(Color online). (a) Scanning electron micrograph of SiC NEMS. This first family of submicron doubly clamped beams exhibits fundamental flexural resonant frequencies from . They were patterned at Caltech from 3C–SiC epilayers grown at Case Western Reserve University (see Ref. 5). (b) Surface nanomachining of NEMS. Fabrication starts on a semiconductor heterostructure such as the one shown in (i) with structural (top) and sacrificial (middle) layers on top of a substrate (bottom). (ii) First an etch mask is defined via electron beam lithography. (iii) Then, the pattern is transferred into the sacrificial layer using an anisotropic etch such as a plasma etch. (iv) Finally, the sacrificial layer under the structure is removed using a selective etch. The structures can be metallized after or during the process depending upon the specific measurement requirements.
(Color online). Schematic representation of a multiterminal electromechanical device.
(Color online). Frequency versus effective geometry for doubly clamped beams made from single-crystal SiC, Si, and GaAs (see Refs. 5 and 9). The inset shows the doubly clamped beam geometry with length , width , and thickness . The fundamental out-of-plane (in-plane) flexural resonance frequency of this structure is given by the expression, . In the plot, values have been normalized to remove the effect of additional stiffness and mass loading due to electrode metallization.
(Color online). Measurements of internal strain in doubly clamped nanomechanical beam resonators. Here, the beam is subjected to a static force in addition to a small ac excitation force around the resonance frequency . The net effect is a shift in . was generated by passing a dc current along the length of the beam in a static magnetic field. The frequency shift data are plotted against per the beam’s unit length for three different magnetic field strengths . The apparent curvature at the lowest field value of , can be attributed to heating effects since to obtain the same , a larger is required at lower . A simple analysis using elasticity theory indicates that is positive and symmetric around in an unstrained beam resonator. A resonator with an internal strain, however, exhibits a sign change in —consistent with the displayed data.
(Color online). Block diagram of a NEMS–transducer–amplifier cascade. In the noise analysis, the nanomechanical resonator is assumed to respond to a drive force localized at , an intrinsic force noise, and the backaction force noise of the transducer. The resonator’s mechanical response is characterized by its (linear, one-dimensional) transfer function . The resonator converts the input drive force into a displacement and subsequently, the (linear) transducer–amplifier converts this displacement into an electrical voltage. We assume that the transducer–amplifier cascade is noisy and has a displacement responsivity . At the output of the cascade, all the noise power as well as the drive force is converted to a voltage and integrated over the measurement bandwidth . Note that for the DR calculation, one can convert all the signals to a displacement (referred to as the input of the transducer–amplifier) or to a voltage (referred to as the output of the transducer–amplifier).
(Color online). Effect of surrounding gas pressure upon the resonance parameters of a GaAs doubly clamped NEMS beam with . The intrinsic of the device was . and of the beam were measured as a function of the gas pressure in the measurement chamber. was later extracted using . The measurement was done with and gases. Note the crossover in the plots from the ideal gas regime to the viscous regime.
(Color online). Maximum reported factors in monocrystalline mechanical resonators varying in size from the macroscale to nanoscale. The data follow a trend showing a decrease in factor that occurs roughly with linear dimension, i.e., with increasing volume-to-surface ratio.
(Color online). Magnetomotive displacement detection scheme. The displacement sensitivity in the text is estimated by assuming that the dominant source of noise is the noise generated in the first stage amplifier.
(Color online). (a) Schematic diagram of the free space optical setup in use at Boston University (see Ref. 89). The optical interferometer is mounted on a translation stage (not shown). The interferometer comprises various beamsplitters (BSs), a reference mirror (RM), and a photodetector (PD). The probe beam used for NEMS displacement detection is tightly focused on the device by a objective lens (OL) with numerical aperture . The light reflecting from the NEMS is collected by the same lens and interferes with the reference beam on the PD. A constant reference path length is maintained by using a low-pass filter (LPF) and a piezoelectric actuator (PZA). The dashed line indicates the portion of the setup used for the Fabry–Perot cavity measurements; the reference arm of the optics is simply blocked in the measurements. (b) Top view and center cross section of a doubly clamped NEMS beam in relation to the optical spot with Gaussian profile.
