Analytical solutions of electroelastic fields in piezoelectric thin-film multilayer: applications to piezoelectric sensors and actuators

Abstract

Analytical expressions are derived for the stresses and electric fields induced in a piezoelectric multilayer deposited on a substrate with lattice misfit and thermal expansion coefficient mismatch. The piezoelectric multilayer can be subjected to an externally applied force, moment and electric potential. The derived formulations can model any number of layers using recursive relations that minimize the computation time. A proper rotation matrix is utilized to generalize the derived expressions to accommodate various orientations allowing each layer to have hexagonal crystal symmetry. The influence of lattice misfit and thermal expansion coefficient mismatch in the presence of externally applied force, moment and electric potential on the state of the electroelastic fields in each layer is evaluated for various applications. Comparison with finite element analysis results shows excellent agreement. The analytical expressions developed here can be useful in designing electromechanical sensors, actuators and optoelectronic devices made from piezoelectric multilayers.

Appendices

Appendix A1

Constitutive relations for piezoelectric materials with the z-direction as c-axis exhibiting transversely isotropic behavior (hexagonal symmetry) can be written in the form [44]:

$$\begin{aligned} \left\{ \begin{array}{l}\sigma _{xx}\\ \sigma _{yy}\\ \sigma _{zz}\\ \sigma _{zy}\\ \sigma _{zx}\\ \sigma _{xy}\end{array}\right\}= & {} \left[ \begin{array}{cccccc}c_{11} &{}\quad c_{12} &{}\quad c_{13} &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ c_{12} &{}\quad c_{11} &{}\quad c_{13} &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ c_{13} &{}\quad c_{13} &{}\quad c_{33} &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad c_{44} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad c_{44} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \frac{1}{2}(c_{11}-c_{12})\end{array}\right] \left\{ \begin{array}{l} \varepsilon _{xx} \\ \varepsilon _{yy} \\ \varepsilon _{zz} \\ 2\varepsilon _{zy} \\ 2\varepsilon _{zx} \\ 2\varepsilon _{xy} \end{array}\right] \,\, - \,\, \left[ \begin{array}{ccc} 0 &{}\quad 0 &{}\quad e_{31} \\ 0 &{}\quad 0 &{}\quad e_{31} \\ 0 &{}\quad 0 &{}\quad e_{33} \\ 0 &{}\quad e_{15} &{}\quad 0 \\ e_{15} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \end{array}\right] \left\{ \begin{array}{l} E_{x} \\ E_{y} \\ E_{z} \end{array}\right\} , \qquad \qquad \nonumber \\ \left\{ \begin{array}{c}D_{1} \\ D_{2} \\ D_{3} \end{array}\right\}= & {} \left[ \begin{array}{cccccc} 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad e_{15} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad e_{15} &{}\quad 0 &{}\quad 0 \\ e_{31} &{}\quad e_{31} &{}\quad e_{33} &{}\quad 0 &{}\quad 0 &{}\quad 0 \end{array}\right] \left\{ \begin{array}{c} \varepsilon _{xx} \\ \varepsilon _{yy} \\ \varepsilon _{zz} \\ 2\varepsilon _{zy} \\ 2\varepsilon _{zx} \\ 2\varepsilon _{xy} \end{array}\right\} \,\, + \,\, \left[ \begin{array}{ccc}\epsilon _{11} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad \epsilon _{11} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \epsilon _{33} \end{array}\right] \left\{ \begin{array}{c}E_{x} \\ E_{y} \\ E_{z} \end{array}\right\} + \left\{ \begin{array}{c}0 \\ 0 \\ P_\mathrm{sp} \end{array}\right\} , \end{aligned}$$

(A1.1)

Appendix A2

The layer-wise stresses and electric displacements calculation procedures have been listed in this section as follows:

