Introduction

Discovered in 20191, the non-magnetic kagome metal series AV3Sb5 (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 CsV3Sb5, the CDW instability forms below TCDW = 94 K2 (102 K for RbV3Sb53 and 78 K for KV3Sb51), while superconductivity (SC) is observed below Tc = 2.5 K2 (0.92 K for RbV3Sb53 and 0.93 K for KV3Sb54).

Recently, several electrical transport, magnetic susceptibility, NMR, XRD, and μSR studies demonstrated the tunability of both orders in the AV3Sb5 series under external pressure5,6,7,8,9,10,11,12,13,14,15,16,17. At low pressures (p < 2 GPa), superconductivity in CsV3Sb5 exhibits a double-dome feature, comprising an enhancement of Tc to ~7 K at p1 ~ 0.6 GPa, and a second peak in Tc at p2 ~ 2 GPa with Tc ~ 8 K, whereas the superconducting gap evolves almost monotonically18. Simultaneously, TCDW is gradually suppressed and vanishes at p25,7,10. Experimental studies revealed a non-trivial evolution of CDW across p1, suggesting that changes in the nature of CDW may cause the double-dome behavior of superconductivity10,16,19. 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 GPa8,11,15. 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 pairs20,21,22. On the other hand, first-principle calculations revealed essential changes in electron-phonon (e-ph) coupling around the QCP, suggesting a conventional phonon-mediated mechanism23,24,25,26.

The CDW state in CsV3Sb5 is unique within the AV3Sb5 series as both the 2 × 2 × 2 and 2 × 2 × 4 orders comprising stacked 2 × 2 in-plane star-of-David and trihexagonal kagome superlattices are possible at ambient pressure27,28,29. Moreover, the experimentally observed CDW state features multiple anisotropic energy gaps30,31, large anomalous Hall effect32,33, intrinsic chirality33,34,35, and a temperature-driven re-arrangement of the order parameters including an electronic nematic phase below ~35 K16,36,37. Some studies discuss electron-phonon coupling as the driving force behind the CDW instability in AV3Sb524,38,39, whereas others point towards a more complex electronic origin40,41. It has been shown that band saddle points (van Hove singularities) close to the Fermi energy lead to a CDW instability in kagome metals42,43,44, making the pressure evolution of the saddle points a potentially important ingredient for understanding the phase transformations of CsV3Sb5 under pressure.

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 calculations11,23,24,25,45,46,47,48. The experimental confirmation of the proposed changes in the electronic band structure, however, remains lacking because the powerful ambient-pressure 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-hydrostaticity49,50. It has been shown that under such conditions, CsV3Sb5 transforms from hexagonal to monoclinic symmetry above ~10 GPa46 as sketched in Fig. 1a. The current reflectivity study has been performed with the quasi-hydrostatic CsI51 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 previously46, this monoclinic distortion affects the band structure only marginally.

Fig. 1: Pressure evolution of the crystal structure and Fermi surface.
figure 1

a Hexagonal (left) and monoclinic (right) crystal structures of CsV3Sb5 as determined by XRD measurements at ambient pressure1 and 14.2 GPa under non-hydrostatic conditions46, respectively, visualized by VESTA73. b, c Representative Fermi surfaces, at 0 and 20 GPa, respectively, created with FermiSurfer74. Hexagonal structures45 were used for the calculations. The color code represents the band velocity.

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Our results reveal a drastic change in the interband absorption around p2 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 px + py band and the shrinkage of the Fermi surface (FS) around the M-point. We further demonstrate that the spectral weight due to localized carriers is strongly suppressed around p2, 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 CsV3Sb5.

Results

Intraband contributions

The decomposed real part of the in-plane optical conductivity at selected pressure points is given in Fig. 2a–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 Fig. 2e, 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.

Fig. 2: Decomposition of optical conductivity and intraband spectral weight analysis.
figure 2

a–d 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 study52. e, f Pressure evolution of the Drude and localization peaks, respectively. g Calculated spectral weight of the two intraband contributions (Drude and localization peaks). 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 CsV3Sb5 as determined by several electrical transport, magnetic susceptibility, NMR, XRD, and μSR studies5,6,7,8,9,10,11,12,13,14,15,16,17. 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.

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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 peaks according to53

$${{{\rm{SW}}}}=\frac{1}{{\pi }^{2}{\varepsilon }_{{{{\rm{0}}}}}c}\int\nolimits_{0}^{{\omega }_{{{{\rm{c}}}}}}{\sigma }_{1}(\omega ){{{\rm{d}}}}\omega ,$$
(1)

with ε0 being the permittivity in vacuum and c the speed of light. The cut-off frequency is chosen as ωc = 50,000 cm−1 considering the high-energy tail of the localization peak. Figure 2g shows the pressure evolution of the Drude and localization peaks’ spectral weight, as well as 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 peaks is observed.

The plasma frequencies are obtained via ωp = \(\sqrt{{{{{\rm{SW}}}}}_{{{{\rm{Drude}}}}}+{{{{\rm{SW}}}}}_{{{{\rm{loc}}}}}}\) and normalized to the value at ambient pressure from a previous study52 (see Supplementary Note 3 for details). As displayed in Fig. 2h, 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 experiment and ab initio calculations over a wide pressure range suggests that electrons in CsV3Sb5 are almost uncorrelated, consistent with the previous assessment of the correlation strength at ambient pressure52. Hence, the main effect beyond the pure band picture is the damping of charge carriers manifested by the localization peak.

