Iterative Phase Retrieval Algorithms for Scanning Transmission Electron Microscopy

Scanning transmission electron microscopy (STEM) has been extensively used for imaging complex materials down to atomic resolution. The most commonly employed STEM imaging modality of annular dark field produces easily-interpretable contrast, but is dose-inefficient and produces little to no contrast for light elements and weakly-scattering samples. An alternative is to use phase contrast STEM imaging, enabled by high speed detectors able to record full images of a diffracted STEM probe over a grid of scan positions. Phase contrast imaging in STEM is highly dose-efficient, able to measure the structure of beam-sensitive materials and even biological samples. Here, we comprehensively describe the theoretical background, algorithmic implementation details, and perform both simulated and experimental tests for three iterative phase retrieval STEM methods: focused-probe differential phase contrast, defocused-probe parallax imaging, and a generalized ptychographic gradient descent method implemented in two and three dimensions. We discuss the strengths and weaknesses of each of these approaches using a consistent framework to allow for easier comparison. This presentation of STEM phase retrieval methods will make these methods more approachable, reproducible and more readily adoptable for many classes of samples.Comment: 25 pages, 11 figures, 1 tabl

Mesoscopic finite-size effects of unconventional electron transport in PdCoO₂

A wide range of unconventional transport phenomena has recently been observed in single-crystal delafossite metals. Here, we present a theoretical framework to elucidate electron transport using a combination of first-principles calculations and numerical modeling of the anisotropic Boltzmann transport equation. Using PdCoO2 as a model system, we study different microscopic electron and phonon scattering mechanisms and establish the mean free path hierarchy of quasiparticles at different temperatures. We treat the anisotropic Fermi surface explicitly to numerically obtain experimentally-accessible transport observables, which bridge between the “diffusive,” “ballistic,” and “hydrodynamic” transport regime limits. We illustrate that the distinction between the “quasiballistic” and “quasihydrodynamic” regimes is challenging and often needs to be quantitative in nature. From first-principles calculations, we populate the resulting transport regime plots and demonstrate how the Fermi surface orientation adds complexity to the observed transport signatures in micrometer-scale devices. Our work provides key insights into microscopic interaction mechanisms on open hexagonal Fermi surfaces and establishes their connection to the macroscopic electron transport in finite-size channels

Anisotropic phonon-mediated electronic transport in chiral Weyl semimetals

The discovery and observations of exotic, quantized optical and electrical responses have sparked renewed interest in nonmagnetic chiral crystals. Within this class of materials, six group V transition metal ditetrelides, that is, XY2 (X = V, Nb, Ta and Y = Si, Ge), host composite Weyl nodes on high-symmetry lines, with Kramers-Weyl fermions at time-reversal invariant momenta. In addition, at least two of these materials, NbGe2 and NbSi2, exhibit superconducting transitions at low temperatures. The interplay of strong electron-phonon interactions and complex Fermi-surface topology presents an opportunity to study both superconductivity and hydrodynamic electron transport in these systems. Towards this broader question, we present an ab initio theoretical study of the electronic transport and electron-phonon scattering in this family of materials, with a particular focus on NbGe2 vs NbSi2, and the other group V ditetrelides. We shed light on the microscopic origin of NbGe2's large and anisotropic room-temperature resistivity and contextualize its strong electron-phonon scattering with a presentation of other relevant scattering lifetimes, both momentum relaxing and momentum conserving. Our work explores the intriguing possibility of observing hydrodynamic electron transport in these chiral Weyl semimetals

Electron hydrodynamics in anisotropic materials

Rotational invariance strongly constrains the viscosity tensor of classical fluids. When this symmetry is broken in anisotropic materials a wide array of novel phenomena become possible. We explore electron fluid behaviors arising from the most general viscosity tensors in two and three dimensions, constrained only thermodynamics and crystal symmetries. We find nontrivial behaviors in both two- and three-dimensional materials, including imprints of the crystal symmetry on the large-scale flow pattern. Breaking time-reversal symmetry introduces a non-dissipative Hall component to the viscosity tensor, and while this vanishes for 3D isotropic systems we show it need not for anisotropic materials. Further, for such systems we find that the electronic fluid stress can couple to the vorticity without breaking time-reversal symmetry. Our work demonstrates the anomalous landscape for electron hydrodynamics in systems beyond graphene, and presents experimental geometries to quantify the effects of electronic viscosity. In some materials electrons can behave hydrodynamically, exhibiting phenomena associated with classical viscous fluids. In this theory work, the authors show that the symmetries of the crystal lattices in which the electrons reside can lead to additional unique hydrodynamic effects

Generalized design principles for hydrodynamic electron transport in anisotropic metals

