Topological phases in the TaSe3 compound

This is the accepted manuscript made available via CHORUS. The article has been published as: Topological phases in the TaSe_{3} compound Simin Nie, Lingyi Xing, Rongying Jin, Weiwei Xie, Zhijun Wang, and Fritz B. Prinz Phys. Rev. B 98, 125143 — Published 26 September 2018 DOI: 10.1103/PhysRevB.98.125143 Topological phases in the TaSe3 compound Simin Nie1 , Lingyi Xing2 , Rongying Jin2 , Weiwei Xie3 , Zhijun Wang4 ,∗ and Fritz B. Prinz1 1 Department of Materials Science and Engineering, Stanford University, Stanford, California 94305, USA 2 Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA, 70803, USA 3 Department of Chemistry, Louisiana State University, Baton Rouge, LA, 70803, USA and 4 Department of Physics, Princeton University, Princeton, New Jersey 08544, USA (Dated: September 10, 2018) Based on first-principles calculations, we show that stoichiometric TaSe3 , synthesized in space group P 21 /m, belongs to a three-dimensional (3D) strong topological insulator (TI) phase with Z2 invariants (1;100). The calculated surface spectrum shows clearly a single Dirac cone on surfaces, with helical spin texture at a constant energy contour. To check the stability of the topological phase, strain effects have been systematically investigated, showing that many topological phases survive in a wide range of the strains along both the a- and c-axes, such as strong TI (STI), weak TI (WTI) and Dirac semimetal phases. TaSe3 provides us an ideal platform for experimental study of topological phase transitions. More interestingly, since superconductivity in TaSe3 has been reported for a long time, the co-existence of topological phases and superconducting phase suggests that TaSe3 is a realistic system to study the interplay between topological and superconducting phases in the future. I. INTRODUCTION The layered transition-metal trichalcogenides MX3 (M=Nb, Ta; X=S, Se) have attracted lots of interest because of the appearance of the charge-density-wave (CDW) states at low temperature (LT)1–4 . For instance, both TaS3 3 and NbSe3 5 are metals at room temperature (RT), and undergo two different CDW transitions as decreasing temperature. But TaSe3 is an exception, in which no CDW transition has been found yet6 . Instead, TaSe3 remains semimetallic from RT to LT and becomes superconducting at Tc = 2.3 K7–9 . The earlier theoretical studies10 suggest that it could be either a semiconductor or a semimetal, depending on the relative energy level of an anti-bonding Se p band to that of the Ta dz2 band. So far, first-principles calculations of TaSe3 have not been reported yet, and the electronic structures, Fermi surfaces and topological properties are still unrevealed. Meanwhile, topological superconductors have attracted much interest due to the emergence of Majorana fermions11,12 and the potential application in quantum computation13 . The previous work by Fu et al.14 has proposed that the topological superconductivity can be realized on the interface between a TI and a BCS superconductor by proximity effect15–19 . Very recently, topological superconductivity has been observed on the surface of iron-based superconductors below Tc 20 , which suggests that a more promising approach to engineer TSCs is to propose superconducting materials with nontrivial topology in their electronic structures16 . Herein, this approach would avoid the structural compatibility and overcome the fabrication challenges related to the interfaces or heterostructures16–19,21–23 . However, most superconductors do not have non-trivial bulk topology in electronic structures15,24 , and some topological candidates do need doping to induce superconductivity, like Cux Bi2 Se3 25,26 , FeTex Se1−x 16,20 , etc. Therefore, it is challenging to find a superconductor with non-trivial electronic topology. In this work, based on first-principles calculations, we show that the single crystal TaSe3 , known as a superconductor for many years7–9 , has non-trivial electronic structure. The crucial band inversion happens at B point even without spin-orbit coupling (SOC). This is different from the situation in Bi2 Se3 27,28 , in which the band inversion is due to the strong SOC effect. Our detailed analysis indicates that the band inversion is attributed to the “broken” type II chains, especially the Se3-Ta1 and Se6-Ta2 bonds. To shorten these bonds by compressive strains would enlarge the band gap and remove the band inversion. Inclusion of SOC in TaSe3 opens a continuous direct gap in the entire Brillouin zone (BZ), but doesn’t change the energy ordering of the bands at the timereversal invariant momentum (TRIM) points. The Z2 invariants (ν0 ;ν1 ,ν2 ,ν3 )29,30 are calculated to be (1;100). The strong topological index ν0 = 1 guarantees the existence of the Dirac-cone states on surfaces31 , which has been further confirmed by our surface calculations. To check the stability of the topological phase, strain effects have been systematically investigated, showing that many topological phases survive in a wide range of the strains along both the a- and c-axes, such as STI, WTI and Dirac semimetal phases. It’s an ideal platform for experimental study of topological phase transitions. As it becomes superconducting below Tc , TaSe3 also provides us a realistic system to investigate the interplay between the topological surface states and superconductivity. This paper is organized as follows. In Sec. II we will introduce the details of first-principles calculations. In Sec. III, the calculation results are presented. Finally, Sec. IV contains the discussion and conclusion. 2 (b) (c) 6 4 3 2 2 1 51 5 1 1 2 2 3 4 6 (a) 1 (b) 1 0.5 0.5 0 0 -0.5 -0.5 Energy (eV) b-axis (a) 0 0.05 0.1 c -1 Z Γ o a FIG. 1. (Color online) Crystal structure and BZs of TaSe3 . (a) The side view of the type I and type II chains colored in blue and red, respectively. Two neighbor prisms are differed by b/2 in the b-direction, making the Ta atoms in the planes of the Se3 triangles of the neighbor prism. (b) The projection view along the chain direction (i.e., the b-direction). The primitive unit cell is shown in light black parallelogram. The nonequivalent Ta atoms (large balls) and Se atoms (small balls) in the primitive unit cell are labeled by yellow and light blue numbers, respectively. The bonds (green dashed lines) of Se5-Ta1-Se3 and Se5-Ta2-Se6 make the prisms form a layer (spanned by the b-axis and (a + c)-axis). (c) Bulk BZ, projected surface BZs, and high-symmetry points. II. CALCULATION METHOD The first-principles calculations were performed within the framework of full-potential linearizedaugmented plane-wave (FP-LAPW) method implemented in WIEN2K simulation package32 . Modified Becke-Johnson exchange potential together with local density approximation for the correlation potential was used to obtain accurate band structures33 . SOC was included as a second variational step self-consistently. The radii of the muffin-tin sphere (RMT) were 2.5 Bohr for Ta and 2.38 Bohr for Se, respectively. The k-points sampling grid of the BZ in the self-consistent process was 7 × 19 × 6. The truncation of the modulus of the reciprocal lattice vector Kmax , which was used for the expansion of the wave functions in the interstitial region, was set to RM T × Kmax = 7. The geometry optimization including SOC interaction was carried out within the framework of the projector augmented wave (PAW) pseudopotential method implemented in Vienna ab initio simulation package (VASP)34,35 . The ionic positions were re−1 laxed until force on each ion was less than 0.005 eV Å . PHONOPY was employed to calculate the phonon dispersion through the DFPT method36 . III. RESULTS TaSe3 crystallizes in the monoclinic layered structure with space group P 21 /m. The basic building blocks of TaSe3 are parallel trigonal-prismatic chains along the baxis, as shown in Fig. 1(a). Each chain is made by a linear stacking of irregular prismatic cages, which consists of six DOS (/eV/unit cell) (c) 24 20 16 Y A B D EC 2 Total Ta d Ta d z Se p Se pz -1 Z Γ Y A B D E C (d) 1 2 0 -0.1 0 0.1 0.2 12 8 4 0 −6 −4 −2 0 2 Energy (eV) 4 FIG. 2. (Color online) Electronic structures of TaSe3 without (a) and