⇦Chinese Physics Letters, 2017, Vol. 34, No. 7, Article code 077101 Topological Nodal Line Semimetal in Non-Centrosymmetric PbTaS$_2$ * Jian-Peng Sun(孙建鹏)** Affiliations SKLSM, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083 Received 30 March 2017 *Supported by the National Natural Science Foundation of China under Grant No 11504366, and the National Basic Research Program of China under Grant Nos 2015CB921503 and 2016YFE0110000.
**Corresponding author. Email: jpsun@semi.ac.cn Citation Text: Sun J P 2017 Chin. Phys. Lett. 34 077101 Abstract Topological semimetals are a new type of matter with one-dimensional Fermi lines or zero-dimensional Weyl or Dirac points in momentum space. Here using first-principles calculations, we find that the non-centrosymmetric PbTaS$_2$ is a topological nodal line semimetal. In the absence of spin-orbit coupling (SOC), one band inversion happens around a high symmetrical $H$ point, which leads to forming a nodal line. The nodal line is robust and protected against gap opening by mirror reflection symmetry even with the inclusion of strong SOC. In addition, it also hosts exotic drumhead surface states either inside or outside the projected nodal ring depending on surface termination. The robust bulk nodal lines and drumhead-like surface states with SOC in PbTaS$_2$ make it a potential candidate material for exploring the freakish properties of the topological nodal line fermions in condensed matter systems. DOI:10.1088/0256-307X/34/7/077101 PACS:71.15.Mb, 73.20.At, 71.55.Ak © 2017 Chinese Physics Society Article Text The exploring of topological materials has been initiated from the discovery of topological insulators in HgTe/CdTe quantum wells,[1] the Bi$_2X_3$ family ($X$=Se, Te,$\ldots$)[2,3] and even conventional semiconductors,[4,5] which have attracted broad research interests in modern condensed matter physics. They are characterized by possessing a bulk energy gap induced by strong spin-orbit coupling (SOC) between inverted bands but with gapless boundary states that are protected by bulk band topology.[2,6] Recently, a great deal of attention has been drawn to topological semimetals (TSMs), which feature robust bulk band degeneracies and the associated topological boundary states. Based on the degeneracy of the band crossing points and their distribution in the Brillouin zone (BZ), TSMs can be classified into Dirac semimetals, Weyl semimetals, and nodal line semimetals. To be specific, a Dirac semimetal[7,8] is characterized by two bands with double degeneracy that cross either at the high-symmetry point or along high-symmetry orientations near the Fermi level, and has to be protected by certain crystalline symmetry. For a Weyl semimetal,[9-13] it is characterized by the crossing of two nondegenerate bands at the Fermi level, and does not require any protection from the crystalline symmetry other than lattice translation. Unlike Dirac and Weyl semimetals possessing isolated bulk band crossing points in the BZ, topological nodal line semimetals possess band touchings along one-dimension (1D) loops or lines in 3D momentum space. To date, there are several theoretical predictions about the topological nodal line semimetals,[14-19] which are featured by bulk nodal lines and drumhead-like surface states. However, for all the above-mentioned works, the nodal lines are not robust and only exist in the absence of SOC. With considering the SOC, each nodal line is gapped or transforms into other types of TSMs due to the interaction of spin components. With regard to real materials, the SOC is immanent and cannot be ignored, which makes it important to study nodal lines semimetals in the presence of SOC. According to the theoretical proof mentioned in Ref. [20], without SOC, inversion plus time reversal symmetry can protect a nodal line, but is insufficient to protect any band crossing in 3D momentum space in the presence of SOC. Therefore, to search stable topological nodal line semimetals with SOC, an additional crystalline symmetry is needed to protect them, such as mirror reflection symmetry. In the presence of SOC, the topological nodal line semimetal state has been theoretically predicted and experimentally realized in some materials with non-centrosymmetric structure and a mirror symmetry.