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 (A = Na, 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 Pr3Cr10−xN11 with Tc = 5.25 K was discovered40. The upper critical field Hc2(0) of Pr3Cr10−xN11 is ~12.6 T, 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 Pr3Cr10−xN11. Although the specific heat coefficient γ = Ce/T in the superconducting state down to 0.5 K is best described by a full gap model, \({C}_{{{{\rm{e}}}}}/T\propto {e}^{-{\Delta }_{0}/({k}_{{{{\rm{B}}}}}T)}\), Δ0/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 “ferromagnetic" here is a broad definition. The temperature dependence of superfluid density measured by transverse-field (TF) μSR down to 0.3 K 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 Pr3Cr10−xN11 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/T = γn + βT2, yielding γn = 0.193(4) mJ g−1 K−2 and β = 1.61(3) μJ g−1 K−4. With a rough estimation of x = 0.5 (see Supplementary Data of ref. 40), we have a large value of γn per mole Cr γn = 21.8(1) mJ K−2 mol-Cr−1. The large γn suggests strong correlations between electrons. Figure 1b shows Hc2(T) determined from specific heat measurements for Pr3Cr10−xN11. Hc2(0) = 20 T or 31 T, 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 HP = 1.84 Tc is only ~9.6 T. This suggests that the superconductivity of Pr3Cr10−xN11 is unlikely to have conventional s-wave pairing symmetry.

Fig. 1: Specific heat data of Pr3Cr10−xN11 under different magnetic fields.
figure 1

a Temperature dependence of electronic specific heat coefficient Ce/(γnT) with different applied magnetic fields are presented in different colors: μ0H = 0 T (black), μ0H = 1 T (red), μ0H = 3 T (blue), μ0H = 5 T (magenta), μ0H = 7 T (wine), μ0H = 9 T (orange). The solid lines are the fits with full-gap-model. The dashed lines indicate Tc for different fields, determined by the middle of the jump. It is worth mentioning that the magenta full-solid dots were measured on an additional sample. To show the difference, the data were named as [5] T. Inset: C/T vs. T2 for different applied fields. The solid line is the fit of the normal state data. b Temperature dependence of Hc2 determined from specific heat measurements are shown in orange. Solid lines are the fits of WHH and GL-model (see the text). Dashed line: the Pauli paramagnetic limit. The error bars represent the estimated uncertainty for Tc under corresponding magnetic fields. c–e Sommerfeld coefficient γ(H) of Pr3Cr10−xN11 (indicated by purple dots), UTe2 (indicated by dark-yellow dots)19, and URhGe (indicated by olive dots)51. Solid lines: fits of \({\gamma }_{0}(H)/{\gamma }_{0}(0) \sim {e}^{-{H}^{* }/H}\).

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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 Δ0 = 0.54 meV. We notice that Δ0/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: l ≠ l0 (l0 = 1 for p-wave pairing system, and the mixing electrons can come from any itinerant electrons with orbital angular momentum l ≠ 1)46. In this case, Ce/T is still exponential at low temperatures for a full-gap superconductor, but ΔCe/(γnTc) is smaller than the BCS value of 1.43 at Tc. This is consistent with our zero-field-specific heat measurement, which gives ΔCe/(γnTc) = 1.0(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 T = 0 K 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 Δ0 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 γ0 ~ 0. 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, γ0 ≠ 0 due to the nodes in the superconducting gap, and γ0(H) changes with \(\sqrt{H}\) for line nodes50. However, γ0(H) of Pr3Cr10−xN11 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 Pr3Cr10−xN11 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 Pr3Cr10−xN11, TF-μSR experiments have been performed. Figure 2a shows the asymmetry spectra in an applied field of 50 mT at T = 0.3 K (vortex state) and T = 8 K (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:

$$\begin{array}{ll}A(t)\,=\,\mathop{\sum }\limits_{i=1}^{2}{A}_{i}{e}^{-\frac{1}{2}{\sigma }_{i}^{2}{t}^{2}}\cos ({\gamma }_{\mu }{B}_{i}t+\varphi )\\\qquad\quad+\,{A}_{{{{\rm{b}}}}}\cos ({\gamma }_{\mu }{B}_{{{{\rm{b}}}}}t+{\varphi }_{{{{\rm{b}}}}}),\end{array}$$
(1)

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:

$${(\sigma /{\gamma }_{\mu })}^{2}=\mathop{\sum }\limits_{i=1}^{2}\frac{{A}_{i}}{{A}_{1}+{A}_{2}}[{({\sigma }_{i}/{\gamma }_{\mu })}^{2}-{[{B}_{i}-\langle B\rangle ]}^{2}],$$
(2)

where

$$\langle B\rangle =\mathop{\sum }\limits_{i=1}^{2}\frac{{A}_{i}{B}_{i}}{{A}_{1}+{A}_{2}}.$$
(3)

