Anal. Chem. 2005, 77, 2690-2699
Magnetically Actuated Complementary Metal Oxide Semiconductor Resonant Cantilever Gas Sensor Systems C. Vancˇura,* M. Ru 1 egg,† Y. Li, C. Hagleitner,‡ and A. Hierlemann
Physical Electronics Laboratory, ETH Zurich, HPT-F16, 8093 Zurich, Switzerland
In the present paper, an electromagnetically actuated resonant cantilever gas sensor system is presented that features piezoresistive readout by means of stress-sensitive MOS transistors. The monolithic gas sensor system includes a polymer-coated resonant cantilever and the necessary oscillation feedback circuitry, both monolithically integrated on the same chip. The fully differential feedback circuit allows for operating the device in selfoscillation with the cantilever constituting the frequencydetermining element of the feedback loop. The combination of magnetic actuation and transistor-based readout entails little power dissipation on the cantilever and reduces the temperature increase in the sensitive polymer layer to less than 1 °C, whereas previous designs with thermally actuated cantilevers showed a temperature increase of up to 19 °C. The lower temperature of the sensitive polymer layer on the cantilever directly improves the sensitivity of the sensor system as the extent of analyte physisorption decreases with increasing temperature. The electromagnetic sensor design shows an almost 2 times larger gas sensitivity than the earlier design, which is thermally actuated and read out using p-diffused resistors. The gas sensor is fabricated using an industrial complementary metal oxide semiconductor (CMOS) process and post-CMOS micromachining. Microcantilevers were initially used as force sensors in atomic force microscopy,1 but they have been also applied to measuring temperature,2 magnetic fields,3 and viscosity.4,5 Another application example has most recently been given by Ono and Esashi.5 In this work, ultrathin microcantilevers were used for magnetic and optical force measurements. Micromachined cantilevers have been applied to chemical sensing, such as the detection of organic * Corresponding author. E-mail:
[email protected]. Phone: ++41 1 633 3927. Fax: ++41 1 633 1054. † Now at Sensirion AG, Zurich, Switzerland. ‡ Now at IBM Research Laboratory, Ru ¨ schlikon, Switzerland. (1) Berger, R.; Gerber, Ch.; Lang, H. P.; Gimzewski, J. J. Microelectron. Eng. 1997, 35, 375-379. (2) Barnes, J. R.; Stephenson, R. J.; Welland, M. E.; Gerber, Ch.; Gimzewski, J. Nature 1994, 372, 79-81. (3) Leichle´, T. C.; von Arx, M.; Allen, M. G. Proc. IEEE MEMS 2001, Interlaken, Switzerland, 2001; pp 274-277. (4) Vidic, A.; Then, D.; Ziegler, Ch. Ultamicroscopy 2003, 97, 407-416. (5) Ono, T.; Esashi, M. Rev. Sci. Instrum. 2003, 74, 5141-5146.
