics/0504091. Shalaev, V. M. and Bozhevolnyi, S. I., “Nanophotonics with Surface Plasmons”, Part II, Photonics Spectra, February 2006, p. 66. Shi, J., Gider, S., Babcock, K. and Awschalom, D. D., “Magnetic Clusters in Molecular Beams, Metals, and Semiconductors,” Science, 271: 937-941, (1996).

166

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

Shimizu, T., Teranishi, T., Hasegawa, S. and Miyake, M., "Size Evolution of AlkanethiolProtected Gold Nanoparticles by Heat Treatment in the Solid State", J. Phys. Chem. B, 107: 2719-2724, (2003). Solin, S. A., “Magnetic Field Nanosensors,” Scientific American, 272: 70 – 77, (2004). Storhoff, J., Elghanian, R., Mucic, R. C., Mirkin, C. A. and Letsinger, R. L., "One-Pot Colorimetric Differentiation of Polynucleotides with Single Base Imperfections Using Gold Nanoparticle Probes", J. Am. Chem. Soc., 120: 1959-1964, (1998). Sun, Y-P., Fu, K., Lin, Y. and Huang, W., “Functionalized Carbon Nanotubes: Properties and Applications,” Acc. Chem. Res. 35: 1096-1104, (2002). Sun, Y-P., Guduru, R., Lawson, G. E., Mullins, J. E., Guo, Z., Quinlan, J., Bunker, C. E. and Gord, J. R., “Photophysical and Electron-Transfer Properties of Mono- and MultipleFunctionalized Fullerene Derivatives,” J. Phys. Chem. B, 104: 4625-4632, (2000). Vossmeyer, T., Katsikas, L., Giersig, M., Popovic, I. G., Diesner, K., Chemseddine, A., Eychmuller, A. and Weller, H., “CdS Nanoclusters: Synthesis, Characterization, Size Dependent Oscillator Strength, Temperature Shift of the Excitonic Transition Energy, and Reversible Absorption Shift,” J. Phys. Chem., 98: 7665-7673, (1994). Williams, D. and Carter, C. B., Transmission Electron Microscopy, Plenum Press, New York, (1996). Yu, Z. and Brus, L., “Rayleigh and Raman Scattering from Individual Carbon Nanotube Bundles”, J. Phys. Chem. B, 105: 1123-1134, (2001). Zach, M. P., Inazu, K., Ng, K. H., Hemminger, J. C. and Penner, R. M., "Synthesis of Molybdenum Nanowires with Millimeter-Scale Lengths Using Electrochemical Step Edge Decoration", Chemistry of Materials, 14: 3206-3216, (2002). Zach, M. P., Ng, K. H. and Penner, R. M., "Molybdenum Nanowires by Electrodeposition", Science, 290 2120-2123, (2000). Zhang, S., Fan, W., Panoiu, N. C., Malloy, K. J., Osgood, R. M. J. and Brueck, S. R. J. Phys. Rev. Lett. 95, 137404 (2005); http://www.arxiv.org/abs/physics/0504208 (2005). Zhang, G. P., Hűbner, W., Beaurepaire, E. and Bigot, J. Y., “Laser-induced Ultrafast Demagnetization: Femtomagnetism, A New Frontier?” Topics in Appl. Phys., 83: 245 289, (2002).

ABOUT THE AUTHORS Luanne Tilstra is an associate professor in the departments of Chemistry and Applied Biology and Biomechanical Engineering at Rose-Hulman Institute of Technology. She teaches courses in physical chemistry, polymer chemistry, spectroscopy, engineering chemistry, and—of course—nanotechnology. Having constructed an instrument to measure the intensity of light scattered by particles separated by capillary electrophoresis, she studies the kinetics of aggregate growth for peptides and other clusters. She received her Bachelor of Science degree from Central College in Pella, IA, and a Ph.D. from Louisiana State University in Baton Rouge, LA. After postdoctoral work at LSU, Dr. Tilstra was a research scientist at the National Institute of Standards and Technology in Gaithersburg, MD. S. Allen Broughton is professor and head of Mathematics at Rose-Hulman Institute of Technology. He has been working with co-author Daniel Jelski—applying his knowledge of geometry and group theory to chemical symmetry problems—ever since he noticed a bucky

