Electrostatic and RF-Properties of MEMS Structures

12

Chapter Twelve Electrostatic and RF-Properties of MEMS Structures Ilkka Tittonen and Mika Koskenvuori Micro and Nanosciences, Micronova, Helsinki University of Technology, Espoo, Finland

12.1 Introduction

– Small size of MEMS and NEMS devices means also

Well-known examples of micromechanical sensors are accelerometers, pressure sensors and cantilevers that are used, for example, as fluid sensors and in various microscopes. In designing and modeling of nano and micromechanical systems the following general aspects should be taken into account:

– Capacitive coupling leads to a novel capacitor of

– The sound velocity depends on the chosen material

and affects the properties of dynamical systems. – Particular design finally determines basic resonance

modes and frequencies and leads to the modification of stiffness. – Any mechanical device has an infinite number of

various mechanical vibrational modes. – Dissipation of mechanical energy is often difficult

to predict, especially since total dissipation is a sum of contributions of intrinsic and external losses of energy. – In case the material is anisotropic (like single crystal

silicon), the mechanical properties depend on the crystal orientation. – Both the capacitive and piezoelectric coupling make

it straightforward to create local forces that can be used in actuation of micromechanical devices. – Capacitive and piezoelectric couplings are also

potential ways to integrate the mechanical systems as part of the electric circuit.

high capacitance values. which value is dynamic and depends on the applied voltage across its gap (bias-voltage UDC). – Small gaps (significantly below 1 μm) are still a

challenge for mass fabrication. – Small (capacitive) mechanical sensors may

easily suffer from charging of surfaces, diffusion, contamination and break through currents and may need to be surface passivated or encapsulated. – RF MEMS devices are highly potential candidates as

reference oscillators, filters and switches. – Finite-element method (FEM) is often more or less

the only way to simultaneously model mechanical, thermal and electric effects. – Often finite-element modeling leads to higher level

circuit simulations by giving the necessary input parameters.

12.2 Model System for a Dynamic Micromechanical Device A most simplified way of modeling a micromechanical moving system is to form a lumped element model that contains a minimum number of physical parameters (Figure 12.1). Resonators moving as a response to an external force can be described using at least three different

221

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Fig 12.1 ● Model system of a dynamic micromechanical system.

parameters: the spring constant k, the effective mass m and the damping constant c. The spring constant depends on the chosen material and the design and in resonant cases also on the particular vibrational mode. The effective mass is a measure of the size of the moving part, but it is also affected by the strength of coupling to the support (substrate, etc.). Damping takes into account both the internal and external sources of energy leakage. Resonators are usually driven using harmonic forces, which are often achieved by capacitive coupling. In such cases the dynamic motion of a micromechanical resonator can be described by the differential equation of motion: mx�� cx� kx  Fˆ cos ωt

(12.1)

The first term on the left describes acceleration that becomes mediated by the effective mass m, the second term originates from damping which is proportional to the velocity x� , and the third term gives the displacement as a response to the external force via the spring constant k. In some cases (e.g., k3x3) higher order nonlinear terms need to be added for more accurate modeling of the response. The right-hand-side of the equation could be any time-dependent or static actuating force. In case of harmonic excitation with the force amplitude Fˆ, the displacement and the solution to the equation of motion can be found in a closed form:

x

Fˆ m

⋅ 2 ⎛ ⎞ cω ω02 − ω

⎜⎜ ⎟⎟⎟ ⎜⎝ m ⎠ ⎡ ⎛ cω ⎞⎟⎤ ⎜⎜ ⎢ ⎟⎟⎥ ⎢ ⎜⎜ m ⎟⎟⎥ cos ⎢ ωt − arctan ⎜ 2 ⎥ 2⎟ ⎜ ⎢ ⎜⎜ ω0 − ω ⎟⎟⎟⎥⎥ ⎢ ⎠⎟⎦ ⎝⎜ ⎣

(

222

)

As is obvious, the phase of the mechanical motion is usually delayed relative to the excitation signal. The maximum amplitude can usually be found at resonance. At this frequency the phase of the oscillation is 90° behind that of the force. The resonance frequency becomes also affected by damping

2 2

(12.2)

2 1 ⎛ c ⎞⎟ ⎟⎟ ωr  ω0 1 − ⎜⎜⎜ 2 ⎜⎝ mω0 ⎟⎠

(12.3)

where ω0 

k meff

(12.4)

where k is the mechanical spring constant of the resonator and meff is the effective mass of the resonator. It should be noted that the effective mass is different from the dimensional mass m  V·ρ that is derived from the volume V of the resonator and the density ρ of the resonator material. The relationship between the two masses is determined by the vibration mode of the resonator. Some examples can be seen from Table 12.1. In principle the modification of the damping could lead to a feasible tuning of the resonance frequency; however, usually this is not easily accomplished in practice. A more detailed effect of damping on the amplitude and phase is shown in Figure 12.2. The dynamic amplitude at the resonance frequency over the static displacement is the Q-factor, Q,

Electrostatic and RF-Properties of MEMS Structures

CHAPTER 12

Fig 12.2 ● Effect of damping on the amplitude and phase response near the resonance frequency that has been normalized to one. Q-factor is varied between 0.1 and 100.