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volatiles6-13 or explosive vapors, such as trinitrotoluene.14 Further applications included humidity sensing,9,13,15 or the detection of mercury vapors,16,17 the investigation of the hydrogen storage capacity of carbon nanotubes,18 and the monitoring of chemical reduction reactions.19 Quantitative detection of the components of a binary mixture has been demonstrated by our group20 and by Kim et al.21 In ref 22 an array of polymer-coated cantilevers with simultaneous resonance frequency and bending readout has been described. Cantilevers can be operated in static and dynamic modes. In the static mode, the deflection of the cantilever upon analyte uptake and stress generation in the sensitive layer is detected. In most cases, optical readout methods such as laser light reflection are used. In the dynamic operation, different actuation schemes can be used to excite the cantilever to its mechanical resonance, such as photothermal, piezoelectric, or capacitive actuation. Photothermal actuation23 has the disadvantage that an external laser is needed, which cannot be integrated on chip and which (6) Hierlemann, A.; Lange, D.; Hagleitner, C.; Kerness, N.; Koll, A.; Brand, O.; Baltes, H. Sens. Actuators, B 2000, 70, 2-11. (7) Lange, D.; Hagleitner, C.; Brand, O.; Baltes, H. Digest of Technical Papers TRANSDUCERS 1999, Sendai, Japan, 1999; pp 1020-1023. (8) Maute, M.; Raible, S.; Prins, F. E.; Kern, D. P.; Ulmer, H.; Weimar, U.; Go ¨pel, W. Sens. Actuators, B 1999, 58, 505-511. (9) Thundat, T.; Chen, G. Y.; Warmack, R. J.; Allison, D. P.; Wachter, E. A. Anal. Chem. 1995, 67, 519-521. (10) Lange, D.; Koll, A.; Brand, O.; Baltes, H. Proc. SPIE 1998, 3224, 233-243. (11) Lange, D.; Hagleitner, C.; Brand, O.; Baltes, H. Proc. IEEE MEMS, Myazaki, Japan, 2000; pp 547-552. (12) Fadel, L.; Lochon, F.; Dufour, I.; Franc¸ ais, O. J. Micromech. Microeng. 2004, 14, S23-S30. (13) Adams, J. D.; Parrott, G.; Bauer, C.; Sant, T.; Manning, L.; Jones, M.; Rogers, B.; McCorkie, D.; Ferrell, T. L. Appl. Phys. Lett. 2003, 83, 3428-3430. (14) Pinnaduwage, L. A.; Wig, A.; Hedden, D. L.; Gehl, A.; Yi, D.; Thundat, T.; Lareau, R. T. J. Appl. Phys. 2004, 95, 5871-5875. (15) Boltshauser, T.; Scho¨nholzer, M.; Brand, O.; Baltes, H. J. Micromech. Microeng. 1992, 2, 205-207. (16) Thundat, T.; Wachter, E. A.; Sharp, S. L.; Warmack, R. J. Appl. Phys. Lett. 1995, 66, 1695-1697. (17) Rogers, B.; Manning, L.; Jones, M.; Sulchek, T.; Murray, K.; Beneschott, B.; Adams, J. D. Rev. Sci. Instrum. 2003, 74, 4899-4901. (18) Esashi, M. Digest of Technical Papers, 2002 Symposium on VLSI Technology, Piscataway, NJ, 2002; pp 6-9. (19) Thundat, T.; Maya, L. Surf. Sci. 1999, 439, L546-L552. (20) Lange, D.; Hagleitner, C.; Hierlemann, A.; Brand, O.; Baltes, H. Anal. Chem. 2002, 74, 3084-3095. (21) Kim, B. H.; Prins, F. E.; Kern, D. P.; Raible, S.; Weimar, U. Sens. Actuators, B 2001, 78, 12-18. (22) Battiston, F. M.; Ramseyer, J. P.; Lang, H. P.; Baller, M. K.; Gerber, Ch.; Gimzewski, J. K.; Meyer, E.; Gu ¨ ntherodt, H. J. Sens. Actuators, B 2001, 77, 122-131. (23) Lavrik, N. V.; Datskos, P. G. Appl. Phys. Lett. 2003, 82, 2697-2699. 10.1021/ac048378t CCC: $30.25
© 2005 American Chemical Society Published on Web 03/29/2005
Figure 1. Micrograph of the cantilever gas sensing system.
