An Approximation for Rapid Simulation of Thin-Film Bulk Acoustic Resonators (FBARs) with Sandwich-Layered Structure

Abstract

Thin-film bulk acoustic resonators (FBARs) operating with essentially thickness-extensional mode have been widely used in communication fields. In this paper, we provide a convenient means for analyzing FBARs with sandwich-layered structure by appropriately neglecting the high-order terms from 3D elasticity equations. First, for straight-crested waves, an approximate method is proposed, which can accurately describe the dispersion relation near the operating frequency range of an FBAR. Using the approximation, the optimum lateral size of a 2D model of frame-like FBAR is obtained, and the results are in good agreement with that obtained by commercial FEM software COMSOL. The approximation is further extended to variable-crested waves in order to analyze the 3D plate models for real devices. The mode shapes of 3D FBARs with and without frame-like structures are obtained. The results show that the approximation presented in this paper is of sufficient accuracy and can be used as an efficient tool for the analysis and design of FBARs.

Appendix: Detailed Derivation of Substituting Displacement Solutions into Boundary Conditions

At interfaces x₃ = h^f and x₃ = 0, substituting displacement solutions into constitutive equations, we have.

$$ \begin{gathered} \left[ {\begin{array}{*{20}c} {u_{1}^{f} } \\ {T_{33}^{f} } \\ {u_{3}^{f} } \\ {T_{31}^{f} } \\ \end{array} } \right]^{h} { = }S^{f} \left[ {\begin{array}{*{20}c} {\alpha_{1h}^{f} } \\ {\beta_{2h}^{f} } \\ {\beta_{1h}^{f} } \\ {\alpha_{2h}^{f} } \\ \end{array} } \right] + \frac{e^s}{{h^{f} }}P\left( {\left[ {\begin{array}{*{20}c} {\alpha_{1}^{f} } \\ {\beta_{2}^{f} } \\ {\beta_{1}^{f} } \\ {\alpha_{2}^{f} } \\ \end{array} } \right] - \left[ {\begin{array}{*{20}c} {\alpha_{1h}^{f} } \\ {\beta_{2h}^{f} } \\ {\beta_{1h}^{f} } \\ {\alpha_{2h}^{f} } \\ \end{array} } \right]} \right) \, \hfill \\ \left[ {\begin{array}{*{20}c} {u_{1}^{f} } \\ {T_{33}^{f} } \\ {u_{3}^{f} } \\ {T_{31}^{f} } \\ \end{array} } \right]^{0} { = }S^{f} \left[ {\begin{array}{*{20}c} {\alpha_{1}^{f} } \\ {\beta_{2}^{f} } \\ {\beta_{1}^{f} } \\ {\alpha_{2}^{f} } \\ \end{array} } \right] + \frac{e^s}{{h^{f} }}P\left( {\left[ {\begin{array}{*{20}c} {\alpha_{1}^{f} } \\ {\beta_{2}^{f} } \\ {\beta_{1}^{f} } \\ {\alpha_{2}^{f} } \\ \end{array} } \right] - \left[ {\begin{array}{*{20}c} {\alpha_{1h}^{f} } \\ {\beta_{2h}^{f} } \\ {\beta_{1h}^{f} } \\ {\alpha_{2h}^{f} } \\ \end{array} } \right]} \right)\hfill \\ \end{gathered} $$

(A1)

where

$$ \begin{gathered} S^{f} = \left[ {\begin{array}{*{20}c} 1 & {r_{1}^{f} \xi } & {} & {} \\ {q_{f} \xi } & { - c_{13}^{f} r_{1}^{f} \xi^{2} + \overline{c}_{33}^{f} \text{i}\eta_{f1} } & {} & {} \\ {} & {} & 1 & { - r_{2}^{f} \xi } \\ {} & {} & { - q_{f} \xi } & {c_{55}^{f} \left( {\text i\eta_{f2} + r_{2}^{f} \xi^{2} } \right)} \\ \end{array} } \right], \hfill \\ P = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & { - r_{2}^{f} \xi } \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right] \hfill \\ \end{gathered} $$

(A2)

$$ \begin{gathered} r_{1}^{f} = {{r^{f} } \mathord{\left/ {\vphantom {{r^{f} } {\eta_{f1} }}} \right. \kern-0pt} {\eta_{f1} }}\hfill \\ r_{2}^{f} = {{r^{f} } \mathord{\left/ {\vphantom {{r^{f} } {\eta_{f2} }}} \right. \kern-0pt} {\eta_{f2} }}\hfill \\ q_{f} = - c_{55}^{f} \left( {\text{i}r^{f0} - 1} \right) \hfill \\ \end{gathered} $$

