Aurélie, Bertrand, Dino, Julien, who welcomed me there; Abdel Alexandre

Alejandro Adrien

Arnaud, Clément Emilie Anthony

This thesis is the result of three years of work at: ST Microelectronics, Rousset; CEMEF (Paris Science and Humanities) MINES ParisTech, Sophia Antipolis and CMP (Centre Microélectronique de Provence) MINES Saint-Etienne, Gardanne, where I have met remarkable people who I wish to acknowledge.

I would first like to thank the members of the jury: Professor Alain Bosseboeuf and Dr. Fabio Coccetti for taking time out of their busy schedules to assess my work. My thanks go also to Professor Roland Fortunier for agreeing to preside the defence. It was a great honour to have you on my thesis committee.

I would like to give special thanks to my academic advisors, Professor Karim Inal and Professor Pierre Montmitonnet. They taught me how to reach good results through methodical and thorough research work. I'm grateful for all their contribution of time, ideas and patience, which contributed to make my Ph.D. a productive and stimulating experience.

I am very much obliged to my industrial advisor, Dr. Pascal Fornara for his implication, time and guidance. My deepest thanks to you for all your support during my four years at STMicroelectronics.

I would also like to thank Dr. Jean-Michel Mirabel, R D manager at Rousset, as well as Dr. Elisabeth Massoni, director of CEMEF, for giving me the opportunity to do my PhD. in their respective departments.

After spending three years mainly at ST Rousset and CMP Gardanne, there are many people I want to thank. First of all, I am deeply indebted to Cristian (El Paisano) and Brice for their guidance and advice on this work. I thank Kevin, Maxime and Mohamed for helping me to improve our device. I'm grateful to the Advanced Technology team: Nando and Antonio, and the TCAD Team: Valerie, Roberto and Julien, for their support and for welcoming me in the team.

I obviously cannot forget to acknowledge the tremendous help from all the technicians, engineers, researchers and interns from ST and the CMP -without whom I would not have achieved such quick and good results. My sincere thanks to: Thierry for his support on HF release; Carole for her help on the first analyses of the SEM/TEM and HF release; Suzanne for her support on the in-situ heating in the SEM test.

Introduction

In the early 1900s, the vacuum tube led to technological advances in electronics. Afterwards, in the 1940s the development of the first transistor established a revolution in the electronic industry. Embedded on a semiconductor, the first silicon transistor (1947) started the race which consists in developing the smallest devices possible. Thus, in 1965, Gordon Moore published a paper forecasting a fast miniaturization of devices that would allow the number of components per integrated circuit to double every year [3]. This proposition establishes the well-known "Moore law", which has been governing the microelectronics development for the last fifty years.

The mechanical machines also took advantage of the miniaturization observed in electronics devices. The first paper about miniaturization in microsystems was written in 1967 by Nathanson et al. [1], who developed and manufactured the resonant gate MOS transistor. The microsystems or Micro-Electro-Mechanical Systems (MEMS) are miniaturized systems incorporating sensors, actuators and information processing devices. These systems, the size of which ranges from a few micrometers to a few millimeters, are made by using various micro/nano fabrication technologies with processes derived from microelectronics or with specific processes.

Manufacturers have made the Moore law the rule in semiconductors industry for developing and miniaturizing circuit boards. However, silicon electronics faces a challenge. The latest circuits are just about 7 nm wide. Yet, the size of individual silicon atoms (around 0.2 nm) appears to be the hard physical limit -considering the possibility of a single-atom-wide circuit. The idea was then to integrate "smarter" components at assembly. Following this new principle, known as "More than Moore", smaller products with more features could still be created.

The technologies Systems on Chip (SoC) were born in response to this context -promising that integrated solutions would add new functionalities in small dimensions. The SoC solution is a technology where several features live together on one single chip. The evolution of this technological solutions is the CMOS-MEMS -i.e. the MEMS is fabricated at the same time as electronics part. The first CMOS-MEMS, following Parameswaran et al. [2], was a poly-silicon micro-bridge. The interest of integrating the MEMS devices in a Complementary Metal Oxide viii Semi-conductor (CMOS) fabrication flow is to reduce cost by avoiding additional fabrication steps with special materials.

The miniaturization of the devices adds new technological challenges, such as the properties changes at small scale. The well-known material properties for bulk evolves at the scale of thin film complexifying its thermal, electrical and mechanical analysis. The goal of this thesis is then to make a functional structure and cope with problems linked to miniaturization.

One of studied devices was originally designed as a passive stress sensor, able to measure the stress level on a film layer. It consists of a metal line inserted in an oxide layer above Silicon. When the oxide is eliminated, the arms of this MEMS are freed and rotate under the effect of initial internal stresses. Fig. 1 It has been suggested that these movements could also be exploited to modify the electrical state of the system in which it is inserted, to change a capacity or to switch on/off. The effect in this case is obtained by an actuating current inducing Joule heating and dilatation. This is the context of this research, which is carried out by Multiphysics Finite Element Method (FEM) simulation and various characterization methods -i.e. SEM nanoprobing, optical profilometer, electrical measurements and in-situ heating SEM.

The challenge then lies in the release of the moving parts by dissolution of the surrounding oxide (out of plane deformation under the effect of residual stress, stiction and residues which prevent contact), in the actuator (current density, repeatability, durability, reliability) and, for ohmic switches, in the ability to establish a real electrical contact with low resistance (real / apparent area of contact with rough surfaces, contact pollution). Of course, an important job is to do the design and simulate of microsystems to overcome these difficulties and / or to study the behavior and measure the effects. This PhD thesis is organized into four chapters: ix • The first chapter is dedicated to a bibliographic study based on four axes: the MEMS process flow; the mechanical stress; the mechanical and electrical contact; and the MEMS reliability.

• The second chapter studies the residual stress in the structures. The stress is studied from the wafer fabrication to the final device. FEM analyses are performed in order to support and explain experimental data.

• The third chapter studies the actuation process on devices following several steps. First a thermal actuation is studied, followed by a Joule effect actuation, finishing with a mechanical contact and its FEM model. The latter considers the release step, the actuation part and the mechanical contact. Born from the need to reduce size and to integrate the mechanical properties in electronics systems, the MEMS have emerged in the 80's. The first MEMS, the resonant gate transistor developed by Nathanson et al. [1] consists in a electrostatically excited cantilever beam employing field-effect transistor.

In this chapter a bibliographical review is presented in order to clarify the complex fabrication process needed by a MEMS device, a CMOS-MEMS [2] in our case. The compatibility between the fabrication of mobile mechanical parts and a classical microelectronics fabrication flow is studied.

Companies in the technology sector are competing to provide better integrated solutions. The last years two trends stood out, the More Moore and the More than Moore.

Contrary to Moore's law1 [3], which aims at increasing the transistor density, the More than Moore (MtM) strategy aims at the integration of applications and functionalities. More Moore (MM) refers to the continued shrinking of horizontal and vertical physical feature sizes to reduce cost and improve performances.

The roadmap for microsystems industry shown in Figure 1.1 -elaborated by the International Technology Roadmap for Semiconductors (ITRS) -put the More than Moore strategy in a central position of the future of microsystems -lead by the internet of things. to perform mechanical functions, they are located in the MtM trend -i.e. as a sensor/actuator device.

Three parts are treated in this chapter. First, an introduction to MEMS fabrication is presented. Second, the issues related to MEMS development are treated in two parts; the residual stress present in thin films and the contact from both mechanical and electrical points of view. Finally, some MEMS reliability issues are given.

MEMS process flow 1.MEMS devices

The word Microsystems, Micro-electro-mechanical systems or just MEMS [4] refers to:

• Micro, related to size of devices in micrometre scale,

• Electro, which suggests either electrical or electronics, and

• Mechanical = existence of mobile or deformable parts.

The MEMS devices according to its functionality are mainly classified as sensors and actuators [4,5].

This definition does not consider all devices or functions currently performed. At present, various physical phenomena are exploited to create new devices. Thus, using physics as:

• information transmission: fluidic [6], optical [7],

• actuator or detector: thermal [8,9], magnetic [10], piezoelectric [11],

The MEMS can be divided in:

• sensor that measures: e.g. pressure [12],

• actuators: with or without moving parts [13].

The conception of MEMS is not only a size reduction of "macro" system, e.g. gear [13], but rather a motivation to solve problems arising in the "small world", e.g. inkjet printer head [14].

A wide variety in the kinds of the MEMS devices, divided between sensors and actuators, was presented. This work studies the switch devices, their integration and fabrication issues.

MEMS Switch introduction

One of the fastest growing markets in the last years is in the MEMS switches, which present various advantages compared with their predecessors, the p-i-n-diode or FET switches: nearzero power consumption (for electrostatic switches); very high isolation (formed by air gap); very low insertion loss [15].

From the first micro-relay developed by Petersen in 1979 [16] these devices have continuously evolved. These last years, the mechanical micro-switch takes an important place as RF-switch device, removing the hegemony of solid state switches [15] in the microelectronics industry.

The RF circuits, where a switch can be commuted at a high frequency, is not the only application. The majority of switches have an electrostatic actuation [15,17], which ensures a high speed commutation; there are also bistable switches [18][19][20][21] and latching switches [22,23], used in low power applications. Among other types of actuation we can find: thermal switches [24,[START_REF] Girbau | A Low-Power-Consumption Out-of-Plane Electrothermal Actuator[END_REF], magnetic switches [START_REF] Taylor | Fully integrated magnetically actuated micromachined relays[END_REF] and capacitive switches [START_REF] Zhang | Wideband CMOS Compatible Capacitive MEMS Switch for RF Applications[END_REF].

In standard fabrication (Figure 1.2), the MEMS device is assembled as a single device, which is put close to the electronic device; the two are wired together and then the assembly is packaged. This method of stacking can be a source of issues in fabrication processes: robustness problems, increase of fabrication cost, longer fabrication time.

The research laboratory and industry actors are working on the technique to improve the embedded multiple functions in a single device. The CMOS-MEMS concept and different fabrication process are then explained in the next section.

CMOS-MEMS

The interest of integrating the MEMS devices in a CMOS fabrication flow is to reduce cost by avoiding additional fabrication steps with special materials and individual packaging processes. The first CMOS-MEMS, due to Parameswaran et al. [2], was a poly-silicon micro-bridge.

The MEMS devices use some of the microelectronics fabrication processes like photolithography or material deposition steps [START_REF] Fedder | Technologies for Cofabricating MEMS and Electronics[END_REF].

The RF MEMS-CMOS development has been studied by Mansour [START_REF] Mansour | RF MEMS-CMOS Device Integration: An Overview of the Potential for RF Researchers[END_REF], who made an in-depth study of the possibilities of integration between CMOS and MEMS technologies. MEMS fabrication processes are classified as:

1.1.3.1 Pre-CMOS: MEMS fabricated before CMOS Sandia National Laboratory [START_REF] Smith | Embedded micromechanical devices for the monolithic integration of MEMS with CMOS[END_REF] proposed a MEMS construction called Pre-CMOS (Figure 1.3), in which the MEMS (Micromechanical Device area) are fabricated before the Back end of Line (BEOL) part of circuit (CMOS Device area). Fig. 1.3 A cross-section schematic of an integrated MEMS-CMOS illustrating pre-CMOS principle [START_REF] Smith | Embedded micromechanical devices for the monolithic integration of MEMS with CMOS[END_REF] In the MEMS fabrication process, annealing at 900 • C should be done to avoid residual stress in polysilicon. At such temperatures, the Al BEOL present in the CMOS circuit would melt -i.e. Al melts at 660 • C. To avoid the melting risk, the annealing is performed before BEOL part.

Intra-CMOS: MEMS fabricated during the CMOS

Infineon Technologies fabricated in the same level as the CMOS device, a membrane in poly-silicon serving as pressure sensor [12]. Fig. 1.4 A cross section schematic of Infineon integrated MEMS technology for a pressure sensor [12] The polysilicon (membrane) is inserted in the BEOL part, thus the stress of this layer should be considered for a successful integration. The fabrication of CMOS is firstly performed, then the MEMS part is fabricated over the CMOS in a specialized MEMS-factory. The principal problem of this technique is the temperature used in MEMS fabrication which may deteriorate the metal lines of the CMOS device. However, Cavendish Kinetics [START_REF] Gaddi | MEMS technology integrated in the CMOS back end[END_REF], Berkeley [START_REF] Franke | Polycrystalline silicongermanium films for integrated microsystems[END_REF], IBM [START_REF] Stamper | Planar MEMS RF capacitor integration[END_REF] (Figure 1.5) and Wispry [START_REF] Natarajan | CMOS integrated digital RF MEMS capacitors[END_REF] develop this sort of CMOS-MEMS. Fig. 1.5 A cross section schematic of IBM/Wispry integrated MEMS technology [START_REF] Stamper | Planar MEMS RF capacitor integration[END_REF] All these possibilities to integrate CMOS and MEMS technologies in a single fabrication flow present various technical integration problems, principally due to fabrication steps in temperature and material choices.

BEOL-MEMS: MEMS fabricated within CMOS

Parameswaran et al. [2] in 1988 created the first BEOL-MEMS. Afterward, Carnegie Mellon university continued the development [START_REF] Fedder | Laminated high-aspect-ratio microstructures in a conventional CMOS process[END_REF][START_REF] Xie | Post-CMOS processing for high-aspect-ratio integrated silicon microstructures[END_REF], which is actually commercialised as Application Specific Integrated MEMS Process Service (ASIMPS) [START_REF] Fedder | Application-Specific Integrated-Microelectromechanical Systems ( MEMS ) process services ( ASIMPS )[END_REF] (Figure 1.6). The principal advantage of this solution is the integration with CMOS fabrication process. Its development needs a Reactive-Ion Etching (RIE) plasma and chemical etching process, both already present in a CMOS standard fabrication flow.

Bibliographical Review

Pre-, Intraand Post-MEMS solutions are intended to insert the "classical MEMS" into a "classical CMOS" device, adding new fabrication steps. All these solutions try to make compatible the MEMS and CMOS fabrication processes.

In conclusion, the CMOS-MEMS principle is a simple way to insert an additional function in an electronic device -e.g. adapting the fabrication steps used in MEMS.

Because of the "simplicity" of integration in a CMOS factory, the BEOL-MEMS principle is used in this work. Later in the manuscript this approach will be developed. Now I will review a few mechanical aspects. The MEMS studied are built from thin films, which have a particular mechanical behaviour due to their small scale (several tens of microns). The device function is not only dependent of constraint but also by their thermomechanical characteristics.

1.2 Mechanical stress in an isotropic thin film 9

Mechanical stress in an isotropic thin film

The thin layers we consider are defined by two characteristics: the film thickness t f is much smaller than the substrate thickness t s (t f << t s ); the substrate thickness t s is much smaller than the film dimensions' d (d >> t s ). In the following, we consider only films of isotropic elastic materials, involving two elastic constants, Young's modulus E and Poisson's ratio ν. In the classical case of constrained film/substrate system, the top surface is free of constraint, σ 3 = 0, which implies in Voigt 1 notation σ 3 = σ 4 = σ 5 = 0. Also, the shear stress in the geometric axes in the plane is generally small, so σ 6 = 0. Although the strain is tri-axial, the plane stress condition applies:

σ 1 = E (1 + ν) • (1 -2ν) [(1 -ν)ε 1 + ν(ε 2 + ε 3 )] (1.1)
σ 2 = E (1 + ν) • (1 -2ν) [(1 -ν)ε 2 + ν(ε 1 + ε 3 )] (1.
2)

σ 3 = E (1 + ν) • (1 -2ν) [(1 -ν)ε 3 + ν(ε 1 + ε 2 )] (1.3)
where E and ν are the isotropic elastic modulus and Poisson's ratio respectively. As

σ 3 = 0 0 = (1 -ν)ε 3 + ν(ε 1 + ε 2 ) → ε 3 = - ν 1 -ν (ε 1 + ε 2 ) (1.4)
Replacing ε 3 , the stress conditions becomes:

1 The stress tensor in matrix notation is

σ =   σ xx σ xy σ xz σ yx σ yy σ yz σ zx σ zy σ zz   the Voigt notation, simplified to a 6-dim. vector is: σ = (σ xx , σ yy , σ zz , σ yz , σ xz , σ xy ) = (σ 1 , σ 2 , σ 3 , σ 4 , σ 5 , σ 6 )
Bibliographical Review

σ 1 = E (1 -ν 2 ) [ε 1 + νε 2 ] (1.5) σ 2 = E (1 -ν 2 ) [νε 1 + ε 2 ] (1.6)
Assuming that the stress is equibiaxial,

σ 1 = σ 2 = σ : σ = E -2ν ε ⊥ with ε ⊥ = ε 3 (1.7) Or σ = Mε ∥ with ε 1 = ε 2 = ε ∥ (1.8) with M the biaxial modulus M = E 1-ν
The residual stress present in a film over a substrate is induced by the sequence of thermal steps in fabrication process and microstructure evolution. The origin of these stresses can be classified as intrinsic and extrinsic.

Origin of stress 1.2.1.1 Intrinsic stress

Generated during the creation of the film [START_REF] Gardner | Mechanical stress as a function of temperature in aluminum films[END_REF][START_REF] Abermann | Measurements of the intrinsic stress in thin metal films[END_REF], the intrinsic stresses depend on: the material deposited; the different growth modes (e.g. columnar grain growth or island growth); the technique used (e.g. Physical Vapor Deposition (PVD), Chemical Vapor Deposition (CVD)); the deposition parameters (e.g. environmental gas, substrate temperature). This intrinsic stress is then considered as fixed in a controlled environment, granted by the microelectronics production line.

The sources of intrinsic stress depend on the deposition process and the involved materials. Some important examples are listed below [START_REF] Klokholm | Intrinsic Stress in Evaporated Metal Films[END_REF]:

• Coalescence of grain boundaries: during the film deposition the grains grow over a substrate. In many cases, the Volmer-Weber island growth mechanism dominates (Figure 1.8). When these crystallites coalesce, a tensile stress is generated.

• Grain growth: due to coalescence of grains and the elimination of its boundaries the film reaches a minimum system energy. Thus, the number of grains decreases as the grains grow. The elimination of the grain boundary densifies the film and a tensile stress is created. The thermo-elastic deformation generated by the thermal steps in the microelectronics fabrication is the main source of extrinsic stress. Due to adhesion, if the substrate is much thicker than the attached films, it imposes its own dilatation over the rest of the layers. The stress is thus built by the difference of expansion coefficients between film and the substrate. Then, for a film over substrate system, the substrate deformation induce a homogeneous biaxial deformation on the film defined by [START_REF] Gardner | Mechanical stress as a function of temperature in aluminum films[END_REF]:

ε f ilm = T 1 T 0 (α sub -α f ilm )dT (1.9)
Where α f ilm and α sub are the coefficients of thermal expansion of the film and substrate respectively. For a limited temperature range, and for α f ilm and α sub independent of temperature, the previous equation can be approximated as:

ε f ilm = (α sub -α f ilm ) △ T (1.10)

Bibliographical Review

Using the previous strain via Hooke's law, the stress can be calculated:

σ f ilm = - E 1 -ν f ilm (α sub -α f ilm ) △ T (1.11)
In this way, the thermal cycles during the fabrication process of the device generate a thermo-elastic deformation. The temperature can produce a microstructure modification and/or can exceed the elastic limit of the film, which finally determines the stress state of the film.

The CMOS fabrication processes include different materials with notably different properties. The residual stress is a result of all the fabrication process steps and the different materials properties (see Table 1 1.1 Coefficients of thermal expansion α, and Young's modulus E of the main materials used in microelectronics [START_REF] Freund | Thin Film Materials[END_REF] The materials used for the structures fabrication and therefore involved in this study are: Al, SiO 2 , TiN and Si. In the fabrication flow, over the silicon substrate a successive alternation of an insulator (SiO 2 ) also called InterMetallic Dielectric (IMD) and a metallic layer -principally Al or Cu -is deposited, thus forming the Back-end-of-line (B). As already mentioned, the differences of material properties between the Si and the added layers, combined with the deposition process and annealing steps, are responsible for residual stress.

Serving to isolate the Si substrate from the following layers, an undoped CVD SiO 2 (Tetraethyl Orthosilicate (TEOS)) is deposited at 700 • C -i.e. CVD deposition. The TEOS oxide has a compressive residual stress of -100 MPa. Furthermore there is the thermal SiO 2 grown at 800 • C to 1200 • C which has a compressive residual stress of -300 MPa [START_REF] Wolf | Silicon Processing for the VLSI Era[END_REF] -mainly used as a protective layer for Si.

The Al film is sputtered using PVD from a target containing 0.5wt% Cui.e. Al (Cu). The addition of copper increases the resistance to electromigration by slowing down the grain 1.2 Mechanical stress in an isotropic thin film 13 boundary diffusion [START_REF] Ames | Reduction of Electromigration in Aluminum Films by Copper Doping[END_REF]. The target reaches a temperature of 90 • C during the deposition stepwhile the wafer is heated to 350 • C. A biaxial residual tensile stress of ∼300 MPa is reported in literature [START_REF] Nix | Mechanical properties of thin films[END_REF][START_REF] Gardner | Mechanical stress as a function of temperature in aluminum films[END_REF] on Al (Cu) layer.

The TiN layer is added on top of the Al layer to avoid the electromigration of Al lines. This TiN is deposited using CVD process with a heater temperature of 450 • C and a wafer temperature of 407 • C. Meanwhile, the TiN layer has a compressive stress near -800 MPa [START_REF] Machunze | Stress gradients in titanium nitride thin films[END_REF][START_REF] Liu | Elimination of initial stress-induced curvature in a micromachined bi-material composite-layered cantilever[END_REF].

These fabrications steps are repeated as many times as there are metal levels in our technology -i.e. the levels that make up the back-end-of the line technology. Thus, the multiple steps which use high temperature make the residual stress in the layer evolve [START_REF] Nix | Mechanical properties of thin films[END_REF]. Figure 1.10 show the stress in an Al layer on a heating-cooling cycle. Fig. 1.10 Stress-temperature plot for an Al film on a Si substrate. Elastic and plastic deformations occur in the film during a thermal cycle [START_REF] Nix | Mechanical properties of thin films[END_REF] The residual stress is harmful in most cases. It is therefore important to measure it. The stress in thin films can be determined from:

• Substrate curvature (optical interferometry [47], X-ray diffraction [START_REF] Bohg | Measurement of stresses in thin films on single crystalline substrates[END_REF], laser scanning [START_REF] Flynn | Measurement and Interpretation of stress in aluminum-based metallization as a function of thermal history[END_REF])

• Lattice spacing (X-ray diffraction [START_REF] Maniguet | X-ray Determination and Analysis of Residual Stresses in Uniform Films and Patterned Lines of Tungsten[END_REF]),

The simplest method is to measure the substrate curvature and use Stoney's approximation [START_REF] Stoney | The Tension of Metallic Films Deposited by Electrolysis[END_REF][START_REF] Freund | Thin Film Materials[END_REF], for a thin substrate-film composite -i.e. the film is thin in front of the thickness of the substrate (<5%) [START_REF] Schwarzer | On the Determination of Film Stress from Substrate Bending: STONEY's Formula and Its Limits[END_REF].

