Abstract
We report results of specific heat and muon spin relaxation (μSR) measurements on a polycrystalline sample of Pr3Cr10âxN11, which shows superconducting state below Tcâ=â5.25âK, a large upper critical field Hc2â~â20âT and a residual Sommerfeld coefficient γ0. The field dependence of γ0(H) resembles γ of the U-based superconductors UTe2 and URhGe at low temperatures. The temperature-dependent superfluid density measured by transverse-field μSR experiments is consistent with a p-wave pairing symmetry. ZF-μSR experiment suggests a time-reversal symmetry broken superconducting transition, and temperature-independent spin fluctuations at low temperatures are revealed by LF-μSR experiments. These results indicate that Pr3Cr10âxN11 is a candidate of p-wave superconductor which breaks time-reversal symmetry.
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Introduction
Spin-triplet superconductivity is rich in physics compared to conventional spin-singlet superconductivity due to the orbital and spin degrees of freedom, and it also has potential application in quantum computation1,2. Therefore, experimental verification of spin-triplet superconductivity has been a long-sought goal. However, intrinsic spin-triplet superconductors are rare. The candidates, such as Sr2RuO43,4,5,6,7,8,9,10,11,12 and UPt313,14,15,16,17, are still in controversy. Recently, unconventional superconductivity was discovered in UTe218,19. The exotic behaviors, including the coexistence of magnetic fluctuations and superconductivity, point nodes in the superconducting energy gap structure20,21,22,23, and time-reversal symmetry breaking inferred from observations of a spontaneous Kerr response in the superconducting state24,25, all point to an odd-parity, spin-triplet pairing superconducting state. The mechanism of spin-triplet pairing is much less understood than that of its counterpart spin-singlet pairing explained by the BCS theory. It is therefore urgent to discover more candidates with spin-triplet superconductivity.
Several Cr-based superconductors have been reported to show unconventional superconductivity. For Cr-based compounds, superconductivity was first discovered by applying external pressure in CrAs, which is on the verge of antiferromagnetic order26,27. Nuclear quadrupole resonance (NQR) measurements reveal that substantial magnetic fluctuations are present in CrAs, and the absence of coherence peak in relaxation rate below Tc indicates an unconventional pairing mechanism28. Further neutron scattering measurements of CrAs29,30 support a direct connection between magnetism and superconductivity. A2Cr3As3 (Aâ=âNa, K, Rb, and Cs)31,32,33 have also attracted much interest. The existence of nodes in the superconducting gap is evidenced by the transport and muon spin relaxation (μSR) measurements34,35. The presence of strong ferromagnetic spin fluctuations is revealed by 75As nuclear magnetic resonance (NMR) measurements36 and NQR37 measurements. Therefore, a possible p-wave superconducting state was suggested in A2Cr3As338,39.
Recently, the first Cr-based nitride superconductor Pr3Cr10âxN11 with Tcâ=â5.25âK was discovered40. The upper critical field Hc2(0) of Pr3Cr10âxN11 is ~12.6âT, which is much larger than the estimated Pauli paramagnetic pair-breaking magnetic field. The correlation between 3d electrons derived from specific heat data is ten times larger than that estimated by the electronic structure calculation40. The enhanced correlation may be induced by the quantum fluctuations41,42. However, the study of the superconducting pairing symmetry is still lacking.
We report detailed muon spin relaxation (μSR) and specific heat measurements of the polycrystalline sample of Pr3Cr10âxN11. Although the specific heat coefficient γâ=âCe/T in the superconducting state down to 0.5âK is best described by a full gap model, \({C}_{{{{\rm{e}}}}}/T\propto {e}^{-{\Delta }_{0}/({k}_{{{{\rm{B}}}}}T)}\), Î0/kBTc is only ~1.19(3). This is much smaller than the weak coupling limit BCS value 1.76. Intriguingly, a large value of γ(0) is discovered, and the field dependence of γ0(H) resembles that of the U-based ferromagnetic superconductors UTe2 and URhGe. It is worth mentioning that UTe2 does not have any long-range ferromagnetic order, and URhGe has a ferromagnetic phase. Thus, the âferromagnetic" here is a broad definition. The temperature dependence of superfluid density measured by transverse-field (TF) μSR down to 0.3âK is consistent with a p-wave pairing symmetry. Furthermore, the zero-field (ZF) μSR experiment reveals the spontaneous appearance of an internal magnetic field below Tc, indicating time-reversal symmetry breaking in the superconducting state. Meanwhile, the temperature-independent spin fluctuations at low temperatures are suggested by the longitudinal-field (LF) μSR experiments.
