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Results for “Vacancy diffusion”

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At least 73 records · Page 4
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Diffusion-mediated chemical concentration variation and void evolution in ion-irradiated NiCoFeCr high-entropy alloy

High-entropy alloys (HEAs) are proposed as potential structural materials for advanced nuclear systems, but little is known about the response of matrix chemistry in HEAs upon irradiation. Here, we reveal a substantial change of matrix chemical concentration as a function of irradiation damage (depth) in equiatomic NiCoFeCr HEA irradiated by 3 MeV Ni ions. After ion irradiation, the matrix contains more Fe/Cr in depth shallower than ~900–1000 nm but more Ni/Co from ~900–1000 nm to the end of the ion-damaged region due to the preferential diffusion of vacancies through Fe/Cr. Preferential diffusion also facilitates migration of vacancies from high radiation damage region to low radiation damage region, leading to no void formation below ~900–1000 nm and void formation around the end of the ion-damaged region at a fluence of 5 × 10 16 cm -2 (~123 dpa, displacements per atom, peak dose under full cascade mode). As voids grow significantly at an increased fluence (8 × 10 16 cm -2 , 196 dpa), the matrix concentration does not change dramatically due to new voids formed below ~900–1000 nm.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2021

Diffusion-mediated chemical concentration variation and void evolution in ion-irradiated NiCoFeCr high-entropy alloy

High-entropy alloys (HEAs) are proposed as potential structural materials for advanced nuclear systems, but little is known about the response of matrix chemistry in HEAs upon irradiation. Here, we reveal a substantial change of matrix chemical concentration as a function of irradiation damage (depth) in equiatomic NiCoFeCr HEA irradiated by 3 MeV Ni ions. After ion irradiation, the matrix contains more Fe/Cr in depth shallower than ~900–1000 nm but more Ni/Co from ~900–1000 nm to the end of the ion-damaged region due to the preferential diffusion of vacancies through Fe/Cr. Preferential diffusion also facilitates migration of vacancies from high radiation damage region to low radiation damage region, leading to no void formation below ~900–1000 nm and void formation around the end of the ion-damaged region at a fluence of 5 × 10 16 cm –2 (~123 dpa, displacements per atom, peak dose under full cascade mode). As voids grow significantly at an increased fluence (8 × 10 16 cm –2 , 196 dpa), the matrix concentration does not change dramatically due to new voids formed below ~900–1000 nm.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Effects of Minor Alloying Elements and Impurities on Long-Range Ordering in Ni-Cr-Based Alloys

There is growing concern about long-range ordered Ni2Cr phase forming during thermal aging of Ni-Cr-based alloys (e.g., 690, 625), which leads to lattice contraction, hardening, and embrittlement, possibly limiting the service life of light water reactor components. The timespan when this phase transformation occurs can vary in orders of magnitude between model and commercial alloys and is likely related to the presence of major (e.g., Fe, Mo) and minor alloying elements (e.g., Ti, Si, Nb). These minor alloying elements have been hypothesized to bind with vacancies and therefore limit diffusion. Contrarily, impurities (e.g., P, S) are believed to enhance vacancy diffusion and accelerate the Ni2Cr phase transformation because of a vacancy drag mechanism. This study implements experimental isothermal aging of model Ni-Cr binary and Ni-Cr-X ternary alloys, where X is Ti, Si, Nb, and P to help understand how each minor alloying element affects the Ni2Cr phase transformation timeline. Synchrotron-based X-ray diffraction and hardness measurements are used to quantify the evolution of the Ni2Cr phase transformation through a variety of short aging times to identify the onset of long-range order. These experimental results are compared to first principles simulations from literature to understand the mechanism by which each element impacts vacancy diffusion and the formation of the long-range ordered phase, Ni2Cr.

