ABINIT: Overview and focus on selected capabilities

Mimicking Real Catalysts: Model Stepped Nickel Surfaces in Furfural Catalysis─Insights into Adsorption, Reactivity, and Defect-Driven Conversion Pathways

The catalytic conversion of furanic compounds into renewable chemicals is essential for sustainable manufacturing. Here, we report a unique self-hydrogenation pathway of furfural to 2-methylfuran on Ni(119) surface, showing how steps and nickel carbides govern reaction selectivity. Thermal desorption and spectroscopic measurements reveal that furfural undergoes decarbonylation to furan on terraces, while step sites act as "hydrogen transfer pumps", abstracting hydrogen from furfural and facilitating its diffusion to terrace-bound molecules, thereby promoting selective hydrogenation to 2-methylfuran. Moreover, the surface-bound hydrogen enhances hydrogenolysis, with product selectivity closely connected to hydrogen concentration. DFT calculations show a preference for the top step edges, where strong bonding and electron redistribution stabilize intermediates and promote catalytic transformations. We further demonstrate how these insights provide a framework for designing advanced catalysts through surface structure optimization. By linking model catalysts with real-world applications, this approach enables the development of efficient and selective catalysts tailored for biomass conversion.

Microhardness, Young's and Shear Modulus in Tetrahedrally Bonded Novel II-Oxides and III-Nitrides

Direct wide-bandgap III-Ns and II-Os have recently gained considerable attention due to their unique electrical and chemical properties. These novel semiconductors are being explored to design short-wavelength light-emitting diodes, sensors/biosensors, photodetectors for integration into flexible transparent nanoelectronics/photonics to achieve high-power radio-frequency modules, and heat-resistant optical switches for communication networks. Knowledge of the elastic constants structural and mechanical properties has played crucial roles both in the basic understanding and assessing materials' use in thermal management applications. In the absence of experimental structural, elastic constants, and mechanical traits, many theoretical simulations have yielded inconsistent results. This work aims to investigate the basic characteristics of tetrahedrally coordinated, partially ionic BeO, MgO, ZnO, and CdO, and partially covalent BN, AlN, GaN, and InN materials. By incorporating a bond-orbital and a valance force field model, we have reported comparative results of our systematic calculations for the bond length d, bond polarity αP, covalency αC, bulk modulus B, elastic stiffness C(=c11-c122), bond-stretching α and bond-bending β force constants, Kleinmann's internal displacement ζ, and Born's transverse effective charge eT*. Correlations between C/B, β/α, c12c11, ζ, and αC revealed valuable trends of structural, elastic, and bonding characteristics. The study noticed AlN and GaN (MgO and ZnO) showing nearly comparable features, while BN (BeO) is much harder compared to InN (CdO) material, with drastically softer bonding. Calculations of microhardness H, shear modulus G, and Young's modulus Y have predicted BN (BeO) satisfying a criterion of super hardness. III-Ns (II-Os) could be vital in electronics, aerospace, defense, nuclear reactors, and automotive industries, providing integrity and performance at high temperature in high-power applications, ranging from heat sinks to electronic substrates to insulators in high-power devices.

Large scale Raman spectrum calculations in defective 2D materials using deep learning

We introduce a machine learning prediction workflow to study the impact of defects on the Raman response of 2D materials. By combining the use of machine-learned interatomic potentials, the Raman-active Γ-weighted density of states method and splitting configurations in independant patches, we are able to reach simulation sizes in the tens of thousands of atoms, with diagonalization now being the main bottleneck of the simulation. We apply the method to two systems, isotopic graphene and defective hexagonal boron nitride, and compare our predicted Raman response to experimental results, with good agreement. Our method opens up many possibilities for future studies of Raman response in solid-state physics.

GPU-Accelerated Solution of the Bethe-Salpeter Equation for Large and Heterogeneous Systems

We present a massively parallel GPU-accelerated implementation of the Bethe-Salpeter equation (BSE) for the calculation of the vertical excitation energies (VEEs) and optical absorption spectra of condensed and molecular systems, starting from single-particle eigenvalues and eigenvectors obtained with density functional theory. The algorithms adopted here circumvent the slowly converging sums over empty and occupied states and the inversion of large dielectric matrices through a density matrix perturbation theory approach and a low-rank decomposition of the screened Coulomb interaction, respectively. Further computational savings are achieved by exploiting the nearsightedness of the density matrix of semiconductors and insulators to reduce the number of screened Coulomb integrals. We scale our calculations to thousands of GPUs with a hierarchical loop and data distribution strategy. The efficacy of our method is demonstrated by computing the VEEs of several spin defects in wide-band-gap materials, showing that supercells with up to 1000 atoms are necessary to obtain converged results. We discuss the validity of the common approximation that solves the BSE with truncated sums over empty and occupied states. We then apply our GW-BSE implementation to a diamond lattice with 1727 atoms to study the symmetry breaking of triplet states caused by the interaction of a point defect with an extended line defect.

Optical Excitation of Coherent THz Dynamics of the Rare-Earth Lattice through Resonant Pumping of f-f Electronic Transition in a Complex Perovskite DyFeO_{3}

Resonant pumping of the electronic f-f transitions in the orbital multiplet of dysprosium ions (Dy^{3+}) in a complex perovskite DyFeO_{3} is shown to impulsively launch THz lattice dynamics corresponding to the B_{2g} phonon mode, which is dominanted by the motion of Dy^{3+} ions. The findings, supported by symmetry analysis and density-functional theory calculations, not only provide a novel route for highly selective excitation of the rare-earth crystal lattices but also establish important relationships between the symmetry of the electronic and lattice excitations in complex oxides.

Colossal Strain Tuning of Ferroelectric Transitions in KNbO3 Thin Films

Strong coupling between polarization (P) and strain (ɛ) in ferroelectric complex oxides offers unique opportunities to dramatically tune their properties. Here colossal strain tuning of ferroelectricity in epitaxial KNbO3 thin films grown by sub-oxide molecular beam epitaxy is demonstrated. While bulk KNbO3 exhibits three ferroelectric transitions and a Curie temperature (Tc) of ≈676 K, phase-field modeling predicts that a biaxial strain of as little as -0.6% pushes its Tc > 975 K, its decomposition temperature in air, and for -1.4% strain, to Tc > 1325 K, its melting point. Furthermore, a strain of -1.5% can stabilize a single phase throughout the entire temperature range of its stability. A combination of temperature-dependent second harmonic generation measurements, synchrotron-based X-ray reciprocal space mapping, ferroelectric measurements, and transmission electron microscopy reveal a single tetragonal phase from 10 K to 975 K, an enhancement of ≈46% in the tetragonal phase remanent polarization (Pr), and a ≈200% enhancement in its optical second harmonic generation coefficients over bulk values. These properties in a lead-free system, but with properties comparable or superior to lead-based systems, make it an attractive candidate for applications ranging from high-temperature ferroelectric memory to cryogenic temperature quantum computing.

