S. Roth, D. Carroll One-Dimensional Metals
One-Dimensional Metals, Second Edition. Siegmar Roth, David Carroll Copyright © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30749-4
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Authors Dr. Siegmar Roth Max-Planck-Institut f r Festk!rperforschung Heisenbergstr. 1 70569 Stuttgart Germany
[email protected] Prof. Dr. David Carroll Laboratory for Nanotechnology at Clemson Clemson University Clemson, SC 29534 USA
[email protected]
Cartoons by Rolf D. Wuthe, Atelier Wuthe, Weinheim
&
This book was carefully produced. Nevertheless, authors and publisher do not warrant the information contained therein to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No. applied for British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie, detailed bibliographic data is available in the Internet at
. 7 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Printed on acid-free paper. All rights reserved (including those of translation in other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publisher. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Printed in the Federal Republic of Germany Composition K&hn & Weyh, Freiburg Printing betz-druck GmbH, Darmstadt Bookbinding Großbuchbinderei Sch0ffer GmbH & Co. KG, Gr&nstadt ISBN
3-527-30749-4
V
Biographies After studying physics at the University of Vienna, Siegmar Roth carried out his thesis work at the reactor center in Seibersdorf, Austria, and received his PhD at the Institute of Professor Erich Schmid. From 1968 to 1970 he worked at the Siemens Research Laboratories in Erlangen, Germany, on the solid-state physics of novel semiconductors. After a three-year stay at the High Flux Reactor of the Institute Laue Langevin and four years at the High-Field Magnet Laboratory, both in Grenoble, France, where his research centered on superconductors, he joined the Max-Planck-Institut fr Festkrperforschung in Stuttgart, Germany. He is currently head of the Synthetic Nanostructures Group in von Klitzing’s department. In addition, he is Senior Visiting Professor at the Shanghai Institute of Technical Physics of the Chinese Academy of Sciences, CEO of Sineurop Nanotech GmbH Stuttgart, and Scientific Advisor to Shanghai Yangtze Nanomaterials. David Carroll carried out his thesis work at Wesleyan University in Middletown, Connecticut, USA, receiving his PhD in 1993. At the University of Pennsylvania in Philadelphia, his postdoctoral work focused on the application of scanning probes to oxide surfaces. After this, he joined Prof. Rhle's group at the Max-PlanckInstitut fr Metallforschung in Stuttgart, Germany. For two years there, his work centered on the application of scanning probes to interface studies and supported nanostructures. From Stuttgart, he became an assistant professor at Clemson University, Clemson, South Carolina, USA. Professor Carroll now heads the Nanotechnology group at Wake Forest University in Winston-Salem, North Carolina.
VII
Table of Contents 1 1.1 1.2 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.3.5 1.4
Introduction
1
Dimensionality 1 Approaching One-dimensionality from Outside and Inside 2 Dimensionality of Carbon Solids 7 Three-dimensional Carbon: Diamond 7 Two-dimensional Carbon: Graphite 8 One-dimensional Carbon: Cumulene, Polycarbyne, Polyene 9 Zero-dimensional Carbon: Fullerene 10 What About Something in Between? 12 Peculiarities of One-dimensional Systems 12
2 One-dimensional Substances 19 2.1 A15 Compounds 22 2.2 Krogmann Salts 26 2.3 Alchemists’ Gold 28 2.4 Bechgaard Salts and Other Charge-transfer Compounds 30 2.5 Polysulfurnitride 33 2.6 Phthalocyanines and Other Macrocycles 34 2.7 Transition Metal Chalcogenides and Halides 36 2.8 Conducting Polymers 38 2.9 Halogen-bridged Mixed-valence Transition Metal Complexes 41 2.10 Miscellaneous 42 2.10.1 Poly-deckers 42 2.10.2 Polycarbenes 43 2.11 Isolated Nanowires 43 2.11.1 Templates and Filler Pores 44 2.11.2 Asymmetric Growth using Catalysts 45 2.11.3 Nanotubes 46 2.11.4 Inorganic Semiconductor Quantum Wires 47 2.11.5 Metal Nanowires 48
VIII
Table of Contents
3 3.1 3.1.1 3.1.2 3.1.3 3.2 3.2.1 3.2.2 3.2.3 3.3 3.3.1 3.3.2 3.3.3
One-dimensional Solid-State Physics
53
3.4
Crystal Lattice and Translation Symmetry 54 Classifying the Lattice 54 Using a Coordinate System 56 The One-dimensional Lattice 58 Reciprocal Lattice, Reciprocal Space 60 Describing Objects by Momentum and Energy 60 Constructing the Reciprocal Lattice 61 Application to One Dimension 61 Electrons and Phonons in a Crystal, Dispersion Relations 63 Crystal Vibrations and Phonons 64 Phonons and Electrons are Different 67 Nearly Free Electron Model, Energy Bands, Energy Gap, Density of States 68 A Simple One-dimensional System 73
4
Electron–Phonon Coupling, Peierls Transition
5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8
General Remarks on Conducting Polymers 85 Conjugated Double Bonds 86 Conjugational Defects 89 Solitons 92 Generation of Solitons 100 Nondegenerate Ground State Polymers: Polarons Fractional Charges 106 Soliton Lifetime 108
6 6.1 6.2 6.3 6.4 6.5 6.6 6.7
General Remarks on Conductivity 113 Measuring Conductivities 118 Conductivity in One Dimension: Localization 126 Conductivity and Solitons 129 Experimental Data 133 Hopping Conductivity 139 Conductivity of Highly Conducting Polymers 145
7 7.1 7.2 7.3 7.4 7.5 7.5.1 7.5.2
Basic Phenomena 153 Measuring Superconductivity 159 Applications of Superconductivity 161 Superconductivity and Dimensionality 162 Organic Superconductors 163 One-dimensional Organic Superconductors 164 Two-dimensional Organic Superconductors 167
Conducting Polymers: Solitons and Polarons
Conducting Polymers: Conductivity
Superconductivity
77 85
113
153
103
Table of Contents
7.5.3 7.6 8 8.1 8.2
Three-dimensional Organic Superconductors Future Prospects 170
168
Charge Density Waves 177
8.5 8.6
Introduction 177 Coulomb Interaction, 4kF Charge Density Waves, Spin Peierls Waves, Spin Density Waves 178 Phonon Dispersion Relation, Phase, and Amplitude Mode in Charge Density Wave Excitations 181 Electronic Structure, Peierls–Frhlich Mechanism of Superconductivity 183 Pinning, Commensurability, Solitons 184 Field-induced Spin Density Waves and the Quantized Hall Effect 188
9 9.1 9.2 9.3 9.3.1 9.3.2 9.3.3 9.3.4 9.4
Miniaturization 193 Information in Molecular Electronics Early and Radical Concepts 197 Soliton Switching 197 Molecular Rectifiers 200 Molecular Shift Register 201 Molecular Cellular Automata 203 Carbon Nanotubes 204
10 10.1 10.2 10.2.1 10.2.2 10.3 10.3.1 10.3.2 10.3.3 10.4 10.5
Introduction 211 Switching Molecular Devices 212 Photoabsorption Switching 212 Rectifying Langmuir–Blodgett Layers Organic Light-emitting Devices 216 Fundamentals of OLEDs 218 Materials for OLEDs 219 Device Designs for OLEDs 220 Solar Cells 220 Organic Field-effect Transistors 223
11 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8
Introduction 227 Superconductivity and High Conductivity Electromagnetic Shielding 227 Field Smoothening in Cables 228 Capacitors 229 Through-hole Electroplating 230 Loudspeakers 231 Antistatic Protective Bags 232
8.3 8.4
Molecular-scale Electronics
193
Molecular Materials for Electronics
Applications
196
211
215
227 227
IX
X
Table of Contents
11.9 11.10 11.11 11.12 11.13 11.14 11.15 11.16 11.17 11.18 11.19
Other Electrostatic Dissipation Applications 233 Conducting Polymers for Welding Plastics 234 Polymer Batteries 235 Electrochemical Polymer Actuators 236 Electrochromic Displays and Smart Windows 237 Electrochemical Sensors 237 Gas Separating Membranes 238 Corrosion Protection 239 Holographic Storage and Holographic Computing 239 Biocomputing 240 Outlook 242
12
Finally
Index
247
245
XI
Preface and Acknowledgments This book originated from lectures on “Physics in One Dimension” given at the University of Karlsruhe in the 1980s. I am grateful to all the students who contributed by asking questions. Some of them later became PhD students in my research group in Stuttgart. The style and content of the book reflect the everyday research work of an interdisciplinary and international research group, where people of different background have to quickly catch up on basic concepts in order to meet on equal terms for discussions. The reader is expected to have some basic knowledge of science or engineering, for example, of physics, chemistry, biology, or materials science. To consolidate this knowledge you will have to consult textbooks on experimental physics, as well as on organic, inorganic, and physical chemistry. But the present book should help to forge links, and with these links the monographs recommended in the Appendix should be accessible. It should also be possible to follow international topical meetings. We hope that some of the aspects of the book are so interesting that they are attractive even to complete neophytes or to outsiders, and that some features will also appeal to the experienced researcher. My thanks are due to all the members of our team (Lidia Akselrod, Tarik Abou-Elazab, Teresa Anderson, Marko Burghard, Hugh Byrne, Claudius Fischer, Thomas Rabenau, Michael Schmelzer, Manfred Schmid, Andrea Stark-Hauser, Andreas Werner). Without constant discussions in the lab’s coffee corner, the book would not have been possible. Particular thanks go to Andrea Stark-Hauser, who not only did all the typing but was also engaged in collecting references and figures. Manfred Schmid assisted in the preparation of technical drawings, and the cartoons were drawn by Gnter Wilk. Teresa Anderson, Hugh Byrne, and Andreas Werner went through my first drafts and generated major inputs to the final phrasing of the text, which was ultimately polished by the experts at VCH-Verlag and with whom it was a pleasure to cooperate. The whole team has benefited from the cooperation within the Sonderforschungsbereich “Molekulare Elektronik physikalische und chemische Grundlagen” of the Deutsche Forschungsgemeinschaft, within the European BRJTE/EURAM Project HICOPOL (comprising groups in Stuttgart, Karlsruhe, Montpellier, Nantes, Strasbourg, and Graz), and within the European ESPRIT Network NEOME (Austria, Belgium, Denmark, England, France, Germany, Italy, The Netherlands, Sweden, and Switzerland). Siegmar Roth
April 1995
XIII
Preface to the Second Edition About ten years have passed since the lectures in Karlsruhe on “Physics in One Dimension” and since the first edition of this book. When we were asked to work on an update for the second edition, we felt that many things had changed – and were surprised that many parts were still valid! New are the Nobel Prizes – to Curl, Smalley, and Kroto for the fullerenes in 1996 – to Kohn and Pople in 1998 (see the soliton as a Pople–Walmsley defect) – and to Heeger, MacDiarmid, and Shirakawa in 2000 for conducting polymers. New are applications of conducting polymers, and in particular, the commercialization of several of these applications, and (almost) new are carbon nanotubes. In fact, these nanotubes are the new toys of the materials scientists, and like locust swarms, they crowd on every tiny bit of these carbon crumbs, producing a new wave in literature statistics (compare Figure 2-3). Since we are part of this swarm, we decided to devote several paragraphs and sections of the second edition of our book to nanotubes and to stress the relationships between conjugated polymers and carbon nanotubes. As was true for the first edition, this second edition would also not have been possible without the support and the many discussions among our teams in Stuttgart, Shanghai, Clemson, and Wake Forest. In particular, we are grateful to Ekkehard Palmer, who did most of the computer work for this edition and to the experts at Wiley-VCH in Weinheim. We enjoy working in this field, we enjoyed working on the book, and we hope that our readers will enjoy reading our modest oeuvre. David Carroll and Siegmar Roth
October 2003
1
1
Introduction 1.1
Dimensionality
Dimensionality is an intellectually intriguing concept, especially when we think of dimensionalities other than three. 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 could then respond with the understatement, We think in only two dimensions, one of which is always time. The other
Figure 1-1 Simultaneously with Herbert Marcuse’s book One-dimensional Man [1], which broadly influenced the youth movement of the 1960s, W.A. Little’s paper on the Possibility of
synthesizing an organic superconductor’ [2] was published, motivating many physicists and chemists to investigate low-dimensional solids.
One-Dimensional Metals, Second Edition. Siegmar Roth, David Carroll Copyright © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30749-4
2
1 Introduction
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). This fictional response implies more than just a sophisticated play 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 which physicists are interested in’ is one property of an object. The object in question is embedded in space, usually in three-dimensional space. Objects may be very flat, such as flounders, saucers, or oil films, which all have greater length and width than thickness, even negligibly small thickness. Such objects can be regarded as (approximately) two-dimensional. In another example, the motion of an object may be restricted to two dimensions, like that of a boat on the surface of the sea (we hope!). In our everyday experience, one- and two-dimensional objects and one- and two-dimensional motions actually seem more common than their three-dimensional counterparts, so low-dimensionality should not be spectacular. Perhaps that is the reason for the introduction of non-integer (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 wellknown peculiarity that, for example, the coastline of Scotland has a fractal dimension of 1.33 and the stars in the universe, 1.23. Solid-state physics treats solids both as objects and as the space within which objects of physics exist, e.g., various silicon single crystals can be compared with each other, or they can be considered as the space within 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 can be discussed. On the other hand they are the two-dimensional space in which electrons move rather freely. Similar considerations apply to the (quasi) onedimensional hydrocarbon chains of conducting polymers.
1.2
Approaching One-dimensionality from Outside and 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 bottomup’ 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? Certainly, thin compared to a microscopic parameter such as, for example, the mean free path of an electron or the Fermi wavelength. Which of the microscopic parameters is the decisive one will depend largely on the chosen measurement system. Does the wire have to be drawn so extensively as to finally become a monatomic chain?
1.2 Approaching One-dimensionality from Outside and Inside
Figure 1-2 An external’ approach to onedimensionality. 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 lm in diameter, but this is still far from
being one-dimensional. (By lithographic processes, semiconductor structures can be made narrow enough to exhibit one-dimensional properties.)
The Fermi wavelength becomes relevant when discussing the eigenstates of the electrons (we will 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): DEL = h2/2m (p/L)2
(1)
where DEL is the spacing and L is the length of the box. 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 (DEL >> kT), 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’s axis, but it is a very large box for motions along the wire. Hence in two dimensions (radially) it represents an insulator, in one dimension (axially) it is a metal!
3
4
1 Introduction
Figure 1-3
Electrons in small and large boxes and energy spacing of the eigenstates.
If there are only very few electrons, the Fermi energy is small and the Fermi wavelength fairly large. This is the case for semiconductors at very low doping concentrations. Wires of such semiconductors are already one-dimensional if their diameter is on the order of some 100 . Such thin wires can be fabricated from silicon or from gallium arsenide by lithographic techniques, and effects typical of 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 of such a bundle is the polyacetylene fiber, consisting of some thousand polymer chains, closely packed with a typical interchain distance of 3 to 4 . Certainly there will be some interaction between the chains; however, when interchain coupling is small, it can be assumed that just the net sum of the individual chains determines the outcome of the experiment (Figure 1-4).
Figure 1-4 Experiments on individual chains are difficult to perform, but bundles of chains are quite common, for example, fibers of polyacetylene.
1.2 Approaching One-dimensionality from Outside and Inside
Figure 1-5 Crystal surfaces are excellent twodimensional systems. The man above tries to improve the crystal face by mechanical polishing. The qualities achieved by this method are
not sufficient for surface science. Surface scientists cleave their samples under ultrahigh vacuum conditions and use freshly cleaved surfaces in their experiments.
Another method of geometrical shaping employs surfaces or interfaces (Figure 1-5). The surface of a silicon single crystal is an excellent two-dimensional system, and there are various ways of confining charge carriers to a layer near the surface. Actually, the physics of two-dimensional electron gases are an important part of today’s semiconductor physics [6], and most of the two-dimensional electron systems are confinements to surfaces or interfaces. The most fashionable effect in a two-dimensional electron gas is the quantized Hall effect or von Klitzing effect [7]. A onedimensional surface, i.e., an edge of a crystal, is much more difficult to prepare and of hardly any practical use. If one argues, however, that exposing a sample to a magnetic field reduces the effective dimensionality by one, then a silicon surface in a magnetic field would be an excellent example of a one-dimensional electronic system. In fact, reducing von Klitzing’s sample to edge channels’ is one way of explaining the von Klitzing effect [8]. The internal’ approach to one-dimensional solids comprises the gradual increase of anisotropy. In crystalline solids the electrical conductivity is usually different in different crystallographic directions. If the anisotropy of the conductivity is increased in such a way that the conductivity becomes very large in one direction and almost zero in the two perpendicular directions, a nearly one-dimensional conductor results. Of course, there is no simple physical way to increase the anisotropy. However, it is possible to look for sufficient anisotropy in already existing solids that could be regarded as (quasi) one-dimensional. Some anisotropic solids are listed in the next chapter of this book. How large should the anisotropy be to be one-dimensional? A possible answer is Large enough to lead to an open Fermi surface’.
5
6
1 Introduction
The Fermi surface is a surface of constant energy in reciprocal space’ or momentum space. The Fermi surface and reciprocal space are discussed in detail in Chapter 3, but for the discussion here it is sufficient to imagine this surface as describing all of the electron states within the solid that are available to take part in electrical transport. For an isotropic solid, the Fermi surface is spherical, meaning that electrons can move in any direction of the solid equally as well. If the electrical conductivity is large in one crystallographic direction and small in the other two, the Fermi surface becomes disk-like. The kinetic energy of the electrons can then be written as E = p2/2m*, resembling the kinetic energy of a free particle (p: momentum, m: mass), except that the mass has been replaced by the effective mass m*. The effective mass indicates the ease with which an electron can be moved by the electric field. If the electrons are easy to move, the conductivity is high. Easy motion is described by a small effective mass (small inertia), and p must also be small to keep E constant. If it is infinitely difficult to move an electron in a specific direction, the effective mass is infinitely large in this direction and the Fermi surface is infinitely far away. However, the extension of the Fermi surface is restricted: if the Fermi surface becomes too large in any direction it will merge with the Fermi surface generated by the neighboring chain (next Brillouin zone’, in proper solid-state physics terminology). This merging opens’ the Fermi surface, similar to one soap bubble merging with another bubble (Figure 1-6).
Figure 1-6 Open Fermi surfaces, analogous to merged soap bubbles, as a criterion of low dimensionality. The Fermi surface belongs to a solid that is essentially two-dimensional. The solid has no electronic states contributing to electrical conductivity along the axial direction but easily conducts radially, normal to the axis.
1.3 Dimensionality of Carbon Solids
1.3
Dimensionality of Carbon Solids
Silicon is outstanding among solids [9]: it is most perfect solid producible – there are fewer imperfections in a single silicon crystal than there are gas atoms in ultrahigh vacuum (per unit volume). It is the solid we know most about, and it is the solid that has greatly influenced the vocabulary of solid-state physics. Carbon is located directly above silicon in the periodic table of the elements, and just as silicon is outstanding among the solids, carbon is outstanding among the elements. Carbon forms the majority of chemical compounds. Much of organic chemistry simply involves arranging carbon atoms (with hydrogen not having any specific properties but just fulfilling the task of saturating dangling bonds). In our context, carbon has the remarkable property of forming three-, two-, one-, and zero-dimensional solids. This is related to the fact that carbon is able to form single, double, and triple bonds. This ability of carbon to form many types of bonds distinguishes it from silicon in another important way – it leads to biology. 1.3.1
Three-dimensional Carbon: Diamond
Since we are accustomed to silicon, diamond appears as the trivial solid form of carbon (Figure 1-7). Diamond has similar semiconducting properties to silicon. Both substances share the same type of crystal lattice. The lattice parameters are different (a = 5.43 in silicon and 3.56 in diamond), and the energy gap between the valence and conduction band is larger in diamond by 5.4 eV, compared to 1.17 eV in silicon. Diamond is more difficult to manufacture and more difficult to purify than silicon, but it has better thermal conductivity and can be used at high temperatures. Since the costs of the raw material affect the final price of electronic equipment only slightly, some people believe in diamond as the semiconductor of
Figure 1-7
Diamond lattice.
7
8
1 Introduction
the future. Silicon is typically used with added dopants to modify its electronic behavior slightly. Doping diamond has proven to be far more difficult, however, and must be better understood before the realization of high quality diamond electronics. Semiconductors’ are often mentioned in this book even though the title promises metals as the main subject. The reason stems from the fact that a doped semiconductor can be regarded as a metal with low electron concentration. Here, metal’ is essentially used as a synonym for electrically conductive solid-state system’. 1.3.2
Two-dimensional Carbon: Graphite
In diamond the carbon atoms are tetravalent, that is, each atom is bound to four neighboring atoms by covalent single bonds. Another well-known carbon modification is graphite (Figure 1-8). Here all atoms are trivalent, which means that in a hypothetical first step only, three valence electrons participate in bond formation and the forth valence electron is left over. The trivalent atoms form the planar honeycomb lattice and the residual electrons are shared by all atoms in the plane, similar to the sharing of the conduction electrons by all atoms of a simple metal (e.g., sodium or potassium). The various graphite layers interact only by weak van der Waals forces. In a first approximation graphite is an ensemble of nearly independent metallic sheets. In pure graphite they are about 3.35 apart, but can be separated further by intercalating various molecules. Charge transfer between the intercalated molecules and the graphitic layers is also possible. Graphite with intercalated SbF5
Figure 1-8
Graphite lattice.
1.3 Dimensionality of Carbon Solids
shows an anisotropy of about 106 in electrical conductivity, conducting a million times better within a layer than between layers. Diamond is a semiconductor and graphite is a metal; in diamond there are very few mobile electrons – in an undoped perfect diamond single crystal at absolute zero there are exactly zero mobile electrons – and in graphite there are many, one electron per carbon atom. This difference is not due to the difference in dimensionality (three in diamond and two in graphite) but to the single and double bonds. Several attempts have been made to build three-dimensional graphite [10]. Theoretically it seems possible [11], but practically it has not yet been achieved. 1.3.3
One-dimensional Carbon: Cumulene, Polycarbyne, Polyene
By using double bonds one can easily construct – at least on paper – one-dimensional carbon: Figure 1-9 shows a monatomic carbon chain. This substance is called cumulene, the name referring to cumulated double bonds. From organic chemistry it is well known that double bonds can be isolated’ (separated by many single bonds), conjugated’ (in strict alternation with single bonds), or cumulated’ (adjacent). Cumulene has not been synthesized, but by using the principles of quantum chemistry we can predict whether cumulene would be stable or would rather transform into polycarbyne, an isomeric structure in which triple bonds alternate with single bonds. Polycarbyne is shown in Figure 1-10. The odds are for polycarbyne being the more stable molecule. Polycarbyne occurs in interstellar dust and in trace amounts within natural graphite but is not yet available for performing experiments [12].
Figure 1-9
Figure 1-10
One-dimensional carbon: cumulene.
One-dimensional carbon: polycarbyne.
If we accept the simplification that, in carbon compounds, hydrogen atoms just have the purpose of saturating dangling bonds and that they do not otherwise contribute to the physical properties of the material, cumulene and polycarbyne are not the only one-dimensional carbon solids. From this point of view, all polymers based on chain-like molecules are one-dimensional. In organic chemistry, the ending -yne’, as in polycarbyne, indicates triple bonds. The ending -ene’ stands for double bonds and -ane’ for single bonds. A polyane is
Figure 1-11
Polyethylene.
9
10
1 Introduction
shown in Figure 1-11. (To confuse the subject, this substance is called polyethylene, ending in -ene’ instead of -ane’. The names of polymers are often derived from the monomeric stating material, which in this case is ethylene, H2C=CH2. Here the monomer contains a double bond. During polymerization the double bond breaks to link the neighboring molecules.) Polyanes are insulators and of little interest in the context of this book. (Insulators are large band-gap semiconductors. Because of the large band-gap it is difficult to lift electrons into the conduction band and therefore the number of mobile electrons is negligible.) Figure 1-12 shows polyacetylene, the prototype polyene, the simplest polymer with conjugated double bonds. The structure shown in Figure 1-12 is often simplified to the one in Figure 1-13, since by convention carbon atoms do not have to be drawn explicitly at the ends of the bonds and protons at the chain end are neglected. Chapter 5 is concerned with conducting polymers and discusses polyacetylene in more detail.
Figure 1-12
Polyacetylene.
Figure 1-13
Polyacetylene, simplified notation.
1.3.4
Zero-dimensional Carbon: Fullerene
If we work our way down in dimensionality from body to plane to line we will finally end up at the point as a zero-dimensional object. Do zero-dimensional solids exist? In semiconductor physics the quantum dot’ is well-known [13]. This is a small disc cut out of a two-dimensional electron gas. It is small compared to the Fermi wavelength, so that the electrons are restricted in all three dimensions of space (the onedimensional analog to a quantum dot is often called a quantum wire’). According to the discussion in Section 1.2, a quantum dot is a zero-dimensional object. The present state of the art is to fabricate quantum dots containing more than one but fewer than ten electrons. Because of the low electron concentration in semiconductors, such quantum dots can have large diameters, up to several hundred Angstroms. The quantum dots of carbon solid-state physics are the famous fullerenes [14]. Under certain conditions, carbon forms regular clusters of 60, 70, 84, etc. atoms. A C60 cluster is composed of 20 hexagons and 12 pentagons and resembles a soccer ball (Figure 1-14). The diameter of a ball is about 10 ; thus it is considerably smaller than a semiconductor quantum dot. However, in carbon compounds the electron concentration is higher than in inorganic semiconductors: in a system of conjugated
1.3 Dimensionality of Carbon Solids
Figure 1-14
A fullerene molecule.
double bonds there is one p electron per carbon atom (more on p-electrons in Chapter 5). In other words, there are 60 p electrons in a fullerene ball of 10 diameter, compared to some five electrons on a 100 GaAs quantum dot. In quantum chemistry and solid-state physics, 60 is quite a large number (we are used to counting one, two, many’). In fact, a 60-particle system is already a mini-solid, and a fullerene ball plays a dual role in solid-state physics: it is a mini-solid and it is a constituent of a macro-solid. For example, we can study electronic excitations in the mini-solid and their mobility and interaction with lattice vibrations. At the same time it is possible to examine unexpected transport properties of the macro-solid, like superconductivity [15], photoconductivity, and electroluminescence [16]. Figure 1-14 shows the graphic representation of a fullerene mini-solid, Figure 1-15 schematically indicates the fullerene macro-solid.
Figure 1-15
The fullerene crystal lattice.
11
12
1 Introduction
1.3.5
What About Something in Between?
Conceptually, we might conceive of a solid that is a combination of dimensions. Imagine, for instance, a single graphene sheet described in the section on graphite. Roll this conductive sheet into a seamless tube in which each atom is three-fold coordinated, as in the sheet. When the diameter of such a tube is between 14 and 200 , we refer to the object as a carbon nanotube’. For such an object, the electron wave function is confined to box-like states around the circumference. Along the axis of the tube, the electrons move in essentially a one-dimensional system. Normally, this would appear to be similar to the semiconductor wires mentioned earlier. However, this circumference (or rolled-up) dimension allows for a set of spiral-like classical trajectories of the electron as it moves down the tube. In this way, if a threedimensional field (like a magnetic field) should penetrate the tube, the phase of the electronic wave function would be altered, resulting in Arahanov–Bohm effects. Thus, although the tube certainly has the character of a one-dimensional system it also has a little more’. However it is clearly not quite two-dimensional. For such systems, the topology of the dimension must be considered. We will learn more about carbon nanotubes in the following chapters.
1.4
Peculiarities of One-dimensional Systems
Theory predicts that strictly one-dimensional systems will behave very unusually, so in this context the word pathological’ is often used. If real systems appear less pathological than predicted, this is because real systems are only quasi and not strictly one-dimensional. Real systems differ from ideal systems by having chains of finite rather than infinite length. In addition, the chains show imperfections such as kinks, bends, twists, or impurities. They are contained in an environment other than perfect vacuum, with neighboring chains at a finite distance and thus a nonzero interaction between them. If you have ever followed a slow truck on a narrow mountain road, you have painfully experienced a very important aspect of one-dimensionality: obstacles cannot be bypassed (Figure 1-16). A rather famous demonstration of one-dimensional conduction studied by solid-state physicists is that of the monatomic metal wire. If one takes a very large number of gold atoms and places them very close to each other so as to form a wire, then the transmission of an electron down this wire is rather easily calculated. Now, we offer a very subtle change to this wire and replace in its center one gold atom with one silver atom and recalculate the transmission probability of the electron traveling its length. What is found is that, even for very small variations in the periodic atomic potential, reflections of the electron wave on the wire become large [17]. Another, more sophisticated, conceptualization of this fact can be made in terms of percolation: the threshold for bond percolation in one dimension is 100%! Perco-
1.4 Peculiarities of One-dimensional Systems
Figure 1-16
A very important aspect of one-dimensionality is that obstacles cannot be bypassed.
lation means macroscopic paths from one side of the sample to the other. Such macroscopic paths are necessary, for example, for electrical conduction. The concept of bond percolation is most easily demonstrated by a grid (Figure 1-17) [18]: bonds are cut at random. In a two-dimensional square lattice a few cuts have little effect on sample properties; in particular, the conductivity drops only slightly, due to holes that appear. When 50% of the bonds are cut, no path is left that connects one side of the sample to the other and the conductivity must be zero. The percentage of intact bonds necessary to establish macroscopic paths is the percolation threshold’. The higher the dimensionality of the sample, the lower the percolation threshold. For a one-dimensional system the threshold – quite simply – is 100%: if we cut one bond, the sample consists of two disconnected pieces.
Bond percolation demonstration on a two-dimensional grid, where bonds are successively cut in a random way [18].
Figure 1-17
13
14
1 Introduction
Another – trivial – aspect of one-dimensional systems is low connectivity. Each atom is connected to two other atoms only: one on the left and one on the right. In three-dimensional solids there are connections to neighbors in the back and front, as well as to neighbors above and below. Connectivity is a topological concept. Chemists usually speak of the coordination number, the number of nearest neighbors. In a one-dimensional chain the coordination number is two. Electron-lattice coupling. If bonds are completely broken, a one-dimensional system separates into two pieces. Usually complete breaking of bonds does not happen, however. Often bonds are only partially cleaved, for example, one component of the double bond in the system, as in Figure 1-9 or 1-13. In chemical terms, this means that a bonding state is excited to form an anti-bonding state. In semiconductor physics it would be described as an electron being lifted from the valence band into the conduction band. Such manipulation of valence electrons is quite common in semiconductors and is the first step for photoconductivity and photoluminescence. In a three-dimensional semiconductor like silicon, the transfer of an electron from the valence to the conduction band creates mobile charge carriers (the electron in the conduction band and the hole left behind in the valence band), but it does not change the arrangement of the atoms in the crystal. This is due to the high connectivity of the silicon lattice, where breaking or weakening one bond does not have much effect. In low-connectivity one-dimensional systems, where each atom is held in place by two neighbors only, each change in bond strength leads to a large distortion of the lattice. In conjugated polymers, the lattice distortion shows as a change in bond length when a double bond is partially broken to yield a single bond. With strong electron-lattice coupling, the electrons moving in the solid create a large distortion which polarizes the lattice. If the effect is distinct enough, the electrons receive a new name: polarons’. Depending on the strength of the coupling and the symmetry of the lattice, a variety of quasi particles results from electron-lattice coupling, the most famous of which (and very typical of one-dimensional systems) is the soliton’ [19]. We will have a closer look at solitons and polarons in Chapter 5. Another peculiarity of one-dimensional systems is band-edge singularities in the electronic density of states. We will learn more about electrons in a solid in Chapter 3, but here we note that in a solid, electrons cannot have any energy (as they can have in vacuum). There are allowed regions (bands) separated by forbidden gaps. The density of states within a band, that is, the number of states per energy interval, is not constant. The form of the density-of-state function depends on the crystal structure. Near the band edge it reflects the dimensionality of the system. This is indicated in Figure 1-18: in three dimensions the density-of-state function N(E) is parabolic, in two dimensions it is step-like, and in one dimension there is a squareroot singularity to infinity. In real systems N(E) never reaches infinity, of course, but there is at least a very high density of states. If the Fermi level is within such a region, high-temperature superconductivity is favored (see “A15 Compounds”, Section 2.1). One-dimensionality also differs from the other dimensionalities in random walk problems. In a high dimension it is very unlikely that a random walker will return
1.4 Peculiarities of One-dimensional Systems
Density-of-state function at the band edge in three-, two-, and one-dimensional electronic systems. Note the singularity that occurs in the one-dimensional system.
Figure 1-18
to the place he started from, whereas in one dimension this happens quite often. Whether or not the random walker comes back to the point of departure is important in discussing the recombination of photogenerated charge carriers and thus the time constants of transient photoconductivity and of luminescence. Luminescent devices might turn out to be the most important practical applications of one-dimensional metals. One-dimensional solids are particularly interesting in the context of fundamental studies of phase transitions. In fact, much of the motivation in this field arises from the hope of finding the key to high-temperature superconductivity. There is the famous theorem of Landau that phase transitions are impossible in one-dimensional systems [20]: long-range order is unstable with respect to the creation of domain walls, because the entropy term in the free enthalpy will always overcompensate the energy needed to form new walls. But whereas phase transitions are impossible, one-dimensional systems might be close’ to a phase transition even at fairly high temperatures. Fluctuations might anticipate’ the phase transition and already prompt the consideration of some technologically useful properties such as, for instance, low-resistance charge transport. Perhaps we could allow for just a little bit’ of three-dimensionality and thus obtain a high-temperature superconductor? Organic superconductors are known, but they are closer to two-dimensionality than to one-dimensionality. Their superconducting transition temperatures reach 12 or 13 K (for fullerene even up to 33 K), still far below the recently discovered inorganic-oxide superconductors with transition temperatures of 100 K and above [21]. Chapter 7 is devoted to organic superconductors. In Chapter 4 we discuss the Peierls transition. This is a transition from a metal at high temperatures to a semiconductor or insulator at low temperatures. It is driven by the strong electron–phonon coupling in one-dimensional systems and related to the formation of conjugated double bonds in conducting polymers (here referring to
15
16
1 Introduction
the strict alternation of double and single bonds with different bond lengths rather than to equidistant one-and-a-half bonds’). The charge density waves (Chapter 8) can be regarded as an incomplete’ or itinerant Peierls transition. Before discussing these phenomena we will focus on the concepts of solid-state physics in greater detail (Chapter 3), after first enjoying a tour of some of the most attractive one-dimensional substances in Chapter 2. Chapter 9 contains an introduction to the field of molecular electronics, which in many aspects is linked to low-dimensional solids. At first glance one might propose to using polymer chains of a one-dimensional metal as ideal molecular wires. This is not very realistic. However, the strong electron–phonon coupling in one-dimensional systems (and in particular in organic systems) leads to a better localization of excitations and thus might be applied to denser packing of information (i.e., to smaller devices and higher integration). Nanoscale molecular-electronic devices certainly are applications of the distant future. But there are also near-future and even present-day applications of one-dimensional solid-state systems. Materials for Molecular Electronics are described in Chapter 10. A survey of this, Chapter 11, concludes this book. To complete this introductory chapter we reprint in Figure 1-19 a haiku’ that was used during the closing ceremony of the International Conference on Science and Technology of Synthetic Metals in Kyoto [22]. Synthetic Metals TC pr 1d Kyoto thank we Figure 1-19
Haiku from the ICSM 1986 closing session in Kyoto [22].
References and Notes
References and Notes 1 H. Marcuse. One-Dimensional Man: Studies
in the Ideology of Advanced Industrial Society. Beacon Press, Boston, 1964. 2 W.A. Little. Possibility of synthesizing an organic superconductor. Physical Review A 134, 1416 (1964). nd 3 C.F. von Weizscker. Aufbau der Physik. 2 Edition, Carl Hauser Verlag, Mnchen, 1986. 4 a) B.B. Mandelbrot. The Fractal Geometry of Nature. Freeman, New York, 1982. b) I. Stewart. Les Fractal. Librairie Classique Eugne Belin, Paris, 1982. (This is a book of popularized science in cartoon form.) 5 C.W.J. Beenakker and H. van Houten. Quantum transport in semiconductor nanostructures. Solid-State Physics 44, 1 (1991). H. van Houten, C.W.J. Beenakker, and B.J. van Wees. Quantum point contacts. Semiconductors and Semimetals 35, 9 (1991). 6 B. McCombe and A. Nurmikko. In: International Conference on: Electronic Properties of Two-Dimensional Systems. Elsevier, Amsterdam, 1994. 7 K. von Klitzing, G. Dorda, and M. Pepper. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Physical Review Letters 45, 494 (1980). K. von Klitzing. The quantized Hall effect. Reviews of Modern Physics 58, 519 (1986). 8 R.J. Haug. Edge-state transport and its experimental consequences in high magnetic fields. Semiconductor Science and Technology 8, 131 (1993). 9 H. Queisser. Kristallene Krisen: Mikroelektronik; Wege der Forschung, Kampf und Mrkte. Piper, Mnchen, 1985. 10 R.H. Baughman and C. Cui. Polymers with conjugated chains in three dimensions. Synthetic Metals 55, 315 (1993). 11 R. Hoffmann, T. Hughbanks, M. Kertesz, and P.H. Bird. Hypothetical metallic allotrope of carbon. Journal of the American Chemical Society 105, 4831 (1983). 12 U.H.F. Bunz. Polyine – faszinierende Monomere zum Aufbau von Kohlenstoffnetzwerken. Angewandte Chemie 106, 1127 (1994); Polyynes fascinating monomers for the construction of carbon networks. Angewandte Chemie International Edition English 33, 1073 (1994).
13 L.J. Geerlings, C.J.P.M. Harmans, and L.P.
Kouwenhoven (Eds.), Proceedings of the NATO Advanced Research Workshop on the Physics of Few-Electron Nanostructures, Noordwijk, The Netherlands, 1992; Physica B: Condensed Matter 189, 1 (1993); Amsterdam, North Holland, 1993. 14 C. Taliani, G. Ruani, and R. Zamboni. In: Fullerenes: Status and Perspectives. World Scientific Advances Series in Fullerenes; World Scientific Publisher, Singapore, 1993. H.W. Kroto, J.E. Fischer, and D.E. Cox. The Fullerenes. Pergammon, Oxford, 1993. D. Koruga, S. Hameroff, J. Withers, R. Loutfy, and M. Sundareshan. Fullerene C60: Physics, Nanobiology, Nanotechnology. Elsevier, Amsterdam, 1993. W.E. Billups and M.A. Ciufolini. Buckminsterfullerenes. VCH-Verlag, Weinheim, 1993. 15 S. Grtner. In: Festkrperprobleme/Advances in Solid State Physics 32, 295; Heidelberg, Springer Verlag 1992. A.F. Hebard. Superconductivity in doped fullerenes. Physics Today 45 (Nov), 26 (1992). D.W. Murphy, M.J. Rosseinsky, R.M. Fleming, R. Tycko, A.P. Ramirez, R.C. Haddon, T. Siegrist, G. Dabbagh, J.C. Tully, and R.E. Wastedt. Synthesis and characterization of alkali metal fullerides: AxC60. Journal of Physics and Chemistry of Solids 53, 1321 (1992). K. Lders. Superconducting intercalation compounds of C60 fullerene and graphite. Chemical Physics of Intercalation II, NATO ASI Series B 305, 31, Plenum Press, New York, 1993. J.E. Fischer, S. Roth, and S.A. Solin, NATO ASI Series, Series B 305, Plenum Press, New York, 1993. 16 A.T. Werner, J. Anders, H.J. Byrne, W.K. Maser, M. Kaiser, A. Mittlebach, and S. Roth. Broadband electroluminescent emission from fullerene crystals. Applied Physics A 57, 157 (1993). 17 This is given as a computational problem in many solid-state courses. To learn more see, for example: S. Datta. Electronic Transport in Mesoscopic Systems. Press Syndicate of the University of Cambridge, Cambridge, 1995. 18 R. Zallen. The Physics of Amorphous Solids. John Wiley & Sons, New York, 1983.
17
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1 Introduction
19 S. Roth and H. Bleier. Solitons in polyacety-
lene. Advances in Physics 36, 385 (1987). A.J. Heeger, S. Kivelson, J.R. Schrieffer, and W.P. Su. Solitons in conducting polymers. Reviews on Modern Physics 60, 781 (1988). Y. Lu. Solitons and Polarons in Conducting Polymers. World Scientific Publisher, Singapore, 1988. 20 L.D. Landau and E.M. Lifshitz. Lehrbuch der Theoretischen Physik V/1. 7. Auflage, Akademie-Verlag, Berlin, 1990. 21 H. Kuzmany, M. Mehring, and J. Fink. In: Electronic Properties of High-Tc Supercon-
ductors. Springer Verlag, Berlin-Heidelberg, 1993. See also the latest issue of Physica C, a journal which since 1988 has been specifically devoted to high-temperature superconductivity. 22 S. Roth. Synthetic Metals 22, 190 (1987) and S. Roth. Erratum: concluding remarks: conductive polymers, Synthetic Metals 18, 869 (1987). (A haiku is a Japanese poem of 17 syllables distributed in 3 lines of 5, 7, and 5 syllables and using a highly symbolic language.)
19
2
One-dimensional Substances During the past twenty or thirty years many physicists, chemists, and materials scientists have been excited by low-dimensional solids. One of the motivations was certainly the quest for high-temperature superconductors. In 1964 W. A. Little [1] suggested synthesizing an organic superconductor by appropriately functionalizing polyacetylene, a polymer with conjugated double bonds (see Figure 1-12), introduced before, when we discussed quasi one-dimensional forms of carbon. We will take a closer look at this substance in the sections on conducting polymers in Section 2.8 and Chapter 5. Little proposed replacing some of the hydrogen atoms in polyacetylene by specifically designed substituents R, as shown in Figure 2-1. In Little’s superconductor, electrons are supposed to move along the polymer backbone and to excite the substituents when passing by: one electron deposits an exciton in the substituent group and the next electrons re-absorb the exciton. This exchange’ of excitons couples’ with the electrons in the same way that phonons couple with electrons in the BCS theory of superconductivity [2]. If the coupling effect is strong enough, high superconducting transition temperatures might be achieved, perhaps even as high as room temperature. The substituent proposed in Little’s paper is shown in Figure 2-2. The arrow indicates the change of the bond arrangement in the substituent during excitation. The hope of room temperature superconductivity has not yet been fulfilled, and no substance has been found in which superconductivity seems to occur according to Little’s mechanism. However, organic superconductors do exist, although with transition temperatures (Tc) considerably below room temperature. Until now the highest Tc in an organic superconductor is about 12 K [3], unless fullerenes are also considered organic substances, in which case the highest Tc is 18 K for K3C60 and 33 K for Cs2RbC60 [4]. Organic superconductors are discussed in Chapter 7.
Figure 2-1 Little’s superconductor [1]. Specially designed groups are attached to polyacetylene chains so that excitations in the substituent pair’ the electrons moving along the chain.
One-Dimensional Metals, Second Edition. Siegmar Roth, David Carroll Copyright © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30749-4
20
2 One-dimensional Substances
Figure 2-2 Suggestion for a substituent R in Little’s superconductor and rearrangement of double bonds upon excitation [1].
Another motivation for studying low-dimensional solids is to obtain fundamental knowledge from extreme cases. After the concepts of solid-state physics had been established in typical’ solids such as alkali metals, potassium chloride, and silicon, the next step is either to look at similar solids such as transition metals and III–V semiconductors or to reach for the limits of solid-state matter and investigate exotics: organic and molecular crystals, liquid crystals, amorphous solids, and low-dimensional solids. Of course, both approaches have been undertaken. Scientists attracted by exotics gradually formed a sort of club, hopping from one fashionable system to the next. The main topics of the International Conferences of Science and Technology of Synthetic Metals – ICSM somewhat reflect this trend (Table 2-1). The discussion of low-dimensional systems in this chapter follows more or less the historic order as the focus of the community’s attention moved along. Summary of the ICSM meetings (International Conferences of Science and Technology of Synthetic Metals – ICSM).
Table 2-1
Date
Place
Proceedings
1976
Siofok, Hungary
1978
Dubrovnik, Yugoslavia
1980
Helsingor, Denmark Boulder, CO, USA
Organic Conductors and Semiconductors ed. L. Pal, G. Grner, A. Janossy, and J. Solyom, Lecture Notes in Physics, Vol. 65, Springer Verlag , Heidelberg, 1977 Quasi-one-dimensional Conductors I and II ed. A. Barisic, A. Bjelis, J.R. Cooper, and B. Leontic, Lecture Notes in Physics, Vol. 95, Springer Verlag, Berlin, 1979 Low-dimensional Synthetic Metals ed. K. Carneiro, Chemica Scripta, Vol. 17, Nos. 1-5 (1981) Low-dimensional Conductors ed. A.J. Epstein, E.M. Conwell, Molecular Crystal Liquid Crystal, Vols. 77, 79, 81, 83, 85, 86 (1982) Low-dimensional Synthetic Conductors and Superconductors ed. P. Bernier, J.J. Andr, R. Comes, and J. Rouxel, J. Physique, Paris, Colloq. C3, Supplement 6, Tome 44 (1983)
1981
1982
Les Arcs, France
2 One-dimensional Substances Table 2-1
Continued.
Date
Place
Proceedings
1984
Abano Terme, Italy
1986
Kyoto, Japan
1988
Santa F, NM, USA
1990
Tbingen, Germany
1992
Gteborg, Sweden
1994
Seoul, Korea
1996
Snowbird, UT, USA
1998
Montpellier, France
2000
Gastein, Austria
2002
Shanghai, China
2004
Wollongong, Australia
International Conference on the Physics and Chemistry of Low-dimensional Synthetic Metals – ICSM ’84 ed. C.P. Pecile, G. Zerbi, R. Bozio, and A. Girlando, Molecular Crystal Liquid Crystal, Vols. 117–121 (1985) International Conference on Science and Technology of Synthetic Metals – ICSM ’86 ed. H. Shirakawa, T. Yamabe, and K. Yoshino, Synthetic Metals, Vols. 17–19 (1987) International Conference on Science and Technology of Synthetic Metals – ICSM ’88 ed. M. Aldissi, Synthetic Metals, Vols. 27–29 (1988/1989) International Conference on Science and Technology of Synthetic Metals – ICSM ’90 ed. M. Hanack, S. Roth, and H. Schier, Synthetic Metals, Vols. 41–43 (1991) International Conference on Science and Technology of Synthetic Metals – ICSM ’92 ed. S. Stafstrm, W.R. Salaneck, O. Ingans, and T. Hjertberg, Synthetic Metals, Vols. 55–57 (1993) International Conference on Science and Technology of Synthetic Metals – ICSM ’94 Synthetic Metals, Vols. 69–70 (1995) International Conference on Science and Technology of Synthetic Metals – ICSM ’96 Synthetic Metals, Vol. 84 (1997) International Conference on Science and Technology of Synthetic Metals – ICSM ’98 Synthetic Metals, Vol. 102 (1999) International Conference on Science and Technology of Synthetic Metals – ICSM ’00 Synthetic Metals, Vol. 121 (2001) International Conference on Science and Technology of Synthetic Metals – ICSM ’02 to be published in Synthetic Metals International Conference on Science and Technology of Synthetic Metals – ICSM ’04
The exotic solids community has often been criticized for moving too quickly and, in particular, with too much fuss. But we should remember that this workstyle introduces a large amount of interdisciplinary exchange of thoughts, making the participants aware of the rich possibilities of solid-state physics and natural science. It saves them from narrow-mindedness and ensures flexibility. Figure 2-3 shows a statistical evaluation of the literature on solitons. Solitons are probably the most popular of the fashionable exotics. They are discussed in Chapter 5. The wave of publica-
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2 One-dimensional Substances
Figure 2-3 Plot of annual numbers of publication on solitons on a background of Katsushika Hokusai’s wood carving View of Mount Fuji from a Wave Trough in the Open Sea off Kanagawa’.
tions rises like a solitary wave. Similarly, the number of contributions to the ICSM meetings increases (Table 2-1): the proceedings of the Gteborg meeting, for example, extended over 5000 pages and weighed 20 pounds.
2.1
A15 Compounds
A15 compounds are intermetallic compounds such as V3Si and Nb3Sn, in which one of the partners is a transition metal. The A15 structure is shown in Figure 2-4. The essential feature of this structure is that the transition metal atoms are arranged
Figure 2-4 Crystal structure of A15 compounds. Filled circles are transition element atoms (vanadium or niobium). They form three sets of mutually perpendicular chains. Open circles symbolize Si, Sn, Ge, or Ga atoms. They serve mainly to support the structure.
2.1 A15 Compounds
in chains. There are three mutually perpendicular sets of chains, pointing to the three directions of three-dimensional space. At first glance the structure seems three-dimensional and it takes some imagination to recognize the chains. The distance between two vanadium or niobium atoms in the same chain is smaller than that between atoms of different chains. In addition, the wave functions of the valance electrons of vanadium are oriented so that the in-chain overlap is considerably larger than the chain–chain overlap. The origin of the name A15 compounds’ is remarkable. It is the classification of the crystal structure, however misleading. Structures starting with the letter A are assigned to structures of elements, not of compounds. A15 would then be the structure of the 15th modification. Since 15 is a fairly large number, and numbering probably would start on the simple side, we would expect a complicated structure. Another name for the A15 structure is b-tungsten structure’, because for some time people thought that this is the structure of a complicated modification of the element tungsten. Later it turned out that b-tungsten is not elementary tungsten but actually a binary compound, a suboxide of tungsten, W3O. The wrong assignment in the early days is easy to understand. X-ray diffraction is the most important tool for structural analysis; the diffraction intensity is proportional to the square of the atomic number, and thus the light oxygen atoms with atomic number 8 are difficult to observe in the presence of the heavy tungsten atoms with atomic number 74. However, with one-dimensional metals, the crystallographic misinterpretation does not matter: we are only interested in one sort of atoms, those forming metallic chains, and these are the heavy atoms. The light atoms act mainly as spacers. A15 compounds are important as superconductors, and for along time the highest superconducting transition temperatures were found among the A15 compounds (V3Si: 17 K; Nb3Sn: 23K; V3Ga: 15 K). Even today, Nb3Sn is the most important material for high-field superconducting magnets. Of course, it was speculated that the high transition temperature of the A15 superconductors might be somehow related to the chain-like arrangement of the transition metal atoms. In addition to superconductivity, another interesting phenomenon is observed in A15 compounds: a structural phase transition from a cubic crystal lattice to a tetragonal lattice. With reference to the crystallography of iron–carbon systems (steel) this transition is often called the Martensit transition’. At the Martensit transition the chains of one set contract and the chains belonging to the other two sets expand. This transformation is accompanied by the softening’ of lattice vibrations because some of the restoring forces (those that stabilize the cubic crystal) fade at the transitions. Perhaps these soft lattice vibration are responsible for the coupling of the conduction electrons to form Cooper pairs and thus for the tendency to superconductivity? In any case, an interesting reciprocal action occurs between the superconducting phase transition and the Martensit phase transition: whichever occurs first upon cooling seems to rule out the other. Exposure to a magnetic field modifies both transitions [5]. (A review of A15 superconductors is available in [6]; for general reading on superconductivity [7] is recommended.) In Figure 2-5 the soft lattice vibrations are illustrated for a superconducting V3Si sample near the Martensit transition, and in a magnetic field. The velocity of sound
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2 One-dimensional Substances
Figure 2-5
Field dependence of the soft-mode sound velocity in V3Si at various temperatures [5].
is plotted against the applied magnetic field. The polarization and propagation direction of the sound waves are chosen so that the sound velocity is determined by the restoring force that stabilizes the cubic structure. At the Martensit transition this force vanishes and the velocity approaches zero. Close to the Martensit transition the sample is superconducting. Superconductivity prevents the Martensit transition. As the magnetic field is turned on, superconductivity is gradually suppressed, and another effect of the magnetic field becomes dominant. From now on the velocity increases, i.e., the sample hardens, because the magnetic field works towards preventing the Martensit transition. The soft-mode behavior of A15 compounds and its modification by superconductivity and by magnetic fields can be traced back to singularities in the electronic structure of these compounds. Figure 2-6 shows the density of states for V3Si. There is a wide band due to the s electrons and three narrow d bands, denoted d3z2–r2, dxz/dyz, and dxy/dx2–y2, respectively. In Chapter 3 we will discuss the electronic density of states of a one-dimensional solid which has square root singularities at the band edges, i.e., the density of states approaches infinity with (E0 – E)–1/2. These singularities are indicated in Figure 2-6a. V3Si has three mutually perpendicular sets of vanadium chains. In the cubic phase all chains are equivalent and the singularity occurs at identical energetic positions for all chains. In the tetragonal phase this degeneracy is lifted, because in the compressed chains the bandwidth increases and the band edge moves downwards; in the expanded chains it moves upwards (Figure 2-6b). If the Fermi level is close to the singularity in the density of states, moving the singularity downwards causes an overall reduction of electronic energy.
2.1 A15 Compounds
Figure 2-6 Electronic density of states of V3Si in the cubic phase (a) and the tetragonal phase (b). In the cubic phase the singularities at the band edges are threefold degenerate, because all three sets of V chains are identical. In the
tetragonal phase this degeneracy is lifted, resulting in a lowering of the Fermi level and, in consequence, of the overall energy of the electrons [8].
Thus the tetragonal phase is energetically favored. At high temperatures there are also occupied states above the Fermi level, and the energy gain by the tetragonal distortion of the lattice vanishes. Therefore the cubic lattice is stable at high temperatures. The Martensit transition is a special case of the more general Jahn–Teller theory. It states that an electron shared by degenerate states can always gain energy when the degeneracy is lifted and the electron moves into the lower state, as schematically indicated in Figure 2-7. Another example of energy gain by symmetry breaking is the Peierls transition, which is discussed in Chapter 4. Similarly, the occurrence of superconductivity also stabilizes the cubic phase. Superconductivity creates a gap around the Fermi level; modifications of the band structure within this gap do not affect the energy balance. The same argument can be used to explain the lattice hardening induced by a magnetic field (in the nonsuperconducting state). In a magnetic field Zeemann splitting of the electron energies occurs. This Zeemann splitting alters the singularity in the density of states, so that a further modification by lattice distortion is less effective.
Figure 2-7 Scheme of the Jahn–Teller effect: an electron shared by degenerate states gains energy upon lifting degeneracy and descends to the lower state.
25
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2 One-dimensional Substances
2.2
Krogmann Salts
As mentioned above, A15 compounds contain three perpendicular sets of chains. Therefore the properties of the material exhibit cubic symmetry. If there was only one set of chains, a much more drastic appearance of one-dimensionality would be expected. Krogmann salts [9] are a good example of solids with only one set of chains. The prototype of Krogmann salts has the stoichiometric formula K2[Pt(CN)4]Br0.3 . 3H2O. Commonly this compound is called KCP, an abbreviation of kalium (= potassium) tetracyanoplatinate. The essentials of the structure are presented in Figure 2-8 [10]. Figure 2-8a shows a planar complex of platinum surrounded by four cyano groups. Two of the electronic orbitals of platinum are above the plane. In Krogmann salts such complexes are stacked on top of one another to generate a columnar structure (Figure 2-8b). For the physicist the important part of the Krogmann salt is the metal chain. The function of the cyano groups is to sterically support the structure. In addition to Pt(CN)4 complexes, the crystals also contain alkali metals for electron balance, with fine tuning’ achieved by halogen. The (001) and (010) projections of the elementary cell of KCP are shown in Figure 2-9 [10]. The one-dimensionality of Krogmann salts becomes apparent when looking at KCP crystals under polarized light. If the light is polarized parallel to the direction of the platinum chains, the crystal appears shiny and lustrous like metals. For perpendicular polarization its appearance is dull like an insulator. Some people say that in solid-state physics each effect has its proper substance in which the effect is best exhibited and most easily studied. If so (in A15 compounds we discussed the Martensit transition) KCP is the substance belonging to the Kohn anomaly. The Kohn anomaly is the precursor of another structural phase transition that is likely to occur in one-dimensional solids: the Peierls transition (more on
Figure 2-8 Electronic orbitals of platinum above and below the plane of the tetracyanoplatinate complex (a); linear chain arrangement of platinum orbitals (b) [10].
2.2 Krogmann Salts
Figure 2-9 Elementary cell of the Krogmann salt KCP. The upper part is a section perpendicular to the metallic chains, the lower part a section parallel to the chains [10].
Peierls transition in Chapter 4). The Peierls transition consists of pairing of atoms in a metallic chain. This pairing – also an example of symmetry breaking, in this case, of translational symmetry – results in a doubling of the elementary cell, containing one atom before the phase transition and two after. Again there are restoring forces, which tend to keep the atoms of the chain equidistant and which become soft when the phase transition is approached. In A15 compounds the softening is observed for long-wavelength shear waves (i.e., transverse polarization), in KCP it occurs for short-wavelength compressional waves (with longitudinal polarization). The atom pairing – or more generally, the rearrangement of the atoms – corresponds to a compressional wave with a wavelength twice the interatomic distance. Figure 2-10 shows the dispersion relation for longitudinal vibrations in KCP. The dispersion relation is the dependence of the vibrational frequency on the reciprocal wavelength, and the vibrational frequency is a measure of the restoring force. A pronounced dip in the dispersion relation of KCP can be noticed. A small dip is predicted for the vibration dispersion relation of any metallic solid, and it occurs where the wavelengths of the vibrations match the wavelength of the electrons at the Fermi level, as is demonstrated in Chapter 4. The small dip is called the general Kohn anomaly. However in KCP, the Kohn anomaly is giant, a dip that actually reaches zero frequency, corresponding to the fading of the restoring force, and leads to a structural phase transition. The velocity of longwavelength lattice vibrations can be measured by ultrasonic techniques, e.g., for V3Si (Figure 2-5). To measure the frequency of other vibrations and to sample the dispersion relation, neutron scattering has to be employed (Figure 2-10). (The above discussion is actually oversimplified: the giant Kohn anomaly does not occur at a
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Longitudinal acoustic phonon branch in [001] direction of KCP showing the Kohn anomaly as a large dip [11].
