Abstract
The discovery of several electronic orders in kagome superconductors AV3Sb5 (A means K, Rb, Cs) provides a promising platform for exploring unprecedented emergent physics1,2,3,4,5,6,7,8,9. Under moderate pressure (<2.2 GPa), the triple-Q charge density wave (CDW) order is monotonically suppressed by pressure, while the superconductivity shows a two-dome-like behaviour, suggesting an unusual interplay between superconductivity and CDW order10,11. Given that time-reversal symmetry breaking and electronic nematicity have been revealed inside the triple-Q CDW phase8,9,12,13, understanding this CDW order and its interplay with superconductivity becomes one of the core questions in AV3Sb5 (refs. 3,5,6). Here, we report the evolution of CDW and superconductivity with pressure in CsV3Sb5 by 51V nuclear magnetic resonance measurements. An emergent CDW phase, ascribed to a possible stripe-like CDW order with a unidirectional 4a0 modulation, is observed between Pc1 ≅ 0.58 GPa and Pc2 ≅ 2.0 GPa, which explains the two-dome-like superconducting behaviour under pressure. Furthermore, the nuclear spin-lattice relaxation measurement reveals evidence for pressure-independent charge fluctuations above the CDW transition temperature and unconventional superconducting pairing above Pc2. Our results not only shed new light on the interplay of superconductivity and CDW, but also reveal new electronic correlation effects in kagome superconductors AV3Sb5.
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The data supporting the findings of this study are available from the corresponding author upon reasonable request. Source data for Figs. 1–4 and Extended Data Figs. 1–10 are provided with this paper.
Change history
10 January 2023
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Acknowledgements
We thank K. Jiang, J.F. He and Y.L. Wang for the valuable discussion. This work is supported by the National Natural Science Foundation of China (grant nos. 11888101, 12034004 and 12074364), the National Key R&D Program of the MOST of China (grant no. 2017YFA0303000), the Strategic Priority Research Program of Chinese Academy of Sciences (grant no. XDB25000000), the Anhui Initiative in Quantum Information Technologies (grant no. AHY160000) and the Innovation Program for Quantum Science and Technology (grant no. 2021ZD0302800).
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T.W. and X.C. conceived the experiments. L.Z., Z.Wu, L.N., M.S., K.S., D.S., J.L., D.Z., S.L., B.K., Y.Z., K.L. and T.W. performed NMR experiments. Y.Y. performed first-principles calculations. F.Y., D.S. and J.Y. grew the single crystals. L.Z., T.W. and X.C. interpreted the results. L.Z., Z.X., Z.Wang, T.W. and X.C. wrote the manuscript. All authors discussed the results and commented on the manuscript.
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Extended data figures and tables
Extended Data Fig. 1 Determination of Tconset, Tc1 and Tc2 from the resonant frequency of the NMR tank circuit.
Tconset is defined as the temperature where the average value deviates from zero. Tc1 and Tc2 are defined by the valley of the second derivation of –Δf/f. Tconset is labelled by a black arrow. Tc1 and Tc2 are labelled by red and blue curves, respectively.
Extended Data Fig. 2 Determination of Tnem under different pressures.
a–d, Temperature-dependent 1/T1T in the triple-Q CDW phase. The 1/T1T was measured on the low-frequency peak of the central transition lines (see Fig. 2a). The nematic transition leads to an additional drop in 1/T1T below Tnem54. With increasing pressure, Tnem is nearly unchanged up to 0.48 GPa. The coloured line is a guide for the eyes. The grey vertical line marks Tnem. e–h, The evolution of the full width at half maximum (FWHM) in the triple-Q CDW phase. The FWHM was also measured on the low-frequency peak of the central transition lines. Similar to 1/T1T, a sudden increase in FWHM is also revealed below Tnem9. The black dashed line is a guide for the eyes. 1/T1T data for 0 GPa come from ref. 33. One of FWHM data for 0 GPa comes from ref. 9. Except for data for 0.08 GPa is measured on sample B, and data for other pressures are measured on sample A.
