Mechanical resonator for hermeticity evaluation of RF MEMS wafer–level packages

Master's Thesis, 2002

124 Pages, Grade: 1.0 (A)



1. Introduction
1.1 What is MEMS ?
1.2 Hermeticity
1.3 Wafer-level sealing technologies
1.3.1 Anodic bonding
1.3.2 Silicon direct bonding
1.3.3 Eutectic bonding
1.3.4 Adhesive bonding
1.4 How to measure hermeticity

2. Theory
2.1 Bending of a beam
2.2 Mechanical vibrations
2.2.1 Free, undamped vibration Flexural mode vibration Shear deformation Torsional vibration Longitudinal vibration
2.2.2 Damping
2.2.3 Quality factor
2.2.4 Pressure dependency of the quality factor
2.3 Electrostatic excitation
2.4 Optical detection
2.4.1 Fabry - Perot - Interferometer
2.4.2 Optical properties of silicon

3. Project management

4. Design
4.1 System requirements
4.2 Design considerations
4.3 Design Process
4.4 Stiction
4.5 Mode coupling
4.6 Material
4.7 Excitation and detection
4.7.1 Electrostatic excitation of the beam
4.7.2 Mechanical stability of the bottom-bottom electrode configuration
4.7.3 Electrical connections
4.7.4 Detection technique
4.8 Designing a basic beam structure
4.9 Design variations and other resonator shapes
4.10 Mask drawing

5. Fabrication
5.1 Flowchart

6. Measurement setup
6.1 Measurement Results and Discussion

7. Final Discussion and Conclusions

8. Acknowledgments

9. References

Appendix A1, Ion-Implantation

Appendix A2, simulation results for considered structures

Appendix A3, Detailed Flowchart

Symbols used

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Abbreviations used

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constants used

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The presented thesis describes the design, fabrication and evaluation of a hermeticity sensor. The thesis is divided into 4 parts. The first part, chapter 1, describes the background to MicroElectroMechanicalSystems (MEMS) and to packaging. Chapter 2 describes the background to the sensor principle and the functionality of the sensor.

The design part, chapter 4, explains the functionality of the sensor and how it is influenced by different parameters. Simulation tools were used to finalize the design. Various designs were made and the corresponding masks drawn.

The fabrication process for the sensor devices is explained in chapter 5. Here the problems and challenges encountered are described.

Chapter 6 discusses the measurements performed and demonstrates that the fabricated device works as a hermeticity sensor. Appendix A1-A3 gives additional information.

Enclosed with this thesis a CD can be found. It contains the thesis as well as additional information including the simulation results and the simulation files.

1. Introduction

1.1 What is MEMS ?

Originating from the microelectronic industry the field of Microsystems, often referred to as (microelectromechanical system) or MST (Microsystems Technology), has been growing and attracted huge interest in recent years. The classical field of MEMS is usually divided into two categories, those devices that detect information, called microsensors, and those devices that can respond to information, or act, called actuators. The use of standard electronic processes to create MEMS devices and the resulting possibility of large scale integration (LSI) leads to a high yield thus to a low price for each device. Recent developments in processing technologies led to (MicroOpticalElectricalMechanicalSystems) and BioMEMS (MEMS for biomedical applications). They are neither actuators nor sensors but MEMS devices with a great innovation potential in this field of technology.

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First developed in the 1970s and then commercialised in the 1990s, MEMS make it possible for systems of all kinds to be smaller, faster, more energy-efficient and less expensive. In a typical MEMS configuration, integrated circuits (ICs) provide the “thinking” part of the system, while MEMS complement this intelligence with active perception and control functions.

MEMS are not single applications or devices, nor are they defined by a single fabrication process or limited to a few materials. More than anything else, MEMS represent an overall strategy combining miniaturization, multiple components and microelectronics to design and fabricate integrated electromechanical systems.

Micromachining and new materials are the keys for creating new devices. Microsystems can combine many subsystems like electrical, mechanical, fluidic, thermal, optical parts on one small chip.

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Microsystems influence our daily life even though nobody can see the tiny devices. Recent efforts have led to new applications such as very small pressure sensors, as shown in fig. 1.1, or microneedles for biomedical applications, fig. 1.2, microfluidical pumps or RF-MEMS for wireless applications.

