Electronic nature of charge density wave and electron-phonon coupling in kagome superconductor KV3Sb5

The Kagome superconductors AV3Sb5 (A = K, Rb, Cs) have received enormous attention due to their nontrivial topological electronic structure, anomalous physical properties and superconductivity. Unconventional charge density wave (CDW) has been detected in AV3Sb5. High-precision electronic structure determination is essential to understand its origin. Here we unveil electronic nature of the CDW phase in our high-resolution angle-resolved photoemission measurements on KV3Sb5. We have observed CDW-induced Fermi surface reconstruction and the associated band folding. The CDW-induced band splitting and the associated gap opening have been revealed at the boundary of the pristine and reconstructed Brillouin zones. The Fermi surface- and momentum-dependent CDW gap is measured and the strongly anisotropic CDW gap is observed for all the V-derived Fermi surface. In particular, we have observed signatures of the electron-phonon coupling in KV3Sb5. These results provide key insights in understanding the nature of the CDW state and its interplay with superconductivity in AV3Sb5 superconductors.

T he newly discovered Kagome superconductors AV 3 Sb 5 (A = K, Rb, Cs) have attracted much attention because they provide an ideal platform to investigate the interplay of topology, electron correlation effects, and superconductivity 1,2 . In the crystal structure of AV 3 Sb 5 (Fig. 1a), the vanadium atoms form a Kagome lattice that is a two-dimensional network of corner-sharing triangles. The metallic Kagome lattice presents a unique electronic structure characterized by a Dirac cone at the Brillouin zone corner, von Hove singularities (VHS) at the zone boundary, and a flat band throughout the entire Brillouin zone 3,4 . Such a Kagome lattice is expected to harbor topological states 3,5 , fractional charges 4,6 , density wave orders 3,7,8 , and unconventional superconductivity [8][9][10][11] . For example, AV 3 Sb 5 family exhibit anomolous Hall effect 12,13 , although there is neither local-moment nor long-range magnetic ordering present in them 1,12,14 ; unconventional charge density wave (CDW) has been revealed in AV 3 Sb 5 [15][16][17] . At present, the pairing symmetry of the AV 3 Sb 5 superconductors has been extensively studied and it is still being debated whether the superconductivity is unconventional [18][19][20][21][22] .
In this work, we carried out high-resolution angle-resolved photoemission (ARPES) measurements to investigate the nature of the CDW instability in KV 3 Sb 5 . Clear evidence of electronic structure reconstruction induced by the 2 × 2 CDW transition is revealed by the observation of the band and Fermi surface foldings. The band splitting and CDW gap opening on multiple bands are observed at the boundaries of both the original and 2 × 2 reconstructed Brillouin zones. We have clearly resolved all the Fermi surface that enables us to map out the Fermi surfaceand momentum-dependent CDW gap. The signature of electron-phonon coupling has been found on the V-derived bands. These results provide key insight in understanding the origin of the CDW and its role on the exotic physical properties and superconductivity in Kagome superconductors.

Results
The measured and calculated Fermi surface. High-quality KV 3 Sb 5 single crystals are prepared by a two-steps self-flux method 1 and characterized by X-ray diffraction (see "Methods" and Supplementary Fig. 1a). The transport and magnetic measurements show that our samples exhibit a CDW phase transition at T CDW~8 0 K, consistent with the previous reports 1, 23 . In the normal state above T CDW , KV 3 Sb 5 crystallizes in a hexagonal structure with the P6/mmm space group, hosting a typical Kagome structure composed of vanadium Kagome net (Fig. 1a). In the CDW phase, a distortion of the V-Kagome lattice engenders the 2 × 2 reconstruction and forms a tri-hexagonal (TrH) structure on the V-Kagome plane ( Fig. 1b) 15,25 . Such a lattice distortion leads to Brillouin zone reconstruction in the reciprocal space which can be described by three wavevectors (Fig. 1d). Figure 1e shows the Fermi surface mapping of KV 3 Sb 5 measured at 20 K in the CDW state. Extended momentum space that includes both the first and second Brillouin zones is covered in our measurements. This is important to obtain a complete Fermi surface since the band structures of KV 3 Sb 5 exhibit The Tri-Hexagonal (TrH) lattice distortion caused by the 2 × 2 CDW transition 15,25 . The K, V, Sb atoms are presented as gray, purple, and blue balls, respectively. c Schematic of the three-dimensional Brillouin zone