Carbon nanostructures

Investigators: V. Osipov, D. Kolesnikov, V. Katkov

Carbon nanocones
Carbon nanotubes
Negatively curved nanostructures
Electronic structure of various nanoparticles


At the present time, carbon nanostructures are regarded as artificially composed structures with the nanometer size. Their properties are the subject of both theoretical and experimental investigation; nowadays they have a very wide range of possible applications (see, for example, [1]).

The history of carbon nanostructures begins in 1985, when the Buckminsterfullerene C60 was discovered by Croto[2]. Since that time, the number of discovered structures is rapidly increasing. The examples of them are: the nanotubes discovered by Ijima [3], the family of fullerenes C70, C76 [4], C84 [5], C60 in a crystalline form, carbon nanocones [6], carbon nanohorns [3], nanoscale carbon toroidal structures [7] and helicoidal tubes [8], periodical carbon structures Schwarzites (proposed in [9]) and Haeckelites [10], etc. These carbon structures could be singlewalled or multiwalled; they may have zero, positive or even negative Gaussian curvature (Schwarzites). Recently, a few types of similar non-carbon structures were discovered: for example, boron nitride nanotubes [11], molybdenum disulfide or tungsten disulfide structures [12] and even the silicon nanotubes [13]! In this review, the topics of doped carbon nanostructures and combined nanostructures (fullerenes in the nanotube or the peapod structures, metal inside fullerenes etc.) will not be discussed in detail, due to the boundlessness of this subject.

As it was well known before 1985, carbon could form two types of lattice: the diamond lattice with sp3 - hybridization, where each atom is connected with four others, and the graphite lattice. Graphite is formed by flat hexagonal layers of carbon atoms, separated by 3.35 Å (angstroms), and the distance between each two nearest carbon atoms in the layer is 1.42 Å. The bonding energy of two atoms located in the same layer exceeds the same energy for different layers, and as a good approximation one can consider these atoms as having sp2 - hybridization.

Figure 1. The bended graphene layer

It is interesting to note that the bonding energy for the graphite between atoms in the same layer is as strong as the bonding in the diamond structure, although the solidity of graphite, determined by interlayer bonding, is very low. All the carbon nanostructures known at the present time could be constructed from an ordinary hexagonal graphene layer. Another key concept of the carbon nanostructures is disclination, which is presented in the lattice as five- or sevenfold, or (scientifically speaking) topological defect of rotation. Let us imagine a plane with hexagonal network on it, where each atom is connected with three others (see Fig.1). The length of each connection has a tendency to remain unchanged, and if we bend the plane, it will straighten up. The simplest way to change the plane’s shape is to cut the sector from the plane and glue its opposite sides.

According to the fact, that our plane is made of the hexagons, we can cut the sector with the angle multiple of 2p /6. When the cut is made, there will appear five-, four- or even threefold. The lattice, trying to minimize its potential energy, will bend into a conical surface. Finally we got the most simple carbon nanostructure - the carbon nanocone (see Fig. 3). On the other hand, we can insert the sector of the lattice between the cuts, which gives the sevenfold, or octagon, etc. In the first case, the Gaussian curvature of the surface (a well-known local geometrical parameter) will be positive. In the second case, it will be negative.

Figure 2. The plain (a), the positively (b) and the negatively (c) curved surfaces

The most common example of the surface with positive Gaussian curvature is the sphere shown on the Fig. 2 (b), while the onesheet hyperboloid (Fig. 2 (c)) has negative curvature. The plain and the cylinder is, according to the differential geometry, non-curved surface with exact zero Gaussian curvature. So, the presence of the pentagons in the hexagonal network leads to the positive curvature, and the inclusion of the hexagons leads to the negative sign of the curvature. It will be shown that carbon nanostructures could have all the geometries listed above.

Carbon nanocones

Carbon nanocones, discovered in 1994 [6], are the most simple example of the nanostructured carbon. They are made, as a rule, of the hexagonal plane with a different number of pentagonal defects, more precisely, from one to five. Each cut, or the pentagonal disclination, has the angle 2 π /6. As it was shown in [14], the fivefold (or positive disclination) could be stable, but the most stable configuration for more than one defect is the configuration, where they are separated by hexagons (the isolated pentagon rule) [15].

The nanocones are produced by carbon condensation on a graphite substrate [6], and by pyrolysis of heavy oil [16]. Another method of their formation is laser ablation of graphite targets [17].

Figure 3. The carbon nanocone

Theessence of the method is heating the graphite surface with intensive short laser pulse, which evaporates some number of atoms from the graphene sheet, and other atoms rearrange into the conical surface described above. So, the laser beam here plays the same role as the cut and glue procedure. One needs to notice that the mechanism of nanostructure growth has not been studied enough yet.

