Modelling of MEMS/NEMS resonators and functional design of a mechanical transistor

The most basic example of charge transport is represented by the collective motion of free charge carriers in a conductive medium. In the context of electromechanical devices, more complex forms of charge transport, labeled with the umbrella term electron shuttle, exist. The simplest system exhibiting a shuttle mechanism comprises a set of three conductors: two fixed electrodes and a vibrating element between them. Under certain boundary conditions a limit-cycle is established, and the oscillator alternatively takes and releases a finite amount of electrons while approaching the electrodes. Since this form of charge transport relies on the presence of mechanical vibrations, the described phenomenon constitutes the archetype of a “mechanical charge carrier”. Although the first concept of electron shuttle is over two centuries old, only in recent times [Gorelik and Isacsson, 1998] it found reinvigorated interest in the field of nanotechnology, producing a novel branch of both theoretical and experimental research. In fact, the introduction of quantum effects in shuttle devices produces interesting motion regimes peculiar of nanoscale. Refer to such systems as Quantum Shuttle Modules (QSMs). Many original architectures have been conceptualized and realized in the last decade. A promising application is contained in a patent [Blick and Marsland, 2008] which proposes a switching element based on electron shuttle and capable of reproducing the main functionalities of a conventional transistor, depicted as composed by an array of mechanically coupled QSM subsystems, whose vibrating elements are nanocantilevers. Refer to this invention as the NanoMechanical Transistor (NMT). At the present day, the NMT is an unexplored concept: no experimental setup has been realized nor theoretical model has been proposed yet. The research work contained in this thesis is intended to provide a first theoretical description and an early design stage for such NMT device. Notice that, since the NMT is composed by a set of QSMs, a preliminary study of these systems is needed. This work is consequently divided in two Parts: the first one focuses on the QSMs with an exquisitely theoretical approach, while the second one is intended to assess the feasibility of a real application, the NMT. A brief description of the thesis contents is presented ahead. Part One begins by introducing the fundamental concepts of quantum tunneling and Coulomb blockade. Then, the general scheme of a QSM is presented. As usual in literature, a concentrated parameters model is used, and the state of the system is reduced to a couple of lagrangian descriptors: the oscillator position and charge. Conductive and dynamical properties are investigated with both analytical and numerical approaches. Then, a systematic study of QSMs is attempted by considering two basic shuttle mechanisms: in the first case, the oscillator is self-excited by a shuttle current between electrodes at different voltages; in the second case, the oscillator vibrates under parametric resonance and a shuttle current is established between two electrodes at the same voltage. The portrayed phenomenologies are complementary, meaning each QSM presents a combination of these two fundamental forms of electron shuttle. Part Two opens with an overview of the conventional transistors and an analysis of the NMT patent. Follows the choice of the characteristic scales of the device and the typology of QSM to be the best candidate as NMT subsystem. Under suitable approximations, closed-form formulae for capacitances, quantum tunneling and electrostatic force are produced. A flexible set of equations is obtained, allowing to perform a large number of numerical experiments. First, a single QSM subsystem is considered: a predictive model is proposed in which a QSM is related to a Turing machine whose admitted states are represented by the feasible motion regimes. Then, more QSM subsystems are arranged to realize the NMT: each module is electrostatically independent but mechanically coupled with its nearest neighbors. A functional analysis of the whole system is presented, in which peculiar motion regimes are investigated, and a set of design and control strategies aimed to correctly reproduce switching and amplification functionalities is proposed. Last, the black-box electrical characterization of the device is outlined. In conclusion, the main original contributions of this research work are: i) the theoretical study of a novel device – the NMT – which has led to synthetize a series of design requirements and control strategies; II) the conceptualization of a parametric resonant QSM, which – differently from the self-excited one – constitutes a new archetype of electron shuttle.