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Methods for enhanced electrical transduction and characterization of micromechanical resonators
Sensors and Actuators A 158 (2010) 263–272
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Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna
Methods for enhanced electrical transduction and characterization of micromechanical resonators A.T-H. Lin a , J.E-Y. Lee a,b , J. Yan a , A.A. Seshia a,∗ a b
Department of Engineering, Trumpington Street, University of Cambridge, Cambridge, United Kingdom Department of Electronic Engineering, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong
a r t i c l e
i n f o
Article history: Received 3 August 2009 Received in revised form 18 January 2010 Accepted 18 January 2010 Available online 25 January 2010 Keywords: Bulk mode resonators Harmonic driving Piezoresistive sensing
a b s t r a c t This paper details the design and enhanced electrical transduction of a bulk acoustic mode resonator fabricated in a commercial foundry MEMS process utilizing 2.5 m gaps. The I–V characteristics of electrically addressed silicon resonators are often dominated by capacitive parasitics, inherent to hybrid technologies. This paper benchmarks a variety of drive and detection principles for electrostatically driven square-extensional mode resonators operating in air via analytical models accompanied by measurements of fabricated devices with the primary aim of enhancing the ratio of the motional to feedthrough current at nominal operating voltages. In view of ultimately enhancing the motional to feedthrough current ratio, a new detection technique that combines second harmonic capacitive actuation and piezoresistive detection is presented herein. This new method is shown to outperform previously reported methods utilizing voltages as low as ±3 V in air, providing a promising solution for low voltage CMOS-MEMS integration. To elucidate the basis of this improvement in signal output from measured devices, an approximate analytical model for piezoresistive sensing specific to the resonator topology reported here is also developed and presented. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Silicon bulk acoustic wave (BAW) microresonators have been demonstrated for high Q solutions to a variety of applications such as timing references, filters and precision sensors. These devices are recognised as a promising alternative to quartz crystal resonators due to their realizable small form factor and the potential for integration with standard IC electronics. However, due to the relatively high stiffness and large motional resistance, the motional signal is often swamped by parasitic feedthrough in hybrid technologies that do not integrate electronics and MEMS on the same substrate. The physical sources of this capacitive feedthrough parasitic includes (1) the direct overlapping capacitance of the transducer, (2) capacitive coupling through the substrate via the bond pads, (3) interconnects, and (4) the electrical package [1]. Although integration with CMOS results in lower capacitive parasitics, the resonator bias voltages are often limited to less than 5 V, which sets a limit for the detectable motional signal. One way to increase the motional signal is to reduce the transduction gap thereby decreasing the motional resistance. However, the realization of sub-micron transduction gaps relies on advanced fabrication
∗ Corresponding author. E-mail address: [email protected] (A.A. Seshia). 0924-4247/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2010.01.024
technology that often requires careful control on the definition of sidewall spacers and/or sacrificial layers. The latter often places significant constraints on the MEMS fabrication process, integration with CMOS and ultimately device yield. Operating the resonator in vacuum often results in lower damping and an increased motional signal at the expense of the added complexity of vacuum packaging. Various studies have been devoted to actuation and detection principles with the aim to enhance the motional to feedthrough current ratio, including harmonic actuation, mixing techniques and piezoresistive sensing [2–5]. This paper benchmarks these principles for an electrostatically driven square-extensional (SE) mode resonator operating in atmospheric pressure and fabricated in a foundry MEMS process utilizing 2.5 m gaps. The aim is to illustrate techniques and principles that enhance the ratio of the motional to feedthrough current at nominal operating voltages. To this end, analytical models are presented to elucidate the relative merits between various methods, which are further validated via actual measurements of fabricated SOI microresonators. Pursuant to the ultimate aim of enhancing motional to feedthrough current ratio, a new detection technique is also introduced that combines second harmonic capacitive actuation and piezoresistive detection. This method outperforms previously reported methods and utilizes voltages an order of magnitude lower, at ±3 V in air. To elucidate the basis of this improvement in signal output from measured devices,
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Fig. 2. Optical micrograph of the square resonator fabricated by the MEMSCAP SOIMUMPs process.
Fig. 1. The square-extensional (SE) mode shape from ANSYS FE simulation.
anchors are lithographically patterned on the top silicon and DRIE etched. A polyimide protective coat is applied to the top silicon surface and the substrate layer is etched through from the backside to release the structures. The remaining exposed oxide is removed using a vapor HF process to prevent stiction, after which, the polyimide layer is removed to fully release the structure. The final step is the metallization of the substrate layer using a shadow mask to provide a topside electrical contact to the substrate.
Table 1 Resonator parameters. Parameters
Symbol
Value
Side length Thickness Transduction gap Resonant frequency Measured quality factor
L H g f Q
2000 m 25 m 2.5 m 2.2 MHz 6771
3. Measurement theory and setup
an approximate analytical model for piezoresistive sensing through the resonator is also presented.
This section presents the theory behind the progression of electrostatic methods for reducing feedthrough and introduces piezoresistive sensing methods for increasing motional signal. This model sets up the six transduction methods that are documented and compared, as summarized in Table 2.
2. Device description and fabrication The tested device is a 2.2 MHz square plate resonator operated in the SE mode, wherein the square plate undergoes an extension and contraction symmetrically on each side of the square, as illustrated in Fig. 1. The suspended square is anchored to the substrate via a T-shape connecting stem located on four corners of the square plate. The resonator parameters are shown in Table 1. The measured transduction gap of the tested device is found to be 2.5 m. The resonator is fabricated in a commercial SOI foundry process (MEMSCAP), with one lateral electrode on each side of the square plate. An electrical connection is provided to each anchor of the resonator body. This enables the capacitive and piezoresistive sensing configurations used in this work. An optical micrograph of the fabricated resonator is shown in Fig. 2. The fabrication process begins with a highly doped 25 m thick n-type device layer, followed by metallization to pattern the bond pads through a lift-off process. The resonators, electrodes and
3.1. Electrostatic actuation and sensing Electrostatic actuation is achieved by applying a voltage across a capacitive gap which results in an electrostatic force. The electrostatic energy stored in a capacitor C with potential difference of V applied, is given by 1 2 CV 2
We =
(1)
For a parallel plate electrostatic actuator, the corresponding actuation force is determined by taking the differential of Eq. (1): 1 F= 2
∂C ∂x
V2
(2)
Table 2 Summary of transduction methods, where ω0 , meq , Q, ε0 , A are the angular resonant frequency, equivalent mass of the resonator, quality factor, permittivity of air and the effective transduction area, respectively. Transduction methods One-port Two-port 2nd harmonic Mixing
Motional current (|Im |)
Actuation force (F0 ) Vdc
V (ε
Vdc
V (ε
ac
ac
0 A/g 0 A/g
2
2
)
Qε2 A2 /m ( V 2 V
)
Qε2 A2 /m ( V 2 V
dc
dc
( Vac 2 /4)(ε0 A/g 2 ) 1 2
VRF
0 A/g
One-port driving and piezoresistive sensing
Vdc
2nd harmonic driving and piezoresistive sensing
( V 2 /4)(ε
ac
ac
2
)
0 A/g
2
ac
dc
2
ac
4
)
eq ω0 g
4
eq ω0 g
0
eq ω0 g
0
V V Qε2 A2 /2m ( V dc
RF
(Rr /Rr )(Vd /R) )
)
0
V 2 Qε2 A2 /4m ( V
VLO (ε0 A/g )
V (ε
ac
Parameters 4
(Rr /Rr )(Vd /R)
LO
0
)
eq ω0 g
4
)
Vdc = 30 V Vac = 0 dBm (50 ) Vdc = 30 V Vac = 0 dBm (50 ) Vdc = 30 V Vac = ±3 V Vdc = 30 V VRF = ±3 V, VLO = ±3 V Vdc = 30 V, Vac = 0 dBm(50 ) Vd = −2 V, Id = 5.9 mA Vac = ±3 V Vd = −2 V, Id = 5.9 mA
Peak height (dB) ∼0.02 ∼0.04 ∼8 ∼9 ∼0.5 ∼13
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265
Fig. 3. Schematic showing measurement setups for (a) one-port, (b) two-port, (c) 2nd harmonic, (d) mixing, (e) piezoresistive sensing and (f) 2nd harmonic drive with piezoresistive sensing.
