⇦Chinese Physics Letters, 2017, Vol. 34, No. 2, Article code 027102Express Letter Molybdenum Carbide: A Stable Topological Semimetal with Line Nodes and Triply Degenerate Points * Jian-Peng Sun(孙建鹏), Dong Zhang(张东)**, Kai Chang(常凯)** Affiliations SKLSM, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083 Received 13 January 2017 *Supported by the National Natural Science Foundation of China under Grant Nos 11504366, and the National Basic Research Program of China under Grant Nos 2015CB921503 and 2016YFE0110000.
**Corresponding author. Email: zhangdong@semi.ac.cn; kchang@semi.ac.cn Citation Text: Sun J P, Zhang D and Chang K 2017 Chin. Phys. Lett. 34 027102 Abstract We propose that the hexagonal crystal form of MoC is a stable and new type of topological semimetal. It hosts an exotic Fermi surface consisting of two concentric nodal rings in the presence of spin-orbit coupling, and possesses four pairs of triply degenerate points (TDPs) in the vicinity of the Fermi energy. The coexistence of the nodal ring Fermi surface and TDPs in MoC leads to extraordinary properties such as distinguishable drumhead surface states and manipulatable new fermions, which make MoC a fertile platform for in-depth understanding of topological phenomena and a potential candidate material for topological electronic devices. DOI:10.1088/0256-307X/34/2/027102 PACS:71.15.Mb, 03.65.Vf, 81.05.Zx, 73.20.-r © 2017 Chinese Physics Society Article Text Materials with topologically nontrivial, conduction symmetry-protected surface/edge states[1-3] have attracted enormous attention in recent years. These materials display novel physical properties such as dissipationless transport, topological magneto-electric effects, and Majorana fermions. The exploring of topological materials was initiated from the discovery of topological insulators,[4-6] in which the band topology of the bulk states possesses an energy band gap induced by spin-orbit coupling (SOC) between inverted bands. Shortly thereafter the topological materials family has been expanded to semimetals with robust Dirac points near the Fermi surface,[7] which cannot be broken due to nontrivial topology, and known as topological semimetals (TSMs). Current TSMs can be classified into Weyl, Dirac, and node line semimetals according to the different band crossing points at the Fermi level and the mechanisms protecting them. Specifically, a Weyl semimetal[8-13] can be characterized by the crossing of two nondegenerate bands at the Fermi level, which does not require any protection from the crystalline symmetry other than lattice translation. A Dirac semimetal[14-16] is characterized by two bands with double degeneracy that cross near the Fermi level, and has to be protected by certain crystalline symmetry either at the high-symmetry point or along high-symmetry orientations. Both Weyl and Dirac semimetals possess Fermi surfaces consisting of a few crossing points in the Brillouin zone (BZ). However, the Fermi surface of a node-line semimetal[17-24] is a closed ring-shaped nodal line arising from the valence and conduction bands crossing along the specific crystallographic orientations in the BZ. In addition to the above versatile Fermi surface signatures, the quasiparticle excitations have three- or six-fold degeneracy proposed and introduced as 'new fermions' by Bradlyn et al.[25] Such exotic physical properties make TSMs a fertile platform to realize particles that remain elusive in high energy physics, exhibit quantum anomalies, host new topological surface states such as the Fermi arc and drumhead surface states, and show exotic transport and spectroscopic behaviors arising from the novel bulk and surface topological band structures. Very recently, TSMs with triply degenerate points (TDPs) were predicted theoretically in materials with tungsten carbide-like crystal structures[26-28] and the alloys InAs$_{0.5}$Sb$_{0.5}$.[29] In the band structure of these materials, both one- and two-dimensional (2D) representations are allowed along a certain high-symmetry axis, which makes it possible to generate band crossing between a doubly degenerate band and a non-degenerate band near the Fermi level and to form a TDP.[30] This new type of three-component fermion can be regarded as the 'intermediate state' between the two-component Weyl and the four-component Dirac fermions. In this work, we theoretically demonstrate that MoC is a new type of material that hosts both the node-line semimetal state and the unique fermion state with triply degenerate crossing points near the Fermi energy. Our calculations are based on the density functional theory (DFT) performed by using the Vienna ab initio simulation package (VASP)[31] within the generalized gradient approximation (GGA) in the Perdew–Burke–Ernzerhof (PBE)[32] type and the projector augmented-wave (PAW) pseudopotential.[33] The kinetic energy cutoff is set to 560 eV for the plane-wave basis, and the BZ integration is sampled by summation over $12\times12\times12$ $k$-point meshes.[34] For the electronic self-consistent calculations to converge, the total energy difference criterion is set to $10^{-8}$ eV. The crystal structures are fully relaxed until the residual forces on atoms are less than 0.01 eV/Å.
cpl-34-2-027102-fig1.png
Fig. 1. Top-view (a) and side-view (b) of the crystal structure of MoC; (c) unit cell of the WC-type materials; (d) the bulk Brillouin zone and its projections onto the (001) surface (blue hexagon) and (010) surface (red square); (e) phonon dispersion curves for MoC.
