A bird’s eye view to carbon nanotubes and fullerenes

by

Serdar Durdagi

Theory Department, Fritz-Haber-Institute of Max-Planck Society

Technical University, Berlin Theoretische Festkörperphysik I & II

1st June - 2005, Berlin

Outline

Part I• Carbon• Natural crystalline forms of carbon

– Graphite– Diamond

Part II• Fullerenes

– Structural Properties– Production and mechanism

Part III• Nanotubes

– Basic Background– Growth mechanism – Mechanical and electronic properties– Applications– Defects

Part I: Carbon

- C: 1s2 2s2 2p2 (C has 4 valence electrons)

- Since the energy difference between the upper 2p energy levels and the lower 2s level in carbon is small compared with the binding energy of the chemical bonds, the electronic wave functions for these four electrons can readily mix with each other. The energetically preferred arrangements of these electrons is to form so called sp-hybridized wavefunctions, which can then form covalent bonds with other atoms.

- sp- hybrid: (linear chain of C), possible but can not easily form a solid.

- sp2- hybrid: (planar bonding)

since the bonds are all in same plane, it is natural to assume these all form a single 2D carbon plane, with an angle of 120o between bonds, this makes naturally a honeycomb network.

- This is called a graphene sheet.

A: top-viewCrystal structure of Graphite. The unit cell is shaded in

green.The hexagonal surface lattice is defined by two unit vectors; u and v, in the xy plane with a length of 2.46 Å. The basis of the lattice consists of two carbon atoms α and β with a distance of 1.42 ÅB: perspective viewthe α atoms are directly above an α atom in the layer directly underneath at a distance of 3.348 Å.; the βatoms are over a hollow sites. The unit vector w is parallel to z- axis with length of 6.696 Å.

From: Applied Physical Sciences, 100, 22, 12539, 2003

Properties:

• Soft, gray-black, and stable• Lubricant• Good electrical conductor.

The sheets are electrically conductive in the in-plane direction).

• Density: 2.26g/cm3

• Low melting and boiling points.• Cohesive energy : 7.692 eV/ atom (theo.)*

7.374 eV/atom (exp)

* S. Durdagi, J. M. Carlsson, M. Scheffler( unpublished result)

Graphite (2D)

If you stack many of these graphene sheets on top of each other, you get graphite, which is the ground state structure of carbon. Sheets are held together by weak van der Waalsforces.

Graphite

• Graphite Intercalation compounds (G.I.C):

Intercalation compounds consist of layers -sandwiches- of different chemical species. Many substances can be intercalated between the layers of graphite, but one of the longest known and best studied is potassium, which can intercalated until a limiting formula of C8K is reached. Upon intercalation, the graphite layers move apart somewhat (205 pm), though less than expected as estimated from the diameter of the potassium ion (304 pm or greater). This indicates that the K+ ion ‘nests’ within the hexagonal carbon net.

Because of the similarity in energies of the valence and conduction bands, graphite can be either an electron donor or acceptor. Intercalation of K atoms into graphite results in the formation of K+

ions in the conduction band. Graphite will react with an electron acceptor such as bromine to form C8Br in which electrons have transferred from the valence band of the graphite to bromine.

Diamond (3D)

- sp3- hybrid:

- The tetrahedral sp3 bonding arrangement can be arranged into a 3D solid in the diamond crystal structure, which looks like follows. All electrons are tightly bound between the carbon atoms in strong covalent bonds. The closely-knit structure helps to explain why diamonds are one of the hardest natural substances known.

• Hard• Insulator (since all electrons are localized in the bonds within the sp3 network)• Transparent• Density: 3.51 g/cm3

• Bond distances: 1.54 Å• Cohesive energy : 7.551 eV/ atom (theo.)*

7.3 eV/atom (exp)

* S. Durdagi, J. M. Carlsson, M. Scheffler( unpublished result)

From figure, it is immediately seen that sp2 -bonded graphite is the ground state phase under ambient conditions. At higher temperatures and pressures, a stability region for sp3 - bonded diamond is shown, whereas other regions show stability ranges for hexagonal diamond and the liquid phase.

