Atomistic Kinetic Monte Carlo Simulations Research Articles

A Stranski–Krastanov to Volmer–Weber transition in the growth mode of self-assembled quantum dots

We present recent experiments for tensile-strained Ge QDs on InAlAs(111)A where we observe that growth proceeds via the Stranski–Krastanov (SK) growth mode at lower temperatures and transitions to the Volmer–Weber (VW) growth mode at higher temperatures. We discuss atomistic kinetic Monte Carlo simulations that show how the SK to VW growth mode transition is affected by model parameters. We find that systems are more likely to transition from a SK to a VW mode when the bond energy between substrate atoms is weaker than the bond energy between atoms of the deposited material. We also show how entropic effects due to intermixing can stabilize a layer-by-layer growth mode in certain parameter regimes.

Cu precipitation in electron-irradiated iron alloys for spent-fuel canisters

In this work, the Cu clustering in Fe under irradiation is investigated using experiments, cluster dynamics and atomistic kinetic Monte Carlo (AKMC) simulations. In experiments, cast iron and model FeCu alloy samples were irradiated with 2 MeV electrons for 143 h at 140 °C. The post-irradiation microstructure was characterized using atom probe tomography. Cluster dynamics and AKMC methods were used to simulate the Cu clustering under the same irradiation conditions. Both simulation methods show satisfactory agreement with experiments, lending strength to the validity of the models. Finally, the Cu clustering in spent-fuel repository conditions for 105 years at 100 °C was simulated using both methods. The results indicate that potential hardening by Cu clustering is insignificant over 105 years.

Kinetic Monte Carlo simulations of Ge–Sb–Te thin film crystallization

Simulation of atomic redistribution in Ge–Sb–Te (GST)-based memory cells during SET/RESET cycling is needed in order to understand GST memory cell failure and to design improved non-volatile memories. However, this type of atomic scale simulations is extremely challenging. In this work, we propose to use a simplified GST system in order to catch the basics of atomic redistribution in Ge-rich GST (GrGST) films using atomistic kinetic Monte Carlo simulations. Comparison between experiments and simulations shows good agreements regarding the influence of Ge excess on GrGST crystallization, as well as concerning the GST growth kinetic in GrGST films, suggesting the crystallized GST ternary compound to be off-stoichiometric. According to the simulation of atomic redistribution in GrGST films during SET/RESET cycling, the film microstructure stabilized during cycling is significantly dependent of the GST ternary phase stoichiometry. The use of amorphous layers exhibiting the GST ternary phase stoichiometry placed at the bottom or at the top of the GrGST layer is shown to be a way of controlling the microstructure evolution of the film during cycling. The significant evolution of the local composition in the amorphous solution during cycling suggests a non-negligible variation of the crystallization temperature with operation time.

What can we learn from the exclusion volume of fluctuations and precipitates?

ABSTRACT In the literature, the volume of a cluster (i.e. a fluctuation of the solid solution or a precipitate) is identified as the volume of a sample of the bulk precipitate phase containing the same number of solute atoms (i.e. of the same size). This approximation is suitable for macroscopic precipitates but is debatable for a rigorous description of microscopic clusters, especially during the nucleation and growth stages. To go beyond this simplification, this work focuses on the notion of ‘exclusion volume’, a volume which accounts for real cluster configurations in a given 3D lattice. The exclusion volume normalised by the cluster size is a complex function of the cluster size, which, like the cluster free energy, evolves during precipitation kinetics and converges toward an asymptote in the long-time limit. Furthermore, this asymptote exhibits a bifurcation which clearly separates fluctuations from precipitates, for a cluster size which is found equal to the minimum value of the critical size for nucleation. With the help of Atomistic Kinetic Monte Carlo (AKMC) simulations, it is shown that the evolution of cluster volume during precipitation can be accurately predicted.

A DFT-driven multifidelity framework for constructing efficient energy models for atomic-scale simulations

The reliability of atomistic simulations depends on the quality of the underlying energy models providing the source of physical information, for instance for the calculation of migration barriers in atomistic Kinetic Monte Carlo simulations. Accurate (high-fidelity) methods are often available, but since they are usually computationally expensive, they must be replaced by less accurate (low-fidelity) models that introduce some degrees of approximation. Machine-learning techniques such as artificial neural networks can be employed to work around this limitation and extract the needed parameters from large databases of high-fidelity data. However, the latter are often computationally expensive to produce. This work introduces an alternative method based on the multifidelity approach. Correlations between high-fidelity and low-fidelity predictions are exploited to make an educated guess of the high-fidelity value based only on quick low-fidelity estimations, to be used for instance as an efficient and reliable source of physical data for atomistic simulations. With respect to neural networks, this approach requires less training data because of the lower amount of fitting parameters involved. The method is tested on the prediction of ab initio formation and migration energies of vacancy diffusion in iron-copper alloys, and compared with the neural networks trained on the same database.