Results and discussion
Specific heat measurements
The specific heat coefficient C/T vs. T2 for Pr3Cr10âxN11 measured with different applied magnetic fields are shown in the inset of Fig. 1a. Sharp superconducting transitions can be seen. The field-independent normal state data are well fitted by C/Tâ=âγnâ+âβT2, yielding γnâ=â0.193(4) mJâgâ1âKâ2 and βâ=â1.61(3)âμJâgâ1âKâ4. With a rough estimation of xâ=â0.5 (see Supplementary Data of ref. 40), we have a large value of γn per mole Cr γnâ=â21.8(1)âmJâKâ2âmol-Crâ1. The large γn suggests strong correlations between electrons. Figure 1b shows Hc2(T) determined from specific heat measurements for Pr3Cr10âxN11. Hc2(0)â=â20âT or 31âT, extrapolated by the fits using an empirical formula40\({H}_{{{{\rm{c2}}}}}(T)={H}_{{{{\rm{c2}}}}}(0)(1-{(T/{T}_{{{{\rm{c}}}}})}^{2})\) or GL-model43, respectively, while the Pauli paramagnetic limit HPâ=â1.84âTc is only ~9.6âT. This suggests that the superconductivity of Pr3Cr10âxN11 is unlikely to have conventional s-wave pairing symmetry.
By subtracting the phonon contribution βT2, the temperature dependencies of Ce/(Tγn) at different applied magnetic fields are obtained and displayed in Fig. 1a. Interestingly, Ce/T in the superconducting states for different applied magnetic fields can still be described by a full gap model44,45\({C}_{{{{\rm{e}}}}}/T\propto {e}^{-{\Delta }_{0}/({k}_{{{{\rm{B}}}}}T)}\) with Î0â=â0.54 meV. We notice that Î0/kBTc is only ~1.19(3), which is much smaller than the weak coupling limit BCS value 1.76. It is worth mentioning that point nodes in the energy gap are easy to be erased by the mixing of electrons with different orbital angular momentum: lââ âl0 (l0â=â1 for p-wave pairing system, and the mixing electrons can come from any itinerant electrons with orbital angular momentum lââ â1)46. In this case, Ce/T is still exponential at low temperatures for a full-gap superconductor, but ÎCe/(γnTc) is smaller than the BCS value of 1.43 at Tc. This is consistent with our zero-field-specific heat measurement, which gives ÎCe/(γnTc)â=â1.0(1).
By applying magnetic fields, C/T shows an increase at low temperatures (shown by magenta solid dots in Fig. 1a). This can be due to a hyperfine-enhanced Pr3+ nuclear Schottky anomaly which was commonly reported in Pr systems47,48,49, and can not be deducted easily. By extrapolating the fitting curves of full-gap model to Tâ=â0âK with entropy balance \(\int\nolimits_{0}^{{T}_{{{{\rm{c}}}}}(H)}{C}_{{{{\rm{e}}}}}/TdT={\gamma }_{{{{\rm{n}}}}}{T}_{{{{\rm{c}}}}}(H)\) imposed, we obtain the field dependence of residual coefficient γ0(H) (Fig. 1c). The fitted gap value Î0 slightly decreases with increasing fields, suggesting a large critical magnetic field Hc2. In the clean limit, the zero temperature γ0 of an s-wave superconductor is γ0â~â0. Impurity phase would contribute nonzero γ0 in the dirty limit, but γ0(H) should be field-independent in this case. For d-wave or p-wave superconductivity, γ0ââ â0 due to the nodes in the superconducting gap, and γ0(H) changes with \(\sqrt{H}\) for line nodes50. However, γ0(H) of Pr3Cr10âxN11 is not consistent with any case mentioned above, but is well described by an exponential function \([{\gamma }_{0}(H)\left.\right)-{\gamma }_{0}(0)]/{\gamma }_{{{{\rm{n}}}}}={e}^{-{H}^{* }/H}\), as shown in Fig. 1c. Similar exponential dependence of low-temperature γ was also found in ferromagnetic superconductors UTe219 and URhGe51, which are considered to hold spin-triplet superconductivity. In these U-based single crystals, the enhanced Sommerfeld coefficient is due to the enhancement of ferromagnetic (FM) fluctuations when approaching the FM instability. The possible origin of the exponential-like increment γ0(H) in Pr3Cr10âxN11 could be the field-enhanced anisotropic gap structure52, or the field-enhanced unpaired electrons correlation18,53.
