Small Supercells Research Articles

An efficient scheme for accelerating the calculation of stacking fault energy in multi-principal element alloys

We present the High-Throughput Computing and Statistical Analysis (HCSA) scheme, which efficiently and accurately predicts the stacking fault energies (SFEs) of multi-principal element alloys (MPEAs). Our approach estimates the SFE of a single complex supercell by averaging numerous SFEs from small supercells, resulting in superior accuracy compared to traditional density functional theory (DFT) calculations. To validate our scheme, we applied it to NiFe and Ni10Co60Cr25W5 alloys, achieving an SFE error of only 11%, in contrast to the 45% error obtained from traditional DFT calculations for NiFe. We observed a strong correlation between the average SFEs of samples with the same valence electron concentration as that of the experimental data. Our scheme provides an efficient and reliable tool for predicting SFEs in MPEAs and holds the potential to significantly accelerate materials design and discovery processes.

Atomistic mechanisms of phase nucleation and propagation in a model two-dimensional system

We present a computational study on the solid–solid phase transition of a model two-dimensional system between hexagonal and square phases under pressure. The atomistic mechanism of phase nucleation and propagation are determined using solid-state Dimer and nudged elastic band (NEB) methods. The Dimer is applied to identify the saddle configurations and NEB is applied to generate the transition minimum energy path (MEP) using the outputs of Dimer. Both the atomic and cell degrees of freedom are used in saddle search, allowing us to capture the critical nuclei with relatively small supercells. It is found that the phase nucleation in the model material is triggered by the localized shear deformation that comes from the relative shift between two adjacent atomic layers. In addition to the conventional layer-by-layer phase propagation, an interesting defect-assisted low barrier propagation path is identified in the hexagonal to square phase transition. The study demonstrates the significance of using the Dimer method in exploring unknown transition paths without a priori assumption. The results of this study also shed light on phase transition mechanisms of other solid-state and colloidal systems.

Frenkel pair formation energy for cubic Fe3O4 in DFT + U calculations

The cubic phase of magnetite is stabilized above the Verwey transition temperature of about 120 K via a complex electron–phonon interaction that is still not very well understood. In this work using the DFT + U method we describe our attempt to calculate point defect formation energies for this cubic phase in the static approximation. The electronic structure calculations and atomic relaxation peculiarities are discussed in this context. Only the cubic phase model with a small band gap and charge disproportionation (Fe2+/Fe3+) gives an adequate point defect formation energies, not the semi-metallic model. The relaxation of the local defect atomic structure and the relaxation of the surrounding crystal matrix are analyzed. Point defects cause only local perturbations of atomic positions and charge-orbital order. After analysis of the supercell size effects for up to 448 atoms, we justify the use of small supercells with 56 atoms to make calculations for the cubic phase. The extensive experimental results of Dieckmann et al on defects in magnetite at high temperature are deployed for comparison of our DFT + U results on Frenkel pair formation energies.

Energetic stability and spatial inhomogeneity in the local electronic structure of relaxed twisted trilayer graphene

We study the energetic stability and the local electronic structure of the general twisted trilayer graphene (TTG) with the top and bottom layers rotated with respect to the middle layer respectively by $\theta$ and $\theta'$. Approximate supercells of the moir\'{e}-of-moir\'{e} superlattices with $\theta$ and $\theta^{\prime}$ within $1^{\circ}\sim 2^{\circ}$ are established to describe the structural and electronic properties of relaxed TTG with the periodic boundary condition. Full relaxation demonstrates that the commensurate TTG with $\theta=\theta^{\prime}$ has the local minimum total energy ($E_{tol}$) at a fixed $\theta$, while $E_{tol}$ first reaches a local maximum and begins to drop with decreasing $\theta^{\prime}$ for $\theta^{\prime} < \theta$. Some regions exhibit enhanced in-plane relaxation in the top and bottom layers but suppressed relaxation in the middle layer and form a hexagonal network with the moir\'{e}-of-moir\'{e} length scale. The stacking configurations with the atoms in the three layers vertically aligned at the origin of the relaxed TTG supercells at $\theta$ around $1.6^{\circ}$ and $\theta^{\prime}$ around $1.4^{\circ}$ have a high density of states (DOS) near the Fermi level ($E_F$), which can reach that of the mirror symmetric TTG with equal twist angles of about $1.7^{\circ}$. In contrast, some other stackings can have rather low DOS around $E_F$. The significant stacking dependence of DOS for some TTG supercells demonstrates that the local electronic structure of TTG can exhibit strong spatial inhomogeneity when the twist angles are slightly away from those of the small supercells with large variations of DOS among different stackings. Moreover, the structural relaxation of TTG plays a crucial role in the high DOS and its strong stacking dependence.