where V is the applied voltage, x is the resonator displacement, C = (ε0 A/g) is the capacitance of the parallel plate, and ε0 , g, A are the permittivity, transduction gap and transduction area, respectively. In a one-port configuration as shown in Fig. 3a, all four electrodes are used for actuation, where the motional signal is detected from the resonator anchors through capacitive coupling between the electrodes and the resonator. For fundamental frequency driving, the applied voltage can be written as the sum of a dc bias Vdc and an ac signal Vac = Vac cos(ω0 t) at resonant frequency ω0 , yielding the following expression for actuator force:
F =
1 2
1 = 2
∂C ∂x ∂C ∂x
×
2 Vdc
[Vdc + Vac cos(ω0 t)]
2
2 2 Vac Vac + 2Vdc Vac cos(ω0 t) + + cos(2ω0 t) 2
2
The force component of interest is that which occurs at the resonant frequency ω0 , this gives
∂C = F0 cos(ω0 t) F = Vdc Vac cos(ω0 t) ∂x
where
∂C ε0 A = Vdc Vac F0 = Vdc Vac g2
∂x
(3)
Although maximizing electrode area maximizes the capacitive motional current in a one-port configuration, the feedthrough current can also be substantially large as the actuator capacitor also serves to capacitively couple the resonator body to the surrounding electrodes. In the two-port configuration, as shown in Fig. 3b, the resonator body is grounded and the electrodes are split in two pairs, one used for driving and the other for sensing to reduce the feedthrough through the direct overlap capacitor between the actuator electrode and the sensing port [3]. Although the effective transduction area is reduced by a factor of two, the reduction in the feedthrough parasitics enabled a clearer measurement. However, feedthrough between drive and sense electrodes could still exist via the substrate or the package, which can mask the motional current. To further reduce these feedthrough parasitics, one approach is to shift the feedthrough current relative to the motional current in the frequency domain, and utilize the high-Q resonant response to filter the feedthrough current at the resonant frequency [2,3]. In second harmonic measurements, the nonlinearity of electrostatic transduction can be utilized to drive the structure at half the resonant frequency:
Vac = Vac cos
ω
0
2
t
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limitations that is required for capacitive sensing, which can be large for a bulk mode resonator.
This results in a force at the resonant frequency, given by F=
2 Vac 4
cos(ω0 t)
where
2 Vac ∂C
F0 =
4
∂x
=
∂C ∂x
= F0 cos(ω0 t)
3.2. Piezoresistive sensing
2 Vac ε A 0
(4)
g2
4
Second harmonic drive can be used in both one-port and two-port configurations, though the feedthrough effect from the harmonics of the driving signal, albeit much smaller, can still be observed. A two-port 2nd harmonic setup (Fig. 3c) therefore has better transduction than one-port 2nd harmonic setup, despite the reduction in transduction area. If two ac input frequencies are available, the device can be driven by two off resonance signals VRF and VLO . From Eq. (1), this generates a force at their sum or difference frequency. By setting the input frequencies ωRF , ωLO such that ω0 = ωRF + ωLO , The actuation force is given by 1 2
F =
1 = 2
∂C ∂x ∂C ∂x
2 [VRF cos(ωRF t) − VLO cos(ωLO t)]
([VRF cos(ωRF t)] − 2 VRF VLO cos(ωRF t)cos(ωLO t) 2
+ [VLO cos(ωLO t)] ) 1 2
=
∂C ∂x
(. . . − VRF VLO cos((ωRF − ωLO )t)
Im ≈
Vd R
R
r
(8)
Rr
where R = Rr + Rs , and Rr , Rs are the resistance of the resonator and the resistance of the externally controlled resistor, respectively. In the one-port capacitive driving and piezoresistive sensing configuration, the output signal is a combination of both capacitive and piezoresistive signal. The measured admittance as a function of driving frequency ω can be expressed as Y (ω) =
2
Apart from capacitive measurement techniques, the mechanical motion of the resonator operating in the SE mode can also be detected through the variation in the electrical resistance of the vibrating structure due to the piezoresistive effect of single crystal silicon. This method has been previously shown to be a promising alternative for increasing the amplitude of the motional signal [4,5]. In this work, the electrical resistance between two corners of the resonator structure is monitored by applying a voltage difference, as shown in Fig. 3e. The motional current measured is proportional to the total change in resistance (Rr /Rr ) of the resonator structure and the applied dc voltage Vd , given by
jωCm 2
1 − (ω/ω0 ) + j(ω/ω0 )/Q
+ gm + jωCf
where Cm , gm , Cf are the motional capacitance, transconductance and feedthrough capacitance, respectively. The transconductance is given by
− VRF VLO cos((ωRF + ωLO )t) + . . .)
Im Vd = Vac Vac R
R
r
The force of interest is that which occurs at the resonant frequency ω0 , given by
gm =
1 F = VRF VLO cos(ω0 t) 2
The change in resistance due to piezoresistivity follows:
where F0 =
1 VRF VLO 2
∂C ∂x
∂C ∂x
= F0 =
= F0 cos(ω0 t)
ε0 A 1 VRF VLO 2 2 g
F0 Q meq ω0 2
(5)
(6)
where Q, meq are the quality factor and equivalent mass of the resonator. The motional current detected using capacitive sensing is the result of the modulation of the capacitance due to changes in the gap size from the vibration motion, given by
im = Vdc
∂C ∂x
x˙
(10)
Rr
Rr = l l + t t Rr
This mixing setup is shown in Fig. 3d. In this approach, if the harmonics of the driving signal do not overlap with the sensing signal, the feedthrough effect is minimized. The maximum displacement at resonance for a given actuation force can be expressed as X0 =
(9)
(7)
Substituting the force Eqs. (3)–(5) and displacement Eq. (6) into (7), the motional current for each capacitive measurement method was derived, as summarized in Table 2. It can be seen from the motional current expressions, that capacitive sensing is largely limited by the available dc bias and the transduction gap size, as the detectable motional current is proportional to Vdc and inversely proportional to g4 . Thus for a resonator design, there is a minimal dc voltage
(11)
where l , t are the longitudinal and transverse pezoresistive coefficients, and l , t are the longitudinal and transverse stresses, respectively. 3.2.1. Piezoresistance of the resonator bulk An approximate analytical model was developed to relate the stresses in the resonator bulk (i.e. suspended square plate) for the SE mode to the change in resistance between the two diagonal corners of the square plate. The side edge of the square is aligned to the 1 1 0 direction, with the current flow diagonally across two anchors in the 1 0 0 direction, where l is maximized for n-type silicon [6]. The square can be thought of as numerous resistors connected in parallel and series diagonally as illustrated in Fig. 4. These resistances can be added up in parallel and series to obtain the overall resistance of the square plate between two corners. The SE mode shape can be approximated using a sine function [7]: ux (x) = Ux sin
x
L
uy (y) = Uy sin
y
L
(12)
where Ux , Uy are the maximum displacement along the x- and yaxis, in the coordinate system shown in Fig. 5(a), and L is the side length of the square. For simplicity in calculation, the coordinate system is transformed to orient the x-axis along the direction of the applied field (i.e. rotated 45 degrees), the mode shape equation
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267
Fig. 4. Diagram illustrating the parallel and series resistance in the square.
Fig. 7. Diagram showing the anchor dimensions.
of the displacement, given by εx =
dux (x, y) Ux cos = 2L dx
duy (x, y) Uy = εy = cos 2L dx Fig. 5. The mode shape plot showing the coordinate system for SE mode with (a) side edge aligned to x-axis and (b) side edge rotated 45 degrees from x-axis.
Uy uy (x, y) = √ sin 2
Ux √ (x + y) − √ sin 2L 2 Uy √ (x + y) + √ sin 2L 2
√ (y − x) 2L
(13)
√ (y − x) 2L
Ux cos √ (x + y) + 2L 2L
U y
cos √ (x+y) + 2L 2L
√ (y−x) 2L
(14)
√ (y−x) 2L
The resistance change in the square plate due to piezoresistivity is given by Rsq = ESE (l εx + t εy ) Rsq
becomes Ux ux (x, y) = √ sin 2
(15)
where ESE is the biaxial modulus of the SE mode. The resistance Rc of a conductor with uniform cross-section is given by Rc = ( l/A), where , l, A are the resistively, length of the material, and the crosssectional area, respectively. The static resistance of the resonator can be approximated (assuming uniform E-field) by integrating the resistance on the right half triangle area of the resonator, and mul-
where x, y now refers to the transformed coordinate axis as shown in Fig. 5(b). The strain along the x- and y-direction is the derivative
Fig. 6. MATLAB plot showing derived motional current density distribution in the resonator bulk.
Fig. 8. ANSYS FE simulation showing the difference in distribution of current density sum before and after applied strain.