36 - MATERIALS SCIENCE↗
DOE OSTI2023

Coupled chemo-mechanical modeling of point-defect diffusion in a crystal plasticity fast Fourier transform framework

Below the yield strength and at moderate-to-high homologous temperatures, the inelastic deformation of metals is mostly governed/rate-controlled by vacancy diffusion-mediated processes. As a function of grain size, stress, temperature and dislocation content, vacancies (or atoms) can adopt preferential diffusion paths across grain interiors, along grain boundaries, or towards and along dislocations, resulting in climb and self-climb. In the steady state and under constant load, grain boundary and grain bulk vacancy diffusion-mediated plasticity have been described in seminal works by Coble and by Nabarro and Herring, respectively. Yet, the interplay between all aforementioned potential diffusion pathways has not been comprehensively mapped. This work presents a thermodynamically-consistent full-field model integrated within a voxel-based elasto-viscoplastic fast Fourier transform framework, which considers the coupling between the diffusion-mediated plasticity mechanisms. In the proposed approach, the kinetics and kinematics of plastic deformation due to vacancy diffusion along grain boundaries and grain bulk, as well as the exchange between grain boundaries and bulk are described explicitly. A homogenization approach at the voxel level is further introduced to simultaneously consider bulk and grain boundary diffusion in a numerically efficient fashion. The new formulation predicts the expected strain rate dependencies and the scaling of the steady-state creep rate with respect to grain size, temperature, and stress. Finally, the model predicts the transition from grain bulk to grain boundary-dominated diffusion with reduction in grain size, a significant step towards capturing transitions in deformation behavior without any phenomenological or ad-hoc adjustments.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Vacancy formation energies and migration barriers in multi-principal element alloys

Multi-principal element alloys (MPEAs) continue to garner interest as structural and plasma-facing materials due to their structural (phase) stability and increased resistance to radiation damage. Despite sensitivity of mechanical behavior to irradiation and point-defect formation, there has been scant attention on understanding vacancy stability and diffusion in refractory-based MPEAs. Using density-functional theory, we examine vacancy stability and diffusion barriers in body-centered cubic (Mo 0.95 W 0.05 ) 0.85 Ta 0.10 (TiZr) 0.05 . The results in this MPEA show strong dependence on environment, originating from local lattice distortion associated with charge-transfer between neighboring atoms that vary with different chemical environments. We find a correlation between degree of lattice distortion and migration barrier: (Ti, Zr) with less distortion have lower barriers, while (Mo, W) with larger distortion have higher barriers, depending up local environments. Under irradiation, our findings suggest that (Ti, Zr) are significantly more likely to diffuse than (Mo, W) while Ta shows intermediate effect. Finally, material degradation caused by vacancy diffusion can be controlled by tuning composition of alloying elements to enhance creep strength at extreme operating temperatures and harsh conditions.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Atomic Diffusion, Segregation, and Grain Boundary Migration in Nickel-Based Alloys from Molecular Dynamics Simulations

Grain boundary diffusion and metal mobility in alloys control material performance in many applications and yet remain poorly understood at a mechanistic level. With advances in accessible time and length scales for computational molecular simulations, and recent force field developments, we now possess tools to help unravel those mechanisms. Using large-scale molecular dynamics simulations, here we examined vacancy-mediated diffusion processes in Ni-5Cr alloy with low and high-energy grain boundaries. We show that atomic diffusion inside the grain boundary plane is about four times higher than bulk diffusion, at any temperature, and exhibits a typical Arrhenius behavior with a very small energy barrier (0~.8 eV for Cr and 0.7 eV for Ni within 1300-1600 K). Additionally, the fastest diffusing species inverts; Cr diffusion was faster than Ni in the bulk but slower in the grain boundaries. This is attributed to the creation of high cohesive energy clusters of Cr at the grain boundary. Grain boundary migration was also observed to be temperature dependent and appears to be two times higher in the 5% Cr alloy than in pure Ni, highlighting the important role of the alloying element on grain boundary motion.

