We present a comprehensive study of the electrochemical capacitance between a one-dimensional electronic material and an electrolyte. In contrast to a conventional, planar electrode, the nanoscale dimension of the electrode (with diameter smaller than the Debye length and approaching the size of the ions in solution) qualitatively changes the capacitance, which we measure and model herein. Furthermore, the finite density of states in these low dimensional electronic systems results in a quantum capacitance, which is comparable to the electrochemical capacitance. Using electrochemical impedance spectroscopy (EIS), we measure the ensemble average, complex, frequency dependent impedance (from 0.1 Hz to 1 MHz) between a purified (99.9%) semiconducting nanotube network and an aqueous electrolyte (KCl) at different concentrations between 10 mM and 1 M. The potential dependence of the capacitance is convoluted with the potential dependence of the in-plane conductance of the nanotube network, which we model using a transmission-line model to account for the frequency dependent in-plane impedance as well as the total interfacial impedance between the network and the electrolyte. The ionic strength dependence of the capacitance is expected to have a root cause from the double layer capacitance, which we model using a modified Poisson-Boltzmann equation. The relative contributions from those two capacitances can be quantitatively decoupled. We find a total capacitance per tube of 0.67-1.13 fF/μm according to liquid gate potential varying from -0.5 to -0.7 V.
Keywords: Gouy−Chapman-Stern model; carbon nanotubes; double layer; electrochemical impedance spectroscopy; modified Poisson−Boltzmann equation; quantum capacitance.