(A3)

$$ \begin{gathered} \left[ {\begin{array}{*{20}c} {\alpha_{1h}^{f} } \\ {\alpha_{2h}^{f} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\text{e}^{{\text{i}\eta_{f2} h^{f} }} } & {\text{e}^{{ - \text{i}\eta_{f2} h^{f} }} } \\ {\text{e}^{{\text{i}\eta_{f2} h^{f} }} } & { - \text{e}^{{ - \text{i}\eta_{f2} h^{f} }} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {A_{1}^{f2} } \\ {A_{2}^{f2} } \\ \end{array} } \right] \, \hfill \\ \left[ {\begin{array}{*{20}c} {\beta_{1h}^{f} } \\ {\beta_{2h}^{f} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\text{e}^{{\text{i}\eta_{f1} h^{f} }} } & {\text{e}^{{ - \text{i}\eta_{f1} h^{f} }} } \\ {\text{e}^{{\text{i}\eta_{f1} h^{f} }} } & { - \text{e}^{{ - \text{i}\eta_{f1} h^{f} }} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {B_{1}^{f1} } \\ {B_{2}^{f1} } \\ \end{array} } \right] \, \hfill \\ \left[ {\begin{array}{*{20}c} {\alpha_{1}^{f} } \\ {\alpha_{2}^{f} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 1 & 1 \\ 1 & { - 1} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {A_{1}^{f2} } \\ {A_{2}^{f2} } \\ \end{array} } \right], \, \left[ {\begin{array}{*{20}c} {\beta_{1}^{f} } \\ {\beta_{2}^{f} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {1} & {1} \\ {1} & { - 1} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {B_{1}^{f1} } \\ {B_{2}^{f1} } \\ \end{array} } \right]\hfill \\ \end{gathered} $$

(A4)

By Taylor's expansion and elimination of higher-order terms, we have

$$ \left( {S^{f} } \right)^{ - 1} = \left[ {\begin{array}{*{20}c} {\tilde{S}_{11} } & {\tilde{S}_{12} } & {} & {} \\ {\tilde{S}_{21} } & {\tilde{S}_{22} } & {} & {} \\ {} & {} & {\tilde{S}_{33} } & {\tilde{S}_{34} } \\ {} & {} & {\tilde{S}_{43} } & {\tilde{S}_{44} } \\ \end{array} } \right] $$

(A5)

where

$$ \begin{gathered} \tilde{S}_{11} = \Delta_{1}^{f} q_{f} r_{1}^{f} \xi^{2} + 1, \, \tilde{S}_{12} = - \Delta_{1}^{f} r_{1}^{f} \xi \hfill \\ \tilde{S}_{21} = - q_{f} \Delta_{1}^{f} \xi , \, \tilde{S}_{22} = \Delta_{1}^{f} \left( {1 - \left( {r_{1}^{f} \xi } \right)^{2} } \right) \hfill \\ \tilde{S}_{33} = \Delta_{2}^{f} q_{f} r_{2}^{f} \xi^{2} + 1, \, \tilde{S}_{34} = \Delta_{2}^{f} r_{2}^{f} \xi \hfill \\ \tilde{S}_{43} = q_{f} \Delta_{2}^{f} \xi , \, \tilde{S}_{44} = \Delta_{2}^{f} \left( {1 - \left( {r_{2}^{f} \xi } \right)^{2} } \right)\hfill \\ \end{gathered} $$

and

$$ \Delta_{1}^{f} = \frac{1}{{\overline{c}_{33}^{f} \text{i}\eta_{f1} }}, \, \Delta_{2}^{f} = \frac{1}{{c_{55}^{f} \text{i}\eta_{f2} }} $$

