# Enhanced Sensitivity of FeGa Thin-Film Coated SAW Current Sensor
## 2 Working Principle  
Under the magnetic field generated by the applied current, the FeGa film produces magnetostrictive strain and so-called \(\Delta E\) effect, which modulates the SAW propagation. Figure 1: The scheme of patterned FeGa thin film coated SAW sensing device. A layered media structure of FeGa/SiO\({}_{2}\)/LiNbO\({}_{3}\) is proposed to demonstrate the sensing mechanism, as shown in Figure 2.  
1. In the piezoelectric media  
The constitutive wave motion equations in a piezoelectric LiNbO\({}_{3}\) can be expressed as  
\[\left\{ \begin{array}{c}\rho_{S}\frac{\partial^{2}u_{i}^{I}}{\partial t^{2}}-c_ {jkl}^{II}\frac{\partial^{2}u_{k}^{I}}{\partial x_{l}\partial x_{j}}-e_{kjl} \frac{\partial^{2}\Phi}{\partial x_{l}\partial x_{j}}=0\\ e_{jl}\frac{\partial^{2}u_{k}^{I}}{\partial x_{l}\partial x_{j}}-e_{jk}^{S} \frac{\partial^{2}\Phi}{\partial x_{l}\partial x_{j}}=0\end{array} \right. \tag{1}\]  
where [PERSON]'s summation rule is used. The indices \(i\), \(j\), \(k\), \(l\) = 1, 2, 3, and \(c^{I}\), \(e\), \(\varepsilon\), \(\rho_{S}\) stand for the elastic, piezoelectric, dielectric constants and the mass density of the piezoelectric substrate, respectively. \(u^{I}\) denotes the mechanical displacements, and \(\Phi\) denotes the electric potential. 1. In the SiO\({}_{2}\) media,  
The SiO\({}_{2}\) media is considered as the isotropic and nonpiezoelectric media. Then, the acoustic wave equation can be written as  
\[c_{i\
eq l}^{II}\frac{\partial^{2}u_{k}^{II}}{\partial x_{l}\partial x_{j}}= \rho_{a}\frac{\partial^{2}u_{i}^{II}}{\partial t^{2}} \tag{2}\]  
where \(u^{II}\), \(c^{II}\) and \(\rho_{a}\) are the mechanical displacements, stiffness constants and density of the SiO\({}_{2}\) film, respectively. 1. In the FeGa media,
2. The acoustic wave equation in the FeGa media can also be written as \[c_{i\
eq l}^{III}\frac{\partial^{2}u_{k}^{III}}{\partial x_{l}\partial x_{j}}= \rho_{b}\frac{\partial^{2}u_{i}^{III}}{\partial t^{2}}\] (3)  
where \(u^{III}\), \(c^{III}\) and \(\rho_{b}\) are the mechanical displacements, the stiffness constants and density of the FeGa film, respectively. 1. Magnetostirctive strain of the FeGa film  
Under the magnetic field generated by the applied current, the FeGa film produces magnetostrictive strain, which leads to changes in thickness \(h\) and density \(\rho_{f}\) of FeGa film. \[\left\{ \begin{array}{c}h=h_{0}(1-\frac{\lambda}{2})\\ \rho_{f}=\rho_{0}/[(1+\lambda)\times(1-\frac{\lambda}{2})\times(1-\frac{ \lambda}{2})]\end{array} \right. \tag{4}\]  
where \(h_{0}\) and \(\rho_{0}\) are the thickness and density of unperturbed FeGa film, respectively, and \(\lambda\) is the magnetostrictive coefficient of the FeGa film. 1. \(\Delta E\) effect on the FeGa film
2. The layered media model (FeGa/SiO\({}_{2}\)/LiNbO\({}_{3}\)). Figure 2: The layered media model (FeGa/SiO\({}_{2}\)/LiNbO\({}_{3}\)). The external magnetic field will also modulate the [PERSON]'s modulus (\(E\)) of the FeGa film, which is so-called \(\Delta E\) effect, and the corresponding elastic coefficients in FeGa film are expressed by the perturbed [PERSON]'s modulus \(E^{{}^{\prime}}\) as  
\[\left\{ \begin{array}{c}c_{11}=\frac{E^{{}^{\prime}}(1-u)}{(1+u)(1-2u)}\\ c_{12}=\frac{E^{{}^{\prime}}u}{(1+u)(1-2u)}\\ c_{44}=\frac{c_{11}-c_{12}}{2}=\frac{E^{{}^{\prime}}(1-2u)}{2(1+u)(1-2u)} \end{array} \right. \tag{5}\]  
where \(u\) is the [PERSON]'s ratio. The relationship between the electromagnetic field intensity (\(H\)) and current (\(A\)) can be described by the Biot-Savart law [24],  
\[H=\frac{\mu_{0}I}{4\pi l} \tag{6}\]  
where \(\mu_{0}\) denotes the permeability in vacuum. \(l\) expresses the distance between the sensor chip and wires, which is set to 1 cm. According to the Formulas (1)-(3), the solution of each layer media can be obtained. Then, combined with the mechanics and electrical boundary conditions, the effective surface permittivity method can be used to obtain the speed of SAW.