Introduction

Unconventional superconductivity always arises in the vicinity of another ordered electronic state, such as a magnetic order1, a nematic order2, or even a charge-density wave (CDW)3. In cuprate high-temperature superconductors, iron-pnictides or heavy-fermion compounds, carrier dopings or externally applied pressures can suppress the magnetic or nematic order4,5,6. Quantum critical points (QCPs) and associated fluctuations were often found around the ending point of these orders and considered by many a key to understanding the mechanism of unconventional superconductivity1,7,8,9,10. However, unconventional superconductivity around a CDW QCP is rarely observed11, and whether CDW fluctuations can also mediate the electron pairing is still a mystery.

Recently, a newly discovered quasi-two-dimensional superconductor AV3Sb5 (A = K, Cs, Rb) with kagome lattice has emerged as an excellent platform to study the interplay between topology, superconductivity, and CDW12. Angle-resolved photoemission spectroscopy combined with density-functional theory reveals a series of non-trivial electronic structures in this compound, including flat band, Dirac point, Van Hove singularity, and topological surface states13,14,15,16,17,18. Meanwhile, many exotic features, including chirality19,20,21,22,23, nematicity24, and time-reversal symmetry-breaking25,26, were found in the CDW state, which was proposed to be driven by electron correlations27,28. Especially the unusual phase diagram of the CDW order and superconductivity with applying hydrostatic pressures attracted a lot of attention29,30,31,32. The CDW transition can be gradually suppressed by applying hydrostatic pressure until Pc ~ 1.9 GPa, and the superconducting transition temperature Tc shows a non-monotonic double-dome-like phase diagram until its disappearance around 10 GPa31,32,33,34. Most remarkably, the maximum Tc is right at Pc, where no Hebel–Slichter coherence peak is seen below Tc in the superconducting state35. All these studies point to a possible QCP at Pc36,37, which makes CsV3Sb5 an ideal compound to study the relationship between unconventional superconductivity and CDW. Although the high-pressure nuclear magnetic resonance (NMR) experiments suggested that the CDW undergoes an evolution to a new phase with a possible stripe-like CDW order with a unidirectional 4a0 modulation in pressurized CsV3Sb535, information about pressure-dependent CDW fluctuations is still lacking, which is of much significance to clarify its interaction with superconducting symmetry. Besides the CDW fluctuations, spin fluctuations were also proposed to play an important role in the high-pressure superconducting phase36,38. But whether spin fluctuations exist or not and how they are affected by pressures are still unclear in the current stage.

Results and discussions

Commensurate-to-incommensurate transition of the CDW order

Figure 1 shows the crystal structure and the temperature dependence of AC susceptibility measured at various pressures by using an in situ NMR coil. The strong diamagnetic signal and the sharp superconducting transition are observed at P = 0.40 GPa and P ≥ 1.90 GPa, indicating the high quality of the sample. As in previous studies, the much broader superconducting transitions are observed at 0.84 GPa ≤ P ≤ 1.72 GPa31,32. The obtained pressure dependence of Tc is consistent with previous transport studies (see Supplementary Fig. 3).

Fig. 1: The crystal structure and AC susceptibility measurements.
figure 1

a The pristine crystal structure of CsV3Sb5 at ambient pressure. b The temperature dependence of the AC susceptibility measured by using an in situ NQR coil at various pressures from 0.40 to 2.43 GPa. Solid arrows represent the superconducting transition temperature Tc.

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There are two types of Sb sites in CsV3Sb5. Sb1 is located in the kagome plane surrounded by the vanadium hexagon, and Sb2 is located between the kagome plane and Cs layer as illustrated in Fig. 1a. Sb has two types of isotopes, 121Sb (I = 5/2) and 123Sb (I = 7/2). The quadrupole frequency νq is defined as \({\nu }_{q}=\frac{3{e}^{2}qQ}{2I(2I-1)h}\), where eq is the electric field gradient (EFG), and Q is the nuclear quadrupole moment. For 121Sb nucleus, the NQR spectrum should have two resonance peaks corresponding to ± 1/2 ↔ ± 3/2 and ± 3/2 ↔ ± 5/2 transitions. For 123Sb nucleus, the NQR spectrum should have three resonance peaks corresponding to ± 1/2 ↔ ± 3/2, ± 3/2 ↔ ± 5/2 and ± 5/2 ↔ ± 7/2 transitions. So a total of 10 lines should be observed in 121/123Sb-NQR spectrum for CsV3Sb5, which is indeed seen in previous NQR studies28,39. Figure 2 displays the temperature dependence of 121Sb-NQR spectra corresponding to ± 1/2 ↔ ± 3/2 transitions at various pressures. For all pressures, there is only one peak for both 121Sb1 and 121Sb2 above TCDW. For P < 1.9 GPa, a clear change of the Sb-NQR spectrum due to the CDW transition can be seen as observed at ambient pressure28,39, but TCDW gradually decreases with increasing pressure. The abrupt jump of the Sb1 line was observed until P = 1.23 GPa, indicating the CDW order is of the first order. But it is hard to determine the type of the CDW transition for P = 1.72 GPa, since the line is too broad in the CDW state (see Supplementary Fig. 5).

