Topological Type-II Nodal Line Semimetal and Dirac Semimetal State

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Topological Type-II Nodal Line Semimetal and Dirac Semimetal State in Stable Kagome Compound MgBi 3

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Xiaoming Zhang, Lei Jin, Xuefang Dai, and Guodong Liu J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.7b02129 • Publication Date (Web): 19 Sep 2017 Downloaded from http://pubs.acs.org on September 19, 2017

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The Journal of Physical Chemistry Letters

Topological Type-II Nodal Line Semimetal and Dirac Semimetal State in Stable Kagome Compound Mg3Bi2 Xiaoming Zhang, Lei Jin, Xuefang Dai, and Guodong Liu∗ School of Materials Science and Engineering, Hebei University of Technology, Tianjin 300130, China. E-mail: [email protected]

Abstract Topological type-II nodal line semimetal (NLS) was proposed quite recently and exhibits distinct properties comparing with conventional type-I NLS. Up to date, no ambient-condition stable candidate material has been reported. Here, we propose that a stable Kagome compound Mg3 Bi2 can host a type-II nodal line state with the protection of time reversal and spatial inversion symmetries. Similar to type-I NLSs, the type-II nodal line in Mg3 Bi2 is characterized by the drumhead surface states, which has not been observed in the previous type-II NLSs. The nodal line in Mg3 Bi2 can open a minor gap and a pair of three-dimensional (3D) Dirac points occur when SOC was included. The SOC-induced gap around the nodal line is quite small and the formation of 3D Dirac points are independent with the nodal line. So, the Mg3 Bi2 compound is expected to play a good candidate to investigate the exotic properties of both type-II NLS and 3D Dirac semimetal states.

∗ To

whom correspondence should be addressed

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Topological semimetals have received broad research interests in current condensed matter physics and materials science. 1,2 These materials are characterized with nontrivial band-crossings between conduction and valence bands in momentum space. Around the band-crossings, the quasiparticles exhibit distinct behavior from the usual Schrödinger-type fermions. For instance, a Dirac semimetal features with fourfold degenerate Dirac points (DPs), around which the low-energy quasiparticles are described by massless Dirac fermions. 3–7 With broken time reversal or inversion symmetry, a DP could split into two Weyl points (WPs) that are charaterized by the pair of Weyl fermions with opposite chiralities. 8–13 In momentum space, both Dirac and Weyl cones can be tilted, by controlling the tilting term in their Hamiltonian. According to the degree of tilting, two types of nodal points (usually termed as type-I and type-II) can be classified. For a type-I nodal point, the cone is only slightly tilted so that the hole-like states and electron-like states are well separated by the energy level of the nodal point. 3,8,12,14–21 However, for a type-II nodal point, the cone is completely tipped over so that the hole-like and electron-like states can coexist in certain energy levels. 11,21–24 The type-II fermions are proposed to possess drastically different properties from the type-I fermions. For example, in a type-II Weyl semimetal, the coexistence of hole and electron states could give rise to novel properties such as direction-dependent chiral anomaly, 25 modified anomalous Hall conductivity, 26 momentum space Klein tunneling and novel quantum oscillations, 27–29 which are detectable in transport and optical experiments. Therefore, type-II fermions attract increasing interests nowadays. In practice, both type-I and type-II fermions have been predicted in several materials, 3–5,7–11,22–24 and some are evidenced by experiments. 13–17,19–21 Under certain crystalline symmetry, the crossing points between bands could form one-dimensional (1D) nodal lines in the lattice momentum space, which is drastically different from the isolate nodal points in Dirac and Weyl semimetals. Such nodal line semimetals have received considerable interests since its first proposal in carbon allotropes. 30 Currently, several nodal line materials are proposed, with exotic physical properties revealed. 31–39 By analogy with the nodal point materials, the nodal line materials could also be classified into type-I and type-II categories, depending on the tilting degree of the band dispersion. Up to now, type-I nodal line materials have been intensively