increases the costs and dimensions of the system. Cantilevers with piezoelectric overlay can be used as “self-sensing” devices:13 A piezoelectric layer is used to actuate the cantilever and, at the same time, to record its vibration. This actuation mode poses the problem that a piezoelectric layer, e.g., ZnO,24 has to be applied to the sensor. Such layers are not available in a standard complementary metal oxide semiconductor (CMOS) process, which has been used as a platform technology for the development of integrated sensor systems. Similar considerations hold for layers with magnetic properties, which allow for cantilever actuation by means of external, alternating magnetic fields.25 Capacitive actuation and readout26 of the cantilevers is CMOS-compatible and enables self-sensing devices in analogy to the piezoelectric approach. Cantilevers with electrothermal actuation have been presented by our group.20 This actuation scheme makes use of the bimorph effect, i.e., the different thermal expansion coefficients of the various layer materials forming the cantilever. In comparison to most other actuation schemes, electrothermal actuation is not symmetric with respect to the initial position of the cantilever; i.e., the cantilever bends only in one direction. The oscillation of such cantilevers has been read out by means of integrated p-diffused piezoresistors. These electrothermally actuated cantilevers have been monolithically integrated with a full feedback circuit and have been fabricated using an industrial CMOS process followed by two subsequent postprocessing steps. CMOS technology is the most common technology for fabricating integrated circuits. CMOS technology even enables the integration of different transducers or arrays of transducers with signal-conditioning and -processing circuitry on the same chip.27 Additionally, well-established industrial CMOS processes are commercially available. In this paper, we present a CMOS cantilever gas sensor system operated in the dynamic mode. An ac current through a metal (24) Lee, S. S.; White, R. M. Sens. Actuators, A 1996, 52, 41-45. (25) Han, W.; Lindsay, S. M. Appl. Phys. Lett. 1996, 69, 4111-4113. (26) Davis, Z. J.; Abadal, G.; Forse´n, E.; Hansen, O.; Campabadal, F.; Figueras, E.; Esteve, J.; Verd, J.; Pe´rez-Murano, F.; Borrise´, X.; Nilsson, G.; Maximov, I.; Montelius, L.; Barniol, N.; Boisen, A. Digest of Technical Papers TRANSDUCERS 2003, Boston, MA, 2003; pp 496-499. (27) Hagleitner, C.; Hierlemann, A.; Lange, D.; Kummer, A.; Kerness, N.; Brand, O.; Baltes, H. Nature 2001, 92, 1-9.
path along the edge of the cantilever gives rise to an oscillating Lorentz force if a static, external magnetic field is provided.28 The Lorentz force in turn actuates the cantilever. The cantilever deflection is detected by a transistor-based Wheatstone bridge located near the cantilever base. The cantilever fabrication by using methods of CMOS-compatible micromachining and a monolithic system comprising the cantilever sensor and the corresponding readout circuitry will be detailed below. SYSTEM DESIGN The electromagnetically actuated cantilever chemical sensor system (Figure 1) comprises the cantilever and dedicated circuitry on the same substrate. The cantilever consists of single-crystal silicon with a stack of dielectric layers of the CMOS process (silicon oxide and silicon nitride) on top. A current path with eight loops, realized by using the two metal layers of the CMOS process, is integrated along the edges of the cantilever (Figure 2) for actuation. A static, external magnetic field is oriented in parallel to the cantilever axis, so that a Lorentz force perpendicular to the cantilever plane is generated by a current through the metal path (Figure 3). The Lorentz force, FL, then effectuates a transverse deflection of the cantilever, since it is acting on the cantilever tip, where the metal lines, and, therefore, the currents, are oriented perpendicularly to the magnetic field. The Lorentz force can be calculated using the following formula:
FL ) NIlBext
(1)
where I denotes the current running through the metal path and l the mean length of the loop fraction that is perpendicular to the external magnetic field, Bext. N is the number of current loops integrated on the cantilever. In our case N ) 8, as four current loops are formed in the first metal layer of the CMOS process and another four are formed using the second metal layer. By applying a sinusoidal current through to the current loop, a transverse oscillation of the cantilever in the external magnetic field is achieved. (28) Lange, D.; Hagleitner, C.; Herzog, C.; Brand, O.; Baltes, H. Sens. Actuators, A 2003, 103, 150-155.
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Figure 5. Block diagram the of integrated feedback circuit.
Figure 2. Micrograph of a magnetically actuated cantilever with transistor-based readout. The Lorentz force is generated at the tip of the cantilever, where the magnetic field lines and the metal path are perpendicular.
Figure 3. Schematic view of the resonating cantilever with magnetic actuation and transistor-based readout.
Figure 4. Closeup micrograph of the transistor-based Wheatstone bridge for readout.
