Radio frequency microelectromechanical systems (RF MEMS) have just entered a new and exciting era, with this previously elusive technology finally appearing on the open market. In 2008, in the United States and Japan, the first real devices were released and made commercially available to all. Today, there is intense research and development (R&D) activity, and at all levels from concept to manufacture, within North America, Europe and Asia.

The first book to be dedicated to RF MEMS was published in 2002, and two others soon followed in 2003. Within these books, the most recent references to be cited were papers published back in January 2003. Therefore, the motivation for another book on the subject is clear. At this point, I would like to pay homage to the groundbreaking book entitled RF MEMS: Theory, Design and Technology, by Gabriel M. Rebeiz. Indeed, all these books collectively represent a major literary milestone in RF MEMS and could be considered as a springboard for the later activities that led to the first commercially available devices.

Introduction

The focus of R&D activities in RF microsystems has been mostly at component and circuit levels. Component and circuit performances have tried to improve those of their semiconductor counterparts. This work has been successful in many cases. From the systems level point of view, RF MEMS technology offers improvements in tuning range, power linearity, insertion loss and scalability compared with other technologies. Tuning range relates to high-capacitance ratio switches, rather than continuous tuneability; a limitation that still exists with RF MEMS varactors. However, systems level improvements and new architectures can take full advantage of RF MEMS technology. Proposals for new architectures are continually being presented, but limited demonstrations have been reported to date.

For many applications, more and new functionalities are being sought by radio front-ends, to meet size and cost downscaling pressures. For example, several radios can be integrated into handsets and software-defined radio (SDR) concepts have been implemented in several applications. RF MEMS technology can be one of the enabling technologies to allow these new concepts to be implemented. The use of RF MEMS technology can offer the largest benefits when there are no directly competing technologies and in-high performance applications.

Introduction

The past five years have seen a dramatic rise in the number of niche switch technologies being reported in the open literature. These include, but are not limited to, the general areas of latching, multiway and high-power. These niche technologies are covered within the same dedicated chapter, because of the synergy found between them. It should be of no surprise that there is a large overlap between these technologies; with latches being used in both multiway and high-power switches and the need for implementing high-power multiway switches.

Latching switches

For some applications, switches are required that are non-volatile and able to remain in any state after the source of actuation energy has been removed. For example, in certain safety-critical applications it is essential for the state of the system to be preserved in the event of a power failure. In other systems, for example, in portable electronic devices, remote sensors and unmanned aerial (or air or aircraft) vehicles (UAVs) or systems (UASs), the available power is limited and non-volatile switches are attractive because they require zero holding power, regardless of their state.

Introduction

Over the past ten years, a large number of RF MEMS switches have been developed by use of various fabrication technologies. When compared with bulk acoustic wave (BAW) filters, which are fabricated in millions of units and by tens of companies, the commercialisation of RF MEMS switches and varactors is just beginning. First, it is a very complex challenge, from a technological point of view. Like inertial sensors, RF MEMS switches are based on fragile movable microstructures. Instead of polysilicon, highly conductive metals (e.g. gold (Au), aluminium (Al) and copper (Cu)) must be used and these result in a decrease in thermal stability. Therefore, common MEMS packaging techniques performed at 400 to 500 °C cannot be applied. Also, RF MEMS switches must be packaged hermetically for them to be protected from environmental influences. In addition, device reliability is still problematic, mainly because surfaces come into contact for billions of switching cycles. Under these circumstances, it is easy to imagine that a successful development presupposes quite significant scientific, engineering and financial resources. Second, the numerous conflicts between the following are easily underestimated: concept and operation, RF design and signal routing and fabrication and packaging.

The choice of switch principle, as well as appropriate fabrication technology, strongly depends on the particular intended application. Several application areas for RF MEMS switches are named in the report of Wicht Technologie Consulting (WTC) [1]. For ATE applications, ohmic (i.e. metal contact) RF MEMS switches are chosen from the outset because of their excellent performance over a frequency that can range from dc to many tens of gigahertz. The possibility of a direct replacement of solid-state switches by RF MEMS devices is another important advantage. It can be expected that ohmic switches will dominate in telecommunications switching matrices because of their low frequencies of operation. Phased-array antennas, probably the main application for aerospace and defence, could be based on both ohmic and capacitive switches. In total, it is predicted that this area of the market could be worth more than $150 million in 2011. These products are high-end systems with moderate cost sensitivity. The crucial factor here is the performance improvement.

