## 2 Theory and Equation  
The schematic of the operation principle of a Love wave device is shown in Figure 1. The layered structure without a loading layer is the basic structure for supporting the Love wave. The piezoelectric substrate was a Mn:0.24 PIN-0.46 PMN-0.30 PT ternary single crystal poled along the \([001]_{c}\) pseudo-cubic direction. It is assumed that the \([100]_{c}\) pseudo-cubic direction of the single crystal was along the \(x\) direction. The guiding layer was the unidirectional CFEC. Both the poled ternary single crystal and unidirectional CFEC are elastically anisotropic. In our design, the CFs in the CFEC are oriented along the \(x\) or \(y\) axis. In both cases, the Love wave always propagates in the \(x\) direction. Both PMN-PT binary single crystal and Mn:PIN-PMN-PT ternary single crystal are perovskite in structure. When the two single crystals are poled along the \([001]_{c}\) pseudo-cubic direction, they all have a \(4 mm\) macroscopic symmetry. Based on the partial wave theory applicable to anisotropic  
Figure 1: Schematic of the sensor configuration and coordinate system. media [37; 38; 39; 40], substituting the electrical and mechanical boundary conditions and material constants listed in Table 1 & Ref. [41] into the elastic wave equations and piezoelectric equations, the dispersion equation of Love mode surface acoustic waves can be derived as follows [26; 42]:  
\[\tan\biggl{\{}kh\hat{a}\bigl{[}\left(v/\hat{v}_{s}\right)^{2}-1\bigr{]}^{1/2} \biggr{\}}=\eta\frac{\alpha}{\hat{a}}\biggl{[}\frac{1-\left(v/\hat{v}_{s} \right)^{2}}{\left(v/\hat{v}_{s}\right)^{2}-1}\biggr{]}^{1/2} \tag{1}\]  
where \(\hat{v}_{s}<v<v_{s}\), \(\eta=c_{44}/c_{44}\), \(\alpha=\left(c_{66}/c_{44}\right)^{1/2}\), \(\hat{a}=\left(c_{66}/c_{44}\right)^{1/2}\), \(v_{S}=\alpha\left(c_{44}/\rho\right)^{1/2}\), \(\hat{v}_{s}=\hat{a}\left(c_{44}/\rho\right)^{1/2}\), \(k=2\pi/\lambda\), \(v\) is the phase velocity of Love wave, \(h\) is the thickness of the waveguide layer, \(\lambda\) is the wavelength of Love wave, \(\rho\) is the density, and the parameters labeled \"\({}^{\ast\ast}\)\" denote the parameters of the guiding layer. In Table 1, for the \(x\)-axis, \(c_{44}\ =\ \left(c_{22}\text{--}c_{23}\right)/2\); for the \(y\)-axis, \(c_{55}\ =\ \left(c_{11}\text{--}c_{13}\right)/2\). The loading layer mass per unit area (\(\Delta m_{s}\) ) induces a frequency shift \(\Delta f\) as shown in Figure 1. According to the perturbation theory, the density and thickness of the loading layer are, respectively, \(\rho_{3}\) and \(d\)[43], and the mass frequency sensitivity formula for a Love mode device is given as follows [4; 15; 26; 42]:  
\[S_{m}^{f}=\frac{1}{f_{0}}\lim_{\Delta m_{S}\to 0}\biggl{(}\frac{\Delta f}{ \Delta m_{S}}\biggr{)}=\frac{1}{\rho_{3}d}\frac{\Delta f}{f_{0}}=-\frac{\left( 1-v_{53}^{2}/v_{0}^{2}\right)}{2\rho\Gamma(h)} \tag{2}\]  
where \(\Delta f=f-f_{0}\),\(f\) is the oscillation frequency for the perturbed case (mass loading), \(f_{0}\) and \(v_{0}\) are the frequency and SH wave velocity without mass loading respectively, \(v_{53}\) is the shear wave speed in the loading layer, and \(\Gamma(h)\) is as follows:  
\[\Gamma(h)=\int_{h}^{0}\biggl{|}U_{(z)}^{(2)}\biggr{|}^{2}dz+\frac{\rho}{\hat {\rho}}\int_{0}^{\infty}\biggl{|}U_{(z)}^{(1)}\biggr{|}^{2}dz \tag{3}\]  
The distributions of SH-type displacements normalized to that of the surface (\(z=h\)) can be written as follows [26; 42]:  
\[U_{(z)}^{(1)}=\cos\Biggl{\{}k\hat{a}\biggl{[}\bigl{(}\frac{v}{\hat{v}_{s}} \bigr{)}^{2}-1\biggr{]}^{\frac{1}{2}}h\Biggr{\}}\exp\Biggl{\{}k\alpha\biggl{[} 1-\biggl{(}\frac{v}{v_{s}}\biggr{)}^{2}\biggr{]}^{\frac{1}{2}}z\Biggr{\}}(z\leq 0) \tag{4}\]  
\[U_{(z)}^{(2)}=\cos\Biggl{\{}k\hat{a}\biggl{[}\bigl{(}\frac{v}{\hat{v}_{s}} \bigr{)}^{2}-1\biggr{]}^{\frac{1}{2}}(z-h)\Biggr{\}}(0\leq z\leq h) \tag{5}\]  
In general, there is no shear wave in liquid (e.g., water).