Charge-density-wave-induced bands renormalization and energy gaps in a kagome superconductor RbV3Sb5

Recently discovered Z2 topological kagome metals AV3Sb5 (A = K, Rb, and Cs) exhibit charge density wave (CDW) phases and novel superconducting paring states, providing a versatile platform for studying the interplay between electron correlation and quantum orders. Here we directly visualize CDW-induced bands renormalization and energy gaps in RbV3Sb5 using angle-resolved photoemission spectroscopy, pointing to the key role of tuning van Hove singularities to the Fermi energy in mechanisms of ordering phases. Near the CDW transition temperature, the bands around the Brillouin zone (BZ) boundary are shifted to high-binding energy, forming an"M"-shape band with singularities near the Fermi energy. The Fermi surfaces are partially gapped and the electronic states on the residual ones should be possibly dedicated to the superconductivity. Our findings are significant in understanding CDW formation and its associated superconductivity.

The theory was put forward early that a two-dimensional (2D) energy band with saddle points in the vicinity of E F is unstable against charge density wave (CDW) formation [18]. The CDW, superconducting and topological phases have been extensively investigated in 2D transition metal dichalcogenides [19] and the underlying microscopic mechanism of the CDW formation is still in controversy. Recently, the CDW state and superconductivity are discovered in a family of layered kagome metals AV 3 Sb 5 (A = K, Rb, and Cs) [20][21][22][23][24], which hosts a Z 2 topological invariant and non-trivial topological Dirac surface states near E F [21]. The CDW state is probably driven by the competing electronic orders at the saddle-point singularity with a high density of states [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. X-ray diffraction and scanning tunneling microscopy (STM) reveal the formation of a three-dimensional (3D) 2 × 2 × 2 superlattice at both CDW and superconducting states [24][25][26][27][28], which energetically favors a chiral charge order and an inverse Star of David distortion in kagome lattice with the shift of van Hove singularity to E F [34,35,40]. The CDW states and double superconducting domes are associated with multiple singularities with different energies and orbital characters near E F [27][28][29][30][31][32][33][34][35][36][37][41][42][43][44], which at present need to be further studied in detail. In addition, a giant anomalous Hall effect with the reversal of Hall sign is observed [22], and magnetic order and local moment are not found by magnetic susceptibility and muon spin spectroscopy [22,23,45]. To fundamentally understand these anomalous behaviors and quantum orders, the investigation of the temperature evolution of the low-energy electronic structure is highly desired.
In this paper, we report on a combined angle-resolved photoemission spectroscopy (ARPES) and first-principles calculations study of the temperature evolution of the low-energy electronic structure in RbV 3 Sb 5 , which has a CDW transition temperature (T CDW ) of about 102 K, a sign change of the Hall coefficient at about 40 K, and a superconducting transition temperature (T c ) of about 0.92 K [23]. As a result of the CDW transition, we found that the bands at the zone boundary (M) are shifted down about 40 meV forming an "M"-shape band with its singularity at about 60 meV below E F . Below T CDW , the energy gap of about 20 meV is opened at the Fermi momentum (k F ) of the band centered atM, and no gap is observed at the band centered atΓ at 10 K within experimental energy resolution. The electronic states on the residual Fermi surfaces should be dedicated to the superconducting pairing. Our findings reveal CDW-induced strong bands renormalization and energy gaps at the zone boundary, implying that they are the multiple singularities at M which play ultimate roles  in the formation of both CDW and its related superconducting phases.