Figure 2-10
wavelength exactly twice the interatomic distance, but at some more general location in the reciprocal lattice (for the reciprocal lattice see Chapter 3), the exact position being determined by the Fermi level and ultimately by the electron balance of the crystal due to the alkali and halogen ions.)
2.3
Alchemists’ Gold
A15 compounds contain three perpendicular sets of metallic chains, Krogmann salts one, and – if searched for – a system with two sets should certainly exist too. Alchemists’ gold’ is such a system. Mercury and gold are direct neighbors in the periodic table of the elements. Perhaps the alchemists thought that exposing mercury to a very aggressive environment would convert it into gold. Arsenic fluoride certainly is aggressive, and mercury can be brought into contact with that substance. Figure 2-11 [12] shows the crystal structure of Hg2.86AsF6. The mercury chains along the crystallographic a direction are clearly seen in this figure. To see both sets of chains, in a and b directions, Figure 2-12 is more appropriate [13]. One of the curious properties of this substance is that the mercury in the chains is said to be liquid’, whereas the AsF6 matrix is a well-defined solid. How does a one-dimensional liquid differ from a one-dimensional solid? The usual definition given is that, in both solids and liquids the distance between next neighbors is fixed, and in solids also the angles are fixed. In a perfectly straight chain there is no angular degree of freedom anyway; therefore the difference between solid and liquid must disappear. (Polymer chains as in Figures 1-11 and 1-12 are not straight but zigzag, and hence they can melt by rotation around the bonds.) If alchemists’ gold is
2.3 Alchemists’ Gold
Crystal structure of alchemists’ gold’, Hg2.86AsF6. The mercury chains follow the crystallographic a and b axes [12]. Figure 2-11
cooled, the AsF6 matrix contracts and the crystal sweats’ mercury: small droplets of liquid mercury appear on the faces. Upon heating these droplets are re-absorbed into the bulk crystal. From the point of view of fundamental science, the mercury chains are very interesting. Bulk mercury is a superconductor. Will one-dimensional mercury chains also become superconducting? Is anisotropy to be expected, say, in the critical magnetic field? Will the transition temperature of the mercury chains be above or below
Figure 2-12
gold [13].
Mercury chains in alchemists’
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the Tc of bulk mercury? Or is there no transition to superconductivity because of Landau’s interdict of phase transitions in one dimension? Unfortunately, the escaped mercury makes experiments and their interpretation more difficult than expected. There is a transition to superconductivity around 4 K, which coincides with the Tc of bulk mercury and apparently is due to superconductivity in the secreted droplets. There is a further transition around 2 K, where the chains become superconducting.
2.4
Bechgaard Salts and Other Charge-transfer Compounds
In the Krogmann salts, planar units are stacked to form columns and the molecular orbitals overlap within these columns to form a one-dimensional metal (Figure 2-8; overlap of the d-orbitals of platinum). In a similar way the p-orbitals of carbon can be forced to overlap by appropriate stacking of planar organic units. Figure 2-13 [14] shows the structure of (TMTSF)2PF6, one example of the socalled Bechgaard salts. In Bechgaard salts, organic donors are combined with inorganic acceptors. The donors are fairly large planar organic molecules, in this example, tetramethylenotetraselenofulvalene. For everyday use the names of these compounds are too lengthy and therefore usually abbreviated as, for example, TMTSF. The organic donors contain large numbers of conjugated double bonds. From the point of view of a simple-minded physicist they are just tiny chiplets of graphite. The chiplets are stacked like honeycomb planes, as in graphite, with a certain overlap of the p electrons along the stacking axis. As a consequence, electronic energy bands are formed. A certain amount of charge transfer from the organic donor to the inorganic acceptor occurs. Because of this charge transfer, the electronic bands are only partially filled and the Bechgaard salts are metallic. At low temperatures some of the Bechgaard salts become superconducting, others undergo a metal-toinsulator transition via the Peierls mechanism (for more details on Peierls transition, see Chapter 4; organic superconductivity is discussed in Chapter 7). As in the A15 compounds, superconductivity often competes with other phase transitions, and it is up to molecular engineering to favor one and suppress the other. Solid-state physics of novel materials should establish guidelines on how to optimize solids for particular applications by appropriate molecular engineering.
Figure 2-13
Crystal structure of the Bechgaard salt (TMTSF)2PF6 [14].
2.4 Bechgaard Salts and Other Charge-transfer Compounds
Crystal structure of (fluoranthenyl)2SbF6 [15].
Figure 2-14
Figure 2-15
Organic donors [16].
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Figure 2-14 [15] shows another Bechgaard salt, (fluoranthenyl)2SbF6. This substance is well known for its very narrow ESR (electron spin resonance) lines, which make it a very sensitive magnetic field probe. The narrow ESR lines arise from the delocalization of the electrons over the fluoranthenyl stacks and thus are typical for good one-dimensional metals. Figure 2-15 [16] summarizes several organic donors, and Figure 2-16 [16], organic acceptors. In Bechgaard salts organic donors are combined with inorganic acceptors. It is also possible to make all-organic charge-transfer salts by combining an organic donor with an organic acceptor. A very well known example is TTF-TCNQ. The crystal structure of this substance is shown schematically in Figure 2-17 [17]. The TTF and TCNQ units are arranged in a herringbone pattern. Because of the tilt, the units are more densely packed, with a larger overlap between the molecules in a stack. It is interesting to note that both TTF and TCNQ also crystallize separately, forming
Figure 2-16
Organic acceptors [16].
2.5 Polysulfurnitride
Crystal structure of the organic charge-transfer salt TTF-TCNQ [17].
Figure 2-17
insulating solids. Thus it is the charge transfer from the TTF stacks to the TCNQ stacks that makes TTF-TCNQ metallic. TTF-TCNQ shows many interesting solid-state phenomena, such as high electrical conductivity and a Peierls transition from metal to insulator, but it does not become superconducting. In the 1970s reports appeared on superconductivity with Tc values as high as 58 K in some of the TTF-TCNQ samples [18]. Of course, these reports attracted an enormous amount of attention. In fact, the first report is most frequently cited of all papers ever published in Solid State Communications. Today, however, many scientists agree that the samples showed tricky artifacts caused by the high, strongly temperature-dependent anisotropy of the electrical conductivity [19].
2.5
Polysulfurnitride
At least for a physicist, it is not evident that sulfur and nitrogen can be arranged in long chains in which sulfur atoms and nitrogen atoms alternate regularly. But apparently it can be done; the substance even crystallizes and shows metallic properties. It furthermore becomes superconducting, although only at extremely low temperatures (Tc = 0.26 K [20]). The structure of polysulfurnitride (SN)x is shown in Figure 2-18 [21]. (SN)x can be doped, for example, with bromine. Doping slightly increases the Tc, possibly due to the Fermi level being moved to a position of a higher density of states, which is more favorable for superconductivity. Probably more important than the doping-induced Tc-shift of (SN)x is that the doping of (SN)x might have stimulated the idea of also doping polyacetylene, (CH)x. This way, a vast new field in materials science has been created, the field of conducting polymers and synthetic metals.
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Crystal structure of the inorganic polymer’ polysulfurnitride, (SN)x [21].
Figure 2-18
2.6
Phthalocyanines and Other Macrocycles
When discussing the Bechgaard salts we became acquainted with the stacking of graphite-like chiplets. No doubt, it is more justified to call phthalocyanine and porphyrin graphite chiplets’. The chemical structure of these macrocycles is shown in Figures 2-19 and 2-20. Both substances contain conjugated systems with 18 p electrons. For a physicist, three is already many and ten is infinite, so these macrocycles are infinitely large plates. Properly stacked they become graphite-like.
Chemical structure of the macrocyclic organic compound phthalocyanine, with a transition metal in the center.
Figure 2-19
2.6 Phthalocyanines and Other Macrocycles
Chemical structure of the macrocyclic organic compound porphyrin, with a transition metal in the center.
Figure 2-20
However, phthalocyanines and porphyrin show a feature not present in graphite platelets: a void in the center which can hold a transition metal atom. The choice of center metal allows for fine-tuning the electronic properties of the substance. A particular example is lead phthalocyanine. Although the other phthalocyanines are planar, because the lead atom is so large, the molecule becomes distorted. It assumes the shape of a badminton shuttlecock, which should be easier to stack than flat plates. In addition, a special type of stacking fault may be observed, as indicated in Figure 2-21. The conductivity of a stack depends largely on the presence of such faults and fault generation could be used for switching [22]. To stack planar macrocycles, the shish kebab’ method has been developed: covalent connections are placed between the central metals. and the macrocycles are pinned onto a polymer backbone like pieces of meat on a barbecue spit (Figure 2-22 [23]).
Figure 2-21
Stacks of lead phthalocyanine with stacking faults [22].
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Figure 2-22
Shish-kebab polymer of metallophthalocyanine [23].
2.7
Transition Metal Chalcogenides and Halides
A large family of transition metal chalcogenides and halides exists, which can be described as condensed clusters [24]. The basic structural unit is a cluster consisting of a transition metal M surrounded by group-VI or group-VII elements X. Figure 2-23 shows a trigonal prism with 6 selenium atoms at the corners and a niobium atom at the center, Figure 2-24 presents a MoO6 octahedron. The clusters condense’ by sharing corners, edges, or faces. Prismatic condensation is shown in Figure 2-25, where edge-sharing leads to the layered structure of NbSe2 and facing-sharing to NbSe3 fibers. NbSe2 is representative of a large group of inorganic layered solids, which have been thoroughly discussed in a series of monographs [25]. Some of these layered solids are metals and superconductors,
Figure 2-23 Trigonal prism of NbSe6 as the basic structural unit of low-dimensional niobium-selenide solids.
Figure 2-24 MoO6 octahedron as the basic unit of blue bronzes.
2.7 Transition Metal Chalcogenides and Halides
Figure 2-25 Condensation of MX6 prisms to form layers by edge-sharing (left) or fibers by face-sharing (right) [26].
others become metallic after intercalation’, i.e., after insertion of organic or inorganic molecules between the layers. NbSe3 is one of the most important inorganic quasi-one-dimensional solids. It is particularly suited for studying charge density wave phenomena (see Chapter 8). An example of a corner-sharing cluster condensation is (MX4)nY, shown in Figure 2-26, where M is a transition metal such as Ta or Nb, X a chalcogenide such as Se or S, and Y represents halogen ions between the fibers. A more complicated cluster condensate is shown in Figure 2-27, representing K0.3MoO3, one of the blue bronzes. We have already encountered another member of this cluster-condensation family: the Krogmann salt KCP (Section 2.2). Here M = Pt and X = CN, the cyano group being a close relative of the halogens. Cluster condensation may lead to metals or to semiconductors (insulators), depending on the number of M and Y atoms. When the number of nonmetal atoms in a compound is not sufficient to completely surround the metal cluster, the clusters link by direct M–M bonds [24], thus forming extended M–M chains or metal
Figure 2-26
Crystal structure of (MX4)nY [27].
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Crystal structure of a blue bronze showing the infinite sheets of MoO6 octahedra separated by the alkali ions. The sheets contain infinite chains [28]. Figure 2-27
planes. The alkali atoms in Figure 2-27 and the halogen atoms in Figure 2-26 help to adjust the electron density and fix the Fermi level. Thus they determine the details of the electronic properties of these compounds and, in particular, the charge density wave behavior.
2.8
Conducting Polymers
No other class of one-dimensional materials has triggered such large numbers of publications as conducting polymers. In 1977 it was first discovered that the conductivity of polyacetylene can be increased by doping, by many orders of magnitude [29]. Since then, conducting polymers have dominated the contributions to the International Conferences of Science and Technology of Synthetic Metals. The proceedings of the 1992 conferences in Gteborg (see Table 2-1) consist of nearly 1000 contributions on more than 5000 pages. In 1979 the journal Synthetic Metals came into being, and most of the articles therein deal with conducting polymers. The reason for the popularity of conducting polymers is threefold: .
.
.
Conducting polymers are considered new materials with great potential for new applications [30,31] (see Chapter 10). Conducting polymers are derivatives of polymers, i.e., of compounds with extended systems of conjugated double bonds. Such systems are subject to quantum chemical concepts and calculations [32–34]. Certain excitations occur in polyenes, which are related to solitary waves and to solitons. Thus, there is an interdisciplinary link to field theory, hydrodynamics, elementary-particle physics, and certain aspects of biology [35–40].
2.8 Conducting Polymers
Figure 2-28
Chemical structures of some of the most important conducting polymers.
Conducting polymers are also a main subject of this book and are discussed in Chapter 5, 6, and 10. The 2000 Nobel Prize in Chemistry was awarded to A. J. Heeger, A. G. MacDiarmid, and H. Shirakawa for the discovery of conducting polymers. Figure 2-28 shows the basic chemical structure of some of the most important conducting polymers. The regular array of alternating single and double bonds, characteristic of polyenes, is clearly visible. (In polyaniline the extra electron pair on the trivalent nitrogen atoms participates in band formation, so that this substance is conjugated as well.) In Figure 2-29 the respective polymers are shown as space-filling models. These polymers become conducting only after doping, in the undoped form they are semiconductors. A large part of interesting physics is not only related to the high conductivity after doping but to the extended system of conjugated double bonds. Figure 2-30 shows the chemical structure of polydiacetylene. Here the bond sequence is single–double–single–triple–single–double–single–triple, etc. From a certain point of view it is justified to say that the system is between polyacetylene and polycarbyne (see Figures 1-10 and 1-12). Polydiacetylene is the only conjugated polymer from which large single crystals can be grown. Actually, these crystals are not grown from the polymer, but from monomeric diacetylene. In the crystal the monomers can be polymerized without disturbing the crystal structure. This is a very rare instance of solid-state polymerization, which can be achieved because in diacetylene crystals the monomer molecules are already in the positions and orientations needed for the final polymer. The polymerization is initiated by light or heat. Figure 2-31 shows a scheme of the solid-state polymerization [41,42]. We should note a structurally unifying way of visualizing conducting polymers. This involves the use of a single sheet of graphite. As most tunneling microscopists will tell you, to create an excellent flat surface on highly oriented pyrolytic graphite, all one needs is a little cellophane tape. By placing the tape on the surface and pulling it off again, a few individual layers are removed. This is due to the extremely
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Computer-generated space-filling models of the polymers shown in Figure 2-28.
Figure 2-29
weak van der Waals forces that hold the layers together. Imagine now only one, atomically thin, layer. We generally call this graphene’ and it can be used as the template for most conducting polymers. To demonstrate this, consider the graphene sheet in Figure 2-32.
Figure 2-30
Idealized chemical structure of polydiacetylene.
Figure 2-31
Scheme of solid-state polymerization of diacetylenes.
2.9 Halogen-bridged Mixed-valence Transition Metal Complexes
Figure 2-32
The conjugated systems map fortuitously onto graphene.
Of course in the present context this is not so important. However, when we come to carbon nanotubes, the concept can be used to see the clear link between conjugated systems and nanotubes.
2.9
Halogen-bridged Mixed-valence Transition Metal Complexes
Many scientists agree that halogen-bridged mixed-valence transition metal complexes are the ideal subject for one-dimensional solid-state physics. The long name of these substances has been abbreviated to HMMC; another short colloquial term is MX chains, where M stands for transition metal atoms and X for halogen atoms. The essentials of the structure are shown in Figure 2-33: in the chain M and X atoms alternate; here, M = Pt and X = Cl. Ligands are attached to the metals; in some cases there are counterions between the chains. Hence, in a certain respect HMMCs are related to Krogmann salts (compare Figure 2-8), even though the
Figure 2-33
Halogen-bridged mixed-valence transition metal complex.
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chains are not made exclusively of metal atoms. An electronic energy band is formed by the dz orbitals of the metal atoms and the pz orbitals of the halogen atoms. Depending on structural details, the electrons in this band can have various degrees of localization. The limit on one side is the valencies of the transition metals oscillating along the chains between II and IV (Figure 2-33). At the limit on the other side, all metal atoms have valance state III. The importance of MX chains arises from the fact that the synthesis of these substances is straightforward, and high-quality single crystals can easily be grown – not only for one particular MX compound, but for a large variety. Typical metals are M = Pt, Pd, or Ni and the halogen X = Cl, Br, or I. Common ligands comprise L = halogens, ethylamines, ethylenediamines, or cyclohexanediamine, and as common counterions between the chains Y = halogens or ClO4–. One particular way of writing the stoichiometric formula of an HMMC is [Mr–dL4][Mr+dX2L4]Y4 Here r denotes the average valence of M and d the deviation from the average. If d is small the X atoms are centered between the metal atoms. For large d the halogens move closer to the metal with the higher valence. In some PtCl chains distortion can be as high as 20%. By changing M, L, X, and Y it is possible to vary the properties of the substance in a well-controlled way. In particular, the dimensionality can be adjusted (interchain coupling) and the electron–electron as well as the electron–phonon interactions can be tuned. Important review articles on MX chains are provided in [43] and [44]. Additional publications are found in the Proceedings of the International Conference on Science and Technology of Synthetic Metals (ICSM), from 1990 onwards (see Table 2-1).
2.10
Miscellaneous
In this section some substances are mentioned that are not mainstream in terms of one-dimensionality’, either because sizable samples are not yet available or because other fields are taking care of the respective research. 2.10.1
Poly-deckers
Figure 2-34 shows the structure of the poly-decker’ poly(g5,l-2,3-dihydro-1,3-diborolyl)nickel [45]. Five-membered heterocycles are stacked by sandwiching Ni atoms in a similar way as Fe is sandwiched in ferrocene. If ferrocene is classified as a doubledecker structure, the compound depicted in Figure 2-34 should be called a polydecker.
2.11 Isolated Nanowires
Poly(g5,l-2,3-dihydro-1,3-diborolyl)nickel as an example of a poly-decker’ structure.
Figure 2-34
2.10.2
Polycarbenes
Carbenes contain divalent carbon atoms with two nonbonding electrons. One way of arranging carbenes in a polymer is shown in Figure 2-35 [46]. If the nonbonding electrons in carbenes are in the singlet state their spins are compensated and they form a lone pair with no net spin. In the triplet state carbenes carry spin = 1. In the polymer there might be a positive (ferromagnetic’) interaction between these spins so that the whole molecule is in a high-spin state. These high-spin carbenes can be interpreted as a first step towards all-organic ferromagnets. Organic ferromagnets’ is also a rapidly emerging field in modern materials research [47,48].
Figure 2-35
Polycarbene as one possible first step toward one-dimensional organic ferromagnets.
2.11
Isolated Nanowires
Up to this point, the materials described have utilized asymmetries in bulk conductivity to approximate one-dimensional behavior. To fully understand the phenomena of electrical conduction in one dimension, of course, it would be most interesting to obtain an individual, isolated nanowire. This could then have contacts applied and
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measurements made directly. Within the past few years, significant advances in the growth and manipulation of such wires have been made. 2.11.1
Templates and Filler Pores
Many solids contain pores, gaps, voids, or channels in which materials can be inserted or intercalated in the liquid or gaseous phase. Examples of such materials
Framework structures and tubular representations of the channel systems of zeolites [50].
Figure 2-36
Nanoporous templates have become a standard in the fabrication of a widely dissimilar array of nanowires. The pores shown here are approximately 20 nm in diameter and 1000 lm in length.
Figure 2-37
2.11 Isolated Nanowires
are zeolites and clays, which provide long narrow channels ideal for metal insertion. [49] (Figure 2-36). Recently, the technology to create long pores of well-defined diameter in a matrix such as alumina (Al2O3) has matured to the point of commercialization. The process is quite simple: first, aluminum foil is anodized forming closely packed pores as shown in Figure 2-37. These pores can then be filled with a substance, e.g., monomers for creating PPV. Alternatively, metals can be added, even alternated to make heterojunction, metallic nanowires. Because the technique relies on the processibility of the filling material and is not as sensitive to the chemical makeup, the types of wires that have been demonstrated vary dramatically. 2.11.2
Asymmetric Growth using Catalysts
Templating methods have progressed recently. They are also based on direct catalytic growth methods. A wide assortment of nano-wide, micrometer-long wires can now be reproducibly created in rather large quantities, by directly assembling atoms from a gas on a catalyst particle (chemical vapor deposition, CVD) within a hightemperature, atmospherically controlled oven. Typically, small catalyst particles are faceted, and diffusion of adsorbed (or absorbed) gases across (or through) the particle is greater along one crystallographic direction than another. As some atomic arrangement assembles on the surface of the particle, therefore, asymmetry in growth direction is introduced, resulting in one-dimensional wires. The catalysts can either be added before growth on some substrate placed inside the growth oven or be introduced simultaneously using a vapor coalescence of metal atoms provided by flowing a metal-containing organic with the growth source gas. For one-dimensional wire CVD growth, there are a few generalities that must be met. Catalyst particles must be small (typically < 20 nm), growth temperature must be sufficient to allow for breakdown of gas stock, and catalyst-growth material systems must be chosen to allow for some solid solubility of the growth material in the catalyst particle. Examples of such growth (Figure 2-38) now include such semiconducting materials as GaN, once quite difficult to control.
A transmission electron micrograph of GaN nanowires grown by chemical vapor deposition (CVD) techniques shows well-defined one-dimensional structures (courtesy of J. Liu, Clemson University).
Figure 2-38
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2.11.3
Nanotubes
Unique to asymmetrically grown materials is the carbon nanotube. We have already mentioned this object in two ways. In Section 1.3.4 the zero-dimensional form of carbon was introduced – fullerene. Figure 1-14 shows a football (soccer ball) composed of 12 pentagons and 20 hexagons. This structural concept can be extended to carbon molecules with more hexagons, or even to long narrow carbon tubes, i.e., cylindrical graphite with fullerene caps at the ends. One such tube is shown in Figure 2-39. This figure served as the logo for the International Winter School on Electronic Properties of Novel Materials: Progress in Fullerene Research in Kirchberg, Austria, March 1994 [51]. Additionally, in discussing the concept of a mixed-dimensional’ or topological object, the carbon nanotube came up again. Because the carbon nanotube will be used extensively in this text as an example of an exotic, conducting polymer system with well defined symmetries, it is best to take time here for a more complete description of this object. To construct (conceptually) a carbon nanotube we begin with a graphene (one layer of graphite) sheet. Such a sheet is infinite in two-dimensional extent, one atom thick, and composed of three-fold coordinated carbon using sp2 hybrid bonds as shown in Figure 2-40. Specifying a unique way to roll the sheet into a tube is done with indices (n,m) that are derived from the number of unit vectors (upper left) required to specify vector R. Alternatively, one can specify the diameter of the tube and the angle that R makes with the darkly shaded hexagons running diagonally on the map (the armchair direction’). This angle is termed the chiral angle but is really nothing more than the helicity of the molecule. In the example given above a (9,7) nanotube will be formed when the sheet is rolled.
Nanotube. Carbon can form cylinders of some 10 in diameter and several lm in length. On the right is an atomic-resolution image taken with a tunneling microscope.
Figure 2-39
2.11 Isolated Nanowires
The standard map for understanding carbon nanotubes. Cut the graphene along the dotted lines on the right. Roll it (along vector R) into a tube connecting A–A¢ and B–B¢. This gives a tube. The unit cell of the
Figure 2-40
tube has been marked on the map with dotted lines across the tube axis. The vector marked T is the translation vector for this system. On the right is how this tube will appear once it is rolled (courtesy Jannik Meyer).
The geometry of tiling the surface with hexagons can be quite useful, for instance, knowing that the above is a (9,7) nanotube tells us that the diameter (DIA) of the nanotube has to be DIA = R/p = (3 ac-c / p) (n2 + m2 + nm)1/2 where ac-c is the carbon–carbon distance (lattice parameter). Further, we know that the chiral angle (h) must be h = Cos–1 [(2n + m) / 2(n2 + m2 + nm)1/2] or, in our case: ~25.8. 2.11.4
Inorganic Semiconductor Quantum Wires
We have already encountered lithographically fabricated silicon or gallium arsenide quantum wires in Section 1.2 in the presentation of the external approach to onedimensionality. A semiconductor device that acts as a quantum wire is shown in Figure 2-41. Since the advent of modern microfabrication technologies, semiconductor quantum wires have been subject to a large number of exciting experiments [52]. As substrate, usually AlGaAs–GaAs heterojunctions grown by molecular beam epitaxy are used. At the AlGaAs–GaAs interfaces, a quasi-two-dimensional electron layer is formed which can exhibit extremely high mobilities. Confinement to a one-dimensional wire is then achieved by imposing a microstructured split Schottky gate on the surface (Figure 2-41). Upon applying a negative bias to the split gate, electrons
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Scheme of a semiconductor quantum wire defined by using a split Schottky gate imposed on a AlGaAs–GaAs heterojunction. The white areas between source and drain electrodes indicate the quasi-two-dimensional electron gas formed at the AlGaAs–GaAs heterointerfaces.
Figure 2-41
in the underlying two-dimensional electron system are depleted, thus forming a narrow (or quasi-one-dimensional) channel in the gap. The split gates can be fabricated by sophisticated technologies such as electron beam lithography. The quantum wires can also be fabricated by etching the semiconductor heterostructure. 2.11.5
Metal Nanowires
Before leaving our taxonomy of one-dimensional systems, there is one last, and recent, set of novel experiments we must consider. Recall that in Chapter 1 we mentioned the famous’ problem of the single line of gold atoms acting as a wire? This has actually been realized! The measurements involve crashing’ the gold tip of the probe in a scanning-tunneling microscope into a gold surface and slowly retracting it (Figure 2-42). During retraction, adhesion between the two surfaces allows for
Retracting the tip of a scanning-tunneling microscope probe from a gold surface after contact leaves an atomically thin wire.
Figure 2-42
References and Notes
creation of a very fine wire between the tip and surface. The further the tip is retracted, the thinner the wire becomes. This simple experiment has allowed some of the deepest insight yet into the electromechanical nature of one-dimensionality.
References and Notes 1 W.A. Little. Possibility of synthesizing an
organic superconductor. Physical Review A 134, 1416 (1964). 2 J. Bardeen, L.N. Cooper, and J.R. Schrieffer. Theory of superconductivity. Physical Review 108, 1175 (1957). 3 J.M. Williams, A.M. Kini, U. Geiser, H.H. Wang, K.D. Carlson, W.K. Kwok, K.G. Vandervoort, J.E. Thompson, D.L. Stupka, D. Jung, and M.H. Whangbo. Organic Superconductivity. V.Z. Kresin and W.A. Little (ed.), Plenum Press, New York, 1990. 4 M.J. Rosseinsky, A.P. Ramirez, S.H. Glarum, D.W. Murphy, R.C. Haddon, A.F. Hebard, T.T.M. Palstra, A.R Kortan, S.M. Zahurak, and A.V. Makhija. Superconductivity at 28 K in RbxC60. Physical Review Letters 66, 2830 (1991). Z. Iqbal, R.H. Baughman, B.L. Ramakrishna, S. Khare, N.S. Murthy, H.J. Borneman, and D.E. Morris. Superconductivity at 45 K in Rb/ TI codoped C60 and C60/C70 mixtures. Science 254, 826 (1991). 5 J.D.N. Cheeke, H. Malli, S. Roth, and B. Seeber. Direct observation of soft mode stiffening in V3Si by high magnetic fields. Solid State Communications 13, 1567 (1973). W. Dietrich, S. Roth, H. Malli, and B. Seeber. In: Proceedings of the 14th International Conference on Low Temperature Physics, Helsinki 1975, p. 36. 6 J. Muller. A15-type superconductors. Reports on Progress in Physics 43, 641 (1980). 7 W. Buckel. Supraleitung. Physik-Verlag, Weinheim, 1984. 8 J. Labbe. Structure de bande fine des composs intermtalliques supraconducteurs du type V3Si et Nb3Sn: instabilit cristalline d’origine lectronique, PhD thesis. Paris,1968.
9 K. Krogmann. Planare Komplexe mit Metall-
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Metall-Bindungen. Angewandte Chemie 81, 10 (1969). H. Wagner. PhD thesis. Karlruhe, 1974. H.R. Zeller. Festkrperprobleme/Advances in Solid State Physics XIII, 31 (1973). B. Renker, H. Rietschel, L. Pintschovius, W. Glser, P. Bresch, D. Kuse, and M.J. Rice. Observation of giant Kohn anomaly in the onedimensional K2Pt(CN)4Br0.3.3H2O. Physical Review Letters 30, 1144 (1973). M. Kaveh and E. Ehrenfreund. Square Fermi surface model for conduction in linear chain mercury compounds: Hg3-dAsF6. Solid State Communications 31, 709 (1979). I.U. Heilmann, J.D. Axe, J.M. Hastings, G. Shirane, A.J. Heeger, and A.G. MacDiarmid. Neutron investigation of the dynamical properties of the mercury-chain compound Hg3-dAsF6. Physical Review B 20, 751 (1979). D. Jerome. Organic superconductivity. Chemica Scripta 17, 13 (1981). M. Mehring and J. Spengler. Locally resolved 13 C Knight shifts in the organic conductor (fluoranthenyl)2SbF6. Physical Review Letters 53, 2441 (1984). C. Hamann. J. Heim, and H. Burghardt. Organische Leiter, Halbleiter und Photoleiter. Friedrich Vieweg & Sohn, Braunschweig, 1981. R. Friend and D. Jerome. Periodic lattice distortions and charge density waves in one- and two-dimensional metals Journal of Physics C: Solid State Physics 12, 1441 (1979). L.B. Coleman, M.J. Cohen, D.J. Sandman, F.G. Yamagishi, A.F. Garito, and A.J. Heeger. Superconducting fluctuations and the Peierls instability in an organic solid. Solid State Communications 12, 1125 (1973). D.E. Schafer, F. Wudl, G.A. Thomas, J.P. Ferraris, and D.O. Cowan. Apparent giant con-
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ductivity peaks in an anisotropic medium: TTF-TCNQ. Solid State Communications 14, 347 (1974). 20 R.L. Greene, G.B. Street, L.J. Suter. Superconductivity in polysulfur nitride (SN)X. Physical Review Letters 34, 577 (1975). 21 W. Mller. PhD thesis. Karlsruhe, 1976. 22 C. Hamann, H.J. Hhne, F. Kersten, M. Mller, F. Przyborowski, and M. Starke. Switching effects on polycrystalline films of lead phthalocyanine (PBPC). physica status solidi (a) 50, K189 (1978). 23 M. Hanack, K. Mitulla, and O. Schneider. Synthesis of anew kind of low dimensional conductors. Chemica Scripta 17, 139 (1981); M. Hanack, M. Lang. Conducting stacked metallophthalocyanines and related compounds. Advanced Materials 6, 819 (1994). 24 A. Simon. Kondensierte Metall-Cluster. Angewandte Chemie 93, 23 (1981); Condensed metal-clusters. Angewandte Chemie International Edition English 20, 1 (1981). 25 E. Moses (general ed.), Physics and Chemistry of Materials with Low-Dimensional Structures, Series A: Layered Structures Vols. 1-6. Reidel, Dordrecht, 1976–1979. 26 D.W. Bullett. Theoretical Aspects of Band Structures and Electronic Properties of Pseudo-one-dimensional Solids. H. Kamimura (ed.), Reidel, Dordrecht, 1985. 27 P. Gressier, A. Meerschaut, L. Guemas, J. Rouxel, and P. Monceau. New one-dimensional Vb transition metal tetrachalcogenides structural and transport properties. Journal de Physique C3, 1741 (1983). 28 C. Schlenker, C. Filippini, J. Marcus, J. Dumas, J.P. Pouget, and S. Kagoshima. Peierls transition in the quasi-one dimensional blue bronzes K0.3MoO3 and RbO0.30MoO3. Journal de Physique C3, 1757 (1983). 29 C.K. Chiang, C.R. Fincher, Jr., Y.W. Park, A.J. Heeger, H. Shirakawa, E.J. Louis, S.C. Gau, and A.G. MacDiarmid. Electrical conductivity in doped polyacetylene. Physical Review Letters 39, 1098 (1977). 30 S. Roth and W. Graupner. Conductive polymers: evaluation of industrial applications. Synthetic Metals 57, 3623 (1993). 31 J.S. Miller. Advanced Materials 2, pp. 98, 378, 495, 901 (1990); Advanced Materials 3, pp. 110, 161, 262, 309, 564 (1991); Advanced Materials 4, pp. 45, 298, 435,498 (1992); Advanced Materials 5, pp. 448, 587, 671 (1993).
32 H.C. Longuet-Higgins and L. Salem. Alterna-
tion of bond lengths in long conjugated chain molecules. Proceedings of the Royal Society of London A 251, 172 (1959). 33 H. Kuhn. Free-electron model for absorption spectra of organic dyes. Journal of Chemical Physics 16, 840 (1948); H. Kuhn. A quantum-mechanical theory of light absorption of organic dyes and similar compounds. Journal of Chemical Physics 17, 1198 (1949). 34 G. Knig and G. Stollhoff. Why polyacetylene dimerizes: results of ab initio computations. Physical Review Letters 65, 1239 (1990). 35 W.P. Su, J.R. Schrieffer, and A.J. Heeger. Solitons in polyacetylene. Physical Review Letters 42, 1698 (1979). 36 M.J. Rice. Charged II-phase kinks in lightly doped polyacetylene. Physics Letters A 71, 152 (1979). 37 S.A. Brazovskii. Self-localized excitations in the Peierls–Frhlich state. Zh. Eksp. Teor. Fiz B 7, 677 (1980); Soviet Physics JETP 51, 342 (1980). 38 Y. Lu (ed.). Solitons and Polarons in Conducting Polymers. World Scientific Publisher, Singapore, 1988. 39 S. Roth and H. Bleier. Solitons in polyacetylene. Advances in Physics 36, 385 (1987). 40 A.J. Heeger, S. Kivelson, J.R. Schrieffer and W.P. Su. Solitons in conducting polymers. Reviews of Modern Physics 60, 781 (1988). 41 G. Wegne. I. Mitt. Polymerisation von Derivaten des 2.4-Hexadiin-1,6-diols im kristallinen Zustand. Zeitschrift fr Naturforschung B 24, 824 (1969). 42 H.J. Cantow. Advances in Polymer Science 63. Springer Verlag, Berlin, Heidelberg, 1984. 43 J.T. Gammel, A. Saxena, I. Batistic, A.R. Bishop, and S.R. Phillpot. Two-band model for halogen-bridged mixed-valence transitionmetal complexes. I. Ground state and excitation spectrum. Physical Review B 45, 6408 (1992). 44 S.M. Wever-Milbrodt, J.T. Gammel, A.R. Bishop, and E.Y. Loh, Jr.. Two-band model for halogen-bridged mixed-valence transitionmetal complexes. II. Electron–electron correlations and quantum phonons. Physical Review B 45, 6435 (1992). 45 T. Kuhlman, S. Roth, J. Rozire, and W. Siebert. Polymeres (g5l-2,3-Dihydro-1,3diborolyl)-
References and Notes nickel die erste Polydecker Sandwichverbindung. Angewandte Chemie 98, 87 (1986). 46 T. Sugawara, S. Bandow, K. Kimura, H. Iwamura, and K. Itoh. Magnetic behavior of nonet tetracarbene as a model for one-dimensional organic ferromagnets. Journal of the American Chemical Society 108, 368 (1986). 47 R.H. Friend. Polymers from the Soviet Union. Nature 326, 335 (1987). 48 J.S. Miller, J.C. Calabrese, R.S. McLean, and A.J. Epstein. Meso-(tetraphenylporphinato)manganese(III)-tetracyanoethenide, [MnIIITPP]::% [TCNE]·%, a new structure-type linear-chain magnet with a Tc of 18 K. Advanced Materials 4, 498 (1992). 49 E. Tresos. Contribution a l’Etude des Effets de Taille Quantique de Particules d’Alcalins dans
des Cavites de Zeolithes: Etudes RPE et RMN. PhD thesis. Montpellier, 1993. 50 V. Ramamurthy. Photochemistry in Organized and Constrained Media. VCH-Verlag, Weinheim, 1991. 51 H. Kuzmany, J. Fing, M. Mehring, and S. Roth. Progress in Fullerene Research. World Scientific Publisher, Singapore, 1994. 52 C.W.J. Beenakker and H. van Houten. Quantum transport in semiconductor nanostructures. Solid State Physics 44, 1 (1991). H. van Houten, C.W.J. Beenakker, and B.J. van Wees. Quantum point contacts. Semiconductors and Semimetals 35, 9 (1991). M. Pepper and D. Wharam. Rewriting the rules of resistance. Physics World 45, 8 (1988).
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One-dimensional Solid-State Physics Today most physicists work in the field of solid-state physics. One reason is that, apart from chemistry, solid-state physics provides the fundamental basis of materials science and thus for most new branches of technology. This book is an introduction to a particular area within solid state physics, namely, one-dimensional solid state physics. The main difference between solid-state physicists and other scientists is their familiarity with reciprocal space (Figure 3-1). This difference is similar to that between scientists and nonscientists, where familiarity with exponentials is the boundary. Every educated person knows the definition of 107 or 10–24, but only scientists use powers and logarithms as readily as journalists use a notepad. The explanation of symbols alone does not help, what counts is familiarity. In the first two chapters, reciprocal space was mentioned a few times. This chapter is an attempt to make you more familiar with the conventions of reciprocal space, as they apply to the concepts of dimensionality. For an in-depth treatment of the topic, please see textbooks on solid state physics (for example [1–3]).
Figure 3-1
Solid-state physicists differ from other people by their familiarity with reciprocal space.
One-Dimensional Metals, Second Edition. Siegmar Roth, David Carroll Copyright © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30749-4
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3.1
Crystal Lattice and Translation Symmetry
The three states of matter – solid, liquid, and gas – can be grouped in several different ways according to their physical properties. Solids and liquids are, when grouped together, called condensed matter’ because they are indeed dense and difficult to compress: the atoms are already close together, with very little room for irregularities in interatomic distances. A gas, on the other hand, is highly compressible. Liquids and gases can be classified as fluids – they take the shape of their container. There are no, or only very small, shear forces in fluids. Solids do not flow, because strong forces keep the distance between the atoms fixed and also lock the relative angular positions of the atoms. We classify two types of solids: crystalline solids and amorphous solids. In crystalline solids the molecules or atoms are arranged in a regular way; in amorphous solids there is a large amount of disorder within some inherent limitations. The existence of shear forces in the solid implies order. The angular positions of the atoms cannot be completely random because of steric reasons and chemical bonding: for example, a silicon atom is found in the center of a tetrahedron with an angle of 109.5 between the bonds to its nearest neighbors. Therefore, there is short-range order even in amorphous solids. In crystalline solids the order is long-range, and it can be assumed that the ground state of all matter is crystalline and that all amorphous solids will crystallize sooner or later, although this process might require geological times of millions of years in some instances. Of course, the degree to which something is amorphous or crystalline can be important for some purposes. How defective must a crystalline solid become before it is amorphous? What is glassy’ behavior? To better understand the degree of organization, physicists frequently resort to statistical measures as the correlation function. This is simply the probability of finding an atom–atom separation within the solid. Some base all discussion of condensed matter on this function, as the most basic sense of order. Here, it is important to understand only that a disordered solid may also reflect symmetries and that there are well-defined ways to describe it. 3.1.1
Classifying the Lattice
What does long-range’ order mean? In single crystals the order covers the entire crystal, which is from a few micrometers up to many centimeters depending on its size. Most solids, however, are polycrystalline. They are composed of microcrystalline domains or grains, which stick together to form the bulk solid. The diameter of a grain can be from several fractions of a micrometer up to several millimeters. The long-range order in a crystal leads to the crystal lattice. In a crystal lattice the regular arrangement of the atoms (or molecules) is periodic; the same pattern is repeated over and over again (e.g., sodium lattice, Figure 3-2; sodium chloride lattice, Figure 3-3 – three repeating units in each direction are shown). To classify the many different lattice types their symmetry properties are used. (Many textbooks on
3.1 Crystal Lattice and Translation Symmetry
Figure 3-2
Crystal lattice of sodium
metal.
crystallography exist; however, Ashcroft and Mermin [1] has long been a standard for understanding crystal symmetries at the elementary level.) The lattices in Figures 3-2 and 3-3 are cubic. A cubic elementary cell, which is a commonly found repeating unit, can be constructed as indicated in bold in the upper-right corner. For the sake of clarity, only the atoms at the surface of the crystal are shown and the atoms inside the crystal are omitted. (Examples of textbooks on crystal structure and chemistry are [4] and [5].) Our first example is a simple cubic system. That is, each site of the mental construct of the lattice (the mathematical arrangement of points in a regular square array) is occupied by an atom to form a real lattice of the solid. The subtlety (and formality) of forming the construct mentally, then filling it in with atoms is an important one. After all, we need not imagine only one atom associated with a given
Figure 3-3
chloride.
Crystal lattice of sodium
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site. Consider associating more than one with each site of the square lattice. The picture could get rather complex. The associated group of atoms is referred to as a basis’. For sodium chloride, a two-atom basis, one of Na and one of Cl, is associated with each site of the square lattice. The crystal of sodium chloride is built of sodium and chloride atoms according the chemical formula NaCl. This crystal does have a unique set of symmetries that arranges neighbors equivalently. There are as many sodium atoms as chlorine atoms; however, a particular sodium atom does not have only one single chlorine atom as a partner, but six equivalent chlorine neighbors. These again are neighbors to other sodium atoms. In other words, there is no uniquely defined NaCl unit in the crystal. This coincidence does not always happen when there is a basis set. (More precisely one should speak of sodium and of chlorine ions, rather than atoms, but this is not important in this context.) The distinction between solids with complex basis sets and solids that are elemental (or nearly so) is frequently used to further classify solids. Atomic crystals and molecular crystals can be distinguished along exactly these lines. Most inorganic materials form atomic crystals; examples include silicon, copper, and sodium chloride. In contrast, most organic compounds form molecular crystals. In a molecular crystal most properties of the solid are determined by the properties of the molecule. Crystallization introduces only small changes. For example, benzene crystal contains well-defined C6H6 units. The interactions between the carbon atoms within such a unit (molecule) are much stronger than those between carbon atoms belonging to two different molecules. Benzene molecules exist as benzene vapor, as liquid benzene, as mixtures of benzene with other solvents, and also as incorporated into a molecular crystal. (For a monograph on organic crystals, see [6].) Fundamentally, molecular crystals are described with the same symmetry language as atomic lattices. Thus, we can speak of all crystal structures in the same way. 3.1.2
Using a Coordinate System
The elementary cell is perhaps best explained with the example of the two-dimensional rectangular lattice in Figure 3-4. The elementary cell is presented as hatched rectangle. The lattice is built up by moving this cell parallel to the crystallographic a axis and the crystallographic b axis. The length a and the width b of the elementary cell are called lattice parameters, and the crosses in the figure mark the lattice points. In primitive’ crystal structures the lattice points are the sites at which the atoms or molecules are located. (Nonprimitive structures have additional atoms.) All lattice points are identical and the lattice is assumed to extend infinitely in all directions of space (i.e., the crystal is composed of an (infinitely) large number of atoms or molecules). Any lattice point can be chosen as the origin of a coordinate system. The origin in the figure is marked by a circled cross. The arrows along the crystallographic a and b axis, with lengths a and b respectively, are called the unit lattice vectors a and b. They are said to span the elementary cell. Any lattice point can be reached by a linear combination of the unit vectors:
3.1 Crystal Lattice and Translation Symmetry
Figure 3-4 Two-dimensional rectangular lattice.
¼ ha þ kb T
(1)
is a lattice vector. The integers h and k are the indices of the lattice point to which T points. These indices are used to characterize directions and planes in the vector T crystals. Because all lattice points are equivalent, the physics at each lattice site is the same, or, in mathematical terms, the physical laws are invariant to the addition of a lattice vector to the special coordinate. This is called the translational invariance of the crystal lattice and of the laws governing the physics within a crystal. We have, in fact, already seen a two-dimensional lattice in Chapter 2. The graphene layer is a hexagonal two-dimensional lattice. The translation vectors we gave were termed (an, am). In fact, the simplest way to describe this lattice is as a triangular lattice with a two-atom basis. How would one know this? That is, how do we know when we have the simplest description for a lattice? The easiest way is to pretend to stand on a given lattice point facing, say, north. As you glance to your left and to your right, the view from any lattice point should be the same. This is not true for the hexagonal system in Chapter 2. Try it – adjacent sites will have the bonds oriented differently relative to a given direction. Once the most simple lattice structure has been determined, it is important to realize how many possibilities exist for distinctively different lattices. When in a rectangular lattice a = b, the rectangular lattice is a square lattice. The three-dimensional analog of the square lattice is, of course, the cubic lattice. In lattices of lower symmetry the dimensions of the elementary cell might differ, a „ b „ c; also the angles between the axes do not need to be 90. In fact, there are 18 possible variations of this most simple lattice. When the symmetries of rotations, translations, etc. are considered, a finite enumeration of all possible lattice symmetries can be made (again, see Ashcroft and Mermin [1]).
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3.1.3
The One-dimensional Lattice
The simplest lattice is a one-dimensional crystal lattice, which is merely a row, or line, of equidistant points (Figure 3-5). In a strictly one-dimensional lattice there is no need to consider the angular positions of the atoms, but many real’ one-dimensional substances have the atoms arranged in a zigzag line, as indicated in Figure 3-6. Such a zigzag chain can melt’ by generating defects as illustrated in Figure 3-7, where the strict alternation is disturbed. Another way of looking at these defects is to allow for rotations around a bond between two atoms.
Figure 3-5
One-dimensional lattice.
Figure 3-6
Zigzag chain.
Figure 3-7
Zigzag chain with defects.
As we saw in Chapters 1 and 2, the conducting polymer system is generally constructed of alternating double and single carbon–carbon bonds. To see an example of how the above applies to a real polymer, consider polyacetylene. The structure is given in Figure 1-12 but is reproduced here in its schematic form (Figure 3-8):
3.1 Crystal Lattice and Translation Symmetry
Figure 3-8 The schematic structure of polyacetylene along with its lattice. Notice that the lattice points do not fall on every carbon; rather, there is a two-carbon (along with some hydrogens) basis.
For the one-dimensional system, however, this is not the end of the story. As mentioned before, rotations about a bond are allowed in many systems. In fact, these can occur in an ordered way, leading to another crystal structure for the same material. Again, for an example let’s turn to polyacetylene. In fact, the structure could occur in several ways (Figure 3-9). As noted in Chapter 2, the one-dimensional nature of such systems as polyacetylene means that the properties of the material, especially the electronic transport characteristics, are extremely sensitive to defects (or mistakes) within the lattice. Consider only an occasional bond rotation within the lattice, as seen in Figure 3-7. It makes sense then that our expectation of the electronic properties of such a sys-
Figure 3-9 The difference between cis and trans polyacetylene is the rotation of a few bonds. This gives different crystal structures in one dimension.
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tem of randomly distributed defects is that they are dominated by the electronics of the individual segments as though they are isolated. In this way the prediction of phenomena in one dimension can easily be as complex as that for a three-dimensional system. We will return to this point in later chapters.
3.2
Reciprocal Lattice, Reciprocal Space
The crystal lattice has two functions. On the one hand it is an idealized model of the crystal. Any novel chemical compound is first characterized by determining its crystallographic structure: the symmetry properties of the lattice and the lattice parameters. The second function of the crystal lattice is to furnish a convenient coordinate system for the description of physical phenomena within the crystal. In simple terms – we prefer to give the location of a coffee shop as 5 miles form exit 12 on highway A81, rather than to specify its geographic longitude and latitude. Once the crystal lattice is accepted as a convenient geometrical tool specifically designed to simplify the description of physical phenomena within a given crystal, it is easy to accept other geometrical (mathematical) tools, which are derived from the first one: the reciprocal lattice, or more generally, the reciprocal space. 3.2.1
Describing Objects by Momentum and Energy
The concept of the reciprocal lattice was developed to describe X-ray diffraction patterns of crystals. Reciprocal space is used to describe wave-like phenomena in a crystal. It was already mentioned Section 1.2 in explaining open Fermi surfaces and in Section 2.2 in the context of the Kohn anomaly. Reciprocal space is not used to describe the positions of objects, but to characterize the nature of mobile objects independent of their actual position. For example, consider cars traveling along a highway. To characterize the cars we would determine their velocity v and their weight (mass) m. To determine the velocity a measure of length would be necessary, with the units being miles, kilometers, or unity vectors of the crystal lattice. A physicist would prefer to use momentum and kinetic energy rather than velocity and mass, because with this choice some of the physical equations become more decent – the transition from the particle picture’ to the wave picture’ is especially eased. The wave quantity related to the momentum of a particle is the wave number, which is the reciprocal of the wavelength, and the wave quantity related to the energy is the frequency, something expressed in oscillations per unit time, i.e., a quantity with the dimension of reciprocal time. So some solid-state phenomena are simplified if described in reciprocal space, but the price of this simplification is that we need to get accustomed to reciprocal space. For most solid-state scientists this intellectual investment is worthwhile.
3.2 Reciprocal Lattice, Reciprocal Space
3.2.2
Constructing the Reciprocal Lattice
There is a simple algebraic procedure to derive the reciprocal lattice from the crystal lattice. The crystal lattice is defined by the dimensions of the elementary cell, or, in other terms, by the three unit vectors a, b, and c , which span the elementary cell. The reciprocal lattice is also defined by three unit vectors, now a, b , and c . The direction of a is perpendicular to the plane defined by b and c . In a cubic lattice, its length is just 2p times the reciprocal of the lattice parameter a, hence the name reciprocal lattice’. For a general lattice type the length of a is obtained by dividing the area between b and c by the volume of the elementary cell, which in vector algebra is expressed as a ¼ 2p ðb c Þ = ða b c Þ
(2)
or 2p bc/abc for a rectangular lattice. For example, if in any given lattice a is 3 , in the reciprocal lattice the length of a* is 2p times one third of a reciprocal Angstrom. The factor 2p was chosen to simplify the equations. Simplification in one particular field of physics often means a complication in another field. The factor 2p is often incorporated into other quantities. Examples are the angular frequency x = 2pm for a wave, used instead of the linear frequency m, or Planck’s constant existing in two versions, h and h = h/2p, etc. In practice, the reciprocal lattice and reciprocal space are mainly used for graphic representations. Crystallographers, for example, are interested in predicting the positions of X-ray reflections. They will draw the reciprocal lattice or rather a particular plane of it, choose a certain scale for it (e.g., cm) to represent one reciprocal Angstrom, and then proceed according to the description above (a* perpendicular to the plane of b and c , and so on). They will mark the wave vectors of the incoming and presumed outgoing X-ray beams (the direction to the detector). The difference between these two vectors is the momentum transfer’ associated with the scattering event. A sphere with radius equal to the momentum transfer is the Ewald sphere. If the Ewald sphere touches one of the lattice points on the reciprocal lattice, it will show as a peak in the X-ray intensity (a reflection) at the chosen detector position. If not, the sample (and with it the reciprocal lattice) will have to be rotated until a reciprocal lattice point meets the sphere and the reflection condition is fulfilled. 3.2.3
Application to One Dimension
What does the reciprocal lattice of a one-dimensional crystal look like? Look again at the crystal lattice in Figure 3-5. We anticipate that the reciprocal lattice will also be a linear arrangement of lattice points, the a axis pointing in the same direction as the a axis, and the length of the unit vector a being 2p times the reciprocal of the length a.
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Tetragonal lattice, the limit b fi ¥ and c fi ¥ is the infinite anisotropy’ approach to one-dimensionality and demonstrates that the reciprocal lattice of a linear chain is also a linear chain.
Figure 3-10
However, the previously described procedure to design a reciprocal lattice works only if we use all three dimensions: we absolutely need all unit vectors a, b, and c , to find a unit vector of the reciprocal lattice. With some mediation over Figure 3-10 we try the anisotropy approach of Section 1.2 to find the reciprocal lattice in one dimension. We start with an anisotropic tetragonal lattice, having three axes at right angles, but two lattice parameters much larger than the third (b >> a and c >> a), and increase b and c to infinity. The direction of a is perpendicular to the plane defined by b and c . Because the crystal lattice is already tetragonal, a is perpendicular to that plane and hence a is pointing in the same direction as a. What then is the length of a ? 2p times the area (b·c) divided by the volume (a·b·c) gives 2p/a, a finite value for any value of b and c , valid also for the infinite b and c . What about the lengths of b and c ? b* = 2p(a·c)/(a·b·c) = 2p/b and similarly c = 2p/c. Both approach zero as b and c take infinite values. So the infinitely anisotropic’ onedimensional tetragonal lattice indeed consists of a linear arrangement of points as the reciprocal lattice. In Section 2.1 we mentioned that experiments are usually not carried out on isolated chains, but on bundles’. Bundles are arrays of parallel chains. Of course, a three-dimensional crystal is a bundle, but the term bundle is more general. It also covers arrays of chains with random origin and random distances between the chains. In other words, there is no order in the b and c directions and elementary cells with many different b and c vectors are conceivable. In this case we still get a* = 2p(b·c)(a·b·c) = 2p/a, but b* and c* are undefined. They are not zero as with the infinitely anisotropic lattice when b fi ¥ and c fi ¥. In a disordered bundle b* and c* can take any value. Consequently, the reciprocal lattice of a disordered bundle is not a linear arrangement of equidistant points, but an arrangement of equidistant planes (Figure 3-11). A construction like this has a very practical background. It
3.3 Electrons and Phonons in a Crystal, Dispersion Relations
Reciprocal lattice of a disordered bundle (order in one dimension only) consisting of equidistant planes.
Figure 3-11
actually describes the intensity of X-rays (or neutrons) diffracted by real samples. When the X-rays are detected by the blackening of a photographic film, the planes create lines on the film. Figure 3-12 shows the diffraction pattern of KCP (see Section 2.2). The Peierls distortion (see Chapter 4) leads to reorganization of the platinum atoms within the chains. At room temperature the chains act independently of each other and, as far as the Peierls distortion is concerned, they form a disordered bundle. The X-ray evidence for a disordered bundle is continuous streaks, as shown on the left side of the diffraction diagram. At low temperatures the chains interact, a coherent threedimensional order is established, and the streaks disintegrate into spots [7].
X-ray diffraction pattern of KCP [7]. The streak indicates disordered bundle behavior of the Peierls distortion in the platinum chains.
Figure 3-12
3.3
Electrons and Phonons in a Crystal, Dispersion Relations
We can already see how the constructs of the reciprocal lattice might relate to the wave-like phenomena in a crystal. After all, an electron will have specific wavelengths within the crystal. If the crystal has a characteristic lattice spacing that is some rational fraction of one of those wavelengths, it is natural to expect unusual’ behavior at that electron energy.
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3.3.1
Crystal Vibrations and Phonons
The regular arrangement of atoms or molecules in a crystal is caused by forces that keep the atoms in their positions. The forces are, of course, the chemical bonding forces. In a mechanical model they are represented as elastic springs (Figure 3-13). When an atom is deflected from its position of equilibrium and then left alone, the spring pulls the atom back. Due to inertia it overshoots its initial position; the springs now push it in the opposite direction. The process repeats, and the atom oscillates. The strain in the spring causes adjacent atoms to oscillate as well: a wave will develop. These waves are nothing other than the well-known sound waves. The Greek word for sound is phonos and the particles corresponding to the sound waves (according to the quantum mechanical wave–particle dualism) are called phonons. The human ear perceives sound up to some 10–100 kHz. The phonon frequencies, however, lie in the GHz range; thus most of the phonons in solids are in the ultrasonic region. To characterize a wave, two basic quantities are necessary, the frequency m and the wavelength k. (An additional quantity is the amplitude. In the particle picture this corresponds to the number of phonons.) Frequency and wavelength are related by the velocity t. m= t /k
(3)
The wave velocity depends on the properties of the medium in which the wave propagates, for example, on the force constant of the spring in Figure 3-13. For small values of frequency and amplitude, the force constant is independent of these quantities – otherwise we would not call it a constant. However, this is not true for very high frequencies. Here, waves with different frequencies move with different velocities and a wave package disperses. Similarly, in optics a prism disperses a polychromatic light beam because the refractive index of the prism material (and hence the light velocity) depends on the light frequency. Because of possible frequency–velocity dependence, Eq. (3) is called dispersion relation. For a complete description of a wave, the wavelength is not sufficient. The direction of the wave propagation is also an important factor. Both pieces of information, the direction of propagation and the reciprocal wavelength, are combined in the wave vector k, which points in the direction of propagation and has the length of the reciprocal wavelength multiplied by 2p: |k| = 2p / k
(4)
(It counts the number of waves within a unit length, multiplied by 2p.) Using the wave vector, Eq. (3) becomes m = tk / 2p
(5)
3.3 Electrons and Phonons in a Crystal, Dispersion Relations
From quantum mechanics and the wave–particle duality we know that a wave with frequency m and wave vector k corresponds to a particle with energy E = hm and momentum p = hk/2p, so the dispersion relation can be written in yet another form: E = tp
(6)
For a free particle is p = mt, where m is the mass, so that finally mt2/2 = p2/2m
(7)
In a crystal, phonons (and electrons) are not free particles; they are bound to the crystal and strongly influenced by the crystal lattice. Therefore hk/2p does not really represent the momentum, it is just a quasi-momentum’ or a crystal momentum’. The peculiarity of the crystal momentum is that its value cannot exceed the value of a reciprocal lattice vector. If the momentum of a quasiparticle is increased, the momentum value hs will be transferred to the crystal as a whole (s is a reciprocal lattice vector, see above). The recoil energy associated with the momentum and transferred to the crystal as a whole does not show up in the energy balance because of E = p2/2M. M is the total mass of the crystal, which is almost infinite compared to the mass of an electron (there are typically 1021 atoms in a crystal of 1 cm3 and an atom is about 104 times as heavy as an electron). Similarly, waves in the discrete lattice of a crystal also differ from waves in a continuum. Because of the discreteness of the lattice there is a minimum wavelength. Waves with shorter wavelengths than the interatomic distance are not possible, because there is nothing between two atoms that could oscillate. Minimum wavelength means maximum wave vector and, consequently, maximum momentum. A fancy way of expressing this is to say that the momentum is only defined modulo hs’, where s is a reciprocal lattice vector. The dispersion relation for phonons in the linear model of Figure 3-13 can easily be calculated (refer to textbooks on solid-state physics). The dispersion relation is shown graphically in Figure 3-14, with the energy as ordinate and the wave vector k as abscissa. The dispersion relation is plotted up to the aforementioned maximum value for the wave vector k. The graph does not have to end at that point, but there is no additional information for extending it any further, because the energy does not change if integer multiples of a reciprocal lattice vector are added to the wave vector.