with (b) SOC. The calculated parity eigenvalues of highest valence band (HVB) and the lowest conduct band (LCB) at B and D points are given explicitly. The size of red and blue circles in (a) represents the weights of Se3 pz and Ta1 dz2 , respectively. The inset in (b) shows the zoom-in band structure around the band crossing point located on the A-B line. The red dashed line in (b) indicates the existence of the continuous direct gap. (c) The calculated total DOS and PDOS of TaSe3 . The inset in (c) shows the DOS around the EF . (d) The Fermi surface of TaSe3 . selenium atoms at the corners and one tantalum atom at the center. In the top view of Fig. 1(b), these chains are classified as type I (i.e., dIs = d24 =2.57Å) and type II (i.e., dII s = d16 =2.9Å), depending on the shortest bond (ds ) of an irregular triangle which is formed by three Se atoms. TaSe3 has two type I and two type II chains in a unit cell. The two chains of each type are related by inversion symmetry. The four prismatic chains have strong bonds (i.e., Se5-Ta1-Se3 and Se5-Ta2-Se6) along the a+c direction, forming the TaSe3 layers as depicted by green dashed lines. The interlayer hoppings are relatively weak compared with the intralayer hoppings, suggesting 2D TaSe3 can be easily produced by exfoliation methods. Next, we discuss the qualitative electronic structure of TaSe3 by way of a simple Zintl-Klemm concept37 . In the type I chain of TaSe3 , where the distance between two Se atoms is short enough (i.e., dIs =2.575 Å), a strong covalent p-p bond is formed. Thus, the oxidation state of the Se3 triangle is (Se2− + Se2− 2 ). That’s why ZrSe3 is a semiconductor with Zr4+ and only type I chains38 . However, in the type II chain, the distance is not short enough; namely, the p-p bond is broken. These Se atoms exhibit the normal oxidation states (3Se2− ). At the ionic limit, the chemical valence of tantalum is about 5+ (Ta5+ ). Thus, (TaSe3 )4 can be formulated as 5+ 2(4Se2− +Se2− ) in a primitive unit cell. Accord2 +2Ta ing to this electron counting model, TaSe3 would be a 3 (b) Spacial-inversion ZrTe3 Bonding “a shift” Ta1, Ta2 d z 2 (a) 0.2 TaA d z 2 TaB d z 2 ∆E _ SeA p z 0.1 (c) SeB p z 0.1 Γ -0.1 50 Z B D Γ 200 -0.2 0 160 400 0 100 (d) -0.3~ Y 0.1 ~ Γ ~ ~ S X 0.4 SeA p z+ Se5 pz 0.3 80 0 800 120 K y (1/Å) ( TaSe3 µ1 µ2 2 -0.2 6 2 0 600 0.2 150 -0.1 TaB d z+ EF SeA p z 1 3 (b) 0.3 2 Se3, Se5, Se6 p z “distortion” 200 _ TaB d z E-EF (eV) (a) 0 400 40 0.2 FIG. 3. (Color online) The evolution of the crystal structures and the schemetic diagram of the band-inversion mechanism. (a) The upper pattern is a layer of the ZrSe3 structure, which is a semiconductor and only contains type I chain with Zr4+ , Se2− and (Se)2− states. The middle pattern is the interme2 diate structure generated by a shift in the a-direction (the black dashed line), which breaks the previous bonds of Se3 atoms, denoted by the symbol “x”. The lower pattern is a layer of the TaSe3 structure. The distortion happens in the chains in the box, changing the type I chains to the type II chains. The new bonds between Se6 and Ta2 are built. (b) Schematic diagram of the band evolution in TaSe3 , starting from the atomic orbitals dz2 of Ta1 and Ta2, and pz of Se3, Se5 and Se6. FIG. 4. (Color online) Surface states of TaSe3 . (a) and (b) are the surface band structures of TaSe3 on the (010) and (101̄) surfaces, respectively. The chemical potentials at Fermi level (EF ) and 40 meV below the Fermi level are represented by µ1 and µ2 , respectively. (c) The constant energy contour (E=EF ) of the topological surface states and corresponding spin texture on the (010) surface of TaSe3 . The inset shows the zoom-in Fermi surface around the upper left corner. (d) Zoom-in surface band structures around the Dirac point on the (101̄) plane. semiconductor. However, TaSe3 is metallic according to the transport measurements39,40 , which implies band inversion may occur in the electronic band structure. In order to fully understand the semimetallic propertes of TaSe3 , the first-principles calculations have been performed systematically. When