Introduction

Carbon is an old but new material. It has been used forcenturies going back to antiquity, but yet many new crystalline forms of carbon have only recently been experimentally discovered in the last few decades. These newer crystalline forms include buckyballs, carbon nanotubes (CNTs), and graphene, where the latter two are illustrated in Figure 1.1. Furthermore, carbon nanotubes come in two major flavors, the single-wall and multi-wall varieties, as shown in Figure 1.1a and b respectively. The newer forms of carbon have significantly contrasting properties compared with the older forms of carbon, which are graphite and diamond. In particular, they share in common a hexagonal lattice or arrangement of carbon atoms. In addition, CNTs and graphene occupy a reduced amount of space compared with their older siblings; hence, they are often referred to as reduced-dimensional or low-dimensional solids or nanomaterials for short. To give a comparative (order of magnitude) idea of the critical size scales of these nanomaterials, nanotubes are about 10 000 times thinner than human hair, and graphene is about 300 000 times thinner than a sheet of paper. The typical diameter of nanotubes range from about 1 to 100 nm, and graphene ideally has the thickness of a single atomic layer (∼3.4 Å). Fundamentally, it is the combination of the reduced dimensions and the different lattice structure that leads to the fascinating properties unique to nanotubes and graphene.

Introduction

There are two families of CNTs, namely single-wall CNTs and multi-wall CNTs (MWCNT) as shown in Figure 4.1. A single-wall CNT is a hollow cylindrical structure of carbon atoms with a diameter that ranges from about 0.5 to 5 nm and lengths of the order of micrometers to centimeters. An MWCNT is similar in structure to a single-wall CNT but has multiple nested or concentric cylindrical walls with the spacing between walls comparable to the interlayer spacing in graphite, approximately 0.34 nm. The ends of a CNT are often capped with a hemisphere of the buckyball structure. Carbon nanotubes are considered 1D nanomaterials owing to their very small diameter that confines electrons to move along their length. The central goal of this chapter is to understand the physical structure of CNTs, and to determine their electronic band structures, which will enable us to gain insight into the properties and performance of CNT devices.

The discussion in this chapter requires familiarity with the concepts developed in Chapters 2 and 3, such as crystal lattice and the band structure of graphene. Much of the content of this chapteris essential for the subsequent material presented throughout the book. We will use the acronym CNT to refer specifically to the single-wall variety unless explicitly stated otherwise. MWCNTs will be discussed mostly in the context of interconnect wires in Chapter 7.

Introduction

This chapter explores the equilibrium and thermodynamic electronic properties of single-wall CNTs. Equilibrium refers to the state of a system in the absence of external forces, and thermodynamics accounts for the evolution of the macroscopic properties of the system with temperature. The system we are referring to here is of course CNTs, and the properties of interest include the DOS, group velocity, effective mass, and charge carrier density. These properties are of central importance for understanding and predicting the electrical, optical, and thermal behavior of CNTs. Our learning path will utilize the development of analytical expressions of these properties to provide insight and understanding regarding the inherent solid-state behavior. Additionally, analytical expressions are especially desirable for compact modeling of nanotube devices.

In order to be comfortable with the discussion of the equilibrium properties, the reader should be familiar with the band structure of CNTs developed in the previous chapter. In fact, it will be worthwhile to have in hand a copy of all the band structure figures shown in Chapter 4 as one goes through this chapter. The equilibrium properties will be employed repeatedly in subsequent chapters to describe transport in CNTs under the quasi-equilibrium assumption applicable at low energies. We begin by discussing the DOS of the free-electron gas in 1D space. This provides perspective, allowing us to appreciate and relate to the actual DOS in CNTs which are quasi-1D solids. In general, the DOS is of fundamental importance in understanding crystalline solids, and many of the other equilibrium properties can be derived from this important parameter.

Introduction

The central purpose of this book is to understand the properties of electrons in CNTs and graphene. A good understanding is of utmost importance because it enables us to make electronic devices and engineer the performance of the devices to satisfy our desires. These devices can include, for example, sensors, diodes, transistors, transmission lines, antennas, and electron emission devices. In addition, the devices made out of carbon nanomaterials are being considered as building blocks for future applications broadly referred to as nanoelectronics, which includes circuits and systems. The technology to make nanomaterials and related devices is called nanotechnology.

