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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
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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
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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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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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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.
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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
39
40
2 One-dimensional Substances
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.
41
42
2 One-dimensional Substances
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
43
44
2 One-dimensional Substances
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
45
46
2 One-dimensional Substances
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
47
48
2 One-dimensional Substances
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.
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51