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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. Prominent examples include the anomalous quantization of magneto-electric coupling, half-fermion states that are their own antiparticle, and charge fractionalization in a Bose-Einstein condensate, all of which are not possible with conventional Dirac fermions of the graphene variety. 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 or bismuth. It has recently been proposed that helical Dirac fermions may exist at the edges of certain types of topologically ordered insulators-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. However, helical Dirac fermions have not been observed in existing topological insulators. 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 non-trivial 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 spin-texture in stoichiometric Bi(2)Se(3).M(x) (M(x) indicates surface doping or gating control) paves the way for future graphene-like studies of topological insulators, and applications of the observed spin-polarized edge channels in spintronic and computing technologies possibly at room temperature.

Topological crystalline insulators are new states of matter in which the topological nature of electronic structures arises from crystal symmetries. Here we predict the first material realization of topological crystalline insulator in the semiconductor snTe by identifying its non-zero topological index. We predict that as a manifestation of this non-trivial topology, snTe has metallic surface states with an even number of Dirac cones on high-symmetry crystal surfaces such as {001}, {110} and {111}. These surface states form a new type of high-mobility chiral electron gas, which is robust against disorder and topologically protected by reflection symmetry of the crystal with respect to {110} mirror plane. Breaking this mirror symmetry via elastic strain engineering or applying an in-plane magnetic field can open up a continuously tunable band gap on the surface, which may lead to wide-ranging applications in thermoelectrics, infra-red detection and tunable electronics. Closely related semiconductors PbTe and Pbse also become topological crystalline insulators after band inversion by pressure, strain and alloying.

Weyl fermions are massless chiral fermions that play an important role in quantum field theory but have never been observed as fundamental particles. A Weyl semimetal is an unusual crystal that hosts Weyl fermions as quasiparticle excitations and features Fermi arcs on its surface. Such a semimetal not only provides a condensed matter realization of the anomalies in quantum field theories but also demonstrates the topological classification beyond the gapped topological insulators. Here, we identify a topological Weyl semimetal state in the transition metal monopnictide materials class. Our first-principles calculations on TaAs reveal its bulk Weyl fermion cones and surface Fermi arcs. Our results show that in the TaAs-type materials the Weyl semimetal state does not depend on fine-tuning of chemical composition or magnetic order, which opens the door for the experimental realization of Weyl semimetals and Fermi arc surface states in real materials.

Symmetry-broken three-dimensional (3D) topological Dirac semimetal systems with strong spin-orbit coupling can host many exotic Hall-like phenomena and Weyl fermion quantum transport. Here, using high-resolution angle-resolved photoemission spectroscopy, we performed systematic electronic structure studies on Cd 3 As 2 , which has been predicted to be the parent material, from which many unusual topological phases can be derived. We observe a highly linear bulk band crossing to form a 3D dispersive Dirac cone projected at the Brillouin zone centre by studying the (001)-cleaved surface. Remarkably, an unusually high in-plane Fermi velocity up to 1.5 Â 10 6 ms À 1 is observed in our samples, where the mobility is known up to 40,000 cm 2 V À 1 s À 1 , suggesting that Cd 3 As 2 can be a promising candidate as an anisotropic-hypercone (three-dimensional) high spin-orbit analogue of 3D graphene. Our discovery of the Dirac-like bulk topological semimetal phase in Cd 3 As 2 opens the door for exploring higher dimensional spin-orbit Dirac physics in a real material.

The first-principles band theory paradigm has been a key player not only in the process of discovering new classes of topologically interesting materials, but also for identifying salient characteristics of topological states, enabling direct and sharpened confrontation between theory and experiment. We begin this review by discussing underpinnings of the topological band theory, which basically involves a layer of analysis and interpretation for assessing topological properties of band structures beyond the standard band theory construct. Methods for evaluating topological invariants are delineated, including crystals without inversion symmetry and interacting systems. The extent to which theoretically predicted properties and protections of topological states have been verified experimentally is discussed, including work on topological crystalline insulators, disorder/interaction driven topological insulators (TIs), topological superconductors, Weyl semimetal phases, and topological phase transitions. Successful strategies for new materials discovery process are outlined. A comprehensive survey of currently predicted 2D and 3D topological materials is provided. This includes binary, ternary and quaternary compounds, transition metal and f-electron materials, Weyl and 3D Dirac semimetals, complex oxides, organometallics, skutterudites and antiperovskites. Also included is the emerging area of 2D atomically thin films beyond graphene of various elements and their alloys, functional thin films, multilayer systems, and ultra-thin films of 3D TIs, all of which hold exciting promise of wide-ranging applications. We conclude by giving a perspective on research directions where further work will broadly benefit the topological materials field.

