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At least 19 records

Excessive charge, beam loading, and impedance collapse thresholds for a velvet emitter

Excess emission has been observed from velvet cathodes with total emission times >150 ns. In this diode experiment, we produce a sub-relativistic electron beam with the ability to consistently change γ from 1.2 to 1.5 and β = 0.5–0.75. Electron emission in this particular diode geometry requires electric fields >40 kV/cm. The current increases at steady rates >0.05 A/ns after the head of the pulse, indicating an expansion of the emission surface and reduction in the effective AK gap. Small transients of excess emission (or arcs) are consistently observed for current pulses exceeding 150 ns. The excess emission results in beam loaded levels ≥10 kV on the diode voltage. The effects described here are compounded as diode voltage is increased. The principal objective of these experiments is to quantify electric field emission thresholds, current ramps, excess emission delays, effective beam loading, and impedance collapse.

43 PARTICLE ACCELERATORS↗

Competing electronic states emerging on polar surfaces

Excess charge on polar surfaces of ionic compounds is commonly described by the two-dimensional electron gas (2DEG) model, a homogeneous distribution of charge, spatially-confined in a few atomic layers. Here, by combining scanning probe microscopy with density functional theory calculations, we show that excess charge on the polar TaO 2 termination of KTaO 3 (001) forms more complex electronic states with different degrees of spatial and electronic localization: charge density waves (CDW) coexist with strongly-localized electron polarons and bipolarons. These surface electronic reconstructions, originating from the combined action of electron-lattice interaction and electronic correlation, are energetically more favorable than the 2DEG solution. They exhibit distinct spectroscopy signals and impact on the surface properties, as manifested by a local suppression of ferroelectric distortions.

36 MATERIALS SCIENCE↗

Surface chemistry on a polarizable surface: Coupling of CO with KTaO 3 (001)

Polarizable materials attract attention in catalysis because they have a free parameter for tuning chemical reactivity. Their surfaces entangle the dielectric polarization with surface polarity, excess charge, and orbital hybridization. How this affects individual adsorbed molecules is shown for the incipient ferroelectric perovskite KTaO 3 . This intrinsically polar material cleaves along (001) into KO- and TaO 2 -terminated surface domains. At TaO 2 terraces, the polarity-compensating excess electrons form a two-dimensional electron gas and can also localize by coupling to ferroelectric distortions. TaO 2 terraces host two distinct types of CO molecules, adsorbed at equivalent lattice sites but charged differently as seen in atomic force microscopy/scanning tunneling microscopy. Temperature-programmed desorption shows substantially stronger binding of the charged CO; in density functional theory calculations, the excess charge favors a bipolaronic configuration coupled to the CO. These results pinpoint how adsorption states couple to ferroelectric polarization.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Photoelectron Spectroscopy and Theoretical Study on Monosolvated Cyanate Analogue Clusters ECX-· Sol (ECX- = NCSe-, AsCSe- and AsCS-; Sol = H2O, CH3CN)

Six monosolvated cyanate analogue clusters ECX-?Sol (ECX– = NCSe-, AsCSe-, and AsCS-; Sol = H2O and CH3CN) were investigated using negative ion photoelectron spectroscopy (NIPES). NIPES experiments show that these clusters possess overall similar spectra compared to their respective isolated ECX- anions but shift to higher electron binding energy with solvent CH3CN stabilizing the excess electrons slightly more than H2O. For the ECX-·H2O series, vertical detachment energies and their increments relative to the bare species are measured to be 3.700 / 0.370, 3.085 / 0.425, 3.085 / 0.430 eV for NCSe-, AsCSe- and AsCS-, respectively, while the corresponding values in ECX-·CH3CN series are 3.835 / 0.505, 3.145 / 0.475, and 3.135 / 0.480 eV. Ab initio electronic structure calculations indicate the excess charges located at the terminal N and Se atoms in NCSe- and migrated to the central C atom in AsCSe- and AsCS-. For NCSe-, the solvation is driven by the interactions with the two negatively charged terminal ends, while for AsCSe- and AsCS-, the solvation revolves around the interactions with the central C atom, where all the excess negative charge is concentrated. Two nearly degenerate isomers for NCSe-·H2O are identified, one forming a single strong N…H-O hydrogen bond (HB) and the other featuring a bidentate HB with two hydroxyl H atoms pointing to N and Se ends. In contrast, the negative central C atom in AsCSe-/AsCS- allows the formation of a bifurcated HB with H2O. Similar effects are observed for the acetonitrile case, in which the three H atoms of methyl group interact with the two negatively charged terminal ends in NCSe-, while prefer binding to the central negative carbon atom in AsCSe-/AsCS-. The different binding motifs derived in this work may suggest distinctly different solvation properties in NCSe- versus AsCSe-/AsCS-, with the former anion leading to asymmetric solvation at the N end of the solute, while the latter species creating more ‘isotropic’ solvation around the central C equatorial plane. This work was supported by U.S. Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences, and performed using EMSL, a national scientific user facility sponsored by DOE’s Office of Biological and Environmental Research and located at Pacific Northwest National Laboratory, which is operated by Battelle Memorial Institute for the DOE. The theoretical calculations were conducted on EMSL’s “Cascade” Supercomputer.

