**3. Energy considerations and switching times**

In this section we shall derive useful formulas for the evaluation of the switching times, based on energy computations. The electrostatic energy given by a voltage generator to a capacitive shunt switch in the ON state (UP position of the bridge), at the threshold for the electrostatic actuation, is given by:

$$E\_{initial} = \frac{1}{2} \mathcal{C}\_{ON} V\_{threshold}^2 \tag{28}$$

The threshold value *Vthreshold* of the voltage has been used to account for the full actuation of the switch when such a voltage is imposed. The system is non-conservative, and the final energy of the actuated beam (corresponding to the value of the OFF capacitance) will be changed by the contact and dielectric charging contributions, which have to be greater than the restoring mechanical energy for maintaining the beam in the DOWN position:

$$E\_{final} = \frac{1}{2} C\_{OFF} V\_{threshold}^2 + E\_c + E\_{charge} \ge \frac{1}{2} \,\mathrm{kg}^2 + \frac{1}{4} k\_s g^4 \tag{29}$$

Where *COFF* is the capacitance when the switch is in the OFF state (bridge DOWN), and *(1/2)kg2+(1/4)ksg4* is the mechanical spring energy including the stretching contribution. As an implementation of the treatment given in classical literature about this topic [3], we have included additional terms [5],[7] and the contribution coming from the charging of the dielectric [41]. In fact, when the bridge is in the DOWN position, two more effects have to be

Dynamics of RF Micro-Mechanical

central pull-down, and (b) lateral one.

written as:

Capacitive Shunt Switches in Coplanar Waveguide Configuration 211

(a)

(b) Fig. 8. Simulation of the actuation of the bridge. The curves describing the change of the capacitance (in F) during the actuation time (in s) are: (i) full curve, Ctime1(t), with the entire response by using *k*, (ii) flat full reference line Cmax, i.e. the maximum value obtained when the switch is fully actuated, (iii) Cref=0.9Cmax. Both possible actuations are described: (a)

It is worth noting from Fig. 8 the significant change of the oscillations induced by the different actuation choices. On the other hand, we can simplify the prediction for the actuation time by using the evolution of the envelope describing the actuation, which is

studied: (i) the contact energy *Ec* (repulsive and van der Waals), and (ii) the energy due to the charging process of the dielectric *Echarge*. To maintain the bridge in the DOWN position, accounting also for the above two additional terms, a holding down voltage *Vmin* less than the threshold one can be applied. In this case the balance is obtained between the mechanical restoring energy and the electrostatic energy for the capacitance in the OFF state, plus the contact and charge terms, as:

$$\frac{1}{2}\text{kg}^2 + \frac{1}{4}k\_{\text{s}}\text{g}^4 = \frac{1}{2}\text{C}\_{\text{OFF}}V\_{\text{min}}^2 + E\_c + E\_{\text{charge}}\tag{30}$$

And the maintenance voltage *Vmin* can be written as:

$$V\_{\rm min} = \sqrt{\frac{\text{kg}^2 + \frac{1}{2}k\_s g^4 - 2\left(E\_c + E\_{\rm charge}\right)}{C\_{\rm OFF}}}\tag{31}$$

i.e., *Vmin* can be determined by the spring constant *k*, the gap *g*, and the value of the capacitance in the OFF state (*COFF*). An estimation of the terms like *Ec* and its dependence on the position of the bridge is available[5],[42]. When successive actuations are performed, and considering *Ec* as an offset contribution, *Echarge* will increase up to a value where the switch will remain stuck also for *Vmin*=0, unless to choose properly materials and geometry for the uni-polar bias scheme. Actually, *Vmin*=0 will correspond to the sticking of the bridge, i.e. when 2 4 arg <sup>1</sup> 2 0 <sup>2</sup> *<sup>s</sup> c ch e kg k g E E* . The threshold voltage,

obtained by assuming that the mechanical structure collapses after trespassing the critical distance (1/3)*g*, is given by Eq.(11)[3],[17]. In the cases studied in this paper, and by using the same geometrical and physical data defined for our test structure, the change from the definition of *k* for central actuation to that for the lateral one can correspond to more than doubling the threshold voltage. From the above Eq (31), simple considerations can be anticipated about the contribution of charging processes and possible sticking of the bridge. It is known that successive applications of the voltage correspond to an accumulation of charge in the dielectric and in an increase of the actuation voltage[43], with an asymptotic trend up to its maximum value[44],[45]. In fact, if it is not given time enough to the charges to be dissipated, they grow up to the maximum contribution allowed by the geometry and by the properties of the dielectric material used for the capacitive response (in the case of central actuation) or by the dielectric used for the actuation pads (lateral actuation). The substrate too, when it is oxidized for improving the isolation, is a source of charges.

The switching times can be evaluated by means of the capacitance dynamics. Actually, exception done for the oscillations superimposed to the exponential decay, the envelope of the z-quote when the threshold voltage is applied will describe the change of the capacitance up to the full actuation of the device. As an approximation, we can say that the switch is *actuated* when the capacitance is 90% of that obtained from the full actuation procedure. In Fig. 8 the curve containing the oscillations of the capacitance, the maximum value when the bridge is in the DOWN position, and the reference value corresponding to 0.9*Cmax* are shown.

studied: (i) the contact energy *Ec* (repulsive and van der Waals), and (ii) the energy due to the charging process of the dielectric *Echarge*. To maintain the bridge in the DOWN position, accounting also for the above two additional terms, a holding down voltage *Vmin* less than the threshold one can be applied. In this case the balance is obtained between the mechanical restoring energy and the electrostatic energy for the capacitance in the OFF state,

min arg

2 4

2 *<sup>s</sup> c ch e OFF*

<sup>1</sup> <sup>2</sup>

i.e., *Vmin* can be determined by the spring constant *k*, the gap *g*, and the value of the capacitance in the OFF state (*COFF*). An estimation of the terms like *Ec* and its dependence on the position of the bridge is available[5],[42]. When successive actuations are performed, and considering *Ec* as an offset contribution, *Echarge* will increase up to a value where the switch will remain stuck also for *Vmin*=0, unless to choose properly materials and geometry for the uni-polar bias scheme. Actually, *Vmin*=0 will correspond to the

obtained by assuming that the mechanical structure collapses after trespassing the critical distance (1/3)*g*, is given by Eq.(11)[3],[17]. In the cases studied in this paper, and by using the same geometrical and physical data defined for our test structure, the change from the definition of *k* for central actuation to that for the lateral one can correspond to more than doubling the threshold voltage. From the above Eq (31), simple considerations can be anticipated about the contribution of charging processes and possible sticking of the bridge. It is known that successive applications of the voltage correspond to an accumulation of charge in the dielectric and in an increase of the actuation voltage[43], with an asymptotic trend up to its maximum value[44],[45]. In fact, if it is not given time enough to the charges to be dissipated, they grow up to the maximum contribution allowed by the geometry and by the properties of the dielectric material used for the capacitive response (in the case of central actuation) or by the dielectric used for the actuation pads (lateral actuation). The substrate too, when it is oxidized for improving the

