Carbon nanotubes have come a long way since their modern rediscovery in 1991. This time period has afforded a great many scholars across the globe to conduct a vast amount of research investigating their fundamental properties and ensuing applications. Finally, after two decades, the knowledge and understanding obtained, once only accessible to select scholars, is now sufficiently widespread and accepted that the time is ripe for a textbook on this matter. This textbook develops the basic solid-state and device physics of carbon nanotubes and to a lesser extent graphene. The lesser coverage of graphene is simply due to its relative infancy, with a good deal of the device physics still in its formative stage.

The technical discourse starts with the solid-state physics of graphene, subsequently warping into the solid-state physics of nanotubes, which serves as the foundation of the device physics of metallic and semiconducting nanotubes. An elementary and limited introduction to the device physics of graphene nanoribbons and graphene are also developed. This textbook is suitable for senior undergraduates and graduate students with prior exposure to semiconductor devices. Students with a background in solid-state physics will find this book dovetails with their physics background and extends their knowledge into a new material that can potentially have an enormous impact in society. Scholars in the fields of materials, devices, and circuits and researchers exploring ideas and applications of nanoscience and nanotechnology will also find the book appealing as a reference or to learn something new about an old soul (carbon).

Introduction

The goal of this chapter is to explore the excitation and motion of electron waves under ideal conditions in a metallic conductor. By ideal conditions, we mean that electrons can be excited and transported without any scattering or collision involved. The excitation of electrons can be achieved by applying an external potential to energize the electron waves to oscillate more frequently, which can result in a net electron motion in the presence of a driving electric field, say between two ends of a metallic conductor. It is advisable to commit to memory that the absence of electron scattering is technically called ballistic transport; as such, the metallic conductor in this case would be referred to as a ballistic conductor.

Electrically, the ideal excitation and motion of electrons in low dimensions, such as in 1D space, is manifest in the form of a quantum conductance, quantum capacitance, and kinetic inductance, which represents a different paradigm from our classical electrostatic and magnetostatic ideas. The conductance and inductance reflect the electrical properties of traveling electron waves which lead to charge transport and energy storage, while the quantum capacitance accounts for the intrinsic charge storage that comes about from exciting electrons with an electric potential. In macroscopic bulk metals, the quantum electrical properties are not readily observable or accessible owing to the large number of mobile electrons at hand and the frequent collisions involved.

Introduction

The objective of this chapter is to describe the physical and electronic structure of graphene. Familiarity with concepts such as the crystal lattice and Schrödinger's quantum mechanical wave equation discussed in Chapter2 will be useful. The electronic band structure of graphene is of primary importance because (i) it is the starting point for the understanding of graphene's solid-state properties and analysis of graphene devices and (ii) it is also the starting point for the understanding and derivation of the band structure of CNTs. We begin by broadly discussing carbon and then swiftly focus on graphene, including its crystal lattice and band structure. This chapter concludes on the contemporary topic of GNRs.

Carbon is a Group IV element that is very active in producing many molecular compounds and crystalline solids. Carbon has four valence electrons, which tend to interact with each other to produce the various types of carbon allotrope. In elemental form, the four valence electrons occupy the 2s and 2p orbitals, as illustrated in Figure 3.1a. When carbon atoms come together to form a crystal, one of the 2s electrons is excited to the 2pz orbital from energy gained from neighboring nuclei, which has the net effect of lowering the overall energy of the system. Interactions or bonding subsequently follow between the 2s and 2p orbitals of neighboring carbon atoms.

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.