On the other hand, we can obtain the following relationship by eliminating undetermined amplitude coefficients in Eq. (A4):

$$ \left[ {\begin{array}{*{20}c} {\alpha_{1h}^{f} } \\ {\beta_{2h}^{f} } \\ {\beta_{1h}^{f} } \\ {\alpha_{2h}^{f} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {cs^{f2} } & {} & {} & {sn^{f2} } \\ {} & {cs^{f1} } & {sn^{f1} } & {} \\ {} & {sn^{f1} } & {cs^{f1} } & {} \\ {sn^{f2} } & {} & {} & {cs^{f2} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\alpha_{1}^{f} } \\ {\beta_{1}^{f} } \\ {\alpha_{2}^{f} } \\ {\beta_{2}^{f} } \\ \end{array} } \right] = {E}^{f} \left[ {\begin{array}{*{20}c} {\alpha_{1}^{f} } \\ {\beta_{1}^{f} } \\ {\alpha_{2}^{f} } \\ {\beta_{2}^{f} } \\ \end{array} } \right] $$

(A6)

where

$$ \begin{gathered} sn^{f1} = \text{i}\sin \left( {\eta_{f1} h^{f} } \right), \, sn^{f2} = \text{i}\sin \left( {\eta_{f2} h^{f} } \right) \hfill \\ cs^{f1} = \cos \left( {\eta_{f1} h^{f} } \right), \, cs^{f2} = \cos \left( {\eta_{f2} h^{f} } \right) \hfill \\ \end{gathered} $$

For two electrodes, $S^{d} ,E^{d} ,S^{g}$ and $E^{g}$ can be obtained easily through the replacement of material constants, the detailed derivation process of which is therefore omitted herein. The continuity conditions at interfaces x₃ = h^f and x₃ = 0 are as follows:

$$ \begin{gathered} S^{d} \left[ {\begin{array}{*{20}c} {\alpha_{1}^{d} } \\ {\beta_{2}^{d} } \\ {\beta_{1}^{d} } \\ {\alpha_{2}^{d} } \\ \end{array} } \right] = S^{f} \left[ {\begin{array}{*{20}c} {\alpha_{1h}^{f} } \\ {\beta_{2h}^{f} } \\ {\beta_{1h}^{f} } \\ {\alpha_{2h}^{f} } \\ \end{array} } \right] + \frac{e^s}{{h^{f} }}P\left( {\left[ {\begin{array}{*{20}c} {\alpha_{1}^{f} } \\ {\beta_{2}^{f} } \\ {\beta_{1}^{f} } \\ {\alpha_{2}^{f} } \\ \end{array} } \right] - \left[ {\begin{array}{*{20}c} {\alpha_{1h}^{f} } \\ {\beta_{2h}^{f} } \\ {\beta_{1h}^{f} } \\ {\alpha_{2h}^{f} } \\ \end{array} } \right]} \right) \hfill \\ S^{g} \left[ {\begin{array}{*{20}c} {\alpha_{1}^{g} } \\ {\beta_{2}^{g} } \\ {\beta_{1}^{g} } \\ {\alpha_{2}^{g} } \\ \end{array} } \right] = S^{f} \left[ {\begin{array}{*{20}c} {\alpha_{1}^{f} } \\ {\beta_{2}^{f} } \\ {\beta_{1}^{f} } \\ {\alpha_{2}^{f} } \\ \end{array} } \right] + \frac{e^s}{{h^{f} }}P\left( {\left[ {\begin{array}{*{20}c} {\alpha_{1}^{f} } \\ {\beta_{2}^{f} } \\ {\beta_{1}^{f} } \\ {\alpha_{2}^{f} } \\ \end{array} } \right] - \left[ {\begin{array}{*{20}c} {\alpha_{1h}^{f} } \\ {\beta_{2h}^{f} } \\ {\beta_{1h}^{f} } \\ {\alpha_{2h}^{f} } \\ \end{array} } \right]} \right) \hfill \\ \end{gathered} $$

(A7)

The traction-free boundary conditions of the top and bottom surfaces of the plate are as follows:

$$ \begin{gathered} \left[ {\begin{array}{*{20}c} {T_{33}^{d} } \\ {T_{31}^{d} } \\ \end{array} } \right]^{h} { = }S^{d} \left( {\left[ {2,4} \right],:} \right)\left[ {\begin{array}{*{20}c} {\alpha_{1h}^{d} } \\ {\beta_{2h}^{d} } \\ {\beta_{1h}^{d} } \\ {\alpha_{2h}^{d} } \\ \end{array} } \right] = 0 \hfill \\ \left[ {\begin{array}{*{20}c} {T_{33}^{g} } \\ {T_{31}^{g} } \\ \end{array} } \right]^{h} { = }S^{g} \left( {\left[ {2,4} \right],:} \right)\left[ {\begin{array}{*{20}c} {\alpha_{1h}^{g} } \\ {\beta_{2h}^{g} } \\ {\beta_{1h}^{g} } \\ {\alpha_{2h}^{g} } \\ \end{array} } \right] = 0 \hfill \\ \end{gathered} $$

