͑ Color online ͒ Calibration curve relating resonant frequency shift to 

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... micro- and nanofabricated resonant structures have been employed as resonant mass sensors due to their 1 miniscule sizes which yield high mass sensitivities and for their characteristically high quality factors. Mass loading on the resonant structure causes a characteristic linear downshift in the resonant frequency of the device that can be approxi- mated by f = − f 0 2 M eff m , 1 where ␦ m is the accreted mass, f 0 is the resonant frequency, M eff is the effective proof mass, ␦ f is the corresponding frequency shift, and ␦ m Ӷ M eff . The negative sign in the ex- pression indicates a downshift in the frequency with an in- crease in mass. The quartz crystal microbalance ͑ QCM ͒ is the best example of a resonant sensor that has been commer- cialized for film thickness monitors and other mass sensing 2 applications. The QCM, however, is limited in its mono- lithic integration with electronics as well as in the scaling of detection to include a multitude of sensors on a single platform. Several groups have previously demonstrated micro/ nanomechanical resonant mass sensors based on flexural 3–5 mode resonators. Flexural mode resonators have limited quality factors ͑ Q ͒ in air, thereby limiting the achievable mass resolution. Further miniaturization using nanofabrica- tion techniques allows for the construction of nanoscale me- 6 chanical resonators with high mass sensitivity. However, nanoscale resonators are significantly influenced by manu- facturing tolerances that limit device-to-device performance reproducibility. In addition, it is often difficult to electrically interface to nanoscale resonators as most forms of electromechanical transduction do not scale effectively with dimen- sion. The majority of reported mass sensing techniques use an optical readout system. The complexity and size of such setups render these methods difficult to realize for real-world applications. Mass sensing based on single-crystal silicon bulk mode resonators was previously demonstrated wherein a length- 7 extensional mode is excited. This resonator had a reported Q of 4000 in air, a fraction of that achieved by the resonator described in this letter under similar operating conditions. Additionally, near global system for mobile communications ͑ GSM ͒ standards for low phase noise oscillators have been realized by incorporating a single crystal silicon square- extensional ͑ SE ͒ mode resonator as the timing element. 8 The SE mode may be described as a square plate contracting and extending symmetrically on all four sides. Figure 1 provides an illustrated description of the SE mode. The resonator is laterally driven from electrodes on each side of the square plate symmetrically. The motion of the resonator is moni- tored through the sense current that is produced as a result of the changes in capacitance across the transduction gap. The resonant frequency in the SE mode may be approximately predicted by f 0 = E 4 L , 2 where E denotes the Young’s modulus of silicon ͑ 180 GPa ͒ , ␳ the density ͑ 2330 kg / m 3 ͒ , and L is the length of each side of the plate ͑ 2 mm for this device ͒ . This gives a predicted resonant frequency of 2.18 MHz, which agrees with the solutions obtained from finite-element analysis as well as ex- perimental measurement. The high structural Q in a SE mode resonator improves the frequency stability of the realized oscillator and lowers the equivalent electromechanical loss, both key figures of merit for a mass sensing application. We have designed and fabricated a SE mode resonator in a commercial foundry silicon-on-insulator micro electro mechanical system ͑ MEMS ͒ process through MEMSCAP. The minimum capacitive transduction gap was limited to 3 ␮ m, slightly wider than the minimum gap size allowed by the foundry process. A scanning electron micrograph ͑ SEM ͒ of a corner section of the device is given in Fig. 2. The resonator was initially measured in an open loop using a network analyzer together with a transresistance pre- amplifier to obtain its transmission characteristics. We have measured the transmission of the device at atmospheric pressure and vacuum, deducing the Q at each pressure using a simple and effective method for extracting resonator parameters. 9 The highest Q measured was 1.16 ϫ 10 6 under vacuum ͑ Ͻ 10 mTorr ͒ , and 15 000 at atmospheric pressure. To demonstrate functionality of the device for mass sensing, successive layers of chrome ͑ Cr ͒ were evaporated over the front side of the resonator in incremental thick- nesses of 5 nm at a time, and the transmission of the device was measured in air after each deposition step. A corresponding shift in the resonant frequency can be clearly observed after each deposition step, as shown in Fig. 3. A calibration curve fit relating this frequency shift to the corresponding mass loaded on the resonator was obtained, as shown in Fig. 4. The slope of the best-fit line defines the sensitivity of the mass sensor, and has a measured value of 94 Hz / nm of Cr, equivalent to 3.3 Hz / ng ͑ 132 Hz cm 2 / ␮ g ͒ for the first four data points. The deposition of Cr on the surface of the plate increases the effective stiffness of the structure, thus has the effect of reducing the magnitude of the actual resonant frequency downshift due to mass loading. Equation ͑ 1 ͒ may be modified to take into account changes in the structure stiffness due to the deposited Cr layer such that f = k K eff − m M eff f 0 2, 3 where ␦ k is the change in stiffness, K eff is the effective stiffness of the silicon resonator, and ␦ k Ӷ K eff . Equation ͑ 3 ͒ pre- dicts a sensitivity of 3.5 Hz / ng, which agrees well with the measured calibration curve. The resonator was embedded in the feedback loop of an amplifier to implement an oscillator for our application ͑ Fig. 5 shows the output ͒ . The oscillator circuit employs a hard voltage limiter to accurately control the actuation signal am- plitude that is applied to the resonator in order to avoid mechanical or material nonlinearities. The standard deviation of the output frequency was measured using a frequency counter ͑ Agilent 53132A ͒ . The short term frequency fluctuations were found to be less than 7 ppb ͑ with an averaging time of 50 ms over 100 samples ͒ . This yields a measured mass noise floor of 125 pg / cm 2 , almost an order of magni- 2 tude better than a QCM at the same frequency. This result suggests that it is possible to achieve mass resolutions com- parable with a QCM using this device. This corresponds to a film thickness resolution of 5 mÅ for chrome samples with a density of 7.2 g / cm 3 . The theoretical minimum for fre- 10 quency fluctuations in the limit of thermomechanical noise is in This the order research of 10 was supported for this device by the in U.S. a 1 Hz Army bandwidth, Solider Systems which corresponds Center. The to authors a thermomechanical thank Ibraheem noise Haneef limited for mass wire bonding floor of 20 the fg resonators / cm 2 . In practice, reported the in performance this letter. of the sensor will also depend on other noise sources in the oscillator loop ͑ e.g., sustaining amplifier noise ͒ , and environmental factors ͑ e.g., temperature fluctuations and gas adsorption-desorption processes on the resonator element ͒ that can substantially influence both the short-term and long-term frequency sta- bilities of the oscillator. However, much progress has been in recent years in addressing the temperature sensitivity of resonant frequency for silicon-based microresonators through a 11 variety of process, device, and system-level solutions. In conclusion, a silicon SE mode bulk acoustic resonator has been demonstrated for use in mass sensing, with a measured mass noise floor of 125 pg / cm 2 . This result opens up new possibilities for multiarrayed parallel mass detection. This research was supported by the U.S. Army Solider Systems Center. The authors thank Ibraheem Haneef for wire bonding the resonators reported in this ...