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

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,

Dynamics of RF Micro-Mechanical

which helps in minimizing the charging effects.

actuation.

Capacitive Shunt Switches in Coplanar Waveguide Configuration 223

Fig. 12. Lateral actuation of the 2D structure (COMSOL simulation). The same force used for the central actuation was necessary for having a full collapse of the bridge, but applied on smaller lateral pads (50 μm width). A more uniform actuation is obtained, as evidenced by the colour intensity in the central part. Moreover, the beam is contact-less on the pads,

An example of the 3D response is given in the following Fig. 13 and Fig. 14, where the actuation has been performed by means of the same force used for the 2D case, with details about the full device. The result is coherent with the prediction performed by using the analytical approach and the 2D actuation. In fact, the mechanical force per unitary area imposed in the simulation for obtaining the full collapse of the bridge corresponds to a pressure of 1500 N/m2 , i.e. to a force of 45 x 10-6 N applied onto an area of 300 x 100 x 10-12 m2. This is the value of the mechanical restoring force applied in the centre of the double clamped beam, and by calculating again the value of / *C z* when *z = 2(d+g)/3* and substituting in *V FCz* 2 / the voltage *V*=20 volt is obtained also in this case. So, the actuation of the bridge can be easily predicted by using the uni-dimensional approach, also including rough evaluations about the holes contribution, but avoiding long term simulations, exception done for those structures where the distribution of the holes is very complicated or a significant shape tailoring is present and the analytical approach should be forced by the introduction of effective quantities not really matching the actual situation. By properly choosing the dimensions and the materials, lateral actuation is possible with voltages in the same order of magnitude used for the central one, as from the comparison between the results coming from data in Fig. 13 and Fig. 14, with evidence for contact-less

Only in simple cases the presence of holes can be approximated by defining an effective stiffness for the metal beam. For the above reason, the 3D simulation is really useful, as already stressed, in the case of configurations which have a very peculiar shape. Also in the case of a moving mesh, i.e. a mechanical solver to be used for the dynamical response of the

covered by SiO2, 0.2 μm thick. The bridge width is 100 µm, along the direction normal with respect to the 2D view. In the case of Fig. 11, the parametric force needed to get the fully deformed shape of the beam is given by *F = -kg =* -46 µN, corresponding to a pressure of 1.5x103 N/m2 ca. This is in agreement with the value obtained by using the 2D simulation. On the other hand, the nonlinearity of the mechanical problem allows for a full actuation when the structure begins to be unstable, i.e. when the applied force is sufficient to push the bridge down (1/3) of the gap. This happens at *z = 2(d+g)/3* = 2 µm, corresponding to a mechanically simulated structure subjected to a force *F* = -15 µN and to a pressure of 500 N/m2 ca. By calculating the value of *C z* / when *z = 2(d+g)/3* and substituting in *V FCz* 2 / obtained from the definition of the electrostatic force, the voltage *V*=20 volt is obtained for the threshold, thus demonstrating that the actuation of the bridge can be easily predicted by using the uni-dimensional approach. In this and in the following simulations holes are not included, but from the uni-dimensional simulations, it turns out that V=18 volt is obtained when holes are present, while V=20 volt is the expected threshold voltage for the structure with holes, i.e. only a 10% difference.

Fig. 11. 2D simulation for the central actuation of the MEMS switch (COMSOL simulation). A mechanical force per unitary area on the central conductor of the CPW has been applied as high as 1500 N/m2 to obtain the full actuation.