Q

xˆr  x st

ω02

(

ω02 − ω 2

)

2

⎛ cω ⎞2

⎜⎜ ⎟⎟⎟ ⎜⎝ m ⎠

≈

mω0 c

(12.5)

The last approximation of Eq. 12.5 is valid, when Q is high (i.e., the frequency shift is negligible). We notice that high damping (low Q) makes the resonance invisible as a function of frequency. Often some acoustic microsystems are heavily gas-damped and do not show any typical resonant behavior. The same holds true also for certain sensors where the operation principle is the detection of the mechanical displacement. Typical of such examples are accelerometers. As an example we study a simple (silicon) beam that is fixed from one end and also a bridge that is being supported from both ends (Figure 12.3). By using FEM one can create exaggerated vibrational mode patterns that are useful in understanding the motion and especially the frequency response. Any (micro)mechanical device can be excited to vibrate in an infinite number of modes. What is maybe not so well known is that FEM calculations give us mode-dependent parameters to be used in lumped models in further analysis of the microsystems. In this case we take a piece of single crystal silicon and assume that it is of isotropic material with length L  100 μm, width w  10 μm and height h  10 μm. For Young’s

modulus for silicon we use E  131 GPa, for density ρ  2330 kg/m3 and for Poisson ratio ν  0.27. These examples reveal that the way of supporting and fixing a micromechanical device drastically affects the boundary conditions and the corresponding dynamic response of the system (Figures 12.4–12.5). One should also note that rather simple equations of motion can be used in many cases to predict response by including the effect of the particular vibration mode on the effective mass (Table 12.2 and Figure 12.6). One should especially notice that the dimensional mass obtained by density times volume does not give the correct resonant frequencies since the coupling to the support modifies the free mass response.

12.3 Electrical Equivalent Circuit The mechanical model presented in the previous chapter is useful in many analyses. Often in practice an electrical circuit simulator is used to obtain detailed information of the behavior of this model as a part of an electronic circuit. It is useful to transform the previous information into a form that is readily usable in circuit design calculations. In electrical domain, the basic resonator consists of reactive components such as inductance and capacitance. Since the mechanical oscillator is in practice always damped, a finite Q-factor requires a resistive 223

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Fig 12.3 ● A mechanical bridge that is fixed from both ends. The force is acting in the transverse direction at the center point of the bridge. This system shows also rather strong mechanical nonlinearity, since the stiffness in the longitudinal (horizontal) direction increases with increasing the strength of the force (see Figure 12.6).

load to be included in the equivalent circuit. A series electrical resonance circuit (Figure 12.7) for resonator obeys a differential equation

Table 12.2 Resonant mode frequencies and the corresponding effective mass values of a micromechanical bridge

Mode #

d2q

dq q L 2 R

 uˆ ⋅ cos ωt dt C dt

(12.6)

where q stands for charge. This equation is of analogous form with the equation of mechanical motion. If we scale the electrical equations with the electromechanical coupling factor η, where q  ηx, we can calculate the corresponding values of the components in this equivalent circuit as

Frequency, f (Hz)

Effective mass, meff (kg)

Bridge supported from both ends (Figure 12.4) 1

7331179

9.46E-12

2

18898040

1.42E-12

3

34302020

4.32E-13

4

52256380

1.86E-13

Beam supported only from one end (Figure 12.5)

Rm  Lm 

c η2 m

≈

η2 η2 Cm  k F and u η 224

mω0 Qη 2 (12.7)

q  ηx

1

1228723

3.37E-10

2

7394308

9.3E-12

3

19509020

1.34E-12

4

35484160

4.04E-13

The effective masses are calculated from Eq. 12.4 based on the simulated resonance frequency f and the spring constant k (Figure 12.6). The corresponding resonant modes are shown in Figures 12.4 and 12.5.

Electrostatic and RF-Properties of MEMS Structures

CHAPTER 12

Fig 12.4 ● The first four fundamental resonance modes of a beam that is fixed only at one end. The movement of the beam is fixed to take place only in-plane. The amplitudes in each sub-figure (a–d) are individually scaled and are not therefore mutually comparable.

The strength of the coupling factor η depends on the coupling scheme. Here we use only electrostatic (capacitive) coupling between the mechanical and electrical domain (Figure 12.8). When a voltage difference U is applied between the electrodes of a capacitor C, the electrostatic energy E stored in the capacitor is E=

1 CU 2 2

(12.8)

The value of the capacitance C (in Figures 12.8 and 12.9) depends on the separation of the electrodes (socalled gap size) d, which is varied due to the displacement of the moving resonator, denoted by x and on the capacitance with zero displacement C0  A0/d C

ε0 A C0 d  d−x d−x

(12.9)

where A is the effective area between the electrode and the resonator, usually determined by the thick-

ness of the device layer of the wafer and by the width of the electrodes used to excite and detect the movement.

12.4 Electrostatic Force If the biasing voltage used is a DC voltage UDC, the electrostatic force can be calculated as a negative gradient of energy. Using Taylor’s expansion, the electrostatic force can be written as � U 2C0 U 2 ∂C � . F− ux ≈ − 2 ∂x 2d 2 3 n⎤ ⎡ ⎢1 2⎛⎜ x ⎞⎟⎟ 3⎛⎜ x ⎞⎟⎟ 4 ⎛⎜ x ⎞⎟⎟ � ( n 1)⎛⎜ x ⎞⎟⎟ ⎥ u� ⎜⎜ ⎟ ⎥ x ⎜⎜ ⎟ ⎜⎜ ⎟ ⎜⎜ ⎟ ⎢ ⎝d⎠ ⎥ ⎝d⎠ ⎝d⎠ ⎝d⎠ ⎢⎣ ⎦ (12.10) where the capacitance C is defined by Eq. 12.9. 225