TF-μSR experiments
To further probe the superconducting gap symmetry of Pr3Cr10âxN11, TF-μSR experiments have been performed. Figure 2a shows the asymmetry spectra in an applied field of 50âmT at Tâ=â0.3âK (vortex state) and Tâ=â8âK (normal state). Faster damping is observed at base temperature compared to the normal state data. The data are fitted well by the following function form:
where the first and second terms correspond to muons that stop in the sample and silver sample holder, respectively. The silver background signal is temperature-independent, and A1 and A2 are also temperature-independent, even above Tc. The second moment of the composite field distribution in the sample can be calculated as54:
where
The relaxation rate induced by vortex lattice can be extracted by \({\sigma }_{{{{\rm{sc}}}}}={[{\sigma }^{2}-{\sigma }_{{{{\rm{nm}}}}}^{2}]}^{\frac{1}{2}}\), where Ïnm is originated from nuclear dipole moments, which is temperature-independent and determined at Tâ>âTc. The temperature dependence of Ïsc is plotted in Fig. 2b. Since the applied field μ0Haâ=â50mTââªâHc2, the penetration depth λL in vortex state has a relation with the second moment of the inner field distribution55,56: \({({\sigma }_{{{{\rm{sc}}}}}/{\gamma }_{\mu })}^{2}=\bigtriangleup {B}^{2}=0.00371\frac{{\Phi }_{0}^{2}}{{\lambda }_{{{{\rm{L}}}}}^{4}}\), where γμâ=â2ÏâÃâ135.5âMHzâTâ1 is the muon gyromagnetic ratio, Φ0â=â2.07âÃâ10â15âWb is the magnetic-flux quantum, and the magnetic penetration depth λL is related to the superfluid density ns and effective mass of electron m* by the London equation \(1/{\lambda }_{{{{\rm{L}}}}}^{2}=4\pi {n}_{{{{\rm{s}}}}}{e}^{2}/{m}^{* }{c}^{2}\). Considering the situation for the full-gap model or point nodes model, we have Ïsc(0âK) ~0.403âμsâ1. Then, we can derive λLâ=â516(3)ânm. The small difference in A(t) above and below Tc is attributed to the large penetration depth λL, which is consistent with the observation of large γ40. The normalized superfluid density ns(T)/ns(0)â=âÏsc(T)/Ïsc(0) is obtained and plotted in Fig. 2b. The temperature dependence of Ïsc(T)/Ïsc(0) can be fitted by following function57,58,59:
where
Î0 is the maximum superconducting energy gap value at Tâ=â0âK, f is the Fermi function, kB is the Boltzmann constant, g(Ï,âθ) describes the angular dependence of the gap, and Ï and θ are the angular coordinates in k-space. It is worth noting that Eq. (4) is applied for three-dimensional situation, since Pr3Cr10âxN11 has a typical three-dimensional cubic structure.
During the fitting procedure, a traditional s-wave superconductivity with g(Ï,âθ)â=â1 (s-wave, red points), a two-gap structure fitted with α model58 (sâ+âs-wave, black points), a unitary p-wave state kxâ±âiky with \(g(\varphi ,\theta )=| \sin \theta |\) (p-wave, cyan points), a p-wave state kx which conserve the time-reversal-symmetry (TRS) with \(g(\varphi ,\theta )=| \sin \theta \cos \varphi |\) (pTRS-wave, orange points), a nonunitary p-wave gap structure with \(g(\varphi ,\theta )={{{\rm{NU}}}}(\theta ,\varphi )=a| \sin \theta \cos \varphi +\sin \theta \sin \varphi | +(1-a)| \sin \theta \cos \varphi -\sin \theta \sin \varphi |\) (pNU-wave, dark-yellow points), and a \({d}_{{x}^{2}-{y}^{2}}\) gap structure were considered (d-wave, wine points). The fitting parameters are listed in Table 1. We find that the normalized superfluid density of Pr3Cr10âxN11 can be well described by both the s-wave model and p-wave model. However, for the s-wave model, \({\Delta }_{0}^{s}/{k}_{{{{\rm{B}}}}}{T}_{{{{\rm{c}}}}}=1.02(3)\), which is much smaller than the BCS value of 1.76. This is reflected in the very narrow plateau of low-temperature superfluid density as shown in Fig. 2b, which is not consistent with conventional isotropic superconductivity60,61. The two-gap model with sâ+âs gives a similar small gap value for the dominant gap, and an extremely small gap with large uncertainty. This suggests that the second gap is not necessary to better describe the data. As can be seen from Fig. 2b and Table 1, p-wave model with point nodes in the gap function not only describes the data well but also gives reasonable gap values. Another possibility is the anisotropic s-wave superconductivity, which would result in a smaller superconducting gap. However, it is worth mentioning that Eq. (2)54 is applicable to single-quanta vortex lattice, while the vortex lattice in p-wave superconductors can be very different62.
ZF- and LF-μSR experiments
ZF-μSR is a powerful method to detect the spontaneous but extremely small internal magnetic field in the superconducting state, which is evidence of TRS breaking63,64,65,66,67. Figure 3a shows representative μSR asymmetry spectra measured in ZF for Pr3Cr10âxN11, above and below Tc. A non-relaxing background signal originating from muons that miss the sample and stop in the silver sample holder has been subtracted. The absence of early-time oscillations or fast relaxation excludes the existence of strong static magnetism. However, the small additional relaxation below Tc can be seen.
