Infrared spectroscopy study of kagome material CsTi$_3$Bi$_5$
Liye Cao Xiangqi Liu Jiayi Cheng Bixia Gao Xiaoting Zhang Yanfeng Guo Fengjie Ma Rongyan Chen











Center for Advanced Quantum Studies and School of Physics and Astronomy, Beijing Normal University, Beijing 100875, China
Key Laboratory of Multiscale Spin Physics (Ministry of Education), Beijing Normal University, Beijing 100875, China


School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China


Center for Advanced Quantum Studies and School of Physics and Astronomy, Beijing Normal University, Beijing 100875, China
Key Laboratory of Multiscale Spin Physics (Ministry of Education), Beijing Normal University, Beijing 100875, China


Center for Advanced Quantum Studies and School of Physics and Astronomy, Beijing Normal University, Beijing 100875, China
Key Laboratory of Multiscale Spin Physics (Ministry of Education), Beijing Normal University, Beijing 100875, China


Center for Advanced Quantum Studies and School of Physics and Astronomy, Beijing Normal University, Beijing 100875, China
Key Laboratory of Multiscale Spin Physics (Ministry of Education), Beijing Normal University, Beijing 100875, China


School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China
ShanghaiTech Laboratory for Topological Physics, Shanghai 201210, China


fengjie.ma@bnu.edu.cn
Center for Advanced Quantum Studies and School of Physics and Astronomy, Beijing Normal University, Beijing 100875, China
Key Laboratory of Multiscale Spin Physics (Ministry of Education), Beijing Normal University, Beijing 100875, China


rychen@bnu.edu.cn
Center for Advanced Quantum Studies and School of Physics and Astronomy, Beijing Normal University, Beijing 100875, China
Key Laboratory of Multiscale Spin Physics (Ministry of Education), Beijing Normal University, Beijing 100875, China


§ ABSTRACT 

The kagome material CsTi_3Bi_5, which is isostructural to the extensively studied charge density wave (CDW) compound CsV_3Sb_5, exhibits intriguing electronic features within its two-dimensional kagome lattices of titanium atoms. Here, we perform optical spectroscopic measurements together with the first-principles calculations on single-crystalline CsTi_3Bi_5 to investigate its electronic properties comprehensively. It is found that the overall optical spectra are very similar to those of CsV_3Sb_5, but the existence of CDW instability is ruled out in CsTi_3Bi_5. Via careful comparison to the optical responses of CsV_3Sb_5, we attribute this difference to a significant reduction in the itinerant carrier density of CsTi_3Bi_5, which is associated with the absence of van Hove singularity near the Fermi level at M point. This result supports the scenario that the CDW in CsV_3Sb_5 is driven by the nesting of van Hove singularity. Additionally, we unveil some exotic low-lying absorption features, which provide clear evidence of flat bands in CsTi_3Bi_5. Our findings contribute to a deeper understanding of exotic phenomena in CsTi_3Bi_5 and provide valuable insights into the role of van Hove singularity in CsV_3Sb_5.





Infrared spectroscopy study of kagome material CsTi_3Bi_5
    Rongyan Chen
    Accepted XXX. Received YYY; in original form ZZZ
=========================================================



§ INTRODUCTION


Recently, kagome material AV_3Sb_5 (A = K, Rb, Cs) has attracted extensive attention in condensed matter physics. There are various exotic physical phenomena observed in the AV_3Sb_5 system, such as unconventional superconductivity (SC) <cit.>, charge-density-wave (CDW) <cit.>, pair density wave <cit.>, electronic nematicity <cit.>, non-trivial topological surface state <cit.>, anomalous Hall effect with the absence of magnetic order <cit.> and so on. Such rich quantum phenomena are rooted in the unique electronic structure of AV_3Sb_5, which contains the general features of kagome lattice materils, such as flat bands <cit.>, Dirac cones with symmetry-protected Dirac points <cit.>, and saddle points <cit.>. This system has provided an intriguing research platform for studying the interplay between SC, CDW order, and nontrivial topological electronic states.