Yuan, Qinqin↗

Plasmon-Assisted Direct Interfacial Charge Transfer Enables Molecular Photodissociation on Metal Surfaces

Plasmonic photocatalysis is promising for driving reactions out of equilibrium, leading to enhanced rates and selectivity, but the energy and charge transfer mechanism at the interface remains to be explored. Here, we revisit visible-light plasmon-assisted photodissociation of dimethyl disulfide molecules (S–S bond cleavage) over single-crystalline silver and copper surfaces, using density functional theory calculations and the delta self-consistent field method. We investigate different excitation mechanisms and show that the direct intramolecular excitations, as suggested earlier to explain the experimental findings, cannot address the observed catalytic bond activation. Instead, the calculated excitations mediated by the metal Fermi level agree well with the threshold energy and maximum-yield energy in experiments and suggest that the reaction follows a direct interfacial charge transfer mechanism. Furthermore, we show that the excitation energy and the charge transfer can be modulated by introducing excess charges in the systems. Here, this study thus provides a fundamental understanding of interfacial charge transfer for driving plasmonic photocatalysis.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Characterization of Engineered Pinholes in Dielectric Stacks of High-Performance Poly-Silicon Passivating Contacts

Passivating contact structures are expected to be implemented in > 50 % of industrially manufactured silicon photovoltaics produced over the next decade. One type of passivating contact utilizes a c-Si/dielectric/doped poly-Si layer stack. In cases where the dielectric layer/s are thick enough to impede tunneling of charge carriers, it was shown that disruptions (pinholes) in the dielectric are required to facilitate transport.[1] Pinholes in SiOx layers have typically been formed via high temperature annealing that allows localized breakup of this layer. This approach has drawbacks; 1) high temperature annealing increases cost and 2) pinhole formation is sensitive to both the SiOx thickness and wafer surface morphology making control of density and dimensions difficult. Recently, an alternative method of pinhole formation relying on metal assisted chemical etching (MACE) was shown to allow control over pinhole areal densities.[2] Such control is critical as both the density and dimensions of the pinholes determine saturation current density and contact resistivity. The contacts formed with the MACE process are known as polysilicon on locally etched oxide or PLEO and also result in low saturation current densities and low contact resistivity due to SiOx passivation and engineered pinhole enabled charge carrier transport, respectively. Photovoltaic devices with efficiencies greater than 20 % have recently been produced with these contacts, demonstrating the viability of contacts with engineered pinholes. Experimentally, preferential local collection of excess charge carriers has been observed and definitively connected to thermally induced pinholes in thick SiOx layers within c-Si/SiOx/poly-Si contact structures.[3] Recent work has shown that enhanced local collection of excess carriers can result from enhanced diffusion of dopants through pinholes in SiOx layers.[4] This study was restricted to simulation and characterization of poly-Si with the same dopant type as the c-Si substrates. In this work we focus our analysis on boron doped poly-Si with different stacks of dielectric layers (PLEO and PLENO where the N refers to a SiNy layer on top of the SiOx) on n-type crystalline silicon. This provides insight into how engineered pinholes affect charge collection across the p-n junction. Like previous observations of locally enhanced carrier transport through thermally induced pinholes, the EBIC images in Figure 1 shows that engineered pinholes in PLENO (1a) and PLEO (1b) also exhibit locally enhanced carrier transport at pinhole locations (bright spots in the images). We will also present detailed correlative nanoscale structural and chemical analysis using TEM imaging and EELS analysis with sample preparation guided by EBIC imaging. This allows us to connect the density and dimensions of engineered pinholes with charge carrier transport and measured photovoltaic device parameters. Our findings will accelerate the adoption of PLEO and PLENO contacts in industrially manufactured photovoltaic devices.

