Coexistence of New Fermions in Topological Semimetal TaS

TL;DR

This paper predicts that TaS is a stable topological semimetal hosting various topological fermions, including nodal lines, Weyl points, and triply degenerate points, with unique surface states and potential for studying their interplay.

Contribution

It introduces TaS as a new topological semimetal hosting coexistence of multiple topological fermions and analyzes their electronic structures with and without spin-orbit coupling.

Findings

TaS hosts ring-shaped nodal lines and triply degenerate points without SOC.

SOC causes TDP splitting and transforms some nodal rings into Weyl points.

Unique Fermi arcs and Lifshitz transition are observed on certain surfaces.

Abstract

We theoretically propose that, the single crystal formed TaS is a new type of topological semimetal, hosting ring-shaped gapless nodal lines and triply degenerate points (TDPs) in the absence of spin-orbit coupling (SOC). In the presence of SOC, the each TDP splits into four TDPs along the high symmetric line in the momentum space, and one of the nodal ring remains closed due to the protection of the mirror reflection symmetry, while another nodal ring is fully gapped and transforms into six pairs ofWeyl points (WPs) carrying opposite chirality. The electronic structures of the projected surfaces are also discussed, the unique Fermi arcs are observed and the chirality remains or vanishes depending on the projection directions. On the (010) projected surface, one may observe a Lifshitz transition. The new type of topological semimetal TaS is stable and experimentally achievable, and the coexistence of topological nodal lines, WPs and TDPs states in TaS makes it a potential candidate to study the interplay between different types of topological fermions.

Equations

$${\hat{v}}_{\alpha (\beta \gamma )}=\left(i/\hslash \right)\left[\hat{H},{\hat{r}}_{\alpha (\beta \gamma )}\right]=\frac{1}{\hslash }\frac{\partial \hat{H}(\mathbf{k})}{\partial k_{\alpha (\beta \gamma )}},$$

$${\Omega }_{n,\alpha \beta }(\mathbf{k})=-2\mathrm{Im}\sum _{m\ne n}\frac{⟨u_{n\mathbf{k}}|{\hat{v}}_{\alpha }|u_{m\mathbf{k}}⟩⟨u_{m\mathbf{k}}|{\hat{v}}_{\beta }|u_{n\mathbf{k}}⟩}{{\left[{\omega }_m(\mathbf{k})-{\omega }_n(\mathbf{k})\right]}^2},$$

$${\Omega }_{n,\gamma }(\mathbf{k})={ϵ}_{\alpha \beta \gamma }{\Omega }_{n,\alpha \beta }(\mathbf{k}),$$

$$\phi (\theta )=\mathrm{Im}\{\ln [\det \mathcal{W}(\mathcal{C})]\},$$

Full text

I INTRODUCTION

Topological semimetals (TSMs) have stimulated tremendous interests in recent years, not only for they are metallic mimic of the topological insulators Hasan and Kane (2010); Qi and Zhang (2010); Moore (2010); Qi and Zhang (2011), but also for they provide fertile platforms to realize unique particles in high-energy particle physics. Although the searching for novel fermions in condensed matters can stem back to 1937 Herring (1937), C. Herring proposed accidental two-fold degeneracy of bands crossing in three-dimensional (3D) lattice and found it robust even without any symmetry protection within Weyl function. And later G. Volovik proposed Weyl fermions in superfluid 3He-A Volovik (1999). Till 2011, X. Wan et al. proposed the first specific compounds $\text{Re}_{2}\text{Ir}_{2}\text{O}_{7}$ (Re=rare earth element) Wan et al. (2011) to be Weyl semimetal. After that, abundant TSMs candidates have been proposed.

