5 p nanotubes chiara

NANOTUBI DI CARBONIO : struttura, proprietà, sintesi, applicazioni….. (SEMINARIO di CHIARA CASTIGLIONI)

Here we have what is almost certainly the strongest, stiffest, toughest molecule that can ever be produced, the best possible molecular conductor of both heat and electricity. In one sense the carbon nanotube is a new man-made polymer to follow on from nylon, polypropylene, Kevlar. In another, it is a new “graphitic” filler, but now with the ultimate possible strength. In yet another, it is a new species in organic chemistry, and potentially in molecular biology as well, a carbon molecule with the almost alien property of electrical conductivity, and super-steel strength.

R.E. Smalley, Chemistry Nobel 1996

Pres

sure

(GPa

)

A: commercial synthesis of diamond fromgraphite by catalysis;

B: P=T threshold of very fast (<1 ms) solid-solid transformation of graphite to diamond;

C: P=T threshold of very fast transformation of diamond to graphite;

D: single crystal hexagonal graphite transformsto retrievable hexagonal-type diamond;

E: upper ends of shock compression/quenchcycles that convert hex-type graphite particlesto hex-type diamond;

F: upper ends of shock compression/quenchcycles that convert hex-type graphite to cubic-type diamond;

B, F, G: threshold of fast P=T cycles, howevergenerated, that convert either graphite or hexagonal diamond into cubic-type diamond;

H, I, J: path along which a single crystal hex-type graphite compressed in the c-direction at room temperature loses some graphitecharacteristics and acquires propertiesconsistent with a diamond-like polytype, butreverts to graphite upon release of pressure.

Phase diagram of carbon emphasizing graphite, cubic diamond, and hexagonal diamondphases, as well as liquid carbon. Solid lines represent equilibrium phase boundaries.

OTHER CARBON MATERIALS

disordered carbons

“graphitic”

– micro and nanocrystalline graphites– carbon fibers– glassy carbon– porous graphites– carbon black

mixed sp2, sp3, spC atoms

– amorphous carbons– diamond like carbons (DLC)

fullerenes

nanotubes

D. Donadio, L. Colombo, P. Milani, G. Benedek, Phys. Rev. Lett., 83, 776-779 (1999)

Carbon nanotubes, M.S. Dresselhaus, G. Dresselhaus,Ph. Avouris (Eds.) Springer (2001)

– fullerenes– nanotubes– amorphous carbons

– carbon nanotubes– porous graphites

– carbon fibers, amorphous carbons and DLC hard coatings

APPLICATIONS

electronics

energy storage, batteries, sensors

mechanical and tribologicalapplications

Legame σ tra orbitali atomici di tipo np (pz)

Legame π tra orbitali atomici di tipo np (px,py)

L'ibridazione nel carbonioC (Z = 6)configurazione elettronica: 1s2 2s2 2p2

1s2 shell Kalto potenziale di ionizzazionenon e' interessata allaformazione del legame chimico

2s2 2p2 shell Lincompleta, a piu' alta energia(minore potenziale di ionizzazione)Responsabile del legame chimico

Ibrido sp3: lobi diretti nello spazio secondo i vertici di un tetraedro il cui centro corrisponde al nucleo del carbonio

( )zyx ppps 222221

1 +++=ψ

( )zyx ppps 222221

2 −−+=ψ

( )zyx ppps 222221

3 −+−=ψ

( )zyx ppps 222221

4 +−−=ψ

Si ottengono 4 orbitali ibridi dalla combinazione di 1 orbitale scon 3 orbitali p notazione sp3

x

y

z

(-1,-1,1)

(0,0,0)

(1,1,1)

(-1,1,-1)

(1,-1,-1)

Giustificazione dell’orientamento spaziale degli orbitali ibridi sp3

( )zyx ppps 222221

1 +++=ψ

( )zyx ppps 222221

2 −−+=ψ

( )zyx ppps 222221

3 −+−=ψ

( )zyx ppps 222221

4 +−−=ψ1

23

4

Ibrido sp2

3 elettroni di valenza

1 elettrone di valenza

( )xps 2223

11 ⋅+=ψ

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅+⋅−= yx pps 2

232

212

31

2ψ

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅−⋅−= yx pps 2

232

212

31

3ψ

zp2

Ibrido sp

yp2xp2

Restano non ibridizzati( )zps 22

21

1 +=ψ

( )zps 222

12 −=ψ

Ibrido sp

Esempio di molecole con carbonio in stato diibridazione sp3: metano

Esempio di molecole con carbonio in stato diibridazione sp3: etano

Esempi di ibridi sp2: copresenza di legame σ e π

σ

π

Etilene,

H2C=CH2

Esempi di ibridi sp2: copresenza di legame σ e π

σ

π

Butadiene,

H2C=(CH)-(CH)=CH2

Esempi di ibridi sp2: copresenza di legame σ e π

σ

π

Benzene,

C6H6

Esempi di ibridi sp: copresenza di legame σ e π

Acetilene,H-C≡C-H

Triplo legame:1 di tipo σ, 2 di tipo π

Struttura del diamante (ibridazione sp3)

metano (n=1)

etano (n=2)

propano (n=3)

esano (n=6)

butano (n=4)

pentano (n=5)

