Two-Dimensional Coupling Vibration Analysis of Laterally Acoustically Coupled Two-Port Thin-Film Bulk Acoustic Resonators

Abstract

In this paper, we present an approach to studying the mode coupling vibrations in two-port thin-film bulk acoustic wave resonator (FBAR) devices with two pairs of electrodes deposited on the zinc oxide film. The two-dimensional plate theory established in our previous work is employed, which takes into account the coupling of the operating thickness-extensional mode with the extensional, flexural, fundamental and second-order thickness-shear modes. The propagation of straight-crested waves in the plate is studied, and the state-vector approach is successfully used to simplify the derivation process. For a structurally symmetric device, the modes are separated into quasi symmetric and antisymmetric ones. Frequency spectra and corresponding mode shapes are obtained under the stress-free boundary conditions, respectively, and then coupling effects and energy trapping phenomenon are discussed in detail. Some results for structures with asymmetric electrode distributions are also shown. It is found that the choice of aspect ratio has a great effect on mode couplings of FBAR devices. This study will be useful for the design of FBAR filters and sensors.

Appendix A: Expressions for the Constants in Equations of Motion (Eq. (3)) and Constitutive Relations (Eq. (4))

$$\begin{aligned} {{\bar{c}}}_{11}^{(0)}= & {} c_{11}^{\left( 0 \right) } +c_{13}^{(1)} \gamma _{3110} +c_{13}^{(2)} \gamma _{3210} ,\;{{\bar{c}}}_{13}^{(0)} =c_{13}^{\left( 0 \right) } +c_{33}^{(1)} \gamma _{3110} +c_{33}^{(2)} \gamma _{3210} \nonumber \\ {{\bar{c}}}_{11}^{(1)}= & {} c_{11}^{\left( 1 \right) } +c_{13}^{(2)} \gamma _{3110} +c_{13}^{(3)} \gamma _{3210} ,\;{{\bar{c}}}_{11}^{(2)} =c_{11}^{\left( 2 \right) } +c_{13}^{(3)} \gamma _{3110} +c_{13}^{(4)} \gamma _{3210} \nonumber \\ {{\bar{e}}}_{31}^{(0)}= & {} e_{31}^{\left( 0 \right) } +e_{33}^{(1)} \gamma _{3110} +e_{33}^{(2)} \gamma _{3210} \end{aligned}$$

(A1)

$$\begin{aligned} {{\bar{c}}}_{31}^{(0)}= & {} c_{13}^{\left( 0 \right) } +c_{13}^{(1)} \gamma _{3130} +c_{13}^{(2)} \gamma _{3230} ,\;{{\bar{c}}}_{33}^{(0)} =c_{33}^{\left( 0 \right) } +c_{33}^{(1)} \gamma _{3130} +c_{33}^{(2)} \gamma _{3230} \nonumber \\ {{\bar{c}}}_{31}^{(1)}= & {} c_{13}^{\left( 1 \right) } +c_{13}^{(2)} \gamma _{3130} +c_{13}^{(3)} \gamma _{3230} ,\;{{\bar{c}}}_{31}^{(2)} =c_{13}^{\left( 2 \right) } +c_{13}^{(3)} \gamma _{3130} +c_{13}^{(4)} \gamma _{3230} \nonumber \\ {{\bar{e}}}_{33}^{(0)}= & {} e_{33}^{\left( 0 \right) } +e_{33}^{(1)} \gamma _{3130} +e_{33}^{(2)} \gamma _{3230} \end{aligned}$$

(A2)

$$\begin{aligned} {{\bar{c}}}_{55}^{(0)}= & {} c_{44}^{\left( 0 \right) } -\frac{c_{44}^{(2)} c_{44}^{\left( 2 \right) } }{c_{44}^{\left( 4 \right) } },\;{{\bar{c}}}_{55}^{(1)} =c_{44}^{\left( 1 \right) } -\frac{c_{44}^{\left( 2 \right) } c_{44}^{(3)} }{c_{44}^{\left( 4 \right) } } \end{aligned}$$

(A3)

$$\begin{aligned} {\tilde{c}}_{11}^{(1)}= & {} c_{11}^{\left( 1 \right) } +c_{13}^{(1)} \gamma _{3111} +c_{13}^{(2)} \gamma _{3211} ,\;{\tilde{c}}_{13}^{(1)} =c_{13}^{\left( 1 \right) } +c_{33}^{(1)} \gamma _{3111} +c_{33}^{(2)} \gamma _{3211} \nonumber \\ {\tilde{c}}_{11}^{(2)}= & {} c_{11}^{\left( 2 \right) } +c_{13}^{(2)} \gamma _{3111} +c_{13}^{(3)} \gamma _{3211} ,\;{\tilde{c}}_{11}^{(3)} =c_{11}^{\left( 3 \right) } +c_{13}^{(3)} \gamma _{3111} +c_{13}^{(4)} \gamma _{3211} \nonumber \\ {\tilde{e}}_{31}^{(1)}= & {} e_{31}^{\left( 1 \right) } +e_{33}^{(1)} \gamma _{3111} +e_{33}^{(2)} \gamma _{3211} \end{aligned}$$