Current TSMs can be classified as Weyl semimetals, Dirac semimetals and node-line semimetals according to the different band crossing points at the Fermi level and the mechanisms protecting them. To be more specifically, a Dirac semimetal Young et al. (2012); Young and Kane (2015); Wang et al. (2012); Wieder et al. (2016) is characterized by hosting Dirac points of linear crossing of two doubly degenerate bands in momentum space near the Fermi level. The Dirac points are protected by certain crystalline symmetry, as a result, which makes the quadruple degenerate points located either at the high-symmetry point or along high-symmetry lines. The node-line semimetal Burkov et al. (2011); Yu et al. (2015); Kim et al. (2015); Fang et al. (2015); Bian et al. (2016a, b); Chan et al. (2016); Takane et al. (2016); Huang et al. (2016); Zhu et al. (2016a); Sun et al. (2017) is another type of TSM where the valence and conduction bands cross along one-dimensional (1D) lines in momentum space which results in forming a ring-shaped nodal line. Different from the Dirac and node-line semimetals, the Weyl semimetal Wan et al. (2011); Burkov and Balents (2011); Halász and Balents (2012); Weng et al. (2015); Xu et al. (2015a); Sun et al. (2015); Autès et al. (2016); Sánchez-Barriga et al. (2016); Deng et al. (2016); Tamai et al. (2016) is classfied by possessing Weyl points (WPs) that are point-like band crossings of two nondegenerate bands with linear dispersion near the Fermi level. Weyl semimetal does not require any protection from the crystalline symmetry other than lattice translation, which makes WPs possibly located at generic $\mathbf{k}$ points. These WPs with opposite chirality represent sources or sinks of Berry curvature and have to appear in pairs. They are associated with the topological Chern number $\mathcal{C}=\pm 1$ and positive (negative) Chern number is equivalent to a source (sink) of Berry curvature. 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 is a closed ring-shaped nodal line arising from the valence and conduction bands crossing along the specific crystllagraphic orientations in the BZ.

Besides the above-mentioned topological semimetals, another class of topological semimetal materials characterized by three- or sixfold band crossings have been proposed and are named as “new fermion” by Bradlyn $et$ $al$ Bradlyn et al. (2016). Very recently, the TSMs with triply degenerate points (TDPs) have been predicted theoretically in some materials with hexagonal lattice structure Zhu et al. (2016b); Weng et al. (2016) and the CuPt-ordered alloys $\text{InAs}_{0.5}\text{Sb}_{0.5}$ Winkler et al. (2016). In the band structures of these materials, both one- and two-dimensional representations are allowed along a certain high-symmetry axis, which makes it possible to generate band crossing between a doubly degenerate band and a nondegenerate band near the Fermi level and form a TDP Zaheer et al. (2013). This new type of three-component fermions can be regarded as the “intermediate state” between the four-component Dirac and the two-component Weyl fermions.

As the new fermions emerge, the interplay among the fermions reveals underlying physics. For example, the Dirac fermions can split into one pair of Weyl fermions with opposite chirality by breaking spatial inversion or time-reversal symmetry Yang et al. (2014); Wieder et al. (2016), and the ring-shaped nodal lines can be gapped Huang et al. (2016); Zhu et al. (2016a) or split into Weyl fermions Weng et al. (2015, 2016). From this point of view, hunting for materials hosting versatile topological particles is of particular interests.

In this work, we propose that TaS is a new type of material that hosts topological nodal lines, Weyl fermion states and unique fermion states with triply degenerate crossing points near the Fermi level simultaneously. There will be six pairs of WPs near the mirror plane $k_{z}=0$. Every pairs of WPs possess opposite chirality (+1 or -1) and each WP can be viewed as a singular point of Berry curvature or “magnetic monopole” in momentum space Wan et al. (2011). Since TaS owns unique $D_{3h}$ symmetry, all WPs are located at the same energy. Along the $\Gamma-A$ line in the BZ, four TDPs are presented in the vicinity of the Fermi energy. These TDPs are protected by the $C_{3}$ rotation symmetry along $\Gamma-A$ direction. The coexistence of topological nodal lines, WPs and TDPs states in TaS makes it a potential candidate to study the interplay between different types of topological fermions.

