Infrared nano-spectroscopy of ferroelastic domain walls in hybrid improper ferroelectric Ca3Ti2O7

Ferroic materials are well known to exhibit heterogeneity in the form of domain walls. Understanding the properties of these boundaries is crucial for controlling functionality with external stimuli and for realizing their potential for ultra-low power memory and logic devices as well as novel computing architectures. In this work, we employ synchrotron-based near-field infrared nano-spectroscopy to reveal the vibrational properties of ferroelastic (90\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^{\circ }$$\end{document}∘ ferroelectric) domain walls in the hybrid improper ferroelectric Ca\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_{3}$$\end{document}3Ti\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_{2}$$\end{document}2O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_{7}$$\end{document}7. By locally mapping the Ti-O stretching and Ti-O-Ti bending modes, we reveal how structural order parameters rotate across a wall. Thus, we link observed near-field amplitude changes to underlying structural modulations and test ferroelectric switching models against real space measurements of local structure. This initiative opens the door to broadband infrared nano-imaging of heterogeneity in ferroics.

(1) fully relaxed (2)  structures with symmetry A2 1 am as starting points for our lattice-dynamical calculations: (1) a structure that is fully relaxed (lattice parameters and atomic positions) within density functional theory (DFT), and (2)    to the totally symmetric A 1 modes, these projections are characteristic of the modes of the material -again not a function of the measurement method. We use these calculations to 3 guide our thinking.

Supplementary Note 4: Projection of phonon eigenvectors
The bulk Ca 3 Ti 2 O 7 structure with symmetry A2 1 am can be expressed as R A2 1 am = R I4/mmm + u, where R is a vector containing the atomic positions of all the atoms, and u is the displacement vector. The distortion u can be decomposed into symmetry adapted modes that transform like the irreducible representations of I4/mmm: Here e iτ is a symmetry-adapted mode that transforms like irreducible representation τ = {Γ + 1 , Γ − 5 , X − 3 , X + 2 } and i sums over the number of modes that transform like each irreducible representation τ . The coefficients are the overlaps A iτ = u · e iτ , note that these are the symmetry adapted mode amplitudes that are reported in Supplementary Table 1. In an analogous manner, the A 1 phonons can be decomposed into the same basis of symmetry (2) The overlaps A iτ = (e A 1 · e iτ ) are reported in Fig. 4 (a) in the main text.
Supplementary Table 2. Experimental and calculated phonon frequencies for The three right columns report the calculated phonon frequencies for structures (1) and (2) and their symmetry. The left three columns report the experimentally measured frequencies with Raman, infrared, and near field infrared (NFIR) spectroscopy.  Contour plot of the third harmonic near-field amplitude across DW 1 and 2. The color scales are slightly different due to the signal size. (c, d) Height profile (above) and fixed frequency cuts of the third harmonic near-field amplitude (below) as a function of distance across DW 1 and 2.

Supplementary Notes 5: Comparing second and third harmonic response
The synchrotron infrared nano spectroscopy setup enables simultaneous collection of higher harmonic data. Although the higher harmonic data is generally considered to have more sensitivity to the near-field, it also suffers from having lower signal-to-noise ratios. Prior work with this technique indicate that second harmonic detection is an appropriate compromise between the signal-to-noise and near-field sensitivity with good rejection of far-field artifacts as detailed by Supplementary Reference [4] in the paper. Within signal-to-noise limits, we generally find no (or very few) differences between the second and third harmonic signals 8 and thus believe the second harmonic channel to be mostly free from far-field artifacts. That said, we wanted to test this expectation with a direct comparison. Comparing contour plots in the main text with the data in Supplementary Figure 3 [5]. Unfortunately, the low frequency limit of the near-field setup (330 cm1) precludes following the behavior of the Ca-containing modes (which contribute most of the polarization amplitude) along with the octahedral rotations and tilts that are so central to the trilinear coupling mechanism [3,[6][7][8], although if needed, one could project the appropriate irreducible representations onto the higher frequency modes to gain some insight. We therefore decided to try a line scan across a ferroelectric wall. We identified candidate ferroelectric walls for analysis and near-field line scans using a combination of AFM, piezoforce microscopy, and a careful examination of the ridges and topography of the crystal surface as illustrated in Fig. 2(c, d). We were not, however, able to discern a unique infrared signature across any of the six or seven ferroelectric domain walls that we examined. This places clear constraints on the local lattice distortions at ferroelectric walls in Ca 3 Ti 2 O 7 and suggests that they are well localized -if not atomically thin -as previously supposed [5,9,10]. We estimate that the 180 • ferroelectric walls in Ca 3 Ti 2 O 7 are no more than 5 nm thick. This observation reinforces our interpretation that the ferroelastic walls are thick because, if they are not, we would not expect a signal from them -by analogy to the ferroelectric walls described here. This finding raises the interesting question of how two relatively thick ferroelastic walls combine to form such a thin, Néel-like ferroelectric wall [5].
Is there a unique cancellation of the structural distortions? Or does the interplay between charge and structure mitigate the charge divergence [5] and reduce the natural width of the structural distortion? More detailed near-field infrared imaging combined with other local probe techniques and theoretical modeling may be able to differentiate between these and other models.