We present a review of experimental and theoretical studies of the anomalous Hall effect (AHE), focusing on recent developments that have provided a more complete framework for understanding this subtle phenomenon and have, in many instances, replaced controversy by clarity. Synergy between experimental and theoretical work, both playing a crucial role, has been at the heart of these advances. On the theoretical front, the adoption of Berry-phase concepts has established a link between the AHE and the topological nature of the Hall currents which originate from spin-orbit coupling. On the experimental front, new experimental studies of the AHE in transition metals, transition-metal oxides, spinels, pyrochlores, and metallic dilute magnetic semiconductors, have more clearly established systematic trends. These two developments in concert with first-principles electronic structure calculations, strongly favor the dominance of an intrinsic Berry-phase-related AHE mechanism in metallic ferromagnets with moderate conductivity. The intrinsic AHE can be expressed in terms of Berry-phase curvatures and it is therefore an intrinsic quantum mechanical property of a perfect cyrstal. An extrinsic mechanism, skew scattering from disorder, tends to dominate the AHE in highly conductive ferromagnets. We review the full modern semiclassical treatment of the AHE together with the more rigorous quantum-mechanical treatments based on the Kubo and Keldysh formalisms, taking into account multiband effects, and demonstrate the equivalence of all three linear response theories in the metallic regime. Finally we discuss outstanding issues and avenues for future investigation.Comment: 53 pages, 44 figure

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.

Dirac semimetals and Weyl semimetals are 3D analogs of graphene in which crystalline symmetry protects the nodes against gap formation [1-3]. Na3Bi and Cd3As2 were predicted to be Dirac semimetals [4, 5], and recently confirmed to be so by photoemission [6-8]. Several novel transport properties in a magnetic field H have been proposed for Dirac semimetals [2, 10, 11, 16]. Here we report an interesting property in Cd3As2 that was unpredicted, namely a remarkable protection mechanism that strongly suppresses back-scattering in zero H. In single crystals, the protection results in ultrahigh mobility, 9 × 10 6 cm 2 /Vs at 5 K. Suppression of backscattering results in a transport lifetime 10 4 × longer than the quantum lifetime. The lifting of this protection by H leads to a very large magnetoresistance. We discuss how this may relate to changes to the Fermi surface induced by H.

2Charge density wave (CDW) transitions are a frequent occurrence in transition metal chalcogenides due to their low structural dimensionality. Layered MX 2 compounds and chain-based MX 3 compounds, where M is a group 4 or 5 metal and X = S, Se, or Te, are the best known examples [1][2][3][4][5][6][7]. These transitions arise to allow electronic systems to minimize their energy by removing electronic states at the Fermi level. This is achieved by introducing a new structural periodicity at the Fermi wave vector, inducing a band gap. Superconductivity and the CDW state are two very different cooperative electronic phenomena, and yet both occur due to Fermi surface instabilities and electron-phonon coupling. A number of CDW-bearing materials are also superconducting [8][9][10][11][12][13], and the idea that superconductivity and CDW states are competing electronic states at low temperatures is one of the fundamental concepts of condensed matter physics. Surprisingly, no system has yet been reported in which the emergence of a superconducting state after a charge density wave state has been suppressed via doping has been studied in detail: a transition that implies a deep connection between the two states, i.e., that the same electrons are participating in both transitions. TiSe 2 was one of the first CDW-bearing compounds known, and is also one of the most frequently studied as the nature of its CDW transition has been controversial for decades. The CDW transition, at approximately 200 K, is to a state with a commensurate (2a,2a,2c) wavevector without an intermediate incommensurate phase [3,16,17]. The commensurate CDW wavevector and electronic structure calculations indicate that, unlike the case in most materials, the CDW in TiSe 2 is not driven by Fermi surface nesting. The normal state is presently believed to be either a semimetal or a semiconductor with a small indirect gap [3, 16, 18 -22] (Fig. 1a, inset). This results in a systematic expansion of the unit cell with Cu content in Cu x TiSe 2 , as evidenced by the lattice parameters shown in Fig. 1a. The expansion of the cell parameters is maintained up to x = 0.11. For higher Cu contents, both a and c remain unchanged from their value at x = 0.11. It can therefore be concluded that the solubility limit for Cu in TiSe 2 is x = 0.11 ± 0.01.Of particular interest is the evolution of the charge density wave with Cu doping.Electron and X-ray diffraction studies of pure TiSe 2 at low temperatures show the presence of reflections corresponding to the basic trigonal structure and also the 2a, 2c superstructure reflections associated with the CDW state [3,19]. increases with Cu content. This suggests that the Cu doping introduces carriers into the conduction band in TiSe 2 , increasing the electronic density of states and therefore the Pauli paramagnetism. This is further confirmed by specific heat measurements, described below. A drop in the susceptibility of pure TiSe 2 is seen as the temperature is lowered below the CDW transition at 200 K, consistent with th...

