Luis Elcoro

Topological quantum chemistry

Since the discovery of topological insulators and semimetals, there has been much research into predicting and experimentally discovering distinct classes of these materials, in which the topology of electronic states leads to robust surface states and electromagnetic responses. This apparent success, however, masks a fundamental shortcoming: topological insulators represent only a few hundred of the 200,000 stoichiometric compounds in material databases. However, it is unclear whether this low number is indicative of the esoteric nature of topological insulators or of a fundamental problem with the current approaches to finding them. Here we propose a complete electronic band theory, which builds on the conventional band theory of electrons, highlighting the link between the topology and local chemical bonding. This theory of topological quantum chemistry provides a description of the universal (across materials), global properties of all possible band structures and (weakly correlated) materials, consisting of a graph-theoretic description of momentum (reciprocal) space and a complementary group-theoretic description in real space. For all 230 crystal symmetry groups, we classify the possible band structures that arise from local atomic orbitals, and show which are topologically non-trivial. Our electronic band theory sheds new light on known topological insulators, and can be used to predict many more.

Intergrowth polytypoids as modulated structures : the example of the cation deficient oxides LaTi1-xO3

A new, essentially composition-independent structural model is proposed for the perovskite-related compound series LanTin-δO3n, with δ ≤ n/4, based on superspace formalism. In the standard 3-dimensional crystallographic approach, one must specify space group and cell parameters separately for each composition. In the superspace approach, the superspace group is invariant while the cell parameters and primary modulation wave-vector vary slowly and continuously with composition. To a first approximation, the structure at any particular composition can be described in terms of mixed cubic and hexagonal stacking of close-packed LaO3 (and Ti) layers along the z-axis. The actual layer stacking sequence varies abruptly with composition, and the titanium vacancies occur as entire layers between neighbouring hexagonal close-packed LaO3 layers. The continuously variable, composition-dependent character of reciprocal space indicates that the various layer stacking sequences must be intimately correlated and shows that all the compounds can be interpreted in terms of a structural modulation over a common average structure. It is shown that a very simple model in superspace is sufficient to describe the layer stacking sequences for any particular composition. The atomic surfaces representing the atoms in superspace are shown to be well described by means of crenel functions. The cation displacive modulations are approximately described by sawtooth functions. The only variable parameters are the composition-dependent (but smoothly varying) primary modulation wave-vector and associated Ti occupation domain width. Conventional space groups can be easily determined from the superspace group and the composition. From a practical point of view, this means that we have a very good single starting point to refine the actual atomic positions for any composition, including incommensurate or long period structures. Even more importantly, the superspace approach has the flexibility to represent in a simple manner the symmetry-allowed deviations from the ideal model. A comparison with the scarce number of refined structures in this system and in the homologous (Ba,La)nTin-1O3n system, demonstrates the suitability of the formalism for the analysis of these compound series and for the analysis of compositionally flexible systems in general.

Intergrowth polytypoids as modulated structures: a superspace description of the Srn(Nb,Ti)nO3n + 2 compound series

A new, unified superspace approach to the structural characterization of the perovskite-related Sr(n)(Nb,Ti)(n)O(3n + 2) compound series, strontium niobium/titanium oxide, is presented. To a first approximation, the structure of any member of this compound series can be described in terms of the stacking of (110)-bounded perovskite slabs, the number of atomic layers in a single perovskite slab varying systematically with composition. The various composition-dependent layer-stacking sequences can be interpreted in terms of the structural modulation of a common underlying average structure. The average interlayer separation distance is directly related to the average structure periodicity along the layer stacking direction, while an inherent modulation thereof is produced by the presence of different types of layers (particularly vacant layers) along this stacking direction. The fundamental atomic modulation is therefore occupational and can be described by means of crenel (step-like) functions which define occupational atomic domains in the superspace, similarly to what occurs for quasicrystals. While in a standard crystallographic approach, one must describe each structure (in particular the space group and cell parameters) separately for each composition, the proposed superspace model is essentially common to the whole compound series. The superspace symmetry group is unique, while the primary modulation wavevector and the width of some occupation domains vary linearly with composition. For each rational composition, the corresponding conventional three-dimensional space group can be derived from the common superspace group. The resultant possible three-dimensional space groups are in agreement with all the symmetries reported for members of the series. The symmetry-breaking phase transitions with temperature observed in many compounds can be explained in terms of a change in superspace group, again in common for the whole compound series. Inclusion of the incommensurate phases, present in many compounds of the series, lifts the analysis into a five-dimensional superspace. The various four-dimensional superspace groups reported for this incommensurate phase at different compositions are shown to be predictable from a proposed five-dimensional superspace group apparently common to the whole compound series. A comparison with the scarce number of refined structures in this system and the homologous (Nb,Ca)(6)Ti(6)O(20) compound demonstrates the suitability of the proposed formalism.

