Single-Electron Phenomena in Electrochemistry of Biosensors

??In this work, we modelled Coulomb blockade and Coulomb staircase phenomena starting from first-principles and taking advantage of Schrödinger-like problems knowledge. ??Coulomb blockade and staircase phenomena are physical effects related to the quantization of levels in a nanoparticle. In the past, experimental results concerning these phenomena have been modelled with either semiclassical or circuit-like models, which derive electrical parameters such as capacitances and resistances in order to reproduce the measured currents trends. ???? ??Our approach blossoms in the framework of quantum mechanics. As a first step, we studied a simplified model, which is often used to describe a similar quantum structure, in which discretized energy levels exist in the region enclosed between two one-dimensional potential energy barriers. In this model, when electrons are forced to drift across the system under the effect of a bias voltage, they are able to tunnel across the first energy barrier to move to one of the available energy levels, eventually leaving the system by tunneling across the other barrier. ??When dealing with coherent electron transport collisions are not phase-breaking and quantization of the available conducting channels emerges either as a staircase-like or an oscillating resonant current-voltage characteristics, depending on the nanoparticle size. ???? ??Literature provides analytic expressions for quantum tunneling across one, two or multiple barriers when no external bias is applied to the system. However, as our goal was to model the effect of bias voltage we were forced to revert to a numerical approach. To this purpose, we implemented the solution of the out-of-equilibrium Schrödinger equation using Finite Element Method (FEM) and Non-Equilibrium Green's Functions (NEGF) approaches. ???? ??The FEM solution, although computationally less expensive than the NEGF algorithm, proves to be viable only at equilibrium and small-bias conditions, while the NEGF solution provides reliable results over a wider bias voltage range. Hence, we focused on a NEGF approach. The obtained results have been analyzed in order to identify the role that each geometrical parameter plays in determining the current-voltage curve shape. ???? ??Once the first step completed, as our target was to apply the model to the simulation and design of nanoparticle-enhanced electrochemical biosensors, we modified the geometrical structure in order to emulate a nanoparticle with an attached molecule. Now, quantum tunneling can only occur when an energy level within the nanoparticle sits at an energy close enough to that of an unoccupied energy level of the molecule, or vice versa. Current-voltage characteristics obtained for these kind of systems have been found to be very similar both in shape and order of magnitude to those available in literature. ???? ??As the problem to tackle is rather challenging, the present work, although leading to satisfactory results, represents a first critical step in the direction of a comprehensive tool for the design of nanoparticle-enhanced electrochemical biosensors. Improvements can be made in optimizing the solving algorithm, making it more efficient and less time-consuming, as well as including other relevant physical phenomena, thus obtaining a more complete description of the electron transfer. An example could be the hopping phenomenon, which plays a role most probably as relevant as tunneling, but for incoherent electron transport.