Siegmar Roth and David Carroll

One-Dimensional Metals

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Siegmar Roth and David Carroll

One-Dimensional Metals

Conjugated Polymers, Organic Crystals, Carbon Nanotubesand Graphene

Third Completely Revised and Enlarged Edition

The Authors

Siegmar RothMPI f.Festkörperforschung, StuttgartFRG

David CarrollDept. of PhysicsWake Forest University

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V

Contents

About the Authors XIPreface to the Third Edition XIIIPreface to the Second Edition XVPreface to the First Edition XVII

1 Introduction 11.1 Dimensionality 11.2 Approaching One-Dimensionality from Outside and from Inside 21.3 Dimensionality of Carbon Solids 71.3.1 Three-Dimensional Carbon: Diamond 71.3.2 Two-Dimensional Carbon: Graphite 81.3.3 One-Dimensional Carbon: Cumulene, Polycarbyne, Polyene 91.3.4 Zero-Dimensional Carbon: Fullerene 111.3.5 What about Something in between? 121.4 Peculiarities of One-Dimensional Systems 13

References 17

2 One-Dimensional Substances 192.1 A15 Compounds 232.2 Krogmann Salts 272.3 Alchemists’ Gold 292.4 Bechgaard Salts and Other Charge Transfer Compounds 312.5 Polysulfurnitride 342.6 Phthalocyanines and Other Macrocycles 362.7 Transition Metal Chalcogenides and Halides 382.8 Conducting Polymers 402.9 Halogen-Bridged Mixed-Valence Transition Metal Complexes 442.10 Miscellaneous 452.10.1 Poly-deckers 452.10.2 Polycarbenes 462.11 Isolated Nanowires 462.11.1 Templates and Filled Pores 462.11.2 Asymmetric Growth Using Catalysts 48

VI Contents

2.11.3 Carbon Nanotubes 492.11.4 Inorganic Semiconductor Quantum Wires 512.11.5 Metal Nanowires 522.12 Summary 53

References 53

3 One-Dimensional Solid-State Physics 573.1 Crystal Lattice and Translation Symmetry 573.1.1 Classifying the Lattice 593.1.2 Using a Coordinate System 623.1.3 The One-Dimensional Lattice 633.1.4 Carbon Nanotubes as One-Dimensional Lattices 653.2 Reciprocal Lattice, Reciprocal Space 673.2.1 Describing Objects Using Momentum and Energy 673.2.2 Constructing the Reciprocal Lattice 683.2.3 Applying This to One Dimension 693.3 The Dynamic Crystal and Dispersion Relations 713.3.1 Crystal Vibrations and Phonons 713.3.2 Quantum Considerations with Phonons 793.3.3 Counting Phonons 813.4 Phonons and Electrons Are Different 833.4.1 Electron Waves 843.4.2 Electron Statistics 853.4.3 The Fermi Surface 863.4.4 The Free Electron Model 873.4.5 Nearly Free Electron Model; Energy Bands, Energy Gap, and Density

of States 913.4.6 The Molecular Orbital Approach 973.4.7 Returning to Carbon Nanotubes 983.5 Summary 102

References 102

4 Electron–Phonon Coupling and the Peierls Transition 1054.1 The Peierls Distortion 1074.2 Phonon Softening and the Kohn Anomaly 1114.3 Fermi Surface Warping 1124.4 Beyond Electron–Phonon Coupling 113

References 114

5 Conducting Polymers: Solitons and Polarons 1175.1 General Remarks 1175.2 Conjugated Double Bonds 1195.3 A Molecular Picture 1225.3.1 Bonding and Antibonding States 1235.3.2 The Polyenes 123

Contents VII

5.3.3 Translating to Bloch’s Theorem 1285.4 Conjugational Defects 1325.5 Solitons 1365.6 Generation of Solitons 1445.7 Nondegenerate Ground-State Polymers: Polarons 1465.8 Fractional Charges 1515.9 Soliton Lifetime 153

References 156

6 Conducting Polymers: Conductivity 1596.1 General Remarks on Conductivity 1596.2 Measuring Conductivities 1646.2.1 Simple Conductivity 1646.2.2 Conductivity in a Magnetic Field 1686.2.3 Conductivity of Small Particles 1696.2.4 Conductivity of High-Impedance Samples 1716.2.5 Conductivity Measurements without Contacts 1716.2.6 Thermoelectric Power – the Seebeck Effect 1726.3 Conductivity in One Dimension: Localization 1756.4 Conductivity and Solitons 1786.5 Experimental Data 1826.6 Hopping Conductivity: Variable Range Hopping vs.

