1

The Influence of Gas Damping on the Harmonic and Transient Response of MEMS Devices J. De Coster1 , H.A.C. Tilmans2 , P.G. Steeneken3 and R. Puers1

1

KULeuven, Department ESAT-MICAS, Kasteelpark Arenberg 10, B-3001, Heverlee, Belgium 2

IMEC, Kapeldreef 75, B-3001 Leuven, Belgium

model is in agreement with both the harmonic and transient measurements. Thanks to this twofold validation, the model allows for estimating the switching time of a given design with reasonable accuracy. II. H ARMONIC E XCITATION

3

Philips Research, Prof. Holstlaan 4, 5656AA Eindhoven, the Netherlands {jdecoste|puers}@esat.kuleuven.be Abstract— The influence of air damping on the mechanical response of MEMS devices is investigated in this paper, using capacitive RF-MEMS switches with perforated parallel-plate electrodes as test vehicles. A twofold approach is used to analyse the damping coefficient as a function of ambient pressure. First, the up-state damping coefficient is extracted from resonance frequency measurements. Next, the position-dependent damping coefficient is extracted from closing-time measurements on the RF-MEMS switches. With these measurements, a Simulink model is adjusted to correctly simulate the closing time and trajectory of the switch.

A. Basic Model Fig. 1 shows a SEM image of a capacitive RF-MEMS switch fabricated in the PASSITM process [7]. Air damping occurs as the perforated membrane moves up and down, thereby squeezing the air between the bottom electrode and the membrane and inducing a flow of air through the perforated holes. As a consequence, the behaviour of the switch can be captured in an equivalent mass-spring-damper system, as depicted in Fig. 2 [8].

Index Terms— gas damping, MEMS switches, switching time, mechanical resonance

I. I NTRODUCTION Even though air damping does not generally influence the steady-state performance of MEMS devices, it does have a significant impact on their dynamic behaviour. In principle, the damping can be reduced or eliminated altogether by operating the MEMS device in a vacuum. This is, however, not always possible and it may not even be desirable. Therefore, air damping and its influence on e.g. the pull-in time of MEMS switches has been studied by several authors [1], [2], [3], [4]. Taking a somewhat different approach, this paper investigates the influence of air damping on two operation modes of a capacitive RF-MEMS switch: harmonic excitation and step responses are studied, and the measurement results are correlated. Although harmonic excitation as such is not relevant for the operation of RF-MEMS switches, this can be of practical interest for pressure and leak rate measurements on wafer-level MEMS packages [5], [6]. From impedance measurements on harmonically driven switches, the damping coefficient can be determined as a function of ambient pressure.The measured damping coefficient is then introduced in a Simulink model describing the transient response of the switch. Next, the switching time is simulated for varying ambient pressures. Finally, the simulated switching times are validated by measurements. It will be shown that the Simulink

Fig. 1. Capacitive RF-MEMS switch fabricated in the PASSITM process (Philips) [7].

When a DC voltage Vdc and an AC voltage Vac are applied simultaneously to the system, the membrane makes excursions around the static deflection zst . If the frequency of Vac is swept around the mechanical resonance frequency f0 of the device, the resonance shows up in the electrical admittance Y = G + jB [9]. This is shown in Fig. 3. The curves in the top half of the figure are raw measurement data; the bottom half shows the conductance G after subtraction of the parasitics from the high-resistivity silicon (HRSi) substrate. The measurement of G and B allows for determining the resonance frequency f0 and the mechanical quality factor Qm , cf. [9]. For instance, the quality factor Qm can be derived graphically as the ratio of the resonance frequency f0 and the half-power bandwidth BW as indicated in Fig. 3. It is worth noting that if the upper curve for G (including the substrate parasitics) were used for this graphical method, the resulting Q would not be the mechanical Qm of the device, but rather a combination of the mechanical and electrical Q

2

4

m

Vdc C0

Vac

−a

a

x

3

10 Q m [−]

g0

−9

−zst

zd

10

b [10 Ns/m]

b

k

z

2

10 Fig. 2. Schematical cross-section of the electrostatically driven RF-MEMS switch from Fig. 1.

1.6

b

−6

x 10

1

10 −2 10

G, HRSi

1.4

G , B [ µ S]

1.2

−1

0

10

10

Pa [mbar]

1

10

Fig. 4. Measured pressure-dependence of the Qm factor and damping coefficient.