Exchange-correlation potential built on the derivative discontinuity of electron density

Electronic structures are fully determined by the exchange-correlation (XC) potential. In this work, we develop a new method to construct reliable XC potentials by properly mixing the exact exchange and the local density approximation potentials in real space. The spatially dependent mixing parameter is derived based on the derivative discontinuity of electron density and is first-principle. We derived the equations for solving the mixing parameter and proposed an approximation to simplify these equations. Based on this approximation, this new method gives reasonable predictions for the ionization energies, fundamental gaps, and singlet-triplet energy differences for various molecular systems. The impact of the approximation on the constructed XC potentials is examined, and it is found that the quality of the XC potentials can be further improved by removing the approximation. This work demonstrates that the derivative discontinuity of electron density is a promising constraint for constructing high-quality XC potentials.

Giant Enhancement of Hole Mobility for 4H-Silicon Carbide through Suppressing Interband Electron-Phonon Scattering

4H-silicon carbide (4H-SiC) possesses a high Baliga figure of merit, making it a promising material for power electronics. However, its applications are limited by low hole mobility. Herein, we found that the hole mobility of 4H-SiC is mainly limited by the strong interband electron-phonon scattering using mode-level first-principles calculations. Our research indicates that applying compressive strain can reverse the sign of crystal-field splitting and change the ordering of electron bands close to the valence band maximum. Therefore, the interband electron-phonon scattering is severely suppressed and the electron group velocity is significantly increased. The out-of-plane hole mobility of 4H-SiC can be greatly enhanced by ∼200% with 2% uniaxial compressive strain applied. This work provides new insights into the electron transport mechanisms in semiconductors and suggests a strategy to improve hole mobility that could be applied to other semiconductors with hexagonal crystalline geometries.

Efficient determination of Born-effective charges, LO-TO splitting, and Raman tensors of solids with a real-space atom-centered deep learning approach

We introduce a deep neural network (DNN) framework called theReal-spaceAtomicDecompositionNETwork (radnet), which is capable of making accurate predictions of polarization and of electronic dielectric permittivity tensors in solids and aims to address limitations of previously available machine learning models for Raman predictions in periodic systems. This framework builds on previous, atom-centered approaches while utilizing deep convolutional neural networks. We report excellent accuracies on direct predictions for two prototypical examples: GaAs and BN. We then use automatic differentiation to efficiently calculate the Born-effective charges, longitudinal optical-transverse optical (LO-TO) splitting frequencies, and Raman tensors of these materials. We compute the Raman spectra, and find agreement withab initioresults. Lastly, we explore ways to generalize the predictions of polarization while taking into account periodic boundary conditions and symmetries.

Second-harmonic generation tensors from high-throughput density-functional perturbation theory

Optical materials play a key role in enabling modern optoelectronic technologies in a wide variety of domains such as the medical or the energy sector. Among them, nonlinear optical crystals are of primary importance to achieve a broader range of electromagnetic waves in the devices. However, numerous and contradicting requirements significantly limit the discovery of new potential candidates, which, in turn, hinders the technological development. In the present work, the static nonlinear susceptibility and dielectric tensor are computed via density-functional perturbation theory for a set of 579 inorganic semiconductors. The computational methodology is discussed and the provided database is described with respect to both its data distribution and its format. Several comparisons with both experimental and ab initio results from literature allow to confirm the reliability of our data. The aim of this work is to provide a relevant dataset to foster the identification of promising nonlinear optical crystals in order to motivate their subsequent experimental investigation.

Predicting the One-Particle Density Matrix with Machine Learning

Two of the most widely used electronic-structure theory methods, namely, Hartree-Fock and Kohn-Sham density functional theory, require the iterative solution of a set of Schrödinger-like equations. The speed of convergence of such a process depends on the complexity of the system under investigation, the self-consistent-field algorithm employed, and the initial guess for the density matrix. An initial density matrix close to the ground-state matrix will effectively allow one to cut out many of the self-consistent steps necessary to achieve convergence. Here, we predict the density matrix of Kohn-Sham density functional theory by constructing a neural network that uses only the atomic positions as information. Such a neural network provides an initial guess for the density matrix far superior to that of any other recipes available. Furthermore, the quality of such a neural-network density matrix is good enough for the evaluation of interatomic forces. This allows us to run accelerated ab initio molecular dynamics with little to no self-consistent steps.

Ab initio-simulated optical response of hot electrons in gold and ruthenium

Optical femtosecond pump-probe experiments allow to measure the dynamics of ultrafast heating of metals with high accuracy. However, the theoretical analysis of such experiments is often complicated because of the indirect connection of the measured signal and the desired temperature transients. Establishing such a connection requires an accurate model of the optical constants of a metal, depending on both the electron temperature Te and the lattice temperature Tl. In this paper, we present first-principles simulations of the two-temperature scenario with Te ≫ Tl, showing the optical response of hot electrons to laser irradiation in gold and ruthenium. Comparing our simulations with the Kubo-Greenwood approach, we discuss the influence of electron-phonon and electron-electron scattering on the intraband contribution to optical constants. Applying the simulated optical constants to the analysis of ultrafast heating of ruthenium thin films we highlight the importance of the latter scattering channel to understand the measured heating dynamics.

Macroscopic Polarization from Nonlinear Gradient Couplings

We show that a lattice mode of arbitrary symmetry induces a well-defined macroscopic polarization at first order in the momentum and second order in the amplitude. We identify a symmetric flexoelectric-like contribution, which is sensitive to both the electrical and mechanical boundary conditions, and an antisymmetric Dzialoshinskii-Moriya-like term, which is unaffected by either. We develop the first-principles methodology to compute the relevant coupling tensors in an arbitrary crystal, which we illustrate with the example of the antiferrodistortive order parameter in SrTiO_{3}.

