Three-dimensional energy gap and origin of charge-density wave in kagome superconductor KV3Sb5

Kagome lattices offer a fertile ground to explore exotic quantum phenomena associated with electron correlation and band topology. The recent discovery of superconductivity coexisting with charge-density wave (CDW) in the kagome metals KV3Sb5, RbV3Sb5, and CsV3Sb5 suggests an intriguing entanglement of electronic order and superconductivity. However, the microscopic origin of CDW, a key to understanding the superconducting mechanism and its possible topological nature, remains elusive. Here, we report angle-resolved photoemission spectroscopy of KV3Sb5 and demonstrate a substantial reconstruction of Fermi surface in the CDW state that accompanies the formation of small three-dimensional pockets. The CDW gap exhibits a periodicity of undistorted Brillouin zone along the out-of-plane wave vector, signifying a dominant role of the in-plane inter-saddle-point scattering to the mechanism of CDW. The characteristics of experimental band dispersion can be captured by first-principles calculations with the inverse star-of-David structural distortion. The present result indicates a direct link between the low-energy excitations and CDW, and puts constraints on the microscopic theory of superconductivity in alkali-metal kagome lattices.


Introduction
Kagome lattice is at the forefront of exploring exotic quantum states owing to its peculiar geometry characterized by a two-dimensional (2D) network of corner-sharing triangles. The representative geometrical effect appears as quantum magnetism in insulating kagome lattice, where the strong magnetic frustrations inherent in the triangular coordination lead to a quantum spin liquid [1,2]. In the metallic counterparts, the electronic states originating from the kagome-lattice symmetry are of particular interest, as they consist of a nearly flat band, Dirac-cone band, and saddle-point van Hove singularity which often dominate the physical properties of strongly correlated electron systems and topological materials. When the electron filling is tuned for the flat band, ferromagnetism [3] or charge fractionalization [4,5] may appear, whereas the tuning of 3 Dirac-cone band is expected to create strongly correlated Dirac semimetal, topological insulator, and Weyl semimetal phases [6][7][8][9]. Recent experiments reported several model kagome materials with the properly tuned electron-filling and the realization of predicted exotic properties [10][11][12][13][14][15][16][17][18][19][20][21]. On the other hand, the electron filling at the saddle point in kagome materials has been scarcely realized despite several intriguing theoretical predictions such as unconventional density wave orders [6, [22][23][24], superconductivity [23][24][25][26][27], and nematic instability [24].
Despite accumulating experimental and theoretical studies [31][32][33][34][35][36], the origin of CDW in AV3Sb5 is highly controversial. A fundamental issue is the type of structural distortion responsible for the CDW formation. This is essential to unveil the mechanism of superconductivity, because the superconductivity appears in the distorted phase. First-principles calculations proposed two types of distortions sharing the same space group of P6/mmm (No. 191) to account for the in-plane 2×2 periodicity (Fig. 1a). 4 One is the "Star-of-David" (SoD) distortion of V atoms which has a close connection with a well-known motif of a strongly correlated CDW state in transition-metal dichalcogenides [61]. The other is an inverse type of the SoD distortion, where V atoms show an opposite displacement compared with the SoD case, resulting in a periodic arrangement of triangular and hexagonal patterns, called "Tri-Hexagonal" (TrH; i.e., inverse SoD) structure. Although both phases are energetically more stable than the undistorted (1×1) phase in the calculations [54], it is experimentally highly controversial which distortion actually takes place [31][32][33][34][35][36]53]. Besides the type of distortion, it is also unclear how the distortion influences the electronic states and how the 3D nature of CDW manifests itself in the electronic states.
In this study, we provide insights into these key questions through the investigation of low-energy excitations in full k space by utilizing photon-energy-tunable ARPES. We demonstrate the appearance of a 3D pocket due to CDW-induced electronic reconstruction and an anisotropic CDW gap maximized around the saddle point of kagome V band. These characteristics can be reproduced by first-principles calculations assuming TrH structural distortion, and further suggest the importance of inter-saddlepoint scattering for the occurrence of CDW.