Appendix: Activities for Nanotechnology

167

ball in Dan’s office. Professor Broughton received his Bachelor of Science degree (1976) with a major in mathematics from the University of Windsor. He earned Master of Science and Doctor of Philosophy degrees (1978, 1982) at Queen’s University. He has held academic positions at Memorial University of Newfoundland, the University of Wisconsin–Madison, Cleveland State University, and then the last dozen years at Rose-Hulman. Robin S. Tanke is associate professor of chemistry the University of WisconsinStevens Point. She teaches courses in general and organic chemistry as well as in advanced synthesis and nanotechnology. In addition to her role as instructor, she is an active researcher in chemistry, specializing in synthesis of organic and inorganic compounds as well as nanomaterials. Professor Tanke received a Bachelor of Science degree (Phi Beta Kappa) in 1986 with a major in chemistry from the University of Notre Dame. She earned a Doctor of Philosophy degree in 1990 in chemistry from Yale University, followed by a N.I.H. postdoctoral appointment at the University of Wisconsin- Madison. Daniel Jelski is currently professor of chemistry and dean of science and engineering at the State University of New York at New Paltz. Until 2007, heserved as professor and head of thechemistry department at Rose-Hulman Institute of Technology. He does computational research on the vibrational motions of buckminsterfullerene. In the past he has studied the properties and structure of small silicon clusters. He received a Bachelor of Science degree from the University of Chicago, and a Ph. D. from Northern Illinois University. He did postdoctoral work at the University at Buffalo, and then taught for many years at SUNY College at Fredonia. Valentina French is associate professor of physics at Indiana State University. Her research interests include optical properties of solids, nonlinear systems and physics education. She received a Bachelor of Science degree with a major in physics from the University of Bucharest, Romania and M.S. and Ph.D. degrees in physics from Oklahoma State University. Guoping Zhang is assistant professor of physics at Indiana State University. His research area includes nanoscience and in particular C60, ultrafast phenonmena, nonlinear optics, laser-induced ultrafast demagnetization in ferromagnets, high harmonic generation, strong correlated materials, soft x-ray spectroscopy and resonant inelastic x-ray scattering. He received a Ph. D. from Fudan University, Shanghai, China. He has published nearly sixty papers in Physical Review and other journals, and coedited two books. He is a reviewer for Physical Review and Physical Review Letters. Alexander K. Popov is research professor of physics at the University of WisconsinStevens Point. His research area is resonant laser interaction with atoms and molecules, laserdriven processes in metal nanocomposites and in negative-index nanometamaterials. His work has led to 450 articles/chapters and 3 books. He received a Master’s Degree (Electrical Engineering) from Tomsk University and a Doctoral Degree (Physics and Mathematics) from Krasnoyarsk Institute of Physics of the Russian Academy of Sciences. He headed The Department of Optics at Krasnoyarsk State University and The Research Department of Coherent and Nonlinear Optics at The Institute of Physics. He taught for many years in Russia, China, and the U.S.A. He has been named Soros Professor by the International Science Education Program. Arthur B. Western, professor of physics and optical engineering, is vice president for academic affairs and dean of the faculty at Rose-Hulman Institute of Technology. His background includes research and development in phase transitions in hydrogen-bonded

168

Luanne Tilstra, S. Allen Broughton, Robin S. Tanke et al.

semiconductors, magneto-hydrodynamics, and holographic interferometry. He is co-author, with William P. Crummett, of the introductory text, University Physics: Models and Applications. He received his bachelor’s degree from Rollins College and M. S. and Ph. D. degrees in physics from Montana State University. Prior to arriving at Rose-Hulman, he was associate professor and head of the department of physics and geophysical engineering at Montana Tech, Butte, Montana. Thomas F. George has been chancellor and professor of chemistry and physics at the University of Missouri St. Louis since 2003. His research area is chemical/materials/laser physics, and his work has led to 645 articles/chapters, 5 textbooks, 14 edited books and 185 conference abstracts. He serves as editor of the International Journal of Theoretical Physics, Group Theory and Nonlinear Optics. He has received the Marlow Medal and Prize from the Royal Society of Chemistry in Great Britain and fellowships from the Guggenheim, Sloan and Dreyfus foundations. He has been named a fellow of the New York Academy of Sciences, American Physical Society, Society of Photo-Optical Instrumentation Engineers and American Association for the Advancement of Science. In 2004 he was elected as a foreign member of the Korean Academy of Science and Technology.

INDEX

A absorption spectra, 53, 55 absorption spectroscopy, 47, 112, 150 acid, 2, 12, 157 adhesion, 163 AFM, 12, 36, 37, 47, 50, 62, 150 age, 2 agent, 51, 52, 55, 56, 100, 160 aggregates, 103, 127, 129, 131, 132, 138, 139, 142 aggregation, 52, 56, 139, 140, 154, 157, 160 alters, 160 aluminum, 16, 24, 68, 122, 161, 163 aluminum surface, 16 ammonia, 148, 152 amplitude, 21, 110, 113, 114, 121, 125, 126, 129, 130, 155, 156 animals, 57, 65 antibody, 100 antigen, 100 appendix, viii aqueous solutions, 61 argon, 30, 50, 150 aromatic rings, 17 arsenic, 24 artificial atoms, 127, 162 assumptions, 114 atomic force, 37, 62, 91, 147 atomic force microscope, 37, 62 atomic orbitals, 22, 23 atomic positions, 76 atoms, 3, 5, 6, 7, 9, 10, 15, 16, 17, 18, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 34, 36, 41, 45, 46, 49, 52, 53, 57, 62, 65, 67, 76, 80, 83, 84, 92, 93, 95, 100, 126, 127, 132, 133, 140, 141, 154, 155, 156, 158, 166 attachment, 146

attention, 117, 127, 134 Au nanoparticles, 52, 55 automobiles, 14 averaging, 89 awareness, 13

