Mapping Local Topology, Defects and Emergent Order of Picometer-Scale Periodic Lattice Modulations

In complex condensed matter systems, patchworks of order and disorder can form in competing order parameters, with localized pockets of emergent symmetry breaking which may be mere nanometers across. Measuring the real space variation of emergent order and its defects is therefore essential. By combining in-situ cryogenic (77K) high-angle annular dark-field scanning transmission electron microscopy, fast-acquisition low-SNR referenceless image registration with lattice-hop-error correction, picometer precision atomic column gaussian fitting, and isolation and reconstruction of both the underlying and symmetry broken lattices in Fourier space, it becomes possible to map modulations of the lattice parameter at each atomic site with picometer precision. Alternatively, real space measurement of periodic lattice modulations is possible using phase lock-in methods popular in scanning tunneling microscopy, geometric phase analysis more common in transmission electron microscopy, pairwise atomic distance histograms in one or two dimensions, and other methods. The advantages and drawbacks of these various approaches to uncovering the structure and topology of lattice symmetry breaking in real space will be discussed and compared, with examples drawn from complex oxides and transition metal dichalcogenides.