(Color online). (a) Variable temperature, UHV microwave cryostat for NEMS mass sensitivity measurements. The sample chamber (SC) is inserted into the bore of a superconducting solenoid (So) in liquid helium. The radiation baffles (RB) establish a line of sight along the axis from a room temperature thermal Au evaporation source to the bottom of the cryostat. The NEMS resonators are carefully placed in this line-of-sight, some away from this thermal-evaporation source. A calibrated quartz crystal monitor (QCM) at a distance of and a room temperature shutter (Sh) are employed to determine and modulate the Au atom flux, respectively. With knowledge of the exposed NEMS surface area (determined from careful scanning electron microscopy measurements), and the mass flux of the evaporator as measured by the QCM, one can determine the exact mass of the adsorbed Au atoms on the NEMS as . In this system, the geometric factor is . (b) Scanning electron micrographs of nanomechanical doubly clamped beam sensor elements. (c) Conceptual diagram of the phase-locked loop NEMS readout. The principal components are: voltage controlled radio frequency oscillator (VCO); four-port power splitter (PS) (with three 0° and one 180° output ports); NEMS mass sensor with radio frequency bridge readout; mixer ; phase shifter (Ø); variable gain amplifier ; and low pass filter (LPF); frequency counter .
(Color online). Frequency shifts (bottom) induced by sequential gold atom adsorption upon a silicon carbide doubly clamped beam resonator. The (initial) fundamental frequency is . The accreted mass of gold atoms in the upper plot is measured by a separate quartz crystal detector. The rms frequency fluctuations of the system (i.e., the noise level in the lower trace) correspond to a mass sensitivity of for the averaging time employed. for this system appears to be limited by the noise in the transducer–amplifier cascade, i.e., obtained upon setting [see Table II and Eq. (6)]. The system parameters determined from separate measurements were , , and . This leads to the approximate result that —quite close to what we experimentally attain.
(Color online). Data from three additional SiC devices with , 56, and displayed with linear fits. The measured and calculated device properties are presented in Table III.
Important attributes for a family of doubly clamped Si beams with at . The effective force constant is defined for point loading at the beam’s center. Nonlinear onset amplitude has been characterized using the criterion described in the text. The thermomechanically limited linear dynamic range (DR) is calculated for the natural bandwidth of the beam and , where . Effective mass for the fundamental mode is where is the total mass of the beam.
Calculated and experimental values of the mass responsivity in doubly clamped beam resonators.
Expressions for NEMS phase noise and frequency fluctuations for different noise mechanisms. are given for a measurement bandwidth . The resonator is assumed to be driven to its critical drive amplitude characterized by an energy . In the first row, the (white) voltage noise generated in the transducer–amplifier cascade is the dominant noise source; the symbols used have been defined in Sec. II F. The DR here is amplifier limited, and . In the second row, we present frequency fluctuations due to thermomechanical noise. Here, the measurement scheme effectively determines , although the obtained is scheme independent. In the third row, temperature fluctuations are considered. Here, is the temperature dependent speed of sound; is the linear thermal expansion coefficient; and and are the thermal conductance and the thermal time constant for the nanostructure, respectively. In the fourth row, the noise presented arises from the adsorption–desorption of gas molecules upon the resonator. To determine the adsorption–desorption noise, the surface is modeled as comprising sites for the adsorption of molecules of mass with representing the variance in the occupation probability of a site; is the correlation time for an adsorption–desorption cycle. In the bottom row, the momentum exchange noise from impinging gas molecules has been calculated for the low pressure limit with the parameters defined in Sec. III A.
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