Fluctuation-Assisted Tunneling 1866.7 Conductivity of Highly Conducting Polymers 1956.8 Magnetoresistance 197

References 202

7 Superconductivity 2097.1 Basic Phenomena 2097.2 Measuring Superconductivity 2167.3 Applications of Superconductivity 2187.4 Superconductivity and Dimensionality 2197.5 Organic Superconductors 2207.5.1 One-Dimensional Organic Superconductors 2227.5.2 Two-Dimensional Organic Superconductors 2257.5.3 Three-Dimensional Organic Superconductors 2277.6 Future Prospects 229

References 231

8 Charge Density Waves 2358.1 Introduction 2358.2 Coulomb Interaction, 4kF Charge Density Waves, Spin Peierls Waves,

Spin Density Waves 2368.3 Phonon Dispersion Relation, Phase and Amplitude Mode in Charge

Density Wave Excitations 240

VIII Contents

8.4 Electronic Structure, Peierls–Fröhlich Mechanism ofSuperconductivity 242

8.5 Pinning, Commensurability, Solitons 2438.6 Field-Induced Spin Density Waves and the Quantized Hall

Effect 248References 250

9 Molecular-Scale Electronics 2539.1 Miniaturization 2539.2 Information in Molecular Electronics 2579.3 Early and Radical Concepts 2589.3.1 Soliton Switching 2589.3.2 Molecular Rectifiers 2619.3.3 Molecular Shift Register 2629.3.4 Molecular Cellular Automata 2649.4 Carbon Nanotubes 265

References 269

10 Molecular Materials for Electronics 27110.1 Introduction 27110.2 Switching Molecular Devices 27210.2.1 Photoabsorption Switching 27210.2.2 Rectifying Langmuir–Blodgett Layers 27410.3 Organic Light-Emitting Devices 27710.3.1 Fundamentals of OLEDs 27710.3.2 Materials for OLEDs 28410.3.3 Device Designs for OLEDs 28410.3.4 Performance and Outlooks 28510.3.5 Field-Induced Organic Emitters 28610.3.6 Organic Lasers and Organic Light-Emitting Transistors 28910.4 Solar Cells 29410.5 Organic Field Effect Transistors 29810.6 Organic Thermoelectrics 30010.7 Summary 302

References 303

11 Even More Applications 30711.1 Introduction 30711.2 Superconductivity and High Conductivity 30711.3 Electromagnetic Shielding 30811.4 Field Smoothening in Cables 30811.5 Capacitors 30911.6 Through-Hole Electroplating 31011.7 Loudspeakers 31111.8 Antistatic Protective Bags 311

Contents IX

11.9 Other Electrostatic Dissipation Applications 31311.10 Conducting Polymers for Welding of Plastics 31411.11 Polymer Batteries 31411.12 Electrochemical Polymer Actuators 31611.13 Electrochromic Displays, Smart Windows, and Transparent

Conducting Films 31711.14 Electrochemical Sensors 31911.15 Gas-Separating Membranes 32011.16 Hydrogen Storage 32111.17 Corrosion Protection 32111.18 Holographic Storage and Holographic Computing 32211.19 Biocomputing 32311.20 Outlook 325

References 325

12 Finally 327Reference 328

Glossary and Acronyms 329

Index 335

XI

About the Authors

Prof. David Carroll earned his PhD in Physics atWesleyan University in Connecticut (1993) working onthe excited-state dynamics of charged defects induced incomplex metal oxides under electron beam irradiation.His postdoctoral work at the University of Pennsylvaniain Philadelphia was under the direction of Prof. D.Bonnell and focused on the application of scanningprobes to transition metal oxide surfaces. From UPENN,Dr Carroll joined the group of Prof. M. Rühle at the

Max-Planck-Institut für Metallforschung in Stuttgart, Germany, where his workcentered on the application of scanning probes to interface studies of supportednanostructures. It was at MPI that Dr Carroll obtained a lifelong fascination withdimensionality in solid-state physics. Following Stuttgart, Dr Carroll becamean assistant professor then associate professor at Clemson University in SouthCarolina, where his work expanded to include organic devices and organicelectronics. In 2003, Dr Carroll and his research team moved to Wake ForestUniversity in Winston-Salem NC, USA, to establish a center dedicated to theresearch and development of electroactive, matrix nanocomposites. His researchcontinues to focus on the role of dimensionality in the thermal, electrical, andoptical phenomena of nanoscale structures and their meso-assemblies. Dr Carrollis currently a Professor of Physics at Wake Forest University, Director of theCenter for Nanotechnology and Molecular Materials at Wake, and a Fellow ofthe American Physical Society.