SOC is ignored, the calculated band structure along high symmetric lines in the BZ is shown in Fig. 2(a). There is always a direct gap between the conduction bands and valence bands, except two crossing points on the AB and DE lines. To elucidate the mechanism of the band inversion, we have calculated the projected weights on six nonequivalent Se atoms (i.e., Se1, Se2, · · · , and Se6, as shown in Fig. 1(b)) and two nonequivalent Ta atoms (i.e., Ta1 and Ta2, as shown in Fig. 1(b)), respectively. In the fatted-band plot of Fig. 2(a), we denote the weights of Se3 pz orbital (ẑ||~b) and Ta1 dz2 orbital by the size of the red and blue circles, respectively. We can clearly see that the pz band and the dz2 band have an overlap at the Fermi level (EF ). Further calculated results indicate that the up-going dz2 band mainly comes from the dz2 states of both Ta1 and Ta2, consistent with the ligand crystal splitting of five d orbitals in a prismatic cage. However, the down-going pz band is mostly from the pz orbitals of Se3 and Se6 atoms (both of them belong to the “broken” type II chains). To some extent, it suggests that the metallic band structure has to do with the type II chains. These similar results are also obtained by our calculated partial density of states (PDOS) in Fig. 2(c), which show that the Se p states and Ta d states are mainly located below and above the EF , respectively, with the hybridization between them. Near EF , the states are dominated by pz states of Se3 and Se6 atoms and dz2 states of Ta1 and Ta2 atoms. To fully understand the metallic electronic band structure, we first investigate the crystal structure evolution beginning with the layered structure of ZrSe3 . In ZrSe3 type I chains are arranged, as shown in Fig. 3(a), as a perfectly layered structure. It can be changed to the crystal structure of TaSe3 by the following two steps. First, a shift may occur along the a-direction for every four prismatic chains (MX3 )4 , as shown in Fig. 3(a). As such, the bonds between selenium and tantalum atoms crossing the black dashed line are broken, as depicted by the symbol “x” in Fig. 3(a). Second, the distortion can happen in the prisms of the box by breaking the shortest p-p bonds between Se atoms. In addition, the new bonds between Ta2 and Se6 are built. Together, one can find that the crystal environment of Se3 and Se6 atoms of type II chains can change dramatically. The band inversion can be understood from the bonds (green dashed lines) of Se5-Ta1-Se3 and Se5-Ta2Se6, which support the layered structure of TaSe3 (see Fig. 1(b) and Fig. 3(a)). In the atomic limit, the energy levels of the Ta d orbitals are higher than Se p orbitals. Under the crystal field of the prismatic cage, the dz2 orbitals are lower than other d orbitals. The pz orbitals of Se atom are higher than other p orbitals, since it doesn’t orient toward the Ta atoms. Therefore, only pz orbitals of Se3, Se5 and Se6, and dz2 orbitals of Ta1 and Ta2 are considered in the schematic diagram of the band inversion, as shown in Fig. 3(b). Starting from the atomic limit, the energy level of the Ta dz2 orbitals is higher -0.2 0 0.2 K (1/Å) 0.4 0 -0.1~ Y ~ Γ ~ X ~ 0 S x 4 (b) 1 8 P3 δc (%) 4 WTI Energy (eV) (a) STI 0 P2 -4 P1 NI 0.5 P1 0 -0.5 -8 -4 Energy (eV) 0.5 0 δa (%) 4 (d) 1 P2 0 0.5 YA BD EC BD EC P3 0 -0.5 -0.5 -1 Z Γ -1 Z Γ 8 Energy (eV) -8 (c) 1 YA BD EC -1 Z Γ YA FIG. 5. (Color online) Phase diagram of TaSe3 with strain along both a- and c-directions. (a) The blue, green and red regions represent NI, STI and WTI phases, respectively. (b), (c) and (d) are band structures of TaSe3 with SOC for c = 0.95c0 (P1 ), c = 0.97c0 (P2 ) and c = 1.05c0 (P3 ), respectively. The parities of HVB and LCB at three TRIMs (B, D and Z) are given. than that of the Se pz orbitals. In step I, because Se5 forms two bonds while Se3 (or Se6) forms only one bond, the pz state of Se5 is pushed much lower than the pz state of Se3 (or Se6). The hybridization makes the two pz orbitals of Se3 and Se6 form two mixed states, called SeA and SeB , respectively. Similarly, the two mixed states of Ta dz2 