Experimental hallmarks of the charge-ordered state

The BSCMO orthorhombic perovskite lattice (space group Pnma, Fig. 1a) is imaged in projection along the b-axis with aberration-corrected high-angle annular dark-field (HAADF)-STEM (Fig. 2a), which is sensitive to the Coulomb potential of the atomic nuclei; heavier Bi/Sr/Ca atomic columns (A-sites) appear brighter than lighter Mn columns (B sites) in the Z-contrast image. Temperature-dependent resistivity and magnetic susceptibility measurements on the host BSCMO crystal reveal an anomaly associated with charge ordering at T c = 315 K and 318 K, respectively (Supplementary Figs. 1 and 2). Transport curves measured at zero field are nearly identical to those measured under application of a 2 T magnetic field, comparable to that of the microscope objective at the position of the specimen (Supplementary Fig. 1). Reflective polarized optical microscopy reveals approximately 100 μm twin domains (Supplementary Fig. 3); STEM and electron diffraction are performed within a single twin domain.

Fig. 1 Periodic lattice displacements in reciprocal space. a The perovskite structure of Bi 0.35 Sr 0.18 Ca 0.47 MnO 3 (BSCMO) and the projection of the unit cell along the b-axis. b Electron diffraction over a 1 μm selected area and the Fourier transform of a 30 nm field of view scanning transmission electron microscopy image of BSCMO along the b-axis. Satellite peaks corresponding to two transverse and displacive modulations with perpendicular wavevectors q 1 ≈ 1/3 a* and q 2 ≈ 1/3 c* are indicated by blue and red arrows, respectively. c, d Schematic of the Fourier transform of a square lattice (for simplicity) displaced by transverse modulations along x and y, respectively. The intensity of a satellite peak is reduced when its reciprocal vector, k = (k x , k y ), is not parallel to the modulation polarization A i and vanishes when k · A i = 0. e Stripe states contain locally unidirectional modulations, while checkerboard states contain overlapping bidirectional modulations. Both stripe and checkerboard order are consistent with the reciprocal space data, which reflects the spatially averaged structure and cannot definitively determine the local symmetry Full size image

Fig. 2 Mapping picometer scale, periodic displacements of atomic lattice sites. a High-angle annular dark-field scanning transmission electron microscopy projection image along the b-axis. The heavier (Bi, Sr, Ca) sites (green) appear brighter than the lighter Mn sites (red). b Mapping picometer scale periodic lattice displacements (PLDs) Δ 1 (r) at each atomic lattice site in response to a single modulation wavevector q 1 . PLD maps indicate a displacive modulation rather than an intensity modulation (cation order, charge disproportionation) with transverse polarization and 3a periodicity. Triangles represent displacements, with the area scaling linearly with displacement amplitude. The color represents the angle of the polarization vector, A 1 , relative to the modulation wavevector, q 1 , where blue (yellow) correspond to 90° (−90°) as indicated in the colorbar. c Map of Δ 2 (r) displacements at each atomic lattice site in response to q 2 in the same region as a, b. The significantly weaker Δ 2 (r) response is characteristic of locally striped, rather than checkerboard, ordering. The scale bar corresponds to 1 nm Full size image

Electron diffraction (Fig. 1b) shows a constellation of satellite peaks indicating two transverse, displacive PLDs (Fig. 1c, d) offsetting the atomic lattice with displacements

$${\bf{\Delta }}_i({\bf{r}}) = {\bf{A}}_i{\kern 1pt} {\mathrm{sin}}\left( {{\bf{q}}_i \cdot {\bf{r}} + \phi _i} \right),\quad i \in \left\{ {1,2} \right\}$$ (1)

where A i , q i , and ϕ i are the PLD amplitude vector, wavevector, and phase, respectively, and \(\left| {{\bf{q}}_i} \right| \approx \frac{1}{3}\) reciprocal lattice units (Supplementary Figs. 4 and 5). Note that valence modulations have been found to be minimal here (Supplementary Fig. 6) and elsewhere15, therefore the state giving rise to the observed satellite peaks and accompanying the resistivity anomaly is referred to empirically as the charge-ordered or CDW state, agnostic to a particular underlying model. Field-free electron diffraction, with the objective lens turned off, showed no discernible changes in the superlattice structure (Supplementary Fig. 7), consistent with the magnetic field-dependent resistivity measurements and suggesting the charge-ordered state is robust to the applied magnetic field. Diffraction shows coexistence of the two orthogonal PLDs within a 1 μm selected area. A STEM Fourier transform (Fig. 1b) shows coexistence within a 30 nm field of view. In order to further investigate the local PLD structure, we extract the displacement vectors associated with each of the two modulations at every atomic site to generate the PLD maps shown in Fig. 2.