The $\gamma$-MoC[35] (denoted as MoC for simplicity in this study), fabricated by carburizing molybdenum with carbon monoxide has superior stability over its allotropies such as $\beta$-MoC and $\gamma'$-MoC. Its single crystal has a hexagonal tungsten carbide-like structure,[36] the same as Ta(Nb)N[37-39] and ZrTe,[40] and possesses the same symmetry $D_{3h}^{1}$ with space group $P\bar{6}m2$ (No. 187). The Mo and C atoms are located in the $1a(0,0,0)$ and $1d(\frac{1}{3},\frac{2}{3},\frac{1}{2})$ Wyckoff positions, respectively, as shown in Fig. 1. The optimized lattice constants of MoC are $a=b=2.918$ Åand $c=2.827$ Å, which fit the experimental data ($a=b=2.898$ Åand $c=2.809$ Å) excellently. To further prove the stability of the optimized structural parameters, we calculate the phonon dispersions using the frozen phonon method[41] as implemented in the PHONOPY code,[42] as shown in Fig. 1(e). No imaginary frequencies are observed in the phonon dispersions throughout the whole BZ, confirming the dynamically structural stability. Therefore, we adopt the optimized structural parameters to perform the following band structure calculations.
cpl-34-2-027102-fig2.png
Fig. 2. Band structures of MoC without (a) and with (b) the SOC. Enlarged band structures including SOC along (c) ${\it \Gamma}$–$K$–$M$ and (d) ${\it \Gamma}$–$A$, in which the four TDPs are denoted by red dots. (e) The gap between two touching bands of MoC in the $k_z=0$ plane of the Brillouin zone. The gap around the $K$ point is enlarged in (f).
The calculated band structures without the SOC are shown in Fig. 2(a). Because the $s$ electrons of the transition metal Mo are transferred to the carbon ions, as a consequence, near the Fermi energy, the valence and conduction bands are mainly generated and dominated by the strongly correlated $d$ electrons of Mo. In the absence of SOC, a band inversion occurs around the $K$ point between the bands arising from the $d_{x^2-y^2}+d_{xy}$ and $d_{xz}+d_{yz}$ levels of the Mo atoms. Since the $k_z=0$ plane is a mirror plane and the two bands have opposite mirror eigenvalues, the band inversion leads to a double degenerate nodal ring centered around the $K$ point. In addition to the band inversion near the $K$ point, we also find a band crossing between one non-degenerate band composed of an Mo $d_{z^2}$ orbital and a double degenerate band coming from the contribution of the $d_{x^2-y^2}$ and $d_{xy}$ orbitals of Mo atoms along the ${\it \Gamma}$–$A$ direction in the BZ. The crossing point is a TDP with no spin-orbit coupling, protected by the $C_{3v}$ symmetry with its counterpart located at the time-reversal point. Generally, the SOC effect becomes important for $d$ electrons in heavy elements such as Mo, and can modify the energy spectrum of nodal line semimetals in two different ways:[43] either they can open up a full gap in the spectrum in most of the reported materials, or they can split the nodal ring into two concentric nodal rings if the systems carry an additional nonsymmorphic symmetry, for example, a mirror reflection symmetry in MoC. By introducing the SOC effects in our band structure calculation, due to the lack of inversion symmetry, the splitting of bands in the presence of spin-orbit coupling at specific momenta are shown in Fig. 2(b). We find that the SOC cannot make the nodal lines fully gapped or give rise to other types of semimetals; the lines split into two concentric nodal rings around the $K$ point,[21,22] and this feature is quite different from other node-line semimetals, as clearly shown in Fig. 2(c). To further prove the existence of the two concentric nodal rings in MoC, we compute the energy difference between two touching bands for a full 2D BZ at the $k_z=0$ plane, as illustrated in Fig. 2(e). One can clearly see that the brighter color indicates the larger gap and the gapless states are mainly situated near the high symmetrical $K(K')$ point. In Fig. 2(f), around the $K(K')$ point, there are two concentric circles where gapless ${\boldsymbol k}$ points are located, which provide reliable and direct evidence for the existence of two nodal rings in the Fermi surface of MoC. As for the dispersions along the ${\it \Gamma}$–$A$ line, there are four triply degenerate crossing points emerging near the Fermi level, which are the result of band splitting induced by SOC and are strictly protected by the threefold rotation symmetry individually.
cpl-34-2-027102-fig3.png
Fig. 3. Evolution of Wannier centers for MoC along $k_y$ in the (e) $k_z=0$ and (f) $k_z=\pi$ planes.
