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Weyl semimetal phase in non-centrosymmetric transition metal monophosphides

Based on first principle calculations, we show that a family of nonmagnetic materials including TaAs, TaP, NbAs and NbP are Weyl semimetal (WSM) without inversion center. We find twelve pairs of Weyl points in the whole Brillouin zone (BZ) for each of them. In the absence of spin-orbit coupling (SOC), band inversions in mirror invariant planes lead to gapless nodal rings in the energy-momentum dispersion. The strong SOC in these materials then opens full gaps in the mirror planes, generating nonzero mirror Chern numbers and Weyl points off the mirror planes. The resulting surface state Fermi arc structures on both (001) and (100) surfaces are also obtained and show interesting shapes, pointing to fascinating playgrounds for future experimental studies.

magnetic materials, where the spin degeneracy of the bands is removed by breaking time reversal symmetry. As mentioned, the WSM can be also generated by breaking the spatial inversion symmetry only, a method which has the following advantages. First, compared with magnetic materials, nonmagnetic WSM are much more easily studied experimentally using angle resolved photo emission spectroscopy (ARPES) as alignment of magnetic domains is no longer required. Second, without the spin exchange field, the unique structure of Berry curvature leads to very unusual transport properties under strong magnetic field, unspoiled by the magnetism of the sample.
Currently, there are three proposals for WSM generated by inversion symmetry breaking. One is a super-lattice system formed by alternatively stacking normal and topological insulators. [18] The second involves Tellurium or Selenium crystals under pressure [19] and the third one is the solid solutions LaBi 1−x Sb x Te 3 and LuBi 1−x Sb x Te 3 [20] tuned around the topological transition points. [21] In the present study, we predict that TaAs,TaP, NbAs and NbP single crystals are natural WSM and each of them possesses a total of 12 pairs of Weyl points. Compared with the existing proposals, this family of materials are completely stoichiometric, and therefore, are easier to grow and measure. Unlike in the case of pyrochlore iridates and HgCr 2 Se 4 , where inversion is still a good symmetry and the appearance of Weyl points can be immediately inferred from the product of the parities at all the time reversal invariant momenta (TRIM), [22][23][24] in the TaAs family parity is no longer a good quantum number. However, the appearance of Weyl points can still be inferred by analyzing the mirror Chern numbers (MCN) [25,26] and Z 2 indices [22,27] for the four mirror and time reversal invariant planes in the BZ. Similar with many other topological materials, the WSM phase in this family is also induced by a type of band inversion phenomena, which, in the absence of spin-orbit coupling (SOC), leads to nodal rings in the mirror plane. Once the SOC is turned on, each nodal ring will be gapped with the exception of three pairs of Weyl points leading to fascinating physical properties which include complicated Fermi arc structures on the surfaces.

CRYSTAL STRUCTURE AND BAND STRUCTURE
As all four mentioned materials share very similar band structures, in the rest of the paper, we will choose TaAs as the representative material to introduce the electronic struc-tures of the whole family. The experimental crystal structure of TaAs [28] is shown in Fig. 1(a). It crystalizes in body-centered-tetragonal structure with nonsymmorphic space group I4 1 md (No. 109), which lacks inversion symmetry. The measured lattice constants are a=b=3.4348Å and c=11.641Å. Both Ta and As are at 4a Wyckoff position (0, 0, u) with u=0 and 0.417 for Ta and As, respectively. We have employed the software package OpenMX [29] for the first-principles calculation. It is based on norm-conserving pseudopotential and pseudo-atomic localized basis functions. The choice of pseudopotentials, pseudo atomic orbital basis sets (Ta9.0-s2p2d2f1 and As9.0-s2p2d1) and the sampling of BZ with 10 × 10 × 10-grid have been carefully checked. The exchange-correlation functional within generalized gradient approximation (GGA) parameterized by Perdew, Burke, and Ernzerhof has been used. [30] After full structural relaxation, we obtain the lattice constants a=b=3.4824Å, c=11.8038Å and optimized u=0.4176 for As site, in very good agreement with the experimental values. To calculate the topological invariant such as MCN and surface states of TaAs, we have generated atomic-like Wannier functions for Ta 5d and As 4p orbitals using the scheme described in Ref. 31 We first obtain the band structure of TaAs without SOC by GGA and and plot it along the high symmetry directions in Fig. 1(c). We find clear band inversion and multiple band crossing features near the Fermi level along the ZN, ZS and ΣS lines. The space group of the TaAs family contains two mirror planes, namely M x , M y (shaded planes in Fig.1(b) ) and two glide mirror planes, namely M xy , M −xy (illustrated by the dashed lines in Fig.1(b)).
The plane spanned by Z, N and Γ points is invariant under mirror M y and the energy bands within the plane can be labeled by mirror eigenvalues ±1. Further symmetry analysis shows that the two bands that cross along the Z to N line belong to opposite mirror eigenvalues and hence the crossing between them is protected by mirror symmetry. Similar band crossings can also be found along other high symmetry lines in the ZNΓ plane, i.e. the ZS and NS lines. Altogether these band crossing points form a "nodal ring" in the ZNΓ plane as shown in Fig.2(b). Unlike for the situation in ZNΓ plane, in the two glide mirror planes (M xy and M −xy ), the band structure is fully gaped with the minimum gap of roughly 0.5 eV.
The analysis of orbital character shows that the bands near the Fermi energy are mainly formed by Ta 5d orbitals, which have large SOC. Including SOC in the first principle calculation leads to a dramatic change of the band structure near Fermi level, as plotted in Fig. 1(d). At first glance, it seems that the previous band crossings in the ZNΓ plane are  all gaped with the exception of one point along ZN line. Detailed symmetry analysis reveals that the bands "2" and "3" in Fig.1(d) belong to opposite mirror eigenvalues, indicating the almost touching point along the ZN line is completely accidental. In fact there is a small gap of roughly 3 meV between bands "2" and "3" as illustrated by the inset of Fig.1(d).
The ZNΓ plane then becomes fully gapped once SOC is turned on.

