Observation of flat band, Dirac nodal lines and topological surface states in Kagome superconductor CsTi3Bi5

Kagome lattices of various transition metals are versatile platforms for achieving anomalous Hall effects, unconventional charge-density wave orders and quantum spin liquid phenomena due to the strong correlations, spin-orbit coupling and/or magnetic interactions involved in such a lattice. Here, we use laser-based angle-resolved photoemission spectroscopy in combination with density functional theory calculations to investigate the electronic structure of the newly discovered kagome superconductor CsTi3Bi5, which is isostructural to the AV3Sb5 (A = K, Rb or Cs) kagome superconductor family and possesses a two-dimensional kagome network of titanium. We directly observe a striking flat band derived from the local destructive interference of Bloch wave functions within the kagome lattice. In agreement with calculations, we identify type-II and type-III Dirac nodal lines and their momentum distribution in CsTi3Bi5 from the measured electronic structures. In addition, around the Brillouin zone centre, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{Z}}}_{2}$$\end{document}Z2 nontrivial topological surface states are also observed due to band inversion mediated by strong spin-orbit coupling.

Quantum materials with layered kagome structures have drawn enormous attentions because such a two-dimensional (2D) network of corner-sharing triangle lattice can give rise to many exotic quantum phenomena, such as spin liquid phases [1][2][3][4], topological insulator and topological superconductor [5][6][7], fractional quantum Hall effect [8], quantum anomalous Hall effect [9,10] and unconventional density wave orders [11][12][13][14][15][16].All these exotic quantum phenomena are thought to originate from the unique electronic structure of the kagome lattice including flat bands, Dirac cones and saddle points when the spin orbital coupling, magnetic ordering or strong correlation are taken into consideration.Nevertheless, the definitive identification of such unique electronic structures in the kagome materials are still scarce and the underlying mechanism to induce those exotic quantum phenomena from such electronic structures remains illusive.For example, the kagome superconductors AV 3 Sb 5 (A=K, Rb or Cs) [17], which have been the focus of recent extensive research, exihibit anomalous Hall effect [18], unconventional charge density wave (CDW) [15,[19][20][21][22][23][24][25], pairing density wave [14] and possible unconventional superconductivity and nematic phase [13,18,25,26].However, the nature and origin of these noval physical properties are still in hot debates.Even for the clear identification of the flat band, it still needs further investigations.It is significant to establish a relationship between the unique electronic structures of the kogome lattice and its novel quantum phenomena.
CsTi 3 Bi 5 is a newly discovered kagome superconductor which is isostructural to the AV 3 Sb 5 superconductors (Fig. 1a) [27].The titanium atoms form a kagome network with the bismuth atoms lying in the hexagons and above and below the triangles (Figs.1a and 1b).
Magnetic susceptibility and electrical resistivity measurements of CsTi 3 Bi 5 indicate that there is no phase transition observed down to the superconducting transition at 4.8 K [28].This is different from CsV 3 Sb 5 that exhibits a CDW transition around 94 K [17].The similar crystal structure but the absence of the CDW order in CsTi 3 Bi 5 provide a good opportunity to study the intrinsic electronic structure of the kagome lattice with reference to CsV 3 Sb 5 and understand the origin of various quantum phenomena in kagome materials.
In this paper, we investigate the electronic structure of the newly discovered kagome superconductor CsTi 3 Bi 5 .By using high resolution laser-based angle-resolved photoemission spectroscopy (ARPES), in combination with the band structure calculations, we have directly observed the characteristic electronic features of the kagome lattice.We directly observed the flat band derived from the destructive interferences of the Bloch wave func-tions within the kagome lattices.We also identify the Dirac nodal loops and nodal lines in three dimensional momentum space.The Z2 nontrivial topological surface states are also observed.Such coexistence of multiple nontrivial band structures in one kagome superconductor provides a new platform to study the rich physics in the kagome lattice.