The band structure calculations reveal the exotic Fermi surface and bands crossing in MoC, but we still have to prove the nontrivial topological property of the actual material. Considering that the electronic structure at $k_z=0$ and $k_z=\pi$ planes can be viewed as 2D subsystems with time-reversal symmetry, the band topology can be characterized by $\mathbb{Z}_2$ topological invariants. Here we apply the method of the evolution of Wannier charge centers (WCCs)[44,45] to calculate the $\mathbb{Z}_2$ number by counting how many times the evolution lines of the Wannier centers cross the arbitrary reference line. The computed $\mathbb{Z}_2$ indexes based on each WCC sheet indicate that MoC is a topological nontrivial node-line semimetal. From Figs. 3(a) and 3(b), we see explicitly that both $k_z=0$ and $k_z=\pi$ planes are topologically nontrivial with $\mathbb{Z}_2=1$. We can expect the strong side of the surface states of MoC on specific projected surfaces to exhibit the same topologically protected properties as in surfaces of weak topological insulators.[46] For specific projected surface simulation, a numerical tight-binding model based on the maximally localized Wannier function (MLWF) method[47,48] has been constructed to investigate the projected surface states obtained from the surface Green's function of the semi-infinite system. The projected BZs are illustrated in Fig. 1(d).
cpl-34-2-027102-fig4.png
Fig. 4. (a) The projected surface density of states for the (001) surface. (b) The enlarged surface states near the $\bar{K}$ in (a). The Fermi surfaces for the (001) surface states at (c) $E=43$ meV and (d) $E=50$ meV.
For the (001) projected surface, the two ring-shaped node lines are projected and centered around $\bar{K}$, and the bulk states around TDPs are projected onto a rather tiny area around the $\bar{\it \Gamma}$ point because the TDPs are located on the ${\it \Gamma}$–$A$ line within the bulk BZ. The projected (001)-surface band structures and Fermi surfaces are calculated and plotted in Fig. 4, from which we can find two bright states distinguishable from the bulk states, indicating the so-called 'drumhead' surface states. The drumhead surface state either inside or outside the projected nodal rings is one of the most significant signatures of topological node-line semimetals, and its location is strictly protected by the surface band topology. At the (001) surface, these states connecting nodal rings are outwards and determined by the Berry phase evolution in the folded BZ. Moreover, they are clearly separated from the bulk states, which make them experimentally verifiable by angle-resolved photoemission spectroscopy. For better observation of the two ring-shaped nodal lines (NL1 and NL2), the details of the surface band structures are enlarged in Fig. 4(b). To obtain information about the band topology of the nodal ring in the surface states, we plot the Fermi surfaces around the $K$ point at $E=43$ meV and $E=50$ meV, respectively, as shown in Figs. 4(c) and 4(d). In both cases, we see two concentric nodal rings. The (010) surface states and Fermi surfaces are shown in Fig. 5(a). Since the nodal rings around the $K$ point are located in the $k_z=0$ plane, the surface bands projected onto the (010) surface are plotted along the $\tilde{\it \Gamma}$–$\tilde{X}$ line and are buried in the bulk states. Meanwhile, the TDPs along the ${\it \Gamma}$–$A$ line in the bulk band are projected onto the $\tilde{\it \Gamma}$–$\tilde{Z}$ path, and are merged into the projected bulk states too. However, we can find a pair of surface states crossing the surface Fermi level. They are still topologically protected, since as shown in Figs. 3(e) and 3(f), both $k_z=0$ and $k_z=\pi$ are topologically nontrivial with $\mathbb{Z}_2=1$. As a consequence, there will be a Kramers doublet of surface states appearing along the $\tilde{\it \Gamma}$–$\tilde{X}$ and $\tilde{Z}$–$\tilde{M}$ lines. In addition, the two surface states are degenerate at an anisotropic surface Dirac cone at the point $\tilde{M}$ with a chemical potential of 1 eV in the folded BZ; the appearance of the surface Dirac cone at $\tilde{M}$ is due to $\mathbb{Z}_2=1$ of the $k_z=\pi$ plane. Like the surface states in the MoC (001) surface, those in MoC (010) are experimentally distinguishable. The Fermi surface of MoC (010) is relatively complicated and is plotted in Fig. 5(b). Along the $-\tilde{Z}$–$\tilde{\it \Gamma}$–$\tilde{Z}$ path, we find a dumbbell-shape Fermi surface, which is mainly created by the bulk states. However, the Fermi surface around the $\tilde{X}$ is rooted in the topological nontrivial surface states and contains two pairs of Fermi arcs connecting the bulk state pockets, as shown in Fig. 5(c).