B bachelor’s degree, 166 background information, 85 backscattering, 42, 43 band gap, 25, 40 barriers, 64 base pair, 53, 64 batteries, 12 beams, 8, 36, 106 behavior, 13, 15, 21, 31, 40, 87, 96, 110, 137, 157, 158 bending, 82, 107, 159 benefits, 18 bias, 97 biological systems, 103 blood, 91, 145 Boltzmann constant, viii, 93 bonding, 22, 23, 53, 64, 157, 158, 163 bonds, 8, 15, 16, 17, 64, 70, 74, 76, 80, 160, 161 branching, 127 breakdown, 128 breathing, 43 Brownian motion, 139 bulk materials, 92

C C60, 7, 8, 9, 10, 12, 49, 68, 154, 163, 165 calculus, vii, viii, 84

170 cancer, 14, 18 cancer treatment, 14 cancerous cells, 14 candidates, 90 capillary, 132, 165 carbohydrate, 2, 157 carbon, viii, 4, 6, 7, 8, 9, 10, 11, 12, 15, 18, 23, 35, 43, 49, 50, 51, 57, 60, 66, 67, 68, 69, 70, 71, 72, 82, 83, 84, 88, 89, 90, 91, 101, 143, 154, 155, 156, 157, 159, 160, 162 carbon atoms, 6, 7, 8, 10, 23, 43, 49, 67, 68, 69, 83, 84, 88 carbon materials, 67 carbon nanotubes, viii, 4, 7, 11, 18, 35, 50, 51, 57, 66, 68, 82, 89, 90, 91, 101, 154, 157, 159 carrier, 8 casting, 59 catalyst(s), 12, 50, 51 cell, 7, 18, 24, 25, 44, 47, 74, 76, 77, 78, 79, 80, 83, 84, 100, 150, 153, 160 cell death, 18 ceramics, 91 channels, 55 chemical bonds, 64 chemical properties, 7, 16 chemical reactions, 21, 51 chemical vapor deposition, vii, 49, 60, 154 Chicago, 165 China, 165, 166 chirality, 4, 12, 18, 40, 43, 51, 69, 84, 155 chlorophyll, 27 classes, vii, 81, 128, 156 classical mechanics, 21, 31, 113 classification, 82 classroom, vii, 92 clustering, 9 clusters, 7, 8, 9, 10, 40, 43, 52, 90, 103, 165 coatings, 152, 153 cobalt, 49, 94 coherence, 133 collisions, 8, 9, 27, 35, 45 colloids, 13, 18, 51, 66, 139, 150, 162 community, 127 complementary DNA, 53 components, 2, 89, 91, 99, 103, 104, 134, 159, 160 composites, 89, 91 composition, 133 compounds, 165 computation, 84 concentrates, vii concentration, 4, 9, 39, 40, 119, 129, 132, 150 conception, 3 conduction, 24, 46, 82, 159

Index conductivity, 11, 18, 24, 69, 82, 83, 142 conductor, 15, 24, 46, 69, 81, 82, 116 configuration, 63, 92 confinement, 18, 40, 103, 109, 112, 113, 117, 126, 133, 137, 140, 142 Congress, iv construction, 66, 78, 89, 90 contrast agent, 100 control, 6, 11, 12, 34, 49, 51, 52, 55, 57, 61, 65, 66, 99, 101, 103, 130, 133 conversion, 108, 110, 116, 132, 141 copper, 5, 22, 23, 68, 69, 116 Coulomb interaction, 95 counsel, viii counterbalance, 139 coupling, 95, 115 covalent bond, 53 covering, viii, 80 credit, vii, 3 critical value, 116 crossing over, 72 crystal structure, 35 crystalline, 24, 51, 76, 133, 141 crystals, 23, 24, 120, 155 currency, 99 CVD, 49, 50, 57, 60, 154 cycles, 158

D damping, 111, 112, 113, 114, 116, 125, 155, 157 data analysis, 43 decay, 30 definition, 6, 68, 75, 79, 85, 89, 97, 140 deformation, 82, 85, 86, 87, 89, 160 degenerate, 92, 93, 155 density, 8, 9, 61, 68, 85, 99, 100, 101, 106, 107, 108, 113, 114, 116, 119, 121, 129, 141, 152, 155, 156, 158, 159 deposition, 58, 59, 152 deposits, 153 desire, vii, 1, 4, 64, 89 destruction, 35 detection, 36, 41, 46, 91, 122, 123, 158 diamond, 6, 7, 10, 15, 16, 17, 23, 24, 67, 68, 120, 154, 160, 163 dielectric materials, 114, 115 dielectric permittivity, 134 dielectrics, 120 diffraction, vii, 34, 43, 44, 106, 117, 119, 120, 122, 124, 129, 132, 135, 141, 155, 157, 162 dimensionality, 15, 85, 97, 156 diodes, 36

