Correlation effects in the capacitance of a gated carbon nanotube

For a capacitor made of a semiconducting carbon nanotube (CNT) suspended above a metallic gate, Coulomb correlations between individual electrons can lead to a capacitance that is much larger than the geometric capacitance. We argue that when the average spacing $n^{-1}$ between electrons within the low density 1-dimensional electron gas (1DEG) in the CNT is larger than the physical separation $d$ between the CNT and the gate, the enhancement of capacitance is expected to be big. A recent Coulomb blockade experiment[1], however, has observed no obvious increase of capacitance even at very low electron density. We show that this smaller capacitance can be understood as the result of the confining potential produced by the potential difference between the source/drain electrodes and the gate, which compresses the 1DEG when the electron number decreases. We suggest that by profiling the potential with the help of multiple split gates, one can return to the case of a uniform 1DEG with anomalously large capacitance.