Experimental perspective on three-dimensional topological semimetals

B. Q. Lv, T. Qian, and H. Ding

Rev. Mod. Phys. 93, 025002 – Published 26 April 2021

Abstract

Topological semimetals (TSMs) are characterized by bulk band crossings in their electronic structures, which are expected to give rise to gapless electronic excitations and topological features that underlie exotic physical properties. The most famous examples are Dirac and Weyl semimetals, in which the corresponding low-energy fermionic excitations, i.e., the Dirac and Weyl fermions, are direct analogs of elementary particles in quantum field theory. The last decade has witnessed an explosion of research activities in the field of TSMs thanks to precise theoretical predictions, well-controlled material synthesis, and advanced characterization techniques including angle-resolved photoemission spectroscopy, scanning tunneling microscopy, magnetotransport measurements, optical spectroscopy, etc. Here recent progress in three-dimensional TSMs is reviewed with an emphasis on their characteristic bulk electronic structures, including dimensionality (such as zero-dimensional nodal points, one-dimensional nodal lines, and two-dimensional nodal surfaces), degeneracy (twofold, threefold, fourfold, sixfold, or eightfold) of the band crossing, the slope (type I and type II) and order (linear, quadratic, or cubic) of the band dispersion near the crossing, the characteristic topological invariants (such as monopole charges), and the crystallographic symmetries that stabilize the band crossings. The distinct signatures of the various topological semimetal phases, such as the nontrivial surface states (including Fermi arcs of Dirac and Weyl semimetals) and the unique transport and optical responses (such as chiral anomaly-induced negative magnetoresistance in Dirac and Weyl semimetals), are also reviewed.

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Received 3 December 2019

DOI:https://doi.org/10.1103/RevModPhys.93.025002

Physics Subject Headings (PhySH)

Dirac semimetal Node-line semimetals Topological materials Weyl semimetal

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

B. Q. Lv

Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

T. Qian and H. Ding

Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China

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Vol. 93, Iss. 2 — April - June 2021