TOPOLOGICAL INVARIANTS FOR MIRROR PLANE AND WEYL POINTS
Since the material has no inversion center, the usual parity condition [22][23][24] can not be applied to predict the existence of WSM. We then resort to another strategy. As previously From the above analysis of the MCN and Z 2 index of several high-symmetry planes, we can conclude that Weyl points exist in the TaAs band structure. We now determine the total number of the Weyls and their exact positions. This is a hard task, as the Weyl points are located at generic k points without any little-group symmetry. For this purpose, we calculate the integral of the Berry curvature on closed surface in k-space, which equals the total chirality of the Weyl points enclosed by the given surface. Due to the four fold rotational symmetry and mirror planes that characterize TaAs, we only need to search for the Weyl points within the reduced BZ -one eighth of the whole BZ. We first calculate the total chirality or monopole charge enclosed in the reduced BZ. The result is one, which guarantees the existence of, and odd number of Weyl points. To determine precisely the location of each Weyl point, we divide the reduced BZ into a very dense k-point mesh and compute the Berry curvature or the "magnetic field in momentum space" [31,33] on that mesh as shown in Fig.3. From this, we can easily identify the precise position of the Weyl points by searching for the "source" and "drain" points of the "magnetic field". The Weyl points in TaAs are illustrated in Fig.2(a), where we find 12 pairs of Weyl points in the vicinity of what used to be, in the SOC-free case, the nodal rings on two of the mirror  Fig. 1(b). The position is given in unit of the length of Γ-Σ for x and y and of the length of Γ-Z for z.  Table. I. circle. By varying the radius of the cylinder, it is easy to show that such FSs must start and end at the projection of two (or more) Weyl points with different "monopole charge"