Fermi surface reconstruction by CDW
At first, we present the Fermi surface (FS) topology of KV 3 Sb 5 . Figure 1c shows the ARPES-intensity mapping at E F as a function of k x and k y at T = 120 K (above in KV3Sb5). A circular pocket centered at the G point and two (small and large) triangular shaped intensity patterns centered at each K point are resolved, as also visualized in Fig.   1d. According to the band structure calculations ( Supplementary Fig. 2), they are attributed to the 5pz band of Sb atoms embedded in the kagome-lattice plane (Sb1 in Fig.   1a) and the kagome-lattice band with mainly the 3d character of V atoms (V in Fig. 1a) [31,35], respectively (note that the small and large triangular features have the dominant K due to the CDW-gap opening. Also, the intensity of triangular pockets is strongly distorted at T = 20 K to show a discontinuous behavior at particular k points (black arrows), in contrast to that at T = 120 K which shows a smooth intensity distribution. This indicates the reconstruction of FS due to the strong modulation of band dispersions by the periodic lattice distortion associated with the CDW, which has not been resolved in previous ARPES studies of AV3Sb5 [29,37,[43][44][45][46][47]49,50]. Plot of second-derivative 7 ARPES intensity at T = 20 K in Fig. 1f signifies that the discontinuous intensity distribution (Fig. 1e) is accompanied with the emergence of a small pocket-like feature near the K point (white dotted ellipse). This pocket is associated with the CDW because it is absent at T = 120 K (Fig. 1d). The appearance/absence of the pocket-like feature below/above TCDW is more clearly seen from a direct comparison of momentum distribution curves (MDCs) at several ky slices in Fig. 1g; the peaks in MDCs marked by triangles, which correspond to the pocket-like feature, appear only below TCDW. Therefore, our observation supports the Fermi-surface reconstruction due to CDW.
To obtain further insights into the FS reconstruction, we look at the change in the To clarify the three-dimensional (3D) electronic states above TCDW, we have also mapped out the FS at kz ~ p by using hn = 130 eV photons, and show the ARPES intensity and corresponding second-derivative plots at T = 120 K in Figs. 1j-1m. While the overall 8 intensity at kz ~ p (Fig. 1j) is similar to that at kz ~ 0 (Fig. 1c) as seen in the existence of a circular pocket at the G ____ point and triangular FSs at the K ____ point, the bright intensity around the M point seen at kz = 0 (Fig. 1c) is absent at kz ~ p (Fig. 1j). This is because the saddle-point band is located well above EF at kz ~ p [29, 35,37,43] due to the kz dispersion.
To see the influence of CDW at kz ~ p, we have also mapped out the ARPES intensity at a large CDW gap and stays near EF. These observations demonstrate that the most prominent change in the energy bands across TCDW occurs in the SP1 band. When the k cut is chosen so as to pass slightly away from the M point (cut 2 in Fig. 2a), the SP1 band (black circles) moves closer to EF and stays at EB = 50-80 meV (Figs. 2g and 2h), signifying that the magnitude of CDW gap is sensitive to the in-plane wave vector k//.
Along cut 3 which passes the electron pocket at G (Figs. 2i and 2j), the band crosses EF with no gap opening, suggesting that this FS pocket (Sb 5p orbital) is not a main player of CDW. To elucidate the k dependence of CDW gap, we measured ARPES data along various k cuts parallel to the cuts 1-3 at T = 20 K and extracted EDCs at the k points where the leading edge shows a smallest shift relative to EF (called minimum gap locus [62]) along each cut. The result plotted in Fig. 2k shows that the gap gradually becomes smaller on approaching the G point (cut 3; |Dkx| = 1) from the M point (cut 1; |Dkx| = 0). This is also seen from the Lorentzian fitting of the EDCs in Fig. 2l  It is noted that, although the kz resolution in the present study (~0.18 Å -1 ; see Method) is about twice larger than that in soft x-ray ARPES [63], a complete loss of the resolution in kz is avoided and it is possible to perform ARPES measurements by selecting kz to some extent. When hn is set to probe kz = 0 or p, the electronic states at around kz = 0 or p are dominantly probed although the electronic states within ±0.25 p/c centered at kz = 0 or p are partially involved. This condition, which is similar to the previous vacuumultraviolet ARPES study on the kz-dependent gap [64], should ensure the validity of our qualitative argument that the gap at around k z = 0 is larger than that at around k z = p.