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Figure 1
Timeline of developments of topological quantum phases in condensed-matter systems. IQHE, integer quantum Hall effect; FQHE, fractional Hall effect; QAHE, quantum anomalous Hall effect; QSHE, quantum spin Hall effect; TI, topological insulator; TCI, topological crystalline insulator; TKI, topological Kondo insulator; DSM, Dirac semimetal; WSM, Weyl semimetal; TNLSM, topological nodal-line semimetal; HOTI, high-order topological insulator; HOTSM, high-order topological semimetal.
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Figure 2
Schematic plots of the band structures of Schrödinger, massive Dirac, massless Dirac, and Weyl fermions. The curves with mixed and uniform colors represent doubly degenerate and nondegenerate bands, respectively.
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Figure 3
(a) Schematic drawing of one pair of Weyl nodes of opposite chirality in 3D momentum space. (b),(c) Consider a 2D cylinder enclosing a Weyl node; a chiral edge state appears due to the nonzero Chern number. (d)–(f) Connection of surface states to bulk Weyl nodes. A Weyl node behaves as a magnetic monopole (MMP) in momentum space, and the helicoid edge states appear as a Fermi arc on the surface connecting projections of two Weyl nodes of opposite chirality. Adapted from [479], [307], and [124].
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Figure 4
(a) Schematic band dispersions of one pair of Weyl nodes of opposite chirality. (b) Illustration of the chiral anomaly based on a Landau-level spectrum of Weyl fermions in parallel electric and magnetic fields in the quantum limit. Filled (empty) circles denote occupied (unoccupied) states, while red and green areas indicate the right-handed ( $+$ ) electrons and left-handed ( $-$ ) electrons, respectively. (c) Schematic diagram for the planar Hall effect measurement geometry ( $V_{H}$ is the Hall voltage). (d) Angular dependence of planar Hall conductivity for $B = 5 T$ and $γ = 0$ ( $γ$ is the tilt parameter). Adapted from [308], [336], and [559].
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Figure 5
Schematic illustration of possible band-inversion-induced topological states. The curves with mixed and uniform colors represent doubly degenerate bands and nondegenerate bands, respectively. WNL, Weyl nodal line; TPSM, triple point semimetal.
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Figure 6
Schematic plot of the band structure for space group 16 with and without SOC. From [73].
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Figure 7
Schematic plot of the band structure of various symmetry-enforced band degeneracies in condensed-matter systems. WF, Weyl fermion; WP, Weyl point; TCF, three-component fermion; TP, triple point; DF, Dirac fermion; DP, Dirac point; CFF, charge-2 fourfold fermion; CFP, charge-2 fourfold point; RSWF, Rarita-Schwinger-Weyl fermion; RSWP, Rarita-Schwinger-Weyl point; DTCF, double class-II three-component fermion; DTP, double triple point; DDF, double Dirac fermion; DDP, double Dirac point. Adapted from [310].
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Figure 8
Schematic plot of the band dispersion and Fermi surfaces of type-I, type-II, and type-III Dirac and Weyl points. From [185].
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Figure 9
Band crossings with linear, quadratic, and cubic dispersions.
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Figure 10
(a) Calculated band structure of ${Na}_{3} Bi$ with and without including the SOC. (b) Bulk and projected $(010)$ surface BZs of ${Na}_{3} Bi$ .The red arrows indicate the momentum location of the Dirac points. (c) Schematic band dispersions near the Dirac points in the $k_{x}^{D} − k_{y}^{D}$ and $k_{x}^{D} − k_{z}^{D}$ planes, respectively. (d) 3D ARPES intensity plots, showing the band dispersion of the bulk Dirac cone along the in-plane ( $k_{x}^{D} − k_{y}^{D}$ -plane) and out-of-plane ( $k_{y}^{D} − k_{z}^{D}$ -plane) directions, respectively. (e) ARPES intensity plot shows the band dispersion of the upper Dirac cone along the $\bar{Γ} − \bar{M}$ direction after the in situ $K$ doping. Adapted from [505], and [294].
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Figure 11
(a) Calculated Fermi-arc surface states and their spin textures (in-plane components) for the $(010)$ surface of ${Na}_{3} Bi$ . (b) Similar to (a), but obtained from the fitted effective Hamiltonian with additional exchange field $h_{1} = 6 meV$ . (c) Calculated band structure of ${Na}_{3} Bi$ for the $(100)$ surface. (d) ARPES intensity plot at $E_{F}$ in the $(100)$ surface of ${Na}_{3} Bi$ recorded at $h υ = 55 eV$ . SS, surface states. (e) ARPES intensity plot for 2D $k$ slices shown in (d). Adapted from [505], and [551].
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Figure 12
