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Equivalent imperfect interface model of PN junction of piezoelectric semiconductor for the multi-field coupled waves propagation

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A Correction to this article was published on 08 November 2023

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Abstract

In this paper, an equivalent imperfect interface model of PN homojunction/ heterojunction of piezoelectric semiconductor for the multi-field coupled wave propagation is proposed firstly. PN junction is a special structure formed by the contact of two different types of doped semiconductors, which has been used extensively in semiconductor devices. Due to the gradient distribution of the electric potential and carrier concentration in PN junction of finite thickness, the reflection and the transmission will arise when the coupled waves propagate through PN junction. The effects of the PN junction on the wave propagation will be much more complicated by the accurate estimation. An equivalent imperfect interface model without thickness but with seven interface parameters is established to simulate the effects of PN junction which largely reduces the calculation cost. The numerical examples are provided and compared with the state transfer matrix method and the piecewise homogenization method. Energy flux of the reflected and transmitted waves are estimated, and the energy conservation is checked to verify reliability of the numerical results.

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Acknowledgements

The work is supported by National Natural Science Foundation of China (No. 11872105, NO.12072022, NO.11911530176 and NO.12202039), Fundamental Research Funds for the Central Universities (FRF-TW-2018-005, FRF-BR-18-008B, FRF-TP-18-077A1).

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Authors and Affiliations

  1. Department of Applied Mechanics, University of Science and Technology Beijing, Beijing, 100083, China

    Zibo Wei, Peijun Wei, Chunyu Xu & Xiao Guo

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  1. Zibo Wei
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  2. Peijun Wei
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  3. Chunyu Xu
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  4. Xiao Guo
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Correspondence to Peijun Wei.

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Appendices

Appendix A

$$\begin{gathered} {\mathbf{c}} = \left[ {\begin{array}{*{20}c} {c_{11} } &\quad {c_{12} } &\quad {c_{13} } &\quad 0 &\quad 0 &\quad 0 \\ {c_{12} } &\quad {c_{11} } &\quad {c_{13} } &\quad 0 &\quad 0 &\quad 0 \\ {c_{13} } &\quad {c_{13} } &\quad {c_{33} } &\quad 0 &\quad 0 &\quad 0 \\ 0 &\quad 0 &\quad 0 &\quad {c_{44} } &\quad 0 &\quad 0 \\ 0 &\quad 0 &\quad 0 &\quad 0 &\quad {c_{44} } &\quad 0 \\ 0 &\quad 0 &\quad 0 &\quad 0 &\quad 0 &\quad {\tfrac{1}{2}(c_{11} - c_{12} )} \\ \end{array} } \right],\quad {\mathbf{e}}_{{\mathbf{1}}} = \left[ {\begin{array}{*{20}c} 0 &\quad 0 &\quad {e_{31} } \\ 0 &\quad 0 &\quad {e_{31} } \\ 0 &\quad 0 &\quad {e_{33} } \\ 0 &\quad {e_{15} } &\quad 0 \\ {e_{15} } &\quad 0 &\quad 0 \\ 0 &\quad 0 &\quad 0 \\ \end{array} } \right],\quad {{\varvec{\upvarepsilon}}} = \left[ {\begin{array}{*{20}c} {\varepsilon_{11} } &\quad 0 &\quad 0 \\ 0 &\quad {\varepsilon_{11} } &\quad 0 \\ 0 &\quad 0 &\quad {\varepsilon_{33} } \\ \end{array} } \right], \hfill \\ {\mathbf{e}}_{{\mathbf{2}}} = \left[ {\begin{array}{*{20}c} 0 &\quad 0 &\quad 0 &\quad 0 &\quad {e_{15} } &\quad 0 \\ 0 &\quad 0 &\quad 0 &\quad {e_{15} } &\quad 0 &\quad 0 \\ {e_{31} } &\quad {e_{31} } &\quad {e_{33} } &\quad 0 &\quad 0 &\quad 0 \\ \end{array} } \right],\quad {{\varvec{\upmu}}}^{{\mathbf{p}}} = \left[ {\begin{array}{*{20}c} {q\overline{p}_{N} \mu_{11}^{p} } &\quad 0 &\quad 0 \\ 0 &\quad {q\overline{p}_{N} \mu_{11}^{p} } &\quad 0 \\ 0 &\quad 0 &\quad {q\overline{p}_{N} \mu_{33}^{p} } \\ \end{array} } \right], \hfill \\ \end{gathered}$$
$${\mathbf{d}}^{{\mathbf{p}}} = \left[ {\begin{array}{*{20}c} {qd_{11}^{p} } &\quad 0 &\quad 0 \\ 0 &\quad {qd_{11}^{p} } &\quad 0 \\ 0 &\quad 0 &\quad {qd_{33}^{p} } \\ \end{array} } \right],\quad {{\varvec{\upmu}}}^{{\mathbf{n}}} = \left[ {\begin{array}{*{20}c} {q\overline{n}_{N} \mu_{11}^{n} } &\quad 0 &\quad 0 \\ 0 &\quad {q\overline{n}_{N} \mu_{11}^{n} } &\quad 0 \\ 0 &\quad 0 &\quad {q\overline{n}_{N} \mu_{33}^{n} } \\ \end{array} } \right],\quad {\mathbf{d}}^{{\mathbf{n}}} = \left[ {\begin{array}{*{20}c} {qd_{11}^{n} } &\quad 0 &\quad 0 \\ 0 &\quad {qd_{11}^{n} } &\quad 0 \\ 0 &\quad 0 &\quad {qd_{33}^{n} } \\ \end{array} } \right],$$