Fig. 2: Temperature-dependent NQR spectra.
figure 2

a–h represent the temperature dependence of 121Sb1 (blue color) and 121Sb2 (red color) NQR spectra at various pressures. The black dotted line indicates the temperature where the CDW phase transition occurs.

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Inside the CDW state, we further found that the line shape of 121Sb2-NQR spectra experienced a remarkable change with increasing pressure, as shown in Fig. 3. For P ≤ 0.4 GPa, a simple splitting of 121Sb2 lines was observed, indicating that the CDW order is still commensurate and the lattice distortion should be still star-of-David (SoD) pattern28. With increasing pressure, at P = 0.84 GPa, both Sb1 and Sb2-NQR lines start to broaden, and new lines emerge at low frequencies. Below P = 1.23 GPa, some sharp lines can still be seen between f = 72 MHz to 75 MHz, but only broad lines remain at P = 1.72 GPa.

Fig. 3: Pressure evolution of NQR spectra.
figure 3

The red peaks are 121Sb-NQR spectra in the CDW state at T ~ Tc from ambient pressure to 1.72 GPa. The blue peaks are the 121Sb-NQR spectra at T ~ Tc above P = 1.90 GPa. The solid lines are the guides to the eye.

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The observed broadening and emergence of new lines imply that the CDW modulation is totally different from the modulation at ambient pressure. Below we show that an incommensurate (IC) CDW order with superimposed SoD and TrH pattern stacking along the c-axis can consistently account for the observed results. Generally speaking, in a commensurate CDW state, the NQR line reflects the small number of physically non-equivalent nuclear sites in the unit cell so that the spectrum with discrete peaks was observed. In an incommensurate state, however, since the translational periodicity is lost, the number of non-equivalent nuclear sites becomes much larger and leads to a larger broadening40. A modulation due to CDW order will cause an additional term in the resonance frequency at the Sb site (x, y). In our model, we consider both one-dimensional (1D) and two-dimensional (2D) incommensurate modulations. In the 1D case, we assume that the charge modulation along one in-plane direction is incommensurate and introduce an additional cosine function as \(\cos (\frac{2\pi }{a}{q}_{x}\cdot x)\)40. In the 2D case, we assume that the incommensurate modulation is in-plane and introduce an additional term as \(\cos (\frac{2\pi }{a}{q}_{x}\cdot x)+\cos [\frac{2\pi }{b}({q}_{y}\cdot x\cdot \cos \beta +{q}_{y}\cdot y\cdot \sin \beta )]\)40. qx and qy are the wave vectors along a and b-axis, respectively. β is the angle between the two in-plane wave vectors qx and qy, which is π/3 for the kagome lattice studied in this work. For the 1D incommensurate case, we propose that the SoD and TrH patterns could be either superimposed, as illustrated in Fig. 4a41, or formed two different domains, as illustrated in Fig. 4b. For the 2D incommensurate case, we assume an additional charge modulation on top of the superimposed SoD and TrH pattern, either along the a-axis or c-axis, as illustrated in Fig. 4d, e. By only considering the structural distortion in the plane and convoluting with a Lorentz function (details about NQR spectra simulation are present in Supplementary Note 5), we can reproduce the spectra at P = 1.72 GPa for both 1D and 2D incommensurate modulation as shown in Fig. 4c, f, respectively.