Comparison between ARPES results and first-principles calculations
A next important step is to pin down the structural distortion responsible for the FS reconstruction and CDW gap. For this sake, we have carried out first-principles bandstructure calculations for the CDW phase of KV3Sb5 by assuming the SoD or TrH distortion. Figures 3a and 3b Table 1); this comparison that properly stands on the experimental data would be meaningful because the calculations with lattice relaxation do not always reflect correct band parameters (note that the calculations with fully relaxed lattice parameters underestimate the CDW-gap size in the present case, as shown in Supplementary Fig. 7).
One can see from Figs. 3a and 3b several similarities between the two calculations. For example, both calculations predict main bands with a strong intensity showing a broad correspondence with the original bands in the normal state (dashed green curves) as well as several weak sub-bands associated with the CDW-induced band folding. Thus, in order to distinguish an appropriate model that reproduces the experimental data, it may be useful to look at the (E, k) region where two models show a critically different spectral behavior, rather than to examine the overall agreement/disagreement of the valence-band dispersion. We found that the prominent difference in the calculated band structure is seen around the M point near EF. This is reasonable because the proximity of the saddle point to EF plays a crucial role to reduce the total energy of system and thereby the band 13 FIG. 3. Calculated band structure and distinction of structural distortions. a, b Calculated band dispersions along the GMKG cut for the SoD and TrH distortions, respectively. Calculated band dispersions for the undistorted 1×1 phase is shown by dashed green curves. Calculations have been performed with the 2×2×1 superlattice because the influence from the change of unit-cell (doubling or quadrupling) along the c axis was not observed in the ARPES data, as seen from the 2p c -1 periodicity of the CDW-gap anisotropy along kz in Fig. 2o. c, ARPES intensity at T = 20 K measured along the MK cut at hn = 114 eV. d, e Calculated band dispersions along the MK cut for the SoD and TrH models, respectively. Red dashed curves are a guide for the eyes to trace the SP1 and SP2 bands. f, Schematics of k-dependent CDW gap on the SP1 band, together with the inter-saddle-point scattering vectors Q1-Q3 and three M points, M1-M3. g, Kagome lattice of V atoms with three different sublattices (V1-V3) shown with different coloring (red, blue, and green spheres, respectively). Calculated Wannier orbital for the SP1 band at one of the M points, which selectively places the V1 sublattice, is also indicated. h, Kagome lattice under the TrH distortion in which longer and shorter V-V bonds are highlighted by thick red and thin black lines, respectively. structure associated with the saddle point is expected to be sensitive to the type of distortions.
To specify the appropriate model, we have chosen the MK cut in which the energy dispersion of SP1 and SP2 bands is well visible in the experiment. The ARPES-derived band dispersion along the MK cut at T = 20 K in Fig. 3c signifies an M-shaped structure below EF due to the SP1 band. The M-shaped feature does not exist in the calculated band dispersion in the normal state (green curves in Fig. 3a and 3b) because the SP1 band is associated with the V-3dx2-y2-derived saddle point which is strongly modified by the kdependent CDW gap as discussed in Figs. 2c-2f. One can recognize in Fig. 3c another rapidly dispersive L-shaped saddle-point band with the V-3dxz/yz character (SP2 band).  [36], suggesting a tendency that the experimental data in Fig. 3c is more likely reproduced by the TrH distortion. Therefore, the TrH distortion may actually take place in the CDW phase of KV3Sb5. To obtain a decisive conclusion on this point, additional effects which are not taken into account in the present study, e.g., band renormalization and resultant change in the gap value as well as more complex structural distortions, should be investigated. 15 The 3D CDW gap and the possible TrH structural distortion discussed above put a strong constraint on the mechanism of CDW and superconductivity, as well as on the possible topological nature of superconducting state. In general, the electronic energy gain associated with the CDW is governed by the k region where a large CDW gap opens at EF. The observed strong k-dependent CDW gap, which takes the maximum at the M point as a function of both kz and k//, strongly suggests that the proximity to EF of the saddle-point band (SP1 band) with the V-3dx2-y2 character plays a major role in lowering the electron kinetic energy. In addition, the absence of a clear folding of the kz-dependent CDW gap suggests that, despite the existence of unit-cell doubling or quadrupling along the c axis [33,35,36], the V electrons in the kagome plane do not feel so strongly the superlattice potential. Therefore, the electron scattering by the in-plane Q vector connecting two different M points is a leading factor to stabilize the CDW in KV3Sb5 (Fig. 3f). While such Q vector is unidirectional, the existence of three equivalent Q vectors (Q1-Q3 in Fig. 3f) connecting three different M points (M1-M3) in the Brillouin zone leads to the 3Q-CDW state that satisfies the 2×2 periodicity. Besides the 3Q-CDW, the possible TrH distortion can be also explained by the bond-order state. Namely, the SP1 band at the M1-M3 points in the normal state is predominantly occupied by electrons in three different sublattices composed of V1-V3 atoms (see sublattice-selective distribution of the Wannier orbital for the SP1 band in Fig. 3g), and the inter-saddle-point electron scattering via the Q1-Q3 vectors enhances the sublattice interference effect [26].
This results in shortening of neighboring V-V bonds indicated by thick red lines in Fig.   3h (note that other V-V bonds indicated by thin black lines become longer), leading to tiling of hexagonal and triangle bond patterns consistent with the TrH distortion [55,65].
Also, the lattice distortion responsible for the 2×2 CDW is critically important to pin 16 down the superconducting pairing symmetry because the location of gap nodes and the topological nature of superconductivity are directly linked to the type of structural distortion in the ground state [66]. The present study further suggests that the CDWinduced reconstructed FS, but not the normal-state FS, must be considered to construct a microscopic theory of superconductivity in AV3Sb5.

