Piezoresistive Sensing in a SOI Mechanically Coupled ... - IEEE Xplore

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 59, NO. 11, NOVEMBER 2012

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Piezoresistive Sensing in a SOI Mechanically Coupled Micromechanical Multiple-Resonator Array Abid Iqbal and Joshua E.-Y. Lee, Member, IEEE

Abstract—This paper presents the first results from mechanically coupling multiple resonators in an array in silicon-oninsulator while also employing piezoresistive sensing. By careful design of the couplers, the resonators in the array are synchronized to all resonate in phase and also at the same frequency. By collectively summing the currents from each of the resonators in the array, the net output current is thereby increased. By using piezoresistive sensing, the benefit of arraying is further improved by exploiting the higher electromechanical conversion gain provided by the piezoresistive effect. Further to this enhancement in transduction, our results also show that Q was increased with the coupling. A synchronized array of up to three extensional mode square-plate resonators is demonstrated here, although the concept can be further extended to larger size arrays. We have also formulated a semi-analytical model (with the aid of finite-element analysis) to describe the electromechanical transfer function of the device. Close agreement between our measurements and the model confirms that the observed enhancements afforded by arraying are within theoretical expectations. Index Terms—Mechanical coupling, micromechanical, piezoresistive, resonator, silicon-on-insulator (SOI).

I. I NTRODUCTION

Q

UARTZ crystal resonators have traditionally been the favored choice in providing precision timing references owing to their high quality factor and temperature stability. However, the difficulty in realizing integration with CMOS technology and their macroscopic form factor continues to open up applications where micromechanical (MEMS) resonators are recognized as viable alternatives [1], [2]. The most common method of actuating these devices is by applying an electrostatic force via a capacitive transducer. The output is then sensed through a motional current as a result of a modulation in the gap of the capacitive transducer. Due to the small form factor of these devices, this generally leads to a large motional resistance, which is given as follows: mω0 (1) Rx = 2 . η Q

Manuscript received May 3, 2012; revised June 27, 2012; accepted August 6, 2012. Date of publication October 3, 2012; date of current version October 18, 2012. This work was supported by a grant from the City University of Hong Kong (Project No. 7002690). The review of this paper was arranged by Editor F. Ayazi. The authors are with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong (e-mail: aiqbal3@styudent. cityu.edu.hk; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TED.2012.2213094

In the aforementioned equation, m denotes the dynamic mass, ω0 is the angular resonant frequency, Q is the quality factor, and η is the transduction factor which, for a parallel gap capacitor, is given by η=

ε0 A Vdc . g2

(2)

ε0 is the permittivity in air, A is the overlapping area of the transducing capacitor, g is the gap of the capacitor, and Vdc is the dc voltage applied across the gap. From (1) and (2), it can be seen that the motional resistance (Rx ) of the device scales with the fourth order of the transducer gap. Hence, to reduce Rx , various groups have reported results on fabricating narrow gaps smaller than 100 nm [3] or gaps that are partially filled with a solid high-k dielectric that are 10 nm wide [4], [5], as well as replacing the gap with a solid dielectric [6], [7]. One drawback in these approaches is the complexity of the process. Particularly for air gap transducers, electrostatic nonlinearity is more apparent when scaling the gap to these extremes. Dielectric breakdown is one complication observed in nanometer-thin dielectric transducers. One fresh approach that is a paradigm shift away from gap narrowing methods to lower Rx uses multiple resonators to boost the output. By mechanically coupling the resonators together, it has been demonstrated that the resonance of the individual resonators could be synchronized with each other. By collecting the currents from individual resonators, the net output current is thereby increased. This approach has been demonstrated in a polycrystalline silicon process for both inplane [8] and out-of-plane vibration modes [9]. These results have shown that the motional resistance could be reduced roughly in proportion with the number of resonators in the array [10] leading to lower insertion loss and lower phase noise. In fact, by increasing the net size of the device beyond that of a single resonator, the power handling limit could be increased overall. Furthermore, this approach can be applied to gap narrowing techniques to further extend their enhancement on transduction. In this paper, we present a mechanically coupled resonator array fabricated in silicon-on-insulator (SOI). The concept of mechanically coupled resonator arrays in SOI for filtering applications has been reported previously [11]. In this paper, we focus on mechanically coupling resonator arrays in SOI technology for synchronized oscillation. It is also the first demonstration of piezoresistive sensing to be used in a mechanically coupled resonator array. The thicker structural layer afforded by SOI has the advantage of larger electrostatic coupling and higher Q [12]. By applying piezoresistive sensing for

