Introduction

Materials that can display superconductivity are extremely rare in nature. Although some elements are found in metallic form, superconductivity has only been reported in meteorites that contain alloys of tin, lead, and indium1. Superconducting compounds are even scarcer, and only the mineral covellite, CuS, shows superconductivity in samples that occur naturally2, a discovery that occurred many decades after superconductivity was first detected in laboratory-grown CuS crystals3. We know of only three other minerals where synthetic analogs are superconductors: parkerite, Ni3Bi2S2, with superconducting transition temperature, Tc ≈ 0.7 K4,5, and two isostructural compounds, miassite, Rh17S15 (Tc = 5.8 K)6, and palladseite, Pd17Se15 (Tc = 2.2 K)7. Here, we study the superconducting properties of synthetic miassite, which is also one of the few rhodium-containing minerals. Initially believed to have Rh9S8 composition, this compound was first synthesized in the 1930s8, and superconductivity in polycrystals was reported in 1954 by Matthias et al.6. Stoichiometry was refined to Rh17S15 in the early 1960s9. A mineral with the same composition was discovered significantly later in the placers of the Miass River in the Ural Mountains in Russia, from which it derives its name10,11. Natural miassite is found in isoferroplatinum deposits as small rounded inclusions up to 100 μm in diameter12. The natural mineral contains a large amount of impurities, such as iron, nickel, platinum, and copper, at a level of a few atomic percent13.

The superconducting properties of miassite display a number of remarkable features. It is exceptional among the naturally occurring superconductors, showing an anomalously high upper critical field greater than 20 T14, exceeding the Pauli limit of about 10 T. In contrast, the upper critical field of palladseite is about 3.3 T, below the Pauli limit7, while in elemental superconductors, covellite2,15 and parkerite4 the upper critical field is orders of magnitude smaller. The heat capacity jump at Tc is reported to significantly exceed the prediction of the weak-coupling Bardeen–Cooper–Schrieffer (BCS) theory11. The electronic heat capacity in the normal state shows a large Sommerfeld coefficient11,16, comparable to heavy-fermion superconductors17 and probably originates from Rh d-electrons18. The low-temperature variation of the heat capacity measured in single crystals deviates from the exponential attenuation expected in a fully gapped superconductor7, contradicting previous findings in polycrystalline samples16. The Hebel–Schlichter peak is notably absent in Rh17S15, contrary to expectations for s-wave superconductivity16. We note that the experimental results in the isostructural palladseite are much more consistent with the BCS s-wave theory7. Finally, there is an order of magnitude difference between the coherence length of about 4 nm derived from Hc2(0), and the BCS length scale ξ0 ≈ 39 nm, which is at least ten times greater. In the weak-coupling BCS theory, in the clean limit (which we prove is the case here), these two lengths are of the same order, ξ = 0.87ξ0 for isotropic s-wave19 and similarly, with a slightly different numerical prefactor, for arbitrary k-dependent order parameter, including d-wave20. Therefore, an extremely high Hc2(0) alone represents a significant departure from the BCS theory.

The unusual experimental observations in Rh17S15 motivated us to clarify its superconducting pairing state. In this report, we present strong evidence for an unconventional gap structure in Rh17S15. Our discovery is based on measurements of the temperature-dependent London penetration depth, down to ~Tc/100, and the response to non-magnetic disorder. London penetration depth shows T-linear variation below Tc/3, consistent with the line nodes in the superconducting order parameter21,22. Further evidence of a nodal gap is provided by the significant suppression of Tc and Hc2 by non-magnetic defects induced by 2.5 MeV electron irradiation. Our results are consistent with an extended s-wave state that has circular line nodes. The existence of gap nodes is a hallmark of unconventional superconductivity, observed in cuprates21,23, some iron pnictides24,25,26,27, heavy-fermions17, and possibly organic superconductors28. All of these materials are products of synthetic solid-state chemistry and are not found in nature. Our work establishes Rh17S15 as a unique member of the unconventional superconductors, being the only example that occurs as a natural mineral.

