41

CHAPTER 3

ANALYSIS OF MEMS BASED 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 Amplifier (PA) and receiver Low Noise

Amplifier (LNA).Low ON-state insertion loss switching is required in order

to minimize degradation in power-added efficiency and noise figure

performance in PA and LNA 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

42

the latter tries to exploit 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. Also, 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).

A 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

Allen 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,

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 (Admschik 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. A 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

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. An 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 Aided Design (CAD) 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 extensively 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 relaxation methods (Cai et al 1993,

Gilbert et al 1995), surface Newton method (Bachtold et al 1995, Yie

45

et al 1994), coupled Newton method (Aluru and White 1996, Aluru and White

1997), multilevel Newton method (Aluru 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 relaxation, 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).A hybrid methodology combining the Finite

Element-Boundary Integration (FE-BI) method for analyzing the fixed 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. A surface

meshing/2-D meshing elements on the same micro device are required to

perform exterior electrostatic analysis based on boundary element analysis

(Anantha Suresh et al 1996). Further to add the complexity, 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 complexity of mesh generation in micro device

an efficient approach is to consider mesh less methods for the modeling and

46

design of MEMS devices (Belytschko et al 1996).Reproducing Kernel

Particle Method (RKPM) was first proposed for the analysis of fixed -fixed

and cantilever micro beams over the ground planes by Aluru (Aluru 1999,

Aluru 2000, Aluru and Li 2001, Li and Aluru 2003a,b). This chapter proposes

the formulation of mesh less method using RKPM to analyze the RF MEMS

switches realized on an ECPW platform. Also an equivalent circuit model for

elevated coplanar waveguide switch has been proposed and the correctness of

RKPM formulation is verified

3.3 PROBLEM STATEMENT 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 fixed 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 A fixed-fixed ECPW Switch with boundary conditions

47

3.4 MATHEMATICAL FORMULATION

3.4.1 Governing Equation

The governing Euler Bernoulli’s equation of a beam subjected to

electrostatic forces is given by (Aluru 1999)

wg

EIgVw

xu

tu

EIo

~65.012

~2

2

4

4

2

2 (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, o is the permittivity of free space, V is

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 (Aluru 1999).

3.4.2 Boundary Conditions

Boundary conditions at the fixed and the free end are given as

(i) ,0u represents a fixed end.

(ii) 0, xudxdu , represents a slope.

(iii) 03

3

2

2

xu

xu represents no connection (no restraint) and no load.

(iv) FxuEI

x

2

2represents the application of a point load F.

(3.2)

48

The boundary conditions on the gradient of the displacement (i.e.

the slope) are treated through a Lagrange multiplier technique (Aluru 1999).

3.4.3 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

0)()( ,,4

4

2

2

nduuduvPd

xuvd

tu

EIv xx

(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

xxu , , xxv,

The weak formulation of equation (3.3) is summarized as,

nduvduvPnduvndwvdwvd

tw

EIv xxxxxxxxxxxxx ,,)(,,,,,,2

2

(3.4)

To obtain a matrix form from the equation (3.4), the displacement

u and the function v are approximated by using the RKPM shape function,

49

i.e. B

NP

BBuNu

1 (NP=1 to 101)

A

NP

AAvNv

1

(NP=1 to 101) (3.5)

Substituting (3.5) in (3.4)

NP

BxxBxxB

NP

AxxAxxAttB

NP

BB

NP

AAA duNvNduN

EIvN

1,,

1,,,

11

NP

BxBxB

NP

AxxAxxA

NP

BxxBxxB

NP

AxAxA nduNvNnduNvN

1,,

1,,

1,,

1,,

nduvNduPvN x

NP

AxxAxxA

NP

AAA ,

1,,

1)( (3.6)

where AN , BN are the RKPM shape functions, Au and Av are the unknowns

associated with particle A. For any particle A, a nonlinear residual equation

can be written as

)()()( uRuRuR statA

dynAA (3.7)

)(uR statA can be written from equation (3.6) as

dnuNNdnuNNduNNuR

NP

BBxBxxA

NP

BBxxBxA

NP

BBxxBxxA

statA

1,,

1,,

1,,)(

nduNduPN xxxAA ,,)( (3.8)

3.4.4 Static Analysis For static analysis, the dynamic residual term in equation (3.7) is

not considered and the residual )(uRA is simply the static residual.