Index dipole, 27, 28, 41, 47, 64, 65, 117, 118, 119, 125, 126, 137, 141, 155, 156, 157, 159 dipole moment(s), 27, 28, 41, 47, 117, 155, 159 direct measure, 105 dispersion, 35, 112, 154, 158 displacement, 72, 75, 86, 89, 109, 113, 127, 140 distribution, 9, 22, 23, 25, 93, 118, 119, 127, 137, 139, 156 DNA, 38, 39, 53, 64 doping, 24, 97 Doppler, 134 double helix, 64 dream, 91 dyes, 132

E earth, 7, 145, 161 Education, 162, 165, 166 Einstein, 20, 93 elasticity, 89 electric charge, 117, 155 electric current, 58 electric field, 19, 27, 28, 33, 41, 106, 108, 109, 110, 113, 114, 117, 118, 119, 120, 121, 134, 137, 155, 157, 159 electric potential, 109 electrical conductivity, 17, 69 electrical properties, 14, 83 electrical resistance, 97 electricity, 11, 157 electrodes, 58, 152, 153 electrolysis, 153 electromagnetic, 19, 20, 21, 27, 46, 103, 104, 105, 107, 108, 109, 114, 116, 117, 119, 125, 129, 130, 134, 135, 140, 141, 155, 158, 160, 161 electromagnetic fields, 130 electromagnetic wave(s), 19, 103, 104, 105, 108, 114, 117, 119, 125, 134, 135, 158, 160, 161 electron, 2, 10, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 30, 31, 33, 34, 35, 36, 38, 40, 41, 45, 46, 47, 51, 58, 59, 69, 92, 93, 95, 96, 97, 98, 99, 103, 109, 110, 112, 113, 114, 115, 117, 125, 126, 127, 129, 130, 133, 137, 139, 140, 141, 142, 155, 156, 157, 158, 159, 160 electron charge, 113, 158 electron density, 27, 155 electron diffraction, 34 electron microscopy, 35 electrophoresis, 165 electrostatic interactions, 157 emission, 26, 27, 41, 45, 109, 125, 126, 132, 155, 160

171

enantiomers, 3 energy, 8, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 30, 31, 33, 35, 40, 41, 45, 46, 47, 53, 69, 91, 92, 93, 95, 96, 97, 98, 105, 107, 110, 126, 127, 129, 133, 134, 139, 141, 149, 150, 154, 155, 156, 158, 161 entropy, 4, 64 environment, 7, 8, 15, 46, 110, 111, 112, 133, 160 enzymes, 64 epoxy, 89 equilibrium, 21, 76, 109, 113, 126, 137, 140, 160 equipment, 149, 152 erosion, 14, 90 etching, 2, 157 evanescent waves, 120, 124, 141, 156 evidence, 10, 51, 90, 91 excitation, 9, 25, 27, 28, 41, 43, 132, 133 exciton, 25, 133 exclusion, 92 exercise, 46, 76, 84, 101 expertise, vii exploitation, 6 exposure, 60, 140, 158 extinction, 39, 154, 156

F fabrication, 4, 49 failure, 87 fatigue, 91 feedback, 62, 124 Fermi level, 23, 25 ferritin, 57 ferromagnets, 165 fibers, 90 film, 38, 39, 50, 152, 153 filtration, 148 flex, 62, 91, 146 flexibility, 90, 101 flight, 8, 9 float, 147 floating, 148 fluctuations, 26, 100 fluorescence, 26, 27, 28, 41, 45, 47, 156 focusing, 30, 67, 103, 105, 117, 122, 135, 155 football, 143, 144, 145 four-wave mixing, 130 fractal dimension, 127 fractal properties, 127 free energy, 52 friendship, viii fullerene, 10, 154 funding, vii, viii

172

Index

G gas phase, 26, 49 gases, 50, 106, 113, 129, 154 generation, 165 germanium, 23, 46 glass(es), 2, 26, 33, 120, 121, 132, 147, 152, 153, 158 gold, 13, 14, 18, 38, 39, 51, 52, 53, 54, 55, 62, 63, 140, 150, 152 gold nanoparticles, 14, 18, 38, 39, 51, 52, 53, 54, 55 graphene sheet, 7, 10, 58, 67, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 83, 89, 150, 155, 160 graphite, 4, 6, 7, 8, 9, 10, 11, 15, 16, 17, 49, 58, 67, 68, 69, 88, 89, 90, 152, 153, 154, 156 grass, 27 Great Britain, 13, 51, 166 groups, 15, 49, 53, 54, 61, 76, 97 growth, 50, 51, 52, 110, 127, 143, 150, 156, 165