Point defect evolution under irradiation: Finite size effects and spatio-temporal correlations

The evolution of point defect concentrations under irradiation is controlled by their diffusion properties, and by their formation and elimination mechanisms. The latter includes the mutual recombination of vacancies and interstitials, and the elimination of point defects at sinks. Two models are traditionally used to predict the defect concentration evolution, the standard rate theory (SRT) and the atomistic kinetic Monte Carlo (AKMC). We show in this work a large discrepancy in the defect concentrations between both methods when the average number of defects in the AKMC simulation box is close or lower than one. The reason is that AKMC naturally captures strong space and time correlations between vacancies and interstitials generated by the finite size of the periodic simulation box. These correlations strongly affect the recombination rate and the point defect concentrations. SRT fails to predict such correlations, and the corresponding solution deviates from the more accurate solution given by the AKMC simulation under similar conditions. These finite size correlation effects are strong when the elimination of point defects occurs by recombination only, but can still be significant in the presence of sinks. In order to account for the spacio-temporal correlation in a continuum framework, we introduce a Correlated Pair Theory (CPT). This theory fully takes into account the correlations between vacancy and interstitial pairs and predicts point defect concentrations in good agreement with AKMC simulations. Inversely, if the goal is for the AKMC to reproduce defect concentrations in bulk using small simulation boxes, the naturally occurring correlations need to be corrected. We show here that the CPT can be used to modify the elimination rates in the AKMC simulations, so as to yield point defect concentrations in agreement with SRT.

Efficient Calculation Methods for the Diffusion Coefficient of Interstitial Solutes in Dilute Alloys.

In the example of oxygen diffusion in dilute ferritic iron alloys it is shown that the calculation of the diffusion coefficient can be separated into a contribution related to the migration in the interaction region between oxygen and the substitutional solute and a part related to diffusion in pure body centered cubic (bcc) Fe. The corresponding diffusion times are determined by analytical expressions using Density-Functional-Theory (DFT) data for the respective binding energies. The diffusion coefficient in the interaction region must be determined by atomistic kinetic Monte Carlo (AKMC) simulations with DFT values for the migration barriers as input data. In contrast to previous calculations, AKMC simulation must only be performed for one concentration of the substitutional solute, and the obtained results can be employed to obtain data for other concentrations in a very efficient manner. This leads to a tremendous decrease of computational efforts. Under certain conditions it is even possible to use analytical expressions where merely DFT data for the binding energies are needed. The limits of applicability of the presented calculation procedures are discussed in detail. The methods presented in this work can be generalized to interstitial diffusion in other host materials with small concentrations of substitutional solutes.

The impact of alloying elements on the precipitation stability and kinetics in iron based alloys: An atomistic study

Iron based industrial steels typically contain a large number of alloying elements, even so-called low alloyed steels. Under irradiation, these alloying elements form clusters that have a detrimental impact of the mechanical properties of the steel. The stability and formation mechanisms of such clusters are presently not fully understood. Therefore, in this work, we study the thermal stability and formation kinetics of small solute clusters in the bcc Fe matrix. We use density functional theory (DFT) to characterize the binding energy of vacancy/solute clusters containing Cr, Mn, Ni, Cu, Si and P, thereby exploring >700 different configurations. The constructed DFT data base is used to fit a cluster expansion (CE) for the vacancy-FeCrMnNiCuSiP system. In turn, the obtained CE is applied in atomistic kinetic Monte Carlo simulations to study the effect of Mn, Ni, Cr, Si and P on the precipitation formation in the FeCu alloy. We conclude that the addition of Mn and Ni delay the precipitation kinetics by an order of magnitude. The additional alloying with traces of P/Si further delays the kinetics by an additional order of magnitude. We found that Si plays an essential role in the formation of spatially mixed MnNiCuSi cluster.

