Abstract
Firstprinciples calculations have recently been used to develop comprehensive databases of nonmagnetic topological materials that are protected by timereversal or crystalline symmetry. However, owing to the low symmetry requirement of Weyl points, a symmetrybased approach to identifying topological states cannot be applied to Weyl semimetals (WSMs). To date, WSMs with Weyl points in arbitrary positions are absent from the wellknown databases. In this work, we develop an efficient algorithm to search for Weyl points automatically and establish a database of nonmagnetic WSMs with Weyl points near the Fermi level based on the experimental noncentrosymmetric crystal structures in the Inorganic Crystal Structure Database (ICSD). In total, 46 Weyl semimetals were discovered to have nearly clean Fermi surfaces and Weyl points within 300 meV of the Fermi level. Nine of them are chiral structures which may exhibit the quantized circular photogalvanic effect. In addition, the nonlinear optical response is studied and the giant shift current is explored. Besides nonmagnetic WSMs, our powerful tools can also be used in the discovery of magnetic topological materials.
Introduction
Since the discovery of topological insulators^{1,2,3,4,5,6,7}, topological band theory has been widely applied to solid state materials^{8,9,10,11} and leads to the further classification of both insulators^{12,13,14,15,16,17,18} and semimetals^{19,20,21,22,23,24,25,26,27,28,29} based on the quantum topological invariants. The development of the topological semimetals stimulated extensive research in understanding a variety of topological properties. Very recently, some comprehensive databases for the symmetry protected nonmagnetic topological states were established by three independent works^{30,31,32}. These useful databases contain not only the topological insulators, but also the semimetals with degeneracy points located at highsymmetric lines or points. However, in these databases, an important category of topological materials, e.g., the Weyl semimetal (WSM) whose degeneracy point can locate at any position in the entire Brillouin zone, is still absent.
Generally speaking, topological semimetals are the systems whose conduction and valence bands cross with each other in the Brillouin zone and cannot be removed by perturbations of Hamiltonian without breaking specific symmetries. Topological WSM is such a material with double degeneracy Weyl points. Compared with other topological states, the WSM is extraordinary since the topology is not protected by any crystalline symmetry. It can only appear in the system whose spin degeneracy is removed by breaking time reversal or spacial inversion symmetry. In this case, the bands that create the Weyl points take the 2 × 2 Weyl equation form, \(H=\pm {v}_{F}\overrightarrow{k}\cdot \overrightarrow{\sigma }\), which can be viewed as half of the massless Dirac equation with defined chirality in threedimensional space^{33}. Because a twobands model Hamiltonian can be described by a linear combination of three Pauli matrices generally, the linear band crossing can be obtained by tuning the parameters without considering any additional conditions. Therefore, besides time reversal or spacial inversion symmetry, the lattice periodic boundary condition is the unique requirement for Weyl points. Accordingly, the identification of topology based on the characteristics of highsymmetry points does not work anymore for WSMs, and that is why the WSMs are still absent in the comprehensive databases^{30,31,32}.
In this work, we only focus on the experimental structures to complete the databases by including nonmagnetic WSMs. The total information about their topological properties is listed in the Supplementary Information. At the same time, two effective tools used for generating Wannier function automatically and searching Weyl points are developed in Fig. 1. The accuracy of them is essential for finding materials with observable properties of Weyl points, Some of them have already been theoretically predicted and experimentally confirmed, such as the family of TaAs^{23,24,25,26}. Besides typeI and typeII WSMs, nine chiral WSMs with inversion and mirror symmetries breaking are also listed in our database. For chiral WSMs, the truly quantized circular photogalvanic effect (CPGE)^{34,35,36,37} could be induced by the energy difference of Weyl points with opposite chirality. Moreover, we explored the bulk photovoltaic effect (BPVE)^{38} of all these noncentrosymmetric WSMs. A giant BPVE which reach up to 700 μA/V^{2} is predicted, and some of them can even extend to the regime of visible light.
