Measurement of the quantum capacitance of interacting electrons in carbon nanotubes

The electronic capacitance of a one-dimensional system such as a carbon nanotube is a thermodynamic quantity that contains fundamental information about the ground state(1). It is composed of an electrostatic component describing the interactions between electrons and their correlations, and a kinetic term given by the electronic density of states. Here, we use a field-effect transistor geometry to obtain the first direct capacitance measurement of individual carbon nanotubes, as a function of the carrier density. Our measurements detect the electrostatic part of the capacitance as well as the quantum corrections arising from the electronic density of states. We identify the van-Hove singularities that correspond to the one-dimensional electron and hole sub-bands and show that the measured capacitance exhibits clear electron - hole symmetry. Finally, our measurements suggest the existence of a negative capacitance, which has recently been predicted to exist in one dimension as a result of interactions between electrons(2-4).