II MODELS AND METHODS

As shown in Fig. 1, TaS has a hexagonal lattice structure Slovyanskikh et al. (1984), with the same as Ta(Nb)N Friedrich et al. (2013); Litinskii (1989, 1990) and ZrTe Örlygsson and Harbrecht (2001) which host same symmetry $D_{3h}^{1}$ with space group $P\bar{6}m2$ (No. 187). Ta and S atoms are located in $1a(0,0,0)$ and $1d(\frac{1}{3},\frac{2}{3},\frac{1}{2})$ Wyckoff positions, respectively. The optimized lattice constants are $a=b=3.273$ Å and $c=3.337$ Å, which reasonably agree with experimental datas $a=b=3.280$ Å and $c=3.145$ Å. To further prove the structural stability of the optimized crystal, we calculate the phonon dispersions by using the frozen phonon method Parlinski et al. (1997) as implemented in the PHONOPY code Togo et al. (2008), as shown in Figure. 1(d). No imaginary frequencies are observed throughout the whole BZ in phonon dispersions, indicating its dynamically structural stability. Therefore, in this work, we adopt the relaxed lattice parameters for all the consequential calculations.

To explore the electronic structures of TaS, we perform the band structure calculation by using the Vienna $ab$ $initio$ simulation package (VASP) Kresse and Furthmüller (1996) within the generalized gradient approximation (GGA) in Perdew-Burke-Ernzerhof (PBE) Perdew et al. (1996) type and the projector augmented-wave (PAW) pseudopotential Blöchl (1994). The kinetic energy cutoff is set to be 560 eV for wave-function expansion, and the $k$-point grid is sampled by sums over $12\times 12\times 12$ Monkhorst and Pack (1976). 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 0.01 eV/Å. In order to investigate the projected surface states and Fermi surfaces, a tight-binding model Hamiltonian based on the maximally localized Wannier functions (MLWF) method Marzari and Vanderbilt (1997); Souza et al. (2001) has been constructed. Then, we apply an iterative method Sancho et al. (1984, 1985) implemented in the software package WannierTools Wu et al. to obtain the surface Green’s function of the semi-infinite system and the imaginary part of the surface Green’s function is the local density of states (LDOS) at the surface.

III RESULTS AND DISCUSSIONS

III.1 Overview of electronic structures of the bulk material

The calculated band structures excluding the spin-orbit coupling (SOC) is shown in Fig. 2(a). Near the Fermi level, the valence and conduction bands are mainly attributed by the $d$ orbitals of Ta. In the absence of SOC, there is a band inversion around the $K$ point between $d_{x^{2}-y^{2}}+d_{xy}$ and $d_{xz}+d_{yz}$ of Ta atom, which leads to a gapless nodal ring centering at $K$ point in the horizontal $k_{z}=0$ mirror plane protected by $\sigma_{h}$ symmetry. The similar nodal ring band topology is observed in ZrTe Zhu et al. (2016b); Weng et al. (2016) and MoC Sun et al. (2017), but is missing in TaN Zhu et al. (2016b). Since the three materials share the same space group with TaS, the existence of nodal ring is not simply determined by the crystal symmetry. Unlike the character of inverted band structures around the $K$ point, we find a band crossing between one nondegenerate band composed by $d_{z^{2}}$ orbital of Ta atom and a double degenerate band composed by the combination of $d_{x^{2}-y^{2}}$ and $d_{xy}$ orbitals of Ta atom along the $\Gamma-A$ line in the reciprocal lattice. The crossing point is a TDP protected by the $C_{3v}$ symmetry in the absence of SOC. Since TaS hosts the heavy element Ta, the SOC can not be simply ignored. By introducing the SOC, due to the lack of inversion symmetry, the energy bands happen to split in momentum space, as demonstrated in Fig. 2(b). From Fig. 2(b), one can find out that, the nodal ring centering at $K$ point is gapped and the original TDP splits into four discrete crossing points along the $\Gamma-A$ symmetric line. Considering that the electronic structures within the $k_{z}=0$ and $k_{z}=\pi$ planes can be decoupled and regarded as individual 2D subsystems with time-reversal symmetry, we can give a well defined $\mathds{Z}_{2}$ number to determine the band topology for all the occupied bands. Since TaS has no inversion symmetry, we use the method of the evolution of Wannier charge centers (WCCs) Soluyanov and Vanderbilt (2011a, b); Taherinejad et al. (2014) to calculate the $\mathds{Z}_{2}$ number by counting how many times the evolution lines of the Wannier centers cross an arbitrary reference line. In Fig. 2(c) and 2(d), we find that both $k_{z}=0$ and $k_{z}=\pi$ planes are topologically nontrivial with $\mathds{Z}_{2}=1$.

