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Spatial symmetries in crystals may be distinguished by whether they preserve the spatial origin. Here we study spatial symmetries that translate the origin by a fraction of the lattice period, and find that these non-symmorphic symmetries protect an exotic surface fermion whose dispersion relation is shaped like an hourglass; surface bands connect one hourglass to the next in an unbreakable zigzag pattern. These 'hourglass' fermions are formed in the large-gap insulators, KHgX (X = As, Sb, Bi), which we propose as the first material class whose band topology relies on non-symmorphic symmetries. Besides the hourglass fermion, another surface of KHgX manifests a three-dimensional generalization of the quantum spin Hall effect, which has previously been observed only in two-dimensional crystals. To describe the bulk topology of non-symmorphic crystals, we propose a non-Abelian generalization of the geometric theory of polarization. Our non-trivial topology originates from an inversion of the rotational quantum numbers, which we propose as a criterion in the search for topological materials., Spatial symmetries are ubiquitous in crystals. A basic geometric property that distinguishes these symmetries concerns the ways in which they transform the spatial origin: rotations, inversions and reflections preserve the [...]

Magnetoresistance is the change in a material's electrical resistance in response to an applied magnetic field. Materials with large magnetoresistance have found use as magnetic sensors (1), in magnetic memory (2), and in hard drives (3) at room temperature, and their rarity has motivated many fundamental studies in materials physics at low temperatures (4). Here we report the observation of an extremely large positive magnetoresistance at low temperatures in the non-magnetic layered transition-metal dichalcogenide W[Te.sub.2]: 452,700 per cent at 4.5 kelvins in a magnetic field of 14.7 teslas, and 13 million per cent at 0.53 kelvins in a magnetic field of 60 teslas. In contrast with other materials, there is no saturation of the magnetoresistance value even at very high applied fields. Determination of the origin and consequences of this effect, and the fabrication of thin films, nanostructures and devices based on the extremely large positive magnetoresistance of W[Te.sub.2], will represent a significant new direction in the study of magnetoresistivity., Large magnetoresistance (MR) is an uncommon property, mostly of magnetic compounds. Giant magnetoresistance (GMR) (5) and colossal magnetoresistance (CMR) (6,7) occur in thin-film metals and manganese-based perovskites, for example. In [...]

Topological insulators are insulating materials that display massless, Dirac-like surface states in which the electrons have only one spin degree of freedom on each surface. These states have been imaged by photoemission, but little information on their transport parameters, for example, mobility, is available. We report the observation of Shubnikov--de Haas oscillations arising from the surface states in nonmetallic crystals of [Bi.sub.2][Te.sub.3]. In addition, we uncovered a Hall anomaly in weak fields, which enables the surface current to be seen directly. Both experiments yield a surface mobility (9000 to 10,000 [centimeter.sup.2] per volt-second) that is substantially higher than in the bulk. The Fermi velocity of 4 x [10.sup.5] meters per second obtained from these transport experiments agrees with angle-resolved photoemission experiments. 10.1126/science.1189792

Topological insulators are a new class of insulators in which a bulk gap for electronic excitations is generated because of the strong spin-orbit coupling (1-5) inherent to these systems. These materials are distinguished from ordinary insulators by the presence of gapless metallic surface states, resembling chiral edge modes in quantum Hall systems, but with unconventional spin textures. A key predicted feature of such spin-textured boundary states is their insensitivity to spin-independent scattering, which is thought to protect them from backscattering and localization. Recently, experimental and theoretical efforts have provided strong evidence for the existence of both two- and three-dimensional classes of such topological insulator materials in semiconductor quantum well structures (6-8) and several bismuth-based compounds (9-13), but so far experiments have not probed the sensitivity of these chiral states to scattering. Here we use scanning tunnelling spectroscopy and angle-resolved photoemission spectroscopy to visualize the gapless surface states in the three-dimensional topological insulator [Bi.sub.1-x][Sb.sub.x], and examine in detail the influence of scattering from disorder caused by random alloying in this compound. We show that, despite strong atomic scale disorder, backscattering between states of opposite momentum and opposite spin is absent. Our observations demonstrate that the chiral nature of these states protects the spin of the carriers. These chiral states are therefore potentially useful for spin-based electronics, in which long spin coherence is critical (14), and also for quantum computing applications, where topological protection can enable fault-tolerant information processing (15,16)., Angle-resolved photoemission spectroscopy (ARPES) experiments on the (111) surface of [Bi.sub.1-x][Sb.sub.x] crystals have been used to identify surface states within the bulk bandgap of these compounds (9,10). The shape of [...]

Helical Dirac fermions--charge carriers that behave as massless relativistic particles with an intrinsic angular momentum (spin) locked to its translational momentum--are proposed to be the key to realizing fundamentally new phenomena in condensed matter physics (1-9). Prominent examples include the anomalous quantization of magneto-electric coupling (4-6), half-fermion states that are their own antiparticle (7, 8), and charge fractionalization in a Bose-Einstein condensate (9), all of which are not possible with conventional Dirac fermions of the graphene variety (10). Helical Dirac fermions have so far remained elusive owing to the lack of necessary spin-sensitive measurements and because such fermions are forbidden to exist in conventional materials harbouring relativistic electrons, such as graphene (10) or bismuth (11). It has recently been proposed that helical Dirac fermions may exist at the edges of certain types of topologically ordered insulators (3, 4, 12)--materials with a bulk insulating gap of spin-orbit origin and surface states protected against scattering by time-reversal symmetry--and that their peculiar properties may be accessed provided the insulator is tuned into the so-called topological transport regime (3-9). However, helical Dirac fermions have not been observed in existing topological insulators (13-18). Here we report the realization and characterization of a tunable topological insulator in a bismuth-based class of material by combining spin-imaging and momentum-resolved spectroscopies, bulk charge compensation, Hall transport measurements and surface quantum control. Our results reveal a spin-momentum locked Dirac cone carrying a nontrivial Berry's phase that is nearly 100 per cent spin-polarized, which exhibits a tunable topological fermion density in the vicinity of the Kramers point and can be driven to the long-sought topological spin transport regime. The observed topological nodal state is shown to be protected even up to 300 K. Our demonstration of room-temperature topological order and non-trivial spintexture in stoichiometric [Bi.sub.2] [Se.sub.3]. [M.sub.x] ([M.sub.x] indicates surface doping or gating control) paves the way for future graphene-like studies of topological insulators, and applications of the observed spinpolarized edge channels in spintronic and computing technologies possibly at room temperature., Unlike conventional Dirac fermions as in graphene, helical Dirac fermions possess a net spin and are guaranteed to be conducting because of time-reversal symmetry (2-5), allowing the unique possibility of [...]

A topologically ordered material is characterized by a rare quantum organization of electrons that evades the conventional spontaneously broken symmetry--based classification of condensed matter. Exotic spin-transport phenomena, such as the dissipationless quantum spin Hall effect, have been speculated to originate from a topological order whose identification requires a spin-sensitive measurement, which does not exist to this date in any system. Using Mott polarimetry, we probed the spin degrees of freedom and demonstrated that topological quantum numbers are completely determined from spin texture--imaging measurements. Applying this method to Sb and [Bi.sub.1-x][Sb.sub.x], we identified the origin of its topological order and unusual chiral properties. These results taken together constitute the first observation of surface electrons collectively carrying a topological quantum Berry's phase and definite spin chirality, which are the key electronic properties component for realizing topological quantum computing bits with intrinsic spin Hall--like topological phenomena.