and the two-dimensional Brillouin zone projected on the (001) surface in the pristine phase in a. High-symmetry points and high-symmetry momentum lines are marked. d The original (black lines) and 2 × 2 reconstructed (blue lines) Brillouin zones. The Γ, K, and M ( Γ 0 , K 0 , and M 0 ) are the high-symmetry points of the pristine (2 × 2 reconstructed) Brillouin zones. The arrows indicate three wavevectors (marked as Q 1 , Q 2 , and Q 3 ) of electronic structure reconstruction. e Fermi surface mapping of KV 3 Sb 5 measured at T = 20 K. Four Fermi surface sheets are observed marked as α (orange lines), β (red lines), γ (blue lines), and δ (green lines). For convenience, high-symmetry points are labeled with different indexes such as Γ 1 , Γ 21 , M 11 , M 12 , M 21 , K 11 , K 12 , and K 21 . f Symmetrized Fermi surface mapping of KV 3 Sb 5 . It is obtained from taking the second derivative with respect to momentum on the second Brillouin zone data in e. g The calculated Fermi surface at k z = π/c corresponding to pristine crystal structure in a. significant photoemission matrix element effects in different momentum space ( Fig. 1e and Supplementary Figs. 2, 3). The Fermi surface mapping in Fig. 1e, combined with the analysis of the related constant energy contours ( Supplementary Fig. 2) and band structures ( Supplementary Fig. 3), gives rise to a Fermi surface topology that is mainly composed of a circular electronlike pocket around Γ (α), a large hexagon-shaped hole-like sheet centered around Γ (β), a triangular hole-like pocket around K (γ) and a triangular electron-like pocket around K (δ) as marked in Fig. 1e. The γ pocket is clearly visualized around K 21 but is weak around K 12 ; its size increases with increasing binding energy in the measured constant energy contours ( Supplementary Fig. 2). On the other hand, the δ pocket is clearly observed around K 12 but is weak around K 21 ; its size decreases with increasing binding energy in the constant energy contours ( Supplementary Fig. 2). The quantitatively extracted Fermi surface is shown in Fig. 1f, which agree well with the calculated Fermi surface of KV 3 Sb 5 at k z = π/c in its pristine structure in Fig. 1g.
CDW-induced Fermi surface reconstruction and band folding. The CDW-related 2 × 2 lattice reconstruction is expected to generate electronic structure reconstruction, as illustrated in Fig. 1d. However, no signature of such electronic reconstruction has been detected in the previous ARPES measurements 2,37-44 . We have observed clear evidence of electronic structure reconstruction induced by the 2 × 2 CDW transition in KV 3 Sb 5 both in the measured Fermi surface and the band structure. Figure 2a replots the Fermi surface mapping of KV 3 Sb 5 shown in Fig. 1e, focusing on the first Brillouin zone. In addition to the main Fermi surface, some additional features are clearly observed, as marked by the arrows in Fig. 2a. Figure 2b shows the effect of the 2 × 2 lattice reconstruction on the Fermi surface as induced by one of the three wavevectors, Q 1 . The reconstructed Fermi surface sheets (dashed lines in Fig. 2b) are produced from shifting the original α, β, γ, and δ main Fermi surface (solid lines in Fig. 2b) by the wavevector of ± Q 1 . As shown in Fig. 2a, the extra features can be attributed to the reconstructed Fermi surface because the observed features (1, 2), (3,4), and 5 agree well with the reconstructed δ, β, and α Fermi surface, respectively. Under the measurement geometry we used, the observed folded bands are mainly from Q 1 while those from Q 2 and Q 3 are rather weak. This is due to the photoemission matrix element effect. By changing the measurement geometry, the folded bands from other Q 2 or Q 3 wavevectors can also be observed ( Supplementary Fig. 4). The electronic reconstruction is also directly evidenced in the measured band structure in Fig. 2c, in which the band measured along the Γ-M direction coincides with the direction of Q 1 wavevector. As shown in Fig. 2c, in addition to the main β bands, some extra bands are clearly observed around Γ (β 0 L and β 0 R ). The extra feature around Γ resembles the strong β band at M. Quantitative analysis of the momentum distribution curve (MDC) at the Fermi level in Fig. 2d indicates that the two extra features at Γ (β 0 L and β 0 R ) are separated from the β band at M (β L and β R ) by exactly a wavevector of Q 1 . Further analysis of the photoemission spectra (energy distribution curves, EDCs) at M and Γ in Fig. 2e indicates that they have similar lineshape near the Fermi level within an energy range of~0.3 eV. Figure 2f shows the Fermi surface mapping of KV 3 Sb 5 measured by high-resolution