XII About the Authors

Siegmar Roth has obtained his PhD in physics at theUniversity of Vienna, Austria, for experimental work atthe Forschungszentrum Seibersdorf (Moessbauer Effect)and his Habilitation at the University of Karlsruhe,Germany (Solitons in Conducting Polymers). After afew years with the Siemens Research Laboratories inErlangen, Germany (Novel Semiconductors), he joinedthe Institut Laue Langevin in Grenoble, France (NeutronScattering) and then the High Field Magnet Lab in

the same town (High Temperature Superconductors), from where he moved toStuttgart to become leader of the Research Group on Synthetic Nanostructuresat the Max Planck Institute for Solid State Research (Conducting Polymers,Fullerenes, Carbon Nanotubes). From here he retired in summer 2008. From2009 to 2012 he joined the School of Electrical Engineering of Korea Universityin the frame of the WCU Project on Flexible Nanosystems, and since 2012 he is a“free-lance scientist” in München, Germany.

XIII

Preface to the Third Edition

Ten more years have passed: Karlsruhe Lectures (1980s), First Edition (1995),Second Edition (2003), and Third Edition (2015): three 10-year intervals. Thegrasshoppers have swarmed from conjugated polymers over high-temperaturesuperconductors and fullerenes to carbon nanotubes and now they exploitgraphene. We, the authors of this textbook, are proud to hop with the hoppers,and while writing we were surprised how many of the early concepts havesurvived, although sometimes in new costumes: see, for example, our remarks on“solitons” and “pseudo-spin.”

The Third Edition differs from the previous two by having more algebra andby using “Exploring Concepts” boxes. Mathematic equations might frighten somereaders, and we advise them to skip over these parts. But the algebra will also makethis edition more useful as a textbook for classes of solid-state physics or materialsscience. We hope we have retained enough of the vividness of the coffee-cornerdiscussions in our research groups to make the textbook also an enjoyable fireplacereader for long winter nights.

January 2015 David Carroll and Siegmar RothWinston-Salem and München

XV

Preface to the Second Edition

About 10 years have passed since the lectures in Karlsruhe on “Physics in OneDimension” and since the first edition of this textbook. When we were askedto work on an update for the second edition, we felt that many things hadchanged – and we were surprised that many parts were still valid.

New are the Nobel Prizes – to Curl, Smalley, and Kroto for the fullerenes in1996; Kohn and Pople in 1998 (see the soliton as a Pople–Walmsley defect); andHeeger, MacDiarmid, and Shirakawa in 2000 for conducting polymers. New areapplications of conducting polymers, and in particular, the commercialization ofseveral of these applications, and (almost) new are carbon nanotubes. In fact, thesenanotubes are the new toys of the materials scientists, and like locust swarms, theycrowd on every tiny bit of these carbon nanotubes, producing a new wave in liter-ature statistics (compare Figure 2.3). Since we are part of this swarm, we decidedto devote several paragraphs and sections of the second edition of our textbookto nanotubes and to stress the relationships between conjugated polymers andcarbon nanotubes.

As was true for the first edition, this second edition would also not have beenpossible without the support and the many discussions among our teams inStuttgart, Shanghai, Clemson, and Wake Forest. In particular, we are grateful toEkkehard Palmer, who did most of the computer work for this edition, and to theexperts at Wiley VCH-Verlag in Weinheim.

We enjoy working in this field, we enjoyed working on this textbook, and wehope that our readers will enjoy reading our modest oeuvre.

October 2003 David Carroll and Siegmar RothWinston-Salem and Stuttgart

XVII

Preface to the First Edition

This book originated from lectures on “Physics in One Dimension,” that I havegiven at the University of Karlsruhe in the 1980s. I am grateful to all the studentswho contributed by asking questions. Some of them later became PhD studentsin my research group in Stuttgart.

