Bending and breaking of stripes in a charge ordered manganite

Abstract

In charge-ordered phases, broken translational symmetry emerges from couplings between charge, spin, lattice, or orbital degrees of freedom, giving rise to remarkable phenomena such as colossal magnetoresistance and metal-insulator transitions. The role of the lattice in charge-ordered states remains particularly enigmatic, soliciting characterization of the microscopic lattice behavior. Here we directly map picometer scale periodic lattice displacements at individual atomic columns in the room temperature charge-ordered manganite Bi_0.35Sr_0.18Ca_0.47MnO₃ using aberration-corrected scanning transmission electron microscopy. We measure transverse, displacive lattice modulations of the cations, distinct from existing manganite charge-order models. We reveal locally unidirectional striped domains as small as ~5 nm, despite apparent bidirectionality over larger length scales. Further, we observe a direct link between disorder in one lattice modulation, in the form of dislocations and shear deformations, and nascent order in the perpendicular modulation. By examining the defects and symmetries of periodic lattice displacements near the charge ordering phase transition, we directly visualize the local competition underpinning spatial heterogeneity in a complex oxide.

Fig. 1

Periodic lattice displacements in reciprocal space. a The perovskite structure of Bi_0.35Sr_0.18Ca_0.47MnO₃ (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

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

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

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

Fig. 5

Topological singularity coincident with a PLD grain boundary. a A complete Δ = Δ ₁ + Δ ₂ map of the interface between a Δ ₁-dominant region and coexisting Δ ₁ and Δ ₂ order. b A Δ ₁ map of the same region reveals a dislocation in Δ ₁, with a Burgers vector of one PLD wavelength, ${\lambda }_{\mathrm{PLD}}\widehat{{\mathbf{q}}_1}$. Analogous to the elastic deformation of an atomic lattice about a crystal dislocation, the elastic response of the PLD includes bending and compression of wavefronts and local displacement rotations. Some attenuation of Δ ₁ is apparent in the mixed region. c–e The phase ϕ ₁, |Δ ₁|, and |Δ ₂|, respectively, in the same region. Δ ₁ weakens and Δ ₂ grows within ~λ _PLD of the defect core, where ϕ ₁ exhibits an expected 2π winding. A narrow inlet of |Δ ₁| amplitude collapse extends from the upper left to the singularity. The scale bar corresponds to 2 nm