1

B, HRSi

0.8

The damping force Fd on the moving switch can be expressed as:

f ≅ f0

0.6 0.4

G, no parasitics

Fd =

BW

0.2 0 f3dB /f0 0.9994 0.9996 0.9998

Fig. 3.

Qm

1

f/f 0

b=

Measured admittance of the switch for frequencies around f0 .

factor. Repeated measurements with varying ambient pressure Pa yield the Qm factor as a function of pressure. This is depicted in Fig. 4. Qm is seen to saturate as the pressure drops below 10−1 mbar, which means that air damping is not the dominant loss-mechanism at low pressures. On the other hand, there is a clear correlation for pressures above 10−1 mbar. f0 and Qm are related to the lumped parameters m, k and b of the spring-mass-damper system from Fig. 2 according to:

f0

=

Qm

=

(2)

The damping coefficient b as defined in (1) and used in Fig. 4, is therefore equal to:

1.0002 1.0004

r k 1 2π m 2πf0 m b

bg z˙ (g0 + zst )3

(1)

Hence, the damping coefficient b can be derived from the measured Qm ’s, as shown by the circles in Fig. 4. It should be noted that the values of b that are presented in this figure, are valid only for the position z = zst of the membrane, while the harmonic excursions around z = zst are neglected. B. Geometry-dependent damping model An analytical model for the damping coefficient b of a MEMS switch with perforated membrane was proposed in [3].

bg · η(Pa , zst ) 3

(g0 + zst )

(3)

In the above expression, the geometry-dependence and the pressure-dependence of b are separated: bg only depends on the geometry whereas the factor h/(g0 +zst )3 brings the pressureand position-dependence into account. The value of bg is given by: b

= with

bskvor

=

brectangle

=

bskvor + brectangle 12A2t Ns2 π

Ah A2 − h2 2At 8A   t  Ah −0.25log − 0.375 At (Le − 0.6We ) We3 

(4)

where Ah is the cumulative area of the holes in the membrane, At is the membrane area that is enclosed by the holes and Ae is the area around the circumference of the membrane (more details on these quantities can be found in [3]). In (3), it is seen that the pressure-dependence of the damping coefficient is caused by the varying viscosity η of air. In [4], the following pressure-dependence of the effective viscosity η was proposed for the squeezed are film in an accelerometer: η (Pa , zst ) =

1 1 + 9.638Kn1.159

(5)

3

4

10

−9

z [ µ m]

10

2

10

1.5

1.5

1

1

0.5

0.5

Measured Calculated

1

10 −2 10

−1

10

0

Pa [mbar]

10

0 0

1

10

Fig. 5. Calculated and measured damping coefficient for the RF-MEMS switch shown in Fig. 1.

where the viscosity of air at standard conditions, η0 , is equal to 1.8 · 10−5 P a m/s. The Knudsen number Kn is the ratio of the mean free path λ to the height g0 + zst of the air gap. Fig. 5 shows a comparison between the damping coefficient b that was measured (Fig. 4) and the calculated values according to equations 3-5. Although the measured and calculated values are in rather good agreement, it should be emphasised that the values of b from Fig. 5 are valid only for the up-state position z = zst of the membrane. It will be shown in the next paragraph that a further modification of (5) is required in order to model the measured switching times correctly. This may be due to the fact that (5) was originally derived for the squeezefilm effect of a moving body without perforations, causing an over-estimation of the pressure-dependence of the damping coefficient. III. T RANSIENT EXCITATION A. Model Using the damping model that was described and validated above, a SimulinkTM model was implemented in order to solve the mechanical equation of motion for the RF-MEMS switch, at varying ambient pressures. In the subsequent text, the switching transient of the device is considered, rather than the harmonic vibration around a static deflection zst. In each time step of the Simulink model, the value of b is updated according to equation (3) with the current displacement z and pressure P . As an example, Fig. 6 shows the simulated closing and opening trajectories of the membrane for an ambient pressure Pa = 0.5 atm and a DC voltage Vdc = 15 V . The pull-in voltage of the switch is Vpi = 14 V . B. Measurements In order to validate the simulated closing trajectory, the response of the switch was measured using the setup shown in Fig. 7. A DC square wave and a 100MHz-signal are applied

Opening

2

2

3

b [10 Ns/m]

