⇦Chinese Physics Letters, 2021, Vol. 38, No. 7, Article code 077101 Dual Topological Features of Weyl Semimetallic Phases in Tetradymite BiSbTe$_{3}$ Z. Z. Zhou (周字桢)1, H. J. Liu (刘惠军)2, G. Y. Wang (王国玉)3, R. Wang (王锐)1*, and X. Y. Zhou (周小元)1* Affiliations 1Center for Quantum Materials and Devices, College of Physics, Chongqing University, Chongqing 401331, China 2Key Laboratory of Artificial Micro- and Nano-Structures of Ministry of Education, and School of Physics and Technology, Wuhan University, Wuhan 430072, China 3Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Science, Chongqing 400714, China Received 11 March 2021; accepted 30 April 2021; published online 3 July 2021 Supported by the National Natural Science Foundation of China (Grant Nos. 11604032, 11674040, and 51672270), and the Fundamental Research Funds for the Central Universities (Grant No. 106112016CDJZR308808).
*Corresponding authors. Email: rcwang@cqu.edu.cn; xiaoyuan2013@cqu.edu.cn Citation Text: Zhou Z Z, Liu H J, Wang G Y, Wang R, and Zhou X Y 2021 Chin. Phys. Lett. 38 077101    Abstract Based on first-principles calculations and symmetry arguments, we reveal that the non-centrosymmetric ternary tetradymite BiSbTe$_{3}$ possesses exotic dual topological features of Weyl semimetallic phases with $Z_{2}$ index (1:000). The results show that the helical Dirac-type surface states protected by the time-reversal symmetry are present in the vicinity of the Brillouin zone center, which is consistent with the experimental report. Furthermore, we show that four pairs of Weyl points reside exactly at the Fermi level, which are guaranteed to be located on high-symmetry planes due to mirror symmetries. The helical surface states and the projected Weyl nodes are well separated in the momentum space, facilitating their observations in experiments. This work not only uncovers a unique quantum phenomenon with dual topological features in the tetradymite family but also paves a fascinating avenue for exploring the coexistence of multi-topological states with wide applications. DOI:10.1088/0256-307X/38/7/077101 © 2021 Chinese Physics Society Article Text Materials with a nontrivial band topology have guided significant advances over the past decade due to the fundamental scientific importance and immense potential applications.[1–3] A representative example is the surface states of topological insulators (TI), in which the suppressed backscattering channels favor dissipationless electronic conduction.[4,5] Subsequently, a new class of topological quantum materials named Weyl semimetals (WSMs) have been discovered,[6–9] in which the band crossed points (i.e., Weyl points) with linear dispersions near the Fermi level, and can be characterized by the Weyl equation.[10] The Weyl points with opposite chirality always appear in pairs,[10] which are separated in momentum space via breaking either the inversion (${\cal I}$) or time-reversal (${\cal T}$) symmetry. Intriguingly, the exotic bulk and surface states in WSMs have induced various manifestations such as negative magnetoresistance[11] and chiral anomaly,[12] which also stimulated extensive investigations for the functionalities including valley polarization[13] and magnetization switching.[14] Since the binary Bi$_{2}$Te$_{3}$, Bi$_{2}$Se$_{3}$, and Sb$_{2}$Te$_{3}$ were demonstrated to be strong TIs both theoretically and experimentally,[15,16] tetradymite compounds have received increasing attention owing to the intimate link between the band topology and realistic applications.[17–19] For instance, topological tetradymites provide ideal platforms for exploring the coupling mechanism between the nontrivial surface states and thermoelectric effects.[20–22] To date, many tetradymites have been identified to be strong TIs, e.g., Sb$_{2}$Te$_{2}$S, Sb$_{2}$Te$_{2}$Se, Bi$_{2}$Te$_{2}$Se, Bi$_{2}$Se$_{2}$Te, and BiSbSeTe$_{2}$.[23–25] In particular, Cao et al.[26] proposed a two-dimensional descriptor to scan a huge number of TIs in the tetradymite family based on high-throughput prediction. Despite the rapid progress in screening TIs, tetradymites with WSM features have rarely been observed. One important reason is that the band extrema of tetradymite with strong spin-orbital coupling (SOC) do not locate at the high-symmetry lines,[27] making the possible nodal points hidden in the Brillouin zone (BZ). In contrast to conventional materials with the individual topological phase, tetradymite compounds are expected to host multiple topological features at distinct positions in momentum space, which are highly desirable for the study of coexisting topological features. Hence, a previously uncharted and much meaningful material space with unconventional topological quantum phenomena remains largely unexplored in the tetradymite family. In this Letter, we show that the non-centrosymmetric ternary tetradymite BiSbTe$_{3}$ satisfies the criteria mentioned above. Using first-principles calculations combined with a ${\boldsymbol k}$$\cdot$${\boldsymbol p}$ model analysis, we demonstrate the presence of exotic dual topological features, which has not been reported in realistic material up to now. Apart from the topological inverted gap near the $\varGamma$ point, we reveal that there are four pairs of Weyl points located on high-symmetry planes rather than high-symmetry lines. This study offers a feasible platform for investigating the coexistence of multi-topological phases with unconventional nontrivial surface states. Methods. Based on density functional theory (DFT), the electronic band structures of BiSbTe$_{3}$ were calculated with the projector augmented wave approach,[28,29] as implemented in the Vienna ab initio simulation package.[30–32] The Perdew–Burke–Ernzerhof (PBE) functional with the generalized gradient approximation[33] was employed in structural relaxation and electronic calculations. A $15\times 15\times 15$ Monkhorst–Pack ${\boldsymbol k}$-mesh was adopted for the BZ integrations with the kinetic-energy cutoff set to be 550 eV. The SOC effect and the van der Waals (vdW) interactions with the form of optB86b-vdW[34] were explicitly included in our calculations. In addition, partially self-consistent $GW_{0}$ calculations with a $6\times 6\times 6 \varGamma$ centered ${\boldsymbol k}$-grid were performed to accurately predict the electronic band structures of BiSbTe$_{3}$. Based on the maximally localized Wannier functions,[35] the topological surface states were studied via the iterative Green's function method,[36] as implemented in the so-called WANNIERTOOLS package.[37] To ensure the reliability, all the calculated results were rechecked by the MBJ method as shown in Fig. S3 of the Supplemental Material.
cpl-38-7-077101-fig1.png
Fig. 1. (a) The primitive cell and (b) conventional cell of BiSbTe$_{3}$. (c) The BZ of the primitive cell as well as the surface BZs of (001) and (110) planes.
Results and Discussions. Tetradymite with the ${\cal I}$ symmetry such as Bi$_{2}$Te$_{3}$ crystalizes in a rhombohedral structure belonging to the space group $R\bar{3}m$ (No. 166). By substituting one Bi atom with Sb element for each quintuple layer shown in Figs. 1(a) and 1(b), the ${\cal I}$ symmetry breaking induces that the space group of the ternary BiSbTe$_{3}$ transforms to be $R3m$ (No. 160). In this case, the ${\cal T}$ symmetry and three reflection symmetries with respect to the (100), (010), and (110) mirror planes are preserved. Figure 1(c) plots the bulk BZ and the surface BZs in (110) and (001) plane. The high-symmetry points are also projected to the corresponding surface. The optimized lattice parameters of BiSbTe$_{3}$ are calculated to be $a = 4.344$ Å and $c = 30.518$ Å for the conventional cell, which nicely agree with the experimental results.[38] It is noteworthy that the single crystal of BiSbTe$_{3}$ in a disorder phase has already been synthesized experimentally due to its excellent thermoelectric performance,[39] which ensures the existence of the tetradymite with such a stoichiometry. Additionally, we have calculated the phonon dispersion relations of BiSbTe$_{3}$ as plotted in Fig. S4 of the Supplemental Material. There is no imaginary frequency, guaranteeing the dynamic stability. The formation energy of BiSbTe$_{3}$ is also evaluated by Sb-doping in Bi$_{2}$Te$_{3}$. The calculated formation energy is $-$28 meV/primitive-cell, which indicates the energetic stability. Moreover, we have calculated the total energy of the noncentrosymmetric BiSbTe$_{3}$, which is 46 meV/primitive-cell lower than that of the inversion symmetric system. Here we achieve the inversion symmetric BiSbTe$_{3}$ by constructing a $1\times 1\times 2$ supercell with nominal formula of Bi$_{2}$Sb$_{2}$Te$_{6}$. Therefore, it is convincing that the noncentrosymmetric BiSbTe$_{3}$ is stable, implying that BiSbTe$_{3}$ will be verified further in experiments.
cpl-38-7-077101-fig2.png
Fig. 2. The orbital-decomposed band structures of BiSbTe$_{3}$ (a) without and (b) with SOC. The ${P}$ and ${T}$ are not the high-symmetry points with the coordinates of (0.402, 0.500, 0.210) and (0.402, 0.210, 0.500), respectively. (c) Enlarged view of the Weyl points framed in (b). (d) The band structures along the direction containing $W_{2+}$ and $W_{2-}$ points. The coordinates of ${A}$ and ${B}$ are (0.094, 0.094, 0.000) and (0.594, 0.594, 0.500), respectively.
Figures 2(a) and 2(b) illustrate the orbital-decomposed band structures of BiSbTe$_{3}$ without and with SOC, respectively, where the vdW interactions and quasiparticle $GW_{0}$ approximation are explicitly considered. In the absence of SOC, BiSbTe$_{3}$ is predicted to be a normal semiconductor with a direct bandgap of 0.13 eV at the $\varGamma$ point. By taking into account the SOC, we see from Fig. 2(b) that visible band inversion arises near the $\varGamma$ point, which yields topologically protected surface states as demonstrated experimentally.[17] Combining the calculated topological $Z_{2}$ invariants of $(v_{0}:v_{1} v_{2} v_{3})=(1:000)$ (see Fig. S5 in the Supplemental Material), the nontrivial topological nature of BiSbTe$_{3}$ can be sufficiently testified as also observed in binary tetradymites. Furthermore, it is found that the band structures exhibit remarkable spin splitting except for the time-reversal invariant ${\boldsymbol k}$-points ($\varGamma$, $L$, and $F$). Such a fact also reveals the broken ${\cal I}$ but reserved ${\cal T}$ symmetry, which allows the realization of the Weyl semimetallic phase.[40] Indeed, a total of four pairs of type-I Weyl points can be screened by a careful search in the whole BZ. The corresponding ${\boldsymbol k}$-points are listed in Table 1, as well as their chirality determined via the Wilson-loop method.[37] Interestingly, it is found that all the Weyl points almost exactly reside at the Fermi level, which could maximize the transport phenomena caused by chiral anomaly. In Fig. 2(b), we plot the band structures along the ${\boldsymbol k}$-path containing two Weyl points with the same chirality of $+$1, among which obvious band inversion is taken place. Hence, BiSbTe$_{3}$ is a unique quantum material possessing both helical Dirac-type surface states and Weyl semimetallic states. Note that all the Weyl points locate at the genetic ${\boldsymbol k}$-points, where large Zeeman-type splitting is induced by the strong SOC of Bi element in the non-centrosymmetric space group $R3m$. Figure 2(c) is the enlarged view of the Weyl points highlighted by the black rectangle in Fig. 2(b), where the top valance band and bottom conduction band at $W_{2+}$ exhibit giant Zeeman-type splitting of 0.14 eV and 0.12 eV, respectively. The total splitting value is larger than the band gap without SOC, which in turns leads to the band cross at the Fermi level. The dispersions around the band touching points are linear in momentum space implying the Weyl semimetallic state. Additionally, the insulator-metal transition originating from spin splitting in a nonmagnetic system is highly appealing for the applications in spintronics.[41]
Table 1. The coordinates in momentum space and the chirality of eight Weyl points.
Coordinate Chirality
$k_{x}=k_{y}$ plane $W_{1-}$ ($-0.422$, $-0.422$, $-0.328$) $-1$
$W_{1+}$ ($-0.402$, $-0.402$, $-0.308$) $+1$
$W_{2-}$ (0.422, 0.422, 0.328) $-1$
$W_{2+}$ (0.402, 0.402, 0.308) $+1$
$k_{x}=k_{z}$ plane $W_{3-}$ ($-0.422$, $-0.328$, $-0.422$) $-1$
$W_{3+}$ ($-0.402$, $-0.308$, $-0.402$) $+1$
$W_{4-}$ (0.422, 0.328, 0.422) $-1$
$W_{4+}$ (0.402, 0.308, 0.402) $+1$
To comprehend the band topology inherent to the crystal symmetry, we show in Fig. 2(d) the band structures with ${\boldsymbol k}$-points turned to $W_{2+}$ and $W_{2-}$ presented at $k_{x}=k_{y}$ plane, where the coordinates can be denoted as $(k_{x}, k_{x}, k_{z})$. Besides the ${\cal T}$ symmetry, the little group belongs to ${\cal M}$ mirror symmetry, which preserves two irreducible representations of $\varGamma_{3}$ and $\varGamma_{4}$ derived from the character table. The crossed bands adjacent to the Fermi level possess the eigenvalues $+i$ for $\varGamma_{3}$ and $-i$ for $\varGamma_{4}$. Based on a two-band ${\boldsymbol k}$$\cdot$${\boldsymbol p}$ model ignoring the kinetic term, the general $2\times 2$ Hamiltonian can be written as $$ H(k_{x}, k_{y}, k_{z})=\sum\limits_{i=x,y,z} {f_{i} (k_{x}, k_{y}, k_{z})\sigma_{i} },~~ \tag {1} $$ where $f_{i}({k_{x}, k_{y}, k_{z}})$ and $\sigma_{i}$ are the real functions and Pauli matrices, respectively. For the $k_{x}=k_{y}$ plane, we consider the combination of ${\cal T}$ and ${\cal M}$. The Hamiltonian is thus constrained by $$ [H(k_{x}, k_{y}, k_{z}), {\cal{TM}}]=0.~~ \tag {2} $$ Here ${\cal M}$ can be expressed as $i\sigma_{z}$, and the ${\cal T}$ operator can be represented as $-i\sigma_{y} K$ in the presence of SOC, where $K$ is a complex conjugate operator. The product ${\cal{TM}}$ expressed as $i\sigma_{x} K$ denotes an antiunitary mirror symmetry. Accordingly, Eq. (2) gives $$\begin{align} &f_{x,y} (k_{x}, k_{y}, k_{z})=f_{x,y} (-k_{y}, -k_{x}, -k_{z}) \\ &f_{z} (k_{x}, k_{y}, k_{z})=-f_{z} (-k_{y}, -k_{x}, -k_{z}) .~~ \tag {3} \end{align} $$ Under this condition, the Hamiltonian for the crossed bands should be given by $$\begin{alignat}{1} H(k_{x}, k_{x}, k_{z})=f_{x} (k_{x}, k_{x}, k_{z})\sigma_{x} +f_{y} (k_{x}, k_{x}, k_{z})\sigma_{y},~~~~~~ \tag {4} \end{alignat} $$ which allows the presence of Weyl points in $k_{x}=k_{y}$ plane. These symmetry-protected Weyl points suggest the robust topological Weyl semimetallic state of BiSbTe$_{3}$. It should be emphasized that the mirror $k_{x}=k_{y}$ plane does not invert the components of $k$-vectors. Thus, the ${\cal{TM}}$ operator in this work is not an axis, and the ${\cal M}$ operator does not change the chirality. To have a better understanding, the graphical representation of the distribution of the symmetry protected Weyl points in the Brillouin zone are plotted in Fig. S6 of the Supplemental Material. Furthermore, we also find that the other four Weyl points in $k_{x}=k_{z}$ plane should be accidental nodal points, which do not correspond to any symmetry and easily annihilate each other. Thus, there are a total of four pairs of Weyl points residing on the Fermi level in BiSbTe$_{3}$. Among them, only the two pairs of Weyl points residing on the $k_{x}=k_{y}$ plane are protected by mirror symmetry. The dual topological features of BiSbTe$_{3}$ can produce unique nontrivial surface states. Figure 3(a) displays the surface state on (001) plane with a topologically protected Dirac cone lying at $\sim $0.2 eV below the Fermi level, and the corresponding Fermi surface map as well as the spin texture at the energy of $-0.1$ eV is plotted in Fig. S7 in the Supplemental Material. Such an observation is in good agreement with the experimental measurement using angle-resolved photoemission spectroscopy[17] and indicates the helical Dirac-type surface states around the $\varGamma$ point as also found in binary tetradymites. Remarkably, we find that the surface states are terminated by a projection of bulk Weyl cone in $\overline {{\varGamma }}$–$\overline {{M}}$ direction. The Fermi surface map in the (001) plane at the energy of 0.015 eV is plotted in Fig. 3(b), where the continuous Fermi arcs form sixfold pockets reflecting the threefold rotational and ${\cal T}$ symmetries. More importantly, it is clear that a discontinuous Fermi arc starts from one Weyl point and ends at the other located along ${\hat{{\!\varGamma }}}$–${\hat{{\!R}}}$ direction on $k_{x}=k_{y}$ surface as shown in Fig. 3(c). The left and right nodes signed by the green and purple point are projected from two Weyl points of $W_{2+}$ and $W_{2-}$, respectively. It should be mentioned that the positions of each pair of two Weyl points are close to each other, leading to the nearly overlapped projections in $k_{x}=k_{y}$ surface. As a result, two Fermi arcs are observed at $W_{2+}$ and $W_{2-}$ points.[42] Additionally, we see from Fig. S8 of the Supplemental Material that the gapless points connected with Fermi arcs are projected from two Weyl nodes of $W_{2+}$ and $W_{4+}$. Figure 3(d) illustrates the Fermi arcs in the $k_{x}=k_{y}$ surface, where a closed loop is clearly visible. It is noted that the surface states corresponding to these two topological features are distinct and well separated in the BZ, which facilitates its experimental measurement.
cpl-38-7-077101-fig3.png
Fig. 3. (a) Surface states projected on the (001) plane and the (b) corresponding Fermi surface. (c) The projected surface states for the $k_{x}=k_{y}$ plane along ${\hat{{\!\varGamma }}}$–${\hat{{\!R}}}$ direction. (d) The Fermi arcs projected on $k_{x}=k_{y}$ surface.
It should be mentioned that the nontrivial band topology of some TIs and WSMs are sensitive to the strains.[43–45] Herein, we consider both the compressive and tensile strains along the $a$- and $c$-directions, which may have a qualitatively different influence on the topological states.[44] Figures 4(a) and 4(b) plot the band structures of BiSbTe$_{3}$ in proximity of valance and conduction band edges under in-plane strains with $a=0.98 a_{0}$ and $a=1.02 a_{0}$, respectively. In both cases, the symmetry protected Weyl points in the $k_{x}=k_{y}$ plane are still reserved with the chirality implied by the evolution of Wannier charge centers, whereas the band crossed points are slightly lifted above the Fermi level. Similar to the cases of the in-plane strain, the Weyl points also reserved for the out-of-plane strains with $c=0.98 c_{0}$ and $c=1.02 c_{0}$ as shown in Figs. 4(c) and 4(d), respectively. These Weyl points (see Table S1) are away from the original positions but they are also distributed on $k_{x}=k_{y}$ plane, since all symmetries remain under the in-plane or out-of-plane strains. Moreover, our additional calculations manifest that the nonzero topological invariant is not affected by either the in-plane or out-of-plane strains (see Fig. S9 in the Supplemental Material), and the band inversions at the $\varGamma$ point are always in existence. Such a fact reveals that the topological $Z_2$ phase protected by the ${\cal T}$ symmetry is robust to the perturbations generated by strains. Consequently, the dual nontrivial topological states in BiSbTe$_{3}$ are impervious to both the in-plane strains and out-of-plane strains.
cpl-38-7-077101-fig4.png
Fig. 4. The band structures of BiSbTe$_{3}$ under 2% in-plane (a) compressive and (b) tensile strains. The insets show the evolution of the Wannier charge centers around the band crossed points. [(c), (d)] The band structures with 2% compressive and tensile strains along interlayer directions, respectively. The coordinates of $A_{1}$, $B_{1}$, $A_{2}$, and $B_{2}$ are (0.100, 0.100, 0.000), (0.600, 0.600, 0.500), (0.091, 0.091, 0.000), and (0.591, 0.591, 0.500), respectively.
In summary, our theoretical study reveals that the ternary tetradymite BiSbTe$_{3}$ possesses both helical Dirac-type surface states protected by the ${\cal T}$ symmetry and ideal Weyl semimetallic features. The discovery of Weyl points lying away from the high-symmetry lines is also reminiscent of the potential WSM concealed in tetradymites without the ${\cal I}$ symmetry, which considerably broadens the avenue for fundamental investigation of multiple topological phases. Intriguingly, the present work proposes a perfect platform to study the interplay between emergent topological nontrivial states and possible applications, especially for the thermoelectricity of tetradymites. References CO Oxidation Facilitated by Robust Surface States on Au-Covered Topological InsulatorsSpin polarization of the quantum spin Hall edge statesPressure-Induced Topological and Structural Phase Transitions in an Antiferromagnetic Topological InsulatorColloquium : Topological insulatorsTopological insulators and superconductorsTopological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridatesWeyl Semimetal Phase in Noncentrosymmetric Transition-Metal MonophosphidesType-II Weyl semimetalsRobust Twin Pairs of Weyl Fermions in Ferromagnetic OxidesThe Adler-Bell-Jackiw anomaly and Weyl fermions in a crystalChiral anomaly and classical negative magnetoresistance of Weyl metalsEvidence for the chiral anomaly in the Dirac semimetal Na3BiTopological Imbert-Fedorov Shift in Weyl SemimetalsVoltage-Driven Magnetization Switching and Spin Pumping in Weyl SemimetalsTopological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surfaceExperimental Realization of a Three-Dimensional Topological Insulator, Bi2Te3Band structure engineering in (Bi1−xSbx)2Te3 ternary topological insulatorsConnecting Thermoelectric Performance and Topological-Insulator Behavior: Bi 2 Te 3 and Bi 2 Te 2 Se from First PrinciplesSwitching of charge-current-induced spin polarization in the topological insulator BiSbTeSe 2 In-Plane Transport and Enhanced Thermoelectric Performance in Thin Films of the Topological Insulators Bi 2 Te 3 and Bi 2 Se 3 Enhanced Thermoelectric Performance and Anomalous Seebeck Effects in Topological InsulatorsMaximizing the thermoelectric performance of topological insulator Bi 2 Te 3 films in the few-quintuple layer regimeAmbipolar field effect in the ternary topological insulator (BixSb1–x)2Te3 by composition tuningFermi level tuning of topological insulator Bi 2 (Se x Te 1−x ) 3 nanoplatesFirst-principles study of the thermoelectric properties of quaternary tetradymite BiSbSeTe 2Artificial intelligence for high-throughput discovery of topological insulators: The example of alloyed tetradymitesEffects of van der Waals interactions and quasiparticle corrections on the electronic and transport properties of Bi 2 Te 3 Projector augmented-wave methodFrom ultrasoft pseudopotentials to the projector augmented-wave methodAb initio molecular dynamics for liquid metalsAb initio molecular-dynamics simulation of the liquid-metal–amorphous-semiconductor transition in germaniumEfficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis setGeneralized Gradient Approximation Made SimpleChemical accuracy for the van der Waals density functionalwannier90: A tool for obtaining maximally-localised Wannier functionsQuick iterative scheme for the calculation of transfer matrices: application to Mo (100)WannierTools: An open-source software package for novel topological materialsAIP Conference ProceedingsAIP Conference ProceedingsPrediction and control of spin polarization in a Weyl semimetallic phase of BiSbZeeman-type spin splitting in nonmagnetic three-dimensional compoundsFerromagnetic Weyl fermions in CrO 2 Symmetry-protected ideal Weyl semimetal in HgTe-class materialsSharp Raman Anomalies and Broken Adiabaticity at a Pressure Induced Transition from Band to Topological Insulator in Sb 2 Se 3 Rhombohedral S b 2 S e 3 as an intrinsic topological insulator due to strong van der Waals interlayer coupling
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