The intimate contact between the electrons in a carbon nanotube and solvated ions in a liquid electrolyte gives rise to a unique electrostatic situation not seen in previous electrochemical cells: electric fields are confined to dimensions comparable to the radius of a solvated ion, and a sharply varying quantum density of states of the electrons directly affects the electrostatics. The corresponding capacitance, in this case, consists of two main types of capacitance in series: a quantum component arising from the electronic density of states and an electrochemical component arising from the ion screening and diffusion.
To quantitatively study these capacitances, in Chapter 2, we start with measuring the ensemble average, complex, frequency dependent impedance between a purified semiconducting nanotube network and an aqueous electrolyte at different ionic concentrations. 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 nanotube network and the electrolyte. In Chapter 3, we push the measurement technique to resolve the liquid-gate capacitance down to a single nanotube level. Due to the atomic dimensions, the small capacitance (~ 100 aF) has been a challenge for researchers to measure and observe directly. In order to resolve the small capacitance (of order 100 aF) above the background stray capacitance (of order 100 pF), we designed, developed, and implemented an integrated, on-chip shield. With this system, we measure the capacitance of one to a few nanotubes quantitatively as a function of both bias potential and ionic concentration at room temperature. The ionic strength dependence of the capacitance is expected to have a root cause from the electrochemical capacitance, which we model using a modified Poisson Boltzmann equation. The relative contributions from those two capacitances can be quantitatively decoupled.
So far, the capacitance measurements are confined in DC-KHz domain, where most Nano-biosensors operate. However, ionic screening effect due to mobile ions prevents target-sensing beyond Debye length. In Chapter 4, we demonstrate capacitance measurement/mapping in GHz domain using scanning microwave microscopy (SMM) and use it to image vital mitochondria in respiration buffer. The SMM is combined with an interferometric and tuned reflectometer to optimize the sensitivity even in an electrophysiologically relevant liquid (hence conducting) environment.