Simonnin, Pauline GN↗
DOE OSTI2026

Model Parameter Development for Complex Materials: Species-Specific Diffusion Barriers in 316 Stainless Steel from Systematic DFT Calculations

Vacancy-mediated diffusion barriers in 316 stainless steel have been systematically calculated using density functional theory to provide essential parameters for mesoscale microstructure evolution models. A statistical sampling approach employing 210 nudged elastic band calculations across multiple special quasi-random structures captures the effects of local chemical environments in this concentrated alloy. The computational methodology addresses challenges specific to chemically disordered systems, including proper magnetic treatment throughout multi-step calculations and validation against experimental structural properties. The calculated activation barriers reveal clear species-dependent diffusion behavior with the hierarchy Ni >> Fe ˜ Cr >> Mo. Nickel exhibits the highest barriers (0.74–1.31 eV, mean 1.045 eV), confirming its role as the slowest-diffusing major component. Iron and chromium show similar moderate barriers averaging 0.587 eV and 0.522 eV, respectively. Remarkably, molybdenum demonstrates exceptionally low barriers (0.12–0.28 eV, mean 0.194 eV), suggesting much higher mobility than previously recognized and potentially significant implications for precipitation kinetics and microstructure evolution. The barrier ranges remain consistent across different 316 SS compositions, supporting parameter transferability for modeling applications. The overall mean barrier of 0.64 eV provides a practical approximation for phase field simulations, while species-specific values enable detailed treatments of diffusion-controlled processes. This systematic approach establishes a validated framework for generating diffusion parameters in other concentrated alloys where experimental data are limited, while providing the first systematic set of species-specific barriers for predictive modeling of 316 stainless steel microstructure evolution.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Edge-Mediated Annihilation of Vacancy Clusters in Monolayer Molybdenum Diselenide (MoSe 2 ) under Electron Beam Irradiation

Here annihilation of vacancy clusters in monolayer molybdenum diselenide (MoSe2) under electron beam irradiation is reported. In situ high-resolution transmission electron microscopy observation reveals that the annihilation is achieved by diffusion of vacancies to the free edge near the vacancy clusters. Monte Carlo simulations confirm that it is energetically favorable for the vacancies to locate at the free edge. By computing the minimum energy path for the annihilation of one vacancy cluster as a case study, it is further shown that electron beam irradiation and pre-stress in the suspended MoSe2 monolayer are necessary for the vacancies to overcome the energy barriers for diffusion. The findings suggest a new mechanism of vacancy healing in 2D materials and broaden the capability of electron beam for defect engineering of 2D materials, a promising way of tuning their properties for engineering applications.

2D materials↗
DOE OSTI2021

Impacts of vacancy-induced polarization and distortion on diffusion in Solid Electrolyte Li 3 OCl

We report that Lithium-rich oxychloride antiperovskites are promising solid electrolytes for enabling next-generation batteries. Here, we report a comprehensive study varying Li + concentrations in Li 3 OCl using ab initio molecular dynamics simulations. The simulations accurately capture the complex interactions between Li + vacancies (V$^'_{Li}$), the dominant mobile species in Li 3 OCl. The V$^'_{Li}$ polarize and distort the host lattice, inducing additional non-vacancy-mediated diffusion mechanisms and correlated diffusion events that reduce the activation energy barrier at concentrations as low as 1.5% V$^'_{Li}$. Our analyses of discretized diffusion events in both space and time illustrate the critical interplay between correlated dynamics, polarization and local distortion in promoting ionic conductivity in Li 3 OCl.

25 ENERGY STORAGE↗
DOE OSTI2025

First-principles investigation of cerium and neodymium diffusion in BCC chromium and vanadium via vacancy-mediated transport

Lanthanide transport plays a crucial role in the performance and longevity of metallic nuclear fuels. This study examines the diffusion behavior of Ce and Nd—two major fission products—in body-centered cubic (BCC) Cr and V, which are potential liner or coating materials for mitigating fuel-cladding chemical interactions (FCCI). Using density functional theory (DFT) calculations and self-consistent mean-field (SCMF) analysis, the vacancy-mediated diffusion coefficients are evaluated. Our findings reveal that Ce and Nd act as oversized solutes and are strongly bound to vacancies in BCC Cr and V, with diffusivities in Cr significantly lower than in V and in hexagonal closed-packed (HCP) Zr, as investigated in our previous work. The activation energies for Ce and Nd diffusion are 3.39 and 3.32 eV, respectively, in BCC Cr, and 2.56 and 2.33 eV, respectively, in BCC V. Analysis of vacancy drag and partial diffusion coefficient ratios indicates a strong tendency for lanthanide enrichment at vacancy sinks in BCC Cr, and to a lesser extent in BCC V, with this effect persisting up to the melting point in Cr and remaining substantial for Nd in V at high temperatures. Under irradiation, the increase in vacancy concentration is expected to enhance lanthanide transport, potentially accelerating interactions at liner-cladding interfaces. Although BCC Cr exhibits relatively low lanthanide diffusivities under equilibrium conditions, the expected segregation tendencies under irradiation suggest that Zr liners may be a more favorable option. Further investigations using rate theory, cluster dynamics, and phase-field modeling are required to quantitatively assess the performance of these materials in reactor environments.