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studied, 31–39 while there is only one report on type-II nodal line, proposed in compound K4 P3 quite recently. 40 To be noted, previous K4 P3 seems to be not a good candidate material for further experimental investigations and practical applications. Firstly, as an intermediate product of other K-P phases, K4 P3 is highly unstable under air and moisture conditions, and even explosive when exposed to the light. 41 Secondly, K4 P3 is quite different from a rigorous nodal line semimetal, because its band-crossing factually forms a pair of nodal loops (not closed nodal lines) traversing the whole Brillouin zone. Thirdly, K4 P3 is a topological metal rather than a semimetal, so that the characteristics of the nodal lines are most hidden by nearby trivial band intertwining. For example, the drumhead-like surface states, a typical signature of nodal line materials, are not evident in K4 P3 . Therefore, there is urgent need to explore stable topological materials with clear signature of type-II nodal line state. In current work, based on first-principles calculations, we report that Mg3 Bi2 , an ambientstable Kagome compound, 42 presents a clear type-II nodal line state close to Fermi level when spin-orbit coupling (SOC) is absent. Along the nodal line, the type-II band dispersion is observed. Unlike the type-II nodal loops which traverse the whole Brillouin zone in former K4 P3 , the bandcrossing in Mg3 Bi2 factually forms a closed type-II nodal line nesting inside the Brillouin zone. Benefiting from the clean band structure around the nodal line, one can observe the drumhead surface states either inside or outside the nodal line on the (001) surface depending on surface termination. When SOC is included, the nodal line is weakly gapped, and a pair of three-dimensional (3D) DPs that are independent with the nodal line appears. Therefore, we expect the topological semimetal Mg3 Bi2 could provide a good platform to study both properties of type-II nodal line and 3D Dirac semimetal states. To investigate the topological properties of Mg3 Bi2 compound, first-principles calculations are carried out in the framework of density functional theory (DFT), 43 as formulated in the Vienna ab initio simulation package (VASP). 44 The projector augmented wave (PAW) method is adopted for the ionic potentials. Exchange-correlation potential is treated by the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) functional. 45 Plane waves with a kinetic energy


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Figure 1: Crystal structure of Kagome compound Mg3 Bi2 in (a) top and (b) side views. (c) Bulk Brillouin zone and the projected surface Brillouin zones of the (001) and (010) surfaces with highsymmetry points labelled. cutoff of 600 eV were adopted as basis set. The Brillouin zone is sampled with 11×11×5 Γcentered k-mesh for the structural optimization and with 19×19×9 k-mesh for the self-consistent calculations. The energy and force convergence criteria are set to be 10−6 eV and 0.001 eV/Å, respectively. The electronic structure calculations are performed both without and with SOC. The topological features of surface states are studied by constructing the maximally localized Wannier functions, 46,47 making use of the WANNIER-TOOLS package by Wu, 48 with combination of an iterative Green’s function approach. 49 According to the previous experimental studies, 42 the Mg3 Bi2 compound is a semimetal phase and stable under ambient conditions. It possesses a layered Kagome lattice structure with the space group of P3m1 (No. 164). As shown in Figure 1a and 1b, the Mg and Bi atoms form alternating layers of triangular sheets. Three Mg atoms situate at 1a (0, 0, 0) and 2d (1/3, 2/3, z) Wyckoff sites, and two Bi atoms occupy 2d’ (1/3, 2/3, z’) sites, respectively. The optimized lattice constants are a=b= 4.702Å, c= 7.436Å, and well match the experimental ones (a=b= 4.666Å, c= 7.401Å) 42 with the deviation less than 1%. The values of z and z’ from lattice relaxation are 0.6286 and 0.2212, respectively. The optimized lattice structure is adopted for the following calculations. We start with the electronic band structure of Mg3 Bi2 compound in the absence of SOC. As shown in Figure 2a, we find that the material manifests a semimetal band structure: conduction and valence bands overlap along the Γ-M and K-Γ directions, and are well separated around the 5 ACS Paragon Plus Environment


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Figure 2: (a) Calculated electronic band structure of Mg3 Bi2 compound without SOC. (b) Enlarged band structure along K-Γ and Γ-M paths. (c) Illustration of the type-II nodal line in the bulk Brillouin zone. Noted, the nodal line dose not exactly situate at the kz =0 plane. (d) The Fermi surface of Mg3 Bi2 compound, where electron and hole pockets are denoted as red and blue colours, respectively. (e) The top view of the Fermi surface.