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Fig 12.5 ● The first four fundamental resonance modes of a bridge having the two ends fixed. The movement of the beam is fixed to take place only in-plane. The amplitudes in each sub-figure (a–d) are individually scaled and are not therefore mutually comparable.

drive voltage frequency. This can be seen by calculating the second power of voltage U as

0.12 0.1

y = 1.5077·1014·x3 + 2.0237·107·x2 + 20082·x + 3·10–6

F (N)

0.08

2 U 2  (U DC ûac cosωt)2  U DC

0.06

1 2

ûac cos 2ωt 2U DC ûac cosωt 2

0.04 0.02

(12.11)

and by calculating the force F by inserting Eq. 12.11 into Eq. 12.10 as

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

x (μm)

Fig 12.6 ● Deflection amplitude of the bridge shown in Figure 12.3 as the transverse force is increased. As is obvious, weak forces cause a rather linear response, but the system shows also a nonlinear spring constant when strong forces are present. This property can be used to fine tune the resonance frequencies.

From Eq. 12.10 it is evident that the force is proportional to the second power of the voltage. Thus, if the excitation of the resonator is performed with AC voltage, a DC bias voltage is required to create the force that excites the resonator at the frequency of the AC 226

1 2 ûac 2

� 1 ∂C F− . 2 ∂x ⎛ 2 ⎞� 1 2 2 ⎜⎜U DC 1 uac

uac cos 2ωt 2U DC uac cos ωt⎟⎟⎟ ux ⎜⎝ ⎠ 2 2 (12.12) From Eq. 12.12 we can see that the electrostatic force can have components at zero frequency, at ω and at 2ω, when the frequency of the exciting AC voltage is ω. Since we aim at having an excitation at ω frequency, the DC voltage amplitude should typically be stronger than the AC voltage amplitude. Eq. 12.12 also reveals that

Electrostatic and RF-Properties of MEMS Structures

Rs

Rm

Cm

where the factor multiplying x can be interpreted as the electrical spring constant ke

Lm

ZL

Fig 12.7 ● An electrical equivalent circuit presentation of a simple mass-spring system presented in Section 12.2. The shaded parts represent the source and load that are used to excite and detect the resonance, respectively. x Static electrode

m

Y C(t)

Fig 12.8 ● Model system of a dynamical micromechanical device coupled by a time-dependent capacitance C(t).

Static electrode

U F

x

Fig 12.9 ● Electrostatic force depends on the voltage over the coupling capacitance that is generated by the static and movable electrodes.

the DC voltage significantly amplifies the excitation strength. As the generated displacement x is typically small compared with the gap size d (i.e., the resonator operates in the linear region), the force can be approximated as ⎛ x ⎞⎤ C ⎡ C U2 F ≈ − 0 ⎢1 2⎜⎜ ⎟⎟⎟⎥ U 2 ≈ − 0 DC ⎝⎜ d ⎠⎥⎦ 2d ⎢⎣ 2d C U uˆ C U2 x − 0 DC ac cos ωt − 0 2DC d d

2 C0U DC

d2

(12.14)

We thus obtain a very useful result that the DC voltage directly leads to one-directional tuning of the resonance frequency of the mechanical resonator. The strength of this tuning can be calculated by replacing the mechanical spring constant k with the sum of electrical and mechanical spring constants k → km  ke in case of no damping: ke C U2  ω0 1 − 0 Dc km km d 2 ⎛ C U 2 ⎞⎟⎟ ≈ ω0 ⎜⎜⎜1 − 0 Dc ⎟ ⎜⎝ km d 2 ⎟⎟⎠

F

d

ke  −

ω  ω0 1 −

k

Movable electrode

CHAPTER 12

(12.13)

(12.15)

This spring softening effect can be seen from Figure 12.10. In various RF applications, this is a very useful property. From Eq. 12.15 it appears, that the electric field which is generated by the applied DC bias-voltage UDC can be used to tune the spring constant of the resonator. This opens at least two options: (i) by applying a constant voltage (U  UDC) the resonant frequency of the resonator can be tuned and (ii) by applying a periodically changing voltage (U  uac), the spring constant can be modulated, this can be used to realize mechanical parametric amplification of the electromechanical device [1]. Additional effects caused by the capacitive nonlinearity will be discussed later in more detail. We can thus summarize that for the electrostatic coupling realized via voltage-biased parallel plate transducers the magnitude of the electrostatic force at the excitation frequency ω depends both on the applied AC and DC voltages and also on the gradient of capacitance. It should also be noted, that even when high DC bias voltage is used, the electrostatic coupling leads to a minimal power consumption. If the nonlinear contribution to the excitation force is unacceptable, another way to perform the biasing of a micromechanical device is to keep the charge q instead of the voltage of the capacitor constant. In this case the force becomes F−

q2 ∂ ⎛⎜ 1 ⎞⎟ � ⎜ ⎟ ux 2 ∂x ⎜⎝ C ⎟⎠

(12.16)

As a difference to the voltage biasing, the charge biasing leads to the force amplitude F that is independent of the actual displacement x. 227

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180

–20 150 V 112.5V 75 V

90

S21 (dB)

37.5 V –60

0

–80

–90

–100

Phase (degree)

–40

–180

5.100M

5.225M

5.350M

5.475M

5.600M

Frequency (Hz) Fig 12.10 ● The bias-voltage UDC is varied between UDC  [0…150] V. The effects of increased electromechanical coupling are evident by (i) the increased transmission S21 and (ii) lowering of the resonance frequency (spring softening). For UDC  0 V, there is no output signal.