Results

London penetration depth

London penetration depth measurements were carried out using the tunnel diode resonator (TDR) technique29 in a dilution refrigerator to access temperatures significantly below Tc, as low as ~ Tc/10030 (see Supplementary Note 1 for details). Figure 1a shows the variation of London penetration depth, Δλ(T), in an as-grown single crystal (S-1) measured in both a 3He cryostat (black full symbols) and a dilution refrigerator30 (blue open symbols), showing almost perfect agreement between the two measurements. As shown in the inset, the pristine sample exhibits a sharp superconducting transition at Tc = 5.31 K, determined by the temperature of a maximum of dΔλ(T)/dT. It is sufficiently close to the temperature of zero resistance and is consistent with that determined by heat capacity and magnetization measurements. A photograph of an as-grown single crystal is also shown in the inset in Fig. 1a. The main panel of Fig. 1a presents the behavior of Δλ(T) below 0.3Tc, where it shows a linear dependence on the temperature, Δλ(T) ∝ T. In this temperature range, the magnitude of the superconducting gap is nearly constant, and the temperature dependence of the penetration depth reflects the gap anisotropy22. Our results are inconsistent with a full superconducting gap in Rh17S15, which would imply an exponential saturation of the penetration depth at low temperatures22,31. Instead, the T-linear variation of Δλ(T) in Rh17S15 is characteristic of superconductors with line nodes in the clean limit, similar to the experimental observations in the high − Tc cuprates21. Notably, in our experiments, this T-linear behavior extends down to very low temperatures, ~Tc/100, placing a strenuous upper limit on possible deep gap minima.

Fig. 1: London penetration depth and superfluid density in Rh17S15.
figure 1

a The temperature variation of London penetration depth, Δλ(T), in an as-grown single crystal Rh17S15. The main panel shows a T-linear variation of Δλ(T) in a characteristic temperature range below 0.3Tc. The open and closed symbols represent data taken by TDR setups in the dilution refrigerator (DR-TDR) and 3He system (3He-TDR), respectively. Inset: the full temperature range graph of Δλ(T)/Δλ(6 K) and the photograph of a typical single crystal on a 1 mm-scale paper. The dashed vertical line indicates Tc determined by using dλ/dT maximum criterion. b Normalized superfluid density ρs = λ2(0)/λ2(T) in Rh17S15. The lines represent the theoretical curves for the full-gap s-wave (green), the line-nodal d-wave in-plane (violet) and out-of-plane (magenta), and the anisotropic s-wave (gray and red) pairing types. A parameter r is defined in Eq. (1). Note that the curves for d-wave and extended s-wave are nearly identical and cannot be distinguished on the figure.

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The near-perfect T-linear behavior observed in our experiments implies that our samples are in the clean limit. Indeed, the very short coherence length ξ(0) ≈ 4 nm, due to a very large Hc2(0) ≈ 20 T, is smaller than the mean free path, which can be estimated using Hall effect measurements of carrier mobilities and concentrations at Tc32, which yields ℓ ≈ 86 nm for ρ(Tc) ≈ 10 μΩcm supporting the clean limit of our Rh17S15 crystals.

Superfluid density and superconducting energy gap

The superconducting gap structure can be analyzed using the temperature-dependent normalized superfluid density, ρs(T) = λ2(0)/λ2(T). The absolute value of λ at T = 0 cannot be directly determined from our measurements of Δλ(T). The reported values of λ(0) in the literature vary from 490 nm determined from Hc1(0) = 30 Oe14 to 750 nm from μSR studies33. Taking into account such a significant uncertainty in λ(0), we used the thermodynamic Rutgers relation, [dρs(t = 1)/dt]/λ2(0) = 16π2TcΔC/ϕ0[dHc2(t = 1)/dt], where t = T/Tc, ϕ0 = 2.07 × 10−7 G cm2 is a magnetic flux quantum34, and heat capacity jump at Tc, ΔC, was taken from Uhlarz et al.35, see Supplementary Note 2 for details. We obtain λ(0) = 550 nm, which is between the two values in the literature. As can be seen in Fig. 1b, the normalized superfluid density, ρs, of Rh17S15 is very different from the expectations of full-gap s-wave superconductors (green line). The choice of λ(0) does not qualitatively alter the overall temperature dependence, which is the main criterion to probe the gap anisotropy; see the supporting material. We do not observe any characteristic features of a multigap superconducting state, such as the concave curvature of ρs(T) at elevated temperatures. Although Rh17S15 is a multiband material, as shown by band structure calculations36 and Hall effect measurements32, its superconducting state is characterized by a single gap, consistent with conclusions from heat capacity measurements16. Similar behavior is observed in other materials, for example, SrPd2Ge2, in which a sign-changing Hall effect suggests multi-band transport in the normal state37, but the superconductivity is still characterized by a single isotropic gap38.

