Phase field simulations of ferroelectrics domain structures in PbZrxTi1−xO3 bilayers
Introduction
Multilayered heterostructures with unique functional properties have attracted significant attention. The properties of such multilayered systems can be tuned through control of the composition, the thickness of each layer, and the number of alternating layers [1], [2], [3]. For example, it has been shown that multilayered ferroelectric thin films demonstrate both pronounced polarization and high dielectric permittivity compared with single layer films [4], [5], [6], [7], [8]. Many factors can lead to these enhanced properties, including electric field-induced coupling [1], morphotropic phase boundary effects [5], epitaxial strains [8], [9], [10], and unique electrostatic interactions between the two layers [4], [11]. Recently, piezoresponse force microscopy (PFM) and cross-sectional transmission electron microscopy (TEM) analyses revealed a unique “herringbone” ferroelastic domain structure in PbZr0.3Ti0.7O3/PbZr0.7Ti0.3O3 bilayers. The ferroelastic domains were shown to be mobile under an applied external electric field, leading to an electromechanical coefficient up to three times larger than that in constrained single layered PbZr0.3Ti0.7O3 thin films [12]. These results were supported by a thermodynamic theory which found that the out-of-plane piezoelectric coefficients d33 in bilayers can be an order of magnitude larger than the corresponding single layer constituents due to electrical and electromechanical interdomain and interlayer interactions [13].
A number of theoretical investigations of the domain stabilities and ferroelectric properties of multilayered heterostructured thin films [11], [12], [14], [15], [16] have discussed the critical role of electrostatic interactions in the electromechanical response of ferroelectric multilayers. Recently phase field simulations have been applied to understand domain structure evolution during the ferroelectric transition [17], including bulk single crystals [18], [19], [20] and confined nanoferroelectrics [21]. Li et al. [22] extended the model by incorporating the elastic solution for thin films in an elastically anisotropic system combining Khachaturyan’s microscopic elasticity theory [23], [24] and Stroh’s formalism of anisotropic elasticity [25], [26]. This thin film model has been successfully applied to predicting the domain structures in a number of constrained three-dimensional (3-D) ferroelectric thin film systems [27], [28], [29]. In particular, the phase field model was applied to a barium titanate–strontium titanate (BT–ST) superlattice based on the bulk model with various coefficients and properties along the thickness direction. Results from simulations of the dependence of the phase transition temperature on the number of perovskite unit cells along the growth direction of each layer showed quantitative agreement with ultraviolet Raman spectroscopy, and the polarization distribution across the superlattice thickness was calculated under different interfacial coherencies in the superlattice simulation using the phase field model [30], [31].
In this work the phase field method is extended to study the domain structure in a PZT bilayer system with differing compositions in the two layers. We studied the strain effect in each individual layer and electrostatic interaction between the two layers on the domain structures. These were compared with the single layer case for the same composition and thickness. We finally constructed strain–temperature phase diagrams for the bilayer system and compared them with the single layer film diagram. These results were also compared with experimental observations and prior thermodynamic analyses. It was found that mechanical and electrostatic interactions between the two layers may lead to significantly different domain structures from the single layer, resulting in fundamentally different physical properties from the individual layer systems.
Section snippets
Framework of the phase field model
We consider a ferroelectric bilayer system which comprises two layers with different compositions. Fig. 1 shows a schematic plot of the bilayer system, wherein the z-axis represents the out-of-plane direction. The bottom layer is (0 0 1) Pb(Zr0.7Ti0.3)O3 (R), while the top layer is (0 0 1) Pb(Zr0.3Ti0.7)O3 (T). In the phase field model temporal evolution of the polarization vector field is described by the time-dependent Ginzburg–Landau equations, i.e.where L is
Modeling the interactions of two layers
The variation in the composition-dependent coefficient ζ within the bilayer is described as follows:where ζtop and ζbott represent the Landau coefficients or electrostrictive coefficients of the top and bottom layers, respectively, and φ takes the value of 0.0 in the top layer and 1.0 in the bottom layer. In order to solve the electrostatic equilibrium equations and the mechanical equilibrium equations for the entire bilayer the elastic coefficients and dielectric
Numerical simulations
In this work the semi-implicit Fourier spectral method for both time stepping and spatial discretization is employed to solve Eq. (1) [39]. The initial film is in a paraelectric state, which was created by assigning a value of 0 to each component of polarization at each lattice site plus a small amount of random noise. Here we take a ferroelectric bilayer consisting of a T layer and a R layer of the same thickness, 20.0 nm. The pseudocubic lattice parameters of the T layer and the R layer are
Domain structures in a PZT bilayer and single layer
We first studied the domain configurations of a PZT bilayer film grown on a silicon substrate at room temperature, 25 °C, as a specific example. It is assumed that the bilayer system is incoherent with the substrate and the strain for the bilayer film with the substrate is 0.2% due to the different thermal expansion coefficients of the film and substrate. The in-plane lattice parameter of (0 0 1) silicon is 3.840 Å, while the bottom R layer has a quasi-cubic lattice parameter of 4.1032 Å. If the
Interactions between the two layers and comparison with experiments
As shown in Section 5.1, when the two layers are coherent the misfit strain between the two layers plays a critical role in determining the domain structures. We also performed simulations for Pb(Zr0.3Ti0.7)O3 (T)/Pb(Zr0.55Ti0.45)O3 (R) bilayers and obtained similar domain structures, shown in Fig. 2a, when the two layers are coherent and incoherent with the substrate. Based on the simulation results and analysis we can make the following predictions.
- i.
If the thickness of both the layers is less
Conclusion
A three-dimensional phase field model was employed to study ferroelectric bilayer domain structures. It was demonstrated that at room temperature (much lower than the Curie temperatures) the misfit strain caused by the lattice parameter differences between the two layers leads to domain structures significantly different from that of the single layer. The domain sizes in the rhombohedral bilayer are reduced compared with the single layer case under the effect of a tetragonal layer. Above the
Acknowledgements
This work was supported by the NSF MRSEC under Grants Nos. DMR-0820404, DMR-1006541, and DMR-1210588. The work at UNSW was supported by the ARC Discovery Project scheme.
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