2.5

Closing

z [ µ m]

2.5

0 50

Time [ µ s]

100

0

50

100

150

Time [ µ s]

Fig. 6. Simulated closing and opening trajectories of a switch at Pa = 0.5 atm.

bias T Vac

MEMS

bias T

Vo

scope

Vdc

Fig. 7. Measurement setup used for determining the transient response of RF-MEMS switches.

to the switch using bias tees and RF wafer probes. As the capacitance of the switch varies during the switching transient, the transmission coefficient S12 of the switch varies and the measured voltage V0 changes accordingly. An oscilloscope is used to detect the output voltage V0 . Since the sampling rate of the scope is lower than the signal frequency, the envelope of the signal is measured. In order to derive the height z + g0 of the membrane from the measured voltage V0 , the S-parameters of the cables, bias tees and wafer probes were measured individually and Agilent’s ADS software was used to simulate the output voltage of the setup as a function of the switch capacitance. With this calibration curve (C, V0 ), the trajectory from Fig. 8 was extracted. Fig. 8 shows the measured and simulated trajectories of the membrane. The ambient pressure was Pa = 600 mbar and the applied bias voltage Vdc was 18V , roughly 25% above the pullin voltage Vpi of 14V . The measurement shows that the switch is fully closed after Ton = 160 while the simulated closing time is 170µs. When the membrane is in the up-position, the capacitance is very low and this results in significant measurement noise for small membrane displacements. The limited number of quantisation levels of the scope further increases this effect. When the ambient pressure Pa is varied, the closing time changes as depicted in Fig. 9. The dark line that interconnects the circles, represents the measured closing times. The lightest, dotted curve is the simulated closing time if expression (5)

4

3

−6

x 10

As mentioned above, this may be due to the fact that equation (5) was originally derived for the squeeze-film damping of an accelerometer mass without perforations. Since the perforations reduce the squeeze-film effect, this may be taken into account by modifying the exponent n of the Knudsen number Kn in equation (5). Choosing a value that is 30% lower than the initial value, yields the dashed line interconnecting the triangles in Fig. 9.

z+g 0 [ µ m]

2.5 2 1.5 1 0.5

IV. C ONCLUSIONS

Simulation Measurement

0 0

50

100

150

Time [ µs]

Fig. 8. Simulated and measured trajectories of the membrane during closing.

200 180

Ton [ µ s]

160

V. ACKNOWLEDGMENT

140

This work was carried out with the financial support of the European Commission, IST-project MEMS2TUNE (IST-200028231).

120 100

Measurement

80

R EFERENCES

Simulation 1 Simulation 2

60 40 0

The influence of air damping on the mechanical response of RF-MEMS switches is modelled and measured using two approaches. First, the air damping coefficient for the up-state of the device is extracted from measurements of the resonance frequency at varying ambient pressures. The measured values correspond well with theory. Next, this damping coefficient is used to simulate the transient response of the device to a DC voltage step input. Measurements show that the model over-estimates the pressure-dependence of the switching time. A modification to the effective viscosity of the air yields a much better agreement between measured and simulated closing times in a wide pressure interval.

200

400

600

Pa [mbar]

800

1000

Fig. 9. Simulated and measured switching times for varying ambient pressure.

is used for the effective viscosity. There is clearly a large difference between the measured and simulated closing times.

[1] R. K. Gupta et al, IEEE MEMS1997 26-30 Jan, 1997, Nagoya, Japan, pp. 290-294 [2] T. Veijola et al, IEEE MTT-S 3-7 June, 2002, Seattle, USA, pp. 12131216 [3] P.G. Steeneken et al, J. Micromech. Microeng. 15 (2005) 176-184 [4] T. Veijola et al, Sensors and Actuators A 48 (1995) 239-248 [5] T. Corman et al Sensors and Actuators A 66 (1998) 160-166 [6] J. De Coster et al, EMPC2005, 12-15 June, 2005, Brugge, Belgium, pp. 599-603. [7] J.T.M. van Beek et al, MRS fall meeting 1-5 Dec, 2003, Boston, Mat. Res. Soc. Symp. Proc. Vol.783, pp. B3.1.1-B3.1.12 [8] H. A. C. Tilmans, J. Micromech. Microeng. 6 (1996) 157-176 [9] H.A.C. Tilmans, Int. J. Mod. and Sim. 19 (1999) 312-321