The penetration depth measurements clearly indicate that a gap with line nodes is realized in Rh17S15. Since Knight shift experiments indicate a spin-singlet order parameter16, the orbital component of the pair potential must be even parity. A possible gap consistent with the evidence is a sign-changing state in the A1g irreducible representation (irrep) of the point group. Although nodes are not required by the symmetry of the order parameter, “accidental” nodes depending on the microscopic details of the system are possible, as is the case in some pnictide compounds39. To realize this, we propose a gap function with both isotropic and anisotropic components,

$$\Delta (\hat{{{{{{{{\boldsymbol{k}}}}}}}}})={\Delta }_{0}{C}_{r}[r+(1-| r| )({\hat{k}}_{x}^{4}+{\hat{k}}_{y}^{4}+{\hat{k}}_{z}^{4})]$$
(1)

where −1 ≤ r ≤ 1 tunes the relative strength and sign of the isotropic and anisotropic gap components, and Cr is a normalization constant chosen so that the Fermi-surface average of \(| \Delta (\hat{{{{{{{{\boldsymbol{k}}}}}}}}}){| }^{2}\) is independent of r (see Supplementary Note 3 for details)). This gap structure is visualized in Fig. 2 for r = −0.4. For a spherical Fermi surface, Eq. (1) includes the lowest power of k that gives an anisotropic A1g gap, and has line nodes for −0.5 < r < −0.25. As shown in Fig. 1b, the calculated superfluid density22 is in quantitative agreement with the experiment for −0.45 ≤ r ≤ −0.4, which corresponds to circular line nodes centered about the crystal axes (see Supplementary Note 3 for details).

Fig. 2: Superconducting gap in Rh17S15.
figure 2

Suggested superconducting gap in Rh17S15 with r = −0.4 in Eq.(1). The solid black lines represent the nodes. The color scale bar indicates the relative gap magnitude.

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Other nodal states are strongly constrained by the cubic crystal symmetry. For example, although the three-dimensional d-wave state \(\Delta (\hat{{{{{{{{\boldsymbol{k}}}}}}}}})=\sqrt{15/4}{\Delta }_{0}({\hat{k}}_{x}^{2}-{\hat{k}}_{y}^{2})\) in the Eg irrep is consistent with the penetration depth data, it reduces the symmetry from cubic to tetragonal40. Such a nematic superconducting state is highly exotic, having so far only been observed in the Bi2Se3 family41. The nematicity is reflected in the superfluid density, which only fits the data for an in-plane field. In general, a nematic superconductor possesses multiple nematic domains with different orientations of the order parameter. We expect this to lead to a measured superfluid density in between the extreme limits of the in-plane and out-of-plane responses. However, our data is only consistent with a monodomain sample. Although we cannot fully exclude a nematic state in the Eg irrep, the isotropic superfluid density of the A1g state makes the latter a more conservative scenario. Pairing states in other nontrivial irreps are also unlikely since they imply nematic or high-angular-momentum pairing states.

Effect of disorder on the superconducting transition temperature

An independent test for a sign-changing superconducting gap function was made by studying the effect of nonmagnetic disorder on Tc. This  disorder was introduced in a controlled manner by low-temperature electron irradiation, which is a known method to introduce a metastable population of vacancies42,43, see Supplementary Note 4 for details. Figure 3 shows the temperature-dependent resistivity of single crystalline sample S-2 of Rh17S15 before irradiation in the pristine state (black line) and after electron irradiations with two doses of 0.912 C/cm2 (red line) and 2.912 C/cm2 (blue line), respectively. The upper inset shows the temperature-dependent difference where pristine-state data were subtracted from the data after the irradiation. A clear violation of the Matthiessen rule is observed below about 100 K. The lower inset in Fig. 3 zooms in on the superconducting transition region. A clear suppression of Tc is evident, with a reduction of 26% and 40% after irradiation with 0.912 C/cm2 and 2.912 C/cm2, respectively.

