Weyl semimetals are a class of materials that can be regarded as three-dimensional analogs of graphene breaking time reversal or inversion symmetry. Electrons in a Weyl semimetal behave as Weyl fermions, which have many exotic properties, such as chiral anomaly and magnetic monopoles in the crystal momentum space. The surface state of a Weyl semimetal displays pairs of entangled Fermi arcs at two opposite surfaces. However, the existence of Weyl semimetals has not yet been proved experimentally. Here we report the experimental realization of a Weyl semimetal in TaAs by observing Fermi arcs formed by its surface states using angle-resolved photoemission spectroscopy. Our first-principles calculations, matching remarkably well with the experimental results, further confirm that TaAs is a Weyl semimetal.
The recent discoveries of Dirac fermions in graphene and on the surface of topological insulators have ignited worldwide interest in physics and materials science. A Weyl semimetal is an unusual crystal where electrons also behave as massless quasi-particles but interestingly they are not Dirac fermions. These massless particles, Weyl fermions, were originally considered in massless quantum electrodynamics but have not been observed as a fundamental particle in nature. A Weyl semimetal provides a condensed matter realization of Weyl fermions, leading to unique transport properties with novel d…
Chiral pumping from optical electric fields oscillating at THz frequencies is observed in the Weyl material TaAs with electric and magnetic fields aligned along both the a- and c-axes. Free carrier spectral weight enhancement is measured directly for the first time, confirming theoretical expectations of chiral pumping. A departure from linear field-dependence of the Drude weight is observed at the highest fields in the quantum limit, providing direct evidence of field-dependent Fermi velocity of the chiral Landau level. Implications for the chiral magnetic effect in Weyl semimetals from the o…
Model for charge carrier spectra in topological semimetals of the TaAs family
B. Verkin Institute for Low Temperature Physics & Engineering of the National Academy of Sciences of Ukraine（乌克兰国家科学院B.维尔金低温物理与工程研究所）；Institute of Low Temperature and Structure Research, Polish Academy of Sciences（波兰科学院低温结构研究所）
Detecting the spectroscopic signatures of Dirac-like quasiparticles in emergent topological materials is crucial for searching their potential applications. Magnetometry is a powerful tool for fathoming electrons in solids, yet its ability for discerning Dirac-like quasiparticles has not been recognized. Adopting the probes of magnetic torque and parallel magnetization for the archetype Weyl semimetal TaAs in strong magnetic field, we observed a quasi-linear field dependent effective transverse magnetization and a strongly enhanced parallel magnetization when the system is in the quantum limit…
Three-dimensional (3D) topological Weyl semimetals (TWSs) represent a novel state of quantum matter with unusual electronic structures that resemble both a "3D graphene" and a topological insulator by possessing pairs of Weyl points (through which the electronic bands disperse linearly along all three momentum directions) connected by topological surface states, forming the unique "Fermi-arc" type Fermi-surface (FS). Each Weyl point is chiral and contains half of the degrees of freedom of a Dirac point, and can be viewed as a magnetic monopole in the momentum space. Here, by performing angle-r…
We investigate the magnetic properties of TaAs, a prototype Weyl semimetal. TaAs crystals show weak diamagnetism with magnetic susceptibility of about -7 * 10^{-7} emu/(g*Oe) at 5 K. A general feature is the appearance of a minimum at around 185 K in magnetization measurements as a function of temperature. No phase transition is observed in the temperature range between 5 K and 400 K. The magnetic properties indicate that the intrinsic Fermi level in TaAs is not located at the Weyl nodes, in agreement with the theory prediction.