Results
Our automatic workflow for searching nonmagnetic WSMs is shown in Fig. 1. It includes two crucial algorithms, the Wannier function generation and Weyl points search. Constrained by the accuracy of the density functional theory (DFT), we don’t consider the systems with strongly correlated felections, namely Lanthanide and Actinide series except element La, while beginning our work with the experimental noncentrosymmetric crystal structures in the Inorganic Crystal Structure Database (ICSD)^{39}. Here we remain some systems with 3delections, which may also be strongly correlated systems, to make sure that we don’t omit some potential candidates. After removing alloys and repeated structures, totally 8896 compounds left. Among them, all the systems with the total magnetic moments exceeding 0.05 μ_{B} per unit cell and magnetic elements V, Mn, Fe, Co, and Ni are regarded as magnetic compounds. Furthermore, the gap and the charge carrier density (N) of the nonmagnetic phases are checked. Because we are only interested in WSMs with observable topological Weylpointsrelated properties, the insulators and good metals (with N > 2 × 10^{22}/cm^{3}, which is obtained by the comparison with Co_{2}VGa^{40}) will not be considered in the further calculations.
After all the screening processes above, we obtain the list of the nonmagnetic noncentrosymmetric experimental crystal structures who may possess Weyl points near Fermi level. For these compounds, the atomicorbitallike Wannier functions are constructed by an effective algorithm automatically as shown in Fig. 1b. The crucial problem to sole for this step is the setting of the fitting windows with appropriate projection orbital basis sets. In the nonmagnetic firstprinciples calculations, we select s orbitals of alkalineearth metals, d orbitals of transition metals and s + p orbitals of pblock elements as the minimum basis sets. Thus, we can label the maximum weight value among the selected band as r. The weight represent the ratio of selected basis projection. If the weight r is smaller than 0.8, the minimum basis sets will be insufficient to produce the DFT band structure; hence, we switch the bases to the maximum basis sets which include s + p + d orbitals of all the elements. With the selected basis sets, the inner projection energy window can be easily fixed as 90% of the maximum weight value r. Subsequently, we loop over the outer projection energy window and check the mean absolute error between the DFT and Wannier band structures around the Fermi level until the error is smaller than 0.02 eV. Meanwhile, it is noteworthy that a good projection can be always guaranteed because of the large band gap between valence and core states.
Using the highsymmetry tightbinding (TB) model Hamiltonians obtained in the construction of Wannier functions, the Weyl points can be scanned. Since the existence of Weyl points in noncentrosymmetric nonmagnetic system only requires the translational symmetry, the fact that the Weyl points would locate at generic kpoints without any special symmetry renders the search difficult. The typical way to search for the Weyl points use the band gap as a criterion. However, this method is much easy to reach the local minimum value, thereby losing some Weyl points. In this work, we use the Berry phase of the parallelepiped corresponding to the kmesh of the reciprocal space as the criterion. Initially, based on the TB model Hamiltonian, we genera;te a relatively sparse 51 × 51 × 51 kmesh along the three reciprocal vectors to calculate the Berry phase of each parallelepiped. If the value is larger than 0.01 × 2π, Weyl points might be present inside the parallelepiped. For this type of parallelepiped, we increase the kmesh finer and repeat calculating Berry phase. Eventually, the Berry phase will increase in some fine parallelepiped and finally reach a value n × 2π, where n is an integer, corresponding to a single Weyl point. In our calculations, the parallelepiped is refined to around 1/1600^{3} of the reciprocal lattice to obtain the Berry phase with n ≥ 0.99.
After searching the Weyl points, we have a raw list of the WSMs with all the information of Weyl points. It may also have some magnetic materials such as antiferromagnets. Thus, we process a double check here to modify the raw list, and the list of nonmagnetic WSMs are obtained.
Despite having as many as 8896 noncentrosymmetric candidates, we only find 46 nonmagnetic WSMs with Weyl points near the Fermi level within 300 meV, including nine chiral crystal structures. This result express that the good nonmagnetic WSMs are difficult to find. The details of every WSMs that we find by our algorithm are given in Supplementary Table 1. Until now, the Weyl points have been classified into typeI and typeII according to the shape of the linear band crossings. The Fermi surface would shrink to a singularity point for the typeI Weyl point, while the typeII Weyl point presents a Fermi surface with a linear crossing which is composed of electron and hole^{41}. On the other hand, another main difference is the precondition of chiral anomaly. It can exist along any direction in the typeI WSM as long as electric and magnetic fields are not perpendicular to each other, but only exist when the electric and magnetic fields point to some selected direction in the typeII WSM.
