Discovery of the Type-II Superconductor Ta₄Rh₂C_1-δ with a High Upper Critical Field

We find Ta₄Rh₂C_1-δ to crystallize in the $\eta$-carbide type structure, with the cubic space group $Fd\bar{3}m$ (no.227) with the cell parameter of $a=$ 11.7947(1) Å. This structure can be rationalized as related compounds, e.g. Nb₄Rh₂C_1-δ ($a=$ 11.8527(2) Å) crystallize also in it. Here, the powder X-ray diffraction technique was employed to identify the phase purity and cell parameters of the obtained samples.

The powder X-ray diffraction (PXRD) pattern and the corresponding Rietveld refinement of the obtained Ta₄Rh₂C_1-δ sample are presented in Figure 1(a). Energy-dispersive X-ray spectroscopy (EDS) analysis reveals a Ta:Rh ratio of 1.9(5):1 for the sample, which is close to the ideally stoichiometric value of 2:1. Reliable quantification of the carbon content by EDS is not possible and is challenging by X-ray diffraction as well [23]. Assuming only a negligible carbon loss during the arc-melting process, the carbon content in Ta₄Rh₂C_1-δ should be close to 0.85, i.e. the nominal composition, for the amount of carbon used for the purest obtained sample. Rietveld refinement analysis determined that the main phase, Ta₄Rh₂C_1-δ, constitutes 96.5 % of the sample, with a minor impurity phase of Ta₂O₅ at 3.5 %. Notably, the formation of TaC as an impurity phase was not observed for these synthesis conditions. Details on the Rietveld refinements of Ta₄Rh₂C are summarized in Table 1, assuming a model $\eta$-carbide structure type, with the cubic space group $Fd\bar{3}m$ and Ta₄Rh₂C stoichiometry.

In Figure 1(b)-(e), we show the crystal structure of Ta₄Rh₂C_1-δ in an ideal chemical stoichiometric general formula of Ta₄Rh₂C. In this $\eta$-carbide structure, Ta atoms occupy the $16c$ and the $48f$ Wyckoff positions, Rh atoms occupy the $32e$ Wyckoff positions, and C atoms occupy the $16d$ Wyckoff positions. Even though there are only 4 Wyckoff positions required to describe the crystal structure, the unit contains nevertheless results in a total of 112 atoms and a formula of Ta₆₄Rh₃₂C₁₆ for one unit cell. In Figure 1(b)&(c) the unit cell with all atoms are shown in two orientations. In 1(d), the connectivity of the Ta1 and Rh atoms are shown: the Ta1 atoms form a network of tetrahedra resulting in a stella quadrangla structure, while the Rh atoms arrange in isolated tetrahedra. The Ta 2 atoms form a network of octahedra in which every second one is slightly distorted, as shown in figure 1(e), with the C atoms filling the octahedral voids.

To understand the physical properties of $\eta$-carbide compound Ta₄Rh₂C_1-δ, we performed temperature dependent magnetic susceptibility, resistivity, and specific heat capacity measurements.

In Figure 2(a), we observe a superconducting transition at a critical temperature of $T_{\rm c}$ = 6.3 K in the temperature dependence of the magnetic susceptibility in zero-field cooled (ZFC) and field-cooled (FC) modes under an external field of $\mu_{0}H=$ 2 mT, respectively. The difference between the FC and ZFC measurements in the superconducting state are prototypical for a type-II superconductor. When dealing with the magnetic susceptibility data, a demagnetization factor N was estimated using the relationship -b = 1/[4$\pi$ (1-N)] [25]. Here, we obtain a value of N = 0.53 for our sample by fitting the field-dependent measurements of the magnetization to a line ($M$ = b$H$ + a) in the low-field region [25].

