Observation of superconductivity and enhanced upper critical field of η-carbide-type oxide Zr4Pd2O

We report the first observation of bulk superconductivity of a η-carbide-type oxide Zr4Pd2O. The crystal structure and the superconducting properties were studied through synchrotron X-ray diffraction, magnetization, electrical resistivity, and specific heat measurement. The superconducting transition was observed at Tc = 2.73 K. Our measurement revealed that the η-carbide-type oxide superconductor Zr4Pd2O shows an enhanced upper critical field μ0Hc2(0) = 6.72 T, which violates the Pauli-Clogston limit μ0HP = 5.29 T. On the other hand, we found that the enhanced upper critical field is absent in a Rh analogue Zr4Rh2O. The large μ0Hc2(0) of Zr4Pd2O would be raised from strong spin–orbit coupling with Pd-4d electrons. The discovery of new superconducting properties for Zr4Pd2O would shed light on the further development of η-carbide-type oxide superconductors.


Introduction
Transition metal oxides are well known as one of the most fascinating solids because of their variety of physical properties [1] such as metal-insulator transition [2], giant magnetoresistance [3], ferroelectricity [4], and high-temperature superconductivity [5].In transition metal oxides, anisotropic-shaped d-orbital electrons and strong electron correlation effect caused by Coulomb interaction between electrons play important roles for emerging the various physical phenomena [6].The first discovery of superconductors in oxide compounds is a perovskite-type structural SrTiO 3-x [7], and the discovery has led to the development of many kinds of oxide superconductors, for example, Ba1-xKxBiO3 [8], YBa2Cu3O7 [9], Li1+xTi2-xO4 [10] and so on.Among the discoveries of oxide superconductors, the η-carbide-type superconductors A 4 B 2 X have attracted attention in recent studies; here, A and B are transition metals and X is a light element such as carbon, nitrogen, or oxygen [11,12].Ma et al. reported the bulk superconductivity of Zr4Rh2Ox (x = 0.7 and 1) [12] and Nb4Rh2C1-δ [13] at transition temperatures of Tc = 2.8 K (x = 0.7), 4.8 K (x = 1) and 9.75 K, respectively.They also found that Ti 4 M 2 O with M = Co, Rh, and Ir show superconductivity at Tc = 2.7 K, 2.8 K, and 5.4 K, respectively [14].The significant discovery of them was not only founding superconductors but also observing a large upper critical field μ 0 H c2 (0) violating the Pauli-Clogston limit (Pauli limit) for Nb4Rh2C1-δ, Ti4Co2O, and Ti4Ir2O.The value of Pauli limit μ0HP is determined with a certain magnetic field at which a gain of paramagnetic Zeeman energy at a normal state is equal to a superconducting condensation energy, given in the following formula [15,16]: where g = 2 is a g-factor for free electron and μB ≈ 9.27×10 -24 J T -1 is a Bohr magneton.The Δ(0) is a superconducting gap energy at 0 K described as Δ(0) = 1.76kB T c (k B ≈ 1.38×10 -23 J K -1 is a Boltzmann constant) in the single gap Bardeen−Cooper−Schrieffer (BCS) model [17].The large μ0Hc2(0) overwhelming the μ0HP can arise from special electronic states and structural properties such as multi-band effect [18][19][20], spin-triplet cooper pairing [21], Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconducting state [22,23], global or local inversion symmetry breaking [24,25], and strong spin-orbit coupling (SOC) [26,27].Particularly, spin-orbit scattering originating from SOC suppresses a cooper pair breaking by the Pauli paramagnetic effect because SOC destroys spin as a good quantum number, and makes spin susceptibility of the superconducting state close to that of a normal state, described in Werthamer-Helfand-Hohenberg (WHH) theory [28,29].Therefore, the strong SOC has the potential to achieve a large μ 0 H c2 (0) superconducting state, and the strength can be controlled

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by chemical elemental substitution [30].The strength of SOC, ξ can be approximately calculated using a hydrogen-like atom model [31]: where Z, n, and l are atomic number, principal quantum number, and orbital angular momentum, respectively.From the expression, we can understand the strength of SOC proportions to Z 4 within the same electronic orbital.In the case of the Ti 4 Ir 2 O superconductor, we can expect the Ir-5d orbital hosting enhanced SOC should play an important role for the large μ0Hc2(0), and it was found that Ti-3d and It-5d orbitals hybridize near its Fermi level and the violation of the Pauli limit is a result of a combination of strong-coupled superconductivity, SOC, and strong electron correlation [32].Furthermore, the large SOC splitting a band structure along Γ-K lines due to the Ir-5d electrons was weakened by applying pressures, and large μ 0 H c2 (0) undergoes a crossover at 35.6 GPa from well beyond to less than the μ0HP [33].As mentioned above, the η-carbide-type superconductors have been studied from the points of view of the large μ 0 H c2 (0) and SOC effect based on d-block transition metals.Table 1 shows a list of the η-carbide-type superconductors with Tc, μ0HP, and μ0Hc2(0).Some kinds of η-carbide-type oxide superconductors have been reported; however, it is not sufficient as of now, and developing new examples of them is important for a deeper understanding of the η-carbide-type oxide superconducting properties.
Herein, we focus on a Zr 4 B 2 O system because superconductivity was solely confirmed in Zr 4 Rh 2 O in the system to the best of our knowledge.We report the discovery of an unrevealed superconducting nature of Zr4Pd2O known as a hydrogen storage material [34,35].Polycrystalline samples of Zr 4 Pd 2 O were obtained by arc melting followed by annealing in an evacuated quartz tube.We performed synchrotron X-ray diffraction (SXRD) measurement at the beamline BL13XU in SPring-8 and checked chemical composition by means of the energy dispersive X-ray spectroscopy (EDX) method to characterize obtained samples.The bulk superconductivity was confirmed through magnetic susceptibility, electrical resistivity, and specific heat measurement, resulting in T c = 2.8 K, 2.73 K, and 2.6 K, respectively.We discuss the μ 0 H c2 (0) for Zr 4 Pd 2 O and Zr 4 Rh 2 O using the electrical resistivity and specific heat data measured at several magnetic fields.We find that Zr 4 Rh 2 O shows the μ0Hc2(0) = 6.16T, lower than the μ0HP = 7.59 T; however, Zr4Pd2O shows the large μ0Hc2(0) = 6.88 T, violating the μ0HP = 5.29 T different from Zr 4 Rh 2 O.The violation of the Pauli limit for Zr 4 Pd 2 O can be attributed to the larger strength of SOC derived from Pd-4d electrons.

Compound
Tc

Crystal structures of Zr4Pd2O and Zr4Rh2O
Schematic images of the crystal structure for Zr 4 Tr 2 O (Tr = Pd or Rh) are shown in Fig. 1(a).These compounds crystalline a cubic η-carbide crystal structure with a space group Fd3 m (No. 227).The metal atoms Zr occupy Wyckoff positions 48f (labeled as Zr1), 16d (labeled as Zr2) and Tr occupies Wyckoff position 32e position.The O atom occupies Wyckoff position 16c, and the occupying can be regarded as void filling in a Ti 2 Ni-type structure.The complicated η-carbide crystal structure consists of Zr1 octahedra centered by O (Fig. 1 (b)) and a geometrically frustrated stella quadrangula lattice (Fig. 1 (c)) [36].The Zr1 octahedra caging the O atom at the center are arranged in the unit cell sharing the corner as seen in a pyrochlore structure.The stella quadrangula lattice can be formed by inserting a small tetrahedron into each tetrahedron making the pyrochlore lattice, and the unit consists of nested Tr and Zr2 tetrahedra with the same center of gravity [37,38] [12,34,35].We found a small amount of impurity phases such as ZrO 2 and Zr in both of them, and reliability factors Rwp were 8.438% for Zr4Pd2O and 16.550% for Zr4Rh2O.We also refined the atomic coordinates and isotropic atomic displacement parameter Uiso for each atom, as summarized in Table 2.The Uiso of the O atom was fixed to be 0.004 because the value approximately close to be zero within the errors on the fitting quality.The chemical compositions of Zr 4 Pd 2 O and Zr 4 Rh 2 O confirmed through EDX were to be 2.10(2) for Zr:Pd and 2.2(2) for Zr:Rh.

Magnetization, Electrical resistivity, and Specific heat
We measured temperature-and magnetic field-dependent magnetizations for Zr4Pd2O and Zr4Rh2O with polished rectangular cuboid samples.The value of a demagnetizing factor N with a rectangular cuboid sample applied a vertical magnetic field can be calculated using dimensional information of the sample: length l, width w, and thickness t [39]: The calculated values of N were 0.66 for Zr 4 Pd 2 O (l = 1.40 mm, w = 1.50 mm, t = 0.51 mm) and 0.81 for Zr 4 Rh 2 O (l = 0.96 mm, w = 1.37 mm, t = 0.18 mm).An actual magnetic field in samples should be modified to an effective inner magnetic field as described in Heff = H -4πMN, where H is an applied magnetic field and M is a magnetization.Thus, magnetic susceptibility χ taken to account for the demagnetizing effect is defined as follows: The horizontal axis is displayed in μ 0 H eff instead of H for precise estimation of the lower critical field.In Zr 4 Pd 2 O, the measurement was carried out in a range of 1.8 K < T < 2.8 K with increments of 0.1 K.In the case of Zr4Rh2O, the data was taken at 1.8 K, 1.9 K, and 2.0 K, then taken with increments of 0.2 K up to 3.6 K.We observed the convex downward curve of M as functions of μ0Heff at each temperature, and the minimum points gradually shifted lower filed as increasing temperature.In a low-filed region, the linear responses of M corresponding to the Meissner state were observed, and the behavior can be described as resulting μ0Hc1(0) = 11.3 mT and 8.2 mT for Zr4Pd2O and Zr4Rh2O, respectively.Zr4Pd2O showed a higher μ0Hc1(0) than that of Zr4Rh2O even lower Tc, suggesting that Zr4Pd2O tends to be more robust against magnetic field rather than Zr4Rh2O.3(a) and 3(b), respectively.In a low-temperature region, we observed a drop of ρ(T) to zero, suggesting a superconducting transition.Zr 4 Pd 2 O showed a sharp transition, and the zero resistivity was observed at T c zero = 2.73 K. Zr 4 Rh 2 O, however, showed a broad transition, and the zero resistivity was observed at T c zero = 3.73 K.The T c obtained from the zero resistivity agreed with the result from the magnetic susceptibility measurement.The ρ(T) exhibits a metallic behavior in a normal state for both Zr4Pd2O and Zr4Rh2O.In the low-temperature normal state, the ρ(T) curve can be fitted using the power-law model: where ρ0, A, and nPL are residual resistivity, temperature-independent coefficient, and power exponent respectively.The ρ(T) curve of Zr 4 Pd 2 O in a range of 4 K < T < 80 K was fitted using the model, providing ρ 0 = 0.22 mΩ cm, A = 0.00011 mΩ K -2 , and n PL = 1.2.For the ρ(T) curve of Zr 4 Rh 2 O, we fitted in a range of 6 K < T < 80 K, obtained ρ 0 = 0.29 mΩ cm, A = 0.00017 mΩ K -2 , and nPL = 1.3.The values of nPL close to 1 suggest that the ρ(T) show linear-temperature dependence in the lowtemperature normal state for Zr4Pd2O and Zr4Rh2O, and the behavior was observed in some η-carbide-type superconductors [12,13].In a high-temperature region (80 K < T < 300 K) where electron-phonon interaction is a dominant mechanism of electron scattering, the ρ(T) shows a convex upward curve with increasing temperature.A similar trend can be found in many superconductors consisting of d-block element [40][41][42][43], and the convex upward curve of ρ(T) in the high-temperature region can be fitted using the following parallel resistor model, yielded by Wiesmann et al. [44]: . 7 The temperature-independent term ρsat corresponds to the saturation of ρ(T) in high temperature.Fisk and Webb found the saturation of resistivity in A-15 superconductors such as Nb 3 Sn [45], and the saturation can be realized when a mean free path becomes comparable to interatomic separations of the material, called as Ioffe-Regel condition [46].The temperature-dependent component ρideal is described with the Bloch-Grüneisen model [47]: where ρ ideal,0 , B, Θ D , and n BG are ideal temperature-independent residual resistivity, temperature-independent coefficient, Debye temperature, and power exponent with the Bloch-Grüneisen model, respectively.The n BG usually takes 2, 3, or 5 depending on the scattering nature.The best fit was obtained when nBG = 5 for both Zr4Pd2O and Zr4Rh2O, and the calculation provided ρsat = 0.52 mΩ cm, ρ ideal,0 = 0.41 mΩ cm, B = 1.49mΩ cm, and Θ D = 261 K for Zr 4 Pd 2 O, and ρ sat = 0.82 mΩ cm, ρ ideal,0 = 0.49 mΩ cm, B = 1.68 mΩ cm, and Θ D = 198 K for Zr 4 Rh 2 O.The ρ 0 can be calculated using ρ sat and ρ ideal,0 as given in ρ 0 = ρ sat ρ ideal,0 /(ρ sat + ρideal,0), and the obtained values of ρ0 are 0.23 mΩ cm and 0.31 mΩ cm, for Zr4Pd2O and Zr4Rh2O, respectively.These ρ0 values are consistent with those obtained by the power-law model.The fitted curves using the power-law model and parallel resistor model are displayed as dashed lines and solid lines, respectively, in Figs.3(a) and 3(b).Resistivity at 300 K, ρ 300 K , was found to be 0.32 mΩ cm for Zr 4 Pd 2 O and 0.47 mΩ cm for Zr 4 Rh 2 O. Residual resistivity ratio, RRR = ρ 300 K /ρ 0 was calculated to be 1.42 and 1.61 for Zr4Pd2O and Zr4Rh2O, respectively.The small RRR value is also seen in other η-carbide-type superconductors [12][13][14], and the poor metallic behavior is a common feature of polycrystalline metallic oxide compounds whose grain-boundary scattering is significant [48].Figures 3(c) and 3(d) show the ρ(T) curves at several magnetic fields for Zr 4 Pd 2 O and Zr 4 Rh 2 O, respectively.The magnetic fields are applied with an increment of 0.2 T for Zr 4 Pd 2 O up to μ 0 H = 3.8 T. For Zr4Rh2O, the magnetic fields are increased by 0.2 T up to μ0H = 4.0 T and then increased by 0.5 T up to μ0H = 6.5 T. The T c zero shifted lower temperature with increasing magnetic field as we expected.We used typical 10%, 50%, and 90% criteria defined with ρ 0 to determine temperature dependences of the upper critical field for Zr 4 Pd 2 O and Zr 4 Rh 2 O (discussed later).
Figures 3(e) and 3(f) show temperature dependences of total specific heat C(T) at several magnetic fields for Zr 4 Pd 2 O and Zr 4 Rh 2 O, respectively.Magnetic fields were applied with increments of 0.2 T up to μ 0 H = 2 T and also measured at μ 0 H = 9 T. We observed clear specific heat jumps, suggesting superconducting transition, up to μ0H = 2 T, and the temperature at which the jumps observed shifted to a lower temperature, consistent with the results from electrical resistivity.Zr4Rh2O showed broader transitions than that of Zr 4 Pd 2 O, as seen in magnetic susceptibility and electrical resistivity measurements.To calculate Sommerfeld coefficient γ and Θ D , we fitted C(T) using the following formula: where β and δ are the coefficients of the phonon contributions for the harmonic and anharmonic terms, respectively.As a result of the fitting, we obtained the values of γ to be γ = 32.5 mJ K -2 mol -1 for Zr 4 Pd 2 O and γ = 18.1 mJ K -2 mol -1 for Zr 4 Rh 2 O.For the phonon contribution coefficients, we obtained β = 1.87 mJ K -4 mol -1 and δ = 0.0016 mJ K -6 mol -1 for Zr4Pd2O, and β = 1.12 mJ K -4 mol -1 and δ = 0.014 mJ K -6 mol -1 for Zr4Rh2O.The fitting curves are shown as solid lines in Figs.3(e) and 3(f).The values of Θ D can be calculated using the β as the following formula: , 10 where N = 7 is the number of atoms per formula unit and R ≈ 8.31 J K -1 mol -1 is an ideal gas constant.The values of the jump were similar and slightly higher than 1.43, which is the expected value by the weak-coupling BCS theory [17].This result suggests that Zr4Pd2O and Zr4Rh2O are electron-phonon coupling superconductors with a little strong-coupling nature.
We can calculate an electron-phonon coupling constant λ el-ph using the McMillan formula [49]: where μ* = 0.13 is a Coulomb coupling constant and the value is used empirically for similar materials containing transition metals.We obtained the values of λel-ph to be 0.60 for Zr4Pd2O and 0.61 for Zr4Rh2O.An electronic density of states at the Fermi energy D(E F ) is proportional to a term (1+λ el-ph ) when we consider the electron-phonon coupling.Therefore, D(E F ) with spin degeneracy can be expressed in the following: The measured γ and calculated λ el-ph provide D(E F ) = 8.61 states eV -1 per formula unit (f.u.) and 4.76 states eV -1 per f.u. for Zr 4 Pd 2 O and Zr 4 Rh 2 O, respectively.

Discussion
Here, we discuss the upper critical fields and other superconducting parameters of Zr4Pd2O and Zr4Rh2O.Figures 4(a) and 4(b) are temperature dependences of upper critical field μ 0 H c2 (T) for Zr 4 Pd 2 O and Zr 4 Rh 2 O, respectively.The data points were taken from temperature dependences of ρ(T) with 10%, 50%, and 90% criteria, and C(T) under several magnetic fields.The upper critical field at 0 K, μ0Hc2(0) can be calculated by fitting the data using the Ginzburg-Landau (GL) model: We obtained the values of μ 0 H c2 (0) for Zr 4 Pd 2 O to be 7.18 T for ρ(T) 10% criterion, 6.88 T for ρ(T) 50% criterion, 6.72 T for ρ(T) 90% criterion, and 9.17 T for C(T).For Zr 4 Rh 2 O, the obtained μ 0 H c2 (0) to be 6.27 T for ρ(T) 10% criterion, 6.16 T for ρ(T) 50% criterion, 5.91 T for ρ(T) 90% criterion, and 7.74 T for C(T).We found that the whole values of μ0Hc2(0) for Zr4Pd2O were higher than that of μ 0 H P = 5.29 T calculated with T c of ρ(T) 50% criterion.On the other hand, for Zr 4 Rh 2 O, the values of μ 0 H c2 (0) derived from ρ(T) criterion were lower than that of μ 0 H P = 7.59 T calculated with T c of ρ(T) 50% criterion.The value of μ 0 H c2 (0) derived from C(T) was close to the μ0HP.The absence of violation of the Pauli limit for Zr4Rh2O is consistent with the previous study [12].The violation of the Pauli limit observed in Zr4Pd2O is an unreported superconducting nature, and a similar violation was reported in other η-carbide-type superconductors as mentioned in the Introduction part.A quasiparticle mean free path l at a normal state near the superconducting state can be estimated using the following formula derived from Singh et al. [50]: where m e , m * , and V M are free-electron mass, effective mass of the individual quasiparticles, and molar volume.The l is in cm unit when we take V M , D(E F ), and ρ 0 are in cm 3 mol -1 , states eV -1 per f.u., and Ω cm, respectively.If we assume m * /m = 1, we obtain l = 0.76 Å for Zr4Pd2O using VM = 72.8cm 3 mol -1 , D(EF) = 8.61 states eV -1 per f.u., and ρ0 = 0.22 mΩ cm.Likewise for Zr4Rh2O, we obtain l = 1.84 Å using VM = 71.7 cm 3 mol -1 , D(EF) = 4.76 states eV -1 per f.u., and ρ0 = 0.29 mΩ cm.A GL coherence length ξ GL can be calculated using the GL model with μ 0 H c2 (0) as the following: where Φ0 ≈ 2.07 ×10 -15 Wb is a magnetic flux quantum.The values of ξGL for Zr4Pd2O and Zr4Rh2O were calculated to be 69 Å and 73 Å, respectively, using the μ 0 H c2 (0) obtained from ρ(T) 50% criterion.The values of ξ GL were found to be much longer than that of l for both Zr 4 Pd 2 O and Zr 4 Rh 2 O. Therefore, both Zr 4 Pd 2 O and Zr 4 Rh 2 O are supposed to be in the dirty limit.The orbital limit μ0Horb can be estimated with WHH theory without considering spin-orbit scattering [28,29].In the dirty limit, μ0Horb is expressed in the following formula: The slope of μ0Hc2(T) at Tc was estimated to be -2.72 TK -1 and -1.81 TK -1 for Zr4Pd2O and Zr4Rh2O, respectively, when using the μ 0 H c2 (T) data of the ρ(T) 50% criterion.These obtained values yielded μ 0 H orb = 5.36 T for Zr 4 Pd 2 O and 5.11 T for Zr 4 Rh 2 O.For Zr 4 Pd 2 O, the value of μ 0 H c2 (0) determined by ρ(T) 50% criterion was found to be larger than that of both μ 0 H p and μ 0 H orb .On the other hand, for Zr4Rh2O, the value of μ0Hc2(0) determined by ρ(T) 50% criterion was higher than that of μ0Horb but lower than that of μ0Hp.The enhanced μ0Hc2(0) larger than both μ0Hp and μ0Horb for Zr4Pd2O implies the importance of spin-orbit scattering caused by strong SOC because the strong SOC can suppress the Pauli paramagnetic pair-breaking effect [28,29] and calculation of μ 0 H orb in Eq. ( 16) does not consider the SOC.The importance of SOC was pointed out by Ruan et al. [32] and Shi et al. [33] in Ti4Ir2O, exhibiting the violation of the Pauli limit.Similarly, we can expect that the enhanced μ0Hc2(0) of Zr4Pd2O would be raised from strong SOC.The strength of SOC proportions to Z 4 within the same electronic orbital as expressed in Eq.
(2), therefore the absence of enhanced μ 0 H c2 (0) for Zr 4 Rh 2 O may be explained by the lower strength of SOC because of the number of d electron configuration: Rh consists of 4d 8 , but Pd consists of 4d 10 .A GL penetration depth λ GL can be obtained using μ0Hc1 and ξGL in the following formula: We obtained the values of λGL to be 2250 Å for Zr4Pd2O and 2697 Å for Zr4Rh2O.GL parameters κGL = λGL/ξGL were estimated to be 33 and 37 for Zr4Pd2O and Zr4Rh2O, respectively.The calculation results agree with the nature of the type-Ⅱ superconductor, shown in Figs.2(a) and 2(b).A thermodynamic critical field μ 0 H c can be estimated using the following expression: The calculation provided the values of μ 0 H c (0) to be 150 mT and 118 mT for Zr 4 Pd 2 O and Zr 4 Rh 2 O, respectively.Finally, we summarized the whole obtained superconducting properties in Table 3.
In summary, we have discovered the bulk superconductivity in Zr 4 Pd 2 O.The crystal structure was found to be the ηcarbide-type structure with a space group Fd3 m (No. 227) through SXRD measurement.The bulk superconductivity was measured by magnetic susceptibility, electrical resistivity, and specific heat measurement, resulting in Tc = 2.8 K, 2.73 K, and 2.6 K, respectively.Zr 4 Pd 2 O was found to belong to the type-II superconductor by magnetic susceptibility measurement in ZFC and FC processes.The upper critical field was determined from electrical resistivity and specific heat data under several magnetic fields.We found that Zr4Pd2O exhibited an enhanced upper critical field μ0Hc2(0) = 6.72 T violating the Pauli limit μ 0 H P = 5.29 T, whereas the absence of the property in isostructural η-carbide-type oxide superconductor Zr 4 Rh 2 O.The enhanced upper critical field can be raised from strong SOC.

Sample preparation
Polycrystalline samples of Zr 4 Pd 2 O and Zr 4 Rh 2 O were prepared by reaction of the Zr plate (99.2%, Nilaco Corporation), ZrO2 powder (98.0%,Wako Special Grade), Rh powder (99.9%,Kojundo Chemical), and Pd powder (99.9%,Kojundo Chemical).These starting materials were weighed to a stoichiometric ratio, and the powders of that were pressed into a pellet.At first, the obtained pellet and Zr plate were melted together by means of an arc melting method on a water-cooled copper stage.Gas inside the arc furnace was replaced by pure argon gas 3 times and then filled with pure argon gas.Before melting the sample, a titanium ingot was melted to reduce residual oxygen gas in the furnace.The sample was melted at least 6 times and turned over at each melting for homogeneity.We observed a negligible 1-2% mass loss after the melting.Second, we crushed the as-cast sample into fine powder and pressed it into a pellet.Subsequently, we sealed the pellet into an evacuated quartz tube and treated an annealing process for 10 days at 800 ℃.A mass loss was not observed after the annealing, implying oxygen in the sample was maintained.

Crystal structure and composition
The phase purity and crystal structure of Zr4Pd2O and Zr4Rh2O were checked by XRD with Cu-Kα radiation using θ-2θ method.The XRD measurement was performed on a Miniflex 600 (Rigaku) equipped with a high-resolution semiconductor detector D/tex-Ultra.For further investigation, we also performed SXRD measurement at the beamline BL13XU in SPring-8 (proposal no.2023B1669) with a wavelength of λ = 0.354367 Å.The obtained SXRD patterns were refined by means of the Rietveld method using RIETAN-FP [51].Schematic images of the crystal structure were depicted using VESTA [52].The chemical compositions of Zr and Tr (Pd or Rh) were examined by EDX on a scanning electron microscope TM-3030plus (Hitachi High-Tech) equipped with computer software SwiftED (Oxford).The chemical composition of oxygen was not considered because of the difficulty of detecting light elements with X-ray spectroscopy.

Measurement of superconducting properties
Temperature and magnetic field dependence of magnetization were measured using a superconducting quantum interference device (SQUID) on a Magnetic Property Measurement System 3 (MPMS3, Quantum Design) equipped with a 7 T superconducting magnet.The measurement was performed using a vibrating sample magnetometry (VSM) mode with polished rectangular cuboid samples to estimate precise demagnetizing factors.The samples were placed in a vertically applied magnetic field.Temperature dependence was measured under μ0H = 1 mT in both zero-field cooling (ZFC) and field cooling (FC) processes.The magnetic field dependence was measured up to μ 0 H = 30 mT at several temperatures.Temperature and magnetic field dependence of Electrical resistivity and specific heat measurements were performed using a physical property measurement system (PPMS Dynacool, Quantum Design) equipped with a 9 T superconducting magnet.Electrical resistivity was measured by a four-probe DC method using silver paste and gold wires for the contact between a polished rectangular cuboid sample and sample puck.The measurement was performed using an excitation current of 1 mA.The specific heat measurement was carried out by means of a thermal relaxation method.The sample was mounted on a stage with N-grease for good thermal connection.
. The SXRD patterns and Rietveld refinement results of Zr 4 Pd 2 O and Zr 4 Rh 2 O at Room temperature are shown in Figs.1(d) and 1(e), respectively.The SXRD patterns were well fit to cubic η-carbide crystal structure and the lattice constants of Zr4Pd2O and Zr 4 Rh 2 O were determined to a = 12.4617(1) Å and 12.3977(3) Å, respectively.The values of a for Zr 4 Pd 2 O and Zr 4 Rh 2 O were confirmed to be close to the previous report

Figure 1 .
Figure 1.Schematic images of the cubic η-carbide crystal structure.(a) Zr 4 Tr 2 O (Tr = Pd or Rh) unit cell.(b) Zr1 octahedron centered by O atom.(c) Unit of stella quadrangla lattice.Rietveld refined Room temperature SXRD patterns of (d) Zr4Pd2O and (e) Zr 4 Rh 2 O.The red points and cyan lines represent obtained SXRD data and calculated data, respectively.Lower solid lines show the Bragg peak positions.The lower blue lines are differences between obtained SXRD data and calculated data.
Figures 2(a) and 2(b) show temperature dependences of the χ for Zr 4 Pd 2 O and Zr 4 Rh 2 O, respectively.We observed a clear superconducting transition at Tc = 2.8 K for Zr4Pd2O and Tc = 3.5 K for Zr4Rh2O.The temperature width in the superconducting transition for Zr4Rh2O was broader than Zr4Pd2O.Moreover, the Tc of that was lower than the previous report (Tc = 4.3 K) and it would be based on the vacancy of oxygen[12].The superconducting state at 1.8 K reached perfect diamagnetism in a zerofield cooling (ZFC) process different from a case for field cooling (FC) process.The hysteresis of temperature-dependent χ with the cooling process reflects the nature of type-Ⅱ superconductor.Temperature dependence of a lower critical field μ0Hc1 can be obtained through magnetic field dependence of M as shown in Figs.2(c) and 2(d).The horizontal axis is displayed in μ 0 H eff instead of H for precise estimation of the lower critical field.In Zr 4 Pd 2 O, the measurement was carried out in a range of 1.8 K < T < 2.8 K with increments of 0.1 K.In the case of Zr4Rh2O, the data was taken at 1.8 K, 1.9 K, and 2.0 K, then taken with increments of 0.2 K up to 3.6 K.We observed the convex downward curve of M as functions of μ0Heff at each temperature, and the minimum points gradually shifted lower filed as increasing temperature.In a low-filed region, the linear responses of M corresponding to the Meissner state were observed, and the behavior can be described as M fit = a * H eff + b * , where a * and b * are numerical constants.The fitting was carried out in a range of 0 mT < μ0Heff < 5 mT.Figures2(e) and 2(f) are differences between M and Mfit for Zr4Pd2O and Zr4Rh2O, respectively.The dashed lines represent the value of μ0Heff which deviates from the linear behavior of M. Temperature dependences of lower critical field μ 0 H c1 (T) collected from the M-M fit are shown in Figs.2(g) and 2(h) for Zr 4 Pd 2 O and Zr 4 Rh 2 O, respectively.The lower critical field at 0 K, μ 0 H c1 (0) can be obtained from the following empirical formula: Figures 2(a) and 2(b) show temperature dependences of the χ for Zr 4 Pd 2 O and Zr 4 Rh 2 O, respectively.We observed a clear superconducting transition at Tc = 2.8 K for Zr4Pd2O and Tc = 3.5 K for Zr4Rh2O.The temperature width in the superconducting transition for Zr4Rh2O was broader than Zr4Pd2O.Moreover, the Tc of that was lower than the previous report (Tc = 4.3 K) and it would be based on the vacancy of oxygen[12].The superconducting state at 1.8 K reached perfect diamagnetism in a zerofield cooling (ZFC) process different from a case for field cooling (FC) process.The hysteresis of temperature-dependent χ with the cooling process reflects the nature of type-Ⅱ superconductor.Temperature dependence of a lower critical field μ0Hc1 can be obtained through magnetic field dependence of M as shown in Figs.2(c) and 2(d).The horizontal axis is displayed in μ 0 H eff instead of H for precise estimation of the lower critical field.In Zr 4 Pd 2 O, the measurement was carried out in a range of 1.8 K < T < 2.8 K with increments of 0.1 K.In the case of Zr4Rh2O, the data was taken at 1.8 K, 1.9 K, and 2.0 K, then taken with increments of 0.2 K up to 3.6 K.We observed the convex downward curve of M as functions of μ0Heff at each temperature, and the minimum points gradually shifted lower filed as increasing temperature.In a low-filed region, the linear responses of M corresponding to the Meissner state were observed, and the behavior can be described as M fit = a * H eff + b * , where a * and b * are numerical constants.The fitting was carried out in a range of 0 mT < μ0Heff < 5 mT.Figures2(e) and 2(f) are differences between M and Mfit for Zr4Pd2O and Zr4Rh2O, respectively.The dashed lines represent the value of μ0Heff which deviates from the linear behavior of M. Temperature dependences of lower critical field μ 0 H c1 (T) collected from the M-M fit are shown in Figs.2(g) and 2(h) for Zr 4 Pd 2 O and Zr 4 Rh 2 O, respectively.The lower critical field at 0 K, μ 0 H c1 (0) can be obtained from the following empirical formula:

Figure 2 .
Figure 2. (a,b) ZFC and FC temperature-dependent magnetic susceptibility under μ 0 H = 1 mT for (a) Zr 4 Pd 2 O and (b) Zr 4 Rh 2 O.The insets are enlarged view near Tc.(c,d) Effective inner magnetic field dependence of magnetic susceptibility for (c) Zr4Pd2O and (d) Zr 4 Rh 2 O.The solid lines are fit of M fit = a * H eff + b * in a range of 0 mT < μ 0 H eff < 5 mT.(e,f) A difference between M and M fit for (e) Zr 4 Pd 2 O and (f) Zr 4 Rh 2 O.The dashed lines correspond to the field where M begins to deviate from the linear behavior.The dashed lines are used to determine temperature dependence of the lower critical field.(g,h) Temperature

Figure 3 .
Figure 3. (a,b) Temperature dependences of electrical resistivity under zero field for (a) Zr4Pd2O and (b) Zr4Rh2O.The insets are enlarged view near T c .The solid and dashed lines are fit to parallel resistor model and power-law model, respectively.(c,d) Temperature dependences of electrical resistivity under several magnetic fields for (c) Zr 4 Pd 2 O and (d) Zr 4 Rh 2 O.The dashed lines represent the 10%, 50%, and 90% criteria to determine temperature dependence of the upper critical field.(e,f) Temperature dependences of total specific heat under several magnetic fields for (e) Zr 4 Pd 2 O and (f) Zr 4 Rh 2 O.The solid lines

Figure 4 .
Figure 4. Temperature dependence of upper critical field for (a) Zr 4 Pd 2 O and (b) Zr 4 Rh 2 O.The solid lines are fit to the GL model.The value of μ 0 H P was calculated using μ 0 H P = 1.86T c with ρ(T) 50% criteria data.

5/12 dependence
of lower critical field for (g) Zr 4 Pd 2 O and (h) Zr 4 Rh 2 O.The solid lines are the fit of μ 0 H c1 (T) = μ 0 H c1 (0)[1-(T/T c ) 2 ].The values of μ 0 H c1 (0) were calculated to be 11.3 mT and 8.2 mT for Zr 4 Pd 2 O and Zr 4 Rh 2 O, respectively.Temperature dependences of electrical resistivity ρ(T) for Zr 4 Pd 2 O and Zr 4 Rh 2 O at zero field are shown in Figs.
The calculated ΘD was Θ D = 194 K and 230 K for Zr 4 Pd 2 O and Zr 4 Rh 2 O, respectively, and the values were close to the calculation result obtained by the parallel resistor model in electrical resistivity measurement.Temperature dependences of the electron contribution of the specific heat Cel(T) estimated by subtracting phonon contributions βT 3 + δT 5 from C(T) are shown in Figs.3(g) and 3(h) for Zr 4 Pd 2 O and Zr 4 Rh 2 O, respectively.T c determined from C el (T) at zero field was 2.6 K for Zr 4 Pd 2 O and 3.3 K for Zr 4 Rh 2 O.The normalized jumps of C el (T), ΔC el /γT c , were estimated to be 1.58 and 1.57 for Zr 4 Pd 2 O and Zr 4 Rh 2 O, respectively.

Table 3 .
The measured superconducting properties of Zr4Pd2O and Zr4Rh2O.