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RF (Radio Frequency) MEMS devices are starting to play a more and more important role in the field of MEMS, especially for wireless applications.

RF-MEMS is a collective name for microswitches, electromechanical resonance filters, tunable capacitors or inductors which operate in the radio frequency region.

RF-MEMS components show superior performance compared to traditional electronic devices[28]. Until now, the commercialisation of such components was limited due to the expensive packaging required for the electromechanical part of such devices, which nowadays is often up to about 80% of the whole device costs. Efforts have been made to achieve a cheap and reliable wafer-level encapsulation technique.

Not only for RF-MEMS applications but for almost every field in MEMS a package is needed which ensures a certain gas tightness since many microdevices only work at low pressure.

1.2 Hermeticity

A hermetic package is defined as an enclosure with an internal cavity demonstrating a certain gas tightness.

It is very often the case that micromechanical devices must work in a certain atmosphere like vacuum, that they must be protected from influences in the outside world or vice versa. Examples are absolute pressure sensors, which require a constant reference pressure, resonant sensors or microsystems for biomedical applications, where hermeticity plays an important role.

To achieve a certain hermeticity the micromechanical device is packaged and sealed under a special atmosphere during fabrication. The sealing technology must ensure a certain gas tightness which enables long term stability for the microsystem. Also, the connections to the outer world, to excitation, detection or signal processing units must be provided by the package.

With the size of the microsystems decreasing, the size of the cavities in which they are enclosed must also decrease. Especially for very small volumes, the requirement of a hermetically sealed package is becoming more and more of a challenge because even very small leak rates have a huge influence on the pressure inside the cavity. This leakage may lead to malfunction and failure of the system. The pressure change per unit time in a device is proportional to the ratio of leak rate to volume:

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where L is the leak rate, with the dimension mbar ⋅l ⋅ s and V the volume in l of the cavity. Many known hermetic sealing technologies are shown in chapter 1.3.1-1.3.4, they vary in process temperature, used materials etc.[38]

All sealing technologies presented in this work are L1 sealing technologies. They consist of one wafer, which carries the devices to be sealed and another wafer, which is bonded onto that wafer for sealing. Packaging on wafer-level entails protecting the devices during process steps to follow, to separate the devices from the outer world while using the device and connecting them to other parts of the microsystem respectively. The requirements on the package depend on the kind of microsystem to be packaged in terms of used materials, interface problems, process temperatures, process compatibility, demands on reliability and long-term stability, working environment and also fabrication costs.

1.3.1 Anodic bonding

Anodic bonding is widely used for hermetic sealing of micromachined devices, it is also known as electrostatic bonding. It is based on joining an electron conducting material and an ion conducting material. With this technology silicon to glass wafers as well as silicon to silicon wafers are bonded where one wafer carries a thin silicon dioxide (SiO2) layer. The thermal expansion coefficient of the glass used should match the thermal expansion coefficient of the silicon to avoid the generation of thermal stress. Also, the glass has to be slightly conductive at the chosen bonding temperature. That’s why special bond glasses are needed, like borosilicatglass Pyrex 7740, TEMPAX from Schott or SD-2 from HOYA.

The bonding process is assisted by heating up to 180-500°C and applying a voltage of about 200 to 1500 Volts. The bonding temperatures are near the glass-softening temperature point but well below its melting point. The electrical field which is applied at the bonding temperature leads to a migration of the ions in the glass towards the cathode, fig. 1.5. A depletion layer in the glass near the silicon surface is created. The voltage drop over this depletion layer creates a large electrical field which pulls the wafer into intimate contact. The elevated temperature results in the forming of covalent bonds between the surface atoms of the glass and the silicon[8].

Anodic bonding leads to strong and hermetic bonds. In case of using an intermediate glass layer the obtained bond strength is slightly lower. But also other intermediate layers are possible such as silicon dioxide, aluminium, silicon nitride or polysilicon[38]. Anodic bonding has good groove closing properties, it was shown that grooves with a depth of 50 nm are perfectly sealed after anodic bonding[33]. One major drawback of Anodic Bonding is the high voltage which is needed to reach a high bond strength. It can destroy microelectronic devices, which makes Anodic Bonding not compatible for packaging wafers with integrated electronic devices. Another drawback of this technique and all other techniques using glass wafers is the outgassing from the glass during the bonding and encapsulation process which degrades the vacuum quality. The outgassing is caused by desorption of moisture or gases adsorbed on the glass surface and outdiffusion of gases which are resident in the glass. It was reported that an initial bonding pressure of 10-[5]mbar resulted in a final internal cavity pressure of 1 mbar[38].