Fig. 3: Effect of electron irradiation on electrical resistivity of Rh17S15.
figure 3

Temperature-dependent resistivity of single crystalline sample, S-2, before irradiation in the pristine state (black line) and after 0.912 C cm−2 (red line) and 2.912 C cm−2 (blue line) electron irradiation. The upper inset shows the temperature-dependent difference between irradiated and pristine curves, revealing a clear Matthiessen’s rule violation below 100 K. The lower inset zooms in on the superconducting transition region.

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We have no evidence that the irradiation of Rh17S15 introduces magnetic impurities, and therefore, here we deal only with the effect of nonmagnetic impurities. Their effect in a superconductor with an anisotropic gap is to smear out the anisotropy, with the order parameter reaching its average value in the dirty limit. This is accompanied by a suppression of the critical temperature; at low disorder strength, the suppression is approximately linear44,45,46

$${t}_{c}=\frac{{T}_{c}}{{T}_{c0}}\approx 1-\left(1-{\langle {\hat{\Delta }}_{{{{{{{{\bf{k}}}}}}}}}\rangle }^{2}\right)\frac{{\pi }^{2}}{4}\Gamma$$
(2)

where \(\langle {\hat{\Delta }}_{{{{{{{{\bf{k}}}}}}}}}\rangle\) is the Fermi surface average of the normalized form factor of the pairing potential, and Γ = ℏ/(2πkBTc0τ) is the dimensionless scattering rate, Tc0 is the transition temperature in the clean limit, and τ is the scattering time in the normal state. In the two extreme limits of a uniform gap (\(\langle {\hat{\Delta }}_{{{{{{{{\bf{k}}}}}}}}}\rangle =1\)) there is no suppression of Tc, whereas the suppression is maximal for a sign-changing d-wave gap (\(\langle {\hat{\Delta }}_{{{{{{{{\bf{k}}}}}}}}}\rangle =0\)); the sign-changing A1g state Eq. (1) lies between these two limits. Thus, the slope dtc/dΓ gives an independent check on the gap structure inferred from the superfluid density.

In the Drude model, the resistivity is ρ = m/(ne2τ), and in London electrodynamics, the penetration depth is λ2 = m/(μ0ne2), where μ0 = 4π × 10−7 H/m is the vacuum magnetic permeability. Therefore, we can express the dimensionless scattering rate via measurable parameters,

$$\Gamma =\frac{\hslash }{2\pi {k}_{B}{T}_{c0}\tau }=\frac{\hslash {\rho }_{0}}{2\pi {k}_{B}{\mu }_{0}{T}_{c0}{\lambda }^{2}(0)}$$
(3)

In our case Tc0 = 5.4 K and λ(0) = 550 nm. As can be seen in the lower insert, the ρ(T) curves saturate approaching Tc from above, and so the values of the resistivity at Tc are good approximations for ρ0.

Unfortunately, the experimental ρ(T) in Rh17S15 is qualitatively different from the conventional Bloch–Gruneisen theory47. Moreover, around 100 K, the Hall effect changes sign, and a notable non-linearity of its field dependence emerges11,32. The analysis of the field-dependent resistivity and Hall effect finds at least two groups of carriers with notable differences in properties conducting in parallel. The low-temperature transport is dominated by high mobility carriers with mobilities up to 600 cm2/(V s), while carriers with normal metallic mobilities of order 1 cm2/(V s) are responsible for resistivity at high temperatures32. The difference in mobilities naturally explains a significantly greater increase in resistivity at low temperatures after irradiation48,49. This extreme difference in the properties of the carriers makes it harder to determine a single effective scattering rate from resistivity change44. The origin of the high-mobility carriers is unclear. Although the shoulder-like feature in Fig. 3 is rather smooth and broad, somewhat sharper features similar to this are observed in three-dimensional materials undergoing charge density wave instability50 (See Supplementary Note 5 for details). High mobility carriers there may arise from small pockets forming upon Fermi surface reconstruction. To eliminate this possibility, we measured X-ray diffraction and did not find any additional peaks down to 5 K (see Supplementary Note 6 for details).