laser-ARPES at 20 K. In addition to the main α Fermi surface sheet, additional features are observed which can be attributed to the reconstructed Fermi surface from the original β, γ, and δ Fermi surface sheets, as marked by the dashed guidelines. Figure 2g-i shows the temperature-dependence of the band structure measured along a momentum cut around the Γ point. Clear band foldings, not only from the β band around M, but also from the γ 2 band around M, can be observed at~40 K (Fig. 2g) and~80 K (Fig. 2h). As shown in Fig. 2a, f, the observed α Fermi surface, and the folded β Fermi surface are quite similar in the measurements using 21.2 and 6.994 eV photon energies. The observed α bands in Fig. 2c, g are also similar although the momentum cut for the 6.994 eV measurement in Fig. 2f is slightly off the Γ-M cut for the 21.2 eV measurement in Fig. 2a. We note that the folded band of γ 2L and γ 2R is prominent in Fig. 2g while it is rather weak in Fig. 2c. This difference may be attributed to the photoemission matrix element effects. These folded bands become significantly suppressed at 120 K ( Fig. 2i) above the CDW temperature of~80 K. These results strongly demonstrate that the extra features at Γ are replicas of the β and γ 2 bands at M caused by the 2 × 2 CDW modulation.
CDW-induced band splitting and gap opening. Besides the electronic structure reconstruction, the manifestations of the CDW transition involve the opening of the CDW gap, both at the Fermi level and away from the Fermi level. We have clearly observed the CDW gap openings for both cases. Figure 3 shows band structures of KV 3 Sb 5 measured along high-symmetry directions Γ-M (Fig. 3a, d), K-M-K (Fig. 3b, e) and Γ-K (Fig. 3c, f) at 20 K. For comparison, we also present the calculated band structures for both pristine (Fig. 3g) and reconstructed ( Fig. 3h) crystal structures. For the reconstructed crystal structure, we calculated the effective band structure of 2 × 2 × 1 TrH CDW phase of KV 3 Sb 5 by unfolding its eigenstates in supercell Brillouin zone into primitive cell Brillouin zone (see "Calculations" section in "Methods" for more details). In the calculated band structure for the pristine lattice structure (Fig. 3g), the bands around the Fermi level originate mainly from the 5p orbitals of Sb (α band from the in-plane Sb while γ 2 band from the out-of-plane Sb) and the 3d orbitals of V (β, γ 1 , and δ bands). The δ band originates from the V-Kagome lattice with the prototypical Dirac point at K and von Hove singularities at M. The β band also comes from V-Kagome lattice with different orbital character. The 2 × 2 lattice reconstruction causes significant modifications of the band structures, manifested mainly by the band splitting and the associated CDW gap opening at M in the original Brillouin zone and M 0 in the reconstructed Brillouin zone. As shown in the calculated band structure for the reconstructed lattice in Fig. 3h, within the energy of interest, three CDW gaps open at M: MG1 from δ 1 band, MG2 from ζ band, and MG3 from δ 2 band. In the meantime, four CDW gaps open at M 0 point: MPG2 from γ 1 band, MPG3 from β 2 band, and MPG4 from δ 1 band. We note that all the bands shown in Fig. 3h are primary bands; it does not include folded bands. Therefore, all the gaps marked in Fig. 3h are real CDW gaps. In addition, the spin-orbit coupling (SOC) is expected to open a gap at the Dirac point formed from the δ bands at K, as marked by DG in Fig. 3g, h. The expected band splittings and CDW gap openings at M and M 0 below the Fermi level are clearly observed in the measured band structures of KV 3 Sb 5 . Figure 3d, e shows the CDW gap openings at the M point where the ζ band opens a gap labeled as MG2 and the δ 2 band opens a gap MG3. In the corresponding EDCs at M, signatures of these two gap openings can also be clearly visualized with the gap size of~150 meV for MG2 and~125 meV for MG3. Figure 3f shows the CDW gap openings at the M 0 point where the β 2 band opens a gap labeled as MPG3 and the δ 1 band opens a gap MPG4. In the corresponding EDC at M 0 in Fig. 3j, the MPG4 gap can be clearly determined with a gap size of~150 meV. The MPG3 gap is present as seen from the dip in EDC near the binding energy of 300 meV that corresponds to the spectral weight suppression in the region pointed out by the arrow in Fig. 3c. However, the related band is weak; its gap size is hard to be determined precisely but estimated to be~150 meV. The SOC gap opening of the Dirac point at K can be seen from the EDCs in Fig. 3k; the measured gap size is~80 meV. The measured band splittings and gap openings agree well with those from band-structure calculations.