To overcome this drawback a baking of the glass under vacuum and the use of a metal coating on the recessed area as a diffusion barrier has been proposed[34]. Further disadvantages of this technique are the difference in the thermal expansion coefficient in higher temperatures and the higher price of these special glasses compared to silicon.

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1.3.2 Silicon direct bonding

With silicon direct bonding (SDB) two silicon wafers are bonded onto each other. Silicon direct bonding can be divided into bonding of hydrophobic wafers and bonding of hydrophilic wafers. This distinction is related to the way the surfaces of the wafers are prepared. The SDB technology requires no additional material. The process can be divided into three steps which can be seen in the following schematic fig. 1.6.

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The direct bonding process is based on reactions between OH-groups in the case of hydrophilic bonding or HF-groups present at the surface in the case of hydrophobic bonding. The higher the density of OH- or HF-groups, the higher the surface energy and the bond strength. Furthermore, flatness requirements are much more stringent than for anodic bonding, the average value of roughness of the initial wafers has to be lower than 1 nm, which is usually given for CMP-polished wafers. A good flatness ensures a large bonding area, but due to bow and warp of wafers an additional pressure is needed.

To create a hydrophobic surface the wafer surface is etched in hydrofluoric acid (HF) to etch away the native silicon dioxide on the surface. To create a hydrophilic surface the wafers are e.g. soaked in a H2O2-H2SO4 mixture, but also other dry and wet chemical possibilities are available[7],[43]. For hydrophilic bonding both wafer surfaces need to have a dioxide layer on top of the bonding surface.

The Prebond consists of contacting the wafers at room temperature where a self bonding throughout the wafer is formed with considerable bonding forces. During the Prebond process capillary forces pull the wafers into intimate contact, this can be supported by mechanically pressing the wafers onto each other. Due to this only a few OH- or HF-groups gap the distance between both wafers.

The postbond step is a high temperature annealing which increases the bond strength by more than on order of magnitude. During the postbond the OH- and HF-groups diffuse away from the surface and covalent bonds over the wafer surfaces are built. Hydrophobic and hydrophilic wafer bonding differs in the annealing temperature needed to achieve a specific surface energy or bond strength respectively.

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Advantages of this technique are the fact that both wafers are silicon wafers, which makes a processing of both wafers possible, no intermediate layer is needed and thermal induced stress can be excluded even for higher temperatures. Major drawbacks are the very high demands on the silicon surface, the need for a surface activation for hydrophilic bonding and the relatively high temperatures.

1.3.3 Eutectic bonding

Eutectic bonding uses the eutectic point in the metal-silicon or metal- metal phase diagram to connect wafers. Gold has been deposited on the wafer surface which formes an eutectic melt with silicon already at 363°C, which is well below the critical temperature for Al components. During the bond process a silicide as an intermediate layer is formed. Other possibilities are the use of Pb/Sn which melts already at 183°C or Ti/Ni as an intermediate layer to connect glass wafers with silicon wafers. Eutectic bonding has the potential to become a very important bonding technology due to its relatively low process temperatures and the use of metals, which enables a very high hermeticity.

1.3.4 Adhesive bonding

Adhesive bonding uses an adhesive intermediate material to join two substrate materials. As adhesives, epoxies or polymers are usually used. After cleaning and drying the wafers an adhesion promoter can be applied to improve the final bond strength. Hereafter follows the spinning of the polymer on one of the wafers and a procuring step to evaporate the solvents. After that the two wafers are joined in the requested atmosphere applying pressure and temperature.

The advantages of this technique are the very low bonding temperatures, which is often below 100°C, different wafer materials can be joined, elastic properties of the polymers reduce the stress between the bonded structures and low processing costs. Surface nonuniformities up to 1 µm can easily be levelled. Adhesive bonding is an appropriate technique to protect MEMS devices during processes such as dicing. A major drawback of that technique is that there are so far no adhesive materials with a low permeability for gas or moisture, which means that no hermetic seal can be achieved with this technique. Another disadvantage is the limited long term stability of many polymer materials.

