Carbon nanotubes and Graphene - Bolyai Kollégiumpeople.bolyai.elte.hu/~hagymasi/nanotubes_graphene1.pdf · Band structure of carbon nanotubes Carbon nanotubes and Graphene Imre Hagymási

History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes

Band structure of carbon nanotubes

Carbon nanotubes and Graphene

Imre Hagymási

16 October, 2008

Solid State Physics Seminar

Imre Hagymási Carbon nanotubes and Graphene



Main points

1 History and discovery of Graphene and Carbon nanotubes

2 Band structure of the GrapheneTight-binding approximationDynamics of electrons near the Dirac-points

3 Structure of Carbon NanotubesProperties of carbon nanotubes

4 Band structure of carbon nanotubesZone-folding approximationOutlook




History and experimental discovery

History of Graphene

Wallace (1947): The band theory of graphite → graphene

McCure (1956): electrons can be described as Dirac-fermionswith zero mass

Geim’s research group (Manchaster, 2004): first experimentalobservation of 2D graphite layer

Geim et al. (2006): Observation of Klein tunneling in graphene

History of Carbon Nanotubes and Fullerene

Curl, Kroto, Smalley (1985): discovery of fullerene

Ijima (1991): discovery of carbon nanotubes (first unambigousexperiment)




Tight-binding approximationDynamics of electrons near the Dirac-points

Structure of the Graphene

a 1

a 2

A B

Figure: The honeycomb structure of the Graphene





Band structure of the Graphene I.

the Bloch-function in tight-binding approximation:

ψk(r) =1√N

∑

R

e ikR[CA(k)ϕA(r −R) + CB(k)ϕB(r − R− d)]

R = n1a1 + n2a2, d = a1+a2

3

taking into account the first neighbours, the dispersionrelation:

E (k) = ǫ0 + |γ0|√

3 + 2 cos ka1 + 2 cos ka2 + 2 cos k(a1 − a2)ǫ0 onsite energy, γ0 =

∫

ϕ∗A(r)HϕB (r − d)d3r , a hopping integral





Band structure of the Graphene II.

-5

0

5

kx

-5

0

5

ky

-2

0

2

ΕHkL

Figure: The valence and the conductionband.

Figure: Band structure’s contour plot





Band structure of the Graphene III.

Behaviour near the K points

the conduction and the valence band form conically shapedvalleys that touch at the six corners of the Brillouin zone

the Fermi level passes through the K - or Dirac-points

the dispersion relation near the K -points:

|E | = ~vF |δk|, δk = k − K, vF ≈ 106m

s

→ special theory of relativity





Conical structure at the Dirac-points

Figure: Dispersion relation near the Dirac-points





Hamiltonian of the graphene

electrons near the Dirac-points can be treated as maslessexcitations, governed by a Dirac-Hamiltonian:

Hgraphene = −i~v

(

σx∂x + σy∂y 00 σx∂x − σy∂y

)

→ Dirac-fermions

in 2D the two subblocks can be transformed to each other bya unitary transformation → valley degeneration





Relativistic effects in graphene

Zitterbewegung → position operator has an oscillating partbeside the motion with a constant velocity

Klein paradox (if V0 ≫ 2mc2 T ≈ 1), it is hard to point out,E > 1016V/cm

in graphene E > 105V/cm is enough to observe thephenomenon

electron scattering on a potential step V0 (p − n junction) →graphene has a negative refractive index!

sinα

sin β=: n = −|E − V0|

E

n can be tuned by varying the gate voltage → electron lenses





Relativistic effects




Properties of carbon nanotubes

Carbon nanotubes





What are the carbon nanotubes?

Basic properties

hollow cylinders of graphite sheets → single-walled nanotube

a tube consisting of several concentrical cylinders → multiwallnanotube (MWNT)

∼ nm diameter, ∼ µm length → quasi 1D crystals

nanotubes are metallic or semiconducting

properties of the nanotubes depend crucially on the way theyare rolled up

Synthesis of single-walled nanotubes

laser ablation

high-pressure carbon-monoxide conversion

arc-dischargeImre Hagymási Carbon nanotubes and Graphene




Two main types of carbon nanotubes

Figure: HRTEM images of a semiconducting and a metallic nanotube.





