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Carbon nanotubes (CNTs) are thin, hollow cylinders, which can be envisioned as being rolled up from graphene a two-dimensional honeycomb lattice with a carbon atom in each site. Multiwall carbon nanotubes were discovered by Japanese scientist Sumio Iijima in 1991 (Iijima, 1991) and, two years later, individual single wall carbon nanotubes (SWCNTs) were reported (Iijima & Ichihashi, 1993). Their diameter is as little as 1 nanometer. Carbon nanotubes provide the ultimate limit formicroelectronicminiaturization. Soon after discovery carbon nanotubes attracted tremendous interest from fundamental science and technological perspectives and thousands of papers have been published on this subject (for a review see e.g. (Dresselhaus et al., 2001)). Amazing aremechanical properties of CNTs e.g. their high strength and high flexibility. Since the C-C bonds within CNTs are one of the strongest bonds in the nature these tubes are predicted to be by far the strongest fibres that can be made. SWCNT was tested to have tensile strength of order of 60 GPa (Wei, 2003). Nanotubes exhibit also exceptional electrical properties, some of them will be discussed in this chapter. The unique electrical properties of CNTs stem fromunusual electronic properties of graphene (Novoselov, 2005; Wallace, 1947). The sp2 hybridization between one s-orbital and two p-orbitals leads to a trigonal planar structure with a formation of a σ-bond between carbon atoms what is responsible for robustness of the lattice in all carbon allotropes. The unaffected p-orbital, which is perpendicular to the planar structure binds covalently with neighboring carbon atoms leading to the formation of half filled π-bands. The low energy band structure of graphene has two bands that join at the corners of the Brillouin zone (two distinct Fermi points at K and K′, called Dirac points (Fig. 1a)). The name of these points reflects the fact that the energy bands disperse linearly away from the touching points, so the dispersion relation for electrons (holes) is described by an isotropic cone that opens upward (downward) near K, K′ points (massless fermions). Graphene is a gapless semiconductor. This is changed when rolling up a piece of graphene to form a carbon nanotube. From the bandstructure of graphene one can obtain the bandstructure of nanotube by imposing appropriate boundary conditions along the circumference. Quantization of momentum in the circumferential direction is given by condition πd · k⊥ = 2πi (i = 1, 2...). Spacing in k⊥ is thus Δk⊥ = 2/d, where d is the diameter of the tube. The quantization cuts discrete slices of two-dimensional Dirac dispersion of graphene. Each line corresponds to a 1D subband for conduction along the nanotube (Fig. 1b, c). Depending on the way graphene is rolled up, carbon nanotube can be either metallic or semiconducting. The former occurs if a slice happens to pass through 13

Spin Dependent Transport through a Carbon Nanotube Quantum Dot in the Kondo Regime