The cantilever vibration is detected by a piezoresistive Wheatstone bridge integrated on the cantilever, which consists of four diode-connected PMOS transistors (Figure 4). The transistor source-drain currents are dependent on the mechanical stress in the transistor channel region.29 The current of these stresssensitive transistors is controlled by the resistive region of the channel, when operated in strong inversion in either the linear 2692 Analytical Chemistry, Vol. 77, No. 9, May 1, 2005
or the saturation regime. The Wheatstone bridge is located at the clamped edge of the cantilever, where the bending and, consequently, the stress are maximal, so that a large piezoresistive output signal can be achieved. In comparison to a Wheatstone bridge consisting of diffused resistors (resistive load below 950 Ω), the transistor configuration as described above is smaller and shows a higher resistive load (20 kΩ) to the power supply. This leads to less power dissipation on the cantilever and, hence, to a lower temperature increase in the sensitive polymer layer. As the extent of physisorption of the analyte molecules in a polymeric matrix is strongly dependent on the temperature, a reduction of the power dissipation on the cantilever yields an enhanced chemical sensitivity of the device. The sensitivity has been increased by almost a factor of 2 in comparison to a previously developed system with electrothermal actuation and resistor-based readout. To facilitate the operation of the device, a dedicated oscillator circuit, with the cantilever as the frequency-determining element, has been integrated on the same substrate. A block diagram of the sensor system is shown in Figure 5. The output signal of the piezoresistive Wheatstone bridge is amplified in two stages and high-pass filtered to remove the offset coming from the mismatch of the bridge transistors. The key element of the feedback circuit is a nonlinear transconductance, which is based on a nonlinear, amplitude-limiting conductance.30 The gain of this stage is decreasing with increasing input signal magnitude to avoid saturation of the overall feedback system during operation. After this stage, the signal is fed via a buffer to the current path on the cantilever. The buffer is needed to drive the low resistive load of the current path (50 Ω equivalent to 40 µW). The whole feedback loop features a fully differential architecture for better suppression of noise and interferences, e.g., coming from the supply lines, and for reduction of the signal distortion as a consequence of nonlinear effects in the circuitry. During operation, the system oscillates at the mechanical resonance frequency of the cantilever. FUNCTIONAL PRINCIPLE The sensor is essentially functioning as a balance, as its resonance frequency is prevailingly dependent on the cantilever mass loading. An increase of the cantilever mass results in a decrease of the resonance frequency. To use the cantilever as a chemical sensor, a sensitive layer has been deposited on the resonating cantilever. Polymers that absorb volatile organic (29) Akiyama, T.; Tonin, A.; Hidber, H. R.; Brugger, J.; Vettiger, P.; Staufer, U.; de Rooij, N. F.; Sens. Actuators, A 1998, 64, 1-6. (30) Krummenacher, F.; Joehl, N. IEEE J. Solid-State Circuits 1988, 23, 750758.
compounds (VOCs) have been used as sensitive materials. For these polymeric films, physisorption and bulk dissolution of the analyte molecules within the polymer matrix are the predominant mechanisms. Absorption of analyte molecules in the polymeric film on the cantilever leads to a mass increase and concurrently to a decrease of the resonance frequency. The frequency shift is directly proportional to the mass change. The amount of analyte absorbed in the polymeric film on the cantilever is determined by the partition coefficient, Kc, of the respective polymer/analyte combination. It is a thermodynamic equilibrium constant and is defined as the ratio of the analyte concentration in the polymeric phase, cPoly (µg/L), and the analyte concentration in the gas phase, cA (µg/L):
Kc ) cPoly/cA
(2)
The partition coefficient strongly depends on the nature of polymer and analyte, the analyte saturation vapor pressure, and on the operation temperature. The sensitivity, S, of a resonant gas sensor is defined as follows:
S ) ∂f0/∂cA
(3)