Despite intensive investigations focused on AV_3Sb_5, there are still a number of intractable problems yet to be solved, particularly regarding the underlying mechanism of CDW order in . In this compound, some studies reveal the formation of a 2 × 2 × 2 superlattice in accompany with the CDW transition  <cit.>, while others report the presence of a 2 × 2 × 4 superlattice <cit.>. Moreover, anomalous Hall effect linked to time-reversal symmetry-breaking <cit.> and electronic nematicity characterized by rotation symmetry-breaking <cit.> are observed in the CDW state, hinting at a possible influence of CDW order on symmetry breaking and nontrivial topology <cit.>. Fermi surface nesting is believed to play a crucial role in the development of CDW, as the van Hove singularities (vHSs) are located very close to the Fermi level (E_F)  and could be perfectly connected by the CDW wave vector q <cit.>. However, the expected divergence of the electronic susceptibility corresponding to this Fermi surface nesting is not observed in theoretical calculation results <cit.>. At the same time, the effect of electron-phonon coupling is suggested to significantly contribute to the lattice distortion and thereby CDW instability <cit.>. Consequently, whether the origin of CDW stems from electron instabilities facilitated by vHSs <cit.> or lattice structure instability <cit.> is not clear yet.

Investigating materials featuring analogous lattice structures as is beneficial to revealing the mysteries in this compound. Specifically, is a newly discovered kagome material that crystallizes in a layered structure with a space group P6/mmm, which is isostructural to  <cit.>. As revealed by experiments and density function theory (DFT) calculations <cit.>, the electronic structure of is highly similar to that of , except that the vHSs of are situated at about 0.15 eV and 0.75 eV above E_F <cit.>, instead of being close to E_F as in  <cit.>. Meanwhile, neither magnetic susceptibility, resistivity, or heat capacity measurements exhibit any discernible anomalies associated with CDW phase transitions in  <cit.>, in contrast to the results of  <cit.>. This observation is further confirmed by subsequent studies, such as scanning tunneling microscopy (STM) and low-temperature x-ray diffraction measurements <cit.>. It is proposed that the absence of CDW order is related to the vSHs being pushed far away above the E_F in  <cit.>.


Furthermore, a superconducting transition at ∼ 4 K is reported in  <cit.>, although there are some controversies concerning its credibility <cit.>. Electronic nematicity associated with orbital ordering is unveiled by some experiments compared with DFT calculations <cit.>, which is analogous to  <cit.>. Further investigations are required to understand the underlying physics of these exotic phenomena thoroughly. Optical spectroscopy is a powerful technique for probing the charge dynamics and detecting possible gap-opening phase transitions. Here, we perform infrared spectroscopy measurements on single-crystalline from 300 K to 10 K, which reveal a very similar optical response to that of . The most prominent difference is that the broader Drude component of is greatly suppressed compared to , which is intimately related to the development of CDW in the latter. Additionally, we also observe some peculiar low-lying absorption characteristics at around 3100 , which have no counterpart in . Combined with our DFT based first-principles calculations, these characteristics are attributed to interband transitions associated with flat bands about 0.35 eV below the E_F.





§ METHODS


Single crystals of were grown by using the flux method. The precursor CsBi was prepared by reacting Cs (purity 99.75%) and Bi granules (purity 99.99%) at 300 ^∘C. Starting materials of CsBi, Bi and Ti powder (purity 99.99%) were loaded inside a Canfield crucible with a molar ratio of 3: 9: 1, which were then sealed in a quartz tube. The assembly was heated up to 1100 ^∘C in a furnace within 12 h and kept at this temperature for 10 h. It was subsequently slowly cooled down to 800 ^∘C at a temperature decreasing rate of 50 ^∘C/h and kept for 5 h, and then further slowly cooled down to 400 ^∘C at 2 ^∘C/h. Finally, the assembly was taken out from the furnace and decanted with a centrifuge to separate single crystal from the flux.


Infrared spectroscopy studies are performed on a Fourier transform infrared spectrometer (Bruker Vertex 80v) in the frequency range from 60 to 23 000 . The single crystal with a size of 5.0 × 6.0 × 0.8 mm^3 is used. The surface of this sample is newly cleaved in order to remove the oxide layer. An in situ gold and aluminium overcoating technique is used to get the reflectivity R(ω). The real part of the optical conductivity σ_1(ω) is obtained by the Kramers-Kronig transformation of R(ω). The Hagen-Rubens relation is used for the low-frequency extrapolation of R(ω). We employ the x-ray atomic scattering functions in the high-energy side extrapolation <cit.>.

