Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “electron localization”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 595 records · Page 33
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Comparison of the canonical transformation and energy functional formalisms for ab initio calculations of self-localized polarons

In materials with strong electron-phonon (e-ph) interactions, charge carriers can distort the surrounding lattice and become trapped, forming self-localized (small) polarons. We recently developed an ab initio approach based on canonical transformations to efficiently compute the formation and energetics of small polarons A different approach based on a Landau-Pekar energy functional has been proposed in the recent literature. Here in this work, we analyze and compare these two methods in detail. We show that the small polaron energy is identical in the two formalisms when using the same polaron wave function. We also show that our canonical transformation formalism can predict polaron band structures and can properly treat zero- and finite-temperature lattice vibration effects, although at present using a fixed polaron wave function. Conversely, the energy functional approach can compute the polaron wave function, but as we show here, it neglects lattice vibrations and cannot address polaron self-localization and thermal band narrowing. Taken together, this work relates two different methods developed recently to study polarons from first-principles, highlighting their merits and shortcomings and discussing them both in a unified formalism.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Magnetochiral tunneling in paramagnetic Co 1/3 NbS 2

Electric currents have the intriguing ability to induce magnetization in nonmagnetic crystals with sufficiently low crystallographic symmetry. Some associated phenomena include the non-linear anomalous Hall effect in polar crystals and the nonreciprocal directional dichroism in chiral crystals when magnetic fields are applied. In this work, we demonstrate that the same underlying physics is also manifested in the electronic tunneling process between the surface of a nonmagnetic chiral material and a magnetized scanning probe. In the paramagnetic but chiral metallic compound Co 1/3 NbS 2 , the magnetization induced by the tunneling current is shown to become detectable by its coupling to the magnetization of the tip itself. This results in a contrast across different chiral domains, achieving atomic-scale spatial resolution of structural chirality. To support the proposed mechanism, we used first-principles theory to compute the chirality-dependent current-induced magnetization and Berry curvature in the bulk of the material. Our demonstration of this magnetochiral tunneling effect opens up an avenue for investigating atomic-scale variations in the local crystallographic symmetry and electronic structure across the structural domain boundaries of low-symmetry nonmagnetic crystals.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Quasiparticle self-consistent GW band structures and phase transitions of LiAlO 2 in tetrahedrally and octahedrally coordinated structures

In this work, a first-principles computational study is presented of various phases of LiAlO 2 . The relative total energies and equations of state of the α, β, and γ phases are determined after structural relaxation of each phase. The β and γ tetrahedral phases are found to be very close in energy and lattice volume with the γ phase having the lowest energy. The octahedral α phase is a high-pressure phase and the transition pressure from the γ and β phases to α is determined to be about 1 GPa. The electronic band structures of each phase at their own equilibrium volume are determined using the quasiparticle self-consistent (QS) GW method as well as using the 0.8 Σ approach in which the QS GW self-energy is reduced by a factor of 0.8 to correct for the underscreening of W in QS GW. The effective masses of the band edges and the nature of the band gaps are presented. The lowest energy γ phase is found to have a pseudodirect gap of 7.69 eV. The gap is direct at Γ but corresponds to a dipole forbidden transition. The imaginary part of the dielectric function and the absorption coefficient are calculated in the long-wavelength limit and the random phase approximation, without local field or electron-hole interaction effects for each phase, and their anisotropies are discussed. Si doping on the Al site is investigated as a possible n-type dopant in γ-LiAlO 2 using a 128-atom supercell corresponding to 3.125% Si on the Al sublattice in the generalized gradient approximation and a smaller 16-atom cell with 25% Si in the QS GW approximation. The Si is found to significantly perturb the conduction band and lower the gap but a clearly separated deep donor defect level is not found. However, the donor binding energy is still expected to be relatively deep, on the order of a few tenths eV in the hydrogenic effective mass approximation.