Optoelectronic Readout of Single Er Adatom's Electronic States Adsorbed on the Si(100) Surface at Low Temperature (9 K)

Integrating nanoscale optoelectronic functions is vital for applications such as optical emitters, detectors, and quantum information. Lanthanide atoms show great potential in this endeavor due to their intrinsic transitions. Here, we investigate Er adatoms on Si(100)-2×1 at 9 K using a scanning tunneling microscope (STM) coupled to a tunable laser. Er adatoms display two main adsorption configurations that are optically excited between 800 and 1200 nm while the STM reads the resulting photocurrents. Our spectroscopic method reveals that various photocurrent signals stem from the bare silicon surface or Er adatoms. Additional photocurrent peaks appear as the signature of the Er adatom relaxation, triggering efficient dissociation of nearby trapped excitons. Calculations using density functional theory with spin-orbit coupling correction highlight the origin of the observed photocurrent peaks as specific 4f→4f or 4f→5d transitions. This spectroscopic technique can facilitate optoelectronic analysis of atomic and molecular assemblies by offering insight into their intrinsic quantum properties.

Differentiating α-moganite, silanol and α-quartz by Raman spectroscopy

The silica phases quartz, silanol and moganite are widely prevalent and consequential in industrial applications and natural science. However, methods for differentiating these important phases are few. Using Raman spectra simulated by density function and perturbation expansion after discretization theory, representative spectra could be obtained and the comingling of diagnostic Raman Bands for the three phases identified in samples. On this basis new methods to identify moganite in Raman spectra are proposed.

Evidence for phonon hardening in laser-excited gold using x-ray diffraction at a hard x-ray free electron laser

Studies of laser-heated materials on femtosecond timescales have shown that the interatomic potential can be perturbed at sufficiently high laser intensities. For gold, it has been postulated to undergo a strong stiffening leading to an increase of the phonon energies, known as phonon hardening. Despite efforts to investigate this behavior, only measurements at low absorbed energy density have been performed, for which the interpretation of the experimental data remains ambiguous. By using in situ single-shot x-ray diffraction at a hard x-ray free-electron laser, the evolution of diffraction line intensities of laser-excited Au to a higher energy density provides evidence for phonon hardening.

On the sign of the linear magnetoelectric coefficient in Cr2O3

We establish the sign of the linear magnetoelectric (ME) coefficient,α, in chromia, Cr2O3. Cr2O3is the prototypical linear ME material, in which an electric (magnetic) field induces a linearly proportional magnetization (polarization), and a single magnetic domain can be selected by annealing in combined magnetic (H) and electric (E) fields. Opposite antiferromagnetic (AFM) domains have opposite ME responses, and which AFM domain corresponds to which sign of response has previously been unclear. We use density functional theory (DFT) to calculate the magnetic response of a single AFM domain of Cr2O3to an applied in-plane electric field at zero kelvin. We find that the domain with nearest neighbor magnetic moments oriented away from (towards) each other has a negative (positive) in-plane ME coefficient,α⊥, at zero kelvin. We show that this sign is consistent with all other DFT calculations in the literature that specified the domain orientation, independent of the choice of DFT code or functional, the method used to apply the field, and whether the direct (magnetic field) or inverse (electric field) ME response was calculated. Next, we reanalyze our previously published spherical neutron polarimetry data to determine the AFM domain produced by annealing in combinedEandHfields oriented along the crystallographic symmetry axis at room temperature. We find that the AFM domain with nearest-neighbor magnetic moments oriented away from (towards) each other is produced by annealing in (anti-)parallelEandHfields, corresponding to a positive (negative) axial ME coefficient,α∥, at room temperature. Sinceα⊥at zero kelvin andα∥at room temperature are known to be of opposite sign, our computational and experimental results are consistent.

Carbon Kagome nanotubes-quasi-one-dimensional nanostructures with flat bands

In recent years, a number of bulk materials and heterostructures have been explored due their connections with exotic materials phenomena emanating from flat band physics and strong electronic correlation. The possibility of realizing such fascinating material properties in simple realistic nanostructures is particularly exciting, especially as the investigation of exotic states of electronic matter in wire-like geometries is relatively unexplored in the literature. Motivated by these considerations, we introduce in this work carbon Kagome nanotubes (CKNTs)-a new allotrope of carbon formed by rolling up Kagome graphene, and investigate this material using specialized first principles calculations. We identify two principal varieties of CKNTs-armchair and zigzag, and find both varieties to be stable at room temperature, based on ab initio molecular dynamics simulations. CKNTs are metallic and feature dispersionless states (i.e., flat bands) near the Fermi level throughout their Brillouin zone, along with an associated singular peak in the electronic density of states. We calculate the mechanical and electronic response of CKNTs to torsional and axial strains, and show that CKNTs appear to be more mechanically compliant than conventional carbon nanotubes (CNTs). Additionally, we find that the electronic properties of CKNTs undergo significant electronic transitions-with emergent partial flat bands and tilted Dirac points-when twisted. We develop a relatively simple tight-binding model that can explain many of these electronic features. We also discuss possible routes for the synthesis of CKNTs. Overall, CKNTs appear to be unique and striking examples of realistic elemental quasi-one-dimensional materials that may display fascinating material properties due to strong electronic correlation. Distorted CKNTs may provide an interesting nanomaterial platform where flat band physics and chirality induced anomalous transport effects may be studied together.

In-Plane Flexoelectricity in Two-Dimensional D_{3d} Crystals

We predict a large in-plane polarization response to bending in a broad class of trigonal two-dimensional crystals. We define and compute the relevant flexoelectric coefficients from first principles as linear-response properties of the undistorted layer by using the primitive crystal cell. The ensuing response (evaluated for SnS_{2}, silicene, phosphorene, and RhI_{3} monolayers and for a hexagonal BN bilayer) is up to 1 order of magnitude larger than the out-of-plane components in the same material. We illustrate the topological implications of our findings by calculating the polarization textures that are associated with a variety of rippled and bent structures. We also determine the longitudinal electric fields induced by a flexural phonon at leading order in amplitude and momentum.