36 - MATERIALS SCIENCE↗
DOE OSTI2026

Elucidating Lithium Transport Mechanisms in Disordered LiF from Machine-Learning Molecular Dynamics Simulations

Lithium fluoride (LiF) is a ubiquitous component of solid- and cathode–electrolyte interphases, yet its functional role remains unclear under the structural and chemical heterogeneity typical of cycling batteries. Here, we systematically quantify how structural disorder, off-stoichiometry, and strain govern Li-ion transport in LiF. Using a machine-learning potential to enable extensive molecular-dynamics sampling, we compare crystalline and amorphous LiF, Li 0.95 F, and LiF 0.95 , and evaluate the impact of small homogeneous deformations. Defect-free crystalline LiF is effectively ion-blocking at 300−500 K, whereas amorphization generates free-volume–assisted percolation pathways that facilitates Li-ion diffusion. At elevated temperatures, thermodynamically driven crystallization disrupts these pathways, leading to non-Arrhenius behavior. In crystalline phases, Li deficiency activates vacancy-mediated diffusion, while in amorphous LiF, transport is governed primarily by network connectivity. Strain is found to have only a marginal effect on Li mobility in both crystalline and amorphous structures.

Batteries↗
DOE OSTI2025

The Crucial Role of Vacancy Concentration in Enabling Superatomic Diffusion in Lithium Intermetallics

Anode-free solid-state Li batteries promise significant increases in energy densities compared to current commercial batteries that rely on liquid electrolytes. Major challenges persist in controlling morphological evolution during the plating and stripping of lithium metal at the anode current collector. Elemental additives that alloy with lithium have been found to modify the plating and stripping behavior of lithium. Many alloying elements form intermetallics with lithium and the mobility of Li through these intermetallics is believed to have an important effect on morphological evolution. This study shows that Li transport coefficients through intermetallics span a wide range in values, with the B32 LiAl intermetallic predicted to have a Li tracer diffusion coefficient as high as 10 –6 cm 2 /s at room temperature, which is 8 orders of magnitude larger than that of isostructural B32 LiZn. This work demonstrates the crucial role of vacancy concentration in controlling the mobility of Li atoms through intermetallics. While the migration barriers for Li-vacancy exchanges in both LiAl and LiZn are remarkably low, the superatomic conductivity in LiAl is shown to arise from the unique electronic structure of the B32 LiAl compound, which favors high concentrations of vacancies.

25 ENERGY STORAGE↗
DOE OSTI2022

Oxygen diffusion in garnet: Experimental calibration and implications for timescales of metamorphic processes and retention of primary O isotopic signatures