ARPES measurements
High-quality single crystals of KV3Sb5 were synthesized by the self-flux method [40].
Photon-energy-tunable vacuum ultraviolet ARPES measurements were performed with the MBS-A1 analyzer at BL5U in UVSOR. We used linearly polarized light of 80-140 eV. The energy resolution was set to be 10-30 meV. The angular resolution was set to be 0.3º, which corresponds to the resolution in the momentum parallel to the kagome plane (k//) of 0.03-0.04 Å -1 . The resolution in the momentum perpendicular to the kagome plane (kz) is estimated to be ~0.18 Å -1 , which corresponds to a half of the G-A distance in the bulk Brillouin zone (~0.5 p c -1 ) [note that, according to the universal curve [67], the electron mean-free path l at the present hn range (~100 eV) is ~5.5 Å, leading to the kz broadening dkz expressed as l -1 of ~0.18 Å -1 ]. Samples were cleaved in situ along the (0001) plane of the hexagonal crystal in an ultrahigh vacuum of 1×10 -10 Torr, and kept at T = 20 or 120 K during the measurements.

Band calculations
First-principles band-structure calculations were carried out by using the Quantum Espresso code package [68] with generalized gradient approximation [69]. Spin-orbit coupling and D3 corrections were included in the calculations unless otherwise stated.
The plane-wave cutoff energy and the k-point mesh were set to be 50 Ry and 11×11×5, respectively. Supercell calculations were carried out on a 2×2×1 supercell with a 5×5×5 k mesh. The unfolding of calculated bands was performed using the BandUP code [70].
Wannier orbital was calculated by using the Wannier90 code [71].

DATA AVAILABILITY
The data sets generated/analyzed during the current study are included in the published article and its Supplementary Information file. The numerical data sets are available from the corresponding author on reasonable request.

CODE AVAILABILITY
Details on the numerical fittings and band-structure calculations are available from the corresponding author on reasonable request.