[21-23] In this work, we theoretically demonstrate that the non-centrosymmetric PbTaS$_2$ hosts a topological nodal line semimetal state with robust bulk nodal lines near the Fermi energy even with the inclusion of SOC. Our calculations are based on the density functional theory (DFT) performed by using the Vienna ab initio simulation package (VASP)[24] within the generalized gradient approximation (GGA) in Perdew–Burke–Ernzerhof (PBE)[25] type and the projector augmented-wave (PAW) pseudopotential.[26] The kinetic energy cutoff for the plane-wave basis is fixed at 560 eV and a $12\times12\times8$ ${\it \Gamma}$-centered $k$-mesh is used for the BZ sampling.[27] For the convergence of the electronic self-consistent calculations, the total energy difference criterion is set to $10^{-8}$ eV. The crystal structure is fully relaxed until the residual forces on atoms are less than $10^{-3}$ eV/Å. To investigate the projected surface states and Fermi surfaces, a tight-binding model Hamiltonian in the basis of the $p$ orbits of Pb and S atoms and $d$ orbits of the Ta atom can be obtained by using the maximally localized Wannier function (MLWF) method.[28,29] Then, we apply an iterative method[30,31] to obtain the surface Green function of the semi-infinite system and the imaginary part of the surface Green function is the local density of states (LDOS) at the surface. PbTaS$_2$ crystallizes in the hexagonal $P\bar{6}m2$ structure with space group $D_{3h}^{1}$ (No.187), as shown in Figs. 1(a) and 1(b). In this structure, the Pb layer intercalates between adjacent TaS$_2$ layers with Pb atoms aligned with Ta atoms in the vertical direction. The lattice structure possesses mirror reflection symmetry with regard to Pb and Ta planes. Since the experimental parameters of the non-centrosymmetric PbTaS$_2$ are missing, we relax the lattice structure without and with the inclusion of SOC, respectively, as listed in Table 1. We find that the two relaxed structure parameters are almost the same; therefore, we will adopt the optimized lattice parameters without the SOC for all of our calculations. To check the stability of optimized lattice parameters, the phonon spectrum is calculated using the frozen phonon method[32] as implemented in the PHONOPY code,[33] as shown in Fig. 1(d). We find that no imaginary frequencies exist along the high-symmetry line in the whole BZ, confirming its dynamically structural stability.
cpl-34-7-077101-fig1.png
Fig. 1. The top-view (a) and side-view (b) for the hexagonal lattice structure of PbTaS$_2$. The dashed lines represent the primitive cell of PbTaS$_2$. (c) Bulk Brillouin zone and its projections onto the (001) surface. (d) The phonon dispersion curves for PbTaS$_2$ along the high-symmetry direction.
The calculated band structures and orbital characteristics of PbTaS$_2$ without and with SOC are presented in Figs. 2(a) and 2(b), respectively. In the vicinity of the Fermi level, the states mainly come from the contribution of Pb $p$ and Ta $d$ orbitals. In the absence of SOC, a band inversion occurs between an electron pocket derived from Pb-$p$ orbitals and a hole pocket from Ta-$d$ orbitals around the $H$ and $K$ points, as shown in Figs. 2(c) and 2(e). More interesting, the inverted band structure around the $H$ point forms a gapless nodal ring protected by the crystalline symmetry on the mirror plane $k_z=\pi$, as shown in Fig. 2(c). However, in the case with SOC, each band splits into two spin-polarized branches with opposite mirror reflection eigenvalues, since the system lacks space inversion symmetry.[21,22] It is noteworthy that two crossing bands with opposite mirror eigenvalues and different orbital components form a pair of nodal rings around the $H$ point on the $k_z=\pi$ plane as a result of mirror symmetry protection, as clearly shown in the enlarged Fig. 2(d). By comparing the band structures without and with the SOC, we can obtain that the estimation of the magnitude of SOC is about 0.32 eV. Moreover, the strong SOC also leads to generating a third nodal ring around the $K$ point on the $k_z=0$ plane, as shown in Fig. 2(f). The three nodal lines are located at $-$0.37, $-$0.23 and $-$0.39 eV, respectively, and the first two are around $H$ on the $k_z=\pi$ plane, while the third one lies on the $k_z=0$ plane around $K$. They are associated with NL1, NL2 and NL3, as presented in Figs. 2(d) and 2(f), respectively.