Extended Data Fig. 3 Analysis of pressure-dependent NMR spectra with different CDW models.
a, Different V sites in a stripe-like CDW order with unidirectional 4a0 modulation. Different colored dots represent different V sites. b–d, Analysis of NMR spectra with stripe-like CDW order with a unidirectional 4a0 modulation. Black circles are experimental data points. Red lines are the sum of all fitting curves. The two colourful peaks with the same intensity in b and f represent two 51V sites in triple-Q CDW order. The five colorful peaks with an intensity ratio of 1:1:2:4:4 in c are the fitting result with stripe-like CDW order. The spectrum in d can be fitted by a combination of triple-Q CDW order and stripe-like CDW order. The fitting constraint for both CDW orders in d is the same as that in b and c. e, different V sites in a superimposed tri-hexagonal and star-of-David model. f–h, Analysis of NMR spectra with the superimposed tri-hexagonal and star-of-David model. The NMR spectrum for triple-Q CDW at ambient pressure is well explained by the tri-hexagonal (or star-of-David) model33. The star-of-David (or tri-hexagonal) modulation also splits the NMR spectrum into two peaks but the position should be different from that in tri-hexagonal modulation. Therefore, the superimposed tri-hexagonal and star-of-David model should exhibit two sets of double peaks in the NMR spectrum. Obviously, this kind of model cannot fit the NMR spectrum under 1.05 GPa in g. Moreover, the NMR spectrum in h cannot be fitted by a combination of two kinds of triple-Q CDW orders in f and g. All data come from sample B. Each table below the figure shows the fitting parameter and fitting constraint used
Extended Data Fig. 4 Pressure-dependent electronic structure and phonon spectra.
a–e, Pressure-dependent DFT calculation of electronic structures without CDW modulation. Both Fermi surface topology and van Hove singularities remain nearly unchanged below 3 GPa. f–j, Pressure-dependent DFT calculation of phonon spectra without CDW modulation. The pressure is the same as the electronic structures on each phonon spectrum. The imaginary frequency at M point disappears above 2 GPa, but the imaginary frequency at L point remains at 3 GPa and finally disappears at 5 GPa. This result suggests that the CDW transition is expected up to 3 GPa, which is inconsistent with the observation in the present NMR experiment.
Extended Data Fig. 5 Comparison of NQR measured 1/T1T at 0.08 GPa and 2.00 GPa.
a, Low-temperature 1/T1T at 0.08 GPa by 123Sb NQR measurement on Sb1 sites. The dashed line is the guide for the eyes; b, temperature-dependent 1/T1T at 2.00 GPa by 123Sb and 121Sb NQR measurements on Sb1 sites. The filled and open pentagons represent 1/T1T measured on 123Sb and 121Sb respectively. The longitudinal axis scale on the right-hand side belonging to 121Sb NQR is 3.41 times larger than that on the left-hand side belonging to 123Sb NQR. Usually, the relaxation rate for the T1 process in NQR has two kinds of relaxation channels. One is the magnetic relaxation channel and the other is the quadrupole relaxation channel. These two relaxation channels can be identified by checking the ratio between the relaxation rates at different isotopes. The magnetic relaxation channel requires \({T}_{1}^{-1}\propto {\gamma }_{n}^{2}\), where γn is the nuclear gyromagnetic ratio. The expected ratio between nuclear spin-lattice relaxation rate at 121Sb and 123Sb through the magnetic relaxation channel is expressed as \(\frac{{T}_{1M}^{-1}(121)}{{T}_{1M}^{-1}(123)}=\frac{{(10.189)}^{2}}{{(5.51756)}^{2}}=3.41\). The quadrupole relaxation channel requires \({T}_{1}^{-1}\propto 3(2I+3){Q}^{2}/[10(2I-1){I}^{2}]\), where Q is the quadrupole moment. Thus, the expected ratio between the nuclear spin-lattice relaxation rate at 121Sb and 123Sb through the quadrupole relaxation channel is \(\frac{{T}_{1Q}^{-1}(121)}{{T}_{1Q}^{-1}(123)}=1.5\) (ref. 66). We checked this ratio at two temperatures above Tc and found that it is very close to 3.41. Therefore, the magnetic relaxation channel dominates T1 in the NQR measurement.