III.2 Nodal lines and triply degenerate points

In order to understand the role of SOC in the nodal ring gapping and TDP splitting, we calculated the detailed band structures of TaS in the absence and presence of SOC within specific BZ regions.

The enlarged band structures around the $K$ and $H$ points are shown in Fig. 3. In the absence of SOC, there are two nodal lines appearing on the mirror plane of $k_{z}=0$ and $k_{z}=\pi$, respectively. The nodal line on the $k_{z}=0$ plane is located about 1.1 eV below the Fermi level centering at the $K$ point, while the other one on the $k_{z}=\pi$ plane is located about 2.8 eV above the Fermi level centering at the $H$ point, as shown in Fig. 3(a) and 3(c). By turning on the SOC, the nodal ring around the $K$ point is fully gapped, as shown in Fig. 3(b). However, different from the $K$ point, the nodal ring around the $H$ point still preserves and splits into two nodal rings marked by red and blue dots respectively in Fig. 3(d) in the presence of SOC. The preserving of nodal ring under SOC is strictly protected by the mirror symmetry $\sigma_{h}$. Along the $\Gamma-A$ direction, there are four triply degenerate crossing points emerging near the Fermi level due to the band splitting induced by SOC, as shown in Fig. 4. Due to the the time-reversal symmetry, these TDPs appear in pairs and are protected by threefold rotation symmetry $C_{3}$ and vertical mirror symmetry $\sigma_{v}$. The four TDPs located within a relatively large energy scale ranging from -1.141 eV to 0.812 eV are a direct result stemming from the large SOC in the heavy element Ta. For a more direct view of the degeneracy and separation of energy bands, the 3D Fermi surface at $E=0$ eV is presented in Fig. 4(c).

III.3 Weyl nodes

Although the nodal ring preserves around the $H$ point on the $k_{z}=\pi$ plane, the nodal ring centering at the $K$ point on the $k_{z}=0$ plane is fully gapped, and the gapped nodal ring may produce a transition into hidden Weyl points. Generally, it is difficult to identify WPs within the momentum space because the WPs are not protected by any additional crystalline symmetry besides the lattice translation and are possibly located at general $\mathbf{k}$ points. However, detailed electronic structure calculations give us some clues. As shown in Fig. 5(a) and 5(b), by tuning the isoenergy reference into the opened gap, one can observe vanishing connections between the three-fold rotational symmetric arcs centering at the $K$ point, which indicates the DOS of the connections become zero. However, by further calculations of the energy gap between blue and red bands in Fig. 2(b), we find that the arc connections are simultaneously bands crossing points, which are demonstrated in Fig. 5(c) and 5(d). As a consequence, we find six pairs of zero mass band crossing points off the $k_{z}=0$ plane within the gap opened in the original nodal ring, and one pair of typical crossing points are located at $(0.27449,0.27449,\pm 0.01006)$ in unit of reciprocal lattice vectors marked as W+ and W-, respectively.

Following, we will check that whether these band crossing points are WPs or not. As an important clue, WPs can be viewed as monopoles with opposite chirality corresponding to the sources and sinks of Berry curvature. Besides, the integral of the Berry curvature on any closed surface in the BZ equals the total chirality of the WPs enclosed by the given surface, which will help us chase down and check WPs quickly. Therefore, to further verify the connecting points host monopole feature and chirality, we apply Kubo formula to calculate the Berry curvature by the tight-binding Hamiltonian based on the MLWF Xiao et al. (2010); Wang et al. (2006),

[TABLE]

[TABLE]

where $\hat{v}_{\alpha(\beta\gamma)}$ is the velocity operator with $\alpha,\beta,\gamma=x,y,z$, $\omega_{m}(\mathbf{k})=\varepsilon_{n}(\mathbf{k})/\hbar$, $\varepsilon_{n}(\mathbf{k})$ and $u_{n\mathbf{k}}$ are the eigenvalue and eigenvector of the Hamiltonian $\hat{H}(\mathbf{k})$, respectively. Finally, we obtain the Berry curvature in momentum space for a given band $n$ by

[TABLE]

where $\epsilon_{\alpha\beta\gamma}$ is the Levi-Civita tensor. By summing over all occupied valence bands, we give the Berry curvature vector $\mathbf{\Omega=}(\Omega_{x},\Omega_{y},\Omega_{z})$.

