Pressure evolution of electron dynamics in the superconducting kagome metal CsV$_3$Sb$_5$

The coexistence of the charge-density wave (CDW) and superconducting phases and their tunability under external pressure remains one of the key points in understanding the electronic structure of $A$V$_3$Sb$_5$ ($A$ = K, Rb, Cs) kagome metals. Here, we employ synchrotron-based infrared spectroscopy assisted by density-functional calculations to study the pressure evolution of the electronic structure at room temperature up to 17 GPa experimentally. The optical spectrum of CsV$_3$Sb$_5$ is characterized by the presence of localized carriers seen as a broad peak at finite frequencies in addition to the conventional metallic Drude response. The pressure dependence of this low-energy peak reflects the re-entrant behavior of superconductivity and may be interpreted in terms of electron-phonon coupling, varying with the growth and shrinkage of the Fermi surface. Moreover, drastic modifications in the low-energy interband absorptions are observed upon the suppression of CDW. These changes are related to the upward shift of the Sb2 $p_x+p_y$ band that eliminates part of the Fermi surface around the $M$-point, whereas band saddle points do not move significantly. These observations shed new light on the mixed electronic and lattice origin of the CDW in CsV$_3$Sb$_5$.


Introduction
Discovered in 2019 1 , the non-magnetic kagome metal series AV 3 Sb 5 (A = K, Rb, Cs) provides an exciting playground for studying a plethora of fascinating electronic phenomena, including the interplay between a charge-density wave (CDW) phase and superconductivity.In CsV 3 Sb 5 , the CDW instability forms below T CDW = 94 K 2 (102 K for RbV 3 Sb 5 low pressures (p < 2 GPa), superconductivity in CsV 3 Sb 5 exhibits a double-dome feature, comprising an enhancement of T c to ∼ 7 K at p 1 ∼ 0.6 GPa, and a second peak in T c at p 2 ∼ 2 GPa with T c ∼ 8 K, whereas the superconducting gap evolves almost monotonically 18 .Simultaneously, T CDW is gradually suppressed and vanishes at p 2 5, 7, 10 .Experimental studies revealed a non-trivial evolution of CDW across p 1 , suggesting that changes in the nature of CDW may cause the double-dome behavior of superconductivity 10,16,19 .For p > 2 GPa, superconductivity is gradually suppressed and vanishes around 9 GPa.Surprisingly, at higher pressures (p > 12 GPa), superconductivity re-emerges, with this second SC phase persisting up to at least 100 GPa 8,11,15 .
This re-entrant behavior, along with the presence of a quantum critical point (QCP) at 2 GPa, has sparked considerable debate on the superconducting pairing mechanism.Some studies propose electronic correlations at ambient pressure and CDW fluctuations around 2 GPa to be the main driving force behind the formation of Cooper pairs [20][21][22] .On the other hand, first-principle calculations revealed essential changes in electron-phonon (e-ph) coupling around the QCP, suggesting a conventional phonon-mediated mechanism [23][24][25][26] .
The CDW state in CsV 3 Sb 5 is unique within the AV 3 Sb 5 series as both a 2 × 2 × 2 and 2 × 2 × 4 order comprising stacked 2 × 2 in-plane star-of-David and trihexagonal kagome superlattices are possible at ambient pressure [27][28][29] .Moreover, the experimentally observed CDW state features multiple anisotropic energy gaps 30,31 , large anomalous Hall effect 32,33 , intrinsic chirality [33][34][35] , and a temperature-driven re-arrangement of the order parameters including an electronic nematic phase below ∼ 35 K 16,36,37 .Some studies discuss electron-phonon coupling as the driving force behind the CDW instability in AV 3 Sb 5 24, 38, 39 , whereas others point towards a more complex electronic origin 40,41 .It has been shown that band saddle points (van Hove singularities) close to the Fermi energy lead to a CDW instability in kagome metals [42][43][44] , making the pressure evolution of the saddle points a potentially important ingredient for understanding the pressure-phase diagram of CsV 3 Sb 5 .
Linking the pressure-induced changes in the electronic properties to modifications in the electronic band structure has been the subject of several high-pressure XRD studies, along with density functional theory calculations 11,[23][24][25][45][46][47][48] . The exprimental confirmation of the proposed changes in the electronic band structure, however, remains lacking because the powerful ambientpressure probes, such as ARPES and STM, are incompatible with pressure environment.Hence, we utilize high-pressure Fourier-transform infrared spectroscopy at room temperature to experimentally probe the pressure-induced modifications in the electronic structure.Assessing these changes in the normal state is crucial for understanding the changes in low-temperature instabilities of the system, because those instabilities are driven by the evolution of the electronic structure in the normal state.
As most high-pressure transport measurements are performed using silicone or Daphne oil as pressure transmitting medium, increasing pressure leads to increasing non-hydrostaticity 49,50 .
It has been shown that under such conditions, CsV 3 Sb 5 transforms from hexagonal to monoclinic symmetry above ∼ 10 GPa 46 as sketched in Fig. 1 (a).The current reflectivity study has been performed with the quasi-hydrostatic CsI 51 as a pressure transmitting medium, ensuring a good sample-diamond interface, which is crucial for reliable measurements.Hence, here, a similar phase transition can also be envisaged.However, as discussed previously 46 , this monoclinic distortion affects the band structure only marginally.
Our results reveal a drastic change in the interband absorption around p 2 where the CDW is suppressed.Using ab initio calculations of the band structure and optical conductivity, we show that these changes can be understood in terms of the upward shift of the Sb2 p x + p y band and the shrinkage of the Fermi surface around the M -point.We further demonstrate that the spectral weight due to localized carriers is strongly suppressed around p 2 , suggesting the reduction in the e-ph coupling.This spectral weight is partially restored above 10 GPa, where re-entrant superconductivity is observed.Our data suggest that changes in the e-ph coupling caused by the reconstruction of the FS should be crucial for the pressure evolution of electron dynamics in CsV 3 Sb 5 .