The switching times can be evaluated by means of the capacitance dynamics. Actually, exception done for the oscillations superimposed to the exponential decay, the envelope of the z-quote when the threshold voltage is applied will describe the change of the capacitance up to the full actuation of the device. As an approximation, we can say that the switch is *actuated* when the capacitance is 90% of that obtained from the full actuation procedure. In Fig. 8 the curve containing the oscillations of the capacitance, the maximum value when the bridge is in the DOWN position, and the reference value corresponding to

*kg k g E E*

*C* 

24 2 *<sup>s</sup> OFF c ch e kg k g C V E E* (30)

arg

arg

<sup>2</sup> *<sup>s</sup> c ch e kg k g E E* . The threshold voltage,

<sup>1</sup> 2 0

(31)

24 2

11 1

plus the contact and charge terms, as:

isolation, is a source of charges.

0.9*Cmax* are shown.

And the maintenance voltage *Vmin* can be written as:

min

sticking of the bridge, i.e. when 2 4

*V*

Fig. 8. Simulation of the actuation of the bridge. The curves describing the change of the capacitance (in F) during the actuation time (in s) are: (i) full curve, Ctime1(t), with the entire response by using *k*, (ii) flat full reference line Cmax, i.e. the maximum value obtained when the switch is fully actuated, (iii) Cref=0.9Cmax. Both possible actuations are described: (a) central pull-down, and (b) lateral one.

It is worth noting from Fig. 8 the significant change of the oscillations induced by the different actuation choices. On the other hand, we can simplify the prediction for the actuation time by using the evolution of the envelope describing the actuation, which is written as:

Dynamics of RF Micro-Mechanical

to the full collapse of the beam.

From Eq. (36) it turns out:

Capacitive Shunt Switches in Coplanar Waveguide Configuration 213

effect of *ks* is marginal if we are far from the actuation region, but it becomes important close

The above modeling is valid under the assumption that we are very close to the full actuation of the switch, but far enough to be obliged in considering the van der Waals and the contact contributions. The initial velocity of the bridge is obtained by *v(d+g)=vin*=0

2 ( )

22 4 <sup>1</sup>

*C CVkg k g*

*OFF ON s*

*m*

i.e. the velocity *v* of the bridge subjected to the force imposed by means of the applied voltage is roughly linearly dependent on the applied voltage *V*. So far, Eq. (37) describes the

22 4 <sup>1</sup>

*C C V kg k g*

 

*OFF ON s*

22 4

 Actually, the dissipation causes a decrease in the velocity of the actuation by means of a

dependence of the actuation velocity on the applied voltage *Va* is shown in Fig. 9, where the

When the bridge is released, no electrostatic force has to be included, and only the change of the potential energy and the contribution of the damping have to be considered for calculating the final velocity of the de-actuated beam, obtaining the result in Eq. (39) for the

<sup>2</sup> ( ) *<sup>s</sup>*

*m*

From the analysis of the above figures, a significant change is expected comparing the results obtained by using the threshold value *Vthreshold* and a voltage higher than the

2 4 1

*kg k g*

term depending on *α*. It is like to substitute the mass of the bridge *m* with *m'=m+(2α/*

de-actuation velocity *vde-act*, with *v(d)=*0 as the initial condition for the velocity:

*de act*

*v d*

<sup>2</sup> *C C V kg k g OFF ON <sup>s</sup> m*

*m*

2

1

2 ( )

2

energy has been calculated accounting for the dissipated power <sup>2</sup> *P Fv v E d d*

0

(39)

 *k m* . The dissipated

*<sup>d</sup>* . The

(37)

(38)

*).* The

Where *Ek* is the kinetic energy and *Ed* is the dissipated one, while

because it is at rest before the application of the voltage.

*act*

*act*

*v d*

Exception done for charging contributions.

dependence of *vact* on *Va* for the exploited device is presented.

*v d*

velocity an instant before the bridge is collapsed. We can also write:

$$C\_{act}(t) = \frac{C\_{OFF}}{1 + \varepsilon \frac{\mathcal{G}}{d} \exp\left(-\frac{\mathcal{B}}{2}t\right)}\tag{32}$$

The actuation occurs when *Cref* crosses the envelope described by the previous Eq. (32). In formulae, after simple algebraic passages, it will be:

$$\tau\_{act} = -\frac{2}{\beta} \ln \left( \frac{1}{9} \frac{d}{\varepsilon \mathcal{g}} \right) \tag{33}$$

By using again the values used for the exploited structure, and assuming that the dielectric is SiO2, it is *ε*=3.94 and *τact*35 µs. The same evaluation can be done for the de-actuation, by using the results coming from Fig. 7 and the related analytical treatment. Actually, the evolution of the capacitance can be written, by using the exponential law for the envelope:

$$\mathcal{C}\_{dead}(t) = \mathcal{C}\_{ON} \frac{d + \varepsilon \mathcal{g}}{d + \varepsilon \mathcal{g} \left[1 - \exp\left(-\frac{\mathcal{J}}{2}t\right)\right]} \tag{34}$$

From which it turns out, if *Cdeact(τdeact)*=0.9*CON*:

$$\pi\_{\text{deact}} = -\frac{2}{\beta} \ln \left[ \frac{1}{9} \left( 1 - \frac{d}{\varepsilon g} \right) \right] \tag{35}$$

By using the same values imposed for the actuation, we obtain *τdeact*12 µs. It has to be stressed that both evaluations do not include the contact energy neither the charging contributions, so they should be corrected, but the order of magnitude should not change, exception done for some delay in the restoring mechanism when the switch is de-actuated, which should increase the *τdeact* value.

