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

TL;DR

This paper extends Stoney's equation to account for flexoelectric and piezoelectric effects in thin films on elastic substrates, enabling more accurate inference of film properties from curvature measurements.

Contribution

The work generalizes Stoney's equation to include electromechanical coupling effects, considering various configurations and film properties, which was not previously addressed.

Findings

Flexoelectric and piezoelectric effects significantly influence film curvature.

The generalized equation applies to both open and closed circuit conditions.

Electromechanical coupling alters strains and electric polarization in the film.

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.