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Fig. 9. Measured frequency response for a 2.2 MHz SE mode resonator in air using (a) one-port, (b) two-port, (c) second harmonic, (d) mixing measurements, (e) one-port capacitive drive with piezoresistive sensing, (f) second harmonic drive with piezoresistive sensing, (g) comparison between the capacitive and piezoresistive sensing methods. The parameters for each measurement are shown in Table 2.
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269
Fig. 11. Restored frequency response curves using one-port drive and piezoresistive sensing in terms of (a) magnitude and (b) phase for a range of applied dc bias.
Fig. 10. Measured frequency response curves using one-port drive and piezoresistive sensing in terms of (a) magnitude and (b) phase for a range of applied dc bias.
Rsq,d = Rsq,s [1 + ESE ε(l + t )] tiply by two to get the expression for the total static resistance:
Rsq
=2× H
˛
dx
=
√ (−2x + 2L)
H
dx dy
+ cos
(16)
0
1 + ESE
U ( + t ) cos 2L l
√ (y − x) 2L
√ (x + y) 2L
(18)
The total dynamic resistance for the square is given by summing up the resistance for all points in the square plate:
˛ Rsq,s + Rsq,s =
√ −x+(L/ 2) 0
dx
H
1 + ESE U (l + t ) cos 2L
√ x−(L/ 2)
where H is the thickness of the resonator, and ˛ is the length from the centre of the resonator to the point where it meets the anchor. The dynamic resistance at each node is therefore given by Rsq,d = Rsq,s + Rsq,s = Rsq,s + Rsq,s ESE (l εx + t εy ) = Rsq,s [1 + ESE (l εx + t εy )]
(17)
where Rsq,d , Rsq,s is the dynamic and static resistance at a given node. From Eq. (14), we can see that, the strain in the x- and y-direction for SE mode is equal (i.e. εx = εy = ε), the expression for dynamic resistance can be simplified to
√ (x 2L
(19)
+ y) + cos
√ (y 2L
−1
− x)
dy
The above integral was solved using Mathematica by plugging in the parameters and the dynamic resistance for the square is obtained. The calculated resistance change of the square corresponding to an actuated displacement of U = 9.601 × 10−11 [m] is (Rsq /Rsq ) = −3.015 × 10−6 . Fig. 6 shows the relative motional signal plot of the resonator bulk in MATLAB. The motional signal is maximum at the centre of the square as the stress is maximum at the centre for the SE mode. 3.2.2. Piezoresistance of the resonator anchors As the square extends in all directions, maximum displacement occurs at each of the four corners, straining the T-shaped anchors connecting to the corners of the square. The geometry of the T-
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Fig. 12. Restored frequency response curves using one-port drive and piezoresistive sensing in terms of (a) magnitude and (b) phase for a range of applied drain current.
anchor is shown in Fig. 7, the ‘stem’ part is aligned to the 1 0 0 direction where l is maximized for n-type silicon. A finite element simulation was done to approximate the resistance change in the anchors for the actuated displacement value using ANSYS. A plot of the change in current density sum before and after displacement load is shown in Fig. 8. The extracted value for the resistance change in the anchors from the FE simulation corresponding to the actuated displacement above is (Ranchor /Ranchor ) = 3.921 × 10−5 . This is about an order of magnitude larger than the bulk resistance change. The overall change in resistance is the sum of resistance change in the bulk and anchors, given by Rsq + Ranchor Rr = Rr Rsq + Ranchor
(20)
Due to the high resistance and high current density distribution at the anchors, this result in majority of the piezoresistive signal came from the anchors as oppose to the bulk. The model for the bulk assumes that current density at every elemental resistor is the same, while in reality current density is maximum in the middle and goes to zero towards the edges. However, since the bulk is not the major contributor to the measured motional current, we did not give further detailed model of the bulk. An approximate analytical model is also developed to relate the piezoresistive signal to the stress induced in the anchors. It is assumed that the piezoresistive signal from the ‘cap’ is very small and can be neglected compare to the ‘stem’. This is because as the ‘cap’ bends, the piezoresistive change due to compression at the inner side cancels out with piezoresistive change due to tension on
Fig. 13. Measured and analytical model for (a) transconductance and (b) Rr /Rr as a function of dc bias and drain current.
the outer side. This can also be seen from the ANSYS simulation in Fig. 8. To calculate the amount of stress in the stem part, the T-shaped anchor can be thought of as a beam (stem part) that is anchored to the centre of a clamped-clamped beam (cap part). A displacement force from the corner of the resonating square causes the beam to compress and the cap to bend. The displacement ratio in the stem and cap is inversely proportional to their spring constant: kcap xstem = xcap kstem
(21)
Therefore, the stiffer the cap, the larger the stress induced in the stem, and hence larger the piezoresistive signal for a given displacement. However, note that the stiffer the cap, the higher the energy loss through the anchors, and hence lower the quality factor of the resonator. There is therefore a trade off between piezoresistive signal enhancement and Q of the system. Using this simplified model, for a given displacement U at the centre edge of the square, the stem is compressed by xstem =
√
2U
kcap kstem + kcap
(22)
The resultant change in resistance in the stem is given by Rstem xstem = E100 ( − t ) Rstem Lstem l
(23)
where E100 , Lstem , are the Young’s modulus in the 1 0 0 direction, length of the stem and the Poisson ratio, respectively. Substituting into equation for gm , assuming the resistance change in the stem contribute to all of the motional transconductance, and neglecting
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the other parts we get: gm
= = = = = =
2R
271
current, as shown in Fig. 9e. This is an improvement to the one/twoport capacitive sensing scheme. Note that in this configuration, the detected signal contains both the capacitive and piezoresistive components of the motional current.
Vd 2Rstem Vd Rstem stem = Vac R Rr Vac R Rstem Rr
Vd 2Rstem E100 ( − t )xstem Vac R Rr Lstem l √ kcap Vd 2Rstem E100 ( − t ) 2U Vac R Rr Lstem l kstem + kcap 2R E √ F0 /keff (24) kcap Vd stem 100 ( − t ) 2 Vac R Rr Lstem l kstem + kcap 1 − (ω/ω0 )2 + j(ω/ω0 )/Q 2R E √ kcap Vd ε0 Le 1 1 stem 100 (l − t ) 2 Vac Vdc 2 Vac R Rr Lstem kstem + kcap g 2 ESE 1 − (ω/ω0 )2 + j(ω/ω0 )/Q
√ kcap Vd 2Rstem E100 ε0 Le 1 1 ( − t ) 2 V R Rr Lstem l kstem + kcap dc g 2 2 ESE 1 − (ω/ω0 )2 + j(ω/ω0 )/Q
where Le , ε0 are the length of electrode on each side of the square, permittivity of air, and keff = 2 ESE H is the effective spring constant of the SE mode. With piezoresistive sensing, the motional signal can be enhanced without necessarily scaling the transduction gap and the drive signal, and can be significantly larger than capacitive measurements. The current through the resonator can be tuned by setting Vd , and is ultimately limited by the maximum current density ceiling for single crystal silicon and the electrical power dissipated in the resonator. Since the motional current is detected via the resonator body, all electrode areas can be utilized for electrostatic actuation of the SE mode. Piezoresistive sensing can be combined with second harmonic drive (Fig. 3f) to utilize both advantages of increased motional signal and reduced parasitic feedthrough for further signal enhancements at nominal operating voltages. 4. Results and discussion Measurements results for each transduction scheme employed for the same square-extensional mode resonator are shown in Fig. 9. All the measurements are done at atmospheric pressure. For capacitive sensing of the motional current, a dc bias of 30 V was applied between the resonator and the electrode.
Further signal enhancement is achieved by combining second harmonic drive and piezoresistive sensing. This allows the rejection of feedthrough currents as well as eliminates the need of a large dc bias for driving and sensing. Fig. 9f demonstrates the measured signal with peak height of 13 dB, an ac actuation voltage of ±3 V and drain current of 5.9 mA. The quality factor measured by taking the 3 dB bandwidth of the measured signal is 6550, which is close to the extracted quality factor of 6771 with a two-port capacitive measurement, showing that the signal distortion from the parasitic feedthrough is largely eliminated. Fig. 9g illustrates the difference in signal strength for capacitive and piezoresistive sensing on the same graph, a clear motional signal enhancement can be seen for the case corresponding to 2nd harmonic drive and piezoresistive sensing. Comparing all the measurement results, we can see that the distortion of the measured resonant peak due to parasitic feedthrough is visible in the one/two-port measurement, and is substantially rejected in the harmonic driving and mixing measurements, and almost unnoticeable in the piezoresistive sensing measurements. Thus, piezoresistive sensing combined with second harmonic drive is a viable transduction principle for electrically interrogated BAW resonators with micron-scale gaps and limited supply voltage in air. As compared to [4], there is no modification to the structural design of the resonator for bulk mode sensing and a large transduction area is available for actuation.
