Introduction

In this chapter we will turn to the arena of quantum optics for illustrative examples of the use of the master equation discussed in Chapters 3 and 6. In fact, quantum optics examples have already been utilized in Chapter 2 as introduction to the density matrix. The principal focus here will be quantum damping in these systems, that is, the damping effect on an atom in interaction with the electromagnetic field as a reservoir. Damping is discussed extensively in Chapter 17 in connection with decay-scattering systems. For this system general phase space distribution functions will be reexamined. To some degree, this has already been done in Chapter 2 with the introduction of the Glauber–Sudarshan P (α a*) function. The micromaser will be discussed as a modern and interesting example of the dynamic interaction of an atom with an electromagnetic cavity not in equilibrium. For use of the student, an appendix to this chapter will briefly review the quantization of the free electromagnetic field and its atomic interaction.

There is no possibility of reviewing this extensive and growing field here. Our desire in this chapter is to connect the general topics of this book to this example. The books of Louisell (1973) and Scully and Zubairy (1997) are excellent. We are also indebted to the work of Nussenzweig, Schleich, and Mandel and Wolf (Nussenzweig, 1973; Mandel and Wolf, 1995; Schleich, 2001). We also recall the fine early introduction to this topic by Agarwal (1973).

Atomic damping: atomic master equation

In this section we shall consider the so-called quantum optics master equation for the reduced atomic density operator, ρA (t), in the Born approximation. The elements of this derivation have already been discussed in Chapter 3 with the derivation of the Pauli equation for 〈α|ρA |α〉. Here the generalization to off-diagonal contributions only adds complication.

The preparation of carbon nanotubes by catalysis has a number of potential advantages over the arc and laser methods discussed in the previous chapter. In particular, catalysis (or chemical vapour deposition, as the process is often called) is much more amenable to scale-up than arc- or laser-evaporation, and many successful processes for the large-scale catalytic synthesis of both SWNTs and MWNTs have been developed. Catalytic techniques also enable nanotube synthesis to be achieved under relatively mild conditions, giving more control over the growth process. Thus, it is possible, using catalytic methods, to grow arrays of aligned nanotubes on substrates. Such arrays are showing great promise as field-emission displays. It may also be possible to construct nano-electronic circuits by using catalysis to grow defined networks of nanotubes. It is widely believed that the main disadvantage of catalytic methods is that nanotubes produced in this way are structurally inferior to those made by the high-temperature arc and laser techniques. While this may still be true for multiwalled tubes, there is evidence that catalyticallyproduced SWNTs can have a high degree of structural perfection, as noted at the end of the last chapter. This may suggest that a common growth mechanism is involved.

The long history of filamentous carbon production by catalysis was outlined in Chapter 1. The present chapter begins with a brief summary of the work carried out in the 1970s and 1980s by Baker, Endo and others, but concentrates mainly on post-1991 research. Methods for growing aligned MWNTs on substrates, and for producing nanotube yarns are described. The structure and possible growth mechanisms of catalytically produced MWNTs are then discussed.

The development of new methods for imaging, measurement and sensing is an important theme in modern research, and the unique properties of carbon nanotubes give them great potential in these areas. For example, nanotubes’ outstanding mechanical properties and unique geometry suggest that they should be ideal tips for atomic force microscopy (AFM). Currently, AFM tips typically consist of microfabricated pyramids of silicon or silicon nitride mounted on cantilevers. These probes can be relatively ‘blunt’ on the scale of the features that are being imaged, and are thus often unable to probe narrow crevices on the specimen surface. Carbon nanotubes, with their elongated shape and tiny diameter not only offer the possibility of much higher resolution imaging, but are also capable of probing the narrowest of fissures. The potential advantages of carbon nanotube tips are illustrated in Fig. 11.1, from the work of Arvind Raman of Purdue University and colleagues (11.1), which shows a nanotube attached to a conventional microfabricated probe. Preparing nanotube AFM tips is not straightforward, however. Two methods can be used: attaching previously produced tubes to the probes, or growing the tubes in situ. The first part of this chapter summarizes the methods available for preparing nanotube AFM tips. The performance of nanotube AFM tips is then discussed.

Gas sensing is another area where the properties of carbon nanotubes can be exploited. The discovery by Alex Zettl's group that the electronic properties of carbon nanotubes are highly sensitive to the presence of oxygen (11.2) was mentioned in Chapter 6 (p. 167).

