Theoretical Discovery/Prediction: Weyl Semimetal states in the TaAs material (TaAs, NbAs, NbP, TaP) class

The recent discoveries of Dirac fermions in graphene and on the surface of topological insulators have ignited worldwide interest in physics and materials science. A Weyl semimetal is an unusual crystal where electrons also behave as massless quasi-particles but interestingly they are not Dirac fermions. These massless particles, Weyl fermions, were originally considered in massless quantum electrodynamics but have not been observed as a fundamental particle in nature. A Weyl semimetal provides a condensed matter realization of Weyl fermions, leading to unique transport properties with novel device applications. Here, we THEORETICALLY identify the first Weyl semimetal in a class of stoichiometric materials (TaAs, NbAs, NbP, TaP), which break crystalline inversion symmetry, including TaAs, TaP, NbAs and NbP. Our first-principles calculation-based predictions on TaAs reveal the spin-polarized Weyl cones and Fermi arc surface states in this compound. We also observe pairs of Weyl points with the same chiral charge which project onto the same point in the surface Brillouin zone, giving rise to multiple Fermi arcs connecting to a given Weyl point. Our results show that TaAs is the first topological semimetal identified which does not depend on fine-tuning of chemical composition or magnetic order, greatly facilitating an exploration of Weyl physics in real materials. (Note added: This theoretical prediction of November 2014 (see paper in Nature Communications) was the basis for the first experimental discovery of Weyl Fermions and topological Fermi arcs in TaAs recently published in Science (2015) at http://www.sciencemag.org/content/early/2015/07/15/science.aaa9297.full.pdf)


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The rich correspondence between high energy and condensed matter physics has led to a deeper understanding of spontaneous symmetry breaking, phase transitions, renormalization and many other fundamental phenomena in nature, with important consequences for practical applications using magnets, superconductors and other novel materials [1][2][3]. Recently, there has been considerable progress in realizing particles previously considered in high energy physics as emergent quasiparticle excitations of crystalline solids, such as Majorana fermions and Dirac fermions [4][5][6][7][8][9][10][11][12]. Materials that host these exotic particles exhibit unique properties and hold promise for applications such as fault-tolerant quantum computation, low-power electronics and spintronics. Weyl fermions were originally considered in massless quantum electrodynamics, but have not been observed as a fundamental particle in nature [1]. Recently, it was theoretically understood that Weyl fermions can arise in some novel semimetals with nontrivial topology [4][5][6]13]. A Weyl semimetal has an electron band structure with singly-degenerate bands that have bulk band crossings, Weyl points, with a linear dispersion relation in all three momentum space directions moving away from the Weyl point. These materials can be viewed as an exotic spin-polarized, three-dimensional version of graphene. However, unlike the two-dimensional Dirac cones in graphene [8], the three-dimensional Dirac cones in Na 3 Bi and Cd 3 As 2 [11,12] or the two-dimensional Dirac cone surface states of Bi 2 Se 3 [9,10], the degeneracy associated with a Weyl point depends only on the translation symmetry of the crystal lattice. This makes the unique properties associated with this electron band structure more robust. Moreover, due to its nontrivial topology, a Weyl semimetal can exhibit novel Fermi arc surface states. Previously, Fermi arcs were only found in high-T c superconductors due to strong electron correlation effects.
Recent theoretical advances in topological physics have now made it possible to realize Fermi arc surface states in a weakly interacting semimetal. Both the Weyl fermions in the bulk and the Fermi arc states on the surface of a Weyl semimetal are predicted to show unusual transport phenomena, which can be used in future device applications. Weyl fermions in the bulk can give rise to negative magnetoresistance, the quantum anomalous Hall effect, non-local transport and local non-conservation of ordinary current [14][15][16][17]. Fermi arc states on the surface are predicted to show novel quantum oscillations in magneto-transport and quantum interference effects in tunneling spectroscopy [18][19][20]. Because of the fundamental and practical interest in Weyl semimetals, it is crucial that robust candidate materials be found.