EBIC↗

Wide Dynamic Range, 10 kHz Framing Detector for 4D-STEM

Hybrid pixel array detectors (PADs) have expanded the fidelity, frame rate and dynamic range for diffractive imaging in electron microscopes. With their high detective quantum efficiencies (DQE), the noise performance limited is by counting statistics. Consequently, improving the precision and quality of high-speed images requires increasing the linear dynamic range and the maximum usable beam current. Here we describe a new prototype hybrid PAD, our next-generation electron microscope pixel array detector (NG-EMPAD) that expands both the framing speed and dynamic range of our earlier EMPAD design. Like the EMPAD, electrons are detected directly in a 500 μm thick silicon sensor bonded pixel-by-pixel to a custom signal processing integrated circuit (IC). Each pixel in the IC integrates charge carriers produced by the high-energy incident electrons. Here, in-pixel circuitry uses a combination of techniques to extend the dynamic range, including adaptive gain switching and the quantitative excess charge dumping similar to the EMPAD. In-pixel buffering of signals during readout allows acquiring a new image while reading out the previously collected image, yielding an image collection duty cycle of nearly 100%.

36 MATERIALS SCIENCE↗

Adsorption, charge transfer and a coverage-driven transition of alkali metals on rutile TiO 2 (110)

The interaction of alkali metals with metal oxide surfaces is central to tuning surface reactivity in heterogeneous catalysis and photocatalysis. Here we present a comprehensive DFT+U study of the adsorption of alkali metals (Li, Na, K, Rb, Cs) on the (110) surface of rutile TiO 2 . At low coverage (θ = 1/8), all alkali metals bind preferentially to bridging oxygen sites with adsorption energies in the range −4.06 to −3.33 eV, transferring nearly one full electron (0.92–0.99 |e|) to the substrate and inducing Ti 4+ → Ti 3+ reduction. The excess charge localizes preferentially at subsurface Ti sites in the form of small polarons. Diffusion barriers indicate facile motion along bridging-oxygen rows, whereas inter-row hopping is strongly hindered. Coverage effects were examined systematically for potassium: adsorption energy and charge transfer per K atom decrease monotonically with increasing θ. Strikingly, a sharp energy discontinuity occurs between θ = 4/8 and θ = 5/8 (ΔE ≈ 1 eV per atom), which we identify as a coverage-driven structural transition arising from steric packing constraints and enhanced K–K electrostatic repulsion once every (1×1) surface cell is occupied. This structural transition perfectly correlates with a dramatic drop in the work function down to an ultra-low minimum of 0.84 eV at θ=5/8, followed by a metallization- driven recovery at higher coverages. Ab initio molecular dynamics simulations confirm zigzag K arrangements at moderate coverage (θ = 1/3), while at high coverage (θ = 2/3) short-range K–K correlations emerge without long-range order. These results provide atomistic insight into the structure–activity relationships underlying alkali promotion effects on oxide-supported catalysts.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Scaling Theory of Neutral Sequence-Specific Polyampholytes

