Fractal Characterization of Heat and Mass Transfer Processes in Nanostructured Materials

The relationship between the fractal self-organization of structural-phase inhomogeneity in nanostructured materials and the processes of heat and mass transfer in them is considered. It is characterized by a non-Euclidean relationship between the rate of increase in the number of the considered elements of the medium and an increase in the scale of their consideration and is described by the transfer equations in fractional derivatives with respect to coordinate and time.

Nature and evolution of incommensurate charge order in manganites visualized with cryogenic scanning transmission electron microscopy

Incommensurate charge order in hole-doped oxides is intertwined with exotic phenomena such as colossal magnetoresistance, high-temperature superconductivity, and electronic nematicity. Here, we map, at atomic resolution, the nature of incommensurate charge–lattice order in a manganite using scanning transmission electron microscopy at room temperature and cryogenic temperature (∼93 K). In diffraction, the ordering wave vector changes upon cooling, a behavior typically associated with incommensurate order. However, using real space measurements, we discover that the ordered state forms lattice-locked regions over a few wavelengths interspersed with phase defects and changing periodicity. The cations undergo picometer-scale (∼6 pm to 11 pm) transverse displacements, suggesting that charge–lattice coupling is strong. We further unearth phase inhomogeneity in the periodic lattice displacements at room temperature, and emergent phase coherence at 93 K. Such local phase variations govern the long-range correlations of the charge-ordered state and locally change the periodicity of the modulations, resulting in wave vector shifts in reciprocal space. These atomically resolved observations underscore the importance of lattice coupling and phase inhomogeneity, and provide a microscopic explanation for putative “incommensurate” order in hole-doped oxides.