Appendix B

$$\begin{aligned} K_{11} & { = }c_{11} k_{x}^{2} + c_{44} k_{z}^{2} - \rho \omega^{2} ,\;K_{12} { = }K_{21} = \left( {c_{13} + c_{44} } \right)k_{x} k_{z} ,\;K_{33} = - \varepsilon_{11} k_{x}^{2} - \varepsilon_{33} k_{z}^{2} , \\ K_{13} & { = }K_{31} = \left( {e_{13} + e_{44} } \right)k_{x} k_{z} ,\;K_{22} = c_{44} k_{x}^{2} + c_{33} k_{z}^{2} - \rho \omega^{2} ,\;K_{23} { = }K_{32} = e_{15} k_{x}^{2} + e_{33} k_{z}^{2} , \\ K_{34} & = q,K_{35} = - q,\;K_{43} = \overline{p}_{N} \left( {\mu_{11}^{p} k_{x}^{2} + \mu_{33}^{p} k_{z}^{2} } \right),\;K_{44} = d_{11}^{p} k_{x}^{2} + d_{33}^{p} k_{z}^{2} - i\omega , \\ K_{53} & = - \overline{n}_{N} \left( {\mu_{11}^{n} k_{x}^{2} + \mu_{33}^{n} k_{z}^{2} } \right),\;K_{55} = d_{11}^{n} k_{x}^{2} + d_{33}^{n} k_{z}^{2} - i\omega . \\ \end{aligned}$$

Appendix C

Appendix D

$$\begin{aligned} A_{1s} &= c_{44} \left( {k_{3}^{\left( s \right)} + G_{s} k_{1}^{\left( s \right)} } \right) + e_{15} H_{s} k_{1}^{\left( s \right)} ,\;A_{2s} = c_{13} k_{1}^{\left( s \right)} + c_{33} k_{3}^{\left( s \right)} G_{s} + e_{33} k_{3}^{\left( s \right)} H_{s} , \hfill \\ A_{3s} &= c_{11} k_{1}^{\left( s \right)} + c_{13} k_{3}^{\left( s \right)} G_{s} + e_{31} k_{3}^{\left( s \right)} H_{s} ,\;A_{4s} = e_{31} k_{1}^{\left( s \right)} + e_{33} k_{3}^{\left( s \right)} G_{s} - \varepsilon_{33} k_{3}^{\left( s \right)} H_{s} , \hfill \\ A_{5s} &= e_{15} \left( {k_{3}^{\left( s \right)} + k_{1}^{\left( s \right)} G_{s} } \right) - \varepsilon_{11} k_{1}^{\left( s \right)} H_{s} ,\;A_{6s} = - iq\left( {\overline{p}_{N} \mu_{33}^{p} H_{s} + d_{33}^{p} Q_{s} } \right)k_{3}^{\left( s \right)} , \hfill \\ A_{7s} &= - iq\left( {\overline{p}_{N} \mu_{11}^{p} H_{s} + d_{11}^{p} Q_{s} } \right)k_{1}^{\left( s \right)} ,\;A_{8s} = - iq\left( {\overline{n}_{N} \mu_{33}^{n} H_{s} - d_{33}^{n} L_{s} } \right)k_{3}^{\left( s \right)} ,\\ A_{9s} &= - iq\left( {\overline{n}_{N} \mu_{11}^{n} H_{s} - d_{11}^{n} L_{s} } \right)k_{1}^{\left( s \right)} , \hfill \\ \end{aligned}$$

