Contribution of substrate to converse piezoelectricresponse of constrauned thin films
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The converse piezoelectric response of a thin film constrained by a substrate is analyzed in different geometries under various boundary conditions. We consider the effects of elastic deformation of the substrate on the total displacement of the film surface induced by the electric field. The change of film thickness and the bending curvature of a film/substrate couple are calculated. For a thin film island clamped on a large thick substrate, the theoretical estimation of the piezoresponse, including a local bending in the vicinity of the island/substrate interface, is in agreement with the finite element calculation.
I. INTRODUCTION
Understanding of the piezoresponse of piezoelectric thin films is essential in the design and fabrication of microsensors, actuators, and other electromechanical devices.1–9 In principle, the converse piezoresponse may be characterized by applying a voltage (field) and measuring the field-induced strain. For example, if a thin film has a tetragonal structure with the polarization along the c axis normal to the film (along the z direction), and epitaxially oriented on a cubic substrate layer with the interface along (001)T //(001)C, the field-induced strain in the film is ⑀ˆ 0 =
冋
d31E 0 0 0 d31E 0 0 0 d33E
册
,
(1)
where d33 and d31 are the longitudinal and transverse piezoelectric coefficients of a free-standing piezoelectric film, and E is the applied electric field normal to the film. From the calculation of the total strain in the film clamped by a substrate, it has been concluded that the effective piezoelectric coefficient of the thin film follows1 d f33
= d33 −
2d31 Sf13 共Sf11 + Sf12兲
,
(2)
where Sfij are the mechanical compliances of the film under constant electric field. Equation (2) demonstrates
a)
Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/JMR.2004.0367 J. Mater. Res., Vol. 19, No. 10, Oct 2004
http://journals.cambridge.org
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that df33 measured by the strain in the film is independent of elastic properties of a substrate if the film is completely clamped by the substrate. However, the piezostrain of a constrained thin film cannot be directly measured by the standard resonance method due to the substrate clamping.2 Most of the static or quasi-static techniques such as cantilever deflection,3 laser interferometry4,5 and atomic force microscopy5–9 provide the effective piezoelectric coefficients through the measurement of the displacement of the film or the thickness change of the sample (from the top surface of the film to the bottom surface of the substrate) instead of the strain in the film. To understand the experimental measurements, we studied the total piezoresponse of a film/substrate couple, including changes of thickness of the film and the substrate, as well as a bending deflection of the film/substrate couple. The theoretical calculation shows that df33 measured through the surface displacement still depend on the substrate elastic compliances and the boundary conditions. Equation (2)
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