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the output readout, better electromechanical conversion could be achieved [13]–[15], thereby also further extending the existing enhancements provided by arraying through mechanical coupling. Implementation of the proposed scheme in SOI additionally exploits the higher piezoresistive coefficients available in single-crystal silicon compared to polycrystalline silicon. The next section of this paper presents the details of the mechanical coupling scheme to achieve synchronization of resonance while also allowing implementation of a piezoresistive sensing readout at the output. Section III describes a semianalytical electromechanical model for the transfer function of the device. In Section IV, the measured results from the electrical characterization of fabricated devices are laid out. These measurement results are compared against predictions by the electromechanical model (from Section III) in Section V. It will be shown in Section V that good agreement between the model and measurements is found, hence indicating that the experimentally observed enhancements gained from piezoresistive sensing with arraying are within theoretical expectations.

Fig. 1. FE simulation of the mode shape for a single square-plate resonator vibrating in the SE mode.

II. S YNCHRONIZED R ESONANCE VIA M ECHANICAL C OUPLING The primary objective of arraying multiple resonators is to the increase the net output current compared to a single resonator through summing up the output currents from individual resonators in the array. This in turn increases the overall transduction. However, to achieve a collective summing of currents, the resonant frequencies of each resonator in the array must be exactly identical to the other resonators in the array. Given Q’s of at least in the thousands reported for MEMS resonators, this corresponds to a tolerance of less than 0.1%—a tight specification that is difficult to meet. Nonetheless, for arraying to produce the desired outcome of realizing collective outputs, exact synchronization between the resonant frequencies of individual resonators is a strict prerequisite. As shown previously by [9], synchronization of resonance could be achieved by coupling the resonators to each other. By mechanically coupling the resonators together, the collection of individual resonators transforms to a multimode higher order composite structure. In each of the modes, the resonators in the array each resonates at the same frequency with the rest. The unit-cell resonator design used in this work is a squareplate resonator excited in the square-extensional (SE) mode described by finite-element (FE) simulation shown in Fig. 1. As can be seen from Fig. 1, the SE mode can be described as a square-plate that expands and contracts symmetrically on all four sides. The resonator is fixed to the substrate through T-shaped tethers at each corner of the square plate to reduce energy dissipation via the anchors. The square-plate resonator is designed with its sides aligned along the 110 direction. The resonator is driven into resonance through four parallel gap electrodes placed on each side of the square-plate. The resonant frequency is governed mainly by the length (L) of the squareplate and approximated by E 1 . (3) f= 2L ρ(1 − ν)

Fig. 2. FE simulation of the mode shape for a dual-resonator coupled array synchronized to vibrate in phase through the λ/2 coupling beam connecting the two square-plate resonators.

Fig. 3. FE simulation of the mode shape for a triple-resonator coupled array synchronized to vibrate in phase through the λ/2 coupling beams connecting the three square-plate resonators.

E and ν are the Young’s modulus and Poisson’s ratio given by the direction of displacement, respectively, while ρ is the density of silicon. The length of the resonator corresponds to half the wavelength of the acoustic wave. To realize a mechanically coupled array, individual resonators are coupled to each other through their corners as shown in Fig. 2. In Fig. 2, an array of two resonators is simulated, while Fig. 3 shows a simulation for an array of three resonators. As can be seen from Figs. 2 and 3, the squareplate resonators are thus arranged diagonally to each other. The resonators are actuated in phase through gap electrodes placed on each side of the square as determined by the arrangement of the electrical routing. As such, the in-phase mode of vibration in the array can be seen to coincide with the phase and arrangement of the electrodes. The corners of the SE mode resonator correspond to points of maximum displacement given by the summation of displacements in the orthogonal axes. The SE mode resonators are mechanically coupled to each