Fig. 4: Simulation of 121Sb-NQR spectrum at P = 1.72 GPa.
figure 4

a and b display the two possible CDW patterns with a one-dimensional (1D) incommensurate (IC) modulation, in which a represents the superimposed SoD and TrH pattern stacking along the c-axis and b represents the coexistence of SoD and TrH domains, respectively. d and e illustrate the superimposed two-dimensional (2D) incommensurate SoD and TrH pattern with an additional charge modulation along the a-axis and c-axis, respectively. The gray level represents the charge density. c and f show the comparison of the 121Sb-NQR spectra at P = 1.72 GPa (gray circle), the simulated incommensurate spectra (black dotted line) and the calculated convolution (orange area for TrH pattern and blue area for SoD pattern) for 1D and 2D incommensurate CDW modulations (details about NQR spectra simulation are present in Supplementary Note 5), respectively. The peaks corresponding to Sb1 and Sb2 sites are marked by the red dashed arrows.

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However, for the 1D incommensurate modulation, the Sb1 NQR spectra should have two peaks of equal intensities at 74.5 and 75.4 MHz, respectively (see Fig. 4c), which is not observed at P = 0.84 GPa or 1.23 GPa (see Fig. 3). We note that a stripe CDW order was proposed by the previous 51V-NMR study35, which is similar to our assumption of the additional modulation along the a-axis. However, the incommensurability of CDW and the coexistence of SoD and TrH patterns were not caught by the 51V-NMR. This might be because Sb nuclei are sensitive to charge modulation from the Sb 5p-orbitals, which was suggested to be different from the CDW originated from the V 3d-orbitals42. In addition, 121/123Sb-NQR spectra were found to have a much larger response to the CDW order compared to the 51V-NMR spectra28,39. In any case, our results suggest that CDW modulation gradually changes from the commensurate CDW at ambient pressure to the incommensurate CDW with increasing pressure. However, the NMR line shape is independent of the value of the CDW wave vector q for incommensurate modulations. Moreover, the present experimental results do not rule out the possibility of more complex CDW patterns beyond the proposed structures in Fig. 4. To further resolve this issue, high-pressure X-ray scattering measurements at 1.72 GPa are needed in the future.

In the range of 0.84 GPa ≤ P ≤ 1.23 GPa, we found that the NQR spectra consist of both narrow and broad peaks (see Supplementary Fig. 7), indicating the coexistence of the commensurate and incommensurate CDWs. Then, there will be a large number of CDW domain walls between the commensurate and incommensurate CDWs in this pressure region. The enhanced interaction and scattering at the domain walls can strongly affect the superconductivity32,43, which is likely responsible for the inhomogeneous superconductivity, as we found (see Supplementary Fig. 8 for the comparison between the commensurate CDW volume fraction and the superconducting transition width).

A commensurate-to-incommensurate transition with increasing pressures, as we found, was recently proposed theoretically36, but a superimposed SoD and TrH pattern was not predicted. In CsV3Sb5, instead of electron–phonon coupling, electron correlations were suggested to be an important factor in forming the CDW order27. Most interestingly, the incommensurate modulation was also reported in Sn-doped CsV3Sb544. Then, one possible scenario is that the ordering wave vector connects parts of the Fermi surface or the hot spots. Also, with increasing pressure, due to the change of the Fermi surface, the ordering wave vector gradually becomes incommensurate. Such a scenario was proposed for the CDW order in cuprates, and the wave vector was found to have a monotonous doping dependence45. It would be interesting to measure the doping dependence of the wave vector by the high-pressure X-ray scattering.

Possible CDW quantum critical point

Next, we turn to the fluctuations above TCDW. By fitting the 121Sb2 and 123Sb2 spectra with the Lorentz function (see Supplementary Fig. 9 for 123Sb-NQR spectra of the Sb2 site), we deduced the linewidth δν121 and δν123 at various pressures as shown in Fig. 5a, b, respectively. Both δν121 and δν123 increase with decreasing temperature until T ~ TCDW, indicating the existence of the short-range CDW order due to CDW fluctuations pinned by quenched disorders, which was also observed in 2H-NbSe2 and underdoped cuprate YBa2Cu3Oy46,47,48. Our observation is consistent with the recent X-ray scattering and specific heat measurements at ambient pressure, which also show the existence of a short-range CDW order above TCDW49. Moreover, we find that the temperature dependence of δν also follows the Curie–Weiss behavior as observed in YBa2Cu3Oy50 and fits both δν121 and δν123 by the Curie–Weiss formula as,

$$\delta {\nu }^{121,123}(T)=\frac{{A}^{121,123}}{T-{\theta }^{121,123}}+{C}^{121,123}$$
(1)

