On the use of symmetry in configurational analysis for the simulation of disordered solids

t 2 g d orbital ordering patterns in KBF 3 (B = Sc, Ti, Fe, Co) perovskites

The orbital ordering (OO) resulting from the partial occupancy of thet 2 g d subshell of the transition metals in KBF3 (B = Sc, Ti, Ffe, Co) perovskites, and the many possible patterns arising from the coupling between the B sites, have been investigated at the quantum mechanical level ( all electron Gaussian type basis set, B3LYP hybrid functional) in a 40 atoms supercell. The numerous patterns are distributed into 162 classes of equivalent configurations. For each fluoroperovskite, one representative per class has been calculated. The four compounds behave similarly: an identical dependence of the energy and volume (or cell parameters) on the OO pattern; the spanned energy interval is small (1 to 2 mEh per formula unit), suggesting that most of the configurations are occupied at room and even at low temperature. A linear model, taking into account the relative orbital order in contiguous sites, reproduces the energy order in the full set for each compound, suggesting that it could be used for studying OO in larger supercells.

Unraveling the Role of Li and Mg Substitution in Layered Sodium Oxide Cathodes for Sodium-Ion Batteries

Substituting electrochemically active elements such as Li and Mg in P2-type layered sodium oxide is an effective strategy for developing competitive cathode materials for sodium-ion batteries. However, the lack of atomic-level understanding regarding the distribution of substitution positions complicates the comprehension of the roles of substituting atoms and the mechanism of sodium-ion intercalation. In this study, we identified the stable configurations of Na in Na0.75Ni0.3Mn0.7O2 and Na0.75Li0.15Mg0.05Ni0.1Mn0.7O2 materials using the site exclusion method. Through simulating the complete charging process for both materials, the structure evolution of the cathodes during the cycling and the impact of the partial substitution of Ni elements by Li and Mg atoms were comprehensively elucidated. Our findings revealed that Mg atoms effectively regulate the distribution of forces within the materials, essentially serving as supportive pillars within the cathode. Meanwhile, Li atoms efficiently mitigated electron localization, consequently diminishing volume fluctuations during the charging process. More importantly, the substitution with Li and Mg atoms could synergistically reduce the interaction between transition metals and sodium ions, thereby reducing the diffusion energy barrier of Na ions. This study not only enhances the comprehension of substituted metal atoms in P2 layered oxides but also offers new insights for the development of sodium-ion cathode materials.

Disorder induced band gap lowering in kesterite type Cu2ZnSnSe4and Ag2ZnSnSe4: a first-principles and special quasirandom structures investigation

Quaternary chalcogenides, i.e. Cu2ZnSnS4, crystallising in the kesterite crystal structure have already been demonstrated as potential building blocks of thin film solar cells, containing only abundant elements and exhibiting power conversion efficiencies of about 14.9% so far. However, due to the potential presence of several structurally similar polymorphs, the unequivocal identification of their ground state crystal structures required the application of more elaborate neutron diffraction experiments. One particular complication arose from the later identified Cu-Zn disorder, present in virtually all thin film samples. Subsequently, it has been shown experimentally that this unavoidable Cu-Zn disorder leads to a band gap lowering in the respective samples. Additional theoretical investigations, mostly based on Monte-Carlo methods, tried to understand the atomistic origin of this disorder induced band gap lowering. Here, we present theoretical results from first-principles calculations based on density functional theory for the disorder induced band gap lowering in kesterite Cu2ZnSnSe4and Ag2ZnSnSe4, where the Cu-Zn and Ag-Zn disorder is modelled via a supercell approach and special quasirandom structures. Results of subsequent analyses of structural, electronic, and optical properties are discussed with respect to available experimental results, and will provide additional insight and knowledge towards the atomistic origin of the observed disorder induced band gap lowering in kesterite type materials.

