Example Substrate curvature due to an electric field

In a transducer for use in a hearing aid, a PZT film of thickness h = 5 fim is deposited on a polysilicon substrate of thickness hg = 150 fim. The tetragonal film is prepared with its c—axis coincident with the 2 3—axis of the system coordinate frame, which is perpendicular to the film-substrate interface. The array of piezoelectric parameters for the PZT film material is 

In this case, 1=82 = 0 and S3 = 10 volts/m-coulomb. It follows from (3.86) that 

The film tends to contract along the interface but is not able to do so because of the substrate. Consequently, a state of equi-biaxial tensile mismatch strain develops in the film with respect to the substrate. 

The in-plane biaxial modulus for PZT is approximately M = 6.1 x 10 N/m. Thus, the membrane force in the fully constrained film is 

According to the Stoney formula (2.7), the substrate curvature is given in terms of this force, the biaxial modulus of the substrate and the thickness of the substrate. The biaxial modulus is approximately Mg = 76 x 10 N/m for polysihcon. The resulting estimate of radius of curvature is = 1.56m. 

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