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VHF enhancement of micromechanical resonator via structural modification
Sensors and Actuators A 96 (2002) 67±77
VHF enhancement of micromechanical resonator via structural modi®cation Y.C. Loke, K.M. Liew* Centre for Advanced Numerical Engineering Simulations, School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore Accepted 9 October 2001
Abstract For the CMOS compatible micromechanical resonators in VHF transceiver applications, it is paramount to have a high quality factor (Q-factor) and high stiffness as it directly in¯uences the dynamic range of the circuits. The resonator also needs to have a high natural frequency in order to reduce the effect of the external noise. A reduction in the size of the resonator may satisfy the above requirements. However, beyond a limit, this may increase its sensitivity to noise. In this paper, structural design improvements on an existing resonator have been carried out, and a prototype based on the new design has been fabricated and tested. The new design has demonstrated an improvement in both the natural frequency (typically 43%) and the Q-factor (typically 25%) as compared to the original design. The increase in natural frequency is due to the addition of reinforcements to the proof mass, and the improvement in Q-factor is due to a reduction in the squeeze ®lm damping coef®cient arising out of a higher driving frequency. # 2002 Elsevier Science B.V. All rights reserved. Keywords: MEMS; Oscillator; RF transceiver; Wireless communication
1. Introduction Recently, an alternative strategy for transceiver miniaturization has surfaced in which the high performance heterodyning architecture is retained, and miniaturization is achieved by replacing the prevailing off-chip, high quality factor (Q-factor), mechanical resonator components with IC-compatible, micromechanical versions. In the VHF range of applications, it is essential for a resonator to achieve high Q and high stiffness, as it is paramount for the communication grade resonators where the stiffness directly in¯uences the dynamic range of the circuits. However, for some large stiffness resonator such as the clamped±clamped beam [1,2], it often comes at the cost of increased anchor dissipation thus lowering the resonator Q-factor. The concept of a virtual free±free beam used by Kun et al. [3] considerably reduces the former problem. To further increase the frequency of a free±free beam, the structure size may have to be reduced. However, there is a limit for this reduction due to fabrication constraint, and also smaller *
Corresponding author. Present address: Nanyang Centre for Supercomputing and Visualisation, School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore. Tel.: 65-790-4076; fax: 65-790-6763. E-mail address: [email protected] (K.M. Liew).
structures are very sensitive to external noise. This chapter therefore discusses the extension of the high frequency range to a next higher level. Two designs of micromachined resonator have been studied. To modify the structural shape, a stiffener is added to the simple free±free resonator membrane in order to ensure an enhancement in the natural frequency. In addition to enhancement of natural frequency, this modi®ed structure displays a capability of reducing the squeeze ®lm damping effect, thus resulting in an improvement in the Q-factor of this micro-device. In general, the center frequency from this two modi®ed structure ranges from 90 to 130 MHz. The scope of this paper is to look into the resonator design, modi®cation of the structure and structure modeling aspects. This paper also presents the electrical interface between the mechanical structure and the applied voltage; it includes electrical equivalence circuit and concludes with the experimental results and discussions. 2. Resonator design Fig. 1a and b shows two schematic view of the resonator in an initial (unmodi®ed) design and a modi®ed design with a single stiffener added to it. The structure comprises a resonator membrane suspended by four torsional beams, each of which is anchored to the substrate. In the former, a single
0924-4247/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 4 2 4 7 ( 0 1 ) 0 0 7 5 6 - 7
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Fig. 1. (a) Schematic view of the resonator at an initial (unmodified) design. (b) Schematic view of the resonator with an addition stiffener (modified) design.
stiffener is added at the top surface of the resonator plate thus modifying the structural shape in order to enhance the natural frequency. A driving electrode runs underneath the center portion of the resonator structure. The applied ac drive voltage causes an alternating electrostatic force that actuates the resonator structure. The sensing current, which is induced by the sense capacitance, is converted to the frequency domain via a spectrum analyzer during the physical testing. Corresponding to the fundamental mode of vibration, the nodal point, i.e. the zero displacement point of the resonant structure is located. A support beam is then attached to this nodal point. The support beam is designed in such a way that it has a length equal to the quarter-wavelength corresponding to the desired resonant frequency. This is to isolate the free± free beam from the rigid anchor. Underneath the resonator are four dimples, strategically located along the zero displacement nodal point. This is to prevent contact between the resonator structure and the drive or sense electrode. An initial drive voltage will cause the dimple to sit on the substrate. This created a 250 nm gap between the resonator and the electrode, thus enhancing the sensitivity of the sense capacitance. Thereafter, the resonator will commence vibrating at the frequency of the applied ac source.
3. Resonator topology and analytical model 3.1. Equation of motion For the free undamped vibration of a beam, the equation based on Timoshenko Theory [4] is expressed as @4x @2x rA 2 0 (1) 4 @x @t where x is the deflection of the center line of the beam, E the Young's modulus of the material, I the cross-sectional moment of inertia, A the cross-sectional area and r is the density of the material. The mode shape for the free±free boundary condition is expressed as
EI
Zmode y cosh b cos b
x sinh b sin b
(2)
with x
cosh bLr sinh bLr
cos bLr sin bLr
(2a)
and b4
rA 2 o EI 0
(2b)
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where o0 is the angular natural frequency of the structural vibration and bLr 4:73 for the first mode. From the mode shape, the zero displacement points of the resonant structure are located at 0.22Lr and 0.78Lr where Lr is the membrane length. From Eq. (2b), the natural frequency can be expressed as 4:732 f0 2pL2r
s s EI 3:561 Et2 2 rA Lr 12r
(3)
where t is the thickness of the resonator membrane which is a fixed parameter due to the design rule of the fabrication process [5]. From the above equation, the only varying quantity is the length, Lr of the membrane, and hence the frequency of this simple structure varies only by the length and not the width of the resonator. Fig. 2 displays a curve pattern of the membrane length affecting the resonator frequency performance. Eq. (3), however, is only valid for this structure which has a simple rectangular crosssection. When the structure undergoes a shape modification, it not only alters the stiffness characteristic but also causes a corresponding change in the natural frequency. 3.2. Structural modification In the case of a modi®ed structure, the general problem that governs the structural modi®cation can be formulated by starting with the vibration characteristics of an initial design given as, K ln Mfn 0 where ln is an eigenvalue matrix, fn the corresponding mode shape vector, and K and M the stiffness and mass matrices, respectively. Using an approximate numerical analysis based on Raleigh method.