Quantification of Crystalline Phases in Hf0.5Zr0.5O2 Thin Films through Complementary Infrared Spectroscopy and Ab Initio Supercell Simulations

In the quest for thinner and more efficient ferroelectric devices, Hf0.5Zr0.5O2 (HZO) has emerged as a potential ultrathin and lead-free ferroelectric material. Indeed, when deposited on a TiN electrode, 1-25 nm thick HZO exhibits excellent ferroelectricity capability, allowing the prospective miniaturization of capacitors and transistor devices. To investigate the origin of ferroelectricity in HZO thin films, we conducted a far-infrared (FIR) spectroscopic study on 5 HZO films with thicknesses ranging from 10 to 52 nm, both within and out of the ferroelectric thickness range where ferroelectric properties are observed. Based on X-ray diffraction, these HZO films are estimated to contain various proportions of monoclinic (m-), tetragonal (t-), and polar orthorhombic (polar o-) phases, while only the 11, 17, and 21 nm thick are expected to include a higher amount of polar o-phase. We coupled the HZO infrared measurements with DFT simulations for these m-, t-, and polar o-crystallographic structures. The approach used was based on the supercell method, which combines all possible Hf/Zr mixed atomic sites in the solid solution. The excellent agreement between measured and simulated spectra allows assigning most bands and provides infrared signatures for the various HZO structures, including the polar orthorhombic form. Beyond pure assignment of bands, the DFT IR spectra averaging using a mix of different compositions (e.g., 70% polar o-phase +30% m-phase) of HZO DFT crystal phases allows quantification of the percentage of different structures inside the different HZO film thicknesses. Regarding the experimental data analysis, we used the spectroscopic data to perform a Kramers-Kronig constrained variational fit to extract the optical functions of the films using a Drude-Lorentz-based model. We found that the ferroelectric films could be described using a set of about 7 oscillators, which results in static dielectric constants in good agreement with theoretical values and previously reported ones for HfO2-doped ferroelectric films.

The d Orbital Multi Pattern Occupancy in a Partially Filled d Shell: The KFeF3 Perovskite as a Test Case

The occupancy of the d shell in KFeF3 is t2g4eg2, with five α and one β electrons. The Jahn-Teller lift of degeneracy in the t2g sub-shell produces a tetragonal relaxation of the unit cell (4.09 vs. 4.22 Å, B3LYP result) not observed experimentally. In order to understand the origin of this apparent contradiction, we explored, with a 2 × 2 × 2 supercell (40 atoms per cell), all possible local structures in which contiguous Fe atoms have a different occupancy of the t2g orbitals with the minority spin electron. A total of 6561 configurations (with occupancies from (8,0,0) to (3,2,2) of the 3 t2g orbitals of the 8 Fe atoms) have been explored, with energies in many cases lower (by up to 1550 μEh per 2 Fe atoms) than the one of the fully ordered case, both for the ferromagnetic and the anti-ferromagnetic solutions. The results confirm that the orientation of the β d electron of Fe influences the electrostatics (more efficient relative orientation of the Fe quadrupoles of the d shell) of the system, but not the magnetic interactions. Three hybrid functionals, B3LYP, PBE0, and HSE06, provide very similar results.

Shry: Application of Canonical Augmentation to the Atomic Substitution Problem

A common approach for studying a solid solution or disordered system within a periodic ab initio framework is to create a supercell in which certain amounts of target elements are substituted with other elements. The key to generating supercells is determining how to eliminate symmetry-equivalent structures from many substitution patterns. Although the total number of substitutions is on the order of trillions, only symmetry-inequivalent atomic substitution patterns need to be identified, and their number is far smaller than the total. Our developed Python software package, which is called Shry (Suite for High-throughput generation of models with atomic substitutions implemented by Python), allows the selection of only symmetry-inequivalent structures from the vast number of candidates based on the canonical augmentation algorithm. Shry is implemented in Python 3 and uses the CIF format as the standard for both reading and writing the reference and generated sets of substituted structures. Shry can be integrated into another Python program as a module or can be used as a stand-alone program. The implementation was verified through a comparison with other codes with the same functionality, based on the total numbers of symmetry-inequivalent structures, and also on the equivalencies of the output structures themselves. The provided crystal structure data used for the verification are expected to be useful for benchmarking other codes and also developing new algorithms in the future.