Despite the significant temperature dependence in the resistivity change by electron irradiation, it is still possible to use Eq. (3) to estimate the scattering rate. Specifically, we use two limiting values, one estimated at Tc where high mobility carriers dominate and another at room temperature determined by normal carriers, giving the results in Fig. 4 by open and filled symbols, respectively. The corresponding rates of suppression dtc/dΓ are approximately −2 and −4 in these two cases. The straight lines in Fig. 4 show the expected Tc suppression for the order parameter of Eq.(1). The isotropic s-wave case corresponds to r = −1 (black solid line), while the dotted line is for r = −0.45 and the dashed line is for r = −0.4. The latter values are most consistent with the fit to the superfluid density. The agreement with the experimental data is closer for r = −0.4, which is also close to a d-wave order parameter shown in Fig. 4 by a green solid line. The strong suppression of the transition temperature, therefore, supports the existence of a sign-changing superconducting gap function in Rh17S15 and is consistent with the fits to the superfluid density.

Fig. 4: Suppression of Tc/Tc0 vs. Γ for various r in Rh17S15 compared to typical gap symmetries.
figure 4

See text for the definitions of Γ and r. The closed and open star symbols represent the calculated Γ with Δρ (see Fig. 2) at room temperature and low temperature, respectively.

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Effect of disorder on the upper critical field

The high upper critical field of Rh17S15, Hc2(0) ≈ 20 T7, suggests that the carriers involved in superconducting pairing should be rather heavy, since \({H}_{c2} \sim {v}_{F}^{-2}\), where vF is Fermi velocity19,51. On the contrary, the London penetration depth and resistivity are dominated by light carriers. To access the properties and response to the disorder of the heavier carriers in the condensate, we measured the upper critical field of the pristine and irradiated samples, with the results shown in Fig. 5a. In the pristine state, Hc2(T) as determined from the onset of resistivity in our measurements (complete symbols) is perfectly consistent with the definition of entropy balance from the specific heat measurements shown by open triangles35. After irradiation with the doses of 0.912 C (red squares) and 2.912 C (blue triangles), the Hc2(T) curves are shifted to lower temperatures with a slight decrease of the slope at Tc. See Supplementary Note 7 for details.

Fig. 5: Temperature-dependent upper critical field in Rh17S15.
figure 5

a Temperature-dependent upper critical field in a pristine single crystal of Rh17S15 (black curve) and electron-irradiated samples with doses of 0.912 C cm−2 (red squares) and 2.912 C cm−2 (blue triangles). The onset of resistivity was used as a criterion. The complete dataset is shown in Supplementary Fig. 6. Measurements were taken in the magnetic fields parallel to [100] crystallographic direction. The green open triangles show Hc2 determined from heat capacity measurements in polycrystalline samples35. b–d The evolution of Hc2(T) with the pairbreaking (Γm) and potential (Γ0) scattering19,54. The corresponding scattering rates are indicated in each panel.

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In our analysis of the upper critical field Hc2, we utilize the fact that the behavior of an isotropic s-wave superconductor with magnetic impurities is practically the same as the behavior of a d-wave superconductor with non-magnetic impurities52,53. This reflects the sensitivity of anisotropic pairing to potential disorder and considerably simplifies the calculation of the upper critical field19,54. Since the suppression of the Tc reveals a gap with anisotropy comparable to a pure d-wave state, this should be a good approximation for Rh17S15. We therefore consider an s-wave pairing state with distinct scattering rates for the nonmagnetic and magnetic disorder, Γ0 and Γm, respectively. The latter is usually associated with scattering on point-like magnetic impurities where the spins of a scattered electron and the impurity flip. In a singlet superconductor, where the spins in a Cooper pair are antiparallel, the reversal of one of them breaks the pair. Nonmagnetic scattering between regions of the Fermi surface with opposite gap signs in Rh17S15 has a comparably deleterious effect on the superconductivity, justifying the magnetic scattering rate in our theory.

