We calculate the conductance variation of several metallic carbon nanotubes as their end is being dipped into a liquid metal electrode, where experiments have shown an achievable conductance close to 1 quantum of conductance. The calculated conductance for a (40,40) nanotube indicates that the current flows almost entirely through the pi mode, and not the pi* mode. The calculation also predicts that for narrower nanotubes (~1 nm in diameter), in a weak coupling regime, the saturation of both pi and pi* modes should be observable. An experiment is proposed to verify this point.
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