Disordered Rock-Salt Type Li2TiS3 as Novel Cathode for LIBs: A Computational Point of View

The development of high-energy cathode materials for lithium-ion batteries with low content of critical raw materials, such as cobalt and nickel, plays a key role in the progress of lithium-ion batteries technology. In recent works, a novel and promising family of lithium-rich sulfides has received attention. Among the possible structures and arrangement, cubic disordered Li2TiS3 has shown interesting properties, also for the formulation of new cell for all-solid-state batteries. In this work, a computational approach based on DFT hybrid Hamiltonian, localized basis functions and the use of the periodic CRYSTAL code, has been set up. The main goal of the present study is to determine accurate structural, electronic, and spectroscopic properties for this class of materials. Li2TiS3 precursors as Li2S, TiS2, and TiS3 alongside other formulations and structures such as LiTiS2 and monoclinic Li2TiS3 have been selected as benchmark systems and used to build up a consistent and robust predictive scheme. Raman spectra, XRD patterns, electronic band structures, and density of states have been simulated and compared to available literature data. Disordered rock-salt type Li2TiS3 structures have been derived via a solid solution method as implemented into the CRYSTAL code. Representative structures were extensively characterized through the calculations of their electronic and vibrational properties. Furthermore, the correlation between structure and Raman fingerprint was established.

Revisiting the Cu-Zn Disorder in Kesterite Type Cu2ZnSnSe4 Employing a Novel Approach to Hybrid Functional Calculations

In recent years, the search for more efficient and environmentally friendly materials to be employed in the next generation of thin film solar cell devices has seen a shift towards hybrid halide perovskites and chalcogenide materials crystallising in the kesterite crystal structure. Prime examples for the latter are Cu2ZnSnS4, Cu2ZnSnSe4, and their solid solution Cu2ZnSn(SxSe1−x)4, where actual devices already demonstrated power conversion efficiencies of about 13 %. However, in their naturally occurring kesterite crystal structure, the so-called Cu-Zn disorder plays an important role and impacts the structural, electronic, and optical properties. To understand the influence of Cu-Zn disorder, we perform first-principles calculations based on density functional theory combined with special quasirandom structures to accurately model the cation disorder. Since the electronic band gaps and derived optical properties are severely underestimated by (semi)local exchange and correlation functionals, supplementary hybrid functional calculations have been performed. Concerning the latter, we additionally employ a recently devised technique to speed up structural relaxations for hybrid functional calculations. Our calculations show that the Cu-Zn disorder leads to a slight increase in the unit cell volume compared to the conventional kesterite structure showing full cation order, and that the band gap gets reduced by about 0.2 eV, which is in very good agreement with earlier experimental and theoretical findings. Our detailed results on structural, electronic, and optical properties will be discussed with respect to available experimental data, and will provide further insights into the atomistic origin of the disorder-induced band gap lowering in these promising kesterite type materials.

The Effect of Interface Diffusion on Raman Spectra of Wurtzite Short-Period GaN/AlN Superlattices

We present an extensive theoretical and experimental study to identify the effect on the Raman spectrum due to interface interdiffusion between GaN and AlN layers in short-period GaN/AlN superlattices (SLs). The Raman spectra for SLs with sharp interfaces and with different degree of interface diffusion are simulated by ab initio calculations and within the framework of the random-element isodisplacement model. The comparison of the results of theoretical calculations and experimental data obtained on PA MBE and MOVPE grown SLs, showed that the bands related to A₁(LO) confined phonons are very sensitive to the degree of interface diffusion. As a result, a correlation between the Raman spectra in the range of A₁(LO) confined phonons and the interface quality in SLs is obtained. This opens up new possibilities for the analysis of the structural characteristics of short-period GaN/AlN SLs using Raman spectroscopy.