In our DFT calculations, the plane-wave basis based method and Quantum-ESPRESSO software package were used <cit.>. We adopted the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof formula for the exchange-correlation potentials <cit.>. Spin-orbit coupling is included in the simulations by using fully relativistic optimized norm-conserving Vanderbilt pseudopotentials from the Pseudo-Dojo library with a kinetic energy cutoff of 85 Ry <cit.>. A mesh of 12×12×9 k-points grid was used for sampling the Brillouin zone, and a denser uniform k-points grid of 20×20×20 was adopted for the non-self-consistent calculations of optical properties. During the simulations, all structural geometries were fully optimized to achieve the minimum energy.




§ RESULTS AND DISCUSSION


The reflectivity spectra R(ω) of at different temperatures are shown in the main panel of Fig. <ref>(a), while an expanded range up to 23 000 is plotted in its inset. Across the entire measured temperature range, R(ω) increases as frequency decreases and approaches a unit at zero frequency, and the low-frequency reflectivity increases upon cooling, indicating a metallic nature of , which is consistent with previous reports <cit.>. 
Furthermore, the overall profile of the R(ω) spectra is very similar to that of at low temperatures in the CDW state, which exhibit a pronounced dip structure at about 1500 along with a hump at around 6500  <cit.>. Here for , the dip and hump features are detected as well, as marked by black arrows in Fig. <ref>(a). However, the dip structure is much weaker and broader, which shifts to about 3000 . Moreover, several even weaker mini-dips could be observed on top of the broad dip at 10 K and 100 K, which is absent in . 

It is worth to note that the dip of only appears below T_CDW, thus intimately related to the CDW phase transition <cit.>. Therefore, it is tempting to explore whether the resembling dip character observed in is related to any kind of CDW condensate. In addition, the origin of mini-dips is intriguing as well.




With temperature decreasing, the dip structure of seems to get more obvious, as expected for the CDW-related absorption. However, it is difficult to tell if there is an onset temperature where the absorption begins to show up. To get a better perspective, we further resort to the real part of optical conductivity σ_1(ω) of , as shown in Fig. <ref>(b). It can be seen that the low-frequency σ_1(ω) consistently exhibits clear Drude components throughout the measured temperatures, i.e. low-energy peaks centered at zero energy, which reaffirms the metallic nature of  <cit.>. The width at half maximum of the Drude peak represents the scattering rate of free carriers γ, which reduces monotonically with temperature decreasing, as can be seen in Fig. <ref>(b). In correlation with the dip and hump structures presented in R(ω), several Lorentz-type peaks show up in the optical conductivity of . The most interesting one is the one located at around 3100 with a series of mini-peaks superposed on top, which corresponds to the dip structure donated in Fig. <ref>(a). For comparison, there is also a peak in the optical conductivity of associated with the dip feature of its reflectivity, which is identified to stem from the opening of a CDW gap. This similarity again puts forward the possibility of a CDW phase in .






In infrared spectroscopy experiments, CDW orders manifest themselves in two complementary ways. The first one is the emergence of a Lorentz-type peak below T_CDW, whose central position shifts to higher energy, and spectral weight (SW) piles up upon cooling. The second one is the transfer of SW from the itinerant Drude component to the emerging Lorentz peak, due to the loss of density of state accompanying the opening of the CDW gap. To prove the existence of a CDW transition, it is essential to provide evidence of these two manifestations. Here, it can be clearly seen from Fig. <ref>(b) that the 3100 peak persists up all the way to room temperature, which suggests either no CDW transition or a CDW transition with T_CDW higher than room temperature. As for the SW redistribution, we have plotted the SW as a function of frequency in Fig. <ref>. The SW generally increases with temperature increasing in a wide energy range and finally merges together roughly above 10 000 . To further illustrate its relative change with temperature, we plot the SW(T) renormalized by SW(300 K) in the inset of Fig. <ref>. If there is a CDW-induced redistribution, the SW below T_CDW is supposed to be smaller than that above T_CDW at frequencies lower than the gap energy, which usually evidences as a dip structure in the renormalized SW <cit.>. However, no such features are observed in the inset of Fig. <ref>, which means the absorption characteristics around 3100 are not related to a CDW gap. 







In order to further reveal the underlying mechanism of this exotic absorption, it is important to track down the quantitative variation of the charge dynamics. Therefore, we decompose the optical conductivity according to the Drude-Lorentz model,

    σ_1(ω) = ∑_iω_Pi^2/4πγ_Di/ω^2+γ_Di^2 + ∑_jS_j^2/4πγ_jω^2/(ω_j^2-ω^2)^2+ω^2 γ_j^2.