(A8)

Substituting Eqs. (A6) and (A7) into Eq. (A8), we have

$$ \left[ {\begin{array}{*{20}c} {S^{d} \left( {\left[ {2:4} \right],:} \right)E^{d} \left( {S^{d} } \right)^{ - 1} \left( {S^{f} E^{f} + \frac{e^s}{{h^{f} }}P\left( {I - E^{f} } \right)} \right)} \\ {S^{g} \left( {\left[ {2:4} \right],:} \right)E^{g} \left( {S^{g} } \right)^{ - 1} \left( {S^{f} + \frac{e^s}{{h^{f} }}P\left( {I - E^{f} } \right)} \right)} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\alpha_{1}^{f} } \\ {\beta_{2}^{f} } \\ {\beta_{1}^{f} } \\ {\alpha_{2}^{f} } \\ \end{array} } \right] = 0 $$

(A9)

Then neglecting higher-order terms, we have

$$ \begin{gathered} S^{d} \left( {\left[ {2:4} \right],:} \right)E^{d} \left( {\left( {S^{d} } \right)^{ - 1} S^{f} E^{f} + \frac{e^s}{{h^{f} }}\left( {S^{d} } \right)^{ - 1} P\left( {I - E^{f} } \right)} \right) \hfill \\ = \left[ {\begin{array}{*{20}c} {W_{11} \xi } & {M_{12} \xi^{2} + N_{12} } & {M_{13} \xi^{2} + N_{13} } & {W_{14} \xi } \\ {M_{21} \xi^{2} + N_{21} } & {W_{22} \xi } & {W_{23} \xi } & {M_{24} \xi^{2} + N_{24} } \\ \end{array} } \right] \hfill \\ \end{gathered} $$

(A10)

where

$$ \begin{gathered} W_{11} = w_{11}^{d} cs^{f2} + w_{14}^{d} sn^{f2} + g_{{{11}}}^{d} ; \, W_{14} = w_{11}^{d} sn^{f2} + w_{14}^{d} cs^{f2} + g_{{{14}}}^{d} \hfill \\ W_{22} = w_{22}^{d} cs^{f1} + w_{23}^{d} sn^{f1} ; \, W_{23} = w_{22}^{d} sn^{f1} + w_{23}^{d} cs^{f1} \, \hfill \\ M_{12} = s_{12}^{d} cs^{f1} + s_{13}^{d} sn^{f1} + h_{{{12}}}^{d} ; \, M_{13} = s_{12}^{d} sn^{f1} + s_{13}^{d} cs^{f1} + h_{{{13}}}^{d} \hfill \\ M_{21} = s_{21}^{d} cs^{f2} + s_{24}^{d} sn^{f2} + h_{{{21}}}^{d} ; \, M_{24} = s_{21}^{d} sn^{f2} + s_{24}^{d} cs^{f2} + h_{{{24}}}^{d} \hfill \\ N_{12} = r_{12}^{d} cs^{f1} + r_{{1{3}}}^{d} sn^{f1} + k_{{{12}}}^{d} ; \, N_{13} = r_{12}^{d} sn^{f1} + r_{{1{3}}}^{d} cs^{f1} + k_{{{13}}}^{d} \hfill \\ N_{21} = r_{21}^{d} cs^{f2} + r_{24}^{d} sn^{f2} ; \, N_{24} = r_{21}^{d} sn^{{f{2}}} + r_{24}^{d} cs^{{f{2}}} \hfill \\ \end{gathered} $$