The primary sequence of charged monomers in a polyampholyte has a profound effect on its conformational properties. In this work, theory and simulations are used to predict how the sequence influences the single-chain behavior of globally neutral polyampholytes (PAs) under salt-free conditions. We consider PAs with Markov statistics of charges, where charge “blockiness” is defined by the correlation λ along the chain. These PAs cover a wide spectrum of primary sequences ranging from alternating (λ = –1) to ideally random (λ = 0) to diblock (λ → 1). In a theta solvent, sufficiently long PAs of any primary sequence form globules, but their internal structure, as well as the position and sharpness of the coil–globule transition, crucially depends on λ. Further, using scaling arguments, we demonstrate that the excess charge of the globule correlation volume (blob) increases with increasing λ, thereby inducing stronger Coulomb attractions between neighboring blobs of opposite charge and leading to higher density PA globules. Scaling predictions are supported by the results of the random phase approximation (RPA) and coarse-grained molecular dynamics simulations. A generalization to good and poor solvents is presented that enables the construction of the resulting scaling diagram of PA conformations in terms of solvent quality and charge blockiness, thereby providing a comprehensive molecular understanding of the role of the PA primary sequence in the chain structure.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Extremely Overdoped Superconducting Cuprates via High Pressure Oxygenation Methods

Within the cuprate constellation, one fixed star has been the superconducting dome in the quantum phase diagram of transition temperature vs. the excess charge on the Cu in the CuO2-planes, p, resulting from O-doping or cation substitution. However, a more extensive search of the literature shows that the loss of the superconductivity in favor of a normal Fermi liquid on the overdoped side should not be assumed. Many experimental results from cuprates prepared by high-pressure oxygenation show Tc converging to a fixed value or continuing to slowly increase past the upper limit of the dome of p = 0.26–0.27, up to the maximum amounts of excess oxygen corresponding to p values of 0.3 to > 0.6. These reports have been met with disinterest or disregard. Our review shows that dome-breaking trends for Tc are, in fact, the result of careful, accurate experimental work on a large number of compounds. This behavior most likely mandates a revision of the theoretical basis for high-temperature superconductivity. That excess O atoms located in specific, metastable sites in the crystal, attainable only with extreme O chemical activity under HPO conditions, cause such a radical extension of the superconductivity points to a much more substantial role for the lattice in terms of internal chemistry and bonding.

Sederholm, Linda (ORCID:0000000172090509)↗

Solar cell contacts: quantifying the impact of interfacial layers on selectivity, recombination, charge transfer, and V oc

Interfacial layers (IFLs) are ubiquitous in solar cells, but their precise impact on carrier transfer rates and the relation of these rates to performance metrics and the concepts of selectivity and recombination is lacking. We report the use of a well-defined interdigitated back-contact (IBC) silicon solar cell to determine the precise role of such IFLs. We characterize the action of the common IFL spiro-OMeTAD by making it a third contact to the IBC cell. This architecture creates three solar cells within a single structure that, with numerical simulation, provide the exchange current densities (i.e., charge transfer rates) for electrons ( J 0n ) and holes ( J 0p ) and the quasi Fermi-level splitting in the absorber, which measures the balance of generation and recombination. Further, we describe the relation of V oc to contact recombination, the asymmetry in electron/hole collection rates at a single contact (contact selectivity), and the asymmetry in collection rates of the same carrier at separate contacts (carrier selectivity). Relative to bare gold, neat spiro-OMeTAD reduces J 0n and J 0p (their geometric mean ( J 0n J 0p ) 0.5 decreases by 10 4 ), decreasing contact recombination. Addition of the common dopant Li-TFSI and air increase J 0p / J 0n by 10 6 with little effect on ( J 0n J 0p ) 0.5 , increasing contact hole selectivity. The significant increase in V oc observed by introducing spiro-OMeTAD/Li-TFSI IFLs into the cells studied, however, is due to an increase in the carrier selectivity rather than the contact selectivity or recombination of the spiro-OMeTAD-modified contact. Operando measurements further show voltage-dependent changes in the J 0 s, demonstrating that spiro-OMeTAD contributes to hysteresis. Broader context : Solar cells operate by photogenerating excess charge carriers in an absorber material and, in competition with recombination, asymmetrically extracting them at so-called carrier-selective contacts, one that ideally collects only electrons and the other, only holes. Particularly in emerging photovoltaics such as perovskites, thin layers of organic semiconductors or related materials are introduced between the absorber and contact to improve power conversion efficiency. In terms of interfacial charge transfer, a prevailing view is that such interfacial layers improve performance by helping block the collection of the undesired carrier, considered a form of recombination. We use a novel platform to study the simultaneous impact of spiro-OMeTAD, a common interfacial layer, on the collection of electrons, the collection of holes, and on the recombination of electrons and holes. We quantitatively demonstrate that spiro-OMeTAD layers indeed passivate the contact toward recombination, but that larger improvements in the open-circuit voltage, a key cell metric, can come not only from this but also from increasing the asymmetry of the collection of electrons in the system. Further, operando measurements show transient changes in the properties of spiro-OMeTAD which suggest that it contributes to hysteresis phenomena commonly observed in perovskite and other solar cells.