Limits to Hole Mobility and Doping in Copper Iodide

Over one hundred years have passed since the discovery of the p-type transparent conducting material copper iodide, predating the concept of the "electron-hole" itself. Supercentenarian status notwithstanding, little is understood about the charge transport mechanisms in CuI. Herein, a variety of modeling techniques are used to investigate the charge transport properties of CuI, and limitations to the hole mobility over experimentally achievable carrier concentrations are discussed. Poor dielectric response is responsible for extensive scattering from ionized impurities at degenerately doped carrier concentrations, while phonon scattering is found to dominate at lower carrier concentrations. A phonon-limited hole mobility of 162 cm2 V-1 s-1 is predicted at room temperature. The simulated charge transport properties for CuI are compared to existing experimental data, and the implications for future device performance are discussed. In addition to charge transport calculations, the defect chemistry of CuI is investigated with hybrid functionals, revealing that reasonably localized holes from the copper vacancy are the predominant source of charge carriers. The chalcogens S and Se are investigated as extrinsic dopants, where it is found that despite relatively low defect formation energies, they are unlikely to act as efficient electron acceptors due to the strong localization of holes and subsequent deep transition levels.

Constrained DFT-based magnetic machine-learning potentials for magnetic alloys: a case study of Fe-Al

We propose a machine-learning interatomic potential for multi-component magnetic materials. In this potential we consider magnetic moments as degrees of freedom (features) along with atomic positions, atomic types, and lattice vectors. We create a training set with constrained DFT (cDFT) that allows us to calculate energies of configurations with non-equilibrium (excited) magnetic moments and, thus, it is possible to construct the training set in a wide configuration space with great variety of non-equilibrium atomic positions, magnetic moments, and lattice vectors. Such a training set makes possible to fit reliable potentials that will allow us to predict properties of configurations in the excited states (including the ones with non-equilibrium magnetic moments). We verify the trained potentials on the system of bcc Fe-Al with different concentrations of Al and Fe and different ways Al and Fe atoms occupy the supercell sites. Here, we show that the formation energies, the equilibrium lattice parameters, and the total magnetic moments of the unit cell for different Fe-Al structures calculated with machine-learning potentials are in good correspondence with the ones obtained with DFT. We also demonstrate that the theoretical calculations conducted in this study qualitatively reproduce the experimentally-observed anomalous volume-composition dependence in the Fe-Al system.

A Perspective on Sustainable Computational Chemistry Software Development and Integration

The power of quantum chemistry to predict the ground and excited state properties of complex chemical systems has driven the development of computational quantum chemistry software, integrating advances in theory, applied mathematics, and computer science. The emergence of new computational paradigms associated with exascale technologies also poses significant challenges that require a flexible forward strategy to take full advantage of existing and forthcoming computational resources. In this context, the sustainability and interoperability of computational chemistry software development are among the most pressing issues. In this perspective, we discuss software infrastructure needs and investments with an eye to fully utilize exascale resources and provide unique computational tools for next-generation science problems and scientific discoveries.

Electronic Properties of Zn2V(1-x)NbxN3 Alloys to Model Novel Materials for Light-Emitting Diodes

We propose the Zn2V(1-x)NbxN3 alloy as a new promising material for optoelectronic applications, in particular for light-emitting diodes (LEDs). We perform accurate electronic-structure calculations of the alloy for several concentrations x using density-functional theory with meta-GGA exchange-correlation functional TB09. The band gap is found to vary between 2.2 and 2.9 eV with varying V/Nb concentration. This range is suitable for developing bright LEDs with tunable band gap as potential replacements for the more expensive Ga(1-x)In(x)N systems. Effects of configurational disorder are taken into account by explicitly considering all possible distributions of the metal ions within the metal sublattice for the chosen supercells. We have evaluated the band gap's nonlinear behavior (bowing) with variation of V/Nb concentration for two possible scenarios: (i) only the structure with the lowest total energy is present at each concentration and (ii) the structure with minimum band gap is present at each concentration, which corresponds to experimental conditions when also metastable structures are presents. We found that the bowing is about twice larger in the latter case. However, in both cases, the bowing parameter is found to be lower than 1 eV, which is about twice smaller than that in the widely used Ga(1-x)In(x)N alloy. Furthermore, we found that both crystal volume changes due to alloying and local effects (atomic relaxation and the V-N/Nb-N bonding difference) have important contributions to the band gap bowing in Zn2V(1-x)NbxN3.

Natural Optical Activity from Density-Functional Perturbation Theory

We present an accurate and computationally efficient first-principles methodology to calculate natural optical activity. Our approach is based on the long-wave density-functional perturbation theory and includes self-consistent field terms naturally in the formalism, which are found to be of crucial importance. The final result is expressed exclusively in terms of response functions to uniform field perturbations and avoids troublesome summations over empty states. Our strategy is validated by computing the natural optical activity tensor in representative chiral crystals (trigonal Se, α-HgS, and α-SiO_{2}) and molecules (C_{4}H_{4}O_{2}), finding excellent agreement with experiment and previous theoretical calculations.

Embedded Localized Molecular-Orbital Representations for Periodic Wave Functions

To more straightforwardly provide local chemical-bonding reasoning in crystalline matter, we introduce a new approach to generate a real-space analogue of periodic electronic structures using "exact" top-down frozen-density embedding calculations. Based on the obtained real-space electronic structure, we then construct localized molecular orbitals and evidence that our technique compares favorably against the commonly used Wannier method, both in terms of numerical efficiency and details of chemical bonding. The new method has been implemented into the LOBSTER software package and designed as a black-box approach, digesting any periodic electronic structure from the currently supported codes, i.e., VASP, Quantum ESPRESSO, and ABINIT.

Stochastic and mixed density functional theory within the projector augmented wave formalism for simulation of warm dense matter

Stochastic density functional theory (DFT) and mixed stochastic-deterministic DFT are burgeoning approaches for the calculation of the equation of state and transport properties in materials under extreme conditions. In the intermediate warm dense matter regime, a state between correlated condensed matter and kinetic plasma, electrons can range from being highly localized around nuclei to delocalized over the whole simulation cell. The plane-wave basis pseudopotential approach is thus the typical tool of choice for modeling such systems at the DFT level. Unfortunately, stochastic DFT methods scale as the square of the maximum plane-wave energy in this basis. To reduce the effect of this scaling and improve the overall description of the electrons within the pseudopotential approximation, we present stochastic and mixed DFT approaches developed and implemented within the projector augmented wave formalism. We compare results between the different DFT approaches for both single-point and molecular dynamics trajectories and present calculations of self-diffusion coefficients of solid density carbon from 1 to 50 eV.