Enumeration of nonequivalent substitutional structures using advanced data structure of binary decision diagram

A derivative structure is a nonequivalent substitutional atomic configuration derived from a given primitive cell. The enumeration of derivative structures plays an essential role in searching for the ground states in multicomponent systems. However, it is computationally difficult to enumerate derivative structures if the number of derivative structures of a target system becomes huge. In this study, we introduce a novel compact data structure of the zero-suppressed binary decision diagram (ZDD) for enumerating derivative structures much more efficiently. We show its simple applications to the enumeration of structures derived from the face-centered cubic and hexagonal close-packed lattices in binary, ternary, and quaternary systems. The present ZDD-based procedure should contribute to computational approaches based on derivative structures in physics and materials science.

The CRYSTAL code, 1976-2020 and beyond, a long story

CRYSTAL is a periodic ab initio code that uses a Gaussian-type basis set to express crystalline orbitals (i.e., Bloch functions). The use of atom-centered basis functions allows treating 3D (crystals), 2D (slabs), 1D (polymers), and 0D (molecules) systems on the same grounds. In turn, all-electron calculations are inherently permitted along with pseudopotential strategies. A variety of density functionals are implemented, including global and range-separated hybrids of various natures and, as an extreme case, Hartree-Fock (HF). The cost for HF or hybrids is only about 3-5 times higher than when using the local density approximation or the generalized gradient approximation. Symmetry is fully exploited at all steps of the calculation. Many tools are available to modify the structure as given in input and simplify the construction of complicated objects, such as slabs, nanotubes, molecules, and clusters. Many tensorial properties can be evaluated by using a single input keyword: elastic, piezoelectric, photoelastic, dielectric, first and second hyperpolarizabilities, etc. The calculation of infrared and Raman spectra is available, and the intensities are computed analytically. Automated tools are available for the generation of the relevant configurations of solid solutions and/or disordered systems. Three versions of the code exist: serial, parallel, and massive-parallel. In the second one, the most relevant matrices are duplicated on each core, whereas in the third one, the Fock matrix is distributed for diagonalization. All the relevant vectors are dynamically allocated and deallocated after use, making the code very agile. CRYSTAL can be used efficiently on high performance computing machines up to thousands of cores.

Terahertz Fingerprints of Short-Range Correlations of Disordered Atoms in Diflunisal

This work proposes a terahertz (THz) spectroscopy approach to the investigation of one of the outstanding problems in crystallography-the structure analysis of a crystal with disorder. Form I of diflunisal, in which the two ortho sites on one phenyl ring of diflunisal show occupational disorder, was used for an illustration. THz radiation interacts with the collective vibrations of correlated disorder, thus providing a promising tool to examine the symmetry of short-range correlations of disordered atoms. Through a thorough examination of the selection rule of THz vibrations in which the disordered atoms are involved to different extents, we deduced that only four short-range correlation possibilities of disorder exist and all of them display unambiguous fingerprint peaks in the 50-170 cm^-1 frequency region. We finally proposed an alternating packing model in which the correlation lengths of disorder are on the nanometer scale.

Modeling short-range substitution order and disorder in crystals: Application to the Ga/Si distribution in a natrolite zeolite

Atomic substitutions are a central feature of the physicochemical properties of an increasing number of solid-state materials. The complexity that this chemical disorder locally generates in otherwise crystalline solids poses a major challenge to the understanding of the relationships between the structure and properties of materials at the atomic and molecular level. Strategies designed to efficiently explore the ensemble of local chemical environments present in disordered crystals and predict their signatures in local spectroscopies such as solid-state nuclear magnetic resonance (NMR) are therefore essential. Focusing on the Ga/Si disorder in the framework of rubidium-exchanged gallosilicate natrolite zeolite (Rb-PST-1) with a high Ga content (SiGa=1.28), we show how the structure-generation approach implemented in the new program supercell (Okhotnikov et al. [26]) provides an excellent basis for the understanding of complex experimental spectroscopic data. Furthermore, we describe how exhaustive explorations of atomic configurations can be performed to seek local structural ordering and/or disordering factors. In the case of Rb-PST-1, we more specifically explore the possibility to form and to detect the presence of thermodynamically unfavorable Ga-O-Ga connectivities. While particularly adapted to the description of dense materials, we demonstrate that this approach may successfully be used to reproduce and interpret the distributions of local structural distortions (i.e., the geometrical disorder) resulting from the chemical disorder in systems as complex as microporous zeolites.

