New type of Weyl semimetal with quadratic double Weyl fermions

Significance

We predict a new Weyl semimetal candidate. This is critically needed for this rapidly developing field as TaAs is the only known Weyl semimetal in nature. We show that SrSi₂ has many new and novel properties not possible in TaAs. Our prediction provides a new route to studying the elusive Weyl fermion particles originally considered in high-energy physics by tabletop experiments.

Abstract

Weyl semimetals have attracted worldwide attention due to their wide range of exotic properties predicted in theories. The experimental realization had remained elusive for a long time despite much effort. Very recently, the first Weyl semimetal has been discovered in an inversion-breaking, stoichiometric solid TaAs. So far, the TaAs class remains the only Weyl semimetal available in real materials. To facilitate the transition of Weyl semimetals from the realm of purely theoretical interest to the realm of experimental studies and device applications, it is of crucial importance to identify other robust candidates that are experimentally feasible to be realized. In this paper, we propose such a Weyl semimetal candidate in an inversion-breaking, stoichiometric compound strontium silicide, SrSi₂, with many new and novel properties that are distinct from TaAs. We show that SrSi₂ is a Weyl semimetal even without spin–orbit coupling and that, after the inclusion of spin–orbit coupling, two Weyl fermions stick together forming an exotic double Weyl fermion with quadratic dispersions and a higher chiral charge of ±2. Moreover, we find that the Weyl nodes with opposite charges are located at different energies due to the absence of mirror symmetry in SrSi₂, paving the way for the realization of the chiral magnetic effect. Our systematic results not only identify a much-needed robust Weyl semimetal candidate but also open the door to new topological Weyl physics that is not possible in TaAs.

Analogous to graphene and the 3D topological insulator, Weyl semimetals are believed to open the next era in condensed matter physics (1–8). A Weyl semimetal represents an elegant example of the correspondence between condensed matter and high-energy physics because its low-energy excitations, the Weyl fermions, are massless particles that have played an important role in quantum field theory and the standard model but have not been observed as a fundamental particle in nature. A Weyl semimetal is also a topologically nontrivial metallic phase of matter extending the classification of topological phases beyond insulators (3–6). The nontrivial topological nature guarantees the existence of exotic Fermi arc electron states on the surface of a Weyl semimetal. In contrast with a topological insulator where the bulk is gapped and only the Dirac cones on its surfaces are of interest, in a Weyl semimetal, both the Weyl fermions in the bulk and the Fermi arcs on the surface are fundamentally new and are expected to give rise to a wide range of exotic phenomena (9–22).

For many years, research on Weyl semimetals has been held back due to the lack of experimentally feasible candidate materials. Early theoretical proposals require either magnetic ordering in sufficiently large domains (3, 23–26) or fine-tuning of the chemical composition to within 5% in an alloy (23, 25–27), which proved demanding in real experiments. Recently, our group and a concurrent group successfully proposed the first, to our knowledge, experimentally feasible Weyl semimetal candidate in TaAs material class (28, 29). The key is that TaAs is an inversion symmetry-breaking, stoichiometric, single-crystalline material, which does not depend on any magnetic ordering or fine-tuning. Shortly after the prediction, the first, to our knowledge, Weyl semimetal state was experimentally discovered in TaAs via photoemission spectroscopy (30, 31). Later, other photoemission works confirmed the discovery in several members of the TaAs material class (32–35).

In this paper, we propose a new type of Weyl semimetal in an inversion-breaking, stoichiometric compound strontium silicide, SrSi₂. Our first-principles band structure calculations show that SrSi₂ is already a Weyl semimetal even in the absence of spin–orbit coupling. After including spin–orbit coupling, two linearly dispersive Weyl fermions with the same chiral charge are bounded together, forming a quadratically dispersive Weyl fermion. We find that such a quadratically dispersive Weyl fermion exhibits a chiral charge of 2 (compared with 1 in the TaAs family). Furthermore, because SrSi₂ lacks both mirror and inversion symmetries, the Weyl nodes with opposite charges are located at different energies. This property may facilitate the realization of the chiral magnetic effect (10–14). This effect has attracted much theoretical interest, because it seems to violate basic results of band theory by suggesting that a Weyl semimetal can support dissipationless currents in equilibrium. Recent theoretical works have clarified the apparent contradiction (see ref. 15 and references therein and, e.g., ref. 16). Our prediction of the Weyl semimetal state in SrSi₂ is of importance because it identifies a much-needed robust Weyl semimetal candidate, paves the way for realizing quadratically dispersive Weyl fermions with higher chiral charges, and allows one to test the theoretical debates of the chiral magnetic effect by direct experimental measurement of a real material.