Interband transitions

Having discussed the intraband contributions in great detail, we now turn to analyzing the low-energy interband transitions. It has been shown that, compared to its sibling compounds, CsV3Sb5 possesses inverted band saddle points, leading to very different low-energy interband transitions52,54. At ambient pressure, optical spectroscopy has revealed the existence of characteristic interband absorptions below 0.06 eV (marked by the red arrow in Fig. 3c), which could be well reproduced by DFT52.

Fig. 3: Pressure evolution of the band structure and interband transitions.
figure 3

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.

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Upon applying pressure, the experimental data show a sudden reduction of the low-energy transitions at around 0.03 eV, already at the lowest measured pressure of 1.7 GPa as depicted in Fig. 3d. This trend is corroborated by the calculated optical conductivity plotted in Fig. 3g. Note that the difference in intensity between theoretical and experimental interband absorptions is due to the absence of electron scattering in the zero-temperature calculations. Band-resolved optical conductivity calculations displayed in Fig. 3b for 4.9 GPa, where the spectral weight below 0.06 eV is almost suppressed, reveal that these low-energy absorption arises from transitions between bands B and C according to the labeling in Fig. 3a. A closer look at the knot of bands that cross the Fermi level along Γ−M shown in Fig. 4a suggests that this part of the band structure changes significantly already at low pressures. With the Sb2 px + py band becoming steeper and pushing vanadium bands above the Fermi level, several bands crossing the Fermi level disappear, and parts of the Fermi surface shrink around 3 GPa, illustrated in Fig. 4c. This effect is even better visible away from the Γ–M line (Fig. 4b), where interband transitions at 0.03–0.06 eV are clearly suppressed by pressure, and only interband transitions at ω > 0.1 eV remain.

Fig. 4: Low-pressure band structures and pressure evolution of the Fermi surface.
figure 4

a Band structure along Γ–M at low pressures. b Band structure away from the Γ–M line shows the suppression of the low-energy interband transitions highlighted by black arrows. c Pressure evolution of the Fermi surface at kz = 0, illustrated using FermiSurfer74 with the band velocity as color scale.

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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 Fig. 3d–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. 3h, further supports the use of the uncorrelated band picture for the description of CsV3Sb5 over a broad pressure range.

Moreover, due to the increasing slope of the Sb2 px + py 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. 3g).

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 AV3Sb5 series52,54,55. On the other hand, the localization peak, a common feature among several kagome metal compounds52,54,55,56,57, can not be explained within the simple band picture. Free charge carriers interacting with low-energy 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 peak58,59. 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 suppressed52. Moreover, optical studies of KV3Sb5 and RbV3Sb5 find strong phonon anomalies, which are associated with the phonon modes coupling to the electronic background54,55. In the magnetic rare-earth kagome metal series RMn6Sn6, a close link between the behavior of phonon modes and the localization peak was also observed56.

It is then natural to use the spectral weight of the localization peak as an experimental gauge of the electron-phonon coupling. Our data suggest that this coupling should be strongly suppressed around 2 GPa, where the CDW disappears, and Tc starts decreasing. 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 picture24 and reveal a strong coupling of the V-V bond-stretching and V-Sb bond-bending phonon modes to the V 3\({d}_{xy,{x}^{2}-{y}^{2},{z}^{2}}\), V 3dxz,yz, and Sb1 5pz bands26. 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. 1b, c. Here, Sb2 pz 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. 2i. Given the unusual non-monotonic behavior of Tc under pressure, several other scenarios influencing superconductivity in CsV3Sb5 can be excluded. Due to only weak changes in the plasma frequency (Fig. 2h), as well as a very different pressure evolution of the electronic density of states at EF (see Supplementary Fig. 3d), the behavior of Tc cannot be explained by modifications in the electronic structure. Moreover, the continuous evolution of the crystal structure45,46,48, as well as of phonons according to previous DFT calculations23,24, 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 Tc.

Apart from high-pressure investigations, CsV3Sb5 was subject to various doping studies60,61,62,63,64,65,66. We would like to point out that while in some cases, for instance, through partial substitution of V-atoms 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 Tc, 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 px + py 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 points62,66. 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 low-energy absorption peak at around 400 cm−1 shifts to lower energies upon Nb-substitution65.

Our study revealed that the reduction in the e-ph coupling around 2 GPa is likely the main reason for the CDW suppression in CsV3Sb5. 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 px + py band, which was so far almost disregarded in the context of the CsV3Sb5 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 CsV3Sb5 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 150–10,000 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 \(\tilde{\sigma }(\omega )\), or the dielectric permittivity \(\tilde{\varepsilon }(\omega )\), 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 code68,69 and cross-checks have been conducted with the FPLO code70. In all cases, the Perdew-Burke-Ernzerhof flavor of the exchange-correlation potential71 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 were used. Considering that the monoclinic distortion does not have a fundamental impact on the band structure46, we used hexagonal structure throughout the pressure range for a simpler comparison. Optical conductivity was calculated using the optic module72 on the dense k-mesh with up to 100 × 100 × 50 points. Spin-orbit coupling was included in all the calculations.