Knowledge of oxygen diffusion in garnet is crucial for a correct interpretation of oxygen isotope signatures in natural samples. A series of experiments was undertaken to determine the diffusivity of oxygen in garnet, which remains poorly constrained. The first suite included high-pressure (HP), nominally dry experiments performed in piston-cylinder apparatus at: (1) T = 1050–1600 °C and P = 1.5 GPa and (2) T = 1500 °C and P = 2.5 GPa using yttrium aluminum garnet (YAG; Y3Al5O12) cubes. Second, HP H2O-saturated experiments were conducted at T = 900 °C and P = 1.0–1.5 GPa, wherein YAG crystals were packed into a YAG + Corundum powder, along with 18O-enriched H2O. Third, 1 atm experiments with YAG cubes were performed in a gas-mixing furnace at T = 1500–1600 °C under Ar flux. Finally, an experiment at T = 900 °C and P = 1.0 GPa was done using a pyrope cube embedded into pyrope powder and 18O-enriched H2O. Experiments using grossular were not successful. Profiles of 18O/(18O+16O) in the experimental charges were analyzed with three different secondary ion mass spectrometers (SIMS): sensitive high-resolution ion microprobe (SHRIMP II and SI), CAMECA IMS-1280, and NanoSIMS. Considering only the measured length of 18O diffusion profiles, similar results were obtained for YAG and pyrope annealed at 900 °C, suggesting limited effects of chemical composition on oxygen diffusivity. However, in both garnet types, several profiles deviate from the error function geometry, suggesting that the behavior of O in garnet cannot be fully described as simple concentration-independent diffusion, certainly in YAG and likely in natural pyrope as well. The experimental results are better described by invoking O diffusion via two distinct pathways with an inter-site reaction allowing O to move between these pathways. Modeling this process yields two diffusion coefficients (D values) for O, one of which is approximately two orders of magnitude higher than the other. Taken together, Arrhenius relationships are:logDm2s-1=-7.2(±1.3)+(-321(±32)kJmol-12.303RT) for the slow pathway, andlogDm2s-1=-5.4(±0.7)+(-321(±20)kJmol-12.303RT) for the fast pathway. We interpret the two pathways as representing diffusion following vacancy and inter-stitial mechanisms, respectively. Regardless, our new data suggest that the slow mechanism is prevalent in garnet with natural compositions, and thus is likely to control the retentivity of oxygen isotopic signatures in natural samples. The diffusivity of oxygen is similar to Fe-Mn diffusivity in garnet at 1000–1100 °C and Ca diffusivity at 850 °C. However, the activation energy for O diffusion is larger, leading to lower diffusivities at P-T conditions characterizing crustal metamorphism. Therefore, original O isotopic signatures can be retained in garnets showing major element zoning partially re-equilibrated by diffusion, with the uncertainty caveat of extrapolating the experimental data to lower temperature conditions.

Geochemistry & Geophysics↗
DOE OSTI2025

Atomic-Scale Insights into Carbon Dissolution in α-, γ-, and θ-Al 2 O 3 : Phase-dependent Transport Dynamics from First-Principles Calculations

α-Al 2 O 3 exhibits superior carburizing corrosion resistance compared to metastable γ-Al 2 O 3 and θ-Al 2 O 3 phases in high-temperature CO 2 environments, yet its atomic-scale origins remain unclear. Using first-principles density functional theory, we systematically investigate carbon dissolution and diffusion in α-Al 2 O 3 , γ-Al 2 O 3 , and θ-Al 2 O 3 , including the effects of oxygen (O) and aluminum (Al) vacancies. Our results show that α-Al 2 O 3 consistently exhibits higher carbon solution enthalpies than γ-Al 2 O 3 and θ-Al 2 O 3 in both pristine and defective structures, indicating lower intrinsic carbon solubility in α-Al 2 O 3 . Vacancies significantly enhance carbon incorporation: O vacancies reduce solution enthalpy, while Al vacancies further amplify this effect, with a strong preference for carbon at Al vacancy sites. Carbon diffusion barriers are also highest in α-Al 2 O 3 , reflecting slower carbon mobility. Al vacancies increase diffusion barriers across all phases, while O vacancies raise barriers in α- and γ-Al 2 O 3 but slightly lower them in θ-Al 2 O 3 . These results reveal a dual mechanism behind the carburizing resistance of α-Al 2 O 3 : reduced carbon solubility and elevated diffusion barriers. Furthermore, this work provides atomic-scale insights to guide the design of alumina-based materials with improved carburizing resistance through phase selection and defect engineering.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Atomic Dynamics of Multi‐Interfacial Migration and Transformations

