Weyl and Dirac semimetals in three-dimensional solids

Weyl and Dirac semimetals are three-dimensional phases of matter with gapless electronic excitations that are protected by topology and symmetry. As three-dimensional analogs of graphene, they have generated much recent interest. Deep connections exist with particle physics models of relativistic chiral fermions, and, despite their gaplessness, to solid-state topological and Chern insulators. Their characteristic electronic properties lead to protected surface states and novel responses to applied electric and magnetic fields. The theoretical foundations of these phases, their proposed realizations in solid-state systems, and recent experiments on candidate materials as well as their relation to other states of matter are reviewed.

Figure 1

Net chirality of Weyl nodes must be zero, which is a consequence of the fact that the net Berry flux integrated over the Brillouin zone (a closed volume) must vanish.

Figure 2

(Left to right) Energy spectra of ${ϵ}_{s\mu }(0,p_y,p_z)$ for the Dirac semimetal ($m=b=b^{\prime }=0$), magnetic semiconductor ($m=1,b=0.5,b^{\prime }=0$), Weyl semimetal ($m=0.5,b=1,b^{\prime }=0$), and line node semimetal ($m=0,b=0,b^{\prime }=1$) for the Hamiltonian equation (8). From [204].

Figure 3

(Left) Band structure from Eq. (8) for values of $m/b$ for the $s=+$ and $\mu =\pm$ bands for increasing $m/b$ in the WSM phase. For finite $m$ the $\mu =+$ band is gapped, while $\mu =-$ contains two Weyl nodes. (Center) The $s=+$ and $\mu =-$ bands near the phase transition at $m/b=1$. (Right) Phase diagram of Eq. (8). At $m/b<1$, the system is a WSM, while $m/b>1$, a gapped semimetal exists. Along $b=0$, a degenerate massive DSM is observed. At $m=b=0$, massless degenerate Dirac fermions exist. From [378].

Figure 4

A heterostructure model of a Weyl semimetal of topological and normal insulators. Doped magnetic impurities are shown by arrows. $d$ is the real space periodicity of the lattice. ${\mathrm{\Delta }}_S$ and ${\mathrm{\Delta }}_D$ are tunneling between topological surface states on the same topological insulator layer and between different layers, respectively. From [53].

Figure 5

(Left) Conventional type I Weyl point with pointlike Fermi surface. (Right) Type II Weyl point is the touching point between electron and hole pockets. Red and blue (highlighted) isoenergy contours denote the Fermi surface coming from electron and hole pockets with chemical potential tuned to the touching point.

Figure 6

(Top left) Chern number, Weyl points, and surface Fermi arcs. (Top right) Connection of surface states to bulk Weyl points. (Bottom) Evolution of the Fermi arc with chemical potential in a particular microscopic model on raising the chemical potential from the nodal energy ($E=0$). Fermi arcs are tangent to the bulk Fermi surface projections and may persist even after they merge into a trivial bulk Fermi surface. From [24], [410], and [143].

Figure 7

(a) Opposite Weyl nodes in a T-breaking WSM in the absence of fields. (b) Spectrum in a magnetic field along the $z$ axis displaying Landau levels that disperse along the field. The zeroth Landau levels are chiral. In addition, an electric field along $z$ generates valley imbalance. (c) Anomalous Hall effect from chiral Fermi arc surface states whose magnitude is determined by the Weyl node separation in momentum space, when the chemical potential is at the Weyl nodes. (d) On moving the chemical potential away from the Weyl nodes, the anomalous Hall conductivity changes but only marginally in the model considered by [58] up to the chemical potential when the Fermi surfaces enclosing the two Weyl points merge.

Figure 8

Weyl semimetal slab in an applied magnetic field. (a) Weyl cyclotron orbits depicted in hybrid real space ($z$) and momentum space ($k_x$, $k_y$). Electrons slide along the surface Fermi arcs and are absorbed by the chiral bulk Landau level which propagates them to the opposite surface. From [315]. Numerically calculated (b) wave function of Weyl orbits, showing their hybrid surface-bulk character and (c) quantum oscillations in density of states from Weyl orbits. From [492].