From: Carbon, 34, 141-153,1996

Phase Diagram of Carbon

Part II: Fullerenes

What is Fullerene?

• Fullerenes are a form of carbon molecule that is neither graphite nor diamond. They consist of a spherical, ellipsoid, or cylindrical arrangement of dozens of carbon atoms. Fullerenes were named after Richard Buckminster Fuller, an architect known for the design of geodesic domes-great circles- which resemble spherical fullerenes in appearance. (A spherical fullerene looks like a soccer ball.) Spherical fullerenes are often called "buckyballs“.

• Even though the ball like C20H20 was already found in 1983, it still came as a big surprise when Harold Kroto, Robert Curl, and Richard Smalley made the completely unexpected discovery (Nature 318, 162, 1985) that the element carbon can also exist in the form of very stable spheres. They termed these new carbon balls as fullerenes.

• Pure fullerenes : C60,C76,C86,C116, etc.

• The carbon atoms in fullerenes are spm (2<m<3). The free valence electrons on the cage building a strong localized π-electron system. This π-electron system influences chemical reactions of the fullerenes. In chemical reactions fullerenes are not reacting aromatic, they show aliphatic behaviour.

First idea of fullerenes

• 1+1= 4 (Triangle and Tetrahedron Synergy or 'explorations in the geometry of thinking' ):Two open-ended triangle may be combined in such a manner as to create the tetrahedron, a figure volumetrically embraced by four triangles.

+

Euler’s theorem of polyhedronsEuler’s theorem of polyhedrons

Leonhard Euler (1707-1783)

The Euler characteristic is a number, C, which characterized the various classes of geometric figures. It depends only on the topology, not the specific shape of the figure. For an arbitrary polyhedron it can be calculated from the number of vertices V, edges E, and faces F:

C = V - E + F

Euler showed that for all simple polyhedra the Euler characteristic is C=2. [e.g, cube: F=6, E=12 , V=8, so C=8-12+6=2]

• For the special case of a simple polyhedron made from only pentagons and hexagons

(P= number of pentagons , H= number of hexagons). Then,

F= P+HE=(5P+6H)/2V=(5P+6H)/3

Inserting all this into Euler’s equation and using C=2,

12 = P + 0.H a simple polyhedron entirely made from pentagons and hexagons must contain exactly 12 pentagons

V = 20 + 2 H (For a fullerene the number of vertices is just the number of carbon atoms in the molecule).

We have thus found that fullerenes must have an even number of carbon atoms, and that there is a lower limit for the size of the fullerenes: C20 is the smallest fullerene. It looks like a dodecahedron.

Structural Properties of Fullerenes

• The pentagons within the fullerenes are needed to introduce curvature, since a network consisting of hexagons only is planar.

• C60, contains 12 pentagons and 20 hexagons, 30 double bonds.

• Diameter of C60 is 7.1 Åall 12 pentagons, are isolated by hexagons and the bonds at the junctions of two hexagons (1.38 Å ) are shorter than the bonds at the junctions of a hexagon and a pentagon (1.45 Å ).

• The length of single bonds is in the range 1.42 Å (graphite) < 1.44 Å (fullerene)

• This molecule is exceptionally stable and strong due to its high symmetry, and the fact that the C-bonds are still pretty close to the ideal sp2 bonding arrangement.

• Angle between a double bond and its adjacent pentagonal faces: α = cos-1[0.5*cos(3/5) π ]

• Angle between two adjacent hexagonal faces:β=cos-1[(8/3)*sin(3/10) π 2-1]

Angle between adjacent hexagonal and pentagonal faces:γ= 0.5*(π- β)- α

Physical constants for C60 and C70 molecules

• The heat of formation of C60 and C70 is 10.16 kcal/mol per C atom, 9.65 kcal/mol per C atom, respectively. The fullerenes are therefore thermodynamically less stable than graphite and diamond.

• It is expected that upon increasing the size of the fullerene the energy content of the spheres asymptotically reaches that of graphite.

• The binding energy of C70 is about 0.2 eV greater than that of C60.

Production of Fullerenes

• The initial producing of fullerenes was pretty much an accident. The authors aim were to determine whether the sort of carbon chain compounds such as HC7N (cyanohexatriyne) found by radioastronomy in the interstellar medium (they used high-power laser to produce hot vapors of carbon to mimic the conditions in interstellar space) could be synthesized by mixing carbon vapor with a suitable reagent such as ammonia and to find the conditions needed for a study of the low temperature electronic spectra of carbon chain compounds using resonance –enhanced two photon ionization.

• In the course of this research they found the fullerene. (They, actually failed to answer to question they first set out to study, but since they got the Nobel prize for the fullerene, this probably doe not bother them too much).

Smalley's cluster apparatus for the laser evaporation of graphite

“Minute quantities of fullerene”

The vaporization laser beam (30-40 mJ, at 532 nm in a 5- ns pulse) is focused through the nozzle, striking a graphite disk which is rotated slowly to produce a smooth vaporization surface. Helium carrier gas provides the thermalizing collisions necessary to cool, react and cluster the species in the vaporized graphite plasma.

production of fullerenes in larger quantitiesproduction of fullerenes in larger quantities• Five years after the discovery of the fullerenes,

Huffmann and Krätschmer managed to produce fullerenes in larger quantities.

• When two graphite rods are heated to a high temperature by an electric arc discharge in an atmosphere of hellium at apressure of 13 kPa the Graphite rods are slowly consumed and soot is formed. Approximately 10% of the soot is made of C60 and C70. The soot is collected and treated with benzene to dissolve the fullerenes, which can then be separated using chromatographic methods.

• Fullerenes are soluble in many organic solvents and they can separate from the non- Fullerene soot using filtration.

Arc Synthesis

• After the discovery of the macroscopic generation of fullerene, we can produce certain amounts of fullerene by following certain technical procedures. However, many questions still unresolved such as why the energetically less stable C60 structure is preferred to graphite or larger fullerenes, and why the extracted higher fullerenes show the magic numbers.

Mechanism of fullerene formation

Dangling bonds: + -

Strain energy : - +

“ A spherical shape distributes the strain as evenly as possible and minimizes the anisotropic contribution to the strain energy”.

• A MD Demonstration of Annealing to a Perfect C60

Mechanism of fullerene formation

Effect of temperature in arc-dischareg method :

Tc < 2500 K Graphite-like structures2500 K < Tc < 3500 K Fullerene-like structuresTc > 3500 K Chaotic- 3D structures

500 carbon atoms in gas phase with random positions and velocities distributed in a cubic box with full periodic boundary conditions. The system is

controlled toward a 3000 K temperature.

Mechanism of fullerene formation

From: http://www.photon.t.u-tokyo.ac.jp/~maruyama/agallery/agallery.html

• Solubles: C60, C70, C76, C78, C84, etc.-all have large HUMO-LUMO gaps

• Insolubles: C74 (D3h), C80 (Ih), etc.-have small HUMO-LUMO gaps.

Different classes of empty fullerenes

Endohedral Fullerenes

• C60 can easily accept electrons and form negative ions. With alkali metals (e.g. potassium), C60 forms a new superconducting crystalline material built from a C60 ion with three charges and three positive potassium ions (K3C60). The material becomes superconducting at 19 K. Because C60can accept and then donate electrons reversibly, the fullerenes can well become catalysts in chemical processes and replace expensive and poisonous metals.

Part III: Nanotubes

• Nanotubes are tubular structures that are typically several nanometers in diameter and many microns in length.

• The aspect length to diameter ratio can be very large (greater than 104), thus leading to a prototype 1D system.

• Nanotubes can be derived from the buckyball molecule C60 by adding belts of atoms or by rolling a two-dimensional graphene sheet cut at various angles with respect to the hexagonal lattice

• In 1991,Sumio Iijima of the NEC Laboratory in Japan reported * the first observation of the multi-walled carbon nanotubes (MWNT) in carbon soot made by arc-discharge. About two years later he made the observation of single-walled nanotubes(SWNT)**.