36 MATERIALS SCIENCE↗
DOE OSTI2024

A Straightforward Model for Quantifying Local pH Gradients Governing the Oxygen Evolution Reaction

The production and consumption of protons by an electrocatalyst will, under certain conditions, generate localized microenvironments with properties distinct from those of the bulk solution. These local properties are particularly impactful for reactions involving proton-coupled electron transfer, where the generation of locally basic or acidic environments may significantly influence the energy efficiency and reaction selectivity of the electrocatalyst. Whereas local pH environments have been observed and characterized in reductive half-reactions, including the CO 2 reduction and hydrogen evolution reactions, the incompatibility of conventional techniques and materials has limited studies in oxidative half-reactions, including the oxygen evolution reaction (OER), which provides the reducing equivalents for solar-to-fuels electrolysis. With the straightforward parameters bulk pH, buffer composition and pK a , and mass transport, we develop a model for describing local pH as a function of current density regardless of the microscopic details of the mechanism. Using an acid-stable PbO x OER catalyst, we observe the formation and dissipation of pH gradients during the OER and validate the model with voltammetric and potentiometric studies. Here, the model predicts how local acidic environments can develop over a narrow OER current density window, thus providing further motivation for the development of OER catalysts that are stable to acid, even when operating in basic aqueous conditions. More generally, the model is not restricted to the OER and is useful for determining the onset of local pH gradients for other electrocatalytic reactions that involve the consumption or generation of protons in energy conversion reactions.

Anions↗
DOE OSTI2021

Disentangling electronic, lattice, and spin dynamics in the chiral helimagnet Cr 1/3 NbS 2

We investigate the static and ultrafast magneto-optical response of the hexagonal chiral helimagnet Cr 1/3 NbS 2 above and below the helimagnetic ordering temperature. The presence of a magnetic easy plane contained within the crystallographic $ab$-plane is confirmed, while degenerate optical pump-probe experiments reveal significant differences in the dynamic between the parent, NbS 2 , and Cr-intercalated compounds. Time-resolved magneto-optical Kerr effect measurements show a two-step demagnetization process, where an initial, subpicosecond relaxation and subsequent buildup (τ > 50ps) in the demagnetization dynamic scale similarly with increasing pump fluence. Despite theoretical evidence for partial gapping of the minority spin channel, suggestive of possible half-metallicity in Cr 1/3 NbS 2 , such a long demagnetization dynamic likely results from spin-lattice relaxation as opposed to minority state blocking. However, comparison of the two-step demagnetization process in Cr 1/3 NbS 2 with other 3d intercalated transition metal dichalcogenides reveals a behavior that is unexpected from conventional spin-lattice relaxation, and may be attributed to the complicated interaction of local moments with itinerant electrons in this material system.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2022

Elucidating facet dependent electronic and electrochemical properties of Cu 2 O nanocrystals using AFM/SCEM and DFT

Cuprous oxide (Cu 2 O) has extensively been studied owing to its excellent optical, magnetic, and catalytic properties. Many of these properties are facet-dependent and have not been well elucidated. This work synthesized cubic, cuboctahedral, octahedral, and rhombic dodecahedral shaped Cu 2 O nanocrystals of ~300 nm in size to evaluate the facet-dependent electrochemical activities. Here, cyclic voltammetry (CV) and electrochemical impedance spectroscopy (EIS) were firstly used to reveal the average electrochemical activities at the ensemble level. Atomic force microscopy-scanning electrochemical microscopy (AFM-SECM) was further used to assess the electrochemical activities of different Cu 2 O nanocrystals at the facet level. Hexaammineruthenium (III) chloride ({Ru(NH 3 ) 6 }Cl 3 ) was employed as the probe molecules that reacted with four different Cu 2 O nanocrystals under —400mV and yielded ~300 pA current between the probing tip and the nanocrystal surface. The tip-current mapping results indicate that rhombic dodecahedral Cu 2 O exhibits higher electrocatalytic activity than other shaped Cu 2 O, due to the presence of dominant exposed facet of {110} as indicated by the relatively high tip current. Density-functional theory (DFT) calculations confirmed the facet dependence of local surface energy and electronic structure of Cu 2 O nanocrystals. Besides electrochemical activity, the surface work function and adsorptive properties were both observed to vary with the shape and dominant exposed facets of Cu 2 O. This study presented a unique experimental and computational chemistry approach to analyze surface electrochemical properties of Cu 2 O crystals at a crystalline facet level.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Electronic thermal transport measurement in low-dimensional materials with graphene non-local noise thermometry

