Patent Publication Number: US-7584649-B2

Title: Sensor with microelectro-mechanical oscillators

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of U.S. Provisional Application No. 60/810,783, filed on Jun. 2, 2006. The disclosure of the above application is incorporated herein by reference. 
    
    
     FIELD 
     The present disclosure relates to an analyte sensors and, more particularly, to a multi-analyte sensor which uses microelectro-mechanical oscillators to detect the presence individual analyte. 
     BACKGROUND 
     The statements in this section merely provide background information related to the present disclosure and may not constitute prior art. 
     Chemical and biological sensors based on microresonantors have been considered viable alternatives to modern sensing systems for some time, undoubtedly because they consume less power and space than their macroscale counterparts. Traditionally, these systems track resonance shifts in single-degree-of-freedom oscillators induced by mass or stiffness changes. As these changes are caused by local bonding, stress stiffening, or a similar chemical-mechanical process, the resonance shifts, in turn, indicate the presence of a given analyte. Existing sensors require the measurement of the response of an individual resonator for the detection of a specific compound, or a class of compounds. Large sensor arrays composed of isolated microresonators can be used to broaden detection capabilities, but the addition of the attendant electronics (arising from a larger number of system outputs) adds to the complexity of such sensor systems. 
     Existing methods of resonant microscale sensing detect a single target analyte by measuring induced resonance shifts in a single, isolated microresonator, which features independent actuation and sensing mechanisms. Multiple analytes can be detected with arrays of microresonators that have either collective or independent actuation; however, current techniques require independent sensing mechanisms for each resonator or a single sensing mechanism which operates in a scanning mode to recursively measure the response of each individual resonator. The first approach limits the number of analytes that can be detected and the second yields a relatively complex system with multiple inputs and outputs. 
     SUMMARY 
     It is an object of the present invention to overcome the disadvantages of the prior art. As such, a sensor is provided which is configured to detect the presence of multiple analytes (e.g., chemical or biological compounds) through the measurement of induced resonance shifts in a coupled array of microelectromechanical or micromechanical resonantors. 
     In one embodiment, an analyte sensor is provided having a sensing element with at least two vibrating members, each with a respective first resonant frequency, wherein the vibrating members are configured to detect the presence of a plurality of analytes. The analyte sensor further has a drive element coupled to the sensing element, the drive element being configured to apply force to the sensing element to induce a vibration within the sensing element. A vibration sensor is configured to measure changes of a resonant frequency of at least one of the vibrating members. 
     In another embodiment, an analyte sensor is provided having first and second masses vibrationally coupled to a shuttle mass. The first mass has a first resonant frequency in the absence of a first analyte, and a second resonant frequency after the exposure to the first analyte. The second mass has a third resonant frequency in the absence of a second analyte, and a fourth resonant frequency after the exposure to the second analyte. The analyte sensor further has a vibration sensor configured to detect vibration of the shuttle mass. 
     In another embodiment, a material detection sensor is provided having a base mass and first and second beams vibrationally coupled to the base mass. The first beam having a first resonant frequency in the absence of a first analyte, and a second resonant frequency after the exposure to the first analyte; the second beam having a third resonant frequency in the absence of a second analyte, and a fourth resonant frequency after the exposure to the second analyte. A vibration sensor configured to detect vibration of the base mass. An actuator configured to drive the shuttle mass. 
     In yet another embodiment, a sensor is provided having a support mass vibrationally coupled to a base. First and second cantilevered beams vibrationally are coupled to the support mass, with the first beam having a first resonant frequency in the absence of a first analyte, and a second resonant frequency in the presense the first analyte. The second beam having a third resonant frequency in the absence of a second analyte, and a fourth resonant frequency in the presence of the second analyte. A vibration sensor is further provided which is configured to detect vibration of the shuttle mass. An actuator provided to drive the shuttle mass at a series of predetermined frequencies. 
     Further areas of applicability will become apparent from the description provided herein. It should be understood that the description and specific examples are intended for purposes of illustration only and are not intended to limit the scope of the present disclosure. 
    
