**6. Thermal Noise**

The sensor performance is limited by the unavoidable noise caused by the thermal fluctuations (Brownian movement) which set the resonator in motion. Therefore, it must be considered to construct a high-performance sensor. The maximum displacement of the cantilever caused by the Brownian noise *wnoise*(*ωn*) [42] is given by the fluctuationdissipation theorem:

$$w\_{\rm noise}(\omega\_n) = \sqrt{\frac{4k\_b T \Delta f Q}{\omega\_n^3 m\_n}} = \sqrt{4k\_b T \Delta f} \sqrt{\frac{w\_n}{\omega\_n F\_{\rm PA}}} \tag{26}$$

where *kb*, *T*, and Δ*f* are the Boltzmann constant, cantilever temperature, and detection bandwidth, respectively. The plot for thermal noise as a function of cantilever geometry is presented in the Appendix A.3, Figure A4.

The signal-to-noise ratio at the resonance pulsation *ω<sup>n</sup>* is given by:

$$SNR = \frac{w\_{\text{fl}}(\omega\_{\text{fl}})}{w\_{noise}(\omega\_{n})} = \sqrt{\frac{w\_{\text{fl}}\omega\_{n}F\_{\text{PA}}}{4k\_{b}T\Delta f}}\tag{27}$$