Adsorption Motifs and Molecular Orientation at the Ionic Liquid/Noble Metal Interface: [C2C1Im][NTf2] on Pt(111).

In solid catalysts with ionic liquid layers (SCILLs), ionic liquid (IL) thin films are used to modify the activity and selectivity of catalytic materials. In this work, we investigated the adsorption behavior of the IL 1-ethyl-3-methyl-imidazolium bis(trifluoromethylsulfonyl)imide [C2C1Im][NTf2] on Pt(111) by combining experimental and theoretical studies. Under ultrahigh vacuum (UHV) conditions, the IL was deposited onto a Pt(111) single crystal surface by physical vapor deposition (PVD) at different surface temperatures (200, 300, and 400 K). The adsorption process was monitored by in situ infrared reflection absorption spectroscopy (IRAS). Complementary to the IRAS studies, we performed density functional theory (DFT) calculations and analyzed the adsorption motifs and orientation of the IL ions. In total, we calculated four different systems: (a) [C2C1Im]+ and [NTf2]- ions in the gas phase; [NTf2]- anions in (b) small (4 × 4) and (c) large (6 × 6) Pt(111) supercells; and (d) a complete ion pair of [C2C1Im][NTf2] in a (6 × 6) Pt(111) supercell. Based on DFT, we simulated IR spectra and compared them to the experimental data. Our results suggest that the binding motif and orientation of the IL is strongly dependent on the actual IL coverage. In the monolayer (ML), [NTf2]- interacts strongly with the metal surface and adopts a specific orientation in which it interacts with the Pt surface via the SO2 groups. Also the [C2C1Im]+ cations adopt a preferential orientation up to coverages of 1 ML. Upon transition to the multilayer region, the specific orientation of the ions is gradually lost.

Misalignment instability in magic-angle twisted bilayer graphene on hexagonal boron nitride

We study the stability and electronic structure of magic-angle twisted bilayer graphene on the hexagonal boron nitride (TBG/BN). Structural relaxation has been performed for commensurate supercells of the heterostructures with different twist angles () and stackings between TBG and BN. We find that the slightly misaligned configuration with and the AA/AA stacking has the globally lowest total energy due to the constructive interference of the moiré interlayer potentials and thus the greatly enhanced relaxation in its 1 × 1 commensurate supercell. Gaps are opened at the Fermi level (E F ) for small supercells with the stackings that enable strong breaking of the C 2 symmetry in the atomic structure of TBG. For large supercells with close to those of the 1 × 1 supercells, the broadened flat bands can still be resolved from the spectral functions. The is also identified as a critical angle for the evolution of the electronic structure with , at which the energy range of the mini-bands around E F begins to become narrower with increasing and their gaps from the dispersive bands become wider. The discovered stablest TBG/BN with a finite of about 0.54∘ and its gapped flat bands agree with recent experimental observations.