Figure 3-13
Model of a one-dimensional crystal. The lattice forces are represented as elastic springs.
Figure 3-14 shows a typical representation in reciprocal space. The wave vector is used which has a reciprocal length (the wave number) and to which reciprocal lattice vectors can be added. Reciprocal space can either be regarded simply as a convenient
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Figure 3-14
Dispersion relation for the model in Figure 3-13.
tool for discussing dispersion relations or as a generalization of the reciprocal lattice. The reciprocal lattice consists of discrete lattice points, connected by reciprocal lattice vectors with the origin of the reciprocal lattice. Reciprocal space also includes the space between the points of the reciprocal lattice. For most purposes, only the space between the origin and the reciprocal lattice points nearest to the origin is important. This part of reciprocal space is the so-called first Brillouin zone. In Figure 3-14 the dispersion relation is plotted for only half of the first Brillouin zone, from the zone center 0 to the edge of the right-hand zone at k = p/a. For symmetry reasons, the other half is just the mirror image. To introduce the concept of Brillouin zones we look at Figure 3-15. It shows the reciprocal lattice of a linear chain. The origin of the reciprocal lattice lies in the center of the figure. The interval between the origin and the first lattice points on both sides is divided into halves; the region 1. BZ’ marks the first Brillouin zone. It is surrounded by the second Brillouin zone and so forth. The concept can easily be generalized to two- and three-dimensional space. For phonons, only the first Brillouin zone is relevant. In discussing the wave vector and the quasi momentum, we noted that this quantity was only defined as modulo hs. A wave vector in the second Brillouin zone would correspond to a wavelength shorter than a lattice period.
Figure 3-15
Brillouin zones.
3.3 Electrons and Phonons in a Crystal, Dispersion Relations
3.3.2
Phonons are Electrons are Different
Usually phonons are associated with waves (sound waves’); however, from quantum mechanics we know that waves also have particle aspects. The reverse is true for electrons: they are looked upon as particles although they also have wave character. The wave character of electrons is particularly important for electrons in a crystal lattice. There are many electrons in a crystal. Most of them circulate around atomic nuclei in inner shells. These electrons are not important in the present context. Here we are interested only in electrons that do not belong to a particular atom but which are delocalized all over the crystal: the conduction electrons of metals as well as, in some respect, the valence electrons of doped or of photoexcited (occasionally also thermally excited) semiconductors. These electrons are described as Bloch waves (the theory of Bloch waves is not discussed in detail here). The Bloch theorem states that, in a periodic lattice, electrons behave (can move) pretty much like phonons. They have a k vector and an energy value, and a dispersion relation connects the two. The shape of the dispersion relation, however, is different. For phonons the dispersion relation initially is linear at the origin of the reciprocal space, resembling a classical wave with constant velocity: v = tk/2p. The electron dispersion relation initially takes a quadratic course E = h2k2/2m, as expected for a classical particle with kinetic energy of E = p2/2m. In this equation m is usually not identical with the free electron mass. It is called the effective mass and is designated by m*. In general, the effective mass is different for k pointing in different directions. To keep the formula E = h2k2/2m*, the effective mass is even allowed to change with k. These tricks with m* allow us to still treat the electrons in a crystal almost as if they were free particles. Another difference between electrons and phonons is that they obey different statistics. Phonons are bosons, and for bosons the Pauli exclusion principle does not exist. It is possible to have several phonons in the same quantum state, i.e., more than one phonon with the same k-vector in one crystal. Electrons, however, are fermions, which follow Fermi statistics and have to observe the Pauli exclusion principle. They cannot populate the same quantum state in duplicate. For each k-value there are only two electrons possible in a crystal, one with spin-up and one with spin-down. At absolute zero, all phonons are in the lowest energy state at k = 0, and the atoms just vibrate in their zero-point motion but the electrons fill the k-states of the Bloch waves up to very high energies. The energy of the highest occupied state is the Fermi energy EF, or the Fermi level. The corresponding wave vector is the Fermi wave vector kF and its reciprocal is the Fermi wavelength kF (divided by 2p). In most metals the Fermi energy is very high, several eV (electron volts), much higher than the thermal energy kT (k: Boltzmann constant; T: temperature), which is only about 30 meV at room temperature. Only the electrons close to the Fermi energy can be thermally excited into an unoccupied k-state. The same is true for acceleration of electrons by an applied electric field, which also involves a change of the k-state, and for many other responses of the electron system to outside influ-
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ences. Consequently almost all properties of an electron system are determined by the electrons at the Fermi energy, and that is why Fermi energy, Fermi wave vector, and Fermi wavelength are so important. The dispersion relation for the electrons is different in different directions, but in each direction there is a well-defined Fermi vector. The tips of these vectors define a surface in reciprocal space. This surface is called the Fermi surface, which reflects h2kF2/ the symmetry of a crystal. For an isotropic solid it is spherical. Because EF = 2m*, the symmetry of the Fermi surface is related to the symmetry of the effective mass. To construct the Fermi surface of a one-dimensional metal we again use the anisotropy approximation introduced in Section 1.2. A small effective mass means that it is easy to move the electrons. If, in a bundle of chains, the electrons move easily along the chains but much less easily perpendicular to the chains, then the effective mass tensor and the Fermi surface look like a lens (a flat ellipsoid) with a short axis in the chain direction and two long axes perpendicular to the chains; the Fermi surface distorts into two planes, as indicated in Figure 3-16. In Section 1.2 (Figure 1-6) open’ Fermi surfaces were mentioned. The Fermi surface in Figure 3-16 definitely is open, because it consists of two parallel planes. Even if the Fermi surface is slightly curved it can still remain open. This effect can be understood with the help of the Brillouin zones. If the Fermi surface is a very flat lens, its diameter might be larger than the dimension of the first Brillouin zone. The Fermi surface would close only in the second or even a higher Brillouin zone. However, then we would be allowed to subtract the reciprocal lattice vectors from kF, thus yielding an open Fermi surface, which is a criterion for one-dimensionality.
Figure 3-16
1D Fermi surface.
3.3.3
Nearly Free Electron Model, Energy Bands, Energy Gap, Density of States
It has been mentioned that in contrast to the phonon dispersion relation, which is linear for small wave vectors, the dispersion relation of electrons is quadratic, E = h2k2/2m. The quadratic behavior is characteristic of classical particles, where E = mv2/2. The free electron model treats the electrons in a crystal as classical particles. This model is surprisingly successful in explaining the electronic transport properties of a solid, such as electrical conductivity and thermal conductivity. We emphasized the word surprisingly, because it is not easy to understand that electrons can
3.3 Electrons and Phonons in a Crystal, Dispersion Relations
freely move between oppositely charged metal ions. However, the Bloch theory justifies the free electron model. As often happens in science, the theoretical justification came years after the successful application. The classical free electron model fails to explain specific heat and magnetic susceptibility. The Pauli exclusion principle has to be taken into account, i.e., Fermi statistics have to be applied to the electrons, leading to the free electron Fermi gas’. To include the specific properties of a given solid and to explain, e.g., why silicon is a semiconductor or sodium is a metal, we have to go one step further and allow for some influence of the crystal lattice on the electron gas. This led to the nearly free electron gas’ model, which differs from the free electron Fermi gas in two ways: .
.
The free electron mass m is replaced by the effective mass m*. The effective mass can be different for electron motions in different crystallographic directions (see above, discussion on the Fermi surface for a one-dimensional solid). Consequently, m* is a tensor, not a scalar (which, when concentrating on one dimension, does not matter, of course). Moreover, m* depends on the magnitude of the wave vector k, and although the dispersion relation is given as E = h2k2/2m*, the functional dependence of E on k deviates from the parabolic shape. The dispersion relation can even have an inflection point at which m* changes sign and becomes negative. The dispersion relation of a nearly free electron gas is not continuous at the edge of a Brillouin zone but has jumps. Because of these discontinuities the allowed energy values are confined to energy bands separated by forbidden gaps.
In Figure 3-17 the dispersion relation for free electrons and the nearly free electron Fermi gas are compared. In the first case we have a simple parabola, in the latter the parabola is distorted and shows jumps where k passes from the first into the second Brillouin zone. At the zone edge (k = p/a) the dispersion function has two different energy values for the same k value. The energy values are separated by Eg, the width of the forbidden energy gap. The energy gap can be explained as an interference phenomenon, which is very simple in a one-dimensional lattice. The electrons (or Bloch waves) are scattered by the electrostatic potential of the positively charged metal ions at the lattice points. The lattice points are separated by the lattice constant a. Waves back-scattered from adjacent ions have a path difference of 2a, and hence waves with this wavelength
Figure 3-17
Electron dispersion relation for free Fermi gas and nearly free Fermi gas.
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interfere constructively. Wavelength a implies a wave vector k = p/a. At this point the discontinuities in the dispersion relations occur. Waves with wave vector k = p/a are in geometrical resonance with the crystal lattice. They are standing waves, not propagating waves. Two types of standing waves with k = p/a can be formed: one type has nodes at the lattice points, the other has maxima at the lattice points. In one case the electrons are between’ the positive ions; in the other they are at’ the ions. Evidently the energy for the two cases is different so that there are two energy values for k = 2p/a. Certain electron energies are not allowed in a crystal for resonance reasons. The resonance is a consequence of the periodic potential of the positive ions in the solid. Here an analogous example might be helpful. A similar situation exists on certain unpaved roads in the Sahara Desert. Because of the camels traveling along it, there are periodic bumps in the road. French 2CV cars resonate at 30 2 km h–1. So the driver is forced to either stay below 28 or above 32 km h–1. In between there is a forbidden gap (Figure 3-18). An important quantity for solids is the electronic density of states (DOS), which is the number of electronic states per energy interval. Evidently, in the forbidden gap the DOS is zero. Figure 3-19 shows the electronic dispersion relation between k = 0 and k = p/2 (as explained above, one half of the first Brillouin zone), together
The structure of certain Sahara roads allows cars to ride well only below or well above 30 km h–1. Between 28 and 32 km h–1 there is a forbidden’ gap. Figure 3-18
3.3 Electrons and Phonons in a Crystal, Dispersion Relations
Figure 3-19
1D electron dispersion relation and DOS.
with the DOS. The number of allowed k values in a Brillouin zone is large, but finite. It is discrete because the electronic states are discrete when the electrons are confined in a box. A crystal can be understood as such a box in which the electrons are confined. The allowed k values are equally spaced along the x axis (Figure 3-19). To get the DOS the dispersion relation is projected onto the energy axis. At places where the dispersion relation is flat, the DOS is high and vice versa. At horizontal parts of the dispersion relation the DOS is infinite. The points of infinite density of states are called van Hove singularities and play an important role in one-dimensional solid-state physics. Figure 3-19 exhibits van Hove singularities for k = 0 and for k = p/a, i.e., the bottom and the top of the energy band. The DOS in Figure 3-19 shows van Hove singularities for a one-dimensional solid. If the solid is three-dimensional, k-vectors point to all directions of space and the number of k-values within a k interval increases with k, thus compensating for the van Hove singularity at k = 0. This k count finally leads to the characteristic dimensionality behavior of DOS as indicated in Figure 1-18 (Chapter 1): parabolic in shape in three dimensions, step function in two dimensions, and square root singularity in one dimension. Many properties of a solid are determined by the DOS at the Fermi energy, N(EF). We recall that electrons are fermions and obey Fermi statistics, i.e., we can accommodate only one electron in a quantum state (labeled by k). In simple words: we pour’ the electrons into the crystal, they fill up the dispersion relation and at places where we run out of electrons there is the Fermi level, the highest occupied electronic state. If there are as many electrons as there are k values in the first Brillouin zone, the Fermi level is at the top of the band. Adding the next electron causes it to jump to the bottom of the next band. Strictly speaking, this is only true for absolute zero, where the Fermi distribution is sharp and where there are no thermal excitations in the electron system. Otherwise the Fermi level is defined in a somewhat more complicated way, and for a completely filled band the Fermi level is placed in the center of the gap above the filled band. A solid with only completely filled and completely empty energy bands is an insulator. Electrons can only be excited across the gap’ and no small energy excitations are possible. For electrical conductivity, however, such small excitations are neces-
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sary. The electric field has to change the k values (and hence the energy) of the electrons to transport electrons form one side of the solid to the other. To have a metal in which an electric field can accelerate electrons so that a electric current becomes possible, partially filled bands are needed. Metals have partially filled bands, semiconductors and insulators only completely filled and completely empty bands. The difference between semiconductors and insulators is just the size of the gap that separates the occupied and the unoccupied bands. In semiconductors the gap is small enough to allow for thermal excitations across the gap. Similarly, electrons can be lifted from a filled into an empty band by photoexcitation. Furthermore, the band filling can be influenced by doping, i.e., by replacing some of the atoms in the crystal with atoms having more or less electrons than the host atoms (impurities). When, for example, silicon with four electrons is replaced by arsenic with five, electrons are added to the crystal, or, using gallium with three electrons, electrons are withdrawn. In addition to the substitutional type of doping, interstitial doping can also be applied, e.g., Li in Si, in which the introduced atom is located between the regular lattice sites. Most dopants in polymers are interstitial, e.g., iodine or sodium in polyacetylene. In chemistry, usually withdrawing of electrons is called oxidation, and adding is reduction. Thus doping is a redox process that causes the Fermi level to shift (the Fermi level is comparable to the chemical potential). The dispersion relation in Figure 3-19 is symmetrical with respect to the bottom and the top of the energy band. It initially proceeds parabolically at either side. This type of symmetry is the so-called electron hole symmetry. Holes indicate missing electrons. If a band is nearly filled it is more convenient to look at those (few) states that are unoccupied than to look at all the many occupied states. Holes move’ in a semiconductor pretty much in the same way as empty seats move’ in a concert hall from the expensive front rows to the cheap back rows: people move forwards, the empty seats backwards. Holes are positively charged quasiparticles, which are responsible for the conductivity in p-doped (acceptor doped’, oxidized’) semiconductors. In n-doped (donor-doped’, reduced’) semiconductors the electric current is carried by excess electrons. At the point where a p- and an n-doped zone of a semiconductor meet, a p–n junction results. A p–n junction has rectifying properties (for textbooks on semiconductor physics, see [8–10]). More complicated arrangements of p- and n-conducting zones lead to all kinds of marvelous semiconductor devices, from laser diodes to megabyte chips, from resistance with negative temperature coefficient to fractional quantum Hall effect. Figure 3-17 shows the approach to the electron dispersion relation from the nearly free particle. The opposite approach is the molecular orbital approximation, which for many organic solids (and also for many inorganic solids) is more appropriate. It is based on the assumption that the electrons are localized at atoms or molecules. If these molecules approach close enough their orbitals interact, resulting in energy splitting. In molecular pairs this is known as Davidov splitting. In a manyparticle system like a crystal the splitting leads to an energy band. The band is as wide as the overlap of the wave functions (several meV to several eV) and the disper-
3.4 A Simple One-dimensional System
sion relation (band structure) is obtained by passing s-shaped curves through the end points (parabolic approach as in Figure 3-19). Often, such a rough estimate is sufficient for an approximate orientation. To calculate the actual band structure, extended expertise in quantum chemistry and fairly large computer power are essential.
3.4
A Simple One-dimensional System
Now that we have been reminded of a few of the tenants of solid-state physics and how they apply to one-dimensional systems, let’s return to an object that we introduced in Chapter 2: the carbon nanotube. Recall that we built the nanotube in our imagination by mapping it onto a graphene sheet and then rolling the sheet to form a seamless cylinder. In fact as we shall see, there is something very special about doing this. It allows there to be an unusual degree of decoupling between the carbon lattice and the mobile charge. That is not to say that no coupling exists, it is just that it is easy to approximate the charge as being free carriers. This may be the only onedimensional carbon system for which this is true. In other systems the charge and the lattice are intimately locked in step, forming a much more complicated object. Thus, the carbon nanotube allows us to understand the simplest system first. Perhaps most astonishingly, a little knowledge of the electronic structure of graphite and of the geometry of the tube leads one to fascinating conclusions with regards to the electronic nature of these objects. Recall that the extra, small’, dimension of circumference was an issue earlier. In fact, it is quite easy to understand why it should be. A half integral number of electron waves, for wave functions of electrons on the tube’s surface, must fit around the circumference. The part of the wave function heading down the tube axis, however, looks like a plane wave. Now one asks: in what states do the electrons exist in the graphene before rolling it (because it is these electrons that are available for constructing the tube)? The Fermi surface of the graphene looks like six points on the edge of a hexagon (Figure 3-20). Notice that the hexagon is tilted with respect to the axial direction of the tube. The amount of tilt is given by the chiral angle. However, the states associated with the circumference appear as straight lines along the direction of vector R. The points on the vertices of the hexagon (the Fermi surface) are the only electrons allowed to participate in conduction of the object. The states associated with the circumference are the only ones that are allowed so that the wave function fits on the tube. The angular position of the hexagon is determined by the angle with which the graphite is oriented relative to the rolling axis. When the hexagon lies at an angle such that the hexagonal vertex intersects one of the circumference states, the tube has electrons that are mobile and it is a rather good conductor. However, when the hexagon vertex misses the circumference state, the electrons at the vertex energy must be given some energy to make a transition into a conducting state. Does that mean the tube acts like a semiconductor? That is right, the nanotube can be either a semicon-
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The Fermi points of graphene sit at the points of the hexagon shown. However, this hexagon is oriented with respect to the axis of the tube.
Figure 3-20
ductor or a metal depending on its chiral angle. Furthermore, the gap depends on the angle and the diameter of the tube. Since we can know the chiral angle in terms of (n,m), we can use a geometric argument to see that if (n,m) is not divisible by 3, then the tube is a semiconductor, otherwise it is a metal. Furthermore, 2/3 of the possible nanotubes are semiconductors and only one third metals. Though we do not show it here, it is clear that as the tube diameter grows, the spacing between the circumference state gets more narrow. Thus, the bandgap of a nanotube Eg ~ 1/DIA. Last, our example above is clearly a semiconductor. Perhaps you can see why people have acquired such a fascination for this beautifully symmetric system? Is this system truly one-dimensional? In what way can we tell? As mentioned earlier, the electronic signature of truly one-dimensional behavior is the occurrence of van Hove singular points in the electronic density of states. Do we in fact observe these? There is a way to measure such density of states on nanoscale objects, that is, by using scanning tunneling microscopy and spectroscopy. In tunneling microscopy, the image is collected by scanning the surface with an atomically sharp tunneling tip. The image contrast is generated through subtle variations in the tunneling current. When one wants to know the electronic structure at a specific place in the image, the tip is stopped from scanning, feedback systems are disengaged, the voltage is ramped and current is collected. This tunneling spectrum is then differentiated to yield a differential conductivity. The tunneling differential conductivity is normalized by a term that varies as the tip height above the object, and the result of this normalization is proportional to the density of electronic states in the area where the spectrum was taken. In a stunning set of experiments two sets of researchers, one at Delft, one at Harvard, correlated the electronic structure of single-walled carbon nanotubes with the atomic structure as determined using scanning tunneling microscopy. Since the tunneling microscope can image at atomic resolution, the researchers were able to show that the set of van Hove singularities predicted for a given set of (n,m) were exactly seen in tunneling spectra (Figure 3-21). In fact, van Hove singular points were identified in multiwalled carbon nanotubes earlier. These objects are simply increasingly larger diameter single-walled
3.4 A Simple One-dimensional System
Tunneling micrographs and density of electronic states showing the occurrence of van Hove singularities in single-walled carbon nanotubes.
Figure 3-21
nanotubes placed in a concentric configuration, giving more than one wall. The interesting point here is that it appears as though there is little cross-talk between the shells. In other words, the outer shell of the multiwalled nanotube is a onedimensional object as well, without regard for what is inside (Figure 3-22). This leaves very little doubt that carbon nanotubes can reflect a one-dimensional electronic nature. How this one-dimensional nature has manifest itself in thermal, optical, and transport properties of both single-walled and multiwalled carbon nanotubes has been intensely studied over the past 10 years [11]. Furthermore, the role of symmetry-breaking in such objects through defects [12], kinks [13] and bends [14], and dopants [15] has also been of significant interest to the scientific community. In each case however, the one-dimensional nature of the nanotube system must be modified with respect to the extra-dimensional degree of freedom around the waist
The tunneling spectra, converted to electronic density of states, for a multiwalled carbon nanotube. On the graph below the spectra, a most simple tight binding prediction for this tube is plotted. (Courtesy of D. Tekleab.)
Figure 3-22
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76
of the tube geometry. This topological modification is especially clear for thermal transport, where the vibrational degrees of freedom must include the so called twiston’, or the twisting of the tube about the axis [16]. In some sense one might claim that a polymer system could exhibit a twisting of the atoms about the axis, and this would be analogous to the twiston. This serves to demonstrate the limited applicability of the concept of one-dimensional’ materials.
References and Notes 1 N.W. Ashcroft and N.D. Mermin. Solid State 2 3
4
5 6
7
8 9
10
11
12
Physics. Sounders, New York, 1976. K.H. Hellwege. Einfhrung in die Festkrperphysik. Springer Verlag, Heidelberg, 1976. C. Kittel. Introduction to Solid State Physics. 6th Edition, John Wiley & Sons, New York, 1986. R.C. Evans. An Introduction to Crystal Chemistry. 2nd Edition, Cambridge University Press, Cambridge, 1966. U. Mller. Anorganische Strukturchemie. B.G. Teubner, Suttgart, 1982. E.A. Silinsh. Organic Molecular Crystals. In: Springer Series in Solid-State Sciences 16. Springer Verlag, Heidelberg, 1980. R. Comes, M. Lambert and H.R. Zeller. A lowtemperature phase transition in the onedimensional conductor K2Pt(CN)4Br0.30·H2O. physica status solidi (b) 58, 587 (1973). K. Seeger. Semiconductor Physics. Springer Verlag, Wien, 1973. K.W. Ber. Survey of Semiconductor Physics. 2 Vols., Van Nordstrand Reinhold, New York, 1993. S.M. Sze. Semiconductor Devices – Physics and Technology. John Wiley & Sons, New York, 1985. For a detailed and easy to understand introduction to these studies, we recommend R. Saito, G. Dresselhaus, and M. S. Dresselhaus. Physical Properties of Carbon Nanotubes. Imperial College Press, London, 1998. D.L. Carroll, Ph. Redlich, P.M. Ajayan, J.C. Charlier, X. Blase, A. De Vita, and R. Car. Elec-
tronic structure and localized states at carbon nanotube tips. Physical Review Letters 78, 2811 (1997). 13 D. Tekleab, R. Czerw, P.M. Ajayan, and D.L. Carroll. Electronic structure of kinked multiwalled carbon nanotubes. Applied Physics Letters 76, 3594 (2000). 14 D. Tekleab, D.L. Carroll, G.G. Samsonidze, and B.I. Yakobson. Strain-induced electronic property heterogeneity of a carbon nanotube. Physical Review B 64, 035419 (2001). 15 D.L. Carroll, S. Curran, P. Redlich, P.M. Ajayan, S. Roth, and M. Rhle. Effects of nanodomain formation on the electronic structure of doped carbon nanotubes. Physical Review Letters 81, 2332 (1998). X. Blase, J.-C. Charlier, A. De Vita, R. Car, Ph. Redlich, M. Terrones, W.K. Hsu, H. Terrones, D.L Carroll, and P.M. Ajayan. Boron-mediated growth of long helicity-selected carbon nanotubes. Physical Review Letters 83, 5078 (1999). R. Czerw, M Terrones, J.-C. Charlier, R. Kamalakaran, N. Grobert, H. Terrones, B. Foley, D. Tekleab, P.M. Ajayan, W. Blau, M. Rhle, and D.L. Carroll. Identification of electron donor states in n-doped carbon nanotubes. Nano Letters 1, 457 (2001). 16 C.L. Kane, E. J. Mele, R.S. Lee, J.E. Fischer, P. Petit, H. Dai, A. Thess, R.E. Smalley, A.R.M. Verschueren, S.J. Tans, and C. Dekker. Temperature-dependent resistivity of singlewall carbon nanotubes. Europhysics Letters 41, 683 (1998).
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Electron–Phonon Coupling, Peierls Transition In Chapter 3 a solid was characterized as a rigid structure of atoms. We then poured’ electrons into this solid and the structure forced the electrons to arrange in a certain way, forming bands and gaps, etc. In this chapter we will change the electronic arrangement and observe how the atoms react to this perturbation. In threedimensional solids the reaction is very small. Because of the high connectivity the atoms are held strongly in their original positions by bonding forces exerted by its neighbors. Atoms in a one-dimensional solid however have only two neighbors. If the bond to one of these atoms is split, the solid breaks apart. If the bond is only weakened, the lattice is heavily distorted. In this respect a one-dimensional lattice is very soft. The situation is illustrated in Figure 4-1, which shows a simplified structure of polyacetylene as an extended chain instead of zigzagged, with a bond angle of 180 instead of 120. In polyacetylene single and double bonds alternate, and the bonds keep the carbon atoms in their positions. Double bonds are stronger than single bonds (although not twice as strong), and consequently the length of a double bond is shorter than that of a single bond. Splitting a single bond means breaking the chain. If one part of a double bond is cleaved (the solid lines in the figure symbolize electron pairs), the double bond is weakened by its transformation into a single bond. As a result, the lattice is distorted due to the lengthening of the respective interatomic distance. A chemist or a molecular physicist would not be surprised by the large lattice distortion. If there is any surprise at all, it is because of the solid-state terminology. Cleaving a double bond – for example by the absorption of light – means raising the molecule into an excited state (note that chemists speak of molecular states whereas
Figure 4-1 Bond scission is polyacetylene, demonstrating the strong electron-lattice coupling in one-dimensional solids.
One-Dimensional Metals, Second Edition. Siegmar Roth, David Carroll Copyright © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30749-4
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Figure 4-2 Two configurations of bianthrone, demonstrating a complicated example of electron-lattice coupling.
in the Chapter 3 we used the term electronic states, which chemists call orbitals). The shape of the molecule in the excited state is different from its configuration in the ground state. For solids this configuration corresponds to the crystal lattice. A complicated example of electron-lattice coupling is illustrated in Figure 4-2. Here a bianthrone molecule is shown, which twists and folds when it changes state [1]. Scientists in the field of molecular electronics’ (see Chapter 9) discuss whether organic materials can beat silicon in achieving miniaturized devices. Strong electron-lattice coupling might help to better localize excitations, thus smaller devices should be possible. In this respect, electron-lattice interactions in organic components are often referred to as coupling to conformational degrees of freedom’. Excitations would be confined to a molecule, which extends over some 10– 20 (the oxygen–oxygen distance in bianthrone is 11 ). In Section 5.4 we will discuss solitons as a very exciting example of localized excitations. An important type of electron-lattice coupling is the Peierls distortion. This lattice distortion belongs to a more general class of symmetry-breaking or degeneracy-lifting transitions, of which the Jahn–Teller effect is the prototype (Figure 2-7). In 1955, Peierls [2] showed that a monatomic metallic chain is unstable and undergoes a metal-to-insulator transition at low temperatures. Figure 4-3 depicts such a chain for example – a linear equidistant arrangement of sodium atoms. Of course, this is just a thought experiment, because sodium atoms do not arrange in chains. They tend to form clusters. To keep them in line like beads on a necklace, some sort of a string to thread them would be necessary (Figure 4-4). That is not so easy and requires a lot of chemistry. Actually all the chemistry of Chapter 2 is just an attempt to approach this idea. Although we cannot synthesize an isolated chain of equidistant monatomic sodium, it is possible to calculate the electronic dispersion relation and the electronic density of states (DOS) by following the outlines in the previous chapter. An S-shaped dispersion relation, parabolic on both sides, is the result. The DOS has square root singularities at the top and at the bottom of the band (which are of no significance in this particular context). The band is half-filled, because each sodium atom contributes one delocalized electron to the solid, and two electrons – spin up and down – can be accommodated in each state. The Fermi wave vector kF is halfway
4 Electron–Phonon Coupling, Peierls Transition
Figure 4-3 Peierls distortion in an isolated chain of equidistant monatomic sodium. Crystal lattice, electronic density of states, and electron dispersion relation without distortion (a) and with distortion (b).
between 0 and p/a, and the Fermi energy EF is at the band center. There is a finite DOS at the Fermi level, N(EF) „ 0. To calculate the total energy of the electrons we have to sum over all electron states up to EF. Now we change the arrangement of the atoms. We call for a Maxwellian demon who approaches the atoms in pairs. He displaces every second atom by an amount d, so that the atoms are no longer equidistant: short distances a – d and long distances a + d alternate. Again the arrangement is periodic, but with repeat distance 2a instead of a. Consequently, the reciprocal lattice changes from a*old = 2p/a to a*new = p/a. In the former lattice the border of the first Brillouin zone was at p/a, now it is at half that distance, just at the Fermi wave vector kF. The former lattice caused the electrons to have a gap at p/a, the new lattice creates an additional gap at p/2a. The demon has transformed the system from a metal with no gap at the
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Figure 4-4 Alkali atoms do not arrange in chains but prefer to form clusters. CH. radicals can be lined up on the r-bonded backbone of a polymer – like beads on the string of a necklace.
Fermi level into a semiconductor with a gap. All states below the gap are filled at absolute zero, all above are empty. If the demon leaves, will the system stay semiconducting (insulating), or will it relax and return to the undistorted lattice? It is easy to see that in the distorted configuration the electronic energy is lower, because the states in the gap have been accommodated above and below the gap (states cannot disappear, there are as many states as atoms in the lattice, because the states are formed from atomic orbitals). Consequently, summing the electronic energies from zero to EF shows that the creation of the gap had reduced the electronic energy. But to achieve this, the demon had to exert work: there are springs’ between the atoms; the demon has alternately compressed and expanded them. The analysis shows that the electronic energy is approximately linear with respect to d, whereas the elastic energy depends quadratically on d. For sufficiently small displacements the gain in electronic energy predominates over the elastic term. Consequently under Peierls assumptions, there is always a gap at absolute zero. How large it is and at what temperature enough electrons are excited to cross the gap so that it becomes ineffective depend on the parameters of the system (spring constants, bandwidth, etc.). As a result, the demon is allowed to leave. We don’t even need him to start the Peierls distortion. At low enough temperatures the system knows’ (by fluctuations) that it gains energy upon distortion and thus it does distort. Electron-lattice coupling drives the one-dimensional metal into an insulator. The monatomic metal chain of Figure 4-3 exists only in a thought experiment. Real one-dimensional metals are more complicated, as we saw in Chapter 2. The Peierls transition, however, has been observed in several systems (for a review, see, e.g., [3]). In Figure 2-10 experimental evidence for the Peierls distortion was presented: the giant Kohn anomaly in KCP. The restoring force stabilizing the equidistant lattice decreases, and the respective phonons get soft. These are phonons with wave vector 2kF. In KCP the conduction band is not half-filled as it would be for an
4 Electron–Phonon Coupling, Peierls Transition
Figure 4-5 Phonon dispersion and Kohn anomaly in a hypothetical inelastic neutron-scattering experiment on an ideal monatomic alkali chain: (a) before distortion and (b) after distortion.
ideal alkali chain (Figure 4-3). There is charge transfer from the platinum chain to the bromine ions and band-filling is more complicated. Therefore in Figure 2-10 the wave vector 2kF does not correspond to the border of the first Brillouin zone, as it would in the alkali chain. Figure 4-5 shows what to expect, from an inelastic neutron-scattering experiment on the alkali chain: the zone-end phonons soften and finally condense into a new lattice point at p/a. Therefore the Brillouin zones have to be altered. From the segment between p/2a and p/a we can subtract a lattice vector of the new reciprocal lattice and transfer the whole segment to the left of the origin. The soft phonon at 2kF also corresponds to the requirement of conservation of energy and momentum in electron–phonon scattering. This pronounced electron– phonon scattering is the high-temperature precursor of the Peierls transition. Energy conservation requires that only electrons at the Fermi surface are involved: the Fermi energy is much larger than the phonon energies and so only quasi-elastic scattering processes can occur. The Fermi surface of a one-dimensional metal consists of two planes perpendicular to the metal chain (Figure 3-15). The phonons scatter the electrons from one plane to the other; this means they just reverse the electron momentum, from +kF to –kF and vice versa. Such momentum flipping is
Figure 4-6 Warped Fermi surface of a one-dimensional metal with slight two-dimensional character.
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Figure 4-7
Projection of warped Fermi surface and nesting’.
called an Umklapp process’. If a phonon wants to change the electron momentum from +kF to –kF, it must have the wave vector 2kF (note, this is sloppy terminology, neglecting that the wave vector is k and the corresponding momentum is hk). Electronic Umklapp processes are also encountered in Chapter 6 where we discuss the theoretical limit of the conductivity of polyacetylene. In an ideal polyacetylene crystal, electron–phonon scattering is again Umklapp scattering, and this scattering is the only contribution to electrical resistivity in an ideal one-dimensional crystal. In the alkali metal chain the 2kF phonons become soft and drive the system from metal to insulator. If the softening is blocked by some other interaction, the 2kF phonons can be frozen out at low temperatures and there is no contribution to resistivity, thus turning the metal into an ideal conductor at T = 0 (but not into a superconductor, which would require a collective state and a finite transition temperature T > 0, not simply vanishing resistance at zero temperature). In an ideal one-dimensional metal the Fermi surface consists of two parallel planes (Figure 3-15). If it is less ideal and some interaction with adjacent chains occurs, the Fermi surface tries’ to become spherical or cylindrical – it bends. Figure 4-6 shows the warped Fermi surface of a system that has a slight two-dimensional admixture, and Figure 4-7 shows the projection of the warped Fermi surface. With this graph we want to demonstrate the concept of nesting’, which makes quasi one-dimensional metals look more one-dimensional than expected. Umklapp processes scatter electrons from one sheet of the open Fermi surface to the other. With parallel sheets all Umklapp processes involve parallel vectors of momentum transfer. For a spherical Fermi surface there are no parallel transfer vectors. If the surface is appropriately warped, however, there are some or even many parallel transfer vectors, as indicated by the arrows in Figure 4-7. In this situation phonons moving along the chain can flip the electron momentum by conserving the projection of the momentum onto the chain. If there are enough’ such parallel transfer vectors, the Peierls transition can occur even if the Fermi surface is not planar. The Peierls transition and nesting play an important role in charge density wave phenomena, discussed in Chapter 8. In the complicated systems mentioned in Chapter 2 all sorts of interactions occur in addition to the electron-lattice interaction. In particular, we need to refer to the important issue of correlation energy. The correlation energy plays an important role at the stage of the free electron model (Section 3.4). This model assumes that electrons behave like molecules in an ideal gas, not being affected by each other
References and Notes
except for momentum exchange during collisions. In a real gas, however, two molecules cannot occupy the same place, and within the molecular volume the molecules have to avoid each other. The molecular volume is very small and in many instances the ideal gas is a good approximation. But electrons do not interact as small hard cores, they interact by long-range Coulombic forces. An electron at a specific lattice site repels other electrons – unless the Coulombic interaction is screened by polarization of the surrounding environment. In our thought experiment we can easily immerse our monatomic chain into such a screening medium. In many real onedimensional systems, electron–electron interaction is as large as the electron-lattice coupling and the above concept of the formation of a Peierls gap is not valid even as a rough approximation. This is particularly true for conjugated polymers [4]. Nevertheless, in the next chapter we discuss polyene chains and conductive polymers as if the electron-lattice coupling were the dominant interaction. This simplification allows us to introduce many relevant concepts in a didactic way (ad usum Delphini’ [5]) and acts as precursor for more sophisticated discussions.
References and Notes 1 K. Schaumburg, J.-M. Lehn, V. Goulle, S. Roth,
H. Byrne, S. Hagen, J. Poplawski, K. Brunfeldt, K. Bechgaard, T. Bjornholm, P. Frederiksen, M. Jorgensen, K. Lerstrup, P. Sommer-Larsen, O. Goscinski, J.-L. Calais, and L. Eriksson. Nanostructures based on molecular materials, p. 153. W. Gpel and Ch. Ziegler (Eds.), VCH-Verlag, Weinheim, 1992. 2 R.E. Peierls. Quantum Theory of Solids. Clarendon Press, Oxford, 1955. 3 R. Friend and D. Jerome. Periodic lattice distortions and charge density waves in one- and
two-dimensional metals. Journal of Physics C: Solid State Physics 12, 1441 (1979). 4 D. Baeriswyl. Conjugated conducting polymers. In: Springer Series in Solid-State Sciences 102, H. Kiess (Ed.), Springer Verlag, Heidelberg, 1992. 5 Dauphin was the title of the French crown prince. Obscene parts of classical books were censored to protect the child’s soul. Such books were marked ad usum Delphini’.
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Conducting Polymers: Solitons and Polarons 5.1
General Remarks on Conducting Polymers
In Section 2.8 conducting polymers were introduced. Figure 2-28 shows the chemical structure of some of the most important conducting polymers; space-filling models are presented in Figure 2-29. Tradition says that Confucius was once asked to name the duties of a good king. First of all, put terminology in order’, the master replied (Figure 5-1). Regarding conducting polymers, the king would have had a hard time. None of the polymers shown in Figures 2-28 and 2-29 are good electrical conductors when chemically pure. They are insulators, or, at best, semiconductors. Only after treatment with oxidizing or reducing agents do they become conducting. This procedure is called doping; but many people don’t like the term doping’ in this context. In semiconductor physics, undoped semiconductors are intrinsically conducting, doped semiconductors extrinsically conducting. In contrast, doped polymers are often referred to as intrinsically conducting polymers’. This is to distinguish them from polymers which acquire conductivity by loading with conducting particles, such as carbon black, metal flakes, or fibers of stainless steel. We call these materials
Figure 5-1 First of all, put terminology in order’ Confucius told the king when asked about the duties of a wise ruler.
One-Dimensional Metals, Second Edition. Siegmar Roth, David Carroll Copyright © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30749-4
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conjugated polymers’; however, one must admit that this name sounds too scientific to acquire financial assistance or attract the attention of application engineers. Fortunately, we can also use the expression synthetic metals’. An enormous amount of literature on conducting polymers already exists. As examples, we refer just to the series of ICSM Proceedings (International Conference of Science and Technology of Synthetic Metals) in Table 2-1, to the KirchbergIWEPP Volumes (International Winter Schools on Electronic Properties of Novel Materials [1–4], to the Proceedings of the Polish Autumn Schools [5–9], to the NATO Advanced Study Institute in Burlington [10], to the Lofthus Meeting [11], to Skotheim’s Handbook of Conducting Polymers [12], to the Springer publication [13] edited by Kiess, and to Przyluski’s [14] and Chien’s [15] monographs. A book devoted to both intrinsic and particle-loaded conducting polymers was published by Mair and Roth [16]. Chapter 5 is focused on physical concepts of conjugated polymers. We approach the subject from the electron-lattice-coupling point of view, as indicated in the previous chapter on the Peierls transition and ignore the equally important influence of electron correlation. Relevant review articles have been published by Roth and Bleier [17] and by Heeger et al. [18], and a recommended monograph was published by Lu [19]. In Chapter 6 we will discuss the electrical conductivity of conducting polymers, and in Chapter 10, aspects of industrial applications.
5.2
Conjugated Double Bonds
From organic chemistry we know several types of double bonds: isolated, cumulated, and conjugated. Conjugated double bonds are double bonds separated by single bonds. In looking at Figure 2-28 we notice strict alternation of double and single bonds in all polymers shown (with the exception of polyaniline – but in polyaniline there is an extra electron pair on the nitrogen atoms and conjugation passes through these extra pairs). Conjugated double bonds behave quite differently from isolated double bonds. As the word implies, conjugated double bonds act collectively, knowing’ that the next-nearest bond also is a double bond. To stress this, a colleague once quoted the Chinese proverb: Pull a hair and you excite the whole body’ [20] (Figure 5-2). To discuss the physics of conjugated double bonds we look at the conjugated polymer with the highest symmetry: polyacetylene. In a thought experiment (Figure 5-3) polyacetylene can be prepared from polyethylene, which is the simplest of the saturated linear-chain polymers (saturated’ organic compounds are those without double bonds). Polyethylene is a zigzag chain of carbon atoms with two hydrogen atoms per carbon atom. Removing one hydrogen from each carbon would leave us, in a first step, with unbonded electrons everywhere. We would obtain a chain of CH. radicals. A CH. radical has some similarity with an alkali atom: both have an extra electron. The poly-CH. chain in Figure 5-3 thus resembles the sodium chain in Figure 4-3. (In alkali metals the extra electrons are s electrons. In a CH. radical, the
5.2 Conjugated Double Bonds
Figure 5-2 Pull on a hair and you excite the whole body’. In a conjugated polymer a double bond knows’ that next-nearest bonds are also double bonds.
extra charge is a p electron. For the discussion in this chapter the difference between s and p electrons does not matter.) At first glance a metal could be expected, and it is not surprising that polyacetylene is the synthetic metal par excellence’. However, from the previous chapter we know that electron-lattice coupling will cause a Peierls transition and drive the metal into an insulator, at least at low temperatures. (Later, we will see that doping suppresses the Peierls transition.) In Figure 5-4 the metallic state’ of polyacetylene (top) and the transition to the insulating state (bottom) are illustrated in a slightly different way. In the metallic state the electrons are depicted as delocalized over the entire chain. The dashed line should allude to the Bloch-wave behavior of the electrons. Delocalization, not unfamiliar to the organic chemist, is found in aromatic rings; often delocalization is symbolized by resonance structures (Figure 5-5). Unlike benzene, however, a sufficiently long, extended chain is not aromatic, and so this metallic state will not actually be realized. In fact, an alternation of short and long bonds will occur, as indicated at the bottom of Figure 5-4 (where short bonds are drawn as double bonds and
Figure 5-3 Thought experiment, producing polyacetylene by dehydrogenation of polyacetylene. In the first step a chain of CH. radicals is obtained analogous to a chain of alkali atoms.
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As in alkali metals the electrons of the CH. radicals delocalize all over the solid. The Peierls distortion then leads to bond alternation. Figure 5-4
long bonds are single bonds). Due to this alternation, there is a gap in the electronic density of states. All states below the gap are occupied and form the valence band, the states above the gap are empty and form the conduction band. Chemists call these bands the p and p* bands. For the discussion of polyacetylene they would prefer an approach via molecular orbitals rather than via solid-state physics. They would first link two CH* radicals to a (CH)2 pair with a double bond between the two CH groups that consists of a r and a p bond. (In a polymer chain s electrons form r bonds, p electrons form p bonds.) There will be a bonding p and an antibonding p* orbital. Forming a macromolecular chain with these (CH)2 pairs, the p and the p* orbitals split to give bands. What solid-state physicists call the top of the valance band’ translates in chemical language to the highest occupied molecular orbital (HOMO)’. The bottom of the conduction band’ is then called the lowest unoccupied molecular orbital (LUMO)’. The solid-state physicist starts with a band, the width of which is given by the average interaction between single carbon atoms, and then bond alternation opens a gap; the chemist starts with the gap, followed by broadening the p and p* orbitals through interaction between carbon pairs. The p–p* gap in polyacetylene is about 1.7 eV. This is well in the region of known inorganic semiconductors (diamond: 5.4 eV; GaAs: 1.43 eV; Si: 1.14 eV; Ge: 0.67 eV). As for a semiconductor, the gap can be determined by optical absorption. An example is shown in Figure 5-6. The polyacetylene band gap is much larger than the Peierls gap in KCP (Section 2.2) or in charge-transfer salts (Section 2.4). The Peierls transition occurs at a temperature Tp that is in the same order of magnitude as the gap energy Eg (just as with superconductivity Eg ~ 4kBTp), because above that temperature many electrons are thermally excited across the gap and the solid does not notice’ the gap. Since
Figure 5-5 Electron delocalization in a benzene ring to illustrate that metallic behavior’ is not unexpected in organic chemistry.
5.3 Conjugational Defects
Figure 5-6 Optical absorption in polyacetylene. Like a classical semiconductor, the sample is transparent to light with quantum energy smaller than the band gap. From the onset of the absorption the band gap can be determined.
1 eV corresponds to a temperature of about 10 000 K, at room temperature polyacetylene is far below the Peierls transition. We cannot heat polyacetylene into the metallic phase, because the polymer decomposes at some hundred degrees Celsius.
5.3
Conjugational Defects
In polyacetylene there is strict bond alternation. For a chemist this is trivial, because dimers (pairs) were polymerized. For a physicist dimerization is a phase transition from a metallic to a semiconducting state. Although admitting that this phase transition is hypothetical (Tp » 10 000 K) the physicist can imagine that the phase transition nucleates at several points of the chain with domains growing around these nucleation centers. Finally, misfits are created at touching domains, as indicated in Figure 5-7. Such misfits are very interesting. In Figure 5-7a the bond alternation is interrupted by two adjacent single bonds. To keep the carbon atom in the domain wall’ (at the defect) tetravalent, a dangling bond must exist (two bonds join neighboring carbons, one the hydrogen atom; the fourth bond has no partner). Dangling bonds are not unusual in semiconductor physics; for example, in amorphous silicon they are quite common. In polymer chemistry they are also known. The defect shown in Figure 5-7 is a radical on a polyene chain. It can be detected by electron spin resonance (ESR) because of its unpaired spin. Are domain walls with two double bonds touching possible, as in Figure 5-7b? Technically this would lead to a carbon atom with 5 bonds and therefore the defect would preferably be shown as in Figure 5-7c.
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Figure 5-7 Misfit in a conjugated chain: (a) two adjacent single bonds plus a dangling bond; (b) two adjacent double bonds, leading formally to a pentavalent carbon atom; (c) three dangling bonds on a misfit.
But is there really a difference between Figures 5-7a and 5-7c? In Figure 5-7c, which of the three electrons form a pair and which one is left alone is left open. (As we will see later, misfits can be neutral or positively or negatively charged, depending on the electron occupation. Therefore you should not object to seeing two electron pairs in Figure 5-7b and three dangling bonds in Figure 5-7c.) Actually, the misfits are not as well localized as shown in Figure 5-7 and a defect extends over some ten bonds, modifying the bond alternation gradually. p¼
left bond right bond average bond
We define a bond length alternation parameter as the parameter p. It changes sign at the defect (Figure 5-8). The Peierls transition discussed in Chapter 4 is associated with a charge density wave. Figure 5-9 illustrates the charge density wave for polyacetylene. If we interpret the bonding dashes as electron pairs, then the charge density wave is trivial: the pelectrons are located at the double bonds. Due to the bond alternation there is a periodic modulation of the p electron density. At that misfit the charge density wave has a 180 phase slip. Here we should again stress that the Peierls transition in polyacetylene is oversimplified. In addition to the electron-lattice coupling there are additional, equally important interactions, often referred to correlation effects. To take these into account the parameter p can be generalized as bond order parameter’ and the charge density wave as bond order wave’ [21]. Conjugated double bonds have been studied for about half a century, not because of their importance in conducting polymers but because they are essential for dyestuffs and interesting for fundamental considerations in quantum chemistry. Pioneering work was done be Kuhn [22] and later by Longuett-Higgins and Salem [23].
5.3 Conjugational Defects
Figure 5-8
Bond alternation parameter changing sign at a misfit on a conjugated chain.
Figure 5-9
Conjugated double bonds as charge density wave in a p-electron system.
Figure 5-10
The soliton thesaurus in conjugated
polymers.
Early in the 1960s Pople and Walmsley [24] investigated the conjugational misfit. In Figure 5-10 several synonyms for this defect are arranged, each one in a different context and stressing different aspects. We deal with the soliton aspect in the following sections of this chapter.
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Optical absorption of polyacetylene showing the emergence of a midgap state upon doping. Note that the absorption curves cross at an isosbestic point, because the midgap peak grows at the expense of the band states.
Figure 5-11
Can conjugational defects be observed experimentally? The ESR signal of unpaired electrons has already been mentioned. Later we will see that there are charged and uncharged defects (just as there are charged and neutral lattice defects in an inorganic semiconductor) and that polyacetylene has the peculiarity that charged defects have no spin. Consequently, charged defects are invisible in ESR experiments – but charged and neutral defects can be seen in optical absorption experiments as states in the gap (midgap states’). If the gap is explained as the separation of bonding and antibonding states, a dangling bond, which is neither bonding nor antibonding but nonbonding, must be in the gap. Similarly, if the gap is a Peierls gap originating from bond length alternation, the alternation parameter is zero at the misfit; there is a chain section without Peierls distortion, and there must be some part of the chain without a gap. We will see in Section 5.5 how conjugation defects are generated by doping. The dopinginduced emergence of the midgap absorption is shown in Figure 5-11.
5.4
Solitons
In Figure 5-10 solitons’ are listed as one of the names the conjugational defect in polyacetylene, and Figure 2-3 shows a superposition of a literature survey counting papers per year having the keyword soliton’ on the Japanese woodcut of Mount Fuji behind a breaking wave. Our intention was to imply that the wave of papers rises soliton-like. First indications of similarity between polyacetylene defects and solitary
5.4 Solitons
waves can be obtained from Figure 5-8, showing the bond alternation parameter resembling a tidal wave. The wave concept in physics has been developed from water waves. But water waves are very complicated; only at small amplitudes are they harmonic waves, looking like a sine or cosine function. Sound waves in solids or electromagnetic waves would be better examples of harmonic waves. At any rate, a harmonic wave is the solution of the wave equation: c2uxx – utt = 0
(1)
This is a linear differential equation of second order. It states that the second derivative with respect to time utt must be proportional to the second spatial derivative uxx of the function u(x,t). This condition is met for the trigonometric functions sine and cosine: u(x,t) = sin (kx – xt)
(2)
In water, the wave equation (1) is only a small amplitude approximation. For large amplitudes, nonlinear differential equations must be used, for example, the Korteweg–deVries equation: ut + 6 uux + uxxx = 0
(3)
This equation is nonlinear, because it contains the product of u and its derivative ux. Other nonlinear differential equation are the Sine–Gordon equation c2uxx – utt = x2 sin u
(4)
and the equation of u4 potentials c2uxx – utt = x2(u3 – u)
(5)
The wave equation (1) applies to motions in a parabolic potential. A chain of balls coupled by springs, as shown in Figure 3-12, would be a good mechanical model. If the potential is nonparabolic, other equations have to be used, e.g., the Sine–Gordon equation (4) for a sinusoidal potential or Eq. (5) for double-well potentials. The alternation of short and long bonds in polyacetylene can be imaged by double-well potentials, as indicated in Figure 5-12. This figure also shows a defect at the center of the chain.
Figure 5-12
Coupled double-well potentials to illustrate conjugational defects in polyacetylene.
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The wave equation (1) has harmonic waves, i.e., a sine or cosine function as solution. The nonlinear Eqs. (3), (4), and (5) also have analytical solutions; however, these are not periodic waves but pulse- or step-like functions. For example, the solution of the double-well equation (5) is u(x,t) = u(x – tt)
(6)
with u(x) = tanh (x/f)
(7)
and f2 = (c2 – t2) / x2
(8)
Again a trigonometric function is found as solution, not the periodic sine or cosine but the step-like tanh function. The step is visible in Figure 5-8. Being used to harmonic waves, we would like to construct steps and pulses as wave packages. Any shape can be Fourier-synthesized by appropriate superposition of harmonic components. However, as mentioned in Chapter 3, wave packages disperse if the velocity of a wave depends on the frequency. For this reason Eq. (2) in Chapter 3 is called the dispersion relation. The particular steps and pulses we are discussing here, however, do not disperse. Therefore they are called solitary waves’. In hydrodynamics, solitary waves have been studied for more than one and a half centuries. They were observed even earlier, and they have been feared because of their nondispersive properties! In fact, Hokusai’s wood cutting in Figure 2-3 has been chosen not only because of its aesthetic value, but also because it is related to tsunamis, disastrous tidal waves generated by earthquakes under the Pacific Ocean, which have destroyed many human settlements along the Pacific coast. In Europe spring tides are known to propagate far inland into river estuaries on the Atlantic; for example, before the Seine was dredged they even went up to Paris! When we encounter an entity that looks like a pulse or a step it is not easy to decide whether it is a wave package or a solitary wave. Experimentally, it would have to be examined in motion to see whether or not it will disperse. But it might well turn out that the ocean is too small, so that the entity does not move far enough for us to see any difference between dispersion and nondispersion. Analogously, polymer chains might be too short, and in addition it would be difficult to observe the motion (although in principle, with magnetic resonance techniques it could be possible). If the properties of the medium are known, the amplitude of the entity can be measured. One could discuss whether an excitation with such a large amplitude is still compatible with the validity of the harmonic approximation. However, usually our knowledge of the medium is not sufficient. In 1977 it was discovered that polyacetylene can be doped and that doped polyacetylene has very high electrical conductivity [25]. Soon afterwards Rice [26]; Su, Schrieffer, and Heeger (SSH’) [27]; and Brazovskii [28] noticed that nonlinear differential equa-
5.4 Solitons
La chasse aux solitons’, reminiscent of the butterfly hunt in a popular song by George Brassens (however, what is hunted there does not seem to be butterflies).
Figure 5-13
tions with solitary wave solutions can be formulated for polyacetylene and that there is a relation between conjugational defects and solitary waves. This observation triggered a long and vivid discussion on whether or not solitons exist in polyacetylene (La chasse aux solitons’) (Figure 5-13) [29–30]. People also speculated whether these solitons might be responsible for the high electrical conductivity (see Chapter 6). If today you enter the term soliton’ in a literature search, many papers on solitons in telecommunications turn up. If we codify a message in pulses and send these pulses along a cable, it is desirable that the pulses do not disperse (Figure 5-14). In conventional cables, pulse-shaping devices are inserted every few kilometers, for example, nonlinear amplifiers which amplify the high pulse center more than the low tails. In glass fibers, dispersion (frequency dependence of the velocity) is compensated by nonlinear optical effects (intensity dependence of the refractive index). It is not surprising that glass fibers are described in a continuous’ way, by differential equations and material parameters (dielectric constant) entering these equations. We would tend to describe cables with amplifiers in a discontinuous way.
Figure 5-14
The advantage of nondispersing solitons in telecommunications.
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However, with a long-enough cable, it would be possible to average over the amplifiers and to incorporate the mean value into the material parameters. There is a solid-state analog to the continuous cable-plus-amplifier description: the jellium model. In jellium the positive ions of a solid are not localized at the lattice sites but homogeneously spread over the volume. Takayama et al. [31] have studied polyacetylene solitons in the continuum model. Solitons are quasiparticles corresponding to solitary waves, similar to phonons corresponding to sound waves and photons to electromagnetic waves (light waves). Many properties of solitons remind us of those of elementary particles, for example, the existence of solitons and anti-solitons. Speaking in terms of step-functions, particles and anti-particles are easily identified as up-steps and down-steps, as illustrated in Figure 5-15. In charge-density waves they correspond to phase-slip centers of +180 and –180. Figure 5-16 shows a soliton and an anti-soliton for polyacetylene chains. Note that it is acceptable to call the first-noticed conjugational defect a soliton or an anti-soliton, but once we have made our choice all further particles on the same chain are defined either as solitons or as anti-solitons. There will always be some solitons on a polyacetylene chain, as a consequence of synthesis (typical ESR results are 400 radicals per 106 carbon atoms). Additional solitons can only be created in pairs, as soliton/anti-soliton pairs (conservation of particle number). From a chemical point of view, creation in pairs is evident, because it involves the breaking of a bond, which leaves two dangling bonds (or the splitting of an electron pair which leaves two radicals). Particle/anti-particle annihilation corresponds to closing the bond again. Figure 5-17 shows soliton migration. A soliton moves by pairing to an adjacent electron and leaving its previous partner unbound. Note that in this way the soliton always occupies odd-numbered sites; the even-numbered sites are reserved for anti-solitons. Figure 5-18 shows the results of a quantum chemical calculation on a polyacetylene chain with 61 carbon atoms [32]. Because polyacetylene synthesis starts with
Figure 5-15
Particles and anti-particles in step functions.
Figure 5-16
Soliton and anti-soliton in polyacetylene.
5.4 Solitons
Solitons and anti-solitons. The presence of a soliton on a polyene chain allows the classification of the lattice sites as even and odd, and of all further conjugational defects as solitons and anti-solitons.
Figure 5-17
Charge distribution of a conjugational defect on a polyacetylene chain of 61 carbon atoms [32].