4.1. One-port and two-port measurements A comparison between one-port and two-port measurements for fundamental frequency capacitive drive and sensing demonstrate a small signal enhancement utilizing the two-port scheme as compared to the one-port measurement, as shown in Fig. 9a and b. The extracted quality factor using Lorentzian function fit from the two-port measurement is 6771. The detailed comparisons between the one-port and two-port measurements have been previously reported in [3]. 4.2. Harmonic driving and mixing measurements Measurements for two-port 2nd harmonic and mixing techniques successfully allows feedthrough currents to be rejected by separating them from the motional current in the frequency domain and showed enhanced signal amplitudes of 8 dB (Fig. 9c) and 9 dB (Fig. 9d), respectively. The difference is due to the fact that 2nd harmonic driving is limited by feedthrough from the harmonics of the drive signal, whereas the drive signal for the mixing method can be selected to reject harmonics overlapping with the detection frequency. 4.3. Piezoresistive sensing measurements The one-port capacitive driving and piezoresistive sensing measurements showed a peak height of 0.5 dB with 5.9 mA drain
4.4. Comparison between experimental, analytical and simulated piezoresistive data The transconductance gm can be extracted from the measured data using Lorentzian function fit. By considering the phase and magnitude plots of the measured frequency response with and without dc bias, the contribution from parasitic feedthrough can be isolated. Similarly, by considering the measured frequency response with and without the drain current, the motional capacitance can be isolated. These can be subtracted from the measured frequency response to extract the motional conductance, and fit the Lorentzian function to extract the resonator parameters. This procedure is applied to the experimental data for one-port drive and piezoresistive sensing. The measured and extracted frequency response curves for a range of dc bias are shown in Figs. 10 and 11. The extracted frequency response curves for a range of drain current are shown in Fig. 12. The extracted gm and (Rr /Rr ) versus dc bias and drain current is plotted in Fig. 13, together with the analytical model from Eq. (24). From the graphs, it can be seen that the (Rr /Rr ) increases linearly with dc bias and remains the same for different drain currents, where as transconductance gm increases linearly with dc bias and drain currents, respectively. This is expected as (Rr /Rr ) is dependent on resonator displacement which is proportional to dc bias, but independent of drain current, whereas gm is proportional to dc bias as well as the drain current as in Eq. (24). Although the measured signal magnitude is smaller
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Acknowledgements This work was supported by the US Army Soldier Systems Center. A. Lin is a Cambridge Commonwealth Trust Scholar, and she would like to thank S. Ghosh and V. Ostanin for useful discussions and help with Mathematica. References
Fig. 14. Measured, modelled and simulated piezoresistance change as a function of resonator displacement.
than that predicted by the analytical model, the trends with dc bias and drain current is in agreement. A comparison between the (Rr /Rr ) extracted from the experimental data, simulated using ANSYS and the analytical model described above for various resonator displacement at the centre edge of the square plate is shown in Fig. 14. The piezoresistance change is proportional to displacement in all three sets of data for small displacements. The magnitude of the experimentally measured response is ∼7% larger than simulation, and ∼14% smaller than predicted by the analytical model. The differences between the calculated and measured value may be due to fabrication tolerances, as a slight change in the anchor dimension would cause a large difference in the relative strain energy distribution and hence the piezoresistive signal output. The analytical model is a simplified one that does not consider the strain at the junction between the stem and the cap, which may have a substantial effect on the piezoresistive signal output. Overall the experimental data is in good agreement with the simulation and the analytical model, considering that the trend is the same, and the data fits well within the experimental variations in the model. 5. Conclusions This paper documents and compared six transduction methods for electrically characterizing a square-extensional mode resonator. We have demonstrated that piezoresistive sensing combined with second harmonic drive substantially rejects the effect of capacitive feedthrough parasitics thereby enhancing readout of the motional signal, which enables strong motional signal to feedthrough current ratios at nominal operating voltages. The measured quality factor from the 3 dB bandwidth method for the second harmonic drive and piezoresistive sensing is comparable to the extracted quality factor using the traditional two-port method, which shows that this is a viable transduction principle for electrically interrogated BAW resonators. At micron-scale transduction gaps, this method outperforms previously reported methods utilizing voltages an order of magnitude lower, at ±3 V in air, providing a promising solution for low voltage CMOS-MEMS integration.
[1] J.E.-Y. Lee, Y. Zhu, A.A. Seshia, A bulk acoustic mode single-crystal silicon microresonator with a high quality factor, J. Micromech. Microeng. 18 (2008) 064001. [2] V.J. Logeeswaran, F.E.H. Tay, M.L. Chan, F.S. Chau, Y.C. Liang, First harmonic (2f) characterisation of resonant frequency and Q-factor of micromechanical transducers, in: Analog Integrated Circuits and Signal Processing, vol. 37, 2003 Oct, pp. 17–33. [3] J.R. Clark, W.T. Hsu, M.A. Abdelmoneum, C.T.C. Nguyen, High-Q UHF micromechanical radial-contour mode disk resonators, J. Microelectromech. Syst. 14 (2005 Dec) 1298–1310. [4] J.T.M. Van Beek, P.G. Steeneken, B. Giesbers, A 10 MHz piezoresistive MEMS resonator with high Q, in: Proceedings of the 2006 IEEE International Frequency Control Symposium and Exposition, vols. 1 and 2, 2006, pp. 475–480. [5] D. Weinstein, S.A. Bhave, Piezoresistive sensing of a dielectrically actuated silicon bar resonator, in: Solid-State Sensors, Actuators, and Microsystems Workshop, Hilton Head Island, South Carolina, 2008, pp. 368–371. [6] Y. Kanda, A graphical representation of the piezoresistance coefficients in silicon, IEEE Trans. Electron. Dev. 29 (1982) 64–70. [7] V. Kaajakari, T. Mattila, A. Qja, J. Kiihamaki, H. Seppa, Square-extensional mode single-crystal silicon micromechanical resonator for low-phase-noise oscillator applications, IEEE Electron. Dev. Lett. 25 (2004 Apr) 173–175.
Biographies Angel T.-H. Lin received her B.Sc. and M.Sc. degree in Electrical and Computer Engineering from the University of Cape Town, South Africa, in 2005 and 2007, respectively, where she studied Electroencephalogram for use in Brain Computer Interfaces. She is currently a Ph.D. student in the University of Cambridge, UK, affiliated with the Nanoscience Centre. Her research interest is in the design and fabrication of MEMS sensors. She is a student member of the IEEE. Joshua E.-Y. Lee received the B.A. (Hons) and M.Eng. (with distinction) degrees in 2005, and most recently the Ph.D. degree, all from the University of Cambridge, UK. He has previously undertaken research attachments at the Institute of Microelectronics, Singapore on two occasions in 2003 and 2004, working on electrochemical biosensors, microfluidics, and FEA modeling of impact mechanics problems. He is currently an Assistant Professor with the Department of Electronic Engineering at the City University of Hong Kong. His current research interests include analogue circuit design for microstructures and sensors, and developing silicon-based microelectromechanical resonators for mass and charge sensing, with extensions to oscillator and filter applications. He is a member of IEEE and ASME, and an associate member of IoP and IET. In January 2008, he was awarded a Research Student Development Fellowship by the Royal Academy of Engineering, UK. Jize Yan received the B.S. degree from Tsinghua University, China, 2003, and the Ph.D. degree from the University of Cambridge, UK, 2007. He is currently a Research Associate with the Engineering Department and Nanoscience Centre, University of Cambridge. His research interests include Radio Frequency Microelectromechanical Systems (MEMS), MEMS/NEMS Sensors, Power Harvesting and Micro-/Nano-Fabrication. He is a member of IEEE. Ashwin A. Seshia received the B.Tech. in Engineering Physics in 1996 from IIT Bombay, and M.S. and Ph.D. degrees in Electrical Engineering and Computer Science from the University of California, Berkeley in 1999 and 2002, respectively and a MA from the University of Cambridge in 2008. During his time at the University of California, Berkeley he was affiliated with the Berkeley Sensor & Actuator Center. He joined the faculty of the Engineering Department at the University of Cambridge in October 2002 where he is presently a Lecturer in Micro- Electro-Mechanical Systems, a Fellow of Queens’ College and affiliated with the Nanoscience Centre. His research interests include the design and fabrication of micro- and nano-scale sensory systems with applications to the monitoring and study of the natural and built environment. He is a member of the IEEE and ASME and in 2008 was appointed as Fellow of the ERA Foundation.