orbitals are called TaA and TaB . The energy levels are illustrated in the middle of Fig. 3(b). In step II, the inversion symmetry is taken into consideration. Each state can split into two hybridized states, one bonding state and one antibonding state, according to the parity. The band inversion in TaSe3 happens between the bonding state of TaB and antibonding state of SeA , which is consistent with our fatted band calculations and PDOS. This band inversion mechanism is further confirmed by the phase diagram under strains, as will be shown later. The inclusion of SOC leads to gap opening at the band crossing points, as seen in the band structure of Fig. 2(b). However, the maximum of the valence bands is still higher than the minimum of the conduction bands, which gives rise to the semimetallic properties of TaSe3 . The calculated Fermi surfaces consist of a large hole pocket enclosing Γ point and two electron pockets near B point, as shown in Fig. 2(d). Further parity analysis indicates that the inverted two bands have different parity eigenvalues at B point, while they have the same eigenvalue (i.e., −1) at D point. Note that the band inversion has already happened even without SOC. The band inversion is attributed to the unique structure of the type II chains, especially the Se3-Ta1 and Se6-Ta2 bonds, which is distinct from the SOC-induced band inversion of the well-known Bi2 Se3 family. To classify TIs, one needs to compute four Z2 topological indices (ν0 ; ν1 , ν2 , ν3 ), where ν0 is a strong topological index and (ν1 , ν2 , ν3 ) are three weak topological indices. Since there is a continuous direct SOC gap, the Fu-Kane Z2 invariants are well defined for the occupied bands below the gap in TaSe3 . Due to the existence of inversion center in TaSe3 , they can be easily calculated by Fu-Kane parity criterion31 at eight TRIM points (Γi , i = 1, 2, . . . 8). The strong topological inQ8 dex is given by (−1)ν0 = i=1 δ(Γi ), where δ(Γi ) is the product of parity eigenvalues of the bands at Γi without counting their time reversal partners. Three weak topological indices are defined at the four TRIM points in a plane offset from Γ point. Explicitly, (−1)ν1 = δ(A)δ(B)δ(D)δ(E), (−1)ν2 = δ(A)δ(C)δ(E)δ(Y ) and (−1)ν3 = δ(C)δ(D)δ(E)δ(Z). The space group of TaSe3 is non-symmorphic with a screw symmetry C̄2b , which is a twofold rotation about the ~b-axis followed by a half lattice translation in the same direction (~b/2). It satisfies the relation C̄2b I = t(~b)I C̄2y , where I is inversion symmetry and the lattice translation t(~b) can be represented by a phase factor in the Bloch basis: t(~b) = exp(−i~b · ~k). Thus, at the TRIM points in the ~k2 = π/b plane, i.e., A, C, E and Y, with the expression t(~b) = −1, C̄2b anticommutates with I. In combination of time reversal symmetry, the anticommutating relation leads that all the states are four-fold degenerate, consisting of two parity +1 bands and two parity −1 bands (i.e., “+ + −−”). Considering that the total number of valence bands is 68 = 4 × 17, we get δ(A)=δ(C)=δ(E)=δ(Y )=(−1)17 =−1. Therefore, the strong topological index ν0 is determined by band inversion at other four TRIMs (Z, Γ, B and D). Their parity products are calculated to be δ(Z)=δ(Γ)=δ(D)=−1 and δ(B)=1, which are extracted from standard numerical calculations based on first-principles calculations. Therefore, the Z2 topological invariants of TaSe3 turn out to be (1;100). In view of the fact that the hallmark of topological non-trivial property is the existence of topological nontrivial surface states, the tight-binding Hamiltonians of semi-infinite samples are constructed by the maximally localized Wannier functions (MLWFs) for all the Ta d and Se p orbitals, which are generated from the firstprinciples calculations. The surface Green’s functions of the semi-infinite sample are obtained using an iterative method41 . The local DOS (LDOS), extracted from the imaginary part of the surface Green’s function, is used to analyze the surface band structures. For STIs, the existence of an odd number of Dirac cones on the surface is ensured by the strong