cpl-34-2-027102-fig5.png
Fig. 5. (a) Projected surface density of states for the (010) surface. (b) Fermi surfaces for the (010) surface. The Fermi surface around the $\tilde{X}$ point in (b) is enlarged in (c).
In conclusion, we have reported that the single-crystal MoC is a stable topological node-line semimetal. These semimetals form a distinct class of topological materials beyond topological insulators and Weyl semimetals. Different from previously proposed materials, the nodal lines in MoC are robust even with the SOC included due to the protection of the mirror reflection symmetry. By calculating the energy difference between two touching bands and the projected surface states, we uncover two nodal rings which are more distinctive than other node-line semimetals. Moreover, the new fermion state with TDPs can be observed in MoC. The appearance of the triply degenerate crossing point is protected by the rotation symmetry and mirror symmetry. The coexistence of the nodal ring Fermi surface and TDPs in MoC lead to extraordinary properties, such as distinguishable drumhead surface states and new fermions that can be manipulated. In light of these novel properties, MoC provides an ideal platform to study unique topological phenomena such as chiral anomalies and non-local transport, and we can expect MoC to be a potential candidate in TSM-based electronic devices. References Topological Insulators in Three DimensionsTopological invariants of time-reversal-invariant band structuresTopological phases and the quantum spin Hall effect in three dimensionsColloquium : Topological insulatorsThe quantum spin Hall effect and topological insulatorsTopological insulators and superconductorsStability of Fermi Surfaces and K TheoryTopological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridatesChern Semimetal and the Quantized Anomalous Hall Effect in HgCr 2 Se 4 Weyl Semimetal Phase in Noncentrosymmetric Transition-Metal MonophosphidesExperimental Discovery of Weyl Semimetal TaAsObservation of Weyl nodes in TaAsDiscovery of a Weyl fermion semimetal and topological Fermi arcsDirac Semimetal in Three DimensionsDirac semimetal and topological phase transitions in A 3 Bi ( A = Na , K, Rb)Three-dimensional Dirac semimetal and quantum transport in Cd 3 As 2 Topological nodal semimetalsTopological node-line semimetal in three-dimensional graphene networksTopological Node-Line Semimetal and Dirac Semimetal State in Antiperovskite Cu 3 PdN Dirac Line Nodes in Inversion-Symmetric CrystalsTopological nodal-line fermions in spin-orbit metal PbTaSe2Drumhead surface states and topological nodal-line fermions in TlTaSe 2 Ca 3 P 2 and other topological semimetals with line nodes and drumhead surface statesDirac-node arc in the topological line-node semimetal HfSiSBeyond Dirac and Weyl fermions: Unconventional quasiparticles in conventional crystalsTopological semimetals with triply degenerate nodal points in θ -phase tantalum nitrideTriple Point Topological MetalsCoexistence of Weyl fermion and massless triply degenerate nodal pointsTopological Phases in InAs 1 − x Sb x : From Novel Topological Semimetal to Majorana WireSpin texture on the Fermi surface of tensile-strained HgTeEfficient 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 integrationsA New Molybdenum CarbideA New Phase formed by High-Pressure Treatment : Face-centred Cubic Molybdenum MonocarbideIn situ study of the high pressure high-temperature stability field of TaN and of the compressibilities of ϑ-TaN and TaONThe band structure of hexagonal NbNOn Fermi's surfaces of hexagonal carbides and nitridesStructure, Properties, and Bonding of ZrTe (MnP Type), a Low-Symmetry, High-Temperature Modification of ZrTe (WC Type)First-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 pressuresTopological nodal line semimetals with and without spin-orbital couplingComputing topological invariants without inversion symmetryEquivalent expression of Z 2 topological invariant for band insulators using the non-Abelian Berry connectionStrong side of weak topological insulatorsMaximally localized generalized Wannier functions for composite energy bandsMaximally localized Wannier functions for entangled energy bands
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