Cluster Expansion Method Research Articles

Convenient Interface to Inverse Ising (ConIII): A Python 3 Package for Solving Ising-Type Maximum Entropy Models

ConIII (pronounced CON-ee) is an open-source Python project providing a simple interface to solving the pairwise and higher order Ising model and a base for extension to other maximum entropy models. We describe the maximum entropy problem and give an overview of the algorithms that are implemented as part of ConIII ( https://github.com/eltrompetero/coniii ) including Monte Carlo histogram, pseudolikelihood, minimum probability flow, a regularized mean field method, and a cluster expansion method. Our goal is to make a variety of maximum entropy techniques accessible to those unfamiliar with the techniques and accelerate workflow for users. Funding Statement: EDL was supported by an NSF Graduate Fellowship under grant no. DGE-1650441. This research was supported in part by a congressional research grant provided by the Dirksen Congressional Center.

Cu-Zn disorder in stoichiometric and non-stoichiometric Cu2ZnSnS4/Cu2ZnSnSe4

Cu-Zn disorder is unavoidable but plays an important role in high-efficiency Cu2ZnSnS4 and Cu2ZnSnSe4 solar cells. Using the cluster expansion method along with Monte Carlo (MC) simulations, we study the Cu-Zn disorder, considering cases both with and without vacancies. We find that the 2a, 2c, and 2d Wyckoff sites all show order-disorder transitions for both cases, in agreement with recent experiments supporting disorder at all 2a, 2c and 2d sites, but, in contrast to early experiments, supporting Cu-Zn disorder only at 2c and 2d sites. Below the transition temperature in non-stoichiometric cases, we find that excess Zn prefers to occupy 2c over 2a sites due to the greater similarity of 2c sites to 2d sites. Such site preferences indicate that Cu-Zn occupations exhibit some new kind of ordering rather than randomly distributed at 2a and 2c sites. We find that while Cu-Zn disorder reduces the band gap, the site preferences in non-stoichiometric samples increase the band gaps by suppressing Cu-Zn disorder. Generally, lowering annealing temperatures, while increasing Zn and vacancies, will lead to larger band gaps.

Mayer’s convergence and thermodynamics of ideal Bose gas

The equation of state for a collection of ideal bosons in both the low-density and high-density regions is found using the method of cluster expansion with a new generating function. The importance of the radius of convergence in the cluster expansion and its connection to the Bose–Einstein condensation phenomenon are studied. The radius of convergence of the partition function is calculated and the values of critical density, fugacity and other thermodynamic properties at condensation are obtained using Mayer’s convergence method.

Discovery of new ground state structures for Li4Mn2O5 and V2O5 from first principles

Computational tools are often useful in resolving the crystal structure of newly discovered materials as well as in predicting existence of the undiscovered materials. The current study illustrates the application of a well-known method of cluster expansion from alloy thermodynamics to resolve the crystal structure of a novel promising cathode compound for Li-ion batteries, Li4Mn2O5. We systematically narrow down the possible space of site orderings through the construction of a gradually improved cluster expansion based on the first principles calculation of total energy to obtain the ground state structure of Li4Mn2O5. The new-found ground state is 66 meV/f.u. lower than the previously determined ground state for this material. On the basis of the new-found Li4Mn2O5-structure, we compute the phase stability of the family of the isostructural 3d transition metal oxides and find that the oxides are metastable in both the lithiated and delithiated states. The exception to this general trend is exhibited by vanadium pentoxide, which is found to be robustly stable in both lithiated and delithiated states. The V2O5 phase in the new structure is more stable by over 1 eV/f.u. compared to the well-known ground state polymorph in Pmmn phase. The delithiation of the lithiated phase can provide a plausible synthesis route to this hitherto unknown ground state polymorph of vanadium pentoxide. The current study not only resolves the crystal structure of Li4Mn2O5, but also predicts the ground state polymorph of V2O5 through a computational route.

High density equation of state and condensation of ideal Bose gas using Mayer’s generating function

The equation of state of an ideal collection of bosons in the low-density and high-density regime are found using the method of cluster expansion with Mayer's generating function. The saturation density and the other thermodynamic properties are calculated by the application of Mayer's convergence of the partition function. By calculating the value of saturation density from the singularity of the partition function series, the differences between the Mayer series convergence and the virial series convergence for ideal bosons are also established.