Figure 9

Theoretical proposals of nonlocal transport in WSMs. Weyl cyclotron orbits lead to (a) a voltage difference on the lower pair of contacts when the current is injected between contacts on the top surface and (b) resonances in transmission of electromagnetic waves at frequencies controlled by the magnetic field. From [30]. (c) An alternate proposal for nonlocal transport utilizing the choral anomaly. A source-drain current $I_{\mathrm{sd}}$ is injected into a WSM slab of thickness $d$ via tunneling contacts of thickness $L_g$. In the presence of a local generation magnetic field $B_g$, a valley imbalance $\mathrm{\Delta }\mu$ is created via the chiral anomaly and diffuses a distance $L\gg d$ away. If a “detection” field $B_d$ is applied, the valley imbalance can be converted into a potential difference $V_{\mathrm{nl}}$ between top and bottom contacts of size $L_d$. From [305].

Figure 10

Development of a Dirac semimetal in an inverted band structure. The band inversion transition reverses the parities ($\pm$) of the $\mathbf{k}=0$ eigenstates in the (a) uninverted and (b) inverted level orderings. (b) The inverted bands are twofold degenerate and undergo an avoided crossing at $\mathbf{k}\ne 0$ which gaps the spectrum. (c) The mixing is forbidden along a symmetry line by the different rotational symmetries of the intersecting bands. This leaves two points each with a fourfold point degeneracy at $\mathbf{k}=\pm {\mathbf{k}}_D$ along the symmetry line that is lifted to linear order in $\mathbf{k}-{\mathbf{k}}_D$. Uninverting the bands produces a pairwise annihilation of the Dirac points and the system reverts to the conventional insulating state as shown in (a).

Figure 11

Formation of a Dirac semimetal by the band inversion mechanism in ${\mathrm{Na}}_3\mathrm{Bi}$. Band structure calculations (left) without and (right) with spin-orbit coupling both show a band inversion transition at the zone center illustrated by tracking the $\mathrm{Na}(3s)$ character of the eigenstates [bold (red) circles]. An expanded view of the dispersion along the $\mathrm{\Gamma }-A$ direction (inset) shows the symmetry-protected intersection of two twofold degenerate branches with distinct $J_z$ rotational eigenvalues at a Dirac point along the symmetry axis. Adapted from [421].

Figure 12

Possible linear dispersions of energy (vertical) vs momentum (horizontal) near a symmetry-enforced Dirac point at the zone boundary of a lattice with a nonsymmorphic space group. (a) The FDIR occurs at a TRIM and the fourfold degeneracy is lifted to form two twofold degenerate branches (bold). (b) Inversion symmetry is absent and four linearly dispersing branches merge at a FDIR. (c), (d) The degeneracy of the FDIR is lifted to form a twofold degenerate and two nondegenerate branches. (c) The FDIR does not occur at a TRIM while in (d) the twofold degenerate branch has zero slope indicating quadratic dispersion of the FDIR occurs at a TRIM. Adapted from [479].

Figure 13

Fermi arcs on the surface of DSMs. (a) A schematic of a DSM showing Dirac nodes along the $k_z$ axis in the bulk BZ and double Fermi arcs on the surface BZs. Note that surfaces perpendicular to the $z$ axis have no arcs. A 2D slice of the bulk BZ perpendicular to the $k_z$ axis is shown as a shaded (green) plane, which projects to the dashed (green) line on the surface BZ. (b)–(d) A symmetry-allowed mass term at the surface admits backscattering between these branches at the contact point which dissociates the surface band from the projected Dirac point. These surface branches can be deformed but not removed from the time reversal symmetric plane at $k_z=0$. If the chemical potential is not aligned with the bulk Dirac points the surface Fermi arcs disappear by merging with the bulk continuum (e), (f). Adapted from [188].

Figure 14

(Left) Calculated band gaps as a function of the average nuclear charge $⟨Z⟩$ for various half-Heusler phases. (Right) Schematic band ordering of normal and inverted bands in cubic semiconductors. From [271].

