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# New type of Weyl semimetal with quadratic double Weyl fermions

Edited by Philip Kim, Harvard University, Cambridge, MA, and accepted by the Editorial Board December 6, 2015 (received for review July 23, 2015)

## 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 SrSi_{2} 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, SrSi_{2}, with many new and novel properties that are distinct from TaAs. We show that SrSi_{2} 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 SrSi_{2}, 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, SrSi_{2}. Our first-principles band structure calculations show that SrSi_{2} 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 SrSi_{2} 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 SrSi_{2} 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 *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 (*s* and *d* and Si *s* and *p* orbitals without performing the procedure for maximizing localization.

SrSi_{2} crystallizes in a simple cubic lattice system (37, 39, 40). The lattice constant is *A*, the crystal lacks inversion and mirror symmetries. The bulk and (001) surface high-symmetry points are noted in Fig. 1*B*, where the centers of the square faces are the *M* points, and the corners of the cube are the *R* points.

We understand the electronic properties of SrSi_{2} at a qualitative level based on the ionic model. The electronic configuration of Sr is ^{+2}. This means that, in SrSi_{2}, Si has an ionic state of Si^{−1}, which is different from the most common ionic state of Si, Si^{+4}, as in SiO_{2}. This situation resembles another well-known semimetal, Na_{3}Bi. 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 SrSi_{2} and Na_{3}Bi is that the SrSi_{2} crystal breaks space-inversion symmetry. Based on the above picture, we expect that the valence electronic states mainly arise from the *C* 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 _{2} 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 *D*. The two crossings along *E* and *F*. We denote the band crossings after the inclusion of spin–orbit coupling as W

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. 2*D*). 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, SrSi_{2} 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. 2*A*, 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 *X*–*Y*–*M* and *Z*–*R*–*M* planes (Fig. 2*B*).

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 SrSi_{2}. Each of the W3 (respectively, W4) (Fig. 2*C*) nodes splits into two Weyl nodes with the same chiral charge, W*E*). We have calculated the chiral charge associated with the quadratic Weyl nodes and our results reveal a chiral charge of

We can understand how spin–orbit coupling turns the doubly degenerate linear W1 node into a quadratic node by examining the action of *z* axis are represented by *i*) *ii*) *iii*) _{2}. More precisely, if we add *ii*) to *H*, the two bands that touch are given by

To test the symmetry protection of the quadratic touching, we apply a uniaxial pressure along the *A*), which breaks the *B*) show that each quadratically dispersive Weyl node located on the _{2}Se_{4} (24, 44). However, the experimental realization of the Weyl semimetal state in HgCr_{2}Se_{4} 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.

Besides the quadratically dispersive Weyl fermions, our calculations show another interesting property of SrSi_{2}. The Weyl nodes with opposite charges are located at different energies. We note that among all Weyl semimetal candidate materials, SrSi_{2} is the first one, to our knowledge, to have this property. It arises from the lack of any mirror symmetry in the SrSi_{2} 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 SrSi_{2} 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. 4*A*. The bigger circles correspond to the quadratic Weyl nodes with chiral charges of *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 SrSi_{2}. 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.

We now study the energy dispersion of the surface states. The dispersion along cut 1 (the black dotted line in Fig. 4*A*) is shown in Fig. 4*B*. 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. 4*A*) must be 0 because it goes through the Kramers points

In summary, we have presented first-principles band structure calculations that predict SrSi_{2} as a Weyl semimetal candidate. Similar to TaAs, SrSi_{2} 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 Bi_{2}Se_{3} (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 SrSi_{2} also offers many distinctive properties not present in the TaAs family of materials.

## Band-Gap Correction in SrSi_{2}

There have been a few optical and transport measurements on SrSi_{2}. These measurements did not reach a consensus about its ground state. Whereas ref. 46 reported an

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 *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, Cd_{3}As_{2} 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 _{3}Bi 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 SrSi_{2}. Fig. S1*A* 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. S1*B* 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 Bi_{2}Se_{3} (45) and TaAs (28), both of which have now been experimentally realized (30, 45).

## Acknowledgments

H.-T.J. thanks the National Center for High-Performance Computing, Computer and Information Network Center National Taiwan University, and National Center for Theoretical Sciences, Taiwan, for technical support. Work at Princeton University was supported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES) under DE-FG-02-05ER46200. Work at the National University of Singapore was supported by the National Research Foundation (NRF), Prime Minister’s Office, Singapore, under its NRF fellowship (NRF Award NRF-NRFF2013-03). T.-R.C. and H.-T.J. were supported by the National Science Council, Taiwan. The work at Northeastern University was supported by the US DOE/BES Grant DE-FG02-07ER46352, and benefited from Northeastern University’s Advanced Scientific Computation Center (ASCC) and the National Energy Research Scientific Computing Center (NERSC) supercomputing center through DOE Grant DE-AC02-05CH11231. S.-M.H., G.C., T.-R.C., and H.L.’s visits to Princeton University are funded by the Gordon and Betty Moore Foundation’s Emergent Phenomena in Quantum Systems (EPiQS) Initiative through Grant GBMF4547 (to M.Z.H.).

## Footnotes

↵

^{1}S.-M.H., S.-Y.X., and I.B. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: suyangxu{at}princeton.edu, nilnish{at}gmail.com, or mzhasan{at}princeton.edu.

Author contributions: S.-M.H., S.-Y.X., I.B., C.-C.L., G.C., T.-R.C., B.W., N.A., G.B., M.N., D.S., H.Z., H.-T.J., A.B., T.N., H.L., and M.Z.H. contributed to the intellectual contents of this work; preliminary material search and analysis were done by S.-Y.X. and I.B. with help from N.A., G.B., M.N., D.S., H.Z., and M.Z.H.; S.-Y.X. designed research; S.-M.H., C.-C.L., G.C., T.-R.C., B.W., H.-T.J., A.B., T.N., and H.L. performed theoretical analysis and computations; S.-M.H., S.-Y.X., C.-C.L., G.C., B.W., N.A., G.B., M.N., D.S., H.Z., H.-T.J., A.B., T.N., H.L., and M.Z.H. analyzed data; S.-M.H., S.-Y.X., I.B., T.N., H.L., and M.Z.H. wrote the paper; H.L. supervised the theoretical part of the work; and M.Z.H. was responsible for the overall physics and research direction, planning, and integration among different research units.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. P.K. is a guest editor invited by the Editorial Board.

↵*Haidemenakis ED, et al. (1977)

*Proceedings of the 8th International Conference on Physics of Semiconductors, Kyoto*189 (1966).This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1514581113/-/DCSupplemental.

Freely available online through the PNAS open access option.

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