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

Abstract

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 T_c = 2.73 K. Our measurement revealed that the η-carbide-type oxide superconductor Zr₄Pd₂O shows an enhanced upper critical field μ₀H_c2(0) = 6.72 T, which violates the Pauli-Clogston limit μ₀H_P = 5.29 T. On the other hand, we found that the enhanced upper critical field is absent in a Rh analogue Zr₄Rh₂O. The large μ₀H_c2(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 properties¹ such as metal-insulator transition², giant magnetoresistance³, ferroelectricity⁴, and high-temperature superconductivity⁵. 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 phenomena⁶. The first discovery of superconductors in oxide compounds is a perovskite-type structural SrTiO_3-x⁷, and the discovery has led to the development of many kinds of oxide superconductors, for example, BaPb_1-xBi_xO₃⁸, YBa₂Cu₃O₇⁹, Li_1+xTi_2-xO₄¹⁰ and so on. Among the discoveries of oxide superconductors, the η-carbide-type superconductors A₄B₂X 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 Zr₄Rh₂O_x (x = 0.7 and 1)¹² and Nb₄Rh₂C_1-δ¹³ at transition temperatures of T_c = 2.8 K (x = 0.7), 4.8 K (x = 1) and 9.75 K, respectively. They also found that Ti₄M₂O with M = Co, Rh, and Ir show superconductivity at T_c = 2.7 K, 2.8 K, and 5.4 K, respectively¹⁴. The significant discovery of them was not only founding superconductors but also observing a large upper critical field μ₀H_c2(0) violating the Pauli-Clogston limit (Pauli limit) for Nb₄Rh₂C_1-δ, Ti₄Co₂O, and Ti₄Ir₂O. The value of Pauli limit μ₀H_P 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:

$$\begin{array}{c}{\mu }_{0}{{H}}_{\text{P}} \, \text{=} \, \frac{\Delta \text{(}{0}\text{)}}{\sqrt{g}{\mu }_{\text{B}}} \, \text{=} \, \text{1.86}{T}_{\text{c}}\text{,}\end{array}$$

(1)

where g = 2 is a g-factor for free electron and μ_B ≈ 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) = 1.76k_BT_c (k_B ≈ 1.38×10^–23 J K^–1 is a Boltzmann constant) in the single gap Bardeen−Cooper−Schrieffer (BCS) model¹⁷. The large μ₀H_c2(0) overwhelming the μ₀H_P can arise from special electronic states and structural properties such as multi-band effect^18,19,20, spin-triplet cooper pairing²¹, 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 μ₀H_c2(0) superconducting state, and the strength can be controlled by chemical elemental substitution³⁰. The strength of SOC, ξ can be approximately calculated using a hydrogen-like atom model³¹:

$$\begin{array}{c}\xi \propto \, \frac{{Z}^{4}}{{n}^{3}{l}\left({l}\,+\,\frac{1}{{2}}\right)\left({l}\,+\,1\right)}\text{,}\end{array}$$

(2)

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 Z⁴ within the same electronic orbital. In the case of the Ti₄Ir₂O superconductor, we can expect the Ir-5d orbital hosting enhanced SOC should play an important role for the large μ₀H_c2(0), and it was found that Ti-3d and Ir-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 correlation³². 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 μ₀H_c2(0) undergoes a crossover at 35.6 GPa from well beyond to less than the μ₀H_P³³. As mentioned above, the η-carbide-type superconductors have been studied from the points of view of the large μ₀H_c2(0) and SOC effect based on d-block transition metals. Table 1 shows a list of the η-carbide-type superconductors with T_c, μ₀H_P, and μ₀H_c2(0). Some kinds of η-carbide-type oxide superconductors have been reported; however, it is not sufficient as of now, and develo** new examples of them is important for a deeper understanding of the η-carbide-type oxide superconducting properties.

Table 1 A list of η-carbide-type superconductors.

Herein, we focus on a Zr₄B₂O system because superconductivity was solely confirmed in Zr₄Rh₂O 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 Zr₄Pd₂O 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 μ₀H_c2(0) for Zr₄Pd₂O and Zr₄Rh₂O using the electrical resistivity and specific heat data measured at several magnetic fields. We find that Zr₄Rh₂O shows the μ₀H_c2(0) = 6.16 T, lower than the μ₀H_P = 7.59 T; however, Zr₄Pd₂O shows the large μ₀H_c2(0) = 6.88 T, violating the μ₀H_P = 5.29 T different from Zr₄Rh₂O. The violation of the Pauli limit for Zr₄Pd₂O can be attributed to the larger strength of SOC derived from Pd-4d electrons.

Results

Crystal structures of Zr₄Pd₂O and Zr₄Rh₂O

Schematic images of the crystal structure for Zr₄Tr₂O (Tr = Pd or Rh) are shown in Fig. 1a. These compounds crystalline a cubic η-carbide crystal structure with a space group Fd $\stackrel{\mathrm{-}}{3}$ 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 Ti₂Ni-type structure. The complicated η-carbide crystal structure consists of Zr1 octahedra centered by O (Fig. 1b) and a geometrically frustrated stella quadrangula lattice (Fig. 1c)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. The SXRD patterns and Rietveld refinement results of Zr₄Pd₂O and Zr₄Rh₂O at Room temperature are shown in Fig. 1d and e, respectively. The SXRD patterns were well fit to cubic η-carbide crystal structure and the lattice constants of Zr₄Pd₂O and Zr₄Rh₂O were determined to a = 12.4617(1) Å and 12.3977(3) Å, respectively. The values of a for Zr₄Pd₂O and Zr₄Rh₂O were confirmed to be close to the previous report^12,34,35. We found a small amount of impurity phases such as ZrO₂ and Zr in both of them, and reliability factors R_wp were 8.438% for Zr₄Pd₂O and 16.550% for Zr₄Rh₂O. We also refined the atomic coordinates and isotropic atomic displacement parameter U_iso for each atom, as summarized in Table 2. The U_iso 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 Zr₄Pd₂O and Zr₄Rh₂O confirmed through EDX were to be 2.10(2) for Zr:Pd and 2.2(2) for Zr:Rh.

Figure 1

Schematic images of the cubic η-carbide crystal structure. (a) Zr₄Tr₂O (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) Zr₄Rh₂O. 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. The impurity contents are shown as mass fraction.

Table 2 Crystalline parameters obtained from Rietveld refinement using SXRD patterns at Room temperature for Zr₄Pd₂O and Zr₄Rh₂O.

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 t³⁹:

$$\begin{array}{c}{N} \, \text{=} \, \frac{4lw}{4lw\,+\,3t\left(l\,+\,w\right)}\text{.}\end{array}$$

(3)

The calculated values of N were 0.66 for Zr₄Pd₂O (l = 1.40 mm, w = 1.50 mm, t = 0.51 mm) and 0.81 for Zr₄Rh₂O (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 H_eff = 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:

$$\begin{array}{c}\chi \, \text{=} \, \frac{M}{H_{\text{eff}}} \, \text{=} \, \frac{M}{H\,-\,4\pi MN}\text{.}\end{array}$$

(4)

Figure 2a and b show temperature dependences of the χ for Zr₄Pd₂O and Zr₄Rh₂O, respectively. We observed a clear superconducting transition at T_c = 2.8 K for Zr₄Pd₂O and T_c = 3.5 K for Zr₄Rh₂O. The temperature width in the superconducting transition for Zr₄Rh₂O was broader than Zr₄Pd₂O. Moreover, the T_c of that was lower than the previous report (T_c = 4.3 K) and it would be based on the vacancy of oxygen¹². The superconducting state at 1.8 K reached perfect diamagnetism in a zero-field 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 μ₀H_c1 can be obtained through magnetic field dependence of M as shown in Fig. 2c and d. The horizontal axis is displayed in μ₀H_eff instead of H for precise estimation of the lower critical field. In Zr₄Pd₂O, 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 μ₀H_eff 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 < μ₀H_eff < 5 mT. Figure 2e and f are differences between M and M_fit for Zr₄Pd₂O and Zr₄Rh₂O, respectively. The dashed lines represent the value of μ₀H_eff which deviates from the linear behavior of M. Temperature dependences of lower critical field μ₀H_c1(T) collected from the M-M_fit are shown in Fig. 2g and h for Zr₄Pd₂O and Zr₄Rh₂O, respectively. The lower critical field at 0 K, μ₀H_c1(0) can be obtained from the following empirical formula:

$$\begin{array}{c}{\mu }_{0}{{H}}_{\text{c1}}\text{(}{T}\text{)} \, \text{=} \, {\mu}_{0}{{H}}_{\text{c1}}\left({0}\right)\left[1\,-\,{\left(\frac{T}{T_{\text{c}}}\right)}^{2}\right],\end{array}$$

(5)

resulting μ₀H_c1(0) = 11.3 mT and 8.2 mT for Zr₄Pd₂O and Zr₄Rh₂O, respectively. Zr₄Pd₂O showed a higher μ₀H_c1(0) than that of Zr₄Rh₂O even lower T_c, suggesting that Zr₄Pd₂O tends to be more robust against magnetic field rather than Zr₄Rh₂O.

Figure 2

(a,b) ZFC and FC temperature-dependent magnetic susceptibility under μ₀H = 1 mT for (a) Zr₄Pd₂O and (b) Zr₄Rh₂O. The insets are enlarged view near T_c. (c,d) Effective inner magnetic field dependence of magnetic susceptibility for (c) Zr₄Pd₂O and (d) Zr₄Rh₂O. The solid lines are fit of M_fit = a^*H_eff + b^* in a range of 0 mT < μ₀H_eff < 5 mT. (e,f) A difference between M and M_fit for (e) Zr₄Pd₂O and (f) Zr₄Rh₂O. 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 dependence of lower critical field for (g) Zr₄Pd₂O and (h) Zr₄Rh₂O. The solid lines are the fit of μ₀H_c1(T) = μ₀H_c1(0)[1-(T/T_c)²]. The values of μ₀H_c1(0) were calculated to be 11.3 mT and 8.2 mT for Zr₄Pd₂O and Zr₄Rh₂O, respectively.

Temperature dependences of electrical resistivity ρ(T) for Zr₄Pd₂O and Zr₄Rh₂O at zero field are shown in Fig. 3a and b, respectively. In a low-temperature region, we observed a drop of ρ(T) to zero, suggesting a superconducting transition. Zr₄Pd₂O showed a sharp transition, and the zero resistivity was observed at T_c^zero = 2.73 K. Zr₄Rh₂O, 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:

$$\begin{array}{c}\rho \left({T}\right) \, \text{=} \, {\rho }_{0} \, + \, {A}{T}^{n_{\text{PL}}}\text{,}\end{array}$$

(6)

where ρ₀, A, and n_PL are residual resistivity, temperature-independent coefficient, and power exponent respectively. The ρ(T) curve of Zr₄Pd₂O in a range of 4 K < T < 80 K was fitted using the model, providing ρ₀ = 0.22 mΩ cm, A = 0.00011 mΩ K^–2, and n_PL = 1.2. For the ρ(T) curve of Zr₄Rh₂O, we fitted in a range of 6 K < T < 80 K, obtained ρ₀ = 0.29 mΩ cm, A = 0.00017 mΩ K^-2, and n_PL = 1.3. The values of n_PL close to 1 suggest that the ρ(T) show linear-temperature dependence in the low-temperature 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:

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) Zr₄Pd₂O and (d) Zr₄Rh₂O. 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) Zr₄Pd₂O and (f) Zr₄Rh₂O. The solid lines are fit to C(T)/T = γ + βT² + δT⁴. (g,h) Temperature dependences of electronic specific heat zero field for (g) Zr₄Pd₂O and (h) Zr₄Rh₂O. The solid lines are used to estimate T_c and dashed lines represent γ value.

$$\begin{array}{c}\rho \left({T}\right) \, \text{=} \, {\left[\frac{1}{{\rho }_{\text{sat}} \, }+\frac{1}{{\rho }_{\text{ideal}}\left({T}\right)}\right]}^{-1}\text{.}\end{array}$$

(7)

The temperature-independent term ρ_sat corresponds to the saturation of ρ(T) in high temperature. Fisk and Webb found the saturation of resistivity in A-15 superconductors such as Nb₃Sn⁴⁵, and the saturation can be realized when a mean free path becomes comparable to interatomic separations of the material, called as Ioffe-Regel condition⁴⁶. The temperature-dependent component ρ_ideal is described with the Bloch-Grüneisen model⁴⁷:

$$\begin{array}{c}{\rho }_{\text{ideal}}\left({T}\right) \, \text{=} \, {\rho }_{\text{ideal,0}} \, + \, {B}{\left(\frac{T}{{\Theta }_{\text{D}}}\right)}^{n_{\text{BG}}}{\int }_{0}^{\frac{{\Theta }_{\text{D}}}{T}}\frac{{t}^{n_{\text{BG}}} \, }{\left(e^{t}- 1\right)\left(1 - e^{-t}\right)}{\text{dt}}\text{,}\end{array}$$

(8)

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 n_BG = 5 for both Zr₄Pd₂O and Zr₄Rh₂O, and the calculation provided ρ_sat = 0.52 mΩ cm, ρ_ideal,0 = 0.41 mΩ cm, B = 1.49 mΩ cm, and Θ_D = 261 K for Zr₄Pd₂O, 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 Zr₄Rh₂O. The ρ₀ can be calculated using ρ_sat and ρ_ideal,0 as given in ρ₀ = ρ_satρ_ideal,0/(ρ_sat + ρ_ideal,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 Fig. 3a and b. Resistivity at 300 K, ρ_{300 K}, was found to be 0.32 mΩ cm for Zr₄Pd₂O and 0.47 mΩ cm for Zr₄Rh₂O. Residual resistivity ratio, RRR = ρ_{300 K}/ρ₀ 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 significant⁴⁸. Figure 3c and d show the ρ(T) curves at several magnetic fields for Zr₄Pd₂O and Zr₄Rh₂O, respectively. The magnetic fields are applied with an increment of 0.2 T for Zr₄Pd₂O up to μ₀H = 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 ρ₀ to determine temperature dependences of the upper critical field for Zr₄Pd₂O and Zr₄Rh₂O (discussed later).

Figure 3e and f show temperature dependences of total specific heat C(T) at several magnetic fields for Zr₄Pd₂O and Zr₄Rh₂O, respectively. Magnetic fields were applied with increments of 0.2 T up to μ₀H = 2 T and also measured at μ₀H = 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 Zr₄Pd₂O, 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:

$$\begin{array}{c}{C}\left({T}\right) \, \text{=} \, \gamma {T} \, + \, \beta {T}^{ 3}+\delta {T}^{5}\text{,}\end{array}$$

(9)

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 Zr₄Pd₂O and γ = 18.1 mJ K^–2 mol^–1 for Zr₄Rh₂O. 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 Fig. 3e and f. The values of Θ_D can be calculated using the β as the following formula:

$$\begin{array}{c} \, {\Theta }_{\text{D}} \, \text{=} \, {\left(\frac{{12}{\pi}^{4}{NR}}{{5}\beta}\right)}^{\frac{1}{{3}}}\text{,}\end{array}$$

(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 calculated Θ_D was Θ_D = 194 K and 230 K for Zr₄Pd₂O and Zr₄Rh₂O, 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 C_el(T) estimated by subtracting phonon contributions βT³ + δT⁵ from C(T) are shown in Fig. 3g and h for Zr₄Pd₂O and Zr₄Rh₂O, respectively. T_c determined from C_el(T) at zero field was 2.6 K for Zr₄Pd₂O and 3.3 K for Zr₄Rh₂O. The normalized jumps of C_el(T), ΔC_el/γT_c, were estimated to be 1.58 and 1.57 for Zr₄Pd₂O and Zr₄Rh₂O, respectively. The values of the jump were similar and slightly higher than 1.43, which is the expected value by the weak-coupling BCS theory¹⁷. 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 formula⁴⁹:

$$\begin{array}{c}{\lambda }_{\text{el-ph}} \, \text{=} \, \frac{1.04\,+\,\mu^{\text{*}}{\text{ln}}\left(\frac{{\Theta }_{\text{D}}}{1.45{T}_{\text{c}}}\right)}{\left(1\,-\,0.62\mu^{\text{*}}\right){\text{ln}}\left(\frac{{\Theta }_{\text{D}}}{1.45{T}_{\text{c}}}\right)\,-\,1.04}\text{,}\end{array}$$

(11)

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 λ_el-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:

$$\begin{array}{c}{D}\left({E}_{\text{F}}\right) \, \text{=} \, \frac{3\gamma }{{\uppi }^{2}{k}_{\text{B}}^{2}\left(1\,+\,\lambda_{\text{el-ph}}\right)}.\end{array}$$

(12)

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 Zr₄Pd₂O and Zr₄Rh₂O, respectively. The higher T_c of Zr₄Rh₂O than that of Zr₄Pd₂O may be based on higher Θ_D as explained in BCS theory¹⁷.

Discussion

Here, we discuss the upper critical fields and other superconducting parameters of Zr₄Pd₂O and Zr₄Rh₂O. Figure 4a and b are temperature dependences of upper critical field μ₀H_c2(T) for Zr₄Pd₂O and Zr₄Rh₂O, 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, μ₀H_c2(0) can be calculated by fitting the data using the Ginzburg-Landau (GL) model:

Figure 4

(a,b) Temperature dependence of upper critical field for (a) Zr₄Pd₂O and (b) Zr₄Rh₂O. The solid lines are fit to the GL model. The value of μ₀H_P was calculated using μ₀H_P = 1.86T_c with ρ(T) 50% criteria data.

$$\begin{array}{c}{\mu }_{0}{{H}}_{\text{c2}}\left({T}\right) \, \text{=} \, {\mu }_{0}{{H}}_{\text{c2}}\left({0}\right)\left[\frac{\text{1}\,-\,{\left(\frac{T}{T_{\text{c}}}\right)}^{2}}{\text{1}\,+\,{\left(\frac{T}{T_{\text{c}}}\right)}^{2}}\right]\text{.}\end{array}$$

(13)

We obtained the values of μ₀H_c2(0) for Zr₄Pd₂O 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 Zr₄Rh₂O, the obtained μ₀H_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 μ₀H_c2(0) for Zr₄Pd₂O were higher than that of μ₀H_P = 5.29 T calculated with T_c of ρ(T) 50% criterion. On the other hand, for Zr₄Rh₂O, the values of μ₀H_c2(0) derived from ρ(T) criterion were lower than that of μ₀H_P = 7.59 T calculated with T_c of ρ(T) 50% criterion. The value of μ₀H_c2(0) derived from C(T) was close to the μ₀H_P. The absence of violation of the Pauli limit for Zr₄Rh₂O is consistent with the previous study¹². 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:

$$\begin{array}{c}{l} \, \text{=} \, 2.372 \, \times {10}^{-14}\frac{{\left(\frac{{m}^{\text{*}}}{m_{\text{e}}}\right)}^{2}{V_{\text{M}}^{2}}}{{D}{\left({E}_{\text{F}}\right)}^{2}{\rho}_{0}}\text{,}\end{array}$$

(14)

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 ρ₀ are in cm³ 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 V_M = 72.8 cm³ mol^–1, D(E_F) = 8.61 states eV^-1 per f.u. and ρ₀ = 0.22 mΩ cm. Likewise for Zr₄Rh₂O, we obtain l = 1.84 Å using V_M = 71.7 cm³ mol^–1, D(E_F) = 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 μ₀H_c2(0) as the following:

$$\begin{array}{c}{\mu }_{0}{H}_{\text{c2}}\left({0}\right)\text{=}\frac{{\Phi }_{0}}{{2}\pi {\xi }_{\text{GL}}^{2}}\text{,}\end{array}$$

(15)

where Φ₀ ≈ 2.07 ×10^-15 Wb is a magnetic flux quantum. The values of ξ_GL for Zr₄Pd₂O and Zr₄Rh₂O were calculated to be 69 Å and 73 Å, respectively, using the μ₀H_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 Zr₄Pd₂O and Zr₄Rh₂O. Therefore, both Zr₄Pd₂O and Zr₄Rh₂O are supposed to be in the dirty limit. The orbital limit μ₀H_orb can be estimated with WHH theory without considering spin-orbit scattering^28,29. In the dirty limit, μ₀H_orb is expressed in the following formula:

$$\begin{array}{c}{\mu }_{0}{{H}}_{\text{orb}} \, \text{=} \, -0.693{T}_{\text{c}}{\left.\frac{{d}{\mu }_{0}{H}_{\text{c2}}\left({T}\right)}{dT}\right|}_{{T}\,=\,{T}_{\text{c}}}\text{.}\end{array}$$

(16)

The slope of μ₀H_c2(T) at T_c was estimated to be − 2.72 TK^–1 and − 1.81 TK^-1 for Zr₄Pd₂O and Zr₄Rh₂O, respectively, when using the μ₀H_c2(T) data of the ρ(T) 50% criterion. These obtained values yielded μ₀H_orb = 5.36 T for Zr₄Pd₂O and 5.11 T for Zr₄Rh₂O. For Zr₄Pd₂O, the value of μ₀H_c2(0) determined by ρ(T) 50% criterion was found to be larger than that of both μ₀H_p and μ₀H_orb. On the other hand, for Zr₄Rh₂O, the value of μ₀H_c2(0) determined by ρ(T) 50% criterion was higher than that of μ₀H_orb but lower than that of μ₀H_p. The enhanced μ₀H_c2(0) larger than both μ₀H_p and μ₀H_orb 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 μ₀H_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 μ₀H_c2(0) of Zr₄Pd₂O would be raised from strong SOC. The strength of SOC proportions to Z⁴ within the same electronic orbital as expressed in Eq. (2), therefore the absence of enhanced μ₀H_c2(0) for Zr₄Rh₂O may be explained by the lower strength of SOC because of the number of d electron configuration: Rh consists of 4d⁸, but Pd consists of 4d¹⁰. For deeper understanding the observed enhanced μ₀H_c2(0) in Zr₄Pd₂O, further investigations such as density functional theory (DFT) calculation considered SOC effect and measurement of electronic state are needed. A GL penetration depth λ_GL can be obtained using μ₀H_c1 and ξ_GL in the following formula:

$$\begin{array}{c}{\mu }_{0}{{H}}_{\text{c1}}\text{(}{0}\text{)} \, = \, \frac{{\Phi }_{0}}{{4}\pi {\lambda }_{\text{GL}}^{2}}{\text{l}}{\text{n}}\left(\frac{{\lambda }_{\text{GL}}}{{\xi }_{\text{GL}}}\right)\text{.}\end{array}$$

(17)

We obtained the values of λ_GL to be 2250 Å for Zr₄Pd₂O and 2697 Å for Zr₄Rh₂O. GL parameters κ_GL = λ_GL/ξ_GL 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 Fig. 2a and b. A thermodynamic critical field μ₀H_c can be estimated using the following expression:

$$\begin{array}{*{20}c} {\mu _{0} {H}_{{{\text{c1}}}} {\text{(}}0{\text{)}} \cdot {\mkern 1mu} \mu _{0} {H}_{{{\text{c2}}}} {\text{(}}0{\text{)}}{\mkern 1mu} = {\mkern 1mu} \left[ {\mu _{0} {H}_{{\text{c}}} {\text{(0)}}} \right]^{2} {\text{ln}}\kappa _{{{\text{GL}}}} {\text{.}}} \\ \end{array}$$

(18)

The calculation provided the values of μ₀H_c(0) to be 150 mT and 118 mT for Zr₄Pd₂O and Zr₄Rh₂O, respectively. Finally, we summarized the whole obtained superconducting properties in Table 3.

Table 3 The measured superconducting properties of Zr₄Pd₂O and Zr₄Rh₂O.

In summary, we have discovered the bulk superconductivity in Zr₄Pd₂O. The crystal structure was found to be the η-carbide-type structure with a space group Fd $\stackrel{\mathrm{-}}{3}$ m (No. 227) through SXRD measurement. The bulk superconductivity was measured by magnetic susceptibility, electrical resistivity, and specific heat measurement, resulting in T_c = 2.8 K, 2.73 K, and 2.6 K, respectively. Zr₄Pd₂O 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 μ₀H_c2(0) = 6.72 T violating the Pauli limit μ₀H_P = 5.29 T, whereas the absence of the property in isostructural η-carbide-type oxide superconductor Zr₄Rh₂O. The enhanced upper critical field can be raised from strong SOC. DFT calculation and measurement of electronic state are future works for further understanding the enhanced upper critical field.

Methods

Sample preparation

Polycrystalline samples of Zr₄Pd₂O and Zr₄Rh₂O 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-FP⁵¹. Schematic images of the crystal structure were depicted using VESTA⁵². 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 μ₀H = 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.