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It is theoretically known that a Weyl semimetal can only arise in a crystal where timereversal symmetry or inversion symmetry is broken. A number of magnetically ordered or inversion symmetry breaking materials have been proposed as Weyl semimetal candidates [21][22][23][24][25]. However, despite extensive effort in experiment, a Weyl semimetal has yet to be realized in any of the compounds proposed thus far. One concern is that the existing proposals require either magnetic ordering in sufficiently large domains [21][22][23][24] or fine-tuning of the chemical composition to within 5% in an alloy [22,24,25], which are demanding in real experiments. Here we propose the first Weyl semimetal in a stoichiometric, inversionbreaking, single-crystal material, TaAs. Unlike previous predictions, our proposal does not depend on magnetic ordering over sufficiently large domains, because our material relies on inversion symmetry breaking rather than time-reversal symmetry breaking. The compound we propose is also stoichiometric and does not depend on fine-tuning chemical composition in an alloy. Single crystals of TaAs have been grown [26]. We believe that this material is promising for experimental realization for the Weyl semimetal phase in this compound.
Tantalum arsenide, TaAs, crystalizes in a body-centered tetragonal lattice system (Fig. 1a). The lattice constants are a = 3.437Å and c = 11.646Å, and the space group is I4 1 md (#109, C 4v ). The crystal consists of interpenetrating Ta and As sub-lattices, where the two sub-lattices are shifted by a 2 , a 2 , δ , δ ≈ c 12 . There are two Ta atoms and two As atoms in each primitive unit cell. It is important to note that the system lacks a horizontal mirror plane and thus inversion symmetry. This makes it possible to realize an inversion breaking Weyl semimetal in TaAs. We also note that the C 4 rotational symmetry is broken at the (001) surface because the system is only invariant under a four-fold rotation with a translation along the out-of-plane direction. The bulk and (001) surface Brillouin zones (BZ) are shown in Fig. 1b, where high symmetry points are also noted.
The ionic model would suggest that the Ta and As atoms are in the 3 + and 3 − valence states, respectively. This indicates that the lowest valence band arises from 4p electrons in As and 5d 2 electrons in Ta, whereas the lowest conduction band primarily consists of 5d electrons in Ta. However, we may expect Ta 5d electrons to have a broad bandwidth because of the wide extent of the atomic orbitals. This leads to strong hybridization with the As 4p states, which may suggest that the conduction and valence bands are not entirely separated in energy and have a small overlap, giving rise to a semimetal. In Fig. 1c, we present the bulk band structure in the absence of spin-orbit coupling. The conduction and valence bands cross each other along the Σ − N − Σ 1 trajectory, which further indicates that TaAs is a semimetal. In the presence of spin-orbit coupling, the band structure is fully gapped along the high symmetry directions considered in Fig. 1d. However, Weyl points which are shifted away from the high symmetry lines arise after spin-orbit coupling is taken into account. Below, we consider the Weyl points in the bulk BZ. We also note that the double degeneracy of bands is lifted in the presence of spin-orbit coupling except at the Kramers' points, which confirms that TaAs breaks inversion symmetry but respects time-reversal symmetry.
To better understand the Weyl points in TaAs to a line node crossing. Next, we include spin-orbit coupling. This causes each line node to vaporize into six Weyl points shifted slightly away from the mirror planes, shown as small circles in Fig. 2a. There are 24 Weyl points in total: 8 Weyl points on the k z = 2π/c plane, which we call W 1 , and 16 Weyl points away from the k z = 2π/c plane, which we call W 2 .