where \(s = I,R.\)

$$\begin{aligned} A_{1T}^{\prime } &= c_{44}^{\prime } \left( {k_{3}^{\prime (T)} + G_{T}^{\prime } k_{1}^{\prime (T)} } \right) + e_{15}^{\prime } H_{t}^{\prime } k_{1}^{\prime (T)} ,A_{2T}^{\prime } = c_{13}^{\prime } k_{1}^{\prime (T)} + c_{33}^{\prime } k_{33}^{\prime (T)} G_{T}^{\prime } + e_{33}^{\prime } k_{33}^{\prime (T)} H_{T}^{\prime } , \hfill \\ A_{3T}^{\prime } &= c_{11}^{\prime } k_{1}^{{^{\prime (T)} }} + c_{13}^{\prime } k_{33}^{\prime (T)} G_{T}^{\prime } + e_{31}^{\prime } k_{3}^{\prime (T)} H_{T}^{\prime } ,\;A_{4T}^{\prime } = e_{31}^{\prime } k_{1}^{\prime (T)} + e_{33}^{\prime } k_{3}^{\prime (T)} G_{b}^{\prime } - \varepsilon_{33}^{\prime } k_{3}^{\prime (T)} H_{b}^{\prime } , \hfill \\ A_{5T}^{\prime } &= e_{15}^{\prime } \left( {k_{33}^{\prime (T)} + k_{1}^{\prime (T)} G_{T}^{\prime } } \right) - \varepsilon_{11}^{\prime } k_{1}^{\prime (T)} H_{T}^{\prime } ,A_{6T}^{\prime } = - iq\left( {\overline{p}_{P} \mu_{33}^{\prime p} H_{T}^{\prime } + d_{33}^{\prime p} Q_{T}^{\prime } } \right)k_{3}^{\prime (T)} , \hfill \\ A_{7T}^{\prime } &= - iq\left( {\overline{p}_{P} \mu_{11}^{\prime p} H_{T}^{\prime } + d_{11}^{\prime p} Q_{T}^{\prime } } \right)k_{1}^{\prime (T)} ,A_{8T}^{\prime } = - iq\left( {\overline{n}_{P} \mu_{33}^{\prime n} H_{T}^{\prime } - d_{33}^{\prime n} L_{T}^{\prime } } \right)k_{33}^{\prime (T)} ,\\ A_{9T}^{\prime } &= - iq\left( {\overline{n}_{P} \mu_{11}^{\prime n} H_{T}^{\prime } - d_{11}^{\prime n} L_{T}^{\prime } } \right)k_{1}^{\prime (T)} . \hfill \\ \end{aligned}$$