Synthesis of Single Crystals of ε-Iron and Direct Measurements of Its Elastic Constants

Seismology finds that Earth's solid inner core behaves anisotropically. Interpretation of this requires a knowledge of crystalline elastic anisotropy of its constituents-the major phase being most likely ε-Fe, stable only under high pressure. Here, single crystals of this phase are synthesized, and its full elasticity tensor is measured between 15 and 33 GPa at 300 K. It is calculated under the same conditions, using the combination of density functional theory and dynamical mean field theory, which describes explicitly electronic correlation effects. The predictive power of this scheme is checked by comparison with measurements; it is then used to evaluate the crystalline anisotropy in ε-Fe under higher density. This anisotropy remains of the same amplitude up to densities typical of Earth's inner core.

Features of Helium-Vacancy Complex Formation at the Zr/Nb Interface

A first-principles study of the atomic structure and electron density distribution at the Zr/Nb interface under the influence of helium impurities and helium-vacancy complexes was performed using the optimised Vanderbilt pseudopotential method. For the determination of the preferred positions of the helium atom, the vacancy and the helium-vacancy complex at the interface, the formation energy of the Zr-Nb-He system has been calculated. The preferred positions of the helium atoms are in the first two atomic layers of Zr at the interface, where helium-vacancy complexes form. This leads to a noticeable increase in the size of the reduced electron density areas induced by vacancies in the first Zr layers at the interface. The formation of the helium-vacancy complex reduces the size of the reduced electron density areas in the third Zr and Nb layers as well as in the Zr and Nb bulk. Vacancies in the first niobium layer near the interface attract the nearest zirconium atoms and partially replenish the electron density. This may indicate a possible self-healing of this type of defect.

Investigating the Optoelectronic and Thermoelectric Properties of CdTe Systems in Different Phases: A First-Principles Study

CdTe is a potential material for making efficient and stable solar cells. The present study aimed to systematically investigate the electronic, optical, and thermoelectric properties of different structural phases of CdTe using density functional theory. The electronic properties were calculated using the modified Becke-Johnson potential with the local density approximation (LDA) correlation. The band structure profiles showed a direct band at the Γ-point for α-cubic, β-hexagonal, γ-orthorhombic, and an indirect band type for the δ-trigonal phase from the A-point at valence band maximum to the Γ-point at conduction band minimum. Hybridization between Te-p and Cd-s bands in the main valence region was observed in the partial density of states plots for all the studied phases. The real component static values of the dielectric function showed a slight decrease with increasing photonic energy after an initial small increase. The intensity of the imaginary component increased above the threshold energy for each phase, with the δ-phase showing a higher reflectivity spectrum than the other phases due to its intense peaks, making it ideal for protecting against high energy radiations. The results indicated that our computed band gaps and refractive index n(ω) were inversely related. The thermoelectric parameters calculated for these phases suggest that they have potential to be used in thermoelectric devices.

Chiral phonons: circularly polarized Raman spectroscopy and ab initio calculations in a chiral crystal tellurium

Recently, phonons with chirality (chiral phonons) have attracted significant attention. Chiral phonons exhibit angular and pseudoangular momenta. In circularly polarized Raman spectroscopy, the peak split of the Γ 3 $$ {\Gamma}_3 $$ mode is detectable along the principal axis of the chiral crystal in the backscattering configuration. In addition, peak splitting occurs when the pseudoangular momenta of the incident and scattered circularly polarized light are reversed. Until now, chiral phonons in binary crystals have been observed, whereas those in unary crystals have not been observed. Here, we observe chiral phonons in a chiral unary crystal Te. The pseudoangular momentum of the phonon is obtained in Te by an ab initio calculation. From this calculation, we verified the conservation law of pseudoangular momentum in Raman scattering. From this conservation law, we determined the handedness of the chiral crystals. We also evaluated the true chirality of the phonons using a measure with symmetry similar to that of an electric toroidal monopole.

Assessing the accuracy of hybrid exchange-correlation functionals for the density response of warm dense electrons

We assess the accuracy of common hybrid exchange-correlation (XC) functionals (PBE0, PBE0-1/3, HSE06, HSE03, and B3LYP) within the Kohn-Sham density functional theory for the harmonically perturbed electron gas at parameters relevant for the challenging conditions of the warm dense matter. Generated by laser-induced compression and heating in the laboratory, the warm dense matter is a state of matter that also occurs in white dwarfs and planetary interiors. We consider both weak and strong degrees of density inhomogeneity induced by the external field at various wavenumbers. We perform an error analysis by comparing with the exact quantum Monte Carlo results. In the case of a weak perturbation, we report the static linear density response function and the static XC kernel at a metallic density for both the degenerate ground-state limit and for partial degeneracy at the electronic Fermi temperature. Overall, we observe an improvement in the density response when the PBE0, PBE0-1/3, HSE06, and HSE03 functionals are used, compared with the previously reported results for the PBE, PBEsol, local-density approximation, and AM05 functionals; B3LYP, on the other hand, does not perform well for the considered system. Additionally, the PBE0, PBE0-1/3, HSE06, and HSE03 functionals are more accurate for the density response properties than SCAN in the regime of partial degeneracy.

Ab Initio Static Exchange-Correlation Kernel across Jacob's Ladder without Functional Derivatives

The electronic exchange─correlation (XC) kernel constitutes a fundamental input for the estimation of a gamut of properties such as the dielectric characteristics, the thermal and electrical conductivity, or the response to an external perturbation. In this work, we present a formally exact methodology for the computation of the system specific static XC kernel exclusively within the framework of density functional theory (DFT) and without employing functional derivatives─no external input apart from the usual XC-functional is required. We compare our new results with exact quantum Monte Carlo (QMC) data for the archetypical uniform electron gas model under both ambient and warm dense matter conditions. This gives us unprecedented insights into the performance of different XC functionals, and it has important implications for the development of new functionals that are designed for the application at extreme temperatures. In addition, we obtain new DFT results for the XC kernel of warm dense hydrogen as it occurs in fusion applications and astrophysical objects. The observed excellent agreement to the QMC reference data demonstrates that presented framework is capable to capture nontrivial effects such as XC-induced isotropy breaking in the density response of hydrogen at large wave numbers.

Non-empirical Mixing Coefficient for Hybrid XC Functionals from Analysis of the XC Kernel

We present an analysis of the static exchange-correlation (XC) kernel computed from hybrid functionals with a single mixing coefficient such as PBE0 and PBE0-1/3. We break down the hybrid XC kernels into the exchange and correlation parts using the Hartree-Fock functional, the exchange-only PBE, and the correlation-only PBE. This decomposition is combined with exact data for the static XC kernel of the uniform electron gas and an Airy gas model within a subsystem functional approach. This gives us a tool for the non-empirical choice of the mixing coefficient under ambient and extreme conditions. Our analysis provides physical insights into the effect of the variation of the mixing coefficient in hybrid functionals, which is of immense practical value. The presented approach is general and can be used for other types of functionals like screened hybrids.

