The Helmholtzian matrix of a graph G=(V(G),E(G)) is a graph-theoretic analogue of the vector Laplacian (or Helmholtz operator) [S. Li, L. Lu, J.F. Wang, A graph discretization of vector Laplacian, 379 (2026) 446–460]. Motivated by the applications of graph Helmholtzian in simplicial networks, we will investiagte its basic spectral properties. As the first graph matrix indexed by edge set, we find that Helmholtzian matrix is positive semi-definite and its non-negativity correlates with the odd cycles in G and the orientation on E(G), while its irreducibility relates to the signed graphs with loops. We show that the eigenvalues of Helmholtzian matrix are independent of the orientation and further investigate the eigenvalue interlacing under edge additions. One of striking findings is that the non-zero eigenvalues of the Laplacian matrix are those of Helmholtzian matrix of every graph. All these discoveries reveal that the Helmholtzian spectrum of G balances and bridges the oriented graphs, weighted graphs and signed graphs as well as their adjacency or Laplacian spectra.
Diamond demonstrates significant application potential in electronic devices owing to its excellent properties, yet its practical application is often limited by interface challenges in precisely controlling its interfacial electronic properties. This study demonstrates that surface termination engineering serves as a potential strategy for effectively modulating the electron affinity (EA) of diamond, thereby tuning the band offsets and Schottky barrier heights (SBHs) at its interfaces. By constructing 24 distinct diamond termination models, the EA of diamond can be tuned from -3.95 to 2.71 eV, governed by the combined effects of surface dipole moment and gap states. Furthermore, the terminated diamond forms a charge-blocking layer at the heterojunction interface, thereby largely preserving the original band positions. Consequently, a linear modulation of the band offset from 0.58 to 4.34 eV is accomplished by integrating beta-Ga2 O3 with 12 different diamond terminations. For metal/diamond contacts, an SBH adjustment from -0.78 to 2.14 eV is achieved at Pt/diamond interfaces by employing different terminations. Notably, the introduction of surface-state-free terminations effectively suppresses metal-induced gap states, thereby achieving significant Fermi-level depinning. This work elucidates the underlying mechanism of termination-induced EA tuning and establishes a paradigm for quantitative band offset and SBH control, providing critical guidance for the rational design of high-performance diamond-based devices. (c) 2026 Published by Elsevier Ltd on behalf of The editorial office of Journal of Materials Science & Technology.
As an ultrawide-bandgap semiconductor for extreme-environment applications, ,B-Ga2O3 exhibits remarkable radiation resistance. However, its behavior under extreme electron irradiation exceeding conventional energy regimes remains largely unexplored. Here, we combine 10 MeV electron irradiation (fluences: 1 & times; 1017 and 5 & times; 1017 cm-2), first principles calculation, and machine-learning molecular dynamics (ML-MD) simulations parameterized to match these conditions to systematically reveal the anisotropic structural limits and phase evolution mechanisms of ,B-Ga2O3. Electron irradiation induces pronounced anisotropic lattice degradation, electronic and optical responses across different crystal planes (100), (010), (001), and (2 & strns; 01). The (100) plane undergoes severe near-surface degradation via the collapse of GaO4 tetrahedra, evidenced by the disappearance of the B(2) g , A(8g) and A(9) g Raman modes. For the (010) orientation, the pronounced channeling effect reduces scattering cross-sections and favors dispersed point defects, broadening the bulk X-ray diffraction (XRD) rocking curve full width at half maximum (FWHM) from similar to 42 to over 155 arcsec. Across the four planes, irradiation induces a similar to 690 nm infrared emission while differentially suppressing their intrinsic blue-violet luminescence. The (001) plane demonstrates the highest overall tolerance. ML-MD simulations reveal that electron irradiation-generated point defects reorganize into a low-fraction, dispersed gamma-phase inclusion that remains structurally coherent with the defective ,B-phase matrix. These insights into the orientation-dependent structural, electronic, optical properties, and phase evolution establish a strategic framework for designing radiation-resistant ,B-Ga2O3 devices with sustained performance under extreme conditions. (c) 2026 Published by Elsevier Ltd on behalf of The editorial office of Journal of Materials Science & Technology.
Rapid recombination of charge carriers remains a critical bottleneck limiting the efficiency of conventional photocatalytic systems. Piezoelectric assisted photocatalysis has emerged as a promising strategy to address this challenge, yet maximizing the synergistic effect between piezoelectricity and photocatalysis is largely restricted by rational catalyst architecture design. Herein, a series of core-shell BaTiO3@TiO2 (BTO@TO) heterojunctions with tunable TiO2 shell thicknesses were rationally fabricated via a sol-gel assisted controlled deposition method, aiming to boost the synergy between the heterojunction built-in electric field and the piezoelectric field for efficient hydrogen evolution. The TiO2 shell thickness exerts a decisive regulatory effect on the charge transfer kinetics in BTO@TO composites, and an optimized moderate shell thickness is indispensable to strike an excellent balance between the heterojunction built-in electric field and piezoelectric field coupling, thereby maximizing the dual-field synergistic effect. Impressively, the optimized BTO@TO-1 composite with a TiO2 shell thickness of similar to 180 nm exhibits exceptional piezo-photocatalytic performance (2.42 times that of photocatalysis alone), with a synergy factor of 2.4 and a synergy efficiency of 58.5%, far exceeding the sum of its individual photocatalytic and piezocatalytic performances, which confirms a prominent synergistic effect. This work highlights the critical role of optimal shell thickness in balancing heterojunction charge separation and piezoelectric field coupling, offering a promising design strategy for ferroelectric-based core-shell catalysts in synergistic piezo-photocatalytic water splitting.
Foliar iron delivery requires simultaneous control of ferrous-state stability, nutrient transport, and leaf-surface residence, yet these functions are rarely integrated within a single formulation. Here, we developed a carbon dot-stabilized and coordination-gated nanocarrier formulation (NFCT) by co-confining Fe2+ and carbon dots (CDs) within amino-functionalized mesoporous silica and depositing an external tannic acid-Fe (TA-Fe) coordination network. Electron spin resonance measurements and density functional theory calculations support complementary contributions of the CDs to reactive-oxygen-species attenuation and Fe2+ coordination, whereas the outer metal-phenolic network provides pH-dependent regulation of nutrient transport. Under the tested storage conditions, NFCT retained 93.0% of recoverable Fe2+ after 7 d and showed substantially greater recoverable Fe2+ availability under mildly acidic than near-neutral or alkaline conditions. On model waxy leaf surfaces, NFCT improved droplet spreading and achieved 72% of the pre-rain total-Fe deposit after a 70.7 mm simulated-rainfall challenge. These redox, transport, and interfacial attributes were associated with greater foliar Fe accumulation, chlorophyll index, biomass production, and root-system remodeling in Fe-deficient rice. In a single-site, single-season proof-of-concept field trial, no statistically significant differences were detected between NFCT and commercial EDTA-Fe for the measured agronomic traits under the tested application program. The results define a carbon-dot-centered structure-property-function framework in which internal Fe2+ protection, external coordination-gated transport, and rainfast foliar residence provide complementary delivery functions.