The ZF asymmetry time spectra A(t) over the whole measured temperature range are best described by the exponentially damped Lorentzian KuboâToyabe (KT) function plus a background signal from the silver sample holder:
where As is the initial asymmetry for the sample signal, Ab is the background signal which is temperature-independent and originated from muons stopping in the sample holder. The expression in brackets is the Lorentzian KT function, which represents a Lorentzian distribution inner field. In general, the field distribution of internal fields such as nuclear dipole moments is Gaussian-like shape68. In Pr3Cr10âxN11, there are different muon-stopping sites inside the sample, as indicated by the TF-μSR experiments. The superposition of the independent Gaussian distributions due to different muon sites is approximated to be a Lorentzian distribution. The same origin for the Lorentzian nuclear field distribution has also been reported in La4Ni3O869 and Sr2Ir0.9Rh0.1O470. The damping rate λd is often interpreted as a dynamic relaxation rate usually originating from spin fluctuations. The temperature dependence of λd is shown in Fig. 3b. Within error bars, λd is constant around 0.22âμsâ1. Such spin fluctuations may be responsible for the enhanced electron correlations observed in the specific heat measurements41,42.
Figure 3c shows the temperature dependence of ÎZF with λd fixed at 0.22âμsâ1. Above Tc, ÎZF is temperature-independent, as expected for muon depolarization by quasi-static nuclear moments68. It is worth mentioning that the large value of În and Ïnm above Tc might be induced by hyperfine-enhanced Pr3+ nuclear moments71. Below Tc, ÎZF has a significant increase due to the onset of static local fields, which is evidence for broken TRS4,72. The increase is of electronic origin, so the electronic and random nuclear contributions (ÎSC and În, respectively) are uncorrelated and added in ÎZF(T)â=âÎSC(T)â+âÎn. In the inset of Fig. 3c, ÎSC(T) is plotted as a function of normalized temperature T/Tc. The data can be fitted with the approximate empirical expression73,74
assuming that ÎSC has the temperature dependence of a BCS-like order parameter. The values of the fitting parameters are listed in Table 2. The zero temperature ÎSC(0) is ~0.042(5)âμsâ1, corresponding to the enhanced field ÎSC(0)/γμâ=â0.49(5)âG. It is interesting to note that a similar magnitude of enhanced relaxation rate was reported in Sr2RuO43,75 and Lu5Rh6Sn1876. The coefficient bâ=â0.95(5) is much smaller than 1.74 for an isotropic BCS superconductor in the weak coupling limit. This indicates a relatively weak electronâphonon coupling strength77, therefore the large upper critical field Hc2 is not likely induced by the strong coupling effect. Alternatively, ÎSC is attributed to other mechanisms besides BCS theory.
It is noted that the static and dynamic field contributions to the muon spin relaxation rate are hard to disentangle experimentally at zero-field. For LF-μSR measurements, where the applied magnetic field HL is parallel to the initial muon spin polarization, and HL is much greater than static local fields at muon sites, the muon spin polarization is âdecoupledâ from the static local fields, and any remaining relaxation for high HL is dynamic in origin78. LF-μSR experiments were carried out in both the superconducting state at Tâ=â0.4âK and normal state at Tâ=â7.5âK. The asymmetry spectra are shown in Fig. 4a, b. The field dependence of A(t) shows characteristics of both decoupling and field-dependent dynamic relaxation: the small-amplitude oscillation with frequency ÏL at small fields is a feature of decoupling68, but decoupling alone would not account for the nonzero field dependence of the overall relaxation rate.
The data are well-fitted using the following equation
where \({{{{\rm{P}}}}}_{Z}^{{{{\rm{LF.LKT}}}}}({B}_{{{{\rm{ext}}}}},\Lambda )\) is the depolarization function for a Lorentz field distribution of static local fields in an applied uniform magnetic field Bext68, Î/γμ is the half-width at half-maximum of the field distribution, which is field-independent and was fixed at the value of ÎZF(T). The field dependence of dynamical rate λd in Fig. 4c are well described as \({\lambda }_{{{{\rm{d}}}}}({H}_{{{{\rm{L}}}}})=a{{H}_{{{{\rm{L}}}}}}^{-n}\), where aâ=â0.45(8), nâ=â0.54(5) for Tâ=â0.4âK, and aâ=â0.43(4), nâ=â0.53(1) for Tâ=â7.5âK. The power-law behaviors of λd(H) can be interpreted as a signature of low-energy spin dynamics79. The overlapping of λd(H) curves, i.e., same fitting parameters a and n within error bars, for Tâ=â0.4âK and Tâ=â7.5âK indicates that λd has the same field dependence at different temperatures and confirms the temperature-independent behavior of λd observed in ZF-μSR experiment. Thus the enhanced relaxation ÎZF below Tc is due to the static magnetism.
