Coexistence of Type-I and Type-II Weyl Points in the Weyl-Semimetal

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Coexistence of Type-I and Type-II Weyl Points in the Weyl-Semimetal OsC2 Minping Zhang, Zongxian Yang, and Guangtao Wang J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b00920 • Publication Date (Web): 30 Jan 2018 Downloaded from http://pubs.acs.org on February 3, 2018

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The Journal of Physical Chemistry

Coexistence of Type-I and Type-II weyl points in the Weyl-Semimetal OsC2 Minping Zhang, Zongxian Yang, and Guangtao Wang∗ College of Physics and Electronic Engineering, Henan Normal University, Xinxiang, Henan 453007, People’s Republic of China (Dated: January 28, 2018) The topologically nontrivial Weyl semimetals have two different types: (i) the standard Weyl cones with point-like Fermi surfaces (type-I) and (ii) tilted Weyl cones that appear at the contact of electron and hole pockets (type-II). These two types of Weyl semimetals have significantly different physical properties in their thermodynamics and magnetotransport. Here we presented a compound OsC2 with both types Weyl Points (WPs) at the equilibrium volume. It has 24 planes around K (or K’) points and 12 type-II WPs in the type-I WPs in the Kz =±0.0241× 2π c Kz =±0.4354× 2π planes at projected K (K’)-point, respectively. The type-I WPs are connected by c the Helix-Trimer Fermi arcs.

Since the theoretical and experimental discovery of topological insulators1–3 , the study of topological states of matter has become one of the major topics in condensed matter physics. The topological materials have been expanded to the phono systems4,5 . Apart from the triumphs of systems with full energy gaps, the gapless systems, such as Dirac6–9 and Weyl10–31 semimetals have stimulated intensive activities in understanding the band topology. Weyl semimetals (WSM) are topologically nontrivial conductors in which the spin non-degenerate valence and conduction bands touch at isolated points in the Brillouin zone, the so called ‘Weyl nodes’15–31 . The nodes occur in pairs of opposite chirality15,24,25 . The WSM phase requires either time-reversal (TR) or inversion symmetry (or both) to be broken15,18 . Recently, it has been further realized that topological Weyl semimetals can be classified into type-I15–22 , which respects Lorentz symmetry, and type-II26–31 , which does not. The TaAs family15–22 of WSMs exhibit ideal Weyl cones in the bulk band structure and belongs to the type-I class, i.e., the FS shrinks to some isolated points, i.e. Weyl Points (WPs). While the layered transition-metal dichalcogenides WTe2 and MoTe2 belong to the type-II class26–31 , with their tilted Weyl cones appearing at the contact of electron and hole pockets26 . Type-II WSMs are expected to show different properties from type-I WSMs, such as thermodynamics, magneto-transport33–38 and topological superconductivity39 . 15–22

For all the previous works on type-I and typeII26–32 WSMs, the two Weyl nodes in pair with opposite chiralities are of the same type26 . We wonder whether it is possible to have a WSM that have both type-I and type-II WPs in the same compound. Li40 use a tightbinding model to realize a “hybrid Weyl semimetal”, in which one Weyl node is of type-I while the other is of type-II. By adding the tilted Hamitonian to the type-I Weyl Hamitonian, Xie41 investigated the disor-

* Corresponding author. E-mail addresses: [email protected]

der induced multiple phase transitions among the threedimensional quantum anomalous Hall (3D-QAH), type-I Weyl Semimetal, type-II Weyl Semimetal, and Normal insulator. By applying pressure on the PdTe2 , Sun42 found that a pair of type-II Dirac points disappeared at 6.1 GPa and a new pair of type-I Dirac points from the same two bands emerged at 4.7 GPa. Under the proper pressure (4.7∼6.1 GPa), the two types of Dirac points can coexist in PdTe2 . So far, a Weyl semimetal with both type-I and type-II WPs had not been found in one compound at the equilibrium volume. Here, we report a new Weyl Semimetal OsC2 with the coexistence of type-I and type-II WPs .

FIG. 1: ((a) The primitive cell of OsC2 with P¯ 6m2 (No.187) symmetry and its top view. (b) Brillouin zone of bulk and the projected surface Brillouin zones of (001) and (100) planes, as well as high-symmetry points.