STM measurements of ${Cd}_{3} {As}_{2}$ . (a) Fourier transforms of $d I / d V$ conductance maps of ${Cd}_{3} {As}_{2}$ at different energies. The red and cyan dots show the scattering of the bulk conduction band and a second electronlike band, respectively. (b) Landau-level point spectra measured at 400 mK ( $I = 400 pA$ , $V = - 250 mV$ , $V_{osc} = 0.8 mV$ ). Indices correspond to the Landau-level assignment. (c) Summary of the Dirac conduction band deduced from the quasiparticle interference pattern (indicated by red dots) and the Landau-level spectra (indicated by yellow circles). Blue and green curves are guides for the eye. Adapted from [202].
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Figure 13
SdH oscillations of ${Cd}_{3} {As}_{2}$ . (a) SdH oscillatory component $Δ R_{x x}$ as a function of $1 / B$ at various temperatures. The current is in the $(112)$ plane, and the magnetic field is perpendicular to the $(112)$ plane. (b) Landau index plot $n$ vs $1 / B$ for two ${Cd}_{3} {As}_{2}$ samples. The filled circles denote the integer index (valley), and the empty circles indicate the half-integer index (peak). From [162].
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Figure 14
(a) Resistance measured at 4.5 K with the applied magnetic field direction changing from perpendicular ( $ϕ = 90 °$ ) to parallel ( $ϕ = 0 °$ ) to the electric-field direction in the $x − y$ plane ( $[001]$ surface) of ${Na}_{3} Bi$ (inset). (b) Longitudinal magnetoresistance of ${Na}_{3} Bi$ measured at temperatures ranging from 4.5 to 300 K, with $B ∥ I ∥ x$ . (c) Angle-dependent magnetoresistance of the ${Cd}_{3} {As}_{2}$ microribbon measured at 2 K. Here the applied constant current is along the [110] growth direction. (d) Corresponding temperature-dependent longitudinal magnetoresistance of the ${Cd}_{3} {As}_{2}$ microribbon. Adapted from [532], and [258].
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Figure 15
PHE of ${Cd}_{3} {As}_{2}$ nanoplates. (a) Scanning electron microscope (SEM) image of a ${Cd}_{3} {As}_{2}$ nanoplate device. Lower left inset: schematic of the measurement configuration. Lower right inset: the thickness of the nanoplates determined by SEM is $\sim 90 nm$ . (b),(c) Measured planar Hall resistivity and negative longitudinal magnetoresistivity at different temperatures. (d) Corresponding temperature-dependent amplitudes of both the planar Hall resistivity and the negative longitudinal magnetoresistivity. Adapted from [523].
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Figure 16
(a) Weyl orbits in a slab of WSM of a finite thickness ( $L$ ) in the presence of a perpendicular magnetic field. (b) Fourier transform of magnetoresistance measured on the ${Cd}_{3} {As}_{2}$ microplates (150 nm) at 2 K, with magnetic fields parallel (90°, blue curve) and perpendicular (0°, red curve) to the $(010)$ surface, respectively. (c) Aharonov-Bohm oscillations in conductance as a function of magnetic flux, measured at variable temperatures in a ${Cd}_{3} {As}_{2}$ nanowire with a diameter of $\sim 115 nm$ . (d) Magnetic field dependence of Hall resistance $R_{x y}$ measured at the three pairs of Hall electrodes, in a wedge-shaped ${Cd}_{3} {As}_{2}$ nanostructure, as illustrated in the inset. Scale bar, $15 μ m$ . (e) Magnetic field dependence of $R_{y x}$ measured in $(112)$ -oriented ${Cd}_{3} {As}_{2}$ thin films with thicknesses of 12 and 23 nm. The corresponding filling factors are labeled, with determined degeneracy factors of 2 and 1 for 12 and 23 nm thick films, respectively. (f) Gate voltage ( $V_{G}$ ) dependence of $R_{x y}$ under a rotating magnetic field from 0° (along the $[112]$ crystal direction) to $54.7 °$ (along the $[001]$ crystal direction). Adapted from [378], [324], [491], [469], [281], and [603].
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Figure 17
(a) Real part of the optical conductivity of a $[001]$ -oriented ${Cd}_{3} {As}_{2}$ sample measured at different temperatures. The black dashed line represents $σ_{1} (ω) \propto ω^{1.65}$ , whereas the green line represents the linear fit. Inset: deduced Fermi velocity at $10 K$ . (b) $\sqrt{B}$ dependence of the Landau-level transition energy measured at $1.8 K$ , on $(001)$ - and $(112)$ -oriented ${Cd}_{3} {As}_{2}$ samples. The black dashed and dotted lines show the calculated curves based on the gapless Kane and Dirac models, respectively. Adapted from [5], and [345].
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Figure 18
Type-II Dirac semimetal in ${PtTe}_{2}$ . (a) Calculated bulk band structures of ${PtTe}_{2}$ along the in-plane ( $S$ - $D$ - $T$ ) and out-of-plane ( $A$ - $Γ$ - $A$ ) directions through the Dirac point with SOC included. (b) Bulk BZ and the projected $(001)$ surface BZ, with high-symmetry points indicated. Red dots (labeled as $D$ ) mark the positions of type-II Dirac points. (c) Calculated bulk bands of ${IrTe}_{2}$ along the $S$ - $D$ - $T$ direction through the Dirac point with SOC included. (d),(e) ARPES intensity plots showing the band dispersion of the bulk Dirac cone along the (d) in-plane and (e) out-of-plane directions, respectively. (f) Second derivative intensity plot showing the Dirac point lying at $E_{F}$ for the $x = 0.1$ ( ${Ir}_{1 - x} {Pt}_{x} {Te}_{2}$ ) sample. Dashed lines are guides for the eye indicating the linear Dirac bands. Adapted from [560], and [128].