Exceptional piezoelectricity, high thermal conductivity and stiffness and promising photocatalysis in two-dimensional MoSi2N4 family confirmed by first-principles

Chemical vapor deposition has been most recently employed to fabricate centimeter-scale high-quality single-layer MoSi2N4 (Science; 2020;369; 670). Motivated by this exciting experimental advance, herein we conduct extensive first-principles based simulations to explore the stability, mechanical properties, lattice thermal conductivity, piezoelectric and flexoelectric response, and photocatalytic and electronic features of MA2Z4 (M = Cr, Mo, W; A = Si, Ge; Z = N, P) monolayers. The considered nanosheets are found to exhibit dynamical stability and remarkably high mechanical properties. Moreover, they show diverse electronic properties from antiferromagnetic metal to half metal and to semiconductors with band gaps ranging from 0.31 to 2.57 eV. Among the studied nanosheets, the MoSi2N4 and WSi2N4 monolayers yield appropriate band edge positions, high electron and hole mobilities, and strong visible light absorption, highly promising for applications in optoelectronics and photocatalytic water splitting. The MoSi2N4 and WSi2N4 monolayers are also predicted to show outstandingly high lattice thermal conductivity of 440 and 500 W/mK, respectively. For the first time we show that machine learning interatomic potentials trained over small supercells can be employed to examine the flexoelectric and piezoelectric properties of complex structures. As the most exciting finding, WSi2N4, CrSi2N4 and MoSi2N4 are found to exhibit the highest piezoelectric coefficients, outperforming all other-known 2D materials. Our results highlight that MA2Z4 nanosheets not only undoubtedly outperform the transition metal dichalcogenides family but also can compete with graphene for applications in nanoelectronics, optoelectronic, energy storage/conversion and thermal management systems.

Machine Learning Enabled Prediction of Stacking Fault Energies in Concentrated Alloys

Recent works have revealed a unique combination of high strength and high ductility in certain compositions of high-entropy alloys (HEAs), which is attributed to the low stacking fault energy (SFE). While atomistic calculations have been successful in predicting the SFE of pure metals, large variations up to 200 mJ/m2 have been observed in HEAs. One of the leading causes of such variations is the limited number of atoms that can be modeled in atomistic calculations; as a result, due to random distribution of elements in HEAs, various nearest neighbor environments may not be adequately captured in small supercells resulting in different SFE values. Such variation further increases with the increase in the number of elements in a given composition. In this work, we use machine learning to overcome the limitation of smaller system sizes and provide a methodology to significantly reduce the variation and uncertainty in predicting SFEs. We show that the SFE can be accurately predicted across the composition ranges in binary alloys. This capability then enables us to predict the SFE of multi-elemental alloys by training the model using only binary alloys. Consequently, SFEs of complex alloys can be predicted using a binary alloys database, and the need to perform calculations for every new composition can be circumvented.

The minimal supercells approach for ab-initio calculation in 2D alloying transition metal dichalcoginides with special quasi-random structure

Density functional theory (DFT) is used to investigate MoS 2 and WS 2 monolayers, which are direct bandgap semiconductors. We study alloying between MoS 2 and WS 2 by using special quasi-random structure (SQS), through a comparison of the computed pair distribution functions with various sizes of supercells. Our calculations show that a model 3 × 3 × 1 supercell structure of pseudobinary alloy Mo 1−x W x S 2 can be correctly performed for energy and electronic band structure calculations. DFT is combined with SQS and reveals that alteration of the W concentration supports the band edges and energy gap. The electronic structure of Mo 1−x W x S 2 clearly supports the results from the experimental observation as well as Monte Carlo simulation. Consequently, our model suggests that the generated alloy monolayer with small supercells via SQS can clearly explain the behaviour of this material, using a low computational time but achieving good agreement with the experiment.