OsC2 crystallizes43 in the hexagonal space group P¯6m2 (No. 187). Os and C atoms occupy the 1f ( 32 , 13 , 0) and 2g (0, 0, 0.18) Wyckoff sites as shown in Figs. 1(a) and 1(b). The lattice constants are a=b=2.87 ˚ A, and c=4.02 ˚ A43 , which are consistent with our optimized parameters. The first-principles calculations were performed by using the Vienna abinitio simulation package (VASP)44 , with the generalized gradient approximation (GGA) of PerdewBurke-Ernzerhof (PBE). Spin-orbit coupling (SOC) is

ACS Paragon Plus Environment


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FIG. 2: (a) The projected band structure of OsC2 without SOC, where the symbol size corresponding to the projected weight of the Bloch states onto the Os d-orbit and C p-orbit. The Os-d-liked band and C-p-liked band cross around K-point and H-point. (b) The band structure with SOC, where the crossing bands around K-point open a gap, but the crossing point along H-K line keeping. The left inserted figure is bulk band structure along the two WPs (0.397, 0.206, ±0.024, in the unit of reciprocal lattice). The right inserted figure is the zoom-in figure in the red circle. The 3D band structures of these points are presented in Fig.S2 in the Supplementary. (c) Without SOC, the crossing points in the Kz=0 plane form node-lines. (d) The Fermi surface in the BZ.

taken into account self-consistently. The k-point grid was 15×15×11. The possible underestimation of band gap within GGA was checked by the nonlocal HeydScuseria-Ernzerhof (HSE06) hybrid functional45 . To get the tight-binding model Hamiltonian, we used the wannier90 package to obtain maximally localized Wannier functions 46,47 of Os d-orbitals and C-p orbitals. The surface Green’s function of the semi-infinite system, whose imaginary part is the local density of states to obtain the dispersion of the surface states, can be calculated through an iterative method48–50 . All the surface states in this paper are C-atoms terminated. Because that the Os-atoms terminated surfaces are unstable, in principle. The O atoms in the air will oxidize the Os atoms on the surface. The projected band structure of OsC2 without SOC is presented in Fig.2(a), where the ‘band-inversion’ 1,2,6 takes place at K-point. Along M-K, Γ-K, and H-K lines, there are three crossing points. Since the kz = 0 plane is a mirror plane and the two crossing bands have opposite mirror eigenvalue24,25,51–53 , the band inversion leads to a circle of node-line centering K point ( see Fig.2(c) ), which is similar to the case in TaAs15 and ZrTe51 . The ‘band-inversion’ and node-line can be tuned by the strain (see the detail results in Fig.S1 in the Supplementary).

When the SOC effect is included, the node-lines in the Kz =0 plane shrink into isolated crossing points24,25,52 , i.e. the so called ‘Weyl Points ’, in the Kz =±0.0241× 2π c planes. But the crossing point around H-point still exists in Fig.2(b) with SOC. In the left inserted figure of Fig.2(b), we presented the band structure along the two WPs (0.397, 0.206, ±0.024, in the unit of reciprocal lattice). Here, the two crossing bands have opposite slopes, which indicates the type-I WPs 24–26 . The 3D bulk band structure of these two WPs is presented in the Fig.S2a. We zoom in the crossing band structure along the H-K line in the right inserted figure of Fig.2(b). We can see that both crossing bands have negative slope, indicating the tilted contact Weyl cones26,27 (see Fig.S2b ). So, this Weyl point is type-II. The calculated Fermi surface is shown in Figs. 2(d), where the surface with purple color (around K or K’ points ) comes from the valence band and the green part ( along H-K line) comes from the conduction band.

FIG. 3: The cross section of the Brillouin zone (BZ) of OsC2 , (a) type-I WPs at Kz =0.0241× 2π and (c) type-II WPs at c . In Fig.a and Fig.c, the WPs in the Kz =Kz =0.4354× 2π c 0.0241× 2π and Kz =-0.4354× 2π planes are hidden. (b) and c c (d) are the projections onto the (100) surface. In the (100) view, we can see the hidden WPs. Here, the red and blue circles present the ‘positive’ and ‘negative’ WPs.