Appendix A2.1

The in-plane and the out-of-plane shear strains are obtained from Eq. (11) as

$$\begin{aligned} \varepsilon _{zy}^{\mathrm{Tot}^i}= & {} -{1\over 2A_{{II}_{1}}^i} \left[ A_{{II}_2}^i E_x^{\mathrm{Tot}^{i}}+ A_{{II}_3}^i E_y^{\mathrm{Tot}^{i}} + A_{{II}_4}^i E_z^{\mathrm{Tot}^{i}} \right] , \nonumber \\ \varepsilon _{zx}^{\mathrm{Tot}^i}= & {} -{1\over 2A_{{II}_{1}}^i} \left[ A_{{II}_5}^i E_x^{\mathrm{Tot}^{i}}+ A_{{II}_6}^i E_y^{\mathrm{Tot}^{i}} + A_{{II}_7}^i E_z^{\mathrm{Tot}^{i}} \right] , \nonumber \\ \varepsilon _{xy}^{\mathrm{Tot}^i}= & {} -{1\over 2A_{{II}_{1}}^i} \left[ A_{{II}_8}^i E_x^{\mathrm{Tot}^{i}}+ A_{{II}_9}^i E_y^{\mathrm{Tot}^{i}} + A_{{II}_{10}}^i E_z^{\mathrm{Tot}^{i}} \right] , \end{aligned}$$

(A2.1)

where

$$\begin{aligned} A_{{II}_{1}}^i= & {} A_{{II}_{11}}^i A_{{II}_{23}}^{i^2} + A_{{II}_{22}}^i A_{{II}_{13}}^{i^2} +A_{{II}_{33}}^i A_{{II}_{12}}^{i^2} -2 A_{{II}_{12}}^i A_{{II}_{13}}^i A_{{II}_{23}}^i - A_{{II}_{11}}^i A_{{II}_{22}}^i A_{{II}_{33}}^i, \\ A_{{II}_2}^i= & {} P_{{II}_{11}}^i A_{{II}_2}^i+ P_{{II}_{21}}^i A_{{II}_3}^i + P_{{II}_{31}}^i A_{{II}_4}^i, \qquad A_{{II}_3}^i = P_{{II}_{12}}^i A_{{II}_2}^i + P_{{II}_{22}}^i A_{{II}_3}^i +P_{{II}_{32}}^i A_{{II}_4}^i, \\ A_{{II}_4}^i= & {} P_{{II}_{13}}^i A_{{II}_2}^i+P_{{II}_{23}}^i A_{{II}_3}^i+P_{{II}_{33}}^i A_{{II}_4}^i, \qquad A_{{II}_5}^i = P_{{II}_{11}}^i A_{{II}_5}^i+ P_{{II}_{21}}^i A_{{II}_6}^i + P_{{II}_{31}}^i A_{{II}_7}^i, \\ A_{{II}_6}^i= & {} P_{{II}_{12}}^i A_{{II}_5}^i + P_{{II}_{22}}^i A_{{II}_6}^i +P_{{II}_{32}}^i A_{{II}_7}^i, \qquad A_{{II}_7}^i = P_{{II}_{13}}^i A_{{II}_5}^i+P_{{II}_{23}}^i A_{{II}_6}^i+P_{{II}_{33}}^i A_{{II}_7}^i, \\ A_{{II}_8}^i= & {} P_{{II}_{11}}^i A_{{II}_8}^i+ P_{{II}_{21}}^i A_{{II}_9}^i + P_{{II}_{31}}^i A_{{II}_{10}}^i, \qquad A_{{II}_9}^i = P_{{II}_{12}}^i A_{{II}_8}^i + P_{{II}_{22}}^i A_{{II}_9}^i +P_{{II}_{32}}^i A_{{II}_{10}}^i, \\ A_{{II}_{10}}^i= & {} P_{{II}_{13}}^i A_{{II}_8}^i+P_{{II}_{23}}^i A_{{II}_9}^i+P_{{II}_{33}}^i A_{{II}_{10}}^i, \qquad A_{{II}_{11}}^i = A_{{II}_{23}}^i (A_{{II}_{22}}^i - A_{{II}_{23}}^i), \\ A_{{II}_{12}}^i= & {} A_{{II}_{13}}^i A_{{II}_{23}}^i -A_{{II}_{12}}^i A_{{II}_{23}}^i, \qquad \qquad A_{{II}_{13}}^i = A_{{II}_{12}}^i A_{{II}_{23}}^i - A_{{II}_{13}}^i A_{{II}_{22}}^i, \\ A_{{II}_{14}}^i= & {} A_{{II}_{23}}^i A_{{II}_{13}}^i -A_{{II}_{12}}^i A_{{II}_{33}}^i, \qquad \qquad A_{{II}_{15}}^i = A_{{II}_{11}}^i A_{{II}_{33}}^i - A_{{II}_{13}}^{i^2}, \\ A_{{II}_{16}}^i= & {} A_{{II}_{12}}^i A_{{II}_{13}}^i -A_{{II}_{11}}^i A_{{II}_{23}}^i, \qquad \qquad A_{{II}_{17}}^i = A_{{II}_{12}}^i A_{{II}_{23}}^i - A_{{II}_{13}}^{i} A_{{II}_{22}}^{i}, \\ A_{{II}_{18}}^i= & {} A_{{II}_{12}}^i A_{{II}_{13}}^i -A_{{II}_{11}}^i A_{{II}_{23}}^i, \qquad \qquad A_{{II}_{19}}^i = A_{{II}_{11}}^i A_{{II}_{22}}^i - A_{{II}_{12}}^{i^2}. \end{aligned}$$