Bandgap prediction by deep learning in configurationally hybridized graphene and boron nitride

It is well-known that the atomic-scale and nano-scale configuration of dopants can play a crucial role in determining the electronic properties of materials. However, predicting such effects is challenging due to the large range of atomic configurations that are possible. Here, we present a case study of how deep learning algorithms can enable bandgap prediction in hybridized boron–nitrogen graphene with arbitrary supercell configurations. A material descriptor that enables correlation of structure and bandgap was developed for convolutional neural networks. Bandgaps calculated by ab initio calculations, and corresponding structures, were used as training datasets. The trained networks were then used to predict bandgaps of systems with various configurations. For 4 × 4 and 5 × 5 supercells they accurately predict bandgaps, with a R2 of >90% and root-mean-square error of ~0.1 eV. The transfer learning was performed by leveraging data generated from small supercells to improve the prediction accuracy for 6 × 6 supercells. This work will pave a route to future investigation of configurationally hybridized graphene and other 2D materials. Moreover, given the ubiquitous existence of configurations in materials, this work may stimulate interest in applying deep learning algorithms for the configurational design of materials across different length scales.

Symmetry and thermodynamics of tellurium vacancies in cadmium telluride

The equilibrium geometries and thermodynamic properties of anion vacancies in cadmium telluride, as predicted by density functional theory, are revisited using semilocal and hybrid density functionals. We find that stable configurations in different charge states can only be found after a systematic search considering several starting geometries. The stable charge states, 0 and 2+, display closed-shell electronic configurations, without deep bandgap levels. The 2+ charge state has a Td symmetry with an outward relaxation, while the neutral state is a mixture of configurations with C2v and C3v symmetries, both with the same energy and a negligible energy barrier. Therefore, the neutral charge state presents an effective Td symmetry. Configurations with different symmetries, e.g., D2d, can exist as metastable states. We show that certain configurations may seem falsely stable due to several facts: the bandgap error of generalized gradient approximation, the k-point sampling used in small supercells, or the use of a restricted set of starting geometries. We believe that the HSE06 hybrid functional allows to obtain accurate formation energies and geometries. We analyze the effect of the spin-orbit coupling and GW quasiparticle corrections to the HSE06 results, and find no qualitative differences. The spin-orbit coupling and GW corrections to the HSE06 energies partially cancel each other. Finally, we investigate the divacancy VCdVTe. The obtained formation energies suggest that isolated tellurium vacancies in neutral charge state can be found only in Te-poor growth conditions, coexisting with divacancies.

Assessing Density Functionals Using Many Body Theory for Hybrid Perovskites.

Which density functional is the "best" for structure simulations of a particular material? A concise, first principles, approach to answer this question is presented. The random phase approximation (RPA)-an accurate many body theory-is used to evaluate various density functionals. To demonstrate and verify the method, we apply it to the hybrid perovskite MAPbI_{3}, a promising new solar cell material. The evaluation is done by first creating finite temperature ensembles for small supercells using RPA molecular dynamics, and then evaluating the variance between the RPA and various approximate density functionals for these ensembles. We find that, contrary to recent suggestions, van der Waals functionals do not improve the description of the material, whereas hybrid functionals and the strongly constrained appropriately normed (SCAN) density functional yield very good agreement with the RPA. Finally, our study shows that in the room temperature tetragonal phase of MAPbI_{3}, the molecules are preferentially parallel to the shorter lattice vectors but reorientation on ps time scales is still possible.

Computing energy barriers for rare events from hybrid quantum/classical simulations through the virtual work principle

Hybrid quantum/classical techniques can flexibly couple ab initio simulations to an empirical or elastic medium to model materials systems that cannot be contained in small periodic supercells. However, due to electronic non-locality a total energy cannot be defined, meaning energy barriers cannot be calculated. We provide a general solution using the principle of virtual work in a modified nudged elastic band algorithm. Our method enables the first ab initio calculations of the kink formation energy for <100> edge dislocations in molybdenum and lattice trapping barriers to brittle fracture in silicon.

