Micromechanical modeling of ferroelectric films
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Micromechanical modeling of ferroelectric films J.E. Hubera) Department of Engineering Science, University of Oxford, Oxford OX1 3PJ, United Kingdom (Received 10 August 2005; accepted 30 November 2005)
Ferroelectric films are growing in significance as non-volatile memory devices, sensors, and microactuators. The stress state of the film, induced by processing or constraints such as the substrate, strongly affects device behavior. Thus, it is important to be able to model the coupled and constrained behavior of film material. This work presents a preliminary study of the application of micromechanical modeling to ferroelectric films. A self-consistent micromechanics model developed for bulk ferroelectrics is adapted for thin film behavior by incorporating several features of the microstructure, mechanical clamping by the substrate, residual stresses, and the crystallographic orientation of the film.
I. INTRODUCTION
II. OVERVIEW OF CONSTITUTIVE MODEL
A major issue in the development of thin film ferroelectric devices is understanding the effects of the residual stresses that arise during processing. By varying the substrate, deposition process, and film material, a variety of microstructures can be produced. The films also typically exist in a state of in-plane residual stress.1 For memory applications, it is important to achieve a sharply defined coercive behavior and a high value of saturation polarization. By contrast, in sensor applications, the dielectric, piezoelectric, or pyroelectric coefficients are the target. Therefore, an ability to predict the effects of residual stress on all of these properties is desirable. In this work, a previous constitutive model for polycrystalline ferroelectric behavior is adapted to capture the behavior of thin films. The model uses a micromechanics approach that allows crystal structure, orientation, and grain shape to be incorporated in a straightforward way. Boundary conditions representative of a clamped film are imposed. The influence of domain nucleation on the film behavior is incorporated as a controlling factor determining the coercive field, and surface effects are included using a simple series capacitance model.
The constitutive model is described in detail elsewhere2,3; in this model, the state of individual ferroelectric grains is characterized by the volume fractions of each type of ferroelectric domain present. The key feature of the model is the representation of ferroelectric switching by the conversion of material from one domain type to another by the operation of a switching system (labeled ␣). The driving force for this transformation, G␣, is given by3
a)
Address all correspondence to this author. e-mail: [email protected] This paper was selected as the Outstanding Meeting Paper for the 2005 MRS Spring Meeting Symposium CC Proceedings, Vol. 881E. DOI: 10.1557/JMR.2006.0082 J. Mater. Res., Vol. 21, No. 3, Mar 2006
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G␣ = ⭈⌬⑀␣ + E⭈⌬P␣ + ⭈⌬d␣⭈E ,
(1)
where and E are the local stress and ele
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