Figure 5-18
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5 Conducting Polymers: Solitons and Polarons
acetylene, with the carbon atoms already paired, it is unlikely that polymer chains with an odd number of carbon atoms occur; however, calculations are certainly possible for these chains. Evidently, an odd-numbered chain must have an unpaired electron, i.e., there must be a soliton in the ground state. Figure 5-18 shows the spatial distribution of the electron density at the midgap state as well as the density in the valence (p) and the conduction (p*) bands. Note that the density at the midgap state is built up at the expense of the p and p* densities. Furthermore, we see that the soliton extends over about 20 lattice sites (this is the thickness’ of the domain wall); in addition, the midgap density accumulates only on every second lattice site, in this case on the even sites, leaving the odd sites for anti-solitons. Several attempts have been made to obtain experimental information on the spatial extension of the soliton wave function. The most reliable results are obtained from ENDOR (electron nuclear double resonance) and from pulsed triple resonance experiments. Grupp et al. [33] have fitted their data to the following spin distribution function: 1 j [g cos2 (jp / 2) – u sin2 (jp / 2)] (9) ri = sec h2 l l where j is the lattice site index, 2l is the full width at half maximum of the soliton wave function, and g and u represent the population of the even- and odd-numbered lattice sites, respectively. The fit yields l = 11 and u/g = 0.43. The corresponding wave function is shown in Figure 5-19. If there is electron-lattice coupling only (see Chapter 4), u/g indicates that the soliton wave function is non-zero also at the sites reserved for anti-solitons. The observed anti-solitonic admixture can be taken as experimental evidence of electron correlations. A soliton is free to move, because the total energy of the system does not depend on the position of the soliton (if the chain is long enough – in short chains, end effects push the soliton to the center). A more detailed analysis shows that the soliton has to overcome an energy barrier when moving from one site to the next (more exactly, to the next-but-one, because it moves in double steps). The existence of this kind of barrier leads to self-trapping’ of the soliton on the chain. Free soliton
Distribution of the spin density of a soliton defect on a polyene chain with l = 11 and u/g = 0.43 used for simulations of ENDOR spectra [33]. Figure 5-19
5.4 Solitons
motion is possible only in polyacetylene. Only here does the energy of the system not depend on the position of the soliton; in the other polymers shown in Figure 2-28 motion of a soliton changes the energy. Polyacetylene is said to have a degenerate ground state’. Conjugational defects in nondegenerate ground state polymers’ are discussed in Section 5.6. One can estimate the effective mass of a soliton. Evidently this is not the free electron mass, because it is not the electron that is moving. The defect is moving, and this means that the electrons change their partners, not their positions. Of course, this requires a slight displacement of the ions to invert short and long bond lengths. These ionic displacements determine the effective mass of the soliton. Within this model Su et al. [27] estimated the effective mass of a polyacetylene soliton to amount to about six free electron masses. There are neutral as well as positively and negatively charged solitons. So far, we have encountered only neutral solitons, for example, in Figure 5-16. The dangling bond consists of an electron sitting on a lattice site that is also occupied by a positively charged ion (carbon nucleus plus closed electron shell). Electronic and ionic charges compensate at the defect as they do in the undisturbed part of the chain. However, in redox reactions the defect is more sensitive than the rest of the chain. The first electron to be removed during oxidation is that of the dangling bond, and the first additional electron in a reduction reaction also goes to the dangling bond. Figure 5-20 shows the p and p* bands (hatched areas) and the soliton midgap state. This state can be occupied by up to two electrons (as with any orbital, because of spin-up and spin-down degeneracy). No matter how many electrons occupy the midgap state, there is always a lattice distortion at the defect interrupting the band alternation. The ionic charge is compensated only when there is one electron at the defect. If there is no electron, the ion is left uncompensated and the soliton is positively charged; two electrons overcompensate the ion and the soliton becomes negative.
Spin charge inversion of a conjugational defect. Charged solitons are spinless; neutral solitons carry a magnetic moment.
Figure 5-20
99
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5 Conducting Polymers: Solitons and Polarons
Chemists call the neutral soliton a radical, the positive soliton a carbocation, and the negative soliton a carbanion (note, however, that in our case radical, carbocation, and carbanion are not isolated but incorporated into a conjugated chain, which leads to a modification of the adjacent chain segments, often called relaxation). In Figure 5-20 the spin of the solitons is also marked. The neutral soliton is an unpaired electron with spin either +1/2 or –1/2. On the positive defect there is no electron and hence no spin, and the negative defect has two electrons with opposite spins, so that the net spin is zero again. Notice the famous spin charge inversion’: whenever the soliton bears charge it has no spin and vice versa. This is trivial considering the chemistry, but it is very surprising when thinking of solitons as particles. We have repeatedly emphasized that the approximations in this chapter are an oversimplification; the most exciting result, the soliton, is an oversimplification as well. But simple models have their justifications. Solitons play a similar key role in nonlinear dynamics as the harmonic oscillator does in linear dynamics.
5.5
Generation of Solitons
We have already mentioned that solitons can be generated only in pairs, as solitons and anti-solitons, provided they are not already present from synthesis. During polyacetylene synthesis only very few solitons are created. There should be a soliton on every chain with an odd number of carbon atoms, but there are far fewer odd-numbered chains than even-numbered, because the synthesis starts from carbon pairs (acetylene). In typical polyacetylene there are about 400 radicals per 106 carbon atoms, as shown, for example, by ESR measurements. There are three methods to generate additional solitons: (1) chemical doping, (2) photogeneration, and (3) charge injection. Doping of conjugated polymers is a chemical redox reaction. As mentioned before, some people dislike the term doping’ in this context, because of the apparent difference from doping of silicon or germanium. One difference is the doping level. Polymers are doped up to several percent, whereas typical doping concentrations of classical semiconductors are in the ppm range. Saturation doping of a polymer certainly leads to a new material with another chemical formula (for example, [(CH)7I3]n for iodine-doped polyacetylene instead of (CH)n for undoped polyacetylene), as well as to a new crystallographic structure. In doped silicon we do not describe it as a new material. But in our opinion there are more similarities than differences. Apart from the high doping level, iodine doping of polyacetylene is very similar to lithium doping of silicon: the dopant goes to interstitial sites of the crystalline lattice of the host; charge is transferred from the dopant to the host, thus moving the Fermi level of the host. The main discussion is concerned with the question of whether doping is a continuous or a stoichiometric reaction and whether there is phase segregation between doped and undoped regions, or whether the doped material dissolves’ in the host. Because of the many lattice defects in polymers, which interact with the dopant, probably the problem cannot be solved.
5.5 Generation of Solitons
Figure 5-21
Creation of solitons by chemical doping (oxidation) of a polyene chain.
In Figure 5-21 the process of polyacetylene doping is shown schematically. In the first step a double bond is broken, followed by the transfer of an electron from the polymer chain to the dopant. Thus two solitons are created, one neutral and the other positively charged (p-doping’). The next dopant then reacts with the neutral soliton, because in most p-doping reactions stoichiometry requires the transfer of two electrons, for example: [PA] + 3I2 fi [PA]++ + 2I –3
(10)
In n-doping only one electron is transferred: [PA] + K fi [PA] – + K +
(11)
but nevertheless only very few neutral solitons survive, because they diffuse along the chain and annihilate with anti-solitons from other doping events. In Chapter 6 we will learn that chemical doping changes the conductivity of conjugated polymers by many orders of magnitude. In fact, there is a doping-induced insulator-to-metal transition. This transition is easy to understand: doping generates solitons. Solitons have midgap states. If there are many solitons, there will be many midgap states; the midgap states will interact, and finally the gap will disappear. If a soliton is a local suppression of the Peierls distortion, doping will just reverse the Peierls distortion. From the spatial extension of the soliton wave function over about 7 lattice sites (see Figure 5-19), the stoichiometry of 7:1 for the ratio of CH to I3 in saturation-doped polyacetylene seems quite reasonable. Photogeneration of solitons is illustrated in Figure 5-22. The figure is drawn so as to stress the similarity of electron and hole generation in semiconductors. In a first step, an electron is lifted from the valence (p) band to the conduction (p*) band. Thus an electron hole pair is created. Now the lattice relaxes around the electron
101
102
5 Conducting Polymers: Solitons and Polarons
Figure 5-22
Photogeneration of solitons.
and the hole, i.e., the bond lengths, readjust and a negative soliton S– and a positive + can move in an electric field and + are formed. The solitons S– and S anti-soliton S thus give rise to photoconductivity, or they can recombine. If they recombine radiatively, photoluminescence is generated. Alternatively to what is shown in Figure 522, we could have assumed that in the first step the double bond is cleaved by absorption of light, leading to two dangling bonds. However in this case, it has to be demonstrated that the bond is not only split, but charges are also separated, so that , but two charged solitons S– the final products are not two neutral solitons S and S + [34]. and S Charge injection is also known from semiconductor physics. A semiconductor is sandwiched between two metal electrodes, as shown in Figure 5-23. By choosing
Soliton generation by charge injection. In a first step electrons and holes are injected from the electrodes. These then relax to form solitons and anti-solitons.
Figure 5-23
5.6 Nondegenerate Ground State Polymers: Polarons
metals with appropriate work functions and additionally applying an external voltage, the Fermi levels of the metals are adjusted in such a way that one electrode injects electrons into the p* band and the other extracts electrons from the p band (which is the same as injecting holes into this band). The lattice relaxes around the injected carriers and solitons are formed. Finally the solitons recombine, some of them by emission of light. Provided that the quantum efficiency for radiative recombination is high enough, an organic light emitting device (OLED) is obtained. We should note one very important difference in the band diagram for charge injection with organics as shown, and that is the absence of so-called band bending’ in the system. Because itinerate, or free, charge does not exist within the polymer before it is injected, bands at interfaces are generally represented in a flat-band’ condition. However, defects at the interface or doping in the polymer can drastically alter this situation. Charge injection dynamics and soliton formation across the metal–polymer interface have been studied extensively during the past decade and are still hotly debated. Most importantly though, Friend et al. [35] have been shown – by optical absorption experiments – that solitons are formed before the injected carriers recombine.
5.6
Nondegenerate Ground State Polymers: Polarons
We have already mentioned that degenerate and nondegenerate ground state polyenes exist. Polyacetylene represents the first group, and all other polymers presented in Figure 2-28 belong to the second group. Degenerate’ means that the energy does not change when single and double bonds are interchanged (Figure 5-24). Evidently, for polyacetylene it does not matter whether the double bond is on the left-hand or the right-hand slopes. In polyphenylene, however, interchange of single and double bonds leads from the aromatic state A (three double bonds within the ring) to the quinoidal state B (only two double bonds within the ring but the rings linked by double bonds instead of single bonds). The quinoidal structure has a much higher energy than the aromatic state. In Figure 5-25 a soliton in polyacetylene is compared with a soliton in polyphenylene. Because of the degeneracy in polyacetylene, the position of the soliton does
Figure 5-24
Degenerate and nondegenerate ground state polymers.
103
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5 Conducting Polymers: Solitons and Polarons
A soliton is free to move in polyacetylene, whereas in polyphenylene it is pushed to the chain end by lattice forces.
Figure 5-25
Figure 5-26
Polaron in polyphenylene.
Figure 5-27
Bipolaron in polyphenylene.
not matter energetically. In polyphenylene the soliton separates a low-energy region from a high-energy region. The soliton is of course driven to the chain end, changing the high-energy quinoidal rings into low-energy aromatic rings as it moves. The situation is comparable to magnetic domain walls in an external magnetic field or to dislocations in a crystal under elastic strain: the magnetic or the elastic field drives the defects out of the solid. To stabilize conjugational defects in a nondegenerate ground state polymer we have to create bound double-defects. Such double-defects are called polarons. An example is shown in Figure 5-26. There is a neutral and a positive soliton. These defects are pushed towards each other by the lattice in order to minimize the length of the quinoidal part of the chain. However, solitons and anti-solitons cannot recombine because one single electron cannot form a bond. A defect composed of two positive solitons is shown in Figure 5-27. This species usually is called a bipolaron, since if two polarons meet, the two neutral solitons can form the bond and only the two charged solitons are left over. Figure 5-28 presents the whole particle zoo for polyacetylene. In the particular instance of polyacetylene, there is no evident binding force for polarons (unless we assume that interactions between neighboring chains in a polyacetylene crystal exist). Both the solid-state and the chemical terms for the defects are presented in the figure, for use as a physical-chemical dictionary’. As demonstrated above, the optical signature of a soliton is the midgap state (because of correlation effects, it is not exactly at the midgap position). A polaron is
5.6 Nondegenerate Ground State Polymers: Polarons
Complex conjugational defects constructed from solitons: a physical-chemical dictionary’.
Figure 5-28
characterized by two states in the gap. The emergence of two states can be rationalized to occur through interaction between the midgap states belonging to the soliton components of the polaron (Figure 5-29). The two gap states of the polaron can be occupied by zero, one, or two electrons each, as indicated in Figure 5-30. Depending on the occupancy, the defect is either neutral or charged and does or does not carry a magnetic moment. The two possibilities (low spin and high spin) of accommodating two electrons on a polaron are reminiscent of excitons in molecules and in inorganic semiconductors. Therefore they are called polaron excitons. Inciden-
Two gap states of a polaron, resulting from splitting the solitonic midgap state by the interaction between the two components.
Figure 5-29
105
106
5 Conducting Polymers: Solitons and Polarons
Figure 5-30
The polaron menagerie’.
Soliton confinement is a nondegenerate polymer. If the solitons forming a polaron are pulled apart, a high-energy section of chain is created between the solitons. Soon the energy stored will be large enough to break a double bond and create two new solitons.
Figure 5-31
tally, polarons are also known in semiconductor physics: an electron moves through the lattice by polarizing its environment, thus becoming a dressed’ electron. This is a lattice distortion that is small compared to the polaron defect in conjugated polymers. The fact that solitons cannot move freely in a nondegenerate polymer is often called soliton confinement’. Soliton confinement has an appealing analogy to quark confinement in elementary particle physics. If we tried to separate the two parts of a polaron by pulling them apart (Figure 5-31), we would create a large chain section with high energy. If the high-energy section is sufficiently long (a few quinoidal rings will do), enough energy is generated to split a double bond and to create two new solitons, which then form new and smaller polarons with the previous solitons, so that the major part of the chain can again relax into the low-energy form. This happens far earlier than the possible cutting of the polymer chain into pieces and isolation of a soliton in one of the fragments. Similarly, quarks cannot be pulled out of elementary particles.
5.7
Fractional Charges
Another relation to elementary particles is the concept of fractional charges. Free particles carry integer charges, usually +e or –e, where e is the electronic charge. Quarks can have charges like 2/3e. If we put electrons into a solid, they of course keep their charge, but they use it to compensate the ionic charges, so that the solid
5.7 Fractional Charges
becomes neutral. If Bloch waves and wave packages out of Bloch waves are constructed, is it trivial that a wave package must have an integer charge? And what about solutions of nonlinear differential equations: must they carry integer charges? Figure 5-32 again shows polyacetylene in the extended version. In the next step, two negative solitons (marked by arrows) are added to the chain. The solitons are 180 phase-slip centers and they amount to the insertion of an additional single bond. To keep the chain length constant, two bonds are removed at the end, a single and a double bond. Counting the electrons, we notice that the newly formed chain has two electrons fewer than the former chain. Since the former chain was neutral, the new chain bears a charge of +2e. If this amount of charge is equally distributed over the two defects, each defect carries a charge of +e. For polyacetylene the charge at a defect is evident, so such a complicated electron-counting procedure is not necessary. The lower part of the figure shows the hypothetical polyfractiolene’. This is a hypothetical polymer with double bonds separated by two single bonds. Polyacetylene without a Peierls distortion would have a half-filled band. The band in polyfractiolene would be filled to 1/3 (fewer electrons than in polyacetylene). Insertion of an extra single bond will again be a phase-slip center, in this case, of 120. Counting the electrons of polyfractiolene in the same way as in polyacetylene shows that two charges are to be shared by three defects! It is not quite clear whether looking at individual solitons is allowed. If all solitons created simultaneously must be viewed together, one might argue that fractional charges are just a bookkeeper’s trick. On the other hand, fractional charges on solitons are a nice analogy to fractionally charged quarks. Additional relationships between charge fractionalization in onedimensional metals and relativistic field theory are discussed by Jackiw and Schrieffer [36]. The similarity between fractionally charged solitons and the fractional quantum Hall effect has been pointed out by Schrieffer [37,38]. Polyfractiolene cannot be synthesized. The bond sequence shown in Figure 5-32 can be stabilized only by inserting protons in the chain, and the proton arrangement
Figure 5-32
Electron counting to demonstrate fractional charges on one-dimensional chains.
107
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5 Conducting Polymers: Solitons and Polarons
would break the required symmetry. But in the crystalline one-dimensional metals mentioned in Sections 2.2, 2.4, and 2.7, band filling of 1/3 or 1/4 exists, and in these systems fractional charges might occur. Bak [39] has even proposed an experiment to obtain evidence for their existence: via the current flowing through a conductor the number of charges is measured and the number of carriers is inferred from analysis of the noise spectrum of the current. Divide the number of charges by the number of carriers, and so on. The proposal was made in 1982. Today one might go a step further and propose using the tip of a scanning tunnel microscope as a local probe.
5.8
Soliton Lifetime
Most measurements of conducting polymers are fairly simple: optical spectroscopy, electrical conductivity, magnetic susceptibility. But the theoretical concepts are so exciting that many scientists engage in very sophisticated experiments. Critics then might ask: Why do you waste such beautiful experiments on such ill-defined samples?’ (And compared to perfect silicon single crystals, conjugated polymers certainly are ill-defined.) An example of this type of experiment is picosecond photoconductivity in polyacetylene. We report these experiments in this book, because we want to give you an impression of the variety of the solid-state physics Armada that can be directed towards new materials – and because we were involved in these experiments. The idea of the experiment is easily sketched: Take stretch-aligned polyacetylene, i.e., polyacetylene in which the chains have been more or less aligned by stretching (certainly a large single crystal would be preferred, but we don’t have that – a stretch-aligned film is the best we can get). Electrodes are evaporated onto the sample and an electric field is applied. Short light pulses are effected on the sample, which photogenerate solitons as explained in Section 5.5. Charged solitons are displaced by the electric field, a photocurrent flows, and the decay of the photocurrent with time can be measured. The photocurrent will flow as long as the charge carriers move. If the experiment is carried out on a silicon single crystal the carriers move until they hit the electrodes at the crystal surface. From the decay of the photocurrent, we obtain the time that the carriers need to cross the crystal (time of flight’). From the electrode separation we know the distance the carriers migrate, and their velocity can be calculated. Division by the electrical field yields the mobility, which is an important material parameter. If the crystal has many imperfections that act as traps, the carriers get trapped before they reach the electrodes. If we know the average distance between traps, we can again calculate the mobility from the decay of the photocurrent. In a very imperfect semiconductor the carrier lifetime is only a few picoseconds. Such a fast decay of the photocurrent cannot be measured directly, because no ammeter would respond fast enough. The fastest instrument today is the preamplifier of an oscilloscope, which has about 100 ps rise time. Therefore picosecond
5.8 Soliton Lifetime
Figure 5-33
Optical correlation experiment for fast photoconductivity decay.
photoconductivity experiments use optical correlation methods. The sample is regarded as a fast optical switch; two such switches are put in series. They are hit by light pulses, one delayed with respect to the other (Figure 5-33). If both pulses arrive simultaneously, both switches are on and the oscilloscope records a signal (of course, distorted by the slow response of the scope, but that does not matter). If the delay is so large that the first switch is off already and the second is not yet on, there is no signal, but if the opening time of the switches overlap, there is some signal. By changing the delay between the pulses, fast events can be probed (pump-and-probe technique’). The two pulses are usually generated by shining a laser onto a beamsplitter, followed by change of the optical path length in one of the branches. Picoseconds circuitry is somewhat more complicated than indicated in Figure 533, mainly because 50 X wave resistance has to be maintained throughout the circuit to avoid pulse distortions. Therefore the switches are constructed as part of the 50 X strip-line (shown in Figure 5-34). One of the gaps in the copper strips is covered by ion-bombarded silicon, the other by a polyacetylene flake. The width of the gaps is about 20 lm. The experimental results are shown in Figure 5-35 [40]. The signal intensity is plotted over the pulse delay. The signal rises within 2.5 ps and decays within 6 ps. The analysis shows that the rise time is mainly determined by the silicon switch, the decay by the polyacetylene sample. From this experiment we know that the photoconductive decay time is about 6 ps. From other optical experiments we estimate the Schubweg, i.e., the distance the solitons migrate before they are trapped; we obtain a mobility of about 1 cm2 V-1 s–1 for charge carriers in polyacetylene. This is in remarkable agreement with theoretical calculations for solitons (and polarons) in polyacetylene [41]. The short lifetime of solitons in polyacetylene can be used to build an ultrafast holographic computer. Photogeneration of solitons makes polyacetylene a nonlinear optical material: the refractive index depends on the number of solitons. Consequently, holograms can be stored in polyacetylene. These holograms fade within a few picoseconds, but this time is long enough to compare two holograms (holographic pattern recognition). After a few picoseconds the polyacetylene is ready to analyze the next hologram. This way, a computer holding the world record in data throughput per time can be constructed [42]. We will learn more about this issue in Chapters 9 and 10.
109
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5 Conducting Polymers: Solitons and Polarons
Two fast optical switches built into 50 X strip-lines for a picosecond correlation measurement to study the fast decay of the photocurrent in polyacetylene (soliton lifetime).
Figure 5-34
Optical correlation measurement of picosecond photoconductive decay in polyacetylene.
Figure 5-35
References and Notes
References and Notes 1 H. Kuzmany, M. Mehring, and S. Roth. Elec-
tronic Properties of Polymers and Related Compounds. Springer Series in Solid-State Sciences 63; Springer Verlag, Berlin-Heidelberg,1985. 2 H. Kuzmany, M. Mehring, and S. Roth. Electronic Properties of Conjugated Polymers. Springer Series in Solid-State Sciences 76; Springer Verlag, Berlin-Heidelberg, 1987. 3 H. Kuzmany, M. Mehring, and S. Roth. Electronic Properties of Conjugated Polymers. III. Basic Model and Applications. Springer Series in Solid-State Sciences 91; Springer Verlag, Berlin-Heidelberg, 1989. 4 H. Kuzmany, M. Mehring, and S. Roth. Electronic Properties of Polymers: Orientation and Dimensionality of Conjugated Systems. Springer Series in Solid-State Sciences 107; Springer Verlag, Berlin-Heidelberg, 1992. 5 J. Przyluski and S. Roth. Electrochemistry of Conducting Polymers. Materials Science Forum 21, Trans Tech Publications, Aedermannsdorf, 1987. 6 J. Plocharski and S. Roth. Electrochemistry of Conductive Polymers. Materials Science Forum 42, Trans Tech Publications, Aedermannsdorf, 1989. 7 Selected Contributions are printed in Synthetic Metals 45/3 (1991). 8 J. Przyluski and S. Roth. Conducting Polymers: Transport Phenomena. Materials Science Forum 122, Trans Tech Publications, Aedermannsdorf, 1993. 9 J. Przyluski and S. Roth. New Conducting Materials with Conjugated Double Bonds: Polymers, Fullerenes, Graphite. Materials Science Forum, Trans Tech Publications, Aedermannsdorf, 1994. 10 M. Aldissi. Intrinsically Conducting Polymers: An Emerging Technology. Kluwer, Dordrecht, 1993. 11 W.R. Salaneck, D.T. Clark, and E.J. Samuelsen. Science and Applications of Conducting Polymers. Adam Higler, Bristol, 1991. 12 T.A. Skotheim. Handbook of Conducting Polymers Vols. 1&2, Marcel Dekker, New York, 1986. 13 H. Kiess. Conjugated Conducting Polymers. Springer Series in Solid-State Sciences 102, Springer Verlag, Heidelberg, 1992.
14 J. Przyluski. Conducting Polymers: Electro-
15
16
17 18
19
20 21
22
23
24
25
26
27
chemistry. Sci-Tech Publications, Vaduz, 1991. J.C.W. Chien. Polyacetylene: Chemistry, Physics, and Material Science. Academic Press, Orlando, 1984. H.J. Mair and S. Roth. Elektrisch leitende Kunststoffe. 2nd Edition, Carl Hanser Verlag, Mnchen-Wien, 1989. S. Roth and H. Bleier. Solitons in polyacetylene. Advances in Physics 36, 385 (1987). A.J. Heeger, S. Kivelson, J.R. Schrieffer, and W.P. Su. Solitons in conducting polymers. Reviews of Modern Physics 60, 781 (1988). Y. Lu. Solitons and Polarons in Conducting Polymers. World Scientific Publisher, Singapore, 1988. K.A. Chao. on a seminar tour through Europe, 1984. D. Baeriswyl. Conjugated conducting polymers. H. Kiess (Ed.), Springer Series in SolidState Sciences 102, 7, Springer Verlag, Heidelberg, 1992 H. Kuhn. Free-electron model for absorption spectra of organic dyes. Journal of Chemical Physics 16, 840 (1948); H. Kuhn. A quantum-mechanical theory of light absorption of organic dyes and similar compounds. Journal of Chemical Physics 17, 1198 (1949). H.C. Longuett-Higgins and L. Salem. Alternation of bond lengths in long conjugated chain molecules. Proceedings of the Royal Society of London A 251, 172 (1959). L. Salem. The Molecular Orbital Theory of Conjugated Systems. Benjamin, London, 1966. J.A. Pople and S.H. Walmsley. Bond alternation defects in long polyene molecules. Molecular Physics 5, 15 (1962). C.K. Chiang, C.R. Fincher, Jr., Y.W. Park, A.J. Heeger, H. Shirakawa, E.J. Louis, S.C. Gau, and A.G. MacDiarmid. Electrical conductivity in doped polyacetylene. Physical Review Letters 39, 1098 (1977). M.J. Rice. Charged II-phase kinks in lightly doped polyacetylene. Physics Letters A 71, 152 (1979). W.P. Su, J.R. Schrieffer, and A.J. Heeger. Solitons in polyacetylene. Physical Review Letters 42, 1698 (1979).
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28 S.A. Brazovskii. Electronic excitations in the
29 30
31
32
33
34
35
Peierls–Frhlich state. JETP Letters 28, 606 (1978); S.A. Brazovskii. Self-localized excitations in the Peierls–Frhlich state. Soviet Physics JETP 51, 342 (1980). D. Baeriswyl. Conducting polymers: solitons or not? Helvetica Physica Acta 56, 639 (1983). S. Roth, K. Ehinger, K. Menke, M. Peo, and R.J. Schweizer. Polyacetylene: la chasse aux solitons. Journal de Physique C 3, 69 (1983). H. Takayama, Y.R. Lin-Liu, and K. Maki. Continuum model for solitons in polyacetylene. Physical Review B 21, 2388 (1980). S. Stafstrm and K.A. Chao. Soliton states in polyacetylene. Physical Review B 29, 2255 (1984). A. Grupp, P. Hfer, H. Kss, M. Mehring, R. Weizenhfer, and G. Wegner. Electronic properties of conjugated polymers (Kirchberg II). In: Springer Series of Solid-State Sciences 76, H. Kuzmany, M. Mehring, and S. Roth (Eds.), Springer Verlag, Heidelberg, 1987. W.P. Su and J.R. Schrieffer. Soliton dynamics in polyacetylene. Proceedings of the National Academy of Sciences of the United States of America 77, 5626 (1980). R.H. Friend and J.H. Burroughes. Charge injection in conjugated polymers in semiconductor device structures. Faraday Discussions of the Chemical Society 88, 213 (1989). J. Kanicki. Polymeric semiconductor contacts and photovoltaic applications. In: Handbook of Conducting Polymers. T. Skotheim (Ed.), Dekker, New York, 1986. J.H. Burroughes, D.C. Bradley, A.R. Brown, R.N. Marks, K. Mackay, R.H. Friend, P.L. Burns, and A.B. Holmes. Light-emitting
diodes based on conjugated polymers. Nature, 347, 539 (1990). M.H. Pilkuhn and W. Schairer. Light emitting diodes. In: Handbook of Semiconductors. T.S. Moss (Series Ed.), C. Hilsum (Volume Ed.), Vol. 4, Elsevier-North Holland, Amsterdam, 1993. G. Leising, S. Tausch, and W. Graupner. Fundamentals of electroluminescence in paraphenylene-type conjugated polymers and oligomers. In: Handbook of Conducting Polymers. T. Skotheim, R. Elsenbaumer, and J. Reynolds (Eds.); Marcel Dekker, New York, 1997. 36 R. Jackiw and J.R. Schrieffer. Solitons with fermion number 1/2 in condensed matter and relativistic field theories. Nuclear Physics B 190, 253 (1981). 37 J.R. Schrieffer. The quantum hall effect: relation to quasi one dimensional conductors. Synthetic Metals 17, 1 (1987). 38 S. Kivelson, C. Kallin, D.P. Arovas, and J.R. Schrieffer. Cooperative ring exchange theory of the fractional quantized Hall effect. Physical Review Letters 56, 873 (1986). 39 P. Bak. Fractional charges and frequency-dependent currents in charge-density-wave systems. Physical Review Letters 48, 692 (1982). 40 J. Reichenbach, M. Kaiser, J. Anders, H. Byrne, and S. Roth. Picosecond photoconductivity in (CH)x measured by cross-correlation. Europhysics Letters 18, 251 (1992). 41 S. Jeyadev and E.M. Conwell. Polaron mobility in trans-polyacetylene. Physical Review B 35, 6253 (1987). 42 D.P. Foote. ME-News No. 13, M. Petty (Ed.), Center of Molecular Electronics, Durham, UK, 1992.
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6
Conducting Polymers: Conductivity 6.1
General Remarks on Conductivity
The section on conductivity in any textbook usually starts with Ohm’s law: I=
V R
R=
V I
V=RI
(1)
where I is the current, V the voltage, and the proportionality constant R is the resistance. The reciprocal of the resistance is the conductance. Not all conductors obey Ohm’s law. Gas discharges, vacuum tubes, and semiconductors often deviate from Ohm’s law, as do most of the one-dimensional conductors discussed in this book. When resistivity or conductivity values are quoted, the current or voltage range in which these values were obtained has to be specified. In an ohmic material the resistance is proportional to the length l of the sample and inversely proportional to the sample cross section A: R=
rl A
(2)
where r is the resistivity, measured in Xcm. Its inverse is the conductivity (r–1 = r). The unit of conductance is Siemens (S). Siemens is the reciprocal of ohm and someX times is written as X–1, , or mho. The unit of conductivity is S cm–1 (in SI units m should be used rather than cm, in publications both m and cm are common and attention must be paid to the figure and table captions). In many solids r depends on the crystallographic direction and hence is not a scalar quantity but a tensor. Such solids are said to have anisotropic conductivity. The combination of high conductivity in one direction and zero conductivity in the two perpendicular directions leads to one-dimensional solids (see the anisotropy approach in Section 1.2). In KCP (Section 2.2) the conductivity parallel to the platinum chains is about 200 times larger than the conductivity in the perpendicular direction [1]; in SbF5-intercalated graphite the conductivity within the ab plane can be up to 1 · 106 times that in the c direction [2]. The anisotropy of highly conducting stretch-aligned polyacetylene has been determined to be r||/r^ = 25 [3]. One-Dimensional Metals, Second Edition. Siegmar Roth, David Carroll Copyright © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30749-4
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Most of our concepts of electricity are derived from the theory of fluids. This is already evident in the terminology: current, cross section, source, drain. Ohm’s law is an immediate consequence of this. The fluid (or free-electron gas) model was introduced by Drude [4] only three years after Thomson’s discovery of the electron. Drude applied the kinetic theory of gases to a metal, which he considered a gas of electrons. In his model the conductivity is given by r=
ne2 s m
(3)
where n is the electron density, e the charge of an electron, m its mass, and s an effective collision time (relaxation time). Introducing the mobility: l = es / m
(4)
we get: r = nle
(5)
Replacing the free-electron mass m in Eqs. (3) and (4) with the effective mass m* leads to the nearly free’ electron model (Section 3.4). The terminology of fluids is retained, and most of solid-state physics (in particular the collision time s) is put into the asterisk in m*. The mobility is related to the diffusion constant D by the Einstein relation l = eD/kT. With the help of theoretical physics we can demonstrate why electrons in a solid behave like a fluid. We have already outlined the basic steps in the previous chapters: Bloch waves, wave packages. This is the elegant part of the theory. A detailed explanation of electrical resistance is much more cumbersome. Of course, assuming partially filled bands, the resistance is a result of deviations from the crystal periodicity (defects), interactions with quasi-particles such as phonons and others, and interactions between the electrons themselves (correlation effect). Not all of the ingenious physicists enjoyed this type of research. As an example we reproduce part of a letter from Wolfgang Pauli to Rudolf Peierls (Figure 6-1). Of course, the dirt’ within a conducting wire of very small dimensions, as we are discussing here, has led to a rather important understanding of the role of general scattering in disordered systems. In very small gold wires, a simple application of the Drude model with randomly dispersed impurities can easily explain repeatable fluctuations in the current–voltage characteristics of the system. The analogy between nuclear scattering and these mesoscopic, universal conductance fluctuations is intriguing. [5] Often solids are classified by their conductivity at room temperature: for typical conductors the conductivity is greater than several thousand S cm–1, for typical insulators it is less than some 10–12 S cm–1, and for semiconductors it is in between. Probably more significant is the classification by the temperature coefficient of the resistance. For this, the resistance is plotted versus the temperature: a positive slope indicates metallic’ materials, a negative slope implies a semiconductor (or an insu-
6.1 General Remarks on Conductivity
Figure 6-1 Part of a letter from Wolfgang Pauli to Rudolf Peierls. A copy of this letter can be found in the coffee corner of the Max Planck Institute for Solid State Research in Stuttgart.
The relevant lines read: “Der Restwiderstand ist ein Dreckeffekt und in Dreck soll man nicht whlen.” (Residual resistance is a filth effect, and in filth one should not root about.)
lator). However, this also is only a very rough classification and does not necessarily agree with the band-structure point of view, where metals are characterized by partially filled conduction bands but semiconductors and insulators have only completely filled and completely empty bands. To learn more about the mechanism of electrical conductivity, the functional dependence of the conductivity on temperature must be determined. Generally speaking, this is the first quantitative determination an experimentalist would use when faced with a novel conductor (like the systems discussed in this text) and, indeed, historically this is what was used in conducting polymers. For systems in which it is expected that a small energy barrier exists to allow charge carriers to flow (or be generated), it is reasonable to guess’ a functional dependence expressed by some power law such as r Tn
(6)
In fact, as we will see, this is theoretically expected for intersoliton hopping’ in slightly doped polyacetylene (see Section 6.6). Alternatively, in some systems, such as crystalline semiconductors, where a well-defined charge reservoir is present, an exponential function of activated behavior may be appropriate: r r0 exp [–DE/kT]
(7)
Finally, there are occasions for considering a soft’ exponential functional form for the temperature dependence, such as r r0 exp [–(T0/T)c]
(8)
with c = 1/2, 1/3, or 1/4. This is particularly applicable in amorphous semiconductors, moderately doped polymers, and other systems with hopping conductivity and charge localization (Section 6.6). In fact, a given system may well express all three of these functional forms, and so a careful microscopic analysis must generally be
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coupled with macroscopic transport determinations to decide on the most applicable model. To decide on the functional dependence it is not sufficient to change the temperature by several percent (e.g., to warm the sample from room temperature to 100 C). Temperature changes of several orders of magnitude are needed. Large temperature changes are more easily obtained by cooling than by heating. Cooling from room temperature to the temperature of liquid helium is a change of nearly two orders of magnitude (from 300 K to 4.2 K), heating from room temperature to 1000 K is only a factor of three, and at 1000 K many solids, in particular most organic solids, decompose. As a consequence, electrical transport measurements require the use of liquid nitrogen or, usually, liquid helium. As an example of the temperature dependence of the electrical conductivity of various materials, Figure 6-2 shows a data compilation after Kaiser [6]. The format of the figure demonstrates an important fact about the electrical conductivity: the conductivity is the parameter with the largest variability range in solid-state physics. Although it is a logarithmic plot and the insulators are not included, the graph has to be much greater in height than in width to accommodate the conductivity values. (The ratio between the room-temperature conductivity of insulators and of copper is the same order of magnitude as 1 km to the diameter of the universe in kilometers!)
Figure 6-2 Comparison of the conductivities of metals (solid lines) and various doped polymers (dashed lines) [6].
6.1 General Remarks on Conductivity
The solid lines in Figure 6-2 represent metals’. Copper and platinum are wellbehaved metals. Their conductivity increases upon cooling, up to a saturation point at about 10 K. This increase is a result of freezing out lattice vibrations. The cold lattice is more perfect than the warm lattice and consequently there is less resistance. Below about 10 K, collisions of electrons with impurity atoms are more important than collisions with phonons. It is not possible to freeze out the impurity atoms. This temperature-independent resistance is called the residual resistance. The resistance of platinum above about 20 K is fairly linear, and in this range platinum can be used as a thermometer. Mg–Zn and Ca–Al are amorphous (glassy) metals. They have no crystal periodicity and their thermal properties (lattice vibrations) tend to be dominated by higherenergy phonons. Thus, there is little variation in conductivity or resistivity upon cooling. However, as the temperature is raised significantly, effects of the higher-energy phonons can be observed. The dashed lines in Figure 6-2 refer to conducting polymers at various doping concentrations. Note that highly conducting polymers behave very similarly to glassy metals. Polymers with lower conductivity show a negative temperature coefficient in resistivity (positive in conductivity), as do semiconductors. However, the decrease in conductivity upon cooling is much slower than for crystalline semiconductors. Data for silicon, for example, would be outside the range in Figure 6-2. In crystalline semiconductors phonons are frozen out in the same way as in metals, but the freezing out of phonons (approximately linear with temperature) is by far overcompensated by the freezing out of electrons (exponential course). Conducting polymers behave more like amorphous semiconductors, in which the electrons do not move in bands but are located at specific states in the gap. They hop between these localized states. Hopping’ is a shorter term for phonon assisted tunnel-
Figure 6-3 Temperature dependence of the conductivity of a highly doped (a) and a moderately doped (b) polyacetylene sample. Note the different asymptotic behaviors as the temperature approaches zero.
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ing’ [7]. As in crystalline semiconductors, the assisting effect of the phonons is more important than their destructive effect, but in conducting polymers the temperature dependence is smoother, because the phonons do not have to excite the electrons across the gap. They act between localized states within the gap. The excitation energies are smaller and, in addition, the distribution of excitation energies is continuous. Hopping conductivity is discussed in more detail in Section 6.6. At room temperature there is not much difference between amorphous metals and amorphous semiconductors. Actually, the temperature coefficient is not the correct criterion for distinction. More relevant is the asymptotic behavior as the temperature approaches zero: in a metal the conductivity stays finite at T fi 0, in a semiconductor or an insulator r fi 0 as T fi 0. As an example, the conductivity of moderately doped and highly doped polyacetylene is presented in Figure 6-3. In the moderately doped sample the conductivity vanishes at absolute zero, the sample stays finite, and the sample remains metallic’. Figure 6-2 also shows the behavior of a superconductor (Y–Ba–Cu–O, dotted line). The resistance disappears at Tc ~ 100 K. (In a superconductor theory requires the conductivity to be infinite. Of course, infinity cannot be measured, but r > 1024 S cm–1 has been observed.)
6.2
Measuring Conductivities
To measure conductivity seems to be quite simple: take an ohmmeter from the shelf and connect it with two alligator clips to the sample. In many cases this will do, but in many others there will be difficulties. The major problems are: .
. .
The conductivity, which can vary over a large range, either from sample to sample or within a sample if the temperature, the pressure, or the doping level is changed. Effect of contacts on the measurement. Homogeneity and anisotropy of the sample.
Actually, measurements are simple only when the resistance of the sample is in the range of several kiloohms (kX) to several megaohms (MX). Here, a current of several milli- or microamperes is passed through the sample and the corresponding voltage is around 1 to 10 Volts. Larger currents easily cause the sample to warm up by Joule heating and smaller voltage signals are difficult to separate from the thermopower and contact voltage at interconnects. For a sample with very low resistance, an ohmmeter would just determine the resistance of the leads. With samples of very low conductivity, only leakage currents through the cable insulation would be measured, rather than currents through the sample. To avoid lead and contact resistances for low-resistance samples, the fourlead method’ is favorable, where current leads are separated from voltage leads. An example is shown in Figure 6-4. Four thin gold wires are glued to the sample with silver paste. The current I is applied through the outer leads and the voltage V is picked up at the inner leads (voltage probes).
6.2 Measuring Conductivities
Figure 6-4 Four-lead method on a needle-like sample. Four thin gold wires are glued to the sample with silver paste.
The resistance is calculated using Ohm’s law and the resistivity is obtained from Eq. (2), where l is the distance between the voltage probes. In Figure 6-5 a circuit equivalent to the one in Figure 6-4 is shown. The current I passes through the sample and creates a voltage drop RxI. The equivalent resistors R1 and R2 represent the resistance of the leads and the contact resistance from lead to sample, respectively (which can be much higher than any other resistance in the circuit). There are voltage drops R1I and R2I; the current source sees’ the sum R1I + RxI + R2I. The voltmeter, however, measures the voltage drops R3I’ + Rx(I + I’) + R4I’, where I’ is the current in the voltmeter circuit and I’ << I, so that it actually determines Rx even if R3, R4 = Rx. Naturally, nothing is as easy as it seems. For an organic crystal sample, the contact glue has to be carefully chosen so as not to dissolve the sample. If the sample contains iodine, as in iodine-doped polyacetylene, silver paste should be replaced by gold paste. Silver reacts with iodine to form highly insulating silver iodide, resulting in extremely high values for R1 and R2. Thus, not enough current can be passed through the sample to yield sufficiently large voltage signals. Organic crystals are often very brittle and break on cooling due to elastic tension. These tensions can be avoided by using very thin gold wires and thermally annealing the wires before pasting them to the sample. In conducting polymers brittleness is less severe and it is often sufficient to just press the sample against four equidistant platinum wires (Figure 6-6).
Figure 6-5
Equivalent circuit for four-lead method.
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Figure 6-6 Simple sample holder for measuring the conductivity of conducting polymers by the four-lead method. Typical sample size 2 mm 8 mm, 100 lm thick. The platinum wires are about 0.2 mm in diameter and
squeezed into groves of the bottom plate. For room-temperature measurements, the top and bottom plates can be made of acrylic, at low temperatures ceramic or sapphire plates are advisable.
A fairly large error, from the geometrical factor l/A, affects the determination of resistivity (Eq. (2)) for several reasons: crystals are often oddly shaped, the thickness of thin films is difficult to measure, the distance between two blots of silver paste is ill-defined. Semiconductor physicists have developed more accurate methods – the four-point method and the van der Pauw method [8]. The four-point method is indicated in Figure 6-7. It is a special instance of the four-lead method discussed above. Here, the contacts must be point contacts. Four point contacts are equidistant and in a straight line. They are pressed against a plane surface of the sample. The sample must be much larger than the contact arrangement and it must be infinitely’ thick. The geometrical factor can then be obtained by integrating over all current paths, and the resistivity is r = 2p d
V I
(9)
where I is the current through the outer contacts, V the voltage drop at the inner contacts, and d the distance between two contacts. For the van der Pauw method, a thin sample of any shape is used. Here the only requirement is that it should be singly connected’, i.e., it should not have any holes. Four contacts are applied close to the edge, as indicated in Figure 6-8. The resistivity is then calculated using the equation:
Figure 6-7
Four-point probe method for conductivity measurements.
6.2 Measuring Conductivities
Figure 6-8
Van der Pauw method for conductivity measurements.
pd R12;34 þR23;41 R12;34 r= f ln2 2 R23;41
(10)
where d is the thickness of the sample, and R12,34 is the voltage between contacts 1 and 2 divided by the current through contacts 3 and 4. R23,41 is obtained by cyclic interchange of the contacts. f ( ) is a function of the ratio of the measured resistances, which varies between 1 and 0.2 when R12,34 / R23,41 is between 1 and 103. Tabulated data for f are found in van der Pauw’s original paper [9], as well as in several handbooks [8]. A special instance of the van der Pauw method is the use of a square sample with contacts at the corners (Figure 6-9). Because of apparent symmetry reasons Eq. (10) is reduced to: r=
pd R12,34 ln2
(11)
For measurements in a magnetic field Hall bars’ are often used. An example is shown in Figure 6-10. The current passes through contacts 1 and 2, and a voltage drop proportional to the resistance or magnetoresistance is picked up between probes 3 and 4, 5 and 6. The Hall voltage can be determined between 3 and 5 or 4 and 6.
Figure 6-9
Figure 6-10
Square-sample method to measure the conductivity.
Hall bar’ for magnetotransport measurements.
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Microcontact arrangement for local conductivity measurements. The active area can be as small as a few square micrometers.
Figure 6-11
Physics of one-dimensional materials often require measurements to be carried out on microstructure contact arrangements. In some cases the samples might be just a few micrometers in size (growing larger crystals was not successful), in others the samples are inhomogeneous and conductivity microprobes must be applied at different parts of the sample. For these purposes, a pattern of microcontacts can be arranged on top of a substrate by lithographic processes, and the sample is then laid over the contacts. For a thin film of a conducting polymer, it is often sufficient to wet it in an organic solvent and to lay it onto the structured substrate. After drying it sticks tightly to the contacts. Figure 6-11 shows a microcontact arrangement that was used to measure transport through a thin organic heterolayer of a phthalocyanine and a perylene derivative [10]. The anisotropy of quasi-one-dimensional materials poses special problems in transport measurements. All the above methods have been developed for isotropic materials, and the influence of anisotropy has to be carefully evaluated in each instance. The most straightforward procedure for assessing the anisotropy of the conductivity involves cutting out two thin long strips of a stretched polymer film, one parallel and one perpendicular to the stretching direction. These strips are then placed in a sample holder like the one shown in Figure 6-6. A small misalignment will not significantly influence the determination of ri . However, in r^ , we mainly record the contribution of misaligned ri regions. Montgomery [11] published a procedure to deal with anisotropic samples more accurately. (In Section 2.4 a notorious anisotropy pitfall was already mentioned in context with the quest for high-temperature superconductivity in TTF–TCNQ.) In recent years, lithographic techniques, for the formation of contacts directly onto nanoscale systems, have come into vogue. Electron beam lithography and
6.2 Measuring Conductivities
Figure 6-12
Schematic diagram of the setup used for electrical transport measurements [13].
focused ion beam [12] writing have both been extensively used in the creation of nanoscale versions of Figure 6-6. Figure 6-12 is an example of how a single-walled carbon nanotube can be made into a circuit element. Electron beam lithographic techniques, such as that used to create the circuit element above, have the benefit of not damaging the sample severely. Unfortunately, in this technique, lift-off procedures require that the position of the object of study be well-known throughout several procedures of writing and deposition of metals. A clever way of doing this has been worked out, however, which combines atomic force microscopy to determine the positions of the samples on the substrate after they have been laid down. This position is determined relative to larger markers’ that can be easily imaged in the scanning electron microscope used to do the writing [14]. Curiously, this leads to another complication: What happens when the leads and contacts themselves become as small as the sample? We have discussed contact resistance above; however, this is far more subtle. As the leads from the outside world are whittled down’ to create very small (almost atomic) point contacts with the sample, localization, confinement, and back reflection of carriers might well be expressed in the leads. Such circumstances must be analyzed in a far more formal way as worked out by Landaur and Buttiker, where the leads are included as part of the measured system [15]. The actual application of this analysis has been discussed extensively in several excellent texts. In high-resistivity samples it could happen that leaks in the insulation of the cables are measured rather than the resistance of the sample. It is also possible that the current passes along a humidity layer on the sample surface instead of through the inside of the sample. This source of error can be avoided by using a guard ring
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Figure 6-13
Guard ring method for high-resistivity samples.
electrode, as indicated in Figure 6-13. The central electrode and the guard ring are at the same potential, so that no current flows between these two leads and the current between the guard ring and the back electrode does not pass through the ammeter. Many commercial electrometers are supplied with triaxial cable, so that the guard can extend into the cable. Complications may arise from the time constant. If the capacitance of the cable is 10 pF and the sample resistance is 1012 X, the RC time constant is already 10 s, i.e., it takes ten seconds to charge the cable capacitance. This excludes the application of a lock-in technique. During simple DC measurements on high-resistance samples, on the other hand, space charges build up easily from slow electron trapping and detrapping processes, so that the interpretation of the experiment is by no means straightforward. A very serious problem arises from the fact that many materials can be obtained only as powders. The synthetic chemist usually presents a flask with a black powder in the bottom to his physicist friend to find out whether the substance is interesting’. Only for interesting substances are the efforts of growing single crystals worthwhile. Powders have to be compressed into pellets, and transport in pellets is usually not dominated by bulk properties but by the grain walls, which can act either as lowconductivity blockades or as high-conductivity carrier-accumulation regions. Many polymers can be formed into films. In fact, physicists did not become aware of polyacetylene when it was first synthesized as a powder by Natta et al. in 1958 [16]; it only became famous 16 years later, after Shirakawa had prepared films [17] – as history tells, by a postdoc’s mistake in the directions for the catalyst concentration. Although these films look smooth and shiny to the eye, their morphology is far from pleasing to the physicist, as we will see in Section 6.5 (Figure 6-24). For reasons of contact and powder problems, methods not using contacts have been proposed to determine the conductivity. Most important in this respect is the resonance perturbation of a microwave cavity [18–21]. The sample is put into a small glass tube, which is inserted into a microwave cavity. Microwave absorption in the sample changes the Q value of the cavity and the resonance frequency. From these two parameters, the conductivity of the sample can be evaluated. Of course, the conductivity at microwave frequency (e.g., at 1010 Hz) is not necessarily identical to the DC conductivity. Indeed, when conductivity takes place by a hopping mechanism, large differences have been observed, as would be expected [22]. At even higher frequencies the conductivity can be determined from optical absorption [23]:
6.2 Measuring Conductivities
r = x e2/4p
(12)
where e2 is the imaginary part of the dielectric function. Finally if, as stated above, the temperature-dependent conductivity of a material is the first line of enquiry used by the experimental condensed-matter scientist, the second line of enquiry is frequently the use of thermoelectric power measurements. This measurement is particularly good for understanding how the electrical conductivity might be coupled to the lattice vibrations of a system (phonon scattering for instance) although, unlike that presented above, it does require contacts to the sample. Simply, the thermal electric power (thermopower) is generated in a material by establishing a thermal gradient between two electrical leads attached to the sample. It can be visualized in a simple diagram as in Figure 6-14:
Figure 6-14
Schematic setup for measuring the thermopower.
The procedure is conceptually quite simple: a temperature gradient is established between the current leads of a four-probe measurement. The charge carriers at the warmer end of the sample have more kinetic energy than those of the cooler end and are allowed to migrate. This results in a thermoelectric potential and a measured current when the leads are closed. The current supplied at a given thermovoltage is then the thermopower in simplest terms. It is easy to see why such a measurement could be of use. First, the sign of the carrier is the sign of the thermopower. So it is easy to determine if the sample is a hole-majority carrier or an electron-majority carrier. Second, the thermopower is the largest where there is a small population of phonons and a high carrier mobility and density. In fact, it is quite sensitive to electronic features at the Fermi level of the sample and their coupling to the phonon modes of the system. The expression for thermopower can be obtained as: S¼
1 R @f ðe lÞ rðeÞ de eTr @e
(13)
where T is the temperature; r(e) the density of states at e; f the Fermi function, and l the chemical potential. This expression has been used in conjunction with simple analytical models to predict thermopower behavior in mats of conducting fibers (polymers, nanotubes, etc.). Referred to as heterogeneous conduction’, the approach has become quite powerful in determining the effects of dopants on trans-
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Themopower measurements express carrier majority and dopant levels within fibrillar transport. In the example shown here, various dopants are compared, oxygen (a), boron (b), and nitrogen (c).
Figure 6-15
port in one-dimensional systems. [24]. Typically, such models allow for a metallic, semiconducting, and variable range-hopping contribution of the thermopower to be dominant at different temperatures. The above expression generally simplifies to something with a linear metallic term, a quadratic freeing term, and an exponential to express the variable range-hopping between fibers in a dense mat, as shown in Eq. (14): S = bT + qTp(er T2)–1 exp(Tp/T)[exp(Tp/T)+1]–2 Tp = (e – l) / kb
(14)
As an example of this, consider again mats of multiwalled carbon nanotubes. One might want to know the effects of doping impurities in such materials, and thermopower is a particularly good way to examine these phenomena in detail. Shown in Figure 6-15 is the thermopower as a function of dopant type. Figure 6-15a shows multiwalled nanotubes exposed to oxygen (thought to weakly dope the tube positively). In Figure 6-15b, the tubes are heavily doped with boron. This thermopower curve is positive and linear, indicating a hole majority carrier and nearly metallic behavior of the fibers. Finally, Figure 6-15c shows a mat of nitrogen-doped nanotubes, in which the nitrogen introduces donor states, yielding an electronmajority carrier material (negative curve) and again linear (metallic) behavior.
6.3
Conductivity in One Dimension: Localization
In Eq. (2) the resistivity is a material constant, which can be calculated form the resistance when the sample geometry is known. In very thin wires (external
6.3 Conductivity in One Dimension: Localization
approach to one dimensionality, Section 1.2) and at low temperatures, a new feature occurs: the resistivity depends on the sample geometry, in particular, on the length of the sample. Chaudhari and Habermeier [25] prepared very thin (50 ) and narrow (700 to 5000 ) strips of the amorphous alloy W–Re, some 100 lm long. Current leads at the ends and six voltage probes along the wire were applied (Figure 616). The results indicate that the total resistance of the sample is larger than the sum of resistances between the various leads. In Figure 6-17 the excess resistance is plotted versus the sample length. The excess resistance represents the amount by which the total resistance exceeds the sum of the partial resistances. The sample length is measured by the total resistance, which reaches approximately 3 MX. An excess resistance of 3 kX consequently corresponds to an effect in the order of 10–3. The observed excess resistance is interpreted in terms of localization [26,27]. As already pointed out, the periodicity of the crystal potential is a necessary condition for the existence of Bloch waves. Bloch waves are extended states, delocalized all over the crystal. Occasional defects in a three-dimensional lattice can be treated as scattering centers, giving rise to an ohmic resistance. With many defects, however, the entire concept of Bloch waves fails, and all states in the solid become localized [28]. The electrons move between the localized states by phonon-assisted tunneling (hopping). At low temperatures when the phonons are frozen, the materials act as insulators. We can quantify the disorder in a crystal by introducing the para-
Schematic arrangement of current leads and voltage probes in an experiment to measure excess resistances as a consequence of localization.
Figure 6-16
Figure 6-17
Length dependence of resistance of high-resistance samples measured at 4.2 K [25].
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meter D as the average fluctuation of the crystal potential. If, in a three-dimensional crystal, D becomes comparable to the bandwidth, all states become localized (Anderson–Mott transition’). The lower the dimensionality, the less disorder is needed to localize the electronic states. In strictly one-dimensional systems any finite D localizes all the states. Because there is always some disorder present, all one-dimensional conductors should become insulators at absolute zero. The localization experiment shown in Figures 6-16 and 6-17 was carried out at a finite temperature (4.2 K) and on a sample of finite cross section. The observed excess resistance is interpreted as a precursor to the metal-to-insulator transition. We now give a hand-waving argument for the excess resistance: excess resistance means that the resistance increases faster than linearly with increasing sample length. Theory predicts that it should increase exponentially. If we think of defects as semitransparent mirrors rather than colliding spheres, the exponential decrease of an incoming intensity becomes evident. How could we even have thought that the resistance would increase linearly! In higher dimensionality the electrons can bypass obstacles and their mirror properties are less pronounced. How thick must a wire be for excess resistance to be observed? Thouless [26] has introduced the concept of maximum metallic resistance in thin wires’. There is a maximum resistance that a metallic wire can achieve. Whenever the wire is sufficiently long or thin or disordered that the resistance exceeds a critical value, the wire is no longer metallic – it becomes insulating in the sense discussed at the end of Section 6.1: at absolute zero the resistance is infinite. How large is the critical resistance? Because it is independent of both material and geometry, it can only be a combination of natural constants. It is not difficult to guess that the quantum of electric charge e and the mechanical quantum h will be involved. The combination h/e2 has the dimension of electrical resistance and a value of about 4 kX. Thus, whenever the resistance of a wire is greater than about 4 or 10 kX, the wire is insulating as T fi 0. (Note that here we are content with an order-of-magnitude consideration. However, h/e2 is the famous von Klitzing constant, which, in the quantum Hall effect, can be determined to 8 significant figures. Multiplied by the velocity of light, h/e2 is the reciprocal fine-structure constant 1/a = 137.036, the keystone of quantum electrodynamics.) Let us now take a closer look at polyacetylene. The conductivity of highly conductive polyacetylene is about 105 S cm–1. For a single polymer chain we assume the cross section to be 1 2. Using Eq. (2) a resistance of 104 X for a polyacetylene chain of 10 length is obtained. Thus, individual, noninteracting molecular chains are insulators, not conductors. In Section 6.5 we will demonstrate that polyacetylene samples are neither single crystals nor well-separated individual chains, but spaghetti-like bundles of chains (fibrils). Figure 6-18 shows such a spaghetti network schematically. A fibril consists of about 100 to 1000 chains, and its resistance is about 107 to 108 X cm–1 (if highly doped). A fibril 1 lm long has a resistance of 1 to 10 kX and so is close to the metal-to-insulator transition. The meshes in the network (Figure 6-18) are about 1 lm wide. Interfibrillar contacts favor metallic behavior. Prigodin and Efetov and others [29] have studied the metal–insulator phase dia-
6.4 Conductivity and Solitons
Schematic view of the fibrillar structure of a polymer. The rings indicate interfibrillar crosslinks [29].
Figure 6-18
gram of a fibrillar network by varying the resistance in the fibers and the interfibrillar contacts. The important message of this section is that resistance or conductance cannot be scaled to small dimensions. A very thin wire behaves differently than a thick wire, and a monomolecular chain is not a molecular wire. This has to be kept in mind in speaking of polymer chains connecting molecular devices in the way that macroscopic wires connect transistors (Chapter 9).