ALCANI (ibridazione sp3)

Struttura della grafite (ibridazione sp2)

Vista lungo l'asse c

Ass

ec

1D π conjugated systems:

polyenes

2D π conjugated systems:

PAH, Polycyclic Aromatic Hydrocarbons

POLYCONJUGATED MOLECULES

conjugated 2pz orbitals

Grafite nanostrutturataPAHPolycyclicAromaticHydrocarbons

Nanotubi di carbonio(ibridazione prevalente sp2)

Stable forms of carbon clusters: (a) a piece of a graphene sheet, (b) the fullerene C60, and (c) a model for a carbon nanotube.

Graphene ribbons terminated by (a) armchair edges and (b) zigzag edges, indicated by filled circles. The indices denote the atomic rows for each ribbon.

(a) as obtained from an electricarc deposit, the particles display a well-defined faceted structureand a large inner hollow space

(b) the same particles after beingsubjected to intense electron irradiation. The particles nowshow a spherical shape and a much smaller central empty space.

High-resolution electron micrographs of graphitic particles

Sketch of the cross section of a PAN carbon fiber along the fiber axis direction.

Here the in-plane (La) and c-axis (Lc) structural coherencelengths are indicated.

Schematic model for the microstructure of activated carbon fibers

Fiber after some heattreatment, showing partialalignment of the basic structural units.

High surface area fiberwhere the basic structuralunits are randomlyarranged

D. Donadio, L. Colombo, P. Milani, G. Benedek, Phys. Rev. Lett., 83, 776-779 (1999)

Nanostructured amorphous carbon films

Reference book:Carbon nanotubes, M.S. Dresselhaus, G. Dresselhaus, Ph. Avouris (Eds.) Springer (2001)

S.Ijima, Nature 358, 220 (1991)Nanotubi cresciuti sul catodo durante una scarica ad arco tra 2 elettrodi di grafite (T≈ 3000 K)

Multi-walled carbon nanotubes

Multi-walled carbon nanotubes

Fullerenes within SWNTs: peapods

La@C82

Heat treatment of peapods producesdouble-wall NT

Carbon nanotubes, M.S. Dresselhaus, G. Dresselhaus,Ph. Avouris (Eds.) Springer (2001)

(5,5)

(10,5)

(9,0)

Ch = 4 a1+ 2 a2 nanotubo (4,2)

Rosso: 3,3 armchair, θ=45°

Rosso: (5,0) zig-zag θ=0°

The unrolled honeycomb lattice of an Armchair nanotube

Ch = Chiral vectorT = Translation vector (k)

structural unit

Electronic 1D density of states per unit cellof a 2D graphene sheet for two (n,0) zigzag nanotubes:

(a) the (10,0) nanotube which hassemiconducting behaviour,

(b) the (9,0) nanotube which has metallicbehaviour.

Also shown in the is the density of states for the 2D graphene sheet (dotted line).

Derivative of the current-voltage dI/dVcurves obtained by scanning tunnellingspectroscopy on various isolated single-wallcarbon nanotubes with diameters near 1.4nm. Nanotubes #1 - 4 are semiconducting and #5 - 7 are metallic.

Ch = 4 a1+ 2 a2 nanotube (n,m) → (4,2)

1 2

1 21

1 1

2

2 2

(0,-1) (1,0)

(-1,0) (0,1)

Hamiltoniano elettronico H = H(θ1,θ2) alla Hückel(i.e. tight-binding ristretto a orbitali 2pz)

θi = k•ai

a1

a2

T

Ch

ϕ1τ1

τ2

ϕ2

Curve di dispersione elettronica (4,2)

Funzione del numero quantico μ = 0 .. 25

Ener

gia

in u

nit à

di β

μ = 0

μ = 1

EF

π∗

πμ = N - 1

ξ = π

ξ = -π

Γ0

K

M

μ = 1

μ = 2

μ = 3

ξ = 0

μ = 0

K1

K2

K K

KK

K

Curve di dispersione elettronica (6,3)

