The mathematical field of topology has become a framework to describe the low-energy electronic structure of crystalline solids. A typical feature of a bulk insulating three-dimensional topological crystal are conducting two-dimensional surface states. This constitutes the topological bulk-boundary correspondence. Here, we establish that the electronic structure of bismuth, an element consistently described as bulk topologically trivial, is in fact topological and follows a generalized bulk-boundary correspondence of higher-order: not the surfaces of the crystal, but its hinges host topologically protected conducting modes. These hinge modes are protected against localization by time-reversal symmetry locally, and globally by the three-fold rotational symmetry and inversion symmetry of the bismuth crystal. We support our claim theoretically and experimentally. Our theoretical analysis is based on symmetry arguments, topological indices, first-principle calculations, and the recently introduced framework of topological quantum chemistry. We provide supporting evidence from two complementary experimental techniques. With scanning-tunneling spectroscopy, we probe the unique signatures of the rotational symmetry of the one-dimensional states located at step edges of the crystal surface. With Josephson interferometry, we demonstrate their universal topological contribution to the electronic transport. Our work establishes bismuth as a higher-order topological insulator.

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Higher-Order Topology in Bismuth.

Band structure diagram paths based on crystallography

The discoveries of intrinsically magnetic topological materials, including semimetals with a large anomalous Hall effect and axion insulators1–3, have directed fundamental research in solid-state materials. Topological quantum chemistry4 has enabled the understanding of and the search for paramagnetic topological materials5,6. Using magnetic topological indices obtained from magnetic topological quantum chemistry (MTQC)7, here we perform a high-throughput search for magnetic topological materials based on first-principles calculations. We use as our starting point the Magnetic Materials Database on the Bilbao Crystallographic Server, which contains more than 549 magnetic compounds with magnetic structures deduced from neutron-scattering experiments, and identify 130 enforced semimetals (for which the band crossings are implied by symmetry eigenvalues), and topological insulators. For each compound, we perform complete electronic structure calculations, which include complete topological phase diagrams using different values of the Hubbard potential. Using a custom code to find the magnetic co-representations of all bands in all magnetic space groups, we generate data to be fed into the algorithm of MTQC to determine the topology of each magnetic material. Several of these materials display previously unknown topological phases, including symmetry-indicated magnetic semimetals, three-dimensional anomalous Hall insulators and higher-order magnetic semimetals. We analyse topological trends in the materials under varying interactions: 60 per cent of the 130 topological materials have topologies sensitive to interactions, and the others have stable topologies under varying interactions. We provide a materials database for future experimental studies and open-source code for diagnosing topologies of magnetic materials. High-throughput calculations are performed to predict approximately 130 magnetic topological materials, with complete electronic structure calculations and topological phase diagrams.

High-throughput calculations of magnetic topological materials

Multicomponent rare earth oxide (REO) nanocrystalline powders containing up to seven equiatomic rare earth elements were successfully synthesized in a single-phase CaF2-type (Fm-3 m) structure. The addition of more than six elements resulted in the formation of a secondary phase. Annealing at 1000°C for 1 h led to the formation of a single-phase (Ia-3) even in the 7-component system. In the absence of cerium (Ce4+), secondary phases were observed irrespective of the number of cations or the extent of thermal treatment indicating that cerium cations played a crucial role in stabilizing the multicomponent REOs into a phase pure structure.

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Multicomponent equiatomic rare earth oxides

Ever since the isolation of single-layer graphene in 2004, two-dimensional layered structures have been among the most extensively studied classes of materials. To date, the pool of two-dimensional materials (2DMs) continues to grow at an accelerated pace and already covers an extensive range of fascinating and technologically relevant properties. An array of experimental techniques have been developed and used to characterize and understand these properties. In particular, Raman spectroscopy has proven to be a key experimental technique, thanks to its capability to identify minute structural and electronic effects in nondestructive measurements. While high-frequency (HF) intralayer Raman modes have been extensively employed for 2DMs, recent experimental and theoretical progress has demonstrated that low-frequency (LF) interlayer Raman modes are more effective at determining layer numbers and stacking configurations and provide a unique opportunity to study interlayer coupling. These advantages are due to...

Low-Frequency Shear and Layer-Breathing Modes in Raman Scattering of Two-Dimensional Materials.

In recent years, two important advances have opened new doors for the characterization and determination of magnetic structures. Firstly, researchers have produced computer-readable listings of the magnetic or Shubnikov space groups. Secondly, they have extended and applied the superspace formalism, which is presently the standard approach for the description of nonmagnetic incommensurate structures and their symmetry, to magnetic structures. These breakthroughs have been the basis for the subsequent development of a series of computer tools that allow a more efficient and comprehensive application of magnetic symmetry, both commensurate and incommensurate. Here we briefly review the capabilities of these computation instruments and present the fundamental concepts on which they are based, providing various examples. We show how these tools facilitate the use of symmetry arguments expressed as either a magnetic space group or a magnetic superspace group and allow the exploration of the possible magnetic o...

Symmetry-Based Computational Tools for Magnetic Crystallography

Received January 7, 2011; Revised April 19, 2011 The Bilbao Crystallographic Server is a web site with crystallographic databases and programs available online (www.cryst.ehu.es). It has been operating for more than ten years and new applications are being added regularly. The programs available on the server do not need a local installation and can be used free of charge. The server gives access to general information related to crystallographic symmetry groups (generators, general and special positions, maximal subgroups, Brillouin zones, etc.). Apart from the simple tools for retrieving the stored data, there are programs for the analysis of group-subgroup relations between space groups. There are also software package studying specific problems of solid-state physics, structural chemistry and crystallography.

