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Intrinsic ferroelectric switching from first principles

Abstract

The existence of domain walls, which separate regions of different polarization, can influence the dielectric1, piezoelectric2, pyroelectric3 and electronic properties4,5 of ferroelectric materials. In particular, domain-wall motion is crucial for polarization switching, which is characterized by the hysteresis loop that is a signature feature of ferroelectric materials6. Experimentally, the observed dynamics of polarization switching and domain-wall motion are usually explained as the behaviour of an elastic interface pinned by a random potential that is generated by defects7,8, which appear to be strongly sample-dependent and affected by various elastic, microstructural and other extrinsic effects9,10,11,12. Theoretically, connecting the zero-kelvin, first-principles-based, microscopic quantities of a sample with finite-temperature, macroscopic properties such as the coercive field is critical for material design and device performance; and the lack of such a connection has prevented the use of techniques based on ab initio calculations for high-throughput computational materials discovery. Here we use molecular dynamics simulations13 of 90° domain walls (separating domains with orthogonal polarization directions) in the ferroelectric material PbTiO3 to provide microscopic insights that enable the construction of a simple, universal, nucleation-and-growth-based analytical model that quantifies the dynamics of many types of domain walls in various ferroelectrics. We then predict the temperature and frequency dependence of hysteresis loops and coercive fields at finite temperatures from first principles. We find that, even in the absence of defects, the intrinsic temperature and field dependence of the domain-wall velocity can be described with a nonlinear creep-like region and a depinning-like region. Our model enables quantitative estimation of coercive fields, which agree well with experimental results for ceramics and thin films. This agreement between model and experiment suggests that, despite the complexity of ferroelectric materials, typical ferroelectric switching is largely governed by a simple, universal mechanism of intrinsic domain-wall motion, providing an efficient framework for predicting and optimizing the properties of ferroelectric materials.

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Figure 1: Domain-wall velocity from molecular dynamics simulations.
Figure 2: LGD model of nucleation at domain walls.
Figure 3: Hysteresis loops and coercive fields for several materials simulated using first-principles data.

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Acknowledgements

S.L. was supported by the NSF through Grant DMR-1124696, Grant CBET-1159736, and the Carnegie Institution for Science. I.G. was supported by the US ONR under Grant N00014-12-1-1033. A.M.R. was supported by the US DOE under Grant DE-FG02-07ER46431. Computational support was provided by the US DOD through a Challenge Grant from the HPCMO, and by the US DOE through computer time at NERSC.

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Contributions

S.L., I.G. and A.M.R. designed and analysed the simulation approaches. S.L. performed the molecular dynamics simulations. All authors discussed the results and implications of the work and commented on the manuscript at all stages.

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Correspondence to Shi Liu or Andrew M. Rappe.

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Extended data figures and tables

Extended Data Figure 1 Large-scale molecular dynamics simulations of 90°-domain-wall motions.

a, Schematic of a 40 × 40 × 40 supercell with 90° domain walls used in molecular dynamics simulations. The colours of the domains correspond to the polarization (P) wheel shown at the bottom. White arrows represent the polarization directions of domains. b, Simulated domain evolution under a [100]-oriented electric field (E). The dashed yellow lines show the positions of 90° domain walls. The electric field is turned on at time t0. The domain-wall velocity vDW along [110] (yellow arrows) is estimated on the basis of the change in the supercell dimension (Lx) along [100] from t0 to t0 + Δt. The black arrows scale with the local dipole of each unit cell. The domain wall motion is achieved via the 90° switching of dipoles to dipoles.

Extended Data Figure 2 Lattice constants of supercells used in molecular dynamics simulations.

a, Pb (orange) and Ti (blue) sublattices in a PbTiO3 supercell with 90° domain walls. The boundaries are marked by green lines. aX and aY are effective lattice constants of the domain-wall unit cell defined in the transformed XY coordinates and shown by the red rectangle. When dipoles in one layer of unit cells switch by 90° (c → a), the wall moves by (a2 + c2)1/2/2 along the [110] direction. b, Temperature (T) dependence of obtained from molecular dynamics simulations (squares). It depends on temperature weakly (blue line). c, Plot of polarization change (dPx/dt) versus cell-dimension change (vx). The solid curves show linear fits at 100 K (blue) and 240 K (red).