36 MATERIALS SCIENCE↗

Fermi Level Engineering of Passivation and Electron Transport Materials for p-Type CuBi 2 O 4 Employing a High-Throughput Methodology

Metal oxide semiconductors are promising for solar photochemistry if the issues of excessive charge carrier recombination and material degradation can be resolved, which are both influenced by surface quality and interface chemistry. Coating the semiconductor with an overlayer to passivate surface states is a common remedial strategy but is less desirable than application of a functional coating that can improve carrier extraction and reduce recombination while mitigating corrosion. Here, a data-driven materials science approach utilizing high-throughput methodologies, including inkjet printing and scanning droplet electrochemical cell measurements, is used to create and evaluate multi-element coating libraries to discover new classes of candidate passivation and electron-selective contact materials for p-type CuBi 2 O 4 . The optimized overlayer (Cu 1.5 TiO z ) improves the onset potential by 110 mV, the photocurrent by 2.8×, and the absorbed photon-to-current efficiency by 15.5% compared to non-coated photoelectrodes. It is shown that these enhancements are related to reduced surface recombination through passivation of surface defect states as well as improved carrier extraction efficiency through Fermi level engineering. This work presents a generalizable, high-throughput method to design and optimize passivation materials for a variety of semiconductors, providing a powerful platform for development of high-performance photoelectrodes for incorporation into solar-fuel generation systems.

36 MATERIALS SCIENCE↗

Unraveling the Stable Cathode Electrolyte Interface in all Solid‐State Thin‐Film Battery Operating at 5 V

Abstract Spinel‐type LiNi 0.5 Mn 1.5 O 4 (LNMO) is one of the most promising 5 V‐class cathode materials for Li‐ion batteries that can achieve high energy density and low production costs. However, in liquid electrolyte cells, the high voltage causes continuous cell degradation through the oxidative decomposition of carbonate‐based liquid electrolytes. In contrast, some solid‐state electrolytes have a wide electrochemical stability range and can withstand the required oxidative potential. In this work, a thin‐film battery consisting of an LNMO cathode with a solid lithium phosphorus oxynitride (LiPON) electrolyte is tested and their interface before and after cycling is characterized. With Li metal as the anode, this system can deliver stable performance for 600 cycles with an average Coulombic efficiency >99%. Neutron depth profiling indicates a slight overlithiated layer at the interface prior to cycling, a result that is consistent with the excess charge capacity measured during the first cycle. Cryogenic electron microscopy further reveals intimate contact between LNMO and LiPON without noticeable structure and chemical composition evolution after extended cycling, demonstrating the superior stability of LiPON against a high voltage cathode. Consequently, design guidelines are proposed for interface engineering that can accelerate the commercialization of a high voltage cell with solid or liquid electrolytes.

25 ENERGY STORAGE↗

Uncoupling Electrokinetic Flow Solutions

The continuum-scale electrokinetic porous-media flow and excess charge redistribution equations are uncoupled using eigenvalue decomposition. The uncoupling results in a pair of independent diffusion equations for “intermediate” potentials subject to modified material properties and boundary conditions. The fluid pressure and electrostatic potential are then found by recombining the solutions to the two intermediate uncoupled problems in a matrix-vector multiplication. Expressions for the material properties or source terms in the intermediate uncoupled problem may require extended precision or careful rewriting to avoid numerical cancellation, but the solutions themselves can typically be computed in double precision. The approach works with analytical or gridded numerical solutions and is illustrated through two examples. The solution for flow to a pumping well is manipulated to predict streaming potential and electroosmosis, and a periodic one-dimensional analytical solution is derived and used to predict electroosmosis and streaming potential in a laboratory flow cell subjected to low frequency alternating current and pressure excitation. The examples illustrate the utility of the eigenvalue decoupling approach, repurposing existing analytical solutions or numerical models and leveraging solutions that are simpler to derive for coupled physics.

