Theoretical investigation of polarization scaling in ultrathin epitaxial PbZrxTi1−xO3 films

Abstract

We present a theoretical analysis of the scaling of the polarization and the static dielectric susceptibility through a mean-polarization approach for ultrathin epitaxial PbZrxTi1−xO3 thin films. We use the traditional Euler-Lagrangian framework applied to a Landau-Ginzburg-Devonshire (LGD) nonlinear thermodynamic treatment. The novelty of our approach is that the model hinges on using experimentally measured correlation lengths and temperature scaling relationships to give the size-dependent expansion parameters of the nonlinear thermodynamic potential. These are then used in a Taylor series expansion of the polarization at the center of the film. We show that this method is able to correctly predict experimentally observed scaling without the need for the so-called extrapolation length (which is impossible to measure experimentally). Furthermore, as no implicit correlation between the correlation length (ξ) and the coefficient of the gradient term in the LGD potential (g11) is assumed, the model thus involves fully experimentally measurable parameters and their systematic temperature dependence rather than implicit assumptions. The model finds that the Curie temperature in ultrathin films is more sensitive to epitaxial strain as compared to the polarization and that the critical thickness is strongly dependent on the “temperature-epitaxial strain” parameter space. Interestingly, while it finds that at lower temperatures the depolarization field does play a strong role in the thickness dependence as well as spatial profile of the polarization, with increasing temperature, a significant weakening of the role of depolarization fields occurs. Consequently the interface-induced suppression is lower and, as a result, the polarization profile is more homogenous at higher temperatures. This indicates that systematic temperature dependent studies are fundamental to further understanding of size effects in ferroelectrics.

Surface effect on domain wall width in ferroelectrics

“…1,2,3,4,5,6,7 Laminar domain structure formation in the thick films with free surfaces was considered in the classic papers by Kittel 8 for ferromagnetic media and Mitsui and Furuichi 9 for ferroelectric media. In the classical papers of Cao and Cross 10 and Zhirnov 11 , a single boundary between two domains was considered in the bulk ferroelectrics, allowing for electrostriction contribution.…”

Misfit strain–film thickness phase diagrams and related electromechanical properties of epitaxial ultra-thin lead zirconate titanate films

“…(12) over z in the limit between 0 and z and the details of this approach were discussed in detail elsewhere [25,44]. The solution is given by:…”

“…where P z is the polarization along the [0 0 1] direction at the electrode interface ignoring the depolarizing field, <P 3 > is the average polarization along the z-axis, k is the screening length of the electrode, and e e is the dielectric constant of the electrode [25,[36][37][38][39]. While it was recently proposed that the background dielectric constant may have some implications on the free energy and depolarization field [40][41][42], here we follow the approach of Kretschmer and Binder [28] where the "background" is free space with a dielectric response of e 0 .…”

“…Similar to the short-range strain fields of an array of periodic misfit dislocations at the film-substrate interface, there are alternating localized short-range strain fields resulting from the periodic deviation of the actual misfit strain from the average misfit on the interface between the polydomain layer and the substrate in the c/a/c/a structure. To determine the critical film thickness of the c/a/c/a polydomain, we start off with the out-of-plane polarization P 3 of a monodomain c-phase incorporating the polarization variation in the film due to the electrical boundary conditions resulting in a depolarization field [25,44]:…”

“…In addition to the phase stability as a function of thickness, we also compute variations in the dielectric and electromechanical properties as a function of a multi-parameter space consisting of the temperature, film thickness, effective misfit strain, critical thicknesses for misfit dislocation and polydomain formation, interface-induced polarization gradients, and electrode screening length. The approach builds upon the previously published methodology of free energy functions for possible mono-and polydomain states where the most stable phase is determined as the one with the lowest free energy [25]. Since the film is clamped between two symmetrical electrodes, we assume the polarization inside the film is centrally symmetric along a z-axis along the film thickness resulting in zero gradient of polarization at the center.…”