    
     
       DRAWINGS 
       The drawings described herein are for illustration purposes only and are not intended to limit the scope of the present disclosure in any way. 
         FIGS. 1A-1D  represents a micrograph of a translational, SISO, multi-analyte sensor according to the present teachings; 
         FIGS. 2A and 2B  represent response curves of the sensor shown in  FIG. 1 ; 
         FIG. 3  represents an image of a platinum patch added to the sensor shown in  FIG. 1 ; 
         FIGS. 4A and 4B  represent experimentally-obtained frequency response associated with the sensor shown on  FIG. 1 ; 
         FIG. 5  represents a mass-spring-dashpot analog of the sensor shown in  FIG. 1 ; 
         FIG. 6  represents a desirable form of the frequency response for a representative sensor design; 
         FIG. 7  represents an experimentally-obtained frequency response near resonance of the sensor depicted in  FIG. 1 ; 
         FIG. 8  represents a frequency response of near resonance for the sensor depicted in  FIG. 1 ; 
         FIG. 9  represents another embodiment configured to detect small amounts of a single analyte; 
         FIGS. 10-12  represent alternate sensor configurations; 
         FIG. 13  represents an alternate sensor configured to detect the presence of a material in a fluid; 
         FIGS. 14A-14E  represent an alternate sensor configured to measure the presence of a material in a fluid; 
         FIGS. 15-19  represent sensors driven by a magnetic field; and 
     
    
    