Discussion
Representative materials
Here, we choose two typical candidates covering each class of WSM as examples. For typeI WSM, we highlight an orthorhombic sulfide TaAgS_{3} with space group Cmc2_{1} (No. 36) as shown in Fig. 2a. The energy dispersion calculated by DFT is given in Fig. 2d. It presents a semimetallic structure with small electron and hole pockets located around Γ and Y points, respectively. With inversion symmetry breaking, the spin degeneracy is lifted. Finally, two pairs of Weyl points, a minimum number of Weyl points in timereversalsymmetric systems, would be generated. Because of the C_{2z} rotation and the time reversal symmetry, the Weyl points are constrained in the k_{z} = 0 plane as shown in Fig. 2b. All the Weyl points are related with each other and located at the same energy, only ~30 meV below the charge neutral point, owing to the existence of the mirror planes m_{x} and m_{y}, On the other hand, low charge carrier density can make the Weyl points observable by transport measurements. The Weyl points are further confirmed by the (001) surface state calculation. When these Weyl points are projected into (001) surface, the Weyl points with different chirality are projected into different position. If the chemical potential is set at the energy of the Weyl points, the Fermi arcs connecting the Weyl points can be observed as shown in Fig. 2e. The Fermi arc length can be ~15% of the reciprocal lattice vector, which makes it easy to detect by surface detection methods, such as angleresolved photoemission spectroscopy (ARPES) and scanning tunneling microscopy (STM).
The other example Ag_{2}S is a hybrid WSM^{42} containing both two kinds of Weyl points. It also has an orthorhombic crystal structure with space group P2_{1}2_{1}2_{1} (No. 19), as shown in Fig. 3a. Different from TaAgS_{3}, without considering spinorbital coupling (SOC), the band inversion between the p orbital of S and the d orbital of Ag forms two Dirac points on the highsymmetric lines Γ − X and Γ − Y which is protected by the C_{2} rotation symmetry^{30,31,32}. Furthermore, turning on SOC leads to the separation of the Dirac points and four pairs of Weyl nodes appear, since SOC breaks the spin rotation symmetry. The locations of Weyl points in the Brillouin zone shown in Fig. 3b connected with each other by time reversal, C_{2x} and C_{2y} rotation symmetries. Through the band structure along the kpath crossing one pair of Weyl points in Fig. 3d, we find these Weyl points belong to two different classes, the typeI Weyl points with chirality +1 (W1) and the typeII Weyl points with chirality −1 (W2). They are distinguished by different color in Fig. 3. At the same time, the typeII Weyl points W2 can also be located by the linearcrossinglike Fermi surface in Fig. 3e if we put the chemical potential at W2.
One point we need to emphasized is that Ag_{2}S does not present any mirror symmetry. Although the mirror symmetry is not necessary for the existence of Weyl points, the chirality of Weyl points must be odd with mirror operations. Therefore, the mirror symmetry forces one pair of Weyl points with opposite chirality locate at the same energy. In another words, WSM without mirror symmetry exhibits a net chirality if the Fermi energy is located between one pair of Weyl points with opposite chirality. One of the most important properties for mirrorsymmetrybroken WSMs is the quantized CPGE, which is the only possible quantized signal induced by the monopole Weyl point so far. This can also be applied to Ag_{2}S with W1 ~23 meV above the charge neutral point and W2 ~28 meV below the charge neutral point, as illustrated in Fig. 3d. Because the charge carrier density is only around 5 × 10^{20}/cm^{3} and the contribution of the trivial Fermi surfaces can be neglected, Ag_{2}S may be an ideal candidate to observe the quantized CPGE.
The chiral WSM Ag_{2}S is further confirmed by the chiral surface Fermi arcs. Fixing the energy at W1 and W2, we can obtain the extremely long chiral Fermi arcs in (001) surface in Fig. 3f, g, respectively. Owing to the large spin splitting, the spin texture should be easily observed by both ARPES and STM. Since two W2 Weyl points overlap with each other when they are projected into (001) surface, two Fermi arcs start from same projected Weyl point. These extremely clear topological properties offer a good platform for both surface detection and bulk nonlinear optical transports.
Shift current analysis
Very recently, a giant shift current was observed experimentally in WSM TaAs^{43}, which suggests a topological effect in BPVE and the practical applications of WSMs in optical detectors and solar energy conversion. Inspired by this work, we screened the nonlinear optical response for all the nonmagnetic WSMs in our database. Half of them can host a large shift current above 100 μA/V^{2} with photon energy larger than 0.1 eV (Supplementary Table 2 and Supplementary Figs. 1–3). Our database shows that huge shift current can be obtained in both infrared and visible spectrum in noncentrosymmetryic WSMs, and it provides exotic material candidates for the study of interplay of different functions based on WSMs.
