  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.