The in-plane electric field components ($E_x$ and $E_y$) calculated in terms of $\varepsilon _{ij}$ and $E_z$ with the boundary conditions $D_x = D_y = 0$ (Eq. (12)) can be expressed as

$$\begin{aligned} E_x^{\mathrm{Tot}^{i}}= & {} {1\over P_1^i} \left[ P_2^i E_z^{\mathrm{Tot}^i} + P_3^i + P_4^i \varepsilon _{xx}^{\mathrm{Tot}^i} + P_5^i \varepsilon _{yy}^{\mathrm{Tot}^i} + P_6^i \varepsilon _{zz}^{\mathrm{Tot}^i}\right] , \nonumber \\ E_y^{\mathrm{Tot}^{i}}= & {} {1\over P_1^i} \left[ P_7^i E_z^{\mathrm{Tot}^i} + P_8^i + P_9^i \varepsilon _{xx}^{\mathrm{Tot}^i} + P_{10}^i \varepsilon _{yy}^{\mathrm{Tot}^i} + P_{11}^i \varepsilon _{zz}^{\mathrm{Tot}^i}\right] , \end{aligned}$$

(A2.2)

where the constants are

$$\begin{aligned} P_1^i= & {} \left( A_2^i A_4^i - A_1^i A_5^i\right) , \quad \quad \qquad P_2^i = \left( A_3^i A_5^i -A_2^i A_6^i \right) \quad \quad P_3^i =\left( A_2^i P_{{sp}_{yy}}^i -A_5^i P_{{sp}_{xx}}^i \right) ,\\ P_4^i= & {} \left( A_2 P_{I_{12}} - A_5 P_{I_{11}}\right) , \qquad \qquad P_5^i = \left( A_2 P_{I_{22}} -A_5 P_{I_{21}}\right) \qquad P_6^i = \left( A_2^i P_{I_{32}}^i - A_5^i P_{I_{31}}^i \right) ,\\ P_7^i= & {} \left( A_1^i A_6^i -A_3^i A_4^i\right) , \qquad \qquad P_8^i= \left( A_4^i P_{{sp}_{yy}}^i -A_1^i P_{{sp}_{xx}}^i \right) , \qquad \qquad P_9^i = \left( A_4 P_{I_{11}} - A_1 P_{I_{12}}\right) , \\ P_{10}^i= & {} \left( A_4^i P_{I_{21}}^i -A_1^i P_{I_{22}}^i\right) , \qquad \ P_{11}^i = \left( A_4^i P_{I_{31}}^i - A_1^i P_{I_{32}}^i\right) . \end{aligned}$$