Ab initio methods for the computation of physical properties and performance parameters of electrochemical energy storage devices

With the rapid development of electric vehicles and mobile technologies, there is a high demand for electrochemical energy storage devices and electrochemical energy conversion devices. Devices meeting these needs include metal-ion batteries (MIBs), supercapacitors (SCs), electrochromic devices (ECDs), and multifunctional devices such as electrochromic batteries and supercapatteries. Currently, the goal has been the enhancement of operational parameters and physical properties that results in a higher performance of these devices. In the case of batteries, SCs, and supercapatteries, scientists seek to improve the equilibrium voltage, energy density, power, capacitance, and charge rate. In the case of ECDs, the focus is on improvement of the optical modulation and coloration efficiency. However, synthesis and characterization of new materials, or of materials with optimized properties, is time consuming and highly expensive. Computational simulation of materials can expedite the experimental endeavor by modelling novel atomic structures and predicting device performance. This is possible using ab initio theories and applying physical principles that allow us to understand the underlying mechanisms governing the behavior of materials in these devices. Taking as a point of departure density functional theory (DFT), in this review, we discuss the first principles methods used for the computation of physical properties and performance parameters of electrochemical energy storage devices. A wide coverage of DFT is given, dealing with the strengths and weaknesses of the most popular functionals used in the field of electrochemical energy storage. With these tools, ab initio methods for the computation of basic properties such as effective mass, mobility, optical band gap, transmissivity, conductivity (ionic and electronic), and criteria for structure stability (cohesive energy, formation energy, adsorption energy, and phonon frequency) are addressed. We also highlight the first principles techniques for the calculation of performance parameters in MIBs, SCs, and ECDs.

Microstructure and Hydrogen Permeability of Nb-Ni-Ti-Zr-Co High Entropy Alloys

Hydrogen separation membranes are one of the most promising technologies for hydrogen purification. The development of high-entropy alloys (HEAs) for hydrogen separation membranes is driven by a "cocktail effect" of elements with different hydrogen affinities to prevent hydride formation and retain high permeability due to the single-phase BCC structure. In this paper, equimolar and non-equimolar Nb-Ni-Ti-Zr-Co high entropy alloys were fabricated by arc melting. The microstructure and phase composition of the alloys were analyzed by scanning electron microscopy and X-ray diffraction, respectively. The hydrogen permeation experiments were performed at 300-500 °C and a hydrogen pressure of 4 bar. In order to estimate the effect of composition and lattice structure on hydrogen location and diffusivity in Nb-Ni-Ti-Zr-Co alloy, ab initio calculations of hydrogen binding energy were performed using virtual crystal approximation. It was found that Nb-enriched and near equimolar BCC phases were formed in Nb₂₀Ni₂₀Ti₂₀Zr₂₀Co₂₀ HEA while Nb-enriched BCC and B2-Ni(Ti, Zr) were formed in Nb₄₀Ni₂₅Ti₁₈Zr₁₂Co₅ alloy. Hydrogen permeability tests showed that Nb₂₀Ni₂₀Ti₂₀Zr₂₀Co₂₀ HEA shows lower activation energy and higher permeability at lower temperatures as well as higher resistance to hydrogen embrittlement compared to Nb₄₀Ni₂₅Ti₁₈Zr₁₂Co₅ alloy. The effect of composition, microstructure and hydrogen binding energies on permeability of the fabricated alloys was discussed.

Theoretical and Experimental Studies of Al-Impurity Effect on the Hydrogenation Behavior of Mg

In this paper, we study the influence of hydrogen concentration on the binding energies in magnesium hydrides. The impact of aluminum atom addition on the hydrogenation behavior of magnesium was theoretically and experimentally defined. Doping Al into the Mg lattice allows the uniform hydrogen distribution in both the fcc and bcc Mg lattice at a low hydrogen concentration (H:Mg < 0.875) to be more energetically favorable. In addition, this leads to bcc Mg lattice formation with a uniform hydrogen distribution, which is more energetically favorable than the fcc Mg lattice when the atomic ratio H:Mg is near 0.875. In addition, compared with the pure Mg, in the Al-doped Mg, the phase transition from the hcp to the fcc structure with a uniform distribution of H atoms induces less elastic strain. Thus, the uniform hydrogen distribution is more favorable, leading to faster hydrogen absorption. Pure magnesium is characterized by cluster-like hydrogen distribution, which decreases the hydrogen diffusion rate. This leads to the accumulation of a higher hydrogen concentration in magnesium with aluminum compared with pure magnesium under the same hydrogenation regimes, which is confirmed experimentally.

Simulation of XRD, Raman and IR spectrum for phase identification in doped HfO2 and ZrO2

Fluorite-structured hafnium and zirconia require different, complementary characterization methods to identify the numerous metastable phases. This is because of the many possible positions of the oxygen ions, which are difficult to observe directly. Ab initio simulations are useful to probe the corresponding XRD, Raman, and infrared spectra for fingerprints. However, the predictive power of theoretical methods is limited both by model errors and by boundary conditions such as defects, stresses, and morphology that are difficult to detect. We first consider the calculation of Raman and infrared spectra of the most interesting undoped phases of HfO2 and ZrO2, compare the results with known results, and discuss the uncertainties. Next, we consider the possibilities of classifying the phases using X-ray diffraction. To this end, we introduce the effects of doping, which increases the uncertainty due to structural disorder. For illustration, we examine a large data set of doped structures obtained with ab initio calculations. To make an unbiased assignment of phases, we use machine learning methods with clusters. The limits of X-ray diffraction spectroscopy are reached when phase mixtures are present. Resolution of single-phase polycrystalline samples may only be possible here if these three characterization methods are used.

