**Dynamics of RF Micro-Mechanical Capacitive Shunt Switches in Coplanar Waveguide Configuration**

Romolo Marcelli1, Daniele Comastri1,2, Andrea Lucibello1, Giorgio De Angelis1, Emanuela Proietti1 and Giancarlo Bartolucci1,2 *1CNR-IMM Roma, Roma, 2University of Roma "Tor Vergata", Electronic Engineering Dept., Roma, Italy* 

#### **1. Introduction**

Micro-electromechanical switches for Radio Frequency applications (RF MEMS switches)[1]- [4] are movable micro-systems which pass from an ON to an OFF state by means of the collapse of a metalized beam. They can be actuated in several ways but, generally, the electrostatic actuation is preferred because no current is flowing in the device nor power absorption has to be involved in the process.

The bias DC voltage signal is usually separated with respect to the RF signal for application purposes. Anyway, in the simplest mechanical model, a voltage difference *V* is imposed between the metal bridge, connected to the ground plane of a coplanar waveguide (CPW) structure, and the central conductor of the CPW, which also carries the high frequency signal. Under these circumstances, the switch will experience an electrostatic force which is balanced by its mechanical stiffness, measured in terms of a spring constant *k*. The balance is theoretically obtained until the bridge is going down approximately (1/3) of its initial height. After that, the bridge is fully actuated, and it needs a value of *V* less than the initial one to remain in the OFF (actuated) position, because contact forces and induced charging effects help in maintaining it in the down position. A general layout of the switch is diagrammed in Fig. 1a, with its simplified equivalent lumped electrical circuit. In Fig. 1b the cross-section of the device is shown, with the quantities to be used for the definition of the geometry and of the physical properties of the structure.

The actuation as well as the de-actuation are affected also by the presence of a medium (typically air, or preferably nitrogen for eliminating humidity residual contributions in a packaged device) which introduces its own friction, causing a damping, and altering the speed of the switch [5]-[7]. Several models are currently available to account for a detailed treatment of the damping, including also the presence of holes in the metal beam [8]-[11]. Moreover, the damping modifies the natural frequency of oscillation for the bridge. In particular, the actuation and de-actuation mechanisms will be consequently affected, leading to *simple oscillations* (no fluid damping contribution) or *damped oscillations* (fluid contribution) up to *over-damping* for particular values of the bridge dimensions or material properties. Experimental problems related to the dynamic characterization of

Dynamics of RF Micro-Mechanical

Capacitive Shunt Switches in Coplanar Waveguide Configuration 195

Fig. 1. (a) Schematic diagram of a RF MEMS capacitive shunt switch in coplanar

(FM) to be used for improving the capacitance definition in the down position, (iii) a dielectric layer with thickness *d* deposited onto (iv) the metal M of the central conductor of the CPW, and finally (v) the SiO2 thermally grown layer onto the high

resistivity silicon wafer.

where the metal bridge is suspended by means of dielectric anchors on a multilayer composed by: (i) the air gap *g* with respect to+ (ii) a metal thin layer at a floating potential

configuration (left) and its equivalent circuit (right); (b) cross section of the switch structure,

electrostatically actuated switches have been also considered elsewhere to predict and interpret experimental results [12]. Vibrations and damping are key issues in specific applications, and modeling as well as experimental determination are needed [13]. Optical analysis is one of the most advanced techniques for the characterization of movable Microsystems [14][15]. Other contributions to the motion of the switch are given by the repulsive "contact forces" of the bridge with respect to the plane, when it is already actuated or very close to the plane of the CPW, i.e. close to the substrate. They are due to the interaction between the two surfaces and to the local re-arrangement of the charges. The van der Waals force having an attractive effect has to be also included [5][7]. Both last contributions are, of course, very important when the bridge is close to the bottom RF electrode, in contact with the dielectric used for the capacitance needed to get the best RF isolation for the switch, and close to the actuation pad surface. It is difficult to manage the theory involving all of these contributions, and usually a phenomenological approach is followed, trying to individuate the most important parameters useful to describe the required mechanical and electrical response. For instance, higher is the ratio between the bridge thickness and the bridge length, higher will be the spring constant value and, consequently, the robustness of the switch.

In this chapter it will be demonstrated the usefulness of a fully analytical approach, as compared to the commercial software predictions, thanks to the implementation of all the contributions needed for describing the mechanical response of the double beam structure. In particular, further to predict the relevant quantities useful for the dynamical and electrical characteristics of the switch (actuation and release times, capacitance dynamics, …) a possible optimization of the structure will be proposed for no-contact actuation of the device, in order to minimize the surface and charging effects. One more contribution of this chapter is in the analytical derivation of the actuation voltage, depending on the strength of the applied voltage and the biasing area. This is very important when non-centered actuation voltage is applied, and simplified approaches are no more valid. In fact, this is the real case for devices implemented in RF configurations using the switch as a building block because, for application purposes, the RF and the DC paths have to be distinguished between them. Currently, many papers about the linear and non-linear dynamics of the switch are available in literature [3],[16]-[18] , including also possible collateral effects due to the Casimir force [19][20] or self-actuation mechanisms due to the level of the RF power [21]-[23]. Actually, power is a quantity to be carefully considered for potential applications in bolometers, and for this reason self-actuation was also proposed in nano- and microsystems to get the power value from the actuation onset [24]. Nonlinear response of a double clamped beam under vibrations induced by RF fields have been recently studied, leading to the onset of chaos under specific solicitations and boundary conditions [25].

Analytical approaches have been also used for the circuital modeling of the switches, by fitting data [3][25][26] or deriving in closed form the expressions for the capacitance and for the resistors and inductors involved in the equivalent circuit [27][28]. Methods with EM considerations in circuit derivation have been discussed [30][31]. Inter-modulation products in the response of RF MEMS capacitors have been also investigated [32]. Thermal effects due to the power handling have to be considered as an additional issue.

A comprehensive study involving analytical and numerical predictions has been performed to obtain a full modeling of the electro-mechanical response of shunt capacitive microelectro-mechanical (MEMS) switches for radio-frequency (RF) signal processing. The analytical approach was based on uni-dimensional equations, and it has been settled up for

electrostatically actuated switches have been also considered elsewhere to predict and interpret experimental results [12]. Vibrations and damping are key issues in specific applications, and modeling as well as experimental determination are needed [13]. Optical analysis is one of the most advanced techniques for the characterization of movable Microsystems [14][15]. Other contributions to the motion of the switch are given by the repulsive "contact forces" of the bridge with respect to the plane, when it is already actuated or very close to the plane of the CPW, i.e. close to the substrate. They are due to the interaction between the two surfaces and to the local re-arrangement of the charges. The van der Waals force having an attractive effect has to be also included [5][7]. Both last contributions are, of course, very important when the bridge is close to the bottom RF electrode, in contact with the dielectric used for the capacitance needed to get the best RF isolation for the switch, and close to the actuation pad surface. It is difficult to manage the theory involving all of these contributions, and usually a phenomenological approach is followed, trying to individuate the most important parameters useful to describe the required mechanical and electrical response. For instance, higher is the ratio between the bridge thickness and the bridge length, higher will be the spring constant value and,