This book was originally conceived as a second edition of my earlier work Carbon nanotubes and related structures: new materials for the twenty-first century (Cambridge University Press, 1999). However, the field has expanded rapidly since 1999, and the tale grew in the telling, to the point where I realized I had essentially written a new book. The new title reflects this, as well as the fact that most of the material concerned with ‘related structures’ has been omitted: the book now focuses almost entirely on carbon nanotubes themselves. As with the first book, I have benefited enormously from the freely given assistance of colleagues from around the world, many of whom have also provided copies of images and preprints. The following list almost certainly fails to include all who have helped me, so I apologize for any omissions. I also stress that any errors which remain in the book are my responsibility alone.

I wish to thank: Pulickel Ajayan, Lizzie Brown, Marko Burghard, Hui-Ming Cheng, Hongjie Dai, Walt De Heer, Cees Dekker, Chris Ewels, John Gallop, Jason Hafner, Michael Holzinger, Martin Hulman, Kaili Jiang, Hiromichi Kataura, Ian Kinloch, Ralph Krupke, Alan Lau, Cheol Jin Lee, Jannik Meyer, Geoff Mitchell, Pasha Nikolaev, Henk Postma, Zhifeng Ren, Daniel Resasco, Andrew Rinzler, Milo Shaffer, Wenhui Song, Kazu Suenaga, Sander Tans, Kenneth Teo, Edman Tsang, Daniel Ugarte, Bruce Weisman and Karen Winey.

I would also like to thank Cambridge University Press for their encouragement and patience.

Most importantly, I want to thank my wife, Elaine, and daughters Katy and Laura for their continuing love and support.

As we saw in the last chapter, the excitement surrounding carbon nanotubes was originally sparked by Iijima's production of highly perfect multiwalled tubes using arc-evaporation in 1991. Although the catalytic production of carbon tubules had been known for decades, the structures discovered by Iijima displayed a degree of perfection much greater than those seen in catalytic tubes. The first synthesis of single-walled nanotubes in 1993 also involved arc-evaporation, this time with metal-impregnated electrodes. Arc-evaporation remains an important method of nanotube synthesis, and will be discussed in detail in this chapter. The chapter begins with a description of the practical aspects of the arc synthesis of multiwalled nanotubes. A summary of the various models that have been put forward for the growth of multiwalled tubes in the arc is then given. This is followed by a discussion of the production of multiwalled nanotubes by high-temperature heat treatment of disordered carbon. The synthesis of single-walled carbon nanotubes by arc-evaporation and by laser-vaporization is then covered, and the possible mechanisms of nanotube formation in these processes summarized. Finally, the arc synthesis of double-walled nanotubes is described.

Production of multiwalled nanotubes by arc-evaporation

Early work

The original method used by Iijima to prepare nanotubes (2.1) differed slightly from the Krätschmer–Huffman technique for C60 production in that the graphite electrodes were held a short distance apart during arcing, rather than being kept in contact. Under these conditions, some of the carbon which evaporated from the anode recondensed as a hard cylindrical deposit on the cathodic rod. It was the central part of this deposit that Iijima found to contain both nanotubes and nanoparticles.

Soon after the discovery of fullerene-related carbon nanotubes it became clear that a completely new framework would be needed to analyse the structures and symmetries of these new materials. Although theoretical methods have been developed for analysing cylindrical arrays in biology (5.1), these are insufficient for a full analysis of nanotube structure. The challenge of formulating the new approaches necessary for classifying nanotube structure was taken up by Mildred Dresselhaus and co-workers, and by Carter White and colleagues, among others. The techniques developed by these groups have been essential in determining the electronic and vibrational properties of nanotubes, as discussed in later chapters. Theoretical discussions have also been given of the layer structure of multiwalled tubes, of tube capping and of other aspects of nanotube structure such as elbow connections.

Experimental studies of nanotube structure have mainly been carried out using microscopy. X-ray and neutron diffraction have generally been of less value, since samples of nanotubes always contain tubes with a wide range of different structures. High-resolution transmission electron microscopy (HRTEM) has been by far the most widely used and most valuable technique. Recent improvements in the resolution of HRTEM have meant that the atomic network which makes up individual tubes can be imaged directly, as well as the layer structure, opening the way for a deeper understanding of their structure. The use of spectroscopic techniques to probe nanotube structure is discussed in Chapter 7.

The present chapter begins with a brief discussion of bonding in graphite and fullerenes. Theoretical models of carbon nanotube structure are then summarized.