where ∂f0 is the change of the fundamental resonance frequency and ∂cA is the change of the analyte concentration in the gas phase. The fundamental resonance frequency, f0, of a composite beam of length, L, and cross section, F, is given by
f0 )
λ02 2πL
2
x
E ˆ SiISi + E ˆ OxIOx + E ˆ NiINi + E ˆ LIL FmeanF
(4)
Here, Eˆ denotes the apparent Young’s modulus of each material and I is the respective moment of inertia. Fmean is the average mass density of the four-layered composite beam, consisting of silicon (Si), silicon nitride (Ni), silicon oxide (Ox), and polymer (L). λ0 is an integration constant resulting from the solution of the underlying Euler-Bernoulli differential equation. Its value is λ0 ) 1.875 for the fundamental oscillation mode of a clamped free beam. Using eqs 2-4, the sensitivity can be rewritten as20
S)-
1 f0 tL K 2 Fmean h c
(5)
Here, tL denotes the thickness of the polymeric layer on the cantilever and h is the total thickness of the composite beam. Except for the partition coefficient, Kc, all parameters in this equation are depending on the polymer layer thickness, tL. At low polymer thickness, the increase of the sensitivity, S, with increasing polymer thickness is expected to be almost linear since f0 changes only slightly. At larger polymer thickness, the sensitivity characteristics will be dominated by the decrease of the fundamental resonance frequency, f0, owing to the extra load of the polymer coating. A saturation-like behavior is, therefore, expected. FABRICATION The cantilever chips are fabricated using a 0.8-µm double-poly, double-metal CMOS process of austriamicrosystems (Unterprem-
Figure 6. Packaging concept: schematic cross section of the cantilever chip packaged in a modified CERDIP.
sta¨tten, Austria) in combination with post-CMOS micromachining steps.31 After the completion of the CMOS process, the cantilevers are released by two postprocessing steps that are conducted on wafer level. First, silicon membranes are formed by anisotropic wet etching of the bulk silicon in KOH from the wafer backside. To ensure a defined membrane thickness, an electrochemical etchstop technique32,33 is used, so that the etching stops at the n-well, which has been created by ion implantation during the CMOS process flow. Thereafter, the cantilevers are released by reactiveion etching (RIE) from the wafer front side. The membrane around the designated cantilever shape is removed by front-side RIE of silicon oxide and silicon. The resulting cantilevers are composed of the dielectric layers of the CMOS process on top of a silicon layer (n-well) with a thickness of ∼4.5 µm. The thicknesses of the silicon oxide and the silicon nitride layer are 2.2 and 1 µm, respectively. The total cantilever thickness is 7.7 µm. The cantilevers as used in the sensor system are referred to as sensor cantilevers, and are 150 µm long and 140 µm wide. Additionally, slender test cantilevers with a width of 35 µm and different lengths between 50 and 200 µm were fabricated in order to assess the dependence of the resonance frequency upon the cantilever length. The resonance frequency of the sensor cantilevers is in the range of 400 kHz with a quality factor of better than 1000 in air. The quality factor of the resonant cantilevers at atmospheric pressure is dominated by viscous damping of the surrounding air.20 SYSTEM PACKAGING The polymer-coated front side of the chip featuring the cantilever has to be exposed to sample gas, and an external magnetic field is required to operate the electromagnetically actuated cantilever. Therefore, a permanent magnet has been included in the package underneath the chip to achieve a small footprint of the device. A standard ceramic dual in-line package (CERDIP) has been modified to accommodate the magnet and the chip (Figure 6). The bottom part of the CERDIP has been removed and replaced by an aluminum block. Two neodymium disk magnets (Northwest Magnet Inc., Portland, OR) have been placed in a cavity of the aluminum block. Each magnet has a diameter of 1.5 mm and a height of 0.8 mm. The chip is glued on top of the aluminum block (31) Baltes, H.; Brand, O. Sens. Actuators, A 2001, 92, 1-9. (32) Tabate, O.; Asahi, R.; Funabashi, H.; Shimaoka, K.; Sugiyama, S. Digest of Technical Papers TRANSDUCERS 1991, San Francisco, CA, 1991; pp 811814. (33) Mu ¨ ller, T.; Brandl, M.; Brand, O.; Baltes, H. Sens. Actuators, A 2000, 84, 126-133.