In the normal state, Ta₄Rh₂C_1-δ shows a Pauli paramagnetic behavior, as confirmed by magnetization measurement between 10 K to 300 K in an external field of $\mu_{0}H=$ 1 T (see supporting information S-Fig.3) [23]. To estimate the lower critical field $H_{\rm c1}$, we performed a series of field-dependent measurements of the magnetization in low fields below the critical temperature $T_{\rm c}$, as shown in S-Fig 4.(a) of the supporting information [23]. Here, we used the magnetic-field point where the $M$($H$) curve first deviates from linearity as the measure for $H_{\rm c1}$ [26]. With this approximation, the obtained $H_{\rm c1}$ values are fitted using the semi-empirical formula:

The lower critical field at $T=$ 0 K is determined to be $\mu_{0}H_{\rm c1}(0)$= 9.4 mT as shown in the S-Fig 4 (b) (see supporting information) [23]. After taking the demagnetization factor N = 0.53 into account, the lower critical field is corrected to be $\mu_{0}H_{\rm c1}^{*}(0)$= 20 mT.

We find the resistivity of Ta₄Rh₂C_1-δ to decrease with decreasing temperature, showing a metallic behavior. The temperature dependent electrical resistivity measurement of the polycrystalline Ta₄Rh₂C_1-δ sample from 300 to 1.8 K is shown in the supporting information (see S-Fig. 5) [23]. At a critical temperature $T_{\rm c,onset}$ of 7.2 K, Ta₄Rh₂C_1-δ starts to undergo a transition to a superconducting state and the resistivity completely drops to zero at 6.0 K. Here, the residual resistivity ratio (RRR) value of the annealed polycrystalline sample is defined as $\rho$(300 K)/$\rho$(10 K) $\approx$ 1.22, corresponding to a poor metal behavior. This small RRR value may arise from the polycrystalline nature of the sample. In Figure 2(b), we show the temperature- and field-dependent resistivity $\rho$($T$,$H$) in a temperature range from $T$ = 2 to 9 K and magnetic fields between $\mu_{0}H=$ 0 T and 9 T. We show that the resistivity drops to zero at the transition to the superconducting state for all applied fields. In zero field, we determine the critical temperature to be $T_{\rm c}$ = 6.4 K with a 50 % criterion. As expected, the critical temperature decreases steadily as the applied magnetic field increases. However, the critical temperature is only suppressed to $T_{\rm c}=$ 3.9 K in the maximal applied field of 9 T, which is already evidence of the remarkably high upper critical field of this superconductor.

In Figure 2(c), we present the temperature- and field-dependent specific heat $C$($T$,$H$) in the vicinity of the superconducting transitions, where the data are plotted as $C/T$ versus $T$ in magnetic fields between $\mu_{0}H=$ 0 T and 9 T. The specific heat jumps corresponding to the superconducting transitions are well-pronounced in all measured fields, and shift to lower temperatures, which are in good agreement with the results from the resistivity measurements. At zero field, the specific heat jump temperature corresponding to the superconducting transition is determined to be $T_{\rm c}$ = 6.0 K based on an entropy-conserving construction, as shown in the inset of Figure 2(c).

In the normal state – close to the superconducting transition – the specific heat can be fitted according to the expression:

where $\gamma$ is the Sommerfeld coefficient, corresponding to the electronic contribution to $C$($T$), and $\beta$$T^{3}$ is the phonon contribution to the specific heat. Here, we obtain the $\gamma$ to be 20.9 mJ mol^-1 K^-2, and the $\beta$ to be 0.707 mJ mol^-1 K^-4 for Ta₄Rh₂C_1-δ (see supporting information S-Fig.6) [23]. With above obtained $T_{\rm c}$ and $\gamma$ values, the normalized specific heat jump is found to be $\Delta C/\gamma T_{\rm c}$ = 1.56 in zero field, which is slightly larger than the weak-coupling BCS value of 1.43 and evidence for the bulk nature of the superconducting state in Ta₄Rh₂C_1-δ.

We determined the Debye temperature to be $\Theta_{D}=$ 268 K, using the following relationship:

Here n = 7 is the number of atoms per formula unit, and R = 8.314 J mol^-1 K^-1 is the ideal gas constant.