In this chapter it will be demonstrated the usefulness of a fully analytical approach, as compared to the commercial software predictions, thanks to the implementation of all the contributions needed for describing the mechanical response of the double beam structure. In particular, further to predict the relevant quantities useful for the dynamical and electrical characteristics of the switch (actuation and release times, capacitance dynamics, …) a possible optimization of the structure will be proposed for no-contact actuation of the device, in order to minimize the surface and charging effects. One more contribution of this chapter is in the analytical derivation of the actuation voltage, depending on the strength of the applied voltage and the biasing area. This is very important when non-centered actuation voltage is applied, and simplified approaches are no more valid. In fact, this is the real case for devices implemented in RF configurations using the switch as a building block because, for application purposes, the RF and the DC paths have to be distinguished between them. Currently, many papers about the linear and non-linear dynamics of the switch are available in literature [3],[16]-[18] , including also possible collateral effects due to the Casimir force [19][20] or self-actuation mechanisms due to the level of the RF power [21]-[23]. Actually, power is a quantity to be carefully considered for potential applications in bolometers, and for this reason self-actuation was also proposed in nano- and microsystems to get the power value from the actuation onset [24]. Nonlinear response of a double clamped beam under vibrations induced by RF fields have been recently studied, leading to the onset of chaos under specific solicitations and boundary conditions [25]. Analytical approaches have been also used for the circuital modeling of the switches, by fitting data [3][25][26] or deriving in closed form the expressions for the capacitance and for the resistors and inductors involved in the equivalent circuit [27][28]. Methods with EM considerations in circuit derivation have been discussed [30][31]. Inter-modulation products in the response of RF MEMS capacitors have been also investigated [32]. Thermal effects due

to the power handling have to be considered as an additional issue.

A comprehensive study involving analytical and numerical predictions has been performed to obtain a full modeling of the electro-mechanical response of shunt capacitive microelectro-mechanical (MEMS) switches for radio-frequency (RF) signal processing. The analytical approach was based on uni-dimensional equations, and it has been settled up for

consequently, the robustness of the switch.

Fig. 1. (a) Schematic diagram of a RF MEMS capacitive shunt switch in coplanar configuration (left) and its equivalent circuit (right); (b) cross section of the switch structure, where the metal bridge is suspended by means of dielectric anchors on a multilayer composed by: (i) the air gap *g* with respect to+ (ii) a metal thin layer at a floating potential (FM) to be used for improving the capacitance definition in the down position, (iii) a dielectric layer with thickness *d* deposited onto (iv) the metal M of the central conductor of the CPW, and finally (v) the SiO2 thermally grown layer onto the high resistivity silicon wafer.

Dynamics of RF Micro-Mechanical

and Eq. (1) can be re-written as:

which can be transformed in:

where:

where:

Capacitive Shunt Switches in Coplanar Waveguide Configuration 197

<sup>0</sup> ( ) ; [, ] ( )

Where *ε0*=8.85×10-12 is the vacuum dielectric constant in MKS units and *εr* is the relative dielectric constant of the material covering the bottom electrode. The derivative of *C z*( ) , to

> 2 *<sup>e</sup> <sup>C</sup> F V z*

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

23 2 ( ) *<sup>s</sup> <sup>k</sup> B V m*

2

 

2 0

*r r*

1 1 ; <sup>2</sup> 2 2 *<sup>c</sup> c c K K <sup>L</sup> L L*

*L L L*

 

 

 

(4)

 

*mz k z d g k z d g z V*

2 0 <sup>2</sup> [ ( )] *r r <sup>C</sup> <sup>A</sup> z d zd* 

is given by:

2

*A*

*r*

*r*

*d zd* 

2

*wr* (7)

3 2 0

(5)

(2)

(3)

(6)

(8)

*<sup>A</sup> C z zdd <sup>g</sup> d zd*

*r r*

 

*s*

 

*t r L*

An approximated definition of it for central actuation can be given by [16]:

( )

*z dg*

*m k m*

 

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

*B A m d g*

The spring constant *k* is a measure of the potential energy of the bridge accumulated as a consequence of its mechanical response to the electrical force due to the applied voltage *V*.

> <sup>3</sup> 1 2 *k K Ewr K* 32 8 1

1 2 2

be used in the definition of the electric force <sup>1</sup> <sup>2</sup>

studying the dynamics of RF MEMS shunt capacitive switches in a coplanar waveguide (CPW) configuration. The capacitance change during the electrostatic actuation and the deactuation mechanism have been modeled, including the actuation speed of the device and its dependence on the gas damping and geometry. Charging and surface effects have been discussed, in order to select the physical and geometrical parameters useful for obtaining a reliable double clamped structure, eventually proposing simple configurations having no contact, thus minimizing surface and charging effects on the pads used for the electrostatic actuation. The model has been implemented by means of a MATHCAD program. Energy considerations have been included to account for the voltage necessary to maintain the switch actuated, by using a DC bias lower than the actuation voltage. The contribution of the charging mechanism has been discussed to describe both the maintenance voltage and the effects on the device reliability. Moreover, two- and three-dimensional models have been developed by using the commercial software package COMSOL Multi-physics, and they have been compared with the analytical and numerical results.

The above approach has been done for predicting the dynamical response of a capacitive shunt switch, but it is generally valid for resistive series as well as for capacitive shunt devices, because it has been implemented for a double clamped beam. Experimental findings confirm the validity of the proposed analytical approach, which is especially useful when a fast preliminary design of the device is needed.

#### **2. One-dimensional model of the mechanical response of shunt capacitive RF MEMS switch**

The equation which accounts for the most part of the above introduced physical mechanisms can be written as:

$$m\ddot{z} = F\_e + F\_s + F\_p + F\_d + F\_c \tag{1}$$

where: *m At* is the bridge mass, computed by means of the material density , the area *A* and the thickness of the bridge *t* ; 1 <sup>2</sup> 2 *<sup>e</sup> <sup>C</sup> F V z* is the electric force due to the applied voltage *V* and to the change of the capacitance along the direction of the motion *z*; [ ( )] *F kz d <sup>p</sup> g* is the force due to the equivalent spring characterized by its constant *k*, acting against the electrical force to carry back the bridge to the equilibrium position; <sup>3</sup> [ ( )] *F kz d s s g* is the nonlinear stretching component of the spring constant [3]; *F z <sup>d</sup>* is the damping force due to the fluid, dependent on the bridge velocity *z* and on the damping parameter , which, in turns, is related to the geometry of the bridge and to the viscosity of the medium; *Fc* is the contact contribution, which can be divided in the van der Waals and surface forces, the first acting as attractive and the second one as repulsive, with a possible equilibrium position at a given distance from the bottom electrode of the

switch [5]. The total capacitance of a shunt capacitive MEMS switch can be described in terms of two series capacitors, each of them having its own dielectric constant. This is only a formal way to approach the problem, because the intermediate plate is a dielectric interface and not a metal one. From the above considerations, and with reference to Fig. 1b, the total capacitance will be:

$$\mathbf{C}(z) = \frac{\varepsilon\_0 \varepsilon\_r A}{d + \varepsilon\_r (z - d)}; \qquad z \in \begin{bmatrix} d, & d + \mathbf{g} \end{bmatrix} \tag{2}$$