IQBAL AND LEE: PIEZORESISTIVE SENSING IN A MULTIPLE-RESONATOR ARRAY

other through these points of maximum displacement which also correspond to high-velocity locations [16]. This in turn pushes the other undesired modes far away from the designed mode of interest on the frequency spectrum. In short, the sides of the resonators are aligned along the 110 direction, while the coupling beams between the resonators are along the 100 direction. This coupling arrangement is also particularly convenient for an implementation in SOI by allowing greater ease of electrical routing. The coupler beams were designed to be a half-wavelength (λ/2) for the purpose of pushing the modes further apart [16]. In deriving the equivalent physical length of the coupler beam, differences between the square plate and beam had to be accounted for. Differences include the change in elastic properties associated with different crystal orientations as well as in changing from a 2-D breathing (ref square-plate) mode to a 1-D longitudinal mode (ref coupling beams). Accounting for these differences, the following relation was found by keeping the frequencies in the coupling beam and SE mode resonator equal: E100 λcoupler = 2 (1 − ν). (4) λsquare E110

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Fig. 4. SEM image of a corner of the square-plate resonator, showing one of the supporting T-shaped tethers for anchoring the resonator to the substrate.

Since the presence of T-shaped tethers changes the resonant frequency of the SE mode resonator, the aforementioned relation was further fine tuned with the aid of FE simulations for ensuring that the resonant frequencies of the coupling beam and square-plate resonator matched each other. The simulations for the mechanically coupled array indicated that only the in-phase mode could be identified around the resonant frequency of a single resonator, while spurious modes could not be found. III. P IEZORESISTIVE S ENSING Single-crystal silicon is piezoresistive. Its piezoresistive coefficients can vary between different crystal orientations. For instance, the piezoresistive coefficient in the 100 direction can be as high as three times that in the 110 direction for n-type doping [17], same as that used in this work. Apart from anchoring the resonator to the substrate, the T-shaped tethers here also function as built-in piezoresistors. As the square plate expands and contracts, the stems of the tethers are subjected to axial stress. Since the stems of the tethers are aligned along the diagonals of the square plate, they are conveniently found along the 100 direction where the longitudinal piezoresistive coefficient is maximum as shown in Fig. 4 compared to other in-plane directions. It should be noted here that the SE mode is observable only in the 110 and 100 directions in the (100) plane according to FE simulations once the anisotropy of silicon is considered. This, in turn, allows maximum electromechanical conversion between stress and resistance and, ultimately, the output current. By applying a steady dc current through the resonator, a motional current is produced. The circuit schematic in Fig. 5 describes the one-port transduction configuration applied to all the measurements presented later in Section IV. Each square-plate resonator is driven into resonance using electrostatic actuation through applying a dc

Fig. 5. Top-view circuit schematic of a triple-resonator coupled array showing the associated one-port transduction configuration.

bias voltage (Vdc ) with an ac modulation voltage (Vin ) on the electrodes placed on each side of the resonator F =4

ε0 Le h Vdc Vin . g2

(5)

Le refers to the length of the electrode, and h is the structural thickness of the device. The imposed electrostatic force results in a displacement at resonance given by x=

F Q. k

(6)

Here, k denotes the spring constant of the square-plate resonator which approximates to E 2 k=π h. (7) 1−ν Combining (5)–(7) thus yields 1−ν 4ε0 Le QVdc Vac . x= g2 π2 E

(8)

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The displacement, in turn, gives rise to axial stress in the stem of the tether according to the following relationship: σl = αx.