* Iijima S., Nature, 354, 56, 1991** Iijima S., Nature, 363, 603, 1993

Nanotubes

Multi Wall Carbon Nanotubes (MWNTs) Single Wall Carbon Nanotubes (SWNTs)

Why are they interesting?

• - Extremely Strong(tensile strength of 45 billion pascals)

• - High Current Capacity(1 billion amps/cm2)

• - Good temperature stability(stable up to 2800oC in a vacuum, 750 0C in air)

• - Conducting or semi-conducting characteristics

Structure of Carbon Nanotubes / Basic background

• The structure of a single walled carbon nanotube is conveniently explains in terms of its 1D unit cell, defined by vectors Ch (chiral vector) and T (lattice vector).The circumference of any carbon nanotube is expressed in terms of the chiral vector Ch

Ch = nâ1+mâ2 ≡ (n,m)

The chiral vector is defined on the honeycomb lattice of carbon atoms by unit vectors â1, â2 and the chiral angle Ө with respect to the zigzag axis (Ө=0).

(T is the lattice vector of nanotube unit cell).

3 types of nanotube structure:

• Ө= 0 zigzag

Ө= 30 armchair

0< Ө <30o chiral

Structure of Carbon Nanotubes / Basic background

Ө= 0Ө= 30o0< Ө <30o

(n,0) or (0,m)(n,n) (n,m)

• Both armchair and zigzag nanotubeshave a mirror plane thus are considered as achiral.

•Differences in the nanotube diameter d and chiral angle Ө give rise to differences in the properties of the various carbon nanotubes.

•Nanotube diameter d is given by:

d= (3)1/2 .a c-c .(m2+mn+n2)1/2/ π = Ch / π

where Ch is the length of Ch,

and

Ө=tan-1[(3)1/2/ (2m+n)]

Growth mechanisms of Nanotubes / From membrane to tubes

• Carbon Arc-Discharge• Laser Ablation• Chemical Vapor Deposition

• Carbon arc-discharge method is still the most practical and gives high yield graphitized tubes.

• CVD technique has the potential for making possible large scale production of nanotubes.

Growth mechanisms of Nanotubes / From membrane to tubes

• Carbon Arc-Discharge and Laser ablation

Both methods involve the condensation of carbon atoms, generated from evaporation of solid carbon sources. The temperature involved in these methods are close to the melting temperature of graphite, 3000-4000 oC.

•In arc-discharge, carbon atoms are evaporated by plasma of helium gas ignited by high currents passed through opposing carbon anode and cathode.

MWNTs can be obtained by controlling the growth conditions such as the pressure of inert gas and the arcing current.

[Carbon arc apparatus has two graphite rod, one serves as an anode and other one as a cathode During the process, the anode rod is consumed and forms a carbonaceous deposits (nanotubes and graphitic particles) on the cathode rod)].

Purification of MWNTs can be achieved by heating the grown material in oxygen environment to oxidize away the graphitic particles. (Graphitic particles exhibit higher oxidation rate than MWNTs).

Growth mechanisms of Nanotubes / From membrane to tubes

• Carbon Arc-Discharge

For the growth of SWNTs, a metal catalyst is needed in the arc-discharge system. (e.g., carbon anode contains a small percentage of cobalt catalyst).

• MWNTs and SWNTs

• Relatively cheap

• Many side-products

Growth mechanisms of Nanotubes / From membrane to tubes

• Laser ablation

•The growth of high quality SWNTs at the 1-10 g scale was achieved by Smalley et al., by using a laser ablation (laser oven) method.

•The method utilized intense laser pulses to ablate a carbon target containing 0.5 atomic percent of Ni or Co. The target was placed in a tube-furnace heated to 1200oC. During laser ablation, a flow of inert gas was passed through the growth chamber to carry the grown nanotubes downstream to be collected on a cold finger.

•The produced SWNTs are mostly in the form of ropes consisting tens of individual nanotubesclose-packed into hexagonal crystal via VDW interactions.