Importance of elastic finite-size effects: Neutral defects in ionic compounds

Small system sizes are a well-known source of error in density functional theory (DFT) calculations, yet computational constraints frequently dictate the use of small supercells, often as small as 96 atoms in oxides and compound semiconductors. In ionic compounds, electrostatic finite-size effects have been well characterized, but self-interaction of charge-neutral defects is often discounted or assumed to follow an asymptotic behavior and thus easily corrected with linear elastic theory. Here we show that elastic effects are also important in the description of defects in ionic compounds and can lead to qualitatively incorrect conclusions if inadequately small supercells are used; moreover, the spurious self-interaction does not follow the behavior predicted by linear elastic theory. Considering the exemplar cases of metal oxides with fluorite structure, we show that numerous previous studies, employing 96-atom supercells, misidentify the ground-state structure of (charge-neutral) Schottky defects. We show that the error is eliminated by employing larger cells (324, 768, and 1500 atoms), and careful analysis determines that elastic, not electrostatic, effects are responsible. The spurious self-interaction was also observed in nonoxide ionic compounds irrespective of the computational method used, thereby resolving long-standing discrepancies between DFT and force-field methods, previously attributed to the level of theory. The surprising magnitude of the elastic effects is a cautionary tale for defect calculations in ionic materials, particularly when employing computationally expensive methods (e.g., hybrid functionals) or when modeling large defect clusters. We propose two computationally practicable methods to test the magnitude of the elastic self-interaction in any ionic system. In commonly studied oxides, where electrostatic effects would be expected to be dominant, it is the elastic effects that dictate the need for larger supercells: greater than 96 atoms.

First principle calculation of accurate native defect levels in CaF2

We report on the first principle density functional calculation of the charge transition levels of native defects (vacancies and interstitials) in CaF2 structure. The transition level was defined as the Fermi level where two charge states of given defect have the same formation energy. The common error in the band gap inherited to semiclocal density functional has been accounted for by incorporating the hybrid density functional method, leading to correct placement of the transition levels within the band gap. The band gap size from hybrid calculation has been validated using the full potential, Linearized Augmented Planewave method with the Modified-Becke-Johnson exchange potential. Prior to level calculations, we ensured that an agreement between the formation energies from small (95–97 atoms) and large (323–325 atoms) supercells was achieved after applying the Makov-Payne correction method. Our calculated transition level for the anion vacancy was 2.97 eV below the conduction band, agreeing with the experimental optical absorption band at 3.3 eV associated with the electron transition from the ground state F-center to the conduction band in CaF2.

Clustering/anticlustering effects on the GeSi Raman spectra at moderate (Ge,Si) contents: Percolation scheme vs. ab initio calculations

We test a presumed ability behind the phenomenological percolation scheme used for the basic description of the multi-mode Raman spectra of mixed crystals at one dimension along the linear chain approximation, to determine, via the Raman intensities, the nature of the atom substitution, as to whether this is random or due to local clustering/anticlustering. For doing so, we focus on the model percolation-type GeySi1−y system characterized by six oscillators {1×(Ge−Ge),3×(Ge−Si),2×(Si−Si)} and place the study around the critical compositions y ∼ (0.16, 0.71, and 0.84) corresponding to nearly matching of intensities between the like Raman modes from a given multiplet (Ge−Si triplet or Si−Si doublet). The interplay between the GeySi1−y Raman intensities predicted by the percolation scheme depending on a suitable order parameter κ of local clustering/anticlustering is found to be consistent with ab initio calculations of the GeySi1−y Raman spectra done with the Ab Initio Modeling PROgram code using large (64-, 216-, and 512-atoms) disordered cubic supercells matching the required (y,κ) values. The actual “percolation vs. ab initio” comparative insight at moderate/dilute-(Ge,Si) limits, with an emphasis on the κ-induced intra-bond transfer of oscillator strength, extends a pioneering one earlier achieved at an intermediate composition (y ∼ 0.50) by using small (32-atom) supercells [O. Pagès et al., J. Appl. Phys. 114, 033513 (2013)], mainly concerned with the inter-bond transfer of oscillator strength, providing altogether a complete picture.