Contents lists available at ScienceDirect
Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna
Methods for enhanced electrical transduction and characterization of micromechanical resonators A.T-H. Lin a , J.E-Y. Lee a,b , J. Yan a , A.A. Seshia a,∗ a b
Department of Engineering, Trumpington Street, University of Cambridge, Cambridge, United Kingdom Department of Electronic Engineering, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong
a r t i c l e
i n f o
Article history: Received 3 August 2009 Received in revised form 18 January 2010 Accepted 18 January 2010 Available online 25 January 2010 Keywords: Bulk mode resonators Harmonic driving Piezoresistive sensing
a b s t r a c t This paper details the design and enhanced electrical transduction of a bulk acoustic mode resonator fabricated in a commercial foundry MEMS process utilizing 2.5 m gaps. The I–V characteristics of electrically addressed silicon resonators are often dominated by capacitive parasitics, inherent to hybrid technologies. This paper benchmarks a variety of drive and detection principles for electrostatically driven square-extensional mode resonators operating in air via analytical models accompanied by measurements of fabricated devices with the primary aim of enhancing the ratio of the motional to feedthrough current at nominal operating voltages. In view of ultimately enhancing the motional to feedthrough current ratio, a new detection technique that combines second harmonic capacitive actuation and piezoresistive detection is presented herein. This new method is shown to outperform previously reported methods utilizing voltages as low as ±3 V in air, providing a promising solution for low voltage CMOS-MEMS integration. To elucidate the basis of this improvement in signal output from measured devices, an approximate analytical model for piezoresistive sensing specific to the resonator topology reported here is also developed and presented. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Silicon bulk acoustic wave (BAW) microresonators have been demonstrated for high Q solutions to a variety of applications such as timing references, filters and precision sensors. These devices are recognised as a promising alternative to quartz crystal resonators due to their realizable small form factor and the potential for integration with standard IC electronics. However, due to the relatively high stiffness and large motional resistance, the motional signal is often swamped by parasitic feedthrough in hybrid technologies that do not integrate electronics and MEMS on the same substrate. The physical sources of this capacitive feedthrough parasitic includes (1) the direct overlapping capacitance of the transducer, (2) capacitive coupling through the substrate via the bond pads, (3) interconnects, and (4) the electrical package [1]. Although integration with CMOS results in lower capacitive parasitics, the resonator bias voltages are often limited to less than 5 V, which sets a limit for the detectable motional signal. One way to increase the motional signal is to reduce the transduction gap thereby decreasing the motional resistance. However, the realization of sub-micron transduction gaps relies on advanced fabrication
∗ Corresponding author. E-mail address: [email protected] (A.A. Seshia). 0924-4247/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2010.01.024
technology that often requires careful control on the definition of sidewall spacers and/or sacrificial layers. The latter often places significant constraints on the MEMS fabrication process, integration with CMOS and ultimately device yield. Operating the resonator in vacuum often results in lower damping and an increased motional signal at the expense of the added complexity of vacuum packaging. Various studies have been devoted to actuation and detection principles with the aim to enhance the motional to feedthrough current ratio, including harmonic actuation, mixing techniques and piezoresistive sensing [2–5]. This paper benchmarks these principles for an electrostatically driven square-extensional (SE) mode resonator operating in atmospheric pressure and fabricated in a foundry MEMS process utilizing 2.5 m gaps. The aim is to illustrate techniques and principles that enhance the ratio of the motional to feedthrough current at nominal operating voltages. To this end, analytical models are presented to elucidate the relative merits between various methods, which are further validated via actual measurements of fabricated SOI microresonators. Pursuant to the ultimate aim of enhancing motional to feedthrough current ratio, a new detection technique is also introduced that combines second harmonic capacitive actuation and piezoresistive detection. This method outperforms previously reported methods and utilizes voltages an order of magnitude lower, at ±3 V in air. To elucidate the basis of this improvement in signal output from measured devices,
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Fig. 2. Optical micrograph of the square resonator fabricated by the MEMSCAP SOIMUMPs process.
Fig. 1. The square-extensional (SE) mode shape from ANSYS FE simulation.
anchors are lithographically patterned on the top silicon and DRIE etched. A polyimide protective coat is applied to the top silicon surface and the substrate layer is etched through from the backside to release the structures. The remaining exposed oxide is removed using a vapor HF process to prevent stiction, after which, the polyimide layer is removed to fully release the structure. The final step is the metallization of the substrate layer using a shadow mask to provide a topside electrical contact to the substrate.
Table 1 Resonator parameters. Parameters
Symbol
Value
Side length Thickness Transduction gap Resonant frequency Measured quality factor
L H g f Q
2000 m 25 m 2.5 m 2.2 MHz 6771
3. Measurement theory and setup
an approximate analytical model for piezoresistive sensing through the resonator is also presented.
This section presents the theory behind the progression of electrostatic methods for reducing feedthrough and introduces piezoresistive sensing methods for increasing motional signal. This model sets up the six transduction methods that are documented and compared, as summarized in Table 2.
2. Device description and fabrication The tested device is a 2.2 MHz square plate resonator operated in the SE mode, wherein the square plate undergoes an extension and contraction symmetrically on each side of the square, as illustrated in Fig. 1. The suspended square is anchored to the substrate via a T-shape connecting stem located on four corners of the square plate. The resonator parameters are shown in Table 1. The measured transduction gap of the tested device is found to be 2.5 m. The resonator is fabricated in a commercial SOI foundry process (MEMSCAP), with one lateral electrode on each side of the square plate. An electrical connection is provided to each anchor of the resonator body. This enables the capacitive and piezoresistive sensing configurations used in this work. An optical micrograph of the fabricated resonator is shown in Fig. 2. The fabrication process begins with a highly doped 25 m thick n-type device layer, followed by metallization to pattern the bond pads through a lift-off process. The resonators, electrodes and
3.1. Electrostatic actuation and sensing Electrostatic actuation is achieved by applying a voltage across a capacitive gap which results in an electrostatic force. The electrostatic energy stored in a capacitor C with potential difference of V applied, is given by 1 2 CV 2
We =
(1)
For a parallel plate electrostatic actuator, the corresponding actuation force is determined by taking the differential of Eq. (1): 1 F= 2
∂C ∂x
V2
(2)
Table 2 Summary of transduction methods, where ω0 , meq , Q, ε0 , A are the angular resonant frequency, equivalent mass of the resonator, quality factor, permittivity of air and the effective transduction area, respectively. Transduction methods One-port Two-port 2nd harmonic Mixing
Motional current (|Im |)
Actuation force (F0 ) Vdc
V (ε
Vdc
V (ε
ac
ac
0 A/g 0 A/g
2
2
)
Qε2 A2 /m ( V 2 V
)
Qε2 A2 /m ( V 2 V
dc
dc
( Vac 2 /4)(ε0 A/g 2 ) 1 2
VRF
0 A/g
One-port driving and piezoresistive sensing
Vdc
2nd harmonic driving and piezoresistive sensing
( V 2 /4)(ε
ac
ac
2
)
0 A/g
2
ac
dc
2
ac
4
)
eq ω0 g
4
eq ω0 g
0
eq ω0 g
0
V V Qε2 A2 /2m ( V dc
RF
(Rr /Rr )(Vd /R) )
)
0
V 2 Qε2 A2 /4m ( V
VLO (ε0 A/g )
V (ε
ac
Parameters 4
(Rr /Rr )(Vd /R)
LO
0
)
eq ω0 g
4
)
Vdc = 30 V Vac = 0 dBm (50 ) Vdc = 30 V Vac = 0 dBm (50 ) Vdc = 30 V Vac = ±3 V Vdc = 30 V VRF = ±3 V, VLO = ±3 V Vdc = 30 V, Vac = 0 dBm(50 ) Vd = −2 V, Id = 5.9 mA Vac = ±3 V Vd = −2 V, Id = 5.9 mA
Peak height (dB) ∼0.02 ∼0.04 ∼8 ∼9 ∼0.5 ∼13
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265
Fig. 3. Schematic showing measurement setups for (a) one-port, (b) two-port, (c) 2nd harmonic, (d) mixing, (e) piezoresistive sensing and (f) 2nd harmonic drive with piezoresistive sensing.