Double crystallographic groups and their representations on the Bilbao Crystallographic Server

A new section of databases and programs devoted to double crystallographic groups (point and space groups) has been implemented in the Bilbao Crystallographic Server (this http URL). The double crystallographic groups are required in the study of physical systems whose Hamiltonian includes spin-dependent terms. In the symmetry analysis of such systems, instead of the irreducible representations of the space groups, it is necessary to consider the single- and double-valued irreducible representations of the double space groups. The new section includes databases of symmetry operations (DGENPOS) and of irreducible representations of the double (point and space) groups (REPRESENTATIONS DPG and REPRESENTATIONS DSG). The tool DCOMPATIBILITY RELATIONS provides compatibility relations between the irreducible representations of double space groups at different k-vectors of the Brillouin zone when there is a group-subgroup relation between the corresponding little groups. The program DSITESYM implements the so-called site-symmetry approach, which establishes symmetry relations between localized and extended crystal states, using representations of the double groups. As an application of this approach, the program BANDREP calculates the band representations and the elementary band representations induced from any Wyckoff position of any of the 230 double space groups, giving information about the properties of these bands. Recently, the results of BANDREP have been extensively applied in the description and the search of topological insulators.

Revision of pyrrhotite structures within a common superspace model

The structure of pyrrhotite (Fe(1 - x)S with 0.05 < or = x < or = 0.125) has been reinvestigated in the framework of the superspace formalism. A common model with a centrosymmetric superspace group is proposed for the whole family. The atomic domains in the internal space representing the Fe atoms are parametrized as crenel functions that fulfil the closeness condition. The proposed model explains the x-dependent space groups observed and the basic features of the structures reported up to now. Our model yields for any x value a well defined ordered distribution of Fe vacancies in contrast to some of the structural models proposed in the literature. A new (3 + 1)-dimensional refinement of Fe(0.91)S using the deposited dataset [Yamamoto & Nakazawa (1982). Acta Cryst. A38, 79-86] has been performed as a benchmark of the model. The consistency of the proposed superspace symmetry and its validity for other compositions has been further checked by means of ab initio calculations of both atomic forces and equilibrium atomic positions in non-relaxed and relaxed structures, respectively.

Modular crystals as modulated structures: the case of the lillianite homologous series.

The use of the superspace formalism is extended to the description and refinement of the homologous series of modular structures with two symmetry-related modules with different orientations. The lillianite homologous series has been taken as a study case. Starting from a commensurate modulated composite description with two basic subsystems corresponding to the two different modules, it is shown how a more efficient description can be achieved using so-called zigzag modulation functions. These linear zigzag modulations, newly implemented in the program JANA2006, have very large fixed amplitudes and introduce in the starting model the two orientations of the underlying module sublattices. We show that a composite approach with this type of function, which treats the cations and anions as two separate subsystems forming a misfit compound, is the most appropriate and robust method for the refinements.