Another way to reduce the nonlinearity is to use comb-shaped electrodes instead of the parallel plate electrodes. Then the capacitance becomes ε h( Le − x) C 0 d

12.5 Electromechanical Coupling The relation between the AC-voltage and the force is ∂C uac ∂x

U DC

(12.17)

where Le is the length of the comb finger.

F  U DC

since UDC  uac. If we assume that the motion is sinusoidal, we get

(12.18)

In addition to the voltage and the vibration amplitude, the motional current also depends on the frequency and the gradient of the capacitance between the electrode and the resonator. By dividing the force with the spring constant and noting that the force is actually multiplied by the quality factor of the device, we get the vibration amplitude at resonance xˆR  Q

where the electromechanical coupling factor is η  U DC

C ∂C ≈ U DC 0 ∂x d

(12.19)

Therefore, the coupling between electrostatic and mechanical domains depends on the coupling capacitance and the gap size between the electrodes as well as on the DC bias voltage over the gap. Thus, narrow gaps can be used to increase the electromechanical coupling.

12.6 Sensing of Motion The oscillating MEMS resonator can be sensed using a DC-biased electrode. To analyze the motional current induced by this vibration, we can start by writing down the expression for the electrical current im  228

dq d(CU ) ∂C  ≈ U DC dt dt ∂t

∂C � ∂C ∂(x� ⋅ sin ωt)  U DC ωx ⋅ (12.21) ∂x ∂t ∂x cos ωt  η 2ωûac ⋅ cos ωt

(12.20)

ηuˆac C U uˆ ≈ Q 0 dc ac k kd

(12.22)

Thus, the amplitude of the motional current at resonance can be written in a form η 2 uˆac ω U 2 uˆ ωC2 ˆi ≈ Q DC ac2 0 m, R  Q k kd

(12.23)

The motional current depends strongly on the electromechanical coupling factor (i.e., DC bias voltage and the gradient of the coupling capacitance). On the other hand, the gradient of the capacitance C/x in Eq. 12.20 can again be approximated by Taylor’s series as dq ∂C ∂C ∂x  U Dc  U Dc dt ∂t ∂x ∂t ⎛ ⎞⎟ ∂x ⎛ x ⎞2 C ⎜ x  U DC 0 ⎜⎜1 2 3⎜⎜ ⎟⎟⎟ �⎟⎟⎟ ⎜⎝ d ⎠ d ⎜⎝ d ⎟⎠ ∂t

imot 

(12.24)

Electrostatic and RF-Properties of MEMS Structures

From Eq. 12.24 it is clear that even linear vibrations can result in higher harmonics in the motional current. Therefore, for practical purposes the amplitude of vibration should be maintained at a reasonable level.

The total force exerted onto the resonator is a sum of the electrostatic force (Eq. 12.10) generated by the applied voltage and the mechanical spring restoring force, Fm  kx, due to the displacement of the resonator. When the electrostatic force is increased and the resonator becomes more displaced, the spring restoring force also increases leading finally to the equilibrium position (Figure 12.11). However, when comparing the electrical and mechanical forces in Figure 12.11 it is evident that when the applied voltage exceeds a certain threshold value, the electrical force will always be greater than the mechanical force and no equilibrium position exists. The exact voltage corresponding to this phenomenon is called the pull-in voltage, Upi. The pull-in behavior is observed when the mechanical and electrical forces (and simultaneously also their derivatives) cancel each other. In a static case with one electrode it is given by U pi 

km d 2 2C0

(12.26)

From Eqs 12.25 and 12.26 one can observe that the pull-in voltage depends on the coupling strength and on the mechanical spring constant. For capacitive switches the pull-in phenomenon is desired and thus such devices are usually designed for low pull-in voltages—however, low pull-in voltage usually results in a mechanically soft structure which on the other hand reduces the operating speed (frequency) of the switch. For capacitively coupled resonators the pull-in voltage limits the tunability of the resonance of the device. This limit can be seen by inserting the value of the pull-in voltage (Eq. 12.25) into the equation describing tuning (Eq. 12.15). In case of one electrode the maximum frequency tuning is only in the order 16%, since

 ω0

2 C0U Dc kmd 2

 ω0 1  UDCUpi

8 ≈ 0.84ω0 1 27

|Fe|, U > Upi |Fm|

2

Pull-in point

1.5 1

Equilibrium points

0.5 0

0

0.2

0.4

0.6

0.8

Relative displacement, x/d

Fig 12.11 ● Comparison between the electrostatic force (black curves), Fe and mechanical spring restoring force (gray line), Fm. When the bias-voltage is kept below the pull-in voltage, increasing the displacement the mechanical spring restoring force also increases and the system reaches equilibrium when the forces are equal to each other. However, it is evident that with increasing bias-voltage, the electrostatic force eventually becomes larger than the mechanical spring restoring force and the pull-in occurs.