Local structure of periodic lattice displacements

To calculate the PLD fields Δ i (r) shown in Fig. 2, we first fit all atomic positions in our STEM data with ~2 picometer precision, an approach which has recently emerged as a powerful, quantitative characterization tool22,23,24. However, in contrast to prior STEM atom tracking work, the key challenge in mapping PLDs is defining an appropriate reference lattice, which is complicated by the presence of local PLD phase variations and multiple interpenetrating modulations. Our approach generates a reference image in which the contribution of a single modulation has been selectively removed, by damping all of the relevant satellite peaks from the Fourier transform of the original image. Fitting and subtracting corresponding lattice positions from the image pair yields Δ i (r) quantitatively. Damping the q 1 satellite peaks (Fig. 1b, c, blue arrows) generates a map of Δ 1 (r) (Fig. 2b), while damping the q 2 satellite peaks (Fig. 1b, d, red arrows) maps Δ 2 (r) (Fig. 2c). Simulations indicate that our method accurately reconstructs the PLD structure everywhere except at lattice sites directly adjacent to atomically sharp discontinuities in the PLD field. Analytical and algorithmic details, simulations, and error analysis are found in Supplementary Note 1 and Supplementary Figs. S8–15.

The microscopic structure of charge-ordered phases in manganites remains contested16,17,18,19; here the Δ 1 (r) map in Fig. 2b furnishes real-space evidence for displacive lattice modulations of both the Bi/Sr/Ca sites and the Mn sites, with respective amplitudes of 6.2 and 8.2 pm on the maximal sites (see Supplementary Fig. 16). The displacements are transverse to the modulation wavevector and generate a tripled unit cell. The historically prevailing model conjectures the localization and ordering of Mn3+–Mn4+ ions, which in turn activates an alternating compression and expansion of oxygen octehedra (Jahn–Teller effect)13. Other works propose the formation of Mn pairs (Zener polarons) with minimal valence modulations15,16. Our data suggest a different model. The strong structural modulation shown in Fig. 2b is consistent with the softening of a phonon mode, and the pattern of displacements provides a structural model to further investigate the microscopic origin of the modulated state.

The superposition of multiple modulations can further mask the underlying microscopic mechanism behind PLD formation. For instance, distinguishing overlapping modulations (checkerboards) from spatially anti-correlated unidirectional domains (stripes) is essential but challenging, as both have the same spatially averaged symmetry (Fig. 1b–d)14,20,21,25,26. Our data clearly indicates that locally, BSCMO forms striped states: where one PLD is suppressed, the other is strong, starkly illustrated in the Δ 1 (r) and Δ 2 (r) maps of identical regions in Fig. 2b, c.

Zooming out, Fig. 3a maps the combined displacement field Δ(r) = Δ 1 (r) + Δ 2 (r) over a 30 nm field of view, in which a Δ 1 -dominant region, readily identified by its transverse polarization relative to q 1 (blue/yellow triangles), occupies the right side of the frame, while a Δ 2 (r)-dominant region occupies the upper left corner (red/green triangles). Mapping the displacement magnitudes |Δ 1 (r)| and |Δ 2 (r)| visualizes the striped domain structure, revealing a complex domain morphology with islands of strong modulations (6–11 pm) and basins of PLD suppression (0–3 pm) (Fig. 3b, c). Notably, regions in which both Δ 1 (r) and Δ 2 (r) are present are also observed, such as the bottom left corner of Fig. 3a–c. Quenched disorder tends to broaden phase transitions and favors enhanced isotropy in the nascent-ordered state, and theoretically has been shown to induce apparent fourfold symmetry in 2D striped phases25,26,27. We believe the checkerboard-like regions we observe may result from quenched disorder; varying intensity of atomic columns clearly indicates frozen cation disorder in our data (Supplementary Fig. 17). Alternatively, checkerboard-like ordering could result from projection through stacked Δ 1 (r) and Δ 2 (r) domains in the out-of-plane (b-axis) direction. In either case, the two modulations are predominantly anti-correlated in our data, and we conclude that the symmetry breaking in the disorder-free “clean” limit in this system is very likely striped.