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It is found that the latter yield better estimate to the fundamental frequency of a system, which contain ¯exible elements and multiple degrees of freedom [6]. For an original structure with the natural frequency, on, and the associated mode shape, jn, by applying Raleigh equation, the modi®ed eigenvalue matrix gn is given by gn on Don 2
jTn K DKjn jTn M DMjn
(4)
Assuming Don /on to be small and after some simplification gives, Don jTn DKjn o2n jTn DMjn on 2jTn Kjn
(5)
This is an expression for the relative change in the natural frequency due to the structural modification that only involves the vibration characteristic of the initial structure. Eq. (5) shows that the mode shape of the initial structure provides sensitivity for a simple point mass or stiffness modification. In view of the constraints of fabrication process, there are limitations in designing a rather complex modi®ed structure. As the process uses three layers polysilicon surface micromachining, only two polysilicon layers can be used for the structural layer. The ®rst polysilicon layer is used as the electrode. The second layer comprises the four support beams and a single resonator membrane. The third layer can be stacked onto the second layer thus modifying the structural shape. This third layer is a short strip and term as a stiffener as it stiffened the otherwise unaltered resonator membrane. The process therefore controls the width dimension of the stiffener. Two modi®ed structures are designed and based on the
Fig. 2. Plot of frequency vs resonator length for the initial resonator design prior to structural modification.
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structure shows an improvement in the frequency by 55% from the initial design. 3.3. Support beam design As discussed in Section 2, four beams are ®rst located at the zero-dsplacement point of the resonator support the resonator. Recognizing that the vibrating resonator may undergo a torsional vibration with these support points, it is desirable for the latter to create a virtual zero impedance to the free±free resonator by suitably tuning the length of the support beam such that it can result in a matching system frequency close to the operating frequency of the resonator. With changes in the mode shape (due to the difference in the modi®ed structure), a more well de®ned theory need to be applied to ensure better convergence of the system frequency close to the operating frequency of the resonator. Using a torsional vibration studies on the support beam and coupled with the Timoshenko torsional studies on a rectangular cross-section [7], the length of the support beam is therefore obtained and expressed as s 1 Gg (6) Ls 4fn rJ where g is the torsional constant of the rectangular crosssection, G the shear modulus, r the density of the structure, fn the resonance frequency of the resonator and J is the polar moment of inertia based on either Timoshenko's formulation on the rectangular cross-section or the classical approach. The Timoshenko's formulation is expressed as ! 1 1 3 192 ws X 1 pnt JT w s t 1 tanh (7) 3 p5 t n1 n5 2ws where ws is the width of the support beam and t is the thickness. The classical expression for J would otherwise be written as Jc w s t Fig. 3. Finite element model depicting the first mode shape of the three resonators.
shape it is named as inverse-T and inverse-Pi designs. Fig. 3a shows the FE mode shape for the original design corresponding to the fundamental mode of frequency. Attached to it are four support beams located at the zero displacement node point. Fig. 3b and c shows the corresponding mode shapes for the inverse-T and inverse-Pi designs, respectively. Fig. 4 shows the analytical plot of natural frequency (Eq. (3)) against the variation in the width of the stiffener of the modi®ed structure. The two modi®ed structures display that higher frequency is obtained in the range 90± 130 MHz. The inverse-Pi structure has a higher frequency at 105±125 MHz as the stiffener width xa increases from 0.5 to 1.5 mm. As compared to its predecessor [3], whose performance frequency is in the range of 30±90 MHz, the modi®ed
w2s t2 12
(8)
The difference between the two emanates from the assumption of torsion in a plane. The classical theory Coulomb (1784) and Navier (1864) is based on the assumption that the cross-section of the bar will remain plane and undergone no warping during torsion. Timoshenko's formulation considers warping effects. To obtain a desired suspension beam length between the two approaches, Table 1 displays the ®nite element simulation results that compare the two methods. Second and third columns of the table show the natural frequency computed with free±free boundary condition without any physical support at the nodal (zero-displacement) points. This will be the desired or ideal natural frequency that we aim for with supports. Fourth and ®fth columns of the table deal with the actual natural frequencies with suspension beam support. Comparisons of third and fourth columns suggest that the
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Fig. 4. Comparison plot of the three resonators design depicting the significance increase in the frequency performance via structural modification.
resonating device with the beam length (Lclassical) based on the classical theory has a frequency 2% different from the desired value. However, the design with the beam length based on Timoshenko's formulation (LTimoshenko), yields a frequency that is only about 0.1773% different from the desired value, thus showing that the latter approach is more accurate and therefore is adopted for the design. 3.4. Squeeze film damping Viscous air damping has a dominating effect on the Q-factor of the resonator performance. For a plate under
transverse vibration, the dissipation mechanism is mainly from the effect of squeeze ®lm damping. The Q-factor of this microresonator is simply expressed as Q
Cc 2Csq
(9)
where p Cc 4MK ; Z L=2 MA rf2 dx; L=2
(8a) Z K
2 2 L=2 d f EI dx dx2 L=2
(8b)
Table 1 Vibration characteristics of the resonator design Resonator dimensions (Lr wr t in mm)
Frequency (MHz) (ideal scenario)
Frequency (MHz) (actual scenario in FEM results)a
Theoretical
Lclassical
Values in parentheses represent beam lengths (mm).
LTimoshenko
70
67.68
66.3 (43.86 mm)
67.56 (58.32 mm)
107
99.85
99.009 (29.73 mm)
99.61 (39.53 mm)
122.41
a
FEM
111.12
107.5 (26.71 mm)
111.0 (35.52 mm)
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Fig. 5. 3-D plot of plate base dimension against squeeze film damping.
in which A is the cross-sectional area of the resonator, M the mass, K the stiffness coefficient, f the mode shape of the vibration plate,and Csq is the squeeze film damping. Over the years, extensive research has been carried out on the
squeeze film damping effect on a vertically oscillating planar microstructure [8±11]. It is therefore not the concern of this paper to further analyze on this aspect. Based on Bleach [11], the expression for the squeeze film damping
Fig. 6. Plot of squeeze film damping vs variation in stiffener width xa.