• Catalyst / no catalyst• MWNTs / SWNTs• Yield <70%• Use of very strong laser• Expensive (energy costs)

From: R. E. Smalleyet al., Science, 273, 483 (1996)

Growth mechanisms of Nanotubes / From membrane to tubes

•Chemical Vapor Deposition (CVD)

Growth process involves heating a catalyst material to high temperatures in a tube furnace and flowing a hydrocarbon gas through the tube reactor for a period of time. Material grown over the catalyst are collected upon cooling the system to room temperature.

• Gas phase deposition• Large scale possible• Relatively cheap• SWNTs / MWNTs• Aligned nanotubes• Patterned substrates

Growth mechanisms of Nanotubes / From membrane to tubes

The general growth mechanism in CVD process involves the dissociation of HC molecules catalyzed by the transition metal, and dissolution and saturation of carbon atoms in the metal nanoparticle. The precipitation of carbon from the saturated metal particle leads to the formation of tubular carbon solids.

Tubule formation is favored over other forms of carbon such as graphitic sheets with open edges. This is because a tube contains no dangling bonds and therefore is in a low energy form.

From S. E. Thompson, EEL, 6935

Cohesive energies of different forms of carbon

The energy cost for curving a graphene sheet into a cylinder of a tube with a rollup vector (10,10) is known to be only 0.045 electron-volts (eV) (or 0.08 eVnm2/d2 for a tube of any diameter d). Therefore the carbon atoms in such a tube, neglecting the ends, have 99.4 percent of the cohesive energy that they would have in perfect crystalline graphite. This is far better than C60, which is a major feedstock and has a cohesive energy that is only 92 percent that of graphite.

Stability of SWNTs/ How small they can be?

• It is well established that freestanding isolated tubes present circular sections since the circular shape minimizes the strain energy of tubes.

• Estrain ~ 1/r2 (tube radius)• Estrip ~ 1/r (tube radius)

• While the CNT with diameters smaller than 0.4 nm are energetically less favorable than a graphene sheet.

From: IEEE, Transactions on Nanotech., Vol 3, No:2, 2004 and Topics in Appl. Physics , Carbon Nanotubes, M.S. Dresselhaus, Springer, 1996

Mechanical Properties of CNTs

Carbon nanotube based, and other nanodesigned materials are expected to provide extraordinarily strong but light-weight composites for future structural applications.

To realize these applications, we first need to know a lot more about these materials: especially, - how strong is this nanotube?- how stiff ? - what happens if you bend it ? - twist it ? - stretch it ? - compress it ?

Video clips show single-wall and multi-wall nanotubes undergoing axial compression, bending and torsional twisting.

Simulations and experiments have shown that single-wall nanotubes are strongest known fiber so far and can withstand 10-50 times more deformation before they break.

Simulations also show that nanotubes under extreme deformation remain elastic to a large extent.

For further reading:Physical Review Letters, Vol. 83 (15), pp. 2973-2976 (1999)

Mechanical Properties of CNTs

• From: http://www.applied-nanotech.com

Young's modulus, tensile strength and density of carbon nanotubes compared with some other materials:

MaterialYoung's

modulus (GPa)

Tensile Strength (GPa)

Density (g/cm3)

Single wall nanotube 1054 150

Multi wall nanotube 1200 150 2.6

Steel 208 0.4 7.8

Epoxy 3.5 0.005 1.25

Wood 16 0.008 0.6

• Further Reading: PRL, 83. 15, 2974 (1999)

Electronic Properties of CNTs

• Theoretical calculations have shown early that the electronic properties of the carbon nanotubes are very sensitive to their geometric structure.

• Although graphene is a zero gap semiconductor, theory has predicted that the carbon nanotubes can be metals or semiconductors with different size energy gaps, depending very sensitively on the diameter and the helicity of the tubes, i.e., on the (n, m).

• (n, n) tubes …… …..metal

• (n, m) tubes metal if………n-m=3k, (k, non-zero integer)

(Strictly within within the band-folding scheme, the n-m=3k tubes would all be metals, but because of tube curvature effects very tiny gap opens for the case where k is non-zero).