Fig. 3 Nanoscale domain structure and local symmetry of periodic lattice displacement (PLD) stripes. a Combined PLD map showing the displacements Δ(r) = Δ 1 (r) + Δ 2 (r) at all ~9000 atomic sites in the 30 nm field of view. Colors indicates the displacement polarizations relative to q 1 following the colorbar in Fig. 2, and triangle areas scale linearly with the displacement magnitudes. b, c Maps of the magnitudes |Δ 1 (r)| and |Δ 2 (r)| of the displacements due to each PLD individually reveals that the two PLD strengths are anti-correlated: when one is strong, the other is weak. The PLDs are stripe ordered, segregated into nanoscopic domains. The regions indicated by white delimiters contain local defect structures, which are further analyzed in Figs. 4 and 5. The scale bars correspond to 4 nm Full size image

Nascent order coincident with PLD defects

CDW domain nucleation near T c remains a poorly understood process, particularly in the presence of disorder27,28. We observe PLD defects coincident with both domain boundaries and nascent domain structures, suggesting their involvement in mediating domain growth and termination. Figure 4 magnifies the region containing a ~5 nm island of Δ 2 order embedded in a Δ 1 domain (Fig. 3, upper white delimiters). Inspection of the Δ 1 + Δ 2 map (Fig. 4a) reveals shearing in Δ 1 as it passes through the Δ 2 island, evident in the offset of the wavefronts by ~2 atomic rows. Mapping Δ 1 only (Fig. 4b) accentuates the shear deformation, and exposes Δ 1 attenuation in the strained region, along with rotation of the displacement vectors to roughly align with the local wavefront orientation. To quantify these observations, we map the elastic shear strain field, ε s (r), reflecting local bending in the Δ 1 PLD, along with the magnitudes of the two modulations |Δ 1 | and |Δ 2 | (Fig. 4c–e). ε s (r) is calculated by extracting the local PLD phase (ϕ → ϕ(r) in Eq. 1)29 then computing \(\varepsilon _s({\bf{r}}) = \frac{1}{2}\frac{{\widehat {{\bf{q}}_ \bot }}}{{\left| {\bf{q}} \right|}} \cdot

abla \phi ({\bf{r}})\) (see Supplementary Note 2)30,31. The shear defect plainly coincides with abatement of Δ 1 , and strengthening of Δ 2 .

Fig. 4 Shear deformation coincident with a nascent periodic lattice displacement (PLD) grain. a A complete Δ = Δ 1 + Δ 2 map of a ~5 nm region of incipient Δ 2 order, and a coinciding shearing of the Δ 1 modulation. b A Δ 1 map of the same region highlights the bending wavefronts, and reveals attenuation of the PLD amplitude and some rotation of the displacement vectors in the defective region. c–e The shear strain ε s , |Δ 1 |, and |Δ 2 |, respectively, in the same region. The maximal shearing aligns with attenuation of Δ 1 and emergence of Δ 2 . The scale bar corresponds to 2 nm Full size image

Figure 5 magnifies a domain boundary (Fig. 3, lower white delimiters). Exclusive Δ 1 order occupies the right side of the frame in Fig. 5a, while the displacements to the left suggest an intricate interweaving of the two modulations. Mapping Δ 1 only (Fig. 5b) reveals a prominent dislocation in the PLD, in which a single wavefront abruptly terminates. Analogous to edge dislocations in crystalline solids, where the abrupt termination of a row of atoms is accompanied by elastic deformation in the surrounding lattice, we observe elastic deformation of the PLD about the singularity, evident in the warped wavefronts flanking the dislocation core. No defects in the underlying lattice are observed (Supplementary Fig. 17), and the PLD phase ϕ(r) exhibits an expected 2π winding about the discontinuity (Fig. 5c). The interface between the Δ 1 -dominant domain and the mixed region occurs within a single PLD wavelength of the defect core, as once again disorder in one modulation accompanies commencement of order in the other. Maps of the PLD magnitudes |Δ 1 | and |Δ 2 | (Fig. 5d, e) reinforce these observations. Moreover, theory predicts modulation amplitude collapse at singularities to prevent divergence of the energy density, and the |Δ 1 | map exhibits a narrow inlet of collapsed amplitude extending from the upper left to the defect core, suggesting complex domain restructuring to accommodate the high-energy feature30,31,32,33. While displacements at atomic sites directly adjacent to a true singularity will not be accurately reconstructed, we believe the displacements extracted by our method are valid everywhere, because damping and distortion in the defect’s central region yields reasonably smooth variations of the displacements (see Supplementary Note 1 and Supplementary Figs. S10, S12, and S13).