     Table 1 represents non-dimensional parameter definitions. 
     DETAILED DESCRIPTION 
     The following description is merely exemplary in nature and is not intended to limit the present disclosure, application, or uses. 
       FIG. 1  represents a single input-single output sensor  30  which is configured to detect the presence of multiple analytes (e.g., chemical or biological compounds) through the measurement of induced resonance shifts in a coupled array of microelectromechanical or micromechanical resonators. 
     The sensor  30  exhibits the ability to detect multiple analytes with only a single input and a single output (SISO), that is a single excitation signal is all that is required to drive the device and a single output signal contains all of the information necessary for multiple analyte detection (and potentially identification). The sensor  30  exploits induced resonance shifts in a coupled microresonator array  33 . The sensor has a number of secondary (frequency mistuned) resonators  32  coupled to a primary microresonator or shuttle mass  36 . This primary resonator mass  36  is used for both actuation and sensing purposes. Induced resonance shifts in each of these secondary resonators  34  are detectable through the response of the primary resonator mass  36 . This allows for rapid detection, and potentially the identification, of a variety of analytes on a single sensor platform in an efficient (SISO) manner. 
     While the examples herein are directed to the detection of gaseous chemicals, the system is equally usable for gaseous or liquid chemical or biological detection. It is envisioned these mass sensors  30  based on resonant microsystems are usable for (and, in some cases, implemented in) security, medical, research, and industrial applications. The sensor system is principally proposed for microscale development. As such, it can be constructed through any of a large number of microfabrication processes (i.e., standard silicon on insulator processing, SCREAM processing, etc. . . ). Additionally details relating to such processes can be found in the literature (see, for example, Madou, 1997). 
       FIG. 1A  depicts a resonant single input-single output (SISO) sensor  30  based on a coupled array of microelectromechanical or micromechanical oscillators  32  according to the teachings herein. The sensor  30  exploits resonance shifts induced by mass and/or stiffness changes, resulting from local bonding, stress stiffening, or a similar chemical-mechanical process, to indicate the presence of (and potentially identify) a particular analyte or group of analytes. The sensor  30  allows for detection of multiple analytes is accomplished using a single excitation signal and, more significantly, the measurement of a single output signal. This results in a resonant sensor  30  that maintains the inherent benefits of microscale sensor arrays, namely small size and high sensitivity, yet does so in the context of a simple system that requires significantly less attendant hardware and signal processing than other known alternatives. 
     With general reference to  FIGS. 1A-1D , which represent a sensor  30  found within a silicon substrate  41 , the sensor  30  has a plurality of mistuned masses or microbeams  32   a - 32   d . These masses,  32   a - 32   d , take the form of beams  33  coupled to the shuttle mass  36 . The beams  32   a - 32   d  can be cantilevered or can be coupled at two ends to the shuttle mass  36 . The shuttle mass  36  is vibrationally coupled to a base or ground  43  through at least one support member  38 . Additionally associated within the sensor  30  are an actuator  42  and a system vibration sensor  44 . The actuator  42  is configured to drive the shuttle mass at discrete frequencies between a first predetermined frequency and a second predetermined frequency. 
     Optionally, the mistuned masses  32   a - 32   d  can be configured to detect different materials. In this regard, the mistuned masses  32   a - 32   d  can be formed of or coated with differing materials which will absorb the analyte at different rates. In doing this, the presence of the first analyte will cause the resonant frequency of one beam to shift away from a first frequency to a second frequency. Additionally, in the presence of a second analyte, the resonant frequency of a second beam  32   b  can shift from a third resonant frequency to a fourth resonant frequency. It is envisioned that this frequency shift can be accomplished with a shift in the spring constant K, damping C, and/or addition of mass M to the mistuned masses  32   a - 32   d . For example, in gas-detection situations, it has been found that the change in spring constant is the primary mechanism to shift the response of the system. 
       FIG. 2A  depicts a frequency response for the device shown in  FIGS. 1A and 1D  obtained experimentally using a laser vibrometer. As shown, the system exhibits a distinct low frequency resonance corresponding to a translational mode of the overall system ( 1 ), two resonances corresponding to out-of-plane modes ( 2 , 3 ), and four resonances corresponding to the localized modes of the microbeams  32   a - 32   d  (A,B,C,D), the latter of which have been verified with a stroboscopic in-plane measurement system and video images. Sweeping through a frequency range that covers the beam resonances allows for the detection of resonance shifts in each of the microbeams  32   a - 32   d.    
     To effectuate the shift in resonant frequency in the microbeams  32   a - 32   d , the detection beams  32  can be formed of or be coated with Silicon, Polycrystalline Silicon, Amorphous Silicon, Polycrystalline Diamond, Amorphous Diamond, Gallium Arsenide, Silicon Nitride, Silicon Carbide, Titanium, Gold, Aluminum, Aluminum Nitride, Nickel, Silicon Dioxide, Pyrex (glass), Chromium, Photoresist, Buffer solutions, Platinum, Titanium oxide, or combinations thereof. Additionally, the detection beams  32  can be formed of or be coated with PMMA and other polymer substances for gas sensing. 
     Mass detection was simulated by obtaining an initial frequency response for the shuttle mass  36  and comparing it to the frequency response of the shuttle mass  36  following the deposition of a small amount of platinum on the highest frequency microbeam. A focused ion beam was used to deposit a 1.57×5.10×0.22 μm patch of platinum, see  FIG. 3 , on the desired microbeam, yielding an approximate mass change of 38 pg. Note that this added mass changed all of the resonance peaks in this coupled set of beams, but that due to the localized nature of the response, the shift in the peak of interest (D) was about 50 times that of the other peaks (see  FIGS. 2   b  and  4   a , 4   b ). Additionally, the sensor exhibited a mass resolution of approximately 3 Hz/pg, a value comparable to alternative sensor designs. It is envisioned absorbing material can be added to exterior surface of the microbeams  32   a - 32   d.    
     Through a number of topologically equivalent geometries can be developed based on the sensing principles discussed herein, the translational design depicted in  FIGS. 1A-1D  was selected as an example. This relatively simple sensor geometry consists of the shuttle mass  36  that is suspended above the substrate by four folded beam flexures  38  and driven by interdigitated electrostatic comb drive actuator  42 . Attached to this shuttle are four microbeam oscillators  32   a - 32   d  each with slight frequency mistuning, as realized through small length variations, to ensure ample separation of the coupled system&#39;s natural frequencies—a desirable precursor for localization, as subsequently described. 
     The relevant response characteristics of the device shown in  FIGS. 1   a - 1   d  can be described through the use of the simple linear lumped-mass model shown in  FIG. 5 . Here the N microbeam oscillators are represented by the smaller masses, m 1 , m 2 , etc., and the comparatively larger shuttle mass  36  is represented by M. It is important to note that while topologies such as that shown in  FIG. 5  are believed to be novel in the sensors community, they have garnered attention in other contexts, namely noise control and vibration suppression. Unfortunately, given the stark contract between these applications and that considered here, as well as the requisite form of each of the systems&#39; frequency responses, the results of the previous works are not directly applicable to the present study. 
     The equations of motion governing the system depicted in  FIG. 5  are given by 
     
       
         
           
             
               
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             where z i  is the relative displacement of the ith subsystem, given by
 
 m   i ( {umlaut over (x)}+{umlaut over (z)}   i )+ c   bi ( {dot over (x)}+ż   i )+ c   i   ż   i   +k   i   z   i   +O, i= 1 , . . . , N,  
 
C b , c i , c bi  represent linear damping coefficients arising from intrinsic and extrinsic dissipation, and F(t) represents the electrostatic force produced by the interdigited comb drives.
 
