S-Wave Superconductivity in Kagome Metal CsV₃Sb₅ Revealed by ^121/123Sb NQR and ⁵¹V NMR Measurements

Kagome lattice has been intensely studied in condensed matter physics due to its geometric frustration and nontrivial band topology. Depending on the electron filling, on-site repulsion U, and nearest-neighbor Coulomb interaction V, several possible states have been proposed as the ground state,^[1,2] such as quantum spin liquid,^[3–5] charge density wave (CDW),^[6] and superconductivity.^[7,8]

Recently, superconductivity was discovered in the kagome metal AV₃Sb₅ (A = K, Rb, Cs), which undergoes a CDW transition before the superconducting transition.^[9–12] Scanning tunneling microscopy (STM) observed a chiral charge order that breaks time-reversal symmetry^[13] and leads to the anomalous Hall effect in the absence of magnetic local moments.^[14–17] This chiral charge order is energetically preferred and tends to yield orbital current.^[18] Inelastic x-ray scattering and Raman scattering exclude strong electron-phonon coupling driven CDW,^[19] while optical spectroscopy supports that CDW is driven by nesting of Fermi surfaces.^[20] High pressure can suppress CDW transition and reveal a superconducting dome in the P–T phase diagram.^[21,22] Further increasing pressure, a new superconducting phase emerges.^[23–25]

STM studies observed possible Majorana modes^[26] and pair density wave,^[27] which points to unconventional superconductivity in the surface state. Signatures of spin-triplet pairing and an edge supercurrent have been observed in Nd/K_{1 – x} V₃Sb₅ devices.^[28] Whether the exotic surface superconducting state is intrinsic or comes from heterostructures, such as Bi₂Te₃/NbSe₂,^[29] needs to clarify the gap symmetry in the bulk state first. Thermal conductivity measurements on CsV₃Sb₅ down to 0.15 K showed a finite residual linear term, which suggests a nodal gap and points to unconventional superconductivity.^[25] On the other hand, magnetic penetration depth of CsV₃Sb₅ measured by tunneling diode oscillator showed a clear exponential behavior at low temperatures, which provides an evidence for nodeless superconductivity but does not rule out fully gapped unconventional superconductivity.^[30]

In this work, we report nuclear magnetic resonance (NMR) and nuclear quadrupole resonance (NQR) investigations on CsV₃Sb₅. The splitting of ⁵¹V spectra and ^121/123Sb spectra demonstrates that a commensurate CDW order forms at 94 K with a first-order transition and coexists with superconductivity. In the superconducting state, Knight shift decreases and a Hebel–Slichter coherence peak appears in the spin-lattice relaxation rate just below T_c, which provides evidences for s-wave superconductivity in CsV₃Sb₅.

Single crystals of CsV₃Sb₅ were synthesized using the self-flux method, as reported in Ref. [21]. Superconductivity with T_c = 2.5 K is confirmed by dc magnetization measured using a superconducting quantum interference device (SQUID). The NMR and NQR measurements were performed using a phase coherent spectrometer. The spectra were obtained by frequency step and sum method which sums the Fourier transformed spectra at a series of frequencies.^[31] The spin-lattice relaxation time T₁ was measured using a single saturation pulse.

Figure 1(a) shows the spectra of ⁵¹V with μ₀ H (= 8.65 T) ∥ c. ⁵¹V with I = 7/2 has seven NMR peaks at 150 K. Below T_CDW = 94 K, the magnitude of original peaks decreases gradually and two sets of new peaks arise, which indicates that the CDW transition is commensurate and there are two different sites of V atoms in the CDW state. The coexistence of two phases between 91 K and 94 K demonstrates that the CDW transition is of first-order due to a simultaneous superlattice transition.^[10] The quadrupole resonance frequencies of both ⁵¹V sites increase with decreasing temperature below 94 K, as shown in Fig. 1(b).

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Knight shift ⁵¹ K is deduced from the central peaks and is summarized in Fig. 1(c). Below T_CDW, two splitting peaks give two Knight shift values, one of which jumps up and the other jumps down. The corresponding ⁵¹(1/T₁ T) jumps to the opposite side or does not change at T_CDW, as shown in Fig. 1(d). We also measured the Knight shift and ⁵¹(1/T₁ T) with H ⊥ c, as shown in Fig. S1 of the Supplemental Material, which also shows an opposite trend at T_CDW. Therefore, the splitting of ⁵¹ K is not caused by different hyperfine couplings, but by the change of the orbital part of Knight shift.