Appendix A2.2

Here, we evaluate the in-plane stresses in different layers of the multilayer due to the input misfit strains ($\varepsilon _{ij}^{M^i}$). The externally applied mechanical load per unit area ($F_z$) at the top and bottom surfaces of the multilayer (Fig. 4) and the potential difference created along the thickness direction (Fig. 5) are also considered. Therefore, the top and bottom surfaces of the multilayer are neither traction nor charge free. The spontaneous polarization along the [0001] crystal direction in the hexagonal crystal are also considered in this section. According to the boundary conditions considered, the normal stress ($\sigma _{zz}^{M^i}$) and the electric displacement ($D_z^{M^i}$) along the thickness direction are continuous at the interface. We can calculate the normal strain ($\varepsilon _{zz}^{M^i}$) and the electric field ($E_z^{M^i}$) in the ith layer of the multilayer by using the equations for normal stress ($\sigma _{zz}^{M^i}$) and the electric displacement ($D_z^{M^i}$) from the constitutive relations (10) as

$$\begin{aligned} \varepsilon _{zz}^{{M}^i}= & {} -{1\over B_1^i} \left[ B_2^i \varepsilon _{xx}^{M^i} + B_3^i \varepsilon _{yy}^{M^i} - \left( D_z^{M^i} - {A_{18}^i \over A_{13}^i} F_{z}\right) + B_4^i\right] ,\nonumber \\ E_z^{{M}^i}= & {} {1\over A_{13}^i} \left[ B_5^i \varepsilon _{xx}^{M^i} + B_6^i \varepsilon _{yy}^{M^i} - B_7^i F_{z} + {A_{12}^i\over B_1^i}D_z^{M^i} - B_8^i \right] . \end{aligned}$$

(A2.3)

Substituting $\varepsilon _{zz}^{{M}^i}$ and $E_z^{{M}^i}$ in Eq. (10), the in-plane normal stresses, $\sigma _{xx}^{{M}^i}$ and $\sigma _{yy}^{{M}^i}$, can be expressed as

$$\begin{aligned} \sigma _{xx}^{{M}^i}= & {} B_9^i \varepsilon _{xx}^{{M}^i} + B_{10}^i \varepsilon _{yy}^{{M}^i} + B_{11}^i F_{z} - B_{12}^i D_z^{{M}^i} - B_{13}^i, \nonumber \\ \sigma _{yy}^{{M}^i}= & {} B_{14}^i \varepsilon _{xx}^{{M}^i} + B_{15}^i \varepsilon _{yy}^{{M}^i} + B_{16}^i F_{z} - B_{17}^i D_z^{{M}^i} - B_{18}^i. \end{aligned}$$

(A2.4)

Appendix A2.3

We would now like to derive the layer-wise electroelastic fields in the multilayer due to the in-plane reference strain components ($\varepsilon _{xx}^N$ and $\varepsilon _{yy}^N$). Externally applied forces, applied electric potential and the spontaneous polarization along the [0001] crystal direction were not considered in this section. Therefore, the top and bottom surfaces of the multilayer were considered to be traction and charge free. This made the normal stress ($\sigma _{zz}^{N^{i=0}}$) and electric displacement ($D_z^{N^{i=0}}$) along the thickness direction to vanish. The electroelastic fields in this case are not position dependent (independent of z), therefore, $\sigma _{zz}^{N^i}$ = $D_z^{N^i} = 0$. This condition leads to

$$\begin{aligned} \varepsilon _{zz}^{{N}^i}= & {} -{1\over B_1^i} \left[ B_2^i \varepsilon _{xx}^{N} + B_3^i \varepsilon _{yy}^{N} \right] \qquad \qquad E_z^{{N}^i} = -{1\over A_{13}^i} \left[ B_5^i \varepsilon _{xx}^{N} + B_6^i \varepsilon _{yy}^{N} \right] . \qquad \end{aligned}$$

(A2.5)

Substitution of $\varepsilon _{zz}^{N^i}$ and $E_z^{N^i}$ into the constitutive relations, the in-plane stresses can be expressed as

$$\begin{aligned} \sigma _{xx}^{{N}^i}= & {} B_9^i \varepsilon _{xx}^{{N}} + B_{10}^i \varepsilon _{yy}^{{N}} \qquad \qquad \qquad \sigma _{yy}^{{N}^i} = B_{14}^i \varepsilon _{xx}^{{N}} + B_{15}^i \varepsilon _{yy}^{{N}}. \end{aligned}$$

(A2.6)