In summary, specific heat measurements reveal a large Hc2 value and a large γ(0) of Pr3Cr10âxN11, which is consistent with our former report40. A power-law behavior of the field dependence of γ0(H) resembles that of the U-based ferromagnetic superconductors UTe2 and URhGe. This needs to be confirmed by the measurements on higher-quality samples, since it was found that γ0(0) has sample dependence for UTe2 single crystals80. Although the temperature dependence of specific heat is described by a full-gap model \({C}_{{{{\rm{e}}}}}/T\propto {e}^{-{\Delta }_{0}/({k}_{{{{\rm{B}}}}}T)}\), a small value of Î0â=â0.54âmeV is obtained. This might be explained by either there is an anisotropic full gap, or the possibility that point nodes in the energy gap are easy to be erased by the mixing of electrons with different orbital angular momentum. Temperature dependence of superfluid density measured by TF-μSR measurements can be described by both unitary p-wave model: pxâ±âipy and nonunitary p-wave model: \(\hat{{{{\bf{x}}}}}{p}_{x}+i\hat{{{{\bf{y}}}}}{p}_{y}\), with gap values similar to the one obtained from specific heat measurements, anisotropic s-wave model can not be excluded either. ZF-μSR experiment suggests a TRSB superconducting transition, and temperature-independent spin fluctuations at low temperatures are revealed by LF-μSR experiments. Such spin fluctuations may be responsible for the enhanced electron correlations observed in the specific heat measurements41,42.
The results reveal that Pr3Cr10âxN11 is a candidate of p-wave superconductor which breaks time-reversal symmetry. However, only polycrystalline samples of Pr3Cr10âxN11 can be obtained so far. Higher-quality samples or single crystals are needed, and additional experimental probes are required to check whether superconductivity such as sâ+âis or sâ+âid is applicable.
Methods
Experimental methods
Polycrystalline samples of Pr3Cr10âxN11 were prepared by solid-state reaction, and the characterization data can be found in ref. 40. Low-field magnetization measurements suggest that magnetic impurities are negligible in the sample. Specific heat measurements down to Tâ=â2âK and magnetic field up to μ0Hâ=â9âT were performed in a Physical Property Measurement System using a standard thermal relaxation technique. μSR measurements were carried out on MuSR spectrometer at the ISIS Neutron and Muon Facility, Rutherford Appleton Laboratory, Chilton, UK, over the temperature range 0.27â20âK. About 1âg samples were glued to a silver holder covering a circle with 1 inch in diameter using dilute GE varnish. Stray fields at the sample position were canceled within 1âμT in three directions.
Data availability
All relevant data are available from the corresponding authors upon request. Part of the data is taken from refs. 19 and 51 with reuse permission. The μSR data were generated at ISIS (Neutron and Muon Facility, Rutherford Appleton Laboratory, UK). Derived data supporting the results of this study are available from the corresponding authors or beamline scientists. The Musrfit software package is available online free of charge at https://lmu.web.psi.ch/musrfit/user/html/setup-standard.html.
Code availability
The methods for the numerically generated points and lines reported in the plots of this paper are described in the main text. The actual codes used to produce the results reported in this paper are available from the corresponding author upon request.
References
Wilczek, F. Majorana returns. Nat. Phys. 5, 614â618 (2009).
Stern, A. Non-abelian states of matter. Nature 464, 187â193 (2010).
Luke, G. M. et al. Time-reversal symmetry-breaking superconductivity in Sr2RuO4. Nature 394, 558â561 (1998).
Mackenzie, A. P. & Maeno, Y. The superconductivity of Sr2RuO4 and the physics of spin-triplet pairing. Rev. Mod. Phys. 75, 657â712 (2003).
Nelson, K. D., Mao, Z. Q., Maeno, Y. & Liu, Y. Odd-parity superconductivity in Sr2RuO4. Science 306, 1151â1154 (2004).
Ishida, K. et al. Spin-triplet superconductivity in Sr2RuO4 identified by 17O Knight shift. Nature 396, 658â660 (1998).
Mackenzie, A. P., Scaffidi, T., Hicks, C. W. & Maeno, Y. Even odder after twenty-three years: the superconducting order parameter puzzle of Sr2RuO4. NPJ Quantum Mater. 2, 40 (2017).
Kim, B., Khmelevskyi, S., Mazin, I. I., Agterberg, D. F. & Franchini, C. Anisotropy of magnetic interactions and symmetry of the order parameter in unconventional superconductor Sr2RuO4. NPJ Quantum Mater. 2, 37 (2017).
Pustogow, A. et al. Constraints on the superconducting order parameter in Sr2RuO4 from oxygen-17 nuclear magnetic resonance. Nature 574, 72â75 (2019).
