Computational study of the response of periodic piezoelectric thin films on substrates

This paper reports a finite element study of periodic, piezoelectric thin film islands on a substrate. The interactions of island radius, interisland periodicity, substrate stiffness, and boundary conditions are examined. It is seen that the degradation of the electromechanical response of the piezoelectric film due to deposition on a substrate is reduced due to two factors: a higher ratio of in-plane area of island to in-plane area of substrate and a lower interisland spacing in the periodic geometry. The effective piezoelectric coefficient of the deposited thin film is different depending on whether the converse or the direct piezoelectric effect is being studied and is larger in the latter case.

Figure 1

Schematic diagram of a $\overline{x}\overline{y}$-periodic structure with circular thin film islands and geometric details of a quarter unit cell.

Figure 2

(Color online) Finite elements mesh with bias ratio (bias $\text{ratio}=G_1∕G_N$).

Figure 3

(Color online) Effect of bias ratio and number of elements along film thickness on the relative normalized displacement when an electric field is applied normal to the surface of the film (the converse piezoelectric effect).

Figure 4

(Color online) The dependence of relative normalized displacement on interisland spacing for soft substrates under $\overline{x}\overline{y}$-periodic boundary conditions and rigid substrates when an electric field is applied normal to the surface of the film (the converse piezoelectric effect).

Figure 5

(Color online) The dependence of relative normalized displacement on interisland spacing for soft substrates under $\overline{x}$-periodic, $\overline{x}\overline{y}$-periodic, and free boundary conditions and rigid substrates (the converse piezoelectric effect).

Figure 6

(Color online) Comparison of displacement along the $z$ axis, ${\overline{u}}_z$, for free boundary condition, $\overline{x}$-periodic boundary condition, and $\overline{x}\overline{y}$-periodic boundary condition.

Figure 7

(Color online) The dependence of the relative normalized displacement with respect to depth along thin film for soft substrates under periodic and free boundary conditions and rigid substrates when an electric field is applied normal to the surface of the film (the converse piezoelectric effect).

Figure 8

(Color online) The dependence of the relative normalized displacement with respect to the relative stiffness ratio $\gamma$. All calculations in this figure are subjected to $\overline{x}\overline{y}$-periodic boundary conditions. The reported data correspond to an electric field that is applied normal to the surface of the film (the converse piezoelectric effect).

Figure 9

(Color online) The dependence of the relative normalized displacement with respect to interisland spacing for different values of the radius of each island. The inset is the relative normalized displacement with respect to island radius at a given value of interisland spacing. All calculations in this figure are for a soft substrate subjected to periodic boundary conditions. The reported data correspond to an electric field that is applied normal to the surface of the film (the converse piezoelectric effect).

Figure 10

The dependence of the effective normalized piezoelectric coefficient corresponding to the converse and direct effects with respect to island radius for given values of interisland spacing. All calculations in this figure are for a soft substrate subjected to periodic boundary conditions.