6.4
Conductivity and Solitons
In Section 5.4 we discussed conjugational defects. We saw that these defects are very exciting particles, that they can be neutral or positively or negatively charged and that they can move along a polymer chain (Figure 5-17) with an effective mass only slightly larger than the free electron mass, m* » 6me. Therefore it is tempting to speculate that the conductivity of polyacetylene is related to the motion of charged solitons. The previous section showed that a single polyacetylene chain is an insulator at low temperatures because of localization of the electronic states. Would a polyacetylene chain also undergo a metal-to-insulator transition if the charge carriers were solitons instead of electrons? Solitons are localized (= not extended) by definition’, but in contrast to electrons, the nonextended soliton wave function moves through the lattice, because the soliton has the same energy everywhere, whereas the electron energy varies from site to site. To understand how important charge transport by solitons might be we look at the Drude model and Eqs. (3)–(5). What is the collision time for solitons? Jeyadev and Conwell [30] analyzed the scattering of solitons on one-dimensional longitudinal acoustic phonons, which are compressional waves along the chain. The soliton energy in the compressed section differs from that in the decompressed section. As electron–phonon scattering in a semiconductor, soliton–phonon scattering can be calculated by means of the deformation potential method [31]. Jeyadev and Conwell
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obtained values in the order of 1 cm2 V–1 s–1 for the mobility of solitons in polyacetylene at room temperature and about 100 cm2 V–1 s–1 at 50 K. These calculations neglect impurity scattering, which reduces the mobility particularly at low temperatures. Below 20 K the solitons are self-trapped’ due to the discreteness of the lattice, so that at the low-temperature limit a simple polyacetylene chain is an insulator, both from the electron localization point of view and from soliton transport considerations. The picosecond photoconductivity experiments discussed in Section 5.8 allow for experimentally checking the calculated mobility [32]. The agreement is surprisingly good. If the solitons are not photogenerated but created by doping, the electrostatic interaction between the charge of the solitons and that of the counterions has to be taken into account. For example, by iodine doping (Eq. (10) in Chapter 5) two positive solitons and two negative I–3 ions are created. The solitons are trapped by the Coulomb field of the I–3 ions; the electrostatic energy is much larger than the thermal energy kT (Boltzmann constant times temperature) or the energy generated by the applied electric field. Hence, in doped polyacetylene, charged solitons are not mobile and do not contribute to conductivity. In heavily doped polyacetylene the midgap states associated with the solitons interact and form a soliton band. The soliton band is partially filled, and band conductivity might occur. If the soliton band is wide enough to overlap with the p and p* bands, little of the exciting soliton particles are left. In Figure 6-19 the development of a soliton band from midgap states is schematically shown, as well as the final vanishing of the p–p* gap upon increasing the soliton density. Band conductivity in polyacetylene is further discussed in Section 6.7. Another conductivity mechanism related to solitons is intersoliton hopping [33]. In this model, charged and neutral solitons are present in (slightly) doped polyacetylene. The charged solitons are trapped by the dopant ions, but the neutral solitons are free to move. Whenever a neutral soliton passes close by a charged soliton, an electron can hop between the midgap states belonging to the solitons (Figure 6-20). This is a special case of hopping conductivity, which is presented in Section 6.6.
Development of a soliton band from midgap states and final suppression of the p–p* gap upon increase of soliton concentration. Figure 6-19
6.4 Conductivity and Solitons
Intersoliton hopping: charged solitons are trapped by the dopant counterions, but neutral solitons are free to move. When a neutral soliton is close to a charged soliton, the electron hops from one defect to the other. Figure 6-20
In Chapter 5 we demonstrated that spin–charge inversions occur for solitons in polyacetylene (Figure 5-20): charged solitons carry no spin, spin-carrying solitons have no charge. There was much discussion on whether or not experimental evidence of spin–charge inversion could be obtained, for example, by relating data of electrical conductivity to those of magnetic susceptibility. In Figure 6-21 the paramagnetic Curie susceptibility and the electrical conductivity of polyacetylene are plotted versus the dopant concentration [34–36]. The susceptibility decreases while the conductivity increases. The experimental data are consistent with the assumption that – at low doping concentrations – doping first converts neutral spin-carrying solitons into charged spinless solitons. There are some 200 ppm of neutral solitons in undoped polyacetylene, and the first electrons to be oxidized upon doping are those clinging to the solitons. Simultaneously to charging of existing solitons, doping creates new solitons, as shown in Section 5.5. At high soliton concentrations the solitons interact, and a soliton band is formed, which widens until the p–p* gap is finally suppressed (Figure 6-19). Noninteracting spin-carrying solitons fully contribute to the Curie susceptibility. The Curie susceptibility, decreasing with 1/T (Curie law), is the susceptibility of isolated particles fighting individually against thermal disorder. Interacting particles tend to compensate their spins, until for particles in a band, Pauli susceptibility finally replaces Curie susceptibility. For Pauli susceptibility the sample temperature T has to be substituted for the Fermi temperature TF = EF/k (Fermi energy divided by Boltzmann constant). As a result, the Pauli susceptibility is temperature-independent. Because of TF >> T the Pauli susceptibility is much smaller than the Curie susceptibility. Consequently, the decrease in susceptibility as shown in Figure 6-21 could also be due to the gradual transition from Curie susceptibility to Pauli susceptibility. In nondegenerate ground state conjugated polymers like polyparaphenylene and polypyrrole, solitons are unstable and there are well-defined compound particles like polarons (Figures 5-26, 5-27, and 5-30). Polarons carry a spin, and bipolarons are spin-compensated. Upon doping, at first polarons are created and the susceptibility increases; then, from a certain doping level onwards, susceptibility decreases due to the formation of bipolarons. An example for electrochemical doping of poly-
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Decrease of Curie paramagnetic susceptibility v (from ESR [34,35]) and increase of conductivity r upon doping of polyacetylene, as a consequence of spin–charge inversion for solitons [36].
Figure 6-21
Susceptibility versus dopant concentration in electrochemical doping of polypyrrole. The susceptibility is measured in spins per six pyrrole rings, the doping concentration in electronic charges transferred per six pyrrole rings [37].
Figure 6-22
pyrrole is shown in Figure 6-22 [37]. It is obvious that the susceptibility reaches a maximum when one electronic charge has been transferred to about every 6th pyrrole ring, but this maximum corresponds to only half as many spins as there are charges.
6.5 Experimental Data
Figure 6-23 Pauli susceptibility vP and electrical conductivity r plotted versus the dopant concentration in polyacetylene [36].
Experimentally, the Pauli susceptibility can be separated from the Curie susceptibility by the temperature dependence. In Figure 6-23 the Pauli susceptibility of polyacetylene is plotted versus the dopant concentration [36,38], and the conductivity is taken from Figure 6-21. There is a pronounced step in the Pauli susceptibility at a dopant concentration of 6%. This step is not consistent with a gradual conversion of isolated solitons into a soliton band. It rather suggests a phase transition [39]. A similar step has been observed in sodium-doped polyacetylene [40]. Kivelson and Heeger [41] introduced a theory for a first-order phase transition driven by soliton interactions. An alternative driving force for this phase transition might be a rearrangement of chain packing due to the accommodation of dopant ions [42]. (It is also possible that both soliton interaction and intercalation of counterions lead to separate phase transitions and that these two phase transitions then interlock.) Despite the origin of the phase transition, it is remarkable that between 2% and 6% doping, high conductivity is observed but no detectable magnetic susceptibility, neither Curie nor Pauli.
6.5
Experimental Data
This section is concerned with typical experimental data regarding the electrical conductivity of polyacetylene. Except for samples with very high conductivity (r > 1000 S cm–1), similar data are obtained for all conducting polymers. A particular problem for the interpretation of the data is the complicated morphology of conjugated polymers. This is perhaps most serious with polyacetylene; however, many other polymers are of similar complexity. Figure 6-24 shows an electron micrograph of polyacetylene film. Note the spaghetti structure already mentioned in Section 6.3. An idealization of this structure
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Scanning electron micrograph of polyacetylene film showing the spaghetti- or fleece-like morphology. The scale marked on the rim corresponds to 1 lm.
Figure 6-24
is presented in Figure 6-18, stressing the interfibrillar contacts. An alternative schematic view of the polyacetylene morphology is shown in Figure 6-25. Small crystalline domains are discernible within the fibers, in which the chains are well-ordered. These areas can be analyzed by X-ray or neutron diffraction to give the structure of the elementary cell. Amorphous regions are situated between crystalline domains. A crystalline domain, however, is not perfectly ordered: there are point defects such as chain ends, crosslinks, etc. The overall electrical conductivity is a superposition of intrachain, interchain, and interfiber charge-transport mechanisms. If the polyacetylene film is stretched, the fibers are aligned and the crystallinity within the fibers is improved. As far as the conductivity is concerned, at present we distinguish between two types of polyacetylene samples: (1) Standard or Shirakawa polyacetylene (type S’).
Schematic view of the fibrillar structure of polyacetylene. The fibers are bundles of chains, containing some 100 or 1000 polymer chains. They intersect in fractions of a 1 lm.
Figure 6-25
6.5 Experimental Data
When saturation-doped with iodine these samples achieve conductivity values of about a hundred S cm–1. (2) New, or Naarmann, or highly oriented polyacetylene [43–45] (type N’), in which the conductivity of doped samples exceeds 103 S cm–1 and may even reach 106 S cm–1. The upper sample (highly doped polyacetylene) in Figure 6-3, which stays metallic at low temperature, belongs to type N whereas the lower sample, becoming insulating as T fi 0, is type S. The conductivity in type S is dominated by hopping, and the relevant physical model is discussed in Section 6.6; the conductivity of type N samples is dealt with in Section 6.7. It is generally assumed that type N samples exist only for polyacetylene, whereas samples of all other conjugated polymers are of type S. However, it is possible that material perfection will also lead to Type N samples in polyaniline, polyphenylene–vinylene, and others. Figure 6-26 presents a conductivity chart that compares the room-temperature conductivity of doped conjugated polymers with that of other materials. Note that the polymer conductivity values extend over the whole region from insulating, such as diamond, to highly conducting, such as copper. The hatched area of the polymer arrow represents type N polyacetylene. In Figure 6-27 the conductivity change upon doping is shown for type S polyacetylene. A sharp initial rise is followed by saturation at some 5–6 mol %. These high doping levels contrast with the dopant concentrations we are used to in conventional semiconductors such as silicon or germanium, which are in the ppm range. (The high doping level is one of the reasons why many scientists dislike the term doping’ in connection with conducting polymers.) The initial rise is even more pronounced, when accidental doping by atmospheric oxygen or other impurities is compensated by treating the sample in ammonia. In this case it is possible to observe conductivity changes of more than ten orders of magnitude. Figure 6-28 shows the temperature dependence of the conductivity of iodinedoped (type S) polyacetylene at various doping levels [22,46]. At all levels of iodine
Comparison of the room-temperature conductivity values of conducting polymers with conductivities of other materials.
Figure 6-26
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Conductivity of polyacetylene as a function of dopant concentration (type S polyacetylene, room temperature, iodine doping).
Figure 6-27
Temperature dependence of polyacetylene conductivity at various concentrations of iodine dopant (type S polyacetylene [46]).
Figure 6-28
6.5 Experimental Data
Temperature dependence of the conductivity of undoped polyacetylene according to Epstein et al. [47]. The solid line corresponds to the power law r T14.
Figure 6-29
concentration, the conductivity decreases upon cooling of the sample, with the decrease being much smaller for highly doped than for slightly doped samples. The whole set of curves is consistent with variable range hopping (Eq. (8)), which is discussed in the next section. At very low doping levels, power laws (Eq. (6)) with powers larger than ten fit the data equally well. Figure 6-29 shows Epstein’s data on undoped (accidentally doped) polyacetylene [47]. The straight line corresponds to r T14.
Negative (metal-like) temperature coefficient of the conductivity of highly conducting iodine-doped type N polyacetylene [48].
Figure 6-30
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Highly doped type N samples do not behave according to variable range hopping. They are metallic in the sense that the conductivity does not vanish as T fi 0. In some cases the conductivity even increases upon cooling. An example of such behavior is plotted in Figure 6-30. Note the maximum at 250 K; between room temperature and 250 K the slope is negative [48]. The highest conductivity values published so far are close to 106 S cm–1. One of the world champions is shown in Figure 6-31 [44]. A type N polyacetylene sample is exposed to iodine and the conductivity climbs to 500 000 S cm–1, which is almost as high as the room temperature conductivity of copper. With few exceptions, it has turned out to be very difficult to synthesize samples with conductivities above 105 S cm–1. On the other hand, it is easy to decrease the
Increase in conductivity of a type N polyacetylene sample during exposure to iodine [44].
Figure 6-31
Conductivity of doped polyacetylene as a function of the defect concentration. Defects interrupt the conjugated system of double bonds (segmented polyacetylene’) [49].
Figure 6-32
6.6 Hopping Conductivity
conductivity and to convert type N samples into type S. One method is the deliberate introduction of defects. The formula in Figure 6-32 shows a O=C–CH group within the polyacetylene chain. Such groups cut the conjugated system into shorter segments, and the conductivity can be studied as a function of the average segment length. The results of such an investigation are shown in Figure 6-32. There are two sets of samples. The samples were segmented first and then doped to 3.5% and 15% iodine, respectively. We see that the conductivity decreases exponentially with the defect concentration, or, in other words, it increases exponentially with the segment length (in the range of lengths studied, i.e., from about 20 to 100 ) [49].
6.6
Hopping Conductivity
As already stated, the experimental data on the conductivity of conjugated polymers can be explained by hopping mechanisms – with the exception of highly conducting highly oriented type N polyacetylene, where metallic (band) conductivity has to be assumed. We also know that hopping means phonon-assisted quantum mechanical tunneling’ [7] and that the temperature dependence of hopping conductivity is described by soft’ exponential equations (Eq. (8)). Hopping, of course, is an anthropomorphic term and reminds us of a man crossing a river by jumping from stone to stone (Figure 6-33). The stones are spread out at random. It is quite obvious: the more stones the higher the conductivity. Also, the AC conductivity is greater than the DC conductivity. Coming to a place where the next stone is far away, the hopper has to rest for a long time to sufficiently recover before he can make the big jump. While he is still waiting, the field reverses and he could do an easy hop backwards. Whenever the field changes with high frequency the hopper just goes back and forth between nearby stones. This is the pair approximation to hopping conductivity [50]. To discuss the temperature dependence of hopping conductivity, we have to assume that, in addition to the random spatial distribution of the stones in the river, there is also a random distribution in height. Furthermore, on a warm day it is easier to hop up onto a high stone. At this point the anthropomorphic model fails and it is more reasonable to look at the band-structure representation in Figure 6-34: there
Figure 6-33
Hopping transport: man trying to cross a river by jumping from stone to stone.
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Electronic level scheme of a disordered solid to demonstrate the hopping conductivity (CB: conduction band; VC: valence band; EF: Fermi energy; W: energetic distance between states; R: local distance between states). Figure 6-34
are localized states in the gap, randomly distributed in space as well as in energy. The Fermi energy level is about at gap center, the states below EF are occupied, those above are empty (except for thermal excitations). Electrons hop (tunnel) from occupied to empty states. Most of the hops have to be upward in energy. At high temperatures, many phonons are available which can assist in upward hopping. As these phonons freeze, the electron has to look further and further to find an energetically accessible state. Consequently, the average hopping distance decreases as the temperature decreases – hence the name variable range hopping’. Because the tunneling probability decreases exponentially with distance, the conductivity also decreases. But, as already mentioned, the decrease is smoother than in a semiconductor with a well-defined gap, because there is a continuous distribution of activation energies (energetic distances between states). The calculation leads to a soft exponential equation (compare Eq. (8)): r = r0 exp [–(T0/T)c]
(15)
where c depends on the dimensionality d of the hopping process: c=
1 1þd
(16)
For three-dimensional variable range hopping, c = 4 and we get Mott’s famous T–1/4 law’ T (ln r proportional to T–1/4) [51]: r = r0 exp[–(T0/T)1/4]
(17)
6.6 Hopping Conductivity
with r0 = e2N(EF)Rmph
(18)
where R is the average hopping distance: R = [8/9 pa N(EF) kT]–c
(19)
and T0 =
83 a3 9pkNðEF Þ
(20)
with N(EF) is the electronic density of states at the Fermi energy; a is the inverse localization length (spatial extension of localized wave function), and mph is the typical phonon frequency. As already stated, Eq. (15) is consistent with the temperature dependence of the conductivity shown in Figure 6-28. It turns out that any c value between 12 and 14 fits the data, so that it is difficult to distinguish between one-, two-, and threedimensional variable range hopping. Eq. (15) includes two general parameters r0 and T0. Both depend on the microscopic parameters a and N(EF) (Eqs. (18)–(20)); the former is the inverse spatial extension of the localized electron wave function, and the latter is the electron density of states at the Fermi level. If doping creates defects and defects give rise to states in the gap, N(EF) is roughly proportional to the doping concentration y and we obtain: ln r y–c
(21)
Figure 6-35 shows a fit of Eq. (21) to the data displayed in Figure 6-27. It is clear that the model of variable range hopping explains the doping dependence of the conductivity quite well. The dependence on the localization length can be studied in segmented polyacetylene. We assume that the localization length is proportional to the segment length [52], leading to an approximately exponential decay for the conductivity when the segments are shortened, as can be observed in Figure 6-32. A large electronic density of states implies both high conductivity at room temperature (large r0 in Eq. (18)) and small temperature dependence of the conductivity (small T0 in Eq. (20)). This correlation is evident from the universal curve’ in Figure 6-36, where T0 is plotted versus the room temperature value of the conductivity for various conducting polymers [53]. We have already mentioned, in connection with the man trying to cross a river by jumping from stone to stone (Figure 6-33), that in hopping transport the AC conductivity is higher than the DC conductivity. Figure 6-37 shows the frequency dependence of undoped polyacetylene (i.e., only accidentally doped by oxygen and other atmospheric contamination). The curves refer to different temperatures. At low frequencies the conductivity is constant; but from a certain point, it increases
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Conductivity of iodine-doped polyacetylene versus dopant concentration. The solid curve corresponds to a fit of the variable range hopping model.
Figure 6-35
more or less linearly with frequency. The higher the conductivity at zero frequency, the higher the onset frequency. This relation is quite general, regardless of whether we change the low-frequency conductivity by heating, by doping, or by using a different sample. The solid lines in the graph correspond to calculations by Ehinger et al. [54], who combined variable range hopping and pair approximation to form the extend pair approximation model’ [55].
Universal curve’ relating the room-temperature value of the conductivity to the parameter T0, which is a measure of the temperature dependence of the conductivity. Data are for various conducting polymers [53]. Figure 6-36
6.6 Hopping Conductivity
Frequency as a function of the conductivity of undoped polyacetylene. Data acquisition by Epstein et al. [47], calculations by Ehinger et al. [54].
Figure 6-37
DC and microwave conductivity of iodine-doped polyacetylene and fit of extendedpair approximation [54]: s 30 Hz; d 9.9 GHz.
Figure 6-38
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The data in Figure 6-37 were taken from undoped polyacetylene. In doped polyacetylene the conductivity is so high that the onset frequency is not in the hertz or kilohertz regime but in the gigahertz region. Therefore the frequency dependence is found in the microwave region (10 GHz). In Figure 6-38 the temperature dependencies of DC and AC conductivity at 10 GHz are shown for two differently doped polyacetylene samples [54]. The abscissa scale is in T–1/4, so that a straight line is obtained when Mott’s law is fulfilled. It can be seen that, as expected, the AC data are located above the DC data. The model parameters for the extended pair approximation can be obtained from a fit to the DC data, allowing the calculation of the microwave conductivity. Consequently, the model predicts the experimental results reasonable well (with the exception of the values at high temperatures, where the approximation in the model overestimates both DC and AC conductivity (Mott’s T–1/4 law is a low-temperature approximation to variable range hopping). The model of variable range hopping assumes a random distribution of localized states. If the distribution is not random and the defects tend to cluster, the model of fluctuation-induced tunneling’ is more appropriate [56]. This model assumes
Temperature dependence of the DC conductivity of heavily (iodine) doped polyacetylene and fit of the model of fluctuation-induced tunneling [22]. Solid curve: theoretical fit.
Figure 6-39
6.7 Conductivity of Highly Conducting Polymers
metallic islands in an insulating matrix. Particles of carbon black or aluminum flakes in a thermoplastic polymer are typical examples. Other examples are the coexistence of doped and undoped regions in a conjugated polymer, crystalline and amorphous domains, or disordered fibers (Figure 6-24). Tunneling between large metallic particles is temperature-independent. The tunneling probability is just a function of the barrier width and the barrier height. The number of electrons on a particle fluctuates thermally, however, and the electrostatic potential also fluctuates when the particles are small. Therefore tunneling between small particles (thousand islands’) is a superposition of phonon-assisted and temperature-independent processes. Hence, the variable range hopping Eq. (15) has to be replaced by Eq. (22), which contains T1 as an additional parameter: T0 (22) r = r0 exp T1 þT This leads to constant conductivity for T << T1 and to activated behavior for T >> T1, with the parameters r0, T0, T1 depending on the geometry of the tunnel barrier and the size of the conducting particles. Experimental data on heavily doped polyacetylene (type S) and a fit of Eq. (22) are shown in Figure 6-39 [22].
6.7
Conductivity of Highly Conducting Polymers
In Section 6.5 we mentioned that there are two types of conducting polymer samples: type S, in which charge transport occurs by hopping, and type N, in which there is band conductivity. Here we confine the discussion to type N polyacetylene, but we should mention that recently other heavily doped polymers have also been found to be metallic [57] (in the sense that the conductivity stays finite as T fi 0). At the ICS meeting in 1986 in Kyoto, Naarmann [43] announced that he had modified the Shirakawa synthesis [58] and prepared a highly stretchable version of polyacetylene. After iodine doping, conductivity values of about 100 000 S cm–1 could be obtained. In the following years he could even further increase the conductivity [59]. Naarmann sent several samples to the University of Bayreuth, and although none of these exhibited conductivity values above 100 000 S cm–1, r > 80 000 S cm–1 was confirmed [3]. Naarmann’s idea was to slow the polymerization process by aging the catalyst and by dissolving it in a viscous solvent such as silicon oil. Slower polymerization should lead to polymer films of higher perfection. Following this route, Tsukamoto et al. [44] further improved the synthesis by using decaline as solvent. Subsequently many research groups have tried to repeat Naarmann’s or Tsukamoto’s syntheses, but never succeeded in achieving conductivity values of 100 000 S cm–1 or more. Today it is generally accepted that 20 000 or 30 000 S cm–1 can be obtained in a reproducible way, but to go beyond requires lucky circumstances that are not yet accessible to quantitative analysis. To obtain 30 000 S cm–1 for polyacetylene, two further routes exist: either the catalyst is dissolved in cummene, followed by removing most
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of the solvent before polymerization (solvent-free route’) [45]; or the solvent is a liquid crystal and the polyacetylene film grows epitactically (or quasiepitactically) on its surface [60]. The aim of a European joint project [61] was to synthesize polyacetylene with conductivities greater than 100 000 S cm–1 following one of the above routes, but so far 40 000 S cm–1 has not yet been surpassed. A further goal is to correlate the conductivity of highly conjugated polyacetylene with other measurable material parameters. No correlation has yet been found. Apparently the conductivity is by far more sensitive to impurities and imperfections than any other parameter. For an ideal single crystal of polyacetylene, we can predict its conductivity by theoretical means. The ideal crystal consists of sufficiently long’ well-ordered polyacetylene chains. It is doped to a high level, the dopant ions are between the chains, and their electrostatic potential is screened so that the electrons moving along the chains do not notice it. Because of the high doping and some interchain coupling, the Peierls transition is suppressed. In this way a very anisotropic metal is obtained. The electrons mainly move along the chains and hop only very seldomly to a neighboring chain. If electrons are scattered by phonons, it is predominantly Umklapp scattering, in which the electron momentum is reversed. Because of energy and momentum conservation, only very few phonons can participate in Umklapp processes, namely those phonons whose wave vector is twice the Fermi wave vector of the electrons kF. Because of the lack of appropriate phonons the theoretical’ conductivity of polyacetylene is very high. Estimates yield as high as 2·107 S cm–1 at room temperature, which is about 30 times the conductivity of copper [62–64]. Moreover, 2kF phonons have a fairly large energy (large compared to the sample temperature), so they can easily be frozen out: the conductivity of ideal polyacetylene will rise exponentially, approaching infinity at absolute zero. Of course, the theoretical limit is not observed in experiments. But at least for some samples the temperature coefficient of the conductivity is negative [65,66]. Figure 6-40
Temperature dependence of the conductivity of highly conducting polyacetylene samples. The curves refer to different iodine concentrations.
Figure 6-40
References and Notes
shows a plot of the temperature dependence of the conductivity for several iodinedoped Naarmann polyacetylene samples [66]. In the inset, the maximum around 250 K is clearly seen. Several models assume the coexistence of highly conducting and less conducting domains in polyacetylene. The model of fluctuation-induced tunneling (Eq. (22)) has already been mentioned. Kaiser [67,68] proposed other models for heterogeneous polyacetylene samples and he also evaluated the thermopower. The high conductivity of doped conjugated polymers has stimulated many ideas for applications of these materials. We review this topic in Chapter 10 of this book. Most present-day applications use only moderately conducting polymers. It seems as if highly conducting polymers are gaining more importance today with respect to understanding basic charge-transport phenomena rather than finding direct applications. One of the most promising applications is polymer light-emitting devices (LED). However, to optimize the yield of electroluminescence, understanding charge transport is a matter of first priority.
References and Notes 1 H. Krogman. Planare Komplexe mit Metall-
Metall-Bindungen. Angewandte Chemie 81, 10 (1969). 2 F.L. Vogel. Some potential applications for intercalation compounds of graphite with high electrical conductivity. Synthetic Metals 1, 279 (1979/1980). 3 Th. Schimmel, W. Riess, J. Gmeiner, G. Denninger, M. Schwoerer, H. Naarmann, and N. Theophilou. DC-conductivity on a new type of highly conducting polyacetylene, N–(CH)x. Solid State Communications 65, 1311 (1988) G.T. Kim, M. Burghard, D.S. Suh, K. Liu, J.G. Park, S. Roth, and Y.W. Park. Conductivity and magnetoresistance of polyacetylene fiber network. Synthetic Metals 105 , 207 (1999). 4 P. Drude. Annalen der Physik 1, 566 (1900); Annalen der Physik 3, 369 (1900). 5 B.L. Altshuler, V.E. Kravtsov, and I.V. Lerner. In: Mesoscopic Phenomena in Solids. B.L. Altshuler, P.A. Lee, and R.A. Webb (Eds.), Elsevier, Amsterdam, 1991. 6 A.B. Kaiser. Electronic Properties of Conjugated Polymers. Springer Series in Solid-State Sciences 76, 2; H. Kuzmany, M. Mehring, and S. Roth, (Eds.), Springer Verlag, Heidleberg, 1987.
7 R. Zallen. The Physics of Amorphous Solids.
John Wiley & Sons, New York, 1983. 8 for example: G. Lautz and R. Taubert (Eds.),
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11
12
Kohlrausch – Praktische Physik Band 2, p. 305. B. B. Teubner Verlag, Stuttgart, 1968. L.J. van der Pauw, Philips Research Reports 13, 1 (1958); and Philips Technische Rundschau 20, 230 (1958/1959). C.M. Fischer, M. Burghard, S. Roth, and K. von Klitzing. Organic quantum wells: molecular rectification and single-electron tunnelling. Europhysics Letters 28, 129 (1994). H.C. Montgomery. Method for measuring electrical resistivity in anisotropic materials. Journal of Applied Physics 42, 2971 (1971). In such devices, an ionized beam of heavy atoms (such as Gd) is created in an acceleration column and focused down to a tiny (30 nm) spot. A background metal–organic gas (a gas containing a metal such as Pt) is introduced into the chamber and the ions at the beam’s focus break down the gas near the surface. This results in deposition of the metal onto the substrate in a very well defined way. Unfortunately, it also frequently leads to damage to the sample, since it is being bombarded with heavy ions. It also may contami-
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nate the sample, leaving some metals and neutral heavy atoms behind on the substrate. 13 P.-W. Chiu. PhD Thesis. Mnchen 2003. 14 K. Liu, S. Roth, M. Wagenhals, G. Duesberg, C. Journet and P. Bernier. Transport properties of single-walled carbon nanotubes. Synthetic Metals 103, 2513 (1999). 15 Y. Imry. Introduction to Mesoscopic Physics. Oxford University Press, Oxford, 1997. 16 G. Natta, G. Mazzanti, and P. Atti. Stereospecific polymerization of acetylene. Acad. Nazl. Lincei Rend. Classe Sci. Fis. Mat. Nat. 25, 3 (1958). [17] T. Ito, H. Shirakawa, and S. Ikeda. Simultaneous polymerization and formation of polyacetylene film on the surface of a concentrated soluble Ziegler-type catalyst solution. Journal of Polymer Science, Part A: Polymer Chemistry 12, 11 (1974). 18 L.I. Buranov and I.F. Shchegolev. Method for measuring the conductivity of small crystals at 1010Hz. Pribory i Tekhnika Eksperimenta 2, 171 (1971). 19 J. Joo and A.J. Epstein. Complex dielectric constant measurement of samples of various cross sections using a microwave impedance bridge. Review of Scientific Instruments 65, 2653 (1994). J. Joo, S.M. Long, J.P. Pouget, E.J. Oh, A.G. MacDiarmid, and A.J. Epstein. Charge transport of the mesoscopic metallic state in partially crystalline polyanilines. Physical Review B 57, 9567 (1998). 20 T. Rabenau, A. Simon, R.K. Kremer, and E. Sohmen. The energy gaps of fullerene C60 and C70 determined from the temperature dependent microwave conductivity. Zeitschrift fr Physik B 90, 69 (1993). 21 G. Schaumburg, H.W. Helberg. Application of a cavity perturbation method to the measurement of the complex microwave impedance of thin super- or normal conducting films. Journal de Physique III 4, 917 (1994). D.N. Peligrad, B. Nebendahl, A. Mehring, A. Dulcic, M. Pozek, and D. Paar. General solution for the complex frequency shift in microwave measurements of thin films. Physical Review B 64, 224504 (2001). 22 K. Ehinger and S. Roth. Non-solitonic conductivity in polyacetylene. Philosophical Magazine B 53, 301 (1986).
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A.B Kaiser and Y.W. Park. Conduction mechanisms in polyacetylene nanofibres. Current Applied Physics 2, 33 (2002). A.B. Kaiser, G.U. Flanagan, D.M. Stewart, and D. Beaglehole. Heterogeneous model for conduction in conducting polymers and carbon nanotubes. Synthetic Metals 117, 67 (2001). I.A. Misurkin. Electric conductivity of polyacetylene in a three-dimensional model of conducting polymers. Zhurnal Fizicheskoi Khimii 70, 923 (1996). C.O. Yoon, M. Reghu, D. Moses, A.J. Heeger, Y. Cao, T.A. Chen, X. Wu, and R.D. Rieke. Hopping transport in doped conducting polymers in the insulating regime near the metal–insulator boundary: polypyrrole, polyaniline and polyalkylthiophenes. Synthetic Metals 75, 229 (1995). L. Genzel, F. Kremer, A. Poglitsch, G. Bechtold, K. Menke, and S. Roth. Millimeter-wave and far-infrared conductivity of iodine- and AsF5doped polyacetylene. Physical Review B 29, 4595 (1984). A. B Kaiser, Y. W. Park, G. T. Kim, E. S. Choi, G Dusberg, and S. Roth. Electronic transport in carbon nanotube ropes and mats. Synthetic Metals 103 2547 (1999). P. Chaudhar and H.-U. Havermeier. Quantum localization in amorphous W–Re alloys. Physical Review Letters 44, 40 (1980). D.J. Thouless. Maximum metallic resistance in thin wires. Physical Review Letters 39, 1167 (1977). R. Krishnan and V. Srivastava. Localization and interactions in quasi one-dimension. Microelectronic Engineering 51, 317 (2000). R. Krishnan and V. Srivastava. Resistance of quasi-one-dimensional wires. Physical Review B 59, R12747 (1999). B. Kramer, G. Bergmann, and Y. Bruynseraede. Localization, Interaction, and Transport Phenomena. Springer Series in Solid-State Sciences 61, Springer Verlag, Heidelberg, 1985. Y.B. Khavin, M.E. Gershenson, and A.L. Bogdanov. Strong localization of electrons in quasione-dimensional conductors. Physical Review B 58, 8009 (1998). P.W. Anderson. Absence of diffusion in certain random lattices. Physical Review 109, 1492 (1958). V.N. Prigodin and K.B. Efetov. Localization transition in a random network of metallic
References and Notes wires: a model for highly conducting polymers. Physical Review Letters 70, 2932 (1993). A.J. Epstein, W.P. Lee, and V.N. Prigodin. Lowdimensional variable range hopping in conducting polymers. Synthetic Metals 117, 9 (2001). V.N. Prigodin and A.J. Epstein. Nature of insulator–metal transition and novel mechanism of charge transport in the metallic state of highly doped electronic polymers. Synthetic Metals 125, 43 (2001). T. Mitra and P.K. Thakur. Electronic transmittance and localisation in quasi-1D systems of coupled identical n-mer chains. International Journal of Modern Physics B 12, 1773 (1998). N. Dupuis. Metal–insulator transition in highly conducting oriented polymers. Physical Review B 56, 3086 (1997). V.N. Prigodin. One-dimensional localization effects in three-dimensional structures of highly-conducting polymers. Synthetic Metals 84, 705 (1997). 30 S. Jeyadev, E.M. Conwell. Soliton mobility in trans-polyacetylene. Physical Review B 36, 3284 (1987). T. Ikoma, S. Okada, H. Nakanishi, K. Akiyama, S. Tero-Kubota, K. Mobius, and S. Weber. Spin soliton in a pi-conjugated ladder polydiacetylene. Physical Review B 66, 014423 (2002). 31 J. Bardeen and W. Schockley. Deformation potentials and mobilities in non-polar crystals. Physical Review 80, 72 (1950). 32 J. Reichenbach, M. Kaiser and S. Roth. Transient picosecond photoconductivity in polyacetylene. Physical Review B 48, 14104 (1993). 33 S. Kivelson and A.J. Heeger. Electron hopping conduction in the soliton model of polyacetylene. Physical Review Letters 46, 1344 (1981). S. Kivelson and A.J. Heeger. Electron hopping in a soliton band: conduction in lightly doped (CH)x. Physical Review B 25, 3798 (1982). A.B. Kaiser and Y.W. Park. Conduction mechanisms in polyacetylene nanofibres. Current Applied Physics 2, 33 (2002). S. Bhattacharyya, S.K. Saha, M. Chakravorty, B.M. Mandal, D. Chakravorty, and K. Goswami. Frequency-dependent conductivity of interpenetrating polymer network composites of polypyrrole–poly(vinyl acetate). Journal of Polymer Science B 39, 1935 (2001). S. Bhattacharyya, S.K. Saha, T.K. Mandal, B.M. Mandal, D. Chakravorty, and K. Gos-
wami. Multiple hopping conduction in interpenetrating polymer network composites of polypyrrole and poly(styrene-co-butyl acrylate). Journal of Applied Physics 89, 5547 (2001). 34 I.B. Goldberg, R.H. Crowe, P.R. Newman, A.J. Heeger, and A.G. MacDiarmid. Electron spin resonance of polyacetylene and AsF5-doped polyacetylene. The Journal of Chemical Physics 70, 1132 (1979). 35 D. Davidov, S. Roth, W. Neumann, and H. Sixl. ESR and conductivity studies of doped polyacetylene. Zeitschrift fr Physik B 51, 145 (1983). 36 M. Peo. PhD Thesis. Konstanz: 1982. T. Ikoma, S. Okada, H. Nakanishi, K. Akiyama, S. Tero-Kubota, K. Mobius, and S. Weber. Spin soliton in a p-conjugated ladder polydiacetylene. Physical Review B 66, 014423 (2002). 37 M. Nechstein, F. Devreux, F. Genoud, E. Vieil, J.M. Pernaut, and E. Genies. Polarons, bipolarons and charge interactions in polypyrrole: physical and electrochemical approaches. Synthetic Metals 15, 59 (1986). 38 S. Ikehata, J. Kaufer, T. Woerner, A. Pron, M.A. Druy, A. Sivak, A.J. Heeger, and A.G. MacDiarmid. Solitons in polyacetylene: magnetic susceptibility. Physical Review Letters 45, 1123 (1980). A.J. Epstein, H. Rommelmann, M.A. Druy, A.J. Heeger, and A.G. MacDiarmid. Magnetic susceptibility of iodine doped polyacetylene: the effects of nonuniform doping. Solid State Communications 38, 683 (1981). 39 T. Masui, T. Ishiguro, and J. Tsukamoto. Spin susceptibility and its relationship to structure in perchlorate doped polyacetylene in the intermediate dopant-concentration region. Synthetic Metals 104, 179 (1999). 40 T.-C. Chung, F. Moreas, J.D. Flood, and A.J. Heeger. Solitons at high density in trans(CH)x: collective transport by mobile, spinless charged solitons. Physical Review B 29, 2341 (1984). T. Masui and T. Ishiguro. Spin gap behavior and electronic phase separation in doped polyacetylene. Synthetic Metals 117, 15 (2001). 41 S. Kivelson and A.H. Heeger. First-order transition to a metallic state in polyacetylene: a strong-coupling polaronic metal. Physical Review Letters 55, 308 (1985).
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42 Q. Zhu, J.E. Fishcher, R. Zuzok, S. Roth. Crys-
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46 47
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tal structure of polyacetylene revisited: An x-ray study. Solid State Communications 83, 179 (1992). H. Naarmann. Synthesis of new conductive polymers. Synthetic Metals 17, 223 (1987,) V.A. Marikhin, E.G. Guk, and L.P. Myasnikova. New approach to achieving the potentially high conductivity of polydiacetylene. Physics of the Solid State 39, 686 (1997). P.Y. Liu and J.F. Chen. Preparation of polyacetylene with a high conductivity. Journal of Materials Science & Technology 15, 387 (1999). J. Tsukamoto, A. Takahashi, K. Kawasaki. Structure and electrical properties of polyacetylene yielding a conductivity of 105 S/cm. Japanese Journal of Applied Physics 29, 125 (1990). K. Akagi, M. Suezaki, H. Shirakawa, H. Kyotani, M. Shimomura, and Y. Tanabe. Synthesis of polyacetylene films with high density and high mechanical strength. Synthetic Metals 28, D1 (1989). D.S. Suh, J.G. Park, J.S. Kim, D.C. Kim, T.J. Kim, A.N. Aleshin, and Y.W. Park. Linear highfield magnetoconductivity of doped polyacetylene up to 30 Tesla. Physical Review B 65, 165210 (2002). G.T. Kim, M. Burghard, D.S. Suh, K. Liu, J.G. Park, S. Roth, and Y.W. Park. Conductivity and magnetoresistance of polyacetylene fiber network. Synthetic Metals 105, 207 (1999). K. Ehinger. PhD Thesis. Konstanz, 1984. A.J. Epstein, H. Rommelmann, M. Abkowitz, and H.W. Gibson. Dependent conductivity of polyacetylene. Molecular Crystals and Liquid Crystals 77, 81 (1981). S. Roth, W. Pukacki, D. Schfer-Siebert, and R. Zuzok. Science and Technology of Conducting Polymers. M. Aldissi (Ed.), CRC Press, Boca Raton. D. Schfer-Siebert. PhD Thesis. Karlsruhe, 1988 D. Schfer-Siebert, C. Budrowski, H. Kuzmany, and S. Roth. Influence of the conjugation length of polyacetylene chains on the DC conductivity. Synthetic Metals 21, 285 (1987). A. Miller and E.A. Abrahams. Impurity conduction at low concentrations. Physical Review 120, 745 (1960).
51 N.F. Mott and E.A. Davis. Electronic Processes
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in Non-Crystalline Materials, 2nd Edition, Clarendon Press, Oxford, 1979. S. Roth. Hopping Transport in Solids 28, 377. M. Pollak and B.I. Shklovskii (Eds.), Elsevier, Amsterdam, 1991. D. Schfer-Siebert, C. Budrowski, H. Kuzmany, and S. Roth. Electronic Properties of Conjugated Polymers 76, 39. Springer Series in Solid-State Sciences, H. Kuzmany, M. Mehring, and S. Roth (Eds.), Springer Verlag, Heidelberg, 1987. K. Ehinger, S. Summerfield, W. Bauhofer, and S. Roth. DC and microwave conductivity of iodine-doped polyacetylene. Journal of Physics C: Solid State Physics 17, 3753 (1984). S. Summerfield, P.N. Butcher. A unified equivalent-circuit approach to the theory of AC and DC hopping conductivity in disordered systems. Journal of Physics C: Solid State Physics 15, 7003 (1982). S. Summerfield and P.N. Butcher. Analysis of AC and DC hopping conductivity in impurity bands and amorphous semiconductors. Journal of Physics C: Solid State Physics 16, 295 (1983). P. Sheng. Fluctuation-induced tunneling conduction in disordered materials. Physical Review B 21, 2180 (1980). C.O. Yoon, M. Reghu, D. Moses, and A.J. Heeger. Transport near the metal–insulator transition: polypyrrole doped with PF6. Physical Review B 49, 10851 (1994). G.T. Kim, M. Burghard, D.S. Suh, K. Liu, J.G. Park, S. Roth, and Y.W. Park. Conductivity and magnetoresistance of polyacetylene fiber network. Synthetic Metals 105, 207 (1999). A.J. Heeger. The critical regime of the metal– insulator transition in conducting polymers: experimental studies. Physica Scripta T 102, 30 (2002). S. Bhattacharyya and S.K. Saha. Room-temperature metallic behavior in nanophase polypyrrole. Applied Physics Letters 80, 4612 (2002). H.C.F. Martens, H.B. Brom, and R. Menon. Metal–insulator transition in PF6 doped polypyrrole: failure of disorder-only models. Physical Review B 64, 201102 (2001). M. Ahlskog, A.K. Mukherjee, and R. Menon. Low temperature conductivity of metallic conducting polymers. Synthetic Metals 119, 457 (2001).
References and Notes H.C.F. Martens, J.A. Reedijk, H.B. Brom, D.M. 62 L. Pietronero. Ideal conductivity of carbon p polymers and intercalation compounds. de Leeuw, and R. Menon. Metallic state in disSynthetic Metals 8, 225 (1983). ordered quasi-one-dimensional conductors. 63 S. Kivelson and A.J. Heeger. Intrinsic conducPhysical Review B 63, 073203 (2001). tivity of conducting polymers. Synthetic MetB. Chapman, R.G. Buckley, N.T. Kemp, A.B. als 22, 371 (1988). Kaiser, D. Beaglehole, H.J. Trodahl. Low-energy M. Kryszewski and J. Jeszka. Charge carrier conductivity of PF6-doped polypyrrole. Physitransport in heterogeneous conducting polycal Review B 60, 13479 (1999). mer materials. Macromolecular Symposia W.G. Clark, K. Tanaka, S.E. Brown, R. Menon, 194, 75 (2003). F. Wudl, W.G. Moulton, and P. Kuhns. High R.S. Kohlman, A. Zibold, D.B. Tanner, G.G. field NMR studies of conduction electron Ihas, T. Ishiguro, Y.G. Min, A.G. MacDiarmid, dynamics in metallic polypyrrole-PF6. Synand A.J. Epstein. Limits for metallic conductivthetic Metals 101, 343 (1999). ity in conducting polymers. Physical Review M. Ahlskog and R. Menon. The localization-inLetters 78, 3915 (1997). teraction model applied to the direct-current conductivity of metallic conducting polymers. [64] A.J. Heeger. Charge transfer in conducting polymers. striving toward intrinsic properties. Journal of Physics: Condensed Matter 10, Faraday Discussions of the Chemical Society 7171 (1998). 88, 203 (1989). T. Fukuhara, S. Masubuchi, and S. Kazama. 65 Y.W. Park. C.O. Yoon, and C.H. Lee. ConducPressure-induced metallic resistivity of PF6tivity and thermoelectric power of the newly doped poly(3-methylthiophene). Synthetic processed polyacetylene. Synthetic Metals 28, Metals 92, 229 (1998). D27 (1989). 58 H. Shirakawa and S. Ikeda. Infrared spectra of J. Plocharski, W. Pukaxki, and S. Roth. Conpolyacetylene. Polymer Journal 2, 231 (1971). ductivity study of stretched oriented new poly59 H. Naarmann, N. Theophilou. New process for acetylene. Journal of Polymer Science B 32. the production of metal-like, stable polyacety447 (1994). lene. Synthetic Metals 22, 1 (1987). 66 W. Pukacki, R. Zuzok, S. Roth, and W. Gpel. H. Naarmann. The development of electriElectronic Properties of Polymers: Orientacally conducting polymers. Advanced Materition and Dimensionality of Conjugated Sysals 2, 345 (1990). tems (Kirchberg IV). Springer Series in Solid H. Naarmann. Electronic Properties of ConjuState Sciences 107, 106. H. Kuzmany, M. gated Polymers. Springer Series of SolidMehring, and S. Roth (Eds.), Springer Verlag, State Sciences 76, 12. H. Kuzmany, M. MehrHeidelberg, 1992. ing, and S. Roth (Eds.), Springer Verlag, Hei67 A.B. Kaiser. Electronic Properties of Polymers: delberg, 1987. Orientation and Dimensionality of Conjugat60 K. Akagi, S. Katayama, M. Ito, H. Shirakawa, ed Systems (Kirchberg IV). Springer Series in and K. Araya. Direct synthesis of aligned cisSolid States Sciences 107, 98. H. Kuzmany, rich polyacetylene films using low-temperaM. Mehring, and S. Roth (Eds.), Springer Verture nematic liquid crystals. Synthetic Metals lag, Heidelberg, 1992. 28, D51 (1989). 68 A.B. Kaiser. Conductivity and thermopower of 61 BRITE/EURAM Action HICOPOL (Highly doped polyacetylene and polyaniline. MateriOriented Highly Conducting Polymers). als Science Forum 122, 13 (1993). Graz, Karlsruhe, Montpellier, Nantes, Strasbourg, Stuttgart, 1990–1995.
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Superconductivity This text is an exploration of how dimensionality, specifically one-dimensionality, expresses itself in the basic phenomena of electrical transport, and to a lesser degree, lattice dynamics. It therefore seems most appropriate to consider how the dimensionality of a solid might influence one of the most stunning observations in transport in the past century; superconductivity. In this chapter, we do not delve too deeply into the theoretical underpinnings of superconductivity in its many incarnations. Instead, a feeling for how dimensionality might play a role and a vision of how organics, or synthetic metal superconductors, might develop is presented. The many references can provide the interested reader with an extensive comparison of the most modern models. Suffice it to say that in most cases, quantitative models for organic systems (20 years after their discovery) are still hotly debated in the scientific community.
7.1
Basic Phenomena
Only a decade after Thomson’s discovery of the electron (1887) and Drude’s model of the free electron gas (1900), Heike Kamerlingh Onnes discovered that the resistivity of mercury suddenly vanishes when the sample is cooled slightly below the boiling point of liquid helium [1]. In Figure 7-1 Onnes’ data are reproduced. Onnes interpreted the phenomenon as a phase transition and he coined the term superconductivity’. Since then, about 40 elements have been found to become superconducting at low temperatures, some only at very low temperatures (e.g., Rh at 3.2 · 10–4 K), and some only at high pressure in addition to cooling (e.g., Si: Tc = 5.4 K at p > 110 kbar; Se: Tc = 6.9 K at p > 130 kbar); far more than 1000 superconducting alloys and compounds are known. The critical temperature’ at which the materials become superconducting is usually denoted Tc. Two other critical parameters for a superconducting material exist: the critical magnetic field, above which superconductivity breaks down, and the critical current density, which is the maximum current density a superconductor can support. In Table 7-1 superconducting materials are classified as elements, alloys, and intermetallic compounds, organic charge-transfer salts, fullerenes, and oxides. For One-Dimensional Metals, Second Edition. Siegmar Roth, David Carroll Copyright © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30749-4
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Figure 7-1 Heike Kamerlingh Onnes’ discovery of superconductivity in 1911. At 4.15 K the resistance of a mercury sample dropped from some tenths of an ohm to less than 10–5 ohms [1].
each class we selected the material with the highest Tc value known to date. An excellent introduction to superconductivity is the textbook by Buckel [2]. Books and review articles on organic superconductors are listed as Refs. [3–11]. Table 7-1
Classes of superconducting materials.
Class
Material with highest Tc
Tc (K)
Ref.
elements alloys and intermetallic compounds polymers organic charge-transfer salts fullerenes oxides
Nb Nb3Ge (SN)x k-(ET)2Cu[N(CN)2]Cl Cs2RbC60 Hg–Ba–Ca–Cr–O at 150 kbar
9.2 23.2 0.26 12.8 33 135 160
12 13 14 15 16 17
In presenting Onnes’ discovery we emphasized the words suddenly vanishes’. The disappearance of the resistance and consequently currents without resistance are very exciting. From our general knowledge on metallic conductivity (Section 6.1), we would not be surprised to find zero resistivity in a perfect, defect-free, three-dimensional metal at absolute zero, when all phonons are frozen out. However, zero resistance at a finite temperature, say, at 4.15 K as in Hg, or at 135 K as in the system Hg–Ba–Ca–Cr–O, or even at room temperature – that would contradict our intuition. But why, actually? Is it because with persistent currents we could build permanent magnets? Permanent magnets are not far beyond our experience. With superconductors we are able to transport energy without losses. This is important, but it is spectacular only after we know that for most of our economic goods – including energy – transport is more expensive than production (a fact most people are not aware of). In Section 6.3 we learned that the resistance of a thin wire does not increase linearly but exponentially with length, so why should we be surprised that in other systems there is no resistance at all?
7.1 Basic Phenomena
We also emphasized the word suddenly’. The superconductive phase transition is remarkably sharp. The German word Sprungtemperatur’ expresses this sharpness better than the English term critical temperature’ (Sprung = jump). It is difficult to imagine that all possible electron scatterers suddenly vanish. It is easier to assume that the electrons change in such a way that they do not scatter any more. A phase transition in the electron gas is essential for the presently accepted theory of superconductivity, which was formulated in 1957 by Bardeen, Cooper, and Schrieffer [18] (note the long time span of almost half a century that passed from Onnes’ discovery to the BCS theory’). The BCS theory assumes an attractive interaction between the electrons. This interaction is the exchange of phonons. Phonons bind’ two electrons together to form Cooper pairs. But Cooper pairs are quite unusual particles: they do not stick together like two protons forming a hydrogen molecule but are paired in a more general way, they are correlated. This means that their spins and momenta are coupled so that the electrons belonging to one pair move in opposite directions, thus pairing momentum p with momentum –p. (In human relationships sometimes such pairs are found, where one partner always does the opposite of what the other one does (Figure 7-2).) The pair correlation is effective over a characteristic length, called the coherence length n. Usually n is between 1000 and 1 lm, whereas the distance between two electrons in a solid is on the order of 1 . Consequently, the Cooper pairs interpenetrate greatly, and it is perhaps not too difficult to understand that an ensemble of interpenetrating Cooper pairs behaves very differently from a gas of noninteracting electrons. In addition to zero resistivity, the model of the coherent Cooper pair soup’ also explains many other salient properties of the superconducting state: specific heat, thermal conductivity (infinite electrical but no thermal conductivity of the electrons),
Figure 7-2 In human relations sometimes couples are found, where one partner does the opposite of what the other one does.
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and ideal diamagnetism. In 1933, Meissner and Ochsenfeld [19] found that a superconductor expels an applied magnetic field so that within the superconductor the magnetic field is zero (Meissner effect). The latter property offers a convenient contact-fee method for measuring Tc. The importance of electron–phonon coupling is reflected in the BCS expression for the critical temperature: hxD 1 exp (1) Tc = 1.13 NðEF ÞV kB h is where xD is the Debye frequency (a characteristic phonon frequency), h = 2p Planck’s constant, kB is Boltzmann’s constant, N(EF) is the electronic density of states at the Fermi energy, and V* is a constant characterizing the electron–phonon interaction. Eq. (1) can be used to explain the high critical temperatures of the A15 superconductors mentioned in the first chapters. Here, the Fermi level is assumed to be within a narrow band dominated by the d orbitals of the transition elements (e.g., Nb in Nb3Ge), and hence the density of states at the Fermi energy is large. Similarly, the increase of Tc in alkali-doped fullerene crystals when going from Na over K to Cs is interpreted along these lines: the fullerene lattice swells’ by alkali intercalation, the more so for dopants with larger ionic radii. This leads to a smaller overlap of the molecular p orbitals, thus to narrower bands and, via Eq. (1), to higher Tc. In the BCS theory the electrons interact by the exchange of phonons. The relevance of phonons can be tested by the isotope effect. In Eq. (1) the critical temperature Tc is proportional to the Debye frequency xD. This frequency should vary with the square root of the atomic mass M. Consequently we expect Tc M –1/2
(2)
(for a harmonic oscillation x = (F/M)1/2, where F is the force constant and M the mass). The mass can be varied without changing the force constant by exchanging the isotope. This test has been carried out for many elements, and in general Eq. (2) has been verified. Deviations can be explained by taking into account the possible isotope dependence of V* in Eq. (1). For one-dimensional and organic superconductors, often other interactions between the electrons are discussed, instead of or in addition to, the exchange of phonons. Arguments for or against such hypotheses can be obtained from studies of the isotope effect. In organic materials, the carbon isotope 12C (natural abundance 99%) can be replaced by 13C (natural abundance 1%), or protons (natural abundance 99.985%) can be replaced by deuterons (natural abundance 1.5 · 10–4%). In Chapter 4 one example of electron–phonon coupling has already been discussed: the Peierls transition. Superconductivity is another example. The Cooper pairs are very abstract entities: momentum-correlated electrons. The Peierls pairs are less exotic: these are electrons paired in direct space, and in particular, for conjugated double bonds, they are just the p-electron pairs symbolized as solid lines in chemical formulae – at least within the rough approximations used in Chapters 4
7.1 Basic Phenomena
Figure 7-3 Density of states for unpaired electrons in a superconductor. The dotted line indicates the density of states when the sample is normally conducting (T > Tc). The formation of Cooper pairs removes states within EF € D and thus a gap is created.
and 5. We learnt that in the Peierls transition electron–phonon coupling leads to a gap at the Fermi level in the electronic density of states. Similarly, in superconductivity there is also a gap at the Fermi level (Figure 7-3). At absolute zero the BCS theory predicts a width of 2D for the gap: 2D(0) = 3.5 kBTc
(3)
In Peierls systems we encountered solitons as new particles. Their energetic states are in the Peierls gap, in the ideal situation, exactly at midgap (e.g., Figure 5-20). In a superconductor there are other novel particles: the Cooper pairs. The question might be posed, where in Figure 7-3 the Cooper pairs ought to be drawn. But unlike solitons, Cooper pairs cannot be depicted in a diagram of the single-electron density of states. In Section 2.1 we briefly alluded to the BCS gap when mentioning that the occurrence of superconductivity stops the Martensit distortion. Competition between various low-temperature instabilities is a salient feature of one-dimensional metals, which we will take up again in Chapter 8 when dealing with g-ology’. The theory and analysis above forms the basis from which superconductivity derives. There are, however, further subtleties that make the parallel with Peierls systems even more compelling. Most important, as already hinted at above, the division of classes of superconductors suggests that different phenomena (other than simply higher Tc) are observed in the compound oxides than in the elemental superconductors. In fact, this is correct – the so-called high-Tc’ materials are very different. Although the fundamental mechanisms of pairing leading to a superconducting gap are valid, other curious characteristics arise in these materials. An example
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is that of the critical magnetic field. As stated above, the elemental superconductor first discovered by Onnes was sensitive to applied magnetic fields with a certain critical field, for a given temperature, destroying the superconducting state. Below that critical field strength Bc, the type-I (elemental) superconductor is a perfect diamagnet. In the type II, or compound superconductors, there are two such distinctive field values: Bc1 and Bc2. For a magnetic field B less than Bc1, the superconductor exhibits a type-I response to the field. For a magnetic field B greater than Bc2, the superconducting state is destroyed, as also for type-I. However, something unique occurs when the magnetic field is between Bc1 and Bc2. Here, the superconductor has zero resistance but allows partial flux penetration. This is referred to as a vortex state and is pictured as cores of normal material in which the magnetic field penetrates, surrounded by material in the superconducting state responding to that field. Naturally, as the applied magnetic field increases, the number of normal cores also increases until, eventually, the material can no longer sustain any more. At this point Bc2 has been reached and the superconducting state collapses. But, why should such an intermediate state exist in one type of superconductor and not in the other? The answer comes from a more careful examination of the structure of the type-II superconductors (and this is where the subtle parallel to Peierls systems occurs). In most of the type-II high-Tc materials, the ceramic’s crystal structure is composed of two-dimensional planes of conducting oxide. In most examples of high-Tc materials, a conducting set of metal oxide planes (such as CuO) is surrounded by a set of rare earth elements – yttrium, lanthanum, neodymium, gadolinium, and erbium. In the compound YBa2Cu3O7-x (a black, orthorhombic material, where x is the variable for oxygen content), for instance, CuO planes act as two-dimensional sheets of conductivity. In fact, the critical temperature of the superconducting state depends intimately on the overall stoichiometry of the CuO sheets, with the Tc rising as the oxygen content rises. Although the YBa2Cu3O7-x (generally abbreviated YBCO) compound is one of the most widely studied, this behavior seems to hold true for the rest of the high-temperature superconductors as well. This is not to say that a layer of CuOx can superconduct on its own. It was recognized early on that a charge reservoir was necessary for the establishment of electron pairing. This is the role of the surrounding crystal structure, a role quite similar to that of the dopant ions in conducting polymer systems [20]. As we will see below, this parallel is not accidental and one may expect superconducting behavior in some organic systems. However, we are not quite finished. Clearly, the example of the ceramic superconducting systems shows that superconductivity can occur under conditions not previously anticipated by scientists, and even under conditions we would have thought prevented it from occurring (after all, gadolinium is magnetic!). This has happened again recently for MgB2 [21]. This common compound becomes superconducting at 39 K, which is amazing for a metallic material.