Figure 15

(a) Crystal structure (space group ${\mathrm{I}4}_{1}md$, No. 109) of the noncentrosymmetric lattice in the TaAs family of compounds. It is a body-centered tetragonal structure consisting of interpenetrating Ta and As sublattices, where the two are shifted by $(a/2,a/2,\sim c/12)$. For TaAs the lattice constants are $a=3.437\AA$ and $c=11.656\AA$. (b) The first Brillouin zone showing 12 pairs of Weyl points. The red and blue spheres (identified by arrows in black and white) represent the Weyl points with $C=\pm 1$ chirality. Note that this pattern of node chiralities may represent the situation more appropriately for the Nb compounds ([168]; [210]; [36]). See the discussion in the text. From [465].

Figure 16

Nodal structure of TaAs. Without SOC, the conduction and valence bands would intersect on the indicated four closed nodal lines on the $k_x=0$ (red) and $k_y=0$ (blue) mirror planes. SOC has the effect of gapping the bands on the mirror planes, but giving nodes slightly displaced from the planes. Black and white denote chiralities of the nodes. From [148].

Figure 17

Energy dispersions for $W1$ and $W2$. (a) For TaAs the Fermi surfaces enclose single chiral Weyl nodes and the Fermi Chern numbers (${\mathrm{C}}_{FS}$) are $\pm 1$. (b) For TaP, one Fermi surface is believed to enclose two $W1$ Weyl nodes with opposite chirality and hence the ${\mathrm{C}}_{FS}$ is zero. Note that the labeling of $W1$ and $W2$ has been changed with respect to how it appeared originally in the literature so as to make it consistent with the convention used elsewhere in the literature and in the text. From [454].

Figure 18

The electronic phase diagram of the ZrBeSi family showing the calculated density of states at $E_F$ as a function of the total $Z$ (atomic number) divided by the electronegativity difference between the large cation (e.g., ${\mathrm{Ba}}^{+2}$) and the average of the anionic honeycomb sublattice (e.g., ${\mathrm{Ag}}^{+1}{\mathrm{Bi}}^{-3}$). The squares (orange) represent compounds having a calculated Dirac cone and circles (red) represent compounds with none. Calculations were performed with a Perdew-Burke-Ernzerhof parametrization of the generalized gradient approximation functional. From [126].

Figure 19

Band structures of (a) ${\mathrm{AsO}}_2$, (b) ${\mathrm{SbO}}_2$, (c) ${\mathrm{BiO}}_2$ in the $\beta$-crystobalite structure, and (d) the Hamiltonian of Eq. (26) for spin-orbit coupled $s$ states on the diamond lattice. All the spectra feature a symmetry-enforced Dirac point in an FDIR at the zone boundary $X$ point. From [479].

Figure 20

(a) Fermi surface map of the ${\mathrm{Na}}_3\mathrm{Bi}$ sample on (100) surface at photon energy 55 eV. BDP1 and BDP2 denote the two bulk Dirac points. (b) Constant energy ARPES spectra as a function of binding energy. (c) ARPES dispersion cuts $\alpha$, $\beta$, and $\gamma$ as defined in (b). (d) Schematic Fermi surface of ${\mathrm{Na}}_3\mathrm{Bi}$. The shaded (red) areas and the gray (orange) lines represent the bulk and surface states, respectively. (e) Calculated band structure along cuts $\alpha$ ($\gamma$) and $\beta$. Adapted from [459].

Figure 21

ARPES spectra near the $\mathrm{\Gamma }$ point on the (001) surface. (e) Fermi surface mapping. (f) Photoemission spectra measured along the seven momentum cuts labeled as red lines 1 to 7 in (e). The gray (red) dashed lines are curves fitted to the Dirac dispersion expectation. (g) 3D view of the evolution of Fermi surface and constant energy contours at different binding energies. The gray (red) lines are guides to the eye. Displayed images are second derivatives of the original data with respect to energy. Adapted from [222].

Figure 22

(Top left) Fermi surface of TaAs ARPES, at incident photon energy $h\nu =90\mathrm{eV}$, on the (001) surface of TaAs. (Top right) Again FS of TaAs but with Weyl points indicated and BZ marked. Two paths in momentum space are denoted as $\mathcal{C}$ and $\mathcal{P}$. (Bottom right) Measured ARPES spectral function along $\mathcal{C}$, with chiralities of edge modes marked by the arrows. The net Chern number appears to be $+2$, inconsistent with the expectation (two $W2$ nodes project to the surface). This can be explained by considering the small separation between the $W1$ Weyl points. (Bottom left) Measured ARPES spectral function along $\mathcal{P}$. The path encloses only the well-spaced Weyl points and one finds a Chern number $+2$, consistent with expectation. From [36].