topological index ν0 =1. On the (010) surface of TaSe3 , a Dirac cone is obtained at B̄ point in Fig. 4(a), which hosts helical spin texture at the energy contour (E=EF ) in Fig. 4(c). On the (101̄) surface, a Dirac cone is found at X̃ point, as shown in 5 Frequency (THz) (c) (b) Frequency (THz) 8 6 4 2 0 Z Γ 8 YA BD EC 6 4 2 0 Z Γ (d) Frequency (THz) Frequency (THz) (a) YA BD E C 8 6 4 2 0 Z Γ 8 YA BD EC YA BD EC 6 4 2 0 Z Γ FIG. 6. (Color online) Phonon dispersion of TaSe3 with c=1.0 c0 , c=0.95c0 (P1), c=0.97c0 (P2) and c=1.05c0 (P3), as shown in (a), (b), (c) and (d), respectively. Fig. 4(b) and its zoom-in plot in Fig. 4(d). The existence of a single Dirac cone on the surfaces is consistent with the STI phase. Since it’s a layered structure with trigonal-prismatic chains going in the b-direction, the lattice parameters in a- and c-directions are supposed to be more sensitive to temperature or strain than that in the b-direction. Therefore, the strain effects along the a- and c-directions have been systematically investigated in our calculations, in order to check the stability of the STI phase. For each given lattice constant, the internal positions of the atoms are fully relaxed until the force on each atom satisfies the required precision. The computed phase diagram is shown in Fig. 5(a) as a function of lattice parameters a (horizontal axis) and c (vertical axis). On the vertical dashed line in Fig. 5(a) (i.e., the strain in the a-direction is zero), when c grows from 0.95c0 to 1.05c0 (with c0 the lattice parameter without strain), it changes from a normal insulating (NI) phase, to STI phase, and then to WTI phase due to band inversion happening at TRIMs successively. Explicitly, at the beginning (c = 0.95c0 ), no band inversion is found in Fig. 5(b) and the system is a trivial insulator. When c = 0.97c0 , two bands of opposite parity become touching at B point, as shown in Fig. 5(c), resulting in a Dirac semimetal phase at the phase transition point. After the transition point, the system enters STI phase. By increasing c further (e.g. c = 1.05c0 ), another band inversion occurs at Z point, the system turns to WTI phase, as shown in Fig. 5(d). As we can see in Fig. 5(a), both STI (green) and WTI (red) phases survive in a considerable region compared with the NI (blue) phase. The phase boundaries are Dirac semimetals, which may also be observed in experiments. The phase diagram provides us an important roadmap to regulate the topological phase transitions in TaSe3 . Compared with the reported topological materials, TaSe3 has two unique features. First, the compressive strain can remove the band inversion, which is different from the common concept that the compressive strain usually enhances the band inversion42–44 . The compressive strain shortens the distances of the bonds of Ta1-Se3 and Ta2Se6, the hybridization becomes stronger and the average gap ∆E is enlarged between SeA pz (SeB pz ) and TaA dz2 (TaB dz2 ) states. Second, many TIs have an sp-type band inversion or a pp-type band inversion42–44 , while TaSe3 has a pd-type band inversion. Third, most TIs only have band inversion at one TRIM point (such as the Γ point)28,45 , while TaSe3 has band inversion at three TRIM points, which leads to fruitful nontrivial phases under different strains, including 3D STI, 3D WTI and Dirac semimetal. Next, we would like to discuss the possible CDW in TaSe3 . It is well-known that no CDW is observed in NbS2 . However, the phonon dispersion shows imaginary frequencies based on first-principles calculations46 . In order to exclude the possibility of the “latent” CDW instability and greatly support our predictions of nontrivial topological properties in TaSe3 , we carefully calculate the phonon dispersion for TaSe3 , which is shown in Fig. 6(a). No imaginary frequency is found in the dispersion. The phonon calculations have also been checked with respect to many parameters, such as k-point grid, type of smearing and size of supercell etc. In addition, we also calculate the phonon dispersion of TaSe3 with c=0.95c0 (P1), c=0.97c0 (P2) and c=1.05c0 (P3), as shown