(12.25)

Similarly, in a static case with the electrodes on both sides of the moving device the pull-in voltage is

ω  ω0 1 

|Fe|, U = Upi

12.8 Parasitic Capacitance

8d 2 km 27C0

U pi 

|Fe|, U < Upi

2.5

Force, F (a.u)

12.7 Pull-in Phenomenon

3

CHAPTER 12

C0U 2pi kmd 2

(12.27)

The capacitive coupling has a significant drawback. The capacitor C0, which is used to couple electrical voltage to the mechanical force, also conducts the drive current across the capacitor. The equivalent circuit presented earlier should therefore be corrected by a parallel capacitor, which has the same value as the coupling capacitance C0 (Figure 12.12). The parasitic current is often much stronger than the induced motional current (it is a problem, because both currents oscillate at the same frequency). The ratio of the induced motional current and parasitic current is U 2 uˆ ωC2 Q DC ac2 0 2 ˆi QC0U DC m, R kd   ˆi ω uˆac C0 kd 2 p

(12.28)

Thus, high-Q, high DC-voltage and a narrow gap help in improving coupling resulting simultaneously in a lower parasitic current. In addition, we can see that the AC-voltage has no effect on the motional to parasitic current ratio. The effect of the parasitic capacitance can also be seen from the frequency response of the resonator (Figure 12.13). There are also other sources of parasitic capacitance associated with the MEMS-resonators. Read-out electronics circuits are not usually directly integrated with 229

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MEMS, which means that we need for example bonded wires and packaging to apply resonators. The result is the increase of the parasitic capacitance. One potential option to reduce the effect of parasitic capacitance could be to use a parametric amplification scheme, which means that the spring constant is modulated with a frequency ωpump  2ω0/n, where ω0 is the resonant frequency of the device and n is an integer. Usually n  1 and the modulation of the spring constant is performed at the frequency of ωpump  2ω0. Then the parasitic current oscillates at frequency which is different from the desired motional current resulting in the situation that the parasitic capacitances may not anymore be such a severe problem. The other possibility is to use separate electrodes for driving and sensing. If the (parasitic) capacitance between the drive and sensing electrodes is low and the resonator is AC-grounded, most of the parasitic current can be eliminated (in the case of integrated MEMS-resonators).

12.9 Effect of Built-in Potential on Capacitively Coupled MEMSDevices As the strength of the capacitive coupling and also the effects of the capacitive nonlinearity are strongly dependent on the bias-voltage, it is clear that the stability of the bias-voltage is essential when considering the stability of electrostatically coupled MEMS-devices. A DC biased capacitive coupling gap is shown in Figure 12.14. Here the system consists of a static electrode and a vibrating electrode (resonator) and the static electrode is coated with a layer of dielectric. The system is biased with a DC voltage UDC. However, in addition to the applied bias voltage, UDC, the material parameters (work-functions of the materials 1 and 2) and the charges residing in the dielectric layer contribute to the total bias voltage experienced by the coupling gap. When calculating the total bias-voltage in the coupling gap d, we get the result

C0

U  UDC  ϕ12  Rs

Rm

Cm

QOX C2

where UDC is the applied bias-voltage, 12 is the difference of the work-functions between the two electrodes, QOX is the charge in the dielectric on top of the static electrode and C2 is the capacitance over the dielectric. The additional term to the applied bias-voltage UDC is usually called the built-in voltage Vbi of the system

Lm

ZL

Fig 12.12 ● Electrical equivalent circuit of a resonator including the drive signal and the parasitic capacitance parallel with the RLC-tanks.

Vbi  ϕ12

QOX C2

(12.30)

180

–10 0.1 pF

90

10 fF

0

–55 1 fF

Phase (deg)

S21 (dB)

–32.5

0.1 fF –90

–77.5 10 aF

–180

–100 5.100M

5.225M

5.350M

5.475M

5.600M

Frequency (Hz)

Fig 12.13 ● Influence of parasitic capacitance C0 on the transmission (frequency and phase) response of the resonator. The feed-through capacitor C0 is varied in a range of [0…1 pF].

230

(12.29)

Electrostatic and RF-Properties of MEMS Structures

the resonance frequency of the device as can be seen from Eq. 12.15. The effect could be further increased if the coupling of the MEMS-devices is improved by introducing the dielectric layers in the coupling gap [13]. On the other hand, the spontaneous built-in voltage can also be useful, one such example is the vibrating power harvester system utilizing built-in voltages [14].

UDC

(a)

Qox

xvib

Q2

ϕ1 V2 Static electrode (b)

Q1

ϕ2

d V1 Vibrating electrode

UDC

ϕ12

C2

C1

V2

V1

CHAPTER 12

Fig 12.14 ● . (a) Schematic view of a DC-biased capacitive RFMEMS device illustrating various contributions to the total bias voltage over the air gap and (b) a simplified electrical equivalent circuit for the system. Source: Reproduced from [7] with permission from IOP Publishing Ltd.

Equation 12.29 reveals that Vbi plays a role in the stability of various MEMS-devices. Even though its origin is rather well known, the effect on the stability has been much less studied. Built-in voltages are generated when two dissimilar materials are brought into contact—it is in principle a direct measure of the difference in the work functions [2]. In practice, however, the observed work functions are modified due to the thin layers of dielectrics present on the surfaces [3, 4] and vary as a function of the temperature and due to the adsorbents on the surfaces [5, 6]. Therefore, for many practical cases experimental methods are needed to unambiguously determine the built-in potentials of the interfaces. A good example of the effect of the built-in potential in case of a capacitive accelerometer is shown in Figure 12.15 [7]. Here the built-in potential causes the displacement of the proof-mass even in the case of zero acceleration of the device. The built-in potential has also been seen to affect the stability in the MEMSbased voltage-references [8, 9] with what some practical solutions have been found to minimize its effect [10]. For RF-MEMS switches the effect of built-in potential could become an issue by modifying the threshold value of the pull-in voltage, but can also lead to stiction due to the charging of the dielectrics [11, 12]. A similar effect takes place in MEMS-resonators where this voltage modifies the electric spring constant and thus changes

12.10 Further Effects of Electrostatic Nonlinearities from Applications Point of View Electrostatic nonlinearities discussed earlier can generate harmful side-effects in capacitively coupled devices. In the following, these effects are studied in detail with respect to the two perhaps most promising application areas of MEMS-devices in RF-electronics which are filters and reference oscillators.