Results

We computed electronic structures using the norm-conserving pseudopotentials as implemented in the OpenMX package within the generalized gradient approximation schemes (36). Experimental lattice constants were used (37). A $12\times 12\times 12$ Monkhorst–Pack k-point mesh was used in the computations. The spin–orbit effects were included self-consistently. For each Sr atom, three, two, two, and two optimized radial functions were allocated for the s, p, d, and f orbitals ($s3p2d2f2$), respectively, with a cutoff radius of 10 Bohr. For each Si atom, $d2p2d1$ was adopted with a cutoff radius of 7 Bohr. A regular mesh of 600 Ry in real space was used for the numerical integrations and for the solution of the Poisson equation. To calculate the surface electronic structures, we constructed a first-principles tight-binding model Hamiltonian. The tight-binding model matrix elements are calculated by projecting onto the Wannier orbitals (38). We use Sr s and d and Si s and p orbitals without performing the procedure for maximizing localization.

SrSi₂ crystallizes in a simple cubic lattice system (37, 39, 40). The lattice constant is $a=6.563$ $\AA$ and the space group is $P{4}_{3}32$ (#212). As seen in Fig. 1A, the crystal lacks inversion and mirror symmetries. The bulk and (001) surface high-symmetry points are noted in Fig. 1B, where the centers of the square faces are the $X,Y,Z$ points ($X,Y,Z$ are equivalent), the centers of the edges are the M points, and the corners of the cube are the R points.

Fig. 1.

Crystal and electronic structure of SrSi₂. (A) Crystal structure of SrSi₂. (B) Bulk BZ of SrSi₂ with the high-symmetry points noted. (C) First-principles band structure calculation without spin–orbit coupling. (D) Zoomed-in version of the band structure near the spinless Weyl nodes W1 and W2. (E and F) The same as C and D but with spin–orbit coupling. Upon the inclusion of spin–orbit coupling, bands with the same rotation eigenvalues are gapped out whereas those with different rotation eigenvalues remain gapless, forming Weyl nodes W1′ and W2′. Δ_i notes the representations of bands along the $\mathrm{\Gamma }-X$ direction.

We understand the electronic properties of SrSi₂ at a qualitative level based on the ionic model. The electronic configuration of Sr is $4s^2$ whereas the electronic configuration of Si is $3s^{2}3p^2$. Each Sr atom has a strong tendency to give out two electrons to achieve a full shell configuration in an ionic compound, leading to an ionic state of Sr⁺². This means that, in SrSi₂, Si has an ionic state of Si⁻¹, which is different from the most common ionic state of Si, Si⁺⁴, as in SiO₂. This situation resembles another well-known semimetal, Na₃Bi. In both compounds, an element that usually forms a positive ionic state (such as Si⁺⁴ or Bi⁺³) in an ionic compound is forced to form a negative ionic state (such as Si⁻¹ or Bi⁻³). However, we emphasize that a key difference between SrSi₂ and Na₃Bi is that the SrSi₂ crystal breaks space-inversion symmetry. Based on the above picture, we expect that the valence electronic states mainly arise from the $3p$ orbitals of Si. Indeed, this is confirmed by our first-principles calculation results. Fig. 1C shows the calculated bulk band structure along high-symmetry directions in the absence of spin–orbit coupling. We observe a clear crossing between the bulk conduction and valence bands along the $\mathrm{\Gamma }-X$ direction, which agrees with our expectation of SrSi₂ being a semimetal. Interestingly, we note that the band crossing does not enclose any high-symmetry or time-reversal invariant Kramers’ points. In the vicinity of the crossings, the bands are found to disperse linearly along the $\mathrm{\Gamma }-X$ direction as shown in Fig. 1D. The two crossings along $\mathrm{\Gamma }-X$ without spin–orbit coupling are denoted as W1 and W2. In the presence of spin–orbit coupling, bands with the same rotation eigenvalues are gapped out whereas those with different rotation eigenvalues remain gapless as shown in Fig. 1 E and F. We denote the band crossings after the inclusion of spin–orbit coupling as W$1\prime$ and W$2\prime$.

