CHAPTER 3 ANALYSIS OF MEMS BASED SWITCHES

Transcription

1 41 CHPTER 3 NLYSIS OF MEMS BSED SWITCHES 3.1 INTRODUCTION The performance of Radio-Frequency (RF) system for wireless communication application can be significantly enhanced by increasing the performance and functionality of the RF switches they use. One important application of the switch is signal routing, which requires low insertion loss and high OFF-state isolation, especially when implementing redundant subsystems for a transmitter Power mplifier (P) and receiver Low Noise mplifier (LN).Low ON-state insertion loss switching is required in order to minimize degradation in power-added efficiency and noise figure performance in P and LN respectively. While implementing a transmit/receive (T/R) module or switched-diversity sectorized antenna (Cetiner et al 2003), a very high OFF-state isolation switching is important to restrict mutual coupling. In the case of digital phase shifter the compact design of high-performance single-pole multiple-throw switches, having good input and output impedance matching is necessary. Over the past few decades, integrated switching in RF circuits has been performed by P-Intrinsic-N (PIN) diodes within Hybrid Microwave Integrated Circuits (HMICs), and also by cold Field-Effect Transistors (cold- FETs) in Monolithic Microwave Integrated Circuits (MMICs) (Robertson and Lucyszyn 2001). The former can deliver superior broadband RF performance with a Single Pole Single-Throw (SPST) reflective switch configuration but

2 42 the latter tries to eploit the inherent switching compatibility of FETs, operating in their triode region. Unfortunately, even with specially fabricated switching-fets, performance can be poorer than that obtained with discrete PIN diodes. lso, with both PIN diodes and cold-fets, inter modulation distortion presents serious limitations at higher RF power levels (Suneat Pranonsatit et al 2006). Radio-frequency Micro Electro Mechanical System (RF MEMS) has been proved as an emerging technology with great promise for reducing cost and improving performance in certain microwave applications (Lucyszyn 2004).RF switch is the basic and the most sought component in communication systems. RF MEMS switches are devices that use mechanical movement to achieve a short circuit or an open circuit on the RF transmission line for switching the RF signal. These RF MEMS switches have demonstrated improved RF performance and figure-of-merit over the conventional PIN diode and FET switches due to their reduced size and inherent functionality (Lucyszyn 2004, Rebeiz and Muldavin 2001). membrane-based switch on silicon was first reported by Peterson in 1979 (Peterson 1979).MEMS switches subject to various actuation designs including electromagnetic (Hosaka et al 1994, Taylor et al 1997, Taylor and llen 1997, Tilmans et al 1999), magneto static (Wright and Tai 1999), electrostatic (Gretillat et al 1999), thermal-electric (Sanders 1998), and various structural designs including a rotating transmission line (Larson et al 1991), surface micro machined cantilevers (Yao and Chang 1995, Schiele et al 1997, De Los Santos et al 1997, Hyman et al 1999, Hyman et al 1999, Zavracky et al 1997, Majumder et al 1997, McGruer et al 1998, Schlaak et al 1997, Suzuki et al 1999), multiple supported or membrane based designs (Yao and Chang 1995, De Los Santos et al 1997, Sovero et al 1999, Muldavin and Rebeiz 1999, Goldsmith et al 1995, Goldsmith et al 1996, Yao et al 1999,

3 43 Pacheco et al 1998), bulk micro machined or wafer bonded designs (Sakata et al 1999, Hiltmann et al 1997, Drake et al 1995), diamond cantilever and contact (dmschik et al 1999), poly silicon switch (Gretillat et al 1995), mercury micro-drop contact (Simon et al 1996), and bi stable micro relays (Sun et al 1998, Kruglick and Pister 1998). Lateral contacting switches (Schiele and Hillerich 1999, Kruglick and Pister 1999) have also been studied. more detailed and classic review on these switches is given in (Gabriel Rebeiz 2003, Varadhan et al 2003).The significant performance improvements that are possible with these RF MEMS devices compared to conventional switches have important implications in system designs for both military and commercial telecommunications at microwave and millimeter wave frequencies. 3.2 ISSUES To design a RF MEMS electrostatic activated switch, the structure of the switch membrane must be chosen so as to produce the lowest possible insertion loss, actuation voltage, the highest possible isolation, and switching frequency. RF MEMS switches using metal membranes with capacitor coupling realized on a CPW platform combines the advantages of MEMS technology and coplanar wave-guide to achieve reduced size and better RF performance (Qian et al 2000). The MEMS switch design involves two stages namely, design of a Coplanar Waveguide (CPW) transmission line for the required centre frequency to provide RF signal path and secondly design of a switch beam with optimized spring constant, materials and membrane height to reduce the activation voltage with reasonable isolation. In CPW the centre conductor width and gap between the centre conductor and the ground conductor play a very important role in respect of properties such as loss and bandwidth and they also play important role on the MEMS switch design. They will also determine the length of the MEMS