12.10.1 Case 1: Capacitively Coupled Filter The narrow resonance of MEMS-resonators gives the possibility of using them as highly selective bandpass filters. With proper biasing, the input signals at the resonance frequency of the resonator will introduce a force that is amplified by the quality factor Q of the resonator and therefore the signals at this frequency will be transmitted to the readout electronics. On the other hand, the signals lying outside the resonance passband will generate significantly weaker force and are in principle rejected (Figure 12.16a). If a broader passband is required, it can be achieved by parallelizing resonators since the capacitive coupling provides a simple method to couple the input and output signals as long as the impedance matching is taken care of. However, the nonlinearity of the capacitive coupling can seriously deteriorate the performance of the filter due to the introduced intermodulation distortion (IM). The intermodulation distortion is responsible for converting the off-resonance signals into the resonance band. This is especially harmful in the case of filter-applications where those signals that are outside the filter passband would be converted inside the passband (Figure 12.16b). The third order nonlinearity that arises from the capacitive coupling brings along intermodulation distortion even when the vibration is in the linear regime. This can be seen by writing a Taylor’s expansion of the coupling gap capacitance C as C

2 ⎛ ⎞⎟ ⎛ x ⎞3 Aε0 x ⎛x⎞ ⎜ ≈ C0 ⎜⎜1 ∓ ⎜⎜ ⎟⎟⎟ ∓ ⎜⎜ ⎟⎟⎟ �⎟⎟⎟ ⎜⎝ ⎝⎜ d ⎠ (d ± x) d ⎝⎜ d ⎠ ⎟⎠ (12.31)

231

Modeling in MEMS

PA R T I I (a) 8.42 8.4

Vbi = Avg(Vpi2 +Vpi1) Vbi = (–1.51+ 0.1) / 2 Vbi = –705 mV

Capacitance (pF)

8.38 8.36 8.34 8.32

Vpi2

Vpi1

y = 0.03x2 + 0.0437x + 8.3028 Vbi = –0.0437/(2*0.03) Vbi = –728 mV Vbi determined by parabolic fit Vbi determined by pull-in

8.3 8.28 –2.5

–1.5

–0.5

0.5

1.5

2.5

Applied voltage (V)

(b)

(c)

(d)

Vbi

Vbi

UDC

UDC

U

Fig 12.15 ● (a) In capacitive sensors the built-in potential manifests itself as a zero-point offset that can be observed by measuring the capacitance of the sensor as a function of the applied bias-voltage UDC. (b) The capacitance is at minimum when UDC  Vbi. (c) When UDC  Vbi the capacitance increases due to the electrostatic force. (d) When the voltage between the electrodes (U  UDC Vbi) exceeds the pull-in voltage, the electrostatic force draws the electrodes into contact. Source: Reproduced from [7] with permission from IOP Publishing Ltd.

fr + Δf f

Fig 12.16 ● (a) Operating principle of a microelectromechanical resonator used as a filter. The signals within the resonance band of the device are transmitted to the readout electronics while the signals outside the resonance band are rejected. (b) The principle of third order intermodulation generation. The cubic mixing due to the capacitive nonlinearity of the electrostatic transduction converts the signals outside the resonance band into the resonance band.

232

where C0 is the coupling gap capacitance at rest or C0  A0/d. In practice if two signals at frequencies ω1  ωres Δω, ω2  ωres 2Δω are coupled to the resonator acting as a filter the cubic (third order) mixing of these signals produces in addition to other contributions a third-order term at frequency 2ω1  ω2 which is identified as ωres. It is the resonant frequency of the device; therefore, the signal possessing this frequency will be transmitted through the system. This process of IM3 (third order intermodulation distortion) product generation is depicted in frequency domain in Figure 12.16b. To quantify the strength of capacitive nonlinearity, a parameter called the third order intercept point (IIP3) can be measured. It is a measure of how strong off-resonance interferers at frequencies ω1 and ω2 are needed in order to produce equally strong signal in the carrier frequency as if the same power was applied directly at the carrier

Electrostatic and RF-Properties of MEMS Structures

CHAPTER 12

Linear fits to the measured IM products 30 Output signal, Vout (dBmV)

10 –10

Direct excitation of the resonator

–30 –50 Measured intermodulation products

–70 Δω = 1 kHz

–90 –110

Δω = 200 Hz

–130 –150 –30

Δω = 20 kHz –20

–10

0

10

20

30

40

50

Input signal,Vin (dBmV)

Fig 12.17 ● Method of determining the IM3 product that is caused by the capacitive nonlinearity due to the electrostatic transduction. The IM3 product is characterized at three different separations Δω  [20, 1000, 20000] Hz from the resonant frequency ω.

frequency. Schematically this is depicted in Figure 12.17. The intermodulation in capacitive devices is studied for example in Refs. [15–18].