where V is the applied voltage, x is the resonator displacement, C = (ε0 A/g) is the capacitance of the parallel plate, and ε0 , g, A are the permittivity, transduction gap and transduction area, respectively. In a one-port configuration as shown in Fig. 3a, all four electrodes are used for actuation, where the motional signal is detected from the resonator anchors through capacitive coupling between the electrodes and the resonator. For fundamental frequency driving, the applied voltage can be written as the sum of a dc bias Vdc and an ac signal Vac = Vac cos(ω0 t) at resonant frequency ω0 , yielding the following expression for actuator force:
F =
1 2
1 = 2
∂C ∂x ∂C ∂x
×
2 Vdc
[Vdc + Vac cos(ω0 t)]
2
2 2 Vac Vac + 2Vdc Vac cos(ω0 t) + + cos(2ω0 t) 2
2
The force component of interest is that which occurs at the resonant frequency ω0 , this gives
∂C = F0 cos(ω0 t) F = Vdc Vac cos(ω0 t) ∂x
where
∂C ε0 A = Vdc Vac F0 = Vdc Vac g2
∂x
(3)
Although maximizing electrode area maximizes the capacitive motional current in a one-port configuration, the feedthrough current can also be substantially large as the actuator capacitor also serves to capacitively couple the resonator body to the surrounding electrodes. In the two-port configuration, as shown in Fig. 3b, the resonator body is grounded and the electrodes are split in two pairs, one used for driving and the other for sensing to reduce the feedthrough through the direct overlap capacitor between the actuator electrode and the sensing port [3]. Although the effective transduction area is reduced by a factor of two, the reduction in the feedthrough parasitics enabled a clearer measurement. However, feedthrough between drive and sense electrodes could still exist via the substrate or the package, which can mask the motional current. To further reduce these feedthrough parasitics, one approach is to shift the feedthrough current relative to the motional current in the frequency domain, and utilize the high-Q resonant response to filter the feedthrough current at the resonant frequency [2,3]. In second harmonic measurements, the nonlinearity of electrostatic transduction can be utilized to drive the structure at half the resonant frequency:
Vac = Vac cos
ω
0
2
t
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limitations that is required for capacitive sensing, which can be large for a bulk mode resonator.
This results in a force at the resonant frequency, given by F=
2 Vac 4
cos(ω0 t)
where
2 Vac ∂C
F0 =
4
∂x
=
∂C ∂x
= F0 cos(ω0 t)
3.2. Piezoresistive sensing
2 Vac ε A 0
(4)
g2
4
Second harmonic drive can be used in both one-port and two-port configurations, though the feedthrough effect from the harmonics of the driving signal, albeit much smaller, can still be observed. A two-port 2nd harmonic setup (Fig. 3c) therefore has better transduction than one-port 2nd harmonic setup, despite the reduction in transduction area. If two ac input frequencies are available, the device can be driven by two off resonance signals VRF and VLO . From Eq. (1), this generates a force at their sum or difference frequency. By setting the input frequencies ωRF , ωLO such that ω0 = ωRF + ωLO , The actuation force is given by 1 2
F =
1 = 2
∂C ∂x ∂C ∂x
2 [VRF cos(ωRF t) − VLO cos(ωLO t)]
([VRF cos(ωRF t)] − 2 VRF VLO cos(ωRF t)cos(ωLO t) 2
+ [VLO cos(ωLO t)] ) 1 2
=
∂C ∂x
(. . . − VRF VLO cos((ωRF − ωLO )t)
Im ≈
Vd R
R
r
(8)
Rr
where R = Rr + Rs , and Rr , Rs are the resistance of the resonator and the resistance of the externally controlled resistor, respectively. In the one-port capacitive driving and piezoresistive sensing configuration, the output signal is a combination of both capacitive and piezoresistive signal. The measured admittance as a function of driving frequency ω can be expressed as Y (ω) =
2
Apart from capacitive measurement techniques, the mechanical motion of the resonator operating in the SE mode can also be detected through the variation in the electrical resistance of the vibrating structure due to the piezoresistive effect of single crystal silicon. This method has been previously shown to be a promising alternative for increasing the amplitude of the motional signal [4,5]. In this work, the electrical resistance between two corners of the resonator structure is monitored by applying a voltage difference, as shown in Fig. 3e. The motional current measured is proportional to the total change in resistance (Rr /Rr ) of the resonator structure and the applied dc voltage Vd , given by
jωCm 2
1 − (ω/ω0 ) + j(ω/ω0 )/Q
+ gm + jωCf
where Cm , gm , Cf are the motional capacitance, transconductance and feedthrough capacitance, respectively. The transconductance is given by
− VRF VLO cos((ωRF + ωLO )t) + . . .)
Im Vd = Vac Vac R
R
r
The force of interest is that which occurs at the resonant frequency ω0 , given by
gm =
1 F = VRF VLO cos(ω0 t) 2
The change in resistance due to piezoresistivity follows:
where F0 =
1 VRF VLO 2
∂C ∂x
∂C ∂x
= F0 =
= F0 cos(ω0 t)
ε0 A 1 VRF VLO 2 2 g
F0 Q meq ω0 2
(5)
(6)
where Q, meq are the quality factor and equivalent mass of the resonator. The motional current detected using capacitive sensing is the result of the modulation of the capacitance due to changes in the gap size from the vibration motion, given by
im = Vdc
∂C ∂x
x˙
(10)
Rr
Rr = l l + t t Rr
This mixing setup is shown in Fig. 3d. In this approach, if the harmonics of the driving signal do not overlap with the sensing signal, the feedthrough effect is minimized. The maximum displacement at resonance for a given actuation force can be expressed as X0 =
(9)
(7)
Substituting the force Eqs. (3)–(5) and displacement Eq. (6) into (7), the motional current for each capacitive measurement method was derived, as summarized in Table 2. It can be seen from the motional current expressions, that capacitive sensing is largely limited by the available dc bias and the transduction gap size, as the detectable motional current is proportional to Vdc and inversely proportional to g4 . Thus for a resonator design, there is a minimal dc voltage
(11)
where l , t are the longitudinal and transverse pezoresistive coefficients, and l , t are the longitudinal and transverse stresses, respectively. 3.2.1. Piezoresistance of the resonator bulk An approximate analytical model was developed to relate the stresses in the resonator bulk (i.e. suspended square plate) for the SE mode to the change in resistance between the two diagonal corners of the square plate. The side edge of the square is aligned to the 1 1 0 direction, with the current flow diagonally across two anchors in the 1 0 0 direction, where l is maximized for n-type silicon [6]. The square can be thought of as numerous resistors connected in parallel and series diagonally as illustrated in Fig. 4. These resistances can be added up in parallel and series to obtain the overall resistance of the square plate between two corners. The SE mode shape can be approximated using a sine function [7]: ux (x) = Ux sin
x
L
uy (y) = Uy sin
y
L
(12)
where Ux , Uy are the maximum displacement along the x- and yaxis, in the coordinate system shown in Fig. 5(a), and L is the side length of the square. For simplicity in calculation, the coordinate system is transformed to orient the x-axis along the direction of the applied field (i.e. rotated 45 degrees), the mode shape equation
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267
Fig. 4. Diagram illustrating the parallel and series resistance in the square.
Fig. 7. Diagram showing the anchor dimensions.
of the displacement, given by εx =
dux (x, y) Ux cos = 2L dx
duy (x, y) Uy = εy = cos 2L dx Fig. 5. The mode shape plot showing the coordinate system for SE mode with (a) side edge aligned to x-axis and (b) side edge rotated 45 degrees from x-axis.
Uy uy (x, y) = √ sin 2
Ux √ (x + y) − √ sin 2L 2 Uy √ (x + y) + √ sin 2L 2
√ (y − x) 2L
(13)
√ (y − x) 2L
Ux cos √ (x + y) + 2L 2L
U y
cos √ (x+y) + 2L 2L
√ (y−x) 2L
(14)
√ (y−x) 2L
The resistance change in the square plate due to piezoresistivity is given by Rsq = ESE (l εx + t εy ) Rsq
becomes Ux ux (x, y) = √ sin 2
(15)
where ESE is the biaxial modulus of the SE mode. The resistance Rc of a conductor with uniform cross-section is given by Rc = ( l/A), where , l, A are the resistively, length of the material, and the crosssectional area, respectively. The static resistance of the resonator can be approximated (assuming uniform E-field) by integrating the resistance on the right half triangle area of the resonator, and mul-
where x, y now refers to the transformed coordinate axis as shown in Fig. 5(b). The strain along the x- and y-direction is the derivative
Fig. 6. MATLAB plot showing derived motional current density distribution in the resonator bulk.
Fig. 8. ANSYS FE simulation showing the difference in distribution of current density sum before and after applied strain.