Suh, H. G. et al. Stabilizing even-parity chiral superconductivity in Sr2RuO4. Phys. Rev. Res. 2, 032023 (2020).
Ishida, K., Manago, M., Kinjo, K. & Maeno, Y. Reduction of the 17O Knight shift in the superconducting state and the heat-up effect by NMR pulses on Sr2RuO4. J. Phys. Soc. Jpn. 89, 034712 (2020).
Petsch, A. N. et al. Reduction of the spin susceptibility in the superconducting state of Sr2RuO4 observed by polarized neutron scattering. Phys. Rev. Lett. 125, 217004 (2020).
Joynt, R. & Taillefer, L. The superconducting phases of UPt3. Rev. Mod. Phys. 74, 235â294 (2002).
Tou, H. et al. Odd-parity superconductivity with parallel spin pairing in UPt3: evidence from 195Pt Knight shift study. Phys. Rev. Lett. 77, 1374â1377 (1996).
Fisher, R. A. et al. Specific heat of UPt3: evidence for unconventional superconductivity. Phys. Rev. Lett. 62, 1411â1414 (1989).
Schöttl, S. et al. Evidence for unconventional superconductivity in UPt3 from magnetic torque studies. Phys. Rev. B 62, 4124â4131 (2000).
Lambert, F., Akbari, A., Thalmeier, P. & Eremin, I. Surface state tunneling signatures in the two-component superconductor UPt3. Phys. Rev. Lett. 118, 087004 (2017).
Ran, S. et al. Nearly ferromagnetic spin-triplet superconductivity. Science 365, 684â687 (2019).
Aoki, D. et al. Unconventional superconductivity in UTe2. J. Phys. Condens. Matter 34, 243002 (2022).
Nakamine, G. et al. Superconducting properties of heavy fermion UTe2 revealed by 125Te-nuclear magnetic resonance. J. Phys. Soc. Jpn. 88, 113703 (2019).
Metz, T. et al. Point-node gap structure of the spin-triplet superconductor UTe2. Phys. Rev. B 100, 220504 (2019).
Kittaka, S. et al. Orientation of point nodes and nonunitary triplet pairing tuned by the easy-axis magnetization in UTe2. Phys. Rev. Res. 2, 032014 (2020).
Bae, S. et al. Anomalous normal fluid response in a chiral superconductor UTe2. Nat. Commun. 12, 2644 (2021).
Hayes, I. M. et al. Multicomponent superconducting order parameter in UTe2. Science 373, 797â801 (2021).
Wei, D. S. et al. Interplay between magnetism and superconductivity in UTe2. Phys. Rev. B 105, 024521 (2022).
Wu, W. et al. Superconductivity in the vicinity of antiferromagnetic order in CrAs. Nat. Commun. 5, 5508 (2014).
Kotegawa, H., Nakahara, S., Tou, H. & Sugawara, H. Superconductivity of 2.2 K under pressure in helimagnet CrAs. J. Phys. Soc. Jpn. 83, 093702 (2014).
Kotegawa, H. et al. Detection of an unconventional superconducting phase in the vicinity of the strong first-order magnetic transition in CrAs using 75As-nuclear quadrupole resonance. Phys. Rev. Lett. 114, 117002 (2015).
Keller, L. et al. Pressure dependence of the magnetic order in CrAs: a neutron diffraction investigation. Phys. Rev. B 91, 020409 (2015).
Matsuda, M. et al. Evolution of magnetic double helix and quantum criticality near a dome of superconductivity in CrAs. Phys. Rev. X 8, 031017 (2018).
Bao, J.-K. et al. Superconductivity in quasi-one-dimensional K2Cr3As3 with significant electron correlations. Phys. Rev. X 5, 011013 (2015).
Tang, Z.-T. et al. Superconductivity in quasi-one-dimensional Cs2Cr3As3 with large interchain distance. Sci. China Mater. 58, 16â20 (2015).
Tang, Z.-T. et al. Unconventional superconductivity in quasi-one-dimensional Rb2Cr3As3. Phys. Rev. B 91, 020506 (2015).
Shao, Y. T. et al. Evidence of line nodes in superconducting gap function in K2Cr3As3 from specific-heat measurements. EPL 123, 57001 (2018).
Adroja, D. T. et al. Superconducting ground state of quasi-one-dimensional K2Cr3As3 investigated using μSR measurements. Phys. Rev. B 92, 134505 (2015).
Zhi, H. Z., Imai, T., Ning, F. L., Bao, J.-K. & Cao, G.-H. Nmr investigation of the quasi-one-dimensional superconductor K2Cr3As3. Phys. Rev. Lett. 114, 147004 (2015).
Yang, J., Tang, Z. T., Cao, G. H. & Zheng, G.-Q. Ferromagnetic spin fluctuation and unconventional superconductivity in Rb2Cr3As3 revealed by 75As NMR and NQR. Phys. Rev. Lett. 115, 147002 (2015).
