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Oak Ridge National Laboratory Pilot Demonstration of an Attestation and Anomaly Detection Framework using Distributed Ledger Technology for Power Grid Infrastructure

This report summarizes the design and pilot demonstration of a framework called Grid Guard that was created to provide increased data and device trustworthiness to electric grid devices by leveraging distributed ledger technology (DLT), specifically blockchain. Grid Guard contains a combination of core cryptographic methods such as the secure hash algorithm (SHA), and asymmetric cryptography, private permissioned blockchain, baselining configuration data, consensus algorithm (Raft) and the Hyperledger Fabric (HLF) framework. The system implements a low energy, fast, and robust enhancement to system trustworthiness within and across electric grid systems such as substations, control centers and metering infrastructures. Blockchain is a distributed database structured that provides a practically unalterable (immutable) timeline of stored transactions. By relying on hashing and the Raft consensus algorithm, if an entity tries to illegitimately alter a record at one instance of the database the other ledger nodes are not altered. They work to cross-reference each other and easily locate any incorrectly added data and remove it. The bulk raw data is stored in an off-chain storage (outside of the blockchain ledger) and a hash of this baseline data is stored in the Blockchain ledger via hashing windows of time-series and configuration data, after aggregation and filtering. The bulk off-chain data repository is then considered to be trust-anchored using the hashes stored in the blockchain. To secure the electric grid testbed devices and data, device configuration baselines were compared to those baselines that had been previously stored in the ledger. Statistical baselines for device configurations, network communication patterns, and high-speed sensor data are calculated and then stored off-chain and hashes stored in the ledger. Measurements such as three-phase voltage and current, frequency, breaker status, protection scheme settings, network configuration settings (and other device configuration artifacts) and network traffic features (packet interarrival times) are compared every minute or other selected time windows. During phase 1 of the Grid Guard DLT project different DLT technologies were studies, and an assessment was performed on DLT technology vulnerabilities, uses, and key characteristics. DLT consensus protocols were studies (e.g., RAFT, named after Reliable, Replicated, Redundant, And Fault-Tolerant). Also, cryptography, public, private and permissioned or permissionless systems were assessed. Grid Guard implements a permissioned private DLT. Consensus algorithm selection and choice of DLT implementation depended heavily on the use-case. For this use-case, parameters were selected to measure performance and existing tools for assessment. Benchmarking was performed theoretically and practically. During phase 2 hashed transactions/blocks were inserted into the ledger every second. During phase 2 of the Grid Guard DLT project, a prototype framework was developed and demonstrated for attestation of critical substation devices and data using precision timing systems that use PTP and IRIG-B protocols) on a testbed of operational devices that emulated a distribution substation, control center, and power metering infrastructure using real Operational Technology (OT). The testbed includes OT devices such as protective relays, human machine interfaces (HMI), and power meters. To determine when to collect and compare system and network baselines, an initial examination of an anomaly detection capability to identify malicious manipulation of data streams was conducted. The resulting anomaly detection was demonstrated in a set of experiments and leveraged to trigger device artifact attestation checks. Attestation checks occur against device configuration baselines when compared with the immutable blockchain-stored baselines, which provided a cryptographically supported means by which to store baselines. The electrical substation-grid testbed was created to test the Grid Guard framework. The testbed emulates the operations of a portion of a power grid and SCADA systems as closely as possible. The testbed integrates real protocols, mainly IEC 61850 standard protocols, such as the Sampled Value (SV) and the GOOSE protocols. The testbed also supports DNP3 and other layer 2 and layer 3 protocols such as Telnet, SSH, SFTP/FTP and other proprietary protocols needed to connect to industrial control system equipment. The testbed emulates real power conditions using the OpalRT hardware-in-the-loop (HIL) device which can create fault situations that cannot be easily tested on real systems. The electrical substation-grid testbed was created using real measurement, communication, and protection devices that electrical utilities commonly use.

24 POWER TRANSMISSION AND DISTRIBUTION↗

Beam Synchronous for the Rest of Us!