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Fig. 9. Measured frequency response for a 2.2 MHz SE mode resonator in air using (a) one-port, (b) two-port, (c) second harmonic, (d) mixing measurements, (e) one-port capacitive drive with piezoresistive sensing, (f) second harmonic drive with piezoresistive sensing, (g) comparison between the capacitive and piezoresistive sensing methods. The parameters for each measurement are shown in Table 2.
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269
Fig. 11. Restored frequency response curves using one-port drive and piezoresistive sensing in terms of (a) magnitude and (b) phase for a range of applied dc bias.
Fig. 10. Measured frequency response curves using one-port drive and piezoresistive sensing in terms of (a) magnitude and (b) phase for a range of applied dc bias.
Rsq,d = Rsq,s [1 + ESE ε(l + t )] tiply by two to get the expression for the total static resistance:
Rsq
=2× H
˛
dx
=
√ (−2x + 2L)
H
dx dy
+ cos
(16)
0
1 + ESE
U ( + t ) cos 2L l
√ (y − x) 2L
√ (x + y) 2L
(18)
The total dynamic resistance for the square is given by summing up the resistance for all points in the square plate:
˛ Rsq,s + Rsq,s =
√ −x+(L/ 2) 0
dx
H
1 + ESE U (l + t ) cos 2L
√ x−(L/ 2)
where H is the thickness of the resonator, and ˛ is the length from the centre of the resonator to the point where it meets the anchor. The dynamic resistance at each node is therefore given by Rsq,d = Rsq,s + Rsq,s = Rsq,s + Rsq,s ESE (l εx + t εy ) = Rsq,s [1 + ESE (l εx + t εy )]
(17)
where Rsq,d , Rsq,s is the dynamic and static resistance at a given node. From Eq. (14), we can see that, the strain in the x- and y-direction for SE mode is equal (i.e. εx = εy = ε), the expression for dynamic resistance can be simplified to
√ (x 2L
(19)
+ y) + cos
√ (y 2L
−1
− x)
dy
The above integral was solved using Mathematica by plugging in the parameters and the dynamic resistance for the square is obtained. The calculated resistance change of the square corresponding to an actuated displacement of U = 9.601 × 10−11 [m] is (Rsq /Rsq ) = −3.015 × 10−6 . Fig. 6 shows the relative motional signal plot of the resonator bulk in MATLAB. The motional signal is maximum at the centre of the square as the stress is maximum at the centre for the SE mode. 3.2.2. Piezoresistance of the resonator anchors As the square extends in all directions, maximum displacement occurs at each of the four corners, straining the T-shaped anchors connecting to the corners of the square. The geometry of the T-
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Fig. 12. Restored frequency response curves using one-port drive and piezoresistive sensing in terms of (a) magnitude and (b) phase for a range of applied drain current.
anchor is shown in Fig. 7, the ‘stem’ part is aligned to the 1 0 0 direction where l is maximized for n-type silicon. A finite element simulation was done to approximate the resistance change in the anchors for the actuated displacement value using ANSYS. A plot of the change in current density sum before and after displacement load is shown in Fig. 8. The extracted value for the resistance change in the anchors from the FE simulation corresponding to the actuated displacement above is (Ranchor /Ranchor ) = 3.921 × 10−5 . This is about an order of magnitude larger than the bulk resistance change. The overall change in resistance is the sum of resistance change in the bulk and anchors, given by Rsq + Ranchor Rr = Rr Rsq + Ranchor
(20)
Due to the high resistance and high current density distribution at the anchors, this result in majority of the piezoresistive signal came from the anchors as oppose to the bulk. The model for the bulk assumes that current density at every elemental resistor is the same, while in reality current density is maximum in the middle and goes to zero towards the edges. However, since the bulk is not the major contributor to the measured motional current, we did not give further detailed model of the bulk. An approximate analytical model is also developed to relate the piezoresistive signal to the stress induced in the anchors. It is assumed that the piezoresistive signal from the ‘cap’ is very small and can be neglected compare to the ‘stem’. This is because as the ‘cap’ bends, the piezoresistive change due to compression at the inner side cancels out with piezoresistive change due to tension on
Fig. 13. Measured and analytical model for (a) transconductance and (b) Rr /Rr as a function of dc bias and drain current.
the outer side. This can also be seen from the ANSYS simulation in Fig. 8. To calculate the amount of stress in the stem part, the T-shaped anchor can be thought of as a beam (stem part) that is anchored to the centre of a clamped-clamped beam (cap part). A displacement force from the corner of the resonating square causes the beam to compress and the cap to bend. The displacement ratio in the stem and cap is inversely proportional to their spring constant: kcap xstem = xcap kstem
(21)
Therefore, the stiffer the cap, the larger the stress induced in the stem, and hence larger the piezoresistive signal for a given displacement. However, note that the stiffer the cap, the higher the energy loss through the anchors, and hence lower the quality factor of the resonator. There is therefore a trade off between piezoresistive signal enhancement and Q of the system. Using this simplified model, for a given displacement U at the centre edge of the square, the stem is compressed by xstem =
√
2U
kcap kstem + kcap
(22)
The resultant change in resistance in the stem is given by Rstem xstem = E100 ( − t ) Rstem Lstem l
(23)
where E100 , Lstem , are the Young’s modulus in the 1 0 0 direction, length of the stem and the Poisson ratio, respectively. Substituting into equation for gm , assuming the resistance change in the stem contribute to all of the motional transconductance, and neglecting
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the other parts we get: gm
= = = = = =
2R
271
current, as shown in Fig. 9e. This is an improvement to the one/twoport capacitive sensing scheme. Note that in this configuration, the detected signal contains both the capacitive and piezoresistive components of the motional current.
Vd 2Rstem Vd Rstem stem = Vac R Rr Vac R Rstem Rr
Vd 2Rstem E100 ( − t )xstem Vac R Rr Lstem l √ kcap Vd 2Rstem E100 ( − t ) 2U Vac R Rr Lstem l kstem + kcap 2R E √ F0 /keff (24) kcap Vd stem 100 ( − t ) 2 Vac R Rr Lstem l kstem + kcap 1 − (ω/ω0 )2 + j(ω/ω0 )/Q 2R E √ kcap Vd ε0 Le 1 1 stem 100 (l − t ) 2 Vac Vdc 2 Vac R Rr Lstem kstem + kcap g 2 ESE 1 − (ω/ω0 )2 + j(ω/ω0 )/Q
√ kcap Vd 2Rstem E100 ε0 Le 1 1 ( − t ) 2 V R Rr Lstem l kstem + kcap dc g 2 2 ESE 1 − (ω/ω0 )2 + j(ω/ω0 )/Q
where Le , ε0 are the length of electrode on each side of the square, permittivity of air, and keff = 2 ESE H is the effective spring constant of the SE mode. With piezoresistive sensing, the motional signal can be enhanced without necessarily scaling the transduction gap and the drive signal, and can be significantly larger than capacitive measurements. The current through the resonator can be tuned by setting Vd , and is ultimately limited by the maximum current density ceiling for single crystal silicon and the electrical power dissipated in the resonator. Since the motional current is detected via the resonator body, all electrode areas can be utilized for electrostatic actuation of the SE mode. Piezoresistive sensing can be combined with second harmonic drive (Fig. 3f) to utilize both advantages of increased motional signal and reduced parasitic feedthrough for further signal enhancements at nominal operating voltages. 4. Results and discussion Measurements results for each transduction scheme employed for the same square-extensional mode resonator are shown in Fig. 9. All the measurements are done at atmospheric pressure. For capacitive sensing of the motional current, a dc bias of 30 V was applied between the resonator and the electrode.
Further signal enhancement is achieved by combining second harmonic drive and piezoresistive sensing. This allows the rejection of feedthrough currents as well as eliminates the need of a large dc bias for driving and sensing. Fig. 9f demonstrates the measured signal with peak height of 13 dB, an ac actuation voltage of ±3 V and drain current of 5.9 mA. The quality factor measured by taking the 3 dB bandwidth of the measured signal is 6550, which is close to the extracted quality factor of 6771 with a two-port capacitive measurement, showing that the signal distortion from the parasitic feedthrough is largely eliminated. Fig. 9g illustrates the difference in signal strength for capacitive and piezoresistive sensing on the same graph, a clear motional signal enhancement can be seen for the case corresponding to 2nd harmonic drive and piezoresistive sensing. Comparing all the measurement results, we can see that the distortion of the measured resonant peak due to parasitic feedthrough is visible in the one/two-port measurement, and is substantially rejected in the harmonic driving and mixing measurements, and almost unnoticeable in the piezoresistive sensing measurements. Thus, piezoresistive sensing combined with second harmonic drive is a viable transduction principle for electrically interrogated BAW resonators with micron-scale gaps and limited supply voltage in air. As compared to [4], there is no modification to the structural design of the resonator for bulk mode sensing and a large transduction area is available for actuation.