and

$$ \begin{aligned} w_{{11}}^{d} & = \;cs^{{f2}} q_{d} + cs^{{f1}} c_{{33}}^{d} {\text {i}}\eta _{{d1}} m_{{21}}^{d} ,w_{{14}} = sn^{{f1}} c_{{33}}^{d} \text{i}\eta _{{d1}} m_{{34}}^{d} + n_{{44}}^{d} sn^{{f2}} q_{d} \\ w_{{22}}^{d} & = \;sn^{{f2}} c_{{55}}^{d} m_{{12}}^{d} \text{i}\eta _{{d2}} - n_{{22}}^{d} sn^{{f1}} q_{d} ,{\mkern 1mu} w_{{23}} = - cs^{{f1}} q_{d} + c_{{55}}^{d} cs^{{f2}} \text{i}\eta _{{d2}} m_{{43}}^{d} \\ s_{{12}}^{d} & = \;cs^{{f2}} q_{d} m_{{12}}^{d} - n_{{22}}^{d} cs^{{f1}} c_{{13}}^{d} r_{1}^{d} + cs^{{f1}} c_{{33}}^{d} \text{i}\eta _{{d1}} l_{{22}}^{d} \\ {\mkern 1mu} s_{{13}}^{d} & = - c_{{13}}^{d} r_{1}^{d} sn^{{f1}} + l_{{33}}^{d} sn^{{f1}} c_{{33}}^{d} \text{i}\eta _{{d1}} + sn^{{f2}} q_{d} m_{{43}}^{d} \\ s_{{21}}^{d} & = sn^{{f2}} c_{{55}}^{d} \left( {\text{i}\eta _{{d2}} l_{{11}}^{d} + r_{2}^{d} } \right) - sn^{{f1}} q_{d} m_{{21}}^{d} \\ {\mkern 1mu} s_{{24}}^{d} & = - cs^{{f1}} q_{d} m_{{34}}^{d} + c_{{55}}^{d} cs^{{f2}} \left( {\text{i}\eta _{{d2}} l_{{44}}^{d} + r_{2}^{d} n_{{44}}^{d} } \right) \\ r_{{12}}^{d} & = n_{{22}}^{d} cs^{{f1}} c_{{33}}^{d} \text{i}\eta _{{d1}} ,{\mkern 1mu} r_{{21}}^{d} = sn^{{f2}} c_{{55}}^{d} \text{i}\eta _{{d2}} ,{\mkern 1mu} r_{{13}}^{d} = sn^{{f1}} c_{{33}}^{d} \text{i}\eta _{{d1}} \\ {\mkern 1mu} r_{{24}}^{d} & = c_{{55}}^{d} cs^{{f2}} \text{i}\eta _{{d2}} n_{{44}}^{d} \\ \end{aligned} $$

$$ \begin{gathered} l_{11}^{d} = \Delta_{1}^{d} r_{1}^{d} \left( {q_{d} - q_{f} } \right); \, l_{22}^{d} = - \Delta_{1}^{d} \left( {q_{d} r_{1}^{f} + \overline{c}_{33}^{f} \text{i}\eta_{f1} \left( {r_{1}^{d} } \right)^{2} + c_{13}^{f} r_{1}^{f} } \right) \hfill \\ l_{33}^{d} = \Delta_{2}^{d} r_{2}^{d} \left( {q_{d} - q_{f} } \right); \, l_{44}^{d} { = } - \Delta_{2}^{d} \left( {r_{2}^{f} q_{d} + c_{55}^{f} \text{i}\eta_{f2} \left( {r_{2}^{d} } \right)^{2} - c_{55}^{f} r_{2}^{f} } \right) \hfill \\ m_{12}^{d} = r_{1}^{f} - \Delta_{1}^{d} r_{1}^{d} \overline{c}_{33}^{f} \text{i}\eta_{f1} ; \, m_{21}^{d} = \Delta_{1}^{d} \left( { - q_{d} { + }q_{f} } \right)\hfill \\ m_{34}^{d} = - r_{2}^{f} + \Delta_{2}^{d} r_{2}^{d} \text{i}\eta_{f2} c_{55}^{f} ; \, m_{43}^{d} = \Delta_{2}^{d} \left( {q_{d} - q_{f} } \right) \hfill \\ n_{22}^{d} { = }\Delta_{1}^{d} \overline{c}_{33}^{f} \text{i}\eta_{f1} {; }\,n_{44}^{d} { = }\Delta_{2}^{d} c_{55}^{f} \text{i}\eta_{f2} \hfill \\ \end{gathered} $$

$$ \begin{gathered} a_{{{11}}}^{d} { = } - r_{1}^{d} r_{2}^{f} sn^{f2} , \, a_{14}^{d} { = } - r_{1}^{d} r_{2}^{f} \left( {cs^{f2} - 1} \right) \hfill \\ a_{22}^{d} { = }\left( {r_{1}^{d} } \right)^{2} sn^{f1} , \, a_{23}^{d} { = } - \left( {1 + cs^{f1} } \right)\left( {r_{1}^{d} } \right)^{2} \hfill \\ b_{12}^{d} { = }r_{1}^{d} sn^{f1} {, } \, b_{13}^{d} { = } - r_{1}^{d} \left( {1 + cs^{f1} } \right) \hfill \\ b_{21}^{d} { = }r_{2}^{f} sn^{f2} , \, b_{24}^{d} { = }r_{2}^{f} \left( {cs^{f2} - 1} \right)\xi \hfill \\ \end{gathered} $$