H Hamiltonian, 22 hands, viii, 12, 155 hardness, 90, 91 healing, 91 heat, 14, 18, 69, 108, 110, 111, 116, 154 heat capacity, 14, 18 heating, 14, 51 height, 31, 36, 37, 38, 39, 63, 86, 93, 124 helium, 8, 9, 49 host, 65, 66, 157 HRTEM, 36 human brain, 4 Huygens Christiaan, 1 hydrogen, 15, 50, 52, 58, 64, 65, 66, 157, 166 hydrogen atoms, 15 hydrogen bonds, 65 hydrophobic interactions, 148 hydroxyl, 15 hypothesis, 7, 9, 10, 83

I identity, 35, 44 illumination, 30, 105, 123 images, 11, 12, 34, 35, 50, 51, 59, 61, 63, 100, 128, 155, 157 imaging, 38, 47, 65, 162 IMF, 157 implementation, 6 incidence, 115, 120, 121, 135

India, 162 indices, 107, 111, 140 industry, 49, 66 inelastic, 28, 35, 41, 45, 156, 165 infinite, 71, 89, 94 information technology, 103 injury, iv insight, 46 instruments, 1, 12, 39, 105 insulators, 25, 46 integrated circuits, 2, 60 integration, 9 intensity, 28, 29, 30, 39, 40, 41, 107, 123, 124, 129, 155, 160, 165 Interaction(s), 8, 16, 19, 23, 26, 27, 30, 31, 36, 51, 53, 65, 66, 95, 96, 97, 107, 109, 110, 112, 113, 114, 133, 137, 156, 166 interface, 16, 107, 115, 120, 121, 123, 135, 141, 156 interference, 20, 28, 29, 34, 105, 106, 129, 132, 133, 141, 155, 157 Internet, 84 interpretation, 33 interstellar dust, 7 interval, 104, 112, 115, 120, 160 intravenously, 91 invariants, 75 ions, 37, 52, 65, 139 iron, 5, 34, 51, 57, 94, 96, 148, 156 irradiation, 61, 140, 150, 159 isotropic media, 107

J justification, 17, 26

K KBr, 141 kinetics, 165

L language, 100, 104 laser(s), vii, 8, 9, 14, 18, 36, 41, 49, 61, 62, 66, 122, 123, 130, 132, 133, 134, 140, 146, 149, 150, 157, 165, 166 laser ablation, vii, 49, 61, 62, 66, 150, 157 laser radiation, 132 laws, 3, 21, 31, 117, 134, 155 learning, 51, 67, 104 lens, 32, 33, 135 lifetime, 113, 157

Index ligand, 64 light beam, 29, 106, 117, 120, 135, 136 light emitting diode, 14 light scattering, vii, 28 limitation, 1, 2, 122 literature, viii, 64 lithography, 2, 59, 60, 61, 63, 66, 122, 157, 158, 161 liver, 100 liver cells, 100 localization, 109, 124 location, 8, 11, 31, 34, 36, 37, 72, 76, 91, 100, 104, 144 Louisiana, 165 low temperatures, 46

M machinery, 99 magnet, 133, 148 magnetic field, 19, 30, 34, 92, 93, 94, 95, 96, 97, 99, 100, 104, 105, 106, 107, 134, 155, 156, 157, 158 magnetic moment, 92, 93, 95, 96, 97, 98, 100, 155, 156, 157, 158, 160 magnetic particles, 149 magnetic properties, viii, 14, 85, 91, 92, 93, 97, 100, 133 magnetic resonance, 100 magnetic resonance imaging, 100 magnetism, 96, 97, 100, 157 magnetization(s), 93, 94, 95, 96, 97, 98, 99, 103, 135, 149, 157 magnetoresistance, 97, 98, 99, 157 manganese, 97 manipulation, 4, 63 manufacturing, 4, 12 mapping, 147 Mars, 144 massive particles, 21 mathematics, vii, 140, 165 matrix, 2, 25, 84, 89, 90, 157 measurement, 105, 112, 160 measures, 11, 38, 111 mechanical properties, 14, 85, 90, 91 media, 107, 120, 121, 134, 135, 136, 156, 157 medicine, 1, 103 melting, 14 melting temperature, 14 membranes, 14 memory, 91, 101 Mercury, 144 meridian, 77, 78, 79 mesoporous materials, 55 metal ions, 51, 55, 65

173

metal nanoparticles, 103, 129, 130, 132, 139 metal salts, 51, 66 metals, 13, 18, 23, 30, 67, 76, 96, 116, 137, 141, 142, 153 methane, 50, 66 mice, 91 micelles, 56 microcavity, 132 micrometer, 2, 70, 157 microscope, 1, 2, 10, 19, 31, 33, 34, 38, 47, 51, 62, 124, 129, 152 microscopy, 31, 32, 36, 38, 47, 91, 122, 124, 143, 147, 157, 159 microtubules, 11, 162 microwaves, 19 minerals, 161 Missouri, 166 mobility, 23 models, 1, 109, 126, 127 modulus, 68, 69, 87, 88, 89, 90, 126, 157, 161 mole, 4, 21 molecular orbitals, 22, 26, 45, 156 molecules, 3, 4, 7, 10, 15, 16, 19, 25, 26, 27, 28, 41, 53, 56, 62, 64, 65, 86, 89, 91, 127, 130, 131, 132, 133, 139, 140, 141, 154, 157, 161, 166 molybdenum, 58 momentum, 92, 160 monomers, 2, 157 Montana, 166 motion, 21, 43, 92, 93, 140, 158, 161 motivation, 85, 143 movement, 26, 36, 39, 106, 159 MRI, 100 multiples, 12

