Diffuse X-ray scattering unveils the nature of nanoscale ferroelectric domain walls

Ferroelectric materials exhibit intrinsic switchable electric polarization along their polar axis, which is typically constrained by symmetry and strain coupling. Historically, ferroelectric domain walls, which separate uniformly polarized regions, were considered to be exclusively of Ising-type, lacking the complexity of Néel- and Bloch-type walls found in ferromagnets (Figure 1a) [1].

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Fig. 1: Equilibrium ferroelectric domain patterns in PbTiO₃. a) Schematic illustrations of Ising-, Bloch- and Néel-type domain walls. Arrows represent the directions of spontaneous polarization. The grey square represents the plane of the domain wall in all cases, while the white square(s) in the Bloch and Néel cases denote the planes of polarization rotation. b) Predicted polarization arrangement in the PbTiO₃ layer of a PbTiO₃/SrTiO₃ superlattice for different domain periods (Λd, given in unit cells). Arrows indicate the direction and magnitude of polarization in the x-z plane, with the background colour representing the y-component of polarization (i.e., the Bloch component). c) Simulation results for bulk PbTiO₃, analogous to the configuration in (b).

However, over the past two decades, theoretical advances revealed that polarization rotations may occur within ferroelectrics, challenging conventional assumptions. These predictions have now been experimentally validated in various ferroelectric systems, leading to the discovery of unconventional polarization patterns such as flux-closure configurations, polar skyrmions, merons, and supercrystals [2].

More specifically, atomistic simulations have proposed that domain walls in the archetypal ferroelectric lead titanate (PbTiO₃) can exhibit a switchable Bloch polarization, and behave as nanoscale ferroelectrics in their own right [3]. These Bloch components are proposed to exhibit switchable chirality, opening up potential applications in nanoelectronics and other technologies. However, experimental confirmation of Bloch-type domain walls has remained challenging due to the nanoscale dimensions and complexity of the structures involved.

In this study, the internal structure of ferroelectric domain walls in PbTiO₃ /SrTiO₃ superlattices was investigated using a combination of diffuse X-ray scattering and state-of-the-art theoretical methods. These superlattices, composed of ultrathin layers of ferroelectric PbTiO₃ and dielectric SrTiO₃, have proven to be excellent systems for studying domain wall behaviour using X-ray diffraction [4] and are central to many experimental reports on unconventional polarization textures [5].

Using diffuse X-ray scattering techniques at beamlines ID28 and ID01, researchers probed the nanoscale structure of domain walls in these superlattices, with the experimental data complemented by first- and second-principles calculations, phase-field simulations, and diffuse scattering modelling. 

Theoretical modelling of bulk PbTiO₃ using a realistic generalized gradient approximation confirmed that ferroelectric domain walls can exhibit Bloch polarization, characterized by significant displacements of Pb atoms. This polarization is expected to emerge between 150 K and 200 K, consistent with a ferroelectric-to-paraelectric transition confined at the domain wall.

However, in superlattices where ultrathin PbTiO₃ layers are sandwiched between SrTiO₃, the behaviour differs significantly. While periodic ferroelectric domains still form, the polarization at domain walls and at interfaces rotates perpendicular to the PbTiO₃ surface, creating flux-closure configurations that minimize depolarization fields (Figure 1b). Notably, Bloch components appear only in domains much smaller than those predicted at equilibrium (Figure 1b), in contrast to bulk PbTiO₃, where Bloch polarization remains size-independent (Figure 1c). This disappearance is attributed to interactions between the Bloch component and the flux-closure arrangement, which dominates in superlattices. 

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Fig. 2: Diffuse X-ray diffraction signatures of Bloch domain walls. Each column represents a different set of diffraction calculations or measurements, as indicated by the column titles. Individual plots are labelled with the indices of the calculated or measured reciprocal space planes (e.g., HK1) or the specific peaks probed (e.g., 0-11). Insets in the leftmost two columns illustrate the domain patterns in the x-y plane, corresponding to the theoretical models used in the diffraction simulations.

These findings indicate that Bloch polarization in PbTiO₃ is highly sensitive to boundary conditions and is unlikely to occur in PbTiO₃/SrTiO₃ superlattices in equilibrium. To test this experimentally, diffuse X-ray scattering was utilized. Phase-field modelling first simulated the equilibrium domain structure of a PbTiO₃/SrTiO₃ superlattice, confirming the presence of flux-closure patterns without Bloch components. A second model was then created with Bloch components for comparison. Simulated diffraction patterns showed distinct features, with HK0 reflections sensitive to Bloch components and HK1 reflections capturing flux-closure arrangements (Figure 2).

Diffuse X-ray scattering experiments at ID28, conducted on superlattices of varying domain periods and layer thicknesses, clearly showed flux-closure signatures in HK1 peaks but no evidence of Bloch components in HK0 peaks (Figure 2). High-resolution measurements at ID01 down to 2.2K confirmed these results, with no signs of Bloch components or domain wall transitions (Figure 2). 

In conclusion, PbTiO₃/SrTiO₃ superlattices exhibit flux-closure domain patterns without Bloch-type walls. This work highlights the importance of boundary conditions and demonstrates that diffuse X-ray scattering is a powerful technique for probing nanoscale polarization textures in ferroelectrics.

 
Principal publication and authors
Assessing the ubiquity of Bloch domain walls in ferroelectric lead titanate superlattices, E. Zatterin (a), P. Ondrejkovic (b), L. Bastogne (c), C. Lichtensteiger (d), L. Tovaglieri (d), D. A. Chaney (a), A. Sasani (c), T. Schülli (a), A. Bosak (a), S. Leake (a), P. Zubko (e,f), P. Ghosez (c), J. Hlinka (b), J.-M. Triscone (d), M. Hadjimichael (g), Phys. Rev. X 14, 041052 (2024); https://doi.org/10.1103/PhysRevX.14.041052
(a) ESRF 
(b) Institute of Physics of the Czech Academy of Sciences, Prague (Czech Republic)
(c) University of Liege, Sart-Tilman (Belgium)
(d) University of Geneva, Geneva (Switzerland)
(e) University College London, London (UK)
(f) London Centre for Nanotechnology, London (UK)
(g) University of Warwick, Coventry (UK) 

References
[1] G. Catalan et al., Rev. Mod. Phys. 84, 119-156 (2012).
[2] M. Hadjimichael et al., Nat. Mater. 20(4), 495-502 (2021). 
[3] J.C. Wojdeł & J. Íñiguez, Phys. Rev. Lett. 112(24), 247603 (2014).
[4] M. Hadjimichael et al., Phys. Rev. Lett. 120, 037602 (2018).
[5] S. Das et al., Nature 568(7752), 368-372 (2019).