Introduction

We now turn to the problem of describing long-wavelength lattice vibrations in multilayered structures. At a microstructural level this problem is solved by using the techniques of lattice dynamics requiring intense numerical computation. This approach is inconvenient, if not impracticable, at the macroscopic level, where the kinetics and dynamics of large numbers of particles need to be described. In this theoretical regime it is necessary to obtain models that transcend those at the level of individual atoms in order to describe electron and hole scattering, and all the energy and momentum relaxation processes that underlie transport and optical properties of macroscopic structures. The basic problem is to make a bridge between the atomic crystal lattice and the classical continuum. We know from the theory of elasticity and the theory of acoustic waves, which hark back to the nineteenth century, that continuum theory works extremely well for long-wave acoustic waves. The case of optical vibrations is another matter. Here, the essence is one atom vibrating against another in a primitive unit cell and it is by no means obvious that a continuum approach can work in this case. This has been highlighted by controversy concerning the boundary conditions that long-wave optical vibrations obey at each interface of a multilayer structure.

This book has grown out of my own research interests in semiconductor multilayers, which date from 1980. It therefore runs the risk of being far too limited in scope, of prime interest only to the author, his colleagues and his research students. I hope that this is not the case, and of course I believe that it will be found useful by a large number of people in the field; otherwise I would not have written it. Nevertheless, knowledgeable readers will remark on the lack of such fashionable topics as the quantum-Hall effect, Coulomb blockade, quantized resistance, quantum tunnelling and any physical process that can be studied only in the millikelvin regime of temperature. This has more to do with my own ignorance than any lack of feeling that these phenomena are important. My research interests have not lain there. My priorities have always been to try to understand what goes on in practical devices, and as these work more or less at room temperature, the tendency has been for my interest to cool as the temperature drops.

Introduction

An electron in a quantum-well subband can be scattered to another state in the same subband or into a state in another subband. Intrasubband and intersubband scattering rates have to be calculated separately since different wavefunction symmetries are involved in the two cases, and this implies correspondingly different symmetries of the optical mode. For simplicity we will assume that the electrons are completely confined within the well and that the interaction is with polar-optical modes. In the case of LO modes in a polar material this interaction is via a scalar potential. However, as we will see, it is possible in the unretarded limit (velocity of light is infinite) to replace the vector potential of the electromagnetic interface wave with a scalar potential via a unitary transformation (not a gauge transformation) and treat the IP mode on the same footing as an LO mode, but with a frequency-dependent scalar potential. We assume the TO mode has no interaction.

No fewer than four different scattering sources exist, in general. Two of these are associated with well modes, two with barrier modes. In general, the LO band of frequencies in either material does not span the range between the LO and TO zone-centre frequencies ωLO and ωTO.

Single Heterostructures

In bulk material, neutralization of the enormous electric fields produced by the spontaneous polarization can be achieved by a relatively modest adjustment of the population of surface states. This is no longer possible in the case of inhomogeneous material exhibiting inhomogeneous spontaneous polarization, such as the AlGaN/GaN heterostructure. A layer of AlGaN grown epitaxially on GaN has not only a spontaneous polarization larger than that of GaN but it has, in addition, a piezoelectric polarization associated with the elastic strain inevitably present as a consequence of the lattice mismatch. No straightforward adjustment of the population of the surface states can eliminate the resultant fields. As a result, nitride heterostructures differ uniquely from heterostructures of cubic semiconductors in exhibiting large, built-in electric fields, and these result in the spontaneous formation of a quasi-2D electron gas at the AlGaN/GaN interface. In the cubic case such a gas must be produced by doping, thereby, ineluctably, introducing charged-impurity scattering even when the system is modulation doped. Doping is unnecessary in nitride structures so, in principle, enormous lowtemperature mobilities are possible and will, no doubt, be achieved as growth techniques evolve.

An estimate of the density of the polarization-induced electron population at the AlGaN/GaN interface can be obtained assuming the existence of a common branch-point Fermi energy and simple electrostatics.

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.