CsTi 3 Bi 5 single crystals were grown using a self flux method [28].Typical CsTi 3 Bi 5 crystals with a lateral size of ∼3 mm and regular hexagonal morphology were obtained.High resolution angle-resolved photoemission measurements were performed using a lab-based ARPES system equipped with the 6.994 eV vacuum-ultra-violet (VUV) laser and a hemispherical electron energy analyzer DA30L (Scienta-Omicron) [29,30].The laser spot is focused to around 10 um on the sample in order to minimize the influence of sample inhomogeneity.The light polarization can be varied to get linear polarization along different directions.In the LV (LH) polarization the electric vector of the laser light is perpendicular (parallel) to the plane formed by the light path and the analyzer lens axis.The energy resolution was set at 1 meV and the angular resolution was 0.3 degree corresponding to 0.004 Å−1 momentum resolution at the photon energy of 6.994 eV.All the samples were cleaved in situ at a low temperature of 20 K and measured in ultrahigh vacuum with a base pressure better than 5 x 10 −11 mbar.The Fermi level is referenced by measuring on clean polycrystalline gold that is electrically connected to the sample.
The exchange correlation functional is treated by Perdew-Burke-Ernzerh (PBE) of parameterization of generalized gradient approximation (GGA) [33].The convergence criterion of atomic forces in structural optimization with VASP is less than 1 meV/ Å total energy convergence threshold of all processes is 10 −6 eV/atom.The cutoff energy of the plane-wave is set as 520 eV.The Γ centered 20×20×12 Monkhorst-Pack k-point grid is used in the self-consistent cycle.Wannier90 package [34] is used to fit Wannier functions and construct tight-binding models, and WannierTools [35] package is used to calculate the surface spectral functions by using the surface Green's function method.Calculations of structures' parity are performed through a combination of the irvsp program [36] and VASP.
Figure 1d shows the Fermi surface mapping of CsTi 3 Bi 5 measured at 20 K.The entire first BZ is covered by our laser ARPES measurements.Five Fermi surface sheets are clearly observed, as quantitatively shown in Fig. 1e.The Fermi surface consists of three electron-like Fermi surface sheets around Γ (α, β and γ 1 in Fig. 1e), an electron-like triangular Fermi pocket around K (γ 2 in Fig. 1e) and a small hole-like Fermi pocket around M (δ in Fig. 1e).
In order to understand the measured electronic structure, we carried out detailed band structure calculations.Figs.1f and 1g show the calculated band structures of CsTi 3 Bi 5 without considering the spin orbit couping (SOC) (Fig. 1f) and considering SOC (Fig. 1g).These calculations project the band structures onto different Ti 3d orbitals along high symmetry directions in the BZ.The low energy bands are mainly from the 3d orbitals of titanium.The characteristic electronic features of a kagome lattice, including the flat band, two saddle points at M and a Dirac point at K, can be clearly observed as marked in Figs.
1f and 1g.These features are mainly from the Ti 3d x 2 −y 2 /xy orbitals (red lines in Figs.1f and   1g) except that the saddle point VHS1 is from Ti 3d z 2 (green line in Figs.1f and 1g).The consideration of SOC shows little effect on the flat band and the saddle points but opens a gap at the Dirac points (Fig. 1g).
The calculated band structures of CsTi 3 Bi 5 (Figs.1f and 1g) are very similar to that of CsV 3 Sb 5 where the kagome lattice related electronic features are mainly from the V 3d orbitals [13,37].The main difference is the band position with respect to the Fermi level.
In  and FeSn [46].However, there is little clear evidence reported about the kagome-derived flat band in the 135 family represented by AV 3 Sb 5 (A=K, Rb or Cs).
Figure 2a and 2b show the band structures of CsTi 3 Bi 5 measured at 20 K along the Γ-M -K-Γ high symmetry directions under the LV (Fig. 2a) and LH (Fig. 2b) light polarizations.
In order to resolve all the band structures more clearly, the corresponding second derivative image is shown in Fig. 2c.There seems to be a dispersionless band across the entire Brillouin zone at the binding energy of ∼220 meV.A careful analysis indicated that this band consists of two different parts.The first part is marked by the red dashed line in Fig. 2c while the rest of the band represent the second part.As compared with the band structure calculations in Figs.2d and 2e, the first part shows a good agreement with the flat band from the band structure calculations.This indicates that it is kagome lattice derived flat band.This band can be attributed to the local destructive interferences of the Bloch wave functions within the kagome lattices (Fig. 2f).The second part of the dispersionless band at ∼220 meV is not expected from the band structure calculations (Figs.2d and 2e).A careful inspection indicates that there is an additional spectral weight buildup in the binding energy range of 220∼500 meV.The two energies happen to coincide with the top and bottom energy positions of the first part flat band.This indicates that the extra spectral weight buildup is closely related to the first part flat band.The second part flat band at ∼220 meV actually represents a spectral weight cut-off at this energy (see Fig. S1 in Supplementary Materials for details).