To accomplish our central purpose, it is essential that we are familiar with the mathematical techniques and physical ideas behind the theory of electrons, particularly in solids. Specifically, in order to understand and describe the behavior of electrons in a solid requires consideration of:

1. (i) The general quantum mechanical wave nature of electrons.

2. (ii) The periodic arrangement of atoms in crystalline solid matter, which is frequently called the crystal structure or lattice.

The introductory discussion of electrons in solids in this chapter will proceed in a manner that is beneficial for developing intuition, by considering an introductory quantum mechanical description of electrons, and subsequently exploring the crystal structure. Our attention throughout the chapter will be focused on the mathematical techniques, central ideas, and main results regarding electrons in solids. In view of the fact that quantum mechanics and solid-state physics are themselves fundamental disciplines of physics of great breadth and depth, this chapter is primarily intended as an elementary review of the minimum basic concepts and techniques in the quantum mechanical description of electrons in solids.

Introduction

This chapter explores electron transport in metallic nanotubes as it relates to interconnect applications. Both single-wall and multi-wall CNTs are considered. The reader will find it beneficial to be familiar with Chapter 4, which discusses the structure of nanotubes, and Chapter 6, which explores ideal nanotube electrical properties, such as the quantum conductance, quantum capacitance, and the kinetic inductance. Employing CNTs as metallic wires to route direct-current (DC) and high-speed signals in an integrated circuit was one of the earliest application ideas promoting nanotubes because of their high current-carrying capability and ballistic transport over relatively long lengths. In this light, we will examine both the low-frequency (lossy) and high-frequency (lossless) transmission line models for single-wall and multi-wall nanotubes in order to elucidate their interconnect properties. These models include bias or field-dependent electron scattering in the nanotube vis-á-vis the mean free path. As such, electron scattering and mean free path will be discussed, although at a somewhat elementary level. Additionally, the temperature and diameter dependence of the electron mean free path and resistance will be highlighted.

At the end of the day, future nanomaterials such asCNTs have to be benchmarked against existing materials to quantify any performance benefits over conventional approaches. For this purpose, we will discuss the performance of CNTs compared with copper, which is the standard metal used in nanoscale integrated circuits today.

Introduction

In analogy to a water pipe that allows the guided flow of water, a transistor is an electronic device that allows for the guided flow of electrons with the key innovation being the influence of a gate that controls the amount of flowing electrons (the gate is similar in concept to a valve controlling the amount of water). The most popular flavor of the transistor is the field-effect transistor (FET), which came to reality in 1960 and forms the cornerstone of modern electronics that has revolutionized computing, communications, automation, and healthcare and fosters today's digital lifestyles. In part due to the continuous miniaturization or scaling of the transistor dimensions, silicon (Si) has evolved to be the de facto semiconductor for making transistors that enable smaller, faster, cheaper, and more power-efficient integrated circuits (also called chips) for an extensive variety of applications. However, transistor scaling and the resulting performance enhancement cannot continue forever owing to both physical and technical reasons. Obviously, the transistor cannot be reduced to a size of zero length for example, and this imposes a physical limit to the miniaturization of devices. Fortunately, we have yet to reach this physical limit. At present, the more pressing issues are technical in nature: relating to the challenges of fabricating small transistors and, in addition, the significance at short size scales of some otherwise undesirable device phenomena which are collectively referred to as short-channel or small-dimension effects.

Introduction

In contemporary fundamental and applied science research, the potential applications and the perceived broader impacts are undoubtedly the primary drivers for expanding the research enterprise. This has certainly been the case for nanotube research. The unique unprecedented properties of CNT, such as their perfect tubular structure, outstanding electrical and thermal conductance, tunable optical properties, and superior mechanical strength and stiffness, have generated great excitement, leading to the pursuit of both fundamental insights of the beauty of nature in reduced dimensions of condensed matter, and the novel applications and technological breakthroughs that can be developed. In essence, the exploration of nanotubes (and other nanomaterials) is to learn about their nature and their interaction with fields and matter that will allow us to synthesize CNTs, design devices, and develop unique materials for next-generation transformative products. This endeavor has brought together many parties across several boundaries of knowledge, from nanomedicine to nanoscience to nanotechnology.