Where *ε0*=8.85×10-12 is the vacuum dielectric constant in MKS units and *εr* is the relative dielectric constant of the material covering the bottom electrode. The derivative of *C z*( ) , to

be used in the definition of the electric force <sup>1</sup> <sup>2</sup> 2 *<sup>e</sup> <sup>C</sup> F V z* is given by:

$$\frac{\partial \mathbb{C}}{\partial z} = -\frac{\left\| \varepsilon\_0 \varepsilon\_r^2 \right\|^2}{\left[ d + \varepsilon\_r (z - d) \right]^2} A \tag{3}$$

and Eq. (1) can be re-written as:

$$m\ddot{z} + k[z - (d + g)] + k\_s[z - (d + g)]^3 + a\dot{z} = -\frac{1}{2} \frac{\varepsilon\_0 \varepsilon\_r^2 A}{\left[d + \varepsilon\_r(z - d)\right]^2} V^2 \tag{4}$$

which can be transformed in:

$$
\ddot{\zeta} + \beta \dot{\zeta} + \alpha^2 \zeta + \frac{k\_s}{m} \zeta^3 = B(\zeta) V^2 \tag{5}
$$

where:

196 Microelectromechanical Systems and Devices

studying the dynamics of RF MEMS shunt capacitive switches in a coplanar waveguide (CPW) configuration. The capacitance change during the electrostatic actuation and the deactuation mechanism have been modeled, including the actuation speed of the device and its dependence on the gas damping and geometry. Charging and surface effects have been discussed, in order to select the physical and geometrical parameters useful for obtaining a reliable double clamped structure, eventually proposing simple configurations having no contact, thus minimizing surface and charging effects on the pads used for the electrostatic actuation. The model has been implemented by means of a MATHCAD program. Energy considerations have been included to account for the voltage necessary to maintain the switch actuated, by using a DC bias lower than the actuation voltage. The contribution of the charging mechanism has been discussed to describe both the maintenance voltage and the effects on the device reliability. Moreover, two- and three-dimensional models have been developed by using the commercial software package COMSOL Multi-physics, and

The above approach has been done for predicting the dynamical response of a capacitive shunt switch, but it is generally valid for resistive series as well as for capacitive shunt devices, because it has been implemented for a double clamped beam. Experimental findings confirm the validity of the proposed analytical approach, which is especially useful

**2. One-dimensional model of the mechanical response of shunt capacitive** 

The equation which accounts for the most part of the above introduced physical

is the bridge mass, computed by means of the material density

1 <sup>2</sup>

voltage *V* and to the change of the capacitance along the direction of the motion *z*; [ ( )] *F kz d <sup>p</sup> g* is the force due to the equivalent spring characterized by its constant *k*,

<sup>3</sup> [ ( )] *F kz d s s g* is the nonlinear stretching component of the spring constant [3];

the viscosity of the medium; *Fc* is the contact contribution, which can be divided in the van der Waals and surface forces, the first acting as attractive and the second one as repulsive, with a possible equilibrium position at a given distance from the bottom electrode of the

The total capacitance of a shunt capacitive MEMS switch can be described in terms of two series capacitors, each of them having its own dielectric constant. This is only a formal way to approach the problem, because the intermediate plate is a dielectric interface and not a metal one. From the above considerations, and with reference to Fig. 1b, the total

is the damping force due to the fluid, dependent on the bridge velocity *z* and on

, which, in turns, is related to the geometry of the bridge and to

2 *<sup>e</sup> <sup>C</sup> F V z*

acting against the electrical force to carry back the bridge to the equilibrium position;

*mz F F F F F e s <sup>p</sup> d c* (1)

is the electric force due to the applied

, the area

they have been compared with the analytical and numerical results.

when a fast preliminary design of the device is needed.

**RF MEMS switch** 

where: *m At*

*F z <sup>d</sup>* 

switch [5].

capacitance will be:

the damping parameter

mechanisms can be written as:

*A* and the thickness of the bridge *t* ;

$$\begin{aligned} \zeta' &= z - (d + g) \\ \beta &= \frac{\alpha}{m} \\ \alpha &= \sqrt{\frac{k}{m}} \\ B(\zeta') &= -\frac{1}{2m} \frac{\varepsilon\_0 \varepsilon\_r^2}{\left[d + \varepsilon\_r \left(\zeta' + g\right)\right]^2} A \end{aligned} \tag{6}$$

The spring constant *k* is a measure of the potential energy of the bridge accumulated as a consequence of its mechanical response to the electrical force due to the applied voltage *V*. An approximated definition of it for central actuation can be given by [16]:

$$k = K\_1 \left( 32Ewr^3 \right) + K\_2 \left[ 8\sigma \left( 1 - \nu \right) wr \right] \tag{7}$$

where:

$$\begin{aligned} K\_1 &= \frac{1}{2 - \left(2 - \frac{L\_c}{L}\right) \left(\frac{L\_c}{L}\right)^2}; K\_2 = \frac{1}{2 - \frac{L\_c}{L}}\\ r &= \frac{t}{L} \end{aligned} \tag{8}$$

Dynamics of RF Micro-Mechanical

Capacitive Shunt Switches in Coplanar Waveguide Configuration 199

Fig. 2. Typical shape of a perforated beam used for RF MEMS double clamped switches. Holes are realized for facilitating the sacrificial layer removal and their position, number

Generalizing the equations introduced in literature about this topic [3] we can write:

; () 6 9 4 <sup>2</sup> <sup>48</sup> ( )

*LEI <sup>k</sup> f a L L a La a*

We included in this contribution a generalization of the mechanical evaluation for *k*. Actually, in the previous definitions, the spring constant is approximated accounting for a load in the center of the beam, but a constant load can be applied in different places along the beam, and the *k* value has to be calculated depending on the real position and extension of the applied force with respect to the bridge area. Two contributions are expected, the first one coming from the inertia moment of the beam and its Young modulus, and the second one having a technological origin due to the residual stress of the released bridge.

32 23

*<sup>A</sup>* (11)

 

(10)

*wt <sup>I</sup>* is the inertia moment classical calculation. Following the above definitions,

3

0

=3.94 (SiO2), *t*=1.5 μm for the gold bridge, *ρ*=19320 kg/m3 for the

we can effectively derive a *k*-value depending on the actuation position and on the size of the pads used to impose the voltage between the beam and the bottom electrode necessary

The threshold voltage, obtained by assuming that the mechanical structure collapses after

8 27 *threshold kg <sup>V</sup>*

For the results coming from all the next simulations, we assume the following parameters for the bridge: *L*=600 μm as the bridge total length, *Lc*=300 μm as the switch length in the RF contact region (width of the central conductor of the CPW), *w*=100 μm as the bridge width, before the contact region, to be considered as an averaged value to fit the real cases if a tapering is present, *wS*=100 μm for the switch width (transversal dimension of the switch, parallel with respect to the CPW direction), *d*=thickness of the dielectric material=0.2 μm,

and dimensions are properly tailored depending on the application.