1.4 How to measure hermeticity

Several technologies exist for measuring the hermeticity of a package or the quality of a hermetic seal.

- To measure the pressure inside a cavity and its change over time
- With IR (infrared-)measurements a change in the atmosphere inside a cavity can be observed
- Bolometers measure the thermal conductivity of the gas in a cavity, which changes with its pressure
- The cavity can also be opened and the outflowing gas molecules can be observed

The leak rate can be measured according to MIL-STD-883D Method 1014.9. According to this method, hermeticity testing requires fine leak and gross leak testing respectively. Gross leaks are tested by using fluorocarbon liquids (FC-84 and FC-40) and are based on the bubble method. The samples are placed in FC-84 (boiling temperature 84 °C) for several hours. After taking the samples out and drying them, they are immediately transferred to FC-40 (boiling temperature 139 - 189 °C) and heated up to 110 °C. Because of the difference in boiling temperature of these fluids, the presence of FC-84, and thus, the existence of a gross leak can be observed as a stream of bubbles[6],[27]. The fine leak test is always carried out first. It consists of pressurizing the sample with a high pressure of He, i.e. 3 bars absolute pressure, for several hours.

Next, the samples are transferred to a He mass spectrometer where the He leak rate is measured. These tests are not very suitable for the field of Microsystems due to the very small cavity volumes which are used in MEMS,[27]although they are widely used in industry.

Instead of measuring the leak rate, the change of the thermal conductivity of the gas inside the cavity can be measured. Thermal conductivity sensors use the fact that heat loss of a hot object to the ambient is related to the pressure of its surrounding gas. Thermal conductivity sensors for different pressure regions are available. Conventional thermal conductivity sensors are used for vacuum measurements ranging from 10-[2]to 10[4]Pa, but they are also available for higher or even lower pressures down to 10-[5]Pa[30],[29]. A major drawback of these sensors is a rather complex structure which leads to an elaborate and costly technology.

Another possibility for determining the hermeticity is to measure the pressure inside the cavity and its change over time. This can be done either by using conventional pressure sensors or by measuring the pressure dependent damping of a resonator structure. The resonant sensor comprises an element vibrating at resonance frequency, which changes its resonance frequency as a function of a physical or chemical parameter. The frequency output of the resonance sensor can be digitised by using standard electronic equipment.

Since the frequency contains the information of the physical or chemical parameter to be measured, the output signal for linear systems is independent of analogue levels such as time and set-up dependent drifts in the excitation voltage. Furthermore, the output is directly connected to the parameter to be measured. For instance, with a mechanical resonance sensor a change in the ambient pressure leads directly to a change in the damping, thus to a change in its resonance behaviour. This will be discussed in chapter 2.2.4.

2. Theory

In this chapter some important fundamentals for deflection of a beam structure and vibration of simple beam structures are discussed further important fundamentals of electrostatic excitation and optical detection techniques are explained. For the following discussion a linear system and also no plastic deformation of the structures are assumed.

2.1 Bending of a beam

If a force or a pressure is exerted to a beam structure it will bend dependent on it Young’s modulus, its geometry and the force or pressure which is applied to the beam. Consider a cantilever beam of length l, width w and thickness t as it is shown in the following figure.

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The general differential equation of the elastic curve is:

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where E is Young’s modulus, I the moment of inertia of the section of the beam with respect to the neutral axis, M represents the bending moment and u z the displacement in z -direction.

To solve this equation, the expression for M is written out and integrated twice. The constants are determined from the boundary conditions of the beam[2].

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The following table shows the maximum deflection for different load types on a single-clamped or double-clamped beam.

A broad discussion can be found in[11].


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The spring constant k is determined by the force which is needed to achieve a certain maximum deflection:

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2.2 Mechanical vibrations

A system which contains two or more energy reservoirs and the energy can be transferred from one reservoir to another one is called an oscillatory system. Any oscillatory system is able to oscillate under appropriate conditions. In linear systems the natural frequencies are independent of the excitation amplitude. Any oscillatory system can be excited by an external force. The response of a mechanical system depends on the geometrical and material properties of the system and on the damping since energy is dissipated by friction and other resistances.