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Fermi level when they are away from these directions. On considering the fact that GGA usually underestimates the band gap width of semimetal and semiconductor, here we check the electronic band structure by using a combination of modified Becke-Johnson exchange potential with local density approximation (MBJLDA), 50 as it predicts band gap and band order in topological materials with favorable accuracy similar to more expensive hybrid functional and GW calculations. 51–54 The results show a similar semimetal signature with the band crossing along the K-Γ direction (see Figure S1 in Supporting Information). Such semimetal character of Mg3 Bi2 has already been evidenced by former electrical conductivity and soft x-ray absorption experiments. 55,56 Because of the same topological nature by the two computations and sufficient experimental evidences, we use electronic structure from GGA for the discussions in the following. By a careful investigation to the band overlaps in Figure 2b, we find that there exist a linear band-crossing along the Γ-K direction and a tiny band gap along the Γ-M direction. Considering the slope of band dispersion, the linear crossing point along Γ-K direction is factually a type-II DP. Since the system possesses both time reversal (T) and spatial inversion (P) symmetries, under which the spinless Hamiltonian is always real valued, 30 this type-II DP cannot be isolated but belongs to a nodal line. After a careful scan of electronic band structure, we find the crossing between the conduction and valence bands indeed produces a closed type-II nodal line around the Γ point, as schematically shown in Figure 2c. To be noted, except for the crossing points along Γ-K direction, the obtained nodal line in Mg3 Bi2 does not exactly lie on the mirror plane of kz =0, but slightly wrinkles in the 3D Brillouin zone. Therefore, the nodal line does not arise from the mirror symmetry, but is merely protected by the coexistence of T and P symmetries. The shape of the wrinkled nodal line can be easily traced from the isoenergy contour by setting an energy level slightly above the nodal line. 38,57 A contour at the Fermi level and the obtained Fermi surface are shown in Figure 2d and 2e for Mg3 Bi2 . The Fermi surface clearly consists of two parts: six ellipsoid-shaped hole pockets and one bagel-shaped electron pocket centered at the Γ point. It can be observed that the bagel-shaped Fermi surface is slightly twisted along kz direction, which reflects the wrinkled nodal line centering the Γ point. It is also worth noting that, Mg3 Bi2 factually possesses a closed type-II



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nodal line, drastically different from that in previous K4 P3 , where band-crossing forms a pair of unclosed type-II nodal loops traversing through the whole Brillouin zone. 40 Topological nodal line materials usually manifest drumhead surface states either outside or inside the projected nodal line. 30–39 However, such important signature is not evident in previous type-II nodal line material K4 P3 , 40 mainly because it inherently possesses an metallic electronic structure so that the nature of nodal line is mostly hidden by nearby trivial band intertwining. Now we turn to the surface states of Mg3 Bi2 compound. For the Mg-terminated (001) surface, it clearly manifests drumhead surface states nestling inside the projected nodal line, as shown in Figure 3a. Its surface band has small dispersion with a bandwidth of 50 meV, being comparable with that of typical type-I nodal line in carbon allotropes. 30 Interestingly, the topological surface character of Bi-terminated (001) surface is drastically different, where the drumhead surface bands situate outside the projected nodal line, as displayed in Figure 3b. Such dependence between topological surface states and surface terminations is previously described to originate from the unique topological properties of 1D atomic chains under PT symmetry. 57 The above results clearly indicate that the type-II nodal-line materials also feature with drumhead surface character, similar to the previous type-I category. In general, SOC can shift the topological properties in materials. For instance, SOC usually drives the nodal line materials into the other topological phases, such as topological insulators, 58 Dirac and Weyl semimetals, 34,59 and also other kinds of nodal line materials. 60,61 For Mg3 Bi2 compound, we display the calculated electronic band structure under SOC in Figure 4a and 4b. It can be seen that a small indirect gap occurs around the type-II nodal line under SOC. Simultaneously a pair of type-I 3D DPs along the Γ-A direction is generated as shown in Figure 4b and 4c, which indicates Mg3 Bi2 compound is a 3D Dirac semimetal. The existence of the 3D DPs under SOC is also confirmed by the MBJLDA computation (see Figure S2 in Supporting Information). The properties of 3D Dirac semimetals have been mostly studied in the well known Na3 Bi 3 and Cd3 As2 materials. 5 However, on considering the facts that Na3 Bi is not stable in air condition and Cd3 As2 naturally contains toxic arsenic element, we argue that current Mg3 Bi2 could serve as a


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Figure 3: (a) Calculated (001) surface band structure without SOC for (a) Mg-terminated and (b) Bi-terminated surface. Drumhead surface states are denoted by arrows.