Dirac Nodal Lines
In some quantum materials, the bands can cross at a discrete point in the momentum space, forming Dirac point with spin degeneracy or Weyl point with spin polarization.The Dirac points can also form nodal lines and nodal loops in three dimensional momentum space [47].The Dirac points can be categorized into three types according to the slopes of the involved bands [48,49].The materials, which have the electronic structure with the type-II (two dispersion branches exhibit the same sign of slope) or type-III (one of two branches is dispersionless) Dirac points, may host exotic properties, e.g., the chiral anomaly [48] and Klein tunneling [50].However, there have been few established cases of the type-II and type-III Dirac point realization in real materials, not to mention their simultaneous observation in one material.share the same sign of slope along both Γ-M (Fig. 3c) and Γ-K (Fig. 3e) directions, forming a type-II Dirac nodal loop.The NL2 point is formed by the crossing of the γ 2 band and the kagome flat band, as shown by the blue and red dashed lines in Figs.3d and 3e.Since the kagome flat band is nearly dispersionless, the NL2 Dirac nodal loop can be categorized into type-III.

Nontrivial Topological Surface States
The spin-orbit coupling is stronger in CsTi 3 Bi 5 than that in CsV 3 Sb 5 because of the heavy element Bi.We also note that the calculated energy bands give rise to a strong topological Z2 index since the PT symmetry is conserved in CsTi 3 Bi 5 (see Fig. S3 in Supplementary Materials) [27,28].This will result in possible topologically nontrivial surface states.Fig. 4a shows the band structure measured around Γ along the M -Γ-M direction under the LV light polarization.The corresponding second derivative image is shown in Fig. 4b.For comparison, Figs.4c and 4d show the calculated band structures without and with SOC, respectively, along the same momentum cut.All the observed bands in Fig. 4b can be well assigned by comparing with the calculated bands (as shown by coloured lines in Figs.4b and   4d) except for one band that is marked as TSS in Fig. 4b.In order to understand its origin, we analyzed the energy bands in details.We find that CsTi 3 Bi 5 has symmetry-protected band degeneracy along the Γ-A path between the γ and β bands, as well as between the β and α bands, giving rise to multiple topological Dirac semimetal states (see Fig. S3 in Therefore, we provide a definitive spectroscopic evidence that nontrivial topological surface states exist in the kagome superconductor CsTi 3 Bi 5 .
In summary, by using high resolution laser based ARPES in combination with the DFT band structure calculations, we investigate the electronic structure of the newly discovered

CsTi 3
Bi 5 the kagome lattice related bands are shifted upwards by ∼1 eV when compared with those in CsV 3 Sb 5 .This is because CsTi 3 Bi 5 has one electron less per Ti per unit cell than that of CsV 3 Sb 5 when Ti is replaced by V.As a result, although the Ti 3d orbitals still dominate the density of states (DOS) around the Fermi level E F , the two van Hove singularities (VHS) in CsTi 3 Bi 5 are above the Fermi level whereas they are close or below the Fermi level in CsV 3 Sb 5[38,39].This provides a possible explanation of the absence of the CDW order in CsTi 3 Bi 5 .The upward band shift also moves the flat band close to Fermi level in CsTi 3 Bi 5 .

Figure
Figure1hshows the calculated Fermi surface in the three dimensional Brillouin zone.The Fermi surface consists of five sheets which are quite two dimensional.This is expected due to the strong in-plane bonding and weak interlayer coupling in CsTi 3 Bi 5 which is similar to that in CsV 3 Sb 5 .The calculated Fermi surface at k z =0 is shown in Fig.1i.To make a better comparison between the measured Fermi surface and band structures with the band structure calculations, we find that the Fermi level of the calculated band structures needs to be shifted downwards by ∼90 meV, as shown in Figs.1f and 1g.The calculated Fermi