The style and content of this textbook reflect the everyday research work of aninterdisciplinary and international research group, where people of different back-ground have to quickly catch up on basic concepts in order to meet on equal termsfor discussions.

The reader is expected to have some basic knowledge of science or engineering,for example, of physics, chemistry, biology, or materials science. To consolidatethis knowledge, you will have to consult textbooks on experimental physics, aswell as on organic, inorganic, and physical chemistry. But this textbook shouldhelp to forge links, and with these links the monographs recommended in theAppendix should be accessible. It should also be possible to follow internationaltopical meetings. We hope that some of the aspects of this textbook are so inter-esting that they are attractive even to complete neophytes or outsiders, and thatsome features will also appeal to the experienced researcher.

My thanks are due to all the members of our team (Lidia Akselrod, TarekAbou-Elazab, Teresa Anderson, Marko Burghard, Hugh Byrne, Claudius Fischer,Thomas Rabenau, Michael Schmelzer, Manfred Schmid, Andrea Stark-Hauser,and Andreas Werner). Without constant discussion in the lab’s coffee corner,the textbook would not have been possible. Particular thanks go to AndreaStark-Hauser, who not only did all the typing but was also engaged in collectingreferences and figures. Manfred Schmid assisted in the preparation of technicaldrawings, and the cartoons were drawn by Günther Wilk. Teresa Anderson,Hugh Byrne, and Andreas Werner went through my first drafts and generatedmajor inputs to the final phrasing of the text, which was ultimately polished bythe experts at Wiley VCH-Verlag and with whom it was a pleasure to cooperate.

The whole team has benefited from the cooperation within the Sonder-forschungsbereich “Molekulare Elektronik – physikalische und chemischeGrundlagen” of the Deutsche Forschungsgemeinschaft, within the EuropeanBRITE/EURAM Project HICOPOL (comprising groups in Stuttgart, Karlsruhe,Montpelier, Nantes, Strasbourg, and Graz), and within the European ESPRIT

XVIII Preface to the First Edition

Network NEOME (Austria, Belgium, Denmark, England, France, Germany, Italy,the Netherlands, Sweden, and Switzerland).

April 1995 Siegmar RothStuttgart

1

1Introduction

1.1Dimensionality

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

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

Solid-state physics treats solids as both objects and the space in which objectsof physics exist, for example, various silicon single crystals can be compared with

One-Dimensional Metals: Conjugated Polymers, Organic Crystals, Carbon Nanotubes and Graphene,Third Edition. Siegmar Roth and David Carroll.© 2015 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2015 by Wiley-VCH Verlag GmbH & Co. KGaA.

2 1 Introduction

Figure 1.1 Simultaneously with HerbertMarcuse’s book “One-Dimensional Man”[1], which widely influenced the youthmovement of the 1960s, Little’s article on

“Possibility of Synthesizing an Organic Super-conductor” [2] was published, motivatingmany physicists and chemists to investigatelow-dimensional solids.

each other, or they can be considered as the space in which electrons or phononsmove. 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 betweenthem that extend into the third dimension. On the other hand, these planes arethe two-dimensional space in which electrons move rather freely. Similar consid-erations apply to the (quasi) one-dimensional hydrocarbon chains of conductingpolymers.

1.2Approaching One-Dimensionality from Outside and from Inside

There are two approaches to low-dimensional or quasi-low-dimensional systemsin solid-state physics: geometrical shaping as an “external” and increase ofanisotropy as an “internal” approach. These are also sometimes termed “top-down” and “bottom-up” approaches, respectively. For the external approach, letus take a wire and draw it until it gets sufficiently thin to be one dimensional(Figure 1.2). How thin will it have to be for being truly one dimensional? Thisdepends a little on exactly what property of the object is desired to expresslow-dimensional behavior. Certainly, thin compared to some microscopicparameter associated with that property. For example, for one-dimensionalelectrical transport properties, the object must have length scales such that themean free path of an electron or the Fermi wavelength is affected by the physicalconfinement of the structure.

1.2 Approaching One-Dimensionality from Outside and from Inside 3

Figure 1.2 An “external approach” to one-dimensionality. A man tries to draw a wireuntil it is thin enough to be regarded asone-dimensional. Metallic wires can be madeas thin as 1 μm in diameter, but this is still

far away from being one-dimensional. (Bylithographic processes, semiconductor struc-tures can be made narrow enough to exhibitone-dimensional properties.)