If the system is excited by a step force it will respond with one or more of its natural frequencies and oscillate freely, only due to the forces inherent to the system itself. The natural frequency is a property of the system, in case of a mechanical resonating system it depends on its mass distribution, its stiffness, materials and also on the damping factor. Depending on the damping of the system the structure will respond with an oscillatory or a nonoscillatory motion.

Vibration that takes place under the excitation of external forces is called forced vibration. When the excitation is oscillatory the system is forced to vibrate at the excitation frequency. If the system is excited by an external frequency the amplitude is also dependent on the excitation frequency. The highest amplitude is achieved if the system is excited with its natural frequency, this is called resonance.

A beam, single- or double-clamped responds to an excitation in different ways as it is shown in the following drawings.

For a mechanical structure not only the time or frequency dependent behaviour but also the spatial dependent behaviour must be considered.

Here it has to be distinguished between flexural, torsional and longitudinal mode of vibration as it is shown in fig. 2.3.

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Furthermore for each mode of vibration different harmonics have to be considered.

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The fundamental wave of flexural vibration mode shows the highest amplitude.

2.2.1 Free, undamped vibration

First of all the undamped, free motion of this prismatic, homogeneous beam will be discussed. The effects of damping will be discussed later. Flexural mode vibration

The first mode of flexural vibration of a beam shows the highest deflection, thus, enables the highest signal to noise ratio when detecting the signal therefore it is the most interesting one for sensor applications.

The free motion of a cantilever beam can be described as follows, where shear deformation is neglected and no distributed loads are present. The following figure shows an infinitesimal piece of a beam structure.

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Master Thesis, Sebastian Fischer KTH, S3, 2002

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Mechanical Resonator for Hermeticity Evaluation of RF MEMS Wafer-level Packages

Where M is the bending moment, Q the shear force, ρ the density of the beam material and A the area in x,y -direction.

If the influence of shear deformation is neglected and if no distributed loads are present, the free motion of this prismatic, homogeneous and undamped cantilever is given by the following equation. As an example a deflection in z -direction is assumed.

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the following equation is found:

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where E is Young’s modulus and I the moment of Inertia.

With E, I, ρ, A = wt = const., it can be written:

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This partial differential equation describes the time and spatial dependent behaviour for a free, undamped oscillating beam structure. The displacement varies with time and also with the position along the beam. To solve this partial differential equation a solution which contains of a product of two solutions is assumed. In general these are the time dependent differential equation T(t) and the spatial dependent equation X(x).

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Both equations can be solved by finding the characteristic equation.

The time dependent differential equation without damping has the form

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For a physical system only the real part of the general solution has to be considered which leads to the following solutions.

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With A, α, β as the amplitude of the vibration, φ as the phase angle and ωn as the n th natural frequency. To find a solution for the spatial dependent differential equation, the following boundary conditions for different types of beams (see EQ. 2.2-2.5) must be considered[2].

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For a single-clamped beam the transcendental eigenvalue equation[2]

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is found. Eq. 2.18 has an infinite number of solutions. By using a numerical approximation the first solution

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The natural frequency in hertz can generally be expressed in the form[2]:

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where λi is a dimensionless parameter which is a function of the boundary conditions applied to the beam, l is the length of the beam, ML is the mass per unit length.

Also the mode shape of the single-clamped beam can be calculated, by using the following solution[2]:

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It is advisable to employ a high numerical accuracy in λi and c i when computing the mode shapes since changes as small as 10-[6]can result in a significant change in the computed mode shape. Various natural frequencies for various boundary conditions can be found in[2]. Shear deformation

Shear deformation can be important in short beams or in higher modes of slender beams.

The natural frequency of a shear beam can be calculated with[2]:

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For a beam with a rectangular cross section the shear coefficient can be calculated by:

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The transverse deformation of a real beam is the sum of flexural and shear deformations. Shear deformations are usually negligible in analysis for slender beams. It can become important in higher modes of slender beams, in analysis of very short beams or in special structures.

As an advice numerical solutions are required to incorporate both flexural and shear deformation in the prediction of the natural frequencies of beams.

The natural frequency of a beam with comparable shear and flexural mode can be estimated by using the Southwell-Dunkerley approximation[2]:

illustration not visible in this excerpt Torsional vibration

Torsional vibrations are the result of local twisting of a beam or shaft about its own axis. Exact closed form solution for torsional vibrations can only be obtained for the case of shafts or tubes with circular cross sections, which can be found e.g. in[4].