Since their discovery, topological insulators have been expected to be ideal spintronic materials owing to the spin currents carried by surface states with spin--momentum locking. However, the bulk doping problem remains an obstacle that hinders such application. In this work, we predict that a newly discovered family of topological materials, the Weyl semimetals, exhibits large intrinsic spin Hall effects that can be utilized to generate and detect spin currents. Our $ab~initio$ calculations reveal a large spin Hall conductivity that is comparable to that of $4d$ and $5d$ transition metals. T…
A Weyl semimetal is a new state of matter that host Weyl fermions as quasiparticle excitations. The Weyl fermions at zero energy correspond to points of bulk band degeneracy, Weyl nodes, which are separated in momentum space and are connected only through the crystal's boundary by an exotic Fermi arc surface state. We experimentally measure the spin polarization of the Fermi arcs in the first experimentally discovered Weyl semimetal TaAs. Our spin data, for the first time, reveal that the Fermi arcs' spin polarization magnitude is as large as 80% and possesses a spin texture that is completely…
We report large magnetoresistance (MR) at low temperatures in single-crystalline nonmagnetic compounds TaAs$_2$ and NbAs$_2$. Both compounds exhibit parabolic-field-dependent MR larger than $5\times10^3$ in a magnetic field of 9 Tesla at 2 K. The MR starts to deviate from parabolic dependence above 10 T and intends to be saturated in 45 T for TaAs$_2$ at 4.2 K. The Hall resistance measurements and band structural calculations reveal their compensated semimetal characteristics. The large MR at low temperatures is ascribed to a resonance effect of the balanced electrons and holes with large mobi…
Weyl semi-metal is the three dimensional analog of graphene. According to the quantum field theory, the appearance of Weyl points near the Fermi level will cause novel transport phenomena related to chiral anomaly. In the present paper, we report the first experimental evidence for the long-anticipated negative magneto-resistance generated by the chiral anomaly in a newly predicted time-reversal invariant Weyl semi-metal material TaAs. Clear Shubnikov de Haas oscillations (SdH) have been detected starting from very weak magnetic field. Analysis of the SdH peaks gives the Berry phase accumulate…
Symmetry plays a central role in conventional and topological phases of matter, making the ability to optically drive symmetry change a critical step in developing future technologies that rely on such control. Topological materials, like the newly discovered topological semimetals, are particularly sensitive to a breaking or restoring of time-reversal and crystalline symmetries, which affect both bulk and surface electronic states. While previous studies have focused on controlling symmetry via coupling to the crystal lattice, we demonstrate here an all-electronic mechanism based on photocurr…
In 1929, H. Weyl proposed that the massless solution of Dirac equation represents a pair of new type particles, the so-called Weyl fermions [1]. However the existence of them in particle physics remains elusive for more than eight decades. Recently, significant advances in both topological insulators and topological semimetals have provided an alternative way to realize Weyl fermions in condensed matter as an emergent phenomenon: when two non-degenerate bands in the three-dimensional momentum space cross in the vicinity of Fermi energy (called as Weyl nodes), the low energy excitation behaves …
TaAs as one of the experimentally discovered topological Weyl semimetal has attracted intense interests recently. The ambient TaAs has two types of Weyl nodes which are not on the same energy level. As an effective way to tune lattice parameters and electronic interactions, high pressure is becoming a significant tool to explore new materials as well as their exotic states. Therefore, it is highly interesting to investigate the behaviors of topological Weyl fermions and possible structural phase transitions in TaAs under pressure. Here, with a combination of ab initio calculations and crystal …
Thin crystalline layers of TaAs Weyl semimetal (9 and 18 nm thick) are grown by molecular beam epitaxy on GaAs(001) substrates. The (001) planes of the tetragonal TaAs lattice are parallel to the GaAs(001) substrate, but the corresponding in-plane crystallographic directions of the substrate and the layer are rotated by 45°. In spite of a substantial lattice mismatch (about 19%) between GaAs(001) substrate and TaAs epilayer no misfit dislocations are observed at the GaAs(001)/TaAs(001) interface. Only stacking fault defects in TaAs are detected with transmission electron microscopy. Thorough X…
The high pressure phase diagrams of TaAs family (NbP, NbAs, TaP and TaAs) are explored systematically using crystal structure search together with first-principles calculations. Our calculations show that NbAs and TaAs have similar phase diagrams and possess the same structural phase transition sequence of I41md $\rightarrow$ P-6m2 $\rightarrow$ P21/c $\rightarrow$ Pm-3m, with a bit different transition pressures. The phase transition sequence of NbP and TaP are somewhat different from that of NbAs and TaAs. In which new structures emerge, for instance, the Cmcm structure in NbP and the Pmmn s…
We report a structural study of the Weyl semimetals TaAs, TaP, NbP, and NbAs, utilizing diffraction techniques (single crystal x-ray diffraction and energy dispersive spectroscopy) and imaging techniques (transmission electron microscopy/scanning transmission electron microscopy). We observe defects of various degrees, leading to non-stoichiometric single crystals of all four semimetals. While TaP displays a large pnictide deficiency with composition TaP$_{0.83(3)}$, and stacking faults accompanied by anti-site disorder and site vacancies, TaAs displays transition metal deficiency with composi…
We report a polarized Raman study of Weyl semimetal TaAs. We observe all the optical phonons, with energies and symmetries consistent with our first-principles calculations. We detect additional excitations assigned to multiple-phonon excitations. These excitations are accompanied by broad peaks separated by 140~cm$^{-1}$ that are also most likely associated with multiple-phonon excitations. We also noticed a sizable B$_1$ component for the spectral background, for which the origin remains unclear.
