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# Inverse transition of labyrinthine domain patterns in ferroelectric thin films

## Abstract

Phase separation is a cooperative process, the kinetics of which underpin the orderly morphogenesis of domain patterns on mesoscopic scales1,2. Systems of highly degenerate frozen states may exhibit the rare and counterintuitive inverse-symmetry-breaking phenomenon3. Proposed a century ago4, inverse transitions have been found experimentally in disparate materials, ranging from polymeric and colloidal compounds to high-transition-temperature superconductors, proteins, ultrathin magnetic films, liquid crystals and metallic alloys5,6, with the notable exception of ferroelectric oxides, despite extensive theoretical and experimental work on the latter. Here we show that following a subcritical quench, the non-equilibrium self-assembly of ferroelectric domains in ultrathin films of Pb(Zr0.4Ti0.6)O3 results in a maze, or labyrinthine pattern, featuring meandering stripe domains. Furthermore, upon increasing the temperature, this highly degenerate labyrinthine phase undergoes an inverse transition whereby it transforms into the less-symmetric parallel-stripe domain structure, before the onset of paraelectricity at higher temperatures. We find that this phase sequence can be ascribed to an enhanced entropic contribution of domain walls, and that domain straightening and coarsening is predominantly driven by the relaxation and diffusion of topological defects. Computational modelling and experimental observation of the inverse dipolar transition in BiFeO3 suggest the universality of the phenomenon in ferroelectric oxides. The multitude of self-patterned states and the various topological defects that they embody may be used beyond current domain and domain-wall-based7 technologies by enabling fundamentally new design principles and topologically enhanced functionalities within ferroelectric films.

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## Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## Code availability

The codes that are used in this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

We acknowledge DARPA grant number HR0011727183-D18AP00010 (TEE programme), ARO grant number W911NF16-1-0227 and DARPA grant number HR0011-15-2-0038 (MATRIX programme). Computations were made using the Arkansas High Performance Computing Center and the Arkansas Economic Development Commission. B.X. acknowledges the startup fund from Soochow University and support from Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions. S. Prosandeev appreciates the support of RMES grant number 3.1649.2017/4.6 and RFBR grant number 18-52-00029_Bel_a. V.G., S.F. and B.D. acknowledge a public grant overseen by the French National Research Agency (ANR) as part of the Investissements d’Avenir programme (reference: ANR-10-LABX-0035, Labex NanoSaclay) and the project EXPAND through ANR-17-CE24-0032, as well as through the PIAF project.

## Author information

Authors

### Contributions

Y.N. conceived the study of the inverse-transition phenomenon as part of a research project about labyrinthine structures initiated by L.B. Y.N. and S. Prokhorenko carried out the simulations and analysed the data. J.F. fabricated the thin films by pulsed-laser deposition, carried out the annealing experiments with the help of C.C., and performed the PFM measurements. B.X. and S. Prosandeev performed additional BiFeO3 simulations. S.F. performed the conducting atomic force microscopy experiments. This experimental work was done under the guidance of V.G. B.D. performed XRD measurements. After a first draft written by Y.N., all authors discussed the results and contributed to the final manuscript.

### Corresponding author

Correspondence to Y. Nahas.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature thanks Anna Morozovska and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Temperature evolution of the parallel-stripe ground state.

af, The evolution with temperature of the ground-state dipolar configuration (parallel stripes) in the middle layer of an 80 × 80 × 5 film of Pb(Zr0.4Ti0.6)O3 upon increasing temperature (10 K (a), 110 K (b), 185 K (c), 260 K (d), 335 K (e) and 410 K (f)), where grey (red) dipoles are oriented along the [001] $$([00\bar{1}])$$ pseudo-cubic direction). g, The temperature variation of the scaled structure factor $$\tilde{S}(a{q}_{s},T)$$. Vertical dashed line indicates the inflection point of $$\tilde{S}(a{q}_{{s}},T)$$ and is taken as the locus of Tc.

### Extended Data Fig. 2 Specific heat of the parallel-stripe and labyrinthine states.

Specific heat C as a function of temperature (in arbitrary units). Data were gathered upon slowly heating the ground-state parallel-stripe domain pattern (1) and the labyrinthine domain pattern (2).

### Extended Data Fig. 3 Spatial distribution of on-site, first and second nearest neighbours and dipole–dipole interaction energies.

ad, The probability density functions of the cell-by-cell energies (on-site energy (a), first nearest neighbours (1NN) interaction energy (b), second nearest neighbours (2NN) interaction energy (c) and dipole–dipole interaction energy (d)) calculated for the labyrinthine domain structure at 10 K for Pb(Zr0.4Ti0.6)O3 within a 64 × 64 × 5 supercell. Each panel provides the contributions stemming from the domains and domain walls, separately. eh, The corresponding mappings of energies onto the middle layer of the film. Blue to red colour gradient shows increasing values of unit-cell energies.

### Extended Data Fig. 4 Spatial distribution of third nearest neighbours, elastic and electrostrictive energies.

ac, The probability density functions of the cell-by-cell energies (third nearest neighbours (3NN) interaction energy (a), elastic energy (b) and electrostrictive energy (c)) calculated for the labyrinthine domain structure at 10 K for Pb(Zr0.4Ti0.6)O3 within a 64 × 64 × 5 supercell. Each panel provides the contributions stemming from the domains and domain walls, separately. df, The corresponding mappings of energies onto the middle layer of the film. Blue to red colour gradient shows increasing values of unit-cell energies.

