One-Dimensional Metals

Conjugated Polymers, Organic Crystals, Carbon Nanotubes and Graphene

Siegmar Roth, David Carroll

• Publisher: Wiley-VCH
• Year: 2015
• ISBN:
  9783527690190
• Edition: 3
• Topic: Technology & Engineering
• Subtopic: Materials Science
• Index: Technology & Engineering

Low-dimensional solids are of fundamental interest in materials science due to their anisotropic properties. Written not only for experts in the field, this book explains the important concepts behind their physics and surveys the most interesting one-dimensional systems and discusses their present and emerging applications in molecular scale electronics. Chemists, polymer and materials scientists as well as students will find this book a very readable introduction to the solid-state physics of electronic materials. In this completely revised and expanded third edition the authors also cover graphene as one of the most important research topics in the field of low dimensional materials for electronic applications. In addition, the topics of nanotubes and nanoribbons are widely enlarged to reflect the research advances of the last years.

Chapter 1
Introduction

1.1 Dimensionality

Dimensionality is an intellectually very appealing concept and speaking of a dimensionality other than three will surely attract some attention. Some years ago, it was fashionable to admire physicists who apparently could “think in four dimensions” in striking contrast to Marcuse's “One-Dimensional Man” [1]. Physicists would then respond with the understatement: “We only think in two dimensions, one of which is always time. The other dimension is the quantity we are interested in, which changes with time. After all, we have to publish our results as two-dimensional figures in journals. Why should we think of something we cannot publish?” (Figure 1.1).

Figure 1.1 Simultaneously with Herbert Marcuse's book “One-Dimensional Man” [1], which widely influenced the youth movement of the 1960s, Little's article on “Possibility of Synthesizing an Organic Superconductor” [2] was published, motivating many physicists and chemists to investigate low-dimensional solids.

This fictitious dialog implies more than just sophisticated plays on words. If physics is what physicists do, then in most parts of physics there is a profound difference between the dimension of time and other dimensions, and there is also a logical basis for this difference [3]. In general, the quantity that changes with time and in which physicists are interested, is one property of an object. The object in question is imbedded in space, usually in three dimensions. Objects may be very flat, such as flounders, saucers, or oil films with length and width much greater than their thickness. In this case, thickness can be negligible. Such objects can be regarded as (approximately) two dimensional. But, in another example, the motion of an object is restricted to two dimensions like that of a boat on the surface of a sea (hopefully). According to our everyday experience, one- and two-dimensional objects and one- and two-dimensional motions actually seem more common than their three-dimensional counterparts, and hence low-dimensionality should not be spectacular. Perhaps that is the reason for the introduction of noninteger (“fractal”) dimensions [4]. Not much imagination is necessary to assign a dimensionality between one and two to a network of roads and streets – more than a highway and less than a plane. It is a well-known peculiarity that, for example, the coastline of Scotland has the fractal dimension of 1.33 and the stars in the universe that of 1.23.

Solid-state physics treats solids as both objects and the space in which objects of physics exist, for example, various silicon single crystals can be compared with each other, or they can be considered as the space in which electrons or phonons move. On one hand, the layers of a crystal, for instance, the ab-planes of graphite, can be regarded as two-dimensional objects with certain interactions between them that extend into the third dimension. On the other hand, these planes are the two-dimensional space in which electrons move rather freely. Similar considerations apply to the (quasi) one-dimensional hydrocarbon chains of conducting polymers.

1.2 Approaching One-Dimensionality from Outside and from Inside

There are two approaches to low-dimensional or quasi-low-dimensional systems in solid-state physics: geometrical shaping as an “external” and increase of anisotropy as an “internal” approach. These are also sometimes termed “top-down” and “bottom-up” approaches, respectively. For the external approach, let us take a wire and draw it until it gets sufficiently thin to be one dimensional (Figure 1.2). How thin will it have to be for being truly one dimensional? This depends a little on exactly what property of the object is desired to express low-dimensional behavior. Certainly, thin compared to some microscopic parameter associated with that property. For example, for one-dimensional electrical transport properties, the object must have length scales such that the mean free path of an electron or the Fermi wavelength is affected by the physical confinement of the structure.

Figure 1.2 An “external approach” to one-dimensionality. A man tries to draw a wire until it is thin enough to be regarded as one-dimensional. Metallic wires can be made as thin as 1 µm in diameter, but this is still far away from being one-dimensional. (By lithographic processes, semiconductor structures can be made narrow enough to exhibit one-dimensional properties.)

But, does the wire have to be drawn so extensively to finally become a monatomic chain? Well, the Fermi wavelength becomes relevant when discussing the eigenstates of the electrons (we learn more about the Fermi wavelength in Chapter 3). If electrons are confined in a box, quantum mechanics tells us that the electrons can have only discrete values of kinetic energy. The energetic spacing of the eigenvalues depends on the dimensions of the box, the smaller the box the larger the spacing (Figure 1.3):

1.1

where ΔE_L is the spacing, L is the length of the box, m is the mass of the electrons, and h is Planck's constant. The Fermi level is the highest occupied state (at absolute zero). The wavelength of the electrons at the Fermi level is called the Fermi wavelength. If the size of the box is just the Fermi wavelength, only the first eigenstate is occupied. If the energy difference to the next level is much larger than the thermal energy (ΔE_L ≫ kT), then there are only completely occupied and completely empty levels and the system is an insulator. A thin wire is a small box for electronic motion perpendicular to the wire axis, but it is a very large box for motions along the wire. Hence, in two dimensions (radially), it represents an insulator and in one dimension (axially) it is a metal! This is simply because ΔE_radially ≫ kT whereas ΔE_lengthwise ≪ kT.

Figure 1.3 Electrons in small and large boxes and energy spacing of the eigenstates.

If there are only very few electrons in the box, the Fermi energy is small and the Fermi wavelength is fairly large. For real materials, these are the electrons that can participate in bonding–antibonding orbitals. This is the case for semiconductors at very low doping concentrations. Wires of such semiconductors are already one dimensional if their diameter is of the order of approximately hundreds of angstroms.

Such thin wires can be fabricated from silicon or from gallium arsenide by lithographic techniques and effects typical for one-dimensional electronic systems have been observed experimentally [5]. Systems with high electron concentrations have to be considerably thinner if they are to be one dimensional. It turns out that for a concentration of one conducting electron per atom, we really need a monatomic chain!

Experiments on single monatomic chains are very difficult, if not impossible, to perform. Therefore, typically, a bundle of chains rather than one individual chain is used. An example for such a bundle is the po...

Table of contents

• Cover
• Related Titles
• Title Page
• Copyright
• Table of Contents
• About the Authors
• Preface to the Third Edition
• Preface to the Second Edition
• Preface to the First Edition
• Chapter 1: Introduction
• Chapter 2: One-Dimensional Substances
• Chapter 3: One-Dimensional Solid-State Physics
• Chapter 4: Electron–Phonon Coupling and the Peierls Transition
• Chapter 5: Conducting Polymers: Solitons and Polarons
• Chapter 6: Conducting Polymers: Conductivity
• Chapter 7: Superconductivity
• Chapter 8: Charge Density Waves
• Chapter 9: Molecular-Scale Electronics
• Chapter 10: Molecular Materials for Electronics
• Chapter 11: Even More Applications
• Chapter 12: Finally
• Glossary and Acronyms
• Index
• End User License Agreement