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Crystallography online: Bilbao Crystallographic Server

The programs hosted in the Bilbao Crystallographic Server (http://www.cryst.ehu.es) are briefly explained along with worked examples on various cases related to different fields of applications. It is our aim to have this text acting as a primer on the various usage of the crystallographic tools in conjunction with each other due to the modular structure of the server. For this reason, diverse topics such as crystallographic groups and their subgroups, pseudosymmetry, extinction conditions, k -vectors and irreducible representations have been discussed in the context.

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An introduction to the tools hosted in the Bilbao Crystallographic Server

Hard-mode Raman spectroscopy was applied to analyze the temperature-induced transformation processes in perovskite-type $({\mathrm{ABO}}_{3})$ single crystals of $(1\text{\ensuremath{-}}x){\mathrm{Na}}_{0.5}{\mathrm{Bi}}_{0.5}{\mathrm{TiO}}_{3}\text{\ensuremath{-}}x{\mathrm{BaTiO}}_{3}$ (NBT-$x\mathrm{BT})$ in a wide temperature range between 100 and 1010 K and a composition range of $x=0--0.074$ across the morphotropic phase boundary (MPB). The results show abundant uncoupled ferroic structural distortions even at 1010 K and coexistence of two types of mesoscopic-scale ferroic order at lower temperatures. Octahedral ${\mathrm{BO}}_{6}$ tilting is typical of pure NBT, while the incorporation of A-site ${\mathrm{Ba}}^{2+}$ suppresses the tilting and promotes the off centering of ${\mathrm{BO}}_{6}$ octahedra. The temperature evolution of the phonon modes clearly reveals the two macroscopically observed critical temperatures ${T}_{m}$ and ${T}_{d}$ as well as, in the case of $x\ensuremath{\ne}0$, two characteristic temperatures ${T}^{\ensuremath{'}}$ and ${T}^{\ensuremath{'}\ensuremath{'}}$ preceding the ${T}_{m}$ and ${T}_{d}$, respectively, which are attributed to mesoscopic-scale antiferroelectric and ferroelectric coupling processes within the A-site-cation subsystem. At $x\phantom{\rule{4pt}{0ex}}l\phantom{\rule{4pt}{0ex}}{x}_{\mathrm{MPB}}$ the two sublattices, off-centered A-site cations and off-centered B-site cations, remain incoherent in the entire temperature range. Only when the amount of Ba reaches ${x}_{\mathrm{MPB}}$ the two subsystems couple dynamically, which can explain the enhancement of properties at the MPB. The overall ferroic distortion, however, has a minimum at the MPB, probably because at ${x}_{\mathrm{MPB}}$ the ferroelectric coupling between B-site ${\mathrm{Ti}}^{4+}$ cations is reduced to a greater extent than the enhancement of ferroelectric coupling between A-site ${\mathrm{Bi}}^{3+}$ cations.

Structural transformations in (1 -x ) Na 0.5 Bi 0.5 TiO 3 -x BaTiO 3 single crystals studied by Raman spectroscopy

Resonance Raman spectroscopy was applied to doped PbSc0.5Ta0.5O3 and PbSc0.5Nb0.5O3 relaxor ferroelectrics, to better understand the effect of composition disorder on the mesoscopic-scale polar order in complex perovskite-type (ABO3) ferroelectrics. The excitation photon energy used was 3.8 eV, which is slightly above the energy gap and corresponds to the maximum of the optical dielectric permittivity. Group-theory analysis reveals that the resonance Raman scattering (RRS) observed under these conditions is allowed only in polar crystal classes. Therefore, RRS is dominated by the atomic dynamics of nanoregions with coherent polar distortions, which considerably facilitates the comparison of polar order in various compounds. The results show that A-site doping (Ba(2+), Sr(2+), La(3+), Bi(3+)) has significantly stronger effect on the structural polarity than the introduction of a third element at the B site (Nb(5+) or Sn(4+) doped in PbSc0.5Ta0.5O3). The A-site substitution by cations that in contrast to Pb(2+) have isotropic outermost electron shells disturbs the system of lone-pair electrons, thus reducing the correlation length of coupled polar distortions and the strength of the electric field associated with the mean polarization of polar nanoregions. A-site doping with larger cations (Ba(2+)) augments the polar deformation of the individual BO6 octahedra due to local elastic fields. As a result, such A-site doping intensifies the initial structural polarity at high temperatures and prevails the enlargement of the polar fraction at low temperatures. A-site doping with smaller cations (Sr(2+), La(3+)), regardless if they are isovalent or aliovalent to Pb(2+), increases the correlation length of antiferrodistortive order (BO6 tilts), which in turn assists the development of double-perovskite structure with coherent local polar distortions. A-site doping with aliovalent cations (Bi(3+)) having the same outermost electron shell and ionic radius as the host A-site Pb(2+) cations leads to stronger coupling along the -B-O-B- bond linkages due to enhanced random local electric fields.

http://iopscience.iop.org/article/10.1088/0953-8984/28/47/475902/pdf

G. de la Flor

Papers

Symmetry-Based Computational Tools for Magnetic Crystallography

Crystallography online: Bilbao Crystallographic Server

An introduction to the tools hosted in the Bilbao Crystallographic Server

Structural transformations in (1 -x ) Na 0.5 Bi 0.5 TiO 3 -x BaTiO 3 single crystals studied by Raman spectroscopy

The effect of chemical variations on the structural polarity of relaxor ferroelectrics studied by resonance Raman spectroscopy.