$$ \begin{gathered} g_{11}^{d} { = }\Delta_{1}^{d} cs^{f1} c_{33}^{d} \text{i}\eta_{d1} b_{21} ,\, g_{14}^{d} { = }\Delta_{1}^{d} b_{{{24}}} cs^{f1} c_{33}^{d} \text{i}\eta_{d1} \xi \hfill \\ g_{{{22}}}^{d} { = }\Delta_{1}^{d} \left( {sn^{f2} c_{55}^{d} \text{i}\eta_{d2} b_{12} + sn^{f1} q_{d} sn^{f1} } \right) \hfill \\ g_{{{23}}}^{d} { = }\Delta_{1}^{d} \left( {b_{12} sn^{f2} c_{55}^{d} \text{i}\eta_{d2} - sn^{f1} q_{d} \left( {1 + cs^{f1} } \right)} \right) \hfill \\ h_{12}^{d} { = }\Delta_{1}^{d} \left( {cs^{f2} q_{d} b_{12} + a_{22} + sn^{f1} cs^{f1} c_{13}^{d} r_{1}^{d} } \right) \, \hfill \\ h_{13}^{d} { = }\Delta_{1}^{d} \left( {cs^{f2} q_{d} b_{13} - \left( {1 + cs^{f1} } \right)cs^{f1} c_{13}^{d} r_{1}^{d} + cs^{f1} c_{33}^{d} \text{i}\eta_{d1} a_{{{23}}} } \right)\hfill \\ h_{21}^{d} { = }\Delta_{1}^{d} \left( {sn^{f2} c_{55}^{d} \text{i}\eta_{d2} a_{11} - sn^{f1} q_{d} b_{21} } \right) \hfill \\ h_{{{24}}}^{d} { = }\Delta_{1}^{d} \left( {a_{{1{4}}} sn^{f2} c_{55}^{d} \text{i}\eta_{d2} - b_{{{24}}} sn^{f1} q_{d} } \right) \hfill \\ k_{12}^{d} { = } - \Delta_{1}^{d} sn^{f1} cs^{f1} c_{33}^{d} \text{i}\eta_{d1},\, k_{13}^{d} { = }\Delta_{1}^{d} \left( {1 + cs^{f1} } \right)cs^{f1} c_{33}^{d} \text{i}\eta_{d1} \hfill \\ \end{gathered} $$

Obviously, for the last two equations in Eq. (A9), we have

$$ \begin{gathered} S^{g} \left( {\left[ {2:4} \right],:} \right)E^{g} \left( {\left( {S^{g} } \right)^{ - 1} S^{f} E^{f} + \left( {S^{g} } \right)^{ - 1} P\left( {I - E^{f} } \right)} \right) \hfill \\ = \left[ {\begin{array}{*{20}c} {W_{31} \xi } & {M_{32} \xi^{2} + N_{32} } & {M_{33} \xi^{2} + N_{33} } & {W_{34} \xi } \\ {M_{41} \xi^{2} + N_{41} } & {W_{42} \xi } & {W_{43} \xi } & {M_{44} \xi^{2} + N_{44} } \\ \end{array} } \right] \hfill \\ \end{gathered} $$

(A11)

The coefficients in Eq. (A11) can also be obtained easily by the replacement of material constants in Eq. (A10). From the condition for a nontrivial solution, we have

$$ \begin{gathered} \left| {\begin{array}{*{20}c} {W_{11} \xi } & {M_{12} \xi^{2} + N_{12} } & {M_{13} \xi^{2} + N_{13} } & {W_{14} \xi } \\ {M_{21} \xi^{2} + N_{21} } & {W_{22} \xi } & {W_{23} \xi } & {M_{24} \xi^{2} + N_{24} } \\ {W_{31} \xi } & {M_{32} \xi^{2} + N_{32} } & {M_{33} \xi^{2} + N_{33} } & {W_{34} \xi } \\ {M_{41} \xi^{2} + N_{41} } & {W_{42} \xi } & {W_{43} \xi } & {M_{44} \xi^{2} + N_{44} } \\ \end{array} } \right| \hfill \\ = W\xi^{2} + \left( {N_{13} N_{32} - N_{12} N_{33} } \right)\left( {N_{21} N_{44} - N_{24} N_{41} } \right) = 0 \hfill \\ \end{gathered} $$

