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