We systematically study the nature of the band crossings in Fig. 2. In the absence of spin–orbit coupling, our calculation shows that the bands disperse linearly along all three directions in the vicinity of each W1 or W2 (Fig. 2D). We have calculated the chiral charge associated with W1 and W2 by calculating the Berry flux through a closed surface that encloses a node. Our calculation shows that both W1 and W2 carry a nonzero chiral charge and that they have opposite charges. These calculations prove that they are Weyl nodes. Therefore, SrSi₂ is already a Weyl semimetal even without spin–orbit coupling. We have calculated the band structure throughout the first Brillouin zone (BZ). As shown in Fig. 2A, in total there are 60 Weyl nodes within the first BZ without spin–orbit coupling. These 60 Weyl nodes can be categorized into four groups. W1 and W2 are the Weyl nodes located on the $\mathrm{\Gamma }$–$X\left(Y,Z\right)$ axes. W3 and W4 are located on the $\mathrm{\Gamma }$–X–Y–M and $\mathrm{\Gamma }$–Z–R–M planes (Fig. 2B).

Fig. 2.

Bulk Weyl nodes in SrSi₂. (A) Locations of the Weyl nodes within the first BZ without spin–orbit coupling. Colors of circles represent the signs of the chiral charge. (B and C) In the absence of spin–orbit coupling, the Weyl nodes can be categorized into four groups. W1 and W2 are located in the $\mathrm{\Gamma }-X\left(Y,Z\right)$ direction. W3 and W4 are located on the $\mathrm{\Gamma }-X-Y$ and the $\mathrm{\Gamma }-Z-M$ planes. Please refer to Fig. 1B for the definition of these planes. In the presence of spin–orbit coupling, each W3 or W4 splits into two linearly dispersive (charge 1) Weyl nodes with the same chiral charge. Each W1or W2 becomes a quadratically dispersive (charge 2) Weyl node. We denote the Weyl nodes with extra primes (e.g., W1 becomes W$1\prime$) after including spin–orbit coupling. (D and E) Energy dispersions of a W1(W$1\prime$) Weyl cone with and without spin–orbit coupling. (F) Schematic illustration of the evolution of Weyl cones upon the inclusion of spin–orbit coupling.

We now consider the band structure with spin–orbit coupling. We note that each Weyl node without spin–orbit coupling should be considered as two degenerate Weyl nodes with the same chiral charge but with the opposite physical spins. In general, spin–orbit coupling is expected to lift the degeneracy of the physical spin due to the lack of inversion symmetry in SrSi₂. Each of the W3 (respectively, W4) (Fig. 2C) nodes splits into two Weyl nodes with the same chiral charge, W$3\prime$ and W$3″$ (W$4\prime$ and W$4″$), which are separated in momentum space. The nodes W1 and W2, in contrast, do not split, but their dispersion changes from linear to quadratic. Specifically, a W$1\prime$ cone on the $k_x$ axis disperses quadratically along $k_y$ and $k_z$ directions whereas its dispersion along $k_x$ remains linear (Fig. 2E). We have calculated the chiral charge associated with the quadratic Weyl nodes and our results reveal a chiral charge of $\pm 2$. Usually, the Weyl cones in a Weyl semimetal disperse linearly and have chiral charges of $\pm 1$. The existence of quadratically dispersive Weyl fermions is interesting in itself. Furthermore, recent theoretical studies have suggested that a quadratic band touching may lead to interesting non-Fermi liquid interaction effects because the long-range tail of the Coulomb repulsion is not screened (41–43).

We can understand how spin–orbit coupling turns the doubly degenerate linear W1 node into a quadratic node by examining the action of $C_4$ rotation symmetry on an effective $\mathit{k}\cdot \mathit{p}$ Hamiltonian. We note that all of the quadratically dispersive Weyl nodes are located on the $k_x$, $k_y$, and $k_z$ axes which are the $C_4$ rotational axes of the crystal, suggesting that they are protected by the $C_4$ symmetry. Using the Pauli matrices ${\sigma }_1,\hspace{0.17em}{\sigma }_2,\hspace{0.17em}{\sigma }_3$ in spin space, $C_4$ rotations around the z axis are represented by $e^{-\text{i}\pi {\sigma }_3/4}$. We choose the same representation $e^{-\text{i}\pi {\tau }_3/4}$ acting on the orbital space Pauli matrices ${\tau }_1,\hspace{0.17em}{\tau }_2,\hspace{0.17em}{\tau }_3$, that is, $R_{C_4}=e^{-\text{i}\pi {\tau }_3/4}\otimes e^{-\text{i}\pi {\sigma }_3/4}$. The doubly degenerate Weyl Hamiltonian that is $C_4$ invariant, i.e., that obeys $R_{C_4}H\left(k_x,k_y,k_z\right)R_{C_4}^{-1}=H\left(k_y,-k_x,k_z\right)$, in the absence of spin–orbit coupling is then given by

$$H=v{k}_x{\tau }_x+v{k}_y{\tau }_y+v\prime k_z{\tau }_z.$$ [1]