Structure of the Carbon Nanotubes

Definitions:

chiral vectorc = n1a1 + n2a2 usuallydenoted by (n1, n2)

T: tube axis, the minimallattice vector ⊥ c

diameter: d = |c|π

=

a0

π

√

n21

+ n1n2 + n22

Figure: Making single-walled nanotube of asingle graphite layer.





Two main types of carbon nanotubes

Figure: Armchair nanotube. Figure: Zigzag nanotube.





Structure of Carbon Nanotubes

The unit cells of different nanotubes, a denotes the translationalperiod





Mechanic and electric properties of carbon nanotubes

Mechanic properties

material Young’s Modulus (TPa) Tensile Strength (GPa)

SWNT 1-5 13-53

Armchair SWNT 0.94 126.2Zig-zag SWNT 0.94 94.5

MWNT 0.8-0.9 150Stainless Steel ∼0.2 0.65-1

Electric properties

Depending on the (n1, n2) vector, a nanotube is

metallic if 3|(n1 − n2)

semiconducting otherwise




Zone-folding approximationOutlook

Band structure of carbon nanotubes I.

the tube is infinitely long→ kz wave vector is continuous in the interval

(

−π

a, π

a

)

a: translational period

along the circumference k⊥ wave vector is quantized(Born-Kármán boundary condition): m · λ = |c| = π · dthe allowed k⊥ vectors (k1, k2 are the reciprocal lattice vectorsof graphene):

k⊥ =2n1 + n2

qnR k1 +2n2 + n1

qnR k2

m = −q

2+ 1, . . . , 0, 1, . . . ,

q

2, n = GCD(n1, n2)

q: the number of hexagonal cells in the nanotube unit cellR = 3 if (n1 − n2)/3n is an integer, R = 1 otherwise





Band structure of carbon nanotubes II.

first approximation (zone folding): electronic properties of carbonnanotube can be obtained by cutting the band structure ofgraphene

Figure: Brillouin zone of a (7,7) armchair and a (13,0) zig-zag tube.Imre Hagymási Carbon nanotubes and Graphene




Band structure of carbon nanotubes III.

Condition for nanotubes being metallic

it can be explained by the Fermi-surface of graphene

if the K point of the Brillouin-zone is a part of the allowedstates → the nanotube is metallic

the K point of graphene is at 1

3(k1 − k2)

K point is allowed if

K · c = 2πm =1

3(k1 − k2)(n1a1 + n2a2) =

2π

3(n1 − n2),m ∈ Z

→ 3m = n1 − n2





Band structure of carbon nanotubes IV.

Figure: Allowed k lines in the Brillouin zone of graphene.





Beyond the Zone-folding approach

Curvature effects

C-C distance for atoms with different ϑ azimuthal angle isreduced

angles of the hexagons are not 60ďż˝

Fermi-point moves away → secondary gaps appear in zig-zagnanotubes





Conclusion, outlook

Carbon nanotubes, conclusion

experimental results are in good agreement with thezone-folding approach

curvature effects modify the properties of the nanotubes

Graphene, outlook

masless fermion wave equation can be mapped to neutrinos(2007)

graphene + superconducting domain → new phenomenon(Beenakker, 2004): specular Andreev reflection





Outlook: specular Andreev reflection (2005)

in an N-S system if E < ∆, the electron can not enter thesuperconducting region → Andreev-retro-reflectionin graphene both of them can occur





Thank you for your attention!


Carbon nanotubes and Graphene - Bolyai Kollégiumpeople.bolyai.elte.hu/~hagymasi/nanotubes_graphene1.pdf · Band structure of carbon nanotubes Carbon nanotubes and Graphene Imre Hagymási

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