Next, we will examine the monopole feature of the crossing points. Since a pair of WPs with opposite chirality represent the source and sink of the Berry curvature, we calculate the Berry curvature distribution in momentum space. From Fig. 6(a) and 6(b), we can clearly see the Berry curvature around W+ and W- is divergent and convergent, respectively, which indicates the W+ and W- act as the feature of the source and sink of the Berry curvature.

In order to confirm the chirality of the crossing points, we need get the flux of Berry curvature through a surface enclosing them. The corresponding flux of Berry curvature is computed by discretizing a closed sphere parametrized by angles $\theta$ and $\varphi$ into 1D-loops, as shown in Fig. 6(c). Then, We calculate the WCCs on longitudinal loops around a sphere enclosing the crossing point. The sum of the WCCs on a loop $\mathcal{C}$ with longitudinal $\theta$ angle corresponds to the accumulated Berry phase along the loop, or equal to the average position $\left\langle\varphi\right\rangle$ of the charge on the loop. It can be determined from the Wilson loop $\mathcal{W}(\mathcal{C})$ as

[TABLE]

where $\mathcal{W}(\mathcal{C})=\prod\limits_{i=0}^{L-1}M^{(\mathbf{k}_{i,}\mathbf{k}_{i+1})}$, $\left[M^{(\mathbf{k}_{i,}\mathbf{k}_{i+1})}\right]_{mn}=\langle u_{m\mathbf{k}_{i}}|u_{n\mathbf{k}_{i+1}}\rangle$ are the overlap matrices along the loop $\mathcal{C}$. Since 1D loops cover a closed surface, the center of charge $\left\langle\varphi\right\rangle$ can only shift by an integer multiple of $2\pi$ when $\theta$ varies from 0 to $\pi$. This multiple is equal to the chirality $\mathcal{C}$ of the crossing point enclosed in the sphere. Therefore, we apply the method to a sphere enclosing W+ or W-. As shown in Fig. 6(d), when the angle $\theta$ varies from 0 to $\pi$, the average position of the Wannier centers $\left\langle\varphi\right\rangle$ is shifted by $2\pi$ or $-2\pi$, which indicates W+ and W- are indeed a pair of WPs with chirality $\mathcal{C}=+1$ and $\mathcal{C}=-1$, respectively.

Around the WPs (W+ and W-), we plot the energy dispersions, as shown in Fig. 7(a) and 7(b). Since the TaS hosts high symmetry $D_{3h}$ including $C_{3}$ rotation and mirror symmetry et al, which results in all WPs are located at the same energy (1.026 eV below the Fermi level). By utilizing above comprehensive techniques, we obtain at once the positions and chirality of all six pairs of WPs in the first whole BZ, which are clearly presented in Fig. 7(c). All these WPs are located near the $K$ point and its time reversa $K^{\prime}$ point and are off the $k_{z}=0$ mirror plane. The Fig. 7(d) displays the projection of all WPs onto the top and slide surfaces. When these WPs are projected onto top (001) surface, each pair of WPs with opposite chirality are projected onto a same position labeled by large green dots. Whereas on the protected (010) surface, the WPs with different chirality are projected onto a different position. The large and small dots represent the projections of two WPs with same chirality and a single WP, respectively.

III.4 Surface states and Fermi arcs

Since different projections bring different expressions of chirality of WPs, we calculate the projected surface states and Fermi surfaces (Fermi arcs) onto the (001) and (010) surfaces. The (001)-surface states and Fermi surfaces are shown in Fig. 8. Around the $\tilde{K}$ point, there is a analogous nodal ring but with a bulk gap about 30 meV due to the spin-orbit interaction along the $\tilde{K}-\tilde{M}$ direction, as shown in Fig. 8(c). In Fig. 8(b), a Weyl node is exactly located at $-1.026$ eV on the $\tilde{\Gamma}-\tilde{K}$ line, which mainly roots in the contribution of the projected WP. Around the symmetric $\tilde{K}$ or $\tilde{K}^{\prime}$ point in the BZ, there are three pairs of projected WPs existing, which will generate threadlike Fermi arcs, as clearly presently in Fig. 8(d). Each green circle represents the projections of two WPs with opposite chirality, which interconnect by the Fermi arcs. Moreover, since the TDPs are located on the $\Gamma-A$ line, they are projected on the $\tilde{\Gamma}$ point surrounded by the bulk state projections.