N nanocomposites, 103, 117, 127, 135, 166 nanocrystals, 13, 17, 40 nanodots, 67 nanolithography, 124 nanomaterials, 14, 59, 65, 85, 90, 91, 103, 117, 133, 165 nanometer(s), 2, 3, 4, 6, 19, 33, 97, 103, 104, 116, 117, 122, 124, 125, 127, 129, 132, 137, 143, 145, 146 nanometer scale, 4, 19, 116, 122, 125 nanoparticle synthesis, 52 nanoparticles, vii, viii, 4, 6, 13, 14, 16, 17, 18, 19, 25, 30, 31, 35, 37, 44, 46, 47, 51, 52, 53, 55, 56, 57, 61, 62, 65, 66, 67, 85, 89, 90, 91, 100, 116, 119, 126, 127, 129, 132, 138, 139, 140, 147, 149, 150

174

Index

nanophotonics, 117 nanorods, 62, 63 nanoscale materials, 14, 15 nanostructured materials, 49, 59, 112 nanostructures, 50, 51, 59, 61, 64, 66, 99, 112, 116, 127, 129, 130, 132, 133, 140, 141, 160, 161 nanosystems, 100 nanotechnology, iv, vii, 2, 3, 4, 6, 7, 10, 12, 18, 35, 40, 46, 55, 67, 100, 104, 126, 134, 135, 140, 141, 165 nanotube(s), 4, 6, 7, 10, 11, 12, 15, 35, 38, 40, 42, 43, 45, 46, 47, 49, 50, 51, 56, 62, 67, 68, 69, 70, 71, 73, 74, 77, 78, 80, 81, 82, 83, 84, 88, 89, 90, 91, 101, 151, 152, 154, 155, 157, 159, 160, 161, 162 nanowires, 58, 59, 68, 135 National Science Foundation, viii neglect, 116 neodymium, 148 New York, iii, iv, 47, 161, 162, 163, 164, 165, 166 nickel, 49, 94 nitrate, 149, 150 nitrogen, 46, 50, 53 NMR, 7, 100 Nobel Prize, 38, 65 noise, 100 nonlinear optics, 134, 157, 165 nonlinear systems, 165 non-metals, 23 North America, 145 NSOM, 122, 123, 125, 157 nuclear charge, 23, 27 nuclear magnetic resonance, 7 nucleation, 52 nuclei, 22, 23, 27, 31, 133, 156 nucleus, 22, 27, 92, 96, 117, 158

O observations, 21, 51 observed behavior, 160 oil, 2, 50, 157 Oklahoma, 165 one dimension, 15, 26, 31, 77 operator, 22 optical fiber, 123, 125 optical microscopy, 122, 123, 157 optical properties, viii, 14, 25, 32, 103, 109, 112, 114, 126, 127, 129, 133, 139, 141, 142, 159, 165 optics, 2, 25, 103, 104, 109, 114, 116, 119, 122, 124, 129, 130, 134, 135, 141, 157 organic compounds, 15 organic solvent, 56

orientation, 28, 29, 63, 97, 98, 119, 133, 134, 137, 154, 158 orientation state, 133 oscillation, 104, 105, 106, 108, 109, 110, 111, 113, 114, 117, 126, 130, 141, 155, 157, 158, 160 oxidation, 52, 66, 153 oxide nanoparticles, 57 Oxygen, 3, 53, 65

P pairing, 53 parameter, 82, 116, 127, 130, 158 Paris, 97 particle mass, 31 particles, 3, 4, 7, 13, 14, 15, 18, 21, 23, 25, 26, 27, 28, 29, 30, 31, 33, 37, 40, 44, 46, 51, 52, 61, 64, 97, 100, 103, 110, 114, 117, 125, 127, 129, 137, 138, 139, 149, 150, 154, 156, 157, 160, 161, 165 passivation, 15, 44, 53, 158 peptides, 165 performance, 99 periodicity, 76, 104, 119 permeability, 134, 157 permit, 140 permittivity, 113, 115, 125, 158 pH, 152 phase transitions, 166 phonons, 21, 27 phosphorescence, 26, 27, 47, 158 phosphorous, 13, 51, 53 photolithography, 2, 158 photoluminescence, 131, 132 photonics, 127, 130, 132 photons, 20, 27, 30, 61, 158 physical chemistry, 165 physical properties, 14, 64 physics, vii, viii, 39, 92, 97, 98, 103, 105, 116, 117, 119, 125, 126, 130, 142, 162, 163, 164, 165, 166 pigments, 27 planets, 144 plasma, 8, 61, 116, 141, 157, 158 platinum, 38, 61, 62, 66, 152, 153 Poland, viii polarizability, 27, 28, 41, 159 polarization, 98, 105, 114, 119, 137, 158, 159 polarized light, 105 polymer(s), 60, 139, 165 polystyrene, 59 population, 93 potential energy, 140 power, 91, 107, 108, 121, 125, 126, 127, 132, 141, 152, 154