1 1 '

*f a da*

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

*<sup>L</sup> L a <sup>k</sup> g a wt g a da*

2 1

*x x*

2 1

*x x*

' "

for such a collapse, without approximations.

trespassing the critical distance (1/3)*g*, is given by [3][17]:

*kkk*

3 12

with dielectric constant

Where

*L* is the bridge total length, *Lc* is the switch length in the RF contact region (width of the central conductor of the CPW), *w* is the bridge width, before the contact region, which can be also an averaged value as an approximation of the real cases if a tapering is present, *t* is the Au thickness of the bridge. The other parameters are the Young modulus *E*, the residual stress and the Poisson coefficient *ν*. As well established, the Young modulus is an intrinsic property of the material, and specifically it is a measure of its stiffness [33]

The Poisson coefficient *ν* is a measure related to the response of a material when it is stretched in one direction, and it tends to get thinner in the other two directions.

 is mainly related to the process for obtaining the mechanical structure, and it must be measured in the real case. Actually, is a measure of the stress which remains after the original cause of the stresses (external forces, heat gradient) has been removed. They remain along a cross section of the component, even without the external cause.

A new, and more accurate definition of *k* has been recently given [35] for the treatment of miniature RF MEMS switches. Actually, it turns out that [35]:

$$\begin{aligned} \mathbf{L}\_{nc} &= \mathbf{L} - \mathbf{L}\_c\\ \mathbf{L}\_d &= \mathbf{L}\_{nc}/2\\ \mathbf{k}\_{new} &= \frac{2Ewt^3}{L\_d^3} + \frac{Ewt\mathbf{g}^2}{L\_d^3} + \frac{2\sigma(1-\nu)tw}{L\_d} \end{aligned} \tag{9}$$

where *knew* is the new definition of the spring constant, to be compared with that given in Eq. (8). Generally, *knew > k* and bridges shorter than 500 µm ca. are better approximated by the *knew* definition, exhibiting higher actuation voltages.

A recent experimental approach was also adopted for evaluating the contribution of the spring constant and for modeling it on the base of nano-indentation techniques [35].

All the quantities previously introduced have to be re-defined because of the presence of holes in the released beam. The holes need to be used for an easier removal of the sacrificial layer under the beam, and for mitigating the stiffness of the gold metal bridge, i.e. for better controlling the applied voltage necessary for collapsing it, to have not values too high because of the residual stress.

In this framework, we have re-calibrated the material properties accounting for the holes distribution on the metal beam. Literature definitions [3][37] are generally accepted for analytically describing the effect of the holes by means of the pitch, i.e. the center-to-center distance *p* between the holes and the edge-to-edge distance *l*. The situation is explained in the Fig. 2. In this way, the *ligament efficiency* will be given by the term (1-(*l/p*)) and such a term will be used in this paper for evaluating the effective quantities which are decreased with respect to the original one. Following this approach, *eff=*(1-(*l/p*)), while *Eeff=E*(1-(*l/p*)), *νeff=ν*(1-(*l/p*)).

For the effective mass, we preferred to use a definition based on the ratio between the area with and without the holes, thus obtaining *meff=m(A/A0)*, where *A0* is the geometrical area of the beam and *A* is the effective one considering the presence of the holes. All the evaluations which will be shown in this chapter are based on the previously defined quantities, calculated accounting for their effective contribution, but they have been written without the *eff*-pedex, for the sake of simplicity.

*L* is the bridge total length, *Lc* is the switch length in the RF contact region (width of the central conductor of the CPW), *w* is the bridge width, before the contact region, which can be also an averaged value as an approximation of the real cases if a tapering is present, *t* is the Au thickness of the bridge. The other parameters are the Young modulus *E*, the residual stress and the Poisson coefficient *ν*. As well established, the Young modulus is an intrinsic

The Poisson coefficient *ν* is a measure related to the response of a material when it is

is mainly related to the process for obtaining the mechanical structure, and it must be

original cause of the stresses (external forces, heat gradient) has been removed. They remain

A new, and more accurate definition of *k* has been recently given [35] for the treatment of

2 2 1

*Ewt Ewtg tw <sup>k</sup> L L L*

where *knew* is the new definition of the spring constant, to be compared with that given in Eq. (8). Generally, *knew > k* and bridges shorter than 500 µm ca. are better approximated by the

A recent experimental approach was also adopted for evaluating the contribution of the

All the quantities previously introduced have to be re-defined because of the presence of holes in the released beam. The holes need to be used for an easier removal of the sacrificial layer under the beam, and for mitigating the stiffness of the gold metal bridge, i.e. for better controlling the applied voltage necessary for collapsing it, to have not values too high

In this framework, we have re-calibrated the material properties accounting for the holes distribution on the metal beam. Literature definitions [3][37] are generally accepted for analytically describing the effect of the holes by means of the pitch, i.e. the center-to-center distance *p* between the holes and the edge-to-edge distance *l*. The situation is explained in the Fig. 2. In this way, the *ligament efficiency* will be given by the term (1-(*l/p*)) and such a term will be used in this paper for evaluating the effective quantities which are decreased

For the effective mass, we preferred to use a definition based on the ratio between the area with and without the holes, thus obtaining *meff=m(A/A0)*, where *A0* is the geometrical area of the beam and *A* is the effective one considering the presence of the holes. All the evaluations which will be shown in this chapter are based on the previously defined quantities, calculated accounting for their effective contribution, but they have been written without the

3 3

spring constant and for modeling it on the base of nano-indentation techniques [35].

<sup>3</sup> <sup>2</sup>

 

*eff=*

(1-(*l/p*)), while *Eeff=E*(1-(*l/p*)),

*d d d*

is a measure of the stress which remains after the

(9)

property of the material, and specifically it is a measure of its stiffness [33]

measured in the real case. Actually,

stretched in one direction, and it tends to get thinner in the other two directions.

along a cross section of the component, even without the external cause.

2

*nc c d nc*

 

*L LL L L*

miniature RF MEMS switches. Actually, it turns out that [35]:

*new*

*knew* definition, exhibiting higher actuation voltages.

with respect to the original one. Following this approach,

because of the residual stress.

*eff*-pedex, for the sake of simplicity.

*νeff=ν*(1-(*l/p*)).

Fig. 2. Typical shape of a perforated beam used for RF MEMS double clamped switches. Holes are realized for facilitating the sacrificial layer removal and their position, number and dimensions are properly tailored depending on the application.