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and covers the cavity. The electrical connections to the CERDIP package have been made by wire bonding. EXPERIMENTAL SECTION Layer Deposition. After completion of the post-CMOS micromachining steps and after packaging, polymeric films have been deposited onto the sensing structures by spray-coating with an airbrush (Badger, model 200-F, Franklin Park, IL) and shadow masks. The airbrush has been fixed at a distance of 10 cm from the chip. For spraying, the polymers are dissolved in dichloromethane (concentrations between 2.5 and 10 mg/mL), and nitrogen is used as carrier gas. A wide range of partial selectivities and sorption properties can be covered by careful selection of the polymeric coating materials. The microsensors investigated have been coated with the slightly polar poly(etherurethane) (PEUT, Thermedics Inc., Woburn, MA), ethyl cellulose (EC, Aldrich Chemical Co., Inc., Milwaukee, WI), and the nonpolar poly(dimethylsiloxane) (PDMS, ABCR, Karlsruhe, Germany). Gas Measurement Setup. For gas tests, the packaged sensor chips are mounted in the measurement chamber of a computercontrolled gas manifold. Vapors are generated from specifically developed, temperature-controlled (T ) 223-293 K) vaporizers using synthetic air as carrier gas and then diluted as desired using computer-driven mass-flow controllers. The vapor-phase concentrations at the respective temperatures can be calculated following the Antoine equation.34 The internal volume of the used vaporizers, which distribute the liquid over a large-area, packed-bed type support to maximize surface/volume ratio is significantly smaller than that of typical gas-washing bottles (“bubblers”).35 By using these vaporizers, the noise in the sensor signals caused by concentration fluctuations or aerosol formation of the liquid analytes is reduced, and the reproducibility of the adjusted gasphase concentrations is significantly enhanced. A photoacoustic detector (infrared light for excitation, 1314 Photoacoustic Multigas Monitor, Innova Airtec Systems) is used as an independent reference to assess the actual analyte gas-phase concentrations. All vapors are mixed and temperature-stabilized before entering the thermoregulated chamber. All gas tubings in the manifold are made of stainless steel. The sensors were mounted inside a flowthrough cell, and the measurements were performed at a temperature of 303 K. The thermostat used for the measuring chamber was a microprocessor-controlled Julabo FP 30 MH (Julabo, Seelbach, Germany). The gas flow rate to the sensors was 200 mL/min at a total pressure of 105 Pa. Typical experiments consisted of alternating exposures to air and vapor. Exposure times of 5-10 min are followed by 5-10 min of purging the chamber with synthetic air. To test for reproducibility, analyte concentrations are ramped up and down. The resonance frequency of the cantilevers is continuously measured using a frequency counter with a gate time of 1 s.
Figure 7. Micrograph a of magnetically actuated slender test cantilever with only one PMOS transistor on the cantilever.
RESULTS AND DISCUSSION Cantilever Properties. Equation 4 describes the dependence of the fundamental resonance frequency of a four-layered composite cantilever beam on the cantilever parameters. The equation
is derived from the Euler-Bernoulli beam theory36 under the assumption that plane cross sections before bending remain plane after bending in a composite beam.37 This assumption is valid for pure bending and small deflections regardless of the material. The Euler-Bernoulli beam theory holds for the one-dimensional case and is, therefore, applicable to long and slender beams. The test cantilevers as used for the measurements feature a width of 35 µm and lengths between 50 and 200 µm. Owing to the narrow cantilevers, the Wheatstone bridge for the cantilever vibration readout had to be reduced in size. Only one of the four diodeconnected PMOS transistors was placed on the cantilever, while the other three were placed on the substrate (Figure 7). During the postprocessing, an underetching of the cantilever support structures occurs. This is a consequence of the misalignment of the backside etching mask that defines the openings for the anisotropic etching of the silicon from the wafer backside and temporal overetching during this wet-etching process. As a result, the cantilevers are not attached to the silicon substrate but protrude from an underetched membranelike structure. The width of this suspension membrane structure depends on the postprocessing. The cantilever arrays used for the measurements featured an underetching of ∼10 µm. The underetching leads to a decrease of the fundamental cantilever resonance frequency as compared to calculations using eq 4, since the less rigid cantilever suspension is not taken into account. As a result of the nonrigid suspension, the vibration modes of the cantilevers do not exhibit a node at the suspension point, and the movement spreads into the membrane. This effect can be modeled by replacing the length of the cantilever, L, in eq 4 by an effective length, Leff. Figure 8 shows a plot of the fundamental resonance frequencies of 16 cantilevers versus 1/Leff2. For each length (50, 100, 150, and 200 µm) four samples were measured. The chosen effective length, Leff, of the cantilevers used for this plot was 10 µm larger than the geometric length, L, of the cantilevers. The straight line in Figure 8 has been obtained using eq 4. The relative deviation from the model was less than 7% for all measured cantilevers. Going to shorter cantilevers, the deviations from the model become more significant, and the suspension of the cantilever from a membrane instead of a rigid support can no longer be described by a simple approximation using an effective length. Additionally,
(34) Riddick, J.; Bunger, A. Organic Solvents. In Techniques of Chemistry, Weissberger, A., Ed.; Wiley-Interscience: New York, 1986; Vol. II. (35) Bodenho ¨fer, K.; Hierlemann, A.; Schlunk, R.; Go ¨pel, W. Sens. Actuators, B 1997, 45 (3), 259-264.