We have performed polarized and temperature-dependent Raman scattering measurements on extremely large magnetoresitance compounds TaAs$_2$ and NbAs$_2$. In both crystals, all the Raman active modes, including six A$_g$ modes and three B$_g$ modes, are clearly observed and well assigned with the combination of symmetry analysis and first-principles calculations. The well-resolved periodic intensity modulations of the observed modes with rotating crystal orientations, verify the symmetry of each assigned mode and are fitted to experimentally determine the elements of Raman tensor matrixes. The b…
We have investigated the spin texture of surface Fermi arcs in the recently discovered Weyl semimetal TaAs using spin- and angle-resolved photoemission spectroscopy. The experimental results demonstrate that the Fermi arcs are spin-polarized. The measured spin texture fulfills the requirement of mirror and time reversal symmetries and is well reproduced by our first-principles calculations, which gives strong evidence for the topologically nontrivial Weyl semimetal state in TaAs. The consistency between the experimental and calculated results further confirms the distribution of chirality of t…
Very recently the topological Weyl semimetal (WSM) state was predicted in the noncentrosymmetric compounds TaAs, TaP, NbAs, and NbP and soon led to photoemission and transport experiments to verify the presumed topological properties such as Fermi arcs (unclosed Fermi surfaces) and the chiral anomaly. In this work, we have performed fully \textit{ab initio} calculations of the surface band structures of these four WSM materials and revealed the Fermi arcs with spin-momentum-locked spin texture. On the (001) polar surface, the shape of the Fermi surface depends sensitively on the surface termin…
Photodetectors with broadband optical response have promising applications in many advanced optoelectronic and photonic devices. Especially, those with the detection range up to mid-infrared at room temperature are very challenging and highly desired. Recently, Weyl semimetal has been discovered and proposed to be favorable for photodetection since in general it breaks Lorentz invariance to have tilted chiral Weyl cones around Fermi level, which leads to chirality dependent photocurrents at arbitrarily long wavelength. Furthermore, the linear dispersion bands in Weyl cones result in very high …
MX monopnictide compounds (M=Nb,Ta, X = As,P) are prototypical three-dimensional Weyl semimetals (WSMs) that have been shown in bulk single crystal form to have potential for a wide variety of novel devices due to topologically protected band structures and high mobilities. However, very little is known about thin film synthesis, which is essential to enable device applications. We synthesize TaAs(001) epilayers by molecular beam epitaxy on GaAs(001) and provide an experimental phase diagram illustrating conditions for single phase, single-crystal-like growth. We investigate the relationship b…
We report a comparative polarized Raman study of Weyl semimetals TaAs, NbAs, TaP and NbP. The evolution of the phonon frequencies with the sample composition allows us to determine experimentally which atoms are mainly involved for each vibration mode. Our results confirm previous first-principles calculations indicating that the A$_1$, B$_1(2)$, E$(2)$ and E$(3)$ modes involve mainly the As(P) atoms, the B$_1(1)$ mode is mainly related to Ta(Nb) atoms, and the E$(1)$ mode involves both kinds of atoms. By comparing the energy of the different modes, we establish that the B$_1(1)$, B$_1(2)$, E$…
Tantalum arsenide is a member of the non-centrosymmetric monopnictides, which are putative Weyl semimetals. In these materials, three-dimensional chiral massless quasiparticles, the so-called Weyl fermions, are predicted to induce novel quantum mechanical phenomena, such as the chiral anomaly and topological surface states. However, their chirality is only well-defined if the Fermi level is close enough to the Weyl points that separate Fermi surface pockets of opposite chirality exist. In this article, we present the bulk Fermi surface topology of high quality single crystals of TaAs, as deter…
The elastic modulus $c_{44}$ of a single crystal of the Weyl semimetal TaAs was investigated by measuring relative changes in the sound velocity under application of a magnetic field up to 10 T. Using an ultrasonic pulsed-echo technique, we studied the shear response of the crystal when the angle between the sound wave propagation and the magnetic field is changed. We observe a broken tetragonal symmetry at fields above 6 T, an anisotropy that is likely related to a longitudinal negative magnetoresistance and therefore might provide evidence of the chiral anomaly, one of the main topological s…