The observation of a large Nernst signal eN in an extended region above the critical temperature Tc in hole-doped cuprates provides evidence that vortex excitations survive above Tc. The results support the scenario that superfluidity vanishes because long-range phase coherence is destroyed by thermally-created vortices (in zero field), and that the pair condensate extends high into the pseudogap state in the underdoped (UD) regime. We present a series of measurements to high fields H which provide strong evidence for this phase-disordering scenario. Measurements of eN in fields H up to 45 T reveal that the vortex Nernst signal has a characteristic "tilted-hill" profile, which is qualitatively distinct from that of quasi-particles. The hill profile, which is observed above and below Tc, underscores the continuity between the vortex-liquid state below Tc and the Nernst region above Tc. The upper critical field (depairing field) Hc2 determined by the hill profile (in slightly UD to overdoped samples) displays an anomalously weak T dependence, which is consistent with the phase-disordering scenario. We contrast the Nernst results and Hc2 behavior in hole-doped and electron-doped cuprates. Contour plots of eN (T, H) in the T -H plane clearly bring out the continuous extension of the low-T vortex liquid state into the the high-T Nernst region in holedoped cuprates (but not in the electron-doped cuprate). The existence of an enhanced diamagnetic magnetization M that survives to intense H above Tc is obtained from torque magnetometry. The observed M scales accurately like eN above Tc, confirming that the large Nernst signal is associated with local diamagnetic supercurrents that persist above Tc. We emphasize implications of the new features in the phase diagram implied by the high-field results, and discuss several theories.

Bi 2 Se 3 is one of a handful of known topological insulators. Here we show that copper intercalation in the van der Waals gaps between the Bi 2 Se 3 layers, yielding an electron concentration of ~ 2 x 10 20

Two general features of a superconductor, which appear at the critical temperature, are the formation of an energy gap and the expulsion of magnetic flux (the Meissner effect). In underdoped copper oxides, there is strong evidence that an energy gap (the pseudogap) opens up at a temperature significantly higher than the critical temperature (by 100-220 K). Certain features of the pseudogap suggest that it is closely related to the gap that appears at the critical temperature (for example, the variation of the gap magnitudes around the Fermi surface and their maximum amplitudes are very similar). However, the Meissner effect is absent in the pseudogap state. The nature of the pseudogap state, and its relation (if any) to the superconducting state are central issues in understanding copper oxide superconductivity. Recent evidence suggests that, in the underdoped regime, the Meissner state is destroyed above the critical temperature by strong phase fluctuations (as opposed to a vanishing of the superfluid density). Here we report evidence for vortices (or vortex-like excitations) in La(2-x)Sr(x)CuO4 at temperatures significantly above the critical temperature. A thermal gradient is applied to the sample in a magnetic field. Vortices are detected by the large transverse electric field produced as they diffuse down the gradient (the Nernst effect). We find that the Nernst signal is anomalously enhanced at temperatures as high as 150 K.

In an electric field, the flow of electrons in a solid produces an entropy current in addition to the familiar charge current. This is the Peltier effect, and it underlies all thermoelectric refrigerators. The increased interest in thermoelectric cooling applications has led to a search for more efficient Peltier materials and to renewed theoretical investigation into how electron-electron interaction may enhance the thermopower of materials such as the transition-metal oxides. An important factor in this enhancement is the electronic spin entropy, which is predicted to dominate the entropy current. However, the crucial evidence for the spin-entropy term, namely its complete suppression in a longitudinal magnetic field, has not been reported until now. Here we report evidence for such suppression in the layered oxide Na(x)Co2O4, from thermopower and magnetization measurements in both longitudinal and transverse magnetic fields. The strong dependence of thermopower on magnetic field provides a rare, unambiguous example of how strong electron-electron interaction effects can qualitatively alter electronic behaviour in a solid. We discuss the implications of our finding--that spin-entropy dominates the enhancement of thermopower in transition-metal oxides--for the search for better Peltier materials.