We included in this contribution a generalization of the mechanical evaluation for *k*. Actually, in the previous definitions, the spring constant is approximated accounting for a load in the center of the beam, but a constant load can be applied in different places along the beam, and the *k* value has to be calculated depending on the real position and extension of the applied force with respect to the bridge area. Two contributions are expected, the first one coming from the inertia moment of the beam and its Young modulus, and the second one having a technological origin due to the residual stress of the released bridge. Generalizing the equations introduced in literature about this topic [3] we can write:

$$\begin{aligned} k' &= -\frac{LEI}{2} \frac{1}{\int\_{x1}^{x2} f(a) da} & \qquad \qquad \qquad f(a) = \frac{1}{48} \left( L^3 - 6L^2 a + 9La^2 - 4a^3 \right) \\\ k'' &= \frac{L}{2} \frac{1}{\int\_{x1}^{x2} g(a) da} & \qquad \qquad \qquad \qquad g(a) = \frac{L-a}{2\sigma \left(1-\nu\right) wt} \\\ k &= k' + k'' \end{aligned} \tag{10}$$

Where 3 12 *wt <sup>I</sup>* is the inertia moment classical calculation. Following the above definitions,

we can effectively derive a *k*-value depending on the actuation position and on the size of the pads used to impose the voltage between the beam and the bottom electrode necessary for such a collapse, without approximations.

The threshold voltage, obtained by assuming that the mechanical structure collapses after trespassing the critical distance (1/3)*g*, is given by [3][17]:

$$V\_{dthreshold} = \sqrt{8 \frac{kg^3}{27 \varepsilon\_0 A}}\tag{11}$$

For the results coming from all the next simulations, we assume the following parameters for the bridge: *L*=600 μm as the bridge total length, *Lc*=300 μm as the switch length in the RF contact region (width of the central conductor of the CPW), *w*=100 μm as the bridge width, before the contact region, to be considered as an averaged value to fit the real cases if a tapering is present, *wS*=100 μm for the switch width (transversal dimension of the switch, parallel with respect to the CPW direction), *d*=thickness of the dielectric material=0.2 μm, with dielectric constant =3.94 (SiO2), *t*=1.5 μm for the gold bridge, *ρ*=19320 kg/m3 for the

Dynamics of RF Micro-Mechanical

conductor of the CPW.

will be studied.

formalism, as:

**2.1 Case V=0, β=0** 

actuation can be predicted only by using Eq. (10).

the well known equation of the harmonic oscillator:

accounts for the geometry used for the actuation of the switch.

Capacitive Shunt Switches in Coplanar Waveguide Configuration 201

distance *l*=10 µm and center-to-center distance *pitch*=20 µm have been imposed. For the position of the actuation pads, we have chosen two possibilities: (i) central actuation, distributing the force over the entire area defined by the switch width and by the width of the CPW; and (ii) lateral actuation, by means of two poly-silicon pads 50 µm wide and long as the bridge width, centered with respect to the slot of the CPW. With reference to the following Fig. 3 and to Table I, the integration for obtaining the value of *k* to be used in Eq. (11) has been performed by choosing x1 and x2 by means of a coordinate system having the origin on the left side of the bridge. In this way, for the central actuation, x1=L/2 and x2=(L/2)+(Lc/2); because of the symmetry of the analyzed structure, the integration is performed only on half of the length where the force is applied, and multiplied by a factor two (see Eq. (10)). For the lateral actuation, the same symmetry is invoked, and we have x1=(L/2)+(Lc/2)+dpad and x2=(L/2)+(Lc/2)+dpad+Lpad, i.e. from the beginning to the end of the dielectric actuation pad for its entire length Lpad , distant dpad from the edge of the central

From the above data and from the *k*-values and actuation voltages *Vact* calculated for both

Central Actuation Eq. (7) and Eq. (10)

*k* [N m-1] 10, 21 208 *Vact* [V] 13, 18 56 Table 1. Spring constant values and actuation voltages for the exploited geometry. It is worth noting that the voltage needed for the central actuation calculated by Eq. (7) is only an approximation with respect to that calculated by the exact Eq. (10); moreover the lateral

It is evident that the actuation requires a voltage applied in the center of the bridge much lower with respect to that needed at the sides of the double clamped structure. This is immediately understandable because of the higher value of the spring constant when the lateral actuation is used. Actually, in this modeling, the value of *k* is the parameter which

Now, the particular and the general cases where voltage and damping are present or absent

In this elementary case a small voltage can be applied just as a perturbation to the bridge, and after that the source is turned off, leaving the bridge moving in a non-dissipative environment. The cubic term due to the stretching is negligible and Eq. (5) is simplified in

2

The simple solution of Eq. (12) for the harmonic oscillator can be found by using a complex

( ) exp[ ( )] *t it*

 

0 (12)

(13)

 

0

Lateral Actuation Eq. (10)

actuation schemes (central and lateral) we obtained the results shown in Table 1:

gold density, *E*=Young modulus=80×109 Pa, *ν*=0.42 for the metal Poisson coefficient and *σ*=18 MPa as the residual stress of the metal (measured on specific micromechanical test structures). The above defined physical quantities, as already discussed in the previous section, have been re-calculated because of the ligament efficiency.

An evaluation of *k* and of the actuation voltage has been given, according to the general geometry introduced in Fig. 3. In particular, holes with 10 µm diameter, side-to-side

Fig. 3. Cross section of the geometry used for the simulation of the pull down of the metal beam: (a) by using the central actuation, and (b) by means of the lateral pads.

gold density, *E*=Young modulus=80×109 Pa, *ν*=0.42 for the metal Poisson coefficient and *σ*=18 MPa as the residual stress of the metal (measured on specific micromechanical test structures). The above defined physical quantities, as already discussed in the previous

An evaluation of *k* and of the actuation voltage has been given, according to the general geometry introduced in Fig. 3. In particular, holes with 10 µm diameter, side-to-side

Fig. 3. Cross section of the geometry used for the simulation of the pull down of the metal

beam: (a) by using the central actuation, and (b) by means of the lateral pads.

section, have been re-calculated because of the ligament efficiency.

distance *l*=10 µm and center-to-center distance *pitch*=20 µm have been imposed. For the position of the actuation pads, we have chosen two possibilities: (i) central actuation, distributing the force over the entire area defined by the switch width and by the width of the CPW; and (ii) lateral actuation, by means of two poly-silicon pads 50 µm wide and long as the bridge width, centered with respect to the slot of the CPW. With reference to the following Fig. 3 and to Table I, the integration for obtaining the value of *k* to be used in Eq. (11) has been performed by choosing x1 and x2 by means of a coordinate system having the origin on the left side of the bridge. In this way, for the central actuation, x1=L/2 and x2=(L/2)+(Lc/2); because of the symmetry of the analyzed structure, the integration is performed only on half of the length where the force is applied, and multiplied by a factor two (see Eq. (10)). For the lateral actuation, the same symmetry is invoked, and we have x1=(L/2)+(Lc/2)+dpad and x2=(L/2)+(Lc/2)+dpad+Lpad, i.e. from the beginning to the end of the dielectric actuation pad for its entire length Lpad , distant dpad from the edge of the central conductor of the CPW.

From the above data and from the *k*-values and actuation voltages *Vact* calculated for both actuation schemes (central and lateral) we obtained the results shown in Table 1:


Table 1. Spring constant values and actuation voltages for the exploited geometry. It is worth noting that the voltage needed for the central actuation calculated by Eq. (7) is only an approximation with respect to that calculated by the exact Eq. (10); moreover the lateral actuation can be predicted only by using Eq. (10).