II. RESULTS AND DISCUSSION
The crystal structure of RbV 3 Sb 5 crystallizes in a hexagonal structure with P6/mmm (No. 191) space group [20][21][22][23][24], in which V-Sb slabs consisting of V kagome nets and interspersing Sb atoms are separated by alkali-metal ions along the c axis, as shown in Fig. 1(a). There are two kinds of Sb sites: the Sb1 site at the centers of V hexagons, and the Sb2 site below and above the centers of V triangles forming hexagon layers. The corresponding original (red lines) and 2 × 2 reconstructed (blue lines) BZs projected on the (001) surface with the high-symmetry points are shown in Fig. 1(b). Figures 1(c) and 1(d) show constant-energy surfaces taken at 140 K at E F and E F -0.28 eV, respectively. The high intensity around theM points at the Fermi surfaces suggests the singularities or the surface states at the proximity of E F . To investigate the 3D character of the Fermi surfaces, we carried out the photon-energy-dependent ARPES measurement. With an empirical value of the inner potential of ∼ 8.2 eV and c = 9.07 Å [24], we found that hv = 86 eV is close to the Γ point and 100 eV close to the A point, according to the free-electron final-state model [46]. Three electronlike pockets (α, β, and κ) along theΓ −K direction are indicated in Figs. 1(g) and 1(h). All of the three bands show weak k z dispersions both at E F and E F -0.28 eV, as shown in Figs. 1(e) and 1(f), which reveal the 2D electronic dispersions alongΓ −K and 2D Dirac cones at theK points. Based on the ARPES data, we estimate that the widths of the α, β, and κ Fermi pockets alongΓ −K are about 0.42, 0.10, and 0.31 Å −1 and the Fermi velocities of them are about 3.25, 3.60, and 1.70 eV Å (1.28 eV Å for another branch of the Dirac bands), respectively. Figure 2 shows the temperature evolution of the bands along theΓ −M direction on a sample cleaved at 120 K (#S1). The intensity plots along theΓ −M direction taken at 120 and 10 K are shown in Figs. 2(a) and 2(b), respectively. Comparing the data taken at the two temperatures, one can see that the α band is shifted up, which is mainly attributed by surface reconstructions along with time [47]. Figure 2(c) shows the momentum distribution curves (MDCs) of the α band taken at 120 K, revealing the two splitting sub-branches. The STM results suggest an isotropic scattering vector connecting different states of the α pocket [26][27][28], while the two Sb sites or k z integration can also cause the bands splitting in the ARPES data. We have carried out substantial experiments on the samples with various conditions, e.g. cleaved at both high and low temperatures then measured them at a few stabilized tempera-   Fig. 4(b).
To check the CDW-induced bands renormalization, we directly compare the data taken on the freshly cleaved samples at low temperature (#S6: Cleaved at 10 K) and high temperature (#S7: Cleaved at 140 K), as shown in the supplementary Fig. S(3) [48]. The sharp contrast between Figs. S(3)(a) and S(3)(b) reveals that the band renormalization is indeed induced by temperature rather than a trivial surface reconstruction. We also measure the energy gaps on a freshly cleaved sample at 10 K (#S3), as shown in Figs. 3(i) and 3(j). The energy positions in Fig. 3(j) show CDW gaps at -20 meV Intensity (a.u.)  and the tip of κ band at about -60 meV, respectively. In addition, the temperature evolution of the Dirac cone at theK point is shown in Fig. 3(g). The Dirac point is almost not moved along with decreasing temperature. The energy gap of the Dirac cone is about 100 meV, which is much larger than the calculated value of ∼ 15-25 meV induced by spin-orbit interaction in the normal state [ Fig. 4(a)] and is close to the calculated value of ∼ 120 meV in the CDW state [ Fig. 4(b)]. The CDW-gaps induced by the band folding at the Dirac cones are needed to be considered.
With the help of the orbital-projection band calculation [ Fig. 4(a) and the supplementary Figs. S(4) [48]], one can find the α band atΓ is mainly contributed by the out-of-plane Sb1-p z (dark), and the κ and γ bands atM are mainly derived from the out-of-plane V-d z2 (green) and V-d xz /d yz (blue) orbitals. The bonding of the out-of-plane orbitals and the interlayer coupling strength are enhanced along with the decreasing of the temperature, which is also revealed by the reduction of the c-axis lattice constant [21]. The renormalization of the bands with the out-of-plane character should be more appreciated with the CDW instability. Coulomb scattering of electrons between the orbital-selective saddle-point singularities at M can give rise to instabilities of the Fermi surfaces and lead to CDW states [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. The 2D Dirac bands atK (κ and κ ) originate from in-plane V-d xy /d x2−y2 (red) orbitals, and hybridized with out-of-plane V-d xz /d yz (blue) orbitals near thē M point. The Dirac bands remain nearly motionless upon the CDW phase transition.
The calculation in CDW state with the inverse Star of David can well describe our experimental observations [ Fig. 4(b) and the supplementary Fig. S(5) [48]]. As marked with the dashed ellipses in Fig. 4(b), the κ band at the L point sinks below E F , forming an "M"-shape band with its tips at about 60 meV. The tips can be viewed as new singularities, which may be further associated with superconducting states. The CDW-induced band renormalization is endowed with an electronic correlation effect. Previous studies provide strong evidence that traversing the singularity to E F is beneficial in the formation of ordering phenomena. For instance, in CDWmaterial TaSe 2 also with 3D 2 × 2 × 2 superstructure, by tuning the energy position of the singularity, the T C is enhanced by more than an order of magnitude [49]. Recently, ther- mal conductivity and high-pressure resistance measurements reveal two superconducting domes and exotic pairing states [29,30,43,44], which may be associated with the optimal positions of the singularities concerning E F match with corresponding bosons. Besides the bands renormalization mentioned above, the CDW-induced energy gap ∆ ∼ 20 meV is opened at k F of the band nearM which is consistent with STM results [25][26][27][28]. The CDW-induced gap is not observed at the band nearΓ at T = 10 K within the experimental energy resolution. Thus, momentum-and orbital-dependence of the electronic states are involved in the CDW formation in RbV 3 Sb 5 . The STM measurements further reveal that the CDW gap is particlehole asymmetric [25][26][27][28], which is previously found in CDWmaterial NbSe 2 [50,51]. As a case of typical quasi-2D materials without a strongly nested Fermi surface, the presence of a particle-hole asymmetric gap in NbSe 2 could be an indication that electron correlation is important in driving the CDW [50,51]. Analogously, combined with the large ratio 2∆/k B T CDW ∼ 4.55 in analogy to strong-coupling superconductors, the CDW formation in RbV 3 Sb 5 is likely mediated by electronic interactions enhanced by low dimensionality. Recent inelastic x-ray scattering studies demonstrate an unconventional and electronic-driven mechanism that couples the CDW and the topological band structure in RbV 3 Sb 5 [33].
In addition, as a Z 2 topological kagome metal, RbV 3 Sb 5 hosts non-trivial topological Dirac surface states at the timereversal-invariantM points and remains the same after the CDW transition, as shown in the supplementary Fig. S(6) [48]. It is possible to realize the Majorana zero-energy modes and their related topological superconductivity in these materials. Because the electronlike bands near theM point show k z dispersions [the supplementary Fig. S(7) [48]], the surface states atM are possibly located above E F . The chemical potential need to be elevated for further studying the surface states in detail.