Introduction

Electrostatic actuation is seemingly the most prevalent method of applying forces to deformable elements in microsystems. Two factors make electrostatic actuation appealing in systems with micron-scale dimensions. First, it is associated with the physics of the system, when practical voltages (e.g. –100 V) are applied across micron-scale gaps; the resulting electrostatic fields are sufficiently high to deform elements which, in turn, are sufficiently flexible due to their micron-scale thickness. Second, electrostatic actuation is perfectly compatible with microfabrication technology used for fabricating ICs. The same fabrication processes may be used to produce isolated conducting elements that are supported by flexible suspensions.

Electrostatic actuation was introduced over four decades ago [1, 2], and has found many applications in actuation and sensing [3, 4]. Electrostatic forces are inversely proportional to the square of the distance between the electrodes and are, therefore, inherently non-linear [5, 6]. Because of this non-linearity, the electromechanical response of electrostatic actuators may become unstable. This instability, known as the pull-in phenomenon, is an unwarranted effect in applications where a large controllable dynamic range is wanted. In such cases, special designs (e.g. comb-drives [7]) or special operation modes (e.g. charge actuation [8]) may alleviate the difficulty. In other applications, the pull-in instability may be utilised to achieve a fast transition between two states of an electromechanical switch [9]. Electromechanical switches have many uses in optical MEMS [10, 11], RF MEMS [12, 13], nanoelectronics [1, 14] and nanologic [15, 16].

Introduction

This chapter aims at provide a medium-term perspective for the future developments of RF MEMS technology up to 2020. This work is essentially based on the final report delivered by the EU-funded FP6 project Applied Research Roadmap for RF Micro/nano Systems Opportunities (ARRRO), a Coordinated Action whose aim was “to assess the current status and develop a vision of the future requirements for RF MEMS and RF NEMS technology, products and applications”. The roadmap described here is uniquely based on approximately 80 interviews with industrial companies (either developing or assessing the potential use of RF MEMS). Unlike other studies, which are based on data gained from the open literature, the findings presented here paint a more realistic future roadmap, from an industrial perspective.

Unlike ARRRO's report, which essentially refers to RF MEMS in the broad sense of RF microsystems technology (MST), only RF MEMS devices with movable parts are considered here. Having said this, reference is occasionally made to fixed micromachined devices (e.g. inductors), BAW and micromechanical resonators. With this in mind, a comprehensive roadmap in graphical form for RF MST components is shown in Fig. 13.1.

This chapter is an introduction to the semiclassical approach for the Helmholtz equation in complex systems originating in the field of quantum chaos. A particular emphasis will be made on the applications of trace formulae in paradigmatic wave cavities known as wave billiards. Its connection with random matrix theory (RMT) and disordered scattering systems will be illustrated through spectral statistics.

Introduction

The study of wave propagation in complicated structures can be achieved in the high-frequency (or small-wavelength) limit by considering the dynamics of rays. The complexity of wave media can be due either to the presence of inhomogeneities (scattering centers) of the wave velocity or to the geometry of boundaries enclosing a homogeneous medium. It is the latter case that was originally addressed by the field of quantum chaos to describe solutions of the Schrödinger equation when the classical limit displays chaos. The Helmholtz equation is the strict formal analog of the Schrödinger equation for electromagnetic or acoustic waves, the geometrical limit of rays being equivalent to the classical limit of particle motion. To qualify this context, the new expression wave chaos has naturally emerged. Accordingly, billiards have become geometrical paradigms of wave cavities.

In this chapter we will particularly discuss how the global knowledge about ray dynamics in a chaotic billiard may be used to explain universal statistical features of the corresponding wave cavity, concerning spatial wave patterns of modes, as well as frequency spectra.

Introduction

For a two-dimensional enclosure, such as a membrane or the cross section of an infinitely long duct, those with the very simplest shapes (circles, rectangles, spheres, boxes, etc.) with simple uniform boundary conditions, the modes and natural frequencies can be determined analytically. For any other shape they may be determined numerically by a range of mature numerical techniques of which finite element and boundary element analyses are the best known and the most widely studied. Knowing how to calculate the modes and natural frequencies for any particular shape, however, is not the same as understanding how those modes and natural frequencies depend on the shape. Suppose, for example, that we wish to improve the design of a component by optimizing some quantity such as weight, while leaving its natural frequencies unchanged. In the course of such an optimization changes will be made to the shape, whereupon the process of calculating the modes and natural frequencies must begin all over again; at best, part of the mesh can be re-used. Such an analysis cannot tell us where effort can be most or least profitably concentrated.