Appendix E

$$\begin{aligned} C_{1j} &= f_{1} ,j = 6,7,8,9,10; \, C_{2j} = f_{2} G_{j} ,j = 6,7,8,9,10; \hfill \\ C_{ij} &= f_{m} A_{mk}^{{^{\prime } }} ,i = 3,4;j = 6,7,8,9,10;m = 1,2;k = 1,3,5,7,9; \\ C_{5j} &= f_{5} A_{4k}^{{^{\prime } }} , j = 6,7,8,9,10; \, k = 1,3,5,7,9; \, \hfill \\ C_{6j} & = f_{6} A_{6k}^{{^{\prime } }} ,j = 6,7,8,9,10; \, k = 1,3,5,7,9; \, C_{7j} { = }f_{7} A_{8k} ,j = 1,2,3,4,5; \, k = 2,4,6,8,10; \, \hfill \\ C_{8j} &= - C_{1} A_{2k}^{{^{\prime } }} - C_{2} A_{2k}^{{^{\prime } }} ,j = 6,7,8,9,10;k = 1,3,5,7,9; \\ {\text{ C}}_{9j} &= - C_{3} A_{6k} , j = 1,2,3,4,5;k = 2,4,6,8,10; \hfill \\ {\text{C}}_{9j} &= - C_{3} A_{6k}^{{^{\prime } }} - C_{4} A_{2k}^{{^{\prime } }} - C_{5} A_{4k}^{{^{\prime } }} ,j = 6,7,8,9,10;k = 1,3,5,7,9; \hfill \\ {\text{C}}_{10j} &= - C_{6} A_{6k}^{{^{\prime } }} - C_{7} A_{2k}^{{^{\prime } }} - C_{8} A_{4k}^{{^{\prime } }} ,j = 6,7,8,9,10;k = 1,3,5,7,9; \, \end{aligned}$$
$${\mathbf{A}}_{{\mathbf{1}}} { = }\left[ {\begin{array}{*{20}c} { - e^{{ - ik_{z}^{\left( 2 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - e^{{ - ik_{z}^{\left( 4 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - e^{{ - ik_{z}^{\left( 6 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - e^{{ - ik_{z}^{\left( 8 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - e^{{{ - ik_{z}{\left( {10} \right)}} z_{{\text{n}}}^{ - } }} } \\ { - G_{2} e^{{ - ik_{z}^{\left( 2 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - e^{{ - ik_{z}^{\left( 4 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - G_{6} e^{{ - ik_{z}^{\left( 6 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - G_{8} e^{{ - ik_{z}^{\left( 8 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - G_{10} e^{{{ - ik_{z}{\left( {10} \right)}} z_{{\text{n}}}^{ - } }} } \\ { - A_{12} e^{{ - ik_{z}^{\left( 2 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - A_{14} e^{{ - ik_{z}^{\left( 4 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - A_{16} e^{{ - ik_{z}^{\left( 6 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - A_{18} e^{{ - ik_{z}^{\left( 8 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - A_{110} e^{{{ - ik_{z}{\left( {10} \right)}} z_{{\text{n}}}^{ - } }} } \\ { - A_{22} e^{{ - ik_{z}^{\left( 2 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - A_{24} e^{{ - ik_{z}^{\left( 4 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - A_{26} e^{{ - ik_{z}^{\left( 6 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - A_{28} e^{{ - ik_{z}^{\left( 8 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - A_{210} e^{{{ - ik_{z}{\left( {10} \right)}} z_{{\text{n}}}^{ - } }} } \\ { - A_{42} e^{{ - ik_{z}^{\left( 2 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - A_{44} e^{{ - ik_{z}^{\left( 4 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - A_{46} e^{{ - ik_{z}^{\left( 6 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - A_{48} e^{{ - ik_{z}^{\left( 8 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - A_{410} e^{{{ - ik_{z}{\left( {10} \right)}} z_{{\text{n}}}^{ - } }} } \\ { - A_{62} e^{{ - ik_{z}^{\left( 2 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - A_{64} e^{{ - ik_{z}^{\left( 4 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - A_{66} e^{{ - ik_{z}^{\left( 6 