Appendix A2.4

The bending strains, $\varepsilon _{xx}^B = K_x (z-\varPi _x)$, $\varepsilon _{yy}^B = K_y (z-\varPi _x)$, are functions of z which varies linearly through the thickness. According to the boundary conditions considered, $\sigma _{zz}^{i = 0} = \sigma _{zz}^{i = n} = D_z^{i = 0} = D_z^{i = n} = 0$. These conditions were used to calculate the normal strain ($\varepsilon _{zz}^{B^{i=0}}$) and the electric field ($E_{z}^{B^{i=0}}$) in the bottom layer, as well as the normal strain ($\varepsilon _{zz}^{B^{i=n}}$) and the electric field ($E_{z}^{B^{i=n}}$) in the top film layer along the thickness direction. Substituting these parameters in the constitutive relations (Eq. (10)), the in-plane stresses can be expressed as

$$\begin{aligned} \sigma _{xx}^{{B}^{i=0}}= & {} B_9^0 K_x \left( z - \varPi _x \right) + B_{10}^0 K_y \left( z - \varPi _y \right) , \qquad \sigma _{yy}^{{B}^{i=0}} = B_{14}^0 K_x \left( z -\varPi _x \right) + B_{15}^0 K_y \left( z - \varPi _y \right) , \nonumber \\ \sigma _{xx}^{{B}^{i=n}}= & {} B_9^n K_x (z - \varPi _x) + B_{10}^n K_y (z- \varPi _y), \qquad \sigma _{yy}^{{B}^{i=n}} = B_{14}^n K_x (z -\varPi _x) + B_{15}^n K_y (z - \varPi _y). \end{aligned}$$

(A2.7)

Utilizing the continuity conditions of the stress and electric displacement in the thickness direction at the interface, we can obtain the normal strain ($\varepsilon _{zz}^{{i+1}}$) and the electric field ($E_z^{i+1}$) as

$$\begin{aligned} \varepsilon _{zz}^{i+1}= & {} \left[ B_5^{i+1} K_x(z-\varPi _x) + B_6^{i+1} K_y (z-\varPi _y) + B_{7}^{i+1}\varepsilon _{zz}^{B^i} - B_8^{i+1} E_z^{B^i}\right] ,\nonumber \\ E_z^{i+1}= & {} {1\over B_4^{i+1}} {\left[ B_1^{i+1}K_x(z-\varPi _x) + B_2^{i+1} K_y (z-\varPi _y) + B_{3}^{i+1}\varepsilon _{zz}^{B^i} - A_{13}^i A_{17}^{i+1} E_z^{B^i} \right] }. \end{aligned}$$

(A2.8)

Substitution of $\varepsilon _{zz}^{i+1}$ and $E_z^{i+1}$ in the in-plane normal stress components of the $(i+1)$th layer gives

$$\begin{aligned} \sigma _{xx}^{i+1}= & {} B_{9}^{i+1} K_x \left( z - \varPi _x\right) + B_{10}^{i+1}K_y \left( z - \varPi _y \right) + B_{11}^{i+1} \varepsilon _{zz}^{B^i} - B_{12}^{i+1} E_{z}^{B^i}, \nonumber \\ \sigma _{yy}^{i+1}= & {} B_{13}^{i+1}K_x \left( z - \varPi _x\right) + B_{14}^{i+1} K_y \left( z - \varPi _y\right) + B_{15}^{i+1}\varepsilon _{zz}^{B^i} - B_{16}^{i+1} E_{z}^{B^i}. \end{aligned}$$

(A2.9)

The constants $A_1^i$ through $A_{28}^i$ and $B_1^i$ through $B_{28}^i$ as well as $B_{1}^{i+1}$ through $B_{16}^{i+1}$ are as follows:

