1. Carbon is a conductor (in most of its modifications).
2. There is a modification of C: Carbin is a linear polymer of carbon, it has semiconductor properties, and its conductivity increases greatly under the influence of light. The first practical application is based on this property - in photocells (see Wikipedia).
3. To include any material for 1)metals, metal alloys, intermetallic compounds (IMC), etc., or 2)semiconductors, a simple experiment is carried out: if, when heated, the material conducts current and increases resistance R linearly from temperature T (for both types of polarity, the sample is metal or IMC;
if, when heated, the material conducts current and reduces its resistance linearly / nonlinearly with temperature T (for both types of polarity to the sample) - this is a semiconductor. https://chem21.info/info/1498733/
This is a very sensitive inquiry. Firstly, there are lots of misconstrued/confusing meaning and nomenclatures of carbon nanomaterials especially the QDs and "dots" as the case may be. So what is the right term to denote carbon nanoparticles that exhibit the quantum the confinement phenomenon? Carbon dots or carbon quantum dots or graphene quantum dots? These materials should be characterized by the nature of the precursors used to prepared them. As you know, allotropes of carbon are amorphous carbon, graphite and diamond. It is only graphite which largely acts as a conductor, as opposed to diamond, which has a different crystal structure and acts as an insulator. So the question is, if carbon dots are prepared using materials that gives amorphous carbon (as it is usually the case), how can the CDs be conductive?
In summary, only CDs/CQDs/GQDs synthesized using graphitic sources may unarguably possess conducting properties.
Conductors and semiconductors are by definition macroscopic systems. Dots are more like atoms, 'large' atoms. As in the case of atoms, systems consisting of an assembly of dots can be conductors or semiconductors / insulators depending on the concentration of the dots. For sufficiently high concentration of the dots, the electronic wave functions centred on the dots overlap and atomic-like spectra of the dots transform into bands (recall that in the tight-binding approximation, the widths of the energy bands are proportional to the overlaps of the relevant dot wave functions).
Generally, carbon is a semiconductor material. It's has four bound electrons in the last orbital because of carbon atom is made covalent bond with others. Pure carbon is exactly semiconductor.
OK great! Semiconducting properties have been reported for CQDs derived from this source (CA). Also, doping with heretero atoms (N, Co, S etc) has influence on the band gaps of CQDs (see attached article).