Fermilab’s Tevatron Clock (TCLK) infrastructure has been an integral part of the accelerator control network since the 1980’s. This 10MHz Manchester encoded protocol has enabled flexible, real-time event distribution for thousands of devices connected to the timing network with a high degree of reliability. Forthcoming upgrades to the Fermilab complex (PIP-II, LBNF, ACORN) necessitate higher levels of precision to maintain inter-bunch timing for Instrumentation and Control purposes. This presents as an opportunity to refine the event distribution protocol for tighter synchronization between machines, experiments, and eventually far-site operations. This paper outlines a method by which beam-synchronous events may be distributed through asynchronous serial protocols via integration with local LLRF and global PPS reference signals. This method is ideal for synchrotron machines with aggressive frequency sweeps (such as Fermilab's 38~53MHz Booster) and allows for precision timing to be maintained across machines without specialized hardware.

43 PARTICLE ACCELERATORS↗

The Integrated Safety-Critical Advanced Avionics Communication and Control (ISAACC) System Concept: Infrastructure for ISHM

Integrated System Health Management (ISHM) architectures for spacecraft will include hard real-time, critical subsystems and soft real-time monitoring subsystems. Interaction between these subsystems will be necessary and an architecture supporting multiple criticality levels will be required. Demonstration hardware for the Integrated Safety-Critical Advanced Avionics Communication & Control (ISAACC) system has been developed at NASA Marshall Space Flight Center. It is a modular system using a commercially available time-triggered protocol, ?Tp/C, that supports hard real-time distributed control systems independent of the data transmission medium. The protocol is implemented in hardware and provides guaranteed low-latency messaging with inherent fault-tolerance and fault-containment. Interoperability between modules and systems of modules using the TTP/C is guaranteed through definition of messages and the precise message schedule implemented by the master-less Time Division Multiple Access (TDMA) communications protocol. "Plug-and-play" capability for sensors and actuators provides automatically configurable modules supporting sensor recalibration and control algorithm re-tuning without software modification. Modular components of controlled physical system(s) critical to control algorithm tuning, such as pumps or valve components in an engine, can be replaced or upgraded as "plug and play" components without modification to the ISAACC module hardware or software. ISAACC modules can communicate with other vehicle subsystems through time-triggered protocols or other communications protocols implemented over Ethernet, MIL-STD- 1553 and RS-485/422. Other communication bus physical layers and protocols can be included as required. In this way, the ISAACC modules can be part of a system-of-systems in a vehicle with multi-tier subsystems of varying criticality. The goal of the ISAACC architecture development is control and monitoring of safety critical systems of a manned spacecraft. These systems include spacecraft navigation and attitude control, propulsion, automated docking, vehicle health management and life support. ISAACC can integrate local critical subsystem health management with subsystems performing long term health monitoring. The ISAACC system and its relationship to ISHM will be presented.

Gwaltney, David A.↗

Enhancing quantum clocks and sensors with randomization and decoherence

This letter shows how incoherent dynamics can lead to metrological advantages in quantum sensing. The results rely on the fact that incoherent dynamics lead to an additive contribution to the quantum Fisher information about time. Such an additive contribution can reduce the error of optimal estimation protocols, as implied by the quantum Cramér–Rao bound. I characterize regimes in which the estimation of a time interval or a frequency is enhanced by decoherence, thereby identifying cases in which incoherent dynamics serve as a metrological resource. The decoherence processes that yield enhanced precision of quantum sensors can be engineered by randomized Hamiltonian dynamics. I illustrate the results with protocols that display improved sensing of time intervals or global fields by qubit and photonic sensors. Enhanced precision of time intervals is achieved with Hamiltonians that include randomized global parameters. Enhanced precision in field estimation is obtained by randomized sensing times.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Nonlocal subpicosecond delay metrology using spectral quantum interference

Precise knowledge of position and timing information is critical to support elementary protocols such as entanglement swapping on quantum networks. While approaches have been devised to use quantum light for such metrology, they largely rely on time-of-flight (ToF) measurements with single-photon detectors and, therefore, are limited to picosecond-scale resolution owing to detector jitter. In this work, we demonstrate an approach to distributed sensing that leverages phase modulation to map changes in the spectral phase to coincidence probability, thereby overcoming the limits imposed by single-photon detection. By extracting information about the joint biphoton phase, we measure a generalized delay—the difference in signal–idler arrival, relative to local radio frequency (RF) phase modulation. For nonlocal ranging measurements, we achieve (2$σ$) precision of ±0.04 ps and for measurements of the relative RF phase, (2$σ$) precision of ±0.7°. We complement this fine timing information with ToF data from single-photon time-tagging to demonstrate absolute measurement of time delay. By relying on off-the-shelf telecommunications equipment and standard quantum resources, this approach has the potential to reduce overhead in practical quantum networks.