The electron-phonon coupling constant $\lambda_{\rm ep}$ can be estimated from the Debye temperature, using the semi-empirical McMillan approximation [27]:

Here, the Coulomb repulsion parameter $\mu^{*}$ is set to be 0.13 according to an empirical approximation that was widely used in superconductors with similar elements (e.g. NbRh₂B₂ and TaRh₂B₂) [28, 25, 29, 30, 31]. Based on these values, the $\lambda_{\rm ep}$ value for Ta₄Rh₂C_1-δ is calculated to be 0.71, which is smaller than the 0.83 for Nb₄Rh₂C_1-δ.

The measured $\gamma$ value corresponds to a density of states at the Fermi-level of $D(E_{\rm F})$ of 5.23 states eV^-1 per formula unit (f.u.) in Ta₄Rh₂C_1-δ, when using the following relationship:

In Figure 3(a), we present the field-dependence of the critical temperatures $T_{\rm c}$ determined from the resistivity measurements using the common 10 %-, 50 %-, and 90 %-criteria , as well as the specific heat capacity as shown in Figures 2(b) and (c), (compare, e.g., references [7, 28, 32]). When using the Ginzburg-Landau (GL) formalism, the zero-temperature upper-critical field $\mu_{0}H_{\rm c2}$(0) for Ta₄Rh₂C_1-δ is determined to be 23.5 T, 20.7 T, 20.1 T, and 21.3 T for the 10 %-, 50 %-, and 90 %-criteria of resistivity and the specific heat capacity, respectively (see supporting information S-Fig.7) [23]. Usually, a Ginzburg-Landau (GL) model fitting will give higher estimated upper-critical field values for the $\eta$-carbide structure type superconductors [7, 8, 19].

Here, we make a conservative estimation of the upper-critical field $\mu_{0}H_{\rm c2}{\rm(0)}$ for Ta₄Rh₂C_1-δ using the Werthamer-Helfand-Hohenberg (WHH) formalism in the dirty limit according to [33, 7, 34]:

with $h^{\ast}_{fit}$ being

where $t$ = $T/T_{c}$ ($T_{c}$ is the transition temperature at zero field), while $C_{1}$ = 0.153 and $C_{2}$ = 0.152 are two parameters [34]. The zero-temperature upper-critical field $\mu_{0}H_{\rm c2}$(0) is determined to be 19.3 T, 17.4 T, 16.9 T, and 17.7 T for the 10 %-, 50 %-, and 90 %-criteria of resistivity and the specific heat capacity, respectively. All of these values exceed the corresponding weak-coupling BCS Pauli paramagnetic limits of $\mu_{0}H_{\rm Pauli}\approx 1.86{\rm[T/K]}\times T_{\rm c}=$ 12.5 T, 11.9 T, 11.1 T, and 11.2 T, respectively.

In Figure 3(b), we present a comparison of Ta₄Rh₂C_1-δ with previously reported superconductors containing Ta in $T_{\rm c}$ and $\mu_{0}H_{\rm c2}(0)$ evaluation [25, 35, 36, 37, 38, 31, 39, 40, 41, 42, 29, 43, 44, 45, 46]. The critical temperature $T_{\rm c}$ of Ta₄Rh₂C_1-δ is higher than most of the reported Ta-based superconductors and its $\mu_{0}H_{\rm c2}(0)$ value is higher than all the listed known superconductors except the highly anisotropic Ta₂Pd_0.92S₆, when this material is measured with the applied field being vertical to the $b$-axis [43]. Until now, reported Ta-based superconductors with $\mu_{0}H_{\rm c2}(0)$ value higher than the weak coupling Pauli limit are limited to: TaRh₂B₂, TaIr₂B₂, Ta₂Pd_0.92S₆, 4$H_{b}$-TaSe₂, Ta₂V_3.1Si_0.9, and the quasi-crystal superconductor Ta_1.6Te [25, 29, 43, 44, 45, 46]. All these superconductors have highly anisotropic crystal structures. In contrary to this, the crystal structure of Ta₄Rh₂C_1-δ is cubic and centrosymmetric, which strongly reflects the unusual nature of the Pauli limit violation in this material. It should be noted that Ta₄Rh₂C_1-δ has the second highest critical temperature $T_{\rm c}$ and upper critical field $\mu_{0}H_{\rm c2}(0)$ values among all reported $\eta$-carbide structure type superconductors, to date [7, 8, 9, 47].