OFF-State Leakage and Performance Variations Associated With Germanium Preamorphization Implant in Silicon–Germanium Channel pFET

Parameter variations in the transistor characteristics with new materials and process steps pose an increasing challenge for CMOS scaling to nanometer feature size. Alternate channel materials such as silicon–germanium (SiGe) for p-type field effect transistor (pFET) at 32 nm and beyond are useful because of higher mobility and lower threshold voltage ( $\text{V}_{T}$ ) but suffer from higher gate-induced drain leakage (GIDL) and could be a source of additional variability. In this paper, experimental results, a noise-like approach called the statistical impedance field method, and atomistic kinetic Monte Carlo simulations are used to report that the elimination of prehalo Ge preamorphization implant (PAI) from the SiGe pFET process flow reduces GIDL and its variation due to systematic variations in gate length and width but increases the time-zero (static) random GIDL and performance variations. This is primarily due to random dopant position fluctuations in the extension region for off-state leakage ( ${I}_{ \mathrm{\scriptscriptstyle OFF}} $ ) variability and in the halo region at the drain sidewall for $\text{V}_{T}$ variability. However, the increase in random variability without Ge PAI reduces for lower supply voltages and, thus, offers advantages of reduced GIDL with the same electrostatics, lower systematic variations, and similar ${I}_{ \mathrm{\scriptscriptstyle OFF}}$ random variability for scaled voltages.

Influence of substitutional atoms on the diffusion of oxygen in dilute iron alloys

A multiscale approach including density functional theory (DFT) and atomistic kinetic Monte Carlo (AKMC) simulations is applied to investigate the diffusion of interstitial oxygen atoms in bcc Fe under the influence of substitutional foreign atoms (Al, Si, P, S, Ti, Cr, Mn, Ni, Y, Mo, and W). The substitutional atoms can be assumed to be immobile since their diffusion coefficient is much smaller than that of oxygen. First, jumps of oxygen in pure bcc Fe between first-, second-, and third-neighbor octahedral interstitial sites are investigated. It is found that the first-neighbor jump is most relevant, with the tetrahedral site as the saddle point. The second-neighbor jump consists of two consecutive first-neighbor jumps. The barrier for a direct third-neighbor jump is too high to be significant for the diffusion process. In the presence of substitutional atoms the most important migration paths are first-neighbor jumps between modified octahedral sites with modified tetrahedral sites as saddle points. Calculations show that Si, P, Ni, Mo, and W cause some modifications of the migration barriers of oxygen and their interaction with O is mainly repulsive. Al, Cr, and Mn have a significant influence on the barriers and they exhibit strong attractive interactions with O. The most important modification of the barriers is found for S, Ti, and Y where deep attractive states exist. Based on the migration energies obtained by DFT, AKMC simulations on a rigid lattice are employed to determine the diffusion coefficient of oxygen in a dilute iron alloy containing the different substitutional atoms. It is found that Si, P, Ni, Mo, and W have almost no influence on the diffusivity of O. The presence of Al, Cr, Mn, S, Ti, and Y causes a significant reduction of the mobility of oxygen. In these cases the temperature dependence of the oxygen diffusion coefficient shows considerable deviations from an Arrhenius law. These phenomena are discussed in detail by considering the occupation time for the different states. The present findings on the strong dependence on the kind of substitutional atoms change the picture of oxygen diffusion in dilute iron alloys substantially.

Classical interatomic potential for quaternary Ni–Fe–Cr–Pd solid solution alloys

In this paper, we present a new quaternary interatomic potential for the NiFeCrPd system, which is an extension on the previous NiFeCr potential. Density functional theory is used to calculate the quantities to be fitted, with particular focus on the energetics of point defects with solutes, for the potential to be used towards understanding radiation damage properties. The potential thus will enable the modeling of multi-elemental solid solution alloys consisting of up to four elements. To test the potential, we have performed atomistic kinetic Monte Carlo simulations to investigate the effect of configurational entropy on the self-diffusion coefficients. The self-diffusion coefficients are found to increase with chemical complexity, contrary to the common postulation of sluggish diffusion in high entropy alloys (HEAs). In addition, we have performed molecular dynamics simulations to elucidate the effect of Pd on vacancy diffusion and clustering in pure Ni and binary alloys. In agreement with recent irradiation experiments, our simulations show that while large vacancy clusters, such as stacking fault tetrahedra, are formed in pure Ni, Ni–Fe and Ni–Cr systems, negligible vacancy clustering is observed in Ni–Pd systems, indicating a possible effect of Pd in reducing cluster sizes. We suggest that this potential will be useful for studying the defect evolution in multi-component HEAs.