Sella, an Open-Source Automation-Friendly Molecular Saddle Point Optimizer

We present a new algorithm for the optimization of molecular structures to saddle points on the potential energy surface using a redundant internal coordinate system. This algorithm automates the procedure of defining the internal coordinate system, including the handling of linear bending angles, for example, through the addition of dummy atoms. Additionally, the algorithm supports constrained optimization using the null-space sequential quadratic programming formalism. Our algorithm determines the direction of the reaction coordinate through iterative diagonalization of the Hessian matrix and does not require evaluation of the full Hessian matrix. Geometry optimization steps are chosen using the restricted step partitioned rational function optimization method, and displacements are realized using a high-performance geodesic stepping algorithm. This results in a robust and efficient optimization algorithm suitable for use in automated frameworks. We have implemented our algorithm in Sella, an open-source software package designed to optimize atomic systems to saddle point structures. We also introduce a new benchmark test comprising 500 molecular structures that approximate saddle point geometries and show that our saddle point optimization algorithm outperforms the algorithms implemented in several leading electronic structure theory packages.

Constrained Density Functional Theory: A Potential-Based Self-Consistency Approach

Chemical reactions, charge transfer reactions, and magnetic materials are notoriously difficult to describe within Kohn-Sham density functional theory, which is strictly a ground-state technique. However, over the last few decades, an approximate method known as constrained density functional theory (cDFT) has been developed to model low-lying excitations linked to charge transfer or spin fluctuations. Nevertheless, despite becoming very popular due to its versatility, low computational cost, and availability in numerous software applications, none of the previous cDFT implementations is strictly similar to the corresponding ground-state self-consistent density functional theory: the target value of constraints (e.g., local magnetization) is not treated equivalently with atomic positions or lattice parameters. In the present work, by considering a potential-based formulation of the self-consistency problem, the cDFT is recast in the same framework as Kohn-Sham DFT: a new functional of the potential that includes the constraints is proposed, where the constraints, the atomic positions, or the lattice parameters are treated all alike, while all other ingredients of the usual potential-based DFT algorithms are unchanged, thanks to the formulation of the adequate residual. Tests of this approach for the case of spin constraints (collinear and noncollinear) and charge constraints are performed. Expressions for the derivatives with respect to constraints (e.g., the spin torque) for the atomic forces and the stress tensor in cDFT are provided. The latter allows one to study striction effects as a function of the angle between spins. We apply this formalism to body-centered cubic iron and first reproduce the well-known magnetization amplitude as a function of the angle between local magnetizations. We also study stress as a function of such an angle. Then, the local collinear magnetization and the local atomic charge are varied together. Since the atomic spin magnetizations, local atomic charges, atomic positions, and lattice parameters are treated on an equal footing, this formalism is an ideal starting point for the generation of model Hamiltonians and machine-learning potentials, computation of second or third derivatives of the energy as delivered from density-functional perturbation theory, or for second-principles approaches.

Thermal Transport Properties of Diamond Phonons by Electric Field

For the preparation of diamond heat sinks with ultra-high thermal conductivity by Chemical Vapor Deposition (CVD) technology, the influence of diamond growth direction and electric field on thermal conductivity is worth exploring. In this work, the phonon and thermal transport properties of diamond in three crystal orientation groups (<100>, <110>, and <111>) were investigated using first-principles calculations by electric field. The results show that the response of the diamond in the three-crystal orientation groups presented an obvious anisotropy under positive and negative electric fields. The electric field can break the symmetry of the diamond lattice, causing the electron density around the C atoms to be segregated with the direction of the electric field. Then the phonon spectrum and the thermodynamic properties of diamond were changed. At the same time, due to the coupling relationship between electrons and phonons, the electric field can affect the phonon group velocity, phonon mean free path, phonon−phonon interaction strength and phonon lifetime of the diamond. In the crystal orientation [111], when the electric field strength is ±0.004 a.u., the thermal conductivity is 2654 and 1283 W·m−1K−1, respectively. The main reason for the change in the thermal conductivity of the diamond lattice caused by the electric field is that the electric field has an acceleration effect on the extranuclear electrons of the C atoms in the diamond. Due to the coupling relationship between the electrons and the phonons, the thermodynamic and phonon properties of the diamond change.

Role of pressure on electronic, magnetic and structural properties at iron's Curie temperature: a DFT + DMFT study

We use the combination of density functional theory and dynamical mean-field theory to compute the Curie temperature of the iron body-centered cubicαphase and probe its pressure dependence. Our calculations reveal thatTCshows a decrease which is very weak over a domain of pressures that is much larger than the stability domain of theαphase. This is consistent with the experimental results. We highlight the importance of the Hund's couplingJnot only on the electronic and magnetic properties but also on the structural properties. Lastly, we analyze the electronic and magnetic properties under pressure and discuss the evolution of magnetic moments in both phases in relation to the change of Curie temperature.

DFT-1/2 and shell DFT-1/2 methods: electronic structure calculation for semiconductors at LDA complexity

It is known that the Kohn-Sham eigenvalues do not characterize experimental excitation energies directly, and the band gap of a semiconductor is typically underestimated by local density approximation (LDA) of density functional theory (DFT). An embarrassing situation is that one usually uses LDA+Ufor strongly correlated materials with rectified band gaps, but for non-strongly-correlated semiconductors one has to resort to expensive methods like hybrid functionals orGW. In spite of the state-of-the-art meta-generalized gradient approximation functionals like TB-mBJ and SCAN, methods with LDA-level complexity to rectify the semiconductor band gaps are in high demand. DFT-1/2 stands as a feasible approach and has been more widely used in recent years. In this work we give a detailed derivation of the Slater half occupation technique, and review the assumptions made by DFT-1/2 in semiconductor band structure calculations. In particular, the self-energy potential approach is verified through mathematical derivations. The aims, features and principles of shell DFT-1/2 for covalent semiconductors are also accounted for in great detail. Other developments of DFT-1/2 including conduction band correction, DFT+A-1/2, empirical formula for the self-energy potential cutoff radius, etc, are further reviewed. The relations of DFT-1/2 to hybrid functional, sX-LDA,GW, self-interaction correction, scissor's operator as well as DFT+Uare explained. Applications, issues and limitations of DFT-1/2 are comprehensively included in this review.

Sound velocities and thermodynamical properties of hcp iron at high pressure and temperature

Sound velocities and thermodynamical properties of hcp iron have been computed usingab initiocalculations over an extended density and temperature range, encompassing the conditions directly relevant for the Earth's inner core. At room temperature, and up to 350 GPa, an excellent agreement is obtained between present results and experimental data for many thermodynamical quantities: phonon density of states, vibrational entropy, heat capacity, Grüneisen parameter and thermal expansion. With increasing temperature, along an isochore, we observe a strong decrease of the phonon frequencies, demonstrating that intrinsic anharmonic effects cannot be neglected. We also carefully compare previous theoretical data for the sound velocities and try to explain the discrepancies observed with experiments. Finally, we propose a temperature dependant Birch's law that we compare with previous experimental work.