Fermi surface-and momentum-dependent CDW gaps. Now we come to the CDW gap on the Fermi surface. To this end, we took high energy resolution (~4 meV) ARPES measurements on KV 3 Sb 5 at 5 K, covering the momentum space around M 21 as shown in Fig. 4l. In this region, in addition to the well-resolved α and β Fermi surface sheets, the γ and δ sheets are also well separated because the former is strong around K 21 while the latter is strong around K 12 . The clearly distinguished four Fermi surface sheets facilitate the extraction of the Fermi surface-dependent and momentum-dependent CDW gaps. Figure 4a-e shows the symmetrized EDCs along the four Fermi surface; the data are taken on the two β sheets on the two sides of M 21 for confirming the data reliability. In the symmetrized EDCs, the gap opening causes a spectral weight suppression near the Fermi level that gives rise to a dip at the Fermi level; the gap size can be determined by the peak position near the Fermi level. The extracted CDW gaps along the four Fermi surface sheets are plotted in Fig. 4f-j. No CDW gap opening is observed around the α Fermi surface as shown in Fig. 4a, f. For the β Fermi surface, both measurements in Fig. 4b, c give a consistent result on the CDW gap in Fig. 4g, h. The CDW gap on the β Fermi surface is anisotropic; it shows a minimum close to zero along the Γ-M and Γ-K directions but exhibits a maximum in the middle between these two directions. The CDW gaps along the γ and δ Fermi surface sheets show similar behaviors, as seen in Fig. 4d, e and i, j. They are both anisotropic, showing a minimum along the Γ-K direction and a maximum along the K-M direction. The EDCs along the γ and δ Fermi surface also show multiple features (Fig. 4d, e); besides the low energy peak, there is another peak at a higher binding energy around 70 meV. As we will show below, such a peak-dip-hump structure can be attributed to the electron-phonon coupling. Figure 4k shows a three-dimensional picture summarizing the Fermi surface-dependent and momentum-dependent CDW gaps we have observed in KV 3 Sb 5 . The CDW gaps measured along the β, γ, and δ Fermi surface sheets show strong momentum anisotropy with only small portions of the Fermi surface ungapped. We have checked on the origin of the CDW gap anisotropy in terms of the Fermi surface folding picture ( Supplementary Fig. 5). The observed CDW gap anisotropy is in a qualitative agreement with the expected results from the Fermi surface folding picture. Signatures of electron-phonon coupling. The CDW transition usually involves electronic structure reconstruction and lattice distortion in which the electron-phonon coupling plays an important role 45 . We have obtained clear evidence of electron-phonon coupling in KV 3 Sb 5 . Figure 5a-c zooms in on the band structures of KV 3 Sb 5 near the Fermi level measured along Γ-K, K-M-K and Γ-M-Γ directions at 20 K in the CDW state. The corresponding EDCs are shown in Fig. 5d-f. For the δ band in Fig. 5a, γ and δ bands in Fig. 5b and β band in Fig. 5c, the peak-dip-hump structure is clearly observed near their respective Fermi momenta as the peaks are marked by triangles and the humps are marked by bars in Fig. 5d-f. Figure 5g shows the expanded view of the δ band in Fig. 5a. A kink in the dispersion can be observed as marked by an arrow in Fig. 5g. The quantitative dispersion is obtained by fitting momentum distribution curves (MDCs) at different binding energies and plotted on top of the observed band in Fig. 5g. Taking a linear line as an empirical bare band, the effective real part of the electron self-energy is shown in Fig. 5h. It shows a clear peak at~36 meV. The observed kink in the energy dispersion and the peak-dip-hump structure in EDCs are reminiscent of those from the electron-boson coupling in simple metal 46 and high-temperature superconductors 47 . The phonon frequency of the vanadium vibrations in AV 3 Sb 5 can reach up to~36 meV 32 that is consistent with the mode energy we have observed. Therefore, we have observed significant selfenergy effects in KV 3 Sb 5 . It can be interpreted in terms of electron-phonon coupling which is present for all the β, γ and δ bands.