Index prediction, 82 president, 166 pressure, 49, 68, 134 probability, 31, 38, 126, 160, 161 probe, 31, 36, 37, 38, 137 production, 2, 57 productivity, 133 promote, 23 propagation, 19, 21, 47, 104, 105, 106, 107, 108, 114, 122, 134, 159, 160, 161 proportionality, 19, 26, 39, 87, 93, 94, 154, 156, 157 protective coating, 49 protein(s), 14, 64, 57 pulse, 8, 14 pure water, 61, 62

Q quanta, 20, 126 quantum computing, 133 quantum confinement, 14, 40 quantum dot(s), 13, 25, 127, 133, 159 quantum mechanics, 17, 19, 21, 27, 31, 156, 159 quantum state, 26, 126, 133, 159, 161 quartz, 50, 132

R Radiation, 19, 27, 28, 30, 41, 45, 103, 104, 105, 107, 108, 109, 110, 116, 117, 119, 120, 122, 125, 126, 129, 132, 133, 141, 142, 154, 155, 156, 157, 159, 160 radio, 19, 104 radius, 4, 14, 18, 40, 88, 90, 125, 127, 129, 144, 156, 157, 159 Raman, 28, 41, 42, 43, 47, 159, 164 Raman spectra, 43 Raman spectroscopy, 41, 42, 43 range, 39, 40, 68, 89, 96, 104, 112, 115, 116, 120, 127, 135, 139 reactant, 49, 50 reading, vii, viii, 72, 97 reagents, 149 reality, 31 recall, 6, 15, 17, 22, 24, 96 recognition, 66 reconstruction, 15, 17, 158 red blood cell, 145 reduction, 13, 51, 58, 66, 91, 139, 153 reflection, 116, 141, 160 reflectivity, 18, 107, 120 refraction index, 134, 135

175

refractive index, 103, 106, 107, 108, 112, 114, 115, 116, 120, 134, 135, 141, 142, 154, 158, 159, 160 refractive indices, 111, 112, 114, 120, 121, 140 regulation, 57 relationship(s), 18, 29, 50, 87, 88, 94, 125 relaxation, 112 resistance, 90, 97, 98, 99, 125, 126, 152, 157 resolution, 2, 5, 30, 33, 36, 38, 60, 61, 65, 90, 117, 119, 120, 122, 123, 124, 157, 159 resonator, 132 returns, 25 rhodium, 38 rigidity, 87 rings, 10, 65, 89, 133, 156 risk, 91 rods, 63 rolling, 71, 73, 88, 151 Romania, 165 room temperature, 17, 94, 97, 100 Royal Society, 166 Russia, 166

S salt(s), 2, 13, 51, 66, 76, 139, 157 sample, 31, 32, 33, 34, 35, 39, 41, 43, 47, 86, 87, 88, 122, 123, 124, 137, 139, 140, 148, 156, 158, 159, 160 satisfaction, 35 saturation, 15, 16, 17, 49, 94, 155, 159 scanning tunneling microscope, 158 scatter, 27, 28, 32, 125 scattered light, 28, 29, 159 scattering, 28, 41, 43, 98, 155, 159, 165 Schrödinger equation, 22, 31 science, vii, 140, 142, 165 seed, 149, 150 self-assembly, vii, 60, 64, 148, 150, 159 semiconductor(s), 13, 18, 23, 24, 25, 46, 49, 67, 82, 97, 127, 133, 141, 159, 166 sensing, 99, 130, 132 sensitivity, 100 sensors, 99, 100 separation, 15, 100, 117, 137, 155 series, 33, 34, 150, 157 shape, 13, 22, 23, 28, 35, 36, 38, 43, 51, 55, 74, 91, 114, 117, 119, 124, 127, 133, 137, 138, 139, 158, 160 shear, 86, 87, 159 shock, 99 sign, 95, 135, 157 signaling, 64 silicon, 16, 24, 38, 39, 46, 60, 97, 158, 165