Figure 23

(a) Schematic showing the projection of a pair of Weyl points on the (001) surface BZ and the Fermi arc (gray curves) connecting them for materials with increasing SOC strength. (ii)–(iv) A comparison of the calculated (left) and ARPES measurement (right) of the spoonlike FSs. The red and blue dots denote the chirality of Weyl points. (b)–(d) ARPES measurements of the (i) spoonlike FS and (ii), (iii) band dispersions for NbP, TaP, and TaAs, respectively. The positions of the band dispersions presented in (ii) and (iii) are indicated by the dotted red lines in (i). (e) Summary of the extracted $\mathrm{\Delta }K1$ and $\mathrm{\Delta }K2$ [from (b)–(d)] from the three compounds, plotted against a rough measure of the strength of SOC. $\mathrm{\Delta }K1$ and $\mathrm{\Delta }K2$ represent the separation between the Weyl points and Fermi arcs, respectively. From [231].

Figure 24

(a) Spin-integrated FS map near $\mathrm{\Gamma }-Y$ recorded with a spin-resolved ARPES system for TaAs. The gray (red) arrows indicate the direction of measured in-plane spin polarizations of the Fermi arc b2 at C4. (b) Corresponding theoretical spin texture of surface states. Dashed circles (red and yellow) indicate the Weyl nodes $W1$ inferred to have negative and positive chiralities, respectively. From [243].

Figure 25

(a) (Left) Crystal structure of NbP that has two different surfaces. (Right) Schematics of the experimental ARPES momentum space mapping near the $X$ point for the Nb- and P-terminated surfaces. The bulk Weyl nodes ($W1$ and $W2$) projections are illustrated by circles. (b) A comparison of experimental FS between Nb-terminated (central horizontal loop, blue) and P-terminated (central vertical loop, red) surfaces. Projection of Weyl nodes $W2$ at the intersection of FSs for opposite surfaces is shown by filled circles, whereas other intersections are shown by open circles. Weyl nodes $W1$ are indicated with diamonds. Fermi surfaces $S2$, $S8$, and $S9$ are believed to be Fermi arcs, whereas $S1$, $S3$, $S4$, $S6$, and $S7$ are from trivial Fermi surfaces. From [361].

Figure 26

Conductivity calculated within the linearized Boltzmann formalism of a single Weyl node with disorder, in units of $e^2{v}_F^2/h\gamma$ as a function of frequency at different temperatures. Frequencies and temperatures are in units of ${\omega }_0=2\pi v_F^3/\gamma$. One can see that the functional dependence changes at $\omega \sim T$ (given by the thin black line). Note that at the highest temperature plotted deviations from the temperature-independent dc result are found. From [161].

Figure 27

(Top) The optical conductivity of ${\mathrm{ZrTe}}_5$ at 8 K at frequencies below $1200{\mathrm{cm}}^{-1}$. The dotted (red) line is the linear fitting of ${\sigma }_1(\omega )$. From [81]. (Bottom) Optical conductivity for TaAs at 5 K. The steep (blue) and shallow (black) solid lines through the data are linear guides to the eye. The steep (blue) line shows the Weyl part of the spectrum, while the shallow (black) line comes from higher energy non-Weyl states. The inset shows the spectral weight as a function of frequency at 5 K (solid red curve), which follows an ${\omega }^2$ behavior (dashed blue line). From [452].

Figure 28

Interband optical conductivity from the Weyl semimetal to gapped semimetal phase transition. Here $m$ is a Dirac mass parameter and $b$ is an intrinsic Zeeman-field-like parameter. The Hamiltonian is the same as appears in Eq. (8). From [378].

Figure 29

The quantum oscillation phase shift $\beta$ (labeled $\varphi$ and given in units of $1/8$) vs $E_F$ for different relative strengths of linear ($E_A$) and quadratic ($E_M=0.05\mathrm{eV}$) terms in the energy spectra. The curves break because $\beta$ cannot be fit in the parameter range where beats form. The insets indicate the location of Fermi energy with respect to the model band structure. The vertical dashed line marks the Lifshitz transition where the system goes from two Weyl pockets to a single larger one. From [412].

