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Bulk carbon is an insulator, but at nanoscale (e.g. carbon nanotubes) carbon may become a good electrical conductor. If I am correct this means carbon nanotube band structure is metal-like. E.g. it has partly filled valence band (like in the rightmost case in the figure below).

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At nanoscale materials are influenced by quantum confinement. Because of quantum confinement semiconductor nanoparticle has higher band gap and discrete energy levels compared to bulk semiconductor (see the figure about quantum confinement below). Quantum confinement phenomenon in nanoparticle

I am assuming that quantum confinement phenomenon also occurs for carbon at nanoscale. Then, how certain carbon nanotubes can show good electrical conductivity?

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I am sorry to say, but your question contains several false assumptions, most importantly:

  • Nanotubes are good conductors:

    No, not all of them. Certain types have metallic/semi-metallic conductance, others are semiconductors. Single wall carbon nanotubes are generally indexed with a so-called chirality index (m,n) that tells you how the carbon sheet is wrapped up into a tube. See wikipage.

  • Nanotubes behave exactly like 3D nanoparticles in terms of band structure:

    No, nanosheets (graphene) and nanotubes behave in different ways and your description of quantum confinements don't work in that way in 1D infinite (nanotubes) and 2D infinite (graphene) cases.

    You can derive an approximate band structure of graphene/nanotubes using tight-binding models, which in most cases predicts correctly, whether the structure is metallic or semiconducting. See simulation of nanotube band structure.

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