$$\begin{aligned} A_1^i= & {} {1\over A_{{II}_1}^i} \left( P_{{II}_{11}}^i A_{{II}_2}^i + P_{{II}_{21}}^i A_{{II}_5}^i + P_{{II}_{31}}^i A_{{II}_8}^i \right) - \kappa _{11}^i, \\ A_2^i= & {} {1\over A_{{II}_1}^i} \left( P_{{II}_{11}}^i A_{{II}_3}^i + P_{{II}_{21}}^i A_{{II}_6}^i + P_{{II}_{31}}^i A_{{II}_9}^i \right) - \kappa _{12}^i, \\ A_3^i= & {} {1\over A_{{II}_1}^i} \left( P_{{II}_{11}}^i A_{{II}_4}^i + P_{{II}_{21}}^i A_{{II}_7}^i + P_{{II}_{31}}^i A_{{II}_{10}}^i \right) - \kappa _{13}^i, \\ A_4^i= & {} {1\over A_{{II}_1}^i} \left( P_{{II}_{12}}^i A_{{II}_2}^i + P_{{II}_{22}}^i A_{{II}_5}^i + P_{{II}_{32}}^i A_{{II}_8}^i \right) - \kappa _{21}^i, \\ A_5^i= & {} {1\over A_{{II}_1}^i} \left( P_{{II}_{12}}^i A_{{II}_3}^i + P_{{II}_{22}}^i A_{{II}_6}^i + P_{{II}_{32}}^i A_{{II}_9}^i \right) - \kappa _{22}^i, \\ A_6^i= & {} {1\over A_{{II}_1}^i} \left( P_{{II}_{12}}^i A_{{II}_4}^i + P_{{II}_{22}}^i A_{{II}_7}^i + P_{{II}_{32}}^i A_{{II}_{10}}^i \right) - \kappa _{23}^i, \\ A_7^i= & {} {1\over A_{{II}_1}^i} \left( P_{{II}_{13}}^i A_{{II}_2}^i + P_{{II}_{23}}^i A_{{II}_5}^i + P_{{II}_{33}}^i A_{{II}_8}^i \right) - \kappa _{31}^i, \\ A_8^i= & {} {1\over A_{{II}_1}^i} \left( P_{{II}_{13}}^i A_{{II}_3}^i + P_{{II}_{23}}^i A_{{II}_6}^i + P_{{II}_{33}}^i A_{{II}_9}^i \right) - \kappa _{32}^i, \\ A_9^i= & {} {1\over A_{{II}_1}^i} \left( P_{{II}_{13}}^i A_{{II}_4}^i + P_{{II}_{23}}^i A_{{II}_7}^i + P_{{II}_{33}}^i A_{{II}_{10}}^i \right) - \kappa _{33}^i, \\ A_{10}^i= & {} A_{I_{31}}^i - P_{I_{31}}^i {P_4^i \over P_1^i} - P_{I_{32}}^i {P_9^i \over P_1^i}, \qquad \qquad \qquad A_{11}^i = A_{I_{32}}^i - P_{I_{31}}^i {P_5^i \over P_1^i} - P_{I_{32}}^i {P_{10}^i \over P_1^i}, \\ A_{12}^i= & {} A_{I_{33}}^i - P_{I_{31}}^i {P_6^i \over P_1^i} - P_{I_{32}}^i {P_{11}^i \over P_1^i}, \qquad \qquad \qquad A_{13}^i = P_{I_{31}}^i {P_2^i \over P_1^i} + P_{I_{32}}^i {P_7^i \over P_1^i} + P_{I_{33}}^i, \\ A_{14}^i= & {} P_{I_{31}}^i {P_3^i \over P_1^i} + P_{I_{32}}^i {P_8^i \over P_1^i} , \qquad \qquad \qquad \qquad A_{15}^i = P_{I_{13}}^i - A_7^i {P_4^i \over P_1^i} - A_8^i {P_9^i \over P_1^i}, \\ A_{16}^i= & {} P_{I_{23}}^i - A_7^i {P_5^i \over P_1^i} - A_8^i {P_{10}^i \over P_1^i}, \qquad \qquad \qquad A_{17}^i = P_{I_{33}}^i - A_7^i {P_6^i \over P_1^i} - A_8^i {P_{11}^i \over P_1^i}, \\ A_{18}^i= & {} A_9^i + A_7^i {P_2^i \over P_1^i} + A_8^i {P_9^i \over P_1^i}, \\ A_{19}^i= & {} A_{I_{11}}^i - P_{I_{11}}^i {P_4^i \over P_1^i} - P_{I_{12}}^i {P_9^i \over