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A pearl's distinguished beauty and toughness are attributable to the periodic stacking of aragonite tablets known as nacre. Nacre has naturally occurring mesoscale periodicity that remarkably arises in the absence of discrete translational symmetry. Gleaning the inspiring biomineral design of a pearl requires quantifying its structural coherence and understanding the stochastic processes that influence formation. By characterizing the entire structure of pearls (∼3 mm) in a cross-section at high resolution, we show that nacre has medium-range mesoscale periodicity. Self-correcting growth mechanisms actively remedy disorder and topological defects of the tablets and act as a countervailing process to long-range disorder. Nacre has a correlation length of roughly 16 tablets (∼5.5 µm) despite persistent fluctuations and topological defects. For longer distances (>25 tablets , ∼8.5 µm), the frequency spectrum of nacre tablets follows [Formula: see text] behavior, suggesting that growth is coupled to external stochastic processes-a universality found across disparate natural phenomena, which now includes pearls., Competing Interests: The authors declare no competing interest., (Copyright © 2021 the Author(s). Published by PNAS.)

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 ([Formula: see text]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 ([Formula: see text]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., Competing Interests: The authors declare no conflict of interest., (Copyright © 2018 the Author(s). Published by PNAS.)

Charge-density waves (CDWs) and their concomitant periodic lattice distortions (PLDs) govern the electronic properties in several layered transition-metal dichalcogenides. In particular, 1T-TaS 2 undergoes a metal-to-insulator phase transition as the PLD becomes commensurate with the crystal lattice. Here we directly image PLDs of the nearly commensurate (NC) and commensurate (C) phases in thin, exfoliated 1T-TaS 2 using atomic resolution scanning transmission electron microscopy at room and cryogenic temperature. At low temperatures, we observe commensurate PLD superstructures, suggesting ordering of the CDWs both in- and out-of-plane. In addition, we discover stacking transitions in the atomic lattice that occur via one-bond-length shifts. Interestingly, the NC PLDs exist inside both the stacking domains and their boundaries. Transitions in stacking order are expected to create fractional shifts in the CDW between layers and may be another route to manipulate electronic phases in layered dichalcogenides., Competing Interests: The authors declare no conflict of interest.

The layered transition metal dichalcogenides host a rich collection of charge density wave phases in which both the conduction electrons and the atomic structure display translational symmetry breaking. Manipulating these complex states by purely electronic methods has been a long-sought scientific and technological goal. Here, we show how this can be achieved in 1T-TaS2 in the 2D limit. We first demonstrate that the intrinsic properties of atomically thin flakes are preserved by encapsulation with hexagonal boron nitride in inert atmosphere. We use this facile assembly method together with transmission electron microscopy and transport measurements to probe the nature of the 2D state and show that its conductance is dominated by discommensurations. The discommensuration structure can be precisely tuned in few-layer samples by an in-plane electric current, allowing continuous electrical control over the discommensuration-melting transition in 2D.

Bilayer graphene has been a subject of intense study in recent years. The interlayer registry between the layers can have dramatic effects on the electronic properties: for example, in the presence of a perpendicular electric field, a band gap appears in the electronic spectrum of so-called Bernal-stacked graphene [Oostinga JB, et al. (2007) Nature Materials 7:151-157]. This band gap is intimately tied to a structural spontaneous symmetry breaking in bilayer graphene, where one of the graphene layers shifts by an atomic spacing with respect to the other. This shift can happen in multiple directions, resulting in multiple stacking domains with soliton-like structural boundaries between them. Theorists have recently proposed that novel electronic states exist at these boundaries [Vaezi A, et al. (2013) arXiv:1301.1690; Zhang F, et al. (2013) arXiv:1301.4205], but very little is known about their structural properties. Here we use electron microscopy to measure with nanoscale and atomic resolution the widths, motion, and topological structure of soliton boundaries and related topological defects in bilayer graphene. We find that each soliton consists of an atomic-scale registry shift between the two graphene layers occurring over 6-11 nm. We infer the minimal energy barrier to interlayer translation and observe soliton motion during in situ heating above 1,000 °C. The abundance of these structures across a variety of samples, as well as their unusual properties, suggests that they will have substantial effects on the electronic and mechanical properties of bilayer graphene.