By first-principle calculations, we located the positions of all the WPs in the BZ, with the typical type-I WP at (0.397, 0.206, ±0.024) and type-II WP at (0.333, 0.333, ±0.0435). We schematically shown the type-I WPs in the Kz =0.0241× 2π c plane (Fig.3a). Because of the existence of the mirror symmetry of Kz =0 plane, there are opposite chirality type-I WPs in the Kz =-0.0241× 2π c plane. From the top view (Fig.3a) of the BZ, we can only see the WPs in the Kz =0.0241× 2π c plane. While the WPs 2π in the Kz =-0.0241× c plane are hided by the ones on the top-layer, since they have the same Kx and Ky coordinates. The WPs in the Kz =-0.0241× 2π c plane can be seen in the projected (100)-plane (Fig.3b). In the Kz =0.0241× 2π c plane (Fig.3a), there are 12 type-I WPs in the BZ, with ‘positive’ WPs ( red circles ) surround-


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ing K-point and ‘negative’ WPs ( blue circles ) surrounding K’-point. In Fig.3b, the points above the black line are projected from the Kz =0.0241× 2π c plane, while the points below the black line are projected from the Kz =0.0241× 2π c plane. The type-II WPs in the Kz =0.4354× 2π c plane are presented in Fig.3c. There are total six type-II WPs, with three ‘positive’ WPs ( red circles ) at the projected K-points and three ‘negative’ WPs ( blue circles ) at the projected K’-points. Because of the mirror symmetry of Kz =0 plane, each WP in the Kz =0.4354× 2π c plane has its partner with opposite chirality in the Kz =-0.4354× 2π c Due to the high crystalline (Kz =0.5646× 2π c ) plane. symmetry ( the C3 rotation, vertical mirror Γ-M-A, horizontal mirror Γ-M-K ) in OsC2 , all the WPs can be related to each other by the these symmetries. Therefore all the type-I WPs are located at the same energy (71 meV), similar to the situation in ZrTe51 , strained HgTe24 , and chalcopyrites25 . While all the type-II WPs are located at -68 meV. To clearly show the type-I WPs, we calculated the surface states, Fermi arcs on the (001)-plane, and the Berry curvatures of the Kx -Ky plane at Kz =0.0241× 2π c , ¯ −K ¯ line, the bulk bands as shown in Fig.4. Along the Γ are gapped, in consistence with the SOC band structure of Fig.2(b). But, the bulk bands have a touch point be¯ and M ¯ , indicating the Weyl point. Such touch tween K point is zoomed in by the inserted figure. Fig.4(b) clearly shows the two WPs , with two surface states connecting them. The Fermi arcs of the top (001)-surface, at energy 0.072 eV, are presented in Fig.4(c), with the yellow and blue circles indicating the ‘positive’ and ‘negative’ WPs . To further confirm the ‘chirality’ of these WPs , we calculated the Berry curvature, which can be looked as26,54 the ‘Magnetic field’ in the ‘K-space’, as shown in Fig.4(d). There are three magnetic ‘source’ points surrounding K-point, and three magnetic ‘drain’ points circling K’-point. These ‘source’ and ‘drain’ points are the WPs with ‘positive’ and ‘negative’ chirality, and they can be looked as the magnetic monopoles26,54 in the K-space. In Fig.5, we presented the bands structure projected onto the (100)-surface. Along A-B line, the bulk bands are gapped, with the existence of two surface states, in Fig.5(a). From the projected bands structure in Fig.5(b), we can clearly see the two WPs at W1 =(0.3968, 0.2064, -0.0241) and W2 =(0.3968, 0.2064, 0.0241), with one surface state connecting them. The Fermi arcs on the (100)surface and the WPs are consistent with the schematic Fig.3(b). Surrounding K(or K’)-point, there are six typeI WPs , with three in the Kz =+0.0241× 2π c plane and 2π the other three in the Kz =-0.0241× c . How the WPs be connected by the Fermi arcs? Generally, their connection configurations should follow the two constraints: (1) each projection point only has two arcs to connect it and (2) the connection pattern should preserve both D3h point group symmetry and time-reversal symmetry. Here, we

¯ K¯ M ¯ line, FIG. 4: (a) The (001)-surface state along the Γwhere the white line is the Fermi level (EF ), the green line at the energy of the type-I WPs (71 meV above EF ). The inserted figure with white border is the zoom-in figure of the WP. (b) The (001)-surface state through two WPs (W1 and W2 , see Fig.3(a) ). (c) The Fermi arc on the (001)-surface, at the energy (71 meV)of the type-I WPs. (d) The Berry curvature of the Kx -Ky plane at Kz =0.0241× 2π , where the c ‘positive’ and ‘negative’ WPs are guided by the red and green points, respectively.