It is evident that the actuation requires a voltage applied in the center of the bridge much lower with respect to that needed at the sides of the double clamped structure. This is immediately understandable because of the higher value of the spring constant when the lateral actuation is used. Actually, in this modeling, the value of *k* is the parameter which accounts for the geometry used for the actuation of the switch.

Now, the particular and the general cases where voltage and damping are present or absent will be studied.

### **2.1 Case V=0, β=0**

In this elementary case a small voltage can be applied just as a perturbation to the bridge, and after that the source is turned off, leaving the bridge moving in a non-dissipative environment. The cubic term due to the stretching is negligible and Eq. (5) is simplified in the well known equation of the harmonic oscillator:

$$
\ddot{\tilde{\zeta}} + \alpha^2 \tilde{\zeta} = 0 \tag{12}
$$

The simple solution of Eq. (12) for the harmonic oscillator can be found by using a complex formalism, as:

$$
\zeta'(t) = \zeta\_0 \exp[-i(\alpha t + \phi)] \tag{13}
$$

Dynamics of RF Micro-Mechanical

 2 1 or

allowed oscillation before the possible collapse, it will be:

 

It could happen that

where 0 0,max 

valid the condition

condition for the damping is

**2.3 De-actuation of the bridge** 

 ' '3 

such a contribution we can write [16]:

*g* and

 2 1 

shown in Fig. 4b for a beam laterally actuated.

 2 1 

100 nm thick layer of SiO2, from Eq. (2), the ratio is defined as:

Capacitive Shunt Switches in Coplanar Waveguide Configuration 203

frequency values. Consequently, an oscillating response or an over-damping could be

( ) 'Re exp ' exp <sup>0</sup> <sup>2</sup> *t it t*

 

Analogously to the case of non-damped oscillations, and coherently with the maximum

( ) ( ) 'Re exp ' exp <sup>0</sup> <sup>2</sup> *zt d g i t <sup>t</sup>*

geometry can be found in literature [3][5][7] and, as well established, the velocity of the bridge plays a key role together with the geometry and the viscosity of the medium. For

> 2 3

*A g*

1.2566 10 1

A temperature dependence has been included, which accounts for the deformations induced by the change in the total length of the beam [3]. *µ* is the air viscosity, *A* is the full effective area of the bridge, including the ligament efficiency, *T0*=110.33 K and *T* is the absolute temperature [3]. In our case, *T*=300 K. Different formulations appear in other papers [7], and it is the result of an approximation introduced in the equations used by us [3], but it does not change the conceptual approach to the problem. Other available results concern with the operation of the switch in harsh environment [38]. The response of the bridge is given in the following Fig. 4a. It is worth noting that by using the values imposed for the geometry of the exploited device, and considering that, by means of the definition of *k* given in Eq. (10), is

obtained in this case, without oscillations. On the other hand, for lateral actuation, as it is diagrammed in Fig. 3b, the beam exhibits a higher value for the spring constant, and the

The dielectric used for the RF MEMS switch capacitance is often SiO2 or Si3N4, having *εr*=3.94 or 7.6 respectively, or any other dielectric material assuring a high *COFF/CON* ratio. Current results on high-ε materials are very promising for improving the isolation ratio [38]. The thickness of the dielectric material is usually in the order of tens to a hundred of nm, to provide, with the dimension of the gap *g*, the proper ratio of the capacitance between the ON (bridge up) and the OFF position (bridge down). By imposing *g*=2.8 μm and by using a

 

6 0

*<sup>T</sup> <sup>T</sup> T*

depending on the damping and the resonant

(23)

*/2.* Formulations of the damping term for a given

(21)

(22)

1

for the central actuation, an over-damped flat solution is

, thus resulting in damped oscillations. See the results

 2 1 

3 2

 

*=*

obtained. For this reason, the Real Part of the Complex Solution has to be taken:

and the real part will be:

$$
\zeta'(t) = \zeta\_0 \cos(\alpha t + \varphi) \tag{14}
$$

with 0 0, max 1 3 *<sup>g</sup>* , and 2 . The maximum value of the amplitude is given by the

maximum allowed oscillation before a possible collapse induced by the applied voltage, and the phase is given by assuming an initial motion towards the bottom electrode. Then, the time dependence of the vertical coordinate to describe the maximum oscillation can be written as:

$$z(t) = (d+g) - \frac{1}{3}g\sin(\alpha t) \tag{15}$$

This is a well established solution for the un-damped motion equation, but real cases always need the contribution of damped oscillations, which have been studied and will be presented in detail in the following sections.

#### **2.2 Case V=0, β≠0**

The presence of environmental intrinsic damping in the motion of the bridge will cause Eq. (5) to be re-written in the following way:

$$
\ddot{\zeta} + \beta \dot{\zeta} + o^2 \zeta + \frac{k\_s}{m} \zeta^3 = 0 \tag{16}
$$

As already mentioned, there are literature results concerning the correct analytical modeling of the medium in which a metal membrane is moving. By using a phenomenological approach, the effect of the damping can be modeled by defining a complex radian frequency as:

$$
\alpha \flat = \alpha \flat - i \Omega \tag{17}
$$

where, usually but not necessarily, . The complex solution for Eq. (16) will be:

$$
\zeta'(t) = \zeta\_0 \, \text{[exp]} \left[ -i \left( \phi' t + \phi \right) \right] \tag{18}
$$

from which it can be inferred the relation between the natural radian frequency and that modified by the damping in the following form:

$$\alpha \alpha' = \alpha \sqrt{1 - \left(\frac{\beta}{2\alpha}\right)^2} \tag{19}$$

It means that a decrease of the natural frequency is expected when the damping is present. In many physical situations, the condition 2 1 holds. This will depend on the geometry of the bridge and on the intrinsic damping of the medium. The ratio 2 is a measure of the influence of the damping, and it will cause a different dynamical response [3][5][6]. The complex solution for the Eq. (16) with damping is:

$$
\zeta'(t) = \zeta\_0 \, \text{[exp}[-i(\phi't + \phi)]\text{exp}(-\frac{\beta}{2}t) \tag{20}
$$

0

 ( ) cos( ) *t t* 

maximum allowed oscillation before a possible collapse induced by the applied voltage, and the phase is given by assuming an initial motion towards the bottom electrode. Then, the time dependence of the vertical coordinate to describe the maximum oscillation can be

<sup>1</sup> ( ) ( ) sin( ) <sup>3</sup> *zt d g g t*

This is a well established solution for the un-damped motion equation, but real cases always need the contribution of damped oscillations, which have been studied and will be

The presence of environmental intrinsic damping in the motion of the bridge will cause Eq.

As already mentioned, there are literature results concerning the correct analytical modeling of the medium in which a metal membrane is moving. By using a phenomenological approach,

( ) 'exp[ ( ' )] *t it*

from which it can be inferred the relation between the natural radian frequency and that

It means that a decrease of the natural frequency is expected when the damping is present.

measure of the influence of the damping, and it will cause a different dynamical response

<sup>0</sup> ( ) 'exp[ ( ' )]exp( ) <sup>2</sup> *t it t*

 

 2 1 

 

2 

2

the effect of the damping can be modeled by defining a complex radian frequency as:

 

0

' 1

geometry of the bridge and on the intrinsic damping of the medium. The ratio

 

2 3 <sup>0</sup> *<sup>s</sup> <sup>k</sup> m*

 

(16)

. The complex solution for Eq. (16) will be:

(20)

*i* (17)

(18)

(19)

holds. This will depend on the

 2is a

 

(14)

(15)

. The maximum value of the amplitude is given by the

 

[3][5][6]. The complex solution for the Eq. (16) with damping is:

 

and the real part will be:

1 3

*<sup>g</sup>* , and 2

presented in detail in the following sections.