The above evaluations concern with the response of the switch at the threshold voltage for the actuation. On the other hand, by increasing the applied voltage above such a threshold value, the velocity for the actuation can be increased too, as it can be obtained by considering the energy spent in the actuation process by performing the integral of the motion equation without accounting for contact or charging:

$$\begin{aligned} \int\_{d+g}^{d} m \dot{z} dz &= m \Big|\_{v(d+g)}^{v(d)} \dot{z} d\dot{z} = \frac{1}{2} m \Big[ v^2 (d) - v^2 \left( d + g \right) \Big] = E\_k = \int\_{d+g}^{d} \left( F\_e + F\_m + F\_d \right) dz \\ \int\_{d+g}^{d} F\_e dz &= \frac{1}{2} \left( C\_{OFF} - C\_{ON} \right) V^2 \\ \int\_{d+g}^{d} F\_m dz &= -\frac{1}{2} k g^2 \\ \int\_{d+g}^{d} F\_d dz &= -\frac{1}{4} k\_s g^4 \\ \int\_{d+g}^{d} F\_d dz &= -\frac{\alpha}{\phi \nu} \Big[ v^2 (d) - v^2 (d + g) \Big] \end{aligned} \tag{36}$$

Where *Ek* is the kinetic energy and *Ed* is the dissipated one, while *k m* . The dissipated energy has been calculated accounting for the dissipated power <sup>2</sup> *P Fv v E d d <sup>d</sup>* . The effect of *ks* is marginal if we are far from the actuation region, but it becomes important close to the full collapse of the beam.

The above modeling is valid under the assumption that we are very close to the full actuation of the switch, but far enough to be obliged in considering the van der Waals and the contact contributions. The initial velocity of the bridge is obtained by *v(d+g)=vin*=0 because it is at rest before the application of the voltage.

From Eq. (36) it turns out:

212 Microelectromechanical Systems and Devices

*OFF act <sup>C</sup> C t*

1 exp <sup>2</sup>

*d*

The actuation occurs when *Cref* crosses the envelope described by the previous Eq. (32). In

2 1 ln 9 *act*

By using again the values used for the exploited structure, and assuming that the dielectric is SiO2, it is *ε*=3.94 and *τact*35 µs. The same evaluation can be done for the de-actuation, by using the results coming from Fig. 7 and the related analytical treatment. Actually, the evolution of the capacitance can be written, by using the exponential law for the envelope:

2 1 ln 1

By using the same values imposed for the actuation, we obtain *τdeact*12 µs. It has to be stressed that both evaluations do not include the contact energy neither the charging contributions, so they should be corrected, but the order of magnitude should not change, exception done for some delay in the restoring mechanism when the switch is de-actuated,

The above evaluations concern with the response of the switch at the threshold voltage for the actuation. On the other hand, by increasing the applied voltage above such a threshold value, the velocity for the actuation can be increased too, as it can be obtained by considering the energy spent in the actuation process by performing the integral of the

*<sup>g</sup> <sup>t</sup>*

*d g*

1 exp <sup>2</sup>

*d g*

 

*dg t*

 (32)

(33)

(34)

(35)

(36)

( )

( )

motion equation without accounting for contact or charging:

( ) 2 2

<sup>1</sup> ( ) <sup>2</sup>

*k emd d g vd g d g*

*mzdz m zdz m v d v d g E F F F dz*

 

*d v d d*

2

( )

*F dz C C V*

*F dz kg*

*F dz k g*

*s s d g*

*<sup>e</sup> OFF ON d g*

*d*

*d <sup>m</sup> d g d*

*d <sup>d</sup> d g*

2

4

*F dz v d v d g* 

2 2

 

() ( )

*deact ON*

*d g C tC*

9 *deact*

formulae, after simple algebraic passages, it will be:

From which it turns out, if *Cdeact(τdeact)*=0.9*CON*:

which should increase the *τdeact* value.

$$w\_{act}(d) = \sqrt{\frac{(\mathbb{C}\_{OFF} - \mathbb{C}\_{ON})V^2 - \mathrm{kg}^2 - \frac{1}{2}k\_s \mathrm{g}^4}{m + 2\frac{\alpha}{\alpha}}}\tag{37}$$

i.e. the velocity *v* of the bridge subjected to the force imposed by means of the applied voltage is roughly linearly dependent on the applied voltage *V*. So far, Eq. (37) describes the velocity an instant before the bridge is collapsed. We can also write:

 22 4 <sup>1</sup> 2 ( ) 2 *OFF ON s act C C V kg k g v d m* (38) 22 4 1 

$$\rightarrow \left. \left\langle \sqrt{\frac{\left(\mathbf{C}\_{OFF} - \mathbf{C}\_{ON}\right)V^2 - \left\|\mathbf{g}\right\|^2 - \frac{1}{2}k\_s\mathbf{g}^4}{m}} \right\rangle\_{a \rightarrow 0} \right|$$

Actually, the dissipation causes a decrease in the velocity of the actuation by means of a term depending on *α*. It is like to substitute the mass of the bridge *m* with *m'=m+(2α/).* The dependence of the actuation velocity on the applied voltage *Va* is shown in Fig. 9, where the dependence of *vact* on *Va* for the exploited device is presented.

When the bridge is released, no electrostatic force has to be included, and only the change of the potential energy and the contribution of the damping have to be considered for calculating the final velocity of the de-actuated beam, obtaining the result in Eq. (39) for the de-actuation velocity *vde-act*, with *v(d)=*0 as the initial condition for the velocity:

$$w\_{dc-act}(d) = \sqrt{\frac{\text{kg}^2 + \frac{1}{2}k\_s g^4}{m}}\tag{39}$$

Exception done for charging contributions.

From the analysis of the above figures, a significant change is expected comparing the results obtained by using the threshold value *Vthreshold* and a voltage higher than the

Dynamics of RF Micro-Mechanical

dependence on *Va* is expected.

*act*

*v z*

Then, in our case 0 ( )

**4. Contact and van der Waals forces** 

constant.

Capacitive Shunt Switches in Coplanar Waveguide Configuration 215

Using capacitance dynamics considerations at the threshold voltage, we have assumed that the de-actuation and actuation times (*τde-act* and *τact* respectively) are the time values when the capacitance reaches the 90% of its final value. Such a definition is typically used to introduce the electrical response of a lumped circuit, and, by inverting the capacitance equation at these values, we get the phenomenological definitions for the characteristic deactuation and actuation times. When energy and motion considerations are used, *τact* is the

Concerning the de-actuation mechanism, by using the approach to lower the applied voltage for the maintenance of the bridge in the down position up to the *Vmin* value, no

For the actuation, the applied voltage is always present, and, by imposing a generic z-value

2 4 <sup>2</sup> <sup>1</sup> ( ) <sup>2</sup> ( )

*ON s*

 

*m*

<sup>0</sup> ( ) *act d act d g*

and the applied voltage *V* can be imposed as a parameter. As a consequence, the actuation

A similar result can be obtained for the de-actuation time, but it is independent of the actuation voltage, because the de-actuation occurs when the applied voltage is turned off.

In the case of the de-actuation the additional force due to the voltage is no more present, and the structure will respond only to the restoring forces dominated by the value of the spring

The van der Waals and contact energies introduced in Eq. (29-30) and discussed in literature[5],[42] have attractive and repulsive effects respectively. From a physical standpoint they are due to dipolar contributions induced by atomic interactions. The energy associated with the two effects can be written in the form of the Lennard-Jones potential[42]:

> 12 6 12 6 4 *<sup>c</sup> B C <sup>E</sup>*

Where *B* and *C* are positive constant values and *R* is the inter-atomic distance. Sometimes an exponential trend for the repulsive forces is preferred[42]. Because of the analytical formulation of such a potential, a bound state for the atoms is obtained for the minimum of

 

*R R R R*

*dz dt*

*Cz C V k z d g k z d g*

2

(40)

(42)

*v z*

(41)

time for the full mechanical collapse of the beam and not just a time constant.

in the integration for the energy spent in the actuation, it turns out that

To obtain the actuation time, the following equation can be used:

time for the exploited configuration is given in Fig. 10 by means of Eq. (41).

*dz dt*

=18 µs ca.

*v z*

*de act d g*

*de act <sup>d</sup>*

threshold one. The actuation speed is almost doubled by doubling the applied DC voltage. This enhances the dynamical response of the device, but it lowers the lifetime of the switch, tightly related to the mechanical fatigue induced by the electrostatic actuation. It is worth noting that, due also to the increased pull down voltage, the lateral actuation is faster than the central one.

Fig. 9. Actuation speed *vact* in m/s of the RF MEMS capacitive switch Vs the applied voltage *Va* in volt, for the two simulated situations: (a) central actuation, and (b) lateral one.

threshold one. The actuation speed is almost doubled by doubling the applied DC voltage. This enhances the dynamical response of the device, but it lowers the lifetime of the switch, tightly related to the mechanical fatigue induced by the electrostatic actuation. It is worth noting that, due also to the increased pull down voltage, the lateral actuation is faster than

(a)

(b) Fig. 9. Actuation speed *vact* in m/s of the RF MEMS capacitive switch Vs the applied voltage

*Va* in volt, for the two simulated situations: (a) central actuation, and (b) lateral one.

the central one.

Using capacitance dynamics considerations at the threshold voltage, we have assumed that the de-actuation and actuation times (*τde-act* and *τact* respectively) are the time values when the capacitance reaches the 90% of its final value. Such a definition is typically used to introduce the electrical response of a lumped circuit, and, by inverting the capacitance equation at these values, we get the phenomenological definitions for the characteristic deactuation and actuation times. When energy and motion considerations are used, *τact* is the time for the full mechanical collapse of the beam and not just a time constant.

Concerning the de-actuation mechanism, by using the approach to lower the applied voltage for the maintenance of the bridge in the down position up to the *Vmin* value, no dependence on *Va* is expected.

For the actuation, the applied voltage is always present, and, by imposing a generic z-value in the integration for the energy spent in the actuation, it turns out that

$$w\_{\rm act}(z) = \sqrt{\frac{\left[\mathcal{C}(z) - \mathcal{C}\_{ON}\right]V^2 - k\left[z - \left(d + g\right)\right]^2 - \frac{1}{2}k\_s\left[z - \left(d + g\right)\right]^4}{m + 2\frac{\alpha}{\alpha}}\tag{40}$$

To obtain the actuation time, the following equation can be used:

$$
\tau\_{act} = \int\_0^{\tau\_{act}} dt = \int\_{d+g}^d \frac{dz}{v(z)}\tag{41}
$$

and the applied voltage *V* can be imposed as a parameter. As a consequence, the actuation time for the exploited configuration is given in Fig. 10 by means of Eq. (41).

A similar result can be obtained for the de-actuation time, but it is independent of the actuation voltage, because the de-actuation occurs when the applied voltage is turned off.

$$\text{Then, in our case } \tau\_{de-act} = \int\_0^{\tau\_{de-act}} dt = \int\_d^{d+g} \frac{dz}{v(z)} = 18 \text{ } \mu\text{s ca.}$$

In the case of the de-actuation the additional force due to the voltage is no more present, and the structure will respond only to the restoring forces dominated by the value of the spring constant.

#### **4. Contact and van der Waals forces**

The van der Waals and contact energies introduced in Eq. (29-30) and discussed in literature[5],[42] have attractive and repulsive effects respectively. From a physical standpoint they are due to dipolar contributions induced by atomic interactions. The energy associated with the two effects can be written in the form of the Lennard-Jones potential[42]:

$$E\_c = 4\delta \left[ \left( \frac{\sigma}{R} \right)^{12} - \left( \frac{\sigma}{R} \right)^6 \right] = \frac{B}{R^{12}} - \frac{C}{R^6} \tag{42}$$

Where *B* and *C* are positive constant values and *R* is the inter-atomic distance. Sometimes an exponential trend for the repulsive forces is preferred[42]. Because of the analytical formulation of such a potential, a bound state for the atoms is obtained for the minimum of

Dynamics of RF Micro-Mechanical

1 6 *R* 

Capacitive Shunt Switches in Coplanar Waveguide Configuration 217

the energy, i.e. for the value of *R* vanishing the derivative of *Ec*. It will happen for

A completely different approach has been followed elsewhere[45], where, further to the problems caused by the processing, the pinning due to inter-solid adhesion is analytically and experimentally treated for typical MEMS materials and structures. Also in this case a potential with a binding energy is derived, where the critical distance is associated with the contact area and the surface energy, and with the restoring force of cantilevers and double

> 1 4 3 2 <sup>3</sup> \* <sup>2</sup> *<sup>s</sup> Et h*

It depends on the Young modulus *E*, the thickness of the beam *t*, the distance from the

be fitted depending on the system, and the order of magnitude should be between 100 and 300 mJm-2, but lower values have been also obtained in specific experiments on cantilevers and bridges. It is reasonable to assume a difference depending on the materials used for the contact region (metal-to-metal, metal-to-dielectric, …) and for other surface characteristics,

> *U wl s S S*

Being *w* the width of the beam, *l* the length of the cantilever before the contact area and *s* the length of the contact area with the substrate. The situation is illustrated in Fig. 7 of [45]. A particularly important result is the prediction of the distance at which a cantilever or a beam is detached, by using the definition of a *peeling bound number Np*, where *Np>1* means that the beam can be free again after the contact with the substrate, while it remains in contact with the substrate for *Np<1.* The results interesting for MEMS materials which can be used for beams to be electro-statically actuated are summarized in the following Table II[45].

Table 2. Peeling bound numbers for cantilevers and doubly clamped beams. is the residual tensile stress. *l* is the full length of the suspended structure, either the cantilever or the

Jm-2, in a doubly clamped beam configuration, it turns out that *Np*1.42 for our perforated 1.5 µm thick and 600 µm long beam. From a quick analysis of the quantities playing a role in the definition of *Np*, it is worth noting that it is important to have a thick beam in order to

By using the values for the exploited shunt capacitive MEMS device and from [47]

*Structure Np*

 

(43)

*<sup>s</sup>*. In many cases the surface energy has to

(44)

3 2 4

3 2 2 2

*<sup>s</sup>*=0.06

3 8 *<sup>S</sup> Et h l*

4 2 <sup>128</sup> 4 256 <sup>1</sup> 5 21 2205 *<sup>S</sup> Et h l h l Et t* 

can be present in both the processing and the operating conditions of MEMS devices.

clamped bridges. The value for such a critical distance *s\** for a cantilever is:

*s*

like the roughness. The interfacial adhesion energy is given by[45]:

substrate *h* and the surface energy per unit area

Cantilever

Doubly clamped beam

doubly clamped beam.

 2 . Of course, all of the above considerations are free from the ambiguity of possible failures coming from adhesion related mechanisms induced by humidity, which

Fig. 10. Central (a) and lateral (b) actuation time τa as a function of the applied voltage Va for the studied device.

(a)

(b)

Fig. 10. Central (a) and lateral (b) actuation time τa as a function of the applied voltage Va for

the studied device.

the energy, i.e. for the value of *R* vanishing the derivative of *Ec*. It will happen for 1 6 *R* 2 . Of course, all of the above considerations are free from the ambiguity of possible failures coming from adhesion related mechanisms induced by humidity, which can be present in both the processing and the operating conditions of MEMS devices.

A completely different approach has been followed elsewhere[45], where, further to the problems caused by the processing, the pinning due to inter-solid adhesion is analytically and experimentally treated for typical MEMS materials and structures. Also in this case a potential with a binding energy is derived, where the critical distance is associated with the contact area and the surface energy, and with the restoring force of cantilevers and double clamped bridges. The value for such a critical distance *s\** for a cantilever is:

$$\mathbf{s}^\* = \left(\frac{3}{2}\frac{Et^3\hbar^2}{\gamma\_s}\right)^{1/4} \tag{43}$$

It depends on the Young modulus *E*, the thickness of the beam *t*, the distance from the substrate *h* and the surface energy per unit area *<sup>s</sup>*. In many cases the surface energy has to be fitted depending on the system, and the order of magnitude should be between 100 and 300 mJm-2, but lower values have been also obtained in specific experiments on cantilevers and bridges. It is reasonable to assume a difference depending on the materials used for the contact region (metal-to-metal, metal-to-dielectric, …) and for other surface characteristics, like the roughness. The interfacial adhesion energy is given by[45]:

$$\mathcal{L}I\_S = -\gamma\_S w \left(1 - s\right) \tag{44}$$

Being *w* the width of the beam, *l* the length of the cantilever before the contact area and *s* the length of the contact area with the substrate. The situation is illustrated in Fig. 7 of [45]. A particularly important result is the prediction of the distance at which a cantilever or a beam is detached, by using the definition of a *peeling bound number Np*, where *Np>1* means that the beam can be free again after the contact with the substrate, while it remains in contact with the substrate for *Np<1.* The results interesting for MEMS materials which can be used for beams to be electro-statically actuated are summarized in the following Table II[45].


Table 2. Peeling bound numbers for cantilevers and doubly clamped beams. is the residual tensile stress. *l* is the full length of the suspended structure, either the cantilever or the doubly clamped beam.

By using the values for the exploited shunt capacitive MEMS device and from [47] *<sup>s</sup>*=0.06 Jm-2, in a doubly clamped beam configuration, it turns out that *Np*1.42 for our perforated 1.5 µm thick and 600 µm long beam. From a quick analysis of the quantities playing a role in the definition of *Np*, it is worth noting that it is important to have a thick beam in order to

Dynamics of RF Micro-Mechanical

have:

since the first actuation*.*

accumulated by dielectric layers.

(1) (1)

*th ch*

......................

,0

*<sup>T</sup> V dE*

(2) (1) (2)

*th ch ch*

( ) | |exp

,0 ,0

*T T V dE <sup>E</sup>*

*ch*

**5. Charging effects and sticking** 

Capacitive Shunt Switches in Coplanar Waveguide Configuration 219

The maintenance voltage, i.e. the voltage *Vmin* for holding down the bridge after the actuation, is lower than the threshold one. This will depend on the physics of the electrical contact. Actually, considering the results about surface forces commented in the previous paragraph, they should play no role in sticking, unless introducing other contributions depending on metal-to-metal diffusion and local welding due to aging, heating and power handling. So far, charging of the dielectric material used for the actuation pads should be the main effect in increasing the internal electrical field opposing the actuation one, which will

By using Eq. (31), and neglecting the contact contribution previously discussed, we shall

2 4

And the sticking will happen for *kg2+(1/2)ksg4=2Echarge*. The energy accumulated by means of a charge trapping process will have no effect if a long time is left after the release of the bridge and before the successive actuations, but its contribution on *Vmin* is not negligible

A possible phenomenological approach for explaining the increase in the threshold value under a uni-polar scheme for the actuation voltage of an RF MEMS switch can be given accounting for the change induced in the electric field and, consequently, in the charge

dielectric layer and *Ech* is the electric field related to the accumulated charge, directed along the normal with respect to the dielectric plane. The charge will decay after the release of the

,0 *th*( ) | ( )| | |exp *ch ch*

switch again. Such an effect can be formalized by using the following equations:

(3) (1) (2) (3)

*th ch ch ch*

*T T V dE <sup>E</sup> <sup>E</sup>*

( ) ( ) | |exp 2 | |exp

,0 ,0 ,0

( ) ( ) | |exp 3 | |exp 2 | |exp

*ch ch*

( )

 

*ch ch*

*<sup>t</sup> V t dE t dE*

Where *τch* is the time constant of the decay process during the time *t*. By using a uni-polar scheme for the voltage to be applied to the MEMS switch (positive pulses only), and by imposing a pulse train with period *T* and pulse-width *τ*, successive actuations will be affected by a partial decay of the accumulated charge before the next pulse will actuate the

*kg k g E*

*C* 

<sup>1</sup> <sup>2</sup> 2 *s ch e OFF*

arg

(46)

*ch*

(47)

 

> 

 

> 

, where *d* is the thickness of the

*ch T*  (48)

depend on the charging of the device after many actuations [41],[43],[44],[45].

min

*V*

Such an extra-voltage can be written as | | *V dE th ch*

bridge following an exponential trend, in such a way that:

have an effective restoring force. By decreasing the thickness *t* the value of *Np* will decrease too. Moreover, the Young modulus *E* should be increased, and an additional but less relevant contribution comes from the increase of . In conclusion, a mechanically strong structure would be less influenced by the surface adhesion forces.