In low-dimensional systems, the combination of reduced dimensionality, strong interactions and topology has led to a growing number of many-body quantum phenomena. Thermal transport, which is sensitive to all energy-carrying degrees of freedom, provides a discriminating probe of emergent excitations in quantum materials and devices. However, thermal transport measurements in low dimensions are dominated by the phonon contribution of the lattice, requiring an experimental approach to isolate the electronic thermal conductance. In this work we measured non-local voltage fluctuations in a multi-terminal device to reveal the electronic heat transported across a mesoscopic bridge made of low-dimensional materials. Using two-dimensional graphene as a noise thermometer, we measured the quantitative electronic thermal conductance of graphene and carbon nanotubes up to 70 K, achieving a precision of ~1% of the thermal conductance quantum at 5 K. Employing linear and nonlinear thermal transport, we observed signatures of energy transport mediated by long-range interactions in one-dimensional electron systems, in agreement with a theoretical model.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Order-Disorder Ferroelectric Transition of Strained SrTiO 3

SrTiO 3 is an incipient ferroelectric that is believed to exhibit a prototype displacive, soft mode ferroelectric transition when subjected to mechanical stress or alloying. We use high-angle annular dark field imaging in scanning transmission electron microscopy to reveal local polar regions in the room temperature, paraelectric phase of strained SrTiO 3 films, which undergo a ferroelectric transition at low temperatures. These films contain nanometer-sized domains in which the Ti columns are displaced. In contrast, these nanodomains are absent in unstrained films, which do not become ferroelectric. Here, the results show that the ferroelectric transition of strained SrTiO 3 is an order-disorder transition. We discuss the impact of the results on the nature of the ferroelectric transition of SrTiO 3 .

36 MATERIALS SCIENCE↗
DOE OSTI2025

Photocurrent Nanoscopy of Quantum Hall Bulk

Understanding nanoscale electronic and thermal transport of two-dimensional (2D) electron systems in the quantum Hall regime, particularly in the bulk insulating state, poses considerable challenges. One of the primary difficulties arises from the presence of chiral edge channels, whose transport behavior obscures the investigation of the insulating bulk. Using near-field optical and photocurrent nanoscopy, we probe real-space variations of the optical and thermal dynamics of graphene in the quantum Hall regime without relying on complex sample or electrode geometries. Near the charge-neutrality point, we detect strong optical and photothermal signals from resonant inter-Landau-level (LL) magnetoexciton excitations between the zeroth and ± 1 st LLs, which gradually weaken with increasing doping due to Pauli blocking. Interestingly, at higher doping levels and full-integer LL fillings, photothermal signals reappear across the entire sample over an approximately 10 − μ m scale, indicating unexpectedly long cooling lengths and nonlocal photothermal heating through the insulating bulk. This observation suggests that thermal conductivity persists for the localized states even as electronic transport is suppressed—a clear violation of the Wiedemann-Franz law. Our experiments provide novel insights into nanoscale thermal and electronic transport in incompressible 2D gases, highlighting the roles of magnetoexcitons and chiral edge states in the thermo-optoelectric dynamics of the Dirac quantum Hall state. Published by the American Physical Society 2025

36 MATERIALS SCIENCE↗
DOE OSTI2025

Peak‐in‐Valley Metal Nano‐Architectures via E‐Beam‐Guided Metal Oxide Redox

Focused electron beams enable nanoscale material modification via localized etching or deposition. In liquid-phase electron-beam-mediated processing, radiolysis-driven redox reactions present an opportunity to control both etching and deposition simultaneously. Here, this duality using a water-ammonia solvent as a tunable redox mediator on copper surfaces is demonstrated. At lower ammonia concentrations, the oxidation process dominates, etching copper to sub-50-nm depths. The copper ions and ion-complexes released during this initial oxidation step are reduced by solvated electrons resulting in metal deposition into the etched sites, over longer e-beam exposures, producing characteristic peak-in-valley nanostructures. Conversely, at higher ammonia concentrations copper-ammine ion complexation and radiolytic oxidizing species scavenging by ammonia occur at higher rates, creating a reducing environment conducive to rapid beam-guided copper deposition. Reaction-transport simulations and experiments are performed to show the effects of ammonia-mediated radiolysis chemistry, describing the direct influence of solvent concentration on redox balance and the outcome of e-beam guided processing. By uniting both etching and deposition within a single framework, this work provides a versatile route for controlled surface nanostructuring.