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

ΔEL = h2

2m(𝜋

L

)2 , (1.1)

where ΔEL is the spacing, L is the length of the box, m is the mass of the electrons,and h is Planck’s constant. The Fermi level is the highest occupied state (at absolutezero). The wavelength of the electrons at the Fermi level is called the Fermi wave-length. If the size of the box is just the Fermi wavelength, only the first eigenstateis occupied. If the energy difference to the next level is much larger than the ther-mal energy (ΔEL ≫ kT), then there are only completely occupied and completelyempty levels and the system is an insulator. A thin wire is a small box for electronicmotion perpendicular to the wire axis, but it is a very large box for motions alongthe wire. Hence, in two dimensions (radially), it represents an insulator and in onedimension (axially) it is a metal! This is simply because ΔEradially ≫ kT whereasΔElengthwise ≪ kT .

If there are only very few electrons in the box, the Fermi energy is small andthe Fermi wavelength is fairly large. For real materials, these are the electrons that

4 1 Introduction

E

E

ΔE

ΔE

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

can participate in bonding–antibonding orbitals. This is the case for semiconduc-tors at very low doping concentrations. Wires of such semiconductors are alreadyone dimensional if their diameter is of the order of approximately hundreds ofangstroms.

Such thin wires can be fabricated from silicon or from gallium arsenideby lithographic techniques and effects typical for one-dimensional electronicsystems have been observed experimentally [5]. Systems with high electronconcentrations 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, wereally need a monatomic chain!

Experiments on single monatomic chains are very difficult, if not impossible, toperform. Therefore, typically, a bundle of chains rather than one individual chain isused. An example for such a bundle is the polyacetylene fiber, consisting of somethousands of polymer chains, closely packed with a typical interchain distanceof 3–4 Å. Certainly, there will be some interaction between the chains; however,in case of small interchain coupling, it can be assumed that just the net sum ofthe individual chains determines the properties of the bundle (Figure 1.4) and theexperiment becomes one of an ensemble of one-dimensional chains.

Another method of geometrical shaping employs surfaces or interfaces(Figure 1.5). The surface of a silicon single crystal is an excellent two-dimensionalsystem and there are various ways of confining charge carriers to a layer near thesurface. In fact, the physics of two-dimensional electron gases is an important partof today’s semiconductor physics [6] and most of the two-dimensional electron

Dd

d ... Chain diameterD ... Bundle diameter

Figure 1.4 Experiments on individual chainsare difficult to perform. But bundles of chainsare quite common, for example, fibers of poly-acetylene.

1.2 Approaching One-Dimensionality from Outside and from Inside 5

Figure 1.5 Crystal surface are excellent two-dimensional systems. The man above tries toimprove the crystal face by mechanical pol-ishing. The qualities achieved by this method

are not sufficient for surface science. Sur-face scientists cleave their samples underultrahigh vacuum conditions and use freshlycleaved surfaces for their experiments.

systems are confinements to surfaces or interfaces. The most fashionable effect ina two-dimensional electron gas is the quantized Hall effect or von Klitzing effect[7]. A one-dimensional surface, that is, the edge of a crystal, is much more difficultto prepare and hardly of any practical use. But, one can argue that exposing asample to a magnetic field would be an excellent example of a one-dimensionalelectronic system since electrons can be forced into motion along specific pathsdefined by the crystal and the field. 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 gradualincrease of anisotropy. In crystalline solids, the electrical conductivity is usuallydifferent in different crystallographic directions. If the anisotropy of the conduc-tivity is increased in such a way that the conductivity becomes very large in onedirection and almost zero in the other two perpendicular directions, a nearlyone-dimensional conductor will result. Of course, there is no simple physical wayto increase the anisotropy. However, it is possible to look for sufficient anisotropyin the existing solids that could be regarded as (quasi) one dimensional. Someanisotropic solids are compiled in the next chapter of this textbook. How largeshould the anisotropy be to meet one-dimensionality? A possible answer is:“Large enough to lead to an open Fermi surface.”