Discussion
The CDW state is first proposed for a one-dimensional chain of atoms with an equal spacing a which is argued to be inherently unstable against the dimerized ground state 48 . It usually involves one band with a half electron filling. This would open a CDW gap at the Fermi point k F = ±π/2a and produce a lattice reconstruction with a wavevector of π/a. Such a Fermi surface nesting picture is extended to real materials with higher dimensions where the CDW state is realized because segments of the Fermi surface are nearly parallel connected by a wavevector Q CDW 45 . This would give rise to a partial CDW gap opening on the Fermi surface and reconstructions of both the electronic structure and the lattice with a wavevector of Q CDW . In KV 3 Sb 5 , multiple Fermi surface sheets are observed with the estimated filling of 0.17 electrons/unit for the α pocket, 1.10 holes/unit for the β pocket, 0.48 holes/unit for the γ pocket, and 0.27 electrons/unit for the δ pocket (Fig. 1f). Judging from the measured Fermi surface topology, the probable nesting vectors, if exist, are along the Γ-K direction where parts of the β, γ, and δ Fermi surface sheets exhibit nearly parallel Fermi surface. However, these vectors are not consistent with the band folding CDW wavevectors Q 1 , Q 2 , and Q 3 along Γ-M direction directly determined by STM and X-ray diffraction measurements [15][16][17] . Therefore, whether the classical Peierls instability picture can be applied to the CDW formation in KV 3 Sb 5 needs further investigations. Besides the Fermi surface nesting, the CDW phase can also be driven by the concerted action of electronic and ionic subsystems where a qdependent electron-phonon coupling plays an indispensable , and K (c) points, respectively. The measured band structures are indicated by guidelines and the associated CDW gaps and SOC gap are also marked. g Calculated band structure of KV 3 Sb 5 with pristine crystal structure in Fig. 1a at k z = π/c with SOC. h The calculated band structures of KV 3 Sb 5 with reconstructed TrH crystal structure in Fig. 1b at k z = π/c with SOC. In addition to the original high-symmetry points Γ, M and K, new high-symmetry points from the reconstructed Brillouin zone (Fig. 1c) Supplementary Fig. 6. The locations of the momentum cuts, 1, 2, and 3 for a, b, and c, respectively, are shown in i. The Fermi momenta of the β, γ, and δ bands are marked by arrows and labeled by k F 1~k F 6. d-f The corresponding EDCs for the band structures in a-c, respectively. Peak-dip-hump structure can be observed near k F 1, k F 2, k F 4, k F 5, and k F 6. The EDC peaks are marked by triangles while the humps are marked by bars. g Expanded view of the δ band inside the dashed box in a. The MDC fitted dispersion is shown by the green line and the dashed black line represents the calculated band from Fig. 3h as the bare band. h Real part of the electron self-energy extracted from g. It shows a peak at~36 meV. i Schematic of the Fermi surface and the locations of the momentum cuts 1, 2, and 3 for the band structures in a, b, and c, respectively. part 49,50 . In AV 3 Sb 5 system, the driving force for the CDW formation remains under debate 15,17,25,32,33,37,38 . Based on our observations, we found that the electron-phonon coupling plays a major role in generating the CDW phase in KV 3 Sb 5 . Firstly, the measured Fermi surface (Fig. 1f) and band structures (Fig. 3a-c) of KV 3 Sb 5 show a high agreement with the band-structure calculations that do not incorporate the electron-electron interactions, which indicates the electron correlation effect is weak in KV 3 Sb 5 . Note that in the comparison, there is no Fermi level adjustment in the calculated electronic structures. Specifically, the calculated δ band bottom at K in Fig. 3h lies at the binding energy of 270 meV while the measured value of the δ band bottom at K in Fig. 5a is~256 meV, indicating a weak band renormalization due to electron correlation. Secondly, besides the gap opening around the Fermi surface, we have also observed a clear CDW gap opening at M and M 0 with a gap size up to~150 meV (Fig. 3) highly away from the Fermi level. Thirdly, the electron-phonon couplings on the β, γ, and δ bands are directly observed (Fig. 5). All these results indicate that the CDW phase in KV 3 Sb 5 is mainly driven by the electron-phonon coupling induced structural phase transition.