copper nanostructures↗
DOE OSTI2024

Public Data Set: Erratum: “Initial Characterization of Electron Temperature and Density Profiles in PEGASUS Spherical Tokamak Discharges Driven Solely by Local Helicity Injection” [Phys. Plasmas 28, 102504 (2021)]

This public data set contains openly-documented, machine readable digital research data corresponding to figures published in G.M. Bodner et al., ‘Erratum: “Initial Characterization of Electron Temperature and Density Profiles in PEGASUS Spherical Tokamak Discharges Driven Solely by Local Helicity Injection” [Phys. Plasmas 28, 102504 (2021)],’ Physics of Plasmas 31, 129904 (2024).

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2020

Influence of Polymorphism on the Electronic Structure of Ga 2 O 3

The search for new wide-band-gap materials is intensifying to satisfy the need for more advanced and energy-efficient power electronic devices. Ga 2 O 3 has emerged as an alternative to SiC and GaN, sparking a renewed interest in its fundamental properties beyond the main β-phase. Here, three polymorphs of Ga 2 O 3 , α, β, and ε, are investigated using X-ray diffraction, X-ray photoelectron and absorption spectroscopy, and ab initio theoretical approaches to gain insights into their structure–electronic structure relationships. Valence and conduction electronic structure as well as semicore and core states are probed, providing a complete picture of the influence of local coordination environments on the electronic structure. State-of-the-art electronic structure theory, including all-electron density functional theory and many-body perturbation theory, provides detailed understanding of the spectroscopic results. The calculated spectra provide very accurate descriptions of all experimental spectra and additionally illuminate the origin of observed spectral features. This paper provides a strong basis for the exploration of the Ga 2 O 3 polymorphs as materials at the heart of future electronic device generations.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2026

Structural modulation, physical properties, and electronic band structure of the kagome metal UCr 6 ⁢Ge 6

The chemical flexibility of the 𝑅⁢𝑀 6 ⁢𝑋 6 stoichiometry, where an 𝑓-block element is intercalated in the CoSn structure type, allows for the tuning of flatbands associated with kagome lattices to the Fermi level and for emergent phenomena due to interactions between the 𝑓- and 𝑑-electron lattices. Yet, 5⁢𝑓 members of the “166” compounds are underrepresented compared with 4⁢𝑓 members. In this work, we report single-crystal growth of UCr 6 ⁢Ge 6 , which crystallizes in a monoclinically distorted Y 0.5⁢ Co 3 ⁢Ge 3 -type structure. The real-space character of the modulation, which is unique within the 𝑅⁢𝑀 6 ⁢𝑋 6 family, is approximated by a 3×1×2 supercell of the average monoclinic cell. The compound has kagome-lattice flatbands near the Fermi level and a moderately enhanced electronic heat capacity, as evidenced by its low-temperature Sommerfeld coefficient (𝛾=86.5 mJ mol −1 K −2 ) paired with band structure calculations. The small, isotropic magnetization and featureless resistivity of UCr 6 ⁢Ge 6 suggest itinerant uranium 5⁢𝑓 electrons and Pauli paramagnetism. Angle-resolved photoemission spectroscopy results provide evidence for uranium 5⁢𝑓 weight at the Fermi level and for a flatband near the Fermi level associated with the chromium 3⁢𝑑 kagome lattice. The isotropic magnetic behavior of the uranium 5⁢𝑓 electrons starkly contrasts with localized behavior in other uranium 166 compounds, highlighting the high tunability of the magnetic ground state across the material family.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Pauling’s rules for oxide-based minerals: A re-examination based on quantum mechanical constraints and modern applications of bond-valence theory to Earth materials