### Extended Data Fig. 5 Energetics and spatial correlations at play in the inverse transition.

a, Evolution with temperature of dipole–dipole energy density upon heating the ground-state parallel-stripe domain pattern (1) and the labyrinthine domain pattern (2). These two curves meet above 200 K, the temperature at which the inverse transition occurs. The third curve (3) corresponds to what would have been the dipole–dipole energy of domain walls if the labyrinthine domain walls would have gradually wiggled with no reordering of the stripes (fictive labyrinthine evolution). b, Evolution with temperature of the typical size of locally ordered ground-state tiles composing the labyrinthine domain pattern. Data were obtained via the analysis of structure factors of square patches of varying size at each temperature. Specifically, at each temperature, ξ corresponds to the maximal patch size featuring two-peaked structure factor. Solid line is a guide for the eyes.

### Extended Data Fig. 6 Simulations of the inverse transition in thick BiFeO3 films.

a, b, The evolution with temperature of the domain pattern in BiFeO3 in terms of the distribution of the ferroelectric (a) and AFD (b) order parameters. Results were obtained through Monte Carlo simulations using the effective Hamiltonian scheme of a 36 × 36 × 10 film subjected to a −0.16% misfit strain, with periodic boundary conditions. The system was abruptly quenched from 2,000 K down to 10 K and consequently progressively heated up with 40,000 relaxation sweeps at each temperature. It can be seen that the distributions of both ferroelectric and AFD order parameters exhibit the inverse transition with Tinv ≈ 1,100 K and Tc ≈ 1,300 K (these numerically predicted temperatures are in good agreement with our experimental findings). We find that below Tinv, the system exhibits mixed 109° and 71° domain walls, while above Tinv, only 109° domain walls are observed. In a, dipoles are coloured according to their z component. In b, AFD vectors are coloured according to the arctan(Wy/Wx), where Wy and Wx denote the y and x components of the AFD local vectors.

### Extended Data Fig. 7 Simulations of the inverse transition in thin BiFeO3 films.

a, b, The evolution with temperature of the domain pattern in BiFeO3 in terms of the distribution of the dipolar (a) and AFD (b) order parameters. Results were obtained through Monte Carlo simulations using the effective Hamiltonian scheme of a 36 × 36 × 5 film subjected to a −0.5% misfit strain with open boundary conditions, a partial screening at film interfaces (effective screening parameter β = 0.5). The system was abruptly quenched from 2,000 K down to 10 K and consequently progressively heated up with 40,000 relaxation sweeps at each temperature. It can be seen that the distributions of both ferroelectric and AFD order parameters exhibit the inverse transition with Tinv ≈ 525 K and Tc ≈ 650 K. We find that the system exhibits 71° domain walls. In a, dipoles are coloured according to their z component. In b, AFD vectors are coloured according to the arctan(Wy/Wx), where Wy and Wx denote the y and x components of the AFD local vectors.

### Extended Data Fig. 8 Origin of the memory effect.

ad, Structure factor plots (a, b) and bubble configurations (c, d). Panels a and c correspond to the bubble state at 10 K, as obtained upon applying a field of 40 × 107 V m−1 to the parallel-stripe configuration. Panels b and d correspond to the bubble state at 10 K, as obtained upon applying a field of 40 × 107 V m−1 to the labyrinthine configuration.

### Extended Data Fig. 9 Imaging of the domain structure evolution with temperature in the BiFeO3 sample.

al, Topography, in-plane PFM phase and amplitude of the as-grown sample (ac), and the same sample after annealing at 773 K (df), 1,023 K (gi) and 1,073 K (jl). Each annealing step was 1 h long. ‘z-scale’ corresponds to 4 nm (a, d, g) and 10 nm (j).

### Extended Data Fig. 10 Structural properties of the BiFeO3 sample before and after annealing.

2θω XRD patterns of the as-grown BiFeO3 sample and the same sample after the successive annealing up to 1,073 K. a, Full scale. b, Zoom around the (001) peak.

### Extended Data Fig. 11 Evolution with temperature of the lattice parameter of the BiFeO3 sample.

Evolution of the out-of-plane parameter upon heating the parallel-stripe phase of the BiFeO3 sample. Values were obtained by fitting the XRD data and do not reveal any phase transition up to 1,160 K.

### Extended Data Fig. 12 Ferroelectric and elastic domains in the BiFeO3 sample.

Ferroelectric and elastic domain structures in a BiFeO3 thin film grown on a (110)-oriented DyScO3 substrate before and after annealing. a, b, In-plane PFM phase images of a BiFeO3 thin film for an as-grown sample (a) and a sample after annealing at 1,073 K for 1 h (b). Images are 2 × 2 μm2. c, d, Reciprocal space mappings around (002) reflections for the same BiFeO3 thin film for the as-grown sample (c) and the sample after annealing at 1,073 K for 1 h (d). The pink arrows indicate the satellite positions to the left and right of the (002) film peak. The X-ray beam is aligned at Φ = 90°, that is, perpendicular to the stripes. The indices of DyScO3 and BiFeO3 are written in the monoclinic cells. Qx,y and Qz indicate the in-plane and out-of-plane reciprocal space units, respectively.

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Nahas, Y., Prokhorenko, S., Fischer, J. et al. Inverse transition of labyrinthine domain patterns in ferroelectric thin films. Nature 577, 47–51 (2020). https://doi.org/10.1038/s41586-019-1845-4

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