The Fermi surface is a surface of constant energy in “reciprocal space” ormomentum space. While the Fermi surface and reciprocal space will be discussedin detail in Chapter 3, for the discussion here it is sufficient to imagine this surfaceas describing all of the electron states within the solid that are available to takepart 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 smallin the other two, the Fermi surface becomes disk-like. The kinetic energy of theelectrons can then be written as E = p2∕2m∗, resembling that of a free particle(p=momentum, m=mass), with the exception that the mass has been replaced

6 1 Introduction

by the effective mass m*. The effective mass indicates the ease with which anelectron can be moved by the electric field. If the electrons are easy to move, theconductivity is high. Easy motion is described by a small effective mass (smallinertia) and p must also be small to keep E constant. If it is infinitely difficult tomove an electron in a specific direction, its effective mass will become infinitelylarge in this direction and the Fermi surface will be infinitely far away. However,the extension of the Fermi surface is restricted: if the Fermi surface becomesvery large in any direction it will merge with the Fermi surface generated by theneighboring chain or plane (“next Brillouin zone,” in proper solid-state physicsterminology) assuming this hypothetical solid is made up of stacked structuresof some sort. This merging “opens” the Fermi surface, similar to a soap bubblelinking with another bubble (Figure 1.6).

Figure 1.6 Open Fermi surfaces, analogousto merged soap bubbles, as a criterionof low-dimensionality. The Fermi surfacebelongs to a solid that is essentially two

dimensional. The solid will have no electronicstates contributing to electrical conductivityalong the axial direction but will easily con-duct radially, normal to the axis.

1.3 Dimensionality of Carbon Solids 7

1.3Dimensionality of Carbon Solids

Silicon is outstanding among solids [9]: it is the most perfect solid pro-ducible – there are fewer imperfections in a silicon single crystal than there aregas atoms in ultrahigh vacuum (per unit volume). It is the solid we know mostabout, and it has largely influenced the vocabulary of solid-state physics (probablyapparent by a style analysis of this text too). Carbon is located directly abovesilicon in the periodic table of the elements, and just as silicon is outstandingamong the solids, carbon is outstanding among the elements. Carbon forms themajority of chemical compounds. Much of organic chemistry simply involvesarranging carbon atoms (with hydrogen not having any specific properties butjust fulfilling the task of saturating dangling bonds). In our context, carbon hasthe remarkable property of forming three-, two-, one-, and zero-dimensionalsolids. This is related to the fact that carbon has the ability to form single, double,and triple bonds. This characteristic feature of carbon sets it aside from silicon inanother important way, that is, it leads to biology.

1.3.1Three-Dimensional Carbon: Diamond

Beginning with an example from silicon, diamond appears as the trivial solid formof carbon (Figure 1.7). Diamond has similar semiconducting properties to sili-con. Both substances share the same type of crystal lattice. The lattice parametersare different (a= 5.43 Å in silicon and 3.56 Å in diamond), and the energy gapbetween valence and conduction bands is larger in diamond: 5.4 eV, comparedto 1.17 eV in silicon. Diamond is more difficult to manufacture and more diffi-cult to purify than silicon, but it has better thermal conductivity and can be usedat high temperatures. Since the costs of raw material change the final price ofelectronic equipment only slightly, some people believe in diamond as the semi-conductor of the future. Silicon is typically used with added dopants to modify its

616 pm

Figure 1.7 Diamond lattice.

8 1 Introduction

electronic behavior. Doping diamond has proven to be far more difficult however,and must be better understood before the realization of high-quality diamondelectronics.

“Semiconductors” will often be mentioned in this textbook although the titlepromises metals to be the main subject. The reason stems from the fact that adoped 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.”

1.3.2Two-Dimensional Carbon: Graphite

In diamond, the carbon atoms are tetravalent, that is, each atom is bound tofour neighboring atoms by covalent single bonds. Another well-known, nat-urally occurring carbon modification is graphite (Figure 1.8). Here, all atomsare trivalent, which means that in a hypothetical first step, only three valenceelectrons participate in bond formation, and the fourth valence electron is leftover. The trivalent atoms form the planar honeycomb lattice and the residualelectrons are shared by all atoms in the plane similarly to the sharing of theconduction electrons by all atoms of a simple metal (e.g., sodium or potassium).The various graphite layers only interact by weak van der Waals forces. In a firstapproximation, graphite is an ensemble of nearly independent metallic sheets.In pure graphite, they are about 3.35 Å apart, but can be separated furtherby intercalating various molecules. Charge transfer between the intercalatedmolecules and the graphitic layers is also possible. Graphite with intercalatedSbF5 shows an anisotropy of about 106 in electrical conductivity, conducting amillion times better within a layer than between layers.

670 pm

Figure 1.8 Graphite lattice.