Capacitively coupled silicon microresonators have been studied as a replacement for piezoelectric quartz crystals as a central component in reference oscillators for quite some time. It has been shown, that the silicon microresonators are capable of competing with the macroscopic quartz resonators both in terms of short-term stability [19–22] and in long-term stability [23–25]. However, the capacitive coupling used with silicon microresonators brings an interesting contribution to the short-term instability. The short-term stability dictates the spectral purity of the signal. It is usually specified through the phasenoise referenced at carrier power at certain offset from the carrier frequency (Figure 12.18). For example in order to meet the GSM-specifications [26] for the receiver sensitivity and blocking, a 13 MHz reference oscillator must meet the phase-noise specifications Lf  130 dBc/Hz at Δf  1 kHz offset from the carrier and L  150 dBc/Hz far away from the carrier (so called noise floor) [20]. The strict phase-noise requirements arise from the fact that for practical use the output frequency of an oscillator is multiplied in a phase-locked loop (PLL) to match the carrier frequency (e.g., 900 MHz for GSM-900) and naturally the existing low-frequency noise becomes multiplied in a same process.

3 m

Noise floor

L ∝ 1ω

Lf

L (dBc/Hz)

12.10.2 Case 2: Capacitively Coupled Reference Oscillator

Carrier level

1 L∝ ω

1 L∝ ω

2 m

1 m

1 L∝ ω

0 m

Δf 1

Frequency offset from carrier (Hz)

Fig 12.18 ● Definition of SSB (single-side-band) noise. The nonlinear mixing of the low-frequency 1/f noise increases the near carrier noise and changes the slope of the near carrier phase-noise from 1/ω2 to 1/ω3.

Typically the single-sideband phase-noise-to-carrier ratio for an ideal oscillator can be approximated by Leeson’s equation [27, 28] L(Δω) 

2 1 kb T ⎛⎜ ω0 ⎞⎟ ⎟⎟ ⎜⎜ 4π Evib ⎝QΔω ⎟⎠

(12.32)

from which two important factors can be seen: (i) in practice the phase-noise reflects the competition between the mechanical energy Evib of the resonator and thermal noise kbT and (ii) the width of the resonance (1/Q) leads to a bandpass filtering of the noise reducing the noise as the frequency difference to the carrier Δω increases. 233

Modeling in MEMS

PA R T I I

In a more generic form of Eq. 12.32 the noise voltage includes also the noise contribution from the amplifier electronics. L(Δω) 

un

2

uac

2

⎡⎛ ω ⎞2 ⎤ ⎢⎜ ⎥ 0 ⎟

⎟ 1 ⎜ ⎢⎜ ⎥ ⎟ ⎢⎣⎝ 2QΔω ⎟⎠ ⎥⎦

(12.33)

where uac is the signal voltage which is usually limited by the nonlinearities of the reference component (i.e., MEMS resonator) and un is the total noise voltage. From Eq. 12.33 the phase-noise can be seen to decrease as 1/ω2. However, for capacitively coupled devices Eq. 12.33 does not take into account the effect of aliasing of the 1/f noise due to the capacitive coupling. In the capacitive transduction gap, the 1/f noise is mixed with the signal voltage produced by the resonator giving an additional noise term. A more accurate equation for the dual side-band noise has been presented in [29] as L(Δω ) 

un

2

uac

2

Z0

US

ZSW U1

Z0 U2

ZL

UL

Fig 12.19 ● Micromechanical switch (impedance ZSW) connected in series with the transmission line. The transmission line (impedance Z0) is assumed to be perfectly matched to the source and load impedances, ZS and ZL, respectively.

ZS US

Z0

Z0 U1 ZSW

U2

ZL

UL

Fig 12.20 ● A micromechanical switch (impedance ZSW) connected in parallel (shunt) with the transmission line. The transmission line (impedance Z0) is assumed to be matched to the source and load impedances ZS and ZL, respectively.

.

2 ⎡ ⎢⎛⎜ ω0 ⎞⎟⎟ ⎜ ⎢⎜ ⎟ ⎢⎣⎝ 2QΔω ⎟⎠

⎛ 2 2 ⋅ ⎜⎜1 2 Γ Rm uac ⎜⎝

⎤ (12.34) 2 ωc ⎞ ⎟⎟ 1⎥ ⎥ Δω ⎟⎠ ⎥⎦

which leads to Eq. 12.33 using an aliasing factor   0. One can deduce from Eq. 12.34 that the phase-noise will decrease as 1/ω3 instead of 1/ω2 as is the case in Eq. 12.33 due to the mixing of low-frequency amplifier noise present at the resonator input in the capacitive transduction gap. This can easily increase the near carrier noise by more than 30 dB [30]. It is evident from Eq. 12.34 that increasing the vibration amplitude does not help with aliasing of 1/f noise when it comes to the noise that is very close to the carrier frequency as it is produced by mixing of the thermal noise and the signal produced by the resonator. However, the noise floor becomes lower by increasing the vibration energy of the resonator. Therefore, if capacitive transduction is used to realize an oscillator, attention must be paid to minimize the low-frequency noise of the amplifier or filter the noise prior to reaching the resonator.