Funzione del numero quantico μ = 0 .. 41

Ener

gia

in u

nit à

di β

NT conduttore

Curve di dispersione elettronica (17,8)

Ener

gia

in u

nit à

di β

Funzione del numero quantico μ = 0..325

Densità di stati elettronicidi due nanotubi chirali

metallici

(14,5)(11,8)

EF

EF

Van Hovesingularities

Zigzag: (10,0) Ch ≅ 2.42 nm

Armchair: (10,10) Ch ≅ 4.2 nm

( ) ( )( )[ ]{ } 2121cos1cos12cos23, ϕϑϕϕθε ++±+= mp

( ) ( )( )[ ]{ } 2121cos1cos12cos23, ϕϑϑϕθε ++±+= mp

Analytic expressions for the electronic energies have beenobtained with a symmetry treatment of Pz orbitals in the frame of Hückel Theory

ε ε

θ/π θ/π

(10,10) Ch ≅ 4.2 nm

(10,0) Ch ≅ 2.42 nm

Wave function, Van Hove peak at energy -0.95 Beta units

Tube axis

Energy dispersion and density of states for(9,0) zigzag nanotube

Density of states for (150,150) armchair nanotube

(150,150) Ch=63 nm

Figure 5: TEM micrographs of seaweed-like carbon objects produced at 6.5 GPa and 950°C.Figure 4: TEM micrograph (a) at low magnification

and (b), (c) at high magnification of MWNT treated at 5.5 GPa and 950°C.

100

200

300

400

500

600

700

800

900

1000

1100

1200

Abs

orba

nce

1400 1600 1800 2000 Wavenumbers (cm-1)

1580G

100

150

200

250

300

350

400

450

500

550

600

650

700

750

800

850

900

Abs

orba

nce

1400 1600 1800 2000 Wavenumbers (cm-1)

1573

1330D

G

Raman spectra of graphite and amorphous carbon

Crystalline graphite

Disordered graphite

Annealed amorphous carbon courtesy of A.C. Ferrari

Dept. of EngineeringCambridge (UK)

Ram

an Intensity

Ram

an Intensity

Wavenumbers (cm-1)

Wavenumbers (cm-1)

Ram

an Intensity

GD

Wavenumbers (cm-1)

Spettri Raman Risonanti di un campione di nanonotubi singola parete contenente nanotubi di diversi diametri

A.M. Rao, E. Richter, S. Bandow, B. Chase, P.C. Eklund, K. W. Williams, M. Menon, K. R. Subbaswamy, A. Thess, R. E. Smalley, G. Desselhaus, M.S. Dresselhaus, Science 275 (1997) 187

Room temperature RBM spectra for bundlesof SWNTs produced by pulsed

laser vaporization using an Fe/Ni catalyst in a carbon target. Spectra (a)-(d) are

collected at fixed laser excitation energy (1.17 eV; Nd:YAG) from samples grown at

T = 780, 860, 920 and 1000 °C, respectively. Note that the spectral weight shifts to

smaller RBM frequencies with increasinggrowth temperature (Tg) indicating thatdiameter

increases with increasing Tg. The intensitiesand frequencies of the RBM bands in

spectra (e)-(g) collected from the same sample(Tg=1000°C) but with different laser

excitation energies (488nm; 514.5nm; 647 nm; 1064nm) are quite different, demonstrating how different diameter tubesare excited as the excitation energy changes.

Taken from:S.K. Doorn et al., PRL 94, 016802 (2005)

G+

G+G-

Raman spectroscopy is used to characterize carbon nanotubes; the G band brings important structural information

Studying a metal/semiconductor junction in a nanotube using space-resolved Raman

G- is associated tometallic tubes: why ?

Carbon nanotubes:

extended π-conjugated systems

long range electronic and vibrational interactions

crucial dependence of the electronic structure on the geometric structure (n,m)

phonons do experimentally depend on the diameter and electronic structure of the tube

⇒ Fairly challenging system to model !

Polyconjugated carbon systems

Kohn Anomalies and Electron-Phonon Interaction in Graphite (S. Piscanec, M. Lazzeri, F. Mauri, A. C. Ferrari, and J. Robertson, PRL, 93 (2004))

Graphite & Carbon Nanotubes

See poster 39-M, M. Tommasini, A. Milani, A. Lucotti, M. Del Zoppo, C. Castiglioni, G. Zerbi

… Polyynes also !

