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I have been searching around for a while for this but I am having trouble finding any actual figures, all I can seem to find is that it is "very high".

So I am wondering, does anyone have any figures of what the electrical conductivity of a carbon nanotube is, a theoretical or estimated answer is fine. I am preferably looking for the answer in $Sm^{-1}$.

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The numbers will greatly vary depending on the kind of nanotube. The following are some examples from cursory Google searches.

Electrical conductivity was increased by 50 percent to 1,230 siemens per meter.

http://news.ncsu.edu/releases/wms-zhu-cnt-composites/

And that’s not all: colossal carbon tubes are ductile and can be stretched, which makes them attractive for applications requiring high toughness. They also have high electrical conductivities of around 103 siemens per centimetre at room temperature, compared with 102 siemens per centimetre for multi-walled carbon nanotube fibres.

http://physicsworld.com/cws/article/news/2008/aug/08/carbon-nanotubes-but-without-the-nano

The researchers found that the electrical conductivity increased with increasing nanotube content and temperature – in contrast to earlier findings. They observed a maximum conductivity of 3375 siemens per metre at 77°C in samples that were 15% nanotube by volume.

http://physicsworld.com/cws/article/news/2003/aug/20/nanotubes-boost-ceramic-performance

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The conductivity of a nanotube depends on the chirality / diameter of the individual nanotube and the conductivity. The chirality determines whether a nanotube is classified as either metallic or semiconducting.

The highest conductivity that has been measured for a carbon nanotube is ~100 MS/m, which is equivalent to a resistivity of 1 uOhm-cm. This result was reported by McCuen and Park (2004) for a metallic carbon nanotube.

https://doi.org/10.1557/mrs2004.79

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