in Fig. 6(b), 6(c) and 6(d), respectively. No imaginary frequency can be found in any of them. Therefore, it is safe to exclude the effect of CDW in TaSe3 . IV. DISCUSSION AND CONCLUSION To achieve topological superconductivity, it is challenging to find a material with both nontrivial topology of the electronic structure and superconductivity. A prior theoretical proposal of intrinsic non-trivial material is high-Tc superconductor FeTex Se1−x 16 with x = 0.5 supporting topological surface states at EF . Very recently, the topological superconducting phase has been verified on (001) surface by spin-resolved ARPES experiments20 . Here we show another promising candidate TaSe3 , whose superconductivity has been reported for many years7–9 , is topologically non-trivial and has spin-momentum locking states on the surface. Compared to the FeTex Se1−x system, TaSe3 does not require doping to introduce superconductivity, implying this superconductor can be grown in high-quality single crystals. At the Fermi level (µ1 ), as we show in Fig. 4(c), the surface states are well separated from the bulk states. Based on Fu-Kane’s proposal, the superconducting states of the surface states induced by the bulk superconductivity below Tc =2.3 K can be topologically nontrivial. However, if the chemical poten- 6 tial lies 40 meV (µ2 ) below the Fermi level in Fig. 4(a), the surface states merge into the bulk states, which could kill the topological superconducting state and make the surface topological superconducting state sensitive to the position of the chemical potential. The future experimental work is needed to search for the potential topological superconductivity in the system. In conclusion, we have calculated the electronic structure, and topological properties of TaSe3 with P 21 /m crystal structure by using density functional theory. The calculated topological invariants are (1;100), which indicate that it belongs to the STI phase. A single Dirac cone is obtained in the calculated surface spectra. Further systematical calculations of strain effects suggest TaSe3 can realize multiple topological non-trivial phases under strains, including STI, WTI and Dirac semimetal phases. These topological non-trivial phases survive in a wide range of the strain along the a- and c-axes, making TaSe3 an attractive platform to study the topological phase transitions. In addition, the co-existence of the topological phases and superconductivity of TaSe3 suggests that it could be a realistic system to study the in- ∗ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 [email protected] K. Tsutsumi, T. Takagaki, M. Yamamoto, Y. Shiozaki, M. Ido, T. Sambongi, K. Yamaya, and Y. Abe, Physical Review Letters 39, 1675 (1977). J. Wilson, Physical Review B 19, 6456 (1979). R. J. Cava, R. Fleming, R. Dunn, and E. Rietman, Physical Review B 31, 8325 (1985). S. Srivastava and B. Avasthi, Journal of materials science 27, 3693 (1992). T. Ekino and J. Akimitsu, Japanese Journal of Applied Physics 26, 625 (1987). P. Haen, F. Lapierre, P. Monceau, M. N. Regueiro, and J. Richard, Solid State Communications 26, 725 (1978). T. Sambongi, M. Yamamoto, K. Tsutsumi, Y. Shiozaki, K. Yamaya, and Y. Abe, Journal of the Physical Society of Japan 42, 1421 (1977). M. Yamamoto, Journal of the Physical Society of Japan 45, 431 (1978). S. Nagata, H. Kutsuzawa, S. Ebisu, H. Yamamura, and S. Taniguchi, Journal of Physics and Chemistry of Solids 50, 703 (1989). E. Canadell, I. Rachidi, J. Pouget, P. Gressier, A. Meerschaut, J. Rouxel, D. Jung, M. Evain, and M. Whangbo, Inorganic Chemistry 29, 1401 (1990). F. Wilczek, Nature Physics 5, 614 (2009). M. Leijnse and K. Flensberg, Semiconductor Science and Technology 27, 124003 (2012). A. Y. Kitaev, Annals of Physics 303, 2 (2003). L. Fu and C. L. Kane, Physical review letters 100, 096407 (2008). B. Yan, M. Jansen, and C. Felser, Nature Physics 9, 709 (2013). Z. Wang, P. Zhang, G. Xu, L. Zeng, H. Miao, X. Xu, T. Qian, H. Weng, P. Richard, A. Fedorov, et al., Physical Review B 92, 115119 (2015). terplay between topological and superconducting phases in the future. ACKNOWLEDGMENTS We thank Prof. Robert J. Cava for useful discussions. F. B. P. and S. N. were supported by Stanford Energy 3.0. W. X. was supported by LSU-startup funding and the Louisiana Board of Regents Research Competitiveness Subprogram (RCS) under Contract Number LEQSF (2017-20)-RD-A-08. Z. W. was supported by the Department of Energy Grant No. DE-SC0016239, the National Science Foundation EAGER Grant No. NOAAWD1004957, Simons Investigator Grants No. ONRN00014-14-1-0330, No. ARO MURI W911NF-12-1-0461, and No. NSF-MRSEC DMR- 1420541, the Packard Foundation, the Schmidt Fund for Innovative Research, and the National Natural Science Foundation of China (No. 11504117). L.X. and R.J. are supported by Department of Energy Grant No. SC00i6315. 17 18 19 20 21 22 23 24 25 26 27 28 29 30 X. Wu, S. Qin, Y. Liang, C. Le, H. Fan, and J. Hu, Physical Review B 91, 081111 (2015). L. M. Schoop, L. S. Xie, R. Chen, Q. D. Gibson, S. H. Lapidus, I. Kimchi, M. Hirschberger, N. Haldolaarachchige, M. N. Ali, C. A. Belvin, et al., Physical Review B 91, 214517 (2015). G. Xu, B. Lian, P. Tang, X.-L. Qi, and S.-C. Zhang, Physical Review Letters 117, 047001 (2016). P. Zhang, K. Yaji, T. Hashimoto, Y. Ota, T. Kondo, K. Okazaki, Z. Wang, J. Wen, G. Gu, H. Ding, et al., arXiv preprint arXiv:1706.05163 (2017). Q. Xu, Z. Song, S. Nie, H. Weng, Z. Fang, and X. Dai, Physical Review B 92, 205310 (2015). Y. Xing, H. Wang, C.-K. Li, X. Zhang, J. Liu, Y. Zhang, J. Luo, Z. Wang, Y. Wang, L. Ling, et al., npj Quantum Materials 1, 16005 (2016). Y.-Q. Wang, X. Wu, Y.-L. Wang, Y. Shao, T. Lei, J.-O. Wang, S.-Y. Zhu, H. Guo, L.-X. Zhao, G.-F. Chen, et al., Advanced Materials 28, 5013 (2016). A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, and H. Shtrikman, Nature Physics 8, 887 (2012). Y. S. Hor, A. J. Williams, J. G. Checkelsky, P. Roushan, J. Seo, Q. Xu, H. W. Zandbergen, A. Yazdani, N. Ong, and R. J. Cava, Physical review letters 104, 057001 (2010). L. A. Wray, S.-Y. Xu, Y. Xia, Y. San Hor, D. Qian, A. V. Fedorov, H. Lin, A. Bansil, R. J. Cava, and M. Z. Hasan, Nature Physics 6, 855 (2010). H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, Nature physics 5, 438 (2009). W. Zhang, R. Yu, H.-J. Zhang, X. Dai, and Z. Fang, New Journal of Physics 12, 065013 (2010). M. Z. Hasan and C. L. Kane, Reviews of Modern Physics 82, 3045 (2010). X.-L. Qi and S.-C. Zhang, Reviews of Modern Physics 83, 1057 (2011). 7 31 32 33 34 35 36 37 38 L. Fu, C. L. Kane, and E. J. Mele, Physical Review Letters 98, 106803 (2007). P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, and J. Luitz, WIEN2k, An Augmented Plane Wave Plus Local Orbitals Program for Calculating Crystal Properties (TU Vienna, Vienna, 2001). F. Tran and P. Blaha, Physical Review Letters 102, 226401 (2009). G. Kresse and J. Furthmüller, Computational Materials Science 6, 15 (1996). G. Kresse and J. Furthmüller, Physical Review B 54, 11169 (1996). A. Togo, F. Oba, and I. Tanaka, Physical Review B 78, 134106 (2008). R. Nesper, Zeitschrift für anorganische und allgemeine Chemie 640, 2639 (2014). J. O. Island, A. J. Molina-Mendoza, M. Barawi, R. Biele, E. Flores, J. M. Clamagirand, J. R. Ares, C. Sánchez, H. S. van der Zant, R. DAgosta, et al., 2D Materials 4, 022003 (2017). 39 40 41 42 43 44 45 46 R. Buhrman, C. Bastuscheck, J. Scott, J. Kulick, D. Gubser, T. Francavilla, S. Wolf, and J. Leibowitz, in AIP Conference Proceedings, Vol. 58 (AIP, 1980) pp. 207–215. E. Bjerkelund, J. Fermor, and A. Kjekshus, Acta Chemica Scandinavica 20, 1836 (1966). N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Reviews of Modern Physics 84, 1419 (2012). Y. Nie, M. Rahman, D. Wang, C. Wang, and G. Guo, Scientific Reports 5, 17980 (2015). E. T. Sisakht, F. Fazileh, M. Zare, M. Zarenia, and F. Peeters, Physical Review B 94, 085417 (2016). D. Shao, J. Ruan, J. Wu, T. Chen, Z. Guo, H. Zhang, J. Sun, L. Sheng, and D. Xing, Physical Review B 96, 075112 (2017). S.-M. Nie, Z. Song, H. Weng, and Z. Fang, Physical Review B 91, 235434 (2015). C. Heil, S. Poncé, H. Lambert, M. Schlipf, E. R. Margine, and F. Giustino, Physical review letters 119, 087003 (2017).