From the result in the next Fig. 12, where the lateral actuation is imposed, it turns out that by properly choosing the shape and the dimensions of the structure, the actuation occurs without having the bridge touching the lateral pads. This could help in decreasing the charging effects for these devices, mainly due to the dielectric used onto the actuation pads, which dramatically affects their reliability in terms of the charge stored. Moreover, the necessity to separate the RF and DC paths is another important reason for preferring a lateral actuation in actual devices.

covered by SiO2, 0.2 μm thick. The bridge width is 100 µm, along the direction normal with respect to the 2D view. In the case of Fig. 11, the parametric force needed to get the fully deformed shape of the beam is given by *F = -kg =* -46 µN, corresponding to a pressure of 1.5x103 N/m2 ca. This is in agreement with the value obtained by using the 2D simulation. On the other hand, the nonlinearity of the mechanical problem allows for a full actuation when the structure begins to be unstable, i.e. when the applied force is sufficient to push the bridge down (1/3) of the gap. This happens at *z = 2(d+g)/3* = 2 µm, corresponding to a mechanically simulated structure subjected to a force *F* = -15 µN and to a pressure of 500 N/m2 ca. By calculating the value of *C z* / when *z = 2(d+g)/3* and substituting in *V FCz* 2 / obtained from the definition of the electrostatic force, the voltage *V*=20 volt is obtained for the threshold, thus demonstrating that the actuation of the bridge can be easily predicted by using the uni-dimensional approach. In this and in the following simulations holes are not included, but from the uni-dimensional simulations, it turns out that V=18 volt is obtained when holes are present, while V=20 volt is the expected threshold

Fig. 11. 2D simulation for the central actuation of the MEMS switch (COMSOL simulation). A mechanical force per unitary area on the central conductor of the CPW has been applied

From the result in the next Fig. 12, where the lateral actuation is imposed, it turns out that by properly choosing the shape and the dimensions of the structure, the actuation occurs without having the bridge touching the lateral pads. This could help in decreasing the charging effects for these devices, mainly due to the dielectric used onto the actuation pads, which dramatically affects their reliability in terms of the charge stored. Moreover, the necessity to separate the RF and DC paths is another important reason for preferring a

voltage for the structure with holes, i.e. only a 10% difference.

as high as 1500 N/m2 to obtain the full actuation.

lateral actuation in actual devices.

Fig. 12. Lateral actuation of the 2D structure (COMSOL simulation). The same force used for the central actuation was necessary for having a full collapse of the bridge, but applied on smaller lateral pads (50 μm width). A more uniform actuation is obtained, as evidenced by the colour intensity in the central part. Moreover, the beam is contact-less on the pads, which helps in minimizing the charging effects.

An example of the 3D response is given in the following Fig. 13 and Fig. 14, where the actuation has been performed by means of the same force used for the 2D case, with details about the full device. The result is coherent with the prediction performed by using the analytical approach and the 2D actuation. In fact, the mechanical force per unitary area imposed in the simulation for obtaining the full collapse of the bridge corresponds to a pressure of 1500 N/m2 , i.e. to a force of 45 x 10-6 N applied onto an area of 300 x 100 x 10-12 m2. This is the value of the mechanical restoring force applied in the centre of the double clamped beam, and by calculating again the value of / *C z* when *z = 2(d+g)/3* and substituting in *V FCz* 2 / the voltage *V*=20 volt is obtained also in this case. So, the actuation of the bridge can be easily predicted by using the uni-dimensional approach, also including rough evaluations about the holes contribution, but avoiding long term simulations, exception done for those structures where the distribution of the holes is very complicated or a significant shape tailoring is present and the analytical approach should be forced by the introduction of effective quantities not really matching the actual situation.

By properly choosing the dimensions and the materials, lateral actuation is possible with voltages in the same order of magnitude used for the central one, as from the comparison between the results coming from data in Fig. 13 and Fig. 14, with evidence for contact-less actuation.