It's noteworthy that this equation is structured in Gaussian units for the sake of convenience. The first term is the Drude components, which describe the charge dynamics of intraband carriers. The second term is the Lorentz components, which are used to describe excitations across the energy gaps or interband transitions. Here, ω_Pi=√(4π n e^2/m^*) and γ_Di = 1/τ_Di are the plasma frequency and the scattering rate of free carriers of the ith Drude component, where n is the carrier density and m^* is the effective mass. ω_j, γ_j = 1/τ_j, and S_j are the resonance frequency, width, and resonance strength of the jth Lorentz component, respectively. To minimize fitting errors, we use the complex fitting method to fit the real parts of the optical conductivity using a nonlinear least-squares technique. The part of the fitting parameters with error bars at all measured temperatures are listed in Table <ref>. The representative examples of the fitting results of σ_1(ω) at 10 K and 300 K are plotted in Fig. <ref>. Two Drude components with distinct scattering rates are demanded for the fitting, which indicates that the conduction electrons come from different energy bands or Fermi surfaces. Meanwhile, seven Lorentz terms are employed, although the latter five ones are almost temperature-independent and thus not listed. It is worth to remark that the mini-peaks around 3000 are not taken into account in our fitting procedure. The peak positions of these mini-peaks are identified to be at 2362, 2972, 3630, and 4627 at 10 K, as marked by orange arrows in Fig. <ref>(b). 




It is instructive to compare the charge dynamics of free carriers in with that in , where two Drude components with significantly different scattering rates are used as well <cit.>. We found that the scattering rates of generally decrease with temperature decreasing, whereas the plasma frequencies are roughly temperature independent. These characteristics are the same as when it is in the normal phase above T_CDW. Furthermore, the plasma frequency of the narrow Drude term (ω_P1) is roughly comparable between the two compounds, whereas ω_P2 of is much lower than that of  <cit.>, indicating a much smaller carrier density n or much larger effective mass m^*. It is worth noting that for the narrow Drude term is ascribed to the light-electron-like and multiple Dirac bands at Γ and K points, whereas the broad one is attributed to the heavy hole bands near M point. For comparison, we have calculated the electronic structure of by DFT. The band structure with spin-orbit coupling along the high-symmetry k-paths is presented in Fig. <ref>(a). Similar to , there are also very light-electron bands crossing the E_F around Γ point, and Dirac type of conduction band near K point. These bands usually contribute to Drude components with relatively small scattering rates <cit.>, and thus highly likely to be responsible for narrow Drude term (Drude1). In contrast, the band structure of at M point is quite different from that of . The vHSs are pushed up to 0.15 eV and 0.75 eV above E_F instead of near E_F, as indicated by the pink arrows in Fig. <ref>(a). As a result, the density of states near E_F is expected to be much lower than . Assuming that Drude2 of is also related to the bands near M point, this agrees perfectly with the fact that the ω_P2 of is much lower than . 

To discuss the possible contribution of effective mass m^* to the suppression of ω_P2, it is useful to estimate the electron correlation strength, which could be reflected by the ratio of experimental kinetic energy K_exp to theoretical kinetic energy K_band <cit.>. In simple metals with itinerant electrons of band mass, the agreement between the experimental and theoretical results would lead to K_exp/K_band ≃ 1. However, in correlated materials with quasiparticles 
of enhanced effective mass, the strong Coulomb interaction would drastically suppress the experimental kinetic energy, and thus generate a much smaller K_exp/K_band.
 The kinetic energy can be calculated by

    K = 2ħ^2c_0/π e^2∫_0^ω_cσ_1(ω)dω,

where c_0 is the c-axis lattice parameter, and ω_c is the cutoff frequency covering the entire Drude component where σ_1(ω) reaches a minimum value. The theoretical σ_1(ω) calculated by DFT method using the random phase approximation is depicted by the red solid line in Fig. <ref>(b). Here, ω_c = 1700 is used to estimate both K_exp and K_band, yielding a value of K_exp/K_band = 0.74. This value is smaller than that of CsV_3Sb_5, which is about 0.81 <cit.>. Therefore, a stronger correlation is expected in , which is consistent with the results of STM experiments <cit.>. Nevertheless, this slight increase of correlation strength can hardly explain the substantially suppressed ω_P2 of , which is therefore predominantly ascribed to the significant reduction of carrier density, or the shift of vHSs at M point.


It seems that the optical responses of and are highly similar to each other, but this only stands in the normal states. Upon entering the CDW state, the broad Drude component of is significantly reduced, and the lost SW is transferred towards higher frequencies due to the formation of a CDW-related gap. In contrast, the SW of all the Drude and Lorentz terms of are stable against temperature variation, incompatible with a CDW phase transition. Particularly, the suspicious peak located at around 3100 (Lorentz1) is actually robust up until 300 K, whose SW reduces very slightly with warming, as listed in Table <ref>. It only looks obscure at high temperatures in the σ_1(ω) spectra because it overlaps with the broadening Drude peak. Furthermore, the scattering rate of the broad Drude components in shows an abrupt reduction associated with the CDW transition T_CDW <cit.>, which is absent in . Therefore, we believe it is safe to conclude that there is no CDW-related physics in . Nonetheless, a detailed comparison between and would shed light on the origin of CDW in . We found the most prominent difference, from the spectroscopic view, lies in the SW of the broad Drude components, which are linked to the conduction bands near M point. In other words, the itinerant carriers of are substantially reduced compared to , due to the absence of vHSs near the E_F. Therefore, our results strongly suggest that the CDW phase transition in originates from the vHSs near the E_F. 





Finally, we would like to discuss the possible origination of the Lorentz1 peak centered at around 3100 (∼ 0.38 eV). Note that this peak does not have a counterpart in , and there are a series of mini-peaks superposed on it. In , the first Lorentz peak in the normal state is located at around 0.7 eV, which is argued to involve the electronic states near the saddle points at M point. Since the saddle points of are significantly shifted upwards, they are unlikely to be responsible for the Lorentz1 peak of 0.38 eV. Instead, an obvious flat band is identified across the entire Brillouin zone in from our DFT results, which is in good agreement with ARPES measurements and previous calculations <cit.>. Especially, the flat band is located at around 0.35 eV below the E_F, which leads to a pronounced peak in the density of states at this energy, as indicated by a black arrow in the inset of Fig. <ref>(b). Moreover, it closely matches with the peak position of Lorentz1 (∼ 0.38 eV). Therefore, we propose that Lorentz1 corresponds to interband transitions from the flat band to the unoccupied conduction band, as indicated by blue arrows in Fig. <ref>(a). Additionally, slight dispersion of the flat band could lead to additional tiny peaks on top of the main peak, as is also reported in some kagome materials, like Fe_3Sn_2 <cit.>. Hence, the mini-peaks on top of Lorentz1 provide clear evidence of the existence of flat bands in , which is not observed in yet. 






§ CONCLUSION


In conclusion, we have investigated the electronic properties of the kagome material through optical spectroscopic measurements and band structure calculations. Our findings reveal a metallic response in across the temperature range from 300 K to 10 K, consistent with its transport behaviors. Despite the striking similarity between the optical responses of and , there are no signatures of a charge-density-wave gap in . Moreover, the spectral weight of the broad Drude component of is significantly suppressed compared to , which we attribute to the absence of van Hove singularities near the Fermi level in . This result supports the claim that charge-density-wave formation in is linked to the van Hove singularities at M point. Furthermore, the comparison between the experimental and theoretical optical conductivity of points to an electronic correlation a little stronger than . In addition, we have identified low-lying absorption features linked to interband transitions that involve flat bands in . Our work provides valuable insights into the underlying physics of exotic properties of and highlights the importance of vHSs in the development of charge-density-wave in .





ACKNOWLEDGMENTS


R. Y. Chen and F. J. Ma acknowledges the National Key Projects for Research and Development of China (Grant No. 2021YFA1400400), the National Natural Science Foundation of China (Grant Nos. 12074042 and 12074040), the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 11704033), and the Fundamental Research Funds for the Central Universities (Grant No. 2243300003). Y. F. Guo acknowledges the National Key R&D Program of China (Grant No. 2023YFA1406100) and the Double First-Class Initiative Fund of ShanghaiTech University. This work was supported by the Synergetic Extreme Condition User Facility (SECUF). F. J. Ma was also supported by the BNU Tang Scholar.