58 GEOSCIENCES↗

Measuring the flatband potential in 2D semiconductors: Pitfalls and a possible SECCM solution

The flatband potential (V fb ) is a critical parameter in semiconductor electrochemistry, defining the potential at which no excess charge exists at the semiconductor/electrolyte interface. It serves as a key reference for interpreting charge transfer kinetics and current–voltage behavior. However, conventional methods like Mott–Schottky analysis fail for atomically thin 2D materials due to the breakdown of the depletion approximation. This perspective examines the limitations of traditional V fb measurements for 2D semiconductors and the experimental challenges that arise. To address these issues, we propose using scanning electrochemical cell microscopy (SECCM) to spatially resolve the potential of zero charge (V pzc ), equivalent to V fb . This approach mitigates sample heterogeneity issues, such as pinholes or multilayer defects, and offers a pathway to more accurate electrochemical characterization. Ultimately, this method will enhance understanding of current–potential behavior in 2D materials, supporting the design of advanced systems for photoelectrocatalysis, energy conversion, and sensing.

2D semiconductors↗

Infrared Nanoimaging of Hydrogenated Perovskite Nickelate Memristive Devices

Solid-state devices made from correlated oxides, such as perovskite nickelates, are promising for neuromorphic computing by mimicking biological synaptic function. However, comprehending dopant action at the nanoscale poses a formidable challenge to understanding the elementary mechanisms involved. Here, we perform operando infrared nanoimaging of hydrogen-doped correlated perovskite, neodymium nickel oxide (H-NdNiO 3 , H-NNO), devices and reveal how an applied field perturbs dopant distribution at the nanoscale. This perturbation leads to stripe phases of varying conductivity perpendicular to the applied field, which define the macroscale electrical characteristics of the devices. Hyperspectral nano-FTIR imaging in conjunction with density functional theory calculations unveils a real-space map of multiple vibrational states of H-NNO associated with OH stretching modes and their dependence on the dopant concentration. Moreover, the localization of excess charges induces an out-of-plane lattice expansion in NNO which was confirmed by in situ X-ray diffraction and creates a strain that acts as a barrier against further diffusion. Our results and the techniques presented here hold great potential for the rapidly growing field of memristors and neuromorphic devices wherein nanoscale ion motion is fundamentally responsible for function.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Ultrafast Formation of Jahn–Teller Polarons Revealed by State-Selective Excitation in Correlated Spinel Co 3 O 4

Jahn–Teller polarons are quasiparticles that stem from symmetry breaking and strong local electron–phonon coupling. They originate from an excess charge carrier being dressed by a local lattice distortion, caused by the Jahn–Teller effect, and they critically impact electrical, structural, and magnetic properties in transition metal oxides. The observation of the microscopic steps involved in their formation is essential for enabling control over material properties through the targeted activation of local, site-specific modifications with light pulses. While Jahn–Teller distortion associated with polaron formation was predicted to contribute significantly to changes in electronic band gap and optical properties in Co 3 O 4 , its experimental observation remains elusive, requiring signatures of local symmetry reduction. In this work, we demonstrate Jahn–Teller polaron formation in spinel Co 3 O 4 . By exciting electronic transitions at 3.10 eV and 1.55 eV, we target either the Oh Cobalt(III) or the Td Cobalt(II) ions, and drive the subsequent coherent responses of the system through two different pathways. For the former, we demonstrate that ligand-to-metal charge transfer leads to Jahn–Teller polaron formation, which is linked to the deformation potential and magnetoelastic coupling. For the latter, we identify the coherent excitation of a T 2g phonon mode launched by on-site d-d electronic transitions. Key to our observations is the ability to target site-specific electronic excitations in spinel Co 3 O 4 using ultrafast optical pulses, while monitoring the ensuing low-energy collective modes through the coherent time-domain response of the material. Our approach, which combines the comparative analysis of experimental fingerprints with the support from density functional theory calculations, is broadly applicable to systems in which Jahn–Teller polaron physics has been theoretically predicted but remains experimentally unverified, and it underscores the potential of electronic-state targeting as a route to selectively excite and probe quasiparticle dynamics in solids.

Lattices↗