7.2 Measuring Superconductivity
7.2
Measuring Superconductivity
A vast armada of experimental methods has been applied to superconductors. This section focuses on only a few hints for those who want to find out whether or not they have synthesized a new superconductor. Because superconductivity is a sharp phase transition the effect is difficult to miss, when resistance or magnetic susceptibility are followed as a function of temperature. Complications arise because the samples frequently are small, oddly shaped, brittle, and hydroscopic or pyrophorous – and often low temperatures, high pressures, and magnetic fields have to be applied simultaneously. Often the bulk of the sample has turned out to be nonsuperconducting, but there was a small superconducting contamination, which manifested itself as a little kink in the resistance-versus-temperature curve and then – following this hint – a new superconductor was found. For resistance measurements, we refer to Section 6.2. Because we are mainly interested in resistance jumps and not in absolute values, the exact geometry of the sample does not matter. A four-lead method is advisable, but it is not necessary to apply the four points or the van der Pauw technique. Often measurements can be carried out on pressed pellets or even on sufficiently well-squeezed powder samples, because the contact resistance between the grains is bridged by the Josephson effect (superconducting tunneling through barriers, see Buckel’s textbook [2]). In new and exotic materials, however (for example, in organic superconductors and fullerenes), the coherence length might be very small, and the overall resistance of a granular film would not change although the core of the granuli could be superconducting. A superconductivity test independent of external and internal contact resistances is the measurement of the magnetic susceptibility. This method makes use of the Meissner effect, which expels an applied magnetic field from the superconductor. Many chemical laboratories are equipped with SQUID magnetometers, and one can easily determine susceptibility-versus-temperature curves. Simpler alternatives are
Figure 7-4 Simple homemade mutual inductance device to measure superconducting transitions through the neck of a standard helium container.
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mutual induction measurements of small concentric coils (Figure 7-4). It is easy to wind a transformer, say, two concentric coils of 1000 windings of an insulated thin copper wire. The whole device can be made so small that it fits through the outlet of a liquid helium container. The inner coil can be excited by the reference signal of a lock-in amplifier and the induction signal picked up at the outer coil is fed to the amplifier input. The sample is placed inside the coils and, when it becomes superconducting, the induction suddenly changes (the inductance of a coil depends on the susceptibility of the material inside the coil). It is not very difficult to modify the device so that it fits into dilution refrigerators, high-pressure cells, or other sophisticated sample environments. To calibrate the device, a small piece of a known superconductor is placed in the coil together with the sample. Since the inductance change is proportional to the volume becoming superconducting, this method allows one to estimate the superconductive volume fraction of the sample. As a typical example of the use of this technique, consider the work done by S. Roth using the mutual inductance method to measure superconductivity in powder samples of potassium-intercalated graphite. This material is very pyrophorous. Therefore the potassium–graphite powder was mixed with vacuum grease under argon atmosphere. This was sufficient to maintain an oxygen-free environment for further sample handling. The paste was squeezed into the coils, together with a short piece of iridium wire stuck in as a calibrator (Tc of iridium is 0.14 K). The device was put into a dilution refrigerator, and both the iridium and the potassium– graphite transitions were easily observed. Unfortunately, no transitions were observed when several potassium-doped conjugated polymers were treated the same way, but the investigations were not sufficiently systematic to completely exclude the possibility of superconductivity in potassium-doped polymers in the temperature range of a dilution refrigerator. Zero resistance and ideal diamagnetism are convincing evidence of superconductivity. But the experimental methods discussed above do not yield absolute values; they reveal changes in the conductivity and changes in the susceptibility. Therefore, precautions must be taken before relating the observed changes to the occurrence of superconductivity. One (trivial) piece of advice is to measure the empty sample holder or to run controls with nonsuperconducting dummy samples. The inductance method is especially sensitive to the presence of solder (which contains lead) or other superconductive additives in the vicinity of the coil. Another precaution is checking the superconducting state by the application of magnetic fields. Superconductivity is destroyed by magnetic fields. High critical temperature implies high critical field, with the details depending on the type of superconductor. If Tc is very low, in the mK range, the earth’s magnetic field might be strong enough to destroy superconductivity. On the other hand, the critical fields of high Tc superconductors can be as high as 100 Tesla or more. For ET’ salts with Tc » 0 K (see Section 7.5) the critical field is about 20 T. When resistance or susceptibility anomaly is observed that is independent of the applied magnetic field, we should hesitate to assign it to superconductivity.
7.3 Applications of Superconductivity
7.3
Applications of Superconductivity
Superconductivity is an exciting physical phenomenon. In spite of all the excitement, it took a surprisingly long time – half a century – before it could be theoretically explained, and irrespective of all the enthusiasm, it seems to be taking even longer until superconductivity can be commercially used. Today, superconductivity has found important economic niches, but superconductivity certainly is not yet a major part of our technological environment, and regardless of all the new and high-temperature superconductors, the early discovered modest-Tc niobium is still the most commonly used superconducting material. An obvious application is superconducting cables, and both superconducting DC and superconducting AC cables have been developed. Concerning energy transport, not only losses have to be minimized, but the price of the material, ease of processability, cost of maintenance, and land prices have to be considered as well. So far, superconducting cables have not yet become competitive. A soon-to-come transport application of superconductivity might be interconnects in integrated circuits. Here, superconductivity would have the advantage of avoiding Joule heating, which is one of the major problems in today’s high-performance computers (for this purpose, high-Tc superconductors are needed, because at the temperature of liquid helium silicon semiconductor devices fail to work properly). Most laboratory magnets use superconducting coils. This application of superconductivity is well known among scientists, but it does not involve a large market volume. From the economic point of view, magnets for NMR spectrometers are more important, particularly for NMR tomographs in hospitals. Large magnets for bubble chambers as high-energy particle detectors and for beam guidance in accelerators are superconducting, but this application is rather la science pour la science’ than the general economy. In the future, superconducting magnets for levitation trains and for plasma confinement in nuclear fusion reactors are foreseeable. For levitation trains, other alternatives to superconducting magnets exist and are used. However, several more decades will pass before the first commercial fusion reactor is actually running. (Requiring about one century of continuous effort until the economic breakeven point, nuclear fusion is one of the most remarkable technological endeavors.) SQUIDs (Superconducting quantum interferometer devices) are very sensitive magnetometers. They have nearly replaced the Faraday balance in chemical research laboratories. Their extreme sensitivity even allows the magnetic field associated with electric currents in the human brain to be measured; thus SQUIDs open new fields in brain research. Because of the very short switches have been discussed as an alternative to silicon semiconductor technology in electronic data processing, and some companies have pushed the development of computers based on the Josephson effect. As we have seen, there are many promising ideas for the technological application of superconductivity, some promises have been realized, but there was no technological revolution. In many cases, from the economic point of view, superconductivity has turned out to be not more than the second-best solution. Usually this was
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not a question of high Tc, but rather of the mechanical properties of the superconducting material, of its processability, and of reproducible control in microstructuring (Josephson computer). It is not very likely that a one-dimensional metal will develop into a widely applied superconducting material, but we expect that the knowledge gained by investigation of these substances will help to improve other useful materials.
7.4
Superconductivity and Dimensionality
As we have seen, the superconducting state seems to be a delicate interplay of lattice instability and electron correlation. Although we must recognize that dimensionality must play some role in overall allowed correlation lengths in a system, clearly, lattice stability can be more easily tuned in low-dimensional systems, and the quest for high-temperature superconductors has been one of the driving forces in the field of one-dimensional metals. For a long time, intermetallic compounds with A15 structure (Section 2.1) held the world record for high Tc. Because of the van Hove singularities in their one-dimensional band structure, there are sharp peaks in the electronic density of states. If the Fermi level happens to fall in such a peak (which can happen when materials are sufficiently doped), N(EF) is large and Eq. (1) predicts a high Tc. The same singularities lead to lattice instabilities and to soft value for V* in Eq. (1) and again to high Tc. Of course, there should be an upper limit to Tc because at some point the lattice will become too unstable. The search for better compromises between density of states and lattice instability has motivated a number of exotic approaches, including examining materials that superconduct under high pressure. Figure 2-1 shows a reproduction of Little’s proposal [22] of a conjugated chain with appropriate side groups that would superconduct. Electrons moving along the chain would be spin- and momentum-correlated by excitations in the side groups. These excitations take the role of the phonons in a traditional BCS superconductor. Figure 2-2 shows an example of a sidegroup excitation that might be helpful for superconductivity. According to Little’s estimates this mechanism could lead to Tc values as high as room temperature. Synthesis efforts following Little’s course very closely have not led to new superconducting materials. In fact, (SN)x a polymer composed of sulfur–nitrogen chains (Section 2.5) is the only polymer that has become superconducting so far. Its transition temperature is very low (Tc = 0.26 K). In addition, the chains are close: they sufficiently interact to suppress one-dimensionality. However, superconductors in which one-dimensional aspects are most pronounced are the Bechgaard salts (Section 2.4 and 7.5). Here again, the competition with the other instabilities typical of one-dimensional metals limits Tc to a few Kelvin. This same competition exists to some degree in the oxide superconductors [17,23,24], as discussed above, but at least some aspects of one-dimensionality or low-dimensionality also exist in the cuprates.
7.5 Organic Superconductors
7.5
Organic Superconductors
Organic superconductors can be subdivided into one-, two-, and three-dimensional solids (provided that fullerenes are classed with the organics; in Table 7-1 fullerenes are listed as a separate class) [25]. This subdivision is based on the topology of the Fermi surface (Section 3.3). As a reminder, one-, two-, and three-dimensional Fermi surfaces are shown in Figure 7-5. In low-dimensional solids the Fermi surfaces are open; they consist of two parallel planes for one-dimensional solids and cylinders for two-dimensional solids. Weak interactions in higher dimensionality (interchain and interplane) lead to warping and barrel-like distortions until the Fermi surface finally becomes closed; in this case we speak of a three-dimensional solid. The study of the Fermi surface has been nicknamed fermiology. Experimental tools in fermiology are oscillations in the magnetoresistance (Shubnikov–de Haas, SdH’, oscillations) and oscillations in the magnetic susceptibility (de Haas–van Alphn, dHvA’, oscillations). These oscillations occur when the resistance or the susceptibility is measured as a function of the magnetic field. The magnetic field exerts a lateral force on moving electrons (Lorentz force) and with a strong enough field, the electrons move in circles (Landau orbits). The radius of a Landau orbit depends on the magnetic field, and the passage of the Landau orbits through the Fermi surface leads to oscillations. In previous decades high-purity crystals of silver, gold, copper, etc., were studied extensively using this method; a recommended monograph is
Figure 7-5 Different shapes of Fermi surfaces: quasi onedimensional (top), quasi two-dimensional (middle), and anisotropic three-dimensional (bottom) [9].
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Figure 7-6
Shubnikov–de Haas oscillations in bH(ET)2I3 [27].
Shoenberg’s book [26]. Some of the organic charge-transfer salts form very perfect single crystals and these materials have led to a renaissance in fermiology [10]. As an example we show the SdH oscillations of bH(ET)2I3 in Figure 7-6 [27]. The dimensionality can be deduced from the angular dependence of the oscillations when the sample is rotated in the magnetic field. (In one-dimensional metals with totally open Fermi surfaces – top graph in Figure 7-5 – oscillations are not observed because the electrons cannot move in circles.) 7.5.1
One-dimensional Organic Superconductors
(TMTSF)2PF6 is the organic charge-transfer salt in which superconductivity was observed for the first time [28]. A pressure of 12 kbar had to be applied, and superconductivity occurred at Tc » 0.9 K. (TMTSF)2CIO4 is the only member of the onedimensional family that becomes superconducting at ambient pressure (Tc = 1.2 K, [27]). The transition temperatures of TMTSF superconductors are listed in Table 7-2. TMTSF is an abbreviation for tetramethyl tetraselenofulvalene (Figure 7-7). It is related to tetrahiafulvalene TTF (Figure 7-8), the sulfur atoms being replaced by selenium atoms and four methyl groups added. The anions PF6, AsF6, SbF6, TaF6, ReO4, and ClO4 act as electron acceptors, but beyond that they do not participate in metallic conductivity or superconductivity.
Figure 7-7
TMTSF – tetramethyl tetraselenofulvalene.
7.5 Organic Superconductors Table 7-2
Critical temperatures of TMTSF superconductors (after Williams et al. [7]).
Compound
Tc (K)
Pressure (kbar)
(TMTSF)2PF6 (TMTSF)2AsF6 (TMTSF)2SbF6 (TMTSF)2TaF6 (TMTSF)2ReO4 (TMTSF)2FSO3 (TMTSF)2ClO4
0.9 1.1 0.4 1.4 1.3 2.1 1.4
12 12 11 12 9.5 6.5 ambient
Figure 7-8
Figure 7-9 Different overlaps of the electronic wave functions forming the conduction bands in conjugated polymers and in molecular crystals: (a) Overlap of atomic p orbitals forming the p band of conjugated polymers and the molecular p orbitals in aromatic systems
TTF – tetrathiafulvalene.
(benzene, fulvalene, etc.). Two chains are shown. (b) Overlap of the atomic p orbitals (or the molecular p orbitals) forming the conduction band of molecular crystals (e.g., of charge-transfer salts). Two stacks are shown.
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Conductivity is exclusively due to the overlapping p orbitals of the cations. This overlap is different from the p-orbital overlap forming the p bands in conjugated polymers. The different types of overlaps are schematically shown in Figure 7-9. Overlapping in conjugated polymers occurs sidewise, along the polymers axis, and leads to very broad bands, W » 10 eV; overlapping in charge-transfer salts is top-tobottom, along the stacking axis, and leads to rather narrow bands, W » 1 eV. Conjugated polymers are intramolecular one-dimensional conductors, charge-transfer salts are intermolecular conductors. (There is also an intermolecular, for example, interchain overlapping in conjugated polymers with W = 1 eV and an interstack overlap in TMTSF charge-transfer salts with W << 1 eV.) Sometimes it has been argued that superconductivity in seleno organic compounds is not very spectacular, because selenium itself is a superconductor, albeit at very high pressures (Tc = 6.9 K for p > 130 kbar). The ET salts discussed in the next section are sulfur-based organic superconductors and, from that point of view, perhaps more exciting. The importance of the TMTSF compounds lies in their onedimensionality and in the possibility they offer to study the competition between superconductivity and other one-dimensional instabilities. (TMTTF)2Br2 was later found to become superconducting at 26 kbar; Tc = 0.8 K. It is the first sulfur-based superconductor in the (TM)2X series [30]. Figure 7-10 sows the generalized phase diagram of the (TM)2X series, where TM stands for tetramethylene derivatives of thio or selenofulvalene and X denotes inorganic anions (PF6, Br, ClO4). We move along the abscissa by applying external pressure or by changing the chemical composition and thus exerting internal pressure’. The arrows indicate where the respective compounds are located at ambient external pressure. The region in which superconductivity occurs in denoted SC. We discuss
Generalized phase diagram of the (TM)2X series of one-dimensional solids [9,11]. The arrows indicate the position of the respective substances at ambient pressure: 1 = (TMTTF)2PF6; 2 = (TMDTDSF)2PF6 Figure 7-10
(tetramethylene dithiodiselenofulvalene); 3 = (TMTTF)2Br; 4 = (TMTSF)2PF6; 5 = (TMTSF)2ClO4; SC: superconducting; SDW: spin density wave; SP: spin Peierls; LOC: localized states.
7.5 Organic Superconductors
the nonsuperconducting phases in the next chapter. An important feature of diagram is the vicinity of superconductivity (SC) and spin density waves (SDW). At ambient pressure only (TMTSF)2ClO4 shows a normal-to-superconducting transition upon cooling. All other compounds go into an insulating state, which can be suppressed by about 12 kbar for (TMTSF)2PF6. The TMTSF compounds have been extensively reviewed in Refs. [4,31,32] and discussion of additional one-dimensional superconducting systems can be found in [7–8,11]. 7.5.2
Two-dimensional Organic Superconductors
In one-dimensional superconductors the competition with the other instabilities seems to limit the critical temperature. Therefore stronger interstack coupling appears to be desirable. The ET salts exhibit fairly strong lateral coupling between the cations, so that they are two-dimensional rather than linear. As evidenced from fermiology, the Fermi surface is cylindrical and the conductivity is isotropic in the plane of the donor molecules (the in-plane to out-of-plane anisotropy is about 103). Figure 7-11 shows the chemical structure of bis(ethylenedithio)tetrathiafulvalene, BEDT-TTF or, shortened, ET. It is a sulfur-based relative of TTF and contains eight instead of four chalcogen atoms. (ET)2ReO4 was the first sulfur-based organic superconductor (Tc = 2 K at p > 4.5 kbar [33]), k-(ET)2Cu[N(CN)2]Br is the superconductor of the group with the highest ambient pressure Tc (11.6 K) [34], and k(ET)2Cu[N(CN)2]Cl with Tc = 12.8 K at a pressure of 0.3 kbar [15] holds the absolute Tc record in this competition. Table 7-3 summarizes the ET superconductors. Note that the ET donor can be combined with a large variety of acceptors and the system exhibits a rich diversity in structures. Some groupings, like the b and the k families, exist, but for the unskilled observer the structure is not easy to comprehend. We note that ET layers exist in both families (in the b family ET molecules from a honeycomb-like sulfur network, in the k family it is more complicated). These donor layers are separated by layers of acceptor molecules. For further details, please refer to Refs. [7] and [8]. The layered structure of ET salts advances these substances into the vicinity of the high-temperature ceramic copper oxide superconductors, which also consist of metallic’ layers (here, CuO2 layers) separated by inert’ spacers and counterions [35]. In both classes of superconductors the coherence length n is very anisotropic. In k-(ET)2Cu[N(CN)2]Br n^ = 37 within the plane of the donor layer and n|| = 4 perpendicular to the plane. The latter value is remarkable, because it is
Figure 7-11
BEDT–TTF or ET-bis(ethylenedithio)tetrathiafulvalene.
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Critical temperatures of ET superconductors (after Williams [8]).
Compound
Tc (K)
Pressure (kbar)
(ET)2ReO4 b-(ET)2I3 b*-(ET)2I3 c-(ET)3(I3)2.5 h-(ET)2I3 k-(ET)2I3 a (ET)2I3 (a/b)-(ET)2I3 b-(ET)1.96(MET)0.04I3 b-(ET)2IBr2 b-(ET)2AuI2 k-(ET)4Hg3-8Cl8
2.0 1.4 8.0 2.5 3.6 3.6 7–8 2.5–6.9 4.6 2.8 4.98 1.8 5.3 4.3 6.7 2.0 1.15 2.0 10.4 5.0 11.6 12.8
4.5 ambient 0.5 ambient ambient ambient ambient ambient ambient ambient ambient 12 29 ambient 3.5 ambient ambient 16 ambient ambient ambient 0.3
k-(ET)4Hg2.89Br8 (ET)2Hg1.41Br4 a-(ET)2[(NH4)Hg(SCN)4] (ET)3Cl2·2H2O k-(ET)2Cu(NCS)2 k-(ET)2Ag(CN)2·H2O k-(ET)2Cu[N(CN)2]Br k-(ET)2Cu[N(CN)2]Cl
much smaller than the distance (» 15 ) between the organic layers [7], thus stimulating discussions on unconventional interplane coupling mechanisms. Studying ET salts might help to understand copper oxide superconductors and vice versa. 7.5.3
Three-dimensional Organic Superconductors
In Section 1.3.4 we introduced fullerene as a zero-dimensional modification of carbon. We can look at fullerene from many different view points. Because in a C60 ball 60 p electrons are confined in a sphere about 10 in diameter, a fullerene molecule can be regarded as a quantum dot, which is a quasi zero-dimensional object. But 60 is already a fairly large number, and so a C60 molecule is also a minisolid. We can think of it as a rolled-up graphite platelet or as a coiled-up chain of a conjugated polymer. We would expect to find aspects of both polyacetylene and graphite in the behavior of fullerene. The minisolid-like C60 molecules form a regular array, the three-dimensional fullerene crystal, which we could call a macrosolid; the overlap in Figure 7-9a is relevant to the minisolid; the overlap in Figure 7-9b is relevant to the macrosolid. As conjugated polymers and as graphite, fullerene can be intercalated with various dopants. In conjugated polymers and graphite, both acceptors and donors can
7.5 Organic Superconductors
Crystal structure of fullerene. The hatched spheres indicate the interstitial sites, which can be occupied by dopant ions [36].
Figure 7-12
be intercalated, in fullerene so far only donor doping has been possible. Figure 7-12 shows a plane of a fullerene crystal. The hatched circles mark interstitial sites (tetragonal and octahedral), which can be occupied by dopant ions. In Figure 7-13 we present a more sophisticated view of the spatial arrangement of C60 balls and dopant ions. We know that alkali–graphite intercalation compounds become superconducting. Usually the critical temperature is below 1 K, but in the Cs–Bi–graphite system Tc
Summary of structure types of alkali-intercalated C60. (The A15 structure has been observed only in the alkaline earth phase, Ba3C60.) [37] Figure 7-13
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Figure 7-14 Variation of Tc with the lattice parameter a0 for various compositions of A3C60 [36].
values up to almost 5 K have been observed. Alkali-doped fullerenes are also superconductors and, surprisingly, the critical temperatures are as high as 33 K. Thus, they beat the ET charge-transfer salts by a factor of three. For a comparison of graphite and fullerene superconductors, we refer to Lders [36] and to the proceedings of the 1994 Warsaw Autumn School on New Materials with Conjugated Double Bonds [38]. We mentioned in Section 7.1 that the C60 balls in a fullerene crystal move farther apart when the material is doped. This increases the density of states at the Fermi surface and thus Tc increases. This behavior is nicely shown in Figure 7-14, where the critical temperatures of A3C60 are plotted versus the lattice parameter a0 (in A3C60 the symbol A stands for alkali metal). The intensive study of fullerenes has just begun. People now work on chemically modifying the carbon balls. It might well turn out that, with the presently highest critical temperature of 33 K [16], the maximum Tc has not yet been reached for this material class. For reviews and monographs on fullerenes see Refs. [39–45]. We should recognize, however, that recent developments of smaller fullerene derivatives such as C20 [46] and C36 have suggested modifications to our view of threedimensionally-based organic superconductors. Specifically, these smaller cage molecules can have extraordinarily high density of states [47] at the Fermi level. Furthermore, they exhibit significantly stronger electron–phonon coupling constants than their large cage counterparts (C60, C70, C80) [48]. Clearly, there may be more to come from this system.
7.6
Future Prospects
In research and development it is not possible to predict the future by extrapolating from the past. Breakthroughs are unforeseen, otherwise they would not be called breakthroughs. In retrospect, most evolutions begin exponentially, continue linearly,
7.6 Future Prospects Table 7-4
Chronology of organic superconductors (partly after Williams et al. [8]).
Year
Organic Superconductors
1962 1970 1973 1978 1979 1981 1983 1985 1986 1987 1987 1990
synthesis of TCNQ synthesis of TTF TTF–TCNQ (first organic metal’) synthesis of ET Bechgaard salts (TMTSF)2X, superconducting under pressure, Tc » 1 K (TMTSF)2ClO4, first organic superconductor at ambient pressure, Tc » 1.4 K b-(ET)2ReO4, first sulfur based organic superconductor under pressure, Tc » 1 K synthesis of C60 superconductivity in La–Ba–Cu–O Tc above liquid-nitrogen temperature k-(ET)2Cu(SCN)2, Tc » 10.4 K k-(ET)2Cu[N(CN)2]Br, Tc » 11.6 K k-(ET)2Cu[N(CN)2]Cl, Tc » 12.8 K at 0.3 kbar C60 available in large quantities superconductivity in alkali-doped C60 Tc = 133 K in Hg–Ba–Ca–Cu–O Tc = 160 K under pressure in Hg–Ba–Ca–Cu–O
1990 1991 1993 1994
and then saturate. But there is no way of foretelling the saturation level from the exponential start. Nevertheless, it is fascinating to look at the historic developments in a certain field of science and to speculate on the future. In this respect it does not matter whether the selection of historic events is biased or not, because we use the data only as a ladder for our fantasy. A selection of historic events in organic (and high-Tc) superconductivity is given in Table 7-4. The highest Tc values are plotted versus the year of the respective observation. If the various compounds are classed into teams, races can be arranged. Figure 7-15 shows the competition between metallic, organic, oxide [35], and fullerene superconductors. We have the impression that the oxides will win, if the goal is to achieve room-temperature superconductivity. In the race between organics and inorganics shown in Figure 7-16, it appears that the organics will overtake soon.
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History of superconductivity; transition temperatures are plotted versus the year of discovery. On the right side the boiling points of helium, hydrogen, and nitrogen are marked.
Figure 7-15
Figure 7-16 Superconductivity transition temperature Tc versus year of discovery, illustrating the race between inorganic and organic superconductors [8].
References and Notes
References and Notes 1 H.K. Onnes. Communication from the Physi-
cal Laboratory of the University of Leiden 120b (1911). 2 W. Buckel. Supraleitung – Grundlagen und Anwendungen. 3rd Edition, Physik-Verlag, Weinheim, 1984. 3 V.Z. Kresin and W.A. Little. Organic Superconductivity. Plenum, New York, 1990. 4 G. Saito and S. Kagoshima. The Physics and Chemistry of Organic Superconductors. Springer Verlag, Berlin, 1989. 5 T. Ishiguro and K. Yamaji. Organic Superconductivity. Plenum, New York, 1990. 6 A. Khurana. Highest Tc continues to increase among organic superconductors. Physics Today 43 (September), 17, 1990. 7 J.M. Williams, A.J. Schultz, U. Geiser, K.D. Carlson, A.M. Kini, H.H. Wang, W.-K. Kwok, A.-H Whangbo, and J.E. Schirber. Organic Superconductors – New Benchmarks. Science, 252, 1501 (1991). 8 J.M. Williams, J.R. Ferraro, R.J. Thorn, K.D. Carlson, U. Geiser, H.H. Wang, A.M. Kini, and M.-H. Whangbo. Organic Superconductors (Including Fullerenes): Synthesis, Structure, Properties, and Theory. Prentice-Hall, Englewood Cliffs, 1992. J.M. Williams; H.H. Wang, M.A. Beno, T.J. Emge, L.M. Sowa, P.T. Copps, F. Behroozi, L.N. Hall, K.D. Carlson, and G.W. Crabtree. Ambient-pressure superconductivity at 2.7 K and higher temperatures in derivatives of b(ET)2IBr2: synthesis, structure, and detection of superconductivity. Inorganic Chemistry 23, 3839 (1984). A.M. Kini, U. Geiser, H.H. Wang, K.D. Carlson, J.M. Williams, W.K. Kwok, K.G. Vandervoort, J.E. Thompson, D.L. Stupka, D. Jung, M.-H. Whangbo. A new ambient-pressure organic superconductor, k-(ET)2Cu[N(CN)2]Br, with the highest transition temperature yet observed (inductive onset Tc = 11.6 K, resistive onset = 12.5 K). Inorganic Chemistry 29, 2555 (1990). J.M. Williams, A.M. Kini, H.H. Wang, K.D. Carlson, U. Geiser, L.K. Montgomery, G.J. Pyrka, D.M. Watkins, J.M. Kommers, S.J. Boryschuk, A.V. Strieby-Crouch, W.-K. Kwok, J.E. Schirber, D.L. Overmeyer, D. Jung, and M.-H. Whangbo. From semiconductor-semiconductor transition (42 K) to the highest-Tc organic
superconductor, kappa (ET)2Cu[N(CN)2Cl (Tc = 12.5 K). Inorganic Chemistry 29, 3272 (1990). J.A. Schlueter, U. Geiser, J.M. Williams, H.H. Wang, W.K. Kwok, J.A. Fendrich, K.D. Carlson, C.A. Achenbach, J.D. Dudek, D. Naumann, T. Roy, J.E. Schirber, and W.R. Bayless. The first organic cation-radical salt superconductor (Tc = 4 K) with an organometallic anion: superconductivity, synthesis and structure of k-(ET)2M(CF3)4(C2H3X3). Journal of the Chemical Society, Chemical Communications 1994,1599 (1994). U. Geiser, J.A. Schlueter, H.H. Wang, A.M. Kini, J.M. Williams, P.P. Sche, H.I. Zakowicz, M.L. VanZile, J.D. Dudek, P.G. Nixon, R.W. Winter, G.L. Gard, J. Ren, and M.-H. Whangbo. Superconductivity at 5.2 K in an electron donor radical salt of bis(ethylenedithio)tetrathiafulvalene (BEDT-TTF) with the novel polyfluorinated organic anion b-(ET)2SF5CH2CF2SO3). Journal of the American Chemical Society 118, 9996 (1996). J.M. Williams, A.J. Schultz, U. Geiser, K.D. Carlson, A.M. Kini, H.H. Wang, W.-K. Kwok, M.-H. Whangbo, and J.E. Schirber. Organic superconductors: new benchmarks. Science 252, 1501 (1991). J.M. Williams, H.H. Wang, K.D. Carlson, A.M. Kini, U. Geiser, A.J. Schultz, W.K. Kwok, U. Welp, G.W. Crabtree, S. Fleshler, M.-H. Whangbo, and J.E. Schirber. New high-Tc benchmarks for organic and fullerene superconductors. Physica C 185-189, 355 (1991). J.A. Schlueter, K.D. Carlson, U. Geiser, H.H. Wang, J.M. Williams, W.-K. Kwok, J.A. Fendrich, U. Welp, P.M. Keane, J.D. Dudek, A.S. Komosa, D. Naumann, T. Roy, J.E. Schirber, W.R. Bayless, and B. Dodrill. Superconductivity up to 11.1 K in three solvated salts composed of [Ag(CF3)4]- and the organic electrondonor molecule bis(ethylenedithio)tetrathiafulvalene (ET). Physica C 233, 379 (1994). 9 D. Jrome. The physics of organic superconductors. Science 252, 1509 (1991). 10 J.S. Brooks. MRS Bulletin, p. 29, August 1993. 11 D. Jrome. Organic conductors, p. 405. J.-P. Farges (Ed.), Marcel Dekker, New York, 1994. D. Jrome. Advances in Synthetic Metals: Twenty Years of Progress in Science and Technology. P. Bernier, S. Lefrant, and G. Bidan (Eds.), Elsevier, Amsterdam, 1999.
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12 Table on superconducting elements. in: W.
Buckel. Supraleitung – Grundlagen und Anwendungen, p. 20. 3rd Edition, Physik-Verlag, Weinheim, 1984. 13 L.R. Testardi, J.H. Wernick, and W.A. Royer. Superconductivity with onset above 23 K in Nb–Ge sputtered films. Solid State Communications 15, 1 (1974). 14 R.L. Greene, G.B. Street, and L.J. Suter. Superconductivity in polysulfur nitride (SN)X. Physical Review Letters 34, 577(1975). 15 J.M. Williams, A.M. Kini, H.H. Wang, K.D. Carlson, U. Geiser, L.K. Montgomery, G.J. Pyrka, D.M. Watkins, J.M. Kommers, S.J. Boryschuk, A.V. Strieby-Crouch, W.-K. Kwok, J.E. Schirber, D.L. Overmeyer, D. Jung, and M.-H. Whangbo. From semiconductor–semiconductor transition (42 K) to the highest-Tc organic superconductor, k-(ET)2Cu[N(CN)2]Cl (Tc = 12.5 K). Inorganic Chemistry 29, 3272 (1990). 16 K. Tanigaki, T.W. Ebbesen, S. Saito, J. Mizuki, J.S. Tsai, Y. Kubo, and S. Kuroshima. Superconductivity at 33 K in CsxRbyC60. Nature 352, 222 (1991). 17 A. Schilling, M. Cantoni, J.D. Guo, and H.R. Ott. Superconductivity above 130 K in the Hg-Ba-Ca-Cu-O system. Nature 363, 56 (1993). 18 J. Bardeen, L.N. Cooper, and J.R. Schrieffer. Theory of superconductivity. Physical Review 108, 1175(1957). 19 W. Meissner and R. Ochsenfeld. Ein neuer Effekt bei Eintritt der Supraleitfhigkeit. Naturwissenschaften 21, 787 (1933). 20 P.J. Ford, and G.A. Saunders. High temperature superconductivity: ten years on. Contemporary Physics 38, 63 (1997). 21 J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, and J. Akimitsu. Superconductivity at 39 K in magnesium diboride. Nature 410, 63 (2001). 22 W.A. Little. Possibility of synthesizing an organic superconductor. Physical Review A 135, 1416 (1964). 23 J.G. Bednorz and K.A. Mller. Possible high Tc superconductivity in the Ba-La-Cu-O system. Zeitschrift fr Physik B 64, 189 (1986). 24 M.K. Wu, J.R. Ashburn, C.J. Torng, P.H. Hor, R.L. Meng, L. Gao, Z.H. Huang, Y.Q. Wang, and C.W. Chu. Superconductivity at 93 K in a new mixed-phase Yb-Ba-Cu-O compound system at ambient pressure. Physical Review Letters 58, 908 (1987).
25 I. Takehiko, K. Yamaji, and G. Saito. Organic
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30
31 32
33
34
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Superconductors. Springer Series in SolidState Sciences 88, 2nd Edition, December 1998. D. Shoenberg. Magnetic Oscillations in Metals. Cambridge University Press, Cambridge, 1984. W. Kang, G. Montambaux, J.R. Cooper, D. Jrome, P. Batail, and C. Lenoir. Observation of giant magnetoresistance oscillations in the high-Tc phase of the two-dimensional organic conductor b-(BEDT-TTF)2I3. Physical Review Letters 62, 2559 (1989). D. Jrome, A. Mazaud, M. Ribault, and K. Bechgaard. Superconductivity in a synthetic organic conductor (TMTSF)2PF6. Journal de Physique Lettres 41, L95 (1980). K. Bechgaard, K. Carneiro, M. Olsen, F.B. Rasmussen, and C.S. Jacobsen. Zero-pressure organic superconductor: di-(tetramethyltetraselenafulvalenium)-perchlorate [(TMTSF)2ClO4]. Physical Review Letters 46, 852 (1981). L. Balicas, K. Behuia, W. Kang, P. Auban-Senzier, E. Canadell, D. Jrome, M. Ribault, and J.-M. Fabre. (TMTTF)2Br: the first organic superconductor. Advanced Materials 6, 762 (1994). L.N. Bulaevskii. Organic layered superconductors. Advances in Physics 37, 443 (1988). E.M. Conwell. Highly Conducting Quasi-OneDimensional Organic Crystals. Academic Press, San Diego, 1988. S.S.P. Parkin, E.M. Engler, R.R. Schumaker, R. Lagier, V.Y. Lee, J.C. Scott, and R.L. Greene. Superconductivity in a new family of organic conductors. Physical Review Letters 50, 270 (1983). A.M. Kini, U. Geiser, H.H. Wang, K.D. Carlson, J.M. Williams, W.K. Kwok, K.G. Vandervoort, J:E. Thompson, D.L. Stupka, D. Jung, M.-H. Whangbo. A new ambient-pressure organic superconductor, k-(ET)2Cu[N(CN)2]Br, with the highest transition temperature yet observed (inductive onset Tc = 11.6 K, resistive onset = 12.5 K). Inorganic Chemistry 29, 2555 (1990). For documentation on oxidic superconductors see, for example: M.A.G. Aranda. Crystal structures of copperbased high-Tc superconductors. Advanced Materials 6, 905 (1994).
References and Notes A.V. Narlikar. Studies of High Temperature Superconductors, Vols. I–XI, Nova Science Publishers, New York, 1989–1994. D.M. Ginsberg. Physical Properties of High Temperature Superconductors, Vols. I–III, World Scientific, Singapore, 1990–1992. 36 K. Lders. Chemical Physics of Intercalation. P. Bernier, J.E. Fischer, S. Roth, and S.A. Solin (Eds.), NATO ASI Series, Plenum Press, New York, 1993. 37 R.M. Fleming, O. Zhou, and D.W.M.T. Siegrist. Progress in Fullerene Research, p. 211. H. Kuzmany, J. Fink, M. Mehring, and S. Roth (Eds.), World Scientific, Singapore: 1994. 38 J. Przyluski and S. Roth. New conducting materials with conjugated double bonds: polymers, fullerenes, graphite. Materials Science Forum, Trans Tech Publications, Aedermannsdorf, 1994. 39 C. Taliani, G. Ruani, and R. Zamboni. Fullerenes: Status and Perspectives. World Scientific, Singapore, 1993. 40 H.W. Kroto, J.E. Fischer, and D.E. Cox. The Fullerenes. Pergammon, 1993. 41 D. Koruga, S. Hameroff, J. Wither, R. Loutfy, and M. Sundareshan. Fullerene C60: Physics, Nanobiology, Nanotechnology. Elsevier, Amsterdam, 1993.
42 W.E. Billups and M.A. Ciufolini. Buckminster-
fullerenes. VCH, Weinheim, 1993. 43 S. Grtner. Festkrperprobleme. Advances in
Solid State Physics 32, 295 (1992). 44 H. Kuzmany, J. Fink, M. Mehring, and S. Roth.
45
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Electronic Properties of Fullerenes. Springer Series in Solid-State Sciences 117, Springer Verlag, Heidelberg, 1993. H. Kuzmany, J. Fink, M. Mehring, and S. Roth, Progress in Fullerene Research. World Scientific, Singapore, 1994. M.S. Dreselhaus, G. Dresselhaus, and P.C. Ecklund. Science of Fullerenes. Academic Press, New York, 1995. H. Prinzbach, A. Weiler, P. Landenberger, F. Wahl, J. Wrth, L.T. Scott, M. Gelmont, D. Olevano, and B. v. Issendorff. Gas-phase production and photoelectron spectroscopy of the smallest fullerene, C20. Nature 407, 60 (2000). J.C. Grossman, S.G. Louie, and M.L. Cohen. Solid C36: crystal structures, formation, and effects of doping. Physical Review B 60, R6941 (1999). A. Devos and M. Lannoo. Electron–phonon coupling for aromatic molecular crystals: possible consequences for their superconductivity. Physical Review B 58, 8236 (1998).
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8
Charge Density Waves In this section, we explore collective excitations that compete with the superconductivity processes. Intimately related to electron-lattice coupling, the mechanisms of these excitations (charge density waves, spin density waves, etc.) are almost transparent in low-dimensional systems.
8.1
Introduction
In Chapter 4 we discussed the Peierls transition, which is a structural phase transition in which the atoms rearrange and simultaneously the electronic structure changes. The atomic rearrangement leads to new reflections in X-ray and neutron diffraction patterns (Figure 3-12). A precursor of the Peierls transition is the Kohn anomaly, which is a phonon softening at the wave vector q that corresponds to two times the Fermi wave vector kF (Figure 2-10). The change of the electronic structure causes a metal-to-insulator transition. The Peierls transition is a typical one-dimensional phenomenon and occurs provided the Fermi surface is planar or at least contains parallel sections (nesting’). For demonstration purposes, we used the (hypothetical) monatomic alkali chain; in Chapter 5 we applied the Peierls concept to conjugated polymers, replacing the alkali atoms with CH radicals. Thus, bond alternation was explained as a Peierls transition. Figure 5-9 shows the p-electron density along the chain. We see a sinusoidal wave. A charge density wave is a spatial modulation of the electron density: r = r0 cos (qr + u)
(1)
to which the density of the (heavy) positive ions couples so as to minimize the total energy. This coupling is particularly favorable where the phonon modes soften, such as at the Kohn anomaly. The coupling between the electronic density (represented by the electrons) and the mass density (represented by the ions) is the essential feature of the charge density wave (unfortunately the term charge density wave’ does not sufficiently stress the coupling and the term charge density–mass density wave’ would be too clumsy). In conjugated polymers, the modulation of the p-elecOne-Dimensional Metals, Second Edition. Siegmar Roth, David Carroll Copyright © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30749-4
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tron density is rather trivial because there must be a higher electron density at the double bonds. Conjugated polymers, however, are not ideal Peierls systems. In Peierls systems, electron–phonon interactions are dominant, and in conjugated polymers, electron–electron interactions are equally important. To account for this more complicated situation, the bond alternation in polymers is called bond order wave (BOW) rather than charge density wave (CDW). Alkali metal chains and conjugated polymers are one-dimensional systems with half-filled bands (there is one electron per lattice site; in a completely filled band there would be two electrons, one spin-up and the other spin-down). In a half-filled band, the elementary cell doubles at the transition, and the Brillouin zone is reduced by a factor of two. The Kohn anomaly occurs at the phonon vector q = 2kF which is the end of the first Brillouin zone in the undistorted lattice. The Peierls concept, however, is not restricted to half-filled bands. In the hypothetical experiment on polyfraktiolene’ (Figure 5-32), we encountered band filling of 1/3, changing the structure by tripling the unit cell. More complicated is the situation in KCP, where the band filling is 5/6. Consequently, 2kF = 10/6a*, which can be transferred into the first Brillouin zone by subtracting a reciprocal lattice vector a* so that we get the result q = 2/3a*. This fits quite well to the position at which the Kohn anomaly is observed by inelastic neutron scattering (Figure 2-10).
8.2
Coulomb Interaction, 4kF Charge Density Waves, Spin Peierls Waves, Spin Density Waves
The charge density wave with wave vector q = 2kF is just one representative of a large variety of electron-driven lattice distortions in one-dimensional solids. In the phase diagram of the Bechgaard salts in Figure 7-10, spin density waves (SDW) are presented, and in some systems there is a charge density wave with q = 4kF instead of q = 2kF. The 4kF charge density wave can occur when the coulombic interaction between the electrons is large. In the Hubbard model [1], it is assumed that there is coulombic repulsion U between two electrons at the same lattice site. If U is large – on the order of the band width – all electrons are localized, because they cannot pass (see the car on the narrow mountain road in Figure 1-16). If the band is half-filled there is one electron on each lattice site. Adding an additional electron requires the energy U, because this electron has to be at a place where another electron is already situated, and hence it is repelled. This repulsion creates the coulombic gap of width U at the Fermi surface; the system is not a metal but a Hubbard insulator. The coulombic gap is illustrated in Figure 8-1. The electronic band structure E = E(k) is plotted with the filled part of the band indicated by a thick line. The Fermi energy is at point A. An additional electron is found not at A but at B, which is higher by the coulombic repulsion energy U. To discuss the 4kF charge density wave we take a look at a linear chain with one electron per three atoms (Figure 8-2). The electrons tend to be equally spaced. A lattice distortion in which the atoms are grouped by three and brought closer to the
8.2 Coulomb Interaction, 4kF Charge Density Waves, Spin Peierls Waves, Spin Density Waves
Figure 8-1 Creation of a coulombic gap in a half-filled band due to electron–electron interaction: (a) without coulombic interaction; (b) coulombic interaction switched on [2].
electrons will minimize the energy. In contrast to the Peierls distortion with q = 2kF, the distortion shown in Figure 8-2 occurs at q = 4kF. This can be easily explained: the wavelength of the charge density wave is k = 3a and the wave vector is q = 2p/ 3a. The band filling results in a = 1/6 because a completely filled band would have two electrons per site (one spin-up and one spin-down). In our case, however, there is only one electron per three sites. In a linear chain the Fermi vector is proportional to the filling factor kF = a·p/a, hence in our case, kF = (1/6) (p/a) and consequently q = 4kF. A more general way of achieving this result is to say that strong coulombic interaction lifts the spin degeneracy of the electrons. The electron system decouples into two subsystems with the states of one of the subsystems being energetically so far away that they can be disregarded. A subsystem band with one electron per atom is completely filled, not half-filled. All k vectors of the electronic states have to be multiplied by two, and the charge density wave occurs at q = 4kF. The linear array of equidistant electrons in Figure 8-2b still has spin degrees of freedom and the 4kF CDW is not the ground state. The spins can order ferromagnetically (all parallel) or antiferromagnetically (up and down alternating). The magnetically ordered spin chain is known as Heisenberg spin chain. It has collective excita-
Figure 8-2 4kF charge density wave in a chain with one electron per three atoms and strong coulombic interaction: (a) undistorted chain; (b) 4kF charge density wave [2].
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Figure 8-3 Spin Peierls transition in a 4kF charge density wave. The 4kF CDW leads to a chain of equidistant electrons (Heisenberg spin chain), which can order magnetically. The magnetic energy can then be lowered by the approach of the electrons in pairs.
tions, the magnons. If the equidistant electron arrangement is changed into a pair arrangement as indicated in Figure 8-3, the magnon energy can be lowered in a way similar to the lowering of the electron energy in the Peierls transition. The lattice adjusts to the paired electrons, the elementary cell is doubled, and the lattice distortion is again found at q = 2kF. Because of the analogy the transition within the 4kF CDW is called a spin Peierls (SP) transition. The spin Peierls state is different from the spin density wave (SDW), which is accompanied by no lattice distortion [3]. A spin density wave actually is a split charge density wave: it consists of a wave for spin-up electrons and another one for spindown electrons with a phase shift of 180 between the spin-up and the spin-down wave. The splitting is brought about by electron–electron interactions. Because of the phase shift, the corresponding lattice distortions interfere destructively and there is no net distortion. Therefore, a SDW cannot be detected by structural investigations (X-ray or neutron scattering); however, local susceptibility measurements such as ESR (electron spin resonance) and NMR (nuclear magnetic resonance) turn out to be very powerful, the latter because of the hyperfine interaction through which the electronic spins relax the nuclear spins [4]. A summary of the various instabilities in the CDW family is given in Table 8-1. At low temperatures these instabilities compete with each other and with superconductivity. Sometimes this competition is expressed in terms of g-ology’, using the electron-scattering parameters g1 and g2 as axes of a coordinate system and assigning the most divergent instabilities to the respective fields in the g1–g2 plane (Figure 8-4). The parameter g1 is the amplitude for backward and g2 that for forward scattering. In a more complete diagram g3 is used for Umklapp scattering. In this diagram two types of superconductivity occur: the singlet superconductivity SS and the triplet superconductivity TS. Singlet superconductivity is the well known conTable 8-1
Various instabilities in the CDW family (partly after Alcacer [4]).
Acronym:
2kF CDW
4kF CDW
SDW
SP
full name
2kF charge density wave 2kF yes
4kF charge density wave 4kF yes
spin density wave 2kF no
spin Peierls state 2kF yes
wave vector lattice distortion
8.3 Phonon Dispersion Relation, Phase, and Amplitude Mode in Charge Density Wave Excitations
Figure 8-4 g-ology of one-dimensional instabilities. g1 is the amplitude for backward, g2 for forward, and g3 for Umklapp scattering. TS: triplet superconductivity, SS: singlet superconductivity, SDW: spin density wave, CDW: charge density wave.
ventional’ BCS superconductivity, in which the spins of the electrons in a Cooper pair are opposite so that the net spin is zero. In triplet superconductivity the spins are parallel, summing to a total spin S = 1, which can take three orientations relative to an external field (+1, 0, and –1, hence triplet). In the phase diagram of the Bechgaard salt superconductors (Figure 7-10), we noted that superconductivity is adjacent to spin density waves, and in Figure 8-4 we find the superconducting neighbor of SDW to be TS rather than SS. This observation has led to speculations that the superconductivity in Bechgaard salts may be of the triplet type rather than singlet. For more details on g-ology, see Refs. [5–9].
8.3
Phonon Dispersion Relation, Phase, and Amplitude Mode in Charge Density Wave Excitations
Figure 8-5 shows the phonon dispersion relation of a system susceptible to the Peierls transition. In Figure 8-5a we see the dispersion relation far above the Peierls transition. The shape of this curve is known from Figure 3-14. In Figure 8-5b the Kohn anomaly develops at q = 2kF, as demonstrated in the example of KCP in Figure 2-10. In Figure 8-5c the phonons at the Kohn anomaly become completely soft and the lattice rearranges. In this rearrangement 2kF turns into a new reciprocal lattice point and the first Brillouin zone is now limited by €kF. The parts outside the first Brillouin zone are indicated by dotted lines in Figure 8-5d. These outside parts can be transferred into the first Brillouin zone by subtraction of the reciprocal lattice vector (= €kF). The reduced zone scheme is shown in Figure 8-5e. Here, there are two phonons with energy x = 0 at wave vector k = 0. This contradicts one of the fundamental theories of lattice dynamics. Because nicht sein kann, was nicht sein
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Figure 8-5 Phonon dispersion relation in a crystal with Peierls transition: (a) undistorted phonon dispersion relation at high temperature; (b) development of Kohn anomaly; (c) phonons at Kohn anomaly become completely soft; (d) lattice rearrangement (dotted parts of
the dispersion relation lie outside the first Brillouin zone of the new lattice); (e) phonon dispersion relation in the reduced zone scheme; (f) lifting of the q = 0 degeneracy and development of an amplitude mode (A–) and a phase mode (A+) in CDW terminology.
darf’ the lattice lifts this degeneracy and pushes one phonon branch (A+) upwards, which in the terminology of lattice dynamics is called an optical phonon branch (many optical phonon branches are allowed), whereas A– remains the acoustic phonon branch (only one acoustic phonon branch is permitted). From the point of view of lattice dynamics this is not unusual. Rearrangements of the phonon branches occur in many structural phase transitions. But in the world of charge density waves, we call A– the phase mode (phasons) and A+ the amplitude mode [10]. An inspection of the charge density wave reveals that the amplitude mode corresponds to a modulation of the CDW amplitude and the phase mode to a phase modulation. The fact that, for a phason at q = 0, x = 0 turns out to be very important. The term
8.4 Electronic Structure, Peierls–Frhlich Mechanism of Superconductivity
q = 0 corresponds to a sliding of the charge density wave as a whole, and this motion needs no energy. This is the renowned Peierls–Frhlich mechanism of superconductivity. In a real crystal the charge density wave is always associated with lattice imperfections, but a CDW in a perfect crystal would be a Peierls–Frhlich superconductor.
8.4
Electronic Structure, Peierls–Frhlich Mechanism of Superconductivity
Figure 8-6a shows the electronic band of a Peierls system. There is a band gap of width D at the Fermi wave vector kF (see also Figure 4-3). In the Peierls–Frhlich mechanism the charge density wave moves through the crystal lattice. From the previous section, we know that in a perfect crystal this sliding motion does not require energy (for the phason mode x = 0 at q = 0). Strictly speaking, this is true only in the jellium approximation, where the positive ions do not form a discrete lattice but are continuously spread over the solid – or for an incommensurate charge density wave where kF is not a rational fraction of a reciprocal lattice vector (next section). The flow without resistance stems from the fact that the energy is independent of the position (phase) of the charge density wave, as well as from the absence of inelastic scattering due to the Peierls gap. Charge density waves in real crystals, however, are associated with impurities (next section). In Figure 8-6b the band structure of a system with sliding charge density wave is shown. The CDW moves with the sliding velocity ms, leading to a displacement of the planes of the Fermi surface by q:
q = m*ms/h
(2)
where m* is the effective mass of the electrons and corresponds to a current: ICDW = n e ms
(a) Electron band with Peierls gap D at Fermi wave vector kF. (b) Displacement of the electron system with uniform velocity mS and shift of Fermi vector by q, leading to asymmetric gap positions and a positive sum of electron k vectors. Figure 8-6
(3)
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and n is the number of electrons in the sample. The sliding velocity ms corresponds to the drift velocity in the usual derivation of the electric conductivity (see textbooks on solid-state physics). The existence of a gap does not prevent conductivity, provided the electric field can force the gap to become asymmetric (at different positions in the +k and –k directions). In this case the gap excludes loss due to inelastic scattering processes, and hence the current flows without resistance. In an insulator the electric field cannot push the gap into asymmetric positions, because the gap originates from structural features unable to slide along the crystal.
8.5
Pinning, Commensurability, Solitons
Sliding of a charge density wave without resistance is possible only if the following two conditions are fulfilled: . .
The crystal is perfect. The charge density wave is incommensurate or the crystal is made of jellium.
Jellium is an interesting theoretical model. It assumes a continuous distribution of the positive charges instead of discrete atoms. Incommensurability means that the ratio between the wave vector and the reciprocal lattice vectors is an irrational number. In both cases, the charge density wave cannot register’ with the crystal lattice and hence the energy is independent of the phase of the CDW. (Here the meaning of the word register’ is taken from color printing, where the pads have to register to ensure correct superposition. Incommensurability was originally a theological term implying that there is no common measure for the grace of God and human merit, meaning that one cannot be offset against the other.) A mechanical example of registering is the cogwheel and the bicycle chain. The cogs register with the holes of the chain to prevent sliding. In many physical systems registering also occurs when higher harmonics match, but not when there is no rational ratio between the periodicities. An example of incommensurability is the rotation of the earth around its axis and its revolution around the sum. To compensate this incommensurability, we have to add intercalary days in leap years. The charge density wave in polyacetylene (Figure 5-9) registers very strongly. To move it, a very high barrier has to be passed. This barrier corresponds to the cleavage of all double bonds. If the wavelength of the CDW is three times the lattice constant or perhaps 3/5, registering is much less pronounced. If CDW and lattice are incommensurate, no registering is possible. Of course, since the rational numbers are infinitely dense, the exact distinction between commensurate and incommensurate is rather artificial, but for practical purposes a CDW is incommensurate when the attraction to the crystal lattice is smaller than the thermal energy or other energy fluctuations. Impurities and crystal imperfections will pin the charge density wave. For example, a charge density wave cannot slide over chain ends. Any other irregularity also prevents the charge density wave from sliding, because impurities prefer to remain
8.5 Pinning, Commensurability, Solitons
Figure 8-7 Temperature dependence of the conductivity of TTFTCNQ [11]. The ordinate is the conductivity normalized to the room temperature value. Just above the metal-to-insulator transition at 53 K, an unusual increase in conductivity is observed, which in some samples takes the form of giant’ conductivity.
either at the crest or in the valleys of the CDW. The interactions between individual impurities with the CDW sum to a pinning force, and the CDW slides only if the force exerted by an applied electric field is larger than the pinning force. This leads to a threshold field for CDW motion. Below the threshold the CDW does not slide, above threshold it does, however, not without resistance. Moving the CDW in the presence of impurities leads to finite resistivity. Figure 8-7 shows the temperature dependence of the conductivity of TTF-TCNQ [11]. The conductivity increases upon cooling as expected of a metal, but below 80 K the increase grows abnormally large and probably is a precursor of sliding CDW conductivity. At 53 K a transition to a pinned CDW occurs and the sample becomes insulating. In some samples an even greater conductivity increase just above 53 K has been observed [12], and reports of this giant’ conductivity have had an enormous impact on the scientific community. To demonstrate the existence of a threshold field, we chose TaS3 as an example [13]. In TaS3 the Peierls transition to a pinned CDW occurs at 220 K. In Figure 8-8 the electric field dependence of the conductivity at 130 K is shown. The conductivity is normalized to the room-temperature value. Up to the threshold field of ET » 0.3 V cm–1, the conductivity is negligibly small. At ET the conductivity rises abruptly and saturates at high field, at values that correspond to the normal state conductivity. Saturation at normal state values is plausible from Eq. (3), relevant for CDW conductivity but identical to the conductivity in the Drude model of free electrons when the sliding velocity corresponds to the drift velocity. Another consequence of pinning is the frequency dependence of the conductivity. Pinning makes it hard for a CDW to slide, but the CDW can still oscillate fairly easily. This leads to low DC and high AC conductivity. Rice et al. [14] have predicted nonlinear charged excitations in pinned charge density waves. They obtained the following equation for the phase j of the CDW: 2
2
@ u 2@ u 2 dV =0 2 – c0 2 + x0 du @t @x
(4)
where x0 is the oscillation frequency of the CDW (determined by far-infrared measurements), and c0 the phason velocity as derived in the Lee, Rice, Anderson analy-
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Figure 8-8 CDW conductivity of TaS3 as a function of the applied electric field. Below ET the conductivity is negligibly small, above Tc sliding sets in, but the conductivity is not free
of resistance. It remains finite and saturates at values expected for normal state conductivity (~ sRT) [13].
sis [10] of the A– mode in Figure 8-5. The quantity V in the last term of Eq. (4) is the potential energy due to the registering of the CDW to the lattice. Consequently, V is a periodic function and Eq. (4) is the Sine–Gordon equation (Eq. 4 in Chapter 5) mentioned in the general discussion of solitons in Section 5.4. We know that the Sine–Gordon equation has solitonic solutions. A soliton in a CDW is a phase slip center, as depicted in Figure 8-9. It can be demonstrated that its charge amounts to €2e, equivalent to that of two electrons. (The integral over a harmonic wave is zero,
Figure 8-9 Phase slip center (soliton) in a pinned CDW. (a) Spatial change in the phase u. Note the similarity to the bond alternation parameter in polyacetylene in Figure 5-8. (b) Charge density wave with local phase distortion. The dotted line corresponds to the undistorted wave [2].
8.5 Pinning, Commensurability, Solitons
Figure 8-10 Noise spectrum for sliding CDW conduction in TaS3. The narrow band noise is seen as a pronounced peak which shifts to higher frequencies as the sliding velocity of the CDW increases (along with increasing bias voltage) [13].
of course, but at the distortion the negative and positive parts of the wave differ in width, and integrating then leads to €2e.) The conductivity of a CDW system below the threshold is not exactly zero. It is believed that the residual conductivity is at least partially due to solitons. Soliton conductivity in pinned charge density waves is similar to creep phenomena in metallurgy. Long before the shear forces are strong enough to allow for sliding of crystal slabs along lattice planes, the crystal deforms by the motion of dislocations.
Classical washboard model to demonstrate that noise frequency increases with the sliding velocity.
Figure 8-11
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8 Charge Density Waves
Washboard model of sliding charge density wave. ET: threshold field. The ball begins to roll at the threshold. The frequency of the narrow band noise increases with the velocity (which is proportional to the current) (after Grner and Zettl [13]).