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FIG. 6: The (100)-surface state along the A-B line (a) and C-D line (b) (see Fig.3(d)). (c) The Fermi arcs on the (100)surface, at energy -68 meV, i.e. the energy of the type-II WPs . The yellow and blue circles are added to guide the eyes for the positive and negative WPs.

FIG. 5: The (100)-surface state along the A-B line (a) and C-D line (b) (see Fig.3(b)). (c) The Fermi arcs on the (100)surface, at the energy (71 meV) of the type-I WPs. The yellow and blue circles are added to guide the eyes for the positive and negative WPs. (d) The possible connection patterns of Fermi arcs in the (001)-surface. Around each K(K’)-point, the top three points in the Kz =+0.0241× 2π plane, the bottom c three ones in the Kz =-0.0241× 2π plane. From the top-view, c the top-points overlay the bottom-points. In order to draw the Fermi arcs connections, we move them away artificially. , formLeft: the arcs connect directly along Kz =±0.0241× 2π c ing the ‘Dimmer’; Middle: the arcs connect three points in the same Kz plane, forming the Plane-Trimer; Right: the arcs connect blue point (in the Kz =+0.0241× 2π plane ) to the red c ), forming the Helix-Trimer. point (in the Kz =-0.0241× 2π c

propose three possible configurations: (a) the ‘Dimmer’ configuration (Left in Fig.5(d)), where the arc connects directly along Kz =±0.0241× 2π c ; (b) the Plane-Trimer configuration, where the three WPs within the same Kz plane connect each other; (c) the Helix-Trimer configuration, where the WPs (red points) in the Kz =-0.0241× 2π c plane ‘Helically‘ connect with the WPs (blue points) in Kz =+0.0241× 2π c plane. If the ‘Dimmer’ configuration is true, we can not see the surface states connecting W1 and W2 in Fig.4(b). So this configuration can be ruled out. If the Plane-Trimer configuration is correct, we can not see the surface state connecting W1 (0.3968, 0.2064, -0.0241) and W2 (0.3968, 0.2064, 0.0241) in Fig.5(b). So, the Plane-Trimer configuration can be ruled out. Only in the Helix-Trimer configuration, we can see the Fermi


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arcs on both (001) and (100) surfaces. Since the type-II WPs locate in the H-K line (along Kz direction). This type of WPs were ‘buried’ into the bulk bands, if we projected the surface state onto the (001)-surface. The type-II WPs in the Kz =0.4354× 2π c plane have their opposite chirality partners in the Kz =0.5646× 2π c plane. Looking down from the Kz axis, we can only see the WPs in the Kz =0.5646 plane, because that the WPs in the Kz =0.4354 plane were hided. In order to see the WPs in the Kz =0.4354× 2π c plane, we projected them onto the 100-surface. The calculated surface band structure along the A-B line, C-D line and the Fermi Arcs are presented in Fig.6a, b, c. Combining Fig.6a and Fig.6c, we find the S1 surface state connects two WPs in the Kz =0.4354× 2π c plane. In Fig.6b, the tilted bulk band structure touched at the WPs, in consistence with the results in Fig.S2b (see Supplementary). In conclusion, by the first-principle calculations, we

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find a new Weyl semimetal OsC2 . This compound has both type-I and type-II WPs at the equilibrium volume. In the Kz =0.0241× 2π c plane, there are 12 type-I WPs, which have their opposite chirality partners in the Kz =-0.0241× 2π c plane. These type-I WPs are connected by the Helix-Trimer Fermi Arcs. In the Kz =0.4354× 2π c plane, there are 6 type-II WPs, companying with their 2π partners in the Kz =0.5646× c plane, because of the mirror symmetry of Kz =0 plane. The authors acknowledge support from the NSF of China (No.11274095, No.10947001) and the Program for Science and Technology Innovation Talents in the Universities of Henan Province (No.2012HASTIT009, No.104200510014, and No.114100510021). This work is supported by The High Performance Computing Center (HPC) of Henan Normal University.

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Coexistence of Type-I and Type-II Weyl Points in the Weyl-Semimetal

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