Passive stress sensor

Another way to measure the residual stress is to insert a sensor in the thin film, for instance the Rosettes stress sensor [START_REF] Jaeger | CMOS stress sensors on [100] silicon[END_REF][START_REF] Ewuame | Investigations of Thermomechanical Stress Induced by TSV-Middle (Through-Silicon via) in 3-D ICs by Means of CMOS Sensors and Finite-Element Method[END_REF], which measures the electrical response from an array of transistors, and the cross-shape stress sensor [START_REF] Wilson | Direct Measurement of Electromigration Induced Stress in Interconnect Structures[END_REF][START_REF] Kasbari | Direct Local Strain Measurement In Damascene Interconnects[END_REF], which relies on the rotation in a structure due to an initial stress level.

The design of our cross-shape stress sensor is based on the work of Horsfall [START_REF] Horsfall | A novel sensor for the direct measurement of process induced residual stress in interconnects[END_REF] on aluminium and Kasbari [START_REF] Kasbari | Direct Local Strain Measurement In Damascene Interconnects[END_REF] on copper structures. The structure, integrated in a single layer, is composed of two metallic expansion arms connected to a metal central pointer, thus forming a conducting asymmetrical cross (Figure 1.11). These movements could also be exploited to modify the electrical state of the system in which it is inserted, to change a capacity or to switch on/o f fi.e. electro-mechanical switch. The effect in this case is obtained by an actuating current inducing Joule heating and dilatation.

As a switch has been chosen for test implementation, the present study will try to understand the mechanisms and laws related to electro-mechanical contact. The electrical contact term requires a junction between two conductors, which is apt to carry electric current. Next section develops the study of contact, considering both mechanical and electrical properties.

Mechanical and electrical contact

Holm [START_REF] Holm | Electric Contacts[END_REF] has studied the electric contact, and one of the first relations proposed by Holm highlights the need to define first the mechanical contact. A scheme of the contacting parts (Figure 1.12) recalls that surfaces are not perfectly smooth. One undesirable effect of these asperities is that only a few contact spots are available for the electric current, inducing a constriction of the flow, and a resulting increase of the "electric contact resistance". To make a good contact both the mechanical and the electrical contact should be studied and improved. The next section studied the mechanical contact through different models.

Mechanical contact

Mechanical contact between two smooth bodies is described as a relationship between the contact area and the applied pressure applied, involving material characteristics:

• Elastic modulus (E) • Poisson's ratio (ν)

• Elastic limit (R p or σ y ) or Hardness (H) Contact between rough surfaces requires more sophisticated models to predict the overall contact area, the real area in contact, and in some cases the contact spot size distribution and local pressure. The knowledge of the topography in the nanometric scales is indispensable.

The short name aspot for the conducting contact areas, referring to the radius a of a circular contact area, is a widely accepted term. The sum of all these areas or aspots is the load bearing area A b (A b = ∑ n πa 2 n ) upon which the pressure p is applied [START_REF] Holm | Electric Contacts[END_REF] that gives a load P of:

P = ξ c HA b (1.12)
where ξ c is the p/H ratio, usually 0.2 < ξ c < 1 and H is the (macroscopic) contact Hardness, so the spot has only a plastic deformation.

Bibliographical Review

The mechanical contact is imperfect at every scale. The hills and valleys present on the surface make a difference between the real contact and nominal contact area.

Modelling the real contact area requires to establish a relation between the applied macroscopic load and the number and size of contact spots, and for completeness their deformation and contact stress. Each contact spot may be elastic or plastic.

Many authors have tried to model the three basic cases: elastic, elastic-plastic or purely plastic model deformation of spots. Some used analytical, others numerical approaches, the description of a-spots being statistical or deterministic (Table 1.2).

Reference

roughness elastic-plastic statistical analytical description -deterministic -numerical [START_REF] Greenwood | Contact of Nominally Flat Surfaces[END_REF][START_REF] Hertz | Miscellaneous papers[END_REF] Statistic elastic Statistical analytical [START_REF] Majumdar | Fractal Model of Elastic-Plastic Contact Between Rough Surfaces[END_REF] Fractal elastoplastic Statistical analytical [START_REF] Jackson | A multi-scale model for contact between rough surfaces[END_REF] Multiscale elastoplastic Deterministic numerical [START_REF] Yastrebov | Rough surface contact analysis by means of the Finite Element Method and of a new reduced model[END_REF] Real measurement elastoplastic deterministic/ statistical numerical [START_REF] Duvivier | Etude expérimentale et modelisation du contact electrique et mecanique quasi statique entre surfaces rugueuses d'or : application aux micro-relais MEMS[END_REF][START_REF] Arrazat | Maitrise de la Microstructure de films minces d'or par traitements de surface pour l'optimisation du contact mecanique et ohmique des micro-relais MEMS[END_REF] Deterministic plastic Deterministic analytical Table 1.

Resume of contact theories

In the following, the survey is restricted to small to moderate real contact area, as can be expected in our electro-mechanical contact problems ((1 to 6) µN [START_REF] Kataoka | Low contact-force and compliant MEMS probe card utilizing fritting contact[END_REF][START_REF] Broue | Multi-Physical Characterization of Micro-Contact Materials for MEMS Switches[END_REF]). We leave aside the high pressure contacts which can be found in metal forming [START_REF] Zhan | Study of the contact force in free-formsurfaces compliant EDM polishing by robot[END_REF] or mechanical joints [START_REF] Bograd | Modeling the dynamics of mechanical joints[END_REF]. Physically, this implies that plasticity is limited to the smallest scale asperities, or to the top of larger asperities.

Elastic, statistical analytical contact:

Considering roughness peaks as spherical caps, this first analysis is based on Hertz' theory [START_REF] Greenwood | Contact of Nominally Flat Surfaces[END_REF][START_REF] Hertz | Miscellaneous papers[END_REF] of elastic contact of a sphere (radius β ) with a semi-infinite body. Originally, the contact radius a c , area A b , and load P can be expressed in terms of w, the distance by which the sphere has sunk into the half-space, as [START_REF] Greenwood | Contact of Nominally Flat Surfaces[END_REF]

a c = β 1/2 w 1/2 , A b = πβ w, P = 4 3 E ′ β 1/2 w 3/2 (1.13)
Where E ′ is the equivalent elasticity modulus

1 E ′ = 1 -ν 2 1 E 1 + 1 -ν 2 2 E 2 (1.14)
E i and ν i are respectively the Young's modulus and Poisson's coefficient of material i.

Mechanical and electrical contact

From these equations, Greenwood and Williamson derived their model of contact between rough surfaces (Figure 1.13) using several assumptions:

• purely elastic micro-contacts,

• independent isolated sphere/plane contacts (strain and stress fields of neighbouring contacts do not interfere),

• the contact of two rough surfaces can be modelled as a rough and a smooth one in contact, using the concept of combined roughness,

• only one scale of roughness is considered.

Distribution of asperities

Assume that two surfaces come in contact until their reference planes are separated by a distance d. Further, the asperities heights have a random distribution, the probability that asperities height is between z and z + dz above some reference plane is φ (z)dz. Accepting that each asperity deforms elastically according to Hertz model, the number of contact spots n, the total contact area A r and the total load P in general case are:

n = N ∞ d φ (z)dz (1.15)
Since N number of asperities, w = zd and A b = πβ w, then the total area A r of contact is

A r = πNβ ∞ d (z -d)φ (z)dz (1.16)
And the total load P is Greenwood and Williamson (GW) [START_REF] Greenwood | Contact of Nominally Flat Surfaces[END_REF] represent the previous result in a non-dimensional way, describing heights in terms of standard deviation σ a of the height distribution. They also Bibliographical Review introduce the surface density of asperities η, which multiplied by the nominal area A n gives the number of asperities N = ηA n . Then [START_REF] Greenwood | Contact of Nominally Flat Surfaces[END_REF]:

P = 4 3 NE ′ β 1/2 ∞ d (z -d) 3/2 φ (z)dz (1.
Number of contact spots n = ηA n F 0 (h ′ ) (1.18)
Total contact area

A r = 2ηA n β σ F 1 (h ′ ) (1.19) Load P = 4 3 ηA n E ′ β 1/2 σ 3/2 F 3/2 (h ′ ) (1.20)
Where h ′ , the standardized separation, is equal to d/σ and

F n (h ′ ) = ′∞ h (s -h ′ ) n φ * (s)ds (1.21)
Where φ * (s) is the height distribution scaled to make its standard deviation unity.

The model of GW is the first model of rough contact based on statistical method and serves as a basis for the creation of a multitude of other statistical contact models.

Criticism

Although very successful, the GW model has been criticized. Its assumptions have been successively questioned.

1.3.1.2.1 Elastic vs plastic asperities Some argue that plastic asperities should be modelled as well, as suggested by GW themselves. Considering the hertzian stress field inside the solids and expressing their plastic limit, GW [START_REF] Greenwood | Contact of Nominally Flat Surfaces[END_REF] proposed to check for the elastic/plastic state of asperities using the plasticity Index:

ψ = E H h p β (1.22)
Where H is the material hardness and h p is the standard deviation of the distribution of peak heights. If ψ < 0.6 the contact deformation can be considered as purely elastic for all asperities. If ψ > 1 some asperities deform plastically, even under low loads and the contact area is proportional to applied load P:

Mechanical and electrical contact

A n = P H (1.23)
This shows that surfaces, the asperities of which have larger radii of curvature β , wide distribution of peak heights (h p ) or small E/H ratios tend to remain elastic for larger loads.

Elastic-plastic deformation of asperities has later on been introduced in most models. For example, fractal models [START_REF] Majumdar | Fractal Model of Elastic-Plastic Contact Between Rough Surfaces[END_REF][START_REF] Jackson | A multi-scale model for contact between rough surfaces[END_REF] conclude that small spots are in plastic deformation while large spots are in elastic deformation, because small scale peaks have much smaller radii of curvature, an effect neglected by GW. Duvivier [START_REF] Duvivier | Etude expérimentale et modelisation du contact electrique et mecanique quasi statique entre surfaces rugueuses d'or : application aux micro-relais MEMS[END_REF], Arrazat [START_REF] Arrazat | Maitrise de la Microstructure de films minces d'or par traitements de surface pour l'optimisation du contact mecanique et ohmique des micro-relais MEMS[END_REF] and Yastrebov [START_REF] Yastrebov | Rough surface contact analysis by means of the Finite Element Method and of a new reduced model[END_REF] are also cases of elastic-plastic microcontacts, in a classical (mono-scale) context. These models are described below.

1.3.1.2.2
The Multiscale character of surface roughness Others consider that roughness being known as a multiscale stochastic process, multiscale analyses should be elaborated.

In practice, GW considers the existence of a single radius (β ) of curvature, which is the average for a surface profile. The problem is the ambiguity of the measurement scale, because the measured radius evolves with the scale of analysis (length measured, discretization step). This is dealt with in Multiscale and fractal models (see Table 1.2 above). 1.3.1.2.3 Interactions between contact spots Finally, GW assume that asperities are far enough from each other that no interaction of the strain and stress fields occurs. This implicitly assumes a very low real area of contact. This can be questioned as well.

It must be noted that GW model considers an isotropic surface. This probably makes a difference in the intensity of the interactions. For instance, Pennec et al. [START_REF] Pennec | Modelisation du contact metal-metal: Application aux microcommutateurs MEMS RF[END_REF] considered an area of interaction and numerical analysis to define the interaction between asperities.

All three issues are reviewed next. In each case, many papers have been published. I will not go too deeply into this approach, because the main aim of this PhD thesis is to make a device robust and electrically active.

Multiscale, including fractal analysis

The fractal analysis described by Majumdar and Bhushan (MB) defines a surface as a succession of asperities [START_REF] Majumdar | Fractal Model of Elastic-Plastic Contact Between Rough Surfaces[END_REF], observation scale-dependent. They consider a surface profile z(x) as shown in Figure 1.14. They proposed the Weierstrass-Mandelbrot (WM) functions1 as an adequate description of this Multiscale character. Such a roughness is composed of sinusoidal waves of different wavelengths and amplitudes superposed on each other. From the WM function, at a wavelength of l i = γ n i , the asperity profile z(x) before any deformation is given by [START_REF] Miao | A complete contact model of a fractal rough surface[END_REF]:

z(x) = G D-1 l 2-D cos πx l ; - l 2 < x < l 2 (1.24)
where D is the fractal dimension of the surface profile (1 < D < 2), G is the fractal roughness parameter, and l is the length scale of a single asperity (Figure 1.15). In the MB model the length scale l is the same as its real contact length, therefore the interference (or distance of interpenetration) (w) of any asperity is equal to its full height (δ ):

w = δ = G D-1 l 2-D (1.25)
The principal difference with GW model is that the fractal model considers the radius of curvature (β ) of a contact spot to be a function of the area of the spot.

β = a D/2 π 2 G D-2 (1.26)
where a = radius of the area of contact. The critical area of asperities is:

a rc = G 2 K ′ H 2E 2 D-1 (1.27)
Where H is the hardness of the material, E is the elastic modulus, and K ′ is the hardness coefficient related to the Poisson ratio ν of the material by [START_REF] Miao | A complete contact model of a fractal rough surface[END_REF] K ′ = 0.454 + 0.41ν. a r is the real contact area of asperities, therefore when a r < a rc , the contact mode is plastic, and it becomes elastic when a r > a rc . The multiscale contact theory proposed by Jackson and Strator (JS) is also a model of rough surfaces in contact [START_REF] Jackson | A multi-scale model for contact between rough surfaces[END_REF], similar in principle to fractal analysis that tries to model multiscale asperities. This non-fractal multiscale model is applicable to elastic and elastoplastic conditions. A scan length (L) is selected, the surface data is acquired and a FFT is performed. From the resulting Fourier series, the area asperity density and radius of curvature are computed for each frequency level according to [START_REF] Jackson | A multi-scale model for contact between rough surfaces[END_REF]:

R i = 1 4π 2 β i f 2 i (1.28)
where f i denotes the spatial frequency i (i.e. the reciprocal of wavelength) and β i are the asperity amplitude corresponding to the given frequency. Figure 1.16 shows a proportionality between area and load. This observation is valid for Jackson and Green (JG) formulation, a Hertz model (Trunc), the Majumdar and Bhushan (MB) fractal model and the GW statistical contact model using the Jackson and Green (JG) formulation. Fig. 1. 16 Comparison of elasto-plastic surface contact models [START_REF] Jackson | A multi-scale model for contact between rough surfaces[END_REF] Bibliographical Review This proportionality, can be interpreted by the average pressure (p), based on the real contact area (A r ):

p = F A r (1.29)
1.3.1.4 Deterministic description of asperities to be deformed Advances in the computer field have allowed important improvements in FEM, along with the improvements of characterization methods such as Atomic-Force Microscopy (AFM), they have permitted true representations of the sample. This is the basis of "deterministic" (as opposed to statistical) approaches.

A semi-analytical model

The method proposed by Duvivier [START_REF] Duvivier | Etude expérimentale et modelisation du contact electrique et mecanique quasi statique entre surfaces rugueuses d'or : application aux micro-relais MEMS[END_REF] and Arrazat [START_REF] Arrazat | Maitrise de la Microstructure de films minces d'or par traitements de surface pour l'optimisation du contact mecanique et ohmique des micro-relais MEMS[END_REF] mixes a classical elastic model, GW like, with an analytical model of plastically deformed peaks. Furthermore, each asperity is described individually in terms of height and radius of curvature (single-scale however) thanks to an AFM measurement of the surface. At a given penetration of the indenter (hard and smooth compared to the rough surface under study), each peak is notified as elastic or plastic and the relevant model is applied, given the spot size and pressure. The load is finally obtained as the discrete sum of the loads over each asperity.

Here however, the short distance spot-spot interaction is not included; but the large scale elastic sinking of the surface under the indenter load is accounted for and reduces the effective interference of each asperity with the indenter for a given position of the latter.

FEM modelling of interactions between contact spots

The criticism made to GW model related to individual treatment of asperities has been seen by Pennec et al. [START_REF] Pennec | Modelisation du contact metal-metal: Application aux microcommutateurs MEMS RF[END_REF]. They have treated the interaction when several neighbouring asperities are deformed in a contact with a rigid body. As in the previous paper, they obtain a real representation of a bump in a capacitive switch thanks to AFM (Figure 1.17a), then the model is exported to a FEM software with an intermediate Matlab treatment (Figure 1.17a).

Using this data, a FEM model can be performed, this time considering the neighbouring asperities deformation.

Arrazat in his PhD thesis has also worked on the interaction between the contact points. Figure 1.18 shows a comparison of experiments and models of a sphere/plane contact that include or not interaction between spots. The models without interactions between spots overestimate contact force compared with the experimental data. Without interaction, the number of spots in contact is also larger. Due to the small dimensions (0.3 to 3 µm) of the contact between the arm of the MEMS (1 µm cross section) and the pad, few peaks will be involved so that statistical approaches are questionable. Moreover, deterministic methods may now rely on the high precision of AFM scanning on the one hand, of the FEM model on the other hand. Among the asperity contact models, that elastic-plastic ones should be preferred is shown by the application of the plasticity index (ψ) concept.

Mechanical contact area is not necessarily the area of electrical contact (Figure 1.19) as we will see in the next section (subsection 1.3.2). Fig. 1.19 Schema of a real electrical contact, making the difference between apparent, electrical and mechanical contact area [START_REF] Duvivier | Etude expérimentale et modelisation du contact electrique et mecanique quasi statique entre surfaces rugueuses d'or : application aux micro-relais MEMS[END_REF] In this way the studies of mechanical contact reported in this section, are not sufficient to estimate an electrical contact area. Meanwhile the next section focuses on how the electrical current flows through the contact: conduction mode and reliability of the contact.

Electrical contact

Ensuring good electrical conduction means reducing the resistance. The contact resistance at an interface must be adapted from the general concept of electrical resistance (R):

R = ρ • L A (1.30)
where ρ is the resistivity, L is the length and A is the area of the conducting electrical lines. Taking a junction between two solids and measuring the resistance between two points, a certain distance from the interface, it is found that the resistance is larger than the sum of the resistances calculated from Equation 1.30. This introduces the existence of a specific "contact" resistance. Now in most cases, the contact is between rough surfaces as pictured in (Figure 1.12). This also constitutes a resistance to electrical conduction, since only a small fraction of the apparent contact surface is available, in the form of a few contact spots -i.e. Assume a regular array of contact spots of diameter 2a, separated by distances 2r. Electric current lines have to bend and concentrate to go through the spot. The solution of electric field equations has been made in different situations. From a solution of Laplace's equation using appropriate boundary conditions, Timsit defines in [START_REF] Timsit | Electrical contact resistance: properties of stationary interfaces[END_REF] The macroscopic conductors are characterized by Ohm's law, which establishes that the resistance of a given sample is directly proportional to its length L and inversely proportional to its area A. However, Ohm's law is no longer applicable at very small scale. In the present project, typical contact area is 1 µm and individual contact spots may be of the order of tens of nanometers.

In microscopic systems, we can identify different regimes of transport related to length scales. An important length scale is the electron elastic mean free path ℓ, which roughly measures the distance between elastic collisions with static impurities. For Crystalline Al, the electron-movement has a mean free paths of ℓ ∼ 50nm [START_REF] Kanter | Slow-Electron Mean Free Paths in Aluminum, Silver, and Gold[END_REF][START_REF] Mayadas | Intrinsic Resistivity and Electron Mean Free Path in Aluminum Films[END_REF].

Considering the relation between the mean free path ℓ and the contact length, we can identify two principal regimes [START_REF] Agraït | Quantum properties of atomic-sized conductors[END_REF]: if ℓ << L, it is called diffusive (each electron undergoes a number of diffusions on obstacles; Ohm's law); L is a characteristic length, to be specified case by case, if ℓ > L, it is called ballistic (no or few collisions) in which the electron momentum can be assumed constant.

For ballistic transport, Sharvin's model [START_REF] Duvivier | Etude expérimentale et modelisation du contact electrique et mecanique quasi statique entre surfaces rugueuses d'or : application aux micro-relais MEMS[END_REF] calculates the resistance of a circular constriction of radius a in a surface which separates two half-spaces. The Sharvin Resistance (R s ) is characterized by the mean free path ℓ and the resistivity ρ:

R s = 4 3 • ρ ℓ πa 2 (1.33)
In contrast to subsection 1.3.2, now the surface A and the length L are replaced by πa 2 (the surface of a-spot) and ℓ (the mean free path) respectively.

Bibliographical Review

For quasi-ballistic transport, Wexler's model [START_REF] Wexler | The size effect and the non-local Boltzmann transport equation in orifice and disk geometry[END_REF] makes the link between ballistic (a << ℓ) and diffusive (a >> ℓ) regimes (Knudsen number, K * = ℓ/a):

R w = 4K * ρ 3πa + ργ(K * ) 2a = R s + γ(K * )R H (1.34)
where R W is the so-called Wexler resistance, and γ(K * ) is a slowly varying gamma function.

The model involves a function γ(K * ), which weights the addition of the Holm diffusive term to the ballistic term of Sharvin. This function has been calculated by Wexler using Green function integral in the Maxwell limit method, but their calculation is not simple.

Nikolic [START_REF] Nikolić | Electron transport through a circular constriction[END_REF] has proposed a simplified form, where γ(K * ) has the limiting value γ(0) = 1 and R s R H = 0. Nikolic's method is able to compute γ(K * ) numerically to an accuracy of better than 1 %:

γ(K * ) = 1 + 0.83K * 1 + 1.33K * (1.35)
Nikolic's answer for γ differs little from the approximate answer of Wexler Many problems may occur in a metal-metal electrical contact: stiction, creep, sintering, growth of oxide which change the contact resistance. They are particularly significant for small dimension electrical contacts. In the next section the oxidation of aluminium and electrical conduction through this oxide are studied.

Physical evolutions of the surfaces and consequences on electric contact resistance

The models above give a static, geometric view of electric contact. Furthermore, contacts evolve by a number of physical phenomena. In this work, the main impediment to electrical contact is the oxidation of the Al.

A thin layer of oxide grows naturally on almost all metals in contact with air. Growth is faster at higher temperature. Alumina is a very good insulator -electrical resistivity around (10 20 -10 22 ) µΩ cm contrary to aluminium around 2.8 µΩ cm [START_REF] Mercier | Quantitative Evolution of Electrical Contact Resistance between Aluminum Thin Films[END_REF]. It builds a high contact resistance so two possibilities for electric conduction are:

• Generate a tunnel effect; that means using a large electric field to generate a weak current flow.

• Cracking alumina improves contact conduction; it is possible by increasing the load, by imposing sliding, or electrically, by a high current flow leading to breakdown of the oxide. When this process is complete, stable ohmic conduction can be retrieved.

Both solutions, tunnel effect and alumina cracking, have some drawbacks.

• Tunnel effect allows only a weak flow of current (around 1 nA for 5 nm of aluminium oxide).