Extended Data Figure 3 Elastic energy contribution to nucleation energy.

a, Effective lattice constants across 90° domain walls. The inset is the top view of the 40 × 40 × 40 supercell used in molecular dynamics simulations; black arrows indicate the polarization direction. The effective lattice constants (aX and aY) are defined in XY coordinates, as explained in Extended Data Fig. 1. The averaged lattice constants for each layer of cells across the domain wall along the [110] direction are plotted. b, c, Distributions of strain gradient at the domain wall in the presence of a nucleus. and are the effective lattice constants along Y and Z in the absence of nucleus (t = 0 ps in molecular dynamics simulations), respectively.

Extended Data Figure 4 Schematic of a triangular-shaped nucleus, as in the Miller–Weinreich model.

The triangular-shaped nucleus (red) has a polarization direction (white arrows) that is antiparallel to its neighbouring domains (blue). The depolarization charges ρ1,2 at two boundaries are of the same sign, giving rise to repulsive energy penalty. The expressions for nucleation energy (Unuc), depolarization energy (Ud), depolarization-contributed domain-wall energy (σp) and the dimensions for the critical nucleus a* and l* are taken from the original work of Miller and Weinreich, ref. 15; c and b are lattice constants (c ≈ b in PbTiO3 and BaTiO3), e is the base of natural logarithm, and ϵ is the dielectric constant. The σp/σw ratio determines the aspect ratio of the critical nucleus (l*/a*).

Extended Data Figure 5 Distributions of polarization gradient at the domain wall in the presence of a nucleus.

a, b, The polarization gradients (dPY/dY, a; dPZ/dZ, b) are highest at the boundary of the nucleus. The maximum polarization gradient is around 0.08 C m−2 Å−1, much smaller than the value estimated by the classical theories in ref. 15 (0.25 C m−2 Å−1). This difference is due to the diffuse nature of the boundary. The total boundary charge (ρ1 + ρ2 + ρ3 + ρ4) is zero.

Extended Data Figure 6 Results for the Miller–Weinreich model of nucleation on the PbTiO3 90° domain wall using various conditions for the interface boundary.

a, Nucleus energy U as a function of Miller–Weinreich nucleus area (al, given in terms of the number of unit cells (uc)) for the original Miller–Weinreich model (black) and Miller–Weinreich models with s = 0.41, fϵ = 1, Qtot ≠ 0 and fc = 1 (red), s = 0.41, fϵ = 2, Qtot ≠ 0 and fc = 1 (green), s = 0.41, fϵ = 2, Qtot = 0 and fc = 1 (blue), s = 0.41, fϵ = 2, Qtot = 0 and fc = 1/2 (magenta), and s = 0.41, fϵ = 2, Qtot = 0 and fc = 1/3 (cyan). Inset, zoomed-out view showing all the curves. b, Aspect ratio of the Miller–Weinreich nucleus (l*/a*) as a function of the ratio between σp and σw. The Miller–Weinreich assumption that l*a* is not valid for realistic values of σp and σw. c, σp for different interface conditions. The actual σp is much smaller than the estimate used by Miller and Weinreich (MW; ref. 15).

Extended Data Figure 7 Results for the Miller–Weinreich model of nucleation on the PbTiO3 180° domain wall using various conditions for the interface boundary.

a, Nucleus energy U as a function of Miller–Weinreich nucleus area (al, given in terms of the number of unit cells (uc)) for the original Miller–Weinreich model (black) and Miller–Weinreich models with s = 0.41, fϵ = 1, Qtot ≠ 0 and fc = 1 (red), s = 0.41, fϵ = 2, Qtot ≠ 0 and fc = 1 (green), s = 0.41, fϵ = 2, Qtot = 0 and fc = 1 (blue), s = 0.41, fϵ = 2, Qtot = 0 and fc = 1/2 (magenta), and s = 0.41, fϵ = 2, Qtot = 0 and fc = 1/3 (cyan). Inset, zoomed-out view showing all the curves. b, σp for different interface conditions. The actual σp is much smaller than the estimate used by Miller–Weinreich (MW; ref. 15).

Extended Data Figure 8 Test of domain-wall velocity (vx) convergence with supercell size.

The colours of the domains in the bottom panels correspond to those in Extended Data Fig. 1. The error bars are standard deviations.

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Liu, S., Grinberg, I. & Rappe, A. Intrinsic ferroelectric switching from first principles. Nature 534, 360–363 (2016). https://doi.org/10.1038/nature18286

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