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Figure 19
dHvA oscillations for ${PdTe}_{2}$ . (a) Oscillatory components of the out-of-plane magnetization vs $1 / B$ at various temperatures. (b) Temperature dependence of the fast Fourier transformation (FFT) amplitude for six oscillation modes ( $α$ to $ζ$ ). The solid lines represent the LK fits for the effective mass. Inset: FFT spectra of $Δ M$ oscillations. (c) Landau-level fan diagram of the $α$ mode. Inset: fit of the Dingle temperature. (d) The calculated electronic structures of ${PdTe}_{2}$ along high-symmetry lines. The Dirac point and the hole pocket with the $α$ mode are indicated by red arrows. Adapted from [127].
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Figure 20
Symmetry-enforced Dirac semimetal beta-cristobalite ${BiO}_{2}$ . (a) Calculated bulk band structures along high-symmetry lines. (b) Summary of 3D bulk Dirac cones at the center of the three zone faces of the BZ. Adapted from [585].
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Figure 21
(a) Calculated bulk band structures of $α$ -Sn along high-symmetry lines. (b) ARPES spectra on $α − Sn / InSb$ film, measured along the $k_{x}$ direction with 119 eV photon energy, which corresponds to the $Γ$ point of the BZ. (c) Calculated bulk band structures of $Tl (MoTe)_{3}$ as a representative of stable $A (Mo X)_{3}$ compounds. The cubic and symmetry-enforced Dirac point at $A$ , the linear and symmetry-enforced Dirac point at $L$ , and the band-inversion-induced Dirac points along $Γ − A$ are marked with red, blue, and purple circles, respectively. (d) ARPES spectra on $Tl (MoSe)_{3}$ measured along the $k_{z}$ direction with 50 eV photon energy. Adapted from [24], [290], [335], and [608].
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Figure 22
AFM Dirac semimetal in CuMnAs. (a) Calculated bulk band structures of CuMnAs along the high-symmetric lines without (red lines) and with (blue lines) SOC. In the SOC case, the orientation of magnetic moments is along the $z$ direction. (b) Calculated Fermi surface contour on the $(010)$ surface. The red stars indicate the projection of the gapless Dirac points. Adapted from [461].
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Figure 23
(a) DSM showing a pair of Dirac points along the $k_{z}$ axis in the bulk BZ and double Fermi arcs on the side surfaces. Note that the surface perpendicular to the $z$ axis [ $(001)$ surface] has no arcs. (b) Surface spectral density showing the existence of double Fermi arcs connecting the projection of two Dirac points (solid red circles) on the $(100)$ surface. (c) Deformation of double Fermi arcs into a closed Fermi contour by perturbations. (d) Calculated bulk band structures of $β$ -CuI along high-symmetry lines with SOC. The orbital weights are represented by the areas of circles and triangles. (e) The projected surface density of states of $β$ -CuI for the $(100)$ surface. (f) Fermi surface at the energy of bulk Dirac points for the $(100)$ surface. The green arrows indicate the spin textures of the closed nontrivial Fermi pocket centered at the $Γ$ point. From [219], and [254].
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Figure 24
(a) Crystal structure of TaAs. The arrow indicates that the cleavage occurs between the As and Ta atomic layers, producing two kinds of $(001)$ surfaces with either As- or Ta-terminated atomic layers. (b) The calculated band structure of TaAs without SOC. (c) 3D view of the nodal rings (without SOC, green circles) and Weyl points (with SOC, blue and red dots) in the BZ. The red and blue areas represent opposite chiralities of the Weyl nodes. (d) ARPES intensity plots along the $k_{y}$ and $k_{z}$ directions through a pair of W1 Weyl nodes. The white dashed lines are guides for the eye. (e) ARPES intensity plots along the $k_{x}$ and $k_{y}$ directions near the Weyl nodes W2. For comparison, the calculated bands are plotted (red curves) on top of the experimental data. Adapted from [308], [514], and [549].
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Figure 25
(a) Fine $k$ -point sampling calculations of surface states at $E_{F}$ near the WPs. W1 are indicated by dashed circles since the chemical potential is located slightly away from the nodes. Yellow and red areas represent opposite chiralities. (b) ARPES intensity plot at $E_{F}$ along $\bar{Γ} − \bar{Y}$ collected at different photon energies (48–68 eV), showing the observation $k_{z}$ -nondispersive surface states. (c) ARPES intensity plot at $E_{F}$ recorded on the As-terminated $(001)$ surface at $h υ = 20 eV$ . Adapted from [307], and [574].
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Figure 26
(a) Left panel: crystal structure of NbP with two different $(001)$ surface terminations. Right panel: schematics of the corresponding ARPES Fermi surface contours around $\bar{X}$ for the two surface terminations. (b) Summary of measured Fermi surfaces from the Nb-terminated (blue loops) and P-terminated (red loops) surfaces. The projections of the W2 Weyl points at the intersection of Fermi surfaces are overlaid by solid circles, whereas open circles show other intersections. Possible W1 Weyl nodes are also indicated with diamonds. From [439].