Finding critical points and reconstruction of electron densities on grids

The quantum theory of atoms in molecules (QTAIM), developed by Bader and co-workers, is one of the most popular ways of extracting chemical insight from the results of quantum mechanical calculations. One of the basic tasks in QTAIM is to locate the critical points of the electron density and calculate various quantities (density, Laplacian, etc.) on them since these have been found to correlate with molecular properties of interest. If the electron density is given analytically, this process is relatively straightforward. However, locating the critical points is more challenging if the density is known only on a three-dimensional uniform grid. A density grid is common in periodic solids because it is the natural expression for the electron density in plane-wave calculations. In this article, we explore the reconstruction of the electron density from a grid and its use in critical point localization. The proposed reconstruction method employs polyharmonic spline interpolation combined with a smoothing function based on the promolecular density. The critical point search based on this reconstruction is accurate, trivially parallelizable, works for periodic and non-periodic systems, does not present directional lattice bias when the grid is non-orthogonal, and locates all critical points of the underlying electron density in all tests studied. The proposed method also provides an accurate reconstruction of the electron density over the space spanned by the grid, which may be useful in other contexts besides critical point localization.

Spectroscopic signatures of nonpolarons: the case of diamond

Polarons are quasi-particles made from electrons interacting with vibrations in crystal lattices. They derive their name from the strong electron-vibration polar interactions in ionic systems, that induce spectroscopic and optical signatures of such quasi-particles. In this paper, we focus on diamond, a non-polar crystal with inversion symmetry which nevertheless shows interesting signatures stemming from electron-vibration interactions, better denoted "nonpolaron" signatures in this case. The (non)polaronic effects are produced by short-range crystal fields, while long-range quadrupoles only have a small influence. The corresponding many-body spectral function has a characteristic energy dependence, showing a plateau structure that is similar to but distinct from the satellites observed in the polar Fröhlich case. We determine the temperature-dependent spectral function of diamond by two methods: the standard Dyson-Migdal approach, which calculates electron-phonon interactions within the lowest-order expansion of the self-energy, and the cumulant expansion, which includes higher orders of electron-phonon interactions. The latter corrects the nonpolaron energies and broadening, providing a more realistic spectral function, which we examine in detail for both conduction and valence band edges.

Pseudodiagonalization Method for Accelerating Nonlinear Subspace Diagonalization in Density Functional Theory

In density functional theory, each self-consistent field (SCF) nonlinear step updates the discretized Kohn-Sham orbitals by solving a linear eigenvalue problem. The concept of pseudodiagonalization is to solve this linear eigenvalue problem approximately and specifically utilizing a method involving a small number of Jacobi rotations that takes advantage of the good initial guess to the solution given by the approximation to the orbitals from the previous SCF iteration. The approximate solution to the linear eigenvalue problem can be very rapid, particularly for those steps near SCF convergence. We adapt pseudodiagonalization to finite-temperature and metallic systems, where partially occupied orbitals must be individually resolved with some accuracy. We apply pseudodiagonalization to the subspace eigenvalue problem that arises in Chebyshev-filtered subspace iteration. In tests on metallic and other systems for a range of temperatures, we show that pseudodiagonalization achieves similar rates of SCF convergence to exact diagonalization.

Distribution of Hydrogen and Defects in the Zr/Nb Nanoscale Multilayer Coatings after Proton Irradiation

Radiation damage is one of the significant factors limiting the operating time of many structural materials working under extreme conditions. One of the promising directions in the development of materials that are resistant to radiation damage and have improved physical and mechanical properties is the creation of nanoscale multilayer coatings (NMCs). The paper is devoted to the experimental comprehension of changes in the defect structure and mechanical properties of nanoscale multilayer coatings (NMCs) with alternating layers of Zr and Nb under irradiation. Series of Zr/Nb NMCs with different thicknesses of individual layers were fabricated by magnetron sputtering and subjected to H+ irradiation. The evolution of structure and phase states, as well as the defect state under proton irradiation, was studied using the methods of high-resolution transmission electron microscopy (HRTEM), X-ray diffraction analysis (XRD), glow discharge optical emission spectroscopy (GDOES), and positron annihilation spectroscopy (PAS). The layer-by-layer analysis of structural defects was carried out by Doppler broadening spectroscopy (DBS) using a variable-energy positron beam. To estimate the binding energy and the energy paths for the hydrogen diffusion in Zr/Nb NMCs, calculations from the first principles were used. When the thickness of individual layers is less than 25 nm, irradiation causes destruction of the interfaces, but there is no significant increase in the defect level, the S parameter (open volume defects amount) before and after irradiation is practically unchanged. After irradiation of NMC Zr/Nb with a thickness of layers 50 and 100 nm, the initial microstructure is retained, and the S parameter is significantly reduced. The GDOES data reveal the irregular H accumulation at the interface caused by significant differences in H diffusion barriers in the bulk of Zr and Nb multilayers as well as near the interface’s region.

A DFT study of defects in paramagnetic Cr2O3

Cr₂O₃ is not only a promising functional material, but also an essential barrier to protect chromia-forming alloys against high temperature corrosion. The Cr₂O₃ protecting layer grows slowly via defect-mediated diffusion. Several types of point defects could be responsible for the diffusion process depending on the oxidation environment, resulting in different semiconductor characters of chromia. According to the literature, the defect chemistry of Cr₂O₃ in the antiferromagnetic (AFM) state has been well studied using density functional theory (DFT) calculations but not in the paramagnetic (PM) state, which is the fundamental state of Cr₂O₃ above 318 K. PM Cr₂O₃ is simulated in this study using special quasi-random structures (SQS). The formation energies of intrinsic point defects in AFM and PM Cr₂O₃ are calculated to study the defect chemistry and the semiconductor properties in different oxidation environments (temperature and oxygen partial pressure P_O2) using a thermodynamic model. It is found that O vacancies and insulating-type Cr₂O₃, in which commensurate electrons and holes are dominant before atomic defects are more favorable at high temperatures and at low P_O2, while Cr vacancies and p-type Cr₂O₃ are more favorable at low temperatures and at high P_O2, according to the calculations both in AFM and PM Cr₂O₃. However, the limits of dominant zones for defects and for semiconductor characters shift to higher temperatures or lower P_O2 in PM state calculations.