Averaging the intensity of many-layered structures for accurate stacking-fault analysis using Rietveld refinement

Many technologically important synthetic and natural materials display stacking faults which lead to complex peak broadenings, asymmetries and shifts in their powder diffraction patterns. The patterns can be described using an enlarged unit cell (called a supercell) containing an explicit description of the layers. Since the supercell can contain hundreds of thousands of atoms with hundreds of thousands ofhklreflections, a Rietveld approach has been too computationally demanding for all but the simplest systems. This article describes the implementation of the speed-ups necessary to allow Rietveld refinement in the computer programTOPASVersion 6 (Bruker AXS, Karlsruhe, Germany). Techniques implemented include: a peaks buffer that allows hundreds of thousands ofhkl-dependent peak shapes to be automatically approximated by a few hundred peaks; an averaging process for hundreds of large supercells with minimum impact on computational time; a smoothing technique that allows for the use of small supercells which approximate supercells ten to 20 times larger; and efficient algorithms for stacking sequence generation. The result is Rietveld refinement of supercells operating at speeds several thousand times faster than traditional Rietveld refinements. This allows quantitative and simultaneous analysis of structure and microstructure in complex stacking-faulted samples.

Oxygen vacancy diffusion in bulk SrTiO3 from density functional theory calculations

Point defects and their diffusion contribute significantly to the properties of perovskite materials. However, even for the prototypical case of oxygen vacancies in SrTiO3 (STO), the predictions of oxygen vacancy activity vary widely. Here we present a comprehensive and systematic study of the diffusion barriers in bulk STO. Using density functional theory (DFT), we assess the role of different supercell sizes, density functionals, and charge states. Our results show that vacancy-induced octahedral rotations, which are limited by the boundary conditions of the supercell, can significantly affect the computed oxygen vacancy diffusion energy barrier. In addition, we find that the diffusion energy barrier of a charged oxygen vacancy is lower than that of a neutral one. This difference is magnified in small supercells. We demonstrate that with increasing supercell size, the effects of the oxygen vacancy charge state and the type of DFT exchange and correlation functional diminish, and all DFT predicted migration energy barriers asymptote to a range of 0.39–0.49eV, which is smaller than the reported experimental values. This work provides important insights and guidance that should be considered for investigations of point defect diffusion in perovskite materials and in oxide superlattices.

Density functional theory calculations of magnetocrystalline anisotropy energies for ( Fe1–xCox)2B

We present and discuss density functional theory calculations of magnetic properties of the family of ferromagnetic compounds, ( )B, focusing specifically on the magnetocrystalline anisotropy energy (MAE). Using periodic supercells of various sizes (up to 96 atoms), it is shown that the general qualitative features of the composition dependence of the MAE is in agreement with experimental findings, while our predicted magnitudes are larger than those of experiment. We find that the use of small supercells (6 and 12-atom) favors larger MAE values relative to a statistical sample of configurations constructed with 96-atom supercells. The effect of lattice relaxations is shown to be small. Calculations of the Curie temperature for this alloy are also presented.

Does water dope carbon nanotubes?

We calculate the long-range perturbation to the electronic charge density of carbon nanotubes (CNTs) as a result of the physisorption of a water molecule. We find that the dominant effect is a charge redistribution in the CNT due to polarisation caused by the dipole moment of the water molecule. The charge redistribution is found to occur over a length-scale greater than 30 Å, highlighting the need for large-scale simulations. By comparing our fully first-principles calculations to ones in which the perturbation due to a water molecule is treated using a classical electrostatic model, we estimate that the charge transfer between CNT and water is negligible (no more than 10(-4) e per water molecule). We therefore conclude that water does not significantly dope CNTs, a conclusion that is consistent with the poor alignment of the relevant energy levels of the water molecule and CNT. Previous calculations that suggest water n-dopes CNTs are likely due to the misinterpretation of Mulliken charge partitioning in small supercells.