Since their introduction in 1929, Pauling’s five rules have been used by scientists from many disciplines to rationalize and predict stable arrangements of atoms and coordination polyhedra in crystalline solids; amorphous materials such as silicate glasses and melts; nanomaterials, poorly crystalline solids; aqueous cation and anion complexes; and sorption complexes at mineral-aqueous solution interfaces. The predictive power of these simple yet powerful rules was challenged recently by George et al. (2020), who performed a statistical analysis of the performance of Pauling’s five rules for about 5000 oxide crystal structures. They concluded that only 13% of the oxides satisfy the last four rules simultaneously and that the second rule has the most exceptions. They also found that Pauling’s first rule is satisfied for only 66% of the coordination environments tested and concluded that no simple rule linking ionic radius to coordination environment will be predictive due to the variable quality of univalent radii. We address these concerns and discuss quantum mechanical calculations that complement Pauling’s rules, particularly his first (radius sum and radius ratio rule) and second (electrostatic valence rule) rules. We also present a more realistic view of the bonded radii of atoms, derived by determining the local minimum in the electron density distribution measured along trajectories between bonded atoms known as bond paths, i.e., the bond critical point (rc). Electron density at the bond critical point is a quantum mechanical observable that correlates well with Pauling bond strength. Moreover, a metal atom in a polyhedron has as many bonded radii as it has bonded interactions, resulting in metal and O atoms that may not be spherical. O atoms, for example, are not spherical in many oxide-based crystal structures. Instead, the electron density of a bonded oxygen is often highly distorted or polarized, with its bonded radius decreasing systematically from ~1.38 Å when bonded to highly electropositive atoms like sodium to 0.64 Å when bonded to highly electronegative atoms like nitrogen. Bonded radii determined for metal atoms match the Shannon (1976) radii for more electropositive atoms, but the match decreases systematically as the electronegativities of the M atoms increase. As a result, significant departures from the radius ratio rule in the analysis by George et al. (2020) is not surprising. We offer a modified, more fundamental version of Pauling’s first rule and demonstrate that the second rule has a one-to-one connection between the electron density accumulated between the bonded atoms at the bond critical point and the Pauling bond strength of the bonded interaction. Pauling’s second rule implicitly assumes that bond strength is invariant with bond length for a given pair of bonded atoms. Many studies have since shown that this is not the case, and Brown and Shannon (1973) developed an equation and a set of parameters to describe the relation between bond length and bond strength, now redefined as bond valence to avoid confusion with Pauling bond-strength. Brown (1980) used the valence-sum rule, together with the path rule and the valence-matching principle, as the three axioms of bond-valence theory (BVT), a powerful method for understanding many otherwise elusive aspects of crystals and also their participation in dynamic processes. We show how a priori bond-valence calculations can predict unstrained bond-lengths and how bond-valence mapping can locate low-Z atoms in a crystal structure (e.g., Li) or examine possible diffusion pathways for atoms through crystal structures. In addition, we briefly discuss Pauling’s third, fourth, and fifth rules, the first two of which concern the sharing of polyhedron elements (edges and faces) and the common instability associated with structures in which a polyhedron shares an edge or face with another polyhedron and contains high-valence cations. The olivine [α-(MgxFe1–x)2SiO4] crystal structure is used to illustrate the distortions from hexagonal close-packing of O atoms caused by metal-metal repulsion across shared polyhedron edges. We conclude by discussing several applications of BVT to Earth materials, including the use of BVT to: (1) locate H+ ions in crystal structures, including the location of protons in the crystal structures of nominally anhydrous minerals in Earth’s mantle; (2) determine how strongly bonded (usually anionic) structural units interact with weakly bonded (usually cationic) interstitial complexes in complex uranyl-oxide and uranyl-oxysalt minerals using the valence-matching principle; (3) calculate Lewis acid strengths of cations and Lewis base strengths of anions; (4) determine how (H2O) groups can function as bond-valence transformers by dividing one bond into two bonds of half the bond valence; (5) help characterize products of sorption reactions of aqueous cations (e.g., Co2+ and Pb2+) and oxyanions [e.g., selenate (Se6+O4)2- and selenite (Se4+O3)2-] at mineral-aqueous solution interfaces and the important role of protons in these reactions; and (6) help characterize the local coordination environments of highly charged cations (e.g., Zr4+, Ti4+, U4+, U5+, and U6+) in silicate glasses and melts.