Table 1. The relaxed lattice parameters without and with the inclusion of SOC.
Without SOC With SOC
$a=b$ (Å) 3.3746 3.3785
$c$ (Å) 9.0155 9.0210
Atom positions Pb ($0,0,0$) Pb ($0,0,0$)
Ta ($\frac{1}{3},\frac{2}{3},\frac{1}{2}$) Ta ($\frac{1}{3},\frac{2}{3},\frac{1}{2}$)
S($0,0,\frac{1}{2}\pm0.1711$) S($0,0,\frac{1}{2}\pm0.1694$)
An important characteristic of topological nodal line semimetal is the existence of 'drumhead-like' surface states either inside or outside the projected bulk nodal rings, which can be determined by the Berry phase.[18,21,22] Therefore, to further verify the topological nodal line states and associated drumhead-like surface states, the projected (001) surface states and Fermi surfaces are calculated, as shown in Figs. 3 and 4, respectively. Figures 3(a)–3(c) present the projected bulk band structures and surface states with different surface terminations along $\bar{{\it \Gamma}}$–$\bar{K}$–$\bar{M}$–$\bar{{\it \Gamma}}$ in the full surface BZ illustrated in Fig. 1(c), while Figs. 3(d)–3(f) are the scaling figures of Figs. 3(a)–3(c) around the $\bar{K}$ point with the purpose of displaying the nodal lines clearly. From Fig. 3(d), three nodal lines marked by color balls are seen clearly and located on different energies, which correspond with the arrow labels in Figs. 2(d) and 2(f). They originate from the bulk band crossing between electron and hole pocket on the $k_z=0$ and $\pi$ planes, and are protected by mirror reflection symmetry. Due to lacking inversion symmetry in PbTaS$_2$, the surface can be terminated by a Pb layer or an S layer, as shown in Fig. 1(b). The projected surface states associated with two termination surfaces near the $\bar{K}$ point are shown in Figs. 3(e) and 3(f). Unlike the projected bulk band, which is independent of surface termination, the dispersion of surface states is found to be sensitive to the surface condition. However, in the case of two different surface terminations, each nodal line is connected by drumhead-like surface states, which are protected by band topology. In the Pb-terminated surface, the surface state SS1 connects to NL1 and extends outwards into bulk states. On the contrary, the surface state SS2 connecting to NL2 disperses inwards along the edge of the Dirac cone located on the position of the green ball in Fig. 3(e). The surface state SS3 associated with NL3 disperses inwards with a large span with respect to $\bar{K}$. With a great difference from Pb termination, in the case of the S-terminated surface, NL1 and NL3 are interconnected by the surface state SS1. The surface state SS2 from NL2 extended outwards first moves into the bulk gap and then merges into the bulk band region. To obtain a comprehensive view of the nodal ring and drumhead-like surface states, we plot the projected Fermi surface, as shown in Figs. 4(a) and 4(b). The contour energy is set at $E=-0.33$ eV in the vicinity of NL1 for the Pb-terminated surface and $E=-0.21$ eV in the vicinity of NL3 for the S-terminated surface, respectively, as indicated by the blue dashed lines in Figs. 3(e) and 3(f). The drumhead-like surface states and nodal rings are clearly presented on the right panel of Fig. 4. To the Pb-terminated surface, the surface state SS1 and nodal ring nearly overlap around $\bar{K}$ and the surface state SS2 nested in the edge of the bulk states. In the S-termination case, the surface state SS2 is separated from the nodal ring along $\bar{K}$–$\bar{M}$. Under the protection of mirror reflection symmetry, these nodal rings become stable against gap opening.
cpl-34-7-077101-fig2.png
Fig. 2. Bulk band structures of PbTaS$_2$ without (a) and with (b) the inclusion of SOC. The color shows the atomic orbital components. Enlarged band structures around $H$ without (c) and with (d) SOC. Here (e) and (f) are the same as (c) and (d) but for band structures around $K$. The bulk nodal lines (NLs) in (d) and (f) are marked by red arrows.