The obtained upper critical field $\mu_{0}H_{\rm c2}(0)$ value, together with the lower critical field $\mu_{0}H_{\rm c1}^{*}(0)$ value can be used to calculate other relevant superconducting parameters for Ta₄Rh₂C_1-δ. Here, the $\mu_{0}H_{\rm c2}(0)$ value from the 50%-criterion is 17.4 T, and it corresponds to a superconducting Ginzburg-Landau coherence length of $\xi_{\rm GL}$ $=$ 43.5 Å according to the following equation:

where $\Phi_{0}=h/(2e)\approx 2.0678\times 10^{-15}$ Wb is the quantum flux. The superconducting penetration depth $\lambda_{\rm GL}$ can be estimated from the values of $\xi_{\rm GL}$ and $\mu_{0}H_{c1}^{*}$ obtained above by using the relation:

We obtained a value of $\lambda_{\rm GL}=$ 1743 Å for Ta₄Rh₂C_1-δ. The value of $\kappa_{\rm GL}=\lambda_{\rm GL}/\xi_{\rm GL}$ is calculated to be 40. These values demonstrate that Ta₄Rh₂C_1-δ is a type-II superconductor with a short superconducting coherence length $\xi_{\rm GL}$ and a large superconducting penetration depth $\lambda_{\rm GL}$.

In Table 2, we list all the parameters that we have obtained for Ta₄Rh₂C_1-δ and compare them with its isostructural superconductor Nb₄Rh₂C_1-δ. We find the superconducting properties of Ta₄Rh₂C_1-δ are similar to those of Nb₄Rh₂C_1-δ, especially the high upper critical field exceeding the weak coupling Pauli limit. Therefore, Ta₄Rh₂C_1-δ is both isostructural and isoelectronic to its sister compound Nb₄Rh₂C_1-δ. Previously, the isostructural and isoelectronic Nb/Ta superconducting sister compounds pairs NbC - TaC [39], NbRh₂B₂ - TaRh₂B₂ [25], and NbIr₂B₂ - TaIr₂B₂ [29] have been explored and compared. Here, Ta₄Rh₂C_1-δ and Nb₄Rh₂C_1-δ represent a new pair of isostructural and isoelectronic Nb/Ta superconducting compounds.

We performed density functional theory calculations for Ta₄Rh₂C_1-δ based on the structure determined in this work and given in Table 1. Figure 4 shows the band structure and density of states (DOS) of Ta₄Rh₂C_1-δ. We perform the calculation for $\delta=0$, but we determine the amount of hole doping a carbon deficiency of $\delta=0.15$ would entail; it is marked by a pink region in Figure 4. As Ta is chemically similar to Nb, it is not surprising that the electronic structure of Ta₄Rh₂C_1-δ does resemble the electronic structure of Nb₄Rh₂C_1-δ (see Ref. [7]). From the calculated DOS shown in Figure 4 (b), the DOS at $E_{\rm F}$ with $\delta=0.15$ gives a theoretical value of 5.45 states eV^-1/f.u., which is comparable with the value of 5.23 states eV^-1/f.u. calculated from heat capacity measurement (see Table 2). Similarly, in the previous study on Nb₄Rh₂C_1-δ, the DOS at $E_{\rm F}$ with $\delta=0.3$ provided a theoretical value of 9.63 states eV^-1/f.u., which is close to the experimentally derived value of 9.32 states eV^-1/f.u. from heat capacity measurement(see Ref. [7]).

We have also compared the effect of spin-orbit coupling on the band structures of Nb₄Rh₂C and Ta₄Rh₂C, respectively. We find significantly stronger splitting of bands in Ta₄Rh₂C, indicating stronger effects of spin-orbit coupling due to the replacement of the $5d$ transition metal Ta for the $4d$ transition metal Nb (see supplementary information S-Figure 8) [23]. In the supporting information (S-Figure (9) and (10)), we present the calculated Fermi surface of Nb₄Rh₂C_1-δ with $\delta=0$, and $\delta=0.15$, respectively [23].