We present the band structure near one of the Weyl points W 1 in Fig. 2c, where a point band touching is clearly observed. Next, we consider how the Weyl points project on the (001) surface BZ. We show a small region in the surface BZ around theΓ −X line, which includes the projections of six Weyl points, two from W 1 and four from W 2 . A schematic of the projections of all 24 Weyl points on the surface BZ is shown in Fig. 2e. We find that for all points W 2 , two Weyl points project on the same point in the surface BZ. We indicate the number of Weyl points that project on a given point in the surface BZ in Fig. 2e. Lastly, we study the chirality of the Weyl points by calculating the Berry flux through a closed surface enclosing a Weyl point. We call positive the chiral charges which are a source of Berry flux and negative the chiral charges which are a sink of Berry flux. We color the positive chiral charges white and the negative chiral charges black. In Fig. 2f, we show the spin texture of the band structure near two Weyl points W 1 . In this case, the spin texture is found to be consistent with the texture of the Berry curvature. Moreover, it turns out that the points W 2 project on the surface BZ in pairs which carry the same chiral charge.
Another key signature of a Weyl semimetal is the presence of Fermi arc surface states 6 which connect the Weyl points in pairs in the surface BZ. We present calculations of the (001) surface states in Fig. 3. We show the surface states on the top surface in Fig. 3a and the bottom surface in Fig. 3b. We find surface Fermi arcs that connect Weyl points of opposite chirality in pairs. To better understand the rich structure of the Fermi arcs, we show a schematic of the surface states on the top surface in Fig. 3c  Magnetic order can be difficult to predict from first-principles, may be difficult to measure experimentally, and most importantly, may not form large enough domains in a real sample for the properties of the Weyl semimetal to be preserved. Fine-tuning chemical composition is typically challenging to achieve and introduces disorder, limiting the quality of single crystals. For instance, a proposal for a Weyl semimetal in Y 2 Ir 2 O 7 [21] assumes an all-in, all-out magnetic order, which is challenging to verify in experiment [27,28]. Another proposed Weyl semimetal, HgCr 2 Se 4 [23], has a clear ferromagnetic order, but because of the cubic structure there is no preferred magnetization axis, likely leading to the formation of many small ferromagnetic domains. Moreover, a proposal in Hg 1−x−y Cd x Mn y Te [24] requires straining the sample to break cubic symmetry, applying an external magnetic field and finetuning the chemical composition. Another proposal for the inversion breaking compounds LaBi 1−x Sb x Te 3 and LaBi 1−x Sb x Te 3 [25] requires fine-tuning the chemical composition to within 5%. Despite extensive experimental effort, a Weyl semimetal has not been realized in any of these compounds. We believe that in large part the difficulty stems from relying on magnetic order to break time-reversal symmetry and fine-tuning the chemical composition to achieve the desired band structure. We note that, in contrast to time-reversal symmetry breaking systems, large single crystals of inversion symmetry breaking compounds exist, such as the large bulk Rashba material BiTeI, where the crystal domains are sufficiently large so that the Rashba band structure is clearly observed in photoemission spectroscopy [29]. We propose that TaAs overcomes the difficulties of previous candidates because it realizes a Weyl semimetal in a stoichiometric, inversion-breaking, single-crystal material.
We have also studied other members of this class of compounds, TaP, NbAs and NbP.
They have the same crystal structure as TaAs. Our results show that their band structure is qualitatively the same as that of TaAs, in that the conduction and valence bands form line nodes in the absence of spin-orbit coupling and a gap is opened in the presence of spinorbit coupling, leading to a Weyl semimetal. In particular, since Nb and P have quite weak spin-orbit coupling, our calculations suggest that NbP is better described as a topological nodal-line semimetal. Thus NbP might offer the possibility to realize novel nodal line band crossings in the bulk and "drumhead" surface states stretching across the nodal line on the surface [30].