Interactions of charge carriers with lattice vibrations, or phonons, play a critical role in unconventional electronic transport of metals and semimetals. Recent observations of phonon-mediated collective electron flow in bulk semimetals, termed electron hydrodynamics, present new opportunities in the search for strong electron-electron interactions in high carrier density materials. Here we present the general transport signatures of such a second-order scattering mechanism, along with analytical limits at the Eliashberg level of theory. We study electronic transport, using ab initio calculations, in finite-size channels of semimetallic ZrSiS and TaAs2 with and without topological band crossings, respectively. The order of magnitude separation between momentum-relaxing and momentum-conserving scattering length scales across a wide temperature range make both of them promising candidates for further experimental observation of electron hydrodynamics. More generally, our calculations suggest that the hydrodynamic transport regime does not, to first order, rely on the topological nature of the bands. Finally, we discuss general design principles guiding future search for hydrodynamic candidates, based on the analytical formulation and our ab initio predictions. We find that systems with strong electron-phonon interactions, reduced electronic phase space, and suppressed phonon-phonon scattering at temperatures of interest are likely to feature hydrodynamic electron transport. We predict that layered and/or anisotropic semimetals composed of half-filled d shells and light group V/VI elements with lower crystal symmetry are promising candidates to observe hydrodynamic phenomena in the future. © 2022 American Physical Society

Catalytically mediated epitaxy of 3D semiconductors on van der Waals substrates

The formation of well-controlled interfaces between materials of different structure and bonding is a key requirement when developing new devices and functionalities. Of particular importance are epitaxial or low defect density interfaces between 2D materials and 3D semiconductors or metals, where interfacial structure influences electrical conductivity in field effect and optoelectronic devices, charge transfer for spintronics and catalysis and proximity-induced superconductivity. Epitaxy and hence well-defined interfacial structures have been demonstrated for several metals on van der Waals-bonded substrates. Semiconductor epitaxy on such substrates has been harder to control, for example during chemical vapor deposition of Si and Ge on graphene. Here, we demonstrate a catalytically-mediated heteroepitaxy approach to achieve epitaxial growth of 3D semiconductors such as Ge and Si on van der Waals-bonded materials such as graphene and hexagonal boron nitride. Epitaxy is “transferred” from the substrate to semiconductor nanocrystals via solid metal nanocrystals that readily align on the substrate and catalyze the formation of aligned nuclei of the semiconductor. In situ transmission electron microscopy allows us to elucidate the reaction pathway for this process and to show that solid metal nanocrystals can catalyze semiconductor growth at a significantly lower temperature than direct chemical vapor deposition or deposition mediated by liquid catalyst droplets. We discuss Ge and Si growth as a model system to explore the details of such heterointerfacing and its applicability to a broader range of materials.This work was supported by the EPSRC (EP/K016636/1 and EP/P005152/1) and ERC (Grant 279342: InSituNANO

Catalytically mediated epitaxy of 3D semiconductors on van der Waals substrates

The formation of well-controlled interfaces between materials of different structure and bonding is a key requirement when developing new devices and functionalities. Of particular importance are epitaxial or low defect density interfaces between two-dimensional materials and three-dimensional semiconductors or metals, where an interfacial structure influences electrical conductivity in field effect and optoelectronic devices, charge transfer for spintronics and catalysis, and proximity-induced superconductivity. Epitaxy and hence well-defined interfacial structure has been demonstrated for several metals on van der Waals-bonded substrates. Semiconductor epitaxy on such substrates has been harder to control, for example during chemical vapor deposition of Si and Ge on graphene. Here, we demonstrate a catalytically mediated heteroepitaxy approach to achieve epitaxial growth of three-dimensional semiconductors such as Ge and Si on van der Waals-bonded materials such as graphene and hexagonal boron nitride. Epitaxy is “transferred” from the substrate to semiconductor nanocrystals via solid metal nanocrystals that readily align on the substrate and catalyze the formation of aligned nuclei of the semiconductor. In situ transmission electron microscopy allows us to elucidate the reaction pathway for this process and to show that solid metal nanocrystals can catalyze semiconductor growth at a significantly lower temperature than direct chemical vapor deposition or deposition mediated by liquid catalyst droplets. We discuss Ge and Si growth as a model system to explore the details of such hetero-interfacing and its applicability to a broader range of materials.This work was supported by the EPSRC (EP/K016636/1 and EP/P005152/1) and ERC (Grant 279342: InSituNANO

A removal lemma for systems of linear equations over finite fields

We prove a removal lemma for systems of linear equations over finite fields: let X 1, …, X m be subsets of the finite field F q and let A be a (k × m) matrix with coefficients in F q ; if the linear system Ax = b has o(q m−k ) solutions with x i ∈ X i , then we can eliminate all these solutions by deleting o(q) elements from each X i . This extends a result of Green [Geometric and Functional Analysis 15 (2) (2005), 340–376] for a single linear equation in abelian groups to systems of linear equations. In particular, we also obtain an analogous result for systems of equations over integers, a result conjectured by Green. Our proof uses the colored version of the hypergraph Removal Lemma