In general the natural frequency of the torsional vibration can be calculated with:

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With Ip as the polar area moment of inertia of cross section about axis of torsion.


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C is the torsional constant of the cross section. For a rectangle cross

section as shown in fig. 2.1 the torsional constant can be calculated by:

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where w is the beam width and t is its thickness.

The parameter c is determined by the w/t -ratio:

illustration not visible in this excerpt Longitudinal vibration

Longitudinal vibrations arise from stretching and contracting of the beam along its own x -axis as shown in fig. 2.1. The longitudinal deformation is assumed to be uniform over the cross section. The natural frequencies for longitudinal vibration of a cantilever beam can be calculated by:

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2.2.2 Damping

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Vibrating systems are all subjected to damping to some degree because energy is dissipated by friction and other resistances. Damping leads to a change in the amplitude, the bandwidth and the natural frequency of the system. The influence on the natural frequency is only minor, close approximation calculations can be achieved by not taking damping effects into account. The influence on the amplitude and the bandwidth are much larger and have therefore practically a higher significance. Damping limits the vibration amplitude at resonance, it changes the bandwidth of the system and leads to a decay in time of the amplitude for a free vibrating structure after excitation.

When considering the effects of damping of a mechanical resonating structure two different causes have to be considered. It has to be distinguished between a dissipative damping part which is proportional to velocity and an inertial damping part which is proportional to acceleration. For a cantilever beam vibrational energy can be dissipated via coupling to the support structure and by internal friction.

The inertial damping effect arises due to inertial friction within the vibrating structure and due to the gas flow around the oscillator.

In inertial damping due to the gas flow around the structure it has to be distinguished between damping due to a free airflow force, damping due to squeeze film damping[16],[15],[17]and damping due to a close rigid plate[26]as shown in fig. 2.6.

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The damping due to internal friction is material dependent and also depends on the support. If a structure with a very high Q-value is needed, the structure has to be balanced whereby the support losses are minimised. A balanced structure has a fixed centre of gravity and the sum of all forces and moments resulting from the vibration are zero[17]. Single-crystalline silicon has shown to have a very low damping with a damping factor of about 10-[6][16],[17]due to internal friction. Since Microactuators are usually made from a single piece of material energy losses due to support loss can be neglected. As long as the amplitude of the beam is smaller than the thickness of the beam, the damping due to a rigid plate is negligible[26]. The surrounding medium exerts a drag force F 1, which is known for a harmonically oscillating sphere as[15],[16]:

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Taking the effects of damping into account, the free motion of a prismatic, homogeneous beam, which includes a rectangular cross section of the cantilever as it is shown in fig. 2.1, is described by:

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with f 1 as the dissipative damping part and f 2 as the inertial damping part. The second term, f 2, leads to a smaller resonance frequency. The inertial damping part f 2 is much smaller than the mass of the beam and can be neglected for silicon or metal actuators[16].

The following paragraphs show the influence of damping on free vibrating systems and harmonically excited systems.


The mass of the beam is assumed to be concentrated at the tip of a

massless spring, the damped vibration of a cantilever can be modelled by using a mass-spring system as it is shown in the following figure.

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From fig. 2.7 B) the equation of motion can be derived

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The solution of this equation has two parts. If the excitation force F(t) =0, a homogeneous differential equation whose solution corresponds physically to a free-damped vibration is obtained. First the homogeneous equation is examined. To find a solution for the homogeneous differential

equation, a solution of the form

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is assumed, where s is a constant.

Upon substitution into EQ. 2.35, it is obtained

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Which satisfies for all values of t when:

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This equation is called the characteristic equation. The characteristic equation has two roots:

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This leads to a general form of the solution:

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The exact solution has to be evaluated from the initial conditions u ( t = 0 ) and u ( t = 0 ). The first term is an exponentially decay function of time. The behaviour of the terms in the parentheses depends on whether the numerical expression within the radical is positive, zero or negative. The differential equation of motion can be expressed in terms of the damping factor ξ and the natural frequency ωn.

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The inertial damping part f 2 has been neglected.

The three cases of damping can now be expressed on whether the damping factor ξ is greater than, less than or equal to unity.