           
         
       
    
     Assuming ample device thickness (approximately 10 μm or larger) and a harmonic voltage excitation of the form
 
 V ( t )= V   A  cos ω t,   (4)
 
the electrostatis force, F(t), can be approximated by
 
     
       
         
           
             
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     Where ε 0  represents the free space permittivity, n the number of comb fingers on the electrostatic comb drive, g the gap between adjacent comb fingers, and h the device thickness. Nondimensionalizing the resulting equations of motion and redefining the dynamic variables x and z by translation according to 
                 x   ^     =       x   -     x   _       =     x   -       F   0       k   b             ,         z   ^     i     =       z   i     .             
to eliminate the explicit DC forcing term (which is dependent on V A ) results in a new set of governing equations given by
 
                             u   ″     +     2   ⁢           ⁢     ζ   b     ⁢   Λ   ⁢           ⁢     u   ′       +       Λ   2     ⁢   u       =       ⁢       Γ   ⁢           ⁢   cos   ⁢           ⁢   Ω   ⁢           ⁢   τ     -       ∑   i     ⁢           ⁢         m   ^     i     ⁡     (       u   ″     +     υ   i   ″       )         -                     ⁢         ∑   i     ⁢           ⁢     2   ⁢           ⁢     ζ   bi     ⁢       Υ   ^     i     ⁢       m   i     ⁡     (       u   ′     +     υ   i   ′       )           ,           m   ^     i     ⁡     (       u   ″     +     v   i   ″       )       +                       ⁢       2   ⁢           ⁢     ζ   i     ⁢     Υ   i     ⁢       m   ^     i     ⁢     v   i   ′       +         m   ^     i     ⁢     Υ   i   2     ⁢     v   i                       =       ⁢         -   2     ⁢           ⁢     ζ   bi     ⁢     Υ   i     ⁢         m   ^     i     ⁡     (       u   ′     +     v   i   ′       )       ⁢           ⁢   i     =   1       ,   …   ⁢           ,   N                 (   8   )               
with system parameters defined as in Tab. 1. For the sake of analysis, these equations can be compiled into a standard matrix form given by
   MX″+CX′+KX=φ (τ)  (9) 
where X represents the compiled state vector, M the effective mass matrix (which incorporates inertial coupling terms), C the effective damping matrix (which incorporates dissipative coupling terms), K the effective stiffness matrix, and φ(r 1 ) the effective forcing vector, which is sparse except for the first element. From this matrix equation, the system&#39;s response can be easily determined through a variety of standard techniques.
 
     Because of the large number of parameters (4N+4) present in Eq. 9, a number of qualitatively distinct frequency responses can be realized using the sensor topology detailed in  FIG. 5 . Unfortunately, most of these responses are poorly suited for sensing. Several features of this response are worth noting. First, the response includes a low-frequency resonance (1), which approximately occurs at the resonant frequency 
               Ω   0     ≈       1     ω   0       ⁢         k   b       M   +       ∑   i   N     ⁢           ⁢     m   i                     
corresponding to a bulk, in-plane mode where essentially all of the microbeams  32   a - 32   d  are moving together with the shuttle mass  36  as a lumped mass. Additionally, the response includes four comparatively higher frequency resonances: (A), (B), (C), and (D). These resonances occur at frequencies slightly greater than γ i  and correspond to in-plane modes where the system&#39;s energy is largely localized in, or confined to, a single microbeam (as confirmed by noting the comparatively larger resonant amplitudes). As these higher frequency resonances (corresponding to the localized modes) induce a measurable resonance in the response of the shuttle mass  36  M, resonance shifts induced by chemomechanical processes on any of the microbeams  32   a - 32   d  can be detected using solely the shuttle mass&#39; response. Accordingly, sweeping the excitation frequency of the system Ω through a frequency domain that includes γ 1 , . . . , γ N  allows for the detection of up to N distinct analytes, providing proper functionalization.
 