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Figure 27
(a) Photoemission intensity plot at $E_{F}$ of the $(001)$ As-terminated surface of TaAs. The red and yellow arrows indicate the direction of spin polarization on the inner and outer Fermi surfaces at the high-symmetry lines, respectively. (b) Corresponding theoretical spin texture of surface states. Adapted from [306].
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Figure 28
(a) Fourier transform of the $d I / d V$ conductance map measured at $V_{B} = 40 meV$ on the $(001)$ As-terminated surface of TaAs. The red dot indicates the ( $2 π / a$ , $2 π / a$ ) point in the reciprocal space. (b) Weighted Fermi surface (WFS) calculated by projecting the electronic states only to the topmost As layer. The $Q$ vectors indicate the expected scattering wave vectors. (c) Calculated spin-dependent scattering probability (SSP) map based on the Fermi surface shown in (b). The orange rectangles in (a) and (c) indicate the corresponding quasiparticle interference vectors. (d) Similar to (a) but measured at $V_{B} = 0 meV$ . (e) Subtraction of ellipselike QPI peaks at Bragg points [ $G_{\pm y}$ in (d)] times $α = 1.14$ from the central BZ. (f) Calculated SSP of Fermi arcs alone. Inset: contributing scattering processes (Q4a–Q4c) within the Fermi arc along $Γ − Y$ . From [25], and [199].
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Figure 29
(a) Angle-dependent magnetoresistance (MR) of a TaAs single crystal measured at $T = 1.8 K$ , with $I$ along the $a$ axis. Here $θ$ is defined as the angle between $B$ and $z$ . Inset: the extracted minima of MR at different angles in a magnetic field from 1 to 6 T. (b) The measured longitudinal resistance (open circles) and the corresponding fitting curves (red dashed lines) at various temperatures. (c) Longitudinal magnetoresistance of TaAs measured at $T = 2 K$ , with $B ∥ I$ along the $a$ and $c$ axes. The green curves are the fits to the longitudinal MR data in the semiclassical regime. (d) Chemical potential dependence of the chiral coefficient $C_{W}$ . The chiral coefficient is defined as $C_{W} = e^{4} τ_{a} / 4 π^{4} ℏ^{4} g (E_{F})$ , where $g (E_{F})$ is the density of states at $E_{F}$ and $τ_{a}$ is the axial charge relaxation time. $B_{C}$ is a critical field that characterizes the crossover between the weak antilocalization (WAL) regime and the negative magnetoresistance regime, and $C_{WAL}$ is the coefficient describing the magnitude of the 3D WAL effect. Adapted from [191], and [605].
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Figure 30
(a) Magnetic field dependence of the voltage for two different contact configurations (inset) in NbP, measured at 2 K with $B ∥ I$ . The dotted lines are the corresponding simulations of the voltage distribution taking into account the contact configurations and the field-induced inhomogeneous current distribution. (b) Experimental (left panel) and simulated (right panel) magnetic field dependence of the longitudinal MR ( ${MR}^{*} = [V (B) - V_{0}] / V_{0}$ ) for three different geometries in TaP, measured at 1.85 K with $B ∥ I ∥ a$ axis. Adapted from [14], and [110].
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Figure 31
(a) Magnetic torque and resistivity of NbAs measured at 4 K and an angle of 25° off the $c$ axis toward the $a$ axis. The orange line indicates the quantum limit determined as the position of the last quantum oscillations. (b) $ρ_{x x}$ and $ρ_{y x}$ of TaP as a function of the magnetic field at 1.5 and 4.2 K. $t$ and $s$ correspond to the Landau levels from the electronlike and holelike Weyl pockets, respectively, with the subscripts denoting the Landau indices. (c) Longitudinal magnetoresistance of TaAs for $I ∥ B ∥ c$ measured at different temperatures. Inset: microstructured TaAs device. (d) Ultrasonic attenuation of TaAs measured at 315 MHz for $k ∥ B ∥ c$ , where $k$ is the propagation wave vector of the longitudinal sound. Inset: enlargement of the data from 0 to 65 T. Adapted from [325], [606], and [389].
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Figure 32
The mobility–carrier density scaling plot of various 2D and quasi-2D systems. From [602].
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Figure 33
(a) Real part of the optical conductivity of TaAs measured at 5 K. The blue and black lines are guides for the eye. Inset: corresponding spectra weight $S (ω)$ , which is proportional to $ω^{2}$ . (b) Landau-level resonance energy vs $B$ measured at 5 K for NbAs. C1 and C3 (W2) and C2 (W1) denote the interband transitions that involve the zeroth chiral Landau-level of Weyl bands (W1 has a higher Fermi velocity), respectively; the blue and red lines are the corresponding fitting curves. T1 represents the nonzero Landau-level transition, which follows a $\sqrt{B}$ dependence. Adapted from [534], and [595].