42 ENGINEERING↗

Practical Hamiltonian learning with unitary dynamics and Gibbs states

We study the problem of learning the parameters for the Hamiltonian of a quantum many-body system, given limited access to the system. In this work, we build upon recent approaches to Hamiltonian learning via derivative estimation. We propose a protocol that improves the scaling dependence of prior works, particularly with respect to parameters relating to the structure of the Hamiltonian (e.g., its locality k). Furthermore, by deriving exact bounds on the performance of our protocol, we are able to provide a precise numerical prescription for theoretically optimal settings of hyperparameters in our learning protocol, such as the maximum evolution time (when learning with unitary dynamics) or minimum temperature (when learning with Gibbs states). Thanks to these improvements, our protocol has practical scaling for large problems: we demonstrate this with a numerical simulation of our protocol on an 80-qubit system.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Mars Communication Protocols

Over the next decade, international plans and commitments are underway to develop an infrastructure at Mars to support future exploration of the red planet. The purpose of this infrastructure is to provide reliable global communication and navigation coverage for on-approach, landed, roving, and in-flight assets at Mars. The claim is that this infrastructure will: 1) eliminate the need of these assets to carry Direct to Earth (DTE) communications equipment, 2) significantly increase data return and connectivity, 3) enable small mission exploration of Mars without DTE equipment, 4) provide precision navigation i.e., 10 to 100m position resolution, 5) supply timing reference accurate to 10ms. This paper in particular focuses on two CCSDS recommendations for that infrastructure: CCSDS Proximity-1 Space Link Protocol and CCSDS File Delivery Protocol (CFDP). A key aspect of Mars exploration will be the ability of future missions to interoperate. These protocols establish a framework for interoperability by providing standard communication, navigation, and timing services. In addition, these services include strategies to recover gracefully from communication interruptions and interference while ensuring backward compatibility with previous missions from previous phases of exploration.

Kazz, G. J.↗

Ground Based, Millimeter Wave Measurement of Ozone in the Middle Atmosphere

There is a need for highly reliable measurements of stratospheric ozone. Policy makers worldwide concerned with public health rely oil a clear consensus from the scientific community as a basis for ozone-related environmental policy that has a significant impact oil national economies. Tile latest Such consensus was presented in WMO, and used in a 1999 meeting of the parties considering amendments to the Montreal Protocol oil Substances that Deplete the Ozone Layer. Tile scientific community, in turn, needs highly precise and accurate measurements of ozone levels, and small time derivatives of these levels, both in continued - development of its understanding of the physical and chemical processes involved and as clear evidence that these processes are occurring as stated. Over most of the world, changes in ozone levels are small. For example, over the heavily populated northern midlatitudes, the linearized rate of ozone decline is between 0.2% per year and 0.7% per year, depending on altitude. These values are small enough to make measurement requirements technically challenging. Data quality may suffer from imperfections in individual instruments. In one instance, early results from a satellite-borne ozone sensor were later found to be invalid because of calibration drift. Even in the absence of drift, tile absolute calibration of a new sensor may differ slightly from that of its predecessor in service. Most ozone remote sensing instruments operate at ultraviolet or infrared wavelengths where scattering from dust and aerosols must be taken into account; results from these systems may be or are affected following a major volcanic eruption, such as tile one at Mt. Pinatubo in 1991. Given these difficulties, a consensus of measurements from several independent systems is required to insure a reliable understanding of stratospheric ozone levels. Because of the above-described need for highly precise and accurate ozone measurements using several independent techniques, there was interest in developing several techniques which were known but not highly developed in the 1980's into systems capable of being used in ail operational manner to make measurements with the level of quality needed to-detect small trends in ozone levels. A ground-based microwave instrument capable of remotely sensing stratospheric ozone had been designed by tile Principal Investigator of the present project. This instrument was built at tile Millitech Corporation in South Deerfield, Massachusetts before tile present work began. (Funding for design and construction of the instrument came from sources other than the present grant.) Tile instrument measured the spectrum of one of the many emission lines produced by purely rotational transitions of ozone, one at a frequency of 110.8 GHz. The altitude distribution of ozone can. in principle, be retrieved from the details of the pressure-broadened spectrum of the ozone transition. However, the level of contamination of the spectral measurement by instrumentally induced artifacts must be very low in order to retrieve a ozone profile of useful quality from it. The Millitech instrument demonstrated spectral purity at ail adequate level, and there were promising ideas for instrumental improvements and for further development of the technique. The initial objectives of the present project, then, were to develop techniques for calibrating the Millitech instrument, to minimize artifacts in tile spectra it produces, to optimally retrieve ozone profiles from tile spectra, to test tile quality of the microwave profiles by comparing them with profiles obtained using several other, independent techniques over both short and periods of time, and to perform research using the ozone data gathered with the instrument.