Phase diagrams and elastic properties of the Fe-Cr-Al alloys: A first-principles based study

The phase diagrams and elastic properties of the Fe-Cr-Al alloys in full-temperature and all-compositional ranges are calculated. By combining first-principles calculations and cluster variation method, binary and ternary phase diagrams are obtained. A new ternary ordered phase B32 which is different from ternary extension of binary phases appears in the ternary section around temperature of 600 K. The binary FeAl phases show an extremely high solubility for Cr, while the binary CrAl phase solid solution has a low solubility for Fe. By combining first-principles calculations and cluster expansion method, the bulk modulus, shear modulus and Poisson's ratio are calculated. The shear modulus and Poisson's ratio show a strong ordering dependency, while the ordering dependency in bulk modulus is weak. Disordered Fe-Cr alloys with a little Al solvent shows ductile property, the Al-rich corner has brittle property.

Atomic interactions in C15 Laves phases

The C15 phase ( $$Fd\bar{3}m$$ ) has been extensively studied by several methodologies to evaluate its atomic interactions. Ideally ordered at the $$AB_2$$ composition, this Laves phase is known to present a non-stoichiometry domain accommodated by substitutional atomic disorder at high temperature. With the Cu–Mg system as an example, a cluster expansion method study revealed that first neighbors pair interaction is positive yielding a favorable mixing of Cu and Mg atoms in 16d site and a homogeneity extended in the Mg-richer side. To guide the thermodynamic modeling of the C15 phase, special quasi-random structure cells have been generated at several compositions to simulate atom mixing with and without the merging of 8a and 16d sites. Combined with electronic density functional theory, calculation on several systems (Ta–V, Cr–Nb, Mo–Zr and Cr–Ti) was done to estimate the mixing energies on the two different sublattices. The results are compared to published assessments and open a discussion on the acceptability of the traditional thermodynamic model.

Exogenous shock and multifractal random walk

We construct a log-volatility process for Multifractal Random Walk from a discrete time model as a scale limit and consider an exogenous shock and the relaxation process of the volatility. In this construction, taking an effect of exogenous shock into account, we consider a model for trades transacted by traders with different investment time horizons. Using the method of cluster expansion developed in mathematical physics, we obtain the convergence of scale limit of log-volatility process. For this scale limit, we prove the relaxation of the volatility after exogenous shock is given by an inverse power law $$\frac{1}{t^\delta }$$ with exponent $$\delta =1$$ .

Optimizing special quasirandom structure (SQS) models for accurate functional property prediction in disordered 2D alloys

2D materials such as MXenes have garnered attention in a wide field of applications ranging from energy to environment to medical. Properties of 2D materials can be tailored via alloying and in some cases, solid-solutions (disordered alloys) are formed. To predict the disordered alloy properties via first-principles, the model structure needs to imitate the random arrangements of alloyants and yet remains computationally tractable. Using density functional theory and the cluster expansion method, we investigate the accuracy of using of special quasirandom structures (SQSs) for predicting disordered 2D alloy properties, evaluating the effect of SQS supercell size on the prediction quality of formation energies, elastic properties, and structural parameters. We illustrate the findings with 5 different disordered binary MXene alloy systems (where M = Ti and M′ = Zr, Hf, V, Nb, or Ta), demonstrating that SQSs around 6–8 times the primitive cell (N = 6–8) are sufficient to attain convergence in the property predictions versus supercell size. For formation energies, SQSs with N > 4 are found to reproduce the formation energies of the fully disordered phase within ~2.5 meV. For the simulation of the experimentally-synthesized TiNbCO2, we find convergence in structural parameters and elastic tensors at N ~ 6. We traced the convergence of the predictions to the convergence in the band structure-related properties via analysis of the electronic densities-of-states and the projected crystal overlap Hamilton population. Our findings suggest that modest sized SQSs would reproduce the properties of disordered MXene alloys. The results should help guide the investigations of structure-property relationships in other disordered 2D materials as well.

Enhancing the Photocatalytic Performance of MXenes via Stoichiometry Engineering of Their Electronic and Optical Properties.

