Carbon nanotubes and Graphene

Transcription

1 16 October, 2008 Solid State Physics Seminar

2 Main points 1 History and discovery of Graphene and Carbon nanotubes 2 Tight-binding approximation Dynamics of electrons near the Dirac-points 3 Properties of carbon nanotubes 4 Zone-folding approximation Outlook

3 History and experimental discovery History of Graphene Wallace (1947): The band theory of graphite graphene McCure (1956): electrons can be described as Dirac-fermions with zero mass Geim s research group (Manchaster, 2004): first experimental observation 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 unambigous experiment)

4 Structure of the Graphene Tight-binding approximation Dynamics of electrons near the Dirac-points A a 2 B a 1 Figure: The honeycomb structure of the Graphene

5 I. Tight-binding approximation Dynamics of electrons near the Dirac-points the Bloch-function in tight-binding approximation: ψ k (r) = 1 e ikr [C A (k)ϕ A (r R) + C B (k)ϕ B (r R d)] N R R = n 1 a 1 + n 2 a 2, d = a 1+a 2 3 taking into account the first neighbours, the dispersion relation: E(k) = ǫ 0 + γ coska 1 + 2coska 2 + 2cosk(a 1 a 2 ) ǫ 0 onsite energy, γ 0 = ϕ A (r)hϕ B(r d)d 3 r, a hopping integral

6 II. Tight-binding approximation Dynamics of electrons near the Dirac-points Ε k ky 0 kx 5 5 Figure: The valence and the conduction band. Figure: Band structure s contour plot

7 III. Tight-binding approximation Dynamics of electrons near the Dirac-points Behaviour near the K points the conduction and the valence band form conically shaped valleys 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 = v F δk, δk = k K, v F 10 6m s special theory of relativity

8 Conical structure at the Dirac-points Tight-binding approximation Dynamics of electrons near the Dirac-points Figure: Dispersion relation near the Dirac-points

9 Hamiltonian of the graphene Tight-binding approximation Dynamics of electrons near the Dirac-points electrons near the Dirac-points can be treated as masless excitations, governed by a Dirac-Hamiltonian: ( σx H graphene = i v x + σ y y 0 0 σ x x σ y y ) Dirac-fermions in 2D the two subblocks can be transformed to each other by a unitary transformation valley degeneration

10 Relativistic effects in graphene Tight-binding approximation Dynamics of electrons near the Dirac-points Zitterbewegung position operator has an oscillating part beside the motion with a constant velocity Klein paradox (if V 0 2mc 2 T 1), it is hard to point out, E > V/cm in graphene E > 10 5 V/cm is enough to observe the phenomenon electron scattering on a potential step V 0 (p n junction) graphene has a negative refractive index! sin α sin β =: n = E V 0 E n can be tuned by varying the gate voltage electron lenses

11 Relativistic effects Tight-binding approximation Dynamics of electrons near the Dirac-points

12 Properties of carbon nanotubes Carbon nanotubes

13 What are the carbon nanotubes? Properties of carbon nanotubes Basic properties hollow cylinders of graphite sheets single-walled nanotube a tube consisting of several concentrical cylinders multiwall nanotube (MWNT) nm diameter, µm length quasi 1D crystals nanotubes are metallic or semiconducting properties of the nanotubes depend crucially on the way they are rolled up Synthesis of single-walled nanotubes laser ablation high-pressure carbon-monoxide conversion arc-discharge

14 Two main types of carbon nanotubes Properties of carbon nanotubes Figure: HRTEM images of a semiconducting and a metallic nanotube.

15 Structure of the Carbon Nanotubes Properties of carbon nanotubes Definitions: chiral vector c = n 1 a 1 + n 2 a 2 usually denoted by (n 1,n 2 ) T: tube axis, the minimal lattice vector c diameter: d = c π = a 0 π n n 1n 2 + n 2 2 Figure: Making single-walled nanotube of a single graphite layer.

16 Two main types of carbon nanotubes Properties of carbon nanotubes Figure: Armchair nanotube. Figure: Zigzag nanotube.

17 Properties of carbon nanotubes The unit cells of different nanotubes, a denotes the translational period

18 Properties of carbon nanotubes Mechanic and electric properties of carbon nanotubes Mechanic properties material Young s Modulus (TPa) Tensile Strength (GPa) SWNT Armchair SWNT Zig-zag SWNT MWNT Stainless Steel Electric properties Depending on the (n 1,n 2 ) vector, a nanotube is metallic if 3 (n 1 n 2 ) semiconducting otherwise

19 Zone-folding approximation Outlook I. the tube is infinitely long k z 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 d the allowed k vectors (k 1,k 2 are the reciprocal lattice vectors of graphene): k = 2n 1 + n 2 qnr k 1 + 2n 2 + n 1 qnr k 2 m = q 2 + 1,...,0,1,..., q 2, n = GCD(n 1,n 2 ) q: the number of hexagonal cells in the nanotube unit cell R = 3 if (n 1 n 2 )/3n is an integer, R = 1 otherwise

20 Zone-folding approximation Outlook II. first approximation (zone folding): electronic properties of carbon nanotube can be obtained by cutting the band structure of graphene Figure: Brillouin zone of a (7,7) armchair and a (13,0) zig-zag tube.

21 Zone-folding approximation Outlook 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 allowed states the nanotube is metallic the K point of graphene is at 1 3 (k 1 k 2 ) K point is allowed if K c = 2πm = 1 3 (k 1 k 2 )(n 1 a 1 + n 2 a 2 ) = 2π 3 (n 1 n 2 ),m Z 3m = n 1 n 2

22 Zone-folding approximation Outlook IV. Figure: Allowed k lines in the Brillouin zone of graphene.

23 Beyond the Zone-folding approach Zone-folding approximation Outlook Curvature effects C-C distance for atoms with different ϑ azimuthal angle is reduced angles of the hexagons are not 60ďż Fermi-point moves away secondary gaps appear in zig-zag nanotubes

24 Conclusion, outlook Zone-folding approximation Outlook Carbon nanotubes, conclusion experimental results are in good agreement with the zone-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

25 Zone-folding approximation Outlook Outlook: specular Andreev reflection (2005) in an N-S system if E <, the electron can not enter the superconducting region Andreev-retro-reflection in graphene both of them can occur

26 Zone-folding approximation Outlook Thank you for your attention!