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Generalization of Stoney's equation for flexoelectric thin films on elastic substrates

Swarnava Ghosh

TL;DR

The work generalizes Stoney's equation to account for electromechanical coupling in thin films on elastic substrates by formulating a linear, axisymmetric bilayer model and deriving the total enthalpy including elastic, dielectric, piezoelectric, and flexoelectric contributions. Using energy minimization, it yields analytical expressions for the in-plane stretching strain $\varepsilon_0$ and midplane curvature $\kappa$ under open and closed circuit conditions for both uniform and nonuniform film properties, and computes the film polarization. The results reveal nonlinear dependencies of curvature, strain, and polarization on thickness ratios and stiffness ratios, with significant deviations from Stoney’s original predictions in regimes of strong coupling or large gradients, thereby enabling more accurate film-property determination from curvature measurements. The approach lays a foundation for exploring flexoelectric/piezoelectric effects in thin-film systems and suggests extensions to non-axisymmetric, large-deformation, and anisotropic regimes with potential impact on nanoscale device design.

Abstract

When a thin film is deposited on an incompatible elastic substrate, the film develops an elastic mismatch strain, causing the film-substrate system to bend. Stoney's equation relates the curvature of the bent film-substrate system with the residual stress developed in the film, and can be used to infer film properties from curvature measurements. Certain materials exhibit electromechanical coupling, such as piezoelectricity and flexoelectricity, which can alter the curvature and strains. In this work, we generalize Stoney's equation to include flexoelectric and piezoelectric effects in the film. Considering both open and closed circuit configurations, as well as uniform and non-uniform film properties, we compare different cases of electromechanical coupling and discuss their influence on curvature, strains, and electric polarization in the film.

Generalization of Stoney's equation for flexoelectric thin films on elastic substrates

TL;DR

The work generalizes Stoney's equation to account for electromechanical coupling in thin films on elastic substrates by formulating a linear, axisymmetric bilayer model and deriving the total enthalpy including elastic, dielectric, piezoelectric, and flexoelectric contributions. Using energy minimization, it yields analytical expressions for the in-plane stretching strain and midplane curvature under open and closed circuit conditions for both uniform and nonuniform film properties, and computes the film polarization. The results reveal nonlinear dependencies of curvature, strain, and polarization on thickness ratios and stiffness ratios, with significant deviations from Stoney’s original predictions in regimes of strong coupling or large gradients, thereby enabling more accurate film-property determination from curvature measurements. The approach lays a foundation for exploring flexoelectric/piezoelectric effects in thin-film systems and suggests extensions to non-axisymmetric, large-deformation, and anisotropic regimes with potential impact on nanoscale device design.

Abstract

When a thin film is deposited on an incompatible elastic substrate, the film develops an elastic mismatch strain, causing the film-substrate system to bend. Stoney's equation relates the curvature of the bent film-substrate system with the residual stress developed in the film, and can be used to infer film properties from curvature measurements. Certain materials exhibit electromechanical coupling, such as piezoelectricity and flexoelectricity, which can alter the curvature and strains. In this work, we generalize Stoney's equation to include flexoelectric and piezoelectric effects in the film. Considering both open and closed circuit configurations, as well as uniform and non-uniform film properties, we compare different cases of electromechanical coupling and discuss their influence on curvature, strains, and electric polarization in the film.
Paper Structure (20 sections, 86 equations, 10 figures, 1 table)

This paper contains 20 sections, 86 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Constitutive relations in flexoelectricity
  3. Thin flexoelectric film on an elastic substrate
  4. Uniform elastic mismatch strain and elastic properties
  5. Expressions for stretching strain, curvature, and electric polarization
  6. Case I: both piezoelectric and flexoelectric effects are present
  7. Case II: piezoelectric, dielectric and flexoelectric effects are absent
  8. Case III: flexoelectric effect is absent
  9. Case IV: piezoelectric effect is absent
  10. Electric polarization in the film
  11. Discussion
  12. Non-uniform elastic mismatch strain and elastic properties
  13. Linearly increasing elastic mismatch strain and constant material properties
  14. Case I: both piezoelectric and flexoelectric effects are present
  15. Case II: piezoelectric, dielectric and flexoelectric effects are absent
  16. ...and 5 more sections

Figures (10)

  • Figure 1: Top figure shows a film deposited on a flexible substrate. The film has an elastic mismatch and is held in a flat configuration by an applied stress $\sigma_m$. Upon removal of this applied stress, the film-substrate system deforms as shown in the bottom.
  • Figure 2: The film substrate system for (a) converse case with applied voltage $V$ across the height of the film and (b) direct case without any applied voltage is shown.
  • Figure 3: Normalized stretching strains and curvatures of the film-substrate system for various ratios of film-substrate thickness $h_f/h_s$ and stiffness $M_f/M_s$. (a) show the stretching strain and (d) the curvature when only elastic effects are considered (case II). (b), (c) show the stretching strains and (e), (f) the curvatures for converse and direct effects when both piezoelectric and flexoelectric effects are present (case I).
  • Figure 4: Normalized stretching strains and curvatures for the converse case shown as a function of the ratio of the thickness of film and substrate for different cases of stiffness ratios $M_f/M_s$. Figures (a)-(c) show the stretching strains, and (d)-(f) show the curvatures.
  • Figure 5: Normalized stretching strains and curvatures for the direct case shown as a function of the ratio of the thickness of film and substrate for different cases of stiffness ratios $M_f/M_s$. Figures (a)-(c) shows the stretching strains, and (d)-(f) shows the curvatures.
  • ...and 5 more figures