(36) Weaver, W.; Timoshenko, S. P.; Young, D. H. In Vibration Problems in Engineerimg, 5th ed.; Wiley-Interscience: New York, 1990. (37) Gere, J. M.; Timoshenko, S. P. In Mechanics of Materials, 3rd ed.; PWSKent; Boston, MA, 1990.
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Figure 8. Dependence of the cantilever resonance frequency on the effective cantilever length, Leff. Solid line is calculated according to formula 4. Four samples measured for each cantilever length.
Figure 9. Sensor response of a device coated with 0.5-µm PDMS upon absorption of two different analytes (dashed black, toluene; black, methanol). Analyte concentrations ranging from 500 ppm to 2500 ppm in steps of 500 ppm.
the readout transistor occupies almost the whole surface area of such small cantilevers, which compromises the three-layer (silicon, silicon oxide, and silicon nitride) model. Gas Measurements. The sensor response of the polymercoated cantilever upon absorption of VOCs is expected to be linear in the range of analyte concentrations for which Henry’s law holds (less than 2% of the saturation vapor pressure at the respective temperature). Figure 9 shows the response of a cantilever coated with a thin layer (0.5 µm) of poly(dimethylsiloxane) (PDMS) upon exposure to different concentrations (500-2500 ppm in steps of 500 ppm) of methanol and toluene. The frequency change upon exposure to toluene is ∼8 times higher than that upon exposure to methanol, which can be explained by the following considerations. The molecular mass of toluene (MR ) 92.1 g/mol) is ∼3 times larger than that of methanol (MR ) 32 g/mol). Furthermore, the saturation vapor pressure of toluene (pv ) 6.2 kPa at 30 °C) is smaller than that of methanol (pv ) 31.9 kPa at 30 °C). Therefore, a smaller sensor signal of methanol is expected owing to its higher volatility. Toluene also partitions better into the PDMS than methanol since the dielectric coefficients of toluene and PDMS are almost the same. The relative dielectric constant of toluene is ) 2.4 and is close to the value of PDMS, ) 2.7, whereas methanol ( ) 31.9) is much more polar and is, therefore, hardly absorbed by the polymer. The sensitivity of the cantilever gas sensing system is strongly dependent on the thickness of the polymer layer. Figure 10 shows the frequency changes upon absorption of increasing concentrations of toluene for devices coated with PEUT layers of different
Figure 10. Resonance frequency shift of the cantilever gas sensing system coated with different PEUT thicknesses upon absorption of toluene.
Figure 11. Measured (*) and modeled (s) toluene sensitivity of PEUT-coated cantilevers versus polymer thickness.
thickness. For each polymer thickness, the respective sensitivity can be derived from the slope of the data fit. The sensitivity increases with increasing layer thickness. When plotting the sensitivity versus the polymer thickness (Figure 11), a behavior according to eq 5 can be seen. At low polymer thickness (