An example of the evolution of current density with an applied potential is shown in Due to our typical expected contact spot dimensions (< 1 µm) compared with the electron mean free path (50 nm), ballistic transport will have to be considered in addition to diffusive. This means that the fractional real area of contact is not sufficient; the contact mechanics model should give us an estimate of the individual size of contact spots.

Mechanical and electrical contact 29

The question of oxide elimination, which is very sensitive to the yet unknown contact stresses, will also have to be examined. Then, a bibliographical study of possible conduction modes through the insulator is presented in next section.

Electrical conduction modes in the insulators

An Ohmic conduction governs the electrical conduction in the so-called "electrical conductors" -i.e. a linear relation between voltage U and current I (U = I • R). Even if an insulator is supposed to prevent the flow of current, in fact it gives way to small currents when the electrical field or temperature is sufficiently high.

The electrical conduction in a composite material can be represented by the energy band model. In the case of a Metal-Oxide-Metal (MIM) capacitor, the energy band in the rest state and in the case of an external voltage U ext applied at one electrode is shown in Figure 1.24 (both metal electrodes are identical). Figure 1.24a shows the Fermi level E F (electrochemical potential of electrons) in equilibrium, qφ B is the thermodynamic work (required to remove an electron from the material to a state of zero total energy E F ), q is the electron charge and φ B is the junction barrier height.

Figure 1.24b shows an external voltage U ext applied in the electrodes that modifies the band diagram; now the Fermi level E F is not in equilibrium (pointed lines represent the rest position), the -qφ B energy is deformed to conserve the equilibrium.

The conduction mode originates from a single or multiple conduction mechanisms. Two groups of conduction mechanisms can be defined, depending on electrical properties of either the insulator material or the electrode-dielectric contact [START_REF] Chiu | A Review on Conduction Mechanisms in Dielectric Films[END_REF][START_REF] Gloos | Properties of native ultrathin aluminium oxide tunnel barriers[END_REF][START_REF] Sze | Physics of Semiconductor Devices[END_REF]: Bibliographical Review 1.3.2.3.1 Electrode-limited conduction mechanism Electrode-limited conduction mechanisms, depend on electrical properties at the electrode-dielectric contact.

Schottky emission Also known as thermo-ionic emission, an electron flux is obtained by a thermal excitation of carriers -i.e. enough energy to overcome a potential-energy barrier (Figure 1. [START_REF] Girbau | A Low-Power-Consumption Out-of-Plane Electrothermal Actuator[END_REF]. The electrons at one metal side overcome the energy barrier at the interface to go to the other metal electrode -i.e. without the application of a strong voltage. The Schottky emission is the most frequent conduction mechanism at high temperature.

The current density J SE of Schottky emission can be related to the electric field (ξ ) and temperature (T ) by:

J SE = 4πqm * (kT ) 2 h 3 •     -q φ B -qξ 4πε kT     (1.36)
Where q is the electron charge, m * is the effective electron mass in the oxide, k is the Boltzmann's constant, h is Planck's constant, φ B is the junction barrier height, ξ is the electric field across the oxide and ε is the permittivity of the oxide. Fowler-Nordheim (FN) tunnelling If the applied electric field is large enough at an electrode side, the electrons see a triangular barrier, which they can cross to reach the conduction band of the second electrode and generate a tunnel current (Figure 1.26). At low temperature (room temperature), the Schottky emission is suppressed and tunnelling current is dominant.

The current density J FN of Fowler-Nordheim tunnelling can be related to the electric field (ξ ) by: Direct tunnelling Uses the same principle of FN tunnelling, the difference lies in the applied voltage and the insulator thickness. Therefore, direct tunnelling is more dominant in oxide thinner than 3 nm [84] -i.e. FN tunnelling is dominant at thicker oxide and higher electric field. If the voltage across the insulator is small, the electrons see the oxide thickness and the current flows thanks to direct tunnelling through the oxide (Figure 1.27).

J FN = q 2 8πhφ B • exp -8π √ 2qm * 3hξ (1.37)
The current density J DT of direct tunnelling can be approximated by:

J DT ≈ exp -8π √ 2q 3h • m * φ B • κ • t eot (1.38)
Where κ is the relative dielectric constant of the oxide layer and t eot is the equivalent oxide thickness.

The direct tunnelling occurs when the carriers are excited and allowed to cross the band gap (thinner than FN tunnelling case).

Bulk-limited conduction mechanism

The bulk-limited conduction mechanisms depend only on the properties of the dielectric itself. The current density J PF can be expressed as:

Bibliographical Review

J PF = qµN C ξ • exp     -q φ T -qξ πε kT     (1.39)
Where µ is the electronic mobility, N C is the density of states in conduction band and φ T is the depth of traps potential well.

Mechanical and electrical contact

Grain-boundary-limited conduction The grain boundary resistivity is not negligible in the total resistivity of the insulator. In a polycrystalline material the resistivity of the grain boundaries can be higher than that of the grain itself.

The resume of equations and the voltage/temperature dependence for each conduction process are presented in Table 1

.3. Process Expression Voltage(V )& temperature(T )dependence Thermionic emission J = A * • T 2 • exp -q φ B - qφ i 4πε i kT ∝ T 2 • exp q kT (a √ V -φ B FN Tunnelling J ∝ ξ 2 i • exp -4 √ 2m * •(qφ B ) 3/2 3qhξ i ∝ V 2 • exp -b V PF emission J ∝ ξ i • exp   -q φ B - qξ i πε i kT   ∝ V • exp q kT (2a √ V -φ B ) Ohmic J ∝ ξ i • exp -△E ac kT ∝ V • exp -c T Ionic conduction J ∝ ξ i T • -△E ai kT ∝ V T • -d T Space-charge-limited J = 9ε i µV 2 8d 3 ∝ V 2
Table 1.3 Basic conduction processes in insulators [START_REF] Sze | Physics of Semiconductor Devices[END_REF] where, A * = effective Richardson constant (A * = 4πqm * h 3 ), φ B = barrier height, ξ i =electric field in insulator, ε i =insulator permittivity, m * =effective mass, k is the Boltzmann constant, d=insulator thickness, △ E ac =activation energy of electrons, △ E ai =activation energy of ions. V ≈ ξ i d, a = q 4πε i d , b, c, and d are constants. The dielectric is an undesirable material that prevents an electrical contact -in the best case, it allows a weak current flow. The dielectric observed in this work grows from an Al-Cu thin layer. The sample itself is manufactured using photolithography process, which involves several chemical steps -e.g. etching, polishing, and cleaning. All this chemical process contributes to generate a native oxide that is not a standard Al 2 O 3 oxide with well-known characteristics.

MEMS reliability

1.4 MEMS reliability

In order to analyze the reliability of MEMS system, one must understand the possible ways in which the system may fail. Through the study of MEMS reliability, we can ensure a durable functioning of these devices in time. Depending on the type of MEMS, the failure modes can be of diverse origin, and tests to detect and prevent failures will be also different. Therefore, and in accordance with the type of structures studied, many failure sources can be related to failure devices, among others: Mechanical fracture, stiction, wear, delamination, vibration and shocks, electrostatic discharge and dielectric charging, radiation effects, temperature, humidity.

In this work the reliability is studied among three axes: the mechanical, the thermal and the contact issues.

Mechanical issues

The moving parts of MEMS devices makes it vulnerable to material flaws. Thus, material issues in MEMS can be divided into three categories:

The material set: The interest of integrating the MEMS device in the microelectronics component, forces us to use the CMOS facilities -i.e. only a limited range of materials may be used in fabrication flow [15,[START_REF] Gaddi | MEMS technology integrated in the CMOS back end[END_REF]12].

In order to ensure the material compatibility and a reliable process integration, any new material used has to be tested. The fabrication and testing time can take several months. Even a compromise between higher mechanical performances and optimal transducer elements (electrical, thermal, oxidation) should be found.

The principal materials used in CMOS fabrication include: Silicon wafers as the substrate and deposited layers of: Polycrystalline silicon for resistive elements; aluminium and copper as conductor; silicon dioxide, silicon nitride and titanium nitride as electrical insulation and/or passivation.

The manufacturing process: Like in the previous section, the new structure must be compatible with CMOS manufacturing process -i.e. photolithography, additive processes, planarization, bonding. Additionally, to create a free-standing part into the CMOS device, an etching step in fabrication process is then added. This extra manufacturing step, which normally is an etch of sacrificial SiO 2 , adds chemical contamination [START_REF] Chen | Effect of process parameters on the surface morphology and mechanical performance of silicon structures after deep reactive ion etching (DRIE)[END_REF].

Likewise, the stress generated in the different fabrication steps causes reliability problems (Figure 29) and reduces the operating performance in service of electronic devices. The known failure mechanisms are principally: voids [START_REF] Yue | Stress Induced Voids in Aluminum Interconnects During IC Processing[END_REF][START_REF] Yost | Stress-driven diffusive voiding of aluminum conductor lines[END_REF]; hillocks [START_REF] Lahiri | Mechanical stress induced void and hillock formations in thin films[END_REF]; cracks -which impede a proper functioning of the system. The specific design: The simulation tools help to create the device with a reduced number of prototypes, reducing the cost of final product. The mechanical properties of materials may vary considerably between macro-and micro-scale. These differences should be considered in order to create a robust and durable design:

Elastic properties, essential for accurate performance, principally in cantilever beams and diaphragm devices [4,[START_REF] Haque | Deformation mechanisms in free-standing nanoscale thin films: a quantitative in situ transmission electron microscope study[END_REF].

Strength. The MEMS devices being exposed to large deflexion levels. The strength of both ductile and brittle materials can be very dependent on the scale and fabrication flow [START_REF] Chen | Dependence Of Silicon Fracture Strength And Surface Morphology On Deep Reactive Ion Etching Parameters[END_REF].

Even if the thermal issues can be treated in the mechanical part, the thermal actuation of the devices suggests a separate study.

Thermal issues

The electrical actuation by Joule heating induces high temperatures e.g. in the contact zone and in the geometrical constrictions [START_REF] Timsit | Constriction Resistance of Thin Film Contacts[END_REF]. The thermal actuation cycles undergone by the structure along its life generate additional thermal stress [START_REF] Yeo | Characteristics of thermal stresses in Al(Cu) fine lines. I. Unpassivated line structures[END_REF]. As the device is made out of Al, the critical point is the contact zone, which can reach quickly very high temperatures compared with the melting temperatures.

The contact is reached in a "hot switching" actuation mode, on a rough surface, this generates a concentration of electric flow. The anode temperature increase caused by a current flux is not easily calculated. If a semi-infinite solid at ambient temperature is considered and with a constant electron flow -i.e. increase the temperature in the surface -a rough estimation of the temperature (T ) at the surface can be calculated using the Carslaw expression [START_REF] Carslaw | Conduction of Heat in Solids[END_REF]:

T = 2F 0 K Kt πρ m c (1.40)
where F 0 is the constant electron flow, K is the thermal conductivity of electrode material, ρ m is the material density, t is the time and c the specific heat. Poulain et al. [START_REF] Poulain | A Nano-Scale Investigation of Material Transfer Phenomena at Make in a MEMS Switch[END_REF] used this equation to study the temperature in function of time with gold (Au), ruthenium (Ru) and platinum (Pt) as electrode materials. Using Equation 1.40 to calculate the time when an aluminum contact reaches the melting temperature (660 • C) is then estimated. Therefore, for Al electrodes, where K=237 W m -1 K -1 , ρ=2700 kg m -3 and c=900 J kg -1 K -1 [START_REF] Freund | Thin Film Materials[END_REF] (using F 0 = 0.1 × 10 12 W m -2 [START_REF] Poulain | A Nano-Scale Investigation of Material Transfer Phenomena at Make in a MEMS Switch[END_REF]), 660 • C is reached at 20 ns, an almost instantaneous heating.

Contact issues

Gold, ruthenium and platinum are often used in electro-mechanical contacts due to their good electrical conductivity and corrosion resistance [START_REF] Poulain | A Nano-Scale Investigation of Material Transfer Phenomena at Make in a MEMS Switch[END_REF][START_REF] Hou | Al/Au composite membrane bridge DC-contact series RF MEMS switch[END_REF]. However, as already mentioned, in this work, only aluminium is used for its easy integration in a CMOS fabrication and its good electrical conductivity (2.8 × 10 -8 Ω m).

The contact issues are related to the degradation of metal contact during switch operation, which has the effect of degrading the quality of the metal-metal contacts.

Two types of contact exist: "hot switching" is when switching occurs with a signal or potential across the electric contact, "cold switching" when no field exists across the contacts as the switch opens or closes.

The main reliability problems related to the contact are:

1.4.3.

Surface contamination

Usually from the manufacturing stage, the surface contamination prevents good electrical contact [START_REF] Dickrell | Silicone Oil Contamination and Electrical Contact Resistance Degradation of Low-Force Gold Contacts[END_REF]. In some cases, depending on the material used, the contamination can even help the development of insulating oxide in the surface. This is the case with Al, which oxidizes in contact with air. The thickness of this contamination layer depends on the chosen material, the fabrication process and the operating environment [START_REF] Ma | Metal contact reliability of RF MEMS switches[END_REF].

The switch operation can also generate contamination in the surfaces in contact. The friction of surfaces at nanoscale should be considered [START_REF] Johnson | Contact mechanics and the wear of metals[END_REF], this friction can generate polymer formation or accumulation of the contamination [START_REF] Carré | Polymer-Aluminium Adhesion. I. The Surface Energy of Aluminium in Relation to its Surface Treatment[END_REF].

Electric arc

The electrical arc failure is associated to "hot switching" operation mode [15]. Thanks to small separation of electrodes and the applied electrical current the formation of an electrical arc is possible. Therefore, if the voltage between electrodes is larger than the ionization voltage of the gas between the electrodes an arc can be produced.

The arc generation is defined by Paschen's corrected formula [START_REF] Poulain | A Nano-Scale Investigation of Material Transfer Phenomena at Make in a MEMS Switch[END_REF] which links the breakdown voltage (V B ) to the product of the gas pressure (p g ) by the gap distance (d).

V B = B * p g d ln(A * p g d) -ln ln 1 + 1 γ se (1.41)
Where γ se is the secondary electron emission coefficient at the cathode, A * is the saturation ionization in the gas at given U/p g , and B * is a constant related to the excitation and ionization energies. The minimum in this curve occurs when the electronic mean free path is just barely sufficient to allow electrons to gain the ionization energy. Fig. 1.32 Typical evolution of the current and voltage as the function of the contact gap just before contact closure [START_REF] Poulain | A Nano-Scale Investigation of Material Transfer Phenomena at Make in a MEMS Switch[END_REF] In Figure 1.32 (Poulain et al. [START_REF] Poulain | A Nano-Scale Investigation of Material Transfer Phenomena at Make in a MEMS Switch[END_REF]) an electric current is measured although the two Au/Au electrodes are separated by 10 nm. Several irregular current peaks of a few hundreds of nA are observed.

Material transfer phenomena

Material transfer or sintering is another phenomenon observed in "hot switching" (Figure 1.33) [START_REF] Poulain | A Nano-Scale Investigation of Material Transfer Phenomena at Make in a MEMS Switch[END_REF]. If anode and cathode surfaces approach (a), the roughness amplifies the electric field (ξ ) locally by tip effect (b), β ′ is an amplification factor. Then the contact surfaces are sufficiently close to trigger the electron flux by Fowler-Northeim effect (c). The electron flux collides with the anode, locally heating the anode (d) and causing evaporation (e). A compromise can be found on the initial roughness of the contact surface. The rougher the surface is, the worse the electrical contact (too few contact spots). A large contact force can be needed to grant electrical contact by flattening asperities and increasing the real contact area. Nevertheless, if after repetitive contact the surfaces become too smooth, a risk of stiction between moving parts appears.

Stiction

The stiction is an adhesion of surfaces of mobile part -e.g. contacting parts. Stiction occurs either in fabrication process or in operation life of device. The main mechanisms of adhesion are [START_REF] Tas | Stiction in surface micromachining[END_REF][START_REF] Jensen | Adhesion effects on contact opening dynamics in micromachined switches[END_REF]:

Capillary forces, Also known as meniscus forces, they are due to liquid residues. In the MEMS technologies, a poorly controlled packaging is the origin of this liquid. The capillarity force (F m ) for a sphere/plane contact is defined by [START_REF] Chen | Separation modes in microcontacts identified by the rate dependence of the pull-off force[END_REF]:

F m = 4πβ γ 1ν cos(θ ) (1.42)
where γ 1ν is the surface tension of water (72.8 mN m -1 at 20 • C), β the radius of the sphere and θ the contact angle.

Hydrogen bridging, Hydrophilic silicon surfaces contain adsorbed water layers generating hydroxyl groups. When the contact is closing, hydroxyl groups can form strong hydrogen bonds.

Electrostatic forces, They are generated by the Coulomb attraction between charged surfaces in contact -e.g. polarised from contact potential during actuation. It is caused by differences in the local energy states and electron work functions, the energy is:

F(x) = 1 2 ε air S U 2 d 2 (x) (1.43)
Where U is the potential difference between surfaces, S is the electrode surface, d is the initial gap, x is the displacement and ε air the air permittivity. At small separations (∼100 nm), electrostatic forces equal Van der Waals forces.

Van der Waals forces, Van der Waals forces result from the interaction between instantaneous dipole moment of atoms. The attractive force of Van der Waals and the repulsive force of Born are included in this kind of adhesives forces. The Van der Waals force (F V DW ) in given in function of the separation (d) by [START_REF] Tonck | Electrical and mechanical contact between rough gold surfaces in air[END_REF]:

F V DW (d) = - A H 6πd 3 (1.44)
where A H is the Hamaker constant (∼ 2.8 × 10 -19 J [103]).

Figure 1.34 shows the bond strength in function of separation distance between electrodes for all the forces described above. The capillarity energy was calculated for water using a surface tension of 73 mN m -1 and a contact angle of zero degrees with a reference distance d of 2 µm. For the electrostatic energy a voltage difference of 0.5 V was used and for the Van del Waals energy a Hamaker's constant of silicon in air equal to 27 × 10 -20 J has been used.

Summary

This chapter has given a brief description of Microsystems -also known as Micro-electromechanical systems or just MEMS. The MEMS devices are generally designed either to measure a physical variable (sensor) or to perform some functionality (actuator). This kind of devices are in continuous evolution. During the last years, it moved from standalone parts to complex Systems on Chip (SoC). This evolution is justified by More than Moore evolution principle, which tends to integrate additional functionalities in a single chip. This integration is the source of many issues in the fabrication process, which are studied in this chapter.

In the first part of this chapter, the CMOS-MEMS are presented. After presenting several models, the BEOL-MEMS concept is chosen to make our device, because it allows an easy integration on CMOS process flow. Then, a study about mechanical stress in Al metallic layer is carried out. This study helps to understand the mechanisms developed during fabrication process, which cause stress in the final product. This residual stress, normally harmful to the devices is used on our structure as a trigger of a switch function.

A thermo-electro-mechanical switch embedded in the BEOL (Al) of CMOS technologies is presented in chapter 2, which is activated by the Joule effect.

The physics of the contact, both mechanical and electrical, is studied next. The mechanical part of the contact, is explained thanks to different approaches: statistic, fractal, and deterministic models. The contact in the structures is performed by a lateral movement, so the contact surfaces, perpendicular to the plane, are difficult to study.

As the switch is made of aluminium, an oxide layer of (2 to 4) nm will grow on the surface, including the contact area. This oxide prevents an ohmic electric conduction, so other conduction mechanisms are studied.

After reviewing the different electrical conduction modes in the insulators, the Fowler-Nordheim tunnelling effect should be considered.

The final part of this chapter is dedicated to the study of MEMS reliability. Three parts were presented:

• The mechanical issues, in particular the manufacturing problems and the harmful stress.

Both must be considered in the manufacture of the device.

• The thermal issues, the 'hot switching' mode is explained. High temperatures, around 660 • C (melting point of Al) can be achieved in some 50 ns for an electrodes made out of aluminium.

• The contact issues, the risks of electric arcs and sintering are studied for an operation mode. The stiction is also studied, which can be present during manufacturing or operation mode.

This bibliographical chapter gives us the fundamental understanding of the physical principles discussed later in this thesis. Not only for the manufacturing stage, but also for the actuation and robustness studies.

Résumé français

Une description des microsystèmes est réalisée dans ce chapitre -dispositifs connus par leur acronyme anglais MEMS (Micro-electro-mechanical systems). Les MEMS sont généralement conçus soit pour transformer l'état d'une grandeur physique observée en une grandeur utilisable (capteur), soit pour convertir un signal physique en un autre (transducteur).

Ce type de dispositifs est en permanent évolution, et ces dernières années ils sont passés d'un mode de fonctionnement autonome à une participation à un système plus complexe -Système sur une puce (SoC, de son acronyme en anglais System-on-Chip). Une telle transformation est justifiée par le principe "More than Moore", qui contrairement au principe "More Moore" qui cherche à diminuer la taille des dispositifs, vise à intégrer des fonctionnalités supplémentaires dans une seule puce. Cette intégration est une source de nombreux problèmes étudiés dans ce chapitre.

Dans la première partie de ce chapitre, les dispositifs CMOS-MEMS sont présentés. Le concept BEOL-MEMS est choisi parmi d'autres méthodes pour réaliser nos dispositifs, du fait de leur intégration simplifiée dans un procédé de fabrication CMOS. Ensuite, une étude sur la contrainte mécanique dans la couche métallique d'Aluminium est réalisée. Cette étude aide à la compréhension des mécanismes qui provoquent l'état des contraintes dans le produit final. Cette contrainte résiduelle, nocive pour la plupart des applications, est utilisée dans la structure comme déclencheur d'une fonction de commutation.

Un commutateur thermo-électro-mécanique, intégré dans le BEOL (Al) des technologies CMOS et activé par l'effet Joule est étudié dans ce travail de thèse.

La physique du contact, à la fois mécanique et électrique est étudiée ensuite. La partie mécanique du contact est expliquée à travers différents approches : statistique, fractal, et déterministe. Comme nous ne connaissons pas les surfaces étudiées (contact latéral), la partie mécanique du contact reste difficile à modéliser.

Comme l'interrupteur est fait en aluminium, une couche d'oxyde de 2 nm to 4 nm croît sur la surface, notamment dans la zone de contact. Cet oxyde empêche une conduction électrique ohmique ; ainsi d'autres mécanismes de conduction sont étudiés. Après avoir examiné les différents modes de conduction électrique dans les isolants, l'effet tunnel Fowler-Nordheim semble être le plus approprié à étudier.

La dernière partie de ce chapitre est consacrée à l'étude de la fiabilité des MEMS. Trois parties sont traitées ici :

• Les problèmes mécaniques, en particulier les problèmes de fabrication et la contrainte résiduel sont étudiés.

• Les problèmes thermiques : le mode de "commutation à chaud" est expliqué. Pour des électrodes faites en aluminium, des températures d'environ 660 • C (point de fusion de l'Al) peuvent être atteintes en moins de 50 ns.