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For the damping factor three cases have to be distinguished.

A damping of ξ < 1, which is called the underdamped case, leads to an oscillatory motion of the system, as shown in fig. 2.8. The roots of the characteristic equation are conjugate complex.

A damping of ξ > 1, which is called the overdamped case, leads to a nonoscillating motion of the system. The roots of the characteristic remain on the real axis.

A damping of ξ = 1, which is called the aperiodic case, leads to an aperiodic motion. A double root for the characteristic equation is obtained, with s 1= s 2=-ω n.

If no external force is exerted to the beam, the amplitude will decay with time as shown in fig. 2.8. The natural logarithm of the ratio of any two successive amplitudes is called the logarithmic decrement δ.

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When a system is subjected to harmonic excitation, it is forced to vibrate at the same frequency as that of the excitation.

The time dependent differential equation with damping for a harmonically excited system is:

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The solution to EQ. 2.46 consists of the superposition of the solution of the homogeneous equation and the particular integral. The particular solution is a steady-state oscillation of the same frequency ω as that of the excitation. The particular solution can be assumed to be of the form:

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where Uz is the amplitude in z -direction of oscillation and φ is the phase of the displacement with respect to the exciting force.

The nondimensional expressions for the amplitude and phase as shown in fig. 2.9 become:

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These equations indicate that the nondimensional amplitude Uzk/F 0 and the phase φ are functions only of the frequency ratio ω / ω n and the damping factor ξ. In fig. 2.9 it can be seen that the damping factor has a large influence on the amplitude and the phase angle near resonance.

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It can be seen that the vibration amplitude increases with a decreasing damping factor. For higher damping factors the frequency shifts slightly to lower values.

2.2.3 Quality factor

The quality factor (Q -factor) is a very important parameter in evaluating a vibrating system. It is defined as:

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Where δ is the logarithmic decrement. The Q -factor can also be calculated from the amplitude-frequency spectrum of the forced vibration. The resonance frequency divided by the bandwidth at the 3 dB amplitude points determines the Q -factor.

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fig. 2.10 also shows the relation between the damping factor and the Q -factor, thus the damping factor can be calculated by:

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Since there is no possibility to calculate the pressure dependency of the

Q -factor for an arbitrary structure, it can only be given an approximation, which was derived by[15]and confirmed by[20]for a simple cantilever beam structure.

2.2.4 Pressure dependency of the quality factor

There are several damping mechanisms acting on the structure limiting the overall Q -factor. The quality-factor can be written as:

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From EQ. 2.53 it can be seen, that only the dominant damping parameter needs to be calculated to get a good approximation.

In different pressure regions different damping effects have a dominating influence. During the next chapters the pressure range is divided into three regions and the pressure dependent behaviour of the dominating damping effect is shown. [8]. The materials vary in their gas penetration properties as it is shown in fig. 1.4. Metals are the best materials in terms of repelling moisture. Ceramics, such as silicon dioxide is an excellent moisture barrier. Organic polymers such as epoxies and silicones are several orders of magnitude more permeable to moisture.

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Existing standardised methods used for evaluating hermeticity of encapsulated cavities are either unsuitable for very small volumes (<1 mm[3]) or only applicable for special encapsulation techniques.

However, encapsulated resonator structures may be suitable and universally usable test devices to evaluate the hermeticity of encapsulated cavities. 1.3 Wafer-level sealing technologies A wide variety of hermetic sealing technologies are available on wafer level, chip level or device level. The technologies presented in the following subchapters are wafer-level sealing technologies. In wafer level sealing technologies, L0 and L1 level packaging must be defined. In L0 level packaging a die feature is encapsulated where mainly metal evaporation is used for encapsulation[8],[36]. In L1 level packaging entire dies are encapsulated.


Excerpt out of 124 pages


Mechanical resonator for hermeticity evaluation of RF MEMS wafer–level packages
University of Applied Sciences Berlin  (FB1)
1.0 (A)
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Mechanical, MEMS
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Dr. Sebastian Fischer (Author), 2002, Mechanical resonator for hermeticity evaluation of RF MEMS wafer–level packages, Munich, GRIN Verlag,


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Title: Mechanical resonator for hermeticity evaluation of RF MEMS wafer–level packages

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