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Figure 34
(a) 3D bulk BZ and the projected $(001)$ surface BZ of ${MoTe}_{2}$ . Weyl points with positive and negative monopole charges are displayed as green and gray dots. (b) Calculated fine band structure of ${MoTe}_{2}$ along the $K − K^{'}$ direction crossing two types of Weyl points. (c) Calculated spectral intensity maps of ${MoTe}_{2}$ at $E_{F} + 5 meV$ . (d) Calculated spectral intensity maps of ${MoTe}_{2}$ at $E_{F}$ (shifted by $- 0.02 eV$ to account for the slight hole doping). The white dashed boxes indicate the regions of interest shown in (c). (e) ARPES intensity maps of ${MoTe}_{2}$ measured at $E_{F}$ with a 6.3 eV laser. (f),(g) ARPES intensity plot and the calculated electronic structure of ${Mo}_{0.25} W_{0.75} {Te}_{2}$ at $k_{x} \sim k_{w}$ along the $k_{y}$ direction. The red lines indicate the energy positions of the surface band bottom, the W1 point, and the W2 point, respectively. Adapted from [446], [29], and [107].
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Figure 35
(a) The calculated surface and projected bulk states of ${Mo}_{0.66} W_{0.34} {Te}_{2}$ $(001)$ surface at $E = 50 meV$ . The black and white dots represent the projected Weyl nodes with opposing chiralities. (b) Same as (a), but with only the surface states considered. (c) Calculated QPI patterns based on (b). (d) Experimental QPI patterns at $E = 50 meV$ . The white dotted rectangles in (c) and (d) highlight the Fermi-arc-derived QPI patterns. Adapted from [631].
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Figure 36
(a) NLMR of a ${WTe}_{2}$ thin flake measured at various gate voltages under the $B ∥ I ∥ b$ -axis condition. Inset: schematic structure of the gated four-probe devices. (b) Positive longitudinal magnetoresistance with current along the $a$ axis. (c) Chiral coefficient ( $C_{W}$ ) extracted from (a), where the maximum value of $C_{W}$ indicates the crossing of the type-II Weyl points. (d) FFT spectra of the $b$ -axis ${WTe}_{2}$ ribbon. An extra frequency of 78 T shows at $B > 8.0 T$ relative to the bulk ${WTe}_{2}$ . Adapted from [500], and [261].
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Figure 37
(a) 3D bulk BZ (black) and the projected $(001)$ surface BZ (red) of ${Co}_{3} {Sn}_{2} S_{2}$ . The green lines and red and blue dots indicate the locations of the nodal lines and the Weyl points in the BZ. (b) Calculated band structures of ${Co}_{3} {Sn}_{2} S_{2}$ along high-symmetry lines with and without SOC. (c) Calculated spectral intensity maps for the $(001)$ Sn-terminated surface of ${Co}_{3} {Sn}_{2} S_{2}$ at $E_{F} + 60 meV$ . (d) ARPES intensity plots recorded with 124 eV photons. Red arrows indicate surface Fermi arcs (SFA). (e) ARPES intensity plot along the direction parallel to $k_{y}$ [green line in (c)], showing the linear band dispersion across the Weyl point. (f) Temperature-dependent anomalous Hall conductivity measured at zero magnetic field. Inset: logarithmic temperature dependence of $σ_{H}^{A}$ . (g) Fourier transform of a $d I / d V$ map taken at $V_{bias} = 7.5 meV$ on the $(001)$ Sn-terminated surface, showing hexagonal-shaped QPI patterns originating from the scattering processes involving Fermi arcs [pink arrows in (ii)] and surface-projected bands [red arrows in (ii)]. (h) Fourier transform of a $d I / d V$ map taken at $V_{bias} = 70 meV$ on the $(001)$ Co-terminated surface showing QPI broad peaks along $Γ − M$ [marked with pink ellipses in (i)] that originate from the scattering processes involving the Fermi-arc bands [pink arrows in (ii)]. Adapted from [284], [543], [283], and [329].
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Figure 38
(a) Calculated band structure of ${HgCr}_{2} {Se}_{4}$ after including SOC [with majority spin aligning to the $(001)$ direction]. Two kinds of band crossings (labeled A and B) are observed near $E_{F}$ . (b) Band crossings in $k$ space. Crossing A gives rise to two isolated Weyl points along the $Γ − Z$ line, and the trajectory of crossing B is a closed loop surrounding the $Γ$ point along the $k_{z} = 0$ plane. (c) Schematic of the in-plane quadratic band dispersion around the Weyl nodes. Inset: chiral spin texture. (d) Calculated edge states for the plane with $k_{z} = 0.06 π$ . The red and green lines indicate the states located at different edges. (e) Calculated surface Fermi arcs for the ( $k_{y}$ , $k_{z}$ ) side surface. From [538].
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Figure 39
The calculated band structures of the representative compounds ${Pd}_{3} {Bi}_{2} S_{2}$ (SG 199), $K_{3} {BiTe}_{3}$ (SG198), and ${Ta}_{3} Sb$ (SG 223) showing the existence of threefold, sixfold, and eightfold band degeneracies, respectively. From [42].
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Figure 40