• Les problèmes de contact, les risques d'arcs électriques et le frittage sont étudiés. L'adhérence est également étudiée, qui peut être présente lors de la fabrication ou pendant le fonctionnement de la structure.

Ce chapitre bibliographique nous donne les bases pour comprendre les principes physiques discutés plus loin dans ce manuscrit, pour l'étape de fabrication, mais aussi pour les études d'actuation et de robustesse de la structure.

Remember that the operating cycle of the device is composed by the steps of: fabrication; release; actuation and optimization. In this chapter the mechanical stress state existing in structures, before and after release is studied. In order to quantify and understand the stress present in the structures, from their fabrication to released devices, FEM study and physical characterisation are performed.

The stacks of material are those used in microelectronics industry (B) -e.g. the metallic layer containing the structures is composed principally by Al/TiN.

The fabrication process introduces residual stress in the metallic layer. This stress evolves with steps process: deposition of a metallic layer, annealing and film patterning. This evolution is studied in the first part of this chapter.

Once the structures have been manufactured, the release of freestanding parts is performed. The partial relaxation of residual stress generates a rotation in the structure. The angle of this rotation is then measured on Scanning Electron Microscopy (SEM) images. In order to reproduce this rotation in a virtual laboratory, a FEM model is created. First a linear elastic simulation of the rotation at release is performed. The design of this cross-shape CMOS-MEMS is based on the work of Horsfall [1] on aluminium and Kasbari [2] on copper structures. The structure, integrated in a single layer, is composed of two metallic expansion arms connected to a metal central pointer, thus forming a conducting asymmetrical cross Figure 2.1.

From thin film to line stress state

As explained in fabrication process description (B), a thin Al film is patterned to shape our structures. The structure design, basically composed of lines, imposes different levels of stress compared with an infinite 2D film. Thus the metallic biaxial stress measured on unpatterned wafers evolves to a longitudinal uniaxial stress in fabricated structures -i.e. from full wafer to lines.

The stress in thin films can be measured by optical interferometry [5], X-ray diffraction (to measure lattice spacing [6] or to measure substrate curvature [7]), and laser scanning [3]. The simplest method is to measure the substrate curvature and use Stoney's approximation [8,9]. This approximation gives the apparent stress in M3 metallic layer. The technological stack is shown in cross section in Figure 2.3.

The film consists of:

• 0.52 µm of patterned standard CMOS metallic layer (M3) -Ti 7.5 nm / Al 470 nm / TiN 45 nm.

Then the substrate consists of: In this standard fabrication process, the M3 metallic layer (522.5 nm) represents only 0.06 % of the silicon wafer thickness (750 µm). This configuration allows the stress determination according to the film/substrate model proposed by Stoney [8]. A test is then performed to confirm the ∼300 MPa stress value found in literature [4,10].

Managing the stress state in structures

The thermal cycles during the fabrication process of the device generate a thermo-elastic deformation, which is accumulated. Then, the effect of the temperature can produce a microstructure modification and/or a stress exceeding the elastic limit of the film, which finally determine the stress state of the film. The stress is thus built by the difference of expansion coefficients between each film and the substrate -e.g. 23 × 10 -6 K -1 for aluminium and 3 × 10 -6 K -1 for Si [4]. If the expansion coefficients are isotropic, a biaxial stress is generated. After layer deposition, during step-cooling, the Al layer shrinks more than Si which produces a tensile stress state on the Al.

Full Wafer stress approximation: Stoney's formula

In the case of a tensile stress (+), the face of substrate bonded to the film becomes concave. On the contrary, for a compressive stress (-), the film/substrate interface becomes convex.

The relationship between radius of curvature R w f = 1 k w f and film main average stress is:

σ f = k w f • E s • t 2 s 6 • (1 -ν s ) • t f (2.1)
E s being the Young's modulus of the substrate Si; t s the substrate thickness; ν s the Poisson's ratio of the substrate; and t f the thickness of the film (Al).

The Stoney formula has the key property that the relationship between curvature k and thin film force f does not depend on properties of the film, only the Young's modulus and Poisson's coefficient of the substrate are involved.

An experiment on full wafers has been carried out to study the effect of different layer thicknesses of M3 on measured stress (the initial courbure is considered as initial state). The capacitive measurement is used to determine the 8 inch wafer shape. During the measurement, the wafer is positioned horizontally onto three pins separated by 120°, they measure the distance between the wafer and each sensor. Additionally, this study gives the individual stress on Al and TiN sublayer. The Design of Experiment (DOE) involves 6 wafers described in Table 2.1. Using Stoney's formula with equi-biaxial stress assumption, an apparent tensile stress can be found in M3 metallic layer. I consider this M3 layer composed by: a Ti/Al layer, which will be as a single layer, noted Al, and a TiN layer. Results of this measurement are shown in Figure 2.4.

Wafers #04 and #05 have only an Al layer (470 nm of thickness). This allows obtaining a bilayer stress directly by Stoney's approximation. Therefore, the tensile stress of +290 MPa consists in the homogeneous biaxial residual stress on a single Al layer, which is confirmed by the literature [3,4,10]. A tensile stress near 150 MPa is obtained on the standard bimetallic layer (wafers #01 and #02) -i.e. composed by 470 nm of Al and 45 nm of TiN. This difference is explained by Young's modulus of TiN, seven times stiffer (500 GPa) than Al (70 GPa) -TiN has a compressive stress [11,12].

The thin layer of TiN (20 nm) over Al (470 nm), #03, gives a tensile stress level slightly higher (15 %) than standard Al/TiN bilayer (#01 and #02). Thereby, the tension stress in the Al film is more compensated when the TiN film in compression is thicker [11]. However, this stress variation is slight when we compare the thickness of theTiN layer of the Al layer due to TiN being much thinner than Al-i.e. for a TiN layer thickness from 10 to 5 percent of Al thickness the stress is 15 % higher.

With wafers #06 and #03 that have the same TiN thickness, we can study the influence of the Al thickness. Conforming to the principle of "thinner = less compensated", the tensile stress decreases by 45 % for a thick Al layer -compared with #03 of M3 layer configuration (Al 470 nm / TiN 20 nm).

Knowing the average M3 layer stress, furthermore the stress and thickness of Al layer, the TiN layer stress can be approximated with the following formula [9]:

σ TiN = 1 t TiN • (σ M3 (t Al + t TiN ) -σ Al t Al ) (2.2)
σ M3 being the apparent biaxial stress in M3; σ Al the apparent biaxial stress in Al (+290 MPa); σ TiN the apparent biaxial stress in TiN; and t x the thickness in x-layer (Al and TiN).

In this way, considering the wafers #04 and #05 and thus the single aluminium layer stress Increasing the thickness of TiN layer decreases the compressive stress [13] i.e. the stress depends on the thickness, because the structure varies with thickness.

σ Al =+290 MPa,
In order to verify our finding, full wafer samples were prepared with a 90 nm TiN deposition over Si substrate. Bow was measured and Stoney's approximation was made. The compressive stress in TiN of -600 MPa was deduced, confirming our results (Table 2.2), therefore: in accordance with [11,12].

• -2.1 GPa if TiN 20 nm
For the Al layer, these experimental data are compared with previous works by Bostrom [14] and Vayrette [15] in Figure 2.5.

Fig. 2.5 Collected data of full wafer stress measures

The stress values are coherent with data reported in previous works. While Bostrom used a composite metallic layer composed of aluminium sputtered from a target containing 0.5wt%Cu and 1wt%Si, Vayrette and the present work use the same aluminium compound (Al 0.5wt%Cu).

The next step in the fabrication process is to pattern the film in order to create the devices in a single layer. This process creates a structure in the form of beams with free sides. Some of the stress components are thus relaxed. The next section analyses the resulting stress state in the lines.

Stress state in a line

The stress measured in the previous section requires a biaxial condition, considering x and y component of stress tensor as equal. The structures studied here have a beam "1D" shape, specifically the expansion arms and the pointer -30 µm in length, 1 µm in width and 0.47 µm in height. This form factor makes it necessary to redefine the stress component in order to consider the supplementary free surfaces and the subsequent stress relaxation.

Several theoretical and experimental stress studies in patterned line structures have been performed [16,17,6,18]. Efforts will focus on finding a relationship between biaxial stress (σ ) descriptions and realistic residual stress in principal axes on line shapes, where the x-axis is the width of the arm, and the y-axis is along the arm -i.e. transversal σ xx , longitudinal σ yy , and normal σ zz along the arm.

A FEM study in unpassivated Al lines shows that the stress along the longitudinal direction is larger than in the transversal direction. This study was performed by Sauter [17] and confirmed experimentally by Flinn and Yeo [16,18]. Maniguet [6] measured the stress in unpassivated W lines. All these studies confirm that the stress state is unequally biaxial and that stresses decrease as line width decreases. Maniguet ([6] found for his tungsten lines a relationship for longitudinal stress component σ yy , which can be applied in aluminium lines because the relationship uses only material properties.

σ yy = σ • (1 -ν f ) (2.3)
where σ is the biaxial stress component and ν f Poisson's coefficient of film.

The Maniguet experiment was performed by X-ray measurements on striped lines over substrate. A planar stress state is assumed (normal component of stress (σ zz ) is assumed to be zero). Two line widths (w) of 1.7 µm and 4 µm have been studied, with the same homogenous height (h) of 0.65 µm. These lines are created from a uniform film which is then patterned, over 500 µm of Si substrate.

σ xx = σ • 0.08) σ xx =+20 MPa
As in [6], the transversal stress is considered in the following to be equal to zero to simplify the future FEM analysis.

The other bilayer component, the TiN layer, has a compressive stress state of σ =-1.3 GPa for 45 nm of TiN full wafer. The Equation 2.3 is used to extract the longitudinal stress along a line:

• Longitudinal stress component σ yy =-750 MPa • Transversal stress component (from Figure 2.6b:

σ xx = σ • 0.08) σ xx =-100 MPa
The transversal stress, representing 10 % of longitudinal stress, is considered in the following to be equal to zero to simplify the future FEM analysis.

The bending observed in freestanding structures is an obstacle on the way to the expected operational mode -i.e. movement in-the plane of mobile structure, without contact with metal layer M1. This section aims at understanding, modelling and reducing this warp defect. In order to understand and quantify the bending in a composite layer, Timoshenko's analysis is recalled next.

Stress study in multilayer case

The beam deflection is a function of its thickness and the residual stress. It is derived from Timoshenko's analysis [19] of a thermal bimetallic cantilever. The next development considers Liu [20] and Vilms [21] studies, which analyse bending in a three-layer metallic composite. The thin 7.5 nm Ti film (below Al), representing 1.6 % of Al thickness, is not considered in the following calculations. Thus a bilayer case is considered for the study.

The model considers no external forces imposed to the cantilever, assume two concentrated forces applied at the centre of their respective layer and the moment taken from the bottom of the Al layer. Forces (F i ) and moments (M i ) of the cantilever must be zero at equilibrium: Fig. 2.8 Bending representation in a three-layer that became a bilayer for the analysis

F Al + F TiN = 0 (2.4) M Al + M TiN = -F Al • t Al 2 + F TiN • t Al + t TiN 2 (2.5)
From beam theory and Fertis' study [22] M = -

1 ρ • (E Al I Al + E TiN I TiN ) (2.6)
where ρ is the radius of curvature of the cantilever beam, which is a function of its length L and tip deflection d:

d = ρ 1 -cos L ρ (2.7)
Taylor series expansion of cos(θ ) leads to,

ρ ≈ L 2 2 • d (2.8)
For a cantilever beam having a rectangular cross section, the moment of inertia is given by:

I i = w • t 3 i 12 , i = Al, TiN (2.9)
where w is the width of cantilever. Equation 2.5 and Equation 2.6 express the beam equilibrium in terms of axial forces and radius of curvature:

Managing the stress state in structures

1 ρ • ∑ i=Al,TiN E i • I i = -F Al • t Al 2 + F TiN t Al + t TiN 2 (2.10)
For the bilayer, the total strain at the interface of paired layers must be equal. The total strain is due to a change in temperature, to the residual stress, the axial force and the bending deformation,

ε total = ε temp + ε stress + ε shear + ε bend (2.11)
Release is performed at room temperature, hence ε temp = 0

σ Al E Al + F Al E Al W Al t Al + t Al 2ρ = σ TiN E TiN + F TiN E TiN W TiN t TiN + t TiN 2ρ (2.12)
Combining Equation 2.4, Equation 2.10 and Equation 2.12, the tip deflection of bilayer can be written as: The tip deflection depends on t i and σ i , i.e. the thickness and stress of each film layer respectively.

d = 3L 2 E Al t
Using the longitudinal stress estimated in previous section1 , and their thickness:

• TiN, σ xx =-750 MPa, 45 nm

• Al, σ xx =+190 MPa, 470 nm Solving numerically Equation 2.14, for the beam length L = 30 µm, the tip deflection is:

d = 4.36 µm
This value has been verified by FEM analysis and a deflection of 4.35 µm is obtained in the tip of cantilever.

Equation 2.14, shows a heterogeneity generated by the Al layer and the TiN layer. The bilayer, with the tensile and compressive stress for Al and TiN respectively, generates a bending of 4.36 µm. If the TiN layer is "removed" (t TiN → 0), Equation 2.14 shows that the curvature will be zero.

This flexion value is for a cantilever -i.e. anchored on one side -the geometry of the device studied is different. This first model points to the bilayer character as a probable cause of the observed bending. It apparently overestimates bending, and a more quantitative comparison with experiment will be performed by FEM analysis in subsection 2.2.2.1.

This calculation explains and justifies the choice of removing the TiN layer from a standard fabrication flow. Hereinafter, the M3 metallic layer is only composed by aluminium with a uniaxial residual stress of +190 MPa. This introduces a change in the process flow, contrary to our initial intention, however this is not detrimental to the project.

Verification of the elimination of bending

In order to confirm a successful release, tilted images of structures fabricated without the TiN layer have been taken. They confirm that the structures are well suspended (Figure 2.9). This qualitative information has been also validated quantitatively by optical profilometry. The previous calculation, performed in a bilayer with opposite stress state shows a bending in the structure -i.e. Al in tension and TiN in compressive stress state. After removing the TiN layer, the optical profilometry shows that the improvement suggested by the model is confirmed by practical implementation.

FEM analysis

In order to create a robust FEM model, the first stage is modelling the mechanical behaviour. Only the arms and the pointer, the functional parts of the structure, are introduced in COMSOL MULTIPHYSICS V.5.0 FEM software (D). The pads are replaced by no-displacement boundary conditions, as they are bulky and fixed on the M1 layer by the oxide. The "ECAD import" module has been used to define the device geometry (Figure 2.11). The relaxation of residual stress state, responsible for the pointer rotation, is realized at initial step.

In order to confirm the efficiency of the removal of the TiN layer, the release model is studied in two cases: first an Al/TiN bilayer is considered, then a single aluminium layer is studied. The formulation used for structural analysis in COMSOL MULTIPHYSICS is a total Lagrangian one, referred to the material initial configuration. This means that the second Piola-Kirchhoff stress tensor S is used together with the Green-Lagrange strain tensor:

ε = 1 2 [(▽u) T + ▽u + (▽u) T • ▽u] (2.15)
The Duhamel-Hooke's law relates the stress tensor to the strain tensor:

S -S 0 = C : (ε -ε 0 ) (2.16)
where C is the 4 th order elasticity tensor, S 0 and ε 0 are the initial stress and strain. For the boundary conditions, zero displacement is imposed at the arm/pad interface (u = 0) -fixed points in Figure 2.11. The other boundaries without any interaction are free to move in any direction.

A linear elastic and isotropic Aluminium material chart was obtained from COMSOL MULTIPHYSICS materials library; Young's modulus is 70 GPa [9], Poisson's coefficient is 0.35 [9]. In the FEM study, the residual stress before release is supposed uniaxial and constant along the longitudinal direction of expansion arms (S Ayy ) and the central pointer (S Pxx ). Consistently, the estimated stress (+190 MPa) is applied to the device as a longitudinal stress in both arms and pointer (see Figure 2.12). Other assumptions were tested:

• Longitudinal stress in arms (A) and pointer (P) S Ayy =S Pxx =+190 MPa ; S Axx =S Pyy =0 MPa,

• Biaxial stress S Axx =S Ayy =S Pxx =S Pyy =+290 MPa, Managing the stress state in structures

• Biaxial stress on the line S Axx =S Ayy =S Pxx =S Pyy =+190 MPa,

• Homogeneous 'y' uniaxial stress S Ayy =S Pyy =+190 MPa, S Axx =S Pxx =0 MPa,

• Homogeneous 'x' uniaxial stress S Ayy =S Pyy =0 MPa, S Axx =S Pxx =+190 MPa, but uniaxial stress was the choice which fits measurements best -longitudinal stress in arms (A) and pointer (P).

The hinge zone is considered as an extension of the arms, with a longitudinal stress. This simplification may generate an error in rotation estimation, due to the stress concentration due to the narrowing of the hinge at its connection with central pointer. Indeed, the stress relaxation in the arm length prevails -helped by the small contact area with the pointer.

The adaptive mesh option is activated in the FEM software, which first solves on an initial mesh and iteratively inserts elements into regions where the error is estimated to be high, and then re-solves the model. The mesh evolves from 5205 to 47 801 tetrahedral elements (3D) after 2 refinement steps Figure 2.12.

Study of the bending of the cross

A linear elastic and isotropic titanium nitride material was defined in COMSOL MULTIPHYSICS materials -Young's modulus is 500 GPa, Poisson's coefficient is 0.25 [9]. Consistently, the uniaxial stress (σ yy =-750 MPa, subsection 2.1.4) is applied to the device as a longitudinal stress in both arms and pointer in the TiN layer.

The TiN layer (45 nm), 10 times thinner than Al (470 nm), is meshed in the 'swept mode': it copies the surface mesh pattern of the aluminium layer, creating triangular prism elements. In order to identify the thickness of TiN for which the flexion is acceptable, a FEM study is carried out. The thickness of TiN evolve from 45 nm to 1 nm and the deflection is simulated. note that 5 or 1 nm of TiN is not realistic, it would be oxidized into TiO 2 and the constants should be changed. However the purpose is just to understand quantitatively the origin of bending and to guide towards a solution to this problem. This measure is carried out for both a cantilever and the cross-shape structure. Figure 2.14 shows the flexion of central pointer in function of the TiN thickness. An acceptable deflection less than 0.5 µm is obtained by 1 nm of TiN, it means in practice that TiN should be elimitated.

For central pointer an acceptable deflection is obtained for a TiN thickness representing less than 1 % of Al thickness. It is also confirmed that by simple elimination of TiN layer, the bending at release step is avoided and the cross-shape structure stays and rotates in the plane (Figure 2.13b).

Study of the rotation of the cross

Now that the in-plane movement is obtained without perturbation using only an aluminium layer for the structure, rotation can be studied reliably. The apparent stress (+190 MPa) is

a = L 2 • θ = 1.2 µm (2.17)
The next sections compares this numerical result with a theoretical model and with measurement on real structures.

Functional devices and residual stress study 81 2.2.2.3 Analytical model of rotation

The isotropic elastic model proposed by Kasbari [23] describes the mechanical equilibrium of the sensor (Figure 2.16). It assumes that the whole structure remains in its initial plane -which is now true theoretically and in practice. The residual stress is considered uniaxial and constant along the longitudinal direction of expansion arms. An analytical model derived from Kasbari [23] is proposed by Vayrette [15]. The model is formulated as the equilibrium between a force due to the arm relaxation and its bending moment in the plane XOY. The rotation stops when the reaction moment counterbalances the relaxation force. The model gives the distance a (Figure 2.16) between initial position before release and final position after release.

a = L p L • d • L • L 0 • (L -L 0 ) d 2 • (2L -L 0 ) + 4 3 • w 2 a • L 0 • α (2.18)
L is the arm length, before release, L p is the total pointer length, L 0 is the unstressed length of the arm and α is a coefficient accounting for the shape of the anchoring of the pointer on the arms (Figure 2.17).

L 0 = L 1 + σ yy E (2.19) α = w h w a (2.20)
where σ yy is the average longitudinal residual stress in the arms and E the Young's modulus. 2.20 a pointer distance from original position -before release -of a =(1.3 ± 0.2) µm was found. This value of a, is in good agreement with the 1.2 µm obtained in FEM analysis.

Rotation measurement after release

After the fabrication process modifications (single Al layer instead of Al/TiN layer), the functional structure has been measured and characterized after release step. In sample preparation, full wafer is sawn into small parts for easy handling. These parts are released in a HF process. The rotation upon release has been measured by SEM image and compared with the analytical and FEM model predictions.

Released structures are observed in the SEM (Zeiss ultra 55), at a tension of 2 kV to avoid charging electrically the metal surface, in the Secondary Electron (SE) mode to enhance relief.

The rotation of central pointer is measured by the separation between the end of the pointer and the contact pad (Figure 2.18).

The rotation due to stress relaxation in the arms has been found to be identical for all the released samples. The rotation angle is 5.0 ± 0.8°. Likewise, it can be characterised by the pointer-pad distance a * .

a * = a 0 + a = a 0 + L 2 • θ (2.21)
where a 0 is the distance from the pointer to the pad before release (0.24 µm) and θ is the rotation angle in rad.

Summary

The functional configuration of the structure is ensured by its stress relaxation -the stress originated at the fabrication process. Three main parts have been presented in this chapter.

First, the study of the origin of the stress, which is analysed from the wafer fabrication process. In order to do so, Stoney's formula has been used in various full wafer configurations. This study allows identifying the principal stresses in the main layers -i.e. aluminium (Al) and titanium nitride (TiN). Then, the stress has been analysed in the structures, which are principally in the shape of lines. The estimated biaxial stress in a full wafer has thus been redefined as a uniaxial stress using the approximation of Maniguet [6]:

σ line = (1 -ν) • σ wa f er .
Secondly, the functional device has been studied at its release stage. This stage consists in etching away the SiO 2 layer (using HF) which completely prevents any movement of the device (Figure 2.2a). The expansion arms are in tension, they shorten at release -asymmetrically pushing the pointer which rotates clockwise (Figure 2.2b) positioning at 1.2 µm from its original position. This release generates a rotation movement in the plane, but also an undesirable bending in the freestanding part -i.e. the pointer. After a study of the stress level in the composite metallic layer, in which theoretical and numerical models showed that the stress in TiN was the cause of the bending, the TiN layer has been removed and experiments have confirmed that this process change does eliminate bending.

Once the structure correctly released, the last part uses FEM as a virtual laboratory. The structure has been simulated to represent its release (rotation) -i.e. a displacement of 1.2 µm for this particular design, sufficient to actuate the switch. Results agree very well with measurement. The numerical model verifies also the correct release -i.e. the device stays in its plane.

La configuration initiale de la structure est assurée par la relaxation des contraintes résiduelles qui ont été créées lors du processus de fabrication. Trois grandes parties sont abordées dans ce chapitre.

Premièrement, l'origine de la contrainte est analysée depuis son processus de fabrication jusqu'à la libération de la structure. La formule de Stoney est utilisée pour identifier la contrainte présente dans les couches principales. Pour cela, différentes configurations de dépôts pleine plaque sont étudiées. Ceci permet d'identifier la contrainte dans les couches supérieures de l'empilement, c'est-à-dire l'aluminium (Al) et le nitrure de titane (TiN). L'étape suivante consiste alors à trouver la contrainte dans les structures qui sont constituées principalement par des lignes. Ainsi la contrainte bi-axiale mesurée en pleine plaque (σ wa f er ) est transformée en contrainte uniaxiale telle qu'elle doit exister dans une ligne (σ ligne ), grâce à des relations expérimentalement validées extraites de la littérature, Maniguet [6] :

σ ligne = (1 -ν) • σ wa f er .
L'étape de fonctionnement de la structure a été étudiée : l'étape de libération. Elle consiste à dissoudre l'oxyde (SiO 2 ) qui entoure la structure et qui empêche tout mouvement (Figure 2.2a). Les bras d'expansion libèrent leurs contraintes résiduelles (en tension), raccourcissent, puis poussent le pointeur central, ce qui génère une rotation (Figure 2.2b). La libération génère aussi une flexion hors plan indésirable du pointeur central. Après une étude de l'état de contrainte des couches qui composent la structure, dans laquelle des modèles théoriques et numériques ont montré que la flexion est due aux contraintes de compression présentes dans la couche de TiN, la flexion du pointeur a pu être annulée en supprimant du procédé de fabrication le dépôt de la couche de TiN.

Une fois la structure correctement libérée, une simulation par éléments finis, analysée dans la dernière partie de ce chapitre, a été réalisée. Cette simulation permet de faire le lien entre la contrainte résiduelle et la rotation de 1.2 µm obtenue après libération. La bonne corrélation entre ces deux paramètres constitue le premier pas vers la construction d'un laboratoire virtuel. Ce laboratoire virtuel sera amélioré au fil des chapitres pour le convertir en un outil de prédiction fiable. Now that the rotational device has been correctly released and characterised, by FEM analysis and measurements, this chapter focuses on the actuation step. The function requested from the device is to act as a switch activated by a heat source -i.e. an actuating current inducing Joule heating. In order to find the thermal characteristics of the device, this chapter is divided into two parts.

Firstly, in order to relate unambiguously the rotation with the applied temperature, a homogeneous heating is performed. A model of the thermal characteristics of the device during actuation is thus achieved and the FEM model created in the previous chapter is improved to include the thermal properties.

Secondly, in order to achieve the contact by a localized heat source, a Joule heating actuation is performed. This part then begins with a FEM model, used as a virtual laboratory tool. With the results of simulation -e.g. the electric current limits to avoid any mechanical deformationthe Joule actuation test is achieved.

The device must be able to commute between the On and Off states. This is why two devices are analysed in this chapter.

The NORMALLY Off structure, studied earlier, moves away from the pad at release step. Thus, in actuation stage, the NORMALLY Off structure moves to reach contact with the pad. On the other hand, a NORMALLY On structure grants a contact at release step and moves away from pad to lose the contact in actuation step.

The NORMALLY Off structure is used in the first part of this chapter to explain the actuation principle. On the contrary, the NORMALLY On structure is used in the last part to measure the contact force obtained at release step. To finish, the numerical model is used to estimate the contact force in actuation step -using a NORMALLY Off design. The thermal actuation is performed by a hot plate placed below the sample that heats the bottom of sample homogeneously. The top part of the sample is heated by conduction through the different layers of the stack (Si, Al, and SiO 2 ).

An advantage of thermal actuation against Joule heating is to perform a measurement without perturbation by electromigration or temperature gradients for instance.

In situ heating in the SEM

A hot stage embedded in the SEM [1] allows applying a controlled temperature and observing the rotation at the same time.

Experimental description

In practice, a sample is glued on a 50 µm thick Tantalum (Ta) foil. The system is attached to a SEM support. Two thermocouples are fixed to the system (Figure 3.1):

• The first one is in the Ta foil to measure its temperature and close the loop of auto control of the electric current applied,

• The second one is in the top of the sample to verify its temperature, which should be close to the chosen nominal temperature.

The system (the sample on the stage) is put inside the SEM. Then the Ta foil is heated by Joule effect.

The imposed temperature steps are programmed as shown in Figure 3.2. The temperature is increased by steps of 50 • C per minute, then held during 6 min to allow enough time to measure the rotation. Both increase and decrease of temperature are performed. The SEM (FEI/Philips XL30 ESEM), used at high tension of 20 kV in the Back-Scattered Electron (BSE) mode, which charges electrically the surface of the sample. To evacuate this charge, a variable pressure environment (0.2 mbar) is used. The SEM images obtained (Figure 3.3) relate the rotation observed to the temperature imposed.

The temperature applied at the bottom of the sample induces a movement in the pointer. For a NORMALLY Off device, the anti-clockwise rotation is observed when the temperature increases from room temperature (20 • C) (Figure 3.3a) to 200 • C (Figure 3.3b). This rotation proves the operating principle. The sample is then cooled to room temperature; the pointer is expected to return to the original position.

Discussion of results

The pointer-pad distance starts from 1.7 µm at room temperature and reaches contact at 200 • C (Figure 3.4). Except above 150 • C where contact takes place with the pad, a linear relationship between the applied temperature (T) and pointer-pad distance (a * ) is observed.

a * = 0.02 • T -2 [µm] (3.1)
After two heating/cooling cycles, the pointer does not return to its original position. A difference of 0.5 ± 0.2 µm is observed at room temperature -in its rest position. The difference between initial and final state could be caused by the plastic deformation. This deformation will be described in next chapter (chapter 4) using FEM analysis.

In the next section, the theoretical model of heat transfer is studied to check that temperature is homogeneous in the experiment. Then the FEM study is carried out and all the results, experimental, theoretical and numerical, are compared.

Theoretical model, first check of the cross temperature

To perform quantitative analyses, we need to verify that the cross temperature, which governs its movement, is approximately equal to the nominal temperature imposed below the device. The estimation of the heat transferred in the system is, in fact, quite complex. There are three distinct modes of heat transfer: conduction, convection and radiation (Figure 3.5).

* = Q * A W m 2
is the rate of heat transfer (Q * ) per unit area (A).

The heat equation (Fourier's law):

q = -k dT dx [W ] (3.2)
where dT dx is the temperature gradient K m and k is the thermal conductivity W mK . The total steady heat flux [2] is composed by: • CONDUCTION, combining the Fourier's law of heat conduction and conservation of energy, in the steady-state case:

ρC p ∂ T ∂t + ρC p u • ∇T + ∇ • (-k∇T ) = q (3.3)
• CONVECTION, the heat conducted away by the fluid/solid interface to the heat transfer, expressed in terms of a convective heat transfer coefficient (h):

q * conv = h • (T s -T ∞ )[W ] (3.5)
where h = q ∆T W m 2 K , T s is the temperature on the surface and T ∞ is the surrounding medium temperature.

• RADIATION, the heat radiated between a surface A at T s and the surrounding air at T ∞ is:

q * rad = ε • σ • (T 4 s -T 4 ∞ )[W ] (3.6)
where ε is the emissivity of the surface, σ is the Stefan-Boltzmann constant

σ = 5.67 • 10 -8 W m 2 K 4
The studied system does not involve heat generation and the analysis is performed at steady heat transfer. Thus, the thermal resistance K•m 2 W concept can be adopted.

Conduction through a wall of thickness (L):

R cond = L k (3.7)
Convection, fluid/solid:

R conv = 1 h (3.8)
Radiation, surface to environment:

R rad = 1 h rad
,where

h rad = ε • σ (T 2 s + T 2 ∞ )(T s + T ∞ ) (3.9)
Considering a one direction heat flow through the stack (Figure 3.5), with only the top layer exposed to convection, under steady-state condition on the metallic layer (M1):

3.1 Thermal actuation (NORMALLY Off structures)

101 q * = k Si T Si -T Si/SiO 2 L Si = k SiO 2 T SiO 2 -T SiO 2 /Al L SiO 2 = k Al T Al -T Al/air L Al = h Al (T Al -T ∞ ) (3.10) = T Si -T Si/SiO 2 R cond,1 = T SiO 2 -T SiO 2 /Al R cond,2 = T Al -T Al/air R cond,3 = T Al -T ∞ R conv (3.11) = T Si -T ∞ R total (3.12)
where

T Si = 200 • C = 473.15 K, T ∞ = 20 • C = 293.15 K and R total = R cond,1 + R cond,2 + R cond,3 + R conv
For a natural convection on a metallic layer, the h value is between 5 to 250 W m -2 K -1 [3][4][5]. In this manner, considering the previous values as limit values in our case, the heat flux should be from:

q * T = h • (T s -T ∞ ) ≈ 5 • 180 = 900 W m 2 = q 1 (3.13) to q * T = h • (T s -T ∞ ) ≈ 250 • 180 = 45000 W m 2 = q 2 (3.14)
The simplified model considers only the upside convection model and is valid for the pad only. The freestanding structure and M3 metallic level have lateral and backside heat transfers that are not considered in the approximation.

From the two previous q *

T values, the temperature T 2 for the interface Si/ SiO 2 are 1 :

T 2 = T 1 - q 1 • dx k = 473.
T 2 = T 1 - q 2 • dx k = 473.15 - 45000 • 375 × 10 -6 130 ≈ 473.02K (3.16)
where dx is the thickness of the layer, k the thermal conductivity and T 1 the temperature in the bottom (Si) of layer. These two values represent the limits according to the literature -i.e. 5 to 250 W m -2 K -1 [3][4][5].

All the interface temperatures of the stack presented in Figure 3.5 are calculated and summarized in Table 3.1 for both q 1 and q 2 estimated values.

Actuation

Material in the stack

Thickness µm

Thermal conductivity W m -1 K -1 [6] q

Temp. in the bottom (T 1 K), for q 1 and q 2

Temp. in the top (T 2 K), for q 1 and q 2 Al 0. The theoretical model proves a quasi-null variation between the temperature applied and the one estimated in surface of the sample. The next section studies the thermal transfer in the solid by a FEM analysis. The heat flux is also corroborated in the next section with the FEM improved model.

FEM study

The previous mechanical FEM model (chapter 2) is improved in order to take into account the thermal behaviour of the device, considering now its complex geometry. In this way, both physics are applied simultaneously to consider a thermal actuation case.

Under the conditions studied, the heat provided at the bottom of the sample is conducted to the patterned top Al layer -almost 375 µm above the heat source.

In order to model the heat transfer in the structure, the "heat transfer in solid" COMSOL MULTIPHYSICS module uses the heat Equation 3.3. For a steady-state problem and a velocity field (u) defined as zero, the first and second term disappear and the equation becomes:

∇ • (-k∇T ) = 0 (3.17)
So as to reduce the computation time of FEM model, the process is divided into two steps.

First, the 2D model with the entire material stack is performed in order to identify the temperature and Heat Transfer Coefficient (HTC) at the device top surface. Then, a 3D model is built to relate the temperature with the obtained rotation.

Thermal properties in technological stack: 2D approximation

The natural convection cooling is studied by 2D model first. Two different approaches can be considered to describe the thermal dissipation:

• Using heat transfer coefficients, Or • Modelling the convective flow of air around the structure, which solves the total energy and momentum balance between the device and the outside air flow around the structure.

The second approach is complex (many physical properties are needed) and very costly; we therefore consider that the Heat Transfer Coefficient approach has a better quality / cost ratio for our purpose to refine the previous 1D analytical model.

The geometrical 2D model, including the material stack in the structure from the bottom to the top is defined by Figure 3.6:

• Silicon (Si), 375 µm • Silicon dioxide (SiO 2 ), 0.8 µm chosen, the value of L (area/perimeter), the air absolute pressure p A and the external temperature T must be defined. The software obtains HTC (h) by the function h = h air (L, p A , T ext ) 1 [9] where L = 51.4µm. A temperature T 0 =200 • C is applied on the bottom of the sample -i.e. at the bottom of the silicon substrate.

Figure 3.7 shows the temperature and the HTC on the top of M1 metallic layer (red dotted line in Figure 3.7a). In order to simplify the display of results, the M1 layer is divided (vertical black dotted line) into: M1 (free layer), pad and equivalent arms zones.

The 200 • C applied in the bottom of the sample remains almost invariant in the top of M1 metallic layer (199.8 • C). The arm is less heated by conduction than the pads (Figure 3.7b), because it loses heat by radiation and convection with the air. The arm temperature will be treated in next section.

Both the temperature and the HTC (30.2 W m -2 K -1 ) are consistent with the values obtained in the previous section (subsection 3.1.2) -i.e. 199.95 • C and for a HTC from 5 to 250 W m -2 K -1 .

In order to relate the temperature in the structure with the rotation of the arms a 3D model is realized in the next section. An average of the temperature estimate on M1 layer (2D FEM model) is used as initial condition in the next section. Meanwhile, the HTC value will be defined in the same way as for the 2D model.

1 If T > T ext then h = k L 0.54Ra 1/4 L if Ra L ≤ 10 7 k L 0.15Ra 1/3 L if Ra L > 10 7 while if T ≤ T ext then h = k L 0.27Ra 1/4 L
Ra L is the Rayleigh number, L is the plate characteristic length (defined as area/perimeter) is a correlation input, and k is the thermal conductivity. The material data are evaluated at T +T ext 2 . The boundary conditions, that is to say the thermal insulation (n • (k∇T ) = 0) and the applied temperature (T 0 =200 • C) are defined in Figure 3.9a. The initial temperature in all system T i =20 • C is also defined as initial condition. The HTC is calculated by the software as external natural convection (like the 2D FEM model) for the following conditions: the external temperature of 20 • C; the absolute pressure of 1 atm; and the plate diameter (area/perimeter) of diffusive surface of 5.14 × 10 -5 m are defined.

The mesh is composed with 72 313 tetrahedral elements, 139 602 prism elements, 11 250 hexahedral elements and 95 946 triangular elements, which are condensed in thinner metallic layers (Figure 3.9b). • The cross is less heated by conduction, and it loses heat by radiation (h rad ∼ 2W m -2 K -1 ) and convection (h conv ∼ 21W m -2 K -1 , which is near to the 30W m -2 K -1 obtained in 2D FEM Model for the M1 metallic layer). Thus, the centre of the cross has the minimum temperature (165 • C). Meanwhile, the average temperature in the arm can be considered at 180 • C

• When the contact is reached, the ends of the pointer should warm up a bit, but the thermal contact is not handled here. .10a shows the distance between pointer and its original position of 0.48 µm. This distance assures the contact, originally at 0.25 µm from the pointer's original position. Remember that contact is not managed in the pad, so a free rotation of pointer is linear.

We can then compare the experimental data (in-situ SEM test) with the fitted curve (subsection 3.1.1) from the FEM analysis. Figure 3.11 shows this comparison: the three curves are very similarly shaped. The FEM model corroborates the data obtained in in-situ SEM test (subsection 3.1.1). The temperature for the M1 metallic layer is identified by a 2D FEM Model, which is then used in a simplified 3D version. Using the 3D model, a temperature of 180 • C in the arms, required to activate the structure, are validated through the arms rotation. As this actuation temperature is well below Al melting point, the structure can be considered viable at this stage.

In order to simulate a voluntary actuation, the next section studies a Joule heating actuation. The maximum temperature obtained in this actuation model (181 • C) supports the temperature needed in a thermal actuation. Indeed, in the thermal actuation experiments the contact is reached at 195 • C in the pad, while in the centre of cross a temperature of 165 • C is estimated. Thus, for a thermal actuation the hottest part is the pad (Figure 3.10b), while for a Joule actuation, the hottest point is in the centre of the freestanding cross (Figure 3.14).

The pointer-pad distance as a function of the applied current is shown in Figure 3.15. A current of 20 mA is needed to reach the contact, which corresponds to 80 mV. The non-linearity of the rotation, as a function of current, will be explained by the Joule heating temperature [10] i.e. heating of the suspended beam causes its resistance to rise. In order to validate this numerical model an electrical test is performed in the next section. Using a nanoprober system, embedded in a SEM, the rotation of a few tens of nanometers can be measured.

3.2 Joule effect actuation (NORMALLY Off structures) 113

Model verification: SEM nanoprobing

In order to characterize the displacement of the structure as a function of the electric current applied, a SEM nanoprobing test is performed. The accuracy of the displacement is in the range of 50 nm and the measurement uncertainly of electric equipment is ±5 %.

Test performed in NORMALLY Off structure

The test is performed on the NORMALLY Off structure. The SEM stage has four motorized tips connected -shown in Figure 3.16. To ensure a current flux in actuation step, tips C and D are connected to actuation pads. Tip B is connected to the right contact pad to detect an electrical contact of the pointer -chosen randomly. Finally, tip A is connected to the metallic M1 layerthe stop layer of the release process. Tip A evacuates the charges generated by the current flow, which otherwise reduces the quality of SEM images.

Discussion of results

The tested structure, after release, has an initial pointer-pad distance of 1.7 µm (Figure 3.18)i.e. at 0 V and 0 A. The electrical current is increased in the arms, activating the mechanism that generates the rotation of the pointer. The current is increased by steps of 2 mA (instantly), then held during 2 min to allow enough time to measure the rotation -the measurement of pointer-pad distance is performed only when the current is increased (when the structure is heated). Since Senturia's model [11] will be used to derive an estimate of the temperature from the current.

The pointer moves towards the contact pad, which reaches contact at 23 mA/0.56 V (13 mW). Intermediate positions between start and contact, with the corresponding voltage/current pairs, are also shown in Figure 3.18.

The relationship between the voltage applied and the pointer-pad distance obtained in nanoprobing test is shown in Figure 3.19. The relationship established in previous section (subsection 3.2.1) is show as "Simulated, pointer-pad distance". Because the pointer-pad distance at rest position is not equal between simulated (1.4 µm) and nanoprobing test (1.7 µm), the "offset of nanoprobing test" curve is added to the graph for easier comparison.

Both the nanoprobing test and the FEM study have the same tendency. However, the pointer/pad distance was estimated at 1.5 ± 0.2 µm, so the results are in the variability of the pointer-pad distance in the structure.

The current needed to reach the contact is 23 mA measured in electrical test, which is close to the 20 mA needed to reach contact for the FEM model.

The difference found for the voltage measured and the voltage simulated, can be explained by the access resistance generated by the probes. The FEM model is an ideal case without In order to understand the mechanism linking the applied current with the temperature reached on the system, the Senturia model is used next.

Theoretical model

First, using an analytical model on self-heating of a resistor Equation 3.19, the global temperature of the structure is determined. The analytical model from S.D. Senturia ([11]) is used to adjust the self-heating phenomenon and determine the increase in temperature Equation 3.20.

For moderate temperature excursions, the temperature coefficient of resistance (using first order Taylor series expansion), shows a linear variation of resistance with temperature:

R = R 0 [1 + α R (T -T 0 )] (3.19)
where R 0 [Ω] is the resistance at reference temperature T 0 (293.15 K), R [Ω] is the resistance at temperature T , and α R is the temperature coefficient of resistance, 3.75 × 10 -3 K -1 for Al. The electrical resistance along the arms at room temperature R 0 =2.47 Ω was measured under low current condition using a 4-point probes method.

The steady-state temperature (T I ) for a steady current I is [11] 1 :

T I = T 0 + R 0 R T I 2 1 -α R R 0 R T I 2 (3.20)
where R T is the thermal resistor [K W -1 ], representing the heat conduction from the resistor to a thermal reservoir held at temperature T 0 .

Using Equation 3.20, the only unknown parameter is the constant R T . To adjust all the measured values of R in the 0-23 mA range, using Equation 3 The curves fit in the small current densities, but for equivalent higher current, the FEM results has the highest temperature. This difference can be explained by the fitted linear first-order model of Senturia, which is valid for low temperature variations.

The actuation of NORMALLY Off structure was conducted by structure heating both homogeneous and localized. Both show that the temperature needed is far from the melting point of aluminium (660 • C), which shows that actuation is feasible in practice.

The theoretical studies, Senturia model for Joule heating and heat equation for thermal actuation, match well with experiments and with the FEM model.

The FEM study fits well the temperature at which the contact is achieved in both cases. Moreover, the FEM model takes into account the mechanical behaviour and the thermal/electrical part, forming a complete actuation cycle i.e. release and actuation.

In the next section, a test structure will be used to determine de contact force for both NORMALLY On and NORMALLY Off design. Then the FEM model is improved to take account of the mechanical contact. In order to measure the contact force after release, another test structure with identical dimensions is created:

• Inverting the expansion arms, like a NORMALLY On structure, the switch turns counterclockwise at release.

• A flexible beam is used instead of a pad (Figure 3.22). Its deflection gives the contact force.

• The vertical pointer (parallel to actuation arms) is a reference to measure the deflection in the contact beam.

The structure with this operation mode is called NORMALLY Onbecause it stays in contact in its rest position (in the absence of actuation). Despite an apparent contact area of 0.2 µm 2 , for this model, the contact is considered as a point contact. The contact of NORMALLY On structure is modelled as a force applied that bends in the plane a flexible beam (Figure 3.22). To obtain the force F a that generates the displacement w at x position (w(x)) on the pointer of length L, the Euler-Bernoulli beam theory is used in a beam with a point load case.

Mechanical contact

119 w(x) = F a 6EI • x 2 (3L -x) (3.22)
F a = 6EIw(x) x 2 (3L -x) (3.23)
where F a is the force applied at point x, E is the elastic modulus, I is the second moment of inertia of the beam cross section and L is the beam length. Comparing with the measured deflection, an equivalent force F a of 2.0 ± 0.5 µN for the device is obtained. The estimated error of 0.5 µN comes from the approximation made in the deflection measurement of SEM image. This estimate will be useful for electrical contact study (chapter 4).

FEM analysis of mechanical contact at release stage in a "NORMALLY On" structure

The FEM analysis is performed on the standard structures, which at release stage reaches the contact pad -e.g. the NORMALLY On structure.

The boundary conditions (Figure 3.23) are similar to those defined previously in the mechanical study. The definition of a contact pair, which in COMSOL MULTIPHYSICS, is composed of a "destination" and a "source" (Figure 3.24) makes possible the study of the mechanical contact. The "source" is defined as mobile (pointer), while the "destination" is defined as fixed and "receives" the contact (pad).

The simulation results of Figure 3.25 show the contact zone and the contact pressure, which reaches a maximum of 50 MPa -i.e. an average pressure of 40 MPa.

The zoom of the contact area (Figure 3.26) allows measuring the surface in contact 4 × 10 -14 m 2 (0.04 µm 2 ) which multiplied by the contact pressure gives the contact force:

F = P • A = 40[MPa] • 4 × 10 -14 [m 2 ] ≈ 1.6[µN] (3.24)
This value corresponds with the estimation made previously, by the theoretical model of the experiment (2 µN), and confirms again the correctness of the FEM model. Similarly, it confirms once again the correct choice of σ yy =190 MPa along the arms. The FEM model allow to estimate the real contact area, which evidences the difference between the apparent contact area observed on SEM images -e.g. 0.2 µm 2 estimated (subsection 3.1.2) versus 0.04 µm 2 simulated.

Measuring the contact force in a structure during its actuation is difficult, because of the small dimensions of the contact (less than 1 µm) as well as the complexity of the lateral contact. In order to identify the contact force reached at actuation step in a NORMALLY Off structure a FEM analysis is performed in next section.

Contact at actuation step: study of the expansion of the contact (NORMALLY Off structures)

In previous section the contact force is estimated and simulated for a NORMALLY On structure -contact is reached at release step. In this section a contact is modelled by FEM analysis for a NORMALLY Off structure -i.e. the contact is reached at actuation step.

It was demonstrated in previous sections that both the Joule heating and the thermal actuations reached the same temperature at the end of actuation (contact). Therefore, in order to obtain a mechanical contact and measure the contact force, only a Joule actuation is done.

The FEM model is similar to the one built in previous subsection 3.3.1.2, where a multiphysical model considers mechanical, thermal and electrical behaviour. In this section the mechanical contact defined by a contact pair is implemented -the same as Figure 3.24.

The contact is reached by an actuation realised with a current of 25 mA through the actuation arms. Figure 3.27 shows the contact area and the pressure applied by the pointer in the padonly a part of the pad is simulated. When the pointer barely touches the pad at 25 mA, the contact pressure is only 0.06 MPa in an area of around 80 nm × 80 nm (0.0064 µm 2 ). Then, the contact area increases to reach the 470 nm × 85 nm (0.04 µm 2 ) obtained for a NORMALLY On device. The contact pressure increases as the applied current increases.

The simulation results of Figure 3.28 show the contact zone and the contact pressure, which reaches a maximum of 22 MPa in this case.

The zoom of the contact area (Figure 3.28) allows estimating prudently the surface in contact 4 × 10 -14 m 2 (0.04 µm 2 ) -similar to Figure 3.26 -which multiplied by the average contact pressure gives an approximation of the contact force: This value of the contact force is obtained for an actuation current near the maximum allowed by the device (see chapter 4) -i.e. 29 mA. For this reason, the contact force cannot be greater than 0.6 µN. This section has studied the actuation step generated by a heating expansion arms both the Joule heating and the hot stage heating.

The Multiphysics FEM model is now complete, including: the mechanical part, to manage the stress relaxation, material deformation and mechanical force; the thermal module, to consider the thermal properties of material (e.g. thermal conductivity and heat transfer coefficient) and study of the material dilatation; electrical module, to manage the electrical behaviour, material resistivity, ground and electrical terminal and as well as heat source.

Optimization

The knowledge acquired at this point, after the study of the release and the actuation, allows proposing improvement to the structures for the future implementations, in order e.g. to enhance the contact, both electrical and mechanical, and to make the structures insensitive to the action of external temperatures excursions.

Thus the next section will explain the following new design. Temperature invariant structure: to avoid undesirable actuations caused either during fabrication (or encapsulation process) or by an uncontrolled homogeneous heating. This device has been designed, fabricated, simulated and tested.

Temperature invariant structure

In the first part of the research work described before, the only source of movement at release stage, to bring the structure either in the On or Off state, was residual stress. However, in a functional operation cycle and for certain applications, the device should only be enabled through a voluntary actuation. To prevent incidental thermal actuation e.g. by the remaining CMOS fabrication steps -annealing of 400 • C -a design improvement, the X-MEMS, is created (see Figure 3.29). In order to confirm the operating principle, a FEM study was performed. In a first stage (Figure 3.30a) it was confirmed that the structure does not rotate after the release of an initial longitudinal stress state of +190 MPa in the arms. Then, a heat source of 400 • C was applied at the bottom of the sample (Figure 3.30b), producing a homogeneous conductive and radiative heating. Thanks to the symmetry of the arms, a negligible displacement of 0.9 nm at the end of the pointer is observed (Figure 3.30b). Deuxièmement, l'actuation provoquée par une source de chaleur localisée, c'est-à-dire une actuation par effet Joule, est effectuée. Le modèle FEM qui a été développé précédemment est encore amélioré en introduisant un module électrique, qui prend en compte un échauffement par le passage d'un courant.

La position du pointeur central de la structure par rapport au plot de contact, après l'étape de libération, est associée aux deux modes de fonctionnement, "NORMALLY On" et 'NORMALLY Off ". Pour les tests d'actuation précédents, une structure "NORMALLY Off " a été utilisée, cette structure s'éloigne du plot de contact dans l'étape de libération, ainsi la structure rentre en contact à l'étape d'actuation.

Dans la dernière partie, une structure "NORMALLY On" est utilisée ; au contraire, c'est dès l'étape de libération que le pointeur rentre en contact avec le plot de contact. En modifiant cette structure (le plot de contact est remplacé par une poutre mobile), la force de contact après libération est mesurée dans cette partie, elle vaut 1.6 µN. Comme dans les parties précédentes, cette étude est effectuée en trois temps : caractérisation physique, simulation multi physique et modèle théorique. The maximum strain reaches 4 % around the hinge; there is no gradient in the 'z' (thickness) direction. The internal side of the hinge (close to the centre of the pointer) shows a tensile strain which is opposed to the external side which has a compressive strain.

This plasticity observed by FEM (at release step), as well as the temperature actuation produces a thermal stress, which can plastify the structure and that will be present during the device life. The next section studies experimentally the repeatability of the movement in the electrical actuation.

Repeatability of the movement

The reversibility of the movement is essential, as switching is supposed to occur thousands of times. In real operation, both ends of the pointer should therefore enter in contact with their pad at the same time. This is not necessarily the case initially (see Figure 4.3), as the pointer may not be strictly straight. In this case, the effect of plastic deformation is positive as it makes the system symmetric as it should be. This deformation solves the irregularity of release (which could prevent simultaneous contact of pointer on both pads and the current flow), and suggests a running-in step before operation. A pure thermal heating is not able to melt the arm, because in the chapter 'actuation' (chapter 3) it has been proved that the structure does not exceed 200 • C, much lower than the melting temperature of Al, 660 • C -temperature that is similar for AlCu ( [4]).

The reason for fracture is therefore to be found in a combination of thermal and mechanical effects.

The computed displacement of the arm in 'x' direction is plotted as a function of the current applied (Figure 4.1). The arms are fixed to the anchor, so no displacement is allowed at this point (on the right). In the initial state (after release) the arm is slightly curved toward centre of pointer -on the picture, upward in the real device; this is an effect of the stress that remains even after release (Figure 4.1). Under 29 mA, the line is strongly curved. The maximum of curvature is found around 8 µm from the hinge, which corresponds to the observed failure location. The strain ε yy along the arm is showed at the top surface of the arms in Figure 4.6b. It is maximal in an area from 4 to 15 µm from the hinge, the maximum gradient is centred on 8 µm, which fits with the failure location.

Remarks:

1. This part of the work has given the maximum allowed current for these particular structures. To generalize it to all structures, it would be necessary to build a complete stress-based damage model, with a fracture parameter depending on temperature ([5]).

2. In a homogenous heating by an external source (hot stage) a similar break failure is not expected, since the maximal temperature to activate the structure in this case is about 180 • C (section 3.1).

4.2 Electrical switch functionality: contact conductance 141

Electrical switch functionality: contact conductance

To standardize the study of the electrical contact, only the NORMALLY On structures are tested in this section -contact is reached at release stage. First electrical tests have shown very small electrical current (or very high contact resistance). Indeed, aluminium is easily oxidized in contact with air, so that oxidation could be the reason. Therefore, a chemical characterisation of contact surfaces is done to check oxide thickness.

Chemical analysis: contact surfaces oxidation

Aluminium in interaction with the air oxidises within 1 min [6]. The etching step in the fabrication process (B) is performed in an equipment under a vacuum, but the wafer fabrication process continues in a non-protected environment.

In contact with air, 4 to 5 nm of native oxide grows on aluminium surfaces [7][8][9]. It is generally termed alumina (Al 2 O 3 ), although oxides and hydroxides may be mixed. This oxide is a good insulator: for Al 2 O 3 , dielectric permittivity 7.5 to 15 , volumic mass of 3000 to 3400 m -3 , and volume resistivity >10 × 10 17 Ω m [10].

In order to characterise this insulating layer, both X-ray Photoelectron Spectroscopy (XPS) and Auger Electron Spectroscopy (AES) analyses were performed to identify the surface contaminants. XPS provides molecular study to identify accurately the chemical component. Complementary, the AES has a better spatial resolution, allowing the chemical identification on small zones. A Focused Ion Beam (FIB) is used for this purpose -installed on a dual beam TEM, a system with both electron and ion beam columns and a platinum injector. The ion milling is performed by the ion beam, using a gallium source, with an acceleration tension of 30 kV, in a vacuum environment (10 × 10 -6 mbar). Electrically, the aluminium oxide prevents ohmic current through the contact. In order to pass an electrical current from pointer to contact pad, or vice versa, two possibilities are analysed:

• Either ensure a measurable tunnel current through the insulator layer, for applications where low current could be sufficient.

• Or add an oxidation-resistant interfacial material on the contacting surfaces to avoid or minimize the native oxide.

Tunnelling through the oxide

The 6 nm of aluminium oxides, measured on the sidewall of the pointer and the pad, becomes 12 nm in contact. It makes it difficult to expect an ohmic conduction. Instead of this, this section concentrates on the possibility to obtain and control a measurable and repeatable tunnel current.

Definition and electrical model

The cross section view of the contact zone shows a configuration similar to a MIM capacitor for a NORMALLY On structure ( Temperature and the field-penetration effect in the metal electrodes are neglected. Thus, for a voltage higher than the potential energy barrier (V > qφ B ), the well-known Fowler-Nordheim tunnelling equation characterizes the current density J FN through the insulator [8,14]:

J FN = K 1 • ξ 2 • exp -K 2 ξ (4.2)
where Fig. 4.11 Dimensions of the interfacial oxide at contact state (pointer/pad)

K 1 = q 2 8πhφ B = 1.538 × 10 -6 (φ B ) -1 A V 2 (4.3)
and

K 2 = 8π √ 2m * 3hq • (qφ B ) = 6.8 × 10 9 m * m 0 1/2 • (φ B ) 3/2 V m (4.4)
where m * electron effective mass in the dielectric m 0 electron rest mass, q electron charge, qφ B Schottky barrier height (i.e. conduction band offset), ξ electric field in the dielectric, h Planck's constant.

Gloos [8] has studied the same material stack configuration -Al/Al 2 O 3 /Al. The electrical properties of native aluminium oxide vary between different studies [8,9,15]. Thus, considering qφ B =2.9 eV ( [16]) (φ B =2.9 V), m * =3.64 × 10 -31 kg ( [8]), the Fowler-Nordheim current density is:

J FN = 5.3 × 10 -7 • ξ 2 • exp -2.13 × 10 10 ξ (4.5)
An electric current of the order of nano-ampere (1 nA) is obtained for an electric field of 2.5 × 10 9 V m -1 (e.g. 30 V applied on an oxide 12 nm thick), in order to verify the theoretical data, an electrical test is performed. In a FEM section of the previous chapter (subsection 3.3.1.2) the contact surface has been estimated as 0.04 µm 2 (4 × 10 -14 m 2 ). For an applied electric field of 1 × 10 9 V m -1 the measured current density J FN =1 × 10 4 A m -2 , represents a current:

I = J • A = 1 × 10 4 A m 2 • 4 × 10 -14 [m 2 ] = 0.4[nA] (4.6)
The weak current measured (Figure 4.14a) is comparable to the literature ( [15]) (Figure 4.14b). Spahr's study ( [15]) found a I-V characteristics for various aluminium oxide thicknesses (2 nm, 6 nm, 12 nm and 24 nm), the difference between them corresponds to different conduction mechanisms -i.e. Poole-Frenkel, Schottky emission, direct tunnelling and thermionic-field emission ( [14]). Identifying the conduction mechanism and the traps present in the oxide is a complex work, especially for the small contact area. The oxide thickness in the contact state is 12 nm. The breakdown is obtained at 23 V, which represents an electric field of 19 MV cm -1 (1.9 × 10 9 V m -1 ). The literature values of the breakdown electric field ( [17,18]) can be higher than 60 MV cm -1 for 1.2 nm, and 10 MV cm -1 for 6 nm thick of Atomic Layer Deposition (ALD) Al 2 O 3 layer. In Gloos' study ( [8]), on native aluminium oxide, a breakdown electric field of 50 MV cm -1 was found. In the same study, it is explained that for Al 2 O 3 the breakdown voltage decreases strongly with the barrier height and the density of tunnelling electrons.

The differences observed in breakdown electric field have been the subject of many studies. Thus, electric field at breakdown depends on ( [19]): the oxide thickness, the electrode material, the mode of sample preparation, the material characteristics and the electron trap density present in the oxide layer.

When the oxide breakdown is achieved, the pointer and the contact pad or cantilever may melt (Figure 4.15). An ohmic resistance is then achieved but no further movement is possible. In order to minimize the impact of the oxidation, an oxidation-resistant interfacial material is added in contact zone. A tungsten (W) coating on the side of Al structures has been chosen. This layer aims at protecting from oxidation, or at least at avoiding oxide growth. This protective layer is called spacer layer -similar to that used in transistor fabrication in the microelectronics industry ( [20]).

Adding an interfacial material to improve electrical contact

The sidewall spacer concept ( [20]) consists in adding a new material on the sidewall of a patterned structure. Used in the microelectronics industry to optimize the transistors electric performance, we have tested it in our structures to improve the electrical contact.

From the materials easily available in the microelectronics industry which are resistant to oxidation and are fabrication process compatible, tungsten is the material chosen here. It has an acceptable electrical resistivity and thermal conductivity, 1.1 mΩ m and 174 W m -1 K -1 respectively ( [21]). It belongs to refractory metals, therefore withstands high temperatures (melting point 3422 • C) and it does not oxidize below 200 • C ( [22,23]) -i.e. the typical operation temperature in the structures. The fabrication method is a standard used in microelectronics industry ( [21]) -i.e. CVD-W deposition over Ti/TiN as adhesive layer with Al.

In order to check the quality of the spacer, TEM images are made. The fabrication process is represented by Figure 4.16 and Figure 4.17, respectively the sample state after W deposition and the final spacer state. In both images an interfacial layer is observed between Al and W. The 6 nm of aluminium oxide, grows when the aluminium comes into contact with the airduring the manufacturing process.

A wafer with full layer of Al enters the etching machine, in a vacuum chamber, at the end of etching, the wafers move out from the protected environment and wait in the air to make the TiN deposition. In order to ensure an electrical contact, the tungsten spacer is not the only improvement achieved in the fabrication process. After the fabrication process, and in order to promote easy manipulation of samples, the wafer needs to be thinned (backlap) to be cut in the sawing stepe.g. this fabrication step can contaminate the wafer surface. In order to avoid any environment contamination, which could deteriorate the future electrical performances, a protective layer is added at the end of the manufacturing process, namely a Al 2 O 3 layer of 30 nm (Figure 4.17) -∼15 nm on each side -which will be removed at release step.

After release, the pointer rotation is smaller than with standard (W-less) samples. The additional material has a Young's modulus 7 times larger than Al-i.e. 411 GPa versus 70 GPa. Thus, the distance pointer-pad measured in the W-Spacer samples is (1.2 ± 0.2) µm (turn less due to the rigidity of the tungsten) (Figure 4.18), instead of (1.5 ± 0.2) µm measured in a standard samples (subsection 2.2.3). In order to validate a correct first rotation, generated by the relaxation of residual stress in Al layer, FEM analysis of the release of a NORMALLY Off cross is performed. As W is deposited on the sides, symmetrically, it has no bending effect around any axis, and it does not contribute to rotation either. Therefore, we do not need to know the residual stress within W .

The FEM model used in this part is similar to subsection 2.2.2 -i.e. only a release stage is studied.

A linear elastic and isotropic Tungsten material chart was obtained from COMSOL MULTI-PHYSICS materials library; Young's modulus is 411 GPa ( [24]), Poisson's coefficient is 0.28 ([24]).

The W layer (50 nm), 10 times less thick than Al (470 nm), is meshed in the 'swept mode': it copies the surface triangular mesh pattern on the top of Al and W layers, creating triangular prism elements (Figure 4.19). The in-plane movement is obtained without perturbation using an aluminium layer, laterally protected by a W coat for the structure, rotation can be studied reliably. The apparent stress (+190 MPa) is applied to the Al part of device as a longitudinal stress in both arms and pointer, which generates a rotation of central pointer at release stage (Figure 4.20).

In resume, the pointer/pad distance on a W-Spacer sample is:

• in simulation: 1.1 µm versus 1.2 µm obtained in a standard sample, i.e. ∼10 % less than with non-W-coated devices, The fabrication process was successfully achieved, obtaining the correct spacer. The verification of correct operation is then validated. The rotation by a local asymmetric and controlled release is verified. The electrical function on this W-Spacer samples is then tested below.

Electrical characterisation of W-Spacer

As Al is the principal material in the patterned structure, the electrical actuation is identical to that performed in the previous chapter (chapter 3). Then during actuation test, the NORMALLY On and NORMALLY Off structures becomes Off and On respectively at 28 mA instead of 20 mA required by standard Al structure. The difference in current actuation between these two structures can be explained by the Young's modulus -i.e. 411 GPa for W versus 70 GPa for Al. In order to corroborate this value, estimated by optical microscopy observation, an electrical experiment will be performed.

Having proved that the structures moves like a standard one, hereafter a normally On structure is studied.

As Al 2 O 3 is still present below the spacer layer, the requested electrical contact in 'W spacer samples' can be performed in two ways:

• Either, tunnelling through the oxide 4.2 Electrical switch functionality: contact conductance 155 • Or breaking down the oxide, and in this case the W comes in contact with the Al.

The second case has a higher probability to work than in the Al-Al case, because the protective W layer will avoid welding of pointer and pad, while ensuring the electrical contact.

The test is performed on NORMALLY On structures. Similar to standard samples, this device makes contact at release stage and no actuation is needed.

In order to validate an ohmic contact, a particular electrical test is performed. The aim of this test is to measure an electrical contact in the On state, then activate the switch and lose contact (showing welding has not occurred). Figure 4.21 shows the connection used, where three terminals ensure the actuation and measure the contact current. The voltage source (VA) is applied in A terminal and swept from 0 to 500 mV to cause a current in the arms as its actuation. To ensure that all current passes through the arms and avoid a leakage current to the contact until 200mV, the Vcont (in the pad c) source is polarised at 200 mV. In order to avoid arms damage or contact melting the current allowed in the arms is limited to 27 mA. Similarly, the current allowed in the contact is limited to 100 nA. This W-spacer solution has:

• a similar current actuation as standard structures (21 mA),

• much better electrical results with an intensity of 100 nA (compared to 1 nA in standard solution) -it could even be higher if we had not limited it to this value to avoid an increase in current and the risk of melt the structure,

• an additional fabrication step which is compatible with CMOS fabrication flow.

After a series of attempts at improving the electrical contact, this is the first really successful solution. Maybe it can be further improved in the sense of higher intensities. But its mechanical robustness must still be tested before this proposition can be implemented.

This electrical result arrives at the last time of characterisation work. Even if this hopeful result arrives at the end of this PhD work, the characterisation experience that was acquired on the structure development proves that this result can be repeated over time. The development of W-Spacer proves that the electrical contact with a small resistance is possible. Therefore, a thermo-electro mechanical switch is perfectly possible. Finalement, la fonctionnalité de l'interrupteur a été validée électriquement dans une structure NORMALLY On. Un contact électrique est établi lors de l'étape de libération, et perdu lors de l'étape d'actuation, la fonction de commutation est donc bien assurée.

Bibliography

Conclusions

In this work, a class of microelectronic MEMS actuators has been investigated. It is innovative by its positioning inside the BEOL exploiting the fabrication residual stress, on a single Aluminium layer, and the choice to keep a perfect compatibility with the CMOS process line, namely by renouncing any non-standard material, e.g. gold or ruthenium which are used to improve electrical contact properties. This has imposed serious constraints for which alternative solutions had to be found, tested and commissioned.

This has forced us to scan a number of difficulties for which solutions have been elaborated at all the stages of the fabrication and commissioning process:

• Deposition and patterning of the layers • Release of the suspended parts of the MEMS • Actuation of the device to obtain a controlled movement and an electrical contact First, a proper release protocole based on HF Vapor has been worked out in the laboratory to form fully suspended structures without stiction due to capillary effect or dummies which jeopardize the functionality during service life. Second, the residual stress has been exploited to obtain a movement by the release itself, either of the NORMALLY On or NORMALLY Off kind. This of course required measuring the residual stress, which has been done through full wafer curvature. A biaxial stress of 290 MPa has been found, in good agreement with literature. For our application, it had to be transposed to the uniaxial case, giving a line stress of 190 MPa. Third, spurious deformation connected in particular with the bilayer effect had to be eliminated, by suppressing the TiN layer and its estimated -900 MPa compressive residual stress from the suspended area which, as a matter of fact, does not need it.

These actions can be viewed as preliminaries but required a lot of time and care. Then, the precise control of the movement has been achieved thanks to adaptations of the design of the suspended part. The relationship between temperature and rotation angle (or more generally movement amplitude) has been established using an original, in situ hot plate in the SEM, then confirmed by Joule effect actuation. Considering the device used to demonstrate the feasibility of the approach, a switch, the last point was to obtain an ohmic electric contact, certainly the most difficult stage of this development due to the oxidation of aluminium which allows at best a poorly controlled tunnel current -which has been measured. This problem was finally solved by introducing a new step, the deposition of a nanometric W layer -but this metal remains among the standards of CMOS, the initial constraint is respected.

All along the study, on top of mastering the fabrication processes and characterizing the materials and structures formed, we took great care to develop a "virtual fabrication lab" in Bibliography 167 the form of the coupled mechanical -electrical -thermal FEM model based on COMSOL MULTIPHYSICS. In the course of the project, it has allowed understanding the behaviour of our devices and elaborating solutions: attributing pointer bending to the bilayer effect, it suggested to eliminate TiN; irreversibility of the movement upon repeated actuation was explained by the observation of plastic strain at release and the effect of the design of the "hinge" has been clarified; the limit current leading to arm break has been referred thanks to numerical simulation not to melting, but to a combination of stress and high temperature, possibly enhanced by high current density via electromigration.

It must be emphasized that all these methodological developments are not restricted to the "cross" switch illustrated all along the memoir. In the course of the work, other geometries have been investigated in details, such as "H" symmetric structures. Other functions have been thought of such as variable capacities from classical comb-drive or a structure using several metal layer to increase the facing surface. We believe a whole family of devices can be developed faster and at a lower cost now that this combined experimental / numerical approach has been built.

Perspectives

The first industrial perspective consists in multiplying the devices of interest, to be developed in a shorter time thanks to the experimental / numerical methodology built here. Other geometries exploiting the fabrication residual stress can be designed, other functions can be given like a stress sensor or a mechanical security device. The present Joule effect actuation, which requires rather high intensities, in the 20 to 30 mA range, is probably too energy-greedy and could be replaced by an electrostatic actuation. The cross switch itself could certainly be improved in terms of dimensions, of geometry for robustness (hinge, cross section shape. . . ). Models and understanding exist for upscaling or downscaling as well as for better functionality of improved robustness.

The tricky contact of the lateral side of a beam with the lateral side of a pad, which is very sensitive to movement imprecisions, can be facilitated by design ideas which have been fabricated but not fully tested during this PhD (see figures below):

• the notched pointer provides longer contact length and hopefully higher currents at contact thanks to its high flexibility. The FEM study of this design modification (Figure 1b) shows a 25-fold increase of the contact area (from 0.04 to 1 µm) and a contact force increase by 50 %, reaching 3 µN. While the maximum pressure is 5.5 MPa, the medium pressure is around of 3 MPa • the tipped pointer which on the contrary concentrates the contact on a well-defined dimension and allows larger contact pressures. The tip shape is designed as square but is rounded by the fabrication process. The tip shape helps to increase the electric field, which should reduce the electric field of ∼10 GV m -1 needed to obtain a 0.4 nA by tunnelling (subsection 4.2.2.1)

The metallic layer fabrication is performed in three steps, Ti, Al (Cu) and TiN deposition (Figure 2). Fig. 2 Description of the metallic layer, composed by Ti/AlCu/TiN, which is made in three continuous deposition steps A thin layer of 7.5 nm of Ti is deposited over the silicon oxide. In the same equipment, changing the chamber of deposition, the 470 nm of Al film is sputtered using PVD from a target containing 0.5wt% Cui.e. Al (Cu). The 45 nm of TiN layer is added on top of the Al layer to avoid the electromigration of Al lines. This TiN is deposited using CVD process with a heater temperature of 450 • C and a wafer temperature of 407 • C. The TiN is classified as a "barrier metal", because it avoids the diffusion of metal (Al (Cu)) into the silicon and silicon dioxide. The pure titanium is sublimed and react with nitrogen in a high-energy, vacuum environment.

This metallic layer is deposited on full wafer (Figure 3a). In order to isolate the consecutive metal layers, an undoped CVD SiO 2 (TEOS) is deposited (Figure 3b) at 700 • C -i.e. CVD deposition. Over the IMD layer of 1.6 µm a new metallic layer is deposited on full wafer (Figure 4a). In order to draw the motif in this new metallic layer, a photolithography is realized next. A photoresist resin is spread over the M3 layer (Figure 4b).

In the case of a resin called "negative", the part exposed to light through a mask becomes insoluble in developer and allows to create the motives on metallic layer (Figure 5a). An anisotropic plasma etches the M3 layer to create the structures on this layer (Figure 5b).

Finally, the photoresist is removed (Figure 6) and the device is finished. At this stage the wafer leaves the production and will be processed in the laboratory to prepare the samples and release the devices.

Abstract

In recent years the miniaturization of microsystems is reaching the physical limit of its development. Thus, a path of innovation in the semiconductor industry is additional functionalities in existing components.

The project consists to integrate a MEMS, within the same metal interconnect of CMOS layer which, by rotating, can establish an electrical contact.

The obstacles are in the release of the moving parts by dissolution of the surrounding oxide (out of plane deformation under the effect of residual stress, stiction, residues which prevent contact), in the actuation (current density repeatability, durability, reliability) and, for ohmic switches, the ability to establish a real electrical contact with low resistance (real / apparent area of contact with rough surfaces, contact pollution).

The work carried out has focused on conception (design) and simulation of microsystems to overcome these difficulties and / or to study the behavior and measure the effects.

Keywords

Residual stress, contact, Back end of line, Multiphysics simulation, CMOS-MEMS Numero national de thèse : xxxx xxxxxxxx

  process flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 MEMS devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 MEMS Switch introduction . . . . . . . . . . . . . . . . . . . . . 1.1.3 CMOS-MEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Mechanical stress in an isotropic thin film . . . . . . . . . . . . . . . . . 1.2.1 Origin of stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Passive stress sensor . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Mechanical and electrical contact . . . . . . . . . . . . . . . . . . . . . 1.3.1 Mechanical contact . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Electrical contact . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 MEMS reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Mechanical issues . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Thermal issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Contact issues . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fig. 1 .

 1 Fig. 1.1 ITRS roadmap, 2010 version

Fig. 1 . 2

 12 Fig. 1.2 Packaging of MEMS device, the electronic device (ASIC) with the mechanical part (MEMS) bound together in a so-called System on Chip (SoC)

1. 1 MEMS process flow 7 1. 1 . 3 . 3

 17133 Post-MEMS: MEMS fabricated after CMOS

Fig. 1 .

 1 Fig. 1.6 A cross section schematic of Carnegie Mellon BEOL-MEMS technology[START_REF] Fedder | Application-Specific Integrated-Microelectromechanical Systems ( MEMS ) process services ( ASIMPS )[END_REF]

Fig. 1 . 7

 17 Fig. 1.7 Representation of the stress component in a thin film over substrate in Voigt notation

Fig. 1 . 8 Fig. 1 .

 181 Fig. 1.8 Coalescence of deposited grains, a) the islands of material grow and b) become elastically strained generating a tensile stress state in the film

Fig. 1 .

 1 Fig. 1.11 Sensor diagram [58], fabricated in a single metallic layer

1. 3

 3 Mechanical and electrical contact 15 1.

Fig. 1 .

 1 Fig. 1.12 Representation of the contact zone between A (rough) and B (smooth) surface

  )

Fig. 1 .

 1 Fig. 1.13 Contact of two surfaces, a smooth and a rough one.

Fig. 1 .

 1 Fig. 1.14 Qualitative representation of statistical self-affinity for a surface profile in fractal model

Fig. 1 .

 1 Fig. 1.15 Illustration of a contact spot of length scale l according to MB model

Fig. 1 .Fig. 1 .

 11 Fig. 1.17 Topographic survey of a bump in a capacitive switch [71], a) AFM representation, b) FEM representation (treated with Matlab)

Figure 1

 1 .19.

1 (

 1 Figure 1.20). With this complete relation, we can calculate a resistance for all three regimes -regarding to Figure1.20, in the quasi-ballistic regime (K * between 0.1 and 10), both Wexler's and Nikolic's approximation are valid.

Fig. 1 .

 1 Fig.1.20 Gamma function dependence on Knudsen number, Wexler (γ Wex ) and Nikolic (γ) approximation[START_REF] Nikolić | Electron transport through a circular constriction[END_REF]

Figure 1 .

 1 21 shows the cross section of the sample used by Mercier et al. [79], Titanium (Ti) and Aluminium (Al) are deposited over SiO 2 . The native aluminium oxide (Al 2 O 3 ) grows at room temperature over the Al layer.

Fig. 1 .

 1 Fig. 1.21 Cross-section illustrating native oxide in aluminium sample[START_REF] Mercier | Quantitative Evolution of Electrical Contact Resistance between Aluminum Thin Films[END_REF]

Fig. 1 .

 1 Fig. 1.22 I-V plot for films of different thickness of Al 2 O 3 ALD (Atomic layer deposition) [80], until its breakdown

Fig. 1 .

 1 Fig.1.24 Simplified scheme of the band structure of a MIM capacitor consisting of metal, oxide, and metal (both metals are identical) illustrating a) a rest state, and b) an external voltage U ext applied[START_REF] Spahr | Conduction mechanisms in thin atomic layer deposited Al2O3 layers[END_REF]

Fig. 1 .

 1 Fig. 1.25 Scheme of the band structure for a thermo-ionic emission

Fig. 1 .Figure 1 .

 11 Fig. 1.26 Scheme of the band structure for a Fowler -Nordheim tunnelling

Fig. 1 .

 1 Fig. 1.27 Scheme of the band structure for a direct tunnelling

Fig. 1 .

 1 Fig. 1.28 Scheme of the band structure for a Poole-Frenkel emission

Fig. 1 .

 1 Fig. 1.29 Defaults caused by mechanical stress, a) cracking, b) voids in an Al line, c) hillock

Fig. 1 .

 1 Fig. 1.30 Estimation of anode heating, the melting point are indicated and the time to reach it[START_REF] Poulain | A Nano-Scale Investigation of Material Transfer Phenomena at Make in a MEMS Switch[END_REF]

Figure 1 .

 1 Figure 1.31 show a typical curve for air, nitrogen and hydrogen at 1 atm, with a minimum value for air at 327 V -i.e. Paschen minimum.

Fig. 1 .

 1 Fig.1.33 Scheme of a material transfer mechanism in a contact[START_REF] Poulain | A Nano-Scale Investigation of Material Transfer Phenomena at Make in a MEMS Switch[END_REF]

Fig. 1 .

 1 Fig.1.[START_REF] Natarajan | CMOS integrated digital RF MEMS capacitors[END_REF] The Bonding energy in function to separation between two electrodes (sphere-plane model)[START_REF] Legtenberg | Electrostatic actuators fabricated by surface micromachining techniques[END_REF]

2. 1

 1 Studied structure: Cross-shaped CMOS MEMS 63 2.1 Studied structure: Cross-shaped CMOS MEMS 2.1.1 Description of rotational structure

Fig. 2 . 1

 21 Fig. 2.1 Schematics and main dimensions of the cross-shape structure before release. a) Functional structure with contact pads, b) test structure, c) zoom on the hinge

• a 1 .

 1 6 µm of sacrificial SiO 2 layer (IMD1), over • a 0.52 µm metallic layer (M1), used to stop the release, over • a 750 µm of Si (100) wafer.

Fig. 2 . 3

 23 Fig. 2.3 Technological stack used to fabricate the devices

Fig. 2 . 4

 24 Fig. 2.4 Full wafer apparent stress measured in a thin metal layer. The 6 wafers have 4 different metal configurations

2. 1 69 • - 1 . 3

 16913 Studied structure: Cross-shaped CMOS MEMS GPa if TiN 45 nm • -0.6 GPa if TiN 90 nm

Fig. 2 . 6

 26 Fig. 2.6 Variation of normalized stress with aspect ratio in a W line for a) longitudinal and b) transversal direction[6]

Fig. 2 .

 2 Fig. 2.7 Tilted SEM view of structure, with the pointer bent

Fig. 2 . 9

 29 Fig. 2.9 Tilted SEM image of rotated structure without the TiN layer

Figure 2 .Fig. 2 .

 22 Fig. 2.10 a) Optical profilometry and b) profile view of structure after release

Fig. 2 .

 2 Fig. 2.11 Model of cross-shape structure used in release study by FEM

Fig. 2 .Fig. 2 .

 22 Fig. 2.12 Adaptive tetrahedral mesh refined in the hinge vicinity: a) 5205, b) 47 801 elements

Figure 2 .

 2 Figure2.13b, shows that the "cross device" remains in the plane, slightly raised from 0.2 µm, and a difference between left and right cross pointer -i.e. a small tilting occurs in the plane.

Fig. 2 .

 2 Fig. 2.14 Deflection simulated in a cantilever and in the central pointer of the device

Fig. 2 .

 2 Fig. 2.15 3D simulation -illustrating the distance between the final position and the initial position (before release)

Fig. 2 .

 2 Fig. 2.16 Sensor diagram, central pointer before (left) and after (right) release

Fig. 2 .

 2 Fig. 2.17 Hinge design used in the Vayrette model

Fig. 2 .

 2 Fig. 2.18 Measurement of rotation after release

3. 1 95 3. 1

 1951 Thermal actuation (NORMALLY Off structures) Thermal actuation (NORMALLY Off structures)

Fig. 3 . 1

 31 Fig. 3.1 Equipment used in in-situ SEM-heating test

Fig. 3 . 2

 32 Fig. 3.2 Set point of temperature applied in the bottom of sample

Fig. 3 .Fig. 3 .

 33 Fig. 3.3 SEM image of Off structure a) at room temperature (20 • C), b) at 200 • C, when the pointer reaches the contact pad

Fig. 3 .

 3 Fig. 3.5 a) Cross section of cross shape system, and b) schema of heat transfer in hot stage actuation test

3. 1

 1 Thermal actuation (NORMALLY Off structures) 103 3.1.

Fig. 3 . 7 Fig. 3 . 8

 3738 Fig. 3.7 Result of 2D FEM that represented in M1 metallic layer, a) detail of layer and structure descriptive b) temperature, and c) heat transfer coefficient

Figure 3 .Fig. 3 .

 33 Figure 3.10a shows the rotation of the cross-shape structure. Even if a NORMALLY Off structure is studied, no contact conditions are defined in the pad to observe free pointer rotation at all temperatures and compare with the fitted experimental curve of Figure 3.4.As the temperature of 200 • C is applied at the top of M1 layer, it can be seen:

Fig. 3 .Figure 3

 33 Fig. 3.10 FEM model of thermal actuation and rotation induced in the cross-shape structure, a) the displacement in 'y' from its original position and b) temperature gradient on the cross-shape

Fig. 3 .

 3 Fig. 3.11 Nominal temperatures applied and distance pointer-pad obtained -comparison between FEM study and previous in-situ test data

Fig. 3 .

 3 Fig. 3.13 Boundary conditions of electrical module of COMSOL MULTIPHYSICS

Fig. 3 .

 3 Fig. 3.14 a) Mesh composed by 20 535 tetrahedral elements, and b) the temperature in the structure with a maximum in the centre of the cross, which is generated by a flux of 20 mA

Fig. 3 .Figure 3 .

 33 Fig. 3.15 Evolution of pointer-pad distance as a function of a) current applied, and b) average temperature in the arms

Fig. 3 .

 3 Fig. 3.16 Example of tips connections during SEM Nanoprobing test

Fig. 3 .

 3 Fig. 3.17 a) The nodes represented on the device, and b) equivalent circuit obtained by connections used in SEM nanoprobing test

  Fig. 3.18 SEM images of contact zone, where the pointer rotates to the contact pad for a high temperature generated by Joule heating.

Fig. 3 .

 3 Fig. 3.19 Experimental current / displacement relationship for a NORMALLY Off structure obtained in a nanoprobing test -reaching a contact at 23 mA (0.56 V)

Fig. 3 .

 3 Fig. 3.20 Equivalent circuit of nanoprobing test, which consider the ideal FEM electrical values and parasitic resistances of nanoprobing system

Fig. 3 .Figure 3 .

 33 Fig. 3.21 Current applied in function to temperature generated in the arms -Senturia and FEM model

3. 3

 3 Mechanical contact 3.3.1 Contact at release step (NORMALLY On structures)

Fig. 3 .

 3 Fig. 3.22 SEM image of the test structure after release. The cross structure rotates and pushes the horizontal flexible beam 0.9 µm from its original position

Fig. 3 .

 3 Fig. 3.23 Boundary conditions in mechanical model for the NORMALLY On structure, performed to measure the contact force, a)the initial stress along the pointer (σ xx ), b) the initial stress along the arms(σ yy ).

Fig. 3 .

 3 Fig. 3.24 The contact boundary conditions, source and destination of contact pair

Fig. 3 .Fig. 3 .

 33 Fig. 3.25 FEM result of the contact pressure on the pad, which is inflicted by the pointer

Fig. 3 .

 3 Fig. 3.27 FEM image of contact pressure for a current applied of 25 mA and 28 mA

3. 3

 3 Mechanical contact 123 F = P • A = 15[MPa] • 4 × 10 -14 [m 2 ] ≈ 0.6[µN] (3.25)

Fig. 3 .

 3 Fig. 3.28 Zoom of the contact zone (Figure 3.27b) with a pressure applied on the pad

Fig. 3 .

 3 Fig. 3.29 Design of new robust design, invariant to external temperature variations and auto balanced for a residual stress

3. 4

 4 Optimization 125 Moreover, if the four arms are heated homogeneously, the dilation is uniform and the rotation is avoided. Heating the arms heterogeneously by an actuating current, connecting either C and B or D and A, the pointer pulls asymmetrically and rotates clockwise or counter-clockwise respectively (Figure 3.31).

Fig. 3 .

 3 Fig. 3.30 FEM result show no movement in both a) release stage and b) heated sample at 400 • C

Fig. 4 . 1

 41 Fig. 4.1 Demonstration of the bending in the arms, a) SEM image shows a small flexion, b) FEM image that amplifies by 5 the 'x' and 'y' shift generated at release step

Fig. 4 . 2

 42 Fig. 4.2 Computed y-component of strain tensor (ε yy )in the hinge: 4 % of plastic deformation

Fig. 4 .

 4 Fig. 4.3 SEM image with microscopic distance measurements of test structure, a) after release step, and b) after 1110 actuation cycles

Fig. 4 . 4

 44 Fig. 4.4 Cycling test shows stability in pointer-pad distance up to 1110 On-Off cycles, after an initial running-in stage of 100 cycles

Fig. 4 .

 4 Fig. 4.5 SEM image showing the failure at 1/3 of arm from hinge, around 8 µm from hinge

Fig. 4 .Figure 4 .

 44 Fig. 4.6 a) Displacement in 'x' direction of the arm as a function of applied current, b) Top view of FEM study of the strain ε yy on the surface of the arm

Fig. 4 .

 4 Fig. 4.7 XPS spectrum on the side of a pad -Al peaks analysis confirms the presence of aluminium oxides in the structure

Fig. 4 . 8

 48 Fig. 4.8 Principal steps of the preparation of a standard TEM sheet by FIB dual beam machine, a) metallization over 20 µm, b) thinning of the sample and c) manipulation with a W tip

Figure 4 .

 4 Figure 4.9 shows a TEM cross section of the contact interface, which allows visualizing the aluminium oxide layer on the top (4 nm) and on the sidewall (6 nm) of a pad. The difference of Al 2 O 3 thickness can be explained by the fabrication process, where the top of Al layer undergoes anisotropic chemical attack, reducing the thickness of the oxide -i.e. process of photolithography like resin removal and the clean step.

Fig. 4 . 9

 49 Fig. 4.9 Cross-section TEM images of the contact zone in a NORMALLY Off structure, a) top of the contact pad showing 4 nm of Al 2 O 3 , b) the contact pad, and c) contacting side showing an oxide layer of 6 nm

Figure 4 .Fig. 4 .

 44 Fig. 4.10 Representation of MIM capacitor by a NORMALLY On structure after release step, a) cross-section TEM view, and b) schematic view, representing the contact

Fig. 4 .

 4 Fig. 4.12 Representation of Fowler-Nordheim model of Al 2 O 3 native layer obtained in Equation 4.5

Fig. 4 .

 4 Fig. 4.13 a) original device, and b) equivalent circuit created by the connection of the probes

Fig. 4 .

 4 Fig. 4.14 Current density vs electric field strength characteristics of contact, a) this study for 12 nm of Al 2 O 3 , b) Spahr [15] study of Al 2 O 3 thicknesses of 2 nm, 6 nm, 12 nm and 24 nm

Fig. 4 .

 4 Fig. 4.15 SEM image after oxide breakdown and contact fusion

Fig. 4 .

 4 Fig. 4.16 The TEM view after CVD-W deposition of a) pointer-pad interface, b) a zoom in the Al/ W interface and c) schematic with all the layers observed: oxide (∼6 nm), Ti (2.5 nm), TiN (3 nm) and W (50 nm)

Fig. 4 .

 4 Fig. 4.17 The TEM view of W-Spacer in the final step before release a) pointer-pad interface, b) a zoom in the Al/W interface and c) schematic with all the layer observed: oxide (∼6 nm), Ti (2.5 nm), TiN (3 nm), W (∼50 nm) and protective layer (Al 2 O 3 ∼15 nm)

Fig. 4 .

 4 Fig. 4.18 SEM image of contact zone in a structure after the release step. b) NORMALLY On and a) for a NORMALLY Off structure distance pointer-pad of (1.2 ± 0.2) µm is measured.

4. 2

 2 Electrical switch functionality: contact conductance 153 4.2.3.1 FEM model (NORMALLY Off structure)

Fig. 4 .

 4 Fig. 4.19 a) Mesh with 692 587 prism elements, b) zoom of hinge zone with the thin coat of tungsten

Fig. 4 .

 4 Fig. 4.20 Illustration of the distance between the final position and the initial position (before release), a) 3D Simulation and b) SEM measurement

Fig. 4 .

 4 Fig. 4.21 Equivalent circuit created by the connection and b) visualization of contact pad utilized on real structure

Figure 4 .

 4 22 show the current in the arms (I arms) which rises to 24 mA. The 'I contact' current is equal to 100 nA until 150 mV -i.e. the compliance current defined in the pad. Then a current flow is observed until 200 mV, which represents the current passing from the arms to the contact. Once the voltage of the arms reaches the 200 mV imposed in the pad the current

Fig. 4 .

 4 Fig. 4.22 I-V curve that corroborates a loss of the contact in a NORMALLY On structure of W-Spacer solution

Fig. 1

 1 Fig. 1 Notch pointer: a) adapted tetrahedral mesh, refined in the contact vicinity, b) contact area (1 µm 2 ) and contact pressure

Fig. 3 a

 3 Fig. 3 a) Metallisation on full wafer, and b) CVD of sacrificial IMD1

Fig. 4 Fig. 6

 46 Fig. 4 a) PVD metallisation on full wafer of M3 Metallic layer, and b) spread of the photoresist

  .1) -e.g. coefficients of thermal expansion and Young's modulus. The SiO 2 , SiNx and TiN compounds have quite different α and E values. Table1.1 gives some values, although it should be kept in mind that these properties may vary considerably from one process to another.

	Material α × 10 -6 [C -1 ]	E [GPa]
	Si	3	130
	SiO 2	∼ 0.8	∼ 63
	SiNx	∼ 2.2	∼ 164
	W	4.27	385 -500
	Cu	16	130
	Al	23	70
	TiN	∼ 9.35	∼ 500(between 450 -590)
	Table

Table 2 .

 2 2.1 Studied structure: Cross-shaped CMOS MEMS 1 The 6 full wafers used to measure the bow, in a variation of M3 deposition

	67

Table 2 .

 2 it is possible to obtain the stress in each layer. M3 MPa Wafer # t Al nm σ Al MPa t TiN nm σ TiN MPa 2 Estimation of stress in each sublayer -TiN and Al. Measured values in bold

	M3		Al			TiN [11]
	σ 150	1.2	470	290	45	-1300
	190	3	470	290	20	-2100
	290	4.5	470	290	-	-
	100	6	880	150	20	-2100

  Al σ TiN t 2 TiN + E Al t 2 Al σ TiN t TiNσ Al t 2 Al E TiN t TiNσ Al t Al E TiN t 2 TiN + 4E Al t 3 Al E TiN t TiN + 6E Al t 2 Al E TiN t 2 TiN + 4E Al t Al E TiN t 3 • t Al t TiN • t Al + t TiN • E Al σ TiNσ Al E TiN E 2 Al t 4 Al + E 2 TiN t 4 TiN + E Al t Al E TiN t TiN • 4t 2 Al + 6t Al t TiN + 4t 2

	E 2 Al t 4 Al + E 2 TiN t 4	TiN TiN	(2.13)
	d =	3L 2 TiN		(2.14)

Table 2 .

 2

3 Values, from Figure 2.1, used to calculate the pointer deviation a Thus, using the numerical values from Table 2.3 in Equation 2.18 to Equation

Table 3 .

 3 1 Interfacial temperatures in the different layer of the sample produced by a thermal actuation of 200 • C

		47	237	q 2 q 1	473.118 473.098	473.117 473.097
	SiO 2	1.6	1.27 [7]	q 2 q 1	473.138 473.099	473.118 473.098
	Al	0.47	237	q 2 q 1	473.138 473.099	473.118 473.099
	SiO 2	0.8	1.27	q 2 q 1	473.148 473.10	473.138 473.099
	Si	375	130	q 2 q 1	473.15 473.15	473.147 473.02

  La structure a été libérée, caractérisée et modélisée dans le chapitre précédent. Ce chapitre est consacré à l'étude de l'actuation des structures analysées. L'actuation est causée par l'échauffement des bras et leur dilatation, qui par conséquent fait tourner le pointeur central (chapter 2). La fonction attendue des structures est une actuation thermique contrôlée, c'està-dire une actuation par effet Joule. Pour cela deux grandes parties sont abordées dans ce chapitre.Premièrement, et afin d'estimer des caractéristiques thermiques propres au système, une actuation thermique est réalisée. L'échantillon est chauffé à sa surface inférieure dans une platine chauffante, la chaleur est transférée jusqu'aux structures, dilatant les bras et provoquant une rotation. Cette expérience, embarquée dans un Microscope Electronique à Balayage (MEB), permet d'obtenir une relation entre la température imposée et la rotation obtenue. A partir de ce test, le modèle FEM, qui à l'origine ne modélisait qu'un comportement mécanique, est amélioré et maintenant c'est un modèle couplé mécanique/thermique qui a été élaboré.

  Le projet consiste à intégrer, dans une même couche métallique d'interconnexion CMOS, un MEMS capable, par sa rotation, d'établir un contact électrique.Les verrous se situent dans la libération des parties mobiles par dissolution de l'oxyde environnant (déformation hors plan sous l'effet des contraintes résiduelles, stiction, présence de résidus qui empêchent le contact), dans l'actionnement (densité de courant, répétabilité, durabilité, fiabilité) ainsi que, la capacité d'établir un vrai contact électrique à faible résistance (aire réelle / apparente du contact des surfaces rugueuses, pollution du contact).

	188 190	Nomenclature Nomenclature
	d l t λ	cantilever tip deflection (Ch.2) length scale of a single asperity (fractal model) time inelastic mean free path
	E ℓ t f µ	isotropic elastic modulus electron elastic mean free path film thickness electron mobility Résumé
	E ′ M t s ν	equivalent elasticity modulus biaxial modulus substrate thickness Poisson's ratio Ces dernières années la miniaturisation des
	E F m * t eot σ Nomenclature Fermi level effective electron mass in the oxide equivalent oxide thickness biaxial stress component microsystèmes atteint la limite physique de leur développement. Ainsi une de voie d'innovation dans l'industrie des semi-
	F m 0 U φ (z) distribution density function of the measured roughness height force in a contact electron rest mass voltage conducteurs est l'intégration des
		fonctionnalités supplémentaires au sein des
	A F 0 M i U ext external voltage area of the conducting electrical lines constant electron flow moments of the cantilever φ B junction barrier height composants déjà existants.
	a F i N V B φ T	distance from the pointer to its initial position forces of the cantilever number of asperities breakdown voltage depth of traps potential well
	A * f i n w ψ	saturation ionization in the gas spatial frequency i number of contact spots interference of asperity (fractal model) plasticity index
	a * F m N C w ρ	pointer/pad distance capillarity force density of states in conduction band sinking on Hertz' theory electrical resistivity
	a c F V DW Van der Waals force radius of a circular contact area N i volume densities of atoms in the metal and in the oxide z height of asperities ρ radius of curvature of the cantilever beam (Ch.2)
	a f G P z(x) surface profile in contact lattice constants on thin film fractal roughness parameter (fractal model) load in a contact ρ m material density
	H p Greek Symbols macroscopic contact hardness pressure applied in a contact σ substrate thickness
	a s	lattice constants on substrate
	h p g α σ a	Planck's constant gas pressure coefficient of thermal expansion standard deviation of the height distribution
	A b h ′ p β σ m f misfit stress bearing area standardized separation average pressure radius of the sphere in a spot contact
	A H h p q β ′ θ	Hamaker constant distribution of peak heights electron charge Le travail réalisé a porté sur la conception, le amplification factor contact angle design et la simulation des microsystèmes
	A n I i R β i θ	nominal area intensities of the metal and oxide photoelectron peaks (Ch.4) electrical resistance afin de surmonter ces difficultés et / ou asperity amplitude corresponding to the given frequency rotation angle (Ch.2) d'étudier le comportement et mesurer les
	a rc A r I i J r Ra L Rayleigh number effets. critical area of asperities total real contact area moment of inertia current density half distance between two spots δ full height of asperity (fractal model) ε ε electric permittivity strain tensor θ a photoelectron take-off angle Mots Clés
	a r K R H ε th κ	real contact area of asperities thermal conductivity Holm electric resistance Contraintes résiduelles, contact, Back end of emissivity relative dielectric constant of the oxide layer line, Simulation Multiphysique, CMOS-MEMS
	B * k R s η ξ	a constant related to the excitation and ionization energies Boltzmann's constant Sharvin Resistance density of asperities electric field
	C K ′ R w f radius of wafer the 4 th order elasticity tensor hardness coefficient related to the Poisson ratio γ(K * ) gamma function ξ c p/H ratio
	c K * R w γ n	specific heat Knudsen number Wexler resistance frequency spectrum of the surface roughness (fractal model)
	D k w f S γ 1ν	fractal dimension of the surface profile (fractal model) curvature of wafer second Piola-Kirchhoff stress tensor surface tension of water
	d L T γ se	distance between two surfaces that come in contact (Ch.1) length of the conducting electrical lines temperature secondary electron emission coefficient

The complexity for minimum component costs has increased at a rate of roughly a factor of two per year. Certainly over the short term this rate can be expected to continue, if not to increase.

z(x) = G D-1 ∑ n cos(2πγ n x) γ (2-D)n ; 1 < D <

; γ > 1, where, D is the fractal dimension, G is a characteristic length scale of the surface and γ n determines the frequency spectrum of the surface roughness

(a) (b)

The limits of γ(K * ) of Wexler are: γ(0) = 1 an lim K * →∞ γ(K * ) = 9π

128 = 0.694

Space-charge-limited conduction (SCLC) This mechanism is similar to the transport conduction of electron in a vacuum diode. One metallic side is heated, the excited atoms within the metal emit electrons which are accelerated across the insulator between the electrodes.Ionic conduction As the name says, this conduction results from the movement of ions caused by an electric field. The ions move between traps in the dielectric to traverse the insulator thickness.

x represent the longitudinal direction on the beam

the temperature applied at the bottom of sample, i.e. in Si layer, with a thickness of 375 µm=dx, a thermal conductivity k=130 W m -1 K -1 , and T 1 =473.15 K

This equation explains how a metal fuse works. For a sufficiently large current, the denominator vanishes, the temperature rise diverges, and the fuse melts.

A limited trial licence of the non-linear mechanics module of Comsol is used to simulate the plasticity in the hinge.

Acknowledgements

Managing the stress state in structures the measurement of deviation from the original position of the central pointer before release. The ruler is composed of 23 nanobeams 250 nm wide and 1 µm and 2 µm long. Thanks to the 250 nm space between the nanobeams, this ruler allows quick measurement by steps of 500 nm i.e. the distance between the center of two nanobeams.

The cross-shape device is fixed by SiO 2 dielectric layer (Figure 2.2a) at the end of fabrication process. As a result of the deposition process, and principally by the annealing (400 • C) and cooling steps [3,4], the arms and pointer have some residual stress. Before the release stage, which consists in etching away the SiO 2 layer (using Vapor Hydrofluoric acid (HF)(C)) which completely prevents any movement of the conducting lines, the expansion arms are in tension. At release, they shorten asymmetrically, pushing the pointer which rotates clockwise (Figure 2.2b). The top view images allow observing and estimating the rotation using the ruler. Next section examines what causes residual stress. In order to do so, it analyses the fabrication steps from the full wafer film to design of the structure patterns. Actuation if the velocity is set to zero (u = 0), and the analysis is performed at steady state ∂ T ∂t = 0, the equation governing pure conductive heat transfer is then:

The initial temperature in all system T i =20 • C is also defined as initial condition. A radiative flux is defined on the free surfaces. The emissivity of oxidized aluminium at 200 • C (0.11 [8]) is added in the model.

The HTC can be either calculated by the software or defined by the user. In this part the calculation of the HTC value is entrusted to the software. As an external natural convection is Actuation

Joule effect actuation (NORMALLY Off structures)

A controlled actuation of the device is performed using Joule effect. Joule's first law relates the heat produced (q) to the electrical power -the square of current (I) by the resistance (R).

In the first stage, the Joule effect actuation is simulated using the thermal characteristics established in the previous thermal actuation. Then, the experimental electrical characterization is realized with SEM nanoprobing experiment.

Extension of FEM study: Thermal to electrical actuation

The previous FEM model considered the mechanical and the thermal models, now it is upgraded with the electrical model. The material properties of aluminium include this time an electrical conductivity (σ elec ) of 35.5 × 10 6 S m -1 [6].

A steady-state FEM analysis is performed for each current step -0 to 20 mA by step of 5 mA. Thus the temperature and displacement values obtained are measured from equilibrium state.

Boundary conditions are separated according to the studied physics:

• MECHANICAL: The initial stress is applied in arms and pointer that are shown in Figure 3.12. The pads of the structure are defined as blocked surfaces (u = 0). • THERMAL: The entire structure has an initial temperature of 20 • C. The HTC is calculated by the software as external natural convection (like the pure thermal FEM Model) Actuation Moreover, the stiction failure in the lateral contact zone at the 'On' state can be avoided thanks to an opposing actuation. Thus for instance if the pointer is stuck due to the actuation of the arms C and B, a D-A actuation generates a contrary rotation and takes off the pointer, provided it is not welded. This case of reversibility is not present in a standard design -i.e. NORMALLY On or NORMALLY Off structure.

The proper functioning of the structure has been verified in a testing structure, without a contact beam. The rotation of the structure both clockwise and counter-clockwise are observed in the actuation trial test. The current needed to observe a rotation in the semiautomatic prober is slightly higher than for a standard structure -i.e. 28 mA.

Summary

Now that the rotational device has been correctly released and characterised, by FEM analysis and physical measurements, this chapter focuses on the actuation step. In this chapter, a study of actuation is carried out, using both thermal and electrical triggering.

The thermal actuation is performed in a hot stage embedded in a SEM. This experiment, visualizes the rotation of the device in function of the temperature applied. A linear relationship is found between the pointer-pad distance and the temperature applied as the arms dilate linearly. A theoretical model and the FEM study found the temperature on the top full metallic layer (M1) is quasi invariable from the 200 • C applied at the bottom of the sample.

The electrical actuation uses the Joule effect as trigger. Starting by an extension of previous FEM model a temperature of 110 • C is obtained when the pointer reaches the contact pad. The characterisation experiment is achieved by a SEM nanoprobing (a prober embedded in a SEM). The contact in this experiment is reached by 23 mA applied through the arms -i.e. similar to 20 mA obtained in FEM model.

The last part of this chapter is focused on the mechanical contact in a NORMALLY On structure (the structure reaches the contact at release stage). A design modification is carriedout. The contact pad is changed by a flexible beam, which allows to estimate the contact force at release stage. Both the experiment and the FEM model obtained a similar contact force, 2 µN and 1.6 µN respectively.

Finally, a new design to prevent accidental thermal actuation is proposed. This structure has been fabricated, simulated and passes the feasibility test.

Performance

The structure has now been released and activated. A robust FEM model takes into account the multiphysical character of devices -mechanical, electrical and thermal. In this chapter the functionality of the structure (switch) is studied in terms of electrical contact (conduction efficiency) and reliability upon cycling -both electrical and mechanical. An ohmic electrical contact, repeatable and reversible, is desired. Then, in order to optimize the electrical contact resistance, two topics are studied:

• the repeatability of the movement upon Joule actuation (section 4.1),

• characterization of the surface in contact, in particular the oxides (section 4.2) 4.1 Structural reliability of the devices 135

Structural reliability of the devices

To achieve a reliable and durable device in functional life cycle, the mechanical performances are studied in this section. Two problems are addressed:

• The structure, in its life cycle, will undergo thermal cycles from room temperature (25 • C) to 200 • C. Thus, the potential plastic material deformation during thermal actuation cycles is studied ( [1]).

• The current flow through the arms occasionally results in melting and fracture of the arms, possibly triggered or accelerated by electromigration ( [2,3]). Thus, the overheating during the actuation stage is treated in subsection 4.1.3.

Displacement study and residual stress after release step

Residual stress is the driving force of the initial positioning of the switch (NORMALLY On/ NORMALLY Off ). During the release stage, part of this stress is relaxed. The structure is then confronted to cycling and switching, which both generate microstructure modification and stress relaxations. The study of this residual stress is a key element in order to determine the functional life and reliability of device. The analysis of the stress which is not released is treated by non-linear FEM analysis (including plasticity) in this section 1 .

The stress relaxation produces a rotation and a bending respectively in the pointer and in the arms. Figure 4.1 shows both the pointer displacement in 'y' direction (a) and the arms bending in 'x' direction (dx), which is quasi not measurable in SEM observation but can be modelled in FEM study.

The central pointer prevents the complete release of stress in arms, and causes arms to bend along x direction. This flexion in the XOY plane generates additional stress -largest in the hinge zone. The FEM calculus thus shows that on the internal side of the lower arm, next to the pointer centre, a tensile stress state is present. On the contrary, the external side undergoes a compressive stress state.

Using the same FEM model defined in subsection 2.2.2, an initial homogenous longitudinal stress of +190 MPa has been imposed along the arms and the pointer. Almost all of this stress is released.

The 3D steady-state simulations are performed by the elasto-plastic behavior on the "structural mechanics" option of COMSOL MULTIPHYSICS.

Performance

The series of performed chemical measurements are explained in E. They confirm that on top of Al layer, a native Al oxide is present. The detailed analysis shows that it is mainly Al 2 O 3 . On the side, three compounds are detected by XPS analysis (Figure 4.7): aluminium oxide (Al 2 O 3 ), aluminium fluoride (AlF 3 ) and a group of complex aluminium oxides and hydroxides defined as Al 2 O x . The fluoride detected is coming from the etching steps -as HF is utilised. Al 2 O x can be composed by an aluminium oxide hydroxide (AlO(OH)) or aluminium hydroxide (Al(OH) 3 ), which are difficult to distinguish by XPS analysis ( [11]).

The thickness of alumina on the top of the sample could be obtained, from XPS analysis, using the calculation of Carlson ([12]) and Strohmeier ([13]).

where θ a is the photoelectron take-off angle, I represents the intensities (i.e. peak areas) of the metal and oxide photoelectron peaks, λ (IMFP) is the inelastic mean free path, and N is the volume densities of atoms in the metal and in the oxide, respectively. Using the λ and N proposed by Strohmeier ([13]) and the values obtained in XPS analysis (Table 4 In order to observe the insulating layer and verify the thickness obtained by the analytical approximation of Equation 4.1, a physical verification is performed by Transmission Electron Microscopy (TEM) analysis. Figure 4.8 shows a generic sample preparation, which is achieved in three steps:

• First, in order to protect the zone of interest, metal deposition is performed in several layers over a surface of 20 µm × 1 µm:

-First Gold-Palladium, followed by -Platinum deposited electrochemically (to avoid damage on surfaces),

-Platinum deposited ionically (to paste a tip and manipulate the sample).

• Second, in order to obtain an electron-transparent sheet, the sample is thinned (0.1 µm) by ion milling

Summary

In this chapter the functionality of the structure (switch) is studied in terms of electrical contact (conduction efficiency), reliability upon cycling -both electrical and mechanical, and optimization of the structure itself with respect to these properties. As switching is supposed to occur thousands of times, the first part of this chapter studied the reversibility of the movement. After release stage, the pointer-pad distance is not identical on both sides of the cross -a variation of 10 % is observed. The structure has been applied a running-in stage. Activated more than 1000 cycles, the pointer-pad distances evolve to become identical and stable.

Numerical simulation is used to show the stress concentration at sharp corners, which is a source of nonlinearity and non-reversibility. The structural damage caused by excessive heating is studied as well.

An ohmic electrical contact, repeatable and reversible, is desired and studied in the second part of this chapter. Then, in order to optimize the electrical contact resistance a study of the contact surface is realized.

During the manufacturing process, a native oxide layer grows on the aluminium structure when in contact with the air. The chemical analysis (XPS and AES) estimates an oxide thickness of 4 nm. Then, a cross-section TEM has been confirmed the 4 nm of Al oxide on the top of sample and 6 nm on the sidewall Al in the contact zone.

Thus, in order to create an electrical contact between the pointer to the pad, two possibilities are analysed:

• Either ensure a measurable tunnel current through the insulator layer, for applications where nA could be sufficient.

• Or add an oxidation-resistant interfacial material on the contacting surfaces to avoid or minimize the native oxide -namely "W-Spacer" solution.

The switch functionality is validated electrically. An electrical contact is reached at release step and lost in actuation step. Another major subject is the packaging of devices. The structure oxidizes easily in contact with air, so it must be done in a controlled environment. A seal ring defined around the device, which are made using the "metallic vias" (a sort of pillar that connects the different levels of metal) has been thought.

The virtual fabrication laboratory is central in our approach. It can certainly be improved, not only by using more complete versions of COMSOL MULTIPHYSICS. Precision can be gained by deepening the understanding of the heat transfer (convective as well as radiative) in confined, microscale and multibody spaces. Investigating non-reversibility and optimizing e.g. the hinge geometry would benefit from a more sophisticated elastic-plastic model, the parameters of which should of course be identified experimentally in an adequate way. Moreover, the assumption that electromigration could be one of the causes of break at high intensity or long times (repeated actuation) could be tested numerically in a more explicit and complete way.

As all R& D work, this PhD has also raised more fundamental questions and put forward needs which could not be investigated in details but would certainly deserve more attention. Thus, the relationship between intensity, temperature and arm fracture would have been determined with more certainty if a microscale temperature measurement system had been available. Two such apparatuses, an IR camera, have been preliminary tested in Toulouse (LAAS) and in Saint-Etienne (EMSE). The CMOS and the MEMS fabrication flow are very similar. This is because both use photolithography processes. The CMOS fabrication is separated in front end of Line (FEOL) and BEOL. The FEOL is composed of the electrically active devices i.e. transistors, diodes, resistances. The interconnections of the actives devices are realized in the BEOL, mainly made of alternating metal and oxide.

At STMicroelectronics, we use mid-wafer short loop test vehicles for rapid learning cycles i.e. without active devices. Thus, only the fabrication of the interconnection part is explained here. Using a Si substrate, the first step is wet thermal oxidation at 1150 • C to obtain a 3 nm thick thermal SiO 2 protective layer Figure 1. In the fabrication flow, over the silicon substrate a successive alternation of an insulator (SiO 2 ) also called IMD and a metallic layer -principally Al (Cu) or Cu -is deposited, thus forming the BEOL.

Back end of line: fabrication process flow

After sawing the dices, the sacrificial layer is etched in a Vapor-HF process for 40 min. This releases thin beams but leaves larger structures (pad) attached to the M1 layer. The Vapor-HF method prevent stiction of free structures to the M1 layer.

The principal parameters used in material deposition are shown in Table B

Appendix C

A critical process step: the release of structures

The most technological step of this project, the release of the main structure (cross), consists in etching the oxide below (or around) the metal layer to free the expansion arms and the central pointer.

There are three methods to etch the SiO 2 ; the wet, the dry, and the gaseous etch which are also known as chemical, physical, and vapor etching respectively.

• The dry etch (physical or plasma etch) uses oxygen plasma, more precisely a CHF 3 , CHF 3 O 2 , SF 6 or SF 6 O 2 at certain temperature and pressure in a vacuum chamber, to remove silicon dioxide; this technique normally produces an anisotropic profile, so it is not suitable to our objective of getting freestanding parts.

• The wet etch (chemical or HF etch) uses HF, more precisely a 50 % solution of hydrofluoric acid in water, applied during 10 s, giving an "isotropic" profile. The sample is then rinsed by deionized water and isopropanol alcohol.

• A third alternative also useful is to use the gas-based method; the wet HF is heated to 40 • C, it evaporates and etches the oxide without wet process.

Several problems due to imperfect release of wet etch, using pure HF solution, are reported. A general difficulty is that contrary to industrial conditions, wet etching by a manual process is an uncontrollable and poorly repeatable procedure, in particular due to the short etching time which cannot be enforced accurately, but there is a risk of damaging the metals.

Another problem with wet etching is the risk to stiction of mobile parts Figure 1. During the process, the evaporating liquid forms a meniscus; its very small radius of curvature builds A critical process step: the release of structures up a very high attractive Laplace force (due to surface tension of the solution) which may bend the arms towards the floor layer. If contact occurs, adhesive forces cannot be counteracted any more. Therefore, I chose gaseous etching. COMSOL MULTIPHYSICS is composed by a main core, Comsol Multiphysics, and then different modules can be added to deepen a certain study (Figure 1).

Fig. 1 The available modules in COMSOL MULTIPHYSICS

In order to model our device in finite element, COMSOL MULTIPHYSICS proves well adapted. This is because of its performance in the various multiphysics coupling modules and simplicity in the post-processing of the data collected. It offers powerful solvers who can solve problems numerically very large.

The predefined templates permit to simulate physical phenomena such as electrostatics, fluid mechanics, heat transfer and deformation of materials, and coupling them with ease through a graphical interface.

To solve a problem using COMSOL MULTIPHYSICS, the logical sequence of steps is:

1. Creation of geometry;

Contact modelling in Comsol

The augmented Lagrangian method is implemented in COMSOL MULTIPHYSICS contact model. This technique is explained by Simo and Laursen [1] and Zavarise [2]. Extract from Comsol documentation where the augmented Lagrangian method is explained [3]: "In the augmented Lagrangian method, the system of equations is solved in a segregated way. Augmentation components are introduced for the contract pressure T n and the components T ti of the friction traction vector T t . An additional iteration level is added where the usual displacement variables are solved separately from the contact pressure and traction variables. The algorithm repeats this procedure until it fulfils a convergence criterion. Using the special gap distance variable, the penalized contact pressure T np is defined on the destination boundary as where d g is the gap distance between the destination and source boundary, and p n is the user-defined normal penalty factor."

Appendix E Chemical analysis

In order to identify the chemical compounds in the structure, an array of bridges is performed.

To have an accurate data information a dense structure is needed. A dense bridge-shaped structures, placed close to our structures are fabricated (Figure 1). The information given by the top of the pad is a reliable information of the surface condition on the top of the sample.

To obtain the surface composition of the sidewall, the information obtained by a top analysis is compared to the tilted analysis. Thanks to this experiment, an aluminium oxide of 4 nm and 6 nm has been found in the top and in the side of Al structure.