P_1^i}, \qquad \qquad A_{20}^i = A_{I_{12}}^i - P_{I_{11}}^i {P_5^i \over P_1^i} - P_{I_{12}}^i {P_{10}^i \over P_1^i}, \\ A_{21}^i= & {} A_{I_{13}}^i - P_{I_{11}}^i {P_6^i \over P_1^i} - P_{I_{12}}^i {P_{11}^i \over P_1^i}, \qquad \qquad A_{22}^i = P_{I_{13}}^i + P_{I_{11}}^i {P_2^i \over P_1^i} + P_{I_{12}}^i {P_7^i \over P_1^i}, \\ A_{23}^i= & {} P_{I_{11}}^i {P_3^i \over P_1^i} + P_{I_{12}}^i {P_8^i \over P_1^i}, \qquad \qquad \qquad \qquad A_{24}^i = A_{I_{21}}^i - P_{I_{21}}^i {P_4^i \over P_1^i} - P_{I_{22}}^i {P_9^i \over P_1^i}, \\ A_{25}^i= & {} A_{I_{22}}^i - P_{I_{21}}^i {P_5^i \over P_1^i} - P_{I_{22}}^i {P_{10}^i \over P_1^i}, \qquad \qquad A_{26}^i = A_{I_{23}}^i - P_{I_{21}}^i {P_6^i \over P_1^i} - P_{I_{22}}^i {P_{11}^i \over P_1^i}, \nonumber \\ A_{27}^i= & {} P_{I_{23}}^i + P_{{II}_{21}}^i {P_2^i \over P_1^i} + P_{{II}_{22}}^i {P_{7}^i \over P_1^i}, \qquad \qquad A_{28}^i = P_{I_{21}}^i {P_3^i \over P_1^i} + P_{I_{22}}^i {P_{8}^i \over P_1^i}, \\ B_1^i= & {} A_{17}^i - A_{12}^i {A_{18}^i \over A_{13}^i}, \qquad \qquad \qquad \qquad \qquad B_2^i = A_{15}^i - A_{10}^i {A_{18}^i \over A_{13}^i}, \\ B_3^i= & {} A_{16}^i - A_{11}^i {A_{18}^i \over A_{13}^i}, \qquad \qquad \qquad \qquad \qquad B_4^i = P_{{sp}_{zz}} + {A_{14}^i A_{18}^i \over A_{13}^i}, \\ B_5^i= & {} A_{10}^i - {A_{12}^i B_{2}^i \over B_1^i}, \qquad \qquad \qquad \qquad \qquad B_6^i = A_{11}^i - {A_{12}^i B_{3}^i \over B_1^i}, \\ B_7^i= & {} 1 + {A_{12}^i A_{18}^i \over A_{13}^i B_1^i}, \qquad \qquad \qquad \qquad \qquad B_8^i = A_{14}^i + A_{12}^i {B_4^i \over B_1^i}, \\ B_9^i= & {} A_{19}^i - A_{21}^i {B_{2}^i \over B_{1}^i} - {A_{22}^i B_{5}^i \over A_{13}^i}, \qquad \qquad \qquad B_{10}^i = A_{20}^i - A_{21}^i {B_{3}^i \over B_{1}^i} - {A_{22}^i B_{6}^i \over A_{13}^i}, \\ B_{11}^i= & {} {A_{18}^i A_{21}^i \over A_{13}^i B_{1}^i} + {A_{22}^i B_{7}^i \over A_{13}^i}, \qquad \qquad \qquad \qquad B_{12}^i = {A_{21}^i \over B_1^i} + {A_{12}^i A_{22}^i \over A_{13}^i B_{1}^i}, \\ B_{13}^i= & {} {A_{21}^i B_{4}^i \over B_{1}^i} - {A_{22}^i B_{8}^i \over A_{13}^i} + A_{23}^i, \qquad \qquad \qquad B_{14}^i = A_{24}^i - A_{26}^i {B_{2}^i \over B_{1}^i} - {A_{27}^i B_{5}^i \over A_{13}^i}, \\ B_{15}^i= & {} A_{25}^i - A_{26}^i {B_{3}^i \over B_{1}^i} - {A_{27}^i B_{6}^i \over A_{13}^i}, \qquad \qquad \qquad \qquad B_{16}^i = {A_{18}^i A_{26}^i\over A_{13}^i B_1^i} + {A_{27}^i B_{7}^i \over A_{13}^i}, \\ B_{17}^i= & {} {A_{26}^i \over B_{1}^i} + {A_{12}^i A_{27}^i \over