1.3 Dimensionality of Carbon Solids 9

Diamond is a semiconductor and graphite is a metal. In diamond, there are veryfew mobile electrons – in an undoped perfect diamond single crystal at absolutezero, there are exactly zero mobile electrons – and in graphite, there are many,one electron per carbon atom. This difference is not due to dimensionality (threein diamond and two in graphite) but to single and double bonds. Several attemptshave been made to build three-dimensional graphite [10]. Theoretically, it seemspossible [11], but practically it has not yet been achieved.

Of course, since the layers of graphite are very weakly bound together, it israther easy to separate them mechanically to form graphene – a single sheet of thehoneycomb lattice. This lattice is truly two dimensional, since there is nowhereelse for the electrons to go except upon the sheet that essentially defines their“world” for them. Notice though that this two-dimensional sheet “samples” thethree-dimensional world in which it lives. If one takes the sheet and bends it inthe third dimension while applying a field across it, one can induce phase accumu-lation in the wavefunction – a Berry’s phase, which comes from the geometricalintersection of the two- and three-dimensional worlds. Graphene has been stud-ied extensively over the last few years and transport in graphene led to the 2010Nobel Prize in physics [12]. By numbers: Density of graphene is 0.77 mg/m2, itsbreaking strength is 42 N/m, electrical conductivity is 0.96× 106 Ω−1 cm−1, andthermal conductivity is 10 times greater than copper. We will return to graphenein later chapters.

1.3.3One-Dimensional Carbon: Cumulene, Polycarbyne, Polyene

As we have already pointed out, carbon has this amazing ability to bond toitself in many ways. Using double bonds, one can easily construct – at least onpaper – 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 singlebonds), or “cumulated” (adjacent). Cumulene has not been synthesized, butusing the principles of quantum chemistry it can be predicted whether cumu-lene would be stable or would rather transform into polycarbyne, an isomericstructure in which triple bonds alternate with single bonds. Polycarbyne is shownin Figure 1.10. The odds are for polycarbyne being the more stable molecule.Polycarbyne occurs in interstellar dust and in trace amounts within naturalgraphite but is not yet available for performing experiments [13].

C C C C C C- - - - - -

Figure 1.9 One-dimensional carbon: cumulene.

C C C C C C

Figure 1.10 One-dimensional carbon: polycarbyne.

10 1 Introduction

If we accept the simplification that in carbon compounds, hydrogen atoms justhave the purpose of saturating dangling bonds (making them nonactive) and thatotherwise they do not contribute to the physical properties of the material, cumu-lene and polycarbyne are not the only one-dimensional carbon solids. From thispoint of view, all polymers based on chain-like molecules are one dimensional.

As a brief note on naming conventions: in organic chemistry, the ending “-yne,”as in polycarbyne, is used to indicate triple bonds. The ending “-ene” stands fordouble bonds and “-ane” for single bonds. A polyane is shown in Figure 1.11. (Toadd a little confusion to the subject, this substance is typically called polyethylene,ending with “-ene” instead of “-ane”! The reason is simply that the names of poly-mers are often derived from the monomeric starting material, which in this case isethylene, H2C=CH2. Here the monomer contains a double bond, but during poly-merization the double bond breaks to link the neighboring molecules.) Polyanesare insulators and of little interest in the context of this textbook. (Insulators arelarge band-gap semiconductors. Because of the large band gap, it is difficult to liftelectrons into the conduction band and therefore the number of mobile electronsis negligible.)

Figure 1.12 shows polyacetylene, the prototype polyene, the simplest polymerwith conjugated double bonds. The structure shown in Figure 1.12 is often sim-plified to the one in Figure 1.13, since by convention carbon atoms do not have tobe drawn explicitly at the ends of the bonds and protons are neglected. Chapter 5is concerned with conducting polymers, and will discuss polyacetylene in greaterdetail.

C

H2

H

H

H

H

C

C

H

H

C H

H

C

H

H

C

H

H

C

H

H

C

CH2C

H2C

H2

CH2

CH2

Figure 1.11 Polyethylene, shown at the topas we might imagine the polymerization ofethylene, shown at the bottom as we mightimagine the arrangement of bonding.

HC

CH

HC

CH

HC

CH

Figure 1.12 Polyacetylene, the prototypepolyene, the simplest polymer with conju-gated double bonds.

Figure 1.13 Polyacetylene using a simplified notation.