Fig. 7. 3-D plot of driving force vs dc bias and ac drive voltage.
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Fig. 8. Plot of frequency ratio vs applied voltage.
coefficient between two parallel plates is given as Csq
64sPa B X m2 n=b2 p6 dod m;n odd mn2 m2 n=b2 2 s2 =p4
(9)
where B is the base area (Lr wr )m2, d the gap between the plate and the substrate, Pa the ambient pressure, b the aspect
ratio, s the squeeze number defined [11] as, s 12 mLr o0 = Pa d2 , m is the viscosity of air at 1:81 10 5 Ns/m (at 20 8C, 1 atm) and od is the resonance frequency of the resonator in rad/s. Unlike the conventional treatment of squeeze ®lm effect [8,9]. Bleach's equation takes into account the aspect ratio of the plate. Fig. 5, shows a 3-D plot of a plate base dimension
Fig. 9. Plot of dimple-down voltage Vd vs support beam length Ls.
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against the squeeze ®lm damping with the highest damping displayed at a larger plate area. The structurally modi®ed resonator has a higher natural frequency and hence the damping effect reduces as the width of the stiffener increases (Fig. 6.). This results in an increase in the Q-factor, which is desirable for the system performances. 4. Electrical interface 4.1. Principle of vibration sensing The performance of this resonator is dependent on the voltage applied. The electrostatic stiffness due to the applied voltage is an important factor as it modi®es the actual frequency of the resonating structure. The working principle of driving the resonator is based on that of the conductive parallel plate capacitor. The driving electrostatic force fd is expressed as follows: 1 @Cd fd Vp Vi 2 2 @x 2 1 V Vi2 @Cd 2 i Vp 2Vp Vi cos ot cos 2ot 2 2 2 @x
Hence s Vp2 ews wr f00 1 f0 km d 3
(15)
where, f00 is the altered center frequency due to electrical spring effect. Fig. 8 shows a plot of frequency ratio f00 =f0 versus the applied dc voltage. It is observed that for an applied voltage of less than 50 V, the electrical spring has a less significant effect on frequency. Thus, for an applied voltage that does not exceed 50 V, the effect of an electrostatic stiffness can be ignored. The choice of the voltage applied is very crucial in actual operation. At ®rst, suf®cient voltage must be applied for the dimple to ®rst touch the substrate; this pull down voltage is term as Vd. At this point, an ac drive source will provide the
(10)
where Vp and Vi are the dc bias voltages, Vpl is the difference between the two bias, i.e. V pl V p V i , Vi is an ac drive voltage, i.e. V i vi cos ot and Cd is the drive capacitance. When the frequency of the excitation voltage is at the resonance frequency of the beam. The dc and second harmonic component do not receive Q-amplification, and thus can be neglected. Hence, Eq. (10) becomes @Cd (11) @x With an appropriate manipulation and simplification on the differential capacitance per displacement, the driving force is expressed as fd
Vp vi cos ot
C0 C0 vi cos ot Vp2 2 sin otx (12) d d where C0 is the static capacitance between the drive electrode and the membrane. Eq. (12) suggests that the applied voltage has a strong influence on the driving forces. Fig. 7 displays a plot of driving force against the dc bias and ac drive voltage. An increase in the dc voltage results in a gradual increase on the driving force. However, the force shows a reduction as the ac drive increases. The induced electrostatic spring constant is de®ned as
f d Vp
@fd C0 Vp2 2 (13) @x d It is not preferable to have a higher electrical spring as it opposes its mechanical counterpart and alters the center frequency of the resonator as shown in the next equation, r rs Vp2 ews wr 1 km ke km (14) f00 1 2p m m km d 3
Ke
Fig. 10. Scanning electron micrograph of an inverse-T resonator.
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Table 2 Parameters of resonator prototype
Fig. 11. A simplified diagram of experimental set-up.
resonating frequency for the structure to resonate. The expression for the dimple-down voltage Vd takes on the form, s 8 Ews t3 d 3 Vd (16) 27 ewr we L3s Apparently, the thickness of the membrane and the gap will have great influence on the dimple-down voltage. However, these are two parameters that are determined by the design rule of the fabrication process. On the other hand, when the length of the support beam varies, a different effect will result as shown in Fig. 9. A longer support length will have a lower dimple-down voltage. A support beam at twice or triple its time of a quarter wavelength may still be feasible, but will reduce the dimple-down voltage, as the suspension beam stiffness reduces. This may result in the device that only sees the support beam vibration as its stiffness produces
Parameters
Simple resonator membrane (70 MHz)
Inverse-T structure (100 MHz)
Initial gap, d (mm) Dimple height (mm) Dimple-down voltage, Vd (V) Applied voltage, Vp (V) Catastrophic pull-down voltage, Vc (V) Young's modulus, E (GPa) Poisson ratio, n Resonator length, Lr (mm) Resonator width, wr (mm) Stiffener width, xa (mm) Support beam length, Ls (mm) Support beam width, ws (mm)
2 0.75 20 25 300 150 0.29 15.40 6 ± 58.32 2
2 0.75 20 25 300 150 0.29 15.40 6 2 39.53 2
a frequency that supersedes the actual resonating membrane vibration. When the electrostatic stiffness ke is equal to the mechanical stiffness of the resonator km, catastrophic pull-down voltage will occur. This happened from a point when the dimple are down and with further downward movement from the resonator membrane towards the electrode will cause the membrane to be critically deformed and destroy the whole system. The catastrophic pull-down voltage, Vc, is expressed as s d 3 km Vc (17) ewr we Generally, it is desired that the applied voltage Vp to satisfy the relation, Vc > Vp > Vd
Fig. 12. Measured spectrum of a simple resonator membrane (70 MHz).
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Fig. 13. Measured spectrum of inverse-T structure.
5. Experimental testing and results The prototype under test is fabricated from a threelayer polysilicon surface micromachining process. Fig. 10 shows the scanning electron micrography of an inverse-T prototype used for the testing. This test specimen is mounted on a 12,000-series cascade microtech microchamber probe station. Measurement and data reading are done on a Hewlett Packard HP 8510B Network Analyzer and later link to a window style interface on HP 85190A; this is a IC CAP, UNIX-base device modeling program. Fig. 11 shows a simpli®ed diagram of the experimental set-up.
In the cascade probe station, the proper probe set-up is such that, three probes will be situated at their respective electric pads. The ®rst probe, having a ®xed dc bias voltage denoted by V5, of 5 V touches the pad that connects to one of the anchor of the support beam. The second probe, which consist of a ®xed ac drive source, Vi, of 4.472 V peak to peak and a tunable dc bias source, Vp, touches the pad that link to the drive electrode. Finally, from the sense electrode, the third probe touches the former and interfaces with a network analyzer. The testing is performed with an input parameter at a tunable dc voltage of 15 V and an ac drive voltage of 4.472 V peak to peak. Table 2, shows the parameters of the resonator prototype used for the testing.
Fig. 14. Comparison plot of FEM with experimental results.
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The ®rst spectrum of Fig. 12 shows that for a simple free± free beam structure, it resonates at a frequency of 70 MHz. The modi®ed structure of an inverse-T with a stiffener width of xa 1:2 mm display a resonance at 100 MHz as shown in Figs. 13 and 14. These two experimental results show a good agreement with the FEM simulation and the analytical result. By superimposing on the experimental plot of inverse-T structure, the experimental Q-factor of this microstructure can be determined as the ratio of the resonance frequency with the 3 dB bandwidth. Thus the experimental Q-factor value is measured to be 263 at ambient condition. For comparison purposes, a ®nite element harmonic analysis (ANSYS [12]) using a squeeze ®lm damping coef®cient calculates following references [11] has been carried out. High Q will obviously be expected at a vacuum condition. Under an ambient environment, this device exhibits a Q of 263, which is desirable in the aspect of frequency stability and accuracy capability of a resonator.
6. Conclusions Micromechanical resonator offers an alternative set of strategies for transceiver miniaturization and improvement. In this resonator design, the basic idea of having quarterwavelength torsional supports attached to nodal points and electrically activated dimple determine the electrode-toresonator gap. The resonator has demonstrated that, under a free±free (or virtually supportless) vibrating condition, it is possible to remove the anchor dissipation and processing problems that presently hinder the clamped±clamped beam counterpart [13]. In addition, the structurally modi®ed resonators comprising a single stiffener and a double stiffener have demonstrated a signi®cant improvement in the natural frequency from 70 to 100 MHz. For the single stiffener design (approximately 43% increase). And for the double stiffener design, the frequency increases to 111 MHz with a 55% improvement. The impedance implication by the support beam is further reinforced when Timoshenko's formulation is adopted for the design length of the support beam to further reduce the impedance that deters the actual resonant frequency from performing. Squeeze ®lm damping has a considerable effect on the total device Q-factor. It is noticed that as the width of the stiffener increases, there is a reduction in the damping value which is desirable for a higher Q-factor of this resonator device. Under ambient condition, this prototype has been tested and the Q-factor obtained from testing is compared with the analytical result. For an inverse-T structure, it displays a Q-factor of 263, which is quite close to its analytical counterpart, 233.
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Acknowledgements Both authors gratefully acknowledge the advice and support of Dr. A.Q. Liu of the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. References [1] C.T.-C. Nguyen, Micromachining technologies for miniaturized communication devices, in: Proceedings of the SPIE on Micromachining and Microfabrication, Santa Clara, California, 20±22 September 1998, pp. 24±38. [2] A.-C. Wong, H. Ding, C.T.-C. Nguyen, Micromechanical mixer and filters, technical digest, in: Proceedings of the IEEE International Devices Meeting, San Franscisco, California, 6±9 December 1998, pp. 471±474. [3] K. Wang, Y. Yu, A.-C. Wong, C.T.-C. Nguyen, VHF free±free beam high Q micromechanical resonators, technical digest, in: Proceedings of the 12th International IEEE Micro Electro Mechanical Systems Conference, Orlando, FL, 17±21 January 1999, pp. 453±458. [4] S. Timoshenko, D.H. Young, W. Weaver, JR, Vibration Problems in Engineering, Fourth Edition, John Wiley Sons, 1974. [5] MCNC/MUMPS, Consolidated Micromechanical Element Library (CAMEL), Camel User Guides, Website: http://www.memsrus.com/ cronos/. [6] W.T. Thomson, Theory of Vibration with Application, Prentice Hall, New Jersey, 1981. [7] S. Timoshenko, J.N. Goodier, Theory of Elasticity, McGraw-Hill, New York, 1951. [8] G.K. Fedder, Simulation of Microelectromechanical Systems, Ph.D. Thesis, University of California, Berkley, 1994. [9] Y.-H. Cho, Viscous damping model for laterally oscillating microstructures, J. Microlelectromech. Syst. 3 (2) (1994) 81±86. [10] T. Veijola, H. Kuisma, J. LahdenperaÈ, T. RyhaÈnen, Equivalent-circuit model of the squeezed gas film in a silicon accelerometer, Sens. Actuat. A48 (1995) 239±248. [11] J.J. Bleach, On isothermal squeeze films, Trans. ASME: J. Lubrication Technol. 105 (1983) 615±620. [12] ANSYS1, Release 5.6. [13] J.-H. Huang, K.M. Liew, C.H. Wong, S. Rajendran, M.J. Tan, A.Q. Liu, Mechanical design and optimization of capacitive micromachined switch, Sens. Actuat. 93 (3) (2001) 273±285.
Biographies Y.C. Loke holds a Masters degree in Engineering from Nanyang Technological University (NTU), Singapore and Bachelor of Engineering with Honours from University of Manchester Institute of Science and Technology, United Kingdom in 2001 and 1998, respectively. He is currently a graduate student with NTU working on his Doctorate program. K.M. Liew joined the Nanyang Technological University as Lecturer in 1991, and is currently a tenured full Professor at the School of Mechanical and Production Engineering. He is the founding director of both Centre for Advanced Numerical Engineering Simulations (CANES), and Nanyang Centre for Supercomputing and Visualisation (NCSV).
VHF enhancement of micromechanical resonator via structural modi®cation Y.C. Loke, K.M. Liew* Centre for Advanced Numerical Engineering Simulations, School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore Accepted 9 October 2001
Abstract For the CMOS compatible micromechanical resonators in VHF transceiver applications, it is paramount to have a high quality factor (Q-factor) and high stiffness as it directly in¯uences the dynamic range of the circuits. The resonator also needs to have a high natural frequency in order to reduce the effect of the external noise. A reduction in the size of the resonator may satisfy the above requirements. However, beyond a limit, this may increase its sensitivity to noise. In this paper, structural design improvements on an existing resonator have been carried out, and a prototype based on the new design has been fabricated and tested. The new design has demonstrated an improvement in both the natural frequency (typically 43%) and the Q-factor (typically 25%) as compared to the original design. The increase in natural frequency is due to the addition of reinforcements to the proof mass, and the improvement in Q-factor is due to a reduction in the squeeze ®lm damping coef®cient arising out of a higher driving frequency. # 2002 Elsevier Science B.V. All rights reserved. Keywords: MEMS; Oscillator; RF transceiver; Wireless communication
1. Introduction Recently, an alternative strategy for transceiver miniaturization has surfaced in which the high performance heterodyning architecture is retained, and miniaturization is achieved by replacing the prevailing off-chip, high quality factor (Q-factor), mechanical resonator components with IC-compatible, micromechanical versions. In the VHF range of applications, it is essential for a resonator to achieve high Q and high stiffness, as it is paramount for the communication grade resonators where the stiffness directly in¯uences the dynamic range of the circuits. However, for some large stiffness resonator such as the clamped±clamped beam [1,2], it often comes at the cost of increased anchor dissipation thus lowering the resonator Q-factor. The concept of a virtual free±free beam used by Kun et al. [3] considerably reduces the former problem. To further increase the frequency of a free±free beam, the structure size may have to be reduced. However, there is a limit for this reduction due to fabrication constraint, and also smaller *
Corresponding author. Present address: Nanyang Centre for Supercomputing and Visualisation, School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore. Tel.: 65-790-4076; fax: 65-790-6763. E-mail address: [email protected] (K.M. Liew).
structures are very sensitive to external noise. This chapter therefore discusses the extension of the high frequency range to a next higher level. Two designs of micromachined resonator have been studied. To modify the structural shape, a stiffener is added to the simple free±free resonator membrane in order to ensure an enhancement in the natural frequency. In addition to enhancement of natural frequency, this modi®ed structure displays a capability of reducing the squeeze ®lm damping effect, thus resulting in an improvement in the Q-factor of this micro-device. In general, the center frequency from this two modi®ed structure ranges from 90 to 130 MHz. The scope of this paper is to look into the resonator design, modi®cation of the structure and structure modeling aspects. This paper also presents the electrical interface between the mechanical structure and the applied voltage; it includes electrical equivalence circuit and concludes with the experimental results and discussions. 2. Resonator design Fig. 1a and b shows two schematic view of the resonator in an initial (unmodi®ed) design and a modi®ed design with a single stiffener added to it. The structure comprises a resonator membrane suspended by four torsional beams, each of which is anchored to the substrate. In the former, a single
0924-4247/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 4 2 4 7 ( 0 1 ) 0 0 7 5 6 - 7
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Fig. 1. (a) Schematic view of the resonator at an initial (unmodified) design. (b) Schematic view of the resonator with an addition stiffener (modified) design.
stiffener is added at the top surface of the resonator plate thus modifying the structural shape in order to enhance the natural frequency. A driving electrode runs underneath the center portion of the resonator structure. The applied ac drive voltage causes an alternating electrostatic force that actuates the resonator structure. The sensing current, which is induced by the sense capacitance, is converted to the frequency domain via a spectrum analyzer during the physical testing. Corresponding to the fundamental mode of vibration, the nodal point, i.e. the zero displacement point of the resonant structure is located. A support beam is then attached to this nodal point. The support beam is designed in such a way that it has a length equal to the quarter-wavelength corresponding to the desired resonant frequency. This is to isolate the free± free beam from the rigid anchor. Underneath the resonator are four dimples, strategically located along the zero displacement nodal point. This is to prevent contact between the resonator structure and the drive or sense electrode. An initial drive voltage will cause the dimple to sit on the substrate. This created a 250 nm gap between the resonator and the electrode, thus enhancing the sensitivity of the sense capacitance. Thereafter, the resonator will commence vibrating at the frequency of the applied ac source.
3. Resonator topology and analytical model 3.1. Equation of motion For the free undamped vibration of a beam, the equation based on Timoshenko Theory [4] is expressed as @4x @2x rA 2 0 (1) 4 @x @t where x is the deflection of the center line of the beam, E the Young's modulus of the material, I the cross-sectional moment of inertia, A the cross-sectional area and r is the density of the material. The mode shape for the free±free boundary condition is expressed as
EI
Zmode y cosh b cos b
x sinh b sin b
(2)
with x
cosh bLr sinh bLr
cos bLr sin bLr
(2a)
and b4
rA 2 o EI 0
(2b)
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where o0 is the angular natural frequency of the structural vibration and bLr 4:73 for the first mode. From the mode shape, the zero displacement points of the resonant structure are located at 0.22Lr and 0.78Lr where Lr is the membrane length. From Eq. (2b), the natural frequency can be expressed as 4:732 f0 2pL2r
s s EI 3:561 Et2 2 rA Lr 12r
(3)
where t is the thickness of the resonator membrane which is a fixed parameter due to the design rule of the fabrication process [5]. From the above equation, the only varying quantity is the length, Lr of the membrane, and hence the frequency of this simple structure varies only by the length and not the width of the resonator. Fig. 2 displays a curve pattern of the membrane length affecting the resonator frequency performance. Eq. (3), however, is only valid for this structure which has a simple rectangular crosssection. When the structure undergoes a shape modification, it not only alters the stiffness characteristic but also causes a corresponding change in the natural frequency. 3.2. Structural modification In the case of a modi®ed structure, the general problem that governs the structural modi®cation can be formulated by starting with the vibration characteristics of an initial design given as, K ln Mfn 0 where ln is an eigenvalue matrix, fn the corresponding mode shape vector, and K and M the stiffness and mass matrices, respectively. Using an approximate numerical analysis based on Raleigh method.
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It is found that the latter yield better estimate to the fundamental frequency of a system, which contain ¯exible elements and multiple degrees of freedom [6]. For an original structure with the natural frequency, on, and the associated mode shape, jn, by applying Raleigh equation, the modi®ed eigenvalue matrix gn is given by gn on Don 2
jTn K DKjn jTn M DMjn
(4)
Assuming Don /on to be small and after some simplification gives, Don jTn DKjn o2n jTn DMjn on 2jTn Kjn
(5)
This is an expression for the relative change in the natural frequency due to the structural modification that only involves the vibration characteristic of the initial structure. Eq. (5) shows that the mode shape of the initial structure provides sensitivity for a simple point mass or stiffness modification. In view of the constraints of fabrication process, there are limitations in designing a rather complex modi®ed structure. As the process uses three layers polysilicon surface micromachining, only two polysilicon layers can be used for the structural layer. The ®rst polysilicon layer is used as the electrode. The second layer comprises the four support beams and a single resonator membrane. The third layer can be stacked onto the second layer thus modifying the structural shape. This third layer is a short strip and term as a stiffener as it stiffened the otherwise unaltered resonator membrane. The process therefore controls the width dimension of the stiffener. Two modi®ed structures are designed and based on the
Fig. 2. Plot of frequency vs resonator length for the initial resonator design prior to structural modification.
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structure shows an improvement in the frequency by 55% from the initial design. 3.3. Support beam design As discussed in Section 2, four beams are ®rst located at the zero-dsplacement point of the resonator support the resonator. Recognizing that the vibrating resonator may undergo a torsional vibration with these support points, it is desirable for the latter to create a virtual zero impedance to the free±free resonator by suitably tuning the length of the support beam such that it can result in a matching system frequency close to the operating frequency of the resonator. With changes in the mode shape (due to the difference in the modi®ed structure), a more well de®ned theory need to be applied to ensure better convergence of the system frequency close to the operating frequency of the resonator. Using a torsional vibration studies on the support beam and coupled with the Timoshenko torsional studies on a rectangular cross-section [7], the length of the support beam is therefore obtained and expressed as s 1 Gg (6) Ls 4fn rJ where g is the torsional constant of the rectangular crosssection, G the shear modulus, r the density of the structure, fn the resonance frequency of the resonator and J is the polar moment of inertia based on either Timoshenko's formulation on the rectangular cross-section or the classical approach. The Timoshenko's formulation is expressed as ! 1 1 3 192 ws X 1 pnt JT w s t 1 tanh (7) 3 p5 t n1 n5 2ws where ws is the width of the support beam and t is the thickness. The classical expression for J would otherwise be written as Jc w s t Fig. 3. Finite element model depicting the first mode shape of the three resonators.
shape it is named as inverse-T and inverse-Pi designs. Fig. 3a shows the FE mode shape for the original design corresponding to the fundamental mode of frequency. Attached to it are four support beams located at the zero displacement node point. Fig. 3b and c shows the corresponding mode shapes for the inverse-T and inverse-Pi designs, respectively. Fig. 4 shows the analytical plot of natural frequency (Eq. (3)) against the variation in the width of the stiffener of the modi®ed structure. The two modi®ed structures display that higher frequency is obtained in the range 90± 130 MHz. The inverse-Pi structure has a higher frequency at 105±125 MHz as the stiffener width xa increases from 0.5 to 1.5 mm. As compared to its predecessor [3], whose performance frequency is in the range of 30±90 MHz, the modi®ed
w2s t2 12
(8)
The difference between the two emanates from the assumption of torsion in a plane. The classical theory Coulomb (1784) and Navier (1864) is based on the assumption that the cross-section of the bar will remain plane and undergone no warping during torsion. Timoshenko's formulation considers warping effects. To obtain a desired suspension beam length between the two approaches, Table 1 displays the ®nite element simulation results that compare the two methods. Second and third columns of the table show the natural frequency computed with free±free boundary condition without any physical support at the nodal (zero-displacement) points. This will be the desired or ideal natural frequency that we aim for with supports. Fourth and ®fth columns of the table deal with the actual natural frequencies with suspension beam support. Comparisons of third and fourth columns suggest that the
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Fig. 4. Comparison plot of the three resonators design depicting the significance increase in the frequency performance via structural modification.
resonating device with the beam length (Lclassical) based on the classical theory has a frequency 2% different from the desired value. However, the design with the beam length based on Timoshenko's formulation (LTimoshenko), yields a frequency that is only about 0.1773% different from the desired value, thus showing that the latter approach is more accurate and therefore is adopted for the design. 3.4. Squeeze film damping Viscous air damping has a dominating effect on the Q-factor of the resonator performance. For a plate under
transverse vibration, the dissipation mechanism is mainly from the effect of squeeze ®lm damping. The Q-factor of this microresonator is simply expressed as Q
Cc 2Csq
(9)
where p Cc 4MK ; Z L=2 MA rf2 dx; L=2
(8a) Z K
2 2 L=2 d f EI dx dx2 L=2
(8b)
Table 1 Vibration characteristics of the resonator design Resonator dimensions (Lr wr t in mm)
Frequency (MHz) (ideal scenario)
Frequency (MHz) (actual scenario in FEM results)a
Theoretical
Lclassical
Values in parentheses represent beam lengths (mm).
LTimoshenko
70
67.68
66.3 (43.86 mm)
67.56 (58.32 mm)
107
99.85
99.009 (29.73 mm)
99.61 (39.53 mm)
122.41
a
FEM
111.12
107.5 (26.71 mm)
111.0 (35.52 mm)
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Fig. 5. 3-D plot of plate base dimension against squeeze film damping.
in which A is the cross-sectional area of the resonator, M the mass, K the stiffness coefficient, f the mode shape of the vibration plate,and Csq is the squeeze film damping. Over the years, extensive research has been carried out on the
squeeze film damping effect on a vertically oscillating planar microstructure [8±11]. It is therefore not the concern of this paper to further analyze on this aspect. Based on Bleach [11], the expression for the squeeze film damping
Fig. 6. Plot of squeeze film damping vs variation in stiffener width xa.
Fig. 7. 3-D plot of driving force vs dc bias and ac drive voltage.
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Fig. 8. Plot of frequency ratio vs applied voltage.
coefficient between two parallel plates is given as Csq
64sPa B X m2 n=b2 p6 dod m;n odd mn2 m2 n=b2 2 s2 =p4
(9)
where B is the base area (Lr wr )m2, d the gap between the plate and the substrate, Pa the ambient pressure, b the aspect
ratio, s the squeeze number defined [11] as, s 12 mLr o0 = Pa d2 , m is the viscosity of air at 1:81 10 5 Ns/m (at 20 8C, 1 atm) and od is the resonance frequency of the resonator in rad/s. Unlike the conventional treatment of squeeze ®lm effect [8,9]. Bleach's equation takes into account the aspect ratio of the plate. Fig. 5, shows a 3-D plot of a plate base dimension
Fig. 9. Plot of dimple-down voltage Vd vs support beam length Ls.
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against the squeeze ®lm damping with the highest damping displayed at a larger plate area. The structurally modi®ed resonator has a higher natural frequency and hence the damping effect reduces as the width of the stiffener increases (Fig. 6.). This results in an increase in the Q-factor, which is desirable for the system performances. 4. Electrical interface 4.1. Principle of vibration sensing The performance of this resonator is dependent on the voltage applied. The electrostatic stiffness due to the applied voltage is an important factor as it modi®es the actual frequency of the resonating structure. The working principle of driving the resonator is based on that of the conductive parallel plate capacitor. The driving electrostatic force fd is expressed as follows: 1 @Cd fd Vp Vi 2 2 @x 2 1 V Vi2 @Cd 2 i Vp 2Vp Vi cos ot cos 2ot 2 2 2 @x
Hence s Vp2 ews wr f00 1 f0 km d 3
(15)
where, f00 is the altered center frequency due to electrical spring effect. Fig. 8 shows a plot of frequency ratio f00 =f0 versus the applied dc voltage. It is observed that for an applied voltage of less than 50 V, the electrical spring has a less significant effect on frequency. Thus, for an applied voltage that does not exceed 50 V, the effect of an electrostatic stiffness can be ignored. The choice of the voltage applied is very crucial in actual operation. At ®rst, suf®cient voltage must be applied for the dimple to ®rst touch the substrate; this pull down voltage is term as Vd. At this point, an ac drive source will provide the
(10)
where Vp and Vi are the dc bias voltages, Vpl is the difference between the two bias, i.e. V pl V p V i , Vi is an ac drive voltage, i.e. V i vi cos ot and Cd is the drive capacitance. When the frequency of the excitation voltage is at the resonance frequency of the beam. The dc and second harmonic component do not receive Q-amplification, and thus can be neglected. Hence, Eq. (10) becomes @Cd (11) @x With an appropriate manipulation and simplification on the differential capacitance per displacement, the driving force is expressed as fd
Vp vi cos ot
C0 C0 vi cos ot Vp2 2 sin otx (12) d d where C0 is the static capacitance between the drive electrode and the membrane. Eq. (12) suggests that the applied voltage has a strong influence on the driving forces. Fig. 7 displays a plot of driving force against the dc bias and ac drive voltage. An increase in the dc voltage results in a gradual increase on the driving force. However, the force shows a reduction as the ac drive increases. The induced electrostatic spring constant is de®ned as
f d Vp
@fd C0 Vp2 2 (13) @x d It is not preferable to have a higher electrical spring as it opposes its mechanical counterpart and alters the center frequency of the resonator as shown in the next equation, r rs Vp2 ews wr 1 km ke km (14) f00 1 2p m m km d 3
Ke
Fig. 10. Scanning electron micrograph of an inverse-T resonator.
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Table 2 Parameters of resonator prototype
Fig. 11. A simplified diagram of experimental set-up.
resonating frequency for the structure to resonate. The expression for the dimple-down voltage Vd takes on the form, s 8 Ews t3 d 3 Vd (16) 27 ewr we L3s Apparently, the thickness of the membrane and the gap will have great influence on the dimple-down voltage. However, these are two parameters that are determined by the design rule of the fabrication process. On the other hand, when the length of the support beam varies, a different effect will result as shown in Fig. 9. A longer support length will have a lower dimple-down voltage. A support beam at twice or triple its time of a quarter wavelength may still be feasible, but will reduce the dimple-down voltage, as the suspension beam stiffness reduces. This may result in the device that only sees the support beam vibration as its stiffness produces
Parameters
Simple resonator membrane (70 MHz)
Inverse-T structure (100 MHz)
Initial gap, d (mm) Dimple height (mm) Dimple-down voltage, Vd (V) Applied voltage, Vp (V) Catastrophic pull-down voltage, Vc (V) Young's modulus, E (GPa) Poisson ratio, n Resonator length, Lr (mm) Resonator width, wr (mm) Stiffener width, xa (mm) Support beam length, Ls (mm) Support beam width, ws (mm)
2 0.75 20 25 300 150 0.29 15.40 6 ± 58.32 2
2 0.75 20 25 300 150 0.29 15.40 6 2 39.53 2
a frequency that supersedes the actual resonating membrane vibration. When the electrostatic stiffness ke is equal to the mechanical stiffness of the resonator km, catastrophic pull-down voltage will occur. This happened from a point when the dimple are down and with further downward movement from the resonator membrane towards the electrode will cause the membrane to be critically deformed and destroy the whole system. The catastrophic pull-down voltage, Vc, is expressed as s d 3 km Vc (17) ewr we Generally, it is desired that the applied voltage Vp to satisfy the relation, Vc > Vp > Vd
Fig. 12. Measured spectrum of a simple resonator membrane (70 MHz).
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Fig. 13. Measured spectrum of inverse-T structure.
5. Experimental testing and results The prototype under test is fabricated from a threelayer polysilicon surface micromachining process. Fig. 10 shows the scanning electron micrography of an inverse-T prototype used for the testing. This test specimen is mounted on a 12,000-series cascade microtech microchamber probe station. Measurement and data reading are done on a Hewlett Packard HP 8510B Network Analyzer and later link to a window style interface on HP 85190A; this is a IC CAP, UNIX-base device modeling program. Fig. 11 shows a simpli®ed diagram of the experimental set-up.
In the cascade probe station, the proper probe set-up is such that, three probes will be situated at their respective electric pads. The ®rst probe, having a ®xed dc bias voltage denoted by V5, of 5 V touches the pad that connects to one of the anchor of the support beam. The second probe, which consist of a ®xed ac drive source, Vi, of 4.472 V peak to peak and a tunable dc bias source, Vp, touches the pad that link to the drive electrode. Finally, from the sense electrode, the third probe touches the former and interfaces with a network analyzer. The testing is performed with an input parameter at a tunable dc voltage of 15 V and an ac drive voltage of 4.472 V peak to peak. Table 2, shows the parameters of the resonator prototype used for the testing.
Fig. 14. Comparison plot of FEM with experimental results.
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The ®rst spectrum of Fig. 12 shows that for a simple free± free beam structure, it resonates at a frequency of 70 MHz. The modi®ed structure of an inverse-T with a stiffener width of xa 1:2 mm display a resonance at 100 MHz as shown in Figs. 13 and 14. These two experimental results show a good agreement with the FEM simulation and the analytical result. By superimposing on the experimental plot of inverse-T structure, the experimental Q-factor of this microstructure can be determined as the ratio of the resonance frequency with the 3 dB bandwidth. Thus the experimental Q-factor value is measured to be 263 at ambient condition. For comparison purposes, a ®nite element harmonic analysis (ANSYS [12]) using a squeeze ®lm damping coef®cient calculates following references [11] has been carried out. High Q will obviously be expected at a vacuum condition. Under an ambient environment, this device exhibits a Q of 263, which is desirable in the aspect of frequency stability and accuracy capability of a resonator.
6. Conclusions Micromechanical resonator offers an alternative set of strategies for transceiver miniaturization and improvement. In this resonator design, the basic idea of having quarterwavelength torsional supports attached to nodal points and electrically activated dimple determine the electrode-toresonator gap. The resonator has demonstrated that, under a free±free (or virtually supportless) vibrating condition, it is possible to remove the anchor dissipation and processing problems that presently hinder the clamped±clamped beam counterpart [13]. In addition, the structurally modi®ed resonators comprising a single stiffener and a double stiffener have demonstrated a signi®cant improvement in the natural frequency from 70 to 100 MHz. For the single stiffener design (approximately 43% increase). And for the double stiffener design, the frequency increases to 111 MHz with a 55% improvement. The impedance implication by the support beam is further reinforced when Timoshenko's formulation is adopted for the design length of the support beam to further reduce the impedance that deters the actual resonant frequency from performing. Squeeze ®lm damping has a considerable effect on the total device Q-factor. It is noticed that as the width of the stiffener increases, there is a reduction in the damping value which is desirable for a higher Q-factor of this resonator device. Under ambient condition, this prototype has been tested and the Q-factor obtained from testing is compared with the analytical result. For an inverse-T structure, it displays a Q-factor of 263, which is quite close to its analytical counterpart, 233.
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Acknowledgements Both authors gratefully acknowledge the advice and support of Dr. A.Q. Liu of the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. References [1] C.T.-C. Nguyen, Micromachining technologies for miniaturized communication devices, in: Proceedings of the SPIE on Micromachining and Microfabrication, Santa Clara, California, 20±22 September 1998, pp. 24±38. [2] A.-C. Wong, H. Ding, C.T.-C. Nguyen, Micromechanical mixer and filters, technical digest, in: Proceedings of the IEEE International Devices Meeting, San Franscisco, California, 6±9 December 1998, pp. 471±474. [3] K. Wang, Y. Yu, A.-C. Wong, C.T.-C. Nguyen, VHF free±free beam high Q micromechanical resonators, technical digest, in: Proceedings of the 12th International IEEE Micro Electro Mechanical Systems Conference, Orlando, FL, 17±21 January 1999, pp. 453±458. [4] S. Timoshenko, D.H. Young, W. Weaver, JR, Vibration Problems in Engineering, Fourth Edition, John Wiley Sons, 1974. [5] MCNC/MUMPS, Consolidated Micromechanical Element Library (CAMEL), Camel User Guides, Website: http://www.memsrus.com/ cronos/. [6] W.T. Thomson, Theory of Vibration with Application, Prentice Hall, New Jersey, 1981. [7] S. Timoshenko, J.N. Goodier, Theory of Elasticity, McGraw-Hill, New York, 1951. [8] G.K. Fedder, Simulation of Microelectromechanical Systems, Ph.D. Thesis, University of California, Berkley, 1994. [9] Y.-H. Cho, Viscous damping model for laterally oscillating microstructures, J. Microlelectromech. Syst. 3 (2) (1994) 81±86. [10] T. Veijola, H. Kuisma, J. LahdenperaÈ, T. RyhaÈnen, Equivalent-circuit model of the squeezed gas film in a silicon accelerometer, Sens. Actuat. A48 (1995) 239±248. [11] J.J. Bleach, On isothermal squeeze films, Trans. ASME: J. Lubrication Technol. 105 (1983) 615±620. [12] ANSYS1, Release 5.6. [13] J.-H. Huang, K.M. Liew, C.H. Wong, S. Rajendran, M.J. Tan, A.Q. Liu, Mechanical design and optimization of capacitive micromachined switch, Sens. Actuat. 93 (3) (2001) 273±285.
Biographies Y.C. Loke holds a Masters degree in Engineering from Nanyang Technological University (NTU), Singapore and Bachelor of Engineering with Honours from University of Manchester Institute of Science and Technology, United Kingdom in 2001 and 1998, respectively. He is currently a graduate student with NTU working on his Doctorate program. K.M. Liew joined the Nanyang Technological University as Lecturer in 1991, and is currently a tenured full Professor at the School of Mechanical and Production Engineering. He is the founding director of both Centre for Advanced Numerical Engineering Simulations (CANES), and Nanyang Centre for Supercomputing and Visualisation (NCSV).