cpl-34-7-077101-fig3.png
Fig. 3. (a) The projected (001) surface density of states (SDOS) for the bulk bands. [(b), (c)] Pb-terminated and S-terminated surfaces of PbTaS$_2$. [(d)–(f)] Enlargement of surface density of states around $\bar{K}$ in (a)–(c), respectively. Color balls indicate the location of three nodal lines (NLs), and arrows represent the surface states (SS).
cpl-34-7-077101-fig4.png
Fig. 4. (a) The Fermi surfaces of Pb-terminated (001) surface in the first surface Brillouin zone (left) and around the $\bar{K}$ point (right) at energy $E=-0.33$ eV. (b) The same as (a) but for the S-terminated surface at energy $E=-0.21$ eV. The surface states and nodal lines are marked by arrows.
In summary, we have theoretically proposed that the non-centrosymmetric PbTaS$_2$ is a stable topological node-line semimetal. Topological node-line semimetals form a distinct class of topological materials beyond topological insulators, Dirac and Weyl semimetals. Different from previous proposed materials, the nodal lines in PbTaS$_2$ are robust even with SOC. Moreover, the newfangled drumhead-like surface states connecting to bulk nodal lines are presented by calculating the projected (001) surface density of states and Fermi surfaces. With different surface terminations, the drumhead-like surface states are distinguished but they all nestle inside or outside the nodal rings. In light of these novel properties of PbTaS$_2$, we provide an ideal platform to study unique topological phenomena such as chiral anomaly and non-local transport. One can expect PbTaS$_2$ to be a potential candidate in TSM-based electronic devices. References Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum WellsTopological Insulators in Three DimensionsExperimental Realization of a Three-Dimensional Topological Insulator, Bi2Te3Polarization-Driven Topological Insulator Transition in a GaN / InN / GaN Quantum WellInterface-Induced Topological Insulator Transition in GaAs / Ge / GaAs Quantum WellsTopological invariants of time-reversal-invariant band structuresDirac Semimetal in Three DimensionsDirac semimetal and topological phase transitions in A 3 Bi ( A = Na , K, Rb)Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridatesWeyl Semimetal in a Topological Insulator MultilayerA Weyl Fermion semimetal with surface Fermi arcs in the transition metal monopnictide TaAs classDiscovery of a Weyl fermion semimetal and topological Fermi arcsWeyl Semimetal Phase in Noncentrosymmetric Transition-Metal MonophosphidesDirac Line Nodes in Inversion-Symmetric CrystalsTopological Node-Line Semimetal and Dirac Semimetal State in Antiperovskite Cu 3 PdN A new form of Ca 3 P 2 with a ring of Dirac nodes Ca 3 P 2 and other topological semimetals with line nodes and drumhead surface statesTopological nodal-line semimetals in alkaline-earth stannides, germanides, and silicidesTopological semimetal to insulator quantum phase transition in the Zintl compounds B a 2 X ( X = Si , Ge ) Topological nodal line semimetals with and without spin-orbital couplingTopological nodal-line fermions in spin-orbit metal PbTaSe2Drumhead surface states and topological nodal-line fermions in TlTaSe 2 Molybdenum Carbide: A Stable Topological Semimetal with Line Nodes and Triply Degenerate PointsEfficient iterative schemes for ab initio total-energy calculations using a plane-wave basis setGeneralized Gradient Approximation Made SimpleProjector augmented-wave methodSpecial points for Brillouin-zone integrationsMaximally localized generalized Wannier functions for composite energy bandsMaximally localized Wannier functions for entangled energy bandsQuick iterative scheme for the calculation of transfer matrices: application to Mo (100)Highly convergent schemes for the calculation of bulk and surface Green functionsFirst-Principles Determination of the Soft Mode in Cubic ZrO 2 First-principles calculations of the ferroelastic transition between rutile-type and CaCl 2 -type SiO 2 at high pressures
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