With spin–orbit coupling, we can add three $\mathit{k}$-independent terms to the Hamiltonian that commute with $R_{C_4}$: (i) ${\tau }_3\otimes {\sigma }_3$, (ii) ${\tau }_1\otimes {\sigma }_1+{\tau }_2\otimes {\sigma }_2$, and (iii) ${\tau }_2\otimes {\sigma }_1-{\tau }_1\otimes {\sigma }_2$. The first term separates the Weyl nodes along $k_z$, whereas either of the latter two generates the quadratic touching seen in our first-principles calculations. We conclude that the latter terms dominate in SrSi₂. More precisely, if we add $\mathrm{\Delta }$ times the term (ii) to H, the two bands that touch are given by

$${\epsilon }_{k,k_z}=\pm \sqrt{v^2{k}^2+v^{\prime 2}{k}_z^2+2\mathrm{\Delta }\left(\mathrm{\Delta }-\sqrt{v^2{k}^2+{\mathrm{\Delta }}^2}\right)},$$ [2]

with $k=\sqrt{k_x^2+k_y^2}$. They disperse quadratically like $v^2{k}^2/2\mathrm{\Delta }$ for $k_z=0$ and linearly like $v\prime k_z$ for $k=0$.

To test the symmetry protection of the quadratic touching, we apply a uniaxial pressure along the $\widehat{z}$ direction (Fig. 3A), which breaks the $C_4$ symmetries along the $k_x$ and $k_y$ directions but preserves the $C_4$ symmetry along the $k_z$ direction. Under such pressure, our calculations (Fig. 3B) show that each quadratically dispersive Weyl node located on the $k_x$ ($k_y$) axis splits into two linearly dispersive Weyl nodes away from the axis. By contrast, the nodes on $k_z$ remain intact. Previously, quadratically dispersive Weyl fermions were predicted in the Weyl semimetal candidate HgCr₂Se₄ (24, 44). However, the experimental realization of the Weyl semimetal state in HgCr₂Se₄ has been proven to be difficult, because there is no preferred magnetization axis in its cubic structure, which likely leads to the formation of many small ferromagnetic domains.

Fig. 3.

Crystalline symmetry-protected quadratically dispersive Weyl nodes. (A) Uniaxial pressure along the $\widehat{z}$ direction breaks the $C_4$ rotational symmetries along the $k_x$- and $k_y$ axes. (B) Under this pressure, each quadratically dispersive Weyl node with a chiral charge of ±2 on the $k_x$ or $k_y$ axis splits into two linearly dispersive Weyl nodes with a chiral charge of ±1. (C and D) Schematic illustration of the evolution of a quadratically dispersive Weyl node on the $k_x$ axis under crystalline symmetry breaking.

Besides the quadratically dispersive Weyl fermions, our calculations show another interesting property of SrSi₂. The Weyl nodes with opposite charges are located at different energies. We note that among all Weyl semimetal candidate materials, SrSi₂ is the first one, to our knowledge, to have this property. It arises from the lack of any mirror symmetry in the SrSi₂ crystal, because a mirror symmetry operation would reflect a Weyl node on one side of the mirror plane to a Weyl node with the opposite chiral charge at the same energy. Such a property is interesting because it enables the chiral magnetic effect (11–14). We propose that SrSi₂ provides an ideal material platform that allows one to test the chiral magnetic effect.

A key signature of a Weyl semimetal is the presence of Fermi arc surface states. We present calculations of the (001) surface states in Fig. 4. The projected Weyl nodes are denoted as black and white circles in Fig. 4A. The bigger circles correspond to the quadratic Weyl nodes with chiral charges of $\pm 2$ whereas the smaller ones correspond to the linear Weyl nodes with chiral charges of $\pm 1$. Our calculated surface Fermi surface shows the existence of Fermi arcs that terminate at the projections of the Weyl nodes. For example, the k-space region highlighted by the blue dotted box reveals two Fermi arcs terminating at the projection of a quadratically dispersive Weyl node (the black circle). This is consistent with its chiral charge of 2. The two Fermi arcs are close to each other in k space because the spin–orbit coupling is fairly weak in SrSi₂. We stress that because the W1 nodes are far separated in momentum space from any other Weyl node, topologically protected Fermi arcs stretch across a substantial portion of the surface BZ. This has to be contrasted with TaAs, where Weyl nodes of opposite chiral charge are rather close to one another.

Fig. 4.