Parrish, Alan↗

Optimality and Noise Resilience of Critical Quantum Sensing

We compare critical quantum sensing to passive quantum strategies to perform frequency estimation, in the case of single-mode quadratic Hamiltonians. We show that, while in the unitary case both strategies achieve precision scaling quadratic with the number of photons, in the presence of dissipation this is true only for critical strategies. We also establish that working at the exceptional point or beyond threshold provides suboptimal performance. This critical enhancement is due to the emergence of a transient regime in the open critical dynamics, and is invariant to temperature changes. When considering both time and system size as resources, for both strategies the precision scales linearly with the product of the total time and the number of photons, in accordance with fundamental bounds. However, we show that critical protocols outperform optimal passive strategies if preparation and measurement times are not negligible. Our results are applicable to a broad variety of critical sensors whose phenomenology can be reduced to that of a single-mode quadratic Hamiltonian, including systems described by finite-component and fully connected models.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Automated Construction of Artificial Lattice Structures with Designer Electronic States

Manipulating matter with a scanning tunneling microscope (STM) enables the creation of atomically defined artificial structures that host designer quantum states. However, the time-consuming nature of the manipulation process, coupled with the sensitivity of the STM tip, constrains the exploration of diverse configurations and limits the size of the designed features. In this study, we present a reinforcement learning (RL)-based framework for creating artificial structures by spatially manipulating carbon monoxide (CO) molecules on a copper substrate by using the STM tip. The automated workflow combines molecule detection and manipulation, employing deep-learning-based object detection to locate CO molecules and linear assignment algorithms to allocate these molecules to designated target sites. We initially perform molecule maneuvering based on randomized parameter sampling for sample bias, tunneling current set point, and manipulation speed. This data set is then structured into an action trajectory used to train an RL agent. The model is subsequently deployed on the STM for real-time fine-tuning of the manipulation parameters during structure construction. Our approach incorporates path-planning protocols coupled with active drift compensation to enable atomically precise fabrication of structures with significantly reduced human input while realizing larger-scale artificial lattices with the desired electronic properties. Furthermore, using our approach, we demonstrate the automated construction of an extended artificial graphene lattice and confirm the existence of a characteristic Dirac point in its electronic structure. Further challenges regarding the RL-based structural assembly scalability are discussed.

Algorithms↗

CAST Technical Bulletin #002: Time Synchronization for Next Generation Power Delivery

This technical bulletin gives a description of the updated effort on Oak Ridge National Laboratory (ORNL) Center for Alternative Synchronization and Timing (CAST) project’s work investigating a terrestrial-based high precision timing infrastructure intended for time synchronization of the next generation power delivery infrastructures. We will provide a high-level description of the CAST application context, key protocols, critical architectural issues, and how GPS/timing data traverse through the system.