The layered transition metal dichalcogenides (TMDs) MX 2 (M = Mo, W; X = S, Se, Te), a class of graphene-like two-dimensional materials, have attracted significant interest because they demonstrate quantum confinement at the single-layer limit 13 . As with graphene, these layered materials can be easily exfoliated mechanically to provide monolayers 3-7,14-16 and assume a hexagonal honeycomb structure in which the M and X atoms are located at alternating corners of the hexagons. However, unlike graphene, which has a gapless Dirac cone band structure, MX 2 has a rather large bandgap, making these materials more versatile as candidates for thin, flexible device applications and useful for a variety of other applications including lubrication 16 , catalysis 17 , transistors 18 and lithium-ion batteries 19 . Most interestingly, an indirect to direct bandgap transition in the monolayer limit has been predicted theoretically and supported experimentally by optical measurements [3][4][5]9,12 . Because of the direct bandgap, monolayer MX 2 is favourable for optoelectronic applications5 and field-effect transistors 15,16,18 . Furthermore, both the conduction and valence bands have two energy degenerate valleys at corners of the first Brillouin zone, making it viable to optically control the charge carriers in these valleys and suggesting the possibility of valley-based electronic and optoelectronic applications 3,6-8 .Despite these exciting developments, direct experimental verification of the novel band structure at the monolayer limit remains lacking. Furthermore, for many applications, it is vital to manufacture high-quality epitaxial films with controllable methods such as chemical vapour deposition (CVD) or molecular beam epitaxy (MBE) 20,21 .

Three types of fermions play a fundamental role in our understanding of nature: Dirac, Majorana and Weyl. Whereas Dirac fermions have been known for decades, the latter two have not been observed as any fundamental particle in high-energy physics, and have emerged as a much-sought-out treasure in condensed matter physics. A Weyl semimetal is a novel crystal whose low-energy electronic excitations behave as Weyl fermions. It has received worldwide interest and is believed to open the next era of condensed matter physics after graphene and three-dimensional topological insulators. However, experimental research has been held back because Weyl semimetals are extremely rare in nature. Here, we present the experimental discovery of the Weyl semimetal state in an inversion-symmetry-breaking single-crystalline solid, niobium arsenide (NbAs). Utilizing the combination of soft X-ray and ultraviolet photoemission spectroscopy, we systematically study both the surface and bulk electronic structure of NbAs. We experimentally observe both the Weyl cones in the bulk and the Fermi arcs on the surface of this system. Our ARPES data, in agreement with our theoretical band structure calculations, identify the Weyl semimetal state in NbAs, which provides a real platform to test the potential of Weyltronics. W eyl semimetals have received significant attention in recent years because they extend the classification of topological phases beyond insulators, host exotic Fermi arc surface states, demonstrate unusual transport phenomena and provide an emergent condensed matter realization of Weyl fermions, which do not exist as fundamental particles in the standard model 1-21 . Such kind of topologically non-trivial semimetals are believed to open a new era in condensed matter physics. In contrast to topological insulators, where only the surface states are interesting, a Weyl semimetal features unusual band structure in the bulk and on the surface, leading to novel phenomena and potential applications. This opens up unparalleled research opportunities, where both bulk-and surface-sensitive experimental probes can measure the topological nature and detect quantum phenomena. In the bulk, a Weyl semimetal has a band structure with band crossings, Weyl nodes, which are associated with definite chiral charges. Unlike the two-dimensional Dirac points in graphene, the surface-state Dirac point of a threedimensional topological insulator or the three-dimensional Dirac points in the bulk of a Dirac semimetal, the degeneracy associated with a Weyl node does not require any symmetry for its protection, other than the translation symmetry of the crystal lattice. The low-energy quasiparticle excitations of a Weyl semimetal are chiral fermions described by the Weyl equation, well known in highenergy physics, which gives rise to a condensed matter analogue of the chiral anomaly associated with a negative magnetoresistance in transport [16][17][18][19][20][21] . On the surface, the non-trivial topology guarantees the existence of surface states in the f...