Weyl
, i.e. they must be "Fermi arcs". [6,8,15] In the TaAs materials family, on most of the common surfaces, multiple Weyl points will be projected on top of each other and we must generalize the above argument to multiple projection of Weyl points. It is easy to prove that the total number of surface modes at Fermi level crossing a closed circle in surface BZ must equal to the sum of the "monopole charge" of the Weyl points inside the 3D cylinder that projects to the given circle. Another fact controlling the behavior of the surface states is the MCN introduced in the previous discussion, which limits the number of FSs cutting certain projection lines of the mirror plane (when the corresponding mirror symmetries are still preserved on the surface).
By using the Green's function method [5] Fig.4(d).  The nodal ring around Σ-point can be modeled by a two-band k · p-theory, in the absence of SOC: where d i (k) are real functions, and k = (k x , k y , k z ) are three components of the momentum The form of the mirror operator is chosen such that the two bands have opposite mirror eigenvalues, an information obtained from the ab-initio calculation. The mirror reflection dictates that which translates into Eq.
(3) states that on the plane k y = 0, only d z is nonzero, and hence generically the equation and we may choose U T = σ z . This symmetry places additional constraints on d i 's: or Eqs. (3,7) determine the general form of our k · p-model.
Notice that now M 2 = −1 and C 2 2T = 1, as needed. With spin degrees of freedom, each band in the previous spin-orbit coupling free model in Eq. [1] becomes two bands, and nodal ring becomes a four-band crossing. In the vicinity of the nodal ring, the addition of SOC is equivalent to adding coupling between different spin components, i.e., 'mass terms', to the previous model. Here the name 'mass term' simply means that these terms are not required to vanish at the nodal ring by any symmetry.
The symmetry of the nodal ring is just mirror reflection. The mass terms hence must commute with mirror symmetry, and a generic term on the k y = 0 plane is given by Note that these mass terms are in general k-dependent, as their values may change as k moves along the nodal ring, but the C 2 * T symmetry makes them satisfy (on the k y = 0 plane) m 1,2,4,6 (k x , 0, k z ) = m 1,2,4,6 (k x , 0, −k z ) = m 1,2,4,6 (k x , 0, k z ), A complete analysis of the band crossing in the presence of all six mass terms is unavailable as the analytic expressions for the dispersion are involved. However, one may see the qualitative role played by each mass term by analyzing them separately. From Eq.(10), we see that m 3,5 are odd under k z → −k z , while the others are even. This indicates that only m 1,2,4,6 -terms are responsible for band crossings appearing on the k z = 0 plane, while the band crossings away from that plane are attributed mainly to the presence of m 3,5 -terms.
At k y = 0 plane, m 1,2 -terms commute with H 0 , so these terms, if of small strength, will split the doubly degenerate nodal ring into two singly degenerate rings, but not open gaps.
The equations for the two new rings are given by One should note that when m 1 -term (m 2 -term) is added, the two rings are the crossing between two bands with same (opposite) mirror eigenvalues. Therefore, the two rings from adding m 1 term are purely accidental, and the two rings from adding m 2 term are protected by mirror symmetry.
Next we discuss the effect of m 4,6 terms, which should in combination gives rise to the pair of Weyl nodes on the k z = 0-plane shown in the paper. The dispersion after adding m 4,6 terms is where With some straightforward algebraic work, it can be shown that the equation E(k) = 0 (band-touching) is equivalent to the following three equations: When |m 6 | > |m 4 |, these equations have at least one pair of solutions on k z = 0-plane symmetric about k y = 0 with codimension zero: they are Weyl nodes on the k z = 0 plane.
In our simulation, we found only one pair of Weyl nodes appear on this plane, which can only be understood if m 4,6 are k-dependent. The equations d x = 0 and d z = 0 determines a closed loop on the k z = 0 plane. At the same time d y (k x , k y , 0) = ± m 2 6 − m 2 4 has solutions that are symmetric about k y = 0. Since d y (k x , 0, 0) = 0, the solutions do not cross the k y = 0 line if m 4,6 are constants. Therefore, the solutions must be two lines, which make four crossings in total with the solution to d z = 0. However, let us recall that all mass terms can also contain a linear function in k x , so it is possible that m 6 − m 4 vanishes for a particular k x . At that k c , the solution to d y = m 2 6 − m 2 4 = 0 is satisfied at k y = 0. If the point (k c , 0, 0) is inside the loop that solves d z (k x , k y , 0) = 0, then there must be an odd number of crossings of d y = m 2 6 − m 2 4 and d z = 0. We can also understand the pairs of Weyl nodes that are away from k z = 0-plane.
Consider a coexistence of both m 4 -and m 5 -terms. The dispersion is given by Solving E(k) = 0 is equivalent to solving The last two equations together give where d x ≡ uk y k z , m 5 ≡ λk z and d y ≡ vk y .
When vm 4 λ < 0, there is no solution. When vm 4 λ < 0, there are four sets of solutions for (k y , k z ). We can substitute them into d z = 0 to obtain the four Weyl nodes observed in our simulation. When m 4 is small, the four Weyl nodes are close to the crossing point of the nodal ring and the k z = 0 plane.
We summarize the roles played by different mass terms: m 1,2 -terms split the nodal ring into two non-degenerate rings. With m 1 -(m 2 -)term alone the ring is the crossing of two bands with same (opposite) mirror eigenvalues. m 3,5 -terms gap the nodal ring except at k y = k z = 0. m 4 -term alone or coexisting with m 3 -term fully gaps the ring. m 4 -term coexisting with m 5 produces four Weyl nodes away from the k z = 0 plane. m 6 -term creates a pair of Weyl nodes on the k z = 0 plane, symmetric about k y = 0.

II. Distribution of nodal lines and Weyl points
A movie demonstrates the momentum space distribution of nodal lines calculated without spin-orbit coupling (SOC) and Weyl points with SOC. The movie can be download here: http://pan.baidu.com/s/1pJwTsmB.