FDTD Modelling of Nanostructures at Microwave Frequency.

Abstract

The thesis which is hereby presented describes a study of the numerical modelling of the coupled interaction of nanostructures with electromagnetic fields in the range of microwaves. This is a very ambitious task and requires a thorough and rigorous implementation of new algorithms designed to this purpose. The first issue to be encountered is the characterisation and the physical understanding of the behaviour of a nanostructure. The term itself, nanostructure, defines any device which has a nanometric size in at least one dimension, regardless of its material and geometry, hence it is a very wide definition. Carbon Nanotubes (CNT), quantum dots and quantum wells fall into this category, for example, and in electronics these structures are generally composed of semiconductor materials, like Silicon or Gallium Arsenide. The first step to take, in order to model such objects from an electronics point of view, is to solve the Schrodinger equation. The Schrodinger equation is a very general formula, widely used in quantum physics, which, when provided with a certain electrical potential in a material, determines the behaviour of the electrons in this material. Needless to say, the electrical potential is the DNA of a material or, in other words, it is the physical property which affects the propagation of electrons and therefore makes a material conducting or non-conducting. Nanostructures are often composed of several materials, hence the potential is not constant and, with opportune geometries, it is possible, in principle, to guide the electron currents through the device, as, for example, a channel in a MOSFET. This principle holds for very small structures where the electron transport can be considered ballistic, i.e. when the structures are smaller than the free mean path of the particle. The behaviour of the electrons is affected both by external factors, such as temperature or applied electric and magnetic fields, and internal factors, such as the electron mobility or the doping concentration, which are dependent on the used materials. This parameters play a very important role whilst modelling the behaviour of particles such as electrons and in this work the main focus is the study of the impact of external electromagnetic fields. The electromagnetic fields (EM fields) are composed of an electric field component and of a magnetic field component, which can be analysed separately in order to better understand the response of nanostructures to their application. A rigorous analysis is presented by showing numerical results, obtained with the modelling of the Schrodinger equation, compared with the expected theoretical results, exploiting simple structures, where it is possible to calculate the solutions analytically. The second part of thesis focuses on the impact of the EM fields on the nanostructure, hence the combined effect of both electric and magnetic fields affecting the electrons' propagation, and the mutual coupling of the fields with the quantum effects. Indeed the study of nanodevices for microwave applications requires to consider the contribution of a parameter called quantum current density, which accounts for the quantum effects generated by the structure. This is normally ignored in conventional devices because the quantum contributions are negligible but, by using opportune materials and opportune geometries, these currents become relevant and they may have an impact on the propagation of the EM fields. For this reason a consistent part of the thesis is dedicated to investigate the mutual coupling between EM fields and quantum effects, by implementing the Maxwell-Schrodinger coupled model. A chapter is dedicated to the novel approaches taken in order to tackle the issues and the limits of the numerical implementation; in particular two solutions are presented, nonuniform domains and the parallelisation of the algorithm. These approaches are vital whilst modelling numerically such physical problems since the required computational capacity increases with the accuracy requirements. Solving the presented algorithms conventionally would limit the potential of the method and thus a thorough study has been made in order to improve the efficiency of the simulations. In the last chapter, three different scenarios are presented, each one of them showing different features of the coupled model. The results are illustrated and discussed, including the limits due to the chosen approximations. References to the analytical solutions are provided in order to validate the obtained numerical results.