Some metals like Pt have a good *E*-value but they are less robust under continuous solicitations. Some others, like Tungsten Carbide, are very hard, but compatibility with standard CMOS processes and integration with microelectronics have to be fully demonstrated. Recent and promising results about molybdenum are available[34]. Materials selection properties are, in general, a hot topic for RF MEMS applications [48]. The threshold value for the actuation voltage is affected by *E*, which enters the definition of the spring constant *k*, and it could render unacceptable the voltage to be used for the actuation of the bridge. So far, a reasonable compromise involves the definition of the geometry and of the material to be used. Anyway, just doubling the value of *E*, i.e. not using Au but Pt, the actuation voltage passes from 29 volt to 30 volt ca. for the central actuation. It means that *Np* passes from 1.42 to 1.53 ca. Since Pt is softer than Au, it is a negligible advantage and un-practical solution, because it will also cause problems in terms of repeated actuations of the bridge. An estimation of the van der Waals force has been also given by using a different formulation[49]. Repulsive forces, which can influence by a factor 2 or 3 the full evaluation of the contact contributions, have been neglected. As already discussed, such an interaction is responsible for binding in metallic contacts as well as in crystalline materials. The interaction energy due to van der Waals contribution can be written as:

$$
\mathcal{U}\_{\rm VDW} = -\frac{A}{12\pi d^2} \tag{45}
$$

Where *A* is defined as the Hamaker constant (*A*=1.6 eV for Si [49], or *A*=4.410-19 J [47]) and *d* is the separation between the two surfaces. In the case of MEMS, at a microscopic scale, the roughness should avoid the adhesion, because it decreases the effectiveness of the surface contact, and residual gas molecules give a further contribution against a possible stiction [49]. As a further demonstration that it is quite difficult to have non-ambiguous results on the definition of the surface energy constant, other findings [18] have also to be considered, where s=1.37 Jm-2 is given in the case of metal-to-metal contact in ohmic switches.

By using another approach [47], the value of s can be obtained accounting that UVDW when the surfaces are in contact between them corresponds to s, i.e. s= UVDW(d=d0), where d0 is the minimum distance between the two surfaces. Considering that the roughness due to the processing of the device is equivalent to cause a residual air gap in the order of 50-100 nm [51], we can estimate for our purposes

$$\gamma\_s = 4.4 \times 10^{19} / (12 \,\pi \text{d} \text{d} \text{}^2) \text{} = (1.17 \div 4.67) \times 10^{-6} \,\text{J} \cdot \text{m}^2 \text{.} $$

If such a value can be considered reasonable for the MEMS technology, no limits on the full length of the bridge should be found. In any case, from all of the above considerations, and from the last ones concerning the roughness, the surface energy should play a minor or negligible role in the sticking of the cantilevers and doubly clamped beams. Humidity only should generate real problems of sticking for un-properly packaged devices

#### **5. Charging effects and sticking**

218 Microelectromechanical Systems and Devices

have an effective restoring force. By decreasing the thickness *t* the value of *Np* will decrease too. Moreover, the Young modulus *E* should be increased, and an additional but less relevant contribution comes from the increase of . In conclusion, a mechanically strong

Some metals like Pt have a good *E*-value but they are less robust under continuous solicitations. Some others, like Tungsten Carbide, are very hard, but compatibility with standard CMOS processes and integration with microelectronics have to be fully demonstrated. Recent and promising results about molybdenum are available[34]. Materials selection properties are, in general, a hot topic for RF MEMS applications [48]. The threshold value for the actuation voltage is affected by *E*, which enters the definition of the spring constant *k*, and it could render unacceptable the voltage to be used for the actuation of the bridge. So far, a reasonable compromise involves the definition of the geometry and of the material to be used. Anyway, just doubling the value of *E*, i.e. not using Au but Pt, the actuation voltage passes from 29 volt to 30 volt ca. for the central actuation. It means that *Np* passes from 1.42 to 1.53 ca. Since Pt is softer than Au, it is a negligible advantage and un-practical solution, because it will also cause problems in terms of repeated actuations of the bridge. An estimation of the van der Waals force has been also given by using a different formulation[49]. Repulsive forces, which can influence by a factor 2 or 3 the full evaluation of the contact contributions, have been neglected. As already discussed, such an interaction is responsible for binding in metallic contacts as well as in crystalline materials. The interaction energy due to van der Waals

> <sup>2</sup> 12 *VDW <sup>A</sup> <sup>U</sup>*

Where *A* is defined as the Hamaker constant (*A*=1.6 eV for Si [49], or *A*=4.410-19 J [47]) and *d* is the separation between the two surfaces. In the case of MEMS, at a microscopic scale, the roughness should avoid the adhesion, because it decreases the effectiveness of the surface contact, and residual gas molecules give a further contribution against a possible stiction [49]. As a further demonstration that it is quite difficult to have non-ambiguous results on the definition of the surface energy constant, other findings [18] have also to be considered,

By using another approach [47], the value of s can be obtained accounting that UVDW when the surfaces are in contact between them corresponds to s, i.e. s= UVDW(d=d0), where d0 is the minimum distance between the two surfaces. Considering that the roughness due to the processing of the device is equivalent to cause a residual air gap in the order of 50-100 nm

 s= 4.410-19/(12d02) = (1.17 4.67)10-6 Jm-2. If such a value can be considered reasonable for the MEMS technology, no limits on the full length of the bridge should be found. In any case, from all of the above considerations, and from the last ones concerning the roughness, the surface energy should play a minor or negligible role in the sticking of the cantilevers and doubly clamped beams. Humidity only

where s=1.37 Jm-2 is given in the case of metal-to-metal contact in ohmic switches.

should generate real problems of sticking for un-properly packaged devices

*<sup>d</sup>* (45)

structure would be less influenced by the surface adhesion forces.

contribution can be written as:

[51], we can estimate for our purposes

The maintenance voltage, i.e. the voltage *Vmin* for holding down the bridge after the actuation, is lower than the threshold one. This will depend on the physics of the electrical contact. Actually, considering the results about surface forces commented in the previous paragraph, they should play no role in sticking, unless introducing other contributions depending on metal-to-metal diffusion and local welding due to aging, heating and power handling. So far, charging of the dielectric material used for the actuation pads should be the main effect in increasing the internal electrical field opposing the actuation one, which will depend on the charging of the device after many actuations [41],[43],[44],[45].