• all others semiconductors.

Electronic Properties of CNTs

• The Fermi surface of an ideal graphite sheet consists of the six corner K points.

•When forming a tube, owing to the periodic boundary conditions imposed in the circumferential direction, only a certain set of k states of the planar graphite sheet is allowed.

•The allowed set of k’s depends on the diameter and helicity of tube.

•Whenever the allowed k’s include the point K, the system is a metal with non-zero density of states at the Fermi level, resulting in a 1D metal with 3 linear dispersing bands.

•When the point K is not included, the system is a semiconductor with different size energy gaps.

Electronic Properties of CNTs

• Let’s look (5,0) and (6, 0) nanotubes.

(5,0) nanotube (6,0) nanotube

• The only “allowed” energy states line on the intersection of the parallel lines with the Hexagonal Brillouin Zone. Note that for (5, 0) none of the lines intersects K points. So (5,0) is semiconducting. But, for (6,0) two points intersects K points (metal).

Density of States of CNTs

From: Acc. Chem. Res. 35, 1063, 2002

Density of States of CNTs

From: Acc. Chem. Res. 35, 1063, 2002

Effect of bending and twisting to CNT’s electonic behaviour

From: http://www.pa.msu.edu/cmp/csc/nanotube.html

Effect of twisting on the properties of a metallic armchair (6,6) nanotube. Three twisted configurations are shown. Twisting is found to transform the metallic nanotube to a semiconducting one with a band-gap that varies with the twist angle as shown.

Applications of CNTs

From: cover of Science, 9 November 2001

‘ As logic and memory circuits’

- Using nanotups as nanoprobe tips- Field emitters- Storage or filtering media- Nanoscale electronic devices, etc.

Macro applications; where armies of nanotube molecules might line up to form a light, strong wire or a composite that could be unbeatable as a material for making light weight vehicles for space, air and ground. If the costs ever permit, these materials might be used in the elements of bridges, or of tall, earthquake-resistant buildings or towers. Light ammunition and bulletproof vests can be envisaged. All these applications rely on mechanical strength, a property that is essentially straightforward but that requires volume production of the crucial components, defect-free nanotubes of greater length.

• Richard Smalley: ”These nanotubes are so beatiful that they must be useful for something”

Applications of CNTs

Nanotubes have ideal electronic properties for molecular electronic devices. Here is the one possible uses claimed for nanotubes as electronic devices.

Why CNTs?

SWCNTs, which are one-dimensional systems, do not allow the small-angle scattering of electrons or holes by defects or phonons that occurs in a three-dimensional system because carriers in them have only two directions of propagation, forward or backward.

Nanotube Field Effect Transistor

APL, 73, 2447, 1998

The transistor action is due to the el. field around the gate modulating the Schottky barrier at the points where the source and drain terminals contact nanotube. For reasonably short tubes (<1µm), the effects of the gate voltage on channel resistance is minimal. The source and drain terminals should be far apart to prevent electron tunneling between them.( Further reading:http://www.crhc.uiuc.edu/ece497nc/scribe/nanotube1.pdf)

Applications of CNTs/ Defects in CNTs

The possible defective structures can be classified into 4 main groups:

• Topological (introduction of ring sizes other than hexagons)• Rehybridization (ability of carbon atom to hybridize between sp2 and sp3)• Incomplete bonding defects (vacancies, dislocations,...)• Doping (with other elements than carbon)

Due to their finite size, an interesting structural feature occurs near the ends of all tubes from the closure of the graphene cylinders by the incorporation of topological defects such as pentagons in the hexagonal carbon lattice.

Carbon nanotube intermolecular junctions, formed by interposing one or multiple topologic pentagon-heptagon (5/7) defects in the hexagonal structure between two nanotube segments of different helicity..

Metal-semiconductor junctionsMetal-metal junctionsSemiconductor-semiconductor junctions

From: Acc.of Chem. Res., 35,12,2002

Applications of CNTs/ Defects in CNTs

Doping CNTs:

From: Acc.of Chem. Res., 35,12,2002