4 44 bridge used for realization of the shunt switch. The length of the bridge will have great impact on the switch speed, insertion loss and isolation.the Elevated Coplanar Waveguide (ECPW) dealt in Chapter 2 overcomes the constraints in the coplanar waveguides in respect of the insertion loss and isolation. n electro statically actuated shunt switch on an ECPW platform is proposed in Figure 3.1 (Kanthamani et al 2006) which shows better electrostatic performance than the RF MEMS switches realized on CPW platform. (a) Off-state (b) On-state Figure 3.1 Elevated coplanar waveguide RF MEMS shunt switch MEMS structures are geometrically complicated, electromechanically coupled, and are inherently three-dimensional (3-D) structures. Development of fast, efficient and reliable Computer ided Design (CD) systems for the analysis of MEMS is more complicated than for traditional mechanical or electrical systems. The analysis of 3-D electromechanical systems involves two coupled domains, namely elasto mechanics and electrostatics, each of which have been studied etensively in the literature (Bathe et al 1975, Nabors and White 1991, Phillips and White 1994). Coupled domain analysis of MEMS switches has been done using relaation methods (Cai et al 1993, Gilbert et al 1995), surface Newton method (Bachtold et al 1995, Yie

5 45 et al 1994), coupled Newton method (luru and White 1996, luru and White 1997), multilevel Newton method (luru and White 1997).In addition to the above methods Finite element-based elasto static analysis and accelerated boundary element-based electrostatic analysis have been combined using algorithms based on relaation, a form of surface-newton method, and a tightly coupled Newton method. To model the moving parts of MEMS switches with respect to time a Finite Difference Time Domain (FDTD) formulation was proposed (Tentzeris 2002). Later multi resolution time domain technique, which is an adaptive generalization of the FDTD technique with the use of wavelets to alleviate the computational burdens of the FDTD analysis was proposed (Bushyager and Tentzeris 2001). hybrid methodology combining the Finite Element-Boundary Integration (FE-BI) method for analyzing the fied section of the switch, and the Method of Moments (MOM) for analyzing the movable beam has been proposed for modeling RF-MEMS switches(wang et al 2003).This approach is intended to address the large scale variation within a single computational domain. But in all these methods it is essential to generate a uniform or adaptive volume mesh elements/3-d mesh elements on the electromechanical micro device, to perform the finite element based elastic analysis. surface meshing/2-d meshing elements on the same micro device are required to perform eterior electrostatic analysis based on boundary element analysis (nantha Suresh et al 1996). Further to add the compleity, the volume meshing has to be compatible with the surface meshing and also careful selection of interpolation solution for good convergence is required for coupled domain analysis. To circumvent the compleity of mesh generation in micro device an efficient approach is to consider mesh less methods for the modeling and

6 46 design of MEMS devices (Belytschko et al 1996).Reproducing Kernel Particle Method (RKPM) was first proposed for the analysis of fied -fied and cantilever micro beams over the ground planes by luru (luru 1999, luru 2000, luru and Li 2001, Li and luru 2003a,b). This chapter proposes the formulation of mesh less method using RKPM to analyze the RF MEMS switches realized on an ECPW platform. lso an equivalent circuit model for elevated coplanar waveguide switch has been proposed and the correctness of RKPM formulation is verified 3.3 PROBLEM STTEMENT The static analysis of a RF MEMS switch realized on an ECPW platform reduces to that of solving the Euler Bernoulli s equation of a beam subjected to electrostatic forces with appropriate boundary and interface conditions. The geometry of the RF MEMS ECPW switch chosen for analysis is shown in Figure 3.2.It is assumed that the fied ends of the switch has zero displacement variations. Upon the application of the electrostatic potential the beam gets deformed and at a certain voltage namely the pull in voltage the beam becomes unstable and collapses onto the bottom electrode. The problem is to analyze the proposed RF MEMS switch on ECPW platform to obtain the static pull-in voltages. Figure 3.2 fied-fied ECPW Switch with boundary conditions

7 MTHEMTICL FORMULTION Governing Equation The governing Euler Bernoulli s equation of a beam subjected to electrostatic forces is given by (luru 1999) 2 4 u u w~ V o 2 4 EI t 2EIg 2 2 g w ~ (3.1) where is the mass density per unit length of the beam, u is the displacement of the beam, E is the Young's modulus of the material, I is the moment of inertia, w ~ is the width of the beam, is the permittivity of free space, V is o the applied voltage and g is the gap between the beam and the ground electrode. The Euler-Bernoulli equation describes the relationship between the beam's deflection and the applied load. The Euler beam equation arises from a combination of four distinct subsets of beam theory: the kinematics, constitutive, force resultant, and equilibrium. The beam equation contains a fourth-order derivative in u, hence it mandates for four boundary conditions (luru 1999) Boundary Conditions Boundary conditions at the fied and the free end are given as (i) u 0, represents a fied end. du (ii) u, 0, represents a slope. d 2 u u (iii) represents no connection (no restraint) and no load. (iv) 2 u EI F represents the application of a point load F. 2 (3.2)

8 48 The boundary conditions on the gradient of the displacement (i.e. the slope) are treated through a Lagrange multiplier technique (luru 1999) RKPM Formulation The governing equation (3.1) has higher order derivative (strong form) which involves the difficulty in imposing the boundary condition. So the strong form has to be converted to weak form using Lagrange multiplier technique. Multiply the governing equation by an arbitrary function v such that it satisfies the boundary condition and integrate the governing equation over the domain 2 4 u u v d ( ) (,, ) 0 2 v d 4 vp u d u u nd EI t (3.3) where is the domain, is the boundary of the domain, is the Lagrange multiplier, is the variation of the Lagrange multiplier and n is the unit outward normal. Integrating equation (3.3) by parts and noting that u,, v, The weak formulation of equation (3.3) is summarized as, 2 w v d 2 EI t v, w, d v, w, nd v, u, nd vp( u) d v, u, nd (3.4) To obtain a matri form from the equation (3.4), the displacement u and the function v are approimated by using the RKPM shape function,

9 49 i.e. u B1 1 N B u v N v B (=1 to 101) (=1 to 101) (3.5) Substituting (3.5) in (3.4) 1 N 1 N 1 N v EI v B1 N,, B1 N B u u tt d 1 N nd N 1, 1, v v, B1 N N,, B1 u u d nd v P( u) d N, v, u nd (3.6) where N, N are the RKPM shape functions, B u and v are the unknowns associated with particle. For any particle, a nonlinear residual equation can be written as dyn stat R ( u) R ( u) R ( u) (3.7) R stat N ( u) N R stat (u) can be written from equation (3.6) as N, B1 u B d N N, B1 u Bnd N N, B1 u Bnd P( u) d N, u nd (3.8), Static nalysis For static analysis, the dynamic residual term in equation (3.7) is not considered and the residual R (u) is simply the static residual. Equation (3.8) (without the dynamic residual term) can then be solved by

10 50 employing a Newton's method. The displacement increment within each Newton iteration can be computed by solving the following equation R stat u B u B R stat (u) (3.9) stat R where J B (u) u B Equation (3.9) can be modified as J B stat ( u) u R ( u) (3.10) B ( ) where J ( u ( 1) ) R is the Jacobian matri, u R is the B displacement increment vector, and ( 1) ( u) R is the static residual R stat vector. The entries of Jacobian matri is given as B J B ( u) N, N d N, N nd N, N nd N P( u) N u B d (3.11) In matri form equation (3.10) can be written as ( stat B u) u B 1 R ( u) 1 J (3.12) Solving the resulting system of equations (3.12), gives displacements at each point, which in turn can be used to calculate the down state capacitance in the ON state of the switch. 3.5 EQUIVLENT CIRCUIT PPROCH FOR ECPW BSED RF MEMS SWITCH To show the validity of the RKPM analysis of ECPW switch, an equivalent circuit model is proposed and simulated to obtain the RF