For the (010) surface states and Fermi surfaces, which are presented in Fig. 9. From the Fig. 9(a), one can clearly see that there is a surface Dirac cone at $\bar{Z}$ point, which is a result of $Z_{2}=1$ of the $k_{z}=\pi$ plane. The two branches of the surface Dirac cone connect the bulk states of the conduction and valence bands, respectively. In order to observe the fancy Fermi arcs, we plot the Fermi surface at $E=-1.026$ eV, as shown in Fig. 9(b) and 9(c). In the Fig. 9(c), the projected WPs and Fermi arcs are clearly presented, and each Fermi arc connects the projections of the WPs with opposite chirality. The red and blue large circles represent the projections of two WPs with the same chirality and have a monopole charge $+2$ and $-2$, respectively. While the red and blue small circles represent a single projected WP and have a monopole charge $+1$ and $-1$, respectively. This is consistent with the description of Fig. 7(d). In addition, the TDPs located at the $\Gamma-A$ line in the bulk band are projected onto the $\bar{\Gamma}-\bar{Z}$ path, and they can not be detached from other surface states and are buried in the projected bulk states, which is mainly due to the breaking of the $C_{3}$ and vertical mirror plane $\sigma_{v}$ symmetry on (010) slide surface.

III.5 Lifshitz transition

Since the connections between WPs carrying opposite chirality preserve on the (010) projection surface, and each pair of different chiral WPs are located very close to each other in momentum space, they can inevitably hybridize together. By adjusting chemical potential appropriately, one can expect a topological change in the band contours, which is known as the Lifshitz transition Wu et al. (2015); Xu et al. (2015b); Volovik (2017); Zhang and Volovik . In Fig. 10, we present the Lifshitz transition by changing the contour energy. At $E=-1.026$ eV, Fermi arcs connecting the projected WPs with opposite chirality are clearly shown in Fig. 10(a), which has been introduced in detail on the above. As the contour energy decreases, the Fermi arcs disappear and the topological properties will change. The projected surface bands will close to each other, then form two concentric ellipses, finally develop into two ellipses side by side.

IV SUMMARY AND CONCLUSIONS

In conclusion, in this paper we report that, TaS is a stable topological semimetal hosting versatile quasiparticle excitations such as topological nodal rings, Weyl points and three-fold degenerate points protected by the rotation symmetry $C_{3}$ and vertical mirror symmetry $\sigma_{v}$ simultaneously in the presence of SOC. Based on the DFT calculations, we study the topologically nontrivial properties in TaS both in bulk states and projected surface states. We find the nodal ring centering at $H$ point of the moment space remain closed to meet the requirement of the horizontal mirror symmetry $\sigma_{h}$ and the nodal ring centering at $K$ point is fully gapped by introducing the SOC. In the gapped nodal ring, there are six pairs of WPs associating with the point group $D_{3h}$ and locating at the same energy in the first BZ. Chirality and monopole feature of the WPs are verified by the calculation of the Berry curvature, and the characteristic Fermi arcs are found on the particular projected surfaces, which connects the projections of WPs with opposite chirality. By the projection onto the (010) surface and adjustment of chemical potential, a Lifshitz transition is also observed. The coexistence of versatile topological excitations brings extraordinary properties such as drumhead surface states and negative magnetoresistance effects. In light of these properties of the electronic band structures and surface states in TaS, we provide a guiding principle to search the particular topological semimetal, and establish an experimental achievable material platform for studying the unique topological phenomena in solids.

ACKNOWLEDGMENTS

This work was supported by NSFC Grants No. 11504366 and NBRPC Grants No. 2015CB921503 and 2016YFE0110000.