Micromechanical Modelling of Ferroelectric Films
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Micromechanical modelling of ferroelectric films J. E. Huber Department of Engineering, University of Cambridge, Trumpington St, Cambridge, CB2 1PZ, UK. ABSTRACT 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 behaviour. Thus it is important to be able to model the coupled and constrained behaviour of film material. This work presents a preliminary study of the application of micromechanical modelling to ferroelectric films. A self-consistent micromechanics model developed for bulk ferroelectrics is adapted for thin film behaviour by incorporating features such as grain structure, mechanical clamping by the substrate, residual stresses, and crystallographic orientation of the film. INTRODUCTION 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 behaviour 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 behaviour is adapted to capture the behaviour 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 behaviour is incorporated simplistically as a controlling factor determining the coercive field and surface effects are included using a simple series capacitance model. OVERVIEW OF CONSTITUTIVE MODEL The constitutive model is described in detail elsewhere [2, 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 (labelled α). The driving force for this transformation, Gα , is given by Gα = σ · ∆ǫα + E · ∆Pα + σ · ∆dα · E
(1)
where σ and E are the local stress and electric field, while ∆ǫα , ∆Pα and ∆dα are the changes in remanent strain, polarization, and piezoelectric tensor respectively, due to the operation of switching system α. When switching occurs, there is a volume fraction transfer rate f˙α due to each switching system, such that the total volumetric dissipation rate w˙ D due to all switching systems
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