A_{13}^i B_{1}^i}, \qquad \qquad \qquad \qquad \qquad \qquad B_{18}^i = {A_{26}^i B_{4}^i \over B_{1}^i} - {A_{27}^i B_{8}^i \over A_{13}^i}, \\ B_1^{i+1}= & {} A_{12}^{i+1} \left( A_{15}^i - A_{15}^{i+1} \right) - A_{17}^{i+1} \left( A_{10}^i - A_{10}^{i+1} \right) \qquad B_2^{i+1} = A_{12}^{i+1} \left( A_{16}^i - A_{16}^{i+1} \right) - A_{17}^{i+1} \left( A_{11}^i - A_{11}^{i+1} \right) , \nonumber \\ B_3^{i+1}= & {} \left( A_{12}^{i+1}A_{17}^i - A_{12}^i A_{17}^{i+1} \right) , \qquad \qquad \qquad \qquad B_4^{i+1} = \left( A_{13}^{i+1} A_{17}^{i+1} - A_{12}^{i+1} A_{18}^{i+1} \right) , \\ B_5^{i+1}= & {} {1\over A_{12}^{i+1} } \left[ \left( A_{10}^i - A_{10}^{i+1} \right) + {A_{13}^{i+1} B_1^{i+1} \over B_5^{i+1}} \right] , \qquad \qquad \qquad B_6^{i+1} = {1\over A_{12}^{i+1} } \left[ \left( A_{11}^i - A_{11}^{i+1} \right) + {A_{13}^{i+1} B_2^{i+1} \over B_5^{i+1}} \right] , \\ B_7^{i+1}= & {} {1\over A_{12}^{i+1} } \left[ A_{12}^i + {A_{13}^{i+1} B_3^{i+1} \over B_5^{i+1}} \right] \qquad \qquad \qquad \qquad B_8^{i+1} = {1\over A_{12}^{i+1} } \left[ A_{13}^i + {A_{13}^{i+1} B_4^{i+1} \over B_5^{i+1}} \right] , \\ B_{9}^{i+1}= & {} \left[ A_{{19}}^{i+1} + A_{{21}}^{i+1}B_6^{i+1} - A_{{22}}^{i+1} {B_1^{i+1} \over B_5^{i+1}} \right] , \qquad \qquad \qquad B_{10}^{i+1} = \left[ A_{{20}}^{i+1} + A_{{21}}^{i+1}B_7^{i+1} - A_{{22}}^{i+1} {B_2^{i+1} \over B_5^{i+1}} \right] , \\ B_{11}^{i+1}= & {} \left[ A_{{21}}^{i+1}B_8^{i+1} - A_{{22}}^{i+1} {B_3^{i+1} \over B_5^{i+1}} \right] , \qquad \qquad \qquad \qquad B_{12}^{i+1} = \left[ A_{{21}}^{i+1}B_9^{i+1} - A_{{22}}^{i+1} {A_{13}^i A_{17}^{i+1} \over B_5^{i+1}} \right] , \nonumber \\ B_{13}^{i+1}= & {} \left[ A_{{24}}^{i+1} + A_{{26}}^{i+1}B_6^{i+1} - A_{{27}}^{i+1} {B_1^{i+1} \over B_5^{i+1}} \right] , \qquad \qquad \qquad B_{14}^{i+1} = \left[ A_{{25}}^{i+1} + A_{{26}}^{i+1}B_7^{i+1} - A_{{27}}^{i+1} {B_2^{i+1} \over B_5^{i+1}} \right] , \nonumber \\ B_{15}^{i+1}= & {} \left[ A_{{26}}^{i+1}B_8^{i+1} - A_{{27}}^{i+1} {B_3^{i+1} \over B_5^{i+1}} \right] , \qquad \qquad \qquad \qquad B_{16}^{i+1} = \left[ A_{{26}}^{i+1}B_9^{i+1} - A_{{27}}^{i+1} {B_4^{i+1} \over B_5^{i+1}} \right] . \end{aligned}$$

Appendix A3

The material properties of gallium nitride (GaN) [32, 33], aluminum nitride (AlN) [40, 45], PZT 5A [25], stainless steel, aluminum and the adhesive layer, LARC-SI [42], are listed in Table 2.

Table 2 Material properties of materials and alloys used for various types of sensors and actuators