4.1. One-port and two-port measurements A comparison between one-port and two-port measurements for fundamental frequency capacitive drive and sensing demonstrate a small signal enhancement utilizing the two-port scheme as compared to the one-port measurement, as shown in Fig. 9a and b. The extracted quality factor using Lorentzian function fit from the two-port measurement is 6771. The detailed comparisons between the one-port and two-port measurements have been previously reported in [3]. 4.2. Harmonic driving and mixing measurements Measurements for two-port 2nd harmonic and mixing techniques successfully allows feedthrough currents to be rejected by separating them from the motional current in the frequency domain and showed enhanced signal amplitudes of 8 dB (Fig. 9c) and 9 dB (Fig. 9d), respectively. The difference is due to the fact that 2nd harmonic driving is limited by feedthrough from the harmonics of the drive signal, whereas the drive signal for the mixing method can be selected to reject harmonics overlapping with the detection frequency. 4.3. Piezoresistive sensing measurements The one-port capacitive driving and piezoresistive sensing measurements showed a peak height of 0.5 dB with 5.9 mA drain
4.4. Comparison between experimental, analytical and simulated piezoresistive data The transconductance gm can be extracted from the measured data using Lorentzian function fit. By considering the phase and magnitude plots of the measured frequency response with and without dc bias, the contribution from parasitic feedthrough can be isolated. Similarly, by considering the measured frequency response with and without the drain current, the motional capacitance can be isolated. These can be subtracted from the measured frequency response to extract the motional conductance, and fit the Lorentzian function to extract the resonator parameters. This procedure is applied to the experimental data for one-port drive and piezoresistive sensing. The measured and extracted frequency response curves for a range of dc bias are shown in Figs. 10 and 11. The extracted frequency response curves for a range of drain current are shown in Fig. 12. The extracted gm and (Rr /Rr ) versus dc bias and drain current is plotted in Fig. 13, together with the analytical model from Eq. (24). From the graphs, it can be seen that the (Rr /Rr ) increases linearly with dc bias and remains the same for different drain currents, where as transconductance gm increases linearly with dc bias and drain currents, respectively. This is expected as (Rr /Rr ) is dependent on resonator displacement which is proportional to dc bias, but independent of drain current, whereas gm is proportional to dc bias as well as the drain current as in Eq. (24). Although the measured signal magnitude is smaller
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Acknowledgements This work was supported by the US Army Soldier Systems Center. A. Lin is a Cambridge Commonwealth Trust Scholar, and she would like to thank S. Ghosh and V. Ostanin for useful discussions and help with Mathematica. References
Fig. 14. Measured, modelled and simulated piezoresistance change as a function of resonator displacement.
than that predicted by the analytical model, the trends with dc bias and drain current is in agreement. A comparison between the (Rr /Rr ) extracted from the experimental data, simulated using ANSYS and the analytical model described above for various resonator displacement at the centre edge of the square plate is shown in Fig. 14. The piezoresistance change is proportional to displacement in all three sets of data for small displacements. The magnitude of the experimentally measured response is ∼7% larger than simulation, and ∼14% smaller than predicted by the analytical model. The differences between the calculated and measured value may be due to fabrication tolerances, as a slight change in the anchor dimension would cause a large difference in the relative strain energy distribution and hence the piezoresistive signal output. The analytical model is a simplified one that does not consider the strain at the junction between the stem and the cap, which may have a substantial effect on the piezoresistive signal output. Overall the experimental data is in good agreement with the simulation and the analytical model, considering that the trend is the same, and the data fits well within the experimental variations in the model. 5. Conclusions This paper documents and compared six transduction methods for electrically characterizing a square-extensional mode resonator. We have demonstrated that piezoresistive sensing combined with second harmonic drive substantially rejects the effect of capacitive feedthrough parasitics thereby enhancing readout of the motional signal, which enables strong motional signal to feedthrough current ratios at nominal operating voltages. The measured quality factor from the 3 dB bandwidth method for the second harmonic drive and piezoresistive sensing is comparable to the extracted quality factor using the traditional two-port method, which shows that this is a viable transduction principle for electrically interrogated BAW resonators. At micron-scale transduction gaps, this method outperforms previously reported methods utilizing voltages an order of magnitude lower, at ±3 V in air, providing a promising solution for low voltage CMOS-MEMS integration.
[1] J.E.-Y. Lee, Y. Zhu, A.A. Seshia, A bulk acoustic mode single-crystal silicon microresonator with a high quality factor, J. Micromech. Microeng. 18 (2008) 064001. [2] V.J. Logeeswaran, F.E.H. Tay, M.L. Chan, F.S. Chau, Y.C. Liang, First harmonic (2f) characterisation of resonant frequency and Q-factor of micromechanical transducers, in: Analog Integrated Circuits and Signal Processing, vol. 37, 2003 Oct, pp. 17–33. [3] J.R. Clark, W.T. Hsu, M.A. Abdelmoneum, C.T.C. Nguyen, High-Q UHF micromechanical radial-contour mode disk resonators, J. Microelectromech. Syst. 14 (2005 Dec) 1298–1310. [4] J.T.M. Van Beek, P.G. Steeneken, B. Giesbers, A 10 MHz piezoresistive MEMS resonator with high Q, in: Proceedings of the 2006 IEEE International Frequency Control Symposium and Exposition, vols. 1 and 2, 2006, pp. 475–480. [5] D. Weinstein, S.A. Bhave, Piezoresistive sensing of a dielectrically actuated silicon bar resonator, in: Solid-State Sensors, Actuators, and Microsystems Workshop, Hilton Head Island, South Carolina, 2008, pp. 368–371. [6] Y. Kanda, A graphical representation of the piezoresistance coefficients in silicon, IEEE Trans. Electron. Dev. 29 (1982) 64–70. [7] V. Kaajakari, T. Mattila, A. Qja, J. Kiihamaki, H. Seppa, Square-extensional mode single-crystal silicon micromechanical resonator for low-phase-noise oscillator applications, IEEE Electron. Dev. Lett. 25 (2004 Apr) 173–175.
Biographies Angel T.-H. Lin received her B.Sc. and M.Sc. degree in Electrical and Computer Engineering from the University of Cape Town, South Africa, in 2005 and 2007, respectively, where she studied Electroencephalogram for use in Brain Computer Interfaces. She is currently a Ph.D. student in the University of Cambridge, UK, affiliated with the Nanoscience Centre. Her research interest is in the design and fabrication of MEMS sensors. She is a student member of the IEEE. Joshua E.-Y. Lee received the B.A. (Hons) and M.Eng. (with distinction) degrees in 2005, and most recently the Ph.D. degree, all from the University of Cambridge, UK. He has previously undertaken research attachments at the Institute of Microelectronics, Singapore on two occasions in 2003 and 2004, working on electrochemical biosensors, microfluidics, and FEA modeling of impact mechanics problems. He is currently an Assistant Professor with the Department of Electronic Engineering at the City University of Hong Kong. His current research interests include analogue circuit design for microstructures and sensors, and developing silicon-based microelectromechanical resonators for mass and charge sensing, with extensions to oscillator and filter applications. He is a member of IEEE and ASME, and an associate member of IoP and IET. In January 2008, he was awarded a Research Student Development Fellowship by the Royal Academy of Engineering, UK. Jize Yan received the B.S. degree from Tsinghua University, China, 2003, and the Ph.D. degree from the University of Cambridge, UK, 2007. He is currently a Research Associate with the Engineering Department and Nanoscience Centre, University of Cambridge. His research interests include Radio Frequency Microelectromechanical Systems (MEMS), MEMS/NEMS Sensors, Power Harvesting and Micro-/Nano-Fabrication. He is a member of IEEE. Ashwin A. Seshia received the B.Tech. in Engineering Physics in 1996 from IIT Bombay, and M.S. and Ph.D. degrees in Electrical Engineering and Computer Science from the University of California, Berkeley in 1999 and 2002, respectively and a MA from the University of Cambridge in 2008. During his time at the University of California, Berkeley he was affiliated with the Berkeley Sensor & Actuator Center. He joined the faculty of the Engineering Department at the University of Cambridge in October 2002 where he is presently a Lecturer in Micro- Electro-Mechanical Systems, a Fellow of Queens’ College and affiliated with the Nanoscience Centre. His research interests include the design and fabrication of micro- and nano-scale sensory systems with applications to the monitoring and study of the natural and built environment. He is a member of the IEEE and ASME and in 2008 was appointed as Fellow of the ERA Foundation.