(9)

The proportionality factor α that relates the axial stress σ to displacement x can be numerically obtained through FE simulation in COMSOL. From FE simulations, it was also verified that the transverse stresses are negligible compared to longitudinal stresses. Due to the piezoresistive effect, this axial stress in the stem of the tether gives rise to a change in the resistance of this particular section of the structure ΔR = πl σl + πt σt . R

(10)

Since the transverse σt is insignificant compared to the longitudinal σl , the second term in (10) can be neglected. Combining with (9), we obtain 4ε0 Le 1−ν ΔRv = απl (11) QVdc Vac . Rv g2 π2 E Rv refers to the resistance of the stem of the tether whose resistance is modulated by the stress imposed by the resonator’s vibration. When a steady dc drain current Id is applied through the resonator, this yields a modulated motional current given by Rv ΔRv Iout = (12) Id Q. Rv + Rs Rv From (11) and (12), we obtain an overall transfer function at resonance Rv 4ε0 Le 1−ν Iout = απl gm = QId Vdc . Vin Rv + Rs g2 π2 E (13) Rs refers to the total static resistance in the path of the drain current Id . Later in Section V, the aforementioned derived transfer function will be applied to the measured data for validation between the semianalytical model and experiments. A value of −102.2 × 10−11 Pa−1 was used for πl according to [17]. It should be noted that the value of πl is dependent on doping concentration and this value represents the maximum value achieved at the optimal doping concentrations. Since (13) is used to assess if device measurements were within reasonable theoretical expectations, an exact determination of πl was not required, and a reliable estimate was sufficient for this purpose. Dual-resonator coupled arrays as well as triple-resonator coupled arrays were fabricated on the same die containing alongside conventional single square-plate resonators (all with the same length of 360 μm) by using a commercially available standard SOI MEMS process. Fig. 6(a) and (b) shows scanning electron micrographs (SEMs) of a dual-resonator coupled array and a triple-resonator coupled array, respectively. The devices were fabricated on 25 μm-thick SOI with a gap of about 3 μm (measured under the SEM).

Fig. 6. (a) SEM image of the fabricated SOI dual-resonator coupled array; (b) SEM image of the fabricated SOI triple-resonator coupled array. TABLE I C OMPARISON OF FE-C OMPUTED E IGENFREQUENCY AND THE M EASURED R ESONANT F REQUENCY

IV. E LECTRICAL C HARACTERIZATION OF D EVICES The fabricated devices were characterized in a Janis Research cryogenic probe station at around 50 μtorr using GSG microwave probes. The dc bias voltage was maintained at 50 V for all the resonators, while the source power was kept at 0 dBm. Table I compares the measured resonant frequencies for the three devices against predictions by FE simulation using COMSOL when no drain current is applied to the resonators. It can be seen that the measured resonant frequencies agree well with the FE simulation eigenfrequencies to within less than 0.1%. Fig. 7 shows the measured electrical transmission for the triple-resonator coupled array after removing feedthrough [18].

IQBAL AND LEE: PIEZORESISTIVE SENSING IN A MULTIPLE-RESONATOR ARRAY

Fig. 7. Measured electrical transmission for the triple-resonator coupled array with increasing total drain current.

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Fig. 10. Comparison of experimentally measured normalized piezoresistive transconductances against model curve fit as a function of increasing drain current (per resonator) for a single resonator, dual-resonator coupled array, and triple-resonator coupled array; the capacitive motional current has been removed.

transconductances and drain current for all three devices is shown in Fig. 10. V. D ISCUSSION

Fig. 8. Comparison of quality factors with increasing drain current for a single resonator, dual-resonator coupled array, and triple-resonator coupled array.

Fig. 9. Comparison of normalized transconductances (including capacitive motional currents) with increasing drain current (per resonator) for a single resonator, dual-resonator coupled array, and triple-resonator coupled array.

This was achieved by first measuring the feedthrough current which is given by the measured transmission when the dc bias drive voltage is set to zero. This reference was then subtracted from the raw data that are embedded in feedthrough to obtain the frequency response plots in Fig. 7. The measured electrical transmission shows that increasing the drain current leads to an increase in signal due to the increased transduction according to (13). It can be seen from Fig. 8, which shows the extracted Q’s as a function of drain current, that the Q varies with the drain current. Furthermore, we can see that Q varies between the three devices. As such, to compare the electromechanical conversion between the three devices, the transconductances were normalized through dividing each value by its Q factor to produce the plots in Figs. 9 and 10. The relationship between the normalized