11 51 performance. The equivalent circuit model available in (Muldavin and Rebeiz 2000), for RF MEMS switch realized on CPW platform is used as a basis for obtaining the equivalent circuit model for ECPW switch as in Figure 3.3. The static down state capacitance (C) found out using RKPM analysis is used in the proposed equivalent circuit model. The inserted dielectric of ECPW introduces some amount of capacitance, substrate resistance and they are included in the equivalent circuit as C id, R sub. Z c Z 1 Z 2 Z c (a) CPW shunt switch (b) ECPW switch Figure 3.3 Equivalent circuit model of RF MEMS shunt switch The component values in the equivalent circuit model as in Figure 3.3 are calculated as below. Up-State/Down State Capacitance The parallel-plate capacitance of the MEMS shunt switch is C oww td g 0 r (3.13)

12 52 where o is the permittivity, w is the width of the centre conductor,w is the width of the beam, g o is the gap height, t d is the dielectric layer thickness, r is the relative permittivity. Inserted Dielectric Capacitance The inserted dielectric in the ECPW switch contributes an amount of capacitance, which is given as C id o r g id (3.14) where g id is the height of the inserted dielectric, is the contact area. Substrate Resistance The resistance of the inserted dielectric, indicated as R Sub in Figure 3.2 can be given as Rsub e (t / 2m ) 2 s F (3.15) where where s is the gap between the conductors, w is the width of the conductor, g o is the gap between the contacts, t is the thickness of the conductor, m is the metal skin depth and s is the substrate conductivity.

13 RESULTS ND DISCUSSIONS Electrostatic Performance The validity of the proposed analysis procedure is done using a fied fied beam over ECPW platform on a silicon substrate at a frequency of 40 GHz (Muldavin and Rebeiz 2000).The calculated beam parameters at 40GHz are: length 300µm, width 80µm, and thickness 1.5µm. The initial gap (g o ) between the beam and the bottom electrode is 1µm.Since gold metal is used for conductors, Young's modulus of 80 GPa and a mass density of kg/m 3 are used in the analysis. The displacement and the slope are assumed to be constrained at both ends of the beam. The switch is analyzed using RKPM by employing 101 sprinkled/scattered particles. The software code for the analysis procedure has been written in Matlab and the solution to the governing equation (3.1) along with the boundary condition (3.2) is obtained in the form of displacements. Once the displacements are known the downstate capacitance can be calculated using the formulas available in (Muldavin and Rebeiz 2000).The capacitance found is used in the equivalent circuit model to obtain the RF performance of the switch. The deflections of the beam with respect to the length as a function of applied voltages using RKPM analysis are presented in Figure 3.4.From the result, it is found that proposed ECPW switch provides lower pull-in voltage than the conventional RF MEMS switches realized on CPW platform available in (Muldavin and Rebeiz 2000) for the same structural dimensions. Since the gap height between the center conductor of ECPW and beam is reduced the pull in voltage gets reduced. The values of the pull-in voltage obtained using the Intellisuite MEMSCD is shown in Figure 3.5. Table 3.1 gives the electrostatic performance comparison of the proposed switch with the conventional switch.

14 "V = 0" "V = 2" "V = 5" Deflection of the beam in um "V = 8" "V = 11" "V = 14" "V = 17" "V = 18.57" Position along the length of the beam in um Figure 3.4 Deflection of fied- fied beam on ECPW for a series of applied voltages obtained using RKPM analysis. The pull in voltage is volts Figure 3.5 Deflection of fied- fied beam on ECPW for a series of applied voltages obtained using Intellisuite MEMS CD. The pull in voltage is 20 volts

15 55 Table 3.1 Electrostatic performance comparison of CPW and ECPW switch Type of switch CPW Down state Capacitance Voltage Intellisuite RKPM Intellisuite RKPM 2.5pF 2.3pF 32.5 V V (Muldavin and Rebeiz 2000). ECPW 5.8pF 5.002pF 17.5 V V Radio Frequency Performance In order to determine the losses, performances of the proposed ECPW switch in both the UP and DOWN states an equivalent circuit simulation is done. Design goal of RF MEMS shunt switch is to minimize the insertion loss and maimize the down state isolation. It is obtained using the transmission line model as introduced in section 3.4.The proposed ECPW switch model is simulated using DS. The physical dimensions of the ECPW are: width of the centre conductor is 100µm and the gap between the conductors is 60 µm, the thickness of the inserted dielectric is 0.5 µm and the gap between the center conductor and the beam is 1µm.The values of the components in the equivalent circuit are calculated using the formulas available in (Muldavin and Rebeiz 2000).The variation of scattering parameters with respect to frequency for CPW and ECPW shunt switches in both UP and DOWN states are presented in Figure 3.6 (a) and 3.6 (b).