There is also one special class of nanocones, called “nanohorns” (they look like animal’s horns) with exact five defects (fivefolds) on the tip [3], Fig.4. They could be produced by the method similar to the nanocones, by use of laser ablation, even on the open-air [18]. These structures with good electron emission properties are easy to get and stable enough [19]. Both nanohorns and nanocones are believed to have good field emission properties, which determine their usage as electron field emitters.


Figure 4. The carbon nanohorn

As for the theoretical prediction of electronic properties of the nanocones, this is still a discussionable topic (see Chapter 6). One may say, that various models predict the increase of the electronic density of states for the cones. For some classes of the cones, there could be a metallic behavior; for example, the nanohorns could be metallic (even if they have a very big length) due to the calculations described in the Chapter 6. The metallization is in good agreement with the experimental measurements of the emission properties [19]. However, this subject is still left open.


The history of fullerenes begins in 1985, when Kroto with colleagues carried out an experiment to simulate the condition of red giant star formation, when cold carbon clusters are produced. With the use of the mass-spectrometer, they found a large peak commensurate with 60 carbon atoms. After intensive discussions, they concluded that this construction could be a truncated icosahedron, or the Buckminsterfullerene C60 (see Fig. 5). Since that time, the family of fullerenes (closed spherical carbon structures) increased, and now it consists of C70, C76, C84, C240, C540 and so on. A careful reader, when looking at Fig. 5, could find the same hexagons and fivefolds, as in the nanocones.
Figure 5. The Buckminsterfullerene

Indeed, one can produce the fullerene from the graphene layer with the same imaginary cut-and-glue procedure but with one important difference: the number of cuts (fivefolds) should be exactly twelve. Actually, in view of the fact that the fullerene is a closed structure, one could notice, that to produce the bounded piece of lattice, we need to cut the sector with the full angle 2p . Thus the fullerene could be made of two such pieces (the upper and the lower one), and the full cutting angle for the fullerene is 4p , or twelve times by 2p /6. More precisely, we need to take into account the genus of the structure: as it is shown in [15], it determines the number of squares, fivefolds, sevenfolds and octagons in the lattice. For the closed structure with genus zero, when the lattice is composed of pentagons and hexagons, the number of pentagons should be exactly twelve.

The fullerenes, being closed structures with zero genus, differ by their shape and symmetry. The Buckminsterfullerene C60, shown in Fig. 5, has a spherical-like shape and the full group of symmetry of the icosahedron Ih. This fact means that it could be rotated by the angle of 2p /5 around the center of each pentagon and reflected in the mirror located on the each plane of its symmetry. Another class of spherical fullerenes like C140 and C260 (proposed in [15]) lacks the mirror symmetry h, and their maximum symmetry group is icosahedral (I). There is also a class of non-spherical fullerenes, the most famous of them C70 – the “Rugby Ball”, with an ellipsoidal shape. One should also note that all of the fullerenes have the same Gaussian curvature sign (positive) , therefore all of them have a convex surface.

Due to the small size, some individual properties of fullerenes are now under investigation, but as it was shown in [20], the fullerenes could appear also in the crystalline form. According to the authors cited above, the doped structure of C60 crystals is superconductive at 33 K (but this fact is not proved yet well enough).

When one is speaking about the fullerenes, one should mention an interesting property of these closed structures – the ability for holding the molecules inside (hence the fullerenes are potential nano - capsules). The nano – capsule is a closed nanostructure (the fullerene or the nanotube) with one or more atoms of the substance inside the structure. Experimentally there were detected the nano - capsules like a metal – inside - fullerenes La@C60, La@C70, La@C74, La@C82, capsulated radioactive materials U@C28, Gd@C82, the peapod structures (fullerenes inside the nanotube) C60@SWNT, and so on. For the substance enclosed in the fullerene nano-capsule, carbon atoms act like a defense shield: the experiments show that the fullerene containers are good for protecting their contents from water and acid. Some interesting magnetic properties of the ferromagnetic metals (Fe, Co, Ni) inside the fullerenes were observed; namely, the magnetic properties of the metals remain unchanged. As for the capped radionuclides, the stability of these metallic fullerenes could bring the new effective solution of the radioactive waste elimination. There is also an interesting substance called “Technegas” (it is used in the medicine), discovered in 1984. The production method of this gas appears to be very similar to the production method of the fullerenes; actually, it appears to be big (from ten to one hundred nanometers) carbon nanocrystals with the metastable technetium atoms inside. Consequently, this substance was the first commercial application of the filled carbon structures before the discovery of the fullerene in 1985.

Carbon nanotubes

The nanotubes, discovered by Ijima (see [3] and Figs. 6,7), are for now the most studied carbon nanostructures. The nanotubes are the tubes made from graphene plain, with one (Single Walled) or more than one (Multiwalled) layers. Sometimes they could have the cap at their ends (see Fig.6).