Figure 8-12
If the charge density slides (above the threshold), a very peculiar noise is generated and superimposed on the DC current. A spectral analysis reveals that this noise consists of a very narrow peak and its higher harmonics (narrow band noise). For an example, see the noise spectrum of TaS3 in Figure 8-10. Again there is a close analogy to shear deformation of metals. When some metals (for example, tin or cadmium) are bent they emit a characteristic acoustic noise (Zinngeschrei’, tin cry). A classical single-particle model [13] for the interpretation of narrow band noise is the so-called washboard model’ (Figure 8-11). The CDW slides as a rigid entity over rigidly fixed impurities. It corresponds to a ball rolling down a corrugated slope (washboard). This simple model explains both the threshold field and the dependence of the noise frequency on the CDW current (Figure 8-12).
8.6
Field-induced Spin Density Waves and the Quantized Hall Effect
The symmetry of TM2X Bechgaard salts is triclinic, but most physical features can be better explained by assuming orthorhombic symmetry with axes a, b, and c. The direction of highest conductivity is along the stacking axis, which we assign to the a direction (see also Figure 7-9). The crystal is not only anisotropic with respect to the a direction and the bc plane, the bc plane itself is anisotropic. For (TMTSF)2X compounds, the ratio of the band widths is ta : tb : tc = 100 : 10 : 0.3; ta » 0.2 eV (the t’s are the so-called transfer integrals, the bandwidth is W = 4t). Because of the finite values of tb and tc, the Fermi surface is not planar but warped. There is still sufficient nesting so that (TMTSF)2PF6 is in the SDW state at ambient pressure and
8.6 Field-induced Spin Density Waves and the Quantized Hall Effect
Quantum Hall effect in field-induced spin density wave systems: (a) (TMTSF)2ClO4: T = 0.5 K, ambient pressure [19]; (b) (TMTSF)2PF6: T = 0.1 K, p » 8 kbar [20]. Figure 8-13
low temperatures (see Figure 7-10). The application of the modest pressure of 12 kbar increases tb and destroys the nesting condition. Consequently, the CDW is suppressed and the sample becomes superconducting. Applying a sufficiently high magnetic field (above the critical field) suppresses superconductivity, and the sample should behave like a normal metal. But if the field is applied in the c direction (the direction of lowest conductivity), the CDW is reactivated [15]. This means that the magnetic field restores the nesting condition. The effect is called field-induced spin density wave (FISDW). A very amazing property of the field-induced SDW state is a stepwise field dependence of the Hall voltage [16–20], very similar to what is observed in the von Klitzing effect (quantum Hall effect, QHE) [20]. Examples for (TMTSF)2ClO4 and (TMTSF)2PF6 are shown in Figure 8-13 [21]. In ordinary spin density waves, nesting occurs for the wave vector q = 2kF. In high magnetic fields the electronic density of states is modified by the Landau quantization. For example, electrons move in closed orbitals of quantized energy and the density of states between the Landau levels is zero. (Interference between Landau levels and the Fermi surface leads to the de Haas–van Alphn and Shubnikov–de Haas oscillations discussed in the context of fermiology in Section 7.5.) Because of the Landau quantization, the wave vector q of the SDW has to be modified to
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qFISDW = 2kF +
NeHb h
(5)
where N is an integer, H the applied field, and b the lattice spacing in the b direction. For qFISDW there are new nesting conditions, which depend on the magnetic field. A direct consequence of the field-dependent q vector is the quantization of the Hall resistance at rH = h/2Ne2. Because nesting is a fairly crude geometrical overlap, the steps in the FICDW–QHE are far less sharp than in the von Klitzing effect.
References and Notes 1 J. Hubbard. Electron correlations in narrow
energy bands. Proceedings of the Royal Society of London A 276, 238 (1963). J. Hubbard. Electron correlations in narrow energy bands. 3. Improved solution. Proceedings of the Royal Society of London A 281, 401 (1964). G.H. Ding, F. Ye, and B.W. Xu. Charge gap in one-dimensional extended Hubbard model. Communications in Theoretical Physics 39, 105 (2003). 2 S. Kagoshima, H. Nagasawa, and T. Sambongi. One-Dimensional Conductors. Springer Series in Solid-State Sciences 72, Springer Verlag, Heidelberg, 1982. 3 For a basic review on spin density wave physics see: G. Grner. The dynamics of spin-density waves. Reviews of Modern Physics 66, 1 (1994). More modern developments in spin waves and their relationship to low dimensional systems can be found in: R. Konno. The self-consistent renormalization theory of spin fluctuations for itinerant antiferromagnetism in quasi-one-dimensional metals. Physica B 329, 1288 (2003). S. Tomic, T. Vuletic, M. Pinteric, and B. KorinHamzic. Modalities of self-organized charge response in low dimensional systems. Journal de Physique IV 12, 211 (2002). A.M. Gabovich, A.I. Voitenko, and M. Ausloos. Charge- and spin-density waves in existing superconductors: competition between Cooper pairing and Peierls or excitonic instabilities. Physics Reports: Review Section of Physics Letters 367, 583 (2002).
Y. Tomio, N. Dupuis, and Y. Suzumura. Effect of nearest- and next-nearest-neighbor interactions on the spin-wave velocity of one-dimensional quarter-filled spin-density-wave conductors. Physical Review B 64, 125123 (2001). S. Mazumdar, R.T. Clay, and D.K. Campbell. The nature of the insulating state in organic superconductors. Synthetic Metals 120, 679 (2001). K. Sengupta and N. Dupuis. Effective action and collective modes in quasi-one-dimensional spin-density-wave systems. Physical Review B 61, 13493 (2000). A.M. Gabovich and A.I. Voitenko. Non-stationary Josephson tunneling involving superconductors with spin-density waves. Physica C 329, 198 (2000). M. Tsuchiizu and Y. Suzumura. Competition between SDW and SC states in two-Hubbard chains. Synthetic Metals 103, 2189 (1999). 4 L. Alcacer. Organic Conductors, p. 269 (1994). J.P. Farges (Ed.), Marcel Dekker, New York, 1994. 5 L.G. Caron. Organic Conductors, p. 25 (1994). J.P. Farges (Ed.), Marcel Dekker, New York, 1994. 6 D. Jrome. Organic Conductors, p. 405 (1994). J.P. Farges (Ed.), Marcel Dekker, New York, 1994. 7 J. Slyom. The Fermi gas model of onedimensional conductors. Advances in Physics 28, 201 (1979). Y. Fuseya, H. Kohno, and K. Miyake. New varieties of order parameter symmetry in quasione-dimensional superconductors. Physica B 329, 1455 (2003).
References and Notes Y. Fuseya, Y. Onishi, H. Kohno, and K. Miyake. Novel type of pairing in quasi-one-dimensional superconductivity. Physica B 312, 36 (2002). I.J. Lee, S.E. Brown, W.G. Clark, M.J. Strouse, M.J. Naughton, W. Kang, and P.M. Chaikin. Triplet superconductivity in an organic superconductor probed by NMR knight shift. Physical Review Letters 88, 017004 (2002). M. Kohmoto and M. Sato. Spin-triplet superconductivity in quasi-one dimension. Europhysics Letters 56, 736 (2001). J. Gonzalez. Consistency of superconducting correlations with one-dimensional electron interactions in carbon nanotubes. Physical Review Letters 87, 136401 (2001). 8 V.J. Emery. Highly Conducting One-Dimensional Solids, p. 247. J.T. Devreese, R.P. Evrard, and V.E. van Doren (Eds.). Plenum Press, New York, 1979. 9 C. Bourbonnais and L.G. Caron. Renormalization group approach to quasi-one-dimensional conductors. International Journal of Modern Physics B 5, 1033 (1991). S. Haddad, S. Charfi-Kaddour, C. Nickel, M. Heritier, and R. Bennaceur. Renormalization of the hopping parameters in quasi-onedimensional conductors in the presence of a magnetic field. European Physical Journal B 34, 33 (2003). J.I. Kishine and K. Yonemitsu. Dimensional crossovers and phase transitions in strongly correlated low-dimensional electron systems: renormalization-group study. International Journal of Modern Physics B 16, 711 (2002). D. Zanchi and H.J. Schulz. Weakly correlated electrons on a square lattice: renormalizationgroup theory. Physical Review B 61, 13609 (2000). V. Vescoli, L. Degiorgi, W. Henderson, C. Gruner, K.P. Starkey and L.K. Montgomery. Dimensionality-driven insulator-to-metal transition in the Bechgaard salts. Science 281, 1181 (1998). 10 P.A. Lee, T.M. Rice, and P.W. Anderson. Conductivity from charge or spin density waves. Solid State Communications 14, 703 (1974). 11 M.J. Cohen, L.B. Coleman, A.F. Garito, and A.J. Heeger. Electrical conductivity of tetrathiofulvalinium tetracyanoquinodimethan (TTF) (TCNQ). Physical Review B 10, 1298 (1974). 12 L.B. Coleman, M.J. Cohen, D.J. Sandman, F.G. Yamagishi, A.F. Garito, and A.J. Heeger. Super-
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conducting fluctuations and the Peierls instability in an organic solid. Solid State Communications 12, 1125 (1973). G. Grner and A. Zettl. Charge density wave conduction: a novel collective transport phenomenon in solids. Physics Reports 119, 117 (1985). M.J. Rice, A.R. Bishop, J.A. Krumhansl, and S.E. Trullinger. Weakly pinned Frhlich charge-density-wave condensates: a new, nonlinear, current-carrying elementary excitation. Physical Review Letters 36, 432 (1976). J.J. Kwak, J.E. Schriber, R.L. Greene, and E.M. Engler. Magnetic quantum oscillations in tetramethyltetraselenafulvalenium hexafluorophosphate [(TMTSF)2PF6]. Physical Review Letters 46, 1296 (1981). M. Ribault, D. Jrome, J. Tuchendler, C. Weyl, and K. Bechgaard. Low-field and analogous high-field Hall effect in (TMTSF)2ClO4. Journal de Physique Letters 44, 953 (1983). K. Oshima, M. Suzuki, K. Kikuchi, H. Kuroda, I. Ikemoto, and K. Kobayashi. High field magnetoconductivity in (TMTSF)2ClO4. Journal of the Physical Society of Japan 53, 3295 (1984). S. Kagoshima. Low-dimensional organic conductors under high magnetic fields. Journal de Physique I 6, 1787 (1996). P.M. Chaiken, M.-Y. Choi, J.F. Kwak, J.S. Brooks, K.P. Martin, M.J. Naughton, E.M. Engler, and R.L. Greene. Tetramethyltetraselenafulvalenium perchlorate, (TMTSF)2ClO4, in high magnetic fields. Physical Review Letters 51, 2333 (1983). A.V. Kornilov, V.M. Pudalov, Y. Kitaoka, K. Ishida, T. Mito, J.S. Brooks, J.S. Qualls, J.A.A.J. Perenboom, N. Tateiwa, and T.C. Kobayashi. Novel phases in the field-induced spin-density-wave state in (TMTSF)2PF6. Physical Review B 65, 060404 (2002). A.G. Lebed. Does the “quantized nesting model” properly describe the magnetic-fieldinduced spin-density-wave transitions? JETP Letters 72, 141 (2000). W. Kang, H. Aoki, and T. Terashima. Evidence of new split FISDW transitions: transport properties. Synthetic Metals 103, 2119 (1999). S. Valfells, J.S. Brooks, Z. Wang, S. Takasaki, J. Yamada, H. Anzai, and M. Tokumoto. Quantum Hall transitions in (TMTSF)2PF6. Physical Review B 54, 16413 (1996).
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S. Kagoshima. Low-dimensional organic conductors under high magnetic fields. Journal de Physique I 6, 1787 (1996). 19 J.R. Cooper, W. Kang, P. Auban, G. Montambaux, and D. Jrome. Quantized Hall effect and a new field-induced phase transition in the organic superconductor (TMTSF)2PF6. Physical Review Letters 63, 1988 (1989). A.V. Kornilov, V.M. Pudalov, Y. Kitaoka, K. Ishida, T. Mito, J.S. Brooks, J.S. Qualls, J.A.A.J. Perenboom, N. Tateiwa, and T.C. Kobayashi. Novel phases in the field-induced spin-density-wave state in (TMTSF)2PF6. Physical Review B 65, 060404 (2002). T. Vuletic, C. Pasquier, P. Auban-Senzier, S. Tomic, D. Jerome, K. Maki, and K. Bechgaard. Influence of quantum Hall effect on linear and nonlinear conductivity in the FISDW states of the organic conductor (TMTSF)2PF6. European Physical Journal B 21, 53 (2001). J. Eom, H. Cho, and W. Kang. Quantum Hall effect and anomalous transport in
(TMTSF)2PF6. Journal de Physique IV 9, 191 (1999). 20 S.T. Hannahs, J.S. Brooks, W. Kang, L.Y. Chiang, and P.M. Chaikin. Quantum Hall effect in a bulk crystal. Physical Review Letters 63, 1988 (1989). M. Inoue, N. Miyajima, M. Sasaki, H. Negishi, S. Negishi, H. Kadomatsu, and V.A. Kulbachinskii. A new type of bulk quantum Hall effect in Bi2-xSnxTe3 crystals. Physica B 284, 1718 (2000). S. Hill, J.S. Brooks, S. Uji, M. Takashita, C. Terakura, T. Terashima, H. Aoki, Z. Fisk, and J. Sarrao. Bulk quantum Hall effect in n-Mo4O11. Synthetic Metals 103, 2667 (1999). 21 K. von Klitzing, G. Dorda, and M. Pepper. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Physical Review Letters 45, 494 (1980). K. von Klitzing. The quantized Hall effect (Nobel Lecture in Physics, 1985). Reviews in Modern Physics 58, 519 (1986).
193
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Molecular-scale Electronics “Computers in the future may weigh no more than 1.5 tons.” – Popular Mechanics, 1949
9.1
Miniaturization
At least since the Club of Rome published The Limits of Growth’ in 1973 [1], we have been aware that many products of our civilization grow exponentially. It is less well known that some decrease exponentially. An example of the latter is electronic devices. In Figure 9-1 the minimum feature size of electronic components is plotted versus the year of use. A straight line results, provided the size is plotted on logarithmic scale. Such diagrams are known as Moore plots [2]. Originally they were used to demonstrate the continuous decrease of the price per bit in integrated memories.
Figure 9-1 Development of the structural size of electronic devices in the course of time and extrapolation to sizes as small as typical interatomic distances in solids by the year 2020.
One-Dimensional Metals, Second Edition. Siegmar Roth, David Carroll Copyright © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30749-4
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Because the price of a silicon chip remains more-or-less constant (some p Dollars’), and because the number of bits per chip increases continuously (from 124 kbits in the seventies to 16 Mbits in the mid 19902), a bit becomes cheaper and cheaper. Extrapolating the Moore plot in the reverse direction, vacuum tubes and probably also the Chinese abacus might be included. If extended to the future the plot would lead to a feature size of 10 in the year 2020. This increasing density of electronics through miniaturization has become a relatively large part of the economies of western nations. For example, increasing computational power creates machines for applications that were previously not approached with computers. The molecular dimensions of such ultrasmall circuitry certainly justify the name molecular electronics’. From a contemporary point of view, however, the term molecular electronics is not very well defined. It is used in connection with both electronics on a molecular level, which we discuss in this chapter, and molecular materials for electronics, which we discuss in Chapter 10 [3]. In the context of the first meaning, molecular electronics first found expression when, in the 1970s, chemists Mark Ratner and Ari Aviram speculated about using single molecules as rectifiers, allowing electrical currents to pass only in one direction (discussed in more detail below). Throughout the 1970s and 1980s, the fundamental concepts of this field
Figure 9-2
Three-terminal devices make gain possible.
9.1 Miniaturization
were laid down in a variety of topical conferences. Some of the proceedings of these workshops on molecular electronics are listed in the Refs. [4–10]. It was not until the 1990s, however, when a key development came, and Tour and coworkers measured the current flowing between two gold electrodes connected by a single molecule [11]. The basic premise of these early approaches is quite easy to visualize. As shown in Figure 9-2, the conceptualization is of a single molecule between interconnecting leads that performs a signal-processing electrical function, such as rectification, resistance, etc. One can already see that many of the issues of dimensionality discussed previously should apply to such situations. In this conformation, the electrodes are linked to the atoms of the molecule through a chemical bond. In the example cited above, a benzene molecule was linked to Au electrodes via sulfur atoms at each end. The sulfur atoms, which happen to form strong bonds with gold, held the molecule in place between the two gold wires separated by a distance of several nanometers. By using such configurations and doping the molecule in the leads, effects such as molecular conduction, coulomb blockade, and negative-differential resistance’ (when the current decreases as the voltage increases) have been observed. Although these phenomena have provided interesting tests of molecular orbital theory, they fall short of the essential element necessary for information processing electronics: gain. The basis of computer circuits is the transistor. The transistor is essentially nothing more than a variable-state switch. Very simply, it works like this: two contacts are placed on a thin, doped, semiconducting film (like Si), so that charge injection can be accomplished easily. A back electrode is placed on the back of the semiconductor layer. When a voltage is applied to the gate, a space charge or depletion’ region builds up in the active layer. This impedes charge flow through the source and drain electrodes. When the bias reaches some specified large value, the current flow between source and drain is turned off altogether. This is actually a field-effect transistor, of which several types exist. However, they all have a singular property: if the configuration is balanced just right, it is clear that small fluctuations in the gate can result in rather large changes in the source–drain current. Importantly, we could take the point of view of supplying an input current at the gate (Ig) and measuring its output at the drain (Id). If we did this, it would appear as though small inputs of Ig lead to large outputs of Id. In fact, the gain of such a system is generally defined in terms of this concept: b = Id/Ig. It is not a real gain of course; the currents must all sum throughout the circuit. However, the nonlinearity in the circuit response is quite powerful and can be useful in signal amplification, information processing, and memory applications. Essentially, it relies on a three-terminal flow-control’ system. In contrast, early experiments on molecules used two-terminal devices without the capability of exhibiting gain. Trying to achieve gain within a single-molecule device is a significant stumbling block to the overall usefulness of such concepts. In 2000, however, Di Ventra and colleagues showed, through rather complex calculations, that this hurdle is surmountable under some conditions. The predicted the behavior of a three-terminal device, based on the 1997 benzene device, can exhibit gain by resonant tunneling [12]. Quite different from the macroscopic charge-flow device described above, the resonant-tunneling device must express ele-
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ments of quantum mechanical confinement (i.e., be zero- or one-dimensional) to work. Alternatively, we have already seen several much more simple materials in which quantum phenomena are expressed; specifically, quantum dots and quantum wires. Such inorganic systems could also generate gain by similar principles to those described above. We mentioned in Section 1.2 that, in these systems with a small electron concentration (silicon, gallium arsenide), effects of low dimensionality (socalled quantum-size effects) are seen at much larger sizes than in systems with high electron concentration (carbonic systems). In fact, the preparation of inorganic quantum wires and quantum dots is state-of-the-art today, and these systems are subjects of intensive research and development [13]. Quantum dots can be made so small that they contain only a few or even single’ electrons. The possibility of single-electron electronics has been pointed out [14,15], and single-electron transistors [16,17] and single-electron memories [18–20] have been demonstrated. Thus, this can be seen as a second approach to molecular-scale electronics and, incidentally, an approach more easily accomplished by using the lithography tools of topdown assembly. In both approaches, the operational parameters must be taken into account. For single-electron devices, the level spacing of electrons-in-a-box (Section 1.2, Eq. (1)), and also the coulombic energy of the occupants of that box, or coulombic blockade, determine the energy to place another electron into the system. For large’ particles the coulombic blockade can be calculated from the charging energy of a capacitor: Echarging =
e2 2C
(1)
For a platelet of several micrometers in diameter, the capacitance is on the order of 10–15 F and the charging energy is several tenths of a millielectronvolt. Only at temperatures below 1 K can single-electron effects be observed; otherwise, the thermal energy kT exceeds the charging energy. The capacitance is proportional to the cross section of a particle and, in macromolecules several nanometers in diameter, the charging energy is comparable to the thermal energy at room temperature [21]. For individual atomic sites in a crystal lattice, the charging energy will finally correspond to the Hubbard model discussed in Section 8.2 (Figure 8-1). These considerations immediately link electronic device miniaturization and single-electron effects to molecular electronics, and indeed, articles on multiple tunnel junctions’ in GaAs do not fail to point out that their ideas are also applicable to macromolecules [21].
9.2
Information in Molecular Electronics
As described above, the information within a molecular circuit is carried and acted upon by the existence and magnitude of charge. The flow of charge is literally the flow of information within the system. We note here, that within this paradigm there should be some inherent advantages of organic materials. Specifically, in low-
9.3 Early and Radical Concepts
dimensional organic materials, electronic excitations are usually strongly coupled to the underlying crystal structure. The complicated structure of a macromolecule offers additional degrees of freedom, e.g., conformational degrees of freedom, which localize the excitations. Figure 4-2 shows an example of a molecule twisting and puckering when its excitation state is changed. Better localization should allow for closer packing of information than is possible in inorganic semiconductors. A particular case of localization in one-dimensional systems is solitons/polarons in conjugated polymeric chains (Section 5.4). They move as compact form-preserving pulses and not as dispersing wave packages (Figure 5-14). Information transfer in these systems is simply the group velocity of the wave function of the localized charge. The periodic structure of Si or GaAs, however, implies a certain degree of delocalization of electrons (Bloch waves). Under the conditions of low defect density, the charge is well represented by plane waves across the entire molecular structure. In this situation, not only is the modulus of the wave function preserved from one contact to the next, the phase of the wave function also is. In a phase-preserving system, it is conceivable that gain can also be achieved in a three-terminal configuration. However, we may ask if information transfer within this system now occurs at the phase velocity. The answer to this is not so straightforward. Generally, information is transferred when the phase between two states is compared. When these compared states are entangled, then information transfer may not be restricted to the group velocity of the states. These concepts lead naturally to the concepts of quantum bits [22]. Of course, we do not mean to imply that organics cannot be used in such applications. As Teich and colleagues have shown, it is quite possible to build an addressable entangled state with heteropolymers [23]. However, the delocalizedcarrier concept seems to have a natural extension to such applications.
9.3
Early and Radical Concepts
Clearly, from the above discussion, the computational paradigm will ultimately decide the best approach to molecular-scale electronics. Most early work revolved around conjugated organic (synthetic metal) systems used as devices, as discussed above. In fact, this is still the mainstay effort in molecular-scale electronics. Thus, it is helpful to examine detailed proposals for molecular circuit elements in terms of their excitations and their dimensionality. 9.3.1
Soliton Switching
Soon after solitons were discussed for polyacetylene [24–26], F. L. Carter introduced the concept of soliton switching’ [27]. In Chapter 5 a soliton in polyacetylene was described as a domain wall separating chain segments with different arrangements of bond alternation. These arrangements can be interpreted as logical states, and a
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Figure 9-3 Polyacetylene chain with two differently-conjugated segments separated by a domain wall (soliton). The domains can be interpreted as different states, to which different logical values can be assigned.
passing soliton switches’ the chain from one state to the other (Figure 9-3). To read’ the state of a chain segment, two adjacent carbon atoms have to be marked and the bond between them has to be inspected. Carter proposed to incorporate a push–pull olefin (Figure 9-4) into the chain. In the ground state there is a double bond between the two central carbon atoms. The molecule can be excited by absorption of light. The excited molecule has a single bond in the central position and carries a dipole moment. The substituted polyacetylene chain is depicted in Figure 9-5. The push–pull olefin is used either as a soliton valve’ – in the ground state the conjugation along the chain is unaffected and the soliton can pass; in the excited state the conjugation is interrupted and the soliton movement is blocked – or as a soliton detector’ – if the unit is imbedded in a state-B segment, light can be absorbed (double bond between central carbon atoms = ground state), but if in a state-A segment, it already has a central single bond and cannot absorb light. There are also soliton bifurcations’ (Figure 9-6) and soliton reversals’ (Figure 9-7). Carter’s proposals have often been ridiculed as too naive and unfeasible. But their value is not to give instructions on how to synthesize a computer but rather to initiate the development of simple concepts for theoretical investigations.
Figure 9-4 A push–pull olefin, which changes the nature of the bond between the central carbon atoms. It can be incorporated into a polyacetylene chain to serve as a soliton valve’ or as a soliton detector’.
9.3 Early and Radical Concepts
Figure 9-5 A push–pull olefin imbedded in trans-polyacetylene can be switched off by the propagation of a soliton or can be used as a soliton detector.
Figure 9-6 Soliton bifurcations. If the link is more complicated, the switch could be set from the outside, and bifurcations could be used to fan’ information from the macroscopic world into the molecular-electronic device.
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Figure 9-7
Molecular soliton reversals’.
9.3.2
Molecular Rectifiers
Almost ten years before Carter, Aviram and Ratner designed a molecular rectifier’ [28]. An organic donor, for example TTF, is linked via an inert spacer to an organic acceptor, e.g., TCNQ (Figure 9-8) [29]. The spacer is called the r bridge, because it contains only saturated bonds (in Section 9.3.4 we will also meet p bridges). Figure 9-9 depicts the level scheme of a D–r–A molecule. Evidently, it is very asymmetric with respect to electron transfer from donor to acceptor or vice versa. The Aviram–Ratner device is shown in Figure 9-10. The rectifying molecules have to be organized in an ordered way between metal electrodes, so that all donor sites point in the same direction. To extend these ideas from a single molecule, as discussed in the introduction, to a computer in a beaker’, an ordered deposition of organic molecules is necessary. Many researchers have suggested that molecules like the one described may lend themselves well to Langmuir–Blodgett techniques [30,31].
Figure 9-8
Molecular rectifier as proposed by Aviram and Ratner [28], a so-called D–r–A molecule.
9.3 Early and Radical Concepts
Figure 9-9 Level scheme of a D–r–A molecule. Because of the energy difference between D––r–A+ and D+–r–A– it is easier for electrons to move in one direction, and so the molecule acts as a rectifier [29].
Molecular rectifier using D–r–A molecules as proposed by Aviram and Ratner [28]. M1 and M2 are neutral electrodes. Schematic view according to Metzger [29]. Figure 9-10
9.3.3
Molecular Shift Register
In 1989 Hopfield and coworkers [32] proposed a molecular shift register. A shift register is an array of cells (Figure 9-11). At each clock purse, the information of one cell is shifted to the next cell. Hopfield’s proposal is to make a microelectronic memory based on silicon. The information would be stored in capacitors of about 1 lm2 in cross section, and to extend the storage capacity, the third dimension would be used. That is, a molecular shift register would be set on top of each capacitor. The
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Figure 9-11 Schematic view of a molecular shift register. The cells are polymerized and a polymer chain is suspended between two electrodes, which are part of a microelectronic capacitor [32].
shift register would then be loaded by injecting electrons from the metallic capacitor plate into the first cell of the register, and it would be read by depositing the electrons into the channel of a field-effect transistor at the opposite side of the register. A cell of the shift register would have to consist of a donor and an acceptor part, as indicated in Figure 9-12. Electron transfer from cell to cell is shown in the schematic energy diagram in Figure 9-13. An organic molecule that could act as a cell in a molecular shift register is depicted in Figure 9-14. It has not only a donor and an acceptor group, but also a
Schematic representation of the cells in a shift register memory. The cells contain at least one donor and one acceptor functional group [32].
Figure 9-12
Figure 9-13
The donor and acceptor levels in a typical shift register polymer [32].
Organic molecule that could act as a cell in a molecular shift register: upon proper illumination an electron is transferred from the left side to the right side [32].
Figure 9-14
9.3 Early and Radical Concepts
sensitizer in the bridge. The sensitizer acts as a valve. Absorption of light opens the valve, and an electron is transferred from the donor to the acceptor. The process is very fast, e.g., it occurs within 1 ps. When the light source is shut off, the valve closes and the electron cannot go back. However, it can – and will – tunnel to the next cell, e.g., within 1 ns. So if there is a clock pulse every 10 ns, each clock pulse shifts the electrons by one step to the right, because the shift involves fast intramolecular and slow intermolecular charge transfer. The filled and empty cells carry information as 1’ and 0’ (chemically speaking, the molecules are neutral or ionized). The molecule shown in Figure 9-14 is polymerized to a chain length of about 600 repeat units and many identical chains would be placed between the electrodes. The device would be immersed in a solution to shield the coulombic repulsion (see the discussion on single-electron devices in Section 9.1). If the transfer efficiency is 99.9%, at each step about half of the electrons arrive at the receiving electrode after 600 hops, and 5000 chains would provide a signal large enough for detection with the sensitivity of present-day microcircuits. The realization of Hopfield’s proposal does not seem to be impossible, although probably not with 600 repeat-unit polymers. 9.3.4
Molecular Cellular Automata
Figure 9-15 shows a donor and an acceptor linked by a p bridge (in contrast to the r bridge used in Section 9.3.2). Excitation of a D–p–A molecule can be interpreted as soliton switching: in the excited state, single and double bonds are inverted relative to the ground state. In addition, the excited state carries an electronic dipole moment. At the risk of oversimplifying, we could say that in a D–p–A molecule the donor and acceptor communicate by the exchange of solitons. A third functional group in the chain, called the modulable barrier (Figure 9-16), leads to a three-terminal device and allows logical operations. A linear array of D–p–A molecules (Figure 9-17) has been proposed as a molecular cellular automaton [33]. The scheme of a cellular automaton is presented in Figure 9-18. Here, only a one-dimensional array of cells is shown, but there could be two- or three-dimensional arrays, or arrays with even higher connectivities. The cells
Polyene chain with donor and acceptor groups attached at the ends. Upon optical excitation, single and double bonds in this D–p–A molecule are inverted and the excited molecule carries a dipole moment.
Figure 9-15
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Donor–acceptor-substituted polyene with a third functional group incorporated, the barrier’. This three-dimensional device allows for logical operations.
Figure 9-16
Figure 9-17
Donor–acceptor-substituted polyenes as cells of a linear cellular automaton [33].
Figure 9-18
Scheme of a cellular automaton.
must exhibit at least two states to be capable of storing information. When the array is exposed to clock pulses, e.g., to pulses from a laser, the cells switch (change their state), and the probability of switching a particular cell has to depend on the states of the neighboring cells (conditional switching). A cellular automaton is the extreme of a parallel computer. It can be shown that a cellular automaton can carry out all computations of a conventional (sequential) computer, however, often much faster. It is easy to see that the array of D–p–A molecules is a cellular automaton: by the absorption of light the cells switch from state A to state B. The switching probability depends on the state of the neighboring cells, according to the dipolar moment associated with the state of the cells. The field of the dipoles detunes the resonance frequency for absorption. The relevant question is, of course, How good a cellular automaton is a D–p–A chain? Like all the proposals in this section, the D–p–A cellular automaton also is not intended to work as a practical device; it is a model to allow the asking and study of particular questions.
9.4
Carbon Nanotubes
With the advent of novel molecules like the carbon nanotube, another approach to organic molecular-scale circuitry has appeared. Nanotubes are possible to manipulate spatially and, as demonstrated earlier, contacts can be added by several lithographic routes. Because of the unique one-dimensional band structure of the carbon nanotube and the fact that it represents a single crystal (generally visualized without defect), one might expect that charge would be transported ballistically’, that is,
9.4 Carbon Nanotubes
Schematically, the total number of available conduction channels in a mesoscopic wire that lie between the Fermi levels of the contacts all contribute to the current.
Figure 9-19
without scattering from one end to the other [34]. Of course, there are still the leads that touch the tube. There is a simple way to imagine what will happen in such a material and to understand why it would be useful as a device. Imagine a nanowire arranged between two electrodes as in Figure 9-19. Electrodes 1 and 2 have chemical potentials l1 and l2. The wire has length L. We take the electrodes as an unlimited reservoir of charge. If we assume no scattering at the interfaces between electrode and nanowire, and l1 > l2, then only electrons with energies l1 > E > l2 and having a wave vector going from left to right contribute to current in the wire. The pseudo Fermi level (defined by the highest occupied energy level for k > 0 and k < 0, treated as though these are independent because there is no scattering between them) for k > 0 is l1 and for k < 0 is l2. As we saw earlier, the dimensionality of the wire causes the allowed states to spread apart into sub-bands, as shown by the dashed lines. We can denote these energy states as Ei(k), for the ith state or band. So, whatever the current in the system is, we must add up the currents in the individual states between the pseudo Fermi levels to get it. Typically, the sub-bands with k states that have energies l1 > E > l2 are referred to as channels.’ The number of channels is a function of E, generally denoted M(E). Classically, an electron with a velocity given by v = h–1 (¶E/¶k) goes toward contact 2 in the above example, and one with an energy l1 > E > l2 contributes to the net current as I = ev/L. L/v is simply the transit time for the carrier. We can then add up a total current as: I = 2e/p
P Ð i
k>0
(h–1 (¶E/¶k) [ f(Ei – l1) – f(Ei – l2)] dk = 2 {e2/h} M (l1 – l2)/e
Spin degeneracy has of course been added to the sum. The inverse of the level spacing (demonstrated in previous chapters) has been used as the current density per sub-band, and the f’s are the Fermi levels. Now notice that if the width of the wire is small, as in a carbon nanotube, » 1.4 nm, the number of states between l1 and l2, M, is about 2 (spin-up and spin-down), even for large differences in the chemical potentials (i.e., > 1 eV). However, for large wires, say > 1 lm, the number of channels is huge (l1 – l2 = 1 eV, M » 106) [35].
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Schematic view of a nanotube field-effect transistor (taken from the homepage of the Molecular Biophysics Group at Delft University of Technology, C. Dekker www.mb.tn.tudelft.nl).
Figure 9-20
The voltage between the two leads is obviously V = (l2 – l1)/e and the resistance is given by Rc = V/I = h/2e2 (1/M), a very famous result. This is generally referred to as the contact resistance’, and h/2e2 is the quantum of resistance, » 12.9 kX. Thus, a wire with no scattering has a resistance proportional to M–1 or a conductance proportional to M. Because we have M = 2 for a metallic nanotube, as stated above, we should measure 2 · 12.9 kX for its resistance. If the scattering mechanism is time-independent, Landauer has shown that the conductance formula above is modified by adding a probabilistic term (the transfer matrix). This is essentially the probability of scattering between channels as the charge travels between the leads. It looks like this: R = h/2e2 (1/M T) where T is the probability for transmission and is given as P
ij
‰tij‰2
where the sum is over all M, and tij is the probability of scattering from the ith state to the jth state. Thus, for a nanotube, we expect the resistance to be R = 2 · 12.9 kX (1–T)/T Note that the reflected wave function causes a drop in the wire’s potential and that resistance is proportional to (1 – T). Of course, the actual predicted value then depends on a specific model for the scattering phenomena. It is also important to note that the phase relation of the carrier’s wave function can be lost in the scattering events. However, when it is preserved, scattering can lead to interference effects
9.4 Carbon Nanotubes
that result in localization of charge on the nanowire. The most important expression of these phase-preserving interactions is universal conductance fluctuations in which the noise observed in a transport IV curve is not noise at all, but rather interference patterns formed by the impurities and defects within the quantum wire [36]. This formulation holds only when a high degree of delocalization exists in the wire. This is certainly true for nanotubes, and a number of experiments have demonstrated this [34]. Shown in Figure 9-20 is a nanotube placed on two electrodes. Notice too, that the tube has been placed in a field-effect transistor configuration as described above. The behavior of this system as a transistor is shown in Figure 9-21. The conductance is given in the inset, and the numbers are different from our expectations above. Clearly, the contacts here are not quite ideal. Nevertheless, many people have now repeated these experiments and suggest that such performance is more than adequate in transistor design. In fact, carrying the concepts of molecular circuits one step further, several groups, including researchers at Delft and IBM [38] have constructed more complex signal-processing circuits from these basic transistors. Figures 9-22 and 9-23 show circuits developed by the Delft group. The circuits demonstrate the use of molecular electronics in inverters, logic gates, memory systems, and oscillators; all the basic building blocks of computational systems. To date, the question remains: How will this be done on the scale of modern semiconductor chips? However, it is clear the conceptually, there are a number of routes to achieving molecular-scale electronics.
IV characteristics as a function of gate voltage shows that nanotubes would make rather good transistors at room temperature [38].
Figure 9-21
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Figure 9-22
Height image of a single-nanotube transistor, acquired with an atomic force microscope. Whereas in Figure 9-20 the nanotube is placed on a SiO2 layer over doped
silicon, here the nanotube is placed on Al2O3 over Al. The gate contact is Al, which allows better coupling and higher gain.
Demonstration of one-, two-, and three-transistor logic circuits with carbon nanotube FETs. (A) Output voltage as a function of the input voltage of a nanotube inverter. (inset) Schematic of the electronic circuit. The resistance is 100 MX. (B) Output voltage of a nanotubes NOR for the four possible input states (1,1), (1,0), (0,1), and (0,0). A voltage of 0 V represents a logical 0 and a voltage of
–1.5 V represents a logical 1. The resistance is 50 MX. (C) Output voltage of a flip-flop memory cell (SRAM) composed of two nanotube FETs. The output logical stays at 0 or 1 after the switch to the input has been opened. The two resistances are 100 MX and 2 GX. (D) Output voltage as a function of time for a nanotube ring oscillator. The three resistances are 100 MX, 100 MX, and 2 GX [39].
Figure 9-23
References and Notes
References and Notes 1 D.H. Meadows, D.L. Meadows, and J. Randers.
The Limits of Growth, Report to the Club of Rome, 1972. 2 H. Queisser. Kristallene Krisen: Mikroelektronik; Wege der Forschung, Kampf und Mrkte. Piper, Mnchen, 1985. 3 D. Bloor, M. Hanack, A. Le Mhaut, R. Lazzaroni, J.P. Rabe, S. Roth, and H. Sassabe. Conjugated polymeric materials, opportunities in electronics, optoelectronics, and molecular electronics. NATO ASI Series E: Applied Science 182, 587. J.L. Bredas and R.R. Chance (Eds.), Kluwer, Dordrecht, 1990. 4 F.L. Carter. Molecular Electronic Devices. Marcel Dekker, New York, 1982. 5 F.L. Carter. Molecular Electronic Devices II. Marcel Dekker, New York, 1987. 6 F.L. Carter, R.E. Siatkowski, and H. Wohltjen. Molecular Electronic Devices. North Holland, Amsterdam, 1988. 7 A. Aviram. Molecular Electronics – Science and Technology. United Engineering Trustees, New York, 1989. 8 F.T. Hong. Molecular Electronics – Biosensor and Biocomputers. Plenum Press, New York, 1989. 9 G.J. Ashwell. Molecular Electronics. Wiley, New York, 1992. 10 Ch. Ziegler, W. Gpel, and G. Zerbi. Molecular Electronics. Proceedings of Symposium H of the 1993 E-MRS Spring Conference, North Holland, Amsterdam, 1993. 11 J. Chen, M.A. Reed, A.M Rawlett, and J.M. Tour. Large on–off ratios and negative differential resistance in a molecular electronic device. Science 286, 1550 (1999). M.A. Reed, C. Zhou, C.J. Muller, T.P. Burgin, and J.M. Tour. Conductance of a molecular junction. Science 278, 252 (1997). 12 M. Di Ventra, S.T. Pantelides, and N.D. Lang. The benzene molecule as a molecular resonant-tunneling transistor. Applied Physics Letters 76, 3448 (2000). 13 C.W.J. Beenakker and H. van Houten. Quantum Transport in Semiconductor Nanostructures. Solid State Physics 44, 1. H. Ehrenreich and D. Turnbull, Academic Press, San Diego, 1991. S. Sarkar (Ed.), Exotic States in Quantum Nanostructures. Kluwer, Amsterdam, 2003.
P. Harrison (Ed.), Quantum Wells, Wires and Dots: Theoretical and Computational Physics. John Wiley & Sons, New York, 2000. 14 K.K. Likharev. Possibility of creating analog and digital integrated circuits using the discrete, one-electron tunneling effect. Soviet Microelektronics 16, 109 (1987). K.K. Likharev. Correlated discrete transfer of single electrons in ultrasmall tunnel junctions. IBM Journal of Research and Development 32, 144 (1988). 15 D.V. Averin and K.K. Likharev. Mesoscopic Phenomena in Solids. p. 173. B.L. Altshuler, P.A. Lee, R.A. Webb (Eds.), Elsevier, Amsterdam, 1991. D.V. Averin and K.K. Likharev. Single Charge Tunneling. p. 311. H. Grabert and H.M. Devoret (Eds.), Plenum Press, New York, 1992. 16 T.A. Fulton and G.J. Dolan. Observation of single-electron charging effects in small tunnel junctions. Physical Review Letters, 59, 109 (1987). 17 L.S. Kuz’min and K.K. Likharev. Direct experimental observation of discrete correlated single-electron tunneling. JETP Letters 45, 495 (1987). 18 L.J. Geerlings, V.F. Anderegg, R.A.M. Holweg, J.E. Mooij, H. Pothier, D. Esteve, C. Urbina, and M.H. Devoret. Frequency-locked turnstile device for single electrons. Physical Review Letters 64, 2691 (1990). 19 P. Lafarge, H. Pothier, E.R. Williams, D. Esteve, C. Urbina, and M.H. Devoret. Direct observation of macroscopic charge quantization. Zeitschrift fr Physik B 85, 327 (1991). 20 K. Nakazato, R.J. Blaikie, R.J.A. Leaver, and H. Ahmed. Single-electron memory. Electronics Letters 29, 384 (1993). 21 K. Nakazato and H. Ahmed. The multiple tunnel junction and its application to single-electron memories. Advanced Materials 5, 668 (1993). 22 There are now a number of excellent monographs on quantum computing. An excellent hands-on’ treatise comes from C. Williams and S. Clearwater. Explorations in Quantum Computing. Springer Verlag, Berlin, 1997. 23 W.G. Teich, K. Obermayer, G. Mahler. Structural basis of multistationary quantum sys-
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25
26
27 28
29
30
31
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9 Molecular-scale Electronics tems. II. Effective few-particle dynamics. Physical Review B 37, 8111 (1988). M.J. Rice. Charged p-phase kinks in lightly doped polyacetylene. Physics Letters A 71, 152 (1979). W.P. Su, J.R. Schrieffer, and A.J. Heeger. Solitons in polyacetylene. Physical Review Letters 42, 1698 (1979). S.A. Brazovskii. Electronic excitations in the Peierls–Frhlich state. JETP Letters 28, 606 (1978). S.A. Brazovskii. Self-localized excitations in the Peierls–Frhlich state. Soviet Physics JETP 51, 342 (1980). F.L. Carter. Molecular Electronic Devices. p. 51. Marcel Dekker, New York, 1982. A. Aviram and M.A. Ratner. Molecular rectifiers. Chemical Physics Letters 29, 277 (1974). R.M. Metzger. Electricity and Magnetism in Biology and Medicine. p. 175. M. Blank (Ed.), San Francisco Press, San Francisco, 1993. For a monograph, see for example: G. Roberts. Langmuir–Blodgett Films. Plenum Press, New York, 1990. R.M. Metzger. Biomolecular Electronics: Advances in Chemistry Series 240, 81. R.R. Birge (Ed.), American Chemical Society, 1994. J.J. Hopfield, J.N. Onuchic, and D.N. Beratran. A molecular shift register based on electron transfer. Science 241, 817 (1988). J.J. Hopfield, J.N. Onuchic, and D.N. Beratran. Electronic shift register memory based on molecular electron-transfer reactions. Journal of Physical Chemistry 93, 6350 (1989).
D.N. Beratran, J.N. Onuchic, and J.J. Hopfield. Electronics: Biosensor and Biocomputers. p. 352. Plenum Press, New York, 1989. 33 S. Roth, G. Mahler, Y.Q. Shen, and F. Coter. Molecular electronics of conducting polymers. Synthetic Metals 28, C815 (1989). 34 S.J. Tans, M.H. Devoret, H. Dai, A. Thess, R.E. Smalley, L.J. Geerligs, and C. Dekker. Individual single-wall nanotubes as quantum wires. Nature 386, 474 (1997). Z. Yao, H.W. C. Postma, L. Balents, and C. Dekker. Carbon nanotube intramolecular junctions. Nature 402, 273 (1999). H.W.Ch. Postma, T.F. Teepen, Z. Yao, M. Grifoni, C. Dekker. Carbon nanotubes single-electron transistors at room temperature. Science 293, 76 (2001). 35 These numbers are from R. Saito, G. Dresselhaus and M. Dresselhaus (Eds.), Physical Properties of Carbon Nanotubes. p. 142. Imperial College Press, London, 1998. 36 P.A. Lee, A.D. Stone, and H. Fukuyama. Universal conductance fluctuations in metals: effects of finite temperature, interactions, and magnetic field. Physical Review B 35, 1039 (1987). 37 V. Derycke, R. Martel, J. Appenzeller, and P. Avouris. Carbon nanotube inter- and intramolecular logic gates. Nano Letters 1, 453 (2001) 38 S.J. Tans, A.R.M. Verschueren, and C. Dekker. Room-temperature transistor based on a single carbon nanotube. Nature 393, 49 (1998). 39 A. Bachtold, P. Hadley, T. Nakanishi, C. Dekker. Logic circuits with carbon nanotube transistors. Science 294, 1317 (2001).
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Molecular Materials for Electronics As noted in the previous chapter, there is more than one use of the term molecular electronics’. In this chapter we explore the definition that applies to the use of synthetic metals and organic semiconductors to create micrometer-scale electronics. Only a few examples of this broadly developing field are presented here. However, the principles of the use of these materials can be clearly seen.
10.1
Introduction
The models discussed in the previous chapter are electronics on a molecular level’. However, as alluded to, one could also imagine covering large plates with Aviram– Ratner rectifiers, which would be used as large-scale rectifiers rather than as molecular level devices, thus changing the subject to molecular materials for electronics’. In Section 3.1 we learned that the constituents of molecular crystals are molecules and not atoms. The term molecular material is used in this manner. A molecule is designed in such a way that it exhibits the desired feature, for example, rectifying properties. Then the molecules are assembled to form a material, for example, a Langmuir–Blodgett film. The molecules keep their properties in the film. This opens up possibilities of tailoring material properties by molecular engineering. Molecular materials for electronics are not new. Even the etymological roots of electronics lead to a molecular material – to the natural resin amber, which in Greek is electron’ and in which electric phenomena were first observed (charging by friction). Commonly, molecular materials serve as insulators. Other well known molecular materials for electronics are photoresists used in the photolithographic process for the production of microelectronic devices. Often, molecular materials must be electroactive’ or photoactive’ to qualify for molecular electronics’.
One-Dimensional Metals, Second Edition. Siegmar Roth, David Carroll Copyright © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30749-4
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10.2
Switching Molecular Devices
There are an enormous number of switching molecules’. In fact, every molecule has several excited states, and transitions between these states can be regarded as switching. Some molecules are bistable and are switched back and forth by external triggers; others need a trigger to switch only one way – they switch back spontaneously. The latter phenomenon is rather trivial. An example is the absorption of light and the subsequent decay of the excited state. But even this trivial behavior can be used for data processing (see the polyacetylene holographic computer in the next chapter). In this section we present two switching molecules, one because of its relationship to conjugated polymers and the other because of its complicated conformational changes during switching. 10.2.1
Photoabsorption Switching
In Figure 10-1 is shown the chemical structure of PCNC (7-piperonyl-7¢,7¢-diapocarotene-7¢-nitrile), a D–p–A molecule with a piperonyl group as donor and a cyano group as acceptor. As indicated in Figure 9-15, the excitation of this molecule corresponds to pushing an electron from the donor toward the acceptor. An additional electron on a polymer chain behaves like a soliton (see Section 5.5), and quantum chemical calculations indeed show that a phase-slip center in the charge distribution is created [1]. Solitons lead to states in the gap, which can be seen in optical absorption spectra. Figure 10-2 shows the results of an investigation of photoinduced absorption in PCNC [2,3]. The sample is irradiated by light with quantum energy greater than the p–p* gap, so that many molecules are pumped into the excited state. The absorption difference between pump-on and pump-off is recorded. The figure shows a photoinduced peak at about 1.2 eV and a bleaching dip, which is due to the reduction of the ground state population by constant pumping. The peak results from states in the gap that do not exist in the ground state. It is reasonable to assign this peak to solitons. We switch the molecules by irradiating with light, and we read the state of the molecule by absorption spectroscopy. The purpose of this experiment is not, however, to demonstrate the possibility of molecular switching. We want to investigate the excited state. From the position of the photoin-
Figure 10-1
Donor–acceptor polyene, a D–p–A molecule for photoinduced absorption.
10.2 Switching Molecular Devices
Photoinduced absorption of donor–acceptor polyene. The dashed line corresponds to normal absorption. Important features are the bleaching dip at 2.2 eV and the photoinduced peak at 1.2 eV, which corresponds to states within the p–p* gap [3].
Figure 10-2
duced peak, information can be obtained on whether the soliton concept can be applied reasonably well to short-chain polyenes, whether it will survive the attachment of donor and acceptor groups at the chain ends, etc. Short polyenes are usually treated by ab initio calculations [4,5], which do not explicitly make use of solitons or polarons. However, the short-chain approach must continuously fit to the long-chain approach. The bianthrone molecule is shown in Figure 4-2. We learned that the ground state is planar and puckered, and the excited state twisted and unpuckered. Puckering and twisting can be followed from the temporal evolution of the absorption spectrum. For this purpose, the sample is hit with a short, intense light pulse
Temporal evolution of the absorption spectra of unsubstituted (left side, BA) and substituted (right side, BATMF) bianthrone. The time indicated is the delay of the white light pulse with respect to the switching pulse (pump pulse) [6].
Figure 10-3
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Chemical formulas of unsubstituted bianthrone (BA) and bianthrone substituted with tetramethylfulvalene (BATMF).
Figure 10-4
(1 ps, 1 mJ), which triggers the switching, and then wide-band absorption spectra are taken at intervals of several picoseconds or nanoseconds. The experiment requires a fast optical pump-and-probe setup and is about as sophisticated as the soliton lifetime investigation described in Section 5.8 (short intense laser pulses, picosecond and nanosecond delay lines, conversion into picosecond white light pulses, diode array as spectrometer). Figure 10-3 shows a comparison of the time evolution of the absorption spectra of substituted and unsubstituted bianthrone (BATMF and BA, respectively). The chemical structures are shown in Figure 10-4. The features in the absorption spectra can be assigned to twist and pucker changes [6]. As a striking result, we see substitution stabilizing the excited state (Figure 10-5). Systematic studies of this type should finally help to design molecules with time constants optimized for the intended application.
Decay of the excited state in substituted bianthrone (BATMF) compared to that in unsubstituted bianthrone (BA) as an example of molecular engineering for optimizing decay time constants [6].
Figure 10-5
10.2 Switching Molecular Devices
10.2.2
Rectifying Langmuir–Blodgett Layers
An elegant use of heterojunctions based on Langmuir–Blodgett (LB) films has been demonstrated by Fischer et al. [7], utilizing covered gold microelectrodes (Figure 6-11) with a few LB layers of a palladium phthalocyanine derivative followed by several layers of a perylene derivative, who finally succeeded in evaporating gold top electrodes without destroying the delicate organic structure (Figure 10-6). The device structure is so small that several of the diodes placed on a silicon chip are pinhole-free, and current voltage characteristics can be measured. Figure 10-7 shows the characteristics of Au/PcPd/Au (dotted lines) and of Au/ PTCDI/Au (solid line) sandwiches. These characteristics are symmetric and blocking to a pronounced threshold. The threshold is interpreted as due to the coincidence of the Au Fermi level with the p or p* band edge in the organic layer. Figure 10-8 shows the asymmetric characteristics of a Au/PTCDI/PcPd/Au sandwich, with a positive threshold of 0.9 eV and a negative threshold of –0.5 eV. At a working point of 0.6 eV, a rectification ratio of several orders of magnitude is obtained. Additional interesting features are the steps in the IV characteristic shown in Figure 10-8. These are interpreted as single-electron effects along the lines of Section 9.1. The p-electron systems of phthalocyanine and perylene are tentatively assumed to be quantum dots charged by single electrons and blocking further electrodes from following (unless the voltage is raised high enough to surpass the Coulomb barrier, hence the steps).
Rectifying organic heterolayers. (a) Palladium phthalocyanine; (b) perylene derivative; (c) arrangement of sandwiched layer structure. The substituent groups R in (a) and (b) were attached to facilitate film formation in the LB technique.
Figure 10-6
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Symmetric current voltage characteristic of Au/PcPd/Au and Au/PTCDI/Au sandwiches. The structures block up to a threshold voltage and conduct above it.
Figure 10-7
Asymmetric current–voltage characteristic of a Au/PTCDI/PcPd/Au heterostructure. Rectifying properties are observed when the device is operated between 0.5 and 0.9 V. The steps in the characteristic are interpreted as single-electron charging of the organic macromolecules. Figure 10-8
10.3
Organic Light-emitting Devices
Presently it seems as if organic light-emitting devices (OLEDs) are among the most promising molecular electronic systems for short-term realization of marketable technology (in fact, certain brands of electric shavers now use OLEDs). Here we present a schematic view (Figure 10-9) of these devices, along with some of the important features of their operation.
10.3 Organic Light-emitting Devices
Figure 10-9
Schematic setup of an organic light-emitting device (OLED).
The overall design is conceptually quite simple. A thin organic film is sandwiched between two electrodes, one of them semitransparent, one reflective. A voltage is applied: one electrode injects electrons, the other injects holes into the film. The lattice relaxes, and electrons and holes convert into solitons or polarons or whatever electron-lattice coupling requires the charge carriers to do (see Section 5.5 on soliton generation by charge injection). Finally, the charge carriers recombine and light is emitted (luminescence). The process is outlined diagrammatically in Figure 10-10. There are two recombination channels: a radiative and a nonradiative one. For efficient LEDs the radiative channel must be more pronounced. Nonradiative recombination generates heat and limits the lifetime of the device. Furthermore, carriers should not get lost by falling into traps or by hitting the opposite electrode without meeting a partner to recombine with. And finally, the polymer and other parts of the device must be sufficiently transparent to the generated light so that it does not get reabsorbed before leaving the LED. In semiconductor physics LED means lightemitting diode, and the diode is a pn junction or a Schottky barrier. In this book we use the more general expression: light-emitting device. With this interpretation of the letter D’ we want to indicate that the organic film might be a p type or an n type semiconductor, but it also could be an intrinsic semiconductor with an equal concentration of electrons and holes. In particular, both types of carriers can be injected. Then the device would not be rectifying unless electrodes with different work functions are used. There are two classes of OLEDs: OLEDs based on conjugated polymers and multilayer dye, or small molecule, OLEDs [8]. Polymer OLEDs were first reported by Burroughes et al. [9], and since then have been investigated by several groups [10–17]. The emission spectrum can be adjusted by molecular engineering, and presently the whole visible range of colors is available in light-emitting polymers and small molecules. Internal quantum efficiencies in optimized devices range from 1% to 4%, which in many cases is better, for blue light much better, than inorganic competitors (1% is sufficient for most indoor applications). Brightness levels of 100 candela m–2, which are the standard for back-lit liquid crystal displays, are relatively easily achieved, with current densities of about 10 mA cm–2 for a device with 1% efficiency. Still a problem is lifetime: 1000 hours of reliable operation presents a minimum level of performance for a variety of applications, but 20 000 hours would be
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desirable. Lifetimes of several hundred hours without failure have already been achieved [18], and experts are optimistic about the future of polymer OLEDs. 10.3.1
Fundamentals of OLEDs
To more fully understand the operation of the OLED configuration, the problem must be broken into three parts: charge injection, charge transport, and charge recombination. Doing this for double carrier injection has been a difficulty even in solid-state systems with long-range order. However, short-range order and hopping mechanisms make the problem considerably more difficult with organics. Beginning with charge injection, we first imagine what the local potential barriers to injection are after contact has been made. Shown in Figure 10-11 is an energy level diagram of a single-layer OLED system, in which the organic is depicted as a fully depleted semiconductor (i.e., no free charge). Qualitatively, we can view the process of injection as above. Before the contacts are added, the Fermi levels of the cathode and anode are shown as EF. When the contacts are brought into contact with the semiconductor, the Fermi levels of the contacts must align across the device. This results in a built-in potential, Vbi, across the organic layer. For an applied potential Vapp of less than Vbi, the electric field inside the organic opposes charge injection and forward drift currents. When Vapp > Vbi, current is injected over the barrier. Current density and irradiance in the device then increase rapidly. Notice that the level of the cathode to the polymer band edge might be quite different from the level of the anode to the band edge; thus, overall injection barriers for electrons and holes might be different in this scheme. The exact method of injection, that is, how the bare charge of the cathode and anode becomes the more exotic excitation of the organic system, is still not fully understood. However, significant progress has been made in modeling this process
Figure 10-10 The injection of positive and negative charge results in charged polarons. The polaronic states sit within the HOMO–LUMO gap of the active material, as discussed in previous chapters. Radiative recombination yields light.
