## 6 Detectivity  
The frequency-dependent noise floor of a sensor system is usually given by a spectral density that is related to the unit of the physical quantity to be detected. For a physical quantity with the arbitrary unit au, the representation of the sensor system's noise floor could be given as a PSD of the fluctuations of the arbitrary quantity in units of au\({}^{2}\)/Hz. However, in general, it is more common to use the amplitude spectral density (ASD) of the fluctuations of the arbitrary quantity in units of au\({}/\sqrt{\mathrm{Hz}}\), referred to as the limit of detection (LOD) or detectivity [26]. For the magnetic field sensor system considered in this article, the LOD is given in units of T\({}/\sqrt{\mathrm{Hz}}\); thus, it is also referred to as equivalent magnetic noise floor. Depending on the operation mode, the LOD is given by the ratio of the ASD of random amplitude fluctuations and the overall amplitude sensitivity  
\[\mathrm{LOD_{AM}}(f_{\mathrm{x}})=\frac{\sqrt{S_{\mathrm{z}}(f_{\mathrm{x}})} }{|S_{\mathrm{AM}}(f_{\mathrm{x}})|}=\frac{\sqrt{S_{\mathrm{z0}}(f_{\mathrm{ x}})+S_{\mathrm{am}}(f_{\mathrm{x}})}}{|S_{\mathrm{AM}}(f_{\mathrm{x}})|} \tag{53}\]  
or by the ratio of the ASD of random phase fluctuations and the overall phase sensitivity  
\[\mathrm{LOD_{PM}}(f_{\mathrm{x}})=\frac{\sqrt{S_{\varphi}(f_{\mathrm{x}})}}{| S_{\mathrm{PM}}(f_{\mathrm{x}})|}=\frac{\sqrt{S_{\varphi\varphi}(f_{\mathrm{x}})+S_{ \varphi\varphi}(f_{\mathrm{x}})}}{|S_{\mathrm{PM}}(f_{\mathrm{x}})|}. \tag{54}\]  
In each case, the detectivity is limited by the same two physical noise sources. The limit due to the thermal electrical noise of the dielectric material yields  
\[\mathrm{LOD_{AM0}}(f_{\mathrm{x}})=\frac{\sqrt{S_{\mathrm{z0}}(f_{\mathrm{x}} )}}{|S_{\mathrm{AM}}(f_{\mathrm{x}})|}=\frac{R_{\mathrm{m}}\;\sqrt{6k_{\mathrm{ B}}T_{0}}\;(4Q-\sqrt{2})\sqrt{64\;Q^{4}f_{\mathrm{x}}^{4}+16\;Q^{2}f_{\mathrm{x}}^{2}f_{ \mathrm{res}}^{2}+9f_{\mathrm{res}}^{4}}}{8\;S_{\mathrm{mag}}\bar{V}_{\mathrm{ ex}}Q^{2}f_{\mathrm{res}}\sqrt{R_{0}}} \tag{55}\]  
and  
\[\mathrm{LOD_{PM0}}(f_{\mathrm{x}})=\frac{\sqrt{S_{\varphi\varphi}(f_{\mathrm{x} })}}{|S_{\mathrm{PM}}(f_{\mathrm{x}})|}=\frac{R_{\mathrm{m}}\;\sqrt{2k_{ \mathrm{B}}T_{0}}\;\sqrt{4Q^{2}f_{\mathrm{x}}^{2}+f_{\mathrm{res}}^{2}}}{S_{ \mathrm{mag}}\bar{V}_{\mathrm{ex}}Q\;\sqrt{R_{0}}} \tag{56}\]  
whereas the limit due to the thermal mechanical noise of the resonator is given by  
\[\mathrm{LOD_{AMm}}(f_{\mathrm{x}})=\frac{\sqrt{S_{\mathrm{am}}(f_{\mathrm{x} })}}{|S_{\mathrm{AM}}(f_{\mathrm{x}})|}=\frac{\sqrt{6k_{\mathrm{B}}T_{0}R_{ \mathrm{m}}}\;\sqrt{8\;Q^{2}f_{\mathrm{x}}^{2}+3\;f_{\mathrm{res}}^{2}}\;\sqrt {8\;Q^{2}-4\sqrt{2}\;Q+1}}{4\;S_{\mathrm{mag}}\bar{V}_{\mathrm{ex}}Q^{2}} \tag{57}\]  
and  
\[\text{LOD}_{\text{PMm}}(f_{\text{x}})=\text{LOD}_{\text{PMm}}(0)=\frac{\sqrt{S_{ \text{pmm}}(f_{\text{x}})}}{|S_{\text{PM}}(f_{\text{x}})|}=\frac{f_{\text{res}} \sqrt{2k_{\text{B}}T_{0}R_{\text{m}}}}{S_{\text{mag}}\tilde{V}_{\text{ex}}Q}. \tag{58}\]  
Because both noise sources are additive noise (see above), the LOD can be improved in each case by increasing the excitation voltage \(\tilde{V}_{\text{ex}}\). Figure 8 illustrates the detectivities for the exemplary magnetic field sensor considered in this article. For both operation modes, and in agreement with the previous noise analysis (see Figure 7), the low-frequency detectivities are limited by the thermal mechanical noise of the micromechanical resonator and are constant with the frequency. On the contrary, at higher frequencies and for both operating modes, the thermal electrical noise due to the piezoelectric layer limits the detectivity. While the detectivity in AM mode deteriorates with an increasing frequency proportional to \(f_{\text{x}}^{2}\), the detectivity in PM mode increases only proportional to \(f_{\text{x}}^{1}\) (compare the slopes of the associated dynamic sensitivities in Figure 5).