Luo, J. et al. Tuning the distance to a possible ferromagnetic quantum critical point in A2Cr3As3. Phys. Rev. Lett. 123, 047001 (2019).
Yang, J. et al. Spin-triplet superconductivity in K2Cr3As3. Sci. Adv. 7, eabl4432 (2021).
Wu, W. et al. Superconductivity in chromium nitrides Pr3Cr10âxN11 with strong electron correlations. Natl Sci. Rev. 7, 21â26 (2019).
Miyake, A. et al. Enhancement and discontinuity of effective mass through the first-order metamagnetic transition in UTe2. J. Phys. Soc. Jpn. 90, 103702 (2021).
Zou, Y. et al. Fermi liquid breakdown and evidence for superconductivity in YFe2Ge2. Phys. Status Solidi Rapid Res. Lett. 8, 928â930 (2014).
Zhu, X., Yang, H., Fang, L., Mu, G. & Wen, H.-H. Upper critical field, hall effect and magnetoresistance in the iron-based layered superconductor LaFeAsO0.9F0.1âδ. Supercond. Sci. Technol. 21, 105001 (2008).
Singh, D. et al. Time-reversal symmetry breaking in the noncentrosymmetric superconductor \({{{{\rm{Re}}}}}_{6}{{{\rm{Ti}}}}\). Phys. Rev. B 97, 100505 (2018).
Singh, D., Sajilesh, K. P., Marik, S., Hillier, A. D. & Singh, R. P. Superconducting and normal state properties of the noncentrosymmetric superconductor NbOs2 investigated by muon spin relaxation and rotation. Phys. Rev. B 99, 014516 (2019).
Leggett, A. J. A theoretical description of the new phases of liquid 3He. Rev. Mod. Phys. 47, 331â414 (1975).
Aoki, Y. et al. Thermodynamical study on the heavy-fermion superconductor PrOs4Sb12: evidence for field-induced phase transition. J. Phys. Soc. Jpn. 71, 2098â2101 (2002).
Pathak, A. K., Paudyal, D., Mudryk, Y., Gschneidner, K. A. & Pecharsky, V. K. Anomalous Schottky specific heat and structural distortion in ferromagnetic PrAl2. Phys. Rev. Lett. 110, 186405 (2013).
Hargreaves, T. et al. Low-temperature specific heat of the electron-doped superconductor Pr2-xCexCuO4-δ. Phys. C 303, 33â40 (1998).
Kopnin, N. B. & Volovik, G. E. Flux flow in d-wave superconductors: low temperature universality and scaling. Phys. Rev. Lett. 79, 1377â1380 (1997).
Hardy, F. et al. Transverse and longitudinal magnetic-field responses in the Ising ferromagnets URhGe, UCoGe, and UGe2. Phys. Rev. B 83, 195107 (2011).
Anand, V. K. et al. Specific heat and μSR study on the noncentrosymmetric superconductor LaRhSi3. Phys. Rev. B 83, 064522 (2011).
Jiao, L. et al. Chiral superconductivity in heavy-fermion metal UTe2. Nature 579, 523â527 (2020).
Khasanov, R. et al. Muon-spin-rotation measurements of the penetration depth in Li2Pd3B. Phys. Rev. B 73, 214528 (2006).
Brandt, E. H. Flux distribution and penetration depth measured by muon spin rotation in high-Tc superconductors. Phys. Rev. B 37, 2349â2352 (1988).
MacLaughlin, D. E. et al. Muon spin relaxation and isotropic pairing in superconducting PrOs4Sb12. Phys. Rev. Lett. 89, 157001 (2002).
Maisuradze, A. et al. Superfluid density and energy gap function of superconducting PrPt4Ge12. Phys. Rev. Lett. 103, 147002 (2009).
Khasanov, R. et al. Evidence of nodeless superconductivity in FeSe0.85 from a muon-spin-rotation study of the in-plane magnetic penetration depth. Phys. Rev. B 78, 220510 (2008).
Shan, L. et al. Distinct pairing symmetries in Nd1.85Ce0.15CuO4ây and La1.89Sr0.11CuO4 single crystals: evidence from comparative tunneling measurements. Phys. Rev. B 72, 144506 (2005).
Biswas, P. K. et al. Chiral singlet superconductivity in the weakly correlated metal LaPt3P. Nat. Commun. 12, 2504 (2021).
Iguchi, Y. et al. Microscopic imaging homogeneous and single phase superfluid density in UTe2. Phys. Rev. Lett. 130, 196003 (2023).
Garaud, J., Babaev, E., Bojesen, T. A. & Sudbø, A. Lattices of double-quanta vortices and chirality inversion in pxâ+âipy superconductors. Phys. Rev. B 94, 104509 (2016).