⁵¹(1/T₁ T) in a Fermi liquid state is proportional to the square of the density of states (DOS) around the Fermi energy N(E_F). Just below T_CDW, ⁵¹(1/T₁ T) of the right peak jumps to 64% of the value above T_CDW, indicative of the decrease of N(E_F) by 20%. ⁵¹(1/T₁ T) of the left peak is smooth at T_CDW. The difference of ⁵¹(1/T₁ T) between two V sites manifests the modulation of DOS at different V positions, which is consistent with 2 × 2 CDW charge order.^{[27] 51}V has a very small quadrupole moment, Q = –5.2 × 10⁻³⁰ m². Therefore, the relaxation is governed by magnetic fluctuations and can reflect the change in the DOS, even though the electric field gradient (EFG) fluctuates strongly near T_CDW. On the contrary, ¹²¹Sb and ¹²³Sb have larger quadrupole moments, ¹²¹ Q = –54.3 × 10⁻³⁰ m² and ¹²³ Q = –69.2 × 10⁻³⁰ m², respectively, which can be used to detect the EFG fluctuations and perform NQR measurement.

In order to study the superconducting state, we must perform measurement at a magnetic field lower than the upper critical field, μ₀ H_c2 ∼ 0.5 T.^[25] Usually NQR experiments do not need applying magnetic field, however Knight shift cannot be measured at zero field. Here we measure ¹²¹ K by applying a perturbing field of 64 Oe that is larger than the lower critical field, μ₀ H_c1 ∼ 2 mT.^[32] With a perturbing field, the nuclear spin Hamiltonian is the sum of electric quadrupole interactions and the nuclear Zeeman energy^[33]

$$\begin{eqnarray}\begin{array}{ll} & {\mathcal H} ={ {\mathcal H} }_{Q}+{ {\mathcal H} }_{Z}\\ = & \frac{h{\nu }_{zz}}{6}\left[(3{I}_{z}^{2}-{I}^{2})+\frac{\eta ({I}_{+}^{2}+{I}_{-}^{2})}{2}\right]\\ & -\gamma \hslash (1+K){H}_{0}\left({I}_{z}\cos \theta +\frac{{I}_{+}{e}^{-i\phi }+{I}_{-}{e}^{i\phi }}{2}\sin \theta \right).\end{array}\end{eqnarray} \tag{ 1 }$$

Here ν_zz is the quadrupole resonance frequency along the principal axis (c-axis), ${\nu }_{zz}\equiv \frac{3{e}^{2}qQ}{2I(2I-1)}$, with eq = V_zz ; η is an asymmetry parameter of the EFG, $\eta =\frac{{V}_{xx}-{V}_{yy}}{{V}_{zz}}$. V_xx , V_yy , V_zz are the EFGs along the x, y, z directions, respectively; θ and ϕ are the polar and azimuthal angles between the direction of the applied field and the principal axis of EFG. When η is small enough to be negligible, the resonance frequency at a small perturbation field along c-axis is

$$\begin{eqnarray}&&{\nu }_{{\rm{NQR}}}({H}_{0})={\nu }_{{\rm{NQR}}}(0)\mp \gamma (1+K){H}_{0},{I}_{z}=\pm |{I}_{z}|.\end{eqnarray} \tag{ 2 }$$

The perturbing field eliminates the degeneration of ± I_z and results in a spectra splitting. When H₀ ⊥ c, the peak of $|\pm \frac{1}{2}\rangle \leftrightarrow |\pm \frac{3}{2}\rangle$ is split by the perturbing field, while other peaks are not affected.^[34]

$$\begin{eqnarray}\begin{array}{ll}{\nu }_{{\rm{NQR}}}({H}_{0})= & {\nu }_{{\rm{NQR}}}(0)\pm \frac{I+ \frac{1}{2}}{2}\gamma (1+K){H}_{0},\\ & |\pm \frac{1}{2}\rangle \leftrightarrow |\pm \frac{3}{2}\rangle,\\ {\nu }_{{\rm{NQR}}}({H}_{0})= & {\nu }_{{\rm{NQR}}}(0),\\ & |\pm m\rangle \leftrightarrow |\pm (m+1)\rangle,m\ge \frac{3}{2},\end{array}\end{eqnarray} \tag{ 3 }$$

where m is a half-integer.