III. CONCLUSION
In summary, we have studied the electronic structures of a kagome superconductor RbV 3 Sb 5 in both the normal phase and the CDW phase. We observed the CDW-induced bands renormalization and energy gaps on the bands at the zone boundary, where multiple orbital-selective singularities exist. Momentum-and orbital-dependence of the electronic states are involved in the CDW formation and the associated superconductivity. Our findings strongly imply that the singularities near E F play important roles in the formation of ordering phases and the electronic states on the residual Fermi surfaces to the superconducting pairing. Single crystals of RbV 3 Sb 5 were synthesized by the selfflux method as described elsewhere [23]. RbV 3 Sb 5 single crystals are stable in the air. ARPES measurements were performed at the Dreamline and 03U beamlines of the Shanghai Synchrotron Radiation Facility (SSRF). The energy and angular resolutions were set to 10-24 meV and 0.02 Å −1 , respectively. The Fermi cut-off of the samples was referenced to an evaporated gold film on the sample holder. Samples were cleaved in situ, exposing flat mirrorlike (001) surfaces. The pressure was maintained at less than 2 × 10 −10 Torr during temperature-dependent measurements.

IV. ACKNOWLEDGEMENTS
The first-principles electronic structure calculations on RbV 3 Sb 5 were performed by using the projector augmented wave (PAW) method [52,53] as implemented in the Vienna ab initio simulation package (VASP) [54]. The generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) type [55] was used for the exchange-correlation functional. The kinetic energy cutoff of the plane-wave basis was set to be 350 eV. The BZ was sampled with a 10 × 10 × 6 k-point mesh. For the Fermi surface broadening, the Gaussian smearing method with a width of 0.05 eV was adopted.
The zero-damping DFT-D3 method was adopted to describe the interlayer van der Waals (vdW) interactions [56]. The lattice constants and the atomic positions were fully relaxed until the forces on all atoms were smaller than 0.01 eV/Å. The relaxed lattice constants a = b = 5.4333 Å and c = 8.9986 Å are consistent with the experimental result [24]. The surface states in the projected 2D BZ were calculated with the surface Green's function method by using the WannierTools package [57]. The tight-binding Hamiltonian of the semi-infinite system was constructed by the maximally localized Wannier functions [58]. To study the CDW phase of RbV 3 Sb 5 , a 2 × 2 × 1 supercell and a 5 × 5 × 5 k-point mesh for the corresponding BZ sampling were used. The initial atomic distortions were firstly set according to the in-plane structures of the previously reported Star of David and inverse Star of David patterns [34], and then both the lattice parameters and the internal atomic positions were fully relaxed. The band structures of the CDW phases were unfolded in the BZ of the unit cell with the band unfolding method [59] as in the Py-Vaspwfc package [60].