We report a structural study of the Weyl semimetals TaAs and TaP, utilizing diffraction and imaging techniques, where we show that they contain a high density of defects, leading to non-stoichiometric single crystals of both semimetals. Despite the observed defects and non-stoichiometry on samples grown using techniques already reported in the literature, de Haas-van Alphen measurements on TaP reveal quantum oscillations and a high carrier mobility, an indication that the crystals are of quality comparable to those reported elsewhere. Electronic structure calculations on TaAs reveal that the p…
As a fascinating topological phase of matter, Weyl semimetals host chiral fermions with distinct chiralities and spin textures. Optical excitations involving those chiral fermions can induce exotic carrier responses, and in turn lead to novel optical phenomena. Here, we discover strong coherent chiral terahertz emission from the Weyl semimetal TaAs and demonstrate unprecedented manipulation over its polarization on a femtosecond timescale. Such polarization control is achieved via the colossal ultrafast photocurrents in TaAs arising from the circular or linear photogalvanic effect. We unravel …
The resistance of a metal in a magnetic field can be very illuminating about its ground state. Some famous examples include the integer and fractional quantum Hall effects\cite{Klitzing-QHE,Tsui-FQHE}, Shubnikov-de Haas oscillations\cite{SdH}, and weak localization\cite{Lee-WL} \emph{et al}. In non-interacting metals the resistance typically increases upon the application of a magnetic field\cite{Pippard-MR}. In contrast, in some special circumstances metals, with anisotropic Fermi surfaces\cite{Kikugawa-PdCoO2LMR} or a so-called Weyl semimetal for instance\cite{Nielsen-ABJ,Son-ChirAnom}, may …
Weyl semimetals are materials with topologically nontrivial band structure both in the bulk and on the surface, hosting chiral nodes which are sinks and sources of Berry curvature. Weyl semimetals have been predicted, and recently measured, to exhibit large nonlinear optical responses. This discovery, along with their high mobilities, makes Weyl semimetals relevant to a broad spectrum of applications in optoelectronic, nanophotonic and quantum optical devices. While there is growing interest in understanding and characterizing the linear and nonlinear behavior of Weyl semimetals, an ab initio …
Weyl semimetals provide the realization of Weyl fermions in solid-state physics. Among all the physical phenomena that are enabled by Weyl semimetals, the chiral anomaly is the most unusual one. Here, we report signatures of the chiral anomaly in the magneto-transport measurements on the first Weyl semimetal TaAs. We show negative magnetoresistance under parallel electric and magnetic fields, that is, unlike most metals whose resistivity increases under an external magnetic field, we observe that our high mobility TaAs samples become more conductive as a magnetic field is applied along the dir…
Magnetism in Weyl semimetals is desired to investigate the interaction between the magnetic moments and Weyl fermions, e.g. to explore anomalous quantum Hall phenomena. Here we demonstrate that proton irradiation is an effective tool to induce ferromagnetism in the Weyl semimetal TaAs. The intrinsic magnetism is observed with a transition temperature above room temperature. The magnetic moments from d states are found to be localized around Ta atoms. Further, the first-principles calculations indicate that the d states localized on the nearest-neighbor Ta atoms of As vacancy sites are responsi…
A Weyl semimetal (WSM) features Weyl fermions in its bulk and topological surface states on surfaces, and is novel material hosting Weyl fermions, a kind of fundamental particles. The WSM was regarded as a three-dimensional version of "graphene" under the illusion. In order to explore its promising photoelectric properties and applications in photonics and photoelectronics, here, we study the anisotropic linear and nonlinear optical responses of a WSM TaAs, which are determined by the relationship and balance between its topological surface states and Weyl nodes. We demonstrate that topologica…