Results
Intraband contributions.The decomposed real part of the in-plane optical conductivity at selected pressure points is given in Figs. 2 (a-d) (see Supplementary Note 2 for details on the fitting process).While the high-energy contributions (ω > 4000 cm −1 ) are not notably affected by the applied pressure, major changes occur at low energies (ω < 2000 cm −1 ).The optical spectrum at ambient pressure (p = 0 GPa), reproduced from ref. 52 , is characterized by several interband absorptions and two clearly separated intraband contributions (i) a sharp Drude peak due to free charge carriers and (ii) an additional broad peak centered at finite frequencies corresponding to the response of localized carriers.Both of these intraband features are found to be highly sensitive to the applied pressure, as illustrated in Figs. 2 (e) and (f).Already at moderate pressures (p < 2.3 GPa), the intensity of the localization peak is drastically reduced and continues to decrease up to ∼ 9 GPa.Upon further increasing the pressure, the localization peak becomes more pronounced again.On the other hand, the opposite trend is observed for the Drude contribution, signaling an interplay between localized and free charge carriers.
To further explore this interplay, we calculate the spectral weight (SW) by integrating the real part of the optical conductivity of the fitted Drude and localization peak according to 53 with ε 0 being the permittivity in vacuum and c the speed of light.The cut-off frequency is chosen as ω c = 50000 cm −1 considering the high-energy tail of the localization peak.Fig. 2 (g) shows the pressure evolution of the Drude and localization peak spectral weight, as well as of the total spectral weight of the intraband processes (Drude + localization peak).While the total intraband SW is found to be almost unaffected by pressure, a redistribution of spectral weight between the Drude and the localization peak is observed.
The plasma frequencies are obtained via ω p = √ SW Drude + SW loc and normalized to the value at ambient pressure from a previous study 52 (see Supplementary Note 3 for details).As displayed in Fig. 2 (h), a good match between the experimental and DFT plasma frequency is observed.Here, even small features in the experimental plasma frequency like dips around 2.5 GPa, 8 GPa, and 11 GPa are well reproduced by our calculations.This good agreement between exper-iment and ab initio calculations over a wide pressure range suggests that electrons in CsV 3 Sb 5 are almost uncorrelated, consistent with the previous assessment of the correlation strength at ambient pressure 52 .Hence, the main effect beyond the pure band picture is the damping of charge carriers manifested by the localization peak.With this suppression of absorption below 0.1 eV, the experimental interband optical conductivity at low energies can be described by two absorption peaks as plotted in Figs. 3 (d-f).One of these peaks only slightly changes with pressure (black dotted line), while the other reveals a shift to higher energies (solid green line).This behavior is in line with our calculations showing a systematic shift of band C around the M -point away from the Fermi level, leading to a blue shift of the absorption peak related to the B-C transitions.The high consistency of the energy shift between calculations and experiment, as depicted in Fig. 3 (h), further supports the use of the uncorrelated band picture for the description of CsV 3 Sb 5 over a broad pressure range.
Moreover, due to the increasing slope of the Sb2 p x + p y band under pressure, band A comes closer to the Fermi level and even crosses it at higher pressures (see Supplementary Note 4).
Consequently, the contribution of the A-B transitions increases upon applying pressure, compared to the B-C transitions that were dominant at ambient pressure, leading to an increase of spectral weight at around 0.2 eV (see Fig. 3 (g)).

