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Higher order topological insulators (HOTIs) are a new class of topological materials which host protected states at the corners or hinges of a crystal. HOTIs provide an intriguing alternative platform for helical and chiral edge states and Majorana modes, but there are very few known materials in this class. Recent studies have proposed Bi as a potential HOTI, however, its topological classification is not yet well accepted. In this work, we show that the (110) facets of Bi and BiSb alloys can be used to unequivocally establish the topology of these systems. Bi and Bi$_{0.92}$Sb$_{0.08}$ (110) films were grown on silicon substrates using molecular beam epitaxy and studied by scanning tunneling spectroscopy. The surfaces manifest rectangular islands which show localized hinge states on three out of the four edges, consistent with the theory for the HOTI phase. This establishes Bi and Bi$_{0.92}$Sb$_{0.08}$ as HOTIs, and raises questions about the topological classification of the full family of Bi$_{x}$Sb$_{1-x}$ alloys.
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Recent theoretical works on effective, four-band models of three-dimensional, Dirac semimetals suggest the generic planes in momentum space, orthogonal to the direction of nodal separation, and lying between two Dirac points are higher-order topological insulators, supporting gapped, edge-states. Furthermore, the second homotopy classification of four-band models shows the higher-order topological insulators support quantized, non-Abelian Berrys flux and the Dirac points are monopoles of $SO(5)$ Berrys connections. Due to the lack of suitable computational scheme, such bulk topological properties are yet to be determined from the emph{ab initio} band structures of Dirac materials. In this work, we report first, comprehensive topological classification of emph{ab initio} band structures of Na$_3$Bi, by computing Wilson loops of non-Abelian, Berrys connections for several, Kramers-degenerate bands. Our work shows the quantized, non-Abelian, Berrys flux can be used as a stable, bulk invariant for describing higher-order topology and topological phase transitions.
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.
We used low-energy, momentum-resolved inelastic electron scattering to study surface collective modes of the three-dimensional topological insulators Bi$_2$Se$_3$ and Bi$_{0.5}$Sb$_{1.5}$Te$_{3-x}$Se$_{x}$. Our goal was to identify the spin plasmon predicted by Raghu and co-workers [S. Raghu, et al., Phys. Rev. Lett. 104, 116401 (2010)]. Instead, we found that the primary collective mode is a surface plasmon arising from the bulk, free carrers in these materials. This excitation dominates the spectral weight in the bosonic function of the surface, $chi (textbf{q},omega)$, at THz energy scales, and is the most likely origin of a quasiparticle dispersion kink observed in previous photoemission experiments. Our study suggests that the spin plasmon may mix with this other surface mode, calling for a more nuanced understanding of optical experiments in which the spin plasmon is reported to play a role.
The field of topological materials science has recently been focussing on three-dimensional Dirac semimetals, which exhibit robust Dirac phases in the bulk. However, the absence of characteristic surface states in accidental Dirac semimetals (DSM) makes it difficult to experimentally verify claims about the topological nature using commonly used surface-sensitive techniques. The chiral magnetic effect (CME), which originates from the Weyl nodes, causes an $textbf{E}cdottextbf{B}$-dependent chiral charge polarization, which manifests itself as negative magnetoresistance. We exploit the extended lifetime of the chirally polarized charge and study the CME through both local and non-local measurements in Hall bar structures fabricated from single crystalline flakes of the DSM Bi$_{0.97}$Sb$_{0.03}$. From the non-local measurement results we find a chiral charge relaxation time which is over one order of magnitude larger than the Drude transport lifetime, underlining the topological nature of Bi$_{0.97}$Sb$_{0.03}$.
Three-dimensional topological insulators (3D-TIs) possess a specific topological order of electronic bands, resulting in gapless surface states via bulk-edge correspondence. Exotic phenomena have been realized in ferromagnetic TIs, such as the quantum anomalous Hall (QAH) effect with a chiral edge conduction and a quantized value of the Hall resistance ${R_{yx}}$. Here, we report on the emergence of distinct topological phases in paramagnetic Fe-doped (Bi,Sb)${_2}$Se${_3}$ heterostructures with varying structure architecture, doping, and magnetic and electric fields. Starting from a 3D-TI, a two-dimensional insulator appears at layer thicknesses below a critical value, which turns into an Anderson insulator for Fe concentrations sufficiently large to produce localization by magnetic disorder. With applying a magnetic field, a topological transition from the Anderson insulator to the QAH state occurs, which is driven by the formation of an exchange gap owing to a giant Zeeman splitting and reduced magnetic disorder. Topological phase diagram of (Bi,Sb)${_2}$Se${_3}$ allows exploration of intricate interplay of topological protection, magnetic disorder, and exchange splitting.
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