176 silver, 37, 129, 132, 135, 137, 138, 139, 140, 142, 149, 150, 162 Singapore, 161 sites, 28, 30, 52, 159 skeleton, 46 skills, 2 skin, 116, 141, 142, 160 sludge, 148 soccer, 7, 154 sodium, 52, 58, 149 solid state, 2, 92, 117, 125 solvent, 3, 59, 60 sorting, 99 species, 7, 8, 9, 10, 17, 39, 64, 156 spectroscopy, 4, 35, 40, 42, 45, 47, 112, 114, 132, 156, 158, 160, 165 spectrum, 7, 9, 19, 20, 21, 30, 39, 43, 45, 46, 126, 138, 139, 141, 150 speech, 6, 18, 91 speed, 19, 30, 99, 106, 107, 108, 112, 114, 125, 160, 161 speed of light, 19, 30, 106, 112, 114, 125, 160, 161 spelling, 5 spin, 27, 92, 93, 95, 96, 97, 98, 133, 155, 156, 158, 160 sputtering, 152 St. Louis, 166 stability, 10, 70, 158 stabilization, 53, 139 stages, 139 stimulus, 153 STM, 38, 39, 47, 158 stoichiometry, vii, 3, 155 storage, 85, 91, 97, 99, 100 strain, 16, 85, 86, 87, 89, 90, 157, 159, 160 strength, 26, 27, 39, 40, 68, 69, 85, 89, 90, 91, 93, 95, 97, 100, 105, 107, 118, 119, 120, 157 stress, 68, 69, 85, 86, 87, 89, 90, 155, 157, 159, 160 stretching, 87 strikes, 27, 120, 141 students, vii, viii, 140, 146 suicide, 3 sulfur, 53, 62 Sun, 35, 144, 162, 164 supply, 152 surface area, 15, 16, 148 surface energy, 160 surfactant, 56, 61, 62 susceptibility, 93, 94, 155, 157 suspensions, 13 Switzerland, 10 SWNTs, 11, 35, 38, 40, 43, 47, 49, 50, 51, 67, 68, 69, 72, 89, 90

Index symmetry, 127, 156, 165 symptoms, 91 synthesis, 3, 4, 10, 55, 139, 143, 150, 165 systems, 6, 18, 49, 64, 65, 85, 90, 97, 99, 133, 140, 159

T tantalum, 14, 90 teaching, vii technical assistance, viii technology, 1, 2, 3, 4, 6, 117, 119 telecommunications, 135 TEM, 35, 36, 37, 43, 44, 47, 50, 51, 52, 61, 90, 150 temperature, 18, 21, 23, 93, 94, 95, 96, 100, 104, 149, 155, 160 tensile strength, 12, 89, 160 tensile stress, 85, 87, 160 terraces, 58 textbooks, viii, 166 thallium, 141 theory, vii, 19, 21, 40, 46, 68, 69, 82, 89, 90, 97, 125, 165 thermal energy, 21, 95, 97 thermalization, 9 thermodynamics, 52 thin films, 61, 143, 154 thinking, 99 time, vii, 1, 2, 3, 4, 6, 7, 8, 9, 13, 15, 18, 20, 27, 31, 35, 51, 62, 83, 89, 100, 103, 104, 105, 108, 117, 125, 126, 133, 134, 140, 144, 155, 158, 160 timing, 99 tin, 77, 152 tin oxide, 152 tissue, 100 titanium, 14, 89, 90 toluene, 10 total internal reflection, 121, 123, 156, 158, 160 transducer, 40 transistor, 2 transition(s), 27, 30, 39, 40, 41, 94, 126, 127, 133, 156, 159 transition metal, 94 translation, 76, 77, 78, 160 transmission, 34, 36, 47, 51, 160 Transmission Electron Microscopy (TEM), 33, 36, 164 transmits, 26, 60 transparency, 70, 150 transparent medium, 141 transport, 25, 98 tungsten, 12, 30, 38, 68, 90 tungsten carbide, 90

Index tunneling, 10, 31, 38, 47, 160

U undergraduate, vii, 140 uniform, 12, 55, 89, 155 uranium, 30 urea, 3 urine, 91 UV, 40, 60, 140 UV light, 140

V vacuum, 16, 30, 106, 108, 112, 122, 125, 148 valence, 22, 23, 24, 25, 30, 31, 40, 46, 69, 96, 109 validity, 19, 83 values, 21, 40, 43, 68, 70, 74, 75, 80, 82, 87, 92, 99, 105, 106, 108, 117, 120, 126, 142, 144 variability, 68 variable(s), 4, 152 variation, 23, 38, 112 vector, viii, 29, 72, 73, 74, 75, 78, 79, 80, 83, 86, 87, 104, 105, 117, 134, 151, 154, 155, 159, 160 velocity, 21, 30, 47, 104, 106, 108, 109, 110, 112, 116, 117, 126, 134, 141, 158, 159, 160, 161 Venus, 144 vibration, 93 viruses, 65 vision, 61

177

W watches, 1 wave vector, 108, 134 wavelengths, 27, 30, 40, 104, 116, 120, 129, 139 wear, 14, 90 web, 147, 154 weight ratio, 68 wind, 8, 143 wires, 2, 91, 133, 135, 159 Wisconsin, vii, 165, 166 wood, 68, 110 writing, 11

X x-ray(s), 19, 28, 29, 30, 33, 34, 35, 41, 43, 44, 45, 47, 161, 165 X-ray diffraction, 28, 34, 44, 161

Y yield, 3, 4, 6, 12, 20, 55, 64, 66

Z zeolites, 55