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The null space of the k-th order Laplacian Lk, known as the {\em k-th homology vector space}, encodes the non-trivial topology of a manifold or a network. Understanding the structure of the homology embedding can thus disclose geometric or topological information from the data. The study of the null space embedding of the graph Laplacian L0 has spurred new research and applications, such as spectral clustering algorithms with theoretical guarantees and estimators of the Stochastic Block Model. In this work, we investigate the geometry of the k-th homology embedding and focus on cases reminiscent of spectral clustering. Namely, we analyze the {\em connected sum} of manifolds as a perturbation to the direct sum of their homology embeddings. We propose an algorithm to factorize the homology embedding into subspaces corresponding to a manifold's simplest topological components. The proposed framework is applied to the {\em shortest homologous loop detection} problem, a problem known to be NP-hard in general. Our spectral loop detection algorithm scales better than existing methods and is effective on diverse data such as point clouds and images. 

Recent advancement in the switching of perpendicular magnetic tunnel junctions with an electric field has been a milestone for realizing ultra-low energy memory and computing devices. To integrate with current spin-transfer torque-magnetic tunnel junction and spin–orbit torque-magnetic tunnel junction devices, the typical linear fJ/V m range voltage controlled magnetic anisotropy (VCMA) needs to be significantly enhanced with approaches that include new materials or stack engineering. A possible bidirectional and 1.1 pJ/V m VCMA effect has been predicted by using heavily electron-depleted Fe/MgO interfaces. To improve upon existing VCMA technology, we have proposed inserting high work function materials underneath the magnetic layer. This will deplete electrons from the magnetic layer biasing the gating window into the electron-depleted regime, where the pJ/V m and bidirectional VCMA effect was predicted. We have demonstrated tunable control of the Ta/Pd(x)/Ta underlayer's work function. By varying the Pd thickness (x) from 0 to 10 nm, we have observed a tunable change in the Ta layer's work function from 4.32 to 4.90 eV. To investigate the extent of the electron depletion as a function of the Pd thickness in the underlayer, we have performed DFT calculations on supercells of Ta/Pd(x)/Ta/CoFe/MgO, which demonstrate that electron depletion will not be fully screened at the CoFe/MgO interface. Gated pillar devices with Hall cross geometries were fabricated and tested to extract the anisotropy change as a function of applied gate voltage for samples with various Pd thicknesses. The electron-depleted Pd samples show three to six times VCMA improvement compared to the electron accumulated Ta control sample. 

Abstract Plasmonic structural color, in which vivid colors are generated via resonant nanostructures made of common plasmonic materials, such as noble metals have fueled worldwide interest in backlight-free displays. However, plasmonic colors that were withstanding ultrahigh temperatures without damage remain an unmet challenge due to the low melting point of noble metals. Here, we report the refractory hafnium nitride (HfN) plasmonic crystals that can generate full-visible color with a high image resolution of ∼63,500 dpi while withstanding a high temperature (900 °C). Plasmonic colors that reflect visible light could be attributed to the unique features in plasmonic HfN, a high bulk plasmon frequency of 3.1 eV, whichcould support localized surface plasmon resonance (LSPR) in the visible range. By tuning the wavelength of the LSPR, the reflective optical response can be controlled to generate the colors from blue to red across a wide gamut. The novel refractory plasmonic colors pave the way for emerging applications ranging from reflective displays to solar energy harvesting systems. 

Many manifold embedding algorithms fail apparently when the data manifold has a large aspect ratio (such as a long, thin strip). Here, we formulate success and failure in terms of finding a smooth embedding, showing also that the problem is pervasive and more complex than previously recognized. Mathematically, success is possible under very broad conditions, provided that embedding is done by carefully selected eigenfunctions of the Laplace-Beltrami operator Δ_M. Hence, we propose a bicriterial Independent Eigencoordinate Selection (IES) algorithm that selects smooth embeddings with few eigenvectors. The algorithm is grounded in theory, has low computational overhead, and is successful on synthetic and large real data. 

Complex oxide films stabilized by epitaxial growth can exhibit large populations of point defects which have important effects on their properties. The site occupancy of pulsed laser‐deposited epitaxial terbium iron garnet (TbIG) films with excess terbium (Tb) is analyzed, in which the terbium:iron (Tb:Fe)ratio is 0.86 compared to the stoichiometric value of 0.6. The magnetic properties of the TbIG are sensitive to site occupancy, exhibiting a higher compensation temperature (by 90 K) and a lower Curie temperature (by 40 K) than the bulk Tb₃Fe₅O₁₂garnet. Data derived from X‐ray core‐level spectroscopy, magnetometry, and molecular field coefficient modeling are consistent with occupancy of the dodecahedral sites by Tb³⁺, the octahedral sites by Fe³⁺, Tb³⁺and vacancies, and the tetrahedral sites by Fe³⁺and vacancies. Energy dispersive X‐ray spectroscopy in a scanning transmission electron microscope provides direct evidence of Tb_Feantisites. A small fraction of Fe²⁺is present, and oxygen vacancies are inferred to be present to maintain charge neutrality. Variation of the site occupancies provides a path to considerable manipulation of the magnetic properties of epitaxial iron garnet films and other complex oxides, which readily accommodate stoichiometries not found in their bulk counterparts.