To put CNT in a broader perspective, over the last decade, nanotube applied research and development in academic and industrial laboratories across the world has enjoyed a substantial rise, reflecting a rise in the deeper understanding of the material. Figure 9.1 shows the increase in CNT patent applications and patents issued in the United States. It is an indicator of the growing effort to employ nanotubes in innovative applications. Invariably, many of the applications of CNTs take advantage of their inherent nanoscale dimension, large surface-to-volume ratio, and unique combination of electrical, optical, thermal, and structural properties.

Introduction

Granular materials are ubiquitous throughout nature. From the beauty of sand dunes and the rings of Saturn to the destructive power of snow avalanches and mudslides, the flow of ice floes, and the manner in which plate tectonics determine much of the morphology of the Earth [1–4]. These phenomena arise from the interplay between structural and dynamical properties that result in the collective behavior of a vast number of smaller, distinct entities that we call “grains.” From a technological point of view, granular materials play a dominant role in numerous industries, such as mining, agriculture, civil engineering, pharmaceuticals manufacturing, and ceramic component design. Even apparently the most mundane of activities from coffee bean bag filling at the grocery store to pouring salt onto our dinner plates at night involve various aspects of granular matter mechanics that continue to puzzle us. It is estimated that particulate media are second only to water as the most manipulated material for human usage [4], amounting to trillions of dollars per annum in the US alone. The importance of granular materials to our daily lives cannot be overstated.

In the previous chapter we have considered the properties of strongly correlated electrons. The systems we mostly had in mind were the compounds of transition metal and maybe rare earth elements with partially filled inner d or f shells. We have discussed only the correlated electrons themselves, the prototype model being the Hubbard model (12.1).

When turning to real materials, several extra factors missing in the model (12.1) are important. One of them is the possible influence of orbital degrees of freedom, especially in cases with orbital degeneracy, treated in Section 12.9.

In many situations there is yet another very important factor. There may exist in a system, besides correlated electrons, also electrons of other bands, e.g. electrons in wide conduction bands, responsible for ordinary metallic conductivity. Such is for instance the situation for magnetic impurities in metals, or in the concentrated systems like rare earth metals and compounds in which localized f electrons coexist with the metallic electrons in broad spd bands. The interplay between localized, or, better, strongly correlated electrons and itinerant electrons of the wide bands can lead to a number of very interesting consequences; these will be discussed in this chapter.

Localized magnetic moments in metals

When we put transition metal impurities in ordinary metals (e.g. Mn or Fe in Cu, Au), the result may be two-fold. In certain cases the impurities retain their magnetic moment, but in others they lose it.

Foreword

There are many good books describing the foundations and basics of solid state physics, such as Introduction to Solid State Physics by C. Kittel (2004) or, on a somewhat higher level, Solid State Physics by N. W. Ashcroft and N. D. Mermin (1976). However there is a definite lack of books of a more advanced level which would describe the modern problems of solid state physics (including some theoretical methods) on a level accessible for an average graduate student or a young research worker, including experimentalists.

Usually there exists a rather wide gap between such books written for theoreticians and those for a wider audience. As a result many notions which are widely used nowadays andwhich determine ‘the face’ of modern solid state physics remain ‘hidden’ and are not even mentioned in the available literature for non-specialists.

The aim of the present book is to try to fill this gap by describing the basic notions of present-day condensed matter physics in a way understandable for an average physicist who is going to specialize in both experimental and theoretical solid state physics, and more generally for everyone who is going to be introduced to the exciting world of modern condensed matter physics – a subject very much alive and constantly producing new surprises.

In writing this book I tried to follow a unifying concept throughout. This concept, which is explained in more detail below, may be formulated as the connection between an order in a system and elementary excitations in it.