Fermi arc surface states surface states on the (001) surface of SrSi₂. (A) Calculated surface-state Fermi surface of the (001) surface of SrSi₂. The sharp red lines show the surface states, the shaded areas show the surface projections of bulk bands, and the black and white circles show the projected Weyl nodes. The larger circles are the quadratically dispersive Weyl nodes with a chiral charge of $\pm 2$, whereas the smaller ones are the linearly dispersive Weyl nodes with a chiral charge of $\pm 1$. Cut 1 and cut 2 are two line-cuts through the surface BZ along the $k_x$ direction. They correspond to the projection of two 2D $k_x,k_z$ slices in the bulk BZ. (B) E-k dispersion along cut 1, noted by the dotted line in A. The red lines represent surface states from the top surface whereas the blue ones are from the bottom surface. The shaded areas are the projection of the bulk bands.

We now study the energy dispersion of the surface states. The dispersion along cut 1 (the black dotted line in Fig. 4A) is shown in Fig. 4B. On each surface, there are four copropagating and two counterpropagating surface states. Therefore, the Chern number associated with the 2D k slice (cut 1) is 2. The Chern number associated with cut 2 (the green dotted line in Fig. 4A) must be 0 because it goes through the Kramers points $\overline{Y}$ $\left(0,\pi \right)$ and $\overline{M}$ $\left(\pi ,\pi \right)$. This result agrees with the total chiral charge, 2, which is enclosed by the surface formed by cut 1 and cut 2 in the bulk. (It encloses eight W4 nodes, eight W3 nodes, and one W2 node with the chiral charge +1, −1, and +2, respectively.)

In summary, we have presented first-principles band structure calculations that predict SrSi₂ as a Weyl semimetal candidate. Similar to TaAs, SrSi₂ is an inversion-breaking, stoichiometric single crystal, which highlights its experimental feasibility. Our prediction here is reliable and robust, to the same extent as our previous predictions of the topological insulator state in Bi₂Se₃ (45) and topological Weyl semimetal states TaAs (28), both of which have now been experimentally realized (30, 45). We note that the Weyl semimetal state can be further stabilized by applying external pressure to compress the lattice. Our results identify a much-needed robust Weyl semimetal candidate. The predicted Weyl semimetal state in SrSi₂ also offers many distinctive properties not present in the TaAs family of materials.

Band-Gap Correction in SrSi₂

There have been a few optical and transport measurements on SrSi₂. These measurements did not reach a consensus about its ground state. Whereas ref. 46 reported an $\sim 30$-meV band gap, more recent papers like ref. 47 reached the opposite conclusion, i.e., a gapless semimetal.

This is not surprising at all as transport and optical experiments are not precise in determining the band gap, especially for narrow band-gap semiconductors (gap $<0.5$ eV) and semimetals (gap $=0$). The imprecise nature of transport and optical measurements on this issue is quite intuitive. Transport and optical measurements do not have momentum(k) resolution. For the Dirac/Weyl semimetals considered here, the energy gap only vanishes at a few discrete k points. In other words, band structure at “99.999%” of the bulk BZ is fully gapped. Also, the density of states vanishes at the Fermi level. Therefore, it is extremely difficult for transport and optical measurements to correctly reveal the gapless nature of Dirac/Weyl semimetal materials. There have been multiple examples that a compound is now fully established as a gapless semimetal using angle-resolved photoemission spectroscopy and other modern techniques but was wrongly reported as a gapped semiconductor with a narrow energy gap.

For example, Cd₃As₂ has now been confirmed as a gapless Dirac semimetal (48–52). However, “ancient” transport and optical papers in 1960–1980 reported a finite band gap of $\simeq 0.2-0.4$ eV* (53–56). Similarly, Na₃Bi has now been confirmed as a gapless Dirac semimetal (57–60). However, ancient transport and optical papers also reported a finite band gap (61–63).

We check the effects of band-gap corrections (64, 65) on our first-principles calculations on SrSi₂. Fig. S1A shows the calculation in the absence of band-gap correction, which is obtained from the generalized gradient approximation (GGA) and is the same as what we show in the main text. Fig. S1B shows the calculation using the Heyd–Scuseria–Ernzerhof (HSE) exchange-correlation functional (66), which improves the band-gap prediction in GGA. However, it can be seen that the semimetal ground state remains robust. We note that the Weyl semimetal state can be further stabilized by applying external pressure to compress the lattice. This further confirms that our prediction here is reliable and robust to the same extent as our predictions of the topological insulator and topological Weyl semimetal states in Bi₂Se₃ (45) and TaAs (28), both of which have now been experimentally realized (30, 45).

Fig. S1.

Band structure of SrSi₂ without SOC before and after band-gap corrections. (A) Band structure obtained from GGA shows a gapless semimetallic ground state. (B) Band structure obtained from HSE still shows the gapless semimetallic ground state.