Effects of interface dislocations on properties of ferroelectric thin films
Introduction
Much work has been done recently on nano-scale ferroelectric thin films because of their remarkable properties that make them very useful in electro-optic, pyroelectric, and piezoelectric devices, as well as ultra-high-density nonvolatile memories (Haeni et al., 2004; Fong et al., 2004; Streiffer et al., 2002; Balzar et al., 2004). It is well known that properties of ferroelectrics in the bulk and thin-film forms may be significantly different, depending on the combined effects of film surfaces, depolarization field, film/substrate interfaces, epitaxial stress and the external electric field, etc (Catalan et al., 2005; Sinnamon et al., 2002; Wesselinawa et al., 2005; Scott et al., 2005; Wang et al., 2003). In this regard, misfit dislocations introduces a large stress field superimposing on the epitaxial stresses to significantly affect the physical properties of the film. Related investigations constitute an active area of thin-film research.
Dislocation structure formed in BaTiO3 grown on SrTiO3 substrate has been analyzed by using X-ray and transmission electron microscopy, and determined the critical thickness of the misfit dislocation generation (Suzuki et al., 1999). They also confirmed that the misfit relaxation depended on the presence of dislocations. Similar investigations have been carried out and result showed that misfit dislocations were very important on the stability of the polarization field (Sun et al., 2004; Chu et al., 2004). The dielectric properties of ferroelectric thin film also depended on internal stresses and dislocation-type defects (Li et al., 2001; Canedy et al., 2000). Using the phase-field model, effects of interfacial dislocations on the polarization distribution and domain structure of ferroelectric thin film have been investigated (Hu et al., 2003). They also developed a method to predict the evolution of a domain structure in a ferroelectric thin film with an arbitrary spatial distribution of dislocations. However, the effects of the depolarization field and the near-surface eigenstrain relaxation could not be readily taken into account within this approach. Another concern also arose from the non-linear nature of the model, which did not always admit a unique solution.
In our previous work (Wang and Woo, 2005; Zheng et al., 2006a), we adopted the time-dependent Ginzburg–Landau equation approach to investigate the self polarization distribution near misfit dislocations, successfully including the effects of relaxation of the surface effect and the depolarization. The existence of a dead layer near the film/substrate interface was confirmed (Haun et al., 1987). However, the dielectric constant was assumed to be spatially uniform. This assumption is inconsistent with the spatially varying polarization field, on which the dielectric constant is functionally dependent. Besides, among many of the most important properties of a ferroelectric thin film, only the spontaneous polarization distribution was discussed.
In this paper, effects of interfacial dislocations on the self-polarization, Curie temperature and dielectric constant are considered as functions of temperature and film thickness. The spatial variations of the self-polarization, the depolarization, and the extrapolation length are specifically taken into account. This is done via the finite-difference solution of the evolution equation derived from a thermodynamic model in which the free energy is constructed with the help of the Landau–Devonshire functional. The interface dislocations effects on the remnant polarization and the coercive field of ferroelectric thin film also is investigated. The results are discussed and the observations concluded.
Section snippets
The free energy equation
We consider a single-domain ferroelectric thin film on a compliant substrate, which undergoes a cubic to tetragonal (tetragonal to cubic) phase transformation on cool-down (heat-up). We use a coordinate system in which the x-axis is parallel to [1 0 0], the y-axis to [0 1 0], and the z-axis to [0 0 1], where the film in the x- and y-directions extend to infinity, the film surface and the film/substrate interface are on the z=h and z=0 planes, respectively, h being the film thickness (Fig. 1). A
The finite difference method and the Runge–Kutta method
The evolution of the self-polarization field P is obtained by numerically solving the time-dependent Ginzburg–Landau Eq. (2.5) subject to the electric and mechanical boundary conditions. We adopt the second-order finite difference method for spatial integration and the fourth-order Runge–Kutta method for time integration to solve Eq. (2.5) in this work (Wang and Zhang, 2006). The ferroelectric thin film is considered as a stack of layers, each of which has a finite thickness Δz with physical
Simulation results and discussions
To be specific, we consider a PbTiO3 thin film on a rigid LaAlO3 substrate. We use as an approximation material constants for the Landau free energy, the electrostrictive coefficients and the elastic properties for bulk materials from the literatures (Lakovlev et al., 2002; Pertsev et al., 1998; Streiffer et al., 2002; Li et al., 2002b). We consider that this approximation should be sufficient for our discussion.
We first consider the polarization field near a single edge dislocation,
Summery and conclusions
The time-dependent Ginzburg–Landau equation of a ferroelectric thin film with interfacial dislocations is solved numerically using a finite-difference scheme. Taking into account the inter-dependence of the polarization and depolarization fields, as well as the surface effect, effects of interfacial dislocations on ferroelectric thin films are investigated. We obtain the self-polarization field around a single edge misfit dislocation at different temperatures, and determine the dead region
Acknowledgments
This project was supported by Grants PolyU5312/03E, 5322/04E and GU164. Co-author BW is also grateful for grants from the National Science Foundation of China (Nos. 50232030, 10172030 and 10572155) and the Science Foundation of Guangzhou Province (2005A10602002).
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