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Figure 4: (a) Calculated electronic band structure of compound Mg3 Bi2 with SOC. (b) Enlarged band structure along and K-Γ and Γ-A paths. (c) The position of the pair of 3D DPs in the bulk Brillouin zone. good 3D Dirac material for further experimental investigations and practical applications. Next, we turn to the surface states of Mg3 Bi2 compound under SOC. When projected on (001) surface, the whole Γ-A path superposes into Γ point on the surface (see Figure 1c). In this case, the projected bulk Dirac cone that situates at the Γ point is visible (see Figure 5a), while we cannot draw the configuration of its nontrivial surface states because the Fermi surface at the Dirac cone (at E= 0.251eV) is just a point (see Figure 5b). Then we consider (110) surface projection, where e Ze path in the side surface Brillouin zone (see Figure 1c). The the bulk Γ-A path projects to Γprojected surface band structure and Fermi surface at E= 0.251eV are shown in Figure 5c and 5d, where the exact positions of DPs are indicated. Although the projection of Dirac cone is hidden by the other bulk states (see Figure 5c), the profile of two half-circle-shaped Fermi arcs could be faintly visible in the given Fermi surface (see Figure 5d). Before closing, we would like to make further discussions on the topological properties of Mg3 Bi2 compound. Because of the inherent type-II band dispersion, SOC opens an indirect gap


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Figure 5: (a) Calculated (001) surface band structure with SOC and (b) corresponding Fermi surface at the chemical potential of 0.251 eV. (c) and (d) are similar with (a) and (b), but for the case of side (110) surface. The position of bulk DP is denoted by the cyan points. In (d), the arrows point the profile of two half-circle-shaped Fermi arcs connecting the DPs. around the nodal line, which is different from the direct gap in previous type-I nodal line materials. Importantly, we note that the SOC-induced gap is only 36 meV along Γ-K direction (see Figure 4b), quite smaller than that of most typical nodal line materials such as antiperovskite Cu3 NPd (>60meV), 33,34 hexagonal pnictide CaAgBi ( 80meV), 51 and BaSn2 category (50-160meV). 50 Benefiting from the small SOC-induced gap, the type-II nodal line semimetal state and its unique properties are promising to be observed in Mg3 Bi2 compound. The SOC strength may be further diminished by doping lighter elements such as Sb or As, potentially making the type-II nodal-line state more prominent. Tuning the SOC strength by alloying with lighter elements has already been proved effectively in former topological Weyl and Dirac semimetals . 62,63 The pair of 3D DPs along the Γ-A direction is found to be independent with the existence of type-II nodal line, therefore both type-II nodal line and 3D Dirac states are expected to be achievable in current Mg3 Bi2 system. In summary, on the basis of first-principles calculations, we predict the existence of a type-II nodal line state in a stable Kagome compound Mg3 Bi2 when SOC is absent. The nodal line is



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formed by continuous type-II nodal points and is protected by the time reversal and spatial inversion symmetries. Different from previous unclosed type-II nodal loops in the unstable compound K4 P3 , we firstly report a closed type-II nodal line in ambient-condition stable Mg3 Bi2 material. Depending on surface termination, the drumhead surface states either inside or outside the nodal line were observed on the (001) surface projection of Mg3 Bi2 . The nodal line is weakly gapped and a pair of 3D DPs emerges when SOC is included. So, Mg3 Bi2 compound is also a 3D Dirac semimetal. On considering the facts that the SOC-induced gap around the nodal line is much smaller than typical nodal line materials and the formation of 3D Dirac points is independent with the nodal line, the Mg3 Bi2 is promising to serve as a good candidate material to study both properties of type-II nodal line semimetal and 3D Dirac semimetal states.

Acknowledgement This work is supported by the Natural Science Foundation of Hebei Province (No. E2016202383), the Program for Leading Talents in Science and Technology Innovation of Chongqing City (No. cstckjcxljrc19), and the Basic and Frontier Research Project of Chongqing City (No. cstc2014jcyja50005)

Supporting Information Available Electronic band structures with modified Becke-Johnson potential for Mg3 Bi2 ; electronic band structures for Mg3 Sb2 and Mg3 As2 ; some discussions and references.

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Topological Type-II Nodal Line Semimetal and Dirac Semimetal State

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