Through perturbative analysis, the expression of shift current^{44} is written as
where i, j, k = x, y, z. The second order optical conductivity tensor can be computed as
where n and m are band indexes, f_{nm} = f_{n} − f_{m} is the FermiDirac distribution difference, \({r}_{nm}^{i}={A}_{nm}^{i}(1{\delta }_{n,m})\) is the interband Berry connection, \({r}_{nm}^{i;j}={\partial }_{{k}_{j}}{r}_{nm}^{i}i({A}_{nn}^{j}{A}_{mm}^{j}){r}_{nm}^{i}\) depends on intermediate virtual states, and E_{nm} = E_{n} − E_{m} is the energy difference between two bands. The delta function \(\delta \left(\hslash \omega {E}_{mn}\right)\) is disposed by a Gauss distribution with smearing factor 0.01 eV. To avoid the numerical divergence with zero denominator, we replace the transitionenergyrelated term \(\frac{1}{{E}_{mn}}\) by \(\frac{{E}_{mn}}{{E}_{mn}^{2}+{\alpha }^{2}}\) with α = 0.04 eV, which is stable for the shift current with photon energy above 0.1 eV. After a kgrid convergence test, we find that the change is less than 5% with kgrid increasing from 100 × 100 × 100 to 200 × 200 × 200. Here, we use a grid 200 × 200 × 200 in our calculations. The calculated tensors of shift current are doublechecked by the symmetry. For convenient, we set the tensor at each kpoint as
In the experimental measurement of WSM TaAs (ref. ^{43}), an apparent measured value σ_{xxz} = 34 ± 3.7 μAV^{−2} with photon energy ћω = 0.117 eV is obtained. Our calculation gives a wellfitting theoretical value σ_{xxz} = 27 μAV^{−2} with the same photon energy. Furthermore, ref. ^{43} added the reflectance of TaAs at 10.6 μm to the apparent measured value manually and the giant shift current σ_{xxz} = 154 ± 17 μAV^{−2} is obtained in the end. At the same time, σ_{zzz} has the biggest value for TaAs in both our calculation and ref. ^{43}.
The largest shift current in WSMs was found in BiTeI with space group P3, in which the component of σ_{xxz} can reach up to 750 μA/V^{2} at a photon energy 0.66 eV. It is around five times larger than that in TaAs. Moreover, four independent tensor components in BiTeI are above 200 μA/V^{2} in a large photon energy window of 0.3–0.7 eV (see Fig. 4a), which allows large shift current in different experimental setup. Hence the new WSM BiTeI offers a more ideal candidate for the generation of strong BPVE. Considering the layered structure of BiTeI stacking along z, this component provides a natural advantage to integrate the material into devices.
Beside the largest shift current candidate BiTeI, we also find another two kind of candidates. One is multifunctional candidates combining topology, strong shift current, and superconductivity. As indicated in Fig. 4b, c, the shift current tensor can reach up to σ_{zzz} = 564 μA/V^{2} and σ_{zzz} = 250 μA/V^{2} for the new found WSMs LaIrP and LaRhP, respectively. In addition, since both LaIrP and LaRhP are superconductors^{45} with transition temperatures T_{c} = 5.3 and 2.5 K, these two materials will provide good platform for the study of the interplay among topology, superconductivity and nonlinear optical responses. The other one is the candidates with large photon energy. For all these WSMs with large shift current in Supplementary Table 2 and Supplementary Figs. 1–3, we find that the peak value of σ_{ijk} locates at an energy below 1 eV, except SrHgSn, CaBiAg and CaCdPb. Fig. 4d shows the shift current tensor of SrHgSn with space group P6_{3}mc (No. 186) as an example. One can find a peak value σ_{zzz} = 165 μA/V^{2} at photon energy ~1.8 eV, which is already located at the range of red light.