(5) to be re-written in the following way:

'

modified by the damping in the following form:

In many physical situations, the condition

where, usually but not necessarily,

with 0 0, max

**2.2 Case V=0, β≠0** 

 

written as:

It could happen that 2 1 or 2 1 depending on the damping and the resonant frequency values. Consequently, an oscillating response or an over-damping could be obtained. For this reason, the Real Part of the Complex Solution has to be taken:

$$\zeta'(t) = \zeta\_0 \,' \text{Re}\left\{ \exp\left(i(\,\phi't + \phi)\right) \right\} \exp\left(-\frac{\beta}{2}t\right) \tag{21}$$

Analogously to the case of non-damped oscillations, and coherently with the maximum allowed oscillation before the possible collapse, it will be:

$$z(t) = (d+g) - \zeta\_0 \, ' \text{Re}\left\{ \exp\left(i(o^\circ t + \phi)\right) \right\} \exp\left(-\frac{\beta}{2}t\right) \tag{22}$$

where 0 0,max ' '3 *g* and *=/2.* Formulations of the damping term for a given geometry can be found in literature [3][5][7] and, as well established, the velocity of the bridge plays a key role together with the geometry and the viscosity of the medium. For such a contribution we can write [16]:

$$\begin{aligned} \alpha &= \frac{3}{2\pi} \mu \frac{A^2}{g^3} \\ \mu &= 1.2566 \times 10^{-6} \sqrt{T} \left( 1 + \frac{T\_0}{T} \right)^{-1} \end{aligned} \tag{23}$$

A temperature dependence has been included, which accounts for the deformations induced by the change in the total length of the beam [3]. *µ* is the air viscosity, *A* is the full effective area of the bridge, including the ligament efficiency, *T0*=110.33 K and *T* is the absolute temperature [3]. In our case, *T*=300 K. Different formulations appear in other papers [7], and it is the result of an approximation introduced in the equations used by us [3], but it does not change the conceptual approach to the problem. Other available results concern with the operation of the switch in harsh environment [38]. The response of the bridge is given in the following Fig. 4a. It is worth noting that by using the values imposed for the geometry of the exploited device, and considering that, by means of the definition of *k* given in Eq. (10), is valid the condition 2 1 for the central actuation, an over-damped flat solution is obtained in this case, without oscillations. On the other hand, for lateral actuation, as it is diagrammed in Fig. 3b, the beam exhibits a higher value for the spring constant, and the condition for the damping is 2 1 , thus resulting in damped oscillations. See the results shown in Fig. 4b for a beam laterally actuated.

#### **2.3 De-actuation of the bridge**

The dielectric used for the RF MEMS switch capacitance is often SiO2 or Si3N4, having *εr*=3.94 or 7.6 respectively, or any other dielectric material assuring a high *COFF/CON* ratio. Current results on high-ε materials are very promising for improving the isolation ratio [38]. The thickness of the dielectric material is usually in the order of tens to a hundred of nm, to provide, with the dimension of the gap *g*, the proper ratio of the capacitance between the ON (bridge up) and the OFF position (bridge down). By imposing *g*=2.8 μm and by using a 100 nm thick layer of SiO2, from Eq. (2), the ratio is defined as:

Dynamics of RF Micro-Mechanical

using the following formulas:

0

recovery times of the RF MEMS devices.

mechanism.

**2.4 Actuation of the bridge** 

*g*

Capacitive Shunt Switches in Coplanar Waveguide Configuration 205

the nominal ratio is *R*56 for the above imposed values of the parameters useful for describing the bridge mechanics by using SiO2. Actually, as it has been discussed elsewhere[28], the OFF position of the switch is not characterized by a completely flat bridge fully adapted to the bottom electrode, thus contributing to an isolation between the two states lower than expected, unless to use different technological solutions like the floating metal one[39]. In the case of de-actuation, the full solution can be deduced by

<sup>0</sup>

 

( ) Re exp ' exp <sup>2</sup> *deact z t dg it t*

All of the above described analytical solutions of the motion will involve a dynamical response of the capacitance *C*, which shall change according to the variation of *z*. Some detailed plots for the change of the capacitance as a function of time will be presented as a novel contribution with respect to the purely mechanical considerations. This is very important in the evaluation of the transient times useful to define the effective response and

In the following Fig. 5 and Fig. 6, an example of the de-actuation response and of the *C*response are shown, quite in agreement with similar results obtained elsewhere[29] but in this case we want to stress the extension of the analytical approach to the lateral actuation

The actuation process is in principle more complicated, because the computation should involve the presence of the external force and of the contact forces when the bridge is very close to the contact area with the substrate. At a first approximation, without including the

<sup>0</sup>

This is the solution for the full equation with the initial conditions given by *z(0)=d+g*, i.e. with the bridge in the up position, pushed down by a voltage over the threshold. Actually, only positive values are allowed, because of the constrain due to the presence of the

( ) |Re exp ' |exp <sup>2</sup> *act z t dg it t*

Intuitively, it means that the beam is expected to bounce before the full, final collapse, and the bridge will attempt to restore its initial position, but the applied voltage will force it to

 

( ) Re exp ' exp <sup>2</sup> *act z t d it t*

*t it t*

 

above contributions, we can write: 0 ( ) 'exp[ ( ' )]exp( ) <sup>2</sup>

0

*g*

substrate. In particular, the solution for the actuation *zact(t)* will be:

 

, which will lead to:

(27)

*g d* . In our case,

(25)

(26)

where the last equation is valid in most practical cases, being 1 *<sup>r</sup>*

Fig. 4. Damped oscillations around the equilibrium position for (a) a centrally actuated, and (b) a laterally actuated beam. Higher values of *k* (lateral actuation) correspond to enhanced oscillations before obtaining again the equilibrium position. The time t is in sec and the vertical coordinate is in m.

(a)

(b) Fig. 4. Damped oscillations around the equilibrium position for (a) a centrally actuated, and (b) a laterally actuated beam. Higher values of *k* (lateral actuation) correspond to enhanced oscillations before obtaining again the equilibrium position. The time t is in sec and the

vertical coordinate is in m.