Extended Data Fig. 6 Field dependence of the 51V NMR spectrum at 2 K.
a, Central lines of the 51V NMR spectrum in 5.1 T, 9.0 T and 11.9 T. The horizontal axis is the Knight shift. If the NMR spectrum is fully paramagnetic, the spectrum is unchanged under different applied magnetic fields. b, NMR frequency vs applied magnetic field. The NMR frequency is taken as the maximum intensity (marked by blue arrows in a). The dashed line is a linear fitting nearly across the zero point with an error bar of 0.0003 T. No significant evidence of a magnetic moment is expected here. All data points are collected in sample C.
Extended Data Fig. 7 Correlation between Tc and 1/T1T under pressure.
a, Pressure-dependent 1/T1T and Tc. 1/T1T under each pressure is calculated by the weighted average method that sums up 1/T1T measured on each peak with its spectrum weight. The spectrum weight is determined by the normalized integration area of 1/T1T as shown in Extended Data Fig. 9. All 1/T1T data were collected at 10 K above the maximum value of Tc. To avoid effect from the phase transition, only 1/T1T belonging to a pure phase is plotted here. All data points are collected in sample B. b, Plot of 1/T1T vs Tc under the same pressure in a. All data points can be separated into two groups. It is obvious that the data in different CDW phases (triple-Q CDW and stripe-like CDW) exhibit a similar linear behaviour, supporting a strong correlation between Tc and 1/T1T in the CDW phase. In the kagome phase, the data seem to follow a similar linear behaviour with the same slope but the disappearance of the CDW order leads to a constant shift in 1/T1T
Extended Data Fig. 8 Determination of Tstripe above Pc1.
a–e, The temperature-dependent central lines of 51V NMR under different pressures. The spectrum configuration continuously changes from 0.54 GPa to 0.76 GPa at 2 K, which reveals the first order quantum phase transition. Meanwhile, the transition temperature of the stripe-like CDW Tstirpe rapidly increases above 0.64 GPa and converges to TCDW in 0.76 GPa. f–j, The temperature-dependent renormalized intensity ratio of stripe-like CDW under different pressures. The renormalized intensity ratio is expressed as A1/(A1+A2), where A1 and A2 are the integral spectrum weights on the characteristic peaks belonging to triple-Q CDW and stripe-like CDW respectively. The characteristic peaks of triple-Q CDW and stripe-like CDW are labelled by blue and magenta bars, respectively. The spectral integration on both triple-Q CDW and stripe-like CDW is performed on the corresponding peak frequency with a width of 10 kHz. All data were collected from sample A
Extended Data Fig. 9 Integral area for nuclear spin-lattice relaxation measurement in different CDW states as shown in Fig. 3.
Integral area of 1/T1T analysis for a 0.08 GPa at 2.02 K; 0.67 GPa b at 44.8 K and c at 2.0 K; d 1.05 GPa at 2.0 K. Each colorful block on the left represents the frequency integral region for each value of 1/T1T on the right with the same color. Here, 1/T1T is the same as that in Fig. 3
Extended Data Fig. 10 Analysis of pressure-dependent phase volume.
a–b, Pressure dependence of the phase volume for sample A and sample B. The phase volume is determined through the weight of the characteristic peak belonging to different phases. The weight of the characteristic peak is calculated by integrating the intensity on part of the peak and then divided by the whole area of the spectrum. Finally, the phase volume is determined by renormalizing the pressure-dependent weight of each phase to unity (value from 0 to 1). Here, the phase volumes near ambient pressure and those above 2.00 GPa are manually defined as 0 or 1 since they are pure phases. The standard weight of the characteristic peak belonging to the pure stripe-like phase is estimated to be near 1.2 GPa for the two samples since the weight at approximately 1.2 GPa is nearly unchanged with pressure. The integration areas of each phase are shown in c and d. c–d, The integrating areas for the calculation of the phase volume. The red block is the integration area belonging to the 3Q CDW phase. The yellow block is the integration area belonging to stripe-like CDW
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Zheng, L., Wu, Z., Yang, Y. et al. Emergent charge order in pressurized kagome superconductor CsV3Sb5. Nature 611, 682–687 (2022). https://doi.org/10.1038/s41586-022-05351-3
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DOI: https://doi.org/10.1038/s41586-022-05351-3
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