Next, we provide some general comments on the nature of the phase transition between a trivial insulator, a Weyl semimetal and a topological insulator, illustrated in Fig. 4a. When a system with inversion symmetry undergoes a topological phase transition between a trivial 8 insulator and a topological insulator, the band gap necessarily closes at a Kramers' point. If we imagine moving the system through the phase transition by tuning a parameter m, then there will be a critical point where the system is gapless. In a system which breaks inversion symmetry, there will instead be a finite range of m where the system remains gapless, giving rise to a Weyl semimetal phase. In this way, the Weyl semimetal phase can be viewed as an to produce a bulk insulator. We consider the bottom surface, and annihilate the Weyl points in pairs in the obvious way to remove all surface states alongΓ−X. Then, we can annihilate the remaining Weyl points to produce two concentric Fermi surfaces around theX ′ point.
Since this is an even number of surface states, we find that this way of annihilating the Weyl points gives rise to a trivial insulator. If, instead, we annihilate the remaining Weyl points to remove the Fermi arc connecting them, only the closed Fermi surface would be left aroundX ′ , giving rise to a topological insulator. A similar analysis applies to the top surface.
Finally, we highlight an interesting phenomenon regarding how an electron travels along a constant energy contour in our predicted Weyl semimetal, TaAs. Fig. 4b shows the calculated Fermi surface contours on both the top and the bottom surfaces near theX point. We consider an electron initially occupying a state in one of the red Fermi arcs, whose wavefunction is therefore localized on the top surface in real space. We ask how the wavefunction evolves as the electron traces out the constant energy contour. As the electron moves along the Fermi arc (the red arc on the left-hand side of Fig. 4c), it will eventually reach a Weyl point, where its wavefunction will unravel into the bulk. The electron will then move in real space to the bottom surface of the sample. Then it will follow a Fermi arc on that surface (the blue arc on the left-hand side of Fig. 4b), and reach another Weyl point.
The electron will next travel through the bulk again and reach the top surface, returning to the same Fermi arc where it began. This novel trajectory is predicted to show exotic surface transport behavior as proposed in Ref. [20]. This is one example of the unique transport phenomena predicted in Weyl semimetals. To study these effects both from fundamental interest and for novel devices, here we provides a promising candidate for a Weyl semimetal in a stoichiometric, inversion-breaking, single-crystal material, as represented by TaAs. We hope that our theoretical prediction will soon lead to the experimental discovery of the first Weyl semimetal in nature.

Methods
First-principles calculations were performed by OPENMX code based on norm-conserving pseudopotentials generated with multi-reference energies and optimized pseudoatomic basis functions within the framework of the generalized gradient approximation (GGA) of density functional theory (DFT) [31,32]. Spin-orbit coupling was incorporated through j-dependent pseudo-potentials [33]. For each Ta atom, three, two, two, and one optimized radial functions were allocated for the s, p, d, and f orbitals (s3p2d2f 1), respectively, with a cutoff radius of 7 Bohr. For each As atom, s3p3d3f 2 was adopted with a cutoff radius of 9 Bohr. A regular mesh of 1000 Ry in real space was used for the numerical integrations and for the solution of the Poisson equation. A k point mesh of (17 × 17 × 5) for the conventional unit cell was used and experimental lattice parameters were adopted in the calculations. Symmetry-respecting Wannier functions for the As p and Ta d orbitals were constructed without performing the procedure for maximizing localization and a real-space tight-binding Hamiltonian was obtained [34]. This Wannier function based tight-binding model was used to obtain the surface states by constructing a slab with 80-atomic-layer thickness with Ta on the top and As on the bottom. † Electronic address: suyangxu@princeton.edu ‡ Electronic address: nilnish@gmail.com § Electronic address: mzhasan@princeton.edu   As the electron proceeds around the Fermi arc, it will eventually approach a Weyl point and the wavefunction will unravel into the bulk. The electron will then move in real space to the bottom surface of the sample, and there it will follow the Fermi arc on that surface. The electron will eventually approach another Weyl point, pass again through the bulk and remerge in the same Fermi arc where it began, completing the circuit. d, By contrast, a topological insulator has a full bulk band gap, so an electron tracing out a constant energy contour in the surface states will not encounter a bulk state and the wavefunction will always remain localized on the same surface.