The carbon nanotubes (CNTs) are produced using four main methods: arc discharge of graphite electrodes in inert atmospheres [3], pyrolysis of hydrocarbons over catalysts [21], laser vaporization of graphite targets [22], and electrolysis of graphite electrodes in the molten salts [23]. Generally speaking, the number of techniques used to produce CNTs is growing day by day, and according to the recent data [24], even the methane burning in our kitchen produces some quantity of nanotubes and other carbon nanocrystalls! This fact raises an important question about the nano-pollution of the environment and the influence of the nanostructures over our health, and this is the question still to be solved.

Figure 6. The capped carbon nanotube

The main parameters determining their type are their diameter and chirality. The chirality, or the orientation of six-folds in the nanotube, is an “internal” property, but it determines their stability (their quantity in the experiment) and electronic properties (theoretically shown by Hamada in [25]). The plain graphene, being a semiconducting material, in the form of the nanotubes could have the dielectric, and even the metallic properties. This fact determines the usage of the metallic CNTs as field emitters, even their commercial application – in the flat panel displays (the prototype of flat panel display for TV on the carbon nanotubes was presented by Samsung in 1999), and as emitters in the electronic microscopy. It is interesting to note that recently the similar silicon nanotubes were produced [26], and they have the common dependence between chirality and metallization. One can speculate, that this fact could be applied also for other two-dimensional systems; but unfortunately the other materials suitable for nanotubes, like boron nitride and molybdenum disulfide, are dielectric.

Figure 7. Some types of nanotubes: (a) armchair, (b) zigzag, and (c) chiral tubes. From the review [15]

Another interesting property of carbon nanotubes is their strength. Indirect measurements [27] revealed that multi-walled carbon nanotubes possess a Young’s modulus around 1.8 TPa, i. e. 100 times larger than steel! The robustness of the nanotubes, in combination with their low weight, leads to the rising of some fantastic technical projects, such as a space lift. However, the challenge of building a super-strong composite material out of nanotubes is still underway.

There are also many other possible applications of the carbon nanotubes - for example, the quantum wires, chemical sensors (because of their high specific surface, the electrical resistance of the tubes is distinctly changing in the presence of some chemical compounds), and so on. The future developments in the nanotechnology promise the revolution in all the scopes of human’s activity; although this revolution could be only a myth, the carbon nanotubes are for now the major breakthrough in the technological development.

Negatively curved nanostructures

Figure 8. The nanotube with two sevenfolds (black)

Figure 10. The “high genus fullerene”. From the review [10]

As it was mentioned earlier, the inclusion of the heptagons in the hexagonal lattice leads to the appearance of negative curvature.

Figure 9. The four cells of Schwarzite (a) and the TPMS (b). From the review [10]

The single sevenfold in the plain graphene lattice was theoretically studied in [14], but this situation, unfortunately, has not been observed in the experiment yet. The heptagons were observed in the nanotubes ([28], Fig. 8), and in the work [29] the magnetic properties of negatively curved structures were calculated. The heptagon included into the graphene layer bends it into a buckled surface; and symmetrical adding of the heptagons into the nanotube leads to the geometry of one-sheet hyperboloid, as it is shown in Fig.8.

Another type of negatively curved periodical nanostructures was proposed by Terrones [10]; he supposed that a mathematical object called “Triple Periodical Minimal Surface” (TPMS) could be found in the nature, in the form of carbon zeolite-like structures; he called them “Schwarzites” (Fig.9). They appear to be some kind of cubic lattice, but they are two-dimensional structures and thus their properties differ from those we have in the ordinary crystalline materials. The energetic calculations performed for Schwarzites in the work [30], shows, that they are more stable than C60. Some possible applications of Schwarzites, according to Terrones [10], are semiconducting nanodevices, new catalysts and molecular sieves. So, the presence of the heptagons in the lattice could lead to the increase of stability for such structure. That’s why Terrones proposed also the existence of such exotic structures, like “high genus fullerenes” (Fig. 10). Although this strange structure has a quasi-spherical shape, it contains only hexagons and heptagons, and therefore its curvature is negative. It is topologically similar to the sphere with twenty-one handles. An important feature of the complex graphitic structures is that they exhibit holes of labyrinths, in which molecules can be inserted. The calculations showed that around the holes (necks), the electronic behavior is metallic [31].

Among other types of exotic structures, containing heptagons as well as pentagons with hexagons, one can mention the toroidal structure [7], helicoidal graphitic tube [8] and the Haeckelites [10]. The Haeckelites, named in honor of German zoologist Ernst Haeckel, are the structures with variable shape geometries, including planar; they consist of fivefolds and hexagons with the required number of sevenfolds added to negate the curvature. One should note, that according to [10] the Haeckelites are metallic (they have non-zero density of states on the Fermi energy).