(A4)

$$\begin{aligned} {\tilde{c}}_{55}^{(1)}= & {} c_{44}^{\left( 1 \right) } -\frac{c_{44}^{(2)} c_{44}^{\left( 3 \right) } }{c_{44}^{\left( 4 \right) } },\;{\tilde{c}}_{55}^{(2)} =c_{44}^{\left( 2 \right) } -\frac{c_{44}^{(3)} c_{44}^{\left( 3 \right) } }{c_{44}^{\left( 4 \right) } } \end{aligned}$$

(A5)

$$\begin{aligned} {{\hat{c}}}_{11}^{(2)}= & {} c_{11}^{\left( 2 \right) } +c_{13}^{(1)} \gamma _{3112} +c_{13}^{(2)} \gamma _{3212} ,\;{{\hat{c}}}_{13}^{(2)} =c_{13}^{\left( 2 \right) } +c_{33}^{(1)} \gamma _{3112} +c_{33}^{(2)} \gamma _{3212} \nonumber \\ {{\hat{c}}}_{11}^{(3)}= & {} c_{11}^{\left( 3 \right) } +c_{13}^{(2)} \gamma _{3112} +c_{13}^{(3)} \gamma _{3212} ,\;{{\hat{c}}}_{11}^{(4)} =c_{11}^{\left( 4 \right) } +c_{13}^{(3)} \gamma _{3112} +c_{13}^{(4)} \gamma _{3212} \nonumber \\ {{\hat{e}}}_{31}^{(2)}= & {} e_{31}^{\left( 2 \right) } +e_{33}^{(1)} \gamma _{3112} +e_{33}^{(2)} \gamma _{3212} \end{aligned}$$

(A6)

where $c_{pq}^{( n )}$, $e_{kp}^{( n )}$ and $\rho ^{( n )}$ are the integrals of usual elastic constants, piezoelectric constants and mass density along the thickness of plate, defined by

$$\begin{aligned} c_{pq}^{(n)} =\int _{-h}^{ h} {c_{pq} x_{3}^{n} \hbox {d}x_3 } ,\;e_{kp}^{(n)} =\int _{ -h}^{ h} {e_{kp} x_3^n \hbox {d}x_3 } ,\;\rho ^{(n)}=\int _{-h}^{h} {\rho {}x_3^n \hbox {d}x_3 } \end{aligned}$$

(A7)

and we define

$$\begin{aligned} \gamma _{3110}= & {} \frac{c_{33}^{\left( 3 \right) } c_{13}^{\left( 2 \right) } -c_{33}^{\left( 4 \right) } c_{13}^{\left( 1 \right) } }{c_{33}^{\left( 2 \right) } c_{33}^{\left( 4 \right) } -c_{33}^{\left( 3 \right) } c_{33}^{\left( 3 \right) } },\;\gamma _{3130} =\frac{c_{33}^{\left( 3 \right) } c_{33}^{\left( 2 \right) } -c_{33}^{\left( 4 \right) } c_{33}^{\left( 1 \right) } }{c_{33}^{\left( 2 \right) } c_{33}^{\left( 4 \right) } -c_{33}^{\left( 3 \right) } c_{33}^{\left( 3 \right) } } \nonumber \\ \gamma _{3111}= & {} \frac{c_{33}^{\left( 3 \right) } c_{13}^{\left( 3 \right) } -c_{33}^{\left( 4 \right) } c_{13}^{\left( 2 \right) } }{c_{33}^{\left( 2 \right) } c_{33}^{\left( 4 \right) } -c_{33}^{\left( 3 \right) } c_{33}^{\left( 3 \right) } },\;\gamma _{3112} =\frac{c_{33}^{\left( 3 \right) } c_{13}^{\left( 4 \right) } -c_{33}^{\left( 4 \right) } c_{13}^{\left( 3 \right) } }{c_{33}^{\left( 2 \right) } c_{33}^{\left( 4 \right) } -c_{33}^{\left( 3 \right) } c_{33}^{\left( 3 \right) } } \nonumber \\ \gamma _{3210}= & {} \frac{c_{33}^{\left( 3 \right) } c_{13}^{\left( 1 \right) } -c_{33}^{\left( 2 \right) } c_{13}^{\left( 2 \right) } }{c_{33}^{\left( 2 \right) } c_{33}^{\left( 4 \right) } -c_{33}^{\left( 3 \right) } c_{33}^{\left( 3 \right) } },\;\gamma _{3230} =\frac{c_{33}^{\left( 3 \right) } c_{33}^{\left( 1 \right) } -c_{33}^{\left( 2 \right) } c_{33}^{\left( 2 \right) } }{c_{33}^{\left( 2 \right) } c_{33}^{\left( 4 \right) } -c_{33}^{\left( 3 \right) } c_{33}^{\left( 3 \right) } } \nonumber \\ \gamma _{3211}= & {} \frac{c_{33}^{\left( 3 \right) } c_{13}^{\left( 2 \right) } -c_{33}^{\left( 2 \right) } c_{13}^{\left( 3 \right) } }{c_{33}^{\left( 2 \right) } c_{33}^{\left( 4 \right) } -c_{33}^{\left( 3 \right) } c_{33}^{\left( 3 \right) } },\;\gamma _{3212} =\frac{c_{33}^{\left( 3 \right) } c_{13}^{\left( 3 \right) } -c_{33}^{\left( 2 \right) } c_{13}^{\left( 4 \right) } }{c_{33}^{\left( 2 \right) } c_{33}^{\left( 4 \right) } -c_{33}^{\left( 3 \right) } c_{33}^{\left( 3 \right) } } \end{aligned}$$

(A8)