There are two isotopes of Sb nuclei which are ¹²¹Sb with I = 5/2 and ¹²³Sb with I = 7/2. The NQR spectra should contain two ¹²¹Sb peaks and three ¹²³Sb peaks. Figures 2(a) and 2(b) show the spectra with a perturbing field of 64 Oe along c-axis, which eliminates the degeneration of ± I_z and results in a small spectral splitting. There are two different crystallographic sites for Sb in CsV₃Sb₅ with the atomic ratio of Sb1 : Sb2 = 1 : 4.^[9] The relative intensities of different sites are proportional to their atomic ratio,^[35] so Sb2 site has larger spectral intensity than Sb1 site. The spectra in Figs. 2(a) and 2(b) are of Sb2 atoms which encapsulate the kagome layer with a graphite-like network. Sb1 atoms located in the kagome plane with V atoms have weak spectra, which are presented in the Supplemental Material. The ν_zz and η deduced from the spectra above and below T_CDW are summarized in Table 1, where η is zero above T_CDW reflecting isotropy in the xy-plane. Below T_CDW, η of Sb2 changes to finite values, which indicate the breaking of in-plane symmetry and two unequal sites of Sb2 with the ratio of 1 : 2. The slightly decreasing ν_zz implies that the main change is not the EFG strength, but the EFG direction at Sb2 sites. On the other hand, the EFG at Sb1 site changes frequency obviously without asymmetry.

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T₁ was measured at $|- \frac{1}{2}\rangle \leftrightarrow |- \frac{3}{2}\rangle$ peak of ¹²¹Sb2 and $|- \frac{3}{2}\rangle \leftrightarrow |- \frac{5}{2}\rangle$ peak of ¹²³Sb2, respectively. After spectra splitting below T_CDW, T₁ was measured at the right-side peaks, marked by arrows in Fig. 2(b). The degeneration is eliminated by the magnetic field, so the NQR recovery curves are no longer applicable. The recovery curve used for $|- \frac{1}{2}\rangle \leftrightarrow |- \frac{3}{2}\rangle$ transition of ¹²¹Sb is

$$\begin{eqnarray}\begin{array}{ll}1-\frac{M(t)}{M(\infty )}= & \frac{1}{35}{e}^{- \frac{t}{{T}_{1}}}+\frac{3}{56}{e}^{- \frac{3t}{{T}_{1}}}+\frac{1}{40}{e}^{- \frac{6t}{{T}_{1}}}\\ & +\frac{25}{56}{e}^{- \frac{10t}{{T}_{1}}}+\frac{25}{56}{e}^{- \frac{15t}{{T}_{1}}}.\,\end{array}\end{eqnarray} \tag{ 4 }$$

The recovery curve used for $|- \frac{3}{2}\rangle \leftrightarrow |- \frac{5}{2}\rangle$ transition of ¹²³Sb is

$$\begin{eqnarray}\begin{array}{ll}1-\frac{M(t)}{M(\infty )}= & \frac{1}{84}{e}^{- \frac{t}{{T}_{1}}}+\frac{1}{21}{e}^{- \frac{3t}{{T}_{1}}}+\frac{1}{132}{e}^{- \frac{6t}{{T}_{1}}}+\frac{25}{308}{e}^{- \frac{10t}{{T}_{1}}}\\ & +\frac{100}{273}{e}^{- \frac{15t}{{T}_{1}}}+\frac{49}{132}{e}^{- \frac{21t}{{T}_{1}}}+\frac{49}{429}{e}^{- \frac{28t}{{T}_{1}}}.\end{array}\end{eqnarray} \tag{ 5 }$$

We use the same fitting function for both above and below T_CDW, since the η is small. Figure 2(c) shows the temperature dependence of 1/T₁ T of ^121/123Sb. Upon cooling, ^121/123(1/T₁ T) starts to increase a little and drops sharply below T_CDW for both ¹²¹Sb and ¹²³Sb. With further cooling, ^121/123(1/T₁ T) becomes constant from 70 K to T_c, which obeys the Korringa relation and suggests that the system is in a Fermi liquid state. ⁵¹(1/T₁ T) of ⁵¹V has shown that DOS loss is only 20%. Here the large drop implies that the EFG fluctuations play an important role above T_CDW and are suppressed after CDW transition. Moreover, ^121/123(1/T₁ T) decreases continuously without sudden jump like ⁵¹(1/T₁ T). This implies that the superlattice transition has strong effect on V atoms that locate in the kagome plane, but has very small effect on Sb2 atoms that locate outside the kagome plane. Therefore, ^121/123(1/T₁ T) of Sb2 shows a second-order CDW transition behavior.