Equation (3.8) (without the dynamic residual term) can then be solved by

50

employing a Newton's method. The displacement increment within each

Newton iteration can be computed by solving the following equation

)(uRuu

R statAB

B

statA

(3.9)

where )(uJu

RAB

B

statA

Equation (3.9) can be modified as

)()( uRuuJ statABAB (3.10)

where )()( NPNPAB uJ R is the Jacobian matrix, )1( NP

Bu R is the

displacement increment vector, and )1()( NPuR statA R is the static residual

vector. The entries of Jacobian matrix is given as

dN

uuPNndNNndNNdNNuJ BAxBxxAxxBxxAxxBxxAAB

)()( ,,,,,,

(3.11)

In matrix form equation (3.10) can be written as

11 )()( NPstat

ANPBNPNPAB uRuuJ (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 EQUIVALENT CIRCUIT APPROACH FOR ECPW BASED 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

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 Cid, Rsub.

(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

r

d

o

tg

wWC

0

(3.13)

Zc Zc Z2 Z1

52

where o is the permittivity, w is the width of the centre conductor,W is the

width of the beam, og is the gap height, dt 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

idgAro

idC (3.14)

where gid is the height of the inserted dielectric, A is the contact area.

Substrate Resistance

The resistance of the inserted dielectric, indicated as RSub in

Figure 3.2 can be given as

Fs

tesubR

m

2

)2/( (3.15)

where

where s is the gap between the conductors, w is the width of the conductor,

og 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.

53

3.6 RESULTS AND DISCUSSIONS

3.6.1 Electrostatic Performance

The validity of the proposed analysis procedure is done using a

fixed –fixed 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

(go) 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 19300

kg/m3 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 MEMSCAD is shown in Figure 3.5. Table 3.1

gives the electrostatic performance comparison of the proposed switch with

the conventional switch.

54

0 50 100 150 200 250 300-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Position along the length of the beam in um

Def

lect

ion

of th

e be

am in

um

"V = 18.57"

"V = 17"

"V = 14"

"V = 11"

"V = 8"

"V = 5"

"V = 2""V = 0"

Figure 3.4 Deflection of fixed- fixed beam on ECPW for a series of

applied voltages obtained using RKPM analysis. The pull in

voltage is 18.57 volts

Figure 3.5 Deflection of fixed- fixed beam on ECPW for a series of applied voltages obtained using Intellisuite MEMS CAD. The pull in voltage is 20 volts

55

Table 3.1 Electrostatic performance comparison of CPW and ECPW switch

Type of switch

Down state Capacitance Voltage Intellisuite RKPM Intellisuite RKPM

CPW 2.5pF (Muldavin and Rebeiz 2000).

2.3pF 32.5 V 34.08 V

ECPW 5.8pF 5.002pF 17.5 V 18.57 V

3.6.2 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 maximize 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 ADS. 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).

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

57

The simulation results of Figure 3.6 (b) show an isolation between

the input and output ports of the switch as 42.01 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

MEMSCAD. The results obtained using RKPM analysis method agrees well

with the Intellisuite MEMSCAD.

Figure 3.7 Performance comparison of ECPW Switch in down state with the static capacitance obtained using Intellisuite and RKPM

3.6.2.1 Effect of various inserted dielectrics

Figure 3.8 presents the variation of scattering parameters for

various inserted dielectric layer materials such as Air (єr2=1), Alumina

58

(єr2=9.8), and Silicon (єr2=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.

3.6.2.2 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 Lm=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

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

3.6.2.3 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). As 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 ~f0/2).Once the capacitance is

determined, the inductance value controls the resonant frequency location.

The inductance has a strong effect on the slope of S21 after fo/2 and this can be

used to fit an accurate model of the switch inductance.

60

(a)

(b)

Figure 3.10 Down state S parameters for an 80µm wide beam showing

the effect of inductance and capacitance

61

3.6.2.4 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

Cd= 3.5 pF , L=5.9 pH. The response for Rs=0.07, 0.25, 0.5ohms are included

for comparison. It is seen that as the series resistance gets smaller, the

resonance in S21 gets sharper and deeper (-50,-40,-33dB respectively).Also

the series resistance has virtually no effect at f < 3fo/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

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 CAD. The equivalent RLC model is

simulated using ADS 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.