24 POWER TRANSMISSION AND DISTRIBUTION↗

Onboard early detection and mitigation of lithium plating in fast-charging batteries

Fast-charging is considered as one of the most desired features needed for lithium-ion batteries to accelerate the mainstream adoption of electric vehicles. However, current battery charging protocols mainly consist of conservative rate steps to avoid potential hazardous lithium plating and its associated parasitic reactions. A highly sensitive onboard detection method could enable battery fast-charging without reaching the lithium plating regime. Here, we demonstrate a novel differential pressure sensing method to precisely detect the lithium plating event. By measuring the real-time change of cell pressure per unit of charge (dP/dQ) and comparing it with the threshold defined by the maximum of dP/dQ during lithium-ion intercalation into the negative electrode, the onset of lithium plating before its extensive growth can be detected with high precision. In addition, we show that by integrating this differential pressure sensing into the battery management system (BMS), a dynamic self-regulated charging protocol can be realized to effectively extinguish the lithium plating triggered by low temperature (0 °C) while the conventional static charging protocol leads to catastrophic lithium plating at the same condition. We propose that differential pressure sensing could serve as an early nondestructive diagnosis method to guide the development of fast-charging battery technologies.

25 ENERGY STORAGE↗

Improving noisy free-energy measurements by adding more noise

Estimating free-energy differences using nonequilibrium work relations, such as the Jarzynski equality, is hindered by poor convergence when work fluctuations are large. For systems governed by overdamped Langevin dynamics, we propose the counterintuitive approach of adding noise in order to increase the precision of such calculations. Here, by introducing additional stochastic fluctuations to the system and rescaling its potential energy accordingly, we leave the thermodynamics of the system unchanged while increasing its relaxation rate. For a given time-dependent protocol this modification reduces the dissipated reduced work, leading to more accurate free-energy estimates. The method is designed to be used in experiment, and we illustrate its operation using computer simulations applied to two model systems. However, the regime of applicability of this strategy is likely limited, because it requires control of the system's potential energy in a way that is feasible in only a few experimental settings.

Whitelam, Stephen [Lawrence Berkeley National Labo↗

Time-Resolved Hanbury Brown–Twiss Interferometry of On-Chip Biphoton Frequency Combs Using Vernier Phase Modulation

Biphoton frequency combs (BFCs) are promising quantum sources for large-scale and high-dimensional quantum information and networking systems. In this context, the spectral purity of individual frequency bins will be critical for realizing quantum networking protocols like teleportation and entanglement swapping. Measurement of the temporal autocorrelation function of the unheralded signal or idler photons comprising the BFC is a key tool for characterizing their spectral purity and in turn verifying the utility of the biphoton state for networking protocols. Yet the experimentally obtainable precision for measuring BFC correlation functions is often severely limited by detector jitter. The fine temporal features in the correlation function—not only of practical value in quantum information, but also of fundamental interest in the study of quantum optics—are lost as a result. We propose a scheme to circumvent this challenge through electro-optic phase modulation, experimentally demonstrating time-resolved Hanbury Brown–Twiss characterization of BFCs generated from an integrated 40.5-GHz Si 3 N 4 microring, up to a 3×3-dimensional two-qutrit Hilbert space. Through slight detuning of the electro-optic drive frequency from the comb’s free spectral range, our approach leverages Vernier principles to magnify temporal features, which would otherwise be averaged out by detector jitter. We demonstrate our approach under both continuous-wave and pulsed-pumping regimes, finding excellent agreement with theory. In conclusion, our method reveals not only the collective statistics of the contributing frequency bins but also their temporal shapes—features lost in standard fully integrated autocorrelation measurements.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Entanglement swapping systems toward a quantum internet

We demonstrate conditional entanglement swapping, i.e. teleportation of entanglement, between time-bin qubits at the telecommunication wavelength of 1536.4 nm with high fidelity of 87%. Our system is deployable, utilizing modular, off-the-shelf, fiber-coupled, and electrically controlled components such as electro-optic modulators. It leverages the precise timing resolution of superconducting nanowire detectors, which are controlled and read out via a custom developed graphical user interface. The swapping process is described, interpreted, and guided using characteristic function-based analytical modeling that accounts for realistic imperfections. Our system supports quantum networking protocols, including source-independent quantum key distribution, with an estimated secret key rate of approximately 0.5 bits per sifted bit.