Aoki, Y. et al. Time-reversal symmetry-breaking superconductivity in heavy-fermion PrOs4Sb12 detected by muon-spin relaxation. Phys. Rev. Lett. 91, 067003 (2003).
Luke, G. M. et al. Muon spin relaxation in UPt3. Phys. Rev. Lett. 71, 1466â1469 (1993).
Hillier, A. D., Quintanilla, J. & Cywinski, R. Evidence for time-reversal symmetry breaking in the noncentrosymmetric superconductor LaNiC2. Phys. Rev. Lett. 102, 117007 (2009).
Biswas, P. K. et al. Evidence for superconductivity with broken time-reversal symmetry in locally noncentrosymmetric SrPtAs. Phys. Rev. B 87, 180503 (2013).
Hillier, A. D. et al. Muon spin spectroscopy. Nat. Rev. Methods Prim. 2, 4 (2022).
Hayano, R. S. et al. Zero-and low-field spin relaxation studied by positive muons. Phys. Rev. B 20, 850â859 (1979).
Bernal, O. O. et al. Charge-stripe order, antiferromagnetism, and spin dynamics in the cuprate-analog nickelate La4Ni3O8. Phys. Rev. B 100, 125142 (2019).
Tan, C. et al. Slow magnetic fluctuations and critical slowing down in Sr2Ir1âxRhxO4. Phys. Rev. B 101, 195108 (2020).
MacLaughlin, D. E. et al. Weak quasistatic magnetism in the frustrated kondo lattice Pr2Ir2O7. Phys. B: Condens. Matter 404, 667â670 (2009).
Sigrist, M. & Ueda, K. Phenomenological theory of unconventional superconductivity. Rev. Mod. Phys. 63, 239â311 (1991).
Zhang, J. et al. Broken time-reversal symmetry in superconducting Pr1âxLaxPt4Ge12. Phys. Rev. B 100, 024508 (2019).
Gross, F. et al. Anomalous temperature dependence of the magnetic field penetration depth in superconducting UBe13. Z. Phys. B: Condens. Matter 64, 175â188 (1986).
Hicks, C. W. et al. Limits on superconductivity-related magnetization in Sr2RuO4 and \({{{{{\rm{PrOs}}}}}_{4}{{{\rm{Sb}}}}}_{12}\) from scanning squid microscopy. Phys. Rev. B 81, 214501 (2010).
Bhattacharyya, A. et al. Broken time-reversal symmetry probed by muon spin relaxation in the caged type superconductor \({{{{\rm{Lu}}}}}_{5}{{{{\rm{Rh}}}}}_{6}{{{{\rm{Sn}}}}}_{18}\). Phys. Rev. B 91, 060503 (2015).
Coombes, J. M. & Carbotte, J. P. Dependence of 2Î0/kBTc on the shape of electron-phonon spectral density. J. Low. Temp. Phys. 63, 431â446 (1986).
Blundell, S. J. Spin-polarized muons in condensed matter physics. Contemp. Phys. 40, 175â192 (1999).
Keren, A., Mendels, P., Campbell, I. A. & Lord, J. Probing the spin-spin dynamical autocorrelation function in a spin glass above Tg via muon spin relaxation. Phys. Rev. Lett. 77, 1386 (1996).
Aoki, D. et al. First observation of the de Haas-van Alphen effect and fermi surfaces in the unconventional superconductor UTe2. J. Phys. Soc. Jpn. 91, 083704 (2022).
Acknowledgements
We thank Prof. Douglas E. MacLaughlin for the fruitful discussion. We are very grateful to the ISIS Cryogenics Group for valuable help during the μSR experiments (ISIS.RB 1910277). This research was supported by the National Key Research and Development Program of China, Nos. 2022YFA1402203 and 2021YFA1401800, the National Natural Science Foundations of China, Nos. 12174065 and 12134018, and the Shanghai Municipal Science and Technology Major Project Grant No. 2019SHZDZX01. J.C. acknowledges support by the Swiss National Science Foundation (Projects No. 200021-188564). C.S.C. acknowledges support by the China Scholarship Council.
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L.S. designed the experiments. W.W. and J.L.L. grew the samples. C.S.C., Z.H.Z., Y.X.Y., C.T., and A.D.H. performed the μSR experiments. Specific heat measurements were performed by C.S.C., Q.W., and M.Y.Z., C.S.C., and L.S. analyzed the data. All authors participated in the discussion. The manuscript was written by C.S.C., J.C., and L.S.
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Chen, C.S., Wu, Q., Zou, M.Y. et al. Unconventional superconductivity in Cr-based compound Pr3Cr10âxN11. npj Quantum Mater. 9, 22 (2024). https://doi.org/10.1038/s41535-024-00634-6
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DOI: https://doi.org/10.1038/s41535-024-00634-6