C. Castiglioni, et al. Phyl. Trans. R. Soc. Lond. A., 362(2004)

PolyenesRaman dispersion with chain length

- Structural unit: 2 atoms- Screw axis symmetry- Real (curved) geometry

Modeling electrons and phonons in carbon nanotubes

A general treatment forany carbon nanotube

(n,m)

Bloch theorem and nanotube boundary

conditions

- band structure- DOS

Calculation of phonons on the basis of valence

coordinatesGFL = Lω2

(with curved geometry)

- phonon dispersion- vibrational displacements- phonon DOS

Ch = 4 a1+ 2 a2 (4, 2) nanotube

Describing the geometry of a generic (n,m) nanotube

μ = N - 1

ξ = π

ξ = -π

Γ0

K

M

μ = 1

μ = 2

μ = 3

ξ = 0

μ = 0

K1

K2

K K

KK

K

Electronic band structure of semiconducting (4,2) nanotube

Function of the quantum numbers μ,ξ

Ener

gy

(units

of β )

μ = 0

μ = 1

EF

π∗

π

(14,5)

EF

Ohno’s three parameters force field (1) generalised to graphite (2)(1) K. Ohno, J. Chem. Phys. 95, 5524 (1995)

(2) C. Mapelli, C. Castiglioni, G. Zerbi, K. Müllen, Phys. Rev. B (1999)

semiempirical parameters

( ){ }

'')','(),(

..)]','(*),()','(*),()][','(),(*)','(),(*[2

12121

21210

212102121021210212104, ϑϑϑϑ

ϑϑεϑθεϑϑϑθϑϑϑθϑϑϑθϑϑϑθ

π

π

π

λμμλνσσνπ

π

π

π

π

πνσλμ dddd

cccccccccc

e

eeee∫∫∫∫−−−− −

+++=Π

electronic structure (Hückel)

The vibrational force fieldis coupled to the

electronic structure

bond stretching force constants

bondorder

bond-bondpolarizability

jiij

Eββπ

∂∂∂

≡Π2

S. Piscanec, M. Lazzeri, F. Mauri, A. C. Ferrari, and J. Robertson, PRL, 93 (2004)Kohn Anomalies and Electron-PhononInteraction in Graphite

jiij

Eββπ

∂∂∂

≡Π2 Long range

stretching force constants

Phonon dispersion curves of graphiteKohn anomaly and long range

interactions

Ohnoforce field;

variablethreshold on

fij

Geometrical parameters of the (n,m) tube

Boundary conditions:

Brillouin zone integration

The correct long range behavior of the force field is dictated by the electronic-structure dependent bond-bond polarizabilities Π:

Generalization of the Ohno Force Field to nanotubesof any diameter and chirality

Method based on graphene cell (2 atoms) + screw axis symmetry

Metallic: slow decaySemiconducting: fast decay

Bond-bond polarizabilities Πij

jiij

Eββπ

∂∂∂

≡Π2

It is directly related to stretchingforce constants

The G matrix is specific for any given nanotube:

G = G(n,m)

tube curvature

The F matrix is specific for any given nanotube

electronic structure (Πij):

F = F(n,m)

All data shown are taken from:A. Jorio, A. G. Souza Filho, et al., Phys. Rev. B, 65, 155412 (2002)

G band:different frequency dispersion law(while changing the tube diameter) observed for metallic and semiconducting nanotubes

Raman spectra of individualsingle wall nanotubes

(18,9)

(19,1)

(11,2) (17,7)

metallic

semiconducting

(17,3)

(15,2)

G- transversal ?(dramatically

diameterdependent…)

G+ longitudinal ?(diameterindependent…)

Empirical force field(armchair tubes)

Experimental findingsby Jorio et al.

longitudinal G- transversal G+

longitudinal

transversal

A. Jorio, et al., Phys. Rev. B, 65, 155412 (2002)

Large longitudinal/transversalsplitting: favourably compareswith experiments and…

independent theoretical worksby M. Lazzeri et al. PRB 73, 155426 (2006)

μ=1

μ=0

Dispersion of the G line Full symbols: longitudinal phononsOpen symbols: transversal phononsCold colours: metallic CNTsWarm colours: semiconducting CNTs

Phonons of the chiral (6,3) metallic tube

Conclusions

1. Carbon nanotubes share long range interactionphysics similarly to other π-conjugated systems(polyacetylene, graphite)

2. A successful and general model of phonons in nanotubes has been introduced which couplesto the electronic structure of the given (n,m) tube

3. The correct longitudinal/transversal splitting of the Gphonon as a function of tube diameter is found. The assignment of the long./transv. character of G phonons for general tubes is proposed