By using Eq. (31), and neglecting the contact contribution previously discussed, we shall have:

$$V\_{\rm min} = \sqrt{\frac{kg^2 + \frac{1}{2}k\_s g^4 - 2E\_{\rm charge}}{\mathcal{C}\_{\rm OFF}}}\tag{46}$$

And the sticking will happen for *kg2+(1/2)ksg4=2Echarge*. The energy accumulated by means of a charge trapping process will have no effect if a long time is left after the release of the bridge and before the successive actuations, but its contribution on *Vmin* is not negligible since the first actuation*.*

A possible phenomenological approach for explaining the increase in the threshold value under a uni-polar scheme for the actuation voltage of an RF MEMS switch can be given accounting for the change induced in the electric field and, consequently, in the charge accumulated by dielectric layers.

Such an extra-voltage can be written as | | *V dE th ch* , where *d* is the thickness of the dielectric layer and *Ech* is the electric field related to the accumulated charge, directed along the normal with respect to the dielectric plane. The charge will decay after the release of the bridge following an exponential trend, in such a way that:

$$
\Delta V\_{th}(t) = d \mid \vec{E}\_{ch}(t) \mid = d \mid \vec{E}\_{ch,0} \mid \exp\left(-\frac{t}{\tau\_{ch}}\right) \tag{47}
$$

Where *τch* is the time constant of the decay process during the time *t*. By using a uni-polar scheme for the voltage to be applied to the MEMS switch (positive pulses only), and by imposing a pulse train with period *T* and pulse-width *τ*, successive actuations will be affected by a partial decay of the accumulated charge before the next pulse will actuate the switch again. Such an effect can be formalized by using the following equations:

$$\begin{split} &\Delta V\_{\rm{th}}^{(1)} = d \mid \bar{E}\_{\rm{ch},0}^{(1)} \mid \exp\Big(-\frac{\left(T-\tau\right)}{\tau\_{\rm{ch}}}\Big) \\ &\Delta V\_{\rm{th}}^{(2)} = d \left[ \mid \bar{E}\_{\rm{ch},0}^{(1)} \mid \exp\Big(-2\frac{\left(T-\tau\right)}{\tau\_{\rm{ch}}}\Big) + \mid \bar{E}\_{\rm{ch},0}^{(2)} \mid \exp\Big(-\frac{\left(T-\tau\right)}{\tau\_{\rm{ch}}}\Big) \right] \\ &\Delta V\_{\rm{th}}^{(3)} = d \left[ \mid \bar{E}\_{\rm{ch},0}^{(1)} \mid \exp\Big(-3\frac{\left(T-\tau\right)}{\tau\_{\rm{ch}}}\Big) + \mid \bar{E}\_{\rm{ch},0}^{(2)} \mid \exp\Big(-2\frac{\left(T-\tau\right)}{\tau\_{\rm{ch}}}\Big) + \mid \bar{E}\_{\rm{ch},0}^{(3)} \mid \exp\Big(-\frac{\left(T-\tau\right)}{\tau\_{\rm{ch}}}\Big) \right] \\ &\vdots \end{split} \tag{48}$$

Dynamics of RF Micro-Mechanical

such an effect [53],[54].

Capacitive Shunt Switches in Coplanar Waveguide Configuration 221

configurations, several solutions are currently studied for characterizing or suppressing

Starting from the evaluations obtained by means of the uni-dimensional approach described in the previous sections, an extension to 2D and 3D structures has been performed by means of the COMSOL Multi-physics software package [55]. Commercial software begins now to be quite popular for simulating physical processes involving mechanical, thermal, high frequency and many other possible (and contemporary) solicitations for the exploited structure. In fact, only simple geometries can be efficiently simulated by using a unidimensional approach, thus estimating actuation times and actuation voltages without using long and complicated simulations with finite element methods. On the other hand, a full simulation is very important especially when the shape of the bridge is tailored in a not simple way. This happens when the cross section has not a constant width, or specific technological solutions, like metal multi-layers for the bridge, and dimples to help the electrical contact in the actuation area are realized. Holes are also present on the beam for improving the sacrificial layer removal and for lowering the spring constant, which is important when the stress induced by the technological process is not acceptable for practical purposes. In all of the above situations, effective quantities can be defined accounting for a re-definition of mass, contact area and beam width. Of course, small changes with respect to the ideal double clamped beam will have a small influence on the response of the entire structure, but more sophisticated geometries and technological solutions need a different evaluation. Moreover, software able to treat combined solicitations of the MEMS device has to be considered if the goal is the definition of a figure of merit for such a technology. For this purpose, 2D and 3D mechanical simulations have been performed to clearly state differences and advantages of such an approach with respect to the uni-dimensional one. An additional consideration is that the deformed shape of the actuated bridge, also in the case of simple geometries, is particularly useful for the prediction of the electrical properties of the device, which could be affected by parasitics for very high frequencies, starting form the millimetre wave range (F > 30 GHz). In the following discussion, parametric and electro-static simulations will be presented, with the aim to compare the central and the lateral actuation, and the expected shape of a simple fixed-fixed beam structure. As the threshold voltage *Vthreshold* is not dependent on the width of the bridge, because it is proportional to the ratio *k/A* between the spring constant *k* and the area *A* of the actuation region, the actuation of a bridge with no holes neither tapering along the width can be considered a 2D problem. Some 2D results are presented in Fig. 11- 14, where the OFF state of the switch has been obtained by using a central actuation (DC signal along the central conductor of the CPW) or a lateral one by means of symmetrical pads. In both cases the electrostatic package implemented in COMSOL has been used, with a parametric simulation performed by changing the value of the applied voltage. A structure having the same dimensions imposed for the uni-dimensional treatment has been simulated: full length *L*=600 μm, width *W*=300 μm for the central conductor of the coplanar structure (corresponding to the bridge length in the actuation region), thickness *t*=1.5 μm for the bridge. The residual stress is again σ=18 MPa. The central conductor is Au, 0.1 μm thick,