Superspace description of NaCa4Nb5O17 (a perovskite-related compound of the type AnBnO3n+2) as a modulated layered structure

The recently determined structure of NaCa(4)Nb(5)O(17) is reanalyzed within the superspace framework. The material follows the general features of the superspace model proposed for the perovskite-related compounds of the general formula A(n)B(n)O(3n+2) [Elcoro et al. (2001). Acta Cryst. B57, 471-484]. It can be described as a commensurate modulated structure with discontinuous atomic domains given by occupational crenel functions. These atomic domains automatically introduce the layered configuration of the actual structure in real space. However, the displacive modulations follow a quite different pattern from that observed in the Sr(n)(Nb,Ti)(n)O(3n+2) series, discussed in the above-mentioned reference. The superspace group relevant in this new case has been identified through a systematic search of all the possible groups and a comparison of the resulting superspace embeddings of the experimental three-dimensional structure. Being a commensurate structure, the fundamental ambiguity of the superspace description was broken by choosing the highest possible symmetry that yields smooth displacive atomic modulations and hence minimizes the number of parameters. The efficiency of the proposed superspace model is demonstrated with a new refinement of the structure. Assuming the models general relevance, symmetry properties for the whole [Ca,Na](n)Nb(n)O(3n+2) series are predicted.

MAGNDATA: towards a database of magnetic structures. I. The commensurate case

A free web page under the name MAGNDATA, which provides detailed quantitative information on more than 400 published magnetic structures, has been made available at the Bilbao Crystallographic Server (http://www.cryst.ehu.es). It includes both commensurate and incommensurate structures. In the first article in this series, the information available on commensurate magnetic structures was presented [Gallego, Perez-Mato, Elcoro, Tasci, Hanson, Momma, Aroyo & Madariaga (2016). J. Appl. Cryst. 49, 1750–1776]. In this second article, the subset of the database devoted to incommensurate magnetic structures is discussed. These structures are described using magnetic superspace groups, i.e. a direct extension of the non-magnetic superspace groups, which is the standard approach in the description of aperiodic crystals. The use of magnetic superspace symmetry ensures a robust and unambiguous description of both atomic positions and magnetic moments within a common unique formalism. The point-group symmetry of each structure is derived from its magnetic superspace group, and any macroscopic tensor property of interest governed by this point-group symmetry can be retrieved through direct links to other programs of the Bilbao Crystallographic Server. The fact that incommensurate magnetic structures are often reported with ambiguous or incomplete information has made it impossible to include in this collection a good number of the published structures which were initially considered. However, as a proof of concept, the published data of about 30 structures have been re-interpreted and transformed, and together with ten structures where the superspace formalism was directly employed, they form this section of MAGNDATA. The relevant symmetry of most of the structures could be identified with an epikernel or isotropy subgroup of one irreducible representation of the space group of the parent phase, but in some cases several irreducible representations are active. Any entry of the collection can be visualized using the online tools available on the Bilbao server or can be retrieved as a magCIF file, a file format under development by the International Union of Crystallography. These CIF-like files are supported by visualization programs like Jmol and by analysis programs like JANA and ISODISTORT.

Structures in superspace of intergrowth polytypoids LaTi1-xO3 and (Ba1-4xLa4x)Ti1-xO3 with x=1/5

Abstract New Rietveld refinements of powder diffraction data of the cation deficient perovskite-related compounds mentioned in the title were performed in 4D using a recently proposed superspace model with discrete atomic occupation domains. The results confirm the validity and efficiency of the superspace description of these layered systems. The resulting 3D structure for La5Ti4O15 corrects a previously reported model.

Symmetry conditions for type II multiferroicity in commensurate magnetic structures

Type II multiferroics are magnetically ordered phases that exhibit ferroelectricity as a magnetic induced effect. We show that in single-k magnetic phases the presence in the paramagnetic phase of non-symmorphic symmetry combined with some specific type of magnetic propagation vector can be sufficient for the occurrence of this type of multiferroic behaviour. Other symmetry scenarios especially favourable for spin driven multiferroicity are also presented. We review and classify known type II multiferroics under this viewpoint. In addition, some other magnetic phases which due to their symmetry properties can exhibit type II multiferroicity are pointed out.