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - A_{68} e^{{ - ik_{z}^{\left( 8 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - A_{610} e^{{{ - ik_{z}{\left( {10} \right)}} z_{{\text{n}}}^{ - } }} } \\ { - A_{82} e^{{ - ik_{z}^{\left( 2 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - A_{84} e^{{ - ik_{z}^{\left( 4 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - A_{86} e^{{ - ik_{z}^{\left( 6 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - A_{88} e^{{ - ik_{z}^{\left( 8 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - A_{810} e^{{{ - ik_{z}{\left( {10} \right)}} z_{{\text{n}}}^{ - } }} } \\ \end{array} } \right],$$
$${\mathbf{B}}_{{\mathbf{1}}} { = }\left[ {\begin{array}{*{20}c} {e^{{ik_{z}^{\prime (1)} z_{p}^{ + } }} } &\quad {e^{{ik_{z}^{\prime (3)} z_{p}^{ + } }} } &\quad {e^{{ik_{z}^{\prime (5)} z_{p}^{ + } }} } &\quad {e^{{ik_{z}^{\prime (7)} z_{p}^{ + } }} } &\quad {e^{{ik_{z}^{\prime (9)} z_{p}^{ + } }} } \\ {G_{1}^{{\prime }{ik_{z}^{\prime (1)} z_{p}^{ + } }} } &\quad {G_{3}^{\prime } e^{{ik_{z}^{\prime (3)} z_{p}^{ + } }} } &\quad {G_{5}^{\prime } e^{{ik_{z}^{\prime (5)} z_{p}^{ + } }} } &\quad {G_{7}^{\prime } e^{{ik_{z}^{\prime (7)} z_{p}^{ + } }} } &\quad {G_{9}^{\prime } e^{{ik_{z}^{\prime (9)} z_{p}^{ + } }} } \\ {A_{11}^{\prime } e^{{ik_{z}^{\prime (1)} z_{p}^{ + } }} } &\quad {A_{13}^{\prime } e^{{ik_{z}^{\prime (3)} z_{p}^{ + } }} } &\quad {A_{15}^{\prime } e^{{ik_{z}^{\prime (5)} z_{p}^{ + } }} } &\quad {A_{17}^{\prime } e^{{ik_{z}^{\prime (7)} z_{p}^{ + } }} } &\quad {A_{19}^{\prime } e^{{ik_{z}^{\prime (9)} z_{p}^{ + } }} } \\ {A_{21}^{\prime } e^{{ik_{z}^{\prime (1)} z_{p}^{ + } }} } &\quad {A_{23}^{\prime } e^{{ik_{z}^{\prime (3)} z_{p}^{ + } }} } &\quad {A_{25}^{\prime } e^{{ik_{z}^{\prime (5)} z_{p}^{ + } }} } &\quad {A_{27}^{\prime } e^{{{ik_{z}{^{\prime } (7)}} z_{p}^{ + } }} } &\quad {A_{29}^{\prime } e^{{ik_{z}^{\prime (9)} z_{p}^{ + } }} } \\ {A_{41}^{\prime } e^{{ik_{z}^{\prime (1)} z_{p}^{ + } }} } &\quad {A_{43}^{\prime } e^{{ik_{z}^{\prime (3)} z_{p}^{ + } }} } &\quad {A_{45}^{\prime } e^{{ik_{z}^{\prime (5)} z_{p}^{ + } }} } &\quad {A_{47}^{\prime } e^{{ik_{z}^{\prime (7)} z_{p}^{ + } }} } &\quad {A_{49}^{\prime } e^{{ik_{z}^{\prime (9)} z_{p}^{ + } }} } \\ {A_{61}^{\prime } e^{{ik_{z}^{\prime (1)} z_{p}^{ + } }} } &\quad {A_{63}^{\prime } e^{{ik_{z}^{\prime (3)} z_{p}^{ + } }} } &\quad {A_{65}^{\prime } e^{{ik_{z}^{\prime (5)} z_{p}^{ + } }} } &\quad {A_{67}^{\prime } e^{{ik_{z}^{\prime (7)} z_{p}^{ + } }} } &\quad {A_{69}^{\prime } e^{{ik_{z}^{\prime (9)} z_{p}^{ + } }} } \\ {A_{81}^{\prime } e^{{ik_{z}^{\prime (1)} z_{p}^{ + } }} } &\quad {A_{83}^{\prime } e^{{ik_{z}^{\prime (3)} z_{p}^{ + } }} } &\quad {A_{85}^{\prime } e^{{ik_{z}^{\prime (5)} z_{p}^{ + } }} } &\quad {A_{87}^{\prime } e^{{ik_{z}^{\prime (7)} z_{p}^{ + } }} } &\quad {A_{89}^{\prime } e^{{ik_{z}^{\prime (9)} z_{p}^{ + } }} } \\ \end{array} } \right],$$