Combining both density functional theory and the cluster expansion method, we investigate 3 binary MXene alloy systems of semiconducting Ti2CO2, Zr2CO2, and Hf2CO2, where the transition metals substitute one another (i.e., Ti2(1- x)Zr2 xCO2, Ti2(1- x)Hf2 xCO2, and Zr2(1- x)Hf2 xCO2). We show that this group of MXene alloys forms the solid-solution phase across all compositions. Special quasirandom structures are generated to model the solid-solution phase of these alloys, using which we demonstrate how their structural, mechanical, electronic, and optical properties are tuned via stoichiometry engineering. These alloys exhibit outstanding mechanical strength and stability. They possess indirect band gaps of 1.25-1.80 eV. For Ti2(1- x)Zr2 xCO2 and Ti2(1- x)Hf2 xCO2, they display higher absorbance in the solar spectrum than their constituent Zr2CO2 and Hf2CO2, respectively. Most of the MXene alloys also show appropriately aligned band edges for water splitting. We predict the Ti2(1- x)Zr2 xCO2 alloy with x = 0.2778 to be the most promising water-splitting photocatalyst among the MXenes studied here, outperforming its constituents, Ti2CO2 and Zr2CO2, when solar absorbance performance and band-edge alignments are simultaneously considered. This work demonstrates that alloying can be used to effectively tune photocatalytic performance.

Error controlling of the combined Cluster-Expansion and Wang–Landau Monte-Carlo method and its application to FeCo

Simulating order–disorder phase transitions in magnetic materials requires the accurate treatment of both the atomic and magnetic interactions, which span a vast configuration space. Taking FeCo as a prototype system, we demonstrate that this can be accurately and efficiently addressed by the combined Cluster-Expansion method and Wang–Landau algorithm. All involved key parameters in this method are carefully tested. Based on these tests, error controlling approaches are also developed. Our method is readily applicable to the study of similar second-order phase transitions in other binary and multi-component systems.

Interpenetrating silicene networks: A topological nodal-line semimetal with potential as an anode material for sodium ion batteries

Motivated by the novel properties of topological nodal-line semimetal and its compatibility with the well-developed Si-based semiconductor technology, we propose a three-dimensional all-silicon topological nodal-line semimetal composed of interpenetrating silicene networks (ISN). Using state-of-the-art first-principles calculations and the cluster expansion method, we find the ISN structure to be dynamically, thermally, and mechanically stable. In addition, it exhibits linearly dispersive band crossings along two perpendicular high-symmetry directions due to the band inversion and shows two nontrivial nodal lines, traversing in the Brillouin zone near the Fermi level. Furthermore, ISN has the potential as an anode material for sodium-ion batteries with an extremely low diffusion energy barrier and a small volume change during the charging/discharging process, which would lead to a high rate and stable cycling performance.

Lithium Adsorption on Graphene at Finite Temperature

The increasing demand for high-energy-density lithium-ion batteries motivates a search for alternative electrode materials. Experimentally obtained graphene-based structures have been suggested to replace the state-of-the-art graphitic anode. For a thorough characterization of Li adsorption on graphene, we study the interaction of Li with graphene both at zero and finite temperatures. The zero-temperature study was carried out by means of density functional theory (DFT), accounting for van der Waals (vdW) interactions, whereas the finite temperature behavior was studied by Monte Carlo techniques with a DFT-derived Li–graphene interaction potential constructed via cluster expansion method. Our calculations reveal two distinct types of orderings of Li on graphene, Li-gas (dispersed Li-ion) and Li-cluster phases. The zero-temperature calculations show that, even when vdW is included, the Li–graphene interaction is mainly electrostatic and phase separation to pristine graphene and bulk Li is energetically most favorable. However, at nonzero temperatures, entropy contribution to free energy allows the lesser-ordered Li-gas and Li-cluster states to be more favorable at sufficiently low concentrations: at temperatures below 400 K and concentrations below 1Li:6C, Li-gas and Li-cluster phases coexist whereas at higher concentrations, only clusters remain stable. At temperatures above 400 K, Li-gas phase can be stabilized with respect to Li cluster or Li bulk at higher concentrations. Furthermore, small variations in chemical potential are shown to be enough to change that concentration threshold. Finally, we show that the Li-cluster phases can have Li-island or Li-stripe ordering; however, Li stripes appear due to the finite size of the simulation cell and therefore the Li-island phase is expected to dominate in the thermodynamic limit instead.