The relaxation rate induced by vortex lattice can be extracted by \({\sigma }_{{{{\rm{sc}}}}}={[{\sigma }^{2}-{\sigma }_{{{{\rm{nm}}}}}^{2}]}^{\frac{1}{2}}\), where σnm is originated from nuclear dipole moments, which is temperature-independent and determined at T > Tc. The temperature dependence of σsc is plotted in Fig. 2b. Since the applied field μ0Ha = 50mT ≪ Hc2, 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 γμ = 2π × 135.5 MHz T−1 is the muon gyromagnetic ratio, Φ0 = 2.07 × 10−15 Wb 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 σsc(0 K) ~0.403 μs−1. Then, we can derive λL = 516(3) nm. 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) = σsc(T)/σsc(0) is obtained and plotted in Fig. 2b. The temperature dependence of σsc(T)/σsc(0) can be fitted by following function57,58,59:

$$\begin{array}{ll}\quad\frac{{\sigma }_{{{{\rm{sc}}}}}(T)}{{\sigma }_{{{{\rm{sc}}}}}(0)}\\=1+\frac{1}{2\pi }\int\nolimits_{0}^{2\pi }\int\nolimits_{0}^{\pi }\int\nolimits_{\Delta (T,\varphi ,\theta )}^{\infty }\left(\frac{\partial f}{\partial E}\right)\frac{E\sin \theta dEd\varphi d\theta }{\sqrt{{E}^{2}-\Delta {(T,\varphi ,\theta )}^{2}}},\end{array}$$
(4)

where

$$f={[1+\exp (E/{k}_{{{{\rm{B}}}}}T)]}^{-1},$$
(5a)
$$\Delta (T,\varphi ,\theta )={\Delta }_{0}\delta \left(\frac{T}{{T}_{{{{\rm{c}}}}}}\right)g(\varphi ,\theta ),$$
(5b)
$$\delta \left(\frac{T}{{T}_{{{{\rm{c}}}}}}\right)=\tanh \left\{1.82{\left[1.018\left(\frac{{T}_{{{{\rm{c}}}}}}{T}-1\right)\right]}^{0.51}\right\}.$$
(5c)

Δ0 is the maximum superconducting energy gap value at T = 0 K, f is the Fermi function, kB is the Boltzmann constant, g(φ, θ) describes the angular dependence of the gap, and φ and θ are the angular coordinates in k-space. It is worth noting that Eq. (4) is applied for three-dimensional situation, since Pr3Cr10−xN11 has a typical three-dimensional cubic structure.

Fig. 2: TF-μSR asymmetry spectra and data analysis with various models for Pr3Cr10−xN11.
figure 2

a Representative TF-μSR asymmetry spectra for Pr3Cr10−xN11 in the normal (circles) and superconducting (triangles) states with an external magnetic field of μ0Hext = 50 mT. Solid curves are the fits to the data with Eq. (1). For clarity, the spectra are shown in a rotating reference frame corresponding to a field of 45 mT. a Error bars indicate the statistical error. b Temperature dependence of muon spin relaxation rate σSC, the lines are the fits to Eq. (4) with five different models, and the fitting parameters are listed in Table 1. Inset: Temperature dependence of muon spin relaxation rate σ determined by Eq. (2). b Error bars are obtained from the fitted equations described in the main text.

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During the fitting procedure, a traditional s-wave superconductivity with g(φ, θ) = 1 (s-wave, red points), a two-gap structure fitted with α model58 (s + s-wave, black points), a unitary p-wave state kx ± iky 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 Pr3Cr10−xN11 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 s + s 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.

Table 1 Fitting parameters for different gap symmetry models
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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 Pr3Cr10−xN11, 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.

Fig. 3: Evidence for time-reversal symmetry breaking by ZF-μSR data.
figure 3

a Representative ZF asymmetry spectra for Pr3Cr10−xN11, measured above and below Tc (triangles and circles, respectively). A constant background signal has been subtracted from the data. Solid curves: fits using the damped Lorentzian KT function Eq. (6). a Error bars indicate the statistical error. b Temperature dependence of the exponentially damping rate λd. The dashed line is the guide of the eye. c Temperature dependence of ZF KT static Lorentzian relaxation rate ΛZF with λd is fixed at 0.22 μs−1. Inset: temperature dependence of electronic relaxation rate ΛSC (see text). Blue dashed line: fit using Eq. (7). b, c Error bars are obtained from the fitted equation Eq. (6).