Electronic structure of various carbon nanoparticles

There are many different methods to calculate the electronic structure of the carbon lattice, including the “tight-binding” and “ab initio” calculations, the k· p approximation, and the evolution of this approach, giving the two-dimensional Dirac equation (one can call it a “sublattice approximation”, since two components of the Dirac equation appeared on the two sublattices). As it was shown above, the main feature of the nanostructures is their geometrical properties. As usual, the “ab initio” and “tight-binding” methods operate with nanostructure as with three-dimensional objects. The k· p and “sublattice” approximations, like a band model for the carbon nanotubes [32], work with the 2D - shape of the structure.

Figure 11. The band structure of the planar graphite (π -orbitals) [34]

As for the single-electron approximation, one can introduce the local density of states (LDoS) as the main parameter describing the electronic properties of the system. The theory for band structure of graphite was developed in 1947 by Wallace [33]. As it is shown in Fig. 11, near the corner K of the Brillouin zone of the planar graphite (and near all the other corners) the dispersion law could be approximated with the linear function, as in the massless Dirac (or Weyl) equation; and this leads to the “sublattice” approximation mentioned above. One should also notice, that at the Fermi energy there are two independent wave functions, located on the two sublattices [34].

In the presence of the five and / or sevenfolds, one needs to take into account two independent things: a boundary condition for the Dirac spinor function, and a topologically non – trivial gauge field. The first factor was mentioned in the works [35,36]. In these articles, an elegant approach to the defect systems was proposed, and the boundary conditions for the initial Schrödinger equation lead to the nontrivial boundary conditions for the spinors, and finally to the fields of the defect. Nevertheless, the second factor was not taken into consideration. This factor was known mostly in the elasticity theory [37]; in the field of electronic structure of nanoparticles, it was introduced in the work [38], and in the recent work [39] it was used to describe the electronic structure of the carbon nanohorns (see Fig.4 in the Chapter 2). The spinor boundary conditions in these articles were not taken into account as in [35,36], and for this reason the proposed model was criticized [40]. An attempt of full description of the system was made in [41]; however, this is still a subject for the discussion.

In contrast to the defect structures, the electronic states of the nanotubes is a well – investigated topic, in both theoretical and experimental aspects. The translational and rotational symmetries of the nanotube, being similar to the symmetries of the solids, result in the band theory [25]. The essence of this theory could be formulated in a very compact and simple form. Actually, let’s plot a Brillouin zone and the lines of constant wave vector in the momentum space (Fig. 12). The orientation of the Brillouin zone depends on the type of the tube (armchair, zigzag etc.), and the “distance” between the lines (its dimension is converse length) depends on the radius of the tube; if the line is crossing the hexagon’s corner, the phase of the wavefunction could take an exact zero value, and this result in the metallization of the (sub)lattice. The similar condition for the zero phase was used to describe a metallization of the carbon nanocones in [36]. An example of the metallic behavior is shown in Fig.12 (a). On the contrary, in Fig. 12 (c,d) this condition is not satisfied, and these tubes are semiconducting. This results in the, respectively, zero or non-zero electronic density of states on the Fermi level. When the lines are close to the hexagon corners, as it is shown in Fig. 12(b), the width of the band-gap is small. The connection of the metallic and semiconducting tube of the type (a) and (b) could lead to the nanoscale semiconducting device like a diode or field transistor (FET).

Figure 12. (a) Armchair, (b,c) zig-zag and (d) chiral tube; (a) metallic, (b) small gap semiconductor, and (c,d) semiconductor. From the review [42]

The realization of the connection, proposed by many different authors [43], may be a chirality-changing pentagon - heptagon pair included into a nanotube structure. The practical use of such devices is expected in the very near future.


At the end of this review, one needs to remember the words of Richard Feynman: “... there is plenty of room at the bottom”. This statement was made in 1959, and its meaning reflects the (possible) significance of the molecular – scale devices. In his lecture at the California Institute of Technology, Feynman proposed the new ideas to be realized on the molecular size level – the information on the nanoscale, miniaturized and quantum computers, and the extensively advertised concept of nano - bots. And what do we see at the present time? On the one hand, there is a popularized idea of nano – robots, which will appear to build any thing in no time and without any cost; but there are no ideas (even in the theoretical aspect) how these systems should look like. There are many enthusiasts of this approach [44], but they lack the practical techniques which are needed to convert their dreams into the scientific and technical reality. On the other hand, there are some types of the nanostructures (the quantum dots, semiconducting heterostructures and the carbon nanostructures described above) which are investigated enough, and some of them have commercial applications (see Chapter 4). The most important thing we need to do now is to develop the present technologies in accordance with the present state of the science and with the requirements of the mankind. The highly advertised nanorobots would be just a little part of the wonders of future engineering. Both the development and use of the novel technologies are in our hands; our duty is to manage them in the worthy way.


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