(a) Calculated band structures of CoSi along high-symmetry lines without spin-orbit coupling. Inset: calculated 3D bulk Fermi surfaces. (b),(c) Curvature intensity plot of the ARPES data together with the calculated band structure (red curves) showing the band dispersions along the $R − Γ − R$ and $M − Γ − M$ directions, respectively. (d) ARPES intensity maps at $E_{F}$ , measured with (i) $h υ = 75 eV$ and (ii) $h υ = 110 eV$ . (e) Surface Fermi arcs extracted from the ARPES intensity maps with 75 (blue dots) and 110 eV (red dots) photon energy showing the connection between the $(001)$ surface projections of the bulk Fermi surfaces at $\bar{Γ}$ and $\bar{M}$ . (f) ARPES intensity plot showing the chiral band dispersions on the closed loop indicated in (d). (g) Experimental QPI patterns taken on the $(001)$ surface of CoSi at $U = 0 mV$ . (h) Corresponding QPI simulations including SOC. The dotted rectangles in (g) and (h) mark the surface’s first BZ, and the yellow and green arrows highlight the Fermi-arc-derived QPI patterns. (i) The calculated surface states of the CoSi $(001)$ surface at $E_{F}$ with SOC included. Adapted from [462], [390], and [593].
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Figure 41
(a) Experimental and calculated real part of the optical conductivity of RhSi. The linear-in-frequency region (below $\sim 600 meV$ ) was assigned to transitions between the linear crossing bands near the $Γ$ and $R$ points. (b) Measured circular photogalvanic effect (CPGE) amplitude of RhSi as a function of photon energy. Inset: schematic photon-energy dependence of the CPGE, which is proportional to the total flux ( $C$ ) of the Berry curvature passing through the surface $S_{W}$ . For $ℏ w < E_{c}$ (blue shaded region), $S_{W}$ corresponds to a single surface enclosing the $Γ$ point, and $C$ is thereby identical to the Chern number at the $Γ$ point, i.e., 4. Above $E_{c}$ , $S_{W}$ encloses two nodes of opposite chirality at the $Γ$ and $R$ points, resulting in a zero $C$ and a vanishing CPGE. From [318], and [392].
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Figure 42
(a) Schematics of the band structures of three-component fermions (left) and the equal-energy contours at three different energy cuts: below, between, and above the two triple points. (b) Calculated band structures of WC near the triple point with SOC. The black dots indicate the triple points. The green and red curves represent doubly degenerate and nondegenerate bands, respectively. (c) ARPES intensity plot of WC along the $k_{y}$ direction with $h υ = 555 eV$ (corresponds to the no. 1 triple point in WC). (d) ARPES intensity plot at $- 200 meV$ recorded with $h υ = 555 eV$ . The white and green rectangles indicate the $(100)$ surface BZ (SBZ) and the bulk BZ (BBZ) on the $k_{x} = 0$ plane, respectively. (e) Schematic of surface Fermi arcs (red and pink curves) connecting the surface projections of bulk TPs (green dots) on the $(100)$ surface as well as the surface-state (SS) bands along the $\tilde{Γ} − \tilde{X}$ and $\tilde{A} − \tilde{R}$ directions. Adapted from [313].
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Figure 43
(a) Temperature-dependent charge-carrier mobility of MoP. Inset: crystallographic directions in the hexagonal representation. (b) Longitudinal magnetoresistance of WC measured at various temperatures for $B ∥ I ∥ x$ (inset). Adapted from [161] and [245].
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Figure 44
(a),(b) Schematic illustration of the topological invariants of mirror-symmetry-protected and $PT$ -symmetry-protected nodal lines in spinless systems, respectively. The blue and red curves in (a) indicate the crossing valence and conduction bands of opposing mirror eigenvalues. The red circles and $ϕ$ in (b) indicate the closed loop ( $L$ ) and the corresponding Berry phases. Adapted from [125].
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Figure 45
(a) 3D bulk BZ and the projected $(001)$ surface BZ of ${Cu}_{3} PdN$ . The orange curves and red points indicate the locations of nodal rings and nodal points without and with SOC included, respectively. (b) Calculated band structures of ${Cu}_{3} PdN$ along high-symmetry lines without SOC. (c) Calculated surface band structures and density of states for $(001)$ surface without SOC. Adapted from [587].
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Figure 46
(a) 3D bulk BZ and the projected $(001)$ surface BZ of ZrSiS. The red dashed lines and yellow planes indicate the momentum locations of nodal lines and nodal surfaces without SOC. The green lines represent the Dirac nodal lines in the presence of SOC. (b),(c) Calculated band structures of ZrSiS along high-symmetry lines without and with SOC, respectively. Differently colored lines in (b) represent different irreducible representations. (d) 3D intensity plot of the ARPES spectra measured with $h υ = 436 eV$ showing the nodal-line structure in the $k_{z} = 0$ plane. (e),(f) ARPES intensity plots showing the band dispersion along the $A − Z$ and $Z − R$ directions, respectively. Adapted from [411], and [133].
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Figure 47