where A represents the amplitude of the Curie–Weiss fit, and C is a constant. As shown in Fig. 5, both δν121 and δν123 are fitted very well, and the obtained 121θ and 123θ are plotted in the phase diagram (see red triangles in Fig. 6). Most surprisingly, we find that both 121θ and 123θ are very close to TCDW from the ambient pressure to P = 1.72 GPa, indicating a divergent behavior of δν. Therefore, our results suggest that the NQR line broadening approaching TCDW is related to the CDW fluctuations. There is no present theory giving the quantitative relationship between the CDW susceptibility and the NQR linewidth δν; however, in analogy with the magnetic and nematic quantum phase transitions7,8, we can take θ as an indicator of the QCP. If θ = 0, it means that the CDW susceptibility diverges at T → 0, indicating a CDW QCP. As shown in Fig. 6, both 121θ and 123θ are almost zero at Pc ~ 1.9 GPa, suggesting a CDW QCP at this pressure. We note that Tc reaches the maximum at Pc, implying the possible relationship between CDW fluctuations and the superconductivity. In order to make a firm conclusion, it will be important to make sure whether the CDW transition is of second order at P > 1.72 GPa, and whether the CDW QCP is beneath the superconducting dome9,51.

Fig. 5: The NQR linewidth.
figure 5

a and b show the full-width at half-maximum (FWHM) δν of 121/123Sb2-NQR spectra at various pressures. The solid lines are Curie–Weiss fits, and the obtained θ values are plotted in Fig. 6. Error bars are s.d. in the fits of the NQR spectra.

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Fig. 6: The obtained phase diagram of CsV3Sb5.
figure 6

The black square is the CDW transition temperature TCDW determined by the temperature dependence of Sb2-NQR spectral intensity (see Supplementary Fig. 4). The blue circle is the superconducting transition temperature Tc × 4 obtained in this work. The blue triangle is Tc × 4 taken from previous transport measurements31,32. The red triangle is the obtained 121/123θ from the Curie–Weiss fitting in Fig. 5. For P = 2.43 GPa, δν has a very weak temperature dependence, which leads to a large error bar ~ 100 K from the Curie–Weiss fitting. So we did not plot 121/123θ at P = 2.43 GPa in the phase diagram. Colors in the normal state represent the evolution of the 1/T1T of 123Sb2. Solid and dashed lines are guides to the eye. The error bar for TCDW represents the temperature interval in measuring the NQR spectra (see Supplementary Fig. 4).

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Spin fluctuations

Lastly, we tried to obtain more information about fluctuations by measuring the spin-lattice relaxation rate 1/T1 at both 121Sb2 and 123Sb2 sites at various pressures, as shown in Fig. 7. At all pressures, 1/T1T increases with decreasing temperature toward TCDW. To further show the evolution of 1/T1T, we make a contour plot in Fig. 6. 1/T1T is almost identical for P < 1.72 GPa, but starts to be enhanced from P = 1.9 GPa after the full suppression of the CDW order, which shows a totally different behavior comparing to the NQR line broadening (see Fig. 5). The nuclear spin-lattice relaxation rate 1/T1 is mainly composed of two contributions including magnetic interaction and quadrupole interaction. If the quadrupole relaxation process is predominant, the 1/T1 ratio between 121Sb and 123Sb is expected to be [121Q2(2 ⋅ 121I + 3)/121I2(2 ⋅ 121I − 1)]/[123Q2(2 ⋅ 123I + 3)/123I2(2 ⋅ 123I − 1)] = 1.4352, in which 121Q = −0.53 × 10−24 cm2 and 123Q = −0.68 × 10−24 cm2 are taken. If the magnetic relaxation process is predominant, the 1/T1 ratio between 121Sb and 123Sb is expected to be \({(^{121}\gamma {/}^{123}\gamma )}^{2}=3.41\), in which 121γ = 10.189 MHz/T and 123γ = 5.51756 MHz/T are taken. As shown in Fig. 8, the 1/T1 ratio 123T1/121T1 is close to 3.41 for all pressures, indicating that 1/T1T is mainly contributed by spin fluctuations. Therefore, our results suggest the existence of spin correlations in CsV3Sb5. With increasing pressure, the spin correlations are significantly enhanced after the complete suppression of CDW order. More interestingly, as reported by previous transport studies, Tc does not drop rapidly for P > 1.9 GPa (see Supplementary Fig. 10 for the complete phase diagram)31,32. Our results suggest that the superconductivity is sustained by the spin fluctuations at high pressures, which seems to be consistent with recent theoretical studies36,38. In passing, we also note that a new superconducting state arises above P ~ 15 GPa with the pressure further increasing33,34. Whether spin fluctuations still play a role for such a higher pressure phase needs high-pressure NMR measurements by using diamond anvils to clarify.