Geochemistry & Geophysics↗
DOE OSTI2024

Zero-Strain Metal-Insulator Transition by the Local Fluctuation of Cation Dimerization

The coupled electronic and structural transitions in metal-insulator transition (MIT) hinder ultrafast switching and ultimate endurance. Decoupling these transitions and achieving a zero-strain electronic MIT can overcome the fundamental limitations of MIT in solid materials. Here, this study demonstrates that iso-valent Ti dopants in supercooled VO 2 epitaxial films cause MIT with minimal hysteresis without changing unit-cell volume and crystal symmetry. The Ti dopants in the VO 2 lattice locally alter the configuration of V-V pairs, where the long-range ordering in V-V pairs is disrupted, and the nano-domains of V-V dimers are formed. Strikingly, these local V-V dimers persist even above the electronic transition temperature (T MI ), facilitating the zero-strain electronic MIT with nanoscale structural heterogeneity. The geometrically compatible interface between insulating and metallic phases drastically enhances switching speed and endurance during electrically and optically driven zero-strain MIT. In conclusion, this discovery offers a fresh perspective on the scientific understanding of MIT and the improved functionality in terms of device speed and reliability by decoupling electronic and structural transitions.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Tetragonal Kondo Insulator EuCd 2 Sb 2 Discovered via High Pressure High Temperature Synthesis

Abstract Magnetic and electronic properties of quantum materials heavily rely on the crystal structure even in the same chemical compositions. In this study, it is demonstrated that a layered tetragonal EuCd 2 Sb 2 structure can be obtained by treating bulk trigonal EuCd 2 Sb 2 under high pressure (6 GPa) and high temperature (600 °C). Magnetization measurements of the newly formed layered tetragonal EuCd 2 Sb 2 confirm an antiferromagnetic ordering with Neel temperature ( T N ) around 16 K, which is significantly higher than that ( T N ≈ 7 K) of trigonal EuCd 2 Sb 2 , consistent with heat capacity measurements. Moreover, bad metal behavior is observed in the temperature dependence of the electrical resistivity and the resistivity shows a dramatic increase around the Neel temperature. Electronic structure calculations with local density approximation dynamic mean–field theory (LDA+DMFT) show that this material is strongly correlated with well‐formed large magnetic moments, due to Hund's coupling, which is known to dramatically suppress the Kondo scale.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2022

In Situ Optical Quantification of Extracellular Electron Transfer Using Plasmonic Metal Oxide Nanocrystals**

Extracellular electron transfer (EET) is a critical form of microbial metabolism that enables respiration on a variety of inorganic substrates, including metal oxides. However, quantifying current generated by electroactive bacteria has been predominately limited to biofilms formed on electrodes. To address this, we developed a platform for quantifying EET flux from cell suspensions using aqueous dispersions of infrared plasmonic tin-doped indium oxide nanocrystals. Furthermore, tracking the change in optical extinction during electron transfer enabled quantification of current generated by planktonic Shewanella oneidensis cultures. Using this method, we differentiated between starved and actively respiring cells, cells of varying genotype, and cells engineered to differentially express a key EET gene using an inducible genetic circuit. Overall, our results validate the utility of colloidally stable plasmonic metal oxide nanocrystals as quantitative biosensors in aqueous environments and contribute to a fundamental understanding of planktonic S. oneidensis electrophysiology using simple in situ spectroscopy.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

Quantum Fluctuations and Multifractally enhanced Superconductivity in Disordered Thin Films

The interplay between electron–electron interactions and weak localization (or anti-localization) phenomena in two-dimensional systems can significantly enhance the superconducting transition temperature. For this work, we develop the theory of quantum fluctuations within such multifractally enhanced superconducting states in thin films. In conditions of weak disorder, we employ the Finkel’stein nonlinear sigma model to derive an effective action for the superconducting order parameter and the quasiclassical Green’s function, meticulously accounting for the influence of quantum fluctuations. This effective action, applicable for interactions of any strength, reveals the critical role of well-known collective modes in a dirty superconductor, and its saddle-point analysis leads to modified Usadel and gap equations. These equations comprehensively incorporate the renormalizations stemming from the interplay between interactions and disorder, resulting in the non-trivial energy dependence of the gap function. Notably, our analysis establishes a direct relation between the self-consistent gap equation at the superconducting transition temperature and the known renormalization group equations for interaction parameters in the normal state.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