12.11 RF-Properties RF-properties are studied from switching point of view. The intention of this chapter is to discuss some aspects related to the RF-properties, but to be in no way a comprehensive study of the RF-properties of MEMS-devices. A micromechanical switch is usually a MEMS-tunable capacitor that operates either in series or in shunt-mode. A series switch, (Figure 12.19) is used to shortcut (in onstate) or isolate (in off-state) the signal pathway between the input and the output in a transmission line. A shunt 234

ZS

switch (Figure 12.20) shortcuts the transmission line to the ground (in off-state) or isolates the transmission line from the ground (in on-state). In the following, some RFparameters are derived for both series and shunt switches. Again, the devices used are capacitively actuated even though the actuation principle does not significantly affect the studied parameters. It should also be noted, that the following is true for a contact switch where in the downstate the switch makes a metal–metal contact between the terminals of the switch and in the up-state the terminals are separated from each other by an air gap showing significant capacitance between the terminals. The RF-properties are usually characterized by such concepts as isolation (ISOL), insertion loss, (IL) and reflection. Actually both the isolation and insertion loss describe the forward power transmission, S21, of the switch, but the difference is that isolation refers to the case when the switch is in the off state and insertion loss is defined when the switch is in the on state. The switch can be represented as a transmission line with impedance ZSW in Figure 12.19. The forward transmission can be calculated as S21 

U2 2U L  U1 US

(12.35)

Equation 12.35 assumes that the transmission line is matched to the source and load impedances or ZS  ZL  Z0. When the voltage at the load UL is calculated the forward power transmission gets the value S21

2



1 Z 1 SW 2Z0

2

(12.36)

Electrostatic and RF-Properties of MEMS Structures

When a MEMS-switch is in the down-state, the switch can be represented by the resistance RS of the metal–metal contact of the switch. With this, the insertion loss of the series switch becomes IL 

(12.37)

In the up-state, the switch becomes a capacitor with a capacitance of CS and the isolation can be calculated as ISOL 

up 2 S21

2

≈ (2ωCS Z0 )

2

1

 1

Z0 2Z SW

2

(12.39)

ISOL 

⎛ 2Z SW ⎞⎟2 ⎟ ≈ ⎜⎜⎜ ⎜⎝ Z ⎟⎟⎠

(12.40)

0

As previously stated, in the down-state, the switch can be represented with a resonant circuit; therefore, the impedance of a down-state switch can be written as Z SW

1  RP jω LP

jωCP

1 2π

1 LP CP

(12.43)

2

(12.44)

The insertion loss for the up-state switch is mainly the loss of the transmission line and, therefore, not determined by the switch itself. However, in addition 2 the power reflection S11 from the switch must be taken into account according to S11

2

1

 1

2Z SW Z0

2

(12.45)

or 2

⎛ ωC Z ⎞2 ≈ ⎜⎜ U 0 ⎟⎟⎟ ⎜⎝ 2 ⎠

(12.46)

where CU indicates the capacitance of the up-state (or open) switch. In addition an important figure of merit in terms of switches, the so called cutoff frequency, fc, should be noted. It is usually determined by the resistance and capacitance of the switch as f 

1 1 2π CS RS

(12.47)

(12.41)

where the resistance, inductance and capacitance denoted with a subscript P indicate the values for downstate switch (connected in parallel). The isolation of the switch can be calculated in three frequency ranges, determined by the LC series-resonant frequency of the switch fLC 

down ISOL  S21

⎪⎪⎧⎛ 2 ⎞2 ⎟⎟ , f � f ⎪⎪⎜⎜ LC ⎪⎪⎝⎜⎜ ωCP Z0 ⎠⎟⎟ ⎪⎪ 2 ⎪⎛ ⎪ 2R ⎞ ≈ ⎨⎜⎜⎜ P ⎟⎟⎟ , f  fLC ⎪⎪⎜⎝ Z0 ⎟⎠ ⎪⎪ 2 ⎪⎛ ⎪⎪⎜ 2ωLP ⎞⎟⎟ , f � f LC ⎪⎪⎜⎜⎜ Z ⎟⎟ ⎝ 0 ⎠ ⎩⎪⎪

S11

By using the condition defined as ZSW  Z0 the isolation becomes down 2 S21

⎧⎪ 1 ⎪ , f � fLC ⎪⎪⎪ jωCP  ⎪⎨ RP , f  fLC ⎪⎪ ⎪⎪ jω LP , f � fLC ⎪⎪ ⎪⎩

and the isolation is the given by

(12.38)

From Eqs 12.37 and 12.38 it is clear that the isolation and the insertion loss of a contact switch can be determined by the contact resistance RS and the upstate capacitance CS of the switch. For a parallel (shunt) switch in Figure 12.20 the operation principle is opposite—when the switch is closed, it provides a large capacitance (usually in the pico farads) to the ground and the source signal is therefore shorted to the ground. When the switch is in the open-state, the capacitance to the ground is small (usually in the femto farads) and the switch does not affect the signal in the transmission line—therefore, the electrical equivalent of a capacitive shunt switch is a parallel capacitor in the up-state and a parallel connected series resonant circuit in the down-state. The forward power transmission is given in Eq. 12.39: S21

Therefore, the impedance of the switch becomes

Z SW

R ≈ 1− S Z0

down 2 S21

CHAPTER 12

(12.42)

The cutoff frequency is the frequency where the ratio of the up-state to the down-state impedance becomes unity. However, as one can deduce from Eq. 12.47 this definition of the cutoff frequency is actually valid for series switches and for MEMS shunt switches the cutoff frequency is not strictly applicable as the switch inductance usually limits the down-state performance before fc. Due to this and because the MEMS shunt switches usually result in acceptable isolation up to twice their LC resonance frequency, this limit or 2·fLC is used instead of fc [31]. 235

PA R T I I

Modeling in MEMS

Acknowledgments The authors would like to thank the following people for the collaboration in the research that has led among other things to the results discussed in this chapter: Nikolai

Chekurov, Osmo Vänskä, Ville-Pekka Rytkönen, Pekka Rantakari, Tuomas Lamminmäki, Tomi Mattila, Aarne Oja, Ari Alastalo, Heikki Seppä, Jyrki Kiihamäki, Hannu Kattelus and Ville Kaajakari.

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