$${\mathbf{A}}_{{\mathbf{2}}} { = }\left[ {\begin{array}{*{20}c} { - H_{2} e^{{ - ik_{z}^{\left( 2 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - H_{4} e^{{ - ik_{z}^{\left( 4 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - H_{6} e^{{ - ik_{z}^{\left( 6 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - H_{8} e^{{ - ik_{z}^{\left( 8 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - H_{10} e^{{{ - ik_{z}{\left( {10} \right)}} z_{{\text{n}}}^{ - } }} } \\ { - Q_{2} e^{{ - ik_{z}^{\left( 2 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - Q_{4} e^{{ - ik_{z}^{\left( 4 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - Q_{6} e^{{ - ik_{z}^{\left( 6 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - Q_{8} e^{{ - ik_{z}^{\left( 8 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - Q_{10} e^{{{ - ik_{z}{\left( {10} \right)}} z_{{\text{n}}}^{ - } }} } \\ { - L_{2} e^{{ - ik_{z}^{\left( 2 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - L_{4} e^{{ - ik_{z}^{\left( 4 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - L_{6} e^{{ - ik_{z}^{\left( 6 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - L_{8} e^{{ - ik_{z}^{\left( 8 \right)} z_{{\text{n}}}^{ - } }} } &\quad { - L_{10} e^{{{ - ik_{z}{\left( {10} \right)}} z_{{\text{n}}}^{ - } }} } \\ \end{array} } \right],$$
$${\mathbf{B}}_{{\mathbf{2}}} { = }\left[ {\begin{array}{*{20}c} {H_{1}^{{\prime }{ik_{z}^{\prime (1)} z_{p}^{ + } }} } &\quad {H_{3}^{\prime } e^{{ik_{z}^{\prime (3)} z_{p}^{ + } }} } &\quad {H_{5}^{\prime } e^{{ik_{z}^{\prime (5)} z_{p}^{ + } }} } &\quad {H_{7}^{\prime } e^{{ik_{z}^{\prime (7)} z_{p}^{ + } }} } &\quad {H_{9}^{\prime } e^{{ik_{z}^{\prime (9)} z_{p}^{ + } }} } \\ {Q_{1}^{\prime } e^{{ik_{z}^{\prime (1)} z_{p}^{ + } }} } &\quad {Q_{3}^{\prime } e^{{ik_{z}^{\prime (3)} z_{p}^{ + } }} } &\quad {Q_{5}^{\prime } e^{{ik_{z}^{\prime (5)} z_{p}^{ + } }} } &\quad {Q_{7}^{\prime } e^{{ik_{z}^{\prime (7)} z_{p}^{ + } }} } &\quad {Q_{9}^{\prime } e^{{ik_{z}^{\prime (9)} z_{p}^{ + } }} } \\ {L_{1}^{\prime } e^{{ik_{z}^{\prime (1)} z_{p}^{ + } }} } &\quad {L_{3}^{\prime } e^{{ik_{z}^{\prime (3)} z_{p}^{ + } }} } &\quad {L_{5}^{\prime } e^{{ik_{z}^{\prime (5)} z_{p}^{ + } }} } &\quad {L_{7}^{\prime } e^{{ik_{z}^{\prime (7)} z_{p}^{ + } }} } &\quad {L_{9}^{\prime } e^{{ik_{z}^{\prime (9)} z_{p}^{ + } }} } \\ \end{array} } \right],$$
$${\mathbf{D}}_{{\mathbf{1}}} { = }\left[ {\begin{array}{*{20}c} {e^{{ - ik_{z}^{\left( I \right)} z_{{\text{n}}}^{ - } }} } &\quad {G_{I} e^{{ - ik_{z}^{\left( I \right)} z_{{\text{n}}}^{ - } }} } &\quad {A_{1I} e^{{ - ik_{z}^{\left( I \right)} z_{{\text{n}}}^{ - } }} } &\quad {A_{2I} e^{{ - ik_{z}^{\left( I \right)} z_{{\text{n}}}^{ - } }} } &\quad {A_{4I} e^{{ - ik_{z}^{\left( I \right)} z_{{\text{n}}}^{ - } }} } \\ \end{array} } \right]^{T} ,$$
$${\mathbf{D}}_{2} { = }\left[ {\begin{array}{*{20}c} {A_{6I} e^{{ - ik_{z}^{\left( I \right)} z_{{\text{n}}}^{ - } }} } &\quad {A_{8I} e^{{ - ik_{z}^{\left( I \right)} z_{{\text{n}}}^{ - } }} } &\quad {H_{I} e^{{ - ik_{z}^{\left( I \right)} z_{{\text{n}}}^{ - } }} } &\quad {Q_{I} e^{{ - ik_{z}^{\left( I \right)} z_{{\text{n}}}^{ - } }} } &\quad {L_{I} e^{{ - ik_{z}^{\left( I \right)} z_{{\text{n}}}^{ - } }} } \\ \end{array} } \right]^{T} .$$

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Wei, Z., Wei, P., Xu, C. et al. Equivalent imperfect interface model of PN junction of piezoelectric semiconductor for the multi-field coupled waves propagation. Acta Mech 235, 73–92 (2024). https://doi.org/10.1007/s00707-023-03643-x

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