Only in simple cases the presence of holes can be approximated by defining an effective stiffness for the metal beam. For the above reason, the 3D simulation is really useful, as already stressed, in the case of configurations which have a very peculiar shape. Also in the case of a moving mesh, i.e. a mechanical solver to be used for the dynamical response of the

Dynamics of RF Micro-Mechanical

Capacitive Shunt Switches in Coplanar Waveguide Configuration 225

being based on the construction of the same geometry. Specifically, as it is the case of the COMSOL software package, thermal, power and charging effects could be considered in the same simulation environment. For the above reasons, this will be very useful to get a figure

As a final demonstration of the validity of the proposed theoretical approach for the electromechanical analysis, one more simulation has been performed by means of the electrostatic

where *Q* is the charge and *E* is the electric field. The result is shown in Fig. 15, where the structure is pushed down (1/3) of the gap, before the final actuation occurs following the mechanical instability of the structure. It happens by imposing a voltage *V* = 22 volt ca., very

Fig. 15. Electrostatic 2D simulation for the exploited double clamped beam. V = 22 volt was

An actual configuration having the same dimensions described in the previous sections has been realized and preliminary tested. A photo of the structure is given in Fig. 16. SU-8 polymeric sides have been realized by photo-lithography to be used as a support for both the ground planes of the CPW, and the suspended metal bridge. Silicon oxide has been deposited as a dielectric, and the actuation has been performed by means of the central

The realization of double-clamped RF MEMS capacitive shunt switches has been performed by means of negative photo-resist SU-8 for the realization of the ground planes of the coplanar configuration, elevated with respect to the wafer, while positive S1818 photo-resist

RF MEMS Switches have been manufactured on a 4 inch high-resistivity (ρ > 5000 ohm cm) silicon wafer <100> oriented, having a thickness of 400 μm. For the realization of the devices, a 4 mask sequence has been considered, and the entire fabrication process is sub-

necessary for having a movement of the beam close to (1/3) of the gap.

**7. Technology, experimental results and discussion** 

conductor of the CPW.

divided in five steps:

has been used as a sacrificial layer.

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

of merit for the RF MEMS technology based on different input conditions.

close to the calculated threshold value by using the uni-dimensional approach.

force directly defined within COMSOL as 1 1 <sup>2</sup>

device, many information can be already obtained from evaluations based on a fully analytical model, without involving cumbersome simulations. A real advantage in having a full 3D modelling of the device is in the combination of mechanical and RF predictions,

Fig. 13. 3D COMSOL simulation of the RF MEMS shunt capacitive switch in the OFF state (bridge in the down position), centrally actuated.

Fig. 14. 3D simulation of the RF MEMS switch in the OFF state (bridge in the down position), laterally actuated when the applied force per unitary area is F=1500 N/m2. The deformation of the bridge is represented by the change in the colours, from the blue (at rest) to the red (fully actuated).

device, many information can be already obtained from evaluations based on a fully analytical model, without involving cumbersome simulations. A real advantage in having a full 3D modelling of the device is in the combination of mechanical and RF predictions,

Fig. 13. 3D COMSOL simulation of the RF MEMS shunt capacitive switch in the OFF state

Fig. 14. 3D simulation of the RF MEMS switch in the OFF state (bridge in the down position), laterally actuated when the applied force per unitary area is F=1500 N/m2. The deformation of the bridge is represented by the change in the colours, from the blue (at rest)

(bridge in the down position), centrally actuated.

to the red (fully actuated).

being based on the construction of the same geometry. Specifically, as it is the case of the COMSOL software package, thermal, power and charging effects could be considered in the same simulation environment. For the above reasons, this will be very useful to get a figure of merit for the RF MEMS technology based on different input conditions.

As a final demonstration of the validity of the proposed theoretical approach for the electromechanical analysis, one more simulation has been performed by means of the electrostatic

force directly defined within COMSOL as 1 1 <sup>2</sup> 2 2 *<sup>e</sup> <sup>C</sup> F V QE z* 

where *Q* is the charge and *E* is the electric field. The result is shown in Fig. 15, where the structure is pushed down (1/3) of the gap, before the final actuation occurs following the mechanical instability of the structure. It happens by imposing a voltage *V* = 22 volt ca., very close to the calculated threshold value by using the uni-dimensional approach.

Fig. 15. Electrostatic 2D simulation for the exploited double clamped beam. V = 22 volt was necessary for having a movement of the beam close to (1/3) of the gap.