Atomistic modeling of α’ precipitation in Fe-Cr alloys under charged particles and neutron irradiations: Effects of ballistic mixing and sink densities

The α′ precipitation in Fe-Cr alloys under electron, neutron and ion irradiations is modelled by cluster dynamics and atomistic kinetic Monte Carlo simulations. The former method provides the evolution of the density of point defect sinks. This density is then introduced in the Monte Carlo which models the diffusion of point defects and atoms. By comparison with previous studies, this provides a better description of the evolution of point defect concentrations and therefore of the balance between two competing irradiation effects: the acceleration of diffusion and the ballistic mixing. The prediction of the model is in agreement with existing atomic-scale experiments already reported after electron, ion and neutron irradiations or obtained for the present study. Irradiation is found to accelerate the precipitation by orders of magnitude, except for the case of ion irradiation at high doses rates, where the simulations predict a ballistic dissolution of α′ precipitates. According to our model, this dissolution is mainly due to a high sink density, which reduces the concentrations of point defects and limits the acceleration of diffusion. This effect may be reinforced by carbon contamination, which reduces the mobility of point defects.

Morphology Determines Conductivity and Seebeck Coefficient in Conjugated Polymer Blends.

The impact of nanoscale morphology on conductivity and Seebeck coefficient in p-type doped all-polymer blend systems is investigated. For a strongly phase separated system (P3HT:PTB7), we achieve a Seebeck coefficient that peaks at S ∼ 1100 μV/K with conductivity σ ∼ 3 × 10-3 S/cm for 90% PTB7. In marked contrast, for well-mixed systems (P3HT:PTB7 with 5% diiodooctane (DIO), P3HT:PCPDTBT), we find an almost constant S ∼ 140 μV/K and σ ∼ 1 S/cm despite the energy levels being (virtually) identical in both cases. The results are interpreted in terms of a variable range hopping (VRH) model where a peak in S and a minimum in σ arise when the percolation pathway contains both host and guest sites, in which the latter acts as energetic trap. For well-mixed blends of the investigated compositions, VRH enables percolation pathways that only involve isolated guest sites, whereas the large distance between guest clusters in phase-separated blends enforces (energetically unfavorable) hops via the host. The experimentally observed trends are in good agreement with the results of atomistic kinetic Monte Carlo simulations accounting for the differences in nanoscale morphology.

An empirical potential for simulating vacancy clusters in tungsten

We present an empirical interatomic potential for tungsten, particularly well suited for simulations of vacancy-type defects. We compare energies and structures of vacancy clusters generated with the empirical potential with an extensive new database of values computed using density functional theory, and show that the new potential predicts low-energy defect structures and formation energies with high accuracy. A significant difference to other popular embedded-atom empirical potentials for tungsten is the correct prediction of surface energies. Interstitial properties and short-range pairwise behaviour remain similar to the Ackford-Thetford potential on which it is based, making this potential well-suited to simulations of microstructural evolution following irradiation damage cascades. Using atomistic kinetic Monte Carlo simulations, we predict vacancy cluster dissociation in the range 1100–1300 K, the temperature range generally associated with stage IV recovery.

Precipitation in a copper matrix modeled byab initiocalculations and atomistic kinetic Monte Carlo simulations

A kinetic Monte Carlo approach is used to study the influence of Cr, Fe, Al, or Mg addition on the precipitation in a Cu–Ni–Si alloy. The simulation method is based on a vacancy diffusion model. The crucial parameters of this method are the pairwise mixing energies of all contained elements which are determined by ab initio calculations. The number of dissolved atoms in equilibrium state is used to estimate the influence of Cr, Fe, Al, or Mg on the electrical conductivity. In order to estimate the influence of the alloying elements on strength ab initio calculations of misfit strain at the Cu/Ni3Si interface are performed. The atomistic kinetic Monte Carlo (AKMC) simulations reveal that Cr, Fe, Al, and Mg atoms are located at different positions: Mg and Al atoms are preferably located at the interface of Ni–Si precipitates and Cu matrix, Cr atoms diffuse into Ni–Si precipitates, and Fe atoms form Fe clusters surrounded by Ni–Si shells. Cr addition leads to a significantly reduced fraction of atoms dissolved in the matrix which indicates an increase of electrical conductivity, whereas Mg addition results in a high misfit strain at the Cu/Ni3Si interface which can contribute to increased strength.