10.3 Organic Light-emitting Devices
since its first discovery. In fact, instead of the simple band as shown in our diagram, it is a little more precise to imagine the organic electronics as a distribution of isolated localized states [19]. This absence of long-range order has profound effects on all aspects of fundamental device operation: injection, transport, and recombination. Yet many concepts of the crystalline system can be transferred. For instance, organic/metal interfaces can be ohmic or injection-limited [20]. With very few exceptions, most attempts to model this injection have relied on Richadson–Dushman thermionic emission [21] or Fowler–Nordheim tunneling [22]. In more advanced treatments, the disorder of the polymeric materials, or the organic material, suggests that significant localization occurs near the injection electrode, resulting in increased reflection probabilities, reduced injection rates due to dipole layers, and a steeper field dependence of the injected current. It is however important to note that not all simulations and models yield the same results. In fact, if injection can be modeled as tunneling for lower currents but as thermionic at higher currents, then injection probabilities will vary widely depending on the regime of conductivity [23]. The second major step in understanding OLEDs is how the charge is transported in the organic. In fact, this is a general statement of transport in such materials and has been investigated in the past in connection with other applications [24]. Charge transport is typically viewed as a hopping process, in which hopping takes place between sites that are statistically different in their surroundings. Formally, theories such as Bssler’s (where disorder is distributed using a Gaussian function) have been quite successful in describing the overall features of transport [25]. The primary feature of such models is that the mobility of the carriers depends on the applied electric field and is usually written: l = lo exp [(q/kT) b (F)1/2] b: Poole–Frenkel factor; F: applied field. Reflecting on our previous model for injection, it is quite easy to see that such a system should result in field-dependent mobilities. States introduced into the gaps will have trapping and detrapping times that depend on the applied field. If the mobility of the carrier is dominated by these trapping–detrapping events, then the overall effect will be to build in a nonlinear field dependence to the time of flight through some part of the material. Finally, there is little doubt that Langevin-type rate equations dominate in organic systems [26]. Yet, again, recombination mechanisms on a microscopic scale are lacking. We do know that, in analogy to crystalline systems, microcavity effects are observed in these systems. Modification spectrally and spatially of spontaneous emission rates, using microcavities, suggest further correlations with solid-state systems [27]. 10.3.2
Materials for OLEDs
Electroluminescence was first discovered in poly(para-penylenevinylene: PPV) by workers at Cambridge in 1990 [9]. Since that time, a vast number of polymers have
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been demonstrated with high efficiencies, multiple colors, and other features. In fact, emissive polymers can be thought of as falling into classes: the phenylenevinylenes, the thiophenes, the pyridines, the polyfluorenes, and more, with each based on a modification of a simpler semiconducting polymer. Side groups and alterations to the chain are used to change the overall performance, coloration, and solubility of the polymers. Whole catalogues are now available from manufacturers of electroluminescent products, which reflect different color options, different balances between hole and electron mobilities at specified applied fields, etc. 10.3.3
Device Designs for OLEDs
We should note that having a really efficient emissive polymer does not always lead to a good OLED. As with any solid-state electronic device, the engineering of the OLED must take into account contact barriers, lifetime, diffusion lengths, and more. Generally, the electrodes must be matched to the polymer of choice by using UPS (or some equivalent technique) measurements of the work function. This is usually easily done to the metal side, because adding a small amount of alkali metal can adjust the metal work-function value so as to match a needed value. However, on ITO, the transparent hole injector, it is a little more difficult. The addition of a PEDOT (polyethylene dioxythiophene) layer on top of the ITO can help with this only a little. Second, hole and electron mobilities are not usually the same. So it is necessary to use layer thicknesses arranged such that the electrons and holes arrive in the center of the luminescent layer at the same time. This is not as easy as it sounds. Really thick luminescent layers lead to nonradiative recombination and a reduction in performance. Finally, not all electrons and holes that make it to the center of the active layer recombine (radiatively or not), but rather – What happens if they miss? This can lead to leakage currents, that are due to the carriers that do not recombine in some way and make it to the opposite electrode. Thus, most designers choose materials that do not allow for the opposite charge to be injected into them. For example, ITO is an excellent hole conductor and quite good at injecting holes. However, it is a poor electron conductor and does not readily allow for electrons to leak through the contact. Alternatively, a blocking layer can be added to the device to prevent this parasitic process. This gives the electrons more time to interact with holes.
10.4
Solar Cells
The organic photovoltaic cell is related to the organic light-emitting diode, and much of the above discussion applies. Highly doped conjugated polymers are metals, but undoped or lightly doped polymers are semiconductors, as pointed out above. Generally, in a crystalline system like silicon, a pn junction is formed be-
10.4 Solar Cells
Figure 10-11 Band diagrams for applying contacts and subsequent injection of charge into organic layers.
tween two thin layers of doped material. Light that enters creates an electron–hole pair that migrates randomly in the film until they cross the pn junction. At this point (at this pn interface), the electron–hole pair is separated, and we retrieve this as usable current. It is not easy to make pn junctions in conjugated polymers, however, because the dopants, being interstitial, are very mobile. So they migrate and compensate. But Schottky barriers can be formed just by evaporating a thin layer of a metal with a proper work function onto the polymer. A Schottky barrier on polyacetylene or another conjugated polymer acts as a solar cell. A device is schematically depicted in Figure 10-12. (A Schottky barrier works as a solar cell because, at the polymer–metal interface, there is a built-in’ electrical field created by electron exchange between metal and polymer. If the incoming light generates electron–hole pairs (or soliton–anitsoliton pairs), these carriers are separated in the field. If the electrodes are connected to an external load, a photocurrent can flow and the field is maintained.) Power conversion efficiencies of » 1%–3% (depending on design) can be achieved presently. There are reasons to believe that this approach will work and other reasons to believe that it will encounter difficulties. It is well known, for instance, that organics can provide exceptionally good chromophores. Indeed, the efficiency of a polymer to convert light into an electron–hole pair [28] is excellent, exceeding all but the bestknown solid-state crystalline systems. However, it is equally well known that it is nearly impossible to remove substantial amounts of these excitations from the polymer as current. Thus, organics couple well to photons but make poor photovoltaic
Figure 10-12
Schematic view of a polyacetylene solar cell.
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conversion materials. The difficulty comes in fundamental materials properties that derive from both the polymer electronics as well as its structure in solid form. Migration lengths for the donor–acceptor excitations [29] are much shorter than the absorption lengths required to create the excitations. Thus, the film thickness must be 500 nm to create an excitation, but the distance to an effective separation barrier must be 50 nm for the electrons and holes to be removed as useful current. In the field of photovoltaics devices as it applies to organics, and specifically polymers (our one-dimensional metals), several basic terms recur. The first is the internal quantum efficiency (IQE) of the device. This is the number of electron–hole pairs actually separated per incoming photon and is spectrally resolved. Second, and most often cited, is the external efficiency, or the power conversion efficiency, ge: ge = (Voc [V] · Isc [A cm–2] · F.F.) / (Pin [W cm–2]) Here Voc, Isc, F.F., and Pin are the open-circuit voltage, the short-circuit current, the filling factor, and the incident power, respectively. The filling factor is determined by calculating the maximum power rectangular area under the I/V curve in the 4th quadrant. Simply, it is given by F.F. = Vp · Ip / (Voc · Isc) where Vp and Ip are the intersections of the I/V curve with the maximum power rectangle. Figure 10-13 shows a typical IV curve of an illuminated and dark organic photocell. The rectangle drawn against the 0 V line is the maximum power rectangle’. In the example shown, we have 0.2 [V] · 8.0 · 10–4 [A] (1.6 · 10–5), or a filling factor F.F. equal to roughly 0.23. Our example here happens to be a thin-film device illu-
Figure 10-13 IV curve of a thin-film photovoltaic cell under illuminated and dark conditions. The maximum power rectangle is drawn as described in the text.
10.5 Organic Field-effect Transistors
minated with a solar standard of 1.5 air masses (am 1.5 g). This simulates the average amount and composition of sunlight throughout the day. This particular example has an external power efficiency of about 2.3%. The offset (or open-circuit voltage) in the system is the difference in work functions of the contact and thin active film. This has been studied extensively by Marks et al. [30]. In such studies comparisons can be made between devices of the structures ITO/PPV/Mg and ITO/PPV/Ca, for instance. The offset voltages in the IV curve, that is, the open-circuit voltages, were 1.2 V and 1.7 V, respectively. These numbers are roughly the same as the difference in work function between Mg and ITO and Ca and ITO. However, in an ITO/PPV/Al device, one also obtains 1.2 V for the open-circuit voltage. Oddly, this is significantly higher than the difference in work function between ITO and Al. Thus, a sizable Schottky barrier has occurred, effectively raising the injection energy cost. It is surprising that this is not seen so strongly in alkali metals for this system. As mentioned above, the essential difficulty, and thus the focus of intense research, is the mismatch of length scales in this system. Some rather novel approaches have been made to addressing this shortcoming of organics. One unusual approach has been the introduction of a highly conducting nanophase’, such as fullerenes [31]. In this scheme, fullerene molecules are added to the interface region of the heterojunction to aid in the separation of charge. The fullerenes can then provide a percolating network of highly conductive material out of the layer. This approach has been tried with conducting polymer phases as well as with carbon nanotubes [32]. Interpenetrating networks have provided some of the most exciting directions in this field. If the charge can be separated at the interface of the nanophase and the host polymer, then one could imagine very efficient harvesting of current from the photoactive layer. A solar cell made of single-crystal silicon has to run for about eight years until it has generated as much energy as had been used for the production of the cell. But because conjugated polymers are synthesized by room-temperature catalytic processes, they are energetically cheap and have a much higher energy-harvest factor. But the harvest factor is not the only economically relevant figure. Land costs and maintenance costs are equally important. Recently, with higher efficiencies being achieved and the need for flexible and mobile power sources growing, it is becoming clear that organic solar cells may be a viable alternative to silicon. For instance, thinfilm based organics could reduce the launch weight of a solar-powered satellite significantly, because of the extraordinarily low density of the materials.
10.5
Organic Field-effect Transistors
In each of the examples above, we have seen how the disordered nature of our onedimensional systems has modified standard notions about standard electronic devices. This applies as well to transistors made from organic semiconducting systems. More specifically, we might well imagine that the field-dependent mobility
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Figure 10-14
All-polymer field-effect transistor [36].
would be expressed rather strongly in an all-polymer field-effect transistor. Polymer field-effect transistors [33–39] have historically been prepared as a tool for investigating material properties of the polymers, but with improving material quality, they are approaching practical use. Prepared in the form of thin films, these semiconductors are ideal for the creation of flexible, inexpensive field-effect transistors. Figure 10-14 shows such an all-polymer field-effect transistor [33–39]. This transistor consists of a polymer film between a source and a drain contact and a gate on the back. To avoid direct contact between the source or drain and the gate, the gate is protected by a thin insulating layer of silicon dioxide. As the back gate of the device is biased, the total amount of charge allowed through the channel is changed. An example of drain current versus drain voltage characteristics is shown in Figure 10-15 [39]. The characteristics demonstrate that all-polymer field-effect transistors work’. More important is the present use of such devices as experimental tools, for example, to determine the mobility of charge carriers in thin organic layers.
Figure 10-15 Drain current versus drain voltage characteristics of an all-organic (sexithiophene) field-effect transistor [39].
References and Notes
The fundamental operation of the device follows in the same manner as the crystalline system [40]. As mentioned above, however, the mobility of charge is dominated by its hopping nature. Furthermore, there is a relatively low number of charge carriers, and it is therefore rather difficult to build a significant depletion region. These factors introduce limitations into the current applicability of the organic fieldeffect transistor structures. However, advances are being made rapidly.
References and Notes 1 Y.Q. Shen, W. Ghring, S. Hagen, S. Roth, and
M. Hanack. NMR and Pariser–Parr–Pople investigations of donor–acceptor substituted polyenes. Journal of Molecular Electronics 6, 31 (1990). 2 M. Filzmoser. PhD thesis. Graz, 1991. 3 M. Filzmoser and S. Roth. Photoinduced absorption investigations on donor–acceptor polyenes. Synthetic Metals 41, 1263 (1991). 4 B.S. Hudson, B.E. Kohler, and K. Schulten. Linear polyene electronic structure and potential surfaces. Excited States 6, 1 (1982). 5 B.E. Kohler. New Conducting Materials with Conjugated Double Bonds: Polymers, Fullerenes, Graphite. Materials Science Forum. J. Przyluski and S. Roth (Eds.), Tans Tech Publications, Aedermannsdorf, in press. 6 J. Anders, H.J. Byrne, J. Poplawski, S. Roth, T. Bjrnholm, M. Jrgensen, P. Sommer-Larsen, and K. Schaumburg. Excited state transient spectroscopy of anthracene based photochromic systems. Synthetic Metals 57, 4820 (1993). 7 C.M. Fischer, M. Burghard, S. Roth, and K. von Klitzing. Organic quantum wells: molecular rectification and single-electron tunnelling. Europhysics Letters 28, 129 (1994). 8 T. Tsutsui and S. Saito. Intrinsically Conducting Polymers: An Emerging Technology, p. 123. M. Aldissi (Ed.), Kluwer, Dordrecht, 1993. 9 J.H. Burroughes, D.D.C. Bradley, A.R. Brown, R.N. Marks, K. Mackay, R.H. Friend, P.L. Burns, and A.B. Holmes. Light-emitting diodes based on conjugated polymers. Nature 347, 539 (1990). 10 D. Braun and A.J. Heeger. Visible-light emission from semiconducting polymer diodes. Applied Physics Letters 58, 1982 (1991). 11 Y. Ohmori, M. Uchida, K. Muro, and K. Yoshino. Visible-light electroluminescent diodes utilizing poly(3-alkylthiophene). Japanese Journal of Applied Physics 30, L1938 (1991).
12 Y. Ohmori, M. Uchida, K. Muro, and K.
13
14
15
16
17
18
19
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Yoshino. Blue electroluminescent diodes utilizing poly(alkylfluorene). Japanese Journal of Applied Physics 30, L1941 (1991). G. Grem, G. Leditzky, B. Ullrich, and G. Leising. Realization of a blue-light emitting device using polyphenylene. Advanced Materials 4, 36 (1992). P.L. Burn, A.B. Holmes, A. Kraft, D.D.C. Bradley, A.R. Braun, and R.H. Friend. Synthesis of a segmented conjugated polymer chain giving a blue-shifted electroluminescence and improved efficiency. Journal of the Chemical Society, Chemical Communications 34 (1992). P.L. Burn, A.B. Holmes, A. Kraft, D.D.C. Bradley, A.R. Braun, R.H. Friend, and R.W. Gymer. Chemical tuning of electroluminescent copolymers to improve emission efficiencies and allow patterning. Nature 356, 47 (1992). G. Gustafsson, Y. Cao, F. Klavetter, N. Colaneri, and A.J. Heeger. Flexible light-emitting diodes made from soluble conducting polymers. Nature 357, 477 (1992). A.R. Brown, N.C. Greenham, R.W. Gymer, K. Pichler, D.D.C. Bradley, R.H. Friend, P.L. Burn, A. Kraft, and A.B. Holmes. Intrinsically Conducting Polymers: An Emerging Technology, p. 87. M. Aldissi (Ed.), Kluwer, Dordrecht, 1993. S. Karg, W. Reiss, V. Dyakanov, and M. Schwoerer. Electrical and optical characterization of poly(phenylene-vinylene) light emitting diodes. Synthetic Metals 54, 427 (1993). H. Bssler. Charge transport in disordered organic photoconductors (a Monte Carlo simulation study). physica status solidi (b) 175, 15 (1993). M. Abkowitz, J.S. Facci, and J. Rehm. Dimension scaling of 1/f noise in the base current of quasi self-aligned polysilicon emitter bipolar junction transistors. Journal of Applied Physics 82, 2671 (1998).
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21 N.W. Ashcroft and N.D. Mermin. Solid State
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23
24
25
26 27
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29
Physics, p. 363. Holt, Rinehart, Winston, New York, 1976. R.H. Fowler and L. Nordheim. Electron emission in intense electric fields. Proceedings of the Royal Society A 119, 173 (1928). Y.N. Garstein and E.M. Conwell. Field-dependent thermal injection into a disordered molecular insulator. Chemical Physics Letters 255, 93 (1996). V.I. Arkhipov, E.V. Emelianova, Y.H. Tak, and H. Bssler. Charge injection into light-emitting diodes: theory and experiment. Journal of Applied Physics 84, 848 (1998). M.N. Bussac, D. Michoud, and L. Zuppiroli. Electron injection into conjugated polymers. Physical Review Letters 81, 1678 (1998). P.R. Emtage and J.J. O’Dwyer. Richardson– Schottky effect in insulators. Physical Review Letters 16, 356 (1966). P.M. Borsenberger and D.S. Weiss. Organic Photoreceptors for Electrophotography. Marcel Dekker, New York, 1998. D. Hertel, U. Scherf, and H. Bssler. Charge carrier mobility in a ladder-type conjugated polymer. Advanced Materials 10, 1119 (1998). M. Redecker, D.D.C. Bradley, M. Inbasekaran, and E.P. Woo. Nondispersive hole transport in an electroluminescent polyfluorene. Applied Physics Letters 73, 1565 (1998). P. Langevin. Ann. Chem. Physics 3, 289 (1903). N. Takada, T. Tsutsu, and S. Saito. Control of emission characteristics in organic thin-film electroluminescent diodes using an opticalmicrocavity structure. Applied Physics Letters 63, 2032 (1993). H.F. Wittman, J. Gruner, R.H. Friend, G.W.C. Spencer, S.C. Moratti, and A.B. Holmes. Microcavity effect in a single-layer polymer lightemitting diode. Advanced Materials 7, 541 (1995). S. Tokito, K. Noda, and Y. Taga. Strongly directed single mode emission from organic electroluminescent diode with a microcavity. Applied Physics Letters 68, 2633 (1996). E.A. Silinsh. On the physical nature of traps in molecular crystals. physica status solidi (a) 3, 817 (1970). N.S. Sariciftci, L. Smilowitz, A.J. Heeger, and F. Wudl. Photoinduced electron transfer from a conducting polymer to buckminsterfullerene. Science 258, 1474 (1992).
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G. Yu, J. Gao, J.C. Hummelen, F. Wudl, and A.J. Heeger. Polymer photovoltaic cells: enhanced efficiencies via a network of internal donor–acceptor heterojunctions. Science 270, 1789 (1995). R.N. Marks, J.J.M. Halls, D.D.C. Bradley, R.H. Friend, and A.B. Holmes. The photovoltaic response in poly(p-phenylene vinylene) thinfilm devices. Journal of Physics: Condensed Matter 6, 1379 (1994). For a review of the field see, for example: G. Hadziioannou and P.F. van Hutten (Eds.), Semiconducting Polymers. Wiley-VCH, Weinheim, 2000. E. Kymakis and G. A. J. Amaratunga. Singlewall carbon nanotube/conjugated polymer photovoltaic devices. Applied Physics Letters 80, 112 (2002). G. Horowitz. Organic superconductors for new electronic devices. Advanced Materials 2, 287 (1990). F. Garnier, G. Horowitz, and D. Fichou. Conjugated polymers and oligomers as active material for electronic devices. Synthetic Metals 28, 705 (1989). H. Burroughes, R.H. Friend, and P.C. Allen. Field-enhanced conductivity in polyacetyleneconstruction of a field-effect transistor. Journal of Physics D 22, 956 (1989). H. Burroughes, C.A. Jones, R.A. Lawrence, and R.H. Friend. Conjugated polymeric materials, opportunities in electronics, optoelectronics, and molecular electronics. J.L. Bredas and R.R. Chance (Eds.). NATO ASI Series E: Applied Science 182, 221. Kluwer, Dordrecht, 1990. J. Paloheimo, P. Kuivalainen, H. Stubb, E. Vuorimaa, and P. Yli-Lathi. Molecular fieldeffect transistors using conducting polymer Langmuir–Blodgett films. Applied Physics Letters 56, 1157 (1990). J. Paloheimo, E. Punkka, P. Kuivalainen, H. Stubb, and P. Yli-Lathi. New dimensions for semiconductor devices: polymeric field-effect transistors. Acta Polytechnica Scandinavica, Electrical Engineering Series 64, 178 (1989). F. Garnier, F. Deloffre, A. Yassar, G. Horowitz, and R. Hajlani. Intrinsically Conducting Polymers: An Emerging Technology, p. 107. M. Aldissi (Ed.). Kluwer, Dordrecht, 1993. See, for example: S.M. Sze. Physics of Semiconductor Devices 2nd Edition, Wiley Interscience, New York, 1981.
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Applications 11.1
Introduction
In this chapter we offer an extended overview of actual and intended applications without attempting to be comprehensive. Since it is not possible to rank the applications chronologically or by their commercial volume, we follow more or less the scale of conductivity, beginning with superconductivity and ending with applications involving properties other than conductivity. A review “Conducting polymers: materials for commerce” with numerous original references has been written by J. S. Miller [1].
11.2
Superconductivity and High Conductivity
At the moment, organic and other exotic superconductors cannot yet compete with conventional superconductors such as Nb metal or A15 intermetallic compounds. The (sometimes considerable) advantage of higher critical temperatures is overcompensated by problems of material control. For cables and magnets, the mechanical properties are not sufficient, and for microelectronic applications (e.g., Josephson computer), structural control is far from sufficient. Synthetic metals for normal conducting cables are a very attractive concept. In principle, polymers should have a high tensile strength and low weight, so that in this respect they would be advantageous over copper or aluminum. Such plastic wires’ are of growing interest to the automotive and aircraft industries.
11.3
Electromagnetic Shielding
Electromagnetic compatibility (EMC) and electromagnetic interference (EMI) are important technological issues. Most electronic components generate electromagnetic waves. With the possible exception of some mobile telephones (handhelds’) One-Dimensional Metals, Second Edition. Siegmar Roth, David Carroll Copyright © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30749-4
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Figure 11-1
Radar camouflage net made from a fabric coated with conducting polypyrrole [3].
the intensity of the electromagnetic radiation is biologically harmless, but the radiation is strong enough to disturb other electronic equipment (remember the request in aircraft to switch off electronic devices during takeoff and landing, because they might interfere with the plane’s navigation system). Consequently, national and international standards require electromagnetic shielding. If the computer housing is of stainless steel or of aluminum the instrument is automatically shielded. But today computers and other pieces of consumer electronics are in plastic cases. For shielding, the plastic has to be metallized by various expensive processes. An alternative is to load the polymer with conductive fillers such as carbon black, carbon nanotubes, aluminum flakes, or steel needles. Even better would be inherently conductive polymers, such as doped polyaniline or polypyrrole. The problem is the processability of the conducting polymers. Engineers would like to injection-mold the housings, but with inherently conducting polymers this is not yet possible. A compromise is a compound composed of an insulating processable polymer and polyaniline or polypyrrole fillers. For reviews on this topic see Ref. [2]. A special example of electromagnetic shielding is radar camouflage. Figure 11-1 shows a camouflage net spread over a military tank. The fibers of the net are made of a polypyrrole-coated fabric [3].
11.4
Field Smoothening in Cables
Today the largest volume of conducting polymers is used in the cable industry. Presently polymers filled with carbon black are used; perhaps they will be replaced by inherently conducting polymers one day. Figure 11-2 shows equipotential lines at the end of a coaxial cable. The abrupt end of the outer lead causes a squeezing of the
11.5 Capacitors
Field inhomogeneities at cable ends. They can be smoothed by a protective shield of moderate conductivity [4].
Figure 11-2
equipotential lines and singularities in the field strength. These field spikes often lead to insulation failures. An extension of the outer lead by a material of some 102 or 103 X cm smoothes the field and prevents failures. Similar, but less pronounced field inhomogeneities occur along the cable, because the inner part of the cable consists of twisted wires. These inhomogeneities can be smoothed by placing a layer of moderately conducting material around the wires.
11.5
Capacitors
For capacitors with a large capacity, we need a large area and a thin dielectric layer. This is usually achieved in electrolyte capacitors. They consist of an electrochemically roughened aluminum surface that is exposed to oxygen to form a thin aluminum oxide layer as dielectric. The second capacitor plate is made of a liquid electrolyte. Using a metal instead of the electrolyte would not work, because small scratches in the oxide layer would short-circuit the capacitor. The electrolyte can be replaced by an organic conductor. Charge-transfer salts (TCNQ) have been used; presently, polypyrrole is mainly used. A polymer-based condenser is schematically shown in Figure 11-3 [1]. An emerging area of technological development that utilizes one-dimensional materials, specifically, carbon nanotubes and nanofibers, is the so-called super capacitor’. A super capacitor can have 20 times the capacitance density of conventional capacitors, and essentially unlimited cycle life, similar to a battery. This high capacitance is the result of electrostatic charge storage at the interface between activated carbon and an aqueous electrolyte in an electric double layer. So the ions are stored electrostatically, rather than through chemical reactions as in a battery. Generally, the heavy ion movement and large surface area of a super capacitor can result in long, stable discharge, however, very much resembling the discharge of a battery
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Figure 11-3
Cross section of a polypyrrole condenser.
over shorter time scales. Super capacitors do not dry up like electrolytic capacitors, have no discharge memory effects as do NiCd batteries, and can be practically charged and discharged indefinitely. To achieve a high volumetric efficiency, the super capacitor is must be designed with extremely high surface-area electrodes. Carbon has proven ideal for this purpose because the naturally occurring double layer created at the interface between the carbon and a liquid electrolyte when a voltage is applied establishes a thin dielectric layer. This allow for very thin plate separations. Capacitance densities of 30 F per gram or more using carbon have been realized, which corresponds to an effective contact surface area of 2000 m2. Recently, the use of carbon nanotubes and other nanofibers as electrode materials has been intensely studied.
11.6
Through-hole Electroplating
After capacitors (more than 560 million TCNQ-based capacitors had been sold by the end of 1992), through-hole plating is the second commercially successful highvolume application of organic conductors (every second piece of consumer electronics produced today in Europe uses polypyrrole in this process). In printed circuits it is often necessary to print structures on both sides of the board and to have conductive connections from one side to the other. Drilling a hole, sticking a wire through and soldering it on both sides is too expensive for an industrial production line. Therefore, the wall of the hole is electrochemically plated with copper (Figure 11-4), but for electroplating a conducting substrate is needed. To this end, conventionally, first a copper layer is electroless deposited. This process leads to environmental problems with the disposal of the catalyst solutions. The next generation of technology used polypyrrole as electroless deposit, which simplifies the process and is environmentally friendlier [5–7].
11.7 Loudspeakers
Through-hole plating on printed circuit boards (PCBs). Before electrochemical copperplating, a thin layer of conducting polypyrrole is deposited on the walls of the hole.
Figure 11-4
11.7
Loudspeakers
In Figure 11-5 a polymer-based electrostatic loudspeaker is presented [8]. A large 6 lm polyester film (2 m high) is suspended between two metal grids. The polyester
Electrostatic loudspeaker using a polyaniline-coated polyester membrane [8].
Figure 11-5
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film is coated with a 0.1 lm layer of a conducting form of polyaniline. High voltage is applied between the grid and the speaker membrane, and an AC audio signal is coupled onto the grids via a transformer. The high-voltage field keeps the membrane centered between the grids, and the AC field moves it with acoustic frequency. The advantage of a conducting polymer over a normal metal is better adhesion. An important aspect of the electrostatic loudspeaker is its low inertia. Combined with low-inertia microphones, electrostatic loudspeakers can be used as antinoise devices that compensate acoustic noise by emitting 180-phase-shifted antinoise.
11.8
Antistatic Protective Bags
Electronics items are easily damaged by static discharge. Consequently, they have to be kept in electrically conducting containers for storage and transport (electrostatic discharge (ESD) protection). Only moderate conductivity is needed for static charges to be dissipated. But it is often desirable that the containers be transparent. Therefore, antistatic transparent films have been developed. One possibility is to load a transparent polymer with conductive fillers. To obtain non-zero conductivity, so many filler particles have to be added that they touch to form a continuous conduction path (percolation threshold). With spherical particles this occurs at a volume filling factor of some 30%. At such high loads the material becomes black and brittle. For needle-shaped fillers the percolation threshold can be kept below 1% and transparency is preserved. Figure 11-6 illustrates how the percolation threshold depends on the shape of the filler particles. Several companies have developed materials that contain crystalline needles of a conducting charge-transfer salt in a transparent matrix. As an example, Figure 11-7 shows a micrograph of a polycarbonate film loaded by 0.7% methylchinolinium–TCNQ [9]. Alternatively, the development of poly(ethylene dioxythiophene), commonly referred to as PEDOT, has now reached the point of significant market impact. This conducting polymer is generally water soluble, easy to process and paint onto sur-
Percolation threshold depending on the shape of the filler particles (i.e., on the aspect ratio = the length-todiameter ratio).
Figure 11-6
11.9 Other Electrostatic Dissipation Applications
Transparent film of polycarbonate loaded with 0.7% of the charge-transfer complex methylchinolinium– TCNQ [9]. Figure 11-7
faces, reasonably inexpensive, and available in a variety of resistivities from 100 MXcm to kX cm. Now available from companies such as Bayer, this material is quickly finding a place in the electronics industry as electrostatic dissipative coatings.
11.9
Other Electrostatic Dissipation Applications
Electrostatic dissipation applications of conducting polymers rank from high-tech to low-tech. A low-tech example would be floor tiles. Conventionally one would add antistatic additives to the polymer. These additives are hygroscopic and form a conductive layer on the surface of the polymer by attracting moisture from the atmosphere. Consequently, the conductivity depends on the humidity of the air. In polymer blends in which one component is inherently conducting, electrostatic dissipation is independent of atmospheric conditions. In offices equipped with floor tiles dissipating electrostatic charge, one would not get an electric shock when touching the doorknob on a dry winter day (Figure 11-8). Pushing the level of technology higher and higher, this family of applications also comprises television screens coated with a transparent conductive layer to prevent tickling when one approaches the TV set too closely, polyaniline-coated computer discs (barium ferrite-based 4 MB discs are on the market [1]), and finally, discharge layers in electron-beam lithography. The resolution of electron-beam lithography is limited by charging effects: insulating resist systems charge during electron-beam exposure and deflect the beam, which leads to pattern displacement. When polyaniline is incorporated into the resist process as a thin conducting layer, no pattern displacements are observed. Actually, polyaniline itself can be made radiation-sensitive and can serve as a conducting resist for electron-beam lithography [10].
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Electrostatic discharge between a scientist walking to the coffee shop and the doorknob of the materials research laboratory.
Figure 11-8
11.10
Conducting Polymers for Welding Plastics
Conducting polymers can be heated by microwave absorption. This effect can be used for welding plastics. A conducting gasket is formed by compression molding a mixture of polyaniline powder and a powder of a plastic similar to that to be welded.
Microwave welding of plastics by using gaskets of polyaniline. The maximum joint strength achieved is compared to the strength of the bulk material for polycarbonate (PC), nylon, and polyester (PETG) [11]).
Figure 11-9
11.11 Polymer Batteries
The gasket is placed between the pieces to be joined, and everything is put into a microwave oven. The joints so formed are, in some cases, as strong as the materials being bonded. Figure 11-9 shows a histogram of the maximum joint strength achieved for polycarbonate, nylon, and polyester [11].
11.11
Polymer Batteries
An electrochemical battery consists of two plates of different metals dipped into an electrolyte (strictly speaking, this is a electrochemical cell; a battery is a unit of several cells, just as a unit of several guns makes an artillery battery). The metals do not need to be natural metals; they can also be synthetic metals, for example, p-doped and n-doped polyacetylene. Figure 11-10 shows a simple polyacetylene battery, useful for lecture hall demonstrations. If a current is established by connecting wires to the polyacetylene films, the battery discharges and the films undope’. By loading, the process is reversed. In more realistic batteries, only one electrode consists of a synthetic metal (usually polyaniline or polypyrrole), the other is a lightweight natural metal like lithium. An example is shown in Figure 11-11. When the polymer battery was first proposed early in the 1980s, the low weight of the polymers relative to the lead used in car batteries was thought to be a major advantage. But the dopants are quite heavy, and there is no factor-of-ten superiority, which would be necessary for rapid economic success of a new material (as pointed out in Section 10.12). Lithium–polymer batteries have been developed by several companies. They are even on the market, but they are as good or as bad as some other batteries, for example, the Ni/Cd battery. So these new batteries mainly wait in stock until the use of cadmium is banned because of environmental considerations.
Figure 11-10
Lecture hall demonstration model of a polyacetylene battery.
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Figure 11-11
Schematic view of a lithium–polyaniline battery.
11.12
Electrochemical Polymer Actuators
A conjugated polymer swells’ when it is doped. The dopant ions move between the polymer chains, pushing them apart, and the volume expands. This expansion can be controlled electrochemically, and it can be used in artificial muscles’ (actuators, as the robotics people call it). The forces that occur when the material swells are surprisingly large, and the fractional length change DL/L is as large as 10%, two orders of magnitude larger than in piezoelectric polymers. A pair of microelectrochemical tweezers’ is shown in Figure 11-12. Electrochemical transfer of dopant from the outer branch causes the tweezers to open [12]. A novel variant of these actuators was recently based on carbon nanotube mats. In these actuators, nanotube mats are submerged in a liquid electrolyte (like saltwater). When the tube mat is biased, doping the carbon nanotubes with ions from a
Figure 11-12 Artificial muscle’ (polymer-based electrochemical actuator). The tweezers open when dopant is transferred from the outer to the inner layers. (P)x symbolizes the undoped and (P+yA–y)x the doped polymer [11].
11.14 Electrochemical Sensors
Figure 11-13 Left: Photograph of an actuator at two positions of its motion. Right: Schematic view of an actuator demonstrator based on bucky paper glued to Scotch tape immersed into salt water, and branched to a torchlight battery [13].
solution results, leading to bond extensions of about 3%. Each tube then pushes against other tubes, extending the sheet length of the nanotubes. If the tube mats are assembled in a bi-morph, and an asymmetric bias is placed on the two leaves of the sandwich, the artificial muscle can be made to curl around, as seen in Figure 11-13 [13].
11.13
Electrochromic Displays and Smart Windows
Most conjugated polymers change color when doped. The closing of the p–p* gap upon doping is responsible for this phenomenon – and in some cases additional conformation changes of the polymer chains. Since the doping level can be controlled electrochemically (by monitoring the current flow), the optical transparency and the color of the polymer can be controlled as well – which can be used for displays and for window shades. Such electrical shades are called smart windows’. Actually, it is possible to use the polymer battery of Figure 11-10 as a model of a smart window: all parts can be made of transparent material, except for the polymer, which is only transparent (for light with energy below the width of the gap) if a polymer is undoped. DaimlerChrysler has announced smart windows in cars based on the electrochromic pair tungsten oxide/polyaniline [14].
11.14
Electrochemical Sensors
As demonstrated in Figure 6-27, the conductivity of a conjugated polymer changes drastically on adding a small amount of an oxidizing or reducing agent. This effect can be used in sensorics. Figure 11-14 shows an electrochemical microsensor, which is the electrochemical analogue of a transistor [12]. A polymer film is placed over a source and a drain contact. In the electrolyte above the film at some distance
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Figure 11-14
Electrochemical microsensor working in a way analogous to a transistor [12].
is a gate contact. A gate voltage is applied between source and gate, and a drain voltage between source and drain. The gate current leads to a redox reaction in the polymer, thus changing the drain current. The device can be microstructured and then it is sensitive to less than 10–10 coulombs of charge with a response time of less than 100 ls. Sensor response to less than 10–15 mole of an oxidant has been demonstrated.
11.15
Gas Separating Membranes
Conjugated polymers are porous. The fleece-like structure of polyacetylene is shown in Figure 6-24. Polyaniline becomes porous by repeated doping and undoping. These pores are considerably smaller than the voids in the polyacetylene fleece. The small
Figure 11-15 The permeability of gases through a polyaniline membrane can be controlled by the doping process and appropriate undoping and redoping cycles [15].
11.17 Holographic Storage and Holographic Computing
pores in polyaniline can be used for gas separation. Figure 11-15 shows the permeabilities of a polyaniline membrane (as cast, doped, undoped, and doped again) for nitrogen and helium [15]. The redoped material has a very high H2/N2 separation factor. By appropriate treatment, polyaniline membranes can be optimized for specific separation tasks. 11.16
Corrosion Protection
Several reports exist on protecting stainless steel and other metals against corrosion by coating with a polyaniline layer [16,17]. A German company plans to put a polyaniline-based rust stop kit’ on the market, and a Swedish group is investigating polypyrrole as corrosion protection for dental implants.
11.17
Holographic Storage and Holographic Computing
A hologram is an optical interference pattern. It can be recorded in any material that is sensitive to light and that has sufficiently high spatial resolution (the wavelength of light is about 1 lm). Holograms can be used for optical data processing and for pattern recognition. A special field of molecular electronics is to use organic materials or even biomaterials for hologram recording. One of the promising materials in this respect is bacteriorhodopsin. Bacteriorhodopsin is the key protein in the photosynthetic system of the bacterium Halobacterium halobium. It is contained in a special part of the membrane of the bacterium, the so-called purple membrane. The chromophore of bacteriorhodopsin is retinal, a chain molecule with six conjugated double bonds and functional endgroups. This molecule establishes the link to conju-
Figure 11-16
Holographic pattern recognition using a polyacetylene film as storage medium [19].
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Figure 11-17 Holographic pattern recognition based on polyacetylene as a storage medium. Upper part: pattern to be analyzed. Lower left: reference pattern. Lower right: result of correlation experiment. Bright spots appear where
the reference pattern is found. Because of the fast decay time of the excited states in polyacetylene the correlation can be carried out within 1 ps.
gated polymers and some other substances treated in this book. Optical storage, pattern recognition, and genetic engineering with respect to bacteriorhodopsin and its mutant variants have been reviewed by Bruchle et al. [18]. Here, polyacetylene is introduced as another organic material for holographic data processing [19]. Figure 11-16 shows the setup of a polyacetylene-based picosecond image processor, and Figure 11-17 presents the results of a pattern-recognition experiment. Because of the fast relaxation of the excited states in polyacetylene, one image can be processed every picosecond. With images of 251 251 pixels, the potential processing speed is more than 1016 pixels per second.
11.18
Biocomputing
Today the label bio’ is used for almost everything, from biodegradable plastic bags via bioyoghurt to biorhythm. So why not bioelectronics (Figure 11-18)? The word bioelectronics’ is used for a large variety of topics: .
Biocompatible electronic devices, such as pacemakers and other electronic implants including wires that pass into human tissue without being rejected (all natural metals are rejected, perhaps we will find some synthetic metals that are not).
11.18 Biocomputing . .
.
.
.
.
. .
.
In-vivo sensors and other sensors for biologically relevant processes. Sensors based on biological materials or biological processes (for example, immobilized enzymes laid over the gate of a field-effect transistor). Biological materials for electronics, such as bacteriorhodopsin. We discussed the use of bacteriorhodopsin for holographic memories in Section 11.17. Materials for electronics synthesized by biological processes rather than by conventional synthesis. Materials inspired by biology and useful for electronics (neural nets and cellular automata). Algorithms inspired by biology (neural nets, multivalent logics, parallel computing, fuzzy logic). Artificial (electronic) eyes, ears, noses, etc. Biological–inorganic hybrids of electronic relevance, such as neurons growing on a silicon chip. Etc.
Figure 11-18
Today the label bio’ is used for almost everything.
Most of these topics have very little in common with one-dimensional metals, except that they are all exotic. The human brain, in fact, is exactly the opposite of a one-dimensional metal. An essential feature of a one-dimensional metal is the low connectivity: each member of the chain has two neighbors only, whereas a neuron interacts with hundreds of other neutrons. In bacteriorhodopsin, however, there exists a structural link on a molecular level: the light-sensitive moiety of rhodopsin is retinal, a conjugated chain molecule, which undergoes conformational change upon light absorption.
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11.19
Outlook
E. Genies once presented the scenario “Inherently Conducting Polymers in Our Homes” [20]: . . . . . . . . . . . . . .
“Roofs covered with conducting polymers for photovoltaic conversion. Walls covered with conducting polymers for electrochemical heat pumps. Optical fibers covered with conducting polymers for many sensors. Antistatic paints and varnishes to avoid the deposition of dust. Filters with conducting polymers to eliminate dust from air. Electrostatic devices for ion injection into air. Smart windows. Large screens on walls for TV and video viewing. Antenna for satellite reception. Antistatic textiles on walls. Textiles for heating systems. Large electrostatic loudspeakers. Regeneration of air and water. Conducting polymers in the wall for many kinds of electrical contacts. Electroluminescent diodes for IR beams and sensors and triggers all over the house.”
It is not likely that conjugated polymers (or any other one-dimensional material) will actually be used for all the applications in Genies’ scenario, but it is very probable that they will be used in some of them. Perhaps one day we will replace the pictures on our walls with large video screens, and whenever we want to see a Drer or a Picasso or a Michelangelo we slide a disk into a slot and admire the classical paintings from our armchair, just as we listen to Beethoven’s concertos today. References and Notes 1 J.S. Miller. Conducting polymers: part A and
B. Advanced Materials 5, 587 and 671 (1993). 2 H.J. Mair and S. Roth. Elektrisch leitende Kunststoffe. 2. Ausgabe, Carl Hanser Verlag, Mnchen, 1989. 3 R.V. Gregory, W.C. Kimbrell, and H.H. Kuhn. Conductivity textiles. Synthetic Metals 28, C823 (1989). H.H. Kuhn. Intrinsically conducting polymers: an emerging technology. M. Aldissi (Ed.). NATO ASI Series E: Applied Science 246. Kluwer, Dordrecht, 1993. 4 G.H. Kleinheins and K.P. Goetze. in Elektrisch leitende Kunststoffe, p. 121. 2. Ausgabe, H.J. Mair and S. Roth (Eds.). Carl Hanser Verlag, Mnchen, 1989.
5 Kunststoffmetallisierung und leitende Poly-
6
7
8 9
mere. Proceedings of 15. Ulmer Gesprche. Leuze Verlag Germany, Saulgau, 1993. K. Beator, B. Bressel, and H.J. Grapentin. Direct electroplating of printed-circuit boards: experience of the user. Metalloberflche 46, 9 (1992). H. Stuckmann. Blasberg-Mitteilungen No. 11, November 1991. J. Hupe. Blasberg-Mitteilungen No. 11, November 1991. Courtesy of B. Wessling, Zipperling Kessler & Co, Ahrensburg, Germany. G. Heywang. in Elektrisch leitende Kunststoffe, p. 347. 2. Ausgabe, H.J. Mair and S. Roth (Eds.). Carl Hanser Verlag, Mnchen, 1989.
References and Notes 10 M. Angelopoulos, J.M. Shaw, and J.J. Ritsko.
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Science and Applications of Conducting Polymers. W.R. Salaneck, D.T. Clark, and E.J. Samuelson (Eds.). Adam Hilger, Bristol, 1991. A.J. Epstein, J. Joo, C.-Y. Wu, A. Benaker, C.F. Faisst, Jr., J. Zegarski, and A.G. MacDiarmid. Intrinsically Conducting Polymers: An Emerging Technology, p. 165. M. Aldissi (Ed.). NATO ASI Series E: Applied Science 246. Kluwer, Dordrecht, 1993. R.H. Baughman and L.W. Shacklette. Science and Applications of Conducting Polymers, p. 47. W.R. Salaneck, D.T. Clark, and E.J. Samuelson (Eds.). Adam Hilger, Bristol, 1991. R.H. Baughman, C. Cui, A.A. Zakhidov, Z. Iqbal, J.N. Barisci, G.M. Spinks, G.G. Wallace, A. Mazzoldi, D. De Rossi, A.G. Rinzler, O. Jaschinski, S. Roth, and M. Kertesz. Carbon nanotube actuators. Science 284, 1340 (1999). Daimler-Benz High Tech Report 3, 44 (1994) (Mosaik).
15 M.R. Anderson, B.R. Mattes, H. Reiss, and R.B.
16
17
18
19
20
Kaner. Conjugated polymer films for gas separation. Science 252, 1412 (1991). D.W. DeBerry. Modification of the electrochemical and corrosion behaviour of stainless steels with an electroactive coating. Journal of the Electrochemical Society 132, 1022 (1985). B. Wessling. Passivation of metals by coating with polyaniline: corrosion potential shift and morphological changes. Advanced Materials 6, 226 (1994). C. Bruchle, N. Hampp, and D. Oesterhelt. Optical applications of bacteriorhodopsin and its mutated variants. Advanced Materials 3, 420 (1991). D.P. Foote. ME-News No. 13, 1992. M. Petty (Ed.). Center of Molecular Electronics, Durham, U.K. E. Genies. Intrinsically Conducting Polymers: An Emerging Technology, p. 75. M. Aldissi (Ed.). NATO ASI Series E: Applied Science 246. Kluwer, Dordrecht, 1993.
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Finally Technological revolutions do exist. They are triggered by major innovations. The steam engine, the railroad, electricity, and artificial fertilizers are examples [1]. Economists speak of Kondratiev cycles. It is unlikely that one-dimensional metals, conducting polymers, or organic superconductors will initiate a new Kondratiev wave. Not even a spray-on room-temperature superconductor would do that. Of course, materials scientists are excited when giant conductivity in TTF-TCNQ, solitons in polyacetylene, superconductivity in cuprates, or mass production of fullerene balls are reported. They will have open-ended discussion meetings till early in the morning, they will file hundreds of research proposals, and they will write thousands of publications. But neither the scientists nor the administrators of funding agencies should be disappointed if five years later electrical cables are still made of copper and computer chips are still made of silicon. The more a new technology differs from the established methods, the higher the innovative step. A new material that is just a little bit better will have a hard time replacing conventional materials, because it is not only competing in regular costs, but also has to compensate for the depreciation of the investments in the old product. Consequently, for quick success, a new material must have at least an order-of-magnitude advantage’ somewhere. Such a clear and evidently overwhelming advantage of one-dimensional metals has not yet been visible (with the possible exception of light-emitting devices for large-scale optical displays). Therefore these materials will slowly spread into economic niches rather than replace existing materials.
References 1 N.D. Kondratiev. The major economic cycles.
Economic Bulletin of the Conjuncture Institute 1, Moscow, 1925.
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Index a A15 compounds 22–25 AC conductivity 139–145 Acceptors, organic 32 Actuators, polymer 236 Alchemist’s gold 28 f Amorphous materials, conductivity 117 Amplitude mode, charge density wave excitations 181 ff Anisotropy, infinite 62 Anti-solitons, polyacetylene 96 f Antistatic protective bags 232 Asymmetric growth, nanowires 45 Aviram–Ratner device 200 f
b BA see Bianthrone BAMTF see Bianthrone Band-edge singularities, one-dimensional systems 14 Batteries, polymer 235 BCS theory 155 Bechgard salts 30–33 BEDT-TTF 167 Benzene ring 88 Bianthrone 78, 213 f Bifurcations, soliton 198 f Biocomputing 240 f Bipolaron, polyphenylene 104 Bloch theorem 67 Bond length alternation parameter, polyacetylene 90 f Bond order parameter, polyacetylene 90 Bond order wave 178 polyacetylene 90 Bond percolation, one-dimensional systems 12 f BOW see Bond order wave
Brillouin zone 66 charge density wave excitations 181 ff
c C60 molecules 168 ff Cables, field smoothing 229 Capacitors, organic 229 f Carbanion 100 Carbocation 100 Carbon nanotubes 12, 204–208 electronic struture 73–76 Carbon solids, dimensionality 7–12 Catalysts, nanowire growth 45 CDW see charge density waves Cellular automata, molecular 203 f Ceramic superconductors 158 Charge density waves 177–190 4kf 178 ff excitations 181 ff polyacetylene 90 f Charge injection, polyacetylene 100, 102 f Charge-transfer compounds 30–33 Charged solitons 129 f Charges, fractional 106 ff Chemical doping, polyacetylene 100 f Commensurability, charge density waves 184 Computing bio- 240 f holographic 239 f Conducting polymers 38–41, 85–110 conductivity 113–147 highly 145 ff Conductivity and solitons 129–133 conducting polymers 113–147 giant 185 hopping 139–145 in one dimension 126–129 Confinement, soliton 106
One-Dimensional Metals, Second Edition. Siegmar Roth, David Carroll Copyright © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30749-4
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Index Conjugated double bonds, conducting polymers 86–89 Conjugational defects, polyacetylene 89–93, 105 Connectivity, one-dimensional systems 14 Cooper pair 155 Coordinate system, crystal lattice 56 Corrosion protection 239 Coulomb interaction 178 f Coulombic blockade 196 Critical temperature, superconductivity 153 Crystal lattice one-dimensional 58 f reciprocal 60–63 translation symmetry 54–59 Crystal vibrations 64 Crystallographic axis 56 Cumulene, dimensionality 9 Curie susceptibility, polyacetylene 131 f
d D-p-A molecule 212 Davidov splitting 72 DC conductivity 139–145 Density of states, electronic 14 f, 68–73 Diamond, dimensionality 7 Dimensionality 1–2 and superconductivity 162 carbon solids 7–12 Disordered bundle, reciprocal lattice 62 f Dispersion relations 63–73 Displays, electrochromic 237 Donor–acceptor polyene 212 f Donors, organic 31 Doped polymers, conductivity 116, 135–139, 142–145 Doping chemical 100 f n- and p- 72 DOS see Density of states Dreckeffekt 115
e Effective mass, electrons in crystals 67 Electrochromic displays 237 Electroluminescence 219 Electromagnetic shielding 227 f Electron hole symmetry 72 Electron–lattice interaction see Electron– phonon coupling Electron–phonon coupling 14, 77–82 Electronics, molecular 16 Electrons, in crystals 63–73
Electroplating, through-hole 230 Electrostatic dissipation 233 Elementary cell 56 Eletrochemical sensors 237 Energy bands 68–73 Energy gap 68–73 ET, two-dimensional organic superconductor 167 f Excess resistance, conducting polymers 127 f
f Fermi energy 67 Fermi gas 69 Fermi surface 68 carbon nanotubes 73 one-dimensional metal 81 f organic superconductors 163 Fibrillar structure, polyacetylene 128 f, 134 Field effect transistors nanotube 206 ff organic 223 f Field-induced spin density waves 188 ff Field smoothing, cables 229 Forbidden gaps 69 Four-lead method 118 ff Four-point probe method 120 Fractional charges 106 ff Free electrons 68–73 Fullerene 168 ff dimensionality 10 f Fully depleted semiconductor 218
g Gain, electronics 195 Gaps, energy 68–73 Gas separating membranes 238 Giant conductivity 185 Graphite, dimensionality 8 Guard ring method, high resistivity samples 124
h Hall bar, magnetotransport measurements 121 Hall effect, quantized 188 ff Halogen-bridged mixed-valence transition metal complexes 41 f Heisenberg spin chain 180 Heterojunctions, Langmuir–Blodgett layers 215 f Hg2.86AsF6 28 f
Index High resistivity samples guard ring method 124 length dependence of resistance 127 High-Tc superconductors 157 f Highest occupied molecular orbital 88 OLED 218 Highly conducting polymers 145 ff HMMC see Halogen-bridged mixed-valence transition metal complexes Holographic computing 239 f Holographic storage 239 f HOMO see Highest occupied molecular orbital Hopping doped polymers 117 intersoliton 130 f Hopping conductivity 139–145
i ICMS see International Conference of Science and Technology of Synthetic Metals Infinite anisotropy 62 Information processing, molecular electronics 196 Inorganic semicondutor quantum wires 47 Internal quantum efficiency, solar cell 222 International Conference of Science and Technology of Synthetic Metals 20 Intersoliton hopping 130 f Iodine doping, polyacetylene 136 ff, 142–145 Isolated nanowires 43 ITO 220, 223
Localization, conductivity in one dimension 126–129 Loudspeakers, polymer-based 231 Lowest unoccupied molecular orbital 88 OLED 218 LUMO see Lowest unoccupied molecular orbital
m Macrocycles 34 ff Magnetic susceptibility, superconductivity measurements 159 f Magnetotransport measurements 121 Meissner effect 156, 159 f Membranes, gas separating 238 Mercury chains 29 Metal nanowires 48 Microcontacts, conductivity measurements 122 f Microwave conductivity 143 Microwave welding, plastics 234 Miniaturization 193–196 Molecular cellular automata 203 f Molecular electronics 16, 193–208 Molecular materials, for electronics 211–225 Molecular rectifiers 200 f Molecular shift register 201 ff MoO6 36 Mutual inductance method, superconductivity measurements 160 MX6 37 (MX4)nY 37
j Jahn-Teller effect 25 Jellium, charge density waves 184
k Kamerlingh Onnes 153 KCP 26 ff Peierls distortion 63 Peierls transition 80 Kohn anomaly 80 f K2[Pt(Cn)4]Br0.3 3 H2O see KCP Krogmann salts 26 ff
l Langmuir–Blodgett layers, rectifiers 215 f Lattice vector 57 LB layers see Langmuir–Blodgett layers LED see Light-emitting device Light-emitting device, organic 216–220 Little’s superconductor 19 f
n Nanoporous templates 44 Nanotubes 12, 46 f carbon 73–76, 204–208 Nanowires asymmetric growth 45 isolated 43 metal 48 Narrow band noise, charge density waves 188 NbSe6 36 f Nb3Sn 22 f Nearly free electron model 68–73 Noise, charge density waves 188 Nondegenerate ground state polymers 103–106
o OLED see Organic light-emitting device One-dimensional carbon 9, 12
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Index One-dimensional organic superconductors 164–167 One-dimensional solid state physics 53–76 One-dimensional substances 19–48 One-dimensionality 2–6 peculiarities 12–16 Optical correlation measurements, solitons 108 ff Organic field-effect transistors 223 f Organic heterolayers 215 f Organic light-emitting device 216–220 Organic photovoltaic cells 220–223 Organic superconductors 163–172
p Pauli susceptibility 133 PCNC 212 f PEDOT 232 Peierls distortion, KCP 63 Peierls–Frhlich mechanism, superconductivity 183 Peierls systems charge density waves 177 f superconductivity 156 ff Peierls transition 77–82 conducting polymers 88 spin 180 Phase mode, charge density wave excitations 181 ff Phase slip center, charge density waves 186 Phase transitions, one-dimensional systems 15 Phasons 182 Phonon disperion relation, charge density wave excitations 181 ff Phonons, in crystals 63–73 Photoabsorption switching 212 ff Photogeneration, polyacetylene 100 f, 108 Photovoltaic cells, organic 220–223 Phtalocyanines 34 ff Pinning, charge density waves 184 Polarons, conducting polymers 85, 103–106 Poly-deckers 42 Poly(para-penylenevinylene) 219, 223 Polyacetylene 86–110 bond scission 77 conductivity 133–139 crystal lattice 59 dimensionality 10 fibrillar structure 128 f, 134 solar cell 221 soliton switching 197–200 Polycarbenes 43
Polycarbyne, dimensionality 9 Polyene, dimensionality 9 Polyethylene 86 f Polyfractiolene 107, 178 Polymer actuators, electrochemical 236 Polymer batteries 235 Polymer sensors 237 Polymers conducting 38–41, 85–110, 113–147 highly conducting 145 ff nondegenerate ground state 103–106 Polyphenylene, polarons 104 Polypyrrole 228–231 susceptibility 132 Polysulfurnitride 33 Porphyrin 35 Power conversion efficiency, solar cell 222 PPV see Poly(para-penylenevinylene)
q Quantized Hall effect, field-induced spin density waves 188 ff Quantum wires, inorganic semicondutor 47
r Radar camouflage 228 Random walk problems, one-dimensional systems 14 f Reciprocal lattice 60–63 Reciprocal space 60, 65 Reciprocal time 60 Rectifier Langmuir–Blodgett layers 215 f molecular 200 f Resistance, conducting polymers 124–128 Restwiderstand 115
s SDW see Spin density waves Sensors, polymer 237 Shift register, molecular 201 ff Shish kebab method, macrocycles 35 f Shubnikov–de Haas oscillations 164 Sine–Gordon equation 93, 186 Singlet superconductivity 181 Singularities, van Hove 71 Sliding, charge density waves 184–188 Smart windows 237 Sodium chloride, crystal lattice 55 Sodium metal, crystal lattice 55 Solar cells, organic 220–223 Solid state physics, one-dimensional 53–76 Soliton bifurcations 198 f
Index Soliton confinement 106 Soliton detector 198 f Soliton–phonon scattering 129 f Soliton switching 197–200 Soliton valve 198 Solitons and conductivity 129–133 charge density waves 186 f conducting polymers 85, 92–99 generation 100–103 in telecommunication 95 lifetime 108 ff Spin density waves 178–181 field-induced 188 ff Spin Peierls transition 180 Spin Peierls waves 178–181 Square-sample method 121 SQUID see Superconducting quantum interference device Storage, holographic 239 f Superconducting materials, classes 154 Superconducting quantum interference device 161 Superconductivity 153–172, 227 measurements 159 f Peierls–Frhlich mechanism 183 singlett 181 triplet 181 Superconductors ceramic 158 high-Tc 157 f one-dimensional 19 f organic 163–172 TMTSF 188 ff Susceptibility Curie 131 f magnetic 159 f Pauli 133 Switching molecular devices 212–216
Three-dimensional carbon 7 Three-dimensional organic superconductors 168 ff Threshold field, charge density waves 185 Through-hole electroplating 230 TMTSF superconductors 164–167 quantized Hall effect 188 ff (TMTSF)2PF6 30–33 TQNC, organic capacitor 229 Transition metal chalcogenides 36 f Transition metal complexes, halogen-bridged mixed-valence 41 f Transition metal halides 36 f Translation symmetry 54–59 Translational invariance 57 Triplet superconductivity 181 TTF see Tetramethyl tetrathiafulvalene TTF-TCNQ 32 f charge density waves 185 Two-dimensional carbon 8 Two-dimensional organic superconductors 167 f Two gap state, polarons 105 Type I superconductor 158 Type II superconductor 158 Type N polyacetylene, conductivity 135–138 Type S polyacetylene, conductivity 135 f
v van der Pauw method 121 van Hove singularities 71 carbon nanotubes 73–75 von Klitzing constant 128 von Klitzing effect see quantized Hall effect V3Si 22–25
w Washboard model, sliding charge density waves 188 Welding of plastics 234
t Telecommunication, solitons 95 Templates, nanoporous 44 Tetramethyl tetraselenofulvalene see TMTSF Tetramethyl tetrathiafulvalene 165, 167 Thermopower measurements, conducting polymers 125 f Thin films see also Langmuir–Blodgett layers photovoltaic cells 222
y YBa2Cu3O7–x 158
z Zero-dimensional carbon 10 f Zigzag chain, crystal lattice 58 Zinngeschrei 188
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