Fig. 7: The quantity 1/T1T of 121/123Sb at Sb2 cite.
figure 7

a and b are temperature-dependent 1/121T1T and 1/123T1T measured at the Sb2 site under various pressures. Solid lines are guides to the eye. Error bars are s.d. in the fits of the nuclear magnetization recovery curve.

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Fig. 8: Temperature dependence of the 1/T1 ratio 123T1/121T1.
figure 8

The horizontal dashed lines represent purely magnetic fluctuations (ratio = 3.41) and EFG fluctuations (ratio = 1.43), respectively. Error bars are s.d. in the fits of the nuclear magnetization recovery curve.

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In conclusion, we have presented the systematic 121/123Sb-NQR measurements on CsV3Sb5 under hydrostatic pressures. We found that the CDW structure gradually changes from a commensurate SoD pattern at ambient pressure to a superimposed incommensurate SoD and TrH pattern at P = 1.72 GPa. Above TCDW, we find that the linewidth of NQR spectra increases with decreasing temperature, indicating the existence of CDW fluctuations pinned by quenched disorders. The linewidth shows a Curie–Weiss temperature dependence and tends to diverge at Pc ~ 1.9 GPa, where Tc shows the maximum. Spin fluctuations are enhanced for P ⩾ Pc, which is probably responsible for the slow decrease of Tc at high pressures. Our results reveal the evolution of CDW structure and an emerged CDW QCP with increasing hydrostatic pressures, providing new insight into the superconducting pairing mechanism in CsV3Sb5.

Methods

Sample preparation and NQR measurement

Single-crystal CsV3Sb5 was synthesized by the self-flux method12. The typical size of the single crystal is around 3 × 2 × 0.1 mm. Several single crystals were mounted inside a piston-cylinder pressure cell made of CuBe alloy. To maintain consistency and ensure the number of quenched disorders remains unchanged, all measurements were conducted on the same single crystals.

The NQR measurements were performed with a phase-coherent pulsed NQR spectrometer. The 121/123Sb-NQR spectra were acquired by sweeping the frequency point by point and integrating the spin-echo signal. Since the EFG principal axis of 121/123Sb is along the c-axis28, we stack the CsV3Sb5 single-crystal flakes along the c direction to obtain a better NQR signal. The nuclear spin-lattice relaxation rate 1/T1 was measured by the saturation-recovery method. The 121T1 was obtained by fitting the nuclear magnetization M(t) with \(1-M(t)/M(0)=\frac{3}{28}\exp (-3t/{T}_{1})+\frac{25}{28}\exp (-10t/{T}_{1})\) and 123T1 was fitted by \(1-M(t)/M(0)=\frac{9}{97}\exp (-3t/{T}_{1})+\frac{16}{97}\exp (-10t/{T}_{1})+\frac{72}{97}\exp (-21t/{T}_{1})\), where M(0)and M(t) are the nuclear magnetization respectively at thermal equilibrium and time t after the comb pulse.

High-pressure NQR measurement

We used a commercial BeCu/NiCrAl clamp cell from C&T Factory Co., Ltd. (Japan) and Daphne oil 7373 as a transmitting medium53. When we applied pressure above 1.7 GPa, we heated the pressure cell up to 315 K to prevent the solidification of the pressure medium Daphne 737353. Although care has been taken, there is still a possibility that the pressure might be uniaxial at higher pressures, which could broaden the NQR lines at high temperatures, as shown in Fig. 5. For 0.4 GPa ≤ P ≤ 2.36 GPa, the applied pressure has been calibrated by the NQR frequency 63νQ of Cu2O54,55. The Cu2O powder and single-crystal CsV3Sb5 were placed together inside the NQR coil. There is a pressure deficit from room temperature to low temperature due to the solidification of Daphne oil 737353, so the pressure cell was pressurized at room temperature, and the NQR frequency of 63vQ was measured at T ~ 5 K (see Supplementary Fig. 1). The νq of Sb2 shows a linear pressure dependence (see Supplementary Fig. 2d). For P = 2.43 GPa, the applied pressure was obtained by the value of νq at T = 100 K.