Davis, Samantha I. [Caltech] (ORCID:00000001999481↗

DSMS investment in support of satellite constellations and formation flying

Over the years, NASA has supported unmanned space missions, beyond earth orbit, through a Deep Space Mission System (DSMS) that is developed and operated by the Jet Propulsion Laboratory (JPL) and subcontractors. The DSMS capabilities have been incrementally upgraded since its establishment in the late '50s and are delivered primarily through three Deep Space Communications Complexes (DSCC 's) near Goldstone, California, Madrid, Spain, and Canberra, Australia and from facilities at JPL. Traditionally, mission support (tracking, command, telemetry, etc) is assigned on an individual-mission basis, between each mission and a ground-based asset, independent of other missions. As NASA, and its international partners, move toward flying fullconstellations and precision formations, the DSMS is developing plans and technologies to provide the requisite support. The key activities under way are: (1) integrated communications architecture for Mars exploration, including relays on science orbiters and dedicated relay satellites to provide continuous coverage for orbiters, landers and rovers. JPL is developing an architecture, as well as protocols and equipment, required for the cost-effective operations of such an infrastructure. (2) Internet-type protocols that will allow for efficient operations across the deep-space distances, accounting for and accommodating the long round-trip-light-time. JPL is working with the CCSDS to convert these protocols to an international standard and will deploy such protocol, the CCSDS File Delivery Protocol (CFDP), on the Mars Reconnaissance Orbiter (MRO) and on the Deep Impact (01) missions. (3) Techniques to perform cross-navigation between spacecrafi that fly in a loose formation. Typical cases are cross-navigation between missions that approach Mars and missionsthat are at Mars, or the determination of a baseline for missions that fly in an earth-lead- lag configuration. (4) Techniques and devices that allow the precise metrology and controllability of tightformations for precision constellation missions. In this paper we discuss the four classes of constellatiodformation support with emphasis of DSMS current status (technology and implementation) and plans in the first three areas.

space operations↗

Critical Contribution of Imbalanced Charge Loss to Performance Deterioration of Si-Based Lithium-Ion Cells during Calendar Aging

Increasing the energy density of lithium-ion batteries, and thereby reducing costs, is a major target for industry and academic research. One of the best opportunities is to replace the traditional graphite anode with a high-capacity anode material, such as silicon. However, Si-based lithium-ion batteries have been widely reported to suffer from a limited calendar life for automobile applications. Heretofore, there lacks a fundamental understanding of calendar aging for rationally developing mitigation strategies. Both open-circuit voltage and voltage-hold aging protocols were utilized to characterize the aging behavior of Si-based cells. Particularly, a high-precision leakage current measurement was applied to quantitatively measure the rate of parasitic reactions at the electrode/electrolyte interface. The rate of parasitic reactions at the Si anode was found 5 times and 15 times faster than those of LiNi 0.8 Mn 0.1 Co 0.1 O 2 and LiFePO 4 cathodes, respectively. Here, the imbalanced charge loss from parasitic reactions plays a critical role in exacerbating performance deterioration. In addition, a linear relationship between capacity loss and charge consumption from parasitic reactions provides fundamental support to assess calendar life through voltage-hold tests. These new findings imply that longer calendar life can be achieved by suppressing parasitic reactions at the Si anode to balance charge consumption during calendar aging.

25 ENERGY STORAGE↗

Controlled Synthesis of SnO 2 Nanocrystals with Tunable Band Gaps

Tin(IV) oxide nanocrystals (SnO 2 NCs) have significant potential in various applications, with their performance closely related to their band gap. The band gap is influenced by the size and shape of the NCs, which can be precisely controlled by adjusting reaction conditions. In this study, we present deliberately designed synthesis protocols to produce high-quality SnO 2 NCs with tunable band gaps using different methods. Key factors affecting the synthesis include control of the oxidizing agent, reaction temperature, solvent selection, and reaction time optimization. The resulting NCs were characterized by using TEM, XRD, XPS, and optical spectroscopy. Notably, SnO 2 NCs synthesized by controlling the oxidizing agent (air injection) in a hot organic solution were smaller in size and exhibited abundant oxygen vacancies. In contrast, extending the reaction time or using ethanol as a solvent in hydrothermal systems facilitated larger spherical or rod-like SnO 2 NCs with fewer oxygen vacancies. Further analysis of the band gap and valence band maximum energy revealed that the abundant vacancies in SnO 2 NCs synthesized with the air-controlled hot organic solution method resulted in a narrower band gap and an upshifted valence band. These synthetic strategies illustrate the potential for deliberately designing SnO 2 NCs with optimized electronic structures for various applications.

band gap↗