Figure 30

(Top) The oscillatory component of the resistance $R_{xx}$ of ${\mathrm{Cd}}_3{\mathrm{As}}_2$ as a function of $1/B$ extracted from $R_{xx}$. A smooth background has been subtracted. (Bottom) Landau index $n$ plotted against $1/B$. The closed circles denote the integer index ($R_{xx}$ valley), and the open circles indicate the half integer index ($R_{xx}$ peak) for two different samples. The index plot can be linearly fitted for both samples measured, giving intercepts of 0.56 and 0.58. Adapted from [150].

Figure 31

(Top) Inferred longitudinal MR for TaAs samples at 2 K for $E$ and $B$ in the $a$ crystallographic direction. The short (green) curve is a fit to the longitudinal MR data in the semiclassical regime based on a chiral anomaly model. (Bottom left) Chemical potential dependence of the chiral coefficient $C_W$. (Bottom right) Angular dependence of the chiral coefficient $C_W$. Adapted from [487].

Figure 32

(Left) Inferred magnetoresistance of ${\mathrm{Na}}_3\mathrm{Bi}$ when $\mathbf{B}$ lies in the $x\text{−}z$ plane at an angle $\theta$ with respect to $\mathbf{E}$ that points in the $x$ direction. Plotted as a function of the field for different angles. (Right) Magnetoconductance plotted as a function of the angle for different fields. Adapted from [451].

Figure 33

Simulated potential distribution for different conductivity anisotropies $A={\sigma }_{zz}/{\sigma }_{xx}$ and a geometry of $0.4\times 0.3\times 2.0{\mathrm{mm}}^3$ ($w\times t\times l$). The lines are contour lines of the equipotentials. The increased MR anisotropy strongly distorts the equipotential lines. From [102].

Figure 34

Demonstration of the current jetting effect in the non-WSM or DSM system ${\mathrm{TaAs}}_2$. (a) Measured apparent longitudinal MR for when the contacts are not fully crossing the sample. (b) The same sample but for contacts placed such that fully cross the sample. From [483].

Figure 35

Angular dependence of magnetoresistance of GdPtBi and YPtBi. Adapted from [348].

Figure 36

(a) Scanning electron microscope image of triangular and rectangular devices used to observe hybrid surface-bulk quantum oscillations. The rectangular sample is $0.8\mu \mathrm{m}$ wide, $3.2\mu \mathrm{m}$ tall, and $5\mu \mathrm{m}$ long. The other device features an equilateral triangular cross section with a base of $a=2.7\mu \mathrm{m}$. Both devices have a similar cross-sectional area and circumference of the cross section. The crystallographic direction perpendicular to the surface of the rectangular device is [102] and [010] parallel to the surface. (b) Sketch of the hybrid surface-bulk Weyl orbits for rectangular and triangular cross sections. (c) Frequency spectrum of the triangular and rectangular samples for field orientations perpendicular to each of the surfaces (0° for the rectangle and 60° for the triangle). From [262].

Figure 37

(Left) ARPES measured dispersion curves along the $k_x$ direction measured at photon energies of 7, 8, 9, and 10 eV. (The corresponding momentum cuts are indicated by the colored lines on the right figure.) The dispersion curve obtained by the band calculations is superimposed (gray curve). The data close to $E_F$ are fitted by a parabolic function shown as the dotted (light blue) curve. The estimated effective mass at the $\mathrm{\Gamma }$ point is 6.3 free electron masses. (Right) The calculated band dispersion in the $k_x-k_{111}$ sheet. The ARPES data from the left panel are overlayed. From [203].

Figure 38

(a) Gyroids can be fabricated by drilling holes along the $x$, $y$, and $z$ directions. Shown is a bcc unit cell in which a single gyroid structure can be approximated by drilling holes. (b) The double-gyroid structure is made by stacking layers along the [101] direction. The structure is made with two inversion counterparts interpenetrating each other. Inversion symmetry is broken by reducing the vertical connections to the thin cylinders for the first and third (red) gyroids. (c) Shown on the left is the assembled structure. A magnified view from top is shown on the right with a centimeter ruler in the background. From [237].