An object kinetic Monte Carlo model for the microstructure evolution of neutron‐irradiated reactor pressure vessel steels

This work presents a full object kinetic Monte Carlo framework for the simulation of the microstructure evolution of reactor pressure vessel (RPV) steels. The model pursues a “gray‐alloy” approach, where the effect of solute atoms is seen exclusively as a reduction of the mobility of defect clusters. The same set of parameters yields a satisfactory evolution for two different types of alloys, in very different irradiation conditions: an Fe–C–MnNi model alloy (high flux) and a high‐Mn, high‐Ni RPV steel (low flux). A satisfactory match with the experimental characterizations is obtained only if assuming a substantial immobilization of vacancy clusters due to solute atoms, which is here verified by means of independent atomistic kinetic Monte Carlo simulations. The microstructure evolution of the two alloys is strongly affected by the dose rate; a predominance of single defects and small defect clusters is observed at low dose rates, whereas larger defect clusters appear at high dose rates. In both cases, the predicted density of interstitial loops matches the experimental solute‐cluster density, suggesting that the MnNi‐rich nanofeatures might form as a consequence of solute enrichment on immobilized small interstitial loops, which are invisible to the electron microscope.

Te homogeneous precipitation in Ge dislocation loop vicinity

High resolution microscopies were used to study the interactions of Te atoms with Ge dislocation loops, after a standard n-type doping process in Ge. Te atoms neither segregate nor precipitate on dislocation loops, but form Te-Ge clusters at the same depth as dislocation loops, in contradiction with usual dopant behavior and thermodynamic expectations. Atomistic kinetic Monte Carlo simulations show that Te atoms are repulsed from dislocation loops due to elastic interactions, promoting homogeneous Te-Ge nucleation between dislocation loops. This phenomenon is enhanced by coulombic interactions between activated Te2+ or Te1+ ions.

Atomic-based phase-field method for the modeling of radiation induced segregation in Fe–Cr

We present a quantitative phase-field modeling of radiation-induced segregation in Fe–Cr alloys. The evolution of chemical and point defect concentration fields are described by an Onsager formalism combined to a Cahn–Hilliard like diffusion equation for the introduction of non-uniform driving forces. Both the Onsager transport coefficients and the driving force parameters are extracted from atomic Monte Carlo simulations with point defect diffusion models fitted on DFT calculations, in a composition range between 0 and 20at.% Cr and in a temperature range between 600 and 1000K. Phase-field simulations are able to quantitatively reproduce the evolution of segregation profiles obtained from direct atomistic kinetic Monte Carlo simulations, while being typically two orders of magnitude faster. It is shown that a precise parameterization of the concentration-dependent Onsager transport coefficients, thermodynamic factors, and equilibrium point defect concentrations is crucial for the phase-field method to be quantitative.

Radiation-accelerated precipitation in Fe–Cr alloys

The kinetics of phase separation in Fe–Cr alloys under irradiation is modeled by Atomistic kinetic Monte Carlo simulations that include the formation, migration and elimination of vacancies and self-interstitials at point defects sinks. The evolution of the sink density is modeled by cluster dynamics, and taken into account in the Monte Carlo simulations by a rescaling of the time. The results are in good agreement with available experimental observations of neutron irradiation at 290 °C. The irradiation is found to accelerate the kinetics of phase separation by orders of magnitude, with an acceleration factor given by the increase in point defect concentrations. The microstructure evolution is qualitatively the same as during isothermal annealing, except in the vicinity of point defect sinks. The effects of equilibrium and radiation induced segregations at grain boundaries are considered. The ballistic mixing occurring in displacement cascades is modeled, and is found to be insufficient to produce the dissolution of chromium rich precipitates at 290 °C, even at high dose rates. Therefore, it cannot explain the absence of precipitation observed during ion irradiations.

Modeling radiation induced segregation in iron–chromium alloys

Radiation induced segregation in ferritic Fe–Cr alloys is studied by Atomistic Kinetic Monte Carlo simulations that include diffusion of chemical species by vacancy and interstitial migration, recombination, and elimination at sinks. The parameters of the diffusion model are fitted to DFT calculations. Transport coefficients that control the coupling between diffusion of defects and chemical species are measured in dilute and concentrated alloys. Radiation induced segregation near grain boundaries is directly simulated with this model. We find that the diffusion of vacancies toward sinks leads to a Cr depletion. Meanwhile, the diffusion of self-interstitials causes an enrichment of Cr in the vicinity of sinks. For concentrations lower than 15%Cr, we predict that sinks will be enriched with Cr for temperatures lower than a threshold. When the temperature is above this threshold value, the sinks will be depleted in Cr. These results are compared to previous experimental studies and models. Cases of radiation induced precipitation and radiation accelerated precipitation are considered.