Weyl semimetals are conductors whose low-energy bulk excitations are Weyl fermions, whereas their surfaces possess metallic Fermi arc surface states. These Fermi arc surface states are protected by a topological invariant associated with the bulk electronic wavefunctions of the material. Recently, it has been shown that the TaAs and NbAs classes of materials harbor such a state of topological matter. We review the basic phenomena and experimental history of the discovery of the first Weyl semimetals, starting with the observation of topological Fermi arcs and Weyl nodes in TaAs and NbAs by ang…
We present a systematic study of both the temperature and frequency dependence of the optical response in TaAs, a material that has recently been realized to host the Weyl semimetal state. Our study reveals that the optical conductivity of TaAs features a narrow Drude response alongside a conspicuous linear dependence on frequency. The width of the Drude peak decreases upon cooling, following a $T^{2}$ temperature dependence which is expected for Weyl semimetals. Two linear components with distinct slopes dominate the 5-K optical conductivity. A comparison between our experimental results and …
TaAs and NbAs are two of the earliest identified Weyl semimetals that possess many intriguing optical properties, such as chirality-dependent optical excitations and giant second harmonic generation (SHG). Linear and nonlinear optics have been employed as tools to probe the Weyl physics in these crystals. Here we extend these studies to address two important points: determining the complete anisotropic dielectric response, and to explore if and how they can reveal essential Weyl physics. For the first time, we determine the complete anisotropic dielectric functions of TaAs and NbAs by combinin…
Electron-phonon coupled hydrodynamics in semimetal TaAs_2
Wuhan National High Magnetic Field Center and School of Physics, Huazhong University of Science and Technology（华中科技大学物理学院和武汉国家高能磁场中心）；Laboratoire de Physique et d’Étude de Matériaux (CNRS) ESPCI Paris, PSL Research University（巴黎高等物理化工大学国立科学研究中心物质物理与研究实验室）
While all media can exhibit first-order conductivity describing current linearly proportional to electric field, $E$, the second-order conductivity, $σ^{(2)}$ , relating current to $E^2$, is nonzero only when inversion symmetry is broken. Second order nonlinear optical responses are powerful tools in basic research, as probes of symmetry breaking, and in optical technology as the basis for generating currents from far-infrared to X-ray wavelengths. The recent surge of interest in Weyl semimetals with acentric crystal structures has led to the discovery of a host of $σ^{(2)}$ -related phenomena…
Unusually high phonon thermal conductivity in the Weyl semimetal TaP: A comparative study with TaAs
College of Physics and Optoelectronic Engineering, Chongqing Normal University（重庆师范大学物理与光电工程学院）；School of Science, Chongqing University of Posts and Telecommunications（重庆邮电大学理学院）；Center for Computational Science and Engineering, Southern University of Science and Technology（南方科技大学计算科学与工程中心）；College of Physics and Electronic Engineering, Chongqing Normal University（重庆师范大学物理与电子工程学院）；College of Physics and Center of Quantum Materials and Devices, Chongqing University（重庆大学物理学院量子材料与器件中心）；Chongqing Key Laboratory for Strongly Coupled Physics, Chongqing University（重庆大学强耦合物理重庆市重点实验室）
We report the measurement of efficient charge-to-spin conversion at room temperature in Weyl semimetal/ferromagnet heterostructures with both oxidized and pristine interfaces. Polycrystalline films of the Weyl semimetal, TaAs, are grown by molecular beam epitaxy on (001) GaAs and interfaced with a metallic ferromagnet (Ni$_{0.8}$Fe$_{0.2}$). Spin torque ferromagnetic resonance (ST-FMR) measurements in samples with an oxidized interface yield a spin torque efficiency as large as $ξ_{\mathrm{FMR}}=0.45\pm 0.25$ for a 8 nm Ni$_{0.8}$Fe$_{0.2}$ layer thickness. By studying ST-FMR in these samples …
Charge transport of topological semimetals has been in the focus of intensive investigations because of their non-trivial band topology. Heat transport of these materials, on the other hand, is largely unexplored and remains elusive. Here we report on an observation of unprecedented, giant magnetic quantum oscillations of thermal conductivity in the prototypical Weyl semimetal TaAs. The oscillations are antiphase with the quantum oscillating electronic density of states of a Weyl pocket, and their amplitudes amount to two orders of magnitude of the estimation based on the Wiedemann-Franz law. …