In general, a spin-lattice relaxation occurs through magnetic and/or electric-quadrupole channels. The magnetic relaxation is related to the gyromagnetic ratio γ_n and gives ¹²³ T₁/¹²¹ T₁ = 3.4, while the electric-quadrupole relaxation is related to the quadruple moment Q and gives ¹²³ T₁/¹²¹ T₁ = 1.5.^[36] Above T_CDW, ¹²³ T₁/¹²¹ T₁ is smaller than 3, as shown in Fig. 2(d), which indicates that both magnetic fluctuations and EFG fluctuations are important. Below T_CDW, the ratio increases with decreasing temperature and approaches to 3.4 at zero temperature limit, which indicates that the magnetic process becomes dominant at low temperature. Therefore, the effect of EFG fluctuations in the superconducting state is negligible.

The Knight shift of Sb can be deduced from the spectra splitting. The field calibrated by a Gauss meter has an uncertainty, so the absolute Knight shift value is not reliable in an ultra-low field. The relative Knight shift Δ K = K – K₀, where K₀ is the average value above T_c, has very weak dependence on field. The Δ K values for all directions are almost unchanged above T_c and decrease below T_c, as shown in Fig. 3, which is consistent with spin-singlet pairing. The perturbing field is much larger than the lower critical field, so the drop below T_c is not from Meissner effect but intrinsic spin susceptibility.

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In an ultra-low field, ¹²¹ K measurements have large errors which preclude further analysis. On the other hand, spin-lattice relaxation rate can be measured accurately. ^121/123(1/T₁ T) shows a clear Hebel–Slichter coherence peak just below T_c and then rapidly decreases at low temperatures as shown in Fig. 4. It evidences that the superconducting gap is of s-wave symmetry. As for d- or p-wave, gap sign changes over the Fermi surface, which suppresses a coherence peak. In iron-based superconductors, s^±-wave also changes gap sign between different Fermi surfaces and suppresses a coherence peak.^[37] So far, the Hebel–Slichter coherence peak has only been observed in s-wave superconductors. Here the coherence peak is small, which may be due to residual EFG fluctuations^[38] and/or a strong electron-phonon coupling.^[39] In other s-wave superconductors, the coherence peaks of Sb are also very small, such as in YbSb₂ ^[40] and BaTi₂Sb₂O.^[36] A recent STM study has suggested the absence of sign-change in the superconducting order parameter.^[41] Magnetic penetration depth measured by tunneling diode oscillator has shown a nodeless behavior.^[30] These results are both consistent with the s-wave gap symmetry. There is also experimentally observed residual density of states in the superconducting state,^[25,41] which may be due to the competition between superconductivity and CDW. Another s-wave superconductor Ta₄Pd₃Te₁₆ also has a CDW order coexisting with superconductivity and shows residual density of states in the superconducting state.^[42]

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¹²¹(1/T₁ T) is measured at the split peak due to the CDW transition and shows sudden changes at both T_CDW and T_c, which indicates that the superconductivity coexists with the CDW state. The coexistence of superconductivity and topological charge order may stimulate exotic excitations in the surface state,^[26,43] which needs further investigations. Recently another superconducting phase at high pressure was found in CsV₃Sb₅.^[24,25] Whether the high-pressure superconducting phase is the same as that at ambient pressure needs further studies.

In conclusion, we have performed ⁵¹V NMR and ^121/123Sb NQR studies on CsV₃Sb₅. The variation of the NMR and NQR spectra below T_CDW can be understood by the occurrence of a first-order commensurate CDW transition. EFG fluctuations are suppressed and magnetic fluctuations become dominant at low temperature. In the superconducting state, ¹²¹ K decreases and ^121/123(1/T₁ T) shows a coherence peak just below T_c, which is the hallmark of the s-wave full gap superconductivity. The superconductivity coexists with the CDW state, which may lead to exotic surface state.

We thank S. K. Su, X. B. Zhou and B. Su for assistance in some of the measurements.