In summary, through our high-resolution ARPES measurements and the density functional theory (DFT) calculations on KV 3 Sb 5 , clear evidence of the 2 × 2 CDW-induced electronic structure reconstruction has been uncovered. These include the Fermi surface reconstruction, the associated band-structure foldings, and the CDW gap openings at the boundary of the pristine and reconstructed Brillouin zone. The Fermi surfacedependent and momentum-dependent CDW gap is measured and strong anisotropy of the CDW gap is observed for all the V-derived Fermi surface sheets. The electron-phonon couplings have been observed for all the V-derived bands. These results indicate that the electron correlation effect in KV 3 Sb 5 is weak and the electron-phonon coupling plays a dominant role in driving the CDW transition. They provide key information in understanding the origin of the CDW state and its interplay with superconductivity in AV 3 Sb 5 superconductors.

Methods
Growth and characterization of single crystals. High-quality single crystals of KV 3 Sb 5 were grown from a two-steps flux method 1 . First, KSb 2 alloy was sintered at 573 K for 20 hours in an alumina crucible coated with aluminum foil. Second, high-purity K, V, Sb, and KSb 2 precursor were mixed in a molar ratio of 1:3:14:10 and then sealed in a Ta tube. The tube was sealed in an evacuated quartz ampoule, heated up to 1273 K, soaked for 20 h, and then cooled down to 773 K at a rate of 2 K/h. Shiny lamellar crystals were separated from the flux by centrifuging with a regular hexagon shape and a size up to 4 × 4 mm 2 (inset of Supplementary Fig. 1a). The crystals were characterized by X-ray diffraction (Supplementary Fig. 1a) and their magnetic susceptibility and resistance were measured ( Supplementary Fig. 1b,  c). The CDW transition temperature, T CDW , is~80K from the magnetic measurement in Supplementary Fig. 1b.
High-resolution ARPES measurements. High-resolution angle-resolved photoemission measurements were carried out on our lab system equipped with a Scienta R4000 electron energy analyzer 51,52 . We use a helium discharge lamp as the light source that can provide a photon energy of hν = 21.218 eV (helium I). The energy resolution was set at~20 meV for the Fermi-surface mapping (Fig. 1) and bandstructure (Figs. 2a, c, 3 and 5) measurements and at 4 meV for the CDW gap measurements (Fig. 4). We also use an ultraviolet laser as the light source that can provide a photon energy of hν = 6.994 eV with a bandwidth of 0.26 meV. The energy resolution was set at~2.5 meV for the measurements in Fig. 2f-i. The angular resolution is~0.3°. The Fermi level is referenced by measuring on a clean polycrystalline gold that is electrically connected to the sample. The sample was cleaved in situ and measured in a vacuum with a base pressure better than 5 × 10 −11 Torr.
Calculations. First-principles calculations are performed by using the Projected Augmented Wave Method (PAW) within the spin-polarized density functional theory (DFT), as implemented in the Vienna Ab Initio Simulation Package (VASP) [53][54][55] . We construct 2 × 2 × 1 supercell to describe the TrH CDW phase of KV 3 Sb 5 . The crystal structures are relaxed by using the Perdew-Burke-Ernzerhof (PBE) functional 56 and zero damping DFT-D3 van der Walls correction 57 until the forces are <0.001 eV/Å. The cutoff energy of plane-wave basis is set as 600 eV and the energy convergence criterion is set as 10 −7 eV. The corresponding Brillouin zones are sampled by using a 16 × 16 × 10 (for primitive cell) and a 8 × 8 × 10 (for supercell) Gamma centered k-grid. The effective band structure is calculated by the band-unfolding method 58,59 proposed by Zunger et al. with BandUP code 60,61 . The 2 × 2 × 1 TrH CDW order reconstruction is an input in our band-unfolding calculation. All the DFT calculations are ab initio without any adjustable parameters except for the standard exchange-correlation functional and pseudopotentials.

Data availability
All data are processed by using Igor Pro 8.02 software. All data needed to evaluate the conclusions in the paper are available within the article and its Supplementary Information files. All raw data generated during the current study are available from the corresponding author upon reasonable request.