     Given the large number of qualitatively distinct frequency responses that can be obtained for the system depicted in  FIG. 5 , it is important to note how responses similar to that shown in  FIG. 6 , which are well suited for resonant mass sensing, can be recovered through careful selection of system parameters. To this end, details on the selection of system inertia ratios (m i ), system frequency ratios (γ i /Λ), and the amplitude of the AC voltage excitation applied to the electrostatic comb drives (V A ), which manifests itself in Γ, are detailed here. 
     To ensure ample separation between resonances (A), (B), (C), and (D), as well as all other resonances corresponding to both in-plane and out-of-plane modes, the frequency ratios (γ i /Λ) must be carefully selected. Specifying (γ i /Λ) too close to unity leads to little or no separation between the low-frequency resonance (1) and resonances (A)-(D). This results in the saturation of the resonant peaks associated with the localized modes, which can hinder the detection of resonance shifts, especially in the presence of additive noise. Similar decreases in resonant amplitudes can occur if the resonances occurring at approximately γ i  are placed too close to resonances corresponding to out-of-plane modes or higher frequency in-plane modes (neither of which are captured with the lumped-mass model detailed here, but both of which could be captured through modal analysis of the structure using finite element techniques). It should also be noted that each (γ i /Λ) value must be well separated from all other (γ i /Λ) values, as placing two resonances too close together can lead to the formation of multi-resonance passband (i.e., the resonance peaks become indistinct), which prevents the detection of individual resonance shifts—a necessary for the SISO mass sensing methods detailed here. 
     Due to the presence of inertial coupling in Eq. 9, specification of the system&#39;s inertia ratios (m i ) is used to control the coupling strength between the microbeams  32   a - 32   d  and the shuttle mass  36 , and thus is used in conjunction with frequency ratio selection to manipulate the degree of localization in the coupled system&#39;s response. For the device considered herein, strong mode localization is a prerequisite for efficient mass sensing. This can be attributed to the fact that while mass and/or stiffness changes in a single microbeam lead to shifts in all of the system&#39;s resonances, in the presence of strong mode localization these mass and/or stiffness changes induce a significantly larger shift in the resonance associated with the altered oscillator. This, providing selective surface functionalization, allows for the rapid identification of a given resonance shift&#39;s source (namely, the particular microbeam, which having undergone a chemomechanical process, has had its mass and/or stiffness altered), which may ultimately facilitate automated analyte identification. 
     Given the linear nature of the system detailed herein, manipulation of the AC excitation voltage amplitude (V A ) is used primarily to control the relative amplitude of the shuttle mass&#39; response. As the ability to operate in low Q environments is an ultimate goal with this sensor, large amplitude responses, realized using a comparatively large excitation amplitude, are generally desirable. These large amplitudes, however, an lead to a nonlinear resonance structure, which proves problematic for sensing. As such, practical magnitude constraints on V A  must be adopted and accounted for in the course of design. 
     To validate the SISO, multi-analyte sensor concept detailed herein, the device depicted in  FIGS. 1   a - 1   d  was fabricated using a standard silicono-on-insulator (SOI) process flow, which included a single lithographic step, followed by a silicon deep etch (performed using DRIE), an O 2  reactive ion etch (RIE) to remove excess polymers, and, finally, a wet hydrofluoric acid (HF) etch to undercut and release the device. Mass detection was then simulated by comparing the sensor shuttle mass&#39; frequency response before and after the deposition of a small amount of mass on the highest frequency microbeam resonator. Actuation can be provided electrostatically through the integrated comb drives and response measurements were obtained optically through single beam laser vibrometry. Note that as the device vibrates in-plane in the desired mode of operation, a 45° mirror was cut into the silicon adjacent to the shuttle mass  36  using a focused ion beam. This permitted the laser to be reflected off the mirror and onto the shuttle mass  36  itself, thus allowing for the direct acquisition of the shuttle mass&#39; velocity. 
       FIG. 7  shows the frequency response acquired for the sensors depicted in  FIGS. 1   a - 1   d , actuated with a 6.2 V AC signal in 275 mTorr pressure, prior to added mass deposition. As evident, the response is largely compatible with the desired form of the frequency response shown in  FIG. 6 . Specifically, the response includes a low frequency resonance  91 ) at approximately 10 kHz, which corresponds to the bulk in-plane mode; two resonances (2) and (3), not predicted by the lumped-mass model, but subsequently predicted via finite element methods, at approximately 49 kHz and 63 kHz with approximately 5 kHz spacing, which correspond to the microbeam-localized modes of the system. It should be noted that the localized nature of the latter modes has been verified visually through the use of a stroboscopic in-plane measurement system with video capture capabilities. 
     Following the acquisition of the frequency response data summarized in  FIG. 7 , a small platinum patch, measuring approximately 1.57×5.10×0.22 μm with an approximate mass of 38 pg, was deposited at the tip of the highest frequency microbeam using a focused ion beam (FIB) deposition system (see  FIG. 3 ). The effect of this added mass is detailed in  FIG. 8 , which shows a close-up view of the sensor&#39;s frequency response near the four resonances corresponding to the localized microbeam modes, obtained prior to and after platinum deposition. As evident, the added mass caused shifts in each of the system&#39;s resonances, but the most pronounced shift, by far, occurred in the resonance associated with the localized mode of the highest frequency microbeam (that to which the platinum patch was added). This is verified through closer examination of the individual resonance peaks shown in  FIG. 2B . Specifically, it is seen that resonance (D), that associated with the localized mode of the altered microbeam  32   a , shifted by approximately 124 Hz and resonance (C), the resonance with the next largest shift, shifted by only approximately 3 Hz. Accordingly, due the localized nature of the sensor&#39;s response, the resonance shift associated with the mass added beam was approximately 40 times greater than that of the next largest resonance shift, a fact that could be used in implementation to identify the altered microbeam. 
       FIG. 9  represents an alternate sensor  65  according to an alternate embodiment. The sensor  65  of the shuttle pass  36  coupled to ground using two pairs of supports beams  38 . Disposed on the shuttle mass  36  is a plurality of cantilevered sensing beams  32 . These beams  32  can have mismatched resonant frequencies or can have identical resonant frequencies. In this regard, the beams  32  having the same or similar resonant frequency can be configured to detect very small amounts of the same analytes. Additionally, it is envisioned that each of the sensing beams  32  can be mistuned masses, as described above, configured to detect varying analytes. Additionally shown are detecting and actuating combs  42  and  44 . 
       FIG. 10  detects an alternate sensor configuration. Shown is the sensor  50  coupled to ground through at coupling members  38 . The sensor  50  is formed of a suspended mass  36  and a plurality of dependent mistuned detection members  32 . Optionally, the suspended mass  36  can be driven using a capacitor or electromagnetic drive. Additionally, the sensor  50  can have associated piezo detection sensor coupled to the shuttle mass  36 . 
       FIGS. 11 and 12  represent sensor  60  and  70  which are rotationally coupled to a base  50  using a central coupling member  38 . The sensors  60  and  70  are configured to be driven by either internal or external drive combs  40 . The drive combs  40  are coupled to a circular mass  36 , which has a plurality of mistuned masses  32  that are radially disposed either inside or outside of the circular mass  36 . As described above, the mistuned masses  32  are configured to individually detect disparate analytes. 
       FIGS. 13 through 14E  represent sensors  80  and  90 . The sensors  80  and  90  are configured to detect the presence of analytes found within flowing fluids. In this regard, the sensors  80  and  90  have members  82  which define through bores  84   a - 84   d  which are configured to accept flowing fluids. Optionally, the sensing beams  82   a - 82   d  are mistuned, having varying lengths. These beams  82   a - 82   d  are coupled at a first end to the base  88  and to a second end at the shuttle mass  36 . In the presence of an analyte, the stiffness or mass of the beams  82   a - 82   d  will change. It should be noted that the beams  82   a - 82   d  represent both support beams and detection beams.  FIGS. 14B-14E  represent multiple vibrational modes of the sensor  90  shown in  FIG. 14A . 
       FIGS. 15-19  represent alternate sensors which are driven using the application of a constant magnetic field and an alternating current. Optionally, the sensors can be driven using a constant current and an alternating magnetic field. The sensors  100   a - 100   e  have a shuttle mass  36  coupled to ground using beams  38 . The mass  36  has exterior or interior sensing beams  32 . These beams  32  can be fixed at both ends ( 100   e ) or cantilevered ( 100   a - 100   d ). 
     To facilitate comparison between the device presented here and other resonant mass sensor designs, it is envisioned a number of pertinent sensor metrics can be modified to affect the sensor response. In the case of mass sensors, the most relevant metric is mass sensitivity, which is the smallest change in mass that can be accurately detected using a given sensor. For mass sensors  30  operating in a resonant mode, mass sensitivity is dependent on two independent metrics: mass responsivity and frequency resolution. The mass responsivity and frequency resolution. The mass responsibility of a system is a largely deterministic quantity that dictates how much the resonant frequency of a system changes with a small mass addition. Frequency resolution, on the other hand, quantifies the smallest frequency shift that can be measured in the presence of noise and uncertainty. In general, if frequency shifts are induced primarily through mass addition, the mass sensitivity, ∂m, of a resonant sensor can be approximated by
 
∂m≈S −1 ∂ω 0  
 
where S is the mass responsivity and ∂ω 0  is the system&#39;s frequency resolution. For the multi-degree-of-freedom sensor detailed here, however, this must be modified to account for N resonance shifts induced by up to N mass changes. This results in a mass sensitivity at the jth microbeam, Δm j , which is approximated by
 
Δ m   j≈S   ij   −1 Δω i   (12)
 
where S ij  represents the i, j component of the mass responsivity matrix, which quantifies the shift in ith resonance of the system due to mass addition at the jth oscillator, and Δω i  quantifies the frequency resolution associated with the Ah resonance.
 
     Using the experimental results detailed in the previous section the mass responsivity matrix associated with the device shown in  FIGS. 1   a - 1   d  can be partially compiled. For example, the mass responsivity associated with resonance (D) and a mass addition to the highest frequency microbeam can be computed to be 3.3 Hz/pg. Similarly, the mass responsivity associated with resonance (C) and a mass addition to the highest frequency microbeam can be computed to be approximately 0.1 Hz/pg. Continuing with these computations will reveal a diagonally dominant mass responsivity matrix. This can be attributed to the localized nature of the response, which, as detailed in the previous section, leads to significantly larger shifts in the resonances associated with the localized modes of the mass loaded microbeams  32   a - 32   d.    
     Though the experimentally-determined mass responsivities reported above are comparable to other reported values for resonant multi-analyte sensors, they are significantly lower than those reported for sensors based on isolated microresonators. While much of this difference can be rectified through device scaling, it is important to note that the responsivities of the device presented here will always be slightly inferior to those of other microsensors. This can attributed to the inter-oscillator coupling, which manifests itself in the off-diagonal terms of the responsivity matrix, which leads to small shifts in each of the system&#39;s resonances, not just that associated with the localized mode of the altered beam. Current design studies are aimed at minimizing these off-diagonal terms while still allowing for the detection of resonance shifts using the shuttle mass&#39; response. 
     Though the impact of thermomechanical noise, temperature fluctuations, absorption-desorption noise, and other effects known to contribute to a system&#39;s frequency resolution have been examined for isolated resonant microsensors (see, for example), the impact of these effects in coupled oscillator systems has thus far not been considered. Accordingly, present understanding facilitates, at best, a conservative estimate of the frequency resolution(s) associated with the sensor considered herein. For present purposes a frequency resolution of approximately 1 Hz is assumed. This results in a sub-picogram mass sensitivity. 
     However, if these changes prove insufficient, exploitation of alternative geometries, including in-plane torsional devices, and/or the coupled system&#39;s phase response may prove fruitful. 
     Mass addition can be realized via the deposition of a small platinum patch on one of the sensor&#39;s microbeams  32   a - 32   d . In practice, however, resonance shifts typically arise from the accretion of a substance onto a functionalized, chemically-selective surface. In recent years, a considerable amount of research has focused on the development of chemically-active surfaces for use in microcantilever sensors. In the course of sensor development, these prior techniques can be adopted to functionalize cantilever surfaces with different chemicals (metals, polymers, etc.), thus allowing for the detection of multiple analytes using the SISO sensor. 
     While it is shown to adjust the resonant frequency of at least one component on the sensor, it is equally envisioned the changes in stiffness of these beams can be used to provide measurable changes to the resonant frequency of the sensor&#39;s microbeams  32   a - 32   d.