**6. Bi-dimensional and three-dimensional mechanical simulations** 

In general, the amount of charge accumulated during successive actuations could be not constant, but we can assume, as a starting point, that (1) (2) | || | ... | | *EE E ch ch* ,0 ,0 *ch*,0 . It turns out in:

$$\begin{split} \Delta V\_{th}^{(n)} &= d \mid \vec{E}\_{ch,0} \mid \exp\left(-n\frac{(\mathsf{T}-\mathsf{T})}{\mathsf{r}\_{ch}}\right) \quad \text{and} \\ \Delta V\_{th} &= d \mid \vec{E}\_{ch,0} \mid \sum\_{n} \exp\left(-n\frac{(\mathsf{T}-\mathsf{T})}{\mathsf{r}\_{ch}}\right) = d \mid \vec{E}\_{ch,0} \mid \sum\_{n} \mathbf{x}^{n} = d \mid \vec{E}\_{ch,0} \mid \frac{\mathbf{1}}{\mathbf{1}-\mathsf{x}} = \\ &= \frac{d \mid \vec{E}\_{ch,0} \mid \mathsf{ }}{\mathsf{1}-\exp\left(-\frac{\mathsf{T}-\mathsf{T}}{\mathsf{r}\_{ch}}\right)} \quad \text{where} \quad \mathsf{x} = \exp\left(-\frac{\mathsf{T}-\mathsf{r}}{\mathsf{r}\_{ch}}\right) < \mathsf{1} \end{split} \tag{49}$$

From Eq. (49) it turns out that a limit in the charge accumulation exists also in the simplified case of constant value for the induced electric field after each actuation.

The above approach is valid only in the case of a uni-polar scheme. When a bi-polar voltage is applied, it will result in induced electric fields having opposite polarization, and, independently of the decay time for the accumulated charge, the original situation will be partially restored after each application of the threshold voltage [44]. It is worth noting that the charge accumulated because of this mechanism sometimes needs very long times for the decay, and the dominant Poole-Frenkel effect is difficult to be prevented. Presently, studies are performed for optimizing materials and geometries, eventually using non-contact actuations. The previously defined quantity | | *ch*,0 *d E* gives the contribution necessary for evaluating the maintenance voltage after the first collapse of the bridge. Eq. (45) has to be rewritten as:

$$\begin{split} V\_{\text{min}} &= \sqrt{\frac{\text{kg}^2 + \frac{1}{2}k\_s \text{g}^4 - 2E\_{\text{charge}}}{\text{C}\_{\text{OFF}}}} = \sqrt{\frac{\text{kg}^2 + \frac{1}{2}k\_s \text{g}^4}{\text{C}\_{\text{OFF}}}} - \Delta V\_{th}^2 = \\ &= d\sqrt{\frac{\text{kg}^2 + \frac{1}{2}k\_s \text{g}^4}{\text{s}Ad}} - |\vec{E}\_{ch,0}|^2 \end{split} \tag{50}$$

Where, from an energetic standpoint, <sup>2</sup> arg 1 2 *E CV ch e OFF th*

An evaluation of <sup>2</sup> | | *Ech*,0 can be obtained by the knowledge of the charging mechanisms in the exploited dielectric. Results are available in literature about such a change in the voltage threshold by assuming that the Poole-Frenkel (PF) effect is the dominant one in the charge trapping of MEMS devices [51], taking into account that the current density due to the PF effect is *PF ch* | |,0 *J E* .and is the conductivity of the material. The role of the above contribution in MIM capacitors and the dependence on the applied voltage and temperature has been studied elsewhere [52].

It has to be stressed that *COFF* will be quite different from the ideal one if residual air gap contributions have to be included, but in the case of floating metal solutions for the realization of shunt capacitive switches the *COFF* value is naturally obtained [28]. As the sticking induced by charging is one of the major problems in the reliability of RF MEMS

In general, the amount of charge accumulated during successive actuations could be not

,0 ,0 ,0

From Eq. (49) it turns out that a limit in the charge accumulation exists also in the simplified

The above approach is valid only in the case of a uni-polar scheme. When a bi-polar voltage is applied, it will result in induced electric fields having opposite polarization, and, independently of the decay time for the accumulated charge, the original situation will be partially restored after each application of the threshold voltage [44]. It is worth noting that the charge accumulated because of this mechanism sometimes needs very long times for the decay, and the dominant Poole-Frenkel effect is difficult to be prevented. Presently, studies are performed for optimizing materials and geometries, eventually using non-contact actuations. The previously defined quantity | | *ch*,0 *d E* gives the contribution necessary for evaluating the maintenance voltage after the first collapse of the bridge. Eq. (45) has to be re-

2 4 2 4

*s ch e s*

*OFF OFF*

arg 1 2 *E CV ch e OFF th*

.and is the conductivity of the material. The role of the above

the exploited dielectric. Results are available in literature about such a change in the voltage threshold by assuming that the Poole-Frenkel (PF) effect is the dominant one in the charge trapping of MEMS devices [51], taking into account that the current density due to the PF

contribution in MIM capacitors and the dependence on the applied voltage and temperature

It has to be stressed that *COFF* will be quite different from the ideal one if residual air gap contributions have to be included, but in the case of floating metal solutions for the realization of shunt capacitive switches the *COFF* value is naturally obtained [28]. As the sticking induced by charging is one of the major problems in the reliability of RF MEMS

1 1 <sup>2</sup> 2 2

*kg k g E kg k g V V C C*

 

> 2 ,0

( ) <sup>1</sup> | | exp | | | |1

*n*

*ch*

arg <sup>2</sup>

can be obtained by the knowledge of the charging mechanisms in

*th*

. It turns out

*x*

(49)

(50)

constant, but we can assume, as a starting point, that (1) (2) | || | ... | | *EE E ch ch* ,0 ,0 *ch*,0


*d E <sup>T</sup> where x*

case of constant value for the induced electric field after each actuation.

*th ch ch ch n n ch*

*ch*

*<sup>T</sup> V dE n dE x dE*

in:

written as:

An evaluation of <sup>2</sup> | | *Ech*,0

effect is *PF ch* | |,0 *J E* 

has been studied elsewhere [52].

( )

*n th ch*

1 exp

,0

( ) | |exp

*<sup>T</sup> V d E n and*

,0

*T*

*ch*

*ch*

min

2 4

*s*

2 | |

*ch*

1

Where, from an energetic standpoint, <sup>2</sup>

*kg k g d E Ad*

  configurations, several solutions are currently studied for characterizing or suppressing such an effect [53],[54].