Attempts to introduce foreign materials into the empty central cavities of carbon nanotubes began soon after the appearance of Iijima's paper in 1991. This work was driven partly by curiosity – the empty cavities represented ideal ‘nano-test-tubes’ for the study of matter in confined spaces – and partly by the idea of using nanotubes as templates for nanowires. There was also interest in the possibility of filling opened tubes with catalytic metals, to produce new size-selective catalysts. Multiwalled nanotubes were first successfully filled in 1993, using a technique that involved carrying out arc-evaporation in the usual way, but with an anode containing some of the material to be encapsulated. This method generally seems to favour the formation of filled nanoparticles rather than nanotubes, and is only applicable to materials that can survive the extreme conditions of the electric arc. A more generally applicable method, in which the tubes were opened and filled by chemical means, was introduced a short time later. This has now been applied to a wide range of materials, including biological molecules. Single-walled tubes were first opened and filled in 1997, and some fascinating work has been carried out on the effect of confinement in SWNTs on the structure of crystalline materials. There is also great interest in filling SWNTs with fullerenes. In the first part of this chapter the methods used to open and fill carbon nanotubes, and the new science and possible applications that are emerging from this work, will be summarized.

The second part of this chapter gives a brief overview of heterogeneous nanotubes, defined as nanotubes whose carbon atoms are partially substituted with hetero-atoms, typically nitrogen and/or boron.

The mechanical properties of carbon nanotubes have attracted just as much interest as their electronic properties, and with good reason. It is now well established that nanotubes are the stiffest and strongest fibres ever produced. Thus, the Young's modulus of the best nanotubes can be as high as 1000GPa, approximately five times higher than steel, while their tensile strength can be up to 63GPa, around 50 times higher than steel. These properties, coupled with their low density, give nanotubes huge potential in a whole range of structural applications. In many cases, exploiting these properties involves incorporating the tubes into composite materials, and this is discussed in Chapter 9. The present chapter covers the mechanical properties of carbon nanotubes in detail, beginning with a review of the theoretical predictions and then giving a summary of experimental measurements on multi- and single-walled nanotubes. Optical properties of carbon nanotubes are considered next, and the application of various forms of spectroscopy to nanotubes is described. This is followed by a brief discussion of the thermal properties of nanotubes, and finally some comments on the physical stability of nanotubes.

Mechanical properties of carbon nanotubes

Theoretical predictions

Before discussing detailed calculations of the mechanical properties of nanotubes, we can carry out some simple calculations to illustrate the relationship between the diameter of a nanotube and its stiffness. Consider first a tube with an inner diameter of 1 nm.

One of the most amazing characteristics of carbon nanotubes is that they can be metallic, like copper, or semiconducting, like silicon, depending on their structure. These properties were first predicted theoretically and then confirmed experimentally by the extraordinarily skilful application of techniques such as scanning tunnelling microscopy. By the late 1990s, nanotube-based devices such as diodes and field effect transistors were being constructed, and in 2001 the first logic gate based on a single nanotube bundle was reported. At the same time, many groups around the world have explored the field emission properties of nanotubes, for potential applications in display devices. Research into the electronic properties of carbon nanotubes represents a remarkably dynamic and fast-moving field, keeping abreast of which is no easy task. Nevertheless, some of the early work in this area, both theoretical and experimental, has stood the test of time, and many of the pioneering papers have become established classics.

This chapter begins with a brief summary of the electronic structure of graphite and then shows how this has been used as a basis for a theory of the electronic properties of carbon nanotubes. Experimental measurements on the electronic properties of nanotubes are then reviewed and the use of nanotubes in electronic devices described. The magnetic properties of nanotubes are summarized. Finally, the field emission properties of carbon nanotubes are discussed.

Electronic properties of graphite

As one would expect from its structure, the electronic properties of graphite are highly anisotropic.

Many of the outstanding properties of carbon nanotubes that were outlined in Chapters 6 and 7 and elsewhere in this book can be best exploited by incorporating the nanotubes into some form of matrix. The exceptional mechanical properties in particular have prompted huge interest in the production of nanotube-containing composite materials for structural applications. In many cases these composites have employed polymer matrices, but there is also interest in other matrix materials such as ceramics and metals. Preparing such composites is not without its difficulties, however, owing to the tendency of nanotubes to stick together, the challenge of forming bonds between tubes and matrix, and problems associated with the physical properties of some nanotube–matrix mixtures. The aim of this chapter is to give an overview of the very large amount of work that has been carried out on carbon nanotube composites, and to assess how successful this work has been in utilizing the full potential of nanotubes. The incorporation of nanotubes into polymer matrices is considered first.