Improved Prediction of Nanoalloy Structures by the Explicit Inclusion of Adsorbates in Cluster Expansions

Density functional theory (DFT) is widely used to predict the properties of materials, but its direct application to nanomaterials of experimentally relevant size can be prohibitively expensive. It has previously been demonstrated that this problem can be addressed through the generation of cluster expansion models trained on DFT calculations. Here, we evaluate the use of the cluster expansion method to calculate the structures of bimetallic Pt–Cu nanoparticles of varying sizes and compositions and in different chemical environments. The predicted surface composition, shape, and lattice parameters of the alloy nanoparticles are found to be in good agreement with experimental characterization. We demonstrate that, to account for adsorbate-induced surface segregation, the best agreement for surface composition can be achieved by constructing a novel cluster expansion for alloy nanoparticles of varying shapes and sizes that explicitly includes adsorbed oxygen.

Stability of SnSe1−xSx solid solutions revealed by first-principles cluster expansion

The configurational thermodynamics of a pseudo-binary alloy SnSe1−xSx in the Pnma phase is studied using first-principles cluster-expansion method in combination with canonical Monte Carlo simulations. We find that, despite the alloy having a tendency toward a phase decomposition into SnSe and SnS at 0 K, the two constituent binaries readily mix with each other to form random SnSe1−xSx solid solutions even at a temperature below room temperature. The obtained isostructural phase diagram of SnSe1−xSx reveals that the alloy is thermodynamically stable as a single-phase random solid solution over a whole composition range above 200 K. These findings provide a fundamental understanding on the alloying behavior of SnSe1−xSx and bring clarity to the debated clustering tendency in this alloy system.

First principles calculations on the effect of interstitial oxygen on phase stability and β–α″ martensitic transformation in Ti–Nb alloys

The effect of oxygen on phase stability and β–α″ martensitic transformation in Ti–Nb alloys has been studied using first principles calculations. Three stable atomic configurations of Ti–Nb (Ti-12.5, 16.6, and 25 at.% Nb) systems, which can transform from β-phase to α″-phase without changing the local atomic position of Nb atoms, were first identified using the cluster expansion method. Phonon calculations indicated that these structures were stable. Next, the martensitic transformation behavior of Ti–Nb–O system was studied using these structures. We observed a significant lattice distortion around oxygen atoms occupying octahedral interstitial sites that resembles a bcc type of stacking. Our results conclusively revealed that while the oxygen interstitials can oppose the atomic shuffle required for martensitic transformation, they can also cooperatively stabilize the β-phase even at 1 at.% oxygen concentrations by inducing local elastic shear strains. Interestingly, the canceling of these fields can stabilize the β-phase by suppressing the β to α″ transformation which decreases the martensitic start temperature (Ms). Our study revealed that the reduction in Ms is higher at lower Nb concentration. The stabilization of β-phase increases with oxygen concentration.

Unconventional pairing symmetry of interacting Dirac fermions on a π -flux lattice

The pairing symmetry of interacting Dirac fermions on the $\pi$-flux lattice is studied with the determinant quantum Monte Carlo and numerical linked cluster expansion methods. The extended $s^*$- (i.e. extended $s$-) and d-wave pairing symmetries, which are distinct in the conventional square lattice, are degenerate under the Landau gauge. We demonstrate that the dominant pairing channel at strong interactions is an exotic $ds^*$-wave phase consisting of alternating stripes of $s^*$- and d-wave phases. A complementary mean-field analysis shows that while the $s^*$- and d-wave symmetries individually have nodes in the energy spectrum, the $ds^*$ channel is fully gapped. The results represent a new realization of pairing in Dirac systems, connected to the problem of chiral d-wave pairing on the honeycomb lattice, which might be more readily accessed by cold-atom experiments.

Unraveling the Inner-Workings of Lithium Transition Metal Oxyflourides Using DFT-Based Cluster Expansion Method

