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
History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes
Band structure of carbon nanotubes
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
Imre Hagymási Carbon nanotubes and Graphene
History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes
Band structure of carbon nanotubes
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)
Imre Hagymási Carbon nanotubes and Graphene
History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes
Band structure of carbon nanotubes
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
Imre Hagymási Carbon nanotubes and Graphene
History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes
Band structure of carbon nanotubes
Tight-binding approximationDynamics of electrons near the Dirac-points
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
Imre Hagymási Carbon nanotubes and Graphene
History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes
Band structure of carbon nanotubes
Tight-binding approximationDynamics of electrons near the Dirac-points
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
Imre Hagymási Carbon nanotubes and Graphene
History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes
Band structure of carbon nanotubes
Tight-binding approximationDynamics of electrons near the Dirac-points
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
Imre Hagymási Carbon nanotubes and Graphene
History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes
Band structure of carbon nanotubes
Tight-binding approximationDynamics of electrons near the Dirac-points
Conical structure at the Dirac-points
Figure: Dispersion relation near the Dirac-points
Imre Hagymási Carbon nanotubes and Graphene
History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes
Band structure of carbon nanotubes
Tight-binding approximationDynamics of electrons 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
Imre Hagymási Carbon nanotubes and Graphene
History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes
Band structure of carbon nanotubes
Tight-binding approximationDynamics of electrons near the Dirac-points
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
Imre Hagymási Carbon nanotubes and Graphene
History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes
Band structure of carbon nanotubes
Tight-binding approximationDynamics of electrons near the Dirac-points
Relativistic effects
Imre Hagymási Carbon nanotubes and Graphene
History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes
Band structure of carbon nanotubes
Properties of carbon nanotubes
Carbon nanotubes
Imre Hagymási Carbon nanotubes and Graphene
History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes
Band structure of carbon nanotubes
Properties of 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
History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes
Band structure of carbon nanotubes
Properties of carbon nanotubes
Two main types of carbon nanotubes
Figure: HRTEM images of a semiconducting and a metallic nanotube.
Imre Hagymási Carbon nanotubes and Graphene
History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes
Band structure of carbon nanotubes
Properties of carbon nanotubes
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.
Imre Hagymási Carbon nanotubes and Graphene
History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes
Band structure of carbon nanotubes
Properties of carbon nanotubes
Two main types of carbon nanotubes
Figure: Armchair nanotube. Figure: Zigzag nanotube.
Imre Hagymási Carbon nanotubes and Graphene
History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes
Band structure of carbon nanotubes
Properties of carbon nanotubes
Structure of Carbon Nanotubes
The unit cells of different nanotubes, a denotes the translationalperiod
Imre Hagymási Carbon nanotubes and Graphene
History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes
Band structure of carbon nanotubes
Properties of carbon nanotubes
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
Imre Hagymási Carbon nanotubes and Graphene
History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes
Band structure of carbon nanotubes
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
Imre Hagymási Carbon nanotubes and Graphene
History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes
Band structure of carbon nanotubes
Zone-folding approximationOutlook
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
History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes
Band structure of carbon nanotubes
Zone-folding approximationOutlook
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
Imre Hagymási Carbon nanotubes and Graphene
History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes
Band structure of carbon nanotubes
Zone-folding approximationOutlook
Band structure of carbon nanotubes IV.
Figure: Allowed k lines in the Brillouin zone of graphene.
Imre Hagymási Carbon nanotubes and Graphene
History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes
Band structure of carbon nanotubes
Zone-folding approximationOutlook
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
Imre Hagymási Carbon nanotubes and Graphene
History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes
Band structure of carbon nanotubes
Zone-folding approximationOutlook
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
Imre Hagymási Carbon nanotubes and Graphene
History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes
Band structure of carbon nanotubes
Zone-folding approximationOutlook
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
Imre Hagymási Carbon nanotubes and Graphene
History and discovery of Graphene and Carbon nanotubesBand structure of the GrapheneStructure of Carbon Nanotubes
Band structure of carbon nanotubes
Zone-folding approximationOutlook
Thank you for your attention!
Imre Hagymási Carbon nanotubes and Graphene