Redox-induced interconversions of metal oxidation states typically result in multiple phase boundaries that separate chemically and structurally distinct oxides and suboxides. Directly probing such multi-interfacial reactions is challenging because of the difficulty in simultaneously resolving the multiple reaction fronts at the atomic scale. Using the example of CuO reduction in H 2 gas, a reaction pathway of CuO → monoclinic m-Cu 4 O 3 → Cu 2 O is demonstrated and identifies interfacial reaction fronts at the atomic scale, where the Cu 2 O/m-Cu 4 O 3 interface shows a diffuse-type interfacial transformation; while the lateral flow of interfacial ledges appears to control the m-Cu 4 O 3 /CuO transformation. Together with atomistic modeling, it is shown that such a multi-interface transformation results from the surface-reaction-induced formation of oxygen vacancies that diffuse into deeper atomic layers, thereby resulting in the formation of the lower oxides of Cu 2 O and m-Cu 4 O 3 , and activate the interfacial transformations. In conclusion, these results demonstrate the lively dynamics at the reaction fronts of the multiple interfaces and have substantial implications for controlling the microstructure and interphase boundaries by coupling the interplay between the surface reaction dynamics and the resulting mass transport and phase evolution in the subsurface and bulk.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Mechanistic insight into copper cation exchange in cadmium selenide semiconductor nanocrystals using X-ray absorption spectroscopy

In terms of producing new advances in sustainable nanomaterials, cation exchange (CE) of post-processed colloidal nanocrystals (NCs) has opened new avenues towards producing non-toxic energy materials via simple chemical techniques. The main processes governing CE can be explained by considering hard/soft acid/base theory, but the detailed mechanism of CE, however, has been debated and has been attributed to both diffusion and vacancy processes. In this work, we have performed in situ x-ray absorption spectroscopy to further understand the mechanism of the CE of copper in solution phase CdSe NCs. The x-ray data indicates clear isosbestic points, suggestive of cooperative behavior as previously observed via optical spectroscopy. Examination of the extended x-ray absorption fine structure data points to the observation of interstitial impurities during the initial stages of CE, suggesting the diffusion process is the fundamental mechanism of CE in this system.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2026

Production of High Yield, Ultra‐Shallow Nitrogen‐Vacancy Centre Ensembles Through High Purity Forming gas Annealing

Dense, ultra‐shallow (<10 nm from the surface) ensembles of nitrogen‐vacancy (NV) centers in diamond are sought‐after for realising practical quantum sensing at the nanoscale. However, these ensembles typically suffer from reduced NV creation efficiency and spin coherence compared to their deeper and bulk counterparts. Consequently, most demonstrations in the literature to date have been restricted to using single NV centers with atypical coherence and charge stability properties not achievable across entire ensembles. Here, in this work, we demonstrate that controlled annealing in a high purity forming gas atmosphere improves near‐surface NV formation efficiency to over 20%. This environment preserves a high quality hydrogen termination throughout the anneal, avoiding material etching over many hours at temperatures up to 1100°C and possibly helping to suppress vacancy out‐diffusion. We show that high temperature annealing is especially vital for improving spin properties in high density, ultra‐shallow implanted samples due to the localised damage profiles produced. By allowing the formation of high yield NV ensembles with improved spin properties, the optimised forming gas anneal can facilitate the reliable production of more sensitive diamond sensors than have previously been accessible.

Healey, Alexander J. [Royal Melbourne Institute of↗
DOE OSTI2023

Thermodynamics and kinetics of Pb intercalation under graphene on SiC(0001)

SiC-supported graphene intercalated by a two-dimensional Pb monolayer can provide an appealing platform for spintronic applications. Such a monolayer structure is thermodynamically ultrastable, as observed in recent experiments. However, important fundamentals such as the structure of intercalated phases, locations of intercalated atoms, thermodynamic preference for intercalation, and intercalation pathways for this system have not yet been understood conclusively. Here, in this work, extensive density functional theory calculations are performed to assess Pb intercalation thermodynamics and kinetics under graphene on SiC(0001). We find that intercalation of isolated Pb atoms is strongly disfavored over adsorption on top of graphene. However, intercalation of complete Pb layers in the gallery between SiC and graphene buffer layer is strongly favored over supported Pb monolayers and moderately favored over formation of supported large three-dimensional Pb islands. We also find that initiation of intercalation either by individual Pb atoms directly penetrating graphene or by hopping under a static graphene step edge is energetically prohibitive at experimental temperatures. Consequently, more complex intercalation pathways are operative and further analyzed. We demonstrated that once an intercalated Pb monolayer forms around a graphene step edge, a facile Pb mass transport by Pb vacancy-mediated diffusion can be triggered for continued growth of the intercalated monolayer.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