The recent discovery of a Weyl semimetal in TaAs offers the first Weyl fermion observed in nature and dramatically broadens the classification of topological phases. However, in TaAs it has proven challenging to study the rich transport phenomena arising from emergent Weyl fermions. The series Mo$_x$W$_{1-x}$Te$_2$ are inversion-breaking, layered, tunable semimetals already under study as a promising platform for new electronics and recently proposed to host Type II, or strongly Lorentz-violating, Weyl fermions. Here we report the discovery of a Weyl semimetal in Mo$_x$W$_{1-x}$Te$_2$ at $x = …
The electronic and superconducting properties associated with the topologically non-trivial bands in Weyl semimetals have recently attracted much attention. We report the microscopic properties of the type-I Weyl semimetal TaAs measured by $^{75}$As nuclear magnetic (quadrupole) resonance under zero and elevated magnetic fields over a wide temperature range up to 500 K. The magnetic susceptibility measured by the Knight shift $K$ is found to be negative at low magnetic fields and have a strong field ($B$) dependence as ln$B$ at $T$ = 1.56 K. Such nonlinear field-dependent magnetization can be …
Discovery of Weyl fermions in semimetallic tansition-metal monopnictides is a major breakthrough in condensed matter physics. A Weyl semimetal is characterized by the existence of robust Weyl points and unclosed topological surface states in the form of Fermi arc. All the four compounds of the Weyl semimetal transition monopnictide family i.e. NbP, TaP, NbAs and TaAs exhibit extremely high mobility and unsaturated high magnetoresistance (MR). For example, MR values are 8.5x10^5% at 1.85 K for NbP and 1.5x10^5% at 3 K in 9 T for TaAs. NbP also achieves very low value of residual resistivity 0.6…
Strong coupling between discrete phonon and continuous electron-hole pair excitations can give rise to a pronounced asymmetry in the phonon line shape, known as the Fano resonance. This effect has been observed in a variety of systems, such as stripe-phase nickelates, graphene and high-$T_{c}$ superconductors. Here, we reveal explicit evidence for strong coupling between an infrared-active $A_1$ phonon and electronic transitions near the Weyl points (Weyl fermions) through the observation of a Fano resonance in the recently discovered Weyl semimetal TaAs. The resultant asymmetry in the phonon …
The transition metal dipnictides TaAs2 , TaSb2 , NbAs2 and NbSb2 have recently sparked interest for exhibiting giant magnetoresistance. While the exact nature of magnetoresistance in these materials is still under active investigation, there are experimental results indicating anisotropic negative magnetoresistance. We study the effect of magnetic field on the band structure topology of these materials by applying a Zeeman splitting. In the absence of magnetic field, we find that the materials are weak topological insulators, which is in agreement with previous studies. When the magnetic field…
A Weyl semimetal possesses spin-polarized band-crossings, called Weyl nodes, connected by topological surface arcs. The low-energy excitations near the crossing points behave the same as massless Weyl fermions, leading to exotic properties like chiral anomaly. To have the transport properties dominated by Weyl fermions, Weyl nodes need to locate nearly at the chemical potential and enclosed by pairs of individual Fermi surfaces with nonzero Fermi Chern numbers. Combining angle-resolved photoemission spectroscopy and first-principles calculation, here we show that TaP is a Weyl semimetal with o…
To investigate the electronic structure of Weyl semimetals Ta$Pn$ ($Pn=$P, As), optical conductivity [$σ(ω)$] spectra are measured over a wide range of photon energies and temperatures, and these measured values are compared with band calculations. Two significant structures can be observed: a bending structure at $\hbarω\sim$85 meV in TaAs, and peaks at $\hbarω\sim$ 50 meV (TaP) and $\sim$30 meV (TaAs). The bending structure can be explained by the interband transition between saddle points connecting a set of $W_2$ Weyl points. The temperature dependence of the peak intensity can be fitted b…
We construct a tight-binding model realizing one pair of Weyl nodes and three distinct Weyl semimetals. In the type-I (type-II) Weyl semimetal, both nodes belong to type-I (type-II) Weyl nodes. In addition, there exists a novel type, dubbed "hybrid Weyl semimetal", in which one Weyl node is of type-I while the other is of type-II. For the hybrid Weyl semimetal, we further demonstrate the bulk Fermi surfaces and the topologically protected surface states, analyze the unique Landau level structure and quantum oscillation, and discuss the material realization.