The theoretical curves, panels b, c, and d in Fig. 5, illustrate the effects of different types of disorder on Hc2 as calculated from the Eilenberger theory19,54. The theory is parameterized by dimensionless non-pair-breaking and pair-breaking scattering rates Γ0 and Γm, respectively; we note that these are distinct from the scattering rate introduced in Eq. (3). Reviewing the results shows that non-pairbreaking scattering always increases Hc2(0) and its slope at Tc, whereas pair-breaking scattering, which is expected in nodal superconductors with non-magnetic disorder, suppresses both Hc2(0) and its slope at Tc. This is consistent with the trend observed in our experiment and provides the third independent evidence that Rh17S15 is a nodal unconventional superconductor.

Conclusions

We have discovered a rare occurrence of nodal superconductivity in the cubic 4d-electron compound Rh17S15. Our conclusions are based on the linear temperature variation of the London penetration depth and the suppression of the transition temperature and the upper critical field by non-magnetic disorder introduced by electron irradiation. The analysis of the superfluid density and the Tc suppression rate is consistent with an extended s-wave superconducting state with accidental line nodes preserving cubic symmetry. Our results suggest that pure Rh17S15 is the first known unconventional superconductor among materials that also exist in mineral form. However, a large amount of impurities, such as iron, nickel, platinum, and copper, known to be present in natural minerals at a level of a few percent13, make it highly unlikely that they will exhibit superconductivity. Nature knows how to hide its secrets.

Methods

Single crystal growth

To investigate the superconducting state in Rh17S15, we synthesized single crystalline samples out of the Rh-S eutectic region by using a high-temperature flux growth technique. In Ref. 5, it has been shown that the high-temperature solution growth technique can be used to grow binary and ternary transition metal-based compounds out of S-based solutions. In refs. 55,56, high-temperature solution growth was expanded to Rh–S–X ternaries. As part of that effort, we re-determined the Rh-rich eutectic composition to be close to Rh60S40. As a result, we were able to create a slightly more S-rich melt, Rh58S42, by combining elemental Rh powder (99.9+ purity) and elemental S in a fritted Canfield Crucible set57, sealing in a silica ampoule, slowly heating (over 12 h) to 1150 °C and then slowly cooling from 1150 to 920 °C over 50 h and decanting58. Millimeter-sized single crystals of Rh17S15 grew readily (see inset to Fig. 1a).

Electron irradiation

The low-temperature 2.5 MeV electron irradiation was performed at the SIRIUS Pelletron facility of the Laboratoire des Solides Irradiés (LSI) at the Ecole Polytechnique in Palaiseau, France. The acquired irradiation dose is conveniently expressed in C/cm2 and measured directly as a total charge accumulated behind the sample by a Faraday cage. Therefore, 1 C/cm2 ≈ 6.24 × 1018 electrons/cm2. The irradiation was carried out with the sample immersed in liquid hydrogen at about 20 K. See Supplementary Note 4 for details.

Electrical transport

Four probe electrical resistivity measurements were performed in Quantum Design PPMS on three samples S-2, S-3, and S-4. Contacts to the samples were soldered with tin59 and had resistance in mΩ range (See Supplementary Note 8 for details). Resistivity measurements were made on the same samples without contact remounting before and after irradiation to exclude the uncertainty of geometric factor determination.

Tunnel diode resonator

The London penetration depth was measured by using a TDR technique60. The shift of the resonant frequency (in cgs units), Δf(T) = − G4πχ(T), where χ(T) is the differential magnetic susceptibility, G = f0Vs/2Vc(1 − N) is a constant, N is the demagnetization factor, Vs is the sample volume and Vc is the coil volume. The constant G was determined from the full frequency change by physically pulling the sample out of the coil. With the characteristic sample size, R, \(4\pi \chi =(\lambda /R)\tanh (R/\lambda )-1\), from which Δλ can be obtained22,60, see Supplementary Note 1 for details.