1 *<sup>r</sup> r r <sup>d</sup> g gg <sup>R</sup> d dd*

 

(24)

where the last equation is valid in most practical cases, being 1 *<sup>r</sup> g d* . In our case, the nominal ratio is *R*56 for the above imposed values of the parameters useful for describing the bridge mechanics by using SiO2. Actually, as it has been discussed elsewhere[28], the OFF position of the switch is not characterized by a completely flat bridge fully adapted to the bottom electrode, thus contributing to an isolation between the two states lower than expected, unless to use different technological solutions like the floating metal one[39]. In the case of de-actuation, the full solution can be deduced by using the following formulas:

$$\begin{aligned} z\_{\text{dead}}(t) &= d + g + \zeta\_0 \operatorname{Re}\{\exp(i\nu^\circ t)\} \exp\left(-\frac{\beta}{2}t\right) \\ \zeta\_0 &= -g \end{aligned} \tag{25}$$

All of the above described analytical solutions of the motion will involve a dynamical response of the capacitance *C*, which shall change according to the variation of *z*. Some detailed plots for the change of the capacitance as a function of time will be presented as a novel contribution with respect to the purely mechanical considerations. This is very important in the evaluation of the transient times useful to define the effective response and recovery times of the RF MEMS devices.

In the following Fig. 5 and Fig. 6, an example of the de-actuation response and of the *C*response are shown, quite in agreement with similar results obtained elsewhere[29] but in this case we want to stress the extension of the analytical approach to the lateral actuation mechanism.

#### **2.4 Actuation of the bridge**

The actuation process is in principle more complicated, because the computation should involve the presence of the external force and of the contact forces when the bridge is very close to the contact area with the substrate. At a first approximation, without including the

above contributions, we can write: 0 ( ) 'exp[ ( ' )]exp( ) <sup>2</sup> *t it t* , which will lead to:

$$\begin{aligned} z\_{act}(t) &= d + \mathcal{L}\_0 \operatorname{Re} \{ \exp \{ io't \} \} \exp \left( -\frac{\beta}{2} t \right) \\ \mathcal{L}\_0 &= \mathbf{g} \end{aligned} \tag{26}$$

This is the solution for the full equation with the initial conditions given by *z(0)=d+g*, i.e. with the bridge in the up position, pushed down by a voltage over the threshold. Actually, only positive values are allowed, because of the constrain due to the presence of the substrate. In particular, the solution for the actuation *zact(t)* will be:

$$z\_{act}(t) = d - g \mid \text{Re}\left\{ \exp\left(i\nu^\circ t\right) \right\} \mid \exp\left(-\frac{\beta}{2}t\right) \tag{27}$$

Intuitively, it means that the beam is expected to bounce before the full, final collapse, and the bridge will attempt to restore its initial position, but the applied voltage will force it to

Dynamics of RF Micro-Mechanical

Capacitive Shunt Switches in Coplanar Waveguide Configuration 207

(a)

(b)

Fig. 6. Change of the capacitance Vs time for the RF MEMS switch (Ctime), in the case of (a)

central actuation and (b) the lateral one.

Fig. 5. De-actuation Vs time for the shunt switch used as an example (1.5 µm thick). The dashed curve (zup) accounts only for the exponential restoring mechanism, while the full one (zodeact) accounts also for the air damping. The dotted curve (ref) is the 1/3 limit of the gap for the collapse of the bridge. In (a) the central actuation is simulated, while in (b) it is shown the response for the lateral one.

(a)

(b) Fig. 5. De-actuation Vs time for the shunt switch used as an example (1.5 µm thick). The dashed curve (zup) accounts only for the exponential restoring mechanism, while the full one (zodeact) accounts also for the air damping. The dotted curve (ref) is the 1/3 limit of the gap for the collapse of the bridge. In (a) the central actuation is simulated, while in (b) it is

shown the response for the lateral one.

Fig. 6. Change of the capacitance Vs time for the RF MEMS switch (Ctime), in the case of (a) central actuation and (b) the lateral one.

Dynamics of RF Micro-Mechanical

geometry. In fact, the condition

contributing to the fatigue of the beam.

electrostatic actuation, is given by:

**3. Energy considerations and switching times** 

 2 1 

beam.

Capacitive Shunt Switches in Coplanar Waveguide Configuration 209

stay down. So far, the damped oscillation is due to the analytical solution of the motion equation, corresponding to a possible bounce before the full actuation of the bridge. Following this approximation, we shall have the situation depicted in Fig. 7a and 7b. It is worth noting that only for a high value of *k*, like it is for the lateral actuation, the bouncing effect is enhanced, while for the central one no bounces are expected for the imposed

 for the central one, characterized by an imaginary solution and by over-damping. Of course, a compromise has to be chosen between the required actuation speed of the switch and its robustness and lifetime. It has also to be considered that the beam will be stressed more times when it is thicker, because also in this case we get an increase in the value of the spring constant and of the number of possible bounces, but a thin beam could be less reliable because of the experienced fatigue. A similar situation is obtained when you will have higher values for the residual stress, contributing again in the stiffness of the

On the other hand, at least for a limited time, the van der Waals and contact forces will try to maintain the bridge in the DOWN position, introducing additional corrections. This situation is well described in other papers[18], where a solution similar to that presented in Fig.7 is given by studying an ohmic switch. Because of this effect, it is also claimed that a reliability analysis based on the nominal number of actuations is not correct, because it should include the effective number of bounces before the full actuation, each of them

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

> 1 <sup>2</sup> 2

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

1 1 1

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

2 2 4 arg

<sup>2</sup> 2 4 *E C V E E kg k g final OFF threshold c ch e <sup>s</sup>* (29)

the restoring mechanical energy for maintaining the beam in the DOWN position:

is valid only for the lateral actuation, while it is

*E CV initial ON threshold* (28)

Fig. 7. Actuation of the bridge without accounting for contact forces. The curve (zoact) is used to describe: (a) the central actuation for the 1.5 µm thick beam and (b) the lateral one. It is worth noting the number of bounces before the onset of the full actuation.

(a)

(b) Fig. 7. Actuation of the bridge without accounting for contact forces. The curve (zoact) is used to describe: (a) the central actuation for the 1.5 µm thick beam and (b) the lateral one. It is

worth noting the number of bounces before the onset of the full actuation.

stay down. So far, the damped oscillation is due to the analytical solution of the motion equation, corresponding to a possible bounce before the full actuation of the bridge. Following this approximation, we shall have the situation depicted in Fig. 7a and 7b. It is worth noting that only for a high value of *k*, like it is for the lateral actuation, the bouncing effect is enhanced, while for the central one no bounces are expected for the imposed geometry. In fact, the condition 2 1 is valid only for the lateral actuation, while it is 2 1 for the central one, characterized by an imaginary solution and by over-damping. Of course, a compromise has to be chosen between the required actuation speed of the switch and its robustness and lifetime. It has also to be considered that the beam will be stressed more times when it is thicker, because also in this case we get an increase in the value of the spring constant and of the number of possible bounces, but a thin beam could be less reliable because of the experienced fatigue. A similar situation is obtained when you will have higher values for the residual stress, contributing again in the stiffness of the beam.

On the other hand, at least for a limited time, the van der Waals and contact forces will try to maintain the bridge in the DOWN position, introducing additional corrections. This situation is well described in other papers[18], where a solution similar to that presented in Fig.7 is given by studying an ohmic switch. Because of this effect, it is also claimed that a reliability analysis based on the nominal number of actuations is not correct, because it should include the effective number of bounces before the full actuation, each of them contributing to the fatigue of the beam.