Lithium transition metal oxyfluoride is one of the most promising cathode materials that can achieve a two- to three-fold increase in the capacity of the Li-ion batteries compared to the current state-of-the-art [1,2]. Consequently, transition metal oxyfluorides are attracting much interest for both Li-ion and beyond Li-ion batteries in recent years. The mixed-anion compounds such as Li2VO2F and Li2CrO2F offer improved capacity, higher voltage and good conductivity compared to their transition metal oxide counterparts. However, one of the key bottlenecks that prevent the large-scale commercialization of the batteries with transition metal oxyfluorides is the poor cyclability [3]. Understanding the structural, mechanical and electrochemical properties of the transition metal oxyfluorides is, therefore, a crucial step in the further development of such technology. The first-principles modeling is known to be a valuable tool that can accurately predict such properties of material once its atomic arrangements are known. In the case of transition metal oxyfluorides, the exact atomic arrangements, particularly at different lithiation levels, are unknown. Furthermore, the number of possible atomic arrangements to be explored is prohibitively high, which are often described with the term “combinatorial explosion” [4]. In this work, we present a DFT-based cluster expansion method [5,6] which enables one to determine the correlation between the atomic arrangements and their formation energies. The simulation results not only show a good agreement with the reported experimental observations but also provide a deeper insight on the inner-workings of the materials that lead to the observed properties. Acknowledgement: This research was funded by the “LiRichFCC” (“A new class of powerful materials for electrochemical energy storage: Lithium-rich oxyfluorides with cubic dense packing”), FET-Open call of the European Union Horizon 2020 program under the contract No. 711792. 1. Chen et al., Adv. Energy Mater., 5, 1–7 (2015).2. Chen et al., RSC Adv., 6, 65112–65118 (2016).3. Ren et al., Adv. Sci., 2, 1500128 (2015).4. S. Morgan, G. L. W. Hart, and R. W. Forcade, Comput. Mater. Sci., 136, 144–149 (2017).5. M. Sanchez, F. Ducastelle, and D. Gratias, Phys. A, 128, 334–350 (1984).6. D. de Fontaine, Solid State Phys., 47, 33–176 (1994).

Cluster Expansion Method for Simulating Realistic Size of Nanoparticle Catalysts with an Application in CO2 Electroreduction

For the last several decades, there has been a rapid development of nanoscience and nanotechnology. In particular, nanoparticles (NPs) are applied in various catalyst problems, where their enormously large surface-to-volume ratio not only is advantageous for high catalytic performance but also allows the quantum size effect to play a key role in modifying their chemical properties. However, when understanding the size effect of NPs on the catalytic properties by employing density functional theory (DFT) calculations, there has been an obvious experiment–theory gap in simulating nanocatalysts with realistic sizes. In this study, we developed a new simulation method based on the cluster expansion model, namely, CE-np, which enables efficient and accurate calculations of the intermediate binding energies for various sizes of NPs. We then applied CE-np to investigate the electrochemical CO2 reduction reaction (CO2RR) of gold NPs (AuNPs). CE-np reproduces not only DFT-level accuracies in predicting the interme...

Electronic-structure and thermodynamic properties of ZnS1−xSex ternary alloys from the first-principles calculations

In this paper, the composition-dependent electronic-structure and thermodynamic properties for hexagonal wurtzite (WZ) and face-centered cubic zincblende (ZB) ZnS1−xSex ternary alloys have been studied utilizing first-principles pseudopotential plane-wave self-consistent field calculations. Special quasi-random structure method was applied to generate disordered structures of 16-atom supercell for x = 0.25, 0.5 and 0.75. We obtained a direct band-gap at Г point and a shrinking tendency of the band-gap with increasing Se content in both phases. The reasons for the band-gap shrinkage have been discussed based on the analysis of electronic density of states and band offsets. Δ-sol method was applied to correct the band gaps calculated by density functional theory. The Δ-sol corrected band-gaps are in good agreement with experimental band-gaps. The thermodynamic properties were calculated by the cluster expansion method. It turns out that, the stretching forces decrease considerably and linearly with the bond length, while the bending forces are dependent insignificantly on the bond length. The phase diagrams reveal that the lattice vibration effects can influence the critical temperature Tc and the solubility of S- and Se-rich ZnS1−xSex alloys, but the influence is not obvious for ZB-ZnS1−xSex. Based on the calculated phase diagrams without lattice vibration effects, the Tc for WZ- and ZB-ZnS1−xSex are 559 K and 421 K, respectively. When the vibration enthalpy is included, the Tc values of both phases are lowered to 543 K and 418 K, respectively. And if the vibration entropy is also taken into account, the Tc values of both phases are further lowered to 529 K and 415 K, respectively. With inclusion of lattice vibration effects, the ZnS1−xSex alloy starts a phase transition from ZB to WZ at a temperature of 1600 K, and the solid solubility of Se increases with the temperature rise.