The shift current can be considered as a joint effect of the frequencydependent density of states (DOS) and the wave function. Taking SrHgSn as an example, we can try to understand the microscopic mechanism in these two aspects. From the DOS of SrHgSn shown in Fig. 5b, one can find two peaks locating bellow and above the charge neutral point, respectively. The energy difference is around 1.8 eV, which is very close to the two peaks of σ_{zzz} in Fig. 4d. \({\sigma }_{zzz}^{\prime}(\overrightarrow{k};0,\omega ,\omega )\) is also calculated for each kpoint with the photon energy 1.8 and 1.95 eV near the two peaks. Thus we obtained that the positive contribution is mainly from the special plane k_{z} ~0.312π/c at photon energy around 1.8 eV, while k_{z} ~0.132π/c plane at photon energy around 1.95 eV has a negative contribution. The distribution of \({\sigma }_{zzz}^{\prime}(\overrightarrow{k};0,\omega ,\omega )\) at photon energy 1.8 eV is dominated by twelve positive blue lines caused by the excitation from band n − 1 to bands n + 3 and n + 4, as shown in Fig. 5c, f. The distribution of \({\sigma }_{zzz}^{\prime}(\overrightarrow{k};0,\omega ,\omega )\) changes sharply when photon energy increasing from 1.8 to 1.95 eV. At photon energy 1.95 eV, \({\sigma }_{zzz}^{\prime}(\overrightarrow{k};0,\omega ,\omega )\) is dominated by two negative red rings as shown in Fig. 5d, e. These two red rings are derived by the light excitation from two occupied bands n and n − 1 to the nonoccupied bands n + 7, n + 8, n + 9 and n + 10, as presented in Fig. 5g, h. Therefore, the two peaks of σ_{zzz}(0, ω, − ω) in Fig. 4d originate from large DOS located around E_{F} − 1.2 eV and E_{F} + 0.6 eV, and the sign changes because of the different band excitation at different kpoints. One point need to emphasize is that the strong shift current at high photon energy is not caused by the Weyl points here.
In summary, through highthroughput calculations, we have complemented the nonmagnetic topological material databases by including the nonmagnetic WSMs. Among them, nine chiral structures, such as Ag_{2}S, which allow the quantized CPGE and have natural long surface Fermi arcs are also recognized. Furthermore, most of them can have a large shift current, especially for BiTeI. Beside the two databases of nonmagnetic WSMs and their nonlinear optical responses, the effective algorithms for searching Weyl points by Berry phase and automatically Wannier function generation provide powerful tools for the study of magnetic topological materials.
Methods
The calculation starts from the experimental noncentrosymmetric crystal structures in the Inorganic Crystal Structure Database (ICSD, http://www2.fizkarlsruhe.de/icsd_home.html). We take these crystal structures as the input of the fullpotential localorbital (FPLO) minimumbasis code^{46}. Here we choose the generalized gradient approximation of the PerdewBurkeErnzerhof functional^{47} as the exchangecorrelation potential. A dense 12 × 12 × 12 kmesh is used in the static nonmagnetic calculation. SOC is also included selfconsistently during the workflow. In order to get the surface states of WSMs, Green’s function method^{48,49} is applied with a halfinfinite boundary condition. During the DFT calculations, we assume that the ground state is nonmagnetic after excluding the materials with magnetic elements and large total magnetic moment. The final database which may contains antiferromagnets is also doublechecked to make sure they are indeed nonmagnetic.
Data availability
The data generated during this study are available from the corresponding author on reasonable request.
Code availability
The codes used in this study are available from the corresponding author on reasonable request.
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Acknowledgements
This work was financially supported by the ERC Advanced Grant No. 291472 ‘Idea Heusler’, ERC Advanced Grant No. 742068 ‘TOPMAT’. We also acknowledge funding by the DFG through SFB 1143 (project ID 247310070) and the W\(\ddot{u}\)rzburgDresden Cluster of Excellence on Complexity and Topology in Quantum Matter – ct.qmat (EXC 2147, project ID 39085490). Some of our calculations are carried on Cobra cluster of MPCDF, Max Planck society.
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Q.X. and Y.Z. contributed equally to this work. Q.X. collected and analysed the data and did all the calculations. Y.Z. designed the workflow and wrote the codes for the search. W.J. and Y.S. wrote some other supporting codes and gave technical advices. K.K., K.B., and C.F. gave scientific advices. Y.S. and Q.X. wrote the manuscript. The project was supervised by C.F.
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Xu, Q., Zhang, Y., Koepernik, K. et al. Comprehensive scan for nonmagnetic Weyl semimetals with nonlinear optical response. npj Comput Mater 6, 32 (2020). https://doi.org/10.1038/s4152402003011
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