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The ZF asymmetry time spectra A(t) over the whole measured temperature range are best described by the exponentially damped Lorentzian Kubo–Toyabe (KT) function plus a background signal from the silver sample holder:

$$A(t)={A}_{{{{\rm{s}}}}}\left[\frac{1}{{{{\rm{3}}}}}+\frac{{{{\rm{2}}}}}{{{{\rm{3}}}}}\left(1-{\Lambda }_{{{{\rm{ZF}}}}}t\right){e}^{-{\Lambda }_{{{{\rm{ZF}}}}}t}\right]{e}^{-{\lambda }_{{{{\rm{d}}}}}t}+{A}_{{{{\rm{b}}}}},$$
(6)

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 Pr3Cr10−xN11, 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 0.22 μs−1. 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 ΛZF with λd fixed at 0.22 μs−1. Above Tc, ΛZF is temperature-independent, as expected for muon depolarization by quasi-static nuclear moments68. It is worth mentioning that the large value of Λn and σnm above Tc might be induced by hyperfine-enhanced Pr3+ nuclear moments71. Below Tc, ΛZF 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 (ΛSC and Λn, respectively) are uncorrelated and added in ΛZF(T) = ΛSC(T) + Λn. In the inset of Fig. 3c, ΛSC(T) is plotted as a function of normalized temperature T/Tc. The data can be fitted with the approximate empirical expression73,74

$${\Lambda }_{{{{\rm{SC}}}}}(T)={\Lambda }_{{{{\rm{SC}}}}}(0)\tanh \left[b\sqrt{\frac{{T}_{{{{\rm{c}}}}}}{T}}-1\right],$$
(7)

assuming that ΛSC has the temperature dependence of a BCS-like order parameter. The values of the fitting parameters are listed in Table 2. The zero temperature ΛSC(0) is ~0.042(5) μs−1, corresponding to the enhanced field ΛSC(0)/γμ = 0.49(5) G. It is interesting to note that a similar magnitude of enhanced relaxation rate was reported in Sr2RuO43,75 and Lu5Rh6Sn1876. The coefficient b = 0.95(5) is much smaller than 1.74 for an isotropic BCS superconductor in the weak coupling limit. This indicates a relatively weak electron–phonon coupling strength77, therefore the large upper critical field Hc2 is not likely induced by the strong coupling effect. Alternatively, ΛSC is attributed to other mechanisms besides BCS theory.

Table 2 Parameters from fit of Eq. (7)
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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 “decoupled” 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 T = 0.4 K and normal state at T = 7.5 K. 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 ωL at small fields is a feature of decoupling68, but decoupling alone would not account for the nonzero field dependence of the overall relaxation rate.

Fig. 4: Temperature-independent fluctuations indicated by LF-μSR data.
figure 4

a, b LF asymmetry spectra for Pr3Cr10−xN11 for T = 7.5 K (a), T = 0.4 K (b). A constant background signal has been subtracted from the data. Curves: Fits of Eq. (8) to the spectra. a, b Error bars indicate the statistical error. c Field dependence of dynamical muon spin relaxation rate λd at 7.5 K (red circles) and 0.4 K (purple circles) respectively. Curves: Fits of \({\lambda }_{{{{\rm{d}}}}}=a{H}_{{{{\rm{L}}}}}^{-n}\). The error bars are obtained from the fitted equation Eq. (8).

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The data are well-fitted using the following equation

$$A(t)={A}_{{{{\rm{s}}}}}{{{{\rm{P}}}}}_{Z}^{{{{\rm{LF.LKT}}}}}({B}_{{{{\rm{ext}}}}},\Lambda ){e}^{-{\lambda }_{{{{\rm{d}}}}}t}+{A}_{{{{\rm{b}}}}},$$
(8)

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, Λ/γμ is the half-width at half-maximum of the field distribution, which is field-independent and was fixed at the value of ΛZF(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 a = 0.45(8), n = 0.54(5) for T = 0.4 K, and a = 0.43(4), n = 0.53(1) for T = 7.5 K. 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 T = 0.4 K and T = 7.5 K 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 ΛZF below Tc is due to the static magnetism.

In summary, specific heat measurements reveal a large Hc2 value and a large γ(0) of Pr3Cr10−xN11, 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 Δ0 = 0.54 meV 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: px ± ipy 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 Pr3Cr10−xN11 is a candidate of p-wave superconductor which breaks time-reversal symmetry. However, only polycrystalline samples of Pr3Cr10−xN11 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 s + is or s + id is applicable.

Methods

Experimental methods

Polycrystalline samples of Pr3Cr10−xN11 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 T = 2 K and magnetic field up to μ0H = 9 T 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 0.27–20 K. About 1 g 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 1 μT in three directions.