16 56 Figure 3.6 (a) Performance comparison of CPW & ECPW Shunt switch in UP state Figure 3.6 (b) Performance comparison of CPW and ECPW Shunt switch in DOWN state

17 57 The simulation results of Figure 3.6 (b) show an isolation between the input and output ports of the switch as db in DOWN condition. The ECPW shunt switch RF performance are compared with the conventional CPW shunt switches and the former has increased isolation and a higher return loss at a frequency of 35GHz.The static capacitance obtained using Intellisuite and the proposed RKPM analysis is used in the equivalent circuit to obtain the radio frequency performance. Figure 3.7 presents the variation of scattering parameters with respect to frequency as a function of the static capacitance found using the proposed method (RKPM) and Intellisuite MEMSCD. The results obtained using RKPM analysis method agrees well with the Intellisuite MEMSCD. Figure 3.7 Performance comparison of ECPW Switch in down state with the static capacitance obtained using Intellisuite and RKPM Effect of various inserted dielectrics Figure 3.8 presents the variation of scattering parameters for various inserted dielectric layer materials such as ir (є r2 =1), lumina

18 58 (є r2 =9.8), and Silicon (є r2 =11.9).Inserted dielectrics greatly influences the RF performance characteristics of the ECPW switch. Figure 3.8 Down state isolation for the proposed switch showing the effect of various inserted dielectric The down state isolation changes well with respect to the change in the inserted dielectric material capacitance and the substrate resistance. From the Figure 3.8 it is observed that the down state isolation gets improved as the dielectric constant increases Effect of variations of width The width variations of the beam and the corresponding scattering parameter variations with respect to frequency are shown in Figure 3.9. The down state isolation for ECPW switch varies with increasing widths (length is kept constant at L m =300µm).For a beam width change from 40µm to 80µm, the inductance changes by a factor of 3.0, indicating that the RF current is concentrated on the first edge of the beam and it is strongly independent on

19 59 the width of the beam. The results follow a similar pattern reported for CPW switch (Muldavin and Rebeiz 2000). Figure 3.9 Down state S parameters for the proposed switch of various beam widths Effect of variations of capacitance and inductance In a RF MEMS Switch, the effect of inductance and resistance is negligible in the up-state position (Muldavin and Rebeiz 2000). s the physical parameters of the switch changes corresponding change in the capacitance and inductance also occurs. Figure 3.10 shows the variations of down state isolation of the proposed switch with respect to frequency as a function of the capacitance and inductance variations. The capacitance solely controls the response from 1 to 20 GHz (upto ~f 0 /2).Once the capacitance is determined, the inductance value controls the resonant frequency location. The inductance has a strong effect on the slope of S 21 after f o /2 and this can be used to fit an accurate model of the switch inductance.

20 60 (a) (b) Figure 3.10 Down state S parameters for an 80µm wide beam showing the effect of inductance and capacitance

21 Effect of series resistance of the beam Figure 3.11 shows the effect of series resistance of the beam on the scattering parameters as a function of frequency for ECPW switch with C d = 3.5 pf, L=5.9 ph. The response for R s =0.07, 0.25, 0.5ohms are included for comparison. It is seen that as the series resistance gets smaller, the resonance in S 21 gets sharper and deeper (-50,-40,-33dB respectively).lso the series resistance has virtually no effect at f < 3f o /4, thus it is important to measure the S parameters of the switch around the resonant frequency. Figure 3.11 Down-state S parameters for an 80µm wide beam showing the effect of series resistance 3.7 CONCLUSION RKPM formulation of ECPW shunt switch is proposed in this chapter. Electrostatic analysis of the proposed switch is also obtained. Since the gap height gets reduced in the ECPW switch due to inserted dielectric the

22 62 pull-in voltage is reduced as compared to the conventional CPW switches. The down state capacitance obtained using RKPM method is used in the proposed RLC equivalent circuit. The pull-in voltage and the contact capacitance obtained using RKPM formulation agreed well with the values obtained using Intellisuite MEMS CD. The equivalent RLC model is simulated using DS to obtain the RF performance of the proposed switch in both UP and DOWN states. The simulation results show that an isolation of 2dB in the DOWN state more than the CPW switches and an insertion loss of 0.08 db. The influence of each component in the equivalent circuit model is also studied to note the variations of the scattering parameters. The effect of various inserted dielectrics of ECPW and the corresponding changes in the isolation is also studied.