Discussion
Previously, several optical studies demonstrated a close link between the low-energy interband absorptions and fine details in the electronic band structure of the AV 3 Sb 5 series 52, 54, 55 .On the other hand, the localization peak, a common feature among several kagome metal compounds 52, 54-57 , can not be explained within the simple band picture.Free charge carriers interacting with lowenergy degrees of freedom, such as phonons, and electric or magnetic fluctuations, can lead to a backscattering of the electrons, causing localization effects manifested in a displaced Drude peak 58,59 .In the absence of magnetism and electronic correlations, interactions between electrons and phonons become the most plausible reason for the appearance of the localization peak.This interpretation is further supported by the gradual shift of the localization peak to low energies on cooling, as phonons are suppressed 52 .Moreover, optical studies of KV 3 Sb 5 and RbV 3 Sb 5 find strong phonon anomalies, which are associated with the phonon modes coupling to the electronic background 54, 55 .In the magnetic rare-earth kagome metal series RMn 6 Sn 6 , a close link between the behavior of phonon modes and the localization peak was observed 56 .
It is then natural to use the spectral weight of the localization peak as a gauge of the electronphonon coupling.Our data suggest that this coupling should be strongly suppressed around 2 GPa, where the CDW disappears.Concurrently, the increase in the spectral weight of the localization peak above 10 GPa indicates that the e-ph coupling becomes more prominent as superconductivity re-appears.On the microscopic level, first-principle calculations confirm this picture 24 and reveal a strong coupling of the V-V bond-stretching and V-Sb bond-bending phonon modes to the V 3d xy,x 2 −y 2 ,z 2 , V 3d xz,yz , and Sb1 5p z bands 26 .Above 2 GPa, a drastic reduction of the coupling strength λ is proposed, corresponding to the suppression of superconductivity.Consequently, above 12 GPa, the increase of the localization peak most probably signals an increase of e-ph coupling due to the new Fermi surface around A, as displayed in Fig. 1 (b) and (c).Here, Sb2 p z electrons appear at the Fermi level for the first time (see Supplementary Fig. 4 for the band structures at higher pressures), and probably cause the re-entrant superconductivity as illustrated in Fig. 2 (i).Given the unusual non-monotonic behavior of T c under pressure, several other scenarios influencing superconductivity in CsV 3 Sb 5 can be excluded.Due to only weak changes in the plasma frequency (Fig. 2(h)), as well as a very different pressure evolution of the electronic density of states at E F (see Supplementary Fig. 3 (d)), the behavior of T c cannot be explained by modifications in the electronic structure.Moreover, the continuous evolution of the crystal structure 45,46,48 , as well as of phonons according to previous DFT calculations 23,24 , eliminates the Debye temperature as a dominant factor, leaving changes in the e-ph coupling as the most likely reason for the observed pressure evolution of T c .
We would like to point out that while in some cases, for instance, through partial substitution of Vatoms with Nb-atoms, the effects on the electronic properties are similar to applying pressure, i.e., the suppression of the CDW order, along with an enhancement of T CDW , doping has a somewhat different effect on the band structure.As revealed by our study, the energies of the band saddle points are only weakly affected by pressure, while the Sb2 p x +p y bands show a significant upward shift already at moderate pressures.On the other hand, Nb-doping leads to modifications near the Γ-and M -points, highly affecting the band saddle points 62,66 .These differences in the band structure evolution entail a distinct behavior of the low-energy interband optical transitions.The application of external pressure leads to a systematic shift of band B away from the Fermi energy around the M -point, resulting in a blue-shifting interband absorption peak.Conversely, this lowenergy absorption peak at around 400 cm −1 shifts to lower energies upon Nb-substitution 65 .
Our study revealed that the reduction in the e-ph coupling around 2 GPa is likely the main reason for the CDW suppression in CsV 3 Sb 5 .The simultaneous change in the interband absorption gives us a strong hint that this behavior is caused by the reduction of the FS along Γ -M .The Sb2 p x + p y band, which was so far almost disregarded in the context of the CsV 3 Sb 5 physics, shows an upward shift and pushes some of the V 3d bands above the Fermi level.This effect is qualitatively different from the scenario of band saddle points at M moving away from the Fermi level and pulling V 3d bands down in energy, consequently increasing the FS.We thus find a delicate interplay between electronic and lattice degrees of freedom in CsV 3 Sb 5 and identify a large tunability of the kagome bands via changes in the Sb sublattice.