Preparation of carbon nanotube/polymer composites

Solution mixing

Perhaps the simplest method for preparing nanotube/polymer composites involves mixing nanotube dispersions with solutions of the polymer and then evaporating the solvents in a controlled way. This method has been used with a range of polymers, including polyvinyl alcohol, polystyrene, polycarbonate and poly(methyl methacrylate). In order to facilitate solubilization and mixing, the nanotubes are often functionalized prior to adding to the polymer solution.

Carbon in its miscellaneous forms has been used in art and technology since prehistoric times (1.1–1.4). Some of the earliest cave paintings, at Lascaux, Altamira and elsewhere, were produced using a mixture of charcoal and soot. Charcoal, graphite and carbon black (a pure form of soot) have been used as drawing, writing and printing materials ever since: photocopier toner is largely composed of carbon black. Coal and charcoal, of course, have been used as fuels for millennia, and charcoal played an important role in what might be considered humankind's first technology, the smelting and working of metals. Charcoal was used in this way right up to the eighteenth century, when it began to be replaced by coke, a development which helped to stimulate the Industrial Revolution. With the development of the electrical industry in the late nineteenth century, a demand developed for graphite. The American Edward Acheson is credited with producing the first synthetic graphite in 1896. In the twentieth century, the importance of activated carbon in purifying air and water supplies grew steadily, and the invention of carbon fibres in the 1950s provided engineers with a new lightweight, ultra-strong material. Diamonds, like graphite, have been known since antiquity, but until quite recently were only used decoratively. The development of a commercial synthetic method at General Electric in the 1950s opened the way for the industrial use of diamonds.

The history of carbon science is littered with illustrious names. Antoine Lavoisier, in a famous experiment in 1772, proved that diamonds are a form of carbon by demonstrating that they produce nothing but carbon dioxide on combustion.

Few materials can have been as intensively studied as carbon nanotubes. As a result of the enormous amount of work carried out on these structures since 1991, we now have a wealth of information on their electronic, mechanical, thermal, optical and other physical properties. We know how to attach things to the outside of tubes and to put things inside. We can manipulate nanotubes into defined arrangements and incorporate them into polymer, ceramic or metal matrices. At present, however, there are still relatively few nanotube-containing products on the market. The few commercial products that have emerged are in highly specialized areas such as AFM tips. Perhaps the most commercially significant application of nanotubes to date has been as components of lithium ion batteries, although these employ relatively poor quality catalytically-produced MWNTs. It would be fair to say, then, that carbon nanotubes have not yet fulfilled their potential. The main reason for this is quite simple: they are still far too expensive. The price of single-walled nanotubes is currently around $100 per gram, compared with about $30 per gram for gold. Clearly there is still much work to be done in improving the quality and yield of nanotube production.

In this final chapter, a brief and subjective summary is given of some of the highlights of carbon nanotube research. Some comments are then made on the areas where progress still needs to be made.

Highlights of carbon nanotube research

Any list of classic papers on carbon nanotubes would probably have to begin with Sumio Iijima's 1991 letter to Nature (12.1).

None of the techniques discussed in the previous two chapters produce pure carbon nanotubes. When multiwalled nanotubes are produced by arc-evaporation, for example, they are always accompanied by nanoparticles and disordered carbon. When produced catalytically, nanotubes are contaminated with catalyst particles and support material. In order to make use of the nanotubes, therefore, it is often necessary to purify them. This chapter begins with a summary of the chemical and physical methods available for purifying nanotubes. Techniques for processing nanotubes are then reviewed. Procedures for aligning tubes and for forming them into fibres and sheets are covered, and methods for sorting them by length and structure outlined. Ways of improving the solubility of nanotubes by functionalization are covered later in the book (Chapter 8), while the incorporation of nanotubes into composite materials is discussed in Chapter 9.