(A12)

where the value of W can be obtained by numerical calculation software. Substituting Eqs. (27) and (34) into the expressions for N₁₂, N₁₃, N₃₂, and N₃₃, we have

$$ \begin{gathered} N_{12} = N_{12}^{0} + N_{12}^{\delta } \delta_{f} + N_{12}^{\xi } \xi^{2} , \, N_{13} = N_{13}^{0} + N_{13}^{\delta } \delta_{f} + N_{13}^{\xi } \xi^{2} \hfill \\ N_{32} = N_{32}^{0} + N_{32}^{\delta } \delta_{f} + N_{32}^{\xi } \xi^{2} , \, N_{{{3}3}} = N_{33}^{0} + N_{33}^{\delta } \delta_{f} + N_{33}^{\xi } \xi^{2} \hfill \\ \end{gathered} $$

(A13)

where

$$ \begin{gathered} N_{12}^{\xi } = sn^{f0} c_{33}^{d} \text{i}sn^{f0} K_{d} , \, N_{13}^{\xi } = cs^{f0} c_{33}^{d} \text{i}sn^{f0} K_{d} \hfill \\ N_{12}^{\delta } = \text{i}\left( \begin{gathered} cs^{f0} \overline{c}_{33}^{f} \left( {cs^{f0} + \eta_{f}^{0} \text{i}h^{f} sn^{f0} } \right) + r_{12}^{d0} h^{f} sn^{f0} + r_{13}^{d0} h^{f} cs^{f0} \hfill \\ - e^s\cos \left( {2\eta_{f0} h^{f} } \right) + \eta_{d}^{0} cs^{f0} sn^{f0} c_{33}^{d} \text{i} + sn^{f0} c_{33}^{d} sn^{f0} \mu^{d} \hfill \\ \end{gathered} \right)\hfill \\ N_{13}^{\delta } = \text{i}\left( \begin{gathered} sn^{f0} \overline{c}_{33}^{f} \left( {cs^{f0} + \eta_{f}^{0} \text{i}h^{f} sn^{f0} } \right) + r_{12}^{d0} h^{f} cs^{f0} + r_{13}^{d0} h^{f} sn^{f0} \hfill \\ + e^{s} \left( {1 + 2cs^{f0} } \right)sn^{f0} + \eta_{d}^{0} cs^{f0} cs^{f0} c_{33}^{d} \text{i} + cs^{f0} c_{33}^{d} sn^{f0} \mu^{d} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

and the expressions for $N_{32}^{\delta } ,N_{33}^{\delta } ,N_{32}^{\xi }$ and $N_{{3{3}}}^{\xi }$ can be obtained by replacement of material constants as before. Then substituting Eq. (A13) into Eq. (A12) and retaining the terms of $\xi$ up to quadratic and the terms of $\delta_{f}$ linear, we have

$$ U + W\xi^{2} { + }HQ\delta_{f} + HR\xi^{2} = 0 $$

(A14)

where

$$ \begin{gathered} L = N_{21} N_{44} - N_{24} N_{41} \hfill \\ U = N_{13}^{0} N_{32}^{0} - N_{12}^{0} N_{33}^{0} \hfill \\ Q{ = }N_{13}^{0} N_{32}^{\delta } { + }N_{32}^{0} N_{13}^{\delta } - N_{{1{2}}}^{0} N_{{3{3}}}^{\delta } { - }N_{{3{3}}}^{0} N_{{1{2}}}^{\delta } \hfill \\ R{ = }N_{13}^{0} N_{32}^{\varepsilon } { + }N_{32}^{0} N_{13}^{\varepsilon } - N_{{1{2}}}^{0} N_{{3{3}}}^{\varepsilon } { - }N_{{3{3}}}^{0} N_{{1{2}}}^{\varepsilon } \hfill \\ \end{gathered} $$

We can also find that Eq. (12) is equivalent to

$$ U = N_{13}^{0} N_{32}^{0} - N_{12}^{0} N_{33}^{0} { = 0} $$

It is well understood that our theory reduce to pure thickness vibration when the in-plane wavenumbers vanish.