Methods
Optical Measurements.High-quality single crystals were grown and prepared according to ref. 1 .
For the optical measurements, a freshly cleaved sample with a surface area of 150 µm × 150 µm and a thickness of ∼ 60 µm was used.
High-pressure reflectivity measurements were performed at room temperature at the SMIS beamline of the SOLEIL synchrotron, France, on a homemade horizontal microscope with custom Schwarzschild objectives (NA = 0.5).A diamond anvil cell (DAC) with type-IIa diamond anvils and a culet of 400 µm diameter was utilized.Finely ground CsI powder served as the pressure transmitting medium, making it possible to reach pressures up to 17.05 GPa.The sample and Ruby spheres used as pressure gauges were placed inside a stainless steel gasket with a 200 µm diameter hole.The pressure was determined by monitoring the calibrated shift of the ruby R1 fluorescence line as described in ref. 67 .
The reflectivity spectra at the sample-diamond interface were recorded in a broad spectral range of 150 -10000 cm −1 by a Thermo-Fisher iS50 interferometer with KBr and solid substrate beamsplitters, using a MCT detector and a liquid helium-cooled bolometer.The reflectivity of a gold foil loaded into the DAC at ambient pressure served as a reference.Other optical quantities like the complex optical conductivity σ(ω), or the dielectric permittivity ε(ω), were obtained using standard Kramers-Kronig (KK) analysis considering the sample-diamond interface as explained in Supplementary Note 1.
Computational Details.Density-functional (DFT) band-structure calculations were performed with the Wien2K code 68,69 and cross-checks have been conducted with the FPLO code 70 .
In all cases, the Perdew-Burke-Ernzerhof flavor of the exchange-correlation potential 71 was used, and self-consistent calculations were converged on the k-mesh with 24 × 24 × 12 points.Experimental crystal structural parameters from refs. 1,45 ere used.Considering that the monoclinic distortion does not have a fundamental impact on the band structure 46 , we used hexagonal structure throughout the pressure range for a simpler comparison.Optical conductivity was calculated using the optic module 72   E n e r g y ( e V ) F r e q u e n c y ( c m -1 )  Decomposed optical conductivity at several pressure points consisting of a Drude peak (green), a localization peak (blue), and multiple interband transitions (orange).The data at ambient pressure were taken from a previous study 52 .e and f Pressure evolution of the Drude and localization peak, respectively.g Calculated spectral weight of the two intraband contributions (Drude and localization peak).h Pressure dependence of the experimental plasma frequency deduced from the total intraband spectral weight as explained in the text and plasma frequency calculated by DFT.i Schematic pressure phase diagram of CsV 3 Sb 5 as determined by several electrical transport, magnetic susceptibility, NMR, XRD, and µSR studies [5][6][7][8][9][10][11][12][13][14][15][16][17] .The gray shaded areas mark critical pressure regions corresponding to (i) the vanishing of CDW state at ∼ 2 GPa (ii) the disappearance of the first superconducting dome at around 9 GPa, and (iii) the re-emergence of superconductivity at approximately 12 GPa.4 .9 G P a E n e r g y ( e V ) Fig. 3 (c)), which could be well reproduced by DFT 52 .