Supercell program: a combinatorial structure-generation approach for the local-level modeling of atomic substitutions and partial occupancies in crystals

Background

Disordered compounds are crucially important for fundamental science and industrial applications. Yet most available methods to explore solid-state material properties require ideal periodicity, which, strictly speaking, does not exist in this type of materials. The supercell approximation is a way to imply periodicity to disordered systems while preserving “disordered” properties at the local level. Although this approach is very common, most of the reported research still uses supercells that are constructed “by hand” and ad-hoc.

Results

This paper describes a software named supercell, which has been designed to facilitate the construction of structural models for the description of vacancy or substitution defects in otherwise periodically-ordered (crystalline) materials. The presented software allows to apply the supercell approximation systematically with an all-in-one implementation of algorithms for structure manipulation, supercell generation, permutations of atoms and vacancies, charge balancing, detecting symmetry-equivalent structures, Coulomb energy calculations and sampling output configurations. The mathematical and physical backgrounds of the program are presented, along with an explanation of the main algorithms and relevant technical details of their implementation. Practical applications of the program to different types of solid-state materials are given to illustrate some of its potential fields of application. Comparisons of the various algorithms implemented within supercell with similar solutions are presented where possible.

Conclusions

The all-in-one approach to process point disordered structures, powerful command line interface, excellent performance, flexibility and GNU GPL license make the supercell program a versatile set of tools for disordered structures manipulations.

Electronic supplementary material

The online version of this article (doi:10.1186/s13321-016-0129-3) contains supplementary material, which is available to authorized users.

Piezoelectric, elastic, structural and dielectric properties of the Si(1-x)Ge(x)O(2) solid solution: a theoretical study

We apply first principles quantum mechanical techniques to the study of the solid solution Si1-xGexO2 of α-quartz where silicon atoms are progressively substituted with germanium atoms, to different extents, as a function of the substitutional fraction x. For the first time, the whole range of the substitution (x = 0.0, 0.1[Formula: see text], 0.[Formula: see text], 0.5, 0.[Formula: see text], 0.8[Formula: see text], 1.0), including pure end-members α-SiO2 and α-GeO2, is explored. An elongated supercell (doubled along the c crystallographic axis) is built with respect to the unit cell of pure α-quartz and a set of 13 symmetry-independent configurations is considered. Their structural, energetic, dielectric, elastic and piezoelectric properties are computed and analyzed. All the calculations are performed using the CRYSTAL14 program with a Gaussian-type function basis set with pseudopotentials, and the hybrid functional PBE0; all geometries are fully optimized at this level of theory. In particular, for each configuration, fourth-rank elastic and compliance tensors and third-rank direct and converse piezoelectric tensors are computed. It has already been shown that the structural distortion of the solid solution increases, almost linearly, as the substitutional fraction x increases. The piezoelectric properties of the Si1-xGexO2 solid solution are found to increase with x, with a similar quasi-linear behavior. The electromechanical coupling coefficients are enhanced as well and the linear trend recently predicted by Ranieri et al (2011 Inorg. Chem. 50 4632) can be confirmed from first principles calculations. These doped crystals do represent good candidates for technological applications requiring high piezoelectric coupling and high thermal stability.

Symmetry and random sampling of symmetry independent configurations for the simulation of disordered solids

A symmetry-adapted algorithm producing uniformly at random the set of symmetry independent configurations (SICs) in disordered crystalline systems or solid solutions is presented here. Starting from Pólya's formula, the role of the conjugacy classes of the symmetry group in uniform random sampling is shown. SICs can be obtained for all the possible compositions or for a chosen one, and symmetry constraints can be applied. The approach yields the multiplicity of the SICs and allows us to operate configurational statistics in the reduced space of the SICs. The present low-memory demanding implementation is briefly sketched. The probability of finding a given SIC or a subset of SICs is discussed as a function of the number of draws and their precise estimate is given. The method is illustrated by application to a binary series of carbonates and to the binary spinel solid solution Mg(Al,Fe)2O4.