On band structures of layered phononic crystals with flexoelectricity

Abstract

Flexoelectricity becomes remarkable in nanoscale dielectrics where strong strain gradients are expected. The effect of flexoelectricity on elastic plane waves propagating in nanoscale layered phononic crystals is discussed in the current work. The fundamental governing equations and boundary conditions are derived from the virtual work principle. Detailed calculations are performed for nanoscale two-layered and three-layered phononic crystals using the transfer matrix method. Numerical results indicate that phononic crystals possess frequency pass bands and stop bands. For nanoscale layered phononic crystals, flexoelectricity increases the middle frequency regardless of the thickness ratio, whereas the flexoelectric effect on the bandwidth depends on the thickness ratio, which implies that there is an optimal thickness ratio to maximize the bandwidth. In addition, the middle frequency and bandwidth decrease with an increase in the unit cell thickness if the thickness ratio is fixed. Hence, considering the flexoelectric effect on band structures of nanoscale phononic crystals may provide guidance in manipulating elastic wave propagation and facilitate potential applications in phononic crystal devices.

Appendices

Appendix A

Below are the analytical expressions for the classical stress, higher-order stress, electric displacement and electric quadrupole needed in the mechanical and electric continuity conditions

$$\begin{aligned} \sigma _{33}= & {} \left\{ {\begin{array}{l} \left[ {c\alpha _1 -L_1 \left( {e\alpha _1 ^{2}+d\alpha _1 } \right) } \right] e^{\alpha _1 z}A \\ \quad -\left[ {c\alpha _1 -L_2 \left( {e\alpha _1 ^{2}-d\alpha _1 } \right) } \right] e^{-\alpha _1 z}B \\ \quad +\left[ {c\alpha _2 -L_3 \left( {e\alpha _2 ^{2}+d\alpha _2 } \right) } \right] e^{\alpha _2 z}C \\ \quad -\left[ {c\alpha _2 -L_4 \left( {e\alpha _2 ^{2}-d\alpha _2 } \right) } \right] e^{-\alpha _2 z}D+dE \\ \end{array}} \right\} \exp \left( {-\,\hbox {i}\omega t} \right) \\ \tau _{333}= & {} \left( {\begin{array}{l} -\,L_1 f\alpha _1 e^{\alpha _1 z}A-L_2 f\alpha _1 e^{-\alpha _1 z}B \\ \quad -\,L_3 f\alpha _2 e^{\alpha _2 z}C-L_4 f\alpha _2 e^{-\alpha _2 z}D+fE \\ \end{array}} \right) \exp \left( {-\,\hbox {i}\omega t} \right) \\ D_3= & {} \left\{ {\begin{array}{l} \left( {L_1 a\alpha _1 +d\alpha _1 +f\alpha _1 ^{2}} \right) e^{\alpha _1 z}A \\ \quad +\left( {L_2 a\alpha _1 -d\alpha _1 +f\alpha _1 ^{2}} \right) e^{-\alpha _1 z}B \\ \quad +\left( {L_3 a\alpha _2 +d\alpha _2 +f\alpha _2 ^{2}} \right) e^{\alpha _2 z}C \\ \quad +\left( {L_4 a\alpha _2 -d\alpha _2 +f\alpha _2 ^{2}} \right) e^{-\alpha _2 z}D-aE \\ \end{array}} \right\} \exp \left( {-\,\hbox {i}\omega t} \right) \\ Q_{33}= & {} \left\{ {\begin{array}{l} \left( {L_1 b\alpha _1 ^{2}+e\alpha _1 } \right) e^{\alpha _1 z}A-\left( {L_2 b\alpha _1 ^{2}+e\alpha _1 } \right) e^{-\alpha _1 z}B \\ \quad +\left( {L_3 b\alpha _2 ^{2}+e\alpha _2 } \right) e^{\alpha _2 z}C-\left( {L_4 b\alpha _2 ^{2}+e\alpha _2 } \right) e^{-\alpha _2 z}D \\ \end{array}} \right\} \exp \left( {-\,\hbox {i}\omega t} \right) \end{aligned}$$

Appendix B

The state matrix $\mathbf{M}_j \left( z \right)$ can be written as

$$\begin{aligned} \begin{array}{l} \mathbf{M}_j \left( z \right) =\left[ {{\begin{array}{ll} {e^{\alpha _{1j} z}}&{} \quad {e^{-\alpha _{1j} z}} \\ {-\,L_{1j} e^{\alpha _{1j} z}}&{} \quad {L_{2j} e^{-\alpha _{1j} z}} \\ {\left[ {c_j \alpha _{1j} -L_{1j} \left( {h_j \alpha _{1j} ^{2}+d_j \alpha _{1j} } \right) } \right] e^{\alpha _{1j} z}}&{} \quad {-\left[ {c_j \alpha _{1j} -L_{2j} \left( {h_j \alpha _{1j} ^{2}-d_j \alpha _{1j} } \right) } \right] e^{-\alpha _{1j} z}} \\ {-\,L_{1j} f_j \alpha _{1j} e^{\alpha _{1j} z}}&{} \quad {-\,L_{2j} f_j \alpha _{1j} e^{-\alpha _{1j} z}} \\ 0&{} \quad 0 \\ {\left( {L_{1j} b_j \alpha _{1j} ^{2}+e_j \alpha _{1j} } \right) e^{\alpha _{1j} z}}&{} \quad {-\left( {L_{2j} b_j \alpha _{1j} ^{2}+e_j \alpha _{1j} } \right) e^{-\alpha _{1j} z}} \\ \end{array} }} \right. \\ \left. {{\begin{array}{llll} {e^{\alpha _{2j} z}}&{} \quad {e^{-\alpha _{2j} z}}&{} \quad 0&{} \quad 0 \\ {-\,L_{3j} e^{\alpha _{2j} z}}&{} \quad {L_{4j} e^{-\alpha _{2j} z}}&{} \quad z &{} \quad 1 \\ {\left[ {c_j \alpha _{2j} -L_{3j} \left( {h_j \alpha _{2j} ^{2}+d_j \alpha _{2j} } \right) } \right] e^{\alpha _{2j} z}}&{} \quad {-\left[ {c_j \alpha _{2j} -L_{4j} \left( {h_j \alpha _{2j} ^{2}-d_j \alpha _{2j} } \right) } \right] e^{-\alpha _{2j} z}}&{} \quad {d_j }&{} \quad 0 \\ {-\,L_{3j} f_j \alpha _{2j} e^{\alpha _{2j} z}}&{} \quad {-\,L_{4j} f_j \alpha _{2j} e^{-\alpha _{2j} z}}&{} \quad {f_j }&{} \quad 0 \\ 0&{} \quad 0&{} \quad {-\,a_j }&{} \quad 0 \\ {\left( {L_{3j} b_j \alpha _{2j} ^{2}+e_j \alpha _{2j} } \right) e^{\alpha _{2j} z}}&{} \quad {-\left( {L_{4j} b_j \alpha _{2j} ^{2}+e_j \alpha _{2j} } \right) e^{-\alpha _{2j} z}}&{} \quad 0&{} \quad 0 \\ \end{array} }} \right] \\ \end{array} \end{aligned}$$

where $j=1$, 2 and 3 represent the first, second and third sublayers, respectively.