Figure 1 :
Figure 1: Pressure evolution of the crystal structure and Fermi surface.a Hexagonal (left) and monoclinic (right) crystal structures of CsV 3 Sb 5 as determined by XRD measurements at ambient pressure 1 and 14.2 GPa under non-hydrostatic conditions 46 , respectively, visualized by VESTA 73 .b and c Representative Fermi surfaces, at 0 and 20 GPa, respectively, created with FermiSurfer 74 .Hexagonal structures 45 were used for the calculations.The color code represents the band velocity.

F r e q u e n c y ( 1 0 3 c m - 1 )
σ 1 , l o c ( 1 0 2 Ω -1 c m -1 )P r e s s u r e ( G P a )

Figure 2 :
Figure 2: Decomposition of optical conductivity and intraband spectral weight analysis.a-dDecomposed optical conductivity at several pressure points consisting of a Drude peak (green), a localization peak (blue), and multiple interband transitions (orange).The data at ambient pressure were taken from a previous study 52 .e and f Pressure evolution of the Drude and localization peak, respectively.g Calculated spectral weight of the two intraband contributions (Drude and localization peak).h Pressure dependence of the experimental plasma frequency deduced from the total intraband spectral weight as explained in the text and plasma frequency calculated by DFT.i Schematic pressure phase diagram of CsV 3 Sb 5 as determined by several electrical transport, magnetic susceptibility, NMR, XRD, and µSR studies[5][6][7][8][9][10][11][12][13][14][15][16][17] .The gray shaded areas mark critical pressure regions corresponding to (i) the vanishing of CDW state at ∼ 2 GPa (ii) the disappearance of the first superconducting dome at around 9 GPa, and (iii) the re-emergence of superconductivity at approximately 12 GPa.

Figure 3 : 9 Figure 4 :
Figure3: Pressure evolution of the band structure and interband transitions.a Calculated band structure at 4.9 GPa.b Band-resolved contributions to the in-plane component of the calculated optical conductivity.c-f Experimental interband transitions at selected pressures.The red arrow marks the low-energy transitions (ω < 0.06 eV) that become suppressed abruptly between 0 and 3 GPa.At low energies, the pressure data are described by two distinct absorption peaks (solid green line and black dotted line).With increasing pressure, a systematic blue shift of the green absorption is observed.g Pressure evolution of the calculated optical conductivity.The red arrow marks the suppression of low-energy absorption and the circles highlight the shifting interband absorption peak.h Pressure evolution of the blue-shifting low-energy interband absorption peak from the experimental and calculated optical conductivities.

4 F 1 Supplementary Figure 1 : 1 )Supplementary Figure 2 :
r e q u e n c y ( c m -1 ) l e c t i v i t y F r e q u e n c y ( c m -1 ) Reflectivity, optical conductivity, and dielectric permittivity.a Absolute reflectivity at the sample-diamond interface, obtained as described in the text.The grey dashed lines mark the area of diamond phonon absorptions in the spectra.The inset shows the fit of the measured reflectivity exemplary at 9 GPa used to extrapolate the data to ω = 0 eV (green shaded area) and to replace the data in the range of the strong phonon absorptions (blue shaded area).b Calculated real part of the optical conductivity.c Calculated dielectric permittivity.The inset displays the zero-crossing corresponding to the screened plasma frequency of the system with ω Fit parameters of the Drude and localization peaks.a-d Spectral weight, dc conductivity, scattering rate, and calculated plasma frequency, respectively, of the Drude component.e-h Spectral weight, scattering rate, backscattering rate, and peak position, respectively, from fits with the Fratini model.The gray shaded areas mark critical pressure regions corresponding to (i) the vanishing of CDW state at ∼ 2 GPa (ii) the disappearance of the first superconducting dome at around 9 GPa, and (iii) the re-emergence of superconductivity at approximately 12 GPa.i and j Pressure evolution of the localization peak.

Supplementary Figure 4 :
Band structure and band-resolved optical conductivity.a and c Calculated band structures at 11 and 18.5 GPa, respectively.b and d Band-resolved in-plane optical conductivity at 11 and 18.5 GPa, respectively.