Effect of the depolarizing field on the domain structure of an improper ferroelectric

TL;DR

This work investigates whether the depolarizing field from bound surface charges can modify domain patterns in improper ferroelectrics with topological defects. Using phase-field simulations of hexagonal manganite thin films, the authors include an electrostatic term $F_{electrostatic} = -P_s E$ and apply Gauss-law–based field calculations under open boundaries to resolve the depolarizing field. They find that the depolarizing field lowers the average polarization, induces polarization decay away from domain walls, and reshapes the domain-size distribution, yielding ring-like Fourier intensities and a narrower pair-correlation peak, while promoting strong wall alignment along the polarization axis. The results demonstrate that depolarizing fields cannot be neglected in thin-film improper ferroelectrics and suggest electrostatic control as a route for domain-pattern engineering complementary to strain effects.

Abstract

We show that, contrary to common belief, the depolarizing electric field generated by bound charges at thin-film surfaces can have a substantial impact on the domain structure of an improper ferroelectric with topological defects. In hexagonal-manganite thin films, we observe in phase-field simulations that through the action of the depolarizing field, (i) the average magnitude of the polarization decreases, (ii) the local magnitude of the polarization decreases with increasing distance from the domain walls, and (iii) there is a significant alteration of the domain-size distribution and average domain size, which is visualized with the pair-correlation function. We conclude that, in general, it is not appropriate to ignore the effects of the depolarizing field for thin film ferroelectrics.

Figures (6)

• Figure 1: (a) Visualization of the structural order of hexagonal manganites. The order is given by a tilt of the MnO$_{5}$ bipyramids parametrized by the zone-boundary mode $\mathrm{K}_3$. The two-dimensional order parameter is illustrated in polar coordinates with $Q$ as amplitude and $\Phi$ as azimuthal angle of the MnO$_5$ tilt. Three neighboring bipyramids form a trimer and tilt towards a common center, which triples the unit cell sim_hexagonal_2016. (b) Visualization of the deformation of the MnO$_5$ bipyramids during a transition from the paraelectric to the ferroelectric phase. Small asymmetries in this process cause polarization onset fennie_ferroelectric_2005. (c) Schematic drawing of a conceptual distortive-ferroelectric domain structure. Topologically protected vortices are formed by the six domain states of the structural order. The circular symbols denote the improper ferroelectric out-of-plane polarization which follows an alternating pattern.
• Figure 2: Dependence of improper-ferroelectric order on depolarizing field, visualized with phase-field simulations in real space. (a) Domain pattern for a simulation that ignores the electrostatic interaction. (b) Domain pattern for a simulation that includes the electrostatic interaction assuming a background dielectric constant $\varepsilon_{\rm b} = 1$ (see text on this choice). (c) Close-up on the green paths through panels (a) (left) and (b) (right). (d) Domain size along the green path for the simulations without (top) and with (bottom) the depolarizing field. (e, f) Polarization profile along a section, shown as yellow line in (a, b). The gradient and resulting dip in polarization is clearly visible in (f).
• Figure 3: Histogram of the polarization per point in the computational mesh for a simulation (a) without and (b) with consideration of the electrostatic term. The mean of positive/negative values are shown with red vertical bars. For both histograms, the data of Fig. \ref{['fig:RealSpaceImages']} have been used. The average magnitude of the polarization is lower in (b) compared to (a), which illustrates the influence of the depolarizing field.
• Figure 4: Fourier-transformed intensity and pair correlation of twenty combined simulations for (a-c) systems without electrostatic interaction and (d-f) systems with electrostatic interaction. (a) Fourier-transformed intensity and (b,c) pair correlation with two scaling ranges. (d-f) Same as (a-c), but with electrostatic interaction for $\varepsilon_{\rm b} = 1$. Note that data augmentation was used for this analysis (see Appendix \ref{['app:NumericalMethodsAppendix']}). The anisotropy of the plot results from the quadratic simulation mesh and the artifacts its size limitation introduces. If electrostatic interaction and hence a depolarizing field is present, the ring-like intensity in (d) and the spherical wave in (e) and (f) indicate a narrowing of the domain-size distribution, whereas a smaller width of the central peak corresponds to a decrease in the average domain size.
• Figure 5: 3D-distribution of domain walls in simulations of thin films with and without electrostatic interaction. The color signifies the magnitude of the polarization in the adjacent domain. (a) Simulation ignoring electrostatic interaction and (b) simulation including electrostatic interaction. In (b), the additional electrostatic penalty for domain walls perpendicular to the z-direction cause a much stronger alignment of the domain walls along the z-axis than in (a).