(a) Calculated surface spectral weight of ZrSiS at $E_{F} - 100 meV$ illustrating the possible scattering vectors. (b),(c) Experimental and $T$ -matrix-calculated QPI patterns of ZrSiS at $E_{F} + 200 meV$ , respectively. (d) Schematic illustration of the bulk nodal lines (black curves) and surface floating band (red curves) of ZrSiS without SOC. (e) ARPES intensity plot of a bare ZrSiS surface showing the high-intensity surface states (SS and ${SS}^{'}$ ) and the bulk bands (BS). (f) Experimental single-defect-induced QPI patterns of ZrSiSe measured at $E_{F} + 400 meV$ . Adapted from [296], [467], and [643].
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Figure 48
(a) Polar plot of the angular magnetoresistance of ZrSiS measured at different magnetic fields. (b) Calculated FSs of ZrSiS in the 3D bulk BZ. (c) Fast Fourier transform of the magnetoresistance sweep at $T = 1.5 K$ . $α$ and $β$ represent the frequencies of the hole and electron pockets [indicated in (b)], respectively. Labels A and B correspond to the “inner” and “outer” breakdown orbits, respectively, as illustrated in the inset. (d) Field-dependent effective masses for the $α$ and $β$ pockets. The dashed and dotted lines are guides for the eye under the assumption that the zero-field mass is unrenormalized or renormalized, respectively. Adapted from [7], and [375].
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Figure 49
(a) Schematic illustrations of the electronic structure and the corresponding $σ_{1} (ω)$ for (a) energy-flat and (b) energy-dispersive nodal lines. $E$ and $k_{line}$ denote the direction of the electric field and the nodal line, respectively. $2 Δ$ in (b) denotes the gap induced by SOC, and $k_{m} a x υ_{∥}$ indicates the threshold of the $ω$ -linear region. (c) $σ_{1} (ω)$ of ZrSiS measured at various temperatures. (d) Real part of the optical conductivity of ${NbAs}_{2}$ for $E ∥ b$ . The blue dotted line is the fit to ${σ_{1}}^{b} (ω)$ at $10 K$ . Gray dashed and solid lines represent the contributions from the energy-dispersive and energy-flat nodal lines, respectively. Inset: ratio ${σ_{1}}^{b} (ω) / {σ_{1}}^{a} (ω)$ above the gap region. Adapted from [407], and [421].
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Figure 50
(a) 3D bulk BZ and the projected $(001)$ surface BZ of ${PbTaSe}_{2}$ . (b) Calculated band structures of ${PbTaSe}_{2}$ along the $A − H − L$ and $Γ − K − M$ lines with SOC. (c) ARPES intensity plot showing band dispersion along the high-symmetry path $\bar{M} − \bar{K} − \bar{Γ}$ . (d) Calculated surface bands of the $(001)$ surface with Pb termination along the same high-symmetry path. Adapted from [38].
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Figure 51
(a) Calculated band structures of paramagnetic ${IrF}_{4}$ along high-symmetry lines with SOC. The solid red lines and dotted green lines correspond to bands determined from density functional theory and a tight-binding model with chiral symmetry, respectively. (b) Nodal-chain structure in ${IrF}_{4}$ . The solid and dotted lines indicate nodal lines in the visible and hidden faces of the box, respectively. Different colors represent different orientations of the nodal loops. Adapted from [54].
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Figure 52
(a) Schematic plots of the characteristic electronic structures of type-I and type-II nodal lines. (b) Calculated electronic band structure of ${Mg}_{3} {Bi}_{2}$ along the $M − Γ − K$ direction without SOC. (c) 3D plot of the type-II nodal line in the bulk BZ of ${Mg}_{3} {Bi}_{2}$ . (d) ARPES intensity plot measured with $h υ = 74 eV$ , showing band dispersion in the $\bar{Γ} − \bar{K}$ direction. (e) Corresponding calculated band structure with SOC for a Bi-terminated $(001)$ surface. Adapted from [620], and [77].
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Figure 53
(a) Calculated electronic structures of ${Co}_{2} MnGa$ in the ferromagnetic state without SOC. The red arrow indicates the Weyl band crossing near the $K$ point. (b) 3D plot of the three types of Weyl nodal lines in the bulk BZ of ${Co}_{2} MnGa$ . The yellow and red nodal lines lie on the $M_{x}$ , $M_{y}$ , and $M_{z}$ mirror planes. The blue nodal lines are located on the $M_{x y}$ , $M_{\bar{x y}}$ , $M_{y z}$ , $M_{\bar{y z}}$ , $M_{x z}$ , and $M_{\bar{x z}}$ mirror planes. (c) ARPES constant-energy contour of ${Co}_{2} MnGa$ at $E_{B} = 0.01 eV$ measured at $ℏ υ = 50 eV$ and $T = 20 K$ . (d) Calculated constant-energy surface at $E_{B} = 0.08 eV$ for the $(001)$ MnGa-terminated surface, showing consistent multiple nodal-line features, as marked by letters a–d. (e) ARPES intensity plot in the $k_{a}$ direction [the white line in (d)] highlighting the bulk line nodes (yellow ovals) and the drumhead surface states (green ovals). Adapted from [32].
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Figure 54
(a) 3D bulk BZ with high-symmetry points indicated. $R$ and $E$ are the midpoints of the paths $A − L$ and $Γ − M$ , respectively. (b) Calculated electronic structure of ${Ta}_{3} {TeI}_{7}$ along the $E − R − H$ path in the presence of SOC. Adapted from [525].
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