As more resonators are added to the array, the drain current at the dc source is shared among more resonators. Hence, with more resonators, each resonator receives a smaller share of the input drain current. The enhancement from arraying comes when the same drain current is applied to each resonator. Hence, to see this enhancement, the normalized transconductance was plotted against the current per resonator. In order to compare the measured trends against the semianalytical model for the transfer function in (13), one more adjustment has to be made. Since the resonator is actuated and sensed using a one-port configuration, the dc bias from the actuation electrodes appears across the modulated gap during vibrations. As such, a capacitive motional current is generated through this gap and sensed at the output port. This capacitive motional current thus adds on to the piezoresistive motional current that is also sensed at the same output port. As such, the transconductance values in Fig. 9 contain both the piezoresistive and capacitive responses. Hence, the capacitive response had to be removed from the curves in Fig. 9. This was achieved by first measuring the capacitive current output (when the drain current is set to zero) and then subtracting it from the combined output as phasors. Once this adjustment has been made, it can be seen from the graph in Fig. 10 that the prediction by the semianalytical model in (13) agrees well with the experimentally measured characteristics of all three resonators. Fig. 11 shows the shift in frequency with increasing current for the three devices. Each of the curves can be fitted to a power square law, indicating that the shifts are due to joule heating in the device as current is increased. As the number of resonators in the array is increased, we see that the magnitude of the frequency shift decreases for the same amount of current. This in turn is due to the sharing of currents between the resonators, which lowers the extent of joule heating in each resonator. These results thus show that, by arraying resonators through mechanical coupling, the total transduction can be improved for the same drain current applied through each resonator.

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Fig. 11. Comparison of the measured resonant frequency with increasing drain current for a single resonator, dual-resonator coupled array, and tripleresonator coupled array.

VI. C ONCLUSION In this paper, a synchronized array of up to three SE mode resonators that also employs piezoresistive sensing has been demonstrated. Synchronization is realized through the use of half-wavelength couplers, which allow the motional currents of individual resonators in the array to be collectively summed. This in turn leads to an increase in the overall transduction in proportion to the number of resonators in the array. By using piezoresistive sensing in addition to capacitive sensing, higher electromechanical gain is achieved. These enhancements due to arraying and use of piezoresistive sensing are confirmed by measurements of fabricated SOI resonators. These measurements also show that, instead of reducing Q, a higher Q was observed in a triple-resonator coupled array compared to a single resonator. In addition, the same resonant frequencies were measured for a single resonator, dual-resonator coupled array, and triple-resonator coupled array. This thus demonstrates that the insertion loss can be reduced without changing the frequency through the use of arraying. Finally, close agreement between our measurements and the semi-analytical model for the transfer function formulated in this paper confirms that the observed enhancements afforded by arraying with piezoresistive sensing are within theoretical expectations. ACKNOWLEDGMENT The authors would like to thank the EPA Center, Department of Electronic Engineering, City University of Hong Kong, for their support for work on the scanning electron micrograph images.

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Abid Iqbal received the M.S. degree from the Ghulam Ishaq Khan Institute, Topi, Pakistan. He is currently working toward the Ph.D. degree in the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong.

R EFERENCES [1] C. T.-C. Nguyen, “MEMS technology for timing and frequency control,” IEEE Trans. Ultrason., Ferroelect., Freq. Control, vol. 54, no. 2, pp. 251– 270, Feb. 2007. [2] J. T. M. van Beek and R. Puers, “A review of MEMS oscillators for frequency reference and timing applications,” J. Micromech. Microeng., vol. 22, no. 1, pp. 013001-1–013001-35, Jan. 2012. [3] S. Pourkamali, A. Hashimura, R. Abdolvand, G. K. Ho, A. Erbil, and F. Ayazi, “High-Q single-crystal silicon HARPSS capacitive beam resonators with self-aligned sub-100-nm transduction gaps,” J. Microelectromech. Syst., vol. 12, no. 4, pp. 487–496, Aug. 2003. [4] T. Cheng and S. A. Bhave, “High-Q, low impedance polysilicon resonators with 10 nm air gaps,” in Proc. IEEE MEMS Conf., 2010, pp. 695–698.

Joshua E.-Y. Lee (S’05–M’09) received the Ph.D. degree from the University of Cambridge, Cambridge, U.K. He is currently an Assistant Professor with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong.