⇦Chinese Physics Letters, 2023, Vol. 40, No. 3, Article code 037102 Flat Band and $\mathbb{Z}_2$ Topology of Kagome Metal CsTi$_{3}$Bi$_{5}$ Yuan Wang (王渊)1†, Yixuan Liu (刘以轩)1†, Zhanyang Hao (郝占阳)1†, Wenjing Cheng (程文静)1†, Junze Deng (邓竣泽)2†, Yuxin Wang (王郁欣)2, Yuhao Gu (顾雨豪)2, Xiao-Ming Ma (马小明)1, Hongtao Rong (戎洪涛)1, Fayuan Zhang (张发远)1, Shu Guo (郭抒)1, Chengcheng Zhang (张成成)1, Zhicheng Jiang (江志诚)3, Yichen Yang (杨逸尘)3, Wanling Liu (刘万领)3, Qi Jiang (姜琦)3, Zhengtai Liu (刘正太)3, Mao Ye (叶茂)3, Dawei Shen (沈大伟)3, Yi Liu (刘毅)4, Shengtao Cui (崔胜涛)4, Le Wang (王乐)1, Cai Liu (刘才)1, Junhao Lin (林君浩)1, Ying Liu (刘影)1, Yongqing Cai (蔡永青)1*, Jinlong Zhu (朱金龙)1, Chaoyu Chen (陈朝宇)1*, and Jia-Wei Mei (梅佳伟)1* Affiliations 1Shenzhen Institute for Quantum Science and Engineering (SIQSE) and Department of Physics, Southern University of Science and Technology (SUSTech), Shenzhen 518055, China 2Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 3State Key Laboratory of Functional Materials for Informatics and Center for Excellence in Superconducting Electronics, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai 200050, China 4National Synchrotron Radiation Laboratory, University of Science and Technology of China, Hefei 230029, China Received 6 January 2023; accepted manuscript online 21 February 2023; published online 2 March 2023 †These authors contributed equally to this work.
*Corresponding authors. Email: caiyq@sustech.edu.cn; chency@sustech.edu.cn; meijw@sustech.edu.cn Citation Text: Wang Y, Liu Y X, Hao Z Y et al. 2023 Chin. Phys. Lett. 40 037102    Abstract The simple kagome-lattice band structure possesses Dirac cones, flat band, and saddle point with van Hove singularities in the electronic density of states, facilitating the emergence of various electronic orders. Here we report a titanium-based kagome metal CsTi$_{3}$Bi$_{5}$ where titanium atoms form a kagome network, resembling its isostructural compound CsV$_{3}$Sb$_{5}$. Thermodynamic properties including the magnetization, resistance, and heat capacity reveal the conventional Fermi liquid behavior in the kagome metal CsTi$_{3}$Bi$_{5}$ and no signature of superconducting or charge density wave (CDW) transition anomaly down to 85 mK. Systematic angle-resolved photoemission spectroscopy measurements reveal multiple bands crossing the Fermi level, consistent with the first-principles calculations. The flat band formed by the destructive interference of hopping in the kagome lattice is observed directly. Compared to CsV$_{3}$Sb$_{5}$, the van Hove singularities are pushed far away above the Fermi level in CsTi$_{3}$Bi$_{5}$, in line with the absence of CDW. Furthermore, the first-principles calculations identify the nontrivial $\mathbb{Z}_2$ topological properties for those bands crossing the Fermi level, accompanied by several local band inversions. Our results suppose CsTi$_{3}$Bi$_{5}$ as a complementary platform to explore the superconductivity and nontrivial band topology.
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 DOI:10.1088/0256-307X/40/3/037102 © 2023 Chinese Physics Society Article Text The intertwining between strong electronic correlation and nontrivial band topology in quantum materials can give rise to exotic phenomena, e.g., topologically protected Majorana fermions.[1,2] The kagome lattice, composed of a two-dimensional network of corner-sharing triangles, has attracted much attention in recent years because of its special lattice geometry and characteristic electronic structure. The kagome lattice naturally hosts intriguing electronic band structure containing Dirac cones, flat band, and van Hove singularities based on the tight-binding model,[3] which provides a promising platform to realize nontrivial topology[4] and novel correlated ground states.[5-7] During the past decade, topological properties of different kagome lattice compounds with various magnetic ground states have been extensively studied and realized in a series of transition-metal kagome compounds.[8-14] For example, massive Dirac fermions and flat bands in ferromagnet Fe$_{3}$Sn$_{2}$,[8] YMn$_{6}$Sn$_{6}$,[9] antiferromagnet FeSn,[10] and paramagnet CoSn,[11] Chern gapped Dirac fermions in ferromagnetic TbMn$_{6}$Sn$_{6}$,[12] magnetic Weyl fermions and chiral anomaly in ferromagnet Co$_{3}$Sn$_{2}$S$_{2}$[13] and non-collinear antiferromagnet Mn$_{3}$Sn[14] are discovered successively. Apart from these magnetic 3$d$ transition metals, non-magnetic kagome compounds can host rich physics as well. Recently, a new family of $A$V$_{3}$Sb$_{5}$ ($A$ = K, Rb, Cs)[15-17] with layered kagome lattice has attracted tremendous interest in condensed matter physics due to its novel properties, such as giant anomalous Hall conductivity,[18,19] pair density wave order,[20] charge density wave (CDW) order,[21] electronic nematic order,[22,23] time-reversal symmetry breaking,[24-26] and possible chiral flux phase.[27] Despite intensive research efforts, the origin of these orders and the interaction between them are still under hot debate. Through the element species substitution, promising CsV$_{3}$Sb$_{5}$-like kagome materials have been proposed to investigate the novel quantum phenomena in kagome compounds.[28] Recent efforts have succeeded in synthesizing the titanium-based kagome metal CsTi$_{3}$Bi$_{5}$.[29,30] On the one hand, there remain outstanding controversies on its superconductivity and $\mathbb{Z}_2$ topology. On the other hand, the absence of the CDW order in this isostructural compound provides valuable chances of designing control experiments to disentangle these above novel properties. Therefore, it is of vital importance to investigate the physical properties and electronic structure of CsTi$_{3}$Bi$_{5}$. In this Letter, we systematically study the transport, electronic structure, and band topology of the newly discovered kagome metal CsTi$_{3}$Bi$_{5}$. The transport measurements reveal Fermi liquid behavior and no signature of superconductivity and CDW order at ambient and high pressure. Angle-resolved photoemission spectroscopy (ARPES) and density functional theory (DFT) calculations have revealed multiple bands crossing the Fermi levels, with nontrivial $\mathbb{Z}_2$ topology and local inverted gaps, which naturally induce topological surface states. Compared to CsV$_{3}$Sb$_{5}$, the flat band is pushed up towards the Fermi level ($\sim $ 0.25 eV) and is observed clearly. In addition, the van Hove singularities are pushed far away from the Fermi level in CsTi$_{3}$Bi$_{5}$, in line with the absence of the CDW order. Combined with CsV$_{3}$Sb$_{5}$, these results of CsTi$_{3}$Bi$_{5}$ provide clues to investigate the superconductivity and nontrivial band topology of kagome lattice.
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Fig. 1. Crystal structure and bulk electronic properties of CsTi$_{3}$Bi$_{5}$ at ambient pressure. (a) Layered crystal structure of CsTi$_{3}$Bi$_{5}$. (b) Top view of crystal structure. Ti atoms form a perfect kagome lattice and Bi2 atoms form simple graphene-like networks. (c) X-ray diffraction pattern of the CsTi$_{3}$Bi$_{5}$ single crystal with the peaks indexed by (00$l$). (d) Temperature-dependent susceptibility $\chi =M/H$ measured with $\mu _{0}H = 1$ T for $H//c$. (e) Zero-field temperature-dependent resistivity $\rho$ down to 1.8 K on sample 1. The inset is the ultra-low temperature resistivity down to 85 mK on sample 2. (f) Zero-field temperature-dependent heat capacity $C_{p}$.
Table 1. Wyckoff sites, coordinates, and equivalent isotropic displacement parameters ($U_{\rm eq}$) for CsTi$_{3}$Bi$_{5}$ at 100 K, determined by single-crystal x-ray diffraction.
Atom Wyckoff site $x$ $y$ $z$ $U_{\rm eq}$
Ti 3$g$ 0.0 0.5 0.5 0.0038(17)
Bi1 1$b$ 0.0 0.0 0.5 0.0046(11)
Bi2 4$h$ 2/3 1/3 0.7622(2) 0.0052(10)
Cs 1$a$ 0.0 0.0 0.0 0.0105(15)
CsTi$_{3}$Bi$_{5}$ crystallizes in the hexagonal crystal system with the space group $P6/mmm$ (#191, $a = 5.8054(7)$ Å, $c = 9.2312(13)$ Å). Two independent Bi Wyckoff positions (Bi1, $1b$; Bi2, $4h$), one independent Ti position (Ti, $3g$), and one independent Cs position (Cs, $1a$) are found in the asymmetric unit of CsTi$_{3}$Bi$_{5}$ (Table 1). As shown in Figs. 1(a) and 1(b), titanium atoms form a corner-shared kagome network in the crystallographic $ab$ plane. The Bi1 atoms locate in the $ab$ plane, filling the hexagon center of the Ti-kagome lattice. The Bi2 atoms locate above and below the Ti-kagome triangles and form the honeycomb-lattice networks nearby the kagome layer. The Cs atoms fill in the space among the adjacent kagome layers and form simple hexagonal networks. More specifically, $d_{\rm Ti-Ti }=d_{\rm Ti-Bi1} = 2.90$ Å, $d_{\rm Bi2-Bi2} = 3.35$ Å. Figure 1(c) presesnts the x-ray diffraction pattern of the CsTi$_{3}$Bi$_{5}$ single crystal with a series of sharp (00$l$) peaks. The atomic coordinates and equivalent isotropic displacement parameters of CsTi$_{3}$Bi$_{5}$ are summarized in Table 1. Figures 1(d) and 1(f) demonstrate the transport properties of CsTi$_{3}$Bi$_{5}$ characterized by the magnetization, electric transport, and heat capacity measurements at ambient pressure. Magnetization data collected under a field parallel to the $c$ axis with $\mu _{0}H = 1$ T is plotted as susceptibility $\chi =M/H$ in Fig. 1(d). It shows a Pauli paramagnetic behavior, weakly dependent on the temperature. At low temperatures, an upturn appears, which is attributed to the impurity Curie–Weiss contribution with a small concentration of less than 5%. Figure 1(e) shows the zero-field temperature-dependent resistivity on sample 1 with current flowing along the $a$-axis and no anomaly is observed down to 1.8 K. The low-temperature ($T < 40$ K) resistivity can be well described by the Fermi liquid behavior, $\rho =\rho_{0 }+ AT ^{2}$ with $\rho_{0} = 4.3\,µ \Omega \cdot$cm and $A = 0.0015\,µ \Omega \cdot$cm$\cdot$K$^{-2}$. The inset in Fig. 1(e) shows the resistance at the ultra-low temperature on sample 2 down to 85 mK with no resistance drop observed, excluding the superconducting impurity component (e.g., CsBi$_{2}$) in our samples. Figure 1(f) presents the zero-field heat capacity in which the low-temperature data can be fitted by the combination of the electronic linear $T$ part and the phonon cubic-$T$ part, $C_{p}=\gamma T + \beta T^{3}$ with $\gamma = 20$ mJ$\cdot$mol$^{-1}$ K$^{-2}$ and $\beta=8.1$ mJ$\cdot$mol$^{-1}$ K$^{-4}$. These properties describe CsTi$_{3}$Bi$_{5}$ as a metal exhibiting typical Fermi liquid behaviors. Furthermore, under hydrostatic pressure applied up to 46.9 GPa (Fig. S1 in the Supplementary Material), signature of superconductivity is found in the resistivity measurements. However, the signal of the superconductivity is likely caused by the Bi impurities,[31] as evidenced by the non-zero resistivity state and weak drop of the resistance which reflects the small inclusion of a superconducting impurity phase. Further transport experiments at high pressure need to be carried out to fully rule out the possible bulk superconductivity CsTi$_{3}$Bi$_{5}$. To reveal the electronic structure of kagome metal CsTi$_{3}$Bi$_{5}$, high-resolution ARPES measurements are carried out. With the hexagonal Brillouin zone and high-symmetry points shown in Fig. 2(a), the clear periodic dispersion obtained by systematic photon energy-dependent measurements enables us to identify the bulk $\varGamma$ and $A$ points [Fig. 2(b)]. ARPES measurements and DFT calculations at $k_{z}=0$ [Figs. 2(c) and 2(d)] and $k_{z}=\pi$ [Figs. 2(e) and 2(f)] are in satisfactory agreement concerning the general geometry. As shown in Figs. 2 and 3, there are three electron-like bands around the $\varGamma /A$ point, named $\alpha$, $\beta$, and $\gamma$. Thereinto, the $\alpha$ band forms a circular pocket, while pockets composed of $\beta$ and $\gamma$ bands exhibit a hexagonal shape. The band $\delta$ crosses the Fermi level between $\varGamma/A$–$K/H$ and $K/H$–$M/L$, forming a closed triangle surrounding the $K/H$ point. Around the $M/L$ point, there is a hole-like band, named $\varepsilon$, which forms a rhombic-like Fermi surface sheet. Both the measured and calculated Fermi surfaces demonstrate the six-fold rotation symmetry of this material. Furthermore, the Fermi surfaces are almost identical to each other at the $\varGamma$–$K$–$M$ and $A$–$H$–$L$ planes, reflecting its 2D nature.
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Fig. 2. ARPES measured and DFT calculated band structure of CsTi$_{3}$Bi$_{5}$. (a) Three-dimensional Brillouin zone. (b) Photon energy-dependent spectra (60–120 eV) for intensity at $E_{\rm B}=0.56$ eV with $k_{x}$ parallel to the $\overline K $–$\overline \varGamma $–$\overline K$ path. (c) Fermi surface map at the $\varGamma $–$K$–$M$ plane with the photon energy of 88 eV. (d) DFT calculated constant energy contour (CEC) at $k_{z}=0$ plane. (e) Fermi surface map at the $A$–$H$–$L$ plane with the photon energy of 100 eV. (f) Same as (d) but at $k_{z}=\pi$ plane.
We note that both ARPES spectra along $\varGamma$–$K$–$M$ ($\varGamma$–$M$) and $A$–$H$–$L$ ($A$–$L$) paths show very similar feature, which is different from the DFT calculated results with clear $k_{z}$ dependence (Fig. S2 in the Supplementary Material). This discrepancy may come from the surface relaxation effect, which can increase the interlayer spacing on the surface, especially for that between the topmost and the second layer, resulting in a nearly isolated surface layer. Considering the ARPES probe depth (1–2 nm) within the photon energy range can only cover 1–2 atomic layers, the surface relaxation effect can lead to the 2D nature of the band structure and reduce the $k_{z}$ dependence.[32] In a kagome lattice, destructive interference of electron hopping around the kagome bracket can produce rather localized states in real space, which naturally leads to the flat band in momentum space.[4] Previous DFT calculations and APRES measurements reveal that the flat band in CsV$_{3}$Sb$_{5}$ locates beyond 1 eV below $E_{\rm F}$, rendering its physical responses hardly observable.[15,33] The complete substitution of V and Sb atoms with Ti and Bi atoms induces considerable hole carriers, making the realization of the low-energy flat band in CsTi$_{3}$Bi$_{5}$ possible. To explore the expected non-dispersive bands, the ARPES spectra along the high-symmetry direction ($\varGamma$–$K$–$M$) and its corresponding 2D curvature are demonstrated in Figs. 3(a) and 3(b). A pronounced low-energy flat band is observed at 0.25 eV below $E_{\rm F}$ across the whole momentum area, which is in good match with our DFT calculated results [Fig. 3(c)]. Compared to CsV$_{3}$Sb$_{5}$, besides the flat band is pushed upwards to the Fermi level, the van Hove singularities are pushed above the Fermi level correspondingly, which leads to the absence of CDW order. The low-energy flat band suggests that the electron correlation effect may play an important role in CsTi$_{3}$Bi$_{5}$ and calls for further investigation of the theoretical analyses and transport experiments.
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Fig. 3. Direct observation of the flat band. (a) ARPES spectra taken along the $\varGamma$–$K$–$M$–$K$–$\varGamma$ direction. (b) Corresponding 2D curvature of (a). (c) DFT calculated band structure along the $\varGamma$–$K$–$M$–$K$–$\varGamma$ direction. The flat band is marked by the light green shaded area.
Based on the band structure calculations including spin-orbit coupling (SOC), we present an analysis of the topological invariants of CsTi$_{3}$Bi$_{5}$. In V-based kagome superconductors, under the consideration of the SOC effect, a continuous, symmetry-enforced direct gap at every $k$ point, as well as the time reversal and inversion symmetry, makes it possible for calculating the $\mathbb{Z}_2$ topological invariant between each pair of bands near the Fermi level by simply analyzing the parity of the wave function at the time-reversal invariant momentum points.[15,16] As shown in Fig. 4(a), the overall band structure of CsTi$_{3}$Bi$_{5}$ shows a similar feature to that of CsV$_{3}$Sb$_{5}$, except for the band filling induced by the hole doping. Unfortunately, near the Fermi level, there is a symmetry-enforced band crossing between $\varGamma$–$A$. The absence of a continuous direct gap makes it difficult to calculate its $\mathbb{Z}_2$ topological invariant. However, by analyzing the band structure of CsTi$_{3}$Bi$_{5}$, we find some band inversions exhibiting nontrivial nature [marked by green circles in Fig. 4(a)]. Figure 4(b) shows that the electronic density of states of CsTi$_{3}$Bi$_{5}$ near $E_{\rm F}$ is mainly contributed by Ti and Bi orbitals. We select five bands near the Fermi energy and find the $\mathbb{Z}_2$ index changes from $Z_{2}=1$ (band #115) to $Z_{2}=0$ (band #123) in Fig. 4(c). According to such a sign change, in combination with the multiple local nontrivial features with band inversion in the band structure shown in Fig. 4(a), we conclude that there exists nontrivial $\mathbb{Z}_2$ topology locally in the bulk band of CsTi$_{3}$Bi$_{5}$, which naturally leads to the emergence of topological surface states.
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Fig. 4. DFT calculated band structure with $\mathbb{Z}_2$ topology. (a) Calculated band structure with SOC along the high-symmetry directions. (b) Calculated density of states. Colors correspond to different atomic contributions. Green circles highlight the band inversion region while black circle highlights the opening of the gap at the band crossing. (c) Product of parity and $\mathbb{Z}_2$ indices of the selected bands.
In summary, we have systematically investigated the electronic structure and band topology of the newly discovered kagome metal CsTi$_{3}$Bi$_{5}$. Thermodynamic property measurements reveal conventional Fermi liquid behavior in the kagome metal CsTi$_{3}$Bi$_{5}$. No signal of superconducting or CDW transition anomaly is observed down to 85 mK. The ARPES measurements reveal multiple bands crossing the Fermi levels, consistent with the DFT calculations. Compared to CsV$_{3}$Sb$_{5}$, the van Hove singularities are pushed far away above the Fermi level in CsTi$_{3}$Bi$_{5}$, in line with the absence of CDW. Furthermore, first-principles calculations suggest local nontrivial $\mathbb{Z}_2$ topological properties for those bands crossing the Fermi level, implying the existence of topological surface states. The flat band formed by the destructive interference of hopping on the kagome lattice is observed at $\sim $ 0.25 eV below the Fermi level. Combined with CsV$_{3}$Sb$_{5}$, these results of CsTi$_{3}$Bi$_{5}$ provide a complementary platform to investigate the superconductivity and nontrivial band topology in the kagome lattice. Note Added. Recently, we became aware that other groups independently studied the electronic structures of this compound.[34-37] Acknowledgements. This work was supported by the National Key R&D Program of China (Grant No. 2022YFA1403700), the National Natural Science Foundation of China (Grant Nos. 12074163 and 12004030), the Guangdong Basic and Applied Basic Research Foundation (Grant Nos. 2022B1515020046, 2022B1515130005, 2021B1515130007, and 2020B1515120100), the Guangdong Innovative and Entrepreneurial Research Team Program (Grant Nos. 2017ZT07C062 and 2019ZT08C044), the Shenzhen Science and Technology Program (Grant No. KQTD20190929173815000), Shenzhen Key Laboratory of Advanced Quantum Functional Materials and Devices (Grant No. ZDSYS20190902092905285), the Shenzhen Fundamental Research Program (Grant No. JCYJ20220818100405013), and China Postdoctoral Science Foundation (Grant No. 2020M682780 and 2022M711495). The authors acknowledge the assistance of SUSTech Core Research Facilities. The calculations were performed at Tianhe2-JK at Beijing Computational Science Research Center. 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