Purification of multiwalled tubes

MWNTs produced by arc-evaporation

The first successful technique for purifying multiwalled tubes produced by arc-evaporation was described by Thomas Ebbesen and co-workers in 1994 (4.1). Following the demonstration that nanotube caps could be selectively attacked by oxidizing gases (see Chapter 10), these researchers realized that nanoparticles, with their defect-rich structures might be oxidized much more readily than the relatively perfect nanotubes. Therefore they subjected raw nanotube samples to a range of oxidizing treatments, in the hope that the nanoparticles would be preferentially oxidized away. They found that a significant relative enrichment of nanotubes could be achieved in this way, but only at the expense of losing of a major proportion of the original sample.

The first insights into the chemistry of carbon nanotubes grew out of efforts to open and fill the tubes in the early 1990s. As described in Chapter 10, this work clearly demonstrated one very basic aspect of nanotube chemistry: they are most reactive at the tips. Indeed, the reaction of nanotubes with acids and other reagents could be exquisitely selective, with attack occurring only at defective regions. Since this early work, interest in the functionalization of carbon nanotubes has grown rapidly. In many studies the aim of functionalization has been simply to solubilize the tubes, by attaching hydrophilic species to these normally hydrophobic structures. However, as will be seen, the chemical modification of nanotubes is proving to be valuable in a wide range of areas, from the preparation of carbon nanotube composites to the production of sensors. As well as covalent functionalization, there is great interest in ‘non-covalent functionalization’, i.e. connecting molecules to nanotubes without actually forming chemical bonds. The advantage of this approach is that it avoids disrupting the structure of the tubes, enabling their full properties to be retained. Non-covalent functionalization can be achieved by forming van der Waals bonds between planar groups and the tube walls, or by wrapping molecules helically round the tubes. This chapter begins with an overview of the methods that can be used to chemically functionalize carbon nanotubes. This is followed by a discussion of functionalization with biomolecules. Interest in this area was greatly stimulated by the demonstration that functionalized nanotubes can cross cell membranes, suggesting that tubes could be used to ferry therapeutic agents into cells.

Single-particle Green's functions, density response functions and other correlation functions are calculated in many different ways in the literature on Bose-condensed gases. An in-depth comparison and classification of different approaches was first given in the classic paper by Hohenberg and Martin (1965), with an emphasis on the various exact identities (conservation laws) that are satisfied. A key feature to be included in any theory is that a Bose broken symmetry leads to a hybridization of the single-particle excitations and the collective density fluctuations in such a way that the two excitation spectra become identical. This key feature is demonstrated in Chapter 5 of Griffin (1993).

How to relate and assess various approximations for correlation functions in a Bose superfluid has been a topic of continual interest (and some controversy) since the late 1950s. These questions were largely resolved by the early 1960s at a conceptual level but the detailed applications of the theory were limited to dilute Bose-condensed gases at T = 0. Since it was difficult to relate the theory to the properties of superfluid. He at a quantitative level, this formalism based on a Bose broken symmetry was of little interest to experimentalists. The creation of superfluid Bose condensed gases in 1995 changed all this and has given new life to the many body theory of dilute weakly interacting Bose condensed gases.

Various approximations for the Beliaev single-particle self-energies Σαβ were derived in Sections 4.3 and 4.4. The discussion in Chapter 4 was somewhat abstract.

Since the dramatic discovery of Bose–Einstein condensation (BEC) in trapped atomic gases in 1995 (Anderson et al., 1995), there has been an explosion of theoretical and experimental research on the properties of Bose-condensed dilute gases. The first phase of this research was discussed in the influential review article by Dalfovo et al. (1999) and in the proceedings of the 1998 Varenna Summer School on BEC (Inguscio et al., 1999). More recently, this research has been well documented in two monographs, by Pethick and Smith (2008, second edition) and by Pitaevskii and Stringari (2003). Most of this research, both experimental and theoretical, has concentrated on the case of low temperatures (well below the BEC transition temperature, TBEC), where one is effectively dealing with a pure Bose condensate. The total fraction of noncondensate atoms in such experiments can be as small as 10% of the total number of atoms and, equally importantly, this low-density cloud of thermally excited atoms is spread over a much larger spatial region compared with the high-density condensate, which is localized at the centre of the trapping potential. Thus most studies of Bose-condensed gases at low temperatures have concentrated entirely on the condensate degree of freedom and its response to various perturbations. This region is well described by the famous Gross–Pitaevskii (GP) equation of motion for the condensate order parameter Φ(r, t). As shown by research since 1995, this pure condensate domain is very rich in physics.

The main goal of the present book, in contrast, is to describe the dynamics of dilute trapped atomic gases at finite temperatures such that the noncondensate atoms also play an important role.