It turns out that the shapes that can be analyzed are (for good reason) quite untypical compared with arbitrary shapes. The situation mirrors the one that used to prevail in the study of dynamical systems, where linear differential equations were most widely studied because of their solubility, and the fact that other systems showed radically different qualitative behavior was, for a time, ignored.

Vector spaces are essentially linear structures. They are so ubiquitous in pure and applied mathematics that their review in a book such as this is fully warranted. In addition to their obvious service as the repository of quantities, such as forces and velocities, that historically gave birth to the notion of a vector, vector spaces appear in many other, sometimes unexpected, contexts. Affine spaces are often described as vector spaces without a zero, an imprecise description which, nevertheless, conveys the idea that, once an origin is chosen arbitrarily, the affine space can be regarded as a vector space. The most important example for us is the Galilean space of Classical Mechanics and, if nothing else, this example would be justification enough to devote some attention to affine spaces.

Vector Spaces: Definition and Examples

One of the most creative ideas of modern mathematics (starting from the end of the nineteenth century) has probably been to consider particularly important examples and, divesting them of as much of their particularity as permitted without loss of essence, to abstract the remaining structure and elevate it to a general category in its own right. Once created and named, the essential commonality between particular instances of this structure, within what may appear to be completely unrelated fields, comes to light and can be used to great advantage.

The short periodic orbit (PO) approach was developed in order to understand the structure of stationary states of quantum autonomous Hamiltonian systems corresponding to a classical chaotic Hamiltonian. In this chapter, we will describe the method for the case of a two-dimensional chaotic billiard where the Schrödinger equation reduces to the Helmholtz equation; then, it can directly be applied to evaluate the acoustic eigenfunctions of a two-dimensional cavity. This method consists of the short-wavelength construction of a basis of wavefunctions related to unstable short POs of the billiard, and the evaluation of matrix elements of the Laplacian in order to specify the eigenfunctions.

Introduction

The theoretical study of wave phenomena in systems with irregular motion received a big impetus after the works by Gutzwiller (summarized in Gutzwiller 1990). He derived a semiclassical approach providing the energy spectrum of a classically chaotic Hamiltonian system as a function of its POs. This formalism is very efficient for the evaluation of mean properties of eigenvalues and eigenfunctions (Berry 1985, Bogomolny 1988), but it suffers from a very serious limitation when a description of individual eigenfunctions is required: the number of used POs proliferates exponentially with the complexity of the eigenfunction. In this way, the approach loses two of the common advantages of asymptotic techniques: simplicity in the calculation and, more important, simplicity in the interpretation of the results.

Based on numerical experiments in the Bunimovich stadium billiard (Vergini & Wisniacki 1998), we have derived a short PO approach (Vergini 2000), which was successfully verified in the stadium billiard (Vergini & Carlo 2000): the first 25 eigenfunctions were computed by using five periodic orbits.

Having already established our physical motivation, we will proceed to provide a precise mathematical counterpart of the idea of a continuum. Each of the physical concepts of Continuum Mechanics, such as those covered in Appendix A (configuration, deformation gradient, differentiable fields on the body, and so on), will find its natural geometrical setting starting with the treatment in this chapter. Eventually, new physical ideas, not covered in our Continuum Mechanics primer, will arise naturally from the geometric context and will be discussed as they arise.

Introduction

The Greek historian Herodotus, who lived and wrote in the 5th century bce, relates that the need to reconstruct the demarcations between plots of land periodically flooded by the Nile was one of the reasons for the emergence of Geometry in ancient Egypt. He thus explains the curious name of a discipline which even in his time had already attained the status and the reputation of a pure science. For “geometry” literally means “measurement of the Earth,” and in some European languages to this very day the practitioner of land surveying is designated as geometer. In the light of its Earth-bound origins, therefore, it is perhaps not unworthy of notice that when modern differential geometers were looking for a terminology that would be both accurate and suggestive to characterize the notion of a continuum, they found their inspiration in Cartography, that is, in the science of making maps of the Earth.