Figure
Figure3aand 3b show our identification of two sets of Dirac nodal loops and one set of Dirac lines in CsTi 3 Bi 5 .Fig.3ashows the calculated band structure along the highsymmetry directions without considering spin-orbit coupling.We can find two groups of linear dispersion crossings in a covered energy region around E F , marked as NL1 and NL2 in Fig.3a.Our DFT calculations reveal that these Dirac nodes are not isolated, but form multiple nodal loops in k z =0 and k z =π/c planes as seen in Fig 3b.The NL1 type-II nodal loops form in-plane hexagons centered on Γ and A while the NL2 nodal loops form in-plane triangles centered on all the K and H points.These nodal loops are protected by the M z mirror symmetry.Detailed band analysis shows that the type-II NL1 of k z =0 and k z =π/c planes are not connected along the k z direction due to the absence of mirror symmetry between 0<k z <π/c.However, for the type-III NL2 in the k z =0 and k z =π/c planes, we find another set of nodal lines in the Γ-K-H-A plane that links them.These nodal lines are type-III and protected by the M x mirror symmetry.Due to the six-fold rotational symmetry, there are six nodal lines and NL2 loops that are symmetrically distributed near K and L points.Slices at different k z positions have similar band structures, which makes these nodal loops in different slices still possible to be captured experimentally in spite of the opening of small gaps.Moreover, after considering SOC, these nodal loops will further open gaps but the gap size remains small (<50meV).

Figure
Figure 3c to 3e show the measured band structures along Γ-M , M -K and K-Γ highsymmetry directions, respectively.For comparison, the corresponding calculated band structures with SOC are presented in Figs.3f to 3h.The calculated bands agree very well with the experimental results.The NL1 point is formed by the crossing of the β and γ 1 bands,

kagome superconductor CsTi 3
Bi 5 .The observed Fermi surface and band structures show excellent agreement with the band structure calculations.We have identified multi-sets of nontrivial band structures in CsTi 3 Bi 5 including the kagome lattice derived flat band, type-II and type-III Dirac nodal loops and nodal lines, as well as Z2 nontrivial topological surface states.Such coexistence of nontrivial band structures in one kagome superconductor provides a new platform to understand the physics and explore for new phenomena and exotic properties in the kagome materials.These people contribute equally to the present work.

FIG. 1 :
FIG. 1: Fermi surface and calculated band structures of CsTi 3 Bi 5 .a Schematic pristine crystal structure of CsTi 3 Bi 5 .b Top view of the crystal structure with a two-dimensional kagome lattice of Titannium.c Three-dimensional Brillouin zone with high-symmetry points and high-symmetry momentum lines marked.d Fermi surface mapping of CsTi 3 Bi 5 measured at a temperture of 20 K.It is obtained by integrating the spectral intensity within 10 meV with respect to the Fermi level and symmetrized assuming six-fold symmetry.Five Fermi surface sheets are clearly observed and quantitatively shown in e.Three Fermi surface sheets are around the Brillouin zone center Γ marked as α (orange line), β (green line) and γ 1 (light blue line).One Fermi surface is around the K point marked as γ 2 (blue line) and one is around the M point marked as δ (dark blue line).f Calculated band structure along high-symmetry directions without considering SOC.Different colors represent different orbital components of Ti 3d .g Same as f but considering SOC.The flat band (FB), two saddle points (VHS1 and VHS2) and a Dirac point (DP) are marked by arrows.To make a better comparison with measured results, the Fermi level is shifted downwards by 90 meV, as shown by the dashed lines in f and g. h Calculated three dimensional Fermi surface based on the first principle DFT calculations.The Fermi surface sheets are quite two dimensional.The calculated Fermi surface at k z =0 is shown in i.The measured Fermi surface (d) shows an excellent agreement with the calculated one (i).

FIG. 2 :FIG. 3 : 18 FIG. 4 :
FIG. 2: Direct observation of flat band in CsTi 3 Bi 5 .a,b Detailed band structures measured along the Γ-M -K-Γ high symmetry directions under two different polarization geometries, LV (a) and LH (b).c Second derivative image with respect to energy obtained from a,b in order to resolve the band structures more clearly.The red dashed line highlights the observed flat band.d The corresponding calculated band structure along the Γ-M-K-Γ high-symmetry directions without considering SOC.Different colors represent different orbital components of Ti 3d .e Same as d but considering SOC.f Orbital textures of the effective Wannier states giving rise to the flat bands with d xy /d x 2 −y 2 orbitals.g Tight-binding band structures of kagome lattice with (red lines) and without (blue lines) SOC.Inclusion of the spin orbit coupling gaps both the Dirac crossing at K and the quadratic touching between the flat band and the Dirac band around Γ.