**4. Common-mode-rejection in resonant mass sensors**

Mode-localized sensors that utilize an array of ideally identical, weakly coupled, vibrating microstructural elements are shown to be relatively immune to the false output. Environmental factors (such as ambient pressure [30, 32], temperature [31, 40]) and/or nonspecific bindings (in case of mass sensing) influence all the vibrating elements uniformly. These factors as mentioned ideally does not affect the eigenmodes of the system, while shifts in the resonance frequencies still occur. One of the early work addressing the common-mode rejection of mode-localized sensor is given in [30]. As shown in **Figure 1**, in a 2-DoF WCR sensor, induced mismatch (*Δm*) affects one of the resonator. This is called a mass differential mode of operation. However, environmental variable such as temperature or pressure affects both the resonators simultaneously, which is termed as common-mode behavior of the micromechanical mode-localized sensors. It is found that, the mode-localized sensor is efficient in rejecting a common-mode noise (temperature and pressure) to the first order.

**Reference Output used Relative sensitivity <sup>a</sup> Minimum detectable mass,** *Δm* **DoF Material used** [36] eigenstates 5% to 7% (higher than frequency shift output) 154 *pg* 2 borosilicate microspheres [26] eigenstates 3 orders higher 13.5 *pg* 2 platinum patches [19, 38] Frequency shift NA NA 4 polymer solutions [41] Frequency shift NA 1.42 <sup>10</sup><sup>14</sup> *kg* 3 NA [42] eigenstates 7000 10 *pg* 3 polystyrene micro-spheres [39] Frequency shift NA 5 *pg* 4 NA [43] AR 34,361 2.1 *ng* 2 NA [24] AR 2.5%/*pg* (estimated via numerical model) 6 *pg* 2 NA [44] AR (Atmospheric pressure test) 25.31 (two orders higher than frequency shift output) 180 *ng* 3 Nanoparticles used [34] Eigenstates/ frequency shifts NA NA 4 Thermally killed bacteria *a Relative sensitivity is the ratio of sensitivity of eigenstate/s or AR output to the sensitivity of frequency shift output